this method would evidently be insupportably tedious, burdensome to the memory, and also much exposed to error. We have, therefore, recourse to another, founded on the principle of notation, and that the sum of several wholes is equal to the sums of their several parts. Ex. 1. Required the sum of 5683 and 4213. 5683 4213 Arranging these numbers, so that units of the same order may stand under one another; we have 3 units and 3 units, which are together equal to 6 units; 1 ten and 8 tens, equal to 9 tens; 2 hundreds and 6 hundreds, equal to 8 hundreds; 4 thousands and 5 thousands, equal to 9 thousands. Hence the sum of the two given numbers is 9 thousands, 8 hundreds, 9 tens, and 6 units. 9896 7845 6238 507 59 Ex. 2. Required the sum of 7845, 6238, 507, and 59. The sum of the units of the first order is 29, which (No. 15.) are equal to 2 units of the second order, and 9 of the first. Writing these 9 units of the first order, and carrying the 2 units of the second order to the next column, we there find the sum of this second order of units to be 14, which are equal to 4 units of the second order, and 1 of the third. Writing the 4, and adding the 1 to the sum of the units in the 14649 next column, or that of the third order of units, we obtain 16, which, in like manner, is equal to 6 units of the third order, and 1 of the fourth. This unit of the fourth order being added to the sum of the units in its proper column, gives 14, which is equal to 4 units of the fourth order, and 1 of the fifth. In the same way may the process be continued throughout any number of columns. Hence the sum of the given numbers is expressed by one unit of the fifth order, four of the fourth, six of the third, four of the second, and nine of the first. Same example by the Abacus, where the large counter denotes 5, or half the index of the scale. The sum of the counters on the first bar is twenty-nine, which are equivalent to nine on the first, and two on the second. The sum of the counters on the second bar is already twelve, which, together with the two carried, makes fourteen: these, again, an equivalent to four on the second, and one on the third. This one on the third bar, together with those already there, makes sixteen, which are equivalent to six on the third, and one on the fourth. The number of counters on the fourth bar, with the addition of the one carried to it, will be fourteen, which are equivalent to one on the fifth bar, and four on the fourth. Hence the sum of the given numbers is expressed on the Abacus, by one counter on the fifth bar, four on the fourth, six on the third, four on the second, and nine on the first. As our limits will not admit of our introducing Palpable Arithmetic through the various rules, we would refer such as would see more of it, to Professor Leslie's Philosophy of Arithmetic. We shall here give an example or two in Addition, on some of the other scales, but afterwards confine ourselves exclusively to the Denary scale. In the example in the Quinary scale, one is carried for every five units that are in any column. In the other, on the Octary scale, one is carried for every eight units that are contained in any column. 18. There are various methods by which Addition may be proved. One is to repeat the operation, beginning at the top instead of the bottom. Another is to add all the numbers, except the uppermost, and to observe whether this sum, when added to the uppermost number, be equal to the total sum formerly found. If the last result agree with the first, the presumption is, that the work is correct; but still the proof is not absolutely certain, because an error in the first result, may have been concealed by an error in a subsequent one. SUBTRACTION. 18. Subtraction is that numerical process by which we discover the difference of two numbers; that is, how much the greater is greater than the less; or, how much the less is less than the greater. From this definition, it is obvious, that the difference of two numbers may be found, either by observing how many units it is necessary to take from the greater, in order that the less may remain; or, how many to add to the less, in order to obtain the greater. Thus, for example, in seeking the difference of 9 units and 4 units, we observe that unity must be taken away five times from 9, that 4 may remain; or, that it must be added five times to 4, in order to obtain 9; thus showing, that 9 exceeds 4 by 5; or, that 4 is less than 9 by 5; that is, in short, that the difference of the two numbers 9 and 4 is 5. This method alone, like the analogous one in addition, (No. 17.) would be very tedious, if the numbers were large, but by the assistance of a few obvious principles, it may be much abridged. The principles employed are, 1st, That of notation; 2d, That the difference of two wholes is equal to the sum of the differences of their several parts; and, 3d, That the difference of two numbers will remain the same, if they are equally increased. For the sake of distinction, the greater of the two numbers is called the Minuend, and the less the Subtrahend. Ex. 1. Required the difference of 7864 and 5321. Placing the subtrahend under the minuend, so that units of the same order may stand under one another, and then subtracting, the result will be two units of the fourth order, five of the third, four of the second, and three of the first. This is so plain, as to require no explanation. 