setting down a line of products for each figure in the multiplier, so as that the first figure of each line may stand straight under the figure multiplying by.-Add all the lines of products together, in the order as they stand, and their sum will be the answer or whole product required. TO PROVE MULTIPLICATION. THERE are three different ways of proving Multiplication, which are as below: First Method. Make the multiplicand and multiplier change places, and multiply the latter by the former in the same manner as before. Then if the product found in this way be the same as the former, the number is right. Second Method.-*Cast all the 9's out of the sum of the figures in each of the two factors, as in Addition, and set down the remainders. Multiply these two remainders together, and cast the 9's out of the product, as also out of 1234567 the multiplicand. is the same thing, directly under the figure multiplied by. And proceeding in this manner separately with all the figures of the multiplier, it is evident that we shall multiply all the parts of the multiplicand by all the parts of the multiplier, or the whole of the multiplicand by the whole of the multiplier: therefore 4938268 these several products being 86419697407402 6172835 7 times the mult. added together, will be equal 5638267489-4567 times ditto. to the whole required product; as in the example an nexed. * This method of proof is derived from the peculiar property of the number 9, mentioned in the proof of Addition, and the reason for the one may serve for that of the other. Another more ample demonstration of this rule may be as follows:-Let P and Q denote the number of 9's in the factors to be multiplied, and a and b what remain; then 9 P+a and 9 Q+ will be the numbers themselves, and their product is (9 PX9Q) + (9 P × 6) + (9 Q × a) + (a × 6); but the first three of these products are each a precise number of 9's, because their factors are so, either one or both: these therefore being cast away, there remains only a Xb; and if the 9's also be cast out of this, the excess is the excess of 9's in the total product: but a and b are the excesses in the factors themselves, and a X 6 is their product; therefore the rule is true. the the whole product or answer of the question, reserving the remainders of these last two, which remainders must be equal when the work is right.-Note, It is common to set the four remainders within the four angular spaces of a cross, as in the example below. Third Method.-Multiplication is also very naturally proved by Division; for the product divided by either of the factors, will evidently give the other. But this cannot be practised till the rule of Division is learned. CONTRACTIONs in MultiplICATION. I. When there are Ciphers in the Factors. If the ciphers be at the right-hand of the numbers; multiply the other figures only, and annex as many ciphers to the right-hand of the whole product, as are in both the factors. When the ciphers are in the middle parts of the mul tiplier; neglect them as before, only taking care to place the first figure of every line of products exactly under the figure multiplying with. II. When the multiplier is the Product of two or more Numbers in the Table; then Multiply by each of those parts separately, instead of the whole number at once. EXAMPLES. 1. Multiply 51307298 by 56, or 7 times 8. 51307298 7 359151086 8 2873208688 The reason of this rule is obvious enough; for any number multiplied by the component parts of another, must give the same product as if it were multiplied by that number at once. Thus, in the 1st example, 7 times the product of 8 by he given number, makes 56 times the same number, as plainly as 7 times 8 makes 56. D VOL. I. 2. Mul 2. Multiply 31704592 by 36. 3. Multiply 29753804 by 72. 4. Multiply 7128368 by 96. Ans. 1141365312. Ans. 2142273888. Ans. 684323328. 5. Multiply 160430800 by 108. Ans. 17326526400. 6. Multiply 61835720 by 1320. Ans. 81623150400. 7. There was an army composed of 104 * battalions, each consisting of 500 men; what was the number of men contained in the whole? Ans. 52000. 8. A convoy of ammunition † bread, consisting of 250 waggons, and each waggon containing 320 loaves, having been intercepted and taken by the enemy; what is the humber of loaves lost? Ans. 80000. OF DIVISION. DIVISION is a kind of compendious method of Subtraction, teaching to find how often one number is contained in another, or may be taken out of it: which is the same thing. The number to be divided is called the Dividend.The number to divide by, is the Divisor.-And the number of times the dividend contains the divisor, is called the Quotient. Sometimes there is a Remainder left, after the division is finished. The usual manner of placing the terms, is, the dividend in the middle, having the divisor on the left hand, and the quotient on the right, each separated by a curve line; as, to divide 12 by 4, the quotient is 3, Dividend Divisor 4) Rule.-Having placed the divisor before the dividend, as above directed, find how often the divisor is contained in as many figures of the dividend as are just necessary, and place the number on the right in the quotient. 12 4 subtr. 8 4 subtr. 4 4 subtr. Mul A battalion is a body of foot, consisting of 500, or 600, or 700 men, more or less. The ammunition bread, is that which is provided for, and distributed to, the soldiers; the usual allowance being a loaf of 6 pounds to every soldier, once in 4 days. In this way the dividend is resolved into parts, and by trial is found Multiply the divisor by this number, and set the product under the figures of the dividend before-mentioned.-Subtract this product from that part of the dividend under which it stands, and bring down the next figure of the dividend, or more if necessary, to join on the right of the remainder.—Divide this number, so increased, in the same manner as before; and so on till all the figures are brought down and used. N. B. If it be necessary to bring down more figures than one to any remainder, in order to make it as large as the divisor, or larger, a cipher must be set in the quotient for every figure so brought down more than one. TO PROVE DIVISION. * MULTIPLY the quotient by the divisor; to this product add the remainder, if there be any; then the sum will be equal to the dividend when the work is right. found how often the divisor is contained in each of those parts, one after another, arranging the several figures of the quotient one after another, into one number. When there is no remainder to a division, the quotient is the whole and perfect answer to the question. But when there is a remainder, it goes so much towards another time, as it approaches to the divisor; so, if the remainder be half the divisor, it will go the half of a time more; if the 4th part of the divisor, it will go one fourth of a time more; and so on. Therefore, to complete the quotient, set the remainder at the end of it, above a small line, and the divisor below it, thus forming a fractional part of the whole quotient. This method of proof is plain enough for since the quotient is the number of times the dividend contains the divisor, the quotient multiplied by the divisor must evidently be equal to the dividend. There are also several other methods sometimes used for proving Division, some of the most useful of which are as follow: Second Method-Subtract the remainder from the dividend; and divide what is left by the quotient; so shall the new quotient from this last division be equal to the former divisor, when the work is right. Third Method-Add together the remainder and all the products of the several quotient figures by the divisor, according to the order in which they stand in the work; and the sum will be equal to the dividend when the work is right, |