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Full text of "The practical model calculator, for the engineer, mechanic, machinist, manufacturer of engine-work, naval architect, miner, and millwright"

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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,) 



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ling to Uw ut of CopethSt Id tbe yc 

HENRT CARET BAIRD, 
le District Court for tho EaEtccn Bist 



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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. 



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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 — 



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■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. 



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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 



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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. 



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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 



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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 



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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. 



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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. 



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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 



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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. 



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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. 



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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. 



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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 



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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. 



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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|. 



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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 ^ 



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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. 



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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. 



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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 



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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. 



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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. 



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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. 



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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. 



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'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. 



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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 



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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. 



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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 



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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 



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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 



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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. 



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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. 



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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 



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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 



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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. 



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■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. 



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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 



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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 



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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 



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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. 



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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. 



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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: 




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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, 




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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. 



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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. '"-,. ; ,.- 



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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. 



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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 


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■099380 


■150 


■073874 


■147719 



b,Google 



AREAS OP THE SEGMENTS AND ZONES OP A CIECLB. 



HriBht, 


Jib or 5,. 


jSjBofKoM. 


HcieU. 


^,«.rse,. 


i«„rw 


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■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, 



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■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 


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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 


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■242121 


816673 


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296830 


868019 


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349752 


■380900 


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■317893 


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303612 


458 


3o0748 


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■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 


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357727 


■384061 


-356 


250715 


823075 


411 


804171 


858258 


466 


358725 


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•857 


251673 


328775 


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305165 


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252631 


324474 


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359392 


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825171 


414 


307125 


3j9954 


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301713 


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■254550 


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416 


808110 


860513 


470 


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■385884 


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■326569 


416 


30B096 


361070 


471 


303715 


■886172 


■362 


-256471 


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417 


310081 


361623 


472 


304713 


■386505 


■363 


■257433 


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418 


811068 


362178 


473 


865712 


•886832 


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■828625 


419 


3130^4 


502720 


471 


300710 


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■365 


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•329310 


420 


818041 


363264 


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8b7T09 


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■260320 


■329992 


421 


314029 


36380O 


476 


368708 


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■261284 


■880678 


422 


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364343 


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364878 


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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. 



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■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. 





D. 


r 




rri..=.™nv 


n, a,^or 




K a. 


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. 



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S^uarss. ' CabDS. 


6iumlk«». 


C«l« Ro.t,. 


H.=i:irr™lj. 


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8-8708706 


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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«. 


Be<;lprofrl=. 


120 


14400 


1T28O00 


10-9544512 


4-9324242 




121 


14641 


1771561 


110000000 


4-9460874 


-008264463 


1-22 


14834 


1815848 


11-0453610 


4-9696767 


■008106721 


123 


15129 


1860867 


11 '0905365 


4-9731898 


-008130081 


124 


15376 


1906624 


111355287 


4-9866310 


-008004516 


125 


15625 


1953125 


11 ■1808399 


5-0000000 


-008000000 


126 


15876 


2000876 


11-2249722 


6-0132979 


-007936508 


127 


16129 


2048383 


11-3694277 


5-0265257 


-007874016 


128 


16884 


2097152 


11-3137085 




-007812500 


129 


16641 


2146689 


11-3678167 


5-0627743 


-007751938 


130 


16900 


2197000 


11-4017643 


6-0657970 


-007092808 


131 


17161 


2248091 


11-4465231 


5-0787531 


-007688588 


182 


17424 


2299968 


11-4891258 


5-0916434 


■007575758 


Ui 


17689 


2852687 


11-5825626 


5-1044687 


-007518797 


134 


17956 


2406104 


11-5758369 


5-1172299 


-007462687 


13u 


18225 


2460375 


11-6189500 


5-1299378 


-007407407 


130 


18496 


2516456 


11-6619088 


5-1426632 


■007352941 


137 


16769 


2571363 


11-7046999 


6-1661867 


-007299270 


U8 


l')044 


2b2S072 


11-7473444 


6-1676493 


■007246377 


139 


19321 


26'<5619 


11-7898261 


5-1801015 


■007194245 


110 


1%00 


2744000 


11-8321596 


6-1924941 


■007142857 


141 


1''881 


29U3221 


11-8743421 


6-2048279 


■007092199 


14. 


20164 


2863288 


11-9163753 


5-2171084 


■007042254 


145 


20449 


2^24207 


11-9582607 


5-2293215 


■006993007 


144 


20736 


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12-0000000 


6-2414328 


■000944444 


145 


21025 


S048625 


12-0415940 


5-2686879 


■006896552 


146 


2131b 


8112136 


12 -0830460 


6-2656874 


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8176523 


12-1248557 


5-2776321 


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148 


21904 


3241792 


12-1656251 


5-2896725 


■006750767 


149 


22201 


8307949 


12-2065556 


6-8014592 


■006711409 


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22o00 


8375000 


12-2474487 


5-3132938 


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22801 


3442951 


12-2882057 


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■006622617 


li2 


23104 


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6-S368033 


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12-8693169 


6-8484812 


■006586048 


1j4 


23^16 


3662264 


12-4096736 


6-3601084 


■006493506 


1j5 


24025 


3723875 


13-4498996 


6-8716854 


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24831. 


87%416 


12-4899960 


5-3832126 


■006410256 


1j7 


24649 


8869898 


12-5299641 


5-3946907 


■006369427 


1j8 


24904 


8944312 


12-5698061 


6-4061202 


-006329114 


U9 


25261 


4019679 




6-4175016 


-006289808 


IGO 


25600 


4096000 


12-6491106 


5-4388352 


■006250000 


181 


25921 


4178281 


12-6885776 


5-4401218 


■006211180 


1d2 


26244 


4251528 


12-727922! 


6-4518618 


■006172840 


lli3 


26569 


4880747 


12-7671453 


6-4626556 


-006134969 


164 


26806 


4410944 


12-8002485 


5-4737087 


-006097561 


Ibj 


27220 


4492125 


12-8452826 


5-4848006 


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2755b 


4574236 


12-8840987 


6-4968647 


■000024096 


167 


27889 


4657463 


12-9228480 


5-5068784 


-005988024 


108 


28224 


4741632 


12-9614814 


5-6178484 


■005953381 


169 


285t.l 


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18-0000000 


6-5287748 


■005917160 


170 


28900 


4918000 


13-0384048 


5-5396583 


■005882853 


171 


29241 


5000211 


13-0766968 


6-5504991 


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172 


29o84 


5088448 


13-1148770 


5-5612978 


■005818953 


1"3 


29929 


6177717 


13-1529464 


5-572a546 


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174 


80276 


52(j»024 


18-1909060 


5-6827702 


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13-2287566 


6-5984447 


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13-2664092 


5-6040787 


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1 7 


81829 


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18-3041347 


5-6146724 


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31684 


5689762 


13-3416641 


6 ■6252263 


■005617978 


170 


82041 


5735339 


13-8790882 


6-6357408 


■005586592 


1^0 


32400 


5832000 


13-4164079 


5-6462162 


■005555556 


HI 


82701 


592a741 


13-4536240 


5-6566528 


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b,Google 



!, CUBES, SQUARE AND CUBE ROOTS. 



