PROOF OF MULTIPLICATION.

The proof of Multiplication is commonly done by Division; but, for incomplex numbers, it is often more expeditious, and equally sure, to do it by Multiplication itself; viz. by making the multiplicand multiplier, and vice versa.

NOTE. The proof by multiplication, is more expeditious, when the multiplicand and multiplier have a number of figures nearly alike,

We do not speak of the proof by 9, because it is in some cases deficient and faulty.

DIVISION.

DIVISION is an operation, by which is found how many times a number is contained in another.

The greater of the two numbers is called Dividend; the smaller, Divisor, and the result of the operation, Quotient. Both dividend and divisor are also denominated the two members or terms of a division.

When a division is to be only indicated, it is written thus,

but when it is to be executed, the two members must be disposed in this manner, and the quotient placed under the divisor.

24 dividend,

6 divisor.

Dividend 24

6 dicis.

4 quot.

6 1st subtraction.

It follows, from the definition of Division, that this operation can be done by Subtraction. If we had, for instance, 24 to divide by 6, the question would then be, to find how many times 6 is contained in 24, which can be known by subtracting 6 from 24 as many times as can be, and the number of subtractions will be the quotient sought for, which will be 4, as it is seen by the number of subtractions made here: But such a process would 247 be too tedious if the divisor was contained a great number of times in the dividend. The art of Division is to shorten this operation, which abbreviation consists particularly in the use of Pythagoras's Table, which shews how many times a divisor of one figure is contained in a dividend, composed, at most, of two.

18

6 2nd subtraction.

6 3d subtraction.

2010010

6

6

4th subtraction.

This being understood, we shall distinguish only two cases in Division, the one when the divisor contains but one figure, whilst the dividend is composed of ever so many; the other, when both members consist of several figures.

DIVISION.-First Case.

The method to be followed in both cases, is, to consider the given dividend as composed of several partial dividends, in which the divisor be contained the fewest times possible, in order to have always but one figure to write in the quotient. The operation is commenced by the left of the given dividend, in order that the excess of the first partial dividend, over the product of the divisor by the quotient, may be added up to the second partial dividend, and so on; which could not be done, without much trouble, if the operation began by the right of the dividend. This will be illustrated by the following example, and by the explications we shall give of our proceedings. I remark, first, that the first figure, at the left of the dividend, containing the divisor, must be the first partial dividend; then I say, in 9 how many times 7? 1; I write 1 in the quotient; and, to find the excess of this first partial dividend, over the product of the divisor by the quotient, I say, once 7 is 7, which I write under the 9, and taking 7 from 9 I have 2 left; I take down the second figure of the dividend, which I write by the remainder 2,

9471 7

....

7

1་་་ ས།

0

1353

and then the second partial dividend is 24; I then

say, in 24 how many times 7? 3 times; I write 3 in the quotient, and then, to find the excess of the dividend, I say, 3 times 7 is 21, which I set down under 24, then subtracting, I have 3 for the difference; by this 3 I take down the third figure of the dividend, and I have, for my third partial dividend, 37; operating as above, I find for quotient 5, with the remainder 2; finally, I take down 1, and 21 is my last partial dividend, which gives 3 for quotient, and nothing remains; therefore I say, that 1353 is the quotient asked for, or that it represents the number of times the divisor is contained in the dividend, since it is the assemblage of all the quotients of the partial dividends; and the better to be convinced of it, you need but observe, that the first partial dividend was not really 9, but 9000; therefore, when we have said, in 9 how many times 7? it is plain that we have made use of a dividend 1000 times too small, which of course must have given a quotient 1000 times too small; for a dividend 1000 times too small, must necessarily contain the divisor 1000 times too little; then the quotient must be made 1000 times bigger; and, instead of 1, we must write -- 1000; you see, likewise, that the second partial dividend, being a number of hundreds, the quotient 3 should also express hundreds, and ought to have been written

the third partial dividend, being a number of tens, its quotient 5 amounts to

In fine, the fourth partial dividend, being a number of units, has for quotient

Casting up these partial quotients, we find the total quotient to be

300;

50.

3.

1353.

EXPLICATION. Here you see, that the

DIVISION.-Second Case.

divisor, being a number of 753 209 | 368

hundreds, the first partial dividend must be composed of at least 3 figures; taking then the 3 first figures on the left of the dividend, we have said, in 753 how many times 368 ? it is twice: I have written 2 in the quotient; and, multiplying the divisor by this quotient, in order to

736

1720
1472

2489

2208

281

281

2046

$68.

subtract it from the dividend, we had 736, which, subtracted from the partial dividend 753, gives 17 for remainder, which I write, and close by it the 2 which I take down, to form the second partial dividend: then I say, in 172 how many times 368 ? no times; I write O in the quotient; then proceeding, I take down a new figure, which gives 1720 for the third partial dividend, and I say, in 1720 how many times 368? or, as it is not easy to find the ratio be tween two numbers when they are of some extent, I content myself, by way of a trial, with compar ing the units of the highest kind in the divisor, with the same in the dividend; here, therefore, comparing the hundreds in both members, I say, in 17 how many times 3? It gives 5 times: but, before I write down 3 in the quotient, I must examine whether what remains of the dividend will also contain 5 times the remainder of the divisor. In general it is enough to try" the 2d. figure of the divisor; I say then, 5 times 6 are 30, (i. e. 30 tens) carrying mentally the 0 under, the figure 2 of the dividend (which is also a number of tens)