56

Of Simple Quantities.

Sect. 1.

the involution of powers, and the extraction of roots. It will not be improper, therefore, in this place, to trace back the origin of these different methods, and to explain the connexion which subsists among them; in order that we may satisfy ourselves whether it be possible or not for other operations of the same kind to exist. This inquiry will throw new light on the subjects which we have considered.

In prosecuting this design, we shall make use of a new character, which may be employed instead of the expression that has been so often repeated, is equal to; this sign is =, and is read is equal to. Thus, when I write a b, this means that a is equal to b; so, for 15.

for example 3 +5

207. The first mode of calculation, which presents itself to the mind, is undoubtedly addition, by which we add two numbers together and find their sum. Let a and b then be the two given numbers, and let their sum be expressed by the letter c, we shall have a + b = c. So that when we know the two numbers a and b, addition teaches us to find the number c.

208. Preserving this comparison a + b = c, let us reverse the question by asking, how we are to find the number b, when we know the numbers a and c.

It is required therefore to know what number must be added to a, in order that the sum may be the number c. Suppose, for example, a = 3 and c = 8; so that we must have 3 + b = 8; b will evidently be found by subtracting 3 from 8. So, in general, to find b, we must subtract a from c, whence arises b = c — ing a to both sides again, we have b + a = c say=c, as we supposed.

Such then is the origin of subtraction.

a; for by add-a+a, that is to

209. Subtraction therefore takes place, when we invert the question which gives rise to addition. Now the number which it is required to subtract may happen to be greater than that from which it is to be subtracted; as, for example, if it were required to subtract 9 from 5: this instance therefore furnishes us with the idea of a new kind of numbers, which we call negative numbers, because 5 9= - 4.

210. When several numbers are to be added together which are all equal, their sum is found by multiplication, and is called a product. Thus ab means the product arising from the multiplication of a by b, or from the addition of a number a to itself b number of

times. If we represent this product by the letter c, we shall have abc; and multiplication teaches us how to determine the number c, when the numbers a and b are known.

211. Let us now propose the following question: the numbers a and c being known, to find the number b. Suppose, for example, a = 3 and c = 15, so that 36 = 15, we ask by what number 3 must be multiplied, in order that the product may be 15 for the question proposed is reduced to this. Now this is division: the number required is found by dividing 15 by 3; and therefore, in general, the number 6 is found by dividing c by a; from which results the equation b =

с

a

212. Now, as it frequently happens that the number c cannot be really divided by the number a, while the letter b must however have a determinate value, another new kind of numbers presents itself; these are fractions. For example, supposing a = 4, c = 3, so that 46 3, it is evident that b cannot be an integer, but a fraction, and that we shall have b = 2.

213. We have seen that multiplication arises from addition, that is to say, from the addition of several equal quantities. If we now proceed further, we shall perceive that from the multiplication of several equal quantities together 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 shown in treating of powers, how to find the power itself, that is, the letter c, when a root a and its exponent b are given. 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.

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 Eul. Alg.

8

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 ab, 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 b a = c.

In the calculation of powers, on the contrary, the same thing does not take place, and we can by no means write ba instead of a'. A single example will be sufficient to illustrate this: let a 5, and b= 3; we have a But ba 35 = 243: two very different results.

53 125.

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 +ƒ).

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+fis 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.

in

218. To make this more evident, we shall consider an example pure numbers. It is proposed to add the expression 15-6 to If we begin by adding 15, we shall have 128 + 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

12

much. Let us, therefore, take this 6 away by writing it with the negative sign, and we shall have the true sum,

12815-6,

which shows 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 d—e—ƒ to a― bc, 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, or 3 a 4a+a: or when two or more terms may be

+ a
reduced to one. Examples of this second reduction :

3a+2a = 5 a 7b 36 =

6c10c

5a-8a3a; -7b+b6b;

+ 46;

+ 4 c;

_ *,

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 2 b3 . which admit of no abridgment.

b=0;

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 + a2 a and b · 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:

За 26

a32aab +2abb
2aab+2abb

56-6c+ a -aab2abb-b3

4a+3b-7ca3-3aab + 4 abb-b3.