1 = the product, which the nature of the thing required. 336. If we multiply the series which we found for the value (a+b)s, by a+b only, the 1 of by a+b only, the product ought to answer to the + a + b 73 or be equal to the series already found, namely, + b4 as, &c. and this the actual multiplication 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 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 × 3, the difference between 6 and 2 × 3 is 0. 341. But when the two numbers are not equal, as 5 and 3, and it 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 3d, 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 6, and the difference d; the difference d will be found by subtracting b from a, so that dab; whence we see how to find d, when a and b are given. S43. 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 b is found by subtracting the difference from the greater number, which gives bad. For if I take the number a--d 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 d = a b; 2d a=b+d; sd bad; and if one of these three comparisons be just, the others must necessarily be so also. Wherefore, generally, if=x+y, it necessarily follows, that y=-x, and x= y. 346. With regard to these arithmetical ratios we must remark, that if we add to the two numbers a and b, a number c assumed at pleasure, or subtract it from them, the difference remains the same. That is to say, if d is the difference between a and b, that number d will also be the difference between a + c and b+c, and between a c and b C. For example, the difference between the numbers 20 and 12 being 8, that difference will remain the same, whatever number we add to the numbers 20 and 12, and whatever numbers we subtract from them. 347. The proof is evident; for if a b d we have also (a + c) — (b+c)= d; and also (ac) — (b — c) = d. 348. If we double the two numbers a and b, the difference will also become double. Thus, when, abd, we shall have, 2a-2b=2d; and, generally, na-nbnd, whatever value we give to n. = CHAPTER II. of Arithmetical Proportion. 349. WHEN two arithmetical ratios, or relations, are equal, this equality is called an arithmetical proportion. Thus, when a bd and p -q=d, so that the difference is the same between the numbers p and q, as between the numbers a and b, we say that these four numbers form an arithmetical proportion; which we write thus, ab=p-q, expressing clearly by this, that the difference between a and b is equal to the difference between p and q. 350. An arithmetical proportion consists therefore of four terms, which must be such, that if we subtract the second from the first, the remainder is the same as when we subtract the fourth from the third. Thus, the four numbers 12, 7, 9, 4, form an arithmetical proportion, because 1279 — 4.* To shew that these terms make such a proportion, some write them thus; 12.. 7 :: 9. .4 351. When we have an arithmetical proportion, as a—b=p -q, we may make the second and third change places, writing apbq; and this equality will be no less true; for, since a-bp-q, add b to both sides, and we have ab+p -q; then subtract p from both sides, and we have a-pb-q. In the same manner, as 12 7=9 4, so also 352. We may, in every arithmetical proportion, put the second term also in the place of the first, if we make the same transposi tion of the third and fourth. That is to say, if a b = p-q we have also b For ba is the a = q p. For b negative of ab, and qp is also the negative of p-q. Thus, since 12-7-9 -4, we have also 7 12=4 9. 353. But the great property of every arithmetical proportion is this; that the sum of the second and third term is always equal to the sum of the first and fourth. This property, which we must particularly consider, is expressed also by saying that, the sum of the means is equal to the sum of the extremes. Thus, since 12. -7=9. 4, we have 7+ 9 = 12+4; and the sum we find is 16 in both. 354. In order to demonstrate this principal property, let a-bp-q; if we add to both b+q, we have a+q=b+p; that is, the sum of the first and fourth terms is equal to the sum of the second and third. And conversely, if four numbers, a, b, p, q, are such, that the sum of the second and third is equal to the sum of the first and fourth, that is, if b + p = a+q, we conclude, without a possibility of mistake, that these numbers are in arithmetical proportion, and that a-bp-q. For, since a + q = b+p, if we subtract from both sides b + q, we obtain a − b = p — q. Thus, the numbers 18, 13, 15, 10, being such, that the sum of the means (13 + 15 = 28,) is equal to the sum of the extremes (1810 28,) it is certain, that they also form an arithmetical proportion; and, consequently, that = 18- 1315- 10. 355. It is easy, by means of this property, to resolve the following question. The three first terms of an arithmetical proportion being given to find the fourth? Let a, b, p, be the three |