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About Google Book Search Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers discover the world's books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web at |http : //books . google . com/| PRACTICAL MODEL CALCULATOR, ENGINEER, MECHANIC, MACHINIST, MANUPACTUEER OF ENGINB-WOEK, NAVAL ARCHITECT, MINER, AND MILLWRIGHT. OLIVER BYRNE, and Editor of Out "DUUtmar^ of Machina, MechaniiSi Sngiiuwyrk, tini Engtrteer nrof "ISe (XmsanOm for MaAinUU, J/wJanici, und Ei\gintar>i' AvOicTimi Incinii if a A'eio &lea«, (crnml " Tin OifcuftM iif Jbrm," a luisliiiite far (fts diffirenlM aiid In/rgral Oolculus ; " The EWmtnts of Euclid by Coiwirs," and numemju othiT Jfff(fiemfl«caZ and Sfwhanicol Works. Sunnyor-Gentral of the E«^h SetOemenls in (As Iblkhad Jjfcs, Profeimr iff Malli£itiatici, G^gt of Cioit Enffmurs, LoR^en^ PHILADBLPniAL PUBLISHED BY HENRY CAREY (SUCCESSOR TO E. L. CAKEY,) hv Google ling to Uw ut of CopethSt Id tbe yc HENRT CARET BAIRD, le District Court for tho EaEtccn Bist hv Google THE PRACTICAL MODEL CALCULATOR. WEIGHTS Am) UEASUBES. TilE UNIT OF LENGTH. The Yaed. — If a pendulum vibrating seconds in vacuo, in Phi- ladelphia, be divided into 2509 equal parts, 2310 of Bueh equal parts is the length of the standard yard ; the measures are taken on brass rods at the temperature of 32° Fahrenheit. This yarJ will not be in error the ten-miiiionth part of an inch, 2310 : 2509 as 1- to 1-086142 nearly. THE trail OP 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 Q-allon, 231 cubic inches, contains 8-3388822 pounds avoir- dupois, equal 58372-1754 grains troy of distilled water, at 39°-S2 Pah. ; the barometer at 30 inches. ■UNIT OP 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 ilfeire = 3-2808992 English /eei linear measure = 39-3707904 inches. For MuUiples the following Greek words are used : De.ea for 10 times. Hecto — 100 times. Kilo — 1000 times. Slyria— 10000 times. Millimetre For Divisors the following Latin words are used : Bed for the lOM part. C'enti — lOOM part. MiUi — 1000i/( part. Thas 3. Kilometre = 1000 metres. metro 1000 The square Deea Metre, called the Are, is the element of land easure in France, which = 1076-42996 square feet English. The Stere is a cubic m_etre = 35-316582 cubic feet English. hv Google 6 THE PRACTICAL MODEL CALCULATOR. The Litre for liquid measure is a cubic decimetre = 1-760TT 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 cuhic centimetre of distilled water, or of a millilitre, and therefore equal to 15'434 grains troy. The kilogramme is the weight of a cuhic 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 cuhic inches ; and the English com bushel contains eight such gallons, or 2218*192 cuhic inches. apothecaries' weight. Grains marked gr. 20 Grains make 1 Scruple ■— sc. or 3 3 Scruples— 1 Dram — dr. or 5 8 Drams — 1 Ounce — oz. or ,? 12 Ounces — 1 1'ound — lb. or ft. gr. sc. 20 = 1 dr. 60 = 3=1 OK. 480 = 24 = 8 = 1 Ih. 5700 = 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. Drams marked dr. 16 Drams make 1 Ounce — oz. 16 Ounces — 1 Pound — lb. 28 Pounds — 1 Quarter — qr. 4 Quarters — 1 Hundred Weight... — cwt. 20 Hundred Weight... — 1 Ton — ton. dr. oz. 16 = 1 lb. 25(5 = 16 = 1 qr. 7168 = 448 = 28 = 1 cwt. 28672 = 1792 = 112 = 4 = 1 ton. 673440 = 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. JWe, that 1 lb. avoirdupois = 14 11 15i troy. 1 0/. — = 18 5i — 1 dr. — = 1 Si — hv Google ■WEIGHTS AND MEASURES. Grains marked Gr. 24 Grains make 1 Pennyweight Divt. 20 Pennyweights 1 Onnce Oz. 12 Ounces 1 Pound Lb. Gr. Dwt. 24 = 1 Oz. 480 = 20 = 1 Lb. 5760 = 240 = 12 = 1 By tliia -weight arc weighed Gold, Silver, and Jewels. LONa MEASURE. Barley-corns make 1 Inch marked In. - - ■ . _ Ft. laches. Feet Feet Yards and Furlongs Miles G Milca nearly.., In. 12 = 198 = 7920 = 63360 = 5280 1 Foot... . ITard 1 Fathom . 1 Pole or Bod.. ■ 1 Furlong ■ IMilo ■ 1 League ■ 1 Degi'ce Yd. Fth. PI. Far. Mile. Lea. Deji- or ° Ml 5^ = 1 Fur. . 220 - 40 = 1 Mile. - 1760 - 320 - 8 - 1 CtOTH HEASl'EE. 2 Inches and a quarter.... make 1 Kail mariied Kl. 4 Kails .. 3 Quarters 4 Quarters 5 Quarters 4 Qrs. 1', Inoli.. Qr. EF. Yd. EE. BS. SQUARE MEASURE. ..make 1 Sq. Foot marked Ft. 1 Quarter of a Yard.. 1 Ell Flemish.. 1 Yard 1 Ell English... 1 Ell Scotch ... ISq. Yard.. 1 Sq. Pole .. lEood 1 Acre Yd. Pole. . Yil. Sq. PI. 144 Square Inches. 9 Square Feet — 30J Square Yards — 40 Square Poles — 4 Roods — Sq. Inc. Sq. Ft. 144 - 1 1296 = 9 = 39204 - 272i . 1568160 - 10890 . 6272640 - 43560 . 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, &e. 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. . 1210 = 40-1 . 4840 - 100 = 4 Acr. b,Google THE PRACTICAL MODEL CALCULATOR. , OR COEd MEASURE. 2 Pints 2 Quarts ... 2 Pottles... 2 Gallons... 8 Bushels..-. 5 Quarters... 2 Weya jl Quart marked Qt. 1 Pottle.. 1 Gallon IPeck 1 Bushel 1 Quarter, 1 Weigh or Load... 1 Last Pec. 1 Bu. Qr. 612 = 64 = 32 = 8 - 1 2560 = 320 = 160 = 40 = 5 = 5120 = 640 = 320 = 80 = 10 ■■ ^YcJ. Pot. Gal. Pec. Wey. Last. 2 Pints. 2 Quarts — 1 G( 42 Gallons — 1 Tierce 63 Gallons or IJ Tier.. ■— 1 Hogshead. i Tierces — 1 Puncheon. .make 1 Quart marked Qt. ■ ■ Gal. Tier. Hhd. Pu 2 Hogsheads... 2 Pipes — 1 Pipe or Butt.. — ITun Pi. Tun. Pts. 2 = Qta. 1 Gal. 4 = 1 Tier. 168 = 42 = 1 Hhd. 252 = 63 = IJ = 1 Pun. 504 = 672 = 1008 = 504 = 126 2016 = 1008 = 252 - IJ = 1 Pi. = 2 = li. = 1 Tun. 2 Pints 4 Quarts 36 Gallons — 1 Barrel and a half.... — 2 Barrels — 2 Hogsheads — 2 Butts ~ Pts. Qt. 2 = 1 8 = 4 = 288 = 144 = ALE AKD BEER IIEASUUE. make 1 Quart marked Qt. 1 Gallon . . 1 Barrel 1 Hogshead... 1 Puncheon.. ]. Butt ITun Gal. Bar. Hhd. Pun. Butt. Tun. Gal. Ear. = 1 Hhd. = 11=1 Butt. =3 -2=1 hv Google OF TIME. 60 Seconds make 1 Minute marked M. or eOMiEutes — 1 Hour — Hr. 24 Hours — 1 Day — Day. 7 Days — 1 Week — AVk. 4 Weeks — l.Montk — Mo. 1 3 Months,! Day, 6 Hours, 1 i t r v v or oo6 Days, b Hours, j See. Min. 60 = 1 Ilr. 3600 = 60 = 1 Day. 86400 = 1440 = 24 = 1 Wk. 604800 = 10080 = 168 = 7=1 Mo. 2419200 = 40320 = 672 = 28 =4 = 1 3155T600 = 525960 = 8T66 = 365^ = 1 Year. Wk.Da.Hr. Mo. Da.Hr. Or 52 1 6 = 13 1 6 = 1 Julian Year. Da. Hr. M. See. 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 aolected fixed star at the same instant of time; although this coincidence may be but momentary, still this coincidence con- tinually 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 = 866-256365160 - 1 or 365-256365160 mean solar days, whence sidereal day : mean solar day : : 365-256365160 : 366-266365160 : : 0-997269672 : 1 or as 1 : 1-002737803, when 23 hours, 56 imnutes 4-0996608 sec. of mean solar time = 1 sidereal day; and 24 hours, 3 minutes, 56-5461797 see. of sidereal time = 1 mean solar day. The true solar day is the interval hetween two successive coinci- dences of the sun with a fixed object on the earth's surface, bring- ing 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 ahont 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. hv Google ARITHMETIC. Arithmetic is the art or Bcience of numbering; being that branch of Matlicmatics which treats of the nature and properties of numbers. When it treats of whole numbers, it is called Com- mon Arithmetic ; but when of broken numbers, or parts of num- bers, it is called Fractiom. 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. JVumher is either simply one, or a compound of severul units ; as one naan, 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 distinguiohed from Fractions, which are broken numbers, or parts of numbers ; as one-half, two-thirda, or three-fourths, NOTATIOIT AND NTTMERATION. NOTATIOH, or Numeration, teaches to denote or express any pro- posed 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, 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: 6 7 6 7 6 5 7 (5 8 7 hv Google NOTATION AKD KUMERATIOS. 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 tea 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 thus — one 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 tri- millions, 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 doc- Periods. Quadrill.; TriUions; Billions; Millio Half-per. th. un. th. un. Figures. 123,456; 789,098; 765,432; 101,234; 667,890. KuMBRATiON 13 the reading of any number in words that is pro- posed or set down in figures. Notation is the setting down in figures any number proposed in words. OF THK EOMAN 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 ; YioT five; Xiorten; Lfor fifty ; Cfor a, hundred; D for five hun- dred ; M for a thousand. The other numbers they expressed by i repetitions and combinations of tliesc, after the following hv Google TUB PRACTICAL MODEL CALCULATOR. - ir. = 111. = nil. = v. = VI. = VIT. = VIII. = IX. = x. = h. >c. = DorI 3 M or C = MM. = Vorl = yi. = L or = LX. = C_^or = M^or = MM. kc. As often as any character is repeated, BO many times is its value repeated, A less character before a greater diminishes its value. A less character after a greater in- creases its value. For every annexed, thia ten times as many. For every C and 0, placed one at end, it becomes ten times as much. A bar over any number incr 1000 fold. eh it CCIOO. 1000. ccciooo. ccccioooo. 50^ 100 = 500 = 1000 = 2000 = 5000 = 6000 = 10000 = 50000 = 60000 = 100000 = 1000000 = 2000000 = &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 phis, or addition. — minus, or subtraction. X multiplication. -j- division. Thus, 5 + 3, denotes that 8 G — 2, denotes that 2 7x3, denotes that 7 8 -i- 4, denotes that 8 : :: : proportion. = equality. V' square root. ^ cube root, kc. to he added to 5 = S. to be taken from 6^4. to be multiplied by 3 = 21. to be divided by 4 = 2. 6, shows that 2 is to 3 as 4 is to 6, and thus, 2x6=3x4. 6 + 4 = 10, shows that the sum of 6 and 4 is equal to 10. ■/Z, or 3 J denotesthesquareroot ofthenumher 3 — 1'7320508. ■^5, or 5*, denotes the cube root of the number 5 = 1-709976. 7^ denotes that the number 7 is to be squared = 4f>. 8', denotes that the number 8 is to be cubed = 512. hv Google ETILE OF THREE. 13 RVIE OP IHE£)£. The Role 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 ia when more requires less, or less requires more. As in this : if 3 men dig a certain quantity of trench in 14 hours, in Low 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 Pro- portion, ia 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 multiply- ing the 2d and 3d terms together, and dividing the product by the 1st term. Of the three given numbers, two of them contain the supposi- tion, and the third a demand. And for stating and working ques- tions 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 ia to be ; making the other number of sup- position 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. hv Google 14 TEE PRACTICAL MODEL CALCULATOR. Note, If the first and third tenna consist of different denomina- tioRB, reduce them both to the same ; and if the second terra 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 denomina- tion. 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 c^uotients 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 t da. men. da. men. As 15 : 5 : : 24 : 8 Ans. 5 15)iM(8Answer, 120 COMPOITND 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 terra ; 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 mid- dle term, in the same order aa 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 .1 stating, in the same order as they stand, finding a fourth hv Google OF COMMON FRACTIONS. 15 term from them ; and so on, as far as there are any numbers in the genera! stating, making always the fourth number resulting from each simple stating to he 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 long in 8 days, when 16 men can dig 54 yards in 6 days ? Creneral stating. yds. 54 ; 16 men ; : 135 yds. days 8 6 days 482 810 16 4860 810 432)12960(30 Ans. by one operation 1296 The same It/ two operatiom. Ist. 2d. As 54 ; 16 : ; 135 : 40 As 8 ; 40 ; ; 6 ; 30 16 6 810 8 ) 240 ( 30 Ans 136 24 54 ) 2160 (40 216 OF commoN 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 numberSj placed one below the other, with a line between them: , 3 numerator ) , . , . , , ^''"''Tdenominator | ^^''"^ '^ "^'"'^ throe -fourths. The Denominator, or number placed below the line, shows how many equal parts the whole quantUy is divided into ; and repre- sents 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. hv Google IG THE PBACIICAIi MODEL CALCULATOR. Fractions are either Proper, Improper, Simple, Compound, or Mixed. A Proper Fraction is when the numerator is less than the deno- minator ; as J, or ^, or f , &c. An Improper Fraction is when the numerator is equal to, or exceeds, the denominator; as |, or f, or I, &c. A Simple Fraction is a single expression denoting any number of parts of the integer ; as ^, or |. A Compovtnd Fraction is the fraction of a fraction, or several fractions connected with the word of between them ; as J of ^, or I of I of 3, &e. A Mixed Number is composed of a whole number and a fraction together; as 3J, or 12|, &c. A whole or integer number may he expressed like a fraction, by writing 1 below it, as a denominator ; so 3 is |, or 4 is *, &e. A fraction denotes division ; and its value is equal to the quo- tient obtained by dividing the numerator by the denominator; so '4^ is equal to 3, and ^/ is equal to 4. Hence, then, if the numerator be less than the denominator, the value of the fraction ia 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 fraction is greater than 1. EEDTJCTION OF FRACTIONS. Reduction 01? Fractions is the bringing them out of one form or denomination into another, commonly to prepare them for the opera- tions of Addition, Subtraction, &c., of which there are several cases. The Common Measure of two or more numbers is that number which will divide them both without a remainder ; so 3 is a com- mon 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 mea- sure 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, divid- ing always the last divisor by the last remainder, till nothing remains ; then shall the last divisor of all he the greatest common measure sought. When there are more than two numbers, find the greatest com- mon 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. hv Google REDUCTIOS OF FRACTIONS. 17 To find the greatest commoE 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 3(S)54(1 108 36 54)108(2 18)36(2 108 36 So ttat 18 is the answer required. To ahhreviate or reduce fractions to their lowest tenm. EULE. — Divide the terms of the given fraction by any number that will divide them without a remainder ; then divide these quo- tients 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 com- mon 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 num- ber, whatever it he, will give another fraction equal to the former, is evident. And when those dirisions are performed as often as can be done, or when the common divisor is the greatest possible, the terms of the resulting fraction must ho the least possible. 1. Any number ending with an even number, or a cipher, is divi- sible, or can be divided by 2. 2. Any number ending with 5, or 0, ia 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 tliree 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. Ali prime numbers, except 2 and 5, have either 1, 3, 7, or 9, in the place of units ; and ail other numbers are composite, or can bo divided. b,Google 18 THE PRACTICAL MODEL CALCULATOB. 10. When numbers, with a sign of addition or subtraction between them, are to be divided bj any number, then each of those num- 10 + 8 — 4 bers must be divided by it. Thus, ^ ' ' ' ~^ + 4 — 2 = 7. 11, But if the numbers have the sign of mnltipli cation between , ,. . , , ^, 10 X 8 X 3 them, only one of tnem must be divided. Thus, — c v 9 — ~ 10 X 4 X 3 10 X 4 X 1 10 X 2 X 1 20 6x1 ~ 2x1 ~ 1x1 - I -^^■ Reduce J|J to its least terms. IM = i^ = n= iS = A = l> the answer. Or thus : 144 ) 240 ( 1 Therefore 48 is the greatest common measure, and 144 48 ) Jll = f the answer, the same as before. "96)144(1 _96 "48)96(2 96 To reduce a mixed number to iU equivalent improper fraction. Exile. — Multiply the whole number by the denominator of the fraction, and add the numerator to the product ; then set that sum above tlie denominator for the fraction required. Reduce 23§ to a fraction. Or, 23 (23 X 5) J-_2 _ 117 ^ 5 " "" a ■ 115 2 Trf 5 To reduce an improper fraction to its equivalent whole or mix-ed number. Exile. — Divide the numerator by the denominator, and the quo- tient will be the whole or mixed number sought. Reduce ^ to its equivalent number. Here V or 12 -- 3 = 4. Reduce y to its equivalent number. Here V or 15 - 7 = 2J. Reduce ^° to its equivalent number. Thus, 17)749(44Jf 68_ 69" So that W = 44A hv Google EEDUOTION OP PEACTIONS. 19 To reduce a loliola number to an equivalent fraction, having a (fiven denominator. Rule. — Multiply tlie ivhole number by the given denominator, then set the product over the said denominator, and it will form the fraction reijuired. Reduce 9 to a fraction whoae denominator shall be 7. Here 9x7 = 63, then f is the answer. For V = 63 -5- 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 wiU 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 arc common, they may be omitted. Reduce I of I of I to a simple fraction. 1x2x3 6 _ 1 ^^^^ 2x3x4~24~4' 1x2x3 1 ' 2x3x4 ~ 4' ^^ oi^itting the twos and threes, f of § of \l to a simple fraction. 2 X 3 X 10 _ ^ _ ^ _ £ ^'^'■'^ 3 X 5 X 11 ~ 1(35 ~ 33 ~ 11" ; 3 X 10 4 , . t- „ -.-I = ^pr, the same as before. To reduce fractions of different denominators to equivalent frac- tions, having a common denominator. Rule, — Multiply each numerator into all the denominators ex- cept its own for the new numerators ; and multiply all the denomi- nators 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 com- pound fractions, they must be reduced, by their proper rules, to the form of simple fractious. Reduce ^, f, and -J to a common denominator. 1 X 3 X 4 = 12 the new numerator for |. 2x2x4 = 16 for2 3 X 2 X 3 = 18 for|. 2 X 3 X 4 = 24 the common denominator. Therefore, the equivalent fractions are |f, ^, and ^|. Or, the whole operation of multiplying may be very well per- formed mentally, and only set down the results and given fractions thus : i, I, f = ^1, i-l, il = -fe ^, ^i, by abbreviation. hv Google 20 THE PRACTICAL MODEL CALCULATOK, Wlicn the denominators of two given fractions have a coniraon measure, let tliem lie divided by it ; then multiply the terms of each given fraction by tho quotient arising from the other's deno- minator. When the lesa denominator of two fractions exactly divides the greater, multiply the terms of that which hatli the less denominator hy the quotient. When more than two fractions are proposed, it is sometimes con- venient first to reduce two of them to a common denominator, then these and a third ; and so on, till they be all redaced to their Q denominator. To find the value of a fraction in parts of the integer. Rule. — Multiply the integer hy tho numerator, and divide the product by the denominator, by Compound Multiplication and Division, if the integer be a compound quantity. Or, if it he a single integer, multiply the numerator by the parts in the next inferior denomination, and divide the product hy the denominator. Then, if any thing remains, multiply it by the parts in tho 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 dwta. What ia the value of ^ of a cwt.? 1 qr. 7 lb. What is the value of | of an acre? 2 ro. 20 po, What is the value of j^ of a day ? 7 hrs. 12 min. To reduce a fraction from one denomination to another. Rule. — Consider how many of tho less denomination make one of the greater; then multiply the numerator hy that num- ber, if the reduction be to a less name, or the denominator, if to a greater. Reduce f of a cwt. to the fraction of a pound. ADDITION OF FRACTIONS, To add fractions together that have a common denominator. RuLT^. — Add all tho numerators together, and place the sura 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 denomi- nation. To add I and \ together. Here I + J = I = If To add I and \ together. I + ^ = ii! + If = lu — la"- To add \ and 7j and ^ of | together. i + 7i -i- J of f == i -I- ij' -h 1 = i + V + ^ ^ V = 8|. hv Google RULE OF THRBB IN FRACTI0H3. 21 SUETRACTION OF FRACTIONS. Rule. — Prepare the fractions tfie same aa for Addition ; then sub- tract the one numerator from tte other, and set the remainder over the common denominator, for the diEFerence of the fractions sought. To find the difference between | and J. Here | - j = | = |. To find the difference between | and f 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 nume- rator, and all the denominators together for a denominator, which will give the product required. Required the product of | and f. Here |- x J = {], ~ }. Or, I X f = ^ X ^ = 1. Required the continued product of |, 3^, 5, and f of f . 2 13 5 3 3 _ 13 X 3 _ 39 _ Here gX ^ ^i^4^5— 4x2 ~ 8 ~ '* DIVISION OF FEACTIONS. 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 ^ by ^. Here \* -^ f = i = I3, by the first method. Divide | by -,%. Here 5 -^ 1 5 = s X '^ = I X ^ = =5= = 4^, 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 tlie first, as in the Rule of Three in whole If ^ of a yard of velvet cost | of a dollar, what will /g of a yard cost? 3 2 5 8 2 5 Hero o '■ T '■ '■ Ti^ '■ Ti y- -B XT-5 = 'Lofa dolkr. 8 5 16 3 5 lo ^ hv Google 22 IHE PRACTICAL MODEL CALCULATOK. DECIMAL FKACTIONS. A Decimal Fraction ia that which has for its denominator a unit (1) with as many ciphers annexed as the numerator has places ; and it is usnally expressed by setting down the nnmerator only, with a point hefore it on the left hand. Thus, f^ is '5, and ^ is ■25, and ^ is "075, and ^^-^ is -00124 ; where ciphers are pre- fixed to make np as many places as are in the numerator, when there is a deficiency of figures. A mixed numher is made up of a whole number with some deci- mal fraction, the one being separated from the other by a point. TiiuSj 3-25 IB the same as 3^, or f|^. 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 = -^ or J. Eut if they are placed on the left hand, they decrease the value in a tenfold proportion. Thus, ■5 is ^ or 5 tenths, but -05 is only jl^ or 5 hundreths, and -005 is but Tiftj or 5 thousandtlis. The first place of decimals, counted from the left hand towards the right, is called the place of primes, or lOths ; the second is the place of seconds, or lOOths ; the third is the place of thirds, or lOOOths ; and so on. For, in decimals, as well as in whole num- bers, 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: ^a i i i i I -U I I I I I 1 3333333 333333 ADDITION OP DECIMALS. Rule. — Set the numbers under each other according to the value of their places, like as in whole numbers ; in which state the deci- mal 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- ■6241T 14-16 5299-29877, the sum. hv Google MULTIPLICATION OF DECIMAL?, 23 The Bum of 376-25 + 86-125 + 637-4725 + 6-5 + 41-02 + S58-865 = 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 DECIJIALS. EuLE. — Place the numbers under each other according to the value of their places, as in the last rule. Then, heginning at the right hand, subtract as in whole numhers, and point off the deci- mals as in Addition. To find the difference hetiveen I 91-73 91.73 and 2.138. 2-138 I 89-592 the difference. The difference between 1-9185 and 2-73 = 0-8115. The difference hetween 214-81 and 4-90142 = 209-90858. The difference between 2714 and -916 = 2713-084. MULTIPLICATION OF DECIMALS. EuLE.- — Place the factors, and multiply them together the same as if they were whole numbers. Then point off in the product just as many places of decimals as there are decimals in both the fac- tors. But if there be not so many figures in the product, then supply the defect by prefixing ciphers. Multiply 79-347 by 23-15, and ire Lave 1836-88305. Multiply -G3478 by -8204, and we have -520773512. Multiply -385746 by -00464, and we have -00178986144. CONTKACTION I. To multipli/ decimals hy 1 with any number of ciphers, as 10, or 100, or 1000, ^e. 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. COSTRACnON II. To- contract the operation, so as to retain only as many decimals in the product as may be thought necessary, when tJie product teouJd naturally contain several tnore 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 th, Multiply -321096 by -2465 1605480 1926576 1284384 642192. ■0791501640 the product. hv Google 24 THB PRACTICAL MODEL CAICTJLATOR. last ill the product ; and dispose of tlie 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 he the product as required, com- monly to the nearest unit in the last figure. To multiply 27-14986 by 9241036, 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 81 44958 2714 986 2715 542997 2 14 2443487 4 2508-9280 2508-9280 650510 DIVISION OF DECIMALS. Rule. — Divide aa in whole numbers ; and point off in the quo- tient 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 re- quires, 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 I -2685)27-00000 (100-55805 1282 15000 294 15750 1150 23250 769 17700 537 15900 000 24750 Divide 234-70525 by 64-25. 3-653. Divide 14 by -7854. 17-825. Divide 2175-68 by 100. 21-75G8. Divide -8727587 by -162. 5-38739. When the divisor is an integer, with any number of ciphers an- nexed ; cut off those ciphers, and remove the decimal point in the hv Google REDUCTION or DECIMALS, dividend as many places farther to the left as there are ciphers cut off, prefixing ciphers if need he ; then proceed as before. Divide 45-5 by 2100. 21-00 ) -455 ( -0210, &e. 14 CONTRACTION II. Hence, if the divisor be 1 with ciphers, as 10, or 100, or 1000, &c. ; then the quotient 'will be found by merely moving the decimal point in the dividend so many places farther to the left as the di- visor has ciphers ; prefixing ciphers if need bo. So, 217-3 -H 100 = 2-173, and 419 -^ 10 = 41-9. And 5-16 H- 100 = -0516, and -21 -^ 1000 = -00021. CONTRACTION III. When there are many figures in tKe 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 quo- tient, and find how many times they may he contained in the first figures of the dividend, as usual. Let each remainder be a new dividend ; and for every such divi- dend, leave out one figure more on the right hand side of the di- visor ; remembering to carry for the increase of the figures cat 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 usaal till the number of figures in the divisor be equal to those remaining to he found in the quotient, after which begin the contraction. Divide 2508-92806 by 92-41035, so as to have only four deci- mals in the quotient, in which case the quotient will contain six figures. Contracted. Common way. 92-4103,5)2508-928,06(27-1498 660721 13849 4608 912 92 -4103,5) 2508-928,06 {27-1408 66072106 13848610 46075750 91116100 70467850 55395T0 REDUCTION OF DECIMALS. To reduce a common fraction to its equivalent decimal. KoLE. — 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. hv Google THE PRACTICAL MODEI, CALCULATOR. 1 decimal. 24 =4 X 6. Tlica4)7- G) 1-750000 ■291066, kc. f reduced to a decimal, is -STo. ^ reduced to a, decimal, is '04. ^ reduced to a decimal, is -015625. iah reduced to a decimal, is -OTloTT, tc. CASE II. To find t/ie value of a decimal in terms of the inferior denominations. Rule. — Multiply tlie decimal by the number of parts in the next lower denomination ; and cut off as many places for a remainder, to the right baud, as there are places in the given decimal. Multiply that remainder by the parts in the next lower denomi- nation 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;— S dwts. What is the value of -4694 lb. troy :— 5 oz. 12 dwt. 15-744 gi What is the value of -625 cwt. :— 2 qr. 14 lb. What is the value of -009943 mOes :— 17 yd. 1 ft. 5-98848 ii What is the value of -6875 yd. :— 2 qr. 3 nl; 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 higher denominations. Rule. — Divide by the number of parts in the next higher de- nomination ; continuing the operation to as many higher denomi- nations 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 1 dwt. 12 0-05 oz. 10-Q04166, &c. lb. Reduce 7 dr. to the decimal of a pound avoird.: — -02734375 lb. Reduce 2-15 lb. to the decimal of a cwt. : — ■01919(3 civt. Reduce 24 yards to the decimal of a mile: — -013636, kc. miles. Reduce -056 poles to the decimal of an acre i — -00035 ac. Reduce 1-2 pints of wine to the decimal of a Lhd. :— -00238 hhd. Reduce 14 minutes to the decimal of a day : — -009722, kc. da. Reduce -21 pints to the decimal of a peck: — -013125 pec. When there are several numbers, to be reduced all to the decimal of the highest. Set the given numbers directly under each other, for dividends, proceeding orderly from the lowest denomination to the highest. hv Google 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 ob- serving to set the quotient of each division, as decimal parts, on the right hand of the dividend nest below it ; so shall the last quo- tient be the decimal required. Reduce 5 oa. 12 dwts. 16 gr. to lbs. :— -46944, &c. lb. RULE OP 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 ivhole numbers. Any of the convenient examples in the Rule of Three or Rule of Five in Integers, or Common I'ractions, 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 I of a yard of velvet cost § of a dollar, what will i\ yd. cost ? yd. S yd. $ 1 = -.370 -375 : -4 :: -312,5 : -333, &c. ^ 2 = -4 -375 ) -12500 ( -833333, 33j cts. 1260 125 A = -3125. DTJODECIMALS. Duodecimals, or Caoss Multiplication, is a rule made use of by workmen and artificers, in computing the contents of their works. Dimensions aie uauiUy taken infect, 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 con- tents. Rule — Set down the two dimensions, to be multiplied together, one undei the other, so that feet stand under feet, inches under inches, i.c Multiply each teim in the multiplicand, beginning at the lowest, by the feet m the multiplier, and set the result of each straight un- der 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 tho multiplicand ; omitting, however, what is below parts of inches, only carrying to these the proper number of units from the lowest denomination. hv Google 28 THE PRACTICAL MODEL CALCULATOR. Or, instead of multiplying by tho inches, take such parts of the multiplicand aa these are of a foot. Then add the two linea together, after the manner of Compound Ad- dition, carrying 1 to the feet for 12 inches, wlien these come to so many. Multiply 4 f. 7 inc. Multiply 14 f. 9 inc. b y 6 4 by _4__ 6 27 6 59 _1 6^ 7 4 | 29 0^ 6^ 4| INTOLVTION. Involution is the raising of Powers from any given number, aa a root. A Power is a quantity produced by multiplying any given num- ber, called the Root, a certain number of times continually by itaelf. Thus, 2 = 2 is the root, or first power of 2. 3x2= 4 is the 2d power, or square of 2. 2x2x2= 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 HB ST NIKE I'OWERS OF KUMBERS. 1st 2d. 3d. 4th. sa. eth. 7t!i. 8tli. OIL. 1 1 1 1 1 1 1 1 1 2 4 8 16 33 64 128 256 512 3 9 27 81 243 729 2187 6561 10083 4 10 64 256 1024 4096 10384 65536 202144 5 25 125 625 3126 16626 78125 890625 1953125 6 36 216 1296 7776 46656 279986 1679616 10077696 7 49 343 2401 16807 117649 823543 5764801 40353607 8 64 512 4096 32768 262144 2097152 16777216 184217728 9 SI 720 6561 59049 531441 4782969 430467:il 387i20180 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 2^ = 4, is the 2d power of 2. 2^ = S, is the 3d power of 2. 2'' = 16, is the 4th power of 2. 540^, is the 4th power of 540 = 85030560000. hv Google 'IThen two or more powers are multiplied together, their product will he 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. 5tli. 6th. 7th. 8th. 9th. 10th. 2 4 8 16 32 64 128 256 512 1024 or, 2^ 2* 2^ 2* 2' 2' 2' 2' 2« 2'" Here, 4x4= 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 Sijuare of § is i- The 3d power of f is ^||. The 4th power of | is •^. EVOLTTTION. EvoLUTios, 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 2^ = 2 X 2 = 4 ; and 3 is the cube root or 3d root of 27, because 3^ = 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 ia 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 \/ before the power, with the index of the root against it. Thus, the third root of 20 is. expressed by -^20; and the square root or 2d root of it ia v'20, the index 2 being always omitted when the square root is designed, AYhcn the power is expressed by several numbers, with the sign + or — between them, a line is drawn from the top of th e 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. hv Google 30 THE I'KACTICAL MODEL CALCULATOR. But all roots are now often designed like powers, with fractional indices : thus, the square root of 8 is 8 , the cuhe root of 25 is 25 , and the 4th root of 45 - 18 is i5'^:;oII^ or, (45 - IS)''. TO EXTKACT THE SQUAKE 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 giv^n number, after the man- ner 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 I 450 4 416 10862 I 21T24 2 I 21724 W7ien the root is to he extracted to many jjldcea of figures, the -ivorl: may he 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 hv Google TO EXTK4CT TUB SQUABE BOOT. 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 I 100 281 1 1 1 400 281 2824 4 11900 11296 ■Mam 2 : 60400 56664 28284) 3836(1356 1008 160 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 rOB THE SQUARE ROOTS OP COMMON rRACTIOiSS AND JIIXED NUMBERS. First, prepare all common fractions hy 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 \/ab a \/b b ^ah' And this rule will serve whether the root be finite or infinite. 3. Or reduce the common fraction to a decimal, and extract its root, 4. Mixed numbers may he 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 |t is \. The root of -^^ is f a'he root of /j is 0-866025. The root of ^-^ is 0-645497. The root of 17| is 4-168333. hv Google 32 THE PRACTICAL MODEL CALCULATOR. By meana of the square root, also, may readily be found the 4th root, or the 8th root, or the 16th root, &e. ; 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 : 21Qzh-8m ( 145-037237 ( 12-0431407, the 4th root. TO EXTRACT THE CUBE ROOT. 1. Divide the page into three columns (i), (ii), (m), 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 (li) 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 (ill) ; 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 hv Google TO EXTRACT THE CUBE ROOT. 33 last root figure, placing the proiJuct as it arises below the cKvidcnd ; subtract it therefrom, annex another period, and proceed precisely as described in (3), for correcting the columns (i) and (ii). Then with the new trial divisor in (ri), and the new dividend in (iir), 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 there- fore one cipher must be annexed to the number in (i) ; two ciphers to the number in (ii) ; and another period to the dividend in (lli). 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 (ill) ; cat off one figure from the right of (il), 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. m (n) ■ ■ (in) 2 4 21035-8 (27-6049105o944 2 8 8 4 T».. 13035 2 469 11683 67 1669 1362800 7 518 1341576 74 2187.. 11224 7 4896 9142444864 816 2 2 3 5 9 6 2081555136 6 4932 2057416281 228528. . . . 331216 2286611216 331232 228594244 7453 24139855 22860923 1278982 1143046 |-8|28112 228601697 7463 2286091511 2286092314 Required the cube roots of the following numbers : — 48228644, 46666, and 16069223. 364, 36, and 247. 64481-201, «nd 28991029248. 40-1, and 3072. 128211191.55125, and -000076766625. 23405, and -0426. HifJ, and 16. |j, and 2-519842. 91i, and7f 4-5, and 1-9 b,Google 34 THE PRACTICAL MODEL CALCULATOR. TO EXTRACT ANT BOOT WHATEVER. Let N be the given power or number, n the index of the power, A the assumed power, r its root. It the required root of N. Then, aa the sum of k + 1 times A and Ji ~ 1 times K, is to the sum of Ji + 1 times N and n — 1 times A, so is the assumed root r, to the required root R. Or, as half the said sum of n + 1 times A and n — 1 times N, ia to the difference between the given and assumed powers, so is tlie assumed root r, to the diflerence between the true and assumed roots ; which difference, added or subtracted, as the case rer.uivcs, gives the true root nearly. Thatis,(w + l).A + (K-l).N:{w + l).N + {ji-l)-A::r:R. Or, (« + 1) . |A + (n - 1) . JN : A 02 N : : »- : R ai 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 Mth power for the assumed power A. To extract the 5th root o/ 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. Hcnco then we have, N = 21035-8; r = 7-3; n = 5; J . (™ + 1) = 3; J.(h - 1) = L'- A = 20730-716 N~A = 305-0S4 A- 20730-710 N = 21035-8 3 2 3 A = 62192.148 42071-6 2N- 42071-6 As 104263-7 : 305-084 : 7-3 ; : 7-3 : -0213605 915262 2186588 104263-7)2327-11321 14184 3758 630 5 ; -0213605, the Jifforeiice. 7-3 = r add 7-321360 = E, the root, true to the last figure. The 6th root of 21035.8 The 6th root of 2 Tho 7th root of 21035-8 The 7th root of 2 The 9th root of 2 is 5-2.:.4037. is 1-122462. is 4-145092. is 1-104089. is l-08i:059. OF RATIOS, PEOPORTIONS, AND PEOGEESSIOKS. Numbers are compared to each other in two different ways : tho one comparison considers the difference of the two numbers, nnd is named Arithmetical Relation, and the difference sometimes Arithmetical Ratio : the other considers their quotient, and is ca!le!.t hv Google ARITHMETICAL PEOPORTION 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 ; hut the geometrical ratio is f 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 arith- metical proportionals, because 4 — 2 = 8 — 6 = 2; and the two cou- plets 4, 2 and 6, 3 are geometrical proportionals, because J = § = 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, 1 = 1, 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 nest 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 ^ = ^ = 2, but § = ^, 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 he continued, and the num- bers 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 ^ = ^ = ^ = 2, all the same ratio. When the following terms of a Progression exceed each other, it is called an Ascending Progression or Scries ; 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. AllITHMETICAL PROPORTION AND PROGRESSION. The first and last terms of a Progression are called the Extremes ; and the other terms lying between them, the Moans. The moat useful part of arithmetical proportions is contained in the following theorems : TiiEOBEM 1. — If four quantities he in arithmetical proportion, the sum of the two extremes will he equal to the sum of the two means. Thus, of the four 2, 4, 6, 8, here 2 + 8=4 + 6 = 10. hv Google 36 THE PRACTICAL MODEL CALCULATOR. Theorem 2. — In any continued arithmetical progression, tlie sum of tlie 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 numher 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 ia 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 com- mon difference is 2, and one leas 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 numher of terms. Theorem 4. — The sum of all the terms of any arithmetical pro- gression 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 tho terms gives double the sum of all the terms in the series. This is made evident by setting tho 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, Ij the sums are, 16 + 16 + 16 + 16 -f- 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 differ- ence, and the sum of all the terms, when any three of them are given, as in the following Problems : PROBLESI I. Criven the extremes and the numher 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 2)198 99 = the sum. hv Google ARITHMETICAL PEOPOIITION ASD PROGRESSION. 37 The strokes a clock strikes in one whole revolution of the index, or in 12 hours, is 78. PROBLEM n. Given the extremes, and the number of terms; to find the common difference. EuLE. — Subtract the less extreme from the greater, and divide the remainder hy 1 less than the numher of terms, for the common difference. The extremes being 3 and 19, and the number of terms 9 ; re- quired the common difference t 19 A r.. 19-3 _1G_ 9 - 1 ~ 8 ~ '^■ 116 0-' - 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 sura is 840. Q-iven one of the extremes, the common difference, and the numher of terms; to find the other extreme, arid the sum of the series. Rule. — Multiply the common difference hy 1 less than the num- ber of terms, and the product will he 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 arith- metical scries of 9 terms ; to find the greatest term, and the sum of the series ? Ill the greatest term. 3 the least. 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 810. PROBtEM IV. To find an arithmetical mean proportional betteeen two c/iven terms. Rule. — Add the two given extremes or terms together, and take half their sum for the arithmetical mean required. Or, subtract hv Google 38 THE PRACTICAL MODEL CALCULATOR. the less extreme from the greater, and half the remainder will be the common difference ; -which, heing added to the less extreme, or subtracted from the greater, will give the mean requh-ed. To find an arithmetical mean hetween the two numbers 4 and 14, Here, 14 Or, 14 Or, 14 _4 ^ 5 2)-iS 2)10' 9 _9 6 the com. dif. "" 4 the less extreme. T So that 9 is the mean required by both methods. PKOBLEM V. Tojind two arithmetical means hetween two given extremes. Rule. — Subtract the less extreme from the greater, and divide the difference by 3, bo will the quotient be the common difference ; which, heing continually added to the less extreme, or taken from the greater, gives the means. To find two arithmetical means between 2 and 8. Hereg 2 Then 2 + 2 = 4 the one mean, 3]^ and 4 + 2 = 6 the other n com. dif. 2 To find any number of arithmetical means hetween two given terms 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. — Then, by adding this com. dif. continually, 6 ) 12 the means are found, 4, 6, 8, 10, 12. com. dif. 2 GEOMETRICAL PROPORTION AND PROGRESSION. The moat 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 hv Google GE0J1ETB.ICAL PROPORTION ASD PBOGKESSION", 39 the above numbers, tlie product of the means 12 -§- 2 = 6 the one extreme, and 12 -i- 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 pro- duct of the two extremes is equal to the product of any tivo 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 geome- trical 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 3, 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 — „— = 512, and 2^ = 512 also. Consequently, the greatest term is equal to the least term multi- plied by the said power of the ratio whose index is one less than the number of terms. Theorem 4. — The sum of all the terms of any geometrical pro- gression 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 -f- ^_^ = 1024 + 1022 = 2046. The foregoing, and several other properties of geometrical pro- portion, are demonstrated more at large in Byrne's Doctrine of Pro- portion. 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, and the ratio 2, what is the greatest term, and the sum of all the teiins ? 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 and 12. First way. Second way. 12 ;^ ) 12 ( 4, its root, is 2, the r;itio. o6(6 the mean. Then, 3x2 = 6 the mean. 36 Or, 12 -^ 2 = 6 also. hv Google ■to TTIE PRACTICAL MOBEL CALCULATOR. PaOBI.EM II. To find two geometrical mean proportionals betiveen any two numbers. Rtjlb. — 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 hy the ratio for the greater mean, and divide this again by the ratio for tbo less mean. To find two geometrical mean proportionals between 3 and 24. Here, 8 ) 24 ( 8, its cube root, 2 is the ratio. Then, 3x2= 6, and 6 x 2 = 12, the two means. Or, 2-i H- 2 = 12, and 12 -4- 2 = 6, the same. That is, the two means between 3 and 24, are 6 and 12. To find any numher of geometrical mean proportionals between two numbers. Rule. — Divide the greater number by the leas, 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 multi- ply continually from the first term, or divide continually from the last or greatest term. To find four geometrical mean proportionals between 3 and ?G. Here, 3 ) 96 ( 32, the 5th root of which is 2, the ratio. Then, 8x2=6,and 6x2=12, andl2x2=24,and24x2=48. Or, 96 -J- 2=48, and 48-r-2=24, and 24-2=12, and 12-^2= 6. That is, 6, 12, 24, 48 are the four means between 3 and 96. OP MUSICAL PROPORTION. There is also a third kind of proportion, called Musical, 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 ; tlien the fiv.-.-t has the same proportion to the fourth, as the difference bctwcei: the first and second has to the difference between the third ai^l fourth. Asia these, 6, 8, 12, 18; whoi-c, C: 18:; S - G: IS - 12, that is, 6 : 18 ;: 2 ; C. hv Google FELLOWSHIP. 41 When numbers are in Musical Progression, their reciprocals are in Arithmetical Progression ; and the converse, that is, when num- bers ai-e in Arithmetical Progression, their reciprocals are in Mu- sical Progression. So, in these Musicals 6, 8, 12, their reciprocals I, I, ^, arc in arithmetical progression ; for J -|- ^^ — ^^ = ^ ; and | + J = I — I; that is, the sum of tl^e extremes is equal to double the mean, which is the property of arithmeticak. PELIOWSHIP, 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 pro- portion to one another. By this rule are adjusted the gains, or losses, or charges of part- ners in company ; or the effects of bankrupts, or legacies in case of a deficiency of assets or effects; or the shares of prizes, or tbe 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 time : and Double, when each portion is to bo proportional to two or more numbers ; as ivhen the stocks of partners are em- ployed for difi'erent 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 prove the work. — 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 4- 2 -f- 3 = 6 the sum of the proportional numbers. Then, as 6 : 240 : : 1 : 40 the 1st part, and, as 6 : 240 : : 2 : 80 the 2d part, also as 6 : 240 :: 3 : 120 the 3d part. Sum of all 240, the proof. Three persons, A, B, C, freighted a ship with 340 tuns of wine ; of which, A loaded 110 tuns, B 97, and C the rest : in a storm, the hv Google THE PEACTICAL MODEL CALCULATOR. a were obliged to throw overboard 85 tuns ; how much must each perBon sustain of the loss ? Here, 110 + 97 = 20T tuns, loaded by A and E ; tberef., 340 — 20T = 133 tuns, loaded by C. hence, as 340 "' '""' and, a : 110 ; 110 : 27^ tuns = A's loss; ; 97 : 24-J tuns = B's loss ; ; 133 :_33i 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 dif- ferent times. EuLE. — Multiply each person's stock by the time of its continu- ance ; 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 INTEEEST, IsTEPiEST is the premium or sum allowed for the loan, or for- bearance of money. The money lent, or forborne, is caiied 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 pre- mium 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. Otherwise. — Take the interest of one dollar for a year, which, multiply by the given principal, and this product again by tlie time hv Google POSITION. 43 of loan or forbearance, io years and parts, for the interest of the proposed sum for that time. "VYhen there are certain parts or years in the time, as quarters, or months, or days, they may be worked for either by taking the ahquot, or like parts of tho interest of a year, or by the Rule of Three, in the usual way. Also, to divide hy 100, is done by only pointing off two figures for decimals. COMPOUND INTEEEST. 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. Pind the amount of the given principal, for the time of the first payment, by Simple Interest. 'I'hen 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 pay- ment, 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 can- not be resolved by the common direct rules. It is sometimes called False Position, or False Supposition, because it makes a supposi- tion 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 triah 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 supposi- tions 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 he 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 per- form 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 hv Google 44 THE PRACnCAL MODEL CALCULATOE. or number assumed, so is the result in the question to the number sought. A person, after spending J and J of his money, has yet remain- ing $60, what had he at first ? Suppose he had at first $120 Proof. Now J of 120 is 40 J of 144 is 48 i of it is _30 ^ of 144 is m their sum is 70 their sum 84 which taiien from 120 taken from 144 leaves 50 leaves tiO as per question. Then, 50 : 120 : : 60 : 144. What number is that, which multiplied by 1, and the product divided by 6, the quotient may be 14 ? 12. PERMUTATIONS AND COJIBINATIONS. The P^rMiMMiWsof any number of quantities signify the changes which these quantities may undergo with respect to their order. Thus, if we take the quantities a, h, c; then, a b e, a c h, h a e, b e a, c ah, ch a, are the permutations of thesa three quantities taken all together; a b, a e, b a, b c, 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, &e. The number of the permutations of the eight letters, a, h, e, d, e, f, g, h, 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 ques- tion, as in Single Position ; aad find how much each result is dif- ferent 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 supposi- tion, namely, the first position by the second error, and the second position by the first error. If the errors are alike, divide the difierence 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 tho sum of the errors, for the answer. The errors are said to be alike, when they are either botli too great, or both too little ; and unlike, ^^ hen one is too grt:i! ■■:..-') rl^c other too little. hv Google MBNSUEAIION OP SUPERFICIES. 45 What number ia that, ■which, being maltiplied by 6, the prodnct 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. 18 30 6 mult. 6 108 180 18 add. 18 9)126 9)198 14 results. 22 . 20 true res. 20 + 6 errors unlike. ^72 2dpos. 30 mult. 18 1st p E,„.{-- H Sum 8 J ai6 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 tlie errors ■which arise from each of them. Multiply the difference of tho t'wo numbers, found by trial, by the least error, and divide the product by tlie 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. MENSTJEATION 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 he a square, a rectangle, a rhombus, or a rhowhoides. — 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 areaof the square ABCD whose ■ side ia 5 feet 9 inches. Here 5 ft. 9 in. = 5-75 : and 5'T5|^ == 6-75 x 5-75 = 33-0625 /eei = 33/e. in. 9pa. = area required. hv Google 46 THE PRACTICAL MODEL CALCULATOR. Required the area of the rectangle AECB, whose length AB ia 13-75 chains, and hi'eadth BC 9-5 chains. Here 13-75 X 9-5 = 130-625; and '^^^ = 13-0625 ac. = \Zac. OmlO fo. = area required. Required the area of the rhombus ABCP, whose length AB is 12 foot 6 inches, and its height DE 9 feet 3 inches. Here 12/e. 6 in. = 12-5, and^fe. 3 in. = 9-25. Whence, 12-5 X 9-25 = 115-625 /«. = 115 /e. 7 in. Q pa. = area required. What is the area of the rhom- boides ABCD, whose length AE is 10-62 chains, and height DE 7-63 chains. Rere 10-52 X 7-63 = 80-2676; / 80-2676 / = 8-02676 acm= 8 rtc. / 10 ro. 4po. area n ^ To find the area of a triangle, — Multiply the base bj the pendicular height, and half the product wil! be the area. The perpendicular height of the triangle is equal to twice 1 area divided by the base. Required the area of the triangle ABC, whose base AB is 10 feet 9 inelies, and height DC 7 feet 3 inches. fferelGfe.^in. =10-75, and Ife. din. = 7-25. Whence, 10-75 x 7-25 = 77-9375, and 77-9375 -^ = 38-96875 feet = 38 fe. 11 in. ^ " 7^ pa. = area required. To find the area of a triangle whose three sides only are crJra From half the sum of the three sides subtract each side severi' Multiply the half sum and the three remainders continu-ally t> ther, and the square root of the product will he the area v^jiaii Required the area of the triangle ABC, □ whose three sides BC, CA, and AB are 24, 36, and 48 chains respectiyely. ,, 24 + 36 + 48 108 Mere s- = -^ = 54 = J sum of the sides. Also, 54 - 24 = 30>s( diff. ; 54-36 "* = 18 second diff.; and 54 — 48 = 6 third diff: hv Google MENSURATION Off SCPERFICIE3. 4T <18> i = V'174£i60 = 418-282 = a Whence, \/ 54 > required. Any two sides of a right angled triangle being given to find the third side. — When the two legs are given to find tlie hjpo- thennse, add the square of one of the lega to the square of the other, and the square root of the sum will he equal to the hypo- thenuse. When the hypothenuae and one of the legs are gtvtin 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 ABO, the n base AB is 56, and the perpendicular EC 33, what is the hypof henuse ? Mere 56' + 33* = 3136 + 1089 = 4225, and \/4225 = 65 = hypothenuse AC. If the hypothenuse AC be 53, and the base AB 45, what is the perpendicular BC ? '^ " Sere 58* — 45« = 2809 — 2025 = 784, and v'784 = 28 = perpendicular BC. To find the area of a trapezium. — 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 BAED, whose diagonal BE is 84, the perpendicular AC 21, and DE 28. Here 28 + 21 X 84 =49 X 84=4116, ,4116 and — q — = 2058 the area required. To find the area of a trapezoid, or a quadrangle, two of kIiosc opposite sides are parallel — Multiply the sum of the parallel side? 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 214-24, and perpendicular DE 171-16. Here 321-51 + 214-24 = 535-75 = 3«,w of the parallel sides AB, DC. Whence, 535-75 x 171-16 (theperp. DE) = 91698-0700, and ^ = 45849-485 the area required. To find the area of a regular polygon. — JluUipIy half the peri- meter of the figure by the perpendicular falling from its centre Hpon one of the sides, and the product will be the area. The perimeter of any figure ia the sum of all its sides, hv Google THE PRACTICAL MODEL CALCULATOa. Required the area of the regular p ABCBE, whose side AB, or UC, , feet, and the perpendicular OP 17'2 feet. Mere — -^ — ■ = 62'5 = half perimeter flwd62-5x lT-2 = Vilb square feet = arci required. To find the area of a regular polygon, when the side i 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 bo the area. ^id.f N^... Mullipliers. "to^ Na„^. SlQlUplfcr^. 3 4 5 6 7 Trigon or equil, A Tetragon or square Pentagon Heiagoa Heptagon 0'4330ia 1000000 1 -720177 2 ■698076 3 -033912 8 10 11 12 Octagon Nonagon Decagon Duodeoagon 4-828427 6-181824 7 ■694200 9-S6.3e40 ll-liim52 The angle OBP, together with its tangent, for any polygon of not more than 12 sides, is shown in tbo following table : m,t NamM. Angle Oiit>. Tangfliils. 3 4 5 T 8 10 11 12 Trigon Tetragon Pentagon Hesagon Heptagon Octagon Nonogon Decagon Undeoagon 30° 45° 54° 60" 64°4 67''i 70'^ 73° jV 75° ■57735 = J v'3 1-00000 = 1x1 1'37C38 = v'l+ f -v^^ 1-73205 = v/3 2-07652 2-41421 = 1 + v'^ 2-74747 8-07768 = ^5 + 2 v-a 3-40568 3-73205 = 2 + ^3 Required the area of a pentagon whose side is 15. The number opposite pentagon in the table is 1-7204V7. Hence 1-720477 x 16= = 1-720477 X 225 = 387-107825 = area required. The diameter of a circle being given to find the circumference, or the circumference being given to find the diameter. — Multiplv the diameter by 3-1416, and the product will be the circumfer- ence, or Divide the circumference by 3-1416, and the quotient will be the diameter. As 7 is to 22, so ia the diameter to the circumference ; or as 22 is to 7, so is the circumference to the diameter. As 113 ia to 355, so is the diameter to the circumference; or, as 352 ia to 115, so is the circumference to the diameter. hv Google MENSURATION OF SUPERFICIES. 49 If the diameter of a circle be 17, what ia the circuraferenco ? Here 3-1416 X 17 = 53-4072 = circimferenee. If the circumference of a circle be 354, what ia the diameter ? 354-000 Here ' o.-j ±-i a = 112-681 = diameter. To find the length of any arc 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 half the arc. 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 arc : the sum will be the length of the arc very nearly, Wlien^e chord of the arc, and the chord of half the arc 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 tho diameter by the versed sine, and the sqnaro root of the product will be the chord of half the arc. Multiply twice the chord of half the arc by 10 times the versed sine, divide the product by the reserved number, and add the quo- tient 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 squaro of tho diameter subtract the square of the chord, aud extract the square root of the re- mainder. Subtract this root from the diameter, and half the re- mainder will give the versed sine of half the arc, Tlie square of the chord of half the arc being divided by the diameter will give tho versed sine, or being divided by the versed sine will give the diameter. Thelength of the arc may alsobe 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 tunes 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. ^ If the chord DE be 48, and the versed sine n / I \ ,. CB 18, what is the length of the arc? / |^ ', Here 48^ X 15 = 34560 \ '■ \ 18^ X 33 ^ 10692 \^ ! 45252 reserved number. '"-,. ; ,.- hv Google 50 THE PRACTICAL MODEL CALCULATOR. 48^ = 2304 = the square of the chord. 18^ X 4 = 1296 = 4 timeB the square of the versed sine. ^ 3600 = 60 = twice the ehord of half the arc. 60 X 18^ X 10 194400 „ , , NoiB — .-ni-g = - .gij-n = 4'2959, which added to twice the chord of half the arc gives 64-2959 = the length of the arc. 50 ) 2514 reserved number. AC = v/SO X 18 = 30 = the chord of half the arc. 30"X2 X 18 X 10 10800 2514 ~ 2514 =.4-2959, which added to t chord of half the arc gives 64-2959 = the length of the arc. To find the area of a ciVeZe.— Multiply half the circumference by half the diameter, and the product will be the area. Or take ^ of the product of the whole circumference and diameter. What is the area of a circle whose diameter is 42, and circum- ference 131-946? 2 ) 131-946 65-973 = J circumference, 21 = |- diameter. "65973 131946 1385'433 ■= area required. What 13 the area of a circle whose diameter is 10 1 and circumference 31 feet 6 inches ? fe. in. 15 9 = 15-75 = ^ eireumference. 5 3 = 5-25 = i 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. Cireumf. X '2821 = side of an equal square. Diameter x -7071 = side of the inscribed square. hv Google MENSDRATION OP STIPERFICIES. 51 Circumf. X -2251 = side of the inscribed square. Area X '6366 = sido 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. "VVhat is the area of a circle whose diameter is 6 ? 7854 25 = square of the diameter. "39270 15708 19'6350 = tJie answer. To find the area of a sector, or that part of a circle wh'ch w hounded hy any two radii and their included are. — 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 are, 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 n the sector, ,ii;^^^-:^^^^^^n "^"""^ 80 - k/W - 50^ \ \'^ / ' 2 — = 8-7750 = the versed \ ; / sine of half the are. \ i / 80 X 60 - 8-7750 x 27 = 4563-0750 = the reserved number. 2 X v'8-7750 X 80 = 52-9906 = twice the chord of half the arc. 52-9906 X 8-7750 x 10 , , , , 4XM^07"'CO ~ I'OlyO, which added to twice the chord of half the arc gives 54-0096 the length of the arc. 54-0096 X 40 And — 2 = 1080-1920 = 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 ? Kere the diameter is 40. Eenee, the area of the circle = 40= X -7854 = 1600 X -7854 ■= 1256-64, Mow, 360° : 22° : : 1256-64 : 76-7947 = area of the sector. hv Google 52 THE FBAOTICAL MODEL CALCULATOK. To find the area of a segment of a circle. — Find the area of the sector, having the same arc with the segment, bj the last pro- lilem. Find the area of the triangle formed by the chord of the seg- ment, 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 re- quired. The difference between the versed sine and radius, multiplied bj \in\f 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 ? ^^ AG^ 100 ^ , CD = -TTpT = "oa" = 5 = the versed of half the arc. 20 X 60 - SlTST = 1065 = the reserved number. rolTa xT ': 1065 to twice the chord of half the arc gives 20'9390 = the length of the arc. 20-9390 X 10 2 = 104-6950 = area of the sector OACB. OD = OC = CD = 5 the perpendicular height of the triangle. AD = v/AO^ - 0D= = v'75 = 8-6603 = i the chord of the arc. 8-6603 X 5 = 43-3015 = the area of the triangle AOB. 104-6950 - 43-3015 = 61-3935 = area of the segment required; it being iu 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 deci- mals ; 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 = area required. hv Google MEH8URATI0N OF S0PERFIOIES. 53 To find the area of a circular zone, or the epace included between any two parallel chords and their intercepted arcs. — From the greater chord subtract half the difference between the two, mul- tiply 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 EGr, and the two chords AB and DC, find the areas of the segments ABBA, and DCED, the difference of which will be the area of tho zone required. The difference of the tabular segments multiplied by the square of the circle's diameter will give the area of the zone. When the larger segment AEB is greater than a semicircle, find the areas of the segments A&B, 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, ,, and their distance Dr IT^ : required the area .'"'""'''-. ^ of the zone ABCD. ' "^ ^ 20 — 15 -^ = 2-5 = J = the difference ietween the chords. ) = DF. 17-5 3 + i And v'20^ + 16^ = \/Q25 = 25 = the diameter of the circle. TJie segment AEB ieinff greater than a semicircle, we find the versed sine o/DCE = 2-5, and that of AGB = 5. Eence -^ = -100 = tabular versed sine of DEC. And 2c = "200 = tabular versed sine of AGE. Mw -040875 X 25' = area of seg. DEC = 25-546875 And -111823 x 25^ = area of seg. AGB = 69-889375 sum 95-43625 •7854 X 25' = 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 space included between the circumference of two concentric circles. — The difference between the areas of the two circles will be the area of 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 a hv Google THE PRACTICAL MODEL CALCULATOR. ilie circular ring, or the s rencea of those circles. the cir cum fe- nfire AB + CD X AE -CD = 85 X 5 = 1T5,«»(?176 x -7854 = 13T"4450 = area of the nng rejuin-d To find the areas of lune% or tie y a es httween the tnt '-rsectinff arcs of two eccentric circles — rm I the ireas of the two <<egmants from which the lune is f rnip] and their difference mil he the area required. The following property is one of the moot eun u If ABC be a right angled t iingle and semicircles he described on the three sides as diameters, then will the ii 1 tri angle be equal to the two lunes D and F taken together. For the semicircles described on AC and BO = the one described on AB, from each take the segments cut off by AC and EC, then will the lunes AFCE and EDC& = the triangle AOB. The length of the chord AB is 40, the height DC 10, and DE 4 : required the area of the lune ACBEA. The diameter of the circle of lohich ACB 20^ + 10^ is a part = "^'Tji = ^^^ Arid the diameter of tJie circle of which KE^ ii a part = = 104, Now having the diameter and versed sines, we find, The area of seg. ACB = -111823 x 50^ = 279-5575 Andareaofseg.AEB = -009955 X 104^-= 107-6T33 Their difference is the area of the lune \ AEBCA required, J To find the area of an irregular polygon, or a figure of any number of sides. — Divide the figure into triangles and trapeziums, and find the area of each separately. Add these areas together, and the si of the whole polygon. Required the area of the irre- gular figure ABCDEFGA, the fol- lowing Enes being given : GB = 30-5 A»i-ll-2, CO = 6 CD = 29 Fo =11 Cs=-6-6 FD=24-8 Ejj=4 ^ An + Go ^^ 11-2 + 6 Eere - — ^ X GB = „ = 171-8842 \ will he equal to the area X 30-5 + 8-6 X 30-5 = 262-3 area of the trapetium ABCG, hv Google DECIMAL APPROXIMATIONS. ,li±9i area of the trapezium GCDF. ., FD X Ep 24-8 X 4 Also, — o — — = s— FDE. Whence 262-3 + 255-2 + 40-6 = figure required. - X 29 = 8-8 X 20 = 265-2 = -ij-= 49-6 = area of the triangle = arua of the whole Lineal feet multiplied by -00019 = mifes. — yards — -000568 = Squm inches — ■007 = square feet. - ,.rds — •0002061 ' =z acres Circular inches — ■00546 = square feet. Cylindrical inches -0004546 i = cubic feet. — feet ■02909 = cubic _ yards. Cubic inclies — -00058 = cubic feet. — feet — ■03704 = cubic yards. — — — C-232 = imperial gallons. — inches ■003607 = Cylindrical feet — 4-895 = — - inches ~ ■002832 — Cubic inches — ■263 = tt)S. avs. of cast iron. — ■281 = wrought do. — — -283 = — Steel. — ■3225 = -_ copper. — — ■3037 = — brass. -26 — ■4108 = 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. — ™ ■3S54 = — 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. — kilogrammef 1 X 2 ■205 = avoirdupois lb. — grammes X ■002205 = avoii-i iupois lbs. b,Google 56 lUE PKACTICAL MODEL CALCULATOR. Diameter of a sphere X '800 = dimensions of equal cube. Diameter of a sphere x -6667 = length of equal cjlimler. Lineal inches X '0000158 = miles, A Prcneh cubic foot = 2093-4T cuhic inches. Imperial gallons X '7977 = New York gallons. The average quantity of water that falls in rain and snow at Piuladelphia is 36 inches. At West Point the variation of the magnetic needle, Nov. 16th, 1839, was 7° 58' 27" West, and the dip 73° 26' 28". One inch, the integer or wliole number. ■96875 J + A -625 j •28125 -1 + 1, ■9376 J + A -69375 j + ^, •25 i ■90625 J -f 3^ ■5625 i + is •21875 S + * ■875 S j ■53126 S i + ,<, ■1875 B 5 + A ■84376 -a i + A ■s -a i •15625 -a S -f ,i ■8125 & j + A ■46875 & f -f * -125 g. s ■78125 : 1 + A -4375 ^ | + A ■09376 : ,1 ■75 i t ■40626 a 1 + ,', -0625 S ■71875 1 + /s -375 f -03125 J, •6875 i -f A -34376 I + ,>, •65625 ii + A '3126 I + A One foot, or 12 inches, the integer. ■9106 „ llinclies.'i ■4166 „ 6 inches.? ^0625 „ S of in. ■6338 - 10 — •3333 - 4 — -0620S " i — ■75 8 9 — ■26 13- -04166 i 1 — •6666 r 8 — ■1666 ra — -03125 II - ■5833 s 7 — •0833 • 1 — -02083 • J — ■5 = 6 — -07291 3 } — -01041 a J — One yard, or 36 inches, the integer. ■9722 35 inches.; -6-389 23 inches., ■3055 1] inelics. ■9444 34 — ! -6111 22 — I ■2778 10 — •9167 S3 — : ■5833 21 — -26 9 - •8889 „ 32 — 1 •5566 o 20 — -2222 „ 8 — ■8611;; 81 — 1' ■5278:; 10 — ■19442 T ^ ■8333 g 30 — i -5 § 18 — -1667 Be — -8066 f 29 — ! -4722? 17 — ' -1389 g* 5 — -7778 S 28 - :', -4444 "16 - -1111 s 4 - -75 "27 — / -4167 ■'15 — !i -0833 » 3 — -7222 26 — -3889 14 — t -0565 2 _ -6944 - 25 — -soil 13 — " -0278 1 — -6667 24 — -3.!::-'! 12 — b,Google 'teontamingthe areumferenaea, Sqimrel, Cube!, and Arm of OircUiJrom 1 to 100, idmneing Ig a tenti. „Google J CALCULATOR. ,v Google CIRCLES, ADVAHCING BY A 1 : iii ^ iltSS i ill luiioLii lik fh SJ „Google TUB PKACnCAL MODEL CALCULATOR. ,v Google 0IEOLE8, ADVAHCING BY A 1 s, «;.; , J •:zs ?.!:;« db,Google : PRACTICAL MODEL CALCULATOR. dhvGoogle TABLE OF THE LENGTH OF CIRCULAR ARCS. Di^ Ci,=^ S,^| Cb.. - "- a,=c^ a,.™. CuSs. AM. 39M 83IiS S74 -BB 976 -44 978 -21 9SW-M lODOO 976191'4SS 7683.1023 Al E / 7 L / fC ; A " >Thu 1 bt mg un ty r«3. w^ Unib. ^ L.n^h 000048 ■^ OO 06 -2 05 0-0005818 K 00O0J7 00 2 9 so 396 6o4 O'OOOS J- ■A 0000145 4 698 32 6 968 ' 4 00011f36 0-0000194 ft 08 665 no 463 93 h 0-0014544 ft OO00D->42 20 2 0943 5 6 0000.91 22 80 fid 0-00208(2 7 0000539 « 396 63 so H 0'00.3-<71 K 0000„88 w 6 96 3666 9 4 H 0'0026180 0000436 240 4 88 902 Ml 00(lfj485 8490660 i 2 8 0-006S1T8 no 0000970 m 88 flOO 52 6 8 8 0-008T-.66 HO 00014o4 4 6 8 41 > 0116355 40 0001139 6 08 2 360 6 28 85 60 014u444 50 0002424 Reiinn ed the length of a cii culai arc of 37° 42' 58" ? 30° = O-5235088 7° = 0-1221730 40' = 0-0116355 2' = 0-0020368 50" = 0-0002424 8" = 0-0000888 The length 0-6582703 required in terms of the raiiius. 1207° Fahrenheit = 1° of Wedgewood'a pyrometer. Iron melts at about 166° 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 20°. 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 124° leas heat, than when under the pressure of the atmosphere. hv Google THE PBACTICAI. DIODEL CALCULATOE. Aeeas of the Segments and Zones of a Circle of which the Diameter is Unity, and supposed to be divided into 1000 equal parts. ^ Araor j™o( HdsM. i^-i "iSe"' , A™^nt i««oi s^iusnl. S^.^M. ■001 •000042 ■001000 ■051 ■016119 ■050912 ■101 ■041476 ■100309 •002 ■000119 -002000 ■052 ■015561 ■051906 ■102 ■042080 ■101288 ■003 ■000219 •003000 ■058 •016007 -062901 •103 -052687 ■102267 ■004 •000337 ■004000 ■064 ■010457 ■053896 ■104 ■043296 ■103246 ■005 •000470 ■005000 ■056 ■016911 -064890 ■105 ■043908 ■104228 ■006 ■000018 ■006000 ■056 ■017369 ■066883 ■106 ■044522 ■105201 ■007 -000779 ■007000 -057 ■017831 ■056877 ■107 ■045139 ■106178 ■008 ■00095J ■008000 •053 ■018296 ■057870 ■108 ■043759 ■107155 ■009 ■001135 ■009000 ■059 ■018766 -068863 ■109 -046881 ■108181 ■010 -001329 ■010000 ■060 -019239 ■059856 ■110 ■047005 ■109107 ■Oil ■001533 ■011000 ■061 ■019716 -060849 ■111 ■047632 ■110083 ■012 -001746 ■011999 ■062 -020199 ■061841 ■112 ■048262 ■111057 ■018 •001968 ■012999 ■063 ■020680 ■113 ■048894 ■112031 ■014 ■002199 ■013998 -064 ■021168 ■063825 ■114 ■049528 ■113004 ■015 ■002488 -014998 ■065 ■021659 ■004817 ■116 ■050165 ■113978 ■016 ■00268S ■015997 ■066 ■022154 ■066807 ■116 ■050804 ■114951 ■017 •002940 -016997 ■007 ■022652 ■000799 ■117 ■051446 ■116924 ■018 ■003202 -017996 -068 -023154 ■067790 ■118 ■052090 ■116806 ■019 ■003471 ■018996 ■069 -028669 ■068782 ■119 ■052736 ■117867 ■030 ■003748 ■019995 ■070 ■034108 ■009771 ■120 ■053386 ■118888 ■021 -004031 ■020994 ■071 ■024680 ■070761 121 064036 lin809 ■022 ■004322 ■021993 ■072 ■025195 ■071761 122 064689 120779 ■023 ■00*618 ■022092 ■073 ■025714 072740 123 055345 121748 ■024 ■004921 023091 ■074 ■026230 073729 124 006003 122717 ■025 ■005230 ■024990 ■075 ■026761 074718 12o OoObbS 133086 ■026 ■005546 ■026989 ■076 ■027289 075707 J20 ■Oo7S26 124054 ■027 ■005867 ■026987 -077 ■027821 076695 127 ■067991 123f21 -0^8 ■006194 ■027986 ■078 -028356 077683 128 ■058658 12C5S8 -029 ■000527 ■0289S4 ■079 ■028894 078670 129 059327 127j35 ■030 ■006866 ■029982 ■080 ■029435 079058 130 059099 128521 ■031 ■007209 ■030980 ■081 ■029979 080645 181 ■060672 12'»480 ■007558 ■031978 ■082 ■030626 081631 132 061348 130451 ■033 -007918 ■032976 ■083 ■031076 082618 062026 131415 ■034 ■008373 ■033974 ■084 ■031629 083604 134 062707 132379 ■035 ■008688 ■034972 ■085 ■032186 084589 180 068389 lo!i342 ■036 ■000008 ■035969 ■086 ■032745 085G74 180 061074 134304 ■037 ■009388 ■036967 -087 ■033307 08(-w9 lu7 ■O64700 1uj2U6 ■038 ■009768 ■037966 ■088 -088872 087544 138 015449 136228 ■039 ■010148 •089 ■034441 131 Oj6140 187189 •040 ■010537 ■039968 ■090 -035011 089512 140 066833 138149 ■0« ■010931 ■040954 ■091 ■035585 ■09049b 141 007628 139109 ■042 ■011330 ■041951 •092 ■036162 091479 142 0682.0 1*1008 ■043 ■011734 ■042947 •098 -036741 092461 143 063924 141026 ■044 ■012142 -043944 ■094 ■037323 ■09C444 ■144 ■009625 ■141984 ■045 ■012554 -044940 •095 ■037909 ■094426 ■145 ■070828 •142942 ■046 -012971 -045985 ■096 ■088496 ■095407 ■146 ■071033 ■143898 ■047 ■013392 ■046931 ■097 .039087 ■096388 ■147 ■071741 ■144854 ■048 ■013318 ■047927 ■098 .039680 •097809 ■148 ■072450 ■145810 ■049 ■014247 ■048922 ■099 .040270 ■093350 ■149 ■073161 ■140705 ■OoO •014681 ■049917 ■100 .040875 ■099380 ■150 ■073874 ■147719 b,Google AREAS OP THE SEGMENTS AND ZONES OP A CIECLB. HriBht, Jib or 5,. jSjBofKoM. HcieU. ^,«.rse,. i«„rw IMjl ^^.,^. x^.,z^ ■151 074689 ■148674 ■306 ■116660 200^15 261 lb8140 ■248608 ■152 075306 ■149625 -207 ■117460 200924 262 164019 ■249461 -153 076026 ■150578 ■208 ■118271 201835 268 164899 ■360212 ■154 076747 ■151530 ■209 ■11908S 202744 264 166780 -251162 ■165 ■077469 -152481 ■210 ■119897 203652 -65 lb6663 -262011 ■156 ■078194 ■163431 ■211 -120712 204559 266 167546 ■252851 ■157 ■078921 ■154381 ■212 -121529 206465 267 1684S0 -263704 ■168 ■079649 -155330 ■213 -122347 206370 Ib9bl5 -264549 ■159 ■080380 ■156278 ■214 ■128167 207274 -.69 170202 ■265392 ■160 ■081112 -167226 ■215 -128988 209178 270 171080 ■256235 ■161 ■081846 ■158173 ■216 ■124810 209080 371 171978 ■257076 ■163 ■082682 ■159119 -217 '125634 -209'381 273 172867 -267915' ■168 ■083820 -160066 ■218 ■130459 •210882 2-3 173758 ■258764 ■164 ■08406S ■161010 -219 ■1272S6 -211782 274 174649 ■259691 ■165 ■084801 -161854 -220 ■138113 212680 375 175542 ■260427 ■166 -086644 ■162898 ■221 ■12^942 213577 276 176436 ■261261 ■167 ■163841 ■222 ■129773 ■214474 277 177830 ■362094 ■168 ■087086 ■165784 ■223 ■180606 -216869 278 178226 -263926 ■169 ■087785 ■165726 ■224 ■131488 ■216264 279 179122 ■268767 •170 ■088535 ■166666 ■225 ■183372 ■217157 280 180019 ■264586 ■171 ■089287 ■167606 ■226 -133108 -218050 281 180918 ■206414 ■172 ■090041 •168549 ■227 ■133946 ■218941 283 181817 ■266240 ■173 -090797 ■160484 ■184784 -219832 233 183718 -267065 ■174 ■091564 ■170422 ■135634 -220721 384 188619 ■267889 ■176 ■092313 ■171359 ■280 ■186466 -221610 2S5 184531 •268711 ■17G ■093074 ■172295 ■231 •187307 ■232497 286 185425 ■269532 ■177 *98886 -173231 -282 ■138150 ■223854 287 186329 ■270362 ■178 ■094601 ■174166 ■2SS -188995 -224269 218 187234 ■271170 ■179 -176100 ■334 -139841 ■225153 389 188140 ■371987 ■180 ■096134 -176033 ■236 ■140688 •22608b 290 189047 ■273802 ■181 ■096908 -176966 ■236 ■141537 ■226919 291 189966 ■273616 ■182 ■097674 ■177897 ■337 -142387 ■237800 392 190864 ■274428 ■183 ■098447 ■178828 ■288 -148238 ■328080 398 191775 ■375339 ■184 -099221 ■179759 ■239 ■144091 •229569 394 192684 ■376049 ■186 -099997 -180688 ■240 ■144944 -280489 295 193596 ■276857 ■188 -100774 ■181617 ■241 ■145799 ■331818 ■296 -194509 ■277664 ■187 -101553 -183546 ■242 ■146665 ■297 ■195423 ■278469 ■188 ■102384 -18847S -243 ■147513 ■233063 -298 ■196337 ■27927S ■189 ■103116 ■184898 ■244 •148371 ■233937 ■197352 ■280075 ■190 ■103900 ■185323 ■245 ■149230 ■234809 ■300 -198168 -280876 -191 ■104685 -186248 ■246 ■150091 ■235680 ■301 ■199085 ■381675 ■192 ■105472 ■187172 ■357 -1S0953 ■286560 ■803 ■200008 -282478 ■198 ■106261 ■188094 ■248 ■151816 ■237419 -200922 ■194 ■107051 ■189016 ■240 -152680 ■238387 -304 ■301841 -284063 ■195 ■107842 -189988 ■250 ■153648 ■289168 ■805 -202761 ■284857 ■196 ■108636 ■J90858 -251 •154412 ■^40019 ■306 ■203683 -285648 ■197 ■109480 -191777 ■252 ■165280 ■240888 -307 -204605 ■286488 ■198 ■110326 ■192696 -263 ■156149 ■241746 ■808 ■205527 -287227 ■199 ■111024 ■193614 ■254 ■167019 ■242608. ■306451 ■288014 ■200 ■194681 ■265 -157890 ■243469 -810 •207876 ■288799 ■201 ■113624 ■195447 -256 ■158762 -244828 -811 ■208301 -289583 ■202 ■118435 -196362 -267 ■245187 -312 ■209237 -290366 ■203 ■114280 ■197277 ■268 ■160510 ■346044 ■313 ■210154 ■391146 ■204 ■116035 ■193190 ■259 ■161886 -246900 -814 -211082 -291925 ■205 ■116842 -199108 ■260 -162263 ■247755 ■315 -212011 -292702 b,Google 'IHB PRACTICAL MODEL OALCULATOB, lI,-U,|Ar.