Thefe examples, and all others of other denominations, are wrought in the fame manner as fubtraction of money; having refpect to the table belonging to the denomination : thus, in the first example of avoirdupois weight, I fay, take ZI pounds, from 20 I cannot, wherefore I borrow 1 quarter of an hundred from the 3 quarters, and add it to the 20, which makes 48 pounds; then I fay, 21 from 48 and there remains 27; next, 1 that I borrowed and 2 is 3, 3 from 3 I cannot, wherefore I fet down o, as before hinted; laftly, 10 from 12, and there remains 2.

The figures at the head of the columns fhew the number of units which each unit of the next higher denomination

contains.

SECT. IV.

OF MULTIPLICATION.

MULTIPLICATION is, perhaps, the most neceffary rule in arithmetic for business, on account of its difpatch in refolving feveral long queftions.

Multiplication teaches how, from two given numbers, to find a third, that fhall contain either of the two given num. bers as often as the other contains units: thus, 3 times 4 is 12; here 3 and 4 are the two given numbers, and 12 is the third number, or product, which contains 3 as often as 4 contains units, viz. 4 times; or it contains 4 as often as 3 contains units, 3 times,

There

There are three ends principally to be answered by mustiplication :

First. It ferves to bring the greater denominations of money, weights, or meafures, into fmall ones; as, pounds into fhillings, pence, and farthings; hundred weights into pounds, ounces, or drams; miles into yards, feet, barley-corns, &c. Secondly. Having the length and breadth of a plain surface, we find its contents.

1

Thirdly. Having the rate or value of any one thing, we know the rate or value of any number of fuch things, however great.

But before the learner can begin this rule, it is abfolutely neceffary that he have the following table perfectly by heart:

The use of this table is, to find the product of any two numbers; thus, to find the product of 6 times 7, I look in the brace which has 6 at the point, and in the line which has 7 at the beginning, oppofite to which is the product, which is 42.

The table is to be thus read ;

Beginning with the first brace, I fay-2 times 2 is 4, 2 times 3 is 6, 2 times 4 is 8, 2 times 5 is 10, 2 times 6 is 12, 2 times 7 is 14, 2 times S is 16, 2 times 9 is 18, 2 times 10 is 20, 2 times 11 is 22, 2 times 12 is 24.

Then, proceeding in the fame manner, I begin with the fecond brace, faying, 3 times 3 is 9, 3 times 4 is 12, 3 times 5 is 15, 3 times 6 is 18, 3 times 7 is 21, 3 times 8 is 24, 3 times 9 is 27, 3 times 10 is 30, 3 times 11 is 33, 3 times 12

is 36.

Again, I begin with the third brace, saying, 4 times 4 is 16, 4 times 5 is 20, 4 times 6 is 24, 4 times 7 is 28, 4 times 8 is 32, 4 times 9 is 36, 4 times 10 is 40, 4 times 11 is 44, 4 times 12 is 48.

Then, I fay, 5 times 5 is 25, 5 times 6 is 30, 5 times 7 is 35, 5 times 8 is 40, 5 times 9 is 45, 5 times 10 is 50, 5 times. 11 is 55, 5 times 12 is 60,

Then, 6 times 6 is 36, 6 times 7 is 42, 6 times 8 is 48, 6 times 9 is 54, 6 times 10 is 60, 6 times 11 is 66, 6 times 12 is 72.

Then, 7 times 7 is 49, 7 times 8 is 56, 7 times 9 is 63, 7 times 10 is 70, 7 times 11 is 77, 7 times 12 is 84,

Then, 8 times 8 is 64, 8 times 9 is 72, 8 times to is 80, 8 times 11 is 88, 8 times 12 is 96.

Again, 9 times 9 is 81, 9 times 10 is 90, 9 times 11 is 99, 9 times 12 is 108,

Then, 10 times 10 is 100, 10 times is 110, 10 times 12 is 120.

And 11 times 11 is 121, 11 times 12 is 132.

Lastly, 12 times 12 is 144.

In multiplication there are three terms: viz.

The Multiplicand, 12 or fum to be multiplied;
Multiplier, ΤΟ

Product, 120 or refult of the whole.

Multiplication confifts of two parts: multiplication of numbers of one denomination, and multiplication of divers denominations.

Multiplication of numbers of one denomination is either fingle or compound.

Singlé multiplication is, when the multiplicand and mul tiplier confift, each of them, of only 12, or less than 12; and may be performed by one operation, and with one product. Hence the greatest product that can arife by single multiplication is 144, as that is the product of 12 times 12.

Compound multiplication is, when the multiplicand or multiplier, or both, confift of more than 12, and requires more than oue operation.

Rule. Place the multiplier under the multiplicand, in the natural order of figures, viz. units under units, &e.; then, if the multiplier confifts of 12 or less, multiply every figure of the multiplicand by the multiplier; beginning at the place of units, and placing the units of the product right under the units of the multiplier; carrying the tens, or tens and, hundreds, if there be any, to be added to the, next product. Proceed in this manner through the whole, and fet down the whole product of the laft figure, as in the annexed example; obferving, for every ten that is carried to the next product, an unit is to be added thereto; and for every hundred, ten is to be added.'

In this example, the multiplier being 12, I fay, 12 times 9 is 108, wherefore I fet down 8, the units of this product, and

carry

carry 10 to be added to the next product, faying, 12 times a, or 12 times nothing, is nothing; thus I have no product from this figure, wherefore I fet down the unit figure of the 10 I carried to be added to this product, which is o, and carry the remaining to be added to the next product; faying, 12 times 12 (as the two next figures in the multiplicand is 12) is 144, and the I carried makes 145; wherefore I fet it down, being the product of the last figures. And the whole produc of 1209, multiplied by 12, is 14508.

But, if the multiplier confift of feveral places, after having multiplied the multiplicand by the first, or two first figures, as before directed, multiply it in like manner by the next, of two next figures, if they be 12 or less, and in like manner by all the other figures; placing the products below each other, Strictly obferving that the product of each new multiplier is to have its unit placed exactly under the unit in fuch new multiplier, and the fubfequent figures in their regular order towards the left hand; and then the different products added together in the order in which they ftand; as in the following examples :

In this example, I begin with the first figure in the mul tiplier (as the two first are more than 12), faying, 4 times 2 is 2 times 4, that is 8, which I place in the first line of the product, and under 4 the multiplier; then + times 1 is 4, which I also set down in the next place; then 4 times o is o,

or nothing, wherefore I fet down o; next, 4 times 7 is 28, therefore I fay, 8 to be fet down, and carry 2; 4 times 9 is

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