COR. 1. If the same hypothesis be made as in the proposition, the excess of the first and fifth shall be to the second, as the excess of the third and sixth to the fourth. The demonstration of this is the same with that of the proposition, if division be used instead of composition. COR. 2. The proposition holds true of two ranks of magnitudes, whatever be their number, of which each of the first rank has to the second magnitude the same ratio that the corresponding one of the second rank has to a fourth magnitude; as is manifest. PROP. XXV. THEOR. IF four magnitudes of the same kind are proportionals, the greatest and least of them together are greater than the other two together. Let the four magnitudes AB, CD, E, F; be proportionals, viz. AB to CD, as E to F; and let AB be the greatest of them, and consequently F the least (A. & 14. 15.). AB together with F, are greater than CD, together with E. D H! Take AG equal to E, and CH equal to F: then, because as AB is to CD, so is E to F, and that AG is equal to E, and CH equal to F; AB is to CD, as AG to B CH. And because AB the whole is to the whole CD, as AG is to CH, likewise the remainder GB shall be to G the remainder HD, as the whole AB is to the whole (19. 5.) CD: but AB is greater than CD, therefore (A. 5.) GB is greater than HD: and because AG is equal to E, and CH to F; AG and F together are equal to CH and E together, If, therefore, to the unequal magnitudes GB, HD, of which GB is the greater, there be added equal magnitudes, viz. to GB the two AG and F, and CH and E to HĎ; AB and F together are greater than CD and E. Therefore, if four magnitudes, &c. Q. E. D. A E F PROP. F. THEOR. RATIOS which are compounded of the same ratios, are the same with one another.* * A. B. C. Let A be to B, as D tò E; and B to C, as E to F: 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. D. E. 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 to F (22. 5.). Next, Let A be to B, as E to F, and B to C, as D to E; there A. B. C. D. E. F. fore, ex æquali in proportione perturbata PROP. G. THEOR. 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.* Let A be to B, as E to F; and C to D, as G to H: and let A be to B, as K to L; and C to D, as L 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 to F, so let N be to O; and as G to H, so let O be to P; then *See Notes. A. B. C. D. K. L. M. - 1 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 htol be that which is compounded of the ratios of h to k, and k to 1, 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 tol is the same with the ratio of m to p, or h is to 1, as m to o p. g; h, k, l, A, B, C, D; E, F. S, T, V, X. G, H; K, L; M, N, O, P; Q, R. Y, Z, a, b, c, d, 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 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 1, as T to X: 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 (11. 5.) h is to 1, 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. "Reciprocal figures, viz. triangles and parallelograms are such "as have their sides about two of their angles proportionals “in such 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 re"maining 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 line drawn from its vertex perpendicular to the base. * See Note. PROP. I. 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 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 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 (38. 1.): 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 E A F also equal to the triangle H G B C ALC (38. 1.); 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 ALC; and if equal, equal; and if less, less; therefore (5. def. 5.) as the base BC is to the base CD, so is the triangle ABC to the triangle ACD. And because the parallelogram CE is double of the triangle ABC (41. 1.) and the parallelogram CF double of the triangle * See Note |