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18-5277498 


5-6774114 


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13-5646600 


5-6377340 


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186 


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6331625 


13-6014705 


5-0980192 


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34596 


6484866 


13-6881817 


6-7082675 


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187 


84969 


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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 


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101 


86481 




13-8203760 


5-7680652 


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102 


S0864 


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13-8564065 


6-7689982 


■0052033S3 


198 


87249 


7189517 


18-8924400 




■005181347 


194 


87686 


7301884 




5-7880604 


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195 


38025 


7414875 


13-9642400 


5-7988900 


-0051^8205 


196 


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7520588 


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6-8087857 


■005102041 


197 


38800 


7645373 


14-0850088 


5-8186470 


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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 


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41610 


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14-2828509 


5-8867653 


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205 


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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 


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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 


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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 



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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. 



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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. 



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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 



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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. 



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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. 



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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 



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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. 



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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. 



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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 



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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 



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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 



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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 



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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. 



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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 



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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- 



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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, 



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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- 



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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. 



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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. 



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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- 



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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. 



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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. 



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.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, 



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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. 



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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 



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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. 



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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. 



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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. 



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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 



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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 



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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. 



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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 



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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. 



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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. 



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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 



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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 



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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 



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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 



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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- 



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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, 



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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 



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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. 



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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 



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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^ ■>::£:£; 



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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 



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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 



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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. 



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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 = 



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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- 



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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 



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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: 



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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 



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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 : 



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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 



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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 



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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. 



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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 



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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/ 



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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 



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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 



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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. 



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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; 



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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 



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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 



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THE PRACTICAL MODEL CALCULATOR, 



III 



Mi?. 

lalJi 



mi" 

llslllis 



P"*i. 



I'S'S 






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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 



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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- 



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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. 



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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. 



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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. 



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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- 



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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 



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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 



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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 



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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 



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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 



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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 



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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- 




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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 



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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 



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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 



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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. 



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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 



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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 



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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. 



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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 



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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 



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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. 













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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- 



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2455 


48 


280 


675 


2100 


OS 


330 


SOI 


24j7 


49 


282 


664 


_304 


qj 


S31 


298 


2460 


. 60 




654 


2308 


100 




295 


2462 



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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. 



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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^^^^ 









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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. 



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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, 



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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. 



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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 



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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, 



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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- 



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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- 



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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. 



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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- 



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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 



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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 



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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- 



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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 




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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. 



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THE PRACTICAL MODEL CALCULATOR. 




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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. 



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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 



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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. 



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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 



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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 ; 



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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. 



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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. 



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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 



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: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. 



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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. 



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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 



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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 



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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. 



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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. 



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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. 



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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 



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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. 



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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; 



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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 



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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- 



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,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. 



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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. 



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■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. 



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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- 



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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 



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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. 



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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 



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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; 



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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. 



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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. 



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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. 



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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. 




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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 




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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 



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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 



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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 



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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 



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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 




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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 

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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 



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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. 



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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 



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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 



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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. 



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! 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. 



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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 




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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. 



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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 



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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. 




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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' 




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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. 




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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 




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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. 



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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. 



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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. 



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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 




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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) 



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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! 


LigoiDS 






1^" 


:S 


nl(''i 


iS^S^r^ 


iA» 


r 


sa,"'tlXl<"' 


UM- 






s 




EkIi iO swSlrd 


ant 


^u 


AramonU, liquid 


.E 


6S-0 


Irancte d 




IS" 


Eihfr mlptmio 






s4l''a'^=°"^ 


^ bS 


4s»e 




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74 


hsnlengd 




uo-o 


M°^mW°^ 


i,°ri 






use 


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a 


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olies 








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i 


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sw 


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l"l 






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Fl^D^ll 


1U31 


is 


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1010 


n 


'uir,^""™ 






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lIWi 


Li 


ChsHT) trta 


s 


65U 








Cmk 


W) 


lufl 








Ebouy Ind an 


IflS 


7i-0 










1:111 


SSi 








f"°rur 




4.-0 




«s 




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li 


is 


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fe 


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8 8 


aoi^ 






iiis; 


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ii?i! 


71? 


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lu'a 


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ETUBES EiBTHB ETC 






Jt ^ieicT^L a 


>i .XI d 




AlahMttr jeUoH 


























11 








mno 


cL.'ii""'' 




lH 


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704 






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i3jo 


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earn n = oiida 




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na 


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(.dlplrarrited maiopn 






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b,Google 



SPECIFIC GRAVITY. 395 

Taelb of the Weight of a Foot in length of Flat and Rolled Iron. 



« 


Bus^BTB m .».H,., .»^ .ARTS or X» ..VCH. ] 


i 


»! 


8* 1 3i 


3 


2i 


^i 


i 


2 


1 IU|li|U|^ 


i 


J_ 


, 


VM 


1-PiT 


V4T 


1-1fl 


vm 


MS 


mn 


n-q+ 




,\ 


?1fl 






*" 


1-89 


?^? 


167 


1M 






o-is 






























S4U 


B-au 


"'^ 


&« 


''"^ 




















































































































































































^■flB 




ifi^ii 


IBSO 


'b * 


















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iJU-lU 






















ill 


4W 


1 

























T B / 



to 



b,Google 



m THE PKACTICAL MODEL CALCL'LATOE. 

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 


2-&7 






2-02 


2 


1-34 




3-3*i 




2-63 


2 


1-66 


H 


4-25 




8-33 


2 


2-01 








4-12 




2-37 


ll 


6-35 




4-08 




2-79 








5-93 


3 


3-24 








6-% 




3-09 


H 


lO'ag 












11-81 




9-27 


•la 


5-35 


2 


13-44 




10-55 


5' 




^ 


17-01 




13 -sg 


51 


7-99 


2I 


al-00 




16-48 




e-51 








19-9o 




11-18 




30-24 




23-73 


7^ 




3* 


41 16 


3^ 


27-85 


7S 


14-78 














H 


68-01 


H 


37-09 


H 


19-21 




84-00 




42-21 






s 


120-96 


H 


53-41 


10 


26-43 


7 


164-64 




65-93 


12 


B!'99 



The following tables are rendered of great utility hj moan's of 
this table :— 




Suppose it he required to ascertain the weight of a east iron 
pipe 26J inches outside and 23| inside, the length being 6J feet. 
Opposite 26J in the table is 

234-8576 X 7-2 x 6-5 = 10991-135. 
And opposite 23| in the t0,b!e is 

192-285G X 7-2 x 6-5 = 8998-966 subtract 
1992-169 lbs. avr. 
The succeeding table contains the surface and solidity of spheres, 
together ivith the edge or dimensions of equal cubes, the length 
of equal cylinders, and the weight of water in avoirdupois pounds : — 



hv Google 



SPECIFIC QEiVITT. 





Surface and Solidity of Spheres 






K.™=.«, 


8.rb==. 


SoUililJ. 


Cuts. Cy 


indet. 


^V.UM.11«. 


lin. 


