Book V. PROP. K. THE O R. 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 firft 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 feveral 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 firft, if but one remain; is the fame with the ratio compounded of ratios which are the -fame with those remaining of the laft, 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 firft 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 G, H; K, L; M, N, O, P; Q, R. Y, Z, a, b, c, d. 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 : Alfo as G to H, fo 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 fame 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 Book V. e 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 hypothefis, S is to X, as Y to d: Alfo 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 hypothefis, 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; alfo 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 fame 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 1 is the fame with the ratio of m to p, or h is to 1, as m to p. h, k, l. A, B, C, D, E, F. G, H; K, L, S, T, V, X. M, N, O, P, Q, R. Y, Z, a, b, c, d. a II. 5. 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 aequali, 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 inverfion, 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 to d: Alfo, because h is to k, as (C to D, that is, as) T to V; and k is to 1, 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 demonftrated, that m is to p, as a to d: And it has been fhewn, that T is to X, as a to d: Thereforeh is to 1, as m to p. Q. E. D. The propofitions G and K are ufually, for the fake of brevity, expreffed in the fame terms with propofitions F and H: And therefore it was proper to fhew the true meaning of them when they are fo expreffed; efpecially fince they are very frequently made ufe of by geometers. THE "Reciprocal figures, viz. triangles and parallelograms, are see N. A ftraight line is faid to be cut in extreme and mean ratio, IV. The altitude of any figure is the ftraight line drawn from its vertex perpendicular to the PROP. Book VL. See N. a 38. I. PROP. I. THEOR. RIANGLES and parallelograms of the fame altitude are one to another as their bafes. TR 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 bafe BC is to the bafe CD, fo is the triangle ABC to the triangle ACD, and the parallelogram EC to the parallelogram CF. E A F Produce BD both ways to the points H, L, and take any number of ftraight lines BG, GH, each equal to the bafe 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 HC is of the base BC, the fame multiple is the triangle AHC of the triangle ABC: For the fame reafon, whatever multiple the bafe LC is of the bafe CD, the fame multiple is the triangle ALC of the triangle ADC: And if the base HC be equal to the bafe CL, the triangle AHC is alfo equal to the triangle ALC; and if the bafe HC be greater than the bafe CL, likewife the triangle AHC is greater than the triangle ALC; and if lefs, lefs: Therefore, fince there are four magnitudes, viz. the two bafes BC, CD, and the two triangles ABC, ACD; and of the bafe BC and the triangle ABC, the firft and third, any equimultiples whatever have been taken, viz. the bafe HC and triangle AHC; and of the bafe 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 fhewn, that if the bafe HC be greater than the bafe CL, the triangle AHC is greater than the triangle ALC; and if equal, 5. def. 5 equal; and if lefs, lefs: Therefore as the bafe BC is to the bafe CD, fo is the triangle ABC to the triangle ACD. HGB C D K L And becaufe the parallelogram CE is double of the triangle ABC, C. d Is. Se ABC, and the parallelogram CF double of the triangle ACD, Book VI. and that magnitudes have the fame ratio which their equimul- in tiples have ; as the triangle ABC is to the triangle ACD, fo 4. 1. is the parallelogram EC to the parallelogram CF: And because it has been shewn, that as the bafe BC is to the bafe CD, fo is the triangle ABC to the triangle ACD; and as the triangle ABC to the triangle ACD, fo is the parallelogram EC to the parallelogram CF; therefore, as the bafe BC is to the bafe CD, to is the parallelogram EC to the parallelogram CF. Where- 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 bafes Let the figures be placed fo as to have their bafes in the fame ftraight line; and having drawn perpendiculars from the vertices of the triangles to the bafes, the ftraight line which joins the vertices is parallel to that in which their bafes are f, because the f 33. I` perpendiculars are both equal and parallel to one another: Then, if the fame conftruction be made as in the propofition, the demonftration will be the fame. IF a ftraight line be drawn parallel to one of the fides of a triangle, it fhall cut the other fides, or these produced, proportionally: And if the fides, or the fides produced, be cut proportionally, the ftraight line which joins the points of fection fhall 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. See N. a 37. I. Join BE, CD; then the triangle BDE is equal to the triangle CDE, because they are on the fame bafe DE, and between the fame parallels DE, BC: ADE is another triangle, and equal magnitudes have to the fame, the fame ratio; there- b 7. 5. fore, as the triangle BDE to the triangle ADE, fo is the triangle CDE to the triangle ADE; but as the triangle BDE to the triangle ADE, fo is BD to DA, becaufe having the fame e 1. 6. altitude, viz the perpendicular drawn from the point E to AB, they are to one another as their bases; and for the fame reason, |