course of the learner, by any thing not quite essential to his progress, but chiefly have I reserved it until now, when, having passed through another, Rule, he may refresh his mind, and confirm his knowledge of Addition, by returning to the few sums set down for him in that rule; by working them over again; which I recommend to him to do; and by further improving himself, by practising the mode of proof which I have here described. 64. As to the method of PROOF in Subtraction, it is simply this. Having completed your sum, you have three lines of figures; that is to say, first, the the larger sum; second, the sum you are to subtract therefrom; and, third, the remainder, or difference. Now, on a moment's consideration you will perceive, that the two latter sums must always, when added together, exactly amount to the larger sum: twenty, for example, is the larger sum, from which you have subtracted five, and there is a remainder of fifteen : the two latter make up the former sum. And thus it always will be. So, that, to put any process of Subtraction to the proof, all you have to do is, to draw a line under the remainder, and add it and the next sum together, and then your fourth line of figures ought to be same as your first. MULTIPLICATION. 65. This name, too, comes from the Latin, And the meaning of it is so familiar, that a definition is scarcely called for. But, as a definition, if duly attended to, will forward us in the knowledge of our art, we must not withhold it. 66. To multiply, as you know, is to increase; and as you likewise know, MULTIPLICATION is, the act of increasing. But so, also, is ADDITION the act of increasing. Let us, then, mark the difference. 67. Addition, as defined in paragraph 31, is the art of joining together numbers, or sums of various amounts; whereas, Multiplication increases by a joining together of the same sum, several times repeated: as, twice 4 are 8; 4 times 4 are 16; 6 times 12 are 72, and so on. This joining together of the same amount, several times repeated, may, of course, be accomplished by Addition; first writing down the figures in a sort of column, in the manner directed in Addition, paragraph 34. But the end is attained much more quickly, more pleasantly, and with less liability to error, by the Rule we are now treating of; and herein consists its value. 68. In this Multiplication, two sums are multiplied together, and they produce a third: now, these three sums have each an appropriate name, with which you will find it useful to be familiarly acquainted. When we say, sums are multiplied together, or, that one is multiplied by the other; we mean the same thing. And when two sums are to be multiplied, it is immaterial, as to the result, whether we multiply by one, or by the other, of them: as, for instance, we have to multiply 6 and 12 together, the result is the same, whether we say six times twelve, or twelve times six. And so it would be with sums of any amount. 69. But, though the result is the same, it is more easy to multiply the larger sum by the smaller; and this, as you will soon find, is more especially the case, when either of the sums has several figures in it. 70. Having ascertained, then, that it is better, because it is easier, to multiply by the smaller sum, when two sums are to be multiplied together, we call the smaller one, the multiplier, whilst the larger, we call the multiplicand, and the sum produced by the process, is termed, the product; and these are the terms used in this rule of Multiplication: namely, MULTIPLICAND, MULTIPLIER, and PRODUCT. 71. However, before we proceed further with this rule, it will be well to have the very useful Table which has been constructed for the purpose of enabling us, with ease and correctness, to work sums in it. This Table, it is customary for learners to commit to memory; and this is but a trifling task, when compared with its great value, in facilitating the operations of Multiplication, and, indeed, of almost all operations in Arithmetic. To learn the Table, to get it off by rote, to make it familiar on the tongue, and as it were, a part of the mind, so that, when we would multiply any one figure by another, the product shall, instantly, and without the trouble of thinking about it, rise in the mind, be breathed from the lips, and, if pen or pencil be held, traced by the fingers; this is what ought to be done; and thus familiar may the Table be made, with but a very small degree of labour, indeed, compared with that which it would save to every one who has any thing to do with Arithmetic. 72. Then, too, there is the certainty, in our reckonings of all sorts, which, without having this Table in us, there are scarcely any other means of attaining. In short, it is profitable, pleasant, and I may add, indispensible, thus to have it. And, as there is not any other thing in Arithmetic, which demands from the learner so irksome an employment, as this of getting off by rote, so, I hope he will not demur thus to learn this Table: the very best mode of doing which, at the same time that it will be improving to him in writing figures, is, not to say it merely with the lips and tongue, but to write the figures down, as he speaks them; and, to do this in as neat and as regular a manner as possible, thereby training up, and enlisting in the useful office, the hand and the eye, as well as the voice. And, 73. The above I deem the more intelligible form of arrangement for the Table, and, therefore, I give it the precedence. There is, however, another form as some teachers carry on the reckoning so high as twenty times twenty, I will do so in this other Table; but, without urging on the learner the task of in use, somewhat more compact than this. getting it higher than twelve times twelve. 2 3 4 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 23456 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 96 102 108 114 120 98 105 112 119 126 133 140 88.96 104 112 120 128 136 144 152 160 99 108 117 126 135 144 153 162 171 180 66 72 78 84 90 77 84 91 |