1

fraction a+b may be considered as a power of a + b, namely, that power whose exponent is 1; and from this it follows, that the series already found as the value of (a + b)" entends also to this

and this is the same series that we found before by division.

1

(a + b)2

331. Further, being the same with (a + b)-2, let us reduce this quantity also to an infinite series. For this purpose, we 2, and we shall first have for the coefficients

must suppose n =

=2; X = 3; × × = 4; × × × = 5, &c. Consequently we have

332. Let us proceed and suppose n —— 3, and we shall have a

333. The different cases that have been considered enable us to

conclude with certainty, that we shall have, generally, for any nega

And by means of this formula, we may transform all such fractions into infinite series, substituting fractions also, or fractional exponents, for m, in order to express irrational quantities.

334. The following considerations will illustrate this subject further.

If, therefore, we multiply this series by a + b, the product ought to be = 1; and this is found to be true, as we shall see by performing the multiplication :

If, therefore, we multiply this series by (a + b)2, the product ought

also to be 1. Now (a+b)2=aa+2ab+bb. See the operation :

1 = the product, which the nature of the thing required.

336. If we multiply the series which we found for the value of

1

(a + b)2

, by a+b only, the product ought to answer to the fraction

1 or be equal to the series already found, namely,

тъ

and this the actual multiplication will confirm.

1 26 3bb 463 564

+

+ &c.

αθ

SECTION III

OF RATIOS AND PROPORTIONS.

CHAPTER I.

Of Arithmetical Ratio, or of the Difference between Two Numbers.

ARTICLE 337. Two quantities are either equal to one another, or they are not. In the latter case, where one is greater than the other, we may consider their inequality in two different points of view: we may ask, how much one of the quantities is greater than the other? Or we may ask, how many times the one is greater than the other? The results, which constitute the answers to these two questions, are both called relations or ratios. We usually call the former arithmetical ratio, and the latter geometrical ratio, without however these denominations having any connexion with the thing itself: they have been adopted arbitrarily.

338. It is evident that the quantities of which we speak must be of one and the same kind; otherwise we could not determine any thing with regard to their equality or inequality. It would be absurd, for example, to ask if two pounds and three ells are equal quantities. So that, in what follows, quantities of the same kind only are to be considered; and as they may always be expressed by numbers, it is of numbers only, as was mentioned at the beginning, that we shall treat.

339. When of two given numbers, therefore, it is required to find how much one is greater than the other, the answer to this question determines the arithmetical ratio of the two numbers. Now, since this answer consists in giving the difference of the two numbers, it follows, that an arithmetical ratio is nothing but the difference be