m and n are integral or fractional, rational or irrational, positive or negative. Hence, if we have the proportion A: B:: C: D, then we shall also have the proportion mA : nB :: mC: nD, without regarding the nature of the numbers that are represented by m and n; and it is clear that, if we please, we may put m = 1 or n= 1, or both of them equal to 1, and return to the proportion A: B::C::D. Consequently, from the proportion A: B:: C: D we get the proportions mA: nB:: mC: nD, mA: B:: mC: D, AnB: C: nD, which are true without any reference to the nature of the numbers that are represented by m and n. = (.) If we multiply A and B in (1) by m, we shall get mA C ; or, which is the same, if we have the proportion m B D A: B:: C: D, then we shall have the proportion mA : mB :: C: D; and in like manner we shall have A: B:: nC: nD; or, more generally, mA : mB:: nC: nD; these proportions being true without reference to the nature of the numbers that are represented by m and n. In words, if in any proportion either antecedent and its consequent are multiplied by the same numerical expression, the proportion will not be affected; neither will it be affected if we multiply the remaining antecedent and its consequent by any numerical expression. (8.) Since A MA = Hence, one of two quantities of the same kind has the same ratio to the remaining one of the two as the product of the first and any numerical expression has to the product of the second and the same numerical expression. mA mC (9.) From the equation (which is the same as = nB nD the proportion mA : nB :: mC : nD) we see that if mA is greater than nB, then mC must be greater than nD; and if mA equals nB, mC must equal nD; also, if mA is less than nB, mC must be less than nD. Hence, in any proportion, if the first antecedent is greater than its consequent, the second antecedent is greater than its consequent; and if the first antecedent equals its consequent, the second antecedent equals its consequent; and if the first antecedent is less than its consequent, the second antecedent is less than its consequent. = we suppose A, B, C, nD' D to be quantities of the same kind, then it is evident that if we have mA greater than mC, we must also have nB greater than nD, for otherwise the equation will not hold true; and if mA equals mC, nB must equal nD; also, if mA is less than mC, nB must be less than nD. Hence, if we have any proportion, as mA : nB :: mCnD, whose terms are all of the same kind, then, if the first antecedent is greater than the second antecedent, the first consequent is greater than the second; and if the first antecedent equals the second, the first consequent equals the second; also, if the first antecedent is less than the second, the first consequent is less than the second. the same thing must equal each other, we shall of course have A E == ; or, which is the same thing, if we have the F proportions A: B:: C: D, and C: D:: E: F, we get ABE: F. Hence, if two ratios are equal to the same ratio, they are equal to each other; or if an antecedent and its consequent in two proportions are the same, then the remaining antecedents and consequents will constitute a proportion. And in like manner it is easy to see that ratios which are equal to equal ratios, are equal to each other. (12.) Since by (1) we have we divide these A C equals by any positive integer n, we shall evidently get A с nD 1 ; and if we divide A and nB by A, we shall get 1 A nB C =nB, and in the same way we shall get =nD; nD A consequently, we must have the equation nBnD. Now, A C if the multiple nB of B contains A a greater or less integral number of times than the equimultiple nD of D contains C, the preceding equation evidently can not hold true. Hence, since nB can not contain A a greater or less integral number of times than nD contains C, it follows from (5) that we shall have the equation which is also imme B = D A 1 1 diately evident from the equation nB = nD. Hence, if we have the equation the equation B D C ; in other terms, if we have the propor tion A B C D; then we shall also have the proportion B: A: D: C. Consequently, the terms of a proportion are proportional by inversion, or when taken inversely; that is, the second is to the first as the fourth is to the third. (13.) If, in the equation A C = B D' we suppose A, B, C, D to be quantities of the same kind, then, by multiplying A and B by the positive integer m, we shall clearly have the that the multiple mA of A can not contain C a greater or less number of times than the equimultiple mB of B con tains D; and of course by (5) we must have Α B Hence, when we have the proportion A: B:: C: D, whose terms are all of the same kind, then we shall also have the proportion A: C:: B : D. In words, when the terms of a proportion are all of the same kind, they will be proportional when taken alternately; that is, the first is to the third as the second to the fourth. The same things being supposed; since A B = we shall C D' or mA nC:: mB: nD, which is true with out reference to the nature of the numbers represented by m and n. Also, from mA nB we get = or mA: mC mC nD A B C D' nB: nD, which is also true without reference to the nature of the numbers represented by m and n; and if we please we may put either m or n or both of them equal to unity in the preceding proportions, and they will continue to be true. (14.) Let A, B, C, D be four quantities, such that A, B are of one kind and C, D of one kind, which may be of the same kind with A and B, if the case requires it. Then suppose A and C to be multiplied by any positive integer m, and B and D to be multiplied by any positive integer n, and the multiples will be expressed by mA, nB, mC, nD. If, now, we find for all the integers that m and n can represent, that when mA is greater than nB, mC is greater than nD; and when mA is equal to nB, mC is equal to nD; also, when mA is less than nB, mC is less than nD; then we shall have the proportion A: B:: C: D, or, which is the A C = B D same For, by (5), if the proportion A: B:: C: D does not have place, we can suppose A and C to be multiplied by some positive integer m, so that mA shall contain B a greater or less integral number of times than mC contains D; for otherwise the proportion A B C D will have place, as was shown in (5). : If we suppose that mA contains B a greater integral number of times than mC contains D, we may evidently put mA = mC B D +p, (1), where p is evidently not less than unity; since mA contains B a greater integral number of times than mC contains D. If, now, we suppose that mC contains D an integral number of times represented by q, then clearly mA will contain B at least as many integral times as there are units in q +1. Hence, if we represent q + 1 by n, and divide both sides of mA mC (1) by n, we shall get the equation p nB nD n Since mA contains B,n times at least, it will contain nB at least once, and of course mA is not less than nB; but since mC contains D a less number of times than there are units in n, it can not contain nD once; and consequently mC is less than nD. Hence, when mA is not less than nB, we have mC less than nD, which is against the hypothesis; which is, that when mA is not less than nB, mC is not less than nD. In a similar way it may be shown if we suppose mC contains D a greater number of times than mA contains B, that the hypothesis will be contradicted. Consequently, we must suppose that the proportion A: B :: CD or the equation proved. = B has place, as was to be Hence, the first and third of four quantities being multiplied by any positive integer whatever, and the second and fourth being multiplied by any positive integer whatever; if it is found, when the first multiple is greater, equal to, or less than the second, that the third multiple is in like manner greater, equal to, or less than the fourth; then the first quantity has the same ratio to the second that the third, has to the fourth. This proposition is substantially the same as the famous Definition V. of Book V. of Euclid; see R. Simson's Euclid, Book V., Def. V. (15.) If we have any number of quantities of one kind, as A, B, C, D, and as many others, E, F, G, H, of one kind (which may be of the same kind as the first, should the case |