gether powers are derived. Those powers are represented in a general manner by the expression a', which signifies that the number a must be multiplied as many times by itself, as is denoted by the number b. And we know from what has been already said, that in the present instance a is called the root, b the exponent, and a the power. 214. Further, if we represent this power also by the letter c, we have a = c, an equation in which three letters a, b, c, are found. Now we have shewn in treating of powers, how to find the power itself, that is, the letter c, when a root a and its Suppose, for example, a = 5, and b = 3, so that c 53; it is evident that we must take the third power of 5, which is 125, and that thus c = 125. exponent b are given. 215. We have seen how to determine the power c, by means of the root a and the exponent b; but if we wish to reverse the question, we shall find that this may be done in two ways, and that there are two different cases to be considered for if two of these three numbers a, b, c, were given, and it were required to find the third, we should immediately perceive that this question admits of three different suppositions, and consequently three solutions. We have considered the case in which a and b were the numbers given, we may therefore suppose further that c and a, or c and b are known, and that it is required to determine the third letter. Let us point out therefore, before we proceed any further, a very essential distinction between involution and the two operations which lead to it. When in addition we reversed the question, it could be done only in one way; it was a matter of indifference whether we took c and a, or c and b, for the given numbers, because we might indifferently write a + b, or b + a. It was the same with multiplication; we could at pleasure take the letters a and b for each other, the equation a b c being exactly the same as bac. In the calculation of powers, on the contrary, the same thing does not take place, and we can by no means write be instead of a. A single example will be sufficient to illustrate this let a = 5, and b = 3; we have a 53 = 125. But ba= 35 = 243: two very different results. = : SECTION II. OF THE DIFFERENT METHODS OF CALCULATION APPLIED TO COMPOUND QUANTITIES, CHAPTER I. Of the Addition of Compound Quantities. ARTICLE 216. WHEN two or more expressions, consisting of several terms, are to be added together, the operation is frequently represented merely by signs, placing each expression between two parentheses, and connecting it with the rest by means of the sign + If it be required, for example, to add the expressions a+b+c and d+e+f, we represent the sum thus: (a+b+c)+(d+e+f). 217. It is evident that this is not to perform addition, but only to represent it. We see at the same time, however, that in order to perform it actually, we have only to leave out the parentheses; for as the number d+e+f is to be added to the other, we know that this is done by joining to it first+d, then +e, and then +f; which therefore gives the sum a+b+c+d+e+f. The same method is to be observed, if any of the terms are affected with the sign; they must be joined in the same way, by means of their proper sign. 218. To make this more evident, we shall consider an example in pure numbers. It is proposed to add the expression 156 to 12 .8. If begin by adding 15, we shall have 12-8+15; now this was adding too much, since we had only to add 15 6, and it is evident that 6 is the number which we have added too much. Let us therefore take this 6 away by writing it with the negative sign, and we shall have the true 12-815-6, sum, which shews that the sums are found by writing all the terms, each with its proper sign. 219. If it were required therefore to add the expression def to a—b+c, we should express the sum thus: a—b+c+d―e-f, remarking however that it is of no consequence in what order we write these terms. Their place may be changed at pleasure, provided their signs be preserved. This sum might, for example, be written thus: c―e+a—f+d — b. 220. It frequently happens, that the sums represented in this manner may be considerably abridged, as when two or more terms destroy each other; for example, if we find in the same sum the terms +a-a, or 3 a - -4a+a: or when two or more terms may be reduced to one. Examples of this second reduction : 3a+2a=5a; 7b-sb=+4b; 2a5a+a=-2a; -sb-5b+2b=-6b. Whenever two or more terms, therefore, are entirely the same with regard to letters, their sum may be abridged: but those cases must not be confounded with such as these, 2 a a+ 3 a, or 2b3b4, which admit of no abridgment. 221. Let us consider some more examples of reduction; the following will lead us immediately to an important truth. Suppose it were required to add together the expressions a + b and a -b; our rule gives a+b+a—b; now a + a2a and b-b0; the sum then is 2 a: consequently if we add together the sum of two numbers (a+b) and their difference (a —b,) we obtain the double of the greater of those two numbers. Further examples: sa-2b-c|a32aab+2abb 4a+3b-7 ca3 3aab + 4 abb — b3. - CHAPTER II. Of the Subtraction of Compound Quantities. 222. If we wish merely to represent subtraction, we inclose each expression within two parentheses, connecting, by the sign -, the expression to be subtracted with that from which it is to be taken. 9 When we subtract, for example, the expression d—e+f from the expression a―b+c, we write the remainder thus: (a− b + c) -- (d — e +f) ; and this method of representing it sufficiently shews, which of the two expressions is to be subtracted from the other. 223. But if we wish to perform the subtraction, we must observe, first, that when we subtract a positive quantity +b from another quantity a, we obtain a-b: and secondly, when we subtract a negative quantity -b from a, we obtain a +b ; because to free a person from a debt is the same as to give him something. 224. Suppose, now, it were required to subtract the expression bd from the expression ac, we first take away b; which gives a c—b. Now this is taking too much away by the quantity d, since we had to subtract only b―d; we must therefore restore the value of d, and we shall then, have whence it is evident, that the terms of the expression to be subtracted must have their signs changed, and be joined, with the contrary signs, to the terms of the other expression. 225. It is easy, therefore, by means of this rule, to perform subtraction, since we have only to write the expression from which we are to subtract, such as it is, and join the other to it without any change beside that of the signs. Thus, in the first example, where it was required to subtract the expression de+f from a b+c, we obtain a · b + c — d + e −f. An example in numbers will render this still more clear. If we subtract 62+4 from 93+2, we evidently obtain for 9-3+2=8; also, 6-2+4=8; now 8—8=0. 226. Subtraction being therefore subject to no difficulty, we have only to remark, that, if there are found in the remainder two, or more terms which are entirely similar with regard to the letters, that remainder may be reduced to an abridged form, by the same rules which we have given in addition. 227. Suppose we have to subtract from a + b, or from the sum of two quantities, their difference ab, we shall then have a+b=a+b; now a-a0, and b+b=2b; the remainder sought is therefore 2 b, that is to say, the double of the less of the two quantities. 228. The following examples will supply the place of further illustrations. aa+ab+bb3a-4b+5 ca3 + saab+3abb + b3 |√√π+245 3aab+3abb—b3 √ā — 3 √√√T bb+ab. a a2b+4c- 6 aa3. 2aa. 19 a 6b+cl --- CHAPTER III. Of the Multiplication of Compound Quantities. 229. WHEN it is only required to represent multiplication, we put each of the expressions, that are to be multiplied together, within two parentheses, and join them to each other, sometimes without any sign, and sometimes placing the sign between them. For example, to represent the product of the two expressions a-b+c and d-e+f, when multiplied together, we write. (a−b+c) × (d—e+f.) This method of expressing products is much used, because it immediately shews the factors of which they are composed. 230. But to shew how multiplication is to be actually performed, we may remark, in the first place, that in order to multiply, for example, a quantity, such as a-b+c, by 2, each term of it is separately multiplied by that number; so that the product is 2a-2b+2 c. |