Book V. PROP. XXIII. THEOR See N. IF there be any number of magnitudes, and as many others, which, taken two and two, in a cross order, have the same ratio, the first shall have to the last of the first magnitudes the same ratio which the first of the others has to the last. N. B. This is usually cited by the words " ex æquali in proportione per" turbata; or, " ex æquo perturbate." First, Let there be three magnitudes A, B, C, and other three D, E, F, which, taken two and two, in a cross order, have the same ratio, that is, such that A is to B, as E to F; and as B is to C, so is D to E: A is to C, as D to F. Take of A, B, D any equimultiples whatever G, H, K; and of C, E, F any equimultiples whatever L, M, N; and because G, H are equimultiples of A, B, and that magnitudes have the same ratio which their equimula 15. 5. tiples havea; as A is to B, so is G to H. And, for the same reason, as E is to F, so is M to N: but as A is to B, so is E to F; as therefore Gis to H, so is M to b 11.5. Nb. And because as B is to C, so is D to E, and that H, K are equimultiples of B, D, and L, M of C, E; as H is to L, so is K to M: and it has been shown, that G is to H, as M to N: then, because there are three magnitudes G, H, L, and other three K, M, N, which have the same ratio taken two and two in a cross order; if G be greater than L, C 4.5. A G ВС DEF HL KMN d 21. 5. K is greater than N; and if equal, equal; and if less, less d; and G, K are any equimultiples whatever of A, D, and L, N any whatever of C, F; as, therefore, A is to C, so is D to F. : 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 Das E to H. A. B. C. D. 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; by 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. Q. E. D. PROP. XXIV. THEOR. IF the first has to the second the same ratio which See N. 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 G the sixth has to F the fourth: 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 H B fourth. E 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 EH: and because these magnitudes are proportionals, they shall likewise be proportionals when taken jointly b: as, therefore, AG is to GB, so is DH to HE; but as GB to C, so is HE to F. Therefore, ex aqualia, as AG is to C, so is DH to F. Wherefore, if the first, &c. Q. E. D. a 22. 5. A CD F b 18. 5. 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 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, and a A. & consequently F the leasta. AB, together with F, are greater 14. 5. than CD, together with E. b 19. 5. CA. 5. 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, thereforec 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 B G-D H AC EF See N. 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; AB and F together are greater than CD and E. Therefore, if four magnitudes, &c, Q. E. D. PROP. F. THEOR. RATIOS which are compounded of the same ra. tios, are the same with one another. Let A be to B as D to E; and B to Cas E to F: the ratio which Book V. is compounded of the ratio 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. A. B. C. 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 æquali, A is to Cas D to Fa. a 22. 5. b 23.5. A. B. C. D. E. F. Next, Let A be to Bas E to F, and B to Cas D to E; therefore, ex æquali in proportione perturbatab, A is to Cas D to F; that is, 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. PROP. G. THEOR. IF several ratios be the same with several ratios, See N. 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. Let A be to Bas E to F; and C to Das G to H: and let A be to Bas K to L; and C to Das 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: and as A. B. C. D. K. L. M. 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 Mas N to P. Because K is to Las (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 is) Book V. O to P: ex æqualia, K is to Mas N to P. Therefore, if several ratios, &c. Q. E. D. a 22. 5. See N. PROP. H. THEOR. IF a ratio 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 is com a Defini- pounded of the first ratios, be the same with the ratio of G to M, which tion of com ra is compounded of the other pounded tios: and besides, let the ratio of A to D, which is compounded of the ratio. b В. 5. c. 22. 5. A. B. C. D. E. F. 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, is 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. Because, by the hypothesis, A is to Das G to K, by inversionb, Dis to A as K to G; and as A is to F, so is G to M; therefore, ex aquali, D is to Fas K to M. If, therefore, a ratio which is, &c. Q. E. D. |