A. B. C. D. Next, Let there be four magnitudes, A, B, C, D, and other Book V four E, F, G, H, which, taken two and two in a cross order, have the same ratio, viz. A to B, as G to H; B to C, as F to G; and C to D, as E to F: A is to D, as E to H. E. F. G. H. Because A, B, C are three magnitudes, and F, G, H other three, which, taken two and two in a cross order, have the same ratio; but the first case, A is to C, as F to H: but C is to D), as E is to F; wherefore again, by the first case, A is to D, as E to H: and so on, whatever be the number of magnitudes. Therefore, if there be any number, &c. 6217 PROP. XXIV. THEOR. Q. E. D. Work IF the first has to the second the same ratio which See Note. the third has to the fourth; and the fifth to the second the same ratio which the sixth has to the fourth; the first and fifth together shall have to the second the same ratio which the third and sixth together have to the fourth. 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: "AG, the G 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. Because BG is to C, as EH to F; by inversion, C is to BG, as F to EH: and because as AB is to C, so is DE to F; and as C to BG, so F to EH; ex æqualia. AB is to BG, as DE to EII: and because these magnitudes are proportionals, they shall likewise be proportionals when taken. jointly b: as, therefore, AC is to GB, so is B E H DH to HE: but as GB to C, so is HE to F. Therefore, ex æqualia, as AG is to C, so is DH to F. 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 a 22. 5. b 18: 5 Book V. 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. PROP. XXV. THEOR. 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, a A. & 14. and consequently F the leasta. AB, together with F, are greater than CD, together with E. 15. b 19. 5. c A. 5. B G D H. Take AG equal to E, and CH equal to F: then, because as AB is to CD, so is E to F, and that AG is equal to E, and CH equal to F; AB is to CD, as AG to CH. And because AB the whole is to the whole CD, as AG is to CH, likewise the remainder GB shall be to the remainder HD, as the whole AB is to the whole b CD: but AB is greater than CD, therefore e GB is greater than HD: and because AG is equal to E, and CH to F; AG and F together are equal to CH and E together. If, therefore, to the unequal magnitudes A GB, HD, of which GB is the greater, E there be added equal magnitudes, viz. to GB the two AG and F, and CH and E to HD; AB and F together are greater than CD and E. Therefore, if four magnitudes, &c. Q. E. D. See Note. PROP. F. THEOR. RATIOS which are compounded of the same ratios, are the same with one another. A. B. C, Let A be to B, as D to E; and B to C, as E to F: the ratio Book V. 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, is 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. D. E. F. 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 quali, A is to C, as D to Fa. A. B. C. Next, Let A be to B, as E to F, and B to C, as D to E; therefore, ex æquali in proportione perturbatab, A is to C, as D to F; that is, the ratio of A to C, which is compounded of the ratios of A to D. E. F. 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. PROP. G. THEOR. a 22. 5. b 23.5. 637 IF several ratios be the same with several ratios, See Note. each to each; the ratio which is compounded of ratios which are the same with the first ratios, each to each, is the same with the ratio compounded of ratios which are the same with the other ratios, each to each. A. B. C. D. K. L. M. 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 ra tios of A to B, and C to D: and as E to F, so let N be to O; 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 O, and O 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. Because K is to L, as (A to B, that is, as E to F, that is, as) N to O; and as L to M, so is (C to D, and so is G to H, and so Book V. is) O to P: ex æqualia, K is to M, as N to P. Therefore, if several ratios, &c. Q. E. D. a 22.5. See Note. PROP. H. THEOR. IF a ratio be compounded of several ratios be the same with a ratio compounded of any other ratios, and if one of the first ratios, or a ratio compounded of any of the first, be the same with one of the last ratios, or with the ratio compounded of any of the last; then the ratio compounded of the remaining ratios of the first, or the remaining ratio of the first, if but one remain, is the same with the ratio compounded of those remaining of the last, or with the remaining ratio of the last. 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, H to K, K to L, and L. to M : also, let the ratio of A to F, which a Definition is compounded of a the first ratios, be of compounded ratio. b B 5. c 22. 5. A. B. C. D. E. F. the same with the ratio of G to M, Because, by the hypothesis, A is to D, as G to K, by inversion b, D is to A, as K to G ; and as A is to F, so is G to M; therefore, ex æquali, D is to F, as K to M. If therefore a ratio which is, &c. Q. E. D. IF there be any number of ratios, and any number See Note. of other ratios such, that the ratio compounded of ratios which are the same with the first ratios, each to each, is the same with the ratio compounded of ratios which are the same, each to each, with the last ratios; and if one of the first ratios, orthe ratio which is compounded of ratios which are the same with several of the first ratios, each to each, be the same with one of the last ratios, or with the ratio compounded of ratios which are the same, each to each, with several of. the last ratios: then the ratio compounded of ratios which are the same with the remaining ratios of the first, each to each, or the remaining ratio of the first, if but one remain; is the same with the ratio compounded of ratios which are the same with those remaining of the last, each to each, or with the remaining ratio of the last. 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 T to V; and E to F, as V to X: therefore, by the definition of compound ratio, the ratio of S to X is compounded ́h, k, l. A, B; C, D; E, F. G, H; K, L, M, N, O, P, Q, R. e, f, g. m, n, o, p. S, T, V, X, of the ratios of S to T, T to V, and V to X, which are the ́same with 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, O 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 U |