We would expect agency problems to be less severe in other countries, primarily due to the relatively small percentage of individual ownership. Fewer individual owners should reduce the number of diverse opinions concerning corporate goals. The high percentage of institutional ownership might lead to a higher degree of agreement between owners and managers on decisions concerning risky projects. In addition, institutions may be better able to implement effective monitoring mechanisms on managers than can individual owners, based on the institutions' deeper resources and experiences with their own management. Show
Page 3 THE PRACTICAL MODEL CALCULATOR, FOR THE ENGINEER, MECHANIC, MACHINIST, MIANUFACTURER OF ENGINE-WORK, NAVAL ARCHITECT, MINER, AND MILLWRIGHT. BY OLIVER BYRNE, CIVIL, MILITARY, AND MECHANICAL ENGINEER. Compiler and Editor of the "Dictionary of Machines, Mechanics, Engine-work, and Engineering;" Author of " The Companion for Machinists, Mechanics, and Engineers;" Author and Inventor of a New Science, termed "The Calculus of Form," a substitute for the differential and Integral Calculus; " The Elements of Euclid by Colours," and numerous other Mathematical and Mechanical Works. Surveyor-General of the English Settlements in the Falkland Isles. Professor of Mathematics, College of Civil Engineers, London. PHI LADELPHI A: PUBLISHED BY HENRY CAREY BAIRD. (SUCCESSOR TO E. L. CAREY.) SOUTH-EAST CORNER MIARKET AND FIFTII STREETS. 1852. Page 4 Entered according to the act of Congress, in the year 1851, by HENRY CAREY BAIRD, in the Clerk's Office of the District Court for the Eastern District of Pennsylvania. STEREOTYPED BY L. JOHNSON AND CO. PHILADELPHIA. PRINTED BY T. K. AND P. G. COLLINS. Page 5 THE PRACTICAL MODEL CALCULATOR. WEIGHTS AND MEASURES. THE UNIT OF LENGTH. THE YARD.-If a pendulum vibrating seconds in vacuo, in Philadelphia, be divided into 2509 equal parts, 2310 of such equal parts is the length of the standard yard; the measures are taken on brass rods at the temperature of 32~ Fahrenheit. This yard will not be in error the ten-millionth part of an inch. 2310: 2509 as 1' to 1'086142 nearly. THE UNIT OF WEIGHT. The Pound, avoirdupois, is 27'7015 cubic inches of distilled water, weighed in air, at the temperature of maximum density, 39~'82; the barometer at 30 inches. THE LIQUID UNIT. The Gallon, 231 cubic inches, contains 8'3388822 pounds avoirdupois, equal 58372'1754 grains troy of distilled water, at 39~'82 Fah.; the barometer at 30 inches. UNIT OF DRY CAPACITY. The Bushel contains 2150'42 cubic inches, 77'627412 pounds avoirdupois, 543391'89 grains of distilled water, at the temperature of maximum density; the barometer at 30 inches. The French unit of length or distance is the metre, and is the ten-millionth of the quadrant of the globe, measured from the equator to the pole. The French Metre = 3'2808992 English feet linear measure = 39'3707904 inches. For Multiples the following Greek For Divisors the following Latin words are used: words are used: Deca for............ 10 times. Deci for the 10th part. Ifecto -............ 100 times. Centi - 100th part. Kilo -........ 1000 times. JIilli 1000th part. ]fyria —............ 10000 times. Thus a Kilometre = 1000 metres. JIlillimetre= metre 1000 The square Deca Metre, called the Are, is the element of land measure in France, which = 1076'42996 square feet English. The Stere is a cubic metre = 35'316582 cubic feet English. A2 5 Page 6 6 TIHE PRACTICAL MODEL CALCULATOR. The Litre for liquid measure is a cubic decimetre = 1'76077 imperial pints English, at the temperature of melting ice; a litre of distilled water weighs 15434 grains troy. The unit of weight is the gramme: it is the weight of a cubic centimetre of distilled water, or of a millilitre, and therefore equal to 15'434 grains troy. The kilogramme is the weight of a cubic decimetre of distilled water, at the temperature of maximum density, 4~ centigrade. The pound troy contains 5760 grains. The pound avoirdupois contains 7000 grains. The English imperial gallon contains 277'274 cubic inches; and the English corn bushel contains eight such gallons, or 2218'192 cubic inches. APOTHECARIES' WEIGHT. Grains..........................marked....... gr. 20 Grains make 1 Scruple - sc. or D 3 Scruples — 1 Dram -.......dr. or 3 8 Drams - 1 Ounce -....... oz. or I 12 Ounces - 1 Pound -. lb. or lb. gr. sc. 20= 1 dr. 60= 3= 1 oz. 480= 24= 8 = 1 lb. 5760 288 =96 = 12 = 1 This is the same as troy weight, only having some different divisions. Apothecaries make use of this weight in compounding their medicines; but they buy and sell their drugs by avoirdupois weight. AVOIRDUPOIS WEIGcT. Drams.................................................marked dr. 16 Drams............... make 1 Ounce......... - oz. 16 Ounces.............. - 1 Pound........... - lb. 28 Pounds............... - Quarter.............. - qr. 4 Quarters............ - 1 Hundred Weight... - cwt. 20 Hundred Weight... - 1 Ton................... - ton. dr. oz. 16 = 1 lb. 256 = 16= 1 qr. 71.68- 448 = 28 = 1 cwt. 28672- 1792= 112 = 4 1 ton. 573440 - 35840 = 2240 = 80 = 20 = 1 By this weight are weighed all things of a coarse or drossy nature, as Corn, Bread, Butter, Cheese, Flesh, Grocery Wares, and some Liquids; also all Metals except Silver and Gold. Oz. Dwt. Gr. NTote, that 1 lb. avoirdupois = 14 11 1551 troy. 1 oz. - 0 18 5 - 1 dr. - - = 0 1 3 Page 7 WEIGHTS AND MEASURES. 7 TROY WEIGHIT. Grains.....................marked Gr. Gr. Diwt. 24 Grains make 1 Pennyweight Dwt. 24 = 1 Oz. 20 Pennyweights 1 Ounce Oz. 480 = 20 = 1 Lb. 12 Ounces 1 Pound Lb. 5760 = 240 = 12 = 1 By this weight are weighed Gold, Silver, and Jewels. LONG MEASURE. 3 Barley-corns............make 1 Inch..............marked In. 12 Inches............. 1- 1 Foot.............. Ft. 3 Feet...................... - 1 Yard............. - Yd. 6 Feet...................... 1 Fathom.. - Fth. 5 Yards and a half...... 1 Pole or Rod..... - P1. 40 Poles.....................- 1 Furlong......... Fur. 8 Furlongs............. - 1 Mile.............- ile. 3 Miles.................... 1 League.......... Lea. 69; Miles nearly............- 1 Degree...... Deg. or In. Ft. 12= 1 Yd. 36 = 3 = 1 P1. 198 = 161-= 5 = 1 Fur. 7920= 660 = 220 = 40 = 1 Mile. 63360 = 5280 =1760 = 320 = 8 = 1 CLOTH MEASURE. 2 Inches and a quarter...make 1 Nail................ marked Nl. 4 Nails............... - 1 Quarter of a Yard.. - Qr. 3 Quarters............... - 1 Ell Flemish.......... - E F. 4 Quarters............... - 1 Yard................. - Yd. 5 Quarters................... - 1 Ell English........ - E E. 4 Qrs. 1- Inch.............. 1 Ell Scotch....... - E S. SQUARE MEASURE. 144 Square Inches........make 1 Sq. Foot.............marked Ft. 9 Square Feet.......... 1 Sq. Yard............ Yd. 30} Square Yards........ 1 Sq. Pole............. Pole. 40 Square Poles......... - 1 Rood.................. Rd. 4 Roods................ 1 Acre.............. Acr. Sq. Inc. Sq. Ft. 144 = 1 Sq. Yd. 1296 - 9 = 1 Sq. Pl. 39204 = 272k = 30-1 = 1 Rd. 1568160 = 10890 = 1210 = 40 = 1 Acr. 6272640 =43560 = 4840 = 160 = 4 = 1 When three dimensions are concerned, namely, length, breadth, and depth or thickness, it is called cubic or solid measure, which is used to measure Timber, Stone, &c. The cubic or solid Foot, which is 12 inches in length, and breadth, and thickness, contains 1728 cubic or solid inches, and 27 solid feet make one solid yard. Page 8 8 THE PRACTICAL MODEL CALCULATOR. DRY, OR CORN MEASURE. 2 Pints.....................make 1 Quart.............marked Qt. 2 Quarts.................. - 1 Pottle............. Pot. 2 Pottles...... - 1 Gallon.............- Gal. 2 Gallons.................. - 1 Peck...............- Pec. 4 Pecks......- 1 Bushel............- Bu. 8 Bushels................. - 1 Quarter........... Q- r. 5 Quarters................ - 1 Weigh or Load... - Wey. 2 Weys................... 1 Last.............. - Last. Pts. Gal. 8-= 1 Pee. 16= 2 = 1 Bu. 64= 8 = 4- 1 Qr. 512= 64= 32 = 8= 1 Wey. 2560 =320 = 160 =40 = 5 = 1 Last. 5120 = 640 = 320 =80 - 10 = 2 = 1 WINE MEASURE. 2 Pints................. make 1 Quart.............marked Qt. 2 Quarts................. - 1 Gallon............ - Gal. 42 Gallons................ - 1 Tierce............. - Tier. 63 Gallons or 11 Tier.. - 1 Hogshead... - Hhd. 2 Tierces.............. - 1 Puncheon......... - Pun. 2 Hogsheads............ - 1 Pipe or Butt..... - Pi. 2 Pipes.................. - 1 Tun............... Tun. Pts. Qts. 2= 1 Gal. 8- 4= 1 Tier. 336= 168= 42 =1 Hhd. 504 = 252 = 63 = 1- = 1 Pun. 672= 336 = 84=2 = 1 = 1 Pi. 1008 = 504= 126 = 3 = 2 =1}=1 Tun.'2016= 1008 =252 =6 = 4 = 3 =2 =1. ALE AND BEER MEASURE. 2 Pints...................make 1 Quart..............marked Qt. 4 Quarts................ - 1 Gallon............ - Gal. 36 Gallons................ - 1 Barrel............. - Bar. 1 Barrel and a half.... - 1 Hogshead......... - Hhd. 2 Barrels................ - 1 Puncheon......... - Pulln. 2 Hogsheads............ - 1 Butt...............- Butt. 2 Butts................ - 1 Tun................- Tun. Pts. Qt. 2= 1 Gal. 8= 4 = 1 Bar. 288 =144 = 36 1 Hhd. 432 =216 = 54 = 1 = 1 Butt. 864-432 = 108 =3 = 2 = 1 Page 9 OF TIMIE. 9 OF TIME. 60 Seconds...................make 1 Minute.......... marked M. or 60 Minutes.....................- 1 Hour............ Hr. 24 Hours....................... - 1 Day............. - Day. 7 Days........................ 1 Week............ - Wk. 4 Weeks....................... - 1.Month.......... - o. 13 Months, 1 Day, 6 Hours, 1 Julian Year... - Yr. or 365 Days, 6 Hours. Sec. Min. 60 = 1 Hr. 3600 = 60 = 1 Day. 86400 = 1440 = 24 = 1 Wk. 6048'00 = 10080 = 168 = 7 = 1 Mo. 2419200 = 40320= 672= 28 =4 =1 31557600 = 525960 = 8766 = 365} = 1 Year. Wk. Da. Hr. Mo. Da. Hr. Dr 52 1 6 - 13 1 6 = 1 Julian Year. Da. Hr. M. Sec. But 365 5 48 48 = 1 Solar Year. The time of rotation of the earth on its axis is called a sidereal day, for the following reason: If a permanent object be placed on the surface of the earth, always retaining the same position, it may be so located as to be posited in the same plane with the observer and some selected fixed star at the same instant of time; although this coincidence may be but momentary, still this coincidence continually recurs, and the interval elapsed between two consecutive coincidences has always throughout all ages appeared the same. It is this interval that is called a sidereal day. The sidereal day increased in a certain ratio, and called the mean solar day, has been adopted as the standard of time. Thus, 366'256365160 sidereal days = 366'256365160 - 1 or 365'256365160 mean solar days, whence sidereal day: mean solar day:: 365'256365160: 366'256365160:: 0'997269672: 1 or as 1: 1'002737803, when 23 hours, 56 minutes 4'0996608 see. of mean solar time = 1 sidereal day; and 24 hours, 3 mninutes, 56'5461797 sec. of sidereal time = 1 mean solar day. The true solar day is the interval between two successive coincidences of the sun with a fixed object on the earth's surface, bringing the sun, the fixed object, and the observer in the same plane. This interval is variable, but is susceptible of a maximum and minimum, and oscillates about that mean period which is called a mean solar day. Apparent or true time is that which is denoted by the sun-dial, from the apparent motion of the sun in its diurnal revolution, and differs several minutes in certain parts of the ecliptic from the mean time, or that shown by the clock. The difference is called the equation of time, and is set down in the almanac, in order to ascertain the true time. Page 10 ARITHMETIC. ARITHMETIC is the art or science of numbering; being that branch of Mathematics which treats of the nature and properties of numbers. When it treats of whole numbers, it is called Common Arithmetic; but when of broken numbers, or parts of numbers, it is called Fractions. Unity, or a Unit, is that by which every thing is called one; being the beginning of number; as one man, one ball, one gun. Nnumber is either simply one, or a compound of several units; as one man, three men, ten men. An Integer or Whole Number, is some certain precise quantity of units; as one, three, ten. These are so called as distinguished from Fractions, which are broken numbers, or parts of numbers; as one-half, two-thirds, or three-fourths. NOTATION AND NUMERATION. NOTATION, or NUMERATION, teaches to denote or express any proposed number, either by words or characters; or to read and write down any sum or number. The numbers in Arithmetic are expressed by the following ten digits, or Arabic numeral figures, which were introduced into Europe by the Moors about eight or nine hundred years since: viz. 1 one, 2 two, 3 three, 4 four, 5 five, 6 six, 7 seven, 8 eight, 9 nine, 0 cipher or nothing. These characters or figures were formerly all called by the general name of Ciphers; whence it came to pass that the art of Arithmetic was then often called Ciphering. Also, the first nine are called Significant Figures, as distinguished from the cipher, which is quite insignificant of itself. Besides this value of those figures, they have also another, which depends upon the place they stand in when joined together; as in the following Table: ~ - ~o o E H 2 H E E &c. 9 8 7 6 5 4 3 2 1 9 8 7 6 5 4 3 2 9 8 7 6 5 4 3 9 8 7 6 5 4 9 8 7 6 5 9 8 7 6 9 8 7 9 8 109 10 Page 11 NOTATION AND NUMERATION. 11 Here any figure in the first place, reckoning from right to left, denotes only its own simple value; but that in the second place denotes ten times its simple value; and that in the third place a hundred times its simple value; and so on; the value of any figure, in each successive place, being always ten times its former value. Thus, in the number 1796, the 6 in the first place denotes only six units, or simply six; 9 in the second place signifies nine tens, or ninety; 7 in the third place, seven hundred; and the 1 in the fourth place, one thousand; so that the whole number is read thusone thousand seven hundred and ninety-six. As to the cipher 0, it stands for nothing of itself, but being joined on the right-hand side to other figures, it increases their value in the same tenfold proportion: thus, 5 signifies only five; but 50 denotes 5 tens, or fifty; and 500 is five hundred; and so on. For the more easily reading of large numbers, they are divided into periods and half-periods, each half-period consisting of three figures; the name of the first period being units; of the second, millions; of the third, millions of millions, or bi-millions, contracted to billions; of the fourth, millions of millions of millions, or trimillions, contracted to trillions; and so on. Also, the first part of any period is so many units of it, and the latter part so many thousands. The following Table contains a summary of the whole doctrine: Periods. Quadrill.; Trillions; Billions; Millions; Units. Half-per. th. un. th. un. th. un. th. un. th. un. Figures. 123,456; 789,098; 765,432; 101,234; 567,890. NUMERATION is the reading of any number in words that is proposed or set down in figures. NOTATION is the setting down in figures any number proposed in words. OF THE ROMAN NOTATION. The Romans, like several other nations, expressed their numbers by certain letters of the alphabet. The Romans only used seven numeral letters, being the seven following capitals: viz. I for one; V for five; X for ten; L for fifty; C for a hundred; D for five hundred; M for a thousand. The other numbers they expressed by various repetitions and combinations of these, after the following manner: Page 12 12 THE PRACTICAL MODEL CALCULATOR. 1- I. 2 = II. As often as any character is repeated, 3 = III. so many times is its value repeated. 4 = IIII. or IV. A less character before a greater 5 = V. diminishes its value. 6 = VI. A less character after a greater in7 = VITI. creases its value. 8 = VIII. 9 = IX. 10 = X. 50.= L. 100 = C, 500 = D or 13. For every 3 annexed, this becomes ten times as many. 1000 = M or CID. For every C and 0, placed one at each 2000 = MM. end, it becomes ten times as much. 5000 = V or 133. A bar over any number increases it 6000 = VI. 1000 fold. 10000 = X or CCI33. 50000 = L or 1333. 60000 = LX. 100000 = C or CCCI333. 1000000 = M or CCCCI003DD 2000000 = MMI. &c. &c. EXPLANATION OF CERTAIN CHARACTERS. There are various characters or marks used in Arithmetic and Algebra, to denote several of the operations and propositions; the chief of which are as follow: + signifies plus, or addition.:::....... proportion. --.......... inus, or subtraction. -.......... equality. x.......... multiplication........... square root. >.......... division........... cube root, &c. Thus, 5 + 3, denotes that 3 is to be added to 5 = 8. 6 - 2, denotes that 2 is to be taken from 6 = 4. 7 x 3, denotes that 7 is to be multiplied by 3 = 21. 8. 4, denotes that 8 is to be divided by 4 = 2. 2: 3:: 4: 6, shows that 2 is to 3 as 4 is to 6, and thus, 2x6=3 x4. 6 + 4 = 10, shows that the sum of 6 and 4 is equal to 10. V3, or 3t, denotes the square root of the number 3 - 1'7320508. V5, or 5, denotes the cube root of the number 5 = 1'709976. 72, denotes that the number 7 is to be squared = 49. 83, denotes that the number 8 is to be cubed = 512. &c. Page 13 RULE OF THREE. 13 RULE OF THREE. THE RULE OF THREE teaches how to find a fourth proportional to three numbers given. Whence it is also sometimes called the Rule of Proportion. It is called the Rule of Three, because three terms or numbers are given to find the fourth; and because of its great and extensive usefulness, it is often called the Golden Rule. This Rule is usually considered as of two kinds, namely, Direct and Inverse. The Rule of Three Direct is that in which more requires more, or less requires less. As in this: if 3 men dig 21 yards of trench in a certain time, how much will 6 men dig in the same time? Here more requires more, that is, 6 men, which are more than 3 men, will also perform more work in the same time. Or when it is thus: if 6 men dig 42 yards, how much will 3 men dig in the same time? Here, then, less requires less, or 3 men will perform proportionally less work than 6 men in the same time. In both these cases, then, the Rule, or the Proportion, is Direct; and the stating must be thus, As 3: 21:: 6: 42, or thus, As 6: 42:: 3: 21. But, the Rule of Three Inverse is when more requires less, or less requires more. As in this: if 3 men dig a certain quantity of trench in 14 hours, in how many hours will 6 men dig the like quantity? Here it is evident that 6 men, being more than 3, will perform an equal quantity of work in less time, or fewer hours. Or thus: if 6 men perform a certain quantity of work in 7 hours, in how many hours will 3 men perform the same? Here less requires more, for 3 men will take more hours than 6 to perform the same work. In both these cases, then, the Rule, or the Proportion, is Inverse; and the stating must be thus, As 6: 14:: 3: 7, or thus, As 3: 7:: 6: 14. And in all these statings the fourth term is found, by multiplying the 2d and 3d terms together, and dividing the product by the 1st term. Of the three given numbers, two of them contain the supposition, and the third a demand. And for stating and working questions of these kinds observe the following general Rule: RuLE.-State the question by setting down in a straight line the three given numbers, in the following manner, viz. so that the 2d term be that number of supposition which is of the same kind that the answer or 4th term is to be; making the other number of supposition the 1st term, and the demanding number the 3d term, when the question is in direct proportion; but contrariwise, the other number of supposition the third term, and the demanding number the 1st term, when the question has inverse proportion. Then, in both cases, multiply the 2d and 3d terms together, and divide the product by the first, which will give the answer, or 4th term sought, of the same denomination as the second term. B Page 14 14 THE PRACTICAL MODEL CALCULATOR. Note, If the first and third terms consist of different denominations, reduce them both to the same; and if the second term be a compound number, it is mostly convenient to reduce it to the lowest denomination mentioned. If, after division, there be any remainder, reduce it to the next lower denomination, and divide by the same divisor as before, and the quotient will be of this last denomination. Proceed in the same manner with all the remainders, till they be reduced to the lowest denomination which the second term admits of, and the several quotients taken together will be the answer required. Note also, The reason for the foregoing Rules will appear when we come to treat of the nature of Proportions. Sometimes also two or more statings are necessary, which may always be known from the nature of the question. An engineer having raised 100 yards of a certain work in 24 days with 5 men, how many men must he employ to finish a like quantity of work in 15 days? da. men. da. men. As 15: 5:: 24 8 Ans. 5 15) 120 (8 Answer. 120 COMPOUND PROPORTION. COMPOUND PROPORTION teaches how to resolve such questions as require two or more statings by Simple Proportion; and that, whether they be Direct or Inverse. In these questions, there is always given an odd number of terms, either five, or seven, or nine, &c. These are distinguished into terms of supposition and terms of demand, there being always one term more of the former than of the latter, which is of the same kind with the answer sought. RuLE.-Set down in the middle place that term of supposition which is of the same kind with the answer sought. Take one of the other terms of supposition, and one of the demanding terms which is of the same kind with it; then place one of them for a first term, and the other for a third, according to the directions given in the Rule of Three. Do the same with another term of supposition, and its corresponding demanding term; and so on if there be more terms of each kind; setting the numbers under each other which fall all on the left-hand side of the middle term, and the same for the others on the right-hand side. Then to work. By several Operations.-Take the two upper terms and the middle term, in the same order as they stand, for the first Rule of Three question to be worked, whence will be found a fourth term. Then take this fourth number, so found, for the middle term of a second Rule of Three question, and the next two under terms in the general stating, in the same order as they stand, finding a fourth Page 15 OF COMMON FRACTIONS. 15 term from them; and so on, as far as there are any numbers in the general stating, making always the fourth number resulting from each simple stating to be the second term of the next following one. So shall the last resulting number be the answer to the question. By one Operation.-Multiply together all the terms standing under each other, on the left-hand side of the middle term; and, in like manner, multiply together all those on the right-hand side of it. Then multiply the middle term by the latter product, and divide the result by the former product, so shall the quotient be the answer sought. How many men can complete a trench of 135 yards longo in 8 days, when 16 men can dig 54 yards in 6 days? Greneral stating. yds. 54: 16 men:: 135 yds. days 8 6 days 432 810 16 4860 810 432) 12960 (30 Ans. by one operation. 1296 0 The same by two operations. 1st. 2d. As 54: 16:: 135: 40 As 8: 40:: 6: 30 16 6 810 8 ) 240 ( 30 Ans. 135 24 54)2160 (40 -0 216 0 OF C01MMON FRACTIONS. A FRACTION, or broken number, is an expression of a part, or some parts, of something considered as a whole. It is denoted by two numbers, placed one below the other, with a line between them: 3 numerator ) thus,- denominator which is named three-fourths. The Denominator, or number placed below the line, shows how many equal parts the whole quantity is divided into; and represents the Divisor in Division. And the Numerator, or number set above the line, shows how many of those parts are expressed by the Fraction; being the remainder after division. Also, both these numbers are, in general, named the Terms of the Fractions. Page 16 16 THE PRACTICAL MODEL CALCULATOR. Fractions are either Proper, Improper, Simple, Compound, or Mixed. A Proper Fraction is when the numerator is less than the denominator; as I, or 3, or 3-, &c. An Improper Fraction is when the numerator is equal to, or exceeds, the denominator; as -, or 5, or -, &c. A Simple Fraction is a single expression denoting any number of parts of the integer; as -X, or -. A Compound Fraction is the fraction of a fraction, or several fractions connected with the word of between them; as ~1 of -, or - of 6 of 3, &c. A Mixed Number is composed of a whole number and a fraction together; as 34, or 124, &c. A whole or integer number may be expressed like a fraction, by writing 1 below it, as a denominator; so 3 is -, or 4 is 4, &c. A fraction denotes division; and its value is equal to the quotient obtained by dividing the numerator by the denominator; so2 is equal to 3, and 2-0 is equal to 4. Hence, then, if the numerator be less than the denominator, the value of the fraction is less than 1. If the numerator be the same as the denominator, the fraction is just equal to 1. And if the numerator be greater than the denominator, the fiaction is greater than 1. REDUCTION OF FRACTIONS. REDUCTION OF FRACTIONS is the bringing them out of one form or denomination into another, commonly to prepare them for the operations of Addition, Subtraction, &c., of which there are several cases. To find the greatest common measure of two or more numbers. The Common Measure of two or more numbers is that number which will divide them both without a remainder: so 3 is a common measure of 18 and 24; the quotient of the former being 6, and of the latter 8. And the greatest number that will do this, is the greatest common measure: so 6 is the greatest common measure of 18 and 24; the quotient of the former being 3, and of the latter 4, which will not both divide farther. RULE. —If there be two numbers only, divide the greater by the less; then divide the divisor by the remainder; and so on, dividing always the last divisor by the last remainder, till nothing remains; then shall the last divisor of all be the greatest common measure sought. When there are more than two numbers, find the greatest common measure of two of them, as before; then do the same for that common measure and another of the numbers; and so on, through all the numbers; then will the greatest common measure last found be the answer. If it happen that the common measure thus found is 1, then the numbers are said to be incommensurable, or to have no common measure. Page 17 REDUCTION OF FRACTIONS. 17 To find the greatest common measure of 1998, 918, and 522. 918) 1998 (2 So 54 is the greatest common measure 1836 of 1998 and 918. 162) 918(5 Hence 54) 522 (9 810 486 108)162(1 36)54(1 108 36 54)108(2 18)36(2 108 36 So that 18 is the answer required. To abbreviate or reduce fractions to their lowest terms. RuLE. —Divide the terms of the given fraction by any number that will divide them without a remainder: then divide these quotients again in the same manner; and so on, till it appears that there is no number greater than 1 which will divide them then the fraction will be in its lowest terms. Or, divide both the terms of the fraction by their greatest common measure, and the quotients will be the terms of the fraction required, of the same value as at first. That dividing both the terms of the fraction by the same number, whatever it be, will give another fraction equal to the former, is evident. And when those divisions are performed as often as can be done, or when the common divisor is the greatest possible, the terms of the resulting fraction must be the least possible. 1. Any number ending with an even number, or a cipher, is divisible, or can be divided by 2. 2. Any number ending with 5, or 0, is divisible by 5. 3. If the right-hand place of any number be 0, the whole is divisible by 10; if there be 2 ciphers, it is divisible by 100; if 3 ciphers, by 1000; and so on, which is only cutting off those ciphers. 4. If the two right-hand figures of any number be divisible by 4, the whole is divisible by 4. And if the three right-hand figures be divisible by 8, the whole is divisible by 8; and so on. 5. If the sum of the digits in any number be divisible by 3, or by 9, the whole is divisible by 3, or by 9. 6. If the right-hand digit be even, and the sum of all the digits be divisible by 6, then the whole will be divisible by 6. 7. A number is divisible by 11 when the sum of the 1st, 3d, 5th, &c., or of all the odd places, is equal to the sum of the 2d, 4th, 6th, &c., or of all the even places of digits. 8. If a number cannot be divided by some quantity less than the square of the same, that number is a prime, or cannot be divided by any number whatever. 9. All prime numbers, except 2 and 5, have either 1, 3, 7, or 9, in the place of units; and all other numbers are composite, or can be divided. i3 2 2 Page 18 18 THE PRACTICAL MODEL CALCULATOR. 10. When numbers, with a sign of addition or subtraction between them, are to be divided by any number, then each of those num10 ~ 8- 4 bers must be divided by it. Thus, 2 5 + 4- 2 = 7. 11. But if the numbers have the sign of multiplication between 10 x8 x 3 them, only one of them must be divided. Thus, 6 x 2 10 x 4 x 3 10 x 4 xl 10 x 2 x 1 20 6x 1 2x1x Reduce 144 to its least terms. 144 -- 7 -- = 6-= 18 -9 the answer. 240 0 6 0 1, the answer. Or thus: 144) 240 (1 Therefore 48 is the greatest common measure, and 1144 48) 144 the answer, the same as before. 96 )144 (1 96 48)96(2 96 To reduce a mixed number to its equivalent improper fraction. RuLE.-Multiply the whole number by the denominator of the fraction, and add the numerator to the product; then set that sum above the denominator for the fraction required. Reduce 23-& to a fraction. Or, 23 (23x5)+2 117 5 5 = 115 2 117 5 To reduce an improper fraction to its equivalent whole or mixed number. RULE.-Divide the numerator by the denominator, and the quotient will be the whole or mixed number sought. Reduce 12 to its equivalent number. Here 12 or 12 - 3 = 4. Reduce 1 to its equivalent number. Here 1, or 15 7 = o? Reduce 749 to its equivalent number. Thus, 17) 749 (441 68 69 So that 79 = 44 1i 68 Page 19 REDUCTION OF FRACTIONS. 19 To reduce a whole nzmber to an equivalent fraction, having a giveni denominator. RULE.-Multiply the whole number by the given denominator, then set the product over the said denominator, and it will form the fraction required. Reduce 9 to a fraction whose denominator shall be 7. Here 9 x 7 = 63, then 673 is the answer. For 6,3 = 63. 7 = 9, the proof. To reduce a compound fraction to an equivalent simple one. RULE.-Multiply all the numerators together for a numerator, and all the denominators together for the denominator, and they will form the simple fraction sought. When part of the compound fraction is a whole or mixed number, it must first be reduced to a fraction by one of the former cases. And, when it can be done, any two terms of the fraction may be divided by the same number, and the quotients used instead of them. Or, when there are terms that are common, they may be omitted. Reduce I of 2 of 3 to a simple fraction. 1x2x3 6 1 Here2 x 3 x4 24- 4' 1x2x3 1 Or, 2 x 3 x 4 = 4 by omitting the twos and threes. Reduce 2 of 8 of {o to a simple fraction. 2 x 3 x 10 60 12 4 Here3 x'5 x 11 165 33= 11' 2x3x10 4 Or, 3 x 5 x 11 = 14' the same as before. To reduce fractions of different denominators to equivalent fractions, having a common denominator. RULE.-Multiply each numerator into all the denominators except its own for the new numerators; and multiply all the denominators together for a common denominator. It is evident, that in this and several other operations, when any of the proposed quantities are integers, or mixed numbers, or compound fractions, they must be reduced, by their proper rules, to the form of simple fractions. Reduce I, 3, and 4 to a common denominator. 1 x 3 x 4 = 12 the new numerator for 2. 2x2x4= 16......................... for -2. 3 x 2 x 3 = 18......................... for 3. 2 x 3 x 4 = 24 the common denominator. Therefore, the equivalent fractions are 12, n6 and A. Or, the whole operation of multiplying may be very well performed mentally, and only set down the results and given fractions thus: 1n I3, X = 12 —, 618, by abbreviation. Page 20 20 THE PRACTICAL MODEL CALCULATOR. When the denominators of two given fractions have a common measure, let them be divided by it; then multiply the terms of each given fraction by the quotient arising from the other's denominator. When the less denominator of two fractions exactly divides the greater, multiply the terms of that which hath the less denominator by the quotient. When more than two fractions are proposed, it is sometimes convenient first to reduce two of them to a common denominator, then these and a third; and so on, till they be all reduced to their least common denominator. To find the value of a fraction in parts of the integer. RULE.-Multiply the integer by the numerator, and divide the product by the denominator, by Compound Multiplication and Division, if the integer be a compound quantity. Or, if it be a single integer, multiply the numerator by the parts in the next inferior denomination, and divide the product by the denominator. Then, if any thing remains, multiply it by the parts in the next inferior denomination, and divide by the denominator as before; and so on, as far as necessary; so shall the quotients, placed in order, be the value of the fraction required. What is the value of ] of a pound troy? 7 oz. 4 dwts. What is the value of -56 of a cwt.? 1 qr. 7 lb. What is the value of 5 of an acre? 2 ro. 20 po. What is the value of -o of a day? 7 hrs. 12 min. To reduce a fraction from one denomination to another. RULE.-Consider how many of the less denomination make one of the greater; then multiply the numerator by that number, if the reduction be to a less name, or the denominator, if to a greater. Reduce 2 of a cwt. to the fraction of a pound. X A X 218 = f?2 ADDITION OF FRACTIONS. To add fractions together that have a common denominator. RULE.-Add all the numerators together, and place the sum over the common denominator, and that will be the sum of the fractions required. If the fractions proposed have not a common denominator, they must be reduced to one. Also, compound fractions must be reduced to simple ones, and mixed numbers to improper fractions; also, fractions of different denominations to those of the same denomination. To add 5 and 6 together. Here + = 4 = 15 To add ] and ~ together. C +. = 30 + 35 = 3 = 1 To add X and 71 and I of together. + 7 + - of = + 8 +4 I =+ + 2 = = 83. Page 21 RULE OF THREE IN FRACTIONS. 21 SUBTRACTION OF FRACTIONS. RULE.-Prepare the fractions the same as for Addition; then subtract the one numerator from the other, and set the remainder over the common denominator, for the difference of the fractions sought. To find the difference between A and I. Here - -- = - 2 To find the difference between 4 and a. 5 21 _20 1 28 28 - 28' MULTIPLICATION OF FRACTIONS. MULTIPLICATION of any thing by a fraction implies the taking some part or parts of the thing; it may therefore be truly expressed by a compound fraction; which is resolved by multiplying together the numerators and the denominators. RULE. —Reduce mixed numbers, if there be any, to equivalent fractions; then multiply all the numerators together for a numerator, and all the denominators together for a denominator, which will give the product required. Required the product of 3 and 8. Here - x - = 1= Or, X 2 X = Required the continued product of 2, 34, 5, and 4 of 5. 2 13 5 3 3 13 x 3 39 Iere X - X X X = 4 x2 8 47. DIVISION OF FRACTIONS. RULE.-Prepare the fractions as before in Multiplication; then divide the numerator by the numerator, and the denominator by the denominator, if they will exactly divide; but if not, then invert the terms of the divisor, and -multiply the dividend by it, as in Multiplication. Divide 2 by 5. Here 2 -- * = - 1=, by the first method. Divide 5 by -. Here ~ -2 = 5 X U = 5 X -- 41, by the latter. RULE OF THREE IN FRACTIONS. RULE.-Make the necessary preparations as before directed; then multiply continually together the second and third terms, and the first with its terms inverted as in Division, for the answer. This is only multiplying the second and third terms together, and dividing the product by the first, as in the Rule of Three in whole numbers. If - of a yard of velvet cost 5 of a dollar, what will 1 of a yard cost? 3 2 5 8 2 5 I-lere x x c -~of a dollar. 8 5 1 Page 22 22 THE PRACTICAL MODEL CALCULATOR. DECIMAL FRACTIONS. A DECIMAL FRACTION is that which has for its denominator a unit (1) with as many ciphers annexed as the numerator has places; and it is usually expressed by setting down the numerator only, with a point before it on the left hand. Thus, -A is'5, and 2z5 is *25, and 75 is'075, and 1-o-4- is'00124; where ciphers are prefixed to make up as many places as are in the numerator, when there is a deficiency of figures. A mixed number is made up of a whole number with some decimal fraction, the one being separated from the other by a point. Thus, 3'25 is the same as 3-25o, or 302 Ciphers on the right hand of decimals make no alteration in their value; for *5, or'50, or'500, are decimals having all the same value, being each = -o or ~. But if they are placed on the left hand, they decrease the value in a tenfold proportion. Thus, *5 is -1o or 5 tenths, but 05 is only -j- or 5 hundreths, and'005 is but 100 or 5 thousandths. The first place of decimals, counted from the left hand towards the right, is called the place of primes, or 10ths; the second is the place of seconds, or 100ths; the third is the place of thirds, or 1000ths; and so on. For, in decimals, as well as in whole numbers, the values of the places increase towards the left hand, and decrease towards the right, both in the same tenfold proportion; as in the following Scale or Table of Notation: Ca 3 8 8 3 3 3 3 3 3 3 3 ADDITION OF DECIMALS. RULE. —Set the numbers under each other according to the value of their places, like as in whole numbers; in which state the decimal separating points will stand all exactly under each other. Then, beginning at the right hand, add up all the columns of number as in integers, and point off as many places for decimals as are in the greatest number of decimal places in any of the lines that are added; or, place the point directly below all the other points. To add together 29'0146, and 3146'5, 29'0146 and 2109, and 62417, and 14'16. 3146'5 2109''62417 14'16 5299'29877, the sum. Page 23 MULTIPLICATION OF DECIMALS. 23 The sum of 376'25 + 86'125 + 637'4725 + 6'5 + 41'02 + 358'865 = 1506.2325. The sum of 3'5 + 47'25 + 2.0073 + 927'01 + 1'5 = 981.2673. The sum of 276 + 54'321 + 112 + 0.65 + 12'5 +'0463 - 455'5173. SUBTRACTION OF DECIMALS. RULE.-Place the numbers under each other according to the value of their places, as in the last rule. Then, beginning at the right hand, subtract as in whole numbers, and point off the decimals as in Addition. To find the difference between 91'73 91.73 and 2.138. 2'138 89'592 the difference. The difference between 1'9185 and 2'73 = 0'8115. The difference between 214'81 and 4'90142 = 209.90858. The difference between 2714 and'916 = 2713'084. MULTIPLICATION OF DECIIMALS. RULE.-Place the factors, and multiply them together the same Multiply'321096 as if they were whole numbers. by'2465 Then point off in the product just 1605480 as many places of decimals as 1926576 there are decimals in both the fac- 1284384 tors. But if there be not so many 642192. figures in the product, then supply'0791501640 the product. the defect by prefixing ciphers. Multiply 79'347 by 23'15, and we have 1836'88305. Multiply'63478 by'8204, and we have'520773512. Multiply'385746 by'00464, and we have'00178986144. CONTRACTION I. To multiply decimals by 1 with any number of ciphlers, as 10, or 100, or 1000, ce. This is done by only removing the decimal point so many places farther to the right hand as there are ciphers in the multiplier: and subjoining ciphers if need be. The product of 51-3 and 1000 is 51300. The product of 2'714 and 100 is 271'4. The product of'916 and 1000 is 916. The product of 21'31 and 10000 is 213100. CONTRACTION II. To' contract the operation, so as to retain only as many decimals i?, the )product as may be thought neeessary, when the product woul( naturally contain several more places. Set the units' place of the multiplier under that figure of the multiplicand whose place is the same as is to be retained for thl Page 24 24 THE PRACTICAL MODEL CALCULATOR. last in the product; and dispose of the rest of the figures in the inverted or contrary order to what they are usually placed in. Then, in multiplying, reject all the figures that are more to the right than each multiplying figure; and set down the products, so that their right hand figures may fall in a column straight below each other; but observing to increase the first figure of every line with what would arise from the figures omitted, in this manner, namely, 1 from 5 to 14, 2 from 15 to 24, 3 from 25 to 34, &c.; and the sum of all the lines will be the product as required, commonly to the nearest unit in the last figure. To multiply 27414986 by 92'41035, so as to retain only four places of decimals in the product. Contracted way. Common way. 27'14986 27'14986 53014'29 92'41035 24434874 13 574930 542997 81144958 108599 27141986 2715 108599 44 81 54299712 14 24434874 2508o9280 2508s9280 650510 DIVISION OF DECIMALS. RuLE. —Divide as in whole numbers; and point off in the quotient as many places for decimals, as the decimal places in the dividend exceed those in the divisor. When the places of the quotient are not so many as the rule requires, let the defect be supplied by prefixing ciphers. When there happens to be a remainder after the division; or when the decimal places in the divisor are more than those in the dividend; then ciphers may be annexed to the dividend, and the quotient carried on as far as required. 179) 48624097 ( 00271643 2685) 27.00000 (100.55865 1282 15000 294 15750 1150 23250 769 17700 537 15900 000 24750 Divide 234.70525 by 64'25. 38653. Divide 14 by'7854. 17 82 5. Divide 2175'68 by 100. 21'7568. Divide'8727587 by'162. 5 38739. CONTRACTION I. When the divisor is an integer, with any number of ciphers annexed; cut off those ciphers, and remove the decimnal point in the Page 25 REDUCTION OF DECIMALS. 25 dividend as many places farther to the left as there are ciphers cut off, prefixing ciphers if need be; then proceed as before. Divide 45'5 by 2100. 21.00).455 (.0216, &c. 35 140 14 CONTRACTION II. Hence, if the divisor be 1 with ciphers, as 10, or 100, or 1000, &e.; then the quotient will be found by merely moving the decimal point in the dividend so many places farther to the left as the divisor has ciphers; prefixing ciphers if need be. So, 217'3 * 100 = 2'173, and 419. 10 = 41'9. And 5'16. 100 = 0516, and'21. 1000 = 00021. CONTRACTION III. When there are many figures in the divisor; or only a certain number of decimals are necessary to be retained in the quotient, then take only as many figures of the divisor as will be equal to the number of figures, both integers and decimals, to be in the quotient, and find how many times they may be contained in the first figures of the dividend, as usual. Let each remainder be a new dividend; and for every such dividend, leave out one figure more on the right hand side of the divisor; remembering to carry for the increase of the figures cut off, as in the 2d contraction in Multiplication. When there are not so many figures in the divisor as are required to be in the quotient, begin the operation with all the figures, and continue it as usual till the number of figures in the divisor be equal to those remaining to be found in the quotient, after which begin the contraction. Divide 2508'92806 by 92'41035, so as to have only four decirnals in the quotient, in which case the quotient will contain six figures. Contracted. Common way. 92'4103,5)2508'928,06(27'1498 924103,5) 2508'928,06 (27'1498 660721 66072106 13849 13848610 4608 46075750 912 91116100 80 79467850 6 5539570 REDUCTION OF DECIMALS. To reduce a common fraction to its equivalent deecimal. RULE.-Divide the numerator by the denominator as in Division of Decimals, annexing ciphers to the numerator as far as necessary; so shall the quotient be the decimal required. C Page 26 26 THE PRACTICAL MODEL CALCULATOR. Reduce I to a decimal. 24 = 4 x 6. Then 4)7' 6 ) 1 750000 *291666, &c. 3 reduced to a decimal, is'375. B reduced to a decimal, is'04. A reduced to a decimal, is'015625. 2 7 5reduced to a decimal, is'071577, &c. CASE II. To find the value of a decimal in terms of the inferior denominations. RULE.- Multiply the decimal by the number of parts in the next lower denomination; and cut off as many places for a remainder, to the right hand, as there are places in the given decimal. Multiply that remainder by the parts in the next lower denomination again, cutting off for another remainder as before. Proceed in the same manner through all the parts of the integer; then the several denominations, separated on the left hand, will make up the value required. What is the value of'0125 lb. troy: — 3 dwts. What is the value of'4694 lb. troy:- 5 oz. 12 dwt. 15'744 gr. What is the value of'625 cwt.: — 2 qr. 14 lb. What is the value of'009943 miles:- 17 yd. 1 ft. 5'98848 in. What is the value of'6875 yd.: — 2 qr. 3 nls. What is the value of'3375 ac. 1 rd. 14 poles. What is the value of'2083 hhd. of wine: 13'1229 gal. CASE III. To reduce integers or decimals to equivalent decimals of highzer denominations. RULE. —Divide by the number of parts in the next higher denomination; continuing the operation to as many higher denominations as may be necessary, the same as in Reduction Ascending of whole numbers. Reduce 1 dwt. to the decimal of a pound troy. 20 1 dwt. 12 0'05 oz. 0'004166, &c. lb. Reduce 7 dr. to the decimal of a pound avoird.:- -02734375 lb. Reduce 2'15 lb. to the decimal of a cwt.:-' 019196 cwt. Reduce 24 yards to the decimal of a mile:- -013636, &c. miles. Reduce'056 poles to the decimal of an acre: — 00035 ac. Reduce 1'2 pints of wine to the decimal of a hhd.: — 00238 hhd. Reduce 14 minutes to the decimal of a day: — 009722, &c. da.. Reduce'21 pints to the decimal of a peck: — 013125 pec. Whten there are several numbers, to be reduced all to the decinmal of the highest. Set the given numbers directly under each other, for dividends, proceeding orderly from the lowest denomination to the highest. Page 27 DUODECIMALS. 27 Opposite to each dividend, on the left hand, set such a number for a divisor as will bring it to the next higher name; drawing a perpendicular line between all the divisors and dividends. Begin at the uppermost, and perform all the divisions; only observing to set the quotient of each division, as decimal parts, on the right hand of the dividend next below it; so shall the last quotient be the decimal required. Reduce 5 oz. 12 dwts. 16 gr. to lbs.: — 46944, &c. lb. RULE OF THREE IN DECIMALS. RULE.-Prepare the terms by reducing the vulgar fractions to decimals, any compound numbers either to decimals of the higher denominations, or to integers of the lower, also the first and third terms to the same name: then multiply and divide as in whole numbers. Any of the convenient examples in the Rule of Three or Rule of Five in Integers, or Common Fractions, may be taken as proper examples to the same rules in Decimals.-The following example, which is the first in Common Fractions, is wrought here to show the method. If - of a yard of velvet cost 2 of a dollar, what will ~5 yd. cost? yd. 8 yd. $ g.'375'375:'4:: 3125': 333, &c..4 =.4.375).12500(.333333, 33~ cts. 1250 125 1= 3125. DUODECIMALS. DUODECIMALS, or CROSS MULTIPLICATION, is a rule made use of by workmen and artificers, in computing the contents of their works. Dimensions are usually taken in feet, inches, and quarters; any parts smaller than these being neglected as of no consequence. And the same in multiplying them together, or casting up the contents. RULE.-Set down the two dimensions, to be multiplied together, one under the other, so that feet stand under feet, inches under inches, &c. Multiply each term in the multiplicand, beginning at the lowest, by the feet in the multiplier, and set the result of each straight under its corresponding term, observing to carry 1 for every 12, from the inches to the feet. In like manner, multiply all the multiplicand by the inches and parts of the multiplier, and set the result of each term one place removed to the right hand of those in the multiplicand; omitting, however, what is below parts of inches, only carrying to these the proper number of units from the lowest denomination. Page 28 28 THE PRACTICAL MODEL CALCULATOR. Or, instead of multiplying by the inches, take such parts of the multiplicand as these are of a foot. Then add the two lines together, after the manner of Compound Addition, carrying 1 to the feet for 12 inches, when these come to so many. Multiply 4 f. 7 inc. Multiply 14 f. 9 inc. by 6 4 by 4 6 27 6 59 0 1 6] 7 4] 29 0O 66 4} INVOLUTION. INVOLUTION is the raising of Powers from any given number, as a root. A Power is a quantity produced by multiplying any given number, called the Root, a certain number of times continually by itself. Thus, 2 = 2 is the root, or first power of 2. 2 x 2 = 4 is the 2d power, or square of 2. 2 x 2 x 2 = 8 is the 3d power, or cube of 2. 2 x 2 x 2 x 2 = 16 is the 4th power of 2, &c. And in this manner may be calculated the following Table of the first nine powers of the first nine numbers. TABLE OF THE FIRST NINE POWERS OF NUMBERS. st 2d. d. 4th. 5th. 6th. 7th. 8th. 9th. 2 4 1 8 16 32 64 128 256 512 3 9 27 81 243 729 2187 6561 19G83 4 16 64 256 1024 4096 16384 65536 262144 5 25 125 625 3125 15625 78125 390625 1953125 6 36 216 1296 7776 46656 279936 1679616 10077696 7 49 343 2401 16807 117649 823543 5764801 40353607 8 64 512 4096 32768 262144 2097152 16777216 134217728 9 — 81- 729 6561 59049 531441 4782969 43046721 387420489 The Index or Exponent of a Power is the number denoting the height or degree of that power; and it is 1 more than the number of multiplications used in producing the same. So 1 is the index or exponent of the 1st power or root, 2 of the 2d power or square, 3 of the 3d power or cube, 4 of the 4th power, and so on. Powers, that are to be raised, are usually denoted by placing the index above the root or first power. So 22 = 4, is the 2d power of 2. 23 = 8, is the 3d power of 2. 24 = 16, is the 4th power of 2. 5404, is the 4th power of 540 = 85030560000. Page 29 EVOLUTION. 29 When two or more powers are multiplied together, their product will be that power whose index is the sum of the exponents of the factors or powers multiplied. Or, the multiplication of the powers answers to the addition of the indices. Thus, in the following powers of 2. 1st. 2d. 3d. 4th. 5th. 6th. 7th. 8th. 9th. 10th. 2 4 8 16 32 64 128 256 512 1024 or, 21 22 23 24 25 26 27 28 29 210 Here, 4 x 4= 16, and 2 2 = 4 its index; and 8 x 16= 128, and 3 + 4= 7 its index; also 16 x 64 = 1024, and 4 + 6 = 10 its index. The 2d power of 45 is 2025. The square of 4'16 is 17'3056. The 3d power of 3'5 is 42'875. The 5th power of'029 is'000000020511149. The square of 2 is 4. The 3d power of 5 is 1.2 The 4th power of 3 is 2.6EVOLUTION. EVOLUTION, or the reverse of Involution, is the extracting or finding the roots of any given powers. The root of any number, or power, is such a number as, being multiplied into itself a certain number of times, will produce that power. Thus, 2 is the square root or 2d root of 4, because 22 = 2 x 2 = 4; and 3 is the cube root or 3d root of 27, because 33 = 3 x 3 x 3 = 27. Any power of a given number or root may be found exactly, namely, by multiplying the number continually into itself. But there are many numbers of which a proposed root can never be exactly found. Yet, by means of decimals we may approximate or approach towards the root to any degree of exactness. These roots, which only approximate, are called Surd roots; but those which can be found quite exact, are called Rational roots. Thus, the square root of 3 is a surd root; but the square root of 4 is a rational root, being equal to 2: also, the cube root of 8 is rational, being equal to 2; but the cube root of 9 is surd, or irrational. Roots are sometimes denoted by writing the character V before the power, with the index of the root against it. Thus, the third root of 20 is. expressed by V20; and the square root or 2d root of it is /20, the index 2 being always omitted when the square root is designed. When the power is expressed by several numbers, with the sign + or - between them, a line is drawn from the top of the sign over all the parts of it; thus, the third root of 45 - 12 is /45 - 12, or thus, -/(45 - 12), enclosing the numbers in parentheses. c2 Page 30 30 THE PRACTICAL MODEL CALCULATOR. But all roots are now often designed like powers, with fractional indices: thus, the square root of 8 is 82, the cube root of 25 is 252, and the 4th root of 45 - 18 is 45 - 1812, or, (45 - 18)'. TO EXTRACT THE SQUARE ROOT. RULE.-Divide the given number into periods of two figures each, by setting a point over the place of units, another over the place of hundreds, and so on, over every second figure, both to the left hand in integers, and to the right in decimals. Find the greatest square in the first period on the left hand, and set its root on the right hand of the given number, after the manner of a quotient figure in Division. Subtract the square thus found from the said period, and to the remainder annex the two figures of the next following period for a dividend. Double the root above mentioned for a divisor, and find how often it is contained in the said dividend, exclusive of its right-hand figure; and set that quotient figure both in the quotient and divisor. Multiply the whole augmented divisor by this last quotient figure, and subtract the product from the said dividend, bringing down to the next period of the given number, for a new dividend. Repeat the same process over again, namely, find another new divisor, by doubling all the figures now found in the root; from which, and the last dividend, find the next figure of the root as before, and so on through all the periods to the last. The best way of doubling the root to form the new divisor is by adding the last figure always to the last divisor, as appears in the following examples. Also, after the figures belonging to the given number are all exhausted, the operation may be continued into decimals at pleasure, by adding any number of periods of ciphers, two in each period. To find the square root of 29506624. 29506624 ( 5432 the root. 25 104 450 4 416 1083 3466 3 3249 10862 21724 2 21724 Whien the root is to be extracted to mgaly Adlaces of figures, t/re ti'w;w: may be considerably shortened, thus: Having proceeded in the extraction after the common method till there be found half the required number of figures in the root. or one figure more; then, for the rest, divide the last remainder by Page 31 TO EXTRACT THE SQUARE ROOT. 31 its corresponding divisor, after the manner of the third contraction in Division of Decimals; thus, To find the root of 2 to nine places of figures. 2 (1'4142 1 24 100 4 96 281 400 1 281 4 11296 28282 160400 2 156564 28284) 3836 (1356 1008 160 19 2 1'41421356 the root required. The square root of'000729 is'027. The square root of 3 is 1'732050. The square root of 5 is 2'236068. The square root of 6 is 2'449489. RULES FOR THE SQUARE ROOTS OF COMMON FRACTIONS AND MIXED NUMBERS. First, prepare all common fractions by reducing them to their least terms, both for this and all other roots. Then, 1. Take the root of the numerator and of the denominator for the respective terms of the root required. And this is the best way if the denominator be a complete power; but if it be not, then, 2. Multiply the numerator and denominator together; take the root of the product: this root being made the numerator to the denominator of the given fraction, or made the denominator to the numerator of it, will form the fractional root required. a iVa Vab a That is, b — Vb - b - vab' And this rule will serve whether the root be finite or infinite. 3. Or reduce the common fractionto a decimal, and extract its root. 4. Mixed numbers may be either reduced to improper fractions, and extracted by the first or second rule; or the common fraction may be reduced to a decimal, then joined to the integer, and the root of the whole extracted. The root of Z5 is A. The root of 49 is 3. The root of a is 0'866025. The root of - is 0'645497. The root of 17- is 4'168333. Page 32 32 THE PRACTICAL MODEL CALCULATOR. By means of the square root, also, may readily be found the 4th root, or the 8th root, or the 16th root, &c.; that is, the root of any power whose index is some power of the number 2; namely, by extracting so often the square root as is denoted by that power of 2; that is, two extractions for the 4th root, three for the 8th root, and so on. So, to find the 4th root of the number 21035'8, extract the square root twice as follows: 21035'8000 (145'037237 (12'0431407, the 4th root. 1 1 24 110 22 45 4 96 2 44 285 1435 2404 i 10372 5 1425 4 9616 29003 108000 24083 75637 6 87009 6 72249 20991 (7237 3388 (1407 687 980 107 17 20 TO EXTRACT THE CUBE ROOT. 1. DIVIDE the page into three columns (I), (II), (III), in order, from left to right, so that the breadth of the columns may increase in the same order. In column (III) write the given number, and divide it into periods of three figures each, by putting a point over the place of units, and also over every third figure, from thence to the left in whole numbers, and to the right in decimals. 2. Find the nearest less cube number to the first or left-hand period; set its root in column (III), separating it from the right of the given number by a curve line, and also in column (I); then multiply the number in (I) by the root figure, thus giving the square of the first root figure, and write the result in (II); multiply the number in (II) by the root figure, thus giving the cube of the first root figure, and write the result below the first or left-hand period in (III); subtract it therefrom, and annex the next period to the remainder for a dividend. 3. In (I) write the root figure below the former, and multiply the sum of these by the root figure; place the product in (II), and add the two numbers together for a trial divisor. Again, write the root figure in (I), and add it to the former sum. 4. With the number in (II) as a trial divisor of the dividend, omitting the two figures to the right of it, find the next figure of the root, and annex it to the former, and also to the number in (I). Multiply the number now in (I) by the new figure of the root, and write the product as it arises in (II), but extended two places of figures more to the right, and the sum of these two numbers will be the corrected divisor; then multiply the corrected divisor by the Page 33 TO EXTRACT THE CUBE ROOT. 33 last root figure, placing the product as it arises below the dividend; subtract it therefromn, annex another period, and proceed precisely as described in (3), for correcting the columns (I) and (ii). Then with the new trial divisor in (II), and the new dividend in (III), proceed as before. When the trial divisor is not contained in the dividend, after two figures are omitted on the right, the next root figure is 0, and therefore one cipher must be annexed to the number in (r); two ciphers to the number in (II); and another period to the dividend in (III). When the root is interminable, we may contract the work very considerably, after obtaining a few figures in the decimal part of the root, if we omit to annex another period to the remainder in (III); cut off one figure from the right of (II), and two figures from (I), which will evidently have the effect of cutting off three figures from each column; and then work with the numbers on the left, as in contracted multiplication and division of decimals. Find the cube root of 21035'8 to ten places of decimals. (I) (II) (III) 2 4 21035'8 (27'60491055944 2 8 8 4 12.. 13035 2 4 6 9 11683 67 1 6 6 9 1352800 7 5 1 8 1341576 74 2 1 8 7. ~ 11224...... 7 4 8 9 6 9142444864 8 16 2 2 3 5 9 6 2081555136 6 4 9 38 2 2057415281 8 22 228528 8.... 241389855 6 33 1 21 6 22860923 8 28 04 2 2 8 5 6112 16 1278932 4 33 1 2 38 2 1143046 82808 2285942448 135886 4 7453 1 114305 1'8128112 22860169 719 21581 7 4 5 311 20575 228609u151 1006 83 914 228609234 92 8 3 91 212181610o91312 1 Required the cube roots of the following numbers:48228544, 46656, and 15069223. 364, 36, and 247. 64481'201, and 28991029248. 40'1, and 3072. 12821119155125, and'000076765625. 23405, and'0425. 38 24 and 16. 24 and 2'519842. 91k, and 7Y. 4'5, and 1'98802366. 3 Page 34 3-4 THE PRACTICAL ~MODEL CALCULATOR. TO EXTRACT ANY ROOT WHATEVER. LET N be the given power or number, n the index of the power, A the assumed power, r its root, R the required root of N. Then, as the sum of n + 1 times A and in - 1 times N.t is to the sum of n + 1 times N and n - 1 times A, so is the assiumed root r, to the required root R. Or, as half the said sum of n + 1 times A and n - 1 times N, is to the difference between the given and assumed powers, so is the assumed root r, to the difference between the true and assumed roots; which difference, added or subtracted, as the case reciuires, gives the true root nearly. That is, (n + 1).A + (n — 1).N: (n + 1).N + (n- 1)' A::,: R. Or, (n + 1). -A + (n - 1). 1N: A m N:: r: R t r. And the operation may be repeated as often as we please, by using always the last found root for the assumed root, and its Untih power for the assumed power A. To extract the 5thl root of 21035'8. Here it appears that the 5th root is between 7'3 and 7'4. Taking 7'3, its 5th power is 20730'71593. Hence then we have, N = 21035.8; r = 73; in = 5; j. (n + 1) = 3;.(n- 1)= 2. A = 20730'716 N-A = 305.O84 A = 20730 716 N = 21035'8 3 2 3 A = 62192.148 42071'6 2 N = 42071'6 As 104263'7: 305'084:: 7'3: *0213605 7.3 915252 2135588 104263 7 ) 2227 1132 ( 0213605, the difference. 14184 7'3 = r add 3758 630 7'321360 = R, the root, true to 5 the last figure. The 6th root of 21035.8 is 5 -20'7. The 6th root of 2 is 1 12 T4 "2. The 7th root of 21035'8 is 4'145132. The 7th root of 2 is 1-104 0!'. The 9th root of 2 is 1l08O5it9. OF RATIOS, PROPORTIONS, AND PROGRESSIONS. NUmBERS are compared to each other in two different ways: tlhe one comparison considers the difference of the two numbers, and( is named Arithmetical Relation, and the difference sometimes Arithmetical Ratio: the other considers their quotient, and is called Page 35 ARITHMETICAL PROPORTION AND PROGRESSION. 35 Geometrical Relation, and the quotient the Geometrical Ratio. So, of these two numbers 6 and 3, the difference or arithmetical ratio is 6 - 3 or 3; but the geometrical ratio is 3 or 2. There must be two numbers to form a comparison: the number which is compared, being placed first, is called the Antecedent; and that to which it is compared the Consequent. So, in the two numbers above, 6 is the antecedent, and 3 is the consequent. If two or more couplets of numbers have equal ratios, or equal differences, the equality is named Proportion, and the terms of the ratios Proportionals. So, the two couplets, 4, 2 and 8, 6 are arithmetical proportionals, because 4 - 2 = 8 - 6 = 2; and the two couplets 4, 2 and 6, 3 are geometrical proportionals, because 4 = - = 2, the same ratio. To denote numbers as being geometrically proportional, a colon is set between the terms of each couplet to denote their ratio; and a double colon, or else a mark of equality between the couplets or ratios. So, the four proportionals, 4, 2, 6, 3, are set thus, 4: 2:: 6: 3, which means that 4 is to 2 as 6 is to 3; or thus, 4:2 = 6: 3; or thus, 4 = ], both which mean that the ratio of 4 to 2 is equal to the ratio of 6 to 3. Proportion is distinguished into Continued and Discontinued. When the difference or ratio of the consequent of one couplet and the antecedent of the next couplet is not the same as the common difference or ratio of the couplets, the proportion is discontinued. So, 4, 2, 8, 6 are in discontinued arithmetical proportion, because 4-2 = 8 - 6 = 2, whereas, 2 - 8 = - 6; and 4, 2, 6, 3 are in discontinued geometrical proportion, because 4 -= = 2, but 6 -, which is not the same. But when the difference or ratio of every two succeeding terms is the same quantity, the proportion is said to be continued, and the numbers themselves a series of continued proportionals, or a progression. So, 2, 4, 6, 8 form an arithmetical progression, because 4 - 2 = 6 - 4 = 8 - 6 = 2, all the same common difference; and 2, 4, 8, 16, a geometrical progression, because 4 = 4 =? = 2, all the same ratio. When the following terms of a Progression exceed each other, it is called an Ascending Progression or Series; but if the terms decrease, it is a Descending one. So, 0, 1, 2, 3, 4, &c., is an ascending arithmetical progression, but 9, 7, 5, 3, 1, &c., is a descending arithmetical progression: Also, 1, 2, 4, 8, 16, &c., is an ascending geometrical progression, and 16, 8, 4, 2, 1, &c., is a descending geometrical progression. ARITHMETICAL PROPORTION AND PROGRESSION. THE first and last terms of a Progression are called the Extremes; and the other terms lying between them, the Means. The most useful part of arithmetical proportions is contained in the following theorems: THEOREM 1.-If four quantities be in arithmetical proportion, the sum of the two extremes will be equal to the sum of the two means. Thus, of the four 2, 4, 6, 8, here 2 + 8 = 4 + 6 = 10. Page 36 36 THE PRACTICAL MODEL CALCULATOR. THEOREM 2.-In any continued arithmetical progression, the sum of the two extremes is equal to the sum of any two means that are equally distant from them, or equal to double the middle term when there is an uneven number of terms. Thus, in the terms 1, 3, 5, it is 1 + 5 = 3 + 3 = 6. And in the series 2, 4, 6, 8, 10, 12, 14, it is 2 + 14 = 4 + 12= 6 + 10 = 8 + 8 = 16. THEOREM 3.-The difference between the extreme terms of an arithmetical progression, is equal to the common difference of the series multiplied by one less than the number of the terms. So, of the ten terms, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, the common difference is 2, and one less than the number of terms 9; then the difference of the extremes is 20 - 2 = 18, and 2 x 9 = 18 also. Consequently, the greatest term is equal to the least term added to the product of the common difference multiplied by 1 less than the number of terms. THEOREM 4.-The sum of all the terms of any arithmetical progression is equal to the sum of the two extremes multiplied by the number of terms, and divided by 2; or the sum of the two extremes multiplied by the number of the terms gives double the sum of all the terms in the series. This is made evident by setting the terms of the series in an inverted order under the same series in a direct order, and adding the corresponding terms together in that order. Thus, in the series, 1, 3, 5, 7, 9, 11, 13, 15;...... inverted, 15, 13, 11, 9, 7, 5, 3, 1; the sums are, 16 + 16 + 16 + 16 + 16 + 16 + 16 + 16, which must be double the sum of the single series, and is equal to the sum of the extremes repeated so often as are the number of the terms. From these theorems may readily be found any one of these five parts; the two extremes, the number of terms, the common difference, and the sum of all the terms, when any three of them are given, as in the following Problems: PROBLEM I. Given the extremes and the number of terms, to find the sum of all the terms. RULE.-Add the extremes together, multiply the sum by the number of terms, and divide by 2. The extremes being 3 and 19, and the number of terms 9; required the sum of the terms? 19 3 2 Or, 19 +- 3 22 22 Or 219 + 3 x 9 2 x 9 = 11 x 9 = 99. 2)198 99 = lthe sum. Page 37 ARITHMETICAL PROPORTION AND PROGRESSION. 37 The strokes a clock strikes in one whole revolution of the index, or in 12 hours, is 78. PROBLEM II. Given the extremes, and the number of terms; to find the common difference. RULE.-Subtract the less extreme from the greater, and divide the remainder by 1 less than the number of terms, for the common difference. The extremes being 3 and 19, and the number of terms 9; required the common difference? 19 3 19 - 3 16 8) 16 Or, 9-1 = 2. 2 If the extremes be 10 and 70, and the number of terms 21; what is the common difference, and the sum of the series? The com. diff. is 3, and the sum is 840. PROBLEM III. Given one of the extremes, the common difference, and the number of terms; to find the other extreme, and the sum of the series. RULE.-Multiply the common difference by 1 less than the number of terms, and the product will be the difference of the extremes: therefore add the product to the less extreme, to give the greater; or subtract it from the greater, to give the less. Given the least term 3, the common difference 2, of an arithmetical series of 9 terms; to find the greatest term, and the sum of the series? 2 8 16 3 19 the greatest term. 3 the least. 22 sum. 9 number of terms. 2) 198 99 the sum of the series. If the greatest term be 70, the common difference 3, and the number of terms 21; what is the least term and the sum of the series? The least term is 10, and the sum is 840. PROBLEM IV. To find an arithmetical mean proportional between two given terams. RULE. —Add the two given extremes or terms together, and take half their sum for the arithmetical mean required. Or, subtract D Page 38 38 THE PRACTICAL MODEL CALCULATOR. the less extreme from the greater, and half the remainder will be the common difference; which, being added to the less extreme, or subtracted from the greater, will give the mean required. To find an arithmetical mean between the two numbers 4 and 14. Here, 14 Or, 14 Or, 14 4 4 5 2)18 2)10 9 9 5 the com. dif. 4 the less extreme. 9 So that 9 is the mean required by both methods. PROBLEM V. To find two arithmetical means between two given extremes. RULE.-Subtract the less extreme from the greater, and divide the difference by 3, so will the quotient be the common difference; which, being continually added to the less extreme, or taken from the greater, gives the means. To find two arithmetical means between 2 and 8. Here 8 2 Then 2 + 2 = 4 the one mean, 3) 6 and 4 + 2 = 6 the other mean. com. dif. 2 PROBLEM VI. To find any number of arithmetical means between two given terims or extremes. RULE.-Subtract the less extreme from the greater, and divide the difference by 1 more than the number of means required to be found, which will give the common difference; then this being added continually to the least term, or subtracted from the greatest, will give the mean terms required. To find five arithmetical means between 2 and 14. Here 14 2 Then, by adding this com. dif. continually, 6 )12 the means are found, 4, 6, 8, 10, 12. corn. dif. 2 GEOMETRICAL PROPORTION AND PROGRESSION. THE most useful part of Geometrical Proportion is contained in the following theorems: THEOREM 1.-If four quantities be in geometrical proportion, the product of the two extremes will be equal to the product of the two means. Thus, in the four 2, 4, 3, 6 it is 2 x 6 = 3 x 4 = 12. And hence, if the product of the two means be divided by one of the extremes, the quotient will give the other extreme. So, of Page 39 GEOMETRICAL PROPORTION AND PROGRESSION. 39 the above numbers, the product of the means 12 - 2 = 6 the one extreme, and 12 —. 6 = 2 the other extreme; and this is the foundation and reason of the practice in the Rule of Three. THEOREM 2.-In any continued geometrical progression, the product of the two extremes is equal to the product of any two means that are equally distant from them, or equal to the square of the middle term when there is an uneven number of terms. Thus, in the terms 2, 4, 8, it is 2 x 8 = 4 x 4 = 16. And in the series 2, 4, 8, 16, 32, 64, 128, it is 2 x 128 = 4 x 64 = 8 x 32 = 16 x 16 = 256. THEOREM 3.-The quotient of the extreme terms of a geometrical progression is equal to the common ratio of the series raised to the power denoted by one less than the number of the terms. So, of the ten terms 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, the common ratio is 2, one less than the number of terms 9; then 1024 the quotient of the extremes is 2 = 512, and 29 = 512 also. Consequently, the greatest term is equal to the least term multiplied by the said power of the ratio whose index is one less than the number of terms. TIsEOREM 4.-The sum of all the terms of any geometrical progression is found by adding the greatest term to the difference of the extremes divided by one less than the ratio. So, the sum 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, (whose ratio 1024 - 2 is 2,)is 1024 + 2- 1 =1024 + 1022 - 2046. The foregoing, and several other properties of geometrical proportion, are demonstrated more at large in Byrne's Doctrine of Proportion. A few examples may here be added to the theorems just delivered, with some problems concerning mean proportionals. The least of ten terms in geometrical progression being 1, ant.L the ratio 2, what is the greatest term, and the sum of all the terims? The greatest term is 512, and the sum 1023. PROBLEM I. To find one geometrical mean proportional between any two numbers. RuLE. —Multiply the two numbers together, and extract the square root of the product, which will give the mean proportional sought. Or, divide the greater term by the less, and extract the square root of the quotient, which will give the common ratio of the three terms: then multiply the less term by the ratio, or divide the greater term by it, either of these will give the middle term required. To find a geometrical mean between the two numbers 3 andi 12. First way. Second way. 12 3 ) 12 (4, its root, is 2, the ratio. 3.36 (6 the mean. Then, 3 x 2 6 the mean. 36 Or, 12 -. 2 =6 also. Page 40 40 THE PRACTICAL MODEL CALCULATOR. PROBLEM II. To find two geometrical mean proportionals between any two numbers. RULE. —Divide the greater number by the less, and extract the cube root of the quotient, which will give the common ratio of the terms. Then multiply the least given term by the ratio for the first mean, and this mean again by the ratio for the second mean; or, divide the greater of the two given terms by the ratio for the greater mean, and divide this again by the ratio for the less mean. To find two geometrical mean proportionals between 3 and 24. Here, 3 ) 24 ( 8, its cube root, 2 is the ratio. Then, 3 x 2 = 6, and 6 x 2 = 12, the two means. Or, 24 -. 2 = 12, and 12 — 2 = 6, the same. That is, the two means between 3 and 24, are 6 and 12. PROBLE3I III. To find any number of geometrical mean plro2ortionals between tw o numbers. RULE.-Divide the greater number by the less, and extract such root of the quotient whose index is one more than the number of means required, that is, the 2d root for 1 mean, the 3d root for 2 means, the 4th root for 3 means, and so on; and that root will be the common ratio of all the terms. Then with the ratio lmultiply continually from the first term, or divide continually from the last or greatest term. To find four geometrical mean proportionals between 3 and 96. Here, 3) 96 ( 32, the 5th root of which is 2, the ratio. Then, 3 x 2= 6, and 6 x 2=12, and 12 x 2=24, and 24- x 2=48. Or, 96 - 2=48, and 48 2 —29 4, and 24 =:-912, and 1 —. = 6. That is, 6, 12, 24, 48 are the four means between 3 and 96. OF MUSICAL PROPORTION. ThERE is also a third kind of proportion, called AMusical, which, being but of little or no common use, a very short account of it may here suffice. Musical proportion is when, of three numbers, the first has the same proportion to the third, as the difference between the first and second has to the difference between the second and third. As in these three, 6, 8, 12; where, 6: 12:: 8- 6: 12 — 8, that is, 6:12:: 2: 4. When four numbers are in Musical Proportion; then the i..n..t has the same proportion to the fourth, as the difference bctwc;: the first and second has to the difference between the third an;.l fourth. As in these, 6, 8, 1O, 18; 1where, 6 18 - 6: 8 - 6', thatis, 6: 18::2: G. Page 41 FELLOWSHIP. 41 Wlhen numbers are in Musical Progression, their reciprocals are in Arithmetical Progression; and the converse, that is, when numbers are in Arithmetical Progression, their reciprocals are in Musical Progression. So, in these Musicals 6, 8, 12, their reciprocals 6, ~, 2, are in arithmetical progression; for i + A1 = - =-; and + -1 - - 1; thlat is, the sum of the extremes is equal to double the mean, which is the property of arithmeticals. FELLOWSHIP, OR PARTNERSHIP. FELLOWSHIP is a rule by which any sum or quantity may be divided into any number of parts, which shall be in any given proportion to one another. By this rule are adjusted the gains, or losses, or charges of partners in company; or the effects of bankrupts, or legacies in case of a deficiency of assets or effects; or the shares of prizes, or the numbers of men to form certain detachments; or the division of waste lands among a number of proprietors. Fellowship is either Single or Double. It is Single, when the shares or portions are to be proportional each to one single given number only; as when the stocks of partners are all employed for the same tinme: and Double, when each portion is to be proportional to two or more numbers; as when the stocks of partners are employed for different times. SINGLE FELLOWSHIP. GENERAL RULE.-Add together the numbers that denote the proportion of the shares. Then, As the sum of the said proportional numbers Is to the whole sum to be parted or divided, So is each several proportional number To the corresponding share or part. Or, As the whole stock is to the whole gain or loss, So is each man's particular stock to his particular share of the gain or loss. To pirove t/he worlc.-Add all the shares or parts together, and the sum will be equal to the whole number to be shared, when the work is right. To divide the number 240 into three such parts, as shall be in proportion to each other as the three numbers, 1, 2, and 3. Here 1 + 2 + 3 = 6 the sum of the proportional numbers. Then, as 6: 240:: 1: 40 the Ist part, and, as 6: 240:: 2: 80 the 2cd part, also as 6: 240:: 3: 120 the 3d part. Sum of all 240, the proof. Three persons, A, B, C, fieighted a ship with 340 tuns of wine' of which, A loaded 110 tuns, B 97, and C the rest: in a storm, the D2 Page 42 42 THE PRACTICAL MODEL CALCULATOR. seamen were obliged to throw overboard 85 tuns; how much must each person sustain of the loss? Here, 110 + 97 = 207 tuns, loaded by A and B; theref., 340 - 207 = 133 tuns, loaded by C. hence, as 340: 85:: 110 or, as 4: 1:: 110: 27 tuns =A's loss; and, as 4: 1:: 97: 241 tuns = B's loss; also, as 4 1:: 133: 33~ tuns = C's loss. Sum 85 tuns, the proof. DOUBLE FELLOWSHIP. DOUBLE FELLOWSHIP, as has been said, is concerned in cases in which the stocks of partners are employed or continued for different times. RULE.-Multiply each person's stock by the time of its continuance; then divide the quantity, as in Single Fellowship, into shares in proportion to these products, by saying: As the total sum of all the said products Is to the whole gain or loss, or quantity to be parted, So is each particular product To the corresponding share of the gain or loss. SIMPLE INTEREST. INTEREST is the premium or sum allowed for the loan, or forbearance of money. The money lent, or forborne, is called the Principal. The sum of the principal and its interest, added together, is called the Amount. Interest is allowed at so much per cent. per annum, which premium per cent. per annum, or interest of a $100 for a year, is called the Rate of Interest. So, When interest is at 3 per cent. the rate is 3;........................ 4 per cent............. 4;........................ 5 per cent............. 5;........................ 6 per cent..............6. Interest is of two sorts: Simple and Compound. Simple Interest is that which is allowed for the principal lent or forborne only, for the whole time of forbearance. As the interest of any sum, for any time, is directly proportional to the principal sum, and also to the time of continuance; hence arises the following general rule of calculation. GENERAL RULE.-AS $100 is to the rate of interest, so is any given principal to its interest for one year. And again, As one year is to any given time, so is the interest for a year just found to the interest of the given sum for that time. Othlerwise.-Take the interest of one dollar for a year, which, multiply by the given principal, and this product again by the time Page 43 POSITION. 43 of loan or forbearance, in years and parts, for the interest of the proposed sum for that time. When there are certain parts or years in the time, as quarters, or months, or days, they may be worked for either by taking the aliquot, or like parts of the interest of a year, or by the Rule of Three, in the usual way. Also, to divide by 100, is done by only pointing off two figures for decimals. COMPOUND INTEREST. COMPOUND INTEREST, called also Interest upon Interest, is that which arises from the principal and interest, taken together, as it becomes due at the end of each stated time of payment. RULES.-1. Find the amount of the given principal, for the time of the first payment, by Simple Interest. Then consider this amount as a new principal for the second payment, whose amount calculate as before; and so on, through all the payments to the last, always accounting the last amount as a new principal for the next payment. The reason of which is evident from the definition of Compound Interest. Or else, 2. Find the amount of one dollar for the time of the first payment, and raise or involve it to the power whose index is denoted by the number of payments. Then that power multiplied by the given principal will produce the whole amount. From which the said principal being subtracted, leaves the Compound Interest of the same; as is evident from the first rule. POSITION. POSITION is a method of performing certain questions which cannot be resolved by the common direct rules. It is sometimes called False Position, or False Supposition, because it makes a supposition of false numbers to work with, the same as if they were the true ones, and by their means discovers the true numbers sought. It is sometimes also called Trial and Error, because it proceeds by tr'ials of false numbers, and thence finds out the true ones by a comparison of the errors. Position is either Single or Double. SINGLE POSITION. SINGLE POSITION is that by which a question is resolved by means of one supposition only. Questions which have their results proportional to their suppositions belong to Single Position; such as those which require the multiplication or division of the number sought by any proposed number; or, when it is to be increased or diminished by itself, or any parts of itself, a certain proposed number of times. RULE.-Take or assume any number for that required, and perform the same operations with it as are described or performed in the question. Then say, as the result of the said operation is to the position Page 44 44 THE PRACTICAL MODEL CALCULATOR. or number assumed, so is the result in the question to the number sought. A person, after spending 1 and i of his money, has yet remaining $60, what had he at first? Suppose he had at first $120 Proof. Now - of 120 is 40 ~of144is 48 4 of it is 30 4L of 144 is 36 their sum is 70 their sum 84 which taken from 120 taken from 144 leaves 50 leaves 60 as per question. Then, 50: 120:: 60: 144. What number is that, which multiplied by 7, and the product divided by 6, the quotient may be 14? 12. PERMUTATIONS AND COMBINATIONS. THE Permutations of any number of quantities signify the changes which these quantities may undergo with respect to their order. Thus, if we take the quantities a, b, c; then, a b c, a c b, b a c, b c a, c a b, c b a, are the permutations of these three quantities taken all together; a b, a e, b a, b e, c a, e b, are the permutations of these quantities taken two and two; a, b, c, are the permutation of these quantities taken singly, or one and one, &c. The number of the permutations of the eight letters, a, b, e, d, e, f, g, A, is- 40320; becomes, 1. 2. 3. 4.5. 6. 7. 8 = 40320. DOUBLE POSITION. DOUBLE POSITION is the method of resolving certain questions by means of two suppositions of false numbers. To the Double Rule of Position belong such questions as have their results not proportional to their positions: such are those, in which the numbers sought, or their parts, or their multiples, are increased or diminished by some given absolute number, which is no known part of the number sought. Take or assume any two convenient numbers, and proceed with each of them separately, according to the conditions of the question, as in Single Position; and find how much each result is different from the result mentioned in the question, noting also whether the results are too great or too little. Then multiply each of the said errors by the contrary supposition, namely, the first position by the second error, and the second position by the first error. If the errors are alike, divide the difference of the products by the difference of the errors, and the quotient will be the answer. But if the errors are unlike, divide the sum of the products by the sum of the errors, for the answer. The errors are said to be alike, when they are either both too great, or both too little; and unlike, -when one is too grclt:? l:o other too little. Page 45 MENSURATION OF SUPERFICIES. 45 What number is that, which, being multiplied by 6, the product increased by 18, and the sum divided by 9, the quotient shall be 20. Suppose the two numbers, 18 and 30. Then First position. Second position. Proof. 18 30 27 6 mult. 6 6 108 180 162 18 add. 18 18 9) 126 9) 198 9)180 14 results. 22 20 20 true res. 20 + 6 errors unlike. - 2 2d pos. 30 mult. 18 1st pos. Errors { 2 180 36 36 Sum 8 ) 216 sum of products. 27 answer sought. Find, by trial, two numbers, as near the true number as possible, and operate with them as in the question; marking the errors which arise from each of them. Multiply the difference of the two numbers, found by trial, by the least error, and divide the product by the difference of the errors, when they are alike, but by their sum when they are unlike. Add the quotient, last found, to the number belonging to the least error, when that number is too little, but subtract it when too great, and the result will give the true quantity sought. MENSURATION OF SUPERFICIES. THE area of any figure is the measure of its surface, or the space contained within the bounds of that surface, without any regard to thickness. A square whose side is one inch, one foot, or one yard, &c. is called the measuring unit, and the area or content of any figure is computed by the number of those squares contained in that figure. To find the area of a parallelogram; whether it be a square, a rectangle, a rhombus, or a rhomboides.-Multiply the length by the perpendicular height, and the product will be the area. The perpendicular height of the parallelogram is equal to the area divided by the base. Required the area of the square ABCD whose side is 5 feet 9 inches. Here 5ft. 9 in. = 5'75: and 5.7512 = 5.75 x 5'75 = 33'0625 feet = 33fe. 0 in. 9 pa. = area required. A B Page 46 46 THE PRACTICAL MODEL CALCULATOR. Required the area of the rectangle D ABCD, whose length AB is 13'75 chains, and breadth BC 9-5 chains. Here 13'75 x 9'5 - 130'625; and 130'625 10 -- 13'0625 ae. = 13 ae. 0 ro. 10 A - 20. = area required. Required the area of the rhombus D ABCD, whose length AB is 12 feet 6 inches, and its height DE 9 feet 3 inches. Here 12fe. 6 in. = 12'5, and 9 fe. 3 in. 9'25. TWhence, 12'5 x 9'25 = 115'625fe. = // 115 fe. 7 in. 6 pa. = area required. A E B What is the area of the rhomboides ABCD, whose length AB is 10'52 chains, and height DE 7'63 chains. Here 10'52 x 7163 = 80'2676; 80-2676 and 10 802676 acres= 8 ac. 0 ro. 4po. area required. A B B To find the area of a triangle.-Multiply the base by thee, perpendicular height, and half the product will be the area. The perpendicular height of the triangle is equal to twice thle area divided by the base. Required the area of the triangle ABC, whose base AB is 10 feet 9 inches, and height DC 7 feet 3 inches. Here 10 fe. 9 in. = 10'75, and 7fe. 3 in. -- 725. Whence, 10'75 x 7'25 = 77'9375, and 77A9375, _ 2 = 38'96875 feet = 38 fe. 11 in. A 7~ pa. = area required. To find the area of a triangle whose three sides onl~y asre I.From half the sum of the three sides subtract each side sevcr;llv. Multiply the half sum and the three remainders continuPll.- t - ther, and the square root of the product will be the area rc~:?tic:' Required the area of the triangle ABC, whose three sides BC, CA, and AB are 24, 36, and 48 chains respectively. 24 + 36 + 48 108 ~ere 7 —- 2 =-54 2 sumg of the sides. Also, 54 - 24 = 30 first dclff.; 54 - 36 A = 18 second diff.; and 54 - 48 = 6 third diff. Page 47 M1ENSURATION OF SUPERFICIES. 47 WhVence, / 54 x 30 x 18 x 6 = v/ 174960 = 418'282 = area required. Any two sides of a right angled triangle being given to find the third side.-When the two legs are given to find the hypothenuse, add the square of one of the legs to the square of the other, and the square root of the sum will be equal to the hypothenuse. Wh/ten the hypothenuse and one of the legs are given, to find the other leg.-From the square of the hypothenuse take the square of the given leg, and the square root of the remainder will be equal to the other leg. In the right angled triangle ABC, the c base AB is 56, and the perpendicular BC 33, / what is the hypofhenuse? Here 562 + 332 = 3136 + 1089 = 4225, and V4225 = 65 = hypothenuse AC. If the hypothenuse AC be 53, and the base AB 45, what is the perpendicular BC? A B Here 532 - 452 = 2809 - 2025 = 784, and V784 = 28 - perpendicular BC. To find the area of a trapeziumn.-Multiply the diagonal by the sum of the two perpendiculars falling upon it from the opposite angles, and half the product will be the area. Required the area of the trapezium D BAED, whose diagonal BE is 84, the perpendicular AC 21, and DF 28. Here 28 + 21 x 84 = 49 x 84=4116, 4116 and 2 = 2058 the area required. A To find the area of a trapezoid, or a qcadranglcle, two o.f irose" opposite sides are parallel.-Multiply the sum of the parallel sides by the perpendicular distance between them, and half the product will be the area. Required the area of the trapezoid ABCD, whose sides AB and DC are 321'51 and c e 214-24, and perpendicular DE 171-16. Here 32151 + 214-24 = 535'75 = suXn of the parallel sides AB, DC. Whence, 535'75 x 171'16 (the perp. DE) /= i 91698'9700 A B 91698'9700, and = 45849'485 the area 1equilecd. To find the area of a regular polygon. —IMultiply half the perimeter of the figure by the perpendicular falling from its centre upon one of the sides, and the product will be the area. The perimeter of any figure is the sum of all its sides. Page 48 48 THE PRACTICAL MODEL CALCULATOR. Required the area of the regular pentagon D ABC1DE, whose side AB, or BC, &c., is 25 feet, and the perpendicular OP 17'2 feet. Here 25 x 5 625 = half perimeter, / and 62'5 x 17'2 = 1075 square feet = area required. A P B To find the area of a regular polygon, when the side only il given.-Multiply the square of the side of the polygon by the number standing opposite to its name in the following table, and'< the product will be the area. N.o Names. Multipliers. Names. Multipliers. sides. sides. 3 Trigon or equil. A 0'433013 8 Octagon 4828-127 4 Tetragon or square 1'000000 9 Nonagon 6.18182'1 5 Pentagon 1[720477 10 Decagon 7.694209 6 Hexagon 2-598076 11 Undecagon 9'365640 7 Heptagon 38633912 12 Duodecagon 11 1916132 The angle OBP, together with its tangent, for any polygon of not more than 12 sides, is shown in the following table: No. of Angle sides. Names. O. Tangents. 3 Trigon 30~ 0 57735 - - i 3 4 Tetragon 450 1'00000 = 1 X 1 5 Pentagon 540 1 37638 -- /1 + - 5- X 6 Hexagon 600 1.73205 = %/3 7 Heptagon 64~0 2'07652 8 Octagon 67O~ 2-41421 = 1+ /2 9 Nonagon 700 2-74747 10 Decagon 720 3.07768 = /5 + 2 /5 11 Undecagon 7317 3.40568 12 Duodecagon 750 3-73205 = 2 -- 3 Required the area of a pentagon whose side is 15. The number opposite pentagon in the table is 1'720477. Hence 1'720477 x 152 = 1720477 x 225 - 387'107325 - area required. Tlhe diameter of a circle being given to find the cir5ctmference. or the circumference being given to find the diacrctcr. —MAultiply the diameter by 341416, and the product will be the circumference, or Divide the circumference by 3'1416, and the quotient will be the diameter. As 7 is to 22, so is the diameter to the circumference; or as 42 is to 7, so is the circumference to the diameter. As 113 is to 355, so is the diameter to the circumference; or, as 352 is to 115, so is the circumference to the diameter. Page 49 MENSURATION OF SUPERFICIES. 49 If the diameter of a circle be 17, what is the circumference? Here 3'1416 x 17 = 53'4072 = circumference. If the circumference of a circle be 354, what is the diameter? 354'000 Here 351416 = 112'681 = diameter. To find the length of any are of a circle.-When the chord of the arc and the versed sine of half the arc are given: To 15 times the square of the chord, add 33 times the square of the versed sine, and reserve the number. To the square of the chord, add 4 times the square of the versed sine, and the square root of the sum will be twice the chord of ha"f the are. Multiply twice the chord of half the arc by 10 times the square of the versed sine, divide the product by the reserved number, and add the quotient to twice the chord of half the are: the sum will be the length of the arc very nearly. When the chord of the are, and the chord of half the are are given.-From the square of the chord of half the arc subtract the square of half the chord of the arc, the remainder will be the square of the versed sine: then proceed as above. When the diameter and the versed sine of half the arc are given: From 60 times the diameter subtract 27 times the versed sine, and reserve the number. Multiply the diameter by the versed sine, and the square root of the product will be the chord of half the are. Multiply twice the chord of half the arc by 10 times the versed sine, divide the product by the reserved number, and add the quotient to twice the chord of half the arc; the sum will be the length of the arc very nearly. When the diameter and chord of the arc are given, the versed sine may be found thus: From the square of the diameter subtract the square of the chord, and extract the square root of the remainder. Subtract this root from the diameter, and half the remainder will give the versed sine of half the arc. The square of the chord of half the arc being divided by the diameter will give the versed sine, or being divided by the versed sine will give the diameter. The length of the arc may also be found by multiplying together the number of degrees it contains, the radius and the number'01745329. Or, as 180 is to the number of degrees in the arc, so is 3'1416 times the radius, to the length of the arc. Or, as 3 is to the number of degrees in the arc, so is'05236 times the radius to the length of the arc. c If the chord DE be 48, and the versed sine D CB 18, what is the length of the arc? B Here 482 x 15 = 34560 182 x 33 = 10692' 45252 reserved number. E 4 Page 50 50 THE PRACTICAL MODEL CALCULATOR. 482 = 2304 = the square of the chord. 182 x 4 = 1296 = 4 times the square of the versed sine. 4 3600 = 60 = twice the chord of half the are. 60 x 182 x 10 194400 Now 45252 W 45252 = 4'2959, which added to twice the chord of half the are gives 64'2959 = the length of the are. 50 x 60 = 3000 18 x 27 = 486 2514 reserved number. AC = V50 x 18 = 30 = the chord of half the are. 30 x 2 x 18 x 10 10800 2514 = 2514 = 42959, which added to twice the chord of half the are gives 64'2959 = the length of the are. To find the area of a circle. —Multiply half the circumference by half the diameter, and the product will be the area. Or take I of the product of the whole circumference and diameter. What is the area of a circle whose diameter is 42, and circumference 131'946? 2) 131'946 65'973 = circumference. 21 = 2 diameter. 65973 131946 1385'433 = area required. What is the area of a circle whose diameter is 10 feet 6 inches, and circumference 31 feet 6 inches? fe. in. 15 9 = 15'75= -- circumference. 5 3 = 5'25 = diameter. 7875 3150 7875 82.6875 12 8'2500 82 feet 8 inches. Multiply the square of the diameter by'7854, and the product will be the area; or, Multiply the square of the circumference by'07958, and the product will-be the area. The following table will also show most of the useful problems relating to the circle and its equal or inscribed square. Diameter x *8862 = side of an equal square. Circumf. x'2821 = side of an equal square. Diameter x'7071 = side of the inscribed square. Page 51 MENSURATION OF SUPERFICIES. 51 Circumf. x'2251 = side of the inscribed square. Area x'6366 = side of the inscribed square. Side of a square x 1'4142 = diam. of its circums. circle. Side of a square x 4'443 = circumf. of its circums. circle. Side of a square x 1'128 = diameter of an equal circle. Side of a square x 3'545 = circumf. of an equal circle. What is the area of a circle whose diameter is 5? 7854 25 = square of the diameter. 39270 15708 19'6350 = the answer. To find the area of a sector, or that part of a circle whieh ie bounded by any two radii and their included arc.-Find the length of the arc, then multiply the radius, or half the diameter, by the length of the arc of the sector, and half the product will be the area. If the diameter or radius is not given, add the square of half the chord of the arc, to the square of the versed sine of half the arc; this sum being divided by the versed sine, will give the diameter. The radius AB is 40, and the chord BC of the whole arc 50, required the area of D the sector. c B 80- V/802 - 502 = 8'7750 = the versed sine of half the are. 80 x 60 - 8'7750 x 27 = 4563'0750 the reserved number. 2 x V/87750 x 80 = 52'9906 = twice the chord of half the are. 52.9906 x 8.7750 x 10 4563.0750 = 10190, which added to twice the chord of half the are gives 54'0096 the length of the are. 54'0096 x 40 And 2 = 108041920 = area of the sector required. As 360 is to the degrees in the arc of a sector, so is the area of the whole circle, whose radius is equal to that of the sector, to the area of the sector required. For a semicircle, a quadrant, &c. take one half, one quarter, &c. of the whole area. The radius of a sector of a circle is 20, and the degrees in its arc 22; what is the area of the sector? Here the diameter is 40. Hence, the area of the circle = 402 X'7854 = 1600 x'7854 1256 64. Now, 360: 22~:: 1256'64: 76'7947 = area of the sector. Page 52 52 THE PRACTICAL MODEL CALCULATOR. To find the area of a segment of a circle.-Find the area of the sector, having the same arc with the segment, by the last problem. Find the area of the triangle formed by the chord of the segment, and the radii of the sector. Then the sum, or difference, of these areas, according as the segment is greater or less than a semicircle, will be the area required. The difference between the versed sine and radius, multiplied by half the chord of the arc, will give the area of the triangle. The radius OB is 10, and the chord AC 10; what is the area of the segment ABC? c AC2 100 CD 5 =the versed sine Aa of half the are. 20 x 60 - 5 x 27 = 1065 -- the reserved nu mber. 10 x 2 x 5 x 10 1065 = -9390, and this added to twvice the chord of half the are gives 20'9390 = the length of the are. 20'9390 x 10 2 = 104'6950 = area of the sector OACB. OD = OC = CD = 5 the perpendicular height of the triacgle. AD = VA02 - OD2 = V75 - 8'6603 = -1 the chord of the are. 8'6603 x 5 = 43'3015 = the area of the triangle AOB. 104'6950 - 43'3015 = 61'3935 = area of the segment redpaired; it being in this case less than a semicircle. Divide the height, or versed sine, by the diameter, and find the quotient in the table of versed sines. Multiply the number on the right hand of the versed sine by the square of the diameter, and the product will be the area. When the quotient arising from the versed sine divided by the diameter, has a remainder or fraction after the third place of decimals; having taken the area answering to the first three figures, subtract it from the next following area, multiply the remainder by the said fraction, and add the product to the first area, then the sum will be the area for the whole quotient. If the chord of a circular segment be 40, its versed sine 10, and the diameter of the circle 50, what is the area? 5.0) 1.0 *2 = tabular versed sine. *111823 = tabular segment. 2500 = square of 50. 55911500 223646 279'557500 = area required. Page 53 MENSURATION OF SUPERFICIES. 53 To find the area of a circular zone, or the space included between any two parallel chords and their intercepted arcs.-From the greater chord subtract half the difference between the two, multiply the remainder by the said half difference, divide the product by the breadth of the zone, and add the quotient to the breadth. To the square of this number add the square of the less chord, and the square root of the sum will be the diameter of the circle. Now, having the diameter EG, and the two chords AB and DC, find the areas of the segments ABEA, and DCED, the difference of which will be the area of the zone required. The difference of the tabular segments multiplied by the square of the circle's diameter will give the area of the zone. lWhen the larger segment AEB is greater than a semicircle, find the areas of the segments AGB, and DCE, and subtract their sum from the area of the whole circle: the remainder will be the area of the zone. The greater chord AB is 20, the less DC 15, E and their distance Dr 17~: required the area of the zone ABCD. 20- 15 71 25 - 1= the difference between / the chords.' xr,, i, - -B(20 - 2.5) x 2.5' + 9.5 17.5 + =17.5 + 2=5 i 17'5 20 = DF. And V202 + 152 = V625 = 25 = the diameter of the circle. Thre segment AEB being greater than a semicircle, we fincd the versed sine of DCE = 2'5, and that of AGB = 5. 2'5 Hence -25 ='100 = tabular versed sine of DEC. 5 And2 ='200 = tabular versed sine of AGB. Now'040875 x 252 = area of seg. DEC = 25'546875 And'111823 x 252 = area of seg. AGB = 69'889375 sum 95'43625'7854 x 252 = area of the whole circle, = 490'87500 Difference = area of the zone ABCD = 395'43875 To find the area of a circular ring, or the A space included between the circumference of two concentric circles.-The difference between the areas of the two circles will be the area of - ---------- T the ring. Or, multiply the sum of diameters by their difference, and this product again by'7854, and it will give the area required. The diameters AB and CD are 20 and 15: required the area of 2 Page 54 54 THE PRACTICAL MODEL CALCULATOR. the circular ring, or the space included between the circumferences of those circles. Here AB + CD x AB - CD =35 x 5=175, and175 x'7854 = 1]37'4450 = area of the ring required. To find the areas of lunes, or the spaces between the intersecting arcs of two eccentric circles.-Find the areas of the two segments firom which the lune is formed, and their difference will be the area required. The following property is one of the most curious: If ABC be a right angled triangle, G and semicircles be described on the three sides as diameters, then will the said tri- C D angle be equal to the two lunes D and F A taken together. For the semicircles described on AC and BC = the one described on AB, from each A B take the segments cut off by AC and BC, then will the lunes AFCE and BDCG = the triangle ACB. The length of the chord AB is 40, the height DC 10, and DE 4: required the area, of the lune ACBEA. TIle diameter of the circle of which ACB 202 + 102 is a part — = 0 -- = 50. A 10 A - 202 + 42 Acd the diamieter of the circle of which AEB is apart = 104. N2ow having the diameter and versed sines, we fibnd, The area of seg. ACB = -111823 x 502 = 279'5575 And area of seg. AEB -- 009955 x 1042- = 107'6733 Th.eir difference is the area of the lune }- 171'8842 AEBCA required, To fnd the area of an irregular polygon, or a figure of any z1m1,)ber of sides.- Divide the figure into triangles and trapeziums, and find the area of each separately. Add these areas together, and the sum will be equal to the area of the whole polygon. a Required the area of the irre- D gular figure ABCDEFGA, the following lines being given: GB = 30'5 An - 112, CO = 6 \ GD = 29 Fq =11 Cs =6 6 FD = 24.8 Ep = 4...... An + Co 11'2 + 6 Hcee 2x GB= 2 x 30.5 + 8.6 x 30.5 = 2623 = area of the trapezium ABCG. A Page 55 DECIMAL APPROXIMATIONS. 55 And F C x GD 11 + 66 x 29 = 8.8 x 29 = 2552 = area of the trapeziumn GCDF. FD x Ep 24'8 x 4 99'2 Also, - - 2 4926 = area of the triangle FDE. Whence 262'3 + 255'2 + 49'6 = 5671 = area of the?vhole figure required. DECIMAL APPROXIMATIONS FOR FACILITATING CALCULATIONS IN MENSURATION. Lineal feet multiplied by'00019 = miles. - yards - 000568 = - Square inches -007 = square feet. - yards 0002067 = acres. Circular inches - 00546 = square feet. Cylindrical inches -0004546 = cubic feet. feet - 02909 = cubic yards. Cubic inches 00058 = cubic feet. feet - 03704 = cubic yards. -- -- -- 6'232 = imperial gallons. - inches *- 003607 = Cylindrical feet - 4'895 inches - -002832 Cubic inches -'263 = lbs. avs. of cast iron. -'281 = - wrought do. -'283 = - steel. -- *3225 = - copper. --'3037 = - brass. *- 26 = - zinc. -- 4103 = - lead. --.2636 = - tin. - *4908 = - mercury. Cylindrical inches -'2065 = - cast iron. -- -'2168 = - wrought iron. -.2223 = - steel. -- *2533 = - copper. - -.2385 = - brass..2042 = - zinc. -- 3223 = lead. 207 = - tin. --'3854 = - mercury. Avoirdupois lbs. - 009 = cwts. - -'00045 = tons. 183'346 circular inches = 1 square foot. 2200 cylindrical inches = 1 cubic foot. French metres x 3'281 = feet. - kilogrammes x 2'205 = avoirdupois lb. - grammes x'002205 = avoirdupois lbs. Page 56 56 THE PRACTICAL MODEL CALCULATOR. Diameter of a sphere x'806 = dimensions of equal cube. Diameter of a sphere x'6667 = length of equal cylinder. Lineal inches x'0000158 = miles. A French cubic foot = 2093'47 cubic inches. Imperial gallons x *7977 = New York gallons. The average quantity of water that falls in rain and snow at Philadelphia is 36 inches. At West Point the variation of the magnetic needle, Nov. 16th, 1839, was 70 58' 21" West, and the dip 730 26' 28". DECIMAL EQUIVALENTS TO FRACTIONAL PARTS OF LINEAL MEASURES. One inch, the integer or whole number. *96875 + 8'625' |28125 + L.9375 7+ -L'59375 2 +'25'90625 7 + 1'56205 + -11' 21875 8 + 3'875 8'53125 - t + 1_'1875 4 8+ 1G r —q1~~~~~~~~~~~~~~1'84375 X'5 w' 15625 - +'8125 -4'46875 + 3'125 78125 - + - 4375 ) 8-'1 09375.'75 -' 40625 ~ I8 *0625 -1'718T75 + 3 375' 1 03125 1'6875 8 + 34375 +'65625 - + -2 i'3125 1 + 6 One foot, or 12 inches, the integer.'9166 11 inches.'4166, 5 inches.'0625 of in. -6338 10 - 333 333 4 - 0 0508 0'75 9 - 25 3 -'04166 2 -'6666' 8 -'1666 2 -'03125 T -'583 3 7 -'0833 1 - *02083 _ *5 6 - -' 07291 -.101041c31 _ _ _ _ _ _ _ _ _ _. __. __ _ 8 _ One yard, or 36 inches, the integer.'9722 35 inches.-i 6389 23 inclhes.'3055 11 inches.'9444 34 6111 22 - 2 i8 10 -'9167 33 - 5833 21 -'5 9 -'8889 32 5' 5556 20 0- 2222 8'8611 31 -578 19 - 1944 7 -'833 30 - P 18 - I 1667 6 - S 05 6 029 - 4722 1 -'1389 Z 5 777 S 28 -'444 16 - 1111 4 5 227 - 4167'15 - 0833 3 3 T2;2 26 - 83 14 -'055 2 6944 -25'3611 3 - 078 12'66Hl 24 - 2:3 9 1 - Page 57 Table containing the Circunmferences, Squares, Cubes, and Areas of Circles, from 1 to 100, advancing by a tenth. Diam. Cireum. Square. Cube. Area. Diam. Circum. Square. Cube. Area. 1 3-1416 1 1 -7834 9 28-2744 81 729 63-6174 -1 3-4557 1'21 1'331 9503 -1 285885 82-81 753-571 650389.2 37699 1'44 1.728 11309 -2 289027 8464 778688 664762.3 4-0840 1'69 2'197 13273 3 292168 8649 804357 679292 4 4-3982 196 2I744 1'5393' 4 295310 88036 830'581 69-3979 5 4'7124 2*25 3-375 17671' 5 29-8452 90'25 857.375 70-8823.6 5-0265 2560 4-096 2-0106' 6 30'1593 92-16 884-736 72'3824.7 5-3407 2.89 4-913 2-2698' 7 30-4735 9409 912'673 73'8982 -8 56548 3-24: 5'832 2-5446 -8 30-7876 96-04 941'192 75-4298 -9 5'9690 3-61 6-859 2-8352 9 31.1018 98-01 970.299 76.9770 2 6-2832 4 8 3-1416 10 31'4160 100 1000 7835400 1 6-5973 4-41 9.261 3-4636 -1 31-7301 102.01 1030.301 80-1186 -2 6-9115 4-84 10'648 3'8013 [ 2 32-0443 104-04 1061'208 81'7130 3 7-2256 5-29 12'167 4'1547' 3 32-3580 106'09 1092'727 83'3230 4 7'5398 5-76 13'824 4-5239 - 4 32'6726 108'16 1124'864 8409188 5 7-8540 6.25 153625 4-9087' 5 32-9868 110-25 1157'625 86'5903.6 8-1681 6-76 17-576 5-3093' 6 33-3009 112'36 11-91'016 88'2475 7 8-4823 7'29 19'683 5'7255 7 33-6151 114'49 1225-043 89-9204 -8 8-7961 7'84 21'952 6'1575 -8 33-9292 116-64 1259.712 91-6090.9 9-1106 8-41 24-389 6-6052' 9 34.2434 118-81 1295029 93I133 3 94248 9 27. 7-0686 11 34-5576 121 1331 95'034 1 9'7389 9-61 29'791 7-5476'1 34-8717 123.21 1367-631 96'7691 2 10-0531 10-24 32'768 8-0424 *2 351859 125,44 1404-928 98-5205 3 10.3672 10.89 35-937 8-5530' 3 35-5010 127-69 1442.897 100-2877 4 10.6814 11-56 39'304 9-0792 -4 35-8142 129.96 1481-544 102-0705 5 10-9956 12-25 42-875 c -6211' 5 36.1284: 132-25 1520-'875 103'8691 -6 11-3097 12.96 46-656 10-'1787 -.6 36-4425 135156 1560-896 /105'6834: 7 11'6239 13-69 50-653 10-7521.7 36.7567 136-89 1601-613 107'5124 8 11-9380 14-44 54-872 11-3411. 8 37-0708 139-24 1643.032 109-3590 9 12-2522 15-21 59-319 11-9459 -9 37-3840 141-61 1685-159 111'2204 4 12-5664 16 64 12-5664 12 37'6992 144 1728 1130976 -1 12-8805 16.81 68-921 13-2025.1 38'0133 146-41 1771'561 114-9904 2 13.1947 17-64 74-088 13-8544 -2 38-3275 148'84 1815e848 116'8989 3 13-5088 18-49 79-507 14-5220.3 38-6416 151-29 1860-867 118'82?1 4 13-8230 19.36 85'184 15'2053 [ 4 38-9558 153-76 1906-624 120.i631 5 14-1372 20-25 91.125 15-9043.5 39.2700 156.25 1953-125 122-7187 -6 14-4513 21.16 97-336 16-6190 [ 6 39-5841 158.76 2000.376 124.6901 7 14-7655 22-09 103-823 17-3494 -7 39-8983 161.29 2048-383 126.6771 8 15-0796 23-04 110.592 18-0956.8 40-2124 163-84 2097-152 1286799 9 15-3938 24-01 117-649 18-8574' 9 40-5266 166-41 2146'689 130-6084 5 15-7080 25 125 19-6350 13 40-8408 169 2197 132'7326 1 16-0225 26-01 132-651 20-4282.1 41.1549 171-61 2248-091 134-7 824 -2 16-3363 27-04 140-608 21-2372'2 41-4691 174-24 2299-968 /36-8480 3 16-6504 28-09 148-877 22-0618' 3 41'-7832 176-89 2352-637 158-9294 4 16'9646 29-16 157-464 22-9022' 4 42-0974 179-56 2406'104 141-0264 5 17-2788 30-25 166-375 23-7583' 5 42-4116 182-25 2460-375 143-1391 -6 17-5929 31-36 175-616 24-6301 -6 42-7257 184-96 2515-456 145-2675 7 17-9071 32-49 185.193 25-5176.7 43-0399 187-69 2571-353 147-4117 8 18-2212 33-64 195-112 26-4208 *8 43-3540 190-44 2628-072 149-5715 -9 18-5354 34-81 205-379 27-3397.9 43-6682 193-21 2685-619 151-7471 6 18-8496 36 216 28-2744 14 43-9824 196 2744 153-9841 -1 19-1637 37-21 226-981 29-2247 -1 44-2965 198-81 2803-221 156-'1453 -2 19-4779 38-44 23s8-328 30-1907 [ 2 44-6107 201-64 2863-288 1S8-7680 3 19-7920 39-69 250-047 31-1725. 3 44-9248 204'49 2924-207 160-6064 4 20-1062 40-96 262-144 32-1699. 4 45-2390 207'36 2985-9841 162-8605 5 20-4204 42-25 274-625 33-1831' 5 45-5532 210-25 3048-625 165-1303 -6 20-7345 43-56 287-496 34-2120 -6 45-8673 213-16 3112-136 167-4158 -7 21-0487 44-89 300-763 35-2566 -7 46-1815 216-09 3176-523 169-7179 -8 21-3628 46-24 314-432 36-3168 -8 46-4956 219-04 3241-792 172-0340 9 2l-6770 47-61 328-509 37-3928 -9 46-8098 222-01 3307-949 174-3666 7 21-9912 49 343 38-4846 15 47-1240 225 3375 1'76-7150 1 22-3053 50-41 357-911 39-5920 1 47-4381 228-01 3442-951 179-0790 -2 22-6195 51-84 373-248 40-7151 *2 47-7523 231-04 3511-808 181-4588 -3 22-9336 53-29 389-017 41-8539 -3 48-0664 234-09 3581-577 183-8542 -4 23-2478 54-76 405-224 430085 -4 48-3806 237-16 3652-264 186-2654 -5 23-5620 56-25 421-875 44-1787 -5 48-6948 240-25 3723-875 188-6923 -6 23-8761 57-76 438-976 45-3647 -6 49-0089 243-36 3796-416 191-1349 -7 24-1903 59-29 456-533 46-5663 -7 49-3231 246-49 3869-893 193-5932 -8 24-5044 60-84 474-552 47-'7837 -8 49-6372 249-64 3944-312 196-0672 -9 24-8186 62-41 493-039 49-0168 -9 49.9514 252-81 4019-679 198-5569 8 25-1328 64 512 50-2656 16 50-2656 256 4096 201-0624 1 25-4469 65-61 531-441 51-5300 -1 50-5797 259'21 4173-281 203-5835 -2 25-7611 67-24 551-368 52-8102.2 50-8939 262-44 4251-528 206-1209 -3 26-0752 68-89 571-787 54-1062' 3 51-2080 265-69 4330-747 208-6723 -4 26-3894 70.56 592-'704 55-4178 -4 51-5224 268-96 4410-944 211-1411 -5 26-7036 72-25 614-125 56-7451 -5 51-8364 272-25 4492-125 213-8251 -6 27-0177 73-96 636-056 58-0881 -6 52-1505 275-56 4574-296 216-4248 -7 27-.3319 75-69 658-503 59-4469 -7 52-4647 278-89 4657-463 -219-0402 -8 27'-640 77'-44 681-472 60-8213 -8 52-7788 282-24 4741-632 221-6712 9 27-9602 79-21 704-969 62-2115 -9 53-0930 285-61 4826-809 224-3180 57 Page 58 58 THE PRACTICAL MODEL CALCULATOR. Diam. Circum. Square. Cube. Area. Diam. Circum. Square. Cube. Area. 17 53 4072 289 4913 22659806 25 78-5400 625 15625 490-8750 -1 53-7213 292-41 5000'211 229'6588 [ 1 78'8541 630'01 1581:3251 494-8098 *2 54 0355 295'84 5088.448 232'3527 *2 7901683 635-04 16003.008 49S87604 *3 54-3496 299-29 5177-717 235-0623 *3 79-4824 640'09 16194-277 502-7266 ~4 54'6038 302-76 5268-024 237-7877, 4 79-7966 645-16 16387-064 50B-7086 *5 54-9780 306-25 5359.375 240.5287 *5 80o8108 650.25 16581.375 510-7063 *6 55'2921 309-76 5451-776 243-2855 *6 80'4249 655'36 16777-216 514'7196 *7 55'6063 313-29 5545233 246.0579 7 807391 660'49 16974.593 518S74S8 *8 55.9204 316-84 5639-752 248.8461 *8 81-0532 665364 17173-512 522-7936.9 56-2346 320'41 5735'339 251'6500.9 81'3674 670'81 17373'979 526S8541 18 56'5488 324 5832 254 4696 26 81 6816 676 17576 530,9304 1 56'8629 327'61 5929-741 257-3048 ] 1 81-9976 681'21 17779'581 535'0223 *2 57-1771 331-24 6028-568 260-1558 *2 82-3099 686-44 17984-728 539-1299 *3 57*4912 334-89 6128-487 263.0226 *3 82-6240 691-69 18191-447 543-2533 *4 57-8054 338-56 6229-504 265.9050 *4 82-9382 696-96 18399744 547'13923 5 58-1196 342'25 6331-625 268'8031 *5 83-2524 702-25 18609-625 551'5471 *6 58-4337 345-96 6434-856 271-7169 *6 83-5665 707-56 18821-096 555'7176 *7 58'7479 349'69 6539'203 274'6465 *7 83 8807 712.89 19034'163 55909038 *8 59-0620 353.44 6644.672 277-5917 *8 84-1948 718t24 19248-832 564-1056 9 59-3762 357-21 6751-269 280-5527 I 9 84-5090 723-61 19465-109 568S3232 19 59 6904: 361 6859 283-5294 27 84-8232 729 19683 572-5566 *1 60-0045 364-81 6967-871 286-5217 H 1 85-1373 734-41 19902-511 5716'8056 *2 60.3187 368-64 7077-888 289-5298 *2 85-4515 739'84 20123-648 581-0703 *3 60.6328 372.49 7189'057 292.5536 3 ~ 85-7656 745-29 20346'417 585-3507 *4 60 9470 376.36 7301-384 295 5931 *4 86 0798 750 76 20570824 589-6469;5 61-2612 380-25 7414-875 298-6483 *5 86-3940 756'25 20796'875 59359587 *6 61.5753 384.16 7529.536 3017192 *6 86-7081 761-76 21024-576 59852863 *7 61-8895 388-09 7645-373 304-8060 *7 87-0223 767-29 21253-933 602-6295 *8 62-2036 392-04 7762-392 307-9082 *8 87-3364 772-84 21484-952 606-9885.9 62'5178 396-01 7880-599 311-0252.9 87-6506 778-41 21717-639 611-3632 20 62-8320 400 8000 314-1600 28 87-9648 784 21952 615-7536 1 63-1461 404 01 8120 601 317 3094 1 88 2789 789-61 22188-041 6201-596 *2 63-4603 408-04 8242-408 320-4746 I 2 88-5931 795'24 22425-768 624-5814 *3 63-7744 412-09 8365-427 323-6554 I 3 88-9072 800'89 22665-187 629-0190 *4 64-0886 416-16 8489-664 326 8520 *4 89-2214 806-56 22906-304 633-4722 *5 64-4028 420-25 8615-125 330-0643 *5 89-5356 812-25 23149-125 637'9411 *6 64-7161 424.-36 8741-816 333-2923 *6 89-8497 817-96 23393-656 642'4257 *7 65-0311 428-49 8869-743 336-5360 *7 90-1639 823-69 23639-903 646-9261 *8 65-3452 432-64 8998-912 339'7954 *8 90-4780 829'44 23887-872 651-4421 *9 65.6594 436-81, 9129-329 343.0705 [ 9 90.7922 835'21 24137'569 655'9739 21 65 9736 441 / 9261 346'3614 29 91-1064 841 24389 660'5214 1 66.2870 445.21 9393'931 349.6679 I 1 91.4205 846.81 24642.171 665'0845 *2 66.6012 449.44 9528 128 352'9901 *2.91 7347 852'64 24897'088 669'6634 *3 66'7916 453.69 9663'597 356.3281 *3 92.0488 858-49 25153'757 674.2580 *4 67 2930 457 96 9800'344 359.6817 *4 92'3630 864'36 25412'184 678'8683 *5 67'5444 462'25 9938'375 363.0511 *5 92'6772 870'25 25672'375 683'4943 6 67'8585 466'56 10077-696 366'4362 *6 92'9913 876'16 25934-336 688'1360 *7 68.1727 470'89 10218[313 369 8370 *7 93'3055 882'09 26198'073 692'7934 *8 68.4868 475'24 10360'232 373'2534 *8 93'6196 888'04 26463'592 697'4666.9 68.8010 479'61 10503'459 376.6856 1 9 93l9338 894 01 26730 899 702 1554 22 69 1152 484 10648 38091336 30 942480 900 27000 706 8600 *1 69'4293 488'41 10793'861 383.5972 I 1 94 5621 90601 27270901 7115802.2 69.7435 492.84 10941-048 387'0765.2 94.8763 912.04 27543'608 716'3162.3 70'0576 497,29 11089'567 390'5751 *3 95'1904 918'09 27818'127 721'0678 *4 70'3718 501'76 11239'424 394'0823 1 4 95-5046 924-16 28094'464 725'8352.5 70'6860 506.25 11390-625 397'6087.5 95.8188 930'25 28372'625 730'6183.6 71,0001 510 76 11543-176 401-1509.6 96.1329 936.36 28652.616 735.4171 *7 713143 515129 11697'083 404.7087 *7 96-4471 942.49 28934.443- 740'2316 *8 71-6284 519'84 11852'352 408'2823 *8 96 7612 948'64 29218'112 745'0618.9 719426 524.41 12008'989 411-8716.9 97.0754 954'81 29503.629 74909077 23 72 2568 529 12167 415-4766 31 97'3896 961 29791 754'7694 *1 72'5709 533'61 12326'391 419.0972 J 1 97'7037 967-21 -30080o231 759'6467 *2 72'8851 538'24 12487-168 422'7336 *2 98 0179 973 44 30371-328 764-5397 *3 73 1992 542-89 12649-337 426-3858 *3 98.3320 979'69 30664'297 769-4485.4 73 5134 547 56 12812 904 430 0536.4 98'6452 985'96 30959'144 774-3729.5 73'8276 552.25 12977.875 433.7371 *5 98.9604 992-25 31255.875 779'3131 *6 7441417 556.96 13144'256 437.4363 *6 99.2745 998'56 31554-496 784'2689 *7 74.4559 561.69 13312.053 441-1511 *7 99.5887 1004-89 31855'013 789'2406 *8 74'7680 566'44 13481-272 444J8819 *8 99'9028 1011'24 32157'432 794-2278 9 75'0852 571-21 13651-919 448.6283 8 9 100.2170 1017.61 32461.759 79902308 24 75 3984 576 13824 452-3904 32 100.5312 1024 32768 804'2496 1 75.7125 580.81 13997.541 456.1681 1 100.8453 1030.41 33076-161 809'2840 *2 76.0267 585,64 14172.488 459.9616 *2 101-1595 1036 84 38386 248 8143341 *3 76 3408 590 49 14348.907 463'7708 *3 101.4736 1043.29 33698.267 819[ 3999 *4 76 6523 595 36 14526 784 467-5957 *4 101-7478 1049.76 34012.224 824.4815 *5 76.9692 600.25 14706.125 471-4363.5 102.1020 1056'25 34328-125 829.5787 *6 77-2833 605 16 14886.936 475.2926 *6 102 4161 1062'76 34645'976 834'6917 *7 77'5975 610'09 15069'223 479-1646 7 102'7303 1069'29 34965'783 839'8203 *8 77'9116 615'04 1-2525992 483'0524 *8 103'0441 1075'84 35287'552 844-9647 i9 78l2258 620 01 15438 249 486 9558.9 103 3586 108241 35611[289 850 1248 Page 59 CIRCLES, ADVANCING BY A TENTH. 59 Diem Circum. Square. Cube. Area. Diam. Circum. Square. Cube. Area. 33 103'6728 1089 35937 8553006 41 1288056 1681 68921 13202574,1 10309869 1095-61 36264-691 8604920 1 129'1197 1689'21 69426'531 1326'7055 *2 104'3011 1102'24 36594'368 8656992 *2 129'4323 1697'44 69934'528 1333'1693 3 104'6151 1108.89 36926'037 8709222 3 129'7480 1705669 70444997 13396489 4 104'9294 1115'56 37259'704 8761608 4 130'0622 1713'96 70957'944 1346'1441 -5 105-2436 1122-25 37595'375 8814151 5 130'3764 1722'25 71473'375 1352'6551 *6 105-5577 1128-96 37933'056 8866851 *6 130'6905 1730'56 71991'296 1359'1818 105'8719 1135'69 38272'753 8919709 7 131'0017 1738'89 72511'713 1365'7242 106'1860 1142.44 38614'472 8972723 *8 131'3188 1747'24 73034'632 1372'2822 9 106'5002 1149'21 38958'219 9025895 9 131'6320 1755'61 73560'059 1378'8560 34 106'8144 1156 39304 9079224 42 131'9472 1764 74088 1385'4456 1 107'1285 1162'81 39651'821 9132709 1 132'2613 1772'41 74618'461 1392'0508 *2 107'4272 1169-64 40001'688 918'6352 *2 132-5755 1780'84 75151'448 1398'6717 3 107'7568 1176.49 40353-607 924-0115 I 3 132-8896 1789.29 75686-967 1405'3083 -4 108-0710 1183'36 40707'584/ 92904109 4 133-2038 1797-76 76225'024 1411'9607 5 108'3852 1190-25 41063'625 934'8223 5 133-5180 1806-25 76765'625 1418-6287 *6 108'693 1197-16 41421'736 940'2494 6 133'8321 1814'76 77308'776 1425'3125,7 109-0352 1205-09 41781'923 945'6922 7 134'1463 1823-29 77854'483 1432'0119 8 109-3076 1211-04 42144-192 951-1508 } 8 134-4604 1831-84 78402-752 1438-7271 9 109'6118 1218'01 42508'549 956'6250 ] 9 134-7746 1840-41 78958-589 t1445-4580 35 109'9560 1225 42875 9621150 43 135'0888 1849 79507 1452'2046 -1 110-2701 1232'01 43243'551 967'6206 ] 1 135-4029 1857'61 80062'991 1458'9668 2 110'58::i 1239'04 43614'208 973'1420 [ 2 135'7171 1866'24 80621'568 1465'7448 3 110'8984 1246'09 43986'977 978'6790 ] 3 136-0332 1874'89 81182'737 1472'5385 4 111'2126 1253.16 44361'864 984'2318 [ 4 136'3454 1883'56 81746'504 1479'03480 5 111'5268 1260.25 44738.875 989.8003.5 136'6596 1892'25 82312-875 1486.1731 6 111'8409 1267.36 45118.016 995'3845 I 6 136'9737 1900'96 82881'856 1493'0139 7 112'1551 1274.49 45499.293 1000'9843 [ 7 137'2879 1909'69 83453'453 1499'8705 8 112'4692 1281.64 45882.712 1006'6000 [ 8 137'6020 1918.44 84027'672 1506'7427'9112'7834 1288.81 46268.279 1012'2313 9 137'9162 1927-21 84604'519 1513'6287 36 113-0976 1296 46656 1017-8784 44 138'2304 1936 85184 1520-5344 1 113'4117 1303'21 47015'831 10235411 1' 138-5445 1944-81 85766'121 15274537 -2 113'7259 1310'44 47437'928 1029.2195 I 2 138'8587 1953'64 86350'888 1534'3888 3 114'0400 1317'69 47832'147 1034'91311 3 139'1728 1962'49 86938'307 1541'3396 4 114-3542 i1324'96 48228'544 1040'6235.4 139-4870 1971'36 87528'384 1548-3061 5 114'6684 133225 48627'125 1046'3491 5 139'8012 1980'25 88121'125 155522883,6 114-9825 1339-56 49027-896 1052'0904 -6 140'1153 1989'16 88716'536 1562'2862 7 115'2567 1346'89 49430.863 1057'8474 7 140'4295 1998'09 89314'623 1569'2998 8 1_15'6108 1354.24 49836-032 1063'6200 8 140'7436 2007'04 89915'392 1576'3292 -9 115-9250 1361-61 50243'409 1069'4084 [ 9 141'0578 2016-01 90518'849 1583'3742 37 116-'2392 1369 50653 1075'2126 45 141'3720 2025 91125 1590'4350 1 116.5533 1376.41 51064.811 1081.0324 l 1 141-6861 2034.01 91733-851 1597.5114 2 116.8675 1383'84 51-78'848 1086-8679 -2 142'0003 2043'04 92345'408 16046036 3 11-71816 1191-29 51895-117 1692-7191 -3 142-3144 2052-09 92959'677 1611'7114 4 11750{,58{ 1398'76 52313'624 1098'5862 I 4 142'6286 2061'16 93576'664 1618'8350 5 117'8103 1406'25 52734'315 1104'4687 ] 5 142'9428 2070'25 941963'75 1625'9743 6 11.81241 1413-76 53157'376 1110'3671' 6 143'2569 2079.36 94818'816 16331293 7 11843843 1421'29 53582'633 1116-2811 7 143.5711 2088'49 95443'993 164093020 -8 11871j24 1428-84- 54010.152 1122'2109 8 143'8852 2097.64 96071.912 1647.4864 9 119-0666 1436-41 5'4439'939 1128'1564 9 144'1994 2106'81 96702'579 16546885 3 119'3808 1444 54872 1134'1176 46 144'5136 2116 97336 1661'9064 -1 119'6949 1451'61 55306'341 1140'0946 1 144'8277 2125'21 97972'181 1669'1399 -2 120-0091 1459-24 55742.968 1146'0870 / 2 1451419 2134'44 98611-128 1676-3891 3 120-3232 1466-89 56181'887 1152'0954 3 145'4560 2143'69 99252'847 1683-6541 4 120'6374 1474'56 56623'104 1158'1194 -4 1457702 2152'96 99897'344 1690'9347 -5 120-9116 1482-25 57-066-625 1164'1591 5 146'0844 2162'25 100544-'625 1698'2311 -6 1212657 1489'96 57512'456 1170'2145 6 1463985 2171'56 101194'696 1705-5432 -7 121j5799 1497'69 57960'603 1176'2857 7 146'7127 2180-89 101847'563 1712-8710 8 121.8940 150544 58411072 11823725 8 147'0268 2190.24 102503'232 17202144 -9 122-2082 1513'21 58863.869 1188'4651 9 147'3410 2199'61 103161'709 1727-5736 39 122'5224 1521 59319 12945394 47 1476552 2209 103823 17349486 -1 1228365 1528'81 59776'471 1200'7273 1 1479693 2218'41 104487'111 1742'3392 2 123'1507 1536'64 60236'288 1206-8770 2 148'2835 2227'84 105154'048 1749'7455 3 123'4648 1544'49 60698'457 1213-0424 3 148'5976 2237'29 105823'817 1757'1675.4 123'7790 1552'36 61162'984 1219'2243.4 1489118 2246.76 106496.424 1764-6045 -5 124-0932 1560-25 61629.875 1225.4203.5 149.2260 2256.25 107171.875 1772-0587 6 124.4073 1568.16 62099.136 1231'6328 ] 6 149.5361 2265.76 107850.176 1779-5279 -7 124-7215 1576.09 62570.773 1237,8610.7 149.8543 2275.29 108531.333 1787.0127 8 125.0356 1584.04 63044-792 1244'1210 *8 150.1684 2284.84 109215'352 1794.5133 -9 325-3498 1592'01 63521'199 1250'3646 9 150-4826 2294'41 109902'239 1802-0296 40 125'6640 1600 64000 1256'6400 48 150'7968 2304 110592 1809'5616 -1 1259781 1608'01 64481'201 1262'9310 1 151'1109 2313'61 111284'641 1817'1092 -2 126'2923 1616'04 64964'808 1269'2388 -2 151'4251 2323'24 111980'168 1824'6726 -3 126-6064 1624.09 65450-827 1275'5602 3 151'7392 2332'89 112678'587 1832'2518 i4l 126-9206 1632-16 65939-264 1281-8984' 4 152'0534 2342'56 113379-904 1839'8466 5 127-2348 1640-25 66430-125 1288'2523' 5 152-3676 2352-25 114084-125 1847-4571 ~6 127-549 1648-36 66923-416 129-56219 -6 152-6817 2361-96 114791-256 1855-0833 -7 127-.8631 1656-49 67419-143 1301-0071 7 152-9959 2371-69 115501303 18627253 8 128-1772 1661-64 67917-312 1307-4082 8 153-3109 2381-44 116214-272 1870-1820.9 1.28-4914 1672-81 68417-929 1313-8249' 9 153-6242 2391-21 116930-169 1878-0563 Page 60 00 THE PRACTICAL MODEL CALCULATOR. Diam. Circum. Square. Cube. Area. Diam. Cirrur. Square. Cube. Area. _~~~~ ~~~~~im I _ _ _ _.. S-..... _ _ 49 153'9384 2401 117649 1885'7454 57 179'0712 3249 185193 25517646 ~1 154.2525 2410.81 118370771 1893.4501.1 179.3853 3260.41 186169.411 2560.72G0 ~2 154'5667 2420'64 119095'488 1901'1706 *2 179-6995 3271'84 187149'248 2569'7031 ~3 154'8808 2430'49 119823'157 1908'9068'3 180'0136 3283'29 188132'517 257806959 ~4 155'1950 2440'36 120553-784 1916'6587'4 180'3278 3294'76 189119'224 2587'7045'5 155'5092 2450'25 121287-375 1924'4263' 5 180'6420 3306-25 190109'375 2596'7287 ~6 155'8233 2460'16 122023'936 1932'2096 *6 180-9561 3317-76 191102'976 2605'7687 ~7 156-1375 2470-09 122763-473 1940-0086' 7 181'2803 3329-29 192100-033 2614-8243 ~8 156-4516 2480-04 123505'992 1947-8234 *8 181-5844 3340-84 193100'552 2623'-8957'9 156-7558 2490-01 124251-499 1955-6538'9 181-8986 3352'41 194104'539 2632-9828 50 157'0800 2500 125000 1963-5000 58 182-2128 3364 195112 2642-0856 ~1 15753941 2510-01 125751-501 1971'3618'1 182-5269 3375'61 196122'941 2651'2046 ~2 157.7083 2520-04 126506-008 1979-2394 -2 182-8411 3387-24 197137-368 2660-3382 ~3 158'0224 2530-09 127263'527 1987'1326'3 183'1552 3398-89 198155'287 266904882 ~4 158.3366 2540.16 128024.064 1995.0416.4 183.4694 3410.56 199176.704 2678-6538 ~5 158-6508 2550.25 128787.625 2002-9663.5 183.7836 3422.25 200201.625 2687.8351 ~6 158-9649 2560'36 129554'216 2010-9067 *6 184'0977 3433-96 201230056 2697'0321 ~7 159'2791 2570-49 130323'843 2018'8628' 7 184'4119 3445'69 202262'003 2706'2449 ~8 159-5932 2580-64 131096'512 20268346 *8 184'7260 3457'44 203297'472 2715-4733 ~9 159.9074 2590'81 131872'229 2034'8770' 9 185'0402 3469'21 204336'469 2724'7175 51 160 2216 2601 132651 2042'8254 59 185'3544 3481 205379 2733'9774 -1 160-5357 2611'21 133432'831 2050'8443'1 185-6685 3492'81 206425'071 2743-2529 2 16008499 2621-44 134217'728 2058'8784 *2 185'9827 3504'64 207474'688 2752'5442.3 1611640 2631'69 135005'697 2066'9293' 3 186-2696 3516'49 208527'857 2761-8512.4 161.4782 2641'96 135796'744 2074'9953'4 186-6110 3528'36 209584-584 2771'1739.5 161-7924 2652-25 136590-875 2083-0771 -5 186-9252 3540-25 210644'875 2780'5123 6 162.1065 2662-56 137388-096 2091'1746 -6 187-2393 3552'16 211708-736 2789-8664.7 162-4207 2672'89 138188'413 2099-2878' 7 187-5535 3564'09 212776'173 2799'2362.8 162-7348 2683-24 138991'832 2107'4166 -8 187'8676 3576'04 213847'192 2808'6218.9 163'0490 2693-61 139798-359 2115'5612'9 188-1818 3588-01 214921'799 2818-0230 52 163-3632 2704 140608 2123-7216 60 188-4960 3600 216000 2827'4400 -1 163-6773 2714-41 141420-761 2131-8976 -1 188-8101 3612-01 217081-801 2836-8726 -2 163-9935 2724-84 142236-648 2140-0893'2 189'1243 3624-04 218167-208 28463210.3 164-3056 2735-29 143055-667 2148-2967'3 189-4384 3636-09 219256'227 2855'7850.4 164-6198 2745-'76 143877-824 2156'5199' 4 189-7526 3648'16 220348-864 2865'2648.5 164-9340 2756-25 144703-125 2164'7587' 5 190-0668 3660-25 221445-125 2874-7603 6 165-2481 2766-76 145531-576 2173-0133 -6 190-3809 3672-36 222545'016 2884'2015.7 165-5623 2777-29 146363-183 2181-2835'7 190-6951 3684-49 223648-543 28937984 8 165.8764 2787-81 147197-952 2189-5695 -8 191-0092 3696-64 224755-712 2903'3410 9 166.1906 2798-41 148035-889 2197-8712 -9 191-3234 3708-81 225866-529 2912'8993 53 166.5048 2809 148877 2206-1886 61 191-6376 3721 226981 2922'4734 1 16608189 2819-61 149721-291 2214-5216 -1 191-9517 3733-21 228099-131 2932'0631 2 167.1331 2830-24 150568-768 2222-8704 -2 192-2659 3745-44 229220-928 2941-6685 3 167.4472 2940-89 151419-437 2231-2350'3 192-5800 3757'69 230346-397 2951'2897 4 167l7614 2851-56 152273-304 2239-6152 -4 192-8942 3769-96 231475-544 2960'9265 5 168.0756 2862-25 153130-375 2248-0111'5 193-2084 3782-25 232608-375 2970'5791 16 bS.3897 2872-96 153990-656 2256-4227 -6 193-5225 3794-56 233744-896 2980'2174 l7 TS.7049 2S83'69 154854-153 2264-8701 -7 193-8367 3806-89 234885-113 2989-9314 8 169.0180 2894-44 155720-872 2273-2931 -8 194-1508 3819-24 236029'032 29099'300 -9 169-3322 2905-21 156590-819 2281-7519 -9 194-4650 3831-61 237176-659 3009'34054 54 169-6464 2916 157464 2290-2264 62 194-7792 3844 238328 3019'0776 1 169-9605 2926-81 158340-421 2298-7165 -1 195-0933 3856-41 239483-061 3028-8244 ~ 2 170-2747 2937-61 159220-088 2307-2224 -2 195-4075 3868-84 240641-848 3038-5809 -3 170.5888 2948'49 160103-007 2315-7440.3 195-'7216 3881-29 241804-367 3048'36-1 ~4 70.u9030 2959-' 6 160989-184 2324-2813.4 196-0358 3893-76 242970'624 3058'1591 -o 171-2172 2697025 161878-625 2332-8343 -5 196-3500 3906-25 244140'625 3067T9687 ]17i.5313 2981'16 162771-336 2341-4030'6 196-6641 3918-76 245314-376 3077'-7901 17 171-845 2952-09 163667-323 2349-9874 -7 196-9783 3931-29 246491'883 3087'3I1 8 I.72.1596 3003'04 164566-592 2358-5876 -8 197'2924 3943-84 247673-152 3097-919 9 1 421735 3014'01 165469-149 2367-2034 -9 197'6066 3956'41 248858'189 3107'3G44 So 2lA7880 3025 1663715 2375-8350 63 197-9208 3969 250047 31172526 11 673-121 3036001 167284-151 2384-4822' 1 198-2349 3981-61 251239-591 3127-1G34 32. 4163 30O47T04 168196-608 2393-1452 -2 198-5491 3994-24 252435-968 3137'0758 ~3 173-71304 3058'09 169112-377 2401-8238 -3 198-8632 4006-89 253636'137 3147'0114 -4 174[0440 3069616 170031-464 24105182.4 199-1774 4019S56 254840 104 3156-9564 5 174358SS 3050'25 170953-875 2419-2283.5 199-4916 4032-25 256047'875 3166092931 6174'6729 3091-36 171879-616 2427-9541 -6 199-8057 4044-96 257259-456 3176-9115 7110749771 3102-49 172808-693 2436-6956 -7 200-1199 4057-69 258474-853 3186'907 8 175.3092 3113-64 173741-112 2445-4528 -8 200-4340 4070-44 259694-072 3196-9235 9 It756154 3124-81 174676-879 2454-2257 -9 200-7482 4083-21 260917-119 3206'9531 56 1t75-92,1 3136 175616 2463-0144 64 201-0624 4096 262144 3216-0684 -1i'2437 3147-21 176558-481 2471-8187 -1 201-3765 4108-81 263374-721 3227-0593 2 176-5579 3158-44 177504-328 2480-6387 -2 201-6907 4121-64 264609-288 3237'1360 ~ 176'8720 3169-69 178453'547 2489-4745 -3 202-0048 4134-49 265847-707 3247-2284 54 1-185'2 3180-96 179406-144 2498-32'9 -4 202-3190 4147-36 267089-984 3257-633( 117'5004 3192-25 180362-125 2507-1931 -5 202-6332 4160-25 268336-125 327'4603 6 177-8145 3203-56 181321-496 2516-0760 -6 202-9473 4173-16 269586-136 3277-5908 7 17-1287 3214-89 182284-263 2524-9736 -7 203-2615 4186-09 270840-023 3287-7550 8 178-4428 3226-24 183250'-432 2533-8888 -8 203-5756 4199-04 272097-792 3 92920 19 i78-7570 3237-61 184220-009 i 2542-8188 -9 203-8898 4212-01 273359-449 13308-112 Page 61 CIRCLES, ADVANCING BY A TENTH. 61 Diam. Circum. Square. Cube. Area. Diam. Circum. Square. Cube. Area. 65 20-2040 4225 274625 3318'3150 73 229'3368 5329 389017 4185'3966 ~1 2045181 42538.01 275894.451 3328.5340 1 229.6509 5343.61 390617.891 4196.8712 ~2 204.8323 4251.04 277167.808 3338.7668 *2 229.9651 5358.24 392223.168 4208.3614'3 205'1464 4264'09 278445'077 3349'0162 3 230'2792 5372'89 393832'837 4219'8678 ~4 205'4606 4277'16 279726'264 3359'2814 4 230'5934 5387'56 395446'904 4231'3S96 ~5 203.7748 4290.25 281011.375 3369.5623.5 230o9076 5402.25 397065.375 4242.9271 *6 f'206-0089 4303-36 282300-416 33798589.6 231.2217 5416.96 398688.256 4254.4803 7 206'4031 4316'49 283593'393 3390'1712 7 231'5359 5431'69 400315'553 4266'0493 *8 206"717"2 4329'64 284890'312 3400'4992'8 231'8500 5446'44 401947'272 4277'6339'9 207'0314 4342'81 286191'179 3410'8129'9 232'1642 5461'21 403583'419 4t289'2343 66 297'3456 4356 287496 3421'2024 74 232'4784 5476 405224 4300'8304 ~1 207'6597 4369'21 288804'781 3431'5775'1 232'7925 5490'81 406869'021 4312'4821 ~2 207'9739 4382'44 290117'528 3441'9633 *2 233'1067 5505'64 408518'488 4 324'1256 ~3 208'2880 4395'69 291434'247 3452'3749 3 233'4208 5520'49 410172'407 4335'7928 ~ 208'6022 4408'96 292754944 3462'7971 4 233'7350 5535'36 411830'784 43474717 ~5 208'9164 4422'25 294079'625 3473'2351 5 234'0192 555025 413493'625 43591663 -6 209-2305 4435-56 295408.296 3483.6888 6 234-3633 5565-16 415160-936 4370.8766 -7 2,)95447 4448-89 296740.963 3494.1640 7 234.6775 5580-09 416832-723 4382.6026.8 209-8588 4462-24 298077'632 3504-6432 *8 234'9916 5595'04 418508-992 4394'3448 9 210'1730 4475'61 299418'309 3515'1430'9 235'3058 5610'01 420189-749 4406-1018 67 210-4872 4489 300763 3525-6606 75 235'6200 5625 421875 4417-8750 -1 210'8013 4502'41 302111-711 3536-1928'1 235'9341 5640'01 423564'751 4429'6638 -2 211-1155 4515-84 303464-448 3546'7407 *2 236'2483 5655-04 425259'008 4441-4684'3 211'4296 4529'29 304821'217 3557-3043 3 236'5624 5670-09 426957-777 4453-2886'4 211-7438 4542-76 306182'024 3567'8837'4 236'8766 5685'16 428661'064 4465'1246 -5 212-0580 4556'25 307546'875 3578'4787' 5 237-1908 5700-25 430368-875 4476-9763 -6 2123'721 4569-76 308915'776 3589'0895 6 237'5049 5715-36 432081-216 44888437 ~7 212-6863 4583-29 310288'733 3599'7159' 7 237'8191 5730-49 433798-093 4500-7268 ~8 213-0004 4596-84 311665-752 3610'3581 -8 238'1332 5745'64 435519-512 4512-6256 ~9 213-3146 4610'41 313046'839 3621'0160'9 238'4474 5760'81 437245'479 4524-5401 68 213'6288 4624 314432 3631'6896 76 238'7616 5776 438976 4536-4704 ~-1 2139429 4637'61 315821'241 3642'3788 1'I 239'0757 5791'21 440711'081 4548-4163 ~-2 214'2571 4651'24 317214-568 3653-0838 -2 239'3899 5806'44 442450-728 4560'3787 ~3 214-5712 4664'89 318611'987 3663'8040'3 239'7040 5821'69 444194'947 45725353 ~4 21.4-8854 4678'56 320013'504 3674-5410'4 240'0182 5836'96 445943'744 4584-3583 ~5 215-1996 4692-25 321419-125 3685'2931'5 240'3324 5852-25 447697'125 4596'3571 -6 215-5137 4705-96 322828-856 3696'0060 -6 240'6465 5867'56 449455'096 4608-3816 7 215-'8279 4719-69 321242-703 370684145 7 240'9607 5882'89 451217-663 4620-4218 ~-8 216-1420 4733'44 325660-672 3717'6437 8 24112748 5898-24 452984-832 4632-4776 ~9 216'4562 4747-21 327082-'769 3728-4587'9 241'5987 5913-61 454756-609 4644-5492 69 216-7704 4761 328509 3739-2894 77 241'9032 5929 456533 4656-6366 ~1 217-0545 4774-81 329939-371 3750-1357'1 242-2173 5944-41 458314-011 4668-7396 ~2 217-3987 4788-64 331373-888 3760-9978 -2 242'5315 5959-84 460099-648 4680-8383 ~.3 217-7128 4802-49 332812-557 3771-8756'3 242-8456 5975-29 461889-917 4692-9927 ~4 218'0270 4816-36 334255-384 3782-7691 -4 243'1598 5990-76 463684'824 4705-1429.5 218-3412 4830-25 335702-375 3793-6783'5 243'4740 6006'25 465184-375 4717-3087 -6 218-6553 4844'16 337153-536 3804-6032 *6 243-7881 6021-76 467288-576 4729-4903.7 218'9695 4858-09 338608-873 3815'5438'7 244'1023 6037-29 469097-433 4741-6875 ~8 219-2836 4872-04 340068-392 3826-5002 *8 244'4164 6052-84 470910-952 4753-9605 ~9 2109-5978 4886-01 341532-099 3837-'4722'9 24-1'7306 6068-41 472729-139 4766-1292 70 219-9120 4900 343000 3848-4600 78 245-0448 6084 474552 4778-3736 ~1 220-2261 4914-01 344472-101 3859-4952'1 245-3589 6099-61 476379-541 4790-6336 ~2 220-5403 4928-04 345948-408 3870-4826 -2 245-6731 6115-24 478211-768 4802-9094 ~3 220-8544 4942-09 347428-927 3881-5174 3 245-9872 6130-89 480048-687 4815-2010 -4 221-1686 4956-16 348913-664 3892-5680'4 246-3014 6146-56 481890-304 4827-5082 ~ 5 221-4828 4970-25 350402-625 3903'6343'5 246-6156 6162-25 483736-625 4839-8311 ~6 221-'7969 4981-36 351895-816 3914-7163 -6 246-9297 6177'96 485587-656 4852-1697 ~7 222-1111 4998-49 353393-243 3925-8140 -7 247-2439 6193-69 487443-403 4864-5241 8 222-4252 5012-64 354894-912 3936-9274 -8 247-5480 6209-44 489303-872 4876-8973 9 222-7391 5026'81 356400-829 3948-0565 -9 247'8722 6225-21 491169-069 4889-2799 71 223-0536 5041 357911 3959-2014 79 248-1864 6211 493039 4901-6814 ~1 223-3677 5055-21 359425-431 3970-3619 -1 248-5005 6256-81 494913-671 4914-0985 ~ 2 223-6819 5069-44 360944-128 3981-5381 -2 248-8147 6272-64 496793-088 4926'5314 ~-3 223-9960 5083-69 362467-097 3992-7301'3 249-1288 6288-49 498677-257 4938-98020 ~4 224-3102 5097-96 363994-344 4003-9373 -4 249-4430 6304-36 500566-184 4951-4443 ~-5 224-6244 5112-25 365525-875 4015-1611 -5 249-7572 6320-25 502459-875 4963-9243 -6 224-9385 5126-56 367061-696 4026-4002 -6 250-0713 6336-16 504358-336 4976-48-10 ~-7 225-2-527 5140-89 368601-813 4037-6550 -7 250-3855 6352-09 506261-573 49588-9314 ~-8 225-5668 5155-24 370146-232 4048-9254 -8 250-6996 6368-04 5081609-502 00, 1'-4586 -9 225-8810 5169-61 371694-959 4060-2116 -9 251-0138 6384-01 510082-399 5014-0014 72 226-1952 5184 373248 4071-5136 80 251-3280 6400 512000 5026-5(00 -1 226-5093 5198-41 374805-361 4082-8332 -1 251-6421 6416-01 513922-401 5039-1342 -2 22'-8235 5212-84 376367-048 4094-1645 -2 251-9563 6432-04 515849-608] 5051-7242 -3 227'1376 5227-29 377933-067 4105-5125 -3 252-2704 6448-09 517781-627 5064-3208 -4 227-4518 5241-76 379503-424 4116-8793 -4 252-5846 6464-16 519718-464 5076-9552 ~-5 227-7660 5256-25 381078-125 4128-2587 -5 252-8988 6480-25 521660-125 5089-5883 -6 228-0801 5270-76 382657-176 4139-6524 -6 253-2129 6496-36 523606-616 5102-2411 -7 228'3943 5285-29 384240-583 4151-0667 -7 253-5271 6512-49 525557-943 5114-9096 -8 228-'7084 5299-84 385828-352 4162-4943 -8 253-8412 6528-64 527514-112 5127-'5938 -9 229-0226 5314-41 387420-489 4173-9376 -9 254-1554 6544-81 529475-129 5140-2937 F Page 62 62 THE PRACTICAL MODEL CALCULATOR. Diam. Circum. Square. Cube. Area. Diam. Circum. Square. Cube. Area. 81 254.4696 6561 531441 5153.0094 89 279.6024 7921 704969 6221/1534 1 254.7837 6577-21 533411.731 5165.7407.1 279.9165 7938-81 707347-971 6235-1413 *2 255-0979 6593-44 535387-328 5178-4877'2 280-2307- 7956'64 709732'288 6249'1450 *3 255'4120 6609'69 537367'797 5191-2505'3 280'5448 7974'49 712121'957 6263'1644 4 255'7262 6625696 539353'144 5204'0285'4 280'8590 7992'36 714516'984 6277'1995 *5 25600404 6642'25 541343'375 5216'8231'5 281'1782 8010'25 716917'575 6291'2035 *6 256-3545 6658'56 543338'496 5229-6330 *6 281'4873 8028'16, 719323'136 6305'3168 *7 256'6687 6674'89 545338'513 5242'4586 -7 281-8825 8046-09 721734-273 6319'3990 *8 256-9828 6691-24 547343'432 5255-2998'8 282'1156 8064-04 724150'792 6333'4970'9 257-2970 6707-61 549353-259 5268-1568'9 282'4298 8082-01 726572'699 6347'6813 82 257'6112 6724 551368 5281'0296 90 282'7440 8100 729000 6361'7400.1 257'9253 6740'41 553387'661 5293'9180'1 283'0581 8118-01 731432-701 6375'8850 *2 258'2395 6756'84 555412-248 5306-8221'2 283'3723 8136-04 733870-808 6390-0458 ~3 258-5536 6773-29 557441'767 5319'7439'3 283'6864 8154'09 736314'327 6404'2222 ~4 258-8646 6789-76 559476-224 5332-6775'4 284-0006 8172-16 738763-264 6418-4144 ~5 259'1820 6806-25 561515-625 5345-6287'5 584'3148 8190-25 741217-625 6432-6223 ~6 259-4961 6822-76 563559'976 5358'5957 *6 284'6289 8208'36 743677'416 6446-8474 ~7 259-8103 6839-29 565609-283 5371-5983 7 284-9431 8226-49 746142-643 6461-0852 ~8 260-1244 6855-84 567663-552 5384-5762'8 285-2572 8244'64 748613-312 6475-3402.9 260-4386 6872-41 569722-789 5397-5908.9 285-5714 8262'81 751089'429 6489'6109 83 260-7528 6889 571787 5410-6206 91 285-8856 8281 753571 6503-8974 -1 261-0669 6905.61 573856-191 5423.6660 1 286-1997 8299-21 756058-031 6518-1995 *2 261-3811 6922-24 575930.368 5436-7272 2 286-5139 8317-44' 758550.528 6532-5173 ~3 261-6952 6938-89 578009-537 5449-8042 3 286-8290 8335'69 761048'497 6546'8909 *4 262-0094 6955-56 580093-704 5462-8968 *4 287-1422 8353-96 763551'944 6561-2081 *5 262-3236 6972-25 582182-875 5476-0051 5 287.4564 8372-25 766060-875 6575-5651 ~6 262-6376 6988-96 584277-056' 5489-1291 -6 287-7705 8390-56 768575-296 6589-9458 ~7 262-9519 7005-69 586376-253 5502-2689 *7 288-0847 8408-89 771095-213 6604-3222 *8 263-2640 7022'44 588480-472 5515-4243 *8 288-3988 8427-24 773620-632 6618-7542.9 263-5802 7039-21 590589-719 5528-5958.9 288-7130 8445-61 776151-559 668331820 84 263-8944 7056 592704 5541-7824 92 289 0272 8464 778688 6647-6256'1 26a42085 7072581 5948235321 5554-9849 41 289 3413 8482141 7812290961 6662 0848 *2 264'5227 7089'64 596947-688 5568-2032 *2 289'6555 8500'84 783777'448 6676'5597 ~3 264-8368 7106'49 599077'107 5581-4372 *3 289-9696 8519'29 786330-467 6691.0161 ~4 265-1510 7123-36 601211'584 5594'6869 -4 2902838 8537'76 788889-024 6705-5567 *5 265-4652 7140-25 603351'125 5607'9523,5 290-5980 8556'25 791453-125 6720-0787 ~6 265-7793 7157-16 605495-736 5621'2334 *6 290-9121 8574-76 794022-776 6734-6165.7 266-0935 7174'09 607645,423 5634-5682 I7 291-2263 8593-29 786597-983 6749-1699 *8 266-4076 7191-04 609800-192 5647-8428 *8 291-5404 8611-84 799178-752 6763-7391 ~9 266-7218 7208601 611960.049 5661-1710 -'9 291-8546 8630-41 801765-089 6778-3240 85 267-0360 7225 614125 5674-5150 93 292-1688 8649 804357 6792'9246 51 267.3501 7242-01 616295-051 5687.8746 91 292-4829 8667-61 806954-491 6807.5408 *2 267-6643 7259-04 618470-208 5701'2500 2 292-7971 8686-24 809557-568 6822-1730 ~3 267-9784 7276-09 620650-477 5714-6410 *3 293-1112 8704-89,812166.237 6836.8206 *4 268-2926 7293'16 622835,864 5728-0478 *4 293-4254 8723-56 814780.504 6851-4840 ~5 268-6068 7310-25 625026'375 5741-4703 *5 293-7396 8742-25 817400-375 6866-1631 ~6 268-9209 7327-36 627222'016 5754-9085 *6 294-0537 8760'96 820025-856 6880-8579 *7 269-2351 7344'49 629422-793 5768-3624 *7 294-3679 8779-69 822656-953 6895-5685 ~8 269-5492 7361-64 631628-712 5781-8320'8 294-6820 8798-44 825293-672 6910-2947 ~9 269-8634 7378-81 633839-779 5795-3173'9 294-9962 8817-21 827936-019 6925-0367 86 270-1776 7396 636056 5808-8184 94 295-3104 8836 830584 6939-7944 ~ 270-4917 7413-21 638277-381 5822-3351 1 295-6245 8854'81 833237'621 6954-5677 -2 270-8059 7430'44 640503'928 5835-8675'2 295-9387 8873-64 835896 888 6969-3568.3 271-1200 7447-69 642735-647 5849-4157'3 296-2436 8892-49 838561-807 6984-1614 4 271-4342 7464-96 64497 2-544 5862-9795'4 296-5670 8911-36 841232-384 6998-9821 *5 271-7484 7482-25 647214-625 5876-5591'5 296-8812 8930'25 843908-625 7013-8183 *6 272-0665 7499-56 649461-896 5890-1541'6 297'1953 8949'16 846590-536 7028-6702 *7 272-3767 7516-89 651714-363 5903-7654'7 297-5095 8968-09 849278-123 7043-5025 ~8 272'6908 7534-24 653972-032 5917-3920'8 297-8236 8987-04 851971-392 7058-4180 ~9 273-0050 7551-61 656234-909 5931-0344'9 298-1378 9006-01 854670-349 7073-3202 87 273-3192 7569 658503 5944-6926 95 298-4520 9025 857375 708582350 1 273-6333 7586-41 660776-311 5958-3644'1 298-7661 9044-01 860085-351 7103-1654 ~2 273-9875 7603-84 663054-848 5972-0559'2 299-0723 9063-04 862801-408 7118'1116 ~3 274-2616 7621-29 665338-617 5985-7691 *3 299-3944 9082-09 865523-177 7133-0734 ~4 274-5758 7638-76 667627-624 5999-4821.4 299-7086 9101-16 868250-664 71480C510 ~5 274-8900 7656-25 669921-875 6013.2187'5 300'0228 9120:25 870983-875 7163'0443 ~6 275-2041 7673-76 672221'376 6026-9711 *6 300-3369 9139'36 873722-816 7178 0533 ~7 275-5183 7691-29 674526-133 6040-7391 -7 300-6511 9158-49 876467'493 7193'C0780 ~8 275-8324 7708-84 676836-152 6054-5149'8 300-9652 9177-64 879217-912 7 208 118' ~9 276-1466 7726'41 679151-439 6068-3224.9 301-2794 9196-81 881974'079 7223 1745 88 276-4608 7744 681472 6082-1376 96 301-5936 9216 884736 7238'2464 ~1 276-7749 7761-61 683797-841 6095-9684'1 301-9077 9235-21 887503-681 7253 3339'2 277-0891 7779-24 686128'968 6109-8150'2 302-2219 9254'44 890277-128 7268 1'-371 ~3 277-4032 7796'89 688465'387 6123'6774'3 302-5360 9273869 893056-347 728335561 ~4 277-7174 7814-56 690807-104 6137-5554' 4 302-8502 9292-96 895841-344 7298-C907 ~5 278-0316 7832-25 693154-122 6151-4491 *5 303-1644 9312-25 898632-125'i3130E;41 1'6 278-3457 7849-96 695506-456 6165-3585'6 303-4785 9331-56 901428-696 7329-0072 *7 278-6599 7867-69 697864-103 6179-2837' 7 303'7927 9350-89 904231-063 7344-1890'8 278-9750 7885-44 700227-072 6193-2245'8 304-1068 9370-24 907039-232 7359 3864'9 279-2882 7903-21 702595-369 6207-1811.9 304-4210 9389-61 909853'209 7374-5936 Page 63 TABLE OF THE LENGTH OF CIRCULAR ARCS. 63 Diana. Circum. Square. Cube. Area. Diam. Circum. Square. Cube. Area. 97 30-4-7352 9409 912673 7389-8286 -6 309-7617 9721-96 958585-256 7635-6273 30-04'93 9428-41 915498.611 7405,0732.7 310-0759 9741.69 961504-803 651 1933 ~12 30O.3635 9447-84 918330'048 7420-3335 *8 310'3960 9761'44 964430'272 7666-6349 ~3 305-6776 94L67.29 921167.317 7435.6095'9 310.7042 9781'21 967361-669 7682-1623 301 5039918 9486-76 924010-424 7450.9013 99 311-0184 9801 970299 7697 7054 i5 306-3060 9506.25 926859-375 7466-2087'1 311.3325 9820-81 973242'271 7713-2641 ~6 306-6201 9525'76 929714-176 7481-5319'2 311-6467 9840'64 976191'488 7728S8386 ~7 306.9363 9545-29 932574-833 7496'8707.3 311.9608 9860-49 979146-657 774l44288 *8 307 2484 9564'84 935441'8352 7512.2253.4 312.2750 9880'36 982107'784 7760.0347.9 307.5626 9584.41 938313'739 7527.5956'5 812'5892 9900 25 985074'875 7775'6563 98 307-8768 9604 941192 7542-9816 *6 312-9033 9920'16 988047'936 7791'2936 1 308-1909 9623'61 944076-141 7558-3832'7 313-2175 9940'09 991026-973 7806'9466 ~2 308-5051 9643-24 946966-168 7573'8006 -8 313-5116 9960)04 994011'992 7822.6154.3 308'8192 9662'89 919862'087 7589'2338 9 313 8458 9980'01 997002'999 7838'2998 *4 309-1334 9682.56 952763'904 7604.6826 100 314'1600 10000 1000000 7854.0000 8 309'4476 9702'25 955671-625 7620'1471 A TABLE of the Length of Circular Arcs, radius beintg unity. Degree. Length. Degree. Length. Min. Length. Sec. Length. 1 0 01745,3 1 60 1.0471976 6 1 0'0002909 1 0'000048 2 0'0349066 70 1'2217305 1 2 0'0005818 2 0'000097 3 005023599 80 1'3962634 1 3 0'0008727 3 0'0000145 4 0 0698132 90 1 5707963 4 0 0011636 4 0 0000194 5 0'0872665 100 1-7453293 5 0.0014544 5 0.0000242 6 0.1047198 120 2.0943951 6 0.0017453 6 00000291 7 0.1221730 150 2.6179939 7 0.0020362 7 0000088339 8 0.1396263 180 381415927 8 0.0023271 8 00000388 9 0'1570796 210 386651914 9 00026180 9 0.0000436 10 0'1745329 240 4.1887902 10 0'0029089 10 0.0000485 20 0'3490659 270 4.7123890 20 0.0058178 20 0.0000970 30 1 05235988 300 5.2359878 30 0'0087266 30 0'0001454 40 0'6981317 330 5.7595865 40 0'0116355 I 40 0.0001939 50 0'8726646 360 6.2831853 50 00145444! 50 0.0002424 Required the length of a circular arc of 370 42' 58"? 30~ = 0-5235988 7~ = 0.1221730 40' = 0.0116355 2' = 0'0020368 50" = 0.0002424 8" = 0.0000388 The length 0'6582703 required in terms of the radius. 12070 Fahrenheit = 1~ of Wedgewood's pyrometer. Iron melts at about 1660 Wedgewood; 200362~ Fahrenheit. Sound passes in air at a velocity of 1142 feet a second, and in water at a velocity of 4700 feet. Freezing water gives out 140~ of heat, and may be cooled as low as 200. All solids absorb heat when becoming a fluid, and the quantity of heat that renders a substance fluid is termed its caloric of fluidity, or latent heat. Fluids in vacuo boil with 1240 less heat, than when under the pressure of the atmosphere. Page 64 64 THE PRACTICAL MODEL CALCULATOR. AREAS of the Segments and Zones of a Circle of whicAh the DIAMETER is Unity, and supposed to be divided into 1000 equal parts. Height. Area of Area of Hht Area of Area of Height Area of Area of Segment. Zone. Segment. Zone. Sement. Zone. 001 000042 -001000'051 -015119 -050912'101 -041476 100309 ~002 *000119 o002000 *052 I015561 *051906 102 o042080.101288 ~003'000219 [003000 -053 -016007 [052901.103'052687'102267'004'000337'004000'054'016457'053895'104'043296.103246'005'000470'005000 ~055 /016911'054890'105'043908 1042 3 ~006'000618 -'006000.056.017369.055883'106'044522'105201 ~007.000779.007000.057'017831.056877 /107'045189.106178 ~008'000951.008000'058 *018296'057870 -108'045759.107156 ~009'001135'009000 /059'018766'058863'109'046381.108131'010'001329'010000'060'019239'059856'110 ~047005'109107 ~011.001533.011000'061'019716'060849 -111'047632'110082 ~012'001746'011999'062'020196'061841'112'048262'111057 ~013'001968'012999'063'020680'062833'113'048894'112031 ~014'002199'013998 064'021168'063825 114'049528.118004'015'002438'014998'065'021659'064817'115 ~050165.113978'016'002685'015997'066'022154'065807'116'050804'114951 ~017'002940'016997'067,'022652'066799'117'051446'115924 ~018 -003202'017996'068'023154'067790'118'052090'116896 ~019'003471.018996'069'023659'068782'119'052736'117867'020'003743'019995'070'024168'069771'120'053385'118838 ~021'004031'020994.071'024680.070761.121'054036.119809 ~022'004322'021993.072'025195 *071751'122'054689'120779 ~023'004618 022992'073'02.5714.072740'123'055345 121 748'024'004921 023991.074'026236.073729.124'056003.122717 025'0052380'024990'075'026761'074718'125 05GG6663'123686 ~026 00 55446 025989 076 - 027289.'075707'126'057326.124054 ~027'005867'026987.077.027821'076695'127'057991.125621 ~028.006194.027986.078'028356'077683'128'058658'126588 ~ 029 006527'028984'079'028894'078670'129'059327'1,27555'030'006865'029982'080'029435'079658'130'059999'128521 ~031'007209'030980.081.029979'080645'131'060672'129486 ~032'007558'031978'082'030526'081631'132 -061348'130451 ~033'007913'032976'083'031076'082618 133'062026.13141 5 ~034'008273'033974'084'031629.083604'134'062707'132379'03v5 008638'034972'085'032186'084589'1305'063389'133342'036 ~009008'035969'086'032745'0855 74'136'064074'134304'037'009383'036967'087'033307'086559'137'064760'185266 ~088'009763'037965 -088'033872'087544'138 0OG65449'136228 ~039.010148'038962'089'034441'088528'139'066140'137189 ~040 ~010537'039958'090 ~035011'089512'140'066833'138149 ~041'010931'040954'091'035585'090496'141'067528'139109 ~042 ~011330'041951'092'036162'091479'142'068225'140068 ~043 ~011734'042947'093'036741 -092461'143 -068924'141026 ~044 ~012142'043944'094'037323'093444'144'069625'141984 ~045'012554 -044940'095'037909'094426'145'070328'142942 ~046 ~012971'045935'096.038496'095407.146'071033'143898 ~047'013392'046931.097.039087 -096388.147'071741'144854 ~048'013818'047927'098.039680 -097369'148'072450 -145810 ~049'014247'048922'099.04027 6'098350'149'073161'146765'050'014681'049917'100.040875'099330 150'073874'147719 Page 65 AREAS OF THE SEGMENTS AND ZONES OF A CIRCLE. 65 Height. Area of Seg. Area of Zone. Height. Area of Seg. Area of Zone. Height. Area of Seg. Area of Zone. ~151.074589 *148674.206'116650'200915.261 *163140.248608 ~152 *075306 *149625.207'117460.200924 *262 *164019.249461 ~153.076026.150578'208'118271'201835'263.164899.250212.154'076747.151530'209 -119083'202744'264'165780.251162.155 *077469 *152481 -210'119897'203652'265 *166663 *252011 ~156.078194.153431'211.120712.204559.266.167546.252851 ~157'078921'154381 -212.121529'205465'267.168430 *253704 ~158'079649.155330 -213'122347'206370'268 -169315.254549 ~159.080380 -156278'214'123167'207274'269 -170202'255392 ~160.081112.157226 215. -123988 -208178'270.171080 -256235'161 -081846'158173'216'124810'209080'271'171978 -257075 ~162.082582.159119'217'125634'209981 *272'172867'257915' ~163.083320'160065'218'126459'210882'273'173758'258754 ~164 -084059.161010'219'127285'211782'274'174649'259591 ~165'084801'161954'220'128113- 212680'275'175542'260427 ~166.085544.162898'221'128942'213577'276'176435'261261 ~167.086289.163841'222'129773'214474'277'177330'262094 ~168'087036.165784 -223'130605.215369'278.178225.262926 ~169'087785'165725 -224'131438'216264 -279'179122'263757 ~170.088535'166666'225.132272'217157 -280.180019 -264586 ~171.089287.167606'226.133108.218050 -281 -180918'265414'172'090041'168549'227'133945.218941.282 -181817.266240'173.090797.160484-'228.134784'219832'283.182718'267065 -174.091554'170422'229.135624'220721'284.183619 -267889'175'092313'171359 -230- 136465'221610.285.184521 -268711.176'093074,172295'231'137307'222497'286'185425'269532'177,093836.173231'232.138150'223354'287.186329'270352 -178'094601'174166'233'.138995'224269'288'187234 -271170'179'095366'175100'234.139841,.225153.289.188140'271987'180'096134'176033'235 -140688'226036 -290.189047'272802.181'096903 -176966'236'141537 -226919'291.189955 273616'182'097674.177897 -237.142387'227800'292.190864'274428'183.098447.178828'238.143238'228680'293'191775 -275239'184'099221.179759'239.144091'229559'294.192684'276049.185'099997,180688 -240'144944'230439'295'193596'276857'186 -100774.181617 -241'145799'231313'296'194509'277664'187 -101553- 182545.242 -146655 -232189'297 -195422'278469'188'102334'183472'243'147512'233063'298'196337'279273 ~189'103116.184398 -244'148371'233937'299'197252'280075 ~190'103900.185323 -245'*149230'234809'300'198168'280876'191.104685.186248'246'150091'235680'301'199085 ~281675'192 -105472 -187172'257'150953'236550'302'200003'282473 ~193'106261'188094'248.151816'237419'303'200922'283269 -194'107051'189016 -249.152680'238287'304'201841'284063'195'107842'189938'250'153546 -239153'305'202761'284857'196'108636 -190858 -251 -154412.2'40019'306'203683'285648.197 -109430'191777'252.155280.240883'307'204605'286438'198'110226 -192696'253'156149'241746'308'205527'287227'199.111024'193614.254.157019'242608. 809.206451 1288014'200.111823'194531'255'157890'243469'310'207376 -288799.201.112624'195447.256'158762'244328'311'208301'289583.202'113426'196362'257'159636.245187 ~312'209227'290365;203'114230'197277.258'160510'246044'313'210154'291146.204'115035'198190'259.161386'246900'314'211082'291925.205'115842'199103'260'162263'247755 -315'212011'292702 r2 5 Page 66 66 THE PRACTICAL MODEL CALCULATOR. lrea of Seg. ArcoaofZone. Ileiht. Area of Se, AreaofhZoie. Height. Aea of Se. Areaof'Zoie. *316 *212940 *293478 3i 71 -265144'3333-72 *4216 3189'70.36j6463 1T7.213871.294252 *372 *266111 1334041 4)2 319959 366tl85 *318.214802'295025'373'267078'334708 *428 *320948 -36 7504 319.215733.295796'374 *268045 -335373 *429 *321938.G68019 32 0 216666 296565 6 375'269013 *336036'430 322928 *368531 ~321 *217599.297333 -376 *269982 /336696 *431.323918.369040 ~3' *218533 -298098'377 -270951'337354'432.324909.369545 ~323 *219468 *298863 -378 *271920'338010 *433'325900 [ 370047 ~324 *220404 *299625 -379 -272890 *338663'434.326892'370545 *325'221340 *300386'380 *273861 *339314 *435'327882'371040 ~326'222277'301145'381'274832'339963 *436'328874'371531 ~327'223215'301902'382 -275803'340609 *437'329866'3i72019 ~328'224154'302658'383'276775'341253'438'330858.37 2503 ~329'225093'303412.384 *277748'341895 *439'331850'372983'330 *226033'304164 *385'278721'342534'440 *332843'3 7 3460 ~331'226974'304914'386'279694'343171 *441'333836 *373933 ~332'227915'305663'387'280668'343805'442'334829'3874403 *333'228858'306410'388'281642'344437 443'33.5822'374868 ~334'229801'307155'389.282617'345067 444'336816'37,5330'335 2380745'307898'390 *283592'345694 445 338810'37 5788 ~336'231689'308640'391'2841568'346318'446'338804'376242 ~337 *232634'309379'392'285544'346940 *447'339798'370G692 ~338.233580'310117'393'286521'347560 448'340793'377138 339'234526'310853 -394'287498 *'348177.449'341787'377580 3-40'235473'311588'395'288476'348791'450'342782'378018 ~341'236421'312319.396.289453'349403'451'343777'378452 ~342'237369'313050'397'290432 -350012 *452'344772.378881 ~343'238318'313778'398.291411'350619.453'345768'379307 ~344 2839268'314505 -399 -292390 -351223 454'346764'379728'340'240218'315930'400'293369'351824 455 8347759'380145 ~346'241169'315952 -401 *294349'352423.456'348755'380357 ~347'242121'316673.402 *295330'353019 *457'349752'380965 ~348'243074'317393'403'296311'353612 458'350748'381369 ~349'244026'318110'404'297292'354202 459'351745 -381768'350'244980'318825'405'298273'354790'460'352742 382 1G2'351'245934'319538'406 -299255'355376 461 - 353739'382551 3-52'246889'320249'407 -300238'355958'462 -354736'382936 ~353'247845'320958'408'301220'356537'463'355732'383316 ~354'248801'321666'409'302203'357114'464' 356730'383691'355'249757'322371'410 -303187'357688 *465 -357727'384061 ~356'250715'323075 1411'304171'358258'466 -358725'384426 ~357'251673'323775'412'305155'358827'467'359723'384786 ~358'252631'324474'413 -306140'359392'468'360721'385144 3-59'253590'325171'414'307125'359954'469 -361719'385490'360'254550'325866 *415 -308110'360513'470 -362717'385834'361'255510'326559'416 1309095'361070'471 -363715'386172 ~362 *256471'327250'417 1310081 1361623'472 -364713'386505 ~ 36'257433'327939'418 -311068'362173'473 -365712'386832 ~361'258395'328625'419'312054'362720'474 -366710'387153 ~365'259357.329310 420'313041'363264'475 -367709 3S87469 ~366 *260320'329992'421 -314029'363805 *476 -368708 387778 ~367'261284'330673.422 1315016 1364343 *477 -369707.388081 ~368 *262248.331351.423'316004'364878'478'370706 -388377 ~369'263213'332027'424'316992'365410'479'3717041 388669'370 2641 78'332700'425 317981 -365939'480'372704'388951 i~~~~~~~~.sg Page 67 RULES FOR FINDING THE AREA OF A CIRCULAR ZONE, ETC. 67 Heightll. Area of Se2g AreaofZone. Hleight. Area of Seg. AreaofZone. Height. Area of Seg. AreaofZone. 4-81 -373703 -389228 -491 383699 391564 496 388699.3923622 ~482.374702.389497 *492.384699.391748 497 [389699 8392480 ~483.375702 -389759.493.385699 ]391920 *498 ]390699 392.580 ~484 -376702.390014.494 -386699 8392081 499 391699 38926457 435 3077701.390261.495.387699 392229 500 392699.392699.486'378701'390500 ~487'379700'390730 To find the area of a segment of a circle. 488 380700 390953 ~489.381699.391166 RULE.-Divide the height, or versed sine, ~490 382699.391370 by the diameter of the circle, and find the quotient in the column of heights. Then take out the corresponding area, in the column of areas, and multiply it by the square of the diameter; this will give the area of the segment. Required the area of a segment of a circle, whose height is 3} feet, and the diameter of the circle 50 feet. 3o = 3825; and 3'25 - 50 ='065. *065, by the Table, = -021659; and'021659 x 502 = 54'147500, the area required. To find the area of a circular zone. RULE 1. —When the zone is less than a semi-circle, divide the height by the longest chord, and seek the quotient in the column of heights. Take out the corresponding area, in the next column on the right hand, and multiply it by the square of the longest chord. Required the area of a zone whose longest chord is 50, and height 15. 15 - 50 ='300; and'300, by the Table, ='280876. Hence'280876 x 502 = 702'19, the area of the zone. RULE 2.-When the zone is greater than a semi-circle, take the height on each side of the diameter of the circle. Required the area of a zone, the diameter of the circle being 50, and the height of the zone on each side of the line which passes through the diameter of the circle 20 and 15 respectively. 20 - 50 ='400;'400, by the Table, ='351824; and'351824 x 502 = 879'56. 15 - 50 ='300;'300, by the Table, ='280876; and'280876 x 502 = 702'19. Hence 879'56 + 702'19 = 1581'75. Alpproximating rule to Jfind the area of a segment of a circle. RULE.-Multiply the chord of the segment by the versed sine, divide the product by 3, and multiply the remainder by 2. Cube the height, or versed sine, find how often twice the length of the chord is contained in it, and add the quotient to the former product; this will give the area of the segment very nearly. Required the area of the segment of a circle, the chord being 12, and the versed sine 2. 24 12 x 2 = 24; - =8;and8 x 2 =16. 23 -- 24 =- 3333. Hence 16 +'3333=16'3333, the area of the segment very nearly. Page 68 GS PROPORTIONS OF THE LENGTHS OF CIRCULAR ARCS. Height Length Height Length Height Length Height Length Height Length of of of of of of o of of of Arce. Arc. Ace. Are. Are. Are. Are. Ae. Ace. ArcAre. ~100 102645'181 1'08519 261 1'17275'341 1'28583'421 142041 ~101 1'02698'182 1'08611 -262 1.17401'342 1'28739 -422 1'42222 ~102 1'02752 *183 1'08704'263 1'17527 343 1'28895 -423 1'42402 ~103 1'02806'184 1'08797 *264 1'17655'344 1'29052 -424 1'42583 ~104 1'02860'185 1'08890'265 1'17784'345 1'29209 *425 1'42701 ~105 1'02914 186 1'08984'266 117912 346 1'29366'426 1'42945 ~106 1'02970 187 1'09079'267 1-18040 347 1'29523'427 1'43127 ~107 1'03026'188 1'09174'2G8 1-18162 348 1'29681 *428 143309 ~108 1'03082'189 1'09269 *269 1'18294'349 1'29839'429 1'43491 ~109 1'03139'190 1'09365'270 1'18428'350 1'29997 430 1'43673 ~110 1'03196'191 1'09461 271 1'18557'351 1'30156 431 1'43856 ~111 1'03254 192 1'09557 *272 1'18688'352 1'30315'432 1'44039 ~112 1'03312'193 1'09654'273 1'18819'353 1'30474'433 1'44222 ~113 1'03371 194 1'09752'274 1'18969'354 1'30634'434 1-44405'114 1'03430 195 1'09850'275 1'19082'355 1'30794'435 1'44589'115 1'03490'196 1'09949'276 1'19214 356 1'30954 436 1'44773 ~116 1'03551'197 1'10048'277 1'19345'357 1'31115'437 1'44957 ~117 1'03611'198 1'10147'278 1'19477 358 1'31276 438 1'45142 ~118 1'03672'199 1'10247'279 1'19610'359 1'31437'439 1'45327 ~119 1'03734'200 1'10348'280 1'19743 360 1'31599 440 1'45512 ~120 1'03797 201 1'10447'281 1'19887'361 1'31761 441 145697 ~121 1'03860 202 1'1.0548'282 1'20011 362 1'31923'442 1'45883 ~122 1'03923 203 1'10650'283 1'20146'363 1'32086'443 1'46069 ~123 1'03987'204 1'10752'184 1'20282 364 1532249'444 1'46255 ~124 1'04051 205 1'10855'285 1'20419 365 1'32413'445 1'46441'125 1'04116'206 1-10958'286 1'20558'366 1532577'446 1'46628'126 1'04181'207 1'11062'287 1'20696 367 1'32741'447 1'46815'127 1'04247'208 1-11165 -288 1'20828 368 1'32905'448 1'47002 ~128 1'04313 209 1'11269'289 1'20967 369 1'33069'449 1'47189 ~129 1'04380'210 1'11374'290 1'21202'370 1-33234'450 1'47377.130 1'04447'211 1'11479'291 1-21239 371 1-33399'451 1'47565 ~131 1'04515 212 1'11584,292 1'21381'372 1'33564 452 1'47753'132 1'04584'213 1'11692'293 1'21520'373 1'33730'453 1'47942 133 1'04652'214 1'11796'294 1'21658'374 1'33896'454 1'48131 ~134 1'04722'215 1.11904'295 1'21794'375 1'34063'455 1'48320'135 104792 216 1.12011'296 1'21926 376 1'34229 456 1'48509 136 1'04862 217 1'12118'297 1-22061'377 1-34396'457 1'48699'137 1'04932'218 1'12225'298 1'22203'378 1'34563'458 1'48889 ~138 i 105003 -219 1'12334'299 1'22347'379 1'34731'459 1'49079 ~139 1-05075'220 1.12445.300 1'22495'380 1'31899 460j 1.49269'140 105147'221 1'12556'301 1'22635'381 1'35068'461 149460'141 1'05220'222 1'12663 -302 1'22776'382 1'35237'462 1'49651 142 1-05293'223 1'12774'303 1'22918 383 1'35406'463 1'49842 ~143 1'05367.224 1'12885'304 1'23061 384 1'35575'464 1.50033 144 1'05441'225 1'12997'305 1'23205'385 1'35744'465 1-50224 ~145 1'05516.226 1'13108 306 1'23349'386 1-35914'466 1.50416 ~146 1'05591'227 1'13219 307 1'23494'387 1'36084 467 1-50608 ~147 1'05667'228 1'13331 308 1'23636'388 1'36254'468 1'50800'148 1'05743'229 1'13444'309 1'23780 389 1'36425'469 1-50992 ~149 1'05819'230 1'13557'310 1'23925 390 1'36596'470 1'51185'150 1'05896 -231 1'13671'311 1'24070'391 1'36767 471 1-51378 ~151 1'05973'232 1'13786'312 1'24216'392 1'36939'472 1'51571 ~152 1'06051'233 1'13903'313 1'24360'393 137111'473 151764.153 1'06130'234 1'14020'314 1'24506.394 1'37283.474 1.51958 154 1'06209 235 1'14136'315 1'24654'395 1'37455'475 1.52152'155 1'06288'236 1'14247'316 1'24801 396 1-37628'476 1.52346 ~156 1'06368'237 1'14363 317 1'24946'397 1'37801'477 1.52541'157 1'06449'238 1'14480'318 1'25095'398 1'37974'478 1.52736'158 1'06530'239 1'14597'319 1'25243'399 1'38148'479 1'52931 ~159 1'06611'240 1'14714 320 1'25391'400 1'38322'480 1-53126 ~160 1'06693'241 1'14831'321 1'25539 401 1'38496 481 1'53322'161 1-06775'242 1.14949'322 1'25686'402 1'38671'482 1.53518 ~162 1-06858.243 1.15067 -323 1'25836 403 1'38846 483 1'53714'163 1'06941.244 1.15186 -324 1'25987'404 1'39021'484 1.53910 ~164 1'07025'245 1'15308'325 1'26137 405 1'39196'485 1'54106'165 1-07109'246 1'15429 -326 1'26286 406 1'39372'486 1'54302 ~166 1'07194'247 1.15549'327 1'26437 407 1'39548'487 1.54499 ~167 1'07279'248 1'15670 -328 1'26588'408 1'39724 488 1.54696 ~168 1'07365.249 1'15791.329 1'26740 409 1'39900 489 1.54893 ~169 1'07451'250 1'15912 -330 1-26892 410 1'40077 490 1.55090 ~170 1'07537.251 1'16033'331 1'27044'411 1'40254'491 1.55288 ~171 1-07624.252 1'16157.332 1'27196'412 1'40432 492 1.55486 ~172 1-07711.253 1.16279 333 1'27349'413 1'40610.493 1,55685 ~173 1'07799.254 1-16402.334 1'27502'414 1'40788 494 1'55854 174 1-07888'255 1.16526 -335 1'27656'415 1'40966 495 1'56083 ~175 1'07977.256 1.16649 -336 1-27810 -416 1'41.145'496 1'56282'176 1-08066.257 1-16774'337 1-27864 -417 1-41324'497 1-56481'177 1'08156.258 1-16899'338 1-28118'418 1-41503 -498 1-56680'178 1-08246'259 1-17024'339 1-28273'419 1-41682 -499 1.56879'179 1-08337'260 1.17150.340 1-28428.420 1-41861 -500 1-57079'180 1-08428 Page 69 PROPORTIONS OF TIIE LENGTHS OF SEII-ELLIPTIC ARCS. 69 PROPORTIONS OF THE LENGTHS OF SEMIELLIPTIC ARCS. H Ieigllt Length of HIeight Length of Height Length of HIeight Lelngtll of H Ileight Lcnith of of Arc. Arc. of Ar. Arc. of Are. Arc. of Are. Arc. of Arc. ALre. ~100 1-04162 -157 1-10113 *214 1-66678.271 1 23835 *328 1.31472 *101 1-04262.158 1-10224.215 1.16799.272 1-23966 *329 1-31610 -102 1 04362.159 1 10335.216 1 16920.273 1-24097.330 I131748I *103 1-04462.160 1-10447.217 1-17041.274 1-24228.331 1-31886 ~104 1-04562.161 1-10560.218 1.17163 *275 1-24359 -332 1.32024 ~105 1.04662 *162 1 10672 -219 1-17285.276 1-24480 *333 1 32162 ~106 1 04762 -163 1 10784.220 1-17407 *277 1-24612.334 1 32300 ~107 1-04862.164 1-10896 *221 1-17529 -278 1-24744.335 134 138 ~108 1 04962'165 1 -11008 *222 1 17651 *279 1 24876 *336 1 -32576 *109 1 05063.166 1-11120.223 1-17774 *280 1 25010 *337 1 3 2715 ~110 1,05164 -167 1-11232 *224 1-17897'281 1 25142 *338 1 32854 ~111 1 05265'168 1-11344'225 1'18020'282 1 -25274.339 1-32993 ~112 1 05366'169 1-11456'226 1-18143 -283 1 25406 -340 1 -33132 ~113 1-05467.170 1-11569'227 1-18266.284 1.25538 341 1 33272.114 1-05568 171 1.11682'228 1.18390'285 1'25670.312 1 o3341_ ~115 1-05669.172 1-11795 *229 1-18514 *286 1'25803.343 1-33552 ~116 1 05770 *173 1-11908 *230 1-18638 *287 1'25936 -344 1{i.:xo 2C ~117 1-05872.174 1-12021.231 1-18762 -288 1.26069 *345 1-.3833 ~118 1-05974.175 1-12134.232 1-18886'289 1-26202.346 1-339, 7 ~119 1-06076'176 1-12247.233 1-19010'290 1-26335.347 1 34115'120 1 06178 -177 1 12360'234 1'19134 291 1-26468.348 1'34236 *121 1-06280 *178 1-12473 ~235 1-19258'292 1-26601.349 1'34397 *122 1-06382 *179 1-12586 *236 1-19382 *293 1-26734 *350 1,3459'*123 1-06484'180 1-12699 237 1-19506.294 1-26867.851 1.34681 ~124 1 -06586'181 1-12813'238 1-19630'295 1 27000 3.52) 1348923 ~125 1-06689.182 1-12927.239 1.19755.296 1-27133.353 1 341965 ~126 1-06792.183 1-13041'240 1-19880 * -297 1'27267 -354 1 35108 ~127 1-06895'184 1-13155 *241 1.20005' *298 1 27401 *355 1 352.A1 ~128 1-06998'185 1-13269.242 1-20130 -299 1-27535.356 1-35,394~129 1-07001 *186 1-13383 *243 1'20255;.300 1-27669.357 1-35537 ~130 1-07204.187 1-13497 *244 1-20380.301 1'27803.358 1-35680.131 1'07308'188 1.13611 -245 1.20506.302 1 27937.359 1 -35823.132 1-07412'189 1'13726'246 1 20632'303 1-28071.360 1 -35 7 -133 1-07516.190 1-13841.247 1.20758'304 1-28205.3G61 1.36111 *134 1-07621 *191 1'13956'248 1-20884'305 1'28339.362 1 362.55.13.5 107726 192 1.14071'249 1.210101 -306 128474T *363 1 363;99 *136 1 07831 -193 1-14186'250 1-21136'307 1 28609.364 1'36513 *137 1 07937'194 1-14301 -251 1'212631'308 1-28744.365 1'36688 *138 1-08043 *195 1-14416 *252 1-21390'309 1-28879.366 1-36833 139 1.08149 *196 1'14531'253 1'21517'310 1'29014.367 1 36978 *140 1-08255 -197 1-14646.254 1.21644'311 1-29149.368 1371293 *141 1-08362.198 1-14762.255 1-21772'312 1-29285.369 1 [37268'142 1-08469.199 1-14888'256 1 21900'313 1'29421.370 1 -37411 *143 1-08576 *200 1-15014'257 1'22028'314 2-295.57.371 1'3 662 *144 1-08684.201 1-15131 *258 1-22156 *315 1-296003.3I,7 7(08I *145 1-08792 *202 1-15248'259 1'22284 L316 1 - 9829.*33 73`1 3,854 *146 1 08901.203 1-15366'260 1 22412'317 1-29965.374 1 388009 -147 1 09010 -204 1-15484.261 1.22541.318 1 3010 2.3;7 1.3814 6 *148 1-09119'205 1-15602 *262 1 226 701 319 1 30239.376 1 -082) 2 *149 1 09228 *206 1-15720 *263 1 22 799'320 1'30376.3,7 1 8 38I)9 *150 1-09330.207 1-15838.264 1-22928'321 1 30513.378 1 38853 1.51 1 09448 *208 1*15957 *265 123057.322 1 306.o50 79 i 1 8' *152 1-09558 *209 1-16076.266 1-23186 *323 1309787 1 380 1 136879 153 1.09669'210 1-16196'267 1 23315 3724 1 30924.381 1 3O,0 1l'154 1-09780 *211 1-16316 |268 1-23445 *325 1 31061 38' 1 39169 -15.5 51-09891 *212 1-1643-)6'269 1.2357 *5'326 1031198'38 3 1 3'14 156 1-10002 *213 1-16557 1'270 1'237051 327 13133..384 13l94595'275;'2'13 34.95 Page 70 70 THE PRACTICAL MODEL CALCULATOR. I ci ht Length of Height Length of Height Length of Height Length of. H-eilt Legth of of Ace. Ac. Of Are. Are of Are. Are. of Are. Are. of Are. Are. -385 1-39605 *447 1-48850 *509 1-58474 *571 1-68195 *633 1 - 7817 2' ~ 86 1-39751.448 1 49003 *510 1 -58629 *572 1-68354 *634 1-78335 ~387 1 *39897.449 1 -49157 *511 1 -58784 573 1-68513.635 1-78498 -38S 1 40043.450 1 -49311.512 1 -58940 *574 1 68672'636 1-78660 ~389 1 40189'451 1 49465 *513 1'59096 *575 1-68831 *637 1-78823 ~390 1-40335 *452 1.49618 *514 1 59252 *576 1-68990 *638 1-78986 ~391 1-40481.453 1'49771.515 1 -59408 I577 1 69149 *639 1-79149 ~ 392 1'40627 *454 1 49924 *516 1-59564 -578 1-69308 *640 1-79312 -393 1 -40773 *455 1 -50077 *517 1 59720 *579 1-69467 *641 1-79475 ~394 1'40919 *456 1-50230 *518 1-59876 *580 1-69626 *642 1'79638 39 5 1 41065.457 1 -50383.519 1-60032 *581 1-69785 *643 1 -79801 ~396 1.41211 *458 1-50536 *520 1-60188 *582 1-69945 *644 1.79964 397 1 41357.459 1-50689.521 1.60344 I583 1.70105.645 1 801 27 *398 1-41504 *460 1 -50842 *522 1 60500 *584 1-70264 /646 1 80290 *399 1-41651.461 1-50996 *523 1-60656 *585 1-70424 *647 1-80454 *400 1-41798 *462 1-51150 *524 1 -60812'586 1. 70584 *648 1 -80617'401 1-41945 *463 1-51304.525 1-60968 -587 1'70745,'649 1 -80780.402 1-42092 *464 1-51458.526 1-61124.588 1-70905 *650 1 80943 403 1-42239.465 1-51612.527 1161280.589 1-71065.651 1-81107.404 1-42386'466 1-51766/.528 1-61436 *590 1-712.5 *652 1-81271 *405 1 -42533.467 1-51920'.529 1-61592'591 /1 71286 653 1 81435 *406 1-42681 *468 1 -52074.530 1-61748 -592 1-71546 *654 1-81599 407 1-42829.469 1 -52229 *531 1-61904.593 1-71 707.655 1 -81 763 *408 1-42977.470 1-52384.532 1-62060.594 1-718681 656 1-81928 *409 1-43125.471 1 -52539.533 1-62216.595 1 72029_1 657 1 -82091 -410 1-43273.472 1-52691.534 1-62372 596 1'72 1901 -658 1-82255 -411 1-43421.473 1-52849.535 1-62528 *597 1-72350 *659 1-82419 412 1-43569.474 1-53004.536 1-62684 *598 t1'72511 I660 1 82583 i 41 3, 14-371.8 *475 1-53159.537 1 -62840.599 1 726 722 *661 1-82747 ~414 1 43867.476 1-533141.538 1-62996 600 1-7'2833 662 1-82911 ~ 415 1.440161.477 1.53469.539 1-63152.601 1-72994 663 1-83075.41C, 1-441651.478 1 -53625.540 1-63309.602 1 7 3150 5.664 1-830 10 I -417 14-14314.479 1-53781 541 1-63465 I *603 1 7 3316 1'6G5 1 -83404'418 1'44463.480 1-53937.542 1-63623'604 1-73-477 666 1'834068 419 1 4461311481 1-54093 543 1637801 60 173638 GG6 1 83568'419~'~ I!.4s~~ ~.aoo~~481.543'667 - 887331 ~420 1'44763.482 1 54249 *544 1'63937 606 8 1 8 83897 ~421 1'44913 483 1'54405.545 1 64094'607 1-73960i'669 1'840611 -422 1 45064 j *484 1-54561.546 1-64251'608 1 74121 6'70 1.84226 i ~ 423' 1-45214.485 1-54718.547 1-64408'609 1-74283'671 184391' 424 145364.486 1.54875.548 1164565 610 l 1.74444i| 6i72 1 84 5 6 ~425 1.45515.487 1 55032.549 1.64722 611 1.74605i 6i3 1 84, 70 ~ 426 1 45665.488 1.55189].550 1.64879|1.612 174767 6.04 1 8488)5 ~42 7 1-45815.489 1 55346.551 1-650306 *613 1 1.749290 675 1 80500 ~428 1-45966/.490 1-55503 *5-52 1165193'614 1,75091 I' GG 1- -''15 1 429 1.46167 I.491 1.55660.553 1653-30'615 1 1 752.52 -677. 85379' 430 1-46268.492 1.55817.554 1 65507 616 1'75414 1 G6 78 1.85544 1'481 146419!.493 155974 -.0 5; 1-65665. 617 1-73576 6I -679 1 85709 ~432 1 46570i.*494 1-56131' 556 16582)310 618 1*'757,38/'680 1 -85874 ~ 433 1-46721 1.495 1-56289 -557 1665981'619 1'73900 681 1 S860;39 ~ 434 1'468721|.496 1-56447'5358 1'66139''620 1-76062 682 1 860 205 4% | 1 47023 *497 1-56605.559 1-66297 |621 1 - 76224 j 683 I-86370'436 1-471741.498 1.56763.560 1 G645 5 6'| 167636 I'6831 1.86 5 437 1-473261.499 1-56921.561 1 -6613.623 1 76548.685 1 -8600 ~ 48s 1 47478 500 1-57089 1562 1 -6771.6'4 1 *70'10 *686 1 8686B ~40I 1.476301/.501 1-57234 563 1 66929 625 16 179 l7.687 1 -8701 1 4i40 1.47782.o502 1.57389 564 11 -7087 I6 1-04.688 1 8719,a i 44 1.47934 503 1.5754-41 5 56 1.67245 t G627 [ 1771.-97I 689 i89 3(6i 4 1' O2 1.48086 j.504 1.57699.5G66; 1.67403 ] 6'28 1 -.77 7 9 690 18, ).44: 1-48 238.505 1-57854 I.56 l1-67561 61 9 151 1 444 - 4391. 506 1'58009 3568 11 6:0 l' 7761i 7 S69 18,S. 9 411o 1'48o544.507 1.58164 569 1 8 617'6G1'177847 i93 1 S8024 508 158319 570 1G680 63 178009 1 90 Page 71 PROPORTIONS OF THE LENGTHS OF SEMI-ELLIPTIC ARCS. 71 leiiglit Lengthof Height Length of Height Lenth of lcight Length of Ieight. Lengthl of i of Are. Are. of Arc. Arc. of Arc. Ar. r. of Ar. rc..695 1-883561'757 1-98794 *818 2'09360'879 22 0 "91 940') 31479 *t-696 1~88522'758 1-98964'819 2-09536J 880''04- 941 12'31666c 697 171-88688'759 1-99134 820 2-09712'881 29. 0656 942 1231852 6'98 1~ 88854'760 1-99305 821 2-09888'882 2 20'9'943'303 8 G'99 1 89020 *761 1-99476 *822 2-10065'883 2"2'10.)' 944'-32224 700~ 1-89186'762 1'99647'823 2-10242 *884 2-21205'945 2 3 O —111 *701 1-89352'763 1-998181 *824 2'10419 -885 2' 13881 -946 2'32598 - 702 1-89519 *764 1-999891'825 2-10596 *886 i2'21 91'94'!7 2 9'31 T703 1-89685 -765 2-00160 *826 2-10773'887 2 21754 9_48 2' )2972 -704 1-89851 *766 2-00331 *827 2.10950.888 2. 1)937.949 2.33160 05 1-90017 *767 2 005022 *828 2.11127 *889 22" 120.95()0 j.73'8 -706 1~90184 -768 2-00673 *829 2-1130)4 890 2 223003 951 2'335 7 -707 1 90350 *769 2-00844'830 2-11481'891 2'2286 -92',3: o'3l 7~08 1-90517'770 2-01016.831 2.11659 *89'2. 70' 95 9o3 33 1915 -709 1 90684 771 2'01187 *832 2'11837 *893 -.'22534 9o 4 0 *710 1-90852 *772 2'01359 *833 2-12015 *894 2-230`8 O' 95 2' 33 *711 1-91019'773 2-01531'834 2-12193 *895 2.2322'212 -95ti 4'' 34-S3 *712 1~91187'774 2-01702 *835 2-12371'896 2.234061.567,.4 6 7o *713 1 91355 1 *775 2-01874 *836 2'12549'897 2.23590' 91 8'4,-':862 *714 1-91523 *776 2'02045 *837 2-12727'898 2'2 3 7 4'99 2 3.5 0 U 71t5 1-91691 -777 2'02217 *838 2.12905 *899 2 -)3958 96O0 2 11'-t41 *716 1.91859 *778 2-02389 *839 2-13083'900;224142.961 2.35 t 1 7* z17 1 _92027 779 2-02561 -840 2-13261 9G01 22-.432:5 *6,2 1. 35621 *718 1.92195.780 2 02733 /841 2-13439 *902 2-24508.963 1');81S -719 12363 781 2 02907'842 2'13618'903 2-24691 *964 1' 30 -7'20 1-925-31 *'782 _203080 *843 2-13797'904 2.24874 965 2'3611 ~721 1'92700'783 2'032521 844 2'13976'905 2-25057 *966 21 S(; 1 ~722 1 92868 *784 2-03'42;51 845 2-14155'906 2-25040'967 2 o6 *723 1 93036 I'785 2'03598'846 2'14334 907 2'25423 *968 2'6T(i' 72- 1 1-932 0-1 786 2-03771 *847 2-14513'908 2-25606 11969 21" 3:, 2 ~ 725 1'93373'787 2-03944 *848 2'1469)2 909 2.25789 0 23 ~ 726 19 3541 788 2-04117'849 2-14871'910 2.25972 1' 91'l 2-3I1 727 1-937101 789 2-04290'850 2-15050'911 2-261a55 9O2 9 2 3.')-i 728 1-93878.790 2-04462.851 2-15229.912 2-26338, *973'2 3771T ~729 1.94046.791 2.0463.5 852 2.15409'913 2 265211 974::,08T *730 019-4215.792 2 04809.853 2-15589.914 2 26704.9t6 2987 i09j'731 1.94383.793 2.04983.854 2.15770.915 2.26888. 97 6 2 j38-1] 732 1 94552'794 20,5157 855 2.159 50 916 2270 1 9,-7 2.8352 ~ 733.194721 795 2.0.53313 856 2.16130 917 2 272541.98.2.0 7 G. ~734 1948901'796 205505'857 216309 -918 2.27437.99 -9 ~0 ~ 735 195059'797 2-05679.858 2-16489 2919 2.27620 6 980'2.::;905 ~ 736 1.9 5228'798 12-05853.859 2-16668 -920 2 278031. 981 2!'.392 I T ~ 737 1953397 799 2.06027'860 2. 16848.921 2.27987 I )8 2'(' ~738 1-95566'800 2 06202'861 2-17028 9922 2 28170'983 2 39 31i ~ 139 1-95735'801 2-06377'862 2-17209'923 2 28O35o4 94-184 2982; ~740 1.959941 802 2-06552'863 2-17389'924 2.28537'985 21 -O016U ~741 1 96(;741 803 2-06727'864 217'570'925 2.'8790 i'986 i' 40 08 ~74 2 1-962414'804 2-06901'865 2 17*751'926 2.28903'1 987' 40 100 ~743 1,96414 -805 2.070761 866 2-17932' 927 2.29086'3988'2 6 09 ~744 i 196583 806 2-07251 867 2 181131|'92 8 2 2927i30'989'. ~7-745 1.96753'807 2.07427 i868 2 18194 929 2 994.33 1'90'2;.d3, ~'46 1j96923 -808 2.07602'869 2918415;930 2-29301'9 91 ~747 1 1970931 809 2-07777'870 2.186356 931 -998'9'0. *!'3 131'i 748 1197432 810 201983'871 2 18837'3''.30004'7':.15 ~ 749 1.94t3l' 811 2 08128'872 2 19018 l 93' 23010- 8. 9!)4 I 19i 750 11.97602'812 2.083041 873 2.19'00 11 934 2' 3033 -99.5 "' 41' i ~751 1!)9,7772'813 2-08480'874 2-1938_2'935 2 9.,307 I996'. 4' 1'36 752' 1 97'3 814 208656'875 2-19564 9 936. 30 741 "'7' 23! ~ 753 l1.98113'815 2-08832 876 2 19746 93 33)092 9'8 2- 9 23 2.28 751- 1 98283'816 2.0'90008'877 2 19928 938') 31111'9II'. — 153 ~ 755 1.984531'817 2 09198 878 2'2.0110, 93 i' 3129.5 -100' 2.1 0s I75 1-9 8623'.I Page 72 72 THE PRACTICAL MODEL CALCULATOR. To find the length of an are of a circle, or the curve of a ridght semi-ellipse. RULE. —Divide the height by the base, and the quotient will be the height of an arc of which the base is unity. Seek, in the Table of Circular or of Semi-elliptical arcs, as the case may be, for a number corresponding to this quotient, and take the length of the arc from the next right-hand column. Multiply the number thus taken out by the base of the arc, and the product will be the length of the arc or curve required. In a Bridge, suppose the profiles of the arches are the arcs of circles; the span of the middle arch is 240 feet and the height 24 feet; required the length of the arc. 24 -.- 240 ='100; and'100, by the Table, is 1'02645. Hence 1'02645 x 24 = 246'34800 feet, the length required. The profiles of the arches of a Bridge are all equal and similar semi-ellipses; the span of each is 120 feet, and the rise 18 feet; required the length of the curve. 28. 120 ='233; and'233 by the Table, is 1'19010. Hence 1'19010 x 120 = 142-81200 feet, the length required. In this example there is, in the division of 28 by 120, a remainder of 40, or one-third part of the divisor; consequently, the answer, 142'81200, is rather less than the truth. But this difference, in even so large an arch, is little more than half an inch; therefore, except where extreme accuracy is required, it is not worth computing. These Tables are equally useful in estimating works which may be carried into practice, and the quantity of work to be executed from drawings to a scale. As the Tables do not afford the means of finding the lengths of the curves of elliptical arcs which are less than half of the entire figure, the following geometrical method is given to supply the defect. Let the curve, of which the length is required to be found, be ABC. c d A_____________ Produce the height line Bd to meet the centre of the curve in g. Draw the right line Ag, and from the centre g, with the distance gB describe an arc Bh, meeting Ag in h. Bisect Ali in i, and from the centre g with the radius gi describe the arc ik, meeting dB produced to k; then i/c is half the are ABC. Page 73 TABLE OF RECIPROCALS OF NUMBERS. 73 A TABLE of the Reciprocals of Numbers; or the DECIMAL FRACTIONS corresponding to VULGAR FRACTIONS of which the Numerator is unity or 1. [In the following Tables, the Decimal fractions are Reciprocals of the Denominators of those opposite to them; and their product is = unity. To find the Decimal corresponding to a fraction having a higher Numerator than 1, multiply the Decimal opposite to the given Denominator, by the given Numerator. Thus, the Decimal corresponding to 1 being -015625, the Decimal to -5 will be -015625 x 15 = -284375.] Fraction or Decimal or Fraction or Decimal or Fraction or Decimal or Numb. Reciprocal. Numb. Reciprocal. Numb. Reciprocal. 1/2.5 1/47 0212766 192 0108695G5 1/3 -333333333 1/48'0208333333 1/93'010752688 1/4'25 1/49'020408163 1/94'010638298 1/5 *2 1/50 -02 1/95 01052'6316 1/6'166666667 1/51'019607843 1/96 -010416667 1/7'142857143 1/52 019230769 1/97'010309,278 1/8'125 1/53 -018867925 1/98' 010204082 1/9 111111111 1/54'018518519 1/99'01010101 1/10'1 1/55 -018181818 1/100'01 1/11'090909091 1/56'017857143 1/101'00990099 1/12 -08333333 1/57'01754386 1/102'0098039'22 1/13'076923077 1/58'017241379 1/103'009708738 1/14'071428571 1/59'016949153 1/104'009615385 1/15'066666667 1/60'016666667 1/105 -00952381 1/16 -0625 1/61'016393443 1/106 -009433962 1/17 -058823529 1/62 -016129032 1/107 -009345794 1/18'055555556 1/63 -015873016 1/108 -009259'59 1/19 -052631579 1/64.015625 1/109.009174312 1/20.05 1/65.015384615 1/110.009090909 1/21.047619048 1/66.0151515115 1/111 009009009 1/22'045454545 1/67'014925373 1/112 -00O898571 1/23 -043478261 1/68.014705882 1/113 -008849,558 1/24'041666667 1/69'014492754 1/114'00877193 1/25'04 1/70'014285714 1/115 -0086956.52 1/26'038461538 1/71'014084517 1/116'00802069 1/27 -0307037037 11/72'013888889 1/117 008,547009 1/28'035714286 1/73'01369863 1/118 -008474576 1/29'034482759 1/74'013513514 1/119 -008403361 1/30'033333333 1/75'013333333 1/120 -008333333 1/31 -032258065 1/76'013157895 1/121.0082i,4463 1/32'03125 1/77'012987013 1/122.008196721 1/33 -030303030 1/78.012820513 1/123 *008130081 1/34 -029411765 1/79'012658228 1/124. 008064O16 1/35 -028571429 1/80'0125 1/125 -008 1/36'027777778 1/81.012345679 1/126'007936508 1/37 -270027027 1/82 *012195122 1/127'007874016 1/38'026315789 1/83'012048193 1/'128'0078125 1/39 -025641026 1/84 -011904762 1/129 -007751938 1/40 -025 1/85 -011764706 1/130 -007692308 1/41'024390244 1/86 -011627907 1/131 -007633588 1/42 -023809524 1/87 -011494253 1/132 -007575758 1/43'023255814 1/88 -011363636 11/133'007518797 1/44 -022727273 1/89 -011235955 1/134'007462687 1/45 -022222222 1/90'011111111 1/135'007407407 1/46 -02173913 1/91 -010989011 1/136 -007352941 G Page 74 74 THE PRACTICAL MODEL CALCULATOR. Fraction or Decimal or Fraction or Decimal or Frctioi or0 Deeirirna or Numb. Reciprocal. Numb. Reciprocal. clrub. Ilecilerocal. 1/137'00729927 1/198'00500050 0 1 259'003861004 1/1 38'007246377 1/199'005025126 1' 260'003846154 1/139'007194245 1/200'005 1 261'008831418 1/140'007142857 1,/201'004975124 11,262'003816794 1/141'007092199 1/202 *004950495 1 9263 003802281 1/142'007042254 1/203'004926108 1,264' 003787879 1/143'006993007 1/204 *004901961 1,65' 0(o 037,27785 1/144'006944444 1/205'004878049 21 i6'0037)'0398 1/145 *006896552 1/206'004854369 1/267'003745318 1/146'006849315 1/207'004830918 1'268'0037 31343 1/147'006802721 1/208'004807692 1/2U'9'0031o7,72 1/148 *006756757 1/209'004784689 1/270'003703704 1/149'006711409 1/210 *004761905 1:271'00)36900 37 1/150'006666667 1/211'004739336 1 27 2 00367C471 1/151'006622517 1/212'004716981 1/ 273'003663004 1/1-52 006578947 1/213'004694836 1/274 003649)635 1/153'006535948 1/214 -004672897 1/275'00363G364 1/154'006493506 1/216'004651163 1/276'0036231SS 1/1.55.006451613 1/216'00462963 1/27 7 003610108 1/156.006410256 1/217'004608295 1/278.003597122 1/157.006369427 1/ 18 -004587156 1/279.003584229 1/158 -006329114 1l'219'00456621 1/280.003571429 1/159 -006289308 1/',0'004545455 1/281.003558719 1/160 -00625 1,'221'004524887 1/282.003546099 1/161 -00621118 1,/222.004504504 1/,283.003533569 1/162.00617284 1/223'004484305 1/2 84.003522127 1/163.006134969 1/224.004464286 1'285.003 508772 1/164'006097561 l1/'25 004444444 1,286 003496503 1/165.006060606 1/226. 004424779 1/287.003-184321 1/166.006024096 1/')7.004405286 1/288.003472222 1/167.005988024 1 /02S8'004385965 1/2)89 003460208 1/'168'005952381 1/)9 *004366812 1 290.003448276 1/169 -00591716 1 230.004347826 10291 003436 26 1/17O0 005882353 1'231'004329004 1,092 0034')4'658 1/1 71'005847953 1/'232.004310345 1/293'003412969 1/172 *005813953 1 233.004291845 1,2 904'003401361 1/173'005780347 1/234.004273504 1/295' 003389!831 1/174.005747126 1/235.004255319 1/296'0033 78378 1/175.005714286 1,236.004237288 1/297'003367003 1/176'005681818 1! 237.004219409 1/'298 003355705 1/177'005649718 1/238.004201681 1/299'003344482 1/178'005617978 11/239 0041841 1/300'003333333 1/179 *005586592 1/240 004166667 1/301 003322259 1/180'005555556 1 241'004149378 1,302 003311258 1/181'005524862 1, 24 *004132231 1/303'00330133 1/182 00549405 1;'/243.00411522G 1/304 003289474 1/183.005464481 1/244.004098361 1,'305.003278689 1/184.005434783 1/245.004081633 1/306.003267974 1/185.005405405 1'246 004065041 1/307 -003257329 1/186.005376344 "1/247.004048583 1/308 003246753 1/187.005347594 1/248.004032258 1'309.003236246 1/188.005319149 1/249.004016064 1/310.003225806 1/189 005291005 1 1/250 /004 11311.003215434 1/190 1005263158 1/251.003984064 1/,312.003205128 1/191.005235602 1/252.003968254 1/313 003194888 1/192.005208333 1/253 6003952569 1/ 314 003184713 1/193.005181347 1/254 0003931008 1 315 -003174603 1/194.005154639 1/255.003921569 1'/316.003164557 1/195.005128205 1/256 00390625 1/317 0031545374 1/196 -005102041 1/257.003891051 1/318.003144654 1/197 -005076142 1/258.003875969 1,319.003134796!. __ Page 75 TABLE OF RECIPROCALS OF NUMBERS. 75 Fraction or Decimal or Fraction or Decimal or Fraction or Decimal or N aub. Reciprocal. Numb. Reciprocal. Numb. Reciprocal. 1, 082 0 0038125 1/381 -002624672 1/442 -002262443 1- -1003115265 1/382 -002617801 11/443 -00225733"6 13,)- -00310559 1/383 -002610966 1/1444 -002252252 1 3 003095975 1/384 *002604167 1/445 -0022471191 I.00308642 1/385 *002597403 1/446 -00224-2152 I8 2>5 -003076923 1/386 *002590674 1/447 0 0223 )7 1386 I 36.003067485 1/387 *002583979 1/448 -002232143 0297 -0030.58104 1/388 -00257732 1/449 0 02 22'7 17 1 1 2 8 00304878 1/389 -002570694 1/450 -00~2222222 1/32~9 -003039514 1/390 -002564103 1/451 0 0 22)1 72 9 5 19,30 -003030303 1/391 -002557545 1/452 -00-22123089 10 3 1 003021148 1/392 -00255102 1/453 -00')')07506 1 /33 8 003012048 1/393 -002544529 1/454 -002202643 13033 -003003003 1/394 -002538071 11/455 -002197802 13r) 4 -002994012 1/395 -002531646 1/456 -002192982 1 34 35 002985075 1/396 -002.5250253 1/457 -002188184 1/336 -00297619 1/397 -0025018892 1/458 -002~183406 1/307 -002967359 1/398 002512563 /40 9 -002178649 1/398 -00295858 1/399 -002506266 1/460 -002173913 1u9 -0029949853 1/400 -0025 1/461 -0021 69197 1 3840 -002941176 1/401 -002493766 1/462 *0021645002 1 341 -002932551 1/402 *002487562 1'/463 -002 1 5982 7 1/34) -002923977 1/403 -00248139 1/464.00215.3172 1 34~ *002915452 1/404 -002475248 1/465.002150.538 1/ 44 0029697 1/405 -002469136 1 /466 -0021459')3 1I,' 02985 1/406 -002463054 1/467 001412 I. 1 46 -002890173 1/407 -002457002 1/468 -0021306752 1 At 347 002881844 1/408 -00245098 1/469 -0021321966 1/948 -0028713563 1/409 -002444988 1/4 70 -002127166 1/0o49 -00286533 1/410 -002439024 1/471 1 00212'-3142 1 /099-0.002857143 1/411 -00243309 1/472 -002118644 1/.1 -002849003 1/412 -002427184 1/473 0 021I1 41 65) I1/3"9-2 -002840909 1/413 -002421308 1/474 -00210970.5 1 4 I53 -002832861 1/414 -0024154,59 1/475 -002105263 13 1)5 4 -002824859 1/415 -002409639 1/476 -00210084 11/35 -002816901 1/416 *002406846 1/477 *002096486 1, 56 -002808989 1/417 -002398082 1/4 78 -00209205 1 5 7 -00 280112 1/418 -00239`9344 1/479 -002087683 I, ))58 -002793"296 1,419 -00238663.5 1/480 0 02 0 8)8333 1- ~9 -002785.515 1 /42.,,0 -002380952 1/481 -002079002 1 mO -002777778 1/421 -002375297 1/8 02749 61 002770083 1/422 -002369668 1/483 -002070393 1 2, 002762431 1/423 -002364066 1/484 -002066116 1; 913 -002754821 1 1/424 -002358491 1/485 -002061856 11 41 -002747235 1/425 -002352941 1/486 -002057613 1 -002739726 1/426 -002347418 1/487 -002053338 11 366 -00273224 1/427 -00234192 1/488 -00204918 10 9j -002724796 1/428 -002336449 1/489 -00204499 I "G8 002717391 1/429 -002331002 1/490 -002040816 06 -9002710027 1/430 *0029325.581 1,491 -002036666 0 002)702703 1/431 00-2320186 1/4,92 -00-203252 1 11 -00269.5418 1/432 -002314815 1/493 *0020280398 172 -002688172 1/433 *002309469 1/494 -002024291 1 708 -002680965 1/434 -0021304147 1/495 -002020202 1e3 4 -002673797 1/435 -0022988.51 1/496 -002016129 I -002666667 1/436 -002293578 1/497 0 02 0 12 0 72 1, 876 -0026.59574 1/437 -00228833 1/498'002008032 11:e'7 -0026.5252 1/438 -002280"10-5 1/499 -002004008 1, 7b 8 0026455013 ) 1/439 -002277904 1/500 -002 1, 79 -002638521 1/440 -002272727 1,501 -001996008 138 0 -002631579 1/441 -002267574 1 1/502 -0019920302 Page 76 76 THE PRACTICAL MODEL CALCULATOR. Fraction or Decimal or Fraction or Decimal or Fraction or Decimal or Numb. lReciprocal. Numb. Reciprocal. Numb. Reciprocal. 1/503 -001988072 1/564.00177305 1/625.0016 1/504'001984127 1/565 *001769912 1/626 *001597444 1/505.001980198 1/566'001766784 1/627.001594896 1/506'001976285 1/567 *001763668 1/628.001592357 1/507'001972387 1/568'001760563 1/629'001589825 1/508 *001968504 1/569'001757469 1/630'001587302 1/509'001964637 1/570'001754386 1/631'001584786 1/510'001960784 1/571 -001751313 1/632'001582278 1/511.001956947 1/572'001748252 1/633.001579779 1/512'001953125 1/573'001745201 1/ 634'001577287 1/513'001949318 1/574'00174216 1/635.001574803 1/514'001945525 1/575'00173913 1/636'001572327 1/515.001941748 1/576'001736111 1/637 -001569859 1/516 *001937984 1/577'001733102 1/638 *001567398 1/517'001934236 1/578 -001730104 1/639 *001564945 1/518'001930502 1/579'001727116 1/640.0015625 1/519'001926782 1/580.001724138 1/641'001560062 1/520'001923077 1/'581.00172117 1/642'001557632 1/521'001919386 1/582.001718213 1/643'00155521 1/522 -001915709 1/583.001715266 1/644'001552795 1/523 -001912046 1/584'001712329 1/645 *001550388 1/524'001908397 1/585.001709402 1/646.001547988 1/525'001904762 1/'586.001706485 1/647.001545595 1/526 *001901141 1/587.001703578 1/648 -00154321 1/527'001897533 1/588.00170068 1/649 -001540832 1/528'001893939 1/589.001697793 1/660.001538462 1/329.001890359 1/590.001694915 1/651 -001536098 1,'530'001886792 1/591.001692047 1/652.001533742 1/531'001883239 1/592.001689189 1/653.001531394 1j'532 001879699 1/593.001686341 1/654.001529052 1/533'001876173 1/594.001683502 1/655.001526718 1/534'001872659 1/595.001680672 1/656.00152439 1/535'001869159 1/596,00167 7852 1/657 *00152207 1/536'001865672 1/597.001675042 1/658.001519751 1/537 *001862197 1/'598.0016172241 1/659.001517451 1,'538'0018.58736 1/599.001669449 1/660.001515152v 1/539 001855288 1/600.001666667 1/661 001512859 1/540 001851852 1/601 *001663894 1/662 001510574 1/541.001848429 1/602 *00166113 1/663 001508296 11/542'001845018 1/603'001658375 1/664'001506024 1/5438 001841621 1/604'001655629 1/665'001503759 1/544.001838235 1/605 o001652893 1/666 001501502 1/545'001834862 1/606'001650165 1/667'00149925 1/546'001831502 1/607'001647446 1/668'001497006 1/547'001828154 1/608'001644737 1/669'001494768 1/548'001824818 1/609'001642036 1/670'001492537 1/549'001821494 1/610'001639344 1/671 -001490313 1/550'001818182 1/611'001636661 1/672'001488095 1/551'001814882 1/612'001633987 1/673'001485884 1/552'001811594 1/613'001631321 1/674 *00148368 1/553'001808318 1/614'001628664 1/675'001481481 1/554'001805054 1/615'001626016 1/676 *00147929 1/555'001801802 1/616'001623377 1/677'001477105 1/556'001798561 1/617'001620746 1/678 -001474926 1/557'001795332 1/618 *001618123 1/679'001472754 1/558'001792115 1/619'001615509 1/680'001470588 1/559'001788909 1/620'001612903 1/681'001468429 1/560'001785714 1/621'001610306 1/682'001469276 1/561'001782531 1/622'001607717 1/683'001464129 1/562'001779359 1/'623'001605136 1/684'001461988 1/563 001776199 1/624'001602564 1/685'001459864 Page 77 TABLE OF RECIPROCALS OF NUMBERS. 77 Fraction or Decimal or Fraction or Decimal or Fraction or Decimal or Numb. Reciprocal. Numb. Reciprocal. Numb. Reciprocal. 