to N: and because there are three magnitudes G, K, M, and other three H, L, N, which, two and two, have the same ratio: if G be greater than M, H is greater than N; and if equal, equal; and if less, less (20. 5.); and G, H are any equimultiples whatever of A, D, and M, N are any equimultiples whatever of C, F. Therefore, (5. def. 5.), as A is to C, so is D to F. GKM HLN A. B. C. D. Next, Let there be four magnitudes, A, B, C, D, and other four, E, F, G, H, which, two and two, have the same ratio, viz. as A is to B, so is E to F, and as B to C, so F to G; and as C to D, so G to H: A shall be to D, as E to H. E. F. G. H. Because A, B, C are three magnitudes, and E, F, G other three, which, taken two and two, have the same ratio; by the foregoing case, A is to C, as E to G. But C is to D, as G is to H; 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. XXIII. THEOR. Ir 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 perturbata ;" or, "ex æquo perturbato."* 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 mag nitudes have the same ratio which their equimultiples have (15. 5.); 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 G is to H, so is M to N (11. 5.). And because as B is to A • See Note. B C D E F THE ELEMENTS OF EUCLID. 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 (4. 5.) 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, K is greater than N; and if equal, equal; and if less, less (21. BOOK V. G HL KM N 5.); 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. A. B. C. D. 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 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; 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. Ir the first has to the second the same ratio which 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 which the third and sixth together have to the fourth.* G 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 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 æquali (22. 5.), AB is to BG, as DE to EH: and because these magnitudes are proportionals, they shall likewise be proportionals when taken jointly (18. 5.): * See Note. B. E H as, therefore, AG is to GB, so is DH to HE; but as GB to C, so is HE to F. Therefore, ex æquali (22. 5.), 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 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. Ir 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 F the least (A. & 14. 15.). AB, together with F, are greater than CD, together with E. 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 (19. 5.) CD: but AB is greater than CD, therefore (A. 5.) 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, 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. A C E F 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 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 D to F, which by the same definition is compounded of the ratios of D to E, and E to F. D. E. F. * See Note. 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 C, as D to F (22. 5.). 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 perturbata (23. 5.), 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 same with the ratio D to F, which is compounded of the ratios of D to E, and E to F: and in like manner the propositions may be demonstrated, whatever be the number of ratios in either case. PROP. G. THEOR. D. E. F. IF several ratios be the same with several ratios, 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 ratios 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 is) O to P: ex æquali (22. 5.) K is to M, as N to P. Therefore, if several ratios, &c. Q. E. D. PROP. H. THEOR. Ir 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.* * See Note. A. B. C. D. E. F. G. H. K. L. M. 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 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, 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 D, as G to K, by inversion (B. 5.), D is to A, as K to G; and as A is to F, so is G to M; therefore (22. 5.), ex æquali, D is to F, as K to M. If therefore a ratio which is, &c. Q. E. D. PROP. K. THEOR. If there be any number of ratios, and any number 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, 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 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 of the ratios of S A, B; C, D; e, f, g. h, k, l, S, T, V, X. O, P; Q, R. Y, Z, a, b, c, d. m, n, o, p. 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; * Definition of compound ratio. + See Note. |