5 1 — —, &c. b4 b + X Al X We therefore obtain (a + b)−2 = × 5, &c. Now, = 33%. Let us proceed and suppose n —— 3, and we shall have 333. The different cases that have been considered enable us to conclude, with certainty, that we shall have, generally, for any negative power of a+b; 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), the product ought also to be = 1. the operation : Now (a+b)a a +2ab+bb. See of 1 = the product, which the nature of the thing required. 336. If we multiply the series which we found for the value (a+b)a, by a + b only, the product ought to answer to the 1 fraction a+b' or be equal to the series already found, namely, 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. 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. We 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 Eul. Alg. 15 two numbers, it follows, that an arithmetical ratio is nothing but the difference between two numbers: and as this appears to be a better expression, we shall reserve the words ratio and relation, to express geometrical ratios. 340. The difference between two numbers is found, we know, by subtracting the less from the greater; nothing therefore can be easier than resolving the question, how much one is greater than the other. So that when the numbers are equal, the difference being nothing, if it be inquired how much one of the numbers is greater than the other, we answer, by nothing. For example, 6 being = 2 x 3, the difference between 6 and 2 x 3 is 0. 341. But when the two numbers are not equal, as 5 and 3, and is inquired how much 5 is greater than 3, the answer is, 2; and it is obtained by subtracting 3 from 5. Likewise 15 is greater than 5 by 10; and 20 exceeds 8 by 12. 342. We have three things, therefore, to consider on this subject; 1st, the greater of the two numbers; 2d, the less; and Sd, the difference. And these three quantities are connected together in such a manner, that two of the three being given, we may always determine the third. Let the greater number =a, the less= b, and the difference d; the difference d will be found by subtracting b from a, so that da b; whence we see how to find d, when a and b are = given. 343. But if the difference and the less of the two numbers, or b, are given, we can determine the greater number by adding together the difference and the less number, which gives a = b+d. For, if we take from b+d the less number b, there remains d, which is the known difference. Let the less number 12, and the difference 8; then the greater number will be 20. 344. Lastly, if beside the difference d, the greater number a is given, the other number 6 is found by subtracting the difference from the greater number, which gives bad. For if I take the number ad from the greater number a, there remains d, which is the given difference. 345. The connexion, therefore, among the numbers a, b, d, is of such a nature, as to give the three following results: 1st da |