Book V.

PROP. K. THEOR.

If there be any number of ratios, and any num- See N, ber of other ratios such, that the ratio 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, 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 same with several of the first ratios, each to each, be the same with one of the last ratios, or with the ratio compounded of ratios which are the same, each to each, with several of the last ratios: Then the ratio compounded of ratios which are the same with the remaining ratios of the first, each to each, or the remaining ratio of the first, if but one remain; is the same with the ratio compounded of ratios which are the same 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

h, k, l.

A, B; C, D; E, F. G, H; K, L; M, N; O, P; Q, R. e, f, g.

m, n, o, p.

S, T, V, X.
Y, Z, a, b, c, d.

compounded of the ratios of S to T, T to V, and V to X, which are the same 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, 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

L

BOOK V. 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 tod: 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 h to 1 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 to 1 is the same with the ratio of m to p, or h is to l, as m to p.

and,

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 æquali, e is to g, as Y to a: And by the hypothesis, A is to В, that is, S to T, as e to g; wherefore, S is to T, as Y to a; by inversion, T is to S, as a to Y; and S is to X, as Y tod; 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 1, 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 11.5. d; therefore a 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 Fand 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 See N.
"such as have their sides about two of their angles pro-
"portionals 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 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.

Book VI.

PROP. I. THEOR.

See N. TRIANGLES and parallelograms of the same altitude are one to another as their bases.

a 58. 1.

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 equala: 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 also equal to the triangle

E

F

ALC; and if the

base HC be greater HG

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, 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 tri

d

angle ABC, and the parallelogram CF double of the tri- Book VI. angle ACD, and that magnitudes have the same ratio 41. 1. which their equimultiples haved; as the triangle ABC is a 15. 5. 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 CD, so is the triangle ABC to the triangle ACD; and as the triangle ABC is to the triangle ACD, so is the parallelogram EC to the parallelogram CF; therefore, as the base BC is to the base CD, so ise 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 bases. Let their figures be placed so as to have their bases in the same straight 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 bases are, because the perpendiculars are both equal and 33.1. parallel to one another. Then, if the same construction be made as in the proposition, the demonstration will be the

same,

PROP. II. THEOR.

IF a straight line be drawn parallel to one of the See N. sides of a triangle, it shall cut the other sides, or these produced, proportionally: And if the sides, or the sides produced, be cut proportionally, the straight line which joins the points of section shall be parallel to the remaining side of the triangle.

Let DE be drawn parallel to BC, one of the sides of the triangle ABC: BD is to DA, as CE to EA.

b

Join BE, CD; then the triangle BDE is equal to the triangle CDEa, because they are on the same base DE, ▪ 37. 1. and between the same parallels DE, BC: ADE is another triangle, and equal magnitudes have to the same, the same ratiob; therefore, as the triangle BDE to the triangle ADE, 7.5. so is the triangle CDE to the triangle ADE, but as the triangle BDE to the triangle ADE, so is BD to DA, be- 1. 6. cause having the same altitude, viz. the perpendicular drawn from the point E to AB, they are to one another as their bases; and for the same reason, as the triangle CDE