2. Begin at the right hand, and multiply the whole multiplicand severally by each figure in the multiplier, setting down the first figure of every line directly under the fig ure tens; the product must be ten times its simple value; and there fore the first figure of this product must be placed in the place of tens; or, which is the same thing, directly under the figare we are multiplying 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 these several products being added together will be equal to the whole required product. Q. E. D. The reason of the method of proof depends upon this proposition, “that if two numbers are to be multiplied together, either of them may be made the multiplier, or the multiplicand, and the product will be the same." A small attention to the nature of numbers will make this truth evident: for 3×7=21=7X3; and in general 3×4×5×6, &c. =4X3X6X5, &c. without any regard to the order of the terms: and this is true of any number of factors whatever. The following examples are subjoined to make the reason of the rale appear as plain as possible. Beside the preceding method of proof, there is another very convenient and easy one by the help of that peculiar property o the number 9, mentioned in addition; which is performed thus: RULE ure you are multiplying by, and carrying for the tens, as in addition. 3. Add all the lines together, and their sum is the product. Method RULE I. Cast the nines out of the two factors, as in addition, and set down the remainder. 2. Multiply the two remainders together, and if the excess of nines in their product is equal to the excess of nines in the total product, the answer is right. EXAMPLE. 4215 3 excess of 9's in the multiplicand. 33720 29505 33720 3700770 6 ditto in the product excess of 9's in 3 X 5. DEMONSTRATION OF THE RULE. Let M and N be the number of 9's in the factors to be multiplied, and a and b what remains then Ma and N+b will be the numbers themselves, and their product is MXN+M×b+Nxa+axb; but the three first of these products are each a precise number of 9's, because one of their factors is so therefore, these being cast away, there remains only axb; and if the 9's are also 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 axb their product; therefore the rule is true. Q. E. D. This method is liable to the same inconvenience with that in addition. Multiplication may also, very naturally, be proved by division for the product being divided by either of the factors, will evidently give the other; but it would have been contrary to good method to have given this rule in the text, because the pupil is supposed, as yet, to be unacquainted with division. 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. 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 number 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 |