Chapter XXXIII: Part XII: Mathematics (1)

_Mathematics_ is the science which treats of all kinds of quantity whatever, that can be numbered, or measured.

_Arithmetic_ is that part which treats of numbering.

_Fractions_ treat of broken numbers, or parts of numbers.

_Algebra_ is the art of computing by symbols.

In this science, quantities of all kinds are represented by the letters of the alphabet.

_Geometry_ is the science relating to measurement. By the assistance of geometry, engineers, &c., conduct all their works, take the distances of places, and the measure of inaccessible objects, &c.

_Characters, marks, or signs_,

which are used in arithmetic, and algebra, to denote several of the operations, and propositions:

+ _signifies_ plus, or addition,
- minus, or subtraction,
× multiplication,
÷ division,
: :: : proportion,
= equality,
√ square root,
∛ cube root,
4^2 _denotes that_ 4 is to be squared.
4^3 _denotes that_ 4 is to be cubed.

ARITHMETIC.

REDUCTION.

_Reduction_ is the method of converting numbers from one name, or denomination to another: or the method of finding the value of a quantity in terms of some other higher, or lower quantity.

_To reduce from a higher to a lower denomination._

_Rule._—Multiply the given number by as many of the lower denomination as make one of the greater;[40] adding to the product as many of the lower denomination as are expressed in the given sum.

_Example._—In £6 15_s._ 5_d._, how many pence?

£. _s._ _d._
6 15 5
20
----
135
12
----
1625 _Answer._
----

_To convert from a lower to a higher denomination._

_Rule._—Divide the given number by as many of the lower denomination as are required to make one of the greater.[41] Should there be any remainder, it will be of the same denomination as the dividend.

_Example._—Convert 1625 pence into pounds, shillings, and pence.

12) 1625 pence
----
20) 135 5
--------
£6 15_s._ 5 _d._ _Answer._

THE RULE OF THREE, OR SIMPLE PROPORTION.

It is called the _Rule of Three_ because three numbers are given to find a fourth. It is also called _Simple Proportion_, because the 1st term bears the same proportion to the 2nd, as the 3rd does to the 4th. Of the three given numbers, two of them are always of the same kind, or name, and are to be the 1st, and 2nd terms of the question; the 3rd number is always of the same name, or kind as the 4th, or answer sought; and in stating the question it is always to be made the 3rd term. If the answer will be _greater_ than the 3rd term, place the least of the other two given quantities for the 1st term; but if the answer will be _less_ than the 3rd term, put the greater of the two numbers, or quantities, for the 1st term.

_Rule._—State the question according to the above directions, and multiply the 2nd and 3rd terms together, and divide this product by the 1st, for the 4th term, or answer sought.

If the 1st and 2nd terms are not of the same denomination, they must be reduced to it; and if the third term is a compound number, it must be reduced to its lowest denomination before the multiplication, or division of the term takes place.

_Note 1._—The operation may frequently be considerably abridged, by dividing the 1st and 2nd, or the 1st and 3rd terms, by any number which will exactly divide them, afterwards using the quotients, instead of the numbers themselves.*

_Example._—If 2 tons of iron for ordnance cost £40, how many tons may be bought for £360?

As £40 : £360 :: 2 tons : 18 tons.
(Thus 360 × 2) ÷ 40 = 18. The answer.

* Or thus, 9 × 2 = 18. The answer.

_Note 2._—_A concise method of ascertaining the annual amount of a daily sum of money._

_Rule._—Bring the daily sum into pence, and then add together as many pounds, half pounds, groats, and pence, as there are pence in the daily sum, for the amount required. For leap year, add the rate for one day.

_Example._—Required the annual amount of 2_s._ 6_d._ per diem.

2_s._ 6_d._ = 30_d._ 30 pounds.
15 = 30 half pounds.
10_s._ = 30 groats.
2_s._ 6_d._ = 30 pence.
-----------------------------
Annual amount (365 days) ... £45 12_s._ 6_d._

_Note 3._—To find the amount of any number of days’ pay, the daily rate (under twenty shillings) being given.

The price of any article being given, the value of any number may be ascertained in a similar manner.

_Rule 1._ When the rate (or price) is an even number, multiply the given number by half of the rate, doubling the first figure to the right hand for the shillings, the remainder of the product will be pounds.

_Example._ Required the amount of 243 days’ pay, at 4_s._ per diem.

4/2 = 2 243
2
-------
£48 12_s._ _Ans._

_Rule 2._ When the price is an odd number, find for the greatest number as before, to which add one-twentieth of the given number for the odd shilling.

_Example._ What is the price of 566 pairs of shoes, at 7_s._ per pair.

566 2/0 ) 56/6
3 -------
-------- 28 6
169 16_s._ -------
28 6
--------
£198 2_s._ _Ans._

FRACTIONS.

_A fraction_ is a quantity which expresses a part, or parts of a unit, or integer. It is denoted by two numbers placed with a line between them.

_A Simple fraction_ consists of two numbers, called the numerator, and denominator; thus,

3 numerator,
--
5 denominator.

_The Denominator_ is placed below the numerator, and expresses the number of equal parts into which the integer is divided.

_The Numerator_ expresses the number of parts of the broken unit, or integer; or shows how many of the parts of the unit are expressed by the fraction.

_A Compound fraction_ is a fraction of a fraction, as ⅔ of ⅘.

_A Mixed number_ consists of a whole number with a fraction annexed to it, as 4⅖.

_An Improper fraction_ has the numerator greater than the denominator, as 6/5.

REDUCTION OF FRACTIONS.

is bringing them from one denomination to another.

_To reduce a fraction to its lowest terms._

_Rule._—Divide the numerator, and the denominator, by any number that exactly divides them, and the quotients by any other number, till they can be no longer divided by any whole number, when the fraction will be in its lowest terms.

_Example._—Reduce 4032/6048 to its lowest terms.

Thus, (4)4032/6048 = (12)/1008/1512 = (6)/84/126 = (7)/14/21 = 2/3.
_Answer._

_To reduce an improper fraction to a whole, or mixed number._

_Rule._—Divide the numerator by the denominator, the quotient will be the whole number; and the remainder (if any) the numerator of the fraction, having the divisor for the denominator.

_Example._—Reduce 114/12 to a whole, or mixed number.

12 ) 114
--------
9-6/12 _Answer._

_To reduce a mixed number to an improper fraction._

_Rule._—Multiply the whole number by the denominator, and add the numerator to the product, under which place the given denominator.

_Example._—Reduce 17⅝ to an improper fraction.