7864 5321 2543 83 9345 6518 Ex. 2. Required the difference of 9345 and 6518. Here, as the simple units of the subtrahend cannot be taken from those of the minuend, in consequence of the former exceeding the latter, we take one unit of the second order, from the four units of the second order that stand in the minuend; this being equal to ten units of the first order, (No. 15.) will, with the five units, make fifteen units; from which, if the eight units of the subtrahend be taken, there will remain seven. Now, as the four units of the second order in the minuend, have thus 2827 been diminished by one, three will remain; from which, if the one in the subtrahend be taken, two will be left. Again, as the units of the third order in the subtrahend, exceed those in the minuend, we, in like manner, in order to effect the subtraction, take one unit from the nine of the fourth order, which unit, being equal to ten of the third order, will, with the three, make thirteen units of this order; now, if from these the five units of the subtrahend be taken, there will remain eight. Lastly, in consequence of one unit being taken from the nine of the fourth order, eight will remain, from which, if six be taken, two will be left. In the same way may the process be continued throughout any number of columns. 6518 19. This mode of procedure, which is certainly the most natural, is adopted by the Continental Arithmeticians, but differs a little from that employed in Britain. To take the same example as before. Here (in order to effect the subtraction) we add ten units of the first order to the 9345 five of the minuend, and from this sum deduct eight, and seven remains. Now, in order that the difference may remain the same, we must also increase the sub- 2827 trahend by ten units of the first order, or, which will amount to the same thing, by one of the second. This being done, we shall have two units of the second order to subtract from four, which will leave a remainder of two. Adding ten units of the third order, in like manner, to the three, we obtain thirteen, from which, if the five in the subtrahend be taken, there will remain eight. Again, as the minuend was increased by ten units of the third order, we must also increase the subtrahend by ten units of this order, or, which is equivalent, by one unit of the fourth order. The units of the fourth order being increased by this one, will become seven, which, subtracted from nine, leave two as a remainder. This process explains the meaning of the terms borrowing and carrying, as used in arithmetic. The one we carry is always equal to the ten we borrow, because it occupies the next higher place on the scale. 20. Subtraction may be proved by itself, or by the help of addition. For it is obvious, that, if the difference as obtained be correct, it must, when subtracted from the greater, leave the less as a remainder; or, when added to the less, reproduce the greater. Still, however, the proof is not absolutely certain, that is, it is not demonstration, but merely affords presumptive evidence of the operation being right. MULTIPLICATION. 21. Multiplication, as already stated (No. 16), is nothing else than a compendious method of performing addition, when the numbers to be added are all equal to one another. Hence it may be defined to be the method of finding the amount of a given number, repeated a certain number of times. This given number is called the multiplicand; the number of times that it is repeated, the multiplier; and the result, the product. The multiplicand and multiplier are also sometimes denominated factors. When the multiplier is small, the product may be obtained by actually performing the addition. Required, for example, the amount of seven taken four times. Here we have only to write the multiplicand, seven, as often as there are units in the multiplier, four, and then add the four sevens together, thus: 7 7 7 7 28 In this way the products of every two digits may be found, which, when tabulated and committed to memory, will greatly facilitate the process of multiplication. A table of this description, called, from its use, the Multiplication Table, was first formed by Pythagoras, a celebrated Grecian philosopher, in honour of whom, it is sometimes called the Pythagorean Table. With a little extension, it is as follows: 1 2 3 4 5 6 7 8 9 10 11 12 |