PRINCIPLE OF DIVISION. 83 time merely to express, not actually to find. Indeed, if the expression bears the most simple form, that is, if the divisor and dividend are each expressed by a single letter, if these letters are different, and if nothing farther is stated than that the one of them is to be divided by the other, there are not data sufficient for finding the quotient as a separate quantity. Thus, if the quotient of a divided by b is sought; and if we merely know that a stands for one quantity and b for another, but do not know what kind of quantities they are, whether of the same kind with each other or of different kinds, we cannot tell whether the quotient is or is not a quantity which we can or cannot express in any other way than by indicating it; and, even if we know that both quantities are of the same kind, so that the quotient must be a number, we are not in a condition for stating whether the quotient shall be greater than the number 1, equal to it, or less, unless we know that the quantities are equal or unequal, and in the case of inequality, which is the greater and which the less. Therefore, all that we can do in such cases is to indicate that there is a division to be performed; and this is done generally by writing the dividend above a line, and the a divisor below the same. Thus indicates the quotient of a b divided by b, though without pointing out what that quotient may be. The quotients of all quantities may be indicated in the same a+b manner: as, indicates the quotient of the upper quantity by the under, whatever may be the forms in which they are expressed. Division may also be indicated by writing the dividend, then the sign, and lastly the divisor. ab have the same meaning, and are read " We shall point out in another section how the division is to be performed so as to obtain the quotient as a separate quantity in all cases where that is possible; but there are some general principles which we can perhaps better explain in this simple view of the matter. We may, for instance, determine the sign of the quotient, whether we can or cannot express its value by a separate quantity. Here we must bear in mind that the dividend is always equal to the product of the divisor and quotient, so that the finding of a quotient resolves itself into the finding of a quantity the product of which and the quotient shall be equal to the dividend. From this it follows, that if the divisor and dividend have the same sign, the sign of the quotient must be + ; and if they have different signs it must be ; but that in the case of the same quantities as divisor and dividend, the expression for the quotient will be the same quantities whatever may be the signs. Let us illustrate this by the simplest case that can occur, the division of a quantity by itself, or a a The quotient of this, in all cases of the signs, will be expressed by the number 1, because any quantity is, of course, just once itself, and nothing either over or wanting. Now, if it is a a or the quotient will be 1; for - a, the divisor in the first case, multiplied by - 1, the quotient, produces +a, the dividend in the first case; and a, the divisor in the second case, multiplied by -1, the quotient, produces -a, the dividend in that case. Hence the quotient of quantities which have the same sign is always a positive quantity, or as much greater than 0 as its with different signs is whole value expresses. quotient of + a + b' or of MULTIPLYING DIVISORS AND DIVIDENDS. 85 whole value expresses; but the quotient of two quantities always as much less than 0 as its Hence, in a compound quantity, the If the divisor were a, and the dividend 3 a, it is evident that the quotient would be 3, +3 if the signs were the same, and 3 if they were different; and generally, if the divisor were a, and the divident na, that is, any number of times a, whether expressible in terms of the arithmetical scale or not, the quotient would be n, that is, the same number, +n if the signs were the same, and -n if they were different. Now 3 is 1 x 3, and n is 1 × n, whatever number n may stand for; therefore, multiplying the dividend by any quantity has the same effect as multiplying the quotient by the same quantity. If the dividend were a, and the divisor 3 a, the quotient would be one-third part of 1; and if the dividend were a, and the divisor na, the quotient would be the nth part of 1, or 1 n ; therefore, multiplying the divisor produces the same effect as dividing the quotient. Now, if multiplying the dividend multiplies the quotient, and multiplying the divisor divides the quotient, multiplying both by the same quantity, whether that quantity be one single factor or any number of factors, will not alter the value of the quotient; or, an a = whatever n may be, whether large or b n b small, simple or compound, provided it is the same in both cases. This principle, which is so simple that it is nearly self 86 ALGEBRAICAL OPERATIONS. evident, is a very important one in practice; and so is the converse of it, namely, that if both divisor and dividend are divided by the same quantity the quotient is not altered. This last is, in fact, the principle upon which we proceed in the common arithmetical division of one number by another; we consider the divisor as divided by itself, and thus reduced to the number 1; and we divide the dividend also by the divisor, in order to have it expressed in terms of the divisor, considered as one. We shall be better able to see the value of these principles afterwards, and shall discover other means of perceiving their truth; so we shall now proceed to show how the elementary operations are performed algebraically. SECTION VI. ELEMENTARY OPERATIONS IN ALGEBRA. As arithmetic is merely the application of the general principles of algebra to those particular cases of quantities which can be expressed by numbers according to the scale and notation of arithmetic, it follows that the elementary operations in the one must be the same as they are in the other, namely, addition, subtraction, multiplication, and division. But if the notation of algebra, as we have attempted to explain it in the preceding section, is properly understood, these operations are far more easily performed by means of the algebraical symbols than the arithmetical ones. If we write down the given quantities with the proper signs, used in the form which indicates the operation, we have an expression for the result of that operation at once; and all that we have to do farther is to find out whether the expression thus obtained can be reduced to fewer or more simple terms. In this, every case must be considered in itself; and thus every operation in algebra is the discovery of something new, instead of the performing of the same sort of drudgery over and over, as is the case in arithmetic. The first and most general consideration is, whether the case before us can or cannot be simplified; and as it would be vain to try the cases which cannot be simplified, the knowledge of them is the first point to which we must direct our attention. Now, the principle here is a very simple and self-evident one: if the quantities are all of different kinds, that is, all expressed by different letters, or by different letters combined with different numbers as co-efficients, we cannot shorten or simplify the expression. Thus, in ADDITION, if the sum is that of a, b, and c, there is no simpler expression for it than a+b+c. Also, if it is 5 a, 3b, and 4c, there is no simpler expression than 5 a + 3b + 4c. But if it is 5 a, 5 b, and 5 c, we can make it 5 times the sum of a, b, and c, that is (a + b + c) × 5. If the letters are the same, we can bring them into one expression; thus 5 a + 3 a + 4a is = 12a. Also, if the letters are the same, and some of them and others —, we can get one expression for the whole by taking the + into one sum and the into another; subtracting the co-efficients and prefixing the sign of the greater co-efficient to the difference. Thus 5 a 5a = 0; 5 a 10 a = 5a; and so in other cases. Also, if we have a + quantity to add to any expression, and there is a quantity of the same kind, that is, expressed by the same letter or letters in the expression, we get rid of as much of the quantity as is equal to the + one. Thus, if we have to add cb, that is + cb, to a cb, the sum becomes 4a = a; 5 a - |