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Examples in Avoirdupois Weight.

1.

3 Qu. What is the weight of 107 chefts of tea; each cheft weighing four hundred, three quarters, and twenty

In this example 100 is the nearest commenfurable number, but the multiplier is 107; I therefore multiply by 10 and 10, and the original multiplicand by 7; which last product is added to the other..

In the proof, because 107 is an odd number, I take the half of 106, which is 53; multiplying by 5 and 10, for 50, and the original multiplicand by 3; then I multiply the whole product by 2, which doubles it to 106; and lastly, add the multiplicand to the product, for the 1 that was wanting.

By these examples it may be seen, that there is no occafion to have always a commenfurable number, or to come very near one; for, if it want ever so much, it may be worked in this manner; always multiplying the original multiplicand by the overplus, and adding fuch product to the other.

It must be obferved, that if the multiplier, in this part of multiplication, confift of more than 144, it must be refolved into three multipliers at leaft; and if it confift of more than 1728, it must be refolved into more than three.

4 Qu. What is the fuperficial content of a piece of ground, whose breadth is 10274 feet, and length 24640?

Here I multiply the length by the breadth (which is the general rule in fuperficial meafure), and the product is 253151360 square feet; for the answer.

24640

10274

98 560 172480 49280 246400

253151360

5 Qu. How many folid feet does a piece of timber contain, that is forty feet in length, four in breadth, and three in depth?

Here the general rule is, to multiply the length length by the breadth, and that product

40

breadth

by the depth. And the product is 480 folid feet, for the answer.

depth

Multiplication teaches alfo how to multiply different denominations of measure by different denominations, called erofs multiplication; of which I fhall fpeak in Menfuration. :.. Multiplication alfo ferves to bring great denominations of money, weights, or measures, into small ones: but this more properly belongs to the rule of Reduction.

*SECT. V.

OF DIVISION.

DIVISION teacheth how, from two given numbers, ⋅ to find a third; that shall be contained in the largest of the two given numbers, as often as the fmalleft contains units: or the third number contains units, as often as the smallest of the two given numbers is contained in the other. Thus, if 15 were to be divided by 3, the answer would be 5: for 3 is contained in 15, 5 times: and 5 is contained in 15, 3 times. As multiplication teacheth how to bring great denomina

tions into finall ones; and having the rate or value of one thing, to know the rate or value of many; and from the length and breadth of a fuperficies, to know its contents, &c. fo, on the contrary, divifion teacheth how to bring small denominations into great ones; and from the rate or value of many things, and the number of them given, to know the rate or value of one; and from the contents of fuperficies, and the length to find the breadth; or from the fuperficies, and the breadth, to find the length; and from the contents of a folid, and one dimension thereof, to find the other two.

In every fum in divifion, there are three parts which are to be particularly remembered, viz.

The Dividend, or number to be divided.

Divifor, or number by which we divide

Quotient, or anfwer to the work, which thews how often the divifor is contained in the dividend. Thus, in the before-mentioned inftance, 15 is the dividend, 3 the divisor, and 5 the quotient.

Befides these three parts, which are in every fum, there is fometimes a remainder, when the work is finished; which will always happen when the dividend does not exactly con tain the divifor a certain number of times; as, if I were to divide 13 by 4; here 4 the divifor is contained 3 times in the dividend 13, and there is a remainder of 1; and in division of divers denominations, it must be noted, that the remainder is always of the fame denomination with the dividend.

Divifion is either fingle, or compound: fingle divifion is when the divifor does not confift of more than 12, and the dividend of not more than 144. Any queftions of this fort may be answered at once by the inultiplication table, without fetting them down: thus, if it were required to divide To by 10, by that table I know that 10 times 11 is no; thus 10 is contained 1 times in 110, and 11 is the quotient.mouth) Compound divifion is when the divifor contains more than 12, or dividend more than 144, or both.

In divifion the dividend is to have a crooked line placed at

each

each end of it; and before the line on the left hand, the divifor is to be placed, and behind that on the right hand, the quotient, or the answer to the work, is placed,

Rules Mark off from the dividend, by a point placed under the figure, as many figures as there are in the divifor; but if the divifor confist of a greater number than the figures To pointed off from the dividend, then another figure is to be pointed off from the dividend; then feek how often the divifor, or (if the divifor confift of many figures) the first, or firft and fecond figures of the divifor, is contained in the fame number of figures pointed off from the dividend; place the answer in the quotient, and multiply the divifor thereby ; then place the product of fuch multiplication under the pointed-off figures, and fubtract the faid product therefrom; but if the product amounts to more than the figures pointed off, the divifor is to be multiplied by a number that is an unit lefs than the former anfwer, and this anfwer placed in the quotient, instead of the former, and this product fubtracted from the aforefaid figures in the dividend (if this product be ftill too large, the dividend must be multiplied by an unit lefs); to the remainder, the next figure in the dividend is to be brought down, and placed on the right hand thereof, which number is to be taken for a new dividend: feek how * often the divifor is contained in fuch new dividend, place the answer in the quotient, multiply the divifor thereby, fubtract the product from the laft dividend, and bring down the next figure in the dividend, as before. Proceed in this manner till the work be finished; as in the following examples

1

Let 4876 be divided into 7 parts;i:

W

In this example, I place the numbers as ben 7)4876(696 fore directed and because 7 the divifor is 42 more than 4, the firft figure in the dividend, I 207 make a point under the fecond figure. 8 in the ac63 dividend, which makes 48 for the first dividend; 1 then ask how oftem7 the divifor is contained ba in 48, the first dividend, which I find is 6 times, to bandiq nail-baldeoro & svar 29 2 gh: gli ro" I therefore

146

421

4

I therefore place 6 in the quotient, and multiply the divisor 7 thereby, and the product 42 I fet under the dividend 48, to be fubtracted therefrom; and to the remainder 6, I bring down the next figure 7, of the original dividend; thus I have 67 for a new dividend. Then I ask how often 7 is contained in 67, which I find is 9 times: I therefore put 9 in the quotient, and multiply the divifor thereby, and the product 63 I place under, and subtract from, the last dividend; and to the remainder 4 I bring down the next and last figure 6 in the original dividend, which makes 46 for a new dividend. Then 1 afk how often 7 is contained in 46, which I find is 6 times: I place 6 in the quotient, and multiply the divifor thereby the product 42 I fubtract from the last dividend 46, and there being no more figures in the original dividend to bring down, there is a remainder of 4 after the work is finished, and which remainder is always less than the divifor, if -the work be right.

:

The remainder may be fet over the divifor as a fraction of an unit; thus in the answer to this question: if 48761. were to be divided equally among 7 men, what would each man's fhare amount to? Anf. 6961. and 4-7ths of a pound.

Divifion is proved by multiplication: thus the foregoing example is proved by multiplying the quotient by the diyi. for, adding thereto the remainder, and if the product be equal to the dividend the work is right; otherwife not.