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 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 firft 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 i, as m to p. A, B, C, G, H; K, L, M, 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 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, as 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 shown, that T is to X, as a to d: Therefore a h is to 1, as in 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 fhow the true meaning of them when they are fo expreffed; especially fince they are very frequently made ufe of by geometers. THE "Reciprocal figures, viz. triangles and parallelograms, are see N. "fuch as have their fides about two of their angles propor"tionals in fuch manner, that a fide of the firft figure is to "a fide of the other, as the remaining fide of this other is to "the remaining fide of the firft." III. A ftraight line is faid to be cut in extreme and mean ratio, when the whole is to the greater fegment, as the greater fegment is to the less. IV. The altitude of any figure is the ftraight line drawn from its vertex perpendicular to the bafe. PROP. Book VI. See N. a 38. I. 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 bafe BC is to the base 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 bafe HC is of the bafe BC, the fame multiple is the triangle AHC of the triangle ABC: For the fame reafon, whatever multiple the base 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 base CL, the triangle AHC is also 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 shown, that, if the base HC be greater than the bafe CL, the triangle AHC is greater than the triangle ALC; and if equal, equal; and if lefs, lefs: Therefore, as the bafe BC is b 5. def. 5. to the bafe CD, fo is the triangle ABC to the triangle ACD. HG BC D K L And because the parallelogram CE is double of the triangle ABC1 Book VI. n ABC, and the parallelogram CF double of the triangle ACD, 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 fo as to have their bases in the fame ftraight line; and having drawn perpendiculars from the vertices of the triangles to the bafes, the straight line which joins the vertices is parallel to that in which their bafes are f, because the f 33. 1. perpendiculars are both equal and parallel to one another: Then, if the fame conftruction be made as in the propofition, the demonstration will be the fame. I' PROP. II. THEOR. F a ftraight line be drawn parallel to one of the fides of See N. a triangle, it shall cut the other fides, or those 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 CE to EA. Join BE, CD; then the triangle BDE is equal to the triangle CDE, because they are on the fame bafe DE, and be a 37. I. tween 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, because having the fame e 1. 6. altitude, viz. the perpendicular drawn from the point E to AB, they are to one another as their bafes; and for the fame reafon, C Book VI. reafon, as the triangle CDE to the triangle ADE, fo is CE to EA. Therefore, as BD to DA, fo is CE to EA d. d II. 5. Next, Let the fides AB, AC of the triangle ABC, or thefe A A E B C e 1. 6. 8 39. I. produced, be cut proportionally in the points D, E, that is, fo that BD be to DA, as CE to EA, and join DE: DE is paral• lel to BC. The fame conftruction being made, because as BD to DA, fo is CE to EA; and as BD to DA, fo is the triangle BDE to the triangle ADE; and as CE to EA, fo is the triangle CDE to the triangle ADE; therefore the triangle BDE is to the triangle ADE, as the triangle CDE to the triangle ADE; that is, the triangles BDE, CDE have the fame ratio to the triangle ADE; and therefore f the triangle BDE is equal to the triangle CDE: And they are on the fame base DE; but equal triangles on the fame base are between the fame parallels &; therefore DE is parallel to BC. Wherefore, if a straight line, &c. Q. E. D. F the angle of a triangle be divided into two equal angles, by a straight line which alfo cuts the bafe; the fegments of the base fhall have the fame ratio which the other fides of the triangle have to one another: And if the fegments of the base have the fame ratio which the other fides of the triangle have to one another, the straight line drawn from the vertex to the point of fection, di vides the vertical angle into two equal angles. Let the angle BAC of any triangle ABC be divided into two equal angles by the ftraight line AD: BD is to DC, as BA to AC. Through |