««fE>g, ^.tznJn^M it .. of Sae A«„rfZ™ lUgM A ^ f-. J. 4.«,orz„»., ■31G ■212940 ■293478 371 265144 33837- 426 318J70 ■3iiU463 ■BIT ■218871 ■294252 872 266111 384041 427 319')59 ■3661185 ■31S ■214802 ■295025 878 267078 884708 428 320948 ■867504 ■310 ■215733 ■29579S 374 268045 3503/3 429 821988 ■368019 ■Z-2{) ■216666 ■206665 376 260018 336086 430 322928 ■368531 ■821 ■217699 ■297338 876 269982 386696 481 328918 ■360040 ■3li2 ■218533 ■298098 377 270051 337354 432 324909 ■369546 ■sas ■219488 ■298868 378 271920 888010 488 82o000 •370047 ■324 '220404 ■299025 379 272890 338668 434 320892 ■370546 •326 ■221340 ■300886 380 273861 339814 485 327882 ■871040 ■320 ■222277 ■30114o 381 274882 839983 486 328874 ■871531 ■327 ■223215 ■301902 882 275803 340609 437 329806 ■372019 ■328 ■224154 ■302658 376775 341208 488 880858 -872503 ■329 ■225093 •303412 384 277748 34189J 3818o0 '372083 ■330 ■226038 '304164 385 278721 342034 440 332843 ■373460 ■331 ■226974 ■804914 886 279694 343171 441 838886 ■378933 ■332 ■227315 ■30o6G3 387 280668 84880^ 442 334829 ■374403 ■228858 ■806410 888 281642 344437 443 83o822 ■884 ■229801 ■807165 889 282017 345007 441 336816 ■375330 ■335 ■230745 ■307698 390 288592 HobU 445 337810 ■375788 •336 ■231689 ■808640 801 284568 346318 446 888804 ■876242 ■337 ■232G34 ■80a379 892 285544 846940 447 339798 ■370693 ■838 ■233380 ■310117 393 286521 847060 448 340793 ■377138 ■389 ■284626 ■810863 894 287498 348177 449 341787 •377680 ■340 ■235473 ■811688 896 288476 348791 450 343782 ■878018 -3H ■236121 ■812819 896 280468 349408 451 343777 ■878452 ■842 ■2378G9 313050 397 290432 8o001i 152 S44772 ■378881 ■343 ■288318 818778 898 291411 350619 453 845768 ■879307 •314 814505 809 3ol22S 464 846764 ■379728 ■345 •240218 316230 400 298869 Sal824 456 347759 ■380145 ■346 ■241169 3159o2 401 294349 302433 156 848765 •380557 ■347 ■242121 816673 40i 296830 868019 ■457 349752 ■380900 ■348 ■248074 ■317893 408 396311 303612 458 3o0748 ■381369 ■349 ■244036 ■818110 404 297292 364202 459 3ol745 ■381768 ■850 ■244980 ■818826 405 298278 854790 460 362742 ■382162 ■851 ■245934 '819538 406 299255 85o876 401 358739 ■382561 ■352 ■246889 ■3:.0249 407 300238 365958 462 854785 ■882086 ■353 ■247845 •320958 408 301220 306537 403 355732 ■388316 ■354 ■248801 3216t.S 409 302308 357114 464 S507S0 ■883691 ■355 ■249757 822371 410 303187 3o7b88 46o 357727 ■384061 -356 250715 823075 411 804171 858258 466 358725 ■384426 •857 251673 328775 4U 305165 308827 467 3j9723 ■38478Q ■358 252631 324474 418 806140 359392 468 8( 0721 ■385144 ■3S9 258690 825171 414 307125 3j9954 4b9 301713 ■385490 ■860 ■254550 ■325866 416 808110 860513 470 .02717 ■385884 ■861 ■255510 ■326569 416 30B096 361070 471 303715 ■886172 ■362 -256471 ■827250 417 310081 361623 472 304713 ■386505 ■363 ■257433 ■327939 418 811068 362178 473 865712 •886832 ■364 ■258895 ■828625 419 3130^4 502720 471 300710 ■387153 ■365 ■259367 •329310 420 818041 363264 47j 8b7T09 -887469 ■366 ■260320 ■329992 421 314029 36380O 476 368708 ■887778 ■3S7 ■261284 ■880678 422 31601b 364343 477 301707 ■262248 ■881351 423 816004 364878 478 870700 -SS8377 ■869 263213 ■332UJ7 424 31flJ93 305410 179 u71704 ■S88C60 ■370 254178 ■33-700 425 817981 365^89 480 872704 ■388951 b,Google RULES rOE, FINDIK& THE AREA OP A CIRCULAR ZONE, ETC. 67 ■B89497 ■389759 -S90011 ■391604 ■891748 ■391920 To find the area of a segment of a circle. Rule. — Divide the height, or versecl sine, hy the diameter of the circle, and finil the quotient in the column of heights. Then take out the corresponding area, in the column of nrcas, and multiply it hy the square of the diameter ; this ivill give the area of the segment. Beqoired the area of a segment of a circle, whose height is 3^ feet, and the diameter of the circle 50 feet. 3i = 3-25; and 3-25 h- 50 = -065. ■065, by the Table, = -021659; and -021659 X 50^ = 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 bythesquareof the longest chord. Required the area of a zone whose longest chord is 50, and height 15. 15 ^ 50 = -300 ; and -300, hy the Table, = -280876. Hence -280876 x 50^ = 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 -5- 50 = -400 ; -400, by the Table, = -351824 ; and -351824 x 50= = 879-56. 15 -^- 50 = -300 ; -300, by the Table, =-280876 ; and -280876 X 50* = 702-19. Hence 879-56 -|- 702-19 = 1581-75. Approximating rule to find the area of a segment of a circle. Rule. — Multiply the chord of the segment by the versed sine, divide the product by 8, and multiply the remainder by 2. Cube the height, or versed sine, find how often twice the length of the chord ia 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 si: 12 ; = 24; 24 = 8 ; and 8 X 2 = 16. 2^ -H 24 = -; Hence 16 -i- -3333-16-3333, the ai a of the segment very nearly. hv Google PROPORTIONS OF THE LESGTHS OF CIRCULAR ARCS. T "Z" T T " t' T "i:" T HcijM '2' ' _^ 61 ■421 s s ■182 j^oBeii fi3 li H -29052 ■424 H -105 ■ozsso ■ISfl ■08800 M ■17912 •346 ■:^l -425 ■126 ■tss ■im ■02970 ■187 ■0fW9 m ■18163 S '43309 ■^ 5mm ■i ■00309 ■00305 ■OMOl \m ■1842S .■350 ■29S39 •20»7 i ■oasM ■00557 ■43 ■13 ■iiS -09054 l74 '13810 ■ISS6B ■SS* ■30684 ■43S ■434 -44405 ■u ■ 16 ■ 16 ■ 18 -0S490 ;19« ■004B ■190S2 ■355 •366 ■357 ■80704 ■436 s ■439 '16142 l'4682r ■IIB ^ItOT ■200 ■0348 isi ■i^ •m •sieee ■31761 ■440 1-45613 ■46697 ■202 ■ 054S ■442 ■203 -20140 ■363 ■443 ■20232 ■33249 1^462^5 •206 ■ 0866 S6 ■20418 ■305 ■33413 ■446 ■461J1 ■m ■041M ■206 ■0968 •20668 ■20696 ■446 ■447 ias ■0W13 loo ■ 1269 lo ■368 ■360 ■32006 ■449 '47O02 ■129 ■370 ■04(4; ■2U m -21239 •371 ■461 l^tHMS ■11634 m -21881 •372 •33664 ■462 ■^ ■Mi8i ■04003 ■a! ■463 ;464 ^1 m ■466 207 ■4S600 ■04932 -12235 »8 ■378 ■34663 ■48SS0 1SS 'osooa ■210 ■22347 Ssu? ■12S60 ■13663 m (01 ■32770 ■S80 ■382 :ii ■46 ■46 ■40460 ;l4a S ■228 -224 MS ■35575 ■46 ■wS ■12991 (05 ■466 ■ill W6WI ■236 -327 ;1M08 ■306 toi -2334B ■23404 Is? ■36084 ;46fl ■soajs 1^S0800 14B ■23780 ;3a» ■86426 ■28925 ■36596 ■ 1 ms96 ^ •13786 is ■3407O ■M3 ■472 ■61571 ^3 I'S ^4 ■13903 ■14020 a4 ■24380 ■S4500 ^ 17263 ■474 -61958 ■235 ■2ffl64 ■39S ■52162 ■ 6 iSs ■2S7 ■i«S 0.1 ■306 ■37801 ■177 ■ J <1S -25006 ■14M7 ■240 ■X4714 -25BB1 ;400 1-53126 ■241 ■2S639 ■00776 ■242 ■26680 ■^ ■15186 K4 ■4M ^oa ■4I4 -M910 ■346 ■16308 ■406 -240 ■26286 ■24T J27 ■SM37 ;407 ■3864S ■M ■07Ke ■16791 ■4I0 ■54IS90 ■1603S 351 'W>64 ■191 ■68288 ■07BM (32 ■27196 ■172 ■ITS ■264 -16402 »4 ■414 ■407S3 ■493 ■494 ■56854 ■0TSS8 ■355 ■16620 85 ■415 ■40960 ■486 1? ■0767J ■27810 ■27884 ■493 :Si ■250 (39 ■300 i-mso SJO ■189 ■08428 b,Google PROrORTIOi^S OP THE LENGTHS Oi' SB.MI-ELLirTIC AEC3. PROPORTIONS OF THE LENGTHS OF SEMI- ELLIPTIC ARCS. 1-16920 1-17041 1-17163 1-17285 1-17407 1-17529 1-17661 1-17774 1 '17897 1-18020 1-18148 1-19010 1-19134 1-19258 120005 1 20130 1 20255 1 21010 1211861 1 212G3 ' 121800, 121517 121644 1 21772 121000 122028 1 22156 122284 122412 1 22o41 1 22b70 12278 1 22928 ■24480 ■24612 ■24744 ■24876 ■25010 ■25142 ■25274 ■25406 1-32163 1-32300 1-82438 1-32576 1-32715 184823 1 S4&C0 1 35108 -^ 44 ulj 1„ b's JKS7VI 3fG 1 3C833 29014 3b7 1 «[,97« 8b8 2<>PRr. KU.) 1 37208 2HV 870 I 87414 29oo7 8(1 29608 ^73 1 —7f>H 1 s + J I o84 1 1 dJliO I hv Google THE PRACTICAL MODEL CALCtTLATOK. m^ LeoslioE] lltieW I^ngUiof Hoislrt Lsnsthof rfS' Lo^tOf HeWit Lonaii, of a™. ' of Ate. Am. Are. ■335 1^390O5 ■447 1-48850 -509 1-58474 -671 1-68395 -633 1-78172 1-83751 -448 1-49003 -510 1-58629 -572 1-68864 -684 1-78885 ■88T 1 ■89897 -449 1-49167 -611 1-58784 ■673 1-68518 ■635 1-78498 ■38S 1-40043 ■450 1-49311 -512 1-58940 ■674 1-68672 ■636 1-78660 -SS9 1-40189 ■451 1.49465 ■513 1-59096 -676 1-68881 ■637 1-78823 ■390 1-40335 -452 1-49618 .514 1-59262 -576 1-68990 ■638 1-78986 ■8B1 1^40481 ■453 1-49771 .615 1-59408 -67T 1-69149 1-79149 ■393 1^40627 -454 1^49934 ■516 1-59564 ■678 1-69308 ■640 1-79312 ■808 1'40778 -455 1^5O077 .617 1-69720 ■579 1-69467 ■641 1 ■79476 ■394 1-40919 -456 1 ■50280 ■518 1-69876 ■580 1-69636 -642 1-79688 ■895 1 ■41065 -457 b5088S -519 1-60033 ■581 1^69785 ■643 1-79801 ■396 1-41211 458 1-50636 -620 1-60188 ■682 1 ■69946 -644 1-79964 ■397 1-41357 -459 1-60689 -521 1-60344 ■583 1 ■70105 ■645 1-80127 1-41504 -460 1-50842 1.60500 ■584 1 ■70264 -646 1-80390 ■399 1 -41661 -461 1-50996 ■538 1.60656 ■585 1'70424 ■047 1-80454 ■400 1 ■41798 -462 1-51150 -524 1-60812 -686 1-70584 -648 1-80617 ■401 1-41945 -463 1-51304 -525 1-60968 -587 1-70745 ■649 1-80780 ■403 1-42092 -464 1-51468 .526 1-61124 -588 1-70905 ■660 1-80943 •403 1-42289 -465 1-51612 .527 1-61280 ■589 1-710B5 ■651 1-81107 ■404 1-42386 466 1-51766 .528 1-61436 ■690 1-71325 -653 1^81271 ■405 1-42533 467 1-51920 .529 1-61692 -691 1-71286 ■633 1 ■81436 ■400 1-42081 468 1-52074 -630 1-61748 •692 1-71546 ■654 1 ■81599 ■407 1-42829 .469 1-52229 .581 1-61904 ■693 1-71707 -665 1-81703 ■408 1-42977 ■470 1-62384 ■582 1-62060 -504 1-71868 -656 1-81928 ■409 1-48125 ■4T1 1-52539 .538 1-62216 -696 1-72039 ■667 1-82091 ■410 1-43278 ■472 1-62691 534 1 ■62873 ■596 1-72190 -668 1-822^J6 ■411 1^43421 ■47S 1-62849 ■685 -597 1-72350 -659 1-82419 ■413 1^43569 474 1-53004 ■536 1-62684 .598 1 ■72511 ■660 1 ■82583 ■4ia 1-48718 .476 1-68169 .537 .600 1-72673 -661 1-82747 ■414 1 ■43867 476 1^53314 ■588 1-63996 .600 1-72888 -662 1-83911 ■415 1^44016 -477 1 ■53469 .689 1-63152 -601 1-72094 -668 1-83075 ■41B 1 ■44165 .478 1-5S025 .540 1-68309 .602 1-73155 ■664 1-83240 ■417 1-44314 .479 1-53781 .541 1-63465 -603 1-73816 ■605 1-83404 ■418 1-44463 .480 1-58937 .542 1-68623 -604 1 ■73477 ■6G6 1 -88568 ■419 1-44018 481 1-54098 -643 1-63780 •605 1-73638 -607 1 ■83738 ■420 1-44763 .492 1-54249 -544 1-68937 •606 1^73T99 -668 1^8S897 ■421 1-1491S .483 1-64405 -546 1-64004 ■607 1^73960 ■660 1-840U1 ■422 1-45064 .484 1-54561 ■646 1-64351 •608 1^74121 -670 1-842-26 ■423 1-45214 ^5 1-64718 .647 1-64408 •609 1-74283 -671 1-84S01 ■424 1^45364 .486 1-64875 .548 1-64565 *10 1-74444 ■673 1-84356 ■425 1 ■45515 .487 1-55032 .649 1-64722 -611 1-74605 ■678 l-847'>0 ■m 1^15660 1-55180 550 164879 612 1 4 67 6 4 18488 ■427 1 ■45815 .489 1-66346 561 1 60086 6 3 1 49 6 5 18 ■428 1-45966 ■490 1-55508 65" 1 65198 614 1 001 6 6 13 U -429 1^49167 .491 1-55660 558 1653 61o 1 o 677 18 9 ■430 1-46268 ■193 1-5-5817 5o4 16o507 613 1 DiU C8 18 44 ■431 1-46419 .498 1-65974 ii6i> 1 6o6bo 617 670 18 709 ■432 1-46570 494 1.56131 056 1658 8 618 118 4 ■433 1-46721 495 1.56289 6j7 1 60981 6 9 1 oOOO 681 18 OBO ■484 1-40872 .496 1.66447 5 8 1 06189 6 1 606 689 18 ■435 1-470-23 .497 1 ■66605 559 16b 97 621 1 1, 4 688 18 ■438 1-47174 498 1.56763 560 1 G64do 019 1 6886 684 18b 3 ■437 1-47326 .499 1-60921 6G1 1 66618 6 3 1 fa 49 186 00 ■438 1^47 478 .500 1.6708B 66'> 16 1 686 1 80«(, 1 h4Sn 1-47630 .501 1.67234 568 08 180 1 ' ■440 1-47782 ■502 1-57880 564 083 18 19 1 ■441 1-47934 -503 1-57544 56 689 18 3fa I -442 1-48086 -504 1 ■67699 OC 600 18 i -44J -505 1-57854 18 ■ ■4J4 1^4SS01 -606 1-58009 t,S 1 1 1 OT ! -410 1 -43544 -507 1-58161 56 16 8 031 1 84 18 U4 ■44(1 1-48697 508 1-58819 1 6-<0 6 '1 ill 4 1 -=1^ b,Google PROPOaTIONS Off THE LENGTHS OF SEMI-ELLIPTIC ARCS. K':!!^^.^'* K! Ls^hof 5S Lsn;.] " l^Uilho „..,..,. 1 S8356 ■757 1-98791 2 '09300 &SO-22 ■758 108064 ■819 2-09586 880 8S688 ■768 1 ■99134 -820 2-0971 881 ■6U8 1 88S54 ■760 1^99806 2-09888 as ■009 1 89020 ■761 1 ■90476 2-100b6 883 ■700 1 89186 ■762 1^99647 -823 2-1024- 881 ■701 1 89862 ■703 I^SOSIS -824 2-10410 883 ■70:i 1 89619 ■764 1 ■99989 2-1059(1 ■703 1 80685 ■765 2.00I60 ■826 2-107(8 887 ■704 1 89851 ■766 2-0OS3I -827 2^10950 ■705 1 90017 ■767 2-00502 2-Ul''- ■708 1 90184 ■768 2^00678 2^11804 890 ■707 I 9085O ■769 2^00841 -630 2-11481 ■708 1 90S17 ■770 2^01016 -881 2<116d9 ■709 1 aoesi ■771 2-01187 -832 2-1183- 898 0862 7 2 2^01359 -833 2^12015 894 9 9 8 2^01631 -884 2-12198 896 8 i 2 01702 -835 2-12Sa 89t> 6 2^01874 2'12549 897 2 ■02045 2-02217 -887 •838 2-12727 2^12905 8 2-02389 2-1808d SOO 2^02561 ■840 2^132bl 901 80 2^02733 -841 2-18439 90'' 2 4 OH 8 2^oaB07 -842 2-18618 90b 2 ^4691 9b4 1 J 00 8 ■08080 -848 2-13797 904 2-48 4 J65 Ul 1 8 ■844 2-139 ti 905 2"d0 7 ' 1 ■0342.J -815 2-14155 906 1 -846 2-14384 907 1 ' -0^771 ■847 2^14513 908 ■0;!944 ■848 2^I4602 909 ■01117 ■849 2-14871 910 ■01290 -850 2^15060 911 90 2^04162 -851 2-15229 912 2 26338 9 3 1 i 9 2^046B5 ■852 2-15109 918 2 26o21 94I ■04809 -853 2-15580 914 2" 04 93 2-04088 -864 2-157 916 91 2^05167 -8B6 2-15960 91Q 96 2-053S1 -856 2-16130 917 ' 96 9 2-05505 2-05679 •857 2-10309 2-16489 918 019 2 0uS5S 8j9 2 16868 S9 2 0eO''7 860 2 16848 921 98 Ub 1 i 800 2-0020 861 2 "8170 80 2^06877 21 "09 ""SR i J8^ ol| 802 06562 863 21 8tl 2 06727 864 21 U 80 ■06901 866 '1 i 4 80 076 860 21 800 07261 867 "1811 80 074 7 "18 4 808 60 869 "184 809 7 7 8 186 6 931 8 8 2 908 2 081^8 8 1 8 2 1888 (8 ?1 I 8 6 08304 2^08480 2 08b66 0888 008 1 8 b 8 ' Olio J 61 J I 1000 u^ bvGoogle 72 THE PRACTICAL MODEL CALCULATOK. To find the length of an arc of a circle, or the curve of a rigid eemi^ellipse. EuLB. — Divide the height by the hase, 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 are, 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 400, 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 H- 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 moro than half an inch ; tlierefore, except where extreme accuracy is required, it is not worth com- puting. 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 ares 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 bo found, be ABC. Produce the height line Bd to meet the centre of the curve in ff. Draw the right line A^, and from the centre g, with the distance gB describe an arc EA, meeting Ag in h. Bisect Ah in i, and from the centre g with the radius gi describe the arc ik, meeting dB produced to k; then ik is half the arc ABC. hv Google TABLE OF RECIPROCALS OP SUMBERS. T3 A Table of the Reciprocals of Mimbers ; or (fie Decimal Frac- tions eori-esponding to Vulgar Feactiojsb of which the Numera- tor is unity or 1. [In the follow n Tabl tl e Decimal fractions are Reciprocals of the Denominat s f tho opposite to them ; and their product is = unity. To find the D mal ponding to a fraction having a higher Numerator than 1 m It j ! the Decimal opposite to the given De- nominator, by th n Numerator. Thus, the Decimal corre- sponding to ^ being -015023, the Decimal to \\ will be -015625 X. 15 = -284375.] Fr«=M™.r Deoimil «t Fracw™ ot Decimal ot Frl=.i.= M Jl^olm-l or KaciptMai. lEtiproMl. 1/2 ■5 1/47 1212766 1,^2 ■010869505 1/3 ■383333333 1/48 1/93 ■010752088 1/4 ■25 1/49 )i04O8163 1/94 ■010038298 1/5 -2 1/50 02 1/95 ■010526310 1/a ■166066667 1/61 01%07843 1/96 ■010416667 1/7 ■142857143 1/52 019230769 1/97 ■010309278 1/8 ■125 1/53 018867925 1/98 ■010204082 1/9 ■111111111 1/54 018518519 1/99 ■01010101 1/10 ■1 155 01818V818 1/100 ■01 1/11 ■090909091 1/56 017867143 lAOI -00090099 112 '088333338 157 01754386 1/102 ■009803922 1/13 ■078B23077 1/68 017241379 1/103 ■009708738 1/14 ■07U28571 1/59 016949153 1A04 ■009616885 1/15 ■066666667 1/60 016666667 1/105 ■00952381 1/lS -0625 1/61 016398448 1/ioe ■009433962 1/17 -058823529 1/62 016129032 1/107 ■009845794 1/18 ■055566556 1/63 015878016 1/108 ■009259239 1/19 ■052G31579 1/64 015625 1/100 ■009174312 1/20 -05 1/65 015884615 1/110 ■009090909 1/21 ■047619048 1/66 015151515 l/lll ■009009000 1/22 ■045454545 1/67 014926378 1/112 •O089-28571 1/23 ■048478261 1/68 014705882 1/113 ■008349558 1,'24 ■041666667 1/09 014492754 1A14 ■00877] 93 1/26 ■04 1/70 014285714 lAis ■0081395052 1/26 ■038461638 1/71 0110S4517 1A16 ■00802069 1/27 ■087037087 1/72 1A17 ■008547009 1/28 ■035714286 1/78 01369863 1A18 ■008474576 1/29 ■034482759 1/74 013513514 1A19 ■008403361 1/30 1/75 01S333S33 1A20 ■003333333 1/31 ■032268065 1/76 0131B7895 1/121 ■008204463 1/32 ■03125 1/77 012987013 1/122 ■008196721 1/83 ■080S08080 1/78 012820618 1A23 ■008130081 1/34 -029411765 1/79 012658228 1A24 ■008064510 1/35 ■028571429 1/80 0125 1A23 ■008 1/33 ■027777778 1/81 012845679 1A26 -007936508 1/37 ■027037027 1/R2 012195122 1/127 ■007874016 1/88 ■026316789 1/83 ■012048193 1/128 ■0078125 1/39 ■025641026 1/84 ■011904762 1A29 ■007751938 I/IO ■025 01I76470S 1A30 ■007692308 1/41 ■024390244 1/88 011627907 1A31 ■007688588 1/42 ■028809524 1/87 ■011404268 1/133 ■007575768 1(43 ■023255814 ■011363036 1A33 ■007518797 | 1/44 ■022727273 ■011235955 1/134 ■007462687 I'l-i ■0222-22222 1'90 ■oniiiin 1A35 ■007407407 ! 146 ■02178913 I'Jl ■010989011 1/136 ■007352941 1 hv Google THE PEACTICAL MOBBL CALCULATOR. Praotto-i™ EtElmal or 1 FmcLloQot Dtcantl m Fr.,ti.,u=t 1 Dwimal M Hural.. ieolprotiU. NuDib. iwiprocul. ''Niimb.'" Ic^il.iucal. 1/137 00720927 1/198 006060505 1,259 X)8801004 1/138 007246377 1/199 005025126 1,260 ■ XI3840154 1A39 007194245 1/200 OOS 1201 03831418 1/140 007142857 1/201 004976124 1262 -t 03S10T94 1/Ul 007092199 1/202 004950495 1263 ■ 03802281 1/142 007042254 1/203 004920108 1,264 ■ [M)3787879 1/143 006993007 1/204 004001061 1,205 0377358.5 1/144 006944444 1/205 004878049 1,.266 J08759898 1/145 006896562 1/206 004854369 1,267 - 03745318 1/146 006849815 1,'207 004830918 003781348 1/147 000802721 1/208 004807092 1)^09 D0371T472 1/148 006756757 1/209 004784689 1/270 UO37037O4 1/149 006711409 1/210 00476] 005 1,271 008090087 1/150 006660067 1/211 004789836 i;272 008076471 1/151 006622517 1/212 004716981 l/'273 003003004 1/1S2 006578947 1,'213 004694836 1/274 003049685 l/i58 006585948 1,-214 004672897 1/273 003636304 1/154 006498506 1/215 004651163 1,'276 003028188 1/165 006451613 1.'216 00462063 J/277 003610108 1/156 006410256 m 004608 Sd 1/278 003697122 1/157 006369427 1 18 00458 166 1/279 003584229 1/168 0OG329114 1219 004560 1 1/280 00357] 429 1/159 006289308 1 004546465 1/281 003538719 1/160 00625 12 1 004524887 1/282 003540099 ywi 00621118 1-2 004504506 1/283 003533509 1/162 00617284 12 00448480O l,'i84 003522127 1/163 006134969 1/24 004464280 1,'285 008508772 1/1G4 000097561 1 6 004444444 1,'286 003400503 1/165 006060606 1 6 ■0044''47 9 1,287 003484321 1/166 006024096 1/2 7 00440528t> 1,288 003472222 1/167 005988024 I -^ 004 SZ-^l- 1/289 003400208 1/168 005952381 1 i 1 l,-290 003448270 1/109 00591710 1,291 003436428 1,170 O058S285S 1/202 O0S424O68 1/171 005817953 1,293 003412909 1/172 005818953 1/294 003401301 1/178 005780847 1 4 J04 14 1/295 003380831 1A74 005747126 1/ 3o 004266319 1,/2B6 003878878 1/175 005714280 004287288 1/297 003367003 1/176 005081818 1237 ■004219409 1/298 008355705 1/177 005049718 1/ 38 004201081 1/299 003844482 1/178 005617978 1 89 0041=441 1/800 003338383 1/179 005586692 1240 0041 66U6 1,301 003822250 1/180 005565550 1 41 0041408 8 1,'S02 003311258 1/181 005524862 1 4 00418 51 00830133 1/183 005494505 1/243 0O4H5220 1/304 003289474 1/183 005464481 1/244 004098801 1/305 008278689 1/184 005434788 V^je 004081033 1/300 003207974 1/185 005405405 1/246 004066041 1/307 008257329 1/186 005876844 1/247 004048583 003240753 1/187 005347594 1/248 004032258 1/300 003280246 1/183 005S19149 1/240 004016064 1.-310 003225803 1/18Q 005291005 1/250 ■004 1/311 008215434 1/190 ■005263158 1/251 003984064 1,'812 003205128 1/191 ■005235602 1/252 003908254 1,313 003194888 1/192 l,/358 ■003962569 1/314 003184713 1/193 ■005181347 1/264 -003937008 1,-315 003174008 1/194 ■006164689 l,e55 ■003921509 1/310 003104557 1/195 ■005128205 1/256 ■00830623 1,'317 003154674 1/196 ■005102041 1/257 ■003891051 1/318 003144054 1A07 ■005076142 1/268 ■0OS8759li9 1,-310 00313479J b,Google T BLE OF REC PROCAL OP hUMEEES. 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OB B cal Nomb li pr,«aL 8 003 25 00292 6 1/442 002262113 003 6266 1/38 00- a 80 1/443 002257336 003 0659 1 83 0026 0960 1444 002252252 80969 5 /884 00 04 6 1,445 002247191 OS 12 386 ■00259 103 1^446 002242152 93 WG 00 5906 4 1447 002217130 06 4S5 00 5839 9 1448 002233143 8 0* 00 5 3 1/443 00^227171 00 048 8 /38 ■00 5 0694 1/450 002222222 9 O0S0896 4 1/890 ■00 564 3 1451 002217296 ■003030308 1/39 ■002 5 54- 1/4 j2 002212889 00302 148 '/39 00265 1/453 002207606 0080 2048 /898 0026445 9 1454 00220--613 003008003 /394 ■00 5880 l/45o 002197802 8 4 0029940 2 ^96 ■00258 646 1466 002192982 36 0029850 6 /8»6 00262 53 1/457 00218tl84 0029 6 9 89 0026 8892 14o8 00218840S ■0029 359 00 612663 I4j9 002178649 8 -00 95868 / 9 00250 66 1400 002173918 1 9 00 949B5S 400 00 5 14W 002160197 00294 40 0024 3 68 1462 OO2IW0O2 00298266 40 10 8 662 146J 0021jns27 3 002B' 89 7 /40 W248 39 1404 0O21djl72 48 00 9 54^2 404 0024 5248 1,465 0O2150o38 00 9089 405 00 469 36 1/406 002146028 00 89855 OB 00 468054 1467 002141328 00 8 3 40 00245 002 1/468 0021J6752 88 8 4 /08 00 45098 1/469 002132199 00 8 8 09 10 444088 1/470 00-12709 86 / 00 48B0 4 1471 0OJ123142 « 8 4 ■0 433 9 1/472 002118644 8480 4 00 4- 8 1/473 0OJ1I4165 8409 00 42 08 1/474 002109705 00 88 8 5 9 1/47S 002105263 5 00 9 1/476 00210084 4 6 06846 1477 •002096486 /4 00289808 1^478 00209200 / 8 00 9 3 4 1/479 002087088 ■00 86 3 l,4b0 002088383 4 ■OO 80 5 1/481 0020790U2 4 00 5 1/482 0020740&9 1 0088 4 002369 68 1/483 002070893 B 3 / 3 00 864066 1/184 0020bbll6 2 548 424 00 5849 1/486 0020bl8o6 4 85 /4. 00285 94 1/486 002057013 "0 89 6 /4 6 00 34 4 8 1/487 0020^3383 4 00 84 92 1/488 0020191 & 24 96 1/4-8 -002 36449 1/489 00204499 9 4 9 00 8 1/490 002040816 00 4 5 8 1/491 ■0020SGG6 05 8 8 80 1/402 ■00203253 8 4 2 2 4 5 1/403 ■0020^8398 S 33 00 4 9 1/494 ■002024291 /4 4 / 5 00 1/495 ■002020202 002 088 1/496 ■002016129 436 00" 035 8 1/497 ■002012072 43 ■0(12008032 /* 8 ■002 8 1/499 ■002001008 43 ■00 1/500 ■002 1 bo 440 1,501 ■001996008 00 6 9 4 00 1/502 -001902032 b,Google THE PRACTICAL MODEL CALCOLATOE. '"^Jis."' utoii)"™™ "fir ^^j^l:f. ^KumE." K^'iS." 1/003 001988072 1/564 00177805 1/625 0016 1/504 001984127 1/565 001769912 1/626 001697444 1/605 001980198 1/566 001766784 1/637 001694896 1/506 001976286 1/567 001768668 1/628 O015B3357 1/507 001972887 1/568 001760568 1/629 001580825 1/S08 001968504 1/569 001757469 1/630 001587302 1/509 001964687 1/570 001754386 1/631 001684780 1/510 001960784 1/571 001751813 1/632 001582378 1/511 001956947 1/572 O01748252 1/638 001579779 1/512 001968125 1/573 001746201 1/634 001577287 1/618 001949818 1/574 00174216 1/6S5 001574803 1/611 001946525 1/575 O0173913 1/636 001672327 1/516 001941748 1/576 001736111 1/687 001569859 1/516 ■001937984 1/677 001783102 1/688 001567398 1/617 ■001934236 1/678 001780104 1/639 001504045 1/518 001980502 1/579 001727116 1/640 0015625 1/519 ■001926782 1/580 001724188 1/641 001560062 1/520 ■001928077 1/581 00172117 1/642 001557632 1/521 001919886 1/582 001718213 1/648 00155521 1/522 001915709 1/583 001715260 1/644 001552795 1/523 001912046 1/584 001712829 1/645 001550888 1/524 001908897 1/585 001709402 1/646 001647988 1/525 001904762 1/586 001706485 1/647 001545596 1/526 001901141 1/587 001708678 1/648 00154321 1/527 001897588 1,«88 00170068 1/649 001640832 1/528 001898939 1/589 001697793 1/650 001538402 1/529 001800359 1/590 001694915 1/651 001536098 1/680 001889792 1/591 001692047 1/652 001583742 1/531 001883239 1/592 001689189 1/658 001631394 1/582 001879699 1/593 001686841 1/654 001520052 1/538 001876173 1/594 001688502 1/656 001526718 1/534 001872659 1/595 001680672 1/656 00152439 1/586 001869169 1/596 001677853 1/657 00152207 1/533 O018Q5672 1/597 001676042 1/658 001519751 1/587 001862197 1/598 001672241 1/659 001517451 1/638 001858736 1/699 001669449 1/660 001515152 1/539 001856288 1/600 001666667 1/661 001512869 1/540 001851852 1/601 001663894 1/662 001510-374 1/541 001848429 1/603 00166113 1/663 001508296 1/542 001846018 1/603 001668375 1/664 001506034 1/548 001841621 1/604 001655629 1/665 001508759 1/644 001838236 1/605 001652893 1/666 001501502 1/545 001884862 1/606 001650165 1/667 00149925 1/540 001881502 1/607 001647446 1/668 001497006 1/547 001828154 1/608 001644737 1/669 001494768 1/548 001824818 1/600 001642086 1/670 001492537 1/549 001821494 1/610 001639844 1/671 001490313 1/550 001818182 1/611 001636661 1/672 00I488O95 1/651 001814882 1/612 001638987 1/673 001485884 1/562 001811694 1/613 001681321 1/674 00148368 1/568 001808318 1/614 001628664 1/676 001481481 1/654 001805054 1/615 001626016 1/676 00147929 1/556 001801802 1/616 001623377 1/677 001477105 1/656 001798561 1/617 O0I620746 1/678 001474926 1/557 001795882 1/618 001618123 1/679 001472754 1/558 001702115 1/619 001615509 1/680 001470588 1/559 001788909 1/620 001612908 1/681 001468429 1/560 001785714 1/621 001010306 1/663 001466276 1/561 001782531 001007717 001464129 1/562 001779359 001006136 l;084 001401988 1/503 001776199 1/624 001602564 1/085 001459854 b,Google TABLE OP BECIPHOCALS OF NUMBERS. TrtelWniT DetlD..! ot ProctLnn nr lN=lm=l M rrsition or Numb. B=dr™..J. 1,686 ■001457726 1/747 ■001338688 1/808 001287624 1/687 -001455604 1/748 ■001336898 1/809 001236094 '001453488 1/749 •001335113 1,810 001234568 1/689 ■001481879 1/750 ■001338333 1,811 001288016 1/690 ■001449275 1/751 ■001331558 1/812 001231627 1/691 ■001447178 1/752 ■001829787 1/813 001280012 1,692 ■001445087 1/753 ■001328021 1/814 001228601 1,-698 ■001443001 1/754 -00132626 1/815 001226994 l/eai ■001440923 1/756 ■O0I8245OS 1/816 001225499 i^-ess ■001436849 1/766 ■001822751 1/817 00122899 1/696 ■001486783 1/767 ■001821004 1/818 001223494 1/697 ■00148472 1/758 -001319261 l/Bia 001231001 1/698 ■001432665 1,759 ■001317623 1,'820 001219512 1/699 ■001480615 1/760 •001816789 1/821 O0J218O27 1/700 ■001428571 I/76I -00181406 001216545 1/701 -001426534 1/762 ■001312386 001215067 1/703 ■001424501 1/763 -001810616 001218592 1/703 -001422475 1/764 ■001308901 001212121 1/704 -001420455 1/765 ■00180719 001210664 1/T05 ■00141844 ■001305483 1,'827 00120919 1,706 ■001410481 1A67 ■001803781 1/828 001207729 1/707 ■001414427 1/768 ■001802088 001206273 1,708 -001412429 1/769 ■00130039 1/830 001204819 1,709 ■001410437 1/770 ■001298701 1/831 001203369 IJIO ■001108451 1/771 ■001297017 1/833 001201923 1/711 ■00140647 1/772 ■001295337 1/833 00120048 1/712 ■001404404 1/778 -0012S3661 1/884 001199041 1/718 ■001402525 1/774 ■00129199 1/885 O011S7605 1/7U ■00140056 1/776 ■001290328 1/836 001196173 1/71B ■001398601 1/776 ■00128866 1/887 001194743 1/716 ■001896648 L777 ■001287001 1/888 001193817 1/-17 ■0013947 1/778 -001285347 1/889 001191895 1/718 -001893758 1/779 ■001288697 1/840 001190476 1/719 ■001390821 1/780 ■O0I28206I 1/841 001189061 1/720 ■001888889 1^781 ■00128041 1/842 001187648 1/721 ■001386963 1/782 ■001278773 1/843 00118624 1/732 -001886043 1/783 ■001277189 1^44 001184834 1/728 ■001888126 1/784 ■00127551 1/845 001188482 1/724 ■001881215 1/785 ■001278885 1/846 001182088 1/725 ■00187981 1/786 -001272265 1/847 001180638 1/726 ■00137741 1/787 •001270648 1/848 001179245 i;727 -001876516 1/788 ■001269036 l,fl49 001177856 1/728 ■001373626 1/789 -001267427 1^ 001176471 1/729 ■001871742 1/790 ■001265823 1/861 001175088 1/780 ■001369863 1/791 ■001264228 1/952 001173709 1/781 •001867989 1/792 •003262626 1/858 ■001172333 1/732 -00136612 1/798 -001261084 1854 ■00117096 1/788 ■001864256 1/794 ■001269446 1^5 ■001169591 1/784 ■001862898 1/795 •001257862 1/856 ■001168224 1/735 -001360644 1/798 ■001266281 001166861 1/736 ■001358696 1/797 ■001264705 001165501 1/737 -001856862 1/798 ■001263138 001164144 1/738 ■001355014 1/799 ■001251364 001162791 1/789 -00185318 1/800 ■00125 1/861 00116144 1/740 ■001861351 1/801 ■001248489 001160093 1/741 ■001349628 1/802 ■001246883 1/868 Wn68749 1/742 ■001847709 1/803 ■00124538 1/864 001157407 1/743 ■001845895 1/804 ■001248781 1/866 001156069 1/744 -001344086 1/805 •001242236 001154734 1/745 ■001342282 1/806 ■001240695 1/807 001153403 1/746 ■001840488 1/807 ■001339157 1/868 001152074 b,Google THE PRACTICAL MODEL CALCULATOR. rt«tti=n.r DeoLiml ot F™ n Ff^i^or »ecim„l or Nnmh. Kmin,o.»i. 1,B69 ■001150748 19U 109 J 1/957 001044932 1/870 ■001149425 1914 00109409" 1/958 001043841 1/871 ■001148106 lIlS O0109'>896 1/959 001042758 1/872 ■001146789 1916 OOIO^I 08 1/960 001041667 1/873 ■001145475 1917 001090518 1/961 001040583 1/874 ■001144165 1918 001089S25 1/962 O0I089501 1/875 ■001142857 1119 001088139 1/968 0010S8422 ■001141553 19 00108 957 1/964 001037344 1/877 ■001140251 Ifi 1 00108 6 1/965 00108C269 1/878 ■001138952 19 2 001084o99 1/566 001085197 ■001187656 1/9 8 0010834 3 1/967 0010S4126 1/880 ■001188364 ld24 00108 ol 1/968 001033058 1^81 ■001135074 1/9'>J 001081081 1/969 001031992 1/882 ■001183787 1/9 6 001U79914 1/970 001080928 ■001182503 1/527 0010 8 49 1/971 001029860 1/881 ■001131222 J/9 8 0010 86 1/972 001028807 1/885 ■001129944 19 9 0010 04''6 1/978 001027749 ■001128668 1980 00107 j269 1/974 O0I020694 1/887 ■001127896 1031 001074114 1/975 001025641 ■001126126 193 0010 961 1/976 00102459 ■O0II24859 0010 1811 1/977 ■O0IO2SS41 1/890 ■001123596 1/934 001070664 1/978 001032495 1/891 ■001122334 1935 001069610 1/979 00102145 ■001121076 19J6 001068 6 1/080 001020408 1/893 ■001119821 1'937 OOlOfa "06 1/981 001019108 1/894 ■001118568 001066098 1/982 00101838 1/895 ■001117818 10o9 001064963 1/983 001017294 1/890 -001116071 1,940 OOlOfi 83 1/984 00101626 1/897 ■001114827 1/941 0010G2699 1,'985 001015228 1/898 ■001118586 1/542 001061571 001014199 1/899 ■001112347 1/913 001080446 1/987 001013171 1/900 ■OOlllim 1/944 001059322 001012146 i;'901 ■001109878 1/945 001058201 1/939 001011122 1/902 ■001108647 1/946 001057082 1,-990 001010101 1/903 ■00110742 1/947 001055966 1/991 001009082 1/904 ■001106195 1/948 0O1054S52 1/992 001008065 1/005 ■001104S72 1/949 001053741 1/D93 001007049 1/906 ■001108753 1/950 001052032 1/994 00100B036 1/907 ■001102536 1/951 001051525 1/995 001005025 1/908 ■001101322 1/962 00105042 1/996 001004010 1/909 ■00110011 1/953 001049318 1/997 001008009 1/910 ■001 098901 1/954 001048218 1/998 00IOO2004 1/^11 ■001091695 1/955 00104712 1/909 O0!001001 1/912 ■001096491 1/956 001046025 1/1000 001 Divide 80000 by 971. By the above Tahleive find thatl divided by 9T1 g and ■001029866 x 80000 = 82-38928. What is the sum of 5J-5 and ^ ? 883 1 <I5S + •001132603 ■001049318 2 953 X 5 = ■005602315 ■002098636 6 883 ■0077S1141 b,Google MENSURATION OP SOLIDS. D. - d™ ^ q™ 4^ JO D« 42 OS 045 43 or 44 68 4) 00 00 00 01 00 "_« 00 01 14 58 to 06 00 00 000 Tof d h s f b h of Us i d J g — M pyh i cube by itself, and that product again by the side, and it will give the solidity required. The side AB, or BO, of the cube ABODFOHE, ^ is 25'5 : what ia the solidity ? Sere AB' = (22-5)|' - 25-5 X 25-5 X 25-5 - 25-6 X 660-25 - 16681'S76, emtent o/tlie ciiS«. ni bvGoogle \ \ ° \ \ 80 THE PRACTICAL MODEL CALCULATOR. To find the eoUdity of a paralUhpipedon — Multiply the Icngtli by the breadth, and that product again by the depth or altitude, and it will give the solidity required. Required the solidity of a parallelopipedon , _ ABODFEHG, whose length AB ia 8 feet, ^ its breadth FD i^ feet, and the depth or altitude AD 6| feet ? Sere 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 ABOF ED, whose length AB is 10 feet, and either of the equal aides, BC, CD, or DE, of one of its equilateral ends BCD, 2| feet? Rere } X 2-5= X ^/3 = J X 6-25 X x/3 = 1-5625 K s/d = 1-6625 X 1-732 = 2-70625 = area of the lase BCD. 0,^ 2-5 + 2.5 + 2-5 ^ r^_ g_^^ ^ ^ ^^^^ ^^ 3 differences. the sides, BC, CD, DB, of the triangle CDB. And 3-75 - 2-5 = 1-25, .-. 1-25, 1-25 and 1-25 = WJience ^/3-75 X 1-25 X 1-25 X 1-25 = s/S'TS X l-2o« = v'7-32421875 = 2-7063 = area of the base as before, And 2-7063 X 10 = 27-063 solid feet, the content of the prism required. To find the convex surface of a cylinder. — Multiply the peri- phery or circumference of the base, by the height of the cylinder, and the product will be the convex surface. What is the convex surface of the right cylinder ABCD, whose length BC is 20 feet, and the diame- ter of its base AB 2 feet ? Here 3-1416 X 2 = 6-2832 = periphery of the base AB. And 6-2832 X 20 = 125-6640 square feet, the , 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 30^ = -7854 X 900 = 706-86 = area of the lase AB. 35343 And 706-86 X 50 = 35343 cubic inches; or solid feet. 1728 = 20-4531 hv Google for finding the su- MENSURATIOX OF SOLIDS. The four following cases contain all the rules perficiea and solidities of cylindrical ungulas. When the section is parallel to the axis of the cylinder. Rule. — Multiply the length of the arc line of the base by the height of the cylinder, and the product will be the curve surface. Jlultiply the area of the base by the height of the ^ cylinder, and the product will be the solidity. ^Vhen the section passes obliquely through the opposite sides of the cylinder. Rule. — Multiply the circumference of the hase of the cylinder by half the sum of the greatest and least lengths of the ungula, and the product will be the curve surface. Multiply the area of the base of the cylinder by half ^ the sum of the greatest and least lengths of the unguU, and the product will be the solidity. WJien tJie section passes through the hase of the cylin- j r ^ der, and one of its sides. ^ ^ '^ Rule.— Multiply the sine of half the arc of the base y- j, hy the diameter of the cylinder, and from this product / ; subtract the product of the arc and cosine. /... J Multiply the difference thus found, by the quotient of isI^At^a the height divided by the versed sine, and the product o will be the curve surface. Prom I of the cube of the right sine of half the are of the base, subtract the prodact of the aroa of the hase 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. ^ When the section passes obliquely through both ends of the cylinder. Rttle. — 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 j arc of the leas 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 lilie 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 circum- ference of the base hj the slant height, or tho 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 EC 15 feet; required the convex surface of the cone ACE. hv Google THE PRACTICAL MODEL CALCULATOR. Here 3-1416 X 3 - 9-4248 . And 9-4248 X 15 141-3T20 ,mf,r. e of the hose AB. ; 70'686 sqiiarefeet, the convex surface required. To find the convex surface of the frustum of a right cone. — Mul- tiply the sam of the perimeters of the two ends, by the slant height of the frustum, and half the product will be the surface required. In the frustum ABDE, the circumferences of the two ends AB and DE are 22-5 and 15-75 respectively, and the slant height BU is 26 ; what is the convex surface t (22-5 + 15-75) X 26 S^re ^ '^^^^'"J "t = 22-5- + 15-75 X 13 = 38-25 X 13 = 497-25 = convex sur- face. To find the solidity of a eone 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. Required the solidity of the cone ACB, whose diameter AB is 20, and its perpendicular height OS 24. Here -7854 x 20= = -7854 x 400 = 314-16 = area of the base AB. 24 And 314-16 X 3- = 314-16 X 8 = 2513-28 / ;.._.^_^ \^ = solidity required. Required the solidity of the hexagonal pyra- mid ECBD, each of the equal sides of its base being 40, and the perpendicular height C8 60, Mere 2-598076 {multiplier when the side is 1) X 40'^ = 2-598076 X 1600 = 4156-9216 = area of the base. 60 And 4156-9216 x -^ = 4156-9216 x 20 = ■ 3 83138-432 soUditi/. To find the aoUdity of a frustum of a cone or pyramid. — For the frustum of a cone, the diameters or circumferencea 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 hv Google MENSURATION OP SOLIBS. S3 diameters ; multiply the sum by -7854, and the product by the height ; ^ of tlie lust product will be the solidity. Or, Add together tiie square of the circumference of the greater end, the square of the circumference of the less end, and the pro- duct of the two circumferences; multiply the sum by -07958, and the product by the height ; J of the last product will be the solidity. For the frustum 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 tivo sides ; mul- tiply the sum by the proper number in the Table of Superficies, and the product by the height ; J 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 tho square root of their product ; multiply the sum by the height, and J of the product will be the solidity. What is the solidity of the frustum of the cone 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 ? (5^ 4- 3' + 5 X 3) X -7854 x 9 346-3614 _ \ 3 - 3 ~ ^ 115'4538 solid feet, the content of the frustum. What is the solidity of the frustum eEDBS of a hexagonal pyramid, the side ED of whose greater end is 4 feet, that eh of the less end 3 feet, and the height Ss, 9 feet ? (4^ + 3^ + 4 X 3) X 2-598076 x 9 865-159308 = 288-38643S solid feet, the solidity required. The following cases contain all tho rules for finding the superficies and solidities of conical ungulas. Wlten the section passes through the opposite extremities of the ends of the frustum. Let D = AB the diameter of the greater end; d = CD, the diameter of the less end ; /( = perpen- dicular height of the frustum, and n = '7854. d'~ d -yjyd nOh Then — -p. _ , — ■ x —5— = solidity of the greater elliptic ungula ADB. D ^Dd — d^ ndh , , — jj _ J — ■ X -^ = solidity of the less ungula ACD. = difi"erencc of these L J)~d ^ 3 ,And jj^ v'"4"A= + {1)-,^) X (D"^ ^— ^/Dd = curve surface of ADB. hv Google 84 THE PEACTICAL MODEL CALCULATOR. WTien the section cute off parts of the base, and makes the angle D)-B less than the angle CAB, Let S = tabular segment, whose v ersed sine i Bj*-^D; s — tab. seg. whose versed sine is -V- d, and the other letters as above. The {SxD'-sXd'x X p _ , = solidity of the elliptic hoof EFBD. . , 1 T-, — '^ ix{D+d)-Ar And^^::-^ ^/4A^ + (D - df x{Beg. FEE- ^, x ^ "-^ _ ^^ - X ■/ . _ i^ X seg. of the circle AB, whose height is D X — t — ) = convex surface of EFBD, When the section is parallel to one of the sides of the frustum. Let A = area of the base FEE, and the other let- ters as before. A XD — . Then i^yzT^ ~ i"^ ^(^ ~ d) X d) X ^h = solidity of the parabolic hoof EFBD. ^ And ^^^ ^W X (D - df X {seg. FEE - % D=d X v'dxJ) — d) = convex surface of EFBD. When 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 = and the area of the circular seg. EEF = a. the hyperbolic ungula EFBD. And ^ ^^ X x/U^ -i- (D- df X (cir. seg. EEF — d^ Br — * (D — d) Br ^ X — ^^Z^-J^' V ^ = curve surface of EFBD, D^ Br -D-d Br-d-H The transverse diameter of the hyp, seg, = -p __ ^ _ -p and the conjugate = d V j. _ j _ t>, ) fi^oin which its area may be found by the former rules. To find the solidity of a euneus 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. hv Google MENSURATION OF SOLIDS. How many solid feet are there in a wedge, Tvhoso base is 5 feet 4 inches loDg, and 9 inches broad, the length of the edge being 3 feet 6 inches, and the perpendicular height 2 feet 4 inches? Sere 170 (64 X 2 + 42) X 28 X 9 6 170 X 28 X 3 ( 128 + 42) X 28 X 9 6 6 170 X 14 ) = 7140 solid inches. And 7140 -^ 1728 = 4-1319 solid feet, the content. To find the solidity of aprismoid. — To the sum of the areas of the two ends add foux times the area of a section parallel to and equally distant from both ends, and this last sum multiplied by J 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 12 inches, and the corresponding aides of the other 6 and 4 inches, and the perpen^cular 30J feet. Here 14 X 12 + e^Ht = 168 + 24 = 192 = c^ sum of the area of the two e 14 + 6 20 Also - •-■^ ='^^ = length of the middle rectangle. 12 + 4 16 = breadth of the middle rectangle. :0 X 4 = 320 = 4 times the area of Whence 10 X 8 X 4 = the middle rectangle. Or (320 + 192) x -g^ = 512 x 61 = 31232 soUd inches. And 31232 -v- 1728 = 18-074 solid feet, the content. To find the convex surface of a sphere. — Multiply the diameter of the sphere by ita circumference, and the product will be tlie 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. What is the convex superficies of a globe BCG- whose diameter BG- is 17 inches ? Here 3-1416 X 17 X 17 = 53-4072 x 17 = 907"9224 square inches. And 907-9224 -;- 144 = 6-305 square feet. hv Google 86 THE PRAOTICAI. MODEL CALCULATOR. To find the solidity of a sphere or ghhe.— Multiply the cube of the diameter by -5236, and the product wjll he the solidity. What is the solidity of the sphere AEBO, whose diameter AB is 17 inches ? Sere IV X -5236 = 17 X 17 X 17 x -5236 = 289 X 17 X 5236 =4913 x -5236 = 2572-4468 '* ^»t(? 2572-4468 h- 1728 = 1-48868 solid feet. To find the solidity of the segment of a sphere. — To three times the square of the radana 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 wiil he the solidity. The radius Cw of the base of the segment *■ CAD is 7 inches, and the height An 4 inches ; what is the solidity ? Sere (7' X 3 + 4^ X 4 X -5236 = (49x3+4^ ^ x4x-5236 = (147 + 4=)x4x-5236 = (147+16) x 4 X -5236 = 163 X 4 x -5236 = 652 x -5236 = 341-3872 solid ineUs. '^--Jl,--'' 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 less diameter CD 15 inches, and the distance nm of the two ends 10 inches ? Here {W + 7-5= + -|^) 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 = 297T-92264 solid inches. To find the solidity of a spheroid. — Multiply the square of the revolving axe by the fixed axe, and this product again by '5236, and it wiil give the solidity required. ■5236 is = i^ of S-1416. In the prolate spheroid ABCD, the transverse, or fixed axe AC is 90, and the conjugate or revolving axe DB is 70 ; what is the solidity ? ^ Here DB^ x AC x -5236 = 70' x 90 X -5236 = 4900 x 90 X -5236 = 441000 X -5236 = 230907-6 = solidity required. hv Google MENSURATION OF SOLIDS. 87 To find the content of the middle frustum of a spheroid, its length, the middle diameter, and 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 miildle 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 tho solidity. Where -2618 = ^ of 3-1416. In the middle frustum of a spheroid EFGH, the middle diameter DB is 50 inches, and that of either of the ends EF or GH is 40 inches, and its length nm 18 inches ; what is its soli- dity? Mere (50' x 2 + 40') X 18 X -2618 = {2500 X 2 + 1600) X 18 X -2618 = (5000 + 1(300) x 18 x •2618 = 6600 X IS X -2618 = 118800 x -2613 = 31101-81 cuIh'; inches. When the ends are elliptieal or perpendicular to the revolving axis. — JIuhiply twice the transverse diameter of the middle sec- tion 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 ohlato « spheroid, the diameters of the middle section EF are 50 and 30, those of the end AD 40 _ and 24, and its height ne 18; what is the e^— solidity ? Here (50 X 2 x 30 -f 40x24) X 18 X -2618 = (8000 + 960) X 18 X -2618 = 3960 X 18 x ■2618 = 71280 X -2618 = 18661-104 = the soliditg. To find the solidity of the segment of a spheroid, when the base is parallel to the revolving axis. — Divide the square of the revolv- ing 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 DEED, the trans- " verse axis 2 DO is 100, the conjugate AC 60, and the height D» of the segment EDF 10 ; what is the s Mere (j^^, X 300 - 20) X 10= X 36 X 280 X lO^' X -5236 = 100-80 5236 = 5277-888 = tlte solidity. ■5236 -- hv Google 88 THE PRACTICAL MODEL CALCULATOB. When the base is perpendieular to the revolving axis. — Divide tlie 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 prodact again by -5236, and it will give the solidity required. In the prolate spheroid aEjr, the trans- verse axe EF is 100, the conjugate ab 60, ! the height an of the segment aKD 12 ; what is the solidity ? ^ Mere 156 (= diff. of ?>ah and 2a«) X If " i (= EF ^ a5 X 144 (= square of an) X -5236 b = ^ "" X 144 X -5236 = 52 X 5 X 144 X -5236 = 260 x 144 X -5236 = 37440 x -5236 = 19603-584 = the soUditij. To find the solidity of aparabolio conoid. — Multiply the area of the base by balf the altitude, and the product will be the content. What is the solidity of the paraboloid ADB, whose height Dra is 84, and the diameter BA of its circular base 48 ? Here 48^ x -7854 x 42 (= J Dm) = 2304 x ■7854 X 42 = 1809-5616 x 42 =76001-5872 ^ = the solidity. 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 tbe diameters of the two ends by the height of the frus- tum, and the product again by -3927, and it will give the solidity- Required the solidity of the parabolic frus- /^T'"--. turn ABCd, the diameter AB of the greater end being 58, that of the less end do 30, and the height no 18. Sere {58^ + 30=) X 18 X -3927 = (3364 + ^ 900) X 18 X -3927 = 4264 x 18 x -3927 = 76752 x -3927 = 30140-5104 = the solidity. To find the solidity of an hyperholoid. — 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 •6236 will give the solidity. In the hyperholoid ACB, the altitude Qr is 10, the radius Ar of the base 12, and the mid- dle diameter nm 15-8745 ; what is the solidity ? < -5236 = Here 15-874 5^ -|- 12^ x 10 251-99975 -|- 144 x 10 x -5236 = lOx -5236 = 3959-9975 x-5236 = = the solidity. 395-99975 x 2073-454691 " hv Google MENSURATION OS SOLIDS. 89 To find the Bolidity of the frustum of an liyperholio conoid. — 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 J?, rs is 20, the diameter AE of the greater end 32, that DC of the less end 24, and the middle dia- meter nm 28'1708; required the solidity. Here {16^ + 12^ + 28-1708=) X 20 X -52359 = (256 + 144 + T93-59S9) x 20 x -52359 = t 1193-5939 X 20 X -52359 = 23871-878 x -62359 = 12499-07660202 = s. To find the solidity of a tetraedron. — Multiply I'j of the cube of the linear side by the square root of 2, and the product will be the solidity. The lineal- side of a tetraedron ABCw is 4 ; what is the solidity ? 4' ^4x4x4 ^4x4 ^16 j2Xv/2=— ^-^x^2 = --3-x^2=y, 16 22-624 X v" 2 = -g- X 1-414 = — g- ■ = 7-5413 = soUditi/. 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. "What is the solidity of the octaedron EGAD, whose linear side is 4 V - X -^ 2 = 21-333, v/2 = 21-333 X 1-414 = 30-16486 = solidity. 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 AECDE is 3; what is the solidity ? 21^/5 + 47 ^„ , 21 X 2-23606+47 e/ solidity. To find the solidity of an icosaedron. — To three times the square root of 5 add 7, and divide the sum by 2 ; then the square root of hv Google 90 THE PRACTICAL MODEL CALCULATOB. this quotient being multiplied by | of the cube of the lineai- side will give the solidity. T + 3 ^/ 5 That is ^ S^ X %/ (■ — —> } = solidity when S is = to the linear side. The linear side of the icosaedron ABCDEF is 3 ; what is the solidity ? 3 n/ 5 + T 5x3^ 3 X 2'23606 + 7 < 5 X 27 6-70818 13-70818 45 6x9 ~2~ s/6-8540y X 22-5 = 2-61803 X 22-5"= 58-9056 = soUdity. The superficies and solidity of any of the five regular bodies may be found aa follows : EULE 1. Multiply the tabular area by the square of the linear edge, and the product will he 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. S5i' K..... ........ ........ I 4 TetraedrOQ 1.Y3205 0.H785 « HesaGdron 6.00000 1.00000 H Octoedron S.46410 0.47140 12 Dodecaedron 20.64578 7.66-312 20 Icosaedron 8.66025 2.18169 To find the convex superficies ofacylindrie ring. — To the thick- ness of the ring add the inner diameter, and this sum being multi- plied by the thickness, and the product again by 9.8696, will give the superficies. The thickness of Ac of a cylindric ring is 3 inches, and the inner diameter cd 12 inches , / what is the convex superficies ? a I . 12-1-3 X 3 X 9-8696 = 15 x 3 x 9 869i = 45 X 9-8696 = 444-132 = superficies To find the solidity of a cylindria ring — To the thickness of the ring add the inner diameter, and this sum bem^ multiplied by the square of half the thickne&s, and the pioduct dgam ly I hbOl , will give the solidity. hv Google MENSURATION OP SOLIDS. 91 T\'hat is the solidity of an anchor ring, whoso inner diameter is 8 inclies, and thicliness in metal 3 inches ? 8"T3 X i]= X 9-8696 = 11 x 1-5^ x 9-8693 = 11 x 2-25 X 9-8696 = 24-75 X 9-8696 = 244-2T26 The inner diameter AB of the cylindrie ring cdef equals 18 feet, and the sectional diameter cA or Be equals 9 inches ; required the convex surface and solidity of the ring. 18 feet X 12 = 216 inekes, and 216 + 9 * X 9 X 9-8696 = 19985-94 square inches. 216 + 9 X 9= X 2-4674 = 44968-365 eubia inches. 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 ; there- fore, the proper diameter by which to dotcrmiDe the length of the iron in an unbent state, ia the distance from centre to centre of the iron of which the hoop is composed : henee the rule to determine the length 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 di- ameter, 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 tbickneas of the iron IJ, 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 Cir- cumferencea. Thus, 33-25 = 2 feet 9| in., opposite to which is 8 feet 8i inches. Again, let it be required to form a hoop of iron ^ inch in thick- ness, and 16^ inches outside diameter. 16-5 — -875 = 15-625, or 1 foot 3g inches; opposite to which, in the Table of Circumferences, ia 4 feet 1 inch, independent of any allowance for shutting. The length for angle iron, of which to form a ring of a given di- ameter, 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- cd cd tract twice the extreme strength ; and the sum or 17^ "^ remainder is the diameter by which to determine ri ii; the length of the angle iron. Thus, suppose two i i angle iron rings similar to the following bo re- l". —?.[ quired, the exterior diameter AB, and interior ^\ |V^ diameter CD, each to be 1 foot 10 J inches, and erf cd the extreme strength of the iron at the root cd, cd, &c, J of an inch; hv Google 92 THE PBACnCAL MODEL CALCULATOR. tTvice ^ = 1|, and 1 ft. lOJ in. + 1|- = 2 ft. ^ in., opposite to which, in the Table of Circumferences, is 6 ft. 4^ in., the length of the iron for CD ; and 1 ft. 10^ in. - 1| = 1 ft. 8f in., opposite to which is 5 ft. 5J 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 locomotive engine tender tank, 2 feet 8 inches in depth, and its superficial di- mensions the following, with reference to the annexed plan : Length, or dist. between A and B = 10 ft. 2| in. or, 122-75 in. K c B >— - ^ CJ jj Breadth Mean breadth of coke- " C and D = and g - ps lOJ 79-5 46-75 IJ 81-25 8i 32-25 18-6 Radius of back corners vx = Then, 122-75 X 79-5 = 9758-525 square inches, as a rectangle. And 18-5^ X -7854 = 268-8 " " area of circle formed by the two ends. Total 10027-325 " " from which de- duct the area of the coke-space, and the difference of area between the semicircle formed by the two back corners, and that of a rect- angle of equal length and breadth ; Then 46-75 x 37-25 = 1731-4375 area of r, n, s, t, in sq. ins. 32-25= X -7854 , , — -g— = 408-4 area of half the circle m. Radius of back corners = 4 inches j consequently 8^ X -7854 = 25-13, the semicircle's area; and 8 X 4 = 32 - 25-13 = 6-87 inches taken off by rounding the c Hence, 1731-4375 + 408-4 + 6-87 = 2146-T07, 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. hv Google MENSURATION OF TIMBER. Tables hj which to facilitate the Mensuration of Timber. 1. Flat or Board Measure. ... Ares, of* Bre.athin A a»of> Bresdth in A esofj inohs.. aJfo^t iniiht.. U» Uftwt. ■0208 i S334 8 6667 ■0417 H 3542 8i 6875 ■0G23 H 375 8J 7084 1 ■0834 4* 8| 7292 U •10i2 6 4167 75 l| ■125 H 4S75 9} 7708 l| ■1450 tl 4583 9J 7917 2* ■1667 4792 9J 8125 n ■1875 6 5 !0 8334 2* ■2084 lOJ 8643 4 ■2292 5416 io| 875 3 ■35 5625 10| 8959 8 ■3708 7 5888 11 9167 3 ■2016 7 6042 11 9375 ■3125 11 9588 7 6458 11 9792 Application, and Use of the Table. Required the number of square feet in a board or plank 16^ feet in length and 9f inches in breadth. Opposite 9f is -8125 x 16-5 = 13-4 square feet. A board 1 foot 2| inches in breadth, and 21 feet in length ; what is its superficial content in square feet ? Opposite 2f ia '2292, to which add the 1 foot ; then 1-2292 X 21 = 25-8 square feet. In a board 151 inches at one end, 9 inches at the other, and 141 feet in length, how many square feet ? '^^'^^ ^ = 12^, 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 meao girth by twice the length, and the product will bo the solidity, very near the truth. A piece of timber is 30 feet long, and the mean girth is 128 in- ches, what is the solidity ? = 273-06 cubic feet. This i ployed. ■ the truth than if one-fourth the girth he em- hv Google THE PRACTICAL MODEL CALCULATOR. 2. Cubic or Solid Measure. BEaanii Co mTt^ Mf»ii"i CuMtfMt maaj- &^iV^ Mein^ CuUtfesl &S. uA ^■fla. SZS. iii°"f«t. ^.^ iii""^6. Si lilK^fcot. 6 25 12 I 2 25 24 4 6 272 m 1.042 18 2S13 24 4 0S4 294 12J 1'085 2 376 24 24 lit 8 6 817 lll 1'129 18f 2 442 4''j4 7 340 1-174 2 506 25 4o4 7 364 1'219 m 2 574 251 4 428 7 39 IS^ 1-265 2 64 2^i 2oJ 4 516 7 417 1-813 i4 2 709 4 60J 8 44i 14 20' 2 777 .6 4 094 472 14^ 1-41 201 2 898 4-S5 el 601 14J 1-46 20J 2 917 24 48 b 531 14J 1-511 20f 4 909 g 502 16 1-562 21 8 '062 27 502 9 694 16^ 1-615 21 i- 3136 27^ 5 1o8 9 626 15* 15| 1-668 21J 3 209 27i 5-.'i2 9 669 1-772 3 285 211 6 818 10 694 16 1-777 22 3 362 25 6 444 10 73 16 16 1-833 8 438 28 5 542 10 766 1-89 22 3 516 28 28 6b4 10 803 16 1-948 8 508 5 "4 n 81 17 2006 23 3 673 29 6 84 17i 2-066 23t ^754 291 5 941 918 17i 2-126 3^i 8 886 29* 6 044 11^ 359 17| 2-187 231 8 917 29| 0146 In the euliic estimation of timbei, custom has established the rule of i, 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 I girt, and the product is the solidity of the given dimensions in cubic feet. Suppose the mean I girt of a tree 21i inches, and its length 16 feet, what are its contents in cubic feet ? 3-136 X 16 = 50-1T6 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 2 J, a deal 9 by 3, and a plank 11 by 3, these being what are tenned 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 fL'c-t, 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 Id 19 X 11 = 209 and 6 21 21 X 6 = 12fi 12 ) 447 ) 37-25 standard deals. hv Google 3IEK8URATI0N ( Table showing the number of Lineal Feet of Seantling of various dimensions, ivhieh are equal to a Cubic Foot. i...i,» Ft lu ln.h« F. in iml-. rt I 2 jb 4^ 9 qi" 2 ( ^ 2& 9 Ji ? 8 7 2 m lu" lOi 2 ^ ih 20 7 Si 6 6 11 2 2 i 18 b a lU 2 1 4i IG ^ 'i 5 b 12^ 2 5 14 5 ^ ^ 1j 1 t. 7i 4 1 ~1~ 2 11 12 1 s' ^i 2 9 1 ej 11 1 8i 4 i 2 6 I 7 10 5 q 7 9 n 9 1? i S 7 1 8} 9i 2 5 2 3 2 2 8i 8 & b ;? 3 ^ 10 lOS 2 1 1 11 "J 7 7 11] 11 1 10 10 10^ u 7 i b 10 12 a 3 Y} a 1 9 1 8 11 b G 5 6 9 1 llj b 4. 5i 5 3 I, s 12 b b £ 4 10 H — 3 1 ?' 3 4 5 4 1 * 9 1 2 1 ]0 18 ? s U 8 12 S ? s ^ 10 3 7 1 10 1()V S 4J 10 8 t ''I 5 5 11 6 ■■J H 7 9 8 1 aj 3 2 3 12 i b 10 a 10 ^ 7 7 4 6 10 j.i 2 9 2 8 J, 9 1 I 1 ^* <> 4 m m 1 7 5 8 6 5 8 5 4 12" 2 4 ! 11" IP 1 5 1 4 I, 4 H 5 ^ H 3 8 12" 1 i 10 4 10 7 lOJ 4 b 1 7J 3 2 ^ 10 1 J 4 4 ^ S i lOJ 1 4 lU 4 2 h^ 2 10 g 11 1 4 Ji" 4 ^ 2 8 o III Hewn and sawed timber are meaaured by the cabio foot The unit of boaid measuio is a superficwl foot one inch thick To measure round timbtr — Multiply the Itngth m feet by the square of J of the mean girth in inches, and the product divided by 144 gives the content in cubic feet. The \ girths of a piece 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 + 3-3 + 35 + 40 5 -^^^■ ™ 32^ X 30 Then — ^3^— = 213^ cubic feet. hv Google 96 THE PRACTICAL MODEL CALCULATOR. Table containing the Superficies and Solid Content of Spheres, fro7n 1 to 12, and advancing By a tenth. Diun. B-P8rat«^ SoUdlly. DU.™. GaperBcisB. SoLiditj. DL»m. SuikiM™. SolWilJ. To 8'14i? ■6286 Tf 69-8979 54-8617 8-4 221-6712 310-3898 1 8-8013 ■6969 ■8 72-3824 57-9059 ■5 226-9606 321-5558 2 4-5289 ■9047 ■9 76-4298 61-6010 -6 282-3527 333-0389 3 6-8033 1-1603 5-0 78-5400 65-4500 ■7 237-7877 844-7921 i 6-1676 1-4367 ■1 81-7180 69-4560 ■8 243-2865 356-8187 fi 7-0686 1^7671 ■2 84-9488 73-6228 -9 248-8461 869-1217 6 8-0424 2-1446 ■3 88-2476 77-9519 9-0 254-4696 381-7044 7 9-0792 2-8724 -4 91-6090 82-4481 -1 260-1558 394-6697 8 tO-1787 8 ■0636 ■5 95-0334 87-1139 ■2 265-9130 407-7210 9 11-3411 3-5913 -6 98-6205 91-9528 271^7169 421-1613 2 12-5664 41888 102-0706 96-9670 ■4 277^8917 434-8937 1 18-8544 4^8490 105-6834 102-1606 ■6 283-5294 448-9215 2 15-2053 5-5752 109-3590 107-5364 ■6 289-5298 463-2477 3 16-6190 6-3706 6-0 113-0976 113-0976 ■7 296-6931 477-7755 4 18-0956 7-2382 1 -1 116-8989 118-8472 ■8 801-7192 492-8081 6 19-6350 8-1812 ■2 120-7681 124-7886 ■9 307-9082 508-0485 « 21-2372 9-2027 124-6901 130-9246 10-0 814-1600 523-6000 7 22-9022 10-8060 -4 128-6799 137-2585 ■1 820-4746 539-4656 24-6300 11^4940 ■6 132-7826 143-7936 326-8520 555-6485 9 26-4208 12-7700 -6 136-8480 150-5329 883-2923 572-1518 3 28-2744 14-1872 ■7 141-0264 157-4795 -4 339-7954 588-978* 1 80-1907 15-5985 145-2675 164-6365 -5 346-3614 606-1324 2 32-1699 17-1573 ■9 149^6715 172-0073 -6 862-9901 623-6169 8 34-2120 18-8166 7-0 I53^9384 179-5948 859-6817 641-4325 4 36-3168 20^5795 ■1 158^3680 187-4021 -8 366-4362 659-6852 5 88-4846 22 '4493 ■2 162-8606 195^4326 -9 378-2534 678-0771 6 40-7161 24^4290 -8 167-4158 203-6898 11-0 380-1836 696-9116 7 43-0085 26^5219 ■4 172-0340 21M752 -1 887-0765 716-0915 8 46-3647 28-7309 ■5 176-7160 220^8987 ■2 394-0823 736^6200 8 47-7837 310594 ■6 181-4588 229^8478 -3 401-1509 755^5008 4 50-2656 33-5104 ■7 186-2654 239-0511 ■4 408-2823 775-7864 1 62-8103 36-0870 .■8 191-1349 248^4754 •5 415-4766 796-3301 2 55-4178 88-7924 -9 196-0672 258-1562 -6 422-7836 817-2861 3 58-0881 41-6298 80 201-0624 268-0832 -7 430-0536 838-6045 4 60-8213 44-6023 -1 206-1203 278-2626 437-4363 860-2915 5 63-6174 47-7130 ■2 211-2411 288-6962 -9 444-8819 882-8492 ■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. =31104 36- =60480 m 69= =119232 53 91584 VII 120960 3 6184 W S45G0 37 54 93312 71 122688 4 6912 VI 86288 65664 55 95040 TA 124416 152064 6 8640 n 38016 67802 56 96768 73 126144 H9 153792 6 10868 39744 40 69120 hV 98496 74 127872 91) 155520 41472 41 70848 J>K 100224 Vh 129600 91 167248 43200 4^ 72576 6>l 101952 V6 131328 9 15552 26 44928 4;i 74304 60 103680 77 133056 10 17280 V7 46656 44 76032 HI 105408 7« 134784 94 162432 48384 46 62 107136 '(9 136512 95 164160 12 20786 V,H 50112 46 79488 63 108864 HO 188240 13 22404 HO 51840 47 81216 64 110593 81 189968 HV 14 24193 31 63568 48 82944 65 112320 K2 141696 HM 15 25920 56296 4H 84672 114048 K3 143424 99 171072 57024 511 86400 67 115776 84 146152 100 17 29876 34 58752 51 88128 6a 11750411 b,Google CriTIXGS Ayo eubaxkmests. CUTTINGS AND EMBiNKMENTS. The angle of repose upon railways, or that incline on which a carnage would rest in whatover situation it was placed, is said to be at 1 ia 280, or nearly 19 feet per mile ; at any greater rise than this, the force of gravity orercomea the horizontal traction, and carriages will not rest, or remain quiescent upon the line, but will of themselves run down tho line with accelerated Telocity. The angle of practical effect ia variously stated, ranging from 1 in To 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 he given to suit the various materials of which the cuttings are composed : thus, rock will generally stand when tho sides are vertical ; chalk varies from -^ to 1, to 1 to 1 ; gravel 1| to 1 ; coal 1|^ 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 ia calculated for the purpose of ascertain- ing 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 SO feet. Slopes Itol. Da [.in Hiilf o,^.. Conlenl ConliM T Half Coattnl Content c,„... liDgin >ridlK S'Stb f Iper- rS vim £S f™u ^.hiS!' ohiin. bmdt). ~~r TT 75 78 2-44 7.33 14-67 "26" "«" 3599-11 68-66 190-67 3S1-33 2 17 156 42 4-89 14-67 29-88 27 42 8762-00 65-99 198-00 393-00 18 242-00 22-00 44-00 28 48 3969-78 205-33 410-67 10 332 44 9-78 2B-33 68-67 29 44 4182-44 70-88 212-67 425-88 5 20 427 78 12-22 36-67 78-83 80 45 4400-00 78-32 440-00 b 21 628 00 14-67 44-00 88-00 31 46 4622-44 76-77 7 22 b3311 17-11 51-38 102-67 47 4849-78 469-33 8 23 74811 19-56 58-67 117-33 83 48 5082-00 80-67 242-01 484-00 24 868 00 22-00 66-00 182-00 34 49 6319-11 83-11 10 25 977 78 24-44 78-33 146-67 35 50 5561-11 85-55 513-88 11 2C 1102 44 26-89 80-67 161-33 36 61 5808-00 88-00 264-00 528-00 12 27 1282 00 29-33 83-00 176-00 37 52 6059-78 90-44 271-88 I542-G7 18 28 1366 44 31-78 96-33 190-67 38 53 6816-44 92-39 278-671567-83 14 29 1505 78 84-22 102-67 206-88 39 54 6578-00 95-88 288-00;572-00 15 80 1650 00 86-66 110-00 220-00 40 55 6844-44 97-77 298-38 586-67 lb 81 1799 11 39-11 117-33 234-67 41 56 7116-78 100-22 300-671601-83 17 82 1963 11 41-55 124-67 249-88 42 57 7392-00 102-06 808-00 616-00 Ig 33 2112 00 43-99 132-00 264-00 43 58 7673-11 :05-ll 315-33630-67 19 84 2276 78 46-44 189-38 278-67 44 69 7959-11 107-55 822-67643-38 20 35 2444 44 48-89 146-67 293-38 45 60 8250-00 109-99 880-00 660-00 2618 00 51-83 154-00 808-00 46 61 8545-78 112-44 3S7-33;874-67 22 87 2796 44 63-77 161-38 822-67 47 62 8846-44 114-88 844-67,689-33 .i5 2979 78 56-21 168-67 337-33 48 63 9152-00 117-88 1362 -00 '704-00 .,4 a-* dl63 00 68-00 17600 352-00 49 64 9462-44 119-77 369-38[718-67 -^ 40 33bl 11 61-10 183-83 8S6-67| 50 65 jD777-78 122-21 366-67733-83 b,Google THE PRACTICAL MODEL CALCULATOR. 1408-00 1578 '00 1746- 1925-00 2112-00 18-00 3717-00 2988-83 81B7-00 OtSBM- 102-e7 117-88 132-00 146-67 5500-00 6797-00 6101 "" 6413-00 0732-00 7058- 7392-00 7738-00 8081-33 8487-00 8800- 9170- 9548-00 9933-00 _0826-88 10735-00 11182'-00 11546-SS 05-f 97-77 100-22 102-66 105-11 107-56 109-r 113-44 114-88 117-88 119-77 123-21 227-£ 284-67 242-00 24S-88 266-67 364-00 271-38 278-67 286-00 293-33 822-67 380-00 387-88 18400 4B8-67 513-33 B28-00 542-67 557-33 572-00 586-67 801-38 31600 0-67 645-88 Slopes 2 to 1. Dapth Half CobK« Gontant ConleU Depth Hllf Content Contsnt CoDUnt or width Qraleut m. t.S£Z ■-¥E of irWth toSS plX" otaper- M Wpin ;^p«°I rMB.In ^r '^L^ 7^P«!r hirtt.li, ~ir n" 78-22 2-44 7-33 14-67 26 ~6f 5211-66 63-55 190-67 381-83 2- 19 166-22 4-89 14-67 29;33 27 69 5544-00 65-99 198-00 396-00 3 21 264-00 22-00 44-00 23 71 6886-22 68-43 206-83 410-67 4 28 371-56 9-78 29-33 68-67 78 70-88 212-67 425-83 5 12-32 86-67 73-38 30 75 6600-00 73-32 220-00 440-00 6 27 616-00 14-67 44-00 8800 31 77 6971-55 75-77 227-33 464-67 7 29 762-89 17-11 51-33 102-67 32 7Q 7352-89 78-22 234-67 469-33 8 81 899-56 19-56 58-67 117-38 81 774400 80-67 242-00 484-00 83 1056-00 22-00 66-00 182 00 84 8144-89 83-11 249-33 10 35 1232-32 24-44 73-33 140-67 35 8555-55 86-65 266,-67 >13-33 11 87 1898-22 36-89 80-67 161-38 86 87 897600 88-00 364-00 528-00 12 1684-00 29-38 88-00 176-00 87 89 9406-2^ 90-44 271-33 642-67 13 11 1779-65 81-78 95-33 190 67 38 91 9846-22 278-67 557-33 U 43 1984-89 84-22 103-67 205-88 80 98 10296-00 96-38 286-00 572-00 1-6 46 2200-00 86-86 110-00 220-00 40 05 10766-55 97-77 393-83 586-67 le 47 2424-89 89-11 117-88 284-87 41 97 11224-89 100-22 800-67 601-83 17 49 41-55 124-67 249-83 42 99 11704-00 102-66 308-00 616-00 18 51 2904-00 43-99 132-00 264-00 43 101 12192-89 105-11 815-33 630-67 19 58 8158-22 46-44 180-33 278-67 44 103 12691-56 107-56 322-67 645-33 20 55 34-2222 48-89 146-67 298-33 45 105 13200-00 109-99 830-00 660-00 21 57 8696-00 51-38 164-00 30800 46 107 13718-22 112-44 887-38 674-67 22 59 3970-55 53-77 161-83 S22-67 47 109 14246-22 114-88 344-67 689-88 ei 4272-89 66-21 168-67 837-33 48 111 14784-00 117-38 353-00 704-00 24 63 4576-00 58-66 176-00 35200 49 118 .5331-65 119-77 859-88 718-67 26 65 4888-89 61-10 183-38 366-67 50 115 122-21 366-07 783-38 b,Google CUTTIJfGS AND EMBANKMENTS. 99 By the fourth, fifth, and sixth cohimns in each table, the niimher of cubic yards is easily ascertained at any other ividth of formatioa level above or below 80 feet, having the same slopes as by the tables, thus : — Suppose an excavation of 40 feet in depth, and 33 feet in width at fovniatiou 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 as- certained 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 tho quotient is the content in cubic yards. Suppose a cutting of any length, and of which take 1 chain, its depth being 14^ feet, width at the bottom 28 feet, and whose sides have a elope of 1| to 1, required the content in cubic yards : 14-5 X 1-25 = 1M25 + 28 x 14 = 645-75 X 66 = 42619-5 — 27 — = 1578-5 cubic yards. 5 { (6 + rV) ¥ + (6 + rf) 4 + 4 [S + r^^-^J — g^| 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 = S; the perpendicular depth of cutting at the higher end = ^; the perpendicular depths of cutting at the lower end = h'; I, the length of the solid ; and rh' the ratio of the perpendicular height of the slope to the horizontal base, mul- tiplied by the height h'. rh, the ratio r, of the perpendicular height of the slope, to the horizontal base, multiplied by the height Ji. Let & = 30 ; A = 50 ; /t' = 20 ; l=M feet ; and 2 to 5 or | the ratio of the perpendicular height of the slope to the horizontal base: ^ I (30 + § X 20) 20 + (30 + 3 X 50) 50 + 4 [30 -ff ^^ t, ^^ ~\ — ^ — I = 14 I 38 X 20 + 50 X 50 + 4 X 44 X 35 I ^ 131880 131880 cubic feet. — 07 — ~ 4884'44 cubic yards. This rule is one of the most useful in tho mensuration of solids, it will give the content of any irregular solid very nearly, whether it be bounded by right lines or not. hv Google THE PRACTICAL MODEL CALCULATOR, Table of Sqiuires, Ouhes, Square and Ouhe Roots of Numbers. Kumbsr. Sq^.tE!. CobES. S,n«.II.Bl.. Cql»H00lS. ReeiptMri.. 1 1 1 10000000 l-OOOOOOO ■lOOOOOOOO 4 1-4142136 1-2599210 -800000000 3 27 1-7320608 1-4422496 ■833333333 4 16 64 2-0000000 1-587I0I1 ■260000000 5 25 125 2-2360680 1-7099769 ■200000000 6 216 2-4494897 1-8171206 ■166666667 49 343 2-6467613 1-9129812 ■142867143 64 512 2-8284271 2-0000000 -125000000 9 81 729 8-0O000OO 2-0800837 ■111111111 10 100 1000 3-1 622777 2-1544847 -lOOOOOOOO 11 121 1331 3-3166248 2-2289801 -090909091 12 144 1728 8-4641016 2-2894280 ■083338333 13 169 2197 3-6065518 2-3513847 -076923077 14 196 2744 3-7416574 2-4101422 ■071428571 16 225 3375 2-4662121 ■066666667 16 256 4096 4-0000000 2-6198421 -062500000 17 289 4913 41231066 2-6712816 ■068823529 18 824 5832 4-2426407 2-6207414 -065555556 19 861 6869 4-3588989 2-6684016 ■062681679 20 400 8000 4-4721360 2-7144177 ■050000000 21 441 9261 4-5825757 2-7589243 ■047619048 22 4S4 10643 4-6904158 2-8020393 '045464646 23 629 12167 4-7958316 2-8438670 ■043478261 U 676 4-8989795 2-8844991 ■041666667 25 625 15625 6-0000000 2-9240177 ■040000000 26 676 17676 5-0990196 2-9624960 ■038461538 27 729 19683 5-1961524 3-0OO000O ■037037037 28 784 21952 6-2915026 8-0365889 ■035714286 29 841 24889 5-8851648 3-0723168 ■034482759 80 900 27000 5-4772256 8-1072325 ■033333333 31 961 29791 5-6677644 3-1413800 -082258065 32 1024 82768 6-6568542 8-1748021 -031260000 1089 36937 6-7445626 3-2075343 -030303030 34 1156 39304 6-8309519 3-2396118 -029411765 35 1225 42875 5-9160798 3-2710663 -028571429 36 1296 46656 6-0000000 8-8019272 ■027777778 87 1369 60653 6-0827625 3-3322218 ■027027027 1444 64872 6-1644140 8-3619754 ■026816789 89 1521 59319 6-2449980 3-3912114 -025641026 40 1600 64000 6-3245558 S-4199&19 -026000000 41 1681 68921 6-4031242 3-4482172 -024390244 42 1764 6-4807407 3-4760266 ■028809624 43 1849 79307 6-5574385 8-50S398I -023255814 44 1936 85184 6-6332496 3-5303483 ■022727273 45 2026 91125 6-7082039 3-6568938 -022222222 46 2116 97336 6-7823300 S-6880479 ■021789130 47 2209 103823 e -8556646 8-6088261 ■021276600 48 2804 110592 3-6342411 ■020833833 2401 117649 7-0000000 8-6593057 ■020408163 50 2500 125000 7-0710678 3-6840814 ■020000000 61 2601 132661 7-1414284 8-7084298 ■019607843 62 2704 140608 7-2111026 8-7325111 ■019280769 53 2809 148877 7-2801099 8-7562858 ■018867925 64 2916 157464 7-8484692 8-7797631 ■018618519 56 3025 166375 7-4161985 8-8029625 ■0181818)8 56 8186 175616 7-4833148 -017867143 57 S249 185193 7-6498344 3-8485011 ■017648860 b,Google TABLE OP SQUARES, CUBES, SQUARE AHD CUBE ROOTS. S«^bzr. S^uarss. ' CabDS. 6iumlk«». C«l« Ro.t,. H.=i:irr™lj. bS 8364 195112 7-6167781 8-8708706 ■017241879 69 3481 205879 7'6811457 3-8929965 -016949153 60 3600 216000 7-7459667 8-9148676 ■016066667 61 8721 226981 7-8102497 8-9304972 -016393443 62 3844 288828 7-8740079 3-9578915 -016129032 8969 250047 7-9372539 3-9790671 ■015873016 61 40B6 262144 8-0000000 4-0000000 -015625000 65 4225 274625 8-0622677 4-0207256 -015384615 4856 287496 8-1240384 4-0412401 -016151515 67 4489 800763 8-1863528 4-0616480 -014925373 68 4624 314433 8-2462118 4-0816561 -014705882 09 4761 328509 8-8066289 4-1015601 -014492754 70 4900 348000 8-3666003 4-1212853 -014285714 71 5041 857911 8-4261498 4-1408178 -014084517 72 6184 378248 8-4852814 4-1601676 -013888889 73 6329 389017 8-5440087 4-1798390 -018G98680 74 6476 405224 8-6033253 4-1983864 -013518514 5625 421875 8-6602540 4-2171633 ■013333833 78 5776 438976 8-7177979 ■018157895 77 5929 456583 8-7749644 4-2543210 -012987018 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 4-32S7487 ■012845679 82 6724 551363 9-0553851 4-8444816 -012195122 83 671787 9-1104336 4-3620707 -012048193 84 7056 592704 9-1651514 4-3796191 ■011904762 85 7225 614125 9-2195445 ■011764706 7396 9-2736185 4-4140049 -011627907 87 7569 658503 9-3273791 4-4310476 -011494253 88 7744 6S1472 9-8808315 4-4470692 -011308630 7921 704969 9-4389811 4-4647461 -011236955 SO 8100 729000 9-4868330 4-4814047 -011111111 91 8281 75S571 9-6893920 4 4979414 010989011 92 8464 778688 9-6916630 4 5148574 010809505 93 8649 804857 9-6436508 4 6300649 0107«2b88 94 830584 9-6953597 4 6468369 010t,3S_ i« 95 9025 857374 9-7467943 4 562002(1 OlOj.i =1 96 9216 884736 9-7979590 4 5788570 01(14! 1 7 97 9409 912673 9-&488578 4 5947009 010 no2-s 9604 941192 9-8994949 4 6104363 01U_j4(l-i2 99 9801 970299 9-9498744 4 6260650 OIOIUIUIO 100 10000 1O000O0 lO-OOOOOOO 4 6415888 OIOOUOOOO 101 10201 1030301 10-0498766 4 6670095 O09't00990 102 10404 1061208 10-0995049 4 6723287 -0un&u?n22 103 10609 1092727 10-1488916 4 6875482 •009708738 104 10816 1124864 10-1980390 4 7036694 -Ou9bl5885 105 11025 1157625 10-2469508 4 7176940 001523810 106 11236 1191016 10-2956301 4 7826236 0094J8962 107 11449 1225043 10-3440804 4 7474594 00934j794 103 11664 1259712 10-3923048 4 7622082 00l2jlii9 109 11881 1295029 10-4403066 4 7768562 00 11 431 110 12100 1331000 10-4830885 4 7914199 oiMtniH) 111 12321 1867631 10-6356538 4 80j8995 112 12544 1404928 10-5830052 4 8202845 one .-. 1 113 12769 1442897 10-6301458 4 8345881 O0'-><4 P)i8 114 12996 1481544 10-6770783 4 8488076 -O087719u0 115 13225 1520875 10-7238053 4 8629443 0086956o2 110 13456 1560896 10-770329G 4 8760990 008020tj90 117 13639 1601613 10-8166538 4-8909732 -008517009 118 18924 1043032 10-8627806 4-9048681 ■003474570 „. 14161 1685159 10-9087121 4-9186847 •008403361 b,Google THE PRACTICAL MODEL CALCULATOR. K„n,t«. Bqn«a». Cb«. S,„,.»E«=... C.T<.= Boo«. 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K^U K.