PROP. F. THEO. Ratios which are compounded of the same ratios are the same Then the ratio which is compounded of the ratios of A: B, B: C, C: D, D: E, or the ratio of AE, is the same as the ratio compounded of the ratios of F: G, G: H, H: K, K: L, or the ratio of F: L. For A x B x C × D and L and or the ratio of A: E is the same as the ratio of F: L. The same may be demonstrated of any number of ratios so Then the ratio which is compounded of the ratios of A: B, B: C, C: D, D: E, or the ratio of A: E, is the same as the ratio compounded of the ratios of K: L, H: K, G : H, F: G, or the ratio of F: L. or the ratio of A: E is the same as the ratio of F .. Ratios which are compounded, &c. : L. PROP. G. THEO. If several ratios be the same to several ratios, each to each, the ratio which is compounded of ratios which are the same to the first ratios, each to each, shall be the same to the ratio compounded of ratios which are the same to the other ratios, each to each. If a ratio which is compounded of several ratios be the same to a ratio which is compounded of several other ratios, and if one of the first ratios, or the ratio which is compounded of several of them, be the same to one of the last ratios, or to the ratio which is compounded of several of them, then the remaining ratio of the first, or, if there be more than one, the ratio compounded of the remaining ratios, shall be the same to the remaining of the last; or, if there be more than one, to the ratio compounded of these remaining ratios. Let A B, B: C, C : D, D: E, E : F, F: G, G: H, be the first ratios, and P: Q, Q : R, R: S, S : T, T : X, the other ratios; also, let A: H, which is compounded of the first ratios, be the same as the ratio of P: X, which is the ratio compounded of the other ratios; and, let the ratio of A: E, which is compounded of the ratios of A: B, B: C, C: D, D E, be the same as the ratio of P: R, which is compounded of the ratios P: Q, Q: R. Then the ratio which is compounded of the remaining first ratios, that is, the ratio compounded of the ratios E : F, F : G, G: H, that is, the ratio of E: H, shall be the same as the ratio of R X, which is compounded of the ratios of R: S, S: T, T: X, the remaining other ratios. PROP. K. THEO. If there be any number of ratios, and any number of other ratios, such that the ratio which is compounded of ratios, which are the same to the first ratios, each to each, is the same to the ratio which is compounded of ratios, which are the same, each to each, to the last ratios-and if one of the first ratios, or the ratio which is compounded of ratios, which are the same to several of the first ratios, each to each, be the same to one of the last ratios, or to the ratio which is compounded of ratios, which are the same, each to each, to several of the last ratios-then the remaining ratio of the first; or, if there be more than one, the ratio which is compounded of ratios, which are the same, each to each, to the remaining ratios of the first, shall be the same to the remaining ratio of the last; or, if there be more than one, to the ratio which is compounded of ratios, which are the same, each to each, to these remaining ratios. Let A: B, C : D, E : F, G: H, K: L, M : N, be the first ratios, and O P, Q: R, S: T, V: W, X: Y, the other ratios; |