Now by the 5th Definition if A is > B, and also > C, then C is > D, and B is > D, therefore both (C – D) and (B — D) are positive; therefore, THEOREM.-If ratios are compounded of the same ratios, they are the same with one another. A B C DE F DEMONSTRATION. Let A be to B, as D is to E; and B to C, as E is to F: then 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 of D to F, which, by the same 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 æquali, A is to C, as D is to F (a). A B C DE F Next, let A be to B, as E is to F, and B to C, as D is to E; therefore, ex æquali in proportione perturbatâ, as A is to C, so is D to F (b); 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. (a) V. 22. SCHOLIUM. This and the three following propositions have been added by Simson; the two last, propositions H and K, are not read at the Universities. The foregoing proposition may be algebraically expressed as follows, and its truth is then evident: THEOREM.-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. be to B, as E is to F; and C to D, as G is to H: and let A be to B, as K is to L; and C to D, as L is 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 is 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 is to P. Because K is to L, as (A is to B, that is, as E is to F, that is, as) N is to 0; and L is to M, as (C is to D, that is, as G is to H, that is, as) O is to P: therefore, ex æquali, K is to M, as N is to P (a). A E B C D K L M F G H N O P (a) V. 22. SCHOLIUM. This proposition may be algebraically expressed as follows: THEOREM.-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. A B C D E F G H K L M DEMONSTRATION. 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, and 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, viz. 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 is to K; by inversion, D is to A, as K is to G (a); and as A is to F, so is G to M; therefore, ex æquali, D is to F, as K is to M (6). SCHOLIUM. follows: (a) V. B. (b) V. 22. The foregoing proposition, algebraically expressed, is as THEOREM. If A: F be compounded of A: B, B: C, C: D, D: E, E : F, and G: M, be compounded of G: H, H; K, K: L, L: M; THEOREM.-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. DEMONSTRATION. 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 is to T; and C to D, as T is to V; and E to F, as V is to X; therefore, by the definition of compound ratio, the ratio S to X is compounded of h, k, 1. C, D; E, F; A, B; 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, let G be to H, as Y is to Z; and K to L, as Z is to a; M to N, as a is to b; O to P, as b is to c; and Q to R, as c is 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 c to d, which are the same, |