Next, Let there be four magnitudes A, B, C, D, and other four E, F, G, H, which, taken two and two in a cross order, have the fame 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. A. B. C. D. Because A, B, C are three magnitudes, and F, G, H other three, which, taken two and two in a cross order, have the fame ratio; by the first cafe, A is to C, as F to H. but C is to D, as E is to F; wherefore again, by the first cafe, A is to D, as E to H. and fo on, whatever be the number of magnitudes. Therefore if there be any number, &c. Q. E. D. PROP. XXIV. THEOR. F the firft has to the fecond the fame ratio which the see N. IF third has to the fourth; and the fifth to the second the fame ratio which the fixth has to the fourth; the first and fifth together shall have to the fecond, the fame ratio which the third and fixth together have to the fourth. Let AB the firft have to C the fecond the fame ratio, which DE the third has to F the fourth ; and let BG the fifth have to C the fecond the fame ratio, which EH the fixth has to F the fourth. AG, the first G and fifth together, fhall have to C the fecond the fame ratio, which DH, the third and fixth together, has to F the fourth. B А С E H DF 2. 22. 5. lyb; as therefore AG is to GB, fo is DH to HE; but as GB to b. 18. 5. C, fo is HE to F. Therefore, ex aequali, as AG is to C, fo is DH to F. Wherefore if the firft, &c. Q. E. D. COR. 1. If the fame Hypothefis be made as in the Propofition, the excefs of the first and fifth fhall be to the fecond, as the excess of the third and fixth to the fourth. the Demonftration of this is the Book V. fame with that of the Propofition, if Division be used instead of Compofition. COR. 2. The Propofition holds true of two ranks of magnitudes, whatever be their number, of which each of the first ranks has to a fecond magnitude the fame ratio that the correfponding one of the fecond rank has to a fourth magnitude; as is manifeft. PROP. XXV. THEOR. IF four magnitudes 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,&14 5. confequently F the leaft . AB together with F are greater than CD together with E. b. 19. 5. c. A. s. See N. G D H Take AG equal to E, and CH equal to F. then because as AB to CD, fo 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; likewife the remain- B der GB fhall be to the remainder HD, as the whole AB is to the whole b CD. but AB is greater than CD, therefore 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 GB, HD, of which GB is the greater, C ACE F 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. RA PROP. F. THEO R. ATIOS which are compounded of the fame ratios, are the fame 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 fame with the ratio of D to F, which, by the fame 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 fame ratio; ex aequali, A is to C, as D to F. A. B. C. Next, Let A be to B, as E to F; and B to C, as D to E; therefore, ex aequali, in proportione perturbata, 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 B, and B to C, is the fame 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 Propofition may be demonstrated whatever be the number of ratios in either cafe. D. E. F. 2. 22. S b. 23. 5. IF F feveral ratios be the fame with feveral ratios, each See Ne to each; the ratio which is compounded of ratios which are the fame with the first ratios, each to each, is the fame with the ratio compounded of ratios which are the fame 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 fame with the ratios of A to B, and C to D. and as E to F, fo let N be to O; and as G to H, fo 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 fame with the ratios of E to F, and G to H. and it is to be fhewn that the ratio of K to M, is the fame 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, fo is (C to D, and fo is G to H, and fo Book V. is) O to P. ex aequali, K is to M, as N to P. Therefore if feveral ratios, &c. Q. E. D. a. 22. 5. PROP. H. THEOR. See N. tio. IF a ratio compounded of feveral ratios be the fame with a ratio compounded of any other ratios, and if one of the first ratios, or a ratio compounded of any of the firft, be the fame with one of the laft ratios, or with the ratio compounded of any of the laft; then the ratio compounded of the remaining ratios of the first, or the remaining ratio of the firft, if but one remain, is the fame with the ratio compounded of those remaining of the laft, or with the remaining ratio of the laft. A. B. C. D. E. F G. H. K. L. M. Let the first ratios be thofe of A to B, B to C, C to D, D to E, and E to F; and let the other ratios be thofe of G to H, H to K, K to L, and L to M. alfo let the ratio of A to F, which is coma. Defini-pounded of the first ratios be the fame tion of com- with the ratio of G to M, which is compounded rapounded 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 fame 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 firft 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 fame 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. b. E. 5. C. 22. 5. Because, by the Hypothefis, A is to D, as G to K, by inverfion b, D is to A, as K to G; and as A is to F, fo is G to M; therefore, ex aequali, D is to F, as K to M. If therefore a ratio which is, &c. QE. D. PROP. K. THEOR. Book V. IF there be any number of ratios, and any number of other ratios fuch, that the ratio compounded of ratios which are the fame with the firft ratios, each to each, is the fame with the ratio compounded of ratios which are the fame, each to each, with the laft ratios; and if one of the first ratios, or the ratio which is compounded of ratios which are the fame with feveral of the first ratios, each to each, be the fame with one of the lait ratios, or with the ratio compounded of ratios which are the fame, each to each, with feveral of the last ratios. then the ratio compounded of ratios which are the fame with the remaining ratios of the first, each to each, or the remaining ratio of the firft, if but one remain; is the fame with the ratio compounded of ratios which are the fame with thofe remaining of the laft, 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 fust 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 of the ratios of S to T, Sce N. T to V, and V to X, which are the fame with the ratios of A to P, K |