3a416 


-5236 


■8060 


6666 


-0190 


t's 


3-54C5 


-6280 


-8603 


7082 


■0227 


i 


3-9760 


-7455 


-9067 


7500 


-0270 


1% 


4-4301 


■8767 


-9571 


7917 


-0317 


\ 


4-9087 


1-0226 


1-0075 


8333 


-0370 


■fs 


5-4117 


1-1838 


1-0578 


8750 


-0428 


i 


5-8395 


1-3611 


1-1082 


9166 


-0500 


I's 


6-4918 


1-5553 


1-1586 


9583 


-0563 


i 


7-0686 


1-7671 


1-2090 1 


0000 


-0640 


ft 


7-6699 


2-0000 


1-2593 1 


0416 


■0723 


t 


8-2957 


2-2467 


1-3097 1 


0833 


-0813 


ii 


8-9461 


2-5161 


1-3601 1 


1349 


-0910 


¥ 


9-6211 


2-8061 


1-4105 1 


1666 


■1015 


n 


10-3206 


3-1176 


1-4608 1 


2083 


■1128 


i 


11-1)446 


3-4514 


1-5112 1 


2500 


-1250 


u 


11-7932 


3-8081 


1-5616 1 


2916 


-1377 


2 m. 


12-5664 


4-1888 


1-6020 1 


3333 


■1516 


t'b 


13-3640 


4-5938 


1-6633 1 


3750 


■1662 


i 


14-1862 


5-0243 


1-7127 1 


4166 


-1818 


A 


15-0330 


6-4807 


1-7631 1 


4582 


■1982 


> 


15-9043 


6-9640 


1-8135 1 


5000 


■2160 


ft 


16-8000 


6-4749 


1-8638 1 


5516 


-2342 


\ 


17-7205 


7-0143 


1-9142 1 


5832 


■2540 


ft 


18-6655 


7-5828 


1-9646 1 


6250 


-2743 


i 


19-6350 


8-1812 


2-0150 1 


6666 


-2960 


ft 


20-6290 


8-8103 


2-0653 1 


7082 


-3187 


8 


21-6475 


9-4708 


2-1157 1 


7500 


■3426 


n 


22-6907 


10-1634 


2-1661 1 


7915 


■3676 


i 


23-7583 


10-8892 


2-2165 1 


8332 


-3939 


H 


24-8505 


11-6485 


2-2668 1 


8750 


■4213 


i 


25-9672 


12-4426 


2-3172 1 


9165 


-4501 


H 


27-1084 


13-2718 


2-3676 1 


9582 


■4800 


Sin. 


28-2744 


14-1372 


2-4180 2 


0000 


■6114 


A 


29-4647 


15-0392 


2-4683 2 


0415 


■5440 


? 


30-6796 


15-9790 


2-5187 2 


0832 


■5780 


ft 


31-0191 


16-9570 


2-5691 2 


1250 


-6133 


? 


33-1831 


17-9742 


2-6195 2 


1665 


■6401 


ft 


35-3715 


19-0311 


2-6698 2 


2082 


■6884 


t 


35-7847 


20-1289 


2-7202 2 


2500 


-7281 


ft 


37-1224 


21-2680 


2-7706 2 


2915 


-7693 


¥ 


38-4846 


22-4493 


2-8210 2 


3332 


■8120 


ft 


39-8713 


23-6735 


2-8713 2 


3750 


■8561 


i 


41-2825 


24-9415 


2-921T 2 


4166 


■9021 




42-7183 


26-2539 


2-9712 2 


4582 


■9496 




44-1787 


27-6117 


3-0225 2 


5000 


■9987 




45-6636 


29-0102 


3-0728 2 


5415 


1^0493 




47-1730 


30-4659 


3-1232 2 


5832 


1^1020 


41 


48-7070 


31-9640 


3-1730 2 


6250 


1-1501 


4in. 


50-2656 


33-5104 


3-2240 2 


6665 


1-1974 


ft 


31-8486 


35-1058 


3-2743 2 


7082 


1-2698 


i 


53-4562 


36-7511 


3-3247 2 


7500 


1-3293 


ft 


55-0884 


38-4471 


3-3751 2 


7915 


1-3906 


Y 


56-7451 


40-1944 


3-4255 2 


8332 


1-4538 


ft 


58-4262 


42-0461 


3-4758 2 


8750 


1-5208 


f 


60-1321 


43-8463 


3-5262 2 


9165 


1-68G0 


ft 


61-8625 


45-7524 


3-5766 2 


9582 


1-0550 



b,Google 



■THE PRACTICAL MODEL CALCULATOR. 



»._..,. 


Bu.f.... 


Saiaitj. 


cub.. 


C,-.iu..r. 


«-..,H..^.| 


i 


63 -6174 


47-7127 


3-6270 


3-0000 


1-7258 


A 


65-3908 


49-7290 


3-6773 


3-0415 


1-7987 


i 


67 ■2007 


51-8006 


3-7277 


3-0832 


1-8735 


a 


090352 


53-9290 


3-7781 


3-1250 


1-9500 


i 


70-8823 


56-1151 


3-8285 


3-1605 


2-0297 


« 


72-7599 


58-3595 


3-8788 


3-2080 


2-1109 


i 


74-6620 


60-6629 


3-9292 


3-2500 


2-1942 


{1 


70-5887 


62-9261 


3-9796 


3-2913 


2-2760 


5m. 


78-5400 


65-4500 


4-0300 


3-3332 


2-3073 


A 


80-5157 


67-9351 


4-0803 


3-3750 


2-4572 


i 


82-5160 


70-4824 


4-1307 


3-4155 


2-6453 


A 


84-5409 


73 0926 


4-1811 


3-4582 


2-6438 


4 


86-5903 


75-7004 


4-2315 


3-5000 


2-7606 


A 


88-6641 


78-5077 


4-2818 


3-5414 


2-8396 


¥ 


90-7627 


81-3083 


4-3322 


3-5832 


2-9407 


A 


92-8858 


84-1777 


4-3820 


3-6250 


3-0447 


i 


95-0334 


87-1139 


4-4330 


3-6665 


3-1509 


A 


97-2053 


90-1175 


4-4633 


3-7080 


3-2595 


T 


99-4021 


93-1875 


4-5337 


3-7500 


3-3706 


14 


101-0233 


96-3304 


4-5841 


3-7913 


3-4843 


5 


103-8691 


99-5412 


4-6345 


3-8330 


3-6004 


li 


106-1394 


102-8225 


4-6848 


3-8750 


3-7191 


1 


108-4342 


106-1754 


4-7352 


3-9163 


3-8404 


H 


110-7530 


109-5973 


4-7856 


3-9580 


3-9641 


eii. 


113-0970 


113-0976 


4-8360 


40000 


4-0907 


A 


115-4060 


1160688 


4-8863 


40417 


4-2200 


¥ 


117-8590 


120-3139 


4-9367 


4-0833 


4-3517 


A 


120-2771 


124-0374 


4-9871 


4-1250 


4-4874 


? 