17⅝
8
---
141
---
8 _Answer._

_To reduce a compound fraction to a simple fraction._

_Rule._—Multiply all the numerators together for the numerator, and all the denominators for the denominator.

_Example._—Reduce ⅜ of ⅙ of ½ of 9 to a simple fraction.

Numerators 3 × 1 × 1 × 9 27 9
----- ----- = -- = -- _Answer._
Denominators 8 × 6 × 2 × 1 96 32

_To reduce fractions of different denominators to equivalent fractions, having a common denominator._

_Rule._—Multiply each numerator by all the denominators except its own for the new numerators, and multiply all the denominators together for a common denominator.[42]

_Example._—Reduce ⅜, ⅔, and ⅘ to fractions having a common denominator.

3 × 3 × 5 = 45
2 × 8 × 4 = 80
4 × 8 × 3 = 96
8 × 3 × 5 = 120 _Answer_, 45/120, 80/120, and 96/120

ADDITION OF FRACTIONS.

_Rule._—Bring compound fractions to simple fractions; reduce all the fractions to a common denominator, then add all the numerators together, and place their sum over the common denominator. When mixed numbers are given, find the sum of the fractions, to which add the whole numbers.

_Example._—Add together ⅚, ¾, and 6½.

5 × 4 × 2 = 40 40/48 + 36/48 + 24/48 + 6 = 8-4/48
3 × 6 × 2 = 36 or, by cancelling, and dividing,[43]
1 × 6 × 4 = 24 10/12 + 9/12 + 6/12 + 6 = 8-1/12. _Answer._
6 × 4 × 2 = 48

SUBTRACTION OF FRACTIONS.

_Rule._—Prepare the quantities, as in addition of fractions. Place the less quantity under the greater. Then, if possible, subtract the lower numerator from the upper; under the remainder write the common denominator, and, if there be whole numbers, find their difference as in simple subtraction. But if the lower numerator exceed the upper, subtract it from the common denominator, and to the remainder add the upper numerator; write the common denominator under this sum, and carry 1 to the whole number in the lower line.

_Example._— From 54-5/6 or 54-25/30
Take 25-5/15 or 25-10/30
-------
29-15/30 _Answer._

MULTIPLICATION OF FRACTIONS.

_Rule._—Reduce mixed numbers to equivalent fractions; then multiply all the numerators together for a numerator, and all the denominators together for a denominator, which will give the product required.

_Example._—Multiply ⅚, ⅜, and 2½ together.
⅚ × ⅜ × (2½ or) 5/2 = 75/96 _Answer._

DIVISION OF FRACTIONS.

_Rule._—Prepare the fractions, as for multiplication; then divide the numerator by the numerator, and the denominator by the denominator, if they will exactly divide; but if they will not do so, then invert the terms of the divisor, and multiply the dividend by it, as in multiplication.

_Example._—Divide 9/16 by 4½.
9/16 ÷ (4½ or) 9/2 = ⅛ _Answer._

RULE OF THREE IN FRACTIONS.

_Rule._—State the terms, as directed in “Simple proportion;” reduce them (if necessary) to improper, or simple fractions, and the _two first_ to the same denomination. Then multiply together the second and third terms, and the first with its parts inverted, as in division, for the answer.

_Example._—If 4⅕ cwt. of sugar cost £19⅞, how much may be bought for £59⅝?

As 19⅞ : 59⅝ :: 4⅕
Or, 159/8 : 477/8 :: 21/5 : 12⅗ _Answer._
8/159 × 477/8 × 21/5 = 80136/6360 = 12⅗ cwt.

DECIMALS.

_A decimal fraction_ is that which has for its denominator an unit (1), with as many ciphers annexed as the numerator has places; and it is usually expressed by setting down the numerator only, with a point before it, on the left hand. Thus, 5/10 is ·5; 25/100 is ·25; 25/1000 is ·025; ciphers being _prefixed_, to make up as many places as are required by the ciphers in the denominator.

_A mixed number_ is made up of a whole number with some decimal fraction, the one being separated from the other by a point, thus 3·25 is the same as 3-25/100 or 325/100.

_Ciphers on the right hand of decimals_ make no alteration in their value; for ·5, ·50, ·500 are decimals having all the same value, each being = 5/10. But when they are placed on the left hand, they decrease the value in a tenfold proportion; thus, ·5 is 5/10; but ·05 is 5/100.

ADDITION OF DECIMALS.

_Rule._—Set the numbers under each other, according to the value of their places, in which state the decimal separating points will all stand exactly under each other. Then beginning at the right hand, add up all the columns of numbers 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.

_Example._—Required the sum of 29·0146, 3146·5, 14·16, and 165.

29·0146
3146·5
14·16
165·
---------
_Answer_ 3354·6746

SUBTRACTION OF DECIMALS.

_Rule._—Place the numbers under each other according to the value of their places. Then, beginning at the right hand, subtract as in whole numbers, and point off the decimals, as in addition.

_Example._—Subtract 4·90142 from 214·81.

214·81
4·90142
---------
_Answer_ 209·90858

MULTIPLICATION OF DECIMALS.

_Rule._—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 factors. But, if there be not so many figures in the product, prefix ciphers to supply the deficiency.[44]

_Example._—Multiply 32·108 by 2·5.

32·108
2·5
------
160540
64216
-------
80·2700 _Answer._

DIVISION OF DECIMALS.

_Rule._—Divide as in whole numbers, and point off in the quotient as many places for decimals as the decimal places in the dividend exceed those in the divisor. When the decimal places of the quotient are not so many as the above rule requires, the deficiency is to be supplied by prefixing ciphers. When there is 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.

_Example._—Divide 234·7052 by 64·25.

64·25 ) 234·7052 ( 3·65 _Answer._
19275
------
41955
38550
------
34052
32125
-----
1927 _Remainder._

REDUCTION OF DECIMALS.

_To reduce a vulgar fraction to its equivalent decimal._

_Rule._—Divide the numerator by the denominator, as in Division of Decimals, annexing ciphers to the numerator as far as necessary: and the quotient will be the decimal required.

_Example._—Reduce 7/24 to a decimal.

24 = 4 × 6. Then 4)7·
-------
6)1·75
-------
·291666, &c.

_To find the value of a decimal, in terms of the inferior denominations._

_Rule._—Multiply the decimal by the number of parts in the next lower denomination, and cut off as many places to the right hand for a remainder, as there are places in the given decimal. Multiply that remainder by the parts in the next lower denomination, again cutting off for another remainder as before. Proceed in the same manner through all the parts of the integer; then the several denominations, separated on the left hand, will make up the answer.