=il,™^;^ 1 18^ S3 124 6028568 13-4907376 6-6070511 ■0051&4&0O 188 38489 6128487 18-5277498 5-6774114 -005464481 18i 88856 6229504 13-5646600 5-6377340 -005484783 186 34225 6331625 13-6014705 5-0980192 -005405405 ise 34596 6484866 13-6881817 6-7082675 -005370344 187 84969 6539203 18-6747943 5-7184791 -005347594 188 35344 6644672 13-7113092 6-7286543 -005319149 189 85721 6751269 13-7477271 6-7387986 -005291005 190 36100 6859000 18-7840488 5-7488971 -005263158 101 86481 13-8203760 5-7680652 -005235G02 102 S0864 7077888 13-8564065 6-7689982 ■0052033S3 198 87249 7189517 18-8924400 ■005181347 194 87686 7301884 5-7880604 -00515 !039 195 38025 7414875 13-9642400 5-7988900 -0051^8205 196 38416 7520588 14-0000000 6-8087857 ■005102041 197 38800 7645373 14-0850088 5-8186470 -005070143 198 39204 7762892 14-0712478 6-8284867 -0050-50505 199 39601 7880590 14-1007360 5-8882725 -005025126 200 40000 8000000 11-1421856 6-8480355 ■005000000 201 40401 8120601 14-1774469 5-8577000 ■004975124 202 40M04 8242408 14-2126701 5-8674678 ■001950495 203 41209 S306437 14-2478068 6-8771307 ■O049261OS ^Ul 41610 8489664 14-2828509 5-8867653 -0041J01901 205 42025. 8015125 U-8178211 -004878049 200 42436 8741816 14-3627001 5-9059106 ■004854369 207 42849 8860743 14-3874046 5-0154817 -004830918 208 48264 8998912 14-4222051 6-9240921 ■004S07602 200 43081 912932B 14-4668323 6-9344731 ■004784089 210 44100 9261000 14-4918767 5-0489220 ■004761905 211 44521 14-5258390 5-9538418 -004730336 213 41944 9528128 14-6602108 5-9627320 -004716081 ai3 45369 9663597 14-5045195 5-9720936 ■004094836 21-1 45706 9800344 14-6287388 6-9814240 ■004072897 216 46225 9938375 11-6628788 5-9907364 ■001651103 218 4665S 10077606 14-6969385 6-0000000 ■004629030 217 47089 10218313 11-7800109 0-0093460 ■00460&295 218 47524 10360232 11-7648231 6-0184617 -004387156 219 47981 10603459 14-7986486 6-0276502 -004500210 220 48400 10048000 14-8323970 6-0368107 -004545455 221 43841 10793861 14-8060687 6-0169435 -004524887 222 49284 10041048 14-8990044 6-0550489 ■004504505 223 49729 11089567 14-9331S45 6-0041270 -004484305 224 50176 11239124 14-9666205 6-0731779 ■0044042S6 22o 50625 11800625 16-0000000 6-0824020 ■004444444 2:;0 61076 11513176 15-0332064 6-099ffi94 ■004424779 227 61539 11607088 15-0665192 6-1001702 ■004405280 228 51984 11852352 15-0906689 6-1091147 ■004385065 229 62441 12008980 15-1327460 6-1180832 -004306812 230 52900 12167000 15-1667509 6-1269257 ■004347820 231 63861 12326391 16-1986842 0-1357924 ■004320004 232 53834 12487168 15-2315462 6-1446387 ■004810345 233 54289 12649887 15-2643376 6-1534495 ■004291845 284 54756 12812904 15-2970585 6-1622401 ■004278501 235 55235 12077875 16-3297097 6-1710058 -004255319 236 55696 13144256 16-3622015 0-1797466 ■00423728S 237 56169 18312053 16-3948013 6-1884628 ■004219409 288 56644 13481272 15-4272486 6-1971544 -004201681 239 57131 18651919 15-4696248 6-2058218 ■OOilSilOO 240 57600 13824000 15-4919334 6-2144650 -004106667 241 58081 13997521 15-6241747 0-2230843 ■0041-10878 242 58504 14172488 15-5-563492 6-2816707 -004132231 243 59049 14348907 16-5884573 6-2402515 -004113226 b,Google THE PRACIICAL MODEL CALCDLATOR. N.=,l«r. S,«ar«, cat.. S,u,^B.^. o„h.a«t„ RdripTMalj. 244 59586 14526784 ]6'6204994 6-2487998 -004098361 245 60025 14706125 15-6524768 e-2673248 -004081633 246 60516 14886936 15-6843871 6-2658266 -004065041 M47 61009 15069223 15-7162330 6-2743054 ■004048588 248 61504 15252992 15-7480157 6-2827618 -004082258 219 62001 15488249 15-7797338 6-2911946 -004016064 250 62500 15626000 15-8118883 6-2996053 ■004000000 251 63001 15818251 15 -8429795 0-3079935 ■003H84064 252 68S04 16003008 15-8745079 6-3163596 ■008968254 253 64009 16194277 15-9069737 6-3247085 ■003952569 254 64616 16887064 15-9873776 6-8330256 -003937008 256 65025 16581875 16-9687194 6-3413257 -003921569 256 65536 16777216 16-0000000 6-3496042 -003906250 257 66040 16974598 16-0312195 6-8578611 ■008891051 258 66664 17178512 16-0623784 6-3660968 -003875969 259 67081 17378979 16-0934769 6-3743111 -003861004 260 67600 17576000 16-1245166 6-8826043 ■008846154 261 68121 17779581 16-1564944 0-39O6765 -003831418 262 68644 17984728 16-1864141 6-8988279 -003816794 69169 18191447 16-2172747 0-4069685 -003802281 264 60696 18899744 16-2480768 0-4150087 -003787879 265 70225 18609625 16-2788206 6-4281588 -003778685 266 7075G 18821086 16-8095064 6-4312276 -003759398 267 71289 19034163 16-3401346 6-4892767 ■008745818 71824 10248882 16-3707055 6-4473057 -003731343 269 72S61 19466109 16-4012195 6-4553148 ■003717472 270 72900 13683000 164316767 6-4638041 ■003703704 271 73441 19902511 16-4620776 0-4712786 ■008690087 272 7B984 20123648 16-4924225 0-4792236 ■003676471 273 74529 20846417 16-5227116 6-4871641 ■003663004 274 75076 20570824 16-5529454 e-4960663 ■008649685 275 75625 20796875 16-5831240 6-6029572 ■003636364 £76 76176 21024576 16-0132477 6-5108300 -008628188 70729 21253933 16-6438170 6-5186839 ■003010108 278 77284 21484952 16-0783320 6-5265189 ■003597122 77841 16-7082981 6-5848861 ■003584229 £80 78400 21952000 16-7832006 0-5431326 -003571420 281 78961 22188041 16-7630646 0-5499116 -003558710 282 79524 22425768 16-7928666 6-6676722 -003546099 213 80089 220b:)l&7 6-6654144 ■008683569 -84 80656 22906804 16-8522995 06731386 -008522127 i8j 81223 23149120 16-8819480 0-6808443 ■003508772 286 81796 23393666 16-9115846 6-6886828 -003496503 287 82369 2363^903 16-9410748 6-5962023 -003484321 82944 , 23887872 16-9706627 6-6038546 -008472222 289 88o21 -41S76U9 17-0000000 6-6114890 ■003400208 290 84100 24389fH>0 17-0298864 6-6191060 -003448276 2J1 84681 24642171 17-0587221 6-6267054 ■00343642G 292 8o264 2189-088 17-0880075 6-6342874 -0034241^58 293 8d849 17-1172428 6-6418522 ■008412969 294 86436 .5412184 17-1464282 6-6493998 -003401301 39j 87026 256-2370 17-1755640 6-6569302 ■003380831 290 87616 259,4886 17-2046505 6-6644487 ■008878378 297 88209 *l lJb073 17-2836879 6-6719403 ■003307003 iOS 88804 2W6J-.02 17-2626766 6-6794200 ■003366705 89401 2W308J9 17-2916165 6-6868831 -003344482 800 90000 270011000 17-8206081 0-6948206 •003388838 301 90G01 272-0 JOl 17-3498516 6-7017693 -003322269 302 S1204 ZIoUM'i 17-3781472 6-7091720 ■003811258 303 91809 27816127 17-4008962 6-7165700 -003301330 J04 92416 2609441,4 17-4356958 6-7239508 -003289474 30a 93026 2%7-i 2j 17-4642492 6-7313155 ■003278689 b,Google TABLE OF SQUARES, CUBES, SQUARE AND CUBE B,00I9. M^Mr s,»»™. C„l>... S^uar«R«.l,. Cube EoolE. 3llb 3o636 28652016 17-4928557 6-7886641 -003267974 •>07 94241 28934443 17-5214155 6-7459967 ■003257329 808 B4864 29218112 17-5499288 6-7533134 -008246763 309 95481 29503609 17-5788958 6-7606148 -008236246 ^10 96100 29791000 17-6068169 6-7678995 -003226806 an %721 80080231 17-0351921 6-7751690 -003215434 812 07344 30371328 17-0636217 0-7824229 ■008206128 ai3 &79(.9 80664207 17-0918000 6-7896613 -008191888 814 9S5t6 30969144 17-7200461 6-7968844 ■003184713 dl5 S92.5 31255875 17-7482393 6-8040921 ■00-3171008 61Q 99356 31554496 17-7768888 6-8112847 ■003164557 317 100489 31855013 17-8044938 6-8184620 ■003151574 318 101124 32157432 17-8325545 0-8256242 -003144654 319 101761 82461769 17-8605711 0-8327714 ■003I34T96 820 102400 32768000 17-8885488 6-8399087 ■003125000 821 103041 38076161 17-9164729 6-8170213 -008115265 322 108084 33380248 17-9143584 6-8541240 -003105590 323 104329 83698267 17-0722008 6-8612120 ■003035975 m 104976 34012224 18-0000000 6-8682855 -008080120 325 105025 81828125 18-0277564 6-8753438 -003076928 326 106276 81645976 18-0564701 ■003007185 327 106929 34965783 18-0831413 6-8891188 ■003058101 328 107584 85287552 18-1107708 6-8964345 ■0030187SO 329 108241 35611289 18-1883671 6-9034369 ■003039514 830 I0S90O 85937000 18-1059021 6-9101232 -003030303 831 109561 86264691 18-ie34054 6-9173964 -003021118 332 110224 36594368 18-2208672 6-9213556 ■0030J2048 110889 36926087 18-2482876 0-9313088 ■003003003 884 111566 37259704 18-2750669 6-9382821 ■002994012 835 112225 87595375 18-8030052 6-9451496 ■002985075 330 112896 37983056 18-3308028 6-9520638 ■002976190 837 113569 38272753 18-8575598 6-9589434 ■002967359 114244 38614472 18-3847763 6-9658198 .-002958580 839 114921 38958219 18-4119526 6-9720826 ■002949858 340 115600 89304000 18-4390889 6-9795821 ■002911176 841 116281 39651821 18-4661858 6-9863681 ■002932551 342 116964 40001688 18-4932420 6-9931906 ■002923977 343 117649 40353607 18-5202592 7-0000000 ■002915452 844 1I83S6 40707684 18-5472370 7-0067962 ■00290G977 846 119025 41063025 18-5741756 70136791 ■002898551 846 119716 41421736 18-6010752 7-0203490 ■002800178 347 120409 41781328 18-0279800 7-0271058 ■002881811 121104 42144192 18-6547581 7-0338497 ■002878563 849 121801 42508519 18-6815417 7-04O5860 ■002865830 850 122500 42876000 18-7082869 7-0472987 ■002857143 351 123201 43243551 18-7349940 7-0540041 ■002849008 852 128904 43614208 18-7016630 7-O0OS967 -002840909 353 124609 43980977 18-7882942 7-0073767 ■0028828C1 354 12531b 44861864 18-8148877 7-0740440 ■002824859 855 120023 44738875 18-8114437 7-0806988 ■002816901 350 126786 45118016 18-8679623 70873411 ■002808989 357 127449 45499293 18-8944436 7-0989709 ■002801120 368 1.8164 46882712 18-9208879 7-1005885 ■002793296 339 128881 40268279 18-9472953 7-1071937 ■002785615 860 129600 46666000 18-9736660 7-1137866 ■002777778 361 130321 47015831 19-0000000 7-1203874 ■002770083 862 131044 47437928 19-0262970 7-1209360 ■002762431 3bS 131769 47832147 19-0525589 7-1384925 -002754821 364 182496 48228544 19-0787840 7-1400370 ■002747253 865 138225 4Kb27r25 191049782 7-1466695 ■002789726 3bb 1339j6 49027890 19-1311265 7-153090! ■002732240 8(.7 ljit>8y 41430863 19-1572441 7-1595988 ■002724796 b,Google THE PHACTICAL MOLiEL CALCULATOR. N,™b«. .,.„.. cub... Sq=W=ft,»tj. Col.s 11.^. R..,l.,oo.!,. "~3li8~ 135424 49830032 19-1883261 7-1060067 -002717391 369 130161 60243409 19'2093727 7-1726809 ■002710027 870 186900 50653000 19-2353841 7-1790644 oo'-O'-os 371 187641 51061811 19-2613608 7-1856102 ■00 e^iis 372 13B384 61478848 19-2878015 7-1919663 ■00''bh81 373 139129 61895117 19-8132079 7-1981060 00"080Jij 1 374 189876 52318624 19-3390796 7-20188'2 O0"fl 3 J 875 110626 62734375 19-8649167 7'211 4 00 bLOor 376 141376 58167376 19-3907104 7-21765'> 00 66Jo 4 377 142129 53582638 19-4164878 7-22401oO 00 6o 378 142881 64010152 19-4422221 7-2304''68 00 blooO" 879 148611 51439939 19-4679223 7-2367072 00 U88d 1 380 144400 54872000 19-4936887 7-2481566 00 (lolj 9 381 145161 55806341 19-5192218 7-249O045 00 I. 40 382 145924 65742968 19-5448208 7-2558415 00201 eoi 883 146689 66181887 19-6708858 7-26216 5 00 GIO 384 147466 66623104 19-5959179 7-2684B21 00 ooiir 385 148225 67066625 19-6214169 7-2747864 00"u i i 886 148996 57512456 19 ■6468827 7-2810794 00 jlO 4 887 149T69 67960608 19-6723156 7-2873617 00 5** , 388 150J44 68411072 19-6977156 7-2Bg6g30 00"o 3 889 161321 53863869 19'72B082B 7-2998936 OD'o 14 3U0 152100 69319000 19-7484177 7-8061486 00 50410 S91 152881 59776471 19-7787199 7-3123828 002 d oi 392 153061 00236288 19-7989899 7-3186114 ■O026dl0 1 398 154449 00(i98457 19-8242276 7-8248296 ■00 6445 9 394 155280 61162084 19-8491882 7-8S10869 0025380 1 , 395 156025 01629875 19-8746069 7-33-2339 00261.164 896 166816 62099136 19-8997487 7-34Sl'0o 00 1 397 157609 62670778 19-9218588 7-8490J66 00 ,>18° . 158404 63044792 19-9499378 7-36676 4 00 } 1 399 159201 63521199 19-9749841 7-8619178 00 oOb 400 160000 64000000 20-0000000 7-3680630 OC 000 i 401 100801 61481201 20.0249811 7-37419 9 00 4 3 403 161004 64964808 20-0490877 7-3803« 7 -00 4 J , 103 162409 65450827 20-0748590 7-88643 3 00 4813 404 163216 C6S39261 20-0997612 7 ■39-1)418 00 4 I 405 164025 66480125 20-1246118 7-3986868 00 4691' 406 164836 66923116 20-1191417 7-404 "06 ■00 4680 4 407 165649 67419143 20-1742410 7-4107950 DO 46 00" 408 166464 67917812 20-1990099 7-4108595 ■0024O0980 409 167281 68417929 20-2237484 7-1228112 002444088 410 168100 68921000 20-2181567 7-4289580 00 1800 4 411 168921 69426581 20-2731849 7-4849938 00 4330d0 112 169711 69934528 20-2977831 7-1410180 00"! 184 413 170569 70444997 20-3224014 7-4470343 00 4 I Ob 414 1718B6 70967944 20-8469899 7-4580399 00 4154 J 416 172225 71473876 20-3715488 7-45903a9 00 4O90b9 416 173066 71991296 20-8960781 7-4850^28 00 406846 117 178889 72511713 20-4205779 7-1709991 00 3J808 418 174721 73034682 20-4460483 7-476966* 419 175661 73560059 20-4694896 7-48 9942 ■00 asorso 120 176400 74088000 20-4980015 7-18887''4 00 38010 421 177241 74618461 20-5182846 7-4948113 ■00"8 6''9 422 178084 75151448 20-5426386 7-600 406 00 869er9 423 178929 75686967 20-5669638 7-6066607 ■O023r40 424 179776 76226021 20-6912603 7-51 ''6715 ■0023084 1 126 180626 76765625 20-6156281 7-6181 30 ■0023O 141 426 181476 77308776 20-6397674 7-521805'' 00 84 418 127 182829 77851183 20-6639788 7-680 48 00 841 20 428 183184 78402752 20-6881609 7-5361 1 00 8 644 429 184041 78953589 20-7128162 7-541 980 00 100 b,Google TABLE OE SQUARES, CUBES, SQUARE AND CUBE ROOTS. 107 K.ml,.r. S^dnr^a. Cubes. SqaA«Roi,^. C01..IU,.K Rfc,pr.«.ilB. 4iiU 184900 79507000 20-7364414 7-5478423 002325681 481 ISoTHl 80062991 20-7605396 7-5536888 002320166 432 186(i24 80621G68 20-7840097 7-5595268 00-2814815 43a 187489 81182787 20-8086520 7-5653648 002309409 4U 188356 81746504 20-8326867 7-5711748 002304147 435 189225 82312675 20-8666586 7-6769849 002298851 430 190096 20-8806180 7-5827865 002293678 437 190969 83453463 20-9046460 7-5886793 002288380 438 191844 64027672 20-9284495 7-5943683 002288105 439 192721 81604519 20-9623268 7-6001S85 002277904 440 193000 85184000 20'9761770 7-6059049 002272727 441 191481 85766121 21-0000000 7-0110626 002267674 442 195304 21-0237960 7-6174116 002262443 443 196249 86938307 21-0475652 7-6231619 002267886 444 197130 87528384 21 ■0718076 7-6288837 002252262 445 108025 86121125 21-0950281 7-6846007 002247191 446 198916 88716536 21-1187121 7-6403218 002242162 44 199809 89314623 21-1423745 7-6160272 002237136 448 "00 04 89915392 21-1660105 7-0517247 002282148 449 201001 90518849 21-1890201 7-6574188 002227171 4o0 ^0 500 91125000 21-2132034 7-6630943 002222222 4ul ^08401 21-2867606 7-6687065 002217295 43 204304 92345408 21-2602916 7-6744303 002212389 4oJ J 09 92959677 21-2887967 7-6809857 «02:!0760G 454 00116 98576064 21-8072768 7-6857328 ■002202648 4oo d 94196375 21-3307290 7-6913717 002197802 4dU 20 J 94818816 21-3541665 7-6970023 002192982 4j 08840 95448903 21-8775583 7-7020246 002188181 4d8 "09 04 90071912 21-4009346 7-7082388 , 002183406 4o9 ii0(81 96702679 21-4242863 7-7188418 002178649 40 llOOO 97336000 21-4476106 7-7194426 002178913 4«1 21 d 1 97972181 21-4709106 7-7260S25 002169197 40 "13444 98611128 21-4941853 7-7306141 ■002104502 403 2libbi 99252847 21-5174848 7-7861877 -002159827 404 U J6 99897344 21-5406592 7-7417532 •002155172 46o 210 "5 100544625 21-5688587 7-7478109 ■002150538 4()6 217156 101194696 21-6870831 7-7528006 ■002145923 407 216069 101847563 21-6101828 7-7684028 ■002141828 408 2190 4 102S03282 21-6883077 7-7689301 002186762 4(,9 219001 108161709 21-0504078 7-7694020 002182196 470 220900 108823000 21-6794834 7-7749801 002127000 471 2 1641 104487111 21-7025314 7-7804904 002123142 4 2 84 105154048 21-7255610 7-7859928 -002118014 4 3 " 3 9 10:^828817 21-7485682 7-7914875 ■002111165 4 4 224676 106406424 21-7715411 7-7969746 ■002109705 4,0 2.O0-5 107171875 21-7944917 7-8024688 002105268 470 220o70 107850176 21-8174242 7-8079254 002100840 477 227529 108531333 21-8403297 7-8133892 002096486 4/8 228484 109215352 21-8682111 7-8188456 002092050 47'> 229441 109902239 21-8860686 7-8242942 002087668 480 230400 110592000 21-9089023 7-8207868 002083883 481 281861 111284641 21-9317122 7-8351688 002079002 482 2u2324 111980168 21-9544984 7-8405949 002074689 483 283289 112678587 21-9772610 7-8430134 002070393 484 234256 118879904 22-0000000 7-8514214 002006116 23d22o 114084125 22-0227166 7-8566281 002001656 486 236190 114791256 22-0464077 7-8622242 002067618 487 237169 11O501808 22-0680765 7-8676180 002053388 488 238144 116214272 22-0907220 7-8729944 002049180 459 239121 116930169 22-1133444 7-8783084 002044990 410 240100 117149000 22-1859436 7-8837352 002040816 411 2410S1 118*70771 22-1685198 7-8890946 002036600 b,Google THE PRACTICAL MODEL CALCULATOR. B«ml=™. a^.«r=,. ColcJ. Sqn>'''J^-t'- C=b.E„^.. I(«dpr„.ul=. 492 342064 119095488 22'1810730 7-8944468 002032520 493 243049 119828157 22-2086033 7-8997917 63784 22-2261108 7-9051294 002024291 2 87375 22-2485955 7-9104599 002020202 t>0 23936 22-2710675 7-9157832 002016129 00 68478 22-2934968 7-9210994 002012072 KO 05992 22-3159136 7-9264085 002008032 M 51499 22-3383079 7-9317104 002004008 00 00 ^ 00000 22-3606798 7-9370063 002000000 (M 61501 7-9422931 001990008 00 6606008 22-4058666 7-9475739 001992032 00 63527 22-4276615 7-9528477 001988072 02406* 22-4499443 7-9581144 001984127 87625 22-4722061 7-9638743 001980198 06 -5 84216 22-4944488 7-9686271 001976285 23843 22-5166605 7-9788731 001972887 25 96512 22-5888553 7-9791122 001968504 22-5610283 7-9843444 001964637 51000 22-5831796 7-9895697 001960784 82881 23-6053091 7-9947883 001956947 17728 22-6274170 8-0000000 001963125 05697 22-6495033 8-0052049 001949318 96744 22-6715681 8-0104032 001945525 90876 22-6936114 8 ■0155946 ■001941748 26 22-7156334 8-0207794 -001937984 88413 22-7876841 8-0259574 ■001934236 ')1882 22-7596184 8-03U287 001980502 98359 22-7815715 8-0362935 001926782 400 08000 22-8086086 8-0414515 001923077 20761 22-8254244 8-0466080 001919386 .^ 36648 22-8473193 8-0517479 001915709 3 56667 22-8691933 001912046 77824 22-8910468 8-0620180 001908397 08125 22-9128785 8-0671432 001904762 31576 8-0722620 001901141 68183 22-9564806 8-0773748 001897538 84 97952 22-9782506 8-0824800 00189S939 48 35889 23-0000000 80875794 001890359 0900 77001 23-0217289 8-0926728 001886793 21291 23-0434372 8-0977589 001883239 63 28-0651252 8-1028390 001879699 19487 23-0867928 8-1079128 -001876173 78804 23-1084400 8-1129803 001872659 30375 23-1800670 8-1180414 001869159 90656 23-1516738 8-1330962 001865672 8o 54153 23-1732605 8-1281447 001862197 20872 23-1948270 8-1831870 001858736 90 90819 23-2168735 8-1382230 001855288 « 64000 23-2879001 8 1432529 001851852 58 40421 23-2594067 8-1482765 001848429 &42 298704 169220088 23-2808935 8-1532939 001845018 543 294849 160103007 23-3028604 8-1583051 001841621 544 295936 160989184 23-3238076 8-1688102 001888235 545 297025 161878625 23-3452351 8-1683092 001834862 546 298116 162771886 23-3666429 8-1788020 001831503 547 299209 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 23-4788892 8-1981753 001814883 552 804704 168196608 23-4946802 8-2031319 001811594 553 305809 169112377 23-5159520 8-2080825 001808318 b,Google TABLE OE BCJUARES, CUBES, SQUARE AHD CUBE ROOTS. Knmh.r. Ecu.r... Cuhc'. Si...»R<,ola. o.i„K,»t.. Bodpr^ool,, 551 806916 170031404 28-5372046 8-3180271 001805064 555 308025 170958876 23-6684380 8-2179657 001801802 55e 309186 171879618 23-5796522 8-2228985 ■001798561 557 S10249 172808693 23-6008474 8-2278264 ■001795882 558 811864 178741112 23-6220286 8-2827463 001792116 559 312481 174676879 23-6431808 8-2376614 001788909 560 818600 176616000 23-6648191 8-2426706 001785714 661 814731 176558481 23-6854386 8-2474740 )01782531 562 815844 177604328 23-7065892 8-2533715 668 316969 178463547 23-7276210 8-2572635 )01776199 564 318096 179406144 23-7486842 8-2631492 001778050 665 819225 180362125 23-7697289 8-2670294 001769912 566 320356 181321498 23-7907546 8-2719039 001766784 567 821489 1822842GS 28-8117618 8-2767726 001768668 568 822624 188250432 23-8327506 8-3816256 001760563 569 828761 184220009 23-8537209 8-2864938 001767469 570 324900 185193000 28-8746728 8-2918444 001754386 671 326041 18G169411 23-8956068 8-2961903 001751313 572 827184 187149248 23-9165215 8-3010304 001748252 57S 828329 188132517 23-9874184 8-3058951 001745201 574 329470 189119224 23-9582971 8-8106941 001742160 575 830625 190109876 23-9791576 8-3155175 001739130 576 331776 191102976 24-0000000 8-3203353 ■001736!11 577 882927 192100088 24-0208243 8-3251475 001733102 578 334084 193100652 24-0416306 8-8299542 001730104 57B 335241 194104539 24-0624188 8-8347653 001727116 580 836400 196112000 24-0831891 8-3396609 -001724138 581 887561 196122941 24-1039416 8-3443410 001731170 888724 107187868 24-1246762 8-3491256 001718213 683 3398SQ 198155287 24-1458929 8-3589047 001716266 684 841056 J99176704 24-1660919 8-3689784 001713329 585 342226 200201625 34-1867782 8-3634466 -001709403 686 843896 201230056 24-2074369 8-3682095 001706485 587 344509 202262003 24-3280829 8-3729668 00170S678 588 845744 203297472 24-2487113 8-3777188 001700680 689 346921 204336469 24-2693222 8-8824658 001697793 590 848100 205879000 24-2899156 8-3872065 001694915 691 349281 206426071 24-3104996 8-3919428 001693047 592 3504fl4 207474688 24-8810501 8-3966729 001689189 593 351649 208527857 24-3616918 8-4013981 001686341 594 852836 209584584 24-3731153 8-4061180 001683602 695 354025 210644876 24-3926218 8-4108326 001680672 596 855216 211708736 24-4131112 8-4155419 -001677852 597 350409 212776178 24-4335834 8-4202460 -001975042 857604 218847192 24-4540886 8-4249448 ■001672241 690 868801 214921799 24-4744765 8-4296883 -001669449 600 860000 216000000 24-4948974 8-4343267 -001696667 601 361201 217081801 24-6163013 84390098 001668894 362404 218167208 24-5856883 8-4436877 -001661180 863609 219256227 24-5660583 8-4483606 001658875 604 864816 220348864 24-6764115 8-4530281 001655629 605 866025 221445125 24-6967478 8-4576906 001652893 606 367236 222546016 24-6170673 8-4628479 001650166 607 368449 228648548 24-6373700 8-4670001 001047446 608 369664 224755712 24-6576560 8-4716471 001644737 609 870881 225866529 24-6779254 8-4792892 001642036 610 372100 226981000 24-6981781 8-4809261 001639844 611 228099181 24-7184142 8-4855579 -001936661 612 374544 24-7386338 8-4001848 001633987 618 376769 230346397 24-7588368 8-4948095 -001631321 614 281475544 24-7790234 8-4094233 001628664 615 378225 232608376 24-7991935 8-5040350 001626016 b,Google THE PRACTICAL MODEL CALCULATOR. N>m,ta.. B,-a>:... C.b=s. B5n.r=E™«. 1 C beltoMa. E.=iJ.v™»l>, , 616 379456 233744896 24-8198478 8 JO80417 -001623377 017 880689 234885113 24'8394847 8 5132485 -001620746 618 881924 24-8696058 8 6178403 -001618128 619, 883161 287176659 24.8797106 8 5224331 ■001615509 384400 ■238828000 34'8997992 8 5270189 -001612908 621 886641 239463061 24-9198716 8 5816009 -001610806 622 38688* 240641848 24-9399278 8 5301780 -001607717 623 888129 241804367 24'9599679 8 5407501 -001606136 624 242970624 24-9799920 8 5453173 -001602664 625 390025 244140626 26-0000000 . 8 5498797 ■OO10OOOOO 891870 245134376 25-0199920 8 6644372 -001697444 627 393129 246491888 25-0399681 8 6589899 -001594890 628 894884 247673152 25-0699282 8 5636377 ■001692357 629 395641 248858189 25-0798724 8 5680807 ■001689825 830 396900 250047000 35-0998008 8 5726189 -001587302 631 89SI61 251289591 26-1197134 8 5771528- -001584786 633 399424 252435968 25-1396102 8 6816809 -001582278 683 400689 268636137 25-1594913 8 5862247 -001579779 684 401956 254840104 26-1798666 8 5907238 ■001677287 635 403226 266047876 26-1992063 8 5953380 -001674803 404496 267259456 26-2190404 8 5997476 ■001573327 637 405709 258474853 25 ■2388689 8 6042525 -001569859 407044 259694072 25-2586619 8 6087526 -00] 567398 408321 200917119 26-2784493 8 6132480 -001564945 640 409000 202144000 25-2982213 8 6177888 ■001562500 641 410881 208874731 25-8179778 8 6222248 ■001560002 6i2 412164 364609288 25-3377189 8 6267063 ■001667632 643 413449 205847707 28.3674447 8 6311830 ■O0156621O 644 414736 ■267089984 26-3771551 8 6856551 -001652796 645 410125 208386125 26-3968502 8 6401220 -001550888 646 417316 269685130 26.4165302 8 6445856 -001547988 647 418609 270840028 26.4861947 8 6490487 ■001646595 648 419904 -272097792 25-4558441 8 6534974 ■001643210 649- 421201 '273859449 25-4764784 8 6579405 ■001540832 660 422500 274635000 25-4960976 8 6623911 ■O01688462 651 423801 276894451 25.5147018 8 6668810 -001636098 652 425104 277167808 26-5342907 8 6712665 ■001533742 653 426409 278445077 25-5538647 8 6756974 ■001531894 654 427716 270726204 25-5734287 8 6801287 -001529052 655 429026 281011375 25-6929678 8 6845456 ■001520718 656 430336 2S23004I6 26-0124969 8 08S963O ■001524890 657 481689 283503393 25.6330112 8 6983760 -001522070 658 482964 284890312 25-6515107 8 6977843 ■001519751 659 434281 280191179 25-6709953 8 7021882 -00161'7461 660 435600 287496000 26-6904662 8 7065877 -001515163 661 436921 288804781 35-7099208 8 7109827 ■001512859 438244 290117628 26-7298607 8 7153784 -001610674 439669 291434247 26.7487804 8 7197596 ■001508296 664 . 410896 292754944 25-7681975 8 7241414 ■001500024 442285 294079625 25.7875989 8 7285187 ■001503759 666 443666 295408296 25.8069758 8 7828918 ■OOI501603 667 444899 290740968 25-8263481 8 7372004 ■001499250 440224 298077032 25-8456960 8 7416246 ■001497006 660 447661 299418809 26.8650343 8 7459846 ■001494768 070 448900 800763000 25-8843692 8 7503401 -001492537 671 460241 302111711 25-9036077 8 7546913 -001490313 672 451584 303464448 25-9229628 8 7590383 -001488095 673 452929 304821217 ,25-9422435 8 7633809 ■001485884 674 454276 806182O24 25-9615100 8 7677193 ■001483680 676 '455625 307546875 25-9807021 8 7720682 ■001481481 676 466976 S08915776 20-0000000 8 -7703830 -001479290 677 468329 310288783 26-0192287 8 -7807084 ■001477105 b,Google TABLE or SQUARES, CUBES, SQUABE AKD CUBE ROOTS. r,„™i«r Cut^. «,u=«H„.«. Cu>«H„«. meim""- 678 450684 811665753 26-0384381 8-7860296 00147492S 079 401041 318046889 26-0576284 8-7893466 001472754 (iSO 402400 314432000 20-0708096 •001470588 681 46S761 315821241 26-0969767 8-7979679 001468429 682 465124 817214568 26-1151297 8-8022721 001406275 683 460489 318611987 26-1342087 8-8065722 001464129 684 467856 320018504 26-1633087 8-8108081 001401988 685 469225 321419125 26-1725047 8-8151598 001459854 Ga3 470596 26-1910017 8-8104474 001457720 687 471969 824242703 26-2106848 8-8237807 001455604 ti88 478344 325060672 26-2297541 8-8280099 001453488 689 474721 327082769 26-2488005 8-83228-50 001451879 690 476100 328509000 26-2678511 8-8865559 001449275 611 477481 829939371 26-2868789 8-8408227 O0I44T178 602 478864 331373B88 26-3068029 8-8450854 001445087 093 480249 382812557 20-3248082 S-8493440 001443001 694 481636 884255384 20-3438797 8-8686985 001440022 695 483025 335702875 20-8628627 8-8578489 48441b 337153536 20-3818119 8-8620952 001430782 607 485800 20-4007676 8-8663876 001484720 698 487204 840068892 26-4106806 8-8705757 001432605 609 488001 341532099 26-4386081 8-8748009 001480615 700 490000 843000000 20-4575131 8-8790400 001428671 701 491401 344472101 26-4704046 8-8882661 001420534 702 492804 84504B40S 26-4952820 8-8874882 -001424501 703 494209 847428927 20-5141473 8-8917063 -001422475 704 495016 348913004 26-5820083 8-8959204 001420465 705 407025 350402025 26-5518361 8-0001804 001418440 703 498436 851805816 20 ■5700605 8-9048366 001416431 707 499849 858393248 26-5804716 8-9085387 001414427 708 501264 364894012 26-6O82094 8-9127869 001412429 700 602681 350400829 26-6270530 8-9169311 001410437 710 504100 357011000 20-0458252 8-9211314 -001408451 711 505521 359425481 26-6646883 8-9253078 -001406470 712 500944 860944128 26-6883281 8-9294902 -001404494 718 362467097 26-7030598 8-9386687 001402625 71* 609796 863994344 20-7207784 8-087S433 -001400500 7 5 511 5 365526875 20-7394839 8-9420140 001398601 16 51 656 8b 061696 26-7681763 8-9401809 001396048 17 614089 868001818 26-7768567 8-9503488 -001804700 7 8 ol5 '>i 3 0146232 26-7955220 8-0545029 001893758 19 510161 371694959 26-8141764 8-9586581 001390821 518400 873 48000 20-8828157 8-9028095 1 619841 874S05361 20-8514432 8-9660670 001366903 6 1284 6367048 26-8700577 8-9711007 001885042 23 52 a 3 7933067 8-9752406 001883120 7 i 5'>4176 9608424 26-9072481 8-9708706 001381215 25 5 5025 881078125 20-9268240 001870810 5 6 8 667176 20-9443872 8-9876373 001377410 7 5 8529 884 40583 26-9629375 8-99I7620 001375516 8 6 9984 885828352 26-9814751 001378626 53144 8 420489 27-0000000 9-0000000 001371742 80 58 900 889017000 27-0185122 9-0041184 001369863 731 534 61 890b 7891 27-0870117 9-0082229 001807089 7 5 8 4 89 >3168 27-0554985 9-0123288 001300120 733 53 289 398832837 27-0730727 9-0164309 001304356 34 58 6 395440904 27-0924344 9-0206298 001362398 3 540 89 065875 27-1108834 9-0246289 001800544 541 60G 398G&8256 27-1293199 0-0287149 0O13-58G96 737 513169 400816553 27-1477149 9-0328021 001350853 738 544644 401 947272 27-1601554 9-0868857 001350014 730 546121 403583419 27-184-5544 9-0409055 001363180 b,Google THE PRACTICAL MODEL CALCULATOR. N„m),«. ScmSTM. Cahes. Sfloste K^ot!. Clw Roola. Ee^Lurotaii. 710 547600 405224000 27-2029140 9-0450119 001351361 741 549801 406869021 27-2218152 9-0491142 001319628 743 550564 408518188 9-0531831 001847709 713 552049 410172407 27-2580263 9-0572432 001345895 741 65S586 1118S0781 27-2763684 90613098 001314086 745 655025 413493625 27-2946881 9-0653677 001342282 7ie 550516 415160936 27-8130006 90694220 001310183 747 658009 416832723 37-3313007 9-0734726 001338688 718 55D504 418508992 27-3495887 9-0775197 001336898 749 561001 420189749 27-3678644 9-0815631 001335118 750 562500 421875000 27-3861279 9-0856030 001388333 751 564001 423564751 27-4043792 9-0896852 001331658 752 565501 425259008 27-4226184 9-0936719 001329787 753 567009 426957777 27-4408465 9-0977010 001328021 754 568616 428661064 27-1590604 9-1017265 001320260 755 570025 480368875 27-4772633 9-1057485 001821503 756 571586 432081216 274954512 9-1097669 001332751 757 573049 138798093 27-5136330 9-1137818 001321004 758 574561 435619612 27-5317998 9-1177931 001319261 759 576081 437245179 27-5499546 9-1218010 001317523 760 677600 438976000 27-5680975 9-1258053 ■001315739 761 579121 140711081 27-5862281 9-1298061 001314060 762 680641 442150728 27-6043476 9-1838084 001312336 763 582169 114194917 27-6224516 9-1377971 001310G16 764 583696 445943744 27-6405499 9-1417874 001808901 765 58522S 447697125 27-6686334 9-J457742 001307190 766 586756 419455096 27-6767050 9-1497576 001305483 767 588289 451217663 27-6947618 9-1687375 001303781 768 689824 152981832 27-7128129 9-1677139 O01802O8S 769 591361 154766609 37-7308492 9-1616669 001300390 770 692900 456533000 21-7488739 9-1656565 001298701 771 594441 458311011 27-7668868 9-1696225 001297017 772 595984 16U099G48 27-7848880 9-1735862 001295837 773 597529 161889917 27-8028775 9-1775115 001293661 774 599076 163684824 27-8208555 9-1815003 001291990 775 600625 465484375 27-8388218 9-1851527 001290323 773 602176 467288576 27-8567766 91894013 001288660 777 608729 469097483 27-8747197 9-1938474 001287001 778 606281 170910952 27-8926514 0-1972397 001285347 779 606841 472729139 27-9105716 9-2012286 001383697 780 608100 474562000 27-9284801 9-2051641 -001282051 781 609961 476879541 27-0163772 9-2090962 ■001280110 782 611624 478211768 27-9642629 9-2130250 001278772 783 4800486S7 27-9821372 9-2169605 001277189 781 614656 481890304 28-0000000 9-3208726 001275510 785 616225 483736626 28-0178515 9-2247914 001273886 786 617796 485587656 28-0356915 9-2287068 001272265 787 619369 487448403 28 ■0586203 9-2826189 001270648 788 620911 489303872 28-0713377 9'2365277 001269086 789 622621 491169069 28-0891438 9-2404333 001267427 790 624100 493039000 28-1069386 9-2443356 001265828 791 625681 491913671 28-1247222 9-2482311 001261223 792 627624 196793088 28-1424916 8-2521300 001262626 793 628849 498677267 28-1602567 9-2560224 001261034 794 680436 500566184 28-1780066 9-2599111 001259146 796 602159875 28-1957444 9-2637973 001257863 796 633616 5043583S6 28-2134720 9-2676798 001256281 797 685209 506261573 28-2311884 9-2716592 001354705 798 636801 508169692 28-2488938 9-2754352 001253183 799 638401 510082399 28-2665881 9-2798081 001251364 800 610000 512000000 28-2842712 9-2831777 001250000 801 641601 513922401 28-8019434 9-2870144 001248439 b,Google TABLE OF SQUARES, CUBES, SQUARE AND CUBE ROOTS. 113 KomW. ^a-™. C,l.,, B^»«» liout^ c=i«e™i* K.ip.^„. 8oa e48:;04 ei584iJ608 28-8196045 9-2903072 -001246883 808 644809 617781627 28-3372546 9-2947671 -001245380 804 646416 619718164 28-3548938 9-2986239 -001243781 805 M8025 521660125 28S7262I9 9-8024776 -001342236 8oe 649686 623606616 28-8901391 9-3063278 ■001240695 807 651249 B25557943 28-4077454 9-3101750 ■001289167 '■ 808 652864 527514112 28-4253408 9-3140190 ■001237624 ! 809 654481 639475129 28-4429258 9-3178599 -001236094 810 656100 631441000 28-4604989 9-3216975 -001234668 811 657721 533411731 28-4780617 9-3256320 -001233046 812 659844 635387328 28-4956187 9-3298634 -001281527 813 eeo969 637367797 28-6131549 9-3331916 -001230013 814 662596 639853144 28-5306862 9-3370167 -001228601 1 81o 664226 541343375 28-5482048 -001226994 i 816 b658o6 543388496 28-6667137 9-3446575 -001225499 : 617 667489 545338513 28-5832119 9-3484731 ■001238990 1 818 669124 647843482 9-3522857 ■001222494 i 819 6707bl 649353259 28-6181761) 9-3560952 -001231001 i 820 6-2400 551868000 28-6356421 9-3599016 ■001219513 ' 821 674041 653387661 28-6530976 9-3637049 -001316037 , 822 675684 555412248 28-6706424 9-S675051 ■001216545 ! 8^3 677329 657441767 28-6879716 9-3718022 ■001216067 : 824 678976 659476224 28-7054002 9-3750963 -001318592 j 8-5 680625 561516625 28-7228183 9-3788873 -001212121 826 682276 668569976 28-7402157 9-3826752 -001210664 1 827 665609288 28-7676077 9-3864600 -001209190 ! 82S 685584 667663552 28-7749891 9-3903419 ■O0I207739 - 829 687241 569722789 28-7923601 9-3940206 ■001206278 i 830 688000 571787000 28-8097206 9-3977964 ■001204819 i 831 6d05(>l 573866191 28-8270706 9-4015691 -00120S3G9 832 692224 28-8444102 9-4058887 ■001201928 883 578009537 28-8617394 9-4091054 -001200480 834 6J555e 580093704 28-8790582 9-4128690 ■001199041 885 617225 682182875 28-8963666 9-4166297 -001197606 83t. 698896 684277066 28-9186646 9-4203873 ■001196172 837 700569 586376253 9-4241430 -001194748 703244 688480472 28-9482297 9-4278986 -001193317 839 703921 690589719 28-9654967 9-4316423 ■001191895 810 70o600 592704000 28-9827535 9-4853800 -001190476 841 707281 594828321 29-0000000 9-4391807 -001189061 842 70'*964 596947688 29-0172363 9-4428704 -001187648 843 710649 699077107 29-0344623 9-4466072 -001186340 844 712336 601211584 29-0516781 9-450841O ■001184834 845 714025 603351126 9-4540719 -001183432 84(j 715716 605495786 29-0860791 0-4677999 ■001182033 847 717409 607645423 29-1082644 9-4615249 -001180638 848 719104 609800192 39-1204396 9-4662470 ■001179245 849 720801 C11900O49 29-1376046 9-4689661 ■001177856 8B0 722500 614125000 29-1547695 9-4726824 -001176471 851 724201 616295061 29-1719043 9-4763967 ■001175088 8a2 725904 618470208 29-1890390 9-4801061 -001178709 853 727609 620660477 29-2061637 9-4888136 -001172333 854 729316 622835864 29-2233784 9-4876182 ■001170960 8o5 731025 626026376 29-2408830 9-4912300 -001169591 86b 732736 627222016 29-2574777 9-4949188 -001166224 857 734449 629422793 29-2746623 9-4986147 ■001166861 868 736164 631628712 29-3916370 9-5028078 ■001166601 8^9 737881 633839779 29-8087018 9-6059980 -001164144 800 739600 636066000 29-3357566 9-5096864 -001162791 861 741321 638277381 29-3428016 9-5133699 ■001161440 862 743044 640503928 29-3598365 9-5170515 -001160093 744769 642738647 29-3768616 9-5207803 -001168749 b,Google ! PRACTICAL MODEL CALCULATOR. Kuml,=r. BqaoKS. Cul.,a. B,,^ R.„». Cb. R..^.. Rooipiotal,. 864 746406 644972544 a9'8938769 9-5244063 001157407 805 748225 64T214625 29-4108823 9-5280704 001156069 8oe 740056 619461896 29-4278779 9-5817407 001164734 SG7 761680 651714868 29-4448687 D-5354172 001163403 763424 053072032 29'4618397 9-5390818 001152074 BGS 755161 656234909 20-4788059 0-5427437 001150748 870 766900 658503000 294957624. 9-5464027 001149425 871 758641 660776311 29-5127091 9-6500589 001148106 872 760384 663064848 29'6296461 9-6537123 001146789 873 762129 666338617 29-5465734 0-5573030 001145475 763876 667627624 20-5634010 9-5610108 001144165 765625 660931875 29-5803989 9-5646659 001142857 870 767876 67222 I37G 29-6972972 9-6683782 001141563 877 769129 674526183 29-6141858 9-6719377 001 140251 878 770884 676836152 28-6810648 9-5755745 001138952 879 772641 679151439 29-6479342 9-5792085 001137656 880 774400 681472000 29-6647939 9-6828897 001136364 8S1 776181 29-0816442 9-5864683 001135074 882 777924 29-6084848 9-6900987 001183787 779680 688465387 29-7153159 9-5037169 001132603 884 781456 690807104 29-7321375 9-5973373 001131222 783226 608164125 20-7489496 9-6000548 001120944 784906 695506456 29-7657521 9-6045606 001128668 887 786769 697864103 20-7826462 9-608181T 001127896 888 788544 T00227072 29-7903289 9-6117911 001126126 88S 790321 702595399 20-8161Q80 9-6158977 00112J850 890 792100 704960000 29-8828678 9-6100017 001128596 8'Jl 793881 707347971 29-8496331 9-6226030 001122384 802 705664 707932288 20-8663600 9-6262016 001121076 803 797449 712121957 29-8831056 0-6207978 001119821 894 799286 714516984 9-6383907 001118568 895 801026 716917875 29-9166506 9-6369812 001117818 89G 802816 719323136 20-0332591 9-6405690 001110071 897 804609 721784278 29-9499683 0-6441642 -001114827 898 806404 724150792 29-9666481 9-6477367 001118686 839 808201 726672699 29-988B287 9-6613166 ■001112347 SOO 810000 729000000 30-0000000 9-6548938 -OOIllllll 901 811801 781432701 300166621 0-6584684 001109878 902 813604 733870808 800333148 9-6620403 001108647 S03 815409 736314827 80-0499584 9-6656096 ■001107420 904 817216 738763264 30-0665028 96691762 001106105 905 819025 741217626 300832179 9-6727403 -001104972 906 820830 743677416 80-0998339 96763017 ■001103753 907 822640 T461 42643 30-1164407 9-6798604 001102536 008 824464 748613312 80-1330383 9-6834166 001101822 909 826281 751089429 80-1496260 9-6809701 -001100110 910 828100 753571000 80-1662063 9-6905211 001098901 9n 820921 756058031 30-1827765 9-6940694 -001097695 912 881744 758550825 30-1993377 9-6076151 ■001006401 913 883669 761048497 80-2158899 9-7011683 001006290 dl4 835396 763551944 30-2324320 9-7046989 001094092 915 837225 766060875 30-2489669 9-7082369 001092803 916 839056 768575296 30-2054910 9-7117723 001001708 917 840889 771096213 30-2820079 9-7153051 001090513 918 842724 773620632 30-2985148 9-7188854 -001089825 919 844561 776161659 30-3160128 0-7223631 ■001088130 920 846400 778688000 30-3316018 9-7258883 001086057 921 848241 78122B961 80-3479818 9-7294109 -001086776 922 860084 783777448 30-3644629 9-7329309 -001084699 923 851929 786330467 30-3809151 9-7364484 OOI08S423 924 853776 30-3078683 9-7399634 001082261 925 856625 791458126 30-4138127 9-7434758 001081081 b,Google TABLE OF SQUAEES, CUBES, SQUARE AND CUBE ROOTS. .%■„,. t=r, 1 s^u^v.,. C.I.... S^wra IU«.ta. Cuba Boou. R8dp.o.al^ 926 857476 794022776 30-1302481 9-7469857 -O010799U 927 8598^9 796507983 80'4466747 0-7504030 ■001078740 928 861184 790178752 80-4630924 9-7539979 -001077586 929 863041 801765089 80-4795013 9-7575002 -001076420 930 864000 804357000 301050014 9-7610001 ■001075269 931 866761 800954191 30-5122926 9-7644971 ■001074114 982 868624 809557508 80-6286750 9-7670922 -001072901 933 870489 812160237 30-5150187 9-7711815 -001071811 931 872356 811780504 80-5614186 0-7749743 ■001070664 935 874225 817400375 30-5777697 9-7781016 ■001060519 936 876006 820025856 80-5941171 0-7829466 ■001068376 937 877969 822656958 30-6101557 9-7854288 ■001067236 879844 825293672 80-6267857 9-7889087 ■001066098 881721 827936019 30-6431069 9-7923861 ■001064063 940 883600 880584000 30-6591194 9-7058611 ■001068830 941 885481 833287621 80-6757283 9-7998886 ■001062690 942 8878G4 80-6920185 9-8028030 -001061571 B48 889249 838501807 30-7083051 0-8062711 ■001060445 m 891136 811232384 30-7245830 9-8097362 -001059322 945 893025 848908625 80-7408523 9-8131989 -001058201 94G 894916 810590530 30'7571180 9-8166591 ■001057082 947 849278123 30-7733651 9-8201169 ■001055960 948 898704 851971392 30-7806086 9-8285723 -001051852 949 900601 854670349 80-8058136 9-8270262 -001053741 950 902500 857375000 30-8220700 9-8304757 ■001062082 &51 904401 800085351 30-8382879 9-8339238 ■001061526 052 906304 862801408 80-8644972 9-8378695 ■001050120 953 908209 865528177 30-8706981 9-8408127 -001040818 954 910116 868250664 30-8868001 9-8442536 ■001048218 955 912025 870983875 80-9030718 9-8476020 ■001047120 956 913036 873722816 30-9192477 9-8511280 -001046025 657 915849 876467493 30-9354166 9-8545617 ■001044932 958 917764 879217912 80-9515751 9-8579929 ■001043841 959 919681 881974079 80-9677251 9-8614218 ■001042753 960 921600 884736000 9-8648483 ■001041667 901 923521 887603681 31-0000000 9-8682721 ■001010583 962 925444 890277128 81-0161248 9-8716911 ■001039501 963 893056347 81-0322418 9-8751185 -001088422 964 895841844 81-0483491 9-8785305 ■001037341 965 931226 898632125 31-0641191 9-8819451 ■001036209 938150 901428696 31-0805405 9-8853571 ■001035197 967 035089 001231063 81-0966236 9-8887673 ■001084126 968 937024 907089232 31-1126984 9-8921710 ■001033058 909 938961 909858209 81-1287648 9-8955801 -001081992 970 940000 912673000 Sl-1448230 -001030928 971 942841 9161986U 81-1608729 9-9023835 ■001029866 972 944731 918330018 31-1769115 9-9057817 •001028807 978 946729 921167317 81-1929479 9-9091776 ■001027749 974 948676 924010424 31-2089781 9-9125712 ■001026694 975 950625 926859375 31-2219900 9-9159624 ■001025641 076 952576 920714176 31-2409087 9-9193513 ■001021590 977 954529 932574833 81-2569092 9-9227379 ■001028541 978 966484 935111352 31-2729915 9-9261222 ■001022195 S79 958441 038318780 81-2889757 9-9205012 ■001021460 980 960400 941192000 31-3049517 9-9328889 ■001020408 981 962301 944076111 31-3209195 9-9862613 ■001019168 961324 946966168 81-3868702 9-9396363 ■001018330 983 066289 949862087 31-3528308 9-9430092 -001017204 98* 968256 952763901 81-3687743 9-0463797 ■001016260 985 970225 955671625 31-3847097 9-9497479 ■001015228 972196 958585256 31-4006369 0-0531138 ■001014199 987 974169 061504803 31-4165561 9-9564775 -001013171 b,Google THE PRACTICAL MODEL CALCUIATOR. Bombst. 1 Biu»™=, Cubs.- S,n^^BocB. cub, ROCU. KsriprcMla. 9 88 076144 964430272 81 ■432*678 9-9598389 -001012146 978121 067361669 814483704 9-9631981 ■001011122 eeo 980100 970299000 81-4642654 9-9065549 -001010101 991 978242271 31-4801525 9-9099065 ■001009082 993 984064 976191488 314060316 9-9782619 -001008065 993 986049 979146657 31-5119025 9-9766120 -0010070*9 994 982107784 81-5277655 9-9799599 -001006086 995 990026 986074875 31-5436206 9-9833065 ■001005025 992016 98S047936 81-6594677 9-9866488 ■001004016 997 994009 991026973 31-6753068 9-9899900 -001003009 998 996004 994011992 81'5911380 9-9933289 -001002004 999 998001 997002999 31-6069618 9-9966656 -001001001 1000 1000000 1000000000 81:6227766 lO-OOOOOOO ■OOIOOOOOO 1001 1000201 1008003001 81-6885840 10-0083222 -0009990010 1002 1004004 1006012008 81-6548866 10-0066622 ■0009980040 1003 1006009 1009027027 81-6701762 10-0099899 -0000070090 1004 1008016 1012048064 10-0188155 ■0009960159 1005 1010025 1015075125 81-7017849 10-0166389 -0009960240 1006 1010036 1018108216 31-7175030 10-0199601 -0009940358 1007 1014049 1021147343 Sl-7382033 10-0232791 -0009930487 1008 1016064 1024192512 81-7490167 10-0265958 -0009920636 1009 1018081 1027248729 31-7647603 10-0299104 1010 1020100 1030301000 81-7804972 10-0332228 -0009900990 1011 1020121 1033364331 31-7062262 10-0865880 ■0009891197 1012 1024144 1036483728 31-8119474 10-0398410 -0009881423 1013 1026169 1039509197 31-8276609 10-0*31469 -0009871668 1014 I028I96 1042590744 31-8433666 100*64506 ■0009801933 1015 1030225 1045678875 31-8590646 10-0497521 -0009862217 1016 1032256 1048772096 81-8747649 10-0630514 -0009842520 1017 1034289 1051871913 31-890437* 10-0563485 -0009832842 1018 1036824 1054977832 81-9061123 10-0696436 ■0009823188 1019 1038861 1058089869 31-921770* 10-0629364 ■0009813543 1020 1W0400 1061208000 81-0374388 10-0662271 ■00O9803922 1021 1042441 1064332261 31-9530906 100695156 ■0009794319 1022 1044484 1067462648 81-9687347 10-0728020 -0000784786 1023 1046529 1070599167 31-9843712 100760868 ■0009776171 1021 1048676 1078741824 82-0000000 10-0798684 ■0000765625 1025 1060625 1076890626 82-0156212 10-0826484 -0000756008 1026 1052876 1080045676 32-0312348 10-0859262 -0009746589 1027 1054729 32-0*68407 10-0892019 -0009787098 1028 1066T84 1086378952 82-062*391 10-092*756 -0009727626 1029 1058841 1089647389 82-0780298 10-0957469 ■O009718173 1030 1060900 1092727000 32-0986181 100090163 -0009708738 1081 1062961 1005912791 82-1091887 10-1022835 ■0009699321 1032 1066024 1099104768 32-1247568 101055487 ■0000689922 10S3 1067089 1102802987 32-1*03173 10-1088117 ■000968a542 103* 1069156 1105507304 32-1558704 10-1120726 •0000671180 1085 1071225 1108717876 82-1714159 10-1158314 ■0009661836 loss 1078296 1111934656 82-1869539 10-1186882 -0009652510 1037 1075369 1115157653 32-202484* 10-1218428 -0000643202 1038 1077444 1118886872 33-2] 8007* 10-1260058 -0009633911 1039 1079521 1121622319 82-2836229 10-1283157 ■0009624639 1040 1081600 1124864000 32-2490810 10-1315041 ■0009615385 1041 1083681 1128111021 82-2646316 10-1348403 -0009606148 1042 1085764 1131366088 32-2800248 10-1880845 ■0009696929 1043 1087849 1134626507 32-2955105 10-1413266 -0009587738 1044 1089936 1137898184 32-8109888 10-1445667 -0009578544 1045 1092026 1141166125 32-3264598 10-14780*7 -0009569378 1043 1094116 1144445836 82-3419233 10-I61O406 -0009560229 1047 1096209 1147730823 32-357879* 10-15427*4 ■0009561098 1048 1098304 1151022592 32-8728281 10-1676062 -0009541985 1049 1100401 1154320640 10-1607359 -0009532888 b,Google TABLE OP SQUAEES, CUBES, SQUARE AND CUBE K0OT3. NambBr. Squire. Cu..^ 6,a.n,I.»«. 0,1,= k™* 1 RKirrorala. 1050 1102500 1157025000 82-4037035 10-1639036 0009523810 1051 1104601 1160986651 32 ■4191301 10-1671898 0009514746 1052 1106704 1164252608 32-4345496 10-1704129 0009505703 1053 1108809 1167676877 82-4499815 10-1736344 0009496676 1054 1110916 1170905464 82-4653862 10-1768539 0O094870U0 1055 1113125 1174241876 32-4807685 10-1800714 0009478673 1056 1116136 1177583616 82'1961636 0009469697 1057 1117249 11809821S8 82-6115364 10-1866003 0009460738 1058 1119SG4 1184287113 32-5260119 10-1897116 0O094517&6 1059 1121481 1187648379 82-5423802 10-1929209 0009442671 1060 1123600 1191016000 32-5576412 10-1961283 0009433962 1061 1125721 1194389981 82-5729949 10-1993836 0009425071 1062 1127844 1197770328 32-5883416 10-2026369 0009416196 1068 1129969 1201157047 32-6086807 10-2057382 0009407338 1064 1132096 1204650144 82-6190129 10-2089875 0009898496 1065 1134226 1207949625 82-6343377 10-2121847 0009889671 1066 1136356 1211355496 32-6496564 10-2153300 00003808G8 1067 1188489 1214767768 32-6649659 10-2185233 0009872071 1068 1140624 1218186432 82-6802698 10-2217146 0009363296 1069 1142761 1221611509 32-6955664 10-2249089 0009354537 1070 1144900 1225043000 32-7108544 10-2280912 0009845734 1071 1147041 1228480911 82-7261868 10-2312766 0009837068 1072 1149184 1231925248 32-7414111 10-2344699 0009a2&^j8 1078 1151829 1285376017 82-7566787 10-2876418 ■0009319664 1074 1153478 128S83S224 82-7719892 10-2408207 0009310987 1075 1155625 1242286875 32-7871926 10-2489981 0009302326 1076 1157776 1245766979 32-8024398 10-2471785 00092936SO 1077 1159929 12492435SS 82-8176782 10-2503470 0009285061 1078 1162084 1252726562 82-8829108 10-2585186 0009276488 1079 1164241 1256216039 32-8481354 10-2666881 0009367841 1080 1166400 1259712000 32-8633535 10-2698557 0009269269 1081 1168561 1268214441 82-8785644 10-2630213 0009250694 1082 1170724 1266723868 32-8987684 10-2661860 ■O009242144 1088 1172889 1270238787 82-9089658 10-3693467 •0009233610 1084 1175056 1273760704 32-9241553 10-2725065 ■0009225092 1085 1177225 1277289125 82-9398882 10-2756644 0009216500 1086 1179396 1280824058 32-9545141 10-2786203 0000208103 1087 1181569 1284365503 82-9696830 10-2819743 0009190032 1088 1183744 1287913472 32-9848450 10-2851264 0009191176 1089 1186921 1291467969 88-0000000 10-2882765 -00091827-36 1090 1188100 1296020000 33-0161480 10-2914247 ■0009174312 lOSl 1190281 1298596571 83-0302891 10-2945709 0009165903 1092 1192461 1S02170688 33-0454333 19-2977158 0009157509 1093 H9464Q 1305751357 83-0605605 10-3008577 0009149131 1094 1196833 1809838684 33-0756708 10-3039983 0009140768 109S 1109025 1812932375 83-0907842 10-8071868 0009132420 1096 1201216 1816582786 33-1058907 10-3102735 0009124008 1097 1203409 1320139673 88-:20990S 10-8184083 0009116770 1098 1205604 1328768192 331360830 10-3165411 0009107468 1099 1207801 1827373299 33-1611689 10-3196721 0009099181 liOO 1210000 133I0O0O00 83-1662479 10-8228013 OOOS000909 1101 1212201 1384638301 83-1813200 10-3289284 000908:;6J2 1102 1214404 1338273208 33-1968868 10-8290537 0009074410 1103 1216609 1341919727 88-2114438 10-8821770 ■00090661B3 1104 1218816 1845572864 83-2266955 10-3353985 0009057971 1105 1221025 1849282625 33-2415403 10-3384181 0009049774 1106 1223236 1352899016 33-2665763 10-3416858 0009041-391 1107 1226449 1356572043 88-2716095 10-8446617 0009088424 1108 1227664 1360251712 33-2866339 10-8477657 0009035271 1109 1229881 1363938029 83-8016516 10-3508778 0009017133 1110 1282100 1367631000 83-3166025 10-3539880 00090090U9 IllI 1234321 1S7I33063I 33-3316666 10-8570964 00090009(10 b,Google THE PRACTICAL MODEL CALCrLATOR. 1112 S,„.r». Culas. Sq»l,tl, nooi!. Cal«I.o«M, ll»cip.„.,l,. 1236544 1875036928 33'34euu40 10-3602029 -0008992806 1118 1288769 1378749897 33-361 O&W 10-3033076 -0008984726 1114 1240996 1382469544 83 -3766885 10-3064103 -0008970661 1115 1243226 1386195875 83-3916157 10-3695118 -0008968610 1116 1245456 1889928896 33-4065862 10-8726103 -0008960753 1117 1247689 1393668613 33-4216499 10-3757076 ■0008952551 1118 1249924 1397415082 33-4365070 10-3788030 ■OD08944544 1119 1252161 1401168159 33-4614673 10-3818965 •0008936550 1120 1254400 1404928000 33-4664011 10-3849883 -0008928571 1121 1256641 1408694561 384813381 10-38807S1 -0008960607 1122 1258884 1412467848 33-4962684 10-3911661 ■0008912656 1123 1261129 1416247867 38-5111921 10-3942527 -0008904720 1124 1263376 1420034624 83-&261092 10-3978806 ■0008896797 1125 1265625 1428828125 83-5410196 10-4004192 1126 1267876 1427628376 33-5559234 10-4034999 ■0008880995 1127 1270129 1431435383 33-5708206 10-4065787 ■0008878114 1128 1272884 1485249152 83-5857112 10-4096557 ■0008865248 1129 1274641 1439069689 33-6005952 10-4127310 -0008857396 1130 1270900 1442897000 33-6164726 10-4158044 -0008849558 1181 1279161 1446731091 33-6308434 10-4188760 ■0008841733 1132 1281424 1460571968 33-6452077 10-4219458 -0008833922 1183 1283689 1454410637 38-6600653 10-4250188 ■0008826125 1134 1285966 1458274104 33-6749165 10-4280800 ■O008818342 1185 1288225 1462185876 SS-6897610 10-4311443 ■0008810573 1186 1290496 1466003450 83-7045991 10-4842069 ■0008802817 1187 1292769 1469878353 83-7174806 10-4872677 ■0008795075 1138 1295044 1473760072 33-7340556 10-440367T ■0008787346 1139 1297321 1477648619 38-7490741 10-4433889 ■0008779631 1110 1299600 1481544000 83-7688860 10-4464308 ■0008771980 lUl 1801831 1485446221 83-7786915 10-4404929 ■0008764242 1142 1304164 1489855288 33-7934905 10-4525448 ■0008756567 1143 1808449 1493271207 33-8082830 10-455-5948 -OO0S7 48900 1144 J30873G 1497193984 38-8280691 10-4586431 ■0008741259 114-5 1811025 1501123025 83-8378480 10-4616896 ■0008733624 1146 1313316 1505000136 38-8626218 10-4647848 -0008726003 1147 1315609 1509008523 10-4077773 ■0008718396 1148 1817904 1612958792 10-4708158 -0008710801 114S 1320201 1516910049 33-8969026 10-4738579 ■0008703220 1150 1322500 1620875000 38-9116499 10-4768955 ■0008695652 1151 1324801 1524845951 33-9268909 10-4799314 ■0008688097 1153 1327104 1528828808 83-9411255 10-4829B56 ■0008680556 1153 1S29409 1582808577 33-9558537 10-4859980 -0008678027 1154 1331716 1536800264 38-9705755 10-4890286 ■0008665511 1155 1834025 1540798875 83-9852910 10-4920575 -0908658009 1156 1544804416 84-0000000 10-4950847 ■0008650619 1157 133SB49 1548816898 34-0147027 10-4981101 -0008043042 1158 1340964 1552830312 34-0293990 10-5011337 -0008636579 1159 1343281 1656862679 34-0440890 10-5041556 -0U08U2812S 1160 1845600 1560896000 84-0587727 10-5071757 -0008620690 1161 1847921 1564936281 31-0734501 10-5101042 -0008613244 11B2 1350244 1568983528 34-0881211 10-5182109 -0008U05852 1163 1352569 1573037749 34-0127858 10-5102259 -0003598152 1164 1354896 1577098944 34-1174442 10-6192391 -0008591065 1105 1357225 1581107125 34-1820963 10-6222506 -0008583091 1166 1859556 1685242296 34-1467422 10-5252604 ■0008676329 1167 1361889 15893:^4403 34-1613817 10-5282085 ■00085689^0 1168 1364224 169841 8682 34-1760150 10-5312749 ■0008501644 1109 1366561 1597509809 34-1906420 10-5342795 ■0008554320 1170 1868900 100161SO0O 34-2052027 10-5872825 -0003547009 1171 1371241 160 j 723211 84-2198773 10-5402837 -000S539710 1172 1373584 1609810448 34-23448y3 10-5433S32 ■0008532423 UTS 1373929 1013964717 34-24WU875 10-5462810 -0008525149 b,Google TABLE OF SQTJAILEe, CUBES, SQUARE AND CUBE ROOTS. Kuml,«r. .,«™. ci-.-. Squ»r»K=^t., C»l« RooU. IM.il>™<.l8, 1174 1878276 1018006024 34'2089884 10-5402771 -0008517888 1175 1880026 1633234376 84-27827B0 10-6622715 -0008510688 1176 13829T6 1020379778 84-2928564 10-5552642 ■0008503401 1177 1385329 183*532233 84-3074386 10-5582662 -0008496177 1178 1887684 1084091762 34-8220046 10-6012445 -0008488064 1179 18B0041 1638858339 84-3365694 10-5642822 -0008431761 1180 1392400 1043082000 34-8511281 10-5073181 -000847] 670 1181 1394761 1647213741 84-3G508O5 10-6702024 -0003407401 1183 1397124 1651400508 84-8802268 10-5781849 -00081602S7 1183 1899489 1655505487 84-3947670 10-5T61658 -0008J530SD 1184 1401856 1060797604 34-4098011 10-5791449 ■0008415040 1185 1404226 1804000826 84-4288289 10-5821225 •00084B8819 118Q 1106696 1008222856 844883507 10-5850988 ■0008431703 1187 1408969 1072446208 344628668 10-5880725 ■0008421000 1188 1411344 1C70076072 84-4078759 10-5910450 -0008417508 1189 1418721 1680914629 84-4818793 10-5940158 -000841 UJ29 1190 1416100 1085159000 34-4093706 10-5909850 ■0008103301 1191 1418481 1089410871 84-5108678 10-5999525 -OOOS800800 1 im 1420804 34-5253530 10-0020184 ■0008389202 1193 1423249 1097930057 81-5398821 10-6058820 ■0008382320 li94 1425639 1702209884 84-5548051 10-6088451 -0008875200 ! 1195 1428025 1706489875 34-5887720 10-6118060 ■0008308201 i 1100 1430416 1710777586 84-6832329 10-0147652 -0008361204 1197 1432809 1716072373 84-5976879 10-6177228 -0008G5-!219 llliS 1-185204 1719374392 84-6121366 10-6200788 -0008347345 111) J 1-137001 1723688599 34-0205794 10-0280331 ■0008340284 1:!00 1440000 1728000000 34-6410192 10-6205867 ■00083333S8 1 l:iUl 1442401 1782323601 84-6654409 10-0295367 ■0008820805 i l:i02 1444804 1736654408 84-6698710 10-6824800 ■00083 194(i8 12U3 1447209 1740992427 31-0842904 10-6854888 ■0003312.152 1204 144961Q 1745837064 84-6987031 10-9383799 -000830J048 laoj 1463025 1749600125 34-7131090 10-6413244 -000829S755 12Uli 1454436 1754049810 84-7275107 10-6442672 ■0008291874 1207 1450849 1768416748 31-7419055 10-6472085 ■000828-JOU4 120S 1459204 1762790912 34-7662944 10-6501480 ■0008273140 1^09 1461081 1707172829 34-7706778 10-0580800 ■0008271299 l:ilO 1464100 1771501000 34-7860643 10-0560233 •0008201463 1211 1406521 1776950931 84-7094363 10-0589570 -0008257088 1212 14(J8944 1780360128 84-8187904 10-6018902 -0W18250S25 1213 1471369 1784770597 84-8281495 10-6648217 •000824402:; 1478796 1789188844 34-8425028 10-0677616 -0008237232 1215 1476225 1703613376 34-8508501 10-6706799 ■0008250453 121G 1478656 1798045996 84-8711916 10-6736060 -0008223684 1217 1481089 1802486318 34-8865271 10-0765317 -00082] 9927 1218 1488524 1806933232 84-8998567 10-9794552 ■0008210181 1219 1485961 1811389469 34-9141805 10-6828771 •0008203445 1220 1488400 1815848000 84-9284984 10-6863978 ■0008190721 1221 1490841 1820316861 34-9428104 10-6882100 ■0008100008 1222 1493284 1824798048 34-0571166 10-0911831 ■0008183306 1223 1495729 1820276567 34-9714169 10-6940486 ■00081 76 Gl 5 1224 1498176 1833764247 84-9857114 10-6969625 ■0008109335 1225 1600626 1838265025 85-0000000 10-0998748 ■0003103205 1220 1503276 1842771176 35-0142828 10-7027855 ■0008156607 1327 1605539 1847284083 85-0285598 10-7056947 -0008140D50 1228 1507984 1851804352 35-0428309 10-7080023 •0008143322 1229 1510441 1850331989 35-0570963 10-7115083 -0008130090 1230 1512000 1800867000 85-0713558 10-7144127 -0008130081 1231 1515361 1865400391 35-0850099 10-7173155 -0006123477 1232 1517824 1809050168 85-0998-576 10-7203108 -0008116883 1233 1530289 I87451G837 85-1140907 10-7231105 -oooaiioaoo 1234 1522756 1879080304 10-7260140 ■0008103728 1235 1526225 1883052876 35-1425508 10-7289113 -000801)7100 b,Google THE PRACTICAL MODEL CALCULATOR. B™b.r. S,».reB. Cnl«=. SinBie Kwtfl. Cuba Bonis. B..lpr*,^,. iMae 1527696 1888282256 85-1667917 10-7818062 ■0008090615 1287 1580169 1892819058 86'1710108 10 7346997 -0008084074 1238 1632644 1897413272 35-1852242 10 7375916 ■0008077544 1289, 1636121 19020U919 85-1994318 10 7404819 ■0008071025 1240 1587600 1906624000 86-2186887 10 7438707 ■0008064516 1241 1540081 1911240521 35-2278299 10 7462579 -0008058018 1242 1642534 1915864488 85-2420204 10 7191436 ■0008051580 - 1243 1645049 1920495907 35-2562051 10 7520277 -0008045052 1214 1547636 1925134784 35-2708842 10 7549108 •0008088686 1245 1660026 1929781125 85-2845575 10- 7577918 -0008082129 1248 1552621 1934484986 85-2987252 10 76Q670B •0008025662 1247 1655009 1939096223 85-3128872 10 7635488 ■0008019216, 1248 1557504 1943764992 85-8270435 10 7664252 -0008012821 1249 1B60OO1 1948441249 35-3411941 10 7698001 ■0008006405 1250 1662500 1968125000 86-8653391 10 7721785 ■0008000000 1251 1565001 1957816251 35-8694784 10 7750458 •0007998605 1252 1667504 1962515008 35-3886120 10 7779156 ■0007987220 1258 1570009 1967221277 85-397'7400 10 7807843 ■0007B80846 1254 1672516 1971935064 35 ■4118624 10 7836516 ■0007974482 1265 1675025 1976656375 35-4269702 7965173 ■0007968127 1577586 1981385216 . 86-4400903 IC 7898816 ■0007961783 1257 1580049 1986121693 85 -4641958 10 7922441 ■0007955449 1258 ,1582564 1990865512 854682957 10 -7951063 ■0007949126 1259 1585081 1995610979 85-4823900 10 7979049 ■0007942812 1260 1587600 2000376000 85-4964787 10 8008230 ■0007986508 1261 1600121 2005142581 , 35-5105618 10 8086797 ■0007930214 1262 1692644 2009916728 85-6246393 10 8065348 ■0007923930 1268- 1695166 2014698447 35-5387113 10 8093884 •0007917656 1264 1697696 2019487744 35-6527777 10 8122404 ■0007911392 1285 1000326 2024284625 85-5668885 10 8150909 ■0007905138 1266 1602766 10 8179400 •0007898894 1267 1605289 2033901163 86-6949434 10 8207876 ■0007892660 1268 1607824 2038720882 35-6089876 10 8286886 - -0007886435 1269 1610361 2043648109 36-623026a 10 8264782 ■0007880221 1270 1612900. 2048883000 85-6370598 10 8293213 ■0007874016 1271 1616441 2058225511 35-6510869 10 8321629 ■0007867821 1272 1617984 2058075648 35-6651090 10 8350080 ■0007861635 1278 1620529 2062983417 85-6791266 10 8376416 ■0007855460 1274 1628076 2067798824 85-6931866 10 8406788 ■0007819291 1275 1625625 2072671875 35-7071421 10 3486144 ■0007848137 1276 1628176 2077552676 ■ 85-7211422 10 8463485 ■0007836991 1277 1680729 2082440938 35-7351867 10 8491812 ■0007830854 1278 1633284 2087886952 35-7491258 10 8520126 •0007824726 1279 1685841 2092240689 85-7681095 10 8548422 ■0007818608 1280 1638400 2097152000 85-7770876 10 8576704 ■0007812500 1281 1640961 2102071841 35-7910603 10 8604972 ■0007806401 1282 1643524 3106997768 85-8050276 10 8633225 ■0007800312 1288 1646089 21,11982187 85 81898B4 10 8861454 ■0007794282 1284 1648656 2116874304 8&-8329457 10 8689687 ^0007788162 1285 1651225 2121824125 35-8468966 10 -8717897 •0007782101 1286 1658796 2126781656 85-8608421 10 ■8746091 ■0007776050 1287 1656369 2131746903 36-8747822 10 -8774271 ■0007770008 ^ 1288 1658944 2136719872 86-8887169 10 -8802436 ■0007763975 1661521 2141700569 86-9026461 10 88805S7 -0007757952 1290 1664100 2140689000 35-9166699 10 -8858723 ■0007751938 1291 1666681 2151685171 85-9304884 10 ■00O7745S3S 1292 1669264 2156689088 85-9444015 10 8914962 -0007789938 1208 1671849 2161700757 35-9588092 10 8943044 ■0007733952 1294 1674436 2166720184 85-9722115 10 8971123 ■0007727975 1295 1677026 2171747875 85-9861084 10 8999186 ■0007722008 1296 1679616 2176782386 36-0000000 10 9027235 ■0007716049 1297 1682209 2181826073 36-01?8862 10-9055269 ■0007710100 b,Google TABLE OP SQUARES, CDBES, SQUAaB AHD CUBE ROOTS. 121 Numbiir. S,«„es. Cubea. Sqnue Bwrts. Cn..E„^.. E=cir.o.^=. 1298 1084804 2186876892 36-0277671 10'9088290 0007704160 vim 1687401 2191983899 SG'04]e426 10-9111296 0007698229 1300 1690000 2197000000 36-0556128 10-9139287 0007692308 1301 1692601 2202073901 86.0698776 10-9167266 ■0007686895 1302 1696204 2207165608 86-0882871 10-9195228 ■0007080492 130a 1697809 2212245127 36-0970913 10-9223177 ■00076743T9 1804 1700416 2217342464 36 -1109402 10-0251111 ■0007668712 1 1305 1703026 2222417626 36-1247887 10-9279081 0007662885 \ isoe 1706636 2227560016 86-1886220 10-9806937 0007656968 i 1307 17082*9 2232681443 36-1624660 10-9384829 0007651109 1808 1710864 2237810112 86-1662826 10-9362706 0007645260 ] 1309 1713481 2242946629 36-1801060 10-9390569 0007089419 1310 1716100 2248091000 36-1989221 10-9418418 0007033588 i 1311 1718721 2253213231 86-2077840 10-9446253 0007627705 1 1812 1721844 2258403328 36 ■2216406 10-9475074 0007021951 : 1313 1723969 2268571297 86-2368419 10-9601880 0O0701644U ; 1814 1726696 2268747144 86-2491879 10<9629673 0007610350 ! 1816 1720225 2273930876 36-26262B7 10-9557461 0007004503 1316 1731856 2279122496 86-2767143 10-9585215 0007598784 , 1317 1781489 2284822013 36-2901246 10-9612965 0007593011 , 1818 1787124 2289529432 36-3042697 10-9640701 0007587253 1319 1739761 2294744769 86-8180896 10-9668423 0007681501 1820 1742400 22999GS00O 86-8818042 10-9696131 0007575758 1331 1745041 2306199161 86-3465687 10-9723825 •0007570023 1322 1747684 2310438248 86<859S179 10-9751605 ■0007564297 1823 1760329 2315685267 86-8780670 10-9779171 0007558579 1824 1752070 2320910224 36-3868108 10-9800828 ■0007552870 1325 1755625 2326203125 86-1005494 10-9831162 ■0007547170 1326 1768276 2831478976 3G-4142829 10-9862086 0007641478 1837 1760929 2336752783 86-4280113 10-9889696 0007536795 1823 1763584 2342089652 38-4117848 10-9917293 0007630120 1829 17G6241 23173312S9 86'4654&23 10-9911876 0007524454 1330 1768900 2852687000 36-4601660 10-9972445 0007618797 1831 1771561 2357917691 86-4828727 11-0000000 ■0007513148 1832 1771224 2363266368 86-4965752 11-0027541 -0007507608 1838 1776889 2368593087 86-5102T26 11-0066069 ■0007501875 1334 1779556 2873927704 36-5239647 n -0082583 ■0007496252 1386 1782225 2879270375 36-6376618 11-0110082 00071B003T 1836 1781896 2884621056 36-5518888 11-0187569 •00074850110 1837 1787689 2889979753 36-5G50106 11-0165041 0007479482 1338 17B0244 23953464T2 86-6786823 11 ■0192500 ■0007173842 1389 1792921 2100T21219 36-5928489 11-0210945 0007408260 1840 1795600 2406104000 86-6060101 11 ■0347877 ■0007462687 1341 1798281 2411494821 8G-61966G8 11-0274796 •0007457122 1342 1800964 2416893688 86-0833181 11 ■0302199 ■O0OT451666 1313 1803640 2422300607 86-6469144 11 -0329590 0007116016 1844 1806836 2427715584 36-6606056 11-0856967 0007440476 1346 1809026 2433138625 86-0712416 11-0381330 0007434044 1846 181171G 2488569786 86-6878726 11-0111680 0007429421 1347 1814409 2444008923 36-7014986 11-0489017 0007423905 1348 1817104 2449466193 36-7161195 11-0466339 0007418398 1849 1819801 2454911549 86-7287863 I1-0193G49 0007112898 1350 1822500 2460876000 36-7423461 11 ■0620945 0007407407 1361 1825201 2165846661 86-7559519 11-0518227 0007401024 1352 1827904 2471826208 86-7695526 11-0676497 000739G450 1368 1830609 2476818977 36-7831483 11-0602752 ■0007390983 1354 1883316 2482309861 36-7967390 11-0629994 €007385524 1856 1886025 2487818875 86-8103216 11-0667222 0007380074 1356 1838730 2493326016 SG-8289063 11-0684437 0007374631 1357 1841449 2498840298 86-8374809 11 '0711689 0007369197 1358 1844104 2504371712 36-8510516 11-0738828 0007303770 1359 1840881 2509911279 86-8646172 11-0766003 00078&8862 b,Google ! PEACTICAL MODEL CALCnLATOIt. Nu..b,r. 8,-ar... CaW^ s,,...™b..l.. (■„;.. ,i,-.L,. I[,-o|.u,.i.-,. 13U0 1849600 2515456000 30-8781778 n-<Jl;:'.iV6r, ■OUOTi! -329-1 1 1801 18&2321 2521008881 36-8917330 11-0:^20314 ■Ol*T347o3i) 1362 1855044 2526569928 86 ■9052842 11-0.^-17449 ■0007K12I41 1S63 1857769 2532339147 36 '9188299 11-0S74571 -0O073:JO7-J7 1804 1860496 2537716344 36-0323706 11-0901079 -00O73ai878 136a 1868225 2548302125 36-9469064 11-0928775 -0007S26OO7 1866 1865956 2548805896 36 ■9594372 11-0955857 -00073200-! 1 1367 1868689 2554497863 86-9729631 11 ■0082020 -0007315289 1868 1871424 2560108082 36-0861840 11-1009982 ■0007309912 1874161 2565726409 87-0000000 11-10371126 ■0007304002 1870 1876900 2571S58000 37-0135110 11-1064031 ■0007291)270 1371 1879041 2576987811 87-0270172 11-1001070 ■0007293940 1372 1882384 2582630848 87-0405184 11-1118073 ■0007288030 1873 1885129 2588282117 37-0510146 11-1116064 -0007288821 1374 1887876 2598041624 87-0676060 11-1172041 •0007278020 1375 18B0625 2599609375 11-1199004 -0007272727 1898376 2605285S76 37-0044740 11-1225956 ■0007207112 1377 1896129 2610969688 37-1079606 11-1262808 ■00O72021C4 1878 1898884 2616662152 37-1214224 11-1279817 ■0007250894 1379 1901641 2622362939 87-1348803 11-1306729 ■0O072510S2 1380 1901400 2628072000 87-1483512 11-1888028 ■0007240377 1S81 1907161 2683789341 37-1618084 U-1360514 ■0007241130 1B09924 26S9514908 87-1762606 11-1887386 •0007285890 1883 1912689 2615248887 37-1887070 11-1414246 ■0007230058 1384 1916466 2650081104 37-2021505 11-1111008 ■0007225134 1385 1918225 2656741626 87-2165881 11 '1407926 ■O00722O217 1886 1920996 2662300156 87-2290209 11^1494747 ■0007215007 1887 1928769 2668207603 87-2424480 11 ■I 521535 ■0007209805 1388 1926644 2674018072 37 '2658720 11 ■1548860 ■0007204011 1389 1929321 2679820869 87 ■2692903 11^157513a ■0007190424 1890 1032100 2685619000 37-2827087 11 ■1001003 ■0007191215 1801 1934881 2601410471 87-2901124 11 ■1628059 ■0007180073 1392 1937664 2697228288 37-3005162 11 ■1055408 ■0007183908 isas 1940449 2703045457 87-8229152 1M682134 ■0007178751 1394 1943236 2708870984 37-8868004 11-1708852 ■0007173601 1395 1946026 2714704875 87'3496988 11-1735558 ■0007168459 1386 1948816 2720547186 373680881 11-1762250 -0007168324 1397 1951609 2726397773 87-8764682 11-1788930 -0007158196 1398 1954404 2732256792 11-1815598 -0007153076 1899 1967201 2738124199 87-4032084 11-1842252 -0007147963 1400 1960000 2744000000 37-4165788 11-1868894 -0007142857 1401 1962801 2749884201 37-42B9846 11-1896528 ■0007187769 1402 1965604 2755776808 87-4432904 11-1922139 ■0007132668 1403 1068409 2761677827 37-4606416 11-1918748 ■O0O71276S4 1404 1971216 2767587264 37-469Q880 11-1975334 -0007122507 1405 1974025 2773505123 87-4883296 11-2001913 -0007117488 1406 ■ 1976836 2779431416 37-4066666 11-2028479 ■0007I12S76 1407 1970649 2785866143 87-5099987 11-2055032 -0007107821 1408 1982464 2791809312 87-5283261 11-2081578 ■0007102273 1409 1985281 2797260920 37-6366487 11-2108101 -0007097232 1410 1988100 2808221000 87-5400667 11-2134617 -0007092109 nil 1990921 2809189581 87-5682799 11-2161120 -0007087172 1412 1993744 2816166528 37-6765885 11-2187611 -0007082158 1413 1996569 2821151997 87-5808922 11-2214089 -0007077141 1414 1999306 2827145044 87^6031913 11-2210054 ■0007072136 1415 2002225 2833148376 87^6164867 11-2267007 ■0007067188 1416 2005056 2889150296 37-6297764 11-2298448 ■0007062147 1417 2007889 2845178718 87-6480604 11-2319876 ■0007067163 1418 2010724 2851206632 37-6563407 11-2346292 ■0007052186 1419 2013561 2857243059 87-6696164 11 '2372606 ■0007047210 1420 2016400 2863288000 37^6828874 11-2800087 -0007042254 1421 2019241 2869341461 37 6961536 11-2425165 -0007037298 b,Google TABLE OE SQUARES, CUBES, SQUARE AND CUBE BOOTS. 123 ^■„.,l..^. Sfloart!. C,*.,- S,u.r.E«,«. cube Hoct,. itoip™,^. U±^ 202J084 2875403448 87-7004153 11-2451831 0007032349 ZU24a20 2881473907 87-7226722 11-2478185 0007027407 i4:;i 2887558024 8T-TS59245 11-2604527 0007022472 20S0025 2898640626 87-7401723 11-2680856 0007017544 ^033476 289B736776 87-7624152 11-2557173 0007012628 U-27 2036320 2005841483 37-7756535 11-2688478 0007007708 14:;8 2089184 2011054762 37-7888873 11-2609770 0007002801 U2'J 2042041 2918076580 87-8021168 11-2686050 0006907901 I4S0 2044900 2924207000 87-8163408 11-2662318 0006903007 1431 20477G1 2980345991 37-8285606 11-2688578 0006988120 1432 2060624 2936408668 87-8417769 11-2711816 0006983210 1433 2053489 2042640737 87-8549864 11-2741047 0000978307 14S1 2056356 2948814504 37-8681924 11 '2767266 0000073501 1485 2050225 2954087875 87-8818988 11-2798472 0006068641 1436 2062096 2061169856 87-8045906 11-2810666 0006063788 1437 2064069 2067360458 37-9077828 11-2845840 0006958942 1488 2067844 2973559672 37-0209704 11-2872019 0000054103 J439 2070721 207B7075I9 87-9341538 11-2808177 0006949270 1410 2078600 2985984000 87 ■9473310 11-2024323 0006944414 1441 207C481 2992209121 37-9605058 11 ■2050457 0006939025 1442 2079364 8008442888 37-9736751 11-2976579 0006934813 1443 2082240 8004685307 87-G808398 11-8002688 O000OSU0O7 1444 2086130 8010986884 38-0000000 11-8028786 0006925208 1445 2088025 8017196125 38-0131556 11-3054871 ■0006920415 1410 2O8OBI0 3023464586 88-0263067 11-3080945 ■0006915629 1417 2098800 8029741623 88-0894582 11-3107006 0006910830 1448 2096704 8030027892 38-0525952 11-8183056 0006906078 14d0 2009601 8042321849 38-0657826 11-8169004 0006901812 1450 2102500 8048625000 88-0788056 11-3185119 0006806652 Hil aio&ioi 8064986851 88-0019980 11-8211182 0006891799 145-J 2108304 3001257408 38-1051178 11-3237134 0006887062 14oa 2111200 8007586777 88-1182871 11-3268124 0006882312 1454 2114116 3073024664 88-181851B n-8289102 0006877579 1465 2117026 80802713/5 88 1444622 11 3315067 0006872852 145(i 2U9936 8086626816 88 1676681 11 8341022 0006868182 1467 212284S 8092990993 38 1706693 11 3866064 0006863412 1458 2125764 8099363912 38188.662 11 8892804 0006858711 1450 2128681 3105-45579 38 196S685 11 8418813 0006854010 1400 2131600 3112186000 88 20004b3 11 8444719 0006849315 1461 2134521 8118685181 88 2230297 n 3470614 0006844627 1463 2137444 8124943128 38 236108O 11 8406497 0006830945 1463 2140369 8131S59847 88 2491829 11 3522368 0006885270 1464 2143206 8187785344 88 2622029 11 3548227 0006830601 14C5 2140225 3144219025 88 2763184 11 8574075 0000826989 1400 2140156 8150662606 88 2883794 11 3599011 0006821282 1107 2152080 8167114503 38 3014360 11 8625786 0006816638 1488 2155024 8163575282 38 8144881 11 3651547 0006811989 1469 2157S61 8170044709 38 3275858 11 8677347 0006807852 1470 2160900 3176523000 88 3405,00 11 3703136 0006802721 1471 2163841 8183010111 38 3536178 11 3728914 0006708097 1472 2166784 8180606048 88-8666522 11-3754670 0006793478 1473 2160720 8106010817 88-8796821 11-3780433 0006788866 1474 2172676 3202524424 88-8927076 11-3806176 0006784261 1475 2175625 3209046875 88-4057287 11-8831906 0006779661 1473 2178576 8215578176 33-4187454 11-3857625 0006775068 1477 2181529 3222118333 88-4317577 11-8883882 0006770481 1478 2184484 3228667862 88-4447666 11-3909028 0006765900 1470 2187441 8285225239 38-4577691 11-8034712 0006761825 1480 2190400 3211792000 88-4707681 11-3960381 0006756757 1481 21933G1 8248367641 11-3080045 ■0006752104 1482 2106324 3254952168 38-4067530 11-4011605 0006747638 1488 2199:^89 3201546587 38-5007300 11-4037332 0006743U88 b,Google THE PRACTICAL MODEL CALCULATOR. Hnmbsr. Squires. Cnl,a,, aiun.-6E«,u^ Cuba Koiits. B8cLp«™la. 1484 2202256 3268147904 38-5227206 11-4062969 ■0006788844 1485 2206225 8274769125 S8-5866077 11-4088574 -0006734007 1486 2208196 8281379256 88'5486705 11-4114177 •0006729474 1487 2211169 3288008303 38-5616389 11-4139769 ■0006724930 1488 2214144 8204646272 88-6746080 11-4165349 ■0006720430 1480 2217121 8301293169 88-5875327 11-4190918 -0006715S17 1400 2220100" 3807949000 88'6005181 11-4206476 ■0006711409 1491 8314613771 88-6184691 11-4242022 ■0006706908 1492 3226004 3321287488 38-6264158 11-4267556 ■0006702418 1493 2229049 8227070157 88-689e682 11-4203079 ■0006697924 1494 2282086 3834661784 38-6522962 11-4318591 -0006698440 1406 2235025 8841362876 88-6652299 11-4344002 ■00066S8963 1496 ,2238016 3348071986 . 38-6T81593 11-4360581 ■0006684492 1407 2241000 8364790473 38-6910843 11-4305059 ■0006680027 1498 2244004 3361517992 88-7040050 114420525 ■0006676667 1499 2247001 8368264490 38-7169214 U -4445980 ■0006671114 1500 2280000 8375000000 S8-7298885 n-4471424 ■0006666667 1501 2253001 3381754601 88-7427412 11-4496867 •0006662225 1602 2256004 3888518008 88-7656447 11-4522278 ■0006657790 1503 225^009 3S06290527 38-7685439 11-4547688 -0006653360 1504 2262016 3402072064 38-7814380 11-4578087 ■0008648036 1505 2265025 38-7948204 11-4508476 ■0006644618 1506 3416662216 88-8072168 11-4628850 ■0006640106 1507 2271048 8423470843 38-8300973 11-4649216 ■0006635700 1508 2274064 3420288612 88-8329757 11-4674568 ■0006631800 1609 ■ 2277081 8436115229 38-8458491 11-4699911 ■0006626905 1510 2280100 3442951000 38-8587184 11-4725242 ■0006622517 1611 2283121 8449795831 38-8715884 11-4760662 ■0006618134 1512 2286144 3466640728 88-8844442 11 ■4776871 -0006613757 1513 2289160 8463512697 38-8973006 11-4801160 ■0006609386 1514 2202196 8470884744 ■ 38-0101520 11-4826455 ■0006605020 1516 2295225 3477265875 38-9280009 11-4851731 ■0006600660 1516 2298256 3484166096 38-0358447 11-4876095 ■0006696806 1517 2301289 349! 