Next, Let there be four magnitudes A, B, C, D, Book V. and other four E, F, G, H, which, taken two and A.B. C.D. two in a cross order, have the fame ratio, viz. A E. F. G. H. 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. 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 fo on, whatever be the number of magnitudes. Therefore if there be any number, &c. Q. E. D. PROP. XXIV. THE O R. IF F the first has to the second the same ratio which the Sce N. third has to the fourth; and the fifth to the second the same ratio which the sixth lias to the fourth; the first and fifth together shall have to the second, the fame ratio which the third and fixth together have to the fourth. Let AB the first have to the second the fame 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 fixth has to F the fourth. AG, the first G and fifth together, shall have to C the fe H cond the fame ratio, which DH, the third and sixth together, has to F the fourth. Because BG is to C, as EH to F; by B. E 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, fo F to EH; ex aequali · AB is to BG, as DE to EH. and because these magnitudes are proportionals, they Mall likewise be proportionals when taken joint AC DF lyb; as therefore AG is to GB, fo is DH to HE; but as GB co b. 18.5. C, fo is HE to F. Therefore, ex aequali“, as AG is to C, so is DH to F. Wherefore if the first, &c. Q. E. D. COR. 1. If the fame 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 fixth to the fourth. the Demonstration of this is the 2. 22. 5. Book V. fame 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 cach of the first ranks has to a second magnitude the fame ratio that the corresponding one of the second rank has to a fourth magnitude ; as is mania fest. IF 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. consequently F the least". AB together with F are greater than CD together with E. 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; likewise the remain- B der GB shall be to the remainder HD, as D the whole AB is to the whole 6 CD. but G HI AB is greater than CD, therefore < GB А СЕ Е 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. . E. D. b. 19. 5 PROP. F. THEO R. ATIOS which are compounded of the same ratios, are the same with one another. 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 ra A. B. C. tio, is the ratio of A to C, is the same with the ra D. E. F. tio of D to F, which, by the fame Definition, is compounded of the ratios of D to E, and E to 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 aequali, A is to C, as D to F. 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 b. 23. 5 to C, as D to F; that is, the ratio of A to C, which A. B. C. 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 D. E. F. 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. 2. 22. Si PROP. G. THEO R. IF feveral ratios be the fame with several ratios, each see Ho 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 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 A.B.C.D. K. L. M. ratios of K to L, and L to M, which are the same with the ratios of A E. F.G. H. N.O.P. to B, and C to D. and as E to F, fo 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 O to P, which are the same with the ratios of E to F, and G to H. and it is to be sewn 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 0; and as L to M, fo is (C to D, and fo is G to H, and so Therefore if le Rook V. is) O to P. ex aequali", K is to M, as N to P. veral ratios, &c. Q. E. D. a. 22. 5. PROP. H. THEOR. Sec N. Fa ratio compounded of feveral 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 É, 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 com2. Defini-pounded • of the first ratios be the same tion of com- with the ratio of G to M, which is compounded ra A. B. C. D. E. F. pounded of the other ratios. and besides, tio. G. H. K. L. M. 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, 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. Because, by the Hypothesis, A is to D, as G to K, by inver. b. B. 5. fion b, D is to A, as K to G; and as A is to F, so is G to N; therefore ', ex aequali, D is to F, as K to M. If therefore a ratio which is, &c. 0. E. D. C. 22. S. Book V. PROP. K. THEOR. Sce.V. IF there be any number of ratios , and any number of other ratios fuch, 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, cach to each, with the lait ratios; and if one of the first ratios, or the 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 lait ratios, or with the ratio compounded of ratios which are the same, cach 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 renaining of the last, each to each, or with the remaining ratio of the lait. Let the ratios of A to B, C to D, E to Fbe the first ratios; ani 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, h, k, 1. m, n, o, P. S, T, V, X. T to V, and V to X, which are the same with the ratios of A to B, K |