_Example._—What is the value of ·775 pounds sterling.

·775
20
------
Shillings 15·500
12
-----
Pence 6·000 _Answer_ 15_s._ 6_d._

_To convert integers, or decimals to equivalent decimals of higher denominations._

_Rule._—Divide by the number of parts in the next higher denomination, continuing the operation to as many higher denominations as may be necessary.

_When there are several numbers, all to be converted to the decimal of the highest_—

Set the given numbers directly under each other for dividends, proceeding from the lowest to the highest; opposite to each dividend, on the left hand, place such a number for a divisor as will bring it to the next higher name. Begin at the uppermost, and perform all the divisions, placing the quotient of each division, as decimal parts, on the right hand of the dividend next below it; so shall the last quotient be the decimal required.

_Example._—Convert 15_s._ 9¾_d._ to the decimal of a pound sterling.

4 | 3·
12 | 9·75
20 | 15·8125
£·790625 _Answer._

_Example._—Convert 1 dwt. to the decimal of a pound, Troy weight.

20 ) 1
-------
12 ) ·05 oz.
-----------
·004166 lb., &c., _Answer._

RULE OF THREE IN DECIMALS.

_Rule._—Prepare the terms, by reducing the fractions to decimals; compound numbers to decimals of the higher denominations, or integers of the lower; also the first, and second terms to the same name. Then multiply, and divide, as in the Rule of Three, in whole numbers.

_Example._—If ⅜ of a yard of cloth cost £⅖, what will 5/16 of a yard cost?

yd. yd. £. _s. d._
⅜ = ·375 As ·375 : ·3125 :: 4 : ·333 &c. or 6 8
4
------
⅖ = ·4 ·375)·12500 (·3333 &c.
1125 20
--------------
5/16 = ·3215 1250_s._ 6·666 &c.
1125 12
--------------
_Answer_, 6_s._ 8_d._ 125_d._ 7·999 &c. nearly 8_d._

DUODECIMALS.

By Duodecimals, artificers, &c., compute the content of their works.

_Rule._—Set down the two dimensions to be multiplied together one under the other, so that feet may stand under feet, inches under inches, &c.

Multiply each term in the multiplicand, beginning at the lowest, by the feet in the multiplier, and set the result of each straight under its corresponding term, observing to carry 1 for every 12, from the inches to the feet. In like manner multiply all the multiplicand by the inches, and parts of the multiplier, and set the result of each term one place removed to the right hand of those in the multiplicand: omitting, however, what is below parts of inches, only carrying to these the proper number of units from the lowest denominations. Or, instead of multiplying by the inches, take such part of the multiplicand as those are of a foot.

Then add the two lines together for the content required.

_Example._—Multiply 14 feet 9 inches, by 4 feet 6 inches.

ft. in.
14 9
4 6
-------
59 0
7 4½
-------
66 4½ _Answer._
-------

TABLES OF WEIGHTS, AND MEASURES.

TROY WEIGHT.

24 grains 1 pennyweight.
480 20 1 ounce.
5760 240 12 1 pound.

AVOIRDUPOIS WEIGHT.

16 drams 1 ounce.
256 16 1 pound.
7168 448 28 1 quarter.
28672 1792 112 4 1 hundred weight.
573440 35840 2240 80 20 1 ton.

_Note._—1 lb. Avoirdupois weight equals 14 oz. 11 dwts. 15½ grs. Troy.
1 oz. ditto 18 dwts. 5½ do.
1 dr. ditto 27·34375 do.

APOTHECARIES’ WEIGHT.

20 grains 1 scruple.
60 3 1 dram.
480 24 8 1 ounce.
5760 288 96 12 1 pound.

WEIGHTS.

_To find the weight, for tonnage._

_Cattle_—
Divide the number by 3, for weight in tons.

_Sheep_ Average 60 lb. each.
Divide by 33, for weight in tons.

_Pigs_ Average 80 lb.
Divide by 15, for tons.

_Beer, or Ale_—
Barrel 3¼ cwt.
Hogshead 5¼ cwt.

_Oats_ Sack—24 stone.
Divide quarters by 5, for tons.

_Rum_—
Divide gallons by 224, for tons.

_Wine_ Cask—12 cwt.

_Rule for ascertaining the weight of Hay._

Measure the length and breadth of the stack; then take its height from the ground to the eaves, and add to this last one-third of the height from the eaves to the top: Multiply the length by the breadth, and the product by the height, all expressed in feet; divide the amount by 27, to find the cubic yards, which multiply by the number of stones supposed to be in a cubic yard (viz., in a stack of new hay, six stones; if the stack has stood a considerable time, eight stones; and if old hay, nine stones), and you have the weight in stones. For example, suppose a stack to be 60 feet in length, 30 in breadth, 12 in height from the ground to the eaves, and 9 (the third of which is three) from the eaves to the top; then 60 × 30 × 15 = 27000; 27000 ÷ 27 = 1000; and 1000 × 9 = 9000 stones of old hay.

LONG MEASURE.

12 inches 1 foot.
36 3 1 yard.
198 16½ 5½ 1 pole, perch, or rod.
7920 660 220 40 1 furlong.
63360 5280 1760 320 8 1 mile.

LAND MEASURE (_Length_).

7·92 inches 1 link.
100 links, or 22 yards 1 chain.
80 chains 1 mile.
69·121 miles 1 geographical degree.

LAND MEASURE (_Surface, or Superficial_).

62·7264 square inches 1 square link.
625 square links 1 square pole, or perch.
10000 square links 1 square chain.
2500 square links 1 square rood, or pole.
10 square chains 1 square acre.
100000 square links 1 square acre.

NAUTICAL MEASURE.

1 nautical mile 6082·66 feet.
3 miles 1 league.
20 leagues 1 degree.
360 degrees the earth’s circumference.

SQUARE MEASURE.

144 s. inches 1 s. foot
1296 9 1 s. yard.
39204 272¼ 30¼ 1 s. pole.
1568160 10890 1210 40 1 rood.
6272640 43560 4840 160 4 1 acre.

CUBIC MEASURE (_Measure of solidity_).

1728 cubic inches 1 cubic foot.
27 cubic feet 1 cubic yard.

_Note._—A cubic foot is equal to 2200 cylindrical inches, or 3300 spherical inches, or 6600 conical inches.

_Timber._

40 feet of round, and 50 feet of hewn timber make 1 _Ton_; 16 cubic feet make 1 _Foot_ of wood; 8 feet of wood make 1 _Cord_.

