the ratio of N to P is compounded of the ratios of N to O, and O to P, which are the same with the ratios of E to F, and G to H and it is to be shown that the ratio of K to M, is the same with the ratio of N to P, or that K is to M, as N to P.

Because K is to L, as (A to B, that is, as E to F, that is, as) N to O; and as L to M, so is (C to D, and so is G to H, and so is) O to P: ex æquali (22. 5.) K is to M, as N to P. Therefore, if several ratios, &c. Q. E. D.

PROP. H. THEOR.

Ir a ratio compounded of several ratios be the same with a ratio compounded of any other ratios, and if one of the first ratios, or a ratio compounded of any of the first, be the same with one of the last ratios, or with the ratio compounded of any of the last; then the ratio compounded of the remaining ratios of the first, or the remaining ratio of the first, if but one remain, is the same with the ratio compounded of those remaining of the last, or with the remaining ratio of the last.*

A. B. C. D. E. F.
G. H. K. L. M.

Let the first ratios be those of A to B, B to C, C to D, D to 'E, and E to F: and let the other ratios be those of G to H, H to K, K to L, and L to M: also, let the ratio of A to F, which is compounded oft the first ratios, be the same with the ratio of G to M, which is compounded of the other ratios; and besides, let the ratio of A to D, which is compounded of the ratios of A to B, B to C, C to D, be the same with the ratio of G to K, which is compounded of the ratios of G to H, and H to K, then the ratio compounded of the remaining first ratios, to wit, of the ratios of D to E, and E to F, which compounded ratio is the ratio of D to F, is the same with the ratio of K to M, which is compounded of the remaining ratios of K to L, and L to M of the other ratios.

Because, by the hypothesis, A is to D, as G to K, by inversion (B. 5.), D is to A, as K to G; and as A is to F, so is G to M; therefore (22. 5.), ex æquali, D is to F, as K to M. If therefore a ratio which is, &c. Q. E. D.

* See Note.

† Definition of compounded ratio.

PROP. K. THEOR.

If there be any number of ratios, and any number 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, 0 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, l,

G, H: K, L, M, N ; 0, 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 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, 0 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 each, with the ratios of G to H, K to L, M to N,

* See Note.

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 h to 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 I is the same with the ratio of m to p, or h is to l, as m to p.

G, H; K, L; M, N; 0, 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 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 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 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.

1

* See Note.

U

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

also equal to the triangle HG BC
ALC (38. 1.); and if the H G B C D

F

K

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.