O to P, which are the same with the ratios of E to F, and G to H and it is to be shewn 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 * K is to M, as N to P. Therefore, 22. 5. if several ratios, &c. Q E. D. PROPOSITION H. THEOR. If a ratio which is compounded of several ratios, See N. 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, shall be 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*, 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. Definition of compound ratio. B. 5. + Hyp. * 22.5. K See N. PROPOSITION K. THEOR.-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 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 h, k, l. A, B; C, D; E, F. G, H; K, L; M, N; O, P ; Q, R. m, n, o, p. S, T, V, X. ratios of S to T, T to V, and V to X, which are the same to 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 c; 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, to 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 to 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 to 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 to 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, to the remaining other ratios, viz. of M to N, O to P, and Q to R: then the ratio of h to shall be the same to the ratio of m to p; or h shall be to l, as m to p. † 11.5. † B. 5. + Hyp. Because e is to f, as (G to H, that is, as) Y to Z; and ƒ is to g, as (K to L, that is, as) Z to a; therefore t ex æquali, † 22. 5. 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: but S is to Xt, 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: in like manner, it may be demonstrated, that m is to p, as a to d; and it has been shewn, that T is to X, as a to d; thereforeh 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 shew the true meaning of them when they are so expressed; especially, since they are very frequently made use of by geometers. + Hyp. 11. 5. THE ELEMENTS OF EUCLID. BOOK VI. DEFINITIONS. I. SIMILAR rectilineal figures are those 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." 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. IV. The altitude of any figure is the straight PROPOSITION I. THEOREM.-Triangles and parallelograms of the same alli- See N. tude, are one to another as their bases. Let the triangles ABC, ACD, and the parallelograms EC, CF, have the same altitude, viz. the perpendicular drawn from the point A to BD: as the base BC, is to the base CD, so shall the triangle ABC be to the triangle ACD, and the parallelogram EC to the parallelogram CF. * HG BC † 3. 1. Produce BD both ways to the points H, L, and + take any number of straight lines BG, GH, each equal to the base BC; and DK, KL, any number of them, each equal to the base CD; and join AG, AH, AK, AL: then, because CB, BG, GH are all equal, the triangles AHG, AGB, ABC are all equal: therefore, 38. 1. 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 38. 1. 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 the triangle AHC; and of the base CD and the triangle ACD, the second and fourth, have been taken any equimultiples whatever, viz. the base CL, and the triangle ALC; and since it has been shewn, that if the base HC be greater than the base CL, the triangle AHC is greater than the triangle ALC; and if equal, equal; and if less, less: therefore *, as the base BC is to the 5 Def. 5. base CD, so is the triangle ABC to the triangle ACD. 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 41. 1. 15. 5. |