_Water._

Maximum density 42 deg. Fahrenheit.

1 cubic foot of water 6¼ imperial gallons.
1 cylindric foot do. about 5 do.
1 cubic foot weighs 62·5 lb. avoirdupois.
1 cylindric do. do. 49·1
1 lineal do. (1 in. square) do. ·434
12·2 imperial gallons weigh 1 cwt.
224 do. do. 1 ton.
1·8 cubic feet do. 1 cwt.
35·84 do. do. 1 ton.

MEASURES OF CAPACITY.

69⅓ cubic in 2 pints 1 quart.
277¼ 8 4 1 gallon.
554½ 16 8 2 1 peck.
2218⅕ 64 32 8 4 1 bushel.
10¼ cubic ft. 512 256 64 32 8 1 quarter.

FRENCH MEASURES.

English
cubic inches.

Millilitre ·06103
Centilitre ·61028
Decilitre 6·10279
Litre, or cubic decimetre 61·02791
Decalitre 610·27900
Hectolitre 6102·79000
Kylolitre 61027·90000
Myrialitre 610279·00000
1 litre is nearly 2⅛ wine pints.
1 kilolitre 1 tun 12¾ wine gallons.
1 stere, or cubic metre 35·3171

English
feet.

Metre 3·281
” French feet, 3·07844
Millimetre. ·03937
Centimetre ·39371
Decimetre 3·93708
Metre 39·37079
Decametre 393·70790
Hectometre 3937·07900
Kilometre 39370·79000
Myriametre 393707·90000
8 kilometres are nearly 5 miles.
1 inch is ·0254 metre.
100 feet are nearly 30·5 metres.

INVOLUTION.

_Involution_ is the raising of powers from any given number, as a root.

A _Power_ is a quantity produced by multiplying any given number, called the _Root_, a certain number of times continually by itself. Thus, 2 × 2 = 4, the 2nd power, or square of 2, expressed thus, 2^2.

_The index, or exponent of a power_ is the number denoting the height, or degree of that power. Thus, 2 is the index of the 2nd power.

Powers that are to be raised, are usually denoted by placing the index above the root, or first power.

Thus 2^2 = 4, the 2nd power of 2.

_Example._—What is the 2nd power of 45?

45 × 45 = 2025 _Answer._

EVOLUTION.

_Evolution_ is the reverse of Involution, being the extracting, or finding the roots of any given powers, or numbers.

_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 2nd root of 4, because, 2^2 = 2 × 2 = 4; and 3 is the cube root, or third root of 27. But there are many numbers of which a proposed root can never be exactly found; by means of decimals, however, the root may be very nearly ascertained.

_Any power of a given number, or root_, may be found exactly by multiplying the number continually into itself.

Those roots which only approximate are called _Surd-roots_; but those which can be found, quite exactly, are called _Rational-roots_. Thus, the square root of 3 is a surd root, but the square root of 4 is a rational root, being equal to 2; also the cube root of 8 is rational, being equal to 2, but the cube root of 9 is surd, or irrational. Roots are sometimes denoted by writing the character √ before the power with the index of the root against it. Thus, the 3rd, or cube root of 20 is expressed by ∛20. When the power is expressed by several numbers with the sign + or - between them, a line is drawn from the top of the sign over all the parts of it; thus the cube (or third) root of 45 - 12 is ∛(45 - 12) or thus ∛(45 - 12).

TO EXTRACT THE SQUARE ROOT.

_Rule._—Divide the given number into periods of two figures each, by setting a point over the place of units, and another over the place of hundreds, and so on over every second figure, both to the left hand in integers, and right hand in decimals. Find the greatest square in the first period on the left hand, and set its root on the right hand of the given number, after the manner of the 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[45] 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 it the next period of the given number, for a new dividend. Repeat the same process over again—viz., 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.

_To extract the square root of a fraction, or mixed number._

Reduce the fraction to a decimal, and extract its root.

Mixed numbers may be either reduced to improper fractions, and the root extracted; or the fraction may be reduced to a decimal, then joined to the integer, and the root of the whole extracted.

_Example._—To find the square root of 29506624.

29506624 ( 5432 The Root.
25
--------
104 | 450
4 | 416
-------------
1083 | 3466
3 | 3249
-------------
10862 | 21724
2 | 21724

TO EXTRACT THE CUBE ROOT.

_Rule 1._—By trials, or by the table of roots (_vide page_ 280), take the nearest rational cube to the given number, whether it be greater, or less, and call it the assumed cube.

2.—Then (_by the Rule of Three_),

As the sum of the given number, and double the assumed cube, is to the sum of the assumed cube, and double the given number, so is the root of the assumed cube, to the root required, nearly.

3.—Or as the first sum,

is to the difference of the given, and assumed cube,
so is the assumed root,
to the difference of the roots, nearly.

4.—Again, by using, in like manner, the cube of the root last found as a new assumed cube, another root will be obtained still nearer. Repeat this operation as often as necessary, using always the cube of the last-found root, for the assumed root.

_Example._—To find the cube root of 21035·8.

By trials it will be found _first_, that the root lies between 20, and 30; and, _secondly_, between 27, and 28. Taking, therefore, 27, its cube is 19683, which will be the assumed cube. Then by No. 2 of the Rule

19683 21035·8
2 2
----- --------
39366 42071·6
21035·8 19683·
------ --------
As 60401·8 : 61754·6 :: 27 : 27·6047 the Root, nearly.

Again for a second operation, the cube of this root is 21035·318645155832, and the process by No. 3 of the Rule will be

21035·318645, &c.
2
------------
42070·637290 21035·8
21035·8 21035·318645, &c.
------------ ------------
As 63106·43729 : diff. ·481355 :: 27·6047 :
: the diff. ·000210560
----------
consequently the root required is 27·604910560

TABLE OF SQUARES, CUBES, AND ROOTS.