055413 38-9486841 11-4902249 -0006691958 1518 2304324 3507968832 88-9616194 114927401 ■0006687615 1619 2307361 8504881359 38-0743505 11-4952722 ■0006688278 1620 2810400 3511808000 38-9871774 11-4977942 ■0006578947 1521 2313441 3518748761 39-0000000 11-6008151 ■0006574622 1523 2316484 8525688648 39 ■0128184 11-5028848 ■0006670302 1523 2S10529 3532642667 11-5053535 ■0006565988 1524 8689605824 390384426 11-5078711 ■0006561680 1525 8546578125 30-O51248S 11-5103876 -0006657877 1526 2828676 3553550576 30-0640490 11-6129030 ■0006553080 1627 2331729 3567549562 39-0768473 11 ■5164178 ■0000548788 1628 2334784 8660668188 ' 39-0896406 11-6179306 ■0006544503 1529 2337841 8674558889 80-1024206 11-5204426 ■0006540222 16S0 2340900 8581677000 39-1152144 11-6229585' ■0006535948 1631 2348961 8688604291 39 1279961 11-5264684 ■0006531679 1532 2347024 8885640768 39-1407716 11 -6279722 ■0006527416 1583 2350080 3602686437 39-1636439 11-8804799 ■0006523157 1634 2358156 8600741804 89-1663120 11-6329865 ■0006618905 1585 2356225 3616805375 30-1700760 11 ■6354920 ■0000514658 1636 2369256 8628878656 39-1018359 11-5379965 ■0006510417 1587 3680961163 89-2045915 n-8404998 ■0006506181 1538 2365444 8638062872 89-2173431 ■ 11-6480021 ■0006601951 1689 2868521 8645163819 89-2800905 11-6456033 ■0006497726 1640 2371600 3652264000 39-2428337 11-5480084 ■0006498506 1541 2374681 3669383421 39-2556728 11 ■6505025 ■0006489293 1542 2377764 3666512083 89-2683078 11-5580004 -O0OS485084 1543 2380849 8678650007 89-2810387 11-5654972 ■0006480881 1544 2388036 3680707184 39-2937654 11-6579931 ■0006476684 1545 2387025 3687068625 39-3084880 11-5604878 -0006472402 b,Google TABLE or SQUARES, CUEES, SQUARE A3D CUBE HOOTS. Number. S,,>=r».. ClCE. S,«ar. n««. QuU R.»l., Ilt^lpr.c.1,. 154lj Ii890116 3695119336 39-8192065 11-5929816 -0000468305 1547 2808209 8T02294323 39-3319208 11-5664740 -0006464124 1518 2396304 8709478592 39-3446311 11-6679655 ■0006459948 154SI 2399401 8716872149 89-3573878 11-5704569 ■0006465778 1550 2102500 3723875000 39-3700394 11-5729458 -0006461613 1551 2405601 3781087161 39-3827873 11-5764336 -0000447453 1552 2408704 8738308608 39-3^54812 11-5779208 -0006443299 1553 2411809 8745639377 89-4081210 11-5804069 ■0006439150 1551 2414916 3752779464 39-4208067 11-5828919 ■0006485006 1555 2418025 3760028875 89-4384883 11-6858759 ■0006430868 1556 2421186 3767287016 39-4461658 11-5878688 ■0006426735 1567 2424249 3774655693 89-4688393 11-5908407 ■0006422603 1658 2437304 8781888112 39-4716087 11-5928215 -0006418486 155S 24S0481 8789119879 39-4841740 11-5953018 -0006414868 1560 2488600 3796416000 39-4968353 11-5977799 ■0006410256 1561 2436721 3808721481 89-6094925 11-6002576 ■0006406150 1562 2439844 8811036828 39-5221457 11-6027342 ■0006402049 1668 2442969 3818360547 89-6347948 11-6052097 ■0006897953 1564 3446096 8825641444 39-5474399 11-6076841 ■0006398862 1565 2449225 3888037126 89-6600809 11-6101575 -0006389776 156B 2462356 3840389496 89-6727179 11-6126299 -0006385696 1567 2455489 8847761268 39-5853608 11-6151012 ■0006381621 loliS 2458624 3855123432 89-6979797 11-6175716 ■0006377551 1569 2461761 886250S009 39-6106046 11-6200407 ■O006373486 1570 2464900 3809883000 39-6232256 11-6225088 -0006369427 1671 2468041 3877292411 39-6868424 11-6249759 -0006365372 1572 2471184 3884701248 89-6484652 11-6274420 ■0006361323 1573 2474329 8892119157 39-6610640 11-6299070 ■0006357279 157* 2477476 8899647224 11-6823710 ■0006353340 1576 2480625 3906984876 89-6862698 11-6348339 ■0006349206 1676 2483776 8914430976 39-6988665 11-6372957 -0006346178 1577 2486929 8921887038 39-7114598 11-6397566 ■0006341154 1578 2490084 3929352552 39-7240481 11-6422164 ■0006837186 1670 2498241 3936827539 89-7366329 11-6446751 ■0006333122 1580 2496400 3944312000 39-7492138 11-6471329 ■0006329114 1581 2499561 8951805941 89-7617907 11-6495895 -0006325U1 1582 2502724 8959809368 39-7743636 11-6520452 ■0006321113 1588 2505889 3966822287 89-7869826 11-6544998 ■0006317119 1584 2509066 3974344704 89-7994976 11-6569584 ■0006313181 16S5 2512225 3981876626 89-8120585 11-6594059 -0006309148 1586 2515896 8989418056 39-8246155 11-6618574 ■0006305170 1587 2518569 3996969003 89-8371686 11-6643079 ■0006301197 1588 2521744 4004529472 89-8497177 11-6667574 -0006297229 1589 2524921 4012099469 39-8622628 11-6692068 ■0006298268 1590 2528100 4014679000 89-8748040 11-6716532 ■0006289308 1591 2631281 4027268071 39-8873413 11-6740996 ■0006285355 1592 2584464 4034866688 39-8998747 11-6765449 ■0006281407 1503 2537649 4042474857 39-9124041 11-6789892 ■0006277464 1594 2540836 4050092584 39-9249295 11-6814325 ■000627852S 1595 2544025 4057719876 89-9374511 11-6838748 -0006269592 1596 2547216 4065356736 89-9499687 11-6863161 -0006285664 1597 2550409 4073003178 89-9624824 -0006361741 1598 2558604 4080659192 89-9749922 11 -691 1055 ■0006257823 1599 2556801 4088324799 39-9874980 11-6936337 -0006253909 1600 2560000 4096000000 400000000 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 mimher, and the root of the next higher integer number, by the decimal part of the given number, and add the hv Google 126 TUB 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 = 4-4721 -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 arithmetteal progression ; or, to extend the foregoing table of squares. Rule. — Find, in the usual way, the squares of the first two num- bers, and subtract the less from the greater. Set down the square of the larger number, in a separate column, and add to it the dif- ference already found, with the addition of 2, as a constant quan- tity ; the product will bo 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 tho answer sought. _ Required the square of 2595. 2595 is three times greater than 865; and tho st^uarc of 805, by the table, is 748225. Then, 748225 x 3^ = 6734025. Rule 2, — If tho number required to be squared be an odd num- ber, and do not exceed twice the amount of any number contained in the table, find the two numbers nearest to each other, which, 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 (932= = 868624) + (933^ = 870489) = 1739113 x 2 = 3478226 - 1 = 3478225. hv Google RULES FOR SQUARES, CUBES, SftUARB BOOTS, ETC. 127 To find the cube of a greater number than is contained in tJie table. Rule. — Proceed, as in squares, to find hoiv many times the num- ber required to he. cubed exceeda 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 tlie pro- duct will be the answer required. Required the cube of 3984. 3984 is 4 times greater than 996; and the cube of 90G, by the table, ia 988047936. Then, 988047936 x 4^ = 63235067904. To find the square or cube root of a highernumber than is in the table. Role. — 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 ia derived will be the answer required, when decimals are not of importance. Required the square root of 542869. In the Table of Squares, the ncareat number ia 543169 ; and the number from 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 aasumed 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 ia the root of the aaaumed 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 tho square root of 321'62. v^321 = 17-9164729, and v'322 = 17-9443584; the difference (■0278855) X -62 + 17-9164729 = 17-9337619. hv Google 128 HIE PRACTICAL MODEL CALCULATOR. To obtain the s(iuare root or cube root of a number containing dect- mak, by inspection. Rule. — The square or cube root of a number containing deci- mals may be found at once by inspection of the tables, by taking the fiourcs cut off in the number, by the decimal point, in ^yairs if for the square root, and in triads if for the cube root. The fol- lowing example will show the results obtaineiJ, by simple inspec- tion of the tables, from the figures 234, and from the numbers formed by the addition of the decimal point or of ciphers. ■0483735465* ■13276143S ■152970585 ■284J: -2340 ■4837354G5 -61622401 2-34 1-52970585 1-32761489 4-88785465 2-860 15-2970585 2340 48-3735465 13-2761439 23400 152-970585 28-60 Tofindthe cubes of numbers in arithmetical progression, or to extend the preceding talle 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 aeries of differences. To form a second series of differences, bring down, in a separate column, tho cube of tlie 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 differeTtees. By Tat). 1500 = 3375000000 6746501 difference. 1490 X 6-1- 6 ^ 9 000 6764501 diff. of ICOO SOOO -I- 6 = 9006 6772510diff. ofl502 Then, 3375000000 Cube of 1500 Diff. for 1500 = 675450 1 3381754501 Cube of 1501 Diff. for 1502= 6772519 8895290527 Cube of 1503 • Deriyed from -002340 by mBans of 2340. t Derived from -002340 by means of 2340. I The nearest result by simple inspeetioa is obtained for -023 by 23. But four places correct oan 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 28808 ia that of 2860, whence -286013 true to the last place, and is after- warda substituted. b,Google TABLE OF THE FOURTH ASD FIFIH POWEKS OF NUllEERS. 129 Table of the Fourth and Fifth Powers of Numhers. BM3058 BBS4198 iwasTB iissiKa 20511149 282176249 Mjoasasi M4696301 002436(43 vmnasu 19S491I6S2 20750n6B3 es5T490i J4M5201 8UM626 SSI5334S9 wgssma 2U14(iaS6 291199921 6277S1916a Ei9D19IHII)0D 62103ai-!61 I7e231lGSS2 19261116S21 20113571876 2100MUe76 21fl3U80SS7 S28n6776«8 hv Google THE PRACTICAL MODEL CALCULAIOK. Table of My^erbolie Logarithm N, I^pLrithm, N. U8=rithm. N. t aesritlim- N. L.,.ri,hm, ToT -0009503 1-58 ■4574248 2-15 7654678 2-72 I -0006318 1-02 ■019802a 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 ■0892207 1-61 •4762341 2-18 7793248 2^75 1^0116008 1'05 ■0487902 1^62 ■4824261 2-19 7839015 2-76 1-0152306 1-06 -0682689 1-63 ■4886800 2-20 7884673 2-77 1-0188473 107 ■0676586 1-64 ■4946962 2-21 7929925 2-78 1-0224509 108 ■0769610 1-65 ■5007752 2-22 7975071 2^79 1-0260415 1-09 ■0861777 1^66 ■6068176 2-23 B020016 2-60 1-0396194 1-10 ■0953102 1-67 ■612S236 2^24 8064758 2'81 10331844 1-11 ■1043600 1-68 ■5187987 2-82 1 ■0367868 1-12 ■1133287 1^60 ■6247286 1-0402766 1-13 ■1222176 1-70 ■5806282 2^84 1-0488040 1-14 ■1810283 1^71 ■5364933 £■86 I -0478189 115 ■1397619 1-72 ■5428242 2-86 1 ■0608216 1-16 ■1484200 1-73 ■5481214 2-87 1-0548120 117 ■1570037 1-74 ■5638851 1-0577902 1-18 ■1655144 1-75 ■5596157 2-89 1-0612561 1-19 ■1739533 1-76 •565813S 2-33 2-90 I ■0647107 1-20 •1828215 1-77 •6709795 2-84 8501509 2-91 1^0681680 1-21 ■1906208 1-78 ■5766133 2-35 8544153 2^92 1-0715836 1-32 ■1988508 1-79 ■5822156 2^86 8586616 2-93 1-0750024 \-2Si ■2070141 1^80 ■5877866 2^37 2-94 1 '0784095 1-24 ■2151118 1-81 •5933268 2-38 8671004 2-95 1 ■0818051 1-25 ■2231485 1-82 ■5988365 2-39 8712983 2-96 1-0851892 1-23 ■2811117 1-83 ■6043159 2-40 8754687 2-97 1-0885019 1-27 ■2390169 1-84 ■6097665 2-41 8796267 2-98 1-0919233 1-28 ■2468600 ■6151856 2^42 8887675 2^99 1-0962733 1-29 ■2546422 ■6205764 2-43 8878912 3^00 1-0986123 1-SO ■2028642 1-87 ■6259884 2-44 8919980 8-01 1-1019400 1-31 ■2700271 1-88 ■6812717 2^45 3-02 1 ■1052568 1-32 ■2776317 1^89 ■6365768 2-46 9001613 3-03 1^1086626 1-83 ■2851789 1-90 ■6418588 2'47 9042181 3-04 MI 18575 1-84 ■2926696 1-91 ■6471082 2^48 9082586 3-06 1-1161416 1-36 ■3001046 1-92 ■6523261 2^49 3^06 1-1184149 ■3074846 1-93 ■6575200 2-50 9162907 3-07 1 ■1216776 1-37 ■8148107 1-94 ■6626879 2^61 0202827 8^08 1^1249295 ■3220884 1-95 ■6678293 2^52 9242689 3-09 1^1281710 ■3293037 1-96 ■6729144 2-63 9282198 3-10 1-1314021 t-40 ■3364722 1^97 ■6780335 2-54 9821640 S^ll 1 ■1316227 1-41 ■3435897 ■6830968 2-55 9360933 512 1 ■1378330 1-42 ■8506568 1-99 ■6881846 2^56 9400072 S^18 1 ■1410830 1-43 ■8676744 200 ■6031472 2-57 9489058 8^14 1^1442227 1-44 ■8646481 2-01 ■6981347 2-68 9477893 8-15 1-1174024 1-45 ■3715635 2-02 ■7030974 2-59 9616578 3-16 1^1506720 1-46 ■3784364 2-03 ■7080867 2-60 9555114 3-17 1^15378I5 1-47 ■3862624 2-04 ■7129497 2^61 9593502 3-18 1 ■1568811 1-48 ■8920420 2-06 ■7178897 2-62 ■9631743 3-19 1-1600209 1-49 ■8987761 2'06 ■7227069 2-63 9669838 3-20 1-1631508 1-50 ■4054651 2-07 ■7275485 2-64 9707789 8-21 1^1662709 1-51 ■4121096 2-08 ■7323678 2-65 9746596 8-22 1-1693818 1-52 •4187103 2^09 ■7371640 2-66 9788261 3^23 1-1721821 1-53 ■4252677 2-10 ■7419373 2 '67 9820784 3-24 1-1755733 1-54 •4817824 2^11 ■7466879 2-68 9858167 3^26 1-1786549 1-55 ■4383549 212 ■7514160 2-69 9895411 8-26 1-1817371 1-56 ■4446858 2-13 -7561219 2-70 9932517 3-27 1-1847899 1-57 ■4510756 1| 2-14 ■7608058 2.,, 9969480 3-28 1-1878434 b,Google TABLE OE HYPEKBOLIC LOOAPJTHMS. X. L^anvithi^. H. L»((lrtU,B. H, I^S^Ubm N Logirillim, 3-29 1-1908875 T9I 1-3635873 "iiT 1 5107219 6 15 1-6880967 1-1939324 3-92 1-3660916 4-54 1 5I^92b9 516 1-6409365 S-Bl 1 '1969481 3-98 1-3686894 4-66 1 6161272 5 17 1-6428726 8-32 1-1999647 3-94 1-3711807 4-56 1 6173226 618 1-6448060 1-2029722 8-95 1-3787156 4-57 1 6196182 519 1-6467886 3-34 1-2059707 3-96 1-8762440 4-53 1 6216990 6 20 1-6486586 3-35 1-2089603 3-97 1-3787661 4-59 1 5238800 5 21 1-6505798 1'211B40S 3-98 1-8812818 4-60 1 5260663 5 22 1-6524974 3'87 1-2149127 3-99 1-3837912 4-61 1 5282278 5 23 1-6544112 3-38 1-2178757 4-00 1-8862943 4-62 1 6803947 5 24 l-65e32U 3-39 1-2208299 4-01 1-3887012 4-68 1 5326568 6 25 1 -6682250 3-40 1-2237764 4-02 1-8912818 4-64 1 684714] 5 26 1-6601310 8-41 1-2267122 4-03 1-3937663 4-65 1 53b8672 6 27 1-6620308 3-42 1-2296405 4-04 1-8962446 4-66 1 6390154 6 28 1 -6630200 3-43 1-2826605 405 1-8987168 4-67 1 6411590 5 29 1-6658182 B-44 1-2364714 4-06 1-4011829 4-68 1 6482'>81 6 80 1-6677068 8-45 1-2388742 4-07 1-4036429 4-69 15454825 6 81 1-6695018 3-46 1-2412685 4-08 1-4060969 4-70 1 6476625 5 32 1-6714733 8-47 1-2141546 4-09 1-4085449 4-71 1 5490879 1-6783512 3-48 1-2470322 4-10 1-4109869 4-72 1 6518087 6 84 1-6752236 3-49 1 -2496017 4-11 14134230 4-78 1 6689262 6 35 1-6770965 8-50 1-2627629 4-12 1-4I58581 4-74 1 5560371 6 36 1-6780030 3-51 1-2566160 4-13 1-4182774 4-75 1 5681446 6 37 1-6808278 3-52 1-2684609 4-14 1-4206957 4-76 1 6602476 6 88 1-6626882 1-2612978 4-16 1-4281083 4-77 1-6623462 5-39 1-6846453 S-54 1-2641268 4-16 14255150 4-78 1-6644405 6-40 1-6863939 8-55 1-2669475 4-17 14279160 4-79 1-5665804 6-41 1-6882491 3-56 1-2G97605 4-18 1-4803112 4-80 1-5686159 5-42 1 -6900058 3-57 1-2725655 4-19 1-4327007 4-31 1-5706971 6-43 1-6919391 3-58 1-2753627 4-20 1-4350846 4-82 1-6727739 5-44 1-6937700 3-59 1-2781521 4-21 1-4374626 4-83 1-6748464 5-45 1-6966163 3-00 1-2809338 4-22 1-4398351 4-84 1-5769147 546 1-6974487 1 3-61 1-2837077 4-23 1-4422020 4-85 1-6789787 6-47 1-6992786 8-62 1-2864740 4-24 1-4445632 1-5810884 6-48 1-7011051 3-68 1-2892826 4-25 1-4469189 4-87 1-5830989 6-49 1-7029282 8-64 1-2919836 4-26 1-4492691 4-88 1-5861452 6-50 1-7047481 8-65 1-2947271 4-27 1-4516138 4-89 1-5871028 5-51 1-7065646 8-66 1-2974681 4-28 1-4639680 4-90 1-5892352 5-52 1-7083778 3-67 1-8001916 4-29 14562867 4-91 1-5912789 5-53 1-7101878 3-68 ■1-8029127 4-30 1-4586149 4-92 1-5938085 6-54 1.7119944 3-69 1-3056264 4-31 1-4609379 1-6953389 5-55 1-7187979 8-70 1-3083328 4-32 1-4632553 4-94 1-59736S8 6-56 1-7156981 1-3110318 4-33 1-4655675 4-95 1-6993675 5-67 1-7173950 a-72 1-3187236 4-34 14678743 4-96 1-6014067 5-58 1-7191887 3-73 1-3164082 4-35 1-4701758 4-97 1-6034198 6-69 1-7209792 8-74 1-3190856 4-36 1-4724720 4-98 1-6054298 5-60 1-7227666 3-75 1-8217658 4-37 1-4747630 4-99 1-6074358 6-61 1-7245507 8-76 1-3244189 1-4770487 5-00 1 -6094379 5-62 1-7268316 8-77 1-8270749 4-39 1-4793292 5-01 1-6114359 5-63 1-7281094 3-78 1-3297240 4-40 1-4816045 502 1-6184300 6-64 1-72988M 8-79 1-8823660 4-41 1-4888746 5-03 1-6154200 6-65 1-7310565 1-8360010 4-42 1-4861396 5-04 1-6174060 5-66 1-73842S8 3-81 1-3876291 4-43 1-4883995 5-05 1-6193882 6-67 1 ■7351601 1-3402504 4-44 1-4906543 5-06 1-6213664 5-68 1-7869612 3-83 1-8428648 4-45 1-4929040 5-07 1-6288408 5-69 1-7387102 8-84 1-3454723 4-46 14951487 5-08 1-6253112 5-70 1-7404661 3-85 1-3480781 4-47 14978883 6-09 1-6272778 5-71 1-7422189 8-86 1-3506671 4-48 1-4996230 510 1-6292405 5-72 1-7439687 3-87 1-3532544 4-49 1-5018527 6-11 1-6311994 6-73 1-7457155 1-8558861 4-50 1-6040774 5-12 1-6881544 5-74 1-7474591 3-89 1-3584091 4-51 1-5062971 6-18 1-6361056 6-75 1-7491998 3-90 ! 1-3609765 4-62 1-5085119 5-14 1-6370630 5-76 1-7609374 b,Google THE PRACTICAL MODEL CALCULATOR, N. I.^earithm, N. ^E="U.m. N. N. Loenritbio. 5-77 17526720 6-39 1 8647342 ToT 1-9473376 7-63 2-0820878 5-78 1-7544086 640 1 8562979 7 02 1-9487682 7-64 2-0333976 6-7S 1-7561823 6-41 1 8578592 7-03 1-9501886 7-66 2-0347066 5-80 1-7578579 6-42 1 8594181 7-04 1-95] 6080 7-66 2-0860119 5-81 1-7895805 6-43 1 8609745 7-05 1-9580275 7-67 2-0873166 1-7613002 6-41 1 8625286 7-06 1-9544449 7-68 2-0366195 5'83 1 ■7680170 6-4S 1 8640801 7-07 1-9558604 7-69 2-0899207 5'84 1-7647308 6-46 1 8656298 7-08 1-9572739 7-70 2-0412203 5-85 1-7664416 6-47 1 8671761 7-09 1-9686853 7-71 2-0425181 o-8ti 1-7681496 6-48 1 8687205 7-10 1-9600947 7-73 2-0438143 5'87 1-7698646 6-49 1 711 1-9615022 7-78 2-0451088 1-7715567 650 1 8718021 7-12 1-9629077 7-74 2-0464016 1-7732559 6-51 1 7-13 1-9648112 7-75 5-90 1-7749523 6-52 1 8748743 7-14 1-9657127 7-76 2-0489823 5-91 1-7766468 6-63 1 8764069 7-15 1-9671123 7-77 2-0502701 5-92 1-7783364 6-54 I 8779371 7-16 1-9686099 7-78 2-0515508 5'93 1-7800242 6-65 1 8794650 7-17 1-9699056 7-79 2-0523408 5'94 1-7817091 6-56 1 7-18 1-9712993 7-80 2-0541237 6'95 1-7833912 6-57 1 8825138 7-19 1-9726911 7-81 2-0654049 S'SG 1-7850704 6-68 1 8840347 7-20 1-9740810 7-82 2-0566845 6-97 1-7807469 6-59 1 8855583 7-21 1-9754689 2-0679624 5-S8 1-7884205 6-60 1 8870696 7-22 1-9768549 7-84 2-0592888 6'99 1-7900914 6-61 1 8885837 7-23 1-9782890 7-86 2-0605135 GOO 1-7917594 6-62 1 8900954 7-24 1-9796212 7-86 2-0617866 (i-Ol 1 -7934247 6-68 1 8916048 7-25 1-9810014 7-87 2-0030580 G-02 1-7950872 6-64 1 8981119 7-26 1-9823798 2-0643278 1-7967470 6-65 I 8946168 7-27 1-9837662 7-89 20666961 6-04 1-7984040 6-66 1 8961194 7-28 1-9861808 7-90 2-0668627 0'03 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 2-0698911 G-07 1-8038686 9006138 7-31 1-9892432 7-93 2-0706580 G-08 1-8050047 6-70 1 9021075 7-32 1-9906108 7-94 2-0719132 09 1-8066481 6-71 1 9085989 7-33 1-8919754 7-95 2-0731719 e-10 1-8082887 6-72 1 9050881 7-84 1-9933387 7-96 2-0744290 6-11 1-8099267 9065751 7 ■35 1-8947002 7-97 2-0766845 e-12 1-8116621 6-74 1 9080600 1-9960699 7-98 2-0769384 613 1-8131947 6-75 1 9096426 7-37 1-9974177 7-99 2-0781907 6-14 1-8148247 6-76 1 9110228 1-9987786 8-00 2-0794415 6-15 1-8164520 6-77 1 9125011 7-89 2-0001278 8-01 2-0806907 6-16 1-8180767 6-78 1 9139771 7-40 2-0014800 8-02 3-0819884 6-17 1-8196988 6-79 1 9154509 7-41 2-0028305 8-08 2-0881845 6-18 1-8218182 9169226 7-42 2-004179O 8-04 2-0844290 6-19 1-8229861 6'81 1 9188921 7-43 2-0065258 8-05 2-0856720 6-20 1-8245498 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 3-0881534 6-22 1-8277699 6-84 1 9227877 7-46 2-0095553 8-08 2-0803918 6-28 1-8293763 6-85 1 9242486 7-47 2-0108949 8-09 2-0906287 6-24 1-8309801 6-86 1 9267074 7-48 2-0122827 8-10 2-0918640 6-25 1-8325814 6-87 1 9271641 7-49 2-0135687 8-11 2-0930984 6-26 1 -8841801 6-88 1 9286186 7-50 2-0149030 8-12 2-0948306 6-27 1-8357768 6-89 1 9300710 7-51 2-0162364 813 2-0966613 6-28 1-8873699 6-90 1 9315214 7-62 2-0175661 8-14 2-0967905 U-29 1-8389610 6-91 1 9329696 7-53 2-0188950 8-15 2-0980182 6-30 1-8405496 6-92 1 9344167 7-54 2-0202221 8-16 2-0992444 6-Sl 1-8421356 6-93 1 9358598 7-55 2-0216475 8-17 2-1004691 6-82 1-8437191 6-94 1 9873017 7-96 2-0228711 8-18 2-1016923 1 -8453002 6-96 1 9387416 7-57 2-0241929 8-19 2-1029140 6-34 1-8468787 696 1 9401794 2-0265131 8-20 2-1041341 6-35 1-8484547 6-97 1 9416162 7-69 2-0268815 8-21 3-1063629 1-8500288 6-98 1 9430489 7-60 2-0281482 2-1065703 6-37 1-8515994 6-99 1 9444805 7-61 2-0294631 8-23 2-1077861 0-38 1-8531680 7-00 1 9469101 7-62 2-0807768 8-24 3-1089998 b,Google TABLE OF HYPSRBOLIC LOQABITHMS. N. toBHlHim. N. Loei,.i,im. N. I^p^lLm. N. Logariam. 8-26 2-1102128 8-69 2-1621729 9-13 2-2115056 9-57 2-2586882 8'2d 2-IU4243 8-70 2-1633230 0-14 2-3126603 9-58 2-2596776 8-27 2-1126348 8-71 2-1644718 9-16 2-2187688 9-69 2-2607209 8-28 2-1188428 8-72 2-1656192 9-16 2-2148461 9-60 2-2617631 8.29 2-1150499 8-73 2-1667653 9-17 2-3159372 9-61 2-2638042 8-80 2-1162555 8-74 2-1679101 9-18 2-2170272 9-62 2-2688442 B'31 2-1174596 8-75 2-1690536 9-19 2-2181160 9-63 2-2648832 8'82 2-1186622 8-76 2-1701959 9-20 3-2192034 9-64 2-2669211 8-33 2.n98684 8-77 2-1713367 9-21 2-2202898 9-65 2-2669579 884 2-1210633 8-78 2-1734763 9-22 2-2218750 9-66 2-2679986 8-85 2-1222616 8-79 2-1786146 9-23 2-2224590 9-67 2-2690282 2-1284584 8-80 21747617 9-24 2-2235418 9-63 2-2700618 8-87 Z-1246539 8-81 2-1758874 9-25 2-2246386 9-69 2-2710944 8-38 2-1258479 2-1770218 9-26 2-2267040 9-70 2-2721258 8-S9 2-1270405 2-1781550 9-27 2-2267833 9-71 2-2781562 8-40 2-1282317 8-84 2-1792868 9-28 2-2278615 9-72 2-2741856 8-41 2-1294214 885 2-1804174 9-29 2-2289S86 9-78 2-2752138 8-42 2-1806098 8-86 2-1815467 9-80 2-2300144 9-74 2-2762411 848 2-1317967 8-87 2-1836747 9-31 2-2310890 9-76 2-2772673 8-44 2-1329822 2-1838015 2-2321626 9-76 2-3782D34 8-45 2-1341664 2-1849270 933 2-2332850 9-77 2-2798165 8-46 2-1353491 8-90 21860512 9-34 2-2348062 9-78 2-3803395 8-47 2-1865304 8-91 2-1871742 9-86 2-2853763 fi-79 3-28136U 8-48 2-1377104 892 2-1882959 2-2364452 9-80 8-4S 2-1388889 8-93 2-1894163 9-37 2-2875180 9-81 2-2834023 8-50 2-1400661 8-94 2-1905356 9-38 2-2885797 2-2844211 8-51 21412419 8-95 21916535 9-39 2-2896452 9-83 2-2854S89 8-52 2-1424163 8-96 2-1927702 9-40 2-2407096 9-84 2-2864556 8-58 2-1435893 8-97 2-1988856 9-41 2-2417729 2-2874714 8'54 2-1447609 8-98 2-1949998 9-42 2-2428850 9-86 2-2884861 8-55 2-1469812 8-99 2-1961128 943 2-2438960 9-87 2-2894D98 8-56 2-1471001 9-00 2-1972245 9-44 2-2449559 3-2905124 8-57 2-1482676 9-01 21983360 e-45 2-2460147 2-2916241 8'58 2-1494339 9-02 2-1994443 9-46 2-2470723 9-BO 2-2925347 8'59 2-1506987 9-08 2-2005523 9-47 2-2481288 9-91 2-2685448 8-6l> 2-1617622 9-04 2-2016591 9-48 2-2491843 2-3945329 8-61 2-1529243 9-05 2-2027647 9-49 2-3502386 9-93 2-2966604 8-62 2-1540851 9-06 2-2088691 9-50 3-2512917 9-94 2-2965670 8-63 2-1652446 9-07 2-2049722 9-51 2-2628438 fi-95 2-2076725 8-M 2-1664026 908 2-2060741 9-52 2-2583948 9-96 2-2985770 8-65 2-1575598 909 2-2071748 9-53 2-2544446 9-97 2-2995800 8-66 2-1587147 9-10 2-2082744 9-54 3-3554934 9-98 2-8005881 8-67 2-1598687 9-11 2-3098727 9-65 2-2665411 9-99 2-3015846 8-68 21610216 9.12 2-2104607 9-56 2-2575877 1000 2-8025851 Logarithms were invented by Juste Byrge, a Frenchman, and not by Napier. See " Biographie Universelie," " The Calculus of Form," article 822, and " The Practical, Short, and Direct Method of Calculating the Logarithm of any given Number and the Number corresponding to any given Logarithm," discovered by OliveirEyrno, the author of the present work. Juste Byrge also invented the proportional compasses, and waa a profound astronomer and ma- thematician. The common Logarithm of a number multiplied by 2 '30 2585052 994 gives the hyperbolic Logarithm of that number. The common Logarithm of 2-22 ia -346353 .-. 2-302585 X -346355 = -7975071 the hyperbolic Logarithm. The application of Loga- rithms to the calculations of the Engineer ivill be treated of here- after. hv Google IS4 THE PHACTICAL MOIJEL CALCULATOR. COMBINATIONS OF ALGEBRAIC QUANTITIES. The following practical examples will serve to illustrate the method of combining or representing numbers or quantities alge- braically ; the chief object of which is, to help the memory with respect to the use of the signa and letters, or symbols. Let a = 6, 6 = 4, c = 3, (? = 2, e = 1, and/ = 0. Then will, (1) 2» + S - 12 + 4 =. 16. (2) «6 + 2« - li = 24 + 6 - 2 = 28. (3) o- - 6' + « +/- 36 - 16 + 1+ = 21. (4) i> X (o - 6) = 16 X (6 - 4) - 16 X 2 - 32. (5) iabe - Ide - 216 - 14 - 202. (6) 2 (o - i) (So - U) - {12 - 8) X {16 - 4) = 44. (7) ^^ X (a - o) - 1^ X (6 - 3) = 4 X 3 = 12. (8) V {«■ - 2S') + (i -/- ,/ (36 - 32) + 2 - = 4. (9) 3«S - (« - 6 - + <i) = 72 - 1 - Tl. (10) 3<iS - (o - S - c - ci) = 72 + 3 = 75. (");7^^)'<(« + ^) = ;n^4'^Tj'<P + ^) = i^- 111 solving the following questiona, the letters a, h, c, &c. are supposed to have the same values as before, namely, 6, 4, 3, tc; hut any other values might have been assigned to them ; therefore, do not suppose that a must necessarily be 6, nor that 6 must be 4, for the letter a may be put for any known quantity, number, or magnitude whatever ; thus a may represent 10 tniles, or 50 pounds, or any number or quantity, or it may represent 1 globe, or 2 eubio feet, &c. ; the same may he said of b, or any other letter. {!) a + b-o = 7. (2) m~d + e = 35. (3) 2a^+ e''-d+f= 79. {i)jx{b-c + d) = 21. (5) 5.'(i - a= + Me = 62. In the use of algebraic symbols, 3 -v tiling as 3 (4(i — 6)'^. 4(c + df [a + i)^, or 4 X e + d' X a + b^, signifies the same thing as 4 V c -\- d • ■$' a + b. (8) 4 („. _ J.) (c - ,) -160. m ^■m-^'-. -62. (8) »/(2(." + 2<p) + fc- -/ = 20. (9) icfb -{d'-d-e 1 - 670. ^ia' - »'= 0. 10) s/tlOd^ — 4«(;j ^c? », 3 ^ 4a — 6 signifies ■ the same b,Google THE STEAM ENGINE. The partienlar esample whieli we sfiall select is tliat 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 m inches by 1'561, and then multiply the sqnare of the diameter of cylinder in inches by '12B5 ; 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 wo 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. and s/4102-3 = 64-05 nearly. 4096 = square of the diameter of the cylinder. 45)_ 5829-97 and -g/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. hv Google ISb THE PRACTICAL MODEL CALCULATOK. Rule. — To find the thickness of large eye of crank. — Multiply the square of the length of the crank in inches by 1*561, and then mul- tiply the square of the diameter of the cyliuder in inches by -ISSo ; multiply the sum of these products by the square of the diameter of the cylinder in inches ; afterwards, divide the prodnct 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^' 4102-3 4096 = square of diameter. 48 ) 16803020-8 1828-28 ) 350062-94 191 -4T and -^191-47 = 5-7t 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 S-OT inches. Rule. — To find the tJdeJcness of the weh of erank 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 hj 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 wc have selected, we have hv Google THE STEAM ENGINE. 48 = length of crank in inches. 2304 1*501 = constant multiplier. 4102-3 64 = diameter of cylinder. 64 4096 1235 = constant multiplier, 505-8 Ana V 4102-3 = 64-05 nearly. 4096 = square of diameter. 728-75 And ^ 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. — Mul- tiply 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 diame- ter of the paddle shaft journal in inches. Thus, to apply this rule to the particular example which we have before selected, wo 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 paddlo 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 inchea. 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 tho cylinder in inchea 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 hv Google ISS THE PEACIICAL MOJJEL CALCL"LATOE. paddlo shaft journal in inches. (The length of the paddle shaft journal is IJ times the diameter.) To apply this rule to the example which we have eclected, we have 64 = diameter of cylinder in inches. 64 4096 48 = length of crank in inches. 196608 and ^ 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 lOj inches. Breadth of the eye of cross-head = 2-64 inches, or between 2J and 2} inches. Depth of the eye of cross-head = 18-37 inches, or very nearly 18J inches. Diameter of the journal of cross-head = 5-5 inches, or 5J inches. Length of journal of cross-head = 6-19 inches, or very nearly Q} inches. Thickness of the web of cross-head at middle = 4-6 inches, or somewhat more than 4| 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 veiy nearly 4 inches. Breadth of web of cross-head at journal = 6-46 inches, or nearly 6^ inches. Diameter of piston rod = 6'4 inches, or 6| inches. Length of part of piston rod in piston = 12-8 inches, or 12i inches. Major diameter of part of piston rod in cross-head = 06-8 inches, or nearly 6^ inches. Minor diameter of part of piston rod in cross-head = 5-76 inches, oP 5| 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 7i and 7J inches. Depth of gibs and cutter through cross-head = 6-72 inches, or very nearly 6| inches. Thickness of gibs and cutter through cross-head = 1-35 inches, or between IJ and 1^ inches. hv Google THE STEAM EKGIiSE. 139 Depth of cutter through piston = 5'45 inches, or nearly 5^ inches. Thickness of cutter through piston = 2*24 inches, or nearly 2| inches. Diameter of connecting roct at ends = 6'08 inches, or neai-ly 6i\, inches. Major diameter of part of connecting rod in cross-tail = G'27 inches, or about 6J inches. Minor diameter of part of connecting rod in cross-tail = 5'76 inches, or nearly 5| 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 2J inches. Mean thickness of strap above cutter = 2-06 inches, or some- what more than 2 inches. Distance of cutter from end of strap = 3-08 inches, or very nearly 3^ inches. Breadth of gibs and cutter through cross-tail = 6-73 inches, or very nearly 6| inches. Breadth of gibs and cutter through hutt = 7-04 inches, or some- what more than 7 inches. Thickness of gibs and cutter through hutt = 1'84 inches, or between If and 2 inches. These results are calculated from the following rules, which give correct results for all sizes of engines. Role 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 crosa-fteat?.— 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 -OSB. The pro- duct is the diameter of the journal in inches. Rule 6. To find the length of the journal of cross-head. — Mul- tiply 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. hv Google 140 THE PRACTICAL MODEL CALCULATOR. EuLB 9, To find the thickness of the web of cross-head at Journal. — Multiply the diameter of the cylinder in inches by 'Ofil, The product is the thickness of the weh of cross-head at journal in inches. Rule 10, To find the breadth of weh of cross-head at journal. — Multiply the diameter of the cylinder in inches by 'lOl. 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. Tho 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 inchea. 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 tho minor diameter of the part of piston rod in erosa-head in inchea KuLB 15. To find the major diamett r cf the pa} t ofp/sl n j od in piston. — Multiply tho diimetei of the cylinder in inches by 14. The product is the major diameter of the pvrt of pistDU lod in piston in inches, Rule 16. To find the minor diamttrr ot the pat t of piste n rod in piston. — Multiply the diimctor of the cylinder in inches by 115. The product ia the minor diimeter of the pait of piston lod in piston. Rule 17. To find the depth of gibi and cuttet though cioss- head. — Multiply the diameter of the cylinder in inches by 105. The product is the depth of the giba and cuttei thiough cross- head. Rule 18. To find the thukness of the qihs and cutter through cross-head. — Multiply the dnmetci of the cylinder m inchea by •021. The product is tho thickness of the gibs and cuttez thiough cross-head. Rule 1&. To find the dipth of euttt.r through pistm — Multiply the diameter of the cylinder m inches by 08 j The product la the depth of the cutter thiough piston m inches Rule 20. To find the Uinkness of cutter through putrn — Mul- tiply the diameter of the cylinder m inchea by 035 The product is the thickness of cutter through piston m inches Rule 21. To find the diameter of connecting rod at end'i- — Mul- tiply the diameter of the cylinder m inches by 0'>5 The proiuet is the diameter of tho connecting rod at ends in inches Rule 22. To find the majir diameter if the pait cf onnednig rod in cross-tail — Multiply the diametet of the cybndei m inches hv Google THE STEAM EKOINE. 141 bj -098. The product is the major diameter of the part of con- necting rod in cross-tail. Rule 23. To find the minor diameter of the part of conneeting rod in cj-oss-tmi.— Multiply the diameter of the cylinder in inches by '09. The product is the minor diameter of the part of con- necting 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 inchea. BuLE 25. To find the tMeknese of the butt. — Divide the diameter of the cylinder in inches by 8. The quotient is the thickness of the butt in inches. BuLE 26. To find the mean thickness of the strap at cutter. — Multiply the diameter of the cylinder in inchea by -043. The pro- duct is the mean thickness of the strap at cutter. Rule 27. To find the mean thickness of the strap above cutter. — JIuItiply 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 tho 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 cj-oss-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 -ll. 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. TJie 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. hv Google 142 THE PRACTICAL MODEL CALCULATOR. Theae results are calculated from the following rules, which give the proper dimensions for engines of all sizes : Kl'LE 1. To find the breadth of the small eye of crank. — Multiply the diameter of the cylinder iu inches hy '063. The product ia 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 'IST. The product is the proper length of the small eye of crank in inches. Rule 3. To find the thickness of the weh of crank at pin centre.— Multiply the diameter of the cylinder in inches by '11. The pro- duct is the proper thickness of the web of crank at pin centre in inches. Rule 4. To find the breadth of the web of erank' at pin centre. — Multiply the diameter of the cylinder in inches by •16. The pro- duct is the proper breadth of crank at pin centre in inches. To illustrate the use of the succeeding rules, let us take the par- ticular 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 = 9'144 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,'^ inches. Breadth of butt = 4-93 inches, or very nearly 5 inches. Thickness of butt = 3'9 inches, or 3^ inches. Mean thickness of strap at cutter = 2-06 inches, or a little more than 2 inches. Mean thickness of strap below cutter = 1'47 inches, or vcry nearly 1^ inches. Depths of gibs and cutter = 5'12 inches, or a little more than 5^ inches. Thickness of gibs and cutter = 1-03 inches, or a little more than 1 inch. Diameter of main centre journal = 11-71 inches, or very neaily 11 J inches. Lengdi of main centre journal = 17'6 inches, or 173 inches. hv Google THE STEAM ENGINE. 143 Depth of eye round end studs of lover = 4-75 inches, or 4^- inelies. Thickness of eye round end studs of lever = 3'33 inches, or 3J inches. Diameter of end studs of lever = 4'48 inches, or very nearly 4^ inches. Length of end studs of lever = 4'86 inches, or betireen 4f and 5 inches. Diameter of air-pump studs = 2-91 inches, or nearly 3 inches. Length of air-pump studs = S*16 inches, or nearly 8] inches. These results were obtained from the folloiving 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. — JIultiply the diameter of the cylinder in inches by '060. 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 thiakness of strap at cutter. — Mul- tiply 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. — Mul- tiply tho 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 findthe thickness of gibs and cutter. — Multiply the diameter of the cylinder in inches by •016. The product ia the thickness of gibs and cutter in inches. Rule 8. To find the diameter of the main centre journal. — Mul- tiply the diameter of the cylinder in inches by -ISS. The product is the diameter of the main centre journal in inches. Rule 9, To find the length of the main centre journal. — Multiply 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 pro- duct 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. — Mul- tiply the diameter of the cylinder in inches by ■07. The product is the diameter of the end studs of lever in inches. Rule 13. To find the length of the end studs of lever. — Multiply hv Google 144 TUG PRACTICAL MODEL CALCrLATOR. tlio iliamctcr of the cylinder in inches by '076. The product is the length of the end studs of lever in inches. UuLB 14. To find the diameter of the air-pump sfuds. — 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 length of the air-pump studs. — Multiply the diameter of the cylinder in inche^by '049. The product is the length of the air-pump studs in inidjes, The next rule gives the proper o^th 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 fength 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 proper dimensions for any size of engine : Rule, — To find the depth aeross^e centre of the side lever. — Multiply the length of the side lever In feet by -7428 ; extract the cube root of the product, and reserve the result for a, multiplier. 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 ive have selofted, we have 20 = length of side lever in feet, •7423 = constant multigiier. 14-846 and -^ 14-S4(> = ^-ISS nearly. 64 = diameter of wlinder in inches. 64 4096 and ,y40M ^16 Hence depth at centre = 16 x 2-458 inches = 30-3-j inches, or between 39j and 39i^ inches. Tlie next set of rules give the dimensions of eeffef.al of the parts of the air-pump machinery which depend upon the ^#etei- of the cylinder only. To illustrate the use of these rules, 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 = 3S-4 inches, or 38f inches. hv Google THE STEAM ENGINE. 145 Thickness of the eye of air-pump cross-head = 1'58 inches, or a little more than 1^ 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 3 J inches. Length of end journals of air-pump cross-head = 3'7 inches, or 3^ inches. Thictness of the web of air-pump cross-head at middle = 2-T6 inches, or a little more than 2J inches. Depth of web of air-pump cross-head at middle = 10'29 inches, or somewhat more than lOJ inches. Thickness of web of air-pump cross-head at journal = 2-35 inches, or about 2f inches. Depth of web of air-pump cross-head at journal = 3'89 inches, or about 3| inches. Diameter of air-pump piston rod when made of copper = 4-27 inches, or about 4J 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 | inch. Depth of cutter through piston = 3-27' inches, or somewhat more than 3J inches. Thickness of cutter through piston = 1'34 inches, or about If inches. These results were obtained from the following rules, and give the proper dimensions for all sizes of engines : Rule 1. To jind 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. — Mul- tiply 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 cross- head. — 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 jour- nals 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 tlie web of air-pump cross- head. — iMultiply the diameter of the cylinder in inches by -IGl. hv Google 146 THE PRACTICAL MODE!. CALCULATOR. The product is the depth at middle of air-pump cross-head Id inches. Rule 8, To find thp thtahiess cf the ueb of air pump cross head atjoumaU — Multijlj the diameter of the cylinder in inches by "037. The proiuct is the thickness of the i^eb of air pump cross-head atjournals m mchea Rule 9. To fitil the dfpth of the air pump u Iiead web at journals. — Multiply the diimetei of the cylinder in inches by 061 The product ia the depth at jourmla of the web of an pump cross head. Rule 10. To Und the dumef^r cf tht, air pump pM n rod lihen of copper. — Multiply the diameter of the cylindei in inches by •067. The product is the diameter of the air pump piston lol, when of copper, in mchea Rule 11. To find tie d ].th ofgih and cittttr though air pump cross-head. — Multiply the diameter of the cylinder in inches by •063. The product is the depth of the ^ibs and cutter through air pump cross-head lu inches Rule 12. To find the fhiokness of the gihs and cutt t through air-pump cross-htai — Multifly the diameter of the cylm lor m inches by -013. The product is the th cLness of the gibs an 1 cutter in inchea. Rule 13. To in I 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 piaton in inches Rule 14. To pnd the tMrlness of eutt r though au pump piston. — Multiply the diameter of the cylinder in inches by 021 The product is the thickness uf 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 dimen- sions 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 ia 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 -6.3 inchea. 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 -OSS. The product is the diameter of the air-pump side rod at ends in inches. Rule 2, Tofindthe breadth of butt for air-pump. — Multiply the 3gle THE STEAM ENGINE. 147 diameter of the cylinder in inches by -046. The product is the breadth of butt in inches. BuLE 3. To find the thickness of hutt for air-pump. — Multiply the diameter of the cylinder in inches by '037. The product is the thickness of hutt for air-pump in inches, KuLE 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 thicknegs of strap below cutter. — Mul- tiply the diameter of the cylinder in inches by 044. 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 hy 0-48. The product is the depth of gibs and cutter for air-pump in inches. Rule 7. Tofindthe thickness of giba 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 tbe engine: — Suppose that we take the particular example of an engine whose stroke is 8 feet, and dia- meter 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 4f 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 2f 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 sqnare inches = 8-37, or about 8f square inches. Sectional area of eccentric hoop in square inches = 8-75, or SJ sqnare inches. Diameter of eccentric pin in inches = 2'24, or 2J inches. Breadth of valve lever for eccentric pin at eye in inches = 5'7, or very nearly 5f inches. Thickness of valve lever for eccentric pin at eye in inches — S. Breadth of parallel motion crank at eye = 4'2 inches, or very nearly 4J inches. Thickness of parallel motion crank at eye = 1'76 inches, or about If 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. Accord- ing to the rule, ttie area of each steam port would be 202'26 square inches. "With regard to the rule, we may remark that the area of the hv Google 148 THE PRACTICAL MODBL CALCULATOR, Bteam port ought to depend principally upon the cnhical 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 atroke of the engine. It is well known, however, that the quantity of steam admitted by a amall hole does not bear so great a proportion to the quantity admitted by a larger one, aa 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 tho length of the stroke in feet ; multiply this product hy 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 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 cir- cumstance 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.— K) find the diameter of Iranah steam pipe. — Multiply together the square of the diameter of the cylinder in inches, the hv Google THE STEAM ENGINE. 149 length of tlie stroke in feet, and "00498; to the prodact 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. 82768 ■00498 = constant multiplier. 163-18 10'2 = constant to he added. 173-38 and v' 173-38 = 18-16. 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 bo 174-3. Hence, according to the rules, we have in succession. Diameter of waste water pipe = 15-87 inches, or between 15f 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 bo about 4f inches. Diameter of feed pipe == 4-12 inches, or between 4 and 4J inches. Diameter of waste steam pipe = 12-17 inches, or nearly 12J inches. Diameter of safety valve, When one is used =14-05 inches. When two are used = 9-94 inches. When three are used = 8-1^ inches. When four are used = 7-04 inches These results were obtained from the f illowmg rules, which will give the correct dimensions fo» all sizes of engines Rule 1. To jind the diamitet o/ nante uater pipe. — Multiply the square root of the nominal horse powei of the engine by 1-2. The product is the diameter of the waste water pipe m inches. Bulb 2. To Jind the area of foot valve passage — Multiply the hv Google 150 THE PRACTICAL MODEL CALCULATOR. nominal horse power of tlic engine by 9 ; divide the product by 5 ; add 8 to the quotient. The sum ia the area, of foot-vaive passage in square inches. BuLE 3. To find the area of injection pipe. — Multiply the nomi- nal 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 nomi- nal 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. Bulb 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 safett/ 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 when four are waeci. -^Multiply the collective nominal horse power of the engines by -125 ; to the product add 5-625 ; extract the square root of the snm. 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. Accord- ing 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 di- ameter 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. hv Google THE STEAM ENGINE. 151 To illustrate this rule we shall take the particular example of an engine wliose main beam is 20 feet long, and the diameter of the cjlinder 64 inches. In this case we have 64 = diameter of cylinder in inches. 64 4096 = square of the diameter. 10 = I length of main beam in feet. 40960 3 = constant multiplier. 122880 ____ 122880(49-714 = ^122880 4 16 64 4 16 58880 8 4800 6231 4 1161 5112 120 5961 119 9 1242 74 129 7203 36 9 10 138 130 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 to- gether 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 = J length of main beam in feet. 4096? '192 = constant multiplier. 7864-32 hv Google 152 THE PEACTICAL MODEL CALCOLATOR, 7864-32 ( 19-89 = -^TSM^ 1 1 1 1 1 6864 1 2 5859 "a 300 1005 X 351 _898 30 651 lOT 9 ^32 "39 1083 _9 J_ 48 112 _9 J_ 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. Sup- pose 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 te 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 -i- 24 = 45-556 square inches; so that we conclude that the diameter is between 7J and 7f 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 feed- pump must be equal to 1093-36 -r- 19-635 = 55-69 inches, or very nearly 55| inches. This rule will be found to give correct dimensions for all sizes 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 cylin- der 50 inches, we have, 50 = diameter of cylinder. 50 2500 = square of the diameter of the cylinder. 6 = length of stroke in feet. Sp y 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- hv Google 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 -4- 5 = I'ii) square feet = 214-56 square inches; we see that the diameter of the pump is about 16J inches. Agaiu, suppose that the diameter of the cold-water pump was 20 inches, and that it waa required to find the length of its stroke. The area of the pump is 314-16 square inches, or 314'16 -i- 144 = 2-18 square feet; so that the stroke of the pump is equal to *r'45 -;- 2-18 = 3-42 feet. The content is calculated from the following rule, which will he 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 5^ feet, and diameter of cylinder tiO inches. In this case we have in succession, 60 = diameter of cylinder in inches. _60 ' S600 = square of the diameter of cylinder. 5J = 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 some- t'm t a tl h ft If course the thickness of the large y h n g when the crank is only keyed on the h f h h ho large eye at all. To illustrate the u f 1 ul w 1 11 pply it to the particular example of an n B wl k 8 f and diameter of cylinder 64 inches. H n d he ul he proper thickness of the large eye f nk 1 n f n 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 5j^ inches. The difference is thus about 2 J 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 thieJcness of the large eye of crank for fly- wheel shaft when of cast iron. — Jlultiply 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, hv Google 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 s,lready selected. 48 = length of crank in inches. 48 2304 = square of length of crank in inches. 1'561 = constant multiplier. 3596^ 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 ^525-397 = 8-07 nearly. To find the hreadth of the web of crank at the centre of the fly- wheel shaft, that is to say, the hreadth 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^ 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 : TiXJiiS. — To find the breadth of the web of cranh at fiy-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 hj show- hv Google THE STEAM ENOINB. 155 ing its application to the particular example of an cnfjine whose stroke is 8 feet, and diameter of cylinder 64 inches. 64 = diameter of cylinder in inches. 64 4096 = square of the diatoeter 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^ [cylinder. 1138 6-66 nearl y, and -5' 11386-66 = 22-49. To determine the thickness of the weh 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 weh 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, tho proper thickness would be 11'26 inches. The proper thickness of web at centre of paddle shaft for a marine engine whoso stroke is 8 feet, and diameter of cylinder 64 inches, is nearly 9 inches. The difi'erence between the two thicknesses is about 2^ inches, which is not too great an allow- ance 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 jiy-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 ; multi- ply 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 ia 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 apply- ing it to the particular engine which we have already selected. hv Google THE PRACTICAL MODEL CALCULATOR. 48 = length of crank in inctes. 48 2304 = square of length of crank. 1-561 = constant multiplier. 3596^ 64 = diameter of cylinder In inches. 64 4096 = square of the diameter of cylinder. 1235 = constant multiplier. 505-8 3596-5 4102-3 = sum and -v/ 4102-3 = 64-05 nearly. 4096 = squaro of diameter. Constant divisor = 184-32) 262348-5 1423-33 and ^ 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 tho diameter of the fly-wheel shaft for au engine whoso 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 tho 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 riJe, the diameter of the paddle shaft journal of a marine engine whose stroke is 8 feet, and dia- meter of cylinder 64 inches, is about 14 inches. The difl'erenee 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. hv Google THE STEAM BNaiNE. 64- 64 4096 - 48 - diameter of cjliucler in inches. square of tlie diameter, length of crank in inches. 196608 5 25 196608 (68-15 - ^ 196608 126 5 6 10 6 150 8 168 25 50 7500 1264 8764 1328 10092 71608 70112 1496 1011 485 166 1011 8 2 174 1013 and 58-15 X -3025 = 1759 Tfhich agrees ivith the number given bj 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 SIS = 119'03. We may remark that this calculation has heen made on the supposition that the fly- wheel 18 BO connected with the engine, as to make exactly one revo- lution 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 cast 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 fiy-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. We shall apply the rule to determine the sectional area of the rim of fly- hv Google 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 heing 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 6'125 = constant multiplier. 1960000 therefore sectional area in square inches = 1960000 -i- 20^ = 1960000 -^ 8000 = 1960 h- 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 n.p. MARINE ENGINES. — DIMENSIOXB OE SEVERAL OF THE PARTS OP 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 = D^ x {•7423 X length of lever in feet} i OF AIR-PUMP 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 = -l?! x D. Diameter of end journals, in inches = -OSl 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 = -ISl x D, Thickness of web at journal = -037 x D. Depth of web at journal = -061 X D. Diameter of air-pump piston-rod, when of copper, in inclies = 067 X D. Depth of gibs and cutter through cross-head, in inches = ■063 X D. hv Google TDK STEAM ENGINE. 15S Thickness of gibs and cutter through crosa-head, iir inches = 013 X D. Depth of cutter through piston, in inches = '051 x D. Thickness of cutter through piston, in inches = '021 x J). PARTS OF THE Diameter of air-pump side rods at ends, in inches = '039 x D. Breadth of butt, in inches = '046 X D. Thickness of hutt, in inches = '037 x D. Mean thickness of strap at cutter, in inches = 'Olft 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 I.AND EN0IKE9, — AREA OF STEAM PORTS. Area of each steam port, in equare inches = 11 x ^ X D^ -i- 1800 + 8. MARINE AND LAND ENGINES. — DIMENSIONS OP BRANCH STEAM PIPES. Diameter of each branch steam pipe = ^/ -00498 x I X D^ X 10-2. MARINE ENGINE.- Diameter of waste .-water pipe, in inches = 1'2 X x/ 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. -f- 3. Diameter of waste steam pipe in inches =v''375xn.P. -I- 16-875. MARINE AND LAND ENGINES. — DIMENSIONS OF SAFETY-VALVES. Diam. of safety-valve, when one only is used =n/-5xH.P.-|-22'5. Diam. of safety-valve, when two are used = s/'SSxH.P.-Ml-aS, Diam. of safety-valve, when three are used = v'"167xH.P.+7-5. Diam, of safety-valve, when four are used =s/-125 X H.P. -f 5-625. LAND ENGINE. — DIMENSIONS OP MAIN REAM. Depth of web of main beam across centre = ■^ 3 X D^ X half length of main beam in feet. Depth of main beam at ends = ■^ -192 X D^ X half length of main beam, in feet. LAND AND MARINE ENGINES. ^CONTENT OP PEBD-PUMP. Content of feed-pump, in cubic inches = D' x Z -^ 30. LAND ENGINES. — CONTEST OF COLD WATER PDMP. Content of cold water pump, in cubic feet = D^ x i -^ 4400. hv Google .60 THE PKACTIOAL MODEL CALCULATOR. LAND ENCilNES. — DIMENaiONS OF CRANK. ThicknGSS of large eye of crack, in inches ^ ^~D^~x~"(r56rx~:^T^1235 D') -^ (K x 666-283}. Breadth of weh of crank at fly-wheel shaft centre, in inches = ^D"^ X •/ (1-561 X R^ + -1235 x D=) -^ 23-04. Tbieknesa of Treb of crank at fly-wheel shaft centre, in inches = ^nD^lw^(r56r^rRM^4235^ri)y=i^l84^. lAND ENGINES.— DIMENSIONS OF FLY-WHEEL SHAri. Diameter of fiy-wheel shaft, when of cast iron = 3025 x ^BxD'. DIMENSIONS or PAETS OF LOCOMOTIVES. DIAMETER OF CYLINDER. In locomotive engines, the diameter of the cylim^er 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 IKDIrCTION PORTa. Rule, — To find the size of the steam ports for the locomotive engine. — Multiply the square of the diameter of the cylinder hy ■068. The product ia 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 porta = -068 x 15 X 15 = -068 X 225 = 15-3 square jnehes, or between 15^ and 15^ square inches. After having determined the area of the porta, we may easily find the depth when the length ia 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 -h 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 -r- 2 = 7'65 inches, or nearly 7| inches. AREA OP EDUCTION PORTS, The proper area for the eduction porta may be found from the fol- lowing 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 porta of a locomotive engine, hv Google THE STEAM ENGINE. 161 when the diameter of the cylinders is 13 inches. In this example we have, according to the rale, Area of eduction port = -128 x 13= = -128 x 169 = 21-632 inches, or between 21J and 21f square inches. BaEADTH OF BRIDGE BETWEEN PORTS, The breadth of the hridgea between the eduction port and the induction ports is usually between f inch and 1 inch, DIAMETER OP BOILER. It is obvious that the diameter of the boiler may vary very con- Bidcrably ; 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 10| inches. tENGTU OP BOILER. The length of the boiler is usually in practice between 8 feet and ^ 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 dume 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 J inches. HEIGHT OE STEAM DOME. The height of the steam dome may vary. Judging from prac- tice, it appears that a uniform height of 2-J feet would answer very well. hv Google 162 THE PRACTICAL MODEL CALCULATOR. rrAMETER OP SAPETY- 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 jind the diameter of the safety-valve. — Divide the diameter of the cylinder in inches by 4. The quotient is the dia- meter 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 J 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 ia entirely founded on practice. Rule. — To findthe 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 or FIRE-r.RATil. The following rule determines the area of the fire-grate usually given in practice. Wo may remark, that the area of the fire-grate in practice follows a more certain rule than any other part of the cDgine 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. Thfere 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 areaof the fire-grate, — Multiply the diameter of the cylinder in inches by '77. The product is the area of the fire- grate 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 llj^ square feet. Though this rule, however, represents hv Google THE STEAM ENGINE. 163 the usual practice, the area of the fire-grate shouUl not be contingent upon the size of the cylinder, but upon tho quantity of steam to be raised. AREA OP UEATING SOUrACE. 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 bnlk and weight aa possible. This object is admirably ac- complished in the construction of the boiler of the locomotive en- gine. This little barrel of tubes generates more steam in an hoar than was formerly raised from a boiler and fire occupying a eon- siderable house. This favourable result is obtained simply by ex- posing 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 sur- face of a locomotive boiler may bo considered to be the four faces of the inside fire-box, plus the sura 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. Htjle. — 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 = 15^ x 5 -J- 2 = 225 x 5 -^ 2 = 1125 -h 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 llj 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 ie 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 If inches ; and therefore the circumference of a section of these tubes, ac- cording to the table, is 6-4978 inches. Hence, supposing the tube to be 8| feet long, the surface of one = 5'4978 x 8J ^ 12 = ■45815 X 8^ = 3-8943 square feet ; and, therefore, the number of tubes = 478-5 -h3-8943 = 123 nearly. The amount of heating sur- face, 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. hv Google 164 THE PRACTICAL MODEL CALCliLATOB. ABEA OF TVATER-LEVEL. This, of course, varies with the different circumstaneea of the boiler. The average area may be found from the following rule. Rule. — To find the area of the water-level. — Multiply the diame- ter 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 thia case we have, according to the rule, Area of water-level = 14 x 2-08 = 29-12 square feet. CTJBICAL CONTEST OF WATER IN B01T.EE. 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 loiter. — Mnltiply 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 wo have, according to the rule, Cubical content of water = 9 X 14' -;- 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 pis- ton. 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. Rdle, — To find the diameter of the feed-pump ™m..— Multiply the square of the diameter of the cylinder in inches by -Oil. The product is the diameter of the ram in inches. Required the diameter of the ram for the feed-pump for a loco- motive engine whose diameter of cylinder ia 14 inches. In this example we have, according to the rule, Diameter of ram = -Oil x 14' = -Oil x 196 = 2-156 inches, or between 2 and 2J inches. COBICAL CONTENT OE STEAil 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. hv Google THE STEAM ENGINE. 165 But as it is desirable that it should he increased, we give the fol- lowing rule. Rule. — To find the cubical content of tfi^ 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 cubic- 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 12^ h- 40 '^ 9 X 144 h- 40 = 32-4 cubic feet. CUBICAL CONTENT OF INSIDE PIRE-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-hox above fire- bars. — Divide the square of the diameter of the cylinder in inches by 4, The quotient is the content of the inside fire-box above fire- bars in cubic feet. Required the content of inside fire-box above fire-bars in a loco- motive engine, when the diameters of the cylinders are each 15 inches. In this example wo have, according to the rule. Content of inside fire-box above fire-bars = 15^ -;- 4 = 225 -s- 4 = 56^ cubic feet. THICKNESS OP THE PLATES OP BOILER. In general, the thickness of the plates of the locomotive boiler is I inch. In some cases, however, the thickness is only f, 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 loeomotivp 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 13= = -03 x 169 = 5-OT inches ; or a very little more than 5 inches. The steam pipe is usually made too small in engines intended for high speeds. DIAMETER OE BEANCH 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 lo- comotive engine. — Multiply the square of the diameter of the cylin- der in inches by -021. The product is the diameter of the branch steam pipe for the locomotive engine in inches. hv Google IbQ TIIH PRACTICAL MODEL CALCULATOR, Rcquiruil the diameter of the branch steam pipes for a locomo- tive engine, when the cylinder's diameter is 15 inches. Here, ac- cording to the rule, diameter of branch pipe = ■021 x 15' = -021 x 225 = 4-726 inches, or about 4f inches. DIAMETER OF TOP OF BLAST PIPE. The diameter of the top of the blast pipe may he found from the following rule. Rule. — To find the diameter of the top of the blast pipe. — Mul- tiply 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 13= = -017 X 169 =2-873 inches, or between 2^ and 3 inches ; but the orifice of the blast pipe should always be made as large as the demands of the blast will permit. DIAMITER 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 tbo 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 example 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 onr gnide, therefore, we have the following rule. Rule. — To find the diameter oftkepiston rod for the loeomotive 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 = 2| inches. TUICKNEKS OP 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 thichness of the piston in the locomotive en- jme.— Multiply the diameter of the cylinder in inches by 2 ; divide hv Google THE STEAM BNG:NE. IdT the product by 7. The quotient is the thickness of the piston in 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 =2x14-^7=4 inches. DIAMETER OF CONNECTING ItOrS AT MinDLE. The following rule gives the diameter of the connecting rod at middle. The rule, we may remark, is entirely founded on practice. Rule.-— To _^m(Z the diameter of the eonneeting rod at middle of the locomotive engine. — Multiply the diameter of the cylinder in inches hy ■21. The product ia 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 i'or this e'^ample ne have, according to the rule. Diameter of coDnecting rods at middle = 12 X '21 — 2-52 inches, or 2 J inches. niAMETElt or BALL ON CROSS-HEAD sriNDLE. The diameter of the ball on the cross-head spindle may he 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 3J inches, niAMETER 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 hear- ings 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 ingido hearing for the loco- motive engine. — Extract the cube root of the square of the diameter of the cylinder in inches ; multiply the result by -Oe. 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 hv Google THE PRACTICAL MODEL CALCULATOR. for a locomotive engine whose cjlimlers are of IS-iKch In this example we have, aecordiog to the rule, 13 169 = = diameter of cylinder in inclies. = aquare of tlie diameter of cylinder. 5 5 5 25 25 60 169(5-6289 - ^169 125 44000 41375 10 5 7500 J75 2625 1820 150 5 8275 800 805 726 and diameter of hearing = 5'5289 X -96 = 5'31 inches nearly; or between 5J and 5J inches. DIAMETER or THE OUTSIDE BEARINGS OF THE CRANK AXLE. The crank axle, in addition to resting upon the inside bearings, is Bometimea also made to rest partly upon outside hearings. These outside bearings are added only for the sake of steadiness, and they do not need to be so strong as the inside hearings. The proper size of the diameter of these bearings may be found from tbe following rule. EuLE. — To find the diameter of outside hearings for the locomo- tive 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 loco- motive engine, the diameter of its cylinders being 15 inches. In this example we have, according to the rule, 15 = diameter of cylinders in inehos. _15 225 = square of diameter of cylinder. ■396 = constant multiplier. hv Google THE STEAM ENGINE. 89-l(4-466 16 64 W 26100 32 21184 4800 3916 496 3528 ■6296 388 512 358 5808 Hence diameter of outside bearing = 4-466 inches, or very nearly 4| inches. diamt:tkr of pi.ain paet op crask axle. It is usual to make the plain part of crank axle of the same sec- tional area as the inside bearings. Hence, to determine the sec- tional area of the plain part when it is cylindrical, we have the fol- lowing 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 axlo for the locomotive engine, whose cylinders' diameters are 14 inches. In this example we have, according to the rule, 14 = diameter of cylinder in inches. 14 196 = square of the diameter of cylinder. -0 196(5-808 = ^196 5 25 125 % 25 71-000 5 50_ T 0-112 10 7500 ^888 5 1_2_64 150 8764 8 1328 158 10092 _8 166 __8 174 hv Google 170 THE PBACTICAI, MODEL CALCULATOK. Hence the plain part of crank axle = 5'808 X -96 — 5-58 nearlj, or a little more than 5^ inchea. DIAMETER 01' ORAKK PIN. The following rule gives the proper diameter of the crank pin. It is ohvious 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. Role. — To find the diameter of the crank pin for the locomotive engine, — Multiply the diameter of the cylinder in inches by '404. The product is the diametor of the crank pin in inches. Required the diameter of the crank pin of a locomotive engine whose cylinders' diameters are 15 inchea. In this example we have, according to the rule. Diameter of crank pin = 15 X '40-1 = 6-06 inches, or about 6 inchea. 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 di- ameter 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 wc 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 aide play. These then are the chief dimensions of locomotive engines ac- cording to the practice most generally followed. The establish- ment of express trains and the general exigencies of steam locomo- tion 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 neces- sity that yet exists of expending a large proportion of the power in the production of a draft ; and this evil is traceable to the inade- quate 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 ar- rangement 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. hv Google THE BTEAM ENGIKE. 171 We bere give the preceding rules in formulas, in tlie belief that those well acquainted with algebraic aymbola 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. — PAETS Or THE CYLINDER. Area, of induction ports, in square inches = '068 X B^. Area of eduction ports, in square inches = -128 x D^. Breadth of bridge between ports between f inch and 1 inch. LOCOMOTIVE ESGINE. — PAETS OJ 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 D' -^ 2. Area of water level, in square feet = 2'08 X D, Cubical content of water in boiler, in cubic feet = 9 x D' -;- 40. Diameter of feed-pump ram, in inches = 'Oil X DK Cubical content of steam room, in cubic feet = 9 X D' -;- 40. Cubical content of inside fire-box above fire bars, in cubic feet = D=-s-4. Thickness of the plates of boiler = | inch, LOCOMOTIVE ENGINE. — DIMENSIOXS OP SEVERAL PIPES. Inside diameter of steam pipe, in inches = '03 X IP. Inside diameter of branch steam pipe, in inches = -021 x D^ Inside diameter of the top of blast pipe = '017 X D^ Inside diameter of the feed pipes = ■141 X D. LOCOMOTIVE ENGINE, — DIMENSIONS OF SEVERAL MOVING PARTS. Diameter of piston rod, in inches = D -4- 7. Thickness of piston, in inches = 2 D -^ 7. Diameter of connecting rods at middle, in inches = '21 X D. Diameter of tho ball on cross-head spindle, in inches = '23 X D. Diameter of the inside bearings of the crank axle, in inches = 96 X -i/ D'. _ __ Diameter of the plain part of crank axle, in inches = "SB x -^ D^ Diameter of the outside bearings of the crank axle, in inches = ^ -396 X D^ Diameter of crank pin, in inches = -404 x D. Length of crank pin, in inches = -233 X D. hv Google THE PEACTICAL MODEL CALCULATOR. Table of tie Pre sure of Steam, in Inches of Meroury, at dif- ferent Temperatures, t™p=- Helt i)^i,.«. lira. V.,.,. Ivotj. Trsdgold, ^m,,.. K,i,L.,». ""'■ 0° 0'08 10 012 20 0-17 O-ll 32 0-26 0-'20 0-18 0-17 0-16 0-00 40 0'34 0-25 0-20 0-24 0-22 0-10 50 0>49 0-36 0-36 0-36 0-37 0-33 0-20 60 0.65 0-52 0-53 0-55 0-48 0-35 70 0-87 0-73 0-75 0-73 0-78 0-6S 0-55 0-77 80 110 1-01 1-05 1-11 0-95 0-82 W 1-30 1-44 1-36 1-53 1-34 118 100 2-12 1-96 1-84 1-60 1-55 110 2'79 2-45 2-62 2-4S 2-79 2-66 2-25 120 3-6S 3-30 8-46 3-46 3-00 130 4-71 4-54 4-41 4-8S 4-43 3-95 140 6'05 6-78 5-88 6-21 5-75 6-15 5-14 ISO 7-53 7-55 7-42 7-94 7-46 6-72 160 fl-79 9-GO 9-62 10-05 9-52 8-65 8-92 170 12-31 12-05 12-14 12"05 12-60 12-14 11-05 11-87 180 16-38 15-16 15-23 15-67 15-30 14-05 12-73 190 18-98 19-00 18-96 18-93 19-00 17-85 19-00 200 28-51 23-60 23-44 23-71 22-65 310 28-82 28-81 28-81 28-86 212 30-00 30-00 80-00 30-00 30-00 30-00 30-00 29-40 220 35-18 85-54 85-19 34-92 35 -S 83-65 230 14-60 43-10 42-47 43-63 42-00 44-5 40 240 53-45 61-70 51-66 50-24 54-9 49-0 Table of the Temperature of Steam at different Pressures i mospheres. I^^^ .^•z. r...u.. r»nne. I..,. TredE^H. s™*™. R„.i»n. w«,. .--"'■ l8t At. 2J2-0° 212"" 212° 312° 212= 212" 212" ^2~ 2cl At. 250-5 250-0 240-8 249 250 250-3 252-5 350-0 8d At. 276-2 275-0 271 274 267 275-2 4th At. 391-5 290 394 298-4 291-5 SthAt. 308-8 804-5 302 809 304-5 euiAt. 320-4 815-5 S22 315-5 TthAt. 331-7 S25-5 326-5 8t1> At. 342-0 336-0 342" 343-6 336-0 9th At. 350-0 345-0 345-0 10th At. 358-9 352-5 11th At. 12th At. 374-0 372' 13th At. I4th At. 336-9 15th At. 892-8 16th At. 398-5 17th At. 403-8 18th At. 408-9 19th At. 413-9 20th At. 418-5 414 405 30th At. 457-2 40th At. 466-6 50th At. 510-6 b,Google THE STEAM ENGIKB. Table of the lExpannion of Air by Heat. FBhmi Fll,™ 90 .... .... 1182 83 1002 63 1071 91 .... .... 1184 34 35 Sb 1004 1007 1009 63 111 65 1073 1075 37 1012 1030 95 .... .... 1142 38 1015 07 1080 90 .... .... 1144 S9 1018 1034 40 41 1031 1023 61 70 10M7 10B9 .... 1150 42 1025 71 1091 100 .... .... 1152 43 1027 72 109J 110 .... .... 1178 44 lOJO 73 lO'^t 120 .... .... 1194 45 1032 74 1097 46 1034 75 1099 47 103G 1101 150 .... .... 1255 48 I0o8 77 1104 100 .... .... 1275 49 60 1040 1013 78 79 1100 1108 180 ..„ .... 1315 51 1045 80 1110 190 .... .... 1334 52 1047 81 1112 200 .... ,... 136* 53 10'.0 82 54 1052 53 1055 84 1118 802 .... .... 1658 S6 10j7 86 1121 892 ... .... 1789 57 lo^y 86 1123 482 .... .... 1919 58 1062 87 1135 572 .... 59 60 1064 1066 89 1128 1180 STKENGTH OF MATERIALS. The chief materiaJa, 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 : — : M^.i.i,, c s E M 163001 omri 36000 1 ^"''* 60000 9000 800O0 69120000 91440000 5530000 6770000 1 ^'^'''^^w^:::::::::::::::. 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 M, 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 Tvere therefore made. The following Table gives, we have no doubt, results as nearly correct as can be required or attained; — hv Google THE PRACTICAL MODEL CALCULATOR, Is the following Table each bar is veducecl to exactly one incli square ; and the transTerse strength, which may be taken aa a criterion of the value of each Iron, is obtained froni a mean between the experiments upon it;^fir5t on bars 4 ft. 6 in. between the supports; and next on,t?iose of half the length, or 2 ft. 3 in. be- tween 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. Ponker, Mo. 3. C OMlwnj, So. S. ; iS WfaiUsh£r Wiilta Sulllfh gri ill End, No. Z. Cold Bloat low Jdoor, Si BnSaty, No. 1 Brimbo, No. S Apedale,No.: Oiabary, Ha, HnliUib, No. 1. tJoia Sltut*. Adfllphi, Mo. 2. Oold KJart-'- BluaiO'o. 3. Oold Bliat -■ " Dercin, Ho. S, CoWSlaat* - .■ Gtattbenie, Noi^'Hot Blul PnKKt, No. 4. Cold Blast Lsne Sad, Ho. S, CBiTon, No. a. Cold Blast*' ■- MBBslag (MMkai Bad) Corbynska]], Ho.2 Pon^poo]» No. 2 '.-■■..■... ■ WBUbrook, No. 3 MlltoiijHr " "-""i^ EBQt No.' Ln^No. o. 1. Bot Bitiat* l.HotBlBn.-. a.HotBlHt..- mloai^ Ho. 2. Cold Bis T»rtiig,No.a.Hot;Blia Coltham, No. 1. HOt Bl Oanoll, Ho.3.ColdB]e Molrktrfc No. 1. Hot B Bierler.Ko.a-- Ooed-T&lou, Ho. 2. Hot Coed-Talon, No. % Cold Monklaad, No. 2. Hot 1 Wb Works, No. 1. Ho PlasbTPaatoo, No. 2. H The irons with asterisks a Cold Blast Iron, 3 taken from Experiments on Hot and hv Google THE STEAM ENGINE. 