122-7187 


127-8320 


6-0375 


4-1060 


4-0236 


A 


125-1852 


131-7053 


6-0878 


4-2083 


4-7638 


¥ 


127-6765 


135-6563 


5-1382 


4-2500 


4-9067 


t 


130-1923 


139-6854 


6-1886 


4-2917 


5-0524 


132-7326 


143-7936 


5-2390 


4-3332 


5-2010 


A 


135-2974 


147-9815 


5-2893 


4-3750 


5-3525 


Y 


137-8867 


152-2499 


5-3377 


4-4165 


5-5069 


li 


140-5006 


150-5997 


6-3901 


4-4583 


5-6780 


i 


143-1391 


161-0315 


6-4405 


4-5000 


5-8245 


ii 


145-8021 


167-5461 


5-4908 


4-5416 


6-0601 


i 


148-4896 


170-1682 


5-5412 


4-5832 


6-1550 


H 


151-2017 


174-8270 


5-5910 


4-6250 


6-3235 


Tin. 


153-9384 


179-5948 


5-6420 


4-0065 


0-4960 


t 


150-6995 


184-4484 




4-7082 


6-6725 


159-4852 


189-3882 


5-7427 


4-7500 


6-8502 


t 


162-2955 


194-1165 


5-7931 


4-7915 


7-0212 


105-1303 


199-5325 


6-8435 


4-8332 


7-2171 


A 


167-9895 


204-7371 


5-8938 


4-8750 


7-4053 


? 


170-8735 


2100331 


5-9442 


4-9166 


7-5970 


tV 


173-7520 


215-4172 


5-9946 


4-9582 


7-7916 


i 


176-7150 


220-8937 


6-0450 


5-0000 


7-9897 


A 


179-6725 


226-7240 


6-0953 


50415 


8-2006 


1 


182-6545 


232-1235 


6-1457 


6-0832 


8-390O 


ii 


185-6611 


237-8883 


6-1961 


5-1250 


8-0044 


¥ 


188-6923 


243-7276 


6-2465 


5-1665 


8-8157 


^S 


191-7480 


249-4720 


6-2968 


5-2082 


9-0234 


i 


194-8282 


255-7121 


6-3472 


5-2500 


9-2491 


H 


197-9330 


261-9673 


0-3976 


5-2913 


9-4753 




2010624 


268-0832 


6-4480 


5-3330 


9-0905 


■}T 


204-2162 


274-4156 


6-4983 


5-3750 


9-9200 



b,Google 



SPECIFIC GRAVITY. 



!.»..„,. 


Surf^,. 


S.li<iitj. 


Cubs, 1 CjUndM. 


w^.im,«. 


i 


207-3946 


280-8469 


6-3487 


5-4164 


10-1583 


A 


210-5976 


287-3780 


6-5991 


5-4581 


10-3944 


? 


213-8261 


294-0095 


6-6495 


5-5000 


10-6343 


A 


217'0770 


300-7422 


6-6998 


5-5414 


10-8778 


? 


220-3537 


307-5771 


6-7502 


5-5831 


111250 


iV 


223-6549 


314-5147 


6-8006 


5-0250 


11-3760 


? 


226-9806 


321-5553 


6-8510 


5-6064 


11-6306 


A 


230-3308 


328-7012 


6-9013 


5-T080 


11-8891 


? 


233-7055 


335-9517 


6-9517 


5-7500 


12-1514 


fj 


2371048 


343-3079 


7-0021 


5-7913 


12-4170 


i 


240-5287 


350-7710 


7-0525 


5-8330 


12-6874 


IS 


243-97T1 


358-3412 


7-1028 


5-8750 


12-9012 


i 


247-4500 


366-0199 


7-1532 


5-9163 


13-2390 


U 


250-9475 


373-8073 


7-2036 


5-9580 


13-5200 




254-4696 


381-7017 


7-2540 


6-0000 


13-8062 


A 


258-0261 


389-7118 


7-3043 


60417 


14-0959 


7 


261-5872 


397-8306 


7-3547 


6-0833 


14-3895 


A 


265-1829 


406-0613 


7-4051 


6-1250 


14-6872 




268-8031 


414-4048 


7-4555 


6-1667 


14-9890 




272-4477 


421-2907 


7-5058 


6-2083 


15-2381 




276-1171 


431-4361 


7-5562 


6-2500 


15-6030 


" 


279-8110 


440-1294 


7-6066 


6-2916 


15-9195 




283-5294 


448-9215 


7-6570 


6-3333 


16-2375 


/a 


287-2723 


437'8500 . 


7-7073 


6-3750 


16-3604 


Y 


291-0397 


466-8763 


7-7537 


6-4166 




ii 


294-8310 


476-0304 


7-8081 


6-4582 


17-2180 


5 


298-4483 


485-3035 


7-8585 


6-500O 


17-5334 


H 


302-4894 


494-6952 


7-9088 


6-5415 


17-8931 


¥ 


306-3550 


504-2094 


7-9592 


6-5832 


18-2373 


il 


310-9452 


513-8436 


8-0096 


6-6250 


18-5837 


lo'i. 


314-1600 


523-6000 


8-0600 


6-6666 


18-6786 


A 


318-0992 


533-4789 


8-1103 


6-7083 


19-2960 


i 


322-0630 


543-4814 


8-1607 


6-7500 


19-C577 




326-0514 


553-6081 


8-2111 


6-7916 


20-0240 




330-0643 


563-8603 


8-2615 


6-8333 


20-3948 




334-1016 


574-2371 


8-3118 


6-8750 


20-6682 ■ 




. 338-1637 


584-7415 


8-3622 


6-9166 


21-1501 


i\ 


342-2503 


595-3677 


8-4126 


6-9582 


21-5344 


i 


346-3614 


606-1318 


8-4630 


7-0000 


21-9238 


A 


350-4970 


617-0207 


8-5133 


7-0416 


22-3176 


? 


354-6571 


628-0387 


8-5637 


7-0833 


22-7162 


il 


358-8418 


639-1871 


8-6141 


7-1250 


23-1194 


( 


363-0511 


650-4666 


8-6643 


7-1666 


23-5274 


*l 


367-2849 


661-8580 


8-7148 


7-2082 


23-9394 


371-5432 


673-4222 


8-7052 


7-2500 


24-3577 


111 


375-8261 


685-099T 


8-8156 


7-2915 


24-7801 


380-1336 


696-9110 


8-8660 


7-3330 


25-2073 


t 


384-4655 


708-9106 


8-9163 


7-3750 


25-6414 


388-8220 


720-9409 


8-9607 


7-4165 


26-0764 


,', 


393-2031 


733-1599 


9-0171 


7-4582 


26-5184 


i 


397-6087 


745-5004 


9-0675 


7-5000 


26-5657 


t 


402-0387 


758-0104 


9-1178 


7-5414 


27-4162 


406-4935 


770-6440 


9-1682 


7-5832 


27-8742 


t 


410-7728 


783-5787 


9-2186 


7-6250 


28-3420 


415-4766 


796-3301 


9-2690 


7-6664 


28-8033 


t 


420OO49 


809-3844 


9-3103 


7-7080 


29-2754 


424-5576 


822-5807 


9-3697 


7-7300 


29-7327 


A 


429-1351 


835-9695 


9-4201 


7-7913 


30-2370 



b,Google 



THE PRACTICAL MODEL CALCULATOR. 



Di«„,.«.. 


