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But A is to B as E is to

[Hypothesis. Therefore G is to H as M is to N.

[V. 11. And because B is to C as D is to E, [Hypothesis. and that H and Kare е н т км л equimultiples of B and D, and L and M are equimultiples of C and E; [Constr. therefore H is to L as K is to M.

[V. 4. And it has been shewn that G is to H as M is to N.

Then since there are three magnitudes G, H, L, and other three K, M, N, which have the same ratio, taken two and two in a cross order; therefore if G be greater than L, K is greater than N; and if equal, equal; and if less, less. But G and K are any equimultiples whatever of A and D, and L and N are any equimultiples whatever of C and F; therefore A is to C as D is to F.

[V. Definition 5. Next, let there be four magnitudes A, B, C, D, and other four E, Ě, G, H, which have the same ratio, taken two A. B. C. D. and two in a cross order ; namely, let E. F. G. H. A be to B as G is to H, and Boto C as F is to G, and C to Das E is to F: A shall be to Das E is to H.

For, because A, B, C are three magnitudes, and F, G, H other three, which have the same ratio, taken two and two in a cross order;

[Hypothesis. therefore, by the first case, A is to C as Fis to H. But C is to D as E is to F;

[Hypothesis. therefore also, by the first case, A is to D as E is to H.

And so on, whatever be the number of magnitudes,
Wherefore, if there be any number &c. Q.E.D.

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PROPOSITION 24. THEOREM. If the first have to the second the same ratio which the third has to the fourth, and the fifth have to the second the same ratio which the sixth has to the fourth, then the first and fifth together shall have to the second the same ratio which the third and sixth together have to the fourth.

Let AB the first have to C the second the same ratio which DE the third has to F the fourth; and let BG the fifth have to C the second the same ratio which EH the sixth has to F the fourth: AG, the first and fifth together, shall have to C the second the same ratio which DH, the third and sixth together, has to F the fourth.

For, because BG is to C as EH is to F,

[Hypothesis. therefore, by inversion, C is to BG as Fis to EH.

(V. B. And because AB is to C as DE is to F,

[Hypothesis. B and C is to BG as F is to EH; therefore, ex æquali, AB is to BG as DE is to EH.

(V. 22. And, because these magnitudes are proportionals, they are also propor- A Ć Ś Ć tionals when taken jointly; [V. 18. therefore AG is to BG as DH is to EH. But BG is to Cas EH is to F;

[Hypothesis. therefore, ex æquali, AG is to C as DH is to F. [V. 22.

Wherefore, if the first &c. Q.E.D.

COROLLARY 1. If the same hypothesis be made as in the proposition, the excess of the first and fifth shall be to the second as the excess of the third and sixth is to the fourth. The demonstration of this is the same as that of the proposition, if division be used instead of composition.

COROLLARY 2. The proposition holds true of two ranks of magnitudes, whatever be their number, of which each of the first rank has to the second magnitude the same ratio that the corresponding one of the second rank has to the fourth magnitude; as is manifest.

PROPOSITION 25. THEOREM. If four magnitudes of the same kind be proportionals, the greatest and least of them together shall be greater than the other two together.

Let the four magnitudes AB, CD, E, F be proportionals ; namely, let ĀB be to CD as É is to F; and let AB be the greatest of them, and consequently F the least:

[V. A, V. 14. AB and F together shall be greater than CD and E together.

Take AG equal to E, and
CH equal to F.
Then, because AB is to CD as
E is to F,

[Hypothesis. and that AG is equal to E, and CH equal to F; [Construction. therefore AB is to CD as AG is to CH.

[V. 7, V. 11. And because the whole AB is to

А с Ё the whole CD as AG is to CH; therefore the remainder GB is to the remainder HD as the whole AB is to the whole CD.

[V. 19. But AB is greater than CD;

[Hypothesis. therefore BG is greater than DH.

[V. A. And because AG is equal to E and CH equal to F, [Constr. therefore AG and F together are equal to CH and E together. And if to the unequal magnitudes BG, DH, of which BG is the greater, there be added equal magnitudes, namely, AG and F to BG, and CH and E to DH, then AB and F together are greater than CD and E together.

Wherefore, if four magnitudes &c. Q.E.D.

BOOK VI.

DEFINITIONS.

1. SIMILAR rectilineal figures are those which have their several angles equal, each to each, and the sides about the equal angles proportionals.

2. Reciprocal figures, namely, triangles and parallelograms, are such as have their sides about two of their angles proportionals in such a 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.

3. 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.

4. The altitude of any figure is the straight line drawn from its vertex perpendicular to the base.

PROPOSITION 1. THEOREM. Triangles and parallelograms of the same altitude are to one another as their bases.

Let the triangles ABC, ACD, and the parallelograms

drawn from the point A to BDas the base BC is to the base CD, so shall the triangle ABC be to the triangle ACD, and the parallelogram EC to the parallelogram CF.

Produce BD both ways;

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straight lines BG, GH, each equal to BC, and any number of straight lines DK, KL, each equal to CD; [1. 3. and join AG,AH, AK, AL.

Then, because CB, BG, GH are all equal, [Construction. the triangles ABC, AGB, AHG are all equal. [I. 38. Therefore whatever multiple the base HC is of the base BC, the same multiple is the triangle AHC of the triangle ABO. For the same reason, whatever multiple the base CL is of the base CD, the same multiple is the triangle ACL of the triangle ACD. And if the base HC be equal to the base CL, the triangle AHC is equal to the triangle ACL; and if the base HC be greater than the base CL, the triangle AHC is greater than the triangle ACL; and if less, less.

[I. 38. Therefore, since there are four magnitudes, namely, the two bases BC, CD, and the two triangles ABC, ACD; and of the base BC, and the triangle ABC, the first and the third, any equimultiples whatever have been taken, namely, the base HC and the triangle AHC; and of the base CD and the triangle ACD, the second and the fourth, any equimultiples whatever have been taken, namely, the base CL and the triangle ACI ;

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