Book V. PROP. K. THEOR. If there be any number of ratios, and any number of see N. other ratios fuch, that the ratio compounded of ratios which are the fame with the first ratios, each to each, is the fame with the ratio compounded of ratios which are the fame, 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 fame with several of the first ratios, each to each, be the fame with one of the last ratios, or with the ratio compounded of ratios which are the fame, each to each, with several of the last ratios : Then the ratio compounded of ratios which are the fame with the remaining ratios of the first, each to each, or the remaining ratio of the first, if but one remain; is the fame with the ratio compounded of ratios which are the fame 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 h, k, 1. S, T, V, X. A, B, C, D, E, F. e, f, g. m, n, o, p. of the ratios of S to T, T to V, and V to X, which are the fame 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 c; and Q to R, asctod: Therefore, by the same definition, the ratio of Y to dis compounded of the ratios of Y to Z, Z to a, a to b, b to c, and c to Book V. c to d, which are the fame, 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 fame 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 fame with the ratios of G to H, and K to L, two of the other ratios; and let the ratio of h to 1 be that which is compounded of the ratios of h to k, and k to 1, which are the fame 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 too, and o to p, which are the fame each to each, with the remaining other ratios, viz. of M to Ν, Ο to P, and Q to R: Then the ratio of h tol is the fame with the ratio of m to p, or his to l, as m to p. a II. 5. G, H; K, L, M, N, O, P, Q, R. Y, Z, a, b, c, d. m, n, o, p. Because e is to f, as (G to H, that is, as) Y to Z; and fis to g, as (K to L, that is, as) Z to a; therefore, ex aequali, e is to g, as Y to a: And, by the hypothefis, 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 aequali, T is to X, es a tod: 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 aequali, 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 shewn, that T is to X, as a tod: Therefore his to l, asm to p. Q. E. D. The propofitions G and K are usually, for the fake of brevity, expressed in the fame terms with propositions F and H: And therefore it was proper to thew the true meaning of them when they are so expressed; especially since they are very frequently made use of by geometers. THE "Reciprocal figures, viz. triangles and parallelograms, are See N. "such as have their fides about two of their angles propor"tionals in such manner, that a fide of the first figure is to "a fide of the other, as the remaining side of this other is to "the remaining fide of the first." 111. A ftraight 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 bafe. PROP. Book VL See N. a 38. 1. T PROP. I. THEOR. RIANGLES and parallelograms of the fame altitude are one to another as their bases. Let the triangles ABC, ACD, and the parallelograms EC, CF have the fame altitude, viz. the perpendicular drawn from the point A to BD: Then, as the base BC is to the base CD, fo 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 bafe CD; and join AG, AH, AK, AL: Then, because CB, BG, GH are all equal, the triangles AHG, AGB, ABC are all equal: Therefore, whatever multiple the base не is of the base BC, the fame multiple is the triangle AHC of the triangle ABC: For the fame reason, whatever multiple the bafe LC is of the qual to the triangle HGBC EA F DKL base CL, likewife the triangle AHC is greater than the triangle ALC; and if less, less: Therefore, fince 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 fecond and fourth, have been taken any equimultiples whatever, viz. the bafe CL and triangle ALC; and that 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, 5. def. 5. equal; and if less, less: Therefore 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, and the parallelogram CF double of the triangle ACD, Book VI. and that magnitudes have the fame ratio which their equimultiples have d; as the triangle ABC is to the triangle ACD, fo41. is the parallelogram EC to the parallelogram CF: And because it has been shewn, that as the base BC is to the base CD, fo is the triangle ABC to the triangle ACD; and as the triangle ABC to the triangle ACD, so is the parallelogram EC to the parallelogram CF; therefore, as the base BC is to the base CD, to is the parallelogram EC to the parallelogram CF. Where-e 11.5. fore triangles, &c. Q.E. D. Cor. From this it is plain, that triangles and parallelograms that have equal altitudes, are one to another as their bases. Let the figures be placed so as to have their bases in the same ftraight line; and having drawn perpendiculars from the vertices of the triangles to the bases, the straight line which joins the vertices is parallel to that in which their bafes are f, because the f 33. י perpendiculars are both equal and parallel to one another: Then, if the fame construction be made as in the proposition, the demonstration will be the fame. PROP. II. THEOR IF a straight line be drawn parallel to one of the fides of see N. a triangle, it shall cut the other fides, or these produced, proportionally: And if the fides, or the fides produced, be cut proportionally, the straight line which joins the points of section shall be parallel to the remaining fide of the triangle. Let DE be drawn parallel to BC one of the fides of the triangle ABC: BD is to DA, as C to EA. a 37, I. Join BE, CD; then the triangle BDE is equal to the triangle CDE, because they are on the fame base DE, and between the fame parallels DE, BC: ADE is another triangle, and equal magnitudes have to the fame, the same ratiob; there- b 7.5. fore, as the triangle BDE to the triangle ADE, so is the triangle CDE to the triangle ADE; but as the triangle BDE to the triangle ADE, so is BD to DA, because having the same c r. 6. altitude, viz the perpendicular drawn from the point E to AB, they are to one another as their bases; and for the same reason, |