175 Rule. — To find from the above Table tbe breakiug weight in rectangular bars, generaDy. Calliiig b and d the breadth and depth in inches, and I the distance between the supports, in feet, and putting 4-5 for 4 ft. 6 in., we have -, ~ = breaking weight in lbs., — the value of 8 being taken from the above Table. IFor example: — What weight would be necessary to break a bar of Low Moor Iron, 2 inches broad, 3 inches deep, and 6 feet be- tween the supports ? According to the rule given above, we have & = 2 inches, <? = 3 inches, ; = feet, S = 472 from the Table. ^, 4-5 X bd'S 4-5 X 2 X 3^ X 472 Then J- ■- g — = 6372 lbs., the break- ing weight. Table of the Cohesive Power of Bodies whose Gross Sectional Areas equal one Square Inch. M.r.,.. toh-si-E Power fewedish hir iron t. 1 OHO RUXSIBD do English do Cast steel 1 4 .iU Blistered do ] 1,J, Shear do Wrought oopper Hartl gOB metal "(. r,(, Cast copper ly,072 Yellow braes, ca'it 17 %S CaBt iron 17 (,.■'* Tin cBBt 4 7)3 Bismuth, oaat 3 250 Lead, cast lB-4 Elastic power or direct tension of wrought iron. medium quality 22 400 Note — A bar of iron is extended 000016, or neailv one ten- thousandth part of ita length, !oi eveiy ton of diiCLt strain per Bquaie inch of sectionil area CENTRE OP 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 con- stituent matter, or so placed that the particles shall be symmetri- cally 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 variea hv Google 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 con- stituent 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 drown 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 conse- quently 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 sup- ported, 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 ia perpen- dicular 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 laid 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 whoro the weight is applied. In a system of three bodies, if a line he 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 hv Google THE STEAM ENGISE. 177 magnitude of the body from which the line is drawn to the sum of the magnitudes of tbe other two ; and, consequently, the single body multiplied by its distance from the common centre of ^rai ity ia equal to the sum of the other bodies multiplied by the distance of their common centre from the common centre of the pjstera If there be taken any point in the straight line or levei joinmg the centres of gravity of two bodies, the sum of the two products of each body multiplied by its distance from that point is erjual to the product of the sum of the bodies multiplied by the distance of their confmon 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 dis- tant 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 propor- tional 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 propor- tional to three times the arc of the sector, the chord of the arc, 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 hv Google 178 THE PRACTICAL MODEL CALCULATOR. tlic position of the centre of gravity is distant from the centre of the circle hy a space which is equal to four times the radius divided by the constant number 8-1416 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 re- spective solids. The centre of gravity of the surface of a spheric seg- ment 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 three- eighths 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. 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 ia 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 ia 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 aa 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 hv Google THE STEAM ENGINE. 179 e¥ery point of tte orbit, in which case the centripetal and centri- fugal forces are equal to one another. In circular orbits the cen- tripetal 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 dis- tance from the centre of forco, ia 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 Js 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 ia 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 perpen- dicularly 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 hv Google loO THE PRACTICAL MODEL CALCrLATOK. the body revolves in a circular orbit, the periodic time, or the time of performing a complete revolution, is expressed by the term nt -/ -i and the velocity or space passed over in the time ( ivill be •/ ds; in which expressions d denotes the diameter of the circular orbit described by the revolving body, « 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 n 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 tlie radii, and inversely as the centripetal forces. CENTRE or GYRATION. The centre of gyration is that point in which, if all the consti- tuent 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 aeting 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 velo- city, and inversely as the distance from the centre of motion ; so that, if the absolute velocities of the revolving bodies be propor- tional to their ridii or distances, the angular velocities will he equil If the axis of motion passes through the centre of gravity, then IS this centre called the principal centre of gyration. The distiuce of the <,entre of gyration from the point of suspen- sion, or the axis of motion in any body or system of bodies, is a geometrical mean betvieen the centres of gravity and oscillation iiom the same point or axis , consequently, having found the dis- tances of these centres m any proposed case, the square root of then product wdl gn e the distance of the centre of gyration. If any part of a system be coneened to be collected in the centre of gyration ol 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, hv Google 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 peri- phery. 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 diametri- cally 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 aide. These are the chief properties con- nected 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 tho centre of gyration from tho 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 |. In the circumference of a circle revolving about the diameter it is equal to the radius multiplied by the square root of J. 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 J. In a solid globe revolving about the diameter it is equal to the radiug multiplied by the square root of |. In the surface of a sphere revolving about the diameter it is equal to the radius multiplied by the square root of f . 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 prin- ciples when we come to treat of fly-wheels in the construction of the different parts of steam engines. When bodies revolving in tho 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 veloci- ties of motion ; and the periodic times, under like circumstances, are directly as the velocities of motion, and inversely as the cen- tripetal 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 dis- tances or radii of the circles. If the times of revolution are as hv Google 182 THE PRACTICAL MODEL CALCULATOR. the radii of the circles, the velocities will be equal, and the cen- tripetal forces reciprocally as the radii. If several bodies revolve in circular orhits round the same or difi'erent centres, the velocities are directly aa 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 propor- tional to the distances or radii of the circles, the centripetal forces ■will bo 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 dis- tances 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 revolu- tion. 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 propor- tional 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 arc 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 hv Google THE STEAM ESGINE. 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 he equal ; for these times are equal to one-fourth of the periodic times, and these times, under the specified condi- tions, 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 eoniaal pendulum, and this species of pendu- lum, or, as it is usually termed, the governor, is of great importance in mechanical arrangements, being employed to regulate the move- ments of steam engines, water-wheels, and other mechanism. As we shall have occasion to show the construction and use of this in- strument 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 sus- pended from a centre, is made to vibrate from side to side in the same vertical plane. PEHDULUMS. If a pendulum vibrates in a small circular arc, the time of per- forming 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 area, 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 ares 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 pro- portional to the chord of the arc through which it descends to ac- quire 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 hv Google 184 THE PRACTICAL MODEL CALCULATOR. roots of the forces. The lengths of pendulums vibrating in dif- ferent places are as the forces of gravity at those places and the squares of the times of vibration. The timea 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 gravi- tating 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 proper- ties of a simple pendulum vibrating in a vertical plane, and the prin- cipal problems that arise in connection with it are the following, viz. : To find the length of a pendulum that shall make any 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 in 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 394015, 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 num- ber of vibrations sought. The number 394015 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 ia 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 CO ; hence we get 60 X 60 X 39-1015 = 140765-4, which, being di- vided by 35 X 35 = 1225, gives 140765-4 -h 1225 = 114-9105 inches for the length required. The length of a pendulum between the centre of suspension and ihe 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 394015 = 140765-4, which, being divided hy 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 numbei' 39-1015 by the square of the time, or 140765-4, is itself a constant quantity, which, heingkept 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 re- volves about a vertical axis in the circumference of a circular plane which is parallel to the horizon. CONICAL PENnuHTM. If a pendulum he suspended from the upper extremity of a ver- tical axis, and be made to revolve about that axis by a conical mo- hv Google THE STEAM ENGINE. 185 fion, ivhicli constrains the revolving body to move in the circum- ference 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 he the pendulum in question, suspended from C, the upper extremity of the vertical axis CD, and let the ball or body E, by revolving about the said asis, describe the circle BE AH, the plane of which is parallel to the horizon ; it is proposeil to assign the time of description, or the time in which the body S performs a revolution about the axis CD, at the distance BD. Conceive the axis CD to denote the weight *( of the revolving body, or its force in the di- rection of gravity; then, by the Compo- sition 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 ED 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 ; bnt, by the laws of gravity, the space fallen perpendicularly from rest, in one second of time, is ^ = IQ^ feet ; consequently we have CD : BD : : 1^12 = ^' -, the space described towards D by the force in BD in one second. Consequently, by the laws of centripetal forces, the periodic time, or the time of t he bod y revolving in the circle BEAH, is expressed by the term jt^/tL^, where « = 3-1416, the circum- ference of a circle whose diameter is unity ; or putting t to denote the time, and expressing the height CD in feet, we get ( — 6-2832 ■J . ■■ , or, by reducing the expression to its simplest form, it becomes t = O-SlflSG-v/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 revo- lution, 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 tiiat vibrates seconds in latitude 51° 30' ? The square root of 39-1393 is 6-2561 ; consequently, by the rule, hv Google 18b THE PRACTICAL MODEL CALCULATOR. we liave, 6'2561 x 0-31986 = 2-0011 seconds for the time of revo- lution 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 revo- lution, corresponding to any given time, can easily he 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 re- quired to find the height of a conical pendulum that would revolve 30 times in a minute. Here tJie time of revolution is 2 seconds for 60 H- 30 = 2; therefore, by division, it is 2 h- 0-31986 = 6-2527, which, being squared, gives 6-2527 = 39-0961 inches, or the )ength of a simple pendulum that vibrates seconds very nearly. In all coDical 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 32i, are constant quantities, eon- ■ sequently * varies as ^/CD. If the heights of the cones, or the distances between the points of suspension and the centres of the planes of revolution, he 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- volving bodies. This will be clearly understood by c the subjoined diagram, where all the pendulums Ca, C5, Ce, Cd, and Ce, having the common axis CD, will revolve in the same time ; and hv Google TMB STSAM ENGISE. 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 ve- locity 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 pendu- lum 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 3-1416' maltipliod by twice the height or axis of the cone. The periodic time, or the time of one revo- lution is equal to the product of 3'1416 */ 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 pen- dulum depends ; but as we shall hereafter have occasion to con- sider it more at large, we need not say more respecting it in this place. Before dismissing the subject, however, it may he proper to put the reader in possession of the rules for calculating the posi- tion 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 moat 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 fre- quent use, suspended from their vertices and vibrating flatwise, are as follow : — In a right lino, 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 ' S n -4- 1 ^ ^ altitude, where n denotes the order of the figure. hv Google 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 ia 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 tho axis, increased by one-third of the parameter ; in a parabola, sus- pended 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 ia three times the arc of tho 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 tho 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 tho other. THE TEMPERATURE AND ELASTIC FOHCB OP BTEAir, In estimating the mechanical action of steam, tho intensity of its elastic force must be referred to somo 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 fiuid 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 2| 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 pro- portionate diminution takes place in the intensity of the emanci- pating elFort 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, hy means of whieh it may he ascertained when a general law exists, and to determine what that law is, in eases where it is known to obtain. — Suppose, for example, that it is required to assign the nature of the law that subsists between the hv Google 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-officient, except the exponent by which the law is indicated. Let E and e be any two values of the elasticity, and T, (, the corresponding temperatures deducted from observation. It is proposed to ascertain the powers of T and i, 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, wo get, T° ; i" : : E : e ; but by the principles of proportion, it is — = — ; 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. 6 — log. E " ^T^. ( - log. r The theorem that we have here obtained is in its form sufH- ciently simple for practical application ; it is of frequent occur- rence in physical science, but especially so in inquiries respecting the motion of bodies moving in air and other resisting media ; and it IB even applicable to the determination of the planetary motions themselves. The process indicated by it in the case that we have chosen, is pimply, 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 tempe- rature 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 843-6 degrees of Fahrenheit's scale, the corresponding elastic forces were 59-6 and 238-4 inches of the mercurial column respec- tively. 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. 2-3084608 Remainder = 0-1 368333 Greater elastic force, 238-4 log. 2-37T3063 Lesser elastic force, 59-6 log. l-"75g463 Remainder =0^60^0600 Let the second of these remainders be divided by the first, as dii'ccted 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. : T":t^ ■>::£:£; hv Google 190 THE PRACnCAL MODEL CALCrLATOR. that is, the elasticities are proportional to the 4'4 power of the temperatures very nearly. Kow this law is rigorously correct, as applied to the particular cases that furnished it ; for if the two temperatures ant! one elas- ticity he 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 io the foregoing investigation. The theorem, however, is valuable to practical men, not only aa 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 eiasticity 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 cSaaa of experi- ments 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 mnst 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 follow- ing 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 modulus of elaeticity of water. The modulus of elasticity of any substance is the measure of its elastic force ; that of water at 60° of tempe- rature 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, then 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 therefore have tlie following proportion. As the expanded volume of water is to the hv Google THE STEAM ENGINE. 191 quantity of e'^pin^mn, =«o in the tnodulus cf ela^ticitj of water to the elaitic It ice of ^team of the sime density as watei Having therefore assjgne I the limit'* beyond whn,h the elastic force of =!team cmnot reach, we shall now proceed to ^pply the principle of our formula to the determination of the general law which connect'' the temperature with the el istic force , and for this purpose, m addition to the notation which wo htve already laid down, kt c denote some constant quantity that a&ects the cl isticity, and d the temperature at which the elaaticitj vanishes , then since this temperature must be applied suhtractively, we h^l e from the foregoing principle c E = (T — i)', and c e = (( — *)" From either of these e ]Uations, therefore the constant quantity e can be determined m terms of the rest when they are known ; thus we have c = ^ — = — i-, and c = i L, and by comparing these two independent values of c, the value of n becomes known ; for ^^ — = — '— = i— — i-, and consequently n = log - e - log. E , , , log. (( - i) - log. (T - 6). ' ■ ■ ■ ^ '' 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, iet the ciaati- cities E and e become E' and e' respectively; while the corre- sponding temperatures T and t assume the values T' and (' ; then by a similar process to the above, we get * — •' = ^ — ^_J, and J°SiJ (B). log. {f - 8} - log. (T' - i] 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 experi- ment ; and in consequence, the law that we derive from them must be strictly empirical ; thus we have log, e — log. E. log, e' - lo g. E .„, log. (i - B) - log. (T - 8) log. {f - 8} - log. (r - «) ■ ; ^ '' We have no direct metbod 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 quan- tity 8 must be found ; and for the purpose of performing the reduc- tion, we shall select experiments performed with great care, and may consequently be considered as representing the law of elas- ticity with very great nicety. T = 212-0 Fahrenheit E = 29-8 inches of mercury. t = 250-3 e = 59-() T'= 293-4 E'= 119-2 ('=343-6 e'= 2.38-4 hv Google 192 THE PRACTICAL MODEL CALCULATOR. Therefore, by substituting these numbers in equation (C), and making a few trials, we find that S = — 50°, and substituting this in either of the equations (A) or (B), we get n — 5'08 ; and finally, by substituting these values of 8 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 1 134'27 / • ■ ■ ■ ^ ^ Where the symbol F denotes generally the elastic force of the Steam in inches of mercury, and ( the corresponding temperature in degrees of rahrcnhcit's thermometer, the logarithm of ttie denominator of the fraction is 2-X279717, 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 experi- ment, and two points near the other extremity ; and by substi- tuting 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 I 135-7«7 / But even here the formula errs in defect, for he has fonnd it necessary to correct it by adding the arbitrary decimal O'l; and thus modified, it b {y^}"-i <^'- 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 prin- ciples 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 : — RoLB, — To the given temperature in degrees of Fahrenheit's thermometer add 51-3 degrees and divide the sum by 135 -1 67 ; to the 5'13 poiver of the quotient add the constant fraction I'fr, and the sum will be the elastic force in inches of mercury. hv Google THE STEAM EHGINE. 193 The process here deserihed 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 diffi- culty. The rule adapted to these numbers is as follows :— Rule eor Loqarithms. — 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-T67, 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 j\; 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 clastic force in inches of mercury ? By the rule it is 250-3 + 51-3 = 301-6 log. 2-4794313 constant den. = 135-767 log. 2-1327940 subtract remainder = 0-3466373 _ SI "5 inverted 17331865 346637 103991 natural number 60-013 log. 1-7782493 If this be increased by ^, 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 ; tlie transformed expression is as follows, viz. : * = 135-767 (F - 0-1)^ - 51-3 .... (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 hj logarithms. Rule. — From the given elastic force in inches of mercury, sub- tract the constant fraction 0-1 ; divide the logarithm of the remain- der 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 tem- perature in degrees of Fahrenheit's thermometer ¥ Here, by proceeding as directed in the rule, we have 238-4 — -O-l = hv Google 194 THE PRACTICAL MODEL CALCULATOE. 238'3, and dividing the logarithm of this remainder hy the coa- staiit exponent 5'1§, we get log. 238-3 -H 5-13 = 2-3771240 ^ 5-13 = 0-4633770 constant co-efficient =135-767 - - log. 2-1327940add natural number =394-61 - - - log. 2-5961710 sum constant temperature = 51-3 subtract required temperature = 343-31 degrees of Fahrenheit's ther- mometer. The temperature by observation is 343-6 degrees, giving a differ- ence of only 0-29 of a degree in defect. For low temperature or low pressure steam, that is, steam not exceeding the simple pres- sure of the atmosphere, M. Pambour gives y- 0-04948 + (135:^) . . .(6). In which equation the symbol p denotes the pressure in pounds avoirdupois per square inch, and ( the temperature in degrees of Fahrenheit's thermometer. When this expression is reduced in reference to temperature, it is ( = 155-7256 (;>- 0-04948) ^-51-3 .... (II). The formula of Tredgold is well known. The equation, in its original form, is 177/* = i + 100. . . .(I): where / denotes the elastic force of steam in inches of mercury, and ( the temperature in degrees of Fahrenheit's thermometer. The same formula, as modified and corrected by M. Millet, becomes 179-0773/^ = * -i- 103 . . . . (K). Dr. Young of Dublin constructed a formula which was adapted to the experiments of his countryman Dr. Dalton : it assumed a form suf&ciently simple and elegant ; it is thus expressed— /= (1 + 0.0029 0' . . . . (L): where the symbol /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 Frencli Academy represented their experiments by means of a formula constructed on the same principles : it is thus expressed — /=(l-l- 0-7153 ()' .... (M): where /denotes the elastic force of the steam expressed in atmo- spheres of 0-76 metres or 29-922 inches of mercury, and ( the tem- hv Google THE STEAM ENGINE. 195 perature estimated above 100 degrees of the centigrade tliermo- metcr ; but when the same formula is SO transformed as to he expressed in the usual terms adopted in practice, it is p = (0-2679 + 0-0067585 i)' . . . . (N): where ^ is the pressure in pounds per square inch, and t the tem- perature 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 tf . . . . (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 tor 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 and for steam above the temperature of 212, it is /=C- 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 trans- late 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 thmk proper to branch the subject into a series of problems, as follows : The temperature of steam being given in degrees of Fahrenheit'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 con- stant 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 temperattu'e 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 hv Google 196 THE PHACXICAI. MODEL CALCULATOR. logaritlira of the elastic force in atmospheres of 30 inches of mer- cury ; 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 rahrecheit, the pro- cess is as follows : 250 -I- 121 = 371. ..log. 2-569374 Constant divisor = 333. ..log. 2-522444 subtract 0-046936 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 : Rule. — Multiply the given temperature in degrees of Fahren- heit's thermometer by the constant coefficient 1-5542, and to the product add the constant number 271-985; then from the loga- rithm of the sum subtract the constant logarithm 2-587711, and multiply the remainder by the exponent 7-71307 ; the natural num- ber 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 ther- mometer 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 re- mainder 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 fol- lows: hv Google 187 X 1'5542 = 290-6354 Constant = 271-985 a dd Sum = 562-6204... log. 2-750216 Constant = 387 log. 2-5 87711 subtract 0162505 X 7-71307 = 1-2 And the natural number answering to this logarithm is 17-923 inches of mercury. By the preceding calculation the result ia 17-802; the slight difference arises from the introduction "of the decimal con- stants, 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 = 20 5-526 add Sum = 630.166 log. 2-709456 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 coefiicient 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 procluct will give the pressure in pounds per square inch, when the temperature is less than 212 de- grees ; 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 eteam 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, there- fore, the temperatures previously employed, the operation is as follows ; 187 X 1-41666 = 264-9155 Constant = 247-91 55 add Sura = 512.8310.log. 2-709974 Constant = 387 log. 2-587711 subtract 0-122263 X 7-71307 = 0-94265S hv Google 198 THE PRACTICAL MODEL CALCULATOR. And tte number answering to thia logarithm is 8'763 lbs. per square inch, and 8-763 X 0-7854 = 6-8824 lbs. per circular incb, the pro- portion 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 = 83-3 log. 2-522444 subtract 0-229031 X 6-42 = 1-470279 And the number answering to this logarithm is 29-568 lbs. per sc[uare inch, or 29568 x 0-7854 = 23-2226 lbs. per circular inch. We have now to reverse the process, and determine the tempera- ture 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. Por the lower temperature, or that which does not exceed the temperature of boiling water, we get ( = 249/^^-175 .... (R). Where t denotes the temperature in degrees of Fahrenheit's ther- mometer, and /the elastic force in inches of mercury, less than 30 inches, or one atmosphere ; hut when the elastic force is greater than one atmosphere, the formula for the corresponding temperature is as follows : i = 196/*^- 121 ... . (S). In the construction of those formulas, we have, for the sake of simphcity, omitted the fractions that obtain in the coefficient of /; 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 he quite sufficient. The rule supplied hy the equations (R) and (S) ia thus expressed : When the elastic force is less than the pressure of the atrflosphere, that is, less than 80 inches of the mercurial column, — Rule. — Divide 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, svihtract the constant temperature 175 degrees, and the remainder will he the temperature sought in degrees of Fahrenheit's thermometer. But when the elastic force exceeds 30 inches, or one atmosphere, the following rule applies : hv Google THE STEAM ENGINE. 199 Divii^e the logarithm of the given elastic force in inches of mer- cury hy the constant index 6-42, and to the quotient add the con- stant logarithm 2'292363 : then, from the natural number answer- ing 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 there- fore 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. 2-896204 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 -H 6-42 = 1-778151 h- 6-42 = 0-276969 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 i^-ith 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 elas- ticity of the steam except by increasing the temperature at the same time ; and, contrariwise, the temperature cannot be increased with- out 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 ease bo increased, it will be found not to possess a higher degree of elasticity than a volume of atmospheric air simi- larly 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 hv Google 200 THE PRACTICAL MODEL CALCULATOR. more steam generated by an increase of temperature ; and conse- quently 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 ex- pand itself. Our nest object, therefore, is to inquire what is the law of elasticity of steam under the conditions that we have here The specific gravity of steam, its density, and the volume which it occupies at difi'erent temperatures, have been determined by ex- periment 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 I'SYS in bulk by 180 degrees of temperature; and again, that steam obeys the law discovered by Boyle and Mari- otte, 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 pres- sure ; 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 calcu- lated by the following theorem, , /(' + 4.')9\ ,™j " ="(f+45s) m where t and t' are the temperatures, and u, 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 tho greater sum by the lesser, and multiply the quotient by the volume at tho lower temperature, and the product will give the expanded volume. If the volume of steam at the temperature of 212 degrees of Pah- renheit be 1711 times the bulk of the water that produces it, what will be its volume at the temperature of 250'8 degrees, supposing the pressure to be the same in both cases ? Here, by the rule, we, have 212 4- 459 = 671, and 250-3 + 459 = 709'3 ; consequently, by dividing the greater by the lesser, and multiplying by the given volume, we get 1^^ 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 cor- responding volume be given, the elastic force at the higher tem- perature 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 wil! occupy under any proposed pressure can readily be found by reversing the pre- ceding process, or by referring to chemical tables containing the hv Google THE STEAM ENQINE. 201 specific gravity of the gases compared witli air as unitj at the same pressure and temperature. Now, air at the mean state of the at- mosphere has a specific gravity of 1| aa compared with water at 1000 ; and the bullts are inversely as the specific gravities, accord- ing to the general laws of the properties of matter previously an- nounced ; hence it follows that air is 818 times the bulk of an equal weight of water, for 1000 -^ If = 818-18. But, by the experiments of Dr. Dalton, it has heen found that steam of the same pressure and temperature has a specific gravity of -GSS com- pared with air as unity ; consequently, we have only to divide the number 818'18 by '625, and the quotient will give the propor- tion of volume of the vapour to one of the liquid from which it is generated ; thus we get 818-18 -5- -625 = 1309 ; that ia, 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 {'X), that when the temperature in- s to t', the volume t /459 + t' \ ' = 1309 X (559^^60) = 2-524(459 + t'); and from this expression, the volume corresponding to any specified elastic force /, and temperature t', may easily he found ; for it ia inversely as the compressing force: that is, /:30: : 2-525(459 + t') : v' ; consequently, by working out the analogy, we get = ;[5:67(459J^). ,j,. f ^ '' By tliis theorem is found the volume of steam as compared with that of the water producing it, when under a pressure correspond- ing to the temperature. The rule in words ia as follows : Rdle. — 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 mul- tiply 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'a thermometer, what ia the volume, compared with that of water ? The temperature being greater than 212 degrees, the force ia cal- culated by the rule to equation (Q), and the process is as follows : 250-3 + 121 = 371-3 log. 2-5697249 Constant divisor = 333 log. 2j;5224442 subtract 0-0472807 x6-42=0-3035421 Atmosphere = 30 inches of mercury log. 1-4771213 add Elastic force = 60-348 log. 1-7806634 ~1 Again it is, 1 , 459 -I- 250-3 = 709-3 log. 2-85083001 ,, r^'^" Constantcoefficient = 75-67 log. 1-8789237 ( 4-7297537 j Volume = 889>39 times that of water, log. 2-94909'03 re- mainder. b,Google 202 THE PEACTICAL MODEL CALCULATOR. Thus we have given the method of calculating the ehistie force of ateam 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 tem- perature 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, ivo 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 (XJ), 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 Pahrenheit, we have only to substitute those values of the elastic force when reduced to inches of mercury, instead of the symbol/ in equation (U), and we obtain, when the temperature is less than 212 degrees, Vol.=75-67(tcm.+459)H-(-004016xtem.-f-702807)'"*" (V). and when tlie temperature exceeds 212 degrees, the expression be- comes Vol. =75-67(tem. -i-459)-H -OOolOl x tem. -f- ■617195f ^ (W.) 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, mul- tiply 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 in- dex 7'71307, and find the natural or common number answcriog to the product, which reserve for a divisor. To the temperature add the constant number 459, and multiply the sum by the coefncient 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 'OOSlOl, and to the product add the constant increment '617195 ; multiply the logarithm of the sum 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 hv Google THE STEAM ENGINE. 203 of water, under the respective temperatures of 187 aud 293-4 de- grees of Fahrenheit's thermometer ? Here, by the rule, we have 187x0-004016=0-750992 Constant incremeiit=0-702807 Sum =14537^ log. -1625043 X 7-71307=l-2534069 and the nUmher answering to this logarithm is 17-92284, the di- visor. But 187 -f 450 = 646, and 646 X 75-07 = 48882-82, the dividend; hence, hy division, we get 48882-82-7-17-92284 = 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. 0-3250696x642=2-0869468; and the number answering to this logarithm is 122-165, the divisor. But 293-4 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 cal- culations connected with the steam engine, it is presumed that our readers will find it to their advantage to render themselves familiar with tlie 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 em- bracing only one of the cases above exemplified ; and conseciucntly 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 baa 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 tlie elastic force is proportional to the density : consequently, if/ denote any other elastic force, and w the weight in grains corresponding thereto, then we have 30:/:: 329-4 : w = 10-98/ hv Google 204 THE PRACTICAL MODEL CALCULATOR. the weight of a, cubic foot of vapour at the force/, and temperature 60 degrees of Fahrenheit. Let t denote the temperature at the <• ^ ,.. u .■ .rrx u 459 + i 459 + ( torce/; then by equation (1), we have v = T^a~sr(in ~ "^iTTq^i the volume at the temperature t, supposing the volume at 60 de- grees to be unity ; that is, one cubic foot. Now, since the den- sities are inversely proportional to the spaces which the vapour oc- (459 + , , 5l9u> , , , cupies, we have — k\q — ■ : 1 : : w : w = T^jf^Ti ! "'^^ "7 ^"^ preceding analogy, the value of w ia 10'98f; therefore, by substi- tution, we get , 6698-62/ •° " 459 + i ■ ■ ■ ■ (^>' This equation expresses the weight in grains of a cubic foot of steam at the temperature ( and force/; and if we substitute the value of /, from equations (P) and (Q), reduced to inches of mer- cury, and modified for the two cases of temperature below and above 212 degrees of Fahrenheit, we shall obtain, in the first case. w' = (0-012324 X temp, -f- 2-155611)"'™^ -i- (temp. + 459). . . . (Y) and for the second case, where the temperature exceeds 212, it is w' = (0-01962 X temp. + 2-37374f ^= ~ (temp. + 459) . . . (Z) These two equations, like those marked (V) and (AV) are suf- ficiently simple in their form, and off'er but httle 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, multi- ply the given temperature, in degrees of Fahrenheit's thermometer, by the fraction 0-012324, and to the product add the constant in- crement 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 2-37374; then multiply the logarithm of the sum by the index 6-42, and from the product subtract the logarithm of the temperature in- creased 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 Natural number :^ 157 '863 grains per cubic foot log. 2-11 b,Google THE STEAM HKQINB. 205 For the higher temperature, it ii 2D3-4 X 0-01962 =r 5-75G508 Constant increment = 2-873740 Sam = 8-130248 log. O-yiOlO-58 X 6-42 = 5-8426664 293-i + 459 = 752-4 .... log. 2-876448S , snbtraot HatutaJ number = 925-59 grains pec 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 OP ELASTIC FLUIDS. The next subject that claims our attention is the Telocity 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 mo- tion of free elastic fluids depends upon the temperature and pres- sure of the atmosphere ; and, consequently, when an elastic fluid is confined in a close vessel, it must he similarly circumstanced with regard to temperature and pressure as it would be in an at- mosphere competent to exert the same pressure upon it. The sim- plest 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 bo 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 tho fluids are originally confined ; and the velocity of motion at any other instant is proportional to the squai-e root of the difl"erence 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 conflrmed by observation and experiment, that oblique action produces very nearly the same efTcct 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 ten- dency to retard the motion : it therefore becomes necessary, in all the calculations of practice, to make some allowance for the retard- ation that takes place in passing the orifice ; and this end is most hv Google 206 THE PRACTICAL MODEL CALCUIATOK. conveniently answered by modifying tho constant coefficient ac- cording to the nature of the aperture through which the motion is made. Numerous experimeota have been made to ascertain the effect of contraction in orifices of different forms and under dif- ferent 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 re- sult from the direct unretarded action of the column of the fluid that produces it, we __^ have 3 V = s/57yA 2. For an orifice or tube in the form of the contracted vein 10 V = </60Sih 3. For wide openings having the sill on a~l level with the bottom of the reservoir ... | 4. For sluices with walls in a line with the V 1 V = v^5929i orifice 5. For bridges with pointed piers J 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 , . .,, ■ ^^ ■ - walls (10\ = s/iiGlh 8. For abrupt projections 9. For bridges with square piers , ^ ^ 10. For openings in sluices without side walls 10 V = s/SCOlA 11. For openings or orifices in a thin plate V = \/25/i 12. For a straight tube from 2 to 3 diameters in length projecting outwards 10 V = </4225 13. For a tube from 2 to 3 diameters in length projecting inwards 10 V = s/2916-25h 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 co- lumn producing it, estimated also in feet. Kos. 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, MOTIOS or STEAM IN AN ENGINE. We have already stated that the best method of estimating tho 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 there- fore the leading step of the inquiry ; and since the elastic force of steam is usually reckoned in inches of mercury, 30 inches being hv Google THE STEAM ENGINE. 207 equal fo the pressure of the atmosphere, the subject presents but little difSculty ; for wo have already seen that the height of a co- lumn 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 he as its relative hulk and elastic force ; hence we have 30 : 34-023 :fv:h = 1-lMlfv, where / is the elastic force of the steam in inches of mercury, v the correspond- ing 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 un- retarded 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 hest form of pipes, or a conical tube in form of the con- tracted vein, the velocity into a vacuum, according to No. 2, be- comes V = 8-307 '•/fv; and for pipes of the usual construction. No. 12 gives V = 6-922 v/V; No. 13 gives V = 5-804 v/^ and in the ease of a simple orifice in a thin plate, we get from No. 11 V = 5-322 ^fv. The consideration of all these equa- tions may occasionally ho required, but our researches will at pre- sent 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 ia below 212 degrees of Fah- renheit ; for the expression which indicates the velocity into a va^ cuum being independent of the elastic force, a separate considera- tion for the two cases is here unnecessary. It has been shown in the equation marked (TJ), that the volume of steam which- is generated from an unit of water, is i; = 75-67 (temp. -|- 459) , _, . ^ , , ,,.,,,.-,. ^^ — ^ ; let this value oi v be substituted lor it in the equation V — 6-922 \/fv, and we obtain for the velocity into a vacuum for the usual form of steam passages, as follows, viz. : V = 60-2143 v/(temp. -f- 459). This is a very neat and simple expression, and the object de- termined 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 motioli 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 Fahren- heit'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. hv Google aUo THE PRACTICAL MODEIi CALCOLATOE. By the rule it is 293-4 -1- 469 = 7524 J log. 1-4382244 Coustant coefficient = 60-2143 log. 1-7797018 add Velocity into a yaeunm in feet per second = 1651-08 log. 3-2179262 This is the velocity into a perfect vacuum, when the motiou is made through a straight pipe of uniform diameter ; but when the pipe is alternately enlarged and contracted, the velocity must ne- cessarily 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 mo- tion : it therefore behoves ns, 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 circum- stances of the case demand ; and the amount of these corrections must be estimated according to the nature of the obstructions them- selves. 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 com- mon thing, especially in steam vessels, for 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 deter- mining 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 simi- lar 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 430, the resulting constant 41'6868 may be employed without introducing an error of any consequence. OP THE ABCEHI OP SMOKE AND UEATED AIR IN CIIIMNEYS. 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 oc- curs. The proper construction of chimneys is a matter of very great importance to the practical engineer, for in a close fireplace. hv Google 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 al- lowed 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, h = 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 6 cubic feet of air must pass when expanded by a change of temperature, V = the velocity of ascent in feet p nl t' = the temperature of the extern I a nd ( = the temperature of the air t b d 1 ged by the chimney. Now the force producing the motion n th manifestly the difference between the weight of a lum f th tmospheric air and another of the air discharged by th b mn y and when the temperature of the atmospheric air at 52 1 of Fahren- heit's thermometer, this difference will b \ t 1 by the term k L, . ^gQ -) ; the velocity of ascent will therefore be V = JGi^ h-l rr-zTTKa f feet per second, and the quantity of air discharged per second will therefore be, « J^* «^ P'+T^fl f' 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 cu- bic feet becomes * = 125-69 .J|E3. This would be the quantity discharged, provided there were no increase of volume in consequence of the change of temperature ; but air expands from h to ■■■ , - , "^t-q- ■■ for t' ~ t degrees of tem- perature, as has been shown elsewhere ; consequently, by compa- rison, we have < + 469 ' - 125-69 fl b,Google 210 THE PRACTICAL MODEL CALCULATOR. From this equation, therefore, any one of the quantities wticli it involves can be found, wlien the others are given : it however supposes that there is no other cause of diminution but th^ contrac- tion at the aperture ; but this can seldom if ever be the ease ; for eddies, loss of heat, obstructions, and change of direction in the chimney, wiU diminish the velocity, and consequently a larger area will be required to suffer the heated air to pass, A sufficient al- lowance 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 b a = h v'((' + 459)« -^ 100 (( + 459) ^/h (t' - t). And if we take the mean temperature of the air of the atmo- sphere at 52 degrees of Fahrenheit, and make an allowance of 16 degrees for the difference of density between atmospheric air and coal emoke, our equation will ultimately assume the form a = b -/((' -I- 459f -i- 51100 s/h (('-(- 16). It has been found by experiment that 200 cubic feet of air of at- mospheric density aro 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 horao power : it therefore ap- pears 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, bnt 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 I 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, wo have, 250 -|- 52 = 302, the temperature at which the air enters Constant increment = 459 [the flue. Sum = 761 log. 2-8813847 2)8-6441541 4-3220770 Number of horse power = 40 log. 1'0020600 5-9241370 hv Google THE STEAM ENGINE, 5-9241 250 - 16 = 234 ... . log. 2-3692159 height = TO feet . . log. l_-8450980 2) 4-2143139 2-10T1569 1 Constant = 25-55 . . log. 1.4073909 J . . . 3-5145478 J 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 he taken to have the data ac- curately determined before the calculation is begun. In the above example the particulars are merely assumed ; but even that is suffi- cient 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 down- wards: but these experiments require further confirmation. The method of calculation adopted above is founded on the prin- ciple of correcting the temperature for the difi'erenee 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, hut 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 : 2757-5 \A(t'- 77-55) where a is the area of the transverse section of the chimney in square inches, h the quantity of atmospheric air required for com- bustion of the coal in cubic feet per hour, h the height of the chim- ney in feet, and (' the temperature at which the air enters the flue after passing through the fire. The rule for performing this pro- cess is thus expressed : Rule. — From the temperature at which the air enters the chim- ney, 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 quo- tient by the square root found as above, and the product will he the number of square inches in the transverse section of the chimney. hv Google 212 THE PEACTICAL MODEL CALODLATOB. Suppose a mass of fuel in a state of comtiustion to require 5000 cubic feet of air per hour, what must be tlie size of the chimney when ita height is 100 feet, the temperature at which the heated air enters the chimney being 200 degrees of Fahrenheit's ther- mometer ? By the rule we have 200-77-55=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 Vadd 3-0773399 2757-5 ar. co. log. 6-5594845 j 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 cre- ating 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 inconvenient 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 = ; / I5tj * '\12000-JAw; but when the side of the base is double the size of the top, the equation becomes 8= A\ V12000-0-42 Aw; where 8 is the side of the base in feet, h the height, and m 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, conse- quently w ought to bo 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 «> = 76 lbs. ; and if this be substituted in the foregoing equations, we get for a chimney of equal size through- out, -T. I 1^^ ' * \1200~25A; hv Google THE STEAM EHGINE. and when the chimney tapers to one-half the size at top, it is -Wn 104 '-4, V 12000 - 32 A; where it may he remarked that 12000 lbs. is the cohesiye 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. 2ffR atW VD-(-2^K{L + H' is the expression given hy M. Pikelet for the velocity of smoke in a chimney, v, the velocity ; t, the temperature, whose maximum value is about 300° centigrade ; g = 321 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 sheet- iron, and — '0025 for cast-iron chimneys, a = '00365. Let L=60; H=15Q; D=5 ; K=-00 5; 2j?=64J; i=300^ / 2ffHa(D a=-00365. Then v= \ jy^2a K(H+L) ^ ^^'^^^ ^^^^- A cnhic 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 ves- sel, 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 as- signed 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 ex- amples 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 hv Google 214 THE PRACTICAL MODEL CALCULATOR. the area of fire-grate, or of the quantity of fuel consumed upon it given area of grate surface, and the ireiglit of fuel burned on a foot of fire-grate per hour varies in diiferent cases in practice from 3J to 80 lbs. Upon the quickness of the draft again hingea the ques- tion 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 Tvill be thereby occasioned ; whereas, if the fire be thick, and the draft he sluggish, much of the useful effect of the coal will be lost by the formation of carbonic oxide. The length of the cir- cuit 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 state- ments. When the flue is too long, or its sectional area is insuffi- cient, 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 depo- sition 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 block- ing 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 water- level 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. hv Google THE STEAM ENGINE. Taele i. — " »- 1- li B fl P { { li li ! I 1 1 3 II i l'li:t:", .„ w^. ris= ^o™. -^1r ■s- Unglb of dicuit nude t>; US best In few ■ AreaoffiregEsteifnsquiM 2S-66 ..,. 72-5 ..... SS4-6 50 14-25 l*-45 see Wetght of fuel lurncH on «ieh«,uRrefcotofgr>te, Cob. ft. of water eyaporated byliaibs-offaS^ Cable feet of wntcr erv poMtol per hour bom Sqaan feet of beatel nir- liu»tbreiu:hcaUcr<»tor water e.sporatM per I3'31 Sqa8r.fcetofli«.teii«ui-- ^"bfXn2pb1n.tte" 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 bottom line. It will be observed, that the order in which the different en- gines stand in respect of superiority of duty is the same as in re- spect of amount of expansion. The Holmbush engine baa a duty of 140,484,848 lbs. raised 1 foot by 1 cwt. of coals, and the steam acts expansively over '83 of the whole stroke; while the water- works' Cornish engine has only a duty of 105,664,118 lbs., and expands the steam over only -687 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 sur- face as effective surface, if the boilers be of a considerable size ; but in the case of small boilers, the proportion is larger. The total hv Google THE PRACTICAL MODEL CALCULATOR, III Mi?. lalJi mi" llslllis P"*i. I'S'S hv Google THS STEAM ENfllNE, 21T 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 capaeity of steam room is 8j cubic feet per horse power, in the two horse power boiler, and 5f 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| cu- bic feet per horse power. The content of water is 18J 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 ori- ginal 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 ; hut inasmuch as in modern engines the nominal considera^ bly 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 ; hut as there are two cylinders, this is equivalent to 8 times the content of both cylinders, which ia the proportion commonly followed in land en- gines, 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 strobe, 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 cu- bic feet per horse power. There ia less necessity, however, that the steam space should be large when the Sow of steam from the boiler is very uniform, as it will be where there are two engines at- tached 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 nigh steam' chest too, by rendering boiling over into the steam pipes, or priming as it is called, more difficult, obviates the neces- sity 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 hv Google 218 THE PRACTICAL MODEL CALCULATOR. largo a proportion of Btcam room as 5 or 6 cubic feet per horse power, and about half that amount more nearly represents the general practice. Manj allow 0-64 of a Bquare foot per nomi- nal 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 convenientlj 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 consump- tion 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 ^th of the area of fire grate, and the smallest part, where it enters the chimney, ^th of the area of the fire grate ; but in some of the boilers proportioned on this plan the maximum sectional area is only ,?j or ^, according to the purposes of the boiler. These proportions are retained whether the boiler is fine or tubular, and from 14 to 16 square feet of tube surface is al- lowed 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 differ- ent 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. JSvaporative Power of different of Goal. kinds K.. ,..,....,0.... "Sl^ 4 6 7 9 10 11 12 The best WeKh Aatiracite \nierican The beat small Pittsburgh Arerage small Newcastle PennaylTam-m Coke m Qaa works W 1 h 9 d Newcastle, mixed ^ and 1 De Ij h Band.maUNewcaatle,iandi A e ag large Newcastle De hj h e Blj he Main, NorthumberlanJ <ti93 <H4 8 520 HU74 10 45 7 908 7 817 7 8tij 7 710 7()5& 6 772 6G0U mgth of boilers. — The extension of the expansile method of am to boilers of eiery denomination, and the gradual introduction in connection therewith of a higher pieaauie than for- hv Google THE STEAM EKGIt 219 merlj, makes the question of the strength of boilera one of great and increasing importance. This topic was very successfully eluci- dated, a few yeara ago, by a committee of the Franklin Institute, Philadelphia, and we shall here recapitulate a few of the more im- portant of the conclusions at which they arrived. Iron boiler plate was found to increase in tenacity as its temperature was raised, un- til it reached a temperature of 550° above the fi'eeaing point, at which point its tenacity began to diminish. The following table exhibits the cohesive streogth at different temperatures. 82" t< 80° (he tenacity w as — 58,000 lbs. ot I-yth below its a 570= = 6(1,500 lbs. the madmum. = 55.000 lbs. the same nearly as = 32,000 lbs. nearly J of the mai: 124IP — 22,000 lbs. nearly + of the ma: laiT" — 0,000 lbs. nearly l-7tli of the 3000° i ■oa becomes fluid. The difference in strength between strips of iron cut in the di- rection of the fibre, and strips cut across the grain, wiis 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 ac- count all these contingencies, it appears expedient to limit the ten- sile force upon boilers in actua.l 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 tho following table : — Table showing the Diminution of Strength of Copper Boiler Plates by additions to the Temperature, the Cohesion at 32° being 32,800 U)s. per Square Ineh. H» TBmi«™w™ DimnotionoT TenpsmniB DLlDiDUliOBOt Strensth. 1 90= 0-0175 g 660= 0-3425 180 0-0540 10 769 0-4398 270 0-0926 812 0-4fi44 4 01513 12 0-5&81 5 450 0-204(i 13 081 0-6691 6 460 0-2183 14 1000 0-6741 7 513 0-244G 1200 0-8801 8 629 0-2558 16 1300 1-0000 In the case of iron, the following are the results when tabulated after a similar fashion. b,Google 220 THE PRACTICAL MODEL CALCULATOR. Table of ExperimenU on Iron Boiler Plate at High Tempera- ture; the Mean Maximum Tenacity leing at 550° = 65,000 ffis. per Square Inch. D,„|„.u<,a=f T«mp«r,(u„ Dim! notion «t olatrveii. ohtetvA. 550= 0-0000 824" 0-2010 570 0-0869 932 0-3324 696 00899 S47 0-3593 600 0096* 1030 0-4478 680 0-1047 1111 0-5514 562 0-1165 1156 0-6000 0'H36 1159 0-6011 732 01491 1187 0-6352 734 01535 1287 O-6022 766 0-1589 1245 0-6715 770 01627 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 f 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 ri- veted, the head of the rivet will in all probability be soon burnt away. The best practice appears to bo to run the stays used for the water spaces in this situation, in a line soniewhat 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 at- tended 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 E of staying than is afforded by those boilers. AREA OF STEAM Rule.— To the temperature of steam in the boiler add the con- stant 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. hv Google THE STEAM ENOINE. 221 Let it be required to determine the diameter of the steam pas- sages in an engine of which the diameter of the cylinder is 48 iDchea, the length of stroke 4J 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 hy the rule we get s/JT025(250 + 459) = 2T95-84 ; the number of strokes ia 26, and the length of stroke 4 J feet ; hence itia 6 = '^\ 27i)5.84 = 0"20456(i = 0-20456 X 48 = 9'819 inches; so that the diameter of the steam passages is a little more than one- fifth 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 or FOHCE BY THE DECKEASE OF TEMPERATrEE IN THE STEAM PIPES. Rule. — From the temperature of the surface of the steam pipes subtract the temperature of the external air ; multiply the remain- der by the length of the pipes in feet, and again by the constant number or coefficient 1'68 ; then divide the product hy 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 tem- perature 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 sup- position 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 sur- face of the steam pipe is 250 — 250 X 0-05 = 237-5 ; hence we get 1-68 X 16(237-5 — 60) -,r,r,AA j /// = -i—K— = 10-044 degrees. 5 X 95 H we examine the manner of the composition of the above equa- tion, 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 frpm a higher to a lower temperature during its passage through the steam pipes, it mnst 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 de- crease the quantity of surface exposed to the cooling effect of the atmosphere. hv Google 222 THE PRACTICAL MODEL CALCULATOR. Rule. — From the temperature of the surface of the steam pipo subtract the temperature of the external air ; multiply the remain- der by the length of the pipo in feet, aud 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 tlio 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 pre- ceding example, Tihat wili be the (quantity of reduction in the clastic force occasioned by the effect of cooling in traversing the steam pipe? Since the elastic force of the steam in the boiler enters the equa- tion from which the above rule is deduced, it becomes necessary in the first place to calculate its value ; and tbis is to be done by a rule already given, ■which answers to the case in which the tempera- ture is greater than 212° ; thus we have 250 X 1-69856 = 424-640 Constant number = 205-526 add Sum = 630-166 log. 2-79945 Constant divisor = 3B3 log. 2-52^444 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 clastic force becomes / = 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 ^';th part of the entire elastic force of the steam in the boiier 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- hv Google THE STEAM ENOINB. Z)LA lout to that abstracted in the form of steam, Tfhich the known rela- tion 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 ia double acting and the feed pump single acting, this proportion will he maintained by mating 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 atmo- sphere, 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 pressnre. If, then, the feed pump he l-240th of the capacity of the cylinder when the total pressure of the steam is 20 lbs., it must be l-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 fol- lowing 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 Lcat 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 tempera- ture 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 23-2-4 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 ^'jth of the capacity of the cylinder, which allows some water to run to waste. As a maximum hv Google 224 THE PEACTICAL MODEL CALCUIATOE. effect is obtained when the temperature of the hot well is about 100°, it will not be advisable to reduce it below that temperature in practice. With the superior vacuum due to a temperature of 70° or 80° 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 Joad 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 low- ness 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. 'i'o apply this to an example — which, however, it must be remem- bered, 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 / = 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 therefore J :f -J '459 -t-( /459 + /(/— 30)~"J 42 X 229-92 12 . 1-168; ■0653 X 1-168 X N = •075T N. b,Google .■043549; therefore J jf^zrin\ = ^ '042549 = -20628. TOE STEAM EXGINE. 225 "We liave stated in a former part of this work that a cubic foot of Tvater 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 / = 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 = constant CO- efficient = 196 log. 2-2 natural number = 427-876 2-631318 constant temperature = 121 required temperature 306-876 degrees of Fahrenheit's scale 459 4- t 459 + 306-876 765-876 765-896 therefore j^ (^f _ 30) " 150(150 - 30) ~ 150 x 120 ~ 18000 459 +t 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 inten- tionally made much larger than what theory requires ; usually -^ 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 hv Google 226 THE PRACTICAL MODEL CALCULATOK. 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 !ead, and the remainder will be the cover required. Rule II, — To find at what part of the stroke any given aviount 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 arc add 90° and from the sum of these two arcs subtract the arc 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 arc, add 1 to it, and multiply the sum by half the stroke of the piston, and tho product is the length of that part of the stroke that will be made by the piston before the steam is cut off. Rule lU.^To find how muck before the end of the strolce, 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 arc 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 tho end of its strobe 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 propelling it iy 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 mul- tiply 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 al! 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 b,Google THE STEAM EKGINB. 227 below each, in the second line, is given the quantity of coyer requi- site 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 neceaaary 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, J. Directly under that, in the second line, you have the decimal '250. Suppose that you think 18 inches will be a con- venient length for the stroke of the valve, multiply the decimal ■250 by 18, which gives 4^. 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. 'Jhe words "cover" and "lap" are synony- mous. Consequently, as the cover, in this case, must be 4^ inches, and as half the stroke of the valve is 9 inches, the breadth of the port cannot be moro than (9 — 4^ = 4^) 4| 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 = 5 inches, and for the breadth of the port 10 — 5 = 5 inches. Table I. Distance of the piston from T tlie termination of its stro];e, wtien tlia steam L is cut off, in parts of tile lengtli of its stroke. J i A 1 A A i ft I A A ■]02 Coyerontliesteamsideof tile valTO, in deeiraal parts of tlielongtliofits strolie. ■289 •2T0 •250 ■228 ■204 ■m ■144 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 hv Google 228 THE PRACTICAL MODEL CALCULATOR. proper quantity of cover to give to the valve. Suppose tbat, in the last example, the valve was to have J inch of lead, wo would subtract J inch from the 5 inches found for the cover by the table : that would leave 4| inches for the quantity of cover that the valve ought to have. Table II. iSi-.fi-.' c. r rtqulTKi "' °'nSd "alS^i'fiv I'lTZZ - — «— ^ .., " Iflow"' I A i i. I J i\ ^h 21 6-94 6-48 6^00 5 -47 4 90 4-25 3-47 2-45 23i 6-79 6-34 4-79 4^16 3^39 23 6-65 G-21 5-75 5'24 4-69 4>07 8-32 2-34 22i 6-50 6-07 5^62 51S 4-59 3-25 2-29 22' 6-36 5-94 5^&0 5-02 ■4'49 3'89 3-13 2-24 2i} 6-21 5-80 5-38 4-90 4-39 3 80 8-10 219 21 6-07 5-67 6-25 4-79 4-28 3^72 3-03 2-14 aoj 5-93 5-53 5-12 4'67 4-18 3-63 2 09 20 5-78 5-40 5-00 4-56 4 08 S^54 2-04 19i 6'64 5-26 4-87 4-45 3-45 2-82 199 Id'' 5-49 518 4-7B 4-33 3-88 8-36 2 -74 1-94 131 5-34 4-62 3-77 3-27 2-67 18* 5-20 4-50 410 8'67 319 2^60 1.83 171 5-oa 4-72 4-37 3-99 a -57 3 10 2-5S 1-78 17'' 4-91 4-00 4-25 3-47 8-01 2^45 1-73 lej 4-77 4-45 412 3'76 8'36 3'92 1-68 16 4-62 4^32 4-00 8-65 8^26 2-83 2-31 1-63 151 4-48 4'18 3 ■53 3^16 2'74 2-24 1-58 15^ 4-05 3-75 3-42 306 265 2-16 1-53 l*i 4-19 3 91 8-62 8'31 2-57 2-09 148 1/ 405 3-78 3-50 3-19 2-86 2-48 2-02 143 iSi 3-90 3-64 8-37 8-08 2-75 239 1-95 137 is' 3.76 8'5I 3^25 2-96 2-65 2-80 1-B2 12i 3-61 3-37 8-12 2-85 £■55 2-21 1>80 1.27 1/ 8-47 3-24 3-00 3 '74 2-45 2-12 1-73 1-23 111 3-32 B'lO 2-87 2^62 2-08 1^66 1^17 11^ 3-18 2-97 2-75 2^51 2-24 1-95 158 1^12 101 S-03 2-83 2-63 2^14 161 1-07 10* 2-70 2-50 2^28 3-04 1-77 1^44 I^02 9i 2-65 3-56 2-37 2^17 1-93 1-88 1-32 •96 9^ 2-60 2-48 2-25 2-05 1-84 1-59 ISO ■92 Si 2-46 2-29 2-12 1-94 1-73 1'50 1^28 ■86 8* 2-31 2-16 2-00 1-82 1-68 1-42 1-15 ■81 'i 2'16 2-02 1-87 1-71 1-63 1-08 ■76 7* 203 1-89 1-75 1-60 1-48 124 1-01 ■71 6J 1-88 1-76 1^62 1-48 1'32 1'15 ■94 -66 6* 1-73 1-62 1-50 1-37 1-22 106 ■61 H 1-48 1-37 1-25 1'12 ■79 ■SB 6 144 1-35 1-25 1^14 102 ■72 -51 n 1.30 1-21 112 1^03 ■92 ■80 ■65 ■46 4' 1-16 1-00 ■91 ■82 ■71 -58 ■41 H 1-01 ■94 -87 ■80 ■71 -62 -50 ■35 3 -86 ■81 ■75 ■68 ■61 -53 -44 ■80 Tablo 11. 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 hv Google THE STEAM ENGTSE. 229 at whicli the steam may be cut off. The first column on the left hand consists of various numbers that represent the different lengths that may bo 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 ^ from the end of the stroke,} you find J 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 I at the top, you will find 3'47, which is the cover required to cause the steam to be cut off at J from the end of the stroke, if the valve has no lead. If you wish to give it lead, (say J inch,) sub- tract the half of that, or | = -ISS inch from 3-47, and you will have S-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, 8^— 3'34.'i=5'155 inches, which is the greatest breadth wo can give to the port with this length of stroke. It is scarcely necessary to observe that it is not at aH essential that the port should be so broad as this ; indeed, where great length of stroke in the valve is not inconvenient, it ia always an advantage to make it travel farther than is just neces- sary 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 down- ward stroke, that the steam is entering the upper part of the cylin- der by the upper steam-port, and escaping from below the piston by the lower exJiaustJng-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 communica- tion 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 cylin- der, 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, be- fore 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 hv Google 2-30 THE PRACTICAL MODEL CALCULATOR. the piston will te closed, and the one hehind it opened, at the same time ; but, if there is any cover on the exhausting siJe, 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 cylin- der, 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 pro port ion ably 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, un- der 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, represent- ing 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 deci- mals, which represent the distance of the piston (in terms of the length of its stroke) from the end of its stroke when the exhausting- port before it is opened, corresponding with the degree of expansion indicated by the fraction at the top of the double column and tbe 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 pis- ton 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 l-3d from the end of its stroke, and that the cover on the exhausting side of the valve is l-8th of the whole length of its stroke. Let the stroke of tho piston be 6 feet, or 72 inches. We wish to know when the exhausting-port before the piston will be closed, and when tho ono behind it will be opened. At tho top of the left-hand double column, the given de- gree of expansion (l-3d) is marked, and in the extreme left column we have at the top the given amount of cover (l-8tb). Opposite the l-8th, in the first double column, we have -ITS 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 iefore 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. hv Google THE STEAM ENGINE. ^ 'I t ^ C'uvcr on the eshoostins siiio of the Tfilvu In parts of Uifl length oC its slrolie. i £ i * DiBtance of the piaton from the end of ite Et/oTiB, when the eihaueting-port before it ia ghat (in ports of the strolte). M lilt Dialanee of the piaton from the end of it! stroke, "hen the eshanslinK-port behind it is opened (in pnrta of the atroke). ■161 ■113 ■101 ■082 Dialnnce of the piston from the end of ita stroke, iFhcn the exhausting-port before it is shut (in parta of the stroke). i i i i Diatoncc of the piaton from the end of ita stroke, when the exhansticg-port behind it ia opened (in parts of the stroke). -143 ■100 ■085 ■067 Distance of the piston from the end of ita stroke, when the exhaosting-port before it ia ahut (in parts of the stroke). mi 'III, * 1 i 1 Distance of the piaton from tho end of its stroke, when the exboo sting-port behind it is opened (in pnrts of the stroke). i i i i Diatonee of the piaton fhjm the end of its stroke, when the eihanating.port before it is shut (in parts of the attcke). 