Book 8
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For the same reason also, as C is to D, so is O to M.
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Again, since E measures M the same number of times that F measures P, therefore, as E is to F, so is M to P; [VII. 13 and Def. 20] therefore N, O, M, P are continuously proportional in the ratios of A to B, of C to D, and of E to F.
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I say next that they are also the least that are in the ratios A : B, C : D, E : F.
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For, if not, there will be some numbers less than N, O, M, P continuously proportional in the ratios A : B, C : D, E : F.
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Let them be Q, R, S, T.
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Now since, as Q is to R, so is A to B, while A, B are least, and the least numbers measure those which have the same ratio with them the same number of times, the antecedent the antecedent and the consequent the consequent, [VII. 20] therefore B measures R.
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For the same reason C also measures R; therefore B, C measure R.
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Therefore the least number measured by B, C will also measure R. [VII. 35]
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But G is the least number measured by B, C; therefore G measures R.
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And, as G is to R, so is K to S: [VII. 13] therefore K also measures S.
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But E also measures S; therefore E, K measure S.
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Therefore the least number measured by E, K will also measure S. [VII. 35]
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But M is the least number measured by E, K; therefore M measures S, the greater the less: which is impossible.
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Therefore there will not be any numbers less than N, O, M, P continuously proportional in the ratios of A to B, of C to D, and of E to F; therefore N, O, M, P are the least numbers continuously proportional in the ratios A : B, C : D, E : F. Q. E. D. 69, 71, 99. the ratios A : B, C : D, E : F. This abbreviated expression is in the Greek ὀ ΑΒ, ΓΔ, ΕΖ λόγοι.
PROPOSITION 5.
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Plane numbers have to one another the ratio compounded of the ratios of their sides.
5
Let A, B be plane numbers, and let the numbers C, D be the sides of A, and E, F of B; I say that A has to B the ratio compounded of the ratios of the sides.
