5397 8423 9467 8528 9726 9453 8597 4867 5453 2847 9993 5728 6958 7273 8576 2398 7597 6385 5255 3446 8976 5843 7555 6827 7878 9247 Addition may be considered the principal, and, indeed, the most laborious operation of the Counting-house. The pupil, therefore, should, by all means, endeavour to render himself as skilful as possible in this most important operation. The old adage— practice makes perfect"—is certainly applicable here, if anywhere; and there can be little doubt that one of the very best methods to adopt is to repeat the same operation many times, writing the result each time, by which the student will not only clearly perceive his liability to err, but also the progress he is hourly making towards perfection. He should also frequently repeat each operation by taking the figures in an inverted or contrary order. The answers to these questions are omitted, because the insertion of them might induce carelessness in the scholar, and because the teacher, having once seen them, will, by a mere glance of the eye, determine whether the work offered for inspection is correct or not; and the scholar should not present his work till he has found the same result by two consecutive additions performed in a contrary order. 9785 8993 6769 2856 5537 4689 3898 7965 8796 9365 5597 4468 6257 3889 7989 6985 7397 8836 9267 2745 3498 9678 6554 4367 5879 7983 8235 9396 9927 7869 73466 58529 47988 53547 88622 33664 82573 77648 92595 74929 44972 62833 25297 34569 87644 55827 33947 62352 86579 23745 87966 25348 85462 92999 55774 86258 23644 37337 24685 94927 39624 57863 85647 92958 83459 27766 88342 27923 99424 55277 83922 95986 24654 83756 77732 83988 65689 49786 32246 85698 97235 44468 52392 84675 39375 47892 35673 43939 56599 99877 52567 89237 45678 92846 56375 28426 88777 33637 89975 49638 27946 83577 25842 89764 55844 97358 64643 77396 99884 59847 32376 99333 48877 55634 84424 58957 89752 25846 39575 88647 SECTION IV. SUBTRACTION. 75. THE word Subtraction is derived from the Latin words sub, out or away, and trahere, to draw, to take; and, therefore, signifies to draw out or take away: 76. There can be no difference between two equals. Also, if the whole of a thing be taken away, there can be no remain der. Subtraction, therefore, as an arithmetical operation, is the method of finding the difference between two unequal numbers, by arranging their orders so as to diminish the greater by as many units as are contained in the less. The remaining part of the greater number is, consequently, called remainder or difference. 77. The greater number, then, is composed of the remainder and the less number; consequently, to prove subtraction, we add the remainder and less number together, the sum of which equals the greater, when the work is right. 78. If we take either of two numbers from their sum, the remainder will be the other. Wherefore, when the work is right, if we subtract the remainder from the greater number, we shall obtain the less. 79. Of two unequal numbers, let 19 be the greater and 10 the less. From 19, to subtract 10, we take away the unit 1, which is ten, and the remainder is 9. If to this remainder 9 we add the less number 10, we have the greater number 19: and this is done by replacing the unit on the left of 9. Again: if from 19 we take the remainder 9, we have the less number 10. Hence, we see that to subtract 10 from any of the numbers 11, 12, 13, &c., 19, is merely to take away the unit in the place of tens. Also, to add 10 to any of the digits 1, 2, 3, &c., 9, is merely to place a unit on the left of it. 80. As the sign minus --- represents a simple subtraction, so the sign ; represents a continual subtraction, which divides the number from which we subtract, into equal parts. In both cases—that is, when either of these signs is placed between two numbers--it is always the number on the right of the sign which is to be subtracted from that on the left. 9 3=6. Here, we say: 3 from 9, six; and, in 9 versely, (78) 6 from 9, three. Again: 9 ; :3 or 3 9 - 3 -- 3, &c., as often as we can. Thus, 9 3 6; 6 3=3, and 3 3=0. Where, having made three subtractions and arrived at 0, we say that 9 contains 3 just 3 times; that is, when 9 is divided into equal parts of 3 units each, the three subtractions show there are 3 of those parts; and this number 3 is thence called the quotient, (quotiès, Lat., how often?) 