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Book V.

C, and 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 I 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; 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 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 l, as m to p.

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Becaufe 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 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 becaufe h is to k, as (C to D, that is as) T to V; and k is to l, as (E to F, that is as) V to X; therefore, ex aequali, h is to l, as T to X. in like manner it may be demonftrated that m is to p, as a to d. and it was fhewn that T is to X, as a to d. a. 11. 5. therefore 2h is to 1, as m to p. Q. E. D.

The Propofitions G and K are usually, 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.

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ἐσ

are thofe which have their feveral angles equal, each to each, and the fides about the equal angles proportionals.

II.

Reciprocal figures, viz. triangles and parallelograms, are fuch as see N. "have their fides about two of their angles proportionals in

"fuch 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 "fide of the first."

III.

A straight 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 lefs.

IV.

The altitude of any figure is the straight line

drawn from its vertex perpendicular to the bafe.

Book VI.

See N.

TRIA

PROP. I. THEOR.

RIANGLES and parallelograms of the fame altitude are one to another as their bates.

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.

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 a. 38. 1. triangles AHG, AGB, ABC are all equal. therefore whatever

EA F

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 bafe LC is of
the bafe CD, the fame mul-
tiple is the triangle ALC of
the triangle ADC. and if the
bafe HC be equal to the
bafe CL, the triangle AHC
is alfo equal to the triangle

ALC; and if the bafe HCH GB C D K
K L
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 first 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 base CL and triangle ALC; and that it has been fhewn that if the bafe HC be greater than the base CL, the triangle AHC is greater than the b.5. Def. 5. triangle ALC; and if equal, equal; and if lefs, lefs. Therefore b as the bafe BC is to the bafe CD, fo is the triangle ABC to the triangle ACD.

And because the parallelogram CE is double of the triangle Book VI. ABC, and the parallelogram CF double of the triangle ACD, C. 41. T. and that magnitudes have the fame ratio which their equimultiples

have ; as the triangle ABC is to the triangle ACD, fo is the pa- d. 15. s rallelogram EC to the parallelogram CF. and because it has been fhewn 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 base CD, fo is the parallelogram EC e. 11. 5. to the parallelogram CF. Wherefore 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 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 bafes are f, because the perpen- f. 33. 5. diculars 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

PROP. II.
II. THEOR.

F a straight line be drawn parallel to one of the fides See N, of a triangle, it fhall cut the other fides, or thefe produced, proportionally. and if the fides, or the fides produced be cut proportionally, the flraight 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 base DE, and between the a. 37. 1. fame parallels DE, BC. ADE is another triangle, and equal magnitudes have to the fame, the fame ratio; therefore as the triangle b. 7. 5. 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 c. 1. 6. DA, because having the fame altitude, viz. the perpendicular drawn from the point E to AB, they are to one another as their bases. and

c

Book VI. for the fame reafon, as the triangle CDE to the triangle ADE, so is CE to EA. Therefore as BD to DA; fo is CE to EA d. Next, Let the fides AB, AC of the triangle ABC, or these pro

d. 11. 5.

c. x. 6.

C. 9. 5.

f. 39. I.

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duced, 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 parallel to BC.

The fame conftruction being made, becaufe as BD to DA, so 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 the triangle BDE is equal to the triangle CDE. and they are on the fame bafe DE; but equal triangles on the fame bafe are between the fame parallels f; therefore DE is parallel to BC. Wherefore if a straight line, &c. QE. D.

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F the angle of a triangle be divided into two equal angles, by a ftraight line which alfo cuts the bafe; the fegments of the bafe fhall have the fame ratio which the other fides of the triangle have to one another. and if the fegments of the bafe have the fame ratio which the other fides of the triangle have to one another, the ftraight line drawn from the vertex to the point of fection divides the vertical angle into two equal angles.

Let the angle BAC of any triangle ABC be divided in two equal angles by the ftraight line AD. BD is to DC, as BA to AC.

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