Method of PROOF. Make the former multiplicand the multiplier, and the multiplier the multiplicand, and proceed as before; and if this product is equal to the former, the product is right. I. When there are cyphers to the right hand of one or both the numbers to be multiplied. RULE. Proceed as before, neglecting the cyphers, and to the right hand of the product place as many cyphers as are in both the numbers. EXAMPLES. 3. Multiply 815036000 by 70306. Ans. 57297030800000. II. When the multiplier is the product of two or more num bers in the table. RULE.* Multiply continually by those parts, instead of the whole number at once. EXAMPLES. 1. Multiply 123456789 by 25. 123456789 5 617283945 5 3086419725 the Product. 2. Multiply * The reason of this method is obvious; for any number multiplied by the component parts of another number must give the same product, as though it were multiplied by that sumber at once: thus in example the second, 7 times the product of 8, multiplied into the given number, makes 56 times that given number, as plainly as 7 times 8 makes 56. D Simple Division teacheth to find how often one number is contained in another of the same denomination, and thereby performs the work of many subtractions. The number to be divided is called the dividend. The number you divide by is called the divisor. The number of times the dividend contains the divisor is called the quotient. If the dividend contains the divisor any number of times, and some part or parts over, those parts are called the remainder. RULE.* 1. On the right and left of the dividend, draw a curved line, and write the divisor on the left hand, and the quotient, as it arises, on the right. 2. Find * According to the rule, we resolve the dividend into parts, and find, by trial, the number of times the divisor is contained in each of those parts; the only thing then, which remains to be proved, is, that the several figures of the quotient, taken as one number, according to the order in which they are placed, is the true quotient of the whole dividend by the divisor; which may be thus demonstrated : DEMON. The complete value of the first part of the dividend, is, by the nature of notation, 10, 100, or 1000, &c. times the 2. Find how many times the divisor may be had in as many figures of the dividend, as are just necessary, and write the number in the quotient. 3. Multiply the divisor by the quotient figure, and set the product under that part of the dividend used. 4. Subtract value of which it is taken in the operation; according as there are 1, 2, or 3, &c. figures standing before it; and consequently the true value of the quotient figure, belonging to that part of the dividend, is also 10, 100, or 1000, &c. times its simple value. But the true value of the quotient figure, belonging to that part of the dividend, found by the rule, is also 10, 100, or 1000, &c. times its simple value : for there are as many figures set before it, as the number of remaining figures in the dividend. Therefore this first quotient figure, taken in its com. plete value, from the place it stands in, is the true quotient of the divisor in the complete value of the first part of the dividend. For the same reason, all the rest of the figures of the quotient, taken according to their places, are each the true quotient of the divisor, in the complete value of the several parts of the dividend, belonging to each; because, as the first figure on the right hand of each succeeding part of the dividend has a less number of figures, by one standing before it, so ought their quotients to have; and so they are actually ordered : consequently, taking all the quotient figures in order as they are placed by the rule, they make one number, which is equal to the sum of the true quotients of all the several parts of the dividend; and is, therefore, the true quotient of the whole dividend by the divisor. Q. E. D. To leave no obscurity in this demonstration, I shall illustrate it by an example. EXAMPLE. 4. Subtract the last found product from that part of the dividend, under which it stands, and to the right hand of the remainder bring down the next figure of EXPLANATION. It is evident, that the dividend is resolved into these parts, 85000+600+00+9: for the first part of the dividend is considered only as 85, but yet it is truly 85000; and therefore its quotient, instead of 2, is 2000, and the remainder 13000; and so of the rest, as may be seen in the operation. When there is no remainder to a division, the quotient is the absolute and perfect answer to the question; but where there is a remainder, it may be observed, that it goes so much toward another time, as it approaches to the divisor: thus, if the remainder be a fourth part of the divisor, it will go one fourth of a time more; if half the divisor, it will go half of a time more; and so on, |