When the multiplier is the product of two or more numbers, multiply by one of the numbers, and the product by the other which will sometimes save labour.

When some of the figures or numbers in the multiplier, are multiples or products of other figures in the same; it is generally easier and shorter to multiply by the least numbers, and then these products by such numbers as will make the greater, particular care being taken, to place the units figure of every line, exactly under the units place of that number in the multiplicand to which it belongs.

Multiplication may also be performed without the aid of the table, by continual additions of the multiplicand, and if the multiplier is large may shorten the work.

This, though more curious than useful, yet furnishes a proof that multiplication is but a compendious addition, as was said at the commencement of the rule,

In making this tablet, if the last line be the same as the first with a cipher annexed, it may be presumed to be right.

There are other contractions, which cannot be taught without the aid of Division, and are to be noticed in that rule

For a demonstration of the general rule, and the several contrac tions here given in Multiplication, I refer the reader to Walker's Philosophy of Arithmetic, page 16 and pages 34 to 37.

D

DIVISION

Is a compendious subtraction, shows how often one number is contained in another, and teaches how to separate a given number into as many equal parts as shall be assigned; it consists of three terms, which are called the dividend, divisor and quotient. The first is the number to be divided, the second that by which we are to divide, and the third is the result of the operation, which expresses the number of times that the divisor may be subtracted from the dividend, or number of equal parts required.

When one number is contained in another a certain number of times exactly, it is said to measure this number, but if it does not, there will be something left, which is called the remainder. This number which is sometimes over, is usually considered the numerator of a fraction of which the divisor is the denominator, having a straight line drawn between them. The remainder is always to be thus disposed when a whole number without any name is divided, but when the number divided has a name as pounds, tons, hundreds, &c. the remainder is considered as so many ones left of the said number or dividend, and by multiplying this remainder, by the number contained in the next inferior denomination, to that of the dividend, we may continue the division as at first, and if any thing again remain, it is to be considered as so many ones of the denomination to which the former remainder was brought by the last multiplication, which may again be multiplied and divided as before, until all the inferior denominations of the first dividend are gone through, and then if any thing finally remain, it is to be annexed to the last quotient, placed over the divisor, with a line drawn between them, as a fraction or part of the last quotient.

This character is the mark of Division, and shows, that the numbers between which it is placed are to be divided; the number which it immediately precedes being considered as the divisor, thus 84 ÷ 7 = 12 signifies that the number 84 is to be divided by 7 and 68975, that the number 68 is to be divided by 9; in the first example 84 is the dividend, 7 is the divisor, and 12 is the quotient arising after performing the division; in the second example 68 is the dividend, 9 is the divisor, 7 the quotient and 5 the remainder.

Division is also sometimes represented, by drawing a line, over which placing the dividend, and under it the divisor, thus 3412 signifies the same as 84 ÷ 7 = 12.

84

Though it is almost impossible to give a rule for division, that a child can understand, yet as this book may fall into the

hands of some to whom it may be useful to have one, I subjoin the following general

RULE

Draw a line or hook at the right, and another at the left of the dividend, and write the divisor outside that on the left.

By trial find how many times the divisor is contained in as many of the nearest figures of the dividend thereto as may be necessary, which will not exceed one figure more than the number of figures in the divisor, place the figure denoting this number of times at the right hand, outside the other line or hook, which will give the highest figure of the quotient, multiply the divisor thereby, and place the product under that part of the dividend which thus contained the divisor, substract it therefrom, and to the remainder bring down the next figure of the dividend; again find by trial how often the divisor is contained in this remainder, with the figure thus annexed, and place the figure denoting the number of times in the quotient beside the former, and multiply and subtract as before; but if the first or any subsequent remainder with the figure of the dividend annexed thereto, be less than the divisor, place a cipher in the quotient and bring down another figure or cipher of the dividend, which annex to the former, and if it should still be less than the divisor place another cipher in the quotient, always taking care to put a figure or a cipher in the quotient, for every figure or cipher of the dividend thus brought down, and so on until the remainder is equal to, or exceeds the divisor, or until all the figures or ciphers in the dividend are exhausted, taking care as at first, that there shall not be more than one figure above the number of figures in the divisor, and thus proceed until all the figures in the dividend are brought down, when if there is a remainder, it is to be added to the quotient, as a fraction as before noticed.*

*The above rule is best illustrated by examples.

1. Let 85394 be divided by 7.

After division is performed, by multiplying the quotient by the divisor, and adding the remainder, if any to the product, the former dividend will be obtained if the work is right; or by subtracting the remainder from the dividend, and dividing this remainder by the quotient found, the former divisor will be obtained, which furnishes two methods of proof.

And the same reasoning will apply to any other numbers.

For a further demonstration, and other observations on Division see Walker's Philosophy of Arithmetic, pages 19 to 29.