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		English
	cubic inches.		feet.
		Metre	3·281
Millilitre	·06103	” French feet, 3·07844
Centilitre	·61028	Millimetre.	·03937
Decilitre	6·10279	Centimetre	·39371
Litre, or cubic decimetre	61·02791	Decimetre	3·93708
Decalitre	610·27900	Metre	39·37079
Hectolitre	6102·79000	Decametre	393·70790
Kylolitre	61027·90000	Hectometre	3937·07900
Myrialitre	610279·00000	Kilometre	39370·79000
1 litre is nearly 2⅛ wine pints.		Myriametre	393707·90000
1 kilolitre 1 tun 12¾ wine gallons.		8 kilometres are nearly 5 miles.
1 stere, or cubic metre	35·3171	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, 2².

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 2² = 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, 2² = 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