11686'001457726 1/747.001338688 1/808'001237624 1/687'001455604 1/748'001336898 1/809 *001236094 1/688'001453488 1/749 *001335113 1/810'001234568 1/689'001451379 1/750'001333333 1,811'001233046 1/690'001449275 1/751 *001331558 1/'812'001231527 1/691'001447178 1/752'001329787 1/813 -001230012 1,692'001445087 1/753 *001328021 1/814'001228501 1,693'001443001 1/754'00132626 1/815 -001226994 1/694'001440922 1/755'001324503 1/'816 -001225499 1/695'001438849 1/756'001322751 1/817'00122399 1/696'001436782 1/757 *001321004 1/818'001222494 1/697'00143472 1/758'001319261 1./819'001221001 1/698'001432665 1/759 -001317523 1/820 -001219512 1/699'001430615 1/760'001315789 1,"821'001218027 1/700'001428571 1/761 *00131406 1/822'001216545 1/701'001426534 1/762 *001312336 1/823'001215067 1/702'001424501 1/763.001310616 1/824'001213592 1/703'001422475 1/764.001308901 1/825'001212121 1/704 001420455 1/765.00130719 1/826.001210654 1/705 -00141844 1/766.001305483 1/827.00120919 1/706 001416431 1/767.001303781 1/828.001207729 1/'707'001414427 1/768 -001302083 1/829'001206273 1i708'001412429 1l/769.00130039 1/830.001204819 1/709'001410437 1/770.001298701 1/831.001203369 1/710'001408451 1/771 001297017 1/832.001201923 1'/711 00140647 1/772.001295337 1/833.00120048 1,712.001404494 1/773.001293661 1/834 *001199041 1/713'001402525 1/774.00129199 1/835 *001197605 1/714'00140056 1/775.001290323 1/836.001196172 1/715'001398601 1/776.00128866 1/837.001194743 /i716'001396648 1/777.001287001 1/838.001193317 1/717'0013947 1/778.001285347 1/839.001191895 1/718'001392758 1/779.001283697 1/840'001190476 1/719'001390821 1/780.001282051 1/841.001189061 1/720.001388889 1/781.00128041 1/842.001187648 1/721'001386963 1/782.001278772 1/843'00118624 1/722.001385042 1/783'001277139 1/844'001184834 1/723'001383126 1/784'00127551 1/845'001183432 1/724.001381215 1/785'001273885 1/846'001182033 1/'725'00137931 1/786'001272265 1/847'001180638 1/726'00137741 1/787'001270648 1/848'001179245 1/727'001375516 1/788'001269036 1/849.001177856 1/728'001373626 1/789'001267427 1/850.001176471 1,729.001371742 1/790 -001265823 1/851.001175088 1/730'001369863 1/791 -001264223 1/952'001173709 1/731'001367989 1/792.001262626 1/853'001172333 1/732.00136612 1/793'001261034 1/854'00117096 1/733'001364256 1/794.001259446 1/855.001169591 1/734.001362398 1/795'001257862 1/856'001168224 1/735.001360544 1/796.001256281 1/857.001166861 1/736.001358696 1/797'001254705 1/858'001165501 1/737.001356852 1/798.001253133 1/859'001164144 1/738'001355014 1/799.001251364 1/860.001162791 1/739.00135318 1/800.00125 1/861.00116144 1/j740.001351351 1/801 -001248439 1/862.001160093 1/741'001349528 1/802.001246883 1/863'001158749 1/742'001347709 1/803'00124533 1/864.001157407 1/743.001345895 1/804'001243781 1/865'001156069 1/744 -001344086 1/805'001242236 1/866'001154734 1/745.001342282 1/806'001240695 1/867'001153403 1/'746.001340483 1/807.001239157 1/868'001152074 G 2 Page 78 78 THE PRACTICAL MIODEL CALCULATOR. Fraction or Decimal or Fraction or Decimal or Fraction or Decimal or Numb. Reciprocal. Numb. Reciprocal. Numb. Reciproeal. 1/869 001150748 1/913 00109529 1/957 *001044932 1/870 *001149425 1/914.001094092 1/958'00104i'841 1/871 *001148106 1/915'001092896 1/959'001042753 1/872 *001146789 1/916 001091703 1/960 001041667 1/873 *001145475 1/917'001090513 1/961.001040583 1/874.001144165 1/918 -001089325 1/962'001039501 1/875 -001142857 1/919'001088139 1/963'001038422 1/876 -001141553 1/920'001086957 1/964'001037344 1/877 *001140251 1/921 001085776 1/965 -001036269 1/878 001138952 1/922'001084599 1/966 0010'35197 1/879 001137656 1/923 001083423 1/967.001034126 1/880.001136364 1/924.001082251 1/968.001033058 1/881.001135074 1/92)5.001081081 1/969.001031992 1/882.001133787 1/926 *001079914 1/970.001030928 1/883.001132503 1/927 001078749 1/971.001029866 1/884 001131222 1/928.001077586 1/972.001028807 1/885.001129944 1/929 001076426 1/973.001027749 1/886.001128668 1/930.001075269 1/974.001026694 1/887 001127396 1/931 -001074114 1/975 001025641 1/888.001126126 1/932.001072961 1/976 *00102459 1/889 001124859 1/933 001071811 1/977.001023541 1/890 *001123596 1/934.001070664 1/978.001022495 1/891.001122334 1/935 *001069519 1/979 00102145 1/892.001121076 1/936.001068376 1/980 001020408 1/893.001119821 1/937.001067236 1/981 *001019168 1/894.001118568 1/938 *001066098 1/982.00101833 1/895.001117818 1/939 -001064963 1/983.001017294 1/896.001116071 1/940 -00106383 1/984.00101626 1/897.001114827 1/941 *001062699 1/985 -001015228 1/898 *001113586 1/942.001061571 1/986.001014199 1/899.001112347 1/943 *001060445 1/987 *001013171 1/900.001111111 1/944.001059322 1/988 001012146 1/901.001109878 1/945.001058201 1/989.001011122 1/902.001108647 1/946.001057082 1/990.001010101 1/903.00110742 1/947.001055966 1/991.001009082 1/904 *001106195 1/948 *001054852 1/992 001008065 1/905.001104972 1/949.001053741 1/993 001007049 1/906.001103753 1/950.001052632 1/994.001006036 1/907.001102536 1/951.001051525 1/995.001005025 1/908.001101322 1/952.00105042 1/996.001004016 1/909.00110011 1/953 *001049318 1/997 *001003009 1/910 *001098901 1/954 *001048218 1,/998.001002004 1/911.001091695 1/955.00104712 1/999 001001001 1/912.001096491 1/956 -001046025 1/1000 -001 Divide 80000 by 971. By the above Table we find thatl divided by 971 gives'001029866, and 001029866 x 80000 = 82'38928. What is the sum of 8 and 2? 1 5 x 883 = 001132503 x 5 = 005662515 1 2 x 91 3 -001049318 x 2 = -002098636 5 2 883 + 953 = 007761141 Page 79 MENSURATION OF SOLIDS. 79 WEIGHTS AND VALUES IN DECIMAL PARTS. AVOIRDUPOIS AVOIRDUPOIS! TROY WEIGHT. WEIGHT. WEIGHT. Dec. parts of a lb. Dec. parts of a cwt. Dec. parts of a lb. Ozs. Decimals. Qrs. Decimals. Ozs. Decimals. 11 916666 3' 10 93 75 10 5833333 2 *5 14'875 9 75 1 25 13'8125 8'666666 lbs. Decimals. 12'75 7 *583333 27'241071 11'6875 6.5 26 -232142 10 625 5 *416666 25.223214 9 5625 4.333333 24'214286 8'5 3 *25 23'205357 7 43075 2 -166666 22.196428 6 375 1 *083333 21'187500 5 3125 Dwts. Decimals. 20 *178572 4.25 19 079166 19.169643 3.18 18 075 18.160714 2.125i 17'070833 17.151785 1 0625 16 066666 16'142856 Drs. Deciials. 15 0625 15.133928.15'08 93 14 058333 14'125 14.054686 13 054166 13'116071 13 050780 12'05 12'107143 12.046874 11'045833 11 -098214 11.042968 10'041666 10'089286 10.039062 9'0375 9'0803.57 9 0351.56 8'033333. 8 071428 8.0312.5 7'029166 7'0625 7'027343 6'025 6.053571 6'023437 5'020833 5'044643 5.019531 4'016666 4'035714 4.015625 3'0125 3'026786 3.011718 2'008333 2'017857 2 00 812 1'004166 1 008928 1.003906 Grs. Decimals. Ozs. Decimals. 15'002604 15 -008370 LOSG MIEASURE. 14'002430 14'007812 13'002257 13'007254 Dec. parts of a foot. 12'002083 12'006696 Iils. Deci.als. 11'001910 11'006138 11 q91666th6 10'001736 10'005580 10'833333 9'001562 9'005022 9 i5 8'001389 8'004464 8.666666 7'001215 7'003906 7'583333 6'001042 6'003348 6 5 5'000868 5'002790 5 -4166 6 4'000694 4'002232 4 -333333 3 *000521 3'001674 3 25 2'000347 2'001116 2 *1 1666G, 1 1000173 1'0005.58 1 083333 To find the solidity of a cube, the heicqht of one of its sides being given.-Multiply the side of the cube by itself, and that product again by the side, D C and it will give the solidity required. The side AB, or BC, of the cube ABCDFGHE,G - I. is 25'5: what is the solidity? Here AB3 = (22 5)-3= 25'5 x 25'5 x 25'5 A B 25'5 x 650'25 = 16581'375, content of the ctube. Page 80 80 THE PRACTICAL MODEL CALCULATOR. To find the solidity of aparallelopipedon. F E -Multiply the length by the breadth, and \ that product again by the depth or altitude, D and it will give the solidity required. Required the solidity of a parallelopipedon ---------------------- ABCDFEHG, whose length AB is 8 feet, "1 its breadth FD 41 feet, and the depth or A B altitude AD 6- feet? Here AB x AD x FD = 8 x 6'75 x 4.5 -= 54 x 4'5 = 243 solid feet, the contents of the parallelopipedon. To find the solidity of a prism.-Multiply the area of the base into the perpendicular height of the prism, and the product will be the solidity. What is the solidity of the triangular prism ABCF E ED, whose length AB is 10 feet, and either of the equal sides, BC, CD, or DB, of one of its equilateral AF..i ends BCD, 2~ feet? Here 1 x 2.52 x 4/3 = 4 X 625 x V3 = 15625 x V3 = 1'5625 x 1'732 = 2'70625 = area of the base BCD. 25 + 2'5 + 2'5 7'5 Or, 2 -- 2 375 = sum of the sides, BC, CD, DB, of the triangle CDB. C And 3'75 - 2'5 = 1'25,. 1'25, 1'25 and 125 =- 3 differences. W/hence v3'75 x 1'25 x 1'25 x 1'25 = v3'75 x 1'253 = v/732421875 = 2'7063 = area of the base as before, And 2'7063 x 10 = 27'063 solid feet, the content of the prisnm required. To find the convex surface of a cylinder.-Multiply the periphery or circumference of the base, by the height of the cylinder, and the product will be the convex surface. D --- What is the convex surface of the right cylinder ABCD, whose length BC is 20 feet, and the diameter of its base AB 2 feet? Wfere 341416 x 2 = 62832 = periphery of the base AB. And 6'2832 x 20 - 125'6640 square feet, the A - convexity required. To find the solidity of a cylinder. —Multiply the area of the base by the perpendicular height of the cylinder, and the product will be the solidity. What is the solidity of the cylinder ABCD, the diameter of whose base AB is 30 inches, and the height BC 50 inches. Here'7854 x 302 = -7854 x 900 = 706'86 = area of the base AB. 35343 And 706'86 x 50 = 35343 cubic inches; or 1728 - 20'4531 solid feet. Page 81 MENSURATION OF SOLIDS. 81 The four following cases contain all the rules for finding the superficies and solidities of cylindrical ungulas. F Tltlen the section is parallel to the axis of the cylinder. I RULE.-Multiply the length of the arc line of the base I by the height of the cylinder, and the product will be the carve surface. Multiply the area of the base by the height of the A [ cylinder, and the product will be the solidity. WVhen the section passes obliquely through the opposite sides of the cylinder. RULE.-Multiply the circumference of the base of the cylinder by half the sum of the greatest and least lengths E of the ungula, and the product will be the curve surface. Multiply the area of the base of the cylinder by half A. B the sum of the greatest and least lengths of the ungula, and the product will be the solidity. Wlhen the section passes through the base of the cylin- I der, and one of its sides. RULE.-Multiply the sine of half the arc of the base D by the diameter of the cylinder, and from this product subtract the product of the arc and cosine. Multiply the difference thus found, by the quotient of e A the height divided by the versed sine, and the product o will be the curve surface. From 2 of the cube of the right sine of half the arc of the base, subtract the product of the area of the base and the cosine of the said half arc. Multiply the difference, thus found, by the quotient arising from the height divided by the versed sine, and the product will be the solidity. c When the section passes obliquely through both ends of the cylinder. D RULE. —Conceive the section to be continued, till it meets the side of the cylinder produced; then say, as the difference of the versed sines of half the arcs of the two ends of the ungula is to the versed sine of half the A arc of the less end, so is the height of the cylinder to the part of the side produced. Find the surface of each of the ungulas, thus formed, and their difference will be the surface. In like manner find the solidities of each of the ungulas, and their difference will be the solidity. To find the convex surface of a right cone.-Multiply the circumference of the base by the slant height, or the length of the side of the cone, and half the product will be the surface required. The diameter of the base AB is 3 feet, and the slant height AC or BC 15 feet; required the convex surface of the cone ACB. 6 Page 82 82 THIE PRACTICAL MODEL CALCULATOR. H/ere 3'1416 x 3 = 9'4248 = circumference of the base AB. 9'4248 x 15 141'3720 And 942482 = 2 70'686 squarefeet, the convex surface required. To find the convex surface of the frustum of a right cone. —ultiply the sum of the perimeters of the two ends, by the slant height of the frustum, and half the product will be the surface required. C In the frustum ABDE, the circumferences of the two ends AB and DE are 22'5 and 15'75 respectively, and the slant height BID is 26; what is the convex surface? E D (22'5 + 15'75) x 26 Here 2 = 225 + 1575 x 13 = 38'25 x 13 = 497'25 = convex surface. A' To find the solidity of a cone or pyramid.-Multiply the area of the base by one-third of the perpendicular height of the cone or pyramid, and the product will be the solidity. C Required the solidity of the cone ACB, whose diameter AB is 20, and its perpendicular height CS 24. Ilere'7854 x 202= -7854 x 400 = 314116 E D area of the base AB. 24 And 31416 x - 31416 x 8 = 2513'28 solidity required. A ---------- Required the solidity of the hexagonal pyramid ECBD, each of the equal sides of its base being 40, and the perpendicular height CS 60. Here 2'598076 (multiplier when the side is 1) x 402 = 2'598076 x 1600 = 4156 9216 = area C b of the base. 60 And 4156'9216 x - = 4156'9216 x 20 = 83138'432 solidity. E - S B P A To find the solidity of a frustum of a cone or pyramid.For the frustum of a cone, the diameters or circumferences of the two ends, and the height being given. Add together the square of the diameter of the greater end, the square of the diameter of the less end, and the product of the two Page 83 MENSURATION OF SOLIDS. 83 diameters; multiply the sum by'7854, and the product by the height; ~- of the last product will be the solidity. Or, Add together the square of the circumference of the greater end, the square of the circumference of the less end, and the product of the two circumferences; multiply the sum by'07958, and the product by the height; 1 of the last product will be the solidity. Por thefrustum of a pyramid whose sides are regular polygons.Add together the square of a side of the greater end, the square of a side of the less end, and the product of these two sides; multiply the sum by the proper number in the Table of Superficies, and the product by the height; 1 of the last product will be the solidity. When the ends of the pyramids are not regular polygons. —Add together the areas of the two ends and the square root of their product; multiply the sum by the height, and 4 of the product will be the solidity. What is the solidity of the frustum of the cone E D EABD, the diameter of whose greater end AB is 5 feet, that of the less end ED, 3 feet, and the perpendicular height Ss, 9 feet? (52 + 32 + 5 x 3) x *7854 x 9 34643614 3 - = 3 A B —--- 115'4538 solid feet, the content of the frustum. What is the solidity of the frustum eEDBb of a e b hexagonal pyramid, the side ED of whose greater end is 4 feet, that eb of the less end 3 feet, and the height Ss, 9 feet? (42 + 32 + 4 x 3) x 2'598076 x 9 865'159308 / ~~~~3 3 E - B = 288'386436 solid feet, the solidity required. D The following cases contain all the rules for finding the superficies and solidifies of conical ungulas. When the section passes through the opposite extremities of tzhe ends of the frustum. Let D = AB the diameter of the greater end; C BD d = CD, the diameter of the less end; h = perpendicular height of the frustum, and n ='7854. d2 -- d VDd nDh Then D -- d x = — solidity of the greater A- B elliptic ungula ADB. D VDd- d2 ndh D -- d dx - = solidity of the less ungula ACD. (D~ — d3) nIt D -d x - = difference of these hoofs. n D d (D-dd curve Andurfa d 4h2ce(D-d2) x (D - 2 Dd ADB curve surface of ADB. Page 84 84 THE PRACTICAL MODEL CALCULATOR. When the section cuts off parts of the base, and makes the angle DrB less than the angle CAB. Let S = tabular segment, whose versed sine is c.kD Br-. D; s - tab. seg. whose versed sine is Br - (D - d) -- d, and the other letters as above. Br Br E The(S x D3-sXdA3 V Br - D - d Br-D-d F D d- solidity of the elliptic hoof EFBD. 1 d2 x(D+d)-Ar And D-d V4h2 + (D - d)2x(seg. FBE- D-2 x2 d - Ar Br d- Ar x Vd - Ar x seg. of the circle AB, whose height is D x d= convex surface of EFBD. When the section is parallel to one of the sides of the frustum. Let A = area of the base FBE, and the other let- C D ters as before. AxD Then (D -- d V(B- d) x d) x ~h -= solidity of the parabolic hoof EFBD. A 1 _ r And - d V/4h2 x (D - d)2 x (seg. FBE - 2D —d x v/d x D - d) = convex surface of EFBD. WVhen the section cuts off part of the base, and makes the angle DrB greater than the angle CAB. Let the area of the hyperbolic section EDF = A, D and the area of the circular seg. EBF = a. _h d x Er Then D hx (a x D — A x = solidity of the hyperbolic ungula EFBD. At B And D d x V4h2 + (D - d)2 x (cir. seg. EBF - d'2 Br-I(D-d) Br D-X d V r-d- = curve surface of EFBD. Br - D - d Br - d - D dx Cr The transverse diameter of the hyp. seg. D- d -Br and the Br conjugate = d VD _ d - Br' from which its area may be found by the former rules. To find the solidity of a cuneus or wedge. —Add twice the length of the base to the length of the edge, and reserve the number. Multiply the height of the wedge by the breadth of the base, and this product by the reserved number; - of the last product will be the solidity. Page 85 MENSURATION OF SOLIDS. 85 How many solid feet are there in a wedge, whose base is 5 feet 4 inches long, and 9 inches broad, the length of the edge being 3 feet 6 inches, i and the perpendicular height 2 feet 4 inches? a B (64 x2 + 42) x 28 x 9 (128 + 42) x28 x 9 Here6 - 6 170 x 28 x 9 170 x 28 x 3 6- - 2 = 170 x 14 x 3 = 7140 solid inches. And 7140. 1728 = 4'1319 solid feet, the content. To find the solidity of a prismoid.-To the sum of the areas of the two ends add four times the area of a section parallel to and equally distant from both ends, and this last sum multiplied by g of the height will give the solidity. The length of the middle rectangle is equal to half the sum of the lengths of the rectangles of the two ends, and its breadth equal to half the sum of the breadths of those rectangles. What is the solidity of a rectangle prismoid, the length and breadth of one end being 14 and F 12 inches, and the corresponding sides of the other 6 and 4 inches, and the perpendicular 30~ feet. Here 14 x12 + 6 x4=168+24 = 192=D sum of the area of the two ends. 14 + 6 20 A Also 2 = -- = 10 = length of the middle rectangle. 12 +4 16 And 2 -- = 8 = breadth of the middle rectangle. Whence 10 x 8 x 4 = 80 x 4 = 320 = 4 times the area of the middle rectangle. 366 Or (320 + 192) x 6 - 512 x 61 = 31232 solid inches. And 31232. 1728 = 18'074 solidfeet, the content. To find the convex surface of a sphere. —Multiply the diameter of the sphere by its circumference, and the product will be the convex superficies required. The curve surface of any zone or segment will also be found by multiplying its height by the whole circumference of the sphere. D What is the convex superficies of a globe c BOCG whose diameter BG is 17 inches? Here 3'1416 x 17 x 17 = 53'4072 x 17 = ( / 907'9224 square inches. G 0 - B And 907'9224 -. 144 = 6'305 square feet. H Page 86 86 THE PRACTICAL MODEL CALCULATOR. To find the solidity of a sphere or globe. —Multiply the cube of the diameter by -5236, and the product will be the solidity. What is the solidity of the sphere AEBC, E whose diameter AB is 17 inches? Here 173 x'5236 - 1 x 17 x 17 x'5236 = 289 x 17 x 5236 = 4913 x'5236 = 2572'4468 A B solid inches. And 2572'4468 -- 1728 - 1'48868 solid feet. To find the solidity of the segment of a sphere.-To three times the square of the radius of its base add the square of its height, and this sum multiplied by the height, and the product again by'5236, will give the solidity. Or, From three times the diameter of the sphere subtract twice the height of the segment, multiply by the square of the height, and that product by'5236; the last product will be the solidity. The radius Cn of the base of the segment A CAD is 7 inches, and the height An 4 inches; what is the solidity? IHere (72 x 3 + 42) x 4 x'5236 = (49x3+42) Co x4 x.5236 = (147 + 42) x4 x 5236 = (147 +16)\ x 4 x'5236 = 163 x 4 x.5236 = 652 x'5236 - 341'3872 solid inches. B To find the solidity of a frustum or zone of a sphere.-To the sum of the squares of the radii of the two ends, add one-third of the square of their distance, or of the breadth of the zone, and this sum multiplied by the said breadth, and the product again by 1'5708, will give the solidity. What is the solid content of the zone ABCD, whose greater diameter AB is 20 inches, the DAC less diameter CD 15 inches, and the distance urnt of the two ends 10 inches? 102 Here (102 + 7.52 + -) x 10 x 1'5708 = -'-'(100 + 56'25 + 33'33) x 10 x 1'5708 = 189'58 x 10 x 1'5708 = 1895'8 x 1'5708 = 2977'92264 solid inches. To find the solidity of a sphleroid. —Multiply the square of the revolving axe by the fixed axe, and this product again by'5236, and it will give the solidity required. *5236 is = - of 3'1416. In the prolate spheroid ABCD, the D transverse, or fixed axe AC is 90, and the conjugate or revolving axe DB is 70; A r _ _ what is the solidity? Here DB2 x AC x'5236 = 702 x 90 x 5236 = 4900 x 90 x'5236 = 441000 B x'5236 = 2.30907.6 = solidity required. Page 87 MIEXSURATION OF SOLIDS. 87 To find the content of thle middle frustum of a slpheroid, its length, the middle diameter, aend that of either of the ends, being given, when the ends are circular or parallel to the revolving axis.To twice the square of the middle diameter add the square of the diameter of either of the ends, and this sum multiplied by the length of the frustum, and the product again by'2618, will give the solidity. Where'2618 = -1 of 3'1416. In the middle frustum of a spheroid e EFGH, the middle diameter DB is e 50 inches, and that of either of the a ends EF or G H is 40 inches, and its A ""': l, i length nm 18 inches; what is its soli- dity? Here (502 x 2 + 402) x 18 x'2618 B = (2500 x 2 + 1600) x 18 x'2618 = (5000 + 1600) x 18 x ~2618 = 6600 x 18 x'2618 ='118800 x'2613 = 31101'84 cubie inclhes. When the ends are elliptical or perpendicular to the revol1vivy axis.-Multiply twice the transverse diameter of the middle section by its conjugate diameter, and to this product add the product of the transverse and conjugate diameters of either of the ends. Multiply the sum thus found by the distance of the ends or the height of the frustum, and the product again by'2618, and it will give the solidity required. In the middle frustum ABCD of an oblate s spheroid, the diameters of the middle section A -- D EF are 50 and 30, those of the end AD 40 and 24, and its height ne 18; what is the i'- -— Fsolidity?.lt,_L Here (50 X 2 x 30 + 40 x 24) x 18 x'2618 = (3000 + 960) x 18 x.2618 = 3960 x 18 x.2618 = 71280 x'2618 = 18661'104 = the solidity. To find the solidity of the segment of a spheroid, when the base is parallel to the revolving axis. —Divide the square of the revolving axis by the square of the fixed axe, and multiply the quotient by the difference between three times the fixed axe and twice the height of the segment. Multiply the product thus found by the square of the height of the segment, and this product again by'5236, and it will give the solidity required. In the prolate spheroid DEFD, the trans- D verse axis 2 DO is 100, the conjugate AC 60, and the height Dnm of the segment EDF 10; what is the solidity? F 602 El_ _ liere (002 X 300 -20) x 102 x 5236= A 0 C'36 x 280 x 102 x.5236 = 100.80 x 100 x.52836 = 10080 x'5236 = 5277'888 = the solidity. Page 88 88 TIIE PRACTICAL MODEL CALCULATOR. When the base is perpendicular to the revolving axis. —Divide the fixed axe by the revolving axe, and multiply the quotient by the difference between three times the revolving axe and twice the height of the segment. Multiply the product thus found by the square of the height of the segment, and this product again by'5236, and it will give the solidity required. In the prolate spheroid aEbF, the trans- a verse axe EF is 100, the conjugate ab 60, and A-D the height an of the segment aAD 12; what, F is the solidity? E'''Here 156 (= diff.of 3ab and 2an) x _ (= EF. ab x 144 (= square of an) x'5236 b 156 x 5 x 144 x.5236 = 52 x 5 x 144 x.5236 = 260 x 144 x *5236 = 37440 x'5236 = 19603.584 = the solidity. To find the solidity of a parabolic conoid.-Multiply the area of the base by half the altitude, and the product will be the content. What is the solidity of the paraboloid ADB, whose height Dm is 84, and the diameter /BA of its circular base 48? E,F Here 482 x *7854 x 42 (= Dm) = 2304 x ~7854 x 42 = 1809'5616 x 42 = 76001'5872 --- = the solidity. A Z B To find the solidity of the frustum of a paraboloid, when its ends are perpendicular to the axe of the solid.-Multiply the sum of the squares of the diameters of the two ends by the height of the frusturn, and the product again by'3927, and it will give the solidity. Required the solidity of the parabolic frustum ABCd, the diameter AB of the greater end being 58, that of the less end de 30, and the / height no 18. Here (582 + 302) x 18 x 3927 = (3364 + At-'900) x 18 x'3927 = 4264 x 18 x'3927 = 76752 x.3927 = 30140.5104 = the solidity. To find the solidity of an hyperboloid.-To the square of the radius of the base add the square of the middle diameter between the base and the vertex, and this sum multiplied by the altitude, and the product again by'5236 will give the solidity. In the hyperboloid ACB, the altitude Cr c is 10, the radius Ar of the base 12, and the middle diameter nm 15'8745; what is the solidity? IHere 158745-2 + 122 x 10 x'5236 = 251'99975 + 144 x 10 x'5236 = 395'99975 x ---- 10 x'5236 = 3959.9975 x.5236 =2073'454691 A = the solidity. Page 89 MENSURATION OF SOLIDS. 89 To find the solidity of the frustum, of an hyp2erbolie eonoid.-Add together the squares of the greatest and least semi-diameters, and the square of the whole diameter in the middle; then this sum being multiplied by the altitude, and the product again by'5236, will give the solidity. In the hyperbolic frustum ADCB, the length rs is 20, the diameter AB of the greater end 32, i that DC of the less end 24, and the middle dia- c meter nm 2841708; required the solidity. / ]_ Here (162 ~ 122 + 28.17082) x 20 x'52359 (256 + 144 + 793-5939) x 20 x -52359 = A. 1193'5939 x 20 x'52359 = 23871'878 x'52359 = 12499'07660202 = solidity. To find the solidity of a tetraedron.-Multiply -- of the cube of the linear side by the square root of 2, and the product will be the solidity. The linear side of a tetraedron ABCn is 4; what is the solidity? /X 43 4x4x4 4 x4 16 2x 2= 12 x / 2 - x 2= A B 16 22.624 x 2 = - x 1'414 3 7'5413 = solidity. To find the solidity of an octaedron.-Multiply ~- of the cube of the linear side by the square root of 2, and the product will be the solidity. D What is the solidity of the octaedron BGAD, whose linear side is 4? c", 43 64 B A - x V2 = - x V/2 = 21'333, x /2 = 21'333 x 1'414 = 30'16486 = solidity. G To find the solidity of a dodecaedron.-To 21 times the square root of 5 add 47, and divide the sum by 40: then the square root of the quotient being multiplied by five times the cube of the linear side will give the solidity. The linear side of the dodecaedron ABCDE is 3; what is the solidity? 21 V/5 + 47 21 x 2'23606+47 E V 40 x 27 x 5=V 40 46.95726 +47 x 27 x 5 = 46957 x 135 = 206.901 40 solidity. To find the solidity of an ieosaedron.-To three times the square root of 5 add 7, and divide the sum by 2; then the square root of H2 Page 90 90 THE PRACTICAL MODEL CALCULATOR. this quotient being multiplied by 5 of the cube of the linear side will give the solidity. That is A S3 x v ( 2 ) = solidity when S is = to the linear side. The linear side of the icosaedron ABCDEF D is 3; what is the solidity? 3 V 5 + 7 5 x 32 3 x 223606 + 7 c E V 2 X 6. -2 5 x 27 6.70818 + 7 5 x 9 X 6 = V 2 X 2 13.70818 45;? 2 x 2-=./6'85409 x 22'5 = 2'61803 A x 22'5 = 58'9056 = solidity. The superficies and solidity of any of the five regular bodies may be found as follows: RULE 1. Multiply the tabular area by the square of the linear edge, and the product will be the superficies. 2. Multiply the tabular solidity by the cube of the linear edge, and the product will be the solidity. Surfaces and Solidities of the Regular Bodies. iNo. of N'ames. Surfaces. Solidities. Sides. 4 Tetraedron 1.73205 0.11785 6 Hexaedron 6.00000 1.00000 8 Octaedron 3.46410 0.47140 12 Dodecaedron 20.64578 7.66312 20 Icosaedron 8.66025 2.18169 To find the convex superficies of a cylindric ring.-To the thickness of the ring add the inner diameter, and this sum being multiplied by the thickness, and the product again by 9.8696, will give the superficies. The thickness of Ac of a cylindric ring is 3 X inches, and the inner diameter cd 12 inches; what is the convex superficies? A...._ 12 + 3 x 3 x 9.8696 = 15 x 3 x 9.8696 \,2,/ = 45 x 9'8696 = 4441382 = superficies. To find the solidity of a cylindric ring.-To the thickness of the ring add the inner diameter, and this sum being multiplied by the square of half the thickness, and the product again by 9 8696, will give the solidity. Page 91 MENSURATION OF SOLIDS. 91 What is the solidity of an anchor ring, whose inner diameter is 8 inches, and thickness in metal 3 inches? 8 + 3 x j}2 x 9'8696 = 11 x 1'52 x 9'8693 = 11 x 2'25 x 9'8696 = 24-75 x 9'8696 = 244'2726 = solidity. The inner diameter AB of the cylindric ring f cdef equals 18 feet, and the sectional diameter cA or Be equals 9 inches; required the convex surface arnd solidity of the ring. l e 18 feet x 12 = 216 inches, and 216 + 9 c -' —--— A 7Bn, x 9 x 9'8696 = 19985'94 square inches. 216 + 9 x 92 x 2'4674 = 44968'365 cubic inches. d In the formation of a hoop or ring of wrought iron, it is found in practice that in bending the iron, the side or edge which forms the interior diameter of the hoop is upset or shortened, while at the same time the exterior diameter is drawn or lengthened; therefore, the proper diameter by which to determine the length of the iron in an unbent state, is the distance from centre to centre of the iron of which the hoop is composed: hence the rule to determine the lenryth of the iron. If it is the interior diameter of the hoop that is given, add the thickness of the iron; but if the exterior diameter, subtract from the given diameter the thickness of the iron, multiply the sum or remainder by 3'1416, and the product is the length of the iron, in equal terms of unity. Supposing the interior diameter of a hoop to be 32 inches, and the thickness of the iron 1{, what must be the proper length of the iron, independent of any allowance for shutting? 32 + 1'25 = 33'25 x 3'1416 = 104'458 inches. But the same is obtained simply by inspection in the Table of Circumferences. Thus, 33'25 = 2 feet 91 in., opposite to which is 8 feet 8} inches. Again, let it be required to form a hoop of iron' inch in thickness, and 161- inches outside diameter. 16'5 -'875 = 15'625, or 1 foot 3A inches; opposite to which, in the Table of Circumferences, is 4 feet 1 inch, independent of any allowance for shutting. The length for angle iron, of which to form a ring of a given diameter, varies according to the strength of the iron at the root; and the rule is, for a ring with the flange outside, add to its required interior diameter, twice the extreme strength of the iron at the root; or, for a ring with the flange inside, sub- c d c d tract twice the extreme strength; and the sum or, -: remainder is the diameter by which to determine ---- I the length of the angle iron. Thus, suppose two angle iron rings similar to the following be re- c quired, the exterior diameter AB, and interior........ diameter CD, each to be 1 foot 10~ inches, and c d c d the extreme strength of the iron at the root ecd, d, &c, 8 of an inch; Page 92 92 THE PRACTICAL MODEL CALCULATOR. twice - = 14, and 1 ft. 1041 in. + 13 = 2 ft. 4 in., opposite to which, in the Table of Circumferences, is 6 ft. 44 in., the length of the iron for CD; and 1 ft. 104 in. - 14 = 1 ft. 83 in., opposite to which is 5 ft. 54 in., the length of the iron for AB. But observe, as before, that the necessary allowance for shutting must be added to the length of the iron, in addition to the length as expressed by the Table. Required the capacity in gallons of a () ci locomotive engine tender tank, 2 feet 8 _"_ I inches in depth, and its superficial di- -.-.. q mensions the following, with reference t. I! to the annexed plan: Length, or dist. between A and B = 10 ft. 2f in. or, 122'75 in. Breadth C and D = 6 77 79.5 Length / i and g = 3 103 46'75 Mean breadth of coke- lin = 3 14 37'25 space or 3 Diameter of circle rn =2 84 32'25 " " F s = 1 64 18'5 Radius of back corners vx = 4 4 Then, 122'75 x 79'5 = 9758'525 square inches, as a rectangle. And 18.52 x'7854 = 268'8 " " area of circle formed by the two ends. Total 10027'325 " " from which deduct the area of the coke-space, and the difference of area between the semicircle formed by the two back corners, and that of a rectangle of equal length and breadth; Then 46'75 x 37'25 = 1731'4375 area of r, n, s, t, in sq. ins. 32.252 x'7854 32252 7854= 408'4 area of half the circle rn. Radius of back corners = 4 inches; consequently 82 x'7854 - 25'13, the semicircle's area; and 8 x 4 = 32 - 25'13 = 6'87 inches taken off by rounding the corners. Hence, 1731'4375 + 408'4 + 6'87 = 2146'707, and 10027'235 - 2146'707 = 7880'618 square inches, or whole area in plan, 7880'618 x 32 the depth = 252179'776 cubic inches, and 252179'776 divided by 231 gives 1091'6873 the content in gallons. Page 93 MENSURATION OF TIMBER. 93 TABLES by which to facilitate the Mensuration of Timber. 1. Flat or Board Measure. Breadth in Area of a Breadth in Area of a Breadth in Area of a inches. lineal foot. inches. lineal foot. inches. lineal foot. ~'0208 4 " 3334 8 6667 4'*0417 4- *3542 8t *6875 4'0625 4'375 84'7084 1'0834 44 *3958 84 *7292 11. 1042 5' 4167 9.75 14. 125 5 *43875 94. 7708 1'1459 54 *4583 94. 7917 2 *1667 54 *4792 94 -8125 21. 1875 6.5 10. 8334 24. 2084 64 -5208 104 *8542 23'2292 64'5416 104 *875 3 *'-25 64 *5625 10 *o8959 31 ~2708 7 4 5833 11. 9167 34. 2916 74 -6042 111. 9375 34 3125 74 *625 114 *9583 74'6458 113 *9792 Application and Use of the Table. Required the number of square feet in a board or plank 16- feet in length and 93 inches in breadth. Opposite 9~ is'8125 x 16'5 = 13'4 square feet. A board 1 foot 23 inches in breadth, and 21 feet in length; what is its superficial content in square feet? Opposite 23 is'2292, to which add the 1 foot; then 1.2292 x 21 = 25'8 square feet. In a board 1.5k inches at one end, 9 inches at the other, and 141 feet in length, how many square feet? 15'+5 - 9 2+ =121, or 1' 0208; and 1'0208 x 14'5 = 14'8 sq. ft. The solidity of round or unsquared timber may be found with much more accuracy by the succeeding Rule: —Multiply the square of one-fifth of the mean girth by twice the length, and the product will be the solidity, very near the truth. A piece of timber is 30 feet long, and the mean girth is 128 inches, what is the solidity? 128 128 25'6. 25.62 x 60 Then 144 = 273'06 cubic feet. This is nearer the truth than if one-fourth the girth be employed. Page 94 94 THIE PRACTICAL MODEL CALCULATOR. 2. Cubic or Solid Measure. Mean 14 Cubic feet Mean 14 Cubic feet Mean,'; Cubic feet Mean 14 Cubic feet girt in in each girt in in each girt in in each girt in in each inches. lineal foot. inches. lineal foot. inches. lineal foot. inches. linedl foot. 6 -25 12 1 18 2-25 24 4 6- * 272 121 12-042 18 2-313 24- 4-084 61 *294 124 1-085 18i 2-376 241 4 168 6`3 317 12{ 1.129 183 2.442 240 4.294 7 1340 13 1-174 19 2.506 25 4-34 71 *364 131 1-219 191 2-574 2 51 4-428 71.39 134 1-265 194 2~64 25- 40516 73 *417 13- 1'313 19a 2-709 25' 4 605 8 4 *-444 14 1-361 20 2-777 26 4-694 8- *4472 14- 1-41 20- 2-898 261- 4.785 84 -501 144 1-46 204 2-917 264 4.876 8-'531 143 1-511 203 2-99 263 4.969 9 *562 15 1-562 21 3-062 27 5 06 9}.594 151 1-615 211 3-136 27- 5.158 94 *626 154 1 668 214 3-209 27 652.52 9a *659 153 1-772 2134 3-285 27- 5-348 10 *694 16 1 777 22 3-362 28 5-444 10.*73 161 1-833 221 38438 28- 5-542 10l *766 164 1-89 22' 3-516 284- 5.64 101o 803 161 1.948 221 3.598 283 5-74 11 / 84 17 2.006 23 3-673 29 5-84 11 *878 171- 2-066 231 3-754 291 5-941 114 2- 918 174 2-126 234 3-835 294 6-044 11- 959 17 2-187 231 3-917 293 6-146 In the cubic estimation of timber, custom has established the rule of 1, the mean girt being the side of the square considered as the cross sectional dimensions; hence,-multiply the number of cubic feet by lineal foot as in the Table of Cubic Measure opposite the girt, and the product is the solidity of the given dimensions in cubic feet. Suppose the mean $ girt of a tree 21- inches, and its length 16 feet, what are its contents in cubic feet? 3'136 x 16 - 50'176 cubic feet. Battens, Deals, and Planks are each similar in their various lengths, but differing in their widths and thicknesses, and hence their principal distinction: thus, a batten is 7 inches by 21), a deal 9 by 3, and a plank 11 by 3, these being what are termed the standard dimensions, by which they are bought and sold, the length of each being taken at 12 feet; therefore, in estimating for the proper value of any quantity, nothing more is required than their lineal dimensions, by which to ascertain the number of times 12 feet, there are in the given whole. Suppose I wish to purchase the following: 7 of 6 feet 6 x 7- 42 feet 5 14 14 x 5= 70 11 19 19 x 11 = 209 and 6 21 21 x 6 =126 12 ) 447 ) 37'25 standard deals. Page 95 -MENSURATION OF TIMBER. 95 TABLE showing the number of Lineal Feet of Scantling of various dimensions, which are equal to a Cubic Foot. Inches. Ft. In. Inches. Ft. In. Inches. Ft. In. 2 3 6 4 90- - 9 2 6 — 42 28 9 4 8 10 2 -3 24 0 5 7 2 c lo 2 3 34 ~20 7 54 6 6 11 2 4 18 0 6 60 1 21 41 16 0 6 5 6 0 2 2 0 5 14 5 7 5 1 m 54 13 1 4 7 74 - 2 11 6 120 8 4 6 7' 2 9 61 6 111 | 84 4 3 8 2 6 3.7 5 10 3 9 4 0 8' 2 5 7 9 7 94 3 9 en 9 2 3 8 9 0 10 Y i 7 av 94 2 2 84 86 104 3 1 0 2 1 9 80 4 1 3 3 5 l. 10 1 4 94 747 117 3 2 1 110 10 7 3 12 3O I1 1 9 _ 104 6 10 1 12 1 8 11 6 6 5 A 5 9 114 1 4 54 5 3 8 2 3 12 6 0 6 b 4m1 s 2 1 64 e4 5 o- 9 i 0 lengt 3610 f w 8 3 7 *o 10 18f 61011 I 11 [ 2 24 + 28 + 3 + 5 9 7 8 3 5 1 5 9 3 11 1 7 54 0h 9 30 12 1 6 26 810 2 10 1. 4 7 1 29 ifP 9 1 9 P 7 6 10 11 2 8. 92 18 7 4 114 26' 10 1 7 8 60 12 2l4 o104 1 6 8' 5 8 11 1 5 COD 9 2 4 6 4 0 1 111 1 4 94 50 R 6' 3 85 12 14 10 4 10 7 3 5 101 4 6,O 74 3 2 P 1 1 5 14 4 8 3. 104 14 114 4 2 84 2 10. 11 1 4 12 4 0 9 2 8 114 1 3 Hewn and sawed timber are measured by the cubic foot. The unit of board measure is a superficial foot one inch thick. To measure round timber.-Multiply the length in feet by the square of 1 of the mean girth in inches, and the product divided by 144 gives the content in cubic feet. The I girths of apiece of timber, taken at five points, equally distant from each other, are 24, 28, 33, 35, and 40 inches; the length 30 feet, what is the content? 24 + 28 + 33 + 35 + 40 =32. 321 x 30 Then 144 = 213~ cubic feet. Page 96 96 THE PRACTICAL MODEL CALCULATOR. TABLE containing the Superficies and Solid Content of Spheres, fromn 1 to 12, and advancing by a tenth. Diam. Superficies. Solidity. Diam. Superficies. Solidity. Diam. Superficies. Solidity. 1 0 3'1416'5236 4-7 69-3979 54-3617 8-4 221'6712 310'3398'1 3-8013.6969 *8 72-3824 57'9059'5 226.9806 321'5558 *2 4'5239'9047'9 75-4298 61-6010'6 232'3527 333'0389'3 5'3093 1'1503 5'0 78-5400 65-4500 *7 237-7877 344'7921'4 6-1575 1 4367'1 81'7130 69-4560'8 243-2855 356-8187'5 7-0686 1-7671 *2 84-9488 73-6223 -9 248.8461 369-1217'6 8-0424 2'1446'3 88'2475 77'9519 9'0 254-4696 381-7044 *7 9'0792 2'5724 *4 91 6090 82 4481 -1 260-1558 394-5697 *8 10'1787 3-0536 *5 95-0334 87-1139 -2 265-9130 40'7-7210'9 11-3411 3-5913'6 98-5205 91-9525 *3 271'7169 421'1613 2-0 12'5664 4-1888 *7 102-0705 96-9670 *4 277'5917 434-8937'1 13'8544 4-8490 *8 105-6834 102-1606 *5 283-5294 448-9215 *2 15-2053 5'5752'9 109-3590 107-5364'6 289-5298 463-2477'3 16-6190 6'3706 6-0 113-0976 113 0976'7 295-5931 477-7755'4 18-0956 7-2382 1 1 116-8989 118-8472 -8 301-7192 492'8081'5 19-6350 8-1812 -2 120-7631 124-7885'9 307-9082 508-0485'6 21-2372 9-2027 -3 124-6901 130-9246 10'0 314'1600 523-6000'7 22 9022 10-3060 *4 128-6799 137-2585'1 320'4746 539'4656'8 24'6300 11-4940 -5 132-7326 143'7936 *2 326-8520 555-6485'9 26'4208 12-7700 *6 136-8480 150'5329'3 333.2923 572-1518 3.0 28-2744 14-1372'7 141-0264 157'4795'4 339'7954 588-9784'1 30-1907 15-5985 *8 145'2675 164-6365'5 346-3614 606-1324 *2 32-1699 17'1573'9 149-5715 172'0073 *6 352-9901 623-6159'3 34-2120 18.8166 7'0 153-9384 179-5948'7 359-6817 641-4325'4 36.3168 20-5795'1 158-3680 187-4021'8 366-4362 659-5852'5 38-4846 22-4493'2 162-8605 195'4326 *9 373-2534 678-0771'6 40-7151 24-4290 *3 167-4158 203-6893 11-0 380-1336 696-9116'7 43-0085 26-5219 *4 172-0340 212-1752'1 387-0765 716-0915'8 45-3647 28.7309'5 176-7150 220'8937'2 394-0823 735-6200'9 47-7837 31-0594 *6 181-4588 229-8478 *3 401'1509 755-5008 4-0 50-2656 33-5104'7 186-2654 239'0511 *4 408-2823 775-7364'1 52-8102 36-0870.8 191'1349 248-4754'5 415'4766 796'3301'2 55-4178 38'7924'9 196'0672 258'1552 *6 422-7336 817-2851 *3 58-0881 41.6298 8-0 201-0624 268-0832 -7 430-0536 838-6045'4 60-8213 44'6023'1 206-1203 278-2625'8 437-4363 860-2915'5 63'6174 47'7130'2 211-2411 288-6962'9 444-8819 882-3492.6 66-4782 50*9651'3 216-4248 299-3876 12'0 452-3904 904-7808 To reduce Solid Inches into Solid Feet. 1728 Solid Inches to one Solid Foot. Feet. Inches. Feet. Inches. Feet. Inches. Feet. Inches. Feet. Inches. Feet. Inches. 1 =1728 18=31104 35= 60480 52 —88956 69=119232 85 —-146880 2 3456 19 32832 36 62208 53 91584 70 120960 86 148608 3 5184 20 34560 37 63936 54 93312 71 122688 87 150336 4 6912 21 36288 38 65664 55 95040 72 124416 88 152064 5 8640 22 38016 39 67392 56 96768 73 126144 89 153792 6 10368 23 39744 40 69120 57 98496 74 127872 90 155520 7 12096 24 41472 41 70848 58 100224 75 129600 91 157248 8 13824 25 43200 42 72576 59 101952 76 131328 92 158976 9 15552 26 44928 43 74304 60 103680 77 133056 93 160704 10 17280 27 46656 44 76032 61 105408 78 134784 94 162432 11 19008 28 48384 45 77760 62 107136 79 136512 95 164160 12 20736 29 50112 46 79488 63 108864 80 138240 96 165888 13 22464 30 51840 47 81216 64 110592 81 139968 97 167616 14 24192 31 53568 48 82944 65 112320 82 141696 98 169344 15 25920 32 55296 49 84672 66 114048 83 143424 99 171072 16 27648 33 57024 50 86400 67 115776 84 145152 100 172800 17 29376 34 58752 51 88128 68 117504 Page 97 CUTTINGS AN D E)IBANKMENTS. 97 CUTTINGS AND EMBANKMENTS. THE angle of repose upon railways, or that incline on which a carriage would rest in whatever situation it was placed, is said to be at 1 in 280, or nearly 19 feet per mile; at any greater rise than this, the force of gravity overcomes the horizontal traction, and carriages will not rest, or remain quiescent upon the line, but will of themselves run down the line with accelerated velocity. Tilhe angle of practical effect is variously stated, ranging from 1 in 75 to 1 in 330. The width of land required for a railway must vary with the depth of the cuttings and length of embankments, together with the slopes necessary to be given to suit the various materials of which the cuttings are composed: thus, rock will generally stand when the sides are vertical; chalk varies from - to 1, to 1 to I; gravel 1~ to 1; coal 11 to 1; clay 1 to 1, &c.; but where land can be obtained at a reasonable rate, it is always well to be on the safe side. The following Table is calculated for the purpose of ascertaining the extent of any cutting in cubic yards, for 1 chain, 22 yards, or 66 feet in length, the slopes or angles of the sides being those which are most in general practice, and formation level equal 30 feet. Slopes 1 to 1. De!,th I Content Content Content Depth Half Content Content Content otf width Content of 1 per- of 3 per- of 6 per- of width Content of 1 per- of 3 per- of 6 percut- at in cubic pendicu- pendicu- pendicu- cut- at in cubic pendicu- pendic.-ipenediecuting in top in yards per lar ft. in lar ft. in lar ft. inll ting in top in yards per lar ft. in lar ft. in lar ft. in flet. feet. chain. breadth. breadth. breadth. feet. feet. chain. breadth. breadth. bredt. 1 16 75-78 2-44 7833 14-67 26 41 3599-11 63-50 190-6, 381 33 2 17 156-42 4-89 14.67 29-33 27 42 3762.00 65-99 19800 1396 00 3 18 242-00 7-33 22-00 44-00 28 43 3969-78 68-43 205.3o410.67 4 19 332-44 9-78 2933 5867 29 44 4182-44 70.88121267 14295 33 5 20 427-78 12.22 36-67 73-.33 30 45 4400-00 73.32 220-001440-00 6 21 528.00 14-67 44-00 88-00 31 46 4622-44 75-77 227-331494-67 7 22 633-11 17-11 51-33 102.67 32 47 4849-78 78-22 234-67 469 33 8 23 743-11 19-56 58-67 117.33 33 48 5082-00 80.67 242.00 484'00 9 24 858-00 22-00 66-00132-00 34 49 5319-11 83-11249.331498'67 10 25 977-78 24-44 73-33 146-67 35 50 5561*11 8555 256.67o13 33 11 26 1102-44 26-89 80-67 161-33 36 51 5808-00 88-00 264. 00528'00 12 27 1232-00 29-33 88'00 176-00 37 52 6059-78 90-44 27133-542'67 13 28 1366-44 31-78 95-331190-67 38 53 6316-44 92-39 278'67557-33 14 29 1505-78 34-22 102.671205.33 39 54 6578-00 95.33 286.00 572.00 15 30 1650 00 36 66 110 00 220 00 40 55 6844 44 97.77 293 33 586 67' 16 31 1799 11 39 11 117 33 234 67 41 56 7115 78 100 22 300 6, 601 33 17 32 1953 11 41 55 124 67 249 33 42 57 7392 00 102 66 308 00 616 001 I 18 33 2112 00 43 99 132 00 264 00 43 58 7673 11 105.111315.33 630 67 19 34 2275.78 46.44 139.331278.67 44 659 7959.11 1075.551322.67 645 33 20 35 2444-44 48.89 146.67 293.33 45 60 8250.00 109.991330.00 660 00 21 36 2618-00 51 33 154.00 308.00] 46 61 8545.78 112.441337.33'674.67 22 37 2796'44 53'77 161'33 322671 47 62 8846'44 114'88 344'67 689 33 23 38 2979.78 56.21 168.67 337.33 48 63 9152.00 117.33352.00 704 00 24 39 3168.00 58.66 176.00 352.00 49 64 9462.44 119.77 359.33 718 67 25 40 336111 6110 1833336667 50 65 977778 122 21366 3333 I 7 Page 98 98 THE PRACTICAL MODEL CALCULATOR. Slopes 1~ to 1. Depth EHalf Content. Content Content Depth Half Content Content Content of width Content of 1 per- of 3 per- of 6 per- of width Content of 1 per- of 3 per- of 6 percut- at in cubic pendicu- pendicu- pendicu- cut- at: in cubic pendicu ppendicu- pendicutin in top in yards per liar ft, in lar ft. in lar ft. in ting in top in yards per lar ft. in lar ft. in lar ft. in feet. feet. chain. lbreadth. breadth. breadth. feet. feet. chain. breadth. ]breadth. breadth. 1 1 6- 77-00 2-44 7-33 14-67 26 54 4385-83 63-55 190-67 381-33 2 18 161 33 4-89 14-67 29-33 27 551 4653-00 65-99 198-00 396.00 3 19'1 25300 7-33 22'00 44-00 28 57 4928-00 68-43 205833 410-67 4 21 852 00 9'78 29-33 05867 29 58' 5210383 7088 212'67 42533 5 22 }l 453 -33 12-22 36-67 73-33 30 60 5500-00 73-32 220-00 440-00 6 24 572-00 14-67 44'00 88-00 31 611 5797-00 75-77 227-33 454-67 7 251 693-00 17-11 51-33 102-67 32 63 6101*33 78'22 234'67 469'83 8 27 821-33 19-56 58-67 117-33 33 64'1 6413800 80'67 242'00 484'00 9 28' 957-00 22'00 66.00 132-0O0 34 66 6732.00 83811 249833 498'67 10 30 1100-00 24.44 73.33 146,67 35 674~ 7058.33 85.55 256.67 513-33 11 314 1250-33 26-89 80-67 161-33 36 69 7392-00 88-00 264'00 528'00 12 33 ]1408.00 29.33 88.00 176.00 37 70'- 7733-00 90.44 271.33 542.67 13 344 1573'00 31'78 95-33 190-67 38 72 808133 92,39 278'67 557-33 14 36 1745 -33 3422 102-67 205-33 39 731 8437 00 95833 286-00 572'00 15 374 1925 00 36-66 110-00 220.00 40 75 8800-00 97-77 293'33 58'6-67 16 39 2112 00 39-11 117-33 234.67 41 76' 9170-33 100-22 30067 601 33 17 404 2306;33 41-55 124-67 2493388 42 78 9548-00 102-66 308 00 616.00 18 42 2508.00 43'99 132-00 264'00 43 791. 9933'00 105 11 315-33 630.67 19 433 2717 00 46.44 139,33 278-67 44 81 10325.33 107-55 322-67 645.33 20 45 2933833 48'89 146'67 293833 45 824 10725'00 109'99 330'00 660'00 21 464 3157.00 51-33 154.00 308.00 46 84 11132*00 112.44 337.33 674'67 22 48 3388.00 53'77 161.33 322'67 47 85' 11546.33 114-88 344'67 689'33 23 49' 3626-33 56-21 168-67 337-33 48 87 11968-00 117-33 352-00 704'00 24 51 3872'00 58'66 176.00 352'00 49, 884 12397'00 119.77 359.33 718'67 2.5 52 4125600 61 10 183'33 366'67( 50 90 12833'33 122'21 366'67 733'33 Slopes 2 to 1. Depth Half Content Content Content Depth Half Content Content Content of width Content of 1 per- of 3 per- of 6 per- of width Content of 1 per- of 3 per- of.6 percut- at in cubic pendicu- pendicu- pendicu- cut- at in cubie pendicu- pendicu- pendicutingin top in yards per lar ft. in lar ft. in ar ft. in tinginltop in yards per lar ft. in lar ft. in lar ft. in feet. feet. chain. breadth. breadth. breadth. feet. feet. chain. breadth. breadth. breadth. i 17 78.22 2'44 7-33 14-67 26 67 5211-55 63.55 190o67 381-33 2- 19 166-22 4-89 14-67 29.33 27 69 5544.00 65-991198-00 39600 3 21 264.00 7-33 22-00 44-00 28 71 5886-22 68-43 205-33 410.67 4 23 371-55 9-78 29-33 58-67 29 73 6238822 70-881212-67 425833 5 25 488'89 12-22 36-67 73-33 30 75 6600.00 73-321220.00 440.00 6 27 616'00 14-67 44-00 88-00 31' 77 6971-55 75'77 227-3'3454.67 7 29 752-89 17-11 51-33 102-67 32 79 7352-89 78'221234-67 469-33 8 31 899'55 19-56 58.67117-33 33 81 7744-00 80-671242-001484.00 9 33 1056'00 22-00 66-00 132-00 34 83 8144.89 83-11 249.33 498'67 10 35 1222-22 24-44 73'33 146.67 35 85 8555.55 85-551256.67 513.33 11 37 1398-22 26-89 80-67 161-33 36 87 8976-00 88.001264-001528.00 1-2 39 1584'00 29-33 88-001176-00 37 89 9406-22 90-44 271-331542-67 13 41 1779-55 31'78 95-33 190 67 38 91 9846.22 92'39 278-67 557-33 14 43 1984-89 34'22 102-67 205-33 39 93110296-00 95-331286-001572-00 15 45 2200-00 86-66 110-00 220-00 40 95 10755-.55 97.77 293-33 586-67 16 47 2424-89 39-11 117.33 234-67 41 97 11224.89 100.22 300'67 601.33 17 49 2.659-55 41-55 124-67 249-33 42 99 11704.00 102'66 308.00 616.00 18 51 2904'00 43.99 132.00 264.00 43 101 12192.891105.11 315.33 630.67 19 53 3158-22 46-44 139.33 278-67 44 103 12691-551107.55 322'67 645-33 20 55 34-2222 48'89 146.67 293.33 45 105 13200.001109'99330.001660.00 21 57 3696'00 51.33 154-00 308.00 46 107. 1-3718-22 112-44 337-33 674'67 22 59 13979-55 15377 161-33 322-67 47 109 14246-22 114-881344-67 689-33 23 61 4272-89 56.21 168-67 337-33 48 111 14784-001117'33 352-00 704-00 24 63 4576'00 58-66 176-00 352.00 49 113 15331.55 119-77 359-33 718-67 25 65 4888'89 61-10 183-33 366-67 50 115 1588889 122.21 366'67 733-33 Page 99 CUTTINGS AND EMBANKMENTS. 99 By the fourth, fifth, and sixth columns in each table, the number of cubic yards is easily ascertained at any other width of formation level above or below 30 feet, having the same slopes as by the tables, thus:Suppose an excavation of 40 feet in depth, and 33 feet in width at formation level, whose slopes or sides are at an angle of 2 to 1, required the extent of excavation in cubic yards: 10755'55 + 293'33 = 11048'88 cubic yards. The number of cubic yards in any other excavation may be ascertained by the following simple rule: To the width at formation level in feet, add the horizontal length of the side of the triangle formed by the slope, multiply the sum by the depth of the cutting, or excavation, and by the length, also in feet; divide the product by 27, and the quotient is the content in cubic yards. Suppose a cutting of any length, and of which take 1 chain, its depth being 14t feet, width at the bottom 28 feet, and whose sides have a slope of 14 to 1, required the content in cubic yards: 14'5 x 1'25 = 18'125 + 28 x 14 = 645'75 x 66 = 42619'5 27- = 1578'5 cubic yards. -{ (b +rh)A+ + ( + rh) h + 4 r + 4 [h + -2 gives the content of any cutting. In words, this formula will be:To the area of each end, add four times the middle area; the sum multiplied by the length and divided by 6 gives the content. The breadth at the bottom of cutting = b; the perpendicular depth of cutting at the higher end = A; the perpendicular depths of cutting at the lower end = h'; 1, the length of the solid; and rh' the ratio of the perpendicular height of the slope to the horizontal base, multiplied by the height h'. rA, the ratio r, of the perpendicular height of the slope, to the horizontal base, multiplied by the height h. Let 6 = 30; h2 = 50; i' =20; 1 = 84 feet; and 2 to 5 or 2 the ratio of the perpendicular height of the slope to the horizontal base: 84 (30 +2 x 20) 20 + (30 + 5 x 50) 50 + 4 o + 0+ 5 20 50 + 20}=14 38x20+50x50+ 4x44x35 =131880 131880 cubic feet. 27 =4884'44 cubic yards. This rule is one of the most useful in the mensuration of solids, it will give the content of any irregular solid very nearly, whether it be bounded by right lines or not. Page 100 100 THE PRACTICAL MODEL CALCULATOR. TABLE of Squares, Cubes, Square and Cube Roots of Numbers. Number. Squares. Cubes. Square Robts. Cube Roots. Reciprocals. 1 1 1 1'0000000 1.0000000 *100000000 2 4 8 1'4142136 1 2599210'500000000 3 9 27 1'7320508 1'4422496'333333333 4 16 64 2'0000000 1'5874011'250000000 5 25 125 2-2360680 1'7099759'200000000 6 36 216 2-4494897 1 8171206 -166666667 7 49 343' 2-6457513 1-9129312'142857143 8 64 512 2-8284271 2-0000000'125000000 9 81 729 3'0000000 2'0800837 -111111111 10 100 1000 3-1622777 2-1544347'100000000 11 121 1331 3-3166248 2-2239801'090909091 12 144 1728 3-4641016 2-2894286'083333333 13 169 2197 3-6055513 2-3513347'076923077 14 196 2744 3'7416574 2-4101422'071428571 15 225 3375 3'8729833 2'4662121'066666667 16 256 4096 4-0000000 2-5198421'062500000 17 289 4913 4-1231056 2'5712816'058823529 18 324 5832 4-2426407 2 6207414'055555556 19 361 6859 4-3588989 2-6684016'052631579 20 400 8000 4-4721360 2-7144177 -050000000 21 441 9261 4-5825757 2-7589243'047619048 22 484 10648 4-6904158 2-8020393'045454545 23 529 12167 4-7958315 2-8438670'043478261 24 576 13824 4-8989795 2-8844991'041666667 25 625 15625 5'0000000 2'9240177 -040000000 26 676 17576 5'0990195 2'9624960'038461538 27 729 19683 5-1961524 3'0000000 *037037037 28 784 21952 5-2915026 3'0365889 -035714286 29 841 24389 5-3851648 3-0723168'034482759 30 900 27000 5-4772256 3-1072325'033333333 31 961 29791 5-5677644 3-1413806 -032258065 32 1024 32768 5-6568542 3-1748021'031250000 33 1089 35937 5'7445626 3-2075343'030303030 34 1156 39304 5-8309519 3-2396118'029411765 35 1225 42875 659160798 3-2710663'028571429 36 1296 46656 6-0000000 3-3019272'027777778 37 1369 50653 6-0827625 3-3322218 -02027027 38 1444 54872 6-1644140 3-3619754'026315789 39 1521 59319 6-2449980 3-3912114 -025641026 40 1600 64000 6-3245553 3-4199519 -025000000 41 1681 68921 6-4031242 3-4482172'024390244 42 1764 74088 6-4807407 3-4760266'023809524 43 1849 79507 6-5574385 3-5033981'023255814 44 1936 85184 6-6332496 3'5303483'022727273 45 2025 91125 6'7082039 3-5568933'022222222 46 2116 97336 6-7823300 3-5830479'021739130 47 2209 103823 6-8556546 3-6088261'021276600 48 2304 110592 6'9282032 3-6342411 -020833333 49 2401 117649 7-0000000 3-6593057'020408163 50 2500 125000 7'0710678 3-6840314'020000000 51 2601 132651 7-1414284 3-7084298'019607843 52 2704 140608 7-2111026 3-7325111 *019230769 53 2809 148877 7-2801099 3'7562858'018867925 54 2916 157464 7-3484692 3'7797631'018518519 55 3025 166375 7'4161985 3'8029525 -018181818 56 3136 175616 7-4833148 3-8258624'017857143 57 3249 185193 7-5498344 3-8485011'017543860 Page 101 TABLE OF SQUARES, CUBES, SQUARE AND CUBE ROOTS. 101 Number. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 58 3364 195112 7'6157731 3-8708766 *017241379 59 3481 205379 7-6811457 3'8929965'016949153 60 3600 216000 7-7459667 3-9148676'016666667 61 3721 226981 7-8102497 3 9304972'016393443 62 3844 238328 7'8740079 3-9578915'016129032 3969 250047 7-9372539 3'9790571 015873016 64 4096 262144 8-0000000 4'0000000 015623000 4225 274625 8'0622577 4-0207256 015384615 66 4356 287496 8'1240384 4-0412401 015151515 67 4489 300763 8-1853528 4-0615480 014925373 68 4624 314432 8-2462113 4-0816551'014-705882 69 4761 328509 8-3066239 4-1015661 014492754 70 4900 343000 8.3666003 4.1212853 014285714 5041 357911 8-4261498 4-1408178 014084517 72 5184 373248 8-4852814 4.1601676'013888889 73 5329 389017 8-5440037 4-1793390 013698630 74 5476 405224 8.6023253 4-1983364'013513514 75 5625 421875 8-6602540 4-2171633 01333333 76 5776 438976 8-7177979 4-2358236 013157895 77 5929 456533 8-7749644 4-2543210'012987013 78 6084 474552 8-8317609 4-2726586'012820513 79 6241 493039 8-8881944 4-2908404'012658228 80 6400 512000 8-9442719 4.3088695'012500000 81 6561 531441 9.0000000 4-3267487'012345679 82 6724 551368 9.0553851 4-3444815 012195122 83 6889 571787 9-1104336 4-3620707 012048193 84 7056 592704 9-1651514 4-3795191 011904762 85 7225 614125 9-2195445 4-3968296 011764706 86 7396.636056 9*2736185 4-4140049 011627907 87 7569 658503 9.3273791 4-4310476'011494253 88 7744 681472 9-3808315 4-4470692 011363636 89 7921 704969 9.4339811 4.4647451'011235955 90 8100 729000 9-4868330 4-4814047'011111111 91 8281 753571 9-5393920 4-4979414'010989011 92 8464 778688 9-5916630 4-5143574'010869565 93 8649 804357 9-6436508 4-5306549'010752 688 94 8836 830584 9-6953597 4-5468359 010638298 95 9025 857374 9.7467943 4-5629026 010.526316 96 9216 884736 9-7979590 4.5788570'010416667 97 9409 912673 9.8488578 4.5947009 010309278 98 9604 941192 9-8994949 4-6104363'010204082 99 9801 970299 9-9498744 4-6260650'010101010 100 10000 1000000 10.0000000 4-6415888'010000000 101 10201 1030301 10-0498756 4.6570095 009900990 10 10404 1061208 10-0995049 46723287'009803922 103 10609 1092727 10-1488916 4-6875482 009708738 104 10816 1124864 10-1980390 4-7026694 009615 385 105 11025 1157625 10.2469508 4-7176940 009523810 106 11236 1191016 10-2956301 4-7326235 009433962 107 11449 1225043 10-8440804 4-7474594'009345794 108 11664 1259712 10-3923048 4-7622032 0092 92 59 109 11881 1295029 10-4403065 4-7768562'009174312 110 12100 1331000 10-4880885 4-7914199 -009090909 1L1 12321 1367631 10.5356538 4.8058995 009009009 1 12544 1404928 10-5830052 4-8202845 008928571 113 12769 1442897 10.6301458 4.8345881 008849558 114 12996 1481544 10-6770783 4-8488076 008771930 115 13225 1520875 10-7238053 4-8629442 008 695652 116 13456 1560896 10-7703296 4-8769990'008020690 117 13689 1601613 10-8166538 4-8909732'008547009 118 13924 1643032 10-8627805 4-9048681 *0084745 76i 119 14161 1685159 10'9087121 4'9186847 -008403361 I2 Page 102 102 THE PRACTICAL MODEL CALCULATOR. lrnumber. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 120 14400 1728000 10.9544512 4.9324242'008333333 121 14641 1771561 11-0000000 4.9460874'008264463 1122 14834 1815848 11 0453610 4-9596757'008196721 123 15129 1860867 11-0905365 4-9731898'008130081 124 15376 1906624 11-1355287 4-9866310 -008064516 125 15625 1953125 11-1803399 5'0000000'008000000 126 15876 2000376 11-2249722 5-0132979'007936508 127 16129 2048383 11-2694277 5-0265257'007874016 128 16384 2097152 11-3137085 5-0396842'007812500 129 16641 2146689 11-3578167 5-0527743'007751938 180 16900 2197000 11-4017543 5-0657970'007692308 131 17161 2248091 11-4455231 5-0787531'007633588 132 17424 2299968 11-4891253 5-0916434 -007575758 133 17689 2352637 11-5325626 5-1044687'007518797 134 17956 2406104 11-5758369 5-1172299'007462687 135 18225 2460375 11-6189500 5-1299278'007407407 136 18496 2515456 11-6619038 5-1425632'007352941 137 18769 2571353 11-7046999 5-1551367'007299270 138 19044 2628072 11-7473444 5-1676493'007246377 139 19321 2685619 11-7898261 5'1801015'007194245 140 19600 2744000 11-8321596 5-1924941'007142857 141 19881 2803221 11-8743421 5-2048279'007092199 142 20164 2863288 11-9163753 5-2171034 007042254 143 20449 2924207 11-9582607 5-2293215'006993007 144 20736 2985984 12-0000000 5-2414828 -006944444 145 21025 3048625 1.2-0415946 5-2535879'006896552 146 21316 3112136 12-0830460 5-2656374'006849315 147 21609 3176523 12-1243557 5-2776321 -006802721 148 21904 3241792 12-1655251 5-2895725'006756757 149 22201 3307949 12-2065556 5-3014592 -006711409 150 22500 3375000 12-2474487 5'3132928'006666667 iA51 22801 3442951 12'2882057 5'3250740'006622517 1,2 23104 3511008 12'3288280 5 3368033 -006578947 1o3 23409 3581577 12-3693169 5 3484812 006535948 15 4 23716 3652264 12-4096736 5-3601084'006493506 155 24025 3723875 12-4498996 5'3716854 -006451613 1 6 24336 3796416 12-4899960 5-3832126 -006410256 157 24649 3869893 12-5299641 5.3946907'006369427 158 24964 3944312 12-5698051 5-4061202'006329114 159 25281 4019679 12-6095202 5-4175015'006289308 160 25600 4096000 12-6491106 5-4288352'006250000 161 25921 4173281 12-6885775 5-4401218'006211180 162 26244 4251528 12-7279221 5-4513618'006172840 163 26569 4330747 12-7671453 5-4625556 -006134969 164 26896 4410944 12-80O2485 5-4737037 -006097561 165 27225 4492125 12.8452326 5'4848066 0060606(06 166 27556 4574296 12-8840987 549-58647'006024096 167 27889 4657463 12-9228480 5-5068784'005988024 168 28224 4741632 12-9614814 5-5178484'005952381 169 28561 4826809 13-0000000 5-5287748'005917160 170 28900 4913000 13-0384048 5-5396583 -005882353 171 29241 5000211 13-0766968 5-5504991'005847953 172 29584 5088448 13-1148770 5-5612978'005813953 173 29929 5177717 13-1529464 5-5720546'005780347 174 30276 5268024 1 31909060 5-5827702'005747126 175 30625 5359375 13-2287566 5-5934447'005714286 176 30976 5451776 13'-2664992 5-6040787 -005681818 177 31329 5545233 13-3041347 5-6146724'005649718 178 31684 5639752 13-3416641 5-6252263'005611798 179 32041 5735339 13-3790882 5-6357408 005586592 180 32400 5832000 13-4164079 5-6462162'00a555556 151 32761 5929741 13-4536240 5-6566528.005524862 Page 103 TABLE OF SQUARES, CUBES, SQUARE AND CUBE BooTS. 103 |Number. Squuare aes. C SqSqare Roots. Cube Roots. Recilprocals. 182 33124 6028568 13-4907376 5-6671011.005494505 183 334S9 6128487 13 527 7493 5-6774114 -0050464481 184 33856 6229504 13.5646600 5.6877340 -005434783 185 342:25 63316 25 13-6014705 5-6980192'005405405| 186 341596 6434856 13-6381817 5-7082675.005376G344 187 34969 6539203 13.6747943 5'7184791 005034759 1 188.,344 6644672 13.7113092 5.7286543.0053131d9 | 189 35721 6751269 13-7477271 5-7387936 005291005 190 36100 6859(00 13-7840488 5'7488971.00526:3158 191 36481 6967871 13-8202750 5 75896-52 00502356 0 2 192 36864 7077888 13-8564065 5'7689982 -0 0208:383 193 37249 7189517 13.8924400 5'7789966 0051813 4 194 37636 7301384 13.9283883 5-7889604.00-515;'39 195 38025 7414875 13-9642400 5-7988900.00512'805 196 38416 7529536 14.0000000 5.8087857.005102041 197 38809 7645373 14-0356688 5-8186479.005076142 198 39204 7762392 14.0712473 5.8284867'00-505050 199 39601 7880599 14-1067360 5.8382725'000, o25 12) 200 40000 8000000 14.1421356 5-8480355'005000000 201 40401 8120601 14-1774469 5-8577660'004'975124 202 40804 8242408 14-212670-1 5.8674673'004950495 203 41209 8365427 14247 8068 5.8771307'004( 261"03! 04 41616 8489664 14-2828569 5.8867653'004!-3019'61 205 42025. 8615125 14.3178211 5-8963685'004878049 1206 42436 8741816 14.3527001 5-90659406'0048,5439i` 207 42849 8869743 14-3874946 5-9154817'004830918S 208 43264 89!38912 14.4222051 5-99249921'004807692' 209 43681 9129329 14.4568323 5.9344721'0047846S9( 210 44100 9261000 14.4913767 5-9439220'004761905 211 44521 9393931 14.5258390 5-9533418'004739336 6 212 44941- 9528128 14-5602198 5.9627320'00-471698 213 45369 9663597 14.5945195 5.9720926 00469,34836 214 45796 9800344 14.6287388 5.9814240 00 43 7 8 97 215 46225 9938375 14.6628783 5 9907264'00-165 113 216 46656 10077696 14.6969385 6.0000000'004 629630 1217 447089 10218313 14-7309199 6-0092450 *O0 0 (4')o 2 218 47524 10360232 14-7648231 6-0184617'00438715 J 219 47961 10503459 14-7986486 6'0276502 -0045662C10 220 48400 10648000 14-832 39 0 6-0368107'004454 54 221 48841 10793861 14-8660687 6'0459435'00452488, 222 49284 10941048 14-89'96644 6'050489'004504050 223 49729 11089567 14-9331845 6.0641270'004489305 224 50176 11239424 14-9666295 6-0731779 00446428(;S 225' 50625 11390625 15-0000000 6-0824020'00444-414-!26' 51076 11543176 15-0332964 6.0991994'00442-'779 227 51529 11697083 15.0665192 6'1001702'004405 286i 298 51984 11852352 15-0996689 6'1091147'004385965 229 52441 12008989 15-1327460 6-1180332'0043G6812 230 52900 12167000 15-1657509 6-1269257'0043147826 231 53361 12326391 15-1986842 6-1357 924'O-0043)20C 232 53824 12487168 15-2315462 6-1446337'00-431 l)3-145 233 54289 12649337 15-2643375 6-1534495'004211845 234 54756 12812904 15-2970585 6-1622401'001427;50I 235 55225 12977875 15-3297097 6'1710058'002-5 5319 236 55696 13144256 15-3622915 6-l'197466' 00 237288 237 56169 13312053 15-3948043 6-1884628'004219409 238 56644 13481272 15-4272486 6-1971544 -0042016J81 239 57121 13651919 15-4596248 6-2058218 -004184i00 240 57600 13824000 15-4919334 6'214 60' t00416666i 241 58081 13997o21 15-5241 747 6.2230843'00-1-9378 i 2492 58564 14172488 15-556349ii 2 6 23:1679 7'001132231 243 i59049 14348907 15-5884573 6 2-I02515'004115226 Page 104 104 THE PRACTICAL MODEL CALCULATOR. Number. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 244 59536 14526784 15-6204994 6-2487998 *004098361 245 60025 14706125 15-6524758 6-2573248 *004081633 246 60516 14886936 15-6843871 6-2658266'004065041 247 61009 15069223 15-7162336 6-2743054'004048583 248 61504 15252992 15-7480157 6-2827613'004032258 249 62001 15438249 15-7797338 6-2911946'004016064 250 62500 15625000 15-8113883 6-2996053'004000000 251 63001 15813251 15-8429795 6'3079935'003984064 252 63504 16003008 15'8745079 6-3163596'003968254 253 64009 16194277 15'9059737 6-3247035'003952569 254 64516 16387064 15-9373775 6-3330256'003937008 255 65025 16581375 15-9687194 6-3413257'003921569 256 65536 16777216 16-0000000 6-3496042'003906250 257 66049 16974593 16-0312195 6-3578611 *003891051 258 66564 17173512 16-0623784 6-3660968'003875969 259 67081 17373979 16-0934769 6-3743111'003861004 260 67600 17576000 16.1245155 6'3825043 003846154 261 68121 17779581 16-1554944 6-3906765'003831418 262 68644 17984728 16-1864141 6-3988279'003816794 263 69169 18191447 16-2172747 6-4069585'003802281 264 69696 18399744 16-2480768 6 4150687 003787879 265 70225 18609625 16-2788206 6-4231583 003778585 266 70756 18821096 16'3095064 6 4312276 003759398 267 71289 19034163 16.3401346 6-4392767 -003745318 268 71824 19248832 16'3707055 6-4473057 -003731343 269 72361 19465109 16.4012195 6-4553148 003717472 270 72900 19683000 16-4316767 6-4633041 003703704 271 73441 19902511 16.4620776 6'4712736 003690037 272 73984 20123643 16.4924225 6-4792236 003676471 273 74529 20346417 16-5227116 6-4871541'003663004 274 75076 20570824 16-5529454 6'4950653'003649635 275 75625 20796875 16-5831240 6-5029572 003636364 i276 76176 21024576 16.6132477 6-5108300'003623188 277 76729 21253933 16 643317-0 6 5186839 00361010S s78 77284 214849)2 16-6783320 6-5265189 0035971 22 I279 7377841 21717639 16- 7032931 6-5343351 0035'84229 -2,80 78400 21952000 167332005 6-5421326 *003571429 281 78961 22188041 16-7630546 6-5499116 *0038558719 282 79524 22425768 16-7928556 6-5576722 003546099 283 80089 22665187 16-8226038 6-5654144'003533569 284 80656 22906304 16-8522995 6-5731385 003522127 285 81225 23149125 16-8819430 6'5808443'003508772 286 81796 23393656 16'9115345 6-5885323'003496503 287 82369 23639903 16-9410743 6'5962023'003484321 288 82944. 23887872 16 9705627 6'6038545 -003472222 289 83521 2-4137569 17-0000000 6-6114890'003460208 290 84100 24389000 17-0293864 6-6191060 003448276 291 84681 24642171 17 0587221 6.6267054 0034364 26 292 85264 24897088 17-0880075 6.6342874 0034246058 293 85849 25153757 17-1172428 6'6418522 003412969 294 86436 2.5412184 17-1464282 6-6493998 003401361 295 87025 25672375 171755640 6-6569302 003389831 296 87616 25934836 17-2046505 6'6644437 00337837 8 297 88209 26198073 17.2336879 6-6719403 003367003 298 88804 264683592 17.2626765 6'6794200'003355705 299 89401 26730899 17-2916165 6-6868831 003344482 300 90000 27000000 17.3205081 6-6943295 00333383 3 301 90601 272709t01 17-3493516 6-7017593'003322209 302 91204 27543608 17'3781472 6-7091 729'003311258 303 91809 27818127 17-4068952 6-7165700'003301330 304 92416 28094464 17 4355958 6-7239508'003289474 3,05 93025'28372625 17.4642492 6-7313155 00327869 L ___ - ____ ______ ______ ______ _____3 Page 105 TABLE OF SQUARES, CUBES, SQUARE AND CUBE ROOTS. 105 Number. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 306 93636 28652616 17-4928557 673886641.003267974 307 94249 28934443 17 -5214155 6-7459967.003257329 308 94864 29218112 17-5499288 6.7533134.003246753 309 95481 29503609 17-5783958 6-7606143.003236246 310 96100 29791000 17-6068169 6-7678995 -003225806 311 96721 30080231 17-6351921 6.7751690.003215434 312 97344 30371328 17-6635217 6.7824229.003205128 313 97969 30664297 17-6918060 6-7896613.003194888 314 98596 30959144 17-7200451 6.7968844.003184713 315 99225 31255875 17-7482393 6.8040921.003174603 316 99856 31554496 17-7763888 6 81 12847.003164557 317 100489 31855013 17-8044938 6.8184620 -003154574 318 101124 32157432 17-8325545 6.8256242. 003144654 319 101761 32461759 17-8605711 6.8327714.003134796 320 102400 32768000 17-8885438 6.8399037.003125000 321 103041 33076161 17-9164729 6.8470213 *003115265 322 103684 33386248 17-9443584 6.8541240.003105590 323 104329 33698267 17-9722008 6.8612120.003095975 3 24 104976 34012224 18-0000000 6-8682855.008086420 325 105625 34328125 18.0277564 6-8753433 0030 7692'3 326 106276 34645976 18.0554701 6'8823888.003067485 327 106929 34965783 18.0831413 6-8894188.003058101 328 107584 35287552 18-1107703 6.8964345.003048780 329 108241 35611289 18.1383571 6.9034359 -003039514 330 108900 35937000 18.1659021 6-9104232.003030303 331 109561 36264691 18.1934054 6.9173964.0030211-48 332 110224 36594368 18-2208672 6.9243556 *003012048 333 110889 36926037 18-2482876 6-9313088 -003003003 334 111556 37259704 18.2756669 6-9382321.002994012 335 112225 37595375 18 3030052 6.9451496.002985075 336 112896 37933056 18.3303028 6-9520533.00297619)0 337 113569 38272753 18.3575598 6.9589434 *002967359 338 114244 38614472 18 3847763 6.9658198.-002958580 339 114921 38958219 18.4119526 6-9726826.002949853 340 115600 39304000 18-4390889 6-9795321.002941176 341 116281 39651821 18.4661853 6-9863681.002932551 342 116964 40001688 18-4932420 6-9931906.002923977 343 117649 40353607 18.5202592 7.0000000.002915452 344 118336 40707584 18.5472370 7.0067962.002906977 345 119025 41063625 18-5741756 7-0135791.002898551 346 119716 41421736 18-6010752 7-0203490.002890173 347 120409 41781923 18-6279360 7 0271058.002881844 348 121104 42144192 18-6547581 7-0338497.002873563 349 121801 42508549 18-6815417 7'0405860.002865330 350 122500 42875000 18.7082869 7'0472987.002857143 351 123201 43243551 18-7349940 7'0540041 -002849003 352 123904 43614208 18-7616630 7'0606967.002840909 353.124609 43986977 18.7882942 7-0673767.002832861 354 125316 44361864 18-8148877 7'0740440.002824859 355 126025 44738875 18-8414437 7-0806988.002816901 356 126736 45118016 18-8679623 7-0873411.002808989 357 127449 45499293 18-8944436 7'0939709.002801120 358 128164 45882712 18-9208879 7-1005885.002793296 359 128881 46268279 18-9472953 7'1071937.002785515 360 129600 46656000 18-9736660 7-1137866'002777778 361 130321 47045831 19-0000000 7'1203674.002770083 362 131044 47437928 19.0262976 7-1269360 *002762431 363 131769 47832147 19.0525589 7'1334925 -002754821 364 1.32496 48228544 19-0787840 7'1400370 -002747253 365 133225 48627r25 19-1049732 7-1465695.002739726 366 133956 49027896 19-1311265 7-1530901 *002732240 367 134689 49430863 19-1572441 7.1595988.002724796 Page 106 106 THE PRACTICAL 5MODEL CALCULATOR. iNumber. Squares. Cubes. Square Roots. Cube Roots. Recip.rocals. 368 135424 49836032 19.1833261 7.1660957 0027173,91 369 136161 50243409 19.2093727 7-1725809.002710027 370 136900 50653000 19-2353841 7-1790544.00,7 02 "3 371 137641 51064811 19.2613603 7-18 55162.0026o!154 8 372 138384 51478848 19-2873015 7.1919663.0026i88 1 373 139129 51895117 19'3132079 7-1984050'0026809u'5 374 139876 52313624 19-3390796 7-2048322'0026 73797 375 140625 52734375 19-3649167 7.2112479'002(;6t7 376 141376 53157376 19'3907194 7 2176522 002659,574 377 142129 53582633 19-4164878 7-2240450'00%2665:)2 378 142884 54010152 19.4422221 7.2304268.00%2;4`45508 379 143641 54439939 19-4679223 7-2367972 -002638521 380 144400 54872000 19'4935887 7-2431565 00 6 3159 381 145161 55306341 19-5192213 7.2495045.002624672 382 145924 55742968 19-5448203 7.2558415 -002'T17801 383 146689 56181887 19-5703858 7-2621675 -0026109'i 384 147456 56623104 19-5959179 7-2684824'0026041 7 385 148225 57066625 19-6214169 7-2747864 -00 2597: 4 386 148996 575124-56 19-6468827 7-2810794'002590674 387 149769 57960603 19-6723156 7'2873617 6002583976)! 388 150544 58411072 19-6977156 7'2936330'0025)77320 389 151321 58863869 19-7230829 7-2998936'00257094OG 390 152100 59319000 19-7484177 7-3061436'0025641 03) 391 152881 59776471 19-7737199 7'3123828'002557545 392 153661 60236288 19'7989899 7-3186114 -002551020 393 154449 60698457 19-8242276 7-3248295'002544529 394 155236 61162984 19-8494332 7-3310369'002538071 395 15i6025 61629875 19-8746069 7-33872339 -002531646 396 156816 62099136 19-8997487 7-3434205'00255 )25 397 15760)9 62570773 19-9248588 7-3495966'002518892 398 158404 63044792 19-9499373 7-3557624'002o12563 399 159201 63521199 19-9749844 7-3619178 00 250620' 400 160000 64000000 20-0000000 7.3680630 002500000 401 160801 64481201 20.0249844 7-3741979'0024 i3G6G 402 161604 64964808 20.0499377 7-3803227' 002487962 403 162409 65450827 20.0748599 7.3864373'0024813993) 404 163216 65939264 20-0997512 7392 5418 00247 52 8 405 164025 66430125 20-1246118 739863633 002469136 406 164836 66923416 20-1494417 7.4047206'002463054 407 165649 67419143 20-1742410 7-4107950 0024.5700 408 166464 67917312 20-1990099 7.4168595'002450980 409 167281 68417929 20-2237484 7-4229142'002444988 410 168100 68921000 20.2484567 7.4289589'002439024 411 168921 69426531 20 -2731349 74349938'002433090 412 169744 69934528 20-2977831 7-4410189'002427184 413 170569 70444997 20-3224014 7.4470343'002421308 414 171396 70957944 20.3469899 7-4530399 -002415459 415 172225 71473375 20.3715488 7 4.590359'002409639 416 173056 71991296 20-3960781 7-4650223'002406849 417 173889 72511713 20-4205779 7-4709991 002398082 418 174724 73034632 20-4450483 7-4769664'002392344 419 175561 73560059 20-4694895 7.4829242 -002386635 420 176400 74088000 20.4939015 7-4888724'002380952 421 177241 74618461 20-5182845 7-4948113'002375297 422 178084 75151448 20-5426386 7.5007406'002369668 423 178929 75686967 20.5669638 7-5066607'002364066 424 179776 76225024 20.5912603 7-5125715 -002358491 425 180625 76765625 20.6155281 7-5184730'002352941 426 181476 77308776 20.6397674 7 5243652'002347418 427 182329 77854483 20.6639783 7-5302482'002341920 428 183184 78402752 20-6881609 7-5361221'002336449 429 184041 78953589 20-7123152 7.5419867.002331002 Page 107 TABLE OF SQUARES, CUBES, SQUARE AND CUBE ROOTS. 107 Number. Squares. Cubes. Square Robts. Cube Roots. Reciprocals. 450 184900 79507000 20-7364414 7 5478423.002325581 4;31 185761 80062991 20-7605395 7-5536888.002320186 432 186624 80621568 20-7846097 7-5595263.002314815 433 187489 81182737 20-8086520 756;53548'002309469 434 188356 81746504 20-8326667 7-5711743.002304147 435 189225 82312875 20'8566536 7-5769849.002298851 436 190096 82881856 20-8806130 7-5827865.002293578 437 190969 83453453 20-9045450 7 5885793 -00288330 438 191844 84027672 20-9284495 7-5943633.002283105 439 192721 84604519 20-9523268 7 6001385.002277904 440 193600 85184000 20-9761770 7-6059049 -0022 72 7 441 194481 85766121 21-0000000 7-6116626.002267574 442 195364 86350888 21-0237960 7-6174116.002262443 443 196249 86938307 21-0475652 7-6231519.002257336 444 197136 87528384 21*0713075 7-6288837.0022.52252 445 198025 88121125 21-0950231 7'6346067 -002247191 446 198916 88716536 21-1187121 7-6403213.002242152 447 199809 89314623 21 1423745 7-6460272.002237136 448 200704 89915392 21-1660105 7-6517247.002232143 449 201601 90318849 21-1896201 7-6574138.002227171 450 2020500 91125000 21 2132034 7 6630943 *0022222-2 4o51 2 40o 1 91733851 21 2367606 7o6687665 *0022''17295 452 204304 92345408'21-2602916 7-6 744303 *002212389'3 453 205209 92959677 21 2837967 7-6800857 *002207506 454'206116 93576664 21 3072758 7-6857328 *002202643 4.5.5 20702 5 941916375 21 -3307290 7 6913717 *002197802 456 207 936 94818816 21 3541565 7 6970023 *002192982 457 208849 95443993 21-3775583 7-7026246 *002188184 458 209764 96071912 21-4009346 7 7082388 *002183406 459 210681 96702579 21-4242853 7-7188448 *002178649 460 211600 97336000 214476106 7-7194426 *002173913 461 21'.21 97972181 21-4709106 77-250325 *002169197 462 213444 98611128 21 4941853 7 7306141 *002164502 463 2143693 99252847 21 5174348 7 7361877 002159827 464 215296 99897344 21-5406592 7-7417532 *002155172 465 216225 100544625 21-5638587 7-7473109 00'2150538 466 217156 101194696 2158710331 7-7528606 -00)145923 467 218089 101847563! 21-6101828 7-7584023 002141328 468 219024 102.503232 21*6333077 7-7639361 *002136752 469 219961 103161709 1 216564078 7 7694620 *00213219G 470 220900 103823000 21-6 794834 7- 749801 *002127660 471 221841 104487111 21-7025344 77-804901 *002123142 472 222784-1 105154048 21 7255610 77 859928 *002118644 473 223729 105828817 21-7485632 7-7914875 *002114165 474 224676 106496424 21-7715411 7-7969745 *002109705 47.5 225625 107171875 21*7944947 7-8024538 *002105263 476 226576 107850176 21-8174242 7-8079254 *002100840 477 227529 108531333 21-8403297 7-8133892 002096486 478 228484 109215352 21 8632111 7 8188456 -002092050 479 229441 109902239 21-8860686 7 8242942 002087683 480 230400 110592000 21-9089023 7 829 7353 002083333 481 231361 111284641 21-9317122 7 8351688 002079002 482 232324 111980168 21-9544984 7-8405949 002074689 483 233289 112678587 21 9772610 7*8460134 |002070393 484 234256 113379904 22-0000000 7-8514244 00206G116 485 235225 114084125 22-0227155 7-8568281 002061856 486 236196 114791256 22-0454077 7-8622242 002057613 487 237169 115501303 22-0680765 7 8676130 002053388 488 238144 116214272 22-0907220 7-8729944 002049180 489 1239121 1116930169 22*1133444 7*8783684 002044990 490 240100 117649000 221359436 78837352 002040816 491 241081 118370771 22-1585198 7 8890946 002036660 ~ ~**~ ~ o ~ ~.~o~ oo~ ~~8353~ 2 ~.,1~ ~.~.o. oo~ ~. 4co14o.~ 231~. 11oo~16 oo~1o33 Page 108 108 THE PRACTICAL MODEL CALCULATOR. Number. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 492 242064 119095488 22-1810730 7-8944468 -002032520 493 243049 119823157 22-2036033 7-8997917'002028398 494 244036 120553784 22-2261108 7-9051294 *002024291 495 245025 121287375 22-2485955 7-9104599'002020202 496 246016 122023936 22-2710575 7-9157832'002016129 497 247009 122763473 22-2934968 7-9210994'002019072 498 248004 123505992 22-3159136 7-9264085'002008032 499 249001 124251499 22-3383079 7-9317104'002004008 500 250000 125000000 22-3606798 7'9370053 -002000000 501 251001 125751501 22'3830293 7-9422931 -001996008 502 252004 126506008 22-4053565 7-9475739'001992032 503 253009 127263527 22-4276615 7-9528477'001988072 504 254016 128024064 22-4499443 7-9581144'001984127 505 255025 128787625 22-4722051 7'9633743'001980198 506 256036 129554216 22-4944438 7-9686271'001976285 507 257049 130323843 22-5166605 7-9738731'001972387 508 258064 131096512 22-5388553 7-9791122'001968504 509 259081 131872229 22-5610283 7'9843444'001964637 510 260100 132651000 22-5831796 7-9895697'001960784 511 261121 133432831 22-6053091 7-9947883'001956947 512 262144 134217728 22-6274170 8'0000000'001953125 513 263169 135005697 22-6495033 8'0052049'001949318 514 264196 135796744 22'6715681 8-0104032'001945525 515 265225 136590875 22'6936114 8'0155946'001941748 516 266256 137388096 22'7156334 8'0207794'001937984 517 267289 138188413 22'7376341 8'0259574'001934236 518 268324 138991832 22'7596134 8'0311287'001930502 519 269361 139798359 22'7815715 8'0362935'001926782 520 270400 140608000 22'8035085 8-0414515'001923077 521 271411 141420761 22'8254244 8-0466030'001919386 522 272484 142236648 22'8473193 8-0517479 -001915709 523 273529 143055667 22'8691933 8'0568862'001912046 524 274576 143877824 22-8910463 8'0620180'001908397 525 275625 144703125 22-9128785 8 0671432'001904762 526 276676 145531576 22'9346899 8-0722620'001901141 527 277729 146363183 22-9564806 8'0773743'001897533 528 278784 147197952 22'9782506 8-0824800'001893939 529 279841 148035889 23'0000000 8'0875794'001890359 530 280900 148877001 23'0217289 8-0926723'001886792 531 281961 149721291 23-0434372 8-0977589'001883239 532 283024 150568768 23-0651252 8-1028390'001879699 533 284089 151419437 23-0867928 8-1079128'001876173 534 285156 152273304 23'1084400 8'1129803'001872659 535 286225 153130375 23'1300670 8'1180414'001869159 536 287296 153990656 23'1516738 8-1230962'001865672 537 288369 154854153 23'1732605 8'1281447'001862197 538 289444 155720872 23'1948270 8'1331870'001858736 539 290521 156590819 23-2163735 8-1382230 001855288 540 291600 157464000 23-2379001 8-1432529'001851852 541 292681 158340421 23'2594067 8'1482765'001848429 542 293764 159220088 23-2808935 8-1532939'001845018 543 294849 160103007 23-3023604 8-1583051'001841621 544 295936 160989184 23'3238076 8-1633102'001838235 545 297025 161878625 23-3452351 8-1683092'001834862 546 298116 162771336 23-3666429 8-1733020'001831502 547 299209 163667323 23'3880311 8-1782888'001828154 548 300304 164566592 23'4093998 8'1832695'001824818 549 301401 165469149 23'4307490 8'1882441'001821494 550 302500 166375000 23'4520788 8'1932127'001818182 551 303601 167284151 23.4733892 8 1981753 -001814882 552 304704 168196608 23'4946802 8'2031319'001811594 553 305809 169112377 23.5159520 8'2080825'001808318 I Page 109 TABLE OF SQUARES, CUBES, SQUARE AND CUBE ROOTS. 109 Number. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 554 306916 170031464 23.5372046 8.2130271.001805054 555 308025 170953875 23-5584380 8-2179657.001801802 556 309136 171879616 23-5796522 8*2228985'001798561 557 310249 172808693 23'6008474 8.2278254 *001795332 558 311364 173741112 23-6220236 8-2327463 *001792115 559 312481 174676879 23.6431808 8-2376614.001788909 560 313600 175616000 23-6643191 8-2425706.001785714 561 314721 176558481 23.6854386 8-2474740.001782531 562 315844 177504328 23.7065392 8'2523715 *001779359 563 316969 178453547 23.7276210 8'2572635.001776199 564 318096 179406144 23-7486842 8.2621492.001773050 565 319225 180362125 23-7697286 8.2670294.001769912 566 320356 181321496 23.7907545 8.2719039 *001766784 567 321489 182284263 23-8117618 8-2767726.001763668 568 322624 183250432 23-8327506 8'2816255.001760563 569 323761 184220009 23.8537209 8.2864928 *001757469 570 324900 185193000 23-8746728 8-2913444.001754386 571 326041 186169411 23-8956063 8.2961903 *001751313 572 327184 187149248 23.9165215 8-3010304 *001748252 573 328329 188132517 23.9374184 8.3058651.001745201 574 329476 189119224 23.9582971 8.3106941.001742160 575 330625 190109375 23.9791576 8'3155175.001739130 576 331776 191102976 24.0000000 8.3203353 -001736111 577 332927 192100033 24-0208243 8-3251475 *001733102 578 334084 193100552 24-0416306 8-3299542 *001730104 579 335241 194104539 24-0624188 8-3347553 *001727116 580 336400 195112000 24-0831891 8-3395509 *001724138 581 337561 196122941 24-1039416 8'3443410 -001721170 582 338724 197137368 24-1246762 8-3491256 *001718213 583 339889 198155287 24-1453929 8-3539047 -001715266 584 341056 199176704 24-1660919 8-3586784 -001712329 585 342225 200201625 24.1867732 8.3634466.001709402 586 343396 201230056 24-2074369 8.3682095.001706485 587 344569 202262003 24-2280829 8.3729668.001703578 588 345744 203297472 24-2487113 8'3777188.001700680 589 346921 204336469 24-2693222 8-3824653 *001697793 590 348100 205379000 24-2899156 8-3872065 *001694915 591 349281 206425071 24-3104996 8-3919428'001692047 592 350464 207474688 24-3310501 8-3966729 *001689189 593 351649 208527857 24-3515913 8-4013981 *001686341 594 352836 209584584 24-3721152 8-4061180 -001683502 595 354025 210644875 24.3926218 8-4108326'001680672 596 355216 211708736 24-4131112 8-4155419'001677852 597 356409 212776173 24-4335834 8-4202460'001675042 598 - 357604 213847192 24-4540385 8.4249448'001672241 599 358801 214921799 24-4744765 8'4296383'001669449 600 360000 216000000 24-4948974 8-4343267'001666667 601 361201 217081801 24-5153013 8-4390098'001663894 602 362404 218167208 24-5356883 8-4436877'001661130 603 363609 219256227 24-5560583 8-4483605'001658375 604 364816 220348864 24-5764115 8-4530281'001655629 605 366025 221445125 24-5967478 8-4576906'001652893 606 367236 222545016 24'6170673 8.4623479'001650165 607 368449 223648543 24-6373700 8-4670001'001647446 608 369664 224755712 24-6576560 8-4716471'001644737 609 370881 225866529 24-6779254 8'4762892'001642036 610 372100 226981000 24-6981781 8-4809261'001639344 611 373321 228099131 24-7184142 8.4855579'001636661 612 374544 229220928 24-7386338 8-4901848'001633987 613 375769 230346397 24-7588368 8-4948065 -001631321 614 376996 231475544 24-7790234 8-4994233'001628664 615 378225 232608375 24-7991935 8-5040350 -001626016 K Page 110 110 THE PRACTICAL MODEL CALCULATOR. Number. Squares. Cubes. Square Rootsoots. ube Roots. Reciprocals. 616 379456 233744896 24-8193473- 8-5086417.001623377 617 380689 234885113 24'8394847 8.5132435.001620746 618 381924 236029032 24'8596058 8-5178403'001618123 619. 383161 237176659 24'8797106 8-5224331.001615509 620 384400 -,238328000 24-8997992 8-5270189.001612903 621 385641 239483061 24-9198716 8-5316009.001610306 622 386884 240641848 24-9399278 8-5361780.001607717 623 388129 241804367 24.9599679 8.5407501 *001605136 624 389376 242970624 24-9799920 8-5453173. 001602564 625 390625 244140625 25.0000000 8-5498797.001600000 626 391876 245134376 25-0199920 8-5544372 *001597444 627 393129 246491883 25.0399681 8.5589899 -001594896 628 394384 247673152'25-0599282 8-5635377.001592357 629 395641 248858189 25-0798724 8-5680807 *001589825 630 396900 250047000 25-0998008 8-5726189'001587302 631 398161 251239591 25-1197134 8-5771523-.001584786 632 399424 252435968 25-1396102'8-5816809.001582278 633 400689 253636137 25.1594913'8.-5862247 -001579779 634 401956 254840104 25.1793566 8-5907238'001577287 635 403225 256047875 25-1992063 8-5952380 *001574803 636 404496 257259456 25-2190404 8-5997476 *001572327 637 405769 258474853 25-2388589 8.6042525 *001569859 638 407044 259694072 25-2586619 8-6087526 -001 567398 639 408321 260917119 25.2784493 8-6132480 *001564945 640 409600 262144000 25-2982213 8-6177388 *001562500 641 410881 2633747,21 25-3179778 8 6222248 001560062 642 412164 264609288 -'25-3377189 8.6267063. 001557632 643 413449 265847707 2-3.3574447 8.6311830.001555210 644 414736'267089984 25.3771551 8-6356551 *001552795 645 416125 268336125 25.39685.02 8-6401226.001550388 646 417316 269585136 25.4165302- 8-6445855 001547988 647 418609 270840023 25.4361947 8.6490437 *001545595 648 419904 -272097792 25-4558441 8-6534974.001543210 649- 421201;273359449 25.4754784 8.6579465 *001540832 650 422500 274625000 25.4950976 8-6623911.001538462 651 423801 275894451 25-5147013 8-6668310.001586098 652 425104 277167808 25-5342907 8-6712665.001533742 653 426409 278445077 25-5538647 86756974 *001531394 654 427716 279726264 25.5734237 1 8-6801237.001529052 655 429025 281011375 25.5929678 8.6845456.001526718 656 430336 282300416 25-6124969 8-6889630 *001524390 657 431639 283593393 25-6320112, 8-6933759.001522070 658 432964 284890312 25.6515107 8-6977843 *001519751 - 659 434281 286191179 25-'6709953 8-7021882.001517451 660 435600 287496000 25-6904652 8-7065877.001515152 661 436921 288804781 25-7099203 8-7109827' -001512859 662 438244 290117528 25-7293607 8-7153734.001510574 663 439569 291434247 25-7487864 8'-7197596 *001508296 664 440896 292754944 25 7681975 8-7241414 *001506024 665 442225 294079625 25-7875939 8.7285187.001503759 666 443556 295408296 25-8069758 8-7328918,001501502 667 444899 296740963 25-8263431 8-7372604'001499250 668 446224 298077632 25-8456960 8.7416246.001497006 669 447561 299418309 25-8650343 8-7459846 -001494768 670 448900 300763000 25-8843582. 8-7503401.001492537 671 450241 302111711 25.9036677 8.7546913.001490313 672 451584 363464448 25-9229628 8'7590383.001488095 673 452929 304821217 25-9422435 8-7633809.001485884 674 454276 306182024 25-9615100 8-7677192.001483680 675 455625 307546875 25-9807621 8-7720532.001481481 676 456976 308915776 26-0000000 8-7763830.001479290 677 458329 310288733'26:0192237 8'7807084.001477105 Page 111 TABLE OF SQUARES, CUBES, SQUARE AND CUBE ROOTS. 111 Number. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 678 4.59684 311665752 26-0384331 8-7850296'001474926 679 461041 313046839 26-0576284 8-7893466'001472754 680 462400 314432000 26-0768096 8-790 36593.001470588 681 463761 315821241 26-0959767 8-7979679'001468409 682 465124 317214568 26'1151297 8-8022721 -001466276 683 466489 318611987 26-1342687 8'806.5722 001464129 684 467856 320013504 26'1533937 8'8108681'001461988 685 469225 321419125 26-1725047 8-8151598'001459854 686 470596 322828856 26-1916017 8-8194474'001457726 687 471969 324242703 26'2106848 8-82378307'001455604 688 473344 325660672 26'2297541 8'8280099'001453488 689 474721 327082769 26'2488095 8-8322850 -001451379 690 476100 328509000 26-2678511 8 836'5559'001449275 691 477481 329939371 26-2868789 8 8408227' 001447178 692 478864 331373888 26-3058929 8'8450854'001445087 693 480249 332812557 26-3248932 8-8493440'001443001 694 481636 334255384 26.3438797 8 -8535985 001440922 695 483025 335702375 26-3628527 8-8578489.001438849 696 484416 337153536 26-3818119 8'8620952.001436782 697 485809 338608873 26'4007576 8'8663375'001434720 698 487204 340068392 26-4196896 8'8705757.001432665 699 488601 341532099 26'4386081 8'8748099 -001430615 700 490000 343000000 26-4575131 8-8790400.001428571 701 491401 344472101 26 4764046 8-8832661.001426534 702 492804 345948408 26-4952826 8-8874882.001424501 703 494209 347428927 26-5141472 8-8917063'001422475 704 495616 348913664 26-5329983 8-8959204.001420455 705 497025 350402625 26-5518361 8-9001304'001418410 706 498436 351895816 26-5706605 8-9043366'001416431 707 499849 353393243 26-5894716 8'9085387'001414427 708 501264 354894912 26-6082694 8'9127369'001412429 709 502681 356400829 26-6270539 8-9169311 -001410437 710 504100 357911000 26-6458252 8-9211214'001408,451 711 505521 359425431 26-6645833 8 9253078'001406470 712 506944 360944128 26-6833281 8-9294902'001404494 713 508369 362467097 26-7020598 8'9336687'001402525 714 509796 363994344 26-7207784 8-9378433'001400560 715 511225 365525875 26'7394839 8-9420140'001398601 716 512656 367061696 26-7581763 8-9461809'001396648 717 514089 368601813 26-7768557 8-9503438'001394700 718 515524 370146232 26-7955220 8 9545029'001392758 719 516961 371694959 26.8141754 8-9586581'001390821 720 518400 373248000 26'8328157 8-9628095'001388889 721 519841 374805361 26.8514432 8-9669570'001386963 722 521284 376367048 26-8700577 8-9711007'001385042 723 522729 377933067 26-8886593 8-9752406'001383126 724 524176 379503424 26-9072481 8-9793766'001381215 725 525625 381078125 26-9258240 8-9835089'001379310 726 527076 382657176 26-9443872 8-9876373' 001377410 727 528529 384240583 26-9629375 8 9917620'001375.516 728 529984 385828352 26-9814751 8-9958899'001373626 729 531441 387420489 27.0000000 9'0000000'001371742 730 532900 389017000 27-0185122 9-0041134'001369863 731 534361 390617891 27-0370117 9 0082229'0013j67989 732 535824 392223168 27 0554985 1 9 0123288'001366120 733 537289 393832837 27-0739727 9.0164309 -001364256 734 538756 395446904 27-0924344 9.0205293'001362398 735 540225 397065375 27 1108834 1 9 0246239'001360544 736:S 541696 398882.56 27-1293199 |9 0287149 001358696 737 513169 400315553 2.7'1477149 9'0328021'0013568.52 738 544644 401947272 i 27'1661554 9 0368857'0013255014 739, 546121 403583419 27-1845544 9.04096,55 001353180 Page 112 112 THE PRACTICAL MODEL CALCULATOR. Number. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 740 547600 405224000 27-2029140 9'0450419'001351361 741 549801 406869021 27-2213152 9-0491142 *001349528 742 550564 408518488 27-2396769 9.0531831.001347709 743 552049 410172407 27-2580263 9-0572482.001345895 744 553536 411830784 27.2763634 9-0613098.001344086 745 555025 413493625 27-2946881 9-0653677.001342282 746 556516 415160936 27-3130006 9-0694220.001340483 747 558009 416832723 - 73313007 9-0734726.001338688 748 559504 418508992 27-3495887 9-0775197 *001336898 749 561001 420189749 27-3678644 9.0815631.001335113 750 562500 421875000 27.3861279 9-0856030.001333333 751 564001 423564751 27-4043792 9.0896352'001331558 752 565504 425259008 27.4226184 9-0936719.001329787 753 567009 426957777 27-4408455 9.0977010 *001328021 754 568516 428661064 27-4590604 9-1017265 -001326260 755 570025 430368875 27-4772633 9-1057485.001324503 756 571536 432081216 27-4954542 9-1097669.001322751 757 573049 433798093 27.5136330 9-1137818.001321004 758 574564 435519512 27-5317998 9-1177931.001319261 759 576081 437245479 27-5499546 9-1218010.001317523 760 577600 438976000 27.5680975 9.1258053.001315789 761 579121 440711081 27.5862284 - 91298061.001314060 762 580644 442450728 27.6043475 9.1338034 -001312336 763 582169 444194947 27.6224546 9-1377971.001310616 764 583696 445943744 27-6405499 9-1417874.001308901 765 585225 447697125 27-6586334 9-1457742.001307190 766 586756 449455096 27-6767050 9-1497576.001305483 767 588289 451217663 27.6947648 9.1537375.001303781 768 589824 452984832 27-7128129 9-1577139'001302083 769 591361 454756609 27-7308492 9'1616869 *001300390 770 592900 456533000 27'7488739 9-1656565'001298701 771 594441 458314011 27'7668868 9-1696225'001297017 7 2 595984 460099648 27'7848880 9'1735852 *001295337 773 597529 461889917 27'8028775 9'1775445.001293661 774 599076 463684824 27.8208555 9.1815003 -001'291990 775 600625 465484375 27-8388218 9-1854527'001290323 776 602176 467288576 27 -8567766 9 -1894018'001288660 777 603729 469097433 27-8747197 9-1933474'001287001 778 605284 470910952 27-8926514 9 -1972897'001285347 779 606841 472729139 27-9105715 9-2012286'001283697 780 608400 474552000 27-9284801 9-2051641'001282051 781 609961 476379541 27-9463772 9-2090962'001280410 782 611524 478211768 27-9642629 9-2130250'001278772 783 613089 480048687 27-9821372 9-2169505 *001277139 784 614656 481890304 28-0000000 9-2208726 *001275510 785 616225 483736625 28-0178515 9-2247914'001273885 786 617796 485587656 28 0356915 9 2287068 001272265 787 619369 487443403 28 0535203 9-2326189 *001270648 788 620944 489303872 28-0713377 9'2365277'001269036 789 622521 491169069 28-0891438 9-2404333'001267427 790 624100 493039000 28-1069386 9-2443355'001265823 791 625681 494913671 28-1247222 9-2482344'001264223 792 627624 496793088 28 -1424946 8-2521300'001262626 793 628849 498677257 28-1602557 9 -2560224'001261034 794 630436 500566184 28-1780056 9-2599114 *001259446 795 632025 502459875 28-1957444 9'2637973'001257862 796 633616 504358336 28 2134720 9-2676798 001256281 797 635209 506261573 28 2311884 9-2715592 001254705 798 636804 508169592 28'2488938 9'2754352'001253133 799 638401 510082399 2892665881 9*2793081'001251364 800 640000 512000000 28 2842712 9 2831777'001250000 801 641601 513922401 28 3019434 9 2870444'001248439 Page 113 TABLE OF SQUARES, CUBES, SQUARE AND CUBE ROOTS. 113 INumber. Squares. CuLbe. Square IRoots. Cube Roots. Reciprocals. 802 648204 51584't608 28-31'6045 9-2909072 1001246883 803 644809 517781627 28 33'72546 9'2947671 *001245330 804 6-16416 519718464 28'3548938 9-2986239 -001243781 8005 648025 521660125 28'3725219 9'3024775.001242236 806 649636 523606616 28-3901391 93063278 -001240695 807 651249 525557943 28-4077454 93101750 *00123917 808 652864 527514112 28'4253408 1 9 3140190.001237624 809 654481 529475129 28'4429253 9.3178599 -001236094 810 656100 531441000 28-4604989 9.3216975 *001234568 811 657721 533411731 28-4780617 9-3255320.001233046 812 659344 535387328 28-4956137 9-3293634.001231527 813 660969 537367797 28-5131549 9-3331916.001230012 814 662596 539353144 28-5306852 9-3370167.001228501 815 664225 541343375 28-5482048 9-3408386.001226994 816 665856 543338496 28-5657137 9-3446575 -001225499 817 667489 545338513 28'5832119 9-3484731.001223990 818 669124 547343432 28-6006993 9'3522857.001222494 819 670761 549353259 28-6181760 9'3560952.001221001 820 672400 551368000 28-6356421 9-3599016.001219512 821 674041 553387661 28-6530976 9-3637049'001218027 822 675684 555412248 28-6705424 9-3675051.001216545 823 677329 557441767 28'6879716 9-3713022 *001215067 824 678976 559476224 28-7054002 9'3750963.001213592 825 680625 561515625 28'7228132 9'3788873.001212121 826 682276 563559976 28-7402157 9-3826752 -001210654 827 683929 565609283 28'7576077 9-3864600'001209190 828 685584 567663552 28.7749891 9-3902419'001207729 829 687241 569722789 28-7923601 9-3940206 -001206273 830 688900 571787000 28.8097206 9-3977964 -001204819 831 690561 573856191 28-8270706 9-4015691 -001203369 832 692224 575930368 28-8444102 9-4053387'001201923 833 693889 578009537 28'8617394 9.4091054'001200480 834 695556 580093704 28'8790582 9.4128690 -001199041 835 697225 582182875 28'8963666 9'4166297'001197605 836 698896 584277056 28'9136646 9.4203873'001196172 837 700569 586376253 28'9309523 9-4241420'001194743 838 702244 588480472 28.9482297 9-4278936 -001193317 839 703921 590589719 28'9654967 9-4316423.001191895 840 705600 592704000 28.9827535 9-4353800'001190476 841 707281 594823321 29.0000000 9.4391307 -001189061 842 708964 596947688 29-0172363 9-4428704'001187648 843 710649 599077107 29-0344623 9-4466072 -001186240 844 712336 601211584 29-0516781 9-4503410'001184834 845 714025 603351125 29-0688837 9'4540719'001183432 846 715716 605495736 29.0860791 9'4577999'001182033 847 717409 607645423 29.1032644 9'4615249'001180638 848 719104 609800192 29.1204396 9'4652470'001179245 849 720801 611960049 29.1376046 9-4689661'001177856 850 722500 614125000 29-1547595 9-4726824'001176471 851 724201 616295051 29-1719043 9-4763957'001175088 852 725904 618470208 29-1890390 9-4801061 -001173709 853 727609 620650477 29-2061637 9-4838136 -001172333 854 729316 622835864 29-2232784 9-4875182'001170960 855 731025 625026375 29-2403830 9'4912200'001169591 856 732736 627222016 29-2574777 9-4949188'001168224 857 734449 629422793 29-2745623 9-4986147'001166861 858 736164 631628712 29-2916370 9-5023078'001165501 859 737881 633839779 29-3087018 95-059980'001164144 860 739600 636056000 29'3257566 9-5096854'001162791 861 741321 638277381 29-3428015 9-5133699'001161440 862 743044 640503928 29-3598365 9'5170515'001160093 863 744769 642735647 29-3768616 9'5207303'001158749,K2 8 Page 114 114 THE PRACTICAL MODEL CALCULATOR. Number. Squares. Cubes. Stquare Roots. Cube Roots. Reciprocals. 864 746496 644'97254_; 29.3938769 9-5244063 -001157407 865 748225 64.7214625 29-4108823" 9-5280794 -001156069 866 749956 649461896 29'4278779 9-5317497 *001154734 867 751689 651714363 29.4448637 9-5354172'001153403 868 753424 653972032 29-4618397 9-5390818'001152074 869 755161 656234909 29-4788059 9-5427437'001150748 870 756900 658503000 29'4957624 9'5464027'001149425 871 758641 660776311 29'5127091 9'5500589'001148106 872 760384 663054848 29-5296461 9'5537123'001146789 873 762129 665838617 29-5465734 9-5573630'001145475 874 763876 667627624 29-5634910 9-5610108'001144165 875 765625 669921875 29-5803989 9-5646559'001142857 876 767376 672221376 29-5972972 9-5682782 -001141553 877 769129 674526133 29-6141858 9-5719377'001140251 878 770884 676836152 2906310648 9-5755745 *001138952 879 772641 679151439 29'6479342 9-5792085'001137656 880 774400 681472000 29-6647939 9-5828397'001136364 881 776161 683797841 29'6816442 9'5864682 -001135074 882 777924 686128968 29'6984848 9'5900937'001133787 883 779689 688465387 29'7153159 9-5937169'001132503 884 781456 690807104 29-7321375 9 5973373'001131222 885 783225 693154125 29'7489496 9-6009548'001129944 886 784996 695506456 29-7657521 9-6045696 -001128668 887 786769 697864103 29-7825452 9-6081817'001127396 888 788544 700227072 29-7993289 9-6117911'001126126 889 790321 709595369 29'8161030 9-6153977'001124859 890 792100 704969000 29-8328678 9-6190017'001123596 891 793881 707347971 29-8496231 9-6226030'001122334 892 795664 707932288 29-8663690 9-6262016'001121076 893 797449 712121957 29-8831056 9-6297975'001119821 894 799236 714516984 29-8998328 9-6333907 -001118568 895 801025 716917375 29'9165506 9-6369812'001117818 896 802816 719323136 29-9332591 9-6405690 001116071 897 804609 721734273 29-9499583 9-6441542 -001114827 898 806404 724150792 29-9666481 9-6477367'001113586 899 808201 726572699 29-9833287 9.6513166'001112347 900 810000 729000000 30.0000000 9-6548938'001111111 901 811801 731432701 30-0166621 9-6584684'001109878 902 813604 733870808 30'0333148 9-6620403'001108647 903 815409 736314327 30-0499584 9-6656096'001107420 904 817216 738763264 30-0665928 9-6691762'001106195 905 819025 741217625 30'0832179 9-6727403'001104972 906 820836 743677416 30'0998339 9'6763017'001103753 907 822649 746142643 30-1164407 9-6798604'001102536 908 824464 748613312 30'1330383 9-6834166'001101322 909 826281 751089429 30-1496269 9'6869701'001100110 910 828100 753571000 30-1662063 9'6905211'001098901 911 829921 756058031 30-1827765 9-6940694'001097695 912 831744 758550825 30-1993377 9-6976151'001096491 913 833569 761048497 30-2158899 9-7011583'001095290 914 835396 763551944 30-2324329 9-7046989'001094092 915 837225 766060875 30-2489669 9-7082369'001092896 916 839056 768575296 30-2654919 9-7117723'001091-703 917 840889 771095213 30-2820079 9-7153051'001090513 918 842724 773620632 30-2985148 9-7188354'001089325 919 844561 776151559 30-3150128 9-7223631'001088139 920 846400 778688000 30-3315018 9-7258883'001086957 921 848241 781229961 30-3479818 9'7294109'00108 5776 929 850084 783777448 30-3644529 9-7329309'001084599 923 851929 786330467 30-3809151 9-7364484'001083423 924 853776 788889024 30-3973683 9-7399634'001082251 925 855625 791453125 30-4138127. 97434758'001081081 Page 115 TABLE OF SQUARES, CUBES, SQUARE AND CUBE ROOTS. 115 Nt! ber. Squares. Cabes. Square Roots. Cube Roots. Reciprocals. 926 8 57476 794022776 30'4302481 9'7469857'001079914 927 859329 796597983 30'4466747 9'7504930'001078749 928 861184 799178752 30-4630924 9'7539979'001077586 929 863041 801765089 30-4795013 9-7575002'001076426 930 864900 804357000 30-4959014 9-7610001'001070269 931 866761 806954491 30-5122926 9-7644974'001074114 932 868624 809557568 30'5286750 9'7679922'001072961 933 870489 812166237 30'5450487 9'7714845'001071811 934 872356 814780504 30.5614136 9-7749743'001070664 935 874225 817400375 30-5777697 9 7784616 *001069519 936 876096 820025856 30-5941171 9.7829466'001068376 937 877969 822656953 30-6104557 9-7854288.001067236 938 879844 825293672 39.6267857 9-7889087'001066098 939 881721 827936019 30-6431069 9-7923861.001064963 940 883600 830584000 30-6594194 9-7958611.001063830 941 885481 833237621 30-6757233 9-7993336.001062699 942 887364 835896888 30-6920185 9-8028036'001061.571 943 889249 838561807 30-7083051 9-8062711'001060445 944 891136 841232384 30-7245830 9-8097362.001059322 945 893025 843908625 30-7408523 9-8131989.001058201 946 894916 846590536 30-7571130 9-8166591.001057082 947 896808 849278123 30-7733651 9-8201169.0010055966 948 898704 851971392 30-7896086 9.8235723.001054852 949 900601 854670349 30-8058436 9.8270252'001053741 950 902500 857375000 30'8220700 9.8304757'001052632 951 904401 860085351 30'8382879 9.8339238.001051525 952 906304 862801408 30-8544972 9-8373695.001050420 953 908209 865523177 30-8706981 9-8408127.001049318 954 910116 868250664 30.8868904 9.8442536.001048218 955 912025 870983875 30'9030743 9-8476920'001047120 956 913936 873722816 30-9192477 9-8511280'001046025 957 915849 876467493 30-9354166 9-8545617 -001044932 958 917764 879217912 30-9515751 9-8579929 -001043841 959 919681 881974079 30-9677251 9-8614218'001042753 960 921600 884736000 30-9838668 9-8648483'001041667 961 923521 887503681 31'0000000 9-8682724'001040583 962 925444 890277128 31-0161248 9-8716941'001039501 963 927369 893056347 31.0322413 9.8751135 *001038422 964 929296 895841344 31'0483494 9.8785305'001037344 965 931225 898632125 31-0644491 9.8819451'001036269 966 933156 901428696 31'0805405 9.8853574'001035197 967 935089 904231063 31'0966236 9'8887673'001034126 968 937024 907039232 31-1126984 9'8921749'001033058 969 938961 909853209 31'1287648 9.8955801'001031992 970 940900 912673000 31'1448230 98989830'001030928 971 942841 915498611 31'1608729 9.9023835'001029866 972 944784 918330048 31-1769145 9.9057817'001028807 973 946729 921167317 31'1929479 9.9091776'001027749 974 948676 924010424 31'2089731 9.9125712'001026694 975 950625 926859375 31.2249900 9.9159624'001025641 976 952576 929714176 31.2409987 9.9193513.001024590 977 954529 932574833 31'2569992 9.9227379'001023541 978 956484 935441352 31'2729915 9.9261222'001022495 979 958441 938313739 31'2889757 9'9295042'001021450 980 960400 941192000 31'3049517 9'9328839'001020408 981 962361 944076141 31-3209195 9'9362613'001019168 982 964324 946966168 31-3368792 9-9396363'001018330 983 966289 949862087 31-3528308 9-9430092'001017294 984 968256 952763904 31'3687743 9'9463797'001016260 985 970225 955671625 31-3847097 9-9497479'001015228 986 972196 958585256 31'4006369 9.9531138.001014199 987 974169 961504803 31-4165561 9 9564775 -001013171 Page 116 116 THE PRACTICAL MODEL CALCULATOR. Number. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 988 976144 964430272 31-4324673 9 9598389'001012146 989 978121 967361669 31-4483704 9'9631981.001011122 990 980100 970299000 31-4642654 9-9665549'001010101 991 982081 973242271 31-4801525 9'9699055'001009082 992 984064 976191488 31'4960315 9'9732619 001008065 993 986049 979146657 31-5119025 9-9766120'001007049 994 988036 982107784 31-5277655 9-9799599'001006036 995 990025 985074875 31-5436206 9-9833055'001005025 996 992016 988047936 31-5594677 9-9866488'001004016 997 994009 991026973 31-5753068 9'9899900'001003009 998 996004 994011992 31-5911380 9'9933289'001002004 999 998001 997002999 31'6069613 9'9966656'001001001 1000 1000000 1000000000 31:6227766 10'0000000'001000000 1001 1000201 1003003001 31'6385840 10'0033222'0009990010 1002 1004004 1006012008 31-6543866 10-0066622'0009980040 1003 1006009 1009027027 31-6701752 10-0099899'0009970090 1004 1008016 1012048064 31-6859590 10-0133155'0009960159 1005 1010025 1015075125 31-7017349 10-0166389'0009950249 1006 1010036 1018108216 31-7175030 10'0199601'0009940358 1007 1014049 1021147343 31'7332633 10'0232791'0009930487 1008 1016064 1024192512 31'7490157 10'0265958'0009920635 1009 1018081 1027243729 31'7647603 10'0299104 -0009910803 1010 1020100 1030301000 31-7804972 10-0332228 -0009900990 1011 1020121 1033364331 31-7962262 10-0365330'0009891197 1012 1024144 1036433728 31-8119474 10-0398410'0009881423 1013 1026169 1039509197 31-8276609 10-0431469 -0009871668 1014 10281-96 1042590744 31-8433666 10-0464506'0009861933 1015 1030225 1045678375 31-8590646 10-0497521 *0009852217 1016 1032256 1048772096 31-8747549 10-0530514.0009842520 1017 1034289 1051871913 31-8904374 10-0563485'0009832842 1018 1036324 1054977832 31-9061123 10-0596435 -0009823183 1019 1038361 1058089859 31'9217794 10-0629364'0009813543 1020 10,40400 1061208000 31-9374388 10.0662271.0009803922 1021 1042441 1064332261 31.9530906 10-0695156.0009794319 1022 1044484 1067462648 31 9687347 10.0728020.0009784736 1023 1046529 1070599167 31-9843712 10.0760863.0009775171 1024 1048576 1073741824 32.0000000 10.0793684.0009765625 1025 1050625 1076890625 32-0156212 10.0826484.0009756098 1026 1052676 1080045576 32-0312348 10-0859262 *0009746589 1027 1054729 1083206683 32-0468407 10-0892019.0009737098 1028 1056784 1086373952 32-0624391 10-0924755 -0009727626 1029 1058841 1089547389 32-0780298 10-0957469 -0009718173 1030 1060900 1092727000 32-0936131 10-0990163.0009708738 1031 1062961 1095912791 32-1091887 10-1022835.0009699321 1032 1065024 1099104768 32-1247568 10-1055487.0009689922 1033 1067089 1102302937 32-1403173 10-1088117.0009680542 1034 1069156 1105507304 32-1558704 10-1120726.0009671180 1035 1071225 1108717875 32-1714159 10-1153314.0009661836 1036 1073296 1111934656 32-1869539 10-1185882.0009652510 1037 1075369 1115157653 32-2024844 10.1218428 -0009643202 1038 1077444 1118386872 32-2180074 10.1250953.0009633911 1039 1079521 1121622319 32-2335229 10-1283457 -0009624639 1040 1081600 1124864000 32-2490310 10-1315941 -0009615385 1041 1083681 1128111921 32-2645316 10-1348403.0009606148 1042 1085764 1131366088 32 2800248 10.1380845.0009596929 1043 1087849 1134626507 32-2955105 10-1413266.0009587738 1044 1089936 1137893184 32-3109888 10-1445667.0009578544 1045 1092025 1141166125 32-3264598 10.1478047.0009569378 1046 1094116 1144445336 32-3419233 10-1510406.0009560229 1047 1096209 1147730823 32.3573794 10-1542744.0009551098 1048 1098304 1151022592 32.3728281 10-1575062.0009541985 1049 1100401 1154320649 32.3882695 10.1607359.0009532888 Page 117 TABLE OF SQUARES, CUBES, SQUARE AND CUBE ROOTS. 117 Number. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 1050 1102500 1157625000 32'4037035 10-1639636.0009523810 1051 1104601 1160935601 32-4191301 10-1671893 *0009514748 1052 1106704 1164252608 32-4345495 10-1704129.0009505703 1053 1108809 1167575877 32'4499615 10-1736344.0009496676 1054 1110916 1170905464 32-4653662 10-1768539.0009487666 1055 1113125 1174241375 32-4807635 10-1800714.0009478673 1056 1115136 1177583616 32-4961536 10-1832868.0009469697 1057 1117249 1180932193 32-5115364 10'1865002.0009460738 1058 1119364 1184287112 32-5269119 10-1897116 -0009451796 1059 1121481 1187648379 32-5422802 10-1929209.0009442871 1060 1123600 1191016000 32-5576412 10-1961283.0009433962 1061 1125721 1194389981 32-5729949 10-1993336.0009425071 1062 1127844 1197770328 32-5883415 10-2025369.0009416196 1063 1129969 1201157047 32'6035807 10-2057382.0009407338 1064 1132096 1204550144 32'6190129 10-2089375.0009398496 1065 1134225 1207949625 32-6343377 10-2121347.0009389671 1066 1136356 1211355496 32*6496554 10-2153300.0009380863 1067 1138489 1214767763 32-6649659 10-2185233.0009372071 1068 1140624 1218186432 32-6802693 10-2217146.0009363296 1069 1142761 1221611509 32 6955654 10'2249039.0009354537 1070 1144900 1225043000 32-7108544 10-2280912.000934.5794 1071 1147041 1228480911 32'7261363 10'2312766.000933 7068 1072 1149184 1231925248 32'7414111 10-2344599 00093283.58 1073 1151329 123o5376017 32 7566787 10'2376413.00093196;64 1074 1153476 1238833224 32-7719392 10-2408207.0009310987 1075 1155625 1242296875 32-7871926 10-2439981.0009302326 1076 1157776 1245766976 32.8024398 10-2471735.0009293680 1077 1159929 1249243533 32-8176782 10-2503470.0009285031 1078 1162084 1252726552 32.8329103 10.2535186.00092 764i38 1079 1164241 1256216039 32-8481354 10-2566881.0009267841 1080 1166400 1259712000 32-8633535 10-2598557 00092592 59 1081 1168561 1263214441 32 8785644 10-2630213 *0009250694 1082 1170724 1266723368 32-8937684 10'2661850.0009242144 1083 1172889 1270238787 32'9089653 10'2693467.0009233610 1084 1175056 1273760704 32'9241553 10.2725065.0009225092 1085 1177225 1277289125 32-9393382 10-2756644.0009216590 1086 1179396 1280824056 32-9545141 10-2788203.0009208103 1087 1181569 1284365503 329696830 10-2819743.0009199362 1088 1183744 1287913472 32-9848450 10-2851264.0009191176 1089 1185921 1291467969 33'0000000 10-2882765.00091827 36 1090 1188100 1295029000 33-0151480 10-2914247.0009174312 1091 1190281 1298596571 33-0302891 10-2945709.0009165903 1092 1192464 1302170688 33.0454233 19-2977153.0009157509 1093 1194649 1305751357 33.0605505 10'3008577.0009149131 1094 1196836 1309338584 33'0756708 10'3039982.0009140768 1095 1199025 1312932375 33'0907842 10.3071368.0009132420 1096 1201216 1316532736 33.1058907 10.3102735.0009124008 1097 1203409 1320139673 33.1209903 10-3134083.6009115770 1098 1205604 1323753192 33.1360830 10.3165411.0009107468 1099 1207801 1327373299 33-1511689 10-3196721 -0009099181 1100 1210000 1331000000 33'1662479 10-3228012.0009090909 1101 1212201 1334633301 33-1813200 10-3259284.0009082OS 5 1102 1214404 1338273208 33'1963853 10'3290537.0009074410 1103 1216609 1341919727 33'2114438 10.3321770.00090661,b 1104 1218816 1345572864 33'2266955 10'3352985.00090 5791 1105 1221025 1349232625 33-2415403 10-3384181.0009049774 1106 1223236 1352899016 33-2565783 10-3415358.0009041191 1107 1225449 1356572043 33-2716095 10.3446517.0009033142 1108 1227664 1360251712 33.2866339 10'347 767.00090252 1 1109 1229881 1363938029 33 3016516 10-3508778.0009017133 1110 1232100 1367631000 33.3166625 10.3539880.00090090u9 1111 1234321 13713300631 33-3316666 10-3570964.0009009900 Page 118 118 THE PRACTICAL MIODEL CALCULATOR. Number. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 11121 1236544 1375036928 33.3466640 10-3602029.0008992806 1113 1238769 1378749897 33'3616546 10'3633076'0008984726 1114 1240996 1382469544 33-3766385 10-3664103'0008976661 1115 1243225 1386195875 33-3916157 10-3695113 -0008968610 1116 1245456 1389928896 33-4065862 10.37261.03.0008960753 1117 1247689 1393668613 33-4215499 10-3757076.0008952551 1118 1249924 1397415032 33'4365070 10 3788030 *0008944544 1119 1252161 1401168159 33-4514573 10.3818965.0008936550 1120 1254400 1404928000 33-4664011 10.3849S82 -0008928-571 1121 1256641 1408694561 33-4813381 10-3883081.0008960607 1122 1258884 1412467848 33-4962684 10-3911661 -0008912656 1123 1261129 1416247867 33'5111921 10-3942527 *0008904720 1124 1263376 1420034624 33-5261092 10-3973366.0008896797 1125 1265625 1423828125 33-5410196 10-4004192.0008888889 1126 1267876 1427628376 33-5559234 10-4034999.0008880995 1127 1270129 1431435383 33-5708206 10-4065787 -0008873114 1128 1272384 1435249152 33-585711.2 10.4096557'0008865248 1129 1274641 1439069689 33-6005952 10-4127310 -0008857396 1130 1276900 1442897000 33-6154726 10'4158044.0008849558 1131 1979161 1446731091 33.6303434 10-4188760.00088417 33 1132 1281424 1450571968 33-6452077 10-4219458 -0008833922 1133 1283689 1454419637 33'6600653 10-4250138.0008826125 1134 1285956 1458274104 33-6749165 10'4280800 *0008818342 1135 1288225 1462135375 33'6897610 10.4311443.0008810573 1136 1290496 1466003456 33-7045991 10.4342069.0008802817 1137 1292769 1469878353 33-7174306 10'4372677.0008795075 1138 1295044 1473760072 33'7340556 10'4403677.0008787346 1139 1297321 1477648619 33'7490741 10'4433839.0008779631 1140 1299600 1481544000 33'7638860 10-4464393.0008771930 1. 41 1301881 1485446221 33-7786915 10-4494929.0008764242 1142' 1304164 1489355288 33.7934905 10-4525448 000876G567 11403 1306449 1493271207 33-8082830 10-4555948 -0008748906 1144 1308736 1497193984 33-8230691 10-4586431 0008741259 114 5 1311025 1501123625 33'8378486 10'4616896.0008733624 i 1146 1313316 1505060136 33'8526218 10'4617343.0008726003 li47 1315609 1509003523 33'8673884 1046( 77 73.0008718396 i 1148 1317904 1512953792 33 8821487 10'47108158.0008710801 1149 1320201 1516910949 33-8969025 10'4738579.0008703220 1150 1322500 1520875000 33-9116499 10'4768955.0008695652 1151 1324801 1524845951 33-9263909 10'4799)314 *0008688097 1152 1327104 15288283808 33'9411255 10'4829656'0008680556 1153 1329409 1532808577 33 9558537 10'4859980'0008673027 1154 1331716 1536800264 33-9705755 10'4890286.0008665511 11-55 1334025 1540798875 33-9852910 10 49205 75 0008658009 1156 1336336 1544804416 34-0000000 10195O081 7 00086t0519 1157 1338649 1548816893 34'0147027 10-4981101.0008643042 1138 1340964 1552836312 34 0293990 10U5011337 OO008635579 1159 1343281 1556862679 340-140890 10.5011556 00U08628128 1160 1345600 1560896000 34-0587727 10-5071757.0008620690 1161 1347921 1564936281 34-0734501 10-51019-12 0008613244 1162 1350244 1568983528 34-0881211 10.5132109 -0008605852 l163 1352569 1573037749 34-0127858 1051622259. 0008598-152 l164 1354896 1577098944 34-1174442 10 51923931.0008391035 1165 1357225 1581167125 34 1320963 10 5222.506 0008583691 l166 1359556 158.5242296 34-1467422 10.525t604.0008576329 1 167 1361889 1589324463 34-1613817 10. 5282'J 85.00085689t;)0 1168 1364224 1593413632 3411760150 105312 749 0008561644 1169 1366561 1597509809 34.1906420 10 53427 95 00U080.4320 1170 1368900 1.6016i3000 34'2052627 10-53728285 00083457009 1 71 1371241 1605723211 34.2198773 10-540z 283o 7'000895710 F 1172 1373584 16098 104 18 34234485.5 10 5432 S3 2. 00085;32423 1173 1375929 1613964717 3X4.2490875 10.5462810.000823)5149 Page 119 TABLE OF SQUARES, CUBES, SQUARE AND CUBE ROOTS. 119 Number. Squares. Cubes. Square Roots. Cube Roots. Rceiprocals. 1174 13782 76 1618096024 34'2636834 10'549'2771 000817888 1175 1380625 1622234375 34'2782730 10'5522715'0008510638 1176 1382976 1626379776 34'2928564 10'5552642'0008503401 1177 1385329 1630532233 34'3074336 10'5582552 0008496177 1178 1387684 1634691752 34-3220046 10-5612445'0008488964 1179 1390041 1638858339 34-3365694 10-564232 0058481, 1180 1392400 1643032000 34-3511281 10 50'72181'0008471576 1181 1394761 1647212741 34-3656805 10-5702024'0008467401 1182 1397124 1651400568 34 3802268 105 731849.0008160 2:7 1183 1399489 1655595487 34-3947670 1057161658 0084-)0OS 1184 1401856 1659797504 34-4093011 10-5791449'000844 5)46 11.85 1404225 1664006625 34-4238289 10 58 2122;5 00084.388819 1186 1406596 1668222856 34-4383507 10'5850983'00084317,03 1187 1408969 1672446203 34 4528663 10s5880725 o00084 -i00 1188 1411344 1676676672 34'4673759 10'5910450 0008417508 1189 1413721 1680914629 34-4818793 10-5940158 0008-l410-29 1190 1416100 1685159000 34-4963766 10'5969850'00081003361 1191 1418481 1689410871 34.5108678 10.5999525. 0008'390300; 1192 1420864 1693669888 34-5253530 10-6029184'0008938202 1193 1423249 1697936057 34.5398321 10-6058826.00088'-"O0 1194 14256.36 1702209381 34.5543051 10-6088451.0008375 ()9) 11.95 1428025 1706489875 34-5687720 10.6118060.00080682 01i 1196 1430416 1710777536 34-5832329 10-6147652.0008361204 1197 1432809 1715072373 34-5976879 10-6177228.00085-419 119(8 1435204 1719374392 34-6121366 10-6206788.0008347245 1199 1437601 1723683599 34-6265794 10-6236331.00083 10S 81200 1440090 1728000000 34-6410162 10-6265857.0008333 33 1201 1442401 1732323601 34-6554469 10-6295367'0008326'95 1202 144-804 1736654408 34-6698716 10-6324860.0008319-1468 12 03 1447209 1740992427 34-6842904 10-6354338.0008312552 1201 1449616 1745337664 34-6987031 10.6383799'00083905-8 1205 1452025 1749690125 34-7131099 1064113244.000829875i 1206 1454436 17540419816 347275107 10-6442672'0008291874 1207 1456849 1758416743 34 7419055 10-6472085.0008285004 1208 1459264 1762790912 34.7562944 10.6501480.000)82714G 1209 1461681 1767172329 34-7706773 10.6530860 00982 1299 1210 1464100 1771561000 34-7850543 10-6560223 -0008Gi2643 1211 1466521 1775956931 34.7994253 10.6589570.0008257638 1212 1468944 1780360128 34.8137904 10 6618902.000825 0235 1213 1471369 1784770597 34-8281495 10.6648217.00082 4402'3 1214 1473796 1789188344 34-8425028 10G6677516.000823 72 32 1215 1476225 1793613375 34-8568501 10-6706799.0008230053 1216 1478656 1798045696 34.8711915 10.6736066.000822;368 I 1217 1.481089 1802485313 34-8855271 10-676 5 317 0008169270 1218 1483524 1806932232 34-8998567 10.6794552.0008210181 1219I 1485961 1811386459 34-9141805 10.6823771.0008203445 1220 1488400 1815848000 34-9284984 10-6852973.0008196721 122)1 1490841 1820316861 34-9428104 10-6882160.0008190008 1222 ) 1493284 1824793048 34-9571166 10 6911331'0008183306 2)23 1495729 1829276567 34-9714169 10-6940486.0008176615 122 24 1498176 1833764247 34-9857114 10.6969625.000816993.3 122.5 1500625 1838265625 35.0000000 10.6998748 00081326 5 1226 1503276 1842771176 35 0142828 10 70278-55 0008156607 1227 1505529 1847284083 33 0285598 10'7056947'00081-1'959 122'8 1507984 1851804352 3.- 0428309 10 7086023. 000814332i9 1229 1510441 18566331989 35 0570963 10.7115083.0008136690 6 1230 1512900 18608670000 3. 0713558 10o7144127.000813008l.12l1 1515361 1865409991 35.0856096 10-7173155.0008123477 1 2') 1i517824 1869959168 35.0998575 1 0 702168.00081168583 1233 11520289 1874516337 35-1140997 10.7231165 0008110300 1234 1522756 1879080904 3.1283361 107260146 0008103 728 12 3.5 1525225 1883652875 35.1425568 10-7289112 0008097166 I Page 120 120 THE PRACTICAL MODEL CALCULATOR. Number. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 1236 1527696 1888232256 35'1567917 10'7318062.0008090615 1237 1530169 1892819053 35'1710108 10'7346997'0008084074 1238 1532644 1897413272 35'1852242 10'7375916'0008077544 1239 1535121 1902014919 35'1994318 10'7404819'0008071025 1240 1537600 1906624000 35'2136337 10'7433707.0008064516 1241 1540081 1911240521 35'2278299 10'7462579 *0008058018 1242 1542564 1915864488 35'2420204 10.7491436'0008051530 1243 1545049 1920495907 35,2562051 10'7520277.0008045052 1244 1547536 1925134784 35.2703842 10,7549103.0008038585 1245 1550025 1929781125 35.2845575 10'7577913 0008032129 1246 1552521 1934434936 35-2987252 10'7606708.0008025682 1247 1555009 1939096223 35'3128872 10'7635488 *0008019246 1248 1557504 1943764992 35.3270435 10.7664252 -0008012821 1249 1560001 1948441249 - 35.3411941 10.7693001'0008006405 1250' 1562500 1953125000 35.3553391 10.7721735.0008000000 1251 1565001 1957816251 35.3694784 10'7750453'0007993605 1252 1567504 19'62515008 35'3836120' 10'7779156 *0007987220 1253 1570009 1967221277 35.3977400 10.7807843'0007980846 1254 1572516 1.971935064 85.4118624 10-7836516 *0007974482 1255 1575025 1976656375 35.4259792 10.7865173.0007968127 1256 1577536 1981385216 9354400903 10-7893815.0007961783 1257 1580049 1986121593 35,4541958 10.7922441'0007955449 1258,1582564 1990865512 35.4682957 10.-7951053.0007949126 1259 1585081 1995616979 35.4823900 10-7979649'0007942812 1260 1587600 2000376000 35.4964787 10.8008230'0007936508 1261 1590121 2005142581 35-5105618 10-8036797.0007930214 1262 1592644 2009916728 35:5246393 10-8065348 *0007923930 1263.' 1595166 2014698447 35.5387113 10.8093884,0007917656 1264 1597696 2019487744 3'5.5527777 10.8122404.0007911392 1265 1600225 2024284625 35.5668385 10.8150909,0007905138 1266 1602756 i 2029089096 35.5808937 10.8179400,0007898894 1267 1605289 2033901163 35-5949434 10-8207876.0007892660 1268 1607824 2038720832 38356089876 10.8236336 *0007886435 1269 1610361 2043548109 35-6230262 10.8264782 -0007880221 1270 1612900. 2048383000 35-6370593 10.8293213 *0007874016 i1271 1615441 2053225511 35.6510869 10.8321629.0007867821 1272 1617984 2058075648 35-6651090 10.8350030.0007861635 1273 1620529 2062933417 35.6791255 10.8378416.0007855460 1274 1623076 2067798824 35.6931366 10 8406788.0007849294 1275 1625625 2072671875 35-7071421 10.8435144'.0007843137 1276 1628176 2077552576 35-7211422 10.8463485 *0007836991 1277 1630729 2082440933 835.7351367 10.8491812 *0007830854 1278 1633284 2087336952 35,7491258 10.8520125.0007824726 1279 1635841 2092240639' 35-7631095 10-8548422 -0007818608 1280 1638400 2097152000 35.7770876 10.8576704.0007812500 1281 1640961 2102071841 35.7910603 10.8604972 *0007806401 1282 1643524 2106997768 35.8050276 10'8633225'0007800312. 1283 1646089 2111932187 35.8189894 10'8661454'0007794232 1284 1648656 2116874304 35-8329457 10'8689687:0007788162 1285 1651225 2121824125 35'8468966 10-,8717897 0007782101 1286 1653796 2126781656 35'8608421 10'8746091'0007776050 1287 1656369 2131746903 35'8747822 10'8774271 *0007770008 1288 1658944 2136719872 35-8887169 10'8802436 *0007763975 1289 1661521 2141700569 35'9026461 10'8830587'0007757952 1290 1664100 2146689000 35'9165699 10-8858723'0007751938 1291 1666681 - 2151685171 35-9304884 10-8886845'0007745933 1292 1669264 2156689088 35-9444015 10-8914952'0007739938 1293 1671849 2161700757 35'9583092 10'8943044.0007733952 1294 1674436 2166720184 35-9722115 10-8971123'0007727975 1295 1677025 2171747375 35'9861084 10'8999186'0007722008 1296 1679616 2176782336 36'0000000 10'9027235'0007716049 1297 1682209 2181825073 36-0138862 10'9055269'0007710100 Page 121 TABLE OF SQUARES, CUBES, SQUARE AND CUBE ROOTS. 121 Number. Squares. Cubes. Square Roots. Cube Iloots. Reciprocals. 1298 1684804 2186875592 86.0277671 10-9083290.0007704160 1299 1687401 2191933899 36.0416426 10'9111296'0007698229 1300 1690000 2197000000 36-0555128 10-9139287'0007692308 1301 1692601 2202073901 36-0693776 10-9167265'0007686395 1302 1695204 2207155608 36-0832371 10-9195228' 0007680492 1303 1697809 2212245127 36'0970913 10'92231 77 0007674579 1304 1700416 2217342464 36-1109402 10-9251111'0007668712 1305 1703025 2222447625 36-1247837 10'9279031 -0007062835 1306 1705636 2227560616 36-1386220 10-9306937'0007656968 1307 1708249 2232681443 36'1524550 10'9384829'0007651109 1308 1710864 2237810112 36-1662826 10-9362706'000764-5260 1309 1713481 2242946629 36-1801050 10-9390569'0007639419 1310 1716100 2248091000 36-1939221 10-9418418 *00076C33588 1311 1718721 2253243231 36-2077340 10-9446253'0007627765 1312 1721344 2258403328 36-2215406 10-9475074'0007621951 1313 1723969 2263571297 36-2353419 10-9501880'0007616446 1314 1726596 2268747144 36 24913879 10'9529673 O0007610350 1315 1729225 2273930875 36-2626287 10-9557451 000O60453OT 1316 1731856 2279122496 36-2767143 10-9585215'00079S 8784 1317 1734489 2284322013 36-2904246 10'9612965'0007593014 1318 1787124 2289529432 36-8042697 10-9640701'00078 7258 1319 1738976 1 2294744759 363-180396 10-9668423 0007581501 1320 1742400 2299968000 8363318042 10'9696131'0007575758 1321 1745041 2305199161 36-3455637 10-9723825'0007570023 1322 1747684 2310438248 3683593179 10'9751505.0007564297 1823 1750329 2815685267 36-37'30670 10-9779171 00075-58579 1324 1752976 2320940224 36-3868108 10-9806823'0007552870 1325 1755625 2326203125 36'4005494 10-9834462'0007547170 1326 1758276 2831473976 36-4142829 10'9862086'0007541478 1327 1760929 23f36752783 36'4280112 10'9889696'00075835795 13286 1763584 2342039552 36-4417343 10-991 7293 o0007530120 1329 1766241 2347334289 36-4554523 10-9944876.0007524454 1330 1768900 2352637000 36-4691650 10-9972445 0007518797 1331 1771561 2357947691 36-4828727 11000000 000000513148 1332 1774224 2363266368 36-4965752 11'0027541'0007507508 1333 1776889 2368593037 36'5102725 11'0055069'0007501875 1334 1779556 2373927704 36'5239647 11 0082583'0007496252 1335 1782225 23792706375 36-5376518 11-0110082'0007400637 1386 1784896 2384621056 36'5513388 11-0137569'0007485080 1337 1787569 2389979753 36'5650106 11-0165041'0007479412 1338 1790244 2395346472 36'5786823 11'0192500'000747384_2 1339 1792921 2400721219 36-5923489 11-0219945 *000,746o8260 1340 1795600 2406104000 36-6060104 11 0247377'0007462687 1341 1798281 2411494821 36-6196668 11'0274795'0007457122 1342 1800964 2416893688 36-6333181 11'0302199'0007451565 1343 1803649 2422300607 36-6469144 11'0329590'0007446016 1344 1806336 2427715584 36'6606056 11 0356967'0007440476 1345 1809025 2433138625 36-6742416 11-0384330.0007434944 1346 1811716 2438569736 36-6878726 11-0411680'0007429421 1347 1814409 2444008923 36'7014986 11-0439017'0007423905 1348 1817104 2449456192 36-7151195 11-0466339'0007418398 1349 1819801 2454911549 36-7287353 11-0493649'0007412898 1350 1822500 2460375000 36-7423461 11-0520945'0007407407 1351 1825201 2465846551 36-7559519 11-0548227'0007401924 1352 1827904 2471326208 36-7695526 11-0575497'00073'96450 1353 1830609 2476813977 36-7831483 11-0602752.0007390983 1354 1833316 2482309864 36'7967390 11 0629994'0007883524 1355 1836025 2487813875 36'8103246 11'0657222'0007380074 1356 1838736 2493326016 36.8239053 11.0684437.0007374631 1357 1841449 2498846293 36.8374809 11.0711639 -0007369197 1358 1844164 2504374712 36-8510515 11 0738828 -0007363770 1359 1846881 2509911279 36-8646172 11 -0766003'0007358352 L Page 122 122 THE PRACTICAL MODEL CALCULATOR. | Number. Squares. Cubes. Square Roots. ci(l,e i{'S. ICJciiPloc;s. I 1360 1849600 2515456000 36'8781778 11 078'3165 -00079)9'21AI 1361 1852321 2521008881 36-8917335 11'0 -314. 0t 007347539 1362 1855044 2526569928 36'052842 11' 47449 -00097,42 144 1363 1857769 2532139147 36'9188299 11l08745i1 0000'-36 7 1364 1860496 2537716544 36-9323706 11'0901679.0007331378 1365 1863225 2543302125 36-9459064 11'0928775 0007,326007 1366 1865956 2548895896 36-9594372 11 09558a7.0007320614 1367 1868689 2554497863 36-9729631 11-09829G6 -00073152890 1368 1871424 2560108032 36-9864840 11-1009982 -000 730'94' L 1369 1874161 2565726409 37'0000000 11-1037025.0007304G02 1370 1876900 2571353000 37-0135110 11-1064054 0000729(270 1371 1879641 2576987811 37-0270172 11'109107 0 000 7293946 1372 1882384 2582630848 37-0405184 11-1118073.00072O8630 1373 1885129 2588282117 37-0540146 11-1145064 000728'321 1374 1887876 2593941624 37.0675060 11-1172041 000 2 7802 0 1375 1890625 2599609375 37-0899924 11-1199004.0001722'' 7 2 1376 1893376 2605285376 37'0944740 11-1225955.00072674211377 1896129 2610969633 37-1079506 11-1252893.000 722164 1378 1898884 2616662152 37'1214224 11'1279817.0007256894 1379 1901641 2622362939 37-1348893 11-1306729 *000251632 1380 1904400 2628072000 37-1483512 11-1333628.000724637 7 1381 1907161 2633789341 37'1618084 11-1360514 00017241130 1382 1909924 2639514968 37'1752606 11 1387386 00072335890 1383 1912689 2645248887 37'1887079 11-1414246.0007230658 1384 1915456 2650991104 37.2021505 11-1441093.000'72 5434 1385 1918225 2656741625 37-2155881 11-14G7926.0007220217 1386 1920996 2662500456 37 2290209 11-1494747 0007'215007 1387 1923'769 2668267603 37-2424489 l1521555.000'7209805 1388 1926544 2674043072 37.2558720 11 154885i0 0007204611 1389 1929321 2679826869 37-2692903 11 1575133 *0007199424 1 390 1932100 2685619000 37 2827037 11-1601903.0007194245 1391 1934881 2691419471 37-2961124 11-1628659.0007189073 1392 1937664 2697228288 37. 3095162 11 1(155403.0007183908 1393 1940449 2703045457 37-3229152 11 1682134.0007178751 1394 1943236 2708870984 37 3363094 11 1708852 0007173601 1 1395 1946025 2714704875 3 73496988 11'1735558.0007168459 1396 1948816 2720547136 37-3630834 11 1762250.0007 163 324 1397 1951609 2726397773 37'3764632 11.1788930 -0007158196 1398 1954404 2732256792 37'3898382 11.1815598.0007153076 1399 1957201 2738124199 37.4032084 11.1842252 *0007147963 1400 1960000 2744000000 37.4165738 11-1868894.0007142857 1401 1962801 2749884201 37.4299345 11.1895523.0007137759 1402 1965604 2755776808 37'4432904 11'1922139 -0007132668 1403 1968409 2761677827 37'4566416 11'1948743'0007127584 1404 1971216 2767587264 37'4699880 11 1975334.0007122507 1405 1974025 2773505123 37'4833296 11 2001913.0007117438 1406 1976836 2779431416 37 4966665 11'2028479'0007112376 1407 1979649 2785366143 37.5099987 11.2055032 -0007107321 1408 1982464 2791309312 37'5233261 11 2081573'0007102273 1409 1985281 2797260929 37.5366487 11.2108101'0007097232 1410 1988100 2803221000 37. 5499667 11.2134617 0007092199 1411 1990921 2809189531 37.5632799 11-2161120.0007087172 1412 1993744 2815166528 37-5765885 11 2187611'0007082153 1413 1996569 2821151997 37.5898922 11-2214089 -0007077141 1414 1999396 2827145944 37.6031913 11'2240054'0007072136 1415 2002225 2833148375 37'6164857 11'2267007'0007067138 1416 2005056 2839159296 37.6297754 11.2203448'0007062147 1417 2007889 2845178713 37.6430604 11'2319876'0007057163 1418 2010724 2851206632 37'6563407 11 2346292'0007052186 1419 2013561 2857243059 37.6696164 11.2372696 -0007047216 1420 2016400 2863288000 37.6828874 11.2399087.00070422.54 1421 2019241 2869341461 37.6961536 11-2425465'0007037298 Page 123 TABLE OF SQUARES, CUBES, SQUARE AND CUBE ROOTS. 123 Iu:,r. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 14_22 20220 84 2875403448 37-7094153 11'2451831'0007032349 142 2024929 2881473967 37-7226722 11-2478185'0007027407 1424 i 2027776 2887553024 37-7359245 11-2504527'0007022472 1425 2030625 2893640625 37'7491722 11'2530856'0007017544 1426 2033476 2899736776 37-7624152 11'2557173'0007012623 142 7 2036329 2905841483 37-7756535 11-2583478'0007007708 1428 2039184 2911954752 37-7888873 11-2609770'0007002801 14293 2042041 2918076589 37-8021163 11-2636050 -0006997901 1430 2044900 2924207000 37-8153408 11-2662318 -0006993007 1431 2047761 2930345991 37-8285606 11-2688573 -0006988120 1432 2050624 2936493568 37-8417759 11-2714816 -0006983240 1433 2053489 2942649737 37'8549864 11-2741047'0006978367 1434 2056356 2948814504 37-8681924 11-2767266'0006973501 14035 2059225 2954987875 37-8813938 11-2793472'0006968641 1436 2062096 2961169856 37-8945906 11-2819666'0006963788 1437 2064969 2967360453 37'9077828 11'2845849.0006958942 1438 2067844 2973559672 37'9209704 11'2872019.0006954103 1439 2070721 2979767519 37-9341538 11-2898177'0006949270 1440 2073600 2985984000 37-9473319 11-2924323'0006944444 1441 2076481 2992209121 37'9605058 11'2950457'0006939625 1442 2079364 3098442888 37-9736751 11-2976579'0006934813 1443 2082249 3001685307 37-9868398 11'3002688'0006930007 1444 20851.36 3010936384 38"0000000 11'3028786'0006925208 1445 2088025 3017196125 38'0131556 11-3054871.0006920415 1446 2080916 3023464536 38-0263067 11-3080945.0006915629 1447 2093809 3029741623 38-0394532 11-3107006'0006910850 1448 2096704 3036027392 38'0525952 11 3133056'0006906078 1449 2099601 3042321849 38'0657326 11.3159094.0006901312 1450 2102500 3048625000 38'0788655 11-3185119 *0006896552 1451 2105401 3054936851 38-0919939 11 -3211132'0006891799 1452' 2108304 3061257408 38'1051178 11-3237134'0006887052 1453 2111209 3067586777 38-1182371 11 3263124'0006882312 1454 2114116 3073924664 38-1313519 11-3289102'0006877579 1455 2117025 3080271375 38'1444622 11'3315067'0006872852 1456 2119936 3086626816 38'1575681 11'3341022'0006868132 1457 2122849 3092990993 38-1706693 11-3366964'0006863412 1458 2125764 3099363912 38-1837662 11-3392894'0006858711 1459 2128681 3105745579 38-1968585 11-3418813'0006854010 1460 2131600 3112136000 38-2099463 11'3444719 *0006849315 1461 2134-521 3118535181 38 2230297 11 3470614.0006844627 146 2 2137444 3124943128 38-2361085 11-3496497'0006839945 146;3 2140369 3131359847 38-2491829 11 3522368'0006835270 1464 2143296 3137785344 38'2622529 11-3548227'0006830601 1465 2146225 3144219625 38-2753184 11 3574075'0006825939 1466 2149156 3150662696 38'2883794 11'3599911'0006821282 1467 2152089 3157114563 38'3014360 11 3625735' 0006816633 1468 2155024 3163575232 38-3144881 11-3651547'0006811989 1469 29157961 3170044709 38-3275358 11-3677347.0006807352 1470 2160900 3176523000 38-3405790 11-3703136 -0006802721 1471 2163841 3183010111 38-3536178 11-372891.4 -0006798097 1472 2166784 3189506048 38-3666522 11'3754679'0006793478 14 73 2169729 3196010817 38-3796821 11-3780433'0006788866 14|74 2172676 3202524424 38-3927076 11'3806175'0006784261 1475 2175625 3209046875 38'4057287 11'3831906'0006779661 1476 2178576 3215578176 38-4187454 11'3857625'0006775068 1477 2181529 3222118333 38 431 577 11 3883332'0006770481 1478 2184484 32286673.52 38'4447656 11-3909028'0006765900 1479 2187441 3235225239 38'4577691 11'39334712 0006761325 1480 2190400 3241792000 38'4707681 11 3960384'0006756757 1481 2'193361 3248367641 38-4837627 11 3986045'0006752194 1482 2196324 34254952168 38'4967530 11 4011695 -0006747638 1483 2199289 3261545.587 38 5097390 11-4037332 -0006743088 Page 124 124 THE PRACTICAL MODEL CALCULATOR. Number. Squares. Cubes. Square Roots. - Cube Roots. Reciprocals. 1484 2202256 3268147904 38-5227206 11-4062959 *0006738544 1485 2205225 3274759125 38-5356977 11-4088574 *0006734007 1486 2208196 3281379256 38-5486705 11-4114177'0006729474 1487 2211169. 3288008303 38'5616389 11'4139769'0006724950 1488 2214144 3294646272 385-746030 11-4165349'0006720430 1489 2217121 3301293169 38-5875627 11'4190918'0006715917 1490 2220100 3307949000 38-6005181 11-4206476 *0006711409 1491 2223081 3314613771 38-6134691 11-4242022 *0006706908 1492 2226004 3321287488 38'6264158 11-4267556 -0006702413 1493 2229049 3227970157 38-6393582 11.4293079 -0006697924 1494 2232036 3334661784 38-6522962 11'4318591 *0006693440 1495 2235025 3341362375 38'6652299 11-4344092,0006688963 1496 2238016 3348071936.88-6781593 11'4369581 *0006684492 1497 2241009 3354790473 38'6910843 11'4395059 *0006680027 l 1498 2244004 3361517992 38-7040050 11-4420525 *0006675567 1499 2247001 3368254499' 38'7169214 11-4445980 *0006671114 1500 2250000 3375000000 38-7298335 11-4471424'0006666667 1501 2253001 3381754501 38'7427412 11'4496857'0006662225 1502 2256004 3388518008 38-7556447 11-4522278'0006657790.1503 2259009 3395290527 38-7685439 11.4547688 -0006553360 1504 2262016 3402072064 38 7814389 11 4573087.0006648936 1505 2265025 3408862625 38'7943294 11'4598476'0006644518 1506 2268036 3415662216 38.8072158 11-4623850.0006640106 1507 2271049 3422470843 38'8200978 11'4649215'0006635700 1508- 2274064 3429288512 38-8329757 11-4674568 *0006631300 1509' 2277081 3436115229 38'8458491 11-4699911 *0006626905 1510 2280100 3442951000 38-8587184 11'4725242 *0006622517 1511 2283121 3449795831 38-8715834 11-4750562.0006618134 1512 2286144 3456649728 38'8844442 11'4775871.0006613757 1513 2289169 3463512697 38-8973006 11'4801169 *0006609385 151.4 2292196 3470384744 38.9101529 11.4826455.0006605020 1515 2295225 3477265875 38'9230009 11'4851731'0006600660 1516 2298256 3484156096 38-9358447: 11-4876995'0006596306 1517 2301289 3491055413 38'9486841 11'4902249''0006591958 1518 2304324 3597963832 38'9615194 11-4927491'0006587615 1519 2307361 3504881359 38-9743505 11'4952722'0006583278 1520 2310400 3511808000' 38'9871774 11'4977942'0006578947 1521'2313441 3518743761 39'0000000 11'5003151 -0006574622 1522 2316484 3525688648 39'0128184 11'5028348'0006570302 1523 2319529 3532642667 39'0256326 115053535'0006565988 1524 2322576 3539605824 39'0384426 11-5078711'0006561680 1525 2325625 3546578125 39'0512483 11'5103876'0006557377 1526 2328676 3553559576' 39'0640499 11'5129030'0006553080 1527 2331729 3567549552 39,0768473 11 5154173'0006548788 1528 2334784 3560558183' 39'0896406 11'5179305'*0006544503 1529 2337841 3574558889 39-1024296: 11-5204425'0006540222 1530 2340900 3581577000 39-1152144 11'5229535'0006535948 1531 2343961 3588604291 39-1279951 11-5254634'0006531679 1532 2347024 3595640768 39 1407716 11'5279722'0006527415 1533 2350089 3602686437 39'1535439 11'5304799'0006523157 1534 2353156 3609741304 39'1663120 11-5329865'0006518905 1535 2356225' 3616805375 39-1790760 11'5354920'0000514658 1536 2359256 3623878656 39-1918359 11-5379965'0006510417 1537 2362369 3630961153 39'2045915 11'5404998'0006506181 1538 2365444 3638052872 39-2173431' 11'5430021'0006.501951 1539. 2368521 3645153819 39-2300905 11'5455033'0006497726 1540 2371600 3652264000 39'2428337 11-5480034 -0006493506 1541 2374681 3659383421 39'2555728 11-5505025'0006489293 1542 2377764 3666512088 39-2683078 11'5530004'0006485084 1543'2380849 3673650007 39'2810387 11'5554972'0006480881 1544 2383936 3680797184 39'2937654 11'5579931'0006476684 1545 2387025 3687953625 39'3064880 11'5604878'0006472492 Page 125 TABLE OF SQUARES, CUBES, SQUARE AND CUBE ROOTS. 125 Number. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 154 6 2390116 3695119336 39-3192065 11.5629815.0006468305 1547 2393209 3702294323 39'3319208 11'5654740 *0006464124 1548 2396304 3709478592 39-3446311 11-5679655'0006459948 1549 2399401 3716672149 39-3573373 11 5704559'00064557788 1550 2402500 3723875000 39'3700394 11'5729453 *0006451613 1551 2405601 3731087151 39'3827373 11-5754336'0006447453 1552 2408704 3738308608 39-3954312 11-5779208 *0006443299 1553 2411809 3745539377 39'4081210 11'5804069 -0006439150 lc554 2414916 3752779464 39'4208067 11'5828919 *0006435006 15o55 2418025 3760028875 39'4334883 11-5853759.0006430868 1556 2421136 3767287616 39'4461658 11 5878588.0006426735 1557 2424249 3774555693 39-4588393 11-5903407'0006422608 1558 2427364 3781833112 39-4715087 11-5928215 *0006418485 1559 2430481 3789119879 39-4841740 11'5953013.0006414368 1560 2433600 3796416000 39-4968353 11'5977799'0006410256 1561 2436721 3803721481 39-5094925 11-6002576.0006406150 1562 2439844 3811036328 39-5221457 11-6027342 *0006402049 1563 2442969 3818360547 39-5347948 11'6052097'0006397953 1564 2446096 3825641444 39'5474399 11-6076841'0006393862 1565 2449225 3833037125 39-5600809 11-6101575'0006389776 1566 2452356 3840389496 39'5727179 11-6126299'0006385696 1567 2455489 3847751263 39-5853508 11-6151012 *0006381621 1568 2458624 3855123432 39-5979797 11'6175715'0006377551 1569 2461761 3862503009 39-6106046 11-6200407 *0006373486 15-70 2464900 3869883000 39-6232255 11'6225088.0006369427 1571 2468041 3877292411 39'6358424 11'6249759'0006365372 15 72 2471184 3884701248 39.6484552 11-6274420.0006361323 1573 24743'29 3892119157 39-6610640 11-6299070'0006357279 1574 2477476 3899547224 39-6736688 11-6323710'0006353240 1575 2480625 3906984375 39-6862696 11-6348339 *0006349206 1576 2483776 3914430976 39-6988665 11-6372957'0006345178 1577 2486929 3921887033 39-7114593 11-6397566'0006341154 1578 2490084 3929352552 39-7240481 11-6422164 *0006337136 1579 2493241 3'936827539 39'7366329 11-6446751'0006333122 1580 2496400 3944312000 39-7492138 11'6471329'0006329114 1581 2499561 3951805941 39-7617907 11'6495895'0006325111 1582 2502724 3959309368 39'7743636 11'6520452.0006321113 1583 2505889 3966822287 39'7869325 11'6544998'0006317119 1584 2509056 3974344704 39-7994976 11-6569534'0006313131 1585 2512225 3981876625 39-8120585 11-6594059'0006309148 1586 2515396 3989418056 39-8246155 11'6618574'0006305170 1587 2518569 3996969003 39-8371686 11-6643079'0006301197 1588 2521744 4004529472 39'8497177 11'6667574'0006297229 1589 2524921 4012099469 39-8622628 11-6692058'0006293266 1590 2528100 4014679000 39-8748040 11'6716532'0006289308 1591 2531281 4027268071 39-8873413 11'6740996'0006285355 1592 2534464 4034866688 39'8998747 11-6765449'0006281407 1593 2537649 4042474857 39-9124041 11'6789892'0006277464 1594 2540836 4050092584 39'9249295 11-6814325'0006273526 1595 2544025 4057719875 39-9374511 11-6838748 -0006269592 1596 2547216 4065356736 39'9499687 11-6863161'0006265664 1597 2550409 4073003173 39-9624824 11'6887563 -0006261741 1598 2553604 4080659192 39-9749922 11'6911955'0006257822 1599 2556801 4088324799 39'9874980 11'6936337'0006253909 1600 2560000 4096000000 40'0000000 11'6960709'0006250000 To find the square or cube root of a number consisting of integers and decimals. RULE.-Multiply the difference between the root of the integer part of the given number, and the root of the next higher integer number, by the decimal part of the given number, and add the L2 Page 126 126 THE PRACTICAL MODEL CALCULATOR. product to the root of the given integer number; the sum is the root required. Required the square root of 20'321. Square root of 21 = 4'5825 Do. 20 = 4'47'21 1104 x'321 + 4'4721 = 4'5075384, the square root required. Required the cube root of 16'42. Cube root of 17 = 2'5712 Do. 16 = 2'5198 0514 x'42 + 2'5198 = 2'541388, the cube root required. To find the squares of numbers in arithmetical progression; or, to extend the foregoing table of squares. RULE.-Find, in the usual way, the squares of the first two numbers, and subtract the less from the greater. Set down the square of the larger number, in a separate column, and add to it the difference already found, with the addition of 2, as a constant quantity; the product will be the square of the next following number. The square of 1500..................= 2250000........ 2250000 The square of 1499..................= 2247001 Difference...... 2999 + 2 = 3001 The square of 1501........................................ 2253001 Difference...... 3001 + 2 = 3003 The square of 1502................................... 2256004 To find the square of a greater number than is contained in the table. RULE 1.-If the number required to be squared exceed by 2, 3,4, or any other number of times, any number contained in the table, let the square affixed to the number in the table be multiplied by the square of 2, 3, or 4, &c., and the product will be the answer sought. Required the square of 2595. 2595 is three times greater than 865; and the square of 865, by the table, is 748225. Then, 748225 x 32 = 6734025. RULE 2.-If the number required to be squared be an odd nulmber, and do not exceed twice the amount of any number contained in the table, find the two numbers nearest to each other, wh-ich, added together, make that sum; then the sum of the squares of these two numbers, by the table, multiplied by 2, will exceed the square required by 1. Required the square of 1865. The two nearest numbers (932 + 933) = 1865. Then, by table (9322 = 868624) + (9332 = 870489) = 1739113 x 2 - 3478226 - 1 = 3478225. Page 127 RULES FOR SQUARES, CUBES, SQUARE ROOTS, ETC. 127 To find the cube of a greater number thlcan is contaiined in the table. RULE.-Proceed, as in squares, to find how many times the number required to be, cubed exceeds the number contained in the table. Multiply the cube of that number by the cube of as many times as the number sought exceeds the number in the table, and the product will be the answer required. Required the cube of 3984. 3984 is 4 times greater than 996; and the cube of 996, by the table, is 988047936. Then, 988047936 x 43 = 63235067904. To find the square or cube root of a higher number than is in the table. RULE. —Refer to the table, and seek in the column of squares or cubes the number nearest to that number whose root is sought, and the number from which that square or cube is derived will be the answer required, when decimals are not of importance. Required the square root of 542869. In the Table of Squares, the nearest number is 543169; and the number firom which that square has been obtained is 737. Therefore, /542869 = 737 nearly. To find more nearly the cube root of a higher number than is in the table. RULE. —Ascertain, by the table, the nearest cube number to the number given, and call it the assumed cube. Multiply the assumed cube, and the given number, respectively, by 2; to the product of the assumed cube add the given number, and to the product of the given number add the assumed cube. Then, by proportion, as the sum of the assumed cube is to the sum of the given number, so is the root of the assumed cube to the root of the given number. Required the cube root of 412568555. By the table, the nearest number is 411830784, and its cube root is 744. Therefore, 411830784 x 2 + 412568555 = 1236230123. And, 412568555 x 2 + 411830784 = 1236967894. Hence, as 1236230123: 1236967894:::744: 744'369, very nearly. To find the square or cube root of a number containing decimals. RULE.-Subtract the square root or cube root of the integer of the given number from the root of the next higher number, and multiply the difference by the decimal part. The product, added to the root of the integer of the given number will be the answer required. Required the square root of 321-62. 321. = 17'9164729, and V322 = 17'9443584; the difference ('0278855) x'62 + 17'9164729 = 17'9337619. Page 128 128 THE PRACTICAL MIODEL CALCULATOR. To obtain the square root or cube root of a number containing decimals, by inspection. RULE.-The square or cube root of a number containing decimals may be found at once by inspection of the tables, by taking the figures cut off in the number, by the decimal point, in pairs if for the square root, and in triads if for the cube root. The following example will show the results obtained, by simple inspection of the tables, from the figures 234, and from the numbers formed by the addition of the decimal point or of ciphers. Number. Square Root. Cube Root..00234.0483735465* *132761439t *0234 -152970585 *284t'2340 *483735465. 61622401 2.34 1'52970585 1'32761439 23-40 4'83735465 2'860 234 15 2970585 6-1622401 2340 48 3735465 13-2761439 23400 152'970585 28'60 To find the cubes of numbers in arithmetiealprogression, or to extend the preceding table of cubes. RULE.-Find the cubes of the first two numbers, and subtract the less from the greater. Then, multiply the least of the two numbers cubed by 6, add the product, with the addition of 6 as a constant quantity, to the difference; and thus, adding 6 each time to the sum last added, form a first series of differences. To form a second series of differences, bring down, in a separate column, the cube of the highest of the above numbers, and add the difference to it. The amount will be the cube of the next general number. Required the cubes of 1501, 1502, and 1503. First series of differences. Second series of differences. By Tab. 1500 = 3375000000 Then, 3375000000 Cube of 1500 1499 = 3368254499 Diff. for 1500 - 6754501 6745501 difference. 3381754501 Cube of 1501 1499 X 6+ 6 - 9000 Diff. for 1501 - 6763507 6754501 diff. of 1500 3388518008 Cube of 1502 9000 + 6 = 9006 Diff. for 1502 6772519 6763507 diff. of 1501 3395290527 Cube of 1503 9006 + 6 = 9012 &c., &c. 6772519 diff. of 1502 &c., &c. * Derived from.002340 by means of 2340. t Derived from'002340 by means of 2340. $ The nearest result by simple inspection is obtained for.023 by 23. But four places correct can always be obtained by looking in the table of cubes for the nearest triad or triads, in this instance for 23400; the cube beginning with the figures 23393 is that of 2860, whence.2860 is true to the last place, and is afterwards substituted. Page 129 TABLE OF THE FOURTH AND FIFTH POWERS OF NUMBERS. 129 TABLE of the Fourth and Fifth Powers of Numbers. Number. 4th Pow. hPower. 5t h P ower. 4th Power. 5th Power. 1 1 1 76 33362176 2535525376 2 16 32 77 35153041 2706784157 3 81 243 78 37015056 2887174368 4 256 1024 79 38950081 3077056399 5 625 3125 80 40960000 3276800000 6 1296 7776 81 43046721 3486784401 7 2401 16807 82 45212176 3707398432 8 4096 32768 83 47458321 3939040643 9 6561 59049 84 49787136 4182119424 10 19000 100000 85 52200625 4437053125 11 14641 161051 86 54708016 4704270176 12 20736 248832 87 57289761 4984209207 13 28561 371293 88 59969536 5277319168 14 38416 537824 89 62742241 5584059449 15 50625 759375 90 65610000 5904900000 16 65536 1048576 91 68574961 6240321451 17 83521 1419857 92 71639296 6590815232 18 104976 1889568 93 74805201 6596883693 19 130321 2476099 94 78074896 7339040224 20 160000 3200000 95 81450625 7737809375 21 194481 4084101 96 84934656 8153726976 22 234256 5153632 97 88529281 8587340257 23 279841 6436343 98 92236816 9039207968 24 331776 7962624 99 96059601 9509900499 25 390625 9765625 100 100000000 10000000000 26 456976 11881376 101 104060401 10510100501 27 531441 14348907 102 108243216 11040808032 28 614656 17210368 103 112550881 11592740743 29 707281 20511149 104 116985856 12166529024 30 810000 24300000 105 121550625 12762815625 31 923521 28629151 106 126247696 13382255776 32 1048576 33554432 107 131079601 14025517307 33 1185921 39135393 108 136048896 14693280768 34 1336336 45435424 109 141158161 15386239549 35 1500625 52521875 110 146410000 16105100000 36 1679616 60466176 111 151807041 16850581551 37 1874161 69343957 112 157351936 17623416832 38 2085136 79235168 113 163047361 18424351793 39 2313441 90224199 114 168896016 19254145824 40 2560000 102400000 115 174900625 20113571875 41 2825761 115856201 116 181063936 21003416576 42 3111696 130691232 117 187388721 21924480357 43 3418801 147008443 118 193877776 22877577568 44 3748096 164916224 119 200533921 23863536599 45 4100625 184528125 120 207360000 24883200000 46 4477456 205962976 121 214358881 25937424601 47 4879681 229345007 122 221533456 27027081632 48 5308416 254803968 123 228886641 28153056843 49 5764801 282475249 124 236421376 29316250624 50 6250000 312500000 125 244140625 30517578125 51 6765201 345025251 126 252047376 31757969376 52 7311616 380204032 127 260144641 33038369407 53 7890481 418195493 128 268435456 34359738368 54 8503056 459165024 129 276922881 35723051649 55 9150625 503284375 130 285610000 37129300000 56 9834496 550731776 131 294499921 38579489651 57 10556001 601692057 132 303595776 40074642432 58 11316496 656356768 133 312900721 41615795893 59 12117361 714924299 134 322417936 43204003424 60 12960000 777600000 135 332150625 44840334375 61 13845841 844596301 136 342102016 46525874176 62 14776336 916132832 137 352275361 48261724457 63 15752961 992436543 138 362673936 50049003168 64 16777216 1073741824 139 373301041 51888844699 65 17850625 1160290625 140 384160000 53782400000 66 18974736 1252332576 141 395254161 55730836701 67 20151121 1350125107 142 406586896 57735339232 68 21381376 1453933568 143 418161601 59797108943 69 22667121 1564031349 144 429981696 61917364224 70 24010000 1680700000 145 442050625 64097340625 71 25411681 1804229351 146 454371856 66338290976 72 26873856 1934917632 147 466948881 68641485507 73 28398241 2073071593 148 479785216 71008211968 74 29986576 2219006624 149 492884401 73439775749 75 31640625 2373046875 150 506250000 75937500000 9 Page 130 130 THE PRACTICAL MODEL CALCULATOR. TABLE of Hyperbolic Loyarithms. N. Logarithm. N. Logarithm. N. Logarithm. N. Logarithm. 1.01.0099503 1'58'4574248 2-15 -7654678 2-72 1-0006318 1-02 ~0198026 1 59.4637340 2.16.7701082 2-73 1-0043015 1-03 ~0295588 1 60.4700036 2.17.7747271 2-74 1-0079579 1-04 *0392207 1'61'4762341 2-18'7793248 2'75 1-0116008 1-05 ~0487902 1 62'4824261 2-19'7839015 2.76 1 0152306 106'0582689 1-63'4885800 2 20'7884573 2-77 1 0188473 1-07'0676586 1-64 *4946962 2-21'7929925 2-78 1-0224509 1-08'0769610 1-65'5007752 2-22'7975071 2-79 1-0260415 1-09'0861777 1'66'5068175 2'23'8020015 2-80 1-0296194 1-10 ~0953102 1'67'5128236 2-24'8064758 2-81 1-0331844 1 11 ~1043600 1 68'5187937 2-25'8109302 2-82 1 0367368 1-12 ~1133287 1-69'5247285 2'26'8153648 2'83 1 0402766 1-13 *1222176 1'70'5306282 2'27'8197798 2-84 1 0438040 1-14 ~1310283 1'71'5364933 2-28'8241754 2-85 1 0473189 1~15 1397619 1'72'5423242 2 - 29 8285518 2'86 1 0508216 1-16 *1484200 1'73 5481214 2.30'8329091 2-87 1-0543120 1-17'1570037 1-74'5538851 2-31'8372475 2-88 1-0577902 118'1655144 1'75'5596157 2-32'8415671 2'89 1 0612564 1-19'1739533 1'76'5653138 2'33'8458682 2-90 1-0647107 1 20'1823215 1'77'57 09795 2'34'8501509 2'91 1 0681530 1-21'1906203 1'78'5766133 2'35'8544153 2-92 1-0715836 122 1988598 1 -7 9'5822156 2-36'8586616 2-93 1 0750024 1 -23'2070141 1-80 *5877866 2-37'8628899 2-94 1 0784095 1 -2f4 *2151113 1 81'5933268 2-38'8671004 2-95 1 -0818051 1 5.2231435 1-82.5988365 2-39.8712933 2-96 1-0851892 1 26'2311117 1-83'6043159 2'40 *8754687 2-97 1 -0885619 1-27'2390169 1 84'6097655 2-41 *8796267 2-98 1-0919233 1-28 *2468600 1 85'6151856 2-42 *883767 5 2-99 1 0952733 1 -29 *2546422 1 -86 6205764 2.43.8878912 3.00 1 0986123 1 -30'2623642 1-87'6259384 2-44'8919980 3-01 1-1019400 131'2700271 1'88'6312717 2-45'8960880 3-02 1-1052568 1-32 *2776317 1-89'6365768 2-46'9001613 3'03 1-1085626 1-33'2851789 1-90'6418538 2-47 *9042181 3'04 1-1118575 1-34'2926696 1-91'6471032 2-48'9082585 3805 1-1151415 1-35 ~3001045 1 92'6523251 2-49'9122826 3 06 1-1184149 1-36 *3074846 1-93'6575200 2'50'9162907 3'07 1-1216775 1-37'3148107 1 94'6626879 2-51'9202827 3-08 1-1249295 1-38 ~3220834 1-95 *6678293 2-52'9242589 3 09 1-1281710 1-39'3293037 1'96'6729444 2-53'9282193 3'10 1-1314021 1-40 ~3364722 1'97'6780335 2-54'9321640 3'11 1 1346227 1-41 *3435897 1-98'6830968 2'55'9360933 3'12 1'1378330 1 42'3506568 1 99'6881346 2'56'9400072 3'13 1'1410330 1 43'3576744 2'00'6931472 2-57'9439058 3-14 1-1442227 1-44 ~3646431 2'01'6981347 2-58'9477893 3-15 1-1474024 1-45 ~3715635 2-02'7030974 2'59'9516578 3-16 1-1505720 1-46 ~3784364 2-03'7080357 2-60'9555114 3-17 1-1537315 1'47 *3852624 2'04'7129497 2-61'9593502 3-18 1 1568811 1-48 *3920420 2-05'7178397 2'62'9631743 3-19 1-1600209 1-49 ~3987761 2'06'7227059 2-63 *9669838 3-20 1'1631508 1-50'4054651 2'07'7275485 2-64'9707789 3-21 1 1662709 51 a'4121096 2-08'7323678 2-65'9745596 3-22 1-1693813 1-52 ~4187103 2'09'7371640 2-66'9783261 3-23 1-1724821 1-53'4252677 2-10'7419373 2-67'9820784 3'24 1-175 5733 1 54'4317824 2-11'7466879 2 68'9858167 3-25 1-1786549 1 -55'4382549 2-12'7514160 2-69'9895411 3-26 1-1817271 1-56'4446858 2-13'7561219 2'70'9932517 3-27 1'1847899 1-57'4510756 2-14'76080-58 271'9969486; 3-28 1-1878434 Page 131 TABLE OF HYPERBOLIC LOGARITHIS. 181 N. Logarithm. N. Logarithm. N. Logarithm. N. Logarithm. 8 29 11908875 3 91 1-3635373 4'53 1-5107219 5-15 1'6389967 130 1-1939224 3'92 1-3660916 4-54 1-5129269 5-16 1-640936.5 13 31 1-1969481 3'93 1'3686394 4-55 1-5151272 5-17 1 64 8726 0.32 1-1999647 3-94 1-3711807 4656 1-5173226 5-18 1-6448050 3-83 1-2029722 3-95 1-3737156 4-57 1-5195132 5'19 1-6467336 3 3.4 1'2059707 3-96 1'3762440 4'58 1'5216990 5-20 1-6486586 3.35 1'2089603 3'97 1'3787661 4'59 1-5238800 5.21 1'650578() i 3.36 1'2119409 3'98 1'3812818 4'60 1'5260563 5.22 1'652497-4 3.37 1'2149127 3'99 1'3837912 4'61 1'5282278 5.23 16544112!3.*38 1.2178757 4.00 1.3862943 4.62 1.5308947 5.24 1.6o632 13 39 1.2208299 4.01 1.3887912 463 1.5325568 5.25 1.658'2280 340 1-2237754 4.02 13912818 4-64 15347143 5-26 1.6601310 3- 41 1-2267122 4.03 1-3937663 4.65 1-5368672 5-27 1.66203(0 3-42 1-2296405 4.04 1.3962446 4-66 1-5390154 5-28 1.6639260 3-43 1-2325605 4.05 1.3987168 4-67 1-5411590 5-29 1.6658152 3.44 1-2354714 4-06 1-4011829 4-68 1-5432981 5.30 1.667 0;J S 345 1-2383742 4.07 1-4036429 4-69 1-5454325 5-31 1.669918 I 3.46 1-2412685 4.08 1-4060969 4.70 1-5475625 5.32 1.67147133 3.47 1-2441545 4.09 1-4085449 4-71 1-5496879 5-33 1 6 7oo51 3.48 1.2470322 4.10 1-4109869 4.72 1-5518087 5-34 1 6752256 3.49 1.2499017 4.11 1-4134230 4-73 1-5539252 5-35 1 67709651 3 50 1 2527629 4.12 1-4158531 4.74 1-5560371 5-36 1 -67896:09 3.51 1.2556160 4-13 1.4182774 4.75 1-5581446 5.37 1-6808278 3.5 2 1-2584609 4-14 1.4206957 4.76 1*5602476 5-38 1.6826880 3-53 1.2612978 4-15 1-4231083 4.77 1.5623462 5.39 1-684154o3 3'54 1-2641266 4.16 1-4255150 4-78 1-5644405 5.40 1-686389' 3o.55 1.2669475 4-17 1-4279160 4-79 1.5665304 5-41 1-6884'01 |3 56 1-2697605 4'18 1-4303112 4-80 1-5686159 5-42 1.69009-58 3537 1272.5655 4.19 1-4327007 4.81 1-5706971 5-43 1 6919,-91 i 358 1.2753627 420 1-4350845 4-82 1 5727739 5-44 1.6937r0 3'59 1-2781521 1 4'21 1'4374626 4.83 1.5748464 5.45 1 69561.5-I 0 12809338 4.22 1!4398351 4-84 1-5769147 5-46 1.6974457 361 1-2837077 4'23 1-4422020 4.85 1-5789787 5.47 1 6992786 3-62 1.2864740 4'24 1-4445632 4.86 1.5810384 5648 1'7011051 3 63 1-2892326 4'25 1.4469189 4-87 1.5830939 5-49 1.7029280 3'64 1-2919836 4-26 1'4492691 4.88 1-5851452 5.50 1-7047481 3'65 1-2947271 4-27 1'4516138 4-89 1-5871923 5-51 1-7065646 3-66 1-2974631 4-28 1.4539530 4.90 1.5892352 5-52 1 08377,,8 3-67 1.3001916 4.29 1-4562867 4-91 1-5912739 5-53 1-71018 78 3.68 1-3029127 4.30 1-4586149 4-92 1-5933085 5-54 1.7119944 3-69 1.3056264 4-31 1-4609379 4-93 1-5953389 5.55 1 7137979 3.70 1 3083328 432 1'4632553 4.94 1-5973653 5-56 1'7155981 371 1.3110318 4-33 1-4655675 4'95 1-5993875 5-57 1'7173950 3' 72 1.3137236 4'34 1-4678743 4'96 1-6014057 5-58 1-7191887 3-73 1-3164082 4-35 1-4701758 4-97 1-6034198 5-59 1-7209792 3-74 1.3190856 4-36 1-4724720 4-98 1-6054298 5-60 1-7227666 3.75 1-3217558 4'37 1-4747630 4-99 1-6074358 5-61 1-7245507 3 76 1.3244189 4-38 1-4770487 5'00 1-6094379 5-62 1 7263316 3.77 1.3270749 4-39 1'4793292 5-01 1'6114359 5.63 1 72810943 78 1-3297240 4.40 1'4816045 5'02 1-6134300 5.64 1'7298840 3'79 1-3323660 4-41 1-4838746 5'03 1-6154200 5.65 1'7316555 3'80 1-3350010 4'42 1-4861396 5'04 1.6174060 5.66 1 7334238 3-81 1-3376291 4-43 1'4883995 5.05 1-6193882 5-67 1- 301891 3'82 1-3402504 4-44 1-4906543 5-06 1-6213664 5-68 1.7369519 3'83 1-3428648 4-45 1-4929040 5'07 1-6233408 5'69 1-73871023'84 1-3454723 4-46 1-4951487 5-08 1-6253112 5'70 1 7404661 3-85 1-3480731 4-47 1'4973883 5'09 1'6272778 5.71 1 7422189 3.86 1 3506671 4-48 1-49962030 5.-10 1-6292405 5-72 1 7439687 3-87 1-3532544 4-49 1-5018527 5*11 1-6311994 5*73 1-745715o 3-88 1-3558351 4-50 1-5040774 5-12 1-6331544 5-74 1-7474591 38S9 1-3584091 4-51 1-5062971 5-13 1-63510.56 5-75 1-7491998 3-90 1 13609765 4-52 1-5085119 5-14 1-6370530 5-76 1-7509374 Page 132 132 THE PRACTICAL MODEL CALCULATOR. N. Logarithm. N. Logarithm. N. I Lo,zarithm. N. Logarithm. 5'77 1-7526720 6-39 1-8547342 7-01 1-9473376 7-63 2-0320878 5'78 1-7544036 6-40 1*8562979 7-02 1-9487632 7-64 2-0333976 5'79 1'7561323 6.41 1:8578592 7-03 1'9501866 7-65 2-0347056 5-80 1-7578579 6-42 1-8594181 7'04 1-9516080 7-66 2-0360119 5'81 1-7595805 6'43 1-8609745 7'05 1-9530275 7-67 2-0373166 5'82 1'7613002 6-44 1-8625285 7'06 1-9544449 7-68 2-0386195 5-83 1'7630170 6'45 1'8640801 7'07 1'9558604 7'69 2'0399207 5'84 1'7647308 6'46 1'8656293 7'08 1'9572739 7'70 2'0412203 5'85 1 7664416 6'47 1'8671761 7'09 1'9586853 7'71 2'0425181 5.86 1.7681496 6.48 1.8687205 7.10 1.9600947 7-72 2-0438143 5'87 1'7698546 6'49 1-8702625 7-11 1-9615022 7'73 2-0451088 5-88 1-7715567 6-50 1-8718021 7-12 1-9629077 7-74 2-0464016 5-89 1-7732559 6-51 1-8733394 7-13 1-9643112 7-75 2-0476928 5'90 1'7749523 6'52 1-8748743 7-14 1-9657127 7-76 2-0489823 5-91 1-7766458 6-53 1-8764069 7-15 1-9671123 7-77 2-0502701 5-92 1-7783364 6-54 1 8779371 7-16 1-9685099 7-78 2'0515563 5-93 1 -7800242 6-55 1 8794650 7'17 1-9699056 7-79 2 0528408 5'94 1-7817091 6'56 1-8809906 7'18 1'9712993 7-80 2-0541237 5'95 1'7833912 6-57 1-8825138 7-19 19726911 7'81 2-0554049 5 96 1'7850704 6-58 1-8840347 7-20 1-9740810 7-82 2-0566845 5-97 1-7867469 6-59 1-8855533 7-21 1-9754689 7-83 2'0579624 5-98 1-7884205 6-60 1-8870696 7-22 1-9768549 7-84 2-0592388 5'99 1 7900914 6'61 1'8885837 7 23 1'9782390 7-85 2-0605135 6'00 1-7917594 6-62 1-8900954 7'24 1'9796212 7-86 2'0617866 6;01 1-7934247 6'63 1 8916048 7 25 1'9810014 7'87 2'0630580 6'02 1'7950872 6'64 1-8931119 7-26 1-9823798 7-88 2-0643278 6;'03 1-7967470 6-65 1-8946168 7-27 1-9837562 7-89 2-0655961 6'04 1 7984040 6'66 1-8961194 7-28 1'9851308 7'90 2-0668627 6;05 1 8000582 6-67 1-8976198 7.29 1'9865035 7'91 2'0681277 6 06 1'8017098 6'68 1.8991179 7'30 1-9878743 7'92 2'0693911 6'07 1-8033586 6-69 1-9006138 7.31 1-9892432 7'93 2-0706530 6'08 1-8050047 6-70 1-9021075 7'32 1 9906103 7-94 2-0719132 G609 1-8066481 6-71 1-9035989 7 33 1 9919754 7'95 2 0731719 6'10 1 8082887 6'72 1 9050881 7.34 1 9933387 7'96 2-0744290 6'11 1 8099267 6'73 1 9065751 7.35 1-9947002 7'97 2 0756845 6'12 1-8115621 6-74 1-9080600 7-36 1'9960599 7'98 2-0769384 6-13 1-8131947 6-75 1 9095425 7'37 1'9974177 7'99 2'0781907 6-14 1 8148247 6'76 1.9110228 7'38 1-9987736 8-00 2'0794415 6'15 1 8164520 6'77 1 9125011 7'39 2'0001278 8'01 2-0806907 6-16 1'8180767 6-78 1-9139771 7-40 2'0014800 8'02 2-0819384 6-17 1-8196988 6-79 1-9154509 7 41 2 0028305 8-03 2'0831845 6'18 1-8213182 6'80 1 9169226 7 42 2'0041790 8'04 2'0844290 6'19 1'8229351 6'81 1'9183921 7'43 2'0055258 8'05 2-0856720 6-20 1'8245493 6'82 1.9198594 7'44 2'0068708 8-06 2-0869135 6'21 1 8261608 6'83 1.9213247 7 45 2'0082140 8'07 2-0881534 6'22 1'8277699 6'84 1'9227877 7-46 2'0095553 8-08 2'0893918 6 23 1'8293763 6-85 1.9242486 7'47 2'0108949 8'09 2 0906287 6'24 1 8309801 6'86 1-9257074 7'48 2'0122327 8-10 2'0918640 6'25 1 8325814 6'87 1-9271641 7'49 2'0135687 8'11 2 0930984 6'26 1-8341801 6'88 1.9286186 7'50 2-0149030 8-12 2'0943306 6'27 1 8357763 6'89 1-9300710 7-51 2'0162354 8'13 2'0955613 6 28 1 8373699 6'90 1 9315214 7-52 2 0175661 8'14 2'0967905 6'29 1 8389610 6'91 1 9329696 7'53 2'0188950 8'15 2'0980182 6'30 1 8405496 6'92 1.9344157 7 54 2 0202221 8-16 2'0992444 6'31 1 8421356 6'93 1'9358598 7'55 2-0215475 8'17 2'1004691 6'32 1-8437191 6'94 1'9373017 7'56 2'0228711 8'18 2-1016923 6'33 1 8453002 6'95 1'9387416 7'57 2'0241929 8-19 2'1029140 6'34 1 8468787 6'96 1 9401794 7-58 2'0255131 8'20 2'1041341 6'35 1'8484547 6'97 1-9416152 7-59 2.0268315 8'21 2'1053529 6'36 1-8500283 6-98 1 9430489 7'60 2 0281482 8 22 2 1065702 6'37 1-8515994 6'99 1 9444805 7*61 2-0294631 8'23 2-1077861 6'38 1-8531680 7'00 1-9459101 7-62 2-0307763 8 24 2-1089998 Page 133 TABLE OF HYPERBOLIC LOGARITHMS. 133 N. Logarithm. N. Logarithm. | N. Logarithm. N. Logarithm. 8 25 2'1102128 8.69 2'1621729 9-13 2.2115656 9'57 2.2586332 8'26 2'1114243 8'70 2'1633230 9'14 2.2126603 958 20-2596776 8'27 2'1126343 8'71 2.164471.8 9'15 2-2137538 9'59 2'2607209 8'28 2'1138428 8'72 2-1656192 9-16 2-2148461 9'60 2'2617631 8'29 2'1150499 8'73 2'1667653 9'17 2.2159372 9'61 2 2628042 8'30 2'1162555 8'74 2.1679101 9'18 2.2170272 9-62 2-2638442 8'31 2'1174596 8'75 2'1690536 9'19 2-2181160 9-63 2-2648832 8'32 2'1186622 8'76 2'1701959 9'20 2-2192034 9-64 2-2659211 8-33 2.1198634 8'77 2'1713367 9'21 2.2202898 9.65 2'2669579 8-34 2'1210632 8'78 2'1724763 9'22 2-2213750 9-66 202679936 8-35 2-1222615 8'79 2-1736146 9-23 2-2224590 9-67 2-2690282 8-36 2'1234584 8'80 2.1747517 9'24 2.2235418 9.68 2-2700618 8'37 2'1246539 8.81 2-1758874 9'25 2.2246235 9.69 2.2710944 8'38 2'1258479 8.82 2-1770218 9 26 2.2257040 9'70 2-272 1258 8'39 2'1270405 8-83 2.1781550 9'27 2 2267833 9-71 2-27 i31562 8'40 2'1282317 8.84 2.1792868 9'28 2.2278615 9'72 2 2741856 8'41 2'1294214 8-85 2-1804174 9-29 2.2289385 9-73 2-2752138 8 42 2-1306098 8-86 2-1815467 9'30 2 2300144 9-74 2-2762411 8-43 2-1317967 8-87 2.1826747 9'31 2-2310890 9-75 2 277 92673 8-44 2-1329822 8-88 2-1838015 9-32 2-2321626 9-76 2 2782924 8-45 2'1341664 8 89 2-1849270 9'33 2-2332350 9-77 2-2793165 8-46 2'1353491 8'90 2-1860512 9'34 2-2343062 9'78 2-2803395 8-47 2-1365304 8-91 2-1871742 9 35 2'2353763 9-79 22 813614 8'48 2'1377104 8-92 2-1882959 9'36 2 2364452 9.80 2 2823823 8.49 2'1388889 8-93 2-1894163 9'37 2.2375130 9.81 2-2834022 8'50 2-1400661 8-94 2-1905355 9'38 2'2385797 9.82 2-2844211 8-51 2-1412419 8-95 2-1916535 9-39 2 2396452 9.83 2-2854389 8-52 2-1424163 8-96 2-1927702 9'40 2-2407096 9.84 2-2864556 8'53 2-1435893 8-97 2-1938856 9-41 2.2417729 9.85 2 2874714 8'54 2-1447609 8-98 2.1949998 9 42 2 2428350 9.86 2-2884861 8'55 2'1459312 8-99 2-1961128 9-43 2-2438960 9-87 2.2894998 8-56 2-1471001 9'00 2-1972245 9-44 2-2449559 9-88 2-2905124 8-57 2'1482676 9'01 2.1983350 9'45 2-2460147 9.89 2.2915241 8-58 2'1494339 9'02 2-1994443 9-46 2-2470723 9'90 2'2925347 8'59 241505987 9'03 2-2005523 9-47 2-2481288 9'91 2-2635443 8'60 2.1517622 9.04 2-2016591 9-48 2-2491843 9.92 2-2945529 8'61 2-1529243 9-05 2-2027647 9'49 2.2502386 9-93 2-2955604 8'62 2-1540851 9-06 2-2038691 9'50 2.2512917 9-94 2 2965670 8-63 2-1552445 9.07 2-2049722 9-51 2.2523438 9.95 2 0975,-o 8'64 2-1564026 9-08 2-2060741 9-52 2'2533948 9-96 2'2985770 8-65 2-1575593 9'09 2-2071748 9-53 2-2544446 9-97 2-2995806 8-66 2-1587147 9-10 2 2082744 9-54 2 2554934 9-98 2'3005831 8-67 2-1598687 9-11 2-2093727 9-55 2-2565411 999 2-3015846 8'68 2-1610215 9'12 2-2104697 9-56 2-2575877 10'00 2-3025851 Logarithms were invented by Juste Byrge, a Frenchman, and not by Napier. See " Biographie Universelle," " The Calculus of Form," article 822, and "The Practical, Short, and Direct Methcd of Calculating the Logarithm of any given Number and the Nrumber corresponding to any given Logarithm," discovered by Oliver Byrne, the author of the present work. Juste Byrge also invented the proportional compasses, and was a profound astronomer and:l:mthematician. The common Logarithm of a number multiplied by 2'302585052994 gives the hyperbolic Logarithm of that number. The common Logarithm of 2'22 is'346353.. 2'302585 x'3463533 -= 7975071 the hyperbolic Logarithm. The application of Logarithms to the calculations of the Engineer will be treated of hereafter. M Page 134 134 THE PRACTICAL MIODEL CALCULATOR. COMBINATIONS OF ALGEBRAIC QUANTITITIES. THE following practical examples will serve to illustrate the method of combining or representing numbers or quantities algebraically; the chief object of which is, to help the memory with respect to the use of the signs and letters, or symbols. Leta=6, b=4, c = 3, d=2, e = 1, andf = O. Then will, (1) 2a+ b = 12 + 4 = 16. (2) ab + 2c- d = 24 + 6 - 2 = 28. (3) a2 - b2 + e +f= 36 - 16 + 1+ 0 = 21. (4) b2 x (a - b)= 16 x (6 -4) = 16 x 2 = 32. (5) 3abe - 7de = 216 - 14 = 202. (6) 2 (a - b) (5c - 2d)= (12 - 8) x (15 - 4) = 44. e2 — e2 9 — 1 (7) +f x (a - c) -2 + 0x (6 - 3) =4 x 3=12. (8) (a2 - 2b2) + d - f = (36 - 32) + 2 - = 4. (9) 3ab - (a - b - e + d) = 72 - 1 = 71. (10) 3ab - (a - 6 - c d) = 72 + 3 = 75. V 2abe V 144 (11) (ab-d) x ( + d) v (24 - 8) x (3 + 2)= 15. In solving the following questions, the letters a, b, c, &c. are supposed to have the same values as before, namely, 6, 4, 3, &c.; blut any other values might have been assigned to them; therefore, do not suppose that a must necessarily be 6, nor that b must be 4, for the letter a may be put for any known quantity, number, or mIagnitude whatever; thus a may represent 10 miles, or 50 pounds, or any number or quantity, or it may represent 1 globe, or 2 cubic feet, &c.; the same may be said of b, or any other letter. (1) a + b- e= 7. (6) 4 (a2 - 62) (c - e) =160. (2) 3be - d + e = 35. (7) + d x (2 + e2) (3) 2a2 + -2 - d + f=79 (8) V (2a2 2d2)+ be -f= 20. (4) x (b-e + d) =27. (9) 4a2b- (c2-d - e) = 570. 0 4a2 a (5) 52d - a2 + 4de 62. (10) V(10da-4ed) X d c In the use of algebraic symbols, 3 V 4a - b signifies the same thing as 3 (4a - b)3. 1 1 1 1 4 (e + d)2 (a + b)3, or 4 x e + d x a + b3, signifies the same thing as 4 / + c ~ v a + b. Page 135 135 THE STEAM ENGINE. THE particular example which we shall select is that of an engine having 8 feet stroke and 64 inch cylinder. The breadth of the web of the crank at the paddle centre is the breadth which the web would have if it were continued to the paddle centre. Suppose that we wished to know the breadth of the web of crank of an engine whose stroke is 8 feet and diameter of cylinder 64 inches. The proper breadth of the web of crank at paddle centre would in this case be about 18 inches. To find the breadth of crank at paddle centre. —Multiply the square of the length of the crank in inches by 1'561, and then multiply the square of the diameter of cylinder in inches by'1235; multiply the square root of the sum of these products by the square of the diameter of the cylinder in inches; divide the product by 45; finally extract the cube root of the quotient. The result is the breadth of the web of crank at paddle centre. Thus, to apply this rule to the particular example which we have selected, we have 48 = length of crank in inches. 48 2304 1'561 = constant multiplier. 3596'5 505'8 found below. 4102'3 64 = diameter of cylinder. 64 4096 1235 = constant multiplier. 505'8 and V4102'3 = 64'05 nearly. 4096 = square of the diameter of the cylinder. 45) 262348'5 5829'97 and /5829'97 = 18 nearly. Suppose that we wished the proper thickness of the large eye of crank for an engine whose stroke is 8 feet and diameter of cylinder 64 inches. The proper thickness for the large eye of crank is 5'77 inches. Page 136 136 THE PRACTICAL MODEL CALCULATOR. RULE.- To find the thickness of large eye of erank. —Multiply the square of the length of the crank in inches by 1'561, and then multiply the square of the diameter of the cylinder in inches by'1235; multiply the sum of these products by the square of the diameter of the cylinder in inches; afterwards, divide the product by 1828'28; divide this quotient by the length of the crank in inches; finally extract the cube root of the quotient. The result is the proper thickness of the large eye of crank in inches. Thus, to apply this rule to the particular example which we have selected, we have 48 = length of crank in inches. 48 2304 1'561 constant multiplier. 3596'5 505'8 4102'3 64 = diameter of cylinder in inches. 64 4096 1235 = constant multiplier. 505-8 4102'3 4096 = square of diameter. 48) 16803020S8 1828'28) 350062'94 191.47 and -v191'47 = 5'77 nearly. The proper thickness of the web of crank at paddle shaft centre is the thickness which the web ought to have if continued to centre of the shaft. Suppose that it were required to find the proper thickness of web of crank at shaft centre for an engine whose stroke is 8 feet and diameter of cylinder 64 inches. The proper thickness of the web at shaft centre in this case would be 8-97 inches. RULE.- To find the thickness of the web of crank at paddle shaft centre.-Multiply the square of the length of crank in inches by 1-561, and then multiply the square of the diameter in inches by 1235; multiply the square root of the sum of these products by the square of the diameter of the cylinder in inches; divide this quotient by 360; finally extract the cube root of the quotient. The result is the thickness of the web of crank at paddle shaft centre in inches. Thus, to apply the rule to the particular example which wve have selected, we have Page 137 THE STEAM ENGINE. 137 48 = length of crank in inches. 48 2304 1'561 = constant multiplier. 3596'5 505'8 4102.3 64 = diameter of cylinder. 64 4096 1235 = constant multiplier. 505'8 And V 4102'3 = 64'05 nearly. 4096 = square of diameter. 360) 262348.5 728'75 And v 782'75 = 9 nearly. Suppose that it were required to find the proper diameter for the paddle shaft journal of an engine whose stroke is 8 feet and diameter of cylinder 64 inches. The proper diameter of the paddle shaft journal in this case is 14'06 inches. RULE.-To find the diameter of the paddle shaft journal.-Multiply the square of the diameter of cylinder in inches by the length of the crank in inches; extract the cube root of the product; finally multiply the result by'242. The final product is the diameter of the paddle shaft journal in inches. Thus, to apply this rule to the particular example which we have before selected, we have 64 = diameter of cylinder in inches. 64 4096 48 = length of crank in inches. 196608 and /196608 = 58'148 but 58'148 x'242 = 14'07 inches. Suppose it were required to find the proper length of the paddle shaft journal for an engine whose stroke is 8 feet, and diameter of cylinder 64 inches. The proper length of the paddle shaft journal would be, in this case, 17'59 inches. The following rule serves for engines of all sizes: RULE. —To find the length of the paddle shaft journal.-Multiply the square of the diameter of the cylinder in inches by the length of the crank in inches; extract the cube root of the quotient; multiply the result by'303. The product is the length of the x 2 Page 138 1[8 THE PRACTICAL MODEL CALCULATOR. paddle shaft journal in inches. (The length of the paddle shaft journal is 14 times the diameter.) To apply this rule to the example which we have selected, we have 64 = diameter of cylinder in inches. 64 4096 48 = length of crank in inches. 196608 and v 196608 = 58'148.'. length of journal = 58'148 x'303 = 17'60 inches. We shall now calculate the proper dimensions of some of those parts which do not depend upon the length of the stroke. Suppose it were required to find the proper dimensions of the respective parts of a marine engine the diameter of whose cylinder is 64 inches. Diameter of crank-pin journal = 90'9 inches, or about 9 inches. Length of crank-pin journal = 10-18 inches, or nearly 101 inches. Breadth of the eye of cross-head = 2'64 inches, or between 2and 2{ inches. Depth of the eye of cross-head = 18'37 inches, or very nearly 181 inches. Diameter of the journal of cross-head = 5'5 inches, or 5~ inches. Length of journal of cross-head = 6'19 inches, or very nearly 61 inches. Thickness of the web of cross-head at middle = 4'6 inches, or somewhat more than 41 inches. Breadth of web of cross-head at middle = 17'15 inches, or between 17 —, and 17 inches. Thickness of web of cross-head at journal = 3'93 inches, or very nearly 4 inches. Breadth of web of cross-head at journal = 6'46 inches, or nearly 62 inches. Diameter of piston rod = 6'4 inches, or 6] inches. Length of part of piston rod in piston = 12-8 inches, or 124 inches. Major diameter of part of piston rod in cross-head = 06'8 inches, or nearly 6-T inches. Minor diameter of part of piston rod in cross-head - 5'76 inches, or 5J inches. Major diameter of part of piston rod in piston = 8'96 inches, or nearly 9 inches. Minor'diameter of part of piston rod in piston = 7'36 inches, or between 71 and 71 inches. Depth of gibs and cutter through cross-head = 6'72 inches, or very nearly 61 inches. Thickness of gibs and cutter through cross-head = 1'35 inches, or between 11 and 11 inches. Page 139 THE STEAM ENGINE. 139 Depth of cutter through piston = 5'45 inches, or nearly 54- inches. Thickness of cutter through piston = 2'24 inches, or nearly 24 inches. Diameter of connecting rod at ends = 6'08 inches, or nearly 6-1 inches. Major diameter of part of connecting rod in cross-tail = 6'27 inches, or about 64 inches. Minor diameter of part of connecting rod in cross-tail - 5'76 inches, or nearly 54 inches. Breadth of butt = 9'98 inches, or very nearly 10 inches. Thickness of butt - 8 inches. Mean thickness of strap at cutter = 2'75 inches, or 24 inches. Mean thickness of strap above cutter = 2'06 inches, or somewhat more than 2 inches. Distance of cutter from end of strap - 3'08 inches, or very nearly 31 inches. Breadth of gibs and cutter through cross-tail = 6'73 inches, or very nearly 63 inches. Breadth of gibs and cutter through butt - 7'04 inches, or somewhat more than 7 inches. Thickness of gibs and cutter through butt = 1'84 inches, or between 14 and 2 inches. These results are calculated from the following rules, which give correct results for all sizes of engines. RULE 1. To find the diameter of crank-pin journal. —Multiply the diameter of the cylinder in inches by'142. The result is the diameter of crank-pin journal in inches. RULE 2. To find the length of crank-pin journal.-Multiply the diameter of the cylinder in inches by'16. The product is the length of the crank-pin journal in inches. RULE 3. To find the breadth of the eye of cross-head.-Multiply the diameter of the cylinder in inches by'041. The product is the breadth of the eye in inches. RULE 4. To find the depth of the eye of cross-head.-Multiply the diameter of the cylinder in inches by'286. The product is the depth of the eye of cross-head in inches. RULE 5. To find the diameter of the journal of cross-head.Multiply the diameter of the cylinder in inches by'086. The product is the diameter of the journal in inches. RULE 6. To find the length of the journal of cross-head.-Multiply the diameter of the cylinder in inches by'097. The product is the length of the journal in inches. RULE 7. To find the thickness of the web of cross-head at middle. -Multiply the diameter of the cylinder in inches by'072. The product is the thickness of the web of cross-head at middle in inches. RULE 8. To find the breadth of web of cross-head at middle.Multiply the diameter of the cylinder in inches by'268. The product is the breadth of the web of cross-head at middle in inches. Page 140 140 THE PRACTICAL MODEL CALCULATOR. RULE 9. To find the thickness of the web of cross-head at journal. -Multiply the diameter of the cylinder in inches by'061. The product is the thickness of the web of cross-head at journal in inches. RULE 10. To find the breadth of web of cross-head at journal.Multiply the diameter of the cylinder in inches by'101. The product is the breadth of the web of cross-head at journal in inches. RULE 11. To find the diameter of the piston rod. —Divide the diameter of the cylinder in inches by 10. The quotient is the diameter of the piston rod in inches. RULE 12. To find the length of the part of the piston rod in the piston.-Divide the diameter of the cylinder in inches by 5. The quotient is the length of the part of the piston rod in the piston in inches. RULE 13. To find the major diameter of the part of piston rod in cross-head.-Multiply the diameter of the cylinder in inches by'095. The product is the major diameter of the part of piston rod in cross-head in inches. RULE 14. To find the minor diameter of the part of piston rod in cross-head.-Multiply the diameter of the cylinder in inches by'09. The product is the minor diameter of the part of piston rod in cross-head in inches. RULE 15. To find the major diameter of the part of piston rod in piston.-Multiply the diameter of the cylinder in inches by'14. The product is the major diameter of the part of piston rod in piston in inches. RULE 16. To find the minor diameter of the part of piston rod in piston.-Multiply the diameter of the cylinder in inches by'115. The product is the minor diameter of the part of piston rod in piston. RULE 17. To find the depth of gibs and cutter through crosshead.-Multiply the diameter of the cylinder in inches by -105. The product is the depth of the gibs and cutter through crosshead. RULE 18. To find the thickness of the gibs and cutter through cross-head.-Multiply the diameter of the cylinder in inches by'021. The product is the thickness of the gibs and cutter through cross-head. RULE 1b. To find the depth of cutter through piston.-Multiply the diameter of the cylinder in inches by'085. The product is the depth of the cutter through piston in inches. RULE 20. To find the thickness of cutter through piston. —Multiply the diameter of the cylinder in inches by'035. The product is the thickness of cutter through piston in inches. RULE 21. To find the diameter of connecting rod at ends.-Multiply the diameter of the cylinder in inches by'095. The product is the diameter of the connecting rod at ends in inches. RULE 22. To find the major diameter of the part of connecting rod in cross-tail.-Multiply the diameter of the cylinder in inches Page 141 THE STEAM ENGINE. 141 by'098. The product is the major diameter of the part of connecting rod in cross-tail. RULE 23. To find the minor diameter of the part of connecting rod in cross-tail.-Multiply the diameter of the cylinder in inches by'09. The product is the minor diameter of the part of connecting rod in cross-tail in inches. RULE 24. To find the breadth of butt.-Multiply the diameter of the cylinder in inches by'156. The product is the breadth of the butt in inches. RULE 25. To find the thickness of the butt.-Divide the diameter of the cylinder in inches by 8. The quotient is the thickness of the butt in inches. RULE 26. To find the mean thickness of the strap at cutter.Multiply the diameter of the cylinder in inches by'043. The product is the mean thickness of the strap at cutter. RULE 27. To find the mean thickness of the strap above cutter.Multiply the diameter of the cylinder in inches by'032. The product is the mean thickness of the strap above cutter. RULE 28. To find the distance of cutter from end of strap.Multiply the diameter of the cylinder in inches by'048. The product is the distance of cutter from end of strap in inches. RULE 29. To find the breadth of the gibs and cutter through cross-tail.-Multiply the diameter of the cylinder in inches by *105. The product is the breadth of the gibs and cutter through cross-tail. RULE 30. To find the breadth of the gibs and cutter through butt. —Multiply the diameter of the cylinder in inches by'11. The product is the breadth of the gibs and cutter through butt in inches. RULE 31. To find the thickness of the gibs and cutter through butt.-Multiply the diameter of the cylinder in inches by'029. The product is the thickness of the gibs and cutter through butt in inches. To find other parts of the engine which do not depend upon the stroke. Suppose it were required to find the thickness of the small eye of crank for an engine the diameter of whose cylinder is 64 inches. According to the rule, the proper thickness of the small eye of crank is 4'04 inches. Again, suppose it were required to find the length of the small eye of crank. Hence, according to the rule, the proper length of the small eye of crank is 11'94 inches. Again, supposing it were required to find the proper thickness of the web of crank at pin centre; that is to say, the thickness which it would have if continued to the pin centre. According to the rule, the proper thickness for the web of crank at pin centre is 7'04 inches. Again, suppose it were required to find the breadth of the web of crank at pin centre; that is to say, the breadth which it would have if it were continued to the pin centre. Hence, according to the rule, the proper breadth of the web of crank at pin centre is 10'24 inches. Page 142 142 THE PRACTICAL MODEL CALCULATOR. These results are calculated from the following rules, which give the proper dimensions for engines of all sizes: RULE 1. To find the breadth of the small eye of crank.-Multiply the diameter of the cylinder in inches by'063. The product is the proper breadth of the small eye of crank in inches. RULE 2. To find the length of the small eye of crank.-Multiply the diameter of the cylinder in inches by'187. The product is the proper length of the small eye of crank in inches. RULE 3. Tofind the thickness of the web of crank at pin centre.Multiply the diameter of the cylinder in inches by'11. The product is the proper thickness of the web of crank at pin centre in inches. RULE 4. To find the breadth of the web of crank at pin centre.Multiply the diameter of the cylinder in inches by'16. The product is the proper breadth of crank at pin centre in inches. To illustrate the use of the succeeding rules, let us take the particular example of an engine of 8 feet stroke and 64-inch cylinder, and let us suppose that the length of the connecting rod is 12 feet, and the side rod 10 feet. We find by a previous rule that the diameter of the connecting rod at ends is 6'08, and the ratio between the diameters at middle and ends of a connecting rod, whose length is 12 feet, is 1'504. Hence, the proper diameter at middle of the connecting rod = 6'08 x 1'504 inches = 9144 inches. And again, we find the diameter of cylinder side rods at ends, for the particular engine which we have selected, is 4'10, and the ratio between the diameters at middle and ends of cylinder side rods, whose lengths are 10 feet, is 1'42. Hence, according to the rules, the proper diameter of the cylinder side rods at middle is equal to 4'1 x 1'42 inches = 5'82 inches. To find some of those parts of the engine which do not depend upon the stroke. Suppose we take the particular example of an engine the diameter of whose cylinder is 64 inches. We find from the following rules that Diameter of cylinder side rods at ends - 4'1 inches, or 4-1inches. Breadth of butt = 4'93 inches, or very nearly 5 inches. Thickness of butt = 3'9 inches, or 385 inches. Mean thickness of strap at cutter = 2'06 inches, or a little more than 2 inches. Mean thickness of strap below cutter =- 147 inches, or very nearly 1- inches. Depths of gibs and cutter = 5'12 inches, or a little more than 51 inches. Thickness of gibs and cutter = 1'03 inches, or a little more tli-nl 1 inch. Diameter of main centre journal = 11'71 inches, or very nearly 11 L inches. Length of main centre journal = 17'6 inches, or 17' inches. Page 143 THE STEAM ENGINE. 143 Depth of eye round end studs of lever = 4'75 inches, or 4- inches. Thickness of eye round end studs of lever = 3'33 inches, or 3inches. Diameter of end studs of lever = 4'48 inches, or very nearly 4~ inches. Length of end studs of lever = 4'86 inches, or between 4Z and 5 inches. Diameter of air-pump studs = 2'91 inches, or nearly 3 inches. Length of air-pump studs = 3'16 inches, or nearly 31 inches. These results were obtained from the following rules, which will be found to give the proper dimensions for all sizes of engines. RULE 1. To find the diameter of cylinder side rods at ends.Multiply the diameter of the cylinder in inches by'065. The product is the diameter of the cylinder side rods at ends in inches. RULE 2. To find the breadth of butt in inches. —Multiply the diameter of the cylinder in inches by'077. The product is the breadth of the butt in inches. RULE 3. To find the thickness of the butt.-Multiply the diameter of the cylinder in inches by'061. The product is the thickness of the butt in inches. RULE 4. To find the mean thickness of strap at cutter.-Multiply the diameter of the cylinder in inches by'032. The product is the mean thickness of the strap at cutter. RULE 5. To find the mean thickness of strap below cutter.-Multiply the diameter of the cylinder in inches by'023. The product is the mean thickness of strap below cutter in inches. RULE 6. To find the depth of gibs and cutter.-Multiply the diameter of the cylinder in inches by'08. The product is the depth of the gibs and cutter in inches. RULE 7. To find the thickness of gibs and cutter.-Multiply the diameter of the cylinder in inches by'016. The product is the thickness of gibs and cutter in inches. RULE 8. To find the diameter of the main centre journal. —Multiply the diameter of the cylinder in inches by'183. The product is the diameter of the main centre journal in inches. RULE 9. To find the length of the main centre journal.-iMultiply the diameter of the cylinder in inches by'275. The product is the diameter of the cylinder in inches. RULE 10. To find the depth of eye round end studs of lever.Multiply the diameter of the cylinder in inches by'074. The product is the depth of the eye round end studs of lever in inches. RULE 11. To find the thickness of eye round end studs of lever. -Multiply the diameter of the cylinder in inches by'052. The product is the thickness of eye round end studs of lever in inches. RULE 12. To find the diameter of the end studs of lever.-Multiply the diameter of the cylinder in inches by'07. The product is the diameter of the end studs of lever in inches. RLrULE 13. To find the length of thle end studs of lever.-Multiply Page 144 144 THE PRACTICAL MODEL CALCULATOR. the diameter of the cylinder in inches by'076. The product is the length of the end studs of lever in inches. RULE 14. To find the diameter of the air-2un?2 stald.s. —Multiply the diameter of the cylinder in inches by'045. The product is the diameter of the air-pump studs in inches. RULE 15. To find the lengyth of the air-pump2 stlds.- Multiply the diameter of the cylinder in inches by'049. The product is the length of the air-pump studs in ins. The next rule gives the proper epth in inches across the centre of the side lever, when, as is generally the case, the side lever is of cast iron. It will be observed that the depth is made to depend upon the diameter of the cylinder and the rength of the lever, and not at all upon the length of the stroke, except indeed in so far as the length of the lever may depend upon the length of the stroke. Suppose it were required to find: the proper depth across the centre of a side lever whose length is 20 feet, and the diameter of the cylinder 64 inches. According to the rule, the proper depth across the centre would be 39'26 inches. The following rule will give the proprer dimensions for any size of engine: RULE.-To find the depth aerossVtYe centre of the side lever. Multiply the length of the side lever'in feet by.7423; extract the cube root of the product, and reserve the result for a muiltiplier. Then square the diameter of the cylinder in inches; extract the cube root of the result. The product of the final result and the reserved multiplier is the depth of the side lever in inches across the centre. Thus, to apply this rule to the particular example which we have selegted, we have 20 =.length of side lever in feet. *7423 = constant multiplier. 14'846 and / 14'846 = t4568 nearly. 64 = diameter of cylinder in inches. 64 4096 and v 40~6 i::16 Hence depth at centre = 16 x 2'458 inches = 39'33 inches, or between 3924 and 391 inches. The next set of rules give the dimensions of seeral of the parts of the air-pump machinery which depend upon the dcF teki of the cylinder only. To illustrate the use of these irules, let us take the particular example of an engine the diameter of whose cylinder is 64 inches. We find from the succeeding rules successively, Diameter of air-pump = 38'4 inches, or 388 inches. Page 145 THE STEAM ENGINE. 145 Thickness of the eye of air-pump cross-head -- 1'58 inches, or a little more than 11 inches. Depth of eye of air-pump cross-head = 11'01, or about 11 inches. Diameter of end journals of air-pump cross-head = 3'29 inches, or somewhat more than 31 inches. Length of end journals of air-pump cross-head = 3'7 inches, or 3I inches. Thickness of the web of air-pump cross-head at middle - 2'76 inches, or a little more than 24 inches. Depth of web of air-pump cross-head at middle = 10'29 inches, or somewhat more than 104 inches. Thickness of web of air-pump cross-head at journal = 2'35 inches, or about 28 inches. Depth of web of air-pump cross-head at journal = 3'89 inches, or about 38 inches. Diameter of air-pump piston rod when made of copper = 4'27 inches, or about 44 inches. Depth of gibs and cutter through air-pump cross-head = 4'04 inches, or a little more than 4 inches. Thickness of gibs and cutter through air-pump cross-head ='81 inches, or about 8 inch. Depth of cutter through piston = 3'27 inches, or somewhat more than 34 inches. Thickness of cutter through piston = 1'34 inches, or about 18 inches. These results were obtained from the following rules, and give the proper dimensions for all sizes of engines: RULE 1. To find the diameter of the air-pump.-Multiply the. diameter of the cylinder in inches by'6. The product is the diameter of air-pump in inches. RULE 2. To find the thickness of the eye of air-pump cross-head. -Multiply the diameter of the cylinder in inches by'025. The product is the thickness of the eye of air-pump cross-head in inches. RULE 3. To find the depth of eye of air-pump cross-head.- Multiply the diameter of the cylinder in inches by'171. The product is the depth of the eye of air-pump cross-head in inches. RULE 4. To find the diameter of the journals of air-pump crosshead. —Multiply the diameter of the cylinder in inches by'051. The product is the diameter of the end journals. RULE 5. To find the length of the end journals for air-pump cross-head.-Multiply the diameter of the cylinder in inches by 058. The product is the length of the air-pump cross-head journals in inches. RULE 6. To find the thickness of the web of air-pump cross-head at middle.-Multiply the diameter of the cylinder in inches by'043. The product is the thickness at middle of the web of air-pump cross-head in inches. RULE 7. To find the depth at middle of the web of air-pump crossltead. —Multiply the diameter of the cylinder in inches by'161. N 10 Page 146 146 THE PRACTICAL MODEL CALCULATOR. The product is the depth at middle of air-pump cross-head in inches. RULE 8. To find the thickness of the web of air-pump crosshead at journals.-Multiply the diameter of the cylinder in inches by'037. The product is the thickness of the web of air-pump cross-head at journals in inches. RULE 9. To find the depth of the air-pump cross-head web at journals.-Multiply the diameter of the cylinder in inches by'061. The product is the depth at journals of the web of air-pump crosshead. RULE 10. To find the diameter of the air-pump piston rod when of copper.-Multiply the diameter of the cylinder in inches by ~067. The product is the diameter of the air-pump piston rod, when of copper, in inches. RULE 11. To find the depth of gibs and cutter through air-pump cross-head.-Multiply the diameter of the cylinder in inches by ~063. The product is the depth of the gibs and cutter through airpump cross-head in inches. RULE 12. To find the thickness of the gibs and cutter through air-pump cross-head.-Multiply the diameter of the cylinder in inches by'013. The product is the thickness of the gibs and cutter in inches. RULE 13. To find the depth of cutter through piston.