Arithmetic, Part 1 · Section 17
A Third way of Diviſion.
There is another kind of diviſion which is very much uſed, and is in moſt requeſt with thoſe who have moſt occaſion to divide great numbers, the manner of working is not much unlike the way before taught, one or two examples will make it plain.
Example 1. … Let it be required to divide 162483 by 1321, ſet down your numbers as you ſee them placed in the margine, viz. Firſt, ſet down 162483 the dividend, then on the left hand thereof ſet the diviſor 1321 with a crooked line between them, then on the right hand thereof make another crooked line which muſt ſerve to ſet the figures of the quotient in, ſo are your numbers placed in due order, then draw a line under the dividend, and make a prick under the figure 4, becauſe ſo far the figures of the diviſor would extend if they had been placed under-neath the dividend, according as in the other examples, this prick ſerves onelyonly to ſhew how far you have proceeded in your work, and muſt at every diviſion be removed a place further, till at length you come to the laſt figure of the dividend: your numbers being thus placed with a line under them, you are ready for the work, which muſt be performed according to the directions of the following rule.
THE RULE.
Demand how often the diviſor may be had in the dividend, and place that number in the quotient, then multiply the diviſor by the quotient, and place the product under the line: then ſubtract this product from the dividend, and ſet the remainer under the product, then make a prick under the next figure of the dividend, and bring that figure down to the remainer, and then proceed as before.
Example, … Your numbers being placed as is before directed, you may begin your work in this manner, firſt, ſay how many times 1321, can I have in 1624? ſay once, place 1 in the quotient, by which 1 multiply the diviſor 1321, beginning at the left hand, ſaying, once one is 1, place 1 under the line, then once 2 is 2, ſet 2 under the line, then once 3 is 3, place 3 under the line, laſtly, once 1 is 1, place 1 under the line, then ſubtract this 1321, from 1624, and there will remain 303, to this 303 bring down the next figure in the dividend, namely 8, ſo will that number be 3038, under which draw a line, and repeat the ſame work again, ſaying, how many times 1321 can I have in 3038, which may be had two times, place 2 in the quotient, by which 2 multiply the diviſor 1321, ſaying 2 times 1 is 2, place 2 under the line, then 2 times 2 is 4, place 4 under the line, then 2 times 3 is 6, place 6 under the line, laſtly, 2 times 1 is 2, place 2 under the line, and ſubtract this 2642 from 3038, and there will remain 396, to this 396 bring down the next figure of the dividend, which is 3, ſo is this number made 3963, under which draw a line, and repeat the work once again, ſaying, how many times 1321 can I have in 3963, which may be had 3 times, by which 3 multiply the diviſor 1321, ſaying 3 times 1 is 3, then 3 times 2 is 6, then 3 times 3 is 9, and laſtly, 3 times 1 is 3, which place under the line, and ſubtract it from the line above, which in this example is the ſame number, therefore there remains nothing, and the work is ended, but if any remainer had been, that ſhould have been ſet under the line, as by the examples following will appear.
Other Examples for Practice.
In this Example where 793058, … is divided by 5624, you may perceive that the quotient is 141, and 74 remaining, ſo that the real quotient is 141 74/5624.
The Proof of this Diviſion.
This kind of diviſion is proved by Addition, for If you adde the ſeveral products ariſing from the multiplication of the ſeveral quotients into the diviſor, and alſo adde thereunto the remainder (if any be) the total of this addition ſhall be equal to the dividend, if there be no errour in the work.
So in the example following, … if you adde 4325 the firſt product, and 30275 the ſecond product, and 2796 the remainder, together, in the ſame order as they now ſtand in the example, you ſhall find the Total of this Addition to be 76321, equal to the dividend, which demonſtrates the work to be true.