Let AB the first have to C the second the same ratio which DE the third has to F the fourth; and let BG the fifth have to C the second the same ratio which EH the sixth has to F the fourth. Then AG, the first and fifth together, shall have to C the second, the same ratio which DH, the third and sixth together, has to F the fourth. Demonstration. Because BG is to C, as EH to F; by inversion, 1. C is to BG, as F to EH (V. B) ; and because, as AB is to C, so is DE to F (hyp.); and as C to BG, so is F to EH; ex æquali, 2. AB is to BG, as DE to EH (V. 22); and because these magnitudes are proportionals when taken separately, they are likewise proportionals when taken jointly (V. 18); therefore 3. As AG is to GB, so is DH to HE; but as GB to C, so is HE to F (hyp.); therefore, ex æquali, 4. As AG is to C, so is DH to F (V. 22). Wherefore, if the first, &c. Q.E.D. Cor. 1. If the same hypothesis be made as in the proposition, the excess of the first and fifth shall be to the second, as the excess of the third and sixth to the fourth. The demonstration of this is the same with that of the proposition, if division be used instead of composition. Cor. 2. The proposition holds true of two ranks of magnitudes, whatever be their number, of which each of the first rank has to the second magnitude the same ratio that the corresponding one of the second rank has to a fourth magnitude; as is manifest. PROPOSITION 25.-Theorem. If four magnitudes of the same kind are proportionals, the greatest and least of them together, are greater than the other two together. Let the four magnitudes AB, CD, E, F be proportionals, viz., AB to CD, as E to F; and let AB be the greatest of them, and consequently the least (V. 14 and A). Then AB together with F shall be greater than CD together with E. Construction. Take AG equal to E, and CH equal to F. Demonstration. Because as AB is to CD, so is E to F, and that AG is equal to E, and CH equal to F, therefore 1. AB is to CD, as AG to CH (V. 11 and 7); and because AB the whole, is to the whole CD, as AG is to CH, likewise 2. The remainder GB is to the remainder HD, as the whole AB is to the whole CD (V. 19); but AB is greater than CD (hyp.); therefore 3. GB is greater than HD (V. A) ; and because AG is equal to E, and CH to F; 4. AG and F together are equal to CH and E together (I. Ax. 2); therefore if to the unequal magnitudes GB, HD, of which GB is the greater, there be added equal magnitudes, viz., to GB the two AG and F, and CH and E to HD; 5. AB and F together are greater than CD and E (I. Ax. 4). Therefore, if four magnitudes, &c. Q.E.D. PROPOSITION F.-Theorem. Ratios which are compounded of the same ratios, are the same to one another. Let A be to B, as D to E; and B to C, as E to F. Then the ratio which is compounded of the ratios of A to B, and B to C, which, by the definition of compound ratio, is the ratio of A to C, shall be the same with the ratio of D to F, which, by the same definition, is compounded of the ratios of D to E, and E to F. Demonstration. Because there are three magnitudes A, B, C, and three others, D, E, F, which, taken two and two, in order, have the same ratio; ex æquali, 1. A is to C, as D to F (V. 22). Next. Let A be to B, as E to F, and B to C, as D to E; therefore, ex æquali in proportione perturbatâ (V. 23), A is to C, as D to F; that is, 1. The ratio of A to C, which is compounded of the ratios of A to B, and B to C, is the same with the ratio of D to F, which is compounded of the ratios of D to E, and E to F. And in like manner, the proposition may be demonstrated, whatever be the number of ratios in either case. PROPOSITION G.-Theorem. If several ratios be the same to several ratios, each to each; the ratio which is compounded of ratios which are the same to the first ratios, each to each, shall be the same to the ratio compounded of ratios which are the same to the other ratios, each to each. Let A be to B, as E to F; and C to D, as G to H; and let A be to B, as K to L; and C to D, as L to M. Then the ratio of K to M, by the definition of compound ratio, is compounded of the ratios of K to L, and L to M, which are the same with the ratios of A to B, and C to D. Again, as E to F, so let N be to 0; and as G to H, so let O be to P. Then the ratio of N to P is compounded of the ratios of N to 0, and 0 to P, which are the same with the ratios of E to F, and G to H. And it is to be shown that the ratio of K to M, is the same with the ratio of N to P; or that K is to M, as N to P. A. B. C. D. K. L. M. E. F. G. H. N. O. P. Demonstration. Because K is to L, as (A to B, that is, as E to F, that is, as) N to 0; and as L to M, so is (C to D, and so is G to H, and so is) O to P: ex æquali, 1. K is to M, as N to P (V. 22). Therefore, if several ratios, &c. Q.E.D. PROPOSITION H.