-Multiply the diameter of the cylinder in inches by'051. The product is the depth of the cutter through piston in inches. RULE 14. To find the thickness of cutter through air-pump piston.-Multiply the diameter of the cylinder in inches by'021. The product is the thickness of the cutter through air-pump piston. The next seven rules give the dimensions of the remaining parts of the engine which do not depend upon the stroke. To exemplify their use, suppose it were required to find the corresponding dimensions for an engine the diameter of whose cylinder is 64 inches. According to the rule, the proper diameter of the air-pump side rod would be 2'48 inches. Hence, according to the rule, the proper breadth of butt is 2'95 inches. According to the rule, the proper thickness of butt is 2'35 inches. According to the rule, the mean thickness of strap at cutter ought to be 1'24 inches. Hence, according to the rule, the mean thickness of strap below cutter is'91 inch. According to the rule, the proper depth for the gibs and cutter is 2'94 inches. According to the rule, the proper thickness of the gibs and cutter is'63 inches. The following rules give the correct dimensions for all sizes of engines: RULE 1. To find the diameter of air-pump side rod at ends.Multiply the diameter of the cylinder in inches by'039. The product is the diameter of the air-pump side rod at ends in inches. RULE 2. To find the breadth of butt for air-pump. -.-Multiply the Page 147 THE STEAM ENGINE. 147 diameter of the cylinder in inches by'046. The product is the breadth of butt in inches. RULE 3. To find the thickness of butt for air-pump.-Multiply the diameter of the cylinder in inches by'037. The product is the thickness of butt for air-pump in inches. RULE 4. To find the mean thickness of strap at cutter.-Multiply the diameter of the cylinder in inches by'019. The product is the mean thickness of strap at cutter for air-pump in inches. RULE 5. To find the mean thickness of strap below cutter.-Multiply the diameter of the cylinder in inches by 0'14. The product is the mean thickness of strap below cutter in inches. RULE 6. To find the depth of gibs and cutter for air-pump.Multiply the diameter of the cylinder in inches by 0'48. The product is the depth of gibs and cutter for air-pump in inches. RULE 7. To find the thickness of gibs and cutter for air-pump.Divide the diameter of the cylinder in inches by 100. The quotient is the proper thickness of the gibs and cutter for air-pump in inches. With regard to other dimensions made to depend upon the nominal horse power of the engine: —Suppose that we take the particular example of an engine whose stroke is 8 feet, and diameter of cylinder 64 inches. We find that the nominal horse power of this engine is nearly 175. Hence we have successively, Diameter of valve shaft at journal in inches = 4'85, or between 4a and 5 inches. Diameter of parallel motion shaft at journal in inches = 3'91, or very nearly 4 inches. Diameter of valve rod in inches = 2'44, or about 2- inches. Diameter of radius rod at smallest part in inches = 1'97, or very nearly 2 inches. Area of eccentric rod, at smallest part, in square inches = 8'37, or about 8 square inches. Sectional area of eccentric hoop in square inches = 8'75, or 8-1 square inches. Diameter of eccentric pin in inches = 2'24, or 2{ inches. Breadth of valve lever for eccentric pin at eye in inches = 5'7, or very nearly 59 inches. Thickness of valve lever for eccentric pin at eye in inches = 3. Breadth of parallel motion crank at eye = 4'2 inches, or very nearly 4- inches. Thickness of parallel motion crank at eye = 1'76 inches, or about 13 inches. To find the area in square inches of each steam port. Suppose it were required to find the area of each steam port for an engine whose stroke is 8 feet, and diameter of cylinder 64 inches. According to the rule, the area of each steam port would be 202'26 square inches. With regard to the rule, we may remark that the area of the Page 148 148 THE PRACTICAL MODEL CALCULATOR. steam port ought to depend principally upon the cubical content of the cylinder, which again depends entirely upon the product of the square of the diameter of the cylinder and the length of the stroke of the engine. It is well known, however, that the quantity of steam admitted by a small hole does not bear so great a proportion to the quantity admitted by a larger one, as the area of the one does to the area of the other; and a certain allowance ought to be made for this. In the absence of correct theoretical information on this point, we have attempted to make a proper allowance by supplying a constant; but of course this plan ought only to be regarded as an approximation. Our rule is as follows: RULE.-To find the area of each steam port. —Multiply the square of the diameter of the cylinder in inches by the length of the stroke in feet; multiply this product by 11; divide the last product by 1800; and, finally, to the quotient add 8. The result is the area of each steam port in square inches. To show the use of this rule, we shall apply it to a particular example. We shall apply it to an engine whose stroke is 6 feet, and diameter of cylinder 30 inches. Then, according to the rule, we have 30 = diameter of the cylinder in inches. 30 900 = square of diameter. 6 = length of stroke in feet. 5400 11 59400. 1800 -= 33 8 = constant to be added. 41 = area of steam port in square inches. When the length of the opening of steam port is from any circumstance found, the corresponding depth in inches may be found, by dividing the number corresponding to the particular engine, by the given length in inches: conversely, the length may be found, when for some reason or other the depth is fixed, by dividing the number corresponding to the particular engine, by the given depth in inches: the quotient is the length in inches. The next rule is useful for determining the diameter of the steam pipe branching off to any particular engine. Suppose it were required to find the diameter of the branch steam pipe for an engine whose stroke is 8 feet, and diameter of cylinder 64 inches. According to the rule, the proper diameter of the steam pipe would be 13'16 inches. The following rule will be found to give the proper diameter of steam pipe for all sizes of engines. RULE.-To find the diameter of branch steam pipe.-Multiply together the square of the diameter of the cylinder in inches, the Page 149 THE STEAM ENGINE. 149 length of the stroke in feet, and'00498; to the product add 10'2, and extract the square root of the sum. The result is the diameter of the steam pipe in inches. To exemplify the use of this rule we shall take an engine whose stroke is 8 feet, and diameter of cylinder 64 inches. In this case we have as follows:64 = diameter of cylinder in inches. 64 4096 = square of diameter. 8 = length of stroke in feet. 32768 ~00498 = constant multiplier. 163'18 10'2 = constant to be added. 173'38 and V 173'38 - 13'16. To find the diameter of the pipes connected with the engine. They are made to depend upon the nominal horse power of the engine. Suppose it were required to apply this rule to determine the size of the pipes for two marine engines, whose strokes are each 8 feet, and diameters of cylinder each 64 inches. We find the nominal horse power of each of these engines to be 174'3. Hence, according to the rules, we have in succession, Diameter of waste water pipe = 15'87 inches, or between 151 and 16 inches. Area of foot-valve passage = 323 square inches. Area of injection pipe = 14'88 square inches. If the injection pipe be cylindrical, then by referring to the table of areas of circles, we see that its diameter would be about 43 inches. Diameter of feed pipe = 4'12 inches, or between 4 and 4} inches. Diameter of waste steam pipe = 12'17 inches, or nearly 12{ inches. Diameter of safety valve, When one is used =14'05 inches. When two are used = 9.94 inches. When three are used = 8'12 inches. When four are used = 7'04 inches. These results were obtained from the following rules, which will give the correct dimensions for all sizes of engines. RULE 1. To find the diameter of waste water pipe.-Multiply the square root of the nominal horse power of the engine by 1"2. The product is the diameter of the waste water pipe in inches. RULE 2. To find the area of foot-valve passage.-Multiply the N 2 Page 150 150 THE PRACTICAL MODEL CALCULATOR. nominal horse power of the engine by 9; divide the product by 5; add 8 to the quotient. The sum is the area of foot-valve passage in square inches. RULE 3. To find the area of injection pipe.-Multiply the nominal horse power of the engine by'069; to the product add 2'81. The sum is the area of the injection pipe in square inches. RULE 4. To find the diameter of feed pipe. —Multiply the nominal horse power of the engine by'04; to the product add 3; extract the square root of the sum. The result is the diameter of the feed pipe in inches. RULE 5. To find the diameter of waste steam pipe. —Multiply the collective nominal horse power of the engines by *375; to the product add 16'875; extract the square root of the sum. The final result is the diameter of the waste steam pipe in inches. RULE 6. To find the diameter of the safety valve when only one is used.-To one-half the collective nominal horse power of the engines add 22'5; extract the square root of the sum. The result is the diameter of the safety valve when only one is used. RULE 7. To find the diameter of the safety valve when two are used.-Multiply the collective nominal horse power of the engines by'25; to the product add 11'25; extract the square root of the sum. The result is the diameter of the safety valve when two are used. RULE 8. To find the diameter of the safety valve when three are used. —To one-sixth of the collective nominal horse power of the engines add 7'5; extract the square root of the sum. The result is the diameter of the safety valve where three are used. RULE 9. To find the diameter of the safety valve whenfour are used.-Multiply the collective nominal horse power of the engines by'125; to the product add 5'625; extract the square root of the sum. The result is the diameter of the safety valve when four are used. Another rule for safety valves, and a preferable one for low pressures, is to allow'8 of a circular inch of area per nominal horse power. The next rule is for determining the depth across the web of the main beam of a land engine. Suppose we wished to find the proper depth at the centre of the main beam of a land engine whose main beam is 16 feet long, and diameter of cylinder 64 inches. According to the rule, the proper depth of the web across the centre is 46'17 inches. This rule gives correct dimensions for all sizes of engines. RULE.-To find the depth of the web at the centre of the main beam of a land engine.-Multiply together the square of the diameter of the cylinder in inches, half the length of the main beam in feet, and the number 3; extract the cube root of the product. The result is the proper depth of the web of the main beam across the centre in inches, when the main beam is constructed of cast iron. Page 151 THE STEAM ENGINE. 151 To illustrate this rule we shall take the particular example of an engine whose main beam is 20 feet long, and the diameter of the cylinder 64 inches. In this case we have 64 = diameter of cylinder in inches. 64 4096 = square of the diameter. 10 = 2 length of main beam in feet. 40960 3 = constant multiplier. 122880 0 0 122880 (49'714 = /122880 4 16 64 4 16 58880 4 32 53649 8 4800 5231 4 1161 5110 120 5961 119 9 1242 74 129 7203 35 9 10 138 730 9 10 147 741 To find the depth of the main beam across the ends. Suppose it were required to find the depth at ends of a cast-iron main beam whose length is 20 feet, when the diameter of the cylinder is 64 inches. The proper depth will be 19'89 inches. The following rule gives the proper dimensions for all sizes of engines. RULE. —TO find the depth of main beam at ends.-Multiply together the square of the diameter of the cylinder in inches, half the length of the main beam in feet, and the number'192; extract the cube root of the product. The result is the depth in inches of the main beam at ends, when of cast iron. To illustrate this rule, let us apply it to the particular example of an engine whose main beam is 20 feet long, and the diameter of the cylinder 64 inches. In this case we have as follows: 64 = diameter of cylinder in inches. 64 4096 = square of diameter of cylinder. 10 = - length of main beam in feet. 40960 ~192 = constant multiplier. 7864'32 Page 152 152 THE PRACTICAL MODEL CALCULATOR. 0 0 786432 ( 19'89 = -7864-32 1 1 1 1 1 6864 1 2 5859 2 300 1005 1 351 898 30 651 107 9 432 39 1083 9 4 48 112 9 4 57 116 so that, according to the rule, the depth at ends is nearly 20 inches. To find the dimensions of the feed-pump in cubic inches. Suppose we take the particular example of an engine whose stroke is 8 feet, and diameter of cylinder 64 inches. The proper content of the feed-pump would be 1093'36 cubic inches. Suppose, now, that the cold-water pump was suspended from the main beam at a fourth of the distance between the centre and the end, so that its stroke would be 2 feet, or 24 inches. In this case the area of the pump would be equal to 1093'36 - 24 = 45-556 square inches; so that we conclude that the diameter is between 72 and 74 inches. Conversely, suppose that it was wished to find the stroke of the pump when the diameter was 5 inches. We find the area of the pump to be 19'635 square inches; so that the stroke of the feedpump must be equal to 1093-36 -- 19'635 = 55'69 inches, or very nearly 553 inches. This rule will be found to give correct dimensions for all sizes of engines: RULE.-To find the content of the feed-pump.-Multiply the square of the diameter of the cylinder in inches by the length of the stroke in feet; divide the product by 30. The quotient is the content of the feed-pump in cubic inches. Thus, for an engine whose stroke is 6 feet, and diameter of cylinder 50 inches, we have, 50 = diameter of cylinder. 50 2500 = square of the diameter of the cylinder. 6 = length of stroke in feet. 30) 15000 500 = content of feed-pump in cubic inches. To determine the content of the cold-water pump in cubic feet. To illustrate this, suppose we take the particular example of an en Page 153 THE STEAM ENGINE. 153 gine whose stroke is 8 feet, and diameter of cylinder 64 inches. Suppose, now, the stroke of the pump to be 5 feet, then the area equal to 7'45 - 5 = 1'49 square feet = 214'56 square inches; we see that the diameter of the pump is about 16-1 inches. Again, suppose that the diameter of the cold-water pump was 20 inches, and that it was required to find the length of its stroke. The area of the pump is 314'16 square inches, or 314'16 -- 144 = 2'18 square feet; so that the stroke of the pump is equal to 7'45 - 2'18 = 3'42 feet. The content is calculated from the following rule, which will be found to give correct dimensions for all sizes of engines: RULE.-To find the content of the cold-water pump.-Multiply the square of the diameter of the cylinder in inches by the length of the stroke in feet; divide the product by 4400. The quotient is the content of the cold-water pump in cubic feet. To explain this rule, we shall take the particular example of an engine whose stroke is 51 feet, and diameter of cylinder 60 inches. In this case we have in succession, 60 diameter of cylinder in inches. 60 3600 square of the diameter of cylinder. 51 = length of stroke in feet. 4400)19800 4'5 content of cold water pump in cubic feet. To determine the proper thickness of the large eye of crank for fly-wheel shaft when the crank is of cast iron. The crank is sometimes cast on the shaft, and of course the thickness of the large eye is not then so great as when the crank is only keyed on the shaft, or rather there is then no large eye at all. To illustrate the use of this rule, we shall apply it to the particular example of an engine whose stroke is 8 feet, and diameter of cylinder 64 inches. Hence, according to the rule, the proper thickness of the large eye of crank when of cast iron is 8-07 inches. For a marine engine of 8 feet stroke and 64 inch cylinder, the thickness of the large eye of crank is about 5- inches. The difference is thus about 2} inches, which is an allowance for the inferiority of cast iron to malleable iron. The following rule will be found to give correct dimensions for all sizes of engines: RULE.-To find the thickness of the large eye of crank for flywheel shaft when of cast iron.-Multiply the square of the length of the crank in inches by 1'561, and then multiply the square of the diameter of the cylinder in inches by'1235; multiply the sum of these products by the square of the diameter of cylinder in inches; divide this product by 666'283; divide this quotient by the length of the crank in inches; finally extract the cube root of the quotient. Page 154 154 THE PRACTICAL MODEL CALCULATOR. The result is the proper thickness of the large eye of crank for fly-wheel shaft in inches, when of cast iron. As this rule is rather complicated, we shall show its application to the particular example already selected. 48 = length of crank in inches. 48 2304 = square of length of crank in inches. 1'561 = constant multiplier. 3596'5 64 = diameter of cylinder in inches. 64 4096 = square of the diameter of cylinder. *1235 =constant multiplier. 505'8 3596'5 4102'3 = sum of products. 4096 = square of the diameter of cylinder. 666'283) 16803020'8 length of crank=48) 25219'045 525'397 and V 525'397 = 8'07 nearly. To find the breadth of the web of crank at the centre of the flywheel shaft, that is to say, the breadth which it would have if it were continued to the centre of the fly-wheel shaft. Suppose it were required to find the breadth of the crank at the centre of the fly-wheel shaft for an engine whose stroke is 8 feet, and diameter of cylinder 64 inches. According to the rule, the proper breadth is 22'49 inches. According to a former rule, the breadth of the web of a cast iron crank of an engine whose stroke is 8 feet, and diameter of cylinder 64 inches, is about 18 inches. The difference between these two is about 4-1 inches; which is not too great an allowance for the inferiority of the cast iron. The following rule will be found to give correct dimensions for all sizes of engines: RULE. — To find the breadth of the web of crank at fly-wheel shaft, when of cast iron.-Multiply the square of the length of the crank in inches by 1'561, and then multiply the square of the diameter of the cylinder in inches by'1235; multiply the square root of the sum of these products by the square of the diameter of the cylinder in inches; divide the product by 23'04, and finally extract the cube root of the quotient. The final result is the breadth of the crank at the centre of the fly-wheel shaft, when the crank is of cast iron. As this rule is rather complicated, we shall illustrate it by show Page 155 THE STEAM ENGINE. 155 ing its application to the particular example of an engine whose stroke is 8 feet, and diameter of cylinder 64 inches. 64 = diameter of cylinder in inches. 64 4096 = square of the diameter of cylinder. ~1235 = constant multiplier. 505'8' 48 = length of crank in inches. 48 2304 = square of the length of crank. 1'561 = constant multiplier. 3596'5 505.8 4102.3 = sum of products. V/ 4102'3 = 64'05 nearly. 4096 = square of the diameter of constant divisor = 23'04) 262348.5 [cylinder. 11386'66 nearly. and v 11386'66 = 22'49. To determine the thickness of the web of crank at the centre of the fly-wheel shaft; that is to say, the thickness which it would have if it were continued so far. Suppose it were required to find the thickness of web of crank at the centre of fly-wheel shaft of an engine whose stroke is 8 feet, and diameter of cylinder 64 inches. According to the rule, the proper thickness would be 11'26 inches. The proper thickness of web at centre of paddle shaft for a marine engine whose stroke is 8 feet, and diameter of cylinder 64 inches, is nearly 9 inches. The difference between the two thicknesses is about 2- inches, which is not too great an allowance for the inferiority of cast iron to malleable iron. The following rule will be found to give correct dimensions for all sizes of engines: RULE.-To find the thickness of the web of crank at centre of fly-wheel shaft, when of east iron.-Multiply the square of the length of the crank in inches by 1'561, and then multiply the square of the diameter of the cylinder in inches by'1235; multiply the square root of the sum of these products by the square of the diameter of the cylinder in inches; divide this product by 184'32; finally extract the cube root of the quotient. The result is the thickness of the web of crank at the centre of the fly-wheel shaft when of cast iron, in inches. As this rule is rather complicated, we shall illustrate it by applying it to the particular engine which we have already selected. Page 156 156 THE PRACTICAL MODEL CALCULATOR. 48 - length of crank in inches. 48 2304 _ square of length of crank. 1'561 = constant multiplier. 3596'5 64 = diameter of cylinder in inches. 64 4096 = square of the diameter of cylinder. 1235 = constant multiplier. 505-8 3596'5 4102'3 = sum of products. and / 4102'3 = 64'05 nearly. 4096 = square of diameter. Constant divisor = 184.32) 262348'5 1423'33 and v 1423'33 = 11'24 To find the proper diameter of the fly-wheel shaft at its smallest part, when, as is usually the case, it is of cast iron. Suppose it were required to find the diameter of the fly-wheel shaft for an engine whose stroke is 8 feet, and diameter of cylinder 64 inches. According to the rule, the diameter would be 17'59 inches. It is obvious enough that the fly-wheel shaft stands in much the same relation to the land engine, as the paddle shaft does to the marine engine. According to a former rule, the diameter of the paddle shaft journal of a marine engine whose stroke is 8 feet, and diameter of cylinder 64 inches, is about 14 inches. The difference betwixt the diameter of the paddle shaft for the marine engine, and the diameter of the fly-wheel shaft for the corresponding land engine is about 3~ inches. This will be found to be a very proper allowance for the different circumstances connected with the land engine. The following rule will be found to give correct dimensions for all sizes of engines. RULE.-To find the diameter of the fly-wheel shaft at smallest part, when it is of cast iron.-Multiply the square of the diameter of the cylinder in inches by the length of the crank in inches; extract the cube root of the product; finally multiply the result by'3025. The result is the diameter of the fly-wheel shaft at smallest part in inches. We shall illustrate this rule by applying it to the particular engine which we have already selected. Page 157 THE STEAM ENGINE. 157 64 = diameter of cylinder in inches. 64 4096 = square of the diameter. 48 = length of crank in inches. 196608 0 0 196608 (58.15 =' 196608 5 25 125 5 25 71608 5 50 70112 10 7500 1496 5 1264 1011 150 8764 485 8 1328 158 10092 8 2 166 1011 8 2 174 1013 and 58'15 x'3025 = 1759 which agrees with the number given by a former rule. To determine the sectional area of the fly-wheel rim when of cast iron. Suppose it were required to find the sectional area of the rim of a fly-wheel for an engine whose stroke is 8 feet, and diameter of cylinder 64 inches, the diameter of the fly-wheel itself being 30 feet. According to the rule, the sectional area of the rim in square inches = 146'4 x'813 = 119'02. We may remark that this calculation has been made on the supposition that the flywheel is so connected with the engine, as to make exactly one revolution for each double stroke of the piston. If the fly-wheel is so connected with the engine as to make more than one revolution for each double stroke, then the rim does not need to be so heavy as we make it. If, on the contrary, the fly-wheel does not make a complete revolution for each double stroke of the engine, then it ought to be heavier than this rule makes it. RULE.-To find the sectional area of the rim of the fly-wheel when of east iron.-Multiply together the square of the diameter of the cylinder in inches, the square of the length of the stroke in feet, the cube root of the length of the stroke in feet, and 6'125; divide the final product by the cube of the diameter of the fly-wheel in feet. The quotient is the sectional area of the rim of fly-wheel in square inches, provided it is of cast iron. As this rule is rather complicated, we shall endeavour to illustrate it by showing its application to a particular engine. WNVe shall apply the rule to determine the sectional area of the rim of fly0 Page 158 158 THE PRACTICAL MODEL CALCULATOR. wheel for an engine whose stroke is 8 feet, diameter of cylinder 50 inches; the diameter of the fly-wheel being 20 feet. For this engine we have as follows: 2500 = square of diameter of cylinder. 64 = square of the length of stroke. 160000 2 = cube root of the length of stroke. 320000 64125 = constant multiplier. 1960000 therefore sectional area in square inches = 1960000 - 203 - 1960000. 8000 = 1960 - 8 = 245. In the following formulas we denote the diameter of the cylinder in inches by D, the length of the crank in inches by R, the length of the stroke in feet, and the nominal horse power of the engine by H.P. MARINE ENGINES.-DIMENSIONS OF SEVERAL OF THE PARTS OF THE SIDE LEVER. Depth of eye round end studs of lever ='074 x D. Thickness of eye round end studs of lever = -052 x D. Diameter of end studs, in inches ='07 x D. Length of end studs, in inches = -076 x D. Diameter of air-pump studs, in inches ='045 x D. Length of air-pump studs, in inches ='049 x D. Depth of cast iron side lever across centre, in inches = D3 x {'7423 x length of lever in feet}l. MARINE ENGINE.-DIMENSIONS OF SEVERAL PARTS OF AIR-PUMP CROSS-HEAD. Diameter of air-pump, in inches ='6 x D. Thickness of eye for air-pump rod, in inches -= 025 x D. Depth of eye for air-pump rod, in inches ='171 x D. Diameter of end journals, in inches -'051 x D. Length of end journals, in inches ='058 x D. Thickness of web at middle, in inches ='043 x D. Depth of web at middle, in inches ='161 x D. Thickness of web at journal ='037 x D. Depth of web at journal = -061 x D. MARINE ENGINE.-DIMENSIONS OF THE PARTS OF AIP-PUM'_P PISTON-ROD. Diameter of air-pump piston-rod, when of copper, in inches = 067 x D. Depth of gibs and cutter through cross-head, in inches -'063 x D. Page 159 THE STEAM ENGINE. 159 Thickness of gibs and cutter through cross-head, inr inches = 013 x D. Depth of cutter through piston, in inches ='051 x D. Thickness of cutter through piston, in inches ='021 x D. MARINE ENGINE.-DIMENSIONS OF THE REMAINING PARTS OF THE AIR-PUMP MACHINERY. Diameter of air-pump side rods at ends, in inches ='039 x D. Breadth of butt, in inches *= 046 x D. Thickness of butt, in inches = -037 x D. Mean thickness of strap at cutter, in inches ='019 x D. Mean thickness of strap below cutter, in inches -014 x D. Depth of gibs and cutter, in inches ='048 x D. Thickness of gibs and cutter in inches = D. 100. MARINE AND LAND ENGINES. —AREA OF STEAM PORTS. Area of each steam port, in square inches = 11 x I x D2 - 1800 + 8. MARINE AND LAND ENGINES. -DIMENSIONS OF BRANCH STEAM PIPES. Diameter of each branch steam pipe = V'00498 x 1 x D2 x 10'2. MARINE ENGINE.-DIMENSIONS OF SEVERAL OF THE PIPES CONNECTED WITH THE ENGINE. Diameter of waste.water pipe, in inches =- 12 x V H.P. Area of foot-valve passage, in square inches = 1'8 x H.P.+ 8. Area of injection pipe, in square inches ='069 x H.P. + 2'81. Diameter of feed pipe, in inches / -04 x H.P. + 3. Diameter of waste steam pipe in inches =-v375 xH.P. +16'a75. MARINE AND LAND ENGINES.-DIMENSIONS OF SAFETY-VALVES. Diam. of safety-valve, when one only is used =,/'5x I.P. +225. Diam. of safety-valve, when two are used = v/'25xH.P.+11'25. Diam. of safety-valve, when three are used = V'167 xH.P. +7'5. Diam. of safety-valve, when four are used = /V125 x H.P. + 5'625. LAND ENGINE.-DIMENSIONS OF MAIN BEAM. Depth of web of main beam across centre - YV 3 x D2 x half length of main beam in feet. Depth of main beam at ends/ /192 x D2 x half length of main beam, in feet. LAND AND MARINE ENGINES.-CONTENT OF FEED-PUMP. Content of feed-pump, in cubic inches = D2 x I - 30. LAND ENGINES. —CONTENT OF COLD WATER PUMP. Content of cold water pump, in cubic feet = D2 x 1. 4400. Page 160 160 THE PRACTICAL MODEL CALCULATOR. LAND ENGINES.-DIMENSIONS OF CRANK. Thickness of large eye of crank, in inches = v D2 x (1.561 x R2 +.1235 D2). (R x 666.283). Breadth of web of crank at fly-wheel shaft centre, in inches - / D2 x V/ (1'561 x R2 +'1235 x D2). 23'04. Thickness of web of crank at fly-wheel shaft centre, in inches = VY D2 x V (1'561 x R2 +'1235 x D2) ~ 184'32. LAND ENGINES.-DIMENSIONS OF FLY-WHEEL SHAFT. Diameter of fly-wheel shaft, when of cast iron = 3025 x YR xD2. DIMENSIONS OF PARTS OF LOCOMOTIVES. DIAMETER OF CYLINDER. IN locomotive engines, the diameter of the cylinder varies less than either the land or the marine engine. In few of the locomotive engines at present in use is the diameter of the cylinder greater than 16 inches, or less than 12 inches. The length of the stroke of nearly all the locomotive engines at present in use is 18 inches, and there are always two cylinders, which are generally connected to cranks upon the axle, standing at right angles with one another. AREA OF INDUCTION PORTS. RULE.-To find the size of the steam ports for the locomotive engine.-Multiply the square of the diameter of the cylinder by'068. The product is the proper size of the steam ports in square inches. Required the proper size of the steam ports of a locomotive engine whose diameter is 15 inches. Here, according to the rule, size of steam ports = -068 x 15 x 15 = -068 x 225 = 15'3 square inches, or between 151 and 151 square inches. After having determined the area of the ports, we may easily find the depth when the length is given, or, conversely, the length when the depth is given. Thus, suppose we knew that the length was 8 inches, then we find that the depth should be 15'3 - 8 = 1'9125 inches, or nearly 2 inches; or suppose we knew the depth was 2 inches, then we would find that the length was 15'3 ~ 2 7'65 inches, or nearly 7A inches. AREA OF EDUCTION PORTS. The proper area for the eduction ports may be found from the following rule. RULE. —To find the area of the eduction ports. —Multiply the square of the diameter of the cylinder in inches by'128. The product is the area of the eduction ports in square inches. Required the area of the eduction ports of a locomotive engine, Page 161 THE STEAM ENGINE. 161 when the diameter of the cylinders is 13 inches. In this example we have, according to the rule, Area of eduction port = -128 x 132= -128 x 169 = 21'632 inches, or between 211 and 21a square inches. BREADTH OF BRIDGE BETWEEN PORTS. The breadth of the bridges between the eduction port and the induction ports is usually between 3 inch and 1 inch. DIAMETER OF BOILER. It is obvious that the diameter of the boiler may vary very considerably; but it is limited chiefly by considerations of strength; and 3 feet are found a convenient diameter. Rules for the strength of boilers will be given hereafter. RULE. —To find the inside diameter of the boiler.-Multiply the diameter of the cylinder in inches by 3'11. The product is the inside diameter of the boiler in'inches. Required the inside diameter of the boiler for a locomotive engine, the diameter of the cylinders being 15 inches. In this example we have, according to the rule, Inside diameter of boiler = 15 x 3'11 = 46'65 inches, or about 3 feet 108 inches. LENGTH OF BOILER. The length of the boiler is usually in practice between 8 feet and 8~ feet. DIAMETER OF STEAM DOME, INSIDE. It is obvious that the diameter of the steam dome may be varied considerably, according to circumstances; but the first indication is to make it large enough. It is usual, however, in practice, to proportion the diameter of the steam dome to the diameter of the cylinder; and there appears to be no great objection to this. The following rule will be found to give the diameter of the dome usually adopted in practice. RULE.-To find the diameter of the steam dome.-Multiply the diameter of the cylinder in inches by 1'43. The product is the diameter of the dome in inches. Required the diameter of the steam dome for a locomotive engine whose diameter of cylinders is 13 inches. In this example we have, according to the rule, Diameter of steam dome = 1'43 x 13 = 18'59 inches, or about 18- inches. HEIGHT OF STEAM DOME. The height of the steam dome may vary. Judging from practice, it appears that a uniform height of 2~ feet would answer very well. o2 11 Page 162 162 THE PRACTICAL MODEL CALCULATOR. DIAMETER OF SAFETY-VALVE. In practice the diameter of the safety-valve varies considerably. The following rule gives the diameter of the safety-valve usually adopted in practice. RULE.-To find the diameter of the safety-valve.-Divide the diameter of the cylinder in inches by 4. The quotient is the diameter of the safety-valve in inches. Required the diameter of the safety-valves for the boiler of a locomotive engine, the diameter of the cylinder being 13 inches. Here, according to the rule, diameter of safety-valve = 13. 4 = 3} inches. A larger size, however, is preferable, as being less likely to stick. DIAMETER OF VALVE SPINDLE. The following rule will be found to give the correct diameter of the valve spindle. It is entirely founded on practice. RULE. —To find the diameter of the valve spindle.-Multiply the diameter of the cylinder in inches by'076. The product is the proper diameter of the valve spindle. Required the diameter of the valve spindle for a locomotive engine whose cylinders' diameters are 13 inches. In this example we have, according to the rule, diameter of valve spindle = 13 x'076 ='988 inches, or very nearly 1 inch. DIAMETER OF CHIMNEY. It is usual in practice to make the diameter of the chimney equal to the diameter of the cylinder. Thus a locomotive engine whose cylinders' diameters are 15 inches would have the inside diameter of the chimney also 15 inches, or thereabouts. This rule has, at least, the merit of simplicity. AREA OF FIRE-GRATE. The following rule determines the area of the fire-grate usually given in practice. We may remark, that the area of the fire-grate in practice follows a more certain rule than any other part of the engine appears to do; but it is in all cases much too small, and occasions a great loss of power by the urging of the blast it renders necessary, and a rapid deterioration of the furnace plates from excessive heat. There is no good reason why the furnace should not be nearly as long as the boiler: it would then resemble the furnace of a marine boiler, and be as manageable. RULE.- To find the area of the fire-grate. —Multiply the diameter of the cylinder in inches by'77. The product is the area of the firegrate in superficial feet. Required the area of the fire-grate of a locomotive engine, the diameters of the cylinders being 15 inches. In this example we have, according to the rule, Area of fire-grate ='77 x 15 = 11'55 square feet, or about 111 square feet. Though this rule, however, represents Page 163 THE STEAM ENGINE. 163 the usual practice, the area of the fire-grate should not be contingent upon the size of the cylinder, but upon the quantity of steam to be raised. AREA OF HEATING SURFACE. In the construction of a locomotive engine, one great object is to obtain a boiler which will produce a sufficient quantity of steam with as little bulk and weight as possible. This object is admirably accomplished in the construction of the boiler of the locomotive engine. This little barrel of tubes generates more steam in an hour than was formerly raised from a boiler and fire occupying a considerable house. This favourable result is obtained simply by exposing the water to a greater amount of heating surface. In the usual construction of the locomotive boiler, it is obvious that we can only consider four of the six faces of the inside fire-box as effective heating surface; viz. the crown of the box, and the three perpendicular sides. The circumferences of the tubes are also effective heating surface; so that the whole effective heating surface of a locomotive boiler may be considered to be the four faces of the inside fire-box, plus the sum of the surfaces of the tubes. Understanding this to be the effective heating surface, the following rule determines the average amount of heating surface usually given in practice. RULE.- To find the effective heating surface.-Multiply the square of the diameter of the cylinder in inches by 5; divide the product by 2. The quotient is the area of the effective heating surface in square feet. Required the effective heating surface of the boiler of a locomotive engine, the diameters of the cylinders being 15 inches. In this example we have, according to the rule, Effective heating surface = 152 x 5 -. 2 = 225 x 5 - 2 = 1125 ~ 2 = 5621 square feet. According to the rule which we have given for the fire-grate, the area of the fire-grate for this boiler would be about 111 square feet. We may suppose, therefore, the area of the crown of the box to be 12 square feet. The area of the three perpendicular sides of the inside fire-box is usually three times the area of the crown; so that the effective heating surface of the fire-box is 48 square feet. Hence the heating surface of the tubes = 526'5 - 48 = 478'5 square feet. The inside diameters of the tubes are generally about 1- inches; and therefore the circumference of a section of these tubes, according to the table, is 5'4978 inches. Hence, supposing the tube to be 81 feet long, the surface of one = 5'4978 x 8~ ~. 12 = ~45815 x 8~ = 3'8943 square feet; and, therefore, the number of tubes = 478'5. 38943 = 123 nearly. The amount of heating surface, however, like that of grate surface, is properly a function of the quantity of steam to be raised, and the proportions of both, given hereafter, will be found to answer well for boilers of every description. Page 164 164 THE PRACTICAL MODEL CALCULATOR. AREA OF WATER-LEVEL. This, of course, varies with the different circumstances of the boiler. The average area may be found from the following rule. RULE. —To find the area of the water-level.-Multiply the diameter of the cylinder in inches by 2'08. The product is the area of the water-level in square feet. Required the area of the water-level for a locomotive engine, whose cylinders' diameters are 14 inches. In this case we have, according to the rule, Area of water-level = 14 x 2'08 = 29'12 square feet. CUBICAL CONTENT OF WATER IN BOILER. This, of course, varies not only in different boilers, but also in the same boiler at different times. The following rule is supposed to give the average quantity of water in the boiler. RULE.-To find the cubical content of the water in the boiler.Multiply the square of the diameter of the cylinder in inches by 9: divide the product by 40. The quotient is the cubical content of the water in the boiler in cubic feet. Required the average cubical content of the water in the boiler of a locomotive engine, the diameters of the cylinders being 14 inches. In this example we have, according to the rule, Cubical content of water = 9 x 142. 40 = 44'1 cubic feet. CONTENT OF FEED-PUMP. In the locomotive engine, the feed-pump is generally attached to the cross-head, and consequently it has the same stroke as the piston. As we have mentioned before, the stroke of the locomotive engine is generally in practice 18 inches. Hence, assuming the stroke of the feed-pump to be constantly 18 inches, it only remains for us to determine the diameter of the ram. It may be found from the following rule. RULE. — Tofind the diameter of thefeed-pump ram.-Multiply the square of the diameter of the cylinder in inches by'011. The product is the diameter of the ram in inches. Required the diameter of the ram for the feed-pump for a locomotive engine whose diameter of cylinder is 14 inches. In this example we have, according to the rule, Diameter of ram ='011 x 142 = 011 x 196 = 2'156 inches, or between 2 and 2{ inches. CUBICAL CONTENT OF STEAMI ROOM. The quantity of steam in the boiler varies not only for different boilers, but even for the same boiler in different circumstances. But when the locomotive is in motion, there is usually a certain proportion of the boiler filled with the steam. Including the dome and the steam pipe, the content of the steam room will be found usually to be somewhat less than the cubical content of the water. Page 165 THE STEAM ENGINE. 165 But as it is desirable that it should be increased, we give the following rule. RULE. —To find the cubical content of the steam room.-Multiply the square of the diameter of the cylinder in inches by 9; divide the product by 40. The quotient is the cubical content of the steam room in cubicj feet. Required the cubical content of the steam room in a locomotive boiler, the diameters of the cylinders being 12 inches. In this example we have, according to the rule, Cubical content of steam room = 9 x 122 -. 40 = 9 x 144 -- 40 = 32'4 cubic feet. CUBICAL CONTENT OF INSIDE FIRE-BOX ABOVE FIRE-BARS. The following rule determines the cubical content of fire-box usually given in practice. RULE.- To find the cubical content of inside fire-box above firebars.-Divide the square of the diameter of the cylinder in inches by 4. The quotient is the content of the inside fire-box above firebars in cubic feet. Required the content of inside fire-box above fire-bars in a locomotive engine, when the diameters of the cylinders are each 15 inches. In this example we have, according to the rule, Content of inside fire-box above fire-bars = 152. 4 - 225 — 4 = 564 cubic feet. THICKNESS OF THE PLATES OF BOILER. In general, the thickness of the plates of the locomotive boiler is 3 inch. In some cases, however, the thickness is only -1 inch. INSIDE DIAMETER OF STEAM PIPE. The diameter usually given to the steam pipe of the locomotive engine may be found from the following rule. RULE. —To find the diameter of the steam pipe of the locoimotive engine.-Multiply the square of the diameter of the cylinder in inches by'03. The product is the diameter of the steam pipe in inches. Required the diameter of the steam pipe of a locomotive engine, the diameter of the cylinder being 13 inches. Here, according to the rule, diameter of steam pipe ='03 x 132 = 03 x 169 = 5'07 inches; or a very little more than 5 inches. The steam pipe is usually made too small in engines intended for high speeds. DIAMETER OF BRANCH STEAM PIPES. The following rule gives the usual diameter of the branch steam pipe for locomotive engines. RULE.- To find the diameter of the branch steam pipe for the locomotive engine.-Multiply the square of the diameter of the cylinder in inches by'021. The product is the diameter of the branch steam pipe for the locomotive engine in inches. Page 166 166 THE PRACTICAL MODEL CALCULATOR. Required the diameter of the branch steam pipes for a locomotive engine, when the cylinder's diameter is 15 inches. Here, according to the rule, diameter of branch pipe ='021 x 152 ='021 x 225 = 4'725 inches, or about 43 inches. DIAMETER OF TOP OF BLAST PIPE. The diameter of the top of the blast pipe may be found from the following rule. RULE.- To find the diameter of the top of the blast pipe.-Multiply the square of the diameter of the cylinder in inches by 0'17. The product is the diameter of the top of the blast pipe in inches. The diameter of a locomotive engine is 13 inches; required the diameter of the blast pipe at top. Here, according to the rule, diameter of blast pipe at top ='017 x 132 ='017 x 169 =2'873 inches, or between 23 and 3 inches; but the orifice of the blast pipe should always be made as large as the demands of the blast will permit. DIAMETER OF FEED PIPES. There appear to be no theoretical considerations which would lead us to determine exactly the proper size of the feed pipes. Judging from practice, however, the following rule will be found to give the proper dimensions. RULE.-To find the diameter of the feed pipes.-Multiply the diameter of the cylinder in inches by'141. The product is the proper diameter of the feed pipes. Required the diameter of the feed pipes for a locomotive engine, the diameter of the cylinder being 15 inches. In this exanple we have, according to the rule, Diameter of feed-pipe = 15 x'141 = 2-115 inches, or between 2 and 2- inches. DIAMETER OF PISTON ROD. The diameter of the piston rod for the locomotive engine is usually about one-seventh the diameter of the cylinder. Making practice our guide, therefore, we have the following rule. RULE.- To find the diameter of the piston rod for the locomotive engine. —Divide the diameter of the cylinder in inches by 7. The quotient is the diameter of the piston rod in inches. The diameter of the cylinder of a locomotive engine is 15 inches; required the diameter of the piston rod. Here, according to the rule, diameter of piston rod =15. 7 = 21 inches. THICKNESS OF PISTON. The thickness of the piston in locomotive engines is usually about two-sevenths of the diameter of the cylinder. Making practice our guide, therefore, we have the following rule. RULE. —To find the thickness of the piston in the locomotive engine.-Multiply the diameter of the cylinder in inches by 2; divide Page 167 THE STEAM ENGINE. 167 the product by 7. The quotient is the thickness of the piston in inches. The diameter of the cylinder of a locomotive engine is 14 inches; required the thickness of the piston. Here, according to the rule, thickness of piston = 2 x 14 _ 7 = 4 inches. DIAMETER OF CONNECTING RODS AT MIDDLE. The following rule gives the diameter of the connecting rod at middle. The rule, we may remark, is entirely founded on practice. RULE. — To find the diameter of the connecting rod at middle of the locomotive engine.-Multiply the diameter of the cylinder in inches by'21. The product is the diameter of the connecting rod at middle in inches. Required the diameter of the connecting rods at middle for a locomotive engine, the diameter of the cylinders being twelve inches. For this example we have, according to the rule, Diameter of connecting rods at middle = 12 x'21 = 2'52 inches, or 2- inches. DIAMETER OF BALL ON CROSS-HEAD SPINDLE. The diameter of the ball on the cross-head spindle may be found from the following rule. RULE.- To find the diameter of the ball on cross-head spindle of a locomotive engine.-Multiply the diameter of the cylinder in inches by'23. The product is the diameter of the ball on the cross-head spindle. Required the diameter of the ball on the cross-head spindle of a locomotive engine, when the diameter of the cylinder is 15 inches. Here, according to the rule, Diameter of ball ='23 x 15 = 3'45 inches, or nearly 31 inches. DIAMETER OF THE INSIDE BEARINGS OF THE CRANK AXLE. It is obvious that the inside bearings of the crank axle of the locomotive engine correspond to the paddle-shaft journal of the marine engine, and to the fly-wheel shaft journal of the land-engine. We may conclude, therefore, that the proper diameter of these bearings ought to depend jointly upon the length of the stroke and the diameter of the cylinder. In the locomotive engine the stroke is usually 18 inches, so that we may consider that the diameter of the bearing depends solely upon the diameter of the cylinder. The following rule will give the diameter of the inside bearing. RULE.- To find the diameter of the inside bearing for t]he locomotive engine. —Extract the cube root of the square of the diameter of the cylinder in inches; multiply the result by'96. The product is the proper diameter of the inside bearing of the crank axle for the locomotive engine. Required the diameter of the inside bearing of the crank axle Page 168 168 THE PRACTICAL MODEL CALCULATOR. for a locomotive engine whose cylinders are of 13-inch diameters. In this example we have, according to the rule, 13 = diameter of cylinder in inches. 13 169 = square of the diameter of cylinder. 0 0 169(5'5289 = V/169 5 25 125 5 25 44000 5 50 41375 10 7500 2625 5 775 1820 150 8275 805 5 800 726 155 9075 79 5 3 160 910 5 3 165 913 and diameter of bearing = 5'5289 x *96 = 5'31 inches nearly; or between 5~ and 5U inches. DIAMETER OF THE OUTSIDE BEARINGS OF THE CRANK AXLE. The crank axle, in addition to resting upon the inside bearings, is sometimes also made to rest partly upon outside bearings. These outside bearings are added only for the sake of steadiness, and they do not need to be so strong as the inside bearings. The proper size of the diameter of these bearings may be found from the following rule. RULE. — To find the diameter of outside bearings for the locomotive engine.-Multiply the square of the diameters of the cylinders in inches by'396; extract the cube root of the product. The result is the diameter of the outside bearings in inches. Required the proper diameter of the outside bearings for a locomotive engine, the diameter of its cylinders being 15 inches. In this example we have, according to the rule, 15 = diameter of cylinders in inches. 15 225 = square of diameter of cylinder. ~396 = constant multiplier. 89'1 Page 169 THE STEAM ENGINE. 169 0 0 89.1(4.466 = -89'1 4 16 64 4 16 25100 4 32 21184 8 4800 3916 4 496 3528 120 5296 388 4 512 358 124 58so8 4 8 128 588 4 8 132 596 Hence diameter of outside bearing = 4'466 inches, or very nearly 4~ inches. DIAMETER OF PLAIN PART OF CRANK AXLE. It is usual to make the plain part of crank axle of the same sectional area as the inside bearings. Hence, to determine the sectional area of the plain part when it is cylindrical, we have the following rule. RULE. —To determine the diameter of the plain part of crank axle for the locomotive engine.-Extract the cube root of the square of the diameter of the cylinder in inches; multiply the result by'96. The product is the proper diameter of the plain part of the crank axle of the locomotive engine in inches. Required the diameter of the plain part of the crank axle for the locomotive engine, whose cylinders' diameters are 14 inches. Ir. this example we have, according to the rule, 14 = diameter of cylinder in inches. 14 196 = square of the diameter of cylinder. 0' 0 196(5'808 /196 5 25 125 5 25 71'000 5 50 70'112 10 7500 *888 5 1264 150 8764 8 1328 158 10092 8 166 8 174 UP Page 170 170 THE PRACTICAL MODEL CALCULATOR. Hence the plain part of crank axle = 5'808 x'96 = 5'58 nearly, or a little more than 5~ inches. DIAMETER OF CRANK PIN. The following rule gives the proper diameter of the crank pin. It is obvious that the crank pin of the locomotive engine is not altogether analogous to the crank pin of the marine or land engine, and, like them, ought to depend upon the diameter of the cylinder, as it is usually formed out of the solid axle. RULE.- To find the diameter of the crank pin for the loconotive engine.-Multiply the diameter of the cylinder in inches by'404. The product is the diameter of the crank pin in inches. Required the diameter of the crank pin of a locomotive engine whose cylinders' diameters are 15 inches. In this example we have, according to the rule, Diameter of crank pin = 15 x'404 = 6'06 inches, or about 6 inches. LENGTH OF CRANK PIN. The length of the crank pin usually given in practice may be found from the following rule. RuLE.-To find the length of the crank pin. —Multiply the diameter of the cylinder in inches by'233. The product is the length of the crank pins in inches. Required the length of the crank pins for a locomotive engine with a diameter of cylinder of 13 inches. In this example we have, according to the rule, Length of crank pin = 13 x -233 = 3'029 inches, or about 3 inches. The part of the crank axle answering to the crank pin is usually rounded very much at the corners, both to give additional strength, and to prevent side play. These then are the chief dimensions of locomotive engines according to the practice most generally followed. The establishment of express trains and the general exigencies of steam locomotion are daily introducing innovations, the effect of which is to make the engines of greater size and power: but it cannot be said that a plan of locomotive engine has yet been contrived that is free from grave objections. The most material of these defects is the necessity that yet exists of expending a large proportion of the power in the production of a draft; and this evil is traceable to the inadequate area of the fire-grate, which makes an enormous rush of air through the fire necessary to accomplish the combustion of the fuel requisite for the production of the steam. To gain a sufficient area of fire-grate, an entirely new arrangement of engine must be adopted: the furnace must be greatly lengthened, and perhaps it may be found that short upright tubes, or the very ingenious arrangement of Mr. Dimpfell, of Philadelphia, may be introduced with advantage. Upright tubes have been found to be more effectual in raising steam than horizontal tubes; but the tube plate in the case of upright tubes would be more liable to burn. Page 171 THE STEAM ENGINE. 171 We here give the preceding rules in formulas, in the belief that those well acquainted with algebraic symbols prefer to have a rule expressed as a formula, as they can thus see at once the different operations to be performed. In the following formulas we denote the diameter of the cylinder in inches by D. LOCOMOTIVE ENGINE.-PARTS OF THE CYLINDER. Area of induction ports, in square inches ='068 x D2. Area of eduction ports, in square inches ='128 x D2. Breadth of bridge between ports between 4 inch and 1 inch. LOCOMOTIVE ENGINE.-PARTS OF BOILER. Diameter of boiler, in inches = 3'11 x D. Length of boiler between 8 feet and 12 feet. Diameter of steam dome, inside, in inches = 1'43 x D. Height of steam dome = 2- feet. Diameter of safety valve, in inches = D *- 4. Diameter of valve spindle, in inches-'076 x D. Diameter of chimney, in inches = D. Area of fire-grate, in square feet ='77 x D. Area of heating surface, in square feet = 5 x D2 2. Area of water level, in square feet = 2'08 x D. Cubical content of water in boiler, in cubic feet = 9 x D2 _ 40. Diameter of feed-pump ram, in inches ='011 x D2. Cubical content of steam room, in cubic feet = 9 x D2. 40. Cubical content of inside fire-box above fire bars, in cubic feet = D2 4. Thickness of the plates of boiler - - inch. LOCOMOTIVE ENGINE.-DIMENSIONS OF SEVERAL PIPES. Inside diameter of steam pipe, in inches ='03 x D2. Inside diameter of branch steam pipe, in inches ='021 x D2. Inside diameter of the top of blast pipe ='017 x D2. Inside diameter of the feed pipes ='141 x D. LOCOMOTIVE ENGINE.-DIMENSIONS OF SEVERAL MOVING PARTS. Diameter of piston rod, in inches = D-. 7. Thickness of piston, in inches = 2 D - 7. Diameter of connecting rods at middle, in inches = -21 x D. Diameter of the ball on cross-head spindle, in inches ='23 x D. Diameter of the inside bearings of the crank axle, in inches = 96 x / ID2. Diameter of the plain part of crank axle, in inches ='96 x v D2. Diameter of the outside bearings of the crank axle, in inches = /'396 x D2. Diameter of crank pin, in inches ='404 x D. Length of crank pin, in inches ='233 x D. Page 172 172 THE PRACTICAL MODEL CALCULATOR. TABLE of the Pressure of Steam, in Inches of Mereury, at different Temperatures. Tenperature, Dalton. Ure. Young. Ivory. Tredgold. Southern. Robison. Watt. Fahren- Dtn heit. 00 008............... 10 0112...... 20 0-17......11 32 0o26 020 018... 017 0'16 0'.00 40 034 0-25 0.20... 0.24 0.22 010 50 049 0.36 0.36 0.36 0.37 0'33 0.20 60 0-65 0'52 053... 055 048 035 70 087 0'73 0'75 0'73 0'78 0'68 0.55 0(77 80 116 1.01 1'05... 1.11 0'95 0.82... 90 159 1.36 1.44 1-36 1.53 1.34 1.18 100 2-12 1'86 1'95... 2'08 1'84 1'60 1i'55 110 279 2.45 2.62 2-46 2.79 2'56 2.25 120 3-63 3.30 3.46... 3.68 3.46 3.00 130 471 4.37 4'54 4-41 4.81 4.43 3.95 140 6-05 5'78 5.88... 6-21 5'75 5'15 5'14 150 773 7.53 7'55 7.42 7.94 746 6'72 160 9799 9.60 9.62... 1005 9'52 865 892 170 1231 12'05 12'14 12'05 12'60 12'14 1105 1137 180 15-38 15.16 1523... 15.67 15'20 14'05 12.73 190 18-98 19-00 18'96 18'93 19'00... 17.85 19.00 200 23-51 23,60 2344... 23.71... 22.65 210 2882 28.88 28.81 28.81 28.86... 2862 212 30-00 30.00 30.00 30o00 30000 30.00 30.00 29'40 220 3518 35'54 35'19... 34'92... 35'8 33.65 230 44-60 43.10 42'47 42-63 42.00... 44'5 40 240 53o45 5170 5166... 50o24... 549 49o0 TABLE of the Temiperature of Steam at different Pressures in Atmospheres. I-Pressure in IFrench Faki Presoure in French Dr. Ure. Young. Ivory. Tredgold. Southorn. Robioon. Watt Franklin Atmospheres. Academy. Instituto. 1st At. 212.00 2120 2120 2120 2120... 212 2120 2120 2d At. 250-5 250-0 240-3 249 250 250-3 252-5 250-0 3d At. 2752 275'0 271... 274 267... 2752 4th At. 29837 2915 288 290 294 293"4..... 291'5 5th At. 3OS8 3045 302... 309....... 304.5 6th At. 3204 315'5...... 3 315'5 7th At. 3317 3255......... 3265 8th At. 342.0 3360... 337 342 3436... 3360 9th At. 3500 345'0......... 3450 10th At. 358-9.................. 352-5 11th At. 3850' 12th At. 374'0......... 372'... 0.. 13th At. 3806'.................. 14th At. 386-9..................... 15th At. 3928.................. 3...838 16th At. 39875....... 7........... 17th At. 4038............... 18th At. 4089................... 19th At. 413-9............... 20th At. 4185......... 414...... 405 30th At. 4572................... 40th At 466.6............... 50th At. 510-6............. Page 173 THE STEAM ENGINE. 173 TABLE of the Expansion of Air hy hIieat. Fahren. Fahren. Fahliren. 32......... 1000 61.. 1069 90........ 1132 33......... 1002 62......... 1071 91......... 1134 34......... 1004 63......... 1073 92......... 1136 35......... 1007 64......... 1075 93......... 1138 36........ 1009 65......... 1077 94......... 1140 37......... 1012 66......... 1030 95......... 1142 38......... 1015 67......... 1080 96........ 1144 39......... 1018 68......... 1034 97......... 1146 40......... 1021 69......... 1087 98........ 1148 41......... 1023 70........ 1089 99......... 1150 42......... 1025 71......... 1091 100......... 1152 43... 1027 72......... 1093 110....... 1173 44......... 1030 73......... 1095 120... 1194 45......... 1032 74......... 1097 130......... 1215 46......... 1034 75......... 1099 140......... 1235 47......... 1036 76......... 1101 150......... 1255 48......... 1038 77......... 1104 160......... 1275 49... 1040 78........ 1106 170......... 1295 50... 1043 79......... 1108 180......... 1315 51.. 1045 80......... 1110 190........ 1334 52......... 1047 81........ 1112 200......... 1364 53......... 1050 82... 1114 210......... 1372 54......... 1052 83......... 1116 212......... 1376 55......... 1055 84......... 1118 302......... 1558 56......... 1057 85......... 1121 392......... 1739 57...... 1059 86.... 1123 482......... 1919 58......... 1062 87......... 1125 572......... 2098 59......... 1064 88........ 1128 680......... 2312 60......... 1066 89......... 1130 STRENGTH OF MATERIALS. The chief materials, of which it is necessary to record the strength in this place, are cast and malleable iron; and many experiments have been made at different times upon each of these substances, though not with any very close correspondence. The following is a summary of them:i Materials. C S E M Iron, cast to 16300 } 8100 69120000 5530000 Iron, cast~to............... 36000 I Malleable........... 60000 9000 91440000 67,70000 - Wire................... 80000 The first column of figures, marked C, contains the mean strength of cohesion on an inch section of the material; the second, marked S, the constant for transverse strains; the third, marked E, the constant for deflections; and the fourth, marked MI, the modulus of elasticity. The introduction of the hot blast iron brought with it the impression that it was less strong than that previously in use, and the experiments which had previously been confided in as giving results near enough the truth, for all practical purposes, were no longer considered to be applicable to the new state of things. New experiments were therefore made. The following Table gives, we have no doubt, results as nearly correct as can be required or attained:P2 Page 174 174 THE PRACTICAL MODEL CALCULATOR. RESULTS OF EXPERIMENTS ON THE STRENGTH AND OTHER PROPERTIES OF CAST IRON. IN the following Table each bar is reduced to exactly one inch square; and the transverse strength, which may be taken as a criterion of the value of each Iron, is obtained from a mean between the experiments upon it;-first on bars 4 ft. 6 in. between the supports; and next onthose of half the length, or 2 ft. 3 in. between the supports. All the other results are deduced from the 4 ft. 6 in. bars. In all cases the weights were laid on the middle of the bar. z~~~~~~~ P-. 7'030 1870000 5.10 532 600 1'530 991 Gray Ponkey, No. 3. Cold Blast...... 7'122 17211000 567 595 581 1.747 992'Wh itish Kray Devon, No. 3. Hot Blast~:....... 7'251- 22473650 537 - 537 1'09 589 White Oldberry, No. 3. Hot Blast...... 7'300 22733400 543 517 530 1'005 549 White Pattison, N.AJ. Hot Blast*s..... 7'06 17873100 520 534 527 1'365 710 Whitish gray Beaufort,. No. 3. Hot Blast...... - 7'069 16802000 505 529 517 1'59,9 807 D~ullish gray Pennsylvania-n................ 7'8 15379500 500 5-15 502 1'815 889- Dark gray Bute, No. 1. Cold Blast.... 7'066 15163000 495 487 491 1'764 872 Bluish gray Wind Mill End, No. 2. Cold Blast 7'071 16490000 483 495 489 1'581 765 Dark gray Old Park, No. 2. Cold Blast...7'049 14607000 441 529 485 1'621 718 Gray Beaufort, No. 2. I-ot Blast.... 7'108 163,01000 4:78 470 474 1'512 729 Dull gray Low Moor,:No. 2. Cold Blast-.... 7'055 14509500 462 483 472 1'852 855 Dark gray Buffery, No. 1. Cold Blast*....- 7'079 15381200 463 463 1'55 721 Gray Brimbo, No. 2. Cold Blast...... 7'017 14911,666 466 453 459 1'748 815 Light gray Apedale, No. 2. Hlot Blast..... 7'017 14852000 457 4,55 456 1'730 791 Ligwht gray Oldberry, No. 2. Cold Blast...7'059 14307500 453 457' 455 1'811 822 Dark gray Pentwyn, No. 2- -......... 7'038 15193000 438 473 455 1'484 650 Bluish gray Maesteg, No. 2......... 7'038 13959500 453 455 454 1'957 886 Dark gray Muirkirk, No. 1.'-Cold Blast*...7'113 14003550 4:43 464 453 1'734 770 Bright gray Adelphi, No. 2. Cold Blast.... 7'080 13815500 441 457 449 1'759 777 Light gray Blania, No. 3. Cold Blast....... 7'159 14281466 433 464 448 1'726 747 Bright gray Devon, No. 3. Cokl!..Blast*...... 7'285 22907700 448 -- 4148'790 353 Light gray Gart;sherrie, No.:~?:-'Iot Blast ~,7'017 13894000 427 467 447 1'557 998 Light gray Frood, No. 2. Cold Blast........- - 7'031 13112666 460 434 447 1'825 841 Light gray La~ne End,. No. 2............... 7'028 15787666 444 -- 444 J'414 629 Dark gray Carron, No. 3. Cold Blast*...... - 7'094 16246966 444 443 443 1'336 593 Gray IDundyvan, No. 3. Cold Blast. 7'087' 16534000 456 430 443 1'469 674 Dull gray iMaesteg (Max'kedi Red)......... 7'038 13971500 440 444 442 1'887 830'Bluish gray (]orbynus Hall, No. 2....... 7' 007 13845866 430 454 442 1.687 727 Gray Pontnypool, No. 2..........'7'080 13136500 439 4,41 i440 1'857 816 Dull blue Wallbrook, No. 3.............. 6'979 15394766 432 449 440 1'443~ 625 Light gray Milton, No. 3. Hot Blast. - -... 7'051 15852500 427 449 438 1'368 585 Gray' Buffery, No. 1. Hot ]]last*....6'998 13730500' 436 -- 4136 1'64 721 Dul1l gray Level, No. 1. Hot Blast......... 7,080 15452500 461 403 432 1'516 699 Light gray Pant, No. 2.................... 6'.975 15280900 408 455 431 1'251 511 Light gray Level, No. 2. Hot Blast......... 7'031 15241000 419 439 429 1'358 570'Dull gray W. S. S.,}No. 2................ ~7'041. 14953333 413 446 429 1-339 554 Light gray ~Eagle Foundry, No. 2. Hot; Blast 7'038 14211000 408 446'427 1'512 618 Bluish gray Elsicar, No. 2. Cold Blast....... 6'928 12586500 446 408 427 2.-224-. 992 Gray Varteg, No. -2. Hot:~Ulast -......-. 7'007'1501-2000 422 430 426 1'450 621. Gray ClottaN..H Blast...... 7.128 15510066 4435211-5132 716 Whitish gray Coltham, No. 1. 1 7 464 385 424~ Carroll, No. 2. Cold Blast...... 7'069 17036000.430 408 419 1'231. 530,Gray Muirkirk, No. 1. Hot; Blast*...~ 6'953 13294400 417 419 418 1'570 656 Bluish gray Bierley, No,-2. -'........ 7'185 16~156133 404 1432 418 1.2221 494 Dark gray Coed-Talon, No. 2. I-ot Blast*..6'969 14322500 409 424 410 1'882' 771 Bright gray Coed-Talon, No. 2. Cold Blast*. 6'955 14304000 408 418 413 1'47 0 600 G ray Monkland, No. 2, Hot Blasi....,6'916 12259500 402 40-1 403 1'762 70 9 Bluish gray Ley's Works, No. 1. Hot Blast. 6'957 11539333 392 - 392 1'890 742 Bluish gmay Milton, No. 1. IHot B]last....... 6.976 11974500 353 386 369 1.525 538 G'ray Plaskynastou, 1No. 2. Hot Blastv 6-916 13341633 378 337 3507 1'366 517 Light gray The irons with asterisks are taken. from Experiments on Hot and Cold Blast Iron. Naa~~E OP IRO+J 0.0 ii "; I~.m~~d~lJ Co~o51 Dickerson's, Newark~, N. J~.~~ 7 030 1847r0000 5~10 532 600 1~530 991 Gray Ponkey, No. 3. Cold Blast. 7~~ ~$122 1_7211000 607 595 581 1~747 992'Whitish gray Devon, No. 3. Hot Blast0: 7~~ 7251 22473090 537'- 537 1~09 589 WVhite Oldberry, No. 3. Hot Blast~~~ ~ 300 p7533400 543 517 530 1~005 5419 W~hite Pattison, N. 3. Hot Blast0.~ i~ ~ 056 178'73100 520 534 527 1~365 710 W~hitish giay Beaufort,. No. 3. Hot Blast. 7 069 16802000 505 529 517 1~599 807 Dulli h giay Pennsylvanian. ~~~~~~~~~78 15579500 500 515 502 1~816 889 Baik gsay Bute, No. 1. Cold Blast... ~ ~~ 7 066 15163000 495 487 491 1~764 872 Bluishgiaty wind Mill End, No. 2. Cold B st 7071 10490000 483 405 489 1~581 76o Bask gray Old Park, No. 2. Cold Blast... 7~~1049 14607000 441 529 485 1~621 718 Gr~ay Beaufort, No. 2. H-ot Blast.~~~7~108 16301000 478 470 474 1~512 729 B~ull gray Low Moor, No. 2. Cold......... 7~055 14509500 462 483 472 1~852 855 B~ark gray Duffory, No. 1. Cold Blast' *... 7'079 15381200 463 - 1463 1~55 721 Gra~y Brimbo, N\o. 2. Cold Blast.~~ ~~7~017 14911666 466 453 459 1~748 815 Light gray Apedals, No. 2. Hot Blast.7~ ~ i017 14852000 457 4,55 456 1~730 791 Light gray Oldberry, No. 2. Cold Blast.... 7'059 14307500 453 457' 455 1~811. 822 Bai~kg~ray Pentwyn, No. 2.~~~~~~~ ~ 7~038 15193000 438 473 455 1~484 650 Blui~sh gray M~aesteg, No. 2.~~~~~~~~7~038 13959500 453 455 454 1~957 880 Bark gray M~uirkirk, No.1.'Cold Blast'.... 7 i113 14003550 443 464 453 1'734 770 Bright gray Adelphi, N~o. 2. Cold Blast. ~~~7~080 13835500 441 457 449 17159 7717 Light gray Blania, No. 3. Cold Blast. 7~~~ i159 14281466 433 464 448 17'26 747 Bright gray Devon, No. 3. Cold'.Blast5.~~~7~285 22907700 448 --- 448'790 353 Light gray Gartsherrie, No.'.~'lIot Blast;. 7.017 13894000 427 467 447 1~557 998 Light gray rrood, No. 2. Cold Blast.~~~~7~031 13112666 460 434 447 1~825 841 Light gray Lape End, No. 2..~~~~~~ ~7~028 15787666 444 - 1444'1~434 629 B~ark grayg Carron, No. 3. Cold Blast*.7~~ 1094 16246960 444 443 443 1~336. 593 Gray Dundyvan, No. 3. Cold Blast.~; ~. 7~087') 16534000 456 430 443 1'469 674 Bull gray Maesteg (Masked Red).~~~~7~038 1397i1500 440 444 442. 1~887 830 I'Bluish gray Corbyns Hall, No. 2.~- ~~-7~007 13845866 430 454 442 1.687 7127 Graty Pontypool, No. 2.~~~~~ ~7.080 13536500 439 4411 440 1.857 816 D~ull blue Walibrook, ~No. 3.~~~~~~~ 6~979 15394766 432 449 440 1~443 625 Light gray Milton, No. 3. hot Blast.~~~ ~~7~051 15852500 427 449 438 1~368 585 Gray Buffery, No. 1. Hot Blast' ~.~~6'998 13730500' 436 --- 436 1~64'723 Bull1 gray Level, No. 1. Hot Blast. 7 ~ ~~''080 15452500 461 403 432 1~516 699 Light gray Pant, No. 2.~~~~~~~~~~6975 15280900 408 455 431 1~251 511 Light gray Level, No. 2. Hot Blast.~~~~7'031 15241000 419 439 429 1~358 570 ('Dull gray W.S S.5, No. 2........ r 7~041.1 14953333 413 446 429 1~339 554 Light gray Eagle Ponudry, No. 2. Hot Blast 7~038 14211000 408 446'42'7 11512 618 Bluish gray lilsicar, No. 2. Cold Blast.~~~6~928 12586500 446- 408 427 i2224 992 Gray varteg, No.2. Hot iBlast.~~~ 7'007'15012000 422 430 426 1450 621~ Gray Coltham, No. 1. Hot Blast. ~~7~128 15510066 464 385 424 1~532 716 W~hitish gray Carroll, No. 2. Cold Blast.~~ ~~7~069 17036000.430 408 419 1~23t 530 ~Gray Muirkirk, No. 1. Hot Blast*.~-~6~953 13294400 417 419 418 1~570 050 B3luish gray Bierley, No.,2. 7~~~~~~~~~$185 10156133 404 432 418 1~222 494 Bazrk gray Coed.Talon, No. 2. Hot Blast~. 63969 14322500 409 424 410 1~882 771 B3rig~ht gray Coed.Talon, No. 2. Cold Blast'~.. 955 14304000 408 438 413 1~470 000 Glray Mionkrland, No. 2. Hot Blaa6... 6 E9~16 12259500 402 404 403 1~762 7109 Bluish gray Lsy's Works, No. 1. Hot Blast.. 0 i957 11539333'392 --- 392 18s90( 7421 Bluish gray Milton, No.1. Hot Blast.~-~- 6076G 13974500 333 386 309 1~525 538 G~ray Plaskynaston, No. 2. Hot Blast.~ 6~910 13341633 378 337 357 1~300 517 Light graytS The irons with asterisks are taken from Experiments on Hot and Cold ~Blast Iron. Page 175 THE STEAM ENGINE. 175 RULE.-To find from the above Table the breaking weight in rectangular bars, generally. Calling b and d the breadth and depth in inches, and I the distance between the supports, in feet, 4'5 x b d2 S and putting 4'5 for 4 ft. 6 in., we have 1 = breaking weight in lbs.,-the value of S being taken from the above Table. For example: —What weight would be necessary to break a bar of Low Moor Iron, 2 inches broad, 3 inches deep, and 6 feet between the supports? According to the rule given above, we have b = 2 inches, d = 3 inches, 1 = 6 feet, S = 4721 from the Table. 4*5 x bd2S 4'5 x 2 x32 x 472 Then - 6 = 6372 lbs., the breaking weight. TABLE of the Cohesive Power of Bodies whose Cross Sectional Areas equal one Square Inch. METALS. Cohesive Power in lbs. Swedish bar iron...................................... 65,000 Russian do.............5.......................... 9,470 English do...................... 56,000 Cast steel................................. 134,256 Blistered do................................................ 133,152 Shear do................................................ 127,632 Wrought copper...........................,33,892 Hard gun-metal........................................... 36,368 Cast copper....................................... 19,072 Yellow brass, cast....................................... 17,968 Cast iron............................................... 17,628 Tin, cast.................................................... 4,736 Bismuth, cast............................................. 3,250 Lead, cast................................................. 1,824 Elastic power or direct tension of wrought iron, medium quality........................................ 22,400 NOTE.-A bar of iron is extended'000096, or nearly one tenthousandth part of its length, for every ton of direct strain per square inch of sectional area. CENTRE OF GRAVITY. The centre of gravity of a body is that point within it which continually endeavours to gain the lowest possible situation; or it is that point on which the body, being freely suspended, will remain at rest in all positions. The centre of gravity of a body does not always exist within the matter of which the body is composed, there being bodies of such forms as to preclude the possibility of this being the case, but it must either be surrounded by the constituent matter, or so placed that the particles shall be symmetrically situated, with respect to a vertical line in which the position of the centre occurs. Thus, the centre of gravity of a ring is not in the substance of the ring itself, but, if the ring be uniform, it will be in the axis of its circumscribing cylinder; and if the ring varies Page 176 176 THE PRACTICAL MODEL CALCULATOR. in form or density, it will be situated nearest to those parts where the weight or density is greatest. Varying the position of a body will not cause any change in the situation of the centre of gravity; for any change of position the body undergoes will only have the effect of altering the directions of the sustaining forces, which will still preserve their parallelism. When a body is suspended by any other point than its centre of gravity, it will not rest unless that centre be in the same vertical line with the point of suspension; for, in every other position, the force which is intended to insure the equilibrium will not directly oppose the resultant of gravity upon the particles of the body, and of course the equilibrium will not obtain; the directions of the forces of gravity upon the constituent particles are all parallel to one another and perpendicular to the horizon. If a heavy body be sustained by two or.more forces, their lines of direction must meet either at the centre of gravity, or in the vertical line in which it occurs. A body cannot descend or fall downwards, unless it be in such a position that by its motion the centre of gravity descends. If a body stands on a plane, and a line be drawn perpendicular to the horizon, and if this perpendicular line fall within the base of the body, it will be supported without falling; but if the perpendicular falls without the base of the body, it will overset. For when the perpendicular falls within the base, the body cannot be moved at all without raising the centre of gravity; but when the perpendicular falls without the base towards any side, if the body be moved towards that side, the centre of gravity will descend, and consequently the body will overset in that direction. If a perpendicular to the horizon from the centre of gravity fall upon the extremity of the base, the body may continue to stand, but the least force that can be applied will cause it to overset in that direction; and the nearer the perpendicular is to any side the easier the body will be made to fall on that side, but the nearer the perpendicular is to the middle of the base the firmer the body will stand. If the centre of gravity of a body be supported, the whole body is supported, and the place of the centre of gravity must be considered as the place of the body, and it is always in a line which is perpendicular to the horizon. In any two bodies, the common centre of gravity divides the line that joins their individual centres into two parts that are to one another reciprocally as the magnitudes of the bodies. The products of the bodies multiplied by their respective distances from the common centre of gravity are equal. If a weight be laidt upon any point of an inflexible lever which is supported at the ends, the pressure on each, point of the support will be inversely as the respective distances from the point where the weight is applied. In a system of three bodies, if a line be drawn from the centre of gravity of any one of them to the common centre of the other two, then the common centre of all the three bodies divides the line into two parts that are to each other reciprocally as the Page 177 THE STEAM ENGINE. 177 magnitude of the body from which the line is drawn to the sum of the magnitudes of the other two; and, consequently, the single body multiplied by its distance from the common centre of gravity is equal to the sum of the other bodies multiplied by the distance of their common centre from the common centre of the system. If there be taken any point in the straight line or lever joining the centres of gravity of two bodies, the sum of the two products of each body multiplied by its distance from that point is equal to the product of the sum of the bodies multiplied by the distance of their conmmon centre of gravity from the same point. The two bodies have, therefore, the same tendency to turn the lever about the assumed point, as if they were both placed in their common centre of gravity. Or, if the line with the bodies moves about the assumed point, the sum of the momenta is equal to the momentum of the sum of the bodies placed at their common centre of gravity. The same property holds with respect to any number of bodies whatever, and also when the bodies are not placed in the line, but in perpendiculars to it passing through the bodies. If any plane pass through the assumed point, perpendicular to the line in which it subsists, then the distance of the common centre of gravity of all the bodies from that plain is equal to the sum of all the momenta divided by the sum of all the bodies. We may here specify the positions of the centre of gravity in several figures of very frequent occurrence. In a straight line, or in a straight bar or rod of uniform figure and density, the position of the centre of gravity is at the middle of its length. In the plane of a triangle the centre of gravity is situated in the straight line drawn from any one of the angles to the middle of the opposite side, and at two-thirds of this line distant from the angle where it originates, or one-third distant from the base. In the surface of a trapezium the centre of gravity is in the intersections of the straight lines that join the centres of the opposite triangles made by the two diagonals. The centre of gravity of the surface of a parallelogram is at the intersection of the diagonals, or at the intersection of the two lines which bisect the figure from its opposite sides. In any regular polygon the centre of gravity is at the same point as the centre of magnitude. In a circular arc the position of the centre of gravity is distant from the centre of the circle by the measure of a fourth proportional to the arc, radius, and chord. In a semicircular arc the position of the centre of gravity is distant from the centre by the measure of a third proportional to the arc of the quadrant and the radius. In the sector of a circle the position of the centre of gravity is distant from the centre of the circle by a fourth proportional to three times the arc of the sect9r, the chord of the are, and the diameter of the circle. In a circular segment, the position of the centre of gravity is distant from the centre of the circle by a space which is equal to the cube or third power of the chord divided by twelve times the area of the segment. In a semicircle 12 Page 178 178 THE PRACTICAL MODEL CALCULATOR. the position of the centre of gravity is distant from the centre of the circle by a space which is equal to four times the radius divided by the constant number 3'1416 x 3 = 9'4248. In a parabola the position of the centre of gravity is distant from the vertex by three-fifths of the axis. In a semi-parabola the position of the centre of gravity is at the intersection of the co-ordinates, one of which is parallel to the base, and distant from it by two-fifths of the axis, and the other parallel to the axis, but distant from it by three-eighths of the semi-base. The centres of gravity of the surface of a cylinder, a cone, and conic frustum, are respectively at the same distances from the origin as are the centres of gravity of the parallelogram, the triangle, and the trapezoid, which are sections passing along the axes of the respective solids. The centre of gravity of the surface of a spheric segment is at the middle of the versed sine or height. The centre of gravity of the convex surface of a spherical zone is at the middle of that portion of the axis of the sphere intercepted by its two bases. In prisms and cylinders the position of the centre of gravity is at the middle of the straight line that joins the centres of gravity of their opposite ends. In pyramids and cones the centre of gravity is in the straight line that joins the vertex with the centre of gravity of the base, and at three-fourths of its length from the vertex, and one-fourth from the base. In a semisphere, or semispheroid, the position of the centre of gravity is distant from the centre by threeeighths of the radius. In a parabolic conoid the position of the centre of gravity is distant from the base by one-third of the axis, or two-thirds of the axis distant from the vertex. There are several other bodies and figures of which the position of the centre of gravity is known; but as the position in those cases cannot be defined without algebra, we omit them. CENTRIPETAL AND CENTRIFUGAL FORCES. Central forces are of two kinds, centripetal and centrifugal. Centripetal force is that force by which a body is attracted or impelled towards a certain fixed point as a centre, and that point towards which the body is urged is called the centre of attraction or the centre of force. Centrifugal force is that force by which a body endeavours to recede from the centre of attraction, and from which it would actually fly off in the direction of a tangent if it were not prevented by the action of the centripetal force. These two forces are therefore antagonistic; the action of the one being directly opposed to that of the other. It is on the joint action of these two forces that all curvilinear motion depends. Circular motion is that affection of curvilinear motion where the body is constrained to move in the circumference of a circle: if it continues to move so as to describe the entire circle, it is denominated rotatory motion, and the body is said to revolve in a circular orbit, the centre of which is called the centre of motion. In all circular motions the deflection or deviation from the rectilinear course is constantly the same at Page 179 THE STEAM ENGINE. 179 every point of the orbit, in which case the centripetal and centrifugal forces are equal to one another. In circular orbits the centripetal forces, by which equal bodies placed at equal distances from the centres of force are attracted or drawn towards those centres, are proportional to the quantities of matter in the central bodies. This is manifest, for since all attraction takes place towards some particular body, every particle in the attracting body must produce its individual effect; consequently, a body containing twice the quantity of matter will exert twice the attractive energy, and a body containing thrice the quantity of matter will operate with thrice the attractive force, and so on according to the quantity of matter in the attracting body. Any body, whether large or small, when placed at the same distance from the centre of force, is attracted or drawn through equal spaces in the same time by the action of the central body. This is obvious from the consideration that although a body two or three times greater is urged with two or three times greater an attractive force, yet there is two or three times the quantity of matter to be moved; and, as we have shown elsewhere, the velocity generated in a given time is directly proportional to the force by which it is generated, and inversely as the quantity of matter in the moving or attracted body. But the force which in the present instance is the weight of the body is proportional to the quantity of matter which it contains-; consequently, the velocity generated is directly and inversely proportional to the quantity of matter in the attracted body, and is, therefore, a given or a constant quantity. Hence, the centripetal force, or force towards the centre of the circular orbit, is not measured by the magnitude of the revolving body, but only by the space which it describes or passes over in a given time. When a body revolves in a circular orbit, and is retained in it by means of a centripetal force directed to the centre, the actual velocity of the revolving body at every point of its revolution is equal to that which it would acquire by falling perpendicularly with the same uniform force through.one-fourth of the diameter, or one-half the radius of its orbit; and this velocity is the same as would be acquired by a second body in falling through half the radius, whilst the first body, in revolving in its orbit, describes a portion of the circumference which is equal in length to half the diameter of the circle. Consequently, if a body revolves uniformly in the circumference of a circle by means of a given centripetal force, the portion of the circumference which it describes in any time is a mean proportional between the diameter of the circle and the space which the body would descend perpendicularly in the same time, and with the same given force continued uniformly. The periodic time, in the doctrine of central forces, is the time occupied by a body in performing a. complete revolution round the centre, when that body is constrained to move in the circumference by means of a centripetal force directed to that point; and when Page 180 180 THE PRACTICAL MODEL CALCULATOR. the body revolves in a circular orbit, the periodic time, or the time of performing a complete revolution, is expressed by the term t V' d, and the velocity or space passed over in the time t will be V ds; in which expressions d denotes the diameter of the circular orbit described by the revolving body, s the space descended in any time by a body falling perpendicularly downwards with the same uniform force, t the time of descending through the space, s and A the circumference of a circle whose diameter is unity. If several bodies revolving in circles round the same or different centres be retained in their orbits by the action of centripetal forces directed to those points, the periodic times will be directly as the square roots of the radii or distances of the revolving bodies, and inversely as the square roots of the centripetal forces, or, what is the same thing, the squares of the periodic times are directly as the radii, and inversely as the centripetal forces. CENTRE OF GYRATION. The centre of gyration is that point in which, if all the constituent particles, or all the matter contained in a revolving body, or system of bodies, were concentrated, the same angular velocity would be generated in the same time by a given force acting at any place as would be generated by the same force acting similarly on the body or system itself according to its formation. The angular motion of a body, or system of bodies, is the motion of a line connecting any point with the centre or axis of motion, and is the same in all parts of the same revolving system. In different unconnected bodies, each revolving about a centre, the angular velocity is directly proportional to the absolute velocity, and inversely as the distance from the centre of motion; so that, if the absolute velocities of the revolving bodies be proportional to their radii or distances, the angular velocities will be equal. If the axis of motion passes through the centre of gravity, then is this centre called the principal centre of gyration. The distance of the centre of gyration from the point of suspension, or the axis of motion in any body or system of bodies, is a geometrical mean between the centres of gravity and oscillation from the same point or axis; consequently, having found the distances of these centres in any proposed case, the square root of their product will give the distance of the centre of gyration. If any part of a system be conceived to be collected in the centre of gyration of that particular part, the centre of gyration of the whole system will continue the same as before; for the same force that moved this part of the system before along with the rest will move it now without any change; and consequently, if each part of the system be collected into its own particular centre, the common centre of the whole system will continue the same. If a circle be described about the centre of gravity of any system, and the axis of rotation be made to pass through any point of the circumference, Page 181 THE STEAM ENGINE. 181 the distance of the centre of gyration from that point will always be the same. If the periphery of a circle revolve about an axis passing through the centre, and at right angles to its plane, it is the same thing as if all the matter were collected into any one point in the periphery. And moreover, the plane of a circle or a disk containing twice the quantity of matter as the said periphery, and having the same diameter, will in an equal time acquire the same angular velocity. If the matter of a revolving body were actually to be placed in the centre of gyration, it ought either to be arranged in the circumference, or in two points of the circumference diametrically opposite to each other, and equally distant from the centre of motion, for by this means the centre of motion will coincide with the centre of gravity, and the body will revolve without any lateral force on any side. These are the chief properties connected with the centre of gyration, and the following are a few of the cases in which its position has been ascertained. In a right line, or a cylinder of very small diameter revolving about one of its extremities, the distance of the centre of gyration from the centre of motion is equal to the length of the revolving line or cylinder multiplied by the square root of ~. In the plane of a circle, or a cylinder revolving about the axis, it is equal to the radius multiplied by the square root of I. In the circumference of a circle revolving about the diameter it is equal to the radius multiplied by the square root of ~. In the plane of a circle revolving about the diameter it is equal to one-half the radius. In a thin circular ring revolving about one of its diameters as an axis it is equal to the radius multiplied by the square root of i. In a solid globe revolving about the diameter it is equal to the radius multiplied by the square root of 2. In the surface of a sphere revolving about the diameter it is equal to the radius multiplied by the square root of 2. In a right cone revolving about the axis it is equal to the radius of the base multiplied by the square root of -. In all these cases the distance is estimated from the centre of the axis of motion. We shall have occasion to illustrate these principles when we come to treat of fly-wheels in the construction of the different parts of steam engines. When bodies revolving in the circumferences of different circles are retained in their orbits by centripetal forces directed to the centres, the periodic times of revolution are directly proportional to the distances or radii of the circles, and inversely as the velocities of motion; and the periodic times, under like circumstances, are directly as the velocities of motion, and inversely as the centripetal forces. If the times of revolution are equal, the velocities and centripetal forces are directly as the distances or radii of the circles. If the centripetal forces are equal, the squares of the times of revolution and the squares of the velocities are as the distances or radii of the circles. If the times of revolution are as Q Page 182 182 THE PRACTICAL MODEL CALCULATOR. the radii of the circles, the velocities will be equal, and the centripetal forces reciprocally as the radii. If several bodies revolve in circular orbits round the same or different centres, the velocities are directly as the distances or radii, and inversely as the times of revolution. The velocities are directly as the centripetal forces and the times of revolution. The squares of the velocities are proportional to the centripetal forces, and the distances or radii of the circles. When the velocities are equal, the times of revolution are proportional to the radii of the circles in which the bodies revolve, and the radii of the circles are inversely as the centripetal forces. If the velocities be proportional to the distances or radii of the circles, the centripetal forces will be in the same ratio, and the times of revolution will be equal. If several bodies revolve in circular orbits about the same or different centres, the centripetal forces are proportional to the distances or radii of the circles directly, and inversely as the squares of the times of revolution. The centripetal forces are directly proportional to the velocities, and inversely as the times of revolution. The centripetal forces are directly as the squares of the velocities, and inversely as the distances or radii of the circles. When the centripetal forces are equal, the velocities are proportional to the times of revolution, and the distances as the squares of the times or as the squares of the velocities. When the central forces are proportional to the distances or radii of the circles, the times of revolution are equal. If several bodies revolve in circular orbits about the same or different centres, the radii of the circles are directly proportional to the centripetal forces, and the squares. of the periodic times. The distances or radii of the circles are directly as the velocities and periodic times. The distances or radii of the circles are directly as the squares of the velocities, and reciprocally as the centripetal forces. If the distances are equal, the centripetal forces are directly as the squares of the velocities, and reciprocally as the squares of the times of revolution; the velocities also are reciprocally as the times of revolution. The converse of these principles and properties are equally true; and all that has been here stated in regard to centripetal forces is similarly true of centrifugal forces, they being equal and contrary to each other. The quantities of matter in all attracting bodies, having other bodies revolving about them in circular orbits, are proportional to the cubes of the distances directly, and to the squares of the times of revolution reciprocally. The attractive force of a body is directly proportional to the quantity of matter, and inversely as the square of the distance. If the centripetal force of a body revolving in a circular orbit be proportional to the distance from the centre, a body let fall from the upper extremity of the vertical diameter will reach the centre in the same time that the revolving body describes one-fourth part of the orbit. The velocity of the descending body at any point of the diameter is proportional to Page 183 THE STEAM ENGINE. 183 the ordinate of the circle at that point; and the time of falling through any portion of the diameter is proportional to the arc of the circumference whose versed sine is the space fallen through. All the times of falling from any altitudes whatever to the centre of the orbit will be equal; for these times are equal to one-fourth of the periodic times, and these times, under the specified conditions, are equal. The velocity of the descending body at the centre of the circular orbit is equal to the velocity of the revolving body. These are the chief principles that we need consider regarding the motion of bodies in circular orbits; and from them we are led to the consideration of bodies suspended on a centre, and made to revolve in a circle beneath the suspending point, so that when the body describes the circumference of a circle, the string or wire by which it is suspended describes the surface of a cone. A body thus revolving is called a conical pendulum, and this species of pendulum, or, as it is usually termed, the governor, is of great importance in mechanical arrangements, being employed to regulate the movements of steam engines, water-wheels, and other mechanism. As we shall have occasion to show the construction and use of this instrument when treating of the parts and proportions of engines, we need not do more at present than state the principles on which its action depends. We must, however, previously say a few words on the properties of the simple pendulum, or that which, being suspended from a centre, is made to vibrate from side to side in the same vertical plane. PENDULUMS. If a pendulum vibrates in a small circular are, the time of performing one vibration is to the time occupied by a heavy body in falling perpendicularly through half the length of the pendulum as the circumference of a circle is to its diameter. All vibrations of the same pendulum made in very small circular arcs, are made in very nearly the same time. The space described by a falling body in the time of one vibration is to half the length of the pendulum as the square of the circumference of a circle is to the square of the diameter. The lengths of two pendulums which by vibrating describe similar circular arcs are to each other as the squares of the times of vibration. The times of pendulums vibrating in small circular arcs are as the square roots of the lengths of the pendulums. The velocity of a pendulum at the lowest point of its path is proportional to the chord of the arc through which it descends to acquire that velocity. Pendulums of the same length vibrate in the same time, whatever the weights may be. From which we infer, that all bodies near the earth's surface, whether they be heavy or light, will fall through equal spaces in equal times, the resistance of the air not being considered. The lengths of pendulums vibrating in the same time in different positions of the earth's surface are as the forces of gravity in those positions. The times wherein pendulums of the same length will vibrate by different forces of gravity are inversely as the square Page 184 184 THE PRACTICAL MODEL CALCULATOR. roots of the forces. The lengths of pendulums vibrating in different places are as the forces of gravity at those places and the squares of the times of vibration. The times in which pendulums of any length perform their vibrations are directly as the square roots of their lengths, and inversely as the square roots of the gravitating forces. The forces of gravity at different places on the earth's surface are directly as the lengths of the pendulums, and inversely as the squares of the times of vibration. These are the chief properties of a simple pendulum vibrating in a vertical plane, and the principal problems that arise in connection with it are the following, viz.: To find the length of a pendulum that shall make arny number of vibrations in a given time; and secondly, having given the length of a pendulum, to find the number of vibrations it will make isn any time given.-These are problems of very easy solution, and the rules for resolving them are simply as follow: —For the first, the rule is, multiply the square of the number of seconds in the given time by the constant number 39'1015, and divide the product by the square of the number of vibrations, for the length of the pendulum in inches. For the second, it is, multiply the square of the number of seconds in the given time by the constant number 39'1393, divide the product by the given length of the pendulum in inches, and extract the square root of the quotient for the number of vibrations sought. The number 39-1015 is the length of a pendulum in inches, that vibrates seconds, or sixty times in a minute, in the latitude of Philadelphia. Suppose a pendulum is found to make 35 vibrations in a minute; what is the distance from the centre of suspension to the centre of oscillation? Here, by the rule, the number of seconds in the given time is 60; hence we get 60 x 60 x 39'1015 = 140765'4, which, being divided by 35 x 35 = 1225, gives 140765'4 -. 1225 = 114'9105 inches for the length required. The length of a pendulum between the centre of suspension and the centre of oscillation is 64 inches; what number of vibrations will it make in 60 seconds? By the rule we have 60 x 60 x 39'1015 = 140765A4, which, being divided by 64, gives 140765'4. 64 = 2199'46, and the square root of this is 2199'46 = 46'9, number of vibrations sought. When the given time is a minute, or 60 seconds, as in the two examples proposed above, the product of the constant number 39'1015 by the square of the time, or 140765'4, is itself a constant quantity, which, being kept in mind, will in some measure facilitate the process of calculation in all similar cases. We now return to the consideration of the conical pendulum, or that in which the ball revolves about a vertical axis in the circumference of a circular plane which is parallel to the horizon. CONICAL PENDULUI. If a pendulum be suspended from the upper extremity of a vertical axis, and be made to revolve about that axis by a conical mo Page 185 THE STEAM ENGINE. 185 tion, which constrains the revolving body to move in the circumference of a circle whose plane is parallel to the horizon, then the time in which the pendulum performs a revolution about the axis can easily be found. Let CD be the pendulum in question, suspended from C, the upper extremity of the vertical axis CD, and let the ball or body B, by revolving C about the said axis, describe the circle BE All, the plane of which is parallel to the horizon; it is proposed to assign the time of description, or the time in which the body B performs a revolution about the axis CD, / I at the distance BD. Conceive the axis CD to denote the weight A' B of the revolving body, or its force in the di- E rection of gravity; then, by the Composition and Resolution of Forces, CB will denote the force or tension of the string or wire that retains the revolving body in the direction CB, and BD the force tending to the centre of the plane of revolution at D. But, by the general laws of motion and forces previously laid down, if the time be given, the space described will be directly proportional to the force; but, by the laws of gravity, the space fallen perpendicularly from rest, in one second of time, is g = 16 2 feet; consequently we have CD: BD: 16~: (ji12.BD the space described towards D by the force in BD CD in one second. Consequently, by the laws of centripetal forces, the periodic time, or the time of the body revolving in the circle BEAH, is expressed by the term 2/1-~C where = 31416, the circumference of a circle whose diameter is unity; or putting t to denote the time, and expressing the height CD in feet, we get t = 6'2832:_ CD C12D32}' or, by reducing the expression to its simplest form, it becomes t = 0'31986V/CD, where CD must be estimated in inches, and t in seconds. Here we have obtained an expression of great simplicity, and the practical rule for reducing it may be expressed in words as follows: RuLE.-Multiply the square root of the height, or the distance between the point of suspension and the centre of the plane of revolution, in inches, by the constant fraction 0'31986, and the product will be the time of revolution in seconds. In what time will a conical pendulum revolve about its vertical axis, supposing the distance between the point of suspension and the centre of the plane of revolution to be 39'1393 inches, which is the length of a simple pendulum that vibrates seconds in latitude 510 30'? The square root of 3941393 is 6'2561; consequently, by the rule, Q2 Page 186 186 THE PRACTICAL MODEL CALCULATOR. we have, 6'2561 x 0'31986 = 2'0011 seconds for the time of revolution sought. It consequently revolves 30 times in a minute, as it ought to do by the theory of the simple pendulum. By reversing the process, the height of the cone, or the distance between the point of suspension and the centre of the plane of revolution, corresponding to any given time, can easily be ascertained; for we have only to divide the number of seconds in the given time by the constant decimal 0'31986, and the square of the quotient will be the required height in inches.'Thus, suppose it were required to find the height of a conical pendulum that would revolve 30 times in a minute. Here the time of revolution is 2 seconds for 60 -- 30 = 2; therefore, by division, it is 2 _ 031986 = 6'2527, which, being squared, gives 6'2527 = 39'0961 inches, or the length of a simple pendulum that vibrates seconds very nearly. In all conical pendulums the times of revolution, or the periodic times, are proportional to the square roots of the heights of the cones. This is manifest, for in the foregoing equation of the periodic time the numbers 6'2832 and 386, or 12 x 326, are constant quantities, consequently t varies as V/CD. If the heights of the cones, or the distances between the points of suspension and the centres of the planes of revolution, be the same, the periodic times, or the times of revolution, will be the same, whatever may be the radii of the circles described by the re-'~ d;, c\ d /" e volving bodies. This will be clearly understood by contemplating the subjoined diagram, where all the pendulums Ca, Cb, Cc, Cd, and Ce, having the common axis CD, will revolve in the same time; and Page 187 TlIE STEAM ENGINE. 187 if they are all in the same vertical plane when first put in motion, they will continue to revolve in that plane, whatever be the velocity, so long as the common axis or height of the cone remains the same. This will become manifest, if we conceive an inflexible bar or rod of iron to pass through the centres of all the balls as well as the common axis, for then the bar and the several balls must all revolve in the same time; but if any one of them should be allowed to rise higher, its velocity would be increased; and if it descends, the velocity will be decreased. Half the periodic time of a conical pendulum is equal to the time of vibration of a simple pendulum, the length of which is equal to the axis or height of the cone; that is, the simple pendulum makes two oscillations or vibrations from side to side, or it arrives at the same point from which it departed, in the same time that the conical pendulum revolves about its axis. The space descended by a falling body in the time of one revolution of the conical pendulum is equal to 3'14162 multiplied by twice the height or axis of the cone. The periodic time, or the time of one revolution is equal to the product of 3'1416 V 2 multiplied by the time of falling through the height of the cone. The weight of a conical pendulum, when revolving in the circumference of a circle, bears the same proportion to the centrifugal force, or its tendency to fly off in a straight line, as the axis or height of the cone bears to the radius of the plane of revolution; consequently, when the height of the cone is equal to the radius of its base, the centripetal or centrifugal force is equal to the power of gravity. These are the principles on which the action of the conical pendulum depends; but as we shall hereafter have occasion to consider it more at large, we need not say more respecting it in this place. Before dismissing the subject, however, it may be proper to put the reader in possession of the rules for calculating the position of the centre of oscillation in vibrating bodies, in a few cases where it has been determined, these being the cases that are of the most frequent occurrence in practice. The centre of oscillation in a vibrating body is that point in the line of suspension, in which, if all the matter of the system were collected, any force applied there would generate the same angular motion in a given time as the same force applied at the centre of gravity. The centres of oscillation for several figures of very frequent use, suspended from their vertices and vibrating flatwise, are as follow:In a right line, or parallelogram, or a cylinder of very small diameter, the centre of oscillation is at two-thirds of the length from the point of suspension. In an isosceles triangle the centre of oscillation is at three-fourths of the altitude. In a circle it is five-fourths of the radius. In the common parabola it is five-sevenths of its altitude. In a parabola of any order it is (3 + 1) x altitude, where n denotes the order of the figure. Page 188 188 THE PRACTICAL MODEL CALCULATOR. In bodies vibrating laterally, or in their own plane, the centres of oscillation are situated as follows; namely, in a circle the centre of oscillation is at three-fourths of the diameter; in a rectangle, suspended at one of its angles, it is at two-thirds of the diagonal; in a parabola, suspended by the vertex, it is five-sevenths of the axis, increased by one-third of the parameter; in a parabola, suspended by the middle of its base, it is four-sevenths of the axis, increased by half the parameter; in the sector of a circle it is three times the arc of the sector multiplied by the radius, and divided by four times the chord; in a right cone it is four-fifths of the axis or height, increased by the quotient that arises when the square of the radius of the base is divided by five times the height; in a globe or sphere it is the radius of the sphere, plus the length of the thread by which it is suspended, plus the quotient that arises when twice the square of the radius is divided by five times the sum of the radius and the length of the suspending thread. In all these cases the distance is estimated from the point of suspension, and since the centres of oscillation and percussion are in one and the same point, whatever has been said of the one is equally true of the other. TIE TEMPERATURE AND ELASTIC FORCE OF STEAM. In estimating the mechanical action of steam, the intensity of its elastic force must be referred to some known standard measure, such as the pressure which it exerts against a square inch of the surface that contains it, usually reckoned by so many pounds avoirdupois upon the square inch. The intensity of the elastic force is also estimated by the inches in height of a vertical column of mercury, whose weight is equal to the pressure exerted by the steam on a surface equal to the base of the mercurial column. It may also be estimated by the height of a vertical column of water measured in feet; or generally, the elastic force of any fluid may be compared with that of atmospheric air when in its usual state of temperature and density; this is equal to a column of mercury 30 inches or 21 feet in height. When the temperature of steam is increased, respect being had to its density, the elastic force, or the effort to separate the parts of the containing vessel and occupy a larger space, is also increased; and when the temperature is diminished, a corresponding and proportionate diminution takes place in the intensity of the emancipating effort or elastic power. It consequently follows that there must be some law or principle connecting the temperature of steam with its elastic force; and an intimate acquaintance with this law, in so far as it is known, must be of the greatest importance in all our researches respecting the theory and the mechanical operations of the steam engine. To find a theorem, by means of whLich it wmay be ascertained when a general law exists, and to determine what that law is, in cases where it is known to obtain.-Suppose, for example, that it is required to assign the nature of the law that subsists between the Page 189 THE STEAM ENGINE. 189 temperature of steam and its elastic force, on the supposition that the elasticity is proportional to some power of the temperature, and unaffected by any other constant or co-efficient, except the exponent by which the law is indicated. Let E and e be any two values of the elasticity, and T, t, the corresponding temperatures deducted from observation. It is proposed to ascertain the powers of T and t, to which E and e are respectively proportional. Let n denote the index or exponent of the required power; then by the conditions of the problem admitting that a law exists, we get, tn e T: t"::E: e; but by the principles of proportion, it is T E; log., and if this be expressed logarithmically, it is n x log. -log. and by reducing the equation in respect of n, it finally becomes = log. e - log. E log. t- log. T The theorem that we have here obtained is in its form sufficiently simple for practical application; it is of frequent occurrence in physical science, but especially so in inquiries respecting the motion of bodies moving in air and other resisting media; and it is even applicable to the determination of the planetary motions themselves. The process indicated by it in the case that we have chosen, is simply, To divide the difference of the logarithms of the elasticities by the difference of the logarithms of the corresponding temperatures, and the quotient will express that power of the temperature to which the elasticity is proportional. Take as an example the following data: —In two experiments it was found that when the temperature of steam was 250'3 and 343'6 degrees of Fahrenheit's scale, the corresponding elastic forces were 59'6 and 238-4 inches of the mercurial column respectively. From these data it is required to determine the law which connects the temperature with the elastic force on the supposition that a law does actually exist under the specified conditions. The process by the rule is as follows: Greater temperature, 343'6...................... log. 2'5352941 Lesser temperature, 250'3........................log. 2'3984608 Remainder...................................... = 01368333 Greater elastic force, 238'4...................... log. 2'3773063 Lesser elastic force, 59.6......................... log. 1 7752463 Remainder....................................... = 0'6(020600 Let the second of these remainders be divided by the first, as directed in the rule, and we get n = 6020600. 1368333 = 4'3998, the exponent sought. Consequently, by taking the nearest unit, for the sake of simplicity, we shall have, according to this result, the following analogy, viz.: T4.4 t4'4:'E e; Page 190 190 THE PRACTICAL MODEL CALCULATOR. that is, the elasticities are proportional to the 4'4 power of the temperatures very nearly. Now this law is rigorously correct, as applied to the particular cases that furnished it; for if the two temperatures and one elasticity be given, the other elasticity will be found as indicated by the above analogy; or if the two elasticities and one temperature be given, the other temperature will be found by a similar process. It by no means follows, however, that the principle is general, nor could we venture to affirm that the exponent here obtained will accurately represent the result of any other experiments than those from which it is deduced, whether the temperature be higher or lower than that of boiling water; but this we learn from it, that the index which represents the law of elasticity is of a very high order, and that the general equation, whatever its form may be, must involve other conditions than those which we have assumed in the foregoing investigation. The theorem, however, is valuable to practical men, not only as being applicable to numerous other branches of mechanical inquiry, but as leading directly to the methods by which some of the best rules have been obtained for calculating the elasticity of steam, when in contact with the liquid from which it is generated. We now proceed to apply our formula to the determination of a general law, or such as will nearly represent the class of experiments on which it rests; and for this purpose we must first assign the limits, and then inquire under what conditions the limitations take place, for by these limitations we must in a great measure be guided in determining the ultimate form of the equation which represents the law of elasticity. The limits of elasticity will be readily assigned from the following considerations, viz.: In the first place, it is obvious that steam cannot exist when the cohesive attraction of the particles is of greater intensity than the repulsive energy of the caloric or matter of heat interposed between them; for in this case, the change from an elastic fluid to a solid may take place without passing through the intermediate stage of liquidity: hence we infer that there must be a temperature at which the elastic force is nothing, and this temperature, whatever may be its value, corresponds to the lower limit of elasticity. The higher limit will be discovered by similar considerations, for it must take place when the density of steam is the same as that of water, which therefore depends on the 0odulut.s of elasticity of water. The modulus of elasticity of any substance is the measure of its elastic force; that of water at 60~ of temperature is 22,100 atmospheres. Thus, for instance, suppose a given quantity of water to be confined in a close vessel which it exactly fills, and let it be exposed to a high degree of temperature, thern it is obvious that in this state no steam would be produced, and the force which is exerted to separate the parts of the vessel is simply the expansive force of compressed water; we therefor8 have the following proportion. As the expanded volume of water is to the Page 191 THE STEAM ENGINE. 191 quantity of expansion, so is the modulus of elasticity of water to the elastic force of steam of the same density as water. Having therefore assigned the limits beyond which the elastic force of steam cannot reach, we shall now proceed to apply the principle of our formula to the determination of the general law which connects the temperature with the elastic force; and for this purpose, in addition to the notation which we have already laid down, let c denote some constant quantity that affects the elasticity, and d the temperature at which the elasticity vanishes; then since this temperature must be applied subtractively, we have from the foregoing principle, c E = (T - a)n, and ce = (t - a)n. From either of these equations, therefore, the constant quantity e can be determined in terms of the rest when they are known; thus we have (T --, and (t -, and by comparing these two independent values of c, the value of n becomes known; for (T - ) =- (t — and consequently E e = log. e - log. E (A). log. (t - a) - log. (T - a ). In this equation the value of the symbol 8 is unknown; in order therefore to determine it, we must have another independent expression for the value of n; and in order to this, let the elasticities E and e become E' and e' respectively; while the corresponding temperatures T and t assume the values T' and t'; then by a similar process to the above, we get (T' -El - (t -- ), and E' el =n =_ log. e' - log. E'. (B) log. (t' — l )- log. (T' — 6) Let the equations (A) and (B) be compared with each other, and we shall then have an expression involving only the unknown quantity 8, for it must be understood that the several temperatures with their corresponding elasticities are to be deduced from experiment; and in consequence, the law that we derive from them must be strictly empirical; thus we have log. e - log. E. _ log. e' - log. E log. (t - a) - log. (T - 8) log. (t'- ) — log. (T' — o) We have no direct method of reducing expressions of this sort, and the usual process is therefore by approximation, or by the rule of trial and error, and it is in this way that the value of the quantity 8 must be found; and for the purpose of performing the reduction, we shall select experiments performed with great care, and may consequently be considered as representing the law of elasticity with very great nicety. T = 212'0 Fahrenheit E = 29'8 inches of mercury. t = 250'3 e = 59'6 T'= 293'4 E'= 119'2 t'= 343.6 e'= 238'4 Page 192 192 THE PRACTICAL MODEL CALCULATOR. Therefore, by substituting these numbers in equation (C), and making a few trials, we find that a = - 50~, and substituting this in either of the equations (A) or (B), we get n- = 5'08; and finally, by substituting these values of 6 and n in either of the expressions for the constant quantity c, we get c = 64674730000, the 5'08 root of which is 134'27 very nearly; hence we have F= t + 50 5'08D) 1. 13427 (D)@ Where the symbol F denotes generally the elastic force of the steam in inches of mercury, and t the corresponding temperature in degrees of Fahrenheit's thermometer, the logarithm of the denominator of the fraction is 2'1279717, which may be used as a constant in calculating the elastic force corresponding to any given temperature. We have thus discovered a rule of a very simple form; it errs in defect; but this might have been remedied by assuming two points near one extremity of the range of experiment, and two points near the other extremity; and by substituting the observed numbers in equation (C), different constants and a more correct exponent would accordingly have been obtained. Mr. Southern has, by pursuing a method somewhat analogous to that which is here described, found his experiments to be very nearly represented by F=.(t + 51'3 5'13 F13- }5717 But even here the formula errs in defect, for he has found it necessary to correct it by adding the arbitrary decimal 0'1; and thus modified, it becomes =' { t + t51. (E). F 135.767 + 0.1E. Our own formula may also be corrected by the application of some arbitrary constant of greater magnitude; but as our motive for tracing the steps of investigation in the foregoing case was to exemplify the method of determining the law of elasticity, our end is answered; for we consider it a very unsatisfactory thing merely to be put in possession of a formula purporting to be applicable to some particular purpose, without at the same time being put in possession of the method by which that formula was obtained, and the principles on which it rests. Having thus exhibited the principles and the method of reduction, the reader will have greater confidence as regards the consistency of the processes that he may be called upon to perform. The operation implied by equation (E) may be expressed in words as follows:RULE.-TO the given temperature in degrees of Fahrenheit's thermometer add 51'3 degrees and divide the sumn by 13576T7; to the 5'13 power of the quotient add the constant fraction i, and the sum will be the elastic force in inches of mercury. Page 193 THE STEAM ENGINE. 193 The process here described is that which is performed by the rules of common arithmetic; but since the index is affected by a fraction, it is difficult to perform in that way: we must therefore have recourse to logarithms as the only means of avoiding the difficulty. The rule adapted to these numbers is as follows:RULE FOR LOGARITHMS.-To the given temperature in degrees of Fahrenheit's thermometer add 51'3 degrees; then, from the logarithm of the sum subtract 2'1327940 or the logarithm of 135'767, the denominator of the fraction; multiply the remainder by the index 5'13, and to the natural number answering to the sum add the constant fraction T'6; the sum will be the elastic force in inches of mercury. If the temperature of steam be 250'3 degrees as indicated by Fahrenheit's thermometer, what is the corresponding elastic force in inches of mercury? By the rule it is 250'3 + 51'3 = 301'6 log. 2'4794313 constant den. = 135'767 log. 241327940 subtract remainder = 0-3466373 31'5 inverted 17331865 346637 103991 natural number 60'013 log. 1'7782493 If this be increased by ~6, we get 60'113 inches of mercury for the elastic force of steam at 250'3 degrees of Fahrenheit. By simply reversing the process or transposing equation (E), the temperature corresponding to any given elastic force can easily be found; the transformed expression is as follows, viz.: 1 t = 135-767 (F- 01) 513.... (F). Since, in consequence of the complicated index, the process of calculation cannot easily be performed by common arithmetic, it is needless to give a rule for reducing the equation in that way; we shall therefore at once give the rule for performing the process by'logarithms. RULE.-From the given elastic force in inches of mercury, subtract the constant fraction 0'1; divide the logarithm of the remainder by 5'13, and to the quotient add the logarithm 2'1327940; find the natural number answering to the sum of the logarithms, and from the number thus found subtract the constant 51'3, and the remainder will be the temperature sought. Supposing the elastic force of steam or the vapour of water to be equivalent to the weight of a vertical column of mercury, the height of which is 238'4 inches; what is the corresponding temperature in degrees of Fahrenheit's thermometer? Here, by proceeding as directed in the rule, we have 2384 — 01 = R 13 Page 194 194 THE PRACTICAL MODEL CALCULATOR. 238'3, and dividing the logarithm of this remainder by the constant exponent 5'13, we get log. 238'3 -. 513 = 2'3771240 _ 5'13 = 0'4633770 constant co-efficient = 135'767 - - log. 2-1327940 add natural number = 394'61 - - - log. 2'5961710 sum constant temperature = 51'3 subtract required temperature - 343-31 degrees of Fahrenheit's thermometer. The temperature by observation is 343'6 degrees, giving a difference of only 0'29 of a degree in defect. For low temperature or low pressure steam, that is, steam not exceeding the simple pressure of the atmosphere, M. Pambour gives p 004948 + (5j6 72)56... (G). In which equation the symbol p denotes the pressure in pounds avoirdupois per square inch, and t the temperature in degrees of Fahrenheit's thermometer. When this expression is reduced in reference to temperature, it is t = 155.7256 (p - 0.04948) 5'13 - 513.... (). The formula of Tredgold is well known. The equation, in its original form, is 177f t + 100....(I) where f denotes the elastic force of steam in inches of mercury, and t the temperature in degrees of Fahrenheit's thermometer. The same formula, as modified and corrected by M. Millet, becomes 179.0773fr = t + 103.... (K). Dr. Young of Dublin constructed a formula which was adapted to the experiments of his countryman Dr. Dalton: it assumed a form sufficiently simple and elegant; it is thus expressedf = (1 + 0.0029 t)7.... (L) where the symbolf denotes the elastic force of steam expressed in atmospheres of 30 inches of mercury, and t the temperature in degrees estimated above 212 of Fahrenheit. This formula is not applicable in practice, especially in high temperatures, as it deviates very widely and rapidly from the results of observation: it is chiefly remarkable as being made the basis of a numerous class of theorems somewhat varied, but of a more correct and satisfactory character. The Commission of the French Academy represented their experiments by means of a formula constructed on the same principles: it is thus expressedf = (1 + 0'7153 t)... (M) where f denotes the elastic force of the steam expressed in atmospheres of 0'76 metres or 29'922 inches of mercury, and t the tem Page 195 THE STEAM ENGINE. 195 perature estimated above 100 degrees of the centigrade thermometer; but when the same formula is so transformed as to be expressed in the usual terms adopted in practice, it is p = (0o2679 + 0o0067585 t)5.... (N) where p is the pressure in pounds per square inch, and t the temperature in degrees of Fahrenheit's scale, estimated above 212 or simple atmospheric pressure. The committee of the Franklin Institute adopted the exponent 6, and found it necessary to change the constant 0'0029 into 0'00333; thus modified, they represented their experiments by the equation p = (0.460467 + 0.00521478 t)6... (0). By combining Dr. Dalton's experiments with the mean between those of the French Academy and the Franklin Institute, we obtain the following equations, the one being applicable for temperatures below 212 degrees, and the other for temperatures above that point as far as 50 atmospheres. Thus, for low pressure steam, that is, for steam of less temperature than 212, it is (t + 175)771307. f 387 (P): and for steam above the temperature of 212, it is (t + 121)642 In consequence therefore of the high and imposing authority from which these formulas are deduced, we shall adopt them in all our subsequent calculations relative to the steam engine; and in order'to render their application easy and familiar, we shall translate them into rules in words at length, and illustrate them by the resolution of appropriate numerical examples; and for the sake of a systematic arrangement, we think proper to branch the subject into a series of problems, as follows: The temperature of steam being given in degrees of tFahrenheit's thermometer, to find the corresponding elastic force in inches of mercury.-The problem, as here propounded, is resolved by one or other of the last two equations, and the process indicated by the arrangement is thus expressed:RULE.-To the given temperature expressed in degrees of Fahrenheit's thermometer, add the constant temperature 175; find the logarithm answering to the sum, from which subtract the constant 2'587711; multiply the remainder by the index 7'71307, and the product will be the logarithm of the elastic force in atmospheres of 30 inches of mercury when the given temperature is less than 212 degrees. But when the temperature is greater than 212, increase it by 121; then, from the logarithm of the temperature thus increased, subtract the constant logarithm 2'522444, multiply the remainder by the exponent 6'42, and the product will be the Page 196 196 THE PRACTICAL MODEL CALCULATOR. logarithm of the elastic force in atmospheres of 30 inches of mercury; which being multiplied by 30 will give the force in inches, or if multiplied by 14'76 the result will be expressed in pounds avoirdupois per square inch. When steam is generated under a temperature of 187 degrees of Fahrenheit's thermometer, what is its corresponding elastic force in atmospheres of 30 inches of mercury? In this example, the given temperature is less than 212 degrees: it will therefore be resolved by the first clause of the preceding rule, in which the additive constant is 175; hence we get 187 + 175 = 362...log. 2'558709 Constant divisor = 387...log. 2'587711 subtract 9'970998 x 7'71307 = 9'773393 And the corresponding natural number is 0'5934 atmospheres, or 17'802 inches of mercury, the elastic force required, or if expressed in pounds per square inch, it is 0'5934 x 14'76 = 8'76 lbs. very nearly. If the temperature be 250 degrees of Fahrenheit, the process is as follows: 250 + 121 - 371...log. 2'569374 Constant divisor = 333...log. 2'522444 subtract 0'046930 x 6'42 = 0'301291 And the corresponding natural number is 2'0012 atmospheres, or 60'036 inches of mercury, and in pounds per square inch it is 2'0012 x 14'76 = 29'54 lbs. very nearly. It is sometimes convenient to express the results in inches of mercury, without a previous determination in atmospheres, and for this purpose the rule is simply as follows: RXuLE.-Multiply the given temperature in degrees of Fahrenheit's thermometer by the constant coefficient 1'5542, and to the product add the constant number 271'985; then from the logarithm of the sum subtract the constant logarithm 2'587711, and multiply the remainder by the exponent 7'71307; the natural number answering to the product, considered as a logarithm, will give the elastic force in inches of mercury. This answers to the case when the temperature is less than 212 degrees; but when it is above that point proceed as follows: Multiply the given temperature in degrees of Fahrenheit's thermometer by the constant coefficient 1'69856, and to the product add the Constant number 205'526; then from the logarithm of the sum subtract the constant logarithm 2'522444, and multiply the remainder by the exponent 6'42; the natural number answering to the product considered as a logarithm, will give the elastic force in inches of mercury. Take, for example, the temperatures as assumed above, and the process, according to the rule, is as follows: Page 197 THE STEAM ENGINE. 197 187 x 1'5542 = 290'6354 Constant = 2711985 add Sum = 562-6204...log. 2'750216 Constant - 387..........log. 2'587711 subtract 0'162505 x 7'71307 = 1'253408 And the natural number answering to this logarithm is 17'923 inches of mercury. By the preceding calculation the result is 17'802; the slight difference arises from the introduction of the decimal constants, which in consequence of not terminating at the proper place are taken to the nearest unit in the last figure, but the process is equally true notwithstanding. For the higher temperature, we get 250 x 1'69856 = 424'640 Constant - 205-526 add Sum = 630.166......log. 2'799456 Constant = 333............log. 2'522444 subtract 0'277011 x 6'42 = 1'778410 And the natural number answering to this logarithm is 60'036 inches of mercury, agreeing exactly with the result obtained as above. It is moreover sometimes convenient to express the force of the steam in pounds per square inch, without a previous determination in atmospheres or inches of mercury; and when the equations are modified for that purpose, they supply us with the following process, Viz.: Multiply the given temperature by the constant coefficient 1_41666, and to the product add the constant number 247'9155; then, from the logarithm of the sum subtract the constant logarithm 2'587711, and multiply the remainder by the index 7'71307; the natural number answering to the product will give the pressure in pounds per square inch, when the temperature is less than 212 degrees; but for all greater temperatures the process is as follows: Multiply the given temperature by the constant coefficient 1'5209, and to the product add the constant number 184'0289; then, from the logarithm of the sum subtract the constant logarithm 2'522444, and multiply the remainder by the exponent 6'42; the natural or common number answering to the product, will express the force of the steam in pounds per square inch. If any of these results be multiplied by the decimal 0'7854, the product will be the corresponding pressure in pounds per circular inch. Taking, therefore, the temperatures previously employed, the operation is as follows: 187 x 1.41666 = 264.9155 Constant = 247'9155 add Sum = 512.8310.1og. 2'709974 Constant 387........ log. 2'587711 subtract 0'122263 x 7'71307 = 0'942656 Page 198 198 THE PRACTICAL MODEL CALCULATOR. And the number answering to this logarithm is 8'763 lbs. per square inch, and 8'763 x 0'7854 = 6'8824 lbs. per circular inch, the proportion in the two cases being as 1 to 0'7554. Again, for the higher temperature, it is 250 x 1'5209 -= 380'2250 Constant - 184'0289 add Sum = 564'2539......log. 2-751475 Constant = 333............. log. 2'522444 subtract 0'229031 x 6'42 = 1'470279 And the number answering to this logarithm is 29'568 lbs. per square inch, or 29568 x 0'7854 = 23'2226 lbs. per circular inch. We have now to reverse the process, and determine the temperature corresponding to any given power of the steam, and for this purpose we must so transpose the formulas (P) and (Q), as to express the temperature in terms of the elastic force, combined with given constant numbers; but as it is probable that many of our readers would prefer to see the theorems from which the rules are deduced, we here subjoin them. For the lower temperature, or that which does not exceed the temperature of boiling water, we get 1 t = 249f-7730~ 175.(n). Where t denotes the temperature in degrees of Fahrenheit's thermometer, and f the elastic force in inches of mercury, less than 30 inches, or one atmosphere; but when the elastic force is greater than one atmosphere, the formula for the corresponding temperature is as follows: t = 196f6 - 121.... (S). In the construction of these formulas, we have, for the sake of simplicity, omitted the fractions that obtain in the coefficient off; for since they are very small, the omission will not produce an error of any consequence; indeed, no error will arise on this account, as we retain the correct logarithms, a circumstance that enables the computer to ascertain the true value of the coefficients whenever it is necessary so to do; but in all cases of actual practice, the results derived from the integral coefficients will be quite sufficient. The rule supplied by the equations (R) and (S) is thus expressed: When the elastic force is less than the pressure of the atmosphere, that is, less than 30 inches of the mercurial column,RULE.-IDivide the logarithm of the given elastic force in inches of mercury, by the constant index 7'71307, and to the quotient add the constant logarithm 2'396204; then from the common or natural number answering to the sum, subtract the constant temperature 175 degrees, and the remainder will be the temperature sought in degrees of Fahrenheit's thermometer. But when the elastic force exceeds 30 inches, or one atmosphere, the following rule applies: Page 199 THE STEAM ENGINE. 199 Divide the logarithm of the given elastic force in inches of mercury by the constant index 6'42, and to the quotient add the constant logarithm 2'292363: then, from the natural number answering to the sum subtract the constant temperature 121 degrees, and the remainder will be the temperature sought. Similar rules might be constructed for determining the temperature, when the pressure in pounds per square inch is given; but since this is a less useful case of the problem, we have thought proper to omit it. We therefore proceed to exemplify the above rules, and for this purpose we shall suppose the pressure in the two cases to be equivalent to the weight of 19 and 60 inches of mercury respectively. The operations will therefore be as follows: Log. 19. 7'71307 = 1'278754 + 7'71307 = 0'165791 Constant coefficient = 249....................log. 2'396204 add Natural number = 364'75................ log. 2'561994 Constant temperature = 175 subtract Required temperature = 189'75 degrees of Fahrenheit's scale. For the higher elastic force the operation is as follows: Log. 60 *- 6'42 = 1'778151 - 6'42 = 0276969 Constant coefficient 196...............log. 2'292363 add Natural number = 370'97............log. 2'569332 Constant temperature = 121 subtract Required temperature = 249'97 degrees of Fahrenheit's scale. All the preceding results, as computed by our rules, agree as nearly with observation as can be desired: but they have all been obtained on the supposition that the steam is in contact with the liquid from which it is generated; and in this case it is evident that the steam must always attain an elastic force corresponding to the temperature; and in accordance to any increase of pressure, supposing the temperature to remain the same, a quantity of it corresponding to the degree of compression must simply be condensed into water, and in consequence will leave the diminished space occupied by steam of the original degree of tension; or otherwise to express it, if the temperature and pressure invariably correspond with each other, it is impossible to increase the density and elasticity of the steam except by increasing the temperature at the same time; and, contrariwise, the temperature cannot be increased without at the same time increasing the elasticity and density. This being admitted, it is obvious that under these circumstances the steam must always maintain its maximum of pressure and density: but if it be separated from the liquid that produces it, and if its temperature in this case be increased, it will be found not to possess a higher degree of elasticity than a volume of atmospheric air similarly confined, and heated to the same temperature. Under this new condition, the state of maximum density and elasticity ceases; for it is obvious that since no water is present, there cannot be any Page 200 200 THE PRACTICAL MODEL CALCULATOR. more steam generated by an increase of temperature; and consequently the force of the steam is only that which confines it to its original bulk, and is measured by the effort which it exerts to expand itself. Our next object, therefore, is to inquire what is the law of elasticity of steam under the conditions that we have here specified. The specific gravity of steam, its density, and the volume which it occupies at different temperatures, have been determined by experiment with very great precision; and it has also been ascertained that the expansion of vapour by means of heat is regulated by the same laws as the expansion of the other gases, viz. that all gases expand from unity to 1'375 in bulk by 180 degrees of temperature; and again, that steam obeys the law discovered by Boyle and MAariotte, contracting in volume in proportion to the degree of pressure which it sustains. We have therefore to inquire what space a given quantity of water converted into steam will occupy at a given pressure; and from thence we can ascertain the specific gravity, density, and volume at all other pressures. When a gas or vapour is submitted to a constant pressure, the quantity which it expands by a given rise of temperature is calculated by the following theorem, VI = V tI,+ 4 9............ v' -Tt + 459(T) where t and t' are the temperatures, and v, v' the corresponding volumes before and after expansion; hence this rule. RULE.-To each of the temperatures before and after expansion, add the constant experimental number 459; divide the greater sum by the lesser, and multiply the quotient by the volume at the lower temperature, and the product will give the expanded volume. If the volume of steam at the temperature of 212 degrees of Fahrenheit be 1711 times the bulk of the water that produces it, what will be its volume at the temperature of 250'3 degrees, supposing the pressure to be the same in both cases? Here, by the rule, we. have 212 + 459 = 671, and 250'3 + 459 = 09'3; consequently, by dividing the greater by the lesser, and multiplying by the given volume, we get 709'3 x 1711 = 1808'66 671 for the volume at the temperature of 250'3 degrees. Again, if the elastic force at the lower temperature and the corresponding volume be given, the elastic force at the higher temperature can readily be found; for it is simply as the volume the vapour occupies at the lower temperature is to the volume at the higher temperature, or what it would become by expansion, so is the elastic force given to that required. If the volume which steam occupies under any given pressure and temperature be given, the volume which it will occupy under any proposed pressure can readily be found by reversing the preceding process, or by referring to chemical tables containing the Page 201 THE STEAM ENGINE. 201 specific gravity of the gases compared with air as unity at the same pressure and temperature. Now, air at the mean state of the atmosphere has a specific gravity of 12 as compared with water at 1000; and the bulks are inversely as the specific gravities, according to the general laws of the properties of matter previously announced; hence it follows that air is 818 times the bulk of an equal weight of water, for 1000 - 1- = 818'18. But, by the experiments of Dr. Dalton, it has been found that steam of the same pressure and temperature has a specific gravity of'625 compared with air as unity; consequently, we have only to divide the number 818'18 by'625, and the quotient will give the proportion of volume of the vapour to one of the liquid from which it is generated; thus we get 818'18 -.- 625 = 1309; that is, the volume of steam at 60 degrees of Fahrenheit, its force being 30 inches of mercury, is 1309 times the volume of an equal weight of water; hence it follows, from equation (T), that when the temperature increases to t', the volume becomes v = 1309 x 2.54(459 + t'); and from this expression, the volume corresponding to any specified elastic force f, and temperature t', may easily be found; for it is inversely as the compressing force: that is, f: 30:: 2525(459 + t'): v; consequently, by working out the analogy, we get = 75.67(459 + t').(U) f By this theorem is found the volume of steam as compared with that of the water producing it, when under a pressure corresponding to the temperature. The rule in words is as follows: RULE.-Calculate the elastic force in inches of mercury by the rule already given for that purpose, and reserve it for a divisor. To the given temperature add the constant number 459, and multiply the sum by 75'67; then divide the product by the reserved divisor, and the quotient will give the volume sought. When the temperature of steam is 250'3 degrees of Fahrenheit's thermometer, what is the volume, compared with that of water? The temperature being greater than 212 degrees, the force is calculated by the rule to equation (Q), and the process is as follows. 250'3 + 121 = 371'3 log. 2'5697249 Constant divisor = 333 log. 2-5224442 subtract 0 0472807 x642=0 3035421 Atmosphere - 30 inches of mercury log. 1'4771213 add Elastic force = 60'348 log. 1'7806634 Again it is, 459 + 250'3 = 709'3 log. 2'8508300 dd sub. Constant coefficient = 75'67 log. 1'8789237 add 47297537 J Volume = 88939 times that of water, log. 2'9490903 remainder. Page 202 202 THE PRACTICAL MODEL CALCULATOR. Thus we have given the method of calculating the elastic force of steam when the temperature is given either in atmospheres or inches of mercury, and also in pounds or the square or circular inch: we have also reversed the process, and determined the temperature corresponding to any given elastic force. We have, moreover, shown how to find the volume corresponding to different temperatures, when the pressure is constant; and, -finally, we have calculated the volume, when under a pressure due to the elastic force. These are the chief subjects of calculation as regards the properties of steam; and we earnestly advise our readers to render themselves familiar with the several operations. The calculations as regards the motion of steam in the parts of an engine to produce power, will be considered in another part of the present treatise. The equation (U), we may add, can be exhibited in a different form involving only the temperature and known quantities; for since the expressions (P) and (Q) represent the elastic force in terms of the temperature, according as it is under or above 212 degrees of Fahrenheit, we have only to substitute those values of the elastic force when reduced to inches of mercury, instead of the symbolf in equation (U), and we obtain, when the temperature is less than 212 degrees, Vol.=75'67(tem.+459). (004016xtem.+ 702807)7713~7 (V). and when the temperature exceeds 212 degrees, the expression becomes Vol. =75'67(tem. +459). 005101 xtem. + 617195)6' (1W.) These expressions are simple in their form, and easily reduced; but, in pursuance of the plan we have adopted, it becomes necessary to express the manner of their reduction in words at length, as follows: RULE. — When the given temperature is under 212 degrees, multiply the temperature in degrees of Fahrenheit's thermometer by the constant fraction'004016, and to the product add the constant increment'702807; multiply the logarithm of the sum by the index 7'71307, and find the natural or common number answering to the product, which reserve for a divisor. To the temperature add the constant number 459, and multiply the sum by the coefficient 75'67 for a dividend; divide the latter result by the former, and the quotient will express the volume of steam when that of water is unity. Again, when the given temperature is greater than 212 degrees, multiply it by the fraction'005101, and to the product add the constant increment'617195; multiply the logarithm of the suml by the index 6'42, and reserve the natural number answering to the product for a divisor; find the dividend as directed above, which, being divided by the divisor, will give the volume of steam when that of the water is unity. How many cubic feet of steam will be supplied by one cubic foot Page 203 THE STEAM ENGINE. 203 of water, under the respective temperatures of 187 and 293'4 degrees of Fahrenheit's thermometer? Here, by the rule, we have 187 x0 004016-0'750992 Constant increment=0'702807 Sum =1453799 log. 1625043 x 7'71307=1'2534069 and the number answering to this logarithm is 17'92284, the divisor. But 187 + 459 = 646, and 646 x 75'67 = 48882'82, the dividend; hence, by division, we get 48882'82 - 17'92284 = 2727'4 cubic feet of steam from one cubic foot of water. Again, for the higher temperature, it is 293'4 x 0'005101 = 1'496633 Constant increment = 0'617195 Sum = 2'113828 log. 0'3250696 x642=2 0869468; and the number answering to this logarithm is 122'165, the divisor. But 293'4 + 459 = 752'4, and 752'4 x 75'67 = 56934'108, the dividend; therefore, by division, we get 56934'108 - 122'165 = 466'04 cubic feet of steam from one cubic foot of water. The preceding is a very simple process for calculating the volume which the steam of a cubic foot of water will occupy when under a pressure due to a given temperature and elastic force; and since a knowledge of this particular is of the utmost importance in calculations connected with the steam engine, it is presumed that our readers will find it to their advantage to render themselves familiar with the method of obtaining it. The above example includes both cases of the problem, a circumstance which gives to the operation, considered as a whole, a somewhat formidable appearance: but it would be difficult to conceive a case in actual practice where the application of both the formulas will be required at one and the same time; the entire process must therefore be considered as embracing only one of the cases above exemplified; and consequently it can be performed with the greatest facility by every person who is acquainted with the use of logarithms; and those unacquainted with the application of logarithms ought to make themselves masters of that very simple mode of computation. Another thing which it is necessary sometimes to discover in reasoning on the properties of steam as referred to its action in a steam engine, is the weight of a cubic foot, or any other quantity of it, expressed in grains, corresponding to a given temperature and pressure. Now, it has been ascertained by experiment, that when the temperature of steam is 60 degrees of Fahrenheit, and the pressure equal to 30 inches of mercury, the weight of a cubic foot in grains is 329'4; but the weight is directly proportional to the elastic force, for the elastic force is proportional to the density: consequently, iff denote any other elastic force, and wv the weight in grains corresponding thereto, then we have 30:f:: 329'4: w = 10'98f, Page 204 204 THE PRACTICAL MODEL CALCULATOR. the weight of a cubic foot of vapour at the force f, and temperature 60 degrees of Fahrenheit. Let t denote the temperature at the 459 +- t 459 +- t force f; then by equation (T), we have v 459 60 = 519' the volume at the temperature t, supposing the volume at 60 degrees to be unity; that is, one cubic foot. Now, since the densities are inversely proportional to the spaces which the vapour occupies, we have (459 + t) W = 519 but by the 519 459 w: +; but by the preceding analogy, the value of w is 10'98f; therefore, by substitution, we get 5698262f = 459 + t (X). This equation expresses the weight in grains of a cubic foot of steam at the temperature t and force f; and if we substitute the value of f, from equations (P) and (Q), reduced to inches of mercury, and modified for the two cases of temperature below and above 212 degrees of Fahrenheit, we shall obtain, in the first case,'= (0'012324 x temp. + 2.155611)7'7307. (temp. + 459)... (Y) and for the second case, where the temperature exceeds 212, it is w' = (0'01962 x temp. + 2'37374)6'42. (temp. + 459)... (Z) These two equations, like those marked (V) and (W) are sufficiently simple in their form, and offer but little difficulty in their application. The rule for their reduction when expressed in words at length, is as follows: RULE. —When the temperature is less than 212 degrees, multiply the given temperature, in degrees of Fahrenheit's thermometer, by the fraction 0'012324, and to the product add the constant increment 2'155611; then multiply the logarithm of the sum by the index 7'71307, and from the product subtract the logarithm of the temperature, increased by 459; the natural number answering to the remainder will be the weight of a cubic foot in grains. Again, when the temperature exceeds 212, multiply it by the fraction 0'01962, and to the product add the constant increment 23737374; then multiply the logarithm of the sum by the index 6'42, and from the product subtract the logarithm of the temperature increased by 459; the natural number answering to the remainder will be the weight of a cubic foot in grains. Supposing the temperatures to be as in the preceding example, what will be the weight of a cubic foot in grains for the two cases? Here, by the rule, we have 187 X 0-012324 = 2.304588 Constant increment = 2-155611 Sum = 4-460199 log. 0.6493542 X 7-71307 = 5-0085143 187 + 459 = 646..... log. 2.8102325, subtract Natural number = 157-863 grains per cubic foot log. 2.1982818 Page 205 THE STEAM ENGINE. 205 For the higher temperature, it is 293.4 X 0.01962 = 5.756508 Constant increment = 2'373740 Sum - 8.130248 log. 0'9101038 X 6.42 = 5.8428664 293'4 + 459 = 752'4.... log. 2.8764488, subtract Natural number = 925-59 grains per cubic foot. log. 2-9664176 Here again the operation resolves both cases of the problem; but in practice only one of them can be required. THE MOTION OF ELASTIC FLUIDS. The next subject that claims our attention is the velocity with which elastic fluids or vapours move in pipes or confined passages. It is a well-known fact in the doctrine of pneumatics, that the motion of free elastic fluids depends upon the temperature and pressure of the atmosphere; and, consequently, when an elastic fluid is confined in a close vessel, it must be similarly circumstanced with regard to temperature and pressure as it would be in an atmosphere competent to exert the same pressure upon it. The simplest and most convenient way of estimating the motion of an elastic fluid is to assign the height of a column of uniform density, capable of producing the same pressure as that which the fluid sustains in its state of confinement; for under the pressure of such a column, the velocity into a perfect vacuum will be the same as that acquired by a heavy body in falling through the height of the homogeneous column, a proper allowance being made for the contraction at the aperture or orifice through which the fluid flows. When a passage is opened between two vessels containing fluids of different densities, the fluid of greatest density rushes out of the vessel that contains it, into the one containing the rarer fluid, and the velocity of influx at the first instant of the motion is equal to that which a heavy body acquires in falling through a certain height, and that height is equal to the difference of two uniform columns of the fluid of greatest density, competent to produce the pressures under which the fluids are originally confined; and the velocity of motion at any other instant is proportional to the square root of the difference between the heights of the uniform columns producing the pressures at that instant. Hence we infer that the velocity of motion continually decreases, —the density of the fluids in the two vessels approaching nearer and nearer to an equality, and after a certain time an equilibrium obtains, and the velocity of motion ceases. It is abundantly confirmed by observation and experiment, that oblique action produces very nearly the same effect in the motion of elastic fluids through apertures as it does in the case of water; and it has moreover been ascertained that eddies take place under similar circumstances, and these eddies must of course have a tendency to retard the motion: it therefore becomes necessary, in all the calculations of practice, to make some allowance for the retardation that takes place in passing the orifice; and this end is most s Page 206 206 THE PRACTICAL MODEL CALCULATOR. conveniently answered by modifying the constant coefficient according to the nature of the aperture through which the motion is made. Numerous experiments have been made to ascertain the effect of contraction in orifices of different forms and under different conditions, and amongst those which have proved the most successful in this respect, we may mention the experiments of Du Buat and Eytelwein, the latter of whom has supplied us with a series of coefficients, which, although not exclusively applicable to the case of the steam engine, yet, on account of their extensive utility, we take the liberty to transcribe. They are as follow:1. For the velocity of motion that would result from the direct unretarded action of the column of the fluid that produces it, we have............................................... 3V = V579h 2. For an orifice or tube in the form of the contracted vein................................ 10 V- = 6084h 3. For wide openings having the sill on a) level with the bottom of the reservoir... I 4. For sluices with walls in a line with the. 10 V = V'5929h orifice.......................................... 5. For bridges with pointed piers............ 6. For narrow openings having the sill on a) level with the bottom of the reservoir... 7. For small openings in a sluice with side 10 V = 4761h walls..................... 8. For abrupt projections...................... 9. For bridges with square piers.............. 10. For openings in sluices without side walls 10 V =,/2601h2 11. For openings or orifices in a thin plate..... V = V25l 12. For a straight tube from 2 to 3 diameters in length projecting outwards................10 V = V4225 13. For a tube from 2 to 3 diameters in length projecting inwards............................10 V = /2976'25h It is necessary to observe, that in all these equations V is the velocity of motion in feet per second, and h the height of the column producing it, estimated also in feet. Nos. 1, 2, 11, 12, and 13 are those which more particularly apply to the usual passages for the steam in a steam engine; but since all the others meet their application in the every-day practice of the civil engineer, we have thought it useful to supply them. MOTION OF STEAM IN AN ENGINE. We have already stated that the best method of estimating the motion of an elastic fluid, such as steam or the vapour of water, is to assign the height of a uniform column of that fluid capable of producing the pressure: the determination of this column is therefore the leading step of the inquiry; and since the elastic force of steam is usually reckoned in inches of mercury, 30 inches being Page 207 THE STEAM ENGINE. 207 equal to the pressure of the atmosphere, the subject presents but little difficulty; for we have already seen that the height of a column of water of the temperature of 60 degrees, balancing a column of 30 inches of mercury, is 34'023 feet; the corresponding column of steam must therefore be as its relative bulk and elastic force; hence we have 30: 34'023: fv: h = 1'1341fv, where f is the elastic force of the steam in inches of mercury, v the corresponding volume or bulk when that of water is unity, and h the height of a uniform column of the fluid capable of producing the pressure due to the elastic force; consequently, in the case of a direct unretarded action, the velocity into a perfect vacuum, according to No. 1 of the preceding class of formulas, is V = 8'542 /f v; but for the best form of pipes, or a conical tube in form of the contracted vein, the velocity into a vacuum, according to No. 2, becomes V = 8'307 vf v; and for pipes of the usual construction, No. 12 gives V = 6'922 vf v; No. 13 gives V = 5'804 If v; and in the case of a simple orifice in a thin plate, we get from No. 11 V = 5'322 Vf v. The consideration of all these equations may occasionally be required, but our researches will at present be limited to that arising from No. 12, as being the best adapted for general practice; and for the purpose of shortening the investigation, we shall take no further notice of the case in which the temperature of the steam is below 212 degrees of Fahrenheit; for the expression which indicates the velocity into a vacuum being independent of the elastic force, a separate consideration for the two cases is here unnecessary. It has been shown in the equation marked (U), that the volume of steam which is generated from an unit of water, is v = 75'67 (temp. + 459) 56 (te. 49); let this value of v be substituted for it in the equation V = 6'922 vf v, and we obtain for the velocity into a vacuum for the usual form of steam passages, as follows, viz.: V = 60'2143,(temp. + 459). This is a very neat and simple expression, and the object determined by it is a very important one: it therefore merits the reader's utmost attention, especially if he is desirous of becoming familiar with the calculations in reference to the motioh of steam. The rule which the equation supplies, when expressed in words at length, is as follows:RULE.-TO the temperature of the steam, in degrees of Fahrenheit's thermometer, add the constant number or increment 459, and multiply the square root of the sum by 60'2143; the product will be the velocity with which the steam rushes into a vacuum in feet per second. With what velocity will steam of 293'4 degrees of Fahrenheit's thermometer rush into a vacuum when under a pressure due to the elastic force corresponding to the given temperature. Page 208 208 THE PRACTICAL MODEL CALCULATOR. By the rule it is 293-4 + 459 = 752-4................... log. 1-4382244 Constant coefficient = 60-2143....................log. 1*7797018 add Velocity into a vacuum in feet per second = 1651'68.............log. 3-2179262 This is the velocity into a perfect vacuum, when the motion is made through a straight pipe of uniform diameter; but when the pipe is alternately enlarged and contracted, the velocity must necessarily be reduced in proportion to the nature of the contraction; and it is further manifest, that every bend and angle in a pipe will be attended with a correspondent diminution in the velocity of motion: it therefore behoves us, in the actual construction of steam passages, to avoid these causes of loss as much as possible; and where they cannot be avoided altogether, such forms should be adopted as will produce the smallest possible retarding effect. In cases where the forms are limited by the situation and conditions of construction, such corrections should be applied as the circumstances of the case demand; and the amount of these corrections must be estimated according to the nature of the obstructions themselves. For each right-angled bend, the diminution of velocity is usually set down as being about one-tenth of its unobstructed value; but whether this conclusion be correct or not, it is at least certain that the obstruction in the case of a right-angled bend is much greater than in that of a gradually curved one. It is a very common thing, especially in steam vessels, f'or the main steam pipe to send off branches at right angles to each cylinder, and it is easy to see that a great diminution in the velocity of the steam must take place here. In the expansion valve chest a further obstruction must be met with, probably to the extent of reducing the velocity of the steam two-tenths of its whole amount. These proportional corrections are not to be taken as the results of experiments that have been performed for the purpose of determining the effect of'the above causes of retardation: we have no experiments of this sort on which reliance can be placed; and, in consequence, such elements can only be inferred from a comparison of the principles that regulate the motion of other fluids under similar circumstances: they will, however, greatly assist the engineer in arriving at an approximate estimate of the diminution that takes place in the velocity in passing any number of obstructions, when the precise nature of those obstructions can be ascertained. In the generality of practical cases, if the constant coefficient 60'2143 be reduced in the ratio of 650 to 450, the resulting constant 41'6868 may be employed without introducing an error of any consequence. OF THE ASCENT OF SMOKE AND HEATED AIR IN CHIMNEYS. The subject of chimney flues, with the ascent of smoke and heated air, is another case of the motion of elastic fluids, in which, by a change of temperature, an atmospheric column assumes a different density from another, where no such alteration of temperature occurs. The proper construction of chimneys is a matter of very great importance to the practical engineer, for in a close fireplace, Page 209 THE STEAM ENGINE. 209 designed for the generation of steam, there must be a considerable draught to accomplish the intended purpose, and this depends upon the three following particulars, viz.: 1. The height of the chimney from the throat to the top. 2. The area of the transverse section. 3. The temperature at which the smoke and heated air are allowed to enter it. The formula for determining the power of the chimney may be investigated in the following manner: Put h = the height in feet from the place where the flue enters to the top of the chimney, b = the number of cubic feet of air of atmospheric density that the chimney must discharge per hour, a = the area of the aperture in square inches through which b cubic feet of air must pass when expanded by a change of temperature, v = the velocity of ascent in feet per second, t' = the temperature of the external air, and t = the temperature of the air to be discharged by the chimney. Now the force producing the motion in this case is manifestly the difference between the weight of a column of the atmospheric air and another of the air discharged by the chimney: and when the temperature of the atmospheric air is at 52 degrees of Fahrenheit's thermometer, this difference will be indicated by the term h (t, + 479); the velocity of ascent will therefore be V -642 h. t' + 459 } feet per second, and the quantity of air discharged per second will therefore be, a 64 { tt' + 459}' supposing that there is no contraction in the stream of air; but it is found by experiment, that in all cases the contraction that takes place diminishes the quantity discharged, by about three-eighths of the whole; consequently, the quantity discharged per hour in cubic feet becomes b =125'69 aJ t' - 459') This would be the quantity discharged, provided there were no increase of volume in consequence of the change of temperature; b (t' + 459) but air expands from b to t + 459 for t' - t degrees of temperature, as has been shown elsewhere; consequently, by comparison, we have b (t' + 459) 12 lh69 a (t' - t) - 125 69 t + 459' s2 14 Page 210 210 TIE PRACTICAL MODEL CALCULATOR. From this equation, therefore, any one of the quantities which it involves can be found, when the others are given: it however supposes that there is no other cause of diminution but tl- contraction at the aperture; but this can seldom if ever be the case; for eddies, loss of heat, obstructions, and change of direction in the chimney, will diminish the velocity, and consequently a larger area will be required to suffer the heated air to pass. A sufficient allowance for these causes of retardation will be made, if we change the coefficient 125'69 to 100; and in this case the equation for the area of section becomes a = b V(t' + 459)3. 100 (t + 459) Vh (t' - t). And if we take the mean temperature of the air of the atmosphere at 52 degrees of Fahrenheit, and make an allowance of 16 degrees for the difference of density between atmospheric air and coal smoke, our equation will ultimately assume the form a = 6 V(t' + 459)3 + 51100 Vh (t' - t - 16). It has been found by experiment that 200 cubic feet of air of atmospheric density are required for the complete combustion of one pound of coal, and the consumption of ten pounds of coal per hour is usually reckoned equivalent to one horse power: it therefore appears that 2000 cubic feet of air per hour must pass through the fire for each horse power of the engine. This is a large allowance, but it is the safest plan to calculate in excess in the first instance; for the chimney may afterwards be convenient, even if considerably larger than is necessary. The rule for reducing the equation is as follows:RuLE.-Multiply the number of horse power of the engine by the - power of the temperature -at which the air enters the chimney, increased by 459; then divide the product by 25-55 times the square root of the height of the chimney in feet,. multiplied by the difference of temperature, less 16 degrees, and the quotient will be the area of the chimney in square inches. Suppose the height of the chimney for a 40-horse engine to be 70 feet, what should be its area when the difference between the temperature at which the air enters the flue, and that of the atmosphere is 250 degrees? Here, by the rule, we have, 250 + 52 = 302, the temperature at which the air enters Constant increment = 459 [the flue. Sum = 761.....................log. 2'8813847 3 2)8_6441541 4,3220770 Number of horse power = 40.........................log. 1.6020600 5.9241370 Page 211 THE STEAM ENGINE. 211 5'9241370' 250 - 16 = 234.... log. 2'3692159 height = 70 feet. log. 1.8450980 2)4.2143139 241071569 7 Constant = 25'55.. log. 1.4073909.. 3'5145478 Hence the area of the chimney in square inches is 256'79, log. 2'4095892; and in this way may the area be calculated for any other case; but particular care must be taken to have the data accurately determined before the calculation is begun. In the above example the particulars are merely assumed; but even that is sufficient to show the process of calculation, which is more immediately the object of the present inquiry. It is right, however, to add, that recent experiments have greatly shaken the doctrine that it is beneficial to make chimneys small at the top, though such is the way in which they are, nevertheless, still constructed, and our rules must have reference to the present practice. It appears, however, that it would be the best way to make chimneys expand as they ascend, after the manner of a trumpet, with its mouth turned downwards: but these experiments require further confirmation. The method of calculation adopted above is founded on the principle of correcting the temperature for the difference between the specific gravity of atmospheric air and that of coal-smoke, the one being unity and the other 1'05; there is, however, another method, somewhat more elegant and legitimate, by employing the specific gravity of coal-smoke itself: the investigation is rather tedious and prolix, but the resulting formula is by no means difficult; and since both methods give the same result when properly calculated, we make no further apology for presenting our readers with another rule for obtaining the same object. The formula is as follows: b(t'+ 459)/ 1 2757'5 h (t'- 77'55) where a is the area of the transverse section of the chimney in square inches, b the quantity of atmospheric air required for combustion of the coal in cubic feet per hour, h the height of the chimney in feet, and t' the temperature at which the air enters the flue after passing through the fire. The rule for performing this process is thus expressed: RuLE.-From the temperature at which the air enters the chimney, subtract the constant decrement 77'55; multiply the remainder by the height of the chimney in feet, divide unity by the product, and extract the square root of the quotient. To the temperature of the heated air, add the constant number 459; multiply the sum by the number of cubic feet required for combustion per hour, and divide the product by the number 2757'5; then multiply the quotient by the square root found as above, and the product will be the number of square inches in the transverse section of the chimney. Page 212 212 THE PRACTICAL MODEL CALCULATOR. Suppose a mass of fuel in a state of combustion to require 5000 cubic feet of air per hour, what must be the size of the chimney when its height is 100 feet, the temperature at which the heated air enters the chimney being 200 degrees of Fahrenheit's thermometer? By the rule we have 200 —7755=122'45. log. 2'0879588 Height of the chimney=100.... log. 2'0000000 4'0879588 2)5.9120412 7.9560206 200+459=659... log. 2.8188854) 5000... log. 3.6989700 add 3'0773399 2757'5 ar. co. log. 6.5594845) 1'0333605 10'798 in. This appears to be a very small flue for the quantity of air that passes through it per hour; but it must be observed that we have assumed a great height for the shaft, which has the effect of creating a very powerful draught, thereby drawing off the heated air with great rapidity. The advantage of a high flue is so very great, that the reader may be desirous of knowing to what height a chimney of a given base may be carried with safety, in cases where it is incoirvenient to secure it with lateral stays; and, as an approximate rule for this purpose is not difficult of investigation, we think proper to supply it here. When the chimney is equally wide throughout its whole height, the formula is 156 1 12000 - hwv; but when the side of the base is double the size of the top, the equation becomes 104 s —h 12000 - 0.42 h w; where s is the side of the base in feet, h the height, and mn the weight of one cubic foot of the material. When the chimney stalk is not square, but longer on the one side than the other, s must be the least dimension. The proportion of solid wall to a given base, as sanctioned by experience, is about two-thirds of its area, consequently w ought to be two-thirds of the weight of a cubic foot of brickwork. Now, a cubic foot of dried brickwork is, on an average, 114 lbs.; consequently w = 76 lbs.; and if this be substituted in the foregoing equations, we get for a chimney of equal size throughout, 11200 56 Page 213 THE STEAM ENGINE. 213 and when the chimney tapers to one-half the size at top, it is 1h / 104 -- 12000- 32 h; where it may be remarked that 12000 lbs. is the cohesive force of one square foot of mortar; and in the investigation of the formulas we have assumed the greatest force of the wind on a square foot of surface at 52 lbs. These equations are too simple in their form to require elucidation from us; we therefore leave the reduction as an exercise to the reader, who it is presumed will find no difficulty in resolving the several cases that may arise in the course of his practice. 2 g HatD =D +2gK(L + H' is the expression given by M. Plclet for the velocity of smoke in a chimney. v, the velocity; t, the temperature, whose maximum value is about 3000 centigrade; g = 32, feet; D, the diameter of the chimney; H, the height; L, the length of horizontal flues, supposing them formed into a cylinder of the same diameter as that of the chimney. K = -0127 for brick, ='005 for sheetiron, and ='0025 for cast-iron chimneys. a = -00365. Let L=60; H=150; D=5; K='005; 2g=644; t=300~; 1/ 2gHatD a='00365. Then v= D+2 g K(H+L) = 26'986 feet. A cubic foot of water raised into steam is reckoned equivalent to a horse power, and to generate the steam with sufficient rapidity, an allowance of one square foot of fire-bars, and one square yard of effective heating surface, are very commonly made in practice, at least in land engines. These proportions, however, greatly vary in different cases; and in some of the best marine engine boilers, where the area of fire-grate is restricted by the breadth of the vessel, and the impossibility of firing long furnaces effectually at sea, half a square foot of fire-grate per horse power is a very common proportion. Ten cubic feet of water in the boiler per horse power, and ten cubic feet of steam room per horse power, have been assigned as the average proportion of these elements; but the fact is, no general rule can be formed upon the subject, for the proportions which would be suitable for a wagon boiler would be inapplicable to a tubular boiler, whether marine or locomotive; and good examples will in such cases be found a safer guide than rules which must often give a false result. A capacity of three cubic feet per horse power is a common enough proportion of furnace-room, and it is a good plan to make the furnaces of a considerable width, as they can then be fired more effectually, and do not produce so much smoke as if they are made narrow. As regards the question of draft, there is a great difference of opinion among engineers upon the subject, some preferring a very slow draft and others a rapid one. It is obvious that the question of draft is virtually that of Page 214 214 THE PRACTICAL MODEL CALCULATOR. the area of fire-grate, or of the quantity of fuel consumed upon a given area of grate surface, and the weight of fuel burned on a foot of fire-grate per hour varies in different cases in practice from 3to 80 lbs. Upon the quickness of the draft again hinges the question of the proper thickness of the stratum of incandescent fuel upon the grate; for if the draft be very strong, and the fire at the same time be thin, a great deal of uncombined oxygen will escape up through the fire, and a needless refrigeration of the contents of the flues will be thereby occasioned; whereas, if the fire be thick, and the draft be sluggish, much of the useful effect of the coal will be lost by the formation of carbonic oxide. The length of the circuit made by the smoke varies in almost every boiler, and the same may be said of the area of the flue in its cross section, through which the smoke has to pass. As an average, about one-fifth of the area of fire-grate for the area of the flue behind the bridge, diminished to half that amount for the area of the chimney, has been given as a good proportion, but the examples which we have given, and the average flue area of the boilers which we shall describe, may be taken as a safer guide than any such loose statements. When the flue is too long, or its sectional area is insufficient, the draft becomes insufficient to furnish the requisite quantity of steam; whereas if the flue be too short or too large in its area, a large quantity of the heat escapes up the chimney, and a deposition of soot in the flues also takes place. This last fault is one of material consequence in the case of tubular boilers consuming bituminous coal, though indeed the evil might be remedied by blocking some of the tubes up. The area of water-level is about 5 feet per horse power in land boilers. In many cases, however, it is much less; but it is always desirable to make the area of the waterlevel as large as possible, as, when it is contracted, not only is the water-level subject to sudden and dangerous fluctuations, but water is almost sure to be carried into the cylinder with the steam, in consequence of the violent agitation of the water, caused by the ascent of a large volume of steam through a small superficies. It would be an improvement in boilers, we think, to place over each furnace an inverted vessel immerged in the water, which might catch the steam in its ascent, and deliver it quietly by a pipe rising above the water-level. The water-level would thus be preserved from any inconvenient agitation, and the weight of water within the boiler would be diminished at the same time that the original depth of water over the furnaces was preserved. It would also be an improvement to make the sides of the furnaces of marine boilers sloping, instead of vertical, as is the common practice, for the steam could then ascend freely at the instant of its formation, instead of being entangled among the rivets and landings of the plates, and superinducing an overheating of the plates by preventing a free access of the water to the metal. We have, in the following table, collected a few of the principal results of experiments made on steam boilers. Page 215 THE STEAM ENGINE. 215 TABLE I. Length of circut made by Weight of fuel burned on) 0 each square foot of grate, 3 Cub. ft. of water evaporated ) Cylindrical Circularor Locomo Cylindrical Waton ~~Cfrom initial temperature oa Lca r81331 alby 112 lbsue. naof fue. nal flue. porateda aa feaed h our- 962 152 342-8 459 3346 798 588 face in square feet of heated sur.] Length of circuit made by 5066 725 52-8 7.0 83-1 78 Area of fire grates in square 2333 2609 3510 703 14 3726 houet.................. Weight of fuel burned on) each square footofgrate, 346 4-00 10-75 2031 7933 4682 1331 per hour, in lbs......... Cub. ft. of water evaporated from initial tematmos pherature in1887 1644 1391 14*11 11-14 by 112 lbs. of fuel. )_- ~ 3'1'5The economubic al feet of water eva-r porated per hour from t 1381 137top,9 3440 907 5518 initial temperaturent enSquare feet of heated surface for each cubic foot of 10 5,664,118 lbs., and water evaporated per hour.................. Square feet of heated sur-o face for each square foot - 40-65 6-51 13-13 13-08 47-59 56-0 15. 78 of grate................ Pressure of'steam above 422 2-5 3-68 1.5 50 15-45 the atmosphere in ibs.lls The economical effects of expansion will be found to be very clearly exhibited in the next table. The duties are recorded in the fifth line from the top, and the degree of expansion in the bottohe line. It will be observed, that the order in which the different engines stand in respect of superiority of duty is the same as in respect of amount of expansion. The Hoimbush engine has a duty of 140,484,848 lbs. raised 1 foot by cwagt. of coals, and the steam acts expensively over -83 of the whole stroke; while the waterworks' Cornish engine has only a duty of 105,664,118 lbs., and expands the steam over only -6887 of the whole stroke. Again. comparing the second and last engines together, the Albion Mills engine has a duty of 25,756,752 lbs., and no expansive action. The water-works' engine, again, acts expansively over one-half of its stroke, and has an increased duty of 46,602,333 lbs. Other causes, of course, may influence these comparisons, especially the last, where one engine is a double-acting, rotative engine, and the other a single-acting pumping one; but there can be no doubt that the expansive action in the latter is the principal cause of its more economical performance. The heating surface per horse power allowed by some engineers is about 9 square feet in wagon boilers, reckoning the total surface as effective surface, if the boilers be of a considerable size; but in the case of small boilers, the proportion is larger. The total Page 216 TABLE II. Holmbush, Cornish, Atmospheric En- Non-expansive rota- condensing En- Noncondensing gine, LongfBen-.... y glne single acg double-actining Lgint, Long Bn- tive condensing gr nsifngl ati dngie, nonex- Cornish Engine, rumping Engine ton, Northum- Engine, Albion foapnmping ater Engine,nonex- East London at East London berland, date Mills, London, Steam acts expan- pansive, Con1772. date 1786. sivelyafter the first gleton, Che- WaterWorks. Water Works. sixth of the stroke. shire. 1823. 1836. Diameter of cylinder in inches.................................. 52 34 50 13 79 59 Length of stroke in feet............................................ 7 8 9.1 4 10 7.91 Number of strokes per minute................................... 1 2 16 4'63 27'5 7 11' 5 Pressure on the piston, above or below the atmosphere Estimated at 30 20 517 215 in lbs., per square inch...................................... -25 Weight in lbs. raised one foot by 112 lbs. of coals......... 12,600,000 25,756,752 140,484,848 12,418,560 105,664,118 46,602,333 bd Do. do. by one pound of water, as steam 14,280 28,489 119,097 15,840 110,716 53,369 Effective power of the engine at time of experiment in 40 5 50.0 26 48 12 0 horse power..................................................... Efficiency of the steam, its efficiency in the Albion }501 1000 4180 *556 389 187 Mills being unity..............................556 389 Efficiency of the fuel, its efficiency in the Albion Mills 480 1|000 5.454 | 482 41 181 c being unity................................,.................. 48 Distance of the piston from the end of its stroke when 0 0 833 0 *687.5 the steam is cut off in parts of the length of stroke. 0 r_.. Page 217 THE STEAM ENGINE. 217 heating surface of a two horse power wagon boiler is, according to Fitzgerald's proportions, 30 square feet, or 15 ft. per horse power; whereas, in the case of a 45 horse power boiler the total heating surface is 438 square feet, or 9'6 ft. per horse power. The capacity of steam room is 83 cubic feet per horse power, in the two horse power boiler, and 53 cubic feet in the 20 horse power boiler; and in the larger class of boilers, such as those suitable for 30 and 45 horse power engines, the capacity of the steam room does not fall below this amount, and indeed is nearer 6 than 5-L cubic feet per horse power. The content of water is 181 cubic feet per horse power in the two horse power boiler, and 15 cubic feet per horse power in the 20 horse. power boiler. In marine boilers about the same proportions obtain in most particulars. The original boilers of one or two large steamers were proportioned with about half a square foot of fire grate per horse power, and 10 square feet of flue and furnace surface, reckoning the total amount as effective; but in the boilers of other vessels a somewhat smaller proportion of heating surface was adopted. In some cases we have found that, in their marine flue boilers, 9 square feet of flue and furnace surface are requisite to boil off a cubic foot of water per hour, which is the proportion that obtains in some land boilers; but inasmuch as in modern engines the nominal considerably exceeds the actual power, they allow 11 square feet of heating surface per nominal horse power in their marine boilers, and they reckon, as effective heating surface, the tops of the flues, and the whole of the sides of the flues, but not the bottoms. They have been in the habit of allowing for the capacity of the steam space in marine boilers 16 times the content of the cylinder; but as there are two cylinders, this is equivalent to 8 times the content of both cylinders, which is the proportion commonly followed in land engines, and which agrees very nearly with the proportion of between 5 and 6 cubic feet of steam room per horse power. Taking, for example, an engine with 23 inches diameter of cylinder and 4 feet stroke, which will be 18'4 horse power-the area of the cylinder will be 415'476 square inches, which, multiplied by 48, the number of inches in the stroke, will give 19942'848 for the capacity of the cylinder in cubic inches; 8 times this is 159542'784 cubic inches, or 92'3 cubic feet; 92'3 divided by 18'4 is rather more than 5 cubic feet per horse power. There is less necessity, however, that the steam space should be large when the flow of steam from the boiler is very uniform, as it will be where there are two engines attached to the boiler at right angles with one another, or where the engines work at a great speed, as in the case of locomotive engines. A high steam' chest too, by rendering boiling over into the steam pipes, or priming as it is called, more difficult, obviates the necessity for so large a steam space; and the use of steam of a high pressure, worked expansively, has the same operation; so that in modern marine boilers, of the tubular construction, where the whole of these modifying circumstances exist, there is no necessity for so T Page 218 218 THE PRACTICAL MODEL CALCULATOR. large a proportion of steam room as 5 or 6 cubic feet per horse power, and about half that amount more nearly represents the general practice. Many allow 0'64 of a square foot per nominal horse power of grate bars in their marine boilers, and a good effect arises from this proportion; but sometimes so large an area of fire grate cannot be conveniently got, and the proportion of half a square foot per horse power seems to answer very well in engines working with some expansion, and is now very widely adopted. With this allowance, there will be about 22 square feet of heating surface per square foot of fire grate; and if the consumption of fuel be taken at 6 lbs. per nominal horse power per hour, there will be 12 lbs. of coal consumed per hour on each square foot of grate. The flues of all flue boilers diminish in their calorimeter as they approach the chimney; some very satisfactory boilers have been made by allowing a proportion of 0'6 of a square foot of fire grate per nominal horse power, and making the sectional area of the flue at the largest part Ith of the area of fire grate, and the smallest part, where it enters the chimney, Ailth of the area of the fire grate; but in some of the boilers proportioned on this plan the maximum sectional area is only 1 or 81, according to the purposes of the boiler. These proportions are retained whether the boiler is flue or tubular, and from 14 to 16 square feet of tube surface is allowed per nominal horse power; but such boilers, although they may give abundance of steam, are generally, perhaps needlessly, bulky. We shall therefore conclude our remarks upon the subject by introducing a table of the comparative evaporative power of different kinds of coal, which will prove useful, by affording data for the comparison of experiments upon different boilers when different kinds of coal are used. TABLE of the Comparative Evapoorative Power of different kinds of Coal. Water evapoNo. Description of Coals. rated per lb. of Coals. Lbs. 1 The best Welsh.............................. 9-493 2 Anthracite American........................ 9-14 3 The best small Pittsburgh................. 8526 4 Average small Newcastle.................. 8-074 5 Pennsylvanian........................... 10-45 6 Coke from Gas-works...................... 7'908 7 Coke and Newcastle, small, i and.... 7897 8 Welsh and Newcastle, mixed ~ and... 7865 9 Derbyshire and small Newcastle, i and' 7-710 10 Average large Newcastle.................. 7-658 11 Derbyshire.................................. 6.772 12 Blythe Main, Northumberland...........6-600 Strength of boilers.-The extension of the expansive method of employing steam to boilers of every denomination, and the gradual introduction in connection therewith of a higher pressure than for Page 219 THE STEAM ENGINE. 219 merly, makes the question of the strength of boilers one of great and increasing importance. This topic was very successfully elucidated, a few years ago, by a committee of the Franklin Institute, Philadelphia, and we shall here recapitulate a few of the more important of the conclusions at which they arrived. Iron boiler plate was found to increase in tenacity as its temperature was raised, until it reached a temperature of 550~ above the freezing point, at which point its tenacity began to diminish. The following table exhibits the cohesive strength at different temperatures. At 320 to 800 the tenacity was = 56,000 lbs., or 1-7th below its maximum. At 5700 --- 66,500 lbs., the maximum. At 7200 --- = 55,000 lbs., the same nearly as at 32~. At 10500 - = 32,000 lbs., nearly ~ of the maximum. At 12400 - = 22,000 lbs., nearly X of the maximum. At 13170 - 9,000 lbs., nearly 1-7th of the maximum. At 30000 iron becomes fluid. The difference in strength between strips of iron cut in the direction of the fibre, and strips cut across the grain, was found to be about 6 per cent. in favour of the former. Repeated piling and welding was found to increase the tenacity and closeness of the iron, but welding together different kinds of iron was found to give an unfavourable result; riveting plates was found to occasion a diminution in their strength, to the extent of about one-third. The accidental overheating of a boiler was found to reduce its strength from 65,000 lbs. to 45,000 lbs. per square inch. Taking into account all these contingencies, it appears expedient to limit the tensile force upon boilers in actual use to about 3000 lbs. per square inch of iron. Copper follows a different law, and appears to diminish in strength by every addition of heat, reckoning from the freezing point. The square of the diminution of strength seems to keep pace with the cube of the temperature, as appears by the following table:TABLE showing the Diminution of Strength of COPPER Boiler Plates by additions to the Temperature, the Cohesion at 32~ being 32,800 lbs. per Square Inch. No. Temperature Diminution of No Temperature Diminution of above 320. Strength. above 320. Strength. 1 900 0'0175 9 6600 0.3425 2 180 0.0540 10 769 0.4398 3 270 0.0926 11 812 0.4944 4 360 0.1513 12 880 0.5581 5 450 0.2046 13 984 0-6691 6 460 0'2133 14 1000 0.6741 7 513 0'2446 15 1200 0'8861 8 529 0'2558 1 16 1300 1.0000 In the case of iron, the following are the results when tabulated after a similar fashion. Page 220 220 THE PRACTICAL MODEL CALCULATOR. TABLE of Experiments on IRON Boiler Plate at High Temperature; the Mean Maximum Tenacity being at 550~ = 65,000 lbs. per Square Inch. Temperature Diminution of Temperature Diminution of observed. Tenacity observed. observed. Tenacity observed. 