S«rf..,. 


SoM^V- 


Cute. 


Cjli-aer, 


«.«n»i.>. 




433-7371 


849-4035 


9-4705 


7-8330 


30-7229 


IJ 


438-3636 


863-0283 


9-5208 


7-8750 


31-2157 


i 




876-7999 


9-5772 


7-9163 


31-3883 




447-6902 


890-7070 


9-6216 


7-9580 


32-2169 


12 in. 


452-3904 


904-7808 


9-6720 


8-0000 


32-7259 




471-4363 


962-5158 


9-8735 


8-1666 


34-8142 




490-8750 


1022-656 


10-0750 


8-3332 


36-9886 




506-7064 


1085-251 


10-2765 


8-6000 


39-2535 


13 in. 


530-9304 


1150-337 


10-4780 


8-6666 


41-6077 




551-5471 


1218-000 


10-6790 


8-8332 


44-0551 




572-5566 


1288-352 


10-8810 


9-0000 


46-5961 




593-9587 


1361-346 


11-0825 


9-1665 


49-2399 




615-7536 


1436-758 


H-2840 


9-3332 


51-9675 


i 


637-9411 


1515-106 


11-4855 


9-5000 


54-8014 




660-5214 


1596-260 


11-6870 


9-6665 


57-7367 


1 


683-4943 


1680-265 


11-8885 


9-8332 


60-7751 


15 in. 


706-8600 


1767-150 


12-0900 


10-0000 


64-0178 




730-6183 




12-2915 


10-1666 


67-1672 




754-7694 


1949-821 


12-4930 


10-3332 


70-5250 


1 


779-3131 


2045-697 


12-6940 


10-5000 


73-9929 


16 in. 


804-2496 


2144-665 


12-8960 


10-6666 


77-5725 



Table containing the Weight of Flat Bar Iron, 1 foot in length, 
of various breadths and thicknesses. 



jl 


TBIC^KISS IH rlRTS or AI. IT^CH. I 


^~ 


A 


1 


A 


i 


A 


1 


LtE. 


J 


1 inch. 


Lb,. 


LbB. 


LW. 


Lbs. 


Lta, 


IM. 


LUs. 


Lts. 


lin. 


0-83 


1-04 


1-25 


1-45 


1-66 


1-87 


2-08 


2-50 


2-91 


3-33 


U ' 


0-9S 


M7 


1-40 


1-64 


1-87 


2-00 


2-34 


2-81 


8-28 






1-04 


1-80 


1-56 


1-82 


2-08 


2-84 


2-60 


3-12 


3-74 


4-16 




1-14 


]-43 


1-71 


2 00 


3-29 


2-57 




3-43 


4-01 


4-58 




1-25 


1-66 


1-87 


2-18 


2-50 


2-81 


3-12 


3-75 


4-87 


5-00 


1-35 




2-03 






8-04 


3-38 


406 




5-41 




1-45 




2-18 


2-55 


2-91 




3-64 


4-37 


5-iO 






1-56 


1-95 


2-34 


2-73 


8-12 


8-51 


3-90 


4-68 


6-46 


6-25 


2 m. 


1-66 


2 08 


2-50 


2-91 


8-38 


3-75 


4-16 


5-00 


5-83 


6-66 


H 


1-77 


2-21 


2-65 


3-09 


3-54 




4-42 


5-81 


6-19 


7-08 


2^ 


1-87 


2-34 


2-81 


3-28 


8-75 


4-21 


4-68 


5-62 


G-66 


7-50 


2 


1-97 


2-47 


2-96 


8-46 


3-95 


4-45 


4-94 


5-93 


e-92 


7-91 




2-08 


2-60 




8-64 


4-16 


4-68 


5-20 


6-25 


7-29 


8-33 


2 


2-18 




3-28 


3-82 


4-37 


4-92 


5-46 


6-66 


7-65 


8-75 


2 


2-29 


2-86 


S-43 


4-01 


4-58 


515 


572 


G-87 


8-02 


0-16 


2 


2-39 


2-99 


3-59 


4-19 


4-79 


6-39 


5-88 


7-18 




9-58 


am. 


2-50 


8 12 


3-75 


4-37 


5-00 


5-62 


6-25 


7-50 


8-76 


10-00 


3 


2-70 


8-38 


4.06 


1-78 


6-41 


6-09 


6-77 


8-12 


9-47 


10-83 


S 


2-91 


3-64 


4-37 


510 


5-83 


6-56 


7-29 


8-76 


10-20 


11-66 


3 
4in. 


8-12 


3-90 


4-68 


5-46 


6-25 


7-08 


7-81 


9-37 


10-98 


12-50 


S-3S 


4-19 


6-00 




6-66 


7-50 


8-33 


1000 


H-66 


]3-33 


4 


3-54 


4-42 


5-31 


6-19 


7-08 


7-96 




10-62 


12-39 


14-16 


4 


3-76 


4-68 


6-62 


6-66 


7-50 


8-43 


9-37 


11-25 


13-12 


15-00 


4 


3-95 


4-94 


5-93 


6-92 


7-91 


8-90 




11-87 


13-65 


15-88 


6 n. 


4-17 


5-20 


6-25 


7-29 


8-83 


9-37 


10-41 


12-50 


14-58 


16-66 


5 


4-37 


5-46 


6-56 


7-65 


8-75 


9-84 


10-93 


13-12 


15-81 


17-50 


6 


4-58 


5-72 




8-02 


9-16 


10-31 


11-45 


13-75 


1604 


18-33 




4-79 


6-98 


7-18 


8-38 


9-58 


10-78 


11-97 


14-37 


16-77 


19-16 


6 n. 


5 00 


6-26 


7-50 


8-75 


1000 


11-25 


12-50 


15-00 


17-50 


20-00 



b,Google 



SPECIFIC QEAvnr. 



(fc Smdik Gramtm and other Projxrtie, of 
'ider the ttxndard of comparinn, or 1000. 







tli illi 



i 




! eonUining the Weight of Column, of Water, 
:» Ungth, artel of VaHota DiomeUn, in Us. am 



each one fool 



s "■■ * '■;■■ J "■■ If , 



„Google 



402 THE PRACTICAL MODEL CALCULATOR. 

Table containing the Weight of Square Bar Iron, from 1 to 10 feet 
in length, and from J of an inch to 6 inches square. 





7T 










II 


!«,>*- 


3 hit. 


iitti. 


4 feel. 


5twt. 


6tt*t. 


7 (eft. 


8 feet 


Sftei. 


10 (Ml. 


-- ^^- 


Ma. 


Lis. 


"=. 


