36, and 2 that I carried is nothing, wherefore I have carried; again, 4 times

38, 8 and carry 3; 4 times o is nothing to fet down but the 3 I is 4: thus I have done with the first figure in the multiplier. Beginning with the next, I fay, 9 times 2, or 2 times 9, is 18, 8 and carry 1; the 8 is to be fet exactly under the multiplier 9; then 9 times 1 is 9, and i I carried is 10, 0 and carry 1; 9 times o is o, wherefore I fet down the 1 I carried; again, 9 times 7 is 63, 3 and carry 6; 9 times 9 is 81, and 6 is 87, 7 and carry 8; 9 times o is 0, but 8 was carried; 9 times I is 9: thus I have done with the second multiplier. The next and last two figures in the multiplier being 12, I multiply by them as by one figure, placing the unit figure of the first product under that of the multiplier, which is the 2; faying, 12 times 2, or 2 times 12, is 24, 4 and carry 2; 12 times 1 is 12, and 2 is 14, 4 and

carry 1; 12 times o is o, but I was carried; 12 times 7 is 84, 4 and carry 8; 12 times 9 is 108, and 8 is 116, 6 and carry 11; 12 times o is o, but i was carried, therefore I fet down 1, and carry 1; 12 times 1 is 12, and 1 is 13; which being the laft, I fet it down. The work being finished, all the feveral products are added together, and the total is the real product.

The work is proved by dividing the multiplier in half, and multiplying the original multiplicand by one half, and thẹ product thence arifing by the number 2, as feen in the foregoing example. But if the multiplier cannot be divided exactly in half, as is the cafe when it confifts of an odd number, then a number that is an unit less than the multiplier is to be divided in half; and the multiplicand multiplied by the one half, and that product by 2, as before, and the original multiplicand added to the last product, as in the following xamples ;

In proving this example, I cannot divide the multiplier 4321 exactly in half, because it contains an odd number in the place of units; I therefore take a number that is an unit lefs, viz. 4320, and divide it in half thus: faying, the balf of four thousand is two thousand, and the half of three hundred and twenty is one hundred and fixty; which, together, is 2160 for the first multiplier in the proof; the product of which is again multiplied by 2, and that product added to the first multiplicand, which gives a product equal to the product in the example: which proves the work right.

When the multiplier has cyphers intermixed with the other figures, the fignificant figures only are to be regarded as multipliers, obferving the directions before given, to place the unit figure of each product under that of the multiplier; as in the following examples :→

When the multiplier, or multiplicand, or both, confist of

9

and that I carried is 38, 8 and carry 3; 4 times o is matting, wherefore I have nothing to fet down but the 3 I ormed: again, imesis 4: thes I have done with the frf figure in the mailer. Beginning with the next, I fay, #times 1, or 2 times g. is 18, 8 and carry 1; the 8 is to be fet emely under the multiplier 9; then 9 times 1 is 9, and i I carried is co, o and carry 1; 9 times o is o, wherefore I fet down the : I carried; again, 9 times 7 is 63, 3 and carry 6; times gis 81, and 6 is 87, 7 and carry 8; 9 times ois o, bue 8 was carried; 9 times 1 is 9: thus I have done with the fecond multiplier. The next and last two figures in the multiplier being 12, I multiply by them as by one figure, placing the unit figure of the firft product under that of the multiplier, which is the 2; faying, 12 times 2, or 2 times 12, is 24, 4 and carry 2; 12 times 1 is 12, and- 2 is 14, 4 and carry 1; 12 times o iso, but I was carried; 12 times 7 is 84 4 and carry 8; 12 times 9 is 108, and 8 is 116, 6 and carry 11; 12 times o is o, but 11 was carried, therefore I fet down 1, and carry 1; 12 times 1 is 12, and 1 is 13; which being the laft, I fet it down. The work being finished, all the feveral products are added together, and the total is the real product.

The work is proved by dividing the multiplier in half, and multiplying the original multiplicand by one half, and the product thence arifing by the number 2, as feen in the foregoing example. But if the multiplier cannot be divided exactly in half, as is the case when it confists of an odd number, then a number that is an unit less than the multiplier is to be divided in half; and the multiplicand multiplied by the one half, and that product by 2, as before, and the ori 1 the original multiplicand added to the laft product, as in the following xamples ;

In proving this example, I cannot divide the multiplier 4321 exactly in half, because it contains an odd number in the place of units; I therefore take a number that is an unit lefs, viz. 4320, and divide it in half thus: faying, the balf of four thousand is two thousand, and the half of three hundred and twenty is one hundred and fixty; which, together, is 2160 for the first multiplier in the proof; the product of which is again multiplied by 2, and that product added to the first multiplicand, which gives a product equal to the product in the example: which proves the work right.

When the multiplier has cyphers intermixed with the other figures, the fignificant figures only are to be regarded as multipliers, obferving the directions before given, to place the unit figure of each product under that of the multiplier; as in the following examples:→→

cyphers towards the right hand, and fignificant figures towards the left, fuch cyphers are omitted in the operation, and placed on the right hand of the product; as follows:

In the proof of the laft example, the multiplicand is added to the product as before directed, when the original multiplier cannot be divided exactly in half.

Multiplication of divers Denominations.

Multiplication of divers denominations is performed by multiplying each denomination by the multiplier; beginning with the leaft denomination, and carrying the units of the next denomination to be added thereto, as in addition of money.

Multiplication of divers denominations is either fingle or compound: fingle, when the multiplier confifts of 12 or lefs, and is performed by one multiplier; compound, when the multiplier confifts of more than 12, and requires more than one multiplier.