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. 7864 5321 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. 2543 8 3 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 The product of any two numbers which the table contains, will be found at the junction of the vertical and horizontal columns, at whose extremities the two numbers respectively appear. Thus, to find the product of 6 and 9, we have only to observe, where the columns, at whose extremities 6 and 9 are respectively written, meet one another, and there we find the product 54. The process in Ex. 1. is sufficiently obvious. In Ex. 2. six times four units are twenty-four units, equal to two units of the second order, and four of the first; six times five are thirty, which, with the two carried, are thirty-two, equal to three units of the third order, and two of the second; six times eight are forty-eight, which, with the three carried, make fifty-one, equal to five units of the fourth order, and one of the third; six times seven are forty-two, which, with the five carried, are forty-seven, equal to four units of the fifth order, and seven of the fourth. 22. To multiply any number by 10, it is merely necessary to annex a cipher to it. For, by doing this, each collection of its units is increased ten times; that is, the units are changed into tens, the tens into hundreds, and the hundreds into thousands, &c. or the whole number is rendered ten times as great as at first. In the same manner, to multiply any number by 100, it is only necessary to annex two ciphers to it. For by annexing one cipher, its value (No. 22) is increased ten times, and, by annexing another cipher, this last value will also be increased ten times; that is, the value, after two ciphers are annexed, will be ten times ten times, or a hundred times as great as at first. Hence we draw this general conclusion, that when the multiplier is expressed by unity with any number of ciphers annexed, it is merely necessary to annex that number of ciphers to the multiplicand, in order to obtain the product. Ex. To multiply 75 by 10, we must annex one cipher, and the product will be 750. To multiply it by 100 we must annex two ciphers, and the product will be 7500, and so on. 23. If the digit of the multiplier be any other than unity, the product will be obtained by multiplying by this digit, and then annexing the cipher. 752 400. Thus, for example, the product of 752 multiplied by 400, will be found, by multiplying 752 by 4, and then annexing two ciphers to this product. This is obvious, for, when 752 is multiplied by 4, its value is, of course, increased four times; and when, to this quadruple sum, two ciphers are annexed, its value is 300800 increased a hundred times. Hence this last value will be a hundred times four times, or four hundred times the first; that is, (Def.) it will be the product of 752, multiplied by 400. 24. We are now prepared to explain the method of multiplying by a number which is wholly expressed by digits. Let it be required, for example, to multiply 789 by 453. The meaning here is, that the multiplicand 789 is to be taken 400 times plus 50 times plus 3 times. Now, it is obvious, that if 789 be mutiplied successively by 400, by 50, and by 4, and these partial products be added together, their sum will express the total product of the two given numbers. Hence the operation will stand as follows: 357417 equal to the sum of the partial products, or the total product of 789 by 453. The first partial product is obtained as in Ex. 2 of No. 21, the second and third as in the Ex. No. 23. Again, let it be required to find the product of 8795 by 5004. Here the multiplicand 8795 is to be taken 5000 times plus 4 times. 44010180 equal to the sum of the partial products, or the total product of 8795 by 5004. From a consideration of the foregoing examples, it is manifest, that the ciphers, on the right of the partial products, cannot affect the sum of the latter, that is, the total product; these ciphers, therefore, which merely indicate the rank of the digits, may be omitted in practice, by observing, that the product of the units of the multiplicand by the tens of the multiplier, will be tens, and will therefore occupy the second place from the right; that the product of the units of the multiplicand, by the hundreds of the multiplier, will be hundreds, and will therefore occupy the third place from the right; or, in general, that the product of the units of the multiplicand, by any one of the digits of the multiplier, will express the same order of units as this digit, and will therefore stand exactly under it. 25. If the multiplier be a composite number, that is, one formed by the continual multiplication of two or more numbers, the product may be obtained by multiplying continually by its factors. Thus, for instance, 95 multiplied by 28, will be equal to 95 multiplied by 7, and this last product by 4. This must be obvious, because 4 times 7 times 95, are evidently 28 times 95. 26. The preceding method will apply, so far, although the multiplier is a prime number, that is, one which cannot be formed by the continual multiplication of any numbers. In this case, we have merely to multiply by the factors of that composite number which is nearest the given multiplier, and to or from this product, add or subtract that number of times the multiplicand, expressed by the units in the difference of the composite number and given multiplier, according as the former is less or greater than the latter. This will be better understood, perhaps, by an example. Suppose the product of 83 by 47 were required; then, since 47 is equal to 45 plus 2, or 48 minus 1, we might either multiply continually by the component factors of 45, and to this product add twice 83; or, multiply continually by the component factors of 48, and from this product subtract 83. This latter method is applied with peculiar effect, when the multiplier is any number of nines. If the multiplier were 999 for example, we would merely require to annex three ciphers to the multiplicand, and then subtract it from the product; because 999 is equal to 1000 minus 1. 27. Multiplication may be proved in a variety of ways, many of which will readily occur to the intelligent student. One method is to make the factors change places, and see |