'Hi ■012 ■030 -042 ■055 DiEtance of the piston from the end of its stroke, nhen the exbousting-port behind it is opened (in parts of the stroke). •109 ■071 -053 ■043 Distance of the piaton from tlie end of its stroke, wlien the eshaueting-port before it ia sliHt (in purle of the stroke). Hi! i i i i DisLnnee of the pistno fi'om the end of its eiiiike, when tlio exhaHstitie-pof'el'md it is opened (in parti of the ptroke). ■093 ■058 ■043 ■033 Distance of the piston fVom the end of ita Etii)ke. nlipn tlie p:itl;aHf iina-port before it ia ahnt (in parts of the stroke). m " g. list Distance of the piston from the end of its stroke, when the cKhausling-port hehind it ia ojicned (in pnrta of the stroke). -074 -043 ■022 Distonee of the piaton from the end of its Etroke, when the oiiiau sting-port before it is ahnt (in parts of the stroke.) Ills ■001 ■008 ■013 ■022 Distonca of the piston Irom the end of its it ia opened (in parte of the stroke). ■053 ■027 ■024 -on Distance of the piston from the end of its it ia abut (in ptu'ta of the etroke). ■001 ■003 ■004 •on Distance of the piston from the end of its it is opened (in pnrts of the stroke). b,Google 2aa THE PRACTICAL MODEL CALCULATOR. To take another example. Let the strolte of the valve he 16 inches, the cover on the exhausting eido J inch, the cover on the steam side 3 J 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 -^ 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 3J under ^ at the head of the column, thus showing that the steam will he out ofi" at J from the end of the stroke. Again, under J at the head of the fifth double column from the left in Table III., and in a horizontal line with -^ 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 ia 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 | of the stroke,) the effect would bo 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 ■OOS 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 oeca- Bioned 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 destroy- ing the whole moving power of the engine,) when the piston is ■092 from the end of its stroke, the steam being cut off at J from the end. Whereas, if the steam is only cut off at ^^ from the end of the stroke, the moving power is not withdrawn till only -Oil of the stroke remains uncompleted. It will also be observed that in- creasing the cover on the exhausting side has the effect of retaining the action of the steam longer heldnd the piston, but it at the same time causes the eduction-port hefore 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 always 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 con- denser 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 hv Google THE STEAM EHGIKB. 2i!3 piston moves tlie 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 18 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 ex- hausting 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 ; hut, when there is even a small degree of cover on the steam side, this plan of taking more than 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- hv Google 234 THE PRACTICAL MODEL CALCULATOR. comotive engines, and in figa. 2 aDd 3 is shown an an-angement for working valves by a shifting cam, by which the amount of ex- piinsion may be varied. This particular arrangement, however, is anti(juated, 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 furtlier amount of expansion than this is wanted, it may be accomplished by wire- drawing the steam, or by so contracting the steam passage, that tiie 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 ex- cept near the beginning of the stroke where the piston travels slowly ; for as the speed of the piston increases, the pressure neces- sarily 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, there- fore, 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 cut- ting off the steam, however, is by means of a sepai-ate valve, termed an expansion valve ; but such a device appears to be hardly neces- sary 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 es- cape, is of course indispensable ; but in common rotative engines, which may realize expansive efiicacy by throttling, a separate ex- pansive valve does not appear to be required. In all engines thero 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 pres- sure of a moderate account. In all high pressure engines, there is hv Google THE STEAM EKGINE. 235 a diminution in the power caused by the counteracting pi^ssure 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 atmo- splieve 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 pres- sure of the atmosphere will become relatively as small in counter- acting the impelling pressure, as the attenuated vapour in the con- denser of a condensing engine is in resisting the lower pressure which is there employed. Setting aside loss from friction, and sup- posing the vacuum to be a perfect one, there would be no benefit arising from the use of steam of a high pressure in condensing en- gines, for the same weight of steam used without expansion, or with the same measure of expansion, would produce at every pres- sure 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 IJ cubic feet of the pressure of two atmo- spheres. But there is the same weight of steam, or the same quan- tity 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 con- siderable 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 con- denser, 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. Sothiug can be easier than to compute the ultimate hv Google 236 TIIE PRACTICAL MODEL CALCULATOR. pressure of expanded stcain, so as to see at what point expansion ceiises to be productive of benefit ; for as tlie pressure of expanded steam is inversely as the space occupied, the terminal pressure nhen the expansion is twelve times is just one-twelfth of wha,t 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 hy 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 atmo- sphere, and the tension of the vapour in the condenser be equiva- lent 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 |th of the power. A high pressure engine, there- fore, 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 con- densing. SPOEEOIDAL 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 red- hot 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 sufl'ered to fall below 350" or 400°, the spheroidal condition being no longer maintainable, steam is generated with rapidity, and the cork will be projected from the hv Google MB STEAM ENOIKB. 237 mouth of t!ie flask with great force. In a hoiler, no doubt, where there is a considerable head of water, the repellant action of the spheroidal globules will be more cifcctuallj 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 efi"ects 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 400°, 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 lii;[uid spheroid is much hotter than the spheroid itself. EXPANSION. At page 131 we have given a table of hyperbolic or Byrgcan 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 pro- portionate, will be the differential of the efficiency, and the 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 + ar). Supposing the pressure of the atmosphere to be 15 lbs., 15 + 35 = 50 lbs,, and if the steam bo cut off at i^th of the stroke, it will be expanded into four times its original volume ; so that at the ter- mination 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 « = 3. In such ease the efficiency is _ 1 -f log. (1 + 3), or 1 + log. 4. The hyperbolic logarithm of 4 is 1-386294, ao that the efficiency of the steam becomes 2-386294 ; that is, by cutting off the steam at J, more than twice the effect is produced with the same consump- tion of fuel ; in other words, one-half of the fuel is saved. hv Google 238 THE PRACTICAL MODEL CALCULATOE. 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 de- tailed tables, to facilitate the computation of the power of an en- gine 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 ^th 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 ^th of the stroke, leaving the steam to expand through the remaining |th, the mean pressure during the whole stroke would be 1'164 pounds per square inch. 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 ^ of the pressure ; but in machines in actual operation, whore there is a film of oil between the rubbing surfaces, the fraction is only about one-third of this amount, or ^'jd of the weight. The tractive re- sistance of locomotives at low speeds, which is entirely made up of friction, is in some cases -jj^th of the weight ; but on the average about ^Jijth of the load, which nearly agrees with mj former state- ment. If the total friction bo jj^th of the load, and the rolling friction be niViith of the load, then the friction of attrition must be ^Jijth 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 bo increased in the proportion of 36 to 3, or 12 times, to represent tho friction of the rubbing surface when moving with the velocity of the carriage, ^ths are about ^jth of the load, which does not differ much from the proportion of ^'^d, as previously stated. While this, however, is the average result, the friction is a good deal less in some cases. Engineers, in some experiments upon the friction, found tho 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 friction 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 sJoth of the weight. The function of lubricating substances is to prevent the rubbing surfaces from coming into contact, where- by abrasion would be produced, and unguents are effectual in this hv Google THE STEAM EXOINB. 239 EXPANDED STEAM. — MEAN PRESSURE AT DIrrEEEXT DENSITIES AND KATE OE EXPAKSIOS. Tli« column headed contains ike initial preasare in lbs., and the remaining columns contain the mean pressure in lbs., teUh different grades of expansion. BIBK^BS 1 i 1 § f 1 1 1 i 1 a 2-96 2-89 2-75 2-53 2-22 1-789 1-154 g-85 3-67 3-38 2-386 1-539 5 4-948 4-818 4-593 4-232 3-708 2-982 1-924 5-937 5-782 5-512 5-079 4-450 8-579 2-809 7 6-927 6-746 6-431 5-925 5-241 4-175 2-694 8 7-917 7-710 7-350 6-772 6-9S4 4-773 3-079 M 8-906 8-673 8-268 7-618 6-675 5-868 8-468 9-637 9-187 B-465 7-417 5-965 3-848 11 10-88-5 10-601 10-106 9-311 8-159 6-561 1*^ 11-875 11-565 10-925 10-158 8-901 7-158 4-618 12-865 12-528 11-943 11-004 9-642 7-754 5-003 14 13-864 13-492 12-863 11-851 10-384 8-631 lf> 14-844 14-456 18-781 12-697 11-126 8-947 5-773 15-834 15-420 14-700 I3>544 ia-823 16-383 15-618 14-390 12-609 10-140 C-542 IK 17-818 17-347 16-537 15-237 13-351 10-787 6-927 IH 18-702 18-811 17-448 10-808 14-093 11-833 7-312 m 19-792 19-275 18-876 16-930 14-835 11-930 7-697 m 24-740 24-093 22-968 21-162 18-543 14-912 9-621 m 29-688 28-912 27-562 25-396 22-253 11-546 34-G36 83-731 88-166 29-627 25-961 20-877 18-470 23-860 15-395 4S 44-533 41-343 38-092 33-378 26-842 17-319 5U 49-481 48-187 45-987 42-325 37-067 29-825 19-243 BiEisam ,.,T=. A 1% A A ^ A A A A a 2-980 2-930 2-880 2-710 2-589 2-299 1-981 1-608 0-990 3-974 8-913 3-780 8-614 8-065 2-612 2-087 1-820 4-892 4-232 5-901 5-870 5-670 5-421 5-079 4-698 8-180 1-981 7 6-9-55 6-848 6-615 6-325 5-925 5-364 4-624 3-652 2-311 7-948 7-560 7-228 6-772 6-131 6-284 4-174 2-641 8-943 7-618 6-897 5-945 10 9-936 9-784 S-450 9-036 8-465 7-664 6-606 5-218 8-802 11 10-929 10-763 10-395 9-989 9-311 8-480 7-266 5-789 3-632 1'', 11-740 11-340 10-843 10-158 10-994 9-963 8-687 11-851 10-729 9-248 15 14-904 14-676 14-175 13-554 12-697 11-496 9-909 16 15-897 15-654 15-120 14-457 18-544 12-263 10-569 8-348 6-283 17 16-801 16-066 16-861 14-051 13-028 11-230 8-870 5-013 17-884 17-611 15-237 13-795 11-890 6-944 1!i 18-878 18-589 17-955 17-168 16-088 14-661 12-551 yi; 19-872 19-668 18-900 18-072 I6-S80 15-828 13-212 10-430 6-600 •'5 24-840 24-460 21-102 19-160 16-515 13-040 8-256 ar 29-808 29-352 28-850 27-108 25-395 22-092 19-818 15-654 3)1 34-776 34-244 33-076 31-626 29-627 26-824 23-121 18-263 11-557 41 S9-744 39-186 87-800 86-144 38-860 30-656 26-224 20-872 13-208 45 44-912 44-028 42-525 38-092 34-888 29-727 23-481 14-859 50 [ 49-680 48-920 47-250 45-180 42-325 88-320 33-030 b,Google 240 THE PRACTICAL MODEL CALCULATOE. respect in the proportion of their viscidity ; but if the viscidity of the unguent he 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 he 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 bear- ing, when the surface is too small, wears rapidly away. For all sorts of machinery, the oil of Patrick SarsSeld Devlan, of Heading, Pa., is the best. HORSE POWEK. A horse power is an amount of mechanical force capable of rais- ing 33,000 lbs. one foot high in a minute. 'Jhe average force ex- erted 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 terras. 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 ac- tual horse power of an engine is ascertained by an instrument called an indicator ; but the nominal power la 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 cylin- der 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 ; 8 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 con- venient method of procedure. To ascertain tho nominal power by this method, multiply the square of the diameter of the cylinder ia inches by the cube root of the stroke in feet, and divide the pro- duet 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 hv Google THE STEdM EKGINE. 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 con- venient 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 np 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 the 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 hj 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 deter- mined : 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 ia relatively a much smaller one where the pressure is high than where it does not much exceed the hv Google 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| Ihs,, and by the speed of the piston in feet per minute, and divide by 42,017 ; the fjuotient is the actual horse power. The norainal horse power of a high pressure engine has never been defined ; but it should obvi- ously 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 main- tained in the following rule, which expresses the nominal horse power of high pressure engines : — Multiply the square of the diame- ter of the cylinder in inches hy 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 Yelocity, 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 poimds 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, wc introduce. PAEALLEL MOTION. Rule I. — In such a combination of two levers as is represented in Figs. 1 and 2, ^age 245, to find the length of radius bar required for any given length of lever (?, and proportion of parts of the link, 6rJE and F E, so as to make the point E move in a perpendicular line. — Multiply the length of O C by the length of the segment G E, and divide the product by the length of the segment F E. The quotient is the length of the radius bar. Rule II. — {Fig. 2, page 245.) The length of the radius har and of OG being given, to find the length of the segment (FF) of the link next the radius bar. — Multiply the length of C G by the hv Google THE STEAM EKGINB. Table of Nominal Horse Power of Low Pressure Engines. i Lem H6FS IROSE » r.Bt. 1 Ij 2 H 1 3 ~H~ i H 5 _5J_ 6 7 4 ^ -59 ^ -46 -49 ■52 ^ -66 ~^ -60 ■62 ■65 ■02 ■9e 1-86 8 1-7S l-9« 2-16 2-26 240 ■47 -60 2-T4 3-04 -30 }: 11? IflS a-M Im S?7 -Bl *25 3-64 4-1 3-76 ■6 ■M -07 -92 a-06 ■M 6-1 6-67 I 3-80 4^ 6^M tea 6-01 frM 8-111 7-1 -3 7-98 4-T7 -OS 8-60 ■27 7-80 7-90 fr2S »«6 7-86 9-61 9-SO 1-76 es lfr47 11-38 7-68 8-79 12-19 12-88 18-56 3-96 £0 10-72 12-27 4-66 15-02 6-48 3-B8 8-71 10-46 26-23 20-96 23-44 9-62 22-6B 22-M 26-sa 28-48 29-14 ;B-B9 27-82 28'07 SO'40 33-80 27-61 31-42 33-08 38-18 MB 43-41 44-69 47-05 38 soia sli? sa-71 SJ-42 1-69 39-77 4&M 48-77 6fr72 62^64 48-67 60-11 STii 42-96 4MB 47-16 66-91 82-64 88-48 88-00 81-81 86-12 6S-S6 90-26 98-40 T8-I7 34-a) 89-48 W-20 98-19 109-6 118-7 f% Tile "la 18-2 14-1 1^1 iIm j^ao 81-79 93-83 124-18 129-81 36-03 39-88 Hi 87-16 »9-M 118^ 1267 132-8 138-3 m E6-8 lSS-8 163-8 168-4 177-3 iaa-9 33-8 141-8 173-6 178-8 188-2 131-S 4fft lM-4 184-0 199-1 M ifrs" 39-4 189-0 lifrT 8^4 92-4 M^2 m^ rs 29-4 K ies-i TM 18« 186-a 198-B 95-0 02-9 10-1 228-5 236-2 i] 171-8 193-4 2404 260-6 ^ao 180-2 244-8 189-1 203-8 247-8 213-8 247-4 268-2 269-1 288-0 2-23-8 B3T-6 250-S 261-6 281-T seo-8 90 ji^ l'"-^ '■a'-t 233-9 ^'-' =^"-2 329-7 length of the link G P, 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.— (K^s. 3 and 4, pages 246 and 247.) To find the length of the radius bar (Fff), the length of CG 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. — (Pigi. 3 and^, pages 246 and 247.) To find the length of the radius bar, the horizontal distance of its centre (S) from the main 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 hv Google THE PRACTICAL MODEL CALCUtATOE. Table o/ Nominal Morse Power of High Pressure Engines. 1 I^OTH «, S™>K« » FEE,. 1 1 _ii_ 2 2J 8 JL 4 ^i 5 JL 6 7 2 ■as ■29 ^ -39 ■37 -38 40 ■42 .44 ■45 ■40 ■49 ■39 ■69 ■10 -95 I-l -49 1-17 41 lis 1-65 2-16 2-28 1-96 2-4S i s 1^ 2-i 2-83 ■M U 2-62 2-93 3-42 3-6 -69 3-30 3-18 3;33 4-05 1-1 «t 3-12 3-6? 3«l 4-2S S-BO 4-oa in *16 m 4-39 6-67 6-16 '( S-M 4-S3 6-lfl 6-66 6-8S 6-21 7^20 H fr27 fl-63 8-9B 7-62 7-80 B-13 6-ie 9-12 B-30 6-60 7-20 J-SO 10-17 8-M 8-67 fl-69 10-53 loi 7^ 11* 1-M 10-83 11-16 12-46 18-20 13-62 »-M 11-78 13-80 16-33 ll)-63 16-71 iw "! *-S7 12^ U-01 10-38 !' 12-1l IS'32 IMS ir^i a-08 17-e7 S-43 IMC 20« MH ^3 W4 23^91 14^ 20-31 22-95 2S10 M-39 16-44 26-10 le-ee 28-8G 31-20 21-ia 26-OIi 26-58 30-45 3B-28 26-0* 2»82 31-41 sasa S4-W Sa-51 34^ 3S'30 40-68 11-88 2S-»3 2»^ 31^ 46-38 3&8Y 33-91 SI 46-32 19-88 ss-01 6S-3i 80-63 10-82 4e-3B «6-62 70-17 W 67-W 86-56 68-3! M-03 eT-4S w-ra 08-22 109-9 M 6fr3T fit iK fl*-3S 90-24 103- 12 1 m2 131^ 126-0 131- 1 1-4 IBOil 162-1 16T-6 18^1 i 1»B 28'B 414 i£* 31 1624 lM-1 183-6 jgl lSS-7 ii 230-3 46 ISM H-a IM'6 223-0 230-0 m-i a46-4 les-s ISO'S 199-6 212-1 22M 2BS-: 242^8 260-a 2ffl'2 ^ im ig^g aoi* 217-4 218-6 280-1 2i*8 282-0 2SS-3 270- 263-1 272-0 s 313I mro 213-0 2S2-6 366-1 3001 22S<1 262-3 2^-6 270-5 291-t aoB-6 ^-S 361-6 sa7-2 60 ^^■S ^p 2S9-5 ^•» ^•^ *°^'^ 117-6 439-6 the beam (C D). Divide the square previously found by thia last sum, and the quotient is the length sought. Rule Y.—{Figs. 5 and 6, pages 247, 248.) — To find the length of the radius bar, C G and P Q heing given. — Square G 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 aide lever (C D). From this product subtract the product of D P into C G, and divide A hy the remainder. The quotient is the length required. Rule YI.—{Figs. 5 and 6, pages 24T, 248.) To find the length of the radius bar ; P Q, and the horizontal distance of the centre Sof the radius bar from the main centre being given. — To the given hori- zontal distance add half the versed sine (D N) of the arc described hv Google by tte estremity (D) of the side lever. Square this sum and mul- tiply the square by the length o? the side rod {P D). Call this pro- duct A. Take the same horizontal distance aa before added to the same half versed sine (D N), and multiply the sum by the length of the side rod (P D) : to the product add the product of the length of hv Google THE PRACTICAL MC LiJ tlie side lever C D into the length of Q D, and divide A by the sum. The <iuotient wiU be the length required. When the centre H of the radius has its position determined, rules 4 and 6 will always give the length of the radius bar F H. To get the length of C G, it will only he necessary to draw through the point F a line parallel to the side rod D P, and the point where that line cuts D C will bo the position of the pin G. In using these formulas and rules, the dimensions must all ho taken in the same measure ; that is, either all in feet, or all in inches ; and when great accuracy is required, the corrections mven in Table (A) must be added to on subtracted from the calculated length of the radius bar, according as it is less or greater than the length of G, the part of the beam that works it. 1. Rule 4. — Let the horizontal distance (M C) of the centre (H) hv Google THE STEAM EKGIS'E. sse a \ "x \ T ■il. / k Jxi''\ -> of tKe radius bar from the main centre be equal to 51 inches ; the half versed sine D N = 3 inches, and I) C = 126 inches ; then by the rule we will have (51 + ?if (54)^ 2916 ,,.. , 51 + 3 + 12 6 = 180 = T80" == ^^'^ '"<^^'^^' «hich h the required length of the radius bar (F H). hv Google THE PRACTICAL MOPEL CALCULATOE. Fig. 0, --f5 \ r \ 1 1 \/ v ^'^/l ^ I / \ / 1 \ 1 ■■■-. / ,-' \ , / ■••"\ --JyP^,^^, ' \ !/^ k ". Rule 5. — The following dimensions are those of the Red RoV' stefimer: C G = 32 DP = 94 QD = 74 C D = 65 P Q = 20. Bj the rule we have, A = (32)^ X 94 = 96256 and 96256 _ 96256 _ .„ , 74 X 65 - 94 X S2 ~ 1802 ~ ^^'^' ■n-hich is the reqiiSred length of the radius bar. 3. Rule 6. — Tate 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 incht Then the half versed sine of the arc D' D D" will he about inches, and we will have by the rule A = (83-5 + 2)^ X 94 = 705963-5 and A 705963-5 . 85-5 X 94 4- 65 X 74 ~ 1284-7 " ^* '^ '°^'"'^' the required length of the radius bar in this case. Table (A). This column gives ^^ when CG is the greater, and jrg when F H is the greater. Correcticn to be added to or subtraeted from the calcu- lated length of the radins bar, in decimal parts of its calcatated leogth. 10 ■9 -8 •7 % ■0034 ■01C.5 -0270 ■0817 b,Google THE STEAM BNGIXE. 249 In both of the last two examples Tip = "6 nearly. Tlic correc- tion found by Table (A), therefore, would be 54 X -027 = 1-458 inches, which must he 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 FH = 51-94 inches. In example 3 FH = 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 tho 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 qttotient 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 m = 3, and P = 10, and therefore The breadth = — D = -03 D. lo It will be observed that our rule gives the least value to the depth. In actual practice, however, it is necessary to make allow- ance 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 tho 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 formula/?' d^ = 6 W ?. To apply the formula to cranks, wo may assume the doptli at the shaft to be equal to n times the diameter of the shaft ; hence, if m X D he the diameter of the shaft, the depth of the crank will be »i X m xD. Substituting this in the formula /6d^ = 6 W^ and it becomes /5 X n^ x m' x D' = 6 W I. Now, as before, W = -7854 X P X D^, so that the formula becomes / x 5 X )t^ x m^ = 4-7124 X P X Z. The value of n is arbitrary. In practice it may be made equal to IJ or 1-5. Taking this value, then, for hv Google 250 THE PRACTICAL MODEL CALCULATOR. cast iron, the formula, becomes 15300 x i x | x m' = 4-7124 x P X i, or t305 971= 6 = P i ; but if L denote tlie Jengtli of the crank in feet, the formula becomes 60^ n^ 6 = PL, and .'. 6 ^ P X L -7- 609 m^ 'Ihia 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 pro- duct 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 ^ 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 S feet. Required the breadth of the crank at the shaft when its depth is equal to 1^ times the diameter of the shaft. In this case m = 1, so that the reserved divisor — tf- = 38 : again, elastic force of steam in lbs. per square inch = 20 lbs. ; 3 X 20 hence width of crank = — hq — = 1'6 inches nearly. Role. — 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. STEE^fOTH Of E 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 esti- mated in lbs. per square inch ; tho 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 <? = y^ ^/ 16 = Tj^ = 2-1 inches. hv Google THE STEAM ENGINE. 251 KuLE. — To find the strength of rods alternately extended and compressed, suck as the piston rods of double acting engines. — Mul- tiply the diameter of the piston in inches hj the square root of the maximum pressure of the etoam 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 liite. The rule may also be applied to determine the strength of connect- ing 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 t? = _ \/ P = = 5 inches nearly ; 47 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 malle- able 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 eases. FORMULAS FOB TDE STEENGTH OF VAKIOCS PARTS OF MARISE 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 -^ ^. We shall substi- tute P for 15 -|- p. Hence Entire pressure on piston in lbs. = '7854 X (15 -|- y) X C = -7854 X P X D^ The dimensions of the paddle-shaft journal may be found from the following formulas, which are calculated so that the strain in ordinary working = | elastic force. Diameter of paddle-shaft journal = -08264 {R X P x D=}* Length of ditto = IJ X diameter. hv Google 252 THE PRACTICAL MODEL CALCULATOR. The dimensiona of the several parts of the crank may be found from the following formulas, whica 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 = § elastic force, the strain at pin end of crank = elastic force. Exterior diameter of large eye = diameter of paddle-shaft + iD[P X 1-561 X E" + -00494 X D^ x P ^j^H Length of ditto = diameter of paddle shaft. Exterior diameter of small eye = diameter of crank pin + 02521 X v''? X D. _ Length of ditto = -0375 X ^/ P x D. Thickness of web at paddle centre = D' X P x ■/ U-561 X R^ + -00494 x D' xT} 11000 Breadth of ditto = 2 x thickness. Thickness of web at pin centre — -022 x v' P x D. Breadth of ditto = f x thickness. As these formulas are rather complicated, we may show what hey become when ^ = 10 or P = 25. Exterior diameter of large eye = diameter of paddle shaft -f \ D^/ (1-561 X R' -h -1235 x^ D^ J. 1 15-12 X ^/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 = f x thickness of web. The dimensions of the crank pin journal may he found from the following formulas, which are calculated so that strain when bear- ing at outer end = elastic force, and in ordinary working strain = one-third of elastic force. Diameter of crank-pin journal = ■028B6 x v' P x D. Length of ditto = I 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 bo as to give the strain of we1 2-22i force, and when bearing at outer end = hv Google THE STEAM ENGINE. 253 Exterior diameter of eye = diameter of hole + '02827 X P*^ X D. Depth of ditto = -0979 x P* x IX_ Diameter of journal = -01716 X s/ P x D. Length of ditto = | diameter of journal. Thickness of ^-ch at middle = -0245 x P^" x D. Breadth of ditto = -09178 x P* x D. Thickness of web at journal = -0122 X P^ X D. Breadth of ditto = -0203 x P« 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 = — — ^^| ' Length of part in piston = -04 X D X P. Major diameter of part in crosshead = -OlS X \/P X D. Minor diameter of ditto = '018 X v/P x D. _ Major diameter of part in piston = -028 X v'P X D. Minor diameter of ditto = -023 X s/P X D. Depth of gibs and cutter through crosahead = '0358 x P-* x D. Thickness of ditto = -007 X P^ X D. Depth of cutter through piston = -017 x \/P x D. Thickness of ditto = -007 x P^ 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. , Diameter of connecting rod at ends = -019 X P^ X D. . Diameter of ditto at middle = {1 + -0035 x length in inches} X -019 X x/P X D. Major diameter of part in crosstail = -0196 x P^ X D. Minor ditto = -018 X P^ x D. Breadth of butt - -0313 X P^ X D. Thickness of ditto = -025 X P^ X D. __ Mean thickness of strap at cutter = '00854 x vT x D. Ditto above cutter = -00634 x ^/P X D. _ Distance of cutter from end of strap = '0097 X x/P x D. Breadth of gibs and cutter through crosstail = -0358 x P* x D. Breadth of gibs and cutter through butt = '022 x P^ x D. Thickness of ditto = -00564 x P^ x D, hv Google 254 THE PRACTICAL MODEL CALCULATOR. Tlib dimensions of the several parta of the side rods may be found fiom the folbwmg foimulas, which aro calculated so as to make the strain of the aide lod = one-eixth of elastic force, and the strams ot strap ind cutter = one-fifth of elastic force. Diameter of cylinder side rods at ends = '0129 x P^ X D. Diameter of ditto at middle = (1 + '0035 X length in inches). X -0129 X P^ X D. Breadth of hutt = -0154 x P^ X D. Thickness of ditto = -0122 x P^ x D. Diameter of journal at top end of side rod = -01716 x P^ x D. Length of journal at top end = | diameter. Diameter of journal at bottom end = '014 X P^ x D. Length of ditto = -0152 x P^ X D. Mean thickness of strap at cutter = -00643 X P^ X D. Ditto helow cutter = -004T X P^ x D. Breadth of gibs and cutter = -016 x P* X D. Thickness of ditto = -0033 x P^ X D. The dimensions of the main centre journal may he 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 P' X D. Length of ditto = j X diameter. The dimensions of tho several parts of the air-pump may bo found from the corresponding formulas given above, by taking for D another number d the diameter of air-pump. TUB BEVEEAL 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 ; hut the determination of these proportions, from theoretical considerations, has hitherto been at- tended with insuperable difficulties, arising principally from our im- perfect 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 difi'erent 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 hv Google THE STEAM ENGIXE. 255 a unit for the measurement of extension. Tlicre is this difFcrencc, however, in the two cases, that whereas the length of a foot is fixed (iefiaitively, and is known to every one, the dimensions proper to an engine horse power difi'er 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 onco defined, the measurement should be kept inviolable. The question now arises, what standard ought to be the accepted one. Por our present pur- pose, it is necessary to connect by a formula the three quantities, nominal horses poiverj length of stroke, and diameter of cylinder. With this intention, Let S = length of stroke in feet, d = diameter of cylinder in inches ; m, , . ., , 1^ X -^S ilien the nominal horse power = ■ ,,- — nearly. I, Area of Fire Q-raU. — The average practice is to give 'SS square feet for each nominal horse power. Hence the following rule: Rule 1. — To find the area of the fire grate. — Multiply the num- ber of horses power by -55 ; the product is the area of the fire grate in square feet. Required the total area of tbe 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. Por this purpose we have, : -^g . , <?= X ^"S — .1. leet = — i,r- — leet. 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 in- ches ; divide the product )iy 86 ; the quotient is the area of Cre 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, _ 50^ X ^8 2500 X 2 total area of fire grate m square feet = ■ ■ ■ „ . ■ — = - >, . — = -n^ = 5y nearly. In order to work this example by the first rule, wo find the nominal horse power of the engine wbose dimensions we have spe- cified is 104-3 ; hence, total area of fire grate in square feet = 1064 X -55 = 58-5. hv Google 256 TUB PRACTICAL MODEL CALCULATOE, Willi regard to these rules we may remark, not only that they are founded on practice, and therefore erapyrical, hut they are only applicable to large engines. When an engine is very aroall, it re- quires a niTich 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 ^Furnace. — The length of the furnace differs con- siderably, even in the practice of the same engineer. Indeed, all the dimensions of the furnace depend to a certain extent npon the peculiarity of its position. From the difBculty of firing long fur- naces 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 Furnaoe above Bars. — This dimension is variable, but it is a common practice to mate the height about two feet. rV. Capacity/ of Furnace Chamber above Bars. — The average per horse power may be taken at I-IT feet. Hence the following rule: Bole, — To find the capacity of furnace- chamber above bars. — Multiply the number of nominal horaes power by 1'17 ; the pro- duct 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 aq. in. Hence we have the fol- lowing 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 en- gine 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 terras of the stroke and the diaraeter of cylinder. Thus, _ . , 11-2 X <f X -^S Total area of tubes or flues m square inches = ■ — ■ -,^- _<P_XJ^S - ■ — 4 ■ VI, Effective Seating 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 FLUB BOILERS. Rule 1. — To find the effective heating surfaee 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 Bi^uare feet. hv Google THE STEAM ENGINE. 257 Required the effective heating snrface of an engine of 400 nomi- nal horse power. In this case, according to the rule, effective heating surface in stjuaro feet = 400 x 5 = 2000. The effective heating surface may he expressed in terms of the length of stroke and the diameter of the cjlinder. Rule 2. — To find the total effective heating surface of marine flue boilers. — Multiply the sijuare of the diameter of cylinder in inches hy the cube root of the length of stroke in feet ; divide the product hy 10 : the quotient expresses the number of square feet of effective heating surface. Required the amount of effective heating surface for an engine irhose stroke = 8 ft., and diameter of cylinder = 50 inches. Here, according to Ruie 2, effective heating surface in square feet 50= X '^B 2500 X 2 5000 ^„„ = 10 = ^LO ^ ^0" = ^^'^- To solve this example according to the first rale, we have the nominal horse power of the engine equal to 106'4. Hence, ac- cording to Rule 2, total effective heating surface in square feet = 106-4 X 4-92 = 523^. ErrECTIVE 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 sur- face. It appears that the total heating surface of flue and tubular ma- rine boilers is about the same, namely, about 10 square feet per horse power. VII. Area of Ohimney. — Rule 1. — To find the area of chimney. — Multiply the number of nominal horse power by 10-23 ; the pro- duct is the area of chimney in square inches. Required the area of the chimney for an engine of 400 nominii.l 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 cyJinder. 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 quo- tient 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, 50= x ^"8 2500 X 2 area of chimney in square inches = f = r = 1000. hv Google 258 THE PEACTICAL MODEL CALCULATOE. To work this example according to the first rule, we find, that the nominal horse power of this engine ia 104-6 : hence, area of chimney in square inches = 104-6 x 10-23 = lOTO. The latter value is greater than the former one by 70 inches. This difference arises from our taking too gr^at a divisor in Rule 2. Either of the values, however, is near enough for all practical purposes. VIII, Water in Boiler.— Iho quantity of water in the boiler diifers not only for different boilers, but differs even for the same boiler at different times. It may he 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 tvater in the 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 cubie 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 cu- bic feet of water usually in the boiler. Required the usual quantity of water in the boilers of an engine whose stroke == 8 feet, and diameter of cylinder 50 inches. Here we have from the rule, 5 X 50= X ^8" ox 2500 x 2 cubic feet of water in boiler = -— f= = jy 25000 ^_ , = — jii — = oo2 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 1, — 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 nominal 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 di- ameter 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 wlioso stroke is 8 feet, and diameter of eyliniler 50 inelics. hv Google THE STEAM EKGINB. 259 lu tliiB case, according to the rule, area of water level m sqnare leet = ■ " 47 ' "" = I'Jo. X. iS'feflJK Room. — It is obvioua that the steam room, like the quantity of -water, is an extremely variable quantity, differing, not only for different hollers, but even in the same boiler at different times. It ia 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 he so modified as to apply when the length of stroke and diameter of cylinder are given. Rl'LE 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 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, , . ^ 50' X -^8 2500 X 2 5000 Steam room m cubic feet = -j - ■ = -,r"" = •■ -..r = 333f We find that the nominal horse power of this engine is 100--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 inva- riable, but merely to bo followed as far as circumstances will per- mit. We give them, indeed, as the medium value of the very va- riable practice of several well-knoira constructors ; consequently, although the proportions given by tho 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 prac- tice 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 re- puted nominal horse power, unless we know its maker ; but the hv Google 260 THE PRACTICAL MODEL OALCTILATOK. same confusion exists, also, to some extent, in the construction of boilers. Indeed, many things may be mentioned, which have hitherto operated aa 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 eva- poration in the boiler are indeed known, yet each engine-maker has his own rule in these matters, and which he seems to think pre- ferable to all others, and there are various circumstances influ- encing the result which render facts incomparable unless those cir- cumstances 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 — and, especially, the difference of effect of these losses of power in engines constructed on different scales of magni- tude. Practice must often vary, to a certain extent, in the con- struction of the different parts of the boiler and furnace of an en- gine ; 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 tho desired minimum of means was that of weight, or bulk, or expense of material. Again, in es- timating the proper proportions for a boiler and its appendages, reference ought to bo 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, tho duty of an engine has reference to the amount of mechanical effect produced by a certain consumption of fuel, and is independent of tho time it takes to produce that effect. In ex- pressmg the duty of engine', it would ha\e prevented much need- less confusion if the duty of the boiiei had been entirely separated from that of the engine as, indeed, thoy nro two very distinct things The duty performed by ordmaiy lind rotative steam en- gines IS — One horse power exerted by 10 llis 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 tho 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 hv Google THE STEAM ENGINE. 261 coal I'aise more than a quarter of a million of lbs. one foot bigli. 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 3i^ 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 ibs. lifted one foot high in the minute for each horse power. Some engines lift 56,000 lbs. one foot high in tjie minute hy each horse power, with a consump- tion of 10 lbs. of coal per horse power per hour; and even this performance has been somewhat exceeded without a recourse to ex- pansive action. In all modern engines the actual performance much exceeds the nominal power ; and reference must he had to this circumstance in contrasting the duty of different engines. MECHANICAL POWEIt OP BTEA)!. We may here give a table of some of the properties of steam, and of its mechanical effects at different pressures. This table may help to solve many problems respecting the mechanical effect of steam, usually requiring much laborious calculation. BIE n.i<,c.H. Effi^t .-■< i^« non Lb. TK! ^caei ^. T.tajll; 1 i i— •a •isr "^ SV3L i 1 i i i I 1-00 IH " " ' "' «a tr 10 1 1 ol 4«r4l, 04Bi Jlrt lie- hiio w J8 7 I"-' fw 1^34-0 1 1 _ J It is quite clear that although there is no theoretical limit to the benefit derivable from expansion, there must he a limit in practice, arising from the friction incidental to the ube of \eiy laige cylin- ders, the magnitude of the deduction due to uncondensed vapour when the steam is of a very low pieisuie, and othei circumstances which it is needless to relate. It is cleiJ, too, that while the effi- hv Google 262 THE PRACTICAL MODEL CALCULATOK. cleney of the steam is increased by expansive action, the cfScieney of the engine is diminished, unless the pressure of the steam or the speed of the piston be increased correspondingly ; and that an en- gine of any given size will not exert the same power if made to ope- rate 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 em- ploying a very low pressure, it would probably be found that tliere 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 aceomplish 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 va- pour in the condenser, so that the urging force might not at that point he sufficient to overcome the friction. In practice, therefore, in particular cases, expansion may he carried too far, though theo- retically 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 knoivn. Rule. — To find the Increase of E^eieney arising from workiag Steam expansively. — 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 ex- pressed in terms of the part of the stroke performed with the full pressure of steam, represents the increase of efSciency due to ex- pansion. 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 effi- ciency 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 quan- tity 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 in- crease of efficiency ? hv Google THE STEAM EKOINE. 263 Here half the stroke is to be reckoned as 1, and tlio whole stroke has therefore to he reckoned as 2, The hyperholic logarithm of 2 is -693, ivhich is the increase of efficieacy, and the total efficiency of the stroke ia 1-693, or 1-7. We may here give a tahle to illustrate the mechanical effect of steam under varying circumstances. The tahle shows the me- ,f^.. We»li»nhal r»."'. Vol«m= rf Mool.n,.^ ..^„d.i. i^z =*"""■ w^t. te° p^™,. VUh WtM. .Mv™. 1 103 20 818 173d 51 284 644 2312 126 10'*74 18U 62 £86 534 2 ]i 8 141 7437 1859 6S 287 525 2820 4 152 St So 19 -.5 54 288 516 23.4 5 161 4617 1J24 55 28J 608 2327 169 3897 1048 56 2 01 500 2331 T 176 3376 1%0 57 292 492 233:) 8 182 2983 1989 293 484 283) 187 2674 2003 69 294 477 2a43 10 102 2426 2022 60 296 470 2o47 11 197 22JI 20SQ 01 297 463 2SA 12 201 2060 20jO 62 208 456 2D35 13 20a 1904 2063 299 449 28o0 14 200- -074 64 300 443 2862 16 213 1669 2086 65 301 437 236o 16 216 1578 2097 66 802 431 2u69 17 220 1488 2107 67 303 4-J 2872 223 1411 2117 68 804 41J -875 19 226 1343 2126 69 305 414 2378 20 228 1281 2135 70 uOtl 408 2382 21 231 1225 2144 71 307 403 238j 22 234 1174 2152 3d8 2188 23 236 1127 2160 7o 309 39o 2391 24 289 1084 LI 68 74 810 888 239i 26 241 1044 2170 75 Sll 383 2317 26 243 1007 2182 76 vl2 3"9 2400 27 245 97S 2189 313 374 2403 28 248 941 2H6 -8 814 370 2400 29 250 911 2202 31o 866 2108 30 252 883 2209 80 31b 31-2 2411 SI 254 857 2215 81 817 3j3 2114 32 265 8u8 2221 82 318 854 2417 1 S3 257 810 2226 83 SIR 350 2419 34 2o3 788 2232 84 319 346 2422 35 261 767 2233 85 320 342 2425 263 748 2243 8b 321 339 2427 37 2M 729 2248 87 822 835 3480 38 266 712 22f>3 8'* 323 882 2482 39 267 695 2259 89 323 328 243j 40 269 0-9 221- i 90 824 82j 2438 41 271 664 91 325 822 2440 42 272 649 2278 02 i2b 319 2443 43 274 636 2,78 <•„ 327 31b 2445 44 275 632 2282 94 827 313 2148 45 276 610 2287 95 3 8 310 2450 46 278 5J8 06 829 307 24o3 47 58b 2-'96 97 3"0 304 2455 48 280 675 2100 OS 330 SOI 24j7 49 282 664 _304 qj S31 298 2460 . 60 654 2308 100 295 2462 b,Google 264 TUB PRACTICAL MODEL CALCULATOR. chaiiical eiFect of the steam generated from a cubic inch of water. Our foriaula gives the effect of a cuhic foot of water ; but it can he 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 8 (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 bolow 212° ; but for temperatures above this the increase is less; and for the temperatures used in practice we may consider, with- out any material error, the mechanical effect as constant. INDICATOR. An instniment for ascertaining the amount of the pressure of steam and the state of the vacuum throughout the stroke of a steam engine. Fitzgerald and Neucuran 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. la peculiarity of construction it is simply a small cylinder truly bored, and into which a piston is inserted and loaded by a spring of suit- able 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 va- cuum ; the length being the amount of revolution given to the paper during the piston's performance of its stroke. 'Xo 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 attach- ment ; 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 pro- perly adjusted, allow the engine to make a few revolutions previous to opening the cock ; by which means a horizontal line will be de- scribed upon the paper hy 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 en- gine he 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. hv Google TUB STEAM ERQINE, 265 At the termination of the stroke and immediately previous to its return, the piston of the indicator is pressed down by the surround- ing atmosphere, consequently the bottom side of the diagram is de- Bcribed, and by the time the engine is about to make another de- scending 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 a |^ clearly by example, take the follow- ing diagram, taken from a marine engine, the steam being cut off after the piston had passed through two- thirds of its stroke, the graduated scale on the indicator, tenths of an inch, as shown at each end of the diagram annexed. Previous to the cock being opened, the atmospheric line AB was formed, and, when opened, the pencil was instantly raised by the action of the steam on the piston to C, or what is generally termed the startiii0 comer; by the move- ment of the paper and at the ter- mination of the stroke the line CD was formed, showing the force of the steam and extent of expansion ; from D to E show the moments of 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 en- gine is exhibited, and every deviation from a rectangle, except that of expansion and lead of the valve show the estent 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 de- fect 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 Engine from 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 tlie indicator, divide the sum by the number of spaces, and the / ■",::n'" z /^" 3 lae TJI 7J a 11. 7.« S iM I.. S m >.. s :z5 t.. 5 11. 7 7.C a^^^^ b,Google 266 THE PKACIICAL MODEL CALCULATOR. quotient is the mean effective pressure on the piaton 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 -h 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 6'23 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. 32^ X -7854 X 15-98 x 226 , , Then ^flfio ~ horses power. The line under the diagram and parallel to the atmospheric line is 3|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 cu- bic inches equal 1 lb., it will of course rise in the barometer at- tached to the condenser about 2 inches for every lb. effect of va- cuum, 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 ia a consider- able difference in extent of vacuum in the cylinder to that in the condenser. To estimate hy means of an indicator the amount of effective power produced iy 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 ^ths 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 in ches x 16-73 lbs. X 238 feet 33000 133-26 X 7 = tq ■ = 93-282 horse power. To determine the proper velocity for the piston of a steam engine. — Multiply the logarithm of the mth part of the stroke at which the steam is cut off by 2-3, and to the product of which add •7- Mul- tiply 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. hv Google WEIGHT COKBIHED WITH MASS, VELOCITY, FOECE, AND WOEK DONE. CALCULATIONS ON THE rEINCIPLE OF "VIS VIVA. — MATERIALS EMPLOYED IN THE CONBTEUCTION OF MACHINES. — STEENGTH OF JIATERIALB, THEIR PROPERTIES. — TORSION, DEFLEXION, ELASTICITY, TENACITIES, COMPRESSIONS, ETC. — FRICTION OS REST AHD OF MOTIOiS, COEFFJCIE.NTS OF ALL SORTS OF MOTION. — BANDS. — ROPES.— WHEELS. — IIYDR.^fJ- Lies. — NEW TABLES FOR THE MOTION AND FRICTION OP WATER. — WATER-IVHEELS. — 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 couvse, 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 hori- zontally and continuously, the motion under such circumstuiicea will be uniformly accelerated : what is the acceleration ? 30 800 X 32-2 = 1-2075 feet the second. 2. IVhat force is necessary to move the above-mentioned heavy body, with a 23 feet acceleration, under the same circumstances ? glrg 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 ear, weighing 1120 lbs., moves with a 5 feet velo- city upon horizontal rails, which, let us suppose, offer no impedi- ment 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 20t!i second, or at the beginning of the 21st second ? 50 5 + 32-2 X ^j20 >^ 20 = 33-75, the velocity. 4. A carriage, circumstanced asip the Ipt question, weighs 4000 lbs. ; its initial velocity is 30 feet the secorid, and its terminal velo- city is 70 feet : with which force is the body impelled, supposing it to be in motion 20 seconds ? (70 - 30) X 4000 ,_ „ 32-2x20 =242-17 lbs. We have before noticed that the weight (W), divided by S2-2, or (?)) gives the mass; that is, hv Google atiO THE PHACTICAL MODEL CALCULATOR. ■ — -^— = mass, y And, force = masa X acceleration. 5. Suppose a railway carriage, weighing 6440 lbs., moves on a horizontal plane offering no impediment, and is uniformly accele- rated 4 feet the secoad, ivbat continuous force is applied ¥ 6440 S2'2 ~ '^^^ ^^^* ™^^^- 200 X 4 = 800 Iha., 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 s = rti + 16-1 ^^- X P. For uniformly retarded motions : F F S = at ~ lG-1 X Tir ^ t^; 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 - 32-2 X ^QQ^ X 10 = lirlT feet velocity. 20 X 10 - 16-1 X ^Qjj^ X (10)^ = 170-85 feet, distance de- scribed. 7. In order to find the mechanical work ivhich a draught-horse performs in drawing a carriage, an instrument called a dynamome- ter, 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 time. Let 126 Ihs. be the initial force; after 40 feet is described, let 130 lbs. be tho 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 two 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. hv Google TFEIOHT COMBINED WITH MASS. VELOCITY, ETC. 1304 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 rec[uired. Rule. — To the sum of the first and last, or extreme ordinatea, 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 includ- ing the extreme ones ; the result multiplied hy ^ tho 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 n = 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. weif^ht, resting on a perfectly smooth railroad, a velo- city of 100 feet ? TV1&2. ^ ^'^^'^ = 466838-2 units. A unit of work is that labour which is equal to the raising of a poand through the space of one foot. A unit of work is done when one pound pressure is esorted 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 S3000 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., ivhich transforms the velocity from 30 to 50 feet ? (30)= 64~4 "^ 13'9907, the height due to 30 feet velocity. (50V g^ = 38-8043, the height due to 50 feet velocity. From 38-8043 Take 13-9907 24-8136 5000 hv Google 270 THE PRACTICAL MODEL CALCULATOR. .-. 124068 are the units of work, and just so much workivill tlio carriage perform if a resistance be opposed to it, and it be gradu- ally 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 tlie last two questions ; Fs = (H - /() W. This simple formula involves the principle tactnieally termed the principle of vis viva, or living fobces. H is the height due to one velocity, say v or H = q— and h, the height due to another a, or 7( = K-. The weight of the mass = \\ ; the force F, and the space 8. To express this principle in words, wo may say, that the working power (Fs) which a mass either acquires when it passes from a lesser velocity {«) to a greater velocity {v), or produces when it is com- pelled 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. When 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 124068 - j)^^ = 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)^ 2000 G4-4 ^,250 ' = 1242-2352 1! We have been calculating upon the principle of vis vtca; but the product of the mass and the squai-e of the velocity, without attach- ing to it any definite idea, is termed the vis viva, or living force. 11. A body weighing 2300 ibs. moves with a velocity of 20 feet the second, required the vis viva? 2300 -^^ = 71-42857 lbs., mass. 71-42857 X (20)= = 28571-428, the amount o? vis vim. Hence, if a mass enters from a velocity a, into another v, the unit of work done is equal to half the difference of the vis vii:a, at the commencement and end of the change of velocity. For if tho mass be put = M, and W the weight, hv Google STRESGTH OP MATERIALS. 271 Then M = — , and tlie vis viva to velocity n = Ma^ = ■ ; and the vis viva to velocity v = Mu' = ■ _T.,»l{f-^-^} = (|l-f;/xW-(II-*)W.f„, q- and q-, give the heights due to the velocities v and a, respec- tively. The useful formula Fs = {H - Ji) W, before given, pago 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 bo applied. STRENGTH OF MATEEIAIS. EMPLOYED IN THE CONSTEUCTIOX OP MACHINES. In theoretical mechanics, ire 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 dy- namics. In practical mechanics, we deal with real material objects, among which we find none which arc perfectly hard, and none, ex- cept 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 deduc- tions of pure mechanics, but superadds to the formulte of the ma^ thematician a multitude of facts deduced from observation, and ex- perimentally elaborates a now code of laws suited to the varied con- ditions 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 he, it is of im- portance that its parts combine lightness with strength, and rigidity ■with uniformity of action ; and that it communicates the power without sliocks and sudden changes of motion, by which the passive resistances may be increased and the effect of the engine dimi- nished. To adjust properly the disposition and arrangement of the indi- vidual members of a machine, implies an exact knowledge and esti- tnate 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 con- tribute to the production of a machine possessing the highest amount of capability attainable with the given conditions. Materials. — The material most commonly employed in the con- hv Google 212 THE PRACTICAL MODEL CALCTJLAIOK. atruction of macliinery is iron, in the two states of cast and wrougld 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 ia the reception and trans- mission 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 direc- tion of the length. To tlie former condition of strain is opposed the tenacity of the material ; to the other ia opposed the resistance to the cruBhing of its mhstance. Rupture, by transverse strain, 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 compres- sion, from 36 to 65 tons. The resistance to extension of the best forged iron may be reckoned at 25 tons per inch ; but the corre- sponding resistance to compression, although not satisfactorily ascer- tained, is generally considered to be greatly less than that of cast iron. Roudelet makes it 31J 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 rnpture, the material being the same ; this principle holds not only in respect of iron, but also of wood. Many inquiries have been in- stituted to determine the commonly received principle, that the strength of rectangular beams of the same width to resist rupturo 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, pos- sess little novelty ; but in directing attention to the elastic proper- ties of the materials experimented upon, it was found that the re- ceived doctrine of relation between the limit of elasticity and weight requires modification. Tho common assumption is, that the de- struction of the elastic properties of a material, that is, the dis- placement 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 mani- fested in a permanent set of the material when the load did not ex- hv Google STRENOTH OP MATEKIALE. 273 ceed one-sixteenth of that necessury to produce rupture. Thus a bar of one inch equare, supported between propa 4J- feet apart, did not break till loaded with 496 lbs. but showed a permanent deflec- tion 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 break- ing weights were respectively 364 lbs. and 1120 lbs. These sets were therefore given by j'jd and j^tb of the breai;ing weights. Since these results were obtained, it has been found that time and the weight of the material itself are sufficient to effect a per- manent 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 founda- tion in practice. The beam yields continually to the load, but witii 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 moat important principle developed by experiments, fa that respecting the compression of supporting columna 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 leas 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 lower limit, the crushing force became greater, and when it was greater than the higher limit, the crushing force became less. It was fur- ther 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 ex- plain each other ; for in every height of column above the limit, the section of rupture being a plane at the same angle to the axis of the colutan, must of necessity bo a plane of the same size, and therefore in each case the cohesion of the same number of particles must bo overcome in producing rupture. And further, the same number of particles being to be overcome under the same circum- stances 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 bead under ita load. This height being surpassed, it follows that a pressure which becomes continually lesa as the length of the column is increased, will be sufficient to break it. hv Google 2V4 TEE PRACTICAL MODEL CALCULATOR. This property, moreover, is not confined to cast iron ; the ex- periments of M.'Rond.elet show that with coluaiiis of wrought iron, wood, and stone, similar results are obtained. Ej-om theae facts then, it appears that if supporting columns be taken of different diameters, and of heights so great aa not to allow of their bending, yet suf&ciently high to allow of a complete sepa- ration of the planes of fracture, that is, of heights intermediate to three times and sis times their radius, then will their strengths be as the number of particles in their planes of fracture ; and the planes of fracture being inclined at eijnal angles to the axes of tbe columns, their areas will be as the transverse sections of the .co- lumns, and consequently the strengths of the columns will be as tbeir transverse sections respectively. Taking the mean of three experiments upon a column \ inch diameter, the crushing force was 6426 lbs. ; whilst the mean of foiu' experiments, Conducted in ex- actly the same manner, upon ^ column of f 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 ia so great that jta fracture ia 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 flat, 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 flat 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. CliMaolMoEtoBSoIniim. MdinglS Length ofths oolumi oM =dii>B30 Solid cylindiioal co-l lumn of oast iron, / HolloiT ojlindricftl co- "I Inmnof oast iron, . / Solid cylindrical ■ 00-1 lumn of wtoaght iron, j W = ^'8l^ rf»- Both ends tst. T 11 "1^"'- d^" W..SS..J- For shorter columns, if W' represent the wfeight in tons which would break the column by bending alone, as given by tbe preced- hv Google STEENGTII OF MATERIALS. 275 ing formulas, and W" tlio weight in tons which would crush the co- lumn without bending it, as determined from the subjoined table, then the absolute breaking weight of the column W, is represeuted in tons by the formula, " ~w + w These rules require the use of logarithm? in their application^. When a beam is deflected bj 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 exteyided. Tbe imaginary surface at which the compression terminates and the extension begins — at which there is supposed to be neither extension nor compression — is 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 he broken. It becomes, however, a question of importance to determine the re- lation 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 metals require a much greater force to crush them than to tear them asander, 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 ob- vious, 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 compres- sion will produce its greatest effect when collected the farthest possi- ble from the neutral axis at the top of the beam ; and, in like man- ner, 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 bot- tom of the beam — joined by a comparatively slender rib. Associat- ing 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 i« ibout to bicik by n pression on the other, and consequently is of the strongest form. Thus, supposing that it is le quired to determine that foim m a girder of cast iron: the ratio of the crushing foice of that metal to the force of extension ra-xj be • taken generally as 6^ to 1, which ii theietore al«o the ratio of the lower to the upper flange, as in the annexed sectional dngrim A series of nine castings weie made, giadually mcieasmg the lower flange at the expense of the uppei one, and m the fiist aight hv Google 276 THE PRACTICAL MODEL CALCULATOE. experiments the beam broke by tbe tearing asunder of the longer flange ; and in the last experiment the beam yielded by the crush- ing 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, nhich was greatly the strongest. The circumstances of these experiments are contained in the following table. No »t siperi Rilio of surfaces of oom 1 1 tol 2-82 2368 2 Ito 2 2-87 2567 3 lto4 3-02 2737 4 lto4i 3-37 3183 5 Ito 4 4-50 3214 fl 4 to 51 6-00 3346 7 Ito 31 4-628 3246 8 4 to 4-3 5-8() 3317 » 1 to 6-1 6-4 4075 To determine the weight necessary to hreak beams cast according to the form described : Multiply the area of the section of tbe 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 multi- plied by 536 when the beams are east 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 con- trary, it is clear that the section ought to have a definite relation to the leverage at which the load acts. From a mathematical con- sideration of the conditions, it indeed appears that the effect of a given load to break the beam varies when it is placed over different yr — points of it, as the products ^—^ of the distances of these points from the points of support of \\\:i beam. Thus the eff'ect of a weight pVced at the point W^ is to the effect of the same weight acting upon the point W^, as the product AWj X W^ B is to the product AW, X W^ E ; the points of sup- port being atA andB. Since then the effect of a weight increnses 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 "75 b,Google STREKKTH OP MATERIALS. 2TY taking this law as a fundamental condition in the distribution of the strength of a beam, whose load we may conceive to be accumu- lated at the middle of ita 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. Kow this form of masi- mum 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 ele- vation, 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 espe- cially, 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 expe- rience 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 sub- jected to pressures approaching two-thirds of that which would ac- tually produce rupture ; but in engines employed to drive bone- mills, 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, fractm'e is not unfrequently produced by irregu- hv Google 278 THE PRACTICAL MODEL CALCULATOR. larity of cooling, and it is at all times better to avoid tlie danger entirely, than to endeavour to obviate it by artifice. For this rea^ son, 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 theo- retical properties of parts will not, under all the varieties of circum- stances which arise in the working of a machine, insure that exact adjustment of material and propriety of form so much desired ia constructive mechanics. Every design ought to take for its basis the mathematical conditions involved, and it would, perhaps, be im- possible to arrive at the best forms and proportions by any more direct mode of calculation ; but it is necessary to superacid to the mathematical demons tratiftn, the exercise of a well-matured judg- ment, 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 modu- lus of elasticity of his material, would, in practice, alike deceii-e 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 bo added often where least needed, merely from the deter- mination to avoid fracture, and in consequence of a want of know- ledge 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 thib a utiaight line n if the part be of consideidble length, vibration ought to be coun teracted by intersection of planes, (technically feathers,) as repre sented in the annexed diagiams, or some such form, consistent with the purpo=ie for i\hich the pirt 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 bo at the point where the force is applied; and when the strain passes hv Google STREKGTH OP 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 bo 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 rcYolution of a semi-parabola about its longer axis. In prac- tice, it Tvill, however, be sufficient to employ the frustum 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 ia 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 ia the only figure which gives an axis the property of off'ering, in every direc- tion, the same resistance to flexure. On the contrary, a square sec- tion 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, accord- ing to M. Cauchy, the modulus of rupture by torsion, T, is con- nected with the modulus of rupture by transverse strain S, by the simple analogy T = J S. 'I'he forms of all the parts of a machine, in whatever situation and under every variety of circumstances, may be deduced from these simple figures ; and, if the calculations of their dimensions be correctly determined, the parts will not only possess the requi- site degree of strength, but they will also accord with the general principles of good taste. In arranging the details of a machine, two circumstances ought 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. hv Google THE PRACTICAL MODEL CALCULATOR. hv Google STaENGTII OP MATERIALS. THE COHESIVE STRENOTH t The following Table contains the result of experiments on the cohesive Btrength of various hodies in avoirdupoie pounds ; also, one-third of the ultimate strength of each body, this being consi- dered sw^cient, in most cases, for a permanent load: Kim^ofada =^aar8Bsr Oiia-th ni Bound KiT Uu. tlinl. 1.00D3. v.. » a. a 2onoo 66b7 15, OS 5230 A-h 17000 5667 13357 4452 Teak 15000 5000 11791 3927 Fir 12000 4000 J424 3141 Beach UjOO 3&ff 90o2 soil Oal: IIUOO 361)7 8tio9 2b80 Ca«tiron 1&656 6219 I46o2 4884 English wrought iron 65872 ia-24 43881 14027 Swedish do <]o 720b4 24021 56599 iBaeo Blistered steel 183152 44381 104d7T 34859 Shear do 124400 418bb 97703 32jG8 Cnst do lS42o6 44T52 10a4o4 85151 10072 6357 14979 4993 Wrought do 3^7J2 112b4 20540 8837 Telbw brass ITOfS 1989 14112 4-04 Cait tm 4- (> 15 9 s-it 1 39 Cast leai IS 4 14 2 4-7 IROLLE'U I Rule — To find thi. ultimate cohem e stt enjth of squ ire, round, and rectangular hats, of any of the vat lous bodies, as specified in the table — Multiply the strength of an incli bai, (as in the table,) of the body required, by the ero?a sectional aiea of square ana rectangulai bar=!, or by the square of the diameter of round bars ; and the product it ill be the ultim do coht'^ne stiengtb A bar of ca-^t iion being 1^ inches square, lequiied its cohesiyo 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 f of an inch in thickness. 2 X -375 X 55872 = 41904 lbs. Required the ultimate cohesive strength of a round bar of wrought copper I of an inch in diameter. ■75^ X 26540 = 14928-75 lbs. PROBLEM ir. Rule, — Tlie weight of a body being given, to find the eross sec- tional dimensions of a bar or rod capable of sustaining that weight. — Por square and round bars, divide the weight given by oue-third of the cohesive strength of an inch bar, (as specified in tlie table,) and the square root of the quotient will he the side of the square, or diameter of the bar in inches. hv Google 282 THE PRACTICAL MODEL OALCL'LATOE, And if rectangular, divide the quotient by tlie breadth, and the result will be the thickness. What must be the side of a square bar of Swedish iron to bus- tain a permanent weight of 18000 lbs ? ■ynTfWV ^ '^^' ^^ nearly | of an inch square. Required the diameter of a round rod of cast copper to carry a weight of 6800 lbs. 6800 "^4993 " A bar of English wrought iron is to be applied to carry a weight of 2760 lbs. ; required the thickness, the breadth being two inclies. 2760 j = 1.16 inches diameter. : = -142 - ■071 of an inch in thickness. A Table skoiving the circumference of a rope equal to a chain made of iron of a given diameter, and the tveight in tonn that each is proved to carry ; also, the weight of a foot of chain made from iron of tlmt d ' ^ Provofl b tjrrj nvisht of . lintal ■in tons. 3 JandJ, 1 1-08 i i 2 1-6 4f land J, 3 2 6i i 4 2-7 6 } md ;, 5 3-3 6J ^ G 4 7 fi and X 8 4'6 n f n 5-5 8 (and J, 111 6-1 9 i 13 7.2 9i {and ft 15 8-4 101 1 inch. 18 9-4 ON THE TEANSVEnSE STRENGTH OF 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 cai-ry 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 hv Google ETEENQTH OP MATEEIALS, 283 tht ame niateiiil, C ttetlong, 4 inclica Lioad, ml 4 iiidies deep, ynll suppoit double that weight, being dnectlj 15 then bieadths, — tut a heim of that mitenal, G ftet long, 2 inches bro"d, and 8 inches deep, will austain a weight of &0U0 Iba , being as the squaie of their depths Fiom a mean ot experiments made, to asceinm the ti ms^trse st-ength of various bodie", it appear'* that the ultimite stieu^th of an inch scjuaie, and an mcii round bar of eath, 1 font long, loaded m the middle, and lying looie at both end-., is neailj a'^ follows, m lb=! a\o]idupoi« Nsmes ru 1 „ ,|a.reEar Ca lied Hnond Usr Ons 11 rd 800 21-7 6 3 201 Ash. 11 " 8=15 519 1 <l 417 !49 Pitch pine S16 SOj 719 239 Deal 4J4 14S Cast iron 2j80 8fO 202r f75 4013 lo3S 31)^ 10^0 PROBLEM I Rl/LC — To find th ultiniate hansietie Birength of any iectan gula) beam, auppoited at bcfk ends, and loaded in the mtddh, or supported tn the middle, and loaded at both ends; also, when the weight is between the middle and the end; likewise when fixed at me 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 8^ 20 — ^ = ^^-^*^ ^^^■ When a beam is supported m the middle, and loaded at each end, it will bear the same i\eight 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 m 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, aud 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 plank 24 feet long, 3 inches broad, 7 inches deep, and the ^veight placed 8 feet from one end. 2 X 16 24 '^ - 21-3 eifective length. hv Google 284 IIIB PRACTICAL MODEL CALCULATOR. Again, wben a beam is fixed at one end and loaded at tlic otlier, it will only bear J- of the weight aa when supported at both eads and loacled 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 ? 506^j<jP _ 22923 + 4 = 5730-7 lb.. The same rules apply aa 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 5^ - -j^ = 21104 lbs. IVhat 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 't v/8^ - 5' = 6-24, and 8= - 6-24^ = 269. 2028 X 269 Then, j^ = '^^^^^ ''^s- 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 uni- formly 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. raoBLEJi II, Rule. — To find the breadth or depth of beams intended to suj)- port a permanent wie^j'Ai.— Multiply the length between the sup- ports, in feet, by the weight to be supported in lbs., and divide tho 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 wilt 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 beam IG feet long, 7 inches deep, and to support a weight of 4 tons in the middle. , 8960 X 16 4 tons = 8960 lbs. and -^^ ^ -.^ ■ = 3-4 inches. What must be the depth of a cast iron beam 3-4 inches broad, 16 feet long, and to bear a permanent weight of four tons in the middle ? 8960 "x 'Tir hv Google STREKGTH OP MATERIALS. 285 When a beam is fixed at both ends, the divisor must be multi- plied by 1-5, on account of it being capable of bearing one-half When a beam is loaded uniformly throughout, and loose at both ends, the divisor must be multiplied by 2, because it nill bear double the weight. If a beam is fast at both ends, and loaded uniformly throughout, the divisor must be muitipled 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 !'