Then, by the definition of a compound ratio, the ratio of a g is compounded of the ratios of a: b, b: c, c: d, d: e, e: f, f:g, which are the same as the ratio of A: B, C: D, E : F, G : H, K: : L, M: N, each to each. Then will the ratio of hp be the ratio compounded of the ratios of hk, k: l, l : m, m : n, n: p, which are the same as the ratios of O: P, Q : R, S: T, V: W, X: Y, each to each. .. by the hypothesis agh: p. Also, let the ratio which is compounded of the ratios of A : B, C: D, two of the first ratios (or the ratios of a : c, for A: B = ab, and C: Db: c), be the same as the ratio of a: d, which is compounded of the ratios of a: b, b: c, c: d, which are the same as the ratios of O: P, Q: R, S: T, three of the other ratios. And let the ratios of hs, which is compounded of the ratios of h: k, k: m, m: n, ns, which are the same as the remaining first ratios, namely, E: F, G: H, K: L, M: N; also, let the ratio of eg, be that which is compounded of the ratios e : f, : g, which are the same, each to each, to the remaining other ratios, namely, V: W, X: Y. Then the ratio of hs shall be the same as the ratio of e: g; or h:se: g. f |