81. The frequent repetition or conning of the following Table will greatly facilitate the progress of the learner. In Thus, each equation, where the remainder and less number are not alike, first subtract the less number and afterwards the remainder. Thus, 2 from 5, three; 3 from 5, two; 2 from 6, four; 4 from 6, two; 2 from 7, five; 5 from 7, two, &c., throughout. This will make the Table, as a preparatory exercise, more complete. Subtraction Table. 4-2 2 6 3 3 8 -4 -4 10 — 5=5 5 2 3 3 4 9 4 5 11 5 6 6 2 8 3 5 10 4 6 5 -7 7 2 5 9 3 6 11 4 -7 13 5 = 8 8 2=6 10 3=7 12 —4=8 14 5 9 9 2=7 11 -3=8 13 -9 10 -2=8 12 3 :9 12 6=6 11 - 2=9 14-T=7 13 — 6=7 16 —8=8 15 -7=8 14 – 6=8 189=9 17 .8 9 16 -7 = 9 15 6 9 82. Of the two unequal numbers, whose difference is required, the less may be any number in the natural scale, from 1, to the greater minus 1. Now, if the numbers consist of several orders, and we require the subtraction of a unit of any order towards the left, -as the value of this (37) is greater than that of all the figures which could stand on the right of it, and (46) ten times less than the unit of the lowest order on the left, the subtraction of it would be impossible in any order, other than that to which it belongs. Hence the necessity, in arranging the numbers, for placing units under units, tens under tens, &c., which is done, as in Addition, by placing the units under units, and the other figures in regular order towards the left. 83. The numbers 10, 100, 1000, &c., being (37) greater than 9, 99, 999, &c. &c., it is evident that in arranging, as we do, the less number under the greater, so that units may be under units, tens under tens, &c., all the figures of the less number may be greater than the corresponding figures of the greater. Also that all the orders of the greater, from which we are required to subtract, may be wanting. But, in either case, the greater number must have, at least, one unit of an order on the left of the highest order of the less number; and, as this (37) is greater than all the figures which can stand on the right of it,--and, consequently, than the whole of the less number,--we can, from the value of this unit, form all the orders which are wanting in the greater number, and subtract those of the less number from them. 84. If, to each of two unequal numbers, we add the same number, this addition has no effect upon their difference. Because, it is evident that, by adding to the greater number, we increase the difference; and by adding to the less, we diminish it. Wherefore, as these two opposites (60) destroy the effect of each other, the difference is still the same. 85. When the lower figure is greater than the one above it, and, consequently, its subtraction impossible, we add ten to the upper figure; that is, (79,) we read it as if it had a unit on the left. From this we subtract the lower figure, and write the remainder underneath. Then, proceeding towards the left, we add a unit to the next lower figure; or, rather, we read it one unit greater than it is, and subtract as before. By this process, the same number is added to each of the given numbers ; because (36) the ten which we add to the upper figure, is equal to the unit which we add to the next lower one; consequently, the difference (84) is not affected. Examples. 4520376 Greater. 831497 Less. 3688879 Remainder. Proof by Addition, 4520376 Gr. (sum of Rem. and Less.) In this example, the lower figures being all greater than those above them, we add ten to each of the upper figures, except 4. We, therefore, read them in succession, beginning at the right, 16, 17, 13, 10, 12, 15. Also, in consequence, we add one to each of the lower figures 9, 4, 1, 3, 8, which we read successively, always going from right to left, 10, 5, 2, 4, 9. The enunciation, then, is: 7 from 16, nine; 10 from 17, Xuven; 5 from 13, eight; 2 from 10, eight; 4 from 12, eight; 9 from 15, six; lastly, as there is no figure under 4, 1 from 4, three. 86. When some of the lower figures are greater than those ubove them, and some less, we must not add a unit to the next lowest figure, when we have not added ten to the preceding |