-Theorem. If a ratio which is compounded of several ratios be the same to a ratio which is compounded of several other ratios; and if one of the first ratios, or the ratio which is compounded of several of them, be the same to one of the last ratios, or to the ratio which is compounded of several of them; then the remaining ratio of the first, or, if there be more than one, the ratio compounded of the remaining ratios, shall be the same to the remaining ratio of the last, or, if there be more than one, to the ratio compounded of these remaining ratios. Let the first ratios be those of A to B, B to C, C to D, D to E, and E to F; and let the other ratios be those of G to H, Ḥ to X, K to L, and L to M; also, let the ratio of A to F, which is compounded of the first ratios, be the same with the ratio of G to M, which is compounded of the other ratios; and besides, let the ratio of A to D, which is compounded of the ratios of A to B, B to C, C to D, be the same with the ratio of G to K, which is compounded of the ratios of G to H, and H to K. Then the ratio compounded of the remaining first ratios, to wit, of the ratios of D to E, and E to F, which compounded ratio is the ratio of D to F, shall be the same with the ratio of K to M, which is compounded of the remaining ratios of K to L, and L to M of the other ratios. A. B. C. D. E. F. G. H. K. L. M. Demonstration. Because, by the hypothesis, A is to D, as G to K, by inversion, 1. D is to A, as K to G (V. B); and as A is to F, so is G to M (hyp.); therefore, ex æquali, 2. D is to F, as K to M (V. 22). If, therefore, a ratio which is, &c. Q.E.D. PROPOSITION K.-Theorem. If there be any number of ratios, and any number of other ratios, such, that the ratio which is compounded of ratios which are the same to the first ratios, each to each, is the same to the ratio which is compounded of ratios which are the same, each to each, to the last ratios; and if one of the first ratios, or the ratio which is compounded of ratios which are the same to several of the first ratios, each to each, be the same to one of the last ratios, or to the ratio which is compounded of ratios which are the same, each to each, to several of the last ratios; then the remaining ratio of the first, or, if there be more than one, the ratio which is compounded of ratios which are the same, each to each, to the remaining ratios of the first, shall be the same to the remaining ratio of the last, or, if there be more than one, to the ratio which is compounded of ratios which are the same, each to each, to these remaining ratios. Let the ratios of A to B, C to D, E to F, be the first ratios; and the ratios of G to H, K to L, M to N, O to P, Q to R, be the other ratios; and let A be to B, as S to T; and C to D, as 7 to V; and E to F, as V to X; therefore, by the definition of compound ratio, the ratio of S to I is compounded of the ratios of S to T, T to V, and V to X, which are the same to the ratios of A to B, C to D, E to F; each to each. Also, as G to H, so let Y be to Z; and K to L, as Z to a; M to N, as a to b; 0 to P, as b to c; and Q to R, as c to d; therefore, by the same definition, the ratio of Y to d is compounded of the ratios of Y to Z, Z to a, a to b, b to c, and c to d, which are the same, each to each, to the ratios of G to H, K to L, M to N, O to P, and Q to R; therefore, by the hypothesis, Sis to X, as Y to d. Also, let the ratio of A to B, that is, the ratio of S to T, which is one of the first ratios, be the same to the ratio of e to g, which is compounded of the ratios of to f, and f to g, which, by the hypothesis, are the same to the ratios of G to H, and K to L, two of the other ratios; and let the ratio of h to 1 be that which is compounded of the ratios of h to k, and k to 1, which are the same to the remaining first ratios, viz., of C to D, and E to F; also, let the ratio of m to p, be that which is compounded of the ratios of m to n, n to o, and o to p, which are the same, each to each, to the remaining other ratios, viz., of M to N, O to P, and Q to R. Then the ratio of h to 1 shall be the same to the ratio of m to p; or h shall be to 1, as m to p. Demonstration. Because e is to f, as (G to H, that is, as) Y to Z; and f is to g, as (K to L, that is, as) Z to a; therefore, ex æquali, |