>' x 1-5 ^ ^'"^ 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 v^ o-ra v fi x ■S'i ~ ^'^^ 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 "" 2C} — ~ ^^'^ eifective length of beam, and 63 cwt. = 7056 lbs. ; hence %/ -iQQ X B ~ I0'24 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 al- lowances must be made for wear ; then the divisor may be changed from 6T5 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 92400 ^ f,f.r. = -^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 -g- = -^224 = 6-07 inches diameter. Required the diameter of a solid cylinder of cast iron, for the shaft of a crane, to he capable of sustaining a weight of 10 tons ; hv Google 286 THE PBACTIOAL MOBEL CALCULATOR. one cnfl of the shaft to he made fast in the ground, the other to projfcct GJ- feet; and the effective leverage of the jih as IJ to 1. 10 tons = 22400 lbs., and 22400 X 6-5 X 1-75 ^^75-^^25 = ^^0^ And -^150y = 11-47 inches diameter. The strength of cast iron to wrought iron, ia this direction, is as 9 is to 14 nearly ; hence, if wrought iron is taken in place of cast iron in the last example, what must be its diameter ? ISUil X y ^„„. , V TT — ■ = y'o9 inches diameter. ON TORSION OK 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 ul- timate strength of the shaft or bar, in lbs, avoirdupois. 2. — Multiply the force applied in pounds by the length of the lever in inches, and divide the product by one-third of tlie 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. The following Table contains the result of experiments on ineh hars, of various metals, in lbs. avoirdupois. N=m»„0^i» Rl.miB:.t. 0.»-UlM. S,=»r. Bar. 0«.>-,I.M. 11943 120G3 11400 20025 20508 211 U 6549 4825 1G98 1206 SflSl 4021 3800 6876 7037 1850 1C08 563 402 15206 15360 14502 25i97 26112 T065 0144 2150 1536 5069 5120 4864 8409 8701 8960 2355 2048 717 English wrouglit iron SweiUsl. <lo. do. Cast do Tin What weight, applied on the end of a 5 feet lever, will wrench asunder a 3 inch round bar of cast iron ? -.Tfj = 5374 lbs. avoirdupois. Required the side of a square bar of wrought iron, capable of re- sisting the twist of 600 lbs. on the end of a lover 8 feet long. ()60~x 116 ^"""51:^0"" =^i inches. hv Google STRESGTn OP MATERIALS. 287 In the case of revolving shafts for machinery, &c., the strength is directly as the cuhes of their diameters, and revolutions, and in- versely C5 the resistance they have to overcome ; hence, From praetiee, we find that a 40 horse power steam engine, making 25 revolutioaa per minute, requires a shaft (if made of •wrouffht-iron) to be 8 inches diameter : now, the cube of 8, multi- plied 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 G5 horse power, making 23 revolutions per minute ? 65 X 3-20 „„„ . ■&" ng = 9'67 inches diameter. James Glenie, the mathematician, gives 400 as a constant mul- tiplier for cast iron shafts tliat are intended for first movers in ma- chinery ; 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. Kerjuired the diameter of bearings for the fly-wheel shaft. 400 X 30 = 8-579 inches diameter. Required the diameter of the hearings of shafts, as second movers from a 30 horse engine ; their velocity being 36 revolutions pet- minute. 200 X 30 ■^ OQ — = 5-5 inches diameter. "When shafting is intended to be of ivrougbt iron, use 100 as the multiplier for second movers ; and 80 for shafts connecting smaller machinery. Table of the Proportionate Length of Searings, c Shafts of V, ■ " ■ Journals for ni^.i.7a=b=». I.en,inl»d.w Bii.fBl»eha La=.i„I»cW 1 li 61 8| 11 2i 1 9-1} 3 71 10 2i H 8 lOJ 21 31 81 111 S 4-L 9 12 31- 4- 91 121 4 5- 10 131 41 6- lOJ 14 6> 11 14} 61 71 11} 15. 6 8i 12 10 b,Google THE PEACTICAL MODEL CALCULATOE. uiti't, Ris fiances to Compression, and other Properties of tlic •.oimnon Materials o/ Construction. Al«tof coDir isd^ltbC =t .c™. ''*"" ^'f-^'" '^ni::t to Ita per Ki. Ita Hrsnilh tiii'/t' I««iff„e«i» A«li uua _ 0-23 2-6 0-089 Betth 12225 8i,48 015 21 0-073 17308 10304 0435 0-9 0-49 Bnok 275 562 Cast iron 13434 80^97 I'OOO 1-0 1-000 Copper (wTOnglit) 3u(J0O Elm 9720 1038 0-21 2-9 0073 Fir, or Pine, white 1J346 2028 0-23 2-4 0-1 - — — red llbOO 6^75 0-3 2-4 01 - - yellDw llS3o 5445 0-25 2-9 0-087 Granite, Aberdeen 10910 Griin metal (copper 8 andtml)' GSSOS 0-65 1-25 0-53B MaUeable iron 56000 1-13 0-86 Laieli 12240 5jb8 0-136 2-3 0058 I*ad 1824 0-096 2-5 0-0385 Mahogany, Honduras 114-5 8000 0-24 2-9 0-487 Majble 5e>1 faOuu - Oak 118M) 1j04 0-25 2-8 O-O08 Rope (1 in m oircum ) 200 - Steel 128000 - Stone Baih 478 _ — Craigleith 772 6410 _ — Dundee 2661 6630 _ — Portland 857 3720 Tin (cBsil 47^1. 0-182 0-75 0-25 Zino (slice c) 0120 — 0-S6i 0-5 0-76 Comparative Strength and Weight of Ropes and Chains. 1 II A ..„, 1 1 ■1 3 a T" '4 :2«;: n ■4 a ii (i » u J3 43 3! 2f ft 5J 1 6S 10 23 i 10 -l', 4>, 8 1 16J XOj 28 if 40 11 11 h S| ;, IIH 2 10 lU 30! lin. hli 13 8 S! 7 s M 3 5i 121 36 lA 63 14 18 61 S)i A IK 4 3., 13 30 n 71 16 14 7 in ■ffl 6 2 13} 46 1* 70 18 11 H IS « •27 6 4J 141 4K1 1-i Kl 20 8 «i i» 1 32 7 7 l.H 66 ift 06 22 13 »i 21 la 37 8 13S 16 60 ■If 106 24 18 It must be understood and also borne in minii, that in estimating the amount of tensile strain to which a body is subjected, the weight of the body itself must also be taken into account ; for according to its position so may it approsimate to its whole weight, in tcnd- hv Google STRENGTH OP MATBMALS. 289 ing to produce tension within itself; as in the almost constant application of ropes and chains to great depths, considerable heights, ko. Alloys that are of greater Tenacity than the sum of their Constitu- ents, as determined hy the Experiments of Musehenbroeh Swedish oopper 6 parts, Malacca tin 1 — teoacitj per aquire jnch 64,000 lbs, CbiU copper 8 parts, Malacca tin 1 60,000 Japan copper 5 parts, Banca tin 1 57,000 Anglesea oopper 6 parts, Cornish tin 1 41,000 Common Mock tin 4 lead 1 lino ! 18,000 Malacca tin 4, regnlus of antimony 1 12,000 Block tin o, lead 1 10.200 Block tin S, imo 1 10,000 Lead 1, aino 1 4,500 tttify and Stiengtk of v trious Species of Timber Species otTimliw. TllUSOfE. V»1.8»fS.|| SpwUsotTiiDber. V^lueofE. Vllneofd. I74-T 122 'Se 105 155-5 . 86-2 70-5 119 98 2463 2221 1672 1766 1457 1383 2026 1556 50-64 88-68 133 158-5 90 76 105-47 1013 1632 1341 1102 1100 1200 800 1474 English oak. Canadian do Dantzic do Adriatic do New England fir Mar Forest do. Norway spruce,. . 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 Iha. 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 When the beam is intended to bo 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-T, and the fourth root of the product la the diameter. The distance hotiveen the supports of a beam of Riga fir is 16 feet, and the weight it must be capable of sustaining in the middle of its length is 8000 ibs., with a deflection of not more than I of an inch ; what must be the depth of the beam, suppos- ing the breadth 8 inches ? 16 X 8000 - 90 X 32 X -7 5 ^ ^^^^^ "^ ^ = "^^^^"^ == 1^'^^ '"• t''^ ''epth. Rule. — To determine the absolute strength of a rectangular beam of timber when supported at both ends, and loaded in the middle 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 emergency. — 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 hv Google 290 THE PRACTICAL MODEL CALCULATOR. in inches, anJ the quotient iviU be the ahsoliite 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 aa permanent load. If the load is to be applied at any other point than the middle, then the strength will he, 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. And 18 feet - — 17 feet, or 204 inches, effective length of beam. 1341 X 4 X 10 X 60 ,_^^ Tabular value of 8, red pme = nnj ' = l-'Ji 1 6 Iba., the absolute strength of each piece of timber at that point. Rule. — To determine the dimensions of a rectangular beam capa- ble of supporting a required weight, with a given degree of deflection, when fixed at one end. — Divide the weight to be supported, in lbs., 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 ia intended to bear a load of 700 lbs. at its ex- tremity ; its length being 5 feet, its breadth 4 inches, and the de- flection not to exceed ^ an inch. Tabular value of E = 119 x 4 x -5 = 238, the divisor ; then 700 -H 238 = -^2^ X 5 = 7-25 inches, depth of the beam. Rule. — To find the absolute strength of a rectangular beam, when 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 quo- tient equal the absolute strength of the'beam in lbs. A beam of Riga fir, 12 inches by 4^, and projecting G} feet from the wail; what ia the greatest weight it will support at the ex- tremity of its length ? Tabular value of S = 1100 12 X 4'5 = 54 sectional area, 1100 X 12 X 54 „ „„ . Then, ■ -yg = 91384 lbs. hv Google STRESGTH OF MATERIALS. 291 "When fracture of a beam is producoil by vertical pressure, the fitires of the lower section of fracture are separated by exteasion, 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 ia distinguished as the point of neutral axis. Therefore, by the law of fracture thus esta- blished, 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 re- sistance to compression is nearly as 6^ to 1 of tenacity ; conse- quently 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 jf being about 6} times that of the upper ; but such is not , [ the case with beams of timber ; for although the tenacity of timber he on an average twice that of its resistance to compres- sion, its flexibility is so great, that any considerable length of beam, where columns cannot be situated to its support, requires to he strengthened or trussed by iron rods, as in the following manner : An 1 these applications of pnnciple not only tend to diminish de- flection lut the required purpose is also more effectively attained, and thit by lighter pieces of timber. Ktjlb — To ascertain tie ah olute strength of a oast iron beam of tkt pj uedinff form, oj that of strongest section. — Multiply the sec- tional area of the bottom flange in inches by the depth of the beam in inches, and divide the product by the distance between the sup- ports 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 demon- strated that the maximum of strength is obtained in cast iron, when the thickness of the annulus or ring amounts to Jth 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 shows the greatest weight that ever ought to be laid upon a beam for permanent load, and if there bo any liability to jerks, &c., ample allowance must be made; also, the weight of the beam itself must be included. KuLE. — To find the weight of a cast iron beam of given dimen- sions. — 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. H X 1-5 X 16 X 3-2 = 844-8 lbs. hv Google Sya THE PRACTICAL MODEL CALCDLATOK. 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 weight will produce. E^asH. et«Dt 7 tea. SM. 9t^el. 10 f«. io'. 3 1278 '24 1089 ■33 954 ■426 855 ■64 765 ■66 H 1739 ■205 1482 ■28 12«8 ■365 1164 ■46 1041 ■57 4 2272 ■18 ■245 1700 ■32 1520 ■405 1360 ■5 H 2875 ■16 2450 ■217 2146 ■284 1924 ■86 1721 ■443 5^ 3560 ■144 3050 ■196 2650 ■256 2375 2125 ■4 6 5112 ■12 4856 ■163 3816 ■213 8420 ■27 8060 T 6958 ■103 5929 ■14 5194 ■183 4655 -23 4165 8 9088 ■09 7744 ■123 6784 ■16 6080 ■203 6440 ■25 9801 ■103 8586 ■142 7696 ■18 ■22 10 12100 10600 ■128 9500 ■162 8500 ■2 li 12826 ■117 11495 ■15 10285 ■18 12 15264 ■107 18680 -186 12240 ■17 13 16100 ■125 14400 ■154 U 6 - - ~ — — - 18600 ■115 16700 ■148 lift liu T. ISftst. 18te=t. Wtett. 1 2548 ■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 8396 ■64 3020 -81 2720 I'OO a 5738 ■33 4914 -44 4302 ■57 8826 -72 8488 ■89 10 7083 6071 ■39 5312 ■51 4722 ■64 4250 11 8570 ■26 7846 6428 ■47 5714 ■59 5142 ■73 12 10192 ■24 8736 7648 ■48 6796 ■54 6120 ■67 IS 11971 ■22 ■81 8978 -39 7980 -49 7182 -CI 14 1SS83 ■21 11900 ■28 10412 9253 ■46 8330 ■57 15 15987 ■19 18660 ■26 11952 ■31 10624 -43 9502 ■53 16 18128 ■18 15586 ■24 18584 ■82 12080 ■10 10880 ■5 17 20500 ■17 17500 ■28 15353 ■8 13647 12282 ■47 IS 22982 ■18 19656 ■21 17208 ■28 15700 ■36 13752 ■44 Resistance of Bodies to Flexure hy Vertical Pressure. — When a piece of tinfber 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 timea its diameter, it is impossible to bend it by any force applied longi- tudinally, 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 cir- cumstanced. Wlien the length of a cast iron column with flat ends c([ual3 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 h^ bending : but, when the column hv Google STREHGTH OP MATERIALS. 293 is enlarged in the middle of its length from one and a half to twice its diameter at the ends, hy heing cast hollow, the strength is greater by ^th than in a solid column coataining the same quantity of material. Rule. — To determine the dimensions of a support or column to hear 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 eolumu'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 5^ Tabular value of E = 105, and i,Q-— jng-^— o = ^8'8tt8 = 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 9= — Tabular value of E = 96, and gn v ' qii " ~ '^^^53 = 4 inches nearly. IHmensions of Oylindrioal Columns of Oast Iron to sustain a t/iven load or pressure with safety. s li L«w orheiBhtlnftM ; 4 6 8 10 12 14 1 le 18 20 1 22 1 24 1 Wtigbloiloaaino™. | fl T" 60 49 40 82 2fi 22 18 15 13 11 n m 10!1 91 77 65 55 47 40 34 29 25 17« loa 145 iy« 111 97 84 7H 64 56 49 M ■Ml 232 2X4 IHJ 172 156 106 94 83 810 28S ■;m 242 220 19H 17fi 160 144 130 400 «;.+ 327 KOI 275 251 229 208 189 fi 4.1'/ 427 365 337 810 285 262 « m 599 673 550 .125 497 469 440 413 3K6 360 7 Mm 1013 989 fift9 924 887 H4K 808 7Hf. » M\m 1315 V/hH im4 1185 1142 1097 1052 1003 959 1H4II 1603 IfiKI 1515 1467 1416 1364 1311 m ?UM 2100 2077 VO+5 2007 1964 1916 1885 1S11 17.15 1697 11 7r>7(i 2550 2520 V4WI 2450 2410 23.18 2m H24H Ml 89 2127 12 SUaO 3040 ■im 2U70 2930 ■i'm 2830 2780 2730 2670 2600 Practical utility of thepreceding 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 hv Google 294 TDE 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 civts. The strength of cast iron as a column being = 1-0000 ~ steel — = 2-518 — wrought iron — = 1-745 — oak (Dantzic) — = -1088 — red deal — = -0785 £lastieity of torsion, or resistance of bodies to twisting. — The angle of flexure by torsion is as the length and extensibilitj 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. Rule. — Multiply the power in lbs. by the length of the shaft in feet, and by the leverage in feet ; divide the product by fifty-fivo 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 circum- ference of a wheel 2^ 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 -.^ . .« . , cr ^ = '>/Vd!i\ = 6-67 inches in diameter. 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 — 1-96 Gun-metal = -55 Cast do = 2-1 Deflexion of Rectasgui^^r Beams. Rule. — To ascertain the ainount of deflexion of a uniform beam of cast iron, supported at both ends, and loaded in the middle to the 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 tho Required the deflection of a cast iron beam 18 feet long between the supports, 12-8 inches deep, 2-56 inches in breadth, and bear- ing a weight of 20,000 lbs. in the middle of its length. 18= X -02 — Yg .Q •• = '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 end hv Google ETKENGTH OP MATEKIAIS. 295 and loaded at the other. — Divide the length in feet of the fixed part of the heam by the bngth in feet of the part which yields to the force, and add 1 to the quotieat ; then multiply the square of the length in feet by the quotient so increased, and also by -13 ; divide this product by the mitldle depth in inches, and the quotient will be the deflection, in inches also. Multiply the deflection so obtained for cast iron by -SG, the pro- duct equal the deflection for wrought iron ; for oak, multiply by 2-8; and for fir, 2-4. A Table of the Depths of Square Beams or Bars of Cast Iron, calculated to mj^^iort from 1 Cwt. to 14 Tons in the Middle, the Deflection not to exceed -^^th of an Inch for each Foot in Length. L«^ .F». * a S 10 12 14 18 1. 20 K 24 ■m .|» i ! 1 i 1 } 4-fl us ■s V. i 1 1 68 ■9 i 4-8 i b-2 i 3% O-l ID-S I dirt. IS 3i 1,'lSO 1,232 1,SU i;4B6 1,6S0 ,oia 2,300 SflHO 3900 MOiO 1 2U0 1 two 1 0>0 £0160 22,400 M.8M 1-9 1 2-8 3'0 3° *9 3D i 3-6 SD SI 6-1 In. i to ST IM )8 3-4 1* 5-7 *■! 4-6 i i 3 84 SO M.Umh.«i» 1 15 ii a ba W«» P„ 10 I 10-6 IS 18(30 2S 21 2. S] '1^ 21 30,810 40 20 4^6110 4iM0 63, «B i hh 5. !„ L'|„. ^ "»■=" ■ "• -5 3 ^ 4 - • 66 « M 3 ■». b,Google THE PRACIICAL MODEL CALCULATOR. „,..., 14 16 13 20 1» » 26 1» 30 a £4 Bli i' i' W'W^ia ■«r" i 1 i 1 i 1 1 i 1 1 it's IS'l 1§'S lB-2 W3 i i 40 BW 107^20 112,009 110,480 IS 131,400 i 2'! S-0 iB-a IS'? In. 14'S 4-1 4« 4-6 4« in 16-1 IM 16^ 16-5 1 -0 1 -I In. ia-1 i 18-e 17-4 7-T 7-« 8-1 18-7 iff-e 19-2 lS-4 201 20-8 :»-0 pi -M ■* ■M ■5 ■8i ■a ■m -T ■78 ■8 ■36 ■9 lis I'U ^ fc3 illustrative of the Table. — 1. To find the depth of a rectangular bar of east iron to support a weight of 10 tons in the middle of its length, the deflection not to exceed ^ 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 J of 15"3 = 2'6 inches, the breadth. 2. To find the diameter for a cast iron shaft or soUd cylinder that will bear a given pressure, the flexure in the middle not to ex- inch for each foot of its length, the distance of the 20 feet, and the pressure on the middle equals 10 ceed ^'jth of bearings be. tons. Constant nltiplier 1-7 for round shafts, then 10 X 1-7 = IT. And opposite 17 tons and under 20 feet is 11'2 inches for the di- ameter. But half that flexure ia quite enough for revolving shafts : hence 17 X 2 = 34 tons, and opposite 34 tons is 13-3 inches for the di- ameter. 3. A body 256 lbs. weight, presses against its horizontal sup- port, 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_ 2,50 ' ■ •203125 = , in this case, the eoeffieient offnciion; .: 8750 X 203125 = 1777-34375 lbs., the force required. This calculation is based upon the law, that friction is propor- tional 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 he 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 '! hv Google STREKGTIl OF MATERIALS. 297 28 „ ^TQ = ■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 uuits of work required. A CNIT OP WORK IB the labour which ia 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 ex- tent 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 leas than that of quiescence. a 5. Required the co- efficient of friction, for a sliditig motion, of cast iron upon wrought, lubricated with Dev- lin's oil, and under the following circum- stances : the load A, and sledge nm, weighs 8420 lbs., and requires a weight W, of 1200 lbs. to cause it to pa*?? 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 ease the coefficient of sliding motion will bo 1200 1200 -I- 8420 2 x 2 2 8420 ~~ 8420 ^ </ x 8"" in which ^ = 32'2 feet; the acceleration of the free descent of bodies brought about by gravity. The above expression becomes 44 142515 - 1-142515 X goM^ = ■118121. Hence the coefficient of the friction of motion is •118121, and the coefficient of the friction of quiescence ia "142515. OF IRICTION, In the years 1831, 1832, and 1833, a very extensive set of ex- periments 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 est.abli3hed : 1. When no unguent is interposed, the friction of any two sur- faces (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. AVhilst this ratio is thus the same for the same hv Google ays THE PEACTICAL MODEL CALCULATOR. surfaces of contact, it is different for JifFercnt surfaces of contact. The particular value of it iti respect to a,ny two given surfaces of contact is called the coefficient of friction in respect to those sur- faces. 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 to- gether 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 velo- city of the motion. 4. That where unguents arc 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 un- guent, 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 con- tinuous 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. M. Morin found that with unguents (hog's lard and olive oil) interposed in a continuous stra- tum 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 be- tween -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 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 ^th of the whole pressure ; when soft soap was ap- plied, it became ^th ; and with the softer unguents applied, such as oil, hog's lard, &c., the ratio of the friction to the pressure in- creased ; 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 weigtit tend- ing to force the surfaces together. hv Google STRERGTH OF MATERIALS. Table of tlm Hesiilts of Experiments on the Friction, of Unctuous Surfaces. By M. Mows. Oak upon oak, the fibres being paralle! to the motion Ditto, the libreB of the moving body being perpendicu- lar to the motion Oak apon elm, fibres parallel Elm upon oafc, do Eeeeh upon oak, do Elm upon elm, do Wrought iron upon cJm, do Ditto npon wrought iron, do Ditto upon oast iron, do Cast iron upon ncoaght iron, do Wrought iron npon brosa, do Brass upon wrougiit iron, do Cast iron upon onk, do Ditto upon, elm, do., tha unguent being tallow. Ditto, do., lie unguent being hog's lard and blaok Elm upon oast iron Ditto upon bcasa , Copper npon oak Yellow copper upon cast iron Leather (oi-Mde), well tanned, upon Ditto upon brass, wetted OIGO O'liiG O'lCT O'lSo 0144 0'132 0-391) 0-314 0'420 In these esperiments, the surfaces, after having been smeared ivith an unguent, were wiped, so that no interposing layer of the unguent prevented intimate contact. Taele of the Results of Experiments on, FHction, ivith Unguents interposed. By M. MoRiK, C06ffl=lsnU of FrlcMon. 0'164 0-440 0-075 0-164 0067 0-083 0-254 0-072 0-250 0-136 0-073 0-178 0-066 0-080 0-098 0-055 0-187 0-41t 0-170 0-142 0-OfiO 0-]30 0-217 0-066 0-2o6 0-649 0-214 ... Oiik upon oak, libres parallel.. upon wrought iron Beeoh npon oak, fibres parallel. Elm upon oak, do Do. do Do. do Elm upon elm, do Do. upon cast iron Wrought iron upon oak, fibres ■) parallel J Dry soap. Tallow. Hog's lard. Tallow. Dry soap. TalloiT. Hog's lard. Dry soap. Tallow. t Greased and s: \ rated wilb wat Dry Boa,p. b,Google THE PIUOTICAL MODEL CALCULATOR. C»fflcl^„l>rfFri«i.n. | Wrought iron upon oak, fibres 1 parallel i 085 0108 Tallow. Do. upon elm, do 0-078 Tallow. Ho. do 0-07S Hos's !a.rd. Do. do 056 Olive oil. Do. upon i.a=t iron do Oi08 Tallow. Do. do 073 Hoe's Inrd. Do. do 00(16 0100 Olive oil. Do. ui on -wiought iron, do 082 Tullow. Do, do 081 Ilog's lard. Do. do 070 011s Olive oil. Wrought irm upon brass, do 0103 Tallow. Do. do 073 Hog's lard. Do. do 078 Oliva oil. Cast iron upon oak, do 0189 Dry soap. Do. do 218 01b f Greased and satu- \rated with water. Do. do 078 0100 Tallow. Do. do 075 Hog's lard. Do. do 075 100 Olive oil. Do. upon elm, do 077 Tdlow. Do. do OObl Olive oil. Do. do 091 ; Hog's lard and \ plumbago. j Do. iipouwrougbt lion 0100 Tallow. Do. upon CJ-.t lion 314 Water. Do. do 197 Soap. Do, do 0100 100 Tallow. Do. do 070 100 Hog's lard. Do, do OOUJ Olive oil. Do. do 0^5 f Hog'a lard and \ plumbago. Do, upon bi a-" 103 Tallow. Do. do 07o Hog's lard. Do. do 078 Olive oil. Copper upon oak, fibres paralle 001 0100 Tallow. Yellow copper upon cast iron. 072 0103 Tallow. Do. do 0l>8 Hog's lard. Do do O'OfjO Olive oil ErasB upon cast lion 086 010b Tallow Do do 077 Olive cid Do upon wrought iron 081 1-illow Do do 089 ( l-ird lud plum- Ihftgo Do do 072 Olive oil Bras? upon brass OOjS Oliie oil Steel upon cast iron 0105 0108 Tallow Do do 0-081 Hog Elm d. Do do 0-070 01i>e oil Du upon wruuglit iron 093 Tallow Do do 076 HogslirJ. Do upon brass 056 Tillow Do do OOjS Obi e ml Ho do 0(,7 f Lard and plum- 1 li i„o Tanned os-hide upon east iron.. o-ati5 ... Jf.i™-elam(satu- 1 Tilted with water. The esteut of the surfaces in these experiments bore such ai-olati"u to II sure as to cause them to be separated from one another throughout by m posed stratum of the unguent. b,Google STKENOTH OF MATERIALS. Table ofthi. Bi&uHs of Expinments on the Fiution ot Gudgeons or Axle-ends, in motion vpon their bearings By M ftlORlH, SurWilnCuBlat saUoniifSurf^es LosfficeaL fFritU™. Conted with oil of olneo ) withhogsUrd tallow, \ OT to 0-08 Cast iion a\!ef jn and soft gome J roitfd with asphaltum 0U54 Creasy Greasy and wetted. Coated with oil of olives, 1 witlihogalard,tallow \ 07 to 0-08 Cast jjon a-ilp'j in cast iron be itiDj^ Greasy Greasy and damped OIG 16 Startely greaev Coated with oil of oiiYss, tallow lioga Inrd oc 07 to 008 ings soft gome Toated with oil of olivea 07 to 0-08 n09 019 Wrought iron a-^ka ii!bia-3 be^rmga hogs lard or tillow, / Coaled with hard gome tieasy ani wetted Iron axles in lignum Coated with oil or hog s \ Oil ■vitiE beoimgs Greasy 019 Coated with oil bearrogs Hithhngc lard Table of Coefficients of Friction under Pressures increased continu- ally up to limits of Abrasion. C™ffi<:i=»U .fFrL,.U»n. 32-61bs. ■140 ■174 ■166 ■157 I'BlJowts. ■250 ■275 ■300 ■225 2-00 ■271 ■292 ■333 -219 2-33 ■E21 ■340 ■329 ■344 ■211 8-O0 ■312 ■333 ■347 ■215 3-33 ■300 ■351 -3S1 ■206 S-66 ■876 ■353 -353 ■205 4-00 ■395 ■365 ■354 ■208 4'33 ■403 ■366 ■856 ■221 iM ■409 ■366 ■357 5'00 ■367 ■858 ■238 ■367 ■351) 5-66 ■367 -367 ■235 6-00 ■376 ■403 ■233 ■434 6-66 ■235 7-00 7-33 ;;;;;; ■232 b,Google 302 THE PRACTICAL MODEL CALCULATOR. Comparative friction of steam engines of different modificntion?, if the beam engine be taken aa the stanctard of comparison : — The vibrating engine has a gainof l-l percent. The direct-action engine, with slides — loss of 1-8 — Ditto, with rollers — gain of 0-8 — Ditto, with a parallel motion — gain of 1'3 — Excessive allowance for friction lias liitherto been made in cal- culating the effective power of engines in general ; as it is found practically, by experiments, that, where the pressure upon the pis- ton is about 12 lbs. per square inch, the friction does not amount to more than 1| lbs. ; and also that, by esperiments with an indi- cator 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 EXPEUlMtlNTS MADE BY 31. MORIS ON TUB STIFFNESS OF EOrEfi, OH. TilE RESISTANCE OP BOrES TO EENDINO UPON A CIECIILAR ARC. The experiments upon which the rules and table following are founded were made by Coulomb, with an apparatus the invention of Amonton, and Coidomb himself deduced from them the follow- ing results : — 1. That the resistance to bending could be represented by an p nut f tw t rms, the one constant for each rope nd a h 11 1 h w hall designate by the letter A, and wl 1 tb jl 1 f h n m d the natural stiffness, because it de- p ! n 1 m d f f b ion of tho rope, and the degree of t f t y n nd t ands ; the other, proportional to the t n n T ft! nd f th ope which is being bent, and which p d >y th p 1 t BT, in which B is also a number ntntf 1 padah roller, 2 Tb t h ta t 1 nding varied inversely as the diame- t ftl 11 Thus the complete resistance is represented by the expression A + BT D " ' where D represents the diameter of the roller. Coulomb supposed that for tarred ropes the stiffness was pro- portional to the number of yarns, and M. Navier inferred, from examination of Coulomb's experiments, that the coeflicionta A and E were proportional to a certain power of the diameter, which de- pended on the extent to which the cords were worn. JI. Morin, however, deems this hypothesis inadmissible, and the following is an extract from his new work, "Lei^ons de &Kcanique 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 tho coefficients. A, were proportional to a cer- hv Google STRENGTH OP MATEKIALS. 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 stiifness 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 M. Navier for the expression of the resistance of ropes to bending cannot be admitted, it is ne- cessary to search for another, and it appears natural to try if the factors A and B cannot be expressed for white ropes, simply accord- ing 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 M. Navier, by the number of yarns, we find for n = dO d = 0"'-200 A = 0-2224GO - = 0'0074153. --lu d = 0^444 A = -^ 6d = 0"-0088 A = 0-010604 - = 0-0042343. = 0-0017673. It is seen from this that the number A is not simply propor- tional to the number of yarns. Comparing, then, the values of the ratio — corresponding to the three ropes, wo find the following results : — ' "-nroa. Dileroiitea of tl.c numVst. of rv'S";/ 80 15 00074153 00042343 O-O0T7G73 From SO to 15, 15 jams — 15 to 6. 9 — 0W31810 0-0024770 0-00ofi400 0-000212 000372 0-000352 Mean difference per jara, 0-000245 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 = M [0-0017673 -I- 0-000245 (« — 6)]. = n [0-0002973 -f 0-000245 k]. An expression relating only to dry white ropes, such as were used by Coulomb in his experiments. With regard to the number E, it appears to be proportional to the number of yarns, for we find for hv Google TOE PRACTICAL MODEL CALCULATOK. = SO d = 0"-0200 B - n = e d = O^-OOSS B = Moan 0-0003630 Whence B = 0-000363 n. Consequently, the results of the experiments of Coulomb on dry white ropes will he represented ivith sufEcient exactoess for prac- tical purposes hy the formula K = M [0-00029r + 0-000245 n + 0>0003G3 T] kil. which will give the resistance to bending upon a drum of a metre in diameter, or by the formula R = ^ [0-000297 + 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 U = n [0-0021508 + 0-00n724 n + 0-00119090 T] lbs. for a drum of a foot in diameter, and E = g [0-0021508 + 0-0017724 «, + 0.00119096 T] lbs. for a clrum 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 tlio stiffness of a rope of a diameter equal to unity the same stiifness 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. TARRKD aOPES. 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 E = 0-0125605 „ = 15 _ A = 0-106003 B = 0-006037 „ = 6 — A = 0-0212012 E = 0-0025997 which diifcr very slightly from those which JI, Navier has given. But, if we look for the resistance corresponding to each yarn, we find hv Google :kgth of materials. 30 yarns ~ = . 0-01166l)3 - = 0-000418683 15 — A - 0-0070662 ? - 0-000402466 6 — A _ = 0-0035335 2 = 0-000433283 Mean 0-000418144 We see by this that the value of B is for tarred ropes, as for white ropes, sensibly proportional to the number of yarns, hut it 13 not BO for that of A, as M. Kavier has supposed. Comparing, as we have done for ivhite ropes, the values of — corresponding to the three ropea of 30, 15, and 6 yarns, wo obtain the following results : — JMM. "'?•' „,._ ;;,:: s-p Ipir' 80 16 6 0'0116603 00070662 00035335 From 30 1. - ISt — eot IS. 15 yarns 25 — 0-0045941 0-0035327 0'0081268 0-000306 0-000392 0-O00339 Mea ..0-000346 It follows from this that the value of A can be represented by the formula A = K [0-0035335 + 0-000346 (n ~ 6)] = n [0-0014575 + 0-000346 m] and the whole resistance on a roller of diameter D metres, by R = j5 [0-0014575 + 0-000346 n + 0-000418144 T] kil. Transforming this expression to the American scale of weights and measures, we have R = ^ [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 differ- ent numbers of yarns are calculated from the data of Coulomb, by the formulas, d cent. = v'0-1338 n for dry white ropes, and d cent, = ^/0.18^ n for tarred ropes, which, reduced to the American scale, become d inches = v^ 0-0207"B9 n for dry white ropes, and d inches = ^/6■02883 for tarred ropes. hv Google 6m THE PRACTICAL MODEL CALCULATOR. Note,— The diameter of the rope is to be included in D ; thus, ivith 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 1 1 67 — ^ «B, ^ "'r.^.T-' -«:= yn 3^ eo 300 776 0-72135T «'37323T f O-OlOMJIJii l+OOOaiKOflnl- flOSOB W* 0KH3T1889B Apjilioatiurh 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, tho values of the quantities A and B corresponding to these given quantities, and knovring the tension, T, of the end to he wound up, we shall have its resistance to bending on a drum of a foot in diameter, by the formula R = A + ET. Then, dividing tbis quantity by the diameter of the i-olbr 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 = 0-5097T B = 0-OT14576 and WO have D = 0'5 + 0'0928 ; and consequently. 0-5928 The whole resistance to be overcome, not including the f:-ictioi on the axis, is then Q + R = 1000 + 128 = 1128 lbs. The stiffness in this case augments the resistance by moi'O thai one-eighth of its value. hv Google STRRKOTH OF MATERIALS. 307 FtllLTHER EUCEKT EXPEIdllENTS MADE BY JI. JIORIN, ON TOE TRAC- TION" OP CAKUIACiES, AND THE DESTUCCTIVE EEEECTS WHICH THEY PRODUCE DPON THE ROADS. The study of the effects which ave produced ivhen a. carriage 13 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 eurfaco are due to Coulomb, who determined the resistance offered by rollers of lignum vitie and elm, on plane oak surfaces placed horizontally. ilis experiments showed that the resistance was directly propor- tional 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 Cou- lomb, be expressed by the formula in which A would be a number, constant for each kind of ground, but varying with different kinds, and with the state of their The results of experiments made at Vineennes 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 con- clusions; and we may allow, at least, as a lavf sufficiently exact for practical purposes, that for woods, plasters, leather, and gene- rally for hard bodies, the resistance to rolling is nearly — 1st. Proportional to the pressure. 2d. Inversely proportional to the diameter of the wheels. Sd. Greater as the breadth of the zone in contact is smaller. S UrON CAEBIAGES TEAVEIJ.ISa ON ORDINAUY ROADS. These experiments were not considered sufBcient to authorise the extension of the foregoing conclusions to the motion of car- riages 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 un- dertaken, first at Metz, in 1837 and 1838, and afterwards at Coor- bevoie, in 1839 and 1841, with carriages of every species ; and attention was directed separately to the influence upon the magni- tude 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 hv Google SOO THE PEACTICAL MODEL CALCULATOR. into consideration, tbe weight of the wheels may be neglected in cornpiirison with the total load ; and the relation between the load and the traction, upon a level road, ia approximately given by the equation — F, 2 (Ax/?-,) p-= — ^ ,, for carnages with four wheels, F,' A X /r, ^ and p-= — ■ for carriages with two wheels, in which F, represents the horizontal component of the traction ; P, the total pressure on the ground ; t' and /' the radii of the fore and hind wheels ; r, the mean radius of the boxes ; / 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: — a.,.,.„ „.,,-. „-..„_ PrMSQia. .™.,. Chariotporte corps d'artillerie. Road from Conrbe- Toie to Colomber, dry, in good re- pair, dusty. Eoad from Courbe- Toie to Bezoua, solid. *harJ gra- vel, very dry. 6140 4580 iso'ti 159-9 113-7 1/88-6 1/39-2 1/40-2 ChariotderonlagE, wjtlioutapriaga. 7126 5458 4450 3430 138-9 115-5 93-2 68-4 1/51-3 1/48-9 1/47-7 1/50-2 Cbariotderoalage, Vfith apringa. Eoad from Colomber to Courbevoie, pitohtd, inordina- ry repair, f mud dy Boad from Courbe- voie to Colomber, deep ruts, with muddy detritus. 1600 3292 4300 39-3 89-2 I860 1/40-8 1,36-9 1/36-8 Cun-isgwi with ail equnl wheels. Tw<icafriag«3nith sii equal wheela, hooked on, one behind the other. 3000 4692 6000 eooo 138-9 224-0 285-8 286-7 I/2I-6 1/21-0 1,21-0 1/21-0 From the examination of this table, it appears that on Jsolid gravel and on pitched roads the resistance of carriages to traction is sensibly proportional to the pressure. • En gcavier dur. f Pav^ en 6tat ordinaire. J En b,Google BTRENSTH OF MATEIilALS. 6\}y We remark that tte 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 waa borne upon one carriage or upon two. It follows thence that the trac- tion is, ca;teris paribus and between certain limits, independent of the number of wheels. Influence of the Diameter of the Wheeh. 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 aa a practical law tl h n n erselj proportional to the diameters of the w C.rri«.>™.r"j..L -■ -- - I S~ 1 i ~: H il Ciffilon. Camion. Road ft. bevoie 701, dn 60 SW St 86 "2 /A OB'6 I8-8( 1/60' 1/45-5 HiSri i/as-8 1-7 6-7 42-41 l>013[ 1 U-IH494 a-osoje l|-029» Influence of the Width of the Felloes. Experiments made upon wheels of different breadths, having the same diameter, show, 1st, tfeat on soft ground the resistance to rolling increases aa 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 velo- cities, 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 follow- ing table : — * Empierrement eolido. t Pav6 en grfes. hv Google THE PKACTICAL MODEL CALCULATOR. c...,.„.,..,.,. — ■ .... .... w t™- "3" Apparatus npon a Ground of the po- 104'> Wall! 3 13 165'0 1/6-32 brass shaft. ]ygoii at Mctz, Trot ti-ati lti8-0 1/8-2 \-Mh Walk S-fifiO 215-0 1/6-21 Trot 7-560 107-0 1/6*8 Asisteen-pounder Road from Metz Walk carriage and to Montigny, 'Brisk walk 3-4110 92- 1/40-8 piece. solid graTel, very eyen and Tery dry. Pitoiied road of 1,^1- fCimter 8450 121- Cbaiiot des Mes- R2Rfi Walk 2-770 144- 1/22-8 Fontainebleau, x^rvA »Brisk walk m-i 153- 1/21-9 eprings. 1/18-3 tBrisk trot. 805 183-5 We see, by these examples, tliat the traction undergoes no sen- Bible 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 ve- locity, the velocities were set ofi' 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 repre- sented by the expression, A = a + 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 houi',) which is that of a very slow walk. d, 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 llpitched road Of Metz, A = 0-01936 X 0'08200 (V — 2). These examples are sufficient to show — Ist. 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 in- creases very rapidly with the velocity. On rough roads the resistance increases with the velocity much more slowly, however, for carriages with springs. t Grand trot. hv Google STEENOTH OP MATERIALS. 311 TIius, for a clianotdcs Messagenes G^n(;rales, on a pitched roaiJ, the experiments gave A = 0-0117 X 0-00361 (V — 2) ; 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 *3olid gravel road when dry and in good repair, the latter lost its superiority when muddy or out of repair. INPLUINCE OF THE INCLINATION OF TIIE TRACES. The inclination of the traces, to produce the maximum effect, is given by the expression — AxO-9 6/r' in which /; = the height of the fore extremity of the trace above tte point where it is attached to the carriage ; 6 = the borizontat 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 Bmall diameter it is much less than the construction generally permits. It follows, from the preceding remarks, that it is a to employ, for all carriages, wheels of as large a diameter be used, without interfering with the other essentials to the pur- poses 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. Kow, by bringing the hind wheels as far forward as possible, and placing 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 PESTEDCnvil El'FECTS PRODUCED BY CAUttIACE3 ON TUB IIOAIIS. If we take stones of mean diameter from 1\ 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, th^ undergo no displacement. From this simple esperiment 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 bat slightly increased when the breadths of the wheels was * En erapierrement. b,Google 312 THE PRACTICAL MODEL CALCULATOK. diminisliecl, we might also conclude that the wear of thi Hir.d wouhl be but slightly increasecl by dinouishing the width nf the felloes. Lastly, the resistance to rolling increasing with the velocity, it was natural to think that carriages going at a trot would do liiore injury to the roads than those going at a walk. But s[>nng=(, 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 carnages upon the roads, have confirmed these conclusions. These experiments ahoived that with equal loads, on a solid gra- vel road, wheels of two inches breadtli produced considerably moi o 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 dia- meters 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 wjtii 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 Deduc- tions. When an aperture is made in the bottom or side of a vessel con- taining 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, de- scend 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 oriflce, 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 hv Google HYDEAtfLICS. 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 contain- ing, 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 foct ? A cubic foot of water weighs 62-5 lbs. avoirdupois. (3000 X 62-5 • on " = 6250, the weight of water raised a minute. 6250 X 300 = 1875000, the units of work each minute. Then ~<^ oqqq = 56-818 = the horse-power required. 2. What quantity of water may be discharged through a cylin- drical moutli-piece 2 inches in diameter, under a head of 25 feet ? 2 1 ^2 = -g- of a foot; ,•. the area of the cross section of the mouth-piece, in feet, is ^ x ^ x -7854 = -021816. Theory gives -021816 \/2 ^ 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 me- chanical 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 velo- city of the discharge is about 97 per cent, of that given hy theory. The theoretic velocity is v/2^~x49 = 7 ^"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 jg = ^ = -25, and (-25)2 X -7854 = -0490875 = arcaof orifice. .-. -64 X -0490875 = -031416, the area of the tr.ansversc section of the contracted vein. b,Google 314 THE paACTICAL MODEL CALCULATOH. Hence, 54>523 X -031416 = 1-7129, the cubic feet of water discharged each second. The later experiments of Poncebt, Bidone, and Leshros give -SSS 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 ; 5^5 of this result aro found to be very near the truth. jgOfl-'ri29 = 1-0541. 4. AYhat 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 -jn" = ■C5583, decimal of afoot. 3-94 -j--^- = -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 bo g X -G5583 >/2y((5-328)^ - (5)') = 3-9268 cubic feet. Table I. gives the coefficient of eflux 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 ? :j-g = -5, decimal of a foot. 12 " , decimal of a foot. The theoretical discharge will be I X -6 v/2^ I (9-25/ - (9f I = 3-033 cubic 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 efi'ective discharge. 6. A weir -82 feet broad, and 4-92 feet head of water, hovr many cubic feet are discharged each second ? The quantity will be c X -82 ^/2i?(4■92)^; g = 32-2; hv Google HYDRAULICS. Table I,-— -The Ooeficients for the JUjKux through rcetangvlar ori- fices in a thin vertical plate. The heads are measured where the water may he considered still. a-a •»» ARM h Height o Ohifke. 8 i 1 4 1 79 J d 619 634 656 686 «0l 620 638 b 4 G8I 60J 6 1 640 653 C b 4 606 6 639 6 11 607 6 7 6oO b66 594 eo9 11 4 6 o 63j 634 649 648 bb" G18 6 3 b3 (47 656 ^M^ 615 6 7 64i> 10 593 610 6 8 6 644 650 bOO f 1 bl m 6 8 6 b 8 6''6 641 638 647 644 GO 616 Ci 601 bl5 621 621 630 635 eo 6 3 613 618 618 6 5 630 fO 611 615 bl 1 6 J bO 601 609 612 613 617 619 fto C bO e 9 4 CIS 10 4 9 1 Ml Table II— The Co ^i t ntsfo tlelJffl z t? oujJ r da juh fices in at! n verti il plat tie lea la o/ ater hei j 8 4 2 1 8 4 ■1 593 613 (37 059 6&> biZ 612 f3b bu6 6»0 K u« bl3 635 653 0-6 694 I 594 bl4 6^4 650 087 5 595 614 633 647 681 597 615 632 644 064 b7o / 615 633 641 660 661 599 616 630 638 6j5 •A 601 616 f29 650 6^7 1) 617 629 632 644 651 ? II 604 617 626 628 040 646 " b05 616 622 f27 6G6 '. II 604 en 618 624 632 636 6 n 604 613 bl6 b21 631 II 603 612 613 618 b24 6.0 7 (1 GOi blO 611 hi 6 020 C.l f (1 608 009 614 C!6 617 ! 1) (01 m 607 C12 bl3 (13 10-0 FOl bOS 606 610 610 t,09 b,Google iSTo TilE PRACTICAL MODEL CALCULATOH. c is termed the cociEcIent of efflux, aad on an average may be taken at 4, It 13 found to varj from -385 to --l-W, Then -4 X -82 v'(ti4-4) (4-92p = 2-670033, the cubic feet dis- charged each second. 7. What 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 *r The breadth will b 10 10 . - • -,, = 4-79C3 feet. c ^2g X (-75)^ -4 X v/64-4 x (-75)^ Changes in the coefiicients of efflux through convergent sides often present themselves in practice : they occur in dams which are inclined to the horizon. Poneelet found the coefficient '8, when the board was inclined 45°, 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 ■§ feet, what quan- tity of water will be discharged, the surface of the water standing 4- feet above the surface of the channel, and the coefficient of efflux taken at '78 ? The height of the aperture = -5 sin. 50° =. -38-50222 ; 4- and 4 3830222 = 3-6169778, ai-e the heads of water. .-. g-x 2-25 X -78 X ^2^Uif - {-3-G17)H = 10'5257 cu- bic 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 con- tracted vein on every side. We shall next calculate in cases where the water flows from one or more sides to the aperture, and hence produces a stream only a partially contracted, ni, n, 0, p, are four orifices in the bottom AECD of a vessel ; the contraction by efflux through the orifice o, in the middle of the bot- tom, isgeneral, as the water can flow to it from all sides ; the contraction c from the efflux through m, n, p, 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 hv Google UYDRAULICS. the lower side of the channel, so tliat ther bottom ? The theoretical discharge will be 4 .^ f,„J .,,l\ 817 contraction at the ( J X -^Ig I (3f - (2)^ I = 50-668 cubic feet. 1 the table page 315, may The coefficient of contraction g be taken at -603. I. — Comparison of the Theoretical withthe Real Dhohargesfror. Orijice. "'Hf2r °SS' '""Slices," ' 4381 2722 1 to 0-62133 619(i 3846 1 to 0-62078 7589 4710 1 to 0-62004 i 8763 6430 1 to 0-6203* 5 9797 6075 I to O'6201O 10732 6654 1 to 0-62000 11502 7183 1 to 0-61965 12392 1 to 0-61911 1 to 0-61892 10 13855 8574 11 14530 8990 1 to 0-61873 12 15180 9381 1 to 0-61819 13 15797 9764 1 to 0-61S10 10130 1 to 0-61795 15 1G968 10472 1 to 0-61716 II. — Comparison of the Tlieoretkal with the Real Discharges fro a Tube. Cr>iiEtant height *.rS6 throuih 1 £'Sfe thsoiiUcS 10 Iha raJ Piris FD^t. Cubit IiwhoB 1 4381 3539 1 to 0-81781 2 6196 5002 1 to 0-80729 7589 6126 I to 0-80724 8763 7070 1 to 0-80681 6 0797 7900 1 io 0-80638 10732 8654 1 to 0-80638 7 11592 9340 1 to 0-80577 8 12392 9976 1 to 0-80496 9 13144 10579 1 to 0-80485 10 13855 11161 1 to 0-80488 11 l'!530 1169S J to 0-80477 12 15180 12205 1 to 0-80403 13 15797 12699 1 to 0-80390 14 16393 13177 1 to 0-80382 16968 13620 1 to 0-80270 b,Google 318 THE PRACTICAL MODEL CALCCLATOR. THE DISOIIAEGB BY DIFFERENT APERTURES AiSD lUEES, UKDER DIF- The velocity of water flowing out of a hoHzontal 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 ve- locity ; for, the quantity flowing out iu any short time is aa 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 ; there- fore, the force must be as the square of the velocity. Or, supposing a very small cylindrical plate of water, imme- diately over the orifice, to be put in motion at each instant, by the pressure of the whole cylinder upon it, employed only in generat- ing its velocity ; this plate would he 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 mul- tiplying 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 con- sequence of the contraction of the stream, we must, in order to ob- tain 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 M, Eytelwein, is as 1 to '619 ; but more recent experiments make it aa 1 to '621 up to ■645. APPLICATION OP THE TABLES IN THE PRECEDINO PAGE. Table I. — To find the quantities of water discharged hy orifices of different sizes under different altitudes of the fluid in the reser- voir. 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 aa the theoretical quantity of water discharged by an oriSce one inch in diameter from a height of 15 feet is, by the second co- lumn of the table, 16968 cubic inches in a minute, we have this proportion ; 1 ^/lb : 9 v/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 ac- tual discharge, according to the fourth column of the table, gives 133896 cubic inches for the actual quantity of water discharged by hv Google HYDRAULICS. S19 tbe given aperture. Hence, the quantity sliould 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 which it is compared, the flowing vein of water becomes rather more contracted. The quan- tity 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 qnantities discharged, the real velocity may be deducted from the theoretical by means of the foregoing results. Table II, — To find the quantities of water disaharged ly 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^l5 : 16 ^/25 : : 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 dis- charged. 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 eoTi- tracta already described. According to Eytelwein, the proportion of the theoretical to tlie 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 varia- tions by altering the form and position of the orifice, the thickness of the plate, the form of the vessel, and the velocity of the issu- ing fluid. Dedtietions Jrom experiments made iy Bossut, MiclieUoti. 1. That the quantities of fluid discharged in equal times from different-sized apertures, the altitude of the fluid in the reser- voir being the same, are to each other nearly as the area of the aper- tures. 2, That the quantities of water discharged in equal times by the same orifice under different heads of water, are i?early as the square roots of the corresponding heights of water in the reservoir above the centre of the apertures. b,Google 320 TUE PRACTICAL MODEL CALCOLATOR. 3. That, in general, the quantities of water discharged, in the same time, bj different apertEres 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. 'Xhat, 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 horiaontal 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. i). That the length of the cylindrical horizontal tube may be increased with advantage to four times the diameter of the onfice. 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 diff'erent 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 propor- tional to the square roots of the head of water, 13. That from the two preceding corollaries it folloivs, in gene- ral, that the dischai'ge during the same time, by ditlerent addi- tional 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 tlie outer diameter of tlie orifice is as '6'd to 2(), is augmented very nearly one-seventeenth and seveu-tcnths wore tlwu by cylindrical tubes, if the enlargement be not curried too far. hv Google HYDRAULICS. 321 DISCHARGE BY COMrOUND TUBES. Deductions from the experiments of M. Venturi. In the discharge by compound tubes, if the part of the addi- tional tube nearest the reservoir have the form of tho contracted vein, the expenditure will be the same as if the fluid were not con- tracted at all ; and if to the smallest diameter of this cone a cylin- drical pipe be attached, of the same diameter as the least section of the contracted vein, the discharge of the fluid will, in a horizon- tal 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 Lave a cone applied to the op- posite extremity of the pipe, the expenditure will, under the same head of water, bo 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 tho proportion of 24 to 10. But if a great length of pipe intervene, this additional tube has little or no efl"oct 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 tho 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 bondings ; right angles should be studiously avoided. The interruption of the discharge by various enlargements of the diameter of the tubes having been investigated by Si. 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 in- crease 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 inter- nal 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 sur- face highly polished throughout its whole length, and it may every- hv Google ,122 THE PRACTICAL MOI>EL CALCULATOE. whore possess a diameter greater than the orifice to wliich 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 contractiona be avoided ; for it may hap- pen, by an intermediate enlargement, that the whole of the other advantage may be lost. This will be obvious from the results in the following table, deduced from experiments with tubes having various enlargements of diameter. Head of wnter in inches. Number of en- iarged parts. Seconds in vliieli 4 cubic feet were disoiiarged. 32-5 32-5 32-5 32-5 1 3 5 109 147 192 240 DISCHARGE BY CONDUIT PIPES. On account of the friction against the sides, the less the dia- meter 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 dis- charges made in equal times by horizontal pipes of different lengths, hut 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 he less than If inch for every ISO feet of the length of the pipe. The ratio of the difl'erence 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 pipo is 6° 31', or the depression of the lower extre- mity of the pipe is one-eighth or one-ninth of its length, the rela- tive gravity of the fluid will ho 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 dimi- nished 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 water, unless prevented by air-holes, or stopcocks. hv Google HYDEAULICS. Table I, — Oomparison of the di»charge by conduit pipes of different lengths, 16 lines in diameter, with the discharge ly additional tubes inserted in the same reservoir. — By M. Eosstjt. 2778 JOO to 43'39 1957 100 to 30-91 1687 100 25-07 1351 100 21-34 1178 100 18-61 1052 100 16-62 4068 100 o 45-18 2888 100 32-31 2353 100 26-31 2011 100 22-50 1762 100 to 19-71 1583 100 17-70 Table II. — Comparison of the di ferent lengths, 24 lines in diameter, tional tubes inserted in the i conduit pipes of dif- ith the discharge by addi- ■By M. BossuT. IJamititj.ofW aim diKbirgt-i wSiwa'i'h, I.«iELhr>r B«Wol»li.aaothe '^m^i^Tl^' IoJ»,24liiJ'An ^■lilS''" F8« F6!l CoWc IntSas. Cabiolnths., 30 14243 100 to 68-92 60 14243 5564 100 to 89-06 90 14243 4534 100 to 31-83 120 14243 3944 100 to 27-69 14243 3486 100 to 24-48 14243 3119 100 to 21-90 80 20113 11219 100 to 55-78 60 20112 8190 100 to 40-72 2 SO 20112 6812 lOO to 83-87 120 20112 5885 100 to 29-23 2 20113 6232 100 to 2601 2 180 20112 4710 100 to 23-41 BISCHAKOE BY WEIKS AND EECTAKGULAS ^ Rectangular orifices in the side of a reservoir, extending to the surface. Tlie velocity varying nearly as tho square root of the height, may here be represented hj tho ordinates of a parabola, and the quantity of water discharged by the area of the parabola, or two-thirda of that of the circumscribing rectangle. So that the quantity discharged may be found by taking two-thirds of the velo- city due to the mean height, and allowing for tlie contraction of the stream, according to the form of the opening. In a lake, for example, in the side of which a rectangular open- ing is made without any oblique lateral walls, three feet wide, and hv Google 324 THE PRACTICAL MODEL CALCULATOR. extending two feet below the surfaco of the water, the coefficient of the velocity, corrected for contraction, is 5'1, and the corrected mean velocity | \/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 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 baa a weir three feet in breadth, and the sur- face of the water stands at the height of five feet above it : it is required how much the weir must be ividened, in order that the water may be a foot lower. Here the velocity ia f v/5 X 5'1, and the quantity of water § \/5 X 5-1 x 3 X 5 ; but the velocity must be re- 2 /^ X 5 '1 X 3 X 5 ducedtof -/i X 5-1, andthen the section will bo ■'^ = — — § ■/4 X 51 = — — = = 7'5 X \/5 ; and the height being 4, the breadth must be -7- \/5 = 4-19 feet. The diacbargc from reservoira, with lateral orifices of consider- able magnitude, and a constant head of water, may be found by determining the difi'erence 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 quan- tity 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 ia a river or stream varies at dif- ferent parts of the same transverse section. It ia found to be greatest where the water is deepest, at somewhat less than one- naif the depth from the surface ; diminishing towards the sides and shallow parts. Jtemtance to bodies moving infinids. — The deductions from the experiments of C. Colles, (who first planned the Croton Aqueduct, Now 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, diminbhes the resist- ance with which it moves, provided the weight in the water remains the same. hv Google HYDRAULICS, 325 4, That the greatest breadth of the moving hody should be placed at tho distance of two-fifths of the whole length from the how, when applied to the ordinary forms in naval architecture. 5. That the bottom of a floating solid should be made triangu- lar ; as in that case it will meet with the least resistance when moving in the direction of ita longest axis, and with the greatest resistance when moving with its broadside foremost. F]-iction. 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 cis- tern 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 esperiments were then repeated without the water, and the following are the angles of deflection and vibration in the two cases. In the ■'—'"■ , luthd 22° 30' 22° 24' 22° 30' 20° 0' 22 10 22 6 21 36 21 3 21 54 21 48 20 48 2016 21 3S 21 30 Sc. 4c. &c. fa. 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 tho 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 momenta being summed by integration and equated with the sum of the moments given by the experiments, we have the value of the con- stant pressure at a unit of depth = ■0000469. This constant, in another experiment, the weight of the model being 197 lbs. avoir- dupois, and consequently the part immersed in the water being dif- ferent 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 capadties of pipes for the drainage of towns, have presented some useful results for the guidance of those who have to make hv Google 326 THE PRACTICAL MODEL CALCULATOR. calculations for a similar purpose. The pipes, of various dia- meters, 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 ?ipes reduced to 50 feet in length, gave very nearly the same result. Vithout the addition of junctions, the transverse sectional area of the stream of water near the discharging end was reduced to one- fifth of the corresponding area of the pipe, and it required a sim- ple head of water of about 212 inches to give the same result as that accruing under the circumstances of the junctions. AVith regard to varying sizes and inclinations, it appears, sufSciently for practical purposes, that the squares of the discharges are as the fifth powers of the diameters ; and again, that In steeper declivi- ties 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-mchpipe at several inclinations. D'itLuaf s in 1*1 lin 60 75 lin 320 49 lin 80 68 lin 400 48-6 1 in 100 63 lin 480 48 1 in 120 69 lin 640 47-5 1 in 160 64 lin 800 47-2 1 in 200 62 1 in 1200 46-7 1 in 240 60 Level 46 The conclusion arrived at is, that the requisite sizes of drains and sewers can be dctcrmLned (near enough for practical purposes, as an important circumstance has to he 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 tho fifth powers of the diameters. That at greater declivities than 1 in 70, tho discharges are as the square roots of the inclinations. hv Google WATER WHEELS. 02 i That at less declivities ttan 1 in 70, titc usual law will not obtain ; but near approximations to the trutli may be obtained by observing the relative discltarges of a pipe laid at various small inciinatioas. That increasing the number of junctions, at intervals, accele- rates 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 flovt' 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 euhic feet per minute, tho lateral streams had a velocity of a few feet per second, and the junctions were placed at an angle of about 35° with the main line. It is needless to say that all junctions should be made as nearly parallel with the main lino as possible, otherwise the forces of the lateral currents may impedo rather than maintain or accelerate the main streams. WATER WHEELS. TUE IJXDtaSIIOT WHEEL. The ratio between the power and effect of an undershot T;heel 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 esti- mated labour of an active man. The velocity of the periphery of the undershot wheel should bo equal to half the velocity of the stream ; the float-boards should be ao 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 he 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 head, hut which discharges 60 cubic feet of water, will grind eight hulls 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 nearly as the square of its velocity ; that is, if a mill, driven by a hv Google 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; anotlici- mill, driven by tho same quantity of water, moving with the velocity of five feet per second, will grind nearly i-^/^ bolls in the hour, because a : 4^ : : 4* : 5^ 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 holla of corn in an hour; another mill, tli-iven by water, moving through the same aperture with the velo- city of five feet per second, will grind 5jg bolls nearly in an hour ; for as 3 : 5^g : : 4' : 5' nearly. The height of the virtual head of water may be easily deter- mined 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 per- pendicular height of the fall of water above the bottom of the mill- course, 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 veloeity of the water. Take one-half of the velocity of the water, and it will be the velocity to be given to the fioat-boards, or the number of feet they must move through in a second, to produce a maximum effect. Divide the circumference of the wheel by the velocity of its fioat- boards 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 millstouo in a minute, so must the number of staves in the trundle bo to the number of teeth in 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 efi'ect of tlie water wheel is a maximum, when its circum- ference moves with one-half, or, more accurately, with three- sevenths of the velocity of the stream. THE BKEAST WUEEL. The efi'ect of a breast wheel is equal to the efi'ect of an under shot wheel, whose head of water is equal to the difference of level hv Google WATER WnBELS. 329 between the surface of water in tlio reservoir, and the part whero it strikes tlie wheel, added to that of an overshot, wlioso Iicight is equal to the difference of level between tlie 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 -1 feet, and an overshot wheel when the fall exceeds 10 feet. Also, i\-hen the fall exceeds 10 feet, it should be divided into two, and two breast wheels be erected upon it. Table for breast wheels. 3i ti Ills 1. IL 111 % hi 34 Is il 111 Ti Mt V.i nt || ss k a |.f {i ^ss FeM. F8«. rsel. FtBl. E=o. IM, svr. CaWs n. 1 017 198 ■§ 0-75 2-18 1-92 4-80 1536 74-30 0-34 35-1 1-50 3-09 6-80 1084 37-15 051 12-7 8 -78 3-33 8-82 886 24-77 0-69 6-2 3-01 4-86 3-84 0-60 762 18-57 5 0-80 3-S7 3-76 4-88 4-28 10-70 680 14-86 1-03 2-25 4-51 5-8S 4-70 11-76 626 12-38 r 1-20 1-53 5-26 5-77 12-70 581 10-61 8 1-37 1-10 6-02 6-17 5-13 18-58 543 9-29 l-5i 0-81 6-77 6-55 5-76 U-40 512 8-26 10 1-71 0-77 7-52 C-90 6-07 15-18 436 7-43 It is evident, from the preceding table, that when the height of the fall is less than 3 feet, the depth of tho float-boarda is so great, and their breadth so small, that the breast wheel cannot well be d ; and, on the contrary, when the height of the full ap- ! 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. Tho ninth column contains the quantity of water necessary for impelling tho 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 aa 10 to 6'6, when the water is delivered above the npex 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 ; con- sequently, the quantity of water expended per second, to produce a mechanical eff'ect equal to that of the aforesaid estimated labour of an active man, is, in the first instance, 15'15 lbs., and in the second instance, 12-5 lbs. Hence, the effect of the overshot wheel, under the same circum- hv Google o30 THE PKACTICAL 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. Worhmg machinery hyhydraulio pressure. — The vertical pressure of water, acting on a piston, for raising weights and driving machi- nery, is coming into use in many places where it can be advantage- ously 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 natm-a! fall, like that of Golway, Ireland, which can be conducted to the engine through pipes, is, of course, the most economical situatioit 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 em- ployed. WTierever the contiguity of a steam engine would be dan- gerous, or otherwise objectionable, a water engine would afford the means of receiving and applying the power from any required dis- tance, 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 open- ing 400 inches. The following table shows the woikmg horse power of both the inward and outward discharging Turbine w ater wheels ; they are calculated to the square inch of opening Disch^S^ne Oalwarinatliiirj D-lariUis He»a. UnnsPoKEr. IlOH-Po^sr. H<.d II Tis rm«T. Horse PowBr. g ■00821 ■012611 22 19d23 ■339972 4 ■01483 ■025145 23 20T87 •864182 6 ■02137 ■038124 24 22315 ■384615 6 -02685 ■045618 25 23367 ■112018 7 -03414 ■058314 2G 25125 ■487519 8 ■04198 ■074113 27 26482 ■455698 ■Oo206 28 28135 -4844:>7 10 -058S3 ■106215 20 2Jot.3 ■610838 11 -06921 ■118127 30 80817 ■537721 12 ■07H51 ■135610 31 32316 ■5GI425 13 ■08882 ■160638 32 83617 ■587148 H -10054 ■173158 o3 34823 ■611018 15 ■11002 ■192234 34 3a»4 ■638174 ■12093 ■211592 35 37123 ■665164 17 ■13196 ■231161 31874 ■692150 18 ■14275 ■2S7U5 37 10118 ■726148 19 ■15613 ■273325 38 41762 ■764115 20 ■le927 ■296618 33 42]oa ■804479 21 ■18109 ■3171S7 ' 40 43718 ■849814 b,Google WATER WHEELS. 331 Opposite 25 in the column marked " Head," tlio working horse power to the square inch is found to lie '25667, which, multiplied by 400, gives fl-l-60S, the horse power required. What is the working horse power of an outward discharging Tui-bine, under the effective head of 20 feet ; the area of all the openings being 32-5 square inches. In the table, opposite 20, wo find -2^6618, then -296618 X 325 = 96-4, the required horse power. What is the number of revolutions a minute of an outward discharging Turbine wheel, the head being 19 feet and tlie dia- meter 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 dis- charging 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. These Turbine tables were calculated by the author's brother, the late John O'Bjrne, C. E., wlio died in Kew York, on the tith of AprU, 1851. TiS — Out'ward discharging Turh me. — — — — 1 1"- ■li ~w "IT ~^ 48 54 -sr -^ 11 78|84 Z •Jb __ 100 so 70 60 52 42 37 o5 82 30 28 27 11 4 111 73 57 49 44 41 37 «4 32 30 28 5 123 100 82 71 62 55 51 4t. 42 37 81 6 135 109 91 78 62 66 60 io il Stj 36 7 146 118 96 84 73 65 59 53 49 47 42 40 JH 8 156 125 105 90 79 71 63 57 5- 49 43 4J 39 S IGG 133 111 95 75 67 bl 67 50 49 41 10 1T5 UO 117 100 87 70 64 59 50 ol 47 40 H 183 147 122 106 92 81 74 b7 62 57 o4 49 4b Vi 191 156 127 110 96 79 70 64 o9 5. •)i 51 IB 200 169 115 100 89 81 7^ 57 5j 6., li 206 166 138 n8 104 92 76 64 69 5j 16 213 171 142 122 107 95 86 78 72 60 bl 58 Ob le 222 177 148 126 111 98 89 8J 74 69 64 59 67 n 227 182 152 131 115 101 91 s-s 77 71 66 02 j9 18 234 187 156 134 117 105 94 85 78 73 t3 61 19 193 161 138 120 107 97 88 bl 74 o4 bJ 20 247 197 164 141 124 110 99 90 84 00 €4 ai 252 202 168 145 126 114 101 "iZ 86 73 (& Oj ' 'Z2 259 208 172 149 129 116 105 94 87 80 74 m 07 1 23 2B3 212 no 151 183 119 106 9b 8> 84 70 21 270 216 180 156 135 120 109 98 92 8^ 7b 74 72 25 277 222 184 158 188 123 111 101 93 &b 80 76 74 20 282 226 189 161 141 125 113 108 9j 87 81 78 27 280 229 191 165 148 129 116 105 97 S3 79 77 28 291 233 195 167 146 ISO 118 107 99 91 85 78 2W7 '2Z7 199 170 149 lis 109 100 81 80 ao 303 241 202 174 162 135 111 102 94 »a i^- _!L b,Google HE PRACTICAL MODEL CALCULATOR. Inward discharging Turbine. 1l ■n 30 30 42 48 54 60 66 72 78 TT 00 "ss ~^ 111 86 74 62 54 48 47 40 30 ~32~ 31 30 "27" i 125 96 83 70 62 55 51 45 41 37 34 31 5 141 112 94 78 61 55 50 46 43 40 36 152 122 101 80 76 67 62 55 5i 47 48 42 88 7 166 181 108 98 82 72 65 60 54 51 47 44 42 S 175 189 U6 87 76 71 67 54 49 47 45 9 188 149 123 106 93 81 74 68 63 57 53 51 47 10 135 136 129 111 99 ee 78 71 61 56 52 49 11 208 167 130 117 102 91 82 74 63 5S 62 12 217 169 142 122 107 97 85 78 71 66 61 57 54 13 221 178 148 127 112 99 82 74 69 64 01 56 14 231 184 153 133 110 104 92 76 71 62 58 15 191 159 130 119 107 05 87 SO 73 68 64 01 16 245 198 105 144 111 09 00 76 71 66 68 17 253 203 108 148 127 114 102 92 85 78 73 64 18 260 209 173 ISO 132 116 104 95 87 82 75 69 60 267 215 176 153 134 120 108 89 77 72 67 20 276 222 183 157 138 122 111 101 93 85 79 74 69 ai 288 226 180 162 141 125 lis 103 95 86 SO 75 71 2-2 200 102 164 145 129 116 107 96 89 77 23 209 236 190 1G7 140 13S 118 100 97 84 79 74 2i 303 210 201 171 151 136 122 HI 101 86 80 75 ■25 810 247 206 176 155 138 123 112 104 88 82 76 20 314 248 210 180 157 189 126 115 100 97 00 84 79 27 SIO 254 218 183 102 142 128 117 108 09 85 80 28 827 261 218 186 164 140 129 119 109 102 87 83 20 833 265 221 189 16« 148 111 103 95 80 30 ;«6 271 '^'2* 1"3 168 151 13(5 124 lit 105 97 00 85 wnrDMins. 1. The velocity of windmill sails, whether unloaileil 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 maximum is neiudy, hut somewhat 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 maximnm, 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 continu- ing the same, — 1st, 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 ef- fects will be nearly as 10 to 27^ ; but, 3dly, when the velocities compared are more than double of that when the given load pro- duces a maximum, the effects increase nearly in the simple ratio of the velocity of the wind. hv Google ■WINDMILL?. 333 6. In sails where the figure and position are similar, and the ve- locity 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 mo- tion, 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 posi- tion, when the radius was 30 feet. Table 2 contains the most efficient angles. Kumler of R.HO bsntfen POM! .f the ■iS' sll "£"" ijsl'jri* Angle ot™tiEr. 3 2 miles o-me 1 2 72° 71 IS- IS 6 4 miles o-soo 3 4 72 74 18 middle 16 6 5 miles 0-833 5 6 77* 83' 121 7 Supposing the radius of the sail to be 30 feet, then the sail* will commence at i, or 5 feet from the axis, where the angle of inclina- tion will be 72 degrees ; at |, 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 grind- ing corn. — By M. Coulomb. A windmill, with four sails, measuring 72 feet from the ex- tremity 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. avoir- dupois 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 mill- stone 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 mil! will grind from 880 to 990 lbs. avoirdupois in an hour, or about 22,000 lbs. in 24 hours. hv Google THE APPLICATION OF LOGAKITHMS. The practice of performing calculations by Logarithms is an ex- ercise so useful to computers, that it requires a more particular ex- planation 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 re- sults can be more safely relied upon and more easily obtained than with common arithmetic. By a slight examination, the student will perceive, in some de- gree, 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 ig- norance of the proper application of logarithms. Logarithms are artificial numbers which staad for natural num- bers, and are so contrived, that if the logarithm of one number be added to the logarithm of another, the sum will be the logarithm of the product of these numbers ; and if the logarithm of one num- ber be taken from the logarithm of another, the remainder is the logarithm of the latter divided by the former ; and also, if the loga- rithm of a number be multiplied by 2, 3, 4, or 5, &c., we shall have the logarithm of the square, cube, &o., of that number ; and, on the other hand, if divided by 2, 8, 4, or 5, &c., we have the logarithm of the square root, cube root, fourth root, &c., of the proposed num- ber ; 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 di- viding 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 !