5500 0'0000 8240 0'2010 570 0'0869 932 0'3324 596 0'0899 947 0'3593 600 0'0964 1030 0'4478 630 0-1047 1111 0-5514 562 0'1155 1155 0'6000 722 0-1436 1159 0-6011 732 0-1491 1187 0'6352 734 0-1535 1237 0'6622 766 0-1589 1245 0-6715 770 0'1627 1317 0'7001 The application of stays to marine boilers, especially in those parts of the water spaces which lie in the wake of the furnace bars, has given engineers much trouble; the 3 plate, of which ordinary boilers are composed, is hardly thick enough to retain a stay with security by merely tapping the plate, whereas, if the stay be riveted, the head of the rivet will in all probability be soon burnt away. The best practice appears to be to run the stays used for the water spaces in this situation, in a line somewhat beneath the level of the bars, so that they may be shielded as much as possible from the fire, while those which are required above the level of the bars should be kept as nearly as possible towards the crown of the furnace, so as to be removed from the immediate contact of the fire. Screw bolts with a fine thread tapped into the plate, and with a thin head upon the one side, and a thin nut made of a piece of boiler plate on the other, appear to be the best description of stay that has yet been contrived. The stays between the sides of the boiler shell, or the bottom of the boiler and the top, present little difficulty in their application, and the chief thing that is to be attended to is to take care that there be plenty of them; but we may here remark that we think it an indispensable thing, when there is any high pressure of steam to be employed, that the furnace crown be stayed to the top of the boiler. This, it will be observed, is done in the boilers of the Tagus and Infernal; and we know of no better specimen of staying than is afforded by those boilers. AREA OF STEAM PASSAGES. RULE.-To the temperature of steam in the boiler add the constant increment 459; multiply the sum by 11025; and extract the square root of the product. Multiply the length of stroke by the number of strokes per minute; divide the product by the square root just found; and multiply the square root of the quotient by the diameter of the cylinder; the product will be the diameter of the steam passages. Page 221 THE STEAM ENGINE. 221 Let it be required to determine the diameter of the steam passages in an engine of which the diameter of the cylinder is 48 inches, the length of stroke 41 feet, and the number of strokes per minute 26, supposing the temperature under which the steam is generated to be 250 degrees of Fahrenheit's thermometer. Here by the rule we get /11025(250 + 459) = 2795'84; the number of strokes is 26, and the length of stroke 41 feet; hence d 117 it is 8 d 2795.84 = 0-20456d = 0-20456 x 48 = 9'819 inches; so that the diameter of the steam passages is a little more than onefifth of the diameter of the cylinder. The same rule will answer for high and low pressure engines, and also for the passages into the condenser. LOSS OF FORCE BY THE DECREASE OF TEMPERATURE IN THE STEAM PIPES. RULE.-From the temperature of the surface of the steam pipes subtract the temperature of the external air; multiply the remainder by the length of the pipes in feet, and again by the constant number or coefficient 1'68; then divide the product by the diameter of the pipe in inches drawn into the velocity of the steam in feet per second, and the quotient will express the diminution of temperature in degrees of Fahrenheit's thermometer. Let the length of the steam pipe be 16 feet and its diameter 5 inches, and suppose the velocity of the steam to be about 95 feet per second, what will be the diminution of temperature, on the supposition that the steam is at 250~ and the external air at 60~ of Fahrenheit? Here, by the note to the above rule, the temperature of the surface of the steam pipe is 250 - 250 x 0'05 = 237'5; hence we get,= 168 x 16(237 - 60) = 10'044 degrees.: 5x95 If we examine the manner of the composition of the above equation, it will be perceived that, since the diameter of the pipe and the velocity of motion enter as divisors, the loss of heat will be less as these factors are greater; but, on the other hand, the loss of heat will be greater in proportion to the length of pipe and the temperature of the steam. Since the steam is reduced from a higher to a lower temperature during its passage through the steam pipes, it must be attended with a corresponding diminution in the elastic force; it therefore becomes necessary to ascertain to what extent the force is reduced, in consequence of the loss of heat that takes place in passing along the pipes. This is an inquiry of some importance to the manufacturers of steam engines, as it serves to guard them against a very common mistake into which they are liable to fall, especially in reference to steamboat engines, where it is usual to cause the pipe to pass round the cylinder, instead of carrying it in the shortest direction from the boiler, in order to decrease the quantity of surface exposed to the cooling effect of the atmosphere. m 2 Page 222 222 THE PRACTICAL MODEL CALCULATOR. RULE.-From the temperature of the surface of the steam pipe subtract the temperature of the external air; multiply the remainJer by the length of the pipe in feet, and again by the constant fractional coefficient 0'00168; divide the product by the diameter of the pipe in inches drawn into the velocity of steam in feet per second, and subtract the quotient from unity; then multiply the difference thus obtained by the elastic force corresponding to the temperature of steam in the boiler, and the product will be the elastic force of the steam as reduced by cooling in passing through the pipes. Let the dimensions of the pipe, the temperature of the steam, and its velocity through the passages, be the same as in the preceding example, what will be the quantity of reduction in the elastic force occasioned by the effect of cooling in traversing the steam pipe? Since the elastic force of the steam in the boiler enters the equation from which the above rule is deduced, it becomes necessary in the first place to calculate its value; and this is to be done by a rule already given, which answers to the case in which the temperature is greater than 2120; thus we have 250 x 1'69856 = 424 640 Constant number = 205'526 add Sum = 6304166......log. 2'79945 Constant divisor = 333............log. 2'522444 subtract 0'277011 x 6'42 = 1'778410, which is the logarithm of 60'036 inches of mercury. Again, we have 250 - 0'05 x 250 = 237'5; consequently, by multiplying as directed in the rule, we get 237'5 x 0'00168 x 16 = 6'384, which being divided by 95 x 5 = 475, gives 0'01344; and by taking this from unity and multiplying the remainder by the elastic force as calculated above, the value of the reduced elastic force becomes f = 60'036 (1 - 0'01344) = 59.229 inches of mercury. The loss of force is therefore 60'036 - 59'229 = 0'807 inches of mercury, which amounts to oath part of the entire elastic force of the steam in the boiler as generated under the given temperature, being a quantity of sufficient importance to claim the attention of our engineers. FEED WATER. The quantity of water required to supply the waste occasioned by evaporation from a boiler, or, as it is technically termed, the " feed water" required by a boiler working with any given pressure, is easily determinable. For, since the relative volumes of water and steam at any given pressure are known, it becomes necessary merely to restore the quantity of water by the feed pump equiva Page 223 THE STEAM ENGINE. 223 lent to that abstracted in the form of steam, which the known relation of the density to the pressure of the steam renders of easy accomplishment. In practice, however, it is necessary that the feed pump should be able to supply a much larger quantity of water than what theory prescribes, as a great waste of water sometimes occurs from' leakage or priming, and it is necessary to provide against such contingencies. The feed pump is usually made of such dimensions as to be capable of supplying 3~ times the water that the boiler will evaporate, and in low pressure engines, where the cylinder is double acting and the feed pump single acting, this proportion will be maintained by making the pump a 240th of the capacity of the cylinder. In low pressure engines the pressure in the boiler may be taken at 5 lbs. above the pressure of the atmosphere, or 20 lbs. in all; and as high pressure steam is merely low pressure steam compressed into a smaller compass, the size of the feed pump relatively to the size of the cylinder must obviously vary in the direct proportion of the pressure. If, then, the feed pump be 1-240th of the capacity of the cylinder when the total pressure of the steam is 20 lbs., it must be 1-120th of the capacity of the cylinder when the total pressure of the steam is 40 lbs., or 25 lbs. above the atmosphere. This law of variation is expressed by the following rule, which gives the capacity of feed pump proper for all pressures:-Multiply the capacity of the cylinder in cubic inches by the total pressure of the steam in lbs. per square inch, or the pressure in lbs. per square inch on the safety valve, plus 15, and divide the product by 4800; the quotient is the capacity of the feed pump in cubic inches, when the feed pump is single acting and the engine double acting. If the feed pump be double acting, or the engine single acting, the capacity of the pump must be just one-half what is given by this rule. CONDENSING WATER. It was found that the most beneficial temperature of the hot well was 100 degrees. If, therefore, the temperature of the steam be 212~, and the latent heat 1000~, then 1212~ may be taken to represent the heat contained in the steam, or 1112~ if we deduct the temperature of the hot well. If the temperature of the injection water be 50~, then 50 degrees of cold are available for the abstraction of heat, and as the total quantity of heat to be abstracted is that requisite to raise the quantity of water in the steam 1112 degrees, or 1112 times that quantity, one degree, it would raise one-fiftieth of this, or 22'24 times the quantity of water in the steam, 50 degrees. A cubic inch of water, therefore, raised into steam, will require 22'24 cubic inches of water at 50 degrees for its condensation, and will form therewith 23'24 cubic inches of hot water at 100 degrees. It has been a practice to allow about a wine pint (28.9 cubic inches) of injection water for every cubic inch of water evaporated from the boiler. The usual capacity for the cold water pump is -tth of the capacity of the cylinder, which allows some water to run to waste. As a maximum Page 224 224 THE PRACTICAL MODEL CALCULATOR. effect is obtained when the temperature of the hot well is about 1000~, it will not be advisable to reduce it below that temperature in practice. With the superior vacuum due to a temperature of 70~ or 800 the admission of so much cold water into the condenser becomes necessary,-and which has afterwards to be pumped out in opposition to the pressure of the atmosphere,-so that the gain in the vacuum does not equal the loss of power occasioned by the additional load upon the pump, and there is, therefore, a clear loss by the reduction of the temperature below 100~, if such reduction be caused by the admission of an additional quantity of water. If the reduction of temperature, however, be caused by the use of colder water, there is a gain produced by it, though the gain will within certain limits be greater, if advantage be taken of the lowness of the temperature to diminish the quantity of injection. SAFETY VALVES. RULE.-Add 459 to the temperature of the steam in degrees of Fahrenheit; divide the sum by the product of the elastic force of the steam in inches of mercury, into its excess above the weight of the atmosphere in inches of mercury; multiply the square root of the quotient by'0653; multiply this product by the number of cubic feet per hour of water evaporated, and this last product is the theoretical area of the orifice of the safety valve in square inches. To apply this to an example-which, however, it must be remembered, will give a result much too small for practice. Required the least area of a safety valve of a boiler suited for a 250 horse power engine, working with steam 6 lbs. more than the atmosphere on the square inch. In this case the total pressure is equal to 21 lbs. per square inch; and as in round numbers one pound of pressure is equal to about two inches of mercury, it follows that f = 42 inches of mercury. It will be necessary to calculate t from formula (S) already given. The operation is as follows:log. 42 -- 6'42 = 1'623249 -- 6'42 = 0'252842 constant co-efficient = 196 2'292363 2'545205 natural number - 350'92 constant temperature = 121 t = 229'92 1459 + t 459 + 229 92 therefore ~/f - 30) 42 x 12 - 6-5094 2 vl 1'3669 = 1'168; therefore x ='0653 x 1'168 x N ='0757 N. Page 225 THE STEAM ENGINE. 225 We have stated in a former part of this work that a cubic foot of water evaporated per hour is equivalent to one horse power; therefore in this case N = 250 and x = 18'925 sq. in. As another example. Required the proper area of the safety valve of a boiler suited to an engine of 500 horse power, when it is wished that the steam should never acquire an elastic force greater than 60 lbs. on the square inch above the atmosphere. In this case the whole elastic force of the steam is 75 lbs.; and as 1 pound corresponds in round numbers to 2 inches of mercury, it follows that f = 150. It will be necessary to calculate the temperature corresponding to this force. The operation is as follows:Log. 150 - 6'42 = 2'176091. 6'42 = 338955 constant co-efficient = 196 log. 2-292363 add natural number = 427'876 2 631318 constant temperature 121 required temperature 306'876 degrees of Fahrenheit's scale 459 + t 459 + 306'876 765'876 765'896 therefore f (f - 30) 150(150 - 30) 150 x 120 18000 = 043549; therefore jf (f 30)-/ V 042549 = 20628. Hence the required area = -0653 x'20628 x 500 ='01347 x 500 = 6'735 square inches. If the area of the safety valve of a boiler suited for an engine of 500 horse power be required, when it is wished the steam should never acquire a greater temperature than 300~, it will be necessary to calculate the elastic force corresponding to this temperature; and by formula for this purpose, the required area ='0653 x'231 x 500 ='0151 x 500 = 7'55 square inches. It will be perceived from these examples that the greater the elasticity and the higher the corresponding temperature the less is the area of the safety valve. This is just as might have been expected, for then the steam can escape with increased velocity. We may repeat that the results we have arrived at are much less than those used in practice. For the sake of safety, the orifices of the safety valve are intentionally made much larger than what theory requires; usually x8 of a square inch per horse power is the ordinary proportion allowed in the case of low pressure engines. THE SLIDE VALVE. The four following practical rules are applicable alike to short slide and long D valves. RULE I.-To find how much cover must be given on the steam side in order to cut the steam off at any given part of the stroke.From the length of the stroke of the piston, subtract the length of that part of the stroke that is to be made before the steam is cut off. Divide the remainder by the length of the stroke of the 15 Page 226 226 TIIE PRACTICAL MODEL CALCULATOR. piston, and extract the square root of the quotient. Multiply the square root thus found by half the length of the stroke of the valve, and from the product take half the lead, and the remainder will be the cover required. RULE II.-To find at what part of the stroke any given amount of cover on the steam side will cut off the steam.-Add the cover on the steam side to the lead; divide the sum by half the length of stroke of the valve. In a table of natural sines find the arc whose sine is equal to the quotient thus obtained. To this are add 900, and from the sum of these two arcs subtract the are whose cosine is equal to the cover on the steam side divided by half the stroke of the valve. Find the cosine of the remaining are, add 1 to it, and multiply the sum by half the stroke of the piston, and the product is the length of that part of the stroke that will be made by the piston before the steam is cut off. RULE III. —To find how much before the end of the stroke, the exhaustion of the steam in front of the piston will be cut off.-To the cover on the steam side add the lead, and divide the sum by half the length of the stroke of the valve. Find the arc whose sine is equal to the quotient, and add 90~ to it. Divide the cover on the exhausting side by half the stroke of the valve, and find the are whose cosine is equal to the quotient. Subtract this are from the one last obtained, and find the cosine of the remainder. Subtract this cosine from 2, and multiply the remainder by half the stroke of the piston. The product is the distance of the piston from the end of its stroke when the exhaustion is cut off. RULE IV.-To find how far the piston is from the end of its stroke, when the steam that is propoelling it by expansion is allowed to escape to the condenser.-To the cover on the steam side add the lead, divide the sum by half the stroke of the valve, and find the arc whose sine is equal to the quotient. Find the arc whose cosine is equal to the cover on the exhausting side, divided by half the stroke of the valve. Add these two arcs together, and subtract 90~. Find the cosine of the residue, subtract it from 1, and multiply the remainder by half the stroke of the piston. The product is the distance of the piston from the end of its stroke, when the steam that is propelling it is allowed to escape to the condenser. In using these rules, all the dimensions are to be taken in inches, and the answers will be found in inches also. From an examination of the formulas we have given on this subject, it will be perceived (supposing that there is no lead) that the part of the stroke where the steam is cut off, is determined by the proportion which the cover on the steam side bears to the length of the stroke of the valve: so that in all cases where the cover bears the same proportion to the length of the stroke of the valve, the steam will be cut off at the same part of the stroke of the piston. In the first line, accordingly, of Table I., will be found eight different parts of the stroke of the piston designated; and directly Page 227 THE STEAM ENGINE. 227 below each, in the second line, is given the quantity of cover requisite to cause the steam to be cut off at that particular part of the stroke. The different sizes of the cover are given in the second line, in decimal parts of the length of the stroke of the valve; so that, to get the quantity of cover corresponding to any of the given degrees of expansion, it is only necessary to take the decimal in the second line, which stands under the fraction in the first, that marks the degree of expansion, and multiply that decimal by the length you intend to make the stroke of the valve. Thus, suppose you have an engine in which you wish to have the steam cut off when the piston is a quarter of the length of its stroke from the end of it, look in the table, and you will find in the third column from the left, 4. Directly under that, in the second line, you have the decimal'250. Suppose that you think 18 inches will be a convenient length for the stroke of the valve, multiply the decimal ~250 by 18, which gives 41. Hence we learn that with an 18 inch stroke for the valve, 4- inches of cover on the steam side will cause the steam to be cut off when the piston has still a quarter of its stroke to perform. Half the stroke of the valve must always be at least equal to the cover on the steam side added to the breadth of the port. By the "breadth" of the port, we mean its dimension in the direction of the valve's motion; in short, its perpendicular depth when the cylinder is upright. The words " cover" and " lap" are synonymous. Consequently, as the cover, in this case, must be 41 inches, and as half the stroke of the valve is 9 inches, the breadth of the port cannot be more than (9 - = 4 4) 41 inches. If this breadth of port is not enough, we must increase the stroke of the valve; by which means we shall get both the cover and the breadth of the port proportionally increased. Thus, if we make the length of valve stroke 20 inches, we shall have for the cover'250 x 20 == inches, and for the breadth of the port 10 - 5 = 5 inches. TABLE I. Distance of the piston from! 8 6 4 3 2 the termination of its | 24 1 24 24 24 stroke, when the steam or o or or or or is cut off, in parts of the 1 length of its stroke. J _ 1 1 1 Cover on the steam side of ] the valve, in decimal parts of the length of its stroke. This table, as we have already intimated, is computed on the supposition that the valve is to have no lead; but, if it is to have lead, all that is necessary is to subtract half the proposed lead from the cover found from the table, and the remainder will be the Page 228 228 THE PRACTICAL MODEL CALCULATOR. proper quantity of cover to give to the valve. Suppose that, in the last example, the valve was to have 4 inch of lead, we would subtract 1 inch from the 5 inches found for the cover by the table: that would leave 47 inches for the quantity of cover that the valve ought to have. TABLE II. Length of Cover required on the steam side of the valve to cut the steam off at any of the the stroke under-noted parts of the stroke. of the valve. Inches. 1 5 1 1 1 1 3 46 8 15 24 6.94 6.48 6.00 5.47 4.90 4.25 3847 2.45 231 6.79 6.34 5.88 5.36 4.79 4.16 3.39 2.39 23 6.65 6.21 5.75 5.24 4-69 4.07 3-32 2.34 22~ 6.50 6.07 5-62 5-13 4.59 3.98 3825 2.29 22 6.36 5.94 5'50 5-02'449 3.89 3-13 2-24 21k 6'21 5'80 5638 4'90 4'39 3'80 3810 2'19 21 6'07 5'67 5'25 4.79 4'28 3'72 3803 2'14 20 5'92 5'53 5' 12 4'67 4'18 3'63 2'96 2'09 20 5'78 5 40 5'00 4'56 4'08 3'54 2-89 2'04 19k 5664 5'26 4'87 4'45 3'98 3845 2'82 1'99 19 5'49 5'13 4'75 4'33 3888 3'36 2'74 1-94 18; 5'34 4'99 4'62 4'22 3'77 3'27 2'67 1'88 18 5'20 4'86 4'50 4'10 3'67 3'19 2'60 1'83 17k 5'06 4'72 4'37 3'99 3'57 3'10 2'53 1'78 17 4'91 4'59 4'25 3-88 3'47 3'01 2'45 1'73 16k 4'77 4'45 4'12 3'76 3836 2'92 2'38 1-68 16 4'62 4'32 4 00 3'65 3'26 2 83 2'31 1 63 151 4'48 4'18 3'87 3 53 3'16 2'74 2'24 1!58 15 4'33 4'05 3 75 3842 3-06 2'65 2'16 1.53 14k 4'19 3'91 3'62 3'31 2'96 2'57 2'09 1'48 14 4'05 3878 3'50 3'19 2'86 2'48 2'02 1'43 13k 3'90 3'64 3'37 3'08 2'75 2'39 1'95 1'37 13 3'76 3'51 3 25 2'96 2'65 2'30 1'88 1 32 12k 3'61 3'37 3'12 2'85 2'55 2'21 1 80 1.27 12 3'47 3 24 3'00 2'74 2'45 2'12 1 73 1-22 11k 3'32 3'10 2'87 2'62 2'35 2'03 1'66 1.17 11 3'18 2'97 2'75 2'51 2'24 1'95 1'58 1'12 10k 3'03 2'83 2'62 2'39 2-14 1'86 1'51 1.07 10 2'89 2-70 2'50 2'28 2'04 1'77 1'44 1.02 9~ 2-65 2-56 2'37 2'17 1'93 1'68 1'32 *96 9 2'60 2'43 2'25 2' 05 1'84 1'59 1'30 *92 8k 2'46 2'29 2'12 1'94 1' 73 1'50 1'23' 86 8 2'31 2'16 2 00 1 82 1'63 1 42 1'15' 81 7k 2'16 2'02 1'87 1'71 1'53 10 33 1 08' 76 7 2'02 1'89 1-75 1'60 1'43 1'24 101' 71 6k 1 88 1 75 1 62 1 48 1 32 1 15' 94' 66 6 1'73 1'62 1 50 1'37 1 22 1 06' 86' 61 II 1-58 1-48 1 37 1'25 1.12 97.79' 56 6 1 44 1.35 1 25 1.14 1 02 -88.72 51 4k 1.30 1.21 1.12 1.03. 92 -80 -65. 46 4 1 16 1 08 1 00 -91 -82 - 71 -58 -41 3k 1.01. 94.87. 80.71. 62. 50.35 3 ~86. 81.75 *68. 61.53. 44. 30 Table II. is an extension of Table I..for the purpose of obviating, in most cases, the necessity of even the very small degree of trouble required in multiplying the stroke of the valve by one of the decimals in Table I. The first line of Table II. consists, as in Table I., of eight fractions, indicating the various parts of the stroke Page 229 THE STEAM ENGINE. 229 at which the steam may be cut off. The first column on the left hand consists of various numbers that represent the different lengths that may be given to the stroke of the valve, diminishing, by half-inches, from 24 inches to 3 inches. Suppose that you wish the steam cut off at any of the eight parts of the stroke indicated in the first line of the table, (say at 1 from the end of the stroke,) you find 1 at the top of the sixth column from the left. Look for the proposed length of stroke of the valve (say 17 inches) in the first column on the left. From 17, in that column, run along the line towards the right, and in the sixth column, and directly under the 1 at the top, you will find 3'47, which is the cover required to cause the steam to be cut off at 1 from the end of the stroke, if the valve has no lead. If you wish to give it lead, (say 4 inch,) subtract the half of that, or 1 ='125 inch from 3'47, and you will have 3-47 -'125 = 3'345 inches, the quantity of cover that the valve should have. To find the greatest breadth that we can give to the port in this case, we have, as before, half the length of stroke, 81 —3'345=5'155 inches, which is the greatest breadth we can give to the port with this length of stroke. It is scarcely necessary to observe that it is not at al essential that the port should be so broad as this; indeed, where great length of stroke in the valve is not inconvenient, it is always an advantage to make it travel farther than is just necessary to make the port full open; because, when it travels farther, both the exhausting and steam ports are more quickly opened, so as to allow greater freedom of motion to the steam. The manner of using this table is so simple, that we need not trouble the reader with more examples. We pass on, therefore, to explain the use of Table III. Suppose that the piston of a steam engine is making its downward stroke, that the steam is entering the upper part of the cylinder by the upper steam-port, and escaping from below the piston by the lower exhausting-port; then, if (as is generally the case) the slide valve has some cover on the steam side, the upper port will be closed before the piston gets to the bottom of the stroke, and the steam above then acts expansively, while the communication between the bottom of the cylinder and the condenser still continues open, to allow any vapour from the condensed water in the cylinder, or any leakage past the piston, to escape into the condenser; but, before the piston gets to the bottom of the cylinder, this passage to the condenser will also be cut off by the valve closing the lower port. Soon after the lower port is thus closed, the upper port will be opened towards the condenser, so as to allow the steam that has been acting expansively to escape. Thus, before the piston has completed its stroke, the propelling power is removed from behind it, and a resisting power is opposed before it, arising from the vapour in the cylinder, which has no longer any passage open to the condenser. It is evident, that if there is no cover on the exhausting side of the valve, the exhausting port before U Page 230 230 THE PRACTICAL MODEL CALCULATOR. the piston will be closed, and the one behind it opened, at the same time; but, if there is any cover on the exhausting side, the port before the piston will be closed before that behind it is opened; and the interval between the closing of the one and the opening of the other will depend on the quantity of cover on the exhausting side of the valve. Again, the position of the piston in the cylinder, when these ports are closed and opened respectively, will depend on the quantity of cover that the valve has on the steam side. If the cover is large enough to cut the steam off when the piston is yet a considerable distance from the end of its stroke, these ports will be closed and opened at a proportionably early part of the stroke; and when it is attempted to obtain great expansion by the slide-valve alone, without an expansion-valve, considerable loss of power is incurred from this cause. Table III. is intended to show the parts of the stroke where, under any given arrangement of slide valve, these ports close and open respectively, so that thereby the engineer may be able to estimate how much of the efficiency of the engine he loses, while he is trying to add to the power of the steam by increasing the expansion in this manner. In the table, there are eight double columns, and at the heads of these columns are eight fractions, as before, representing so many different parts of the stroke at which the steam may be supposed to be cut off. In the left-hand single column in each double one, are four decimals, which represent the distance of the piston (in terms of the length of its stroke) from the end of its stroke when the exhaustingport before it is opened, corresponding with the degree of expansion indicated by the fraction at the top of the double column and the cover on the exhausting side opposite to these decimals respectively in the left-hand column. The right-hand single column in each double one contains also each four decimals, which show in the same way at what part of the stroke the exhausting-port behind the piston is opened. A few examples will, perhaps, explain this best. Suppose we have an engine in which the slide valve is made to cut the steam off when the piston is 1-3d from the end of its stroke, and that the cover on the exhausting side of the valve is 1-8th of the whole length of its stroke. Let the stroke of the piston be 6 feet, or 72 inches. We wish to know when the exhausting-port before the piston will be closed, and when the one behind it will be opened. At the top of the left-hand double column, the given degree of expansion (1-3d) is marked, and in the extreme left column we have at the top the given amount of cover (1-8th). Opposite the 1-8th, in the first double column, we have'178 and'033, which decimals, multiplied respectively by 72, the length of the stroke, will give the required positions of the piston: thus 72x 178=12'8 inches = distance of the piston from the end of the stroke when the exhausting-port before the piston is shut; and 72 x'033 = 2'38 inches = distance of the piston from the end of its stroke when the exhausting-port behind it is opened. Page 231 THE STEAM ENGINE. 231 Cover on the exhausting side of the valve in parts of the length of its stroke. 6.. 4..Distance of the piston from the end of its nc ca _o a stroke, when the exhausting-port before F is tkN.) as C) co it is shnt (in parts of the stroke). a 6666. Distance of the piston fron the end of its a cc o a, cc stroke, when the exhausting-port behind it is opened (in parts of the stroke). 6... Distance of the piston from the end of its co o 0 a stroke, when the exhausting-port before F a C C cO o C~3 it is shut (in parts of the stroke). 6 6 6 6,..Distance of the piston from the end of its i' at, c ~ c- stroke, when the exhausting-port behind a S as La s it is opened (in parts of the stroke). 6 ~ 6 * i Distance of the piston from the end of its s stroke, when the exhausting-port before aD i cJa o cs it is shut (in parts of the stroke). P 6 6 6 6 o o o Distance of the piston from the end of its | cc-, C c stroke, when the exhausting-port behind _ CZ it is opened (in parts of the stroke).: 6 6 * Distance of the piston from the end of its oo C)a co stroke, when the exhausting-port before P` *-' * c- r it is shut (in parts of the stroke). a Distance of the piston from the end of its cP < eCn co a stroke, when the exhausting-port behind it is opened (in parts of the stroke). a 6o 6 6 S Distance of the piston from the end of its | C a c -o stroke, when the exhausting-port before c CS c o Q s it is shult (in parts of the stroke). - 6 6 6 Distance of the piston from the end of its g c o oa ao o stroke, when the exhausting-port behind F P ~ co co Io co it is opened (in parts of the stroke). a Distance of the piston from the end of its _ 0c c Co stroke, when the exhaiustingp-port before Cc it is shut (in parts of the stroke). E 6. 6. 6 Distance of the piston firom the end of its c a oI- os I stroke, when the exhausting-port behind a ~ LO c;a G r it is opened (in parts of the stroke). a I 6:... 6 Distance of the piston from the end of its ~ - c -a a ~ stroke, when the exhausting-port before -D C r c it is shut (in parts of the stroke). Distance of the piston from the end of its a c a |c - ~ OC stroke, when the exhausting-port behind Li) CO CIO it- is opened (in parts of the stroke). Distance of the piston from the end of its | d:> cn ~ stroke, wbhen the exhausting-port before cc 3 0 _ Cj {it is shut (in parts of the stroke). Distance of the piston froni the end of its C c 20 1 stroke, wvhen the exhausting-port behind o = s P t it is opened (in parts of the stroke). r ~~P r3 r it, is openedl (in parts of the stroke). 8 Page 232 232 THE PRACTICAL MODEL CALCULATOR. To take another example. Let the stroke of the valve be 16 inches, the cover on the exhausting side 2 inch, the cover on the steam side 3{ inches, the length of the stroke of the piston 60 inches. It is required to ascertain all the particulars of the working of this valve. The cover on the exhausting side is evidently 1 of the length of the valve stroke. Again, looking at 16 in the left-hand column of Table II., we find in the same horizontal line 3'26, or very nearly 3- under l at the head of the column, thus showing that the steam will be cut off at 1 from the end of the stroke. Again, under 6 at the head of the fifth double column from the left in Table III., and in a horizontal line with 1 in the left-hand column, we have ~053 and'033. Hence,'053 x 60 = 3'18 inches = distance of the piston from the end of its stroke when the exhausting-port before it is shut, and -033 x 60 = 1'98 inches = distance of the piston from the end of its stroke when the exhausting-port behind it is opened. If in this valve the cover on the exhausting side were increased (say to 2 inches, or 1 of the stroke,) the effect would be to make the port before the valve be shut sooner in the proportion of'109 to'053, and the port behind it later in the proportion of'008 to'033 (see Table III.) Whereas, if the cover on the exhausting side were removed entirely, the port before the piston would be shut and that behind it opened at the same time, and (see bottom of fifth double column, Table III.) the distance of the piston from the end of its stroke at that time would be'043 x 60 = 2'58 inches. An inspection of Table III. shows us the effect of increasing the expansion by the slide-valve in augmenting the loss of power occasioned by the imperfect action of the eduction passages. Referring to the bottom line of the table, we see that the eduction passage before the piston is closed, and that behind it opened, (thus destroying the whole moving power of the engine,) when the piston is.092 from the end of its stroke, the steam being cut off at 3- from the end. Whereas, if the steam is only cut off at -2 from the end of the stroke, the moving power is not withdrawn till only'011 of the stroke remains uncompleted. It will also be observed that increasing the cover on the exhausting side has the effect of retaining the action of the steam longer behind the piston, but it at the same time causes the eduction-port before it to be closed sooner. A very cursory examination of the action of the slide valve is sufficient to show that the cover on the steam side should alwa-ys be greater than on the exhausting side. If they are equal, the steam would be admitted on one side of the piston at the same time that it was allowed to escape from the other; but universal experience has shown that when this is the case, a very considerable part of the power of the engine is destroyed by the resistance opposed to the piston, by the exhausting steam not getting away to the condenser with sufficient rapidity. Hence we see the necessity of the cover on the exhausting side being always less than the cover on the steam side; and the difference should be the greater the higher the velocity of the piston is intended to be, because the quicker the Page 233 THE STEAM ENGINE. 233 piston moves the passage for the waste steam requires to be the larger, so as to admit of its getting away to the condenser with as great rapidity as possible. In locomotive or other engines, where it is not wished to expand the steam in the cylinder at all, the slide valve is sometimes made with very little cover on the steam side: and in these circumstances, in order to get a sufficient difference between the cover on the steam and exhausting sides of the valve, it may be necessary not only to take away all the cover on the exhausting side, but to take off still more, so as to make both exhausting passages be in some degree open, when the valve is at the middle of its stroke. This, accordingly, is sometimes done in such circumstances as we have described; but, when there is even a small degree of cover on the steam side, this plan of taking more tthan all the cover off the exhausting side ought never to be resorted to, as it can serve no good purpose, and will materially increase an evil we have already explained, viz. the opening of the exhausting-port behind the piston before the stroke is nearly completed. The tables apply equally to the common short slide three-ported valves and to the long D valves. In fig. 1 is exhibited a common arrangement of the valves in lo-,Fig. 1. Fig. 2. Fig. 3. u2 Page 234 234 THE PRACTICAL MODEL CALCULATOR. comotive engines, and in figs. 2 and 3 is shown an arrangement for working valves by a shifting cam, by which the amount of expansion may be varied. This particular arrangement, however, is antiquated, and is now but little used. The extent to which expansion can be carried beneficially by means of lap upon the valve is about one-third of the stroke; that is, the valve may be made with so much lap, that the steam will be cut off when one-third of the stroke has been performed, leaving the residue to be accomplished by the agency of the expanding steam; but if more lap be put on than answers to this amount of expansion, a very distorted action of the valve will be produced, which will impair the efficiency of the engine. If a further amount of expansion than this is wanted, it may be accomplished by wiredrawing the steam, or by so contracting the steam passage, that the pressure within the cylinder must decline when the speed of the piston is accelerated, as it is about the middle of the stroke. Thus, for example, if the valve be so made as to shut off the steam by the time two-thirds of the stroke have been performed, and the steam be at the same time throttled in the steam pipe, the full pressure of the steam within the cylinder cannot be maintained except near the beginning of the stroke where the piston travels slowly; for as the speed of the piston increases, the pressure necessarily subsides, until the piston approaches the other end of the cylinder, where the pressure would rise again but that the operation of the lap on the valve by this time has had the effect of closing the communication between the cylinder and steam pipe, so as to prevent more steam from entering. By throttling the steam, therefore, in the manner here indicated, the amount of expansion due to the lap may be doubled, so that an engine with lap enough upon the valve to cut off the steam at two-thirds of the stroke, may, by the aid of wire-drawing, be virtually rendered capable of cutting off the steam at one-third of the stroke. The usual manner of cutting off the steam, however, is by means of a separate valve, termed an expansion valve; but such a device appears to be hardly necessary in many engines. In the Cornish engines, where the steam is cut off in some cases at one-twelfth of the stroke, a separate valve for the admission of steam, other than that which permits its escape, is of course indispensable; but in common rotative engines, which may realize expansive efficacy by throttling, a separate expansive valve does not appear to be required. In all engines there is a point beyond which expansion cannot be carried with advantage, as the resistance to be surmounted by the engine will then become equal to the impelling power; but in engines working with a high pressure of steam that point is not so speedily attained. In high pressure, as contrasted with condensing engines, there is always the loss of the vacuum, which will generally amount to 12 or 13 lbs. on the square inch, and in high pressure engines there is a benefit arising from the use of a very high pressure over a pressure of a moderate account. In all high pressure engines, there is Page 235 THE STEAM ENGINE. 235 a diminution in the power caused by the counteracting pressure of the atmosphere on the educting side of the piston; for the force of the piston in its descent would obviously be greater, if there was a vacuum beneath it; and the counteracting pressure of the atmosphere is relatively less when the steam used is of a very high pressure. It is clear, that if we bring down the pressure of the steam in a high pressure engine to the pressure of the atmosphere, it will not exert any power at all, whatever quantity of steam may be expended, and if the pressure be brought nearly as low as that of the atmosphere, the engine will exert only a very small amount of power; whereas, if a very high pressure be employed, the pressure of the atmosphere will become relatively as small in counteracting the impelling pressure, as the attenuated vapour in the condenser of a condensing engine is in resisting the lower pressure which is there employed. Setting aside loss fiom friction, and supposing the vacuum to be a perfect one, there would be no benefit arising from the use of steam of a high pressure in condensing engines, for the same weight of steam used without expansion, or with the same measure of expansion, would produce at every pressure the same amount of mechanical power. A piston with a square foot of area, and a stroke of three feet with a pressure of one atmosphere, would obviously lift the same weight through the same distance, as a cylinder with half a square foot of area, a stroke of three feet, and a pressure of two atmospheres. In the one case, we have three cubic feet of steam of the pressure of one atmosphere, and in the other case 1~ cubic feet of the pressure of two atmospheres. But there is the same weight of steam, or the same quantity of heat and water in it, in both cases; so that it appears a given weight of steam would, under such circumstances, produce a definite amount of power, without reference to the pressure. In the case of ordinary engines, however, these conditions do not exactly apply; the vacuum is not a perfect one, and the pressure of the resisting vapour becomes relatively greater as the pressure of the steam is diminished; the friction also becomes greater from the necessity of employing larger cylinders, so that even in the case of condensing engines, there is a benefit arising from the use of steam of a considerable pressure. Expansion cannot be carried beneficially to any great extent, unless the initial pressure be considerable; for if steam of a low pressure were used, the ultimate tension would be reduced to a point so nearly approaching that of the vapour in the condenser, that the difference would not suffice to overcome the friction of the piston; and a loss of power would be occasioned by carrying expansion to such an extent. In some of the Cornish engines, the steam is cut off at one-twelfth of the stroke; but there would be a loss arising from carrying the expansion so far, instead of a gain, unless the pressure of the steam were considerable. It is clear, that in the case of engines which carry expansion very far, a very perfect vacuum in the condenser is more important than it is in other cases. Nothing can be easier than to compute the ultimate Page 236 236 TIIE PRACTICAL iMODEL CALCULATOR. pressure of expanded steam, so as to see at what point expansion ceases to be productive of benefit; for as the pressure of expanded steam is inversely as the space occupied, the terminal pressure when the expansion is twelve times is just one-twelfth of what it was at first, and so on, in all other projections. The total pressure should be taken as the initial pressure-not the pressure on the safety valve, but that pressure plus the pressure of the atmosphere. In high pressure engines, working at from 70 to 90 lbs. on the square inch, as in the case of locomotives, the efficiency of a given quantity of water raised into steam may be considered to be about the same as in condensing engines. If the pressure of steam in a high pressure engine be 120 lbs., or 125 lbs. above the atmosphere, then the resistance occasioned by the atmosphere will cause a loss of -th of the power. If the pressure of the-steam in a low pressure engine be 16 lbs. on the square inch, or 11 lbs. above the atmosphere, and the tension of the vapour in the condenser be equivalent to 4 inches of mercury, or 2 lbs. of pressure on the square inch, then the resistance occasioned by this rare vapour will also cause a loss of 1th of the power. A high pressure engine, therefore, with a pressure of 105 lbs. above the atmosphere, works with only the same loss from resistance to the piston, as a low pressure engine with a pressure of 1 lb. above the atmosphere, and with these proportions the power produced by a given weight of steam will be the same, whether the engine be high pressure or condensing. SPIEROIDAL CONDITION OF WATER IN BOILERS. Some of the more prominent causes of boiler explosions have been already enumerated; but explosions have in some cases been attributed to the spheroidal condition of the water in the boiler, consequent upon the flues becoming red-hot from a deficiency of water, the accumulation of scale, or otherwise. The attachment of scale, from its imperfect conducting power, will cause the iron to be unduly heated; and if the scale be accidentally detached, a partial explosion may occur in consequence. It is found, that a sudden disengagement of steam does not immediately follow the contact of water with the hot metal, for water thrown upon redhot iron is not immediately converted into steam, but assumes the spheroidal form and rolls about in globules over the surface. These globules, however high the temperature of the metal may be on which they are placed, never rise above the temperature of 205~, and give off but very little steam; but if the temperature of the metal be lowered, the water ceases to retain the spheroidal form, and comes into intimate contact with the metal, whereby a rapid disengagement of steam takes place. If water be poured into a very hot copper flask, the flask may be corked up, as there will be scarce any steam produced so long as the high temperature is maintained; but so soon as the temperature is suffered to fall below 350~ or 400~, the spheroidal condition being no longer maintainable, steam is generated with rapidity, and the cork will be projected fromn the Page 237 THE STEAM ENGINE. 237 mouth of the flask with great force. In a boiler, no doubt, where there is a considerable head of water, the repellant action of the spheroidal globules will be more effectually counteracted than in the small vessels employed in experimental researches. But it is doubtful whether in all boilers there may not be something of the spheroidal action perpetually in operation, and leading to effects at present mysterious or inexplicable. One of the most singular phenomena attending the spheroidal condition is, that the vapour arising from a spheroid is of a far higher temperature than the spheroid itself. Thus, if a thermometer be held in the atmosphere of vapour which surrounds a spheroid of water, the mercury, instead of standing at 205~, as would be the case if it had been immersed in the spheroid, will rise to a point determinable by the temperature of the vessel in which the spheroid exists. In the case of a spheroid, for example, existing within a crucible raised to a temperature of 4000, the thermometer, if held in the vapour, will rise to that point; and if the crucible be made red-hot, the thermometer will be burst, from the boiling point of mercury having been exceeded. A part of this effect may, indeed, be traced to direct radiation, yet it appears indisputable, from the experiments which have been made, that the vapour of a liquid spheroid is much hotter than the spheroid itself. EXPANSION. At page 131 we have given a table of hyperbolic or Byrgean logarithms, for the purpose of facilitating computations upon this subject. Let the pressure of the steam in the boiler be expressed by unity, and let x represent the space through which the piston has moved whilst urged by the expanding steam. The density will then be -,and, assuming that the densities and elasticities are prowill be the differental of the iciency, and the portionate, d x will be the differential of the efficiency, and the i + x efficiency itself will be the integral of this, or, in other words, the hyperbolic logarithm of the denominator; wherefore the efficiency of the whole stroke will be 1 + log. (1 + x). Supposing the pressure of the atmosphere to be 15 lbs., 15 + 35 = 50 lbs., and if the steam be cut off at 4th of the stroke, it will be expanded into four times its original volume; so that at the termination of the stroke, its pressure will be 50 4=12'2 lbs., or 2'8 lbs. less than the atmospheric pressure. When the steam is cut off at one-fourth, it is evident that x = 3. In such case the efficiency is 1 + log. (1 + 3), or 1 + log. 4. The hyperbolic logarithm of 4 is 1'386294, so that the efficiency of the steam becomes 2'386294; that is, by cutting off the steam at 4, more than twice the effect is produced with the same consumption of fuel; in other words, one-half of the fuel is saved. Page 238 238 THE PRACTICAL MODEL CALCULATOR. This result may thus be expressed in words: —Divide the length of the stroke through which the steam expands by the length of stroke performed with the full pressure, which last portion call 1; the hyperbolic logarithm of the quotient is the increase of efficiency due to expansion. We introduce on the following page more detailed tables, to facilitate the computation of the power of an engine working expansively, or rather to supersede the necessity of entering into a computation at all in each particular case. The first column in each of the following tables contains the initial pressure of the steam in pounds, and the remaining columns contain the mean pressure of steam throughout the stroke, with the different degrees of expansion indicated at the top of the columns, and which express the portion of the stroke during which the steam acts expansively. Thus, for example, if steam be admitted to the cylinder at a pressure of 3 pounds per square inch, and be cut off within 8th of the end of the stroke, the mean pressure during the whole stroke will be 2'96 pounds per square inch. In like manner, if steam at the pressure of 3 pounds per square inch were cut off after the piston had gone through 1th of the stroke, leaving the steam to expand through the remaining 8th, the mean pressure during the whole stroke would be 1'164 pounds per square inch. FRCTION. The friction of iron sliding upon brass, which has been oiled and then wiped dry, so that no film of oil is interposed, is about -L of the pressure; but in machines in actual operation, where there is a film of oil between the rubbing surfaces, the fraction is only about one-third of this amount, or -Ld of the weight. The tractive resistance of locomotives at low speeds, which is entirely made up of friction, is in some cases hoth of the weight; but on the average about 3-1th of the load, which nearly agrees with my former statement. If the total friction be hth of the load, and the rolling friction be 1-,-th of the load, then the friction of attrition must be 42 -th of the load; and if the diameter of the wheels be 36 in., and the diameter of the axles be 3 in., which are common proportions, the friction of attrition must be increased in the proportion of 36 to 3, or 12 times, to represent the friction of the rubbing surface when moving with the velocity of the carriage. 1ths are about ~-th of the load, which does not differ much from the proportion of -d, as previously stated. While this, however, is the average result, the fiiction is a good deal less in some cases. Engineers, in some experiments upon the friction, found the friction to amount to less than -th of the weight; and in some experiments upon the friction of locomotive axles, it was found that by ample lubrication the fiiction might be made as little as -th of the weight, and the traction, with the ordinary size of wheels, would in such a case be about 5th of the weight. The function of lubricating substances is to prevent the rubbing surfaces from coming into contact, whereby abrasion would be produced, and unguents are effectual in this Page 239 THE STEAM ENGINE. 239 EXPANDED STEAM.-MEAN PRESSURE AT DIFFERENT DENSITIES AND RATE OF EXPANSION. lThe column headed 0 contains the initialpressure in lbs., and the remaining columns contain the meanz pressure in lbs., with dcifftrent grades of expansion. EXPANSION BY EIGHTHS. 0 1 2 3 4 5 6 7 3 2-96 2-89 2-75 2-53 2-22 1 789 1'154 4 3'95 3-85 3-67 3'38 2-96 2 386 1 539 5 4-948 4-818 4-593 4-232 3'708 2-982 1-924 6 5-937 5-782 5-512 5'079 4-450 3-579 2-309 7 6-927 6-746 6-431 5'925 5-241 4-175 2'694 8 7-917 7-710 7'350 6-772 5'934 4 772 3 079 9 8-906 8-673 8-268 7-618 6'675 5'368 3-463 10 9-896 9-637 9-187 8-465 7-417 5-965 3'848 11 10'885 10-601 10-106 9'311 8-159 6-561 4 233 12 11-875 11-565 10-925 10-158 8-901 7-158 4'618 13 12-865 12-528 11-943 11-004 9-642 7-754 5'003 14 13'854 13-492 12-862 11-851 10-384 8'531 5 388 15 14-844 14-456 13-781 12-697 11-126 8-947 5'773 16 15-834 15-420 14-700 13-544 11-868 9-544 6'158 17 16-823 16-383 15-618 14-390 12'609 10-140 6'542 18 17-813 17-347 16-537 15-237 13-351 10-737 6-927 19 18-702 18-311 17-448 16-803 14-093 11-333 7'312 20 19'792 19'275 18'375 16 930 14-835 11 930 7-697 25 24'740 24'093 22'968 21 162 18-543 14-912 9-621 30 29-688 28-912 27.562 25'395 22-252 17-895 11-546 35 34.636 33-731 33-156 29-627 25-961 20-877 13-470 40 39'585 38'550 36'750 33 860 29'670 23 860 15-395 45 44'533 43.368 41.343 38'092 33 378 26-842 17'319 50 49-481 48-187 45-937 42-325 37-067 29'825 19-243 EXPANSION BY TENTHS. 6 36 4 5 6S 0 6 I 1 0 10 I -10 -10 1-10 T0 3 2-980 2.930 2-830 2-710 2-539 2-299 1.981 1-668 0-990 44 3974 3913 3-780 3-614 3.386 3o065 2-642 2.087 1 320 5 4"968 4-892 4-725 4-518 4-232 3-832 3'303 2-609 1'651 6 5-961 5'870 5-670 5-421 5'079 4-598 3'963 3.130 1-981 7 6-955 6'848 6'615 6'325 5.925 5-364 4-624 3-652 2 311 8 7 948 7'827 7'560 7-228 6-772 6-131 5-284 4-174 2-641 9 8'942 8'805 8'505 8'132 7'618 6-897 5-945 4-696 2-971 10 9-936 9'784 9'450 9'036 8-465 7-664 6.606 5.218 3'302 11 10'929 10'762 10'395 9'939 9'311 8'430 7'266 5'739 3 632 12 11-923 11-740 11-340 10-843 10-158 9-196 7'927 6261 3'962 13 12,856 12'719 12-285 11-746 10.994 9 963 8'587 6'783 4'292 14 13'910 13'967 13'230 12.650 11-851 10-729 9 248 7.305 0 4.622 15 14,904 14'676 14'175 13.554 12.697 11 496 9'909 7'827/ 4' 953 16 15'897 15'654 15'120 14.457 13.544 12 2G2 10 569 8.348 5 283 17 16'891 16'632 16'065 15.361 14.051 13-028 11.230 8870 5 613 18 17,884 17.611 17.010 16.264 15.237 13 795 11.890 9.392 5 944 19 18,8718 18.589 17-955 17-168 16-083 14 561 12.551 9.914 i6 6273 20 19.872 19.568 18.900 18.072 16.930 15 328 13.212 10.436 6.600 25 24.840 24-460 23.625 22-590 21.162 19-160 16-515 13-040 8-255 30 29.808 29-352 28-350 27-108 25-395 22 992 19-818 15-654 9.906 35 34-776 34-244 33.075 31-626 29-627 26 824 23-121 18 263 11.557 40 39.744 391386 37.800 36.144 33 860 30 656 26.224 20.872 138208 45 44-912 44-028 42-525 40.662 38.092 34 888 29.727 23 481 14-859 50 49-680 48.920 47-250 45-180 42-325 38 320 33.030 26-090 16.510] 50 49680 47250 Page 240 240 THE PRACTICAL MODEL CALCULATOR. respect in the proportion of their viscidity; but if the viscidity of the unguent be greater than what suffices to keep the surfaces asunder, an additional resistance will be occasioned; and the nature of the unguent selected should always have reference, therefore, to the size of the rubbing surfaces, or to the pressure per square inch upon them. With oil, the friction appears to be a minimum when the pressure on the surface of a bearing is about 90 lbs. per square inch: the friction from too small a surface increases twice as rapidly as the friction from too large a surface; added to which, the bearing, when the surface is too small, wears rapidly away. For all sorts of machinery, the oil of Patrick Sarsfield Devlan, of Reading, Pa., is the best. HORSE POWER. A horse power is an amount of mechanical force capable of raising 33,000 lbs. one foot high in a minute. The average force exerted by the strongest horses, amounting to 33,000 lbs., raised one foot high in the minute, was adopted, and has since been retained. The efficacy of engines of a given size, however, has been so much increased, that the dimensions answerable to a horse power then, will raise much more than 33,000 lbs. one foot high in the minute now; so that an actual horse power, and a nominal horse power are no longer convertible terms. In some engines every nominal horse power will raise 52,000 lbs. one foot high in the minute, in others 60,000 lbs., and in others 66,000 lbs.; so that an actual and nominal horse power are no longer comparable quantities,-the one being a unit of dimension, and the other a unit of force. The actual horse power of an engine is ascertained by an instrument called an indicator; but the nominal power is ascertained by a reference to the dimensions of the cylinder, and may be computed by the following rule:-Multiply the square of the diameter of the cylinder in inches by the velocity of the piston in feet per minute, and divide the product by 6,000; the quotient is the number of nominal horses power. In using this rule, however, it is necessary to adopt the speed of piston which varies with the length of the stroke. The speed of piston with a two feet stroke is, according to this system, 160 per minute; with a 2 ft. 6 in. stroke, 170; 3 ft., 180; 3 ft., 6 in., 189; 4 ft., 200; 5 ft., 215; 6 ft., 228; 7 ft., 245; 8 ft., 256 ft. By ascertaining the ratio in which the velocity of the piston increases with the length of the stroke, the element of velocity may be cast out altogether; and this for most purposes is the most convenient method of procedure. To ascertain the nominal power by this method, multiply the square of the diameter of the cylinder in inches by the cube root of the stroke in feet, and divide the product by 47; the quotient is the number of nominal horses power of the engine. This rule supposes a uniform effective pressure upon the piston of 7 lbs. per square inch; the effective pressure upon the piston of 4 horse power engines of some of the best makers has been estimated at 6'8 lbs. per square inch, and the pressure Page 241 THE STEAM ENGINE. 241 increased slightly with the power, and became 6'94 lbs. per square inch in engines of 100 horse power; but it appears to be more convenient to take a uniform pressure of 7 lbs. for all powers. Small engines, indeed, are somewhat less effective in proportion than large ones; but the difference can be made up by slightly increasing the pressure in the boiler; and small boilers will bear such an increase without inconvenience. Nominal power, it is clear, cannot be transformed into actual power, for the nominal horse power expresses the size of an engine, and the actual horse power the number of times 33,000 lbs. it will lift one foot high in a minute. To find the number of times 33,000 lbs. or 528 cubic feet of water, an engine will raise one foot high in a minute,-or, in other words, the actual power,-we first find the pressure in thie cylinder by means of the indicator, from which we deduct a pound and a half of pressure for friction, the loss of power in working the air pump, &c.; multiply the area of the piston in square inches by this residual pressure, and by the motion of the piston, in feet per minute, and divide by 33,000; the quotient is the actual number of horse power. The same result is attained by squaring the diameter of the cylinder, multiplying by the pressure per square inch, as shown by the indicator, less a pound and a half, and by the motion of the piston in feet, and dividing by 42,017. The quantity thus arrived at, will, in the case of nearly all modern engines, be very different from.that obtained by multiplying the square of the diameter of the cylinder by the cube root of the stroke, and dividing by 47, which expresses the nominal power; and the actual and nominal power must by no means be confounded, as they are totally different things. The duty of an engine is the work done in relation to the fuel consumed, and in ordinary mill or marine engines it can only be ascertained by the indicator, as the load upon such engines is variable, and cannot readily be determined: but in the case of engines for pumping water, where the load is constant, the number of strokes performed by the engine represents the.duty; and a mechanism to register the number of strokes made by the engine in a given time, is a sufficient test of the engine's performance. In high pressure engines the actual power is readily ascertained by the indicator, by the same process by which the actual power of low pressure engines is ascertained. The friction of a locomotive engine when unloaded, is found by experiment to be about 1 lb. per square inch on the surface of the pistons, and the additional friction caused by any additional resistance is estimated at about'14 of that resistance; but it will be a sufficiently near approximation to the power consumed by friction in high pressure engines, if we make a deduction of a pound and a half from the pressure on that account, as in the case of low pressure engines. High pressure engines, it is true, have no air pump to work; but the deduction of a pound and a half of pressure is relatively a much smaller one where the pressure is high than where it does not much exceed the V 16 Page 242 242 THE PRACTICAL MODEL CALCULATOR. pressure of the atmosphere. The rule, therefore, for the actual horse power of a high pressure engine will stand thus:-Square the diameter of the cylinder in inches, multiply by the pressure of the steam in the cylinder per square inch, less 1 lbs., and by the speed of the piston in feet per minute, and divide by 42,017; the quotient is the actual horse power. The nominal horse power of a high pressure engine has never been defined; but it should obviously hold the same relation to the actual power as that which obtains in the case of condensing. engines, so that an engine of a given nominal horse power may be capable of performing the same work, whether high pressure or condensing. This relation is maintained in the following rule, which expresses the nominal horse power of high pressure engines: —Multiply the square of the diameter of the cylinder in inches by the pressure on the piston in pounds per square inch, and by the speed of the piston in feet per minute, and divide the product by 120,000; the quotient is the power of the engine in nominal horses power. If the pressure upon the piston be 80 lbs. per square inch, the operation may be abbreviated by multiplying the square of the diameter of the cylinder by the speed of the piston, and dividing by 1,500, which will give the same result. This rule for nominal horse power, however, is not representative of the dimensions of the cylinder; but a rule for the nominal horse power of high pressure engines which shall discard altogether the element of velocity, is easily constructed; and, as different pressures are used in different engines, the pressure must become an element in the computation. The rule for the nominal power will therefore stand thus:-Multiply the square of the diameter of the cylinder in inches by the pressure on the piston in poumds per square inch, and the cube root of the stroke in feet, and divide the product by 940; the quotient is the power of the engine in nominal horse power, the engine working at the ordinary speed of 128 times the cube root of the stroke. A summary of the results arrived at by these rules is given in the following tables, which, for the convenience of reference, we introduce. PARALLEL MOTION. RULE I —In such a combination of two levers as is represented inz Figs. 1 and 2, page 245, to find the length of radius bar required for any given length of lever C G, and proportion of parts of thle linkk, G E and _P; E, so as to make the point E move in a perpendicular line.-Multiply the length of G C by the length of the segment G E, and divide the product by the length of the segment FE. The quotient is the length of the radius bar. RULE II.-(PFig. 2, page 245.) The length of the radius bar and of C G being given, to find the length of the segment (PE) of the link next the radius bar.-Multiply the length of C G by the Page 243 THE STEAM ENGINE. 243 TABLE of Nominal Horse Power of Low Pressure Engines. LENGTH OF STROKE IN FEET. 1 1. 22 3 33 4 41 5 5. 6 7 4 *34.39 43 *46 49 *52 *54 *56 *58 60 *62 *65 5 *53 *61 *67.72 *76 *81.84 -88 *91 94.96 1-02 6.76 *87.96 1-04 1 10 1-16 1-22 1-26 1-31 1-35 1-39 1-47 7 1-04 1-19 1-31 1-41 1 50 1-58 1-65 1-72 1 78 1-84 1-89 1-99 8 1-36 1-56 1-72 1-85 1-96 2-07 2-16 2'25 2-33 2'40 2-47 2-60 9 1-72 1-97 2-17 2-34 2-49 2-62 2-74 2-84 2.95 3 04 3-13 3.30 10 2-13 2-44 2.68 2-89 3.07 3-23 3-38 3.51 3-64 3-76 3-87 4.07 11 2-57 2-95 3-24 3.49 3-77 3-91 4-15 4-25 4.40 4-54 4/68 4 92 12 3'06 3-51 3-86 4-16 4-42 4-65 4-86 5-06 5-24 5-41 5-57 5-86 13 3-60 4-12 4-53 4-88 5-19 5-46 5-64 5.94 6-15 6-35 6.53 6.88 14 4-17 4-77 5-25 5-66 6-01 6-33 6-62 6-88 7-13 7-36 7-58 7'98 15 4-77 5-48 603 6-50 6'90 7-27 7-60 7'90 8-19 8-45 8-70 9-16 16 5-45 6-23 6-86 7'39 7-86 8-27 8-65 8-99 9-31 9-61 9'90 10'42 17 6'15 7'04 7-75 8-35 8-86 9.34 9-76 10-15 10-52 10-85 11-17 11'76 18 6-89 7-89 8-68 9-36 9'94 10-47 10-94 11-38 11-79 12-17 12-53 13'19 19 7-68 8-79 9-68 10-42 11-17 11-66 12-19 12-68 13-13 13-56 13-96 14-69 20 8-51 9 74 10-72 11-55 12-27 12-92 13 51 14-05 14-55 15-02 15-46 16-28 22 10-30 11-79 12-97 13-98 14-85 15-63 16-62 17-30 17-65 18-18 18 71 19 70 24 12.26 14 03 15-44 16'63 17-67 18-61 19'45 20-23 20'95 31-63 22-27 23'44 26 14'39 16-46 18'12 19'52 20-75 21'84 22-56 23-75 24'6 25-39 26-14 27.51 28 16-G8 19 09 21-02 22-64 24-06 25 33 26-48 27-54 28-52 29-44 30-31 31-90 30 19-15 21-92 24-13 25-99 27.62 29-07 30'40 31-61 32-74 33-80 34-80 36-63 32 21-79 24-96 27-51 29-57 31-42 33-08 34 59 35.97 37-26 38.46 39'59 41'68 34 24-60 28'16 30'99 33 39 35-44 37-34 39'04 40-60 42'06 43-41 44'69 47'05 36 27'57 31'56 34-74 37-42 39'77 41-87 43'77 45-52 47-15 48-67 50-11 52-75 38 30-72 35-17 38871 41-69 44-66 46-64 48-77 50-72 52-54 54-23 55-83 58.78 40 34,04 38-97 42-89 46-20 49.10 51-69 54'04 56-20 58-21 60-09 61'86 65-12 42 37-53 42-96 47-29 50'94 54-13 56-98 59-58 61-96 64-18 66'25 68'21 71-78 44 41-19 47-15 51-90 55-91 59-38 62-54 66-46 68-00 70'44 72-71 74-85 78-79 46 4502 51-54 56-72 61-10 64-88 68'19 71'43 74-33 76-69 79'47 81-81 86-12 48 49-02 56-11 61-76 6653 7070 74-42 77-82 80-94 83-83 86-53 89-08 93-78 50 53-19 60-89 67-02 72-19 76-71 80-76 84'44 87.82 90.96 93.89 96-65 101-7 52 57-55 65-86 72-48 78-08 83-00 87'35 90'25 94-98 98-40 101-55 104'5 110.0 54 62-04 7102 78-17 84-20 89-48 94-20 98-49 102.4 106-1 109-5 112-7 118-7 56 66-72 76-38 84.07 90-55 96-23 101-30 105.9 110.1 114-1 117-8 121-2 127-6 58 71-58 81-93 90-18 97-14 103-2 108-6 113 6 118-2 122-4 126-3 129-2 136-7 60 76-60 87-68 96.50 103.9 110.4 116-3 121'6 126-4 131-0 135-2 139-2 146-5 62 81-79 93.62 103-04 111'0 117-96 124-18 129'81 135-03 139-86 144-37 148-6 156-7 64 87-15 99.84 110'0 118-3 125-7 132-3 138-3 143-9 149-0 153-82 158-4 166-7 66 92-68 106-1 116-8 125'8 133.6 140-7 147'3 153-0 158-5 163-6 168-4 177-3 68 98-40 112-6 123-9 133-6 141-8 149-4 156-2 162.4 168-2 173-6 178-8 188-2 70 104-26 119-3 131-3 141-5 150-4 158-3 165'5 172-1 178-2 184-0 189-4 199-4 72 110-30 126-2 139-0 149.7 159-1 167-4 175'1 182-1 188-6 194.7 200'4 211-0 74 116-5 133.4 146.8 158-1 167.9 176-7 185-4 192-4 199-2 205-7 211-6 223-4 76 122'9 140-7 154-8 166-8 178-6 186-6 195-0 202-9 210-1 216-9 223-3 235-1 78 129-4 148-2 163-1 175-6 186-7 196-5 205-4 212-1 221-4 228-5 235-2 247-6 80 136-2 155-8 171'6 184-8 196-4 206-7 216-1 224-8 232-8 240-4 247-4 260-5 82 143-0 163-8 180.2 194-2 206-2 217-3 226-9 237-8 244-6 252-5 260-0 273.8 84 150-1 171-8 189-1 203-8 216-5 227-9 238-3 247-8 256-7 265-0 272-8 287-1 86 157-4 1801 198-2 213-6 227-0 237-8 247-4 258-2 269-1 277-8 286-0 301-0 88 164-8 188-6 207-6 223-6 237-5 250-2 261-6 272-0 281-7 290'8 299-4 315-2 90 172-3 197-3 217-1 233-9 248-6 261-7 273-6 284-5 291-7 3042 313-2 329-7 length of the link G F, and divide the product by the sum of the lengths of the radius bar and of C G. The quotient is the length required. RULE III. —(Figs. 3 and4,paes24 and 247.) Tofind the length of the radius bar (PTH), the length of C G being given.-Square the length of C G, and divide it by the length of D G. The quotient is the length required. RULE IV. —(Fgis. 3 and 4, pages 246 and 247.) Tofind the length of the radius bar, the horizontal distance of its centre (H) from the gmain centre being given.-To this given horizontal distance, add half the versed sine (D N) of the arc described by the end of beam (D). Square this sum. Take the same sum, and add to it the length of Page 244 244 THE PRACTICAL MODEL CALCULATOR. TABLE of Nominal Horse Power of High Pressure Engines..'~ LENQGTH OF STROKE IN FEET.'" 1 11 2 21 3 3 4 4j 5 5 6 7 2 *25 *29 *32 *35 -37 *38'40 -42'44' 45 *46'49 21' 39 -45' 50'54'57.60 *63 *66 -68 *70 *72' 76 3' 57 *65'72'78 *83 *87 91 95 98 1'01 1'04 1'10 31'78 *89' 98 1-06 1-13 1'19 1-24 1-29 1-34 1-38 1-42 1-49 4 1-02 1-17 1-29 1-38 1-47 1-56 1-62 1-68 1-74 1-80 1'86 1'95 4 1-29 1-48 1-63 1-75 1-86 1-96 2-05 2-13 2-21 2-28 2-35 2-47 5 1-59 1-83 2-01 2-16 2-28 2-43 2-52 2-64 2-73 2-82 2-88 3-06 5 1-93 2-21 2-43 2-62 2-78 2-93 3'12 34.8 3-50 3-42 3-51 3-69 6 2-28 2-61 2'88 3-12 3'30 3-48 3-66 3-78 3;93 405 4-17 4-41 6} 2-69 3'09 3-39 3-66 3'90 4-08 4-23 4-44 4-6% 4-77 4-89 5-16 7 3-12 3-57 3'93 4-23 4'50 4-74 4'95 5-16 5-34 5-52 5-67 5-97 76 3'60 4-11 4-53 4-86 5-19 5-46 5'70 5'94 6-15 6-33 6-51 6-87 8 4-08 468 5-16 5-55 5-88 6-21 6-48 6-75 6-99 7-20 7-41 7'80 8 4-62 5-28 5'82 6-27 6-63 6-99 7-32 7-62 7'89 8-13 8-37 882 9 5-16 5-91 6-51 7-02 7'47 7-86 8-22 8-52 8-85 9-12 9'39 9'90 9 5-76 6-60 7-26 7-80 8-37 8-76 9-15 9-51 9-84 10-17 10-47 10-01 10 6-39 7-32 8-04 8-67 9-21 9'69 10-14 10'53 10-92 11-28 11-61 12-21 10, 7'05 8-04 8-88 9'54 10-14 10-68 11-16 11-61 12-03 12-42 12-78 1347 11 7-71 8-85 9-72 10-47 11-31 11-73 12-45 12'75 13-20 13-62 14-04 14-76 11l 8-43 9'66 10-62 11-46 12-15 12-78 13-80 13-92 14-61 14-91 15-33 16-14 12 9-18 10-53 11-58 12-41 13-26 13-95 14-58 15-18 15-72 16-23 16'71 17' 58 12' 9-96 11-40 12-57 13-53 14-37 15-15 15-84 16-47 17'04 17-58 18-12 19'08 13 10-80 12-36 13-59 14-64 15-57 16-38 16-92 17-82 18-45 19-05 19'59 21'64 13t- 11-64 13-32 14'64 15-78 16-77 17-67 18-48 19-20 19-89 20-52 21-15 22'26 14 12-51 14-31 15-75 16-98 18-03 18-99 19-86 20-64 21-39 22-08 22-74 23'94 14L 13-41 15-36 16-92 18-21 19-35 20-37 21-30 22-14 22'95 23-70 24-39 25-62 15 14-31 16-44 18-09 19'50 20-70 21-81 22-80 23-70 24-57 25-35 26-10 27-48 16 16-35 18-69 20-58 22-17 23-58 24-81 25-95 26'97 27-93 28-83 29-70 31-26 17 18-45 21'12 23'25 25'05 26'58 28'02 29'28 30'45 31'56 32'55 33'57 35'28 18 20-67 23-67 26-04 28-08 29-82 31-41 32-82 34-14 35-37 36-51 37'59 39'57 19 23-04 26-37 29-04 31-26 33-51 34-98 36'57 38-04 39'39 40-68 41-88 44'07 20 25-53 29-22 32-16 34-65 36-81 38-76 40'53 42-15 43-65 45-06 46-38 48-84 22 30'90 35-37 38-91 41'94 44-55 46-89 49-86 51-90 52-95 54-54 56-13 59-10 24 36-78 42-09 46-32 49-89 53-01 55-83 58-35 60-69 62'85 64-89 66-81 70'32 26 43-17 49-38 54-36 58-56 62-25 65-52 67-68 71-25 73-80 76-17 78'42 82'53 28 50'04 57-27 63-06 67'92 72'18 75'99 79'44 82'62 85'56 88'32 90-93 95'70 30 57'45 65-76 72-39 77'97 82-86 87'21 91-20 94-83 98-22 101'40 104-4 109-9 32 65-37 74-88 82-53 88'71 94-26 99-24 103-7 107-9 111-8 115-4 118-7 125-0 34 73-80 84-48 92-9 100-22 106-3 112-0 117-1 121-8 126-2 130-2 134-0 141-1 36 82-71 94-68 104-2 112-2 119-3 125-6 131-3 136-5 141-4 146-0 150-3 158-2 38 92-16 105-5 116-1 125-0 134-0 136-9 146-3 152-1 157'6 162-7 167'5 17.6-3 40 102-1 116-9 129-6 128-6 147-3 155-1 162-1 168-6 174-6 180-2 185-6 195-3 42 112-6 128-9 141-8 152-8 162-4 170-9 178-7 185-9 192-5 198-7 204-6 215-3 44 123-5 141-4 155-7 167-7 178-1 187-6 199-4 204-0 211-3 218-1 224-5 236-3 46 135-0 154-6 170-1 183-3 194-6 204-6 214'3 223-0 230-0 238-4 245-4 258-3 48 147-0 168-3 185'3 199-6 212-1 223-2 233-4 242-8 251-5 259-6 267-2 281-3 50 159-6 182-6 201-0 216-5 230-1 242-3 253-3 263-4 272-9 281-6 289-9 305-1 52 172-6 197-6 217-4 234-2 249-0 262'0 270-7 284'9 295-2 304-6 313-5 330'0 54 186-1 213-0 234-5 252-6 268-4 282'6 295-4 307'2 318-3 328-5 338-1 356-1 56 200-1 229-1 252-2 271-6 288-7 30329 317-7 330.3 342'3 353-4 363'6 382-8 58 214-7 245-8 270-5 291-4 309-6 325-8 340'8 354-6 367-2 378-9 389-7 410-1 60 229-8 263-0 289-5 311-7 331-2 348'9 364-18 379'2 393'0 405' 6 417'6 439-5 the beam (C D). Divide the square previously found by this last sum, and the quotient is the length sought. RULE V.-(Figs. 5 and 6, pages 247, 248.) —To find the le#ngth of the radius bar, C G and P Q being given.-Square C G, and multiply the square by the length of the side rod (P D): call this product A. Multiply Q D by the length of the side lever (C D). From this product subtract the product of D P into C G, and cdivide A by the remainder. The quotient is the length required. RULE VI. —(Fqs. 5 and 6, pages 247, 248.) To~find the length, of the radius bar; P Q, and the horizontal distance of the centre Hof the radius bar from the main centre being given.-To the given horizontal distance add half the versed sine (D N) of the arc described Page 245 THE STEAM ENGINE. 245 Fig. 1. S /c: 2, IIj'i~~~'F 1 a by the extremity (D) of the side lever. Square this sum and multiply the square by the length of the side rod (P D). Call this product A. Take the same horizontal distance as before added to the same half versed sine (D N), and multiply the sum by the length of the side rod (P D): to thle product add the product of the length of S i) // —--— \-,,,,,, __ l?-. [';\.. —. //F/1~C by heexreit () o te ~eleer Suar tissu ad ul tipy te su~r bythelenth ~ te sde od P D. C1! hispro dutE A.Tk h ehriotldsac s eoeaddt h saehaf~rsdsie( N), nmutpytesmthlngho the s~~~~~~ ~~~~~~iderd( ) otepoutadtepouto h egh v2 I Page 246 246 THE PRACTICAL MODEL CALCULATOR. Fig. 3. IL M I \\~~~~~~~~~~~~~ II~~~I length of C G, the part of the beam that works it. / II 1\ I \, I i II When the centre iI of the radius has its position determined, Page 247 THE STEA~r ~,~aIr. 247 Fig. 4. 1)~~~~~~ ~~~~~~~~~~~ I ~ 1 / -- /' J % iI /',/!', i!," / / / / /, / /',/1/. I /! m -I/ / E/ ~ I I~ \ I // ~' " /I /,-., / -' I', --,,-,: —~-~ Y~!L/ 1; /.. - " P'I; ",",.,',%'III/,:;: ".\ \ / t t', %!-. tI I \', %~\ I i t,'1\ V / tll ~._. r ~ I'.,-., t h r u l we wil a (~ + ) 0; S9 16' inhs 51 + 3 2- 8 8 whc isterqie lnt fteraisbr( ) Page 248 248 TIIE PRACTICAL MODEL CALCULATOR. Fig. 6. y'l I' i l;// 2. RULE 5.-The following dimensions are those of the Red Rov, steamer: C G - 32 D P -94 QD = 74 C D 65 P Q= 20. By the rule we have, A = (32)2 x 94 = 96256 and 96256 96256 74 x 65 - 94 x 32- 1802 =53'' wvhich is the required length of the radius bar. 3. RULE 6.-Take the same data as in the last example, on supposing that C G is not given, and that the centre H is fixed a horizontal distance from the main centre, equal to 83'5 inch( Then the half versed sine of the are D' ID D" will be about inches, and we will have by the rule A = (83'5 + 2)2 x 94 = 705963'5 and A 705963-5 85'5 x 94 + 65 x 74 1284'7 =548 inches, the required length of the radius bar in this case. TABLE (A). F I{ This column gives C when Correction to be added to or CC G subtracted from the calcuC G is the greater andC G lated length of the radius F F H bar, in decimal parts of its when F H is the greater. calculated length. 1.0 0.9 -0034 -8 -0075.7 -0163 ~6 -0270.5.0452.4 -0817 Page 249 THE STEAM ENGINE. 249 C G In both of the last two examples H-F 6 nearly. The correction found by Table (A), therefore, would be 54 x'027 = 1'458 inches, which must be subtracted from the lengths already found for the radius bar, because it is longer than C G. The corrected lengths will therefore be In example 2.....................F H = 51'94 inches. In example 3................... F H = 53'34 inches. RULE.- To find the depth of the main beam at the centre. —Divide the length in inches from the centre of motion to the point where the piston rod is attached, by the diameter of the cylinder in inches; multiply the quotient by the maximum pressure in pounds per square inch of the steam in the boiler; divide the product by 202 for cast iron, and 236 for malleable iron: in either case, the cube root of the quotient multiplied by the diameter of the cylinder in inches gives the depth in inches of the beam at the centre of motion. To find the breadth at the centre.-Divide the depth in inches by 16; the quotient is the breadth in inches. An engine beam is three times the diameter of the cylinder, from the centre to the point where the piston rod acts on it; the force of the steam in the boiler when about to force open the safety valve is 10 lbs. per square inch. Required the depth and breadth when the beam is of cast iron. In this case n - 3, and P = 10, and therefore d-d = 30 -' 53 D. The breadth = 5 D = 03 D. 16 It will be observed that our rule gives the least value to the depth. In actual practice, however, it is necessary to make allowance for accidents, or for faultiness in the materials. This may be done by making the depth greater than that determined by the rule; or, perhaps more properly, by taking the pressure of the steam much greater than it can ever possibly be. As for the dimensions of the other parts of the beam, it is obvious that they ought to diminish towards the extremities; for the power of a beam to resist a cross strain varies inversely as its length. The dimensions may be determined from the formulafb d2 = 6 W 1. To apply the formula to cranks, we may assume the depth at the shaft to be equal to n times the diam6ter of the shaft; hence, if m x D be the diameter of the shaft, the depth of the crank will be n x m x D. Substituting this in the formula fb d2 = 6 W, and it becomes fb x n2 x mn2 x D2 = 6 W 1. Now, as before, WV ='7854 x P x ID2, so that the formula becomes f x b x n2 x in2- 4'71124 x P x 1. The value of n is arbitrary. In practice it may be made equal to 1- or 1'5. Taking this value, then, for Page 250 250 THE PRACTICAL MODEL CALCULATOR. cast iron, the formula becomes 15300 x b x 9 x m2 = 4'7124 x P x 1, or 7305 m2 b = P 1; but if L denote the length of the crank in feet, the formula becomes 609 m2 b = PL, and.. b = P x L - 609 in2. This formula may be put into the form of a rule, thus:RULE. — To find the breadth at the shaft when the depth is equal to 1~- times the diameter of the shaft.-Divide the square of the diameter of the shaft in inches by the square of the diameter of the cylinder; multiply the quotient by 609, and reserve the product for a divisor; multiply the greatest elastic force of the steam in lbs. per square inch by the length of the crank in feet, and divide the product by the reserved divisor: the quotient is the breadth of the crank at the shaft. A crank shaft is 4 the diameter of the cylinder; the greatest possible force of the steam in the boiler is 20 lbs. per square inch; and the length of the shaft is 3 feet. Required the breadth of the crank at the shaft when its depth is equal to 19 times the diameter of the shaft. 609 In this case n = 4, so that the reserved divisor - - = 38: again, elastic force of steam in lbs. per square inch = 20 lbs.; 3 x 20 hence width of crank = 38 = 16 inches nearly. RULE.-TO find the diameter of a revolving shaft.-Form a reserved divisor thus: multiply the number of revolutions which the shaft makes for each double stroke of the piston by the number 1222 for cast iron, and the number 1376 for malleable iron. Then divide the radius of the crank, or the radius of the wheel, by the diameter of the cylinder; multiply the quotient by the greatest pressure of the steam in the boiler expressed in lbs. per square inch, divide the product by the reserved divisor; extract the cube root of the quotient, and multiply the result by the diameter of the cylinder in inches. The product is the diameter of the shaft in inches. STRENGTH OF RODS WHEN THE STRAIN IS WHOLLY TENSILE; SUCH AS TIE PISTON ROD OF SINGLE ACTING ENGINES, PUMP RODS, ETC. RULE.-To find the diameter of a rod exposed to a tensile force only.-Multiply the diameter of the piston in inches by the square root of the greatest elastic force of the steam in the boiler estimated in lbs. per square inch; the product, divided by 95, is the diameter of the rod in inches. Required the diameter of the transverse section of a piston rod in a single acting engine, when the diameter of the cylinder is 50 inches, and the greatest possible force of the steam in the boiler is 16 lbs. per square inch. Here, according to the formula, 50 200 d = 16 = 9= 2-1 inches. 95 ~95 Page 251 THE STEAM ENGINE. 251 RULE.-To find the strength of rods alternately extended and compressed, such as the piston rods of double acting engines. —Multiply the diameter of the piston in inches by the square root of the maximum pressure of the steam in lbs. per square inch; divide the product by 47 for cast iron, 50 for malleable iron. This rule applies to the piston rods of double acting engines, parallel motion rods, air-pump and force-pump rods, and the like. The rule may also be applied to determine the strength of connecting rods, by taking, instead of P, a number P', such that P' x sine of the greatest angle which the connecting rod makes with the direction = P. Supposing the greatest force of the steam in the boiler to be 16 lbs. per square inch, and the diameter of the cylinder 50 inches; required the diameter of the piston rod, supposing the engine to be double acting. In this case for cast iron d = 41 P 50 4 = 5 inches nearly; t47 47 for malleable iron d = / /P = 4 inches. The pressure, however, is always taken in practice at more than 16 lbs. If the pressure be taken at 25 lbs., the diameter of a malleable iron piston rod will be 5 inches, which is the usual proportion. Piston rods are never made of cast iron, but air-pump rods are sometimes made of brass, and the connecting rods of land engines are cast iron in most cases. FORMULAS FOR THE STRENGTH OF VARIOUS PARTS OF MARINE ENGINES. The following general rules give the dimensions proper for the parts of marine engines, and we shall recapitulate, with all possible brevity, the data upon which the denominations rest. Let pressure of the steam in boiler =p lbs. per square inch, Diameter of cylinder =D inches, Length of stroke = 2 R inches.. The vacuum below the piston is never complete, so that there always remains a vapour of steam possessing a certain elasticity. We may suppose this vapour to be able to balance the weight of the piston. Hence the entire pressure on the square inch of piston in lbs. = p + pressure of atmosphere - 15 + p. We shall substitute P for 15 + p. Hence Entire pressure on piston in lbs. -7854 x (15 + p) x D2 -'7854 x P x D2. The dimensions of the paddle-shaft journal may be found from the following formulas, which are calculated so that the strain in ordinary working = 5 elastic force. Diameter of paddle-shaft journal = -08264 {R x P x D21 3 Length of ditto = 14 x diameter. Page 252 252 THE PRACTICAL MODEL CALCULATOR. The dimensions of the several parts of the crank may be found from the following formulas, which are calculated so that the strain in ordinary working = one-half the elastic force; and when one paddle is suddenly brought up, the strain at shaft end of crank = 2 elastic force, the strain at pin end of crank = elastic force. Exterior diameter of large eye = diameter of paddle-shaft + 1D[P x 1'561 x R2 +'00494 x D2 x p2]r } 75'59 x V' 1 Length of ditto = diameter of paddle shaft. Exterior diameter of small eye = diameter of crank pin + 02521 x /P x D. Length of ditto ='0375 x V/ P x D. Thickness of web at paddle centre-. D2 X p x {1.561 x R2 +.00494 x D2 X P} i 9000 Breadth of ditto = 2 x thickness. Thickness of web at pin centre -'022 x - P x D. Breadth of ditto = - x thickness. As these formulas are rather complicated, we may show what they become when p = 10 or P = 25. Exterior diameter of large eye = diameter of paddle shaft + ( D V (1'561 x R2 + -1235 x D2) S 3 15.12 x V R Length of ditto = diameter of paddle shaft. Exterior diameter of small eye = equal diameter of crank pin + 126 x D. Length of ditto ='1875 x D. Thickness of web at pin centre ='11 x D. Breadth of ditto = 3 x thickness of web. The dimensions of the crank pin journal may be found from the following formulas, which are calculated so that strain when bearing at outer end = elastic force, and in ordinary working strain = one-third of elastic force. Diameter of crank-pin journal = -02836 x V P x D. Length of ditto = -- x diameter. The dimensions of the several parts of the cross head may be found from the following formulas, in which we have assumed, for the purpose of calculation, the length = 1'4 x D. The formulas have been calculated so as to give the strain of web = x elastic force; strain of journal in ordinary working -I x elastic 2'33 force, and when bearing at outer end = 1 x elastic force. 1'165 Page 253 THE STEAM ENGINE. 253 Exterior diameter of eye = diameter of hole +'02827 x Pi x D. Depth of ditto ='0979 x P~ x ID. Diameter of journal ='01716 x v P x D. Length of ditto = - diameter of journal. Thickness of web at middle ='0245 x P3 x D. Breadth of ditto = -09178 x P.x D. Thickness of web at journal ='0122 x P2 x D. Breadth of ditto = -0203 x P2 x D. The dimensions of the several parts of the piston rod may be found from the following formulas, which are calculated so that the strain of piston rod = - elastic force. Diameter of the piston rod = VP x D. 50 Length of part in piston = -04 x D x P. iMajor diameter of part in crosshead = 019 x V/P x D. Minor diameter of ditto = 018 x 4P x D. Major diameter of part in piston = 028 x VP x D. Miinor diameter of ditto ='023 x V/P x D. Depth of gibs and cutter through crosshead = -0358 x P3 x D. Thickness of ditto ='007 x P3 x D. Depth of cutter through piston ='017 x /P x D. Thickness of ditto = -007 x P2 x D. The dimensions of the several parts of the connecting rod may be found from the following formulas, which are calculated so that the strain of the connecting rod and the strain of the strap are both equal to one-sixth of the elastic force. 1 Diameter of connecting rod at ends ='019 x P2 x D., Diameter of ditto at middle = {1 +'0035 x length in inches} x'019 x VP x D. Major diameter of part in crosstail = -0196 x P2 x D. Minor ditto = -018 x P1 x D. Breadth of butt ='0313 x P2 x D. Thickness of ditto =- 025 x P2 x D. Mean thickness of strap at cutter ='00854 x /P x D. Ditto above cutter = -00634 x VP x D. Distance of cutter from end of strap ='0097 x /P x D. Breadth of gibs and cutter through crosstail ='0358 x P3 x D. 1 Breadth of gibs and cutter through butt ='022 x P2 x D. 1 Thickness of ditto ='00564 x P2 x D. Page 254 254 THE PRACTICAL MODEL CALCULATOR. The dimensions of the several parts of the side rods may be found from the following formulas, which are calculated so as to make the strain of the side rod = one-sixth of elastic force, and the strains of strap and cutter = one-fifth of elastic force. 1 Diameter of cylinder side rods at ends ='0129 x P2 x D. Diameter of ditto at middle = (1 +'0035 x length in inches). x'0129 x P- x D. Breadth of butt ='0154 x P2 x D. Thickness of ditto ='0122 x P2 x D. Diameter of journal at top end of side rod ='01716 x P2 x D. Length of journal at top end = - diameter. Diameter of journal at bottom end -'014 x P2 x D. Length of ditto = -0152 x P2 x D. Mean thickness of strap at cutter ='00643 x P2 x D. Ditto below cutter' -'0047 x P2 x D. Breadth of gibs and cutter ='016 x P2 x D. Thickness of ditto = -0033 x P2 x D. The dimensions of the main centre journal may be found from the following formulas, which are calculated so as to make the strain in ordinary working = one half elastic force. Diameter of main centre journal = -0367 x p2 x D. Length of ditto = E x diameter. The dimensions of the several parts of the air-pump may be found from the corresponding formulas given above, by taking for D another number d the diameter of air-pump. DIMENSIONS OF THE SEVERAL PARTS OF FURNACES AND BOILERS. Perhaps in none of the parts of a steam engine does the practice of engineers vary more than in those connected with furnaces and boilers. There are, no doubt, certain proportions for these, as well as for the others, which produce the maximum amount of useful effect for particular given purposes; but the determination of these proportions, from theoretical considerations, has hitherto been attended with insuperable difficulties, arising principally from our imperfect knowledge of the laws of combustion of fuel, and of the laws according to which caloric is imparted to the water in the boiler. In giving, therefore, the following proportions for the different parts, we desire to have it understood that we do not affirm them to be the best, absolutely considered; we give them only as the average practice of the best modern constructors. In most of the cases we have given the average value per nominal horse power. It is well known that the term horse power is a conventional unit for measuring the size of steam engines, just as a foot or a mile is Page 255 THE STEAM ENGINE. 255 a unit for the measurement of extension. There is this difference, however, in the two cases, that whereas the length of a foot is fixed definitively, and is known to every one, the dimensions proper to an engine horse power differ in the practice of every different maker: and the same kind of confusion is thereby introduced into engineering as if one person were to make his foot-rule eleven inches long, and another thirteen inches. It signifies very little what a horse power is defined to be; but when once defined, the measurement should be kept inviolable. The question now arises, what standard ought to be the accepted one. For our present purpose, it is necessary to connect by a formula the three quantities, nominal horses power, length of stroke, and diameter of cylinder. With this intention, Let S = length of stroke in feet, d = diameter of cylinder in inches; d2 x /S Then the nominal horse power = 4 nearly. I. Area of Fire Grate.-The average practice is to give *55 square feet for each nominal horse power. Hence the following rule: RULE 1.-To find the area of the fire grate.-Multiply the number of horses power by'55; the product is the area of the fire grate in square feet. Required the total area of the fire grate for an engine of 400 horse power. Here total area of fire grate in square feet - 400 x ~55 = 220. A rule may also be found for expressing the area of the fire grate in terms of the length of stroke and the diameter of the cylinder. For this purpose we have,'55 xd'x -S d2 x ~S total area of fire grate 55 d X feet = 86 feet. 47 86 This formula expressed in words gives the following rule. RULE 2.- To find the area of fire grate.-Multiply the cube root of the length of stroke in feet by the square of the diameter in inches; divide the product by 86; the quotient is the area of fire grate in square feet. Required the total area of the fire grate for an engine whose stroke = 8 feet, and diameter of cylinder = 50 inches. Here, according to the rule, 502x Al8 2500 x 2 total area of fire grate in square feet = 86 = 5000 86 59 nearly. In order to work this example by the first rule, we find the nominal horse power of the engine whose dimensions we have specified is 104'3; hence, total area of fire grate in square feet = 106'4 x'55 = 58'5. Page 256 256 THE PRACTICAL MODEL CALCULATOR. With regard to these rules we may remark, not only that they are founded on practice, and therefore empyrical, but they are only applicable to large engines. When an engine is very small, it requires a much larger area of fire grate in proportion to its size than a larger one. This depends upon the necessity of having a certain amount of fire grate for the proper combustion of the coal. II. Length of Furnace.-The length of the furnace differs considerably, even in the practice of the same engineer. Indeed, all the dimensions of the furnace depend to a certain extent upon the peculiarity of its position. From the difficulty of firing long furnaces efficiently, it has been found more beneficial to restrict the length of the furnace to about six feet than to employ furnaces of greater length. III. Height of Furnace above Bars.-This dimension is variable, but it is a common practice to make the height about two feet. IV. Capacity of Furnace Chamber above Bars.-The average per horse power may be taken at 1'17 feet. Hence the following rule: RULE. —To find the capacity of furnace. chamber above bars.Multiply the number of nominal horses power by 1-17; the product is the capacity of furnace chambers above bars in cubic feet. V. Areas of Flues or Tubes in smallest part.-The average value of the area per horse power is 11'2 sq. in. Hence we have the following rule: RULE.-To find the total area of the flues or tubes in smallest part.-Multiply the number of horse power by 11'2; the product is the total area in square inches of flues or tubes in smallest part. Required total area of flues or tubes for the boiler of a steam engine when the horse power = 400. For this example we have, according to the rule, Total area in square inches = 400 x 11-2 = 4480. We may also find a very convenient rule expressed in terms of the stroke and the diameter of cylinder. Thus, 11'2 x d2 x _9S Total area of tubes or flues in square inches 47 d2 X /S 4 VI. Effective Heating Surface.-The effective heating surface of flue boilers is the whole of furnace surface above bars, the whole of tops of flues, half the sides of flues, and none of the bottoms; hence the effective flue surface is about half the total flue surface. In tubular boilers, however, the whole of the tube surface is reckoned effective surface. EFFECTIVE HEATING SURFACE OF FLUE BOILERS. RULE 1.-To find the effective heating surface of marine flue boilers of large size.-Multiply the number of nominal horse power by 5; the product is the area of effective heating surface in square feet. Page 257 THE STEAM ENGINE. 257 Required the effective heating surface of an engine of 400 nominal horse power. In this case, according to the rule, effective heating surface in square feet = 400 x 5 = 2000. The effective heating surface may be expressed in terms of the length of stroke and the diameter of the cylinder. RULE 2.-Tofind the total effective heating surface of marine flue boilers.-Multiply the square of the diameter of cylinder in inches by the cube root of the length of stroke in feet; divide the product by 10: the quotient expresses the number of square feet of effective heating surface. Required the amount of effective heating surface for an engine whose stroke = 8 ft., and diameter of cylinder = 50 inches. Here, according to Rule 2, effective heating surface in square feet 502 x _/8 2500 x 2 5000 =- -~ = -500. 10 = 10 10 To solve this example according to the first rule, we have the nominal horse power of the engine equal to 106'4. Hence, according to Rule 2, total effective heating surface in square feet = 106'4 x 4'92 = 5232. EFFECTIVE HEATING SURFACE OF TUBULAR BOILERS. The effective heating surface of tubular boilers is about equal to the total heating surface of flue boilers, or is double the effective surface; but then the total tube surface is reckoned effective surface. It appears that the total heating surface of flue and tubular marine boilers is about the same, namely, about 10 square feet per horse power. VII. Area of Chimney.-RULE 1.-To find the area of chimnney. -Multiply the number of nominal horse power by 10'23; the product is the area of chimney in square inches. Required the area of the chimney for an engine of 400 nominal horse power. In this example we have, according to the rule, area of chimney in square inches = 400 x 10'23 = 4092. We may also find a rule for connecting together the area of the chimney, the length of the stroke, and the diameter of the cylinder. RULE 2. —To find the area of the chimney.-Multiply the square of the diameter expressed in inches by the cube root of the stroke expressed in feet; divide the product by the number 5; the quotient expresses the number of square inches in the area of chimney. Required the area of the chimney for an engine whose stroke 8 feet, and diameter of cylinder = 50 inches. We have in this example from the rule, 502 x -Y8 2500 x 2 area of chimney in square inches = 5 -- 5 1000. w2 17 Page 258 258 THE PRACTICAL MODEL CALCULATOR. To work this example according to the first rule, we find, that the nominal horse power of this engine is 104'6: hence, area of chimney in square inches = 104'6 x 1023 = 101 0. The latter value is greater than the former one by 70 inches. This difference arises from our taking too great a divisor in Rule 2. Either of the values, however, is near enough for all practical purposes. VIII. Water in Boiler.-The quantity of water in the boiler differs not only for different boilers, but differs even for the same boiler at different times. It may be useful, however, to know the average quantity of water in the boiler for an engine of a given horse power. RULE 1. —To determine the average quantity of water in tCie boiler.-Multiply the number of horse power by 5; the product expresses the cubic feet of water usually in the boiler. This rule may be so modified as to make it depend upon the stroke and diameter of the cylinder of engine. RULE 2.-To determine the cubic feet of water usually in the boiler.-Multiply together the cube root of the stroke in feet, the square of the diameter of the cylinder in inches, and the number 5; divide the continual product by 47; the quotient expresses the cubic feet of water usually in the boiler. Required the usual quantity of water in the boilers of all engine whase stroke 8 feet, and diameter of cylinder 50 inches. Here we have from the rule, 5 x 502 x /8 5 x 2500 x 2 cubic feet of water in boiler = - 47 25000 532 nearly. - 47 532 nearly. The engine, with the dimensions we have specified, is of 106'4 nominal horse power. Hence, according to Rule 1, cubic feet of water in boiler = 106'4 x 5 = 532. IX. Area of Water Level.-RULE I.- To find the area of water level.-The area of water level contains the same number of square feet as there are units in the number expressing the nominal horse power of the engine. Required the area of water level for an engine of 200 nonminal horse power. According to the rule, the answer is 200 square feet. We add a rule for finding the area of water level when the diameter of cylinder and the length of stroke is given. RULE 2.-To find the area of water level. —Multiply the square of the diameter in inches by the cube root of the stroke in feet; divide the product by 47; the quotient expresses the number of square feet in the area of water level. Required the area of the water level for an engine whose stroke is 8 feet, and diameter of cylinder 50 inches. Page 259 THE STEAM ENGINE. 259 In this case, according to the rule, 502 x /8 area of water level in square feet = 47 = 106. X. Steam Room.-It is obvious that the steam room, like the quantity of water, is an extremely variable quantity, differing, not only for different boilers, but even in the same boiler at different times. It is desirable, however, to know the content of that part of the boiler usually filled with steam. RULE 1. —To determine the average quantity of steam room.Multiply the number expressing the nominal horse power by 3; the product expresses the average number of cubic feet of steam room. Required the average capacity of steam room for an engine of 460 nominal horse power. According to the rule, Average capacity of steam room = 460 x 3 cubic feet = 1380 cubic feet. This rule may be so modified as to apply when the length of stroke and diameter of cylinder are given. RULE 2.-Multiply the square of the diameter of the cylinder in inches by the cube root of the stroke in feet; divide the product by 15; the quotient expresses the number of cubic feet of steam room. Required the average capacity of steam room for an engine whose stroke is 8 feet, and diameter of cylinder 5 inches. In this case, according to the rule, 502 x /8 2500 x 2 5000 Steam room in cubic feet = -5 - 15 15 - 15 15 33313. We find that the nominal horse power of this engine is 106'4; hence, according to Rule 1, average steam room in cubic feet = 106'4 x 3 = 320 nearly. Before leaving these rules, we would again repeat that they ought not to be considered as rules founded upon considerations for giving the maximum effect from the combustion of a given amount of fuel; and consequently the engineer ought not to consider them as invariable, but merely to be followed as far as circumstances will permit. We give them, indeed, as the. medium value of the very variable practice of several well-known constructors; consequently, although the proportions given by the rules may not be the best possible for producing the most useful effect, still the engineer who is guided by them is sure not to be very far from the common practice of most of our best engineers. It has often been lamented that the methods used by different engine makers for estimating the nominal powers of their engines have been so various that we can form no real estimate of the dimensions of the engine, from its reputed nominal horse power, unless we know its maker; but the Page 260 260 THE PRACTICAL MODEL CALCULATOR. same confusion exists, also, to some extent, in the construction of boilers. Indeed, many things may be mentioned, which have hitherto operated as a barrier to the practical application of any standard of engine power for proportioning the different parts of the boiler and furnace. The magnitude of furnace and the extent of heating surface necessary to produce any required rate of evaporation in the boiler are indeed known, yet each engine-maker has his own rule in these matters, and which he seems to think preferable to all others, and there are various circumstances influencing the result which render facts incomparable unless those circumstances are the same. Thus the circumstances that govern the rate of evaporation, as influenced by different degrees of draught, may be regarded as but imperfectly known. And, supposing the difficulty of ascertaining this rate of evaporation were surmounted, there would still remain some difficulty in ascertaining the amount of power absorbed by the condensation of the steam on its passage to the cylinder-the imperfect condensation of the same steam after it has worked the piston-the friction of the various moving parts of the machinery-2and, especially, the difference of effect of these losses of power in engines constructed on different scales of magnitude. Practice must often vary, to a certain extent, in the construction of the different parts of the boiler and furnace of an engine; for, independently of the difficulty of solving the general problem in engineering, the determination of the maximum effect with the minimum of means, practice would still require to vary according as in any particular case the desired minimum of means was that of weight, or bulk, or expense of material. Again, in estimating the proper proportions for a boiler and its appendages, reference ought to be made to the distinction between the " power" or " effect" of the boiler, and its " duty." This is a distinction to be considered also in the engine itself. The power of an engine has reference to the time it takes to produce a certain mechanical effect without reference to the amount of fuel consumed; and, on the other hand, the duty of an engine has reference to the amount of mechanical effect produced by a certain consumption of fuel, and is independent of the time it takes to produce that effect. In expressing the duty of engines, it would have prevented much needless confusion if the duty of the boiler had been entirely separated from that of the engine, as, indeed, they are two very distinct things. The duty performed by ordinary land rotative steam engines isOne horse power exerted by 10 lbs. of fuel an hour; or, Quarter of a million of lbs. raised 1 foot high by 1 lb. of coal; or, Twenty millions of lbs. raised one foot by each bushel of coals. Though in the best class of rotative engines the consumption is not above half of this amount. The constant aim of different engine makers is to increase the amount of the duty; that is, to make 10 lbs. of fuel exert a greater effect than one horse power; or, in other words, to make 1 lb. of Page 261 THE STEAM ENGINE. 261 coal raise more than a quarter of a million of lbs. one foot high. To a great extent they have been successful in this. They have caused 5 lbs. of coal to exert the force of one horse power, and even in some cases as little as 38 lbs.; but in these latter cases the economy is due chiefly to expansive action. In some of the engines, however, working with a consumption of 10 lbs. of coal per nominal horse hower per hour, the power really exerted amounts to much more than that represented by 33,000 lbs. lifted one foot high in the minute for each horse power. Some engines lift 56,000 lbs. one foot high in the minute by eacch horse power, with a consumption of 10 lbs. of coal per horse power per hour; and even this performance has been somewhat exceeded without a recourse to expansive action. In all modern engines the actual performance much exceeds the nominal power; and reference must be lhad to this circumstance in contrasting the duty of dinerent engines. MVECHANICAL POWER1 OF STEAM1. WYe may here give a table of some of the properties of steam, and of its imechanical effects at different pressures. This table may help to solve many problems respecting the mechanical effect of steam, usually requiring much laborious calculation. M 1E-^III ANICAi. EFFECT IN TIORSE POWER OF 1 LB. PRESSURES. Tcme-'C iht OF STEA. n de- Cubic... of Nithout Cond.nsation. Condonsation. Fahresn. E ESit. Epawton. Expansion. Atmo- Lbs. per 1. sphere. Sq. Inch. 0 4 2 1 00 1-4 212 00 0-0364 0 0 32 4 95-2 1 70 5 913 110-1 178-6 194-6 1-25 18 38 223-88 0 0410 873 21.5! 10- 1 323 87-41 95-9 158-7 1906G 209'9 1'503 22059 234''3 0'0592J 11;;1 36;4) 3.' 3 1098 30'6 99-3 165-2 199-6 221-1 1-5l 2o'72j 242' 78 0 009 |12) 5 47'4 G0' 8 l 42'5 11-1 102-0 170-0 206-2 2229)5 I09 2d940 25 2. 9 G09 1 2 J56 47 4 601C 42-5 2-00 290 250;79 C'(J88 l117 6855 9 70 75 (;70 43-2 104-3 174-2 212-0 236'5 2'25 33'08 [257'[0 0-0766 1 911 62.8 90-9 86'5 68-8 106' 17 77" 216'7! 242.4 2-'0 36 75 263 93 0'0344 1556 GX68 4 1(018 102-4 89-6!107'7 180'5 220-5 247-1 2'75 4) 42 209s87 09G92 1 1608 3 7 111'0 il.-8 1071 1 109'3 183 2 224'2 251-6 -090 44'10 4 5' 0 0 009(98 1632 71'1 119&'8 121 1219 106 185'4 2277 255-2 3 3o 47 78 279'86 01073 1930 $0i'7 125'6 137'1 136'7 111'7 187'6 230'0 258.7'5o0 51-45 24-683 0-1148 1722 83 8 131-5 145-6 145-8 112'7 189-4 232-4 261-6 3 75 55 12 2S8866 0,1225 1750 86-5 136 8 l1532 1556 1137 1 190-1 234-7 264.4 4.00 58 18 2-12';1 i 0'238 1 1' 89-0 4 141959 1 160 1641 5 114-6 192-68 236-9 267.0 4-51 6(;-15 300 2; 01445 1816 93'2 149-8 i1715 179-4 116-2 195-6 240-5 271-4 5'00 73 0 307'941 0 15198190 0 968 156 5 1816 192'0 11'77 1983 2L14'1 275-6!60( 88'20 320'00 0-1878 190( 102-5 167-2 1396'5 211'4 1202 202 2026 2-197 282-2 700 102190 331-56; 021153 1945 107'0 175'6 208'4 226-5 12241 2034 254-6 2881 8-00 117 60 390 86 0. 21 36 1978 110-6 182'4 217'9 238'4 124-3 209 258-8 292.1 900 132-30 3.51 2 02:708 2006 113-7 1882 1225-9 248-5 1260 212 262-7 29.3 6 10-00 147'00 359':0 0-2977 2029 116-3 19390 232-5 256-7 127-5 25 1 266 0 301-4 12.50 183.75 377 42 093642 2074 121.5 202.5 245.5 273.0 130.7 220 272-9 3095 15-00 220-50 3929() 0:428| 2109 125.7 210.0 255-6 285-4 133-4 225 27859 316-4 17-50 257-25 406.40 0.4924 2136 129.0 216-0 2|636 295.2 135.7 22 9 283-9 322 20 29400c 418-56 0..5549 21 59 131-8 221-0 270.3 305.3 137-8 233 288-3 327.2 25 367-50 429-34 0.6775 2196 136.3 229-1 281.0 316.2 14132 238 295-7 335.8 30 411O00 4T5716 0.79;0 22:26 1400 235.6 289.5 326-4 1442 1244 302 0 343-1 It is quite clear that although there is no theoretical limit to the benefit derivable from expansion, there must be a limit in practice, arising from the friction incidental to the use of very large cylinders, the magnitude of the deduction due to uncondensed vapour when the steam is of a very low pressure, and other circumstances which it is needless to relate. It is clear, too, that while the effi Page 262 262 THE PRACTICAL MODEL CALCULATOR. ciency of the steam is increased by expansive action, the efficiency of the engine is diminished, unless the pressure of the steam or the speed of the piston be increased correspondingly; and that an engine of any given size will not exert the same power if made to operate expansively without any other alteration that would have been realized if the engine had been worked with the full pressure of the steam. In the Cornish engines, which work with steam of 40 lbs. on the inch, the steam is cut off at one-twelfth of the stroke; but if the steam were cut off at one-twelfth of the stroke in engines employing a very low pressure, it would probably be found that there would be a loss rather than a gain from carrying the expansion so far, as the benefit might be more than neutralized by the friction incidental to the use of so large a cylinder as would be necessary to accomplish this expansion; and unless the vacuum were a very good one, there would be but little difference between the pressure of the steam at the end of the stroke and the pressure of the vapour in the condenser, so that the urging force might not at that point be sufficient to overcome the fiiction. In practice, therefore, in particular cases, expansion may be carried too far, though theoretically the amount of the benefit increases with the amount of the expansion. We must here introduce a simple practical rule to enable those who may not be familiar with mathematical symbols to determine the amount of benefit due to any particular measure of expansion. When expansion is performed by an expansion valve, it is an easy thing to ascertain at what point of the stroke the valve is shut by the cam, and where expansion is performed by the slide valve the amount of expansion is easily determinable when the lap and stroke of the valve are known. RULE.- To find the Increase of Efficieney arising from workicny Steam exjpansively.-Divide the total length of the stroke by the distance (which call 1) through which the piston moves before the steam is cut off. The hyperbolic logarithm of the whole stroke expressed in terms of the part of the stroke performed with the full pressure of steam, represents the increase of efficiency due to expansion. Suppose that the pressure of the steam working an engine is 45 lbs. on the square inch above the atmosphere, and that the steam is cut off at one-fourth of the stroke; what is the increase of efficiency due to this measure of expansion? If one-fourth be reckoned as 1, then four-fourths must be taken as 4, and the hyperbolic logarithm of 4 will be found to be 1'386, which is the increase of efficiency. The total efficiency of the quantity of steam expended during a stroke, therefore, which without expansion would have been 1, becomes 2'386 when expanded into 4 times its bulk, or, in round numbers, 2'4. Let the pressure of the steam be the same as in the last example, and let the steam be cut off at half-stroke: what, then, is the increase of efficiency? Page 263 THE STEAM ENGINE. 263 Here half the stroke is to be reckoned as 1, and the whole stroke has therefore to be reckoned as 2. The hyperbolic logarithm of 2 is'693, which is the increase of efficiency, and the total efficiency of the stroke is 1'693, or 1'7. WVe may here give a table to illustrate the mechanical effect of steam under varying circumstances. The table shows the meTotal Total i i pressure Mechanical pressure ~olume of Mechanical in es. Correspond- Volume of Steam effect of in lbs. Correspond- Steam effect of l Iec ing Tem- compared with Cubic Inch of per ing Tem- collmpared Culic IlchI Square perature. Water. Water. Square perature. with Water. of W'ater. inch. eInch. 1 103 20-868 1789 51 284 544 2312 2 126 10'874 1812 52 286 534!2316 3 141 7437 1859 53 287 525 2320 4 152 5685 1895 54 288 516 2324 4 5 161 4617 1924 55 289 508 92327 6 169 3897 1948 56 29 0- 500 2331 7 176 3376 1969 57 292 492 2335 i 8 182 2983 1989 58 293 484 2339! 9 187 2674 2006 59 294 477 2343 10 192 2426 2022 60 296 470 2347 11 197 2221 2036 61 297 463 2351 12 201 2050 2050 62 298 456 2355 5 13 205 1904 2063 63 299 449 23-5) 14 209 1778 2074 64 300 443 2362 15 213 1669 2086 65 301 437 2365 16 216 1573 2097 66 302 431 2369 1 17 220 1488 2107 67 303 425 2372 18 223 1411 2117 68 304 419 2375 19 226 1343 2126 69 305 414 2378 20 228 1281 2135 70 306 408 23882 21 231 1225 2144 71 307 403 2385 22 234 1174 2152 72 308 398 2388 23 236 1127 2160 73 309 393 291 24 239 1084 2168 74 310 388 2394 25 241 1044 2175 75 311 383 2397 26 243 1007 2182 76 312 379 2400 27 245 973 2189 77 313 374 2403 28 248 941 2196 78 314 370 2405 29 250 911 2202 79 315 366 2408 30 252 883 2209 80 316 362 2411 31 254 857 2215 81 317 358 2414 i32' 255 833 2221 82 318 354 2417 33 257 810 2226 83 318 350 2419 3i 259 788 2232 84 319 346 2422 35 261 767 2238 85 320 342 2425 36 263 748 2243 86 321 339 2427 37 264 729 2248 87 322 335 2430 38 266 712 2253 88 323 332 2432 39 267 695 2259 89 323 328 2435 40 269 679 2264 90 324 325 2438 41 271 664 2268 91 325 322 2440 42 272 649 2273 92 326 319 2443 43 274 635 2278 93 327 316 2445 44 275 62 22082 94 327 313 2448 45 276 610 2287 95 328 310 2450 46 278 598 2291 96 329 307 2453 47 279 586 2296 97 330 304 2455 48 280 575 2300 98 330 301 2457 49 282 564 2304 99 331 298 2460 50 283 554 2308 100 332 295 2462 Page 264 264 TIIE PRACTICAL MODEL CALCULATOR. chanical effect of the steam generated from a cubic inch of water. Our formula gives the effect of a cubic foot of water; but it can be modified to give the effect of the steam of a cubic inch by dividing by 1728. In this manner we find, for the mechanical effect of the steam of a cubic inch of water, about 3 (459 + t) lbs. raised one foot high. The table shows that the mechanical effect increases with the temperature. The increase is very rapid for temperatures below 212~; but for temperatures above this the increase is less; tand for the temperatures used in practice we may consider, without any material error, the mechanical effect as constant. INDICATOR. An instrument for ascertaining the amount of the pressure of steam and the state of the vacuum throughout the stroke of a steam engine. Fitzgerald and Neucumn long employed an instrument of this kind, the nature of which was for a long time not generally known. Boulton and Watt used an instrument acting upon the same principle and equally accurate; but much more portable. In peculiarity of construction it is simply a small cylinder truly bored, and into which a piston is inserted and loaded by a spring of suitable elasticity to the graduated scale thereon attached. The action of an indicator is that of describing, on a piece of paper attached, a diagram or figure approximating more or less to that of a rectangle, varying of course with the merits or demerits of the engine's productive effect. The breadth or height of the diagram is the sum of the force of the steam and extent of the vacuum; the length being the amount of revolution given to the paper during the piston's performance of its stroke. To render the indicator applicable, it is commonly screwed into the cylinder cover, and the motion to the paper obtained by means of a sufficient length of small twine attached to one of the radius bars; but such application cannot always be conveniently effected, more especially in engines on the marine principle; hence, other parts of such engines, and other means whereby to effect a proper degree of motion, must unavoidably be resorted to. In those of direct action the crosshead is the only convenient place of attachment; but because the length of the engine's stroke is considerably more than the movement required for the paper on the indicator, it is necessary to introduce a pulley and axle, by which means the various movements are qualified to suit each other. When the indicator is fixed and the movement for the paper properly adjusted, allow the engine to make a few revolutions previous to opening the cock; by which means a horizontal line will be described upon the paper by the pencil attached, and denominated the atmospheric line, because it distinguishes between the effect of the steam and that of the vacuum. Open the cock, and if the engine be upon the descending stroke, the steam will instantly raise the piston of the indicator, and, by the motion of the paper with the pencil pressing thereon, the top side of the diagram will be formed. Page 265 TIHE STEAM ENGINE. 265 At the termination of the stroke and immediately previous to its return, the piston of the indicator is pressed down by the surrounding atmosphere, consequently the bottom side of the diagram is described, and by the time the engine is about to make another descending stroke, the piston of the indicator is where it first started from, the diagram being completed; hence is delineated the mean elastic action of the steam above that of the atmospheric line, and also the mean extent of the vacuum underneath it. But in order to elucidate more D clearly by example, take the follow-, I ing diagram, taken from a marine / 10.8 engine, the steam being cut off after 2 1. the piston had passed through two- 2.8 s 12 thirds of its stroke, the graduated ----------------- E scale on the indicator, tenths of an 5.6 4 12.8 inch, as shown at each end of the 7.6 V 1.2.8 diagram annexed. -------—. — ----- Previous to the cock being 7.6 9 12.8 opened, the atmospheric line AB ------------------------------ was formed, and, when opened, the 7.6 Z 12.8 pencil was instantly raised by the action of the steam on the piston 7.6 X 12.8 to C, or what is generally termed 12 the starting corner; by the move- -- - F ment of the paper and at the ter- 7.6 12.6 ruination of the stroke the line CD >_ -was formed, showing the force of 7.6 9.8 the steam and extent of expansion; c'.,from D to E show the moments of A eduction; from E to F the quality of the vacuum; and from F to A the lead or advance of the valve; thus every change in the engine is exhibited, and every deviation from a rectangle, except that of expansion and lead of the valve show the extent of proportionate defect. Expansion produces apparently a defective diagram, but in reality such is not the case, because the diminished power of the engine is more than compensated by the saving in steam. Also the lead of the valve produces an apparent defect, but a certain amount must be given, as being found advantageous to the working of the engine, but the steam and eduction corners ought to be as square as possible; any rounding on the steam corner shows a defect from want of lead; and rounding on the eduction corner that of the passages or apertures being too small. RULE.- To compute the power of an EEngine firo, the Indicator.Diagram.-Divide the diagram in the direction of its length into any convenient number of equal parts, through which draw lines at right angles to the atmospheric line, add together the lengths of all the spaces taken in measurements corresponding with the scale on the indicator, divide the sum by the number of spaces, and the x Page 266 266 THE PRACTICAL MODEL CALCULATOR. quotient is the mean effective pressure on the piston in lbs. per square inch. Let the result of the preceding diagram be taken as an example. Then, the whole sum of vacuum spaces = 1220 - 10 = 12'2 lbs. mean effect obtained by the vacuum; and in a similar manner the mean effective pressure of steam is found to be 6'28 lbs., hence the total effective force = 18'48 lbs. per square inch. And supposing 2'5 lbs. per square inch be absorbed by friction, What is the actual power of the engine, the cylinder's diameter being 32 inches, and the velocity of the piston 226 feet per minute? 18'48 - 2'5 = 15'98 lbs. per square inch of net available force. 322 x'7854 x 15'98 x 226 Then -- 88 horses power. The line under the diagram and parallel to the atmospheric line is 1-ths distant, and represents the perfect vacuum line, the space between showing the amount of force with which the uncondensed steam or vapour resists the ascent or descent of the piston at every part of the stroke. As the mean pressure of the atmosphere is 15 lbs. per square inch, and the mean specific gravity of mercury 13560, or 2'037 cubic inches equal 1 lb., it will of course rise in the barometer attached to the condenser about 2 inches for every lb. effect of vacuum, and as a pure vacuum would be indicated by 30 inches of mercury, the distance between the two lines shows whether there is or is not any amount of defect, as sometimes there is a considerable difference in extent of vacuum in the cylinder to that in the condenser. To estimate by means of an indicator the amount of effective power produced by a steam engine.- Multiply the area of the piston in square inches by the average force of the steam in lbs. and by the velocity of the piston in feet per minute; divide the product by 33,000, and 7oths of the quotient equal the effective power. Suppose an engine with a cylinder of 37~ inches diameter, a stroke of 7 feet, and making 17 revolutions per minute, or 238 feet velocity, and the average indicated pressure of the steam 16'73 lbs. per square inch; required the effective power. Area = 1104'4687 inches x 16'73 lbs. x 238 feet 33Q00 133.26 x 7 10 - 93.282 horse power. To determine the proper velocity for the piston of a steam engine.Multiply the logarithm of the nth part of the stroke at which the steam is cut off by 2'3, and to the product of which add -7. Multiply the sum by the distance in feet the piston has travelled when the steam is cut off, and 120 times the square root of the product equal the proper velocity for the piston in feet per minute. Page 267 267 WEIGHT COMBINED WITH MASS, VELOCITY, FORCE, AND WORK DONE. CALCULATIONS ON THE PRINCIPLE OF VTIS VIVA.-MATERIALS EMPLOYED IN TIIE CONSTRUCTION OF MIACHINES.-STRENGTH OF MATERIALS, THEIR PROPERTIES. -TORSION, DEFLEXION, ELASTICITY, TENACITIES, CO'MPRESSIONS, ETC. -FRICTION OF REST AND OF MIOTION, COEFFICIENTS OF ALL SORTS OF MIOTION. — BANDS.-ROPES. —WHIEELS. -IIYDRAULICS.-NEW TABLES FOR THE MIOTION AND FRICTION OF EWATER.WATER-WHEELS.-WINDMILLS, ETC. 1. Suppose a body resting on a perfectly smooth table, and, when in motion, to present no impediment to the body in its course, but merely to counteract the force of gravity upon it; if this body weighing 800 lbs. be pressed by the force of 30 lbs. acting horizontally and continuously, the motion under such circumstances will be uniformly accelerated: what is the acceleration? 30 800 x 32 2 = 1'2075 feet the second. 2. What force is necessary to move the above-mentioned heavy body, with a 23 feet acceleration, under the same circumstances? 23 8322 x 800 = 57'14285 lbs. The second of these examples illustrates the principle that the force which impels a body with a certain acceleration is equal to the weight of the body multiplied by the ratio of its acceleration to that of gravity. The first illustrates the reverse, namely, the acceleration with which a body is moved forward with a given force, is equal to the acceleration of gravity multiplied by the ratio of the force to the weight. 3. A railway car, weighing 1120 lbs., moves with a 5 feet velocity upon horizontal rails, which, let us suppose, offer no impediment to the motion, and is constantly pushed by an invariable force of 50 lbs. during 20 seconds: with what velocity is it moving at the end of the 20th second, or at the beginning of the 21st second? 50 5 + 32'2 x 1120 x 20 = 33'75, the velocity. 4. A carriage, circumstanced as in the last question, weighs 4000 lbs.; its initial velocity is 30 feet the secoidt, and its terminal velocity is 70 feet: with which force is the body impelled, supposing it to be in motion 20 seconds? (70 - 30) x 4000 32 2 = 242'17 lbs. 32'2 x 20 We have before noticed that the weight (W), divided by 32'2, or (g), gives the mass; that is, Page 268 268 THE PRACTICAL MODEL CALCULATOR. Weight g mass, And, force = mass x acceleration. 5. Suppose a railway carriage, weighing 6440 lbs., moves on a horizontal plane offering no impediment, and is uniformly accelerated 4 feet the second, what continuous force is applied? 6440 32.2 = 200 lbs. mass. 32'2 200 x 4 = 800 lbs., the force applied. By the four succeeding formulas, all questions may be answered that may be proposed relative to the rectilinear motions of bodies by a constant force. For uniformly accelerated motions: F v= a + 322- t; F s= at + 16' 1 x t2. For uniformly retarded motions: F v = a - 322 Vx t; s = at - 16'1 x x t2; t = the time in seconds, W = the weight in lbs., F = the force in lbs., a = the initial velocity, and v = the terminal velocity. 6. A sleigh, weighing 2000 lbs., going at the rate of 20 feet a second, has to overcome by its motion a friction of 30 lbs.: what velocity has it after 10 seconds, and what distance has it described? 30 20 - 322 x 200 X 10 = 15'17 feet velocity. 30 20 x 10 - 16-1 x 2000 x (10)2= 175'85 feet, distance described. 7. In order to find the mechanical work which a draught-horse performs in drawing a carriage, an instrument called a dynarnometer, or measure of force, is thus used: it is put into communication on one side of the carriage, and on the other with the traces of the horse, and the force is observed from time to timne. Let 126 lbs. be the initial force; after 40 feet is described, let 130 lbs. be the force given by the dynamometer; after 40 feet more is described, let 129 lbs. be the force; after 40 feet more is passed over, let 140 lbs. be the force; and let the next tw, o spaces of 40 feet give forces of 130 and 120 lbs. respectively. What is the mechanical work done? 126 initial force. 120 terminal force. 2)246 123 mean. Page 269 WEIGHT COMBINED WITH MASS, VELOCITY, ETC. 269 123 + 130 + 129 + 140 + 130 = 1304 130'4 x 40 x 5 = 26080 units of work. The following rule, usually given to find the areas of irregular figures, may be applied where great accuracy is required. RULE.-TO the sum of the first and last, or extreme ordinates, add four times the sum of the 2d, 4th, 6th, or even ordinates, and twice the sum of the 3d, 5th, 7th, &c., or odd ordinates, not including the extreme ones; the result multiplied by f the ordinates' equidistance will be the sum. 126 120 246 sum of first and last. 246 + 4 x 130 + 2 x 129 + 4 x 140 + 2 x 130 = 1844. 1844 x 40 3 = 24586'66 units of work or pounds raised one foot high. This rule of equidistant ordinates is of great use in the art of ship-building. This application we shall introduce in the proper place. 8. How many units of work are necessary to impart to a carriage of 3000 lbs. weight, resting on a perfectly smooth railroad, a velocity of 100 feet? (100)2 2 (x 3)2 x 3000 = 465838'2 units. A unit of work is that labour which is equal to the raising of a pound through the space of one foot. A unit of work is done when one pound pressure is exerted through a space of one foot, no matter in what direction that space may lie. Kane Fitzgerald, the first that made steam turn a crank, and patented it, and the fly-wheel to regulate its motion, estimated that a horse could perform 33000 units of work in a minute, that is, raise 33000 lbs. one foot high in a minute. To perform 465838'2 units of work in 10 minutes would require the application 1'4116 horse power. 9. What work is done by a force, acting upon another carriage, under the same circumstances, weighing 5000 lbs., which transforms the velocity from 30 to 50 feet? (64)4 = 13'9907, the height due to 30 feet velocity. 64'4 = 38'8043, the height due to 50 feet velocity. From 38'8043 Take 13'9907 24-8136 5000 124068'0000 x2 Page 270 270 THE PRACTICAL MODEL CALCULATOR..*. 124068 are the units of work, and just so much work will the carriage perform if a resistance be opposed to it, and it be gradually brought from a 50 feet velocity to a 30 feet velocity. The following is without doubt a very simple formula, but the most useful one in mechanics; by it we have solved the last two questions: Fs = (I - h) W. This simple formula involves the principle technically termed the principle of VIS VIVA, or LIVING FORCES. H is the height due to v2 one velocity,. say v or H = 2 and hI, the height due to another a, a 2 or h = 2. The weight of the mass = VW; the force F, and the space 8. To express this principle in words, we may say, that the working power (Fs) which a mass either acquires when it passes from a lesser velocity (a) to a greater velocity (v), or produces when it is compelled to pass from a greater velocity (v) into a less (a), is always equal to the product of the weight of the mass and the difference of the heights due to the velocities. WVhen we know the units of work, and the distance in which the change of velocity goes on, the force is easily found; and when the force is known,; the distance is readily determined. Suppose, in the last example, that the change of velocity from 30 to 50 feet took place in a distance of 300 feet, then 1_24068 300 = 413'56 lbs. = F, the force constantly applied during 300 feet. 10. If a sleigh, weighing 2000 lbs., after describing a distance of 250 feet, has completely lost a velocity of 100 feet, what constant resistance does the friction offer? Since the terminal velocity = 0, the height due to it = 0, hence (100)2 2000 64_4 X 250 = 1242 2352 lbs. We have been calculating upon the principle of vis viva; but the product of the gmass and the square of the velocity, without attaching to it any definite idea, is termed the vis viva, or living force. 11. A body weighing 2300 lbs. moves with a velocity of 20 feet the second, required the vis viva? 2300 32'2 = 71'42857 lbs., mass. 71'42857 x (20)2 = 28571'428, the amount of vis vita. Hence, if a mzass enters from a velocity a, into another v, the unit of work done is equal to half the difference of the vis vicca, at the commencement and end of the change of velocity. For if the mass be put = M, and W the weight, Page 271 STRENGTH OF MATERIALS. 271 Then 3i - - and the vis viva to velocity a = Ma2 = 9 7' ~~g and the vis viva to velocity v = M v2 = Then l { -v2 AVa2}= X w = (II - h) W, for 2 a2 2 and, give the heights due to the velocities v and a, respectively. The useful formula Fs = (HI - h) W, before given,. page 270, may be applied to variable as well as to constant forces, if, instead of the constant force F, the mean value of the force be applied. STRENGTH OF MATERIALS. ON MATERIAL EMPLOYED IN THE CONSTRUCTION OF MACHINES. IN theoretical mechanics, we deal with imaginary quantities, which are perfect in all their properties; they are perfectly hard, and perfectly elastic; devoid of weight in statics and of friction in dynamics. In practical mechanics, we deal with real material objects, among which we find none which are perfectly hard, and none, except gaseous bodies, which are perfectly elastic; all have weight, and experience resistance in dynamical action. Practical mechanics is the science of automatic labour, and its objects are machines and their applications to the transmission, modification, and regulation of motive power. In this it takes as a basis the theoretical deductions of pure mechanics, but superadds to the formulae of the mathematician a multitude of facts deduced from observation, and experimentally elaborates a new code of laws suited to the varied conditions to be fulfilled in the economy of the industrial arts. In reference to the structure of machines, it is to be observed that however simple or complex the machine may be, it is of importance that its parts combine lightness with strength, and rigidity with uniformity of action; and that it communicates the power without shocks and sudden changes of motion, by which the passive resistances may be increased and the effect of the engine dimlinished. To adjust properly the disposition and arrangement of the individual members of a machine, implies an exact knowledge and estiunate of the amount of strain to which they are respectively subject in the working of the machine; and this skill, when exercised in conjunction with an intimate acquaintance with the nature of the materials of which the parts are themselves composed, must contribute to the production of a machine possessing the highest amount of capability attainable with the given conditions. ZiLcaterials. — The material most commonly employed in the con Page 272 272 THE PRACTICAL MODEL CALCULATOR. struction of machinery is iron, in the two states of cast and wrought or forged iron; and of these, there are several varieties of quality. It becomes therefore a problem of much practical importance to determine, at least approximately, the capabilities of the particular material employed, to resist permanent alteration in the directions in which they are subjected to strain in the reception and transmission of the motive power. To indicate briefly the fundamental conditions which determine the capability of a given weight and form of material to resist a given force, it must, in the first place, be observed, that rupture may take place either by tension or by compression in the direction of the length. To the former condition of strain is opposed the tenacity of the material; to the other is opposed the resistance to the crushing of its substance. Rupture, by transverse strtain, is opposed both by the tenacity of the material and its capability to withstand compression together of its particles. Lastly, the bar may be ruptured by torsion. Mr. Oliver Byrne, the author of the present work, in his New Theory of the Strength of Materials has pointed out new elements of much importance. The capabilities of a material to resist extension and compression are often different. Thus, the soft gray variety of cast iron offers a greater resistance to a force of extension than the white variety in a ratio of nearly eight to five; but the last offers the greatest resistance to a compressing force. The resistance of cast iron to rupture by extension varies from 6 to 9 tons upon the square inch; and that to rupture by compression, from 36 to 65 tons. The resistance to extension of the best forged iron may be reckoned at 25 tons per inch; but the corresponding resistance to compression, although not satisfactorily ascertained, is generally considered to be greatly less than that of cast iron. Roudelet makes it 311 tons on the square inch. Cast iron (and even wood) is therefore to be preferred for vertical supports. The forces resisting rupture are as the areas of the sections of rupture, the material being the same; this principle holds not only in respect of iron, but also of wood. Many inquiries have been instituted to determine the commonly received principle, that the strength of rectangular beams of the same width to resist rupture by transverse strain is as the squares of the depths of the beams. In these respects the experiments, although valuable on account of their extent and the care with which they were conducted, possess little novelty; but in directing attention to the elastic properties of the materials experimented upon, it was found that the received doctrine of relation between the limit of elasticity and weight requires modification. The common assumption is, that the destruction of the elastic properties of a material, that is, the displacement beyond the elastic limit, does not manifest itself until the load exceeds one-third of the breaking weight. It was found, however, on the contrary, that its effect was produced and manifested in a permanent set of the material when the load did not ex Page 273 STRENGTH OF MATERIALS. 273 ceed one-sixteenth of that necessary to produce rupture. Thus a bar of one inch square, supported between props 4~1 feet apart, did not break till loaded with 496 lbs. but showed a permanent deflection or set when loaded with 16 lbs. In other cases, loads of 7 lbs. and 14 lbs. were found to produce permanent sets when the breaking weights were respectively 364 lbs. and 1120 lbs. These sets were therefore given by -ld and -th of the breaking weights. Since these results were obtained, it has been found that time and the weight of the material itself are sufficient to effect a permanent deflection in a beam supported between props, so that there would seem to be no such limits in respect to transverse strain as those known by the name of elastic limits, and consequently the principle of loading a beam within the elastic limit has no foundation in practice. The beam yields continually to the load, but with an exceedingly slow progression, until the load approximates to the breaking weight, when rupture speedily succeeds to a rapid deflection. As respects the effect of tension and compression by transverse strain, it was ascertained by a very ingenious experiment that equal loads produced equal deflections in both cases. Another most important principle developed by experiments, is that respecting the compression of supporting columns of different heights. When the height of the column exceeded a certain limit, it was found that the crushing force became constant, and did not increase as the height of the column increased, until it reached another limit at which it began to yield, not strictly by crushing, but by the bending of the material. The first limit was found to be a height of little less than three times the radius of the column; and the second double that height, or about six times the radius of the column. In columns of different heights between these limits, having equal diameters, the force producing rupture by compression was nearly constant. When the column was less than the loiwer limit, the crushing force became greater, and when it was greater than the higher limit, the crushing force became less. It was further found that in all cases, where the height of the column was exactly above the limits of three times the radius, the section of rupture was a plane inclined at nearly the same constant angle of 55 degrees to the axis of the column. These facts mutually explain each other; for in every height of column above the limlit, the section of rupture being a plane at the same angle to the axis of the column, must of necessity be a plane of the same size, and therefore in each case the cohesion of the same number of particles must be overcome in producing rupture. And further, the salme number of particles being to be overcome under the same circumstances for every different height, the same force will be required to overcome that amount of cohesion, until at double the height (three diameters) the column begins to bend under its load. This height being surpassed, it follows that a pressure which becomes continually less as the length of the column is increased, will be sufficient to break it. 18 Page 274 274 THE PRACTICAL MODEL CALCULATOR. This property, moreover, is not confined to cast iron; the experiments of M.'Rondoelet show that with columns of wrought iron, wood, and stone, similar results are obtained. From these facts then, it appears that if supporting columns be taken of different diameters, and of heights so great as not to allow of their bending, yet sufficiently high to allow of a complete separation of the planes of fracture, that is, of heights intermediate to three times and six times their radius, then will their strengths be its the number of particles in their planes of fracture; and the planes of fracture being inclined at equal angles to the axes of the columns, their areas will be -as the transverse sections of the,columns, and consequently the strengths of the columns will be as their transverse sections respectively. Taking the mean of three experiments upon a column ~ inch diameter, the crushing force was 6426 lbs.; whilst the mean of four experiments, conducted in exactly the same manner, upon a-column of 8 of an inch diameter, gave 14542 lbs. The diameters of the-columns being 2 to 3, the areas of transverse section were therefore 4 to 9, which is very nearly the ratio of the crushing weights. When the length of the column is so great that its fracture is produced wholly by bending of its material, the limit has been fixed for columns of cast iron, at 30 times the diameter when the ends are fiat, and 15 times the diameter when the ends are rounded..In shorter columns, fracture takes place partly by crushing and partly by bending of the material. When the column is enlarged in the middle of its length from one and a half to two times the diameter of the ends,' the strength was found by the same experimenter to be greater by one-seventh than in solid columns containing the same quantity of iron, in the same length, with their extremities rounded; and stronger by an eighth or a ninth when the extremities were fiat and rendered immovable by disks. The following formulas give the absolute strength of cylindrical columns to sustain pressure in the direction of their length. In these formulas D = the external diameter of the column in inches. d = the'internal diameter of hollow columns in inches. L = the length of the column in feet. W = the breaking weight in tons. Charater of the column. Length of the column exceeding 15 Length of the column exceeding 30 Character of the column. times its diameter. times its diameter. Both ends rounded.' Both ends flat. Solid cylindrical co- W 149 D" lumn of cast iron, }W = 1'9 L Hollow cylindrical co-I D =13 dI. I) D-' d 3'5 W 0 W 13 rW 44-34 lumn of cast iron, L L'Solid cylindrical co- 428D W 13375 D lumn of wrought iron, - Lfa L For shorter columns, if W' represent the weight in tons which would break the column by bending alone, as given by the preced Page 275 STRENGTH OF MATERIALS. 275 ing formulas, and W" the weight in tons which would crush the column without bending it, as determined from the subjoined table, then the absolute breaking weight of the column W, is represented in tons by the formula, W' xW" W = W' + W" These rules require the use of logarithms in their applications. When a beam is deflected by transverse strain, the material on that side of it on which it sustains the strain is compressed, and the material on the opposite side is extended. The imaginary surface at which the compression terminates and the extension begins-at which there is supposed to be neither extension nor compressionis termed the neutral axis of the beam. What constitutes the strength of a beam is its resistance to compression on the one side and to extension on the other side of that axis-the forces acting about the line of axis like antagonist force at the two extremities of a lever, so that if either of them yield, the beam will be broken. It becomes, however, a question of importance to determine the relation of these forces; in other words, to determine whether the beam of given form and material will yield first to compression or to extension. This point is settled by reference to the columns of the subsequent table, page 280, in which it will be observed that the mnetals require a much greater force to crush them than to tear them asunder, and that the woods require a much smaller force. There is also another consideration which must not be overlooked. Bearing in mind the condition of antagonism of the forces, it is obvious, that the further these forces are placed from the neutral axis, that is, from the fulcrum of their leverage, the greater must be their effect. In other words, all the material resisting compression will produce its greatest effect when collected the farthest possible from the neutral axis at the top of the beam; and, in like manner, all the material resisting extension will produce its greatest effect when similarly disposed at the bottom of the beam. We are thus directed to the first general principle of the distribution of the material into two flanges-one forming the top and the other the bottom of the beam-joined by a comparatively slender rib. Associating with this principle the relation of the forces of extension and compression of the material employed, we arrive at a form of beam in which the material is so distributed, that at the instant it is about to break by extension on the one side, it is about to break by compression on the other, and consequently is of the strongest form. Thus, supposing that it is required to determine that form in a girder of cast iron: the ratio of the crushing force of that metal to the force of extension may be - taken generally as 6} to 1, which is therefore also the ratio of the lower to the upper flange, as in the annexed sectional diagram. A series of nine castings were made, gradually increasing the lower flange at the expense of the upper one, and in the first eight Page 276 276 THE PRACTICAL MODEL CALCULATOR. experiments the beam broke by the tearing asunder of the lower flange; and in the last experiment the beam yielded by the crushing of the upper flange. In the eight experiments the upper flange was therefore the weakest, and in the ninth the strongest, so that the form of maximum strength was intermediate, and very closely allied to that form of beam employed in the last experiment, which was greatly the strongest. The circumstances of these experiments are contained in the following table. No. of experi- Ratio of surfaces of corn- Area of cross sections Strength per sq. inch nents. pression and extension. in sq. inches. of sections in lbs. 1 1 to 1 2'82 2368 2 1 to 2 2'87 2567 3 1 to 4 3.02 2737 4 1 to 41 3.37 3183 5 1 to 4 4.50 3214 6 1 to 5~ 5.00 3346 7 1 to 31 4.628 3246 8 1 to 4.3 5.86 3317 9 1 to 6.1 6.4 4075 To determine the weight necessary to break beams cast according to the form described: Multiply the area of the section of the lower flange by the depth of the beam, and divide the product by the distance between the two points on which the beam is supported: this quotient multiplied by 536 when the beams are cast erect, and by 514 when they are cast horizontally, will give the breaking weight in cwts. From this it is not to be inferred that the beam ought to have the same transverse section throughout its length. On the contrary, it is clear that the section ought to have a definite relation to the leverage at which the load acts. From a mathematical consideration of the conditions, it indeed appears that the effect of a given load to break the beam varies when it is placed over different wA w2 points of it, as the products L I of the distances of these points from the points of support of thil beam. Thus the effect of a weight ploAced at the point AWV1 is to th t effect of the same weight acting upon the point V,, as the prodlulct ANWV x W0 B is to the product AWYr x W2 B; the points of support being at A and B. Since then the effect of a weight increases as it approaches the middle of the length of the beam, at which it is a maximum, it is plain that the beam does not require to have the same transverse section near to its extremities as in the middle; and, guided by the principle stated, it is easy to perceive that its strength at different points should in strictness vary as the products of the distances of these points from the points of support. By Page 277 STRENGTH OF MATERIALS. 277 taking this law as a fundamental condition in the distribution of the strength of a beam, whose load we may conceive to be accumulated at the middle of its length, we arrive at the strongest form which can be attained under given circumstances, with a given amount of material; we arrive at that form which renders the beam equally liable to rupture at every point. Now this form of maximum strength may be attained in two ways; either by varying the depth of the beam according to the law stated, or by preserving the depth everywhere the same, and varying the dimensions of the upper and lower flanges according to the same law. The conditions are manifestly identical. We may therefore assume generally the condition that the section is rectangular, and that the thickness of the flanges is constant; then the outline determined by the law in question, in the one case of the elevation of the beam and in the other of the plan of the flanges, is the geometrical curve called a parabola —rather, two parabolas joined base to base at the middle between the points of support. The annexed diagram represents the plan of a cast-iron girder according to this form, the depth being uniform throughout. Both flanges are of the same form, but the dimensions of the upper one are such as to give it only a sixth of the strength of the other. This, it will be observed, is also the form, considered as an elevation, of the beam of a steam engine, which good taste and regard to economy of material have rendered common. It must, however, be borne in mind, that in the actual practice of construction, materials cannot with safety be subjected to forces approaching to those which produce rupture. In machinery especially, they are liable to various and accidental pressures, besides those of a permanent kind, for which allowance must be made. The engineer must therefore in his practice depend much on experience and consideration of the species of work which the engine is designed to perform. If the engine be intended for spinning, pumping, blowing, or other regular work, the material may be subjected to pressures approaching two-thirds of that which would actually produce rupture; but in engines employed to drive bonemills, stampers, breaking-down rollers, and the like, double that strength will often be found insufficient. In cases of that nature, experience is a better guide than theory. It is also to be remarked that we are often obliged to depart from the form of strength which the calculation gives, on account of the partial strains which would be put upon some of the parts of a casting, in consequence of unequal cooling of the metal when the thicknesses are unequal. An expert founder can often reduce the irregular contractions which thus result; but, even under the best management, fracture is not unfrequently produced by irreguY Page 278 278 THE PRACTICAL MODEL CALCULATOR. larity of cooling, and it is at all times better to avoid the danger entirely, than to endeavour to obviate it by artifice. [For this reason, the parts of a casting ought to be as nearly as possible of such thickness as to cool and contract regularly, and by that means all partial strain of the parts will be avoided. With respect to design, it is also to be remarked, that mere theoretical properties of parts will not, under all the varieties of circumstances which arise in the working of a machine, insure that exact adjustment of material and propriety of form so much desired in constructive mechanics. Every design ought to take for its basis the mathematical conditions involved, and it would, perhaps, be impossible to arrive at the best forms and proportions by any more direct mode of calculation; but it is necessary to superadd to the mathematical demonstration, the exercise of a well-matured judgment, to secure that degree of adjustment and arrangement of parts in which the merits of a good design mainly consist. A purely theoretical engine would look strangely deficient to the practised eye of the engineer; and the merely theoretical contriver would speedily find himself lost, should he venture beyond his construction on paper. His nice calculations of the " work to be performed," of the -vis viva of the mechanical organs of his machine, and of the vnodulus of elasticity of his material, would, in practice, alike deceive him. The first consideration in the design of a machine is the quantity of work which each part has to perform-in other words, the forces, active and inactive, which it has to resist; the direction of the forces in relation to the cross-section and points of support; the velocity, and the changes of velocity to which the moving parts are subject. The calculations necessary to obtain these must not be confined to theory alone; neither should they be entirely deduced by " rule of thumb;" by the first mode the strength would, in all probability, be deficient from deficiency of material, and by the second rule the material would be injudiciously disposed; weight would be added often where least needed, merely from the determination to avoid fracture, and in consequence of a want of knowledge respecting the true forms best adapted to give strength. To the following general principles, in practice, there are but few real exceptions: I. Direct Strain.-To this a straight line must be opposed, and if the part be of considerable length, vibration ought to be counteracted by intersection of planes, (technically feathers,) as represented in the annexed diagrams, v/E/ or some such form, consistent with the purpose for which the part is intended. II. Transverse Strain. —To this a parabolic form of section must be opposed, or some simple figure including the parabolic form. For economy of material, the vertex of the curve ought to be at the point where the force is applied; and when the strain passes Page 279 STRENGTH OF MATERIALS. 279 alternately from one side of the part to the other, the curve ought to be on both sides, as in the beam of a steam engine. When a loaded piece is supported at one end only, if the breadth be everywhere the same, the form of equal strength is a triangle; but, if the section be a circle, then the solid will be that generated by the revolution of a semi-parabola about its longer axis. In practice) it will, however, be sufficient to employ the firustum of a cone, of which, in the case of cast iron, the diameter at the unsupported end is one-third of the diameter at the fixed end. III. Torsion.-The section most commonly opposed to torsion is a circle; and, if the strain be applied to a cylinder, it is obvious the rupture must first take place at the surface, where the torsion is greatest, and that the further the material is placed from the neutral axis, the greater must be its power of resistance; and hence, the amount of materials being the same, a shaft is stronger when made hollow than if it were made solid. It ought not, however, to be supposed that the circle is the only figure which gives an axis the property of offering, in every direction, the same resistance to flexure. On the contrary, a square section gives the same resistance in the direction of its sides, and of its diagonals; and, indeed, in every direction the resistance is equal. This is, moreover, the case with a great number of other figures, which may be formed by combining the circle and the square in a symmetrical manner; and hence, if the axis, strengthened by salient sides, as in feathered shafts, do not answer as well as cylindrical ones, it must arise from their not being so well disposed to resist torsion, and not from any irregularities of flexure about the axis inherent in the particular form of section. This subject has been investigated with much care, and, according to M. Cauchy, the modulus of rupture by torsion, T, is connected with the modulus of rupture by transverse strain S, by the simple analogy T = 4 S. The forms of all the parts of a machine, in whatever situation and under every variety of circumstances, may be deduced froml these simple figures; and, if the calculations of their dimensions be correctly determined, the parts will not only possess the requisite degree of strength, but they will also accord with the general principles of good taste. In arranging the details of a machine, two circumstances ouTht to be taken into consideration. The first is, that the parts subject to wear and influenced by strain, should be capable of adjustment; the second is, that every part should, in relation to the work it has to perform, be equally strong, and present to the eye a figure that is consistent with its degree of action. Theory, practice, and taste must all combine to produce such a combination. No formal law can be expressed, either by words or figures, by which a certain contour should be preferred to another; both may be equally strong and equally correct in reference to theory; custom, then, must be appealed to as the guide. Page 280 280 THE PRACTICAL MODEL CALCULATOR. TABLES OF THE MECHANICAL PROPERTIES OF THE MATERIALS MOST COMMONLY EMPLOYED IN THE CONSTRUCTION OF MACHINES AND FRAMINGS.......!Weight of Tenacity per, Cloushing Modulus of Mod. of Crushing NAMES. Speoific- 1 cubic ft. square inch force per sq.. elasticity rupture force to Gravity. in nb.. i l in b. in tie, in lbs. tenacity. TABLE I. —techanicaZ Properties of the C ommon Metals. Brass (cast) $-399 525'00 179 18 30304- 890000 - 0.17968 893~~~~~000'0573:1 Copper (coot)..... 80607 537-93 19072 ditto (sheet)....8785 549'06 -- ditto (wire-drawn) 8878 56000 61228 -- ditto (in bolts).... - 48000 - - Iron (English wrought)... 7.700 481.20 25Y, tons 2.4920000 - ditto (in bars).76 70. 7-800 487'00 25y. tons ditto (hammered). 30 tone -- - -' - ditto (Russian) in har. - 27 tons ------ ditto (Swedish) in bars... 32 tons ditto (English) in wire, 10th inch diam. 36to 43tons ditto (Rssian) in wire, 1-20th to 1-30th inch diameter.6.. - ditto (rolled in sheets and cut lengthwise) - - 14 tons ditto cut crosswise: ~ ~ - 18 tons ditto in chains, oval links, 6inches clear, iron i n a e r _o ion 34 inch diameter., 21l34tons ditto (Brunton's) with stay cross link, 25 tons -- - Cast-iron (Old Park).. - 18014400 48240 ditto (Adelphi)... - 18353600 45360 ditto ~Alfreton).,..... 17686400 44046 ditto (scrap)... 18032000 45828 ditto (Carron, No. 2) hot baso 7046 440-37 13105 108540 16005000 07503 037:I ditto ( do do. )old blast 7-066 441,62 16683 106375 17270500 3856 6'376: ditto do. No. 3)... 7'094 443-37 14200 115442 16246966 33980 8-129: 1 ditto do. do. hotblast 7'056 441'00 17755 133440 17873100 42120 7-515:1 ditto Devon, No. 3) cold blast 7'295 455-93 22907700 36288 ditto ( do. do. hot blast 7'229 451'81 21907 145435 22473650 43497 ditto (Buffrey,No, 1 cold blast 7-079 442-43 17466 93366 15M1200 37503 5-346: 1 ditto ( do. do. hot blast 6-998 437-37 13434 8697 13730500 35316 6-431: 1 ditto (Coed-Talon, No. 2) cold blast 6.955 434-06 18855 81770 14313500 33104 ditto ( do. do. ) hot blast 6.968 435,50 16676 82739 14322500 33145 4961:1 ditto ( do., No. 3) cold blast 7-104 449062 17102000 43541 - ditto ( do. do.. ) hot blast 6'9.70 435:62 14707900 40159 ditto (Milton, No. 1) hot blast. 6976 436'00 -1.1974500 285529 ditto (Muirkirk, No. 1) cold, blast 7,113 444'56 14003550 35923 ditto ( do. do. ) hot blast 6.953 434-56 - 13294400 33850 ditto (Elsicar, No. 1) cold blast. 7030 439-37 13981000 34862 Lead (English cs).4 717- 1824 720000 ditto (milled-sheet)... 11'407 712-93 3328 ditto (wire-drawn.... 11.317 705'12 2581 Silver (standard)... 10'312 644'50 40902 Mercury (at 320.... 13.619 851-18 ditto (at 600 )... 13.580 848-75 Steel (soft)... 7-780 486-25 120000 ditto (razor-tempered).. 7.8 490.00 150000 - 29000800 - Tin (cast).. 7291 455-68 5322 - 4608000 - Zinc (cast)... 7-028 4392 -2- 13680000 - ditto (rolled... 7-215 450-9 - -- __ TABLE II.-P-incipal Woods. Acacias (English). ~ 0-71 44.37 36000 -201 112 - Bhf New. 0854 53-'7 15784 7733 1 t Dry.0-690 43212 17850 9363 Birch Common. 0-792 49-00 15000 6402 1562400 10920 0-43: American... 0648 40650 - 11033 1257600 9624 Christiania middle 0-698 42-62 12400 1072080 9864 - Deal Memel middle. i 0-590 36-87 1535200 30386 eNorway spruce 0.34-0 21-25 17680 - [English. 0-470 29-37 7000 Elm (seasoned).. 0588 36-75 13489 10331 699840 8 -79 8 i f New England. 0-553 34-56 -061 0-: Fix~~~~~~~~~~~~~~ —- 2191200 6612 I Riga.. 0-753 47'06 12000 6000 Larch (seasoned) 0-522 32-62. 10220 5568 1052800 6894 Lignutm-vitmt1220 76-25 1180. Mahogany (Spanish). 0800 50'00 1600 8198 English. 0-94 58-37 17300 4 wet 1451200 1023(0-28: 9504sh ry 9 0-57:1 Oak Canadian.0-872.54-50 10233 421,m 211480 1059 0-42: 1 9509 dry, f 0'95:1 DAntzio.0-756 47-24 12780 1191200 8748 - (Pitch., 0-660 41-25 7818. 1225600 9792 - Pins Red.0-657 41-06. - 5375 1340000 8946 Yellow.0-461 28-81 5445 1600000 - Plane-tree.0-64 40-01 06 400 11700 Poplar.'0-333 23-93 7200 3107wt 0-43:1 w124dry. 0-74:1 Teako (dr y).52dr Teakow (dry) 0-657 41-06 15000 12101 2414400 14772 0-81:1 Willow (dryn i sh).~~~ 0-390 24'37 14000 [ - Yew (Spanih). 0-807 50-43 8000 - Page 281 STRENGTH OF MATERIALS. 281 THE COHESIVE STRENGTH OF BODIES. The following TABLE contains the result of experiments on the cohesive strength of various bodies in avoirdupois pounds; also, one-third of the ultimate strength of each body, this being considered sufficient, in most cases, for a permanent load: Names of Bodies. Square Bar. One-third. Round Bar. One-third. WOODS. lb. lbs. lbs. Ib s. Boxwood...................... 20000 6667 15708 5236 Ash............................ 17000 5667 13357 4452' Teak........................... 15000 5000 11781 39 27 Fir.............................12000 4000 9424 3141 Beach....................... 11500 3866 9032 3011 Oak............................ 11000 3667 8639 2880 METALS. Cast iron................... 18656 6219 14652 4884 English wrought iron...... 55872 18624 43881 14627 Swedish do. do....... 72064 24021 56599 18866 Blistered steel............... 133152 44384 104577 34859 Shear do................ 124400 41366 97 703 32568 Cast do.1............... 134 256 44752 105454 35151 Cast copper.................. 19072 6357 14979 4993 Wrought do.................. 33792 11264 26540 8827 Yellow brass............... 17, 968 5989 14112 4704 Cast tin....................... 4736 1579 3719 1239 Cast lead..................... 1824 608 1432 477 PROELEMI I. RULE.-To find the ultimate cohesive strength of square, round, and rectangular bars, of any of the variozus bodies, as specified in the table.-Multiply the strength of an inch bar, (as in the table,) of the body required, by the cross sectional area of square and rectangular bars, or by the square of the diameter of round bars; and the product will be the ultimate cohesive strength. A bar of cast iron being 1~1 inches square, required its cohesive power. 1'5 x 1'5 x 18656 = 41976 lbs. Required the cohesive force of a bar of English wrought iron, 2 inches broad, and - of an inch in thickness. 2 x'375 x 55872 = 41904 lbs. Required the ultimate cohesive strength of a round bar of wrought copper 3- of an inch in diameter..752 x 26540 - 14928'75 lbs. PROBLEMI II. RULE.-Tie weigqht of a body being given, to find the cros.s sectional dimensions of a bar or rod capable of sustainig? that weighlt.For square and round bars, divide the weight given by one-third of the cohesive strength of an inch bar, (as specified in the table,) and the square root of the quotient will be the side of the scquare, or diameter of the bar in inches. Y2 Page 282 282 TIIE PRACTICAL MODEL CALCULATOR. And if rectangular, divide the quotient by the breadth, and the result will be the thickness. What must be the side of a square bar of Swedish iron to sustain a permanent weight of 18000 lbs? 18000 V/24021 -= 86, or nearly 8 of an inch square. Required the diameter of a round rod of cast copper to carry a weight of 6800 lbs. 6800 V4993 = 1.16 inches diameter. A bar of English wrought iron is to be applied to carry a weight of 2760 lbs.; required the thickness, the breadth being two inches. 2760 -18624 = 142 2 = 071 of an inch in thickness. A TABLE showing the circumference of a rope equal to a chain made of iron, of a given, diameter, and the weight in tons that each is proved to carry; also, the weight of a foot of chaizn made from iron of that dimension. I Ropes. Chains. Proved to carry weight of a lineal Cir. in Ins. Diam. in Inches. in tons. foot in lbs. Avr. 3 - and j- 1 1~08 4 _ 2 1'5 54 P_2I 4 2'7 6 2 andQ L 5 3'3 6 4 7 8 and 1- 8 4'6 71 3 9 3 5.5 8 4T CLand 16 11 4 6'1 9 13 7'2 92 8 and j- 15 8'4 10-1 1inch. 18 9'4 ON THE TRANlSVERSE STRENGTH or BODIES. The tranverse strength of a body is that power which it exerts in opposing any force acting in a perpendicular direction to its length, as in the case of beams, levers, &c., for the fundamental principles of which observe the following:That the transverse strength of beams, &c. is inversely as their lengths, and directly as their breadths, and square of their depths, and, if cylindrical, as the cubes of their diameters; that is, if a beam 6 feet long, 2 inches broad, and 4 inches deep, can carry 2000 lbs., another beam of the same material, 12 feet long, 2 inches broad, and 4 inches deep, will only carry 1000, being inversely as their lengths. Again, if a beam 6 feet long, 2 inches broad, and 4 inches deep, can support a weight of 2000 lbs., another beam of Page 283 STRENGTH OF MATERIALS. 283 the same material, 6 feet long, 4 inches broad, and 4 inches deep, will support double that weight, being directly as their breadths; -but a beam of that material, 6 feet long, 2 inches broad, and 8 inches deep, will sustain a weight of 8000 lbs.; being as the square of their depths. From a mean of experiments made, to ascertain the transverse strength of various bodies, it appears that the ultimate strength of an inch square, and an inch round bar of each, 1 foot long, loaded in the middle, and lying loose at both ends, is nearly as follows, in lbs. avoirdupois. Names of Bodi(es. Square Bar. Onc-third. Round Bar. One-tlhird. Oak........................... 800 267 628 209 Ash............................I 117 379 893 As. 1137 379 893 298 Elm.......................... 569 139 447 149 Pitch pine.................. 916 305 719 239 Deal......................... 566 188 444 148 Cast iron...............2....8.. 250 860 2026 675 r-lrought iron........ 40...... 4013 1338 3152 1050 PROBLEM I. RULE. —To find the ultimate transverse strength of any rectangular beam, supzorted at both ends, and loaded in the middle; or supported in the middle, and loaded at both ends; also, when the weight is between the middle and the end; likewise when fixed at one end and loaded at the other.-Multiply the strength of an inch square bar, 1 foot long, (as in the table,) by the breadth, and square of the depth in inches, and divide the product by the length in feet; the quotient will be the weight in lbs. avoirdupois. What weight will break a beam of oak 4 inches broad, 8 inches deep, and 20 feet between the supports? 800 x 4 x 82 20 = 10240 lbs. When a beam is supported in the middle, and loaded at each end, it will bear the same weight as when supported at both ends and loaded in the middle; that is, each end will bear half the weight. When the weight is not situated in the middle of the beam, but placed somewhere between the middle and the end, multiply twice the length of the long end by twice the length of the short end, and divide the product by the whole length of the beam; the quotient will be the effectual length. Required the ultimate transverse strength of a pitch pine planlk 04 feet long, 3 inches broad, 7 inches deep, and the weight placed 8 feet from one end. 32 x 16 24 = 21'3 effective length. 916 x 3 x 72 and L - = 6321 lbs. 1.~3 Page 284 284 THE PRACTICAL'MODEL CALCULATOR. Again, when a beam is fixed at one end and loaded at the other, it will only bear { of the weight as when supported at both ends and loaded in the middle. What is the weight requisite to break a deal beam 6 inches broad, 9 inches deep, and projecting 12 feet from the wall? 566 x 6 x 92 = 22923. 4 = 5730'7 lbs. 12 The same rules apply as well to beams of a cylindrical form, with this exception, that the strength of a round bar (as in the table) is multiplied by the cube of the diameter, in place of the breadth, and square of the depth. Required the ultimate transverse strength of a solid cylinder of cast iron 12 feet long and 5 inches diameter. 2026 x 53 12 = 21104 lbs. WThat is the ultimate transverse strength of a hollow shaft of cast iron 12 feet long, 8 inches diameter outside, and containing the same cross sectional area as a solid cylinder 5 inches diameter? / -- 5_2 = 6'24, and 83- 6243 = 269. 2026 x 969 Then, 12 = 45416 lbs. When a beam is fixed at both ends, and loaded in the middle, it will bear one-half more than it will when loose at both ends. And if a beam is loose at both ends, and the weight laid uniformly along its length, it will bear double; but if fixed at both ends, and the weight laid uniformly along its length, it will bear triple the weight. PROBLEM II. RULE.-To find the breadth or depth of beams intended to su iIort a permanent weight. —Multiply the length between the supports, in feet, by the weight to be supported in lbs., and divide the product by one-third of the ultimate strength of an inch bar, (as in the table,) multiplied by the square of the depth; the quotient will be the breadth, or, multiplied by the breadth, the quotient will be the square of the depth, both in inches. Required the breadth of a cast iron bearm 16 feet long, 7 inches deep, and to support a weight of 4 tons in the middle. 8960 x 16 4 tons = 8960 lbs. and 0x = 34 inches. What must be the depth of a cast iron beam 384 inches broad, 16 feet long, and to bear a permanent weight of four tons in the middle? 8960 x 16 /860 x:-4.- 7 inches. 860 x 3-41 Page 285 STRENGTn OF MATERIALS. 285 When a beam is fixed at both ends, the divisor must be multiplied by 1'5, on account of it being capable of bearing one-half more. When a beam is loaded uniformly throughout, and loose at both ends, the divisor must be multiplied by 2, because it will bear double the weight. If a beam is fast at both ends, and loaded uniformly throughout, the divisor must be multipled by 3, on account that it will bear triple the weight. Required the breadth of an oak beam 20 feet long, 12 inches deep, made fast at both ends, and to be capable of supporting a weight of 12 tons in the middle. 26880 x 20 12 tons = 26880 lbs., and 266 x 122 x 1'5 = 9'7 inches. Again, when a beam is fixed at one end, and loaded at the other, the divisor must be multiplied by'25; because it will only bear one-fourth of the weight. Required the depth of a beam of ash 6 inches broad, 9 feet projecting from the wall, and to carry a weight of 47 cwt. 5264 x 9 47 cwt. = 5264 lbs., and V379 x 6 x 25 9'12 inches deep. And when the weight is not placed in the middle of a beam, the effective length must be found as in Problem I. Required the depth of a deal beam 20 feet long, and to support a weight of 63 cwt. 6 feet from one end. 28 x 12 20 = 16.8 effective length of beam, and 63 cwt. = 7056 lbs.; hence 7056 x 16.8 188 x 6 - 10'24 inches deep. Beams or shafts exposed to lateral pressure are subject to all the foregoing rules, but in the case of water-wheel shafts, &c., some allowances must be made for wear; then the divisor may be changed from 675 to 600 for cast iron. Required the diameter of bearings for a water-wheel shaft 12 feet long, to carry a weight of 10 tons in the middle. 10 tons = 22400 lbs., and 22400 600 = — 448 = 7'65 inches diameter. And when the weight is equally distributed along its length, the cube root of half the quotient will be the diameter, thus: 448 92 -- 224 = 6'07 inches diameter. Required the diameter of a solid cylinder of cast iron, for the shaft of a crane, to be capable of sustaining a weight of 10 tons; Page 286 286 THE PRACTICAL MODEL CALCULATOR. one end of the shaft to be made fast in the ground, the other to project 6- feet; and the effective leverage of the jib as 1V to 1. 10 tons =2 2400 lbs., and 22400 x 6'5 x 1'75 675 x'25 1509 And i1509 = 11'47 inches diameter. The strength of cast iron to wrought iron, in this direction, is as 9 is to 14 nearly; hence, if wrought iron is taken in in place of cast iron in the last example, what must be its diameter? 1509 x 9 5014 = 9' 89 inches diameter. ON TORSION OR TWISTING. The strength of bodies to resist torsion, or wrenching asunder, is directly as the cubes of their diameters; or, if square, as the cube of one side; and inversely as the force applied multiplied into the length of the lever. Hence the rule. —1. Multiply the strength of an inch bar, by experiment, (as in the following table,) by the cube of the diameter, or of one side in inches; and divide by the radius of the wheel, or length of the lever also in inches; and the quotient will be the ultinmate strength of the shaft or bar, in lbs. avoirdupois. 2.-A- ultiply the force applied in pounds by the length of the lever in inches, and divide the product by one-third of the ultimate strength of an inch bar, (as in the table,) and the cube root of the quotient will be the diameter, or side of a square bar in inches; that is, capable of resisting that force permanently. Thefollowing TABLE contains the result of ex)erihznents on zizch bars, of various rmetals, in lbs. avoirdulois. Names of Bodies. Round Bar. One-third. Square Bar. One-third. Cast iron............ 11943 3981 15206 5069 English wrought iron 12063 4021 15360 5120 Swedish do. do. 11400 3800 14592 4864 Blistered steel......... 20025 6675 25497 8499 Shear....... do......... 20508 6836 26112 8704 Cast.........do.......... 21111 703 7 26880 8960 Yellow brass........... 5549 1850 7065 23355 Cast copper........... 4 825 1608 6144 2048 Tin....................... 1688 563 2150 717 Lead..................... 1206 402 1536 510 What weight, applied on the end of a 5 feet lever, will wrench asunder a 3 inch round bar of cast iron? 11943 x 33 11943X 3 5374 lbs. avoirdupois. Required the side of a square bar of wrought iron, capable of resisting the twist of 600 lbs. on the end of a lever 8 feet long. 600 x )96 i/~o " 9610= -21 inches. Page 287 STRENGTH OF MATERIALS. 287 In the case of revolving shafts for machinery, &c., the strength is directly as the cubes of their diameters, and revolutions, and inversely rCs the resistance they have to overcome; hence, From practice, we find that a 40 horse power steam engine, making 25 revolutions per minute, requires a shaft (if made of wrought-iron) to be 8 inches diameter: now, the cube of 8, multiplied by 25, and divided by 40 = 320; which serves as a constant multiplier for all others in the same proportion. What must be the diameter of a wrought iron shaft for an engine of 65 horse power, making 23 revolutions per minute? 65 x 320 I 23 - 9'67 inches diameter. James Glenie, the mathematician, gives 400 as a constant multiplier for cast iron shafts that are intended for first movers in machinery; 200 for second movers; and 100 for shafts connecting smaller machinery, &c. The velocity of a 30 horse power steam engine is intended to be 19 revolutions per minute. Required the diameter of bearings for the fly-wheel shaft. 400 x 30 v- -19 = 8'579 inches diameter. Required the diameter of the bearings of shafts, as second movers from a 30 horse engine; their velocity being 36 revolutions per minute. 200 x 30 i/ -36 = 5'5 inches diameter. W1Then shafting is intended to be of wrought iron, use 160 as the multiplier for second movers; and 80 for shafts connecting smaller machinery. TABLE of the Proportionate Length of Bearings, or Journals for Stafts of various diameters. Dia. in Inches. Len. in Inches. Dia. in Inches. Len. in Inches. 1 13 6- 8-:3 1 a21 7 98 2 3 7~ 10 2s 1- 8 1. l0 -1- 02 81. 11_38 3 4- 9 12,31 4~ 94 l4. 4 5 - 10 13a 41 6' 10-1 14 2 211 14} 6 812 2 1 6 8' 12 16 Page 288 288 THE PRACTICAL MODEL CALCULATOR. Tenacities, Resistances to Comlpression, and other Properties of the common Materials of Construction. Absolute. Compared with Cast,ron. Resistance to Names of Bodies. Tenacity in lbs. compression Its strenogth Its extensi- Its stiffness is per sq. inch. is lbs. per sq. is bility is in. Ash....................... 14130 023 206 0.089 Beech....................... 12225 8548 0.15 21 0.073 Brass....................... 168 10304 0-43 09 049 B3ra1ss~,....,..,, 17-368 10304 0.435 0'9 0'49 Brick....................... 275 562 Cast iron.................. 13434 86397 1'000 1 0 1-000 Copper (wrought) 00....... 3000 El......................... 9720 1033 0 21 2-9 0.073 Fir, or Pine, white..... 12346 2028 0-23 2-4 01 - - red........ 11800 5375 0'3 2-4 01 -- -- yellow.... 11835 5445 0'25 ~29 0'087 Granite, Aberdeen....... 10910 - - Gun-metal (copper 8, and tin 1)............... 35838 0-65 1.25 0 535 Malleable iron........... 56000' 112 086 1.3 Larch....................... 12240 5568 0.136 2-3 0058 Lead................ -1824 - 0096 2-5 0.0385 Mahogany, Honduras.. 11475 8000 0[ 24 2 9 0 487 Marble..................... 551. 6060 - - Oak................... 11880 9504 0t 25 2-8 0 093 Rope (1 in. in circum.) 200 Steel..................... 28 128000 - Stone, Bath............... 478 - - Craigleith........ 772 5490 - - Dundee........... 2661 6630 - Portland.......... 857 3729 - Tin (cast)................ 4736 - 0.182 075 025 Zinc (sheet)............... 9120 0- 35 0.5 0'76 Comparative Strength and Weight of Ropes and CU/hains. 55 4:mi 0 o ~. a- o. 0. Proof strength |. D. a= E.5. a strenthi ill tons & cnnt. 32 2.-7 5' 1 51 10 23 J. 43 c 10 0 5 51 -1 101 2 10 1M 30~ lin. 56 13 8 54 7 1 14 3 51 121 36 1 1 63 14 18 6r 94, 9 18 4 3- 13 39 1 71 16C 14 2 4 16 2 8 7 11 1 5 22 5 2 11 13 45 1_ 79 18 11 8 15 11 27 6 41 141 48 1- 87 20 8 t 87 19 32 7 7 151 56 [1. 96 2 2 13 9-1 21 1- 37 8 13' 16 60 106 2 4 18_ 2 _ It must be understood and also borne in mind, that in estimating the amount of tensile strain to which a body is subjected, the w-eight of the body itself must also be taken into account; for according to its position so may it approximate to its whole -weigllt, in tend Page 289 STRENGTH OF MATERIALS. 289 ing to produce tension within itself; as in the almost constant application of ropes and chains to great depths, considerable heights, &c. Alloys that are of greater Tenacity than the sztm of their Constituents, as determined by the -Experiments of Juschenbroek. Swedish copper 6 parts, Malacca tin 1-tenacity per square inch 64,000 lbs. Chili copper 6 parts, Malacca tin 1......................................... 60,000 Japan copper 5 parts, Banca tin 1................................ 57,000 Anglesea copper 6 parts, Cornish tin 1.................................... 41,000 Common block tin 4, lead 1, zinc 1................................. 13,000 Malacca tin 4, regulus of antimony 1...................................... 12,000 Block tin 3, lead 1.................................. 10,200 Block tin 8, zinc 1......................................................... 10,000 Lead 1, zinc 1....................................... 4,500 TABLE of -Data, containing the Results of Experiments on the Elasticity and Strength of various Species of Timber. Species of Timber. Value of E. Value of S. Species of Timber. Value of E. Value of S. Teak............... 174'7 2462 Elm................. 5064 1013 Poona............. 122'26 2221 Pitch pine...... 88'68 1632 English oak....... 105 1672 Red pine........... 133 1341 Canadian do...... 155'5 1766 New England fir 158-5 1102 Dantzic do....... 86'2 1457 Riga fir............ 90 1100 Adriatic do...... 70'5 1383 Mar Forest do. 63 1200 Ash................. 119 2026 Larch.............. 76 900 Beech............ 98 1556 Norway spruce... 105.47 1474 RULE. — To find the dimensions of a beam capable of sustaining a given weight, with a given degree of deflection, when supported at both ends.-Multiply the weight to be supported in lbs. by the cube of the length in feet; divide the product by 32 times the tabular value of E, multiplied into the given deflection in inches, and the quotient is the breadth multiplied by the cube of the depth in inches. When the beam is intended to be square, then the fourth root of the quotient is the breadth and depth required. If the beam is to be cylindrical, multiply the quotient by 1-7, and the fourth root of the product is the diameter. The distance between the supports of a beam of Riga fir is 16 feet, and the weight it must be capable of sustaining in the micddle of its length is 8000 lbs., with a deflection of not more than 4 of an inch; what must be the depth of the beam, supposing the breadth 8 inches? 16 x 8000 90 x 32 x 75= 15175. 8 = - 1897 = 12.35 in. the depth. RULE.-To determine the absolute strength of a rectangular beamn of timber when supported at both ends, and loaded in the gmiddle of its length, as beams in general ought to be calculated to, so that they may be rendered capable of withstanding all accidental cases of emzergency.-Multiply the tabular value of S by four times the depth of the beam in inches, and by the area of the cross section in inches; divide the product by the distance between the supports Z 19 Page 290 290 THE PRACTICAL MODEL CALCULATOR. in inches, and the quotient will be the absolute strength of the beam in lbs. If the beam be not laid horizontally, the distance between the supports, for calculation, must be the horizontal distance. One-fourth of the weight obtained by the rule is the greatest weight that ought to be applied in practice as permanent load. If the load is to be applied at any other point than the middle, then the strength will be, as the product of the two distances is to the square of half the length of the beam between the supports; or, twice the distance from one end, multiplied by twice from the other, and divided by the whole length, equal the effective length of the beam. In a building 18 feet in width, an engine boiler of 5- tons is to be fixed, the centre of which to be 7 feet from the wall; and having two pieces of red pine 10 inches by 6, which I can lay across the two walls for the purpose of slinging it at each end,-may I with sufficient confidence apply them, so as to effect this object? 2240 x 5'5 2 = 6160 lbs. to carry at each end. 14 x 22 And 18 feet - 7 = 11, double each, or 14 and 22, then 18 =17 feet, or 204 inches, effective length of beam. 13,11 x 4 x 10 x 60 Tabular value of S, red pine = 204 = 15776 lbs., the absolute strength of each piece of timber at that point. RULE. — To determine the dimensions of a reetangular beam calapable of supporting a required weight, with a given degree of deflection, when fixed at one end. —Divide the weight to be supported, in Ibs., by the tabular value of E, multiplied by the breadth and deflection, both in inches; and the cube root of the quotient, multiplied by the length in feet, equal the depth required in inches. A beam of ash is intended to bear a load of 700 lbs. at its extremity; its length being 5 feet, its breadth 4 inches, and the deflection not to exceed -2 an inch. Tabular value of E = 119 x 4 x'5 = 238, the divisor; then 700. 238 = /2'94 x 5 = 7'25 inches, depth of the beam. RuLE.-To find the absolute strenyth of a reetangzclar beamc, wq'hen fixed at one end, and loaded at the other. —Multiply the value of S by the depth of the beam, and by the area of its section, both in inches; divide the product by the leverage in inches; and the quotient equal the absolute strength of the'beam in lbs. A beam of Riga fir, 12 inches by 4-, and projecting 6- feet from the wall; what is the greatest weight it will support at the extremity of its length? Tabular value of S = 1100 12 x 4'5 = 54 sectional area, 1100 x 12 x 54 Then, 78 = 9138'4 lbs. Page 291 STRENGTH OF MATERIALS. 291 When fracture of a beam is produced by vertical pressure, the fibres of the lower section of fracture are separated by extension, whilst at the same time those of the upper portion are destroyed by compression; hence exists a point in section where neither the one nor the other takes place, and which is distinguished as the point of neutral axis. Therefore, by the law of fracture thus established, and proper data of tenacity and compression given, as in the Table (p. 281), we are enabled to form metal beams of strongest section with the least possible material: thus, in cast iron the resistance to compression is nearly as 6~ to 1 of tenacity; consequently a beam of cast iron, to be of strongest section, must be of the form- TB, and a parabola in the direction of its length, the quantity of material in the bottom flange being about 6~ times that of the upper: but such is not the case with beams of timber; for although the tenacity of timber be on an average twice that of its resistance to compression, its flexibility is so great, that any considerable length of beam, where columns cannot be situated to its support, requires to be strengthened or trussed by iron rods, as in the following manner: And these applications of principle not only tend to diminish deflection, but the required purpose is also more effectively attained, and that by lighter pieces of timber. RULE. — To ascertain the absolute strength of a cast iron beam of the preceding form, or that of strongest section.-Multiply the sectional area of the bottom flange in inches by the depth of the beam in inches, and divide the product by the distance between the supports also in inches; and 514 times the quotient equal the absolute strength of the beam in cwts. The strongest form in which any given quantity of. matter can be disposed is that of a hollow cylinder; and it has been demonstrated that the maximum of strength is obtained in cast iron, when the thickness of the annulus or ring amounts to Ith of the cylinder's external diameter; the relative strength of a solid to that of a hollow cylinder being as the diameters of their sections. The following table show table shows the greatest weight that ever ouglht to be laid upon a beam for permanent load, and if there be any liability to jerks, &c., ample allowance must be made; also, the weight of the beam itself must be included. RULE.-To find the weight of a cast iron beam of gizen dienzsions.-Multiply the sectional area in inches by the length in feet, and by 3'2, the product equal the weight in lbs. Required the weight of a uniform rectangular beam of cast iron, 16 feet in length, 11 inches in breadth, and 1- inch in thickness. 11 x 1.5 x 16 x 3-2 = 8448 lbs. Page 292 292 THE PRACTICAL MODEL CALCULATOR. A TABLE showing the Weight or Pressure a Beam of Cast Iron, 1 inch in breadth, will sustain without destroying its elastic force, when it is supported at each end, and loaded in the middle of its length, and also the deflection in the middle which that weiqht will produce. Length. 6 feet. 7 feet. 8 feet. 9 feet. 10 feet. Depth Wt. in Defl. in Wt. in Defl. in Wt. in Defll. in Wt. in Def. in Wt. in Defl. in in in. lbs. in. lbs. in. lbs. in. lbs. in. lbs. in. 3 1278 *24 1089 -33 954.426 855.54 765 -66 31 1739.205 1482 *28 1298 *365 1164 *46 1041.57 4 2272.18 1936.245 1700 *32 1520.405 1360 -5 41 2875.16 2450 -217 2146.284 1924 -36 1721.443 5 3560 -144 3050 -196 2650 *256 2375 *32 2125.4 6 5112.12 4356.163 3816.213 3420 -27 3060 -33 7 6958.103 5929.14 5194.183 4655 *23 4165 -29 8 9088.09 7744 -123 6784.16 6080 -203 5440 -25 9 - - 9801.109 8586.142 7695.18 6885 *22 10 - - 12100.098 10600.128 9500.162 8500.2 11 - - - - 12826.117 11495.15 10285.18 12 - - - - 15264.107 13680.135 12240.17 13 _- - -- -- 16100.125 14400 -154 14 - - - - - - 18600.115 16700 j -143 12 feet. 14 feet. 16 feet. 18 feet. 20 feet. 6 2.548'48 2184'65 1912'85 1699 1 08 1530 1'34 7 3471'41 2975'58 2603'73 2314.93 2082 1-14 8 4532 *36 3884.49 3396 *64 3020.81 2720 1 00 9 5733'32 4914'44 4302'57 3825 -72 3438 *89 10 7083'28 6071'39 5312.51 4722 6-1 4250 *8 11 8570 *26 7346 -36 6428'47 5714'59 5142'73 12 10192 *24 8736'33 7648'43 6796'54 6120 *67 13 11971 *22 10260'31 8978'39 7980.49 7182.61 14 13883.21 11900 -28 10412.36 925. 46 8330.57 15 15937.19 13660 *26 11952.34 10624 -43 9562.53 16 18128 *18 15536 -24 13584.32 12080.40 10880 -5 17 20500.17 17500.23 15353.3 13647.38 12282'47 18 22932.16 19656.21 17208.28 15700 -36 13752.44 Resistance of Bodies to Flexure by Vertical Pressure.-When a piece of tintbeC is employed as a column or support, its tendency to yielding by compression is different according to the proportion between its length and area of its cross section; and supposing the form that of a cylinder whose length is less than seven or eight times its diameter, it is impossible to bend it by any force applied longitudinally, as it will be destroyed by splitting before that bending can take place; but when the length exceeds this, the column will bend under a certain load, and be ultimately destroyed by a similar kind of action to that which has place in the transverse strain. Columns of cast iron and of other bodies are also similarly circumstanced. When the length of a cast iron column with flat ends equals about thirty times its diameter, fracture will be produced wholly by bending of the material;-when of less length, fracture takes place partly by crushing and partly by bending: but, when the column Page 293 STRENGTH OF MATERIALS. 293 is enlarged in the middle of its length from one and a half to twice its diameter at the ends, by being cast hollow, the strength is greater by -th than in a solid column containing the same quantity of material. RULE.-To determine the dimensions of a support or column to bear without sensible curvature a given pressure in the direction of its axis.-Multiply the pressure to be supported in lbs. by the square of the column's length in feet, and divide the product by twenty times the tabular value of E; and the quotient will be equal to the breadth multiplied by the cube of the least thickness, both being expressed in inches. When the pillar or support is a square, its side will be the fourth root of the quotient. If the pillar or column be a cylinder, multiply the tabular value of E by 12, and the fourth root of the quotient equal the diameter. What should be the least dimensions of an oak support, to bear a weight of 2240 lbs. without sensible flexure, its breadth being 3 inches, and its length 5 feet? 2240 x 52 Tabular value of E = 105, and 20 x 105 x 3 8888 2'05 inches. Required' the side of a square piece of Riga fir, 9 feet in length, to bear a permanent weight of 6000 lbs. 6000 x 92 Tabular value of E = 96, and 20 x 96 = 4V/253 = 4 inches nearly. _Dimensions of Cylindrical Columns of Cast Iron to sustain a given load or pressure with safety. Length or height in feet. 1 61 8 10 12 1 14 1 16 18 20 22 24 | I Weight or load in cwts. 2 7 60 49 40 32 26 22 18 15 13 11 |2- 119 105 91 77 65 55 47 40 34 29 25 3 178 163 145 128 111 97 84 73 64 56 49 31~ 247 232 21.4 191 172 156 135 119 106 94 83 4 326 310 288 266 242 220 198 178 160 144 130 41 418 400 379 354 327 301 275 251 229 208 189 5 522 501 479 452 427 394 365 337 310 285 262 6 607 592 573 550 525 497 469 440 413 386 360 I 7 1032 1013 989 959 924 887 848 808 765 725 686 8 1333 1315 1289 1259 1224 1185 1142 1097 105o2 1005 959 9 1716 1697 1672 1640 1603 1561 1515 1467 1416 1364 1311 10 2119 2100 2077 2045 2007 1964 1916 1865 1811 1755 1697 11 2570 2550 2520 2490 2450 2410 2358 2305 2248 2189 2127 12 30501 3040i 3020 2970 2930 2900 2830 27 80 2730 2670 2600 Practical utility of the preceding Table.-Wanting to support the front of a building with cast iron columns 18 feet in length, 8 inches in diameter, and the metal 1 inch in thickness; what weight may z2 Page 294 294 TIIE PRACTICAL MODEL CALCULATOR. I confidently expect each column capable of supporting without tendency to deflection? Opposite 8 inches diameter and under 18 feet 1097 Also opposite 6 in. diameter and under 18 feet = 440 = 657 cwts. The strength of cast iron as a column being = 1'0000 - steel -= 2518 wrought iron - = 1745 -- oak (Dantzic) -= 1088 red deal - = *0785 -Elasticity of torsion, or resistance of bodies to twisting. —The angle of flexure by torsion is as the length and extensibility of the body directly, and inversely as the diameter; hence, the length of a bar or shaft being given, the power, and the leverage the power acts with, being known, and also the number of degrees of torsion that will not affect the action of the machine, to determine the diameter in cast iron with a given angle of flexure. RUTLE.-Multiply the power in lbs. by the length of the shaft in feet, and by the leverage in feet; divide the product by fifty-five times the number of degrees in the angle of torsion, and the fourth root of the quotient equal the shaft's diameter in inches. Required the diameters for a series of shafts 35 feet in length, and to transmit a power equal to 1245 lbs., acting at the circumference of a wheel 21 feet radius, so that the twist of the shafts on the application of the power may not exceed one degree. 1245 x 35 x 2.5 1245 x 15 x 4v/1981 = 6.67 inches in diameter. 55 x I Relative strength of metals to resist torsion. Cast iron............ = 1 Swedish bar iron.... 1'05 Copper............... = 48 English do......= 1'12 Yellow brass........ = 511 Shear steel...........= 196 Gun-metal.......... = 55 Cast do.............= 2'1 DEFLEXION OF RECTANGULAR BEAMIS. RULE. —To ascertain the amount of deflexion of a uniform beam of cast iron, supported at both ends, and loaded in the middle to tlhe extent of its elastic force.-Multiply the square of the length in feet by'02, and the product divided by the depth in inches equal the defiexion. Required the deflection of a cast iron beam 18 feet long between the supports, 12'8 inches deep, 2'56 inches in breadth, and bearing a weight of 20,000 lbs. in the middle of its length. 182 x'02 12.8 ='506 inches from a straight line in the middle. For beams of a similar description, loaded uniformly, the rule is the same, only multiply by'025 in place of'02. RULE.-To find the deflection of a beam when fixed at one enld Page 295 STRENGTH OF MATERIALS. 295 and loaded at the other.-Divide the length in feet of the fixed part of the beam by the length in feet of the part which yields to the force, and add 1 to the quotient; then multiply the square of the length in feet by the quotient so increased, and also by'13; divide this product by the middle depth in inches, and the quotient will be the deflection, in inches also. Multiply the deflection so obtained for cast iron by'S6, the product equal the deflection for wrought iron; for oak, multiply by 2'8; and for fir, 2'4. A TABLE of the Deprths of Square Beams or Blars of Ca.st Iron, calculated to sulpplort firom 1 Cwt. to 14 Tons in the AIriddle, the Deflection not to exceed jth of an ]nch for each, Foot in LCength. l.14 16118 20122 24228 0 } Lengths in Feet IS 20 22 30 Weightin Wdi:. d cwt. lbs. In. Tn. In. In. In. In. In. In. In. In. I n. In. In. I 1 Cwt. 112 12 1'4 17 1'9 2'0 22 2'4 2'5 2'6 2-7 29'90 "-1 2 124 1.4 17. 2 0 2.2 2.4 2.6 28S 3.0 3a1 3.3 3.4 3.6 3vi 3 336 16 19 2-2 241 2j 2-9 3-1 3-3 3-4 3-6 3-8 3-9 4-1 4.2 4 448 1.7 2.0 2.4 2.6 2.9 31 3'3 3-5 7 339 40 42 4 19 4 5 5C60 iS 2-2 2-5 2-8 3 0 3 3 3-5 3-7 3-9 4-1 4.3 4.4 46 4 6 672 1'8 2'2 2'6 2'9 "12 3'4 3'7 309 4'1 43 4'5 4'6 48 50 7 784 1'9 2'3 2'7 3'0 33 3'6 3'8 4'1 4.2 4'4 4.6 4.8 50 2 8 896 2,0 2.4 2.8 3'1 34 3. 7 3'9 4.2 4'4 4.6 4'8 5'0.21 5.4 9 1,008 2.0 2.5 2-9 3.2 8-5 3'8 4'0 43 4.5 407 4.9 5. 1 5'3.5 10 1,120 2'1 2'6 30 3'3 6' 3'9 4'2 4'4 4'7 4-9 5'2 5'3 5'4{s 11 1,232 241 2'6 3.0 3l4 3.7 4'0 4'3 4'5 4.8 50 5'3 5'4 5.6 51 12 1,344 2'2 2-7 3'1'5 3-8 4'1 4'4 4.7 4'9 5'1 5'a3 5.5sl'7 13 1,456 2'2 27 3'1 3'5 3-8 4 4 4' 4 4.7 4.9 5'2 5'4 56 5'9 6' 0 14 1,568 2'3 2'8 3'2 3'6 3.9 4'2 4'5 4'8 50 5'3 5.5 5.7 6'0 6I 1 15 1.680 23 2.8 32 3-6 4.0 4'3 4'6 4'9 5.2 5'4 5.6 5.8 6'1 62 16 1,792 2.4 2.9 3.3 3.7 4.0 4'4 4.7 5.0 5.2 5.5 5.7 5.9 6.2 64 17 1,904 2'4 2'9 3'4 3'8 41 4'4 4'7 5.0 5'3 5'5 5'8 6'0 6'2 65 18 2,016 2.4 3'0 3.4 3'8 4.2 4'5 4'8 5.1 5.4 5'6 5.9 6.1 6'4 G6 19 2,128 2'5 3'0 3'5 3'9 4'2 4'6 4'9 5.2 5.4 57 60 6' 2 65 61 1 ton. 2,240 2'5 3'0 3'5 3'9 4.3 456 4'9 5'2 5.5 5'8 6'0 6-3 6'5 68 1I 2,800 2'6 3.2 3'7 4'1 4'5 4'9 5'2 5.5 5.8 6'1 6.4 6'6;' 9 14 3,920 2-91 1! 3,360/ 2'8/ 3'4/ 3'9 4'3 47 53' / ~'5 5-5 e I 6' 1 64 67 7.0 7)' 7 u 3,920 2'9 35 4'0 4'5 4,9 5'3 5.7 6o0 63 67 6-9 7.2 7. 5 7 2 4,-80 291 35 4'1 4'7 5:1 5.5 56 9 6.2 6'5 68 7'2 7'6 77 80 2' 5,600 3'1 38 44 4'9 5'5 58 6' 2 66 69 73 7'6 7'9 8'2 3 6,720 3-3 4.0 406 51 5.7 641 65i 6'9 7'3 7'6 7'9 8'3 8'6 1 9 3k 7,540 3[ 4 4-1 4.8 5,3 5' 8 6c3 6.7 7.1 7 79 82 86 8-9 9 4 8o960 3'5 4'3 4'9 55 6'0 6'5 7' 0 7.4 7. 8s 2 85 s89.9 4 10.80 -/ 4.4 5'1 5'7 6'2 6'7 7'2 7'6 SOI 8-4 8'8 9'1 S1 9'5 5 11;200 ]- 52 5.8 6'4 6'9 7'4 7'8 5.2 86 90 9'4 9'7 101 6 13,440 6 - 1 64 6. 7 6 72 7'7 8' 2 8' 6 9. 0 9'4 98101,2 10-5 7 15,680 - - 5~7 6-3 6-9 7-5 8-0 ssI 8-9 9-4 9-8 10-2 10 6 1119 8 17,920 - - 95 6-6 7278 8 8 9033 i89-3 19 10.1 10.6 1069 113 9 20,160 - - 6-0 6-8 7-4 8-0 So5 900 9.5 10-0 10-4 10-9 113 1137 10 22,400 69-/ 76 8'2 88 9'3 98 10'3 107{ 11'2 11'6 12 0 11 24,640 71 7'8 8'4 9'0 9'5 10'0 1015 I [11'0 11' 119 312.3 12 26,880 7i I 2 79 8-6 92 9'7 10'2 108 11'2 11'7 12'1 12'5 1: 293,120 7' 4 8'1 8'8 9'4 9 19 10'4 11'0 11i 11'*9 t 4.1.483, 14 31,360 7*3 8-3 8-9 9-5 10-1 10-6 1111 1-7 12 126 110 Deflection in inches.1- -151 - -25 -3 35 - -4 - — i 4 —7 —. o -5; Lengths inFeet 10 12 14 16 18 20 212 24 261 28 0 32 84 36 15 I 33,600 87 1 97 90- 3 1 10.8 11411 1919 3l12,' 13'2 137 14' 1 14'1 16 35,840 7'8 8,5 92 98 i 104 11.0 115 12.0 12 13' 0 13'5 13'9 1143 14-7 17 38,080 7o9 8-7 9.4 10.0 10 i6 1127 11'7 12' 137 137 14 5 14 9 1S 40.120 80 808 9- 101 10 119 12' 4 1219 13'4 1309 114'311;7 15' 1 19 42,560 81 8'9 96 103 109 1 115 122 12:6 15'1 1 46 141 5 154 0 15'4 20 44,800 - 90 97 10'4 11'0 11' 125 127 1:32 8 14'2 14'7 15'1 15'6 22 49,280 - 92 100 1017 1311' 9 12' 3 13'0 136 14'1 14'6 l15' 1 l155 15'9 26 53,760 - 94 102 10'9 11.5 2 13034 139'1" 14' 4 14' 9 1541 15'9 16'3 26 58,240 - 9-6 10-4 11-1 11-8 12'4 13-3 13-6 14-2 1- I711.512 15-7 10)2 l G-7 28 62,720 - 98 10'6 1154 12'0 12'7 135 39 1441' lo0 So- 16 0 16- 11 0 DeflsS ion'a i__ches_ ) _. i; T6 j 77 ) 87 -o i,~,.,~,.~o,,~n,~n~ %7 — ~.-3.` -4.4o.5 6.7,.. —8i5.9 Page 296 296 THE PRACTICAL MODEL CALCULATOR. _tons. _____ _bs_ 2___ _ ____ __ Weight in Weigh t i= A A~ -A A_ _ A_ _ A __ In. In. In. In. In. In. In In In. In. In. In. In. In. 30 67,200 10-8 11-5 12.2 12-9 13-5 141 147 152 15:. 7 163 1168 17' 3 1t-7 18 2 32 71,680 110 111-7 124 13-1 13- 7 143 149 15-5 16-0 16-5 17'0 17-lS 1SO 18-5! 34 76,160 111 119 12'6 13-3 13-9 4-5 15-1 15-7 16-2 16-8 17 3 17.8 18.3 18'8 36 80,640 11-3 12-0 12-8 13-4 14-1 14'7 15-3 15-9 16-5 170 1735 18-0 1835 19'0 38 85,120 11-4 122 130 13'6 14 149 158- 16'1 16-7 172 17 18-3 18-8 19-3 40 89,600 - 12-4 13'1 13-8 145 151 157 16 1619 17 5 18 0 18-5 19-1 19a5 42 94,080 - 12-5 13-3 14-0 14-7 153 159 16-5 17-1 17-7 18 2 187 193 19' S 44 98,560 - 12f7 13-5 14-2 14'9 15-5 161 168 17.4 1709 185 19'0 195 200 i 46 103,040 - 12-S 136 14'3 15-0 15 16-3 170 17-6 1 18-1 18- 192 ) 1S 203 48 107,520 - 130-0 17 114-5 15-2 15-9 165 171 177 183 18-8 19'4 i 00 205 50 112,000 - - 13'8 146 15-3 16.0 16-6 17-3 17- 18' 19' 0 19' 6 20-1 20(7 52 116,480 - - 14'0 14-7 15'5 16-2 169' 17'5 18' 18-7 19-2 19i 20'3 21 0 54 120,960 - - 14-1 14-9 15-7 16-3 170 176 182 18 10194 19-9 205 21'1 56 125,440 - - 14-3 15-0 15-S 16-5 17-1 17- 118-4 190 19-6 20 1 20 7 21' 58 129,920 - - 14'4 15-1 15-9 16'6 17]3 17-9 185 19-2 19-, 20'3 20-9 21 4 60 134,400 - - 14'5 15'3 16'0 516-7 17 4 181 18, 7 193 19 ( 0) 5 21X l 21 - SDeflectionfin inches | j.*4'5'5 *55 | *6j.6a | 7 * 7 1.' 7 85 5 | 2 1 0'6 5 11 Examples illustrative of the Table. —1. To find the depth of a rectangular bar of cast iron to support a weight of 10 tons in the middle of its length, the deflection not to exceed o of an inch per foot in length, and its length 20 feet, also let the depth be 6 times the breadth. Opposite 6 times the weight and under 20 feet in length is 15'3 inches, the depth, and 1 of 153 - 2-6 inches, the breadth. 2. To find the diameter for a cast iron shaft or solid cylinder that will bear a given pressure, the flexure in the middle not to exceed -06th of an inch for each foot of its length, the distance of the bearings being 20 feet, and the pressure on the middle equals 10 tons. Constant multiplier 1'7 for round shafts, then 10 x 1'7 = -17. And opposite 17 tons and under 20 feet is 11'2 inches for the diameter. But half that flexure is quite enough for revolving shafts: hence 17 x 2 = 34 tons, and opposite 34 tons is 13'3 inches for the diameter. 3. A body 256 lbs. weight, presses against its horizontal support, so that it requires the force of 52 lbs. to overcome its friction; if the body be increased to 8750 lbs., what force will cause it to pass from a state of rest to one of motion? 52 256 - 203125 = -, in this case, the coefficient offriction;. 8750 x 203125 = 1777-34375 lbs., the force required. This calculation is based upon the law, that friction is proportional to the normal pressure between the rubbing surfaces. Twice the pressure gives twice the friction; three times the pressure gives three times the friction; and so on. With light pressures, this law may not hold, but then it is to be attributed to the proportionately greater effect of adhesion. 4. If a sleigh, weighing 250 lbs., requires a force of 28 lbs. to draw it along; when 1120 lbs. are placed in it, required the units of work expended to move the whole 350 feet? Page 297 STRENGTH OF MATERIALS. 297 28 250 -'112, the coefficient of friction. Then (1120 + 250) x'112 = 153'44 lbs., the force required to move the whole..'. 153'44 x 350 = 53704, the units of work required. A UNIT OF WORK is the labour which is equal to that of raising one pound a foot high. It is supposed that a horse can perform 33000 units of work in a minute. It may also be remarked that friction is independent of the extent of the surfaces in contact, except with trifling pressures and large surfaces, which is on account of the effect of adhesion. The friction of motion is independent of velocity, and is generally less than that of quiescence. A 5. Required the coefficient of friction, for a sliding motion, of castironuponwrought, n n lubricated with Dev- r s lin's oil, and under the following circumstances: the load A, and sledge nm, weighs 8420 lbs., and requires a weight W, of 1200 lbs. to cause it to pass from a state of rest into one of motion: the sledge and load pass over 22 feet on the horizontal way rs, in 8 seconds. In this case the coefficient of sliding motion will be 1200 1200 + 8420 2 x 22 8420 8420 X g x 82' in which g = 32'2 feet; the acceleration of the free descent of bodies brought about by gravity. The above expression becomes 44 142515- 1'142515 x 20608 =-118121. Hence the coefficient of the friction of motion is'118121, and the coefficient of the friction of quiescence is'142515. OF FRICTION, OR RESISTANCE TO MOTION IN BODIES ROLLING OR RUBBING ON EACH OTHER. In the years 1831, 1832, and 1833, a very extensive set of experiments were made at Metz, by M. Morin, under the sanction of the French government, to determine as nearly as possible the laws of friction; and by which the following were fully established: 1. When no unguent is interposed, the friction of any two surfaces (whether of quiescence or of motion) is directly proportional to the force with which they are pressed perpendicularly together; so that for any two given surfaces of contact there is a constant ratio of the friction to the perpendicular pressure of the one surface upon the other. Whilst this ratio is thus the same for the same Page 298 298 THE PRACTICAL MODEL CALCULATOR. surfaces of contact, it is different for different surfaces of contact. The particular value of it in respect to any two given surfaces of contact is called the coefficient of friction in respect to those surfaces. 2. When no unguent is interposed, the amount of the friction is, in every case, wholly independent of the extent of the surfaces of contact; so that, the force with which two surfaces are pressed together being the same, their friction is the same, whatever may be the extent of their surfaces of contact. 3. That the friction of motion is wholly independent of the velocity of the motion. 4. That where unguents are interposed, the coefficient of friction depends upon the nature of the unguent, and upon the greater or less abundance of the supply. In respect to the supply of the unguent, there are two extreme cases, that in which the surfaces of contact are but slightly rubbed with the unctuous matter, as, for instance, with an oiled or greasy cloth, and that in which a continuous stratum of unguent remains continually interposed between the moving surfaces; and in this state the amount of friction is found to be dependent rather upon the nature of the unguent than upon that of the surfaces of contact. MI. Morin found that with unguents (hog's lard and olive oil) interposed in a continuous stratum between surfaces of wood on metal, wood on wood, metal on wood, and metal on metal, when in motion, have all of them very near the same coefficient of friction, being in all cases included between ~07 and'08. The coefficient for the unguent tallow is the same, except in that of metals upon metals. This unguent appears to be less suited for metallic substances than the others, and gives for the mean value of its coefficient, under the same circumstances,'10. Hence, it is evident, that where the extent of the surface sustaining a given pressure is so great as to make the pressure less than that which corresponds to a state of perfect separation, this greater extent of surface tends to increase the friction by reason of that adhesiveness of the unguent, dependent upon its greater or less viscosity, whose effect is proportional to the extent of the surfaces between which it is interposed. It was found, from a mean of experiments with different unguents on axles, in motion and under different pressures, that, with the unguent tallow, under a pressure of from 1 to 5 cwt., the friction did not exceed -3th of the whole pressure; when soft soap was applied, it became -1-th; and with the softer unguents applied, such as oil, hog's lard, &c., the ratio of the friction to the pressure increased; but with the harder unguents, as soft soap, tallow, and anti-attrition composition, the friction considerably diminished; consequently, to render an unguent of proper efficiency, the nature of the unguent must be measured by the pressure or weight tending to force the surfaces together. Page 299 STRENGTI OF MIATERIALS. 299 TABLE of thle Results of Experinments on t]he Friction of Unctuous Surfaces. By M. MoRIN. Coefliicierts of Friction. Surfaces of Contact. Friction of Friction of Motion. Quiescence. Oak upon oak, the fibres being parallel to the motion 0'018 I 0 390 Ditto, the fibres of the moving body being perpendicular to the motion.......................... 01............3 0 314 Oak upon elm, fibres parallel................................ 0t-16 Elm upon oak, do........................................ 0.119 0'420 Beech upon oak, do........................................ 0330 Elm upon elm, do........................................ 0-140 Wrought iron upon elm, do......1...................... 0138 Ditto upon wrought iron, do................................. 0'177 Ditto upon cast iron, do....................................... 0-118 Cast iron upon wrought iron, do..................1...... 0 Wrought iron upon brass, do............................... 0160 Brass upon wrought iron, do........................1.....c Cast iron upon oak, do..................,...................... 0107 0'100 Ditto upon elm, do., the unguent being tallow....... 0.125 Ditto, do., the unguent being hog's lard and black lead.............................................................. 0'137 Elm upon cast iron............................................. 0135 0098 Cast iron upon cast iron....................................... 0-144 Ditto upon brass................................................. 0132 Brass upon cast iron...................................... 0-107 Ditto upon brass................................................. 0-134 0-164 Copper upon oak............................0........ -.100 Yellow copper upon cast iron....................... 0.115 Leather (ox-hide), well tanned, upon cast iron, wetted 0-o29 02'67 Ditto upon brass, wetted................................ 0-244 In these experiments, the surfaces, after having been smeared with an unguent, were wiped, so that no interposing layer of the unguent prevented intimate contact. TAB3LE of tihe Results of Experiments on Friction, witt Ungueents interposed. By M. MORIN. Coefficients of Friction. Surfaces of Contact. Friction of Friction of Unguents. Motion. Quiescence. Oak upon oak, fibres parallel.... 0-164 0-440 Dry soap. Do. do................... 0-075 0-164 Tallow. Do. do.................. 0-067... Hog's lard. Do., fibres perpendicular.......... 0-083 0-254 Tallow. Do. do.................. 0-072... Hog's lard. Do. do................... 0250... Water. Do. upon elm, fibres parallel..... 0-136... Dry soap. Do. do.................. 0-073 0-178 Tallow. Do. do.................. 0-066 Hog's lard. Do. upon cast iron.................. 0-080... Tallow. Do. upon wrought iron............ 0-098... Tallow. Beech upon oak, fibres parallel.. 0-055... Tallow. Elm upon oak, do.................. 0-137 0-411 Dry soap. Do. do.................. 0-170 0-142 Tallow. Do. do.................. 0-060... Hog's lard. Elm upon elm, do.................. 0-139 0-217 Dry soap. Do. upon cast iron.................. 0-066... Tallow. Wrought iron upon oak, fibres 0256 0-649 Gaeased and satuparallel........................... rGreased and satuparallel I......rated with water. Do. do.................. 0214 l)ry soap. Page 300 300 TIIE PRACTICAL MODEL CALCULATOR. Coefficients of Friction. Surfaces of Contact. Friction of Friction of Unguents. lotion. Quiescence. Wrought iron upon oak, fibres 0085 0108 Tallow. parallel........................... Do. upon elm, do.................. 0078Tallow. Do. do.................. 0-076... Hog's lard. Do. do.................. 05Olie oil. Do. upon east iron, do........... 0103... Tallow. Do. do.................. 0 076 I.. t Hog's lard. Do. do.................. 0066 0100 Olive oil. Do. upon wrought iron, do........ 0... Tallow. Do. do0081.................. 0 Hog's lard. Do. do................. 0070 0115 Olive oil. Wrought iron upon brass, do 103..... 0103.. Tallow. Do. do.................. Hog's lard. Do. do.................. 0078... Olive oil. Cast iron upon oak, do........ 0189... Dry sop. Do. do............ 0-218 0 G-646 Greased and satut rated with water. D)o. do.............. 0 078 0 100 Tallow. Do. do.................. 0075 o'ogs lard. Do. do.................. 0075 0100 Olive oil. Do. upon elm, do............. 0077... Tallow. Do. do............ 0061... Olive oil. ~~~~Do.~~~ d~~~~~o.0~~L~091Io { Grlard and Do. do.................. 0.01 0.1Tllo. Do. upon wrought iron............... 0100 Tallo w. Do. upon cast iron.. 0-314... Water. Do. (1o.0.................. 01975.' Soap. Do. do.................. 0100 0'100 Tallow. Do. do.0070 0100 Hog's lard. Do. do..................... 04... Olive oil. Do. do................. 005... { Hog's lard and tplumbago. Do. upon brass.................. 0103... Tallow. Do. o........................... 0Hog's lared. Do. do...................... 078 Olive oil. Copper upon oak, fibres parallel 0-069 0-100 Tallow. Yellow copper upon cast iron.... 0.072 0.103 TalloT w. Do. do........ 0068... Hogr's lard. Do. 0 066... Olive oil. Brass upon cast iron.....0.. 086 0106 Tallow. Do. do................ 0... Olie oil. Do. upon wrought iron............ 0081... Tallow. Do. do..................... 0'089 Lard and plumD d08 mbago. Do. do......... 0-072... Olive oil. Brass upon brass................... 0058... Olive oil. Steel upon cast iron.............. 0105 0'108 Tallow. Do. do...................... 008 1 Hog's lard. Do. do.......................... 0079... Olive oil. Do. upon wrought iron............. 0'093... Tallow. Do. do.......0... 0w076... Hllo s lard. Do. upon brass................... 0056... Tallow. Do. do.................. 0053... Olive oil. Do. do.0Lard and plum-0G7 i Tanned ox-hide upon cast iron.... 0 36.5... e asedl anl satusure as to cause them to be separated from one another througlhout by an interposed stratum of the unguent. Page 301 STRENGTH OF MATERIALS. 301 TABLE of the Results of Experiments on the Friction of Gudgeons or Axle-ends, in motion upon their bearings. By M. MORIN. Surfaces in Contact. State of the Surfaces. Coefficient of Friction. f Coated with oil of olives,) with hog's lard, tallow, 0'07 to 0-08 Cast iron axles in and soft gome........... cast iron bearings. I. With the same and water... 0.08 Coated with asphaltum..... 0'054 Greasy........................... 0-14 Greasy and wetted.. 0'14 F Coated with oil of olives, ) with hog's lard, tallow, 007 to 0'08 Cast iron axles in I and soft gome........... cast iron bearings. i Greasy.......................... 0'16 Greasy and damped......... 0'16 Scarcely greasy............. 0*19 Wrought iron axles Coated with oil of olives, in cast iron bear- tallow, hog's lard, or 007 to 0'08 ings. soft gome................ Coated with oil of olives, 0 07 to 0 08 Wrought iron axles Ihog's lard, or tallow, J in brass bearings. Coated with hard gome. 009 Greasy and wetted.......... 0'19 Scarcely greasy............... 0-25 Iron axles in lignum Coated with oil or hog's 0.11 lard........................ I vitra bearings. Greasy.......................... 0'19 Brass axles in brass f Coated with oil............... 0-10 bearings. With hog's lard............... 0 09 TABLE of Coefficients of Friction under Pressures increased continually up to limits of Abrasion. Coefficients of Friction. Pressure per Square Inch. Wrought Iron upon Wrought Iron upon Steel upon Cast Brass upon Cast Wrought Iron. Cast Iron. Iron. Iron. 3251lbs.. 140. 174. 166 157 1-66 cwts..250. 275. 300. 225 2.00. 271. 292.333 *219 2'33' 285' 321' 340' 214 2.66 -297. 329.344. 211 3 00' 312' 333' 347. 215 3.33' 350' 351' 351. 206 3'66 -376' 353.353. 205 4'00' 395' 365 -354 *208 4'33. 403' 366' 356 *221 4-66' 409' 366.357 *223 5'00......'367 *358. 233 5'33......'367' 359. 234 5-66......'367 *367. 235 6'00......'376' 403. 233 6'33.......434...... *234 6'66............ -235 7 00.................. *232 7.33 *273 A7-33...................273 2 A Page 302 802 TIHE PRACTICAL MIODEL CALCULATOR. Comparative friction of steam engines of different modifications, if the beam engine be taken as the standard of comparison:The vibrating engine..................has a gain of 1 1 per cent. The direct-action engine, with slides - loss of 1-8 Ditto, with rollers..................... - gain of 0 - Ditto, with a parallel motion......... - gain of 1'3 Excessive allowance for friction has hitherto been made in calculating the effective power of engines in general; as it is found practically, by experiments, that, where the pressure upon the piston is about 12 lbs. per square inch, the friction does not amount to more than 14t lbs.; and also that, by experiments with an indicator on an engine of 50 horse power, the whole amount of friction did not exceed 5 horse power, or one-tenth of the whole power of the engine. RECENT EXPERIMENTS MADE BY MI. MORIN ON TIIE STIFFNESS OF ROPES, OR TItE RESISTANCE OF ROPES TO BENDING UPON A CIRCULAR ARC. The experiments upon which the rules and table following are founded were made by Coulomb, with an apparatus the invention of Amonton, and Coulomb himself deduced from them the following results:1. That the resistance to bending could be represented by an expression consisting of two terms, the one constant for each rope and each roller, which we shall designate by the letter A, and which this philosopher named the natural stiffness, because it depends on the mode of fabrication of the. rope, and the degree of tension of its yarns and strands; the other, proportional to the tension, T, of the end of the rope which is being bent, and which is expressed by the product, BT, in which B is also a number constant for each rope and each roller. 2. That the resistance to bending varied inversely as the diameter of the roller. Thus the complete resistance is represented by the expression A + BT where D represents the diameter of the roller. Coulomb supposed that for tarred ropes the stiffness was proportional to the number of yarns, and I. Navier inferred, firom examination of Coulomb's experiments, that the coefficients A and B were proportional to a certain power of the diameter, which depended on the extent to which the cords were worn. M[. Morin, however, deems this hypothesis inadmissible, and the following is an extract from his new work, " Lemons de ilecanicue Pratique," December, 1846:" To extend the results of the experiments of Coulomb to ropes of different diameters from those which had been experimented upon, M. Navier has allowed, very explicitly, what Coulomb had but surmised: that the coefficients, A, were proportional to a cer Page 303 STRENGTH OF MATERIALS. 303 tain power of the diameter, which depended on the state of wear of the ropes; but this supposition appears to us neither borne out, nor even admissible, for it would lead to this consequence, that a worn rope of a metre diameter would have the same stiffness as a new rope, which is evidently wrong; and, besides, the comparison alone of the values of A and B shows that the power to which the diameter should be raised would not be the same for the two terms of the resistance." Since, then, the form proposed by AM. Navier for the expression of the resistance of ropes to bending cannot be admitted, it is necessary to search for another, and it appears natural to try if the factors A and B cannot be expressed for white ropes, simply according to the number of yarns in the ropes, as Coulomb has inferred for tarred ropes. Now, dividing the values of A, obtained for each rope by MI. Navier, by the number of yarns, we find for A 9, = 30 d = 0m'200 A = 0'222460 - 0 0074153. A 15 = d = 0-m1eg4 A 0'063514 -A 0*0042343. = 6 l = 0m0088 A = 0010604- = 00017673 It is seen from this that the number A is not simply proportional to the number of yarns. A Comparing, then, the values of the ratio A- corresponding to the three ropes, we find the following results:Iifferences of Valles of Differenceo of tile alues of Number of A Differcnces of the numbers of the values of for each yarlrns. yaros. A for each v-rn of difference. 30 0 0074153 From30to 15. 15 yarns 0.00~1810 0.0001 15 0 0042343 - 15 to 6. 9 - 0.0024770 0.000272 6 0.0017673 - 30 to 6. 24 - 0.0056400 0 000252_ Mean difference per yarn, 0 0002_45 It follows, from the above, that the values of A, given by the experiments, will be represented with sufficient exactness for all practical purposes by the formula A = n [0'0017673 + 0'000245 (n - 6)]. = n [00002973 + 0.000245 It]. An expression relating only to dry white ropes, such as were used by Coulomb in his experiments. WVith regard to the number B, it appears to be proportional to the number of yarns, for we find for Page 304 304 TIIE PRACTICAL MODEL CALCULATOR. n = 30 d -= 0'0200 B = 0'009738 - = 0'0003246 n = 15 d = 0m0144 B = 0'005518 B- - 00003678 n n = 6 d = -Om0088 B - 0'002380 - = 0'0003967 Mean.............. 0'0003630 Whence B = 0'000363 n. Consequently, the results of the experiments of Coulomb on dry white ropes will be represented with sufficient exactness for practical.purposes by the formula K = n [0'000297 + 0'000245 n + 0'000363 T] kil. which will give the resistance to bending upon a drum of a metre in diameter, or by the formula R = 1) [0000297 + 0'000245 n + 0'000363 T] kil. for a drum of diameter D metres. These formulas, transformed into the American scale of weights and measures, become R = n [0'0021508 + 0'0017724 n + 0'00119096 T] lbs. for a drum of a foot in diameter, and R =D [0'0021508 + 0'0017724 n + 0.00119096 T] lbs. for a drum of diameter D feet. With respect to worn ropes, the rule given by M. Navier cannot be admitted, as we have shown above, because it would give for the stiffness of a rope of a diameter equal to unity the same stiffness as for a new rope. The experiments of Coulomb on worn ropes not being sufficiently complete, and not furnishing any precise data, it is not possible, without new researches, to give a rule for calculating the stiffness of these ropes. TARRED ROPES. In reducing the results of the experiments of Coulomb on tarred ropes, as we have done for white ropes, we find the following values:n = 30 yarns A 0'34982 B = 0 0125605 n = 15 - A 0106003 B = 0'006037 n - 6 - A = 0'0212012 B = 0'0025997 which differ very slightly from those which M. Navier has given. But, if we look for the resistance corresponding to each yarn, we find Page 305 STRENGTH OF MATERIALS. 305 A B n = 30 yarns - 0 0116603 - = 0000418683 n it n = 15 - - = 0'0070662 -= 0'000402466 n 7n A B n = 6 -- 00035335 - = 0'000433283 n n Mean............0000418144 We see by this that the value of B is for tarred ropes, as for white ropes, sensibly proportional to the number of yarns, but it is not so for that of A, as M. Navier has supposed. Comparing, as we have done for white ropes, the values of - n corresponding to the three ropes of 30, 15, and 6 yarns, we obtain the following results:Differences of Values of Differences of the values of Number of A Differences of the number of the values of for each yarns. -. yarns. A fr a so7~~~~~~'n yarn of difference. 30 0.0116603 From 30 to 15. 15 yarns 0-0045941 0-000306 15 0.0070662 - 15 to 6. 9 - 0-0035327 0-000392 6 0.0035335 - 60 to 6. 25 - 0'0081268 0'000339 [Mean................. 0.000346 It follows from this that the value of A can be represented by the formula A = n [0'0035335 + 0'000346 (n - 6)] = n [0'0014575 + 0'000346 n.] and the whole resistance on a roller of diameter D metres, by R = j [0'0014575 + 0'000346 n + 0'000418144 T] kil. Transforming this expression to the American scale of weights and measures, we have R = n [0'01054412 + 0'00250309 n + 0'001371889 T] lbs. for the resistance on a roller of diameter D feet. This expression is exactly of the same form as that which relates to white ropes, and shows that the stiffness of tarred ropes is a little greater than that of new white ropes. In the following table, the diameters corresponding to the different numbers of yarns are calculated from the data of Coulomb, by the formulas, d cent. = V0'1338 n for dry white ropes, and d cent. = O0.186 n for tarred ropes, which, reduced to the American scale, become d inches = v'0O020739 n for dry white ropes, and d inches = /0'02883 for tarred ropes. 2 A 2 20 Page 306 306 THE PRACTICAL MODEL CALCULATOR. NOTE. -The diameter of the rope is to be included in D; thus, with an inch rope passing round a pulley, 8 inches in diameter in the groove, the diameter of the roller is to be considered as 9 inches. Dry White Ropes. Tarred Ropes. Value f the natural Value of the stiff- V alue of the stiffDiameter. ness proportional Diameter. ness eroportional ~ Dimtr sifesA aueo h sifstiffness, A. stiffn s, A. fts to the tension, B. to the tension B ft. lbs. ft. lbs. 6 0-0293 0'0767120 0'00714157 0'0347 0'153376 0'00823133 9 0.0360 0'1629234 0'0107186 001425 0-297647 0'01234700 12 0-0416 0-2810384 0'0142915 0'0490 0'486976 0-01 C46267 I5 0'0465 0'4310571 0'0178641 0-0548 0'721357 0-020578:34 18 0'0509 0'6129795 0-0214373 0'0600 0-000795 0'02469'00 21 0-0550 0'826805 1 0-0250102 0'0618 1'325289 0 0'2S09; 7 24 0'0588 1.0725350 0'0285831 0'0693 1'634839 0'032925j3 27 0'0622 13.501682 0'0321559 0-0735 2'109444 0'03704100 30 0-0657 106597051 0'0357288 0-0775 2-569105 0'04115-6;7 33 0-0689 2-0011455 090393017 0'0813 3-073821 00 -05272`3' 36 00720 2-3744897 0'0428746 008419 3'623593 0-04938S00 39 0-0749 2-7797375 0-0164475 6-0884 4'218416 0-053503i67 42 0-0778 3'216888' 0- 000203 0-0917 4'858304 0'0570 1'331 45 0-0805 3'6859438 0-0535932 0'0949 5-543242 0'061 i7 501 48 0-0831 4'186902t 0'05716I13 0-0980 6'273237 00 6SO5067 51 0-0857 4-7197647 0-0607390 0-1010 70418287 0-06096t;4 54 0-0882 5'2845306 0-0643119 0'1040 7-868393 0-0740,201 57 0-0908 5-8812001 0-06784-17 0-1070 8-733554 0-078 9 67 fc 0-0926 6-5097733 0-0714576 01039 96-13771 0'08231:334 0-0021503n 0-01054412a, t 1/0'000144n - — 0 001724 - 0 0011909Se?z | /0-00002 + 03 A 0-031319SS9n Application of the preceding Tables or Formulas. To find the stiffness of a rope of a given diameter or number of yarns, we must first obtain from the table, or by the formulas, the values of the quantities A and B corresponding to these given quantities, and knowing the tension, T, of the end to be wound up, we shall have its resistance to bending on a drum of a foot in diameter, by the formula R = A + BT. Then, dividing this quantity by the diameter of the roller or pulley round which the rope is actually to be bent, we shall have the resistance to bending on this roller. What is the stiffness of a dry white rope, in good condition, of 60 yarns, or -0928 diameter, which passes over a pulley of 6 inches diameter in the groove, under a tension of 1000 lbs.? The table gives for a dry white rope of 60 yarns, in good condition, bent upon a drum of a foot in diameter, A = 0-50977 B = 0-0714576 and we have D = 0'5 + 0-0928; and consequently, 6-50977 + 0-0714576 x 1000 R 058 = 128 lbs. 0'5928 The whole resistance to be overcome, not includingc the friction on the axis, is then Q + R = 1000 + 128 1128 lbs. The stiffness in this case augments the resistance by more than one-eighth of its value. Page 307 STRENGTH OF MATERIALS. 307 FURTHER RECENT EXPERIMENTS MADE BY M. IMORTN, ON THE TRACTION OF CARRIAGES, AND THE DESTRUCTIVE EFFECTS WHICH THEY PRODUCE UPON THE ROADS. The study of the effects which are produced when a carriage is set in motion can be divided into two distinct parts: the traction of carriages, properly so called, and their action upon the roads. The researches relative to the traction of carriages have for their object to determine the magnitude of the effort that the motive power ought to exercise according to the weight of the load, to the diameter and breadth of the wheels, to the velocity of the carriage, and to the state of repair and nature of the roads. The first experiments on the resistance that cylindrical bodies offer to being rolled on a level surface are due to Coulomb, who determined the resistance offered by rollers of lignum vitao and elm, on plane oak surfaces placed horizontally. His experiments showed that the resistance was directly proportional to the pressure, and inversely proportional to the diameter of the rollers. If, then, P represent the pressure, and r the radius of the roller, the resistance to rolling, R, could, according to the laws of Coulomb, be expressed by the formula P R R=Ar in which A would be a number, constant for each kind of ground, but varying with different kinds, and with the state of their surfaces. The results of experiments made at Vincennes show that the law of Coulomb is approximately correct, but that the resistance increases as the width of the parts in contact diminishes. Other experiments of the same nature have confirmed these conclusions; and we may allow, at least, as a law sufficiently exact for practical purposes, that for woods, plasters, leather, and generally for hard bodies, the resistance to rolling is nearly1st. Proportional to the pressure. 2d. Inversely proportional to the diameter of the wheels. 3d. Greater as the breadth of the zone in contact is smaller. EXPERIMENTS UPON CARRIAGES TRAVELLING ON ORDINARYa ROADS. These experiments were not considered sufficient to authorize the extension of the foregoing conclusions to the motion of carriages on ordinary roads. It was necessary to operate directly on the carriages themselves, and in the usual circumstances in which they are placed. Experiments on this subject were therefore undertaken, first at Metz, in 1837 and 1838, and afterwards at Courbevoie, in 1839 and 1841, with carriages of every species; and attention was directed separately to the influence upon the magnitude of the traction, of the pressure, of the diameter of the wheels, of their breadth, of the speed, and of the state of the ground. In heavily laden carriages, which it is most important to take Page 308 308 THE PRACTICAL MODEL CALCULATOR. into consideration, the weight of the wheels may be neglected in comparison with the total load; and the relation between the load and the traction, upon a level road, is approximately given by the equationF 2 (A x fr for carriages with four wheels, P~ — rI x rI and F1 A x fr, for carriages with two wheels, in which F1 represents the horizontal component of the traction; P1 the total pressure on the ground; r' and r" the radii of the fore and hind wheels; r1 the mean radius of the boxes; f the coefficient of friction; and A the constant multiplier in Coulomb's formula for the resistance to rolling. These expressions will serve us hereafter to determine, by aid of experiment, the ratio of the traction to the load for the most usual cases. Influence of the Pressure. To observe the influence of the pressure upon the resistance to rolling, the same carriages were made to pass with different loads over the same road in the same state. The results of some of these experiments, made at a walking pace, are given in the following table:Ratio of the Carriages employed. Road traversed. Pressure. Traction. traction to the load. kil. kil. Chariotportecorps Road from Courbe- 6992 180'71 1/38-6 d'artillerie. voie to Colomber, 6140 159'9 1/39'2 dry, in good re- 4580 1137 1/402 pair, dusty. Chariotderoulage, Road from Courbe- 7126 138.9 1/51.3 without springs. voie to Bezous, 5458 115-5 1/48-9 solid, *hard gra- 4450 93-2 1/47-7 vel, very dry. 3430 68.4 1/50.2 Chariotderoulage, Road from Colomber 1600 39.3 1/40.8 with springs. to Courbevoie, 3292 89-2 1/'36.9 pitched, inordina- 4996 136'0 1/36'8 ry repair, t muddy Carriages with six Road from Courbe- 3000 138'9 1/21.6 equal wheels. voie to Colomber, 4692 224'0 1/21'0 Twocarriageswith deep ruts, with 6000 285-8 1 /210 six equal wheels, muddy detritus. 6000 286'7 1/21'0 hooked on, one behind the other. From the examination of this table, it appears that on Isolid gravel and on pitched roads the resistance of carriages to traction is sensibly proportional to the pressure. * En gravier dur. t Pav6 en 6tat ordinaire. ~ En empierrement solide. Page 309 STRENGTH OF MATERIALS. 309 We remark that the experiments made upon one and upon two six-wheeled carriages have given the same traction for a load of 6000 kilogrammes, including the vehicle, whether it was borne upon one carriage or upon two. It follows thence that the traction is, caeteris paribus and between certain limits, independent of the number of wheels. Influence of the Diameter of the Wheels. To observe the influence of the diameter of the wheels on the traction, carriages loaded with the same weights, having wheels with tires of the same width, and of which the diameters only were varied between very extended limits, were made to traverse the same parts of roads in the same state. Some of the results obtained are given in the following table. These examples show that on solid roads it may be admitted as a practical law that the traction is inversely proportional to the diameters of the wheels. Diameter of Diameter of the wheels in the wheels in P;. Ratio of Value Value metres. English feet. the trac- Resist- ofA of A Carriages employed. Roads traversed. tion to anceo for the for the:Ei- 5 the pres-'V rolling, French American Fore Hin d Fore ind sure.. scale. scaie. heelheels'heels whee sl 2r' 2I" 2r' l r" m. I m. I I 1 kil. kil. kil. kil. Chariot porte Road from Cour- 2-029 2 029 6-657 6'657 4928 81-6 1/60' 96 720 0-0148 0201856 corps d'artil- bevoie to Colom- 1453 1-453 i767 4-767 4930 108-6 1/45-5 14[4 94'2 0'0139 1004560 lerie. ber, *solid gra- 0872 0872 2861 2861 4924 1790 1/27 4 2593 153 7 0-0137 U004494 vel, dusty. Porte corps d'ar- 2029 2029 6657 6'657 4692 51-45 1/90'45 9-0 42-45 i00092 0'03018 tillerie. 11453 1'453 4-767 4 767 4594 71-45 1/64'3 13-2 5825 000092 0'03018 Chariotcomto-wheeled itched pave- 1110 1358 3-642 4-455 1871 32-10 1/584 4-7 2 740 0-0089 0-02920 A six-wheeled ment of Foncarriage. tainebleau. 0860,05860 2-822 2 822 3270 8105 1/40 4 97 71-3510 0094 0-03084 The same with0 I 1 four wheels. 0'860 0'860 2'822 2'822 3270 78'80 1/41'5 97 69'10 0 0091 0 02986 Camion. 0'592 0'660 1'942 2'165 1500 52-30 1/28'8 8;1 43'50 0'0091 0'02986 Camion. 0420 0597 1378 1559 1600 68l20 11/224 1161 560600-0089 0 02920 Influence of the Width of the lFelloes. Experiments made upon wheels of different breadths, having the same diameter, show, 1st, tIat on soft ground the resistance to rolling increases as the width of the felloe; 2dly, on solid gravel and pitched roads, the resistance is very nearly independent of the width of the felloe. Influence of the Velocity. To investigate the influence of the velocity on the traction of carriages, the same carriages were made to traverse different roads in various conditions; and in each series of experiments the velocities, while all other circumstances remained the same, underwent successive changes from a walk to a canter. Some of the results of these experiments are given in the following table:* Empierrement solide. t Pav6 en grbs. Page 310 310 THE PRACTICAL MODEL CALCULATOR. Rate of Ratio speed, Trac- of the Carriage employed. Ground passed over. Load. Pace. in miles, trction per tio to the hour. loead. kil. miles. kil. Apparatus upon a Ground of the po- 1042 Walk........ 3.13 165.0 1/6-32 brass shaft. lygon at Metz, Trot........ 6.26 168'0 1/6-2 wet and soft. 1335 Walk........ 2.860 215.0 1/6.21 Trot......... 7 560 197.0 1/4~8 Asixteen-pounder Road from Metz 3750 Walk........ 2820 92 1/4. 8 carriage and to Montigny, *Briskwalk 3-400 92- 1/,40 8 piece. solid gravel, Trot......... 5480 102' 1/36'8 very even and tCanter...... 8.450 121- 1/31very dry. Chariot des Mes- Pitched road of 3288 Walk........ 2-770 144- 1/22'8 sageries, sus- Fontainebleau. pended upon six 3353 *Brisk walk 3.82 153- 1/21-9 springs. Trot......... 5'28 161- 1/20'8 }Brisk trot. 8.05 183-5 1/18-3 We see, by these examples, that the traction undergoes no sensible augmentation with the increase of velocity on soft grounds; but that on solid and uneven roads it increases with an increase of velocity, and in a greater degree as the ground is more uneven, and the carriage has less spring. To find the relation between the resistance to rolling and the velocity, the velocities were set off as abscissas, and the values of A furnished by the experiments, as ordinates; and the points thus determined were, for each series of experiments, situated very nearly upon a straight line. The value of A, then, can be represented by the expression, A= a J d (V - 2) in which a is a number constant for each particular state of each kind of ground, and which expresses the value of the number A for the velocity, V = 2 miles, (per hour,) which is that of a very slow walk. c, a factor constant for each kind of ground and each sort of carriage. The results of experiments made with a carriage of a siege train, with its piece, gave, on the Montigny road, ~very good solid gravel,A = 0-03215 x 0.00295 (V - 2). On the lpitched road of AMetz, A = 0'01936 x 0.08-200 (V - 2). These examples are sufficient to show1st. That, at a walk, the resistance on a good pitched road is less than that on very good solid gravel, very dry. 2d. That, at high speeds, the resistance on the pitched road increases very rapidly with the velocity. On rough roads the resistance increases with the velocity much more slowly, however, for carriages with springs. * Pas allong4. t Grand trot. { Trot allong6. Q En tres bon empierrement. it Pave en gris de Sieack. Page 311 STRENGTII OF MATERIALS. 311 Thus, for a chariot des Messageries G6ne'rales, on a pitched road, the experiments gave A = 0'0117 x 0'00361 (V - 20); while, with the springs wedged so as to prevent their action, the experiments gave, for the same carriage, on a similar road, A = 0'02723 x 0'01312 (V - 2). At a speed of nine miles per hour, the springs diminish the resistance by one-half. The experiments further showed that, while the pitched road was inferior to a *solid gravel road when dry and in good repair, the latter lost its superiority when muddy or out of repair. INFLUENCE OF TIIE INCLINATION OF THE TRACES. The inclination of the traces, to produce the maximum effect, is given by the expressionA x 0'96 f r' hf= r- 0 4f r' in which h = the height of the fore extremity of the trace above the point where it is attached to the carriage; b = the horizontal distance between these two points. r' is the radius of interior of the boxes, and r the radius of the wheel. The inclination given by this expression for ordinary carriages is very small; and for trucks with wheels of small diameter it is much less than the construction generally permits. It follows, from the preceding remarks, that it is advantageous to employ, for all carriages, wheels of as large a diameter as can be used, without interfering with the other essentials to the purposes to which they are to be adapted. Carts have, in this respect, the advantage over wagons; but, on the other hand, on rough roads, the thill horse, jerked about by the shafts, is soon fatigued. Now, by bringing the hind wheels as far forward as possible, and pllacing the load nearly over them, the wagon is, in effect, transformed into a cart; only care must be taken to place the centre of gravity of the load so far in front of the hind wheels that the wagon may not turn over in going up hill. ON THE DESTRUCTIVE EFFECTS PRODUCED BY CARRIAGES ON THE ROADS. If we take stones of mean diameter from 21 to 3- inches, and, on a road slightly moist and soft, place them first under the small wheels of a diligence, and then under the large wheels, we find that, in the former case, the stones, pushed forward by the small wheels, penetrate the surface, ploughing and tearing it up while in the latter, being merely pressed and leant upon by the large wheels, they undergo no displacement. From this simple experiment we are enabled to conclude that the wear of the roads by the wheels of carriages is greater the smaller the diameter of the wheels. Experiments having proved that on hard grounds the traction was but slightly increased when the breadths of the wheels was * En empierrement. Page 312 312 THE PRACTICAL MODEL CALCULATOR. diminished, we might also conclude that the wear of the roadl would be but slightly increased by diminishing the width of the felloes. Lastly, the resistance to rolling increasing with the velocity, it was natural to think that carriages going at a trot would do more injury to the roads than those going at a walk. But springs, by diminishing the intensity of the impacts, are able to compensate, in certain proportions, for the effects of the velocity. Experiments, made upon a grand scale, and having for their object to observe directly the destructive effects of carriages upon the roads, have confirmed these conclusions. These experiments showed that with equal loads, on a, solid gravel road, wheels of two inches breadth produced considerably more wear than those of 4~ inches, but that beyond the latter width there was scarcely any advantage, so far as the preservation of the road was concerned, in increasing the size of the tire of the wheel. Experiments made with wheels of the same breadth, and of diameters of 2'86 ft., 4'77 ft., and 6'69 ft., showed that after the carriage of 10018'2 tons, over tracks 218'72 yards long, the track passed over by the carriage with the smallest wheels was by far the most worn; while, on that passed over by the carriage with the wheels of 6'69 ft. diameter, the wear was scarcely perceptible. Experiments made upon two wagons exactly similar in all other respects, but one with and one without springs, showed that the wear of the roads, as well as the increase of traction, after the passage of 4577-36 tons over the same track, was sensibly the same for the carriage without springs, going at a walk of from 2'237 to 2'684 miles per hour, and for that with springs, going at a trot of from 7'158 to 8'053 miles per hour. HYDRAULICS. THE DISCHARGE OF WATER BY SIMPLE ORIFICES AND TUBES. THE formulas for finding the quantities of water discharged in a given time are of an extensive and complicated nature. The more important and practical results are given in the following Deductions. When an aperture is made in the bottom or side of a vessel containing water or other homogeneous fluid, the whole of the particles of fluid in the vessel will descend in lines nearly vertical, until they arrive within three or four inches of the place of discharge, when they will acquire a direction more or less oblique, and flow directly towards the orifice. The particles, however, that are immediately over the orifice, descend vertically through the whole distance, while those nearer to the sides of the vessel, diverted into a direction more or less oblique as they approach the orifice, move with a less velocity than the former; and thus it is that there is produced a contraction in the size of the stream immediately beyond the opening, designated the vena contracta, and bearing a proportion to that of the orifice of Page 313 HYDRAULICS. 313 about 5 to 8, if it pass through a thin plate, or of 6 to 8, if through a short cylindrical tube. But if the tube be conical to a length equal to half its larger diameter, having the issuing diameter less than the entering diameter in the proportion of 26 to 33, the stream does not become contracted. If the vessel be kept constantly full, there will flow from the aperture twice the quantity that the vessel is capable of containing, in the same time in which it would have emptied itself if not kept supplied. 1. How many horse-power (H. P.) is required to raise 6000 cubic feet of water the hour from a depth of 300 feet? A cubic foot of water weighs 62'5 lbs. avoirdupois. 6000 x 62.5 60 = 6250, the weight of water raised a minute. 6250 x 300 = 1875000, the units of work each minute. 1875000 Then 33000 - 56'818 = the horse-power required. 2. What quantity of water may be discharged through a cylindrical mouth-piece 2 inches in diameter, under a head of 25 feet? 2 1 12 = 6 of a foot;.. the area of the cross section of the I 1 mouth-piece, in feet, is 6 x 6 x'7854 ='021816. Theory gives'021816,/2 g x 25 the cubic feet discharged each second; but experiments show that the effective discharge is 97 per cent. of this theoretical quantity: g = 32'2. Hence, *97 x'021816 V/64'4 x 25 ='84912, the cubic feet discharged each second. *84912 x 62'5 = 53'0688 lbs. of water discharged each second. Effluent water produces, by its vis viva, about 6 per cent. less mechanical effect than does its weight by falling from the height of the head. 3. What quantity of water flows through a circular orifice in a thin horizontal plate, 3 inches in diameter, under a head of 49 feet? Taking the contraction of the fluid vein into account, the velocity of the discharge is about 97 per cent. of that given by theory. The theoretic velocity is V2gy x 49 = 7 V/644 - 56'21. ~97 x 56'21 = 54'523 = the velocity of the discharge. The area of the transverse section of the contracted vein is'64 of the transverse section of the orifice. 3 1 12 = = — 25, and (.25)2 x'7854 = -0490875 = area of orifice..'.'64 x'0490875 ='031416, the area of the transverse section of the contracted vein. 2B Page 314 314 THE PRACTICAL MODEL CALCULATOR. Hence, 54'523 x'031416 = 1'7129, the cubic feet of water discharged each second. The later experiments of Poncelet, Bidone, and Lesbros give *563 for the coefficient of contraction. Water issuing through lesser orifices give greater coefficients of contraction, and become greater for elongated rectangles, than for those which approach the form of a square. Observations show that the result above obtained is too great; s of this result are found to be very near the truth. 8 of 1'7129 = 1-0541. 4. What quantity of water flows through a rectangular aperture 7'87 inches broad, and 3'94 inches deep, the surface of the water being 5 feet above the upper edge; the plate through which the water flows being'125 of an inch thick. 7-87 12 = _65583, decimal of a foot. 3.94 1-2 ='32833, decimal of a foot. 5' and 5'32833 are the heads of water above the uppermost and lowest horizontal surfaces. The theoretical discharge will be x'65583 V2 g ((5328) -- (5)2) = 3'9268 cubic feet. Table I. gives the coefficient of efflux in this case,'615, which is found opposite 5 feet and under 4 inches; for 3'94 is nearly equal 4. 3'9268 x'615 = 2'415 cubic feet, the effective discharge. 5. What water is discharged through a rectangular orifice in a thin plate 6 inches broad, 3 inches deep, under a head of 9 feet measured directly over the orifice? 6 =.5, decimal of a foot. 3 2 = *25, decimal of a foot. The theoretical discharge will be x.5 2/ (9.25) -(9) = 3033 cubi feet. Table II. gives the coefficient of efflux between'604 and'606; we shall take it at'605, then 3'033 x'605 = 1'833 cubic feet, the effective discharge. 6. A weir'82 feet broad, and 4'92 feet head of water, how many cubic feet are discharged each second? The quantity will be e x 82 V,2g (4.92)3; g = 32.2; Page 315 HYDRAULICS. 315 TABLE I. — The Coefficients for the Efftux throug/h rectangular orifices in a thin vertical plate. The heads are zecasurcd where the water may be considered still. Headl of water, or distance of the IHEIGHT OF ORIFICE. surface of the water from the upper side In. In. In. In. In. In. the orifice in feet. 8' 4 2 1 8 I4.1 *579 599 619'634 -656 686.2'582 601 620 *638'G54 G81 ~3'585 603 621 640 653 *676.4'588'605 622 639'652 671.5 *591 607'623 637 650 *666.6 594 609 624 *635 *649 662 ~7 *596 611 625 *634 648 *659.8 597 613 *623 *632'647 656 ~9 598 615 *627 631 *645 *653 1.0'599'616 628'630 644'650 2.0'600'617'628'628'641'647 3.0'601'617'626'626'638'644 4.0'602'616'624'623'634'640 5.0'604'615'621'621'630'635 60'603 613'618'618'625'630 7.0'602'611'615'615 6621 G625 8.0'601'609'612'613'617'619 9.0'600'606'609'610'614'613 10.0'600'604'606'608 611 609 TABLE II. — The Coefficients for the Efflz tlhrough rectangular orifices in a thin verticalplate, the heads of water being measured directly over the orifice. Head of water, or distance of the EIGHT OF ORIFICE. surface of the water from the upper side of In. In. In. Ill. In. In. the orifice in feet. 8 4' 2 1'8 4.1 *593 *613'637'659 *685 *708.2 *593 *612'636'656'680'701.3 *593'613'635'653'676.694.4'594.614 6384'650'672 6(87.5'595'614'633'647'668 681s'6'597'615'632'644'664 *675.7'598 ~615'631'641 G660 *669.8 *599 *616'630'638'655'663.9'601 -616'629 6365 650.657 1.0'603'617'629'632'644'651 2.0. 604 *617'626'628'640'646 3.0. 605'616 622'627'636.641 4.0'604'614'618'624,'632'636 5-0'604'613'616'621'628'631 6-0. 603'612'613'618 624.626 7.0'603'610.611'616'620'621 8'0'602'608'609'614 6i16'617 9'0'601'607'607 612'613 *613 10.0'601'603'606'610 G610 /609 Page 316 316 TIIE PRACTICAL MODEL CALCULATOR. c is termed the coefficient of cfflux, and on an average may be taken at'4. It is found to vary from'385 to'444. Then -4 x *82 V(64'4) (4'92)3 = 2-670033, the cubic feet discharged each second. 7. MWhat breadth must be given to a notch, in a thin plate, with a head of water of 9 inches, to allow 10 cubic feet to flow each second' The breadth will be represented by 10 10 4.7963 feet. c v/2g x (75)3 4 x x /644 x (-75)3 Changes in the coefficients of efflux through convergent sides often present themselves in practice: they occur in dams which are inclined to the horizon. Poncelet found the coefficient'8, when the board was inclined 450, and the coefficient'74 for an inclination of 63~ 34', that is for a slope of 1 for a base, and 2 for a perpendicular. 8. If a sluice board, inclined at an angle of 50~, which goes across a channel 2'25 feet broad, is drawn out'5 feet, what quantity of water will be discharged, the surface of the water standing 4. feet above the surface of the channel, and the coefficient of efltux taken at'78? The height of the aperture ='5 sin. 50 = 38830222; 4' and 4- -3830222 = 3.616(778, are the heads of water. 2 -{ 2 _ 33617)2 V; x -225 x -78 x V, (4) (3.17) - 105257 cubic feet, the quantity discharged. The calculations just made appertain to those cases where the water flows from all sides towards the aperture, and forms a contracted vein on every side. AWe shall next calculate in cases where the water flows from one or more sides to the aperture, and hence produces a stream only A B partially contracted. m, n, o, p, are four orifices in I n the bottom ABCD of a vessel; the contraction by t efflux through the orifice o, in the middle of the bottomn, is general, as the water can flow to it from all- sides; the contraction c D fiom the effiux through m, In,, is partial, as the water can only flow to them from one, two, or three sides. Partial contraction gives an oblique direction to the stream, and increases the quantity discharged. 9. What quantity of water is delivered through a flow 4 feet broad, and 1 foot deep, vertical aperture, at a pressure of 2 feet above the upper edge, supposing the lower edge to coincide with Page 317 HYDRAULICS. 817 the lower side of the channel, so that there is no contraction at the bottom? The theoretical discharge will be 2 4 f x /2g (3) - (2) = 50668 cubic feet. The coefficient of contraction given in the table page 315, may be taken at'603. I.-Comparison of the Theoretical with the Real Discharges from an Orifice. Constant height Theoretical dis- Real discharge of the water in the charge through a in the same time Ratio of the reservoir above circular orificehrouh theoretical to the real the centre of the one inch in di- throu esame discharge. orifice. ameter. orce. Paris Feet. Cubic Inches. Cubic Inches. 1 4381 2722 1 to 0-62133 2 6196 3846 1 to 0'62073 3 7589 4710 1 to 0-62064 4 8763 5436 1 to 0-62034 5 9797 6075 1 to 0'62010 6 10732 6654 1 to 0-62000 7 11592 7183 1 to 0'61965 8 12392 7672 1 to 0-61911 9 13144 8135 1 to 0-61892 10 13855 8574 1 to 0'61883 11 14530 8990 1 to 0-61873 12 15180 9384 1 to 0'61819 13 15797 9764 1 to 0'61810 14 16393 10130 1 to 0-61795 15 16968 10472 1 to 0-61716 II.-Comparison of the Theoretical with the Real Discharges frosm a Tube. Constant height Theoretical dis- Rel discharge of the water in the charge through ain the a ctime Ratio of the reservoir above circular orifice by a cylindrical theoretical to the real the centre of the one inch in di- tube one inch in discharge. orifice. ameter. inches long. Paris Feet. Cubic Inches. Cubic Inches. 1 4381 3539 1 to 0'81781 2 6196 5002 1 to 0.80729 3 7589 6126 1 to 0-80724 4 8763 7070 1 to 0.80681 5 9797 7900 1 to 0.80638 6 10732 8654 1 to 0.80638 7 11592 9340 1 to 0-80577 8 12392 9975 1 to 0.80496 9 13144 10579 1 to 0.80485 10 13855 11151 1 to 0-80483 11 14530 11693 1 to 0-80477 12 15180 12205 1 to 0-80403 13 15797 12699 1 to 0-80390 14 16393 13177 1 to 0-80382 15 16968 13620 1 to 0-80270 2n2 Page 318 318 THE PRACTICAL MODEL CALCULATOR. THE DISCHARGE BY DIFFERENT APERTURES AND TUBES, UNDER DIFFERENT HEADS OF AVATER. The velocity of water flowing out of a horizontal aperture, is as the square root of the height of the head of the water.-That is, the pressure, and consequently the height, is as the square of the velocity; for, the quantity flowing out in any short time is as the velocity; and the force required to produce a velocity in a certain quantity of matter in a given time is also as that velocity; therefore, the force must be as the square of the velocity. Or, supposing a very small cylindrical plate of water, immediately over the orifice, to be put in motion at each instant, by the pressure of the whole cylinder upon it, employed only in generating its velocity; this plate would be urged by a force as much greater than its own weight as the column is higher than itself, through a space shorter in the same proportion than that height. But where the forces are inversely as the spaces described, the final velocities are equal. Therefore, the velocity of the water flowing out must be equal to that of a heavy body falling from the height of the head of water; which is found, very nearly, by multiplying the square root of that height in feet by 8, for the number of feet described in a second. Thus, a head of 1 foot gives 8; a head of 9 feet, 24. This is the theoretical velocity; but, in consequence of the contraction of the stream, we must, in order to obtain the actual velocity, multiply the square root of the height, in feet, by 5 instead of 8. The velocity of a fluid issuing from an aperture is not affected by its density being greater or less. Mercury and water issue with equal velocities at equal altitudes. The proportion of the theoretical to the actual velocity of a fluid issuing through an opening in a thin substance, according to MI. Eytelwein, is as 1 to'619; but more recent experiments make it as 1 to'621 up to'645. APPLICATION OF THE TABLES IN THE PRECEDING PAGE. TABLE I.-To find the quantities of water discharged by orifices of different sizes under different altiticdes of the fluid in the reservoir. To find the quantity of fluid discharged by a circular aperture 3 inches in diameter, the constant altitude being 30 feet. As the real discharges are in the compound ratio of the area of the apertures and the square roots of the altitudes of the water, and as the theoretical quantity of water discharged by an orifice one inch in diameter from a height of 15 feet is, by the second column of the table, 16968 cubic inches in a minute, we have this proportion: 1 V15: 9 30:: 16968: 215961 cubic inches; the theoretical quantity required. This quantity being diminished in the ratio of 1 to'62, being the ratio of the theoretical to the actual discharge, according to the fourth column of the table, gives 133896 cubic inches for the actual quantity of water discharged by Page 319 hYDRAULICS. 319 the given aperture. Hence, the quantity should be rather greater, because large orifices discharge more in proportion than small ones; while it should be rather less, because the altitude of the fluid being greater than that in the table with w-hich it is compared, the flowing vein of water becomes rather more contracted. The quantity thus found, therefore, is nearly accurate as an average. When the orifice and altitude are less than those in the table, a few cubic inches should be deducted from the result thus derived. The altitude of the fluid being multiplied by the coefficient 8'016 will give its theoretical velocity; and as the velocities are as the quantities discharged, the real velocity may be deducted from the theoretical by means of the foregoing results. TABLE II.-To find the quantities of water discharged by tubes of different diameter, and under different heights of water. To find the quantity of water discharged by a cylindrical tube, 4 inches in diameter, and 8 inches long, the constant altitude of the water in the reservoir being 25 feet. Find, in the same manner as by the example to Table I., the theoretical quantity discharged, which is furnished by this analogy. 1 V/15: 16 V25:: 16968: 350490 cubic inches, the theoretical discharge. This, diminished in the ratio of 1 to'81 by the 4th column, will give 28473 cubic inches for the actual quantity discharged. If the tube be shorter than twice its diameter, the quantity discharged will be diminished, and approximate to that from a simple orifice, as shown by the production of the vena contracta already described. According to Eytelwein, the proportion of the theoretical to the real discharge through tubes, is as follows: Through the shortest tube that will cause the stream to adhere everywhere to its sides, as 1 to 0'8125. Through short tubes, having their lengths from two to four times their diameters, as 1 to 0'82. Through a tube projecting within the reservoir, as 1 to 0'50. It should, however, be stated, that in the contraction of the stream the ratio is not constant. It undergoes perceptible variations by altering the form and position of the orifice, the thickness of the plate, the form of the vessel, and the velocity of the issuing fluid. Deductions from experiments made by Bossut, Ji~ic7elloti. 1. That the quantities of fluid discharged in equal times from different-sized apertures, the altitude of the fluid in the reservoir being the same, are to each other nearly as the area of the ip, rtures. 2. That the quantities of water discharged in equal times by the same orifice under different heads of water, are nearly as the square roots of the corresponding heights of water in the reservoir above the centre of the apertures. Page 320 320 THE PRACTICAL MODEL CALCULATOR. 3. That, in general, the quantities of water discharged, in the same time, by different apertures under different heights of water in the reservoir, are to one another in the compound ratio of the areas of the apertures, and the square roots of the altitudes of the water in the reservoirs. 4. That on account of the friction, the smallest orifice discharges proportionally less water than those which are larger and of a similar figure, under the same heads of water. 5. That, from the same cause, of several orifices whose areas are equal, that which has the smallest perimeter will discharge more water than the other, under the same altitudes of water in the reservoir. Hence, circular apertures are most advantageous, as they have less rubbing surface under the same area. 6. That, in consequence of a slight augmentation which the contraction of the fluid vein undergoes, in proportion as the height of the fluid in the reservoir increases, the expenditure ought to be a little diminished. 7. That the discharge of a fluid through a cylindrical horizontal tube, the diameter and length of which are equal to one another, is the same as through a simple orifice. 8. That if the cylindrical horizontal tube be of greater length than the extent of the diameter, the discharge of water is much increased. 9. That the length of the cylindrical horizontal tube may be increased with advantage to four times the diameter of the orifice. 10. That the diameters of the apertures and altitudes of water in the reservoir being the same, the theoretic discharge through a thin aperture, which is supposed to have no contraction in the vein, the discharge through an additional cylindrical tube of greater length than the extent of its diameter, and the actual discharge through an aperture pierced in a thin substance, are to each other as the numbers 16, 13, 10. 11. That the discharges by different additional cylindrical tubes, under the same head of water, are nearly proportional to the areas of the orifices, or to the squares of the diameters of the orifices. 12. That the discharges by additional cylindrical tubes of the same diameter, under different heads of water, are nearly proportional to the square roots of the head of water. 13. That firom the two preceding corollaries it follows, in general, that the discharge during the same time, by diffierent additional tubes, and under different heads of water in the reservoir, are to one another nearly in the compound ratio of the squares of the diameters of the tubes, and the square roots of the heads of water. The discharge of fluids by additional tubes of a conical figure, when the inner to the outer diameter of the orifice is as:33 to 26, is augmented very nearly one-seventeenth and seven-tenths more than by cylindrical tubes, if the enlargement be not carried too far. Page 321 HYDRAULICS. 321 DISCHARGE BY COMPOUND TUBES. Deductions from the experiments of MI. Venturi. In the discharge by compound tubes, if the part of the additional tube nearest the reservoir have the form of the contracted vein, the expenditure will be the same as if the fluid were not contracted at all; and if to the smallest diameter of this cone a cylindrical pipe be attached, of the same diameter as the least section of the contracted vein, the discharge of the fluid will, in a horizontal direction, be lessened by the friction of the water against the side of the pipe; but if the same tube be applied in a vertical direction, the expenditure will be augmented, on the principle of the gravitation of falling bodies; consequently, the greater the length of pipe, the more abundant is the discharge of fluid. If the additional compound tube have a cone applied to the opposite extremity of the pipe, the expenditure will, under the same head of water, be increased, in comparison with that through a simple orifice, in the ratio of 24 to 10. In order to produce this singular effect, the cone nearest to the reservoir must be of the form of the contracted vein, which will increase the expenditure in the ratio of 12'1 to 10. At the other extremity of the pipe, a truncated conical tube must be applied, of which the length must be nearly nine times the smaller diameter, and its outward diameter must be 1'8 times the smaller one. This additional cone will increase the discharge in the proportion of 24 to 10. But if a great length of pipe intervene, this additional tube has little or no effect on the quantity discharged. According to M. Venturi's experiments on the discharge of water by bent tubes, it appears that while, with a height of water in the reservoir of 32'5 inches, 4 Paris cubic feet were discharged through a cylindrical horizontal tube in the space of 45 seconds, the discharge of the same quantity through a tube of the same diameter, with a curved end, occupied 50 seconds, and through a like tube bent at right angles, 70 seconds. Therefore, in making cocks or pipes for the discharge or conveyance of water, great attention should be paid to the nature and angle of the bendings; right angles should be studiously avoided. The interruption of the discharge by various enlargements of the diameter of the tubes having been investigated by M. Venturi, by means of a tube with a diameter of 9 lines, enlarged in several parts to a diameter of 24 lines, the retardation was found to increase nearly in proportion to the number of enlargements; the motion of the fluid, in passing into the enlarged parts, being diverted from its direct course into eddies against the sides of the enlargements. From which it may be deduced, that if the internal roughness of a pipe diminish the expenditure, the friction of the water against these asperities does not form any considerable part of the cause. A right-lined tube may have its internal surface highly polished throughout its whole length, and it may every21 Page 322 322 THE PRACTICAL MODEL CALCULATOR. where possess a diameter greater than the orifice to which it is applied; but, nevertheless, the expenditure will be greatly retarded if the pipe should have enlarged parts or swellings. It is not enough that elbows and contractions be avoided; for it may happen, by an intermediate enlargement, that the whole of the other aldvantage may be lost. This will be obvious from the results in the following table, deduced from experiments with tubes having various enlargements of diameter. Seconds in which Head of water Number of en- cubic feet were in inches. larged parts. discharged. 32-5 0 109 32'5 1 147 32'5 3 192 32'5 5 240 DISCHARGE BY CONDUIT PIPES. On account of the friction against the sides, the less the diameter of the pipe, the less proportionally is the discharge of fluid. And, from the same cause, the greater the length of conduit pipe, the greater the diminution of the discharge. Hence, the discharges made in equal times by horizontal pipes of different lengths, but of the same diameter, and under the same altitude of water, are to one another in the inverse ratio of the square roots of the lengths. In order to have a perceptible and continuous discharge of fluid, the altitude of the water in the reservoir, above the axis of the conduit pipe, must not be less than 1~t inch for every 1SO feet of the length of the pipe. The ratio of the difference of discharge in pipes, 16 and 24 lines diameter respectively, may be known by comparing the ratios of Table I. with the ratios of Table II., in the following page. The greater the angle of inclination of a conduit pipe, the greater will be the discharge in a given time; but when the angle of the conduit pipe is 6~ 31', or the depression of the lower extremity of the pipe is one-eighth or one-ninth of its length, the relative gravity of the fluid will be counterbalanced by the resistance or friction against the sides; and the discharge is then the same as by an additional horizontal tube of the same diameter. A curvilinear pipe, the altitude of the water in the reservoir being the same, discharges less water when the flexures lie horizontally, than a rectilinear pipe of the same diameter and length. The discharge by a curvilinear pipe of the same diameter and length, and under the same. head of water, is still further diainished when the flexures lie in a vertical instead of a horizontal plane. When there is a number of contrary flexures in a large pipe, the air sometimes lodges in the highest parts of the flexures, and greatly retards the motion of the warter, unless prevented by air-holes, or stopcocks. Page 323 HYDRAULICS. 323 TABLE I. —Comparison of the dischargqe by conduit pipes of diferent lengths, 16 lines in diameter, with the discharge by additional tubes inserted in the same reservoir.-By M. BOSSUT. Constant Quantity of Water discharged altitude of the Length of in a minute. Ratio between the Water above the the conduit quantities furnished centre of the pipe. by additional by conduit by tube and pipe. aperture. tube, 16 lines in pipe, 16 lines in diameter. diameter. Feet. Feet. Cubic Inches. Cubic Inches. 1 30 6330 2778 100 to 43'39 1 60 6330 1957 100 to 30'91 1 90 6330 1.587 100 to 25'07 1 120 6330 1351 100 to 21'34 1 150 6330 1178 100 to 18'61 1 180 6330 1052 100 to 16'62 2 30 8939 4066 100 to 45-48 2 60 8939 2888 100 to 32'31 2 90 8939 2352 100 to 26'31 2 120 8939 2011 100 to 22'50 2 150 8939 1762 100 to 19'71 2 180 8939 1583 100 to 17'70 TABLE II.-Comparison of the discharge by conduit pipes of different lengths, 24 lines in diameter, with the discharge by additional tubes inserted in the same reservoir.-By M. BossuT. Quantity of Water discharged Constant Lenth o in a minute. Ratio between the altitude of the Length of uRatio between the Water above the the conduit by quantities furnished centre of the pipe. by additional conduit by tube and pipe. aperture. tube, 24 lines in pipe, 24 lines in diameter. diameter. Feet. Feet. Cubic Inches. Cubic Inches. 1 30 14243 7680 100 to 53'92 1 60 14243 5564 100 to 39-06 1 90 14243 4534 100 to 31'83 1 120 14243 3944 100 to 27'69 1 150 14243 3486 100 to 24'48 1 180 14243 3119 100 to 21-90 2 30 20112 11219 100 to 55'78 2 60 20112 8190 100 to 40'72 2 90 20112 6812 100 to 33'87 2 120 20112 5885 100 to 29'26 2 150 20112 5232 100 to 26'01 2 180 20112 4710 100 to 23'41 DISCHARGE BY WEIRS AND RECTANGULAR APERTURES. Rectangular orifices in the side of a reservoir, extending to the sateface. The velocity varying nearly as the square root of the height, may here be represented by the ordinates of a parabola, and the quantity of water discharged by the area of the parabola, or two-thirds of that of the circumscribing rectangle. So that the quantity discharged may be found by taking two-thirds of the velocity due to the mean height, and allowing for the contraction of the stream, according to the form of the opening. In a lake, for example, in the side of which a rectangular opening is made without any oblique lateral walls, three feet wide, and Page 324 324 THE PRACTICAL MODEL CALCULATOR. extending two feet below the surface of the water, the coefficient of the velocity, corrected for contraction, is 5'1, and the corrected mean velocity a./2 x 5'1 = 4'8; therefore the area being 6, the discharge of water in a second is 28'8 cubic feet, or nearly four hogsheads. The same coefficient serves for determining the discharge over a weir of considerable breadth; and, hence, to deduce the depth or breadth requisite for the discharge of a given quantity of water. For example, a lake has a weir three feet in breadth, and the surface of the water stands at the height of five feet above it: it is required how much the weir must be widened, in order that the water may be a foot lower. Here the velocity is - /5 x 5'1, and the quantity of water 2 V/5 x 5'1 x 3 x 5; but the velocity must be reduced to V V4 x 5 1, and then the section will be i v,X5 2 V4 x 5'1 V5 x 3 x 5 5 = — 3 = -75 x NV5; and the height being 4, the breadth must be V 5 = 4419 feet. The discharge from reservoirs, with lateral orifices of considerable magnitude, and a constant head of water, may be found by determining the difference in the discharge by two open orifices of different heights; or, in most cases, with nearly equal accuracy, by considering the velocity due to the distance, below the surface, of the centre of gravity of the orifice. Under the same height of water in the reservoir, the same quantity always flows in a canal, of whatever length and declivity; but in a tube, a difference in length and declivity has a great effect on the quantity of water discharged. The velocity of water flowing in a river or stream varies at different parts of the same transverse section. It is found to be greatest where the water is deepest, at somewhat less than onehalf the depth from the surface; diminishing towards the sides and shallow parts. Resistance to bodies moving inzfluids.-The deductions from the experiments of C. Colles, (who first planned the Croton Aqueduct, New York,) and others, on this intricate subject, are, as stated, thus: 1. The confirmation of the theory, that the resistance of fluids to passing bodies is as the squares of the velocities. 2. That, contrary to the received opinion, a cone will move through the water with much less resistance with its apex foremost, than with its base forward. 3. That the increasing the length of a solid, of almost any form, by the addition of a cylinder in the middle, diminishes the resistance with which it moves, provided the weight in the water remains the same. Page 325 HYDRAULICS. 325 4. That the greatest breadth of the moving body should be placed at the distance of two-fifths of the whole length from the bow, when applied to the ordinary forms in naval architecture. 5. That the bottom of a floating solid should be made triangular; as in that case it will meet with the least resistance when moving in the direction of its longest axis, and with the greatest resistance when moving with its broadside foremost. Friction of fluids.-Some experiments have been made on this subject, with reference to the motion of bodies in water, upon a cylindrical model, 30 inches in length, 26 inches in diameter, and weighing 255 lbs. avoirdupois. The cylinder was placed in a cistern of salt water, and made to vibrate on knife-edges passing through its axis, and was deflected over to various angles by means of a weight attached to the arm of a lever. The experiments were then repeated without the water, and the following are the angles of deflection and vibration in the two cases. In the salt water. In the atmosphere. Angle of Angle to which Angle of Angle to which Deflection. it vibrated. Deflection. it vibrated. 22~ 30' 22~ 24' 22~ 30' 20~ 0t 22 10 22 6 21 36 21 3 21 54 21 48 20 48 2016 21 36 21 30 &c. &c. &c. &c. Showing that the amplitude of vibration when oscillating in water is considerably less than when oscillating without water. In the experiments there is a falling off in the angle of 24', or nearly half a degree. The amount of force acting on the surface of the cylinder necessary to cause the above difference was calculated; and the author thinks that it is not equally distributed on the surface of the cylinder, but that the amount on any particular part might vary as the depth. On this supposition, a constant pressure at a unit of depth is assumed, and this, multiplied by the depth of any other point of the cylinder immersed in the water, will give the pressure at that point. These forces or moments being summed by integration and equated with the sum of the moments given by the experiments, we have the value of the constant pressure at a unit of depth ='0000469. This constant, in another experiment, the weight of the model being 197 lbs. avoirdupois, and consequently the part immersed in the water being different from that in the other experiment, was'0000452, which differs very little from the former, —indicating the probability of the correctness of the assumption. The drainage of water through pipes.-The experiments made under the direction of the Metropolitan Commissioners of Sewers, on the capacities of pipes for the drainage of towns, have presented some useful results for the guidance of those who have to make'2C Page 326 326 THE PRACTICAL MODEL CALCULATOR. calculations for a similar purpose. The pipes, of various diameters, from 3 to 12 inches, were laid on a platform of 100 feet in length, the declivity of which could be varied from a horizontal level to a fall of 1 in 10. The water was admitted at the head of the pipe, and at five junctions, or tributary pipes on each side, so regulated as to keep the main pipe full. The results were as follow: It was found-to mention only one result-that a line of 6-inch pipes, 100 feet long, at an inclination of 1 in 60, discharged 75 Cubic feet per minute. The same experiment, repeated with the line of pipes reduced to 50 feet in length, gave very nearly the same result. Without the addition of junctions, the transverse sectional area of the stream of water near the discharging end was reduced to onefifth of the corresponding area of the pipe, and it required a simple head of water of about 22 inches to give the same result as that accruing under the circumstances of the junctions. AVith regard to varying sizes and inclinations, it appears, sufficiently for practical purposes, that the squares of the discharges are as the fifth powers of the diameters; and again, that in steeper declivities than 1 in 70, the discharges are as the square roots of the inclinations; but at less declivities than 1 in 70, the ratios of the discharges diminish very rapidly, and are governed by no constant law. At a certain small declivity, the relative discharge is as the fifth root of the inclination; at a smaller declivity, it is found as the seventh root of the inclination; and so on, as it approaches the horizontal plane. This may be exemplified by the following results found by actual experiment: Discharges of a 6-inch pipe at several inclinations. Inclination. Discharges in 100i Ineination. Diselharges in 100 feet per minute. feet per mllite. in 60 75 1 in 320 49 1 in 80 68 1 in 400 48'5 1 in 100 63 1 in 480 48 1 in 120 59 1 in 640 47'5 1 in 160 54 1 in 800 47'2 1 in 200 52 1 in 1200 46'7 1 in 240 50 Level 46 The conclusion arrived at is, that the requisite sizes of drains and sewers can be determined (near enough for practical purposes, as an important circumstance has to be considered in providing for the deposition of solid matter, which disadvantageously alters the form of the aqueduct, and contracts the water-way) by taking the result of the 6-inch pipe, under the circumstances before mentioned as a datum, and assuming that the squares of the discharges are as the fifth powers of the diameters. That at greater declivities than 1 in 70, the discharges are as the square roots of the inclinations. Page 327 WATER WHEELS. 327 That at less declivities than 1 in 70, the usual law will not obtain; but near approximations to the truth may be obtained by observing the relative discharges of a pipe laid at various small inclinations. That increasing the number of junctions, at intervals, accelerates the velocity of the main stream in a ratio which increases as the square root of the inclination, and which is greater than the ratio of resistance due to a proportionable increase in the length of the aqueduct. The velocity at which the lateral streams enter the main line, is a most important circumstance governing the flow of water. In practice, these velocities are constantly variable, considered individually, and always different considered collectively, so that their united effect it is difficult to estimate. Again, the same-sewer at different periods may be quite filled, but discharges in a given time very different quantities of water. It should be mentioned that in the case of the 6-inch pipe, which discharged 75 cubic feet per minute, the lateral streams had a velocity of a few feet per second, and the junctions were placed at an angle of about 350 with the main line. It is needless to say that all junctions should be made as nearly parallel with the main line as possible, otherwise the forces of the lateral currents may impedo rather than maintain or accelerate the main streams. WATER WHEELS. THE UNDERSHOT WHEEL. THE ratio between the power and effect of an undershot wheel is as 10 to 3'18; consequently 31'43 lbs. of water must be expended per second to produce a mechanical effect equal to that of the estimated labour of an active man. The velocity of the periphery of the undershot wheel should be equal to half the velocity of the stream; the float-boards should Ibe so constructed as to rise perpendicularly from the water; not more than one-half should ever be below the surface; and from 3 to 5 should be immersed at once, according to the magnitude of the wheel. The following maxims have been deduced from experiments:1. The virtual or effective head of water being the same, the effect will be nearly as the quantity expended; that is, if a mill, driven by a fall of water, whose virtual head is 10 feet, and which discharges 30 cubic feet of water in a second, grind four bolls of corn in an hour; another mill having the same virtual hecad, but which discharges 60 cubic feet of water, will grind eight bolls of corn in an hour. 2. The expense of water being the same, the effect will be nearly as the height of the virtual or effective head. 3. The quantity of water expended being the same, the effect is ncarly as the square of its velocity; that is, if a mill, driven by a Page 328 328 THE PRACTICAL MODEL CALCULATOR. certain quantity of water, moving with the velocity of four feet per second, grind three bolls of corn in an hour; another mill, driven by the same quantity of water, moving with the velocity of five feet per second, will grind nearly 4-70 bolls in the hour, because 3 417:: 42: 52 nearly. 4. The aperture being the same, the effect will be nearly as the cube of the velocity of the water; that is, if a mill driven by water, moving through a certain aperture, with the velocity of four feet per second, grind three bolls of corn in an hour; another mill, driven by water, moving through the same aperture with the velocity of five feet per second, will grind 53- bolls nearly in an hour; for as 3: 543:: 43: 53 nearly. The height of the virtual head of water may be easily determined from the velocity of the water, for the heights are as the squares of the velocities, and, consequently, the velocities are as the square roots of the height. To calculate the proportions of undershot wheels.-Find the perpendicular height of the fall of water above the bottom of the millcourse, and having diminished this number by one-half the depth of the water where it meets the wheel, call that the height of the fall. Multiply the height of the fall, so found, by 64'348, and take the square root of the product, which will be the velocity of the water. Take one-half of the velocity of the water, and it will be the velocity to be given to the float-boards, or the number of feet they must move through in a second, to produce a mnaximum effect. Divide the circumference of the wheel by the velocity of its floatboards per second, and the quotient will be the number of seconds in which the wheel revolves. Divide 60 by the quotient thus found, and the new quotient will be the number of revolutions made by the wheel in a minute. Divide 90, the number of revolutions which a millstone, 5 feet in diameter, should make in a minute, by the number of revolutions made by the wheel in a minute, the quotient will be the number of turns the millstone ought to make for one turn of the wheel. Then, as the number of revolutions of the wheel in a minute is to the number of revolutions of the millstone in a minute, so must the number of staves in the trundle be to the number of teeth inl the wheel, (the nearest in whole numbers.) Multiply the number of revolutions made by the wheel in a minute, by the number of revolutions made by the millstone for one turn of the wheel, and the product will be the number of revolutions made by the millstone in a minute. The effect of the water wheel is a mzaxinunm, when its circumference moves with one-half, or, more accurately, with threesevenths of the velocity of the stream. THE BREAST WHEEL. The effect of a breast wheel is equal to the effect of an under shot wheel, whose head of water is equal to the difference of level Page 329 WATER WHEELS. 329 between the surface of water in the reservoir, and the part where it strikes the wheel, added to that of an overshot, whose height is equal to the difference of level between the part where it strikes the wheel and the level of the tail water. When the fall of water is between 4 and 10 feet, a breast wheel should be erected, provided there be enough of water; an undershot should be used when the fall is below 4 feet, and an overshot wheel when the fall exceeds 10 feet. Also, when the fall exceeds 10 feet, it should be divided into two, and two breast wheels be erected upon it. TABLE for breast wheels..... Feeft r?. Feet. Feet. See......... lbs. avr. Cubic ft. 1 0.17 198.6 0.75 2.18 1.92 4.80 1536 74.30 2 0.34 35.1 1.50 3.09 2.72 6.80 1084 37 15 3 0.51 12.7 2.26 3.78 3.33 8'32 886 24-77 4 069 62 301 436 384 960 762 1857 5 0.86 3.57 3.76 4.88 4.28 10-70 680 14.86 6 1.03 2.25 4-51 56.35 4 70 11 76 626 12-38 7 1.20 1 53 5.26 5'77 5.08 12 70 581 10 61 8 1-37 1.10 6-02 6-17 5-43 13-58 543 9.29 9 1-54 0-81 6-77 6.55 576 14-40 512 8-26 10 1'71 0'77 752 6 90 6-07 15-18 1 486 7413 It is evident, from the preceding table, that when the height of the fall is less than 3 feet, the depth of the float-boards is so great, and their breadth so small, that the breast wheel cannot well be employed; and, on the contrary, when the height of the fall approaches to 10 feet, the depth of the float-boards is too small in proportion to their breadth; these two extremes, therefore, must be avoided in practice. The ninth column contains the quantity of water necessary for impelling the wheel; but the total expense of water should always exceed this by the quantity, at least, which escapes between the mill-course and the sides and extremities of the float-boards. THE OVERSHOT WHEEL. The ratio between the power and effect of an overshot wheel, is as 10 to 6'6, when the water is delivered above the apex of the wheel, and is computed from the whole height of the fall; and as 10 to 8 when computed from the height of the wheel only; consequently, the quantity of water expended per second, to produce a mechanical effect equal to that of the aforesaid estimated labour of an active man, is, in the first instance, 1515 lbs., and in the second instance, 12'5 lbs. Hence, the effect of the overshot wheel, under the same circum Page 330 330 THE PRACTICAL MODEL CALCULATOR. stances of quantity and fall, is, at a medium, double that of the undershot. The velocity of the periphery of an overshot wheel should be from 6~ to 8~ feet per second. The higher the wheel is, in proportion to the whole descent, the greater will be the effect. And from the equality of the ratio between the power and effect, subsisting where the constructions are similar, we must infer that the effects, as well as the powers, are as the quantities of water and perpendicular heights multiplied together respectively. TWo70king machinery by hydraulic pressure.-The vertical pressure of water, acting on a piston, for raising weights and driving machinery, is coming into use in many places where it can be advantageously applied. At Liverpool, Newcastle, Glasgow, and other places, it is applied to the working of cranes, drawing coal-wagons, and other purposes requiring continuous power. The presence of a natural fall, like that of Golway, Ireland, which can be conducted to the engine through pipes, is, of course, the most economical situationr for the application of such power; in other situations, artificial power must be used to raise the water, which, even under this disadvantage, may, from its readiness and simplicity of action, be often serviceably employed. Wherever the contiguity of a steam engine would be dangerous, or otherwise objectionable, a water engine would afford the means of receiving and applying the power from any required distance, precautions being taken against the action of frost on the fluid. Required the horse power of a centre discharging Turbine water wheel, the head of water being 25 feet, and the area of the opening 400 inches. The following table shows the working horse power of both the inward and outward discharging Turbine water wheels; they are calculated to the square inch of opening. Centre Discharging Outward Discharg-' Centre Discllarging Outward DischargTurbine. ing Turbine. Turbine. ing Turbine. Head. Horse Power. Horse Power. I-lead. Horse Power. Horse Power. 3.00821.012611 22.19523.339972 4.01483.025145 23.20]87.364182 5.02137.038124 24.22315.384615 6.02685.045618 25.23667.412013 7.03414.058314 26.25125 -437519 8.04198.074413 27.26482 45.5698 9.05206.089025 28.28135.484427 10.05883.106215 29 *29563.510833 11.06921.118127 30.30817 537721 12.07851 *135610 31.32316.561425 13.08882.150638 32.33617.587148 14.10054 17 3158 33.34823.611013 15.11002.1922334 4.36154. 638174 16.12093.211592 35.37123 66.5164 17.13196.231161 36.39874.692156 18.14275.257145 37.40118'726148 19.15613 *273325 38 41762 i 764115 20.16927.296618 39 420156.804479 21.18109.317167 -10.43718.849814 Page 331 WATER WIIEELS. 331 Opposite 25 in the column marked " Head," the working horse power to the square inch is found to be *25667, which, multiplied by 400, gives 94-668, the horse power required. What is the working horse power of an outward discharging Turbine, under the effective head of 20 feet; the area of all the openings being 325 square inches. In the table, opposite 20, we find'296618, then'296618 x 325 = 96'4, the required horse power. What is the number of revolutions a minute of anl outward discharging Turbine wheel, the head being 19 feet and the diameter of the wheel 60 inches? In the table for the outward discharging wheel, opposite 19, and under 60 inches, we find 97, the number of revolutions required. What is the number of revolutions a minute of an inward discharging Turbine, under a head of 21 feet, the diameter being, 72 inches? In the table for the inward discharging wheel, opposite 21 feet, and under 72 inches, we find 95, the number of revolutions a minute. Tl'ese Turbine tables were calculated by the author's brother, the late John O'Byrne, C. E., who died in New York, on the 6th of April, 1851. Outward dischargingy Turbine. |V DIDIAIMETER IN I INCHES. 24 30 36 42 48 54 60 66 2 78 7 84 90 96 3 100 80 70 60 52 42 37 35 32 30 28 27 21 4 111 89 73 63 57 49 44 41 37 31 4 32 30 2t8 5 123 100 82 71 62 55 51 45 42 38 37 8 3 31 6 135 109 91 78 68 62 55 50 45 42 38 37 36 7 146 118 96 84 73 65 59 53 49 47 4 i40 38 8 156 125 105 90 79 71 63 57 52 49 4300 42.~ 39 9 166 133 111 95 83 75 67 61 57 50 49 4. 41 10 175 140 117 100 87 79 70 64 59 55 51 47 4B 11 183 147 122 105 92 81 74 67 62 57 54 49 48 12 191 156 127 110 96 85 79 70 64 59 55 53 51 13 200 159 1383,15 100 89 81 73 67 62 57 55 53 14 206 166 138 118 104 92 83 75 69 64 59 57 56 15 213 171 142 122 107 95 86 78 72 66'61 58 56 16 222 177 148 126 111 98 89 82 74 69 64 59 57 17 227 182 152 131 115 101 91 83 77 71 66 I 59 18 234 187 156 134 117 105 94 85 78 73 67 63 6 i 19 238 193 161 138 120 107 97 88 81 74 6 64 6 20 247 197 164 141 124 110 99 90 84 76 71 6 6 21 252 202 168 145 126 114 101 92 85 78 73 68 6 22 259 208 172 149 129 115 105 94 87 0 74 69 6 23 263 212 176 151 133 119 106 96 89 84 77 70 24 270 216 180 155 135 120 109 98 92 85 < 7874 7 25 277 2o22 184 158 138 123 111 101 93 86 80 76 4 26 282 226 189 161 141 125 113 103 95 87 81 78 7 27 286 229 191 165 143 129 116 105 97 88 83 79 7 28 291 233 1 15 167 146 130 118 107 I 99 91 i 85 80 78 29 217 27 I 9)37 9 I 10 149 132 119 109 100 92 8t; 81 ) 1 303 t 203'741 L10 1l 2 135 122 111 1 (i 4 88 82 81 Page 332 332 TtIE PRACTICAL MODEL CALCULATOR. Inward disclarging Turbine.'.H 1 DIAMIETER IN INCHES. 24 30 36 42 48 54 60 66 72 78 9i 90 6 31 111 86 74 62 54 48 47 40 36 32 31 30 27 4 125 96 83 70 62 55 51 45 41 37 36 34 31 5 141 112 94 7 8 69 61 55 50 46 43 40 37 36 6 152 122 101 86 76 67 62 55 51 47 43 42 38 7 166 131 108 93 82 72 65 60 54 51 47 44 42 8 175 139 116 99 87 76 71 63 57 54 49 47 45 9 186 149 123 106 93 81 74 68 63 57 53 51 47 10 195 156 129 111 99 86 78 71 66 61 56 52 49 11 208 167 136 117 102 91 82 74 68 63 58 56 52 12 217 169 142 122 107 97 85 78 71 66 61 57 54 13 221 178 148 127 112 99 89 82 74 69 64 61 56 14 231 184 153 133 116 104 92 85 76 71 66 62 58 15 238 191 159 136 119 107 95 87 80 73 68 64 61 16 245 198 165 144 123 111 99 90 83 76 71 66 63 17 252 203 168 148 127 114 102 92 85 78 73 68 64 18 269 209i 173 150 132 116 104 95 87 82 75 69 66 19 267 21.5 176 153 134 120 108 98 89 83 77 72 67 20 276 22'2 183 157 138 122 111 101 93 85 79 74 69 21 288'226 18; 1;62 141 125 113 103 95 86 80 75 71 2 2 290 230 0 192 164 145 129 116 107. 96 89 83 77 73 23 299 235 196 167 146 133 118 109 97 91 84 79 74 24 303 240 201 171 151 135 122 111 101 93 86 80 75 25 310 247 206 176 155 138 123 112 104 96 88 82 76 26 314 2048 210 180 157 139 126 115 106 97 90 84 79 27 0 319 254 213 183 162 142 128 117 108 99 92 85 80 28 327 2631 218 186 164 146 129 119 109:102 93 87 82 29 33'3 265 221 189 166 148 133 121 111 103 95 8'3 83 30 3 3D)6 2'71 I 224! 19 3 168 151.1 33 1241 1114'105 97 90 85 WINDMILLS. 1. TIHE velocity of windmill sails, whether unloaded or loaded, so as to produce a maximum effect, is nearly as the velocity of the wind, their shape and position being the same. 2. The load at the maximrum is nearly, but somewlhat less than, as the square of the velocity of the wind, the shape and position of the sails being the same. 3. The effects of the same sails, at a maximum, are nearly, but somewhat less than, as the cubes of the velocity of the wind. 4. The load of the same sails, at the maximum, is nearly as the squares, and their effect as the cubes of their number of turns in a given time. 5. When sails are loaded so as to produce a maximum at a given velocity, and the velocity of the wind increases, the load continuing the same,-lst, the increase of effect, when the increase of the velocity of the wind is small, will be nearly as the squares of those velocities; 2dly, when the velocity of the wind is double, the effects will be nearly as 10 to 27~; but, 3dly, when the velocities compared are more than double of that when the given load produces a maximum, the effects increase nearly in thie simple ratio of the velocity of the wind. Page 333 WINDMILLS. 333 6. In sails where the figure and position are similar, and the velocity of the wind the same, the number of turns, in a given time, will be reciprocally as the radius or length of the sail. 7. The load, at a maximum, which sails of a similar figure and position will overcome, at a given distance from the centre of motion, will be as the cube of the radius. 8. The effects of sails of similar figure and position are as the square of the radius. 9. The velocity of the extremities of Dutch sails, as well as of the enlarged sails, in all their usual positions when unloaded, or even loaded to a maximum, is considerably greater than that of the wind. The results in Table 1 are for Dutch sails, in their common position, when the radius was 30 feet. Table 2 contains the most efficient angles. 1. 2. Number of Rati between Parts of the revolutionso f Velocity of velocityof radius which Angle with wind-shaft in the wind in wind d divided into the axis. Angle of weather. Ia minute. an hour. volutiono of r wind-shaft. SX parts. 3 2 miles 0X666 1 72 18 2 71 19 3 72 18 middle 5 4 miles 0 800 7 1 4 74 16 6 5 miles 0-833 2 2 __6 83 7 Supposing the radius of the sail to be 30 feet, then the sailPwill commence at 1, or 5 feet from the axis, where the angle of inclination will be 72 degrees; at 2, or 10 feet from the axis, the angle will be 71 degrees, and so on. Results of Experiments on the effect of Windmill Sails in grinding corn.-By MI. COULOMB. A windmill, with four sails, measuring 72 feet from the extremity of one sail to that of the opposite one, and 6 feet 7 inches wide, or a little more, was found capable of raising 1100 lbs. avoirdupois 238 feet in a minute, and of working, on an average, eight hours in a day. This is equivalent to the work of 34 men, 30 square feet of canvas performing about the daily work of a man. When a vertical windmill is employed to grind corn, the millstone makes 5 revolutions in the same time that the sails and the arbor make 1. The mill does not begin to turn till the velocity of the wind is about 13 feet per second. When the velocity of the wind is 19 feet per second, the sails make from 11 to 12 turns in a minute, and the mill will grind from 880 to 990 lbs. avoirdupois in an hour, or about 22,000 lbs. in 24 hours. Page 334 334 THE APPLICATION OF LOGARITHMS. THE practice of performing calculations by Logarithms is an exercise so useful to computers, that it requires a more particular explanation than could have been properly given in that part of the work allotted to Arithmetic. A few of the various applications of logarithms, best suited to the calculations of the engineer and mechanic, have therefore been collected, and are, with other matter, given, in hopes that they will come into general use, as the certainty and accuracy of their results can be more safely relied upon and more easily obtained than with common arithmetic. By a slight examination, the student will perceive, in some degree, the nature and effect of these calculations; and, by frequent exercise, will obtain a dexterity.of operation in every case admitting of their use. He will also more readily penetrate the plans of the different devices employed in instrumental calculations, which are rendered obscure and perplexing to most practical men by their ignorance of the proper application of logarithms. Logarithms are artificial numbers which stand for natural numbers, and are so contrived, that if the logarithm of one number be addled to the logarithm of another, the sum will be the logarithm of the product of these numbers; and if the logarithm of one number be taken from the logarithm of another, the remainder is the logarithm of the latter divided by the former; and also, if the logarithm of a number be multiplied by 2, 3, 4, or 5, &c., we shall have the logarithm of the square, cube, &c., of that number; and, on the other hand, if divided by 2, 3, 4, or 5, &c., we have the logarithm of the square root, cube root, fourth root, &c., of the proposed number; so that with the aid of logarithms, multiplication and division are performed by addition and subtraction; and the raising of powers and extracting of roots are effected by multiplying or dividing by the indices of the powers and roots. In the table at the end of this work, are given the logarithms of the natural numbers, from 1' to 1000000 by the help of differences; in large tables, only the decimal part of the logarithm is given, as the index is readily determined; for the index of the logarithm of any number greater than unity, is equal to one less than the number of figures on the left hand of the decimal point; thus, The index of 12345' is 4', 1234'5 - 38, 123'45- 2-, 12'345- 1-, 1'2345- 0 Page 335 THE APPLICATION OF LOGARITHMS. 335 The index of any decimal fraction is a negative number equal to one and the number of zeros immediately following the decimal point; thus, The index of'00012345 is -4' or 4 -0012345 is — 3 or 3' - 012345 is — 2 or 2" ~12345 is — 1 or 1' Because the decimal part of the logarithm is always positive, it is better to place the negative sign of the index above, instead of before it; thus, 3' instead of -3. For the log. of -00012345 is better expressed by 4'0914911, than by — 40914911, because only the index is negative-i. e., 4 is negative and'0914911 is positive, and may stand thus, — 4 + -0914911. Sometimes, instead of employing negative indices, their complements to 10 are used: for 4'0914911 is substituted 6'0914911 -'0914911 - 7'0914911 - 2.0914911 8'0914911 &c. &c. When this is done, it is necessary to allow, at some subsequent stage, for the tens by which the indices have thus been increased. It is so easy to take logarithms and their corresponding numbers out of tables of logarithms, that we need not dwell on the method of doing so, but proceed to their application. MULTIPLICATION BY LOGARITHMS. Take the logarithms of the factors from the table, and add them together; then the natural number answering to the sum is the product required: observing, in the addition, that what is to be carried from the decimal parts of the logarithms is always positive, and must therefore be added to the positive indices; the difference between this sum and the sum of the negative indices is the index of the logarithm of the product, to which prefix the sign of the greater. This method will be found more convenient to those in-ho have only a slight knowledge of logarithms, than that of using thie aritLhmetical complements of the negative indices. 1. Multiply 37'153 by 4'086, by logarithms. _Aos. Logs. 37'153.............................1'5699939 4'086.0.61128...................0 61984 Prod. 151'8071....2....................'1812923 2. Multiply 112'246 by 13'958, by logarithms. XNos. Logs. 112'246................................2'0501709 13958....................................1'1448232 Prod. 1566'729..........................3'1949941 Page 336 336 THE PRACTICAL MODEL CALCULATOR. 3. Multiply 46'7512 by'3275, by logarithms. NAos. Logs. 46'7512....................................1'6697928'3275.................... 15152113 Prod. 15'31102......................... 1'1850041 Here the +1 that is to be carried from the decimals, cancels the — 1, and consequently there remains 1 in the upper line to be set down. 4. Multiply'37816 by'04782, by logarithms. NAos. Logs.'37816....................................'5776756'04782................ 2'6796096 Prod. 0'0180836........................2 2572852 Here the +1 that is to be carried from the decimals, destroys the -1 in the upper line, as before, and there remains the -2 to be set down. 5. Multiply 3'768, 2'053, and'007693, together. N os. Logs. 3'768....................... 0'5761109 2'053.................................. 03123889'007693..8...................60957 Prod.'0595108............5...... 2'7745955 Here the +1 that is to be carried from the decimals, when added to — 3, makes -2 to be set down. 6. Multiply 3'586, 2'1046,'8372, and'0294, together. _Nos. Logs. 3'586....................................... 05546103 2'1046.....................................0'3231696 *8372..............1.... 1'9228292 *0294......................................24683473 Prod.'1857618.........................1'2689564 Here the +2 that is to be carried, cancels the -2, and there remains the -1 to be set down. DIVISION BY LOGARITHMS. From the logarithm of the dividend, subtract the logarithm of the divisor; the natural number answering to the remainder will be the quotient required. Observing, that if the index of the logarithm to be subtracted is positive, it is to be counted as negative, and if negative, to be considered as positive; and if one has to be carried from the decimals, it is always negative: so that the index of the logarithm of the quotient is equal to the sum of the index of the dividend, the index Page 337 THE APPLICATION OF LOGARITHMS. 337 of the divisor with its sign changed, and -1 when 1 is to be carried from the decimal part of the logarithms. 1. Divide 4768'2 by 36'954, by logarithms. Nos. Logs. 4768'2........................3'6783545 36'954....................................1'5676615 Quot. 129'032..........................2'1106930 2. Divide 21'754 by 2'4678, by logarithms. Nos. Logs. 21'754....................................13375391 24678...................................03923100 Quot. 8'81514........0...................09452291 3. Divide 4'6257 by'17608, by logarithms. Nos. Logs. 4'6257................................0'6651775 ~17608.................................12457100 Quot. 26'27045.........................1'4194675 Here the — 1 in the lower index, is changed into +1, which is then taken for the index of the result. 4. Divide'27684 by 5'1576, by logarithms. Nos. Logs.'27684................................. 14422288 5'1576....................................7124477 Quot.'0536761..................2..7297811 I-ere the 1 that is to be carried frdni the decimals, is taken as -1, and then added to — 1 in the'' upper index, which gives — 2 for the index of the result. 5. Divide 6'9875 by'075789, by logarithms. Nos. Logs. 6'9875.................................... 08443218 -075789................. 8796062 Quot. 92.1967...........................1.9647156 Here the 1 that is to be carried from the decimals, is added to -2, which makes — 1, and this put down, with its sign changed, is +1. 6. Divide'19876 by'0012345, by logarithms. Nos. Logs. *19876............................*......2983290 *0012345...............................3' 0914911 Quot. 161'0043........................2'2068379 Here — 3 in the lower index, is changed into +3, and this added to 1, the other index, gives + 3 - 1, or 2. 2 D 22 Page 338 338 THE PRACTICAL MODEL CALCULATOR. PROPORTION; OR, THE RULE OF THREE, BY LOGARITHMS. From the sum of the logarithms of the numbers to be multiplied together, take the sum of the logarithms of the divisors: the remainder is the logarithm of the term sought. Or the same may be performed more conveniently, for any single proportion, thus:-Find the complement of the logarithm of the first term, or what it wants of 10, by beginning at the left hand and taking each of the figures from 9, except the last figure on the right, which must be taken from 10; then add this result and the logarithms of the other two figures together: the sum, abating 10 in the index, will be the logarithm of the fourth term. 1. Find a fourth proportional to 37'125, 14'768, and 135'279, by logarithms. Log. of 37'125..........................1696665 Complement..........................8....84303335 Log. of 14'768........................... 11693217 Log. of 135'279.......................2..1312304 Ans. 53'8128............................. 17308856 2. Find a fourth proportional to'05764,'7186, and'34721, by logarithms. Log. of'05764......................... 2'7607240 Complement..................... 11'2392760 Log. of'7186......................... 1'8564872 Log. of'34721......................... 1'5405922 Ans. 4'32868......................... 06363554 3. Find a third proportional to 121796 and 3'24718, by logarithms, Log. of 12'796........................... 1 1070742 Complement.............................. 88929258 Log. of 3'24718.......................... 05115064 Log. of 3'24718.......................... 05115064 Ans.'8240216.................... 1'9159386 INVOLUTION; OR, THE RAISING OF POWERS, BY LOGARITHMS. Multiply the logarithm of the given number by the index of the proposed power; then the natural number answering to the result will be the power required. Observing, if the index be negative, the index of the product will be negative; but as what is to be carried from the decimal part will be affirmative, therefore the difference is the index of the result. 1. Find the square of 217568, by logarithms. Log. of 217568..........................04404053 2 Square 7'599947........................ 08808106 Page 339 THE APPLICATION OF LOGARITHMS. 339 2. Find the cube of 7'0851, by logarithms. Log. of 7'0851...........................0'8503460 3 Cube 355-6625........................25510380 Therefore 355'6625 is the answer. 3. Find the fifth power of'87451, by logarithms. Log. of'87451..........................1'9417648 5 Fifth power'5114695.................. 17088240 Where 5 times the negative index 1, being — 5, and +4 to carry, the index of the power is 1. 4. Find the 365th power of 1'0045, by logarithms. Log. of 1'0045.................... 00019499 365 97495 116994 58497 Power 5'148888...................Log. 0-7117135 EVOLUTION; OR, THE EXTRACTION OF ROOTS, BY LOGARITHMS. Divide the logarithm of the given number by 2 for the square root, 3 for the cube root, &c., and the natural number answering to the result will be the root required. But if it be a compound root, or one that consists both of a root and a power, multiply the logarithm of the given number by the numerator of the index, and divide the product by the denominator, for the logarithm of the root sought. Observing, in either case, when the index of the logarithm is negative, and cannot be divided without a remainder, to increase it by such a number as will render it exactly divisible; and then carry the units borrowed, as so many tens, to the first figure of the decimal part, and divide the whole accordingly. 1. Find the square root of 27'465, by logarithms. Log. of 27'465........................2) 14387796 Root 5'2407.....................7193898 2. Find the cube root of 35'6415, by logarithms. Log. of 35'6415..................... ) 1-5519560 Root 3'29093..............................5173186 3. Find the fifth root of 7'0825, by logarithms. Log. of 7'0825........................5 ) 08501866 Root 1'479235.............................1700373 Page 340 340 THE PRACTICAL MODEL CALCULATOR. 4. Find the 365th root of 1'045, by logarithms. Log. of 1'045......................365) 0'0191163 Root 1'000121..................0.........00000524 5. Find the value of ('001234)3, by logarithms. Log. of'001234..........................3'0913152 2 3) 6.1826304 Ans.'00115047..........................2'0608768 Here the divisor 3 being contained exactly twice in the negative index -6, the index of the quotient, to be put down, will be -2. 6. Find the value of ('024554)2, by logarithms. Log. of'024554................2....3901223 3 2) 61703669 Ans.'00384754......................... 35851834 Here, 2 not being contained exactly in -5, 1 is added to it, which gives -3 for the quotient; and the 1 that is borrowed being carried to the next figure makes 11, which, divided by 2, gives ~5851834 for the decimal part of the logarithm. METHOD OF CALCULATING THE LOGRITHM OF ANY GIVEN NUMBER, AND THE NUMBER CORRESPONDING TO ANY GIVEN LOGARITIM. DISCOVERED BY OLIVER BYRNE, THE AUTHOR OF THE PRESENT WORK. The succeeding numbers possess a particular property, which is worth being remembered. log. 1'371288574238542 = 0'1371288574238542 log. 10'00000000000000 = 1'000000000000000 log. 237'5812087593221 = 2'375812087593221 log. 3550'260181586591 = 3'550260181586591 log. 46692'46832877758 = 4'669246832877758 log. 576045'6934135527 = 5'760456934135527 log. 6834720'776754357 = 6-834720776754357 log. 78974890'31398144 - 7'897489031398144 log. 895191599'8267852 = 8'951915998267839 log. 9999999999'999999- 9999999999999999 In these numbers, if the decimal points be changed, it is evident the logarithms corresponding can also be set down without any calculation whatever. Thus, the log. of 13741288574238542 = 2'1371288574238542; the log. of 35'50260181586591 = 1-550260181586591; log.'002375812087593221- 3'375812087593221; log.'0008951915998267852 = 4'951915998267852; Page 341 THE APPLICATION OF LOGARITHMS. 341 and so on in similar cases, since the change of the decimal point in a number can only affect the whole number of its logarithm. These numbers whose logarithms are made up of the same digits will be found extremely useful hereafter. We shall next give a simple method of multiplying any number by any power of 11, 101, 1001, 10001, 100001, &c. This multiplication is performed by the aid of coefficients of a binomial raised to the proposed power. (x + y)' = x + y, the coefficients are 1, 1. (X + y)2 = X2 + 2xy + y2, the coefficients are 1, 2, 1. (x + y)3 = x3 + 3x2y + 3xy2 + y3, the coefficients are 1, 3, 3 1. The coefficients of(x + y)4 are 1, 4, 6, 4, 1. --- -- (X + y)5- 1, 5, 10, 10, 5, 1. --- -- (x + y)6- 1, 6, 15, 20, 15, 6, 1. - - (x + y)7 — 1, 7, 21, 35, 35, 21, 7, 1. - -- (X + y)8- 1, 8, 28, 56, 70, 56, 28, 8, 1. - -- (x + y)9- 1, 9, 36, 84, 126, 126, 84, 36, 9, 1. Let it be required to multiply 54247 by (101)6. The number must be divided into periods of two figures when the multiplier is 101; into periods of three figures when the multiplier is 1001; into periods of four figures when the multiplier is 10001; and so on. e d c b a 54 24 70 00 00 1 3 25 48 2000 a 6 8 13 70 50 6 15 10 84!94 c 20 8114 d 15 I 3 e 6 (54247) x (101)6 = 57 58 42 83 61, true to 10 places of figures. This operation is readily understood, since the multipliers for the 6th power are 1, 6, 15, 20, 15, 6, 1; we begin at a, a period in advance, and multiply by 6; then we commence at b, two periods in advance, and multiply by 15; at c, three periods in advance, and multiply by 20; at d, four periods in advance (counting from the right to the left), and multiply by 15; the period, e, should be multiplied by 6, but, as it is blank, we only set down the 3 carried from multiplying d, or its first figure by 6. As it is extremely easy to operate with 1, 5, 10, 10, 5, 1, the multipliers for the 5th power, it may be more convenient first to multiply the given number by (101)5, and then by (101)1; because, to multiply any number by 5, we have only to affix a cipher (or suppose it affixed) and to take the half of the result. The above example, if worked in the manner just described, will stand as follows: 2D2 Page 342 342 THE PRACTICAL MODEL CALCULATOR. 54124 70 00 00..... 2171123 50 00.....5..a 5 42 47 00...1O..b 5 42 47...10..e 71.....5..d 1..1 (54247) x (101)5 57101/41 42 19 57o0141142 57 58 42 83 61 = (54247)6 x (101)6. The truth of this is readily shown by common multiplication, but the process is cumbersome. However, for the sake of comparison, we shall in this instance multiply 54247 by (101) raised to the 6th power. 101 101 101 1010 10201 = (101)2. 101 10201 102010 1030301 = (101)3. 101 1030301 10303010 104060301 = (101)4. 101 104060401 1040604010 10510100501 = (101)3. 101 10510100501 105101005010 1061520150601 = (101)6. 54247 7430641054207 4246080602404 2123040301202 4246080602404 5307600753005 575842836019652447 the required product, Page 343 ~HE APPLICATION OF LOGARITHMS. 343 which shows that the former process gives the result true to 10 places of figures, of which we shall add another example. Multiply 34567812 by (1001)8, so that the result may be true to 12 places of figures. 3456 7812 0000.....1 2 76542496..... 9 6790...28..6b 19...56..e 3459 5475 9305 the required product. The remaining multipliers, 70, 56, 28, 8, 1, are not necessary in obtaining the first 12 figures of the product of 34567812 by 10001 in the 8th power. As 28 and 56 are large multipliers, the work may stand thus Cb a 3456 7812 0000...... 1 2 7654 2496..a.. 8 6 9136...b..20 28 2 7654..... 8 28 17....50 56 Result, = 345954759305 the same as before. Perhaps this product might be obtained with greater ease by first multiplying 34567812 by (10001)5, and the product by (10001)3; the operation will stand thus: 345678120000...... 1 172839060...... 5 34568......10 3.......10 345850093631 = 34567812 x (10001)5. 103755298..... 3 10376...... 3 345954759305 = twelve places of the product of 34567812 by (10001)5 x (10001)3 = (34567812) x (10001)8. Although these methods are extremely simple, yet cases will occur, when one of them will have the preference. Our next object is to determine the logarithms 1'1; 1'01; 1'001; 1'0001; 1'00001; &c. It is well known that log. (1 + n) = M (n - In2 + In3 - qn4 + -1n5 - 1n6 + &c.) M being the modulus, ='432944819032618276511289, &c. It is evident that when n is I,- 1, H, &c., the calculation becomes very simple. Page 344 344 THE PRACTICAL MODEL CALCULATOR. M ='4342944819032518 M = 2171472409516259 M = 1447648273010839 M =- 1085736204758130 M M = -0868588963806504 M = 0723824136505420 M -= 0720420788433217 M - 0542868102379065 M = 0482549424336946 AM -- 0434294481903252 &c. &c., are constants employed to determine the logarithms of 11, 101, 1001, 100001, &c. To compute the log. of 1'001. In this case n = + 1o000 = *0004342944819033 positive - (100)2-0000002171472410 negative.0004340773346623 + (1000)3 = 0000000001447648 positive ~0004340774794271 - (1000)4 -0000000000001086 negative -0004340774793185 iM (1000) ='0000000000000001 positive'0004340774793186 = the log. of 1'001; true to sixteen places. It is almost unnecessary to remark, that, instead of adding and subtracting alternately, as above, the positive and negative terms may be summed separately, which will render the operation more concise. Positive Terms. Negative Terms.'0004342944819033'0000002171472410 1447648 1086 1'0000002171473496 +-'0004342945266682 - 000000217473496 -0004340774793186 = log. 1'001. In a similar manner the succeeding logarithms may be obtained to almost any degree of accuracy. Page 345 THE APPLICATION OF LOGARITHMS. 345 Log. 1'1 ='041392685158225 &c. which we call A 1'01 ='004321373782643 - B 1001 ='000434077479319 - C 1'0001 ='000043427276863 - - D 1'00001 =- 000004342923104 - - E 1'000001 ='000000434294265 - F 1'0000001 = -000000043429447 - G 1'00000001 =- 000000004342945 H- 1'000000001 =- 000000000434295 - I 1'0000000001 -'000000000043430 - - J 1,00000000001 ='000000000004343 - - K 1'000000000001 =- 000000000000434 - L 1'0000000000001 =- 000000000000043 - M 1'00000000000001 ='000000000000004 - N &c. &c. &c. Without further formality or paraphernalia, for it is presumed that such is not necessary, we shall commence operating, as the method can be acquired with ease, and put in a clearer point of view by proper examples. Required the logarithm of 542470, to seven places of decimals. 542470. 3254820 8 1371 10 85 8 5 7 5 814 2 8 4 = 6B ='02592824 17275 3 Take 5 7 6 01 5 6 2 = 3D -'00013028. From 5 7 6 0 4 5 6 9 576) * *' 31007 28 8 0 =5 E = 00002171 112 7 1 1 5 = 2 F ='00000087 112 1 2 = 2 G = -00000009 ~02608119 Take 5'76045693 From Hence we have log. 542470 = 5'73437574, which is correct to seven decimal places. 6B is written to represent 6 times the log. of 1'01. The nearest number to 542470, whose log. is composed of the same digits as itself, being 576045'6934, &c., our object was to raise 542470' to 576045'69 by multiplying 542470' by some power or powers of 1'1, 1'01, 1'001, 1'0001, &c. Page 346 346 THE PRACTICAL MODEL CALCULATOR. It is here necessary to remark, that A is not employed, because the given number multiplied by 1'1, would exceed 576045'69; for a like reason C is omitted. Again, when half the figures coincide, the process may be performed (as above) by common division; the part which coincides becoming the divisor; thus, in finding 5 E, 576 is divided into 3007, it goes 5 times, the E showing that there are five figures in each period at this step. For A, there is but one figure in each period; for B, there are two figures; for C, there are three figures in each period, and so on. Let it be required to calculate the logarithm of 2785'9, true to seven places of decimals. It will be found more convenient, in this instance, to bring the given number to 3550'26018, the log. of which is 3'55026908. 2 78 5 9 0 3 3 7 0o9 3!9 0 - 2 A -= 08278537 16815 47 0 3l3709 337 2 3542 8 908 = 5S = *02160687 7 0 858 35 Take 3 5 4 99 8 01 = 2 C = 00086815 From 3 5 5 0 2 6 0 2 355)... 218 01 = E =.00003040 2485 31 6 = 8 F = 00000347 218 4 312 - 9 G- ='00000039 312 Take'10529465 From 3'55026018 log. 2785'9 = 3'44496553 At the Observatory at Paris, g = 9'80896 metres, the second being the unit of time, what is the logarithm of 9'80896? In this example, we shall bring 9'80896 to 9'99999, &c. Page 347 THE APPLICATION OF LOGARITHMS. 347 9810819600 9907049600 = 1 B = *0043213738 891 63 446 356654 832 9 9 9 65 70532 = 9 C = -0039066973 2 9 9 8 9 7 2 300 9999569804 =3 D = =0001302818 3i9 9 9 8 3 6 Take 9999969793=4E= 0000173717 From 1 0 0 0 0 0 0 0 0 0 0.... 30207 From which we have......... 3 F = -0000013029 2 H ='0000000087 7J ='0000000003 Take'0083770365 From 1'0000000000 Log. 9'80896 = 9916229635 As before observed, 9 C might have been obtained in the following manner: 8 9 0 7 0 496010 = 1 B, as above. 41953 52 48 99070 99 5times 995668401 7 39826736 5973 9 40 4times 9996570532 = 9C. A French metre is equal to 3'2808992 English feet, required the log. of 3'2808992. e d c 6 a 32 80 89 92 00...once 2296629 44... 7 times from a 6 88 98 88...21 - b 11 48 31...35 c 11148...35 - d [ 7... 21 e 35 17 56 8018 = B 7. Page 348 348 THE PRACTICAL MODEL CALCULATOR. The manner in which B 7 is obtained is worthy of remark: the multipliers being 1, 7, 21, 35, 35, 21, 7, 1, when 7 times the first line (commencing with the period marked a) is obtained, 21 times the same line (commencing with the period marked b) is determined by multiplying the 2d line by 3. If the 2d line be again multiplied by 5, we have the 4th line of the multiplier 35; but to multiply by 5, we have only to take the half the product produced by multiplying by 7, advancing the result one figure to the right. Hence, to find the result for 35 is almost as easy as to find the result for 5. But the object in this case being to bring the proposed number to 35502601815, the process must be continued. c b a 1 351756 801 8 = 7, as above. 9 3 165 811 2 36 12!663!2 84 29 6 354 935 305 8 = C 9 The 2d (or 9) line is produced by beginning at a, but the multiplication may be performed by subtracting 3517568 from 35175680; the 36 line is produced by beginning at b, observing to carry from the preceding figure, making the usual allowance when the number is followed by 5, 6, 7, 8, or 9. The 36 line may be produced by multiplying the 9 line by 4, beginning one period more to the left. To multiply by 84 is not apparently so convenient, for 84 x 352 = 291568; and as only one figure of the period 568 is required, when the proper allowance is made, the result becomes 2916. But, since 84 is equal to 36 x 21, we have only to multiply the 36 line by 2, and add - of it; with such management, the work will stand thus:3511756 180118 = B 7, as before 31165 81112 = 9 times 12 66312 = 36 times 24 3 = 72 times 84 times 42 = 12 times 4tmes 354 935 305 8 - C 9 This amounts to very little more than adding the above numbers together. Many other contractions will suggest themselves, when the mulpliers are large: thus, to multiply any number 57837 by 9, as alluded to above, is easily effected, by the following well-known process:-Subtract the first figure to the right from 10, the second from the first, the third from the second, and so on. 578370...ten times Thus, 57837 x 9 = 57837... once 52 0533...nine times Page 349 THE APPLICATION OF LOGARITHMS. 349 Such simple observations are to be found in every book on mental arithmetic, and therefore require but little attention here. The whole work of the previous example will stand thus:328018 99200 229662944 6889888 114831 1148 + 7 B 7= 3 517 5 68 0 18 = 0302496165 = 7 B 3165811 12 126632 296 C 9 = 3 5 4 913 5 3 05 8 = 0039066973 = 9 C 7098 71 35 D 2= 3 5 5 0 016 2 9 6 4 = 0000868546 - 2 D 17 75 03 4 Take E 5 3550240471 ='0000217146 = 5 E From 3 5 5 02 6 0 182 3550)... 19711 F 5 " 17 7 5 50 = 0000021715 -- 5 F 1961 G5 1 1775 ='0000002172 5 G 1186 1 5 1 7 8 = -0000000217 = 5 8 12 7 = 0000000009 = 12 J3 11= 0000000001 = J3 Take'0342672944 From 3'5502601816 Log. 3280'8992 35159928972.'. log. 3-2808992 = 05159928972. The constant sidereal year consists of 365'25636516 days; what is the log. of this number? In this case it is better to bring the constant 35502601816 to 36525636516, instead of bringing the given number to the constant, as in the former examples. 2E Page 350 350 THE PRACTICAL MODEL CALCULATOR. 7110052036 3550260 B2 =3 621 6 204112 = -0086427476 -=2 B 289729633 10 14054 2028 8 = 6 5 06 9 4 9I827 = -0034726298 = 8 C 18253475 3651 Take D 5 = 3 6 5 2 5 2 06 9 5 3 = -0002171364 =5 D From 36525636516 36525'2) 429563 E 1 = 365252 -'0000043429 = 1E 64311 F1 = 36525 2= -0000004343 =lF 27786 G 7 = 25 5 6 8=- 0000003040 = 7 G 2218 11 6 21 9 1 = 0000000261 6 H I0 27 J 7 = 215 =.0000000003 = 7J.0123376214 Add 3.5502601816 Hence, log. 3652'5636516 = 3'5625978030.*. log. 365'25636516 = 2'562597803. M. Regnault determined with the greatest care the density of mercury to be 13'59593 at the temperature 0~, centigrade. It is required to calculate the log. of 13'59593, to eight places of decimals. In this case it is better to bring the given number to the constant 1371288574. 1 3 519 5 913 0 0 119 8 7(6 7 4 3s8 0 7 8 C8 = 137050788 ='003472630 = 8C 6 8 5 25 14 Subtract D 5 1 3 7 1 1 9 3 2 8 = 000217136 = 5 D From 137128857 9;5 2 9 ='000026058 = E 6 E6 = 8227 1 3[0 2 F9 = 123 4 = 000003909 = F 9 G68 H 5 = 6 9 ='000000022 = H 5'003719755 Page 351 APPLICATION OF LOGARITHMS. 351 Take'003719755 From'137128857 log. 1'359593 ='133409102.'. log. 13'59593 = 1'133409102. TO DETERMINE THE NUMBER CORRESPONDING TO A GIVEN LOGARITHM. This problem has been very much neglected-so much so, that none of our elementary books ever allude to a method of computing the number answering to a given logarithm. When an operation is performed by the use of logarithms, it is very seldom that the resulting logarithm can be found in.the table; we have, therefore, to find the nearest less logarithm, and the next greater, and correct them by proportion, so that there may be found an intermediate number that will agree with the given logarithm, or nearly so. But although the proportional parts of the difference abridge this process, we can only find a number appertaining to any logarithm to seven places of figures when using our best modern tables. As, however, the tabular logarithms extend only to a degree of approximation, fixed generally at seven decimal places, all of which, except those answering to the number 10 and its powers, err, either in excess or defect, the maximum limit of which is I in the last decimal, and since both errors may conspire, the 7th figure cannot be d pended on as strictly true, unless the proposed logarithm falls between the limits of log. 10000 and log. 22200. Indubitably we are now speaking of extreme cases, but since it is not an unfrequent occurrence that some calculations require the most rigid accuracy, and many resulting logarithms may be extended beyond the limits of the table, this subject ought to have a place in a work like the present. It is not part of the present design to enter into a strict or formal demonstration of the following mode of finding the number corresponding to a given logarithm, as the operation will be fully explained by suitable examples. What number corresponds to the logarithm 3'4449555? The next less constant log. to the one proposed is 2 37581209, or rather, 3'37581209, when the characteristic or index is increased by a unit. Secondly. First from 344496555 213 7 5 8 1 2 0 9 constant take 3-37581209 23 758121 = A1 *06915346 2 6 1 3 9:3 39 0'04139269 = A 15 6 8 03 610 3 92 0 9 *02776077 3j9 200 *02592824 = 6 B 39..183253 **178;31 _ 4 C 2 7 171 109 6 5 8 6 173631=40 11 0 966 8.... 9622 1l6 6 4 8685 =2 D 1..... 937 27851282918= C4 Page 352 352 THE PRACTICAL MODEL CALCULATOR......937 278528298 = C4 869 2 E i5 5 7 0 6...... _68 3 43 = 1F 27858 4 0 0 7= D2 1572 = E 2......25 279=F1 22 =G5 G279 1319 = G5..... 3 1 9- H7 3=7H 278590016... 2785'90016 is the number sought. What number corresponds to the logarithm 5'73437574? When the index of this log. is reduced by a unit, the nearest next less constant is 4'66924683. From 4'73437574 Take 4'66924683'6512891 4139269.........1 A'2373622 21.60687......... B:.212035 173631.........4 C...39304 39085........9 D..... 219 There is neither the equal of 217.........5 F this number, nor a....... 2.........0 G less, obtainable from 2.........4 II E, o.. E 0, or E, is - omitted. Then, 4166924683 46692468.........A 1 5 36 17 15 1 25680858 513 617 5 136 26 5398167 8......... B 5 21 59267 3239 2 54197929 6........ C 4 4877 81 1915 54246172 7 2......... D 9 12,7 12.........F 5 2 2......... H 4 54247000 6... 542470'006 is the number whose logarithm is 5'73437574. Page 353 THE APPLICATION OF LOGARITHMS. 353 Had the given logarithm represented a decimal with a positive index, the required number would be 0'000054247, &c.; or if written with a negative index, as 5'73437574, the result would be the same, for the characteristic 5, shows how many places the first significant figure is below unity. Required the number corresponding to log. 2'3727451. The constant 100000000 is the one to be employed in this case. 1'3727451 the given log. minus 1 in the index. 1.0000000.3727451 3725342.........9 A...2109 1737.........4 D.... 372 347........8 E..... 25 22.........5 F 3 3....... 7 G 1' 0 0 0l0 1 00 Constant. 190000ol00o o 840000 112 6 0 0 0 t2126 0o0 84 0 3 6 9 23579485 A 9 2358 8191118 D 4 181917 E 8 118 F 5 16 G 7 23'590949... 235'90949 is the required number, and the seconds in the diurnal apparent motion of the stars. 235.90949"' - 3' 55'90949". Let it be required to find the hyperbolic logarithm of any number, as 3'1415926536. The common log. of this number is ~49714987269 (33), and the common log. of this log. is 1'6964873. The modulus of the common system of logarithms is -4342944819, &c..1. 1: 4342944819:: hyperbolic log. N: common log. N. 2E2 23 Page 354 354 THE PRACTICAL MODEL CALCULATOR. To distinguish the hyperbolic logarithm of the number N fiom its common logarithm, it is necessary to write the hyp. log. Log. N, and the common logarithm log. N. Hence, 4342944819 x Log. N = log. N; or log. ('4342944819) + log. (log. N)= log. (log. N)..'. log. (Log. N) = log. (log. N) - 1'6377843; for 1'6377843 = log..4342944819. Now, to work the above example, from 1'6964873 take 1'6377843 ~0587030, the number corresponding to this corn. log. will be the hyp. log. of 381415927. *0587030 must be reduced to'0000000 which is known to be the log. of 1.'0587030 1 A = 110 0100 0 0 0413927 1 A 44 000 00.173103 6 60 00 172855 4 B 11... 248 11 217 5E 1144 6644 1 B 4..... 31 810 1 = F 7 30 7 F 423=G2...... 1 2G 114472988.~. 1.14472988 is the hyperbolic log. of 3'1415927, true to the last figure; for the hyp. log. 3'1415926535898 = 1'1447298858494. The reason of this operation is very clear, because 1 x 1.1 x (1.01)4 x (1.00001)5 x (1000001)7 (1.0000001)2 = 1-14472988. This example answers the purpose of illustration, but the hyp. log. of 3'1415927 can be more readily found by dividing its com. log.'49714987269 by the constant'4342944819, which is termed the modulus of the common system of logarithms. Suppose it is known that 1'3426139 is the log. of the decimal which a French litre is of an English gallon. Required the decimal. The index, 1, may be changed to any other characteristic, so as to suit any of the constants, as the alteration is easily allowed for when the work is completed. In this instance, it is best to put + 1 instead of 1. From 1'3426139 110'O 0:0'00 Constant Take 1'0000000 8 0l00 0 0 0'3426139 3311415 = 8 A 5 60000 () ( 70100 0 0 ~0114724 56000..86427 = 2 B 28 28297 8$0 26045 = 6 C 1 2252 2 14 315 88 811 -= A 8 Page 355 THE APPLICATION OF LOGARITHMS. 355 2252 21143i58881 =A 8 2171=-5D 42187 17 8 81 21436 43 = 1 E 218 667 495 = B 2 38 11312 005 35=8F 3 280 3 4 3 =7 G 219982784 =C 6 109991 22 2201 El 117 6 1 = F 8 754 = G 7 220096913.'. The French litre -.2200969 English gallons. In measuring heights by the barometer, it is necessary to know the ratio of the density of the mercury to that of the air. At Paris, a litre of air at 0~ centigrade, under a pressure of 760 millimetres, weighs 1'293187 grammes. At the level of the sea, in latitude 450, it weighs 1'292697 grammes. A litre of water, at its maximum density, weighs 1000 grammes, and a litre of mercury, at the temperature of 0~ cent., weighs 13595-93 grammes: 13595'93 *13292697 = the ratio at 450'1'-292697 Now, log. 13595'93 = 4'133409102 (29) and log. 1'292697 = 0'111496744 (30) 4'021912358 = the log. of the ratio at 450. To find the number corresponding to this log., it is necessary to reject the index for the present, and reduce the decimal part to zero. By this means the necessity of using any of the constants is superseded..021912358 10 ~0 00l0 0 0l20 0 ~021606869 - 5 B 5 00 00 00 0...305489 1000000 303991 = 7 D 0 0 50.....1498 1051101005 =B5 1303 3 F 3 735 7...... 195 22 174 = 4G 0 517 4598 D 7 3 1 6 =F 3....... 21 17=4H 42 G4 4 =H4 4 1=1I9 4= 9 I 1105174961 13595'93. by logarithms, 13292697 = 10517'49, &c., which is easily verified by common division. Page 356 356 THE PRACTICAL MODEL CALCULATOR. M. Regnault found that, at Paris, the litre of atmospheric air weighs 1'293187 grammes; the litre of nitrogen 1'256167 grammes; a litre of oxygen, 1'429802 grammes; of hydrogen, 0-089578 grammes; and of carbonic acid, 1'977414 grammes. But, strictly considered, these numbers are only correct for the locality in which the experiments were made; that is for the latitude of 48~ 50' 14" and a height about 60 metres above the level of the sea; M. Regnault finds the weight of the litre of air under the parallel of 450 latitude, and at the same distance from the centre of the earth as that which the experiments were tried, to be 12'926697. Assuming this as the standard, he deduces for any other latitude, any other distance from the centre of the earth, the formula, 1'292697 (1'00001885) (1 - 0'002837) cos. 2 w 2h l+R Here, w is the weight of the litre of air, R the mean radius of the earth = 6366198 metres, h the height of the place of observation above the mean radius, and X the latitude of the place. At Philadelphia, lat. 390 56' 51'5", suppose the radius of the earth to be 6367653 metres, the weight of the litre of air will be 1'2914892 grammes. The ratio of the density of mercury to that of air at the level of the sea at Philadelphia is 10527'735 to 1; required the number of degrees in an arc whose length is equal to that of the radius. 360 As 3'1415926535898; 1:: 2: the required degrees. Log. 360 = 2-556302500767 log. 3'14159265359 = 0497149872694 2'059452623073 log. 2 = 0301029995664 1'758122632409 = the log. of the number required. When the index of this log. is changed into 4, the nearest next less constant is 4'669246832878. From 4'758122632409 416:69 2 l4 6 813 218 7 8 = Constant Take 4'669246832878 9 3 384 9 31 665 7 6 ~088875799531 416 6912 4 6 83 2 9 2 A = *82785370316 5 614 97 8866 7 7 83 A 2..6090429215 5 6 4 9 7 8 8 616 7;8 1B= 4321373783 5701' 628655441(61 =B1..1769055432 2 2 85 1 4 6 2 1 8 4 C = 1736309917 34 2 3 7 7 1 9....32745515 22 8 215 7 E = 30400462 6.....2345053 572911459611229 = C 4 Page 357 TIHE APPLICATION OF LOGARITHMS. 357..... 2345053 57291'45961'229 C 4 5F = 2171471 4 1 0 040217...... 173582 12 031 3 G = 130288 5729547013477 E 7....... 43294 2, 864773 5 9 H1 39087 57........ 4207 5729575166!11216 9 = F 5 9 I = 3909 117 1 81873 - G 3 5 1 15[6 6 2 = H 9......... 298 5 1566 I 6J= 261 5166 9 83438 =J 6..........37 415 8 = I 8 8K= 35 29 =L5........... 2 5729577951295=the num5 L = 2 ber required. But the original index is 1;.I. 57'29577951295~ are the number of degrees in an arc the length of which is equal to that of the radius. The above result may be easily verified by common division, a method, no doubt, which would be preferred by many, for logarithms are seldom used when the ordinary rules of arithmetic can be applied with any reasonable facility. However, this example, like many others, is introduced to show with what ease and correctness the number corresponding to a given log. can be obtained. The extent, also, by far exceeds that obtainable by any tables extant. Other computations give, r~ = 57'2957795130~ = 57~ 17' 44" -80624 the degrees in an arc = radius. r' = 3437'7467707849' = 3437' 44"'80624 the minutes in an are = radius. r"= 206264-8062470963 the number of seconds in an are - radius. The relative mean motion of the moon from the sun in a Julian or fictitious year, of 365k days, is 12 cir. 4 signs, 12~ 40' 15'977315' = 16029615'977315"..'. 16029615'977315": 1 circumference (= 129600"): 365'25 days 29'5305889216 days = the mean synodic month. This proportion may, for the sake of example, be found by logarithms. Log. 365'25........2'56259022460634 log. 1296000......... 611260500153457 8'67519522614091 log. 16029615'977315 = 7120492311805406 1'47027210808685 Page 358 358 THE PRACTICAL MODEL CALCULATOR. If the index of this log. be made 2 instead of 1, the nearest next less constant will be 2'375812087593221. From 2'47027210808685 2 3 7 5 8 1:2 0 8I7 5 9 3o 92 Const. Take 2'37581208759322 47 5 1 6 2 4 1 715 1 8 6 4 ~09446002049363 2 3 1 581,2 0:8 7 5 93 2 A= 08278537031645 2 817 417 312 6 2 5 987 79 - 2 A.1167465017718 15 714 6j5 5119 76 2 B = 864274756529 8 77 3 6 2 9 312 5114 517 7 5 = 2 B..303190261189 291711:'54 15 =2 6 C= 260446487591 17595081851062 463 9_8i_ 743987228...42743773598 58 50 2 9 9 D= 39084549177 4 3 9 9.... 3659224421 2 8 E= 3474338483 295015 3 8 6 6 9635 = C6..... 184885938 2 65513849803 4 F = 173717706 1062 0 5540 2 4 7 8 1..... 11168232 2 8 2 G = 8685889 2952810087 49763 = D9.......2482343 2i3622480700 5 H = 2171473 826787....... 310870 1 7 7 I = 304006 295304632057267=E8.......... 6863 118121 8 5 2 8 1J = 4343 1 7 72..........2520 2 9 5 3 0 5 81 3 2 7 7 5 67 = F 4 5 I= 2172 5906116:3........... 348 3 - L 347 2 9 5 3 0 5 8 7 2 3 3 8;7 3,3 = G 2 8 L= 347 295305872 33873 =G2 2N= 1 1417 65 2 9!4 = H 5 20,6 71l 41 =I7 291531 = J1 1476 5 =K5 213 62 = L 8 6-N 2 295305889217832... 29.5305889218 is the number required. To perform, by logarithms, the ordinary operations of multiplication, division, proportion, or even the extraction of the square root, except in the way of illustration, is not the design of these pages; for such an application of logarithms, in a particular manner only, diminish the labour of the operator. It is not necessary, however, to examine minutely here the instances in which common arithmetic is preferable to artificial numbers; besides, much will depend on the skill and facility of the operator. Page 359 359 TRIGONOMETRY. ANGULAR MAGNITUDES.-TRIGONOMETRY. —HEIGHT AND DISTANCES. — SPHERICAL TRIGONOMETRY.-THE APPLICATION OF LOGARITHMS TO ANGULAR MAGNITUDES. PLANE TRIGONOMETRY treats of the relations and calculations of the sides and angles of plane triangles. The circumference of every circle is supposed to be divided into 360 equal parts, called degrees; also each degree into 60 minutes, each minute into 60 seconds, and so on. Hence a semicircle contains 180 degrees, and a quadrant 90 degrees. The measure of any angle is an arc of any circle contained between the two lines which form that angle, the angular point being the centre; and it is estimated by the number of degrees contained in that arc. Hence, a right angle being measured by a quadrant, or quarter of the circle, is an angle of 90 degrees; and the sum of the three angles of every triangle, or two right angles, is equal to 180 degrees. Therefore, in a right-angled triangle, taking one of the acute angles from 90 degrees, leaves the other acute angle; and the sum of two angles, in any triangle, taken from 180 degrees, leaves the third angle; or one angle being taken from 180 degrees, leaves the sum of the other two angles. Degrees are marked at the top of the figure with a small o, minutes with', seconds with ", and so on. Thus, 570 30' 12" denote 57 degrees 30 minutes and 12 seconds. The complement of an arc, is what it wants of D L a quadrant or 90~. Thus, if AD be a quadrant, then BD is the complement of the arc AB; and, reciprocally, AB is the complement of BD. So E? that, if AB be an are of 50~, then its complement C BD will be 40~. The supplement of an arc, is what it wants of I V G a semicircle, or 180~. Thus, if ADE be a semicircle, then BDE is the supplement of the arc AB; and, reciprocally, AB is the supplement of the are BDE. So that, if AB be an are of 500, then its supplement BDE will be 130~. The sine, or right sine, of an are, is the line drawn from one extremity of the arc, perpendicular to the diameter passing through the other extremity. Thus, BF is the sine of the are AB, or of the are BDE. Hence the sine (BF) is half the chord (BG) of the double are (B AG). The versed sine of an are, is the part of the diameter intercepted between the are and its sine. So, AF is the versed sine of the are Ar, and EF the versed sine of the arc EDB. Page 360 3860 THE PRACTICAL MODEL CALCULATOR. The tangent of an arc is a line touching the circle in owe extremity of that arc, continued from thence to meet a line drawn from the centre through the other extremity: which last line is called the secant of the same arc. Thus, Al is the tangent, and CH the secant, of the arc AB. Also, EI is the tangent, and CI the secant, of the supplemental arc BDE. And this latter tangent and secant are equal to the former, but are accounted negative, as being drawn in an opposite or contrary direction to the former. The cosine, cotangent, and cosecant, of an arc, are the sine, tangent, and secant of the complement of that arc, the co being only a contraction of the word complement. Thus, the arcs AB, BD being the complements of each other, the sine, tangent or secant of the one of these, is the cosine, cotangent or cosecant of the other. So, BF, the sine of AB, is the cosine of BD; and BK, the sine of BD, is the cosine of AB: in like manner, All, the tangent of AB, is the cotangent of BD; and DL, the tangent of DB, is the cotangent of AB: also, CH, the secant of AB, is the cosecant of BD; and CL, the secant of BD, is the cosecant of AB..Hence several remarkable properties easily follow from these definitions; as, That an arc and its supplement have the same sine, tangent, and secant; but the two latter, the tangent and secant, are accounted negative when the arc is greater than a quadrant or 90 degrees. When the arc is 0, or nothing, the sine and tangent are nothing, but the secant is then the radius CA. But when the arc is a quadrant AD, then the sine is the greatest it can be, being the radius CD of the circle; and both the tangent and secant are infinite. Of any arc AB, the versed sine AF, and cosine BK, or CF, together make 75 up the radius CA of the circle. The radius CA, tangent All, and secant CH, form a right-angled triangle CAH. So also do the radius, sine, and cosine, 70 form another right-angled triangle 70 CBF or CBK. As also the radius, cotangent, and cosecant, another rightangled triangle CDL. And all these 60 right-angled triangles are similar to / K each other. 50 The sine, tangent, or secant of an 3,/,// angle, is the sine, tangent, or secant 8/ of the arc by which the angle is mea-,!o sured, or of the degrees, &c. in the same 1'i arc or angle. The method of constructing the scales (,50 60 0 00 of chords, sines, tangents, and secants, es. Sin. usually engraven on instruments, for practice, is exhibited in the annexed \ figure. b ~ ~ ~ ~ ~ Page 361 TRIGONOMETRY. 361 A trigonometrical canon, is a table exhibiting the length of the sine, tangent, and secant, to every degree and minute of the quadrant, with respect to the radius, which is expressed by unity, or 1, and conceived to be divided into 10000000 or more decimal parts. And further, the logarithms of these sines, tangents, and secants are also ranged in the tables; which are most commonly used, as they perform the calculations by only addition and subtraction, instead of the multiplication and division by the natural sines, &c., according to the nature of logarithms. Upon this table depends the numeral solution of the several cases in trigonometry. It will therefore be proper to begin with the mode of constructing it, which may be done in the following manner:To find the sine and cosine of a given are. This problem is resolved after various ways. One of these is as follows, viz. by means of the ratio between the diameter and circumference of a circle, together with the known series for the sine and cosine, hereafter demonstrated. Thus, the semi-circumference of the circle, whose radius is 1, being 3'141592653589793, &c., the proportion will therefore be, As the number of degrees or minutes in the semicircle, Is to the degrees or minutes in the proposed arc, So is 3'14159265, &c., to the length of the said arc. This length of the arc being denoted by the letter a; also its sine and cosine by s and c; then will these two be expressed by'the two following series, viz.:a3 a5 a7 s= a - 2-2.3.4.5 - 2.3.4.5.6.7 + &c. a3 a a7 = a- +120- 5040 + &c. a2 a4 a6 c = 1 - 2 3. + 4 2.3.4.5.6 a2 a4 a6 =1 — - + &C. 2 24-720 + &c. If it be required to find the sine and cosine of one minute. Then, the number of minutes in 180~ being 10800, it will be first, as 10800: 1:: 3'14159265, &c.::000290888208665 = the length of an arc of one minute. Therefore, in this case, a ='0002908882 and a a3= -000000000004, &c. the difference is s =- 0002908882 the sine of 1 minute. Also, from 1' take la2 = 00000000042079, &c. leaves e ='9999999577 the cosine of 1 minute. 2F Page 362 362 THE PRACTICAL MODEL CALCULATOR. For the sine and cosine of 5 degrees. Here, as 180~: 5:: 3'14159265, &c.,:'08726646 = a the length of 5 degrees. Hence, a ='08726646 a3 = -'00011076 + — oa5 = 00000004 these collected give s ='08715574 the sine of 5~. And, for the cosine, 1 = 1- I2 ---'00380771 + -a4 ='00000241 these collected, give e ='99619470 the consine of 5~. After the same manner, the sine and cosine of any other are may be computed. But the greater the arc is, the slower the series will converge, in which case a greater number of terms must be taken to bring out the conclusion to the same degree of exactness. Or, having found the sine, the cosine will be found from it, by the property of the right-angled triangle CBF, viz. the cosine CF ='CB -- BF2, or c = 1 - s2. There are also other methods of constructing the canon of sines and cosines, which, for brevity's sake, are here omitted. To compute the tangents and secants. The sines and cosines being known, or found, by the foregoing problem; the tangents and secants will be easily found, from the principle of similar triangles, in the following manner:In the first figure, where, of the arc AB, BF is the sine, CF or BK the cosine, AH the tangent, CRI the secant, DL the cotangent, and CL the cosecant, the radius being CA, or CB, or CD; the three similar triangles CFB, CAlH, CDL, give the following proportions: 1. CF: FB:: CA: AII; whence the tangent is known, being a fourth proportional to the cosine, sine, and radius. 2. CF: CB:: CA: CH; whence the secant is known, being a third proportional to the cosine and radius. 3. BF: FC:: CD: DL; whence the cotangent is known, being a fourth proportional to the sine, cosine, and radius. 4. BF: BC:: CD): CL; whence the cosecant is known, being a third proportional to the sine and radius. Having given an idea of the calculations of sines, tangents, and secants, we may now proceed to resolve the several cases of trigonometry; previous to which, however, it may be proper to add a few preparatory notes and observations, as below. There are usually three methods of resolving triangles, or the cases of trigonometry-namely, geometrical construction, arithmetical computation, and instrumental operation. Il: the first methlod. —The triangle is constructed by making the parts of the given magnitudes, namely, the sides from a scale of Page 363 TRIGONOMETRY. 363 equal parts, and the angles from a scale of chords, or by some other instrument. Then, measuring the unknown parts by the same scales or instruments, the solution will be obtained near the truth. In the second method.-Having stated the terms of the proportion according to the proper rule or theorem, resolve it like any other proportion, in which a fourth term is to be found from three given terms, by multiplying the second and third together, and dividing the product by the first, in working with the natural numbers; or, in working with the logarithms, add the logs. of the second and third terms together, and from the sum take the log. of the first term; then the natural number answering to the remainder is the fourth term sought. In the third method.-Or instrumentally, as suppose by the log. lines on one side of the common two-foot scales; extend the compasses from the first term to the second or third, which happens to be of the same kind with it; then that extent will reach from the other term to the fourth term, as required, taking both extents towards the same end of the scale. In every triangle, or case in trigonometry, there must be given three parts, to find the other three. And, of the three parts that are given, one of them at least must be a side; because the same angles are common to an infinite number of triangles. All the cases in trigonometry may be comprised in three varieties only; viz. 1. When a side and its opposite angle are given. 2. When two sides and the contained angle are given. 3. When the three sides are given. For there cannot possibly be more than these three varieties of cases; for each of which it will therefore be proper to give a separate theorem, as follows: TiTen a side and its opposite angle are two of the given plarts. Then the sides of the triangle have the same proportion to each other, as the sines of their opposite angles have. That is, As any one side, Is to the sine of its opposite angle; So is any other side, To the sine of its opposite angle. For, let ABC be the proposed triangle, having D \ AB the greatest side, and BC the least. Take AD = BC, considering it as a radius; and let fall the perpendiculars DE, CF, which will evi- A E F B dently be the sines of the angles A and B, to the radius AD or BC. But the triangles ADE, ACF, are equiangular, and therefore AC: CF:: AD or BC: DE; that is, AC is to the sine of its opposite angle B, as BC to the sine of its opposite angle A. In practice, to find an angle, begin the proportion with a side Page 364 364 THE PRACTICAL MODEL CALCULATOR. opposite a given angle. And to find a side, begin with an angle opposite a given side. An angle found by this rule is ambiguous, or uncertain whether it be acute or obtuse, unless it be a right angle, or unless its magnitude be such as to prevent the ambiguity; because the sine answvers to two angles, which are supplements to each other; and accordingly the geometrical construction forms two triangles with the same parts that are given, as in the example below; and when there is no restriction or limitation included in the question, either of them may be taken. The degrees in the table, answering to the sine, are the acute angle; but if the angle be obtuse, subtract those degrees from 180~, and the remainder will be the obtuse angle. When a given angle is obtuse, or a right one, there can be no ambiguity; for then neither of the other angles can be obtuse, and the geometrical construction will form only one triangle. In the plane triangle ABC, C (AB 345 yards Given, BC 232 yards angle A 37~ 20' Required the other parts. A B Geometricwally.-Draw an indefinite line, upon which set off AB 345, from some convenient scale of equal parts. Make the angle A = 370~. With a radius of 232, taken from the same scale of equal parts, and centre B, cross AC in the two points C, C. Lastly, join BC, BC, and the figure is constructed, which gives two triangles, showing that the case is ambiguous. Then, the sides AC measured by the scale of equal parts, and the angles B and C measured by the line of chords, or other instrument, will be found to be nearly as below; viz. AC 174 angle B 27~ angle C 115"~ or 3741 or 78} or 64~ Arithlnetically.-First, to find the angles at C: As side BC 232........................log. 2'3654880 To sin. opp. angle A 370 20'.................. 97827958 So side AB 345....................... 25378191 To sin. opp. angle C 1150 36' or 64~ 24....... 9'9551269 Add angle A 37 20 37 20 The sum 152 56 or 101 44 Taken from 180 00 180 00 Leaves angle B 27 04or 78 16 Then, to find the side AC: As sine angle A 370 20'...................log. 9.7827 958 To opposite side BC 232................. 236o5488 27 04'.................... 9 6580371 So sine angle B 78 16..................... 9 9t0891 To opposite side AC 174'07..................... 2 240 20 TI3 or, 374'56..................... 25735213 Page 365 TRIGONOMETRY. 365 In the plane triangle ABC, AB 365 poles Given, angle A 57~0 12' angle B 24 45 Ans. angle C 98~ 3' Required the other parts. AC 154'33 BC 309'86 In the plane triangle ABC, AC 120 feet Given, C 112 feet angle B 64 34' 21" 39 angle A 57 27' or, 115 539 Required the other parts. Ans. angle C7 57 21 I AB 112.65 feet ( or, 16'47 feet When two sides and their contained angle are given. Then it will be, As the sum of those two sides, Is to the difference of the same sides; So is the tang. of half the sum of their opposite angles, To the tang. of half the difference of the same angles. Hence, because it is known that the half sum of any two quantities increased by their half difference, gives the greater, and diminished by it gives the less, if the half difference of the angles, so found, be added to their half sum, it will give the greater angle, and subtracting it will leave the less angle. Then, all the angles being now known, the unknown side will be found by the former theorem. Let ABC be the proposed triangle, having E the two given sides AC, BC, including the given angle C. With the centre C, and radius CA, D the less of these two sides, describe a semicircle, meeting the other side BC produced in D and E. Join AE, AD, and draw DF parallel to AE. F B Then, BE is the sum, and BD the difference of the two given sides CB, CA. Also, the sum of the two angles CAB, CBA, is equal to the sum of the two CAD, CDA, these sums being each the supplement of the vertical angle C to two right angles: but the two latter CAD, CDA, are equal to each other, being opposite to the two equal sides CA, CD: hence, either of them, as CDA, is equal to half the sum of the two unknown angles CAB, CBA. Again, the exterior angle CDA is equal to the two interior angles B and DAB; therefore, the angle DAB is equal to the difference between CDA and B, or between CAD and B; consequently, the same angle DAB is equal to half the difference of the unknown angles B and CAB; of which it has been shown that CDA is the half sum. Now the angle DAE, in a semicircle, is a right angle, or AE is perpendicular to AD; and DF, parallel to AE, is also perpendicular 2F2 Page 366 366 THE PRACTICAL MODEL CALCULATOR. to AD: consequently, AE is the tangent of CDA the half sum and DF the tangent of DAB the half difference of the angles, to the same radius AD, by the definition of a tangent. But, the tangents AE, DF, being parallel, it will be as BE: BD:: AE: DF; that is, as the sum of the sides is to the difference of the sides, so is the tangent of half the sum of the opposite angles, to the tangent of half their difference. The sum of the unknown angles is found, by taking the given angle from 180~. In the plane triangle ABC, ( AB 345 yards C Given, AC 174'07 yards tangle A 37 20' A B Required the other parts. Geometrically. —Draw AB = 345 from a scale of equal parts. Make the angle A = 37~ 20'. Set off AC = 174 by the scale of equal parts. Join BC, and it is done. Then the other parts being measured, they are found to be nearly as follows, viz. the side BC 232 yards, the angle B 27~, and the angle C 115~. Arithmetically. As sum of sides AB, AC................... 519'07 log. 2'7152259 To difference of side's AB, AC............. 170'93 2.2328183 So tangent half sum angles C and B..... 71~ 20' 10'4712979 To tangent half difference angles C and B 44 16 9'9888903 Their sum gives angle C 115 36 Their diff. gives angle B 27 4 Then, by the former theorem, As sine angle C 115~ 36', or 640 24'......log. 9'0551059 To its opposite side AB 345.................. 2'5378191 So sine angle A 370 20'..................... 97827958 To its opposite side BC 232................. 2'3654890 In the plane triangle ABC, ( AB 365 poles Given, AC 154'33 (angle A 570 12' B C 309'86 Required the other parts. angle B 24~ 45' tangle C 98~ 3' In the plane triangle ABC, AC 120 yards Given, BC 112 yards tangle C 57~ 58' 39" AB 112'65 Required the other parts. angle A 57~ 27' 0" angle B 64 34 21 Page 367 TRIGONOMETRY. 367 When the three sides of the triangle are given. Then, having let fall a perpendicular from the greatest angle upon the opposite side, or base, dividing it into two segments, and the whole triangle into two right-angled triangles; it will be, As the base, or sum of the segments, Is to the sum of the other two sides; So is the difference of those sides, To the difference of the segments of the base. Then, half the difference of the segments being added to the half sum, or the half base, gives the greater segment; and the same subtracted gives the less segment. Hence, in each of the two right-angled triangles, there will be known two sides, and the angle opposite to one of them; consequently, the other angles will be found by the first problem. The rectangle under the sum and difference of the two sides, is equal to the rectangle under the sum and difference of the two segments. Therefore, by forming the sides of these rectangles into a proportion, it will appear that the sums and differences are proportional, as in this theorem. In the plane triangle ABC, c (AB 345 yards Given, the sides AC 232 BC 174'07 A P B To find the angles. Geometrieally.-Draw the base AB = 345 by a scale of equal parts. With radius 232, and centre A, describe an arc; and with radius 174, and centre B, describe another arc, cutting the former in.C. Join AC, BC, and it is done. Then, by measuring the angles, they will be found to be nearly as follows, viz. angle A 270, angle B 3710, and angle C 1151-. Ar}ithmetically.-Having let fall the perpendicular CP, it will be, As the base AB: AC + BC:: AC - BC: AP - BP that is, as 345: 406'07:: 57'93: 68'18 = AP - BP its half is.................. 34'09 the half base is................. 172'50 the sum of these is............. 206'59 = AP and their difference............ 138'41 = BP Then, in the triangle APC, right-angled at P, As the side AC.....................232........log. 2'3654880 To sine opposite angle............ 90~........ 10'0000000 So is side AP.......................20659........ 2'3151093 To sine opposite angle ACP..... 62~ 56'........ 9'9496213 Which taken from............ 90 00 Leaves the angle A......... 27 04 Page 368 368 THE PRACTICAL MODEL CALCULATOR. Again, in the triangle BPC, right-angled at P, As the side of BC........... 174'07.........log. 2'2407239 To sine opposite angle P... 90~......... 10'0000000 So is side BP................. 138'41......... 21411675 To sin. opposite angle BCP 52~ 40'......... 9'9004436 Which taken from..... 90 00 Leaves the angle B... 37 20 Also, the angle ACP... 62~ 56' Added to angle BCP... 52 40 Gives the whole angle ACB...115 36 So that all the three angles are as follow, viz. the angle A 27~ 4'; the angle B 37~ 20'; the angle C 1150 36'. In the plane triangle ABC, (AB 365 poles Given the sides, AC 154'33 BC 309'86 (angle A 570 12' To find the angles. angle B 24 45 angle C 98 3 In the plane triangle ABC, AB 120 Given the sides, AC 11265 fBC 112 ( angle A 570 27' 00" To find the angles. angle B 57 58 39 (angle C 64 34 21 The three foregoing theorems include all the cases of plane triangles, both right-angled and oblique; besides which, there are other theorems suited to some particular forms of triangles, which are sometimes more expeditious in their use than the general ones; one of which, as the case for which it serves so frequently occurs, may be here taken, as follows:When, in a right-angled triangle, there are given one leg and the angles; to find the other leg or the ]hypothenuse; it will be, As radius, i. e. sine of 90~ or tangent of 450 Is to the given leg, So is the tangent of its adjacent angle To the other leg; And so is the secant of the same angle To the hypothenuse. AB being the given leg, in the right-angled tri- C angle ABC; with the centre A, and any assumed ra- F dius, AD, describe an arc DE, and draw DF perpen- F dicular to AB, or parallel to BC. Now it is evident, from the definitions, that DF is the tangent, and AF the secant, of the arc DE, or of the angle A which A D B is measured by that arc, to the radius AD. Then, because of the parallels BC, DF, it will be as AD: AB:: DF: BC AF: AC, which is the same as the theorem is in words. Page 369 OF HEIGHTS AND DISTANCES. 369 In the right-angled triangle ABC, Given the leg AB 162 48" to find AC and BC. angle A 530 7' 48" Geometrically.-Make AB = 162 equal parts, and the angle A = 530 7' 48"; then raise the perpendicular BC, meeting AC in C. So shall AC measure 270, and BC 216. Arithmetically. As radius..................... tang. 45~.........log. 10'0000000 To leg AB.................... 162......... 2'2095150 So tang. angle A............530 7' 48"........ 10'1249371 To leg BC.................... 216......... 2'3344521 So secant angle A...........530 7' 48"........ 10'2218477 To hyp. AC.................. 270......... 24313627 In the right-angled triangle ABC, Given the leg AB 180 Given i the angle A 620 40' AC 392 0147 To find the other two sides. BC 348'2464 There is sometimes given another method for right-angled triangles, which is this: c ABC being such a triangle, make one leg AB ra- F dius, that is, with centre A, and distance AB, describe an arc BF. Then it is evident that the other G leg BC represents the tangent, and the hypothenuse A AC the secant, of the arc BF, or of the angle A. BE D In like manner, if the leg BC be made radius; E then the other leg AB will represent the tangent, and the hypothenuse AC the secant, of the arc BG or angle C. But if the hypothenuse be made radius; then each leg will represent the sine of its opposite angle; namely, the leg AB the sine of the arc AE or angle C, and the leg BC the sine of the arc CD or angle A. And then the general rule for all these cases is this, namely, that the sides of the triangle bear to each other the same proportion as the parts which they represent. And this is called, Making every side radius. OF HEIGHTS AND DISTANCES, BY the mensuration and protraction of lines and angles, are determined the lengths, heights, depths, and distances of bodies or objects. Accessible lines are measured by applying to them some certain measure a number of times, as an inch, or foot, or yard. But inaccessible lines must be measured by taking angles, or by some such method, drawn from the principles of geometry. When instruments are used for taking the magnitude of the 24 Page 370 370 THE PRACTICAL MODEL CALCULATOR. angles in degrees, the lines are then calculated by trigonometry: in the other methods, the lines are calculated from the principle of similar triangles, without regard to the measure of the angles. Angles of elevation, or of depression, are usually taken either with a theodolite, or with a quadrant, divided into degrees and minutes, and furnished with a plummet suspended from the centre, and two sides fixed on one of the radii, or else with telescopic sights. To take an angle of altitude and depression with the quadrant. Let A be any object, as the sun, A moon, or a star, or the top of a tower,.-* or hill, or other eminence; and let it " be required to find the measure of the angle ABC, which a line drawn from the object makes with the horizontal - line BC. Fix the centre of the quadrant in the angular point, and move it round D there as a centre, till with one eye at H D, the other being shut, you perceive the object A through the sights: then will the arc GH of the quadrant, cut off by the plumb line BIH, be the measure of the angle ABC, as required. E C The angle ABC of depression of any ob- G / ject A, is taken in the same manner; except o. that here the eye is applied to the centre, and Hi the measure of the angle is the arc GIH, on the other side of the plumb line. The following examples are to be constructed and calculated by the foregoing methods, treated of in trigonometry. Having measured a distance of 200 feet, in a direct horizontal line, from the bottom of a steeple, the angle of elevation of its top, taken at that distance, was found to be 470 30': from hence it is required to find the height of the steeple. Construction.-Draw an indefinite line, upon which set off AC = 200 equal parts, for the measured distance. Erect the indefinite perpendicular AB; and draw CB so as to make the angle C = 470 30', the angle of elevation; and it is done. Then AB, measured on the scale of equal parts, is nearly 2184. B Calculation. As radius................................. 100000000 To AC 200............................... 23010300' So tang. angle C 470 30'....... 10'0379475 To AB 218'26 required............... 283389775 c A Page 371 OF HEIGHTS AND DISTANCES. 371 What was the perpendicular height of a cloud, or of a balloon, when its angles of elevation were 35~ and 64~, as taken by two observers, at the same time, both on the same side of it, and in the same vertical plane; their distance, as under, being half a mile, or 880 yards. And what was its distance from the said two observers? Construction.-Draw an indefinite ground line, upon which set off the given distance AB = 880; then A and B are the places of the observers. Make the angle A = 350, and the angle B = 64~; and the intersection of the lines at C will be the place of the balloon; from whence the perpendicular CD, being let fall, will be its perpendicular height. Then, by measurement, are found the distances and height nearly, as follows, viz. AC 1631, BC 1041, DC 936. c Calculation.; First, from angle B 64~ Take angle A 35 Leaves angle ACB 29 Then, in the triangle ABC, A B D As sine angle ACB 290................... 9'6855712 To. opposite side AB 880.................... 29444827 So sine angle A 350................... 9'7585913 To opposite side BC 1041'125.................. 3'0175028 As sine angle ACB 29~................... 9855712 To opposite side AB 880................... 2'9444827 So sine angleB 1160or640................... 9'9536602 To opposite side AC 1631'442................... 3'2125717 And, in the triangle BCD, As sine angle D 90~...................10'0000000 To opposite side BC 1041'125................... 3'0175028 So sine angle B 64~................... 9'9536602 To opposite side CD 935'757................... 2'9711630 Having to find the height of an obelisk standing on the top of a declivity, I first measured from its bottom, a distance of 40 feet, and there found the angle, formed by the oblique plane and a line imagined to go to top of the obelisk 41~; but, after measuring on in the same direction 60 feet farther, the like angle was only 23~ 45'. What then was the height of the obelisk? Construction.-Draw an indefinite line for the sloping plane or declivity, in which assume any point A for the bottom of the obelisk, from whence set off the distance AC = 40, and again CD = 60 equal parts. Then make the angle C = 41~, and the angle D = 23~ 45'; and the point B, where the two lines meet, will be the top of the obelisk. Therefore AB, joined, will be its height. Page 372 372 THE PRACTICAL MODEL CALCULATOR. B Calculationr. From the angle C 41~ 00' Take the angle D 23 45 Leaves the angle DBC 17 15 A Then, in the triangle DBC, D As sine angle DBC 17~ 15'................... 9'4720856 To opposite side DC 60................... 1'7781513 So sine angle D 24 45................... 9'6050320 To opposite side CB 81'488................... 1'9110977 And, in the triangle ABC, As sum of sides CB, CA 121'488..... 2'0845333 To difference of sides CB, CA 41 488..... 1'6179225 So tang. half sum angles A, B 69~ 30'.....10'4272623 To tang. half diff. angles A, B 42 241..... 9-9606516 The diff. of these is angle CBA 27 52 Lastly, as sine angle CBA 27~ 5 21............... 9'6582842 To opposite side CA 40............... 1-6020600 So sine angle C 41~0'............... 9'8169429 To opposite side AB 57'623............... 1'7607187 Wanting to know the distance between two inaccessible trees, or other objects, from the top of a tower, 120 feet high, which lay in the same right line with the two objects, I took the angles formed by the perpendicular wall and lines conceived to be drawn from the top of the tower to the bottom of each tree, and found them to be 330 and 641~. What then may be the distance between the two objects? A Uonstruction.-Draw the indefinite ground line BD, and perpendicular to \ it BA = 120 equal parts. Then draw the two lines AC, AD, making the two angles BAC, BAD, equal to the given B C D angles 330 and 64~~. So shall C and D be the places of the two objects. Calculation.-First, In the right-angled triangle ABC, As radius............................................100000000 To AB....................120..................... 2'0791812 So tang. angle BAC..... 330..................... 9'8125174 To BC..................77929..................... 1'8916986 And, in the right-angled triangle ABD, As radius.......................10'0000000 To AB.12....0...120.................... 2'0791812 So tang. angle BAD.... 642~.10'3215039 To BD.............251'585................ 24006851 From which take BC 77'929 Leaves the dist. CD 173'656 as required. Page 373 SPHERICAL TRIGONOMETRY. 373 Being on the side of a river, and wanting to know the distance to a house which was seen on the other side, I measured 200 yards in a straight line by the side of the river; and then at each end of this line of distance, took the horizontal angle formed between the house and the other end of the line; which angles were, the one of them 68~ 2', and the other 730 15'. What then were the distances from each end to the house? Construction.-Draw the line AB = 200 equal parts. Then draw AC so as to make the angle A 68~ 2', and BC to make the angle B = 730 15'. So shall the point C be the place of the house required. Calculation. To the given angle A 68~ 2' Add the given angle B 73 15 Then their sum 141 17 Being taken from 180 0 / — Leaves the third angle C 38 43 A- B~ A B Hence, As sin. angle C 380 43.................. 97962062 To op. side AB 200.................2'3010300 So sin. angle A 68~ 2'................. 99672679 To op. side BC 296'54.................2'4720917 And, As sin. angle C 38~ 43'.................9'7962062 To op. side AB 200.................2'3010300 So sin. angle B 730 15.................9'9811711 To op. side AC 306'19.................2'4859949 SPHERICAL TRIGONOMETRY. This Article is taken from a short Practical Treatise on Sphzerical Trigonometry, by Oliver Byrne, the author of the present work. Published by J. A. Valepy. London, 1835. As the sides and angles of spherical triangles are measured by circular arcs, and as these arcs are often greater than 90~, it may be necessary to mention one or two particulars respecting them. The arc CB, which when added to C D B AB makes up a quadrant or 90~, is called the complement of the are AB; every arc will have a complement, even those which are themselves greater than 900, provided we consider the arcs measured in the direc- tion ABCD, &c., as positive, and consequently those measured in the /H opposite direction as negative. The complement BC of the arc AB commences at B, where AB terminates, and may be considered as generated by the motion of B, the ex2G Page 374 374 THE PRACTICAL MODEL CALCULATOR. tremity of the radius OB, in the direction BC. But the complement of the arc AD or DC, commencing in like manner at the extremity D, must be generated by the motion of D in the opposite direction, and the angular magnitude AOD will here be diminished by the motion of OD, in generating the complement; therefore the complement of AOD or of AD may with propriety be considered negative. Calling the arc AB or AD, 0, the complement will be 90~ - 0; the complement of 36~ 44' 33" is 530 15' 27"; and the complement of 136~ 27' 39" is negative 46~ 27' 39". The arc BE, which must be added to AB to make up a semicircle or 1800, is called the supplement of the arc AB. If the arc is greater than 1800, as the arc ADF its supplement, FE measured in the reverse direction is negative. The expression for the supplement of any arc o is therefore 1800 - o; thus the supplement of 112~ 29' 35" is 67~ 30' 25", and the supplement of 205~ 42' is negative 25~ 42'. In the same manner as the complementary and supplementary arcs are considered as positive or negative, according to the direction in which they are measured, so are the arcs themselves positive or negative; thus, still taking A for the commencement, or origin, of the arcs, as AB is positive, AH will be negative. In the doctrine of triangles, we consider only positive angles or arcs, and the magnitudes of these are comprised between o = 0 and 0 = 180~; but in the general theory of angular quantity, we consider both positive and negative angles, according as they are situated above or below the fixed line AO, from which they are measured, that is, according as the arcs by which they are estimated are positive or negative. Thus the angle BOA is positive, and the angle AOII negative. Moreover, in this more extended theory of angular magnitude, an angle may consist of any number of degrees whatever; thus, if the revolving line OB set out from the fixed line OA, and make n revolutions and a part, the angular magnitude generated is measured by n times 360~, plus the degrees in the additional part. In a right-angled spherical triangle we are to recognise but five B A= b parts, namely, the three sides a, b, c, and the two angles A, B; so that the right angle C is omitted. Page 375 SPHERICAL TRIGONOMETRY. 375 Let A', c', B,' be the complements of A, c, B, respectively, A... —and suppose b, a, Bl, c', Al, to be s placed on the hand, as in the, annexed figure, and that the / fingers stand in a circular order,, the parts represented by the! fingers thus placed are called b circular parts. - / If we take any one of these as \\ O a middle part, the two which lie - next to it, one on each side, will be adjacent parts. The two parts immediately beyond the adjacent parts, one on each side, are called the opposite parts. 1 Thus, taking A' for a middle part, b and c' will be adjacent parts, and a and B' are opposite parts. If we take c' as a middle part, A' and B' are adjacent parts, and b, a, opposite parts. When B' is a middle part, c', a, become adjacent parts, and A', b, opposite parts. Again, if we take a as a middle part, then B', b, will be adjacent parts, and e', A', opposite parts. Lastly, taking b as a middle part, A', a, are adjacent parts, and c', B', opposite parts. This being understood, Napier's two rules may be expressed as follows: — I. Rad. x sin. middle part = product of tan. adjacent parts. II. Rad. x sin. middle part -- product of cos. opposite parts. Both these rules may be comprehended in a single expression, thus, Rad. sin. mid. = prod. tan. adja. = prod. cos. opp.; and to retain this in the memory we have only to remember, that the vowels in the contractions sin., tan., cos., are the same as those in the contractions mid., adja., opp., to which they are joined. These rules comprehend all the succeeding equations, reading from the centre, R = radius. In the solution of right-angled spherical triangles, two parts are given to find a third, therefore it is necessary, in the application of this formula, to choose for the middle part that which causes the other two to become either adjacent parts or opposite parts. In a right-angled spherical triangle, the hypothenuse c = 610 4' 56"; and the angle A - 61~ 50' 29". Required the adjacent leg? 90~ O' 0 0" 90~ O' 00" -61 4 56 A=61 50 29 28 55 04-'. 28 9 31 - A. Page 376 376 THE PRACTICAL MODEL CALCULATOR. 9. 4,;. ob' A.. b ad, asJ )in h nn i x sin A'o o \rtTan. bxtata. cc. C In this example, Al is selected for the middle part, because then b and el become adjacent parts, as in the annexed figure. Rad. x sin. Al = tan. b x tan. e'. A? * tanel / By Logarithms. I C Rad. -... -10 0000000 Sin. A'-2809'21"- 9'6738628 19'6738628 Tan. c'-28~55'4"- 9'7422808 Tan.b'-40~30'16" —9.9315820 The side adjacent to the given 2 angle is acute or obtuse, according as the hypothenuse is of the same, or of different species with the given angle..'. the leg b -40~ 30 16", acute. Supposing the hypothenuse e = 113~ 55, and the angle A = 31~ 51' then the adjacent leg b would be 117~ 34', obtuse. Page 377 SPHERICAL TRIGONOMETRY. 377 In the right-angled spherical triangle ABC, are given the hypothenuse c = 113~ 55', and the angle A = 104~ 08'; to find the opposite leg a. c = 113~ 55' 90 0 23 55 = c'. A = 1040 08' 90 0 A 14 08 -A'. In this example, a is taken for the middle part, then A' and c' are opposite parts. (See the subjoined figure.) A' From the general formula, we have, Rad. x sin. a = cos. A' x cos. c'. cos. A' x cos.' b.. sin. a = Rad. By Logarithms. a cos. A' - 14~ 08'...... 9'9860509 / cos. c' - 23 55... 9'9610108 19'9476617 / Radius...... 100000000 sin. {b 1170 342... 9'9476617 The obtuse side 117~ 34' is the leg required, for the side opposite to the given angle is always of the same species with the given angle. If in a right-angled spherical triangle, the hypothenuse were 78~ 20', and the angle A = 370 25', then the opposite leg a = 36~ 31', and not 143~ 29', because the given angle is acute. c In a right-angled spherical triangle, are given c = 78~ 20', and A = 37~ 25', to find the angle B. 90~ 0' c=78 20 11 40 = c'. 900 0' A = 37 25 52 35 = A' 2G2 Page 378 378 TIIE PRACTICAL MODEL CALCULATOR. Here the complement of the hypothenuse (c') is the middle part; and the complement of the A - angle opposite the perpendicular (A'), and the complement of the angle opposite the base (B') are the adjacent parts. This will readily be perceived by reference to the usual figure in the margin. Rad. x sin. c' = tan. A' X tan. B'; 4 4 tan. B' = Rad. x sin. c' tan. A/' By Logarithms. Rad..................1000000000 sin. c' - 11~ 40'. 9-3058189 19'3058189 c tan. A' - 52~ 35' 10'1163279.. tan. B' - 8~ 48' 9'1894910 But 90-B =B' hence 90 - B' = B. 90~ 0' (A 8 48 B = 81~ 12'. When the hypothenuse and an angle are given, the other angle is acute or obtuse, according as the given parts are of the same or of different species. In the above example, both the given parts are acute, therefore the required angle is acute; but if one be acute and the other obtuse, then the angle found would be obtuse:-Thus, if the hypothenuse be 113~ 55', and the angle A = 31~ 51'; then will B' = 140 08', and the angle B = 104~ 08'. Given the hypothenuse c = 61~ 04' 56", and the side or leg, a = 40~ 30' 20", to find the angle adjacent to a. C/ B' 90~ 0 0" e=61 04 56 A' 28 55 04" = c". The three parts are here / B connected; therefore the complement of.B is the middle / I part, a and the complement of / e are the adjacent parts. Hence we have, / Rad. x sin. B' = tan. a x tan. c'. / /, tan. a x tan.' / sin. B Rad. 5 Page 379 SPHERICAL TRIGONOMETRY. 379 By Logarithms. tan. a - 40~ 30' 20" = 9'9315841 tan. c' - 28 55 04 = 9'7422801 19'6738642 Rad....................10'0000000 sin. B'....28~ 09' 31"...... 9'6738642 90~ 0O 0" B' = 28 09 31 61 50 29 = B. The angle adjacent to the given side is acute or obtuse according as the hypothenuse is of the same or of different species with the given side. Before working the above example, it was easy to foresee that the angle B would be acute; but suppose the hypothenuse = 700 20', and the side a = 117~ 34', then the angle B would be obtuse, because a and c are of different species. RULE V.-In a spherical triangle, right-angled at c, are given c = 78~ 20' and b = 117~ 34', to find the angle B; opposite the given leg, (see the next diagram.) In this example, b becomes the middle part, and ct and B' opposite parts; and therefore, by the rule, Rad. x sin. b = cos. B' x cos. e'; that is, Rad. x sin. b COS. B'cOS. el 90 - 78~ 20' = 11~ 40' -'. b Hence, by Logarithms. Rad......................... 10000000000 sin. b = sin. 1170 34 9 655 S or sin. 62 26 9'9476655 19'9476655 cos. c' 11~ 40'............. 9.9909338 cos. B' 25~ 09'............ 9'9567317 6 Page 380 380 TIHE PRACTICAL MODEL CALCULATOR. B But since the angle opposite the given side is of the same species with the given side, 90~ must be added to B', to pro- a duce B:-viz. 90~ + 25~ 09' = 1150 09'. Given c = 610 04' 56", and b = 400 30' 20", to find the other c side a. Here c' is the middle part, a and b the opposite parts; hence b by position 4, a = 50~ 30' 30". Given the side b = 48~ 24' 16"1, and the adjacent angle A = 660 20' 40", to find the side a. In this instance, b is the middle part, the complement of A and a are adjacent parts. Consequently, a = 590 38' 27". In the right-angled spherical triangle ABC, { The side a = 590 38' 27" Giiven( Its adjacent angle B = 52~ 32' 55" to find the angle A. Answer, 66~ 20' 40". The required angle is of the same species as the given side, and vice versa. Given the side b = 490 17', and its adjacent angle A = 230 28', to find the hypothenuse. Making A' the middle part, the others will be adjacent parts, and, therefore, by the first rule we have c = 51~ 42' 37". In a spherical triangle, right-angled at C, are given b = 29~ 12' 50", and B = 370 26' 21", to find the side a. Taking a for the middle part, the other two will be adjacent parts; hence by the rule, Rad. x sin. a = tan. b x tan. B' that is, rad. x sin. a = tan. b x cot. B tan. b x cot. B.. sin. a - rad. In this case, there are two solutions, i. e. a and the supplement of a, because both of them have the same sine. As sin. a is necessarily positive, b and B must necessarily be always of the same species, so that, as observed before, the sides including the right angle are always of the same species as the opposite angles. Page 381 SPHERICAL TRIGONOMETRY. 381 In working this example, D we find the log. sin. a = 9'8635411, which corresponds to 460 55' 02", or, 1330 04' 58". It appears, therefore, that a is ambiguous, for,0* there exist two right-angled triangles, having an oblique angle, and the opposite side in the one equal to an oblique angle and an opposite side in the other, but the remaining oblique angle in the one the supplement of the remaining oblique c angle in the other. These triangles are situated with respect to each other, on the sphere, as the triangles ABC, ADC, in the annexed diagram, in which, with the exception of the common side AC, and the equal angles B, D, the parts of the one triangle are supplements of the corresponding parts of the other. In a right-angled spherical triangle are Given the side a........... = 42~ 12', to find the adjacent ven its opposite angle A = 48~ angle B. The complement of the given angle is the middle part; and neither a nor B' being joined to A', they are consequently opposite parts; hence, the angle B = 64~ 35', or 115~ 25'; this case, like the last, being ambiguous, or doubtful. Given a = 110 30', and A = 230 30', to find the hypothenuse c. c = 300, or 1500, being ambiguous. In a right-angled triangle, there are given the two perpendicular sides, viz. a = 48~ 24' 16", b = 590 38' 27", to find the angle A. A = 660 20' 40". Given a = 142~ 31t, b = 540 22', to find c. c = 117~ 33'. Given BA 37 1 12 Required the side a. a = 360 31'. GA = 66~ 20' 40"} Given BA = 652 32 55 to find the hypothenuse c. c = 700 23' 42". Page 382 882 THE PRACTICAL MODEL CALCULATOR. MEASUREMENT OF ANGLES. From the " Civil Engineer and Architect's Journal," for Oct. and Nov. 1847. A NEW METHOD OF MEASURING THE DEGREES, MINUTES, ETC., IN ANY RECTILINEAR ANGLE BY COMPASSES ONLY, WITHOUT USING SCALE OR PROTRACTOR. APPLY AB = x, from B to 1; from 1 to 2; from 2 to 3; from 3 to 4; from 4-to 5. Then take B 5, in the compasses, and apply it from B to 6; from 6 to 7; from 7 to 8; from 8 to 9; and from 9 to 10, near the middle of the arc AB. With the same opening, 4,; ~X 10 13 2 B 5 or A 4, or y, which we shall term it, lay off 4,11, 11,12, and 12,13. Then the arc between 13 and 10 is found to be contained 23 times in the arc AB. Page 383 MEASUREMENT OF ANGLES. 383 Hence, we have, 5 x -y = 360~; 9y + z = x; x 23z =x;or, z=-23. x 22x' 9y+ 23x, *- Y 207' By substituting this value in the first equation, we obtain, 22x 5x - = 360. 1013x 360 x 207 207 = 360, and x = 1013 = 730 33' 82. Apply AB = x, from B to 1; from 1 to 2; from 2 to 3; from 3 to 4. Then take B 4, in the compasses, and apply it on the arc, from B to 4; from 4 to 5; from 5 to 6; from 6 to 7; and from 7 to 8, near the middle of the arc AB. With the same opening, B 4 = y, lay off A 9, 9,10, 10,11, 11,12, 12,13, and 13,14. The arc between 14 and 8 is found to be contained nearly 24 times in the arc AB. Therefore, we have, 4x + y = 360; Ily- z = x; 24z =x; or, z 24 x 25x.. y-4 = x;'" Y = 264' Substituting this value of y in the first equation, 25x 4x+ 264 = 360; 360 x 264 x = 1071 88~ 44I'333. How to lay off an angle of any number of degrees, minutes, ~c., with compasses only, without the use of scale or protractor. Let it be required to lay off an angle of f 36~40' = P. Take any / < small opening of the compasses less than one-tenth of the radius, and lay off any number of equal small 2 arcs, from A to 1; a from 1 to 2; from 2 to 2/ o A 3, &c., until we have h laid off an arc, AB, greater than the one required. Draw Bb b through the centre o, then will the arc ab = c arc AB, which we shall Page 384 384 THE PRACTICAL MODEL CALCULATOR. put = 20 M in this example, and proceed to measure a b as in the first example. Lay off a b from b to c; from c to d; from d to c; from e to f; fromf to g. Putting g a = A,, then, 108 6 x 20q + A/ = 360~ -— 11; because, 3600 21600 108 36~ 401 - 2200 -11* Lay off, as before directed, g a, = A, from a to It, from h to s, and b to t; then calling s t, ZA, we have 3 A, + An/ = 20; and we find that s t is contained 28 times in the arc a b; 108.'.120p + Al = -1 p; 3 A+ ~ =20; and 28 = 20,. Eliminating A1 and A2, we find 29205 3 2268 = 12'9 times q, nearly;. 36' 40' =- A o N is laid off with as much ease and certainty as by a protractor. As a second example, let it be required to lay off an angle of 132~ 27'. From 180~ 0' take 132~ 27' = 470 33', which put = 3. 360~ 2400 - 470 33' —317 when put =, then i = 9360~ —. 29 N e B 29qo A h We have laid off 29 small arcs from A to B; 29 = E. AB a b = b cd = de = f. Andag = b h af= A-; hg= A,... 5 x 29p + A, = 360~ = ---— me a-t, (1) 2 A: - A/ = 29 p, or n A1 ~ A2 = E s (2) 13 A = 29%P, or q A 2= E- (3) Page 385 MEASUREMENT OF ANGLES. 385 Eliminating A, and A/, we have {mnq m - (q:z 1)}a {5'2'13 + (13 + 1)}29'317 = n q 2400'2'13 1323729 62400 3 = 214 times ~ very nearly. Hence the line o N determines the angle a o N = 132~ 27'. In the expression mn = (q 1)} (R) vnq substituting the numerals of the first example, then {6'3'28 + (28 - 1)}20'11 29205 p 108-3-28 2268 = 12. 9 times' nearly, the result before obtained. The ambiguous signs of (R) cannot be mistaken or lead to error, if the manner in which it is deduced from (1), (2), (3), be attended to. From (3) A/S = O --; substituting this value of A,, in (2), n A/ = E P =F A, =: q; which, when substituted for A/ in (1), gives M ~ q ); from which (R) is found. This method of measuring angles is more exact than it may appear; for if, in the first example, we take 5x - y = 360; 9y + z = x; and 20z =x, 64800 then x = 881 = 730 33' 85. The first equations gave 73~ 33' 82 when 23 z = x, so it does not matter much whether 20, 21, 22, 23, 24, or 25 times z make x. This fact is particularly worth attention. Given the three angles to find the three sides. The following formulas give any side a of any spherical triangle. - cos. 1 S cos. ( S -A) 2s n.4 2Ov' sin. B sin. C -,nd cos. (4 s -B) cos. (4 s - C) csO. ~ a vl cos. 2 a=v' =sin. B sin. C. Given the three sides to find the three angles. sin. (- S - b) sin. (~ S - c) sin.2 4A =sin. b sin. c. sin. sin in. (s S - a) COS. ~ A = V sin. b sin. c. 25 Page 386 386 GRAVITY-WEIGHT —MASS. SPECIFIC GRAVITY, CENTRE OF GRAVITY, AND OTHER CENTRES OF BODIES. — WEIGHTS OF ENGINEERING AND MECHANICAL MATERIALS.-BRASS, COPPER, STEEL, IRON) WATER, STONE, LEAD, TIN, ROUND, SQUARE, FLAT) ANGULAR, ETC. 1. IN a second, the acceleration of a body falling freely in vacuo is 32'2 feet; what velocity has it acquired at the end of 5 seconds? 32'2 x 5 = 161 feet, the velocity. 2. A cylinder rolling down an inclined plane with an initial velocity of 24 feet a second, and suppose it to acquire each second 5 additional feet velocity; what is its velocity at the end of 3'7 seconds? 24 + 3'7 x 5 = 42'5 feet. 3. Suppose a locomotive, moving at the rate of 30 feet a second, (as it is usually termed, with a 30 feet velocity,) and suppose it to lose 5 feet velocity every second; what is its velocity at the end of 3'33 seconds? The acceleration is - 3'33, negative.. 30 - 5 x 3'33 = 13'35 feet. 4. If a body has acquired a velocity of 36 feet in 11 seconds, by uniformly accelerated motion; what is the space described? 36 x 11 = 198 feet. 5. A carriage at rest moves with an accelerated motion over a space of 200 feet in 45 seconds; at what velocity does it proceed at the beginning of the 46th second? 