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Book V.

PRO P. K. THE O R.

If there be any number of ratios, and any number of see N.

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 compound. ed of ratios which are the same with several of the firit 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 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.

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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'l; and C to D, as T to V; and E to F, as V to X: Therefore, by the des finition of compound ratio, the ratio of S to X is compounded

h, k, 1. A, B, C, D, E, F.

S, T, V, X. G, H; K, L, M, N;0, P; Q R. Y, Z, a, b, c, d. e, f, g.

m, n, o, p.

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 : Allo 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 R 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

Book V. c to d, which are the same, each to each, with the ratios of m 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 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 same with the ratios of G to H, and K to L, two of the other ratios ; and let the ratio of h to l 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 tol is the same with the ratio of m to p, or h is to l, as m to p.

h, k, 1.
, B; C, D, E, F..

S, T, V, X.
G, H; K, L, M, N, O, P; Q, R. Y, Z, a, b, c, d.

e, f, g. m, n, o, p.

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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 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 to d: Also, because h is to k, as (C to D, that is, as) T to V; and k is to ), 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 & h is to 1, as m top. 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 fhew the true meaning of them when they are so expressed; especially since they are very frequently made use of by geometers.

2 II.5.

THE

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S

I. VIMILAR rectilineal figures

are those which have their several angles equal, each to each, and the fides about the equal angles proportionals.

II. “Reciprocal figures, viz. triangles and parallelograms, are Sce 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 fide 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 seg-
ment is to the less.

IV.
The altitude of any figure is the straight line

drawn from its vertex perpendicular to the
base.

PROP

Book VL

PRO P. I.

THE O R.

See N.

T

RIANGLES and parallelograms of the same altitude

are one to another as their bases.

:

Let the triangles ABC, ACD, and the parallelograms EC, CF bave the same 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 bale 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 a 38. 1.

are all equal * : Therefore, whatever multiple the base HC
is of the base BC, the same multiple is the triangle AHC
of the triangle ABC: For the fame reason, whatever multiple
the base LC is of the
base CD, the same mul-

E A F
tiple is the triangle ALC
of the triangle ADC :
And if the base HC be e-
qual to the base CL, the
triangle AHC is also e-
qual to the triangle
ALC, and if the base

HGBC
HC be greater than the

DK

K L base CL, likewise the triangle AHC is greater than the triangle ALC; and if lefs, lefs : 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 firft 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 shewn, that if the base HC be greater than the base CL, the

triangle AHC is greater than the triangle ALC; and if equal, bs, def. 5. equal ; and if less, less: Therefore b 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,

с

dis. Se

ABC', and the parallelogram CF double of the triangle ACD, Book VI. and that magnitudes have the same ratio which their equimul. riples have d; as the triangle ABC is to the triangle ACD, so !: 1; 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 bale BC is to the base CD, o is the parallelogram EC to the parallelogram CF. Where-cil. s. 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 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 f, because the f 33. 19 perpendiculars are both equal and parallel to one another: Then, if the same construction be made as in the propofition, the de. monstration will be the same.

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IF a straight line be drawn parallel to one of the sides of See N.

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 lection 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 C 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 ratiob; there-by.s. 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, so is © BD to DA, because having the same c 1.6. altitude, viz the perpendicular drawn from the point E to AB, they are to one another as their bases; and for the same

reason,

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