RULE. Multiply the quotient by the divisor, and subtract the product from the dividend, and the result will be the true remainder.

The truth of this is extremely obvious; for if the product of the divisor and quotient, added to the remainder, be equal to the dividend, their product, taken from the dividend, must leave the remainder.

The rule, which is most commonly used, is this.

RULE. Multiply the last remainder by the preceding divisor, or last but one, and to the product add the preceding remainder; multiply this sum by the next preceding divisor, and to the product add the next proceding remainder; and so on till you have used all the divisors and remainders.

To explain this rule from the example, we may observe that every unit of the first quotient may be looked upon as contain

III. To perform division more concisely than by the general

rule.

RULE.*

Multiply the divisor by the quotient figures as before, and subtract each figure of the product when you produce it, always remembering to carry so many to the next figure as were borrowed before,

EXAMPLES.

1. Divide 3104675846 by 833.

833)3104675846(372710171 the quotient;

ing 9 of the units in the given dividend; consequently every unit of it, that remains, will contain the same; therefore this remainder must be multiplied by 9, in order to find the units of the given dividend, which it contains. Again, every unit in the next quotient will contain 4 units of the preceding, or 36 of the given dividend, that is, 9 times 4; therefore what remains must be multiplied by 36; or, which is the same thing, by 9 and 4 continually. Now this is the same as the rule; for instead of finding the remainders separately, they are reduced from the bottom upward, step by step, to the first, and the remaining units of the same class taken as they occur,

* The reason of this rule is the same as that of the general rule.

Reduction is the method of bringing numbers from one name or denomination to another without changing the val

ue.

In order to perform reduction, it is necessary to be acquainted with the relative value of the different denominations of coin, weight, and measure, that are used; for which purpose see the following

24 grains make 1 penny-weight, marked grs. dwt.

By this weight are weighed jewels, gold, silver, corn, bread, and liqours.

APOTHECARIES' WEIGHT.

Apothecaries use this weight in compounding their med

icines; but they buy and sell their drugs by Avoirdupois

weight. Apothecaries' is the same as Troy weight, having only some different divisions.

AVOIRDUPOIS WEIGHT.

16 drams make 1 ounce, marked dr. oz.

By this weight are weighed all things of a coarse or drossy nature; such as butter, cheese, flesh, grocery wares, and all metals, except gold and silver.*

2 weys

NOTE

1 last

L.

The diameter of a Winchester bushel is 18 in

ches, and its depth 8 inches.-And one gallon by dry measure contains 2684 cubic inches.

By this measure, salt, lead, ore, oysters, corn, and other dry goods are measured.

1

ALE AND BEER MEASURE.
Marked

2 pints make 1 quart pts. qts.
4 quarts
1 gallon gal.
8 gallons 1 firkin of Ale fir.
9 gallons 1 firkin of Beer fir

Marked

2 firkins 1 kilderkin kil. 2 kilderkins 1 barrel bar. 3 kilderkins 1 hogshead hhd. 3 barrels 1 butt

butt.

NOTE. The ale gallon contains 282 cubic inches. In London the ale firkin contains 8 gallons, and the beer firkin

9; other measures being in the same proportion.

63 gallons 1 hogshead hhd. 18 gallons 1 runlet 84 gallons 1 puncheon pun. 31 gallons 1 barrel

rund.

bar.

By this measure, brandy, spirits, perry, cyder, mead, vinegar, and oil are measured.

NOTE. -231 cubic inches make a gallon, and 10 gallons make an anchor.