200 x 2 - -= 8.~8889 feet, the velocity at the end of the 45th second. 45 The four fundamental formulas of uniformly accelerated motion are vt p t2 v_ v-pt; s S =2i =2; =2p v the velocity, p the acceleration, t the time, and s the space. 6. What space will a body describe that moves with an acceleration of 11'5 feet for 10 seconds. 11.5 x (10)2 = 575 feet. 2 7. A body commences to move with an acceleration of 5'5 feet, and moves on until it is moving at the rate of 100 feet a second; what space has it described? (loo)2 2 x 5.5 = 909.09 feet. 2 x 5-5 Page 387 GRAVITY-WEIGHT-MASS. 887 8. A body is propelled with an initial velocity of 3 feet, and with an acceleration of 8 feet a second; what space is described in 13 seconds? 3 x 13 (13)2 = 715 feet. 9. What distance will a body perform in 35 seconds, commencing with a velocity of 10 feet, and being accelerated to move with a velocity of 40 feet at the beginning of the 36th second? 10 + 40 2 x 35 = 875 feet, the distance. The formulas for a uniformly accelerated motion, commencing with a velocity c, are as follow:p t2 e + v 22 c2 _ =s = ce + =t2 v=c+ pt; s-ct +-2; s-= 2 t; s- 2 p The succeeding formulas are applicable for a uniformly retarded motion with an initial velocity e. p t2 C + v C2 - V2 v = c —t; s=t —; s - 2 t; s8 2 10. A body rolls up an inclined plane, with an initial velocity of 50 feet, and suffers a retardation of 10 feet the second; to what height will it ascend? 50 10 5 seconds, the time. (5)2 = 125 feet, the height required. 2 x 10 The free vertical descent of bodies in vacuo offers an important example of uniformly accelerated motion. The acceleration in the previous examples was designated by p, but in the particular motion, brought about by the force of gravity, the acceleration is designated by the letter g, and has the mean value of 32'2 feet. If this value of g be substituted for p, in the preceding formula, we have, v = 32'2 x t; v = 8'024964 x Vs; s= 16'1 x t'; s ='015528 x v2; t ='031056 x v; and t = -2492224 x Vs. 11. What velocity will a body acquire at the end of 5 seconds, in its free descent? 32'2 x 5 = 161 feet. 12. What velocity will a body acquire, after a free descent through a space of 400 feet? 8'024964 x V/400 = 160'49928 feet. 13. What space will a body pass over in its free descent during 10 seconds? 16'1 x (10)2 = 1610 feet. Page 388 388 THE PRACTICAL MODEL CALCULATOR. 14. A body falling freely in vacuo, has in its free descent acquired a velocity of 112 feet; what space is passed over? *015528 x (112)2 = 194'783232 feet. 15. In what time will a body falling freely acquire the velocity of 30 feet? *031056 x 30 ='93168 seconds. 16. In what time will a body pass over a space of 16 feet, falling freely in vacuo? *2492224 x V16 = -9968896 seconds. If the free descent of bodies go on, with an initial velocity, which we may call c, the formulas are, v=c+gt; v=c+322xt; v=V/c2+2gs; v=Vc2+64.4xs; t2 2V2 _ ~2 s= t + g et + 161x t2; s8= 2 -015528 (2 -- 2). If a body be projected vertically to height, with a velocity which we shall term c, then the formulas become, ~2 29~ ~2 et - 16'1 x t2; = C2 -='015528 (c2 -_ v2). 17. What space is described by a body passing from 18 feet velocity to 30 feet velocity during its free descent in vacuo. From the annexed table, we find that the height due to 30 feet velocity......................................=.............. 13'97516 The height due to 18................................= 5'03106 Space described............................... 8'94410 Since this problem and table are often required in practical mechanics, we shall enter into more particulars respecting it. v2 -2 v2 C2 As= 2g -2g 2g' if we put h = height due to the initial velocity e; that is,,2 = 2g; and h. = the height due to the terminal velocity v; that is, 2g h -: -;.then, s = h, - h, for falling bodies, as in the last example; and s = h - h,, for ascending bodies. Although these formulas are only strictly true for a free descent in vacuo, they may be used in air, when the velocity is not great. The table will be found useful in hydraulics, and for other heights and velocities besides those set down, for by inspection it is seen that the height'201242 answers to the velocity 3'6; and the height 20'12423 to 36; and the height 2012'423 to 360; and so on. Page 389 WEIGHT-GRAVITY-MASS. 389 TABLE of the Heights corresponding to different Velocities, in feet the second. CORRESPONDING SlEIGHT IN FEET. C 0 1 2 3 4 5 6 7 8; 9 0.000000 -000155.000621.001398.002484.003882.005590.007609.009938' 012578 1 5015528.018789.020652.026242.0304348 0349379 4039752 04476 05011 6056 2 062112.068478.075155.082143.089441.097050.104969.113199.121739 130590D 3.139752.149224.159006.169099.187888.190217.201242.212577.204224I.236180 4 248447 -261025.273913.285714.300621.314441 5328572.343013.357764:.372826' 5 *388199.403882.419877.436180.452795.469720.486956.504503. 522360 -550578 6 559006'577795'596894'616304'636025 6C56060'676397'697050'718013'739286 7 *760870 *782764'804970'827484'850310 *873447'896895'920652'944721l *99C099 8 *993789 1-018790 1'044100 1-069720 1-095652 1-121895 1-148421 1-175311 1-201482 1-22997! 9 1-257761 1-285869 1-314285 1-343012 1-372050 1'401400 1-431055 1-461025 1-491304' 15521894 The following extension is obtained from the foregoing table. by mere inspection, and moving the decimal point as before directed. Corresponding 4' Corresponding h Corresponding 3 Corresonding H leight in Feet. HIeight in Feet.. Height in Feet. iHeight in Feet. 10 1 552795 19 5G0.559 28 12 17392 37 21 25777 11 1 878882 20 6'21118 1 29 13005901 38 22'42236 12 -2065218 21 6 84783 30 13 97516 39 23 61802 13 2'624224 22 7'51553 31 14'92237 40 24'84472 14 30413478 23 8 21429 1 32 1590062 41 26 10249 15 3'49379 24 8'94410 33 16'90994 42 27'39131 16 3'97516 25 9'70497 34 18'78883 43 28-57143 17 4'48758 26 10'49690 35 19'02174 44 30'06212 18 5'03106 27 11 31988 1 36 20'12423 45 31 4441 18. What mnass does a body weighing 30268 lbs. contain? 30268 302680 940 lbs. 32'2 - 322 - For the mass is equal to the weight divided by g. And g is taken equal to 32'2; but the acceleration of gravity is somewhat variable; it becomes greater the nearer we approach the poles of the earth. It is greatest at the poles and least at the equator, and also diminishes the more a body is above or below the level of the sea. The mass, so long as nothing is added to or taken from it, is invariable, whether at the centre of the earth or at any distance from it. If MI be the mass and WV the weight of a body, Then AM = -32 0310559 W. - 32'2 19. What is the mass of a body whose weight is 200 lbs? ~031055 x 200 = 6'21118 lbs. The weight of a body whose mass is 200 lbs. is 32'2 x 200 6440'0 lbs. It may be remarked, that one and the same steel spring is differently bent by one and the same weight at differentl places. The force which accelerates the motion of a heavy body on arn inclined plane, is to the force of gravity as the sine of the inclina Page 390 390 THE PRACTICAL MODEL CALCULATOR. tion of the plane to the radius, or as the height of the plane to its length. The velocity acquired by a body in falling from rest through a given height, is the same, whether it fall freely, or descend on a plane at whatever inclination. The space through which a body will descend on an inclined plane, is to the space through which it would fall freely in the same time, as the sine of the inclination of the plane to the radius. The velocities which bodies acquire by descending along chords of the same circle, are as the lengths of those chords. If the body descend in a curve, it suffers no loss of velocity. The centre of gravity of a body is a point about which all its parts are in equilibrio. Hence, if a body be suspended or supported by this point, the body will rest in any position into which it is put. We may, therefore, consider the whole weight of a body as centred in this point. The common centre of gravity of two or more bodies, is the point about which they would equiponderate or rest in any position. If the centres of gravity of two bodies be connected by a right line, the distances from the common centre. of gravity are reciprocally as the weights of the bodies. If a line be drawn from the centre of gravity of a body, perpendicular to the horizon, it is called the linle qf direction, being the line that the centre of gravity would describe if the body fell freely. The centre of gyration is that part of a body revolving about an axis, into which if the whole quantity of matter were collected, the same moving force would generate the same angular velocity. To find the centre of Gy!ration.-Multiply the weight of the several particles by the squares of their distances from the centre of motion, and divide the sum of the products by the weight of the whole mass; the square root of the quotient will be the distance of the centre of gyration from the centre of motion. The distances of the centre of gyration from the centre of motion, in different revolving bodies, are as follow:In a straight rod revolving about one end, the length x'5773. In a circular plate, revolving on its centre, the radius x 70 71. In a circular plate, revolving about one diameter, the radius x'5. In a thin circular ring, revolving about one diameter, radius x *7071. In a solid sphere, revolving about one diameter, the radius x *6325. In a thin hollow sphere, revolving about one diameter, radius X'8164. In a cone, revolving about its axis, the radius of the base x'5477. In a right-angled cone, revolving about its vertex, the height x'866. Page 391 SPECIFIC GRAVITY. 391 In a paraboloid, revolving about its axis, the radius of the base x'5773. The centre of percussion is that point in a body revolving about a fixed axis, into which the whole of the force or motion is collected. It is, therefore, that point of a revolving body which would strike any obstacle with the greatest effect; and, from this property, it has received the name of the centre of percussion. The centres of oscillation and percussion are in the same point. If a heavy straight bar, of uniform density, be suspended at one extremity, the distance of its centre of percussion is two-thirds of its length. In a long slender rod of a cylindrical or prismatic shape, the centre of percussion is nearly two-thirds of the length from the axis of suspension. In an isosceles triangle, suspended by its apex, the distance of the centre of percussion is three-fourths of its altitude. In a line or rod whose density varies as the distance from the point of suspension, also in a fly-wheel, and in wheels in general, the centre of percussion is distant from the centre of suspension three-fourths of the length. In a very slender cone or pyramid, vibrating about its apex, the distance of its centre of percussion is nearly four-fifths of its length. Pendulums of the same length vibrate slower, the nearer they are brought to the equator. A pendulum, therefore, to vibrate seconds at the equator, must be somewhat shorter than at the poles. When we consider a simple pendulum as a ball, which is suspended by a rod or line, supposed to be inflexible, and without weight, we suppose the whole weight to be collected in the centre of gravity of the ball. But when a pendulum consists of a ball, or any other figure, suspended by a metallic or wooden rod, the length of the pendulum is the distance from the point of suspension to a point in the pendulum, called the centre of oscillation, which does not exactly coincide with the centre of gravity of the ball. If a rod of iron were suspended, and made to vibrate, that point in which all its force would be collected is called its centre of oscillation, and is situated at two-thirds the length of the rod from the point of suspension. SPECIFIC GRAVITY. THE comparative density of various substances, expressed by the term specific gravity, affords the means of readily determining the bulk from the known weight, or the weight from the known bulk; and this will be found more especially useful, in cases where the substance is too large to admit of being weighed, or too irregular in shape to allow of correct measurement. The standard with which all solids and liquids are thus compared, is that of distilled water, one cubic foot of which weighs 1000 ounces avoirdupois; Page 392 392 THE PRACTICAL MODEL CALCULATOR. and the specific gravity of a solid body is determined by the difference between its weight in the air, and in water. Thus, If the body be heavier than water, it will displace a quantity of fluid equal to it in bulk, and will lose as much weight on immersion as that of an equal bulk of the fluid. Let it be weighed first, therefore, in the air, and then in water, and its weight in the air be divided by the difference between the two weights, and the quotient will be its specific gravity, that of water being unity. A piece of copper ore weighs 56- ounces in the air, and 433 ounces in water; required its specific gravity. 56'25 - 43'75 = 12'5 and 56'25. 12'5 = 4'5, the specific gravity. If the body be lighter than water, it will float, and displace a quantity of fluid equal to it in weight, the bulk of which will be equal to that only of the part immersed. A heavier substance must, therefore, be attached to it, so that the two may sink in the fluid. Then, the weight of the lighter substance in the air, must be added to that of the heavier substance in water, and the weight of both united, in water, be subtracted from the sum; the weight of the lighter body in the air must then be divided by the difference, and the quotient will be the specific gravity of the lighter substance required. A piece of fir weighs 40 ounces in the air, and, being immersed in water attached to a piece of iron weighing 30 ounces, the two together are found to weigh 3'3 ounces in water, and the iron alone, 25'8 ounces in the water; required the specific gravity of the wood. 40 + 25'8 = 65'8 - 3'3 = 62-5; and 40 - 62'5 = 0'64, the specific gravity of the fir. The specific gravity of a fluid may be determined by taking a solid body, heavy enough to sink in the fluid, and of known specific gravity, and weighing it both in the air and in the fluid. The difference between the two weights must be multiplied by the specific gravity of the solid body, and the product divided by the weight of the solid in the air: the quotient will be the specific gravity of the fluid, that of water being unity. Required the specific gravity of a given mixture of muriatic acid and water; a piece of glass, the specific gravity of which is 3, weighing 3a ounces when immersed in it, and 6( ounces in the air. 6 - 3'75 = 2'25 x 3 = 6'75 -. 6 = 1-125, the specific gravity. Since the weight of a cubic foot of distilled water, at the temperature of 60 degrees, (Fahrenheit,) has been ascertained to be 1000 avoirdupois ounces, it follows that the specific gravities of all bodies compared with it, may be made to express the weight, in ounces, of a cubic foot of each, by multiplying these specific gravities (compared with that of water as unity) by 1000. Thus, that of water being 1, and that of silver, as compared with it, being 10'474, the multiplication of each by 1000 will give 1000 ounces for the cubic foot of water, and 10474 ounces for the cubic foot of silver. Page 393 SPECIFIC GRAVITY. 393 In the following tables of specific gravities, the numbers in the first column, if taken as whole numbers, represent the weight of a cubic foot in ounces; but if the last three figures are taken as decimals, they indicate the specific gravity of the body, water being considered as unity, or 1. To ascertain the number of cubic feet in a substance, from its weight, the whole weight in pounds avoirdupois must be divided by the figures against the name, in the second column of the table, taken as whole numbers and decimals, and the quotient will be the contents in cubic feet. Required the cubic content of a mass of cast-iron, weighing 7 cwt. i qr. - 812 lbs. 812 lbs. -- 450'5 (the tabular weight) = 19803 cubic feet. To find the weight from the measurement or cubic content of a substance, this operation must be reversed, and the number of cubic feet, found by the rules given under "Mensuration of Solids," multiplied by the figures in the second column, to obtain the weight in pounds avoirdupois. Required the weight of a log of oak, 3 feet by 2 feet 6 inches, and 9 feet long. 9 x 3 x 2'5 = 67-5 cubic feet. And 67'5 x 58'2 (the tabular weight) = 3928'5 lbs., or 35 cwt. Oqr. 81 lbs. The velocity g, which is the measure of the force of gravity, varies with the latitude of the place, and with its altitude above the level of the sea. The force of gravity at the latitude of 450 = 32-1803 feet; at any other latitude L, g = 32'1803 feet - 0'0821 cos. 2 L. If g' represents the force of gravity at the height A above the sea, and r the radius of the earth, the force of gravity at the level of the sea will be g = g' (1 + 4 r). In the latitude of London, at the level of the sea, g = 32'191 feet. Do. Washington, do. do., g 32-155 feet. The length of a pendulum vibrating seconds is in a constant ratio to the force of gravity. - 9.8696044. Length of a pendulum vibrating seconds at the level of the sea, in various latitudes. At the Equator....................................390152 inches. Washington, lat. 38~ 53' 23"................39'0958 New York, lat. 400 42' 40"................39'1017 London, lat. 51~ 31'......................39'1393 lat. 450...........................39'1270 - lat. L.............. 39 1270 in.- 009982 cos. 2 L. Page 394 394 TIlE PRACTICAL MODEL CALCULATOR. Specific Gravity of various Substances. Weight of a Weight of- { We'ght of a Wreight of a cubic foot uifcubic foot cubic fos cbfot in ounce'. in pounds. in ou~ces. in powids. METALS. STONES.- Continilued. Antimony, fused. 6,624 414'0 Grindstone. 2,143 134-10 Bismuth, cast. 9,823 614-0 Gypsum, opaque 2,168 1355 Brass, common, east. 7,824 4890 semi-transparent. 2,306 144'1 cast.. 8,396 524'8 Jet, bituminous 1,259 78-8 wire-drawn 8,544 534'0 Lime-stone. 3,182 199'0[ Copper, cast.. 8,788 549'2 Marble 2,700 160'8 wire-drawn. 8,878 554'9 Mill-stone. 2,4S8 155-2 Gold, pure, cast. 19,258 1203'6 Porcelain, China. 2,3,5 149'1 22 carats, stand. 17,486 1093'0 Portland-stone 2,570 16006 20 carats, trinket. 15,709 982'0 Pumice stone 915 57-2 Iron, cast... 7,207 450-5 Paving-stone 2,416 151'0 bars... 7,788 480-8 Purbeck-stone. 2,601 162-6 Lead, cast.1 11,352 709-5 Rotten-stone 1,981 124'0 litharge. 6,300 393'8 Slate, common.. 2,6i72 1670 Mlanganese. 7,000 437'5 new 2,854 178'4 M~ercury, solid,'L' Mercury, solid, - 15,632 977'0 Stone, common 2,320 1575 400 below 00 J rag.2,470 154-4 rag...]2,470 j~ at 32 deg. Fahr. 13,619 851-2 Sulphur, native 2,033 127-1 at 60 deg... 13,580 848'8 melted... 1,991 124-5 at 212 deg. 13,375 836-8 Nickel, cast... 7,07 4880LIQUIDS. Platina, crude, grains. 15,602 975-1 purified... 19,500 1218-8 Acetic acid 1,007 63-0 hammered... 20,337 1271'1 Acetous acid 1,025 64-1 rolled.... ] 22,069 1379'4 Alcohol, commercial 837 52-3 wire-drawn ~ 21,042 1315'1 highly rectified 829 5i18 Silver, cast, pure.. 10,474 654'6 Ammonia, liquid 897 561 Parisian standard 10,175 63650 Beer.... 1,023 68'0 French coin... 10,048 628'0 Ether, sulphuric 739 462 shilling, Goo. III... 10,534 658'4 Milk of cows 1,032 64'5 Steel, soft... 7,933 4896i Muriatic acid 1,194 74-6 hardened. 7,840 490'0 Nitric acid 1,271 79'5 tempered. 7,816 488-5 highly concentrated 1,583 99'0 tecmpered and'hard. 7,818 488'6 Oil of.lmonds, sweet. 917 574 Tin, pure Cornish.. 7,291 455'6 hemp-seed 926 58'0 Tungsten 6,0u66 379 1 linseed 940 58'8 Uranium.. 6,440 402'5 olives...915 57'3 Wolfram... 7,119 4450 poppies 924 57.8 Zinc, usual state. 6,862 429'0 rape-seed 919 57'5 pure.... 7,191 449'5 turpentine, essence 870 54'4 whales.923 57.8 WOODS. Spirits of wine, 37 52-4 commercial j Ash. 845 529 highly rectified.. 829 51'9 Beech.852 53'2 Sulphuric acid..,841 11, Box, Dutch.. 912 57'0 highly concentrated. 2,125 1310 French. 1,328 83'0 Turpentine, liquid. 91 620 Brazilian.. 1,031 64-5 Vinegar, distilled. 1,010 63:1 Cedar, American. 561 35-1 Water, rain, or distilled 1,000 62-5 Indian... 1,315 82-2 sea. 1,026 641 Clherry-tree.. 715 44-8 Cocoa.. 1,040 65'0 M ISCELLANEOSUS SUBCork. 240 15-0 Ebocy, Indian. 1,209 756 STNCES. American 1,331 83-2 60 Elm...671 42-0 Beeswax....965 60-4 Ellu. ~~~~ 671 42'0 91 9 Butter....942 590 Fir, yellow 657 41-1 Camphor989 62-0 wfhite 569 35'6 hLignu-vite a. 5 35 Fat, beef or mutton 923 57-8 Ligriiun-vitc. 1,343 83-4 Lime-tree. 604 37-8 ogs'937 58 Ione.. 1,450 90'6 Logwood... ] 913 571 oney 93 57-1 ndidgo 769 48'1 Mahogany 1,063 66'5 i.. 6 d Maple...-0 470 Ivory... 1,826 1141 Maple. [750 47'0 Lad.98 52 Lard... [~ 59'2~ Oak, heart of, old 1,170 73-1 Opium... 1,336 835 dry... 932 5 Opiu336 Vine. 9-1,327 8-0 Spermaceti. 9. 5910 Vine 1,327 83-0 ~~~~~Sugar, white.160I 104 Walnut. 671 42-0 STallowr -he.0 10 Willow.... 36-6 Yew... 807 50'5 GASES. ~STONES,~ EA~TIS, ETC. A~Atmois~pheric ail beinig estimated Alabaster, yellow. 2,699 168-8 as 1. white 2,730 170-6 Borax... 1,714 107'1 Atmospheric, or common air 1.. 1000 Brick earth..~ 2,000 125-0 Ammoniacal gas.... 590 Chalk.2,784 174-0 Azote.'969 Coal, Cannel 1,270 79-4 Carbonic acid I.5 2) Newcastle 1,270 79-4 Carbonic oxide.... 960 Staffordshire 1,240 77-5 Carburetted hydrogen'491 Scotch. 1,300 81-2 Chlorine....'470 Emery... 4,000 250-0 Hlydrogen.... 074 Fiint, black. 2,582 162-0 MuriAtic acid gas.... 1278 Glass, flint 2,933 170-9 Nitrous gas.. 1-094 swhite 2,892 168-2 Nitrous acid gas.... 2-427 Granite, Aberd. blue 2,625 164-1 Oxygen.... 1104 Cornish.. 2,662 166-4 Steanm..690 Eyptian, red 2,654 16.-9 Sulphureted hydrogen. 1777 Egy gray. 2,728 1705 Sulphuros acid 193 gray 2,7298 170-5 Sulphurous acid... 2-193 Page 395 SPECIFIC GRAVITY. 395 TABLE of the Weight of a Foot in length of Flat and Rolled Iron. BREADTH IN INCHES AND PARTS OF AN INCH. 4 8 3~ 31 241 2~'2 2 13 1 1 11I 1 3 _ _ _ _ _ _ _ _ _ _ _4 4 4 2 8 4 4 2 1'68 1'57 1'47 136 1'26 115 105 094 0'84 0'73 00638057 0' 52 1042 0'31 0'21 252 236 220 204 189 1'73 1'57 1'41 1'2611.10 0'040'086 0'78 003 )'47 031 3-36 3'15 2'91 2'73 52 2'31 2'10 1'89 168 147 1203118 105 084 063 042 5'04 4'72 4'41 400 378 3'46 3'15 283 2'52 2'20 1l89 1173 1.57 120; 0)94 0'63 6'72 6'30 5'88 5404 504' 402 4'20 378 3'36 2-94 2..52 2.31 2-10 108 1_26 8'40 7'87 7"35 082 630 5'77 5'25 4'72 4'20 367 3.' 15 1288 2'62 210 1'57 10.08 9.45 8.82 819 7.56 693 6030 5.66 5.04 4.41 378134; 3.15 252 11'76 11'02 10'29 945 8'82 8'08 7"35 6061 5'88 5'14 441 4-0436 137294 1 13'44 12'60 1176 1002 10'08 9'24 8'40 7'56 6'72 5'87 5'04 4062 4-'20 11 1512 14'16 13'20 1228 11'34 1039 9'45 8'50 7'56 6'60 5067 5'19 4'72 1 16-80 10575 14'70 1135 12060 11'55 10'50 9'45 8'40 7'35 6030 5'77 1 18'48 17'32 1616 1501 13806 1270 11'55 10'39 924 8'07 I1 20'18 18'0 17'64 1038 15'12 13'86 12'60 11'34 10'08 8'80 1 23'54 2205 20'58 1911 17064 1617 14'70 13'22 2 26-88 25'20 2:352 2184 20'16 1848 16'80 15'12 21 33065 31'50 2340 2739 2020 23'10 3 40321 3780 35'28 3270 31 47.04 TABLE Of the Weight of Cast-iron Pipes, in lengths. Weight. Weight. wight. i L~ ii l II ci l CInch. Inch. Feet. C. qr. lb. Inch. Inch. Feet. C. qcr. lb. Inch. Inch. Feet. C. qr. lb. 1 3~ 12 6I I 9 2 0 16 11 I 9 5 0 7 31 21 9 2 3 20 j 9 6 1 12 2 41 21 9 3 2 21 2 8 41 1 4 9 4 1 21 1 9 10 1 2 2 9 6 1 8 1 9 6 0 14 12 1 9 5 0 24 6 20o 7 1 9 30 7 9 62 8 2' a 6 116 3 9 3320 I 9 7320 6 210 9 43 5 1 9 10 3 0 6 3 10 1 9 6 2 4 12 I 9 5 1 16 3 2 2 220 9 3 1 6 9 6 3 9 9 1 06 9 40122 9 8 1 0 9 112 9 5010 1 9 11 0 21 g 9 1 3 6 1 9 700 13 I 9 15220 9 2 1 0 8 9 3 2 4 9 7 0 14 31 1 9 30 i 9 4125 9 827 9 1 60211 9 5 1 1 1 9 11 2 12 9 1 214 1 9 7 1 16 131 ~ 9 5 3 7 9 2 0 8 8 9 3 32 9 7 1 12 9 9 20 o 9 4 4226 9 8 3 16 4 9 1110 9 5 2 22 1 9 11 3 24 I 9 1312 1 9'7 3 8 14 9 6 0 4 f 9/ 2112 9 9 40 0 9 7216 9 2 221 9 5 0 4 9 9 1 0 4 / 9 1 22 6 6 2 1 9 12 1 14 9 20 4 1 9 8026 141 9 6 0024 9 2214 9 9 4 0 18 9 3 14 9 3021 9 5 1 o 9 92 2 5 9 1222 [ 9 6 16 1 9 12t 3 6 9 2110 I 9 8220 15 1 9 6 1 21 g 9 2317 10 1 9 4110 j 9 937 9 3 124! 9 5 1 26 1 9 13 0 26 9 13 10 9 4 2 14 1 9 16 3 5 9 22 0 1 9 90 8 15 1 9 6211 9 3 018 I 10 9 4 2 14 9 10 9 10 9 3 37 9 5 3 7 1 9 13 2 17 ~~~~1 9 50 ~12 ~~1 9 7 0 0 9 11716 6 9 2 0 0 1 9 9 2 0 16 1 9 0 22 9 2221 11 ~ 9 4 3 4 9 101 20 9 31117 9 60-11 1 9 14 08 9 4 016 1 9 71 11 9 17 3 14 1 9I 5 220 1 9 9 3 20 9 1 21 3 4 Page 396 396 THE PRACTICAL MODEL CALCULATOR. TABLE of the Weight of one Poot Length of Miialleable Iron. SQUARE IRON. ROUND IRON. Scantling. Weight. Diameter. Weight. Circumference. Weight. Inches. Pounds. Inches. Pounds. Inches. Pounds. 0-21 1 0-16 1 0-26 0-47 II 037 041 i 0-84 i 066 11 059 134 8 1.03 14 0 82 8 4 4 189 4 1-48 2 105 -7 257 7 2-02 2- 1-34 1 3-36 1 2-63 2' 1.65 14 4-25 1-s 3.33 24 2 01 14- 5.25 14 4-12 3 2-37 183 6635 13 4-98 34 2-79 1' 7.56 13 5.93 3' 3 24 14 8'87 14 6-96 3- 3-69 14 10.29 14- 8-08 4 4-23 17 11-81 18 9-27 4' 5.835 2 13'44 2 10-55 5 6-61 24 17-01 21 13.35 5~ 7.99 24- 21-00 24 16'48 6 9'51 2a- 25.41 2- 19.95 6~ 11_18 3 30-24 3 23 73 7 12 96 3' 41 16 34- 27.85 71 14.78 4 53-76 3- 32-32 8 16.92 44 68-04 34 37-09 8~ 19-21 5 84.00 4 42 21 9 21 53 6 120-96 4 53-41 10 26.43 7 164-64 5 65 93 12 31-99 The following tables are rendered of great utility by means of this table: The weight of Water being 1' -- Copper 8-8 Brass _ 8-4 Iron, cast 7-2 Lead = 11-3 Zinc 7-2 Gun-metal - 8-7 Sand - 1-5 Coal - 1-25 Brick - 2-0 Stone 2-5 Timber, average = 0-85 Suppose it be required to ascertain the weight of a cast iron pipe 261 inches outside and 233 inside, the length being 6~ feet. Opposite 26} in the table is 234-8576 x 7-2 x 6-5 = 10991-135. And opposite 23- in the table is 192-2856 x 7-2 x 6-5 = 8998-966 subtract 1992-169 lbs. avr. The succeeding table contains the surface and solidity of spheres, together with the edge or dimensions of equal cubes, the length of equal cylinders, and the weight of water in avoirdupois pounds: Page 397 SPECIFIC GRAVITY. 397 Surface and Solidity of Spheres. Diameter. Surface. Solidity. Cube. Cylinder. Water in lbs. 1 in. 3'1416' 5236'8060'6666'0190 -116 3'5465'6280'8563'7082'0227 3-9760' 7455' 9067'7500 0270 136 4-4301'8767' 9571'7917'0317 4 4'9087 1'0226 1'0075 *8333'0370 156 5-4117 1-1838 1'0578'8750'0428 8 59395 1-3611 1-1082'9166 0500 -T 6 664918 1'5553 1-1586'9583'0563 i 7'0686 1-7671 1-2090 1'0000'0640 7'6699 2-0000 1-2593 1-0416'0723 8 8-2957 2-2467 1-3097 1'0833' 0813 I -6- 8-9461 2-5161 1-3601 1'1349'0910 4 9-6211 2-8061 1-4105 1'1666'1015 10-3206 3'1176 1-4608 1-2083'1128 w 11-0446 3'4514 1-5112 1'2500'1250 f] W11-7932 3'8081 1'5616 1'2916'1377 2 in. 12-5664 4'1888 1'6020 1'3333'1516 1l6 13-3640 4-5938 1-6633 1-3750'1662 X 14-1862 5-0243 1-7127 1'4166'1818 -T3&- 15-0330 5A4807 1'7631 1'4582'1982 4 15'9043 6'9640 1'8135 1-5000'2160 156 16-8000 6-4749 1'8638 1-5516'2342 38 17-7205 7-0143 1'9142 1-5832'2540 176 18-6655 7-5828 1'9646 1'6250'2743 19-6350 8-1812 2'0150 1-6666'2960 T69 20-6290 8-8103 2'0653 1-7082'3187'87 21'6475 9'4708 2'1157 1-7500'3426 J6 22-6907 10-1634 2'1661 1'7915'3676 4 23-7583 10'8892 2'2165 1-8332'3939 24-8505 11-6485 2-2668 1-8750'4213 25-9672 12-4426 2-3172 1'9165 4501 X6 S 27-1084 13-2718 2-3676 1'9582 4800 3 in. 28-2744 14-1372 2-4180 2-0000 5114 29-4647 15'0392 2-4683 2'0415 5440 i 30'6796 15-9790 2'5187 2-0832'5780 13 6 31-9191 16-9570 2-5691 2-1250'6133 T 33'1831 17-9742 2'6195 2-1665'6401 35-3715 19-0311 2-6698 2-2082'6884 35-7847 20'1289 2'7202 2-2500 7281 -17 37-1224 21-2680 2-7706 2-2915'7693 38-4846 22'4493 2-8210 2-3332'8120 T-9 39'8713 23-6735 2'8713 2-3750'8561 41.2825 24-9415 2-9217 2-4166'9021 i —* 42.7183 26-2539 2-9712 2-4582 *9496 4 44-1787 27-6117 3-0225 2'5000 *9987 f2 45-6636 29'0102 3-0728 2'5415 1-0493 8 47-1730 30'4659 3-1232 2'5832 1'1020 1 $ 48'7070 31-9640 3-1730 2-6250 1-1561 4 in. 50-2656 33-5104 3-2240 2-6665 1-1974 Y6 61'-8486 35-1058 3'2743 2-7082 1-2698 53'4562 36-7511 3-3247 2-7500 1-3293 l36 55-0884 38'4471 3-3751 2-7915 1'3906 i. 56-7451 40-1944 3-4255 2-8332 1-4538 T -15 58-4262 42-0461 3-4758 2'8750 1 5208 60'1321 43-8463 3-5262 2-9165 1'5860 f 16 1 61-8625 45'7524 3-5766 2-9582 1-6550 Page 398 398 THE PRACTICAL MODEL CALCULATOR. Diameter. Surface. Solidity. Cube. Cylinder. Water in lbs. 63-6174 47'7127 3-6270 3'0000 1'7258 96 65-3968 49-7290 3-6773 3'0415 1-7987 8 67-2007 51-8006 3 7277 3 0832 1'8736 16 69-0352 53'9290 3-7781 3-1250 1'9506 70-8823 56'1151 3'8285 3-1665 2-0297 72-7599 58-3595 3-8788 3-2080 2-1109 74'6620 60-6629 3'9292 3-2500 2'1942 -6 765887 62-9261 3'9796 3'2913 2'2760 5 in. 78-5400 65-4500 4'0300 3'3332 2-3673 T16 80'5157 67-9351 4'0803 3'3750 2-4572 82'5160 70-4824 4-1307 3-4155 2'5453 T,3 84-5409 73'0926 4'1811 3-4582 2-6438 ~~4 86'5903 75-7664 4'2315 3'5000 2'7605 it 88'6641 78'5077 4-2818 3'5414 2' 8396 8 90'7627 81-3083 4-3322 3-5832 2'9407 ~-r7 92'8858 84-1777 4'3820 3-6250 3'0447 Y 95'0334 87-1139 4'4330 3'6665 3-1509 97-2053 90'1175 4-4633 3'7080 3 2595 99'4021 93'1875 4-5337 3'7500 3'3706 1 6 101'6233 96'3304 4'5841 3-7913 3'4843 103-8691 99-5412 4'6345 3'8330 3-6004 6 106-1394 102-8225 46848 3'8750 3'7191 ~68 108-4342 106'1754 4-7352 3'9163 3-8404 16 110'7536 109'5973 4'7856 3'9580 3'9641 6 in. 113'0976 113-0976 4-8360 4'0000 4'0907 1 tT 115-4660 116'6688 4-8863 4'0417 4'2200 -87 117-8590 120-3139 4-9367 4'0833 4-3517 y13S 120'2771 124-0374 4'9871 4-1250 4'4874 122'7187 127-8320 5'0375 4'1666 4-6236 T-5 125-1852 131'7053 5'0878 4'2083 4 7638'8 127-6765 135'6563 5-1382 4-2500 4'9067 T7i 130-1923 139-6854 5-1886 4'2917 5-0524? 132-7326 143'7936 5-2390 4'3332 5'2010 19 135-2974 147'9815 5-2893 4'3750 5'3525 137-8867 152'2499 5-3377 4'4165 5'5069 1- 140-5006 156-5997 5'3901 4-4583 5'6786 4 143'1391 161-0315 5'4405 4'5000 5-8245 la 3 145-8021 167-5461 5-4908 4'5416 6'0601 148484896 170'1682 5-5412 4'5832 6-1550 85 s151-2017 174-8270 5-5916 4'6250 6-3235 7 in. 153-9384 179'5948 5'6420 4'6665 6'4960 1-6 l156-6995 184-4484 5'6923 4'7082 6'6725 ~ 159-4852 189'3882 5-7427 4'7500 6'8502 -$ s162-2955 194-1165 5-7931 4'7915 7'0212 4 165-1303 199-5325 5-8435 4'8332 7-2171 _5 167-9895 204-7371 5-8938 4'8750 7'4053 3 170'8735 210-0331 5-9442 4'9166 7'5970 173-7520 215-4172 5-9946 4'9582 7'7916 g 176-7150 220-8937 6'0450 5'0000 7'9897 179'6725 226'7240 6-0953 5'0415 8'2006 T 182-6545 232-1235 6-1457 5'0832 8-3960 jB & 185-6611 237-8883 6-1961 5'1250 8'6044,a 188-6923 243-7276 6-2465 5-1665 8-8157 3s 191-7480 249'4720 6'2968 5-2082 9'0234 17~ 194-8282 255'7121 6-3472 5-2500 9'2491 8~ 197'9330 261-9673 6-3976 5'2913 9'4753 8 in. 201-0624 268-0832 6-4480 5'3330 9'6965,r, 204-2162 274-4156 6-4983 5 3750 9-9260 20'26 27'16 6 4 8'7099 Page 399 SPECIFIC GRAVITY. 399 Diameter. Surface. Solidity. Cube. Cylinder. Water in lbs. 207'3946 280-8469 6-5487 5-4164 10-1583 T36 210'5976 287-3780 6'5991 5-4581 10-3944 4 213-8251 294-0095 6'6495 5'5000 10-6343 s{Jf 217'0770 300-7422 6-6998 5'5414 10'8778 220-3537 307-5771 6-7502 5'5831 11-1250 I176 223-6549 314'5147 6'8006 5'6250 11'3760 226-9806 321-5553 6'8510 5-6664 11-6306 96- 230-3308 328-7012 6-9013 5'7080 11'8891 233'7055 335-9517 6-9517 5'7500 12-1514 6- 237-1048 343'3079 7'0021 5'7913 12'4170 134 240-5287 350'7710 7'0525 5-8330 12'6874 js a 243-9771 358-3412 7-1028 5'8750 12'9612 w- 247-4500 366'0199 7'1532 5'9163 13'2390 j1 5 250'9475 373-8073 7'2036 5-9580 13'5206 9 in. 254-4696 381-7017 7-2540 6-0000 13'8062 8 258-0261 389-7118 7'3043 6'0417 14'0959 2615872 397-8306 7-3547 6-0833 14'3895,36 265'1829 406-0613 7'4051 6-1250 14'6872 14 268 8031 414-4048 7'4555 6"1667 14'9890 - -272'4477 421-2907 7-5058 6'2083 15'2381 276'1171 431-4361 7'5562 6-2500 15'6050 %17S 279 8110 440-1294 7-6066 6-2916 15'9195 2 283-5294 448-9215 7'6570 6-3333 16'2375 -19 287-2723 457'8500 * 7'7073 6'3750 16'5604 8 291 0397 466-8763 7-7557 6-4166 16'6869 294-8310 476-0304 7'8081 6'4582 17'2180 298-4483 485-3035 7-8585 6-5000 1'5534 1 3 302-4894 494'6952 7'9088 6-5415 17'8931 306 3550 504'2094 7'9592 6'5832 18'2373 1 5 310 9452 513-8436 8'0096 6'6250 18'5857 10 in. 314'1600 523'6000 8'0600 6-6666 18'6786 I1 318-0992 533-4789 8'1103 6'7083 19-2960 i 322'0630 543'4814 8-1607 6'7500 19'6577 -13 326-0514 553-6081 8'2111 6-7916 20'0240 4 330-0643 563'8603 8'2615 6'8333 20'3948 A S 334-1016 574-2371 8'3118 6-8750 20'6682 38.338-1637 584-7415 8'3622 6-9166 21-1501 76 3422503 595'3677 8-4126 6'9582 21'5344 E 346'3614 606-1318 8'4630 7'0000 21'9238 -91 1 350'4970 617-0207 8'5133 7'0416 22'3176 A 354-6571 628-0387 8'5637 7'0833 22-7162 aB &358-8418 639'1871 8-6141 7'1250 23'1194 4 363-0511 650-4666 8'6645 7-1666 23'5274 6a 367-2849 661-8580 8-7148 7'2082 23-9394 7 371-5432 673'4222 8-7652 7-2500 24'3577 375'8261 685'0997 8-8156 7'2915 24'7801 11 in. 380-1336 696'9116 8-8660 7'3330 25 2073 ~Tl4 384-4655 708'9106 8'9163 7'3750 25'6414 388-8220 720'9409 8-9667 7'4165 26'0764 -3A 393-2031 733'1.599 9-0171 7-4582 26-5184 I 397'6087 745'5004 9-0675 7'5000 26-5657 s56 402'0387 758-0104 9-1178 7-5414 27'4162 8 406-4935 770-6440 9'1682 7-5832 27'8742 )T1 410-7728 783'5787 9'2186 7'6250 28-3420 I 415'4766 796-3301 9-2690 7-6664 28'8033 9-4- 420-0049 809'3844 9-3193 7'7080 29-2754 i 424-5576 822-5807 9-3697 7 7500 29-7527 l 42941351 835-9695 9-4201 7'7913 30-2370 A~ ~ ~ ~ 2'87 9'69 4'5029131'7527__ ______-______ -_____ Page 400 400 THE PRACTICAL MODEL CALCULATOR. I Diameter. Surface. Solidity. Cube. Cylinder. Water in lbs. 4 433-7371 849'4035 9'4705 7-8330 30-7229 4. 438-3636 863-0283 9-5208 7-8750 31'2157.8 443-0146 876-7999 9-5772 7'9163 31-3883 {1 447-6902 890-7070 9-6216 7-9580 32-2169 12 in. 452'3904 904-7808 9-6720 8'0000 32-7259 471'4363 962-5158 9-8735 8'1666 34-8142 490'8750 1022-656 10-0750 8-3332 36-9886 4 506-7064 1085'251 10'2765 8'5000 39'2535 13 in. 530'9304 1150-337 10'4780 8-6666 41'6077 4 551'5471 1218-000 10'6790 8-8332 44'0551 4 572-5566 1288'252 10-8810 9'0000 46-5961 4 593-9587 1361-346 11-0825 9'1665 49-2399 14 in. 615-7536 1436'758 11-2840 9-3332 51-9675 4IL 637'9411 1515'106 11'4855 9'5000 54'8014 ~ 660-5214 1596-260 11-6870 9'6665 57 7367 4 683-4943 1680-265 11-8885 9-8332 60-7751 15 in. 706-8600 1767'150 12'0900 10'0000 64'0178 4 730'6183 1856'988 12'2915 10-1666 67'1672 4. 754-7694 1949.821 12-4930 10-3332 70-5250 a4 779-3131 2045-697 12'6940 10-5000 73-9929 16 in. 804'2496 2144-665 12'8960 10-6666 77-5725 TABLE containing the Weight of 7Flat Bar Iron, 1 foot in length, of various breadths and thicknesses..0 * THICKNESS IN PARTS OF AN INCH. 4d TW T1g 4.s 3 T_ f _ 1 inch. FP Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. 1 in. 0.83 1 04 1 25 1 45 1.66 1 87 2.08 2.50 2.91 3 33 14 0.93 1.17 1.40 1.64 1.87 2.00 2.34 2.81 3 28 3.75 14. 1.04 1.30 1.56 1.82 2.08 2.34 2.60 3.12 3.74 4.16 1' 1.14 1]43 1.71 2.00 2.29 2.57 2.86 3.43 4.01 4.58 14 1.25 1.56 1.87 2.18 2.50 2.81 3.12 3.75 4.37 5.00 18 1*35 1 69 2 03 2 36 2 70 3 04 3 38 4 06 47 73 5 41 1 145 1 82 2 18 2 55 2 91 3 28 3 64 4 37 5 10 5 83 17 1.56 1.95 2.34 2'73 3'12 3'51 3.90 4'68 5.46 6.25 2 in. 1.66 2-08 2.50 2.91 3.33 3.75 4.16 5.00 5.83 6-66 24 1.77 2.21 2.65 3-09 3.54 3.98 4.42 5.31 6.19 7.08 21 1'87 2'34 2'81 3'28 3-75 4'21 4.68 5'62 6'56 7'50 243 1'97 2'47 2.96 3'46 3'95 4'45 4.94 5'93 6'92 7'91 24 2.08 2.60 3.12 3'64 4'16 4'68 5.20 6'25 7'29 8.33 25 2.18 2-73 3'28 3 82 4'37 4'92 5.46 6'56 7'65 8.75 24 2.29 2-86 3 43 4 01 4'58 5'15 5.72 6'87 8.02 9.16 27 2'39 2.99 3'59 4.19 4'79 5'39 5.98 7'18 8.38 9.58 3 in. 2 50 3.12 3.75 4.37 5 00 5.62 6 25 7'50 8.75 10.00 31 2.70 3.38 4.06 4-73 5-41 6.09 6.77 8.12 9.47 10.83 34 2.91 3.64 4.37 5.10 5.83 6.56 7.29 8.75 10.20 11.66 34 3.12 3.90 4.68 5.46 6.25 7 03 7-81 9.37 10.93 12.50 4 in. 3.33 4.16 5.00 5.83 6.66 7'50 8.33 10.00 11.66 13 33 41 3.54 4.42 5'31 6'19 7'08 7'96 8.85 10'62 12.39 14.16 41 3-75 4'68 5.62 6'56 7 50 8.43 9'37 11.25 13.12 15.00 44 3.95 4.94 5.93 6'92 7'91 8.90 9.89 11'87 13.85 15.33 5 in. 4.17 5 20 6.25 7'29 8'33 9.37 10.41 12'50 14.58 16.66 54- 4.37 5.46 6.56 7'65 8'75 9.84 10.93 13.12 15.31 17'50 54 4.58 5.72 687 802 916 10.31 11.45 13 75 16.04 18.33 53 4.79 5.98 7.18 8-38 9.58 10.78 11.97 14.37 16.77 19.16 6 in. 5.00 6.26 7 50 8'75 10.00 11.25 12650 15.00 1750 20.00 Page 401 SPECIFIC GRAVITY. 401 TABLA combining the Specific Gravities and other Properties of Bodies. Water the standard of comparison, or 1000. METALS. STONtES, EARTHS, ETr.5 Names. *V' ~ I ~ Names. Ftc;3 O~~~~~~~~~~~~ Z Nms. I n 5" S'~~~~~~S ~~'. _. Platinum.. 19500 3280. 3 5 38 Marble, average 2730 17000 113 9' 5 Pure Gold 19258 2016 1 1 18 3 10'0 Granite, ditto 226s51 16)568! 135 6. 2 Pu re7~7~Y3075 Gold00 1. 19782578 20:....1 Mercury.. 13500... Purbeck stoneo 2601 162 56 13> 90 Lead.... 11352 62 -319 81 8 7' 1- 6 8 Portland ditto. 2570 160'62 14 145 1'0 rubcst.. 2601 62'6 1 3}4~19.0i Pure Silver. 10474 1873 2. 2 2 2-4 2 97 Bristol ditto 2 554 159 62 114i~ Bismuth.. 8923 476'156 1445 20 Millstone.... 2484 155'23 14/~. Copper, cast. 8788 1996'193 8-51 Pavingstone. 2415 150.93 1434,5'7 wrought 8910.. 1508 5 28 8-9 Craigleith ditto 2362 1-17- 621 15 50 Brass, cas~. 7 ~98241 90 - 72150 i 68'016 ~''50 801 fto any Grindstone...2143 1393, 16 696 Brass, east.- 7824 1960 -210 6-. degrce Chalk, Brit... 2781 173'811 123:/0.5 sheet. 8396.... 12-23 6 6.. 86 Brick...... 2000 12500 17 0'8 f~~oanyI ~~~Coal, Scotch..1300 81'15,27Y2. Iron, cast. 7264 2786 125 7-87... to any. degree.. Newcastle 1270 79-371 27 " bar.. 7700.. 137 25'00 4 8 4-7 4 3'7 Staffordsh. 1240 77 50 29. Steel, soft. 7833.. 133 58-91.... Cannel 123 7737 29,hard.7816.. to any degree Tin, cast.. 7291 442 278 211 8 4 1-2 5 3-01 Zinc, cast. 71901 773 329 5-06 7 8 1-6 7 3-6 TABLE containing the Weight of Columns of Water, each one foot 3 in. 3-0672 9 in. 27-6120 15 in. 76-7004 21 in. 130-2376 27 in. 248-5116 33 in. 371-2344 ~n ie~~n~t~, and of Various ])iameters~ in lbs36 i. avoirdu~ois ~ 3-3288 / 28-3848 ~ 77-9844 1 Y 152-1288 34 250-8180 11 374-0520 3.6000 [ 29-1672 ~,/ 79-2792, 153-9348 { ~ 253'1352 ~ 376-8004. 3-8820 29-9604 8 80-5836 155-7396 1255-4632 379459-43 2 4-1748 ~, 30-7657 ~ [ 81-9000 ~, 157-5780 ~ 257-8008 ~ 382.5684 I | 5.4510; H ~ 10~mb2 33.242439;~1)4~~U 8 4-4784 Y, 31-6524 08 83-2260 65 159-4152 % 260-1504 385-4292 ~ 4-7928 04 32-4000 ~ 84-51628 ~ 161-2644 04 262-5006 04 388-2096( ~ 5-1180 ~a 33-2424 ~8 85-9104 Y, 163-1220 ~ j 264-8700 Y, 391 -1820 4 in. 5-41 10 in. 34-0884 16 in. 87-2688 22 in. 164 9928 28 in. 267-2616 34 in. 394-0740 5-7996 [ 34-9464 [ 8 88-6368 / 166-8732' [ 269-6532' 396-976 6-1572 35-8152,. 2 90-0168 168-7632 4 272'05448 4 399-8928' 6-52414 [ 36-6936 ~ 91-4176 /a 170-6652 275-6672 ~ 402-8088 ~ 6-9024 ~ 37-5828 Y. 92-8080 ~! 172-5780 ~ / 276-8916 Y, 405-7500 8 7-2912 38-4828 V 94-2192 Y, 174-5004 sa 279-3252 ~1P 408-6948 7-6908 39 393636 { 95-6412 Y 176'4336 4 281-7708 ~/4 411-4116 | 8-1012 4, 40-3152 ~8 97-0710 | a 178-3776 [ [ 284'2260 Y, 414'6180 4 in. 8-5212 11 in. 41-2476 17 in. 98-5176/ 23 in. 180-3324 29in. 286-6920 35 in. 417-5952 ~ 8-9532 42-1908 a 99-9720 ~ 182-2980 Y, 289-1688 Y 420-5844 |9-398-1 43-1436 101-4372, | 184-2744,/ 291-6564 1 4 423-542-4 -34 Y 8-66 V 9-58 ~ 9-848 1 44.1084 / 102-9120 6 186-16 29415 48 / 426-5928 I~ 10-3t2 [ 45-10828 2 104-3988 188-2584 ~ 296-5548 W 429'-6120% 10-7856 1 % 46-0680 | 105-8952 I [ 190-2672 Y, 299-1828 / 432-6432 11-2704 / ~4 47-0641) / 107-4)24 1 192- 2856 301-7124 0 4 35-6840 11-7660 F, 48-0708 ~ 108-9204 ~ 194-3184 Y, 304'2540 3 438-7368 6 in. 12-2712 I 12 in. 490384 18 in. 110-4492 24 in. 1963548 30 in. 306-8052 36 n. 411-.7982' 12-7884 ~ 50-1168 Y, 111-9888 I 198-4056 ~/ 309-3672 l' 47-19573, 13-8540 2 5202(48 1150992 2035384 3145224 ~/ 1440124 ~ 53'2644 [ 2 116-6712 ~2 204'6216 3:!17-1168 37 in. 46'1(980,/~ 14-9616 ~ 54-3348 ~ 118-2528 ~ 21)6-7144 I! 319-7220 }[ 473-0241) 15-5316 ~|, 55-4760 [ 4 119-8452 F | 208-8192 ~/ 322-33698 479. 39 -. ~~~~~~~~~~~~~~~~~~~~~4~~~/ 485-81178 16-1124 Y, 56-4804 | 121-4484 YI 210-9336 ~/ 324.9624 I{ 485'8078 7 in. 16-7028 13 in. 57-6108 19 in. 123-0624 25 in. 213-0588 31 in. 327'6000 38 in. 492-2637 ~ 17-3052 ~ 58-7244 ~ 124-6872 ~ 215-1948 ~I 330-2472 /A 4198-7821 ]17-9172 Y, 59-8476 ~ 126-3228 I, 217-3416 4~ 332-9052 Y, 505-3032 / 18-541.2 Y, 60-9828 1 527-9680 ~ 219-4980 ~ 335-5798 511-9979. 19-1748 Y, 62-1276 ~ / 129-6252 l ~ 221-6664 ~2/ 338'2524 39 in. 518-4132 ] 19-81112 i 63-28:12 I 131'5320 % ~ 223-8444 Y, 340-9428 ~ 525-1821! ~ 20-4744 ~ 64-4406 ~ 132'969 6 ~ 226-03-4 1 0 343-6428 ~ 531-8936 F ~ 21-1404 65-6268 ~ 13-1-6580, 228-2340 Y, 346-3536 04 538-6478 14 to. 66-8148 20 in. 136-3562 il 26 in.:23-04444 32 in. 3490764 40 in. 545-4445 ~ 2225036 ~ 68-0136 ~ 138-0672'5 232 6644 Y, 351-8088 54 552-2839 ~ 23-2020'1 69-2220 ~ 139-7880 ~ 234-8576 ~ 354-5520, 559-1659 ~ 23-9100 Y, 70-4424 ~ 141-5184 2[ 3 37-1404 ~ 357-3048 3 586-090-1 24.5288 4 71-6724 ~ 143.2608 F ~ 239-3928 ~| 360-0696 41 in. 573-0577 Y2 ~ ~~~~~~~~~~~~~ 2 87119 ~ 25-3%524 % 72-9120 1 145-0128 ~ 241-6572 /~ 362-8452, 587-1199 / 26-0988 ~ 74-1648 ] ~ 146-7756, ~ 243-9312 Y, 365-6304 42 in. 601-3526 ~/ 26 8500 ~ 75-4272 Y, 148-5492 F ~ 246-2160 ~8 / 368'4276 00 in. 799-2426 26 Page 402 402 THE PRACTICAL MODEL CALCULATOR. TABLE containing the Weight of Square Bar Iron, from 1 to 10 feet in length, and from - of an inch to 6 inches square. LENGTH OF THE BARS IN FEET. 2 1 foot. 2 feet. 3 feet. 4 feet. 5 feet. 6 feet. 7 feet. 8 feet. 9 feet. 10 feet. Ibs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. 0-2 0 4 0-6 0 8 1 1 1-3 1.5 1-7 1-9 2-1 0 5 1.0 1.4 1.9 2.4 2-9 3-3 3-8 4/3 4.8 g 0-8 1-7 2-5 3-4 4-2 5-1 5-9 6-8 7-6 8-5 1i3 2-6 4'0 5-3 6-6 7-9 9-2 10-6 11'0 13-2 1*9 3 8 5-7 7 -6 9-5 11-4 13-3 15-2 17-1 19-0 8 2-6 5-2 7-8 10-4 12.9 15-5 18-1 20-7 23-3 25.9 1 in. 3-4 6 8 10 1 13 5 16-9 20-3 23 7 27 0 30 4 33-8 1i 4-3 8.6 12.8 17-1 21-4 25-7 29-9 34.2 38'5 42-8 1t 5-3 10-6 15-8 21-1 26-4 31-7 37'0 42-2 47-5 52-8 1i 6-4 12-8 19.2 25.6 32.0 38.3 44.7 51-1 57-5 63.9 1i 7-6 15 2 22 8 30 4 38-0 45 6 53-2 60-8 68-4 76 0 14 8-9 17-9 26 8 357T 44 6 53-6 62-5 71-4 80-3 89-3 1 10-4 20-7 31-1 41-4 51.8 62.1 72-5 82-8 93-2 103-5 1! 11'9 23'8 35-6 47.5 59-4 71.3 83.2 95-1 106-9 118-8 2 in. 13-5 27.0 40'6 54 1 67-6 81;1 94-6 108-2 121'7 135-2 24 15-3 30 5 45 8 61.1 763 91-6 106 8 122 1 137-4 152-6 21 17-1 34.2 51 -3 684 85'6 102'7 119 8 136'9 154-0 171-1 2[ 19-1 38.1 57 2 76.3 95-3 114.4 133 5 152.5 171-6 190-7 2 { 21-1 42.8 63'4 84'5 105'6 126-7 147'8 169.0 190'1 211 2 2 23 3 46 6 G699 93 2 116 5 139'8 163 0 186.3 209'6 232-9 2I 25-6 51-1 76-7 102.2 127-8 153-4 178-9 204.5 230-0 255.6 2- 27-9 55 9 83 8 111 8 139.7 167.6 195-7 223-5 251.5 279.4 3 in. 30 4 608S 912 1217 1521 182.5 212-9 243-3 273'7 304-2 3}i 33 0 66-0 99l0 132 0 165.1 198.1 231-1 264]1 297.1 330-1 34 35 7 71-4 107-1 142 8 178 5 214-2 249 9 285-6 321 3 357 0 3A 38 5 77 0 115 5 154-0 192 5 231 0 269 5 308 0 846 5 385 0 34 41'4 8268 12482 1656 2070 2484 28981 331'3 372 7 414'1 3 44-4 88-8 133 3 177.7 222.1 266-5 310-9 355-3 399 8 444-2 3- 47.5 95-1 142-6 190.1 237.7 285-2 332.7 380.3 427 8 475 3 37 50'8 101-5 152-3 203-0 253 8 304 5 355-3 406-0 456 8 507 ~6 4 in. 54-1 108-2 162-3 216-3 270.4 324'5 378'6! 432'7 486'8 540 8 44 57-5 115-0 172-6 230'1 287-6 345'1 402-6 460-1 517-7 575 2 41 61-1 122-1 18321 244.2 305-3 366-3 427-4 488-4 549.5 610.6 43 64 7 129-4 194,1 258.8 323-5 388-2 452-9 517.6 582 3 647 0 4i 68 4 136 9 205 3 273'8 342.2 410'7 479 1 547'6 616-0 684'5 4 [72'3 144 6 216 9 289'2 361.5 433'8 506.1 578'4 650.7 723'1 4! 76'3 152[5 228 8 305 1 381'3 457'6 533 8 610'1 686e4 762'6 4' 4-8 80 3 160 7 241 0 3213 4017 4820 56 6427 7230 8033 5 in. 84' 5 169/ 0 253 41337 9 422'41506 9 591'4 675'8 760 3 844'8 5' 93 2 186 3 279-5 372.7 465.8 559 0 652.2 745.3 838 5 931.7 54 102 2 204 5 306 7 409 0 511 -2 613 4 715'7 817.9 920(2 1022 4 54 111 8 223 5 33531 447 0 558-8 670 5 782'3 8940 0 1005 8 1117.6 6 in. 121 7 243 3 365 0 1 486'7 608.3 7300 841 6 973.3 1009'5 1216 6:TABLE of the Weight of a Square Foot of Sheet IZron in lbs. avoirdupois, the thickness being the number on the wire-gauge. No. 1 is 5 of an inch; No. 4, 1; No. 11,, rc. No. onwire-gaugel 1 2 1 3 | 4 5 | 6 | 7 1 8 9 i 10 I 11 Pounds avoir....... j12'51 12 11 10 1 9 8 | 7'5 7 7 6 [5'68| 5 No. on whire-gauge 12 113 114 15 16i 17 118 19 20 21 1 22 Pounds avoir....... 4'621 4311 4 13.951 9312.5 218 193 1 62 1 15 1137 Page 403 SPECIFIC GRAVITY. 403 TABLE of the Weight of a Square Foot of Boiler Plate Iron, fromr 8 to 1 inch th.ick, in lbs. avoirdupois. 6 |i 5i |1 38 | i | 2 l 1~ 61 B 2|- i | 8 l in. 15 175 110112.5 15 17.520122.5j 2527 5 30 1325 35137 5 40 TABLE containing the Weight of Round Bar Iron, from 1 to 10 feet in length, and from 4 of an inch to 6 inches diameter. LENGTH OF TEHE BARS IN FEET. I feet. 2 feet. 3 feet. 4 feet. 5 feet. 6 feet. 7 feet. 8 feet. 9 feet. 10 feet. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Ibs. 1 0 03 0/5 0-7 0.8 1/0 1.2 1.3 1251 127 0-4 0-7 1.1 1-5 1.9 22 2-6 3.0 34 3.7 1 07 1.3 2.0 2.7 343 4.0 4.6 5.3 6.0 66 8. 10 2.1 3'1 4'2 5'2 6'3 7'3 8'3 9'4 10.4 48 15 3-0 4.5 6.0 7.5 90 10 5 11.9 134 149 78 20 4.1 61 8'1 102 12/2 142 163 183 203 1 in. 27 5'3 8'0 10'6 13'3 15'9 18'6 21'2 23.9 26/5 1-1 384 6'7 10'1 13'4 16'8 20'2 235 26'9 302 3361 1 - 4 2 8'3 12-5 16.7 20.9 250i 29.2 33 4 37.5 41'7 1 5]0 10.0 15.1 20.1 25'1 3011 351 402 45.2 50.2 12 6l0 11 9 17 9 23 9 29/9 35.8 418 478 5387 59. 7 1- 7 0 14.0 21.0 28.0 35.1 42-1 49.1 56'1 631 70'1 14 8 1 16 3 24.4 32 5 40.6 48.8 56.9 65 70 73 2 81 3 18 93 187 280 37.3 467 56.0 653 / 47 840 933 2 in. 10o6 21.2 31 8 42.5 53 1 63c7a 74-S3 49 95.5 106.2 21 12'0 24-0 36 0 480 59-9 719 839 95 9 1079 1199 - 134 I 269 40.3 53.8 67-2 80-6 41 1075 1210 1344 23 150 30.0 44.9 600 749 899 1048 1198 1348 1498 23 16 7 33.4 50-1 66.8 83 5 1001 1168 1336 1502 166.9 2-5- 18s3 8366 54.9 7382 91. 5 109 128s1 146l3 16461 1829 2a 20-1 40'2 60'2 80' 3 1004 1205 140-5 1606 1807 2008 2-7 21 9 43 9 65 8 87 8 109 7 131.7 1lo3 1756 1975 2194 3 in. 23 9 47'8 71-7 95-6 119 4 143 167-2 1911 215 0 238' 9 31 2559 51'9 77-8 103 7 196 1 6 181-5 2074 2333 2593 43-} 28'0 561 811. 112 2 140 2 1682 1963 2243 2534 2804 -3 302 605 907 121 0 151 181-4 211 2419 272 3024 3S I.32-5 65.0 97-.5 130 0 162 6 195-1 2276 2601 2926 325 - 135 |34.9 69.8 104.7 139.5 17441 2093 244 2791 3140' 3489 3-4 37 3 74.7 112.0 149.3 i186-7 22-0 2613 2987 360 37 3 37 39.9 79-7 119.6 159 5 199 -3 239 2 279. 0 318.9 358.8 398 6 4 in. 425 849 1274 169 9 212 3 48 2972 3397 382 4246 4'1 45 -2 90.3 1355 180.7 225.9 271.0 316.2 361.4 406.6 4,51-7 48-0 95e-9 143-9, 191.8 239.8 287. 7 335-7 383.6 431 6 479.5 4- 50-8 1016-! 1524 1 2033 2541 304-9 355.7 406 5 4573 508 41 53- 8 107.5 161 3 215.0 268. 8 322. 6 7 6 3 430.1 4S3.8 37. 6 45 568 113.6 170.4 227.2 283.9 340. 397.5 454.3 511 1 567 9 44 60.0G 119.8 179.7 239.6 299. 5 359.4 419.3 479.2 539 1 599.0 4 1 63.1 1262 189.3 252.4 315. 5 378. 6 441.7 504.8 567.8 630 9 1 5 in. 66-8i 133 65 200-3 267 0 333-8 400.5 467.3 534.0 6008 6 67 51 5 173'2 146.3 219.5 292 7 369 4390 5122 5854 6585 731 7 51. 80 3 160'6 240'9 321'2 401' 5 481' 8 562'1 642 4 722'7 803 0 51 87 1756 2633 351 1 4389 5267 614.4 7022 790 0 877'8 6 in. 95.61 191.1 286 7 882.2 477. 8 573 3 668.9 764 4 860.0 955.5 TABLE of the Weight of Cast Iron Plates, per Superficial Foot, fronz one-eighth of an inch to one inlch thick. | inch. / inch. |4 inch. inh. inch. 3| inch. | inch. inch. lbs.. lbs oz. lbs.. lbs. bs. oz. lbs. o. lbs.. lbs. lb 13j 9 101 14 8 19 5 24 2 1 29 0 33 13 1 38 10oa 8 4, -~~~~~~~~if.. Page 404 404 THE PRACTICAL MODEL CALCULATOR. TABLE containing the Weight of Cast Iron Pipes, 1 foot in length. OW~~~~ TEI~THICKNESS IN INCHES. _ - h __ ___ __ 4__ inch. 1 -_ 5 Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. 1j 6 9 9.9.................................... 2 88 123 161 203........................ 2j 10-6 14 7 19 2 239........................ 3 12.4 17.2 222 27-6 33.3 39.3 456...... 31 14.2 19.6 25'3 31.3 37.6 44.2 51.1. 4 16 8 22 1 28'4 35'0 41'9 49'1 56.6 64'4 4' 18.0 24.5 31.4 38.7 46.2 54.0 62.1 70'6 5 19.8 27.0 34.5 42.3 50.5 58.9 67.6 76.7 5~ 21.6 29.5 37.6 46.0 54.8 63.8 73.2 82.8 6 23-5 31.9 40.7 49'7 59.1 68.7 78'7 88[ 8 6j 25 3 34 4 43 7 53 4 63 4 73 4 84 2 95 1 17 027 2 36'8 46 8 56'8 67.7 78.5 89'7 101] 2 7, 29'0 39.1 49.9 60.7 72.0 83.5 95.3 107[ 4 8 30o8 41.7 52 9 64.4 76G2 88.4 100.8 113 5 8~ 32.9 44.4 56.2 68.3 80.8 93.5 106.5 119[ 9 9 34.5 46.6 59.1 71.8 84.8 98.2 111.8 125[ 8 92 36-3 49.1 62.1 75'5 89'1 103.1 117.4 131/ 9 10 38.2 51.5 65 2 79.2 93.4 108.0 122.8 138/ 1 10o...... 54.0 68.2 82.8 97'7 112.9 128.4 144[ 2 11...... 56.4 71.3 86.5 102.0 117.8 133.9 150[ 3 11...... 58.9 74.3 90.1 106.3 122.7 139.4 156] 4 12...... 61.3 77 4 93.6 110.6 127.6 145.0 162 6 13............ 82'7, 101'2 118'2 137'4 154'1 1/735 14............ 89'5 108'2 126'5 146'2 165'3 185/ 2 15........... 952 115'7 135'3 156'2 176'2 198/ 1 16.................. 123.3 143.1 166.1 187.5 211/ 3 17.................. 130.2 152.5 178.5 198.2 223/ 4 18............... 137.0 161'2 185'3 209 1 235, 6 19........................ 169.2 195.7 222.3 24I71 20........................ 178.1 205.2 23382 259 0 21.............................. 214'1 243 5 273.2 22................................ 2 223-0 254'8 285'4 23..... 233.4 265 5 298 3 24.......................... 245 2 277.5 310.6 TABLE containing the Weight of Solid Cylinders of Cast Iron, one foot in length, andfrom 3 of an inch to 14 inches diameter. Diameter in Weight in Diameter in WVeight in Diameter in' Weight in Diameter in Weight in Inches. Lbs. Inches. Lbs. Inches. Lbs. Inches. Lbs. 1'39 27 2048 4 58721 7- 148'87 I7W 1188 3 in. 22-35 5 in. 61'96 8 in. 158.63 1 in. 2-47 3 1 24.20 51 64.66 84- 168.15 1- 3313 3-1 26'18 56 68'31 1 8 179'08 i1 3-87 33 28823 53 71400 8a 189'00 13 4-68 3 30.36 5~1 74.98 in. 200-77 1. 5-57 3- 32.57 55 1 78.65 9}- 211.12 1- 6s654 3 3 4s85 5- 81.95 9~ 223-70 1[ 7.59 37 37-21 5 85-81 1 9 235631 17 8.71 4 in. 39-66 6 in. 89-23 10 in. 247-87 2 in. 9-91 4j 41-80 61} 96-82 10~ 273-27 2j 11-19 41 44-77 6 104-72 11 in. 299'92 21 12'54 43 4700 6 1 112-93 11i 327-81 2~'13-98 4 1 50-19 7 in. 121-45 12 in. 356.93 24 15'49 41 52-71 71 130-28 13 418,90 21 17-08 41 55-92 71 139'42 14 485-83 2i 18-74 Page 405 SPECIFIC GRAVITY. 405 TABLE containing the Weight of a Square Foot of Copper and Lead, in lbs. avoirdupois, from 2 to 1 an, inch in thickness, advancing by _ __ _____ ____ Thickness. Copper. Lead. 1 1'45 1'85 A1 2'90 3.70 33 4.35 5-54 1 5'80 7'39 1+ ~ 7'26 9'24 1 + 1 8-71 11108 8 + 1 10'16 12'93 4 11.61 14'77 4 + 1 13'07 16'62 4 + 1 14'52 18'47 4 + 3 15'97 20'3-l 3 17'41 22'16 8 + 92 18'87 24'00 3 + x -1 20 32 25'85 3 + - 21'77 27'70 1 23'22 29'55 TABLE for finding the Weight of Mfalleable Iron, Copper, and Lead" Pipes, 12 inches long, of various thicknesses, and any diameter required. Thickness. Malleable Iron. Copper. Lead. 2E of an inch.'104'121'1'539 1I6a *'208'2419' 3078 ~32~~ *3108'3628' 4616 8'414' 4838'6155 + 1 *518. 6047. 7694 + 1'621'7258'9232 + -725'8466 1'0771 _ 4_______'828'9678 1'231 RULE.-Multiply the circumference of the pipe in inches by the numbers opposite the thickness required, and by the length in feet; the product will be the weight in avoirdupois lbs. nearly. Required the weight of a copper pipe 12 feet long, o15 inches in circumference, + -1 of an inch in thickness.'7258 x 15 = 10'817 x 12 = 130'644 lbs. nearly. TABLE ofthe Weight of a Square FToot of lfillboard in lbs. avoirdupois. Thickness in inches...... 4 | 15 4 8 Weight in lbs............. 688 1'032 1'376 1'72 2064 Page 406 406 THE PRACTICAL MODEL CALCULATOR. TABLE containing the Weight of Wrought Iron Bars 12 inches long in lbs. avoirdupois. Inch. Round. Square. Inch. Round. Square. 4.163. 208 21 16.32 20.80 a "367.467 25 18-00 22.89 ~ 653. 830 2a 19.76 25.12 - 1-02 1 30 24 21.59 27.46 4 1;47 1.87 3 23.52 29.92 8 2200 2.55 31 27'60 35.12 1 2'61 3 32 34 32200 40'80 14 3.31 4.21 34 83672 46.72 14 4.08 5.20 4 41.76 53.12 13 4.94 6.28 44 47.25 60.00 14 5.88 7.48 44- 52.93 67.24 1 6 90 8 78 44 58.92 74.95 1 8.00 10.20 5 65.28 83.20 14 9.18 11.68 5- 72.00 91.56 2 10.44 13.28 5 i 79.04 100.48 21 11.80 15.00 54 86.36 109,82 2 13-23 16.81 6 94.08 119.68 2 14 73 18 74 7 128.00 163.20 TABLE of tlhe Proportional Dimnensions of 6-sided Nuts for Boltsfrom l to 22 inches diameter. Diameter of bolts........ 1 18 1 Breadth of nuts.......... i 1 11 1| 17 1 11- 21 Breadth over the angles 1 i i 1I - 1 1V- 2 217 Thickness.................. 1- 1 1 Diameter of bolts........ 1 14 1 14 14 2 2 4 2 1 Breadth of nuts.......... 2A 21 241 2 13 8 3 3 1 4 Breadth over the angles 214 27 8 3 16 84,o 34 4fj 4 8 Thickness..... 11........... 2. 211 241 I ~____________ _______ _ 6'... 6 ~ 1 -- TABLE of the Speeific Gravity of Water at different temperatures, that at 62~ being taken as unity. 700 F.. 99913 520 F. 1 00076 68' 99936 50 1 00087 66' 99958 48 1.00095 64' 99980 46 1'00102 62 1. 44 1'00107 58 1.00035 42 1.00111 56 1'00050 40 1.00113 54 1.00064 38 1.00115 The difference of temperatures between 62~ and 39~'2, where water attains its greatest density, will vary the bulk of a gallon rather less than the third of a cubic inch. Page 407 SPECIFIC GRAVITY. 407 TABLE of the WVeig^ht of Cast Iron Balls in pounds avoirdupois, from 1 to 12 inches diameter, advancing by an eighth. Inches. Lbs. Inches. Lbs. Inches. Lbs. 1 *14 41 14.76 8~ 84.56 1k *20 47 15 95 85 88.34 1.27 5 17.12 84 92.24 1.37 5 18-.54 88 96 26 1k. 47 5} 199.3 9 100.39 1t -59 5n 21 39 k9 104 60 14 *74!5 22.91 94 108-98 1~ 7 91 5 24.51 93 113.46 2 1*10 54 26.18 9 118 06 2 [ 1 32 57 27.91 95 122.77 21 1157 6 29.72 93 127'63 28 1.84 6k 31 64 98 132.60 2k 2.15 6[ 33362 10 137 71 25 2.49 60 3.567 10 142.91 2 2.86 6k 37 80 101 148.28 27 3' 27 61 40 10 103 153 78 3 3.72 J 64 42 35 10- 159.40 3k 4 20 67 44*74 10~ 165 16 834 4472 7 47.21 103 171-05 a] 5-29 7-8 49-79 108 17,710 3~ 5.80 71 52.47 11 183.29 35 6 56, 7 6 5523 ilk 189.60 I3 7.26 72- 58.06 111- 196.10 3 7 8801 75 60'04 118 202'67 4 8 81 74 64.09 11i 209.43 4k 9 67 77 67 25 111- 216.32 41 10.57 8 70.49 111 223.40 43 11.53 8s 73 85 11 230 57 4~ 12.55 8 T 77 32 12 237.94 41 13.62 88- 80.88 TABLE of the Weight of F lat Bar Iron, 12 inehes long, in lbs. avoirdupois. Thickness. x | 1. | 1 inch. 4 8 2 -- -I --- — 8 — 4 Iinch. 1 21 31 -42 63 4 *31.47 63'94 1'26 157 1 *42.63 *84 126 1 68 2.10 2.52 2.94 1L *52 *78 1'05 1 57 2210 2'62 3'15 3'67 4'20 13 657 *86 1'18 1*73 2'31 2 88 3'46 4'04 4'62 1, * 63 *0 94 1 26 1'89 2'52 3'15 3'78 4-41 5 04 14 733'10 1'47 2'20 2 94 3 67 4'41 5 14 5 87 o 2 *84 1-26 1 68 2 52 3-36 4'20 5'06 5'88 6'72 24 96 1.41 1'89 2.83 3878 4'72 5.66 6.61 7.56 2 105 1.57 210 315 4.20 525 6 30 785 840 = 23 115 1.73 2.31 346 462 5'77 6.93 8.08 9-24 3 1-26 1.89 2.52 3 78 5 04 6.30 7.56 8.82 10.08 314 1 36 2'04 27 3 409' 5 46 682 8'19 9'55 10'92 Pq 3k 1-47 2.20 2.94 4741 5.88 7 35 8.82 10.29 11 76 33 1 57?236 3 15 4 72 6 30 7.87 9.45 11.02 12 60 4 1 68 2.52 336 5.04 6 72 8.40 10.08 11-76 13.44 4:k 1.89 2.83 3.73 5 67.756 9.45 1134 13.23 15-12 5 210 3815 4'12 630 8'40 1050 12'60 16'70 17'80 G 2 52 3.78 5-04 7156 10.08 12 60 15.12 17.64 20.16 Weight of a copper rod 12 inches long and 1 inch diameter = 3 039 lbs. Weight of a brass rod 12 inches long and I inch diameter = 2.86 lbs. Page 408 408 THE PRACTICAL MODEL CALCULATOR. BRASS. — Weight of a Lineal Foot of Round and Square. Diameter. Weight of Weight of Diameter. Weight of Weight of round. square. round. square. Inches. Lbs. Lbs. Inches. Lbs. Lbs. 4i.17 -22 14 8.66 11.03 ~ 39. 50 1- 9.95 12.66 ~'70' 90 2 11.32 14'41 -8 1.10 1.40 21 12.78 16.27 4 1'59 2'02 24 14-32 18'24 4 2.16 2'75 23 15'96 20-32 1 2.83 38 60 24 17.68 22-53 14 3.58 4.56 24 19.50 24.83 14 4.42 5-63 2 21.40 27-25 14 5.35 6.81 1 2 23-39 29 78 14 6.36 8.00 3 2547 32.43 14 7.47 9-51 i STEEL. — Weight of One Foot of Round Steel. Diameter 1 1414 1 1 14 14 172 paris. Weight in - itb. anddeci-'167 376 669 1'04 1'5 2 05 267 338 418 5 06 6'00 7 07 8'2 9'41 11'71 TABLES OF THE WEIGHTS OF ROLLED IRON, Per lineal foot, of various sections, illustrated in the accompanying cutts, viz. Parallel Angle Iron, equal and unequal sides; Taper Angle Iron; Parallel T Iron, equal and unequal depth and width; Taper T Ironl; Sash I'on; and Permnanent and Temporary Rails. TABLE I.-Parallel Angle Iron, of equal sides. (Fig. 1.) Length of sides Uniform thickness lineal footne AB, in inches. throughout. in lbs. Inches. Inches. 3 4 8.0 Fig. 1. 24 4 7. 0 24 3 5.75 24 5-16ths 4.5 2 full 3.75 z 14' 340 14 1 2.5 1 14 No. 6 wire-gauge 175 14 8 1.5 14 9 1'25 1 10 1.0 10. 875 4 11 6;25 12.,5c Page 409 SPECIFIC GRAVITY. 409 TABLE II.-Parallel Angle Iron, of unequal sides. (Fig. 2.) Uniform Weight of one Length of side A, Length of side B, thickness lineal foot Fig. 2. in inches. in inches. throughout. in lbs. Inches. Inches. Inches. \ 3~ 5 1 9.75 3 5 875 3 4 5-16ths 7.5 21 4 5-16ths 6.75 21 4 5.75 2 4 1 5.5 21 3 4.75 1 2 2 1 3.375 14 2 2.875 1 1~ 2 3-16ths 225 TABLE III.-Taper Angle Iron, of equal sides. (Fig. 3.) Weight of one Length of sides, Thickness of Thickness of root Weight of one A A, in inches. edges at B.in lbs Fig. 3. i - ___ ___-,\B Inches. Inches. Inches. 4 4 14.0 3 8 10 375, 2-1 7-16ths 9-16ths 8.25 A 21 3 6.5 / 2 5 full 5-16ths, full 3.875 l 11 5-16ths 325 4 ___bare 5-16ths, bare 2-625 TABLE IV.-Parallel T Iron, of unequal width and depth. (Fig. 4.) tiformt Weight of one Width f th Uniform thick- Uniform eight top tale depth ness of top thickness of lineal foot A, inche Bs. inches. table C. rib D. in lbs. inches. inces._ Fig. 4. Inches. Inches. Inches. Inches. --- -A —-------- 5 6 1575\ 44 31 9.16ths 13 25 ~4 3 -~ ~ 8.875 3- 3 3 8-25 31 4 4 12.5 B 21 3 8 7.0 21 2 5-16ths 3 full 45 2 1 5-16ths 5-16ths 40 14 2 4 3125 4 4 I2 1 1 2.875 11 14 1; 2375 1 1- 3-16ths 3-16ths 1.5 3 1 I 3-16ths 3-16ths 1 125 4 __________________1____________ Page 410 -110 THE PRACTICAL MODEL CALCULATOR. TABLE V.-Parallel T Iron, of equal depth and width. (Fig. 5.) Width of top Uniform Weight of one table, and total thickness lineal foot depth A A. throughout. in lbs. Inches. Inches. 6 4 Fig. 5. 5 7-16ths 13.75 A --------— A 4 _8- 9.75 / < 34 85 3 8 75 21 5-16ths 4 625 2f 5-16ths 4.5 2 5-16ths 3.75 4 4 3.0.} 14 2.25 11 14 1.75 1 3-16ths 1.0 7.725 ~i4 - 625 TABLE VI. —Taper T Iron. (Fig. 6.) Fig. 6. Widthof Total oThickness of lhickness of Uniform WVeight of one top table t tp table atth thicknessof lineal foot inches. inches root C. edges D. rib E. in lbs. inches. ihs - I Inches. Inches. Inches. Incles. Inches. 3 23 7-16ths6th 8.0 3 2~ a 7-1 6thS 8 2 8- Ot B 24 3 7-16ths 5-16ths 5-16ths 525 2 24 4 6.5 E 2 1]- full 5-16th 3 385 2 11 5-16ths i 1 2875 TABLE VII. —Sash Iron. (Fig. 7.) Total Depth of Width at Greatest leiltof one ______ - C. GFdO~ ~eiglht of one depth A. rebate B. edge C. iidth D. in fbo. _ _-~~ ___^~___~~~~ ~Fig. 7, lc 1lhe. Inches. Inches. A > 2 1 No. 9 wire-gauge 5-8ths 1-75 c 4 4 7 9-l16ths 1625 11 a 6 9-16ths 1 25 81 8 10 9-16ths 1-125 D 14 i 10 9-16ths 1]0 1 4 4 _ *75 Fig. 8. TABLE VIII.-Rails equal top and bottom Tables. (Fig. 8.) _' Depth A, in W and bot top Thickness of Weight of one inches. and bottom 131, rbC lineal foot A in inches. rib C.bs. Inches. Inches. Inches. 5 293 3 25.0 4 44 24 23.33 4 24 __ 21-66 i -B_-_-_- I _ _ Page 411 SPECIFIC GRAVITY. 411 TABLE IX.-Temporary Blails. (Fig. 9.) I 6 -I ~~ Fig. 9. Top widthl Rib width Bed -widthl Total depth Thicklless WVeigEht of one -— A — In B, in C, in B, in of hed I linleal foot inclhes. inches. inches. incises. in lbs. Inches. Inces. nches. Inches Inches. B 11 3 2 7-16ths 9.0 12 3 I 1 120 r ii 4 3 1 16'0 2 I 4: ~ 17.33 3 TABLE of Natural Sines, Co-sines, Tangents, Co-tangents, Secants, and Co-secants, to every degree of the Quadrant. Deg. Sines. Co-sines. Tangents. Co-tangents. Seae nts. Co-secants. Degree. 0 o00000 1 00000 -00000 Infinite. 1 00000 Infinite. 90 1 *01745'99985 *01746 57-2900 1.00015 57'2987 89 2.03490.99939 I03492 286363 1-00061 28.6537 88 3 *05234.99863 *05241 19.0811 1 00137 19 1073 87 4.06976 *99756 *06993 14-3007 1 00244 14 3356 86 5 *08716 *99619 *08749 11-4301 1-00382 11'4737 83 6'10453 *99452 *10510 9-51236 1 00551 9 56677 84 7.12187.99255 *12278 8'14435 1 00751 8.20551 83 8'13917'99027 *14054 7'11537 1 00983 7 18530 82 9 *15643 *98769 *15838 6-31375 1'01246 6-39245 81 10 q17065 *98481.17633 5.67128 1.01543 5.75877 80 11 *19081 *98163 *19438 5-14455 101872 5-24084 79 12.20791 *97815 *21256 4.70463 1 02234 4-80973 78 13 22495'97437 *23087 4.33148 1 02630 4-44541 77 14'24192'97030 /24933 4-01078 1.03061 4-13356 76 15/ 2.5882.96593'26795 3 73205 1-03528 3886370 75 16.27564 *96126 *28675 3-48741 1 04030 3862796 74 17.29237 *95630 *30573 3 27085 1-04569 3-42030 73 0 18.30902 {95106 *32492 3 07768 1 05146 3. 23607 72 19'32557 *94552.34433 2-90421 1-05762 3O07155 71 20'34202'93969'36397 2-74748 1-06418 2-92380 70 21'35837 *93358 *38386 2-60509 1 07114 2.79043 69 22'37461 |92718'40403 2'47509 1'07853 2'66947 68 23 39073 *92050 42447 2-35585 1-08636 2.55930 67 24'40674 *91355 *44523 2-24004 1-09464 2 45859 66 25.42262 /90631.46631 2-14451 1 10338 2 36620 65 26.43837 *89879 *48773 2.05030 1-11260 2-28117 64 27'45399'89101 /50952 1-96261 1-12233 2-20869 63 ~28 46947'88295'53171 1-88073 1.13257 2-13005 62 i 29 48481'87462 [55431 1 80405 1'14335 206266 61 30 50000.86603 [57735 1 73205 1.15470 2.00000 60 31 -51504.85717 [60086 1 66428 1 16663 1.94160 59 32 52992 -84805 /62487 1'60033 1.17918 1.88708 58 33 54464'83867 /64941 1 53986 1 19236 1 8o608 57 34 -55919.82904 /67451 1.48256 1 20622 1 78829 56 35 57358'81915 -70021 1'42815 1 22077 1-74345 55 836 58778.80902.72654 1.37638 1 23607 1 70130 54 37 60181.79863 *75355 1'32704 1.25214 1 66164 53 38 *61566'78801 -78129 1 27994 1.26902 1 62427 5 39 62932.77715.80978 1-23490 1 28676 1.58902 51 40 *64279.76604.83910 1.19175 1.30541 1-55572 50 41 65606.75471.86929 1 15037 1.32511 1 52425 49 42 66913.74314 H90040 1.11061 1.34561 1.49448 48 43.68200.73135.93251 1 07237 1 36706 1.46628 47 44'69466'71934 H96569 1-03553 1 39012 1 43956 46 45 *70711.70711 1.00000 1-00000 1 41421 1 41421 45 Deg. Co-sines. Sines. Co-tangents. Tangents. I Co-secants. Secants. Degree. Page 412 412 MOMENT OF INERTIA. CORDS, KNOTS) NODES, CHAIN-BRIDGE.-ANGULAR VELOCITY.-RADIUS OF GYRATION. 1. IF the cord q NB, be fixed at the extremity B, and stretched by a weight of 500 lbs. at the extremity q, and the middle knot or node N, by a force of 255 lbs. pulling upwards, under an angle a N b of 540; what is the tension and position of NB., rI "I!' \, q' ~ I Angle qNr = 1800 - angle qNP; and 900 - aNb = bN c qNr 36~; cos. 360 -- 80902. /5002 + 2552 - 2 x 255 x 500 x cos. 36~ = 329'7 lbs., the magnitude of the tension. 500 sin. 36~ 329 7 ='891386 = sine of angles b N s, or angle BN r = 630 2'. 2. Between the points A and B, a cord 10 feet in length is stretched by a weight W of 500 lbs. suspended to it by a ring; the horizontal distance AE = 6'6 feet, and the vertical distance BE = 3'2 feet; required the position of the ring C, the tensions, and directions of the rope. The tensions of the cords AC, CB are equal, and angle AC b = angle b CB. Page 413 MOMENT OF INERTIA. 413 ~ ~ /! I B hAD -AC + CB- 10 feet. V(102 - 6.62) = 75126 = ED; BD = 7-5126 - 32 = 43126 4-3126 21-563 Dn = -2 = 21563; 75126: 21563 10 7'5126 = 287 = CD = CB; and CA = 10 - 287 = 713. -Bn cosine bCB 21563 75132. Be 287=71 LW 500 339.~. LbCB = 410 18'; 2 cos. 41 018' = 150264 - 7 bs., the tension on the cord CB, which is equal to the tension on AC. Page 414 414 THE PRACTICAL MODEL CALCULATOR. 3. Let 500,000 lbs. be the whole weight on a chain-bridge whose span AB = 400 feet, and height of the arc CD = 40 feet; required the tensions and other circumstances respecting the chains. The tangent of the angles of inclination of the ends of the chain is equal 40x2 200- -= 40000, the angle answer- ~ -<Xl X ing to this natural tangent is 210 48'. The vertical tension at each point of suspension is = half the weight = 250000; the horizontal tension at the points of suspension = 250000 x cot. 250000 210 48' = 625000 lbs. The whole tension at one end will be V6250002 + 2500002 = 673146 lbs. 4. Suppose the piston of a steam engine, with its rod, weighs 1000 lbs.; it has no velocity at its highest and l I lowest positions, but in the middle the velocity is a maximum and equal 10 ft.; what effect will it accumulate by virtue of its inertia in the first half of its path, and give out again in the second half; and what is the mean force which would be requisite to accelerate the motion of the piston in the first half of its path, which is the same as that which it would exert in the second half by its retardation, the length of stroke being 8 feet. According to the principle of vis viva, the effect which the piston will accumulate by virtue of its inertia in the first half of its path, and give ii out again in the second half - 102 2 x 32.2 x 1000 = 1552.794 units of work. Half the path of the piston = 4 feet; hence, 1552'794 41 — 94= 388'1985 lbs., the mean force. MOMENT OF INERTIA, or the MOMENT OF ROTATION, or the MOMENT OF THE MASS, is the sum of the products of the particles Page 415 MOMENT OF INERTIA. 415 of the mass and the squares of their distances from the axis of rotation. 5. If a body at rest, but capable of turning round a fixed axis A, possesses a moment of inertia of 121 units of work, the measures taken in feet and pounds, made to turn by means of a cord and weight of 36 lbs., lying over a pulley in a path of 10 feet; what are the circumstances of the motion. 2 x 6 x 10= 2439347 feet, the angular velocity of the body, which call v; so that each point at the distance of one foot fromn the axis of revolution will describe, after the accumulation of 121 units of work, 2'44 feet in a second. 6'2832 = circumference of a circle 2 feet in diameter, 6'2832 2'44 = 2' 6 seconds, the time of one revolution. 6. If an angular velocity of 3 feet passes into a velocity of 7 feet; what mechanical effect will a mass produce so moving, supposing the moment of inertia to be 200, the measures taken in feet and pounds. According to the principles of vis viva, 200 (72 - 32) -2 = 4000 units of work, which may be 40 lbs. raised 100 feet, 80 lbs. raised 50 feet, 400 lbs. raised 10 feet; and so on. 7. The weight of a rotating mass B is 500 lbs., its distance OB friom the axis of A rotation 3 feet, the weight W, constituting the moving force, 90 lbs., its arm AO OC = 4 feet; required the circum- stances of the motion that ensues. {90+ 500}32'2 = 11'53 lbs., the inert mass accelerated by the force of W. And it is well known that the force divided by the mass gives the acceleration. w Page 416 416 THE PRACTICAL MODEL CALCULATOR. 90 i153 = 7'806, the acceleration of the motion of W. The T7806 angular acceleration in a circle 1 foot from the axis - 1'9515. After 10 seconds the acquired angular velocity will be 1'9515 x 10 = 19'515. 1'9515 x 102 And the corresponding distance = 2 = 97575 feet, measured on a circle one foot from O. 7'806 x 102 The space described by the weight W is 2 = 390'3 feet, which is the same as the space described by C. The circumference of a circle one foot from C = 31416. 97.575 3'1416 = 31'059 revolutions. In the rotation of a body AB about a fixed axis O, all its points describe equal angles in equal times. If the body rotate in a cer0 tain time through the angle 00, or arc = 180~ n, radius = 1; and hence, -- 3'141592, &c.; the elements of the body, a, b, c, &c., at the distances oa = xa, ob= - AdB " x,, &c. from the, axis, will describe the arcs or spaces aa, =, b b- b /. cX2, CCX - = X, )b, &c. If the angular velocity, that fC is, the velocity of those points of /' the body which, 1/ are distant a unit - A —- I-', of length, a foot, from the axis of revolution, be put <-, A-. —- A — =z, then the si- I a. multaneous velo-, cities of the elements of the mass at the distances x1, x, x3, &c., will be, z 2 X9, ZX zX3, &c. And if a be the mass of the element at a; b the mass of the element at b; c the mass of the element at c, &c., their vis viva will be, (Z X)2 a, (z X2)2 b, (z X3)2 c, &c. And.the sum of the vis viva of the whole body = z2 (2a + X,29 2 C, &c.) Page 417 MOMENT OF INERTIA. 417 According to our definition, x12a + x2 b + X2 c, &c. is the mo_ ment of inertia, which may be represented by R; then z2 R is the vis viva of a body revolving with the angular velocity z. Therefore, to communicate to a body in a state of rest an angular velocity z, a mechanical effect F s, or force x space = I the vis viva, must be expended; that is, F s = _ z2 R, or, which is the same thing, a body performing the units of work F s, passes from the angular velocity z to a state of rest. In general, if the initial angular velocity = v, and the terminal angular velocity = z, the units of work will be, Z2 — V2 Fs- 2 x R. The moment of inertia of a body about an axis not passing through the centre of gravity is equivalent to its moment of inertia about an axis running parallel to it through the centre of gravity, increased by the product of the mass of the body and the square of the distance of the two centres. It is necessary to know the moments of inertia of the principal geometrical bodies, because they very often come into application in mechanical investigations. If these bodies be homogeneous, as in the following we will always suppose to be the case, the particles of the mass MI, M2, &c. are proportional to the corresponding particles of the volume V., V2, &c.; and hence the measure of the moment of inertia may be replaced by the sum of the particles of the volume, and the squares of their distances from the axis of revolution. In this sense, the moments of inertia of lines and surfaces may also be found. If the whole mass of a body be supposed to be collected into one point, its distance from the axis may be determined on the supposition that the mass so concentrated possesses the same moment of inertia as if distributed over its space. This distance is called the radius of gyration, or of inertia. If R be the moment of inertia, M the mass, and r the radius of gyration, we then have AI r2 = R, and hence r -= \M. We must bear in mind that this radius by no means gives a determinate point, but a circle only, within whose circumference the mass may be considered as arbitrarily distributed. If into the formula R. = R + M e2, expressed in the words above printed in italics, we introduce R = M r2 and R1- M r12, we obtain r'2 = r2 + e2; that is, the square of the radius of gyration referred to a given axis = the square of the radius of gyration referred to a parallel line of gravity, plus the square of the distance between the two axes. Wheel and axle.-The theory of the moment of inertia finds its most frequent application in machines and instruments, because in these rotary motions about a fixed axis are those which generally present themselves. 27 Page 418 418 THE PRACTICAL MODEL CALCULATOR. If two weights, P and Q, act on a wheel and axle ACDB, with the arms CA = a and DB = b through the medium of perfectly flexible strings, and if the radius of the gudgeons be so small that their friction may be neglected, it will remain in equilibrium if the statical moments P. CA and Q. DB are equal, and therefore P a = Q b. But if the moment of the weight P is greater than that of Q, therefore P a > Q b, P will descend and Q ascend; if P a < Q b, P will ascend and Q descend. Let us now examine the F-B 3 conditions of motion in the case that P a > Q b. The force corresponding to the weight Q and acting at the arm b generates at the Qb arm a a force a, which acts opposite to the force corresponding to the weight P, and hence there is a residuary moving force P - Q b acting at A. The mass Q is reduced by its transference Q 62 from the distance b to that of a to hence the mass moved by ag, or, if the moment of inertia of P - b is M = (P + a -, or, if the moment of inertia of a sM=( a2, r Page 419 MOMENT OF INERTIA. 419 G12 fore, its inert mass reduced to A -g a2 we have, more exactly, M ( 2Q g = (GP2 + Qb2 Gy2) 2 From thence it follows that the accelerated motion of the weight P, together with that of the circumference of the wheel, namely, Qb moving force a 2 Pa - Q b g a= mass = P a2+ Q b2+ Gy a = Pa2+ Q b2+ G a on the other hand, the accelerated motion of the ascending weight Q, or of the circumference of the axle, is, b Pa —Qb q =aP a2 + Qb2 + Gy2gb. The tension of the string by P is S = P - P - P (1 g that of the string by Q is T = Q (1 + q(); hencethe pressure on the gudgeon is, S+T P~ PQ Qq __-( a -Q b )2 g Q P a2 + Q b2 + Gy2; the pressure, therefore, on the gudgeons for a revolving wheel and axle is less than for one in a state of equilibrium. Lastly, from the accelerating forces p and q, the rest of the relations of motion may be found; after t seconds, the velocity of P is v = p t, of Q is v1 = q t, and the space described by P is s = I p t2, by Q is s = ~qt. Let the weight P at the wheel be = 60 lbs., that at the axle Q = 160 lbs., the arm of the first CA = a = 20 inches, that of the second DB = b = 6 inches; further, let the axle consist of a solid cylinder of 10 lbs. weight, and the wheel of two iron rings and four arms, the rings of 40 and 12 lbs., the arms together of 15 libs. weight; lastly, let the radii of the greater ring AE = 20 and 19 inches, that of the less EG = 8 and 6 inches; required the conditions of motion of this machine. The moving force at the circumference of the wheel is, b 6 P - - Q = 60 - 20160 - 60 - 48 = 12 lbs., a 20 the moment of inertia of the machine, neglecting the masses of the gudgeons and the strings, is equivalent to the moment of inertia W b2 10.62 of the axle - 2 - 2= 180, plus the moment of the smaller R, (r12 + r,2) 12 (82 + 62) ring = 2 = 2 = 600, plus the moment of Page 420 420 THE PRACTICAL MODEL CALCULATOR. 40 (202 + 192) the larger ring 40 (202 + 19= 15220, plus the moment of A (p3 -- p23) A p12 + p1p P + p2) the arms, approximately A 3 (pl _ P) - 3 15 (192 + 19 x 8 + 82) - 2885; hence, collectively, Gy2 = 180 + 18885 600 + 15220 + 2885 = 18885, or for foot measure = 144 = 131'14. The collective mass, reduced to the circumference of the wheel is, Q (p 2 + 1 +6 2 18885 a2 60 + 160 ( 20) + 202 g (60 + 160 x 0.09 + 400 ) 0031 = 121-61 x 0.031 = 337 lbs. Accordingly, the accelerated motion of the weight P, together with that of the circumference of the wheel, is, P-aQ _12 = p + Qb2 + Gy2 - = 3.77 = 38183 feet; on the other a2 b 6 hand, that of Q is q a p = 20 3'183 = 0'954 feet; further, the tension of the string by P is = (1 - P =(1(i- 3j) 6 54'07 lbs.; that by Q, on the other hand, Q (1 + )Q - (1 + 0'925 x 0'032) 160 = 1'030 160 = 164.8 lbs.; and consequently the pressure on the gudgeons S + T = 54'06 + 164'80 218'86 lbs., or inclusive of the weight of the machine = 218'86 + 77 = 295'86 lbs. After 10 seconds, P has acquired the velocity pt = 3'084 x 10 = 30'84 feet, and described the space s -' - 30.84 x 5 = 154.2 feet, and Q has ascended a height a s = 0.3 x 154.2 = 46.26 feet. The weight P which communicates to the weight Q the accelePab - Q b2 rated motion q = p a2 + Q 62 ~ G y2,' may also be replaced by another weight P,, without changing the acceleration of the motion Q, if it act at the arm a,, for which, Pi a, -Qb Pa —Qb P, a' + Q b2 + Gy2 P a + Qb + Gy2 Pa2 + Qb2 + Gy2 The magnitude a Q b, represented by kc, and we obQb (b + k) + GY2, tain a,2, - c a p=, and the arm in question, Page 421 MOMENT OF INERTIA. 421 a q)2 b (b d k) + Gy2 a, =~~l (2 - - We may also find by help of the differential calculus, that the motion of Q is most accelerated by the weight P, when the arm of the latter corresponds to the equation P a2 - 2 Q a b= Q b2 + G yT: therefore, bQ 6b Q 2 + Qb2 + Gy2 a =T- + -) p~ The formula found above assumes a complicated form if the friction of the gudgeons and the rigidity of the cord are taken int: account. If we represent the statical moments of both resistances b by F r, we must then substitute for the moving force P - a Q, thi Qb + Fr value P - whence the acceleration of Q comes out, (Pa —Fr)b —Qb2 Qb+Fr I Qb+F( 2+Qb-2+Gy Q pa2 + Q-b2+ -y 2g and a- P ) The weights P = 30 lbs. Q = 80 lbs. act at the arms a = 2 feet. and b = -- foot of a wheel and axle, and their moments of inertia G y2 amount to 60 lbs.; then the accelerated motion of the ascend.ing weight Q is, 30 x 2 x _ - 80 x ()2 30 - 20 22 q 30 x 22 + 80x(~)2+ 60= 120 + 20 + 60 - 200 1'61 feet. But if a weight P2 = 45 lbs. generates the same acceleration in the motion of Q, the arm of Pi is then, k I k)2 80 x o (x + k) + 60 1 2 - 2 - 145, or as k- = 60 40 l0, aj is = 5: J25 - 5 -_ 1 11.358 = 5:E 3786 = 8' 786 feet, or 1'214 feet. The accelerated motion of Q comes out greatest if the arm oT the force or radius of the wheel amount to, ~xSO 1/40 20 + 60 4 116 24 4 + /40 aX 30'+430 + 30 -3 9 9 3 ( 0x 1207 - 20 3-621 3.4415 feet, and q is = 30 x(34415)2 j+ 80) 4 32 2'339 feet. The statical moment of the friction, together with the rigitity of the string, is F r = 8; then, instead of Q b, we must put Q b + F r = 40 + 8 = 48; whence it follows that, 48 1/40 2 8 a -- 0 + 4() + =3 1'6 + v/5227 = 3'886, and the cor..respondent maximum accelerating force 30 x 1'9438- 8 x - 20 34'29 q =30 8 2+8 g= x 32+2 = 2-071 feet. 30 x (3~886)-9 + 80 53.3 Page 422 422 WEIGHT, ACCELERATION, AND MASS. PARALLELOGRAM OF FORCES. —TIIE PRINCIPLE OF VIRTUAL VELOCITIES. -MECHANICAL POWERS: CONTINUOUS CIRCULAR MIOTION, GEARING, TEETH OF WHEELS, DRUMS, PULLEYS, PUMPING ENGINES, ETC. 1. IF a weight of 10 lbs., moved by the hand, ascends with a 3 feet acceleration, what is the pressure on the hand? 3 10 (1 + 32-2) = 10.93168 lbs. If a weight of 10 lbs., moved by the hand, descends with a 3 feet acceleration, the pressure on the hand will be 9'06832 lbs., for then 3 10 (1 - 322) = 906832. If w be the weight of the mass acted upon by the force of the hand, and also by the force of gravity, as g = 32'2, the mass w moved by the sum or difference of these forces will be = -. If P be the pressure on the hand, and p its acceleration, the body falls with the force - p; it also falls with the force w - P; hence,'W w -P=-P..P=(1 )w. g Y When the body is ascending, then p is negative, andwt + P = ( w(1 -). 2. If a body of 200 lbs. be moved on a smooth horizontal track, by the joint action of two forces, and describes a space of 10 feet in the first second, what is the amount of each of these forces; the first makes an angle of 350 with the track upon which the body moves, and the other an angle of 500? In solving this question, the natural sines of the angles 350, 50~, and of their sum 85~, will be required. We shall first take these from the table: sin. 350 =- 57358 sin. 50~ ='76604 sin. 85~ -= 99619. The acceleration is = 20 feet, that is, twice the space passed over in the first second, 200 200 322 - the mass, and 3-2.2 x 20 = 124'224 lbs., the force of the resultant, in the direction of the track upon which the body moves. Page 423 WEIGHT, ACCELERATION, AND MASS. 423 124'224 sin. 35~ One of the components sin (35 + 50) - 71'52 lbs. 124'224 sin. 500 The other component-sin (35 + 50 - 95'52 lbs. These, and the like results, may be obtained with greater ease by logarithms. Log. 124'224 = 2'0942055 Log. sin. 350 = 9'7585913 11'8527968 Log. sin. 850 = 9'9983442 Log. of 71'52413 = 1'8544526 Log. 124-224 = 2'0942055 Log. sin. 50~ - 9'8842540 11.9784595 Log. sin. (85~) = 9'9983442 ~Log. of 95'5247 - 1'9801153 3. A carriage weighing 8000 lbs. is moved forward by a force fA of 500 lbs. upon a horizontal surface AB; during the motion, two resistances have to be overcome, one horizontal of 100 lbs., the amount of friction, represented in the figure byf, the other f2 of i \ 3 200 lbs. acting downwards; the angles f3 nf, and f, n mn, which the directions of these forces make with the horizon, are 61~ and 21~ respectively: it is required to know what work the force f, will perform by converting a 5 feet initial velocity of the carriage into a 20 feet velocity. If we put x = n m, the distance the carriage moves in passing fiom a 5 to a 20 feet velocity, The work of the force1 =A f x nq - 500 x cos. 21~ x x. The work of the force fJ = (-f3) X nu = - 100 x x. The work of the forceJf = (- f2) x np = - 200 x cos. 61~ x x. Page 424 424 THE PRACTICAL MODEL CALCULATOR. Consequently, the work of the effective force will be 269 828 x x = {500 x'94358 - 100 - 200 x -48481} x, since the natural cosine of 21~ ='93358, and the natural cosine of 61~ ='48481. But according to the principle of vis viva, the work done is equal to 202 - 52 06424 x 8000 = 46589'82. 64'4 46589'82 269828 x x = 46589'82 and x = 269828 = 772'665 feet, the space passed over by the carriage. This question is solved on the PRINCIPLE OF VIRTUAL VELOCITIES, which we shall explain, as it is of essential service in practical mechanics. This explanation depends on what is technically termed the " Parallelogram of _Forces." 01 7 P nV n When a material point 0, is acted upon by two forces f1, Aw, whose directions Of0, OfJ, make with each other an angle, if Of1, Of, represent the magnitudes and directions of the forces, the diagonal of the parallelogram 0 f, f represents the resultant in magnitude and direction; that is, the diagonal represents a single force equal to the combined actions of the forces represented by the sides. And if the sides of the parallelogram represent the accelerations of the forces, the diagonal represents the resultant acceleration. Draw through 0, two axes OX and OY, at right angles to each other, and resolve the forces f, and f, as well as their resultant f1, into components in the directions of these axes; namely, f, into it, and mi; f into n2 and mn,; and f, into n d and i,. The forces in one axis are nl n2, and n,; and those in the other nz,, I and mvh And by the parallelogram of forces it is well known that n3 =n, + n and In3 = MI, + nz2. (E). Now if we take in the axis OX any point P, and let fall from it Page 425 MECHANICAL POWERS. 425 the perpendiculars PA, PB, PC, on the directions of the forces f,, f, f, we obtain the following similar right-angled triangles, namely, OAP and 0 nf, are similar; OBP and 0 nf - OCP andOn2 - n O n, OA n, AO.G. o -P = f and na OP f It is easily seen also that CO BO n, = OP f; and n3 = Gp f If the values be substituted in (E), we obtain BO xf,= CO xf + AO xf,. From the similarity of these triangles, and the remaining equation of (E), we can readily find that PB x f= PA xf,+ PC xf2. The equation becomes more compact by putting OA, OC, OB, respectively equal s,, 8s S3; and PA, PC, PB, - q,, 2 q,. Then f. s, = f, s, + f, s, and f q, = f +f, fq,. The same holds good with any number of forcesf,,f,,f3, &c., and their resultant f5, that is fn A= fi,8, + f 2 + Af,s 8 + &c. and fqn=f,ql+ fq-+fq, + &c. If the point of application 0, move in a straight line to P, then OA = s, is called the space of the force f,A and f, s, the work done by the force f,, in moving the body from 0 to P. OB is the space of the resultant, and the productfs, the work done by it. f2 sl is the work done by f, in moving the material point 0 from 0 to P. Hence the work done by the resultant is equal to all the work done by the component forces, as we have shown, fLS= fiS,+ A+ S + 3 8+ &o. PRINCIPLES AND PRACTICAL APPLICATIONS OF MECHANICAL POWERS. MECHANICAL Powers, or the Elements of Machinery, are certain simple mechanical arrangements whereby weights may be raised or resistances overcome with the exertion of less power or strength than is necessary without them. They are usually accounted six in number, viz. the lever, the wheel and axle, the pulley, the inclined plane, the wledge, and the screw; but properly two of these comprise the whole, namely, the lever and inclined plane,-the wheel and axle being only a lever of the first kind, and the pulley a lever of the second,-the wedge and the screw being also similarly allied to that of the inclined plane: however, although such seems to be the case in these re Page 426 426 THE PRACTICAL MODEL CALCULATOR. spects, yet they each require, on account of their various modifications, a peculiar rule of estimation adapted expressly to the different circumstances in which they are individually required to act. THE LEVER. Levers, according to mode of application, as the following, are distinguished as be- c 1st. B A ing of the first, se- 2nd. 3rd. cond, or third kind; lbs. 8 Ibs. and although levers ofequal lengths pro- l56bs. duce different ef- 8 lbs. fects, the general B A... A principles of esti- C BA mation in all are the same; namely, the power is to the 66 lbs. 56 lbs. weight or resistance, as the distance of the one end to the fulcrum is to the distance of the other end to the same point. In the first kind, the power is to the resistance, as the distance AB is to the distance BC. In the second, the power is to the resistance, as the distance AB is to that of AC; and, In the third, the resistance is to the power, as the distance AB is to that of AC. RULE, first kind. —Divide the longer by the shorter end of the lever from the fulcrum, and the quotient is the effective force that the power applied is equal to. Let the handle of a pump equal 65 inches in length, and 10 inches from the shortest end to centre of motion; what is the amount of effective leverage thereby obtained? 55 65 - 10 = 55, and 1- = 5~ to 1. Required the situation of the fulcrum on which to rest a lever of 15 feet, so that 21 cwt. placed at one end may equipoise 30 cwt. at the other, the weight of the lever not being taken into account. 15 x 2.5 25 + 302 = 1.154 feet from the end on which the 30 cwt. is to 2'5 + 30 be placed. It is by the second kind of lever that the greatest effect is obtained from any given amount of power; hence the propriety of the application of this principle to the working of force pumps, and shearing of iron, as by the lever of a punching-press, &c. RULE, second kind.-Divide the whole length of lever, or distance from power to fulcrum, by the distance from fulcrum to weight, and the quotient is the proportion of effect that the power is to the weight or resistance to be overcome. Required the amount of effect or force produced by a power of Page 427 MECHANICAL POWERS. 427 50 lbs. on the ram of a Bramah's pump, the length of the lever being 3 feet, and distance from ram to fulcrum 41 inches. 36 3 feet = 36 inches, and 45- = 8, or the power and resistance are to each other as 8 to 1; hence 50 x 8 = 400 lbs. force upon the ram. The lever on the safety valve of a steam boiler is of the third kcind, the action of the steam being the power, and the weight or spring-balance attached the resistance; but in such application the action of the lever's weight must also be taken into account. THE WHEEL AND PINION, OR CRANE. The mechanical advantage of the wheel and axle, or crane, is as the velocity of the weight to the velocity of the power; and being only a modification of the first kind of lever, it of course partakes of the same principles. RULE.-To determine the amount of effective power produced from a given power by means of a crane with known peculiarities.Multiply together the diameter of the circle described by the winch, or handle, and the number of revolutions of the pinion to 1 of the wheel; divide the product by the barrel's diameter in equal terms of dimensions, and the quotient is the effective power to 1 of exertive force. Let there be a crane the winch of which describes a circle of 30 inches in diameter; the pinion makes 8 revolutions for 1 of the wheel, and the barrel is 11 inches in diameter; required the effective power in principle, also the weight that 36 lbs. would raise, friction not being taken into account. 0 x 8 = 21'8 to 1 of exertive force; and 21'8 x 36 = 784'8 lbs. RULE. —Given any two parts of a crane, to find the third, that shall produce any required proportion of mechanical effect.-M-ultiply the two given parts together, and divide the product by the required proportion of effect; the quotient is the dimensions of the other parts in equal terms of unity. Suppose that a crane is required, the ratio of power to effect being as 40 to 1, and that a wheel and pinion 11 to 1 is unavoidably compelled to be employed, also the throw of each handle to be 16 inches; what must be the barrel's diameter on which the rope or chain must coil? 16 x 2 = 32 inches diameter described by the handle. 32 x 11 And 40 = 8'8 inches, the barrel's diameter. THE PULLEY. The principle of the pulley, or, more practically, the block and tackle, is the distribution of weight on various points of support; the mechanical advantage derived depending entirely upon the Page 428 428 THE PRACTICAL MODEL CALCULATOR. flexibility and tension of the rope, and the number of pulleys or sheives in the lower or rising block: hence, by blocks and tackle of the usual kind, the power is to the weight as the number of cords attached to the lower block; whence the following rules. Divide the weight to be raised by the number of cords leading to, from, or attached to the lower block; and the quotient is the power required to produce an equilibrium, provided friction did not exist. Divide the weight to be raised by the power to be applied; the quotient is the number of sheives in, or cords attached to the rising block. Required the power necessary to raise a weight of 3000 lbs. by a four and five-sheived block and tackle, the four being the movable or rising block. Necessarily there are nine cords leading to and from the rising block. 3000 Consequently 9- = 333 lbs., the power required. I require to raise a weight of 1 ton 18 cwt., or 4256 lbs.; the amount of my power to effect this object being 500 lbs., what kind of block and tackle must I of necessity employ? 4256 500 = 8'51 cords; of necessity there must be 4 sheives or 9 cords in the rising block. As the effective power of the crane may, by additional wheels and pinions, be increased to any required extent, so may the pulley and tackle be similarly augmented by purchase upon purchase. THE INCLINED PLANE. The inclined plane is properly the second elementary power, and may be defined the lifting of a load by regular instalments. In principle it consists of any right line not coinciding with, but lying in a sloping direction to, that of the horizon; the standard of comparison of which commonly consists in referring the rise to so many parts in a certain length or distance, as 1 in 100, 1 in 200, &c.,-the first number representing the perpendicular height, and the latter the horizontal length in attaining such height, both numbers being of the same denomination, unless otherwise expressed; but it may be necessary to remark, that the inclination of a plane, the sine of inclination, the height per mile, or the height for any length, the ratio, &c., are all synonymous terms. The advantage gained by the inclined plane, when the power acts in a parallel direction to the plane, is as the length to the height or angle of inclination: hence the rule. Divide the weight by the ratio of inclination, and the quotient equal the power that will just support that weight upon the plane. Or, multiply the weight by the height of the plane, and divide by the length, —the quotient is the power. Page 429 MECHANICAL POWERS. 429 Required the power or equivalent weight capable of supporting a load of 350 lbs. upon a plane of 1 in 12, or 3 feet in height and 36 feet in length. 350 350 x 3 H15 = 29-16 lbs., or 3 = 29-16 lbs. power, as before. The weight multiplied by the length of the base, and the product divided by the length of the incline, the quotient equal the pressure or downward weight upon the incline. TABLE showing the Resistance opposed to the Motion of Carriages on different Inclinations of Ascending or Descending Planes, whatever part of the insistent weight they are drawn by. HUNDREDS. =H = 100 200 300 400 500 600 700 800 900 -01 -005 -003,33 -0025 -002'00167 00143 -00125 -00111 10'1 00909'00476'00322 -00244 -00196 -00164 00141 00123 0011 20 -05.00833 -00454.00312.00238 -00192 -00161 00139 00122.00109 30 0333.00769.00435 -00303.00232.00189 00159 -00137 0012 00107 40 *025 -00714 -00417 -00294.00227 01.00185 00156 00135 00119 00106 50 02 00667 004 00286 00222 -00182 -00154.00133 00118 00105 60 0166 -00625 00385 0078.0021.00178.00151 -00131 00116.00104 70 0143.00588.0037 -0027 00213.00175.00149 0013 00115 00103 80 0125 00555 00357 -0023 00208 00172 -00147 00128 00114 -00102 90 -0111 -00526'00345'00256 -00204'00169 00145 00126 00112'00101 Although this table has been calculated particularly for carriages on railway inclines, it may with equal propriety be applied to any other incline, the amount of traction on a level being known. Application of the preceding Table. What weight will a tractive power of 150 lbs. draw up an incline of 1 in 340, the resistance on the level being estimated at I-ooth part of the insistent weight? In a line with 40 in the left-hand column and under 200 is -00417 Also in the same line and under 390 is........................ -00294 Added together ='00711 150 Then -00711 = 21097 lbs. weight drawn up the plane. What weight would a force of 150 lbs. draw down the same plane, the fraction on the level being the same as before? Friction on the level = -00417 Gravity of the plane = -00294 subtract = -00123 150 And -00123 - 121915 lbs. weight drawn down the plane. Example of incline when velocity is taken into account.-A power of 230 lbs., at a velocity of 75 feet per minute, is to be employed for moving weights up an inclined plane 12 feet in height and 163 Page 430 430 THE PRACTICAL MODEL CALCULATOR. feet in length, the least velocity of the weight to be 8 feet per minute; required the greatest weight that the power is equal to. 230 x 75 x 163 2811750 12 x 8 = 96 = 29288 lbs., or 13'25 tons. TABLE of Inclined Planes, showing the ascent or descent per yard, and the corresponding ascent or descent per chain, per mile; and also the ratio. Per yard. Per chain. Per mile. Ratio. Per yard. Per chain. Per mile. Ratio. In parts In dec'ls. Inches. Foet. 1 inch. In parts In decimals Inches. Feet. I inch. of an in. f an inch. of an in. of an inch. /.0156'344 2-29 2304 - 4375 9-625 64-17 82 8'0208 458 3-06 1728.5 11 7333 72'0312 *687 4'58 1152 19'5625 12'375 82'5 64' 0417 917 6-11 864 %17'5833 12-833 85'56 62 i 0625 1-375 9'17 576 / *6 13.2 88 60 T, *0833 11833 12'22 432 / *625 13-75 91-67 58 Pa *1 2'2 14.67 360 6667 14'667 97'78 54 - *125 2.75 18-33 288 I'-6875 15-125 100'83 52 i.1667 3.667 24-44 216 ['7 15-4 102-67 51 - *1875 4-125 27-50 192 l *75 16-5 110 48 [ -2 4-4 29-33 180 A.8 17.6 117'33 45 *'25 5-5 36-67 144 l 3 *8125 17.875 11917 44 - l 3 66 44 120 i *8333 18-333 122-22 43 3125 6-875 45'83 115 875 19-25 128H33 41 i *3333 7-333 48'89 108.l 9 19-8 132 40 a.375 8-25 55 96 1 *-9167 20-167 134-44 39 i 4 8-8 58 67 20 I. *9375 20.625 137.5 38 -4167 9-167 61-11 86 1 1 1 2 146'67 36 THE WEDGE. The wedge is a double inclined plane; consequently its principles are the same: hence, when two bodies are forced asunder by means of the wedge in a direction parallel to its head,-Multiply the resisting power by half the thickness of the head or back of the wedge, and divide the product by the length of one of its inclined sides; the quotient is the force equal to the resistance. The breadth of the back or head of a wedge being 3 inches, and its inclined sides each 10 inches, required the power necessary to act upon the wedge so as to separate two substances whose resisting force is equal to 150 lbs. 150 x 1 5 10 = 22.5 lbs. 10 When only one of the bodies is movable, the whole breadth of the wedge is taken for the multiplier. THE SCREW. The screw, in principle, is that of an inclined plane wound around a cylinder, which generates a spiral of uniform inclination, each revolution producing a rise or traverse motion equal to the pitch of the screw, or distance between two consecutive threads,-the pitch being the height or angle of inclination, and the circumference Page 431 MECHANICAL POWERS. 431 the length of the plane when a lever is not applied; but the lever being a necessary qualification of the screw, the circle which it describes is taken, instead of the screw's circumference, as the length of the plane: hence the mechanical advantage is, as the circumference of the circle described by the lever where the power acts, is to the pitch of the screw, so is the force to the resistance in principle. Required the effective power obtained by a screw of 8 inch pitch, and moved by a force equal to 50 lbs. at the extremity of a lever 30 inches in length. 30 x 2 x 3'1416 x 50 *875 -= 10760 lbs. ~875 Required the power necessary to overcome a resistance equal to 7000 lbs. by a screw of 1~ inch pitch, and moved by a lever 25 inches in length. 7000 x 1'25 25 x 2 x 31416 = 55=73 lbs. power. In the case of a screw acting on the periphery of a toothed wheel, the power is to the resistance, as the product of the circle's circumference described by the winch or lever, and radius of the wheel, to the product of the screw's pitch, and radius of the axle, or point whence the power is transmitted; but observe, that if the screw consist of more than one helix or thread, the apparent pitch must be increased so many times as there are threads in the screw. Hence, to find what weight a given power will equipoise: RuLE.-Multiply together the radius of the wheel, the length of the lever at which the power acts, the magnitude of the power, and the constant number 6'2832; divide the product by the radius of the axle into the pitch of the screw, and thQ quotient is the weight that the power is equal to. What weight will be sustained in equilibrio by a power of 100 lbs. acting at the end of a lever 24 inches in length, the radius of the axle, or point whence the power is transmitted, being 8 inches, the radius of the wheel 14 inches, the screw consisting of a double thread, and the apparent pitch equal 8 of an inch? 14 x 24 x 100 x 6'2832 ~625 x 2 x 8 = 21111'55 lbs., or 9'4 tons, the power sustained. If an endless screw be turned by a handle of 20 inches, the threads of the screw being distant half an inch; the screw turns a toothed wheel, the pinion of which turns another wheel, and the pinion of this another wheel, to the barrel of which a weight W is attached; it is required to find the weight a man will be able to sustain, who acts at the handle with a force of 150 lbs., the diameters of the wheels being 18 inches, and those of the pinions and barrel 2 inches. 150 x 20 x 311416 x 2 x 183 = W x 23 x 1;.'. W = 12269 tons. Page 432 432 THE PRACTICAL MODEL CALCULATOR. CONTINUOUS CIRCULAR MOTION. IN mechanics, circular motion is transmitted by means of wh/eels, drums, or pulleys; and accordingly as the driving and driven are of equal or unequal diameters, so are equal or unequal velocities produced: hence the principle on which the following rules are founded. RULE.- When time is not taken into account. —Divide the greater diameter, or number of teeth, by the lesser diameter, or number of teeth, and the quotient is the number of revolutions the lesser will make for 1 of the greater. How many revolutions will a pinion of 20 teeth make for 1 of a wheel with 125? 125. 20 = 6'25, or 61 revolutions. Intermediate wheels, of whatever diameters, so as to connect communication at any required distance apart, cause no variation of velocity more than otherwise would result were the first and last in immediate contact. RULE. — To find the number of revolutions of the last, to 1 of the first, in a train of wheels and pinions. —Divide the product of all the teeth in the driving, by the product of all the teeth in'the driven, and the quotient equal the ratio of velocity required. Required the ratio of velocity of the last, to 1 of the first, in the following train of wheels and pinions; viz., pinions driving,-the first of which contains 10 teeth, the second 15, and third 18;lwheels driven,-first 15 teeth, second 25, and third 32. 10 x 15 x 18 15 25 32 225 of a revolution the wheel will make to 1 15 x 25 x 32 of the pinion. A wheel of 42 teeth giving motion to one of 12, on which shaft is a pulley of 21 inches diameter, driving one of 6; required the number of revolutions of the last pulley to 1 of the first wheel. 42 x 21 12 x 6 = 12'25, or 12~ revolutions. Where increase or decrease of velocity is required to be commaunicated by wheel-work, it has been demonstrated that the number of teeth on each pinion should not be less than 1 to 6 of its wheel, unless there be some other important reason for a higher ratio. RULE.- When time must be regarded.-Multiply the diameter, or number of teeth in the driver, by its velocity in any given time, and divide the product by the required velocity of the driven; the quotient equal the number of teeth, or diameter of the driven, to produce the velocity required. If a wheel containing 84 teeth makes 20 revolutions per minute, how many must another contain to work in contact, and make 60 revolutions in the same time? Page 433 CONTINUOUS CIRCULAR MOTION. 433 84 x 20 =- 28 teeth. From a shaft making 45 revolutions per minute, and with a pinion 9 inches diameter at the pitch line, I wish to transmit motion at 15 revolutions per minute; what at the pitch line must be the diameter of the wheel? 45 x 9 15 - 27 inches. Required the diameter of a pulley to make 16 revolutions in the same time as one of 24 inches making 36. 24 x 36 16 = 54 inches. RULE. —The distance between the centres and velocities of twyo wheels being given, to find their proper diameters.-Divide the greatest velocity by the least; the quotient is the ratio of diameter the wheels must bear to each other. Hence, divide the distance between the centres by the ratio plus 1; the quotient equal the radius of the smaller wheel; and subtract the radius thus obtained from the distance between the centres; the remainder equal the radius of the other. The distance of two shafts from centre to centre is 50 inches, and the velocity of the one 25 revolutions per minute, the other is to make 80 in the same time; the proper diameters of the wheels at the pitch lines are required. 50 80. 25 = 3'2, ratio of velocity, and 3'2 + 1 11'9, the radius of the smaller wheel; then 50 - 11'9 = 38'1, radius of larger; their diameters are 11'9 x 2 = 23'8, and 38'1 x 2 = 76'2 inches. To obtain or diminish an accumulated velocity by means of wheels and pinions, or wheels, pinions, and pulleys, it is necessary that a proportional ratio of velocity should exist, and which is simply thus attained: —Multiply the given and required velocities together, and the square root of the product is the mean or proportionate velocity. Let the given velocity of h wheel containing 54 teeth equal 16 revolutions per minute, and the given diameter of an intermediate pulley equal 25 inches, to obtain a velocity of 81 revolutions in a machine; required the number of teeth in the intermediate wheel, and diameter of the last pulley. V/81 x 16 = 36 mean velocity. 54 x 16 25 x 36 - 36 = 24 teeth, and 81 - 111 inches, diameter of pulley. To determine the proportion of wheels for screw cutting by a lathe.-In a lathe properly adapted, screws to any degree of pitch, or number of threads in a given length, may be cut by means of a 28 Page 434 434 THE PRACTICAL MODEL CALCULATOR. leading screw of any given pitch, accompanied with change wheels and pinions; course pitches being effected generally by means of one wheel and one pinion with a carrier, or intermediate wheel, which cause no variation or change of motion to take place: hence the following RuLE.-Divide the number of threads in a given length of the screw which is to be cut, by the number of threads in the same length of the leading screw attached to the lathe; and the quotient is the ratio that the wheel on the end of the screw must bear to that on the end of the lathe spindle. Let it be required to cut a screw with 5 threads in an inch, the leading screw being of 1 inch pitch, or containing 2 threads in an inch; what must be the ratio of wheels applied? 5 -- 2 = 2-5, the ratio they must bear to each other. Then suppose a pinion of 40 teeth be fixed upon for the spindle,40 x 2'5 = 100 teeth for the wheel on the end of the screw. But screws of a greater degree of fineness than about 8 threads in an inch are more conveniently cut by an additional wheel and pinion, because of the proper degree of velocity being more effectively attained; and these, on account of revolving upon a stud, are commonly designated the stud-wheels, or stud-wheel and pinion; but the mode of calculation and ratio of screw are the same as in the preceding rule;-hence, all that is further necessary is to fix upon any 3 wheels at pleasure, as those for the spindle and studwheels,-then multiply the number of teeth in the spindle-wheel by the ratio of the screw, and by the number of teeth in that wheel or pinion which is in contact with the wheel on the end of the screw; divide the product by the stud-wheel in contact with the spindlewheel, and the quotient is the number of teeth required in the wheel on the end of the leading screw. Suppose a screw is required to be cut containing 25 threads in an inch, the leading screw as before having 2 threads in an inch, and that a wheel of 60 teeth is fixed upon for the end of the spindle, 20 for the pinion in contact with the screw-wheel, and 100 for that in contact with the wheel on the End of the spindle;-required the number of teeth in the wheel for the end of the leading screw. 60 x 12'5 x 20 25 _ 2 = 12'5, and 100 — 150 teeth. Or, suppose the spindle and screw-wheels to be those fixed upon, also any one of the stud-wheels, to find the number of teeth in the other. 60 x 12'5 60 x 12'5 x 20 150 x 100 - 20 teeth, or 150 = 100 teeth. 150 X-100 ~~150 Page 435 CONTINUOUS CIRCULAR MOTION. 435 TABLE of Change W4rheels for Screw Cutting, the leading screw being of 1 inch pitch, or containing two threads in an inch. Number of Number of teeth in Number of teeth in. teeth in |__ _ _ __ h 54 Cd 1_ -A -v.C o a o 80 40 8 40 55 20 19 50 95 20 100 41: 8 50 89 9 0 8 5 20 90 19~ 80 120 20 140 0 80 70 9 0 90 20 95 20, 40 90'0 90 2 80 0 0 75 20 80 22 60 1oo 20 120 2 80 100 101 50 70 20 75 22~ 80 120 20 150 2 8 110 1 1 60 55 20 120. 22- 80 130 20 140 1 80 120 12 90 5 2020 0 123 40 95 20 100 131,{ 130 812- 60 85 20 90 1924 680 120 20 130 314 80 60 1 60 70 20 70 20 60 100 20 150 8 15 70 913 60 90 20 90 250 30 85 20 90 2 80 90 139 40 6100 20 165 21 80 120 20 140 4- 40 85 14 90 90 20 140 27 40 90 20 120 44 40 90 141 60 90 20 95 274 40 100 20 110 43 40 95 15 90 90 20 150 28 75 140 20 150 5 40 100 16 60 80 20 120 284 30 90 20 95 51 40 110 16- 80 100 20 130 30 70 140 20 150 6 40 120 164 80 110 20 120 32 30 80 20 120 64 40 130 17 45 85 20 90 33 40 110 20 120 7 40 140 174 80 100 20 140 34 30 85 20 120 7i 4 40 150 18 40 60 20 120 35 60 140 20 150 30 120 18- 80 100 20 l- 10 36 30 90 20 120 TABLE by which to determine the Number of Teeth, or Pitch of Small Wheels. Diametral Circular Diametral Circular pitch. pitch. pitch. pitch. 3 1-047 9 349 4 785 10 314 5. 628 12 -262 6 -524 14.224 7 449 16 *196 8. 393 20 157 Required the number of teeth that a wheel of 16 inches diameter will contain of a 10 pitch. 16 x 10 = 160 teeth, and the circular pitch ='314 inch. What must be the diameter of a wheel for a 9 pitch of 126 teeth? 126 9 = 14 inches diameter, circular pitch'349 inch. The pitch is reckoned on the diameter of the wheel instead of the circumference, and designated wheels of 8 pitch, 12 pitch, &c. Page 436 436 TIIE PRACTICAL MODEL CALCULATOR. TABLE of the Diameters of Wheels at their pitch circle, to conrtahil a required number of teeth at a given pitch. PITCHI OF TIlE TEETIH IN INCIHES. in. 1 11 j 11 11 1 11 11 2in.j 21 2 2: 2 —' 3 in. ~~~~~~~~~~~~~~ I DIAMETER AT THE FITCHI CIRCLE IN FEET AND INCHIES. 10 0 3 1-0 30 4 0 41 0 4-70 510 0 a 6 0 61) 610 710 8 0 87'0 9 1110 370, 4 0 41j.0 5-0 5 5 s0 01 61 6~ 70 [ t0 1 - 12:0 ~ ~ 31 4040 1050 6 0 6 60 710 71 40 10 1i 10 4710 ~!Cj o 5 0 5'o 6'0 4o 682 710 710 80 1 A 0 40 50 0 60 6 I 76'0 70 7 9 0 9 0 109 201021 0 10 1I, l 2' i10 561~ 0 -10 610 7 0 70 87 0 9 0 9O~ 10,10 10i<0 111-1 o0l 2 1 17 4.1 0 560 6 0 71 0 8'0 sl o0 1 0 1 11 0 1 1 91 1 18 0 5106710 0: 6 8 0 O 8' -,91010 01L040114 1' 3 11s 190 6 0640 720820 911 910100 &111071, 0~to11 1 };.1 31;1 ~4i 3: 0 0 610O 710 ~ 8 0 88 - 90 10 0 1 10 I 0 11011 4 11 2' 1 4I 1 5. 75 1 910 1' 0 710 810 9 10['01 01 11 0'1 1 1 31 3-il 4.1 7 t 1 o 75o? o 7 9}o 0 7o 810 o 1-00 o0 I o5l 3i 1 1 451 4i1 9& 1 23 0 60 8'0 1 o o01 0 ll1 - 2I 1 31 4I1 61 8 110 240i'74 jt'0 84 910100111 1 11 2 1 311 411 5' 711 o10, I. It 2a 08091010 10 (11 Ii 01 1 II. 2~ 1 2 1 3~11 411 6I 1 8'19 111 24 0 8' j 0 11 1;0 1111 011 1'! 9' 19~ 3 1 411 5 -~1. 19 6118 1 0 - 2706 840 01011 1 71 2-1 3 1 411 571611 7 1 9 1411 0 o) S~ O10 0 11 1 0 1 1i 2' 1 3i 1 41 1 1 6I21 181 10 {2 2 0 U',io'o ii'l 0 1 1 3 1 7 581 011 711ill 2 1' 1 "1 2 01 0 1o0 0 1 It 1 I 2 1 1 2 6 1 1 1 91 12 1 28 26) I0 90 31~0 40111 o'l 141 "141 2401001 021 2~ 1 1 4I oz{1 V-121 7 41 9 81102 02 412 a-080 9010 10;~ I.I I01 11212 71 3 7 4Z 18I24-II 2 1010 91 O1 311 4 1 5121 3I 81 51 9 61 I 911 0 2-11 132 219 -3 230011 l 1ii 21. l {ji 1 a' 1 8, I' 9 1 1-1012 02 4 2 7 10'10-01 1 0[ i'll 23-,1 21 4= 3' 1 7,'.-1 8~l110 11122 1 2 9'44 5 2 910 11 51 O - 461 8i 9 I 21I -1 I497 10'3113 1320 1.t~5 0a 1 211 - 4~j 151 7 Q1 111011 2 1 2' 4:12-I10 249 009i 2'~I 0, LO 1 77 211 914101 610 2'2 19W 52 8'12 2 124 II 444 1 I 1 6 1:8I1 91111'' 1 2-} 2124 i 6I2 3- 9" i 0 7' 3I09; ol 61 8104 02211 2'3 10 11 311: 6 - I 1 71~ 9 Ii101 1812 31 2 5 2. 1043 21-;36 1 1 411 11 3I, 1,1 11 <2 5 2' 9 102- 3 41-31 7 71 21 8 41'I 1 0. 80 10 I24 12 1 i2 9)' 3 8' 2 a=-_, 22 11 - 1 a131 14 1 o12I 20. 312' i lG/1 2ftll tc~~: j I s~/l to ~....19,;~,., o.... 19: 139 1'1 9.1114 912 11 212 1111 3'3 2o 0'" 10} 0 1 t 0 1` 8 3 4[ 9 1 0' 2 112"9 2991 12 4 211 471 01~~~~~~ ~~1 8710 082 24'221130343 0 302 1 41;;1 8110 0 1101i2 4' 1 42 921116513 14 11 ~ ~ ~' 3~3 *p471 < 741'911. 2 721 13 2 303 1014 24i 71 9 11 1219 910 3 0'" 94 3'7l 1 4 51 ~ ~ ~ " " "'t' 151 511 741 I 01<[ 6 1 7: 9 1 1 - 1O2 4zs2 06e52 4: 21 12 3 1: 7404 3: 56 1 t 8' 8110-112 0' 2': 2 7195 9911'3: 41 1.',' 571 6I 5~ 1 811l 10'?2 0'9 3'? 212 72210 l~ 07~43 911 413 9'' 11 8 I 8 4: 8 4l~i Oi2 i 58 1 611 81111 2 0 "2 31902 8'8' 10 2 013 33 1071 2:,1 391 6411 9811.12 14? 2 41?'1l3 1 3 4 3. 0.3...1 3 8 00 1'711 j 91112 2)2' 4"8'7 x 9 11)3 23 1 j3 7'31124 41j 91 I4 I 61 1 7 13 112 1 2 1' 51 9 12 7' 2 10 3 0,1 3 213 5 13 7614 o2 J'1I 01,~~~~~~~~~~~~ 4;62 1 711 10- 112 012 379r 5': 2 2 8'910' 32 51 2 373 76- 7 8 1 641 1631I8 1101121 I2 3j2 6 2 112 3 13 4~11 613 97:421247<5 07 1641811I 1062 11 426129111 23 413 13 9214 2 1 8 11,1 lO7181 1112. 612 4972 983 0; 432 I32 3 2, -; 627Il 9l' 8 2 3 074 3- 44 88 231 i 114 3 6 3 8 3 1<~:4 414 921-,: 5 3 6871 3.71 912 i / 071 2.2 2252 88' 101/3 11 63 94 03 45.4] 1 05 - I 68 912 12 I.3:2 5:2 8 111i3 113 43 731310 4 4 61411 3.' I II9: 3 69i 511 912 012 34'.2 6102 801221113 213 54,3 7 3 1074 1.;734 7 5 0425 6 701101 212 31 6 4 [, 1:}2 91 2 3: 3 5213 83 4I 2 14 2614 715 11,O 5 613 I t~ 541 1 5k 5t 7~ 41 ~ 9 1112 IO9123! 1[2 2~-!21:30' 425 [2 7 " }:2; 11.'43; 1551~~~~~~~~~~~~~~~~~~~~~V19 551 7-~21 9122 2~2 ~ 4 42' _} ~2110 3";.. 51~~~~~~~)2'~~~,) ~4~~ { 7',t25~t 4~I 10.~ 2 0}/ I2 4 1 71212 6n: 29,~,a......3 e - I [ I0i2,2 I:t,2 10 3i 04 - 1~3(; 57~~~~~~~~~~~~~~~5 1 42 1 13,..)3 i 58 9 I t 6~ 1 8.1 11 ['2 12 2 1-[238:t2 10~'230&!2932;~ 5.'-,3310142fi: 64 11-~l3148 9;411-1181 1 01~2[ 2 4 8 1'5 014 61 t 7~J1 9-~. 0:} 2 2~[2 5~12 7.~/2~~~~~~2 ~26 2 12 i3 0-~-!34235~' 7 ~ ~~~~~~~ o -i2 75~ 42 3 39 1o:I 53 621 7~:- 611 10:;~: 0~1 3~[ 5;12 8 2/1}3. 9 1,- 6:~:411~ 0,4 i 8 1 10}f, }:.12 6- 2 2 211 3 1 3:f,33 11'21,A: 0i4 i 3 91 2$7:o1 5 51 8 1 1t 7I-2:;2' 7 2 9fi3 0k s32!: 53$ 10.)-4s3:i ~; [ 06 II~~~~ ~ ~~~~. 9 1 113: 1-2 4 822 72 101 3O 3 "" I~~~~~~~~~~~~~~~~1! 519 6}}2~~~~~~ ~ ~~~~~ 4-/1i3211 5~37- 101411 75; }7 1610~ II1 2 1 2 3jr2 62812, 91 011 3 3,53 8~33'11-5~34 0,'2{47};k 9L61~~~~~ ~ ~ ~ 91'-1,2-! 272 3114 I3 14 8 Page 437 CONTINUOUS CIRCULAR MOTION. 4~7 PITCH OF THE TEETH IN INCHES., ~ 11in. l 1~ ~~~~~J]L I 1-a [ }- 1, i1' I Ijj 12in. 21t 2-LI 2.} 2j [3in. DIAMETER1 AT TH1E PITCI{ CIRCLE IN FEEF AND INCHES. l1l0 l2 4-.2729 73 03 3-13 6394O49' 49l.1 132571 4~ ~ ~ ~ ~~~i 3 x4 0 4.,4,2 —o7 72110' 12 4t2 7-21003 I'3 4.3 6 3 91 40 91 5 "5 9 73112 212, 5 2 8 2 10 3 1{.3 4t13 713 10'~4 1 4114100 3-7 ( 1 11_2 212 5~2 8211.,3 "'3 S 3 24 521 <110 4'1I 10 502 80211k/ 2{3 53 83114 4 4 -1IILa 5511n 7602 0L2 24 620 9 4'3 01.3 3. 63 9g,4 0o4 3.4 (> - 5 6 601 772 0:12 3,2 62 93 013 3>3 61i3 9{ 4 1 4 4-4 7'[5 140 {16 11; 782'32 7210'13 13 43 7131 1 31 4 L 7 7 2 0 84 2' 79 2 41 -2 24.'; 1'3 13 43 8'4 34 83 12% 91, 802 1~ 2 72,211 3 2,13 5-3 8213 1.14 3 14 6 I4 95 3oO15 1064 812 11.2 5 2 ~~~~~~~~8''112'-3 23 53 9 8" 1 0' 3' 4 6- 11 4' 12 K 822 2'2 5 2 8 521113 3<13 6-S3 9fi14 0{4 4T4 7 - 110~ _ 53 114}6 6~ 8-2 5 0 -I3 6i-' 124 4,4 87111.045 0 }6 01:0 I 8 1 2 93 03 4'13 75310{4 21-4 4l "I 0 6d51 85 2 2 6125` 913 1 2> 61, 91 000 43 O 4)9 S62 ~"2 64210113 103<3 833110443: 01410810 5 Oa 66 1,,10 8g 84 2 - 76 2103:3 2- ~3 513 9~1 4 2' 4 3 4 S410 0 9 8 1.06 411 82;4j2 6 434 89 40 72 11:1 }23 6430{:1'1224 5> 4 S ~-]a''o0~ 9 8 9 &' o113 133 7& 3102 <4 5~.4 96 1'-} 44-111q0 ~ 89 2 3}2' 62 8'3 0' 3 3113 4' 3 4 2'14 -1 4 4 9 5 1K i SS 05;6 77 211 9' 2 5 3 1 3 4.13 46 0 9 4 942 55) 213 11.3 5~~~~~~~1. 3 8% I0,11 4487 15 3o7'60 4 S6 2 6' 203 0 53 5' 1 s 4 )- 4S: A1'HI S ~ 1 8892 4~2 7~[21;2 11 3 2 2'-3 63 914 4' 1 4 Il 1, 3 0 2(1 1 73 70~ 6'210'3'113 6: 3 91.4 L 5 4 10 1> 5 4 4 0 97 2 6811. 0 6 131'42 1 5 O11. 5;j 9-l!~6 5aj7 78 98 2 8 k 21~ 3 3 4 104 210 41 52' 6%0-7 0' 7 1`1.17 93 2 42 3 38 3 311 34 7.41o 5 / 5 7 oil 0 27 1002 i~~~~~~2111.3 31.'3 7>434 101~~~~~~~~~~~:V -Lo 3` l1 17 o711 I_ 101283 3 41 3 81404 0 4 2;..4 S -104 a1 8116 01.0 21 91 2 42 9 82 6 10 2:2 81j. 03 4 311380 14 04 11 41431 5 0 94 9751 1;5 -8 St~ength of6 s;I t ilp 2 4t 92 M'z ~715 5~,16~~~~~~~~ 1 6.,73 s210? 3o2~3 4310-1412-~- 614 70 41o,;'5 4I~70"7 Pitch9Thickness Breodth 5 — 2_______ _ 6- a 92_6_ 9of teeth of teeth of teeth 3 feet per [ 7~-4 feet-pe 0I fet5er 8fet7e 399 2 7~ 1 9 73 6 205 2743 4114 548 95 2 712 17 68:;3 73! 1 -4:4'4 I I195 94 5 8 6 3 92-87I6/~ 315 15 6&I 1012-! 1350 202f-t s;4 7 0 26982 5 822i _t' _ 109 14 I 2392 97 2 5 2 3 2 13 48 ~ 54 6 4 10315 - 5. 6 138 1 I7 92312 1 1 3 43 399 0215 10.'5 7 95 8 1067 49 2 l10 4 30 4 0 6 6 00 14~ ~~~ ~~~ ~~~~~ ~~~~~~7a:7 28 107 137 20. 27 10026 *6 2 i-a313 7i1:4. 6~4. 86 13 184~53-0 11 10 53 5 4 4 3 7 I{!5:[ 5 76 5 10 t i4 0 1102 2 82 02t/3 4~!3 Si 4 04 4i s $I51t ~ 6O675I ~ TABLE. of the Strength of tlhe Teeth of Cast 2:rot W~teels at a given velocity. Strength of teeth in horse power,:at Pitch Thickness Breadth of teeth of teth of teeth 3 feet per 4 feet per 6 feet per 8 feet per in incites. in inches. in inaches. second. second. second. second. 3-.5 1- 6 1012 13.50 20.24 26.08 2.1 1.0 4 8.00 4.00 6.00 8.00 1-89.9 3.6 2-18 2a.1 4-36 5. 1-68.8 3.2 1.53 2.04 3.06 3.08 1.47.7 2.8 1.027 1.37 2.04 - 1.05.5 2. 37. 50 -7:.1_00 Page 438 438 THE PRACTICAL MIODEL CALCULATOR. ADDITIONAL EXAMPLES ON THE VELOCITY OF WHEELS, DRUMS, PULLEYS, ETC. IF a wheel that contains 75 teeth makes 16 revolutions per minute, required the number of teeth in another to work in it, and make 24 revolutions in the same time. 75 x 16 24 =50 teeth. A wheel, 64 inches diameter, and making 42 revolutions per minute, is to give motion to a shaft at the rate of 77 revolutions in the same time: required the diameter of a wheel suitable for that purpose. 64 x 42 6477 4= 34'9 inches. Required the number of revolutions per minute made by a wheel or pulley 20 inches "diameter, when driven by another of 4 feet diameter, and making 46 revolutions per minute. 48 x 46 20 --'110'4 revolutions. A shaft, at the rate of 22 revolutions per minute, is to give motion, by a pair of wheels, to another shaft at the rate of 15~; the distance of the shafts from centre to centre is 451 inches; the diameters of the wheels at the pitch lines are required. 45'5 x 15'5 y~ + 15*5= 18'81 radius of the driving wheel. 45.5 x 22 And 22 - 15x 22 26'69 radius of the driven wheel. Suppose a drum to make 20 revolutions per minute, required the diameter of another to make 58 revolutions in the same time. 58. 20 = 2'9, that is, their diameters must be as 2'9 to 1; thus, if the one making 20 revolutions be called 30 inches, the other will be 30 -. 29 = 10'345 inches diameter. Required the diameter of a pulley, to make 12~ revolutions in the same time as one of 32 inches making 26. 32 x 26 -125 = 66'56 inches diameter. A shaft, at the rate of 16 revolutions per minute, is to give motion to a piece of machinery at the rate of 81 revolutions in the same time; the motion is to be communicated by means of two wheels and two pulleys with an intermediate shaft; the driving wheel contains 54 feet, and the driving pulley is 25 inches diameter; required the number of teeth in the other wheel, and the diameter of the other pulley. Page 439 VELOCITY OF WHEELS, ETC. 439 V81 x 16 = 36, the mean velocity between 16 and 81; then, 16 x 54 36 x 25 36 -- 24 teeth; and 81 -- 1111 inches, diameter of pulley. Suppose in the last example the revolutions of one of the wheels to be given, the number of teeth in both, and likewise the diameter of each pulley, to find the revolutions of the last pulley. 16 x 54 24 = 36, velocity of the intermediate shaft; 36 x 25 and 1111 81, the velocity of the machine. TABLE for finding the radius of a wheel when the pitch is yiven, or the pitch of a wheel when the radius is given, that shall contain from 10 to 150 teeth, and any pitch required. Number Radi. Number Number dius. Number Radius. of Teeth. d f Teeth. of Teeth. a f Teeth. 10 1'618 46.7327 81 12'895 116 18'464 11 1'774 47 7'486 82 13'054 117 18'623 12 1.932 48 7645 83 13.213 118 18.782 13 2 089 49 7.804 84 13.370 119 18.941 14 2.247 50 7.963 85 13.531 120 19.101 15 2.405 51 8.122 86 13.690 121 19.260 16 2.563 52 8.281 87 13.849 122 19.419 17 2721 53 8.440 88 14.008 123 19 578 18 2.879 54 8.599 89 14.168 124 19.737 19 3.038 55 8.758 90 14.327 125 19.896 20 3.196 56 8-917 91 14.486 1 126 20.055 21 38355 57 9-076 92 14-645 127 20-214 22 3.513 58 9.235 93 14'804 128 20-374 23 3-672 59 9-394 94 14-963 129 20-533 24 3.830 60 9.553 95 15.122 130 20-692 25 3'989 61 9-712 96 15-281 131 20-851 26 4-148 1 62 9872 97 15-440 132 21-010 27 4.307 63 10-031 98 15-600 133 21-169 28 4-465 64 10'190 99 15'759 134 21-328 29 4-624 65 10-349 100 15.918 135 21-488 30 4-788 66 10-508 101 16-077 136 21-647 31 4.942 67 10-667 102 16-236 137 21 806 32 5.101 68 10.826 103 16-395 138 21-965 33 5-260 69 10-985 104 16-554 139 22-124 34 5-419 70 11.144 105 16-713 140 22.283 35 5.578 71 11-303 106 16-873 141 22.442 36 5-737 72 11.463 107 17-032 142 22.602 37 5-896 73 11-622 108 17-191 143 22.761 38 6.055 74 11-781 109 17 350 144 22-920 39 6-214 75 11-940 110 17-509 145 23-079 40 6 373 1 76 12-099 11 111 17 668 146 23 238 41 6'532 77 12'258 112 17-827 147 I 23-397 42 6'691 78 12'417 113 17'987 148 23'556 43 6-850 79 12'576 114 18'146 149 23'716 44 7'009 80 12'735 115 18 305 150 23-875 45 7-168 RuLE.-Multiply the radius in the table by the pitch given, and the product will be the radius of the wheel required. Page 440 440 THE PRACTICAL MODEL CALCULATOR. Or, divide the radius of the wheel by the radius in the table, and the quotient will be the pitch of the wheel required. Required the radius of a wheel to contain 64 teeth, of 3 inch pitch. 10'19 x 3 = 30'57 inches. What is the pitch of a wheel to contain 80 teeth, when the radius is 25'47 inches? 25'47 T 12-735 = 2 inch pitch. Or, set off upon a straight line AB seven times the pitch AC given; divide that, or another exactly the same length, into eleven equal parts; call each of those divisions four, or each of those divisions will be equal to four teeth upon the radius. If a circle be made with any number (20) of these equal parts as radius, AC the pitch will go that number (20) of times round the circle. Were it required to find the diameter of a wheel to contain 17 teeth, the construction would be as follows:A 1 C 2 |3 |4 5 |6 7 1 2 3 4 5 6 7 8 9 10 11 a' E l l i.11 I T 77TT-T 1C I 4 8 12 i' 2; 0 24 8 X32 23 40 4t Thus, 4 divisions and 1- of another equal the radius of the wheel, that is al b- = ab, and Al C1 = AC. Page 441 VELOCITY OF WHEELS, ETC. 441 -Regular approved proportions for wheels with flat arms in the mziddle of the ring, and ribs or feathers on each side.-The length of the teeth =6 the pitch, besides clearance, or 7 the pitch, clearance included. Thickness of the teeth........................ 4 the pitch. Breadth on the face...................-...........21 Edge of the rim.................................. 4 Rib projecting inside the rim................... 9 Thickness of the flat arms...................... 9 Breadth of the arms at the points = 2 teeth and 4 the pitch, getting broader towards the centre of the wheel in the proportion of - inch to every foot in length. Thickness of the ribs, or feathers, 4 the pitch. Thickness of metal round the eye, or centre, 7 the pitch. Wheels made with plain arms, the teeth are in the same proportion as above; the ring and the arms are each equal to one cog or tooth in thickness, and the metal round the eye same as above, in feathered wheels. These proportions differ, though slightly, in different works and in different localities; but they are the most commonly employed, and are besides the most consistent with good and accurate workmanship. For the m f sake of more easy b q reference, we collect them into aP table, which the --- --- annexed diagram. j<' -C' will serve fully to explain. They stand thus: a b = Pitch of teeth = 1 pitch. m n Depth to pitch line, PP, = _n s + n m -= Working depth of tooth, = 6 -. C b - ns = Bottom clearance, 1_10 fh = Whole depth to root, =7 -. p q = Thickness of tooth, = -_ r p = Width of space, = 6The use of the following table is very evident, and the manner of applying it may be rendered still more obvious by the following examples:- 3'1416. 1. Given a wheel of 88 teeth, 2t inch pitch, to find the diameter of the pitch circle. Here the tabular number in the second column answering to the given pitch is'7958, which multiplied by 88 gives 70'03 for the diameter required. 2. Given a wheel of 5 feet (60 inches) diameter, 24 inch pitch, to find the number of teeth. Here the factor in the third column Page 442 442 THE PRACTICAL MODEL CALCULATOR. corresponding to the given pitch is DP X 1'1333, which multiplied by 60 gives t r f 68 for the number of teeth. Pitchin RULE.-To find RULE. —To find 68 for the number of teeth. inches and the diameter in the numberof It may, however, so happen that the parts of an ches, multi- teeth gimutipyaoftehb the meter in inches answer found in this manner contains a tabular numbthe tahilar tc thte givene swing tc the fraction-which being inadmissible by pi h giveern pitch. the nature of the question, it becomes __ necessary to alter slightly the diameter Values of P Values of- Values ofp of the pitch circle. This is readily ac- 6 19095 *5236 complished by taking the nearest whole 5 15915.6283 number to the answer found, and find- 44 1~4270 6981 ing the modified diameter by means of 3 12732 78956 34 11141.8976 the second column. The following0l;,ct 3 9547 1.0472 will fully explain what is meant: 24 *8754 1.1333 3. Given a wheel 33 inches diameter, 24 Y 7958 1 266 13la~~~~~~~~ 2'.7135 1.3963!1 inch pitch, to find the number of 2 *6366 1'5708 teeth. The corresponding factor is 1 5937 1-6755 1'7952, which multiplied by 33 gives 1 5570 17926 59'242 for the number of teeth, that is, / 514774 2.0944 ij *4774 2-0944 591 teeth nearly. Now, 59 would here 1 4377 2 2848 be the nearest whole number; but as a 1 3979 2'5132 wheel of 60 teeth may be preferred for 1 i 35868 279146 y'3183 3'1416 convenience of calculation of speeds, we 7 2785 3 5904 may adopt that number and find the di- 4.1987 4.1888 ameter corresponding. The factor in I 592 6 2832 the second column answering to 13 pitch 3. 1194 8.3776 is'557, and this multiplied by 60 gives 4 0796 12.5664 33'4 inches as the diameter which the wheel ought to have. RULE. —To find the power that a cast iron wheel is capable of transmitting at any given velocity.-Multiply the breadth' of the teeth, or face of the wheel, in inches, by the square of the thickness of one tooth, and divide the product by the length of the teeth, the quotient is the strength in horse power at X velocity of 136 feet per minute. Required the power that a wheel of the following dimernions ought to transmit with safety, namely, Breadth of teeth............... 7 inches, Thickness.......................1'4 And length......2.............2 1.42 = 1'96, and 7'5 x 1'96 2 ~= -735 horse power. The strength at any other velocity is found by multiplying the power so obtained by any other required velocity, and by'0044, the quotient is the power at that velocity. Suppose the wheel as above, at a velocity of 320 feet per minute. 7'35 x 320 x'0044 = 10'3488 horse power. Page 443 MAXIMUM VELOCITY AND POWER OF WATER WHEELS. 443 ON THE MAXIMUM VELOCITY AND POWER OF WATER WHEELS. OF UNDERSHOT WHEELS. THE term "undershot" is applied to a wheel when the water strikes at, or below, the centre; and the greatest effect is produced when the periphery of the wheels moves with a velocity of'57 that of the water; hence, to find the velocity of the water, multiply the square root or the perpendicular height of the fall in feet by 8, and the product is the velocity in feet per second. Required the maximum velocity of an undershot wheel, when propelled by a fall of water 6 feet in height. V/6 = 2'45 x 8 = 19'6 feet, velocity of water. And 19'6 x'57 = 11'17 feet per second for the wheel. OF BREAST AND OVERSHOT WHEELS. Wheels that have the water applied between the centre and the vertex are styled breast wheels, and overshot when the water is brought over the wheel and laid on the opposite side; however, in either case the maximum velocity is -a that of the water; hence, to find the head of water proper for a wheel at any velocity, say: As the square of 16'083, or 258'67, is to 4, so is the square of the velocity of the wheel in feet per second to the head of water required. By head is understood the distance between the aperture of the sluice and where the water strikes upon the wheel. Required the head of water necessary for a wheel of 24 feet diameter, moving with a velocity of 5 feet per second. 5x3 2 = 7'5 feet, velocity of the water. And 258'67: 4:: 7'52:'87 feet, head of water required. But one-tenth of a foot of head must be added for every foot of increase in the diameter of the wheel, from 15 to 20 feet, and ~05 more for every foot of increase from 20 to 30 feet, commencing with five-tenths for a 15 feet wheel. This additional head is intended to compensate for the friction of water in the aperture of the sluice to keep the velocity as 3 to 2 of the wheel; thus, in place of'87 feet head for a 24 feet wheel, it will be'87 + 1'2 = 2'07 feet head of water. If the water flow from under the sluice, multiply the square root of the depth in feet by 5'4, and by the area of the orifice also in feet, and the product is the quantity discharged in cubic feet per second. Again, if the water flow over the sluice, multiply the square root of the depth in feet by 5'4, and a of the product multiplied Page 444 444 THE PRACTICAL MODEL CALCULATOR. by the length and depth, also in feet, gives the number of cubic feet discharged per second nearly. Required the number of cubic feet per second that will issue from the orifice of a sluice 5 feet long, 9 inches wide, and 4 feet from the surface of the water. /4 = 2 x 5'4 = 10'8 feet velocity. And 5 x'75 x 10'8 = 40'5 cubic feet per second. What quantity of water per second will be expended over a wear, dam, or sluice, whose length is 10 feet, and depth 6 inches? 1'20744 x 2 4/5 ='2236 x 5'4 = -3'80496 feet velocity. Then 10 x'5 = 5 feet, and'80496 x 5 = 4024:8 cubic feet per second nearly. In estimating the power of water wheels, half the head must be added to the whole fall, because 1 foot of fall is equal to 2 feet of head; call this the effective perpendicular descent; multiply the weight of the water per second by the effective perpendicular descent and by 60; divide the product by 33,000, and the quotient is the effect expressed in horse power. Given 16 cubic feet of water per second, to be applied to an undershot wheel, the head being 12 feet; required the power produced. 6 x 16 x 62'5 x 60 12 —. 2 = 6 and 33000 10'9 horse power nearly. Given 16 cubic feet of water per second, to be applied to a high breast or an overshot wheel, with 2 feet head and 10 feet fall; required the power. 1 + 10 x 16 x 62.5 x 60 2'2=land = 20 horse power. Only about two-thirds of the above results can be taken as real communicative power to machinery. OF TIIE CIRCLE OF GYRATION IN WATER WHIEELS. The centre or circle of gyration is that point in a revolving body into which, if the whole quantity of matter were collected, the same moving force would generate the same angular velocity, which renders it of the utmost importance in the erection of water wheels, and the motion ought always to be communicated from that point when it is possible. RULE.-TO find the circle of gyration.-Add into one sum twice the weight of the shrouding, buckets, &c., multiplied by the square of the radius, 3 of the weight of the arms, multiplied by the square of the radius, and the weight of the water multiplied by the square of the radius also; divide the sum by twice the weight of the shrouding, arms, &c., added to the weight of the water, and the square root of the quotient is the distance of the circle of gyration from the centre of suspension nearly. Page 445 MAXIMUM VELOCITY AND POWER OF WATER WHEELS. 445 Required the distance of the centre of gyration from the centre of suspension in a water wheel 22 feet diameter, shrouding, buckets, &c. = 18 tons, arms = 12 tons, and water = 10 tons. 22 l2 = 11 and 112 = 121 Then, 18 x 2 = 36 x 121 = 4356 2of12= 8 x 121 = 968 water = 10 x 121 = 1210 6534 And 18 + 12 x 2 = 60 + 10 = 70; hence, 6534 V/ 70 -= 9'6 feet from the centre of suspension nearly. TABLE of Angles for Windmill Sails. Number. Angle with the Plane of Motion. 1 180 240 2 19 21 3 18 18 4 16 14,5 121 9 6 1 7 3 extremity. The radius is supposed to be divided into six equal parts, and - from the centre is called 1, the extremity being denoted by 6. The first column contains the angles according to an old custom; but experience has taught us that the angles in the second column are preferable. THE VELOCITY OF THRESHING MACHINES, MIILLSTONES, BORING IRON, ETC. The drum or beaters of a threshing machine ought to move with a velocity of about 3000 feet per minute; hence, divide 11460 by the diameter of the drum in inches; or 955 by the diameter of the drum in feet; and the quotient is the number of revolutions required per minute. And the feeding rollers must make half the revolutions of the drum, when their diameters are about 3- inches. If the machine is driven by horses, their velocity ought to be from 2- to 3 times round a 24 feet ring per minute. Divide 500 by the diameter of a millstone, in feet, or 6000 by the diameter in inches, and the quotient is the number of revolutions required per minute. In boring cast iron the cutters ought to have a velocity of about 108 inches per minute, or divide 36 by the diameter in inches, the quotient is the number of revolutions of the boring head per minute. And divide 100 by the diameter in inches, the quotient is the number of revolutions per minute, for turning wrought iron in general, and about half that velocity for cast iron. Page 446 446 THE PRACTICAL MODEL CALCULATOR. OF PUMPS AND PUMPING ENGINES. PUMPS are chiefly designated by the names of lifting and force pumps; lifting pumps are applied to wells, &c., where the height of the bucket, from the surface of the water, must not exceed 33 feet; this being nearly equal to the pressure of the atmosphere, or the height to which water would be forced up into a vacuum by the pressure of the atmosphere. Force pumps are applicable on all other occasions, as raising water to any required height, supplying boilers against the force of the steam, hydrostatic presses, &c. The power required to raise water to any height is as the weight and velocity of the water with an addition of about 1 of the whole power for friction; hence the RULE. —Multiply the perpendicular height of the water, in feet, by the velocity, also in feet, and by the square of the pump's diameter in inches, and again by'341; (this being the weight of a column of water 1 inch diameter, and 12 inches high, in lbs. avoirdupois;) divide the product by 33,000, and I of the quotient added to the whole quotient will be the number of horse power required. Required the power necessary to overcome the resistance and friction of a column of water 4 inches diameter, 60 feet high, and flowing with a velocity of 130 feet per minute. 60 x 130 x 42 x *341 1'3 33000 -- 5-'26 + 1P3 = 156 horse power nearly. Hot liquor pumps, or pumps to be employed in raising any fluid where steam is generated, require to be placed in the fluid, or as low as the bottom of it, on account of the steam filling the pipes, and acting as a counterpoise to the atmosphere; and the diameter of the pipes to and from a pump ought not to be less than 2 of the pump's diameter. RULE. —The diameter of a pump and velocity of the water given, to find the- quantity discharged in gallons, or cubic feet, in any given time.-Multiply the velocity of the water, in feet per minute, by the squa.:c or the pump's diameter in inches, and by'041 for gallons, or'GuC 54 for cubic feet, and the product will be the number of gallons, or cubic feet, discharged in the given time nearly. What is the number of gallons of water discharged per hour by a pump 4 inches diameter, the water flowing at the rate of 130 feet per minute? 130 x 60 = 7800 feet per hour. And, 7800 x 42 x'041 = 5116'8 gallons. RULE 1.-The length of stroke and number of strokes given, to find the diameter of a pump, and number of horse power that will discharge a given quantity of water in a given time.-Multiply the Page 447 OF PUMPS AND PUMPING ENGINES. 447 number of cubic feet by 2201, and divide the product by the velocity of the water, in inches, and the square root of the quotient will be the pump's diameter, in inches. 2. Multiply the number of cubic feet by 62'5, and by the perpendicular height of the water in feet, divide the product by 33,000, then will 1 of the quotient, added to the whole quotient, be the number of horse power required. Required the diameter of a pump, and number of horse power, capable of filling a cistern 20 feet long, 12 feet wide, and 61 feet deep, in 45 minutes, whose perpendicular height is 53 feet; the pump to have an effective stroke of 26 inches, and make 30 strokes per minute. 20 x 12 x 6'5 = 1560 cubic feet, and 1560 45 34'66 cubic feet per minute. Then, 34'66 x 2201 426 x 30 - 9'89 inches diameter of pump. And 34'66 x 62'5 x 53 3'48 33000 =- 5 — 69 + 3'48 - 4'17 horse power. RULE. —To find the time a cistern will take in filling, when a known quantity of water is going in, and a known portion of that water is going out, in a given time.-Divide the content of the cistern, in gallons, by the difference of the quantity going in, and the quantity going out, and the quotient is the time in hours and parts that the cistern will take in filling. If 30 gallons per hour run in and 221 gallons per hour run out of a cistern capable of containing 200 gallons, in what time will the cistern be filled? 30 - 22'5 = 7'5, and 200 -- 7'5 = 26'666, or 26 hours and 40 minutes. To find the time a vessel will take in emptying itself of water.Mr. O'Neill ascertained, from very accurate experiments, that a vessel, 3'166 feet long and 2'705 inches diameter, would empty itself in 3 minutes and 16 seconds, through an orifice in the bottom, whose area is'0141 inches; and another 6'458 feet long, the diameter and orifice, as before, would do the same in 4 minutes and 40 seconds; hence, from these experiments, a rule is obtained, namely, Multiply the square root of the depth in feet by the area of the falling surface in inches, divide the product by the area of the orifice, multiplied by 3'7, and the quotient is the time required in seconds, nearly. How long will it require to empty a vessel of water, 9 feet high, and 20 inches diameter, through a hole 3 inch in diameter? V9 = 3, the square root of the depth, 314'16 inches, area of the falling surface, ~4417 inches, area of the orifice; Page 448 448 THE PRACTICAL MODEL CALCULATOR. Then, 314'16 x 3 ~ 4417 x 37 = 5767 seconds, or 9 minutes and 36 seconds. On the pressure qoffluids. —The side of any vessel containing a fluid sustains a pressure equal to the area of the side, multiplied by half the depth; thus, Suppose each side of a vessel to be 12 feet long and 5 feet deep, when filled with water, what pressure is upon each side? 12 x 5 = 60 feet, the area of the side, 2.5 feet = half the depth, and 62'5 lbs. = the weight of a cubic foot of water. Then, 60 x 2'5 x 62'5 = 9375 lbs. RULE.-To find the weight that a given power can raise by a hydrostatic press.-Multiply the square of the diameter of the ram in inches by the power applied in lbs., and by the effective leverage of the pump-handle; divide the product by the square of the pump's diameter, also in inches, and the quotient is the weight that the power is equal to. What weight will a power of 50 lbs. raise by means of a hydrostatic press, whose ram is 7 inches diameter, pump 8, and the effective leverage of the pump-handle being as 6 to 1? 72 x 50 x 6 -.8752 19200 lbs., or 8 tons 11 cwt. In the following rules for pumping engines the boiler is supposed to be loaded with about 2~ lbs. per square inch, and the barometer attached to the condenser indicating 26 inches on an average, or 13 lbs., = 151 lbs., from which deduct ~1 for friction, leaves a pressure of 10 lbs. nearly upon each square inch of the piston. RULE. —To find the diameter of a cylinder to work a pump of a given diameter for a given depth.-Multiply the square of the pump's diameter in inches by ~ of the depth of the pit in fathoms, and the square root of the product will be the cylinder's diameter in inches. Required the diameter of a cylinder to work a pump 12 inches diameter and 27 fathoms deep. V/(122 x 9) = 36 inches diameter. RULE.- Tofind the diameter of a pump, that a cylinder of a given diameter can work at a given depth. —Divide three times the square of the cylinder's diameter in inches by the depth of the pit in fathoms, and the square root of the quotient will be the pump's diameter in inches. What diameter of a pump will a 36-inch cylinder be capable of working 27 fathoms deep? 2 X 3= 12 inches diameter. RULE.- ZTofind the depth from which a pump of a given diameter will work by means of a cylinder of a given diameter. —Divide three Page 449 OF PUMPS AND PUMPING ENGINES. 443 times the square of the cylinder's diameter in inches by the square of the pump's diameter also in inches, and the quotient will be the depth of the pit in fathoms. Required the depth that a cylinder of 36 inches diameter will work a pump of 12 inches diameter. i362 x 3 6144 -= 27 fathoms. An inelastic body of 30 lbs. weight, moves with a 3 feet velocity, and is struck by another inelastic body having -a 7 feet velocity, the two will then proceed, after the blow, with the velocity 50 x 7 + 30 x 3 350 + 90 44 11 V = 50~80 = =8=2= 51 feet. 50 +- 30 80 - 8 - 2 -5 feet. To cause a body of 120 lbs. weight to pass from a velocity c, = 1-} feet into a 2 feet velocity v, it is struck by a heavy body of 50 lbs., what velocity will the body acquire? Here (v - c), - 2 + (2 - 1'5) x 120 = 6 el =2+ A2+ 5= 3'2 1~J~, 50 feet. Two perfectly elastic spheres, the one of 10 lbs. the other of 16 lbs. weight, impinge with the velocities 12 and 6 feet against each other, what will be their velocities after impact? Here AIl = 10 and c, = 12 feet, but M, = 16 and c2 = - 6 feet, hence the loss of velocity of the first body will be 2 x16 (12+ 6) 2 x16 x 18 c1 - v0 = -10 +1- 2 6 22'154 feet; and 2 x 10 x 18 the gain in velocity of the other, v, - C = G = 13'846 feet. From this the first body after impact will recoil with the velocity v, = 12 - 22'154 =- 10'154 feet; and the other with that of - 6 + 13'846 = 7,846 feet. Moreover, the measure of,vis viva of the two bodies after impact = MIv,2 + Mv22 = 10 x 10 1542 + 16 x 7'8462 = 1031 + 985 = 2016, as likewise of that before impact, namely: M~c12 + M[~c,2 = 10 x 122 + 16 x 62 = 11440 + 576 = 2016. Were these bodies inelastic, the first would only lose in velocity e2- - = 11'077 feet, and the other gain V2 - C2 = 6'923 feet; the first would still retain, after impact, the velocity 12 - 11'077 = 0'923 feet, and the second take up the velocity - 6 + 6'923 = 0'923, and the loss of mechanical effect would be (2016 - (10 + 16) 0.9232) -- 2 = (2016 2'22) x 0'0155 = 29'35 ft. lbs. 29 Page 450 450 CENTRIPETAL AND CENTRIFUGAL FORCE. 1. WIAT is the centrifugal force of a body weighing 20 lbs. that describes a circle of 10 feet radius 200 times in a minute? ~000331 x 2002 x 20 x 10 = 2648 lbs., the centrifugal force. ~00331 is a constant number. It is a well established fact that the centrifugal force is to the weight of the body as double the height due to the velocity is to the radius of revolution. Hence, this question may be thus solved: 20 x 3-1416 = 62'832, the circumference of the circle of 10 feet radius. 62'832 x 200 = 12566'4 feet, the space passed over by the weight in one minute. 12566'4 BO-= 209'44 feet, the space described in a second, which is called the velocity. (209'44)2 (62494 =- 681'136 feet, the height due to the velocity. If F be the centrifugal forceF: 20:: 1362' 272: 10. 1362'272 x 20.. F = 10 = 2724'544 lbs. The former rule gives 2648 lbs. 2. What is the centrifugal force at the equator on a body weighing 300 lbs., supposing the radius of the earth = 21000000 feet, and the time of rotation = 86400" -- 24 hours? 21000000 x 300 21000000 x 300 103298 lbs., or one pound 864002 very nearly. 1.224 is a constant multiplier. 3'1416 x 21000000 = 65973600 feet, 1 the circumference of the earth at the equator. 2 x 65973600 860-0 =- 1527'16 feet, the velocity of the weight each second. (1527.16)2 (;64*4 -= 36214.56, the height due to the velocity, F: 300:: 72429'12: 21000000. 72429'12 x 800 F = - — 21000000 = 1'0347 nearly, as by the former approximate method. 3. If a body weighing 100 lbs. describe a circle of 10 feet radius 300 times a minute, what is the diameter of a cast iron cylindrical Page 451 CENTRIPETAL AND CENTRIFUGAL FORCE. 451 rod, connecting the body with the axis, that will safely support this weight? The centrifugal force will be, -000331 x 3002 x 100 x 10 = 29790 lbs. From the' strength of materials, page 281, we find that the ultimate cohesive strength for each circular inch of cross sectional area is 14652 lbs.; but one-third of this weight, or 4884 lbs., can only be applied with safety. 129790 8.. 2840 = 2'46982 inches, the diameter of the cylindrical rod. 4. The dimensions, the density, and strength of a millstone ABDE are given; it is required to find the angular velocity v, in consequence of which rupture will take place on account of the centrifugal force. z i z If we put the radius of the millstone =r, = 24 inches = CG; the radius = CK of its eye = r- = 4 inches; the height P! - GH = 1 = ]2 inches; the density = t = 2500 = specific gravity of the millstone; and the mod,us. of strength = = 750 lbs. = the ultimlate cohesive strength of eaclh s;qiuare inch of cross sectional area in the section Ph, supposing the centrifugal forces - F and ~ F to cause the separation in this section. (r1 - r2) 1 = area of parallelogram GR. Hence, the force in lbs. required to cause rupture will be, 2 (r1 - ra) I x K; the weight of the stone G =- (r12 - r22) ly, and the radius of gyration of each half of the stone, i. e. the distance 4 7. 3 _ 3 of its centre of gravity from the axis of rotation r = x 1_ r At the moment of rupture, the centrifugal force of half the stone is equivalent to the strength; we hence obtain the equation of con Page 452 452 THE PRACTICAL MODEL CALCULATOR. Gr l3 lition X X 2 (r, - r,) IK, i. e. ( x - (r3-r3) = 2 (r, - r%) 1K; or leaving out 2 1 on both sides, it follows that = 3g (r1 -r) K 3gK X (r13-r23) 2 + r, r. + r 2) 7 If rl = 2 feet = 24 inches, r, = 4 inches, K - 750 lbs., and the specific gravity of the millstone = 2'5; therefore the weight 62.5 x 2.5 of a cubic inch of its mass = 1728 = 0'0903 lbs.; it follows that the angular velocity at the moment of rupture is, /3 x 12 x 32'2 x 750 1869400 4 -- 688 x 0'9903 - 6211264- 1121 inches. If the number of rotations per minute = n, we have then 2 n 30 o 30 x 112'1 60; hence, inversely, n =-, but here = = 1070. The average number of rotations of such a millstone is only 120, therefore 9 times less. With what velocity must a body of 8 lbs. impinge against another at rest of 25 lbs., in order that the last may have a velocity of 2 feet? Were the bodies inelastic, we should then have to put: Mt, el 8 x el 33 v = M c+MRX i. e. 2- 8+2hencecl 4 -8 8~ feet, the re2 M1 c2 quired velocity; but were they elastic, we should have v2 -- ~Al e lI'; 33 hence, c — 8 = 41 feet. If in a machine, 16 blows per minute take place between two in1000 1200 elastic bodies I1 g lbs. and AI2 = lbs., with the velog g cities ce = 5 feet, and c -2 feet, then the loss in mechanical ef16 (5 - 2)2 1000o1200 feet from these blows will be: L -- 60 x 2 x 200 4 1 6000 400 15 9 x 44 x 11= 0'576 x — = 20'94 units of work per second. If two trains upon a railroad of 120000 lbs. and 160000 lbs. weight, come into collision with the velocities c =- 20, and c2 = 15 feet, there will ensue a loss of mechanical effect expended upon the destruction of the locomotives and carriages, which in the case of perfect inelasticity of the impinging parts, will amount to (20 + 15)2 120000 x 160000 1 1920000 - _2 x 280000 = x 644 x 28 - 1344000 ft. lbs., or units of work. Page 453 453 SHIP-BUILDING. AND NAVAL ARCHITECTURE. Two rules, by which the principal calculations in the art of shipbuilding are made, may be employed to measure the area or superficial space enclosed by a curve, and a straight line taken as a base. RULE I.-If the area bounded by the curve line ABC and the straight line AC is required to be estimated, by the rule, the base AC is divided into an even number of equal parts, to give an odd number of points of division.? 910 11 12 13 115 1 A 1 2 3 4 5 6 7 8 9 1O 111 1;J 14 la 16 17 18 19'u z[ C IWhere the base AC is divided into twenty equal parts, giving twenty-one points of division, and the lines 11, 2c2, 3'3, &c., ar( drawn from these points at right angles or square to AC, to mevet the curve ABC, these lines, 1'1, 2'2, 3:3, &c., ard denominated ordinates, and the linear measurement of them, on a scale of parts, is taken and used in the following general expression of the rule. Area= {A + 4P + 2Q}. Where A = sum of the first and last ordinates, or 1'1 and 21'21. 4 P = sum of the even ordinates multiplied by 4. Or,., {2d + 4th + 6th + 8th + 10th + 12th + 14th + 1Gth + 18th + 20th} x 4. 2 Q = sum of the remaining ordinates; or, {3d + 5th + 7th + 9th + 11th + 13th + 15th + 17th + 19th} x 2. And r is equal to the linear measurement of the common interval between the ordinates, or one of the equal divisions of the base AC. This rule, for determining the area contained under the curve and the base, may be put under another formi'; for as the Area = {A + 4 P + 2 Q} x 3; it may be transferred into A 2r Area = j 2 + Q }x The practical advantages to be derived from this modification of the general rule will appear when the methods of calculation are further developed. Page 454 454 TIIE PRACTICAL MIODEL CALCULATOR. B A 1 2 3 4 5 6 7 8 9 1U 11 12 13 14 15 16 C RULE II.-If the base AC be so divided that the equal intervals are in number a multiple of the numeral 3, then the total number of the points of division, and consequently the ordinates to the curve, will be a multiple of the numeral 3 with one added, and the area under the curve ABC, and the base AC, can be determined by the following general expression: 3r Area = (A + 2 P + 3 Q} x 8Where A = sum of the first and last ordinates, or 1 and 16. 2 P = sum of the 4th, 7th, o10th, 13th, multiplied by 2, or ordinates bearing the distinction of being in position as multiples of the numeral 3, with one added. 3 Q, the sum of the remaining ordinates, multiplied by 3, or of the 2d, 3d, 5th, 6th, 7th, 8th, 9th, 11th, 12th, 14th, and 15th, multiplied by 3. The number of equal divisions for this rule must be either 3, 6, 9, 12, or 15, &c., being multiples of the numeral 3, whence the ordinates will be in number under such divisions, multiples of the numeral 3, with one added. This rule admits also of a modification in form, to make it more convenient of application. 8 For area= {A + 2P+3Q} x r. As before advanced for the chainne adopted in the general expression for the first rule, the utility of this modification of the second rule will be observable when the calculations on the imnersed body are proceeded with. The rules are formed under the supposition that in the first rule the curve ABC, which passes through the extremities of the ordinates, is a portion of a common parabola, while in the second rule the curve is assumed to be a cubic parabola; the results to be obtained from an indiscriminate use of either of these rules, differ from each other in so trifling a degree, (considered practically and inot mathematically,) as not to sensibly affect the deductions derived by them. William O'Neill, or, as English writers term himi, WTilliam Neael, Twas the first to rectify a curve of any sort; this curve was the semi-cubical parabola; these rules, of such use in the art of shipbuilding, were first given by him, but as is usual, claimed by English pretenders. The foregoing rules, when applied to the measurement of the Page 455 SHIP-BUILDING AND NAVAL ARCtIITECTURE. 4 r immerse(l portion of a floating body, as the displacement of a ship, are used as follows. The ship is considered as being divided longitudinally by equidistant athwartship or transverse vertical planes, the boundaries of which planes give the external form of the vessel at the respective stations, and therefore the comparative forms of any intermediate portion of it. c E o F If the ship be immersed to the line AB, considered as the line of the proposed deepest immersion or lading, the curves IILO and KMF would give the external form of the ship at the positions G and I in that line; and tile areas GIILO, IKAMF cont-ained under the curves HLO, KMF, the right lines GEI, IIK, (the half-breadths of the plane of proposed flotation AB at the points G and I,) and the right lines GO, IF, the immersed depths of the body at those points are the areas to be measured; and if the areas obtained be represented by linear measurements, and are set off on lines drawn at right angles to the line AL at their respective stations, a curve bounding the representative areas would be formed, and the measurement by the rules of the area contained under this curve, and the right line, AB, or length of the ship on the load-water line, would give the sum of the areas thus represented, and thence the solid contents of the immersed portion of the ship in cubic feet of space. In accordance with this application of those rules to measure the displacement of the ship, the usual practice is to divide the ship into equidistant vertical and longitudinal planes, t1he longitudinal planes being parallel to the load-water section or horizontal section formed by the proposed deepest immnlersion. To measure the areas of these planes after they have been delineated by the draughtsman, the constructor divides the depth of each of the vertical sections, or the length of each horizontal section, into such a number of equal divisions as will make either one or the other of the rules 1 or 2 applicable. If the first rule be preferred, the equal divisions must be of an even number, so that there may be an odd nulmber of ordinates; while the use of the second rule, to measure the area, will require the equal divisions of the base to be in number a multiple of the numeral 3, which will make the ordinates to be in number a multiple of the numneral 3, with one added. From the points of equal divisions in the respective sections thus determined, perpendicular ordinates are drawn to meet the curve, or the external form of the transversa planes of the body; and a table for the ordinates thus obtained, having been made, as shown page 467, the measures by scale of the respective ordinates are therein inserted. Page 456 456 TILE PRACTICAL MODEL CALCULATOR. For the area IKMF, the linear measurements of IN, 1'1, 2'2, 3, 4-4, are taken by a scale of parts, and inserted in the column marked 5, page 467, the whole length AB of the load(-water line being divided into 10 equal divisions, and the area IKSIF being supposed as the fifth from B, the fore extreme of the load-water line. To apply the first rule to the measurement of the area of No. 5 section, the ordinates are extracted from the table, page 467, and operated upon as directed by the rule; viz. Area = {A + 4 P + 2 Q} x. IK, or first, 1'1, or 2d, 2-2 or 3d, 4'4, or last, 3'3, or 4th, x 2. added together or 2 Q. added together = A. and x 4 = 4 P. By rule, area = {A + 4P + 2 Q} x 0. r Whence area = {(II + 4'4) + (1'1 + 3'3) 4 + 2'2 x 2} x 3 = area IIKMFF; and, in a simnilar manner, may the several areas of the other transverse sections be determined. Whlen these areas have all been thus measured, they are to be summed by the same rules; the areas themselves being considered as lines, and the result will give the solid for displacement in cubic feet. To shorten this tedious ajpplication of the formula, the arrangement of having double-columned tables of ordinates was introduced, as shown on page 484, and for the more ready use of this enlarged table, the modifications in the formula 467, before alluded to, were adopted, that of A JA2V xrfA, 2r Area-{A +4P + 2Q x 3 = +2- Q} x and that of Area= A+2Pt 3Q} x 3 { +P +1-5Q }x r, as rendering the required number of figures much less, whereby accuracy of calculation is insured and time is saved. In using a table of ordinates constructed for this method of calculation, the linear measurement of the several ordinates of vertical section 5 and the corresponding ones of all the others would be inserted in the double columns prepared for them, in the following order: In the first and last lines of the enlarged table for the ordinates, A distinguishable by -, in the left-hand column of each pair, the measurements of the first and last ordinates of the respective areas are placed, and in the right-hand column of each pair one-half of such measurements, as being one-half of the first and last ordinates of each vertical section or area. In the lines distinguished by 2 P, in the left-hand column, the measurements of the even ordinates Page 457 SHIP-BUILDING AND NAVAL ARCHITECTURE. 457 of each respective area are placed, which having been multiplied by two, the result is placed in the respective right-hand' columns prepared for each vertical section; while in those lines of the table distinguished by Q, the measurements of the ordinates themselves are placed in the right-hand columns, as not requiring by the modification of the rules any operation to be used on them, before being taken into the sum forming the sub-multiple of the respective areas. It may here with propriety be suggested, that in practice the insertion of the linear measurements of the ordinates in the table in red ink will be found useful, and that after such has been done, by the upper line of figures in the table of ordinates thus arranged, being divided by two, the second line of figures being multipled by two, and so on with the others as shown by the table, and the results thus obtained being inserted in their respective right-hand columns as before described, great facility and despatch of calculation are afforded to the constructor. That this method will yield a correct measurement of the areas will be evident by an inspection of the terms of the general expresA 2r sion of area = { + 2 P + Q x -,-, which are placed against. the several lines of the table of ordinates. And it will be equally apparent, that the sum total of the figures inserted in the righthand columns appropriated to each section is a sub-multiple of the area of each section, and that these results arising from the use of the form for area of 2 + 2 P + Q will be one-half of those that would be obtained by abstracting the ordinates from the table, page 467, and using them in the expression A + 4 P + 2 Q and therefore to complete the calculation for the areas by the rule. 2r the first results for the areas must be multiplied by -, and the last by w, where r is equal to the common interval or equal division of the base in linear feet; or the part of the expression for areas of { + 2P + must be multiplied by 2r equivalent to {A + 4P + 2 Q} x 3. The sub-multiples of the areas of the vertical sections thus determined, require to be summed together for the solid of displacement, and by considering the sub-multiples of the areas to be, as before stated, represented by lines or proportionate ordinates, O'Neill's rules, by the same table of ordinates with an additional column, may be made available to the development of the solid of displacement. For the sectional areas being represented by lines, by the first rule, one-half the first and last areas, added to the sum of the products arising from multiplying the even ordinates or representative areas by two, together with the odd ordinates or the areas as given by Page 458 458 THE PRACTICAL MODEL CALCULATOR. the tables, and these being placed in the additional column of the table prepared for them, the sub-multiple of the solid of displacement will be given. The operation will stand thus: Sub-multiple of each of the areas - 2{ + 2 P + Q }, or each area will be T less than the full result, and the representative lines for the areas will be diminished in that proportion; and having used these sub-multiples of the areas thus diminished in the second operation for obtaining the sub-multiple of the solid of displacement under the same rule, the 2 r' results will again be - less than the true result; therefore the sum thus determined will have to be multiplied by the quantity 2r 2 r' -3x -, to give the solid required. In this expression, of 2r 2r' 3 x —, r = the equal distances taken in the vertical planes to obtain the respective vertical areas; r' = the equal distances at which the vertical areas are apart on the longitudinal plane of the ship. The displacement being thus determined, by an arrangement of an enlarged table of ordinates, the functions arising from the submultiples of the areas of the vertical sections being placed in O'Neill's rules to ascertain the displacement, may be used in the table of ordinates to find the distance of the centre of gravity of the immersed body fiom any assumed vertical plane; and also the distance that the same point-" the centre of gravity of displacement" -is in depth from the load-water or line of deepest immersion, and that from the considerations which follow: In a system of bodies, the centre of gravity of it is found by multiplying the magnitude or density of each body by its respective distance from the beginning of the system, and dividing the sum of such products by the sum of the magnitudes or densities. The displacement of a ship may be considered as made up of a succession of vertical immersed areas; and if it be assumed that the moments arising from multiplying the area of each section by its relative distance from an initial plane may be represented by successive lineal measurenments, the general rules will furnish the summation of such moments; and the displacement or sum of the areas has been obtained by a similar process, from whence, by the rule for finding the centre of gravity of a system as before given, the distance of the common centre of gravity from the assumed initial plane would be ascertained, by dividing the sum of the moments of the areas by the sum of the areas or the solid of displacement. To extend this reasoning to the enlarged table of ordinates used for the second method of calculation: The sub-multiples of the respective areas, when put into the formulas to obtain the proportionate solid of displacement, are relatively changed in value to give that solid, and consequently the moments of such functions of Page 459 SIIIP-BUILDING AND NAVAL ARCHITECTURE. 459 the vertical areas ewill be to each other in the same ratio; nand the sum of these proportionate moments, if considered as lines, can be ascertained by multiplying the functions of the areas by their relative distances from the assumed initial plane, or by the number of the equal intervals of division they are respectively friom it, and afterwards, by the rules, summing these results, forming the sum of the moments of the sub-multiples of the functions of the vertical areas: and the proportionate sub-multiple for the displacement is shown on the table; the division therefore of the former, or the sum of the proportional moments of the functions of the areas, by the proportionate sub-multiple for the displacement, will give the distance (in intervals of equal division) that the centre of gravity of the displacement is from the initial plane, which being multiplied by the value in feet of the equal intervals between the areas, will give the distance in feet from the assumed initial plane, or firom the extremity of the base line of the proportional sectional areas for displacement. This reasoning will apply equally to finding the position of the centre of gravity of the body immersed, both as respects length and depth, and on the enlarged tables for construction given, (pages 484 and 485,) the constructor, by adopting this arrangement, will at once have under his observation the calculations on0, and the results of, the most important elements of a naval construction. The foregoing tabular system, for the application of O'Neill's rules to the calculations required on the immersed volume of a ship's bottom, led to a lineal delineation of the numerical results of the tables, and thence the development of a curve of sectional areas, on a base equivalent to the length of the immersed portion of the body, or of the length at the load-water line. To effect this, the sub-multiples of the sectional areas, taken from the tables for calculation, are severally divided by such a constant number as to make their delineation convenient; then these thus further reduced sub-multiples of the areas, being set off at their respective positions on the base, formed by the length of the load-water line, a curve passed through the extreme points of these measurements, ewill bound an area, that to the depth used for the common divisor would form a zone, representative of the solid of displacement. The accuracy of such a representation will be easily admitted, if the former reasoning is referred to. To obtain the solid of displacement from this representative:arac, the load-water line or plane of deepest immersion is considered as being divided lengthwise into two equal parts, which assumption divides the base of the curve of sectional areas also into two equal portions: the line of representative area to that medial point is then drawn to the curve, and triangles are formed on each side of it by joining the point where it meets the curve with the extremities of the base line; this arrangement divides the representative area into four parts, two triangles which are equal, viz. I and 2, and two other areas which are contained under the hypothenuse of Page 460 460 THE PRACTICAL MODEL CALCULATOR. these triangles and the curves of sections, or 3 and 4 of the annexed diagram. iDiagram of a Curve of Sectional Areas. B A D C ABCDA equal sectional area, representative of the half displacement as a zone of a given common depth. AC equal the length of the load-water section from the fore-side of the rabbet of the stem to the aft-side of the rabbet of the post, and D the point of equal division. BD, the representative area of half the immersed vertical section at the medial point D, joining 13 with the points A and C, will complete the division of the representative area ABCDA. ABD and CBD, under such considerations, are equal triangles. BECB, BFAB, areas, bounded respectively by the hypothenuse AB or IBC of the triangles and the curve of sectional areas; and, supposing the curves AFB and BEC to be portions of common parabolas, the solid of displacement will be in tlhe following terms: The area of each of the triangles is equal to r- of AC x BD: hence the sum of the two = ~ of AC x BD1: the hypothenuse AB or BC = 2 )[ + BD2], and the area of BECB if considered as approximating to a common parabola = 2( 2 ) ~ BDj x -3 of the greatest perpendicular on the hypothenuse BC. Area ofBFAB under the same assumption =J[( + BID'] x - of the greatest perpendicular on the hypothenuse AB; whence the whole displacement will be expressed by ~ AC x BD x 2[(A)2 )+ BD2] x 3- of the greatest perpendicular on the hypothenuse B C + [(AC + BD2] x 2 of the greatest perpendicular on the hypothenuse AB. [By a similar method, from the light draught of water, or the depth of immersion on launching the ship, the light displacement, or the weight of the hull or fabric, may be delineated and estimated; and the representative curve for it being placed relatively on the same base as that used for the representative curve for the load displacement, the area contained between the curve bounding the representative area for the load displacement, and the curve bounding the representative area for the light displacement, will be a representative area of the sum of the weights to be received on board, and point out their position to bring the ship from the light line Page 461 SHIP-BUILDING AND NAVAL ARCHITECTURE. 461 of flotation, or the line of immersion due to the weight of the hull when completed in every respect, to that of the deepest immersion, or the proposed load-water line of the constructor —a representation that would enable the constructor to apportion the weights to be placed on board to the upward pressure of the water, and thence approximate to the stowage that would insure the easiest movements of a ship in.a sea. By an inspection of the diagram of the curve of sectional areas, it will clearly be seen that the representative area for displacement under the division of it, into the triangles 1 and 2, and parabolic portions of the area 3 and 4, will point out the rel-tive' capacities of the displacement, under the fore and after half-lengths of the base or load-water line; for, by construction, the triangles ABD and CBD are equal, and therefore the comparative values of the areas BECB and BFAB, or of J[(A2 + BD ] x 2 of the greatest perpendicular on.the hypothenuse BC, compared with (AC)2 + BD2] x 3 of the greatest perpendicular on the hypothenuse AB, or of the relative' alues of the greatest perpendiculars on the hypothenuses BC and A1i, will give the relative capacities of the fore and after portions of the immersed body or the displacement. The representative area ABCDA admits also of a measurement by the second rule. Let BD, as before, be the representative area at the middle point. 5_ I 12 A6 5 4 1) 3 2 1C Divide AD or DC into three equal portions, then the equal divisions being a multiple of 3, the second rule is applicable to measure the areas ABDA or BCDB; for the area ABDA{ 6,6 + BD + 2 x 0 + 3 {44 + 5,5} }8; 6,6 = 0; = { BD + 3 {44 + 5,5} 8; and area BCDB = 1,1 + BD + 2 x O + 3 x {2,2 + 3,3} } 8,where,1=0 { BD + 3 x {22 + 3,3}}-8 BCDB, and the displacement- BD+3 x {4,4+5,5} 8 + BD 3 x {23,3} AD x 8 x by the constant divisor of the areas, or the depth of the zone in feet. Page 462 462 THE PRACTICAL MODEL CALCULATOR. The rules given by O'Neill' for the measurement of the immersed portion of the body of a ship, having been theoretically stated, the practical application of them will be given on the construction. The immersed part of a ship, being a portion of the parallelopipedon formed by the three dimensions; —length on the load-water line, from the foreside of the rabbet of the stemr to the aftside of the rabbet of the stern-post; extreme breadth in midships of the load-water section; and depth of immersion in midships from the lower edge of the rabbet of the keel;-it would seem that the first step towards the reduction of the parallelopipedon, or oblong, into the required form, would be to find what portion of it would be of the same contents as the proposed displacement of the ship-a knowledge of which would enable the constructor, by a comparison of the result with a similar element of an approved ship, to determine whether the principal dimensions assumed would (under the form intended) give an immersed body equal to carrying the proposed weights or lading. The relative capacities of the immersed bodies contained under the fore and after lengths of the load-water line must next be fixed, and the constructor in this very important element of a construction will find little to guide him from the results of past experience and practice. From deductions on approved ships of rival constructors it will be developed, that in this essential element, " the relative difference between the two bodies," they vary from 1 to 13 per cent. on the whole displacement. The relative capacities of the fore and after bodies of immersion under the proposed load-water line would seem at the first glance of the subject to be a fixed and determinate quantity, as being a conclusion easily arrived at from a kno+wledge of the proportiions due to the superincumbent weights-under such a consideration, the weight of the anchors, bowsprit, and foremast would necessarily be supposed to require an excess in the body immersed under the fore lalftlength of the load-water line over that immersed under the after half-length of the same element. In a ship, the necessary arrangement of the weights, to preserve the proposed relative immersion of the extremes or the intedlrcd draught of water, would be pointed out by a delineated curve of sectional areas, described as before directed; but a want of that system, or of sonic other, has often caused an errorl in the actual draught of water, and that under a great relative excess of the volumes of displacement in the fore and after portions of the Irmmersed body. The men-of-war brigs built to a construction-draught of water 12 ft. 9 in. forward, 14 ft. 4 in. abaft, giving 1 ft. 7 in. diiTerence, had under such a construction a difference of displacement between the immersed bodies under the fore and after half-lengths of the load-water line that was equivalent to 10'4 tons for every 100 tons of the vessel's total displacement or weight; but these ships, when Page 463 SHIP-BUILDING AND NAVAL ARCHITECTURE. 463 stowed( and equipped for sea, came to the load-draught of water of 14 ft. 2 in. forward, 14 ft. 3 in. aft, —difference 1 inch, or an immersion of the fore extreme of 18 inches more than was intended by the constructor. The reason of this practical departure from the proposed line of flotation of the constructor was, that the internal space or hold of the ship necessarily followed the external form, giving a hold proportionate to the displacement contained under the several portions of the body; but an injudicious disposal of the stores (in placing the weights too far forward) made them more than equivalent to the upward pressure of the water at the respective portions of the proposed immersion of the body, and thence arose the error or excess in the fore immersion by giving a greater draught of water than was designed. The stowage of a ship's hold, under a reference to the representative area for the displacement, contained between the curves of sectional areas developed for the light and load displacements, would prevent similar errors under any extent to which the relative capacity of the two bodies might be carried. This relative capacity of the two bodies will affect the form of the vessel's extremes, giving a short or long bow, a clear or full run to the rudder; for the whole displacement being a fixed quantity, if the portion of it under the fore half-length of the loadwater line be increased, it must be followed by a proportionate diminution of the portion of the displacement under the after hayf-length of the load-water line, so that the total volutme of the displacement may remain the same, which arrangement will give a proportionately full bow and clean run, and vice versd. The curve of sectional areas under the foregoing considerations is also applicable to a comparison of the relative qualities of ships of the same rate, by showing at one view the distribution of the volume of displacement in each ship, under the draught of water which has been found on trial to give the greatest velocity; based on which, deductions may be made from the relative capacities of the bodies pointed out by the sectional curves, that will serve to guide the naval constructor in future constructions. The curve of sectional areas is also available for forming a scale to measure the amount of displacement of a ship to any assumed or given draught of water. To effect this, on the sheer draught or longitudinal plan of the ship between the load-water line, or that of deepest immersion, and the line denoting the upper edge of the rabbet of the keel, draw intermediate lines parallel to the loadwater line as denoting lines of intermediate immersion between the keel and load-water line; these lines may be placed equidistant from each other, but they are not necessarily required to be so. Find the curve of sectional areas, due to each immersion of the ship denoted by these lines, and measure the areas bounded respectively by these curves, in the manner as before directed for the load displacement: these results will give the magnitudes of the immersed portions of the body in cubic feet, which being divided by 35, the mean of the number of cubic feet of salt or fresh water that Page 464 O464 THE PRACTICAL MODEL CALCULATOR. are equivalent to a ton in weight, will give their respective weights in tons. Assume a line of scale for depth, or mean draught of water, the lower part of which is to be considered the underside of the false keel of the ship, and set off on this line, by means of a scale of parts, the depths of the immersions at the middle section of the longitudinal plan; draw lines (at the points thus obtained) perpentdicular to this assumed line for depth or draught of water, and having determined a scale to denote the tons, set off on each line by this scale the tons ascertained by the curves of sectional areas to be due to the respective immersions of the body; then a curve passed through these points will be one on which the weights in tons due to the intermediate immersions of the body may be ascertained; or, the displacement of a ship to the mean of a given draught may be found by setting up the mean depth on the scale, showing the draught of water-transferring that depth to the curve for tonnage, and then carrying the point thus obtained on the curve for tonnage to the scale of tons, which will give the number of tons of displacement to that depth of immersion or draught of water. Description of the several plans to be delineated by. the draughtsman, previous to the commencement of the calculations. Sheer Plan.-A projection of the form of the vessel on a longitudinal and vertical plane, assumed to pass through the middle of the ship, and on which the position of any point in her may be fixed with respect to height and length. Half-breadth Plan. —The form of the vessel projected on to a longitudinal and horizontal plane, assumed to pass through the extreme length of ship, and on which the position of any point in the ship may be fixed for length and breadth. Body Plan.-The forms of the vertical and athwartship sections of the ship, projected on to a vertical and athwartship plane, assumed to pass through the largest athwartship and vertical section of her, and on which plan the position of any point in the ship may be fixed for height and breadth. These plans conjointly will determine every possible point required; for, by inspection, it will be foundThat the sheer and half-breadth plans have one dimension common to both, viz.:..........Length. Half-breadth and body plane.......................Breadth. Sheer and body plane................................Height. For sheer plan gives length and height......of the same Half-breadth plan gives length and breadth point. Body plan gives breadth and height......... point. Which dimensions form the co-ordinates for any point in the solid, and must determine the position of it. The point C in the load-water section-AB, has for its co-ordinates to fix its position, Page 465 SHIP-BUILDING AND NAVAL ARCHITECTURE. 465 The length, 1'5 of the half-breadth plan. Height, 5'C of the sheer plan, And the breadth, 1 C of the body plan of section. And the same for any other point of the solid or of the ship. In the sheer plan, AB represents the line of deepest immersion, a a, b b, c c, d d, lines drawn parallel to that line at a distance of ~9 feet, making with AB an odd number of ordinates for the use of the first general rule for the area, where area = A + 4 P + 2 Q} x, and A = the sum of the first and last ordinates. P = the sum of the even ordinates, as 2, 4. Q = the sum of the odd ordinates, as 3, &c. The line AB, or length of the load-water line, is bisected at C, and AC, CB are thence equal; C being the middle point of the load-water line, the spaces BC, AC are again divided into four equal divisions, giving five ordinates for each space, at a distance apart of 5'5 feet. This arrangement will give the immersed body of the vessel, as being divided into two parts under an equal division of the loadwater line, and an odd number of ordinates in each section of the body for the application of the first general rule given for finding the areas of the vertical sections and thence the displacement. The half-breadth plan delineates the form of the body immersed for length and breadth, the line AB of the sheer plan being represented in the half-breadth plan by the line marked AB, and a a, b, c c, d d, of the sheer plan by the lines similarly distinguished in the half-breadth plan. The body plan gives the form of the body in the depth, the lines distinguished 5'5 in the sheer and half-breadth plans being in the body plan developed by the curve 5'5'5, giving the external form of the ship at the section 5'5; the same reasoning applies to the other divisions of the load-water line AB. A pile of 400 lbs. weight is driven by the last round of 20 blows of a 500 lbs. heavy ram, falling from a height of 5 feet; 6 inches deeper, what resistance will the ground offer, or what load will the pile sustain without penetrating deeper? 0' 5 Here G = 400, G1 = 700 lbs., II = 5, and s = 20 = 0-025 feet, whereby it is supposed that the pile penetrates equally far for each blow. 700 2 400 x 5 7 2 (700 400) 025 (A)2 x 80000 2400 lbs., the ram, not during penetration, remaining upon the pile. 7002 x 5 4900 P 1100 x 0025 =11 x 200 = 89100 lbs., the ram remaining upon the pile during penetration. For duration, with security, such piles are only loaded from 0o to -L of their strength. 30 Page 466 466 THE PRACTICAL MODEL CALCULATOR. 1st. i,~~~~~~~~~~~~~~~~~ it: z~~~~ z I4 o!0!~~~~~~~~~~~~~~ i l r~~~~~~~~~1~~1 ~~~~~~~~~~~~~ C)~~~~~~4C [., I In i 0.,d Ii -r, 0 ~o 0~~~~~~~~~~~~~~~~~~ 1-. - i. I ~I a I - 0 F I d i ~~~~~~~~4Z o tt.'t k LI on I. CS o,c 0Y:.j _I. 0O CO 1 12I I o..) o AB, Load-water Line. I I b b ~Lines parallel to AB at the Principal ~ r~ Diesin. c c distance of'92 feet apart. ~~~~~~dr Length for Tonnage......45 0DrgtofWe. Keel for Tonnage.......36 10~ Ft. In. Breadth for do........13 6 Afore.............4 6 Burthen in T ons..35 Abaft.7 6 ~........... 9z..........''''' Page 467 SHIP-BUILDING AND NAVAL ARCHITECTURE. 467 Calculations required for the construction drawing of a yacht of 36 tons. —lst. Usual mode of 4 calculating the displacement by vertical and horizontal sections. TABLE of Ordinates for Yacht of 36 Tons.` i Distinguish-i 3 4h(51 6 7 89 ing No. of - 1 2 3 4 (5) 6 7 8 9 the sections.) 1' A 4 3 0 5 06 0 6 3 6 5 4 3.7 4 rthedistanebenates used for 2' P 35 2 4 4!2 5.6 5,6 585 4,4 2,6,35 the erticalsection = c92 feet. I d CO 3' Q 3 17 3 2 4 4 5 0 4 6 3 4 1 7 3 r?=thedistnncebetween the ordi154I 1t 4' P *25 1D 0 22 3a2 38 34 24 11 25 otes used for the horizontal %3 - - - - - - - - - ~~~ sections = 5'5 5' A -2 -4 1 13 2501 24 2 0 14 -6 *2 feet. From this Table the following application of O'Neill's rule, No. 1, is usually made to obtain the volume of displacement to the draught of water shown on the drawing as the load-water line, or line of proposed deepest immersion, de_ signated by AB. General terms of the rule::I r Area = A+4P + 2Q} x To find. the area of vertical section 1, fore body:3\i~ \ \ A=sum of'4 14 P=four times the sum ).35 the first of the even ordinates, - and last'2 or of (2) and (4)......) 25 \6 = A'60=P 42'4 =4 P 2 Q = twice the sum of the odd 3 = Q ordinates, or of (3) x 2.60 = 2 Q Whence the area, which is equal to { A+4P+2Q} x={'6+2'4+'6 x 3' 3'6 x - = 1'2 x'92 = 1'104 = ~ area of section 1. Which sum is half the area of the section 1, and is kept in that form of the half-measurement for the convenience of calculation. Page 468 468 THE PRACTICAL MODEL CALCULATOR. FORE BODY. Vertical Section 2. 3-0 2'4 1V7.4 1'0 2 3'4 = A 34 =- P 3'4 = 2 Q 4 13.6 - 4P 3'4 = A 34 = 2 Q 20.4 = A + 4P + 2Q *92 = r 408 1836 3) 18'768 6'256 = ~ area of Section 2. Vertical Section 3. 5'0 4'2 3'2 1'3 2-2 2 6.3 = A 64 =- P 6.4 = 2 Q 4 25'6 - 4P 6'3 = A 6.4 = 2 Q 38-3 =A + 4P + 2Q'92 r 766 3447 3) 35.236 11'745 - A + 4 P + 2 Q x 3 = area of Section 3. Vertical Section 4. 6'0 5'6 4.4 2~0 3'2 2 80 = A 8%8 = P 8.-= 2Q 4 35-2 = 4P 8'0 = A 88 = 82 Q 6520-=A + 4P + 2Q'92 = r 1040 4680 3) 47.840 15'946 = A + 4P+ 2 Q x 3 = area of Section 4. Page 469 SHIP-BUILDING AND NAVAL ARCHITECTURE. 469 Vertical Section 5. 6'3 5'6 5'0 2'4 3'8 2 87 - A 9.4 = P 10o0 = 2Q 4 37T6 = 4P 8'7 = A 10O0 = 2Q 56'3 = A + 4P + 2 Q ~92 = r 1126 5067 3) 51'796 17'265 = A + 4 P + 2 Q x =1 area of Section 5. Half areas of Vertical Sections 1, 2, 3, 4, and 5. No. 1............................... 1104 feet. 2............................... 6256 3...............................11'745 4...............................15946 5...............................17'265 Displacement of the body under the fore half-length of the loadwater line by the vertical sections, or the summation of the vertical areas 1, 2, 3, 4, and 5, by the formula for the solid, as being equal to {Al + 4 P' + 2 Q' }x where A' = sum of 1st and 5th areas. P' = " 2d and 4th areas. Q = " 3d area. And r' = distance between the vertical sections, or 5'5 feet. 1... 1'104 2... 6'256 3...11'745 = Q' 5...17'265 4...15'946 2 18'369 = A' 22.202 = P' 23'490 = 2 Q' 4 88.808 = 4 P' 18.369 = A' 23.490 = 2 Q' 130.667 = A' + 4 P' + 2 Q' 5'5 = r' 653335 653335 3) 718'6685 r' 239'556 = A' + 4P'+ 2Q' x -cubic ft. of 2 space in ~ fore-body. 3 479'112 = cubic feet of space in fore-body. Page 470 470 THE PRACTICAL MODEL CALCULATOR. Displacement of the body immersed under the after half-length of the load-water line by the vertical areas 5, 6, 7, 8, and 9 of the Table of ordinates. Vertical Section 6. 5, as fore body. 6'1 5'5 4'6 = Q 17.265 2.0 3.4 2 81 = A 8.9 = P 9- = 2 2Q 4 35'6 = 4 P 8'1 — A 9.2=2Q 52.9-=A + 4P + 2Q.92-r 1058 4761 3) 48-668 r I area of 16222 = A+4P+2Qx= arSection 6 Vertical Section 7. 5.4 4.4 3.4 = Q 1.4 2'4 2 6.8=A 6.8=P 6'8= 2Q 4 27-2 =4P 6.8=2Q 6-8 = A 40'8 = A + 4P + 2 Q ~92= r 816 3672 3)37.536 12'512 = A+4P+2Qx= area ofn 7. Vertical Section 8. 3.7 2.6 1'7 = Q.6 1.1 *2 43 -A 37 = P 34 = 2 Q 4 14'8 = 4P 4'3 =A 34 =2 Q 22'5 = A + 4 P + 2 Q ~92 = r 450 2025 8). =0A04P 70x0r= { 1 area of 6' 9 =+Section 8. Page 471 SHIP-BUILDING AND NAVAL ARCHITECTURE. 471 Vertical Section 9..4.35.3 = Q ~2'25 2 ~6 = A 60 = P *6 = 2 Q 4 2'4 = 4P ~6 - A.6 =2Q 3.6 = A + 4P + 2Q ~92 - r 72 324 3) 3312 r 1'104 =A + 4 P + 2 Q x = area of Section 9. Half areas of the vertical sections 5, 6, 7, 8, and 9. Sections. Areas. 5.................................17265 6.................................16'22 7.................................12'512 8................................. 6'9 9........... 1'104 Displacement of the after-body under the after half-length of the load-water line by the vertical sections, or the summation of the immersed areas of the vertical sections 5, 6, 7, 8, and 9 by the formula for the solid as being equal to A' + 4 P' + 2 Q' x where A' = sum of the 5th and 9th areas. P' = " 6th and 8th areas. Q' = " 7th area. and r' = the distance between the vertical sections, or 5'5 ft. 5...17'265 6...16'22 7...12'512 = Q' 9.. 1'104 8... 6'900 2 18'369 - A' 23.120 = P' 25.024 = 2 Q' 4 92.480 = 4 P' 25.024 = 2 Q' 18.369 = A' 135'873 = A' + 4 P' + 2 Q' 55 -- r 679'365 67'936 3) 747'3015 r' 249'1005 =A'+4P'+2 Q'x =cubic ft. 2 in - after-body. 498'2010 After-body in cubic ft. of space. Page 472 472 THE PRACTICAL MODEL CALCULATOR. Displacement of Fore-body by Horizontal Sections. Horizontal Section 1'. 0o4 6.0 5.0 = Q 6'3 3'0 2 6-7 = A' 90 = P 100 Q 4 3600 = 4 P 0o00 = 2 Q 6'70 = A 5270 = A + 4P + 2Q 5'5 = r 2635 2635 3) 289'85 96'61 = A + 4 P + 2 Q x = 1 area of Section 1'. 32 Horizontal Section 2'. ~35 5-7 4'2 = Q 5'60 2'4 2 5.95 = A 8:1 = P 8-4 = 2 Q 4 32=4 - 4 P 8.4 = 2Q 5'95 - A 46.75 = A + 4P + 2Q 5.5 = r 23375 23375 3) 257.125 85'708 = A + 4 P + 2 Q x ~ = ~ area of Section 2'. Horizontal Section 3'..3 4'4 3'2 = Q 5.0 1'7 2 5.3=A 61 1= P 6.4 = 2 Q 4 24'4 = 4P 5.3 = A 6'4 = 2Q 36.1 = A + 4P + 2Q 5'5 = r 1805 1805 3) 198.55 66'18 = A + 4 P + 2 Q x = - area of Section 3'. Page 473 SHIP-BUILDING AND NAVAL ARCHITECTURE. 473 Horizontal Section 4'. ~25 3.2 2.2 = Q 388 1'0 2 4'05-A 4'2 =P 44- =2Q 4 16'8 =4P 4'05 = A 4'40 = 2 Q 25.25=A + 4P + 2Q 55 -=r 12625 12625 3) 1388875 46'291=A + 4P + 2Q x 3= Section4'. Horizontal Section 5'. *2 2-0 1P3 = Q 2'4 *4 2 2'6=A 2'4=P 26 =2Q 4 9'6 =4P 2'6 =A 2.6 =2Q 14.8=A + 4P + 2Q 740 740 3)81'40 2713 = A + 4P + 2Q x = Section 5'. Displacement of the fore-body under the fore half-length of the load-water line by horizontal sections, or the summation of the horizontal sections of the fore-body 1', 2', 3', 4', and 5', by the formula for the solid, as being equal to A'+4P'+2Q' x3; where A' = sum of the 1'st and 5'th areas; P' =," 2'd and 4'th areas; Q' = " 3'd area; and r = the distance between the horizontal sections, or'92 feet. Half areas of the Horizontal Sections 1', 2', 3', 4', and 5'. 1' = 96'61. 4' = 46'29. 2' = 85'708. 5' = 27413. 3' = 66'18. Page 474 474 THE PRACTICAL MODEL CALCULATOR. Areas. Areas. Areas. 1'...9661 2'...85.708 31...6618 = Q' 5'...27'13 4'...46'290 2 123'74 = A' 131'998 = P' 132'36 = 2 Q' 4 527'992 = 4 P' 123'740 = A' 132'360 = 2 Q' 784.092 = A' + 4 P' + 2 Q' ~92 -r 1568184 7056828 3) 721.36464 240'45 = A'+4P'+2Q' x3 ubiC ft in 2 -- A4P+Qx -fore-body. 480'90 = fore-body by horizontal sections in cubic feet of space. Displacement, by horizontal sections of the body immersed under the after half-length of the load-water line, or by the horizontal areas 1', 2', 3', 4', and 5', of the table of ordinates. Calculated areas of 1', 2,' 3', 4', and 5'. Section 1' After-body. 6.3 6'1 5.4 = Q.4 3.7 2 67 =A 9.8 = P 10'8 = 2Q 4 39'2 = 4P 10o8 = 2 Q 6'7 = A 567 = A + 4 P + 2 Q 5'5 = r 2835 2835 3) 311.85 103'95 = A + 4 P + 2 Q x 3 = S etion of Section 1'. Page 475 SHIP-BUILDING AND NAVAL ARCHITECTURE. 475 Section 2' After-body. 5'6 5'5 4.4 ~35 2'6 2 5'95 A 8-1 = P 88 = 2Q 4 32'40 = 4 P 5'95 = A 8.80 = 2 Q 4715= A + 4P + 2Q 5.5 = r 23575 23575 3) 259.325 86'441 = A + 4P + 2 Q x - = area of Section 2'. Section 31 After-body. 5'0 4'6 3.4 = Q ~3 1.7.2 5-3=A 63 =P 68s 2Q 4 25.2 = 4P 5'3 = A 6.8 = 2 Q 37'3 = A + 4P + 2Q 5'5 = r 1865 1865 3) 205'15 r' 68'38 =A+ 4P 2Qx 3 - area of Section 3'. Section 41 After-body. 3-8 3'4 2'4 = Q'25 1'1 2 4.05=A 45=-P 4.8=2Q 4 18.00 = 4 P 4.05 = A 4.80 = 2 Q 26.85 = A + 4P + 2Q 5'5 = r' 13425 13425 3) 147'675 r_ 49'225 = A + 4 P + 2 Q x.3- = area of Section 4'. Page 476 476 THE PRACTICAL MODEL CALCULATOR. Section 51 After-body. 2.4 2.0 1.4 = Q ~2' 6 2 26= A 26 = P 2-8 = 2Q 4 10=4 = 4P 2s8 = 2Q 2'6 = A 15'8 = A + 4P +.2Q 5'5 = r' 790 790 3) 86'90 r' 28'96 = A + 4P + 2 Q x 3 = - area of Section 5'. Displacement by horizontal sections of the after-body under the after half-length of the load-water line, or the summation of the horizontal sections of the after-body, 1', 2', 3', 4', and 5', by the formula of the solid, as being equal to A' + 4 P' + 2 Q' x - Half areas of the After Horizontal Sections, 1', 2', 3', 4', and 5'. Sections. Areas. 1/...............................103' 95. 2............................... 86'44. 31............................... 68'38. 4............................... 49'22. 5............................... 28'96. Areas. Areas. Areas. 1'...103*95 2'...86.44 3'...68.38 = Q' 5'... 28'96 4'...49'22 2 132'91 = Al 135'66 -P' 136'76 = 2 Q' 4 542'64 = 4 P 132'91 - A' 136'76 = 2 Q' 812'31 = A' + 4 P' + 2 Qt ~92 r 162462 731079 3) 747'3252 r 249'1084 = A' + 4 P' + 2 Q' x 3 =cubic ft. of 2 ~ after-body by horizontal sections. 498'2168 = After-body by horizontal sections in cubic feet of space. Page 477 SHIP-BUILDING AND NAVAL ARCHITECTURE. 477 DISPLACEMENT. By Vertical Sections. By Horizontal Sections. Cubic Feet. Cubic Feet. Fore-body (p. 469) 479411 Fore-body (p. 474) 480'9-00 After-body (p. 471) 498'20 After-body (p. 476) 498.216 Sum 977.30 Sum 9794116 Half 488'65 Half 489'558 Cubic Feet. By Horizontal Sections...................9794116 By Vertical Sections.....................977'300 Difference.................. 1'816 cubic feet. Cubic Feet. 979'49 = capacity or displacement in cubic feet of space. The mean weight of salt and fresh water gives 35 cubic feet of space, when filled with water, to be equivalent to a ton avoirdupois; thence the displacement in cubic feet of space being divided by 35 will give the weight of the volume displaced in tons avoirdupois; or 979'49 being divided by 35 gives 5)979'49 7) 195.898 27'985 tons, the weight of the calculated immersed body in tons. AREA OF THE MIDSHIP SECTION, OR OF THE GREATEST TRANSVERSE SECTION. Section at 5. 11...6'3 2'2...6'0 3'3...4'8 = Q 5'5...'2 4'4...2'3 2 65=A 83 = P 96 = 2Q 4 33.2 = 4P 6-5 = A 96 = 2 Q 49.3 - A + 4P + 2Q 1'25 = - where r = the depth, from I to 5, divided by 4 = 5 ft. by 4 2465 1'25 ft. 986 493 3) 61'625 3)620541 A + 4___ _P + 2{ x area of mid2054=+~ +x =3 ship section. 41'082 = Area of midship section without keel. Page 478 478 THE PRACTICAL MODEL CALCULATOR. LOAD-WATER LINE. Area of the load-water line, or area of the assumed deepest plane of immersion, delineated on the half-breadth plan, and marked by the curve AB. From the table of ordinates, p. 467, we have~4 3'0 5'0 ~4 6'0 6'3 - 6'1 5.4 ~*8A 3 7 16'7 = Q 188 -P 2 33.4 =2Q 75.2 = 4P 8 A 33.4 = 2 Q 109'4 = A + 4P + 2 Q 5.5 = rl 5470 5470 3)601.70 200.56 = A + 4 P + 2 Q x - f area of loado'3 water line. 200'56 = 2 area of load-water section in superficial feet. 2 401'12 = area of load-water section, which amount of area being divided by 12, will give the number of cubic feet of space that would be contained in a zone of that area of an inch in depth, and that result being again divided by 35, as the number of cubic feet of water equivalent to a ton in weight, will give the number of tons that will immerse the vessel one inch at that line of immersion. 12)401'12 = area of load-water section in superficial feet. 5) 33'42 = cubic feet in zone of one inch in depth. 7) 6684'955 = tons to the inch of immersion at load-water line. CENTRE OF GRAVITY OF THE DISPLACEMENT. Estimated from Section 1, considered as the Initial Plane. Distinguishing Y, Vertical No. of the Areas. Areas. Moments. 1......... 1104 x 0...................000.000 2..... 6'256 x 1................. 6256 3.......11745 x 2...... 23'490 4.........16069 x 3................... 48.207 5.........17'265 x 4.......69 060 6......... 16'222 x 5................... 81110 7...... 12'512 x 6.................. 75072 8..... 6.900 x 7...... 48.300 9..... 1.104 x 8. 8832 Page 479 SHIP-BUILDING AND NAVAL ARCHITECTURE. 