~IM~ 


Lti»- 


~^ 


~LI)=, 


Lbs. 


j,t.. 


i 0-2 
0-5 


04 


0-6 


0-8 


1-1 


1-3 


1-5 


1-7 


1-9 


2-1 


1-0 


14 


1-9 


24 


2-9 


3-3 


3-8 


4-3 


4-8 




0>8 


1-7 


2-6 


84 


4-2 


5-1 


5-9 


6-8 


7-6 


8-5 




1-8 


2-6 


4-0 


6-3 


6-6 


7-9 


8-2 


10-6 


11-0 


13-3 




1'9 




5-7 


7-6 


9-5 


114 


13-3 


15-2 


17-1 


19-0 




2-6 


6-2 


7-8 


104 


12-9 


16-6 


18-1 


20-7 


23-3 


25-9 




34 


6-8 


10-1 


18-5 


16-9 


20-3 


23-7 


27-0 


80-4 




1 




4-3 


8-6 


12-8 


17-1 


214 


25-7 


29-9 


84-2 


38-5 


42-8 


1 




5-3 


10-6 


15-8 


21-1 


264 


31-7 


87-0 


42-2 


47-5 




1 




64 


12-8 


19-2 




82-0 


88-3 


44-7 


51-1 


57-5 


63-9 


1 




7-6 


15-2 


22-8 


804 


38-0 


46-6 


63-2 


60-8 




76-0 


1 




8-9 


17-9 




36-7 


44-6 




62-6 


714 


80-3 




1 




104 


20-7 


31 -1 


414 


51-8 


62-1 


72-5 




93-2 


103-5 


l| 


11-9 


23-8 


35-6 


47-5 


594 


71-3 


83-2 


95-1 


106-9 


118-8 


a'in. 


13-6 


27-0 


40-6 


64-1 


67-6 


81 -J 


94-6 


108-2 


121-7 


135-2 


2* 

21 


15-3 


80-5 


45-8 


61-1 


76-8 


91-G 


106-8 


122-1 


1374 


152-6 


174 


34-2 


51-3 


G84 


86-6 


102-7 


119-8 


136-9 


164-0 


171-1 


iif 


19-1 




57-2 


76-3 


95-3 


1144 


133-5 


162-5 


171-6 


lSO-7 


U 


21-1 


42-8 


63-4 


84-5 


105-6 


126-7 


147-8 


169 


190-1 


211-2 


2| 


23-3 


46-6 


69-9 


93-2 


116-5 


139-8 


163-0 


18G-3 


209-6 


232-9 


2J 


a6-6 


61-1 


76-7 


102-2 


127-8 


1684 


178-9 


204-5 


230-0 


265-6 




27-0 


55'9 






189-7 


167-6 


195-7 




251-5 


2794 




304 


60-8 


91-2 


121-7 


152-1 




212-9 


248-8 


273-7 


304-2 


3i ' 


33 


66-0 


99-0 


182-0 


165-1 


198-1 


231-1 


264-1 


297-1 


830-1 




86-7 


714 


107-1 


142-8 


178-6 


214-2 


249-9 


285-6 


821-3 


857-0 


3 


38-6 


77 


115-6 


154-0 


192-6 


231-0 




808-0 


846-5 


885-0 


8 
■3 


414 




124-2 


165-6 


207-0 


2484 




331 -S 


372-7 


414-1 


44-4 


88-8 


188 -S 


177-7 


2221 


266-5 


310-e 


365-3 


309-6 


444-2 


3 
3 


47 '5 


95-1 


142-6 


190-1 


237-7 


285-2 


332-7 


880-3 


427-8 


476-3 


50-8 


101-5 


162-3 


203-0 


253-8 


304-5 


856-3 


406-0 


456-8 


507-6 


/in. 


64-1 


108-2 


162-3 


216-8 


2704 


824-6 


878-6 


432-7 


486-8 


540-8 


^i 


67-5 


115-0 


172-a 


280-1 


287-6 


346-1 


402-6 


460-1 


517-7 


575-2 


4i 
4| 


611 


122-1 


183-2 


344-2 


805-3 


886-3 


4274 


4884 


549-5 


610-6 


64-7 


1294 


194-1 


258-8 


328-5 


388-2 


452-9 


517-6 


682-3 


647-0 


4| 


684 


186-S 


205-3 


278-8 


342-2 


410-7 


479-1 


547-6 


616-0 


684-5 




72-3 


144-6 


216-9 


289-2 


861-5 


433-8 


606-1 


5784 


650-7 


723-1 


4| 


76'3 


152-5 


228-8 


806-1 


381-8 


457-6 


533-8 


610-1 




762-6 




80-3 


160-7 


241-0 


321-3 


401-7 


483-0 


562-3 


642-7 


728-0 


803-3 


5 m. 


84-5 


169-0 


2534 


837-9 


4224 


506-9 


6914 


675-8 


760-3 


844-8 


6 




186-3 


279-6 


872-7 


465-8 


659-0 


652-2 


746-8 






6 


102'2 


204-5 


806-7 


409-0 


611-2 


6184 


715-7 


817-9 


920-2 


10224 


5 

6 Q. 


ni-8 


223-5 


335-8 


447-0 


558-8 


670-6 


782-3 


894-0 


1005-8 


1117-6 


121-7 


243-3 


365-0 


486-7 


608-3 


780-0 


841-6 


973-3 


1009-5 


1216-6 



;Tablb of the Weight of a Square Foot of Sheet Iron in lbs. avoirdu- 
pois, the thickness being the number on the wire-gauge. No. 1 
is 1*5 of an inch; No. 4, \; No. 11, \, ^e. 



No. on wire-gauge | 1 j 2 


3 


4 5 1 6 7 


8 





10 


11 


Pounds nvoir [12-5 12 


11 


10 1 9 8 7-5 


7 


6 


5-68 


5 


No. on wire-gauge | 12 13 


14 


16 |16[ 17 18 


19 


20 


21 


32 


Pounds avoir. l4-62|4-3l 


* 


8-a&| 3 |2-5|2-18 


1-93 




1-5 


1-37 



b,Google 



SPECIFIC GBAVITT. 



Table of the Wdylit of a Square Foot of Boiler Plate Iro 
^ to 1 inch thick, in lbs. avoirdupois. 



ilMH fV M A 


i A i\\i flu i\ U\i-in. 


5|7-5]lCI|12-5 15| 


7 '5 


20 22-6 •J6|27-6 30 1 32-5 35|87-5| 40 



Table containing the Weight of Bound Bar Iron, from 1 to 10 f 
in length, and from J of an inek to 6 inches diameter. 





1 


1 


^1 


ifi,. 


xtReb 


3f<,^ 


4fMt. 


Efrt. 


6 fast. 


7f«l. 


Btet 


^1 


10f>sl- 


T^ 


' m. 


"ib^ 


~1^^ 


0-8 


Lbs. 


Lbs. 


Lb=. 




Lh.. 







2 O'a 


0-5 


0-7 


1-0 


1-2 


1-3 


^I! 


1-7 







1 07 


I-l 


1-5 


1-9 


2-2 


2-6 


80 


3-4 









- 1-3 


2-0 


2-7 




4-0 


4-6 


6-8 


60 


6-6 




1 


2-1 


3-1 


4-2 


6-2 


6-3 


7-3 


8-3 


9-4 


10-4 




1 


5 3-0 


4-5 


6-0 


7-5 


0-0 


10-6 


11-9 


13-4 


J4-9 


2 


41 


a-1 


8-1 


10-2 


12-2 


14-2 


16-3 


18-3 


20-3 


in. 


2 


7 5-a 


8-0 


10-6 


13-8 


15-9 


18-6 


21-2 


23-9 


26-5 






4 6-7 


101 


13-4 


lfl-8 


30-2 


28-5 


26-9 


^2'? 