■ 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 num- ber of figures on the left hand of the decimal point ; thus, The index of 12345- is i; 1234-5 _ Z; 12g-4o - 2; 12-345 - 1-, 1-2345 - 0- hv Google THE APPLICATION OP LOGAEITIIMS. 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 ^- -012345 ii -2- or 2- -12345 is -1- or 1- Eecanse 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 — 4'0914911, 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 comple- ments to 10 are used : for J-0914911 is substituted 6-0914911 — 3-0914911 7-0914911 — 2-0914911 8-0914911 &c. ko. "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 LOGAEITHMS. Take the logarithms of the factors from the tabic, 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 differencehe- tween 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 who have only a slight knowledge of logarithms, than that of using tJic aritli- metical complements of the negative indices. 1. Multiply 37-153 by 4-086, by logarithms. 37-153 1-56&9 4-086 .0-611298 4 Prod. 151-8071 2-1812923 2. Multiply 112-246 by 13-958, by logarithms. 112-246 2-0501709 13-958 1-1448232 Prod. 1566-729 3-1949941 hv Google 33S THE PRACTICAL MODEL CALCULATOR, 3. Multiply 46-7512 by -3275, by logarithms. Kbs. Logs. 46-7512 1-6697928 ■3275 r51.52113 Prod. 15-31102, 1-1850041 Here the +1 that is to be carried from the decimals, eanccla the —1, and consequently there remains 1 in the upper line to be set down. 4. Multiply -37816 by -04782, by Jogarithraa. Nos. Logs. ■37816 .T57T6756 ■04782 2^6796096 Prod. 0^018083G .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 -007G93, together. 3-768 0-576lioD 2-053 0-3123889 ■007693 3-8S60957 Prod. -0595108 .2-7745955 Here the +1 that is to be carried from the decimals, ivhon ad- ded to —3, makes —2 to be set down. 6. Multiply 3-586, 2-1046, -8372, and -0294, together. 3-586 0-5546103 2-1046 0-3231696 -8372 .1-9228292 -0294 2-4683473 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 EY LOGAKITHMS. 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 bo con- sidered 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 hv Google THE APPLICATION OP LOGARITHMS. 661 of the divisor -witli its sign changed, and —1 when 1 is to be carried from the decimal part of the logarithms. 1. Divide 4768-2 hj 36-954, by logarithms. Nbs. Logs. 4768-2 3-6783545 36-954 1-5676615 Qaot. 129-032 .2-1106930 2. Divide 21-754 by 2-4678, hy logarithms. Nos. Logs. 21-754 1-3375391 2-4078 . 0-3923100 Quot. 8-81514 0-9452291 3. Divide 4-6257 by -17608, by logarithms. J\^. Logs. 4-6257 0-6651775 ■17608 1-2457100 Quot. 26-27045 1-4194675 Here the —1 in the lower index, is changed into +1, which ia then taken for the index of the result. 4. Divide -27684 by 5-1576, by logarithms. Nos. _ Logs. ■27684 1-4422288 5-1576 0-7124477 Quot. -0536761 2-7297811 Here the 1 that is to be carried frtfm the decimals, is taken as —1, and then added to —1 in the'' upper index, which gives —'I for the index of the result. 5. Divide 6-9875 by -075789, by logarithms. Nos. Logs. 6-9875 .0-8443218 -075789 . 2-8796062 Quot. 92-1967 1-9647156 Here the 1 that is to he carried from the decimals, is added to —2, which maltes — 1, and this put down, with its sign ehangeii, is +1. 6. Divide -19876 hy -0012345, bj logarithms. Nos. Logs. •19876 r2983290 -0012345 3-0914911 Quot. 161-0043 2-2068379 Here — 3 in the lower index, is changed into +3, and this ad- ded to 1, the other index, gives + 3 — 1, or 2. b,Google 338 THE PRACTICAL MODEL CALCULATOE. PRO PORTION; 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 re- mainder is the logarithm of the term sought. Or the same may be performed more conveniently, for any single proportion, thus — Find the complement ot the logarithm of the hrst term, or what it ^ants of HI, by hegmnmg at the left hand and taking each of the figures fiom 9, except the last figure on the right, which must he taken from 10 , then add this result and the logaiithma of the other two figures togethei : the sum, abating 10 m the mdex, will be the kgaiithm of the fourth term. 1. Fmd % fourth proportional to >1 12:,, 14 768, and 135-279, by logarithms Log of 37 125 1 5b966C5 Complement 8 4303335 Log of 14 7b8 1 16t3217 Log of 135 279 2 1312304 An« 53 812h 1 73U'<856 2. Pind a fourth propoitional to 0j7H 7186, and -34721, by logarithms Log. of -05764 2-7607240 Complement 11-2392760 Log. of -7186 1-8564872 Log. of -34721 1-5405922 Ana. 4-32868 0-6363554 3. Find a third proportional to 12-796 and 3-24718, by logarithms, Log. of 12-796 1-1070742 Complement 8-8929258 Log. of 3-2471S 0-5115064 Log. of 3-24718 0-5115064 Ans. -8240216 .T-9159386 INVOLUTION; OB, 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 nega- tive, 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 2'7568, by logarithms. Log. of 2-7568 0-4404053 2 Square 7-599947 hv Google THE APPLICATION OF LOGARITHMS. 339 2. Find the cube of 7-0851, by logarithms. Log. of 7-0851 0-8503460 Cube 355-6625 2-5510380 Therefore 355-6625 is the answer. 3. Find the fifth power of -87451, by logaritbms. Log. of -87451 r-9417648 5 Fifth power -5114695 1^7088240 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 0-0019499 365 97495 116994 58497 Power 5-148888 Log. 0-7117135 EVOLUTION; OR, THE EXTRACTION OP 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 denomina- tor, 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 } 1-4387796 Root 5-2407 -7193898 2. Find the cube root of 35-6415, by logarithms. Log. of 35-6415 S ) 1-5519560 Root 3-29093 -5173186 3. Find the fifth root of 7-0825, by logarithms. Log. of 7-0825 5 ) 0-8501866 Root 1-479235 -1700373 hv Google 340 THE PRACTICAL MODEL CALCULATOR. 4. Eiad the 365th root of 1-045, by logarithma. Log. of 1-045 365) 0-0191163 Koot 1-000121 0-0000524 6. Find the value of (-001234)* by logarithms. Log. of -001234 3-0913152 3) 6-1826304 Ans. -00115047 .2-0608768 Here the divisor 3 being coataiaed exactly twice in the negative index —6, the index of the quotient, to be put down, will he —2. 6. Find the value of (-024554)^, by logarithms. Log of -024554 .1-3901223 2) 6-1703669 Ans. -00384754 .'3-5851834 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 or CALCULATING THE LOGARITHM OF ANY GIVEN NUMBER, AND THE NUMBER CORRESPONDING TO ANY GIVEN WGARITHM. DIS- COVERED BY OUVER BYRNE, THE AUTHOR 01- TUE PRESENT WORK. The succeeding numbers possess a particular property, which is worth being remembered. log. 1-371288574238642 - 0-1371288574238542 log. 10-00000000000000 - 1-000000000000000 log. 237-5812087593221 = 2-375812087593221 log. 3550-260181586691 - 3-550260181686591 log. 46692-46832877758 = 4-669246832877758 loo. 576045-6934135527 = 5-760456934135627 log. 6834720-776754367 = 6-834720776764357 log. 78974890-31398144 - 7-897489031398144 log. 896191599-8267852 - 8-951915998267839 In these numbers, if the decimal points be changed, it is evident the logarithms corresponding can also be set down without any cal- culation whatever. Thus, the log. of 1S7-1288574238542 - 2-1371288574238542; the log. of 35-50260181586591 = 1;550260181586591; log. -002375812087693221 = 3-376812087593221 ; log. -0008951915998267852 - 4-951916998267852; b,Google THE APPUCATION OP L0GAE1THM3. 341 and St) on in similar cases, since the change of the decimal point in a number can only affect the whole number of its loga- rithm. These numbers whose logarithms are made up of the same digits will bo found extremely useful hereafter. We shall next give a simple method of multiplying any number by any power of 11 , 101, 1001, 10001, 100001, kc. This multiplication is performed by the aid of coefficients of a binomial raised to the proposed power. ix + yV ^x + ^, the coefficients are 1, 1. X -h ^f =x^ + 2xy + y^, the coefficients are 1, 2, 1. x-iryY = 7? -\- Z3?y + Zxy^ + f, the coefficients are 1, 3, 3 1. The coefficients of fa; + ?/)''are 1, 4, 6, 4, 1, — ~ {x-\-yf~ 1, 5, 10, 10, 5, 1. — — {x + yf~ 1, 6, 15, 20, 15, 6, 1. — — h-\-yY— 1, 7, 21, 35, 35, 21, 7, 1. — — (a; + 3')=— 1,8, 28, 56, 70, 56, 28, 8,1. — — {x+ yf— 1, 9, 36, 84, 126, 126, 84, 36, 9, 1. Let it be required to multiply 54247 by (101)'. 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 ct b "1 54 24 70 0000 1 3 25 48 2000 6 8 13 70150 I 15 10 84'94 c 20 1 8|14 d 15 1 3 e 6 true to 10 pi (542«) X (101)' = m 58 42 83 61 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 ad- vance, 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)', and then by (101)' ; 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 : hv Google THE PRACTICAL MODEL CALCULATOR. 4247t 2 712: (54247) X (101)' . 570141142 57|0141 ..5..0 )...10..6 ! ...1D..C ..5..d 67 58 42 83 61 - (54247)' X (lOl)'. The truth of tliis 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)-. 101 1010 10201 = 101 10201 102010 1030301 - (101)'. 101 10S0301 10303010 104060301 . 101 104060401 1040604010 10510100501 . 101 10510100501 105101005010 1061520150601 . 54247 7430641054207 4246080602404 2123040301202 4246080602404 5307600753005 575842836019652447 the required product, (101)<. (101)'. (101)-. b,Google I'HB APPLICATION OF LOGAKITlIJtS. 343 which shows that the former process gives the result true to 10 places of figures, of whiuh we shall add another example. Multiply 34567812 hy (1001)', so that the result may he true to 12 places of figures. 3456 I 2 78120000 76542496 96790 19 ....8,.a ..28..S 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 3456 ...b.. 8/ ...C..501 Result, = 345954759305 the same as before. Perhaps this product might be obtained with greater ease by first multiplying 34567812 by {lOOOlJ^ and the prodact by (10001)^ the operation will stand thus : 345850093631 103755298. 10376. = 34567812 x (10001)'. 345954759305 = twelve places of the product of 34567812 by (10001)* X (10001)^ = ^34567812) X (10001)^ Although these methods are extremely simple, yet cases will oc- cur, wbea one of them will have the preference. Oar next object is to determine the logarithms 1*1 ; 1-01 ; 1-001 ; 1-0001; 1-00001; &c. It is we!i known that log. (1 + ») = M (n - ^n? + in' ~ ^n' + in' — Jn* + &c.) M being the modulus, = -432944819032618276511289, &c. It is evident that when n is -^i j^j, jiga, rMay ^c-j the calcula- tion becomes very simple. hv Google 844 THE PEACTICAL MODEL CALCULATOR. M = -4342944819032518 J M - -2171472409516269 1 M - -1447648273010889 J M = -1085736204758130 i M - -0868588968806504 J M = -0723824136505420 { M = -0720420788433217 i M - -0542868102379065 i M = -0482549424336946 i,M - -0434294481903252 &c. &e., are constants employed to determine tte logarithms of 11, 101, 1001, 100001, ke. To compute the log. of 1-001. In this ease n = jJp. M + j-jjg - -0004342944819033 positive - TlOMp " -0000002171472410 negative ■0004340773346623 + njjjjji - -0000000001447648 positive -0004340774794271 iM - rgjfjffa = -0000000000001086 negative -0004340774793185 jM ■ (1000)' -OQOOOOQOOOOOOOOl positive -0004840774793186 - the log. of 1-001 ; true to sixteen places. It is almost unnecessary to remark, that, instead of adding and suhtracting alternately, as above, the positive and negative terms may be summed separately, which will render the operation more concise. Posilii^ Terms. -0004342944819038 1447648 + -0004342945266682 — 000000217473496 Negative Terms. -0000002171472410 1086 -0000002171473496 ■0004340774793186 = log. 1-001. In a similar manner the succeeding logarithms may be obtained to almost any degree of accuracy. b,Google THE APPLICATIOH OF LOGARITHMS. 345 Log. 1-1 1-01 1-001 1-0001 1-00001 1-000001 1-0000001 1-00000001 1-000000001 1-0000000001 1-00000000001 1-000000000001 - -041392685168225 &c. which ti - -004321373782643 = -000434077479319 - -000043427276863 = -000004342923104 - -000000434294265 =. -000000043429447 = -000000004342945 : -000000000434296 : -000000000043430 : -000000000004343 . -000000000000434 1-0000000000001 - -000000000000043 1-00000000000001 - -000000000000004 &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. " ' 1 the logarithm of 542470, to seven places of decimals. e call A — B — — C D E F G H I J K L ke. 5 412 4 32 5 i 8 70 48 13 10 20 71 85 8 - 6B = 3D 5 7 5 8)4 2 8 4 i72 7 5 3 = -02692824 Tale 57601562 Prom 5 7 6 4 5 6 9 = -00013028. 576) ■ ■ • • 310 7 2|8 8 = 6E = -00002171 112 7 l|l5 _2i' - -00000087 12 12 -2G = -00000009 -02608119 Tske 6-76045693 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 nuKuber to 542470, whose log. is composed of the same digits as itself, being 576045'6934, &c,, onr 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. hv Google 346 THE PEACXICAL MODEL CALCULiTOE. It 13 here necessary to remark, that A is not employed, because the given number multiplied by I'l, would exceed 576045'69 ; for a like reason C is omitted. Again, ■when half the figures coincide, the process may be per- formed (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 figui'es in each period at this step. For A, there is but one figure in each period ; for B, there are two figures ; for 0, 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[7|8|5I9|0IOjO 5571800 278590 3 317 0!9 3,9 = 2 A = -08278537 16854170 33 7 09 3:3 7 3 5 4I2 8 9|0 8 - It 85 8 3 5 Take 354 919 801 = 20 = -00086815 Prom 3 5 5 2 6 2 ■ ■ • -aisoi 2]4 85 -TE- . -00003040 3116 2;8 4 = 8P = = -0000034r % - 9G- Tako From 785-9 - . -00000039 ■10529465 3-55026018 log. 2' 3-44496653 At the Observatory at Paris, g = 9*80896 metres, the second being the unit of time, what is the logarithm of 9-80896 1 In this example, we shall bring 9-80896 to 9-99999, &c. hv Google THE APPLICATION OF LOGARITHMS. 98|OS[9 6|00|00 |9 8l0 8|9 6i0 990:70419600 - IE - -0043213; 8|9163446 85|6654 832 9996:5705 29989 99995:69804 3 9 9 9 8 3 = 3 D = -0001302818 Tike 9999969793 -4E = -0000173717 From 10000000000 30207 From which wo hwe 3F = -0000013029 2 H - -0000000087 7 J - -0000000003 Take -0083770365 Log. 9 From 1-0000000000 •9016229635 As before observed, 9 C might have been obtained in the follow- ing manner : 8 9 017 49 6 0;0 = 1 B, as above. 4l9 5 3l5 24!8 9|9 7|0 9P) 5 times 9 9 5:6 6 8'4 0117 8|982673|6 597 39 ^ 4|0 4times9996570632 - 9 C. A French metre is equal to 3-2808992 English feet, required the log. of 3-2808992. e\ d 192 00. ..once i2944... 7 times from ffl 198 88.. .21 — b .4831... 35 — e 1148. ..35 — d L7...21 — e B7. 3517 56 8018 ■ b,Google 348 THE PEACirCAL MODEL CALCULATOR. The manner in which B 7 ia obtained is worthy of remarli : 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 6) is determined by multiplying the 2d line by 3. If the 2d lino be again multiplied by 5, we have the 4th line of the multiplier 35; but to multiply by 5, we have only to take tho half the product produced by mul- tiplying 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\ 1 351 ToOiSOl 8 = B 7, as above. 9 3|l6581l!2 36 12 663 2 84 29'6 354 935 3058 = C9 The 2d (or 9) line is produced by beginning at a, but the multi- plication may be performed by subtracting 3517568 from 35175680 ; the 36 line is produced by beginning at f>, 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 tho 9 line by 4, beginning one period more to the left. To multiply by 84 is not apparently so convenient, for 84 x 352 = 29|568 ; and as only one figure of the period 568 is required, when the proper allowance is made, the result becomes 29|6. But, since 84 is equal to 36 x 2^, we have only to multiply tho 36 line by 2, aud add J of it ; with such management, the work will stand thus : — 351]756r801l8 = E 7, as before 3|165|Sll|2 = 9 times 12|60.3j2 = 36 times 24 3 = 72 times \ 42 = 12 times/ 84 times 354 035 305 8 = C 9 This amounts to very little more than adding tho above numbers together. Many other contractions will suggest themselves, when the mul- pliers 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. C 578370. ..ten times Thus, 57837 x 9 =-! 57837 . ..once 520533. ..nine times hv Google THE APPLICdTION OS LOOjIEITHMS. 349 Such simple observations are to be found in eyerj bool; on men- tal arithmetic, and therefore require but little attention here. The whole work of the previous esampie will stand thus : — 8 018 9:920 944 22 96 62 9 618 89 8 1]|4 8 11 31 + T 351175618 018 = -0302496165 - Sll 6 68 112 1 2|6 6 8 2 296 ■354913530 17098 5 8 = -0039066973 - 71 36 = 365006 2964 = -0000868546 = 2 D 1:7 7 5 3 TaheE5 = 3550240471 = -0000217146 - 5E From 3550260182 3550) 119 7 1 1 F 6 ll7 7 6 - -0000021715 - 5 F 119 6 1 a 5 ll77_5 - -0000002172 - 5 G 118 6 H 6 ll7 8 - -0000000217 - 5 H 12 l7 - -0000000009 - I 2 II J 3 ll - -0000000001 = J 3 Take -0342672944 From 3-5502601816 Log. 3280-8992 = 3-6159928972 .-. log. 3-2808992 = 0-5159928972. 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 con- stant, as in the former examples. b,Google THE PRACTICAL MODEL CALCULATOE. 35 50;2 6 018 16 710 0520 36 3660260 ■0086427476 = ■0034726298 = 32 = 36 2116204112- 2|8 9 7 2 9 6 3 3 1014054 2028 C 8 = 3 6 5 016 9 4 9;8 2 7 = lis 2 5 3476 36 61 .2U = 80 Take D6 = 36626206963- Ftom 36625636516 •0002171364 = ■0000043429 •0000004343 . = 5D S6626-2 El- 429563 365262 = -IE Fl - 16 4 3 1 1 18 6 52 6- -IP G7 - 27786 25568 - •0000003040 = »7e H6 = 2218 2;i 9 1 - •0000000261 - . 6H 10 J7 - 27 2|5 = ■0000000003 = = 7J •0123376214 Add 3-5502601816 Hence, log. 3652-5636516 = 3-5625978030 .-. log. 365-25636516 = 2-56259T803. M. Regnault determined with the greatest care the density of mercury to be 13-59593 at the temperature 0°, centigrade. It ia required to calculate the log, of 13-59593, to eight places of decimals. In thia case it is better to bring the given number to the constant 1371288574. 13 5'969|300 1,0 8 7 6 7 4 38 7 C8 -1370|6078'8 . l6 8 5 2'5 14 ■003472030 -80 Subtract D6-137119328- ■000217136 = 5 D From 137128867 9,5 2 9 - -000026058 = S 6 E6- 8|2 2 7 r3|02 F 9 - 1 213 4 - ■000003909 = F 9 H5. 68 69 . ■ 000000022 -H5 ■003719765 b,Google APPLICATION OB LOGARITHMS. 351 Take -003719755 from -137128857 log. 1-359593 = -133409102 .-. log. 13-59593 = 1-133409102. TO DETBBMIHB THE NUMBER CORRESPOKDISG TO A GIVEN LOGARITHM. This problem has been very mueli neglected — so much so, that none of our elementary books ever allude to a method of comput- ing the number answering to a given logarithm. When an opera- tion is performed by the use of logarithms, it ia very seldom that the resulting logarithm can be found in .the table ; we have, there- fore, to find the nearest less logarithm, and the next greater, and correct them by proportion, so that there may be found an inter- mediate number that will agree with the given logarithm, or nearly 80. But although the proportional parts of the difference abridge thia process, we can only find a number appertaining to any loga- rithm 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 ^ in the last decimal, and since both errors may conspire, the 7th figure cannot be depended on as strictly true, unless the proposed logarithm falls between the Hmits of log. 10000 and log. 22200. Indubitably we are now speaking of extreme cases, but since it is not an unfrequcnt occurrence that some calculations require the most rigid accuracy, and many resulting logarithms may be ex- tended 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 follow- ing 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, r rather, 3-37581209, when the characteristic or index is ii a unit. First from 3-44486555 talis 3-37681209 213 7 5 8 1 2 9 constant |2 3758121=.A1 -06915346 •04139269 -lA ■02776077 •02592824- 6E 2 6.1 3 3 9 3 3 l|6 6 8 0'3 6 3 9 20 6!2 2 3 2 7 714 1 69 6 ijl 96 6 lie 6 9 7 9 5 = B0 8 4 1 8- C4 . .183253 17S631 - 4 C .... 9622 8686 = 2 D 937 2 7 8 6|2 8 2 'J b,Google THE PRACTICAL MODEL CALCULATOR. . . . 937 22 - 5 a S-7H 2786282 918 -C4 |65706 Isi 27 8 6 8|4|0|0l7'-D2 5!5 7 2 = E 2 |2l79 -El l'3 9 -05 lis = 117 278690016 .-. 2786-90016 is the number sought. What number correapouds to the logarithm 6-73437674 i When the index of this log. is reduced by a unit, the next less constant is 4-66924683. From 4-73437674 Take 4-66924683 •6512891 4139269 1 A 2160687 6B ••212035 173631 4 . . . 39804 39085 9D 219 There is neither the Cfjuaj of 217 5 F this number, nor a 772 Cr less, obtainable from 2 4 H ^^ •'■ E9, or B, is omitted. Then, 4166924683 46692468 A] 6113 6117 2|5680 613 5 1611 868 617 136 26 5 3 918 1617 8 8 B5 2|l 5 92 6 7 32 3 9 2 6419|7 9 2 9I6 4 48 7 7 81 19!5 64246721712 D9 12712 F6 I2I2 H4 542470006 . 542470-006 is the number whoso logarithm ia 6-73437574. b,Google THE APPLICATION OF lOQARITHMS. 353 Had the given logarithm represented a decimal with a positive index, the required number would bo 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 ■3727461 3726342.. . . . 2109 1737 . 4D ....372 347 8K .26 22.. ..5F 1:010 90 7 Constant. 86 AS I9 4B2 B 81911 189 l|l 1 D4 E8 E5 6 G7 23690949 .-. 235-90949 is the required number, and the seconds in the di- urnal apparent motion of the stars. 235-90949" = 3' 55-90949". Let it be required to find the hy^jerhoUc logarithm of any number, as 3-1415926536. The common log. of this number is -49714987269 (33), and the common log. of this log. isT-6964873. The modulus of the common system of logarithms is -4342944819, kc. .-. 1 : 4342944819 : : hyperbolic log. N : common log. N. b,Google 354 THE PRACTICAL MODEL CALCULATOIl. To distingmsh the hyperbolic logarithm of the number N from its common logarithm, it is necessary to tvrite the hyp. log. Log. Jy, and the common logarithm log. N. Hence, 4342941819 X Log. N = log. N ; or log. (-4342944819) + log. (log. N) - log. (log. N). .-. log. (Log. N) = log. (log. N) - 1-637784.3 ; for 1-6377843 - log. -4342944819. Now, to work the above example, from 1-6964873 take F6377843 "-0587030, the number corresponding to this com. log. will be the hyp. log. of 3-1415927. -0587030 must be reduced to -0000000 which is known to be the log. of 1. ■0587030 0413927 lA . 173103 172855 4B 248 217 5E 1 A - 1 1!0 OiO 0:0 0:0 440 0,0 6 6'0 14 40 .31 30 71? ..1 2G . 1-14472988 i 1144(1,64411 = B4 !6,7 2'3 = E 5 18 01 =F7 !2i3 - G2 114472988 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)' X (1-00001)' X (1-000001)' X (1-0000001)' - 1-14472988. This example answers the purpose of illustration, but the hyp. log. of S-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. Reiiuii-ed the decimal. The index, 1, may be changed to any other characteristic, so as to suit any of thi ecmstants, as the alteration is easily allowed for when the work is completed. In this instance, it is best to put ■f 1 instead of 1. From 1 Take 1-0000000 -3426139 3311415 - 8 A •0114724 . .86427 - 2 B 28297 26045-60 2262 1'0'0'0 0|0|0;0^0 Constant l8'0 0lo0 0'0 280 0:0 0'0'0 66 700 0,0:0 000 00 800 8:0 1 21:4315 8:8 8,1 = A8 b,Google THE APPLICATION OF LOGAKITHMS. 22o2 2171 - ^81 5D IE 8P 7(J 2 1)4 3|5 S 4 2l8 7 21 881 = A8 178 436 43 - 38 35 = "3 3 = 218,667495-B2 l'3120 5 ' 3280 4 2198,82784 = 06 10 9 9 91 22 2 2 9|2;7 7 97-D 5 01 =E1 61 -F8 54 = 67 220096913 .-. The French litre = -2200969 English gaUons. In measuring heights hy 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 45°, it weighs 1-292697 grammes. A litre of water, at its maximum density, weighs 1000 grammes, and a litre of mer- cury, at the temperature of 0° cent., weighs 13595-93 grammes; 13595-93 ■"■ 1-292697 ~ ratio at 46" Now, log. 13695-93 - 4-133409102 (29) and log. 1-292697 - 0-111496744 (30) 4-021912358 = the log. of the ratio at 45°. To find thenumber corresponding to thislog., 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 •021606869 - 5 B , 305489 303991 = 7D . . . 1498 1303 = 3 F lOIOOjOO 'sOiOO 10 OOjOO 00 00 000 10511010 7 8 57 .195 174 = - 91 13595-93 1-292697 veritied hy common division. 10517 415 98 -D7 |3 1 6 = F 3 4|2 =G4 4 = 114 1-19 . hy logarithms. 105174 9 61 = 10517-49, &c., which is easily b,Google 366 THE PRACTICAL MODEL CALCULATOK. M. Regnault found that, at Paris, the litre of atmospheric air weighs 1'293187 grammes ; the litre of nitrogen. 1'256167 grammes; & litre of oxygen, 1'429802 grammes; of hydrogen, 0-089578 grammes ; and of carhonic 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. Eeg- nault finds the weight of the litre of air under tho parallel of 45° 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, ho 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^ Hero, te is the weight of the litre of air, R the mean radius of the earth = 6366198 metres, A the height of the place of observa- tion above the mean radius, and x tho latitude of the place. At Philadelphia, lat. 39° 56' 51-5", suppose the radius of the earth to be 6367653 metres, the weight of the litre of air will he 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 : ; -g- : the required degrees. Log. 360 = 2-556302500767 log. 3-14159265859 = 0-497149872694 2-059452623073 log. 2 = 0-301029995664 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. = Constant From 4-758122e32409 Take 4-669246832878 ■088875799531 4|6;6 92I4 6 813 9!3:38493 6 4|6 6|9 24!6 2878 6676 8329 7;7 83 6;6 7;8 2 A - ■82786370816 . . 6090429216 6 614 9,7 88 6,6 6e|497 8!8 IB- 4321373783 ..1769055432 4 0- 1736309917 ....32745615 7 E = 30400462 6701628,655 2128 2514 3J4 2 3 2 1461 6218 7719 2825 6 2345053 5729114 696 1|2 2 9 b,Google .298 261 THE APPLICATION OP LOGARITHMS, 357 ...2345053 5729i;459(il;229 - 04 2171471 4|01040J217 ...! 173582 ISjQSl 130288 5729547013477 -E7 ....^43294 218647735 "" 39087 51 '^^ 5729576!6|6il2l69-F5 3509 l|718873-e3 5|l 5 6 6 2 = H 9 S'l 566 = 19 3 438 -J 6* 37 45S = K8 8K- 35 ^-L6 T2 5729577951295 -tie num- 5 L = 2 ber required. But the original index is 1; .-. 57-29577951295» are tlie num- ber 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 loga- rithms 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 correct- ness the number corresponding to a given log. can be obtained. The extent, also, hj far exceeds that obtainable by any tables extant. Other computations give, i-" - 67-2957795130° - 57° 17' 44" -80624 the degrees in an arc = radius. / =. 3437-7467707849' = 3437' 44" -80624 the mmutes in an are = radius. /' = 206264-8062470963 the number of seconds in an arc = radius. The relative mean motion of the moon from the sua in a Julian or fictitious year, of 365J days, is 12 eir. 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 loga- rithms. Log. 365-25 2-56259022460634 log. 1296000 6-112605001534 57 8-67619522614091 log. 16029615-977315 = 7-20492311805406 1-47027210808685 b,Google 358 THG PRACTICAL MODEL CALCULATOR. If the index of this log. be made 2 instead of 1, the neai-est nest less constant will be 2-376812087693221. Prom 2-47027210808685 Take 2-37681208769S22 2A = 09446002049363 08278537031645 2B = .1167466017718 864274758529 6C = . . 303190261189 260446487591 9D- ...42743778598 39084549177 8E = ....3669224421 3474338483 4P = 184886938 173717706 2G = 11168232 8686889 5H = 2482348 2171478 71 = 310870 304006 IJ = 4343 5K = 2520 2172 348 8L = 2N- 317 1 213 7,61811:2 018|7:5 9 3i2|2 Const. 4 7 6 1 6:24 1 716:1 8 64 2|3|7|5(8il|2|o;8i75;93 .2A 2 817417 3 2 6(2 6 9 817 7|9 J67494 6 6 25ll9 7 6 2 874i7 3;26!2 6;0 293|251475]177 015 1759508861 {)6 2 43981772 128 5i8 6 5 2!) 4 399 2960|1538|8669 2|6 55 1016205 35-06 80S 640 29628|10087|49763- 2|3622480700 8|26787 17 2953041632057 l|l81218 1 7 7 2 2963 58J1327756J7 =F4 2953058712138:8,73 114 7 6'6:2 9 2|0 61711 4 29163 ll47 6 236 3-02 H6 1 = 17 1-Jl 6-K6 2-L8 6-N2 295305889217832 .-. 29-6305889218 is the number required. To perform by logarithms the ordinary operations of multipli- cation, divisim proportion oi even the e\ti-iction of the square root, except in the ^ ay of illustiatio is not the design of these pages ; for such an appl cat on ot logarithms m a particular man- ner only dimin »h the labour of the operator It is not necessary, however to examine mi utely hcie the instauces in which common arithmet c is piefeiable to a till al numbe s besides, much will depend on tl e sL li and fac 1 ty oi the p atoi. b,Google TRIQONOMETEY. SPHERICAL TRIGONOJIETEY.- ANGULAR MAGNITUDES. Plane trigonometry treats of the relations and calculations of the sidesand 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 de- grees. The measure of any angle is an arc of any circle contained be- tween 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 de- grees. 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 ", mi- nutes with ', seconds with ", and so on. Thus, 57° 30' 12" de- note 57 degrees 30 minutes and 12 seconds. The complement of an arc, is what it wants of a quadrant or 90°, Thus, if AD he a quadrant, then BD is the complement of the arc AB ; and, reciprocally, AB is the complement of BD. So ^ that, if AB he an arc of 50°, then its complement BD will be 40°. The supplement of an arc, is what it wants o a semicircle, or 180°. Thus, if ADE be a semicircle, then BDE is the supplement of the arc AB ; and, reciprocally, AB is the sup- plement of the arc BDE. So that, if AB he an arc of 50°, then its supplement BDE will be 130°. The sine, or right sine, of an arc, is the line drawn from one extremity of the arc, perpendicular to the diameter passing tiirough the other extremity. Thus, BF is the sine of the arc AB, or of the arc BDE. Hence the sine (BF) is half the chord (BG) of the double arc (BAG). The versed sine of an arc, is the part of the diameter intercepted between the arc and its sine. So, AF is the versed sine of the arc AB, and EF the versed sine of the arc EDB. hv Google 3{!0 THE PRACTICAL MODEL CALCULATOE. The tangent of an are is a line touching the circle in one ex- tremity of that arc, continued from thence to meet a Ime Jidivu from the centre through the other extremity: fthich last line is called the secant of the same arc. Thus, AH is the tangent, anJ CII the secant, of the arc AB. Also, EI is the tangent, ami CI the secant, of the euppleiaental are 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, ED being the complements of each other, the sine, tangent or se- cant of the one of these, is the cosine, cotangent or cosecant of the other. So, BF, the sine of AE, is the cosine of ED ; and BK, the sine of ED, is the cosine of AB : in iike manner, AH, the tangent of AB, is the cotangent of ED ; 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 ED, 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 are is greater than a quadrant or 90 degrees. When the arc is 0, or nothing, the sine and tangent are nothing, but tho 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 ra- dius CD of the circle ; and both the tangent and secant are infinite. Of any arc AE, the versed sine AF, and cosine BK, or CF, together make up the radius CA of the circle. The radius CA, tangent AH, and secant CH, form a right-angled triangle CAH. So also do the radius, sine, and cosine, form another right-angled triangle CBF or CEK. As also the radius, cotangent, and cosecant, another right- angled triangle CDL. And all these right-angled triangles are similar to each other. The sine, tangent, or secant of an angle, is the sine, tangent, or secant of the arc by which the angle is mea- . sured, or of the degrees, &c. in the same ^ arc or angle. The method of constructing the scales of chords, sines, tangents, and secants, usually engraven on instruments, for practice, is exhibited in the annexed hv Google TEIOONOMETRT. 301 A trigonometrical canon, is a table exhibiting the length of the sine, tangent, and secant, to every degree and minute of the quad- rant, with respect to the radius, which is expressed by unity, or 1, and conceived to he divided into 10000000 or more decimal parts. And further, the logarithma 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 tho multiplication and division by the natural sines, kc, 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 tho mode of constructing it, which may he done in the following 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 cir- cumference 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 3-1415926535897S3, kc, 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 g ami c; then will these two be expressed by the two following series, viz. :— ^ = '^- 2l'^2:SAl ~ 2.3.4.5.6.7 + ^^• = "^ ~ "6 + 120 ~ 5040 + ^'^• ■ 2.3.4 2.3.4.5.6 ' + ^-TM + ^ 24 720 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 ■000000000004, &c. ■0002908882 the sine of 1 minute. and ^a^ = the dilFerence is s = Also, from take Jd^ = leaves c = ■0000000423079, &c. 9999999577 the cosim hv Google 3G2 THE PRACTICAL MODEL CALCL^LATOR. For the sine and cosine of 5 degrees. Here, as 180° : 5° : : 3-14159265, &c., : -08720 length, of 5 degrees. Hence, a = -08726646 _ ifflS = - -00011076 + Jjffl* = -00000004 these collected give s = -08715074 the si And, for the cosine, 1=1- - la^ = _ -00380771 + la-' = -00000241 these collected, give c = -99619470 the consine of 5°. After the same manner, the sine and cosine of any other arc may he computed. But the greater the arc is, the slower the series will converge, in ivhich 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 CEF, viz. the cosine CF = ^/CB' - EF^ or e = v'T^^=. There are also other methods of constructing the canon of sines and cosines, which, for brevity's sake, are here omitted. To conijiute 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 are AB, BF is the sine, CF or BK the cosine, AH the tangent, CH the secant, DL the cotangent, and CL the cosecant, the radius being CA, or CB, or Cl>; the three similar triangles CFB, CAH, CJDL, give the following pro- portions : 1. CF : FB ; : CA : AH ; Trhence 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 : FO ; : CD : DL ; whence the cotangent is known, being a fourth proportional to the sine, cosine, and radius. 4. BF : EC : : 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 trigo- nometry; 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, arith- metical computation, and instrumental operation. In tlie first method. — The triangle is constructed by making the parts of the given magnitudes, namely, the sides from a scale of hv Google laiGOKOMETET. 6b6 equal parts, and the angles from a scale of cliords, 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 propor- tion according to the proper rule or theorem, resolve it like any other proportion, in which a fourth term is to he found from three given terms, by muItiplyiDg the second and third together, and dividing the product by the first, in working with the natural num- bers; 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 re- mainder is the fourth term sought. In the third method. — Or ins t rumen tally, aa suppose by the log. lines on one side of the common two-foot scales ; extend the com- passes from the first terra 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 vari- eties 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 aides 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 sepa- rate theorem, as follows : When a side and its opposite angle are two of the given parts. Then the sides of the triangle have the same proportion to each other, aa 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 aide, To the sine of its opposite angle. ^ ¥or, let ABO he the proposed triangle, hai AB the greatest aide, and BO the least. Take AD = EC, considering it as a radius ; and let ^ fall the perpendiculars DE, CF, which will evi- a ^ • □ dently be the sines of the angles A and B, to the radius AD or BO. But the triangles ADE, ACF, are equiangular, and there- fore AO : 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 hv Google 364 THE PILAGirCAL MODEL CALCULATOR. opposite a given angle. And to find a side, begin ivitli 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 mag- nitude be such as to prevent the ambiguity ; because the sine an- swers 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 am- biguity ; for then neither of the other angles can be obtuse, and the geometrical construction will form only one triangle. In the plane triangle ABC, rAB 345 yards Given, -^BC 282 yards I angle A 37° 20' Required the other parts. G-eometrically . — Draw an indefinite line, upon which set off AB = 345, from some convenient scale of equal parts. Make the angle A = 37^°. 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 in- strument, will be found to be nearly as below; viz. AC 174 angle B 27° angle C 115-|° or 374^ or 78i or 64i Arithmetically. — First, to find the angles at : As side BO 232 log. 2-3654880 To sin. opp. angle A 37° 20' 9-7827958 So side AB345 2-5378191 To sin. opp. angle C 115° 36' or 64° 24 9-9551269 Add angle A 37 20 37 20 The sum 152 56 or 101 44 Taken from 180 _00 _1?0_00 Leaves angle B 2T 04 or 78 16 Then, to find the side AC : As sine angle A 37° 20' log. 9-7827958 To opposite side BC 2-32 2-.365488 ^ . , -r, /27°04' 9-6580371 So sine angle E | .^g ^^ 9-9908291 To opposite side AC 174-07 2-2407293 or, 374-56 2-5735213 hv Google TRIGONOMETRY. In the plane triangle ABO, r AB 365 poles Given, < angle A 57° 12' (. angle B 24 45 Required the other parts. In the plane triangle ABC, ( AC 120 foot Given, -^ BC 112 feet Ungle A 67° 27' Required the other parts. Ans. r angle C 98° 3' { AG 154-33 I BC 309-86 igle B 64=" 34' 21" or, 115 25 39 iglc C 57 58 39 or, 7 7 21 AB 112-65 feet or, 16-47 feet When two aides Then it will be, As the sum of those two sides. their contained angle are given. Is to the difference of the s So ia 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 quan- tities increased by their half difference, gives the greater, and di- minished 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 bj the former theorem. Let ABG be the proposed triangle, having r the two given sides AG, BC, including the given , angle C. With the centre 0, and radius CA, | the less of these two sides, describe a semicircle, meeting the other side BC produced in D and E, Join AE, AU, and draw DF parallel to AE. Then, BE is the sum, and BD the difference c sides CB, GA, Also, the sum of the two angl . . _ equal to the sum of tho two CAD, CBA, these suras 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, CI): hence, either of them, as CD A, 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 AU, is also perpendicular 2r2 hv Google Sbb THE PRACTICAL MODEL CALCULATOR. to AD : consequently, AE is the tangent of CDA tte half aum 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 tan- gents AE, DF, being parallel, it will he as BE : BD : : AE : DF ; that ia, as the sum of the sides is to the difference of the sides, so is the tangent of half the sum of the opposito angles, to the tan- gent of half their difference. The sum of the unknown angles is found, by taking tho given angle from 180°. In the plane triangle ABO, f AE 345 yards 5. Given, J AC 174-OT yards y^ \, (angle A 37° 20' ^ ' i, Required the other parts. Qeometrically. — Draw AE = 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 he nearly as follows, viz. the side EC 232 yaa-da, the angle B 27°, and the angle C 115^°. Arithmetically. As sum of sides AE, AC 519-07 log. 2-7152259 To difference of sides AB, AC 170-93 2-2328183 So tangent half sum angles C and E 71° 20' 10-4712970 To tangent half difference angles C and E 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 115° 36', or 64° 24' log. 9-0551259 To its opposite side AB 345 2-5378191 So sine angle A 37° 20' 9-7827958 To its opposite side EC 232 2-3654890 In the plane triangle ABC, r AB 365 poles Given, -^ AC 154-33 [angle A 57° 12' ( BC 309-86 Required the other parts. < angle E 24° 45' (angle C 98° 3' In the plane triangle ABC, f AC 120 yards Given,^ EC 112 yards (angle C57°58'39" ( AE 112-65 Required tho other parts. < angle A 57° 27' 0" igle B 64 34 21 f ^ \ angle (angle Hr,.t9<ihvGoogle TRIGONOMETRY. oOT 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, la to the sura 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 ; conse- quently, 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 seg- ments. Therefore, by forming the sides of these recta.ngles into a proportion, it will appear that the sums and differences are pro- portional, as in this theorem. In the plane triangle AEO, fAB 345 yards Given, the sides ^ AC 232 (_BG 1T4-07 To find the angles. G-eometrieally. — 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, via. angle A 27°, angle B 37J°, aad angle C 115J°. Arithmetically. — ^Ilaving let fall the perpendicular OP, it will be. As the base AB : AG + 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. 2-3054880 To sine opposite angle 90° 10-0000000 So is side AP 206-59 2-3151093 To sine opposite angle AGP 62° 56' 9-9496213 Which taken from 90 00 Leaves the angle A 27 04 hv Google 368 THE TEACTICAL MODEL CALCULATOR. Again, in the triangle BPC, riglit-arc'led at P, As the side of BC 174-07 log. 2-240T239 To sine opposite angle P.. . 90° 10-0000000 So is side BP 138-41 2-1411675 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 E 3T° 20'; the angle C 115° 36'. In the plane triangle ABC, fAB 365 poles GiveD the sides, -^ AC 154-33 (EC 309-86 (angle A 57° 12' To find the angles. ■{ angle B 24 45 U»gleC98 3 In the plane triangle ABC, (AB 120 Given the sides J AC 112-65 (BC 112 (-angle A 57° 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 tri- angles, both right-angled and obiiqae ; besides which, there are other theorems suited to some particular forms of triangles, ivhich 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 hy^othenuse; it will be. As radius, i. e. sine of 90° or tangent of 45° 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- angle ABC ; with the centre A, and any assumed ra- dius, AD, describe an arc DE, and draw DF perpen- 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 j 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. hv Google OE HEIGHTS AND DISTANCES. In the right- ai Given < iglcd triangle ABC, Geometrically. — Make AB =162 equal parts, and the angle A = 53° 7' 48" ; then raise the perpendicular BO, meeting AC in C So sliall AC measure 270, and BC 216. Arithmeticalh/ . ..log. lO-OOOOOOO 2-2096150 10-1249371 2-3344521 10-2218477 2-4313627 As radius tang. 45° TolegAB 162 So tang, angle A 53° 7' 48" TolegEC 216 So secant angle A 53° 7' 48" Tohjp. AC 270 In the right-angled triangle ABC, c-™/ the leg AB 180 ''"™ t the angle A 62° 40' , ^„ jg^-OMT To find tiie other two sides. < Tjri qio-Odi^A There 13 sometimes given another method for right-angled tri- angles, which is this : o ABC being such a triangle, make one leg AB ra- dius, that is, with centre A, and distance AB, de- scribe an arc BF. Then it is evident that the other leg BC represents the tangent, and the hypother AC the secant, of the arc EF, or of the angle A. In tike manner, if the leg BC be made radius ; then the other leg AB will represent the tangent, and the hypo- thenuse AC the secant, of the arc EG or angle C. But if the hypothonuse he made radius ; then each leg will re- present the sine of its opposite angle ; namely, the log 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 hear to each other the same propor- tion as the parts which they represent. And this is called. Making every side radius. OP HEIGHTS AND DISTANCES. Bt the mensuration and protraction of lines and angles, are de- termined 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 in- accessible 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 hv Google 370 THE PRACTICAL MODEL CALCULATOR. angles in degrees, the lines are then calculated bj trigonometry : in the other methods, the linos are calculated from the princi- ple of similar triangles, without regard to the measure of the angles. Angles of elevation, or of depression, are usually taten either with a theodolite, or with a quadrant, divided into degrees and mi- nutes, 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 y'' 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 there aa a centre, tilt with ono eye at D, the other being shut, jou perceive the object A through the sights : then will the arc Gil of tho quadrant, cut off by the plumb line BH, be the measure of the angle ABC, as required. "<F The angle ABC of depression of any ob- ject A, is taken in the same manner ; except that here the eye is applied to the centre, and tho measure of the angle is the arc GH, on \ the other side of the plumb line. \ \. The following examples arc 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 47° 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 ™ 47° 30', the angle of elevation; and it is done. Then AB, mea- sured on the scale of equal parts, is nearly 218j. n Calculation. /R As radius 10-0000000 / 1 1| , To AC 200 2-3010300 / iAWy So tang, angle C 47° 30' 10-0379475 / P.mm To AB 218-26 required 2-3389775 ^ ' " ' T hv Google 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 tirae, both on the same side of it, and in the same vertical plane ; tlieir distance, as under, being half a mile, or 880 yards. And what was its distance from the said two ob- Oonstruction. — Draw an indefinite ground line, upon which set off the given distance AB = 880; then A and E are the places of the observers. Make the angle A = 35°, and the angle E = 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'aleulation. .--''/^ First, from angle B 64° ,--'' Take angle A 35 ^•"'' ,•'' Leaves angle ACB 29 .--"' Then, ia the triangle ABC, abb Assine angleACB 29° 9-6855712 To opposite side AB 880 2-9444827 Sosine angleA 35° 9-7585913 To opposite side BC 1041-125 3-0175028 Assine angleACB 29° 9-6'855712 To opposite side AB 880 2-9444827 Sosine angleB n0°or64° 9-9536602 TooppositesideAC1631-442 3-2125717 And, in the triangle BCD, Assine angle D 90° 10-0000000 To opposite side BC 1041-125 3-0175028 Sosine angleB 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. hv Google ! THE PRACTICAL MODEL CALCUIATOR. Calculation. From the angle C 41° 00' Take the angle D 23 45 Leaves the angle DEC 17 15 Then, in the triacgle DBC, As sine angle DBG 17° 15' To opposite side DC 60 So sine angle D 24 45 To opposite side CB Sl^S'^ 9 4720856 1 7781513 y eo-josao 1 ')110977 1 6179225 10 4272623 9 9606516 9 6582842 1 6020000 9 8169429 1 7(;0Y187 And, in the triangle ABO, As sum of sides CB, C 4 121 488 To difference of sidca CB, CA 41 488 So tang, half sum angles A, B 69° 30' To tang, half diff. angles A, B 42_24J The diff. of these is angle CBA 27 5J Lastly, as sine angle CBA 27° 5y To opposite side CA 40 So sine angle G 41° 0' To opposite side AB 57 633 Wanting to know the distance hetween 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 tho angles formed hy 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 33° and 64^°. What then may be the distaiice between the two objects? * Oonstruetion. — 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 EAC, BAD, equal to the given angles 33° and 64^°. So shall C and D bo the places of the two objects, Qaleulation. — First, In the right-angled triangle ABC, As radius 10-0000000 ToAB 120 2-0791812 So tang, angle BAG 33° 9-8125174 ToBG 77-929 1-8916986 And, in the right-angled triangle ABD, As radius 10-0000000 ToAB 120 2-0791812 So tang, angle BAD.... 64i° 10-3215039 ToBD 251-585 2-4006851 From which take BO 77-929 Leaves the dist. CD 173-656 as required. hv Google SPHERICAL TBXGOSOMETKY. 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 73° 15'. What then were the distances from each end to the house ? Construction. — Draw the line AB = 200 equal parts. Then draw AC so aa to make the angle A = 68° 2', and BC to make the angle B = 73° 15'. So shall the point C be the place of the house reqiuired, Calculation. To the given angle A (58° 2' Add the given angle B 73 16 Then their sura 141 17 Being taken from 180 Leaves the third angle 38 43 Hence, As sin. ang!e C 38° 43'. 9-7962062 To op. side AB 200 So sin. angle A 68° 2' To op, " To op. So sin. To op. 3010300 ■9672679 BC296'54 2-4720917 !e C 38° 43' 9-7962062 AB 200 2-3010300 le B73°15' 9-9811711 AC 306-19 2-4859949 SPHEEICAL TRIGONOMETEY. TkU Arttde is taken Jrom a short Practical Treatise on Sphei-ical Trigonomeirii, III/ Oliver Byrne, the author of the present v>a-k. PiihUshed by J. A. Valpy. 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 AB makes up a quadrant or 90°, is called the complement of the arc AB ; every arc will have a complement, even those which are themselves greater than 90°, provided wo con- sider the ares measured in the direc- tion ABCD, &e., as positive, and consequently those measured in the opposite direction as negative. The complement BC of the arc AE com- mences at B, where AB terminates, and may be considered aa generated by the motion of B, the ex- 2G hv Google 374 TUE PRACTICAL MODEL CALCULATOR. tremity of the radius OB, iu the direction EC. But the eomplo- meat of the arc AD or DC, commencing in like manner at the ex- tremity D, must be generated by the motion of D in the opposite direction, and the angular magnitude AOD will hero be diminished by the motion of OD, in generating the complement ; therefore the complement of AOD or of AD may with propriety be consi- dered negative. Galling the arc AE or AD, e, the complement will he 90° — e ; the complement of 36° 44' 83" is 53° 15' 27" ; and the complement of 136° 2T' 39" is negative 46° 27' 39". The arc BE, which must be added to AB to make up a semi- circle or 180°, is called the supplement of the arc AB. IF the arc is greater than 180°, as the arc ADP its supplement, FE mea- sured in the reverse direction is negative. The expression for the supplement of any arc o is therefore 180° ~ o; thus the supple- ment 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 di- rection 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, All will be negative. In the doctrine of triangles, we consider only positive angles or arcs, and the magnitudes of these are comprised between o = and o = 180° ; hut 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 ares hy which they are estimated are posi- tive or negative. Thus the angle BOA is positive, and the angle AOH negative. Moreover, in this move extended theory of angular magnitude, an angle may consist of any number of degrees what- ever ; thus, if the revolving line OB set out from the fixed line OA, and make n revolutions and a part, the angular magnitude gene- rated is measured by n times 360°, plus the degrees in the ad- ditional part. In a right-angled spherical triangle we are to recognise but five parts, namely, the three sides a, 5, c, and the two angles A, , BO that the right angle C is omitted. hv Google SPHERICAL Tr.IGOKOMETKT. Let A', c', B,' be the comple- ments of A, c, B, respectively, and suppose b, a, W, c', A', to be 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 * circular parts. If we take any one of these as a middle part, the two which lie next to it, one on each side, will be adjaaent parts. The two parts immediately beyond the adjacent parts, one on each side, are called the opposite parts. Thus, taking A' for a middle part, h 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 6, a, opposite parts. When B' is a middle part, c', a, become adjacent parts, and A', h, opposite parts. Again, if we take « as a middle part, then B', h, will be adjacent parts, and c', A', opposite parts. Lastly, taking 6 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. Ead. 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 pa,rts or opposite parts. In a right-angled spherical triangle, the hypothenuse e = 61" 4' 56" ; and the angle A =■ 61° 50' 29". Required the adjacent leg? 90° -61 0' P0° 0' A = 01 50 hv Google THE PEACTICAI. MODEL CALCULATOR. , « ' ?v. .# 'i ■ • ^ -SOO 'DSOJ - Tan.i'tait.':- In this example, A' is selected for the middle part, because then 6 and v' become adjacent parts, as in the annexed figure. Kad. X sin. A' = tan. h X tan. c'. rad. X sin. A' By Logarithms. Kad. - -lO-OOOOOOO Sin. A^-28°9'21" - 9-6738628 19'6738628 Tan. c'-28°55'4" - 9-74228 08 Tan.5'-40°30'16"--9-9315820 'Ihe side adjacent to the given angle is acute or obtuse, accord- ing as the hypothenuse is of the same, or of different species with the given angle, .-. the leg h = 40° 30' 16", acute. Supposing the hypothenuse c = 113° 65', and the angle A = 31° 51', then tiie adjacent leg 6 would he 117° 34', obtuse. hv Google SPUEKICAL TltlGONOMEIKT. In the right-angled spherical triangle ABC, are given tlie hypo- thenuse c = 113° 55', and the angle A = 104° 08'; to find the opposite leg a. 14 08= A'. In this example, a is taken for the middle part, then A' and c' are opposite parts. (See the subjoined £ From the general formula, we have, Rad. X sin. a = cos. A' X cos. c'. COS. A' X COS. c' .-. sm, a = f,— 5 . Had. Sy Logarithms. COS. A' - 14° 08' 9-9860509 cos.c' -2fJ 55 9-9610108 19-9476617 / L - Radius 10-0000000 ^ \\ sin «/l^^°^*'l 9.947661T '^ The obtuse side 117° 34' is the leg required, for the side oppo- site 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 = 37° 25', then the opposite leg a = 36° 31', and not 143° 29', because the given angle is acute. In aright-angled spherical tri- angle, are given c = 78° 20', and A = 37° 25', to find the angle B. 90° 0' c = 78_20 11 40 = c'. 90° 0' A = 37 25 i>2 35= A' hv Google 378 THE PHAGTICAL MODEL CALCULATOR. Here the complement of the hypothenuse {«') is the middle part; and the complement of the ^^ ' angle opposite the perpen- dicular (A'), and the com- plement of the angle oppo- site the base (B'J are the adjacent parts. This wUl readily be perceived by reference to the usual figure in the margin. Rad. X sin. c' = tan. A' X tao. B' ; _ Rad, X sin. c' .■.tan.B' = tan. A' ' B^ Logarithms. 10-0000000 - 11° 40'. 9-3058189 Bad.- 8in. c 19-3058189 tan. A' - 52° 35' 10-1163279 .-.tan. B'- 8° 48' 9-1894910 But 90 — B = B' hence 90 - E' = B. 90° 0' 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 ob- tuse, then the angle found would be obtuse : — Thus, if the hypo- thenuse be 113° 55', and the angle A = 31° 51' ; tlien will B' = 14° 08', and the angle B = 104° 08'. Given the hypothenuse e = 61° 04' 56", and the aide or leg, a = 40° 30' 20", to find tlie angle adjacent to a. c' •&■ 90° 0' 0" e = 61 04 56 28 55 04" = c". The three parts are here connected ; therefore the com- plement of .B is the viiddle part, a and the complement of are the adjacent parts. Hence we have. Bad. X sin. B' = tan. a X tan. Rad. hv Google SPHERICAL TRIGONOMETEY, By Logarithms. tan. a - 40° 30' 20" = 9-9315841 tan. c' ~ 28 55 04 = 9 •74-22801 19-6738642 Bad 10-00000 00 sin. B'....28° 09' 31" 9-6738642 90° 0' 0" E' = 28 09 31 61 50 29 = B. The angle adjacent to the given side is acute or ohtuse accord- ing as the hypothenuse is of the same or of different species with the given side. Before ivorking the above example, it was easy to foresee that the angle B would he acute ; but suppose the hypothcnuse = 70° 20', and the side a = 117° 34', then the angle E 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 E ; opposite the given leg, (see the next diagram.) In this example, b becomes the middle part, and e' and B' oppo- site parts ; and therefore, by the rule, Rad. X sin. b = cos. E' X cos. c' ; that is, Ead. X sin. b cos. B' = ■ i ■. cos. c' 90° - 78° 20' = 11° 40' = c'. Sence, hy Logarithms. i ai i i . Ead 10-0000000 / '^l l\ ^ sin. 6 = sin. lir 34' I 9.947^555 or Bin. 62 2o J 19-9476655 / ' ^^J COS. «• 11° 40' 9-9909338 ' ^T COS. B'25°09' 9-956731T b,Google 380 THE PRACTICAL MODEL CALCULATOR. But since the angle opposite tlie given Bide is of the same species with the given side, yO° must be added to W, to pro- duce E :— via. 90° + 25° 09' = 115° 09'. Given c= 61° 04' 56", and 6 = 40° 30' 20", to find tte other side a. Here c' is the mid- dle part, a and b the opposite parts ; hence by position i, a = 50° 30' 30". Given the side 5 = 48° 24' 10", and the adjacent angle A = 66° 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 = 59° 38' 27". In the right-angled spherical triangle ABC, r,. ( The side a = 59° 38' 27" 1 , e ^ ^i- i * Given -i Ti. 3- I 1 It cfio ■>!)/ ccvi ?to find the angle A. [^ Its adjacent angle B = 52° 32' 55" J ° Answer, 66° 20' 40". The required angle is of the same species as the ^iven side, and Given the side l = 49° 17', and its adjacent angle A = 23° 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 = 37° 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 arc two solutions, i. e. a and the supple- ment 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. hv Google SPHERICAL TKIGONOMETKT. 381 In working this example, we find the log. sin. a — 9-8635411, which corre- sponds to 46° 55' 02", or, 133° 04' 58". It appears, therefore, that a is ambiguous, for J there exist two right-angled I triangles, having an ohlique I angle, and the opposite side 1 in the one equal to an oblique angle and an oppo- site side in the other, but the remaining oblique angle in the one the supplement of the remaining oblique angle in the other. These triangles are situated with respect to each other, on the sphere, as the triangles AEO, ADC, in the annexed diagram, in which, with tho 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 f,- /the side a = 42° 12', \ to find the adjacent \ its opposite angle A = 48° J angle B. The complement of the given angle is the middle part; and neither a nor E' being joined to A', they are consequently opposite parts ; hence, the angle E = 64° 35', or 115° 25' ; this case, like the last, being ambiguous, or doubtful. Given a = 11° 30', and A = 23° 30', to find the hypothenusc c. e = 30°, or 150°, being ambiguous. In a right-angled triangle, there are given the two perpendicu- lar sides, viz. a = 48° 24' 16", b = 50° 38' 27", to find tho angle A. A = 66° 20' 40". Given a = 142° 31', b = 54° 22', to find c c = 117° 33'. Given ■! -d _ oi ig > Required the sidaa. n- fA = 66° 20'40"K c 4 ^1, 1, .1. Given ■{ Tj ~ i^2 '!'' SS ( hypothenuso c. hv Google THE PRACTICAL MODEL CALCULATOR. MEASUREMENT OY ANGLES, " CibU Engineer and Archileet's Jotirnai," Jbr Oct. and Kov. 1847. A NEW METHOD OF MEASUEING THE DEGKEE8, MINUTES, ETC., IN ANY EECTILINEAR ANGLE BY COMPASSES ONLY, WtTHOOT USING SCALE OR PROTRACTOR . Apply AB = z, from B to 1 ; from 1 to 2 ; from 2 to 8 ; 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 AE, With the same opening, B 5 or A 4, or ^, -which we shall terra it, lay off 4,11, 11,12, and 12,13. Then the arc betiveen 13 and 10 is found to be contained 23 times in the arc AB. hv Google MEASUREMENT OF ANGLES. Hence, we have, 5»-y . 23 z = "23- - - 73° 33'-82. a —- 22 » Ev substituting tilis value in the iirst equation, we obtain, 22 a: 6.-25^^ = 360. 1013 a; „„„ , 360 X 207 -Wr " ^™' '"* " lOlT Apply AB = X, from B to 1 ; from 1 to 2 ; from 2 to 3 ; from 3 to 4. Then take E 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 are AE. With the same opening, B4 = ^, lay 6ff A9, 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 are AB. Therefore, we have, ix + y = Zm; 11^-2 =^; X 24 e = x; or, 2 = gl- X mx ■■■ll!'-24 = ^' ■■•!' = 264- Substituting this value of y in the first equation, 2b X 4aT4- 9fiT =360; 860 ) 264 " 1071 - = 88° 44'-333. How to lay off an angle of any number of degrees, minutes, §■€.. with compasses only, without the use of scale or protractor. Let it bo required to lay off an angle of 36° 40' = 3. Take any small opening of the compasses less tban one-tenth of the ra- dius, and lay off any number of equal email , ares, from A to ' from 1 to 2 ; from 2 to 3, &c., until we have laid off an arc, AB, greater than the one required. Draw B b through the centre o, then will the are a 5 = arc AB, which we shall hv Google 384 THE PRACTICAL MODEL CALCULATOR. put = 20 * in this example, and proceed to measure ah aa in tKe (irs^ example. Lay off a 6 from 6 to c ; from c? to d ; from t? to e ; from c to/; from /to ^. Patting g a = A„ t'len, 108 6 X 20* + A. = 360° = TjP; because, 360° _ 21600 _ 108 36° 40' ~ 2200 ~ 11 ■ Lay off, as before directed, g a, = Ai, from a to li, from /( to s, and btot; then calling a (, A„ we have 3 A, + A, = 20 f ; and we find that s ( is contained 28 times in the a 108 iah; .-. 120 ^> + A. = -jr ^ ; 3 Ai + As = 20 ?> ; and 28 A^ = 20 1. Eliminating A, and As) we find 29205 ^„ . 3 = ^29gg"^ = 12-9 times ^ nearly ; ,■, 36° 40' = /, A N is laid off with a3 much ease and certainty as by a protractor. As a second example, let it be required to lay off an angle of 132° 2T'. From 180° 0' take 132° 27' = 47° 33', -which put = s. 360° 2400 , V ^ 7, = "oiy' ivhen put = ;, then « 3 = oo'J = "■ We have laid off 29 small arcs from A to B ; 29 = j. AB = ib = be = cd = de = cf. And a g = hh = af = l\^; hg = Ab- .-.5 X 29* + Ai = 360° = ^u = me?± A, (1) 2 A, - A, = 29<., or m A. ± A, = ^* (2) 13 Aa = 29 f, or ^ A. = '^ (3) hv Google MBAStTREMBNT OF ANGLES. 385 Eliminating Ai smd A™t we have _ {mnq±{q=pl)}U _ {5-2-13 + (13 + 1)}29-317 ^ ~ ^nq *~ 2400-2-13 *■ "" 1323729 ■ irifAoo ? = 21 J times t very nearly. Hence the line o N deter- mines the angle aoN = 132° 27'. In the expression ^ ■ ,„g ?■ (^) substituting the numerals of the first example, then {6-3-28 + (28 - 1)120-11 29205 ^=' 108-3-28 ^ = ^268" ^ = ^^'^ ^''^'^^ ^ "'^''y- 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) Ao = ~~ i substituting this value of Aai '■i (2), M A.= f* q^ A, = in (1), gives -ji = mi^±-{^i^=P —) ; from which (K) is found. This method of measuring angles is more exact than it may ap- pear ; for if, in the first example, we take 5x — y = 360; 9 J/ + z = a^ ; and 20 z = 3:, 64800 „„„ „„, ,. then X = gg-. = 73° 33' 8a. The first equations gave 73° 33' 82 when 23 s = r, so it does not matter much whether 20, 21, 22, 23, 24, or 25 times z make X. This fact is partieiilarly worth attention. Given the three angles to find the three sides, The following formulas give any side a of any spherical triangle. — cos. J S cos. {^ S — A) on. i a = y sGTluErc > "'' cos - » = ,55!iiiA^B)_coMi_s-q ■ ^ sin. B sin, C. Given tile tliree sides to iind tlie three angles. sin. (1 S - 6) sin. (} S - c) sm. jA-^ sin. Ssin. «." . sin. } S .in. (i S - <■) COS. J A = </ sin b sin b,Google 6RATITY-WEIGHT-MASS. BPECIFIC GRAVITY, CENTRE OJ GRAVITY, AND OTTITB CKNTRES Or BODIES. — WEIGHTS or ENGINEERING AND MECHANICAL MATERIALS. — ItKASS, COPl'EE, STEEL, IRON, "WATER, BTONE, LEAB, TIN, ROUND, SQUARE, PLAT, ANGDLAR, ETC. 1. In a second, the acceleration of a body falling freely in vacuo ia 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 velo- city of 24 feet a second, and suppose it to acquire each second 5 ad- ditional feet velocity ; what is its velocity at the end of 3-7 seconds ? 24 + 3-T 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 2^— = 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 — 7P — = 8-8889 feet, the velocity at the end of the 45th second. The four fundamental formulas of uniformly accelerated motion are v=pt; 8=2-; « = T' '^^' V the velocity, p the acceleration, * the time, and s the space. 6. What space will a body describe that moves with an accele- ration of 11-5 feet for 10 seconds. ii:^">* = 5T5fes.. 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 ? ^-ir? = 909-09 feet. 2 X 0-5 hv Google GEAVITY — WEIGHT — MASS. 387 8. A body is propelled witli an initial velocity of 3 feet, and with an acceleration of 8 feet a second ; what space is described in 13 seconds? 8 X 13 + ^^ = 715 fee.. 9. What distance will a body perform in 35 seconds, commenc- ing with a velocity of 10 feet, and being accelerated to move with a velocity of 40 feet at the beginning of the 36th second ? 2 X 35 = 8Y5 feet, the distance. The formulas for a uniformly accelerated motion, commencing with a velocity c, are as follow : — pe c + V if — e The succeeding formulas are applicable for a uniformly retarded motion with an initial velocity c. pe c + V e- ~v^ 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 ? 60 10 = 5 seconds, the time. q- — y-y = 125 fcet, thc height required. 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 mo- tion, 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, v = 'd2-2xt; I! = 8-024964 xv^; 8 = 16-1 X (^; s= -015528 Xv^; t = -031056 X v; a.iiit= -2492224 X ^/3. 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 ? 8-024064 X >/400 = 160-49928 feet. 13. What space will a body pass over in its free descent during 10 seconds? 16-1 X (10)= = 1610 feet. hv Google 388 THE PEAOTICAL 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)' = 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 oyer a space of 16 feet, fall- ing freely in vacuo ? ■2492224 x ^/TQ = -9968896 seconds. If the free descent of bodies go on, with an initial velocity, which we may call c, the formulas are, ft if — ,^ s = ct + g-^ = ct + lQ-lxe; « = ~^ = -015528 {^^'^-c^). If a body be projected vertically to height, with a velocity which wc shall term c, then the formulas become, V = e — 32-2 X ( ; « = v^e' — 64-4 xs; s = ct — g ^ = ct - 16-1 X i=; s = ^ ~'" . = -015528 (c= - v% 17. What space is described by a body passing from 18 feet velo- city 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 = 5-08106 Space descrihcd 8-94410 Since this problem and table are often required in practical me- chanics, we shall enter into more particulars respecting it. ^^~ 2g -2g~2g' if we put h = height due to the initial velocity c; that is, A == 2^ ; and ft, — the height due to the terminal velocity v ; that is, ft, = o~ ; then, s = ftj — ft, for falling bodies, as in tlie last example ; and g = A — ftj, 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 heiglits and veiocities 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. hv Google WEIGHT— GRAVIIY— Table of the Heights corresponding to different Velocities, i the second. ',feet fi CossEsrasDisD llnoiT in t^m. | I 2 3 ^ 5 6 8 1 9 1 ?ii ■0X8789 ■169006 -S9aS94 'Os:iii3 -IflBOOB mm '0349378 ■097050 ■C6D060 ■B9B89ft ■044571 ■113189 ■21257] ■Moeli ■060311 -06 5066: ■121730 -130590: ■K4224 ■238180; •6^360 -6605-3 1-4SI304; l-S21S0i The following extension is obtained from the foregoing table, by mere inspection, and moving the decimal point as before di- rected. is, CorrMnondiTig HUslll>»F«t. 1^ CMTfaponatnr Jfi S^sj'^^t' : p-a 71) 1-552796 li) 5-00559 23 12^17392 87 21 ■25777 i 11 1-878882 ?0 6-21118 23 18^O50O1 88 12 2-065218 21 6-84783 SO 13 ■97616 23-01802 1 13 2-824224 22 7-51553 SI 14-92337 40 24-84472 i 14 3 ■013478 23 8-21429 82 15-90062 15 8^4e379 24 8-04410 16 ■60994 1H 8^975I6 25 9-70497 S4 18-78883 43 28-57143 i 17 4-18753 '^11 1O-49G00 85 ]9^02174 41 30-06212 1 13 5-0310G 27 11-31988 20-12423 45 31-1441 ! 18. AVhat mass does a body weighing 30268 lbs. contain 1 80268 302680 , „ "32T = ~3^^ = ^**' ^^^■ For the mass is equal to the weight divided by^r. And g U 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 fron.' it,' is invariable, whether at the centre of the earth or at any dis- tance from it. If M be the mass and W the weight of a body, Then M = - W W ■03105 i9 W. 200 lbs ? " 32-2 " 19. What is the mass of a body whose weight ii ■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 different places. The force which accelerates the motion of a heavy body on aii. inclined plane, is to the force of gravity as the sine of the inclina- hv Google 390 THE PRACTICAL MODEL CALCULATOR. tion of the plane to the radius, or as the height of the plane to its lenjrth. The velocity acquired by a body in falling from rest through a given height, is the same, whether it fail 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 aa the lengths of those chords. If the body descend in a curve, it suffers no loss of velocity. The centre of gravity of a hody is a point about which all its parts are in equilihrio. 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, there- fore, 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 aa the weights of the bodies. If a line be drawn from the centre of gravity of a body, perpen- dicular to the horizon, it is called the line of 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 Gyration. — 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 mo- tion, in different revolving bodies, are as follow : — In a straight rod revolving about one end, the length. X -STTS. In a circular plate, revolving on its centre, the radius X -TOTl. In a circular plate, revolving about one diameter, the radius X 'O. 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 ■547T. In a right-angled cone, revolving about its vertex, the height X ■86ti. hv Google 8PECIPI0 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 hody revolving about a jkced axi», 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, he 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 sus- pension, 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 sus- pended 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 iu which all its force would be collected is called its centre of oscil- lation, 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 ail solids and liquids are thus compared, is that of distilled water, one cubic foot of which weighs 1000 ounces avoirdupois ; hv Google 392 THE PRACTICAL MODEL CALCULATOR. and the specific gravity of ii solid body is determined by the dif- ference 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 wil! 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 quo- tient will be its specific gravity, that of water being unity. A piece of copper ore weighs 56J ounces in the air, and 43| ounces in water ; required its specific gravity. 66-25 — 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 quantityoffloidequal toit in we^^Ai, 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 tlie 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 -j- 62-5 = 0-64, the specific gravity of the fir. The specific gravity of s, fluid may be determined by taking a, solid body, heavy enough to sink in the fluid, and of known spe- cific gravity, and weighing it both in the air and in the fluid. The difference between the two weights must be multiplied by the spe- cific 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 3J ounces when immersed in it, and 6 ounces in the air. 6 - 3-75 = 2-25 X 3 = 6-75 -r- 6 = 1-125, the specific gravity. Since the weight of a cubic foot of distilled water, at the tem- perature 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 gra- vities (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, tho multiplication of each by 1000 will give 1000 ounces for the cubic foot of water, and 10474 ounces for the cubic foot of silver. hv Google SPECIFIC GRAVITY, 393 In the following tables of specific gravities, the niimbers in the first column, if taken as whole numbers, represent the weight of a, cubic foot in ounces ; but if the last three figures arc taken aa deci- mals, 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 ijuoticnt will be the contents in cubic feet. , Required the cubic content of a mass of cast-iron, weighing 7 cwt. 1 qr. = 812 lbs. 812 lbs. -^ 450-5 (the tabular weight) = 1-803 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 pounda avoirdupois. Required the weight of a log of oak, 3 feet by 2 feet 6 inches, and 9 feet long. 9x3x2-5 = C7-5 cubic feet. And 67-5 x 58-2 (the tabular weight) = 3928-5 lbs., or 35 cwt. qr. 8| 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 45° = 32-1803 feet ; at any other latitude L, g = 32-1803 feet — 0-0821 cos. 2 L. If ff' represents the force of gravity at the height h above the sea, and r the radius of the earth, the force of gravity at the level of the sea will be ^=^'(1 + j^). In the latitude of London, at the level of the sea, </ = 32-191 feet. Do. Washington, do. do., y = 32-155 feet. The length of a pendulum vibrating seconds is in a constant ratio to the force of gravity. I = 9-8696044. Length of a pendulum vibrating seconds at the level of the sea, in various latitudes. At the Equator ,...39-0152 inches. Washington, lat.38° 53' 23" 39-0958 — New York, lat. 40° 42' 40" 39-1017 — London, iat.51°31' 39-1393 — lat.45° 39-1270 — kt.i 39-1270 in.— 0-09982 cos. 2 X. hv Google 394 THE PRACTICAL MODEL CALCULATOR. Specific G-ravity of various Substances. "^Hr* "^£ Hr ■SH'iii' « "^L?I Onnast ce 66.! 414 am 1«0 Jl^ut°' Ji fs* B ■uas umuioii out ^a sS'tt^VMiOt Jat, Utiimiiu,ua 3JjB 3,113 7 M BUB i4sl IkiS 0'ld,p»r«c^t la^ laue P reelain China a*to J401 ?.T,uaK,Btt,i.d^ lUJt-U Porlluid etona Pnmceatona 2u70 ri Ire,,.°^t ll^ it SS' ss K ■ E Mus^ium Ma UP sol il s-,ta s JT-O ^fe""" s L"f HbkeUo^ ^ uSb SI! 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Table of the Weight of one Mot Length of Malleable Iron. EUC.«E ,BOK. Eov»D IBM. 1 SCMltliO?. Wfieu. Di.™«t. Weislit Ci.™mf=r»,u.,. Weijht. loche!. FounSa. lothsa. Founds jBChfi PouDja O'ai 0-16 0-26 0-47 0-37 084 0-6G 4 1-34 1-03 H 0-82 1 1-89 1-48 1-Oa i