479 Moments placed in the Rule. Sum = A + 4P + 2 Q x 000'000 6'256 23'490 8'832 48'207 69'060 -8832 - A 81'110 75'072 8~832 =~A 48'300 167'622 = Q 183'873 = P 2 335'244 = 2 Q 735.492 = 4 P 8'832 - A 335'244 = 2 Q 1079.568 = A + 4P + 2Q 5'5 = r' 5397840 5397840 3) 5937.6240 1979'208 = A + 4P + 2Q xsum of the moments of half the displacement from section 1, in intervals of space of 5'5 ft.; and the half displacement in cubic feet by vertical sections is 488'650 (p. 477) cubic ft.; whence it is found, by dividing the moment 1979'208 by 488'650, that the distance of the centre of gravity of displacement from the section 1 is as follows:488'65) 1979'208 (4'05 intervals from 1. 195460 interval = 5-5 ft. 246080 244325 1755 therefore 4'05 x 5'5 = 22'27 ft. = distance of the centre of gravity of the calculated immersed body from 1. DEPTH OF THE CENTRE or GRAVITY OF THE DISPLACEMENT BELOW THE LOAD-WATER SECTION. Fore-body. After-body. Section. Areas. Areas. Sum of the Areas. Moments. 1' 9661 10395...........20056 x 0 = 000'000 2' 1 85 7081 86'44............172148 x 1 = 172'148 3' 66,18 J 68'38..........134'56 x 2 = 269'12 4' 46'29, 49'22.......... 95'51 x 3 = 286'53 5' [27'13 28'96........... 56'09 x 4 - 224'36 Page 480 480 THE PRACTICAL MODEL CALCULATOR. 000'00 172'148 269'12 = Q 224.36 286.530 2 224'36 - A 458.678 = P 538.24 = 2 Q 4 1834.712 = 4 P 224.360 = A 5388240 = 2 Q 2597.312 = A + 4P + 2 Q *92 r 5194624 23375808 3) 2389'52704 796'509 =A+4P + 2 Q x = 3 sum of the moments of the half displacement calculated from the load-water line: the half displacement by horizontal sections is 489'588 (p. 477) cubic feet; the sum of the moments of the half displacement 796'509 ft., being divided by that quantity, will give the distance in intervals of *92 ft.; the centre of gravity of displacement is below the load-water line. Half solid of displacement. Moments. 489'558 ) 796'509 ) 1'62 intervals of'92 feet; therefore 489558 1'62 x'92 3069510 2937348 324 1458 1321620 979116 1'4904 ft. = the distance the centre of gravity of the calcu342504 lated immersed body is below the load-water section. DISTANCE OF THE CENTRE OF GRAVITY OF THE AREA OF THE LOADWATER SECTION FROM SECTION 1. Ordinates of Section I Distances of them in Moments; being the ProNo. of Section. intervals of 5-5 ft. duct of the Areas by the from Section 1. respective Distances. 1.4 0 000.00 2 3.0 1 3-0 3 5.0 2 10.0 4 6.0 3 18.0 5 6.3 4 25-2 6 6.1 5 30.5 7 5.4 6 32-4 8 3.7 7 25-9 9.4 8 3-2 Page 481 SHIP-BUILDING AND NAVAL ARCHITECTURE. 481 The moments, for summation, put into the rule. 00'0 3.0 10.0 3.2 18.0 25.2 3- A30.5 32.4 25-9 3@2 -A 32~~559 67'6 = Q 77'4 = P 2 4 1352 -- 2 Q 309.6 = 4 P 3.2 = A 135.2 = 2 Q 448'0 = A + 4P + 2Q 55 = r 2240 2240 3) 2464.0 8213 = A + 4 P + 2 Q x = 3 sum of the moments of the half area of the load-water section reckoned from 1; the half area of the load-water section is 200'56 feet (p. 478); the distance, therefore, of the centre of gravity of the load-water section from 1 will be found in intervals of space of 5'5 feet, by dividing the sum of these moments by the half area, thus:Half Area. Moments. No. 200'56) 821'3333 (4'09 intervals, each 80224 5'5 ft. in length. 190933 180504 10429 and 4'09 x 5'5 = 22'5 ft. gives the distance of the centre of gravity of the load-water section from section 1 of the drawing. Relative capacities of the bodies immersed under the fore and after lengths of equal division of the load-water lineBy former calculations. After-body immersed contains........497'79 cubic ft. of space. Fore-body " "........481'70 cubic ft. of space. Difference........ 16'09 = the excess in cubic feet of space of the body displaced under the after half-length of the load-water line over that under the forehalf of the same lineSum of the bodies (by former calculation) or whole 97949 displacement in cubic feet of space (p. 477)...... equal to 9'7949 hundreds of cubic feet of space, whence 16'09, or the difference between the two bodies in cubic feet, being divided by 9'7949, or the displacement expressed in terms of the hundreds 31 Page 482 482 THE PRACTICAL MOTDEL CALCULATOR. of cubic feet of space, will give the excess for every hundred cubic feet of the whole displacement. Displacement in Excess in Hundreds of Cubic Cubic Feet Feet of Space. of Space. 9'7949 ) 16'09000 (1'6 = Ratio of the excess of 97949 the after-body of dis629510 placement over the 587694 fore-body of the same, -41816 denoted by a per-centage of the whole disMETACENTRE. placement. A measure of the comparative stability of a ship, or the height of the metacentre above the centre of gravity of displacement estiY 3 dx mated, from the expression -3, D in which f is the sign of integration and signifies sum:y = the ordinates of the half-breadth load-water section. dx = the differential increment of the length of load-water section. D = displacement of the immersed portion of the body in cubic feet of space. Ordinates from the table. Cubes of the Ordinates..4 4.................4................. 00-064 3*0.. 27.000 50....................................125000 60.....................................216.000 63.....................250.047 = 6 4.. 226'981 54...................................1571464 3.7..................................... 50'653'4.................................... 0'064 Cubes placed in O'Neill's rule for summation of Area = (A. + 4P + 2Q) x 3 00'064 27'000 125'000 00'064 216'000 250'047 128 = A 226'981 157'464 50'653 532'511 = Q 520'634 = P 2 4 1065'022 = 2 Q 2082.536 = 4 P 1065.022 = 2 Q 000.128 = A 3147'686 = A + 4P + 2 Q 5'5 = r' 15738430 15738430 3) 17312'2730 r_ fy3dx = 57707576= A + 4P + 2 Q x Page 483 SHIP-BUILDING AND NAVAL ARCHITECTURE. 48") summation of the cubes of the ordinates of the load-water section; and the height of the metacentre above the centre of gravity of displacement is expressed by 2 f 8'D, in which expression y3 dx = 5770.75 5770'75 and D = 979'1 (p. 477) whence:- x 9791 - 3'98 feet is the height of the metacentre above the centre of gravity of the displacement. RESULTS OF THE CALCULATIONS. 1st Jlietltod. Displacement in cubic feet of space = 979'149. Displacement in tons of 35 cubic _ 27'974. feet of water to a ton............... Area of midship section............... = 41'08 superficial feet. Area of load-water line or plane at 40112 superficial feet. the proposed deepest immersion.. Tons to one inch of immersion at _' 955 tons. that flotation.......................... Longitudinal distance of the centre) of gravity of displacement from = 22'22 feet. section 1. J Depth of the centre of gravity of) displacement below the load-water - = 1'4904 feet. section............................... Distance of the centre of gravity of) the load-water section from verti- -= 22'5 feet. cal section 1.......................... Relative capacity of the after-body in excess of the fore-body in cubic'= 16'09 feet of space.......................... Per-centage on the whole displace- = 1.06. ment................................... Height of the metacentre above the) centre of gravity of displacement, I estimated from the expression -= 3-98 feet. y3 dx 2yf. D 9 The young naval architect has thus been led through the essential calculations on the immersed portion of a ship considered as a floating body. The term essential has here been used under a, knowledge that the table of results might have been swollen to; small volume by a lengthened comparison of the elements of the~ naval construction, such as the ratio of the area of the midship sec-,ion to the area of the load-water section, and that of the area of the midship section to the circumscribing parallelogram; data that will always suggest themselves to the mind, and furnish salutary exercise for his judgment, while the introduction of such comparisons into these rudiments might deter the novice from entering Page 484 FORE-BODY. ~. O0 MQoments t Suma for Cmentres Func- Moment for Cubes of Summation for Centre tions of C Centre of the Ordi- of the of Gravity the Areas Gravity of notes of Cubes for the of Displace- for theO O O Load-water Load-water Value of ment. Solid.. Ca c. OZ Line. Section. fy 3 dx. 000I 00 0 0090 0'10 0'12 0.15 0-17 0'20 0 000'00 0064 | 00032 # _ _ O _l 14 6 O O~ Oa O O 20-40 1 20-4 2 -0.80 200'40 4.80 100 1 6-00 27-000 2 64 000 3830 2 1915 1 1 30 220 3-20 420 500 2 10'00 125000 1 125000 I 156 00 1 3 5200 1 2 _ 14'00 6'40' 880 11' 20 12'00 3 36'00 216-000 2 432'000 1 1| C CO H Solid by Ver-, I tical Areas ) 106 52 I 1- -- I0c~oI ca at I IG D co I I | o:| 9 | | C' | |........ t0C~C Page 485 AFTER-BODY. Moment for Fu - c — C. Mome.t for Cubes 0Ummation Momeootsfor F - -- of Ofthe oft Centre of tis of 0 entre of of the of the Grvity.of tb.Areas Grvt o Ordinates of Cubts for/ Grl~t fteBes "t ~ n(ravity of the'a~lue of frthe 7Are'IMitl Displace- L o ter Load-wter' Diepi fdl for the 0Lio. Slttidl. meit. Zjo ao in.,cton f 3 dx. 53't28O 14li.7 1' 1-20 d 190 2-50 2'80 3-15 4 12-60 250'047 1 55-28 4 14-07 /2 1 __90 __250__047_ 1~~ ~3 dl ~ 01 4, oF doc ~6~.5>o 52.9o1 264-50 5 52-90 2 4.00 6-80 9-20.o11-00 12-20 5 6100 226981 2 453962 to r3 iPn, I II tt30 4~J1 dl ~ —-- AA 122-40 6 20-40 1 1-40 2-40 3-40 440 540 6 32-40 157464 1 1577d20 dl -9 2 d 157-50 7 225~0 2~ 1-2~0 I 2320 40 4'520 7-40 77 1180 50653 2 191-40603 220 573'843 8d8o86 Function) L~D r; 5 1'80 500'65 ~32 Solid.tCj C C5, iS A r f92 feet. rf-~~~~~~~~~~~~~~~~~~~~~~~34 5'20 feet. -4 tSoiid. w~~~~~~~~~0 1o 24 4~ cl dl o dl dl dl 7-20~ ~ ~ ~ 8 00 ~ 0-1 0-201 -702 -0 0-6. 0-032 -20I -9 87886 Function dl 2240~~~~~'100'1 1'5734 IL. l.j~~- ( dl CP~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~9 C1D' 00- 6 3 ] _____ l____o O____c dl tO~~~~~~ t~~~~Q ~ ~ ~ 240 1573'843 878'86 Funel ion) Page 486 486 THE PRACTICAL MODEL CALCULATOR. on a task that would thence seem to be involved in such voluminous results. For the second method of calculation, the table of ordinates is in two portions, viz. the fore and after-bodies under the division of the load-water section into two equal parts, the length of such section being restricted to the distance from the fore-edge o)f the rabbet of the stem to the after-edge of the rabbet of the post. The enlarged tables are shown at pages 484 and 485, and the directions for the working of these tables have been given at page 459, observing only that the ordinates have not been herein inserted in red, as it was there suggested, to insure perspicuity and accuracy. RESULTS FRO3I THE TABLES. By modified rule. Area = A + 2P + Q 2r y I 3 And solid = areas for ordinates } P'Q' 2 r+ summed by rule 2 3 Functions of the areas marked B = { +2 P + Q } Function of the solid equal to B, placed in O'Neill's rules A' + 2 P' + Q' = E 2r 2 r' Whence displacement = E x - x -, in the example r- = 92 r? = 5'5. 2 r 2 r' 1'84 11 Therefore displacement E x x E x x - 2UI~rlLYVVIII~L~V UI~ 3 3 3 3 20 24 Ex. VALUF OF E FROM THE TABLES BY VERTICAL SECTIONS. Table 1...106'50 = submultiple of the fore-body by vertical sections. Table 2....11077 = " after-body " " 217'27 =-sum of the submultiples = E. 20'24 217'27 x 20'24 1 displacement =E x 2024 217- 4 = 414 x 20'24 - V9 - 9 488'5936 = ~ solid of displacement by the summation of the 2 vertical areas given in cubic feet of space. 5)977-1872 7 ) 195'4374 27'92 = Displacement by vertical sections in tons of 35 cubic feet of space. VALUE OF E FROM THE TABLES BY HORIZONTAL SECTIONS. Table 1...106'50 = submultiple of the fore-body by horizontal sections. Table 2...110'75 = submultiple of the after-body by horizontal sections. From whence the same results will be obtained. Page 487 SHIP-BUILDING AND NAVAL ARCHITECTURE. 487 AREA OF MIDSHIP SECTION. From table 1...28'15 = submultiple of the area of Section 5. 1'84 -= 2r 11260 22520 2015 3) 517960 17.265 = - area of upper space of midship section. 3'275 = I area of the lower " " below d d, 20'540 = area of midship section. 2 41'08 = area of midship section. AREA OF THE LOAD-WATER LINE. From table 1...26'35 = submultiple of the area of the fore-body. From table 2...28'35 = " after-body. 54'70 = submultiple for 1 area of load-water line. 11 = 2r' 3) 601'7 A 2r' 200'56 = area- + 2P + Q x 2 12) 401'12 = area of load-water line. 5) 33.42 7)6 684 ~955 = tons per inch of immersion at the loadwater line. POSITION OF THE CENTRE OF GRAVITY OF DISPLACEMENT. By table 2...878'86 = moments from Section 1. and E........217'27 = corresponding function of the displacement. 217127) 878'86 ( 404 intervals of 5'5 feet, giving 4'04 x 869'08 5'5 = 22'22 feet as the distance of the centre of gravity of the die97800 placement from Section 1. 86908 10892 Page 488 488 THE PRACTICAL MODEL CALCULATOR. DEPTH OF THE CENTRE OF GRAVITY OF THE DISPLACEMENT BELOW THE LOAD-WATER SECTION. By table 2...353'72 = moments from load-water line. and E........217'25 = corresponding function of the displacement. 217'25) 353'72 (1-62 intervals of'92 feet, giving 1'62 x 217'25' 92 - 1'4904 as the distance that 136-470 the centre of gravity of displace130-350 lment is below the load-water line. 61200 43450 17750 POSITION OF THE CENTRE OF GRAVITY OF THE LOAD-WATER LINE OF DEEPEST IMMERSION. From table 1.......26'35 ft. From table 2...224-000 = moments c" 2.......28-35 from 1st section. Function for area..54'7 ) 224'0 (4-09 intervals of 5'5 feet, giving 218'8 4-09 x 5-5 = 22-495 feet 5200 as the distance that the 4923 centre of gravity of the load-water section is from *277 vertical section 1. RELATIVE CAPACITIES OF THE CALCULATED IMMERSED BODIES CONTAINED UNDER THE FORE AND AFTER-LENGTHS OF EQUAL DIVISION OF THE LOAD-WATER LINE. Feet. From table 1...Function for the fore-solid......106'50 From table 2...Function for the after-solid...... 11075 4-25 Sum of the functions......217'25 The difference, 4'25 feet, expresses the excess in cubic feet of space of the body, displaced under the after half-length of the load-water line, over that under the fore half-length of the same line, and the sum of the functions, 217-25, is equal to 2'1725 hundreds of cubic feet of space; whence, 4'25 feet, or the difference between the functions for the two bodies, being divided by the function 2'1725, or the function for the displacement of the calculated body expressed in terms of hundreds of cubic feet of space, will give the excess for every hundred cubic feet of that displacement: Function of Excess in Displace- Cubic Feet ment. of Space. 2-1725 ) 4-25000 ( 19 ratio of the excess of the after2-1725 body of calculation over the 207750 fore-body of'the same, de195525 noted by a per-centage of the -- 1222displacement calculated by ~12225 the table of ordinates. Page 489 SHIP-BUILDING AND NAVAL ARCHITECTURE. 489 HEIGHT OF THE METACENTRE ABOVE THE CENTRE OF GRA7ITY OF DISPLACEMENT. From table 2...The summation of the functions) of the cubes of the ordinates for the value of -= 1573'843. the Sf 3 dx........................................... The corresponding function for the solid.........= 217'25. from whence the height of the metacentre above the centre of 2 y3 dx gravity of displacement, expressed by 3f D is as follows: 2 r' fy3 dx = 1573'843 x 3 where r' = 5'5 feet = 1573.843 x 11 17312.273 3 =:. = 5770.75 feet., 3 2r 2 r' (Page 485) 21727 x - x - = displacement = 488'5936 feet, whence displacement or D = 977'1872; and thence 2 y3dx 2 5770'75 11541.53 f yD -= 3 977'1872 2931-5616 = 3'98 feet. RESULTS OBTAINED UNDER THE TWO METHODS OF CALCULATION CONTRASTED. Old Method. Second Method. Displacement in cubic feet of space... 979'139 977'187 Displacement in tons of 35 cubic feet of water to a ton...................... 27'985 27'92 Superficial ft. Superficial ft Area of midship section................. 41'08 41'08 Area of load-water line or plane at the proposed deepest immersion..... 401'12 401'12 Tons to one inch of immersion at line of flotation........,................. 9526 tons.'955 tons. Longitudinal distance of the centre of gravity of the displacement from section 1................................ 22'22 ft. 22'22 ft. Depth of the centre of gravity of displacement below the load-water section....................................... 14812 ft. 1'4904 ft. Relative capacities of the bodies....... 1'6 per cent. 1'9 per ct. Height of the metacentre above the centre of gravity of displacement... 3'98 ft. 3'98 ft. THIRD METHOD OF CALCULATION. CALCULATIONS ON THE DRAUGHT OF THE YACHT OF 36 TONS USING THE CURVE OF SECTIONAL AREAS. The load-water line AB, in the sheer plan, is divided into two equal parts at the point C, and those equal parts are again subdivided at the points D and E; at the points C, D, and E, Page 490 490 THE PRACTICAL MODEL CALCULATOR. I q I-.~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~I F - 0 I I eq cq cq Ca~~~~~~~~~~e o,~~~~~~~~~~c It req i I I Ii ( t — I i 0 r 3 wl=41 C - FG = 44 I 2-7 PK I= 2-45 EO 4-2 FI = 22 FQ = 22-37 99 Q ~~~~~~~~I I II 0 P.o to (O~~~t R =24ee. DN = 5- et A 4fetiG=2 et QI 4 " I -_= PK =25 E02 " FL = ~I EI -r~~~~~~~ F~~~~if i/5 i Q ~~R i:,'Y II I X~:: ~ ~ ~ ~ ~ ~ ~; i\ I u~~~~~~~~~~~~~ I I I O~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ i; /k Ordincates. RtI _ — 2.4 feet. I)N — 5.8 feet. AB -- 44 feet. IOf -- 22 feet. QI -—.4.,, CM —5.0,, FG — 44',E3 2~7~ PK ~2.45,, E0 ~_4'2 "' FI _- 22 " rQ —-- 22.37,, Page 491 SHIP-BUILDING AND NAVAL ARCHITECTURE. 491 thus obtained, the transverse vertical sections of the vessel are delineated. The length of the load-water line from the fore edge of the rabbet of the stem B, to the after edge of the rabbet of the post A, is next drawn below and parallel to the base line SF of the sheer plan; this line, FG, becomes the base line of the curve of the sectional areas. The common sections of the transverse vertical sections of C, D, and E, (which will be straight lines,) with this horizontal and longitudinal plan, are drawn from their respective points of division, H, I, and K, in half-breadth plan. The areas of these transverse vertical sections at D, C, and E, are then calculated, as before, thus:Area A +4P + 2Q x = 2P + Q x;or, Area ={A +2P+3Q 8r{ 2 +P+1.56Qx x r. Half Area of Transverse Vertical Section, at C, by Rule 1, A 2r or, Area= 2 + 2P + Q x 3 1st....6'3 2d...6'0 3d...4'8 = Q Last...'2 4th... 23 2)6.5 8'3 = P 3-25= A 2 16'60 = 2 P A 3'25 = 4.80 = Q 24'65 = +2P+ Q 2r *83= 7395 19720 A 2r 20'4595 = + 2 P +Q x 3 = area of section C in feet. CM 5'0 CM, or depth = 5'0 feet, whence —, or -- = 1 2- = r 2r 2x1- 25 2-5 distance between the ordinates, and -=- = - - 83 feet. Page 492 492 THE PRACTICAL MODEL CALCULATOR. Half Area of Section C, by Rule 2, or, 2 area + P 15 Q x 4 r. 1st....6'3 P=0 5'6 2d. Last...'2 3'05 3d. 2)6.5'A 8.65 = Q. 25=432 = 32 Q. 3.25 =. ~12.97 = 1.5 Q. 3'25} A 12-97 =-2 + P + 15 Q 16 22 5 = 3r = CM = 5'0 feet. 4)8110 3 A 3 20'275 = —area- P +P+1.5Q x4r Half Area of the Transverse Vertical Section at E. 1st....5'0 2d....4'2 3d....2'9 = Q Last...'2 4th....1'7 2) 52 59 = P - A 2 2-6 2 A 2 118 = 2 P A 2.6 2 2.9 = Q A 17.3 =I + 2P + Q EO 4'2 EO, or depth = 4'2 feet, whence 4 - 4 105 = r = dis2r 1'05 x 2 2.1 tance between the ordinates, and 3 = 3- = 3' 7feet; therefore, Area ~ + 2 P + Q x 3 = 17'3 x'7 = 12'11 = half area of transverse vertical section at E. Half Area of the Transverse Vertical Section at D. 1st....5'40 2d....3'5 3d....1'46 = Q Last...9'2 4th....0 7 2) 5.6 42 = P A 2 2 8.4 2P A 2 1.46 = Q 1266 + 2P + Q Page 493 SHIP-BUILDING AND NAVAL ARCIHITECTURE. 493 DN 5-8 DN, or depth = 5'8 feet, whence = = 145 feet = 2r 2 x 1-45 2'9 r = distance between the ordinates, and 3 -- ~97 feet; therefore, Area = { 2 + P + Q } x = 12'66 x 97 =12'28 feethalf area of transverse vertical section at D. Half Areas of the Transverse Vertical Sections. Feet. Feet. (E =12 11) Divided by 5 as the depth assumed for (2 42 At C = 20'20 the zone, give the ordinates for the curve 4'04 D = 12'28) of sectional areas, as.......................... 245 of which 2'42 is set off from H as HR, 4'04 feet from I as IQ, and 2'45 feet from K as KP; the curve IRQPG, passing through the extremities P, Q, and R of the ordinates PKI, QI, and RHI, is the curve bounding the area of a zone, which, to the depth of 5 feet for a solid, will give in cubic feet of space the half displacement of the immersed body, or the displacement of the yacht to the line AB of proposed deepest immersion. To measure this representative area, and from thence the solid, join the points Q, G, and I by the straight lines QG, QF, dividing the curvilinear area FRQPGF into the two triangles QGI, QFI, and the two areas GPQG, FRQF. The triangles by construction are equal, and the area of each one of them is equivalent to GI x QI GI x QI 2, or the whole area GQFIG = 2 x 2 = GI x QI or FI x IQ, FI being equal to IG, each being the half-length of the same element, the load-water line or line of deepest immersion. The areas QPGQ, QRFQ, are bounded by the curve lines QPG, QRF, which are assumed as portions of common parabolas, and under such an assumption their respective areas are equal to 2 of the circumscribing parallelograms, or the area QPGQ = 2 of GQ x x, and the area FRQF = 3- of FQ x x', where x and x' are the greatest perpendiculars that can be drawn from the bases QG and QF to meet the curves QPG, QRF. DISPLACEMENT. AB by a scale of parts = 44 feet, whence FI or IG equal AB 44 2 = — 2 feet = 22 feet; ordinate QI of the medial section = 4'04 feet; and QG = FQ, being the respective hypothenuses of the equal triangles QGI, QFI, are each equal to V/IG2 + Q12= V222 + 4'042 = V484 + 16'32 = v/500'32 = 22'37 feet; and x, by measurement with a scale of parts, ='6 foot, and x' also ~6 foot, from which the half displacement in cubic feet of space will be obtained as follows: — Page 494 494 THE PRACTICAL MODEL CALCULATOR. Area FQGIF = GI X IQ. Cubic fect. Solid under the GI x IQ x 5 = 22 x 41 x 5 = 45100 area FQGIF -G x 4 Area QPGQ = 8 of GQ x x Solid under the 2 44'74 Solid under the =-of GQxx x 5=-2 x 2237x 6x5= 44 x4 area QPGQ Area FRQF = 2 of FQ x x' Solid under the }= 2 of Q xx' x5= x =2237 x6 x5 44'74 area FRQF 540'48 or area of the triangle QGI + area of the triangle QFI + area of the space QPGQ + area of the space FRQF = to the representative area FRQPG, which being multiplied by the assumed depth of 5 feet for the zone of half displacement gives 540.48 cubic feet of space, which divided by 35, as the number of such cubic feet that are equivalent to one ton of medium water, gives 3)540.48 7) 108o09 15'44 tons for half displacement, and that the whole weight of the body is equal to 15'54 x 2 = 30'88 tons = displacement to the line of proposed deepest immersion AB. RELATIVE CAPACITIES or THE BODIES IMMERSED UNDER THE FORE AND AFTER HALF-LENGTHS OF THE LOAD-WATER LINE) AS GIVEN BY THE DELINEATED CURVE OF SECTIONAL AREAS. The triangles QGI and QFI being equal, the relative capacities of the fore and after-bodies will be determined by the proportion that the area QPGI bears to the area QRFI; and as these areas involve two equal terms, or that the base FQ = the base QG, it follows, that the relation of these areas to each other will be expressed by the proportion that the perpendiculars x and x' bear to each other. In the example given, the fore and after-bodies, or the displacements under the fore and after half-lengths of the loadwater AB, are equal; as the perpendiculars x and x' taken from the diagram, on a scale of equal parts, are each *6 of a foot. The area of the midship section is denoted relatively by the medial ordinate of the curve of sections QI, and the full amount of it is obtained by multiplying the function QI by the depth of the zone M. In the example: M = 5; QI = 4-04; then half area of medial section = 4'04 x 5 5 Area of midship section......20'20 Page 1 TABLES OF LOGARITHMS. Page 2 Page 3 LOGARITHMS OF NUMBERS. 3 No. Prop. No. Prop. Part Prop. < | < Part. No L Part. PartLo, r N. Part. 1000 000000 1060 025306 1120 049218 1180 071882 1 000434 43 1 025715 41 1 049606 39 11072250 37 2 000868 86 2 026124 82 2 049993 77 2 072617 73 3 001301 130 3 026533 122 3 050380 116 3 072985 110 4 001734 173 4 026942 163 4 050766 154 4 078352 147 5 002166 216 5 027350 204 5 051152 193 5 073718 183 6 002598 259 6 027757 245 6 051538 232 61 074085 2'20 7 003029 303 7 028164 286 7 051924 270 7 074451 256 8 003460 346 8 028571 326 8 052309 309 8 074816 293! 9 003891 389 9 028978 367 9 052694 347 9 075182 330 1010 004321. 1070 029384 1130 053078[ 1190 075547 1 004751 43 1 029789 40 1 053463 38 I 675912 36 2 005180 86 2 030195 81 2 053846 77 2 076276 73' 3 005609 128 3 030600 121 3 054230 115 3 076640 109 4 006038 171 4 031004 162 4 054613 153 4 077004 145 5 006466 214 5 031408 202 5 054996 191 5 077368 181 6 006894 257 6 031812 242 6 055378 230 6 077731 218 7 007321 300 7 032216 283 7 055760 268 7 078094 254 8 007748 343' 8 032619 323 8 056142 306 8 078457 290 9 008174 385 - 9 033021 364 9 056524 345 9 078819 327 1020 008600 I 1080 033424 1140 056905 1200 079181 1 009026 42 1 033826 40 1 057286 38 1 079543 36 2 009451 85 2 034227 80 2 057666 76 2 079904 72 3 009876 1'27 3 034628 120 3 058046 114 3 080266 108 4 010300 170 4 035029 160 4 058426 1521 4 080626 144 5 010724 212 5 035430 200 5 058805 1901 5 080987 180 6 011147 254l6l 6 035830 240 6 059185 228 6 081347 216 7 011570 297 7 036229 280 7 059563 266 7 081707 252 8 011993 339 8 036629 321 8 059942 304 8 082067 288 9 012415 3821 9 037028 361 9 060320 342 9 082426 324 1030 012837 1090 037426 1150 060698 1210 082785 1 013259 42 1. 037825/ 40 1 061075 38 1 083144 36 2 013680 84 2 038223 79 2 061452 75 2 083503 71 3 014100 126 l 3 038620 119 3 061829 113 3 083861 107 4 014520 168 4 039017 159 4 062206 160 4 084219 143 5 014940 210'5 039414 198 5 062582 188 5 084576 179 6 015360 252 6 039811 238 6 062958 226. -6 084934 214 7 015779 294 7 040207 278 7 063333 263 7 085291 250 8 016197 336 8 040602 318 8 063709 301 8 085647 286 9 016615 378. 9 040998 357 9 064083 338 9 086004 322 1040 017033 1100 041393 1160 064458 1220 086360 1 1 017451 42 l 1 041787 39 1 064832 37 1 086716 35 2 1 017868 83 1 2 042182 79 2 065206 75 2 087071 71 3 1018284 125 3 042575 118 3 065580 112 3 087426 106 4 018700 166i 4 042969 157 4 065953 149i 4 087781 142 5 019116 2081 5 014362 196 5 066326 186i 5 088136 177 6 019532 250 6 043755 236 6 066699 224 6 088490 213 7 019947 291 7 044148 1275 7 067071 261 7 088845 248 8 020361 333 8 014540 18I14 8 067443 298 8 089198 284 9 020775 374 9 044931 354 9 067814 1336 9 089552 319 1050 021189 1 111110 0453231 11 17 0 068186 1 1230 089905 1 021603 41 1 045714 39 1 068557 37 1 090258 35 2 022016 82 2 046105 78 2 068928 74 2 090611 70 3 022428 124 3 016495 117 31 069298 111 3 090963 106 1 4 02)841 1651 4 046885 156 4 069S668 1148 4 091315 141 5 023252 206 5 0i7275 195 5 070038 0 1801 5 091667 1176 61 023664 247 6 047664 234 6 070407 2 2 2 6 0920181211 7 024075 288 7 048053 273 7 070776 259 7 092370 246 8 024486 330 8 048442 312 8 071145 296 8 092721 282 9 024896 371 9 048830 351 9 071514 333 9 093071 317 32 Page 4 4 LOGARITHMS OF NUMBERS. No. Log. Prop. o. Log. Pro. No. Log. Prop. No.. L rog p.r. Part. Pat. g Part. Part. 1240 093422 1300 113943 1360 133539 1420 152288 1 093772 35 1 114277 33 1 1.33858 32 1 152594 30 2 094122 70 2 114611 67 2 134177 64 2 152900 61 3 094471 105 3 114944 100 3 134496 96 3 153205 91 4 094820 140 4 115278 133 4 134814 127 4 153510 122 5 095169 175 5 115610 167 5 135133 159 5 153815 152 6 095518 210 6 115943 200' 6 135451 191 6 154119 183 7 095866 245 7 116276 233 7 135768 2231 7 154424 213 8 096215 280 8 116608 267 8 136086 255' 8 154728 244 9 096562 315 9 116940 300! 9 136403 2871 9 155032 274 1250 096910 1310 117271 1370 136721 1430 155336 1 097257 35 1 117603 33 1 137037 321 1 155640 30 2 097604 69 2 117934 66 2 137354 63 2 155943 60 3 097951 104 3 118265 99 3 137670 94 3 156246 91 4 098297 138 4 118595 1321 4 137987 126 4 156549 121 5 098644 173 5 118926 165!j 5 138303 158 5 156852 151 6 098990 208 6 119256 198 6 138618 189 6 157154 181 7 099335 242 7 119586 231 7 138934 221 7 157457 211 8 09'3681 2 {77 8 119915 264! 8 139249 252 8 157759 242 9 100026 311 9 120245 297 9 139564 284 9 158061 272 1260 100370 1320 120574 1380 139879 1440 158362 1 100715 34 1 120903 33 1 140194 31 1 158664 30 2 101059 69 2 121231 66 2 1.40508 63 2 158965 60 3 101403 103 3 121560 98I 3 140822 94 3 159266 90 4 101747 137 4 121888 131 4 141136 125 4 159567 120 5 102090 172 5 122216 164 5 141450 1571 5 159868 150 6 102434 206 6 122543 197 6 141763 188 6 160168 180 7 102777 240 7 122871 2301 7 142076 219 7 160468 210 8 103119 275 8 123198 262| 8 142389 251 8 160769 240 9 103462 309 9 123525 295 9 142702 282 9 161068 2770 1270 103804 1330 123852 1390 143015 1450 161368 1 104146 34 1 124178 33 1 143327 31! 1 161667 30 2 104487 68 2 124504 65 2 143639 62 2 161967 60 3 104828 102 3 124830 98 3 143951 93 3 162266 89 4 105169 136 4 125156 130 4 144263 12.5 4 162564 119 5 105510 170 5 125481 163 5 144574 156 5 162863 149 6 105851 204 6 125806 195 6 144885 187 6 163161 179 7 106191 238 7 126131 228 7 145196 218 7 163460 209 8 106531 272 8 126456 260 8 145507 249 8 163757 239 9 106870 306 9 126781 2(03 9 145818 280 0 9 164055 269 1280 107210 1340 127105 1400 146128 1460 164353 1 107549 34 1 127429 32 1 146438 31 1 164650 30 2 107888 67 2 127752 6.5 2 146748 62 2 164947 59 3 108227 101 3 128076 971 3 147058 93 3 165244 89 4 108565 135 4 128399 129 4 147367 1241 4 165541 119 5 108903 169 5 128722 161 5 147676 155 5 165838 148 6 109241 203 6 129045 194 6 147985 186 6 166134 178 7 109578 237 7 129368 226 7 148294 217 7 166430 207 8 109916 270 8 129690 2.58 8 148603 248 8 166726 237 9 110253 304 9 130012 291 9 148911 279i 9 167022 267 1290 110590 13850 130334 1410 149219 1470 167317 1 110926 34 1 130655 32 1 149527 31 1 167613 29 2 111262 67 2 130977 64 2 149835 61! 2 167908 59 3 111598 101 3 131298 96 3 150142 921 31168203 88 4 111934 134 4 131619 128 4 150449 123 41168497 118 5 112270 168 5 131939 160 5 150756 154 5 168792 147 6 112605 201 6 132260 192 6 151063 18-1 6 169086 177 7 112940 235, 7 132580 224 7 151370 18 7 169380 206 8 113275 268' 8 132900 2561 8 151676 24611 8 169674 236 9 113609 3021 9 1332191 288 11 9 151982 2977j 9 169968 265 Page 5 LOGARITHMS OF NUMBERS. 5 No. Log. Part. Log. No. Log. PPart. 1480 170262 1540 187521 1600 204120 1660 220108 1 170555 29 1 187803 28 1 204391 27 1 220370 26 2 170848 58 2 188084 56 2 204662 54 2 220631 52 3 171141 88 3 188366 84 3 204933 81 83 220892 78 4 171434 117 4 188647 113I 4 205204 108 4 221153 104 5 171726 146 5 188928 141 5 205475 133 5 121414 130 6 172019 175 6 189209 169 6 205745 1, 6 221675 157 7 172311 204. 7 189490 197 7 206016 189 7 221936 183 8 172603 234 8 189771 225 8 206286 216 8 222196 209 9 172895 263 9 190051 253 9 206556 243 9 222456 235 1490 173186 1550 190332 1610 206826 1670 222716 1 173478 29 1 190612 28 1 207095 27 1 222976 26 2 173769 58 2 190892 56 2 207365 54 2 223236 52 3 174060 87 3 191171 84 3 207634 81 3 223496 78 4 174351 116 4 191451 112 4 207903 108 4 223755 104 5 174641 145 5 191730 140 5 208172 135 5 224015 130 6 174932 175 6 192010 168 6 208441 162 6 224274 156 7 175222 204 7 192289 196 7 208710 188 7 224533 182 8 175512 233. 8 192567 224 8 208978 215 8 224792 208 9 175802 261 9 192846 252 9 209217 241 9 225001 231 1500 176091 1560 193125 1620 20951-5 1680 225309 1 176381 29 1 193403 28 1 209783 27 1 225568 26 2 176670 58 2 193681 56 2 210051 54 2 225826 52 3 176959 86 3 1933959 83 3 210318 80 3 226084 77 4 177248 115 4 194237 111 4 210586 107 42260342 103 5 177536 144 5 194514 139 5j 210853 134 5 226600 129 | 6 177825 173 6 194792 166f 6 211120 161 6 226858 155 7 178113 202 7 195069 194 71211388 187 7 227115 181 8 178401 231 8 195346 222 8 211654 214 8 227372 206 9 178689 2593 9 195623 2,50 9 211921 240 9 227630 232 1510 178977 178977 1570 195900 16 0 212188 1690 227887 1 179264 291 1 196176 27 1 212464 27 1 228144 26 2 179552 57 1 2196452 55 2 212720 53 2 228400 51 3 179839 86 3! 196729 83 3 212986i 80 3 228657 77 4'180126 11 1 4 197005 110 4 215 9 106 4 228913 102 5 180413 144 5 197281 138 5 213G18133 5 229170 128 6 180699 1172 6 197556 1661 6 2131783 l 1 59 6'94'26 154 7 180986 201 197832 191 7 2 186 7 201 19968 179 8 181272 1230 1 8 198107 221 8 214314 12 8 229938 205 9 181558 258 9 198382 248 1 9 2145 9 1391 9 23 0193 231 1520 181844 11580 198657 1640 214844 1700 230449 1 182129 28 1 19893'2 27 1 215109 26 1 230704 2 2s -844 01 7 2199 12 215373 53 230960 51 3 182700 8611 3 199481 8 215G68, 79l 3 231215 76 4182985 114 4 199755 110l 4 215902 106 4 -1470 1 02 5 183270 143 5 200029 1371 5 216166 11321 5 231724 127 6 183554 1171 6 2000303 1641 6 216430 158l 6 231979 153 7 183839 200 7 200577 192 7 216694 185 7 232233 1,Th 8 184123 228 8 200850 219 1 8 216957 211 8 232488 204 9 184407 256 91201124 247 9 217221 238 9 232742 229 1530 184691 1590 201397 1650 217484 1710 232996 1 184975 28 1 201670 27 1 217747 26 1 233250 25 2 185259 57 2 201943 54 2 218010 52 1 223504 51 31 185542 85 3 202216 82 3 218273 79 3 333'7 726 411858251113 41202488 109 l 4 218535 10.5, 4 234011 101 65 186108 142 51202761 136 ) 5 218798 1311 5 234264 127 6 186391 170 6 203033 163 6 219060 157 6 234517 152 71186674 198 7 203305 191 7 219322 1183 7 1234770 177 8 186956 227 8 203577 218 8 219584 2101 8 235023 202 9 187239 255 9 20384812451 9 219846 236 9 235276 228 Page 6 LOGARITHMS OF NUMBERS. No. Log. Part. No. Log. rNo. Log. Pro No. Prop. 1720 235528 / 1780 250420 1840 264818 1900 278754 1 235781 25 1 250664 24 1 265054 23 1 278982 23 2 236033 50 2 250908 49 2 265290! 47 2 279210 45 3 236285 76 3 251151 7 83 3 26.55251 70 3 279439 68 4 236537 101 4 251395 97 4 2675761 94 4 279667 91 5 236789 126 5 251638 121 5 265996 117 5 279895 114 6 237041 1 5 6 251881 146 6 266232 141 6 280123 137 7 237292 1 176 7 266467 164 7 280351 160 8 237544 202 8 252367 195 8 266702 188 8 280578 182 9 237795 227 9 252t610 219 9 266937 211 9 280806 205 1730 238046 1790 252853 i850 267172 1910 281033 1 238297 25 1 253096 24 1 267406 23 1 281261 23 2 238548 50 2 253338 48 2 267641 47 2 281488 45 3 238799 75 3 2,5358 0 73 3 267875 70 3 281715 68 4 239049 100 4 253822S 97 4 268110 94 4 281942 91 5 289299 125 5 254064 121I 5 268344 117 5 282169 113 6 239550 1.50 6 254306 145 6 268578 141 6 282395 136 7 239800 175 7 254548 170 7 268812 164 7 282622 159 8 240050b 209 8 254790 194 8 269046 188 3 282849 181 9 240300 225 9 25.5031 218 9 269279 211 9 283075 204 1740 240549 1800 255273 1860 269513 1920 283301 1 240799 25 1 255514 24 1 1 269746 23 1 283527 23 2 241048 50 2 25575.5 48 2 2 69980 47 2 283753 45 3 241297 75 3 1 255996 72 3 270213 70 3 283979 68 4 241546 100 4 256236 96 4 270446 93 4 284205 90 5 241795 124 5 256477 120 5 270679 116 5 284431 113 6 242044 149 6 256718 144 6 270912 140 6 284656 135 7 242293 174 T 2.56958 168 7 271144 163 7i 284882 158 8 2492541 199 8 257198 192 8 271377 186 8 285107 180 9 242790 223 9 257439 216 9 271609 210 9 8533 2 03 1750 243038 1810 257679 1870 271842 1930 285557 1 |243286t; 25 1 2.57918 24 I2 1 272 97 4 23 1 285782 22 2 243534 50 2 258158 48 2 272306 46 2 286007 45 3 243782 74 3 258398 72 3 272538 70 3 286232 67 4 244030 99 4 2.58637 9t6 4 272776 93 4 286456 89 5 244277 124 5 258877 120 5 273001 116 5 286681 112 6 244524 149 6 259116 144 6 273233 139 6 286905. 137 244772 174 7 259355 167 7 273464 162 7 287130 157 8 243019 198 8 259594 192 8 273696 1861 81287354 179 9 245266 222 9 259833 215 9 273927 209 9 287578 202 176012455613 1820 260071 I1880 274158 1 940 28 7802, 1 245759 25 1 260310 24 1 1274389 23 1 288025 22 2 246006 49 1 260548 48 2 274620 46 2 288249 40 3 2462521 74 3 260787 71 3 274850 69 3 288473 67 4 246499 98 4 261025 95 4 |275081 92 4 288696 89 5 246745 123 5 261263 119 5 275311 115 5 288920 112 6 246991 148 6 261501 143 6 275542 138 6 289143 134 7 247236 173 7 261738 167 7 ]275772 161 7 289366 156 8 247482 197 8 261976 1]91 8 276002 184 8 289589 178 9 247728 221 9 262214 214 *9 276232 207 9 289812 201 1770 247973 1830 262451 1890 276462 1950 1290035 1 248219 25 1 2622688 24 1 276691 23 1 290257 22 2 248461 49 2 262925 47 2 276921 46 i 2 290480 44 3 248709 74 3 2631162 171 3 277151 69 3 2907027 67 4 948954 98 4 263399 95 4'277380 9 11 4 290925 89 5 249198 123 5 263636 118 5 1277609 115 5 291147 111 6 249443 147 6 263873 142 6 [277838 138 6 291369 133 7 249687 172 7 261109 166 7 278067 161 1 7 291591 156 8 249932 196 8 264345 190 8 278296 183 8 291813 178 9 250176 220 9 264582 213' 9 278525 206 9 292034 200 Page 7 LOGARITHMS OF NUMBERS. 7 No. Log. Prop. Lo. Prop No. Log. Pro Part. Pao. Loo.gr. NPart. 1960 292256 2020 305351 2 080 1 318063 2140 330414 1 232478 22 1 305566 21 1 1 18'72 21 1 330617 20 2 292699 44 2 305781 43 2 318481 42 2 301 0 40 3 2929"30 66 3 305996 64 3 318689 63 3 I31022 61 4 293141 88 4 306211 86 4 318898 83 4 331225 81 5 293363 110 5 306425 1071 5 319106 104 5 31427 101 6 293583 133 6 306639 12 9 6 319314 125 6 33ltI60 121 7 293804 155 7 3068654 150 7 0319522 146 7 881832 141 8 294025 177 8 037068 172 8 319730 167 8 332034 162 9 294246 199 9 307282 13 9 319938 188 9 3322836 182 1970 294466 2039 307z496 2090 320146 2150 33I2438 1 294687 221 1 307710 21 1 320354 21 1 33264 0 20 2 294907 44 2 307924 43 2 320562 41 2 332842 40 3 295127 66 3 8 308137 64 1 3 3320769 62 3 333044 60 4 295347 88 4 308351 8 4 3 2 0977 83 4 333246 81 5 295567 110 5 30856- 107 3221184 104 5 3 3447 101 6 295787 13 02 6 30877 8 1t8 6 3213931 125 6 3335649 121 129600 154 708991 149 7 321698 145 7 3338501 141 8 296226 1761j 8 8309204 1716 8 321805 1 8 4 161 9 296446 191'8 91 3097417 19 92 9 i 322012 187 9 334253 181 1980 2966651 040 30960 I 2100 3822219 2160 334454 1 296884 22 1 009843 1 21 1 322426 21 1 i 3346550 20 2 197104 44 2 310056 43 2 226i3 41 2'34856j 40 3 297323 66 3 3 10 i8 64 8 322839 62 3 3133605 60 4 2972542 88 4 310481 85337 4133046 8 80 5 297761 109 5 310693 106 5 323252 103 5 o3.5458 10t 6 209979 131 6 310906 127 6 323458 1 112 6 335658 1'20 7i 2t83198 1653 7 311118 148 7 323665 144 335859 140 8 0298416 175 8 311330 170 8 323871 165 8 3360.59 160 9 98630 197 9 311542 191 9 324077 186 9 33 626 180 1990 298853 2050 311754 32110 324282 217 320 33460 1 299071 22 1 311966 21 1 324488 21 1 33S66iO6 20 2 299'3289 44 2 31177 4 324694 41 2 33686) 40 3 2995017 6 31 342'89 63 3 324899 6f2 3 397060 60 4 299725 87 4 312600 84 4 32 105 82 4 3326;3) 80 53 29943 109 5 312812 106 5 325310 1035 51 3945 9 1. 6 300160 131 6 31302 127 6 2551 11 3 6'33759 1020) 7 300378 103 7 313234 148 7 325721 144 1 337848 11 40 8 300595 174 8 313445 160 8 325926 164 8 338038 1560 9 300813 196 313656 1'DO 9 326131 118.5 9 382576 180 2000 301030 2060 313867 2120 326336 2180 338456 1 301247 22 1 314078 21 1 326541 20 1 338u660 0 2 301464 43 2 314289 42 2 32 67405 41 2 3388 5 40 3 301681 65 3 314499 63 8 326950 61 3 33D054 160 4 3018'38 87 4314710 84 4 327155 82 4 339925) 1 s80 5 302114 108 5 314920 105 6 327359 31024 5 3'34i1 lo 6 302331 130 6 315130 126 6 327.563 123 6 33939W0 1 119 7 302547 152 7 315340 147 7 327767 143 7 33984' 9 130 i9 8 302764 1173 8 315550 168 8 327972 164 8 340047 1 O31 9 302980 195 9 315760 189 9 328176 184 9 340-16 179 2010 303196 2070 315970 2130 3283801 2190 i340444 1 303412 22 1 316180 91 1 328583 20 1 3406-42 20 2 303628 43 2 316r390 42 2 328787 41 2 340841 40 3 303844 65 3 316599 63 3 3289'391 61 3 341039 59, 4 304059 86 4 316809 84 4 329194 81 4 341237 79 5 304275 108 5 317018 10-5 5 329398 10 5 341435) 99 6 3044901129 6 3172271126 6 329i601 12 6 341632 119 7 304706 j 151 7 317436 147 7 329805 142 1 341830 139 8 304921 172 8 317645 168 8 3300908 1 163 1 8 342028 158 9 305136 194 9 317854 189 9 330211 183 9 342225 178 Page 8 8 LOGARITHMS OF NUMBERS. No. LObr.~ Prtp.jl No. Log. Prop. No. Lo, Prop. Prop. Part. Part. i Part. 1i PartP ar.. 2200 342423 2260 354108 2320 365488 1 2380 376577 1 342620 20 1 854301 19 1 365675 19 1 3767 59 18 2 342817 39 2 354493 38 2 3865862 37 2 376942 36 3 343014 59 3 354686 58 3 366049 56 3 377124 55 4 343212 79 4 354876 77 4 366236 75 4 377306 73 5 343409 99 i 5 355068 96 5 366423 93 5 377488 91 6 343606 118 6 355260 115 6 366610 112 6 37 7670 109 7 343802 138 j 7 355452 1341 7 366796 131 7 377852 127 8 343999 1581 8 355643 154 8 366983 150 8 378034 146 9 344196 178 9 355834 173 9 367169 168 9 378216 164 2210 344392 2270 356026 2330 367356 2390 378398 1 344589 20 1 356217 19 1 367542 19 1 378580 18 2 344785 39 2 356408 38' 2 367729 37 2 378761 36 3 344981 59 3 356599 57 3 367915 561 3 378943 55 4 345178 78 4 356790 76 4 368101 75 4 379124 73 5 345374 981 5 356981 95 5 368287 93 5 379306 91 6 345570 118 6 357172 115' 6 368473 112 6 379487 109 7 345766 137 7 357363 134 7 368659 130 7 379668 127 8 345962 157 8 357554 153 8 368844 149 8 379849 146 9 346157 176 9 3057 744 172 9 369030 167 9 380030 164 2229 0 346353 2280 357935 2340 369216 2400 380211 1 346549 19 1 358125 19 1 369401 19 1 380392 18 2 346744 39 2 358316 8 2 369587 37 2 380573 36 3 346939 58 3 358506 57 3 369772 56 3 380754 55 4 347135 78 4 358696 76 4 369958 74 4 3809341 73 5 347330 97 5 358886 95 5 370143 93 5 381115 91 6 34 7525 117 6 359076 114 6 370328 111 6 381296 109 7 t 347720 137 7 359266 1303 7 370513 130 i 7 381476 127 8 347915 156) 8 359456 152 8 370698 148 8 381656 145 9 348110 175 9 359646 171 9 370883 167 9 381837 163 2230 818305 2290 3.59835 2 3.50 3 71068 2410 382017 1 818500 19 1 360025 19 1 371253 18 1 382197 18 2 848694 391 2 360215 38 2 3871437 37' 2 382377 36 3 348889 58 3 360404 57 3 371622 55 3 382557 54 4 349083 78 4 360593 76i 4 371806 74 4 382737'72 5 3-49278 97 5 360783 95 5 371991 92' 5 382917 90 6 349472 117 6 0972 114 6 372175 111 6 383097 108 7 349666 137 7 361161 133 1 372360 129 1 383277 126 8 3-49860 156 8 361350 1.52 8 37 2544 148' 8 383456 144 9 3500.54 175 9 361539 171 9 372728 166 9 383636 1 62 2410 350248 12300 361728 2360 372912 1 24 0 383815 | 1 35044| 19 1 361917 19 1 373096 18 1 38399o 18 2)350636 39 2 36210.5 38 2 373280 37 1 2 384174 36 5 351216 97 51362671 94 5 373831 92 5 384712 90 6 3;51410 116 6 362859 113 6 374015 110 6 384891 108 7 351603 135 7 8363048 132 7 3741.98 129 7 385070 1 26 8 351796 1.55 8 3632'36 151 8 374382 147 8 385249 144 9 351989 174 9 363424 170 9 3744565 166 9 385428 162 2250 3852182 2310 363612 2370 374748 i 2430 385606 1 352375 19 1 363800 19 1 374932 18 i 1 385 785) 18 2 352568 38 2 363988 37 2 3751.15 37 2 385964 35 3 352761 58 3 364176 56 3 375298 55 3 386142 53 4 352954 77 4 364363 75 4 375481 73 4 386321 71 5 353147 96 5 364551 94 5 375664 92 5 386499 89 6 353339 115 6 364739 112 6 375846 110 6 386677 107 7 3563532 134 7 364926 131 7 376029 128 7 386856 125 8 353724 154 8 365113 150 8 376212 147 8 387034 143 9 353916 173. 9 365301 169 9 376394 165 9 387212 161 Page 9 LOGARITHMS OF NUMBERS. 9 No. Log. Prop. No. Log. Prop. No. Lg Prop No. Log Prop. PLog. art. Part. Part Part. 2440 387390 2500 397940 25600 408240 i 260 418301 1 387568 18 11398114 17 1 408410 17, 1 418467 17 2 387746 36 2 398287 3.5 2 408579 34!1 2 418633 33 3 387923 53 3 398461 53 3 408749 51 3 418798 50 4 388101 71 4 398634 691 4 408918 68 i 4 418964 6 6 5 388279 89 5 898808 87 5 409087 851 5 419129 83 6 388456 107 6 398981 104'! 6 409257 102I1 6 419295 99 7 388634 125 7 399154. 1211 7 409426 1191 7 419460 116 8 388811 142 8 399327 138 8 409595 1368 8 419625 1 32 9 388989 160 9 399501 156 9 409764 153 0 9 419791 149 2450 389166 2010 399674 29570 409933 l12630 419956 1 389343 18 1 399847 17 1 410102 17 1 420121 16 2 389520 36 2 400020,51 2 410271 34' 2 420286 33 3 389697 53 3 400192 53 8 3 410440 50 3 420451 49 4 389875 71 4 400365 69l 4 410608 67 4 420616 66 5 390051 89 5 400538 87 5 [ 410777 84 5 420781 82 6 390228 107 6 400711 104i 6 410946 101 6 420945 99 7 390405 125 7 400883 121 7 411114 1181 7 421110 115 8 390582 1442 8 40105(; 138, 8 411283 1351 8 421275 132 9 390759 160 9 401228 1561 9 411451 152 9 421439 148 2460 390935'2.520 401400 i 2580 411620 1'2640 421604 1 391112 18 1 4015573 17ll 1 411788 1 71 1 421768 16 2 391288 35 2 401745 34, 23 411956 34 2 4219331 33 3 391464 53 3 401917 52| 3 412124 501 3 422097 49 4 391641 70 4 402089 69 4 412292 6711 4 422261 66 5 391817 88 5 402261 86 5 412460 84 5 422426 82 6 391993 106 6 402433 1031 6 412628 1011 6 422590 99 7 392169 123 71402605 120, 7 412796 118t1 7 422754 115 8 392345 141 8 402777 138 8 412964 1351 8 422918 132 9 392521 158 9 402949 155 1 9 413132 152' 9 423082 148 2470 392697 2,530 4031201 20590 413300 2650 423246 1 392873 18 1 403292 17 1 418467 1' 14'3410 16 2 393048 G35 2 403464 33-1 2 413635 33! 2 423573 33 3 393224 53 3 1 40363.5 52 3 413802 50 3 4237387 49 4 393400 70 4 403807 6 4 1 413970 67 4 423001 65 5 1393575 88 5 403978 8G r) 1 414137[ 84 5 424064 81 6 393751 106( 404149 103 6 414305 101 6 424228 98 7 393926 121"3 7 4041320 120 7 414472 117 71 424392 114 8 394101 141 8 404192 137 I 8 414639 134t 1 8 424555 131 9 394276 1811 9 404663 154 9 414806 1511 9 424718 147 2480 3944521 1540 4014834 22600 414973 1l2660 424882 1 394627 17 1 405005 17 1 415140 17j 1 425045 16 2 394802 35 2 405175 34 2 415307 33 2 425208 33 3 394977 53 3 405346 51 3 415474 50 1 3 425371 49 4 395152 70 4 405517 68 4 415641 671 4 42534 65 5 395326 87 5 405688 85 5 415808 841 5 425697 81 6 395501 104 6 405858 102 6 415974 101 6 425860 98 7 395676 122 7 406029 119 7 416141 117 7 426023 114 8 395850 139 8 406199 136 8 416308 134 8 426186 130 1 9 396025 157 91406370 153 9 416474 150 9 426349 147 2490 396199 12550 4065460 2610 416640 22 70 426511 1 396374 17I 1 406710 1ll 1 416807 717 1 426674 16 2 396548 35 2 406881 34 2 416973 33 2 426836 33 3 396722 53 3 407051 51 3 417139 50 3 426999 49 4 396896 70 4 407221 68 4 417306 66 i 4 427161 65 5 397070 87 5 407391 8.5 5 417472 83:1 5 427324 81 6 397245 104 6& 407561 102 6 417638 100 6 427486 98 7 397418 122 7 407731 I 119 7 -417804 116 7 427648 114 8 397592 139! 8 1 407900 136 8 417970 133 8 427811 130 9 397766 157 9 408070 1,53 9 418135 149 9 4279783 147 Page 10 10 LOGARITHMS OF NUMBERS. No. Log. No. Log. Popt.. No. Log. Polp. Part. Prrt.. Poart. 2680 428135 2740 437751 2800 447158 2860 456366 1 428297 16 1 437909 16 1 447313 15 1 456518 15 2 428459 32 2 438067 32 2 447468 31 2 456670 30 3 4'28621 48 3 438226 47 3 4476z23 46 3 456821 46 4 428782' 65 4 438384 63 4 447778 62 4 456973 61 5 428944 81 5 438542 79 5 447933 7 7 5 457125 6 6 429106 97 6 438700 95 6 448088 93 6 457276 91 7 429268 113 7 438859 111 7 448242 108 7 457428 106 8 429429 129 8 4'3017 127 8 448397 124 8 457579 122 9 429591 145 9 439175 143 9 448552 13'9 9 45'7730 137 2690 429752 2750 439333 2810 448706 2870 457882 1 429914 16 1 439491 16 1 448861 15 1 458033 15 2 430075 32 2 439648 32 2 449015 31 2 4581841 )30 3 430236 48 3 489806 47 3 449170 46 3 4588336- 45 4 430398 65 4 43992J64 62 4 4934 62 4 458487 61 5 430559 81 5 440122 79 5 449478 7 7 5 458638 7 6 430720 97 6 440279 95 6 4496833 92 6 458789 91 7 430881 113 7 440437 111 7 449787 108 1 7 458940 106 8 431042 129') 8 440594 126 8 449941 123 } 8 4.59091 121 9 431203 145 9 440752 142 9 450095 139| 9 459242 136 2700 431364 2'60 440909 2820 450249 2S80c 459392 1 431525 16 1 441066 16 1 450403 15 1 459543 15 2 431685 32 2 44122'4 31 2 450557 31 2 459694 30 8 431846 48 3 441381 47 3 450711 46 3 459845 45 4 432007 64 4 441538 63 4 450865 62 4 459995 61 5 432167 80 5 441695 78 5 4,51018 77 5 460146 76 6 432328 96 6 441852 94 6 451172 92 6 40296 91 1 7 432488 112 I 7 442009 110 7 451326 108 7 460447 106 8 432649 128 8 442166 126 8 451479 1 23 8 460597 121 9 432809 14-1 9 442323 141 9 41 a1633 1139 460747 1 6 2710 432969 2770 442480 2830 451786 289'0 1460898 1 433129 16 1 4423Ga6 16 1 451940 15 1 461048 15 2 4332'J0 32 2 442793 31 2 452093 31 2 461198 30 3 433450 48 3 442950 47 3 452247 46 3 461348 45 4 433610 64 1 4 443106 63 4 452400 61 4 461498 60O 5 433770 80 5 443263 78 5 452553 77 5 4616491 7 6 433930 96 6 443419 94l 6 452706'32 6 461799 90 7 434090 112 7l 443576 110 7 4528593 107 11 7 461948 105 8 434249 128 8 443732 126 8 453012 12311 8 462(098 120 9 434409 144 9 443888 141 9 453165 138 9 462248 135 2720 434569 2780 444045 2840 4.53318 29001 462398 1 434728 16 1 444201 16 1 453471 15 1 462548 15 2 434888 32 2 444357 31 1 2 453624 31 2 4626971 30 3 435048 48 3 444513 47 3 453777 46 3 1462847 45 1 4 435207 64 4 444669 62 1 4 453930 61 4 462997 60 5 435366 80 5 444825 78 5 454082 77 5 463146 75 6 435526 96i 6 444981 94 i 6 454235 924 G 4634296 90 7 435685 112 7 445137 109 i 7 454387 107 7 1463445 105 8 435844 128 8 445293 1 25 8 454540| 123 8 463594 120 9 436003 144 9 445448 140 9 454692 138 9 463744 135 2730 436163 2790 445604 12850 454845 2910 463893 1 436322 16. 1 445760 16 1 454997 15 1 464042 15 2 436481 32 2 445915 31 2 455149 30 2 464191 30 3 436640 47 3 446071 47 l 3 1455302 46 3 464340 45 4 436798 63 4 446226 62 4 4.55454 61 4 464489 60 5 436957 79 5 446382 78 5 455606 7 6 11 5 464639 75 6 437116 95 6 446537 94 6 455758 91 6; 464787 90 7 437275 111 7 446692 1091 7 455910 106 7 464936 105 8 437433 127 8 446848 1251 8 456062 122 8 1465085 120 9 437592 143 9 447003 140 I 9 456214 137 9 465234 135 Page 11 LOGARITHMS OF'NUMBERS. 1 No. Log. Partop. No. Log. ar No. Log. Prop No. Log. Prt PaI Part. _ Pa 2920 465383 2980 474216 3040 482874 3100 491362 1 46 5532 15 1 474362 15 1 488016 14 1 491502 14 2 465680 30 2 474508 29 2 483159 28 2 491642 28 3 465829 44 3 474653 44 3 483302 43 3 491 82 42 4 465977 59 4 474799 58 4 4483445 571 4 491922 56 5 466126 74 5 474944 73 5 483587 71 5 492062 70 6 466274 89 6 475090 88 6 483730 85 6 4922 01 84 7 466423 104 7 475235 102 7 483872 99 7 492341 98 8 466571 118H 8 475381 1171 8 484015 114 8 492481 11,2 9 466719 133 9 475526 1311 9 484157 128 9 492621 126 2930 466868 2990 475671 ] 3050 484300 3110 492760 1 467016 15 1 475816 15 1 484442? 14 1 492900 1 4 2 467164 30 2 475962 29 2 484584 28 2 493040 28 3 467312 44 3 476107 43 3 484727 43 3 4931 79 42 4 467460 59 4 476252 58 4 484869 57 4 493319 56 5 467608 74 5 476897 72 5 485011 71 5 493458 0 i 6 467756 89 6 476542 87 6 485153 85 6 493597 84 7 467904 104 7 476687 101 7 485295 99 7 493737 98 8 468052 118 8 476832 116 8 485437 114 8 4938476 14 ) 9 468200 133 9 476976 130 4 9 485579 128 9 494015 126 2940 468347 3000 477121 3060 485721 3120 494155 1 468495 15 1 477266 14 1 485863 14 i 1 494294 14 i 2 468643 30 2 477411 29 2 486005 28 2 494433 28 3 468790 44 3 477555/ 43 3 486147 43 3 49T4572 41 4 868938 59 4 477700 58 4 486289 57 4 494711 56 5 469085 74 5 477844 72 5 486430 71 5 494850 69 6 469233 89 6 477989 87 6 486572 85 6 494989 83 7 469380 104 7 478133 101 7 486714 99 7 495128 97 I 8 469527 118 8 478278 116 8 486855 114 8 495267 111 9 469675 133 9 478422 130 9 486997 128 9 495406 125 2950 4698221 3010 478566 3070 487138 3130 495544 1 469969 15 1 478711 14 1 487280 14 1 1 495683 14 2 470116 291 2 4788-55 29 2 487421 28 2 495822 28 i 3 470263 44 3 478999] 43 3 487563 42 3 495960 41 4 470410 59 4 4791431 58 4 487704 57 4 496099 56 5 4705.57 74 5 479287 72 5 487845 711 5 496237 69 6 47-0704 88 6 1479431 86' 6 487986 85 1 6 496376 83 7 470851 103 7 479575 101 7 488127 99 7 496-514 97 8 470998 118 8 479719 115 8 488269 113 8 496653 111 9 471145 132 9 479863 130 9 488410 127 9 496791 122 5 2960 471292 3020 1480007 3080 4885-51 3140 4969301 1 471438 15 1 480151 14 1 488692 14 1 497068 14 2 471585 29 2 480294 29 1 2 488833 28 2 497206 28 3 471732 44 3 480438 43 3 488973 42 3 497344 41 4 471878 59 4 480582 58 V 4 489114 56 4 4974821 55 5 472025 73 1 5 480725 72 5 489255 70 5 497621! 69 6 472171 88 6 480869 86 6 489396 84 6 497759 83 7 472317 102 7 481012 101 7 489537 98 7 49789'7 97 i 8 472464 117 8' 481156 115 8 489677 112 8 498035 110 9 472610 132 9 481299 130 9 489818 126 9 498173 124 2970 1 472756 3030 481443 3090 489958 3150 498311 1 472903 15 1 4815861 14 1 490099 14 1 498448 1 4 2 473049 29 2 481729 29 2 490239 28 2 4985861 28 3 473195 44 3 1481872 43 3 490380 42 3 4987241 41 4 473341 59 4 482016 57 4 490520 56 4 498862 55 5 473487'73 5 482159 711 5 40661 70 5 498999! 69 6 473633 88 6 482302 86 6 490801 84 61499137, 83 7 47,3779 102 7 482445 100 1 7 490941 98 71 499275 97 8 473925 117 8 482588 114 8 491081 112 8 499412 110 9 1474070 132 9 482731 129 1 9 491222 16 I 499550 1246 _______ 12 Page 12 12 LOGARITHMS OF NUMBERS. No. Log r. Log. No. Log. Prp. No. Log. t Part. Part: Port. 3160 499687 13220 507856 3280 515874 3340 523746 1 499824 14 1 507991 13 1 516006 13 1 5238,76 13 2 499962 27 2 508125 27 2 516139 26 2 524006 26 3 500099 41 3 508260 401 3 516271 40 3 524136 39 4 500236 55 4 508395 54 4 516403 53 4 524266 52 5 500374 68 5 508530 67 5 516535 66 5 524396 65 6 500511 82 6 508664 81 6 516668 79 6 524526 7 8 7 500648 96 7 6508799 94 7 516800 92 7 524656'91 8 500785 110 8 508933 I108 8 516932 106 8 52478.3 104 9 500922 123 9 509068 121 9 517064 119 9 524915 117 3170 501059 3230 509202 3290 517196 3350 525045 1 501196 14 1 509337 13 1 517328 13 1 52 174 13 2 501333 27 2 509471 27 2 517460 26 2 525304 26 3 501470 41 3 509606 40 3 517592 40 3 525434 39 4 501607 55 4 509740 54 4 517724 53 4 525563 52 5' 501744 68 5 509874 67 5 517855 66 5 5256093 65 6 501880 82 6 510008 81 6 517987 79 6 52582 8 7 50201 96 7 7510143 94 7 518119 92 7 525951 91 8 502154 110 8 510277 108 8 518251 106 8 526081 104 9 502290 123 9 510411 121 9 518382 119 9 526210 117 3180 502427 3940 5105451 3300 518514 3360 526339 1 502564 14 1 510679 13 1 518645 13 1 526468 13 2 502700 27 2 510813 27 i 2 518777 26 1 2 526598 26 3 502837 41 3 510947 40 t 3 518909 39 3 526727 39 4 502973 54 4 511081 54 4 519040 52 4 5268.;6 52 5 503109 68 5 511215 67 5 519171 66 5 526985 65 6 503246 82 6 511348 80/ 6 519303 79 6 52,114 78 7 503382 95 7 511482 94 7 519434 92 7 527243 91 8 503518 109 8 511616 107 8 519565 10.5 8 5273-72 104 9 503654 123 9 511750 121 9 519697 118 9 527501 117 31901503791 13250 511883 13310 519828 3370 527630 1 503927 14 1 12017 13 1 5199 13 1 527759 13 2 504063 27 1 2 5121.50 27 2 520090 26 2 527888 26 3 504199 41 3 512284 40 1 3 520221 39 3 528016 38 4 504335 54 4 512417 53 4 520352 52 4 528145 51 5 504471 68 5 512.551 67 1 5 520483 66 5 5 28274 64 6 504607 82 1 51268-41 80 6 520614 791 6 528401 77 7 504743 951 7 512818 93 7 520745 92 17 52)8.31 90 8 504878 1091 8 512951 107 8 520876 105 8 528660 103 9 505014 122 9 513084 120 9 521007 118 9 528788 116 3200 505150 3260 513218 3320 521138 3380 528917 1 505286 14 1 5133-51 13 1 1 521269 13 1 529045 13 2 505421 271 2 513484 27 2 521400 26 2 529174 26 3 505557 41 3 513617 40 3521530 39 3 529302 38 4 505692 54 4 513750 53 4 521661 52 4 5294300 51 5 505828 68 5 513883 66 5 521792 65 5 52 559 64 6 505963 82 6 514016 80 6 521922 68 6 529687 77 7 506099 95 7 514149 93 7 522053 97 7 529815 90 8 506234 1091 8 514282 106 8 522183 104 8 529943 103 9 506370 122 9 514415 120 9 522314 117 9 530072 116 3210 506505 3270 514548 3330 522444 3390 530200 1 506640 13 1 514680 13 1 522575 13 I 1 530328 13 1 2 506775 27 2 514813 27 2 522705 261 2 530456 26 3 506911 40 3 514946 40 3 522835 39 3 530584 38 41507046 54 4 515079 53 4 522966 52 4 530712 51 5 507181 67 5 515211 66 5 523096 65 5 530840 64 6 507316 81 6 515344 80 6 523226 78 i 6 530968 77 7 507451 94 7 515476 93 7 523356 971 7 531095 90 8 507586 108 8 515609 106 8 523486 104 8 531223 102 9 1 50721 121 9 515741 120 9 523616 117 9 531351 115 Page 13 LOGARITHMS OF NUMBERS. 13 No. Log. Propt No. Log. Prop.. IN. Log. Prop. No. Log. Prop. | __a____ _t_____ 3- art____ ____ i0()_ _____ rt__.i__ _ |Part. a r.j 3400 531479 3460 539076 3520 546.543 ~3580 553883 1 531607 13 1 539202 13 1546666 12K 1 554004 12 2 531734 25 2 539327 25 21546789 25/ 22 554126 24 3 531862 38 3 539452 38 3 546913 ) 37 3 554-247 36 4 531990 51 41539578 50 4 547036 49 4 554368 49 5 532117 63 5 539703 63 5 547159 62 5 5.5544-189 61 6 532245 76 6 539829 75 6 547282 74 6 554610 73 7 532372 89 7 539954 88 7 547405 86 7 554731 85 8 532500 102 8 540079 100 8 54 529 99 8 554852 97 9 532627 114 9 540204 1131 9 547652 111 9 554973 109 3410 532754 3470 540329!3530 547775 3590 555094 1 532882 13 1 540455 12 1 547898 12 1 555215 12 2 533009 25 2 540580 25 2 548021 25 2 555336 24 3 533136 38 3 540705 37 3 548144 37 3 555457 36 4 533263 51 4 540830 50 4 548266 49 4 555578 48 5 533391 63 5 540955 62 5 548389 61 5 555699 60 6 533518 76 6 541080 75 6 548512 74 6 555820 72 7 533645 89 7 541205 87 7 548635 86 7 555940 84 8 533772 102 8 541330 100, 8 548758 98 8 556061 96 9 533899 114 9 541454 112 9 548881 111 9 556182 108 3420 534026 3480 541579 3540 549003 3600 556302 1 534153 13 1 541704 12 1 549126 12 1 556423 12 2 534280 25 2 541829 25 2 549249 25 2 556544 24 3 534407 38 3 541953 37i; 3 549371 37 3 556664 36 4 534534 51 4 542078 501 4 549494 49 4 556785 48 5 534661 63 5 542203 62', 5 54961.6 61 5 556905 60 6 534787 76 6 542327 75 6 549739 74 6 557026 72 7 534914 89 7 542452 87 7 549861 86 7 557146 84 8 535041 102 8 642576 100 8 549984 98 8 557267 96 9 535167 114 9 542701 112 9 550106 111 9 557387 108 3430 535294 3490 542895 3550 550228 3610 557507 1 535421 13 1 542950 12 1 550351 12 1 557627 12 2 535547 2.5 2 543074 25 i 2 550473 241 2 557748 24 3 535674 38 3 543199 37 3 550595 37 3 557868 36 4 535800 501 4 543323 501 4 550717 49 4 557988 48 5 535927 63 5 543447 62 5 550810 61 5 558108 60 6 536053 761 6 543571 75 6 550962 73M 6 558228 7 2 7 536179 88 7 543696 87 7 551084 86~ 7 558348 84 8 536306 101 8 543820 100 8 551206 98 8 558469 96 9 536432 114 9 543944 112 9 551328 110i 9 558589 108 3440 536558 3500 544068 35GO 551450 l3620 558709 1 536685 13 1 544192 12 1 551572 12 1 558829 12 2 536811 25 2 544316 25 2 551694 24 2 558948 24 3 536937 38 3 544440 37 3 551816 37 3.559068 36 | 4 537063 50 4 544564 50 4 551938 49L 4 559188 48 5 537189 63 5 544688 62 5 552059 61 5 559308 60 6 537315 76 6 544812 74 6 552181 73i 6 559428 72 7 537441 88 7 544936 87 7 552303 86 7 559548 84 8 537567 101 8 545060 99 8 552425 98| 8 559667 96 9 537693 114 9 545183 112 9 552546 110 9 559787 108 3450 537819 3510 545307 1 3.570 552668 3630 559907 1 537945 13 1 545431 12 1 552,790 121 1 560026 121 2 538071 25 2 545554 25 2 552911 241 2 560146 24 3 538197 38 3 545678 37 3 553033 36i 3 560265 36 4 538322 50 4 545802 49 4 553154 491 4 560385 48 5 538448 63- 5 545925 62 5 553276 61 5 560504 60 6 538574 76 6 546049 74 6 555398 73 6 560624 7 2 7 538699 88 7 546172 86 7 553519 85 7 560743 84 8 538825 101 8 1546296 99 8 553640 971 8 560863 96 9 538951 114| 9 1 546419 1111 9 i553762 109 j 9 5G0982 108 Page 14 14 LOGARITHMIS OF NUMBERS. No. Log. No. Log. Prop. No. Log. P No. Log. Prort Part.. pPart. Part. Part. 3640 561101 3700 568202 3760 575188 i 3820 582063 1 561221 12 1 568319 12 1 575303 129 1 582177 11 2 561340 24 2 568436 23 2 575419 231 2 582291 23 3 561459 36 3 568554 35 3 575534 35 3 582404 84 4 561578 48 4 568671 47 4 575650 46 4 582518 45 5 561698 60 5 568788 58 5 575765 58 5 582631 56 6 561817 72 6 568905 70 6 575880 69 6 5827451 68 7 561936 84 7 569023 82 7 575996 80 7 582958 79 8 562055 96 8 5369140 94 8 576111 92 8 58297-2 90 9 562174 108 9 69257 106 9 576226 10 9 583085 10 3650 562293 3710 569374 3770 576341 3830 5831099 1 562412.12 1 569491 12 1 576457 121 1 58331 1 11 2 562531 24 2 569608 23 2 5 76572 23~ 2 583426 23 3 562650 36 3 569725 35 3 576687 35 3 6583539 34 4 562768 48 4 569842 47 4 576802 46 4 583652 45 5 562887 60 5 5699.59 58 5 576917 58 5 58376.5 56 6 563006 71 6 570076 70 6 577032 69 6 5838879 68 7 56312.5 83 7 57(193 82 7 577147 80 7 583992 79 8 563244 95 8 570309 94 8 577262 92 8 584105i 00 9 563362 107 9 570426 106 9 577377 104 9 584218! 102 3660 563481 3720 570543 3780 577492. 3840 584331 1 563600 12 1 570660 12 1 577607 11 1 584444' 11 2 563718 24 2 570776 23 2 577721 23 2 5845571 23 3 563837 36 3 570893 35 3 577836 341 3584670 34 4 563955 48 4 571010 47 41 577951 46 4 584783 45 5 564074 60 5 571126 58 5 578066 57 5 5848W96 56 6 64192'3 71 t6 571243 70 6 578181 68 I 6 585009 G8 57 64311 83 571359 81 7 80 7" 7 585122 79 8 564429 95 8 571476! 93 8 578410'1 8 585235 9 0 9 564548 107 9 571592 105 9 578525 103 9 585348 102 3 6j 70 564666 3730 5717090 3790 578639 3850 585461 1 564784 l 12 1 571825 12 1 578754 11 1 585.574 11 2 564903 24 2 571942 2 23 2 578868 23 1 ) 585686,) 3 565021 36 3 5720.58 35 3 578983 34 31585799 34 4 565139 47 4 572174 47 4 579097 46 4 585912 45 5 565257 69 5 572291 58 5 579212 57 I7 5 586024 56 6 56.5376 71 6 572407 70 6 579326 68 6 586137 1 67 7 565,494 83 7 572523 81 7i 579441 80 7 586250 78 8 565612 95 8 572639 93 8 579,555 91 8 5868362 90 9 565730 107 9 572755 105 9 579669 103 9 586475 1 01 3680 565848 3740 572872 | 3800 5797584 3 860 586587 1 565966 12 1 5729'388 1 1 579898 11 1 5867800 11 2 566084 24 2 573104 23 2 580012 23 2 5868172 22 3 566202 35 3 573220 35 3 580126'7 34 3 58692 5i 34 4 566320 47 4 57336 1 46 4 5802401 4 4 587T037 45 5 566437 59 5 573452 58 5 580355 57 5 5871492 56 6 566555 71 6 573568 70 6 580469 68 6 58762 1 67 7 566673 83 7 573684 81 7 580583 80 7 58873749 78 8 566791 94 8 573800 93 8 580697 91 8 587486( 9 0 9 566909 106 9 573915 1041 9 5808111103 9 587599 i101 3690 567026 3750 574031 3810 580925 3870 587711 1 567144 12 1 574147 121 1 581039 11 1 5878'30 11 2 567262 24 2 5742631 23 2 581153 23 2 58 793.5 22 3 567379 35 3 574379 35 3 581267| 34 3 588047 1 34 4 567497 47 4 1574494 46 4 581381 46 4 588160i 1 5| 5 567614 59 5 574610 58 5 581495 57 5 5882721 56 6 567732 7 1 6 574726 70' 6 581608 68 6 588384' 67 7 567849 83 7 574841| 81 7 581722 80 7 588496 8 8 567967 |94 8 574957 93 8 581836 91 8 588608 90 1 568084 106 9 575072 1104 9 581950 103 9 588720 101 Page 15 LOGARITHMS OF NUMBERS. 15 No. Log. Prop. No. Log. Prop. i No. Log. Prop. No. Log. Prop. Part. - _ _ _ Part. Part. 3880 588832 3940 595496 4000 602060 4060 608526 588944 11 1 595606 11 1 602169 11 1 608633 11 2 589056 22 2 595717 22 2 602277 22 2 608740 21 3 5893167 33 3 595827 33 3 602386 33 3 608847 32 4 589279 44 4 595937 44 4 1602494 43 4 608954 43 5 589391 56 5 596047 55 5 602603 54 5 609061 53 6 589503 67 6 59615*7 66 6 602711 65 6 6091.67 64 7 589615 78 7 596267 77 7 602819 76 7 609274 75 8 589726 89 8 596377 88 8 602928 87 8 609381 86 9 589838 100 9 596487 99 9 603036 98 9 609488 96 3890 589950 3950 596:59 7 4010 603144 4070 609594 1 590061 11 1 5966707 11 1 603253 11. 1 609701 11 2 590173 22 2 596817 22 2 603361 22 2 609808 21 3 590284 33 3 596927 33 3 603469 33 609914 32 4 590396 44 1 4 1 597037 44 4 603577 43 4 610021 43 5 590507 56 5 597146 55 5 603686 54 5 610128 53 6 590619 67 6 597 56 66 6 1 603794 65 6 610234 64 7 590730 78 7 5973t6j 77 7 603902 76 7 610341 75 8 590842 89 8 59D7476 88 8 604010 87 8 610447 86 9 590953 1001 9 597585 99 9 604118 98 9 610554 96 3900 59106.5 3960 597695 4020 604226 4080 610660 1 591176t 11 1 597805 11 1 604334 11 1 610767 11 2 591287 22 2 597914 22 2 604442 22 2 610873 21 3 591399 33 1 3 5,98024 33 3 604550 32 3 610979 32 4 591510 44 4 598134 44 4 604658 43 4 611086 42 5 591621 56 5 598243 55 5 604766 54 5 611192 53 6 591732 67 6 598353 66 6 604874 65 6 611298 64 7 591843 78 11 7 598462 77 7 60-4982 7 6 7 611405 74 8 591 951 89 8 598-o72 88 81 605089 86 8 611511 8-5 9 592066 100 9 598681 99 9 605197 97 9 611617 95 3910 592177 3970 598790 4030 605305 4090 611723 1 592288 11 1 598900 11 1 605413 11 1 611829 11 2 592399( 22 2 2.599009 22 2 605521 22 2 611936 21 3 1593510 33 1 3 599119 33 3 605628 32 3 612042 32 4 592621 44 4 599228' 44 4 605736 43 4 612148 42 5 593232 55 5 599337 55 1 5 605844 54 5 612254 53 6 592843 67 G 1599446 66 6 605951 65 6 612360 64 7 592954 |78 7 599.556 77 7 606059 76 7 6124636 74 8 593064 89 8 5996i65 88 1 8 1606166 86 8 612572 85 9 593175 100 9 599774 99 9 606274 97 9 612678 95 3920 593286 3980 599883 4040 606381 4100 612784 1 593397 11 1 599992 11 1 606489 11 1 612890 11 2 593508 22 2) 600101 22 2 606596 21 2 612996 21 3 593618 33 3 600210 33 3 606704 32 3 613101 32 4 593729 44 4 600319 44 4 606811 43 4 613207 42 5 593840 55. 51600428 54. 5 606919 54 5 613313 53 6 593950 66 6 600537 65 6 607026 64 6 613419 64 7 594061 77 7 600646 76 7 607133 75 7 613525 74 8 594171 88 8 600755 87 8 607241 86 8 613630 85 9 594282 99 9 600864 95 9 607348 96 9 6137836 95 3930 594393 3990 600973 4050 607455 4110 618842 1 594503 11 1 601082 11 1607562 11 1 613947 11 2 594613 22 2 601190 22 21 607669 21 62 14053 21 3 594724 33 3 601299 33 3 607777 32 3 614159 32 4 594834 44 4 601408 44 4 607884 403 4 614264 42 5 594945 55 5 601517 54 5 607991 4 5 614370 53 6 595055 66 6 601625' 6.5 6 608098 64 6 614475 63 7 595165j 77 7 61734 76 7 608205 5 7T 614681 74 8 5952761 88 8 601843 87 8 608312 86, 8 614686 84 9 5953861 99 9 601951 98 9 608419 96 9 614792: 95 Page 16 16 LOGARITHMS OF NUMBERS. No. Log. Part. No. Log. Part. No. Log. Part. No. Log. Part. P___. ___._art. Par_ 4120 614897 4180 621176 j 4240 627366 4300 633468 1 615003 11 1 621280 10. 1 627468 10 I 1 633569 10 2 615108 21 2 621384 21 2 627571 20 2 633670 20 3 615213 31 3 621488 31 3 627673 31 3 6383771 30 4 615319 42 4 621592 42 4'627775 41 4 6338 72 40 5 615424152 5 621695 52 5 627878 51 5 633973 50 6 615529 63 6 621799 62 i 6627980 61 6 63407 4 61 7 615634 73 7 621903 73 7 628082 72 7 634175 71 8 615740 84 - 8 622007 83 8 628184 82 8 634276 81 9 615815 95 9 622110 94 9 628287 92 9 634376 91 4130 615950 4190 622214 4250 628389 4310 634477 1 616055 11 1 622318 10 1 628491 10 1 1 6345 8 10 2 616160 21 2 622421 21 2 628593 20 2 634679 20 3 616265 31 3 62252,5 31 3 628695 31 3 634779 30 4 616370 42 4 622628 41 4 628797 41 4 634880 40 5 616475 52 5 622,732 52 5 628900 51 5 634981 50 6 616580 63 6 i62 285 62 6 629002 61 6 635081 61 7 616685 73 7 622939 72 7 629104 72 I 7 635182 71 8 6167,90 84 J 8 623042 83 8 629206 82 8 635283 81 9 616895 95 9 623146 93 9 629308 92 9 635383 91 4140 617000 4200 623249 4260 629410 4320 635484 1 617105 10 1 623353 10 1 629511 10 1 6385584 10 2 617210 21 2 623456 21 2 i 629613 20 2 635685 20 3 617315 31 Il 3 623559 31 3 629715 30 3 635785 30 4 617420 42 4 623663 41 4 629817 41 4 635886 40 5 617524 52 5 623766 52 5 629919 51 5 1635986 50 6 617629 63 6 623869 62[ 6 630021 61 6 636086 60 7 61.7734 73, 7 623972 72 7 630123 1 7 1636187 70 8 617839 84 8 624076 83 8 630224 81 8 636287 80 9 617943 94 9 624179 93 9 630326 91 91 636388 90 41501 618048 4210 624282 4270 630428 4330 636488 1 618153 10 1 624385 10 1 630530 10 1 636588 10 2 618257 21 2 624488 21 2 630631 20 i 2 66688 20 3 618362 31 3 624591 31 3 630733 30 l 3 636789 30 4 618466 42 4 624694 41 l 4 630834 41 4 636889 40 5 618571 52 5 624798 51 5 630936 51 5 636989 50 6 618675 62 6 624901 62 6 631038 61 6 637089 60 7 618780 73 7 625004 72 7 631139 71 i/ 7 637189 70 8 618884 83 8 625107 82 8 631241 81 8 637289 80 9 618989 94 9 625209 93 9 631342'31 9 637390 90 4160 619093 4220 625312 14280 631444! 4340 637490 1 619198 10 1 625415 10 1 631545 10 1 637590 10 2 619302 21 2 625518 21 2 631647 20 [ 2:637690 20 3 61'9406 31 3 625621 31 3 631748 30 3 637790 30 4 / 619511 42 4 625724 41 4 631849 41 4 637890 40 5 619615 52 5 625827 51 5 631951 51 5i 637990 50 6 619719 62 6 625929 62 6 632052 61 6 638090 60 71 619823 73 7. 626032 72 7 632153 71 7 638190 70 8 619928 83 8 626135 82 8 632255 81 8 638289 80 9 620032 94 9 626238 93 9 63,2356 91 9 638389 90 4170 620136 4230 626340 114290 632457 4350 638489 1 620240 10 1 626443 10 1 632558 10 1 638589 10 2 620344. 21 2 626546 21 2 632660 20 2 638689 20 3 620448 31 3 626648 31 3 632761 30 3 638789 30 4 6 62055 2' 42 4 i 626751 41 4 632862 41 4 638888 40 5 620656 52 5 626853 51 5 632963 51 5 638988 50 6 1620760 62 6 626956 62 6 633064 61 6 639088 60 7 620864 73 7 627058 72 7 633165 71 7 639188 70 8 620968 83 8 627161 82 8 633266 81 8 639287 80 9 621072 94 9 627263 93 9 633367 91 9 639387 90 Page 17 LOGARITHMS OF NUMBERS. 17 No. Log. Prop No. Log. Pp No. og. P o. Logog. Prop. tLog. Part. P art. Part. 43601639486 4420 645422 4480 651278 4540 657056 1 639586 10 1 645520: 10 1 651375 10 1 657151 10 2 1689686 20 2 645619 20 2 651472 19 0 2 657247 19 3 639785 30 3 645717 30 0 3 651569 29 3 65 733 28 4 639885 40 4 645815 39 4 651666 38 4 657438 38 5 639984 50 5 645913 49 5 651762 48 5 657534 47 6 640084 60 6 646011 59 6 651859 58 6 657629 57 7 640183 70 7 646109 69 7 651956 67 7 657725 67 8 640283 80 8 646208 79 8 652053 77 8 6.57820 76 9 640382 90 9 646306 89 9 652150 87 9 657916 86 4370 640481 4430 646404 4490 652246 4550 658011 1 940581 10 1 646502 10 1 652343 10 1 6.58107 10 2 640680 20 2 646600 20 2 652440 19 2 658202 19 3 640779 30 3 646698 29 3 652536 29 3 658298 28 4 640879 40 4 646796 39 4 652633 38 4 658393 38 5 640978 50 5 646894 49 5 652730 48 5 658488 47 6 641077 60 6 646991 59 6 652826 58 6 658584 57 7 641176 70 7 647089 69 7 1 652923 67 7 658679 67 8 641276 80 8 647187' 78 81653019 77 8 658774 [76 9 641375 90 9 647285 88 9 653116 87 9 658870 86 4380 641474 4440 647383 4500 653213 4560 658965 1 641573 10 1 647481 10 1 653309 10 1 659060 10 2 641672 20 2 647579 20 2 653405 19 2 659155 19 3 641771 30 3 647676 29 3 653502 29 3 659250 28 4 641870 40 4 647774 39 4 1653598 38 4 659346 38 5 641970 50 5 647872 49 5 653695 48 5 659441 47 6 642069 59 6 647969 59 6 653791 58 6 659536 57 7 642168 69 7 648067 69 7 653888 67 7 659631 67 8 642267 79 8 648165 78 8 653984 77 8 659726 76 9 642366 89 9 648262 88 9 654080 87 9 659821 86 4390 642464 4450 648360 4510 654176 4570 659916 1 642563 10 1 648458 10 1 654273 10 1 660011 10 2 642662 20 2 648555 19 2 664369 19 2 660106 19 3 642761 30 3 648653 29 3 654465 29 3 660201 28 4 642860 40 4 648750 39 4 654562 38 4 660296 38 5 642959 49 5 648848 49 5 654558 48 5 660391 47 6 643058 59 6 648945 58 6 654754 58 6 660486 57 7 643156 69 7 649043 68 7 654850 67 7 660581 67 8 643255 79 8 649140 78 8 654946 77 8 660676 76 9 643354 89 9 649237 88 9 655042 86 9 660771 86 4400 643453 4460 649335 4520 655138 4580 660865 1 643551 10 1 649432 10 1 655234 10 1 660960 9 2 643650 20 2 649530 19 2 655331 19 2 661055 19 3 643749 30 3 649627 29 3 655427 29 3 661150 28 4 643847 39 4 649724 39 4 655523 38 4 661245 38 5 643946 49 5 649821 49 5 655619 48 5 661339 47 6 644044 59 6 649919 58 6 655714 58 6 661434 57 7 644143 69 7 650016 68 7 655810 67 7 66r529 66 8 644242 79 8 65 0113 78 8 655906 77 8 661623 76 9 644340 89 9 650210 88 9 6.56002 86 9 661718 85 4410 644439 4470 650307 4530 656098 4590 661813 1 644537 10 1 650405 10 1 656194 10 1 661907 9 2 644635 20 2 650502 19 2 656290 19 2 1 662002 19 3 644734 30 3 650599 29 3 656386 29 i 3 662096 28 4 644832 39 4 650696 39 11 4 656481 38 i 41662191 38 5 644931 49 5 650793 49 5 6565577 48 5 6 62285 47 6 645029 59 6 650890 58 6 656673 58 11 6 662380 57 7 645127 69 7 650987 68 1 7 656769 67 7 662474 66 8 645226 79 8 651084 7 8 8 656864 77 8 662569 76 9 645324 89 9 651181 88 1 9 656960 86 9 662663 85 Page 18 18 LOGARITHMS OF NUMBERS. No. Log. Prop. No. Log. Prop No. Log. Prop. No. Log. Prop. P.rL. Part. Parc. Part. 4600 662758 4660 668386 4720 673942 4780 679428 1 662852 9 1 668479 9 1 674034 9 1 679519 9 2 662947 19 2 668572 19 2 674126 18 2 679610 18 3 663041 28 3 668665 28 3 674218 28 3 679700 27 4 663135 38 4 668758 37' 4 674310 37 4 679791 36 5 663230 47 5 668852 47 5 674402 46. 5 679882 45 6 663324 57 6 668945 56 6 674494 55 6 679973 55 7 663418 66 7 669038 65 7 674586 64 7 680063 64 8 663512 76 8 669131 74 8 674677 74 8 680154 73 9 663607 85 9 669224 84 9 674769 83 9 6802 45 82 4610 663701 4670 669317 4730 674861 4790 680335 1 663795 9 1 669410 9 1 674953 9 1 680426 9 2 663889 19 2 669503 19 2 675045 18 2 680517 18 3 663983 28 3 669596 28 3 675136 28 3 680607 27 4 664078 38 4 669689 37 4 675228 37 4 680698 36 5 664172 47 5 669782 47 5 675320 46 5 680789 45 6 664266 56 6 669875 56 6 675412 55 6 680879 55 7 664360 66 7 669967 65 7 675503 64 7 680970 64 8 664454 75 8 670060 74 8 675595 74 8 681060 73 9 664548 85 9 670153 84 9 675687 83 9 681151 82 4620 664642 4680 670246 4740 675778 4800 681241 1 664736 9 1 670339 9 1 675870 9 1 681332 9 2 664830 19 2 670431 18 2 675962 18 2 681422 18 3 661921 28 3 670524 28 3 676053 27.3 681513 27 4 665018 38 4 670617 37 4 676145 36 4 681603 36 5 665112 47 5 670710 46 5 676236 46 5 681693 4 5 6 665206 56 6 670802 55 6 6763 28 55 6 681784 54 7 665299 66 7 670895 64 7 676419 64 7 681874 63 8 665393 75 8 670O88 74 8 676511 73 8 681964 72 9 6653487 85 9 671080 83 9 676602 82 9 682055 81 4630 665581 4690 671173 4750 676694 4810 682145 1 665675 9 1 671265 9 1 676785 9 1 682235 9 2 665'769 19 2 671358.18 2 676876 18 2 682326 18 3 665862 28 3 671451 28 3 676968 27 3 682416 27 4 665956 38 4 6715431 37 4 677059 36 4 682506 36 5 666050 47 5 671636i 46 5 677151 46 5 682596 45 6 666143 56 6 6717281 55 6 677242 55 6 682686 54 7 666237 66 7 671821 64 7 677333 64 7 682777 63 8 666331 75 8 671913 74 8 677424 73 8 682867 72 9 666424 85 9 672005 83 9 677516 82 9 682957 81 4640 666518 4700 672098 4760 677607 4820 683047 1 666612 9 1 672190 9 1 677698 9 1 683137 9 2 66670.5 19 2 672283 18 2 677789 18 2 683227 18 3 666799 28 3 672375 28 3 677881 27 3 683317 27 4 666892 37 4 672467 37 4 677972 36 4 683407 36 5 666986 47 5 672560 46 5 678063 45 5 1683497 45 6 667079 56 6 6672652 55 6 678154 55 6683S87 54 7 667173- 65 7 672744 64 7 678245 64 7 683677 63 8 667266 74 8 672836 74 8 678336 73 8 683767 72 9 667359 84 9 672929 83 9 678427 82 9 683857 81 4650 667453 4710 673021 4770 678518 4830 683947 1 667546 9 1 673113 9 1 678609 9 1 684037 9 2 667640 19 2 673205 18 2 678700 18 2 684127 18 1 3 667733 28 3 67329.7 28 3{ 1678791 27 3 684217 27 4 667826 37 4 673390 37 4 678882 36 4 684307 36 5 667920 47 5 673482 46 5 678973 4 5 5 684396 45 6 668013 56 6 673574 55 6 679064 5.5 6 684486 54 7 668106 65 7 673666 64 7 1679155 64 7 684576 63 8 668199 74 8. 673758 74 8 679246 73 8 684666 72 9 668293 84 9 673850 83 9 679337 82 9 684756 81 Page 19 LOGARITHMS OF NUMBERS. 19 No. Log. Prop. No. Log. Prop. No. Log. Prop. Log. Prop. Part. Part. Part. Part. 48410 68484.5 4900 690196 4960 695482 5020 700704 1 684985 9 1 690285 9 1 695569 9 1 700790 9 2 685O025 18 2 6903731 18 2 695657 17 2 700877 17 3 685114 27 3 690462 27 3 695744 26 3 700963 26 4 685204 36 4 6930550 3 4 695832 35 4 701050 85 5 685294 45 5 690639 44 5 695919 44 5 701136 43 6 685383 54 6 690727, 53 6 696007 52 6 701'22 52 7 68547t 3 63 7 6'90816 62 7 69360'34 61 7 701309 61 8 685563 72 8 690905 71 8 696182 70 8 701395,70 9 685652 81 9 690993 80 9'3 696269 79 9 70148 ) 78 4850 685742 4910 691081 4970 696356 5030 701568 1 t685831 9 1 691170t 9 1 696444 9 1 701654 9 2 685921 18 2 691258 18 2 696531 17 2 701741 17 3 686010 27 3 691347 27 3 696618 26 3 701827 26 4 686100 36 4 691435 30 4 696 70 35 4 701913 35 5 686189 45 5 691524 44 5 t6967937 44 5 701999 43 6 686279 54 6 691612 53 6 1696880 52 6 702086 52 7 686368 63 7 691700 62 7 1696968 61 7 702172 61 8 686457 72 8 691789'3 71 8 697055 70 8 702258 70 9 686547 81 9 6918771 80 9 697142 79 9 7023'44 8 4860 686636 4920 691965 4980 697229 5040 702430 1 686726 9 1 692053 9 1 697317 9 1 702517 9 2 686815 18 2 692142 18 2 697404 17 2 702603 17 3 686904 27 3 692230 27 3 697491 26 3 702689 26 4 686994 36 4 692318 35 4 697578 35 4 702776 34 5 68708<3 45 5 692406 44 5 697665 44 5 702861 43 6 687172 64 6 69249.53 6 697752 52 6 702947 52 7 687261 63 7 692583 6 7 697839 61 7 703033 60 8 687351 72 8 692671 71 8 697926 70 8 703119 69 9 687440 81 9 6927591 80 9 698013 79 9 703205 7 4870 687529 4930. 692817 4990 698100 5050 703291 1 687618 9 1 [692935 9 1 698188 9 1 703377 9 2 687707 18 2 693023 18 2 698275 17 21 703463 17 3 687796 27 3 693111 2)6 3 698362 26 i 3 703549 26 4 687886 3 4 693199 35 4 698448 35 4 703635 34 5 687975 45 5 693287 44 5 69853.5 44 5 703721 43 6 688064 64 54 693375 53 6 6I98622 50 2 6 703807 52 7 688153 62' 7 6934631 62 7698709 61 7 703893 60 8 688242 72 869 8 69351 70 8 68796 70 8 703979 69 9 688331 80 9 6 9036,39 79 9 3 698883 79 9 704065 77 4880 6881200 4940 693727t 5000 698970 5060 104150. 1 688509d 9 1 693815 9 1 6990357 9 1 704236 9 2 68853 98t 18 2 6'9t390t3 18 2 699144 17 2 704-322 17 c3688687 l27 3 6939'91 2 6 3 699231 26 3 704408.1 6 4 1688776 3,6 4 6940788 3 4 699317 135 4 704494 34.5 688865 45 i 5 6{964166/ 44 5 699401 43 5 704579 43 6 688953 54 6i 6 194254] 53 1 6 699491 52 6 704 4(;65 52) 7 68904') 62 6794342 62 699578 61 7 704751 6t0 8 689131 71 8 694430 70 8 699664 70 8 701837 69.) 689220 80 / 9 694517 79, 9 699751 78 9 70-922 77 4890 689309 4950 694605[ 15010 6998381 5070 705008 1 689398 9 1 116940'639 9 1 699924 9 1 105094 9 32 689486 18 2 6914781 18 2l "700011 17 2Kv05179 17 3689575 27 l 3 694868 2 3 700098 3 705265 26 4 689&'6t4 36 4 46949561 3. 5 4 700184 3 4 705350 3414 689753 45 5 695044 44 11 5 700271 43 5 705436 43 6 68!)841 54 6 695181l 53 6 7 00358 52 6 705522 52 7 68(99830 62 2 7 695219 62 7 700-1444 61 7 705607 60 8 6900193 72 8 695307 70 1 8 700531 70 8 70569 69 | 69!0107 80 9 695394 79 9 700617 78 9 705778 77 3; Page 20 20 LOGARITHMS OF NUMBERS. No. Log. Prop N No. Log. Prop. No Log. Part. Pat Part., Part. Part. 5080 705864 5140 710963 5200 716003 5260 720986 1 705949 9 1 711048 8) 1 716087 8 1 721068 8 2 706035 17 2 711132 17 2 716170 17 2 721151 16 3 706120 26 3 711216 25 - 3 716254 25 3 721-238 25 4 706206 34 4 711301 34 4 716337 34 4 721316 33 5 706291 43 5 711385 42 5 716421 42 5 721398 41 6 706376 51 6 711470 51 6 716504 50 6 721481 49 7 706462 60 7 711504 59 7 716588 59 7 721563 58 8 706547 68 8 711638 68 8 716671 67 8 721646 66 9 706632 77 9 711723 7 9 716754 76 9 721728 74 5090 706718 5150 711807 l 5210 716838 50 721 811 1 706803 9 1 711892 8 1 716921 8 1 721893 8 2 706888 17 2 711976 17 2 717004 17 2 721975 16 3 706974 26 3 712060 25 1 3 717088 2.5 3 7 22058[ 25 4 707059 34 1 4 712144 34 4 717171 33 4 722140 33 5 707144 43 5 712229 42 5 717254 42 5 7'22222 41 6 707229 51 1 6 712313 51 6 717338 50 6'722305 49 7 707315 60 7 712397 59 7 717421 58 7 722387 58 8 707400 68 8 712481 68 8 717504 66 8 722469 66 9 707485 77 9 712366 76 9 717587 75 9 722552 74 5100 07570 5160 712650 5220 717671 5280 722634 1 707655 9 1 712734 8 1 717754 8 1 722716 8 t 2 707740 17 i 2 712818 17 717837 17 1 2 72'798 16 31 707826 26 1 3 712902 25 1 717920 25 8 31722881 25 4 707911 34 ii 4 712986 34 1 4 718003 33 1 4 722963 33 j 5 707996 43 3 5 713070 4' 1 5 718086 42 5 723045 41 6 1 70808 1 51 6 71315-1 50 6 718169 50 6 723127 49 7 1708166 60 7 713238 59 7 718253 58 7 723 209 58 8 708251 68 1 8 711332 6 8 7136 66 8 23291 9 708336 77 9 713406 6 9 718419 75 9 728334 174 5110 708421 | 5170 713490 | |30! 718502 5290 723456G 1708506 9 1 713574 8 1718585 8 12 17235'38 8 2 708591 17 2 713658 17 1 718668 17 2 7236201 16 3 708676 26 3 5 71874 25 3 718751 25 31 723702 25 4 708761 34 4 713826 34 4 718834 33 4 723784 33 5 708846 43 5 701310 42 5 718917 42 55 73866 41 6 708931 51 6 713994 50 6 719000 50 6 723948 49 7 709015 0C 7 714078 59 7 719083 58 7 74030 57 8 709100 68 7 8 714162 67 8 719165 66 8724112 66 9 709185 77 9 714246 076 9 719248 5 2 9 74194 74 5120)709270 5180 714330 5240 719331 <503001 724 6| 11709355 8 1 714414 8 1 719414 8. jl 11724358 8 21709440 17 2 714497 17 2 7194197 17 21724440 16 3 K 709524 25 3 714581 25 3 719580 25 3 174522 25 4 )709609 3-4 4 714665 34 4 719663 33 41 724603 33 5 709694 42 5 714749 42 5 719745 41 5 724685 41 G6709779 51 6 714832 50 6 719828 50 6 724761 49 7 709863 59 7 714916 59 7 719911 58 7 724849 57 8' 09948 68 8 715000 67 8 719994 66 8 1 724931 66 9 710033 76 91 715084 76 91720077 75 9 725013 74 5130 710117 5190 715167 5250 720159 5310 725095 1 710202 8 1 715251 8 1 720242 8 1 725176 8 2 710287 17 2 715335 17 2 720325 171 2 725258 16 3 710371 25 3 715418 25 3 720407 25 3 725340 25 4;710456 34 4 715502 34 41 720490 33 4 7254221 33 5 710540 42 51715586 42 5 7520573 41 11 5 725503 41 6 710625 51 6 715669 50 6 72065.5 50 6 725585 49 7 7110710 59 7 715753 59 7 |720738 581 7 725667 I 57 8 710794 67 8 715836 67 8 720821 66 1 8 727481 6 6 9 1710879 76 9 715920 76 11 9 720903 75 | 9 725830 74 Page 21 LOGARITHMS OF NUMBERS. 21 Prop L Prop. Nop. No. Log. Prop. No. Pro.No Log.. No. Lo Prop. g. r. Lg. Part. PaNo.. Prt. 5320 725912 5380 730782 1 5140 735599 5500 740363 1 725993 8 1 730863! 8 1 735679 8 1 74044 8 2 726075 16 2 7309441 16 2i 735759 16 2 740521 18 3 726156 24 3 7310241 24 3 735838 24i 3 740599 204 4 726238 33 4 731105 32 4 735918 32 4 1740678 32 5 726320 41 5 731186 40 5 735998 40 5 740757 40 6 726401 49 6 731266 49 6 736078 48 6 740836 47 7 726483 57 7 731347 57 7 736157 56 7 740915 55 8 726564 65 8 8731428 65 1 8 736237 64 8 740994! 63 9 726646 73 9 731508 73 9 736317 72 9 741073 7 1 5330 726727 5390 731589 5450 736396 5510 741152 1 726809 8 1 731669 8 1 736476 8 1 741230 8 2 726890 16 2 731750 16 2 7 736556 16 2 741309 16 3 726972 24 3 731830 4 31 7366,35 24 3 741388i 24 4 7270.53 33 4 731911 32 41736715 32 4 741467 32 5 727134 41 5 731991 40 5 736795 40 5 741516 40 6 727216 49 6 732072 48 6 736874 48 7416924 47 7 727297 57 7 732152 56 7 736954 56 7 741703 55 8 727379 65 8 732233 64 8 737034 64 8 741 782 63 9 727460 73 9 732313 72 9 737113 72 9 741860 71 5340 727541 5400 732394 5460 737193 51 520 741939 1 727623 8 1 732474 8 1 1731272 8 1 742018 8 2 727704 16 2 732555 16 2 737352 16 2 742093 16 3 727785 24 3 73263.5 24 3 737431 24 3 742175 2.3 4 727866 33 4 732715 32 4 737511 32 4 74254 31 5 727948 41 5 732796 40 5 737590 40 5 742332 3'9 6 728029 49 6 732876 48 61737670 48 6,1 742411 1 47 7 728110 57 7 732956 56 15 7'1377491 56) 1 71742489 55 8 7281391 65 8 733037 64 8'73'7829 64 1 8174-568 63 9 728273 73 9 73311' 7 2 9 737908 7 2 91 74647 71 5350 728354 5410 1733197 5470 737987 5530 74272.5 1 728435 8 1 733278 8 1 738067 8 1 742804 8 2 728516 16 2 7333.58 16 2 7381461 1() 2742882 16 31728597 24 3 733438 - 3 738225 24 3 42961 23 4 728678 33 4 733518 4 738-0.05 3 4 743039 31 5 5728759 41 1 5 1733598 40 5 7383841 40 I 5 7431118 39 6 728841 49 6 733679 48 6 738-63 48 6 743196 47 7 728922 57 7 733759 56 7 738541 56 7 74327.5 55 8 729003 6.5 8 733839 64 8 7386222 64 8 7 433.53 63 9 729084 73 9 733919 72 9 738701 7 9 743431 71 5360 729165 5420 733999 5480 738781 5540 743510 1 729246 8 11734079 8 1I 7388601 8 1 743588 8 2 72793~227 l1 2 734159 1 2 7389391 16 2743667 16 3 729408 24 3 734240 24 3 7339018 21 3 7-13745 23I 4 729489 32 4734320 32 I 4 739097 3 2 4 7e-)'823 31 5 729570 41 5 734400 40 5 739177 40 75 7 43902 39 6 729651 49 6 734480 48 6 739256 47 6 743980 47 7 729732 57 7 734560 56 7 7 8i39335 I 55 7 7440.58 5.5 8 729813 65 8 734640 64 I 8 739414 63 8 744136 63 9 729893 73 9 734720 72 $ 9 739493 71 9 744215 71 5370 729974 5430 734800 I5490 739572 oo50 744293 1 730055 8 1 734880 8 I 1 739651 8 1 744371 8 2 2730136 16 2 734960 16 2 739730 16 2 744449 16 3 730217 24 3 735040 24 3 739810 24 I 3 744528 23 4 730298 32 4 735120 32 4 739889 32 4 744606 31 5 730378 40 5 735200 40 5 739968 40 5 744684 39 6 730459 49 6 735279 48 6 740047 47 6 744762 47 i 7 730540 57 7 735359 56 7 740126 550 7 744840 i5$ 8 730621 65 8 735439 61 8 740205 63 8 744919 63 9 730702 73 9 735519 72 9 740281 71 - 9 744997 71 Page 22 22 LOGARITHMS OF NUIMBERS.!'o. o Prop. Prop. Prop Prop. Part. Part. P.. Part. 5560 745075 15620 749736 5680 754348 5740 758912 1 745153 8 1 749814 8 1 754425 8 1 758988 8 2 745231 16 2 749891 16 2 754501 15 2 759063 15 3 745309 23 3 749968 23 3 754578 23 3 759139 23 4 745387 31 4 750045 31 4 754654 30 4 759214 30 5 745465 39 5 750123 39 5 5754730 38 5 7 59290 38 6 745543 47 6 750200 47 6 754807 46 6 759366 45 7 745621 55 7 750277 54 7 754883 53 7 759441 53 8 745699 62 8 750354 62 8 754960 61 8 759517 60 9 745777 70 9 750431 70 9 755036 69 9 759592 68 557-0 7458551 5630 750508 5690 755112 5750 759668 1 745933 8 1 75058f 8 1 755189 8 1 759743 8 2 746011 16 2 750663 16 2 755265 15 2 759819 15 3 746089 23 3 750740 2:1 3 755341 23 3 759894 23 4 746167 31 4 750817 31 4 755417 80 4 759'70 30 5 746245 39 5 750894 39 5 755494 38 5 760045 38 6 746323 6 750971 47 6 5095570 46 6 760121 45 7 746401 55 7 751048 54 7 755646 53 7 760196 53 8 74647'd 62 8 751 12 62 8 75.5722 61 8 760272 60 9 746556 70 9 751202 70 9 755799 69 9 760347 68 5580 746f634 5640 7 5127 9 65700 755875 5760 760422 1 746712 8 1 7.51356 8 1 755951 8 1 760498 8 2 7467 90 16 2 751433 15 2 756t027 15 2 760573 15 3 746868 23 3 751510 23 3 756103 23 3 7 60649 2 746945 31 4 751587 30 4 756180 30 4 760724 30 - 747023 39 5 751664 38 5 756256 38 5 760799 38 6 7474101 47 6 751741 46 6 75633' 46 6'70875 45 7 747179 55 7 7,51818 54 7 756408!53 1 7 6 09 0 53 8 747256 62 8 751895 62 8 756484 61 61 8 761025 60 9 747334- 70 9 751972 70 9 756560 69 9 61100 68 5590 747412 5650i 752048 5710 7566036 5770 761176 1 747489 8 1 752125 8 1 75'6712 8 1 76125 8 2 7'47567 16 2 7220'2 15 2 756788: 15 2 761326 15 3 7476415 23 3 7.52279 23 3 756864 2"3 3 761402 23 4 747722 31 4 752356 30 4 756940 30 4 761477 30 5 747800 39 5 752433 38 5 757016 38 5 761552 38 6 747878 47 6 752509 4t6 6 757092 46 6 761627 45 7 7 4 7 5; 7 752586 54 7 757168 53 7 761702 53 8 748033 62 8 752663 62 8 757244 61 8 761778 60 9 748110 70 9 752740 70 9 757320 6 9 9 76183 68 5600 748188 5660 752816 5720 757396 5780 761928 1 748266 8 1 752893 8 1 757472 8 1'i2003 8 2 748343 16 2 752970 757548 15 2 763078 15 3 748421 23 3 753047 2 3 757624 23 3 762153 2t 4 748498 31 4 753123 30 4 7T57700 30 4 762228 30 5 748576 39 5 75329'0 38 5 757775 8 5 7623-03 38 6 7486531 47 l 6 753277 46 6 757851 46 6 76278 45 7 748731 54 7 7538353 54 7 757927 53 7 762453 52 8 748808 62 8 753430 62 8 758003 61 8 7625T29 60 9 748885 70| 9 753506 70 9 758079 68 9 762604 68 5610 748963 5670 753583 5730 758155 5790 762679 1 749040 8 71 753660 8 I 7582:30 8 1 76754 8 2 749118 16 2 r753736 15 2 758306 15 2 762829 15 3 749195 23 3 753813 23 3 758382 23 3 762904 22 4 7492-72 31 4 753889 30 4 758458 0 30 4 762978 30 75 49350 39 5 753966 38 5 758533 38 5 763053 38 G 749427 47 6 754042 46 6 758609 46 6 763128 45 7 749504 54 7 7 54119 54 7 7578685 53 7 763203 52 8 749582 62 8 754195 62 8 758760 61 8 763278 60.9 749659 70 9 754272 70 9 758836 68 9 763353 68 Page 23 LOGARITHMS OF NUMBERS. 23 No. Log. Prop. No. Log. Prop. No. Log. P Prop. No. Log. Prt. Part. Part. Part. 5800 763428 5860 767898 5920 772822 5980 776701 1 76'3503 7 1 767972'7 1 7723950 7 1' 776774 7 2 763578 15 2 768046 15, 2 772468 15 2 776846 14 3 763653 22 3 768120 22 3 772'542 22 3 776919 22 4 763727 30 4 768194 30 4 772615 29 4 776992 29 5 763802' 37 5- 768268 /37,, 772688 37 5 777064 36 6 763877 45 6 768342 45 6 772762 44. 6 7-77137 43 7 7.63952 52 7 768416 52 7 772835 51 7 777209 51 8 764027. 60 8 768490 59' 8 772908 59 8 777282 58 9 764101 67, 9 768564 67 9 77'2981 66 9 77735 4 65 5810 764176 5870 768638 5930.773055 5990 777427 1 764251 7 1 768712 7 1 7 773128 -7 1 777499 7 2 764326- 15.'2 768786 15 2 773201' 15 2 777572 14 3 764400 22 3 7'68860 22'3 773274 22 3 777644 22 4 764475 30 4 768934 30 4 773348 29'4 777717 29 5 7:64550 37 5 769008 37 5 773421 87. 5 777789 36 6 764624 45 6 769082 45 6 773494' 44 6 777862 43 7 764699 52 7 769156 52 7 77'3567 51 7 777934 51. 8 764774 60 8'769230 59 8 773640 -59 8 778006 58 9 76188 67 9 769303 67 9 773713 66 9 778079 65 5820 764923 5880 769377'5940 773786 6000 778151. 1 764998 7 1 769451 7 1 773860 7 1 778224 7 2 765072 15 2 769525 15 2 773933 15 217782996 14 3 765147 22 3 769599 22 3 774006.22 31 778368 22 4 7'65221 30.4 769673 30 4 774079 29' 4 778441 29 5 765296 37.5' 769746 37 5 774152 37 5 778513 36 6 -765370 45 6 769820 45 6 774225 44' 6 778585 43 7 765445 52 7 769894 52 ) 7 774298 51 7 778658 -51 8 765520 60 -8 769968 59 \8 774371 59 8 778730 58 9 765594 67 9 770042 67 9 774444 66'9 778802 65 5830 765669 5890 770115 5950 774517 6010 778874 1 7.65743- 7 1'770189 7 1 774590 7 1 778947 7 2- 765818 15 2 770263 15 2 774663 15 2 779019 14 3 765892 22 3 770336 22 3 774736 222 3 77909-1 22 4 765966 30' 4 770410 30 4 774809 29 4 779163. 29 5 766041 37 5 770484 37. 5 774882 37 5 779236 36 6.7661i5 45 6 770557 45 6 774955 44 6 779308 43 7 766190 52 7 770631 52 7 775028 51 7, 779380 51 8 766264 60 -' 8 770705 59 8 8775100 59 8 779452 58 9 766338 67 9 770778 67 9 775173 66 9 779524 65 5840 766413.- 5900 770852 5960 775246 6020 779596 1'766487 7- 1 770926 7 1 775319' 7'1 779669 7 2 766562 15 2 770999 15' 21775392 15 2 779741 14 3 766636'22 3 771073 22 1 3.775465 22 3 77-981.3 22 4 766710 30 4 771146 30' 4 775538 29 4 779885 29 5 766785 37 5 -771220 37 1 5 775610 37 5 779957 36 6 766859 45 6 771293 45 6 775683 44 6 780029 43 7 766933 52 1' - 7 771367 52 7'775756 51 7 780101 50 8 767007 60 8 771440 59 8'775829 59 8'780173 58 9 767082 67 1 9 771514 67' 9 775902 66' 9 78.0245.65 5850 -767156 5910 771587 5970. 775974 6030 780317 1 767230 7,1 771661 7 1 776047 7 1 780389 7 2'767304 15 1 2 77173,4 15'2 776120 15 2 780461 14 3. 767379 22 3.771808 22 3 776193 22 3'780533 22 4 -767453 30, 4 771881 301 4 776265 29 4 780605 29 5 767527 37'5 771955 37 1 5776338 37'5' 780677' 36 6 767601 45. 6 772028,44 6 77'6411 44 6 780749 43 7 767675 52 7 772102 52:7 776483 51 7 780821 50 8 767749 59- 8 772175 59..8 776556 59 8 780893 58 9 767823 67 9 772248 67 1 9 776629 66 9 780965 65 Page 24 24 LOGARITHMS OF NUMBERS. Log Prop. NPr Lo Popg. P rop. N Prop.!No. |Log. _.N.eePart. P No. Log. Parto. Log. Part. 6040 781037' 6100 785330 6160 789581 6220 793790 1 781109 7- 1 785401 7 1 789651 7 1 793860 7 2 781181 14 2 785472 14 -2 789722 14'. 2.793930 14 3. 781253 22 3 785543 21'3 789792 21 3 794000 21 4 781324 29, 4 785615 28 4 789863 28 4 794070 28 5'781396 36 5 785686 36' 5 789933 35 5- 794139 35 6 781468 -43 6 785757 43'6 790004 42 6 794209 42 7 781540 50 -7 785828 50 7. 790074 49 7 794279 49 8 781612 58 8 785899 57 8 790144 56' 8 794349' 56 9 781684.65 9 785970 64 9 790215 63.9 794418 63 6050 781.7.55 6110 786041 6170 790285 6230 794488 1 781827 7 1 786112 7' 1 790356 7 1 794558 7 2 781899 14 2 786183 14.. 2 790426. 14 2 794627 14' 3 781971 22 3 786254 21 3 790496 21 3 794697 21 4 782042 29.4786325. 28 4 -790567 28 4 794767 28 5 782114 36 5 786396 36 5 790637 35 5 794836 35 6 782186 43 6 786467 43' 6 790707 42 6 794906 42 7 782258 50 7 786538 50- - 7 790778 49 7 794976 49 8 782329 58 8 786609 57 8 790848 56 8 795045 56 9 78"401 65 9 786680 64 9 790918 63 9 795115 63 6060 78P473' 6120 786751 6180 790988 6240 795185 1 782544 7 1 786822 7. 1 791.059 7 1 795254 7 2 782616' 14 2 786893 14 2 791129 14 2 795324 14 3 782688 21 3 786964 21 3 79-1199 21 3 795393 21 4 782759 29. 4 787035 28 4 791269 28. 4 795463 28 5 782831 36 5 7871,06 36 5 791340 35 5 795,532 35, 6 782902 43 6 787177 43 " 6 791410 42 6 795602 42 7 782974 50 7 787248 50 7. 79148'0 49' 7795671 49 8. 783046 57 8 787319 57 8 791550 56 8 i95741 56 9- 783-117 64 9 787390 64 9,791620 63 9' 795810,63 6070' 783189' 6130 787460 116190 791691 6250' 795880 1 783260 7' 1 787531 7 1 791761 7 1 795949 7 2 783'332 14. 2 787,602 14 2 7-91831 14 2 796019 14 3 783403 21 3' 787673 21. 791901 21 3 7.96088 21 4 783475 29 4 787744' 28. 4 791971 28 4 796158 28 5 783546 36 5 787815 35 5 792041 35 5 796227 35 6 783618 43. 6. 787885 42 6 792111 42 1 6 796297 42 7 783689 50' 7 787956 49 7 792181 49 7 796366 49 8 783761 57 8 788027 56 8 792252. 56 8 796436 56 9 783832 64 9 788098.63 9 792322 63 9'796505 638 6080 783904.' 6140 788168' 1 6200'792392 6260 796574 1 783975' 7 1 788239 7 1 792462 7 1 -796644' 7 2 784046 14 2 788310 14 2. 792532 14 2' 796713 14 3 784118 21 3'788381 21 3 792602 21 3 796782 21 4 784189 29 4 788451 28 4 792672' 28 4.796852 27 5 784261 36 1 5 788522 35 1 5 792742 35 5 796921 35 6 784332' 43 6 788593 42 6 792812 42 6 796990. 42 7 784403 50'7 78866.3 49 7 792882 49, 7 797060 49 8 784475 57 8.788734 56 11-8 792952 56 1 8 797129 56 9 t84546 64 9 788804 63 9 793022 63 9 797198 62 6090 784617 6150 788875, 621'0 793092 6270 797268 1 784689 7 788946 7 1 1793162 7 1. 1 797337 7 2 784760 14 2 789016 14 21793231 14' 2 797406 14 3 784831 21.' 3 7890871 21'3 793301. 21 1 3 797475. 21 4 784902.29 41789157 28' 4 793371. 28 1 4'797545 27 5 784974.36'5 789228 35 5'793441 35 |'5 797614 35 6 785045 43:6 7789299 42 1 6 793511. 42 6 797683, 42 7 785116'50 7 789369- 49 7 793681 49 7 79775'2 49 8 785187 57 - 8 789440, 56.8 793651 56 8 797821 56 9 785259 64 9' 789510 63' 9 793721 6311 9 797890 62 Page 25 LOGARITHIMS OF NUMBERS. 25 No. Log. Prop. NO Log Prop. No. Prop. Log. irt. Log.i Part. _____o.- Part. i Part.. o. rt 6280 797960 I 6340 802089 6400 806180: 6460 810233 1 798029 7 1 802158 7 i 1 806248 7 1I 1 810300 7 2 7'98098 14 I 2 802226 14 l 2 806316 14 2 810367 13 3 798167 21 II 3 802295 21 3 806384 20 3 810434 20 4 798236 28 I 4 802363 27 4 806451 27 4 810501 27 5 798305 34') 5 802438 34 5 806519 34- 5 810,569 33 6 798374 41 6 802500 41 6 806587 41 6 810636 40 7 798443 48 j 7 80256 48 7 806655 48 5,810703 47 8 798512 55 8 802637 55 8 806723 54 8 810'770 54 9 798582 62 9 802705 62 9 806790 61 9 810837 60 6290 798651 1 6350 802774 6410 806858 6470 810904 1'798720 7 1 802842'7 1 806926 7 1 8109'71 7 2 7'98'789 14 2' 80'(210 14 2 80699410 1 2 811038 13 3 788,5802979 21 3 807061 20 3 811106 20 47'*)79897 28 4 27 428 807129 27 4 811173 27 51'08996 3-1 5 803116 34 1 5 807197 34 5 811240 33 6 99065 41 6 803184 41 1 6 807264 41 6 811'07 40 7'i99184 48 7 80323o2 48 7 807332 48 7 811374 47 8, 799203 55 8 803320 55 8 807400 54 8 811441 54 9 799272 62 9 803389 62 11 9 807467 61 9 811508 60 63001 799341 6360 803457 6420 807535 6480 811575 11,799409 7 1 803525 7 1 807603 7 1 816462 7 2 799478 14,1 2 803594 14 2 807670 14 2 811709 13 i 3 799547 21 3 803662 21 3 807738 20 3 811776 20 4 799616 28' 4 803730 27 4 807806 27 4 811843 27 5:799685 34: 5 80t3798 34 5 807873 38 5 811910 33 6 709754 41 6 803867 41 6 807941 41 6 811977 40 7 I7998923 48 7 803935 48 7 808008 48 7 812044 47 8 799892 5 8 804003 51 8 808076 54 8 812111 54 9 79939611 62 U 9 804071 62 9 808143 61 9 812178 60 6310 800029 t 6370 804130 6430 808211 490 812245 1 8000981 7 1 804208 7 1 808279 11812312 7 2 800167 14 2 804276 14 2 88()346 14 2 812378 13 3 800236i 21! 3 8081344 21 1 3i8084143 20 3 812445 20 4 800305 8 4804412 27 1 4 808481 2 7 4 812512 27 5 8003783 34 i 804486 34 5 05 80849 34 5 8126579 33 6 800442: 41 6 804548 41 6 808(;616 41 6 S12646 40 7 800511 48 7 8046l16 48 7 808684 48; 7 812713. 47 8 800580 t 5 8 80;4685 5 8 808751 | 5 8 812780 54 9 800648': 9:80-753 62 9 808818 61 9 812847 60 63 20 800717 |:380 804821:6440 808886:6-00 812913 1 800786 7 1 804889 i 1 808953 7 II 812980 7 2 800854 14 2 804957 14 2 809021 13 i 2 8130D47 13 3 800923 21 3 805025 21 0 3 809088 20 3 813114 20 4 800992: 28: 41805093 27 i 4 8091a56 7i 4 813181 27 5 8010;60 3j 4 180516l 34 1. 5 809223 34 1 5 818247'3 6 8011291 41 1 6805229 41 l 6 80929O 40 6 813314 40 7 801198 1 48' 805297 48 1 7 809358 47 81381 47 818012661 55 o 80336t5 54 8 80942.1 54 8 81 4448 54 9 801335: 62 0, 805433 1 9 809492 tl 3 i 818514 O60 6U30 801404 16:0 805501 450 809560 Cl610 813581 | | 1 |801472 7 1 80; 559 |7 1 1 |80.,627| 7 I 11 81830 8 7 2 801541 14 2 8056;37 14 1 2 809694 13 21 813714 13 31801609 21 3 805705 20 1 3 80976'2' 2)0 I 3813781 20 4 801678 27 I 4 805773 27 4 8 09829 27 1 4 813848 1 27 5 801747 34 5 805841 34 5 809896 34 5 813914 33 1 6 801815 41 6 805908 41 6 809964 40 6 813981 40 1 7 801884 48 7 805976 48 7 810031 47 7 8140481 47 8 801952 55 8 806044 54 8 8100Q98 54 8 814114 54 9 802021 62 9 806112 61 1 9 810165 61 91 814181 60 Page 26 26 LOGARITIIMS OF NUMBERS. No. oLog. No. Log. r o. Log. l'Pro. Pan. Part. Port. Log -L C6520 814248 6580 818226 6640 822168 6700 896005 1 814;314'7 1 818292 7 1 8 7 1 826140 6 2 814881 13 2 818358 13 2 8282299 13 2 8t6204 13 3 814447 20 3 818424 20 3 822364 20 3 8262t9 19 4 814514 26 4 818490 26 4 822430 26 4 82 334 6 5 814581 33 5 818556 33 5 822495 33 5 82i6399 32 6 814647 40 6 818622 40 G 822560 39 6 826464 39 7 814714 46 7 818688 46 7 82 2626 46 7 826528 45 8 814780 53 8 818754 53 8 822691 52 8 8 82 65't3 52 9 814847 60 9 818819 59 9 822756 i 59 9 82668 58 G630 814913 6590 818885 G5O0 8228231 6710 8267522' 1 814980 7 1 818951 7 1 822887 7 I 1 86787 6 2 815046 13 2 819017 13 2 822952 13 2!82'6852 13 3 815113 20 3 819083 20 3 823018 20 3 826917 19 4 815179[ 26 4 819149 26 4 823083 26 4 82S981 26 5 815246 3a3 5 819215 33 5 823148 3 5 827046 82 6 815312 40 6 819281 40 8232131 39 6 827111 39 7 815378 46 7 819346 46 7 823279 4 6 1 7 82717 45 8 8154465 53 8 819412 5 8 823344. 52 1 8 8272401 52 9 815511 60 9 819478 59 9 8 3409 40 59 9 8278305 58 6510 815578 6600 819544 6(660 823474, 67 0 82O769 1 815644 7 1 8196130 7 1 823539 7 1 8'2743I4 6 2 815711 13 2 819675 13 2 823605 1 13 2 8274c98 1 13 3 815777 20 3 819741 20 3 82'3"670 20 3 827563' 19 1 4 8158413 2 4 819807 2 4 823735 26 4 8'276;8 26 5 815910 38 5 819878 33 5 238 0 33 5 8)762 382 6 8159706 40 6 81' 9939 40 6 82:38;65 3t 6 827757 [9 7 816042 46 7 820004 46 7 8 )2(0 46 827821 i 45 8 816109 53 8 282!0070 8 8 52 8 82788 6 52 9 816175 6t0 9 82 0136 9 9 8 4061 59'3 9 827,1 58 6.50 816241 6610 820 01 6 60J 8423140C) G6730 828015 1 816308 1 820267 7 1 1 824191 6 1 88080 6 2 816374 13 2 820.333 1 2 824o256 13 2 82814_4 13 3 816440 20 3 820399 i' ) 3 824321 19'3 8 8') 19 4 816506 246 4 8204641 26 4 843 2 4 828273 2,6 5 816573 3 5 82030 1 33 5 8244 1 5 828338 32' 6 816639 40 6 820595 40 0 824516 1 9 6 82840 1 39 7 816705 46 7 820661 46 8241 81 4 828467 45 816771 53 820727 53 82446 8 88531 52 9 816838 60 9 820792 59 9 824711 58 9 828595 58 6560 816904 6620 82085S1 668O 824776 6740 828660 81t6970 7 1 8080924 7T 1 824841 6 1 8'28 724 6 8170361 13 2 8099 I 2 I824906 13 2 828789') 13 817102 20 3 82105 20 3 8491 *I 3 828853 19 4 817169 26 4 8211120 26 4 825036 6 I 4 828918 26 817235 33 5 821186 33 5 825101 32 82898 3 2 6 817301 40 6 8212. 1 40 6 825166 39 296 82 9046i 39 87 817367 46 7 8 7 46 7 825231 45 7 829111 45 817433 53 8 821382 8 825296 52 8 829175 5" 9 817499 59 9 821448 59 i 9 825361 58 9 829233 58 657 0817565 6f630 6821514 f6690 825426 6750 i 82930o 1 817631 7 1 821579 7 1 8 254911 6 1 829368 6 2 817698 13 2 821644 13 1 2 82.5556 1 2 8294321 13 3 817764 20 3 821710 20 825621 19 829497 19 4 817830 26 4 821775 26 4 825686 26 4 8299561 26 5 817896 33 5 821841 33 5 825751 32 5 829B625 32 6 81 7962 40 6 821906 8139 6 8295815 39 6 829690 3 7 818028 46 7 821972 46' 7 825880 45 7 829754 45 8 818094 53 88220371 52' 8 82594.5 52 88298181 52 9 818160 59 9822103 59 9i 826010 58 9 1829882 58 Page 27 LOGARITHMS OF NUMBERS. 27 No. ~Log. Pop. No. Log. No. Log Pore. 0 Log. g Nort. Part. I _$PrpPart. Prp 6760 829947 6820 833784 6880 837588. 6940.841359 1 830011 -6 1833848 6 6 837652 6 I 1 841422 6 2 830075 13. 2 8383912 13 2 837715 13 2 841485 13 3 830139 19. 3 833975 19 3 837778 19 I 3 841547 19 4 830204 26 4 834089 26 4 837841 25 4 841610 25. 5 830268 32 51 834103 32'5- 837904 32 5 841672 31 6. 839332 38 6 834166 38. /6 837967 38 6 841785 ~8 7 830396'45 7 834230 45 7, 8380380 44 7 841797 44 8 830460 51 8, 834293 51 8 838093 50'.8 841860 50 9 830525 58 9 834357 58 9 838156 57 9 841922 56 -;677.01830589 6830 834421 -.6890. 838219 6950 841985 11.83053 6 1 834484 6. 1'838282 6 1 842047 6 2 830717 13. 2' 834548 13 2 838345 13 2 842-110 12 3 830781 1-9 3 834611 19 3 838408 19 1 3 842172 ]19 4 830845 26 4 -834675. 26 44 838471 25' 4 842235 25 5 830909 32 51834739 32 5 838534 32 5 5842297 31 6 830973 38 6 834802 38 6 838597 38 6 842360 37 71831087 45 -7 834866.45 7 838660 44.7 842422. 44 8 831102 51'8 834929'3 51: -8 838723 50 I 8' 842484 50 9 831166 58 /9 834993 58 9 838786 57 9 842547 56 6780 831230 6840 835056. 6900 838849 6960 842609' 1'831294 6 1 835120 6 1 838912 6' 1 842672 6 2 831358 13 2 835183 1 3 2 838975 13. 2 842734 12 3 831422 19 3 835247 19 3 839038 19 3 842796 19 4 831486 26 4;:835310{ 26 4 839101 25, 4 842859'25 5 831550 32 51 83537378 3'2 5 839164 31 5.842921. 31 - 6 831614 38 6 835437 38 6 839'227 38 6 18429831 7 7 7 8-31678 45 7T 835500 45 7 839289- 44 7 843046 44 8 831742 51 8'835564 51 8 8'39352 50 8 843108 50 9 831806 58 9 835627.58 9 839415 57 9 843170 56 6790 831870 6850 835691 6910 839478 6970.843233 1 831934.' 6 1 835754 6 1 839541 6 1 843295 6 2 831998 183 2 835817. 13 2 839604 13 2 843357 12 3 832062 19 3 835881 19 3 839667 19 3 843420 19 4'832126 26 4 835944 26 4 839729 25 - 4 843482 1.25 5 832189. 32 5 836007 32 - 5 839792 31 5 843544 31 6 832253 38 6 186071 38 6 839855 38. 6 843606 37 t 7 832317 45'7 836134 45 7 839918 44 i 843669 4~3 8 832381 51 8 836197' 51 8.839981 50 8 843731.50 9 832445. 58'9836261 58 9 840043.57 9 843793 56 6800 832509 6860 836324 6920 840106 6980.843855 1 1832573 6' 1 836387. 6 1 840169- 6 1 84391-8. 6 2 830637 13,2 836451 13 2 840232 13 2' 8'43980 12 3 832700.19 3 836514. 19' 3 840294 19 3 844042 19 4 83276'4 26' 41836577 26 4 840357.25 4 8441'04 25 5 832828' 32 5 83664,1' 832 5 840420 31 5 844166 31 6.832892 38 6 836704 38 1 6 840482 38 6'844229 37 7' 832956 45 7 836767 45, 7 8405451 44 ] 7 844291 43 8 833020 51 8 836830 51 8' 840608 50 8 844353 50 89 833083 58 9 836894 58 9. 840671 67 9'844415 56 6810 833147 6870 8369571. 6930 840733 6990 844477 1 833211 6. - 18370201 6' 1 840796 6' 1 844539'' 6 2 8332756 13 2 837083 13. 2 1'840859 13 2 844601 12 3 833338 19 3'837146 19 3 8409211 19l' 3 844664 "19 4 833402 26 4 -4 837210 25 1 4 840984 25 1 4 844726 25 58 334866 32 a 5 837273 32 5 841046.1 31l 5 844788 31 6 833530'38 I. 6. 837336 38 1 - 6 841109-' 38 6- 844850' 37 71833593, 45, 7 837399 144 7 841172 44' 7 844912 43 8 833657 51 8 837462-'51 8 841234 50 8' 844974.50 918337211 58 9 887525 57 1 9'1841297 56 9 845036 56 I'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Page 28 28 LOGARITHMS OF NUMBERS. No. Log. Prop. No. Log. Prop No. Log. Pro. N. Log. ~ o Log. Prtart. Pro. Pr. 7000 845098 7(60 848805 7120 852480 7180 856124 1 845160, 6 1 848866 6 1 852541 6 1 856185 6 2 845222 12 2 848928 12 2 852602 12 2 856245 12 3 845284 19 3 84989 18 3 852663 18 3 856306 18 4845346 25 4 849051 25 4 852724 24 4 856366 24 5 845408 3 5 849112 31 5 852785 30 5 856427 30 6 845470 37 6 849174 37 6 852846 37 6 856487 36 7 845532 43 7 849235 43 7 852907 43 7 856548 42 8 845594 50 8 849296 49 8 852968 49 8 856608 48 9 845656 56 9 849358 55 9 853029 55 9 856668 54 7010 845718 7070 849419 7130 853090 7190 856729 1 845780 6 1 849481 6 1 853150 G 1856789 6 2 845842 12 2 849542 12 2 853211 12 2856850 12 3 845904 19 3 849604 18 3 853272 18 3 8-56910 18 4 845966 25 4 849665 25 4 853333 24 4 856970 24 5 846028 31 5 849726 31 5 853394 30 / 5 857031 30 6 846090 37 6 849788 37 6 853455 37 6 857091 36 7 846151 43 7 849849 43 7 853516 43 857151 42 8 846213 50 8 849911 49 8 853576 49 8 85721'2 48 9 846275 56 9 849972 55 9 853637 55 9 857i72 54 7020 846337 7080 850033 7140 853698 7200 857332 1 8463]99 6 1 850095 6 1 853759 6 1 85'5,93 6 2 846461 12 21850156 12 2 853820 12 2 895453 12 3 846523( 19 1 3 850217S 18 8 3 53881 18 3 8551 18 4 846584 25 4 850279 25 4 853941 24 41 857574 24 5846646 31 51850340 31 5 84002 30 1 /Se3- 3 0 6 8467081 37 1 6 8504011 37 6 854063 3 7 618571694 336 7 8467701 43 71 850462 43 i 7 854124 43 11 71857754 42 8 846832 50 8 850524j 49 8 854185 49 8 81857815b 48 9 846894 5 9 850s85 55 9 8-.4245 55 918578; 57030 84955 7 090 850646 7150 854.1306 7 7210 85 95 1 847017 6 1 850707 6 7 1 854367 6 1 857995 6 2 847079 12 2 850769i 1 2 854427 12 2 858056 12 3 847141 1 3 850830 18 3 84488 18 3 858116 18 48472021 25 4850891 25 4 854549 24 4 858176 2) 5 847264 31 5 850952 31 5 854610 30 5 5 8258'36 30 6 847326 37 6 801014 37 6 864670 36 6 85821( 36 71847388 43 7 851075 43 7 85 731 42 7 8583-57 42 8847449 50 8 851136 49 8 854792 48 I 8858417' 48 9 847511 56 9 851197 55 11 9 8548525 54 9858477 54 7040 84 7573 17100 851258 17160 84913 7290 85)8537 1 8476341 6 18513201 6 11854974 4 6 1 1 858697 6 2 847696 12 2 851381 12 285r5034 12 j 29 858671 12 3 847758 18 3 851442 18 3 855095 18 3 858718 18 4 84'7819 25 4 851503 25 4 855156 4 85877 4 5 847881 31 5 851564 31 5 21 30 85883 | 0 6 847943 37 1 16 851625 37 6 855277 8 36 i 6 8388982 36 7 848004 43 11 7 851686 43 7 855337 42 7 838958 42' 8 848066 49 8 851747 49 8 855398 48 8 8 9018 48 9 848127 55 9 851808 55 9 855459 54 9 859078 54 |7050 848189 7110 851870 7170 855519 l 7230 859138 1848251 6 1851931 6 1 855580 6 1 859198 6 2 848312 12 21851992 12 2 855640 12 0 2 859258 12 3 848374 18 i 3 852053 18 3 855701 18 i 3 8.39318 18 4 848435 25 4 852114 25 4 855761 24 i 4 859378 24 5 848497 31 5 852175 31 5 855822 30 1 5 859438 30 6 848559 37 6 85226 37 6 855882 6 6 6 859499 36 7 848620 43 7 852297 43 7 855943 42 7859559 42 8 848682 49 8 852358 49 8 856003 498 8859619 18 9 848743 55 1 9 1852419 55 9 856064 54 9ii 859679 54 Page 29 LOGARITHMS OF NUMBERS. 29 Prop.'7 9 3 8 3 3 3 - i Ppr Pr~~~op.Prop. No. Log. Part. No. Log.' i o. Log. rop. Log. Part. Part. Part. Part. I 7240 859739 7300 8633923 7360 866878 7420 870404 1 859799 6 1 863382 6 1 866937 6 1 870462 6 2 859858 12 2 8683412 12 2 866996 12 2 870521 12 3' 859918 18 3 863501 18 3 867053 18 3870.579 18 4 859978 24 4 863561 24 1 4 867114 24 4 8706B38 24 5 860038 30 5 863620 30 5 87173 29 5 870696 29 6 860098 36 6 86s(3680 36 6 867232 35 6 870755 35 7 860158 42 7 863739 42 7 867291 41 7 870813 41 8 860218 48 1 8 863798 48 1 8 867350 47 8 870872 47 9 860278 54 9 863858 54 9 867409 53 9 870930 53 7250 860338 7310 863917 7370 867467 I7430 870989 1 860398 6 1 86977 6 1 867526 6 1 871047 6 2 860458 12 2 864036 12 2 867585 12 2 871106 12 3 860518 18 3 864096 18 3 867644 18 3 871164 18 4 860578 24 4 864155 24 4 867703 24 4 8712283 24 5 860637 30 5 864214 30 5 867762 29 5 871281 29 6 860697 36 6 864274 36 6 867821 35 6 8713'39 35 7 860757 42 7 864333 42 7 867880 41 7 871398 41 8 860817 48 8 864392 48 8f867939 47 8 871456 47 9 860877 54 9 864452 54 9 867998 53 9 871515 53 7260 860937 7320 864511 7380 868056 7440 871573 1 860996 6 1 1 864570 6 1 868115 6 1 871631 6 2 1 861056 12 I 2 864630 12 2 868174 12 2 871690 12 3 861116 18 3 864689 18 3 868233 18 3 871748 18 4 861176 24 4 864748 24 4 88292 24 4 871806 23 5 861236 30 5 864808 30 5 868350 29 5 871865 29 6 ( 861295 36 6 864867 36 6 868409 35 6 1 871923 35 7 861355 42 7 864926 42 7 868468 41 7 871981 41 8 1 8631415 48 8 864985 48 8 868527 47 8 872040 47 9 861475 54 9 865045 54 9 868586 53 1 9 872098 53 7270 861534 7330 865104 17390 868644 17450 872156 1 861594 6 1 865163 6 1 868703 6 1 872215 6 2 861654 12 2 865222 12 2 868762 121 2 872273 12 3 861714 18 3 1865282 18 3 868821 18 l 3 872331 18 4 861773 24 1 4 1865341 24 4 868879 24. 4 872389 23 5 8618331 30 1 5 865400 30 5 81 68938 29 5 872448 29 6 861893 36 6 865459 36 6 868997 3 5 6 872506.35 7 861952 42 7 865518 42 7 869056 41 8 72 564 41 8 862012 48 8 865578 48 1 8 869114 47 8 876221 47 9 862072 54 9 865637 54 9 869173 53 9 872681 53 71280 862131 7340 865696 7400 869232 1460 872739 1 862191 6 1 865755 6 1 869290 6 1 8 77 97 6 2 862251 12 2 865814 12 2 869349 12 2 872855 1 2 3 862310 18 11 3 865874 18 3 869408 18 3 87 2913 18 4 862370 24 4 865933 24 11 4 869466 24 j 41 872972 23 5 862430 30 5 865992 30 5 869525 29 5 18,3030l 20 6 862489 36 6 866051 36 6 869584 35 6 873088 35 7 862549 42 11 866110 42 7 869642 41 7 873146 41 8 862608 48 8 866169 48 8 869701 47 i 8 83204 47 9 862668 54 9 866228 54 9 869760 53 9 873262 53 7290 1 862728 7350 866287 7410 1 869818 7470 1 8733321 1 862787 6 1 866346 6 1 869877 6 j 11873379 6 2 862847 12 2 866405 12 2 869935 13 2 873437 12 3 862906 18 3 866465 18 3 869994 18 j 31 83495 18 4 8629661 24 4 866524 24 4 870053 24 4 873553 23 5 863025 30 5 8665)83 30 5 870111 29 5 873611 29 6 863085 36 l 6 866642 35 6 870170 3 51 6 8i3669 35 7 863144 42 7 866701 41 7 870228 41 17 T873727 41 8 863204 48 8 866760 47 8 870287 47 8 8 785385 47 9 863263 54 9 866819 53 9 5870345 53 9j838441 53 Page 30 30 LOGARITHMS OF NUMBERS. No. Log. Pro. No Log. |roIO No. Log. ro o.p _. Part__ I IPart. P' art. I 7480 873902 7540 877371 7600 880814 760 884229 1 873960 6 1 877429 6 1 880871 6 1 884285O 6 2 874018 12 2 877486 12 2 880928 11 21884342 11 3 874076 17 3 877544 17 3 880985 17 3 88439)9 17 4 874134 23 4 877602 23 4 881042 23 4 884435 23 5 874192 29 5 877659 29 5 881099 28 5 884512 28 6 874250 35 6 877717 34 6 881156 34 6 884569 34 7 874308 41 7 877774 40 7 881213 40 7 884625 40 8 874366t 46 8 877832 46 8 881270 46 8 884682 46 9 874424 52 9 877889! 52 9 881328 51 9 8847391 51 7490 874482 7550 877947 7610 8813885 7670 884795 1 874540 6 1 878004 6 1 881442 6 1 884852 6 2 874598 12 2 878062 12 2 881499 11 2 184909 11 3 874656 17 3 878119 17 3 881556 17 3 884(865 17 4 874714 23 4 878177 23 4 881613 23 4 885022 23 5 874772 29 5 8.78234 29 5 881670 28 51 885078 28 6 874830 35 6 878292 34 6 881727 34 6 885135 34 7 874887 41 7 878349 40 7 881784 40 7 885192 40 8 874945 46 8 878407 46 8 881841 46 8 885248 46 9 875003 52 9 878464 52 9 881898 51 9 885305 51 75O00 875061 7560 878522 7 7620 881955 7680 885361 1 875119 6 1 878579 6 1 882012 6 1 88541.8 6 2 875177 12 878637 12 21882069 11 21885474 11 3875235 17 3 878ii94 17 3 88216 1 3 885563 17 4 875293 23 4 8787511 23 4 882183| 23 4 885587 23 875351 29 5 8878809 29 5 882240 28 5 8856i44 28 6 87540 35 6 87886fi(i 34l4 6 882297 34 6 8859009 O 87546 I 41 7 878924 40 882;354 40 7 885757 818755241 46 8 8789J81 46 8 882411 46 88813 45 9 875531 82 1 52 9 879038s 52 9 t882468 51 9 s88870 51 7510 8756 10 7570 879096 63 0 882524 7 61 090 88592(i 1 87,5698 6 1 879153 6 I 1 882581 6 1 885983 6 2 87571561 12 2 87911 12 2 8826ti38 1 11 2 886039 11 3 875813 17 3879268 17 3 88826951 17 3886096 17 4 875871 23 4 879325 23 4 882752 23 4 886152 23 5 875929 29 5 87938 29 5 882809 28 5 886209 28 6 875987 3 6 879440 4 6 882866 34 6 88626-5 34 7 876045 41 I 1 879497 40 7 882923 40 71 886321 39 8 876102 46 8 879 5-55 4; I 8 882980 46 8 886378 4.5 9 876160 52 9 879612 5 9 883037 51 9 886434 51 7520 876218 7.580 879669 | 7640 883093i 7700 886491 1 8762761 6 1 879726 6 1 883150 6 1 8865047 6 2 876333 12 2 879784 11 2 883207 7 11 2 886604 11 3 876391 17 3 879841 17 3 883264 17 3 886660 17 4 876449 23 4 879898 23 4 883321 2 4 886716 2 23 5 876507 29 5 879956 28 i 5 883377 1 28 5 886773 28 6 876561 34 6 880013 341 6 883434 1-34 6 886829 34 7 876622 40 7 880070 40 i 7 883491, 40 7 886881 39 8 876680 46 8 880127 46 8 883.548 46 8 886942 4.5 9 876737 52 9 880185 51 9 883605 51 9 886998 51 75030 8767950 880242 7650 883661 7710 887054 1 186853 6 1 880299 6 1 8837181 6 1 887111 C6 2 876910, 12 2 880356 11 2 883775 11 11 2 8871671 11 3 876968 17 3 880413 17 3 883832 17 3 887223 17 4 8770261'3 4880471 23 4 883888s 23 4 887280 23 5 877083 29 5 880528 28 5 883945 28 8873361 28 6 877141 3t 618803585 34 6 884002 34 6 8873921 34 I 877198 40 7 880642 40 iI 7 8840.59 40 7 887449 39 8 877256 46 8 880699 46 8 884115 4- 8 88750.5 4 5 9 877314 52 9 880756 51 9 884172 51 9 887561 51 Page 31 LOGARITHMS OF NUMBERS. 31 No. Log. Pro No. Log. Prop. No. Log. 1 Pr No. Log. Prop. Part * 0980 Part. Paro. Part. 7720 887617 7780 89098 780 894316 7900 897627 1 887674 6 1 891035 6 1 8 i4371 6 1 897682 6 2 887730 ] 1 2 891091 11 2 894427 11 2 897737 11 3 887786 17 3 891147 17 3 894482 17 3 897792, 17 4 887842 23 4 891203 22 4 894538 2:2 4 897847 22 5 887898 28 5 891259 28 5 8945938 27 5 I8(790CJ 27 6 887955 34 6 8913141, 34 6 894648 33 6 8977 3 7 888011 39 7 891370 39 7 894704 39 7 8'9801 " 39 8 888067 45 8 891426)f 45 1 8 894759 44 8 8')S06 44 4 9 888123 51 9 891482 50 1 9 894814 50 9 898122 &0 7730 888179 7790 891537 7850 894870 7910 8981,6 1 888236 6 1 891593 6 1 894925 6 1 898231 6 2 888292 11 2 891649 11 2 894980 11 2 898286 11 3 888348 17 3 891705 17 3 895086 17 3 898341 17 4 888404 22 4 8 91760 22 4 895091 22 4 89896 22 5 8884601 28 5 891816 28 5 895146 27 5 898451 27 6 888516 34 6 891872 33 6 895201 33 6 898506t 3 1 7 888573 39 7 891928 39 I 895257 39 7 898561 39 8 888629 45 8 891983 44 8 895312 44 8 898615 44 9 888685 50 9 8920839 50 9 895367 50 9 898670 0o 7740 888741 78001892095 7860 895423 7920 898725 1 888797 6 1 8921500 6 1 895478 6 1 898780 5 2 88885-31 11 2 892206 11 2 895533 11 2 898835 11 3 888909 17 3 892262 17 3 895588 17 3 898890 17 4 888965 22 4 892317 22 4 895643 22 4 898944 22 5 889021 28 5 819373' 28 5 895'3699 27 5 898990 27 6 889077 34 6 892429 378' 6 895754 33 6 899054 33 7 889134 39 7 892484 39 7 8'5809 39 7 899109 38 8 889190 45 8 892540 44 8I 895.)864 44 8 1 899164 44 9 889246 50 9 8921595 50 9 8 953 50 9 899218 50 7750 889302 7810 89'651 l8li80 895975 7930 899273 1 889j358 6 1 89'32707 6 1 896030 6 1 899328 5 2 889'3414 11 2 89272 11 2896085 11 2 899383 11 3 889470 1 ]7 3 89818 17 3 896140 17 3 899437 17 4 889526 22 4 8928 2 4896195 22 4899492 22 5 889582 28 5 8i 6)9 28,5 8'8i6251 27 5 899547 27 6 889638 34 6 892985 33 6 896306 33 6 899602 33 7 889694 39 7 8930240 39 7 896361 39 7 8996056 38 8 889750 45 8 893096 44 8 89641t 44 8 899711 44 9| 889806 50 9 893151 50 9 896471 50 9 899766 50 7760 889862 7820 89320 1 7 880 896526 7940 899820 1 889918 6 1 893'620 6 1 896581 6 1 899875 5 2 889974 11 2 893318 11 2 896636 11 2 899930 11 3 890030 17 3 893373 17 3 896692 17 3 899985 17 4 890086 22 4 893429 22 4 896747 22 4 900039 22 5 890141 28 5 893481 28 5 896802 27 5 900094 27 6 890197 34 6 893540 33 6 896857 33 6 900149 0 33 7 890253 39 l 7 893595 39 7 896912 39 7 900203 38 8 890309 45 8 893651 44 10 8 896967 44 8 900258 44 9 890365 50 9 893706 50 9 897022 50 9 900312 50 7770 890421 7830 893762 1 7890 897077 7950 900367 1 890477 6 1 8938178 6 1 897132 6 1 900422 5 2 890533 11 2 893873 11 2 89J7187 11 2 900476 11 3 8900589 17 3 893928 17 3 897242 17 3 9300531 17 4 890644 22 4 893984 22 4 897297 22 4 900586 22 5 890700 28 5 894039 28 5 807352 27 5 900640 27 6 890756 341 6 894094 33 6 897407 33 6 900695 33 7 890812 39 7 894150 39 7 897462 39 7 9007491 38 8 890868 45 8 894205 44 8 897517 44 8 900804 44 9 890924 50 9 894261 50 9 897572 50 1 9008581 51 Page 32 32 LOGARITHMS OF NUMBERS. No. Log. Prop. No. Log,. P. Log. No. Po. No. o.Part: Parrt. Pr. 7960 900913 8020 904174 8080 90741.1 8140 910624 1 900968 5 1 904228 5 1 907465 5'5 1 910678 2 901022 11 2 904283 11 2 907519 11 2 910731 11 3 901077 16 3 9043137 16 3 907573 16 3 910784 16 4 9301131 22 4 901391: 22 4 907626 22 4 9108.38 21 5 901186 27 5 904445 27 5 907680 27 5 910891 27 6 901240 33 6 904499 32 6 907734 32 6 910944 32 7 90129 5 38 7 9045t53 38 7 907787 38 7 910998 37 8 901349 44 8 904607 43 8 907841 43 8 911051 43 9 901404 49 9 904661 49 9 907895 49 9 911104 48 7970 901458 8030 904715 8090 907948 8150 911158 1 901513 5 1 904770 5 1 908002 5 1 911211 5 2 901567 11 2 904824 11 2 908056 11 2 91126ti 11 3 901622 16 3 904878 16 3 90810' 16 3 9110317 16 4 901676 22 4 904932 22 4 908163 22 4 911371 1 1 5 901731 27 5 904986 27 5 908217 27 5 911424 27 6 901785 33 6 900040 32 6 908270 32 6 911477 382 7 901840 38 7 905094 38 7 908324 38 7 911530 37 8 901894 44 8 905148 43 8 908378 43 8 911584 42 9 901948 49 9 905202 49 9 908431 49 9 911637 48 7989 902003 8040 9052)56 8100 908485 8160 911690 1 902057 5 1 905310 5 1 908539 5 1 911743 5 2i902112 11 2 905364 11 2 908592 11 2 9117971 11 3 902166 16 3 905418 16 3 908646 16 31 911850 16t 4 902221 22 1 4 905z472 22 4 908699 21 4 911903 21 5 902275 27 5 905526 27 5 908753 27 5 911956 27 6 902329 33 6 905580 32 6 908807 3 6 91200' 32 7 902384 38 7 905634 38 17 908860 37 7 9 12063 37 8 902438 44 8 905688 43 8 908914 43 8 912116; 42 9 902492 49 9 905742 49 9 9t)089 67 48 9 912169 48 7990 902547 8050 905796 8110 909021 8170 91'222 1 902601 5 1 905850 5 1 909074 5 1 91;227.5 5 1 2 902655 l11 2 905904 11 2 909128 11 2 912328 11 3 902710 16 3 005958 16 3 909181 16 3 912381 16 4 9027 64 22 4 906012 22 4 909235 21 4 9124o35 21 5 902818 27 5 906065 27 5 909288 27 5 912488 27 6 902873 33 6 906119 32 6 909342 32 6 912541 32 7 902927 38 7 906173 38 7 909'395 37 7 912594 37 8 902981 44 8 906227 43 8 909449 43 i 8 912647 42 i 9 903036 49 9 906281 49 9 9093502 48 9 912700 48 8000 903090 8060 906335 8120 909556 8180 912753 1 903144 5 1 906389 5 1 909609) 5 1 I1912806 5 2 903198 11 2 906443 11 2 1909663 11 21912859 11 3 903253 16 3 906497 16 3 909716 16 3 912913 16 4 903307 22 4 9065O50 22 4 909770 21 4 912966 21 5 903361 27 5 906601 27 5 909823 27 5 913 019 27 6 903416 32 6 906658 32 6 909877 32 6C913072 32' 7 903470 38 7 906712 38 1 909930 37 7' 913125 37 8 903524 43 8 906766 43 8 909984 43 8 913178 421 9 903578 49 9 906820 49 1 9 910037 48 9 913231 48 8010 903632 8070 906873'8130 910090 8190 913284 1 903687 5 1 906927 5 1 910144 5 913337 5 2 903741 11 2 906981 11 2 910197 11 2! 913390 11 3 903795 16 3 907035 16 3 910251 16 3 913443 16 4 903849 22 4 907089 22 4 910304 21 4 913496 21 5 903903 27 5 907142 27 51910358 27 5 913549 27 6 903958 32 6 907196 32 6 910411 32 11 1913602 32 7 904012 38 7 907250 38 7 910464 37 7 9136 55 37 8 904066 43 8 907304 43 8 910518 43 8 913708 42 9 904120 49 9 907358 49 9 910.571 48 9 913761 48, {903958 32~~~~~~~~~~~~~~~~~~~~~~~~~ Page 33 LOGARITHMS OF NUMBERS. 33 No. Log.. Log. Prrop. No. LL. Prrop. No. Log. Prop. Part. Port. Part. Part. 8200 913814 8260 916980 8320 920123 8380 923244 1 913867 5 1 917033 5 1 920175 5 1 923296 5 2 913920 11 2 917085 11 2 920228 10 2 923348 10 3 913973 16 3 917138 16 3 1920280 16 3 923399 16 4 914026 21 4 917190 21 4 9 20o332 21 4 923451 21 5 914079 27 5 917243'216 5 920384 26 5 5923503 26 6 914131 32 6 917295 31 6 920436 31 6 9235551 31 7 914184 37 7 917348 37 7 920489 36 7 923607 36 38 914237 42 8 917400 42 8,20541 42 8 923658 42 9 914290 48 9 917453 47 9 920593 47 9 923710 47 8210 914343 8270 917505 8330 920645 8390 923762 1 914396 5 1 917558 5 1 92 0697 5 1i 92814 5 2 914449 11 2 917610 11 2 9207,49 10 2 928865 10 3 914502 16 3 917663 16 3 920801 16 3 923917 16 4 914555 21 4 917715 21 4 206853 21 4 923969 21 5 914608 27 5 5917768 26 5 920906 26 56 924021 26 6 914660 32 6 917820 31 6 920958 31 6 9240 21 31 7 914713 37 7 917873 37 7 921010 36 7 924124 36 8 914766 42 8 917925 42 8 921062 42 8 924176 42 9 914819 48 9 917978 47 9 921114 47 9 924228 47 8220 914872 8280 918030 8340 921166 8400 924279 1 914925 5 1 918083 5 1 921218 5 1 924331 5 2 914977 11 2 918135 11 2 921270 10 2 924383 10 3 915030 16 3 918188 16 3 921.322 16 3 924434 15 4 915083 21 4 918240 21 4 921374 21 4 924486 21 5 915136 27 5 918292 26 5 921426 26 5 924538 26 6 915189 32 6 918345 31 6 921478 31 6 924589 31 7 915241 37 7- 918397 37 7 921530 36 7 924641 36 8 915294 42 8 918450 42 8 921582 42 8 924693 41 9 915347 48 9 918502 47 9 921634 47 924744 9 46 8230 915400 82)90 918555 8350 921686 8410 924796 1 915453 5 1 918607 5 1 921738 5 1 924848 5 2 915505 11 2 918659 11 1 921790 10 2 924899 10 3 915558 16 3 918712 16 3 921842 16 3 924951 15 4 915611 21 f 4'9118764 21 4 921894 21 4 925002 21 5 915664 27 5 91881t 26 5 921946 26 5 925054 26 6 915716 32 6 918869 31 6 921998 31 6 925106 31 7 915769 37 7 918921 37 07 922050 36 7 925157 36 8 915822 42 8 918973 42 8 922102 42 8 925209 41 | 9 915874 48 9 919026 47 9 922154 47 9 925260 46 8240 915927 8300 919078 8360 922206 8420 925312 1 915980 5 2919130 5 1 922258 5 1 925364 5 2 916033 11 2 919183 11 11 2 922310 10 2 925415 10 3 916085 16 3 919235 16 3 92232 16 3 925467 15 4 916138 21 4 919287 21 4 922414 21 4 925518 21 5 916191 27 5 919340 26 5 922466 26 5 9255701 26 6 916243 32 6| 919392 31 | 6 | 922518 31 6 925621 31 7 916296 37 7 919444 37 7 922570 86 7 925673 | 36 8 916349 42 8 919496 42 8 922622 42 8 925724 41 9 916401 48 9 919549 47 9 922674. 47 9 925776 46 8250 916454 8310 919601 18370 922725 1j8430 925828 1 916507 5 1 919653 5 1 922777 5 1 925879 5 2 916559 11 2 919705 11. I 2 92'2829 10 2 925931 10 3 916612 16 919758 3 922881 16 3 925982 15 4 916664 21 4 919810 21 4 9229331 21 4 926034 21 5 916717 26 5919862 26 5 922985 26 5 926085 26 6 916770 31 6 919914 31 61923037. 31 G 926137 31 7 916822 37 7919967 37 7 923088 36 7 926188 36 8 916875 42 8! 920019 42 L 8 923140 42 8 926239 41 9 916927 47 9 920071 47 9 923192 47 I 9926291 46 L~~~~~907 I2214 Page 34 341 LOGARITHMS OF NUMBERS. No. Log. I"r- | No. Log. Prop. No. Log. Prt. No. Log. Prop. No. o.Prt. Po Pa r..Lo. Part. 8440 926342 8500 929419 8560 932474 18620 935507 1 926394 5 1 929470 5 1 932524 5 1 935558 5 2 926445 10 2 929521 10 2 932575 10 2 935308 10 3 926497 15 3 929572 15 3 932626 1 3 935658 15 4 926548 21 4 929623 20 4 932677 20 4 9335709 20 5 926600 26 5 929674 26 5 932727 25 5 9357 59 25 6 9266 5 1 31 6 9 2 97 2 5 31 6 932778 30 1 6 9.3809 30 7 926702 36 7 929776 36 7 932829 35 7 9'5860 35 8 926754 41 8 929827 41 8 932879 40 8 9 59 0 r40 9 926805 46 9 929878 46 9 932930 45 9 9,3j0 45 8450 926857 8510 929930 8570 932981 18630 936011 1 926908 5 1 929981 5 1 933031 5j 1'936061 5 2 926959 10 2 930032 10 2 933082 10 2 936ll l 10 3 927011 15 3 930083 15 3 933133 15 3 936162 15 4 927062 21 4 930134 20 4 933183 20 4 93621 2 20 5 927114 26 5 930185 26 5 933234 25 5 9 36262 25 6 927165 31 6 930236 31 6 933283 30 6 936313 30 7 927216 36 7 930287 36 7 933335 35 7 936363 35 8 927268 41 8 930338 41 8 933386 40 8 936413 40 9 927319 46 9 9330389 46 9 933437 45 9 936463 45 8460 927370 8520 930440 8580 933487 8640 936514 1 927422 5 1 930191 5 1 933538 5 1 936564 5 2 927473 10 2 930541 10 2 933.388 10 2 936614 1.0 3'9 2 52 4 15 3 930592 15 3 933639 15 3 9366t64 15 4 927576 21 4 930643 20 4 933690 20 4 936715 20 5 927627 26 5 930694 2.5 5 933740 25 5 936765 25 6 927678 31 6 930743 31 6 933791 30 6 936S15 30 7 927730 36 7 930796 36 7 933841 3;5 7 93196863( 35 8 927781 41 8 930847 41 8 9338'92 40 8 9 6936916 40 9 927832 46 9 9 30898 46 9 933943 45 9 936 966 45 8470 927883 8530 930949 8590 9334993 8650 937016 1 927935 5 1 931000 5 1 934044 5 11937066 5 2 92798 10 2 931051 10 2934091 10 2i937116 10 3 928037 15 3 931102 15 3' 934145 15 3 937167 15 4 928088 21 4 931153 20 4934195 20 4 937217 20 5 928140 26 5 931203 25 5 934246 2 5 5 9 37 267 25 6 92'8191 31 6 93125-41 31 6 934296 30 6 93 317 30 7 928242 36 7 9313065 36 7 934347 35 7 913 7367 91 8 928293 41 8 931356 41 8 934397 40 8 93'7418 40 9 928345 46 9 931407 46 9 934448 45 9 937468 45 8480 928396 8540 931458 8600 934498 |8660 937518 1 928447 5 1 931509 5 1 934549 5 1 937568 5 2 928498 10 2 931560 10 l 2 934599 10 2 937618 10 3 928549 15 3 931610 15 3 934650 15 3 937668 15 4 928601 21 4 931661 209 4934700 20 4 937718 20 5 928652 26 5 931712 25 5 9347-1 25 5 937769 230 6 928703 31 6 931763 31 6 934801 30 6 937819 30 7 928754 36 7 931814 36 7 931852 35 7 937869 35 8 928805 41 8 931864 41 8 6934902 40 8 937919 40 9 928856 46 9 931915 46 9 934953 45 9 937969 45 8490 928908 8550 931966 8610 935003 8670 938019 1 928959 5 1 932017 5 1 935054 5 1 938069 5 2 929010 10 2 932068 10 2 935104 10 2 938119 10 3 92,9061 15 3 932118 15 3 9351541 15 3 938169 15 4 929112 20 4 932169 20 4 935205 20 1 4 938219 20 5 929163 26 5 932220 25 5 93255 252 1 5 9382) 69 25 l 6 9)9214 31 6 932271 30 6 9353061 3) 6 938319 30 7 929266 36 7 932321 35 7 935356 35 7'338 370 8 929317 41 8 9323722 40 8 i 9335406 40 8 938420 40 9 929368 46 9 93 428 45 9935457 5 9 938470 4 Page 35 LOGARITHMS OF NUMBERS. 35 No. Log. Part. No. Log. Part. No. Log. Part. No. Log. Part. No. ~v~. Part. Part. ~u o Part. Part. 8680 1938520 8740 941511 8800 944483 8860 947434 1 938570 5 1 941561 5 1 944532 5 1 947483 5 2 938620 10 2 941611 10 2 944581 10 2 947532 10 3 938670 15 3 941660 15 3 944631 15 3 947581 15 4 938720 20 4 941710 20 4 944680 20 4 947630 20 5 938770 25 5 941760 251 5 944729 25 5 9476-79 25 6 938820 30 6 941809 30 6 944779 30 6 947728 29 7 938870 35 7 941859 35.7 944828 35 7 947777 34 8 938920 40 8 941909 40 8 944877 40 8 947826 39 9 938970 45 9 941958 45 9 944927 45 9 947875 44 8690 939020 8750 942008 8810 944976 1 8870 947924 1 939070 5 1 942058 5 1 945025 5 1 947973 5 2 939120 10 2 942107 10 2 945074 10 2 948021 10 3 939170 15 3 942157 15 3 945124 15 3 948070 15 4 939220 20 4 942206 20 4 945173 20 4 948119 20 939270 25 5 942256 25 5 945222 25 5 948168 25 6 939319 30 6 942306 30 6 945272 30 6 948217 29 7 939369 35 7 942355 35 7 945321 35 7 948266 34 8 939419 40 8 942405 40 8 945370 40 8 948315 39 9 939469 45 9 942454 45 9 945419 45 9 948364 44 8700 939519 8760 942504 8820 945469 8880 948413 1 939569 5 1 942554 5 1 945518 5 1 948462 5 2 939619 10 2 942603 10 2 945567 10 2 948511 10 31939669 15 3 942653 15 3 945616 15 3 948560 15 4 939719 20 4 942702 20 4 945665 20 4 948608 20 51939769 25 5 942752 25 5 945715 25 5 948657 25 6 939819 30 6 942801 30 6 945764 29 6 948706 29 7 939868 35 7 942851 35 7 945813 34 7 948755 34 8: 939918 40 8 942900 40 8 945862 39 8 948804 39 9 939968 45 9 942950 45 9 945911 44 9 948853 44 8710 940018 8770 943000 883Q0 945961 8890 948902 1 1940068 5 1 943049 5 1 946010 5 1 948951 5 2 940118 10 2 943099 10 2 946059 10 2 948999 10 3 940168 15 3 943148 15 3 946108 15 3 949048 15 4 940218 20 4 943198 20 4 946157 20 4 949097 20 5 940267 25 5 943247 25 5 946207 26 5 5 949146 25 6 940317 30 6 943297 30 6 946256 29 6 949195 29 7 940367 35 7 943346 35 7 946305 34 7 949244 34 8 940417 40 8 943396 40 8 9463.54 39 8 949292 39 9 940467 45 9 943445 45 9 946403 44 9 949341 44 8720 940516 8780 943494 8840 946452 8900 949390 1 940566 5 1 943544 5 1 946501 5 1 949439 5 2 940616 10 )2 943593 10 2 946550 10 2 949488 10 3 940666 15 3 943643 15 3 946600 15 3 949536 15 4 9407160 2L 4 943692 20 4 946649 20 4 949585 20 5 940765 25 5 943742| 25 5 946698 25 5 949634 25 6 940815 30 6 943791 30 6 9467471'29 6 949683 29 7 940865 35 7 943841 35 7 1946796 34 7, 949731 34 8 940915 40 8 943890 40 8 946845 39 8 949780 3'39 9 940964 45 9 943939 45 9 946894 44 9 949829 44 8730 941014 8790 943989 8850 946943 8910 949878 1; 941064 5 1 944038 5 1 946992 5 1 9499026 5 2 941114 10 2 944088 10 1 2 947041 10 2 949975 10 3 941163 15 3 9441837 15 3 947090| 15 3 1 950021 1) 4 941213 20 4 944186 20 1 947139 20 4 9.50073 2>0 5 941263 25 5 944236 25 5947189 25 11 51950121 25 6 941313 30 6 94285 30 6 9472388 29 6 950170 29 7 941362 35 7 944335 35 947287 34 7 950219 3t 8 < 941412 4() 8 944384 40 8 947336 39 8 950267 39 i 9 941462 45 9 944433 45 9 947385 44 9 950316 44 34 Page 36 36 LOGARITHMS OF NUMBERS. Part. Iart.P Pa. rt. Part. 8920 950365 8980 953276 9040 956168 19100 959041 1 950413 5 11953325 5 1 956216 5 1 959089 5 2 950462 10 2 953373 10 2 956264 10 2 959137 10 3 950511 15 3 953421 15 3 956312' 14 3 959184 14 4 950560 19 4 95,470 19 4 956361 19 4 959232 19 5 950608 24 5 953518 24 5 956409 24 5 959280 24 6 950657 29 6 953566 29 6 956457 29 6 959328 29 7 950705 34 7 953615 34 7 956505 34 7 959375 34 8 950754 39 8 953663 39 8 956553 38 8 95 9423 38 9 950803 44 9 953711 44 9 956601 43 9 959471 43 8930 950851 8990 953760'9050 956649 9110 959518 1 950900 5 1 953808 5 1 956697 5 1 959566 5 2 950949 10 2 953856 10 2 956745 10 2 959614 10 3 950997 15 3 953905 15 3 956792 14 3 959661 14 4 951046 19 4 953953 19 4 956840 19 4 95970!1 19 5 951095 24 5 954001 24 5 956888 24 5 959757 24 6 951143 29 6 954049 29 6 956936 29 i 6 959804 29 7 951192 34 7 954098 34 7 956984 34 7 959852 34 8 951240 39 8 954146 839 8 957032 3-8 8 959900 38 9 951289 44 9 954194 44 9 957080 43 9 959947 43 8940 951337 9000 954242 9060 957128 9120 959995 1 951386 5 1 954291 5 1 957176 5 1 960012 5 2 951435 10 2 954339 10 2 957224 10 I 2 960090 10 3 951483 115 3 9-54387 14 3 95272 14 3 960138 14 4 951-532 19 4 954435 19 4 957320 19 4 960185 19 5 951580 24 5 954484 24 5 95768 24 5 960233 24 6 951629 29 6 954532 29 6 957416 29 " 6 960280 28 7 951.677 34 7 954580 34 7 957461 34 7 960328 33 8 951726 39 8 954628 38 8 957511 3 8 960376 38 9 951774 44 9 954677 43 9 9oo559 43 9 960423 43 8950 951823 9010 954725 | 9070 957607 89130 960471 1 951872 5 1 954773 5 1 95i|5 7 1 9605181 5 2 951920 10 2 954821 10 2 95703 10 i1 2 960566 10 3 951969 15 3 954869 14 3 957751 14 1 3 960613 14 4 952017 19 J 4 954918 19 4 957799 19 q 4 960661 19 5 9.52036 2)4 5 5954966l 241 5[957847 24 1 5 960709 24; 952114 29 6 955014 29 6 957894 29 6 960756 28 7 952163 34 7 955062 34 7 957942 34 1 - 7 960304 033 8 952211 39 8 955110 38 8 957990 38 8 960851 38 9 952259 44 9 955158 43 9 958038 43 9 960899 43 8960 9521308 9020 55206 9080 958086 9140 960946 1 952356 5 1 955255 5 11 1 9.58134 5 1 9609941 5 2 952405 10 2 955303 10 2 958181 10 2 961041 10 3 952453 15 3 9553501 14 3 95822'3 14 3 961081 14 4 952502 19 4 9553997 19 4 958277 19 4 961136 19 5 952550 24 5 955447 24 5 958325 24 5 961184 24 G 952599 29 6 955495 29 6 958373 29 6 961231 28 7 952647 34 7 955543 34 7 958420 34 7 961279 33 8 952696 39 8 95592 38 8 958468 38 8 961326 138 9 952744 44 9 955640 43 9 958516 43 9 961374 43 8970 952792 9030 955688 9090 9585641 9150 961421 1 952841 5 1 9557836 5 1 958612 5 1 961469 5 2 952889 10 2 955784 10 2 9586.59 10 2 961516 10 3 952938 15 3 955832 14 3 958707 14 3 961563 14 4 952986 19 4 955880 19 4 958755 19 4 961611 19 5 95303- 24 5 955928 24 5 958803 24 5 961658 24 6 953083 29 6 955976 29 6 9588.50 29 6 961706 28 7 953131 34 7 956024 34 7 958898 34 2 7 961753 33 8 953180 39 8 956072 38 8 958946 38 8 961801 38 9 953228 44 9 9561201 43 9 958994 43 9 961848 43 Page 37 LOGARITHMS OF NUMBERS. 37 No. Log. Prop. No. Log r. Prop No. Log Prop No Log. Prop. Part.~! Parto. Parto. L P o g. 9160 961895 /9220 964731 9280 967548 9340 970347 1 961943 5 1 964778 5 1 9675935 5 1 970:393 2 961990 10 2 964825 9 2 96476 42| 9 2 9704401 9 31962038 14 3 964872) 14 i 3 967688 14 3 970486 14 4 9620851 19 4 964919 19 4 967735 19 4 970533; 19 5 962132 24 5 964966 24 5 9767782 253 5 970;579 23 6 962180 28 6 965013 28 6 967829 28! 6 9706216 l 28 7 962227 33 7 965060 33 7 96(7875 35 3 90672 33 8 962275 38 8 965108 38 8 9679221 38 8 9)70719 i 37 9 962322 48 9I 965155 42 9 967969 4'2 9 97 0765 42 9170 962369 9230 965202 9290 968016 [9350 970812 1 962417 5 1 9652'49 5 1 98062 5 1 9708-58 5 2 962464 9 2 965296 9 2 968109 9 i 2 970904 9 3 962511 14 3 965343/ 14 3 968156 14 3 9 700931 14 4 962559 19 4 965390 19 4 968203 19 4 970997 1 9 5 962606 24 5 965437 24 5 968249 2 5 971044( 23 6 9626.53 28 6 965484 28 6 968296 28 6 971090 28 7 962701 33 7 965531 33 7 968343 33 7 971137 2 33 8 962748 38 8 965-578 38 8 968389 38 8 971183 9 962795 42 9 9656j25 42 9 968436 42 9 9 71 9 2) 9 9180 9628-)3 9240 965672 9300 968483 9360 971276 1 96289'0 5 1 965719 5 1 968530 5 1 97132 5 2 9629"37 9 2 965766 9 2 968576 9 2/97 13369 9 9 3 96298O5 14 3 965813 14 3 968623 14 3 971415 14 4 9630321 19 4 9465860O 19 4 968670 19 4 971461 19 5 963079 24 5 965907 24 5 968716 23 5 971508 i3t 6 963126 28 6 965954 28 6 968763 2s8 971554 28 7 96311 4 33 7 966001 33 7 968810 33 7 971600 339 8 9632'1 3 88 966048 38 81 8 968856 37 81971647 37 9 9633268 42 9 966095 421 9 968903 42 91971693 42 9190 96331*5 9250 966142 | 9310o 968950 9370 7140 1 963.-36)3 5 1 966189 5 1 968996 5 11 971786 5 2 963410 9 I 2 9966'2331 9 2 969043 9 2 971832 9! 3 9634.7 14 I 3 966283 14 1 3 969090 14 3 971879 14 4 963504 19 4 966629 1 1 4961361 19 4 971952.3 1) 5 96.35521 24 5 96063t;76 24 5 969183 23 5 971971 23 6 96.3599 28 6 966423 28 6 969229 28 6 97 018 8 2 7 963646 33 7 966470 1969276 33 7 1 9720:)641 I3 8 966363 38 8 966517G 38 8 969323 37 81972101 3 9 9637469369 42 9 966.64 42 1 4 9397215 4.2 9200 963788 9260 966611 1 9320 969416 9380 979203 | 1 99688359 5 1 966658 5 1 969462 5 1 9722491 96,882 9 2 2'9667056959 9 2 90 9 72 2 )9 1 3 963929 14 3 966752 14 31 960.53, 14 39 4 14 4 963977 19 49 966798 19 i6 4 969602 19 4 7' ]l 5 9640224 5 966845 24 5 96969619 23 5 )7294-.t 2. 6 964071 28 6 96(68) 28 6 969695 28 S6 I9l10 7 964118 33 7 966939 3 7 969742 33 7 9 7 2 8 964165 38 81966986 38 8 969788 3 8 972 31 9 964212 42 9 967033 42 9 969835 42 9 932i19 -i 9210 964260 9270 967080 9330 969882 9190 92(;6 C 1 964307 5 1 96712799 5 1 9697928 5 11 2 964354 9 21 967173 9 2,96997.3 9 2 9 87258 79 3 964401 14 3 967220 14 397002114 3972804 11 4 964448 19 4 967267 19 4 970()068 19 i 4 9728-1 ] 5 964495 24 5 967314 24 5 59701141 23 1 59 789 2i / 6 964542 28 6 967361 283 6 970161, 28 [6 9729-13 28 7 964590 33 7 967408 33 7 9702071 33 9IO98 68 964637 38 8 967454 38 8 97024 7 8 97303 3 9 9646847501 42 9 96757001 42 99703001 42 99i73082 1 41 Page 38 38 LOGARITHMS OF NUMBERS. Prop. o. Log. Prt. No. Log. Prop. No. Log. Pt. No. Log. Part. -_ _o._Log._ _ _ Part. o.Prt. Part. I 9400 973128 9460 975891 9520 978637 9580 981365 1 973174 5 1 975937 5 1 978683 5 1 981411 5 2 973220 9 2 975983 9 2 978728 9 2 981456 9 3 973266 14 3 976029 14 3 978774 14 3 981501 14 4 973313 18 4 976075 18 4 978819 18 4 981547 18 5 973359 23 5 976121 23 5 978865 23 5 981592 23 1 6 973405 28 6 976166 28 6 978911 27 6 981637 27 7 973451 32 7 976212 32 7 978956 32 7 981683 32 8 973497 37 8 976-258 37 8 979002 36 8 981'28 36 9 973543 41 9 976304 41 9 979047 41 9 9 981773 41 9410 973590 19470 976350 9530 979093 9590 981819 1 973636 5 1 976396 5 1 979138 5 1 981864O 5 2 973682 9 2 97642 9 2 99184 9 2 981909 9 3 973728 14 3 976487 14 3 979230 14 3981954 14 4 973774 18 4'976533 18 4 979275 18 498'000 18 5 973820 23 5 976579 23 5 979321 23 5 982045 23 6 973866 28 6 976625 28 6 979366 27 6 982090 27 7 973913 32 7 976671 32 7 979412 32 7 982135 32 8 973959 37 8 976717 37 8 979457 36 8 982181 36 9 974005 41 9 976762 41 9 979503 41 9.98226 41 9420 974051 9480 976808 9540 979548 9600 982271 1 974097 5 1 976854 5 1 979594 5 1 982316 5 2 97-143 9 2 976900 9 2 979639 9 2 982362 9 3 974189 14 3 976946 14 3 979685 14 3 982407 14 4 9742035 18 4 976991 18 4 979730 18 4 982452 18 5 974281 23 5 977037 23 5 979776 23 5 982497 23 6 974327 28 6 977083 27 6 979821 27 6 92543 27 7 974373 32 [ 7 977129 32 7 979867 32 7 982588 32 8 974420 37 t 8 977175 37 8 979912 36 8 98263 36 9 974466t 41 9 977220 41 9 979958 41 9 982678 41 1 9430 974512 9490 977266 1 9550 980003 9610 982723 1 974558 1 977312 5 1 980049 5 11982769 5 2 974604 9 2 977358 9 2 980094 9 2 982814 9 3 974650 14 3 977403 14 3 980140 14 3 1982859 14 4 974696 18 4 977449 18 4 980185 18 4 982904 18 5 974742 23 5 977495 23 5 980231 23 5 982949 23 6 974788 28 6 977541 27 6 980276 27 6 982994 27 7 974834 32 7 977586 32 7 9803 22 32 7 9830401 32 8 974880 37 8 977632 3I7 8 980367 36 8 983085 36 9 974932 41 9 977678 41 9 980412 41 9 9831301 41 9440 974972 9500 977724 9560 980458 19620 983175 11975018 5 1 797769 5 1 980503 5 1 983220 5 2 975064 9 2 977815 9 2 980549 9 2 9832665 9 3 975110 14 8 977861 14 3 980594 14 3 983310 14 4 975156 18 4 977906 18 4 980640 18 4 983356 18 5 975202 23 5 977952 23 5 980685 23 5 983401 23 6 975248 28 6 977998 27 6 9807.30 27 11 983446 27 7 975294 32 7 978043 32 7 980776 32 7 983491 32 8 9753-0 37 8 978089 37 8 980821 36 8 983536 36 9 975386 41 9 978135 41 9 980867 41 9 983581 41 9450 975432 9510 978180 9570 980912 9630 983626 1 975478 5 1 978226 5 1 980957 5 1 983671 5 2 975524 9 2 978272 9 2 981003 9 2 983716 9 3 975570 14 3 978317 14 3 981048 14 3 983762 14 4 975616 18 4 978363 18 4 981093 18 4 983807 18 5 975661 23 5 978409 23 5 981139 23 5 983852 23 6 975707 28 6 978454 27 6 981184 27 6 983897 27 7 975753 32 7 978500 32 7 981229 32 7 983942 32 8 975799 37 8 978546 37 8 981275 36 8 983987 36 9 975845 41 9 978591 41 9 981320 41 9 984032 41 Page 39 LOGARITHMS OF NUMBERS. 39 No. Log. Prop. No. Log. Prop. No. Log. Prp. Nd. Log. W Fo;Part. Part. Par.t art9640 984077 9700 986772 9760 989450, 9820 992111 1 984122 5 - I986816- 4 1 989494 4 1 992156 4. 9 24167 9 2 986861 9 2 989539 9 2 992200 9 3 984212 14 3 986906 13 3 989583 13 3 992244 13 4 984257 18 4 986951 18 4 98 96,28.18 4 992288 18 5 984302 23 5 986995 22 5 989672 22 5 9923-33 22 6 984347 27 6. 987040 27-, 989717 27 6 992377. 26 7 984392 32 7 987085 31 7 989761 31 7 992421 31 8 984437 36 8' 987130 6 86 8 989806 36 8 992465 35 9 984482 41. ~ 9 987174 40 9 989850 40 9 992509 40 9650 984527.. 9710 987219.'9770 989895 1 9830 992553 1984572 5 1 987264- 4 I 1989939 4 1 992598'4 2 984617- 9 2- 987B09 9 2 989983'9 2 992642 9 3 984662 14 3 987353 13 3 990028 13 3 992686 13 4 98.4707 18'4 987398' 18 4 990072 18 4 992730 18,5 984752 23'- 5 987443 22 6 9901'7 22 5 992774 22 6 984797 27 6.987487 27.6 990161 27. 6 992818 26 7 984842 32 7 987532 31 7 990206 31 7 992863 31 8 984887 36 8 987577 36 8 990250 36 8 992907 35 9 984932 41 9 987622 40 9 990294 40 9. 992951 40 I9660 -984977.9720 98766,6 9780 990339.9840 992995 1 985022 5 1 987711 4 1 990383 4 1 993039 4 2 985067 9 2 987756 9 2' 990428 9 2 993083 9'3 9851.12 14. 3.987800 13 3 990472 13 3 993127 13 4985157 18'4 987845 18 4 990516 18 4 993172 18 5 985202 23. 5, 987890 22 5 990561 22 5 993216 22 6 985247 27 6 987934 27 6- 990605 27 6 99326.0 26 7'3985292 32 7 987979 31 7 990650 31 7 -'993304 31 8 985337 36 8 988024 36 1 8- 990694 36 8 993348 35 9 985382 41 9 988068 40 - 9 1990738- 40 9993392 40 9670 985426. 9780 988113 97901990783 9850 993436 1 985471 4.1 988157 4 11990827 4 1 993480 4 2 985516 9 2 988202 9 2']i990871 9 2 993524 9 3 9-85561 13 3 988247'13 3 1990916 13 3 993568 13 4 985606.18 4 988291 18. 4199096.0 18 4 993613 18 5 085651 22 5 988336'22 5 |991004 22 5 993657 22 61 985696 27, 6 988q8f 27 61991049 27-. 6 993701 26 7 985741 31 7 988425 31 7 991093 31 7. 993745,31 8 985786 36 8 988470 36 8 991137 36 8 993789 -35 9 985830 40 9 983514 40 9 991182 40 9 -993833 40 9680 985875 9740 988559 - 9800 991226 9860 993877 1.985920 4 1 988603 4 -1 991270 4 1- 99392.1 4 2 985965 -9. 2.988648 9 2 991315 9 2 993965 9 3 986010 13 3 988693 13'3 991359 13 3 994009 13 41986055 18 4. 988737 18 1 4 991403 18 4 994053 18 5 986100 22 5 988782 22 5' 991448 22 5 994097 22 6 986144 27 6.988826- 27 1 6 991492 27 6 994141 26 7 986189 31 7. 988871, 31 7 99153.6 31 7 994185 31 8 986234 36 8 988915.36- 8 991580 36 8 99422'9 35 9. 986279 40 9 988960 40 9 991625 40 9 994273, 40 9690 1 986324. 9750 989005 9810 991669 - 9870 994317 1 986369 4 1 1989049 4 1 1 991713 4 1 994361 ~'4 2 986413 9'2 989094 9 2'991757 9 2 994405 9 3 986458 13 3 989138 13 1 3.991802 13 3 994449 13 4 986503 18 4 989183 18 4 991'846 18 >'4 994493 18 5 986548 22 - 5 989227 22 5 991890 22. 5'994537 22'6 986593 27 6 989272 27 6 991934'27 6 994581 26 I 7 186637 31 7 989316 31 7 991979 31 7- 994625 31 8 986682 36 8 989361 36 1 8' 992023 36 8 994669 35 9'986727 40 9 989405 40 9 1992067 40 9 994713 40 Page 40 40 LOGARITHMS OF NUMBERS. No. Log. Part: No. Log. Part N o. Log. Prop. Lo-. 8801994757 9910 996074 9940 997386 9970 998695 1 994891 4 1 996117 4 1 997430 4 1 998739 4 2 994845 9 2 996161 9 2 997474 9 2 998782 9 3 994889 13 3 996205 13 3 997517 13 3 908826 13 4 994933 18 4 9962'49 18 4 997561 17 4 998869 17 5 994977 22 5 996293 22 5 997605 22 5 998913 22 6 995021 26 6 996336 26 6 997648 26 6 998956 26 7 995064 81 7 996380 31 7 997692 30 7 999000 30 8 995108 35 8 996424 35 8 997736 35 8 999043 35 9 995152 40 9 996468 40 9 997779 39 9 999087 39 9890 995196 9920 996512 9950 9978233 9980 999130 1 995240 4 1 996555 4 1 997867 4 1 999174 4 2 995284 9 2 996599 9 2 997910 9 2 999218 9 3 995328 13 3 996643 13 3 9979.54 13 3 999261 13 4 995372 18 4 996687 18 4 997998 17 4 9993005 17 5 995416 22 5 996730 22 5 998041 22 5 999348! 22 6 995460 26 6 996774 26 1 6 998085 26 6 999392 26 7 995504 31 7 996818 31 7 998128 30 7 999435 30 8 995547 35 8 996862 38 1 8 998172 35 8 999478 35 9 995591 40 9 996905 40 9 9982216 39 9 99522 39 9900 9956035 9930 99694 9 9960 998259 9990 999565 1 995679 4 1 996993 4 1 998303 4 1 999609 4 2 995723 9 2 9970307 9 2 998346 9 i 2 999652 9 3 995767 13 3 997080 13 3 998390 13 3 999696 13 4 99 5811 18 IS 4 997124 18 4 998434 17 4 999739 17 5 995854 22 5 997168 22 5 998477 22 5 999783 22 6 995898 26 1 6 997212 26 i 61998i521 " 26 6 999826 26 7 ( 19 I 7 997255 31 7 998.56- 30 7 999870 31 7 8 30 99.5986 35 8 997')99 3( 5 i 8 998608 35 8 9999131 35 9 i960;)30 40 9 997343 39 99734 99 9 9 99986523 39 99 9 No. LOGARITIUIMS TO 50 DECI.IIAL PLACES. 1 0 o000000000000000000o000 000000000000000 000000000 2' 0 301029'3995663981195213 388947244 - 027 4181881462 11 3 0.4771212547196623l;729:5027,'032'511.30' ()012 88-641906 4 06020593991327962390427477789448860.53536"-. i.7 2,(J4922 5 06'989700043360188047i62611057.,)5506'7ov3o 81011853789 6 0'7781512503836-136325087667979796033'59(3;S3! q74565280 0.84509 804001 42568:30 71o.) 1 > lt;'.585! 33S.'5, 8 i3.3572 3| 8 09030899'86!,91.943 58564121(668-1173-47!OS0304.56964438633 09 0 954242i509493'24874.59005580); (;510230(;l18400257 72838139 10 1 00000000000000000000000000000000000000000000000000 11 10413926851.58225040750199971'4302'424170670219046645 1 107)9181246047624827722505692704101362783650862711491 13 11131'94335230683676920650515794232843082972918838707 14 11146128035678238025925955153317129220'25176227778607 15 1176091259055681242081289008530622282431938982728,59 1 1204119982655924780854955.5788979721070727595'2584843 17 1 230448921378273'92854016989432833703000756737842505 18 11255272505103306069803794701234723645168-4760984350 19 1 278753600)9528289615363334757569293179t5112933739450 201'30102999566398119521373889472449302676818988146211 21 1'32221929473391926800724416184775150268370126051866 22 134242268082220623596393886596751726847489207192856 23 136172783601759287886777711225118954969'1751103433610 24 1 38021124171160602293624458742859438950469850857702 2511-39794000867203760957252221055101394646362023707578 Page 41 LOGARITHMIC SINES, ETC. 41 0 DE_.'Di ff._.__, ii.t'. ~ Sine. ~Cosecant. T1angenyt Cotangent. Secant. Cosine. 100 i i1( ~o 10 0~0000 0 oil Infinite. // I jiininite. i000000 10000000 6i'4637266 3. 53{;274 6'463726 13'51362741/'000000 10,000000 59 216.i0i47561501717 3235244 664.7617 6 01717 13.235244'.000000 10000000 58 3 6 94108471 293485 3059153 6 940847 29348 105915 3 09 1 58000000 10-000000 857 41 7 065786 1082312 2 29334214 7 065786 2'S231 12.'934214.00(0000010. 000000 56 5 7.162696 161517 2837304 7'16269t; 161517 12.837304.00000 10000000 0 5.500 6 7.241877 131968 2.758123;7241878 131969 12.758122.000001. 99 54 i 7 7-308824 111578 2.691176 73808825 11157;12.691175.000001 9.-99 9999 ) 53 8 7.366816 96653 2.633184 7.366817 96653 12-6331831.000001 9I 999999'.2 9 7-41g968 85254 2.582032 7.417970 85254 12.-5820380 0000011 9.999999;1 i 10;7-463726 76262 2.536274:7 463727 76268 12.5362738 000002 9.999998 50 11'7.505118 68988 2.494882 7. 505120 68988 12494880.0000021 9. 9999,8 49 12 7.542906 62981 2.457094 i7.542 909 62981 12457091 -00008 9.9999971 48 13 7577668 57936 2.422332;7577672 57937 112422328 -000003 9.9939997i 47 14 7609853/ 53641: 23890141 7!76098157 5342 12. 390143 000004 -99'99996 46 151 7.639816 49938 2860181 7'639810 49939 12360180 000004 99996 16 7 667845 4667142303215 17661849 46715 12'332151.000005 999999 5!9 44 17 7.694173 43881 230582 7 76941791 43882 12'305821.000005 9.901)999 1;4 318 7-718997 413-87 212281003 7 i719003 413738 122809970 000006, 99099941 1s 19 7.742478 391;5'225752) 7'742484 39136 12257516i.000007 9-999993 41 20 7.764754 37127 2.235247 i I77 6i 37128 12 235239.000007 9.9999993:!40 21 77853943 35315 2.21405 7 778595)1 3o11122140, 9 000008 9.999992i3 2 7.806146 338672 2.193854 7.80615)t 33673112 193845.000009 9.999991 38 73 -82~151 32175 2174549 17825O460 9176121 7 706 74'5400 000010 9 9999 0 37 24 784914 30804 5 21560667 843944 30807 1 208071 5605i G 00001 1 9 999989'36 i 25 7 86 1 662 29547 2-138338 7-86164 2i0()9 12-l'1832(1 -000011 9 699989"n 3. 5 24 7.878695 28388/ 2121305 7-878708 28o390 112'121 9', 0001-1 9'0D 99,88 3 4 27.895085 27317 2'104915 7.89)099 27318! 12104901'.0000131 9 999987 8'I 28 7.910879 26832312'089121 7-910894 26320'25 12.089106 000014i 9-;)9986 9 7.9'6119 253992'073881 7.926134 23401 12 07;861. (00015, 9-9;985 31 30 7 940842 24.538 2059158 7'940858 245-10 12. 09142'1 090001 9 99098.'0 31 7'955082 237033 2'044918 7'95100.23375 12 -'0490 00'0001.! 9'J q99'N9 32 7'968870 22980 2031130i "968889 2298212'9031111' 0000:' 9 2999981; 28 33 79393 2227.,~3 2'017i6 939')8,5 2.,7 - 12 01747 1'0000-1 9'999,98'-7 34 7'995198 21608 2'004802 7'995219 21(;10:1'004781i t000211 99 1'9979 7 3o 8007787 20981 1992213 8 00780 2098311 ~9921941'000,02- 9 999977 5 20981 l'9,)221.13-0i)7~:0.)I.~~~..0,,_3.,]95 5.00to 2 -, 36 8'020021 20890','_ 1i', 797. 780' 004s 0t_ - 2 000 0 2 ) 1':9973 _1: 37 8'081919 1981 1968081 8'03N1915 198833 11'9 80.S I 55 00 02,5 9',so9997? 38 38 8013301 1 930)1 6cr (-8 4'-.2)27 193.3 11,171 -,0000'7 927);i99-.) 39 8-054781 1880 1-9-1 5219 8'0A80t 1880 3! 11 9'91919]0000"s 9 997!97 22I 140 8. 06577 1 182 I1 1-.06,06' 18'327 11 9319'0010.:. 913971:. 41 8'07(00 1787 1 923500 8,_ 07Y0 1.0] 7871,11'92346'0000s1~ 9'999999 19 42 8.086965 174411191310135 80869971 1,-14411. 913003 0000'32i 9'99998 18 43 8.097183 17031 I 902817 8. 09)72171 17034' 11,'-783 -000034 9999617 44 8107167 166391. 89268IS 27 8.) 10 2 164211 89'279 -000031 (: 99994 46 45 8.116926 162653 1 883074 18.116961 16268111882037 000037 9.999963 1b 468:.126471 15908W1 8733529 81265102 1591111.8734900 0000309 9.999961;' 14 47813.5810r 1.5536 18641190 S81358511 1550.;811!.8;41-19 000041 999 13 48'8.144903 13238 1-855004 8-144996 195241111.S35001 000042 1 999 1 49';8 1 5090 7 14924 1.846098o 8.1o539oI 1497119.'l 84604,: o00001 6,, 9,()1 11 50'8 162681 14622 1837319!8 16 2727 14625 11.8372738.00090 9'9 45' 10 51i 8.171280 14333 1.828720 8 171S28 11336 11.828(672's00(191S 9 99 2 9 52 8.179713: 14054 1-820287; 8'179763 14057 11'82028.0000)0 11 8 53 8.187985 13786 1812015 8'188003( 13790'11.811964: 0000s52 ()t.999i48 7 54 8.19610' 135291'803898 8'196156 13532[11.803814:0 OOuo 39. 9999.46 6 538 204070 13280 1795930 8.200416 130284 11.793 874:'00005;'). 9 944} 56( 8.211895 13041 1.788105 8.211953 13044 11.788047 000058 9. 9i89 42 4 57 8. 2195 81 12810 1-780419 82196I1 12814117503391 000069999940 3 58 8.2'71 2 125871.772866 8-227195 125-9111. 772: 0000 0 9'1oi 3 2 5 5)9 8 4.2 57 123721I765443 8 3) 4621 17.21 1768i 73) OO11. 99996"'000 q 1 60 8.241855 121641 1758145 8.241922's I. OS 0000066 9'9993)1 0 I Secanrt. I'Csea,,nt. i -os- st.L 59 DOU. Page 42 42 LOGARITHMIC SINES, ETC. 1 DEG. t Sine. Coecant Tangent. Seant. Cosine O-,8-2418,55 1-758145';8-241921 11-758079 1-00t0066 9999934' 60 1 8.249033 11963 1-7509671 8249102 11967 11-750898.000068 9-999932 59 2!8-256094 11768 1'743906 8'256165 11772 11- 43835'000071 9'999929 58 32 8.263042 11580 1'736958j 82G3115 11584 11 -36885 00007399999971 5 41 8'269881 11397 1'730119'8.269956 11402 11 130044' 00007551 9999925 56 5 8'276614 11221 1'723386 8-276691 11225 11 i233091'000078, 9 9999202 55 6!18'283243 11050 1'716757 18.283323 11054 11-716677'000080! 099'99'10 54 7 8-289773 10883 1-710227i,8-289856 10887 11-710144'000082' 9-999918 3 8 8-296207 10722 1.703793 8.296292 10726 11 03708 s00008o5 9.999915 52 9 8302546 10565 1-6974548.-302634 10570 11-6973661 000087/ 99999913 51 10 18-308794 10413 1691206'8-08884 10418 11691116 000090 99'9910 50 11I 8'314954 10266 1-685046 8315046 10270 116849541'000093' 9-999907 I 49 1218-321027 10122 1'67897318-321122 10126 11)678878'000095 9-999905'48 18 8-327016 9982 1'672984!8-'327114 9987 11672886!0000981 9999902 47 14 8-332924 9847 1-667076: 8.333025 9851 11 666975 000101 9.999899 46 15 8'338753 9714 1 -661247 8. 338856 9719 11 5611441 000103| 9-99989'7! 45 16 8-344504 9586 1-655496'8.344610 9590 11-G55390 I000106(il 9-999894- 4 44 17 8-350181 9460 1649819 8.350289 9465 11)649711 000109 9.99)891 43 181 8'355783 9338 1 644217 8 355895 9343 11 644103'000112 9-99988 I 42 1918'361315 9219 1 638685 8.361430 9224 11 638570 -0001151 9-999885 41 20 i8 366777 9103 1'633223 8S3G6895; 9108 11 633105''000118 9.9998802 40 21'8 372171 8990 1627829 8-372292 8995 11 62708t -000121 9999879 39 221!8 377499 8880 1 62'2501 8-377622 8885 11 62237 8'l 000124 9'999876' 38 9238'382762 8772 16017238 8'382889} 8777 11617111'0001271'9 999873 1 37 2418 8387962 8667 1'61.2038 8-388092 8672 11 611908'000130' 9 999870 36 258 -39o3101 8.564 1'606899 8 -30.9324 8.570 11 6016766' 000133' 9 999867 35 26 8'398179 846t4 1-601_821 8 398315 8470 1 1-601 685;0001361 9 999864 34 2 8'403199 8366 1-596801 8.40333.8 8371 11 5'.666'1'000139 9.9986 33 281 8-408161 8271 1-591839 8-408304 8276 11 o5916 9 6 0001W' 9.999858 1 3-2 299i8418068 8177 1-586932 8.413218 8182 11 o86787t 000146 9 99985 31 30 8 417'919 8086 1-582081 8'418068 8091 11.581932 000149 9. 9998)l 30 31 8'422717 -7996 1' 577283 8 42286 9 8002 1 11315771 0001' 9.99848 29 32_ 8'427462 7909 71 6572585' 1 8 7 914 1-572382- 8000 56 39-),844 9 8 33 8432156 7823 15678441l8.432315 7829 1156i68o 1000159 9999841 27 34! 8'436800 7740 1 1563200 8.4369621 774 5 11 563038i l0001u6l 9.99988 26 3-5 8'441394 7657 11 558606 8.-441560' 76(63,11 5o8440 -000166i 9999834')2 36 8 445941 7577 1'5540359 8-446110 7583 11.553890 l000169' 9 999831 24 37' 8'450440 7499 1.549560ij8.450613l 7505 11, 549387.000173i 9.999827 23 38 8 454893 7422 1 54.51071'8.4550701 7428 11 -544930. 0001 76: 9.999824 2'2 39 8'459301' 7346 il540699l118459481 35 2 11 540.519 000180 99999820 21 40 8-463663:7273 1'536335 8.463849 7279 111536151.00018- 9-9999816 20 411 8'467T985 7200 1 o32015 8.468172 72906 1.1-531828 000187 9 99981.3 19 42 8'472263 7129 1527737 18 472454' 7135 11,527546.000191 9 9993809 18 43,8'476498 7060 11 523,502 8 4766931 7066' 11a523308071-0001965 9699980Oi 17 44i 8'480693 6991 1 519301 8i4808921 6998 1 151 9108;.0001990 9 999801 16 45; 484848 6924 1'51515 l 848503(1 6931 11.514950 1000203' 9999197 15 46i18'488963' 6859 1'511037 8 489170, 6865 11.510830'.0002060 99 99794 i 14 47i18'493040' 6794 1'506960 8G4933250 680(SO 1 1.50 6750 00210 9-999C90!, 13 48 8-497078; 6731 1'5029292 18497298 6738 111502707 I0002141 99997186 12 49 8501080 6669 1 498920;8 -0l298; 6676 11-.498702' 000218 9 999j 8_ 11 0;18'505045. 6608 1 49495518.95267 66111- 1494 733 000222, 9999778 10 51|8'508974' 6548 149106 io( 99200' 6555 11490)800 000226 999s 411 52: 8'12897L 6489 1 -48 71& 3 8.13098' 6496 1148690 003I 9 9769 8 582!8'516726! 1 1 -486902~.00023! O')9'976 53.8516726i (16432 1-483274 118 -51691 64i'19 11 1-3.480393 000235, 999976 7 54 8'520.5.31 6'375 1-47)94,5 8-509t790 688'211-'I79210 -0002'3(. ) 9997961 6 55 8c5244_-13 63 19 11 4 76 7) 8 452458 6326C) 11. -475414 00020 9 3 999,37.5 5 56 8'52810 I 6 6 4 1 4 189'8 5283) 6 1 4 14 i1000247 9 999 53i 4 1 )57 8 2)) 8 6211, 1 46817'2 8.32080 6 18'11 467920 -000252 9 99748 11 3 i58 8-5o: o 6138 1'464477 8.35779 616-i5 1-i.422i1 0002o6 9 999744fl 2 59'8'.49!)185 6106 1 460814 8.5394t447 611c3 114450333'00060 9 999 402 1 i60 8'52)'19 605 1'45 181 57181 -513084 6062 11 -456916 000265 9 999!)735 11 O I I Co siIIe. Sca Cthr. e l, i', (,oecant. S;e. b5 DEG. Page 43 LOGARITHMIC SINES, ETC. 43 2 DEG. Sine..Di 5 Cosecant Tangent. Ih Cotangent. S.cant. |Di,, Cosine. 8542819 181 85430841 11'456)16 -002t)6 9-99979- 35o 60 1 8-546422 6004 1-453578 8-546691 6012 -1 45j39O9 -000oo26); 07 9999731 59 2 8-54999'D(.35 595 1 49000 8-550268 5962 11 44)78 3 )'000274 07 9 999726 5.8 3 -5536539 5906 1 -44 6461 8553581715914 114461883 000'27b 8 93 (d'(.972 57 4 8 657i0054 o 58)8 1-442946 8-55738615866 11.4426t 4 00083] 08 j((999717 156 5 8.5690540 5811 1 439460 8560828 5819 11-48 9172 0( 9287 07 9.999713,55 6 8.563999 5765( 1 4369001 8 564'91 5773 11435709/.00O92 089 9999708 154 7 8-567481 57191 423569 8.567727 5727 11430)2278) 000296 07 9-(t98 4, 8 8-570838 54 142914 1 84571187 5682 1 1-42886;3 9)000301 08 9 99)'69K,15,) 9 8-574214 5630 1.42576t 85745'20 5688 111425480' 008;06 08 O9.,9969941 1 10 8'577566 5587 1422434 8577877 5595 11-422123 000311 08 9*999'068' o0 11I 8-580892 5544 11419108 8-581208 5552 11'418792 9000315 07 9'99i6805' 49 12 85o84133 5502 1l -4165807 8 5845145 610 11'415486 000382 08 9-99;9680A48 13 8.58746d 5460 1-412531 8-587795 ]5468 11-412205 *000325 08 9-9996(j, 747 14I 8-590721 5419 1 409279 8'59105115427 1.1-408949 000330 08 9.999)67l46 15 I 8.593948 5379 1.406952 8 59428305387 119405717 000835108 9 999t65"143 16 8597152 53389 1 402848 8 597492 5847 111402508 0008401 08 9999660;44 17| 8'600932 50)0 1899668 8600677 5308 - 1 18 95)933 000845 08 9 999655 A3 18 86093489 5261 18396511 8-603838915270 11396161 *0003001 08 9-.999650(; 42 19 8-696623 3288 12-32189377 8-60697885282 1211 393022:0008551 08 9 (9( 9;. 415ll41 20 i 8-60099734 518'6118390266 86100945194 118389906 *0003t;860 08 99996400 21 8-61l282sl43 K1-38187177 8-613189 5158 11' 388811 00065 08 9 9990300'59 221 8'615891 5112 18384109 8'616262 5121 113837838 000371 10 9999"29I}38 23 S' 8l618)37 5076181381063 8'619313 5085 11 3880687 |00076 08 999624L 37 24 8;621962 5041 18378038 8-62234315050 111 377657'000381 08 9'999619!36 5 8 624965 5006 1375035 8-625352.501511 374648'000386 08 9-999614 85 26 8 627948'4972 1'372052 8'628340'4981 113'71660 l000892 10 9'999608i184 27 86!8)30911f4938'1' 369089 8-63130834947 11 368692 -000397 08 9'996903 O33 28 8638854149041 c1366146 8'634256 4913 11 365744 1000403 10 9.999597'8i13 29) 863676' 487-1 36832-4 8'63718414880 111362816'000408 08 9'999592'I,31 30 8 3(39680 4889138603820 8'6400934848 11359907 0004141 08 9-99,586 30 31 8t 64256384806C1 857437 8-642982(4816 11-357018 |1000419 10 9.9(95C81.29! 32 8'645428 4775 1'354.572 8-64585834784 118354147 000425 10 9 995o75128 88i 864 82 744743 1351726 8-64870414753 118351296 000430 08 99 9799 0 12 34 8-651102 471211348898 8-651537 4722 11'348463'000436 10 1999956426 35 8.653911 468211'346089 8 654352 4691 1 11'845648 000442 10 9999558125 86 | 8.656702 465"21' 843298 86571494661 11 -3428-51'000447 08 9'9995531 24 37 8-6593475,4622(13 40 525 8'659928 4631 11 34007 2'000453 1(0 9 9(J54 7'23 38 18-662230 4592 133i7770 8 662i689 4602111 373711'000405910 9999541 22 39 8.664968145ti3 1.3 5032 1 8-665:43) 4573 11 3884567 000465 10 1.999 9 %;) 1 40 8 -6 7914535 11 33 2311 8.6681.60 444111.331840.000471 10 999953 )10'9 41 t8-.670o93 4506 1329607 i 8-670870 451611329130 l000476 08 9-9995294 l19 4 2) 8.br080 44 79 1.32692_,0 8-673563 4488 11-326437.000482' 10 9999518 18 43 8.6o5 1144o51 18324249 8676i2394461 11 3239761.000488 10 9-999512'17i 44 1 8;8405 4424 1-321595 8- 890044.34 11 321100.0004941 10 9999506 1 45 8-68104314897 1 318907 8 681544 4407 11'318456 1000500 10 9999500 146 8 683665 4870 1-316335 68417 80 11315828.000.507 10 9 999493 14 47 8 686272434411'818728 i 8'686784 4354111313216 -000518 12 9999487:18 48 8-688863 4318 1-311137 1 8i689381 4328 11-310619'000519 10 9.9994811'K1 j 49 8 691438 4292 1308;562 ( 8 6919363 4303 111308037 -000525 10 9 9994 " 11 50 869399814267 1 306002 8 694529 4277 11-305471 -000531 10 9-999469 i10 51 8-696j434242 1-303457 8-697081 4252 11-302919'000537 12.99)94463 9 52 8-6!9073 4217;1-300927 8699617,4228 11-300383'000544 10 9.999456 8 53 8-701589 4192 19-298411 8 70213942038 11297861 -000550 10 9-999401 7 54 8-704090 4168 1.295910 8'7046464179 11-295354'000557 110 9999448 6 55[ 870O6577 4144 1'293423 8'707140 4155 11-292860 1000563 12 9.9994371 5 56 8.78-09049 4121 1-2909511 8,709618:4132 11 290"82'000569 10 9 999481; 4 5718 8711507 4097 1288493 8 712083 4108 11287917' 000576 12 9.999424 3 58 8 718952 4074 1 286048 8.714534i4085 11.285466 000582 10 9994181 2 59,i 8 71638834052 1.283617 8.7169724062 11 283028 ll000589 12 19999411i 1 60 8 718800 4029;1 -281200 8'71939614040 11'280604 1I 000596 12 19'9994041 0 I tI!Cosine. I1 Secsant. CSecat. t. T 1 angent. I8Cosecant. Sine. S7 DEG. Page 44 44 LOGARITHMIC SINES, ETC. 3 DEG. Diff f I ~~~~tI. ~)~S eul~ j Coine/DiffD Cosecant. TangentD Sine. CDftf. Cotangent. Secant. D Cosine. i IOU,, 100'1C t...lo, 0 8-718800 1-281200" 8-719396 11-280604.0 0 ~0 ~59'-O9' 999404 60 1 8-721204 400611278796 8.72180614017 11.278194.000602 10,9999398!59 2 8723595 39841.276405 8.724204 3995 11.275796.000609 12 9-999891!8 3 8725972 3962 1274028 8.726588 3974 11.273412'000616 12 9999384 57 4i 8.728337 39411-271663 8.728959 3952 11.271041.000622 10 9.999378356 t 8.73068813919 1.269312 8.73131713931 11.268683.000629 12 9.999371 a 68.733027 3898 1.266973 8.733663 3909 11.266337'000636 12 9-9993641'54 7 8.735354t3877 1.264646 8'735996]3889111 264004'000643 12 9999357 53 8 8737667 3857 11262333 8'73831713868'11.261683'0006.50 12 9.999350 52 9 8-739969 38361~260031 8'740626!384811'259374'000657 12 999934:;.51 10 8.74225913816 1.257741 8.74292213827 11.257078 000664 12 9993 50 11 8-744536 379611~255464 8.-7452071 3807 11-254793'000671 12 9999329i 49 12 8746802[3776 1.253198 8.7474793787 11.252521'0 00678 12 9999322 48 13 8-749055 3756 1-250945 8.749740 3768 11.250260.000685 12'.99999315 47 14 875129713737 1-248703 8.751989 3749 11.248011.000692 12 9.999308, 46 15 8753528 3717 1.246472 8.754227 3729 11'245773'000699 12 9.999301 45 16 86755747 3698i11244253 88756453 3710111243547'000706 12 9999294-144 17 8757955 36791242045 8758668 3692 11241382 000713 13 99992871 43 18 8'76015113661 1.239849 8.760872 3673 11239128.000721 12 9 999279 42 19 8'762337 3642 1'237663 8'763065 3655 11 236935'0007281 2 9'999272 41 20 8'7645111362411235489 8'765246 3636 11234754'000735 12 9 999'65 40 21 8'766675136061'233325 8'767417 3618 11232583'000743 13 9.9992571 39 22 8'768828 358811'231172 8'769578 3600 11230422'000750 12 9999250 i38 23 8'770970 35701' 229030' 8'771727 3583 11228273'000758 13 9.999242!37 24 8'773101 3553 1226899 8'773866 3565 11226134 1 000765 12 9.999235&!36 25 8'775223 3535 1'224777 8'775995 3548'11'224005'000773 13 9'999227 35 26 8'777333835181.222667 8'778114 3531 112218861'000780 12 9'999220 34 27 8779434 3501 1.220566 8'780222 3514 11'219778'000788 13 9'999212 33 281 8-781524 3484 1-218476 8'782320 3497' 2176800795 13 9 ~9')' 29 8-783605 3467 1216395 8'784408 3480 11215.592'000803 13 9'9991971'31 30 87S5675 3451 1-214325 8786486 3-161 11 213.514 000811 13 9'999189 30 31 8787736 3434 1-212264 8788554 3447 111 211446'000819 13 9' 999181 29 32 8'789787 34181'2102137 879061313431 11 209387'000826 129.999174 28 33 8791828 340211'208172 8'79266293415 11 2077338 000834 13 9 999166 27 34 8'793859 3386 1206141 8794701 3399 205299 0008 13 999158 26 8'781'000830 1 38(396 9 150 25 35 8795881 33701'204119 8796731338 1 03269 000850 19 999 36 8 3797894 3354 1'202106, 8'79875213368 11'201248'000858 103 99949 1.224 37 899897 3339 1'200103 8800763133522111199237'000866 139 999 13 423 38' 8801892 3323 1198108. 8'802765 333711'19723.5 1000874 13 999916 22 39 8.803876 3308 1196124' 838047588 3322111 195242 11 000882 19 99 )11821 40 8.80585213298 1.19148 8806742 8330611 193258 000890 13 9999110 20 i ~0008!i~i 13 i'3 9.999110 20 41 8'807819 73278 1'192181 8 8808<32I 1 91 1.8 091283'00089811 9999102 19 42 880977738263i1.190223 188106831327711893171'000906 13 9999094 18 431 8811726 3249 1.188274 8.81264113262'111873591 000914139 19990 1 44 8'813667 3234 1.186333 8.8145893 248'111 800091 1539'99908677 16 45'~~~~~~~~85110009231599071 45 8'815599 3219 1.184401 8.816529 3233 11 183471 000931 13 9999069 15 461 8817522 3205 1'182478 8.8184611 3219 11181539 000939 13 9.999061 14 47 8.819436 3111 1180564 8820384 3205 111179616 1000971 5i 9-999053'213 48 8-821343 3177 1178657 8.822298 3191 11177702 000956 13 9.999044 12 49 8'8232401316311'176760 8.82420513177 11175795 000964 13 8 999036'11 50 8.825130 3149 1-174870 8.826103 13163 11173897 000973 15 9.999!027 10 51 8-827011 31351'172989 8.827992!315011.172008K000981 13999 9 652 88288843122}1'171116 8829874i3136 11170126 000990 15 9.999010' 8 53 8830749 31.08 11169251 8.831748 3123 11 168252'000998 13 I9 9990002 7 54 8832607 3095 11167393 8.833613 3109 111663187'001007 15 9 998993 6 655 8'834456 3082 1'165544 8.835471 3096 111645249 1001016 159 9989,84 5 56 8.83629730691'163703 8'837321 3083 11162679'001024 13 9.998976'1 4 57 8'838130 305611161870 8.839163 3070 11160837.001033 15 9 998967 3 58 8-839956.30431 160044 8.840998 3057 11159002 C001042 15 9998958' 2 I 5YiO-!1~ril 0i~01~ 11) 1~(98(l i 59 8841774 3030 158226 8842825304 15717500105013 9989 60 8.84358S13017 1'156415 8.814644 3032l11'15.5356:001059 1l 9.998941 0 i j Cosine. I Secant. I Cot. ogesst. I Taus l. -Cooca8llt. Si n. D6 JELI. Page 45 LOGARITHMIC SINES, ETC. 45 4 DEG. tf. z Sine. lDuff Cosecant. Tangent. MMltf Cotangent. Secant. Cosine. Sine. an e.ie)" Ci 0 8-843585 1-156415 8 844644 111-155356 -001059 9-998941 60 1 8845387 3005 1'154613 8'846455 3019 11'153545.001068 15 9'998932 59 2 8'847183 2992 1'152817 8'848260 3007 11151740 001077 15'9'998923 58 8-848971 2980 1-151029 8 850057 29956111149943 001086 15 9 998914!57 48 850751 2967 1-149249 8'851846 298211148154'001095 15 9.998905 56 5 8852525 2955 1-147475 8853628 9M0 11-146372 001104 15 9998896155 6 8 854291 2943 1145709 8-8o554032958 11-144597 001113 15 9.998887 564'T 8856049 2931 1.143951 885717171 2946 11-142829 001122 15 9-998878 53 8 8.857801 2919 1.142199 8.858932 2935 11.141068 001131 15 9 998869 52a 9 8-859546 2908 1'140454 8'86068t; 2923 11'139314.001140 15'9998860 51 10 8'861283 2896 1'138717 8.862438 2911 11137567 001149 15;9,9988-51 50 11 8.863014 2884 113186986 8.864173 2900 111385827 001159 17 j9.998841 49 12 8 864738 2873 1.135262 8 86590612888 111384094 001168 15 9.9988321 48 131 8 866455 2861 1 133545 8-8676328 2877 111132368 001177 15 9'998823 47 14 8'868165 2850 1'134885 8'869351 2866 11-180649 001187 17 9998813' 46 15 8'869868 2839 11130132 8'871064 2854 111128936.001196 15 9'998804 145 16 8'871565 12828 1'128435 8 872770 2843 11 127280 001205 15 9'998795144 17 8873255 2817 1-126745 8'874469 2832 11'125531 -001215 17 9'998785.43 18, 8874938 2806 1"125062 8'876162 2821 11'123838 001224 15 9'998776 A42 19' 8876615 2795 1'123385 8 877849 2811111'122151.001234 17 19 99'8766i"41 20 18'878285 2784 1-121715 88'879529 2800 11'120471 -0012430 15 9'9987*57 40 21 8879949 2773 1 120051 8-881202 2789 11.118798 1001253 17 19998747V 39 22 8881607 2763 1-118393 8'882869 2779 11117131 -001262 17 9'998738138 23 888325,8 2752 11 16742/ 8.884530 2768 11'115470 |001272 15 9'998728 37 24 8-884903 2742 1115097| 8886185i2758 11.11381.5 001282 17 9'998718i836 25 8886542 2731 j111 13458 8'8878332747 11-112167 |001292 17 9'9987083 5o 26 8 888174 27211 1111826 [8'889-176 2737 11-110524 001301 15 9'998699 34 27 8889801 271111-110199 8-891112 2727 11108888 O001311 17 9'998689 133 28 8'891421 270041 108579 8'892742'2717 11-107258 1001321 17 9' 998679 32 29 8 893035 2690 1-106965 8-894366 2707 11-105634 1001332 17 19998669 31 301 8894643 2680 1105357 8'895984i2697 11104016 -001341.17 9998659 30 31 18896246 2670 11037541 8'897596 2687 111102404 10013851 17 999864901129 32 8'897842 2660 1'102158 8899203 2677111100797 001361 17 9998639' 28 33 8-899432 2651 1'100568 8-900803 2667 11'099197 -001371 17 9-998629 27 34 8'901017 264111-098983 8902398 2658 11 097602 -001381 17 9-998619! 26 35 8-902596 263 1 1-097404 8-903987 2648 11 -096013 -001391 17 19-998609 25 36 8 o90416!26i22 1 09583 1 8 905570 2638 11 094430 001401 17 99809, 24 37 8'9037'36 26121094264 8907147 2629111'092853 -001411 17 9'998589 23 38 8.907297 260311-092703 8 908719 2620111091281.001422 18 9.9985 78' 22 39 8 908853 2598 1'091147 Ji 8'910'28t 2610111 089715.001432 17 9.998568 21 40 8.910404 2584 11089596 1 8 911846 2601 11-088154.001442 17 19'998558 20 41 8.911949 2.575 1.088051 1 8 913401 I2592 11-086599.001452 17 [9.998548 19 42 8913488 25661.086512 8 914951 /583 11.085049 -001463o 18 9.998537 I18 43 8.915022 2556 1084978 8'916495 2574 111083505.0014731 17 /9998527 17 44 8 91655012547 1 083450 8 918034 2565 11'081966 -001484 18 9.998516'16 45 8918073 2538 1 081927 8'919568 2556 11'080432 0014941 17 19998506'l15 46' 8919591 2529 11080409 8-92109612547 11-078904.001505 18 9.99849.5 14 4 78-92110312520 1078897 i 8922619 92538 11-077381.001515 17 9-99848.5 13 48 8.922610 2512 11077390 8 924136 2530 11075864.001526 18 9-998174 12 49 8 924112.2.503 11075888 8- 95649 2521 11 074351 1.001536 17 9.998464 11 50 I 8 912609249411 074391 8!927156 2512 11 072844 -001547 18 i99984 10 51 89-27100 248611-072900 8 9286;58 2503 111071342 i-.0015581 1 8399984491 9 52 8.928587 247 7l1-071413 8'930155 1249511 069845 001569 18 99984,311 8 53 8.93006812469 11069932 8.93164712486 11 068353.001579' 17 9.998421 7 54 8.931544 2460 1 068456 8 l9331342478 11G066866 10(01590 18 9'9984101 6 8'988 ~ i.06.98;016~478 111'0t568 55 8 933015' 452l1 06698 85934G161 247011065384'1001601 18 99398399| 5 5 j 8 93448121143 1-065519 8'936093 2461 11-063907 00)1612 18 99983889 4 57 8 93'94.2 2435 1-06-058 8 937565 2453 11 062-435.0016231 18 9 99837i7 3 u8 18 93739812429711 062602 8'9390322445 - 110G60968, 001(634 18 9'998366 2 53 8.38850 24191 -061150 894090,t4 2437 11'0.59506.001645 18 9 9983551 1 608'940'96 2411 1 059i704 8'941952 2429 11 0580418 001656 18 9'983441 0 Cosine. I se':t. Cntangent. Tangent. I Cosecant. sille. i 85 DEG. Page 46 46 LOGARITHMIC SINES, ETC. 5 DEG. Duff. Diff Cosine Sine. Cosecant. Tatngent. Ctangent. Secant. Cosine. 0 8(9.40296 1-059704 8-941952 11 058048 001656 9 998344';0 1 8'941738 2403 1-058262 8-943404 2421 110565696 i'001667 18 19998333'9 2 8-'43174 2894 1-056826 8-944852 2413 11'055148 1001678 18 9'998322)' 8 3 8 944606 2387 1 055394 8.946295 2405 11.053705 1.001689' 18 9.998311 57 4 8'946034 2379 11053966 8-947734 2397 111052266'001700 18 9'998300:)56 5 8-947456 2371 1-052544 8'949168 2390 11'050832 1001711 18 9'998289 55 6 8-948874 2363 1 051126 8'950597 2382 11 049403'001723120 9'998277'54 7 8'950287 2355 11049713 8-952021 2374 11-047979'001734 18 9'9982G6 53 8 8 951696 2348 1 048304 8'953441 2366 11 046559'001745 18 9'998955! 3 9 8-953100 2340 1-046900 8-954856 2359 11-045144'001757 20 9.998243 A51 10 8'954499 2332 1 045501 8-956267 2351 11-043733. 001768 18 9.998232 >50 11 8'955894 2325 1 044106 8-957674 2344 111042326.001780 20 9.998220!49 12 8-957284 231711-042716 8'959075 2336 11'040925'001791 18 9'998209 48 13 8.958670 2310 1 041330 8-960473 2329 111039527.001803 20 9-998197 47 14 8-960052 2302 11039948 8-961866 2322 11-038134 1001814 18 9-998186 46 15 8-961429 2295 1-038571 8-963255 2314 11-036745.001826 20 9-998174145 161 8962801 2288 1-037199 8-964639 2307 11-035361'001837 18 9998i638144 17 8-964170 2280 1 -035830 8-966019 2300 11-033981'001849 20 9-998151'43 18 8 965534 2273 1 034466 8-9G67394 2293 11-032606 001861 20 9-998139 42 19 8-966893 2266 1-033107 8-968766 2286 11-031234'001872 18 9-998128 41 20 8-968249 2259 1-031751 8-97013312279 11-029867'001884 20 9-998116140 21 8'969600 2252 1-030400 1 8971496 2271 11-028504'-001896 20 9'998104 39 22 8,970947 224511 029053 8'972855 2265 11'027145 1001908 20 9-998092 I38 23 8-972289 22388 1027711 8'974209 2257 111025791'001920 20 9-998080 137 24 8-973628 2231 1 026372 8-975560 2251 11-024440 1001932 20 9 998068i1 36 25 8-9714962K224 1-025038 8-976906 2244 111'023094'001944 20 9 9980-61i35 26 8-976293l22171-028707 8-978248 2' )3 11 021752'l0019-56 20 9-998044 34 27 8-977619 2101-022881 8979586 223011-020414 -001968 20 9-9980382 33 28 18-978941 2i2031-021059 8-980921 2223111-019079 i-0019801 20 9-998020 32 29 8 9802591219711.19'9741 8982251 221711110177491 001992 20 9998008 [31 30 8 981578312190 1-018427 8-983577 221011-016423 i002004 20 9997996 30 31 898883117 8984899 2204111-015101 00216 20 9.997984 29 32 1 898418912177 1-015811 8-986217 2197111-013783 0020281 20 999 7972 28 33 8985491121701.014509 8987532 2191 111012468'002041 22 9.997959 27 34 8-9867892i163 1-013211| 8988842 2184111011158.002053 20 9-997947 26 35 8-988083 2157 1- 011917 8-990149 2178:11-009851'002065 20 9'99793>5 25 36 818989374 12150 1-010626 8-991451 2171 11-008-549 002078 22 9-997'922 24 37' 8 990660 2144 1-009340 8-992750!2165 11-007250 002090 20 9-997910 23 388 189919431 138'1 1008057 8-994045 215811. 005955'002103 22 9-997897 2'2 39! 8.993222 21381 1-006778 8-995337 2152 11-0046603 1 002115 20 9-997885 i21 40 8-994497 212511-005503 8-996624 2146 11-003376 1-002128 -22 19-997872 20 411 8-995768 21191-004232 8-997908 2140 11-002092 1-002140 20 9-997860 19 42 1 8 997036 21121-002964 8-999188 2134111-0081'2 - 002153 22 19-997847 18 143 8-998299 21061-001701 9 000465 2127 10-9995a35 -00 216.5 20 9-997835 17 44 8-999560 2100 1-000440. 9-001738 2121 10-998262'0021a78 22.9-99782 21G 45 11 9-000816 2094 0-999184 9-003007 2115110-906993 I 002191 22 9-997809 15 46 19-00206920880-997931, 9-004272 2109110-995728'1002203 20 9-997797 114 47 i 9-003318120820-996682 1j'J-005534 2103 10-994466 1-002216 22 9-99778-4 13 48 9 -004563 20761 0995437 19-006792 2097 10-993208 1002229 22 9-997771 12 49 9-00580a5207010-994195 119-008047 2091 10-991953 1-002242 22 9-997758 11 50 9-007044 20640 0992956 9009298 2085,10-990702'-002255, 22 19997784- 10 51 9-008278 2058 0-991722 19-010546 2079110-989454 -002268 22 9-997732 9 52 1 9 009510120520-990490 9011790 2074 10-988210 1002281 22 9-997719 8 53 9-010737 2046l 0989263 9-013031 2068 10-986969 1,002294 221 9 -9977106 7 54 9-011962 2040 0-988038 9-01426812062 10-985732 1-002307 22 1999.7693 6 55 9-013182 2034 0-986818 9-015502 2056 10-984498 1-002320 22 9-.997680 5 56 9 014400 2029 0-985600 9-016732 2051 10-983268 -002333 22 9-997667 4 57 9-015613 2023 10984387 9 017959 2045 10-982041 1.002346 22 9-997654 3 58 39-016824120170-9831761 9-019183 2039 10-980817 - 0023591 22 19997641 2 59 9-018031 2012!0-981969.9-020403 2034 10-979597 -002372 22 9-997628 1 60 9-019235 200610-980765 9-02162012028110-978380 -002386 23 9-997614 0 / | Cosine. | Secant. Cotangent. Tangent. 1{ Cosecant. Sinle. 84 DEG. Page 47 LOGARITHMIC SINES, ETC. 47 6 DEG. Sine. Diff Cosecant. Tangent. D uff Cotanbent. i Secant. 1 Cosii ne. 0 9-019235.-80765 9021620 10-978380' 00'2386 9)'997614160 1 9-020435 2000.979565 9-022834 2023 10.977166.002399 22 9.997601 59 2 9-021632 1995 *978368 9.024044 2017 10-97 5956.002412 22 9.9197588! l58 3 9-022825 1989.977175 9.0'25251 2011110.974749.002426 23 93997574 57 4 9024016 1984 975984 90264552006 9735455 200610973545 002439 22 9 9-9975616 5 9025203 1978.974797 9.027655 2001 109723451 -0024531 23 9.997547' 55 6 9-026386 1973 973614 9 028852 1995i10'971148 -002466! 22 9-9975341!54 7 9027567 1967.972433 9'030046 1990.10.969954.002480 23 9-997520 53 8 9-028744 1962.971256 9'031237 1985!10-968763.002493 22 9-997'5071 52 9 9.029918 1957.970082 9'032425 1979 10'967575.002507 23 99971493 51 10 9.031089 1951 968911 9-033609 197410.966391 002520 22 9(997480 50 11 9 032257 1946 *967743 9-034791 1969'10'965209 002534 20 9-997466'49 12 9-033421 1941 *966579 9 035969 1964110-964031'002548 23 9.997462 648 13 9-034582 1936'965418 9'087144 1958 10'962856 *002561 22 9-997439 47 14 9-035741 1930.964259 9'038316 1953 10'961684.002575 23 9'997425146 15 9-036896 1925.963104 9'03948511948 10'960515.002589 23 9.99741 11 45 i 16 9.038048 1920.961952 9.040651 1943110.959349 002603 23 9.-9973971i44 17 9.039197 1915.960803 9.041813 1938,10.958187 1002617 23 9-997383'43 18 9.040342 1910'959658 9-042973 19338 10957027'002631 23 9-997'369 42 19 9-041485 1905'958515 9'044130 1928 10'955870 1002645 23 9'997855 41 20 9-042625 1899 9'57375 9'04528411923 10'954716 1002659 23 9.9973411~40 21 9.043762 1895'956238 9-046434 1918 10'953566 1002673 2.3 9.997327 39 22 9 044895 1889'955105 9'047582 1913 10'952418 1002687 23 9-997313'38 23 9.046026 1884'953974 1 9.048727 190810.951273 1-002701 23 9-997299 37 24 9.047154118793.952846 9'049869 1903 110-950131 1-002715 23 9-997285 36 25 9-048279 1875 *951721 9.051008 1898110.948992 |-002729 23 9.997271! 35 26 9-04940011870.950600 9.052144 1893 10.947856 1-002743 23 9.997257, 34 27 9-050519 1865'949481 90653277 1889 10.946728 |-002758 25 9-997242 33 28 9.051635 1860.948365 1 9.054407 1884 10-945593 1.002772 23 9997228 32 29 9.052749 1855.947251 9.055535 1879 10.944465 1.002786 23 9.997214 31 30 9.053859 1850.946141 9056659 11874 10.943341.002801 25 9.9971991 20 31 9.054966 1845.945034 9.057781 1870 10,942219 |'0028151 23 ]9.997185 29 32 19056071 1841 -943929 9.058900 1865 10.941100 1.002830 25 9.997170 28 33 9.057172 1836 *9428281 9-060016 1860 10.939984 1.002844 23 9.997156 27 34 9.058271 1831.941729 9-061130 1855 10.938870 -002859 25 9-997141 126 I35 9059367 1827.940633 1 9.062240 1851 10.937760.002873 23 9.9971271 25 36 9.060460 1822.939540 9.063348 1846 10.9366521 002888 25 9.997112 24 37 9.061551 1817 *938449 9.064453 1842 10.935547.002902 23 9.997098 ]23 38 19062639 1813.937361 9 065556 1837 10.934444 1.002917 25 9.997083 22 39 | 9.063724 1808 -936276 9 066655 1833 10.933345 |.|002932 25 9 997068 121 40 9.064806 1804'935194 1t 9.067752 1828110.932248 I. 002947 25 9.9970531120 41 9.0658851799'934115 9.068846 1824 10'931154.002961 23 9.997039 119 42' 9.066962 1794'933038 9'069938 1819 10'930062'002976 25 9.997024 118 43 9.068036 1790'931964 9-071027 1815 10'928973 i002991 25 9.997009 17 44 19069107 1786 -930893 9'072113 1810 10'927887 11-003006 25 9.996994 16 45 9.070176 1781.929824 9.073197 1806 10.926803.003021 25 9.996979 15 461 9071242 1777.928758 9.074278 1802110.925722 10030361 25 9.996964/14 47 9.072306 1772'927694 9.075356 1797 10.024644 1003051[ 25 9.996949 13 48 9.073366 1768'926634 9.076432 1793'10 923568 1003066[ 25 9.996934 112 49 9.074424 1763'925576 9.07750511789110.922495'003081125 9.996919 11 50 9-075480 1759'924520 9.07857'61784!10-921424.003096 25 9.996904 10 51 9.076533 1755 -923467 9.07964411780110 920356.003111 25 9.996889 9 52 9.077583 1750'922417 ] 9.080710 177611 0 919290.003126l 25 9,-99'6874[ 8 53 9.078631 1746'921369 9.08177311772 10 918207.003142 27 9 996858 7 54 90796 742920324 9 08831 76 101717 0031571 25 9.996843 55 9-080719 1738'9192811 90889)l1 763 10-916109 -003172'25 9.996828 5 56 9'081759 1733'918241 9.084947 1759 10 910503, 003188 27 9 09681f2 4 57 9.082797 1729['9172103 9.0860001i755 10-914000 I003203 25 9-996797/ 3 58 9083232 1725'9161681 9-087050i1751 10-91.')9D0 -00321.81 25 999'36782 2 59 9.084864 1721 *915136I 9.08809811747110-9119021 003234 2719-996766 1 60 9-085894117171 *9141061 [908921441174310- 10856 1 -003249 25 9-996751 0 Cosine. Secant. 1 Cotangent. I TI'augent. i Cosecant. Sine. i 83 DEG. Page 48 48 LOGARITHMIC SINES, ETC. 7 nDEG. Dill Diff [ Tangngent. S t. Dill Dff''!. |Sine. Cosecant. i Tangent. i Cotangent. Secant. 1;100"'. Cosine. -0-u8) 98584 1410 9089144 109108.56 1-0308243 9 9 996751 160 19 9086922 1713 913078 i 9090187 1738 10-909813 -003265 27 9-9967351 59 2 l9(3087947 1709 *912053 9-091228 1735 10908772 1'003280 25 9996720 58 3 9 088970 1704.911030 9-0922,66 1731 10.907734. 003296 27 9.9967041 o7 4 9.089990 1700.910010 9-093302 1727 10.906698.003312 27 9-996688 iO6 5 9:091008 1696.908992 9-094336 1722 10-905664.0033027 25 9-996:67i 65 6 9 092024 1692.907976' 9.095367 1719 10.904633 1003343 27 9.996657 134 7 9-093037 1688'906963! 9-096395 1'715 10'903605'003359 27 9-9966411;53 8 9.094047 1684.905953 19097422 1711 10.902578 003375'27 9.996625:s% 9 909.5056 1680 904944 i 9-098446 1707 10-901554.003390 25 9-996010 51 10 9-0960t62 1676 -903998 8 9-099468 1703 10-900532'003406 27 9'996594 50 11 9097065 1673'9029358 9'1004871 1699 10'899513 003422 7 19996578t149 12 9'098066 1668'901934 9'101504 1695 10'898496'003438 27 9'996562 48 13'9 099065 1665'900935 9 102519 1691 10'897481 003454 27 9'996546'47 14'9100062 1661 *899938'91083532 1687 10.896468 j003470 27 9.996530 46 15 1 9101056 1657 -898944 19 104542 1684 10'895458.003486 27 9996514j 45 16 9.102048 1653'897952 9.105550 1680 10.894450 i 003502 27 9 996498 44 17, 9'103037 1649'896963 I 9,106556 1676 10.893444.003518 27 9.996482 43 18 9.104025 1645.895975.3107559 1672 10.892441.003535 28 9 996465142 19 9.105010 1642.894990 9 10856011669110.891440 -003551 27 9.996449!141 201 9105992 1638 894008 9109559 1665 10890441 003567 27 9.9964331140 21 9106973 1634.893027 9.110556 1661 10.8894441 008583 27 9.936417 39' 22 9-107951 1630.892049 9111551 1658 10.888449 -003600 28 9.9964001138 23 9 108927 1627.891073 9 112543 1654 10.887457.003616 27 99963841 37 24 9-109901 1623 -890099' 9-113533 1650 10-886467 003632 27 9-996368.136 25 1 9-110873 1619.889127 9-114521 1647 10-885479 1003649 28 9-99635113.5 261 9111842 1616.888158 9 115507 16431 10.884493 *003665 27 9-996335134 27 9-112809 1612'887191 9.116491 1639 10 883509.003682 28 9'996318;133 28 9-113774 1608.886226 9.117472 1636 10-882528'003698 27 9-996302 32 29 9'114737 1605 885263 9-118452 1632 10.881548 003715 28 9'996285 31 30 9115698 1601'884302 9-119429 1629 10 880;571 -003731 27 9'996'269'30 31 i 9'116656 1597 *883344 9-120404 1625 10-879596 -003748 28 9-996252 i29 32 9-11761311594 *882387 Ii-121377l1622 10.878623 -003,765 28 9.9969235:28 33 9'118567,15901 881433 9.122348 161810-8776552'003781 27 9.996219 27 34 9'119519 1587'880481 9 123317 161510.876683'003798 28 19'996202 26 35 19120469 1583'879531 9.124284 11 11i10. 875716.00381.5 28 9'9961851'25 36 9 121417 1580 *878583 9.125249 1608 10874751.003832 28 9 9961068,124 37 9'122362 1576'877638 9-126211 1604 10.873789'003849 28 9.996151' 93 38 9'12330611573'876694 9-127172 1601 10.872828'003866 28 19996131!122 39 9-124248150691 875752 9-128130 1597 10.871870 1003883 28 9.996117 21 40 i 9125187 15661'874813' 9129087 11594 10'870913 -003900 28 9-996100 20 41 9 126125'1562[ 873875 9.130041 1591I10 869959.003917 28 9'9960831 19 42 9-127060 1559 *872940 9.130994 1587 10.869006'003934 28 9.996066 18 43 9127993 15561 872007 9.13194411584 10.868056 1003951 28 9 996049' 17 44 i 9128925i1552'871075 1 9.132893 1581 10 867107.003968 28 9.996032116 45 9.12985411549'870146 9-13383911577110-866161 -003985 28 19996015!15 46 9-130781 1545 *869219 9 134784{1574 10.865216 1004002 28 9.995998 14 47 9.131706 1542 868294 9135726 157110-864274 1004020 30 9.995980I 13 48 9.1326301539,867370' 9.136667[1567 10.863333.004037 28 9.995963112 49 9-13355111535 *866449 9.137605 1564 10.862395 0040541 28 9995946 11 50 9-134470 1532.865530 9-13854211561 10.861458.0040721 30 9-995928 10 51 9.135387 1529.864613 9-139476 15658110860524 l004089 28 9995911 9 52 9-136303 1525 *863697 9.140409 1555 10.859591.004106 28 99958941 8 531 9-137216 1522.8627841 9-141340 1551 10-858660.0041241 30 19995876 7 541 9.138128 1519.861872 9.142269 154810.857731 1004141 28 9995859' 6 55 9413903711516.8609631 9-143196 1545 10.856804 i 00 1159 30 19995841 5 56 9.139944'1512.860056' 9144121 1542 10.855879.004177 30 19995823 4 57 9.140850 1509.859150a 9.145044 1539 10.854956.004194 28 9.9958061 3 58 9.141754 1506 858246 9.145966 1535 10854034.004212 30 9.995788 2 59 9-142655 1503 857345' 9 1468851153210853115 004229 28 19 9957711 1 60 1 9-143555 1500 856445 9 147803 1529110.852197 004247 30 9995753 0 L —-I Cosine. ]-I Secant. Cotaangent. T Tangent. 1 Cosecant. Sine. 82 DEG. Page 49 LOGARITHMIC SINES, ETC. 49 8 DEG. Sine. Diff. Cosecant. Tangent. 10 Cangent. tSecant. Dff Cosine. 0 9-143555'856445 9-147803 10-852197 -004247 i9-99575 60 1 9-144453 1496 855547 9o14871'8 1526 10851282 00426.5 0 9 995735 59 2 9-145349 1493 854651 9-149632 1523 10850368 -004283 30 9 995717 58 3 9-146243 1490 -853757 9-150544 1520 10-849456 -004301 30 9'995699 57 4 9-147136 1487 -852864 9-15144' 1517 10-848546 -004319 30,9-995681 56 5 9-148026 1484 851974 9-1526.3 1514 10-847637 *004336 28 9 995-664- 55 6 9-148915 1481 -851085.9-153269 1511 10-846731. -004354 30 9-995646 54 7 9-149802 1478 -850198 9-154174 1508 10-845826 -004372 30 9 995628 53 8 9-150686 1475 -849314 9-155077 1505 10-844923 -004390 30:9-995610 52 9 9-151569 1472 -848431 9-155978 1502 10-844022 -004409 32 9-995591 51 10 9-152451 1469 -847549 9156877 1499 10-8431,23'004427 30 9'995573 50 11 9-153330 1466 846670 9-157775 1496 10-842225 -004445 30'19 995555 49 12 9-154208 1462 *845792 9-158671 1493 10-841329'004463 30 19-995537 48 13 9-155083 1460 *844917 9159565 1490 10-840435 -004481 30 9-995519 47 14 9-155957 1457 *844043, 9160457 1487 1,0839543'004499 30 9-995501'46 15 9156830 1454 843170 9,-161347 1484 10-838653 -004518 32 9-995482 45 16. 9157700 1451 *842300 9-162236 1481 10-837764'004536 30 9-995464 44 17 9-158569 1448 -841431 9-163123 1478 10-836877'004554 30 9-995446 43 18 9-159435 1445 -840565 9-164008 1475 10-835992 -004573 32- 9995427 42 19 9-160301 1442 -839699 9-164892 1473 10-885108 -004591 30 9-995409 41 20 9-161164 1439 *838836. 9165774 1470 10;834226 -004610 32 9-995390 40 21 9-162025 1436 -837975 9-166654 1467 10-833346 0,04628 30 9-995372 39 22 9.162885 1433 -837115 9167532 1464 10-832468 -004647 32.19.995353 38 28 9-163743 1430 t836257 9-168409 1461 10-831591'004666 32 9-995334 37'24 9-164600 1427 -835400 9-169284 1458 10-830716 -004684 30 9-995316 36 25 9-165454 1424 -834546 9-170157 1455 10-829843 -004703 32 9-995297 35 26 9-166307 1422 -833693 9-171029 1453 10828971 -004722 32 9.995278 34 27 9-167159 1419 -832841 9-171899 145010-828101 -004740 30 9995260 33 28 9-168008 141 *831992 9-172767 1447 10-8217233 004759 3 9-995241 32 29 9-168856 1413 -831144 9-173634 1444 10-826366 -004778 32 9-995222 31 30 9-169702 1410 -830298'9-174499 1442 10-825501 -004797 32 9-995203 30 8, 9-'170547 1407 -829453 9175362 1439 10824638 -004816 32 9-995184 29 32- 9-17,1889 1405.828611 9-17,62241436 10-823776'004835 32 9-995165 28 33 9,172230 1402 -827770 9-177084 1433 10-822916 -004854 32 9-99,5146 27 34 9-173070 1399 -826930 9-177942 1431 10-822058 *004878 82 9-995127 26 35 9-173908 1396 -826092 9'178799 1428 10-821201 -004892 32 9-995108 25 36 9-174744 1.894 -825256 9-179655 1425 10-820345 004911'32 9-995089 24 37 9-175578 1391 -824422 9-180508 1423 10-819492 -004930 32 9-995070 23 38 9-176411 1388 -823589 9,181360 1420 10818640 -004949 32 9-995051 22 3,9 9.177242 1385 -822758 9-182211 1417 10-817789 -004968 32 99,95032 21 40 9-178072 1383 -821928 9-183059 1415 10-816941 -004987 32 9-995013 20 41 9-178900 1380'821100 9-183907 1412 10-816093'005007, 33 9-994998 19 42 9-179726 1377'820274'9184752 1409 10-815248'005026 32 9-994974 18 43 9-180551 1374'819449 9-185597 1407 10-814403 -005045 32 9-9949551 17 44 9181374 1372 -818626 9-186439 1404 10-813561'005065 33 9-994935'16 45 9-182190 1369 -817804 9-187280 1402 10-812720'-005084 32 9-994916 15 46'9183016 1367'816984 9-188120 1399 10811880'005104'33 89994896 14 47 9-183834 1364 -816166 9-188958 1396 10-811042 1005123 82 9-994877 13 48 9,184651 1361'8153491 9189794 1394 10-810206 *00514$ 33 9,994857 12 49 9'185466 1359 -814534 9190629 1391 10-809371'005162 32 9-99483811 50 9-186280 1356 813720 9-191462 1389 10808538 1-005182.83 9994818 10 51 9-1870921135,3 -812908 9-192294 1386 10-807706 1005202 33 9-994798 9 52 9-187903 1351'812097 9-193124 1384 10.806876 1005221 32 9-994779 8 53 9-188712 1348'8112881 9193953 1381 10;806047'005241'33 9'994759 7 54 9'18951911346 -810481 9-194780 1379 10805220'005261 33 9-994739 6 55 919032511343 -8096751 9195606 1376 10-804394'005281 32 9-994720 5 56 9-191130 1341'808870 9.196430 1374 10-803570'005300 833 9-994700 4 57' 19193831338'808067 9-197253 1371 10-802747 -005320 33; 99944801 3 58 9-1927341336 -807266 9-198074, 1369 10-801926 -005340 83 9-9,94660 2 59 9-1935341333 806466 9-198894,1366 10-801106'005360 33 9-994640 1 60 9-194332 1330 *805668 9199713 1364 10-800287 -005380 33 9-994620 0 Cosine. Secant.' Cotangent. Tangent. ICosecant. Sine. I 81 DEG. Page 50 50 LOGARITHMIC SINES, ETC. 9 DEG. / Sine. Duff Cosecant. Tangent. Diff; Cotangent. Secant. 1D1, Cosine. 0 9-194332 -805668 9-199713 10-800287 i-005380 9 994620 60 1 9119512911328 -804871 9-200529 1361 10-799471 -005400 33 9-994600 59 2 9-195925 1326 804075 9-201345 1359 10-798655 *005420 33 9-994580; 58 3 9-196719 1323 803281 9-202159 1356 10 797841 -005440| 33 9-994560 57 4 9 197511 1321'802489 9-202971 1354 10-797029'005460 33 9-994540Q 56 5 9-19830211318 -801698 9-203782 1352 10-796218 005481 35 9-994519, 55 6 9-199091 1316 -800909 9-204592 1349 10-795408 1005501 33 9-9944999 54 7 9-199879 1313 -800121 9-205400 1347 10-794600 -005521 33 9-994479 53 8 9-200666t 1311 799334 9-206207 1345 107979793 1005541 33 9-994459 52 9 9-201451 1308 798549 9-207013 1342 10-792987 1005562 35 9994438 51 10 9-202234 1306 797766 9-207817 1340 10-792183 -005582 33 9-994418 50 11 9-203017 1304 -796983 9-208619 1338 10-791381 0056021 33 9-994398; 49 12 9-203797 1301 796203 9-209420 1335 10-790580 0056231 35 9.994377 48 13 9-204577 1299 *795423 9-210220 1333 10-789780 1 005643 83 9-994357i147 14 9-205354 1296 -794646 9-211018 1331 10-788982 i-005664 35 9-994336 46 15 9-206131 1294 -793869 9-211815 1328 10-78818.5 005684 33 9-9943161 45 16 9-206906 1292 -793094 9-212611 1326 10-787389 1 005705 35 9-994295 44 17 9-207679 1289 792321 9-213405 1324 10786595 1 0057261 35 99-94274 43 18 9-208452 1287 -791548 9-214198 1321 10-785802 10057461 33 9-9942541 42 19 9-209222 1285 -790778 9-214989 1319 10-785011 1-005767 35 9-994233 41 20 9-209992 1282 -790008 9-215780 1317 10-784220 1-005788 35 9-994212 140 21 9-210760 1280- 789240 9-216568 1315 10-783432 1-.005809 35 9-994191139 22 9211526 1278 -788474 9-217356 1312 10-782644 i005829 33 9994171 38 23 9-212291 1275 -787709 9-218142 1310110-781858 005850 35 9-994150 37 24 9-213055 1273 -786945 9-218926 1308 10-781074 1-0058711 35 9-994129 36 25 9-213818 1271 -786182 9-219710 1305 10-780290 -0058921 35 9-994108 35 26 9-214579 1268 -785421 9-220492 1303 10779508 1-0059131 35 99940871134 27 9-215338 1266 -784662 9-221272 1301 10-778728 1.0059341 35 9-994066 33 28 9-216097 1264 -783903 9-222052 1299 10-777948 1-005955; 35 9-994045132 29 9-216854 1261 -783146 9222830 1297 10-777170 10059761 35 9-994024 131 30 9-217609 1259 -782391 9-223607 1294 10-776393 -0059971 35 9-994003 30 31 9.21836311257 781637 9-224382 1292110-775618 l0060181 35 9-993982 29 32 9-219116 1255 -780884 9-225156 1290 10-774844 i-006040 37 9-993960 28 33 9-219868 1253 -780132 9-225929 1288110-774071 i 0060611 3 9-993939 27 34 9-22061811250 *779382 9-226700 1286 10-773300,-006082 35 9-993918 26 35 9-221367 1248 -778633 9-227471 1284 10-772529 i006103 35 9-993897 25 36 9-22211511246 -777885 9-228239 1281 10-771761 -00612.5 37 9-993875 24 37 9-222861 1244 -777139 9-229007 1279 10-770993 -0061461 35 9.993854;23 38 9-22360611242 -776394 9-229773 1277 10-770227 1006168 37 9-9938321122 39 9-22434911239 -775651 9-230539 1275 10-769461 1-006189 35 9'993811l121 40 9-22509211237 -774908 9231302 1273 10-768698 1006211 37 9-993789 20 41 9-22583311235.774167 9-232065 1271 10-767935 1-006232 35 9-993768 19 42 9-22657311233 -773427 9-232826 1269 10-767174 0(06254 37 9-993746 18 43 9-22731111231 -772689 9-233586 1267 10-766414. 006275 35 9-993725117 44 9 228048 1228 -771952 9-234345 1265 10-765655 1006297 37 9-993703 16 45 9-22878411226 -771216 9-235103 1262 10-764897 1-006319 37 9-993681 15 46 9-22951811224 -770482 9-235859 1260110-764141 -006340 35 9993660 14 47 9-230252 1222'769748 9-236614 1258 10-763386 -006362 37 9-993638 13 48 9-230984 1220 -769016 9-237368 1256 10-762632 -1006384 37 9993616 12 49 9-231715 1218'768285 9-238120 1254 10-761880 1-006406 37 9-993594.11 50 9232444 1216'767556 9-23887211252 10-761128 -006428 37 9-993572 10 51 9-233172112141 766828 9239622 1250110-760378 -006450 37 9-993550; 9 52 I 9233899 1212 -766101 9-24037111248110-759629 i-006472 37 9-993528 8 53 9-23462511209 -765375 9-24111811246110-758882 1006494 37 9^993506 7 54 9-235349 1207 -764651 9-241865 1244 10-758135 006516 37 9-993484 6 55 9-236073 1205'763927 9242610 1242 10-757390 -00653o 37 9-993462 5 56 9-23679511203 -763205 9-24335411240 10-756646 1-006560 37 9-993440 4 57 9-23751511201.762485 9244097 1238 10-755903.006582 37 9993418 3 58 9-23823511199 761765 9244839 1236110-755161'006604 37 9-993396 2 59 923895311197'761047 9-245579213410-7544421 I 006626 37 9-993374 1 60 9-23967011195 -760330 9-24631911232110-753681 i 096649 38 9-993351' 0 I Cosine. Secant. I Cotangent. Tangent. i Cosecant. Sine. l 60 DEQ. Page 51 LOGARITHMIC SIrES, ETC. 51 10 DEG. Sine. ^ Cosecant. Tangent, 100 Cotangent. Secant. D1;li Cosine. 0 9-239670 -760330 9-246319 10-753681 -006649 9-993351 60.1 9-240386 1193 -759614 9-247057 1230 10,752943'006671 37 9-993329 59 2 9-241101 1191 -758899 9-247794 1228 10-752206 -006693 37 9-993307 58 3 9-241814 1189 -758181' 9-248530 1226 10-751470 -006715 38 9-993285 57 4 9-24252611'87 -757474 9-249264 1224 10-75073,6 006738 37 9-993262 56 5 9-243237 1185 -756763 9-249998 1222 10-750002'006760 37 9-993240 55 6 9-243947 1183 -756053 9-250730 1220'10-749270 -006783 38 9-993217 54 7 9-244656 1181 -755344 9-251:461 1218 10-748539. 006805 37 9-993195 53 8 9-245363 117-9'754637 9-252191 1217 10-747809 -006828 38 9-993172 52 9 9-246069 1177 -75393i 9-252920 1215 10-747080 -006851 38 9-993149 51 10 9-24677511175 -753225 9-253648 121'3 10-746352'006873 37'9-993127 50 11 9-2474781173 -752522 9-254374 1211l'10-74562S -006896 38 9-993104 49 12 9-248181 1171 -751819 9-255100 1209 10-7.44900' -006919 38 9-993081 48 13 9-248883 1169'-751117 9-255824 1207 10-744176 -006941 37 9-993059 47 14 9-249583 1167 -750417 9-256547 1205 10-743453 -006964 38 9-993036 46 15 9-250282 1165 -749718 9-257269 1203 10-742731 -006987 38 9-993013145 16 9-250980 1163 -749020 9-257990 1201 10-742010 -067010 38 9 -992990 44 17 9-251677 1161 -748323 9-258710 1200 10-741290 -007033 38 9-992967 43 18 9-252373 1159 -747627 9-259429 11.9810-740571 -007056 38 9-992944 42 19 9-258067 1158 -746933 9-260146 1196 10-739854 -007079 38 9-992921 41 20 9-253761 1156.-746239 9 260863 1194 10 739137 -007102'38 9-992898 40 21 9-254453-1154''745547 9-261578 1192 10-738422 -007125 88 9-99287'5 39 22 9-255144 1152 -744856 9-262292 1190 10-737708 -007148 38 9-992852 38 23 9-2558341150 -,744166 9-263005 1189 10-736995 -007171 38 9-992829137 24 9256'523 1148 -743477 9-263717 1187 10-736283 -007194 38, -992806 36 25 9-257211 1146 -742789 9-264428 1185 10-735572 -007217 38 9-992783 35 26 9-257898 1144 -742102 9-26513'8 1183 10-734862 -007241 40 9-992759134 27 9-258583 1142 -741417 9.265847 1181 10-73415g -007264 38 9-992736 33 28 9-259268 1141 740732 -9-266555 117.9 10-733445 -007287 38 9-992713!.32 29 9-259951 1139. 740049'9267*261 1178 10-732739 -007311 38 9-992690-31 30'9'260633 1137 -739367 9-26796711176 10-732033 -007334 40 9-992666130 31 9-261314 1135 -738686 19-268671 1174 10-731329 *007357'38 9-992643 29 -32 9-261994 1133 -738006 9-269375 1172 10-730625 -007381 40 9-9926191 28 33 9-262673 1131 -737327 9-270077 1170 10-729923 -007404 38,9-992596 27 34 9-263351 1130 736649 9.,270779 1169 10-729221 -007428 40 9-992572 26 35 9-264027.1128 -735973 9-271479 1167 10-72852'1 -007451 38-9-992549 25 36',9-264703 1126 -735297 1 9272178 1165 10:727,822 -007475 40' 9-992525 24 37 992-65377 1124 -734623 9-272876 1164 10-727124 -007499 40 9-99250 123' 38 9-266051 1122 -733949 9-273573 1162 10-726427 -007522 38 9-992478 22 39 9-266723 1120 -733277 9-274269 1160 10-725731 -007546 40 -9-992454 21 40 9-2'67395 1149'732605 9-2749641158 10-725036 -007570 40 9-992430120 41 9-268065 1117 731935' 9-275658 1157 10-724342 -00759,4 40 9-992406 19 42 9-268734 1115'731266 9.-276351 1155 10!723649 -007618 40 9-992382 18'43 9-269402 1113 -730598 1 9-277043 1153 10-722957 -007642 38 9-992'358 17 4'4'9.270069 1111 -729931 IT 9-277734 1151 10-722266 007665 40 9-992335 16 45 9-270735 1110'729265 j 9 278424 1150 10-721576 -007689 40 9-99231115 46 9-271400 1108 728600 i 9-2791131148 10-720887 -007713 40 19992287 14 47 9272064 1106 *727936:.9-279801 1146 10-720199 -007737 40 9-992263 13 48 -9-27726 1105 -727274 j 9-280488 1145 10-7195512 -007761 40 9&992239 12 49 9-273388 1103 726612 9-2811741143 10-718826 -007786 42 9-99221411 50 9-274049 1101 -725951 9.281858 1141 10:718142 -007810 40 9-9P2190 10 51 9.-27470811099'.725292 9282542 1140 10'717458 -007834 40 9-992166 9 52 9-275367 1098 -724633, 9-283225 1138 10-71677,5 -007858 40 9-992142 8 53 19-276025 1096 -723975 9 283907 1136 10-716093 -007882 40 9-992118 7 54 9-276681 1094.723319 9-284588 1135 10-715412 -007907 42 9-992093 6.55 9-277337 1092:, -722663 9-285268 1133 104714732 -007931 40 19'992069 5 56 9-277991 1091 *-722009 i-285947 11l1 10-714053 -007956 42 19992044 4 57 9-278645 1089 -7213-55 9-286624 1130 10-713376.007980 40 9-9920203 3 58 9-279297 1087'-720703 9287001 1128 10-712699 -008004 40'9-991996 2 59 91279948i1086 -720052 9-287977 1126-10-712023 1-008029 42 19991971 1 60 9-280599'10841 -719401 9 288652 1125 10711348 1008053 40 9991947 0 Cosine. | Seeant. Cotangent. Tangent.' Cosecant. Sine. 35 79 DEQG Page 52 52 LOGARITHMIC SINES, ETC. 11 DEG. Sine. Dff. Cosecant. Tangent. i f0r; Cotangent. Secant. Dt. iff. Cosine. | - 10" 100". 9-280599 27194011 9288652 10-711348 -.0080531 -991947 60 1 i| 9-281248 1082 -718752 9-289326 1123 10710674 008078 42 999192 2 9281897 1081 -718103 I9-289999 1122 10-710001 -008103 42 1999189 I7!58 3 i 9-282544 1079'717456 9-290;671 1120 10.709329;1-008127 409 -991873 157 4 9-283190 1077 -716810 9291342 1118 10-708658 1-008152 42 9991848 56 5 9-283836 1076 -716164 9-292013 1117 10-707987 1-008177 42 i9-991823'j55 6 9-284480 1074 -715520'9 -292682 1115 10-707318 -008201 40 9'991799 54 7 9-285124 1072 -714876 9-293350 1114 10 706650 i -008226 4299 991774 53 98 9285766 1071 -714234 9-294017 1112 10-705983 008251 429 9 991749 i:2 9 9-286408 1069 -713592 9-2)4684 1111 10.05316 008276 4'9991724 51 10 9-287048 1067 -712952 1 9-295349 1109 10-.704651.008301 42 19-99169.) 50 11 9-287688 1066 -71-2312 9-296013 1107 10-703987 -008326 42 9 991)674 49 12 9-288.326 1064 711674 9 296677 1106 10-703323 -0083531 42 9-991614948 13 9-288964 1063 -711036 9-2 97339 1104 10-702661 [008876 42 9 99162&4t 17 14 9-289600 1061 0710400 9 298001 1103 10- 01999 -008401 42199915999 46 15 9-290236 1059 -7097641 9-298662 1101 10.701338 008426 42 19 99157-1;45 16 9-290870 1058 -709130 | 9299322 1100 10-700678 -008451 42 9-9915491q44 17 9-291504 1056 7084960 9-299980 1098 10-700020 -008476 42 9-991524 43 18 9-292137 1054 707863 9-300638 1096 10-699362 -008502 43 9-991498, 42 19 9-292768 1053 707232 19-301295 1095 10-698705 -008527 42 9-9914731 41 20 9-293399 1051 706601 19-301951 1093 10-698049 -008552 42 9-991448 40 211 9294029 1050 -705971 9-'302607 1092 10-697393 -008578 43 999422 139 22 9294658 1048 -705342 9'303261 1090 10-696739 -008603 42 9991397 38 23 9-295286 10416 -704714' 9-303914 1089 10-696086 - 008628 42 9-991372 37 24 9295913 104.5 704087 9-304567 1087 10-695433 -008654 431 9991346; 36 25 9-296539 1043 -703461 9 305218 1086 10-694782'008679 42 i9-9913) 3 26 9-29716411042 -7028361 9305869 1084 10-694131 -008705 43 L99919 5io34 27 9-2977881040 702212 9 -306519 1083 10-693181 -008730 42) 99912,70!33 281 9298412 1039 -701588 9-307168 1081 10-692832 -008756 43 9991244 32 29 9-299034 1037 700966 9-307815 1080 10-692185 -008782 43 9'991218!31 30 9-299655 1036 700345!1 9-308463 1078 10-691537 -008807 42 19991193i 30 31 9-300276 1034 -699724 9-309109 1077 10-690891 -008833 43 9.991167 29 32 9-300895 1032 -699105 9-309754 1075'10-699246 -008859 43 19-991141!28 33 9-3015141031 -698486 9-310398 1074 10-689602 -008885 43 19991115 2)7 34 9302132 1029 *697868 9-31104211073:10-688958 008910 429-991090'26 35 9-302748 1028 -697252 9-311685 1071110-688315 -008936 43 9-991064' 25 36 9-303364 1026 -696636 9 312327 107010-687673 -008962 43 9-991038 24 371 9-303979 1025 -696021 9-312967 106810-687033 -008988 43 9-991012' 23 388 9 304593 1023 -695407 9-313608 1067 10-686392 -009014 43 9-990986 22 39 9-305207 1022 -694793 9-314247 1065 10-685753 -009040 43 199909602 ]l 40 9-305819 1020 -694181 9-3148851106410-685115 -009066 43 9-990934 t20 41 9-3064301019 -693570 9 315523 1062 10-684477 -009092 43 9-990908 19 42 9-307041 1017 -692959 9-316159 1061.10-683841 -009118 43 9990882:18 43 9-30765010161 6923.50 9 3167951060110683205 -009145 45 9990855'17 44 9-308259 1014 -691741 9-317430 1058,10-682570 -009171 43 19990829 16 45 9-308867 1013 691133 9318064 1057l10-681936 -009197 43 9-990803 15 46 9-309474 1011 -690526 9-318697105510-681303 -009223 43'19990777 14 47 9-310080 1010 -689920 9-319329 1054110-680671 -009250 45 9 990750 113 48 9-310685 1008 689315 9-319961 1053 10-680039 -009276 43 9-99072-1 12 49 9-311289 1007'688711 9-320592105110-679408 0093031 45 9'990697 11 50 9311893 1006| -688107 9321222 1050 10-678778 -009329 43 19-990671 10 51 9-312495 10041 -687505 9-321851 1048110-678149 -009355 43 9 990645 9 5'2' 1 9-313097 1003 -686903'9-322479 1047 10-677521 1009382 4.5 9-990618 8 53'9-313698 10011 -686302 9 323106 1045 10-676894 009409 45 9990591 7 54 I9-314297 1000' -685703 9-323733 1044 10-676267 O 0 009435 43 19-990565 6 55 9-314897 998 -685103 9-324358 1043 10-675642 i009462 45 9 990538i 5 356 9'-815495 997j -684505 9'3241983 1041 1067 5017' 009489 45 9 990511 4 57 9-316092 996' 68390l8 9-325607 1040110-6743939 -009515 43 19990485 8 58 9-316U89 994' -683311 9-32623111039 10.6737619 009542 45 19990458 2 59 9-317284 9933 682716' 9'326853 1037110673 147 I-009569 45 19-990431I 1 60l 9'3789 991'' 682121 l 9327475103610 672525 11009596 45 9-9904041 0' i Cosine. Sec.ant.'