38-6 




4 


2 8-3 


12-u 


16-7 


20-9 


25-0 


29-2 


83-4 




41-7 




a 


100 


ISl 


20-1 


25-1 


30-1 


35-1 


40-2 


45-2 


60-3 






11-0 


17-9 




29-9 




41-8 


47-8 


53-7 


59-7 






14 


210 


28-0 


35-1 


42-1 


49-1 


66-1 




70-1 






1 16-3 


24-4 


82-5 


40-6 


48-8 


56-9 


66-0 


73-2 


81-3 




B 


3 18-7 


28-0 


37-3 


4G-7 


56-0 


65-3 


74-7 


84-0 


93-3 


2ui. 


10 


21-2 


81-8 


42-5 


58-1 


63-7 


74-3 


84-9 


95-6 


106-3 


2J 


la 


24'0 


86-0 


48-0 


59-9 


71 -9 


88-9 


95-9 


107-9 


119-9 


^ 


IS 


4 26-9 


40-3 


63-8 


67-2 


80-6 


94-1 


107-6 


131-0 


134-4 


15 


30-0 


44-9 


600 


74-9 


89-9 


101-8 


119-8 


181-8 


149-8 




16 


7 33-4 


50-1 


60-8 


83-5 


100-1 


116-8 


183-6 


150-2 


166-9 


2 


18 


3 . 86-6 


54-9 


73-2 


91-5 


109-8 


128-1 


146-3 


164-6 


182-9 


2 


20 


1 40-2 


00-2 


80-3 


100-4 


120-5 


140-5 


160-6 


180-7 


200-8 


2 


31 


9 43-9 


6o-8 


87-8 


109-7 


181-7 


158-6 


175-Q 


187-5 


219-4 


Sin. 


28 


9 47-8 


71-7 


95-6 


119-4 


148-8 


167-2 


191-1 


215-0 


238-9 


3 




2S 


9 61 -9 


77-8 


103-T 


129-6' 155-0 


181-5 


207-4 




259-3 


3 




28 


66-1 


84-1 


112-2 


110-2 , J68-2 


196-8 


224-3 


258-4 


280-4 


8 




30 


2 eo-5 


90-7 


121-0 


161-2 1 181-4 


211-7 


241-0 


372-2 


802-4 


8 




82 


5 65-0 


97-5 


180-0 


162-6 


195-1 


227-0 


260-1 


292-6 


825-1 


3 




34 




104-7 


139-5 


174-4 




241-3 


279-1 


314-0 


348-9 


3 




37 


3 74-7 [ 112-0 


149-8 


186-7 


224-0 


261-8 


298-7 


336-0 


873-8 


3 




39 


9 79-7 1 119-6 


159-5 


199-3 


289-2 


2790 


818-9 


858-8 


308-6 


4 in. 


42 


5 84-9 ' 127-4 


169-9 


212-8 


254-8 


297-2 


339-7 


382-3 


421-0 


4 




45 


2 90-3 ; 135-5 


180-7 


325-9 


271-0 


316-2 


861-4 


406-6 


451-7 


i 




48 


95-9 1 143-9 


191-8 


239-8 


287-7 


385-7 


383-6 


431-6 


479-5 


4 




CO 


8 lOl-G, 162-4 


203-3 


254-1 


S04-9 1 855-7 


400-5 


457-3 


508-2 


4 




53 


8 107-B 


lGl-8 


216-0 




322-6 


S76-8 


430-1 


483-8 


537-6 


41 




66 


8 113-0 


170-4 


227-2 


2B8-Q 


840-7 


397-5 


464-3 


511-1 


567-9 


i 




6( 


119-8 


179-7 


230-6 


299-5 


859-4 


419-8 


479-2 


5SB1 


699-0 


4r 






1 126-2 


189-3 


252-4 


316-5 


378-0 


441-7 


604-8 


567-8 


630-9 


bm. 


61 


8 183-5 


200-3 


207-0 




400-5 


467-8 


534-0 


600-8 


GC7-5 


H 


73 


2 146-3 


21S-6 


292-7 


365-9 


439-0 


512-2 


585-4 


658-5 


731-7 


n 

6ln. 


80 


3 160-6 


210-9 


321-2 


401-6 


481-8 


562-1 


643-4 


722-7 


803-0 


87 


8 175-6 


263-3 


351-1 


438-9 




614-4 


702 '2 


7000 


877-8 


95 


6 191-1 


286-7 


382-2 


477-8 


573-3 


668-9 


764-4 


8600 


955-6 



Table of the Weight of Cast Iron Plateg, per Superficial Foot, from 
one-eighth of an inch to one inch thick. 



>im.i,. [ rC'-'l'- 


Jitoch. 


!^in,.h. 


KiMh. 


MK^^t. TiiDch. 


1 tn»h. 


4 i3f 1 a'Voj 


14 8 


19 sj 


24 2S 


29 33 ISf 


lbs. OS. 

88 10= 



b,Google 



404 THE PRACnCAL MOflEL CALCULATOK. 

Table containing the Weight of Cast Iron Pipes, 1 foot in length. 



ut 


TB.CKNESS m I^CHE^. 1 


1 


i 


i 


t 


i 


1 inch. 


li 


n 1 


Lbl. 


Ll«. 


Lta. 


Lbs. 


Lb!, 


Lbs. 


Ll«- 


Lbs. 


J' 


6-9 


9-9 
12-3 


le-'i 


20-3 













10-6 


14-7 


19-2 


23-9 












12-4 


17-2 


22-2 


27-6 


33'3 


39-3 






ai 


14-2 


19-6 


26'3 


81 '8 


87'6 


44-2 


51-1 






16-8 




284 


35'0 


41-9 


49-1 


56-6 


644 




18-0 


24-5 


314 




46'2 


54-0 




70-C 




19-8 


27-0 


34-5 


42'8 


50'5 


58'9 






^ 


21-6 


29-5 


87 '6 


48^0 


54'8 


63-8 








23-5 


31-9 


40'7 


49-7 


69-1 


68'7 


78-7 




6* 


25-3 


844 


43-7 


58-4 


634 


734 


84-2 


95-1 




27-2 




46-8 


56-8 


67-7 


78-5 




101-2 


u 


29'0 


S9'l 


49-9 


60-7 


72-0 


83-5 




107-4 


H 


BO-8 


41-7 


52 '9 


64-4 


76-2 


884 


100-8 


113-5 




29 


44 4 


56" 


68 3 




93-5 


106-5 


119-9 




84 5 


4b b 


591 


718 


848 






125-8 


HI 




49 1 


6 1 


55 




103-1 


1174 


131 -9 




88 


615 


00 2 





98 4 


108-0 


122-8 


188-1 






54 


68 


8 8 


97 7 


112-9 


1284 


144-2 






56 4 




86 5 


10 


117-8 


133-9 


150-3 






5S9 


4 


901 


106 8 






1564 






bl3 


4 


93 6 


110 b 


127-6 




ie2-6 


13 






82 


1012 


118 


1374 


154-1 


173-5 


14 








108 2 


1 65 


146-2 


165-8 


185-2 


15 






95 


115 7 


135 3 


156-2 


176-2 


198-1 










123 3 


145 1 


166-1 


187 -5 


211-3 


17 








130 


15 5 


178-5 


198-2 


2284 


IH 
19 








1 


lbl2 
169 2 


185-3 
195-7 


209-1 
222-3 


235-6 
247-1 


20 










1 81 


205-3 




259-0 


21 












214-1 


243-5 


278-2 


■ 22 












223-0 


254-8 


2854 


1 aa 












2834 


265 -5 


298-3 


1 24 












245-2 


277-5 


810-6 



Table contit7n7tg tie Weight of Solid Cyl nders of Oa»t Iron, one 
foot in length, and from f of an inch to 14 inches diameter. 



