Kitap 5
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For the same reason also, as E is to F, so is M to N. And, as A is to B, so is E to F; therefore also, as G is to H, so is M to N. [V. 11]
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Next, since, as B is to C, so is D to E, alternately, also, as B is to D, so is C to E. [V. 16]
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And, since H, K are equimultiples of B, D, and parts have the same ratio as their equimultiples, therefore, as B is to D, so is H to K. [V. 15]
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But, as B is to D, so is C to E; therefore also, as H is to K, so is C to E. [V. 11]
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Again, since L, M are equimultiples of C, E, therefore, as C is to E, so is L to M. [V. 15]
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But, as C is to E, so is H to K; therefore also, as H is to K, so is L to M, [V. 11] and, alternately, as H is to L, so is K to M. [V. 16]
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But it was also proved that, as G is to H, so is M to N.
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Since, then, there are three magnitudes G, H, L, and others equal to them in multitude K, M, N, which taken two and two together are in the same ratio, and the proportion of them is perturbed, therefore, ex aequali, if G is in excess of L, K is also in excess of N; if equal, equal; and if less, less. [V. 21]
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And G, K are equimultiples of A, D, and L, N of C, F.
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Therefore, as A is to C, so is D to F.
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Therefore etc. Q. E. D.
PROPOSITION 24.
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If a first magnitude have to a second the same ratio as a third has to a fourth, and also a fifth have to the second the same ratio as a sixth to the fourth, the first and fifth added together will have to the second the same ratio as the third and sixth have to the fourth.
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Let a first magnitude AB have to a second C the same ratio as a third DE has to a fourth F; and let also a fifth BG have to the second C the same ratio as a sixth EH has to the fourth F; I say that the first and fifth added together, AG, will have to the second C the same ratio as the third and sixth, DH, has to the fourth F.
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For since, as BG is to C, so is EH to F, inversely, as C is to BG, so is F to EH.