| Cotoagellt.: {i langent.!Cos.cant.| Sis. Si i 78 DEG. Page 53 LOGARITHMIC SINES, ETC. 53 12_DEG._ _ _ _ __ nDEG Sine. 0DffI Cosecant. Ta Cotangent. Cot Secant. Dff Csine. — 0 9-317879 - -6821211 9327474 1O-672526'009596- 9999,404 i601 9-318473 990.681527j 9-328095 1035 10.-71905 -009622{ 43 9.990378 59 2 9-319066 988.680934 9-$328715 1033,10-671285 1-009649 45;9990351,538 3 9-319658 987 -680342/ 9-329334 1032 10-67006 1009676 45 9'990324 57 41 9-320249 986 -679751 i 9-3299531030 10U070047 r'009703 45 39990297 56 5 9-320840 984 -679160; 9-3305701029 10-66q9430 i1 009730 45 9-990270 5 5 6 9-321430 983 -678570: 9-331187 1028 10-668813 i 009757 45 09'90243,54 7 9322019 982 -677981 9331803 102610668197 -009785 47 9'9900215; 5 8 9'322607 980 -677393 198332418 1025310-667582 -009812 40 5 9990188i 152 9 9323194 979 -676806 9 -9-338033 1024 10666967 009839i 45 9-990161 51 10 9-323780 977 -676220 9-333646 1023110-666354 009866 45 9-990134'50 11 9-324366 976 675634 9-334259 1021 10-665741 1009893 45 9-99010749 12 9-324950 975 67503059'9334871 1020110-665129'-009921 47 9-990079 48 13 9-325534 973 674466 9'335482 1019 10-664518 -009948 4.5 9-99005: 47 14 9-326117 972 -673883 | 9-330093 1.017 10-66'3907 -009975 45 9'990025% 46 15 9-326700 970'673300 9'336702 1016 10'663298 -010003 47 9-989997145 16 9-327281 969 -672719 9-337311 1015 10 662689 -010030 45 9-9899,70 44 17 9-327862 968 -672138 9338879191013 10-662081 -010058 47 9.089942! 43 18 9-328442 966 -671558 9-3838527 1012 10'661473 -0100851 45 9989915 4d2 19 9-329021 965 -670979 9-339133 1011 10-660867 -010113 47 9-9898871 41 20 9-329599 964 -6704011 9-339739 1010 10-660261 -010140 45 9-989800 410 211 9-330176 962 -609824;9-340344 1008 10-659656 -010168 47 (T9898321 89 22 9.330753 961 -669247;9-340948 1007 10 659052 -010196 47 9-98804 3'8 23 19331329 960 -6686711 9-341552 1006 10-658448 -010223 45 9-989777 837 24 1 93319030 958 -668097 9-342155 10Q4 10-657845. 0102351 47 9-9897490 36 25 98332478 957 -667522 93427357 1003 10-657243 -010279( 47 9-989721 35 26 19~3330351 9356 -666949! 9.343358 1002 10-656642 -010307 47 9-989G0' 31 27 9-333624 954 -666376' 9-343958 1001 10-656042'010335 47 998966.5 33 28 1 933'4195 953 -665805 19-344558 999 10-655442 -010363 47 9(.89637 32 291 9I334 717 952 6334767 952 - 36652 15 998 10-654843 010390 45 9-989610 31 30 93353337' 950 -664663 1 4 94315755 997l10-654245 4i 010418 47 9-989582130 [31 i9-335906 949 -664094' 9-346353 99610-653647 1'l010447 48 9-98 553,)9 32 9-3364753 948 -663525' 9-346949 994 10-653051 0104753 47:9952528 33 9'3370431 946 662957 9-347545 903 10'652455. " 010503 47 9-989497 27 34 93376100 945 662390 i 9-348141 992 10-651859 i:010531, 47 9'989i469 126 35 9338176944 -661824,9-348735 991 10-651265'-010559 47;9894425 36 9'338742 943 -6612.58 i 9-3493'29 9910 610650(7 1 i010587i 47 -9,) 98413 2 37 9-339307 941 -660693 9-349922 98810 65 008 01061 5 47 9-98385 23 38 9-339871 940 -660129' 9-'350514 987 10 649186'0 0IC14 4 8 9~ 983.5.22 39 9-340434 939 -659566 9-351100 986 C10-648894 1010672 47 19 93.828 21 40 9-340996 937 659004 9 351697 985 10-648303 1.010700 47 90 8300. 2.0 41 9-341558 9 36 -658442 i 9 352287 983 10-647 13 i-010729 48 9 -982"71 19 1 42'"9-342119 935 665881 9 -352876 982 10-647124' 010 771 47 98024'8 4c ~33~Ci'7Oi ~G573%1 ii~~3~346rj l010 4. *43' 9'3:42679 934 65 7321 9 -33463 981 10-646535 010786! 48 9-'989214 17 44 9-.`34;23) 932.635-7;1 )-13540543, 980 10-64597 010814 47 9989186 16 45 9-343797 931 -656203 1i 9-354640 979 10-645860 -01013' 48.98015 7 15 46 9-344355 930.655645 I 93552297 977 10-644773 3-010872 48 9. 80128, i4 47 9-344912} 929 -655088; 9-355813. 97610-644187'-010900' 47'9980100 13 48 9-345469 927 -654531 9-3356398 975 10-643602'-010929! 48 9-'989071 12 49 9-346024 926 -653976 1 9-356982 974 10-643018 -010958 48;9-989042 11 50 9-346579 925 -653421 9-357566 973 10-642434 -010986 47!9989014 i10 51 9-347134 924 -652866 9-358149 971 10-641851 -011015| 48 9-988!85 i 9 j 52 9-347687 922 -652313 9-358731 970 10-641269 -0110441 48 i9988956 8 1 i53 9-348240 921 -651760\ 9 359313 969 10-640687.011073' 48 9-988927 7 54 9-348792 920 -651208 9-359893 968 10-640107 i -011102; 48 9-988898 6i 55 9-349343 919 -650657 9-360474i 967 10.639526 1.011131 48 9-988869 5 56 9-349893 917 -650107 9-361053 966 10-638947.-011160 48 9-.988840; 4 i 57 9-350443 916 -6495571 9-361632 965110-638368 1011189 48 19-988811 3 58 9-350992 915 -649008 93622 0 93 10-637790,-011218 48 i9-988782 2 1 59 9-3515401 914 -648460 93627871 962 10-637213 -011247 48 9-988753 1 60 9-352088 918 -647912 9-363364! 961 10-636636 -011276 48 19-988724 0 Cosine. Secant.'J Cotangent. i Tangent. Cosec ant. Sine. 77 DEG. Page 54 54 LOGARITHMIC SINES, ETC. 1 3 D E G._ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Sine. ~~~~~~~~~~~~~~Diff t Diff~' DSin ff. Cosecant. Tangent. If; Cotangent. Secant. D,. Csine. Sn.100" uth) 0( 9-352088 -647912 9-363364 10-636636 -011276 -9887-24l60 1 9-35263085 911 -647365 9-363940 960 10-636060 -01O1305 48 9 988695 59 2 9-353181 910 -646819 9-364515 959 10-635485 -0113834 48 9-988656(6 58 13 9-35372 909 -646274 9-365090 958 10-634910 -011364 50 9-988636 57 4 9-354271 908 -645729 9-365664 957 10-634336 011393 48 9-988607, 56 5 9-354815 907 -645185 9-366237 955 10-633763 -011422 48 9-988578955 6 9-355358 905 -644642 9-366810 954 10-633190'011452 50 9'988548154 17 9355901 904.644099 9'3673821 953 10-632618 1.011481 48 9-988519i53 18 9-356443 903-643557 9-367953 9.52 10-632047 1-01151l 50 9-988489,152 9 9-356984 902643016 9368524951 10631476 -0115401 48 9-988460 51 10 9-357524 901 -6424769-369094 950 10-630906 -011570 50 9.988430'5o0 11 9-358064 899 -641936 9-369663 949 10-630337.-011599 48 9-98840119 12 93586031 898 -641397 9-370232 948 10-629768 1-011629' 50 9.988371 18 13 9-359141 897 -640859 9-370799 946 10-629201.011658 48 9-988342 17 14 9-359678 896 -640322 9-371367 945 10-628633.011688 50 9-988312146 15 9-360215 895 -639785 9-371933 944 10-628067 -011718!50 9.988282 45 16 9-360752 893.639248 9-372499 943 10-627501 -011748 50 9-988252 14 17 9.361287 892 638713 9.373064 942 10-626936 1-011777, 48.998822343 18 9-361822 891.638178 9-373629 941 10,626371 -011807 50 9-988193; 42 191 9-362356 890 -637644 9-3741931940 10-625807 -011837 50 9-988163 41 20 9-3628891 889 -637111 9-374756 939 10-625244 -011867 50 9-988133'40 21 9-363422 888 -636578 9-375319 938 10-624681 -011897 50 9-988103:39 22 9-363954 887.636046 9-375881 937 10-624119 -011927 50 9-988073138 23 9-364485 885.635515 9-376442 935 10-623558 -011957 50 9 -988043 37 24 9-13650161 884.634984 9-377003 934 10-622997 -011987 50 9-988013136 25 9-3655461 883 -634454 9-377563 933 10-622437 1012017 50 9-987983135 26 9-366075 882.633925 9378122 932 10-621878'012047 50 9 -987953 134 271 9 -366604 881.633396 9-378681 931 10.621319 11-012078 52 9-987922'33 28 9-367131 880 -632869 9-379239 930 10-620761 1-012108 50 9-987892 32 29 |936765 879 632341 9-379797 929 10.620203' -012138.50 9.987862!31 30' 9-368185 878 -631815 9380354 928 10-619646 -0121681 50 9-987832 30 I -81 9-868711 876 -6312899-380910 927 10-619090 -012199' 52 9-9878011 29 32 9'3692361 875 630764 9-381466 926 10.618534 1-012229 50 9-987771!28 9 33 9-3697611 874.630239 9-382020 925 10-617980 -'012260 52 9'987740127 3411 9.37028.5 873 -629715 9-382575 924 10-617425 -012290! 50 9-987710:126 35 9.370808 872 -629192 9-383129 923 10-616871 -012321 52 9987679 125 36 9-371380 871 -628670 9-383682 922 10-616318 1-0123511 50 9-9876419:4 37 9-371852 870.-628148 9-384234 921 10-615766 -012382 52 9-987618:23 38 | 9-372373 869 -627627 9-381786 920 10-615214 1-012412 50 9.98758822 39 19-3728941 867 627106 9-385337 919 i10-614663 10124431 52 9-987557 21 40 9.373414 866 6265869-385888 918 10-614112 |-012474 52 9-987526120 41 9.3739331 865 -626067 9-38(6438 917 10-613562 |-012504 50 9 9.87496 19 42 9.374452! 864 -625548 9-386987 916 10 613013.01)2535 52 9-987465118 43 9-374970 863 -625030 9-387536 914 110-612464.012566! 52 9-987434L 17 44 9-375487 862 -624513 9-388084 913 110-611916 1'012597 52 9-987403116 45 9-376003 861 -623997 9-388631 912 I10 -611369 -012628 52 9-987372 15 46 9-376519 860 -623481 9-389178 911 110-610822'-012(659 52'987341'141 47 9-377035 859 -622965 9-389724 910 10-610276,I012690 52 9-98o7310lo 48 9-377549 858 -622451 9-390270 909 11-6097,30 11-012721 52 I) 9,-o1. 49 9-378063 857 -621937 9-390815 908 0-609185-'-012752 52 9-987248'll 50 9-378577 856 -62142319-391360 907 10-608610'-012783 52 9-987217 10 51 9-379089 854 6209119-391903 906 110-608097 i-012814 52 99871861 9 52 9-379601 853 -620399 9-392447 905 110-607553 -012845 52 9-987155' 8 53 9-380113 852 -619887 9-392989 904 10-607011 -012876 52 19-987124] 7 54 9-380624 851 -619376 9-393531 903 10-606469 -012908 53 9'987092 6 55 9-381134 850 -618866 9-394073 902 10-6059271 -0129139 52 9'987061' 5 56 9-381643 849 -618357 9-394614 901 10.605386 1-012970 52 9-987030 4 57 9-382152 848 -617848 9-395154 900110-601846-013002 5: 9-98,998! 3 58 9-382661 847 -617339 9.395694 899 10-604306!.0130.33 52 9-98069671 2 59 9-383168 846 -616832 9-396233 898 10-603767 013064 52 9 9879_' I I1 60, 9-383675 845.-616325 9-396771 897 10'603229_10130961 -3 9'98',(0t:t 0 i Cosine. I | Secant. II Cotangent. Tao u it.:1Coscat., Sii. 1 7 6 IDU. Page 55 LOGARITHMIC SINES, ETC. 55 14 DEG. | Sine. \ Dff Cosecant. Tangent. Diff Cotngent. Secant. 10ff Cosine. 0 9-383675'616325' 9-396771 10-603229 I013096 9-986904160 9'3384182 814'615818 9-397309 896 10-6,026911 013127 52 -9986873 159 2 9'384687 843 -615313 9-397846 896 10-602154 -013159 53 9-986841 i58 3 9-385192 842 -614808 9398383 895 110601617 -013191 53 9986809!57 4 9-385697 841 *614303T 9-398919 894 10,601081 01i3222 52 9-986778' 56 *5 9.-86201 840 -613799 9-399455 893 10-600545 -013254, 53 9-986746 55 6 9-386704 839 -613296 9-399990 892 10-600010 -013286 53 9-986714 154 7 9387207 838.*612793 9-400524 891 1-0-599476.013317 52 9-986683 53 8 9-3'87709 887'612291 9-401058 890 10-598942 013349 53 9 986651' 52 9 9-388210 836 -611790 9-401591 88 10 49 0-'98 013381 53 9-986619 151 10 9 -389711 835 -611289 9-402124 888 10-597876 -013413 53 9-986587 50 11 9-389211 834.610789 9-402656 887 10-597344 -013445 53 9-986555 49 12 9-389711 8338 610289 9-403187 886 10-596813 -013477 53 9-986523 48 13 9.390210 832 -609790 9-403718'885 10-596282 -013509 53 9:986491 47 14 9.390708 831 -609292 9404249 884 10-595751 -01.35A1 53 9-986459 46 15 9-391206 830'608794 9-40-47,78 883 10-595222 -013573 53 9,986427 45 16 9.391703 828 -608297 9-405308 882 10-594692 1013605 53 9-986395144 17 9.392199 827 -607801' 9405836 881 110594164 1013637 53 9-986363 43 18 9.392695 826 -607305 9 406364 880 1059363636 -013669 53 9-986331 42 19 9.393191 825 -606810 9-406892 879 10-593108 -013701 53 9.986299 41 20 9,3'93685 824 -606315 9-407419 878 10-592581'013734 55 9-986266 40; 21 9-394179 823'605821 9-407945-877 10-592055'013766 53 9-986234 39 22 9-394673 822 -605327 9408471 876 10-591529 -013798 53 9.986202 38 23' 9395166 821 -604834 9-408997' 875 10-591003 -013831 55 9-986169 137 24 9-395658 820 -604342 9 409521 874 10-590479 -013863 53 9:-986137 i36 25 9-396150 819 -603850 9-410045 874 10-589955 -1013896 55 9-986104 35 26 9.396641 818 -603359 9-410569 873 10-589431 -013928 53 9-986072 i34 27 9397132 817 -602868 9-411092 872 10-588908 -013961. 55 9-986039 133 28 9-397621 817'602379 9-411615 871 10-588385 -013993 53 9-986007 132 29 9-398111 816 -601889 9412137 870 10-587863 -014026 55 9.985974'31 30 9,398600 815 -601400 9-412658 869 10-587342 1-014058 53 19-985'9421 30 31. 9-399088 814 -600912 9-413179'868 10-586821 -014091 55 9-985909 29 32 9-399575 813 -600425 9413699 867 10-586301 -014124 55 9-985876 28 33 9400062 812 - 599938 l 9-414219 866 10-585781 -014157 55 19985843' 27 34l 9400549 811 -599451 9-414738 865 10-585262 -014189 53 19985811 26 35 9-401035 810, 598965 9 4152571 864 10-584743 -014222 55 9- 985778 25 36 9-401520 809 -598480 H 9415775 864 10 584225 -014255 55 9-985745 24 37 9-402005 808 597995 1 9416293 863 10-583707 -014288 55 19985712 23 38 9-402489 807'597511 9-416810 862 10-583190 -014321 55 9-985679 22 39 9402972 806 597028 i 9 4173261 861 10-582674 -0.143o54 55 9-985646 21 40 09 403455 805 596545 9-417842 860 1.0582158 -014387 55 9 -985613 l20 41 9-403938 804''596062 9-4183-58 859 10-58,1642 -014420' 55 9-985580 19 42 9-404420 803 -595580 9-418873 858 10-581127 -014453 55 9,985547 18 43 9-404901 802 -595099 19-419387 857 10-580613 ]-014486 55 19985514 17 44 9,405882 801 -594618 1 9419901 856 10-580099 -014520 57 9-985480 16 45. 9-405862 800 -594138 19-420415 855 10-579585 -014553 55 9-985447 15 46 9-406341 799.-593659 9-420927 855 10-579073 -014586 55 9-985414 14 47 9-406820 798 593180 9-421440 854 10-578560 -014619 55 9-985381 13 48 9-407299 797 -592701 9-421952 853 10-578048 -014653 57 9-985347 12 | 49 9'407777' 796 -592223" 9-422463 852 10-577537 -014686 55 9-985314 11 50 9-408254 795'591746 9-422974 851 10-577026 -014720 57 9-985280 10 51 9-408731- 794 591269 9-423484 850 10-576516 -014753 55 9-985247 9 52 9-409207 794'590793 9-4239931 849 10-576007'014787 5.7 19985213 8 53 9-409682 793 -590318 9-424503.!848 10-575497 -014820 55 9.985180 7 54 9-410157T 792 589843 9-425011 848 10-574989 -014854 57 9-985146 6 55 9-410632 791 -589368 9-425519 847 10-574481 -014887 55 9'985113 5 56 9-411106 790'588894 9-426027 846 10-573973'-014921 57 [99850'79 4 57 9-4115791 789 -588421 9-426534 845 10.-573466 l014955 57 9.9850451 3 68 9 4120521 788 -587948 9 427041 844 10-572959.Q14989 57 9-98 011l 2 59 9412524 787 -587476 9-427547i 843 10-5724531- 015022 55 9-984978 1 60 9-412996'786'587004 9428052 843 10-571948 1'015056 57 9-984944' 0 Cosine I Secant. Cotangent. j Tangent., Cosecant.. | Sine. |1 | 75 JOEG. Page 56 56 LOGARITHMIC SINES, ETC. 15 DEG._ Dill. Diffl Diff. Sine Cosnt Tag Cotangent. Secant.. Cosine. 0 9-412996 -587004, 9-428052 10i571948 -015056 9-984944 60 1 9-413467 785 -586533 i 9428557 842 10.571443 -015090 57 9-984910 59 2 9-413938 784 58606 12 9-429062 841 10-570938 -015124 57 9-984876 58 3 9-414408 783 585592 9-429566 840 10-570434,015158 57 9 9842 57 4 9-4'14878 783. -585122 9-430070 839 10-569930'015192 57 9 984808 56 5 9-415347 782 584653 1'9430573 838 10-569427 -015226 57 9-984774 55 6 9-415815 781 584185 i 9-431075 838 10-568925 -015260 57 9-984740 54 7 9-416283 780 *583717 9-431577 837 10'568423 -015294 57 9.984706 53 8 9-416751 779'583249 9-432079 836 10-567921 -015328 57 9-984672 52 9 9-417217 778 582783 9-432580 865 10 567420 -015362 57 9-984638 51 10 9-417684,777 -582316 9-433080 834 10-566920 -015397 58 9-984603 50 11 9-418150 776.581850 -9-433580 833 10-566420 -015431 57 9-984569 49 12 9-418615 775 -581385 9-434080 832 10-565920 -015465 57 91984535 48 13 9-419079 774 580921 9-434579 832 10-565421'015500 58 199845.00 47 14 9-419544 773 *580456 9-435078 831 10-564922 015534 57 9-984466 46. 15 9'420007 773 -579993 9-435576 830 10'564424 -015568 57 9-984432 45 16 9-420470 772 -579530 9-436073 829 10,563927 -015603 58 9-984397 44 17 9-420933 771 *579067 9'436570 828 10-563430 -015637 57 9-984363 43 18 9-421395 770 -578605 9-437067 828 10-562933 -015672 58 9-984328 42 19 9-421857 769 -578143 9-437563 827 10-562437 -015706 58 9-984294 41 20 9-422318 768' -577682 9'438059 826 i0-561941 -015741 57 9-984259 40 21 9-422778 767 -577222 94388554, 825 10-561446 -015776 58 9-984224 39 22 9'423238 767 576762 9-439048 824 10-560952 -015810 57 9-984190 38 23 9-423697 766 576303 9 439543 823 10-560457 -015845 58 9:984155 37 24 9424156 765 -575844- 9-440036 823 10-559964 -015880 58 9-984120 36 25 9- 424615 764 575385 9 440529 822 10-559471'015915. 58 9-984085 35 26 9-425073 763 574927 94410'22 821 10-558978 -015950 58 9-984050 34 27 9-425530 762 574470 1i 9445114 820 10-558486 1015985 58 19-.'84015 33 28 9-425987 76.1 574013 9 442006 819 10-557994 -016019 57 9'983981 32 29 9-426443 760 573557 9 -442497 819' 10-557503 -016054 58 9-983946 31 30 9-426899 760 -573101 i 9-442988 818 10-557012 -016089 58 9'983911 30 31 9-427354 759 57264611 9443479 817 10-556521 -016125 60 9-983875 29 32 9-42,7809 758'-572191 9-443968 816 10-556032 1016160 58 9-983840 28 33 9-428263 757 -571737 9-444458 816 10-555542 1016195 58 9-983805 27 34 9-428717 756 -571283 9 444947 815 i10O555053 -016230 58 9-983770 26 35 9-429170 755 -570830 9-445435 814 10-554565 1016265 58 9-983735 25 36 9-429623 754 -570377 9-445923 813 10-554077 -016300 58 19983700 24 37 9-4300751 753.569925 9-446411'812 10-553589, -016336 60 9-983664 23 38 9-430527, 752 -569473 9 446898 812 10-553102 1016871 58 9-983629 22 -39 9-4309781 752 569022 9-447384 811 10-552616 1016406 58 9-983594 21 40 9 4314'299i 751 -568571 9 447870 810 10-552130 -016442| 60 9-983558 i20 41 9-431879, 750 -568121 9-448356 809 10-551644 1016477'58 19983523.19 421 9-432329 749 1567671, 9-448841 8,09 1'0561159 -016513 60 9-983487 18 43I 9-4327781 749 -567222 9-449326 808 10-550674 1016548 58 9-983452 17 44 9-433226 748 -566774 9-449810 807 10-550190 -016584 60 9-9834.16 16 45 9-433675 747 -566325 9-450294 806 110549706 -016619 58 9-983381 15 46 1 94341227746 -565878 9450777 806 110549223 -016655 60 9983345 14' 47 9-434569 745 565431 9-451260 805 10-548740, -016691 60 9-983309 13 48 9-4350161 744 564984 9-451743 804 10',548257 -016727 60 9'983273 12 49 9-4354621 744 -564538 9-452225 803 10-547775.0167621 58 -9983238 11 50 9.435908 743 *564092 9-452706 802 10-547294 -016798 60 9-983202 10 51 9-4363531 742 -563647 9-453187 802'10'546813 -016834 60 9-983166 9 52 9-436798 741 563202 9-453668 801 10-546332 -016870 60 9-983130 8 531 9437242 740 562758 9-454148 800 10-545852 -016906 60 9 83094 7 541 9437686 740.5623141 9-44628 799 10-545372 -016942 60 9'z983058 6 55 9-438129. 739 5618711 9-455107 799 10-544893 -016978 60 9-983022 5 56 9-438572 738 -561428 9-455586 798 10-544414 -017014 60 9-982986 4 57 9 -39014 737 -560986 9-456064 797 10-543936 -017050 60 9:9829501 3 58 9-439456! 736 -560544 9-456542 796 1.0-543458 -017086 60. 9982914 2 59 9:4398971736 -560103 9.457019 796 10-542981 -017122 60 9-982878 1 60 9 440338 7351 559662 9'457496 795 10-542504 -017158 60 9r982842 0 / I Cosine. I Secant.- Cotangent. Tangent. iCosecant. I Sine.. 74 DEG. Page 57 LOGARITIHMIC SINES, ETC. 57 16 DEG. l O..., ~ _ _ _'_. Sine. Dff GCosecant. Tangent. Cotangenlt. Secallt. C / l 0 9-440338 -559662 9457496 10-542504_ 01 7158 8: i- y2ji4 o 1 9-440778 734 559222 9-457973 794 10-542027 *017195 60 9 9' -80' -9 2 9441218 733 *558782 9-458449 793 |10-541551 017231 60 9 8 7(l; 3 9-441658 732 -558342 9-458925 793 10-541075 -017267 60 9 98273 7 4 9-442096 731 -557904 9-459400 792 10,540600 -017301 62' 9 826~'I) 5 5 9442535 731 557465 9-459875 791 10-540125 -017340 60 9 98926. 15i 6 9-442973 730 557027 9-460349 790 10-539651 -017876 60 9-982)62145-4 7 9-443410 729 -556590 9-460823 790 10-539177 -017413 62 9 982587 J53 8 9 443847 728 -556153 9-461297 789 10'538703 017449 60 9 982551 152 9 9-444284 727 5555716 9-461770 788 10-538230 -017486 62 9.982514,'51 10 9-444720 727 -555280 9 462242 788 10-537758 017523 62'9 982477l 50 11 9-445155 726 -554845 9-462714 787 10-537286 -017559 60 9-982441 49 12 9445590 725 554410 9-463186 786 110536814 -017596 62'9 982404 48 13 9446025 724 -553975 9-463658 785 10-536342 -017633 62 9 9823367 147 14 9446459 723 553541 9-464128 785 10-535872 -017669 60 19982331 46 15 9.446893 723 553107 9-464599 784 10-535401 -017706 629.' 982294 45 16 9447326 722 -552674 9-465069 783 10-534931 -017743 62 19-982257 i44 17 9-447759 721.552241 9-465539 783 10-534461 -017780 62 9 982220 43 18 9-448191 720 -551809 9-466008 782 10-533992 017817 629 9 982183 42 19/ 9448623 720 5513.77 9.466476 781 110-533524 017854 62 9-982146 41 20 9-449054 719 -550946 9-466945 780 10-533055 -017891 62 19-982109 40 21 9-449485 718 -550515 9-467413 780 10-532587 -017928 6219 98207239 22 9 449915 717 550085 91467880 779 10-5321800 017965 62 9-982035 38 23 9450345 716' 549655 9-468347 778 10-531653 -018002 62:9-98199837 24 9450775 716 -549225 9-468814 778 10-531186 1018039 62 19981961136 25 I 9-451204 715 -548796 9-469280 777 10 530720 -018076 62 9 981924' 35 26 19451632 714'548368 9-469746 776 10-530254 -018114 63 9-9818861l34 27 9-452060 713 -547940 9470211 775 10-529789 -018151 62 9-981849 33 28 9-452488 713 -547512 9-470676 775 10-5293241 018188 62 19981812132 29 9-452915 712 -547085 i 9471141 774 10-5288591 018226 63 9-98174131 30 9-453342 711'546658 19 471605 773 10-528395 -018263 62.9981737 430 31: 9-453768 710 -546232 9-472068 773 110-527932 -018300 62'9-981700 29 32911454194 710 1 545806 9-4725321 772 110527468 -018338 63 9-981662 28 33 9-454619 709 -545381 94729951 771 110-527005 O018375 62 9 9816256 27 341'9 455044 708 -544956 9-473457 771 110-526543 018413 631 9981587 126 35 9-455469 707'544531 9 4739191 770 110-526081 -018451 63 9 981549 25 361 9 455893 707 -544107 9-474811 769110-525619 -018488 62 9-981512!24 37 9-456316 706 -543684 9-474842.769 1.0-5251581 018526 63 99814745123 38 9-456739 705 -543261 9-4753031 768 10 524697 -018-564 63 9-9814361;22 39 9-457162 704 -542838 9475763 767' 10524237 -0186011 62 9 981399 21 40 9-457584 704'542416 9 4762231 767 10-523777 -018639 63 9-98136 120 41 9-458006 703 -541994i 9-476683 766 10-523317 -018677 63 9-981323 19 42 9-458427 702.5415731 9-4771421 76 5 10-522858 -018715 63 9-9812851 18 43 9-458848 701 -541152 9-4776017 765 10-522399 018753 63 9-981247||17 44 9-459268 701.540732 9-478059 764 10-521941 -018791 63 19981209'16 45 9-459688 700 -540312 9-478517 763 10-521483 10188291 63 9-981171 15 46 9:460108 699 -539892 9-478975 763 10-521025 -018867 63 9-981133114 471 9460527 698 -539473 9-479432 762 110520568 -018905 63 9-981095 13 48 9 460946 698 -539054 9-479889' 761 10.520111 -0189438 63 9-981057 12 49 9-461364 -697' 538636 9-4803451 761 10-519655 1 018981 63 9-981019 11 50 9461782 696 -538218, 9-480801 760 110-519199 -019019 63 9-980981 10 51 i 9462199 695 -537801 9 481257 759 10.618743 -019058 65'9.980942 9 52 946261-6 695 -537384 9-481712 759 10-518288 -019096 63 9-980904 8 53 9-463032 6941 5369681 9482167 758 10-517833 -019134 63 9-980866 7 54 9-463448 693 -536552 9 482621' 757 10-517379 I -019173 65 9-980827 6 55 9-463864 693 -536136 9-4830.75 757 10516925 -019211 63 9-980789 5 56 9-464279 692 535721 9-483529 756 10-516471 019250 65 9-980750 4 571 9-464694 691 -535306 9-483982 755 10,516018 -019288i 63 9-980712 3 5819'465108 6901 534892 9-484435 755 10-515565.019327 65 9-9806731 2 59i 9-465522 690 -534478 9:484887 754 10-5151113 019365 63 9-980635 1 60i 9-465935 689 534065 9-485339 753 10-514661 -019404165 9-980596 0 jj C('osine. Sc nt. S l. Cotangent. I, Tawneiet. ij Cosecant. I Sine. / 73.)i G, Page 58 58 LOGARITHMIC SINES, ETC. 17 EcG._ Sine. Diff Cosecant. Tangent. jD Cotangent. Secant. Ift; Cosine. / Sine. 100 0 9-465935 534065 9-485339 10-514661 019404 9980596 60 1 9466348 688 533652 9-485791 753 10-514209 -019442 63 9-980558)9 2 9-466761 688 533239.9-486242 752 10 513758 -01941t 65 9 980519 j58 3 9-467173 687 -532827'9-486693 751 10-513307 019520 65 9 980480 57 4 9467585 686 -532415 9-487148 751 110-512857 -019558 63 9 980442 |56 5 9467996 685 532004 9-487593 750 10-512407 -019597 65 9-980403 55 6 946840.7 685 531593 9-488043 749 10-511957 -01963.6 65 9-980364'54 7 9468817 684 531183 9-488492 749 10-511508 -019675 65 9 9803255j53 8 9469227 683 530773 9488941 748 10-511059 -019714 65 9-980286:52 9 9469637 683 530363 9 489390 747 10-510610 -019753 65 9 980247| 51 10 9470046 682 -529954 9-489838 747 10-510162 1019792 65,9-980208s50 11 9470455 681 529545 9- 490286 746 10-509714 i019831 65 i9980169149 12 9470863 680 529137 9-490733 746 10-509267'019870 65 9-980130,48l 13 9471271 680 5287291 9-491180 745 10-508820. -019909 65 i9-980091!'47 14 9-471679 679 -528321 9-491627 744 10-508-373 019948 65 19980052"46 15 9472086 678 527914 9-492073 744 10-507927 -019988 67 19,980012fl45 16 9-472492 678 -527508 i9 492'519 743 10-507481 -020027 65 i9-979973'j44 17 9472898 677 527102 9-492965 743 10-507035'020066 65 91979934,43 18 9-473304 676 526696 19493410 7 42 10-506590 -020105 65 i 9 97,985 42 19 9473710 676 526290 9 493854 741 10-506146 1020145 67 9-97 9855 141 20 9 474115 675 -525885 9 494299 740 10-505701' -020184 65 9-97981. 640 21 9-474-519 674 525481 19-494743 740 10505f257 -020224 67 9 979, 7 6'9 22 9474923 674 525077 9 495186 739 10-504814 -020263 65 9-9.59737; 88 23 9-475327 673 -524673 1 9-495630 739 10-504370 -1020303 67 9 979697;37 24 9-475730 672 -524270 19 496073 738 10-503927 1-020o42 65 9 979658'36 25 9-476133 672 -5 23867 9-496515 737 10-503485 2 020382 B7 9 9796181 35 26 9-476536 671 523464 9 9-496957 737 10-503043 1020421 65 9-979579 l34 27 9'476938 670 523062 9.4973991 736 10-502601 1 020461 67 9-979539'193 28 9477340 661 I52260O 9 4978411 736 10502159 -020501 67 19979499 32 29. 9-477741 66(9 |5.22259 9494(2828 735 10-501718 1-020541 67 9 794459 31 30 9 478142 668 521858 i 9 498722 734 10 501278 /020580 65 9 9794201'30 31 9-478542 667 *521458; 9-499163 734 10-500837 1 020620 7 9-979380',29 32 9 478 942 667 521058 2 3 9.499603 733 10,500397 -020660 67 9 979340'028 33 9-479342 666 -520658' 9500042 733 10-499958/ 020700 6(7 9 9793001 27 34 9479741 665 -520259 9i 9-500481 732 10 2499 19 -020740 67 9 99260 26 35 9-480140 665 519800 9 -5009U20 731 10-499080 02( 78 67 9 I 9720 25 36 9480539 664 -519461 9-35059|731 10498641 -8208o20 67 9 979180;24 37 9-480937 663 -519063 9'5017971 730 10-498203 -020860 67 99i91401 23 38 9-481334 663 -518666 1 9-502235 730 10-497765 -020900 67 9 97910022 3 9 9481731 662| -518269 9-502672 29 10-497328 -020941 68 9 979059j 2 40 9-482128 661 -.517872 9-503109, 728 j10496891 1020981 67 97'9i01920 41 9-482525 661 517475 1 9-503546 728 104i96454 1021021 67 19978979! 19 42 9-482921 660 -5170791 9 503982 727 10-496018 1-021061 67 1997893918 43s 9.483316 659.516684; 9-5044181 727 10 495582 0211021 68 9 978898 17 44 19483712 659 -516288 9-5048;54 726 10-495146 -021142 67 9 -978858 16 45 -9.4841071 658.515893 9-5052891 725 10-494711 -021183 68 9-978817 15 46 9-484501 657 515499 9-505724 725 10-494276 -021223 67 9-978777i 14 471 9-484895 657 -515105: 9-5061591 724 10-498841 -021263 67 9 978737 13 48 9 485289 656 514711 9-506593 724 10-493407 -021304 68 9-978696 12 49 9-485682 655 -514318 9-507097 723 10-492973 -021345 68 9-978655 11I 50 9-486075 6565 513925i 9-507460 722 10-492540 -021385 67 9-78615 10 51 9'486467 654 -513533 9 507893 722 10-492107 -021426 68 99-978574 9 522 9-486860 653.513140 9-508826 721 10-491674 -021467 68 9-9785331 8 53 9-487251 653 *512749 9 508759 721 10-491241 -021507 67 199784931 7 54 9-487643 652.512357 9 509191 720 10-490809 i-021548 68 19978452 6 55 1 9488034 651.5119G6 9-509622 719 10-490378 -021589 68 9-978411 5 56 9-488424 651 -511576 9-510054 719 10-489946 1-021630 68 9-978370 4 57 9,488814 650 -511186 9 510485 718 10-489515 I-021671 68 99.9783291 3 58 9-489204 650 -510796 9-510916.718 10489084 1-021712 68'19-9782881 2 59 9-489593 649 *510407! 9 511346 717 10-488654 1021753 68 9 -9782471 1 60 9-489982 648 -510018 9 511776 717 10-488224 -021794 68 9-978206 0 | Cosine. Secant. h Cotangent Tangent. Cosecant.j Sine. 72 DEG. Page 59 LOGARITHMIC SINES, ETC. 59 18 DEG. is DED. ____________ _________________________ _ ___ ______ _ _ ____ Sine. Dff Cosecant. Tangent. Cotangent. Secant. Di. Cosine. 0' 9-4,'9982 510018 9-511776 10 48822.4 021794 9I978 206 6t0 1 9-490371 648 -509629 9 512206 716 10-487794 -021835 68 9-978165 59 2 9-490759 648 509241 9-512635 716 10-487365 -021876 68 9-978124! 58 3 9491147 647 508853 9-513064 715 10-486936 1021917 68 9-978083i157 4 9-491535 646 508465 9-513493 714 10-486507 -021958 69 9-978042 56 5 9-491922 646 508078 9 513921 714 10-486079 -021999 69 9-978001o55 6 9-492308 645 507692 9-514349 713 10 485651 022041 69 9',979o9l54 7 9-492695 644 -507305 9-514777 713 10-485223 022082 69 9-9779181'53 8 9-493081 644 506919 9-515204 712 10-484796 022123 69 9 -9778 77'52 9! 9493466 643 506534 9-515631 712 10-484369 022165 69 199778i35 51 10 19-493851 642.506149 9-516057 711 10-483943 -022206 69 9-977794 50 11 9-494236 642.505764 9516484 710 10-483516 022248 69 9-977752 49 12 1 9494621 641.505379 9-516910 7110 1483090 022289 69 9.9777111 48 13 i 9-495005 641.504995 9-517335 709 10-482665 022331 69 9-977669 147 14 9-495388 640.504612 9-517761 709 10-482239 022372169 9 977628i46 15 9495772 639.504228 9-518185 708 10 481815 022414 69 9977586 45 16 19-496154 639.503846 9518610 708 10-481390 -0224.56 69 9977544 44 17 9-496537 638.503463 9-519034 707 -10-480966 022497 70 9-9.77503'43 18 1 9496919 637 503081 9-519458 706 10-480542 022539 70 9-977461 42 19 9-497301 637.502699 9-519882 706 10-480118 022581 70 9-977419l41 20 9-497682 636.502318 9-520305 705 10-479695 -022623 70 9-977377 140 21 9498064 636.501936 9-520728 705 110479272 -022665 70 9-97,335 39 22 9498444 63.5.501556 9-521151 704 10-478849 -02270770 9-977293 38 23 9 -498825 634.501175 9-521573 704 10-478427 -022749 70 9-977251"37 24 9-499204 634.500796 9-521995 703 10-478005 -022791 70 9-977209 36( 25 9-499,584 633.500416 9-522417 703 10-477583 -022833 70 9-977167 35 26 1 9-499963 632.500037 9-522838 702 10-477162 -1022875 70 9-977125 34 27' 9.500342632.6499658 9-523259 702 10-476741 022917 70 9-977083'33 28 19-500721 631 1.499279 9-523680 701 10-476320 022959 70 9-977041'32 29 19-501099 631.498901 9-524100 701 10-475900 023001 70 9-976999 31 30 1 9501476 630.498524 9-524520 700 10-475480 023043 70 9-976957 30 31 9-501854 629.498146 i 9524939 699 10-475061 i023086 70 9-976914 29 32 1 9502231 629.497769 9-525359 699 10-474641.023128! 70 9976872 28 33 9-502607 628.497393 9-525778 698 10474'22 1023170 70 9-976830 27 34 1 9-502984 628.497016 9-526197 698 10-473803 1023213 71 9-976787 26 351 9-503360 627.4960 264 615 697 10-473385 10232 55 71 9-976745 25 3 9 503735 626.496265 9-527033, 697 110.472967 1023298 71 997670' 224 37 9-504110 626.495890 9-5274511 696 110-472549 023340 71 9-976660'23 38 9-504485 625.495515 9-5278681 696 110472132) 023383 71 9-976617 22 39 9-504860 625.495140 9-528285 695 10-.471715 -023426 71 9-976574 21 40 9-505234 624.494766 9528702! 695 10-471298 023468 71 9-97653220 41 9-505608 623.494392 9-529119 694 10 470881 1023511 71 9-976489 19 42 9,505981 623.494019 9-529535 694 10-470465 -023554 71 1997644618 43 9-506354 622.493646 9-529950 693 10-470050 023596 71 9-976404,17 44 1 9506727 622.493273 9-530366 693 10-469634 0236 71 9-97636116 45 9-507099 621.492901 99-530781 692 |10.469219.023682 71 9-976318 15 46 19507471 620.492529 9531196 691 110-468804 023725 71 9-9762975 14 47 9-507843 620.492157 9-531611 691 10-468389 -023768 71 997696232 13 48 9-508214, 619.491786 1 9532025 690 10-467975 -023811 72 9-976189 i12 49 9-50858 619.491415' 9-532439 690 10-467561 1023854 72 9-976146 11 50 9508956 618.491044 9-532853 689 10-467147 -023897 72 9976103 10 51 9'509326 618.490674 9'533266 689 10 466734'023940 172 9-976060 9 52 9-509696 617 -490304 9-533679 688 10-466321 1023983 72 9976017 8 53 1 9-510065 616.489935 9-534092 688 10-465908 -024026 72 9-975974' 7 5t11 9-510434 616.489566 95534504 687 10-465496.024070 72 9975930 6 55 9510803 615.489197 9-534916 687 10-465084 1024113 72 9 975887! 5 56 9-511172 6151.488828 9-535328 686 10-446 72I 02-156 72 19975844 4 57 9-511540 614.488460 9535739 686 10-464261 024200 72 9975800 3 58 9 511907 613.488093 1 9536150 685 10-463850 -024243 72 919757 57 2 50', 9 512275 613 -487725: 9-536561 685 10-463439 -024286 72 19'975714 1 6(0 9 51'264. 612 *487358 9-536972 684 10 463028 -024330 72 9-975670 0 I Cosie. I Secant. I Cotallgent. I l ansgent. lCosecant. I ie. i 71 DEG. Page 60 60 LOGARITHMIC SINES, ETC. 19 DEG. DuI0"ff. / Diff I Diff.I Sine. Diff Cosecant. Tangent. Dil}; Cotangent. Secant. i1, Cosine. 0 9512612 -487358 9-536972 10-463028 -0224330 9 975670 60 1 9-518009 612 486991 9-537382 684 10-462618 -024373 72 9.975627)59 2 9-513375 611 486625 9-537792 683 10-462208 024417 73 9-975583 58 3 9 513741 611 -486259 9-538202 683 10-461798.024461 73 9-975539 o75 4 9-514107 610 485893 9-538611 682 110461389 -024504 73 9-975496 56 5 9514472 609.485528 9-539020 682 10-460980; -024548 73 9-975452 55 6 9-514837 609 -485163 9-539429 681 10-460571 -024592 73:9-975408 54 7 9-515202 608 *484798 9-539837 681 10-460163 -024635 73 9-975366 53 8 9'515566 608 *484434 9-540245 680 10,459755 024679 73 19-975321 152 9 9-515930 607 -4840701 9-540653 680 10-459347 -024723 73 199752771 51 10 9-516294 607 483706 9-541061 679 10-458939 -024767 73 19975233 50 11 9-516657 606 483343 95-41468 679 10-458532 -024811 73'9-975189 49 12 9-517020 605 482980 9-541875 678 10-458125 -024855 73 19975145148 13 9-517382 605 482618I 9542281 678 10-457719 024899 73 9975101 47 14 9-517745 604 482255 9-542688 677 110457312 024943' 73 9975057'46 15 9-518107 604 481893 9-543094 677 10-456906 024987 73 9.9750131 45 16 9518468 603 481532 9543499 676 10456501 025031 73 9-974969 44 17 9-518829 603 -481171 9 543905 676 110456095 025075 74 9-974925143 181 9519190 602 *480810 9-544310 675 10-455690 1025120 74 9-974880 42 19' 9-519551 601 480449 9-544715 675 10-455285 -025164 74 9-974836; 41 201 9-519911 601 -480089 9-545119 674 10-454881 -025208 74 9-974792 40 21; 9-520271 600 479729 9-545524 674 10'454476 -025252 74 9-94748 39 22 19-520631 600 -479369 9545928 673 10-454072 - 025297 74 9-974703.38 23 1 9520990 599'479010 9-546331. 673 10-453669'025341 74 9974659137 24 9-521349 599 478651 9546735 672 10-453265 025386 74 9-974614 36 25> 9-521707 598 -478293 9547138 672 10-452862 -025430 74 9-974570!35 26 9'522066 598 -477934 9-547540 671 10-452460 -025475 74 9'974525 31 271 9522424 597 -477576 9547943 671 10-452057 025519 74 9.974481 l33 28 9-522781 596 -477219 9-548345 670 10-451655 1025564 74 19 97443l6 2 29 9 -523138 596 -476862 9-548747 670 10-451253 1-025609 74 19-9743911 31 30 1 9 523495 595 -476505 9 549149 669 10-450851 -025653 74 9.974347 30 311 9-523852 595 476148 9-549550 669 10-450450 -025698 75 19974302 29 32 9-524208 5941'475792 9-549951 668 10-450049 -025743 75 19-974257a 28 33 9-524564' 594 1475436 9-550352 668 10-449648 025788 759 97421227 34 9-5249209 593 475080 9-550752 667 10-449248 -025833 75 9-974167126 35l 95252751 593 -474725 9-5511521 667 10-448848 -025878 75 19974122 25 36 19-525630 592 -474370 9-551552 666 110-444848 025923 75 9-974077 24 37 9-5259841 591 474016 9-551952 666 10-448048 025968 75 9-974032123 38 1 9526339 591 -473661 9-552351 665 10-447649 026013 75 9-973987'22 39 9-526693 590 -473307 9-552750 665 10-447250 1026058 75 9-973942'l1 401 9527046 590 -472954 9-553149 665 10-446851 026103! 75 997389l 20 41 9-527400 589 -472600 9-553548 664 10-446452 026148 75 19-97385219 42 9527753 589 -472247 9-553946 664 10-446054 -026193 75 19-973807 l18 43 I 9-5281051 588 -471895 9-554344 663 10-445656 -0262839 75 9-973761!l7 441 9-5284581 588 -471542. 9554741 663 10-445259 -026284 75.9-973716 16 45 9-528810 587 -471190 9-555139 662 10-444861 -026329 76 9.973671 15 46 9-529161 587 -470839 9-5555361 662 10-444464 1026375 76 9-973625,14 47 9-529513 586 -470487 9-555933 661 10-444067 -026420 76 i9-973580 13 48 9-529864 586 -470136 9-556329 661 10-443671 1026465 76 19-973535 12 49 9-530215 585 469785 9556725 660 10-443275 -0265111 76 9-973489 l11 50 9-5305651 585 -469435 9-557121 660 10-442879 -0265561 76 9-973444 10 51 9-530915 584 -469085 9-557517 659 10-442483 -026601 76 9-973398 9 52 9-531265 584 *468735 9-557913 659 10-442087 -026648 76 19973352 8 53 9-531614 583 -468386 9-558308 659 10-441692 -026693 76 19973307 7 54 9-531963 582 -468037 9 558702 658 10-441298 -026739 76 9-973261 6 55i 9-532312 582 -467688 9-559097 658 10-440903 -026785 76 9-973215 5 56 9-532661 581 -467339 9-559491 657 10-440509 -026831 76 9-9731691 4 57 9-533009 581 *466991 9-559885 657 10-440115'0268776 6 9-9731241 3 58 9533357' 580 466643, 9560279 656 10-439721 1026922 76 9-973078 2 59 9-533704 580 -466629611 9560673 656 10-43937 -026968 76 9-97303 2 1 60 9-534052 579 i 465948 9-561066 655 10-438934 027014 76 9-9'2986' 0 C|Csine. I Secant. 1i Cotangent. I I Tangent. I; Cosecant. I Sine. i / 70 DEG. Page 61 LOGARITHMIC SINES, ETC. 61 20 DEG. - |- | Sine. | ff0; Cosecant T ent. T. f, Cotangent. Secant. Dff Cosine. i I 01 9-534052 -465948' 9-561066 110-438934 027014 1997296!60 1 9-534399 578 -465601 " 9-561459 655 10-438541 -027060 77 9 972940 59 2 9-53474.3 577 465255 i9-561851 654 j10-438149 -027106,77 9-972894 58 3 9-535092 577 -464908, 9 562244 654 10'437706 -027152 77 9-972848i57 10'4334 79.97280256 9-535438 577 -464562 j 9-562636 653 110-4364 07198 77 9-972802 56 5 9-535783 576 -464217 9 5630 28 653 10-436972 -02724T 8 9 972755 55 6 9-536129 576 463871 9563419 653 10 436581 -027291 77:9 972709 54 7'-5364-74 575 -463526 1 96563811 652 10-436189 -0273371 77 9-972663 53 8 9-536818 574 4631821 9.564202 652 10-435798 1-027383 77 9-972617.52 9 9537163 574 462837i 9-564592 651 10-435408 1027430 78i 9 972570 51 10 9-537507 573 -462493 9-564983 651 10-435017 -027476 77 9 972524 150 11 9-537851 573 -462149 i 9-565373 650 10-43462777027522 77 9-972478!49 12 9 538194 572 -461806 l 9565763 650 10-434237 -027569 78 9 9724311!48 139 9-53853 572 461462 9-566153 649 10-433847 -027615 77 ff97'-238547 141 9'538880 571 461120 9566542 649 10-433458 -0927662 78 9.972338 46 151 9 539223 571 -460t77 h 9-566932 649 10-433068 027709 78 9 972291145 16 9539565 570 460435 i 9-567320 648 10-432680 -0275 77 9.972245 44 17 9-539907 570 *460093 9 567709'48 10-432291 -027802 78 9.972198 43 18 9 540249 569 459751 9-568098 647 10431902 027849 78 9-972151 42 19i 9.540590 569.459410 9-568486 647 10-4315141 027895 77 19972105i41 20 9-540931 568 459069 9568873 646 10-431127 -027942 78 9 972058` 40 21 9-541272 568 458728 9-569261 646 10-4307319 -027989 78 i9 972011 139 22 I 9-541613 567.458387 9-569648 645 10 430352 -028036 78 9 971964138 231 9-541953 567 458047 9-570035 645 10-429965 -028083'789 971917 37 241 9542293 566 457707 9-570422 645 10-429578 1028130 78 9-9718701 36 25 9-542632 566 *457368 9-570809 644 10-429191 -028177 78 19971823135 26 9-542971 565 457029 9 571195 644 10-428805 -028224 78 i9-971776 34 271 954331 5 456690 6 90 571581 643 10-428419 -028271 789-971729 33 I 28 9-543649 564 456351 9-571967 643 10-428033 -028318 78 9-9716821l32 1 29 1 9-543987 564 j456013 9-572352 642 10-427648 -028365 78 9-9716351131 301 39544325 563 *455675 9-572738 642 10-427262 -028412 78,9971588!30 31' 9 544663 563 -455337 9-573123 642 10-426877 -028460 80'9971540129 32 9-545000 562 -455000 9-573507 641 10-426493 -028507 78 9-971493828 33 9-545338 562 -454662 9-573892 641 1042'6108 -028554 78 9.971446 / 27 34 9-545674 561 -454326 9-574276l 640 10-425724 -028602! 80 19971398, 26 35 9-546011 561 -453989 9-574660 640 10-425340 -0286491 78 9 971351 25 36 9546347 560' 453653 9S575044 639 10-424956, 028697 80 9-971303'24 37 9-546683 560 -453317 9-5754271 639 10-424573 -028744 78 9-971256 23 38 9-547019 5.59 -452981 9-575810 639 110424190 -0287921 80 9971208 22 39 9-547354 559 -452646 9-576193 638 10-423807 -028839, 78 9'971161 21 40 9-547689 558 -452311 9-576576 638 10-423424 -028887 80 9-971113 20 41 9-548024 558 -451976 9-576958 637 10-423042 1028934 78 9-971066 19 421 9-548359 557 -451641 9.577341 637 10-422659 -0289821 80 9-971018 18 43 19-548693 557 451307 9-577723 636 10-422277 -029030 80 9-970970 17 44 9-549027 556 450973 9-578104 636 10-421896.0290781 80 9-970922 1 451 9-549360 556 -450640 9.578486 636 10-421514.0291261 8 09-9708741 15 461 9549693 565 *450307 9-578867 635 10-421133 -0291731 78 9-970827 14 47 9-550026 555 -449974 9-579248 635 10-420752 -029221 80 19 970779113 48 9-550359 554 449641 9.579629 634 10-420371 -029269 80 9-970731112 49 9-550692 554 -449308 19580009 634 10-419991 -029317 80 9-9706831111 50 9-551024 553 *448976 9-580389 634 10-419611.029365 80 9*9706351'10 51 9-551356 553 -448644 9-580769 633 10-4192311 029414 82 I9.970586 9 52 9-551687 552 -448313 9-581149 633 10-418851 -0294621 80 9-970538' 8 53 9-552018 552 447982 9-581528 632 10 418472 i029510 80 19970490 7 54 9-552349 552 447651 9-581907 632 10-418093 i -0295581 80.9 970442 6 55 9-5.52680 551 447320 9-582286 632 10-417714!0296061 80 9970394, 5 56 9-553010 551 446990 9-5826651 631 10-4178335 -029655 82 9 - 970345 4 571 9 553341 550 -446659 9 583043 631 10-416957 -029703 80'9970297 3 58 9-553670550 -446330 9.5834221 630 10-416578 -029751 80 9970249 2 59 9S5540001 549 446000 9-583800 630 10-416200 -029800 8 9-970200' 1 60 9-554329 549 -445671 1 9584177 629 10 415823 -0298481 80 9970152, 0 / Cosine.. i Secant. j Cotangellt. i | 1 Tangent. Cosecant.! j Sine. 61) DEG. Page 62 62 LOGARITHMIC SINES, ETC. 21 DEG. | Sine. D iff; Cosecant. Tangent. Diff; I Cotangent. Secant. Diff. Cosine. I _ _ _, 0 9-554329 -445671 9-584177 |10-415828 iF029848~ 9970152 60 19-554658 548-445342 9-584555 629 10'415445 -029897| 81 9-970103 59) 2 9-554987 548 -445013 9-584932 629'10-415068 -029945 81 9-970055 58 3 9-555315 547 -444685 9-585309 628 10-414691.029994! 81 9-970006 57 49'555643 547-444357 9-585686 628 10-414314 -030043 81 9-969957 56 59-555971 546 -444029 9-586062 627 10-413938'030091 81!9-96990955 69-556299 546 -443701 9-586439 627 10-413561 -030140 81 9-969860 54 7 9-55662 545 -443374 9-586815 627 110-413185 -030189! 81 9-969811;53 8 9-556953 545 -443047 9-587190 626 |10-412810 -030238 81 9-9697 62 52 9-557280 544 -442720 9-587566 6216 10-412434 -030286 81 9-969714 51 10 9-557606 544 -442394 9-587941 625 10-412059 -030335! 81 9-969665 50 11 9-557932 543 -442068 9-588316' 625 10-411684 -030384! 81 19.969616,49 12 9-558258543 -441742 9-588691 625'10-411309 030433 82 9-969567 48 13 9-558588 543.441417 9-589066' 624 10-410934 -030482 82 9-969518 47 14 9.558909 542 -441091 9-589440 624 10-410560 030531 82 9.969469'1!46 15 9-559234 542.440766 9.5898141 623 10410186 030580 82 9969420 45 16 9-559558 541 -440442 9-590188 623 10-409812 -030630 82 9969370 44 17 9-559883 541.440117 9-590562 623 10-409438 -030679 82 9.969321 43 18 9-560207 540.439793 9-590935 622 10-409065 -030728 82 9'969272'!42 19 9-560531 540.439469 9-591308 622 10-408692 -030777 82 9'969223' 41 20 9560855 539.439145 9591681 622 110-408319 -030827 82 9-969173140 21 9.561178 539.438822 9-592054 621 10-407946 -030876 82 9'969124! 39 229-561501 538.438499 9.592426 621 10-407574 -030925 82 9.969075 38 239.561824 538.438176 9-592798'62 0 10-407202 -030975 82 9-969025'37 24 9-562146 537.437854 9-593171 620 10-406829 -031024 82 9-968976 36 25 9-562468, 537 -437532 9-593542' 620 10-406458 -031074 82 9-968926 35 26 9.562790 536.437210 9.593914' 619 10-406086 -.031123 83 9.968877 34 27 7j9.563112 536.436888 9.594285i 619 10405715 -031173 83 9-968827 33 28 9.563433 536.436567 9 -5 04656 618 10-405344 -031223 83 9-968777 32 29 9.563755 535.436245'9-595027 618 10-404973 -031272 83 9-968728'31 30 9-564075 535.435925 9.595398 618 10-404602 -031322 83 9-968678; 0 31 9.564396 534.435604 9.595768! 617 10-404232.031372 83 9.968628 29 329.564716 534.435284 9-596138i 617 10-403862 -031422 83 9.968578128 33 9.565036 533.434964 9-596508' 616 110-403492 -031472 83 9968528"'27 34 9-565356 533.434644 9.5968781 616 10-403122.031521 83 9-968479 26 359.565676 532.434324 9.597247 616 10-402753 -031571 839.968429 25 36 9.565995 532.434005 9-597616 615 10-402384 -031621 83 9.968379124 38 I 37 9566314 531.433686 959798.5 615'10-402015 -031671 83 9-968329 923 389-566632 531.433368 9-5983541 615 10-401646 -031722 83 9.968278 22 39 9-566951 531.433049 9.5987221 614 10-401278 -031772 83 9-968228!I1 40 9.567269 530.432731 9-599091 614 10-400909 -031822 84 9968178~0 41 9-567587 530.432413 9-599459 613 10-400541 -031872 84 9968128. 19 42 9-567904 529.432096 9-599827 613 10-400173 -031922 84 9968078 118 43 9-568222 529.431778 9-600194 613 10399806 01973 84 9968027i 17 44 9.568539 528 -431461 9.600562! 612 10-399438 K032023 84 9.967977 16 45 9.568856 528.431144 9-600929 61'2 10-399071.032073 84.967921 46! 9-569172 528-430828 9-601296 611 10-398704 -0321241 84 9.967876 14 47 9.569488 527 -430512 9.601662 611 10-398338 1-032174 84 9.967826! 13 48 9-569804 527 430196 9-602029 611 10-397971 -032225 84 9.967775!12 49 9.570120 526.429880 9602395 610 10397605 032275 84 9.967725' I1 50 9-570435 526.429565 9.602761 610 10.397239 -032326 84 9.967674 10 51 9-570751 525.429249 9-603127 610 10-396873 032376 84 9.967624 9 52 9-571066 525 -428934 9-603493 609 10-396507.032427 84 9-967573 8 53 9-571380 524 428620 9-603858 609 10-396142 -032478 84 9.967522 7 54 9.571695 524.428305 9604223 609 10-395777 -032529 85 9-967471 6 559-572009 523 -427991 9-604588 608 10-395412 -032579 85 9.967421 5 56 9.572323 523 -427677 9-604953 608 10-395047'0326?30 85 9-967370 4 57 9-572636 523 -427364 9-605317 607 10-394683'-032681 85 9-967319 3 58 9.572950 522 -427050 9-605682 607 10-394318 -032732 85 9-967268 2 59 9-573263 522.-426737 9-606046 607 10-393954!032783 85 9-967217 1 60 9-573575 521.426425 9.606410 606 10-393590 1.032834 85 9-967166 0 I, Cosine.;' Secant. I! Cotangent.! - Tangent. I Cosecant. I Sine.' 68 DEG. Page 63 LOGARITHMIC SINES, ETC. 63 22 DEG. Sine. Dff Cosecant. Tangent. Diff Cotangent. Secant. | Cosine. 0 9573575 426425 9-606410 10-393590 032834 9967166 60 1 9-573888 521 *426112 9 606773 606 10 -393227 0-0328S5 85 9-967115 59 2 9-574200 520 425800 9 607137 606 10-392863 032936 85 9967064 58 3 9574512 520 -425488 9 607500 605 10-392500 032987 85 9967013 57 4 9574824 519 425176 9-607863 605 10-392137 -033039 85 9-966961 56 5 9575136 519 -424864 9-608225 604 10-391775 x 033090 85 9.966910155 6 9575447 519 424553 9 608588 604 10-391412 -033141 85 99668591l54 7 9575758 518 -424242 9-608950 604 10-391050 033192 85 9966808 53 8 9-576069 518 423931 9-609312 603 10-390688 -033244 85 9-9667561"52 9 9576379 517 423621 9-609674 603 10-390326 033295 86 9-966705!51 10 9-576689 517 -423311 9-610036 603 10-389964 -033347 86 9-966653 50 11 9-576999 516 -423001 9-610397 602 110-389603 -033398 86 9966602 49 12 9-577309 516 -422691 9-610759 602 10-389211 -033450 86 9966550i48 13 9-577618 516 -422382 19-611120 602 10-388880 -033501 86 9966499'47 14 9-577927 515 -422073 9'611480 601 10-388520 033553 86 9966447 46 15 9-578236 515 -421764 9-611841 601 10-388159 033605 86 9966395 45 16 9-578545 514 421455 9-612201 601 10 387799 -033656 86 996'6344 44 17 9-578853 514 -421147 9-612561 600 10-387439 -033708 86 9966292 43 181 9579162 513 420838 9-612921 600 10387079 033760 86 9966240 42 19' 9579470 513 420530 91613281 600 10-386719 1033812 86 9-96618841 20 9-579777 513 -420223 9-613641 599 10-386359 033864 86 9-966136 40.21 1 9580085 512 -419915 9-614000 599 10-386000 033915 86 9966085 39 22 9-580392 512 -419608 9-614359 598 10-385641 -033967 87 19 966033/ 38 23 9580699 511 -419301 9-614718 598 10-385282 -034019 87 9-965981 37 24 9-581005 511 418995 9-615077 598 10-384923 034071 87 9965929 36 25 9581312 511 -418688 9-615435 597 10384565 034124 87 9965876 35 26 9-581618 510 -418382 9-615793 597 10-384207 -034176 87 9965824 34 27 9581924 510 -418076 9 616151 597 10-383849 034228 87 9965772 33 28 1 9-582229 509 -417771 9-616509 596 10-383491 034280 87 9965720 32 291 9582535 509 -417465 9-616867 596 10-383133 034332 87 9965668 31 30 9-582840! 509 417160 9-617224 596 10-382776 -034385 87 9-965615 30 31 9-5831451 508 -416855 9-617582 595 10 382418 -034437 87 9-965563 29 32 9'583449' 508 416551 9617939 595 10-382061 -034489 87 9-96551128 33 9-583754 507 -416246 9-618295 595 10-881705 034542 87 9965458 27 34 9-584058 507 -415942 9-618652 594 10-381348 034594 87 9965406 26 35 j 9584361 506 -415639 9-619008 594 10380992 1034647 87 9-965353 2 36 95846651 06 -415335 1 9-619364 594 1.0-380636 -034699 88 9965301124 37 9584968 506 *415032! 9-619721 5903 10-380279 -034752 88 9-96524823 38 9585272 505 *414728 9-620076 593 10-3799224 -034805 88 9965195122 39 9-585574 505 -414426 9-620432 593 10-379568 -034857 88 9965143 21 40 9585877 504 414123 9.620787 92 10-379213.034910 88 9.965090 20 41 9586179 504 413821 9-621142 592 10-378858 1034963 88 9-96503719 42 9586482 503 *413518 9-621497 592 10-378503 -035016 88 9-964984 18 43 9586783 503 -413217 9-621852 52 1 110378148 -035069 88 9-964931117 44 9.587085 503 -412915 9622207 591 10-377793 1035121 88 9964879 16 45 9587386 502 -412614 9-622561 590 10-377439 -035174 88 19964826 i15 46 9587688 502 412312 9-622915 590 10-377085 1035227 88 9964773'i14 47 9587989l 501 412011 9.623269 590 10-376731 -035280 88 9964720 13 489-588289 501 -411711 9-623623 589 10-376377 -035334 88 9964666 12 49 95885901 501 411410 9623976 589 10-376024 -035387 89 9964613 1l 50 9-588890 500 -411110 9-624330 589 10-375670 -035440 89 996456010 51 9-589190 500 410810 9-624683 588 10.375317 -035493189 9 964507j 9 52 9-589489 499 -410511 9-625036 588 10-374964 -035546 89 9964454' 8 53 9-589789 499 -410211 9625388 588 8 10374612 03560089 19964400 7 54 9-590088 499 -409912! 9-625741 587 10-374259 -035653189 9-9643471 6 55 9-590387 498 -409631 9-626093 587 10373907 -035706189 9964294' 5 56 9-590686 498 -409314' 9-626445 587 10-373555 035760 89 9964240 4 57 9-590984 497 -409016 9-626797 586 10373203 035813 89 9-964187 3 58 9-591282 497.408718 9-627149 586 10372851 -0358671 89 9-964133 2 59 9-5915801497 -408420 9-627501 586 10372499 035920 89 19964080 1 60 9-591878 4961 408122 9-627852 586 10372148 -0359741 89 9964026, 0 / s.05123 I 9~6fS'i8:j2 68. angnt I in. Cosine. | Secant. i1 Cotangent. Tangent. Cosecant. Sine. 67 DEG. Page 64 64 LOGARITHMIC SINES, ETC. 23 DEG. S Diff_ Iiff Stan'ntnt lecmnt Cosne - I| Sine. 1ff- Cosecant. Tangent. i Cotangent. Sent. i, Co e 0 9-591878 408122 9-627852 10-372148 -03o5974 9 96402t 6 l 1 9-592176 496;407824 9-628203 585 10-371797 -036028' 89 9 968972 659 2 9-592473 495 407527 9-628554 585 10-371446;-03606189 9 963919 158 3 9-592770 495 407230 9-628905 585 10-371095 -03613) 89 9 963865) a 4 9-593067'495 406933 9-629255 584 10-370745 -03618) 90 9 963811 15 5 9-593363 494' 406637 9-629606 584 10-370394 1036243o0 90 99636757 )5 6 9-593659 494.406341 9-629956 583 10-370044 8036296| 90 9 963704 A54 7 9-593955 493 406045 9-630306 583 10-369694.036850!9 99 93650 i38 8 9-594251 493 -405749 9-630656 583 10-369344 1 036404 (t 90 9 963596 052 9 9-594547 493.405453 9-631005 583 10-368995 l-0364581 90 9-963542'51 10 9-594842 492.405158 9-631355 582 10-368645 |'030512 9 9963488 50 11 9-595137 492.404863 9-631704 582 10-368296.I036566 90 9 963434'49 12 9-595432 491.404568 9-632053 582 10-367947. 036621 90 9-96o379 48 13 9-595727 491.404273 i19-632401 581 10-367599 1 036675! 90 9-96332';: 47 14 9-596021 491.403979 9-632750 581 10-367250!036729 90 9-9632711 46 15 9-596315 490.403685 9-633098 581 10-366902 -036783j 90 9-963217 45 16 9-596609 490.403391 9-633447 580.10-366553 -036837 90 9-963163 44 17 9-596903 489.403097 9-633795 580 /1066205 -0368921 90 963108,43 18 9-597196 489.402804 9-634143 580 10-365857 -036946191 9 963054 42 19 9-597490 489.402510 9-634490' 579 10-365510 i037001 91' 9962999 41 20 9-597783 488.402217 9-634838 579 10-365162 - 037055 91 19-962945 40 21 9-598075 488.401925 9-635185 579 10 364815 -037110191 9 962890 39 22 9-598368 487.401632 9-635532 578 10-364468'-037164 91 9-962836 38 23 9-598660 487.401340 9-635879 578 10-364121 037219 91 9-962781; 37 24 9-598952 487.401048 9-636226 578 10363774 1037273 91 9-962727 36 25 9 599244 486.400756 9-636572 577 10-363428 1-037328 91 9-962672 35 26 9-599536 486.400464 9-636919 577 10-363081 7-037383 91 9 9626171l34 27 9-599827 485.401739-637265 577 10-362735 -037438 91 9-962562 33 28 9-600118 485.399882 9-637611 577 10-362389 -037492, 91 9-962508:32 29 9-600409 485.399591 96379056 576 10-362044 -0387547 91 9-962453;31 30 9-600700 484.399300 9-638302 576 10-361698 1 037602 91 9-962398 30 31 9-600990 484.399o010 9-638647 576 10-361353 -037657 92 9-9623435129 32 9-601280 484.398720 9-638992 575 10-361008.037712 92 9 962288'28 33 9-601570 483.398430 9-639337 575 10-360663 1-037767 9 92 09622933 27 34 9-601860 483.398140- 9-639682 575 10-360318 -037822 92 9-962178 26 35 9-602150 482.397850 9-6400277 574 10-359973 037877 92 9-962123 i25 36 9-602439 482.397561 9-640371 574 10-359629.-037933 92 9-962067 24 37 9-602728 482.397272 9-640716 574 10-359284 -037988 92 9-962012 23 38 9-603017 481.396983 9-641060 573 10-358940 1-0380431) 9-9619571 22 39 9-603305 481.396695 9-641404 573 10-358596' 038098 92 9-961902.21 40 9-603594 481.396406 9-641747 573 10 358253 -038154 92 9-961846 20 41 9-603882 480.396118 9-642091 572 10-357909 -038209 92 9-961791'19 42 9-604170 480.395830 9-642434 572 10-357566 i -038265 92 9-961735 18 43 9-604457 479.395543 9-642777 572 110-3572'23 l 038320 92 9-961680 17 44 9-604745 479.395255 9-643120 572 103-56880 1-038376 92 9-961624 16 45 9-605032 479.394968 9-643463 571.10-356537 l-038431 93 9-961569'15 46 9-605319 478.394681 9-64380( 571 10-356194 i-038487 93 9-961513 14 47i 9-605606 478 -394394 9-644148 571 10-355852'038542 93 9-961458 13 48 9-605892 478.394108 9-644490 570 10355510.0388598 93 9-961402 12 49 9606179 477 ~393821 9-644832 570 10-35516811 038654 93 9-961346 11 50 9-606465 477 -393535 9-645174 570 10 354826 |038710 93 9-961290 10 51 9-606751 476 -393249 9-645516 570 10 354484 1038765 93 19-961235 9 1 52 9-607036 4761 392964| 9645857 569 10-3541431;038821 93 9-961179 8 53 9-607322 476 1 392678 1 9646199 569 10-353801 i-038877 93 9-961123 7 54 9-767607 475 392393 9646540 569 10-353460 038933 93 9-961067: 6 55 9-607892 475 -392108 9-646881 568 10- 33119 1038989 93 9-961011 5 56 9-608177 474 -391823 9-647222 568 10-35278'1039045 93 9-960955' 4 57 9-608461 474 -391539 9-64762 562 8 10- 35438 -039101 93 9-9608991 3 58' 9608745 474 -391255 9-647903 567 1l0352097 -039157. 93 9-960843 2 591 9-609029 473 -390971 9-648243 567 10-351757. -039214 94 9-960786 1 60 9-609313 473 -390687 i 9648583 567 10-351417 10392070 94 9-960730 0 / Cosine. | I Secant. i Cotangent. Tangent. I Csecnt. Sine. I 66 DEG. Page 65 LOGARITHMIC SINES, ETC. 65 24 DEG., Sine. Diff Cosecant. Tangent. 1 0ff Cotangent. Secant. Djff Cosine. r / rSine. C e t T n I_" 0 9 609313 *390687 | 9-648583 10-351417 039270l 9.960730Jl60 1 9-609597 473 390403 19648923 566 10351077' 039326 94 9.960674 159 2 9-609880 472 390120 19649263 566 10-350737 -039382 94 9-960618.58 3 9-610164 472 *389836 9-649602 566 10-350398 039439 94 9-9605611 57 4 9 610447 472 389553 9-649942 566 10-350058 -0394959j 94 9 960505 "56 5 9-610729 471 389271 9-650281 565 10-349719 0399552 94 l9960448 55 6 9-611012 471 -388988 9-650620 565 10-349380 039608 94 19960392154 7 9-611294 470 388706 9-650959 565 10-349041 039665 94 19'9603835!53 8 9611576 470 *388424 9-651297 564 10-348703 039721 94 9-960279 |52 9 9-611858 470 -388142 9-651636 564 10-348364 -039778 94 9 960222 i51 10 9-612140 469 387860 9-651974 564 10-348026 039835 94 9-960165 50 111 9-612421 469 -387579 9-652312 563 10-347688 1039891 94 9-9601091;49 12 9-612702 469 387298 9-652650 563 10-347350 039948 95 9-960052 148 13 9-612983 468 -387017'9652988 563 10-347012 040005 95 19959995!47 14 9-613264 468 386736 9-653326 563 10-346674 040062 95 9-959938 46 15 9-613545 467 386455 9-653663 562 10-346337 040118 95 9-959882 145 16 9-613825 467 386175 9-654000 562 10-346000 040175 95 9-959825 44 17 9-614105 467 -3858951 9-654337 562 10-345663 040232 95 9-959768; 43 18 9-61438 466 38 44 56 1 14326 895 654674995971142 19 9-614665 466 -385335 9-655011 561 10-344989 -040346 95 9-959654|141 20 9-614944 466 -385056 i 9-655348 561 10-344652 1040404 95 9-959596 40 21 9-615223 465 -384 777 97-655684 561 10-344316 1040461 95 9-959539839 221 9-615502 465 -384498 9-656020 560 10-343980 1040518 95 19959482l 38 23 9615781 465 -384219 9-656356 560 10-343644 -040575 95 9-9594251137 24 9-616060 464 -383940 9-656692 560 10-343308 -040632 95 19959368 36 25 1 9-616338 464 -383662 9-657028 559 10-342972 -040690 95 199593101 35 261 9616616 464 383384 9657364 559 10-342636 -040747 96 9-959253 34 27 9-6168941 463 383106 9657699 59 9 10-342301 -040805 96 9-959195! 33 28 1 9-617172 463 382828 j 9658034 559 10'341966'040862 96 9-9591381 32 29 1 9617450 462 382550 I 9-658369 558 110-341631 -040919 96 19959080 131 30 9-617727 462 -382273 9-658704 558 10-341296'040977 96 9'959023 130 31 9-618004 462 -381996 iJ 9-659039 558 1.0-340961 -041035 96 19958965ll29 32 9-618281 461 381719 9-659373 558 10-340627 -041092 96 19958908 28 33 19-618558 461 -381442 9 659708 557 10-340292 -041150 96 9-958850 27 34 9-6188341 461 -381166 9-660042 557 10-339958 -041208 96 9-958792 26 385 19619110 460' 380890 1 9-660376 557 10-339624 -041266 96 9-95873425 36 9-619386 460 -380614 9-660710556 105339290 1041323 96 9.958677 121 37 9-619662 460 -380338 9-661043 556 10-338957'041381 96 9-958619 23 381 9-6199381 459 -380062 9-661377 556 10-338623 11041439 96 9-958561!i22 39 9-620213 459 379787 9-661710 556 10-338290 1041497 96 9-958503' 21 40 9-620488 459 -379512 9-662043 555 10-337957 041555 97' 99584450620 41 9-6207631 458 379237 9-662376/ 555 10-337624 9041613 97 9-9583871119 42 9-621038! 458 -378962 9-662709 555 10-337291 -041671 97 19958329 18 43 9-621313 457 -378687 9663042 554 10-336958 -041729 97 9-958271'17 44 9-621587 457 -378413 9-663375 554 10-336625 1041787 97 9-958213 16 45 9-621861 457 378139 9.6637071 554 10336293 -041846 97 19958154 15 46 1 9622135 456 -377865 9664039 554 10-335961 -041904 97 19-958096114 47 119 6224091 456 377591 9664371 553 10-335629'041962 97 9-958038 13 48 9'6226821 456 1 377318 9-664703 553 10-335297 -042021 97'9-957979112 49 19622956 455.377044 9-665035 553 10-334965 -042079 97 199579221 11 50 9623229 455 -376771 9-665366 553 10-334634 042137 97'-95783,il0 51 9 623502.455 -376498 9-66.5697 552 10-334303 -042196! 97 9-957804 9 52 9'623774 454 -376226 9-666029 552 110333971 -0422541 97 9-957746 8 53 i 9-624047 454 *375953 9-666360 552 10-333640.-042313i 98 9-957687! 7 54 916243191454 -375681 9-666691 551 10-333309 1-0423721 98 989576281' 6 55 9-624591 453 -375409 9-667021 551 10'332979 1,042430 98 19-957570 5 56 9-624863 453 -375137 9-667352 551 110332648 i 042489 98 9-957511!l 4 57 i96251351 453 374865 9-667682 551 10332318 1042548 98 9-9574521 3 58 1 9-625406' 452 374594 9-668013 5650 10-331987 -042607 98 9-9573S93 2 69 9-.625677 452 -374323 9-668343 550 10-331657 -042665 98 9-9573385 1 601 9-6259481452 *3740521 9668672 550 [10-331328 -042724 98 9-957276 0 Cosine. | Secant. 1 Cot angent. Tgent Cosecant. | Sine. - 65 DEG. Page 66 66 LOGARITHMIC SINES, ETC. 25 DEG. | Sine. Diff. Cosecant. Tangent. Diff Cotangent. Secant. Dif. 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Secant. 100 - 09-641842 -358158 9-688182 10-311818 -046340 9-953660 60 1 9-642101 431 *357899 9-688502 534 10-311498 -046401 103 9 9535991 59 2 9642360 431 -357640 9-688823 534 10-311177 046463 103 9-953537 58 3 9642618 431 -357382 9-689143 534 10-310857 -046525 103 9-953475 57 4 9642877 430 *357123 9-689463 533 10-310537 -046587 103 9-953413 56 5 9-643135 430 -35685 9689783 533 10-310217 -046648 103 9-953352155 6 9643393 430 *356607 9-690103 533 10-309897 -046710 103 9-953290,54 7 9-643650 430 -356350 9-690423 533 10-309577 -046772 103 9-953228, 53 8 9-643908 429 -356092 9-690742 533 10-309258 -046834 10319-95316652 9 9-644165 429 -355835 9-691062 532 10-308938 -046896 103 9-953104 51 10 9-644423 429 -355577 9-691381 532 10-308619 -046958 103'9-953042 50 11 9-644680 428 *355320 9-691700 532 10-308300 -047020 103'9-952980 49 12 9-644936 428 -355064 9-692019 531 10-307981 -047082'104 9.952918 48 13 9645193 428 -354807 9-692338 531 10-307662 -047145 104'9-952855 47 14 9-645450 427 -354550 9-692656 531 10-307344 -.047207 1041 9952793 46 15 9-645706 427 -354294 9-692975 531 10-307025 -047269 10419-952731 45 16 9-645962 427 -354038 9-693293 531 10-306707 -047331 104 9-952669 44 17 9-646218 426 -353782 9-693612 530 10-306388 -047394 104 9-952606 43 18 9-646474 426 -353526 9-693930 530 10-306070 -047456 104 9-952544 42 19' 9-646729 426 -353271 9-694248 530 10-305752 -047519 104 9 952481 41 20 9646984 425.353016 9-694566 530 10-305434 1047581 104 9-952419 40 21 9-647240 425 352760 9-694883 529 10-305117 -047644 104 9-952356139 22 9-647494 425 -352506 9-695201 529 10-304799 -047706 104 9-952294138 23 9-647749 424 -352251 9-695518 529 10-304482 -047769 104 9-952231 37 241 9-648004 424 -351996 9-695836 529 10-304164 -047832 104 9-952168136 251 9-648258 424 -351742 9-696153 529 10-303847 -047894 105 9-952106 35 26 9-648512 424 -351488 9-696470 528 10-303530 047957 105 9-952043 34 27 9-648766 423 -351234 9-696787 528 10-303213 -048020 105 9-951980, 33 28 9-649020 423 -350980 9-697103 528 10-302897 -048083 105 9-951917 32 29 9-649274 423 -350726 9 697420 528 10-302580 -048146 105 9-951854 J31 30 9-649527 422 -350473, 9-697736 527 10-302264 -048209 1051 9951791130 31 9-649781 422 -350219 9-698053 527 10-301947 -048272 105 9-951728 29 32 9-650034 422 -349966 9-698369 527 10-301631 -048335 105 9-951665 28 33 9-650287 422 -349713 9 698685 527 10-301315 -048398 10519-951602127 34 9-650539 421 3494611 9-699001 526 10-300999 -048461 105 19951539 26 35 9-650792 421 -349208 9-699316 526 10-300684 -048524 105 9-951476 25 36 9-651044 421 -348956 9-699632 526 10-300368 -048588 10519-951412 24 37 9-651297 420 -348703 1 969997 1 526 10-300053 -048651 105 9-951349 123 38 9-651549 420 -348451 9-700263 526 10-299737 -0487141106 9-951286 22 39 9-651800 420 -348200 9-7005788 525 10-299422 -04877810619-951222121 40 9-652052 419 -347948 9-700893 525 10-299107 -048841 10619-951159 20 41 9-652304 419 -347696' 9-701208 525 10-298792 -048904 106 19951096 19 42 9-652555 419 -347445 9-701523 525 10-298477 -048968110619-951032 18 43 9652806 418 -347194 9-701837 524 10-298163 -049032110619-950968 17 44 9-653057. 418 -346943 9-7021-52 524 10-297848 -049095 10619-950905 16 45a 9-653308 418 -346692 9-702466 524 10-297534 -049159 106 9-950841 15 46 9-653558 418 -3464421 9-702780 524 10-297220 104922210619-950778114 47 9-653808 417 -346192 9 703095 523 10-296905 1049286 10619-950714 13 48 9654059/ 417 -345941! 9703409 523 10-296591 -049350 106 9-950650 12 49 9-654309 417 -345691 9 703723 523 10-296277 1049414 106 9-9505861 11 50 9-654558 416 -345442 9 704036 523 10-295964 -049478 106 9-950522 10 51 9-654808 416 -345192 9 704350 523 10-295650 1049542 107 9-950458 9 52 9-655058 416 -344942,i 9 704663 522 10-295337 -049606 10719-950394 8 53 9-655307 415 -344693 9-704977 522 10-295023 1049670 107 9-950330 7 54 9-655556 415 -344444 9-705290 522 10-294710 1049734 107 9 9502661 6 55 9655805 415 -344195 9705603 522 10-294397 -049798 107 9-950202i 5 56 9-656054 415 -343946 1 9705916 521 10-294084 -049862 107 9-950138' 4 57 9-656302 414 -343698 1 9-706228 521 10-293772 -0499261107 9-950074' 3 58 9-656551 414 -343449 9-706541 521 10-293459 -0499901107 9-950010' 2 59 9-656799 414 -343201 9-7068654 521 10-293146 -050055 107 9-949945l 1 60 9-657047 413 -342953 9-707166 521 10-292834 -050119 107 9-9498811 0 Cosine. Secant. Cotng Tanent. Cosecant. Sine. 33 63 DEG. Page 68 68 LOGARITHMIC SINES, ETC. 27 DEG. Sine. ff Cosecant. Tangent. Cotangent. Secant. ) ff Cosine. I O 9-657047 -342953 9-707166 10-292834 l 0501 19 l 949881 60 1 9-657295 413 -342705 9-707478 520 10-292522 i-050184 107!9-949816!59 2 9-657542 413 -34Z458.9-7077901 520 10-292210 050248!107 9.949752 58 3 9-657790 412 342210 9-708102 520 10-291898'050312l107;9 949688i!57 4 9-658037 412 *341963 9-708414 520 10291586 0503771108 9*949623!56 5 9-658284 412 *341716 9-708726 519 10-291274 -050442 1089 9949558 55 6 9-658531 412.341469 9-709037 519 10-290963 -050506 108 9-949494 54 7 9-658778 411 -341222 9-709349 519 10290651 -650571 108 9-949429 53 8 9-659025 411. -340975 9 709660 519 10-290340 050636 108 9-949364 52 9 9-659271 411 340729 9-709971 519 10-290029 050700 108 9-949300 51 10 9-659517 410 -340483 9710282 518 10-289718.050765 10819.949235 50 11 9-659763 410 340237 9-710593 518 10-289407 0508301089.949170 49' 12 9-660009 410 -339991 9-710904 518 10-289096 -050895 1081;9949105 48 13 9-660255 409 -339745 9-711215 518 10-288785 -050960 10819-949040 47 14 9-660501 409 339499 9-711525 518 10-288475 051025 108 9948975 46 15 9-660746 409 339254 9-711836 517 10-288164 051090 108 9-948910145 16 9-660991 409 339009 9-712146 517 10-287854 051155 108 9-948845'44 17 9-661236 408 *338764 9-712456 517 10-287544 -051220 108 9-948780'43 18 9-661481 408 *338519 9712766 517 10-287234 051285 109 9.948715 42 19 9-661726 408 *338274 9-713076 516 10-286924 051350 109 9-9486501 41 20 9-661970 407 -338030 9-713386 516 10-286614 -051416 109 9-948584:140 21 966'2214 407.337786 9713696 516 10-286304 -051481 109 9-94851'9139 22 9-662459 407 -337541 97- 14005 516 10-285995.0515461109 9.948454 38 23 9-662703 407 337297 9714314 516 10-285686 -051612 109 9-948388 37 24 9662946 406 337054 9-714624 515 10-285376 051677 109 9-9483231 36 25 9-663190 406 -336810 9-714933 515 10285067 1051743 109 9-948257i 35 269-663433 406 -336567 9-715242 515 10-284758 -051808 109 9.948192 34 27 9663677 405 *336323 9-7155511 515 10-284449 10518741109 9-948126 33 28 9.663920 405.336080 9-715860' 514 10-284140 1051940 1099 9948060 32 29 9.664163 405 *335837 9-716168 514 10-283832 -052005 109 9-9479956 31 30 9-664406 405 3-35594 9716477 514 10-283523 052071 110 9-947929',0 31 9-664648 404 335352 1 9716785 514 10283215 052137 110 9.947863l29 32 9-664891 404 -335109 9-717093 514 10-282907 -052203 110 9-947797128 33 9-665133 404 -334867 9717401 513 10-282599 -052269 110 9-947731 27 34 9-665375 403 -334625 9-7177091 513 10-282291 -052335 110i9-94766526 35 9-665617 403 -334383 9-718017 513 10-281983 -0524001110 9947600'25 36 9665859 403 334141 9-718325 513 10-281675 -052467 110 9-947533 24 37 9666100 402 -333900 9-718633 513 10-281367 -052533110i9-947467 23 38 9-666342 402 -333658 9-718940 512 10-281060 -052599 110 9-947401 22 39 9-666583 402 -333417 9719248 512 10-280752 -052665}110 9-94733521 40 9-666824 402 -333176 9-7195-55 512 10-280445 -05273111109-947269 20 41 9667065 401 *332935 9-719862 512 10-280138 -052797 110 9947203 19 42 9667305 401 -332695 9-720169 512 10279831 -0528641109-94713618 43 9667546 401 *332454 9720476 511 10-279524 0529301119947070 17 44 9667786 401 *332214 9-720783 511 10-279217 1052996 111 9-947004 16 45 9668027 400 *331973 9721089 511 10-278911 -053063 111 9-946937 15 46 9668267 400 331733 9721396 511 10-278604 1053129111 9-946871 14 47 9668506 400 -331494 9721702 511 10-278298 -053196 111 9-946804 13 48 9668746 399 *331254 9-722009 510 10-277991 -053262 111 9-946738 12 49 9668986 399 -331014 9.722315 510 10-277685 11053329 111 9-946671 11 50 9669225 399 3'30775 9-722621 510 10-277379 -053396 111 9-946604 10 51 9669464 399 -330536 9-722927 510 10-277073 -053462 111 9-946538 9 52 9669703 398 -330297 9-723232 510 10-276768 -053529 111 9-946471 8 53 9669942 398 -330058 9-7235381 509 10-276462 -053596 111 9-946404 7 54 9670181 398 -329819 9-723844 509 10-276156 -053663 111 9-946337 6 55 9670419 397 329581 9.724149 509 10-275851 -053730 11.1 9-946270 5 56 9670658 397 329342 9-724454 509 10-275546 -053797 112 9-946203 4 57 9670896 397 -329104 9-724759 509 10-275241 -053864 112 9-946136 3 58 96711341 397 -328866 9-7250651 508 10-274935 -053931 112 9-946069 2 59 9-671372 396 -328628 9-725369 508110-274631 053998112 19946002 1 60 9-671609 396 328391 9725674 508 10274326 054065o1129-945935 0 C I Cosine. I Secant. I Cotangent. Tangent. Cosecant. Sine. 62 DEG. Page 69 LOGARITHMIC SINES, ETC. 69 28 DEG. Sine. Diff Cosecant. Tangent. Dif Cotangent. Secant Diff. Cosine. 0 9-671609 -328391 9-725674 110-274326 054065.9-945935 601 9-671847 896 *328153 9-725979 508 10-274021 054132 112 9-945868' 59 2 9-672084 395 327916 9-726284 508 10-273716 l-054200 112 9-945800 58 3 9-672321 395 -327679 9-726588 507 10-273412 -054267 112 9-945733 57 4 9-672558 395 327442 9-726892 507 10-273108 -054334 112 9-945666 56 5 9-672795 395 -327205 9-727197 507 10-272803 054402 112 9-945598l 55 6 9-673032 394.326968 9-727501 507 10-272499 -054469 112 9-945531 54 7 9-673268 394.326732 9-727805 507 10-272195 -054536 112 9-945464, 53 8 9-673505 394 -326495 9-728109 506 10-271891 -054604 113 9 945396 52 9 9-673741 394 -326259 9-728412 506 10-271588 -054672 113 9 945328 51 10. 9-673977 393.326023 9-728716 506 10-271284 054739 113 9 945261 50 11 9-674213 393.325787 9- 729020 506 10-270980 -054807 113 9-945193 49 12 9-674448 393 -325552 9-729323 506 10-270677 -054875 113 9-9451_25 148 13 9-674684 392.325316 9-729626 505 10-270374'-054942 113 9-945058 47 14 9-674919 392.325081 9-729929 505 10-270071 -055010 113 9-944990 46 15 9-675155 392.324845 9-730233 505 10-269767.055078 113 9-9449221 45 16 9-675390 392 -324610 9-730535 505 10-269465.05-5146 113 9-944854144 17 9675624 391.324376 9-730838 505 10-269162 055214 113 9.944786!143 18 9:675859-.391 -324141 9-731141 504 10-268859 m-055282 113 9-9447,18142 19 9-676094 391.323906 9-731444 504 10-268556 -055350 113 9-944650 41 20 9-676328 391.323672 9-731746 504 10-268254 055418 113 9-944582 40 21 9-676562 390.323438 9732048 504 10-267952 -055486.114 9-944514 139 22 9676796 390.323204 9-732351 504 10-267649 055554 114 9-944446 138 23 9677030 390.322970 9-732653 503 10-267347 -055623 114 9-944377 37 24 9677264 390.322736 9 732955 503 10-267045 -055691 114 9-944309 386 25 9677498 389.322502 9-733257 503 10-266743 -0557591114 9-944241 35 26 9677731 389 -322269 9-733558 503 10-266442 -055828 114 9-944172 34 27 9-677964 389.322036 9-733860 503 10-266140 -055896 114 9-944104 -8)3 28 9-678197 388.321803 9-734162 503 10-265838 -055964 11419-944036 32 29 9678430 388.321570 9-734463 502 10-263537 -056033 114 9-943967i31 30 9678663 388.321337 9-734764 502 10 265236 -056102114 9-943899 30 31 9678895 388.321105 9-735066 502 10-264934 -056170 114 9.943830 29 32 9-679128 387.320872 9-735367 502 110-264633 -056239 114 9-943761 28 33 9679360 387..320640 9-735668 502 10-264332 -056307 114 9-943693 27 34 9-679592 387.320408 9-735969 501 10-264031 -056376 11519-943624!26 35 9679824 387.320176 9-736269 501 10-263731 -056445 115 9-943555125 36 9-680056 386.319944 9-736570 501 10-263430 -056514 11519-943486'24 37 9-680288 386.319712 9 736871 501 10-263129 056583 1159-943417 1'23 38 9-680519 386.319481 9-737171 501 10-262829 *056652 115 9-943348!22 39 9-680750, 385.319250 9-737471 500 10-262529 1-056721 115 9-948279 21 40 9680982 385.319018 9-737771 500 10-262229 -056790 115 9-943210 20 41 9-681213 385.318787 9-738071 500 10-261929 -056859 115 9 943141 19 42 9681443 385.318557 9-738371 500 10-261629 -056928 115 9-943072' 118 43 9-681674 384.318326'9 738671 500 10-261329 -056997 115 9-943003 17 44 9 -81905 384.318095 9-738971 50t) 10-261029 057066 115 994293416 45 9-682135 384.317865 9-73927'1 499 10-260729 0571 6 115 9942864 15 46 9-682365 384.317635 9-739570 499 110-260430.057205 115 9-942795 %14 47 9-682595 383.317405 9-739870 499 10-260130 -057274 116 9-942726'13 48 9-682825 383.317175 9-740169 499 10-259831 -057344 116 9-942656 2, 49 9-683055 383.316945 9-740468 499 110259532 -057413 116 9-942587, ll 50 9-683284 383.316716 i 9'740767 498 10-259233.057483 116 9-942517 10 51' 9683514/382.316486 9-741066 498 10-258934 -057552116 9-942448: 9 52 9-683743 382.316257'9741365 498 10-258635.-057622 116 9-942378 8 538 9-683972 382 -316028 9-741664 498 10-258336 -057692 116 9-942308 7 54 9-684201 382.315799: 9-741962 498 10-258038 -057761 116 9-9422391 6 55 9-684430 381.315570 9-742261 498 10-257739 -057831 116 9-942169'i 5 56 9-684658 381.315342 9-742559 497 10-257441 -057901 116 9-942099,1 4 57 9-684887 381.315113 9-742858 497 10-257142 -057971 116 9-942029 3 58 9-685115 380'-314885 9-743156 497 110-256844 -058041 116 9-9419591 2 59 9-685343 380 314657 9-743454 497-110-256546 -058111116 9-941889 1 1 60 9685571 380 314429 9-743752 497 10-256248-0581811179-9418191 0 / I Cosine. Secant. Ii Cotangent. Tangent. Cosecant. Sine. I' 61 DEG. Page 70 70 LOGARITHMIC SINES, ETC. 29 DEC. Sine. 100T I Cosecant. Tangent. 1Dif Cotangent. Secant. ff Cosine. 01 9 685571 314429 9743752 10-256248'058181 9-941819 60 1 9-685799 380 *314201 9-744050 496 10-255950 058251 117 9-941749 59 2 9-686027 379 -313973 9-744348 496 10-255652 -058321 11719941679 58 3 9-686254 379 -313746 9-744645 496 10-255355 058391 117 9-941609 57 4 9-686482 379 *313518 9-744943 496 10-255057 058461 117 9-941539 56 5 9686709 379 313291 9-745240 496 10-254760 058531 117 9-941469 55 6 9686936 378 -313064 9-745538 496 10-254462 058602 117 9-941398 54 7 9-687163 378 -312837 9-745835 495 10-254165 058672 117 9-941328;53 8 9-687389 378 -312611 9-746132 495 110253868 058742 117 9-941258 52 9 9-687616 378 -312384 9-746429 495 10-253571 058813 11719 941187 51 10 9687843 377 -312157 9-746726 495 10-253274 058883 117 9944117 50 11 9688069 377 311931 9-747023 495 10-252977.058954 117 9.941046 49 12 9-688295 377 -311705 9747319 494 10-252681 059025 118 9-940975 48 13 9688521 377 -311479 9-747616 494 10-252384 059095 11819-940905 47 14 9-688747 376 -311253 9-747913 494 10-252087 -059166 118 9.940834 46 15|| 9-688972 376 311028'9748209 494 10-251791 ~059237 118 9-940763 45 161 9-689198 376 -310802 9-748505 494 10-251495 059367 11819-940693 44 17i 9689423 376 -310577 9-748801 494 10-251199 -059378 118 9-940622 43 18 19689648 375 310352 9-749097 493 10-250903 -059449118 9-940551 42 191 9-689873 375 -310127 9-749393 493 10-250607 059520 1189-940480 41 20;1 9690098 375 309902 9-749689 493 10-250311 059591 118 9.940409 40 211 9-690323 375 -309677 9-749985 493 10-250015 1059662 11819-940338 39 22I 96905481 374 *309452 9 750281 493 10-249719 -059733 118 9 940267 38 23 9-690772 374 309228 9-750576 493 10-249424 059804 118 9-940196 37 241. 9-690996 374 309004 9-750872 492 10-249128 -059875 118|9-940125 36 251 9691220 374 -308780 9-751167 492 10-248833 -059946 119 9-940054135 26 1 9691444 373 308556 97751462 492 10-248538.060018 119 9.939982 34 271 9-691668 373 -308332 9751757 492 10-248243 -060089 119 9-939911 33 281 9 691892 373 -308108 9 752052 492 10-247948 -060160 119 9-939840 32 29 19-692115 373 307885 9-752347 491 10-247653 1060232 119 9-939768 31 30 1 9 692339 372 -307661 9-752642 491 10-247358 1060303 119 9-939697130 31 19-692562 372 -307438 9'752937 491 10 247063 1060375 119 9-939625 29 32 1.9692785 372 -307215 9-753231 491 10-246769 -060446 119 9-939554! 28 33' 9693008 371 -306992 9-753526 491 10-246474 -060518 119 9-939482 27 341 9 693231 371 -306769 9-753820 491 10-246180 1060590 119 9-9394101 26 351 9693453'371 *306547 9-754115 490 10-245885 -060661 119 9-939339 25 361 96936761 371 -306324 9-754409 490 10-245591 -060733 119 9-939267 24 37 I 9-6938981 370 306102 9-754703 490 10-245297 -060805 120 199391951 23 38 9-694120' 370 -305880 9-754997 490 10-245003 1060877 120 9.939123 22 39 19-694342 370 -305658 9-755291 490 10-244709 -060948 1209-939052 21 40 9-694564 370 -305436 9-755585 490 10-244415 -061020 120 9-938980 20 41 9-694786 369 -305214 9-755878 489 10-244122 -061092 120 9-938908119 42 9-695007 369 -304993 9-756172 489 10-243828 1061164 120 9-9388361 18 43 9-695229 369 -304771 9-756465 489 10-243535 -061237 120 9-938763 17 44 1 96954501 369 -304550 9-756759 489 10-243241 1061309 120 9-938691 16 45 1 9695671 368 -304329 9-757052 489 10-242948 -061381 120 9-938619 15 46 9695892 368 -304108 9-757345 489 10-242655 -061453 12019-938547 14 47 9-6961131 368 -303887 9-757638 488 10-242362 -06152511209 938475 13 48 9-6963341 368 -303666 9-757931 488 10-242069 1061598 120 9-938402 12 49 9-696554 367 -303446 9-758224 488 10-241776 -061670 121 9-93833011 50 9-696775' 367 303225 9-758517 488 10-241483 -061742 121 9-938258 10 51 9-6969951 367 -303005 9-758810 488 10-241190 -061815 121 9-938185 9 52 9-697215: 367 -302785 9-759102 488 110240898 1061887 121 9-938113 8 53 9-697435 366 302565 9-759395 487 110240605 -061960 121 9-938040 7 54 9-6976541 366 -302346 9-759687 487 10-240313 -062033 121 9-937967 6 55 9-6978741 366 -302126 9-759979 487 10-240021 -062105 121 9-937895 5 56 9-698094 366 *301906 9-760272' 487 10-239728 -062178 121 9-937822 4 57 9-698313' 365 -301687 9-760564 487 110239436 -062251 121 9-937749 3 58 9-698532' 365 *301468 9-760856 487 10-239144 062324 121 9937676 2 59 9-6987511 365 -301249 9-761148 486 10238852 06239611219-937604 1 60 9-698970 365 -301030 9761439 486 10-238561 062469 121 9937531 0 Cosine. t Secant. Cotangent. Tangent. I1 Cosecant. Sine. 60 DEG. Page 71 LOGARITHMIC SINES, ETC. 71 30 nDEG. Sine. Dff; Cosecant. Tngent. Cotngent. Secant. iff Cosine. i 0 9-698970 301030 9-761439 10-238561 -06246 9-937531 60 1 9-699189 364 *300811 9-761731 486 10-238269 1062542 121 9-937458 59 21 9-699407 364 *300593 9-762023 486 10237977 062615 122 9-937385158 3; 9699626 364 -300374 9-762314 486 10-237686 *0626881122 9-937312i57 4 9'699844 364 -300156 9-762606 486 10-237394 -062762 122 9*9372381 56 5 9700062 363 *299938 9-762897 485 10-237103 *062835 122 9-937165 55 6 9700280 363 *299520 9-763188 485 10-236812 -062908 122 9-937092 54 7 9-700498 36f'299702 9-763479 485 10-236521 -062981 122 9-937019 53 8 9-700716 363 -299284 9-763770 485 10-236230 -063054 122 9-936946 52 9 97'00933 363 -299067 9-764061 485 10-235939 063212812 9936872:l51 10 9701151 362 -298849 9-764352 485 10-235648 -063201 122 9-936799 150 11 9701368 362 -298632 9-764643 485 10-235357 063275 122 99936725149 12 9-01585 362 -298415 9-764933 484 10-235067 -063348 122 9-9366;52 48 13 9701802 362 -298198 9-765224 484 10-234776 -063422 123 9.936578 147 14 9-702019 361 297981 9 765514 484 10-234486 063495 123 9-93650!1146 15 9702236 361 -297764 9-765805 484 10-234195 063569 123 9-936431 45 16 9702452 361 -297548 9 766095 484 10-233905 063643 123 9-936357 44 17 9702669 361 *297331 9-766385 484 10-233615 063716 123 9-9362814'43 18 9702885 360 297115 9-766675 483 10-233325 -063790 123 9-936210142 19 9703101 360 *296899 9-766965 483 10-233035 063864 123 9-936136 41 20' 9703317 360 -296(83 9-767255 483 10-232745 063938 123 9-9360621;40 21 9703533 360 -296467 9-767545 483 10-232455 -064012 123 9-935988L 39 22 9'703749 359 *296251 9-767834 483 10-232166 l-064086 123 9-935914138 23 9-703964 359 *296036 9-768124 483 10-231876 1064160 123 9-935840137 24 9 704179 359 -295821 9-768414 482 10-231586 -061234 123 9-93576636 25 9-704395 359 *2956051 9-768703 482 10-231297 -0643081124 9-935692.35 26 9-704610 359 -295390 9-768992 482 10-231008 1064382 124 9-935618 34 27 9-704825 358 *295175 9-769281 482 10-230719 -064457 124 9-93554333 289-705040 358 -294960 9-769570 482 10-230430 *064531'124 9-935469!832 29 9705254 3'8 294746 9~6c8 39531 299 -705254 3058 294746 9-769:860 482 10-230140 *064605124 9.935395:31 30 9705469 358 2945319-770148 481 10-229852 0646801124 9-935320:30 31 9-705683 357 -294317 9-770437 481 10-229563 0647.54 124 9-935246 29 32 9705898 357 -294102 9-770726 481 10-229274 064829 124 9935171128 33 9706112 357 293888l 9771015 481 10-228985 -064903 124 99350971:27 34 9706326 357 -293674 9771303 481 10-228697 064978 12419-935022 26 35 9-706539 356 2934611 9-771592 481 10-228108 065052 124 9-934948 25 36 9-706753 356 293247 9-771880 481 10-228120 -065127 1249934873 24 37 9-706967 356 293033 9-772168 480 10-227832 -065202 124 9-934798 23 38 9707180 356t 292820 9772457 480 10-227543 -065277 125 9)934723 22 39 9-707393 355 -292607 9-772745 480 10-227255 -065351 125!9-934649 21 40 9-707606 355 *292394 9 773033 480 10-226967 -065426 125 9-934574 20 41 9-,707819 355 - 292181 9-773321 480 10-226679 -065501 1259-934499 19 42 9708032 355 291968 9-773608 480 10-226392 0655761125 9-934424 18 43 1 9-708245 354 -291755 9-773896 480 10-226104 -065651 125 9-934349 17 441 9-708458 354 -291542 9-774184 479 10-22581.6 -065726 1259-934274 16 45 9708670 3541 291330 9-774471 479 10-225529 -065801 125 9934199 15 46 9708882 354 291118 9774759 479 10-225241 -065877125 9-934123 14 47 9-709094 353 -290906 9-775046 479 10-224954.065952112519 934048 13 48 9709306 353 -290694 9-775333 479 10-224667 -066027 125 9-933973 12 49 9-709518 353 -290482 9-775621 479 10-224379.066102 125 I 339898 11 50 9709730 853 -290270 9-775908 478 10 224092 1 066178 2619 9,338 22 10 51 9-709941 353 290059 9-776195 478 10-223805 -066253 16 l9933747I 9 152 9-710153 352 *289847 9-776482 478 10223:518 I 066329,12619-933671 i 8 53 9-710364 352 -289636 9776769 478 10-223231 i066404 126'9 93o396' 7 54 9-710575 352 -289425 9- 777055 478 10-222945 I-066480 126 9 933520 6 55 9710786 352 -289214 9-777342 478 10-222658 066555'126 9 933445 5 56, 9710'97 351 -289003 9-7776281 478 10222372 *06i66311269 9033369 4 57 9-711208 351 -288792 9777915! 477 10222085 -.06707 12'6!9 -3293 3 58 9-711419 351 288581 9778201 477 10221799 066, 83 1 1269 (133217 2 591 9 711629 8 351 -288371 9 778 487 1 477 10.221513 o066859 1269 3 141 1 601 9 711839 350 -288161 9.778774 477 10.221226 06693412 9 3306i 0 C)t Cosine.Secant. I Cuiaiget. I TaCge. oseeant. Sine. I G 2 59 DEQ. Page 72 72 LOGARITHMIC SINES, ETC. 31 nEG. Sine. Diff. Cosecant. Tangent. / Cotangent. Secant.. Cosine. 0 9-711839 288161 9-778774 10-221226 066934 9-933066 60 1 9-712050 350 287950 9-779060 477 10-220940 -067010 126 9932990 59 21 9-712260 350 287740 9-779346 477 10-220654 -067086 1279-932914/58 3' 9-712469 350 287531 9-779632 477 10-220368 l067162 127 9-9328381)57 4 9-712679 349 -287321 9-779918 476 10-220082 -067238 127 9.932762! 56 5 9-712889 349 287111 9 780203 476 10-219797 -067315 127 9-932685|,55 6 9-713098 349 286902 9 780489 476 10 219511 067391 127 9 932609:154 7 7 9-713308 349 -286692 9780775 476 10-219225 -067467 127 9-93253353 8 9-713517 349 286483 9-781060 476 10-218940 -067543 127 9932457;152 9 9-713726 348 -286274 9-781346 476 10-218654 -067620 127 9-932380 51 10 9-713935 348 286065 9781631 476 10-218369 1067696 127 9-932304150 11 9-714144 348 285856 9781916 475 10-218084 -067772 127 9-932228:49 12 9-714352 348 285648 9-782201 475 10-217799 0067849 127 9-932151 48 13 9714561 347 285439 9-782486 475 10-217514 -067925 127 9-932075)'47 14 9-714769 347 285231 9.782771 475 10-217229 -068002 128 9-931998 46 i15 9-714978 347 285022 9 -783056 475 10-216944 -068079 128 9-931921 45 16 9-715i86 347 284814 9-783341 475 10-216659 -068155 128 9 931845'44 17 9 715394 347 284606 9-783626 475 1)0216374 068232 128 9931768 43 18 9 715602 346 -284398 9-783910 474 10-216090 -068309 128 9-931691l!42 19 9-715809 346 -284191 9-784195 474 10-215805 -068386 128 9 931614 41'0 9-716017 346 -283983 9-784479 474 10-215521 -068463 128 9-931537i,40 21 9-716224 346 283776 19 784764 474 10'215236 -068540 128 9-931460 39 22 9-716432 345 -283568 9-785048 474 10 214952 -068617 128 9 931383,138 23 9716639 345 -283361 9-785332 474 10-214668 -068694 1281 931306; 37 24 9-716846 345 -283154 9-785616 474 10-214384 -068771 1289- 9312299 36 25 9-717053 345 -282947 9-785900 473 10-214100 -068848 1299 -931152| 35 26 9-717259 345 -282741 9-786184 473 10-213816 068925 129 9931075 34 27 9-717466 344 -282534 9-786468 473 110213532 -069002 129,9 930998/ 33 28 9 717673 344 -282327 9-786752 473 i10213248 -069079 129 9930921 32 299 9 -717879 344 -j282121, 9 -787 0361 473 10-212964.069157 129 9-930843 31 3V 9-718085 344-1 281915 9-787391 473 10-212681 -069234 129 9930766"30 31 l 9- 18 22911 343 -281709 9-7876031 473 10212397 069312 1219 9j 30688 29 32i 93 84971 343 - 2815031l 9-7878861 472 10-212114 -069389 129 9930611128 33' 9718703 343 281297 9-7881701 472 10-211830 -069467 129 9-930633 27 34 i 9718909 343 281091! 9-788453' 472 10211-547 0695441129 99304561j26 35 1 9-719114 343 -280886 9-788736 472 10211264 -069622 129 9-930378 25 36' 9-719320 342 *280680 9'-789019 472 10-210981.-069700 1299-930300'i24 37 9-7195251 342 -280475 9-7893021 472 10-210698 -OG9'77 130 9-9302231 23 38 9-719730 342-280270i 9-789585 472 1021210415 -069855 1301 99301451'22 39 9-7199351 342 -2980065 1 9.7898681 471 10-210132.06993130 9-l9300675 21 1 40 9-720140 341 -279860 9-790151 471 10-209849 1 070011 130 9-929989)|20 41 9-720345 341 279655 97904331 471 10 209567 -070089 130199 9991l| 19 42 9'720549 341 -279451l 9-790716 471 10-2091284 070167 13019-929833,18 43 9-720754 34 1 *279246 9790999 471 10-209001 -070245 130 9-929755) 17 1 4 9720958 340 -279042 9.791281, 471 100208719 11070323 130 9-929677l16 45 9 721162 340 278838 97 915631 471 10-208437 -070401 130 9-929599 15 46 9-7213066 340 278634 9-791846 470 10-208154 -070479 1309 9295215 14 47 9-7215701 340 -278430 9792128 470 10-207872 -070558 130 9929442' 13 48 9-721774 340 -278226 9-792410 470 10-207590 -070636 1309-929364112 49 9-721978 3391 278022 9-792692 470 10-207308 1070714 131 9-929286 l11 50 9722181 39 27819 977929 40 1020 6 070931 207 10 51 9 -722385 339 -277615 9-793256 470 10-206744 |070871 131 9929129' 9 52 1 9722588 339 277412 9-7993538 470 10-206461 2.0700113 139 9929050 8 53 9-722791 339 -277209 9.793815 469 10206181 071028113119 9289721 7 54 9 722994 338 -277006 9-794101 469 10-205899 -071107 13119 928893 6 55 9.723197 338 -276803 9-79438 3 469 /10-205617 1 071185 131 9 928815' 5 56 9-723400 338 -276600 9-794664 469 10-205336 -071264 13119 928736 4 57 9-723603 338 *276397 9.794915 469 10-205055 -071343 131 9 928671 3 58 1 97238051 337 276195 9-795.227 469 10-204773 j 071422 131 9 928578' 2 591 9-724007 337 -275993 9-7935508 469 10-204492 -071501 131 9 9284990 1 60 9-724210 337.275790 9-795789 468 10-204211 -071580 131 9-9284920 0 Cosine. Secant. Cotangent. | Tanrent. Cosecant. Sine. | 58 DEG. Page 73 LOGARITHMIC SINES, ETC. 73 32 DEG. Sine. Di C ec t T t osi.ff. Sine. f Cosecant. Tangent.; Cotangent. Secant. ff Cosine. 0 19-724210 -275790 9-795789 10-204211 071580 9-928420 60 i 1 9-724412 337 *275588 9-796070 468 10-203930'-071658 132 9-928342 159 2 91-724614 337 -275386 9-796351 468 10-203649 -071737 132 9-928263!58 3 9-724816 336 -275184 9-7966321 468 10-203368 -071817 132 9-928183; 57 41 9-725017 336 274983 9-796913 468 10-203087 071896 132 9-928104 56 51 9-725219 336 274781 9-797194 468 10-20 2 92806 0792 52 55 6 9-725420 336 274580 9-.797475 468 10-202525 -072054 13219-927946' 154 7 9725622 335 274378 9-7977551 468 10-202245 -072133 132 9.9278671 53 8 9 725823 335 -274177 9-798036 467 10-201964 -072213 132!9-927787 52 91 9.726024 335 273976 9-798316s 467 10-201684 -072292 13219-927708 51 10 9-726225 335 -273775 9-798596 467 10-201404 i 0723711132 19927629 50 11 9-726426 335 273574 9-7988771 467 10-201123 -072451 132 9-927549 49 12 9-726626 334 -273374 9 799157 467 10-200843!072530 13219-927470 48 13 9-726827 334 273173 9-799437 467 10-200563 1 072610 133!9-927390 047 14 9-727027 334 -272973 9-799717 467 10-2002831 i072690 13319-927310 46 15 9-727228 334 -272772 9'799997 467 10-200003 1.072769 133l9.927231 45 16 9-727428 334 -272572 9-800277 466 10-199723 -1072849 133,9-927151144 17j 9-727628 333 272372 9-800557 466 10-199443 1-072929 133 9-927071 43 18 i 9-727828 333 -272172 9-800836 466 10-199164 1-073009 133 9-926991 42 19 9-728027 333 271973 9-801116 466 10-198884 *073089 133 9-926911 41 20' 9-728227 333 -271773 93-801396 466 10-198604 -073169 133 9-926831!40 211 9-728427 333 271573 9-801675 466 10-198325 |-073249 133 9-926751 39 2211 9-728626 332 -271374 9-801955 466 10-198045 1-07332911339-926671 38 23 9-728825 332 -271175 9-802234 466 10-197766 -073409 133 9-926591 37 24 9729024 332 -270976 9-802513 465 10-197487'073489 1339-926511 36 25a 9-729223 332 270777 9-802792 465 10197208 *073569 134 9926431 35 26! 9-729422 331 *270578 9-803072 465 10-196928 1-073649 134 9926351 34 271 9729621 331 270379 9-803351 465 10-196649 -073730 134'9-926270 33 28 9-729820 331 -270180 9-803630 465 10196370 073810 149-926190 32 291 9-30018 331 -26982 9-803908 465 10-196092 1073890 134 9-926110 31 30' 9-730217 330 *269783 9804187 465 10-195813 11-073971 134 9-926029,30 1 31; 9-730415 330 269585 9.804466 465 10 195534 9074051 134 9-925949il9 i 32 9-730613 330 -269387 9804745 464 10195255 I 074132 134 9.925868 128 33 9 730811 330.2691891 9 805023 464;10-194977 1 -074212 134 9.925788 27 3-4 9-731009 830 -2689'91 9-805302 464 110 194698' 074293 13419-925707, 26 35 9-731206 329 2687941 9805580 464 10-194420 l-0743741l34-'9'256265l' 1 36 9-731404 329 -268596 9-805859' 464 10 194141 i074-551134 l9925545 124 37; 9-731602 329 -2683981 9806137 464'10193863'074535 135 9-925465'23 3879 731799 329 268201 9806415 464:10.193585 074616 135l9'92538419 i 39 9 -31996 329 268004 9-806693 464 10-193307 I 074697 135!9.92530o 21 40 9-732193 328 1267807. 9-806971 4623 10-193029 1 074778 135 9.925222 20 41! 9-732390 328 -267610 / 9807249 463 10 192751 1 074859 135 9-925141 19 42' 9732587 328 -267413 9&8075027 463 10-]92473 p -0749-40 135 9-925060 18 43 i 9-732784 328 -267216 9-807805 463'10-212195 -075021 1359 -924979 17 44 9 732980 328 -2670201 9808083 463 10-11917 -0 5103 135!9-924897 1 45 9 733177 327 266823 9-808361 463 10-191639 -075184 13519 924816 15 46 91-733373 327 266627 9.808638 463 10-191362. 075265 1359-924735 14 47 9-733569 327 266431 9-808916 463 10-191084 -07534611369(-924654 13 48 9-733765 327 *2662351 9-809193 462 10190807 -0754281136!9 924572' 12 49 9-733961 327 -266039; 9-809471 462 10-190529 -075509 136 9-924491 11 50 9-734157 326 2658431 9-809748 462 110-190252 1075591113693 94409 10 51 9-734353 326 -265647 9810025 462'10-189975!075672 136i9 9243281 9 52 9734549 326 265451 9-810302 462'10-189698'075754 1369 9-242461 8 53 9-734744 3 2625256 9-8105801 462 10-189420 i075836 1369-.924164 7 54 9-734939 325 265061 9-810857 462 110189143 107591713619-924083 6 551 9-735135 320 -264865 9-811134 462) 10188866 -075999 136 9-924001 5 561 9-735330 325 264670 9-811410 461 10-188590 -076081 13619-923919 4 571 9-735525 325 264475i 9-811687 461 t10188313 -076163 13619.923837 3 5811 9735719 325 -264281 1 9-811964 461 10-188036 -076245 13619923755 2 59 97 35914 324 -264086 11 9812241 461 10-187759 -07632713719-923673 1 60; 9-736109 324 1263891 Q 9-812517 461 10-187483 1-076409137 9-9235911 0 I Cosine. Secant. i' Cotangent. Tangent. I Cosecant. Sine. 57 DEG. Page 74 74 LOGARITHMIC SINES, ETC. 33 DEG. I --- t' \ Sine. Dif ICosecant Tangent. |oiff; Cotangent. Secant. 1j0 j Cosine. / 0 99-736109 *263891 9-812517 10-187483 -076409 I —-92359160 1 9-736303 324 -263697 9-812794 461 10-187206 076491 137l 992350 59 2 9-736498 324 2635021 9-813070 461 10-1869~30076573 137 9 923427 58 3 9736692 324 263308 9-813347 461 10'186653 076655 137 9-923345157 4 9-736886 323 -263114 9-813623 460 10-186377 *076737 137 9-923263!56 5 9 737080 323 *262920 9-813899 460 10-186101 076819 137 9-923181 55 6 9-737274 323 -262726 9-814175 460 10-185825 076902 13719 923098 54 7 9-737467 323 -262533 9-814452 460 10-185548 076984 137 9923016 153 8 9-737661 323 *262339 1 9814728 460 10-185272 077067 137 9-92293315 9 9-737855 322 *2621451 9-815004 460 10-184996 -077149 137 9-922851 151 10 9-738048 322 *261952 9815279 460 10-184721 077232 137 9-922768 i50 11 9-738241 322 -261759 ~ 9-815555 460 10-184445 *077314138 9-922686 49 12 9-738434 322 -261566 9-815831 460 10-184169 077397 138 9 922603I48 13 9-738627 322 *261373 9-816107 459 10-183893 -077480 13819 922520 47 14 9-738820 321 -261180 9.816382 459 10-183618 077562 13819-9224388146 15 9 -39013 321 -260987 19-816658 459 10-183342 -077645 138!99229355!45 16 9-739206 321.260794 9816933 459 10-183067 077728 138'9 99'222'44 17 9-739398 321 *260602 9-817209 459 10-182791 077811 138 9-922189 43 18 9-739590 321 -260410 19-817484 459 10-182516 -077894138 9-922106' 42 191 9-739783 320 -260217 l 9-817759 459 10-182241 077977 138 9-922023i41 20 97'39975 320 260025 9818035 459 10-181965 078060 138 9-9219401 40 21 9-740167 320 -259833 9 818310 459 10-181690 078143 138 9-921857i 39 22 9-740359 320'259641 9818585 458 10-181415 1078226 1399-921774. 38 23 1 9-740550 320 -259450 9- 818860 458 10-181140 -078309 139 9-921691 37 24 9.740742 319 -259258 9819135 458 10-180865 -078393 189 9-9216071 36 25 9-740934 319 -259066 1 9819410 458 10-180590 -078476 139 9-9215241135 26 1 9 741125 319 -258875 ii 9819684 458 10-180316 -078559 139 9921441134 271 9-741316 319 258684 9-819959 458 10-180041 078643 13919 921357 3'28 9-741508 319 9258492 19-820234 458 10-179766 078726 139!9921 74 32 29 11 9-741699 318 -283011 9-820508 458 10-1794,92 078810 139 99211909 31 30 i 9-741889 318 -258111' 9 820783 458 101792 i 078893 139 991107 30 31 9-742080l 318 -57920 9-8210571 457 10-178943 08977 19I92103 9;j,;i i 4~~ 10~l'iB-132}19. 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Cotangent. I'Tangent. Co-secant.! n Sine. - 56 DEG. Page 75 LOGARITHMIC SINES, ETC. 75 34 DEG. / Sine. DiHf Cosecant. Tangent. Dff Cotangent. Secant. D100ff, Cosine. lot" itt" Cotangent. 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Secant. i Cotangent. I Tangent. Cosecant. Sine. I 55 DEG. Page 76 76 LOGARITHMIC SINES, ETC. 35 DEG. 100"'.i,, DT c, I _i | Sine. lff- Cosecant. Tangent. 100t Cotangent. Secant. Dff. 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Cosine. 01~ 9769219'2830781 i 9-861261 10-138739 092042!9-907958!60 1 9-769393 290 -20607 9-861527 443 10-138473 -092134 153 9-907866 59 2 9-769566 289 230434 1 9 861792 443 10-138208 0092226 153 9-907774 58 31 9769740 289 230260 19-862058 443 10-137942 -092318 153 99907682157 4 9-769913 289 -230087 9-862323 442 101387677 -092410 153 9-907590,56 a 9-770087 289 -229913 9-862589 442 10-137411 -0925021153 9-907498i55 6 9-770260 289 229740 9862854 442 10-137146 092594 153 9-9074061154 9-770433 288 -22967 9-863119 442 10-136881 092686 153 9-907314 53 8 9.770606 288 229394 19 863385 442 10-136615 -092778 154 9.907222152 9 9-770779 288 229 221 9-863650 442 10-136350 -092871 154 9-907129 51 10 9-7709521 288 2290418 9-863915 442 10-136085 -0929631549-9070371150 11 9.771125 288 228875 9-864180 442 10-135820 -09305511549 906945149 12 9771298 288 22870' 9864445 442 10-135555 093148 154 990;852148 131 9 771470 287 -228530 9864710 442 10-135290 093240 154,9 906760 47 14 9'771643 287 2.28357 9-864975 442 10-135025 1093333 154:9-9066671'46 15 9-771815 287 228185 9-865240 442 10-134760 -0934251154 9-9065757;45 16 19-771987 287 228013 9-865505 441 10-134495 093518 154 9-906482!44 17 9-772159 287 227841 9-865770 441 10-134230 -093611 154 9'906389,43 18 9-772331 287 *2276691 9866035 441 10-133965 1093704 155,9-906296 42 19 9-772503 286 227497 9-866300 441 10-133700.093796 155 9-906204 41 20 9-772675 28 -2273251 9866564 441 10-133436 -093889 15519-906111140 21 9-772847 286 -227153 9-866829 441 10-133171 -093982 155 9-90601]839 22 9-773018 286 226982 9 867094 441 10-132906 -094075 155 9,905925 38 23 9-773190 286 226810 9-867358 441 10-132642 -094168 155 99058329137 24 9773361 286 -226639 9-867623 441 10-132377 -094261 155;19905739 i36 25 9 773533 28 5-26467 9-867887 441 10-132113 -094355 15519'905645 35 26 1 9773704 285 226296 9-868152 441 10-131848 -094448 155 9-905552,34 271 9-773875 285 226125 9-868416 441 10-131584 094541 15519'905459l 33 28 9 -774046 285 -225954 9-868680 441 10-131320 -094634 155 9-9053661!32 t29 9 774217 285 225783 9-868945 440 10-131055 -094728 1569-905272'31 30 9-774388 285 225612 9 869209 440 10-130791 -094821 156 9-905179'30 31 9-774558 284 225442 9-869473 440 10-130527 -094915 15619.905085 29 32 9-7747291 284 2252711 9 869737 440 10-130263 -095008 156 9 9049921 28 33 9-774899 284 -225101 9-870001 440 10-1293999 -095102 156 9-904898 27 34 9-775070 284 -2249301 9870265 440 10-129735 -095196 15619-904804 26 35 9-775240 284 2247640 1440 10-129471 095289 156 990471 125 36 9-775410 284 -224590 9870793 440 10-129207 /0935383 1561 9904617i24 37 1 97'75i5801 283 224420 9 871057 440 )10'128943 j0954771156 9'904523; 2 3 38 9.775750 283.224250 9-871321 440 10-128679 1-095571 1561 99044291 22 39 9-775920 283 -2240801 9871585 440 10-128415 -095665 157 9.904335121 40 9-776090 283I -223910 9-8718491 440 10-128151 1095759 157 9-904241;20 411 9.776259 283 -223741 9-872112 440 10-127888 -095853 157 9.904147l19 42 9-776429 283 -223571 9872376 439'10-127624 -095947 157 9-90-1053'18 43 9.776598 282 -223402 9-872640 439 10-127360 11096041 1579 903959'17 44 9.7767668 282 223232 9-872903 439 10-127097 -096136 157 9-903861 16 45 9-776937 282 - 223063 9-873167 439 110126833 -096230 1571 9903770 15 46 9.777106 282 -222894 9873430 439 10-126570.096324 1579 903676' 14 47 9-777275 282 -222725 9873694 439 10-126306 -096419 157 9-903581!113 48 9-777444 281 222556 9 873957 439 -10-126043 109651311571 9903487; 12 49 9-777613 281 -222387 l 9-874220 439 110125780 -096608 157 9-903392 11 50 9-777781 281 2222191 9 874484 439 10-125516 1096702 158 9 903298 10 51 9-777950 281 -222050 9-874747i 439 10-125253 096797 1589 -9032038 9 52 9-778119 281 -221881 1 9875010 439 10-124990 -096892115819-903108 8 53 9-778287 281 -221713 9-875273 439 10-124727'0969861158 9-9030141 7 54 9-778455 280 *221545 9875536 439 10-124464. 097081 15819 902919 6 55 9-778624 280 -221376 9-875800 439 10-124200 -097176 1589 902824' 5 56 9-778792 280 -221208 9-876063 438 1.0-123937 -097271 1589-902729, 4 57 9-7789601280 -221040 9876326 438 10-123674 1097366115819-902634 3 58 9-779128 280 -220872! 9876589 438 10-123411'097461 158 9 9025391 2 59 9-779295 2801 220705 9-876851 438 10-123149 -097556 159 9-902444 1 60 9-779463 279 220537 9-877114 438 10-122886'097651 15919 9023491 0 I |1 Cosine. I Secant., Cotangent. Ta llnrft. I Cosecant. I Sine. j 53 DEG. Page 78 78 LOGARITHMIC SINES, ETC. 37 DEG. I1 Sine. if0 Cosecant. Tangent. Diff. Cotangent. Secant. Df. Cosine. 0' 9-77 9463 *220537 9-877114 10-122886 *09765 51 9.902349 i60 1 9779631 279 220369 9-877377 438 10-122623 -097747 159 9902253'59 2 9-779798 279 -220202 9877640 438 110122360 /097842 159 9.902158r58 3 9-779966 279 220034 9-877903 438 10-122097 9097937 159 9902063 57 4 9-780133 279 -219867 9-878165 438 10-121835 098033 159 9-901967'56 5 9-780300 279 -219700 9-878428 438 10-121572 *098128 1599-901872 55 6 9-780467 278 219533 9-878691 438 110121309 098224 159 9901776 154 7 9780634 278 -219366 9-878953 438 10-121047 098319 159 9901681;153 8 9-780801 278 -219199 9-879216 437 10-120784 -098415 15919-901585 52 9 9-780968 278 -219032 9-879478 437 10-120522 -098510 159 9-901490 51 10 9-781134 278 -218866 9-879741 437 10-120259 -098606 159 9901394' 50 11 9-781301 278 -218699 9-8800031 437 110-119997 098702 160 9-901298 49 12 9-781468 277 -218532 9-880265 437 10-119735 -098798 160 9-901202 48 13 9'781634 277 *218366 9-880528 437 10-119472 *098894160 9-90110647 14'9-781800 277 -218200 9-880790 437 10-119210 *098990 160 9-901010 46 15 9-781966 277 -218034 9-881052 437 10-118948 -09908 160 9-900914 45 16 9.782132 277 -217868 9-881314 437 10-118686 *099182 160 9-900818 44 17 9-782298 277 *217702 9-881576 437 10-118424 -099278 160 9-900722 43 18 9-782464 276 -217536 9881839 437 10-118161 -099374 160 9-900626 42 19 9-782630 276 -217370 9-882101 437 10-117899 -099471 160 9-900529 41 20 9.782796 276 -217204 9-882363 437 10-117637 1099567 160 9-900433 40 21 9-782961 276 -217039 9-882625 437 10-117375 -099663 161 9.900337 39 22 9-783127 27'6 -216873 9-882887 436 10-117113 -099760 161 9.900240 38 233 9.783292 276 -216708 9-883148 436 10-116852 099856 161 9-900144 37 24 9-783458 275 -216542 9-883410 436 10-116590 10999531161 9-900047 36 25 9-783623 275 -216377 9-883672 436 10-116328 -100049 161 9-899951 35 26 9-783788 275 -216212 9-883934 436 10-116066 l 1001461161 9.899854 34 27 9-783953 2'75 -216047 9-884196 436 110115804 -100243 161 9-899757 33 28 9.784118 275 215882 9-8844571 436 10-115543 -100340 161 9-899660 32 29 9-784282 275 215718 9-884719 436 10-115281 -100436 161 9-899564 31 30 9 784447 274.215553 9-884980 436 10-115020 -100533 161 9-899467i'20 31 i 9784612 274 -215388 9-885242 436 10-114758 -100630 162 9-899370;29 32 9.784776 274 215224 9-885503 436 10-114497 -100727 162 9-899273' 8 33 9 784941 274 -215059 9885765 436 10-114235 100824 162 9-899176 127 34 9-785105 274 -214895 9-886026 436 10-113974 -100922 162 9-8990781.26 35 9-785269 274 *214731 9-886288 436 10-113712 -101019 162 9898981125 36 9-785433 273 214567 9-886549 436 10113451 -101116 1629-898884 24 37 9-785597 273.214403 9-886810 436 10-113190 101213 1629-898787 23 38 9785761 273 -214239 9-887072 435 110112928 101.311 162 9-8986891 22 39 9-785925 273 -214075 9-887333 435 10-112667 -101408 162 9898592121 40 9-786089 273 213911 9887594 435 10-112406 l101506 162 98984941 20 41 9-786252 273 -213748 9-887855 435 110.112145 -101603 163 9-898397| 19 42 9-786416 272.213584 9-888116 435 10-111884 -101701 163 9898299 18 43 9.786579 272.213421 9 9 888377 435 110-111623 101798 163989820217 44 9.786742 272 -213258 9888639 435 10-111361.101896163 9-898104 16 45 9-786906 272 213094 9-888900 435 110111100 -101994 163 9-898006 15 46 i 9787069 272 *212931 9889160 435 10-110840 1020921639-897908 14 47 9-787232 272 -212768 9-889421 435 10-110579 102190 163 9897810 13 48 9-787395 271.212605 9-889682 435 1100318 -102288 163 9-897712 12 49 9-787557 271.212443 9-889943 435 10-110057 1102386 16319-897614 11 50 9787720 271 *212280 9-890204 435 10-109796!102484 163 9-897516 10 51 9787883 271 -212117 9-890465 435 110 109535 102582 163 9 897418 9 52 9-788045 271.211955 9-890725 435 10-109275 l102680164 9-8973201 8 53 9-788208 271, 211792 9-890986 434 109109014 1-1027781164 9-897222 7 54 9-788370 271'211630 9-891247 434 10108753 -102877 16419-897123 6 55 9-788532 270 *211468 9-891507 434 110108.493 1102975 164 9-897025 5 56 9-788694 270 -2113061 9891768 434 10-108232 3103074 164 9-8969261 4 57 9.788856 270 *211144 9.892028 434 10-107972 -10317' 164 9-896828 3 58 9-789018 270 -210982 9-892289 434 10-107711 -103271 16419-896729 2 59 9-789180 270 -210820 9-892549 434 10-10745 1-103369 164 9896631 1 60 9-789342 270 -210658 9-892810 434 10-107190 -103468 164 9896532 0 1 Cosine Secan. S Cotangent. Tangent. Cosecant. Sine. 52 DEG. Page 79 LOGARITHMIC SINES, ETC. 79 38 DEG.._ I Sine. DDff; Dff. Sine. Dif Cosecant. Tangent. Diff Cotangent. Secant. iff Cosinb. 0 9-789342 -210658 9-892810 10-107190 -103468 9-896532 60 1 9-789504 269 -210496 9-893070 434 10-106930 103567 164 9-896433 59 2 9-789665 269 *210335 9-893331 434 10-106669 -103665 165 9-896335 58 3 9-789827 269 -210173 9-893591 434 10-106409 -103764 165 9-896236 57 4 9-789988 269 -210012 9-893851 434 10-106149 -103863 165 9-896137 56 5 9-790149 269 -209851 9-894111 434 10-105889 -103962 165 9-896038 55 6 9-790310 269 -209690 9-894371 434 10-105629 -104061 165 9-895939 54 7 9-790471 268 -209529 9-894632 434 10-105368 -104160 165 9-895840 53 8 9-790632 268.209368 9-894892 434 10-105108 -104259 165 9-895741 52 9 9-790793 268.209207 9-895152 433 10-104848 -104359 165 9-895641 51 10 9-790954 268 -209046 9-895412 433 10.104588 -104458 165 9-895542 50 11 9-791115 268.208885 9-895672 433 10-104328 -104557 165 9-895443 49 12 9-791275 268.208725 9-895932 433 10-104068 -104657 166 9 8953431 48 13 9-791436 267.208564 9-896192 433 10-103808 -104756 166 9 895244 47 14 9-791596 267.208404 9-896452 433 10-103548 -104855 166 9-895145 46 15 9-791757 267.208243 9-896712 433 10-103288 *104955 166 9'895045 45 16 9-791917 267.208083 9-896971 433 10-103029 105055t166 9-894945 44 17 9792077 267.207923 9-897231 433 10-102769 1051541166 9-894846 43 18 9792237 267.207763 9-897491 433 10-102509 1052541166 9-894746 42 19 9-792397 266.207603 9-897751 433 10-102249 1053541166 9 894646 41 20 9-792557 266.207443 9-898010 433 10-101990 -1054541166 9-894546 40 21 9792716 266.207284 9898270 433 10-101730 -105554166 9894446 39 22 9-792876 266.207124 9-898530 433 10-101470 -105654 167 9-894346 38 23 9-793035 266.206965 9-898789 433 10-101211 -105754 167 9894246 37 24 9-793195 266.206805 9-899049 433 10-100951 -105854 167 9894146 36 25 9-793354 265.206646 9-899308 432 10-100692 -105954 167 9-894046135 26 9-793514 265.206486 9-899568 432 10-100432 -106054 167 9-893946 34 27 9-793673 265.206327 9-899827 432 10-100173 106154 167 9-893846 33 28 9-793832 265.206168 9-900086 432 10-099914 -106255 167 9-893745 32 29 9-793991 265.206009 9-900346 432 10 09965 —1 106355 167 9-893645 31 30 9-794150 265.205850 9.900605 432 10-099395 -106456 167 9-893544 30 31 9.794308 264.205692 9-900864 432 10-099136 -106556 167 9-893444 29 32 9-794467 264.205533 9-901124 432 110098876 1106657 168 9-893343 28 33 9-794626 264.205374 9-901383 432 10-098617 -1067571168 9 893243 27 34 9-794784 264.205216 9 901642 432 10-098358 -106858 1689-893142 26 35 9-794942 264.205058 9-901901 432 110098099 -106959 168 9-893041 25 36 9-795101 264.204899 9-902160 432 10-097840 1107060 168 9-892940 24 37 9-795259 264.204741 9-902419 432 10-097581 -107161 168 9-892839 23 38 9-795417 263.204583 9-902679 432 10-097321 *107261 168 9892739 22 39 9-795575 263.204425 9-902938 432 10-097062 -107362 168 9-892638 21 40 9-795733 263.204267 9-903197 432 10-096803 -107464 168 9-892536 20 41 9.795891 263.204109 9-903455 432 10-096545 -107565 168 9-892435 19 42 9-796049 263.203951 9-903714 431 10-096286 107666 169 9-892334 18 43 9.796206 263.203794 9-903973 431 10-096027 1107767 169 9-892233 17 44 9-796364 263.203636 9.904232 431 10-095768 1078681169 9.892132 16 45 9796521 262.203479 9-904491 431 110095509 1107970 169 9892030 15 46 9.796679 262.203321 9-904750 431 10-095250 -108071 169 9891929 14 47 9-796836 262.203164 9-905008 431 10-094992 -108173 169 9-891827 13 48 9-796993 262.203007 9-905267 431 10-094733 1108274 169 9891726 12 49 9-797150 262.202850 9-905526 431 10-094474 1108376 169 9-891624 11 50 9-797307 262.202693 9.905784 431 10-094216 11084771169 9891523 10 51 9-797464 261.202536 9.906043 431 10-093957 1108579 169 9-891421 9 52 9-797621 261.202379 9-906302 431 110093698 1108681 170 9 891319 8 53 9.797777 261 ~202223 9-906560 431 110093440 1108783 170 9-891217 7 54 9797934 261 *202066 9-906819 431 10-093181 11088851170 9-891115 6 55 9.798091 261.201909 9-907077 431 10-092923 1108987 170 9.8910131 5 56 9-798247 261.201753 9'907336 431 10-092664 -109089 170 9-890911 4 57 9-798403 261 *201597 9-907594 431 10-092406 109191 170 9-890809 3 58 9-798560 260 -201440 9-907852 431 110092148 1109293 170 9-890707 2 59 9798716 260 *201284 9-908111 431 10-091889 1109395 1709 890605 1 60 9798872 260.201128 9-908369 431 10091631'1094971170 9890503 0 Cosine. Secant. Cotangent. Tangent. Cosecant. Sine. j 51 DEG. Page 80 80 LOGARITHMIC SINES, ETC. 39 DE3. Sine. D,ff Cosecant. Tangent. 10ff; Cotangent. Secant. Di0 Cosine. 0 9-798872 201128 9-908369 10-091631 109497 98908f50;',!,60 1 9-799028 260 -200972 9-908628 430 10-091372 -109600 1 70,9-890- 0i: 59 2 9-799184 260 *200816 9-908886 430 10-091114 -109702 17119-8902)j58-, 3 9-799339 260 *200661 9-909144 430 10-090856 -109805 171 9-890195J 57 4 9-799495 259 *200505 9-909402 430 10-090598 -109907 171 9-890093 56 5 9-799651 259 *200349 9-909660 430 10-090340 -110010 171 9-889990 55 6 9-799806 259 -200194 9-909918 430 10-090082 110112 171 9 889888 54' 7 9-799962 259 -200038 9-910177 430 10-089823 110215 17119-889785005 8 9-800117 259 *1998838 9-910435 430 10-089565 110318 171l9'889682;'52 9 9-800272 259 -199728 9-910693 430 10-089307 110421 1719-889579 151 10 9-800427 258 *199573 9-910951 430 10-089049.110523 1719.889'4771 50 11 9-800582 258' -199418 9-911209 430 10-088791 -110626 171,9 889374 49 12 9-800737 258 199263 9-911467 430 10-088533 -110729 172'9-889271 l48 13 9-800892 258 *-1991081 9-911724 430 10-088276 110832 172i9.889168!47 14 9-801047 258 198953 9-911982 430 10-088018 110936 172 9-889064 46 15 9-801201 258' 198799 9-912240 430 10-087760 -111039 172 9-888961 45 16 9-801356 258'-198644 1 9912498 430 10-087502 -111142 17219.888858 44 17 9-801511 257 -198489 9-912756 430 10-087244 -111245 17219-88875:513 18 9-801665 257 198335 9-913014 430 10-086986 -111349 17219-888651 42 19 9.801819 257 *198181 9-913271 430 10 086729 -111452 172i9-888548 41 20 9-801973 257 198027 9-913529 429 10-086471 1.111556 172 9-8884441;40 21 9-802128 257 *197872 9-913787 429 10-086213 1.11659 173 9-888341; 39 22 9-802282 257 1977181 9914044 429 10-085956 111763 173 9-888237138 23 9-802436 256 197564 9-914302 429 10-085698 111866 173 9-888134! 37 24 9-802589 256 197411 9-914560 429 10-085440 -11197011739-888030,36 25 9-802743 256 197257 9-914817 429 10-085183 112074 173 9-887926! 35 26! 9 802897 256 197103 9-915075 429 10-084925'112178 173 9-887822 134 27 9-803050 256 196950 9-915332 429 10-084668 -112282 173 9-887718'33 28 9803204 256 196796 9-915590 429 10-084410 *1123861173!9-887614 i32 29 9-803357 256 196643 9-915847 429 10-084153 -112490 17319-887510 31 30 9 803511 255 -196489 1 9916104 429 10-083896 112594 17319.887406.:30 31 9-803664 255 -196336 119-916362 429 10-083638 -112698 174|9-8873002i'29 32 9-803817 255 -196183 9-916619 429 10-083381 -112802 174 9-887198 28 33 9-803970 255 196030 9916877 429 10-083123 -112907 174 9-88709, 7 34 9-804123 255 -195877 9-917134 429 10-082866 -113011 174 9-886989'26 35 9-804276 255 -195724 9-917391 429 10-082609 -113115 174 9-88688 J5 | 36 9-804428 254 *195572 9917648 429 10082352 113220 174 9'8867801 24 37 9-804581 254 *195419 9-917905 429 10-082095 113324 174 9 88667" 2O. 38 9-804734 254 -195266 9-918163 429 10-081837 113429 174 9-886o571 22 39 9-804886 254 -195114 9 9184120 429 10-081580 -11334 174 9-886466.21 40 9-805039 254 -194961 9-918677 429 10-081323 *-113638 174 9-886362 1'0 41 9-805191 254 -194809 9-918934 429 10-081066 113743 173o9 88625; 519 42 9-805349 254 -194657 9-919191 428 10-080809 -1138148 175|9-8861.52!18 43 9-805495 253 -194505 9-919448 428 10-080552 -11395o l7 59 88604 17 44 9-805647 253 194353 9919705 428 10-080295 -11405o817519-88594 16 45 i 9-8057991 253 -194201 9-919962 428 10-080038 114163 1759.88583 7i15 46 E9-805951 253 -194049.9-920219 428 10-079781-114268 175,9-885732!14 47 9-806103 253 -193897 9-920476 428 10-079524 114373 17519 885627 113 48 9-806254 253 -193746 9-920733 428 10-079267 -114478 1759 -885522)!1') 49 9-806406 253 -193594 9-920990 428 10-079010 *114584 17519-885416 11| 50 9-806557 252 -193443 9-921247 428 10-078753 114689;17519-8853 1110 51 9-806709 252 -193291 9-921503 428 10-078497 -114795o176 9-8852051 9 62 9-806860 252| 193140 9-921760 428 10-078240 11490001769-885100 8 53 9-807011 252 -192989 9-922017 428 10-077983 -115006 176 98849941 7 54 9-807163 252 *192837 9-922274 426 i10077726 -115111 176 9-884889 6 55 i 9807314 252 -192686 9-922530 428 10-077470 -115217 176 198847831 5 56 9-807465 252 -192535 1 9.922787 428 10-077213 1115323 176|9-884677' 4 57 9-807615 251 *192385 9-923044 428 10-076956 -115428 1769-884o572 3 58 1 9-807766 251 -192234 9-923300 428 10-076700 -115534 176 9-884466 2 59 9-807917 251 *192083 9-923557 428 110-076443 -11564011769-884360' 1, 60 9-808067 251 -191933 9-923813 428 110-076187 *11574617619-884254|l 0 IlI Cosine. 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Secnt Coie I 0 9-825511 -174489 9954437 10-045563 128927 9-871073L60 1 9-825651 234 -174349 9-954691 423 10-045309 -129040 190 9-870960 59 2 9-825791 233 -174209 9-954945 423 10-045055 -129154 190 9-870846 58 3 9-825931 233 174069 9-955200 423 10-044800 -129268 190 9-870732 57 4 9-826071 233 -173929 9-955454 423 10-044546 *129382 190 9-870618 56 5 9-826211 233 -173789 9-955707 423 10-044293 *129496 190 9-87050455 6 9-826351 233 -173649 9-955961 423 10-044039 -129610 190 9-870390154 7 9-826491 233 173509 9-956215 423 10-043785 -129724 190 9-870276 53 8 9-826631 233 -173369 9-956469 423 10-043531 -1298391190 9-870161L52 9 9-826770 233 -173230 9-956723 423 10-043277 *129953 190 9870047 51 10 9-826910 232 -173090 9-956977 423 10-043023 -130067 191 9-869933 50 11 9-827-049 232 -172951 9-957231 423 10-042769 -130182 191 9-869818 49 12 9-827189 232 -172811 9-957485 423 10-042515 /130296 191 9-869704'48 13 9-827328 232 172672 9-957739 423 10-042261 -130411 191 9.869589147 14 9-827467 232 172533 9957993 423 10-042007 -130526 191 9-869474'46 15 9-827606 232 -172394 9-958246 423 10-041754 -130640 191 9-8693601 45 16 9-827745 232 -172255 9958500 423 10-041500 -130755 191 9-869245 44 17 9-827884 232 -172116 9-958754 423 10-041246 1130870 191 9-869130 43 18 9-8280231231 171977 9959008 423 10-040992 -130985 191 9-869015142 19, 9-8281621231 -171838 9-959262 423 10-040738 -131100 192 9-868900 41 201 9828301 231'171699 9959516 423 10-040484 -131215 192 9-8687851 40 21 9-828439 231 -171561 9-959769 423 10-040231 1131330 192 9-868670139 22 9-828578 231 -171422 9 960023 423 10-039977 -131445 192 9-868555?38 23 9-828716 231 -171284 9-960277 423 10-039723 -131560 192 9-868440'37 24 9-8288551231 -171145 9-960531 423 10-039469 -131676 192 9-868324 36 25i 9-828993 230 -171007 9-960784 423 10-039216 -131791 192 9-868209i 35 26 9-8291311 230 -170869 9-961038 423 10-038962 -131907 19219-868093134 27 98292691230'170731 9-961291 423 10-038709 -132022 192 9-867978, 33 28 9-829407 230 -170593 9-961545 423 10-038455 -132138 193 9'867862' 32 29 I 9829545 230 -170455 9-961799 423 10-038201 -132253 193 9-8677471 31 30 9-8296831 230 170317 9-962052 423 10-037948 -132369 193 9-8676311,20 31 9-829821 230'170179 9962306 423 10-037694 1132485 193 9-867515 29 32 9-829959 229 -170041 9962560 423 10 037440l 132601 193 9 867399 28 331 9-8300971 229 -169903 9962813 423 10-037187 -1327171193 9-867283 27 34 9-830234 229 -169766 9-963067 423 10-036933 -132833/193 9-867167 26 35 9-830372 229 -169628 9963320 423 10-036680 -132949 193 9-867051 125 36 i 9-830509 229 -169491 9-963574 423 10-036426 -133065 193 9-866935 24 37 1 9830646 229 -169354 9963827 423 10-036173 -133181 19419-866819123 38 9-830784 229 -169216 9-964081 423 10-035919 -133297 194 9-866703 22 39 19-830921 229 -169079 9964335 423 10-035665 -133414 19419-866586121 40 9831058 228 -168942 9-964588 423 10-035412 -133530 194 9 866470 20 41 9831195 228 -168805 9964842 423 10-035158'133647 194 9-866353119 42 9-8313321228 -168668 9965095 422 10034905 -133763 19419-866237 18 4311 9-8314691228 168531 9965349 422 10-034651 133880 194 19866120117 44 9831.606 228 -168394 9-965602 422 10-034398 -133996119419866004116 45 9-831742 228 -168258 9965855 422 10-034145 -134113119519-865887 15 46 9831879 228 -168121 9-966109 422 10-033891 -134230 195 9-865770114 47i 9832015 228 -167985 9966362 422 10-033638 -134347 195 9-865653 13 48 9832152 227 -167848 9-966616 422 10-033384 1134464 195 9-865536 12 49 9832288 227 -167712 9-966869 422 10-033131 -134581 19519-865419111 50 9-832425 227 -167575 9-967123 422 10-032877 -134698 195 9 865302 10 51 9-832561 227'167439 9967376 422 10-032624 -134815 195 9-865185 9 52 9832697 227 -167303 9967629 422 10-032371 -134932 1959-865068 8 53 9-832833 227 -167167 9967883 422 10-032117 -135050 195 9864950 7 54 9-8329691227 -167031 9-9681361422 10-031864'135167 1959-864833 6 55 9-833105 226 *166895 9-968389 422 10-031611'135284 196 9864716 5 56 9833241 226 *166759 9-968643 422 10-031357 -135402 196 9-864598 4 57 9833377 226 -166623 9-968896 422 10-031104 -135519 196 9-864481 3 58 j 9833512 226'166488 9-969149 422 110030851'135637 196 9-864363 2 59 1 9833648 226 -166352 9-969403 422 10-030597 -135755 196 9-864245 1 60 9833783 226 -166217 9-969656 422 10-030344'135873 196 9-864127 0 Cosine. ISecant. II Cotangent. Tangent. 1' Cosecant. _ Sine. 87 47 DEG. Page 84 84 LOGARITHMIC SINES, ETC. 43 DEG.. IDiff. n Diff. ff I Sine. 1ff Cosecant. Tangent. D10t Cotangent. Secant. Di, Cosine. 0 9-833783 -166217 9-969656 10-030344 -135873 9-864127 60 1 9-833919 226 -166081 9-969909 422 10-030091 -135990 196 9-864010'59 2 9-834054 225 *165946 9-970162 422 10-029838 -136108 196 9-863892 58 3 9-834189 225 -165811 9-970416 422 10-029584 -136226 197 9-863774 57 4 9-834325 225 -165675 9-970669 422 10-029331 -136344 197 9863656 56 5 9-834460 225 -165540 9-970922 422 10-029078 -136462 197 9863538 55 6 9-834595 225 -165405 9-971175 422 10-028825 -136581 197 9-863419 54 7 9-834730 225 -165270 9-971429 422 10-028571 -136699 197 9-863301 53 8 9-834865 225 *165135 9-971682 422 10-028318 -136817 197 9863183' 52 9 9-834999 225.165001 9-971935 422 10-028065 *136936 197 9-863064 51 10 9-835134 224.164866 9-972188 422 10027812 -137054 197 9-86294650 11 9-835269 224.164731 9-972441 422 10-027559 -137173 198 9-862827 49 12 9-835403 224.164597 9-972694 422 10-027306 -137291 198 9 862709' 48 13 9-835538 224.164462 9-972948 422 10-027052 -137410 198 9-862590 47 14 9-835672 224.164328 9-973201 422 10-026799 -137529.198 9-862471 46 15 9-835807 224.164193 9-973454 422 10-026546 -137647 198 9-862353 45 16 9-835941 224.164059 9-973707 422 10-026293 -137766 198 9-862234 44 17 9-836075 224.163925 9-973960 422 10-026040 -137885 198 9-862115 43 18 9-836209 223.163791 9-974213 422 10-025787 -138004 198 9 861996 42 19 9-836343 223.163657 9-974466 422 10-025534 -138123 198 9-861877 41 20' 9836477 223.163523 9-974719 422 10-025281 -138242 198 9-861758 40 21 9-836611 223.163389 9-974973 422 10-025027 1138362 199 9-861638 39 22 9-836745 223.163255 9-975226 422 10-024774 -138481 199 9-861519 38 23 9-836878 223.163122 9-975479 422 10-024521 138600 199 9-861400 37 24 9-837012 223.162988 9-975732 422 10-024268 *138720 199 9-861280 36 25 9-837146 222.162854 9-975985 422 10-024015 -138839 199 9-861161 35 26 9-837279 222.162721 9-976238 422 10-023762 -138959 199 9-861041 34 27 9-837412 222.162588 9-976491 422 10-023509 -139078 199 9-860922133 28 9-837546 222.162454 9-976744 422 10-023256 -139198 199 9-860802132 29 9-837679 222.162321 9-976997 422 10-023003 1&39318 199 9-860682 31 30 9-837812 222.162188 9-977250 422 10-022750 -139438 200 9-860562 30 31 9-837945 222.162055 9-977503 422 10-022497 -139558 200 9-860442129 32 9-838078 222.161922 9-977756 422 10-022244 1139678 200 9-860322'128 33 9-838211 221.161789 9-978009 422 10-021991 -139798 200 9-8602021 27 34 9-838344 221.161656 9-978262 422 10-021738 -139918 200 9-860082 26 35 9-838477 221.161523 9-978515 422 10-021485 -140038 200 9-859962125 36 9-838610 221.161390 9-978768 422 10-021232 -140158 200 9-859842 24 37 9-838742 221.161258 9-979021 422 10-020979 -140279 200 9-859721 23 38 9-838875 221.161125 9-979274 422 10-020726 -140399 201 9-859601122 39 9-839007 221.160993 9-979527 422 10-020473 -140520 201 9-859480 21 40 9-839140 221.160860 9-979780 422 10-020220 -140640 201 9859360 20 41 9-839272 220.160728 9-980033 422 10-019967 -140761 201 9 859239'19 42 9-839404 220.160596 9-980286 422 10-019714 -140881 201 9-859119 18 43 9-839536 220.160464 9-980538 422 10-019462 *141002 201 9-858998117 44 1 9839668 220.160332 9-980791 422 10-019209 -141123 201 9-8588771116 45 9-839800 220.160200 9-981044 422 10-018956 -141244 201 985875615 46 9-839932 220.160068 9-981297 422 10-018703 -141365 202 9-858635114 47 | 9-8400641 220.159936 9-981550 422 10-018450 -141486 202 9-8585141 13 48 9-840196 219.159804 9-981803 422 10-018197 -141607 202 9-858393| 12 49 9840328 219 -159672 9-982056 422 10-017944 -141728 202 9-858272 11 50 9-840459 219.159541 9-982309 422 10-017691 -141849 202 9858151 10 51 9-840591 219.159409 9-982562 421 10-017438 -141971 202 9-8580291 9 52 9-840722 219.159278 9-982814 421 10-017186 -142092 202 9-857908 8 53 9-840854 219.159146 9-983067 421 10-016933 1142214 202 9.857786 7 54 9-840985 219.159015 9-983320 421 10-016680 1142335 202 9-8576651 6 55 9-841116 219.158884 9-983573 421 10-016427 -142457 203 9-8575431 5 56 9-841247 218.158753 9-983826 421 10-016174 -142578 203 9-857422 4 57 9-841378 218.158622 9-984079 421 10-015921 -142700 203 9-857300 3 58 9.841509 218.158491 9-984331 421 110015669 -142822203 9.857178 2 59 9.841640 218.158360 9-984584 421 10-015416 -142944 203 9-8570561 1 60 9-841771 218 -158229 9.984837 421 10-015163 -143066 203 9-8569341 0 C/osine. Secant. I Cotangent. I Tangent. I Cosecant. Sine. I 46 DEG. Page 85 LOGARITHIMIC SINES, ETC. 85 44 DEG., Sine. Dif Cosecant. Tngent Cotangent. Secant Cosine. 0 9-841771 -15829 i' 9-9848a37 10-015163 -143066( 9-856934 60 1 9-841902 218 15808 9-985090 421 10-014910 -143188 203 9 856812 59 2 9-842033 218 I-15796t |I 9985343' 421 10-014657 -143310 203 8566901 58 3 9842163 218 157837 9-985596 421 10-014404 *143432 2049-856568 57 4 9-842294 217 1-1577061 9-985848 421 10-014152 -143554 204i9-86"46 56 56 5 9-842424 217 -157576 9-986101 421 10013899 1143677 204 9-856323 55 6 9-842555 217 -157445 9-986354 421 10-013646 -143799 04 9 856201 54 7 9-842685 217 -1515 73 9-986607 421 10-018393 143922 904 9 856078 53 8 9-842815 217 1577185 9-986860 421 10-013140 114404 204 92855956 52 9 9842946 217 -157054 9-987112 421 10012888 -144167 204 9-856833151 10 9-843076 217 156924 9-987365 421 10-012635 -144289 2049-855711 50 11 9-843206 217 -156794 1 9-987618 421 10012382 -144412 205 9-8555881 49 12 9-843336 216 -156664 9-987871 421 10-012129 1445351205 9-855465 48 13 9-843466 2161 156534 9-988123 421 10-011877 -1446582059-855342947 14 9-843595 216 -156405 9-988376 421 10-011624 -1447 81 205 98552121 46 15 9-843725 216 -156275 9-988629 421 10-011371 -144904 205 9-8550965 45 16 9-843855 216 156145 9.988882 421 10011118.145027 205 985497344 17 9-843984 216 -156016 9-989134 421 10-010866 -145150 2059-85485043 18 9-844114 216 -155886 9 989387 421 10-010613 -145273 205 9-854727'42 19 9-844243 216 -155757 9-989640 421 10-010360 -145397 206 9-854603 41 20 9-84372 215 -155628 9-989893 421 10010107 -145520 206 9854480140 21 9 844502 215 -1.55498 11 9-990145 421 10-009855 -145644 206 9-854356139 22 9-844631 215 -155369' 9-990398 421 10-009602 -145767 206 9-854233 i38 23 9844760 216 -155240 9-990651 421 10 009349 -145891 206 9-854109l'37 24 9-844889 215 -155111;! 9-990903 421 10-000907 -146014 206 9'853986 036 25 9-845018 215 -15498 2 9-991156 421 10-008844 -146138 206 9-85386273; 26 9 845147 215 -154853 9991409! 421 10008591 -146262 206 9-853738 34 27 9845276 215 -154724 1 99916692 421 10-008338 -146386 206 9-853614133 28 9845405 214 -154.595 99919141 421 10-008086 -146510 207 9-853490 32 29 9-845533 214 -154467 1 9-9921671 421 10-007833 146634 207 9-8533661 31 30 9-845662 214 -154338 9-992420 421 10-007580 -146758 207 9-853242l130 31 9-845790 214 -154210 1 9 992672 421 10-007328 -1468821207 9-853118899 32 9-845919 214 -154081 I 9 992925 421 10-007075 114700k6207 9-852994l8 33 19-8460471 214 1153953 9 993178 421 10 006822 1-1471311207 9-852869 27 34 9-846175 214 - 153825 9-993430 421 10-006570 1147255 207 9-852745 926 85 9846304 214 -1536961 9-993683 421 10-006317 -147380 207 9-852620; 25 36 9.846432 214 153568 9993936 421 10-006064 -147504 207 9'85249624 37 9-846560 213 153440 9-994189 421 10-005811 -147629 208 9-8523710923 38 9 846688 213 -153312 9 994441 421 10-005559 1147753j20809 852247i 22 39 9-846816 213 -153184 9-994694 421 10-005306 -147878|208 9-8521229 21 40 9 846944 213 153056 9-994947 421110005053 1-148003|208 9 8519971 20 41 9 -847071; 213 -152929 9-995199 421 10-004801 *1481281208 9-851872 119 42 9-847199 213 -152801 9-995452 421 10-004548 -148253 208 9-851747'l8 43 9-8473271 213 152673 9-9957051 421 10-004295 1148378 208 9-8516221'17 44 9-847454'213 -152546 9-9959571 421 10-004043.148503 208 9.851497' 1 45 9 847582 212 -152418 9-996210 421 10-003790 l-148628 209 9-851372 15 46 l 9-847709 212 -1522911 9-996463' 421 10-003537 *148754 209!9-851246114 47 9847836 212 -152164 9-996715 421 10-003285 -148879 209|9-851121 13 48 19-847964 212 -152036 993969681421 10003032 1-149004 20919-850996l12 49 19-848091 212 -151909 9-997221 421 10-002779 -149130'209l9-8508701ll 50 9-848218,212 -1517821 9-997473 421 10-00-2527 -149265 20919-850745l10 51 9-848345 212 -15160551 9-997726 421 10-002274 -149381 209;9-850619 9 52 9-848472 212 1515281 9-997979 4211000202 1-149507 20999-850493 8 53 19-848599 211 -151401 9-998231 421 10-001769 -149632 2109-850368 7 54 9-848726'211 -151274 9-998484 421 10-001516'-149758'210 9-850242 6 55l 9-848852 211 -151148' 9-998737 421 10-001263 1.149884210 9-850116 5 56 9-848979 211 -151021' 9-998989 421 10-001011 -1500107210 9-849901 4 57 9-849106 211.150894 9-999242421 10000758 150136 10 9-849864 3 581 9-849232 211 -150768 9-9994956 421 10-000505 11502621210 9-8497381 2 59 9-849359 211 -1506411 9-9997471421 10-000253 1-150389210 9-849611 1 60 9-849485 211 1505105 10-0000000 421 10-000000 -150515 2101 98494851 0 i i Cosine.' Secant. I1Cotangent. I Tangent. Cosecant.!' Sine. 45 DEG. Page [unnumbered] Page 583 INDEX. ABBREVIATION of the reduction of decimals,17. Arithmetical proportion and progression, 35 Abrasion, limits of, 301. to 38. Absolute resistances, 288. Ascent of smoke and heated air in chimneys, Absolute strength of cylindrical columns, 274. 208. Accelerated motion, 386. Atmospheres, elastic force of steam in, 195, Accelerated motion of wheel and axle, 419. 196. Acceleration, 415. Atmospheric air, weight of, 356. Acceleration and mass, 422. Average specific gravity of timber, 396. Actual and nominal horse power, 240. Avoirdupois weight, 6. Addition of decimals, 22. Axle and wheel, 417. Addition of fractions, 20. Axle of locomotive engine, 168, 169. Adhesion, 297. Axle-ends or gudgeons, 301. Air, expansion of, by heat, 173. Axles, friction of, 298, 300. Air that passes through the fire for each horse power of the engine, 210. BALLS of cast iron, 407. Air, water, and mercury, 355. Bands, ropes, &c., 267. Air-pump, 254. Bar iron, 400. Air-pump, diameter of,-eye of air-pump cross Beam, 151. head, 145. Beam, the strongest, 276. Air-pump machinery, dimensions of several Bearings of water wheels, 285. parts of, 144. Bearings or journals for shafts of various Air-pump strap at and below cutter, 147. diameters, 287. Air-pump studs, 144. Beaters of threshing machine, 445. Ale and beer measure, 8. Before and behind the piston, 232. Algebra and arithmetic, characters usedin, 12. Blast pipe, 171. Algebraic quantities, 134. Blistered steel, 281. Alloys, strength of, 287. Blocks, cords, ropes, shelves, 428. Ambiguous cases in spherical trigonometry, Bodies, cohesive power of, 175. 381. Bodies moving in fluids, 324. Amount of effective power produced by steam, Boiler, 171. 266. Boiler plate, experiments on, at high tempeAnchor rings, 90. ratures, 220. Angle iron, 91, 408, 409, 410. Boiler plates, 403. Angles of windmill sails, 445. Boilers, 256 and 257. Angles, measurement of, by compasses only, Boilers of copper and iron, diminution of 382. the strength of, 219. Angular magnitudes, 359. Boilers, properties of, 215. Angular magnitudes, how measured, 373. Boilers, strength of, 218. Angular velocity, 412. Bolts and nuts, 406. Apothecaries' weight, 6. Bolts, screw and rivet, 220. Apparent motion of the stars, 353. Boring iron, 445. Application of logarithms, 334. Bossut and Michelloti, experiments on the Approximating rule to find the area of a seg- discharge of water, 319. ment of a circle, 67. Boyle of Cork, 200. Approximations for facilitating calculations, Bramah's press, 427. 55. Branch steam-pipe, 148. Arc of a circle, to find, 49. Brass, copper, iron, properties of, 280. Arc of one minute, to find the length of, 361. Brass, round and square, 408. Are, the length of which is equal to the ra- Breast wheels, 328. dius, 357. Breast and overshot wheels, maximum veArchitecture, naval, 453. locity of, 443. Arcs, circular, to find the lengths of, 68. Buckets and shrouding of water wheels, 446. Area of segment and sector of a circle, 51. Building, to support with cast iron columns, Area of steam passages, 220. 293. Areas of circles, 57. Bushel, 5. Areas of segments and zones of circles, 6-, Butt for air-pump, 146. 65, 66, 67. Butt, thickness and breadth of, 143. Arithmetic, 10. Butt, to find the breadth of, 141. Arithmetical progression, to find the square Byrne's logarithmic discovery, 340. root of numbers in, 126. Byrne's theory of the strength of materials, Arithmetical solution of plane triangles, 366. 272. 583 Page 584 584 INDEX. CALCULATION in the art of ship-building, 453 Crank axle of locomotive, 169. to 494. Crank pin, 170, 252. Calculation of Friction, 267. Crank pin journal, 252. Carriages, motion of, on inclined planes, 429. Crank pin journal, to find the diameter of, 139. Carriages travelling on ordinary roads, 307. Crank pin journal, to find the length of, 139. Carrier or intermediate wheels, 434. Cross head, 252. Carts on ordinary roads, 311. Cross head, to find the breadth of eye of, 139. Cases in plane trigonometry, 363. Cross head, to find the depth of eye of, 139. Cast iron, 174. Cross multiplication, 27. Cast iron pipes, 404. Cross tail, 253. Centre of effort, 483. Cube, 79. Centre of gravity, 175. Cube and cube roots of numbers, 100 to 116. Centre of gravity of displacement of a ship, Cube root of numbers containing decimals, 456, 457, 458. 128. Centre of gyration, 180. Cube root, to extract, 32. Centre of oscillation, 187. Cubes, 397 to 400. Centres of bodies, 386. Cubes, to extend the table of, 128. Centres of gravity, gyration, percussion os- Curve, to find the length of, by construction,72. cillation, 391. Curves, to find the areas of, 453. Centripetal and centrifugal forces, 178, 450. Cuttings and embankments, 97. Chain bridge, 412. Cylinder side rods at ends, to find the diameChimney, 171, 208, 257. ter of, i43. Chimney, size of, 212. Cylinders, 80, 397 to 400. Chimney, to what height it may be carried Cylinders of cast iron, 404. with safety, 212. Circle, calculations respecting, 48, 49, 50, 53. DAMrs inclined to the horizon, 316. Circle of gyration in water wheels, 444. Decimal approximations for facilitating calCircles, 57 to 61. culations, 55. Circles, areas of, 57 to 63. Decimal equivalents, 56. Circular arcs, 68. Decimal fractions, 22. Circular motion, 422. Decimal fractions, table of, 73. Circular parts of spherical triangles, 375. Decimals, addition of, 22. Circumference of a circle to radius 1, 361. Decimals, division of, 24. Circumferences of circles, 57. Decimals, multiplication of, 23. Cloth measure, 7. Decimals, reduction of, 25, 26. Coefficient of efflux, 314. Decimals, rule of three in, 27. Coefficients of friction, 299. Decimals, subtraction of, 23. Cohesive strength of bodies, how to find, 281. Deflection of beams, 295. Collision of railway trains, 452. Deflection of rectangular beams, 294. Columns, comparative strength of, 294. Depth of web at the centre of main beam, 150. Combinations of algebraic quantities, 134. Destructive effects produced by carriages on Common fractions, 15. roads, 311. Common materials, 280. Devlin's oil, 297. Complementary and supplementary arcs, 374. Diagram of a curve of sectional areas, 460. Compound proportion, 14. Diagram of indicator, 265. Condenser, 226. Diameter of cylinder, 251. Condensing water, 223. Diameter of main centre journal, 143. Conduit pipes, discharge by, 322. Diameter of plain part of crank axle, 169. Cone, 82. Diameter of the outside bearings of the crank Conical pendulum, 185 to 187. axle, 168. Connecting rod, 140, 141, 253. Diameters of wheels at their pitch circle to Continuous circular motion, 432. contain a required number of teeth, 436. Contraction by efflux, 316. Dimensions of the several parts of furnaces Contraction of the fluid vein, 313. and boilers, 254. Contractions in the calculation of loga- Direct method to calculate the logarithm of rithms, 348. any number, 346. Copper boilers, 219. Direct strain, 278. Copper, iron, and lead, 405. Discharge by compound tubes, 321. Cosine, to find, 361. Discharge by different apertures from differCosines, contangents, &c., for every degree ent heads of water, 318. and minute in the quadrant, 540 to 576. Discharge of water, 446. Cosines, natural, 411. Discharges from orifices, 426. Cover on the exhausting side of the valve, Displacement of a ship when treated as a in parts of the length of stroke, 231. floating body, 455. Cover on the steam side, 226. Displacement of ships, by vertical and horiCrane, 427. zontal sections, 460, 494. Crane, sustaining weight of, 285. Distance of the piston from the end of its Crank at paddle centre, 135. stroke, when the exhausting port is shut Crank axle, diameter of the outside bearings and when it is open, 231. of, 168. Distances, how to measure, 369. Page 585 INDEX. 585 Division by logarithms, 336, Force of steam, 188. Dodecaedron, 89. Forces, centrifugal and centripetal, 178, 450. Double acting engines, rods of, 250. Fore and after bodies of immersion, 456, 460. Double position, 44. Form, the strongest, 275. Double table of ordinates, 457. Formulas for the strength of various parts Drainage of water through pipes, 325. of marine engines, 251. Dr. Dalton, and his countryman, Dr. Young, Formulas to find the three angles of a spheof Dublin, rical triangle when the three sides are Drums, 422. given, 385. Drums in continuous circular motion, 432. Formula, very useful, 271. Dry or corn measure, 8. Fourth and fifth power of numbers, 129. Duodecimals, 27. Fractions, common, 15. Dutch sails of windmills, 333. Fractions, reduction of, 16, 17, 18, 19. D. valves, 233. Fractions, addition of, 20. Dynamometer, used to measure force, 269. Fractions, subtraction of, 21. Fractions, multiplication of, 21. EDUCTION ports, 171. Fractions, division of, 21. Effective discharge of water, 314. Fractions, the rule of three in, 21. Effective heating surface of flue boilers, 256. Fractions, decimal, 22. Effects of carriages on ordinary roads, 311. Fractions, table of, 73. Elastic force of steam, 188. Fractions, addition contracted, 78. Elastic fluids, 205. Fracture, 292. Elliptic arcs, 69, 70, 71, 72. Franklin Institute, 172, 219. Embankments and cuttings, 97. French litre, 355. Endless screw, 431. French measures, 5, 6. Engineering and mechanical materials, 386. French metre, 347. Engine, motion of steam in, 206. Friction, 238. Engine tender tank, 92. Friction, coefficents of, 300. Enlargements of pipes, interruption of dis- Friction of fluids, 325. charge by, 321. Friction of rest and of motion, 267. Evolution, 29. Friction of steam engines of different modiEvolution by logarithms, 339. fications, 302. Eye, diameter of, 251. Friction of water againstthe sides ofpipes,321. Eye of crank, 136. Friction of water-wheels, windmills, &c., 267. Eye of crank, to find the length and breadth Friction, or resistance to motion, in bodies of large and small, 142. rolling or rubbing on each other, 297. Eye of round end of studs of lever, 143. Friction, laws of, 298. Examples on the velocity of wheels, drums, Frustums, 83. and pulleys, 438. Frustum of spheroid, 87. Exhaust port, 230. Furnace, 256. Expanded steam, 236. Furnace room, 213. Expansion, 237. Expansion, economical effect of, 216. GALLON, 5. Experiments on the strength and other pro- Gases, 394. perties of cast iron, 174. Geering, 422. Explanation of characters, 12. General and universal expression, 376. Extended theory of angular magnitude, 374. General observations on the steam engine,259. Exterior diameter of large eye, 252. General trigonometrical solutions, 365, 369. Extraction of roots by logarithms, 339. Geometrical construction, 362. Geometrical construction of the proportion FALL of water, 444. of the radius of a wheel to its pitch, 440. Feed pipe, 150. Geometrical proportion and progression, 38. Feed water, 222. Gibs and cutter, 140, 253. Felloes of wheels, 309. Gibs and cutter through air pump cross-head, Fellowship, or partnership, 41. 146, 147. Fire-grate, 171, 214. Gibs and cutter through cross-tail and Fitzgerald, 264, 269. through butt, 141. Flange, 91. Gibs and cutter, to find the thickness and Flat bar iron, 407. breadth of, 143. Flat iron, 400. Girder, 275. Flexure by vertical pressure, 292. Girth, the mean in measuring, 94. Flexure of revolving shafts, pillars, &c., 296. Glenie, the mathematician, 287. Flues, 256. Globe, 85. Flues, fires, and boilers, 217. Grate surface, 255. Fluids, the motion of elastic, 205. Gravity, centre of, 175, 386. Fluids, to find the specific gravity 6f, 392. Gravity, specific, 391. Fluids, the pressure of, 448. Gravity, weight, mass, 386. Fluid vein, contraction of, 313. Gudgeons, 420. Foot-valve passage, 149. Gyration, centre of, 180, 390. Force, 267. Gyration, the centre of different figures and Force, loss of, in steam pipes, 221. bodies, 181. Page 586 586 INDEX. HEADS of water, 318. Iron, strength of, 173. Heating surface, 256. Iron, taper and parallel, angle and T, railHeating surface of boilers, 215. way and sash, 408, 411. Heights and discharges of water, 319. Heights and distances, 359. JET, specific gravity of, 394. Height of chimneys, 210. Journal of cross-head, to find diameter of,139. Hfeight of metacentre, 489, 483. Journal of cross-head, to find the length Hewn and sawed timber, 95. of, 139. Hexagon, heptagon, 48. Journal, the mean centre, to find the diameter High pressure and condensing engines, 234. of, 143. Hollow shafts, to find the strength of, 284. Journal, strain of, 252. Horizontal distance of centre of radius bar, Journals for air-pump cross-head, 145. 246, 247. Journals for shafts of various diameters, 287. Horse power, 240. Julian year, 357. Horse power of an engine, dimensions made Juste Byrge, the inventor of logarithms, 133. to depend upon the nominal horse power of an engine, 147. KANE, Fitzgerald, 269. Horse power of pumping engines, 447. Keel and keelson, 433 to 500. Iorse power, tables of, 243, 244. Kilometre, 5. Hot blast, 174. Kilogramme, 6. Hot liquor pumps, 446. Knots, nodes, &c., 412. Hydraulic pressure working machinery, 330. Hydraulics, 267, 312. LATHE spindle wheel, 435. Hydrogen, weight of, 356. Laying off of angles by compasses only, 384. Hydrostatic press, 448. Leg of a spherical triangle, to find, 377. Hyperboloid, 88. Length of crank pin of locomotive, 170. Hyperbolic logarithms, 130 to 133. Length of paddle-shaft journal, 138. Hyperbolic logarithms, how to calculate, 353. Length of stroke, 227, 251. Hypothenuse of a spherical triangle, to find, Lengths that may be given to stroke of the 378. valve, 229. Hypothenuse, 47. Lengths of circular arcs, 68. Lever, 426. ICOSAEDRON, 89. Light displacement, 459. Immersed portions of a ship, to calculate, Line of direction, 390. 456. Link next the radius bar, 242. Immersion and emersion, 453 to 467. Living forces, or the principle of vis viva,270. Impact, 449. Load immersion, 456, 457. Impinging of elastic and inelastic bodies, Load-water line, 456, 478. 452. Locomotive engine, parts of the cylinder, 171. Inaccessible distances, 372. Locomotive engine, diameter of the outside Inches in a solid foot, 96. bearings for, 163. Inclined plane, 428, 429, 430. Locomotive engine, dimensions of several Inclination of the traces of ordinary car- moving parts, 171. riages, 311. Locomotive engine, dimensions of several Inclinations, discharge of a 6-inch pipe at pipes, 171. several, 326. Locomotive engine, parts of the boiler, 171. Increase of efficiency arising from working Locomotive engine, tender tank, 92. steam expansively, 262. Locomotive and other engines, 233. Index of logarithms, 334. Logarithmic calculations, 376. Indicator, 264, 265. Logarithmic calculations of the force of Indicator, the amount of the effective power steam, 190 to 193. of steam by, 266. Logarithmic sines, tangents, and secants for Induction ports, 171. every minute in the quadrant, 540, 576. Inelastic bodies, 449. Logarithms applied to angular magnitudes, Influence of pressure, velocity, width of fel- 359. loes, and diameter of wheels, 309. Logarithms, hyperbolic, 130. Initial plane, 456, 480, 500. Logarithms of the natural numbers from 1 Initial velocity with a free descent, 388. to 100000 by the help of differences, 503 Injection pipe, 150. to 540. Inside discharging turbine, 330. Logarithms, the application of, 334. Integer, 10. Long measure, 7. Integers, to find the square root of, 125. Longitudinal distance of the centre of gravity Interest, simple, 42. of displacement, 470, 500. Interest, compound, 43. Loss of force by the decrease of temperature Involution, 28. in the steam pipes, 221. Involution, 6r the raising of powers by loga- Low pressure engines, 243. rithms, 338. Lunes, 54. Irregular polygons, 54. Iron, forged and wrought, 272. MACHINERY, elements of, 425. Iron plates, 403. Machinery worked by hydraulic pressure, Iron, properties of, 175. 330. Page 587 INDEX. 587 Major and minor diameters of cross-head, OAK, Dantzic, 280. 253. Obelisk, to find the height of, 371. Main beam at centre, 249. Oblique triangles, 368. Malleable iron, 396. Observatory at Paris g = 9-80896 metres,346. Marble, 288. O'Byrne's turbine tables, 331. Marine boilers, 217. Octagon, 48. Mass, 267. Octaedron, 89. Mass, gravity, and weight, 386. O'Neill's experiments, 447. Mass of a body, to find, when the weight is O'Neill's rules employed in the art of shipgiven, 389. building, 454. Materials employed in the construction of Opium, specific gravity of, 394. machines, 267. Orders of lever, 426. Materials, their properties, torsion, deflexion, Ordinates employed in the art of ship-build&c., 267. ing, 455, 456, 458. Maximum accelerating force, 421. Orifices and tubes, discharge of water by, 312. Maximum velocity and power of water Orifices, rectangular, 314. wheels, 443. Oscillation, centre of, 187, 391. Measures and weights, 5. Outside bearings of crank axle, 168. Measurement of angular magnitudes, 374. Outside discharging turbines, 331. Measurement of angles by compasses only, Overshot wheels, 329. 382. Overshot wheels, maximum velocity of, 443. Mechanical effect, 417. Ox-hide, 299. Mechanical powers, 422. Oxygen, 214, 356. Mechanical power of steam, 261. Mensuration of solids, 79. PADDLE-shaft journal, 137, 251. Mensuration of timber, 93. Paraboloid, 88. Mensuration of superficies, 45. Parabolic conoid, 88. Mercury, density of, 350. Parallel angle iron, 409. Mercury, to calculate the force of steam in Parallel motion, 242 to 246. inches of, 201. Parallelogram of forces, 422. Method to calculate the logarithm of any Parallelopipedon, 80. given number, 340. Partnership, 41. Metacentre, 482. Partial contraction of the fluid vein, 316. Metre, 5. Passages, area of steam, 220. Midship, or greatest transverse section, 460, Peclet's expression for the velocity of smoke 487. in chimneys, 213. Millboard, 405. Pendulums, 183, 391. Millstones, 445. Pendulum, conical, 184. Millstones, strength of, 451. Pendulums, vibrating seconds at the level of Modulus of elasticity, 278. the sea in various latitudes, 393. Modulus of logarithms, 343. Percussion, centre of, 391. Modulus of torsion and of rupture, 279. Periodic time, 179. Moment of inertia, 412. Permanent weight supported by beams, 284. Motion of elastic fluids, 205. Permutations and combinations, 44. Motion of steam in an engine, 206. Pillars, strength of, 293. Multiplication of decimals, 23. Pinions and wheels in continuous circular Multiplication of fractions, 21. motion, 432. Multiplication by logarithms, 335. Pipes, discharge and drainage of water Musical proportion, 40. through, 321, 322, 325. Pipes of cast iron, 395. NATURAL sines, cosines, tangents, cotangents, Pipes for marine engines, 149. secants, and cosecants, to every degree of Piston, 251. the quadrant, 411. Piston of steam engine, 414. Naval architecture, 453. Piston rod, 140, 171, 253. New method of multiplication, 342. Piston rod of air-pump, 146. Nitrogen, weight of, 356. Pitch circle, 436. Nominal horse power, tables of, for high and Pitch of teeth, 441. low pressure engines, 243, 244. Pitch of wheels, 435, 439. Notation and numeration, 10. Plane triangles, solution of, 364, 365. Notation, trigonometrical, 359. Plane trigonometry, 359. Number corresponding to a given logarithm, Planks, deals, 94. 351. Polygons, 47, 48. Number of teeth, or the pitch of small Polygons, irregular, 54. wheels, 435. Port, upper and lower, 229. Numbers, fourth and fifth powers of, 129. Position, double, 44. Numbers, logarithms of, 540, 495. Position, single, 43. Numbers, reciprocals of, 73 to 78. Pound, 5. Numbers, squares, cubes, &c., of, 100 to 116. Power, actual and nominal, 241. Numeral solution of the several cases of Power and properties of steam, 261. trigonometry, 361. Power that a cast-iron wheel is capable of Nuts and bolts, 406. transmitting, 442. Page 588 588 INDEX. Power of shafts, 294. Ropes, stiffness of, resistance of, to bendin Practical application of the mechanical 302. powers, 425. Ropes, tarred and dry, 304, 306. Practical limit to expansion, 261. Rotative engines, 260. Practical observations on steam engines, 260. Rotation, moment of, 414. Principle of virtual velocities, 423. Rotation of a body about a fixed axis, 416. Prism, 80. Rotations of millstones, 452. Prismoid, 85. Round and rectangular bars, strength of, 281. Properties of bodies, 401. Round bar-iron, 403. Proportional dimensions of nuts and bolts, Round steel and brass, 408. 406. Rules for pumping engines, 448. Proportion, 14. Rule of three, 13. Proportion, musical, 40. Rule of three by logarithms, 338. Proportion and progression, arithmetical, 35 Rule of three in fractions, 21. to 38. Rupture, 272. Proportion and progression, geometrical, 38 to 40. SAFETY valves, 149,150, 224. Proportion, or the rule of three by loga- Sails of windmills, 332. rithms, 338: Sash iron, 410. Proportion of wheels for screw-cutting, 433. Scales of chords, how to construct, 360. Proportions of boilers, grates, &c., 213. Scale of displacement, 490. Proportions of the lengths of circular arcs, 68. Scantling, 95. Proportions of undershot wheels, 328. Screw cutting by lathe, 433. Pulleys, 422, 427. Screw, power of, 430. Pump and pumping engines, 446. Screw, to cut, 434. Pumping engines, 422. Sectional area measured, 456 to 468. Pyramid, 82. Segments of circles, 64 to 67. Pyrometer, 63. Shelves, cords, blocks, 428. Ship-building and naval architecture, 453. QUADRANT, 359. Sidereal day, 9. Quadrant, log. sines, cosines, &c., for every Side lever, to find the depth across the centre minute in, 540, 576. of, 144. Quadrant, natural sines and cosines for Side rod, 246, 254. every degree of, 411. Side rod of air-pump, 146. Quadrant, to take angles with, 370. Sines, cosines, &c., 411. Quantities, known and unknown, 134. Sines, tangents and secants, 359. Quantity of water that flows through a cir- Singular phenomena, 237. cular orifice, 313, 319. Sleigh, 268. Quiescence, friction of, 299. Slide valve, 225. Slide valve, a cursory examination of, 232. RADIUS bar, 242. Slopes 1. to 1, 2 to 1, and 1 to 1, 97. Radius bar, length of, corrected, 248. Sluice board, 316. Radius of the earth at Philadelphia, 356. Smoke and heated air in chimneys, 202. Radius of gyration, 412. Solid inches in a solid foot, 96. Radius, length of, in degrees, 357. Solids, mensuration of, 79. Rails, temporary, 411. Space described by a body during a free deRailway carriage, 268. scent in vacuo, 388. Railway iron, 410. Specific gravity, 386, 391. Raising of powers by logarithms, 338. Sphere, 85. Reciprocals of numbers, 73 to 78. Spheres, 397 to 400. Recoil, 449. Spheroid, 86, 87, 88. Rectangle, rhombus, rhomboides, to find Spherical trigonometry, 373. the areas of, 45, 46. Spheroidal condition of water in boilers, 236. Reduction of fractions, 16, 17, to 19, 20. Spindle and screw wheels, 434. Regnault's experiments on oxygen, &c., 356. Square, to find the area of, 45. Regular bodies, 90. Square and sheet iron, 402. Relative capacities of the two bodies under Squares and square roots of numbers, 100 the same displacement, 456, 470. to 116. Relative strength of materials to resist tor- Square root, 30. sion, 294. Square root of fractions and mixed numbers, Revolving shaft, 250. 31. Riga fir, 290. Square measure, 6. Right-angled spherical triangles, 374. Stability, 459. Ring, circular, to find the area of, 53. Stars, apparent motion of, 353. Ring, cylindrical, 90. Statical moment, 417. Roads, traction of carriages on, 307. Steam engine, 135. Rolled iron, 395. Steam dome, 171. Roman notation, 11. Steam passages. 220. Rope, strength of, 282. Steam pipes, loss of force in, 222. Ropes, bands, &c., 267. Steam port, 147, 148. Ropes, blocks, pulleys, 428. Steam room, 259. Page 589 INDEX. 589 Steam, elastic force of, 188 to 202. Table of reciprocals of numbers, or of the Steam, temperature of, pressure of, 172. decimal fractions corresponding to comSteam, volume of, 202 to 206. mon fractions, 71 to 77, 78. Steam, weight of, 204. Table of weights and values in decimal Steel, 408. parts, 79. Steel, cast, 409. Table of regular bodies, 90. Stiffness of ropes, 302, 306. Table of the cohesive power of bodies, 175. Stowage, 503. Table of hyperbolic logarithms, 130 to 133. Stowing the hold of a vessel, 453, 456. Table of the pressure of steam, in inches of Strap at cutter, 141. mercury at different temperatures, 172. Strap, mean thickness of, at and before cut- Table of the temperature of steam at differter, 143. ent pressures, in atmospheres, 172. Strength of bodies, 282. Table of the expansion of air by heat, 173. Strength of boilers, 218. Table of the strength of iron, 173. Strength of materials, 173, 271. Table of the superficial and solid content of Strength of rods when the strain is wholly spheres, 96. tensile, 250. Table of solid inches in a solid foot, 96. Strength of the teeth of cast iron wheels, 437. Table of squares, cubes, square& and cube qnuds of lever, 143. roots, of numbers, 100, 101, 116, 125. I-wheel and pinion, 434. Table of cover on the exhausting side of the Subtraction of decimals, 23. valve in parts of the stroke and distance Subtraction of fractions, 21. of piston from the end of its stroke, 231. Table of the proportions of the lengths of TABLE by which to determine the number of circular arcs, 68. teeth or pitch of small wheels, 435. Table of the fourth and fifth power of numTable containing the circumferences, squares, bers, 129. cubes, and areas of circles, from 1 to 100, Table of the properties of different boiladvancing by a tenth, 57, 58, 59, 60 to ers, 215. 63. Table of the economical effects of expanTable containing the weight of columns of sion, 216. water, each one foot in length, in pounds Table of the comparative evaporative power avoirdupois, 401. of different kinds of coal, 218. Table containing the weight of square bar Table of the cohesive strength of iron boiler iron, 402. plate at different temperatures, 219. Table containing the surface and solidity of Table of diminution of strength of copper spheres, together with the edge of equal boilers, 219. cubes, the length of equal cylinders, and Table of expanded steam, 239. weight of water in avoirdupois pounds, Table of the proportionate length of bearings, 397. or journals for shafts of various diameters, Table containing the weight of fiat bar iron, 287. 400. Table of tenacities, resistances to compresTable containing the specific gravities and sion and other properties of materials, other properties of bodies; water the stand. 288. ard of comparison, 401. Table of the strength of ropes and chains, Table containing the weight of round bar 288. iron, 403. Table of the strength of alloys, 289. Table containing the weights of cast iron Table of data of timber, 289. pipes, 404. Table of the properties of steam, 261. Table containing the weight of solid cylin- Table of the mechanical properties of steam, ders of cast iron, 404. 263. Table containing the weight of a square foot Table of the cohesive strength of bodies, 281. of copper and lead, 405. Table of the strength of common bodies, 283. Table for finding the weight of malleable Table of torsion and twisting of common mairon, copper, and lead, 405. terials, 286. Table for finding the radius of a wheel when Table of the length of circular arcs, radius the pitch is given, or the pitch when the ra- being unity, 63. dius is given, for any number of teeth, 439. Table of experiments on iron boiler plate at Table for the general construction of tooth high temperature, 220. wheels, 442. Table of the absolute weight of cylindrical Table for breast wheels, 329. columns, 274. Table of polygons, 48. Table of flanges of girders, 276. Table of decimal approximations for facili- Table of mean pressure of steam at different tating calculations, 55. densities and rates of expansion, 239. Table of decimal equivalents, 56. Table of nominal horse power of high presTable of the areas of the segments and zones sure engines, 244. of a circle of which the diameter is unity, Table of nominal horse power of low pres64, 65, 66, 67. sure engines, 243. Table of the proportions of the lengths of Table of dimensions of cylindrical columns semi-elliptic arcs, 69, 70, 72. of cast iron to sustain a given load with Table of flat or board measure, 93. safety, 293. Table of solid timber measure, 94. Table of strength of columns, 294L Page 590 590 INDEX. Table of comparative torsion, 294. Table of the weight of round steel, 408. Table of the depths of square beams to sup- Table of parallel angle iron of equal sides, 408. port from 1 cwt. to 14 tons, 295, 296. Table of parallel angle iron of unequal sides, Table of the results of experiments on fric- 409. tions, with unguents interposed, 299, 300. Table of taper angle iron of equal sides, 409. Table of the results of experiments on the Table of parallel T iron of unequal width and gudgeons or axle-ends in motion upon their depth, 409. bearings, 301. Table of change wheels for screw-cutting, Table of friction, continued to abrasion, 301. 435.'Fable of friction of steam engines of differ- Table of the diameters of wheels at their ent modifications, 302. pitch circle, to contain a required number Table of tarred ropes, 303. of teeth, 436. Table of white ropes, 305. Table of the angle of windmill sails, 445. Table of dry and tarred ropes, 306. Table of the logarithms of the natural numTable of the pressure and traction of car- bers, from 1 to 10000, by the help of difriages, 308. ferences, 495 to 540. Table of traction of wheels, 309. Table of log. sines, cosines, tangents, cotanTable of the ratio of traction to the load, gents, secants and cosecants, for every de310. * gree and minute in the quadrant, 540 to Table of the coefficients of the efflux through 576. rectangular orifices in a thin vertical plate, Table of the strength of the teeth of cast iron 315. wheels at a given velocity, 437. Table of the coefficients of efflux, 315. Table of approved proportions for wheels Table of comparison of the theoretical with with fiat arms, 441. the real discharges from an orifice, 317. Table showing the cover required on the Table of discharge of tubes of different en- steam side of the valve to cut the steam off largements, 322. at any part of the stroke, 228. Table of the comparison of discharge by pipes Table showing the cover required, 227. of different lengths, 323. Table showing the resistance opposed to Table of the comparison of discharge by ad- the motion of carriages on different incliditional tubes, 323. nations of ascending or descendingplanes, Table of the friction of fluids, 325. 429. Table of discharges of a 6-inch pipe at seve- Table showing the number of linear feet of ral inclinations, 326. scantling of various dimensions which are Table of the velocity of windmill sails, 333. equal to a cubic foot, 95. Table of outside discharging turbine, 331. Table showing the weight or pressure a beam Table of inward discharging turbines, 332. of cast iron will sustain without destroying Table of peculiar logarithms, 340. its elastic force, 292. Table of useful logarithms, 345. Table showing the circumference of rope Table of the specific gravity of various sub- equal to a chain, 282. stances, 394. Table to correct parallel motion links, 248. Table of the weight of a foot in length of flat Table of parallel T iron of equal depth and and rolled iron, 395. width, 410. Table of the weight of cast iron pipes, 395. Tables of cuttings and embankments, slopes, Table of the weight of one foot in length of 1 to 1; 1 to 1; and 2 to 1, 97. malleable iron, 396. Tables of the heights corresponding to differTable of comparison, 396. ent velocities, 389. Table of the weight of a square foot of sheet Tables of the mechanical properties of the iron, 402. materials most commonly employed in the Table of the weight of a square foot of boiler construction of machines and framings, plate from * of an inch to 1 inch thick, 403. 280. Table of the weights of cast iron plates, 403. Tangents, 360. Table of the weight of mill-board, 405. Tangents and secants, to compute, 362. Table of the weight of wrought iron bars, 406. Taper angle iron, 410. Table of the proportional dimensions pf nuts Teeth of wheels in continuous circular motion, and bolts, 406. 432. Table of the specific gravity of water at dif- Teeth of wheels, 422, 436. ferent temperatures, 406. Temperature of steam, 172. Table of the weight of cast iron balls, 407. Temperature and elastic force of steam, 188. Table of the weight of flat bar iron, 407. Tension of chain-bridge, 414. Table of the weight of square and round Tetraedron, 89. brass, 408. Threshing machines, 445. Table of taper T iron, 410. Throttling the steam, 234. Table of sash iron, 410. Timber measure, 93. Table of rails of equal top and bottom, 410. Timber, to measure round, 95. Table of temporary rails, 411. Time, 7. Table of natural sines, cosines, tangents, co- Tonnage of ships, 461 to 494. tangents, secants, and cosecants, to every Torsion, 279. degree of the quadrant, 411. Torsion and twisting, 286. Table of inclined planes, showing the ascent Traction of carriages, 307. or descent the yard, 430. Transverse strength of bodies, 282. Page 591 INDEX. 591 Transverse strain, 278. Vertical sectional areas, 454. Transverse strain, time weight, 273. Virtual velocities, 424. Trapezium, 47. Vis viva, principle of, calculations on, 276, Trapezoid, 47. 388. Triangle, to find the area of, 46, 47. Volume of a ship immersed, 456. Trigonometry, 359. Volume of steam in a cubic foot of water, Trigonometry, spherical, 373. 202, 205. Troy weight, 7. Trussed beams, 291. WATER, modulus of elasticity of, 190. Tubes, discharge of water through, 312. Water level, 214. Tubular boilers, 257. Water, feed and condensing, 223. Turbine water-wheels, 330. Water, spheroidal condition of, in boilers,236. Water in boiler, and water level, 358. ULTIMATE pressure of expanded steam, 236. Water, discharge of, through different orifiUndecagon, 47. ces, 312, 318. Undershot wheels, 327, 443. Water wheels, 327. Unguents, 299. Water wheels, maximum velocity of, 443. Ungulas, cylindrical, 81. Web of crank at paddle shaft centre, 136. Ungulas, conical, 83, 84. Web of cross-head at middle, 139. Unit of length, 5. Web of crank at pin centre, 142. Unit of weight, 5. Web at paddle centre, 252. Unit of dry capacity, 5. Web of cross-head at journal, 140. Units of liquids, 5. Web of air-pump cross-head, 145. Units of work, 269, 297, 414, 4o1. Wedge, 85. Universal pitch table, 442. Wedge and screw, 430. Upper steam port, 229. Weights and measures, 5. Useful formula, 271. Weights, values of, in decimal parts, 79. Use of the table of squares, cubes, &c., 127. Weight, mass, gravity, 386. Weirs, and rectangular apertures, 314, 323. VACUUM, perfect one, 235. Wheel and axle, 417. Vacuum below the piston, 251. Wheel and pinion, 427. Vacuo, bodies falling freely in, 388. Wheels, drums, pulleys, 438. Valves, different arrangements of, 233. Windmills, 332. Valve, length of stroke of, in inches, 228. Wine measure, 8. Valve shaft, 147. Woods, 280. Valve, safety, 224. Woods, specific gravity, 394. Valve, slide, 225. Work done, weight, 267. Valve spindle, 171. Wrought iron bars, 406. Vapour in the cylinder, 229. Vein, contraction of fluid, 330. YARD, 5. Velocity, force, and work done, 267. Yacht, admeasurement of, 466, 470. Velocity of steam rushing into a vacuum, 207. Yarns of ropes, 303. Velocity of smoke in chimneys, 209, 213. Yellow brass, 281. Velocity of piston of steam engine, 266. Yew, 280. Velocity of threshing machines, millstones, boring, &c., 445. ZINc, 280. Velocity of wheels on ordinary roads, 307. Zinc, sheet, 288. Venturi, experiments of, on the discharge of Zone, spherical, 86. fluids, 421. Zone, to find the area of a circular, 53. Versed sine, tabular, 52. Zones of circles, to find the areas of, 64, 65, Versed sine of parallel motion, 244. 66. Versed sine, 359. THE END. Page [unnumbered] Page [unnumbered] Page [unnumbered] Page [unnumbered] PUBLICATIONS OF HENRY CAREY BAIRD, SUCCESSOR TO E. L. CAREY, South-east corner of Market and Fifth Streets, Philadelphia. SCIENTIFIC AND PRACTICAL. THE PRACTICAL MODEL CALCULATOR, For the Engineer, Machinist, Manufacturer of Engine Work, Naval Architect, Miner, and Millwright. By OLIVER BYRNE, Compiler and Editor of the Dictionary of Machines, Mechanics, Engine Work and Engineering, and Author of various Mathematical and Mechanical Works. Illustrated by numerous Engravings. Now publishing in Twelve Parts, at Twenty-five Cents each, forming, when completed, One large Volume, Octavo, of nearly six hundred pages. It will contain, such calculations as are met with and required in the Mechanical Arts, and establish models or standards to guide practical men. The Tables that are introduced, many of which are new, will greatly economize labour, and render the every-day calculations of the practical man comprehensive and easy. From every single calculation given in this work numerous other calculations are readily modelled, so that each may be considered the head of a numerous family of practical results. The examples selected will be found appropriate, and in all cases taken from the actual practice of the present time. Every rule has been tested by the unerring results of mathematical research, and confirmed by experiment, when such was necessary. The Practical Model Calculator will be found to fill a vacancy in the library of the practical working-man long considered a requirement. It will be found to excel all other works of a similar nature, from the great extent of its range, the exemplary nature of its well-selected examples, and from the easy, simple, and systematic manner in which the model calculations are established. NORRIS'S HAND-BOOK FOR LOCOMOTIVE ENGINEERS AND MACHINISTS: Comprising the Calculations for Constructing Locomotives. Manner of setting Valves, &c. &c. By SEPTIMUS NORRIS, Civil and Mechanical Engineer. In One Volume, 12mo, with illustrations................................ $1.50 A TREATISE ON THE AMERICAN STEAM ENGINE. Illustrated by numerous Wood Cuts and other Engravings By OLIVER BYNE. In One Volume, royal 8vo. (In press.) Page [unnumbered] 2 PUBLICATIONS OF HENRY CAREY BAIRD. THE PRACTICAL COTTON-SPINNER AND MANUFACTURER; OR, THE MANAGER'S AND OVERLOOKER'S COMPANION. This work contains a Comprehensive System of Calculations for Mill Gearing and Machinery, from the first moving power through the different processes of Carding, Drawing, Slabbing, Roving, Spinning, and Weaving, adapted to American Machinery, Practice, and Usages. Compendious Tables of Yarns and Reeds are added. Illustrated by large Working-Drawings of the most approved American Cotton Machinery. Complete in One Volume, octavo.................. $3.50 This edition of Scott's Cotton-Spinner, by OLIVER BYRNE, is designed for the American Operative. It will be found intensely practical, and will be of the greatest possible value to the Manager, Overseer, and Workman. THE PRACTICAL METAL-WORKER'S ASSISTANT; For Tin-Plate Workers, Brasiers, Coppersmiths, Zinc-Plate Ornamenters and Workers, Wire Workers, Whitesmiths, Blacksmiths, Bell Hangers, Jewellers, Silver and Gold Smiths, Electrotypers, and all other Workers in Alloys and Metals. By CHARLES HOLTZAPPFEL. Edited, with important additions, by OLIVER BYRNE. Complete in One Volume, octavo................................... $4.00 It will treat of Casting, Founding, and Forging; of Tongs and other Tools; Degrees of Heat and Management of Fires; Welding; of Heading and Swage Tools; of Punches and Anvils; of Hardening and Tempering; of Malleable Iron Castings, Case Hardening, Wrought and Cast Iron. The management and manipulation of Metals and Alloys, Melting and Mixing. The management of Furnaces, Casting and Founding with Metallic Moulds, Joining and Working Sheet Metal. Peculiarities of the different Tools employed. Processes dependent on the ductility of Metals. Wire Drawing, Drawing Metal Tubes, Solder. ing. The use of the Blowpipe, and every other known Metal-Worker's Tool To the works of Holtzappfel, OLIVER BYRNE has added all that is useful and peculiar to the American Metal-Worker. A COMPLETE TREATISE ON TANNING, CURRYING, AND EVERY BRANCH OF LEATHER-DRESSING. From the French and from original sources. By CAMPBELL MORFIT, one of the Editors of the "Encyclopedia of Chemistry," Author of "Chemistry Applied to the Manufacture of Soap and Candles," and other Scientific Treatises. Illustrated with several hundred Engravings. Complete in One Volume, royal 8vo. (In press.) This important treatise will be issued from the press at as early a day as the duties of the editor will permit, and it is believed that in no other branch of applied science could more signal service be rendered to American Manufacturers. The publisher is not aware that in any other work heretofore issued in this country, more space has been devoted to this subject than a single chapter; and in offering this volume to so large and intelligent a class as American Tanners and Leather Dressers, he feels confident of their substantial support and encouragement. THE MANUFACTURE OF IRON IN ALL ITS VARIOUS BRANCHES: To which is added an Essay on the Manufacture of Steel, by FREDERICK OVERMAN, Mining Engineer, with one hundred and fifty Wood Engravings. A new edition. In One Volume, octavo, five hundred pages............$5.00 We have now to announce the appearance of another valuable work on the subject which, in our humble opinion, supplies any deficiency which late improvements and discoveries may have caused, from the lapse of time since the date of "Mushet" and "Schrivenor." It is the production of one of our transatlantic brethren, Mr. Frederick Overman, Mining Engineer: and we do not hesitate to set it down as a work of great importance to all connected with the iron interest; one which, while it is sufficiently technological fully to explain chemical analysis, and the various phenomena of iron under different circumstances, to the batisfaction of the most fastidious, is written in that clear and comprehensive style as to be available to the capacity of the humblest mind, and consequently will be of much advantage to those works where the proprietors may see the desirability of placing it in the hands of their operatives.-London lMorning JournaL Page [unnumbered] PUBLICATIONS OF HENRY CAREY BAIRD. 3 PRACTICAL SERIES. THE volumes in this Series are published in duodecimo form, and the design is to furnish to Artisans, for a moderate sum, Hand-books of the different Arts and Manufactures, in order that they may be enabled to keep pace with the improvements of the age. There have already appearedTHE AMERICAN MILLER AND MILLWRIGHT'S ASSISTANT. $1. THE TURNER'S COMPANION. 75 cts. THE PAINTER, GILDER, AND VARNISHER'S COMPANION. 75 cts. THE DYER AND COLOUR-MAKER'S COMPANION. 75 cts. THE BUILDER'S COMPANION. $1. THE CABINET-MAKER'S COMPANION. 75 cts. The following, among others, are in preparation:A TREATISE ON A BOX OF INSTRUMENTS. By THOMAS KENTISH. THE PAPER-HANGER'S COMPANION. By J. ARROWSMITH. THE ASSAYER'S GUIDE. By OSCAR M. LIEBER. THE AMERICAN MILLER AND MILLWRIGHT'S ASSISTANT: By WILLIAM CARTER HUGHES, Editor of " The American Miller," (newspaper), Buffalo, N. Y. Illustrated by Drawings of the most approved Machinery. In One Volume, 12mo........................................................... $1 The author offers it as a substantial reference, instead of speculative theories, which belong only to those not immediately attached to the business. Special notice is also given of most of the essential improvements which have of late been introduced for the benefit of the Miller.-Savannah Republican. The whole business of making flour is most thoroughly treated by him.-Bulletin. A very comprehensive view of the Millwright's business.-Southern Literary Messenger. THE TURNER'S COMPANION: Containing Instructions in Concentric, Elliptic, and Eccentric Turning. Also, various Plates of Chucks, Tools, and Instruments, and Directions for using the Eccentric Cutter, Drill, Vertical Cutter, and Circular Rest; with Patterns and Instructions for working them. Illustrated by numerous Engravings. In One Volume, 12mo.....................................................75 cts. The object of the Turner's Companion is to explain in a clear, concise, and intelligible manner, the rudi ments of this beautiful art.-Savannah Republican. There is no description of turning or lathe-work that this elegant little treatise does not describe and illustrate.-Western Lit. Mlessenger. THE PAPER-HANGER'S COMPANION: In which the Practical Operations of the Trade are systematically laid down; with copious Directions Preparatory to Papering; Preventions against the effect of Damp in Walls; the various Cements and Pastes adapted to the several purposes of the Trade; Observations and Directions for the Panelling and Ornamenting of Rooms, &c., &c. By JAMES ARROWSMITH. In One Volume, 12mo. Page [unnumbered] 4 PUBLICATIONS OF HENRY CAREY BAIRD. THE PAINTER, GILDER, AND VARNISHER'S COMPANION: Containing Rules and Regulations for every thing relating to the arts of Painting, Gilding, Varnishing, and Glass Staining; numerous useful and valuable Receipts; Tests for the detection of adulterations in Oils, Colours, &c., and a Statement of the Diseases and Accidents to which Painters, Gilders, and Varnishers are particularly liable; with the simplest methods of Prevention and Remedy. Second Edition. In One Volume, 12mo, cloth...................75 cts. Rejecting all that appeared foreign to the subject, the compiler has omitted nothing of real practical worth.-Hunt's Merchants' Magazine. An excellent practical work, and one which the practical man cannot afford to be without.-Farmer and Mechanic. It contains every thing that is of interest to persons engaged in this trade.-Bulletin. This book will prove valuable to all whose business is in any way connected with painting.-Scott's Weekly. Cannot fail to be useful.-N. Y. Commercial. THE DYER AND COLOUR-MAKER'S COMPANION: Containing upwards of two hundred Receipts for making Colours, on the most approved principles, for all the various styles and fabrics now in existence; with the Scouring Process, and plain Directions for Preparing, Washing-off, and Finishing the Goods. Second Edition. In One Volume, 12mo, cloth............................................................................................... 75 cts. This is another of that most excellent class of practical books, which the publisher is giving to the public. Indeed, we believe there is not, for manufacturers, a more valuable work, having been prepared for, and expressly adapted to their business.-Farmer and Mechanic. It is a valuable book.-Otsego Republican. We have shown it to some practical men, who all pronounced it the completest thing of the kind they had seen.-N. Y. Nation. THE BUILDER'S POCKET COMPANION: Containing the Elements of Building, Surveying, and Architecture; with Practical Rules and Instructions connected with the subject. By A. C. SMEATON, Civil Engineer, &c. Second Edition. In One Volume, 12mo....... $1 CONTENTS.-The Builder, Carpenter, Joiner, Mason, Plasterer, Plumber, Painter, Smith, Practical Geometry, Surveyor, Cohesive Strength of Bodies, Architect. THE ASSAYER'S GUIDE; Or, Practical Directions to Assayers, Miners, and Smelters, for the Tests and Assays by Heat and by Wet Processes of the Ores of all the principal Metals, and of Gold and Silver Coins and Alloys. By OscAR M. LIEBER, Mining Engineer, Geologist to the State of Mississippi, etc. 12mo. (In press.) A TREATISE ON A BOX OF INSTRUMENTS, And the SLIDE RULE, with the Theory of Trigonometry and Logarithms, including Practical Geometry, Surveying, Measuring of Timber, Cask and Malt Gauging, Heights and Distances. By THOMAS KENTISH. In One Volume, 12mo. Page [unnumbered] PUBLICATIONS OF HENRY CAREY BAIRD. 5 THE CABINET-MAKER AND UPHOLSTERER'S COMPANION: Comprising the Rudiments and Principles of Cabinet-making and Upholstery, with familiar Instructions, illustrated by Examples, for attaining a proficiency in the Art of Drawing, as applicable to Cabinet-Work; the processes of Veneering, Inlaying, and Buhl Work; the art of Dyeing and Staining Wood, Bone, Tortoise-shell, &c. Directions for Lackering, Japanning, and Varnishing; to make French Polish; to prepare the best Glues, Cements, and Compositions, and a number of Receipts particularly useful for Workmen generally, with Explanatory and Illustrative Engravings. By J. STOKES. In One Volume, 12mo, with Illustrations...................................................................... 75 cts. A large amount of practical information, of great service to all concerned in those branches of business. -Ohio State Journal. PROPELLERS AND STEAM NAVIGATION: With Biographical Sketches of Early Inventors. By ROBERT MACFARLANE, C. E., Editor of the " Scientific American." In One Volume, 12mc. Illustrated by over Eighty Wood Engravings...................................... 75 cts. The object of this "History of Propellers and Steam Navigation" is twofold. One is the arrangement and descriptipn of many devices which have been invented to propel vessels, in order to prevent many ingenious men from wasting their time, talents, and money on such projects. The immense amount of time, study, and money thrown away on such contrivances is beyond calculation. In this respect, it is hoped that it will be the means of doing some good.-Preface. TABLES OF LOGARITHMS FOR ENGINEERS AND MACHINISTS: Containing the Logarithms of the Natural Numbers, from 1 to 100000, by the help of Proportional Differences. And Logarithmic Sines, Cosines, Tangents, Co-tangents, Secants, and Co-secants, for every Degree and Minute in the Quadrant. To which are added, Differences for every 100 Seconds. By OLIVERi BYRNE, Civil, Military, and Mechanical Engineer. In One Vol. 8vo. cl.... $1 THE FRUIT, FLOWER, AND KITCHEN GARDEN. By PATRICK NEILL, L. L. D., F. R. S. E., Secretary to the Royal Caledonian Horticultural Society. Adapted to the United States, from the Fourth Edition, revised and improved by the Author. Illustrated by fifty Wood Engravings of Hothouses, &c. &c. In One Volume, 12mo........................ $1.25 This volume supplies a desideratum much felt, and gives within a moderate compass all the horticultural information necessary for practical use.-Newark Mercury. A valuable addition to the horticulturist's library.-Baltimore Patriot. This work is the production of a most celebrated British horticulturist, Dr. NEILL, of Scotland, for upwards of thirty years the Secretary of the Caledonian Horticultural Society, and in every way qualified to make a standard book upon the subject it discusses. The careful adaptation of the work to the peculiar circumstances and necessities of our own people, is a subject of congratulation, since good books upon horticulture cannot be too much multiplied. We are pleased with the comprehensiveness of Dr. Nz'ILI treatise.-Southern Literary Gazette ELEMENTARY PRINCIPLES OF CARPETRY. By THOMAS TREDGOLD. In One Volume, quarto, with numerous llustrations................................................................ $2.50 Page [unnumbered] 6 PUBLICATIONS OF HENRY CAREY BAIRD. THE ENCYCLOPEDIA OF CHEMISTRY, PRACTICAL AND THEORETICAL: Embracing its Application to the Arts, Metallurgy, Mineralogy, Geology, Medicine, and Pharmacy. By JAMES C. BOOTH, Melter and Refiner in the United States Mint, Professor of Applied Chemistry in the Franklin Institute, &c.; assisted by CAMPBELL MORFIT, Author of "Chemical Manipu}aptiqns," &c. Complete in One Volume, royal octavo, 978 pages, with numerous Woodcuts and other Illustrations. Second Edition. Full bound....................................... $5 It covers the whole field of Chemistry as applied to Arts and Sciences. * * * As no library is complete without a common dictionary, it is also our opinion that none can be without this Encyclopedia of Chemistry.-Scientific American. A work of time and labour, and a treasury of chemical information.-North American. By far the best manual of the kind which has been presented to the American public.-Boston Courier. An invaluable work for the dissemination of sound practical knowledge.-Ledger. A treasury of chemical information, including all the latest and most important discoveries.-Baltimore American. At the first glance at this massive volume, one is amazed at the amount of reading furnished in its compact double pages, about one thousand in number. A further examination shows that every page is richly stored with information, and that while the labours of the authors have covered a wide field, they have neglected or slighted nothing. Every chemical term, substance, and process is elaborately, but intelligibly, described. The whole science of Chemistry is placed before the reader as fully as is practicable with a science continually progressing. * * * Unlike most American works of this class, the authors have not depended upon any one European work for their materials. They have gathered theirs from works on Chemistry in all languages, and in all parts of Europe and America; their own experience, as practical chemists, being ever ready to settle doubts or reconcile conflicting authorities. The fruit of so much toil is a work that must ever be an honour to American science.-Evening Bulletin. SYLLABUS OF A COMPLETE COURSE OF LECTURES ON CHEMISTRY: Including its Application to the Arts, Agriculture, and Mining, prepared for the use of the Gentlemen Cadets at the Hon. E. I. Co.'s Military Seminary, Addiscombe. By Professor E. SOLLY, Lecturer on Chemistry in the Hon. E. I. Co.'s Military Seminary. Revised by the Author of " Chemical Manipulations." In One Volume, octavo, cloth........................................... 1.25 The present work is designed to occupy a vacant place in the libraries of Chemical text-books. It is admirably adapted to the wants of both TEACHER and PUPIL; and will be found especially convenient to the latter, either as a companion in the class-room, or as a remembrancer in the study. It gives, at a glance, under appropriate headings, a classified view of the whole science, which is at the same time compendious and minutely accurate; and its wide margins afford sufficient blank space for such manuscript notes as the student may wish to add during lectures or recitations. The almost indispensable advantages of such an impressive aid to memory are evident to every student who has used one in other branches of study. Therefore, as there is now no Chemical Syllabus, we have been induced by the excellences of this work to recommend its republication in this country; confident that an examination of the contents will produce full conviction of its intrinsic worth and usefulness.Zditoso Prefoae AN ELEMENTARY COURSE OF INSTRUCTION ON ORDNANCE AND GUNNERY. Prepared for the use of the Midshipmen at the Naval School. By JAMES H. WARD, U. S. N. In One Volume, octavo............................. $1.50 STEAM FOR THE MILLION. An Elementary Outline Treatise on the Nature and Management of Steam, and the Principles and Arrangement of the Engine. Adapted for Popular Instruction, for Apprentices, and for the use of the Navigator. With an Appendix containing Notes on Expansive Steam, &c. In One Volume, 8vo...37~ cts Page [unnumbered] PUBLICATIONS OF HENRY CAREY BAIRD. 7 HOUSEHOLD SURGERY; OR, HINTS ON EMERGENCIES. By J. F. SOUTH, one of the Surgeons of St. Thomas's Hospital. In One Volume, 12mo. Illustrated by nearly fifty Engravings............$1.2. CONTENTS: The Doctor's Shop.-Poultices, Fomentations, Lotions, Liniments, Ointments, Plasters. Surgery.-Blood-letting, Blistering, Vaccination, Tooth-drawing, How to put on a Roller, Lancing the Gums, Swollen Veins, Bruises, Wounds, Torn or Cut Achilles Tendon, What is to be done in cases of sudden Bleeding from various causes, Scalds and Burns, Frost-bite, Chilblains, Sprains, Broken Bones, Bent Bones, Dislocations, Ruptures, Piles, Protruding Bowels, Wetting the Bed, Whitlow, Boils, Black-heads, Ingrowing Nails, Bunions, Corns, Sty in the Eye, Blight in the Eye, Tumours in the Eyelids, Inflammation on the Surface of the Eye, Pustules on the Eye, Milk Abscesses, Sore Nipples, Irritable Breast, Breathing, Stifling, Choking, Things in the Eye, On Dress, Exercise and Diet of Children, Bathing, Infections, Observations on Ventilation. HOUSEHOLD MEDICINE By D. FRANCIS CONDIE, M. D. In One Volume, 12mo. Uniform with, and a companion to, the above. (In immediate preparation.) ELWOOD'S GRAIN TABLES: Showing the value of Bushels and Pounds of different kinds of Grain, calculated in Federal Money, so arranged as to exhibit upon a single page the value at a given price from ten cents to two dollars per bushel, of any quantity from one pound to ten thousand bushels. By J. L. ELWOOD. A new Edition. In One Volume, 12mo.........................................................................$1 To Millers and Produce Dealers this work is pronounced by all who have it in use, to be superior in arrangement to any work of the kind published-and unerring accuracy in every calculation may be relied upon in every instance. IE7 A reward of Twenty-five Dollars is offered for an error of one cent found in the work. PERFUMERY; ITS MANUFACTURE AND USE: With Instructions in every branch of the Art, and Receipts for all the Fashionable Preparations; the whole forming a valuable aid to the Perfumer, Druggist, and Soap Manufacturer. Illustrated by numerous Woodcuts, From the French of Celnart, and other late authorities. With Additions and Improvements, by CAMPBELL MORFIT, one of the Editors of the " Encyclopedia of Chemistry." In One Volume, 12mo, cloth................................... $1 ELECTROTYPE MANIPULATION: Being the Theory and Plain Instructions in the Art of Working in Metals, by Precipitating them from their Solutions, through the agency of Galvanic or Voltaic Electricity. By CHARLES V. WALKER, Hon. Secretary to the London Electrical Society, &c. Illustrated by Woodcuts. From the Thirteenth London Edition. In One Volume, 24mo, cloth................................... 62 cts. Page [unnumbered] 8 PUBLICATIONS OF HENRY CAREY BAIRD. PHOTOGENIC MANIPULATION: Containing the Theory and Plain Instructions in the Art of Photography, or the Production of Pictures through the Agency of Light; including Calotype, Chrysotype, Cyanotype, Chromatype, Energiatype, Anthotype, Amphitype, Daguerreotype, Thermography, Electrical and Galvanic Impressions. By GEORGE THOMAS FISHER, Jr., Assistant in the Laboratory of the London Institution. Illustrated by Wood-cuts. In One Volume, 24mo, cloth............ 62 cts. MATHEMATICS FOR PRACTICAL MEN: Being a Common-Place Book of Principles, Theorems, Rules, and Tables, in various Departments of Pure and Mixed Mathematics, with their Applications, especially to the pursuits of Surveyors, Architects, Mechanics, and Civil Engineers. With numerous Engravings. By OLINTHUS GREGORY, L. L. D., F. R. A. S..........................................................................................$1.50 Only let men awake, and fix their eye, one while on the nature of things, another while on the application of them to the use and service of mankind.-Lord Bacon. SHEEP HUSBANDRY IN THE SOUTH: Comprising a Treatise on the Acclimation of Sheep in the Southern States, and an Account of the different Breeds. Also, a Complete Manual of Breeding, Summer and Winter Management, and of the Treatment of Diseases. With Portraits and other Illustrations. By HENRY S. RANDALL. In One Volume, octavo...........................................................................$1.25 MISS LESLIE'S COMPLETE COOKERY. Directions for Cookery, in its Various Branches. By Miss LESLIE. Forty-first Edition. Thoroughly Revised, with the Addition of New Receipts. In One Volume, 12mo, half bound, or in sheep............................. $1 In preparing a new and carefully revised edition of this my first work on cookery, I have introduced improvements, corrected errors, and added new receipts, that I trust will on trial be found satisfactory. The success of the book (proved by its immense and increasing circulation) affords conclusive evidence that it has obtained the approbation of a large number of my countrywomen; many of whom have informed me that it has made practical housewives of young ladies who have entered into married life with no other acquirements than a few showy accomplishments. Gentlemen, also, have told me of great improvements in the family table, after presenting their wives with this manual of domestic cookery, and that, after a morning devoted to the fatigues of business, they no longer find themselves subjected to the annoyance of an ill-dressed dinner.-Preface. MISS LESLIE'S TWO HUNDRED RECEIPTS IN FRENCH COOKERY. A new Edition, in cloth.....................................25 cts. TWO HUNDRED DESIGNS FOR COTTAGES AND VILLAS, &c. &c., Original and Selected. By THOMAS U. WALTER, Architect of Girard College, and JOHN JAY SMITH, Librarian of the Philadelphia Library. In Four Parts, quarto......................................................................... $10 Page [unnumbered] PUBLICATIONS OF HENRY CAREY BAIRD. 9 STANDARD ILLUSTRATED POETRY. THE TALES AND POEMS OF LORD BYRON: Illustrated by HENRY WARREN. In One Volume, royal 8vo, with 10 Plates, scarlet cloth, gilt edges................................... $5 Morocco extra...............................$........................................$7 It is illustrated by several elegant engravings, from original designs by WARREN, and is a most splendid work for the parlour or study.-Boston Evening Gazette. CHILDE HAROLD; A ROMAUNT BY LORD BYRON: Illustrated by 12 Splendid Plates, by WARREN and others. In One Volume, royal 8vo, cloth extra, gilt edges................................... $5 Morocco extra.......................................................................... $7 Printed in elegant style, with splendid pictures, far superior to any thing of the sort usually found in books of this kind.-N-. Y. Courier. SPECIMENS OF THE BRITISH POETS. From the time of Chaucer to the end of the Eighteenth Century. By THOMAS CAMPBELL. In One Volume, royal 8vo. (In press.) THE FEMALE POETS OF AMERICA. By RUFUS W. GRISWOLD. A new Edition. In One Volume, royal 8vo. Cloth, gilt.............................................................. $2.50 Cloth extra, gilt edges................................................................... $3 M orocco super extra...........................................................................$4.50 The best production which has yet come from the pen of Dr. GRISWOLD, and the most valuable contribution which he has ever made to the literary celebrity of the country.-N. Y. Tribune. THE LADY OF THE LAKE: By SIR WALTER SCOTT. Illustrated with 10 Plates, by CORBOULD and MEADOWS. In One Volume, royal 8vo. Bound in cloth extra, gilt edges...................................................................................................$5 Turkey morocco super extra....................................................................$7 This is one of the most truly beautiful books which has ever issued from the American press. LALLA ROOKH; A ROMANCE BY THOMAS MOORE: Illustrated by 13 Plates, from Designs by CORBOULD, MEADOWS, and STEPHANOFF. In One Volume, royal 8vo. Bound in cloth extra, gilt edges... $5 Turkey morocco super extra........................................................... 7 This is published in a style uniform with the "Lady of the Lake." Page [unnumbered] 10 PUBLICATIONS OF HENRY CAREY BAIRD. THE POETICAL WORKS OF THOMAS GRAY: With Illustrations by C. W. RADCLIFFE. Edited with a Memoir, by HENRY REED, Professor of English Literature in the University of Pennsylvania. In One Volume, 8vo. Bound in cloth extra, gilt edges..............$3.50 Turkey morocco super extra...............................................................$5.50 It is many a day since we have seen issued from the press of our country a volume so complete and truly elegant in every respect. The typography is faultless, the illustrations superior, and the binding superb.Toy WThig. We have not seen a specimen of typographical luxury from the American press which can surpass this volume in choice elegance.-Boston Courier. It is eminently calculated to consecrate among American readers (if they have not been consecrated already in their hearts) the pure, the elegant, the refined, and, in many respects, the sublime imaginings of THOMAS GRAY.-icwhmond Whig. THE POETICAL WORKS OF HENRY WADSWORTH LONGFELLOW: Illustrated by 10 Plates, after Designs by D. HUNTINGDON, with a Portrait. Ninth Edition. In One Volume, royal 8vo. Bound in cloth extra, gilt edges..............................................................................................$5 Morocco super extra............................................................................$7 This is the very luxury of literature-LONGFELLOWSo charming poems presented in a form of unsurpassed beauty.-Neal's Gazette. POETS AND POETRY OF ENGLAND IN THE NINETEENTH CENTURY: By RUFUS W. GRISWOLD. Illustrated. In One Volume, royal 8vo. Bound in cloth..............................................................................$3 Cloth extra, gilt edges.......................................................................$3.50 M orocco super extra...............................................................................$5 Such is the critical acumen discovered in these selections, that scarcely a page is to be found but is redolent with beauties, and the volume itself may be regarded as a galaxy of literary pearls.-Democratic Review. THE POETS AND POETRY OF THE ANCIENTS: By WILLIAM PETER, A. M. Comprising Translations and Specimens of the Poets of Greece and Rome, with an elegant engraved View of the Coliseum at Rome. Bound in cloth..........................................................$3 Cloth extra, gilt edges........................................................................$3.50 Turkey morocco super extra....................................................................$5 THE FEMALE POETS OF GREAT BRITAIN. With Copious Selections and Critical Remarks. By FREDERIC ROWTON. With Additions by an American Editor, and finely engraved Illustrations by celebrated Artists. In One Volume, royal 8vo. Bound in cloth extra, gilt edges.............................................................................................$5 Turkey m orocco..................................................................................... $7 Mr. ROWTON has presented us with admirably selected specimens of nearly one hundred of the most celebrated female poets of Great Britain, from the time of Lady Juliana Bernes, the first of whom there is any record, to the Mitfords, the Iewitts, the Cooks, the Barretts, and others of the present day.-Hunt's.Merchants' Magazine. Page [unnumbered] PUBLICATIONS OF HENRY CAREY BAIRD. 11 THE TASK, AND OTHER POEMS. By WILLIAM COWPER. Illustrated by 10 Steel Engravings. In One Volume, 12mo. Cloth extra, gilt edges................................................$2 M orocco extra......................................................................................$3 THE POETICAL WORKS OF NATHANIEL P. WILLIS. Illustrated by 16 Plates, after Designs by E. LEUTZE. 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Cloth............................................................................................ $1.00 Page [unnumbered] Page [unnumbered] On what Principal Will the compound interest for 3 years at 5%?This is an Expert-Verified Answer
Hence, the principal is ₹ 400.
What will be the compound interest earned on an amount of 16800 in 1 3 4 years at the rate of 6.25% per annum?Therefore, compound interest earned = 1886.71875.
At what rate percent per annum will a sum of 7500 give rupees 927 as compound interest in 2 years?answer is 6%p.a.
What will be the compound interest on 700 for 2 years at 20% per annum?This is Expert Verified Answer
Therefore, compound interest = Amount - Principal = ₹ 931.7 - ₹700 = ₹ 231.7.
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At what percent p.a. will the compound interest on 27000 become 5768 in 3 years
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