+----+-------+---------+------------+----------+
| No.| Sqr. | Cube. | Sqr. root. |Cube root.|
+----+-------+---------+------------+----------+
| 1 | 1 | 1 | 1·0000000 | 1·000000 |
| 2 | 4 | 8 | 1·4142136 | 1·259921 |
| 3 | 9 | 27 | 1·7320508 | 1·442250 |
| 4 | 16 | 64 | 2·0000000 | 1·587401 |
| 5 | 25 | 125 | 2·2360680 | 1·709976 |
| 6 | 36 | 216 | 2·4494897 | 1·817121 |
| 7 | 49 | 343 | 2·6457513 | 1·912933 |
| 8 | 64 | 512 | 2·8284271 | 2·000000 |
| 9 | 81 | 729 | 3·0000000 | 2·080084 |
| 10 | 100 | 1000 | 3·1622777 | 2·154435 |
| 11 | 121 | 1331 | 3·3166248 | 2·223980 |
| 12 | 144 | 1728 | 3·4641016 | 2·289428 |
| 13 | 169 | 2197 | 3·6055513 | 2·351335 |
| 14 | 196 | 2744 | 3·7416574 | 2·410142 |
| 15 | 225 | 3375 | 3·8729833 | 2·466212 |
| 16 | 256 | 4096 | 4·0000000 | 2·519842 |
| 17 | 289 | 4913 | 4·1231056 | 2·571282 |
| 18 | 324 | 5832 | 4·2426407 | 2·620741 |
| 19 | 361 | 6859 | 4·3588989 | 2·668402 |
| 20 | 400 | 8000 | 4·4721360 | 2·714418 |
| 21 | 441 | 9261 | 4·5825757 | 2·758923 |
| 22 | 484 | 10648 | 4·6904158 | 2·802039 |
| 23 | 529 | 12167 | 4·7958315 | 2·843867 |
| 24 | 576 | 13824 | 4·8989795 | 2·884499 |
| 25 | 625 | 15625 | 5·0000000 | 2·924018 |
| 26 | 676 | 17576 | 5·0990195 | 2·962496 |
| 27 | 729 | 19683 | 5·1961524 | 3·000000 |
| 28 | 784 | 21952 | 5·2915026 | 3·036589 |
| 29 | 841 | 24389 | 5·3851648 | 3·072317 |
| 30 | 900 | 27000 | 5·4772256 | 3·107232 |
| 31 | 961 | 29791 | 5·5677644 | 3·141381 |
| 32 | 1024 | 32768 | 5·6568542 | 3·174802 |
| 33 | 1089 | 35937 | 5·7445626 | 3·207534 |
| 34 | 1156 | 39304 | 5·8309519 | 3·239612 |
| 35 | 1225 | 42875 | 5·9160798 | 3·271066 |
| 36 | 1296 | 46656 | 6·0000000 | 3·301927 |
| 37 | 1369 | 50653 | 6·0827625 | 3·332222 |
| 38 | 1444 | 54872 | 6·1644140 | 3·361975 |
| 39 | 1521 | 59319 | 6·2449980 | 3·391211 |
| 40 | 1600 | 64000 | 6·3245553 | 3·419952 |
| 41 | 1681 | 68921 | 6·4031242 | 3·448217 |
| 42 | 1764 | 74088 | 6·4807407 | 3·476027 |
| 43 | 1849 | 79507 | 6·5574385 | 3·503398 |
| 44 | 1936 | 85184 | 6·6332496 | 3·530348 |
| 45 | 2025 | 91125 | 6·7082039 | 3·556893 |
| 46 | 2116 | 97336 | 6·7823300 | 3·583048 |
| 47 | 2209 | 103823 | 6·8556546 | 3·608826 |
| 48 | 2304 | 110592 | 6·9282032 | 3·634241 |
| 49 | 2401 | 117649 | 7·0000000 | 3·659306 |
| 50 | 2500 | 125000 | 7·0170678 | 3·684031 |
| 51 | 2601 | 132651 | 7·1414284 | 3·708430 |
| 52 | 2704 | 140608 | 7·2111026 | 3·732511 |
| 53 | 2809 | 148877 | 7·2801099 | 3·756286 |
| 54 | 2916 | 157464 | 7·3484692 | 3·779763 |
| 55 | 3025 | 166375 | 7·4161985 | 3·802953 |
| 56 | 3136 | 175616 | 7·4893148 | 3·825862 |
| 57 | 3249 | 185193 | 7·5498344 | 3·848501 |
| 58 | 3364 | 195112 | 7·6157731 | 3·870877 |
| 59 | 3481 | 205379 | 7·6811457 | 3·892996 |
| 60 | 3600 | 216000 | 7·7459667 | 3·914867 |
| 61 | 3721 | 226981 | 7·8102497 | 3·936497 |
| 62 | 3844 | 238328 | 7·8740079 | 3·957892 |
| 63 | 3969 | 250047 | 7·9372539 | 3·979057 |
| 64 | 4096 | 262144 | 8·0000000 | 4·000000 |
| 65 | 4225 | 274625 | 8·0622577 | 4·020726 |
| 66 | 4356 | 287496 | 8·1240384 | 4·041240 |
| 67 | 4489 | 300763 | 8·1853528 | 4·061548 |
| 68 | 4624 | 314432 | 8·2462113 | 4·081656 |
| 69 | 4761 | 328509 | 8·3066239 | 4·101566 |
| 70 | 4900 | 343000 | 8·3666003 | 4·121285 |
| 71 | 5041 | 357911 | 8·4261498 | 4·140818 |
| 72 | 5184 | 373248 | 8·4852814 | 4·160168 |
| 73 | 5329 | 389017 | 8·5440037 | 4·179339 |
| 74 | 5476 | 405224 | 8·6023253 | 4·198336 |
| 75 | 5625 | 421875 | 8·6602540 | 4·217163 |
| 76 | 5776 | 438976 | 8·7177979 | 4·235824 |
| 77 | 5929 | 456533 | 8·7749644 | 4·254321 |
| 78 | 6084 | 474552 | 8·8317609 | 4·272659 |
| 79 | 6241 | 493039 | 8·8881944 | 4·290841 |
| 80 | 6400 | 512000 | 8·9442719 | 4·308870 |
| 81 | 6561 | 531441 | 9·0000000 | 4·326749 |
| 82 | 6724 | 551368 | 9·0553851 | 4·344481 |
| 83 | 6889 | 571787 | 9·1104336 | 4·362071 |
| 84 | 7056 | 592704 | 9·1651514 | 4·379519 |
| 85 | 7225 | 614125 | 9·2195445 | 4·396830 |
| 86 | 7396 | 636056 | 9·2736185 | 4·414005 |
| 87 | 7569 | 658503 | 9·3273791 | 4·431047 |
| 88 | 7744 | 681472 | 9·3808315 | 4·447960 |
| 89 | 7921 | 704969 | 9·4339811 | 4·464745 |
| 90 | 8100 | 729000 | 9·4868330 | 4·481405 |
| 91 | 8281 | 753571 | 9·5393920 | 4·497942 |
| 92 | 8464 | 778688 | 9·5916630 | 4·514357 |
| 93 | 8649 | 804357 | 9·6436508 | 4·530655 |
| 94 | 8836 | 830584 | 9·6953597 | 4·546836 |
| 95 | 9025 | 857375 | 9·7467943 | 4·562903 |
| 96 | 9216 | 884736 | 9·7979590 | 4·578857 |
| 97 | 9409 | 912673 | 9·8488578 | 4·594701 |
| 98 | 9604 | 941192 | 9·8994949 | 4·610436 |
| 99 | 9801 | 970299 | 9·9498744 | 4·626065 |
|100 | 10000 | 1000000 | 10·0000000 | 4·641589 |
+----+-------+---------+------------+----------+

PILING OF SHOT, AND SHELL.

