2. Begin at the right hand, and take each figure in the lower line from the figure above it, and set down the remainder. 3. If the lower figure is greater than that above it, add ten to the upper figure; from which figure, so increased, take the lower, and set down the remainder, carrying one to the next lower figure; with which proceed as before, and so on till the whole is finished. Method of PROOF. Add the remainder to the less number, and if the sum is equal to the greater, the work is right. From 3287625 From 5327467 Remaind. 943869 Remain. 4319029 2. When any figure of the greater number is less than its correspondent figure in the less, the ten, which is added by the rule, is the value of an unit in the next higher place, by the nature of notation; and the one that is added to the next place of the less number is to diminish the correspondent place of the greater accordingly; which is only taking from one place and adding as much to another, whereby the total is never changed. And by this means the greater number is resolved into such parts, as are each greater than, or equal to, the similar parts of the less: and the difference of the corresponding figures, taken together, will evidently make up the difference of the whole. Q. E. D. The truth of the method of proof is evident: for the difference of two numbers, added to the less, is manifeftly equal to the greater. 4. From 2637804 take 2376982. 5. From 3762162 take 826541. 6. From 78213606 take 27821890. Ans. 260822. Ans. 2935621. Ans. 50391716. 7. The Arabian method of notation was first known in England about the year 1150: how long to the year 1776? was it thence Ans. 626 years. 8. Sir Isaac Newton was born in the year 1642, and died in 1727: how old was he at the time of his decease ? Ans. 85 years. SIMPLE MULTIPLICATION, Simple Multiplication is a compendious method of addition, and teacheth to find the amount of any given number of one denomination, by repeating it any proposed number of times. The number to be multiplied is called the multiplicand. The number you multiply by is called the multiplier. The number found from the operation is called the product. Both the multiplier and multiplicand are, in general, called terms or factors. MULTIPLICATION AND DIVISION TABLE. 66| 72 77| 84 88| 96 20 25 30 35 40 6|12|18|24| 30 | 36|42|48| 54| 60 | 1212436486072 84|96| 108 | 120 | 132 | 144 1 USE USE of the Table in MULTIPLICATION. Find the multiplier in the left-hand column, and the multiplicand in the uppermost line; and the product is in the common angle of meeting, or against the multiplier, and under the multiplicand. Use of the Table in DIVISION. Find the divisor in the left-hand column, and the dividend in the same line; then the quotient will be over the dividend, at the top of the column. RULE.* 1. Place the multiplier under the multiplicand, so that units may stand under units, tens under tens, &c. and draw a line under them. 2. Begin * DEMON. I. When the multiplier is a single digit, it is plain that we find the product; for by multiplying every figure, that is, every part of the multiplicand, we multiply the whole; and writing down the products that are less than ten, or the excess of tens, in the places of the figures multiplied, and carrying the number of tens to the product of the next place, is only gathering together the similar parts of the respective products, and is, therefore, the same thing, in effect, as though we wrote down the multiplicand as often as the multiplier expresses, and added them together : for the sum of every column is the product of the figures in the place of that column; and these products, collected together, are evidently equal to the whole required product. 2. If the multiplier is a number made up of more than one digit. After we have found the product of the multiplicand by the first figure of the multiplier, as above, we suppose the multiplier divided into parts, and find, after the same manner, the product of the multiplicand by the second figure of the multiplier; but as the figure we are multiplying by stands in the place of 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 tens; 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 figure 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=7×3; and in general 3×4×5×6, &c. =4×3×6×5, &c. without any regard to the order of the terms: and this is true of any number of factors whatever. 1 The following examples are subjoined to make the reason of the rule 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 of 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 X5. 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+6 will be the numbers themselves, and their product is MXN+Mxb+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. 1 |