DliniBWrin 


^S"" 


"'iTh':^'" 


™'^e'" 


KmrtM in 


w.^u. 


■"irbtr 


w.^.. 


1 


1-39 


^ 


■ 2048 


^ 


58-72 


n 


148-87 




1-88 


Bin. 


22-35 


Bin. 


61-96 




158-63 




247 




24-20 


6 


64-66 


^i 


168-15 






8-13 




26-18 


5 


68-31 


8i 


179-08 






3-87 


H 


28-23 


5 


71-00 


8J 


189-00 






4-68 




80-36 


5 


74-98 


9 in. 


200-77 






5-57 


s| 


82-57 


5 
6 


78-65 


3 


211-12 






6-54 


34-85 


81-95 


9 
9 


223-70 






7-69 






5 


85-81 


286-31 








4 m. 


39-66 


6 u. 




10 in. 


247-87 


2 in. 


9-91 


4 


41-80 


6i 


96-82 


lOi 


273-27 


2 


11-19 


4 


44-77 


6* 


104-72 




299-92 


2 


12-54 


i 


47-00 


6^ 


112-93 




327-81 


2 


■13-98 


4 


50-19 




12146 




856-98 




15-49 


4 


52-71 


1\ 




13 


418-90 




17 08 


4 


55-92 




139-42 




485-83 


2% 


18-74 















b,Google 



SPECIFIC GRAVITY. 



Table containing the Weight of a Square Foot of Copper and 
Lead, in lbs. avoirdupois, from ^ta ^ an inch in thickness, ad- 
vancing hy ^. 



11.ick„,.. 


CoiT". 


L,id. 


A 


1-45 


1-85 




2-90 


3-70 


« 


4-35 


5-54 




5-80 


7-39 


I + A 


7-26 


9-24 


i + A 


8-7X 


11-08 


i + A 


10-16 


12-93 


J 


11-61 


14-77 


* + A 


13-07 


16-62 


i + A 


14-52 


18-47 


1 +ft 


15-97 


20-31 


1 


17-41 


22-16 


i + A 


18-87 


24-00 


i + A 


20-32 


25-85 


1 + A 


21-77 


27-70 


i 


23-22 


29-55 



Table for finding theWeight of Malleable Iron, Copper, andLead 
Pipes, 12 inches long, of various thicknesses, and any diameter 
required. 



Thiatnca. tl^ 


ei,M«:r™. 


Coi-per. 


LMd. 


. 5^ of an inch. 


104 


121 


-1539 


A 


208 


2419 


-3078 


A 


3108 


3628 


-4616 




414 


4838 


■6155 


! +ft 


618 


6047 


-7694 


! +1, 


621 


7258 


-9232 


i +A 


725 


8466 


1-0771 


i 


828 


9678 


1-231 



Rdle. — Multiply the circumference of the pipe in inches hj the 
numbers opposite the thickness rei^uired, and by the length in feet ; 
the product will be the weight in avoirdupois lbs. nearly. 

Required the weight of a copper pipe 12 feet long, 15 inches in 
circumference, J + ^ of an inch in thickness. 

•7258 X 15 = 10'817 x 12 = 130-644 Iba. nearly. 



Table of the Weight of a Square Foot ofMillboardin lbs. avoirdupois. 



Ttiokness in incbes 


J 1 A J A 1 i 




-688 j 1-032 1 1-376 1-72 j 2-064 





b,Google 



406 THE PRACTICAL MODEL CALCULATOR. 

Table containing the Weight of Wrought Iron Bars 12 inches long 
in lbs. avoirdupois. 



I„th. 


,^,„„d. 1 S,a.r,, 


Inth, 


Kounii. 


?4U«a. 




■163 


■208 


2 


16-32 


20-80 






■467 


2 


1800 


22-89 




■653 






19-76 


25-12 


1-03 


1-30 


2 


21-59 


27-46 




1-47 


1'87 




23-52 


29-92 




2-00 


2-55 


3 


27-60 


85-12 




2-61 


a -32 




32-00 


40-80 


H 


3-31 


4-21 


3 


86-72 


46-72 


i| 


i08 


5-20 


4 


41-76 


58-12 




4-94 


6-28 


4 


47-25 


60-00 




5-88 


7-48 


4 


52-98 


67-24 




6-80 


8-78 


4; 


58-92 


74-95 




S'OO 


10-20 


5 


65-28 


83-20 




918 


11-68 




72-00 


91-56 




10-44 


13-28 


4 


79-04 


100-48 


2J 


11 -80 


1&-00 


01 


86-36 


109-82 


4 


13-23 


lS-81 




94-08 


119-68 


£| 


14-78 


18-74 


7 


128-00 


163-20 


LEom^ 


Prouortio 


lal Dimen 


torts of 6 


■sided Na 


S for Bolts 



\ to 2J inches diameter. 



Diameter of bolts 


s 


1 


^ 


t 


1 


i 


1 


11 


li 




H 


a 


1 


1ft 


1! 


1* 


1| 


ISi 


2f 




Breadth over the angles 


» 


M 


1, 


1| 


Ift 


IK 


2 


2S 


2A 


I'hickness 


A 


ft 


A 


f 


! 


1 


11 


IJ 


1ft 


Diameter of bolts 


15 


li 


1| 


IJ 


1* 


2 


2i 


2! 






2* 


A 


2Hi2i 


3A 


3! 


3i 


4 






Breadth over the angles 


2ii 


2t 


H SA 


34 


3f 


4A 


n 


- 


Thickness 


lA 


1« 


1« 2 


21 


2i 


2J 


2} 



Table of the Speeifie Q-ravity ofWater at different temperatures, 
that at 62° being taken as unit^. 



70° F, 


99913 


52-= F. 


1-00076 


68 




50 


1-00087 




99958 


48 


1-00095 






46 


1-00102 


G2 1 




44 


1-00107 




30035 


i% 


1-00111 




30O50 


40 


1-00118 


54 1 


00064 


38 


1 -00116 



The difference of temperatures between G2° and S9°-2, where 
water attains its greatest density, will vary the bulk of a gallon 
rather loss than the third of a cubic inch. 



hv Google 



SPECIFIC GILAVITY. 



from 1 to 12 incheB diameter, advancing hy an eighth. 


luthSB. 


IM. 


Inchaa. 


Ll«. 


Inchc.. 


i.b.. 




■14 


ii 


14-76 


8* 


84-56