of com pounded ratio. 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 A. B. C. D. E. F. * Definition 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. + B. 5. 1 22.5. Because, by the hypothesis, A is to D, as G to K, by inversion†, D is to A, as K to G: And as A is to F, so is G to M; therefore‡, ex æquali, D is to F, as K to M. If, therefore, a ratio which is, &c. Q. E. D. PROPOSITION 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 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; and K to L, as Z to a; M to N, as a to b; O to P, as b to e; and Q to R, as c 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, each to each, with the ratios of G to H, K to L, M to N, O to P, and Q to R: therefore, by the hypothesis, S is to X, as Y to d. Also, let the ratio of A to B, that is, the ratio of S to T, which is one of the first ratios, be the same with the ratio of e to g, which is compounded of the ratios of e to f, and f to g, which, by the hypothesis, are the same with the ratios of G to H, and K to L, two of the other ratios; and let the ratio of h to be that which is compounded of the ratios of h to k, and k to l, which are the same with the remaining first ratios, viz. of C to D, and E to F; also, let the ratio of m to p, be that which is compounded of the ratios of m to n, n to o, and o to p, which are the same, each to each, with the remaining other ratios, viz. of M to N, O to P, and Q to R: then the ratio of h to l is the same with the ratio of m to p; or h is to l, as m to p. Because e is to f, as (G to H, that is, as) Y to Z; and f is to g, as (K to L, that is, as) Z to a; therefore, ex æquali, e is to g, as Y to a: And by the hypothesis, A is to B, that is, S to T, as e to g; wherefore, S is to T, as Y to a; and, by inversion, T is to S, as a to Y: And S is to X, as Y to d; therefore, ex æquali, T is to X, as a to d: Also, because h is to k, as (C to D, that is, as) T to V; and k is to l, as (E to F, that is, as) V to X; therefore, ex æquali, h is to l, as T to X. 11. 5. In like manner it may be demonstrated, that m is to p, as a to d: And it has been shown, that T is to X, as a to d; therefore* h is to l, as m to p. Q. E. D. The propositions G and K are usually, for the sake of brevity, expressed in the same terms with propositions F and H; and therefore it was proper to show the true meaning of them when they are so expressed; especially since they are very frequently made use of by geometers. BOOK VI. DEFINITIONS. I. SIMILAR rectilineal figures are those which have their se veral angles equal, each to each, and the sides about the equal angles proportionals. II. "Reciprocal figures, viz. triangles and parallelograms, are "such as have their sides about two of their angles pro"portionals in such a manner, that a side of the first figure is to a side of the other, as the remaining side "of this other is to the remaining side of the first." 66 III. A straight line is said to be cut in extreme and mean ratio, when the whole is to the greater segment as the greater segment is to the less. THEOR.-Triangles and parallelograms of the same altitude are one to another as their bases. Let the triangles ABC, ACD, and the parallelograms EC, CF, have the same altitude, viz. the perpendicular * 3. 1. + 38. 1. ↑ 38. 1. drawn from the point A to BD; then, as the base BC is to the base CD, so is the triangle ABC to the triangle ACD, and the parallelogram EC to the parallelogram CF. Produce BD both ways to the points H, L, and * take any number of straight lines BG, GH, each equal EA F HG BC D K L Then, because CB, BG, GH are all equal, the triangles AHG, AGB, ABC are all equalt: therefore, whatever multiple the base HC is of the base BC, the same multiple is the triangle AHC of the triangle ABC. For the same reason, whatever multiple the base LC is of the base CD, the same multiple is the triangle ALC of the triangle ADC: And if the base HC be equal to the base CL, the triangle AHC is also equal to the triangle ALC:‡ and if the base HC be greater than the base CL, likewise the triangle AHC is greater than the triangle ALC; and if less, less: Therefore, since there are four magnitudes, viz. the two bases BC, CD, and the two triangles ABC, ACD; and of the base BC, and the triangle ABC, the first and third, any equimultiples whatever have been taken, viz. the base HC and triangle AHC; and of the base CD and triangle ACD, the second and fourth, have been taken any equimultiples whatever, viz. the base CL and triangle ALC; and that it has been shown that, if the base HC be greater than the base CL, the triangle AHC is greater than the triangle 1 5 Def. 5. ALC; and if equal, equal; and if less, less: therefore, as the base BC is to the base CD, so is the triangle ABC to the triangle ACD. § 41. 1. 15.5. And because the parallelogram CE is double of the triangle ABC§, and the parallelogram CF double of the triangle ACD, and that magnitudes have the same ratio which their equimultiples have¶; as the triangle ABC is to the triangle ACD, so is the parallelogram EC to the parallelogram CF: And because it has been shown, that, as the base BC is to the base CD, so is the triangle ABC to the triangle ACD; and as the triangle ABC is to the triangle ACD, so is the parallelogram EC to the parallelogram CF; therefore, |