Shot, and shells, are usually piled in horizontal courses, the base being either an equilateral triangle, a square, or a rectangle. The triangular, and square piles terminate each in a single ball, but the rectangular pile finishes in a row of balls.

_To find the number of balls in a complete pile._

_Rule._—Add the three parallel edges together; then one-third of the product of that sum, and of the number of balls in the triangular face, will be the number sought.

_Note 1._—_The parallel edges_ in a _rectangular pile_ are the two rows in length at the base, and the upper ridge. In the _square pile_ the same, except that the upper row is only a single ball. In the _triangular pile_, one side of the base, the single ball at top, and that at the back, are considered the parallel edges.

_Note 2._—_The number of balls in the triangular face_ is found by multiplying half the number in the breadth at the base, by the number in the breadth at the base _plus_ 1.

_Note 3._—In all piles the breadth of the bottom is equal to the number of courses. In the oblong pile, the top row is one more than the difference between the length, and breadth of the bottom.

_Example._—To find the shot in a triangular pile, the bottom row consisting of 12 shot.

Parallel edges. { 12
{ 1 12 ÷ 2 = 6
{ 1 12 + 1 = 13
-------
--- Triangular face 78
3 ) 14 4⅔
--- ----
4⅔ 312
--- 52
----
_Answer_ 364

_Example._—To find the shot in a square pile, the bottom row consisting of 12 shot.

12 12 ÷ 2 = 6
12 12 + 1 = 13
1 -----
---- 78
3 ) 25 8⅓
---- -----
8⅓ 624
---- 26
-----
_Answer_ 650

_Example._—To find the shot in an oblong pile, whose base consists of 18 shot in length, and 12 in breadth.

18 18 - 12 = 6
18 1
7 --
---- 7
3 ) 43
----
14⅓ 12 ÷ 2 = 6
---- 12 + 1 = 13
----
78
14⅓
----
312
78
26
-----
_Answer_ 1118
-----

_Triangular pile._

_Rule._—Multiply the base by the base _plus_ 1, this product by the base _plus_ 2, and divide by 6.

_Square pile._

_Rule._—Multiply the bottom row by the bottom row _plus_ 1, and this product by twice the bottom row _plus_ 1, and divide by 6.

_Rectangular, or oblong pile._

_Rule._—Multiply the breadth of the base by itself _plus_ 1; and this product by three times the length of the base _plus_ 1, _minus_ the breadth of the base, and divide by 6.

_In the following formulæ let the letter_ (L) _denote the number in the bottom row, or the length; and_ (B) _the breadth of the lowest course_.

_Triangular pile_ (L × (L + 1) × (L + 2)) / 6

_Square pile_ (L × (L + 1) × (2L + 1)) / 6

_Oblong pile_ (B × (B + 1) × (3L + 1 - B)) / 6

_The number of shot in any pile,_

(whose base does not exceed 21) may readily be ascertained by referring to _the following Table, page 284_.

_For the square pile._—Look for the number of shot in the base, in the first vertical column on the left hand, and also in the diagonal column; and at their angle of meeting will be found the content required.

Thus 20 base gives 2870.

_For the triangular pile._—Look for the number in the base row in the diagonal column, and opposite to it will be found the content.

Thus 18 base gives 1140.

_For the oblong pile._—Look for the number in the length of the base in the vertical column, and the breadth of the base in the diagonal column, and at their angle of meeting will be found the content required.

Thus 17 length, and 12 breadth, gives 1040.

_To find the number of balls in an incomplete pile._

Compute the number in the pile considered as complete; also the number in the upper pile, or part wanting; and the difference between the two piles thus found will be the number in the frustrum, or incomplete pile.

_Table for computing the Content of any Pile, whose base row does not exceed 21 balls._

+--+--+--------------------------------------------------------------
| 1| 2| 4
| +---+
| 2| 5| 3| 10
| +---+
| 3| 8| 14| 4| 20|
| +---+
| 4|11| 20| 30| 5| 35
| +---+
| 5|14| 26| 40| 55| 6| 56
| +---+
| 6|17| 32| 50| 70| 91| 7| 84
| +---+
| 7|20| 38| 60| 85|112|140| 8|120
| +---+
| 8|23| 44| 70|100|133|168|204| 9|165
| +---+
| 9|26| 50| 80|115|154|196|240|285| 10| 220
| +----+
|10|29| 56| 90|130|175|224|276|330|385| 11| 286
| +----+
|11|32| 62|100|145|196|252|312|375|440| 506| 12| 364
| +----+
|12|35| 68|110|160|217|280|348|420|495| 572| 650| 13| 455
| +----+
|13|38| 74|120|175|238|308|384|465|550| 638| 728| 819| 14| 560
| +----+
|14|41| 80|130|190|259|336|420|510|605| 704| 806| 910|1015| 15| 680
| +----+
|15|44| 86|140|205|280|364|456|555|660| 770| 884|1001|1120|1240| 16|
| +
|16|47| 92|150|220|301|392|492|600|715| 836| 962|1092|1225|1360|1496|
|
|17|50| 98|160|235|322|420|528|645|770| 902|1040|1183|1330|1480|1632|
|
|18|53|104|170|250|343|448|564|690|825| 968|1118|1274|1435|1600|1768|
|
|19|56|110|180|265|364|476|600|735|880|1034|1196|1365|1540|1720|1904|
|
|20|59|116|190|280|385|504|636|780|935|1100|1274|1456|1645|1840|2040|
|
|21|62|122|200|295|406|532|672|825|990|1166|1352|1547|1750|1960|2176|
++--+--+---+---+---+---+---+---+---+--+----+----+----+----+----+----+

15... | 816 ||
+----+ ||
16... | 17| 969 ||
| +----+ ||
17... |1785| 18|1140 ||
| +----+ ||
18... |1938|2109| 19|1330 ||
| +----+ ||
19... |2091|2280|2470| 20|1540 ||
| +----+ ||
20... |2244|2451|2660|2870| 21|1771 ||
| +----+ ||
21... |2397|2622|2850|3080|3311| 22|2024||
-------+----+----+----+----+----+----+----++

CORDAGE.

_Ropes_, _cables_, and all other descriptions of cordage are distinguished by their circumference, thus a two-inch rope means a rope two inches in circumference.

_To find the weight of a rope._

_First method._—Multiply the length in fathoms by the square of the circumference, and divide the product by 480 for the weight in cwts.

_Example._—Required the weight of 110 fathoms of 3-inch rope.

3 × 3 × 110 = 990, which divided by 480, gives 2 cwt. 7 lb.
Weight required.

_Second method._—Divide the square of the circumference by 4, the quotient will give the weight, in pounds, per fathom.

_Example._—What is the weight of a 3-inch rope per fathom?

3^2 ÷ 4 = 2¼ lb. Weight required.

_To find the strength of a rope, or the weight it will support._

_First method._—Square the circumference, and divide by 5, for the number of tons which it will bear suspended from it.[46]

_Example._—What weight will 3-inch rope of the best description support?

(3 × 3) / 5 = 9/5 = 1⅘ ton, or 4030 lb. Weight required.

_Second method._—Multiply the square of the circumference by 2, the product will give the _practical weight in cwts_. that may be lifted by it, or about half the breaking weight.

_Example._—What number of cwts. may be lifted by a 3-inch rope?

3^2 × 2 = 18 cwts. Weight required.

_The strain_, in pounds, _a rope will bear safely_ = girt^2 × 200}
” ” _a cable_ ” ” = girt^2 × 120}
nearly.

CHAINS.

_To find the weight of chains._

The square of the diameter of the link, measured in eighths of inches, will give the weight of the chain, per fathom, in pounds.

_Example._—What is the weight per fathom of a ¾-inch chain?

¾-inch = 6/8; 6^2 = 36 lb. Weight per fathom.

_Or_, the weight per foot of the chain, multiplied by 24, will give the weight per fathom of the chain, _nearly_. A chain cable with a stay across the links will weigh about one-twelfth more than the foregoing examples.

_To find the weight that may be safely lifted by a chain._

Divide the square of the diameter of the links, taken in eighths of an inch by 8, and the quotient will give the number of tons that may be lifted by the chain.

_Example._—What number of tons will a chain made of ¾-inch iron carry with safety?

¾-inch = 6/8 6^2 = 36 36 / 8 = 4½ tons. Weight required.

_The safe strain_ is equal to about 8 tons, per square inch, of the iron of which the chain is made.

_The stay across the link of a chain_ increases its strength about one-sixth.

_When the chain is of great length_, a deduction, from the above rules, must be allowed for the weight of it.

IRON RODS.

_To find the weight of round iron rods._

Divide the square of the diameter, in quarter inches, by 2, and the quotient will give the weight in pounds, per yard.

_Example._—What is the weight of a yard of 1-inch round iron.

1 inch = 4 quarters 4^2 = 16 16 / 2 = 8 lb. Weight required.

_To find the weight of square rods._

The weight of round rods, of similar diameter, divided by ·7854 will give the weight of the square rods.

_To find the weight that may be sustained, or lifted by round iron rods._

Find the weight in pounds, per yard; two-thirds of which will give the safe load, in tons.

A round iron rod of average quality of iron, one inch in diameter, will be torn asunder by 16 tons; it will be perceptibly damaged by half this strain, or 8 tons; its safe load will be one-third, or 5·33 tons.

TIMBER.

_To find the area, or superficial content of a plank._

Multiply the length by the mean breadth.[47]

_Example._—Required the content of a board whose length is 11 feet 2 inches, and breadth 1 foot 10 inches

ft. in. ft. in. ft. in.
11 2 × 1 10 = 20 5. Content required.

_To find the solid content of squared, or four-sided timber._

Multiply the mean breadth by the mean thickness, and the product by the length, for the content, nearly.

_Note 1._—If the tree taper regularly from the one end to the other, either take the mean breadth, and thickness in the middle, or take the dimensions at the two ends, and half their sum will be the mean dimensions; which, multiplied as by the above rule, will give the content, nearly.

_Note 2._—If the piece do not taper regularly, take several different dimensions, add them all together, and divide their sum by the number of them, for the mean dimensions.

_Example._—Required the content of a piece of timber 16 feet long, and side of square 14 inches.

ft. in. ft. in. ft. ft. in.
1 2 × 1 2 × 16 = 21 9. Content required.

_To find the solidity of round, or unsquared timber._

1. Multiply the square of the quarter girt (or the square of ¼ of the mean circumference), by the length, for the content.

_Note._—When the tree is tapering, take the mean dimensions, either by girting it in the middle for the mean girt, or at the two ends, taking half the sum of the two; or by girting it in several places, then adding all the girts together, and dividing the sum by the number of them for the mean girt. But when the tree is very irregular, divide it into several lengths, and find the content of each part separately.

_Example._—Required the content of a tree, whose mean girt is 3·15 feet, and length 14½ feet.

3·15 / 4 = ·7875 ·7875 × ·7875 = ·62015625.
·62015 × 14·5 = 8·9922 feet of solid timber. The content required.

2. Find the mean area of a round tree, and multiply it by the length for the content.

_To find the weight of a tree._

Find its content in feet, and multiply that by the specific gravity of the wood.

(_Vide_ GRAVITY, _and Table of Specific Gravities_. _Page_ 318.)

_Example._—Required the weight of an elm-tree; whose mean girt is 5 feet, and length 60 feet.

5/4 = 1·25 1·25 × 1·25 = 1·5625.
1·5625 × 60 = 93·75. Content in feet.

TONNAGE.

Table of Tonnage, and Weight of _one_ of the following Carriages, Carts, Waggons, Gyns, &c., used in land service.

+--------------------------------------+-------+---------+-------------+
| | Tonn- | Weight. | |
| | age. | | |
+--------------------------------------+-------+---------+-------------+
| |Tons. |Cwt. lb.| |
| | ft.| qrs. | |
| | | |} With |
|Carri- {24 pounder | 6 0|34 0 22|} bullock |
|ages. { 8 inch | 6 0|34 2 12|} pole and |
| T{ {For Iron {18 pounder | 4 39|27 2 9|} chain, |
| r{ {Ordnance.{12 ” 21 cwt. | 4 7|18 3 24|} weighing 2 |
| a{ | | |} qrs. 19 lb.|
| v{ {Howitzer {10 inch| 6 17|39 0 9| |
| ” { e{ { 8 ” | 5 37|33 2 0| Do. do. do.|
| ” { l{ {12 pounder | 5 33|22 0 10| |
| ” { l{ { 9 ” | 5 1|20 2 14| |
| ” { i{For Brass { 6 ” | 4 21|17 3 5| |
| ” { n{Ordnance. { {32 pr.| 5 29|23 3 13| |
| ” { g{ {Howitzer {24 ” | 5 6|21 0 17| |
| ” { { { {12 ” | 4 21|18 3 14| |
| ” { c{Ammunition waggon | 5 36|20 0 3| {For all |
| ” { o{ | | { natures. |
| ” { m{Forge | 5 38|19 1 0| |
| ” { p{Store-waggon (without spare | 5 11|18 1 10| |
| ” { l{ wheel) | | | |
| ” { e{Small arm ammunition waggon | 4 36|14 2 16| |
| ” { t{Rocket {12 pounder | 7 33|20 2 8| |
| ” { e{ { 6 ” | 5 17|20 1 20| |
| ” { .{Pontoon {Large | 3 30|42 2 13| |
| ” { { {Small | |22 2 5| |
| ” {Garrison, wood, common standing }| | | |
| ” { for 32 pounder of 25 cwt. }| 1 8| 8 0 7| |
|Capstan, crab | 0 31| 3 3 26| |
| {Forge, cavalry | 4 32|11 2 3| |
| {Hand | 1 10| 4 3 4| |
| {Hospital, conveyance | 3 16|10 2 20| |
|Carts {Sling | 3 38|16 1 17| |
| {Store | 3 16| 9 1 0| |
| {French | 1 32| 5 2 13| |
|Drugs {Large | 2 7|17 1 24| |
| {Small | 0 29| 5 0 4| |
|Gyns, {Large | 1 23| 9 2 22| |
|Triangle {Small | 1 2| 7 3 3| |
| {For 32 pounder | | | |
| { garrison carriage}| 0 26 | 6 0 12| Fir. |
| {Madras {For traversing }| | | |
|Platform. { { carriage with }| | | |
| { { tail-piece }| 1 23 |14 2 0| Teak. |
| {Mortar, Alderson’s pattern | 0 30 | 8 1 4| |
|Portable forge, and pack saddle, | | |
| in wooden case | 0 17 | 2 1 3| |
| {Flanders | 5 0 |16 1 25| |
|Waggons {Platform | 3 16 |21 3 18| |
| {Sling | 8 11 |31 3 26| |
|Waggons, hospital, Mr. Holmes’ {Large | 9 10 |21 0 0| |
| pattern {Small.| 6 30 |18 0 4| |
+--------------------------------------+-------+---------+-------------+

The calculation of tonnage for baggage, stores, &c., is by measurement: _a Ton_, consisting of 40 cubic feet; but metals, and very heavy articles are estimated by actual weight, without reference to bulk.

_To ascertain the tonnage of sailing vessels, the hold being clear._

_Rule._—Divide the length of the upper deck between the afterpart of the stem, and the forepart of the stern-post, into six equal parts.

_Depths._—At the foremost, the middle, and the aftermost of those points of division, measure in feet, and decimal parts of a foot, the depth from the under side of the upper deck to the ceiling at the limber strake. In the case of a break in the upper deck the depths are to be measured from a line stretched in a continuation of the deck.

_Breadths._—Divide each of those three depths into five equal parts, and measure the inside breadths at the following points—viz., at one-fifth, and at four-fifths from the upper deck of the foremost, and aftermost depths, and at two-fifths, and four-fifths from the upper deck of the midship depth.

_Length._—At half the midship depth measure the length of the vessel from the afterpart of the stem to the forepart of the stern-post; then to twice the midship depth add the foremost, and the aftermost depths for the sum of the depths; add together the upper, and lower breadths at the foremost division, three times the upper breadth, and the lower breadth at the midship division, and the upper, and twice the lower breadth at the after division, for the sum of the breadths: then multiply the sum of the depths by the sum of the breadths, and this product by the length, and divide the final product by 3500, which will give the number of tons for register.

If the vessel have a poop, or half deck, or a break in the upper deck, measure the inside mean length, breadth, and height of such part thereof as may be included within the bulkhead; multiply these three measurements together, and dividing the product by 92·4, the quotient will be the number of tons to be added to the result as above found.

In order to ascertain the tonnage of open vessels, the depths are to be measured from the upper edge of the upper strake.

_To ascertain the tonnage of steam vessels._

_Rule._—In addition to the foregoing rules, when applied for the purpose of ascertaining the tonnage of any ship or vessel propelled by steam, the tonnage due to the cubical content of the engine-room must be deducted from the total tonnage of the vessel, as determined by either of the rules aforesaid, and the remainder will be the true register tonnage of the said ship or vessel.

_To determine the tonnage due to the cubical content of the engine-room._

_Rule._—Measure the inside length of the engine-room in feet and decimal parts of a foot, from the foremost to the aftermost bulkhead, then multiply the said length by the depth of the ship or vessel at the midship division as aforesaid, and the product by the inside breadth of the same division at two-fifths of the depth from the deck, taken aforesaid, and divide the last product by 92·4, and the quotient will be the tonnage due to the cubical content of the engine-room.

_To ascertain the tonnage of vessels when laden._

_Rule._—Measure, _first_, the length on the upper deck between the afterpart of the stem, and the forepart of the stern-post; _secondly_, the inside breadth on the under side of the upper deck, at the middle point of the length; and, _thirdly_, the depth from the under side of the upper deck down the pump-well to the sink; multiply these three dimensions together, and divide the product by 130, and the quotient will be the amount of the register tonnage of such ships.

MECHANICS.

_Mechanics_ is the science of forces, and the effects they produce when applied to machines in the motion of bodies.

_Machine, or engine_, is any mechanical instrument contrived to move bodies.

_Equilibrium_ is an equality of action, or force, between two or more powers, or weights, acting against each other, by which they destroy each other’s effects, and remain at rest.

_The centre of motion_ is the fixed point about which a body moves.

_The axis of motion_ is the fixed line about which it moves.

_The centre of gravity_ is a certain point on which a body (being freely suspended) will rest, in any position.

_The whole momentum_ or quantity of force of a moving body, is the result of the quantity of matter multiplied by the velocity with which it is moved.