Book 8
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Next, let E, H by multiplying M make N, O respectively.
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Now, since A is a solid number, and C, D, E are its sides, therefore E by multiplying the product of C, D has made A.
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But the product of C, D is K; therefore E by multiplying K has made A.
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For the same reason also H by multiplying L has made B.
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Now, since E by multiplying K has made A, and further also by multiplying M has made N, therefore, as K is to M, so is A to N. [VII. 17]
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But, as K is to M, so is C to F, D to G, and also E to H; therefore also, as C is to F, D to G, and E to H, so is A to N.
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Again, since E, H by multiplying M have made N, O respectively, therefore, as E is to H, so is N to O. [VII. 18]
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But, as E is to H, so is C to F and D to G; therefore also, as C is to F, D to G, and E to H, so is A to N and N to O.
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Again, since H by multiplying M has made O, and further also by multiplying L has made B, therefore, as M is to L, so is O to B. [VII. 17]
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But, as M is to L, so is C to F, D to G, and E to H.
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Therefore also, as C is to F, D to G, and E to H, so not only is O to B, but also A to N and N to O.
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Therefore A, N, O, B are continuously proportional in the aforesaid ratios of the sides.
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I say that A also has to B the ratio triplicate of that which the corresponding side has to the corresponding side, that is, of the ratio which the number C has to F, or D to G, and also E to H.
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For, since A, N, O, B are four numbers in continued proportion, therefore A has to B the ratio triplicate of that which A has to N. [V. Def. 10]
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But, as A is to N, so it was proved that C is to F, D to G, and also E to H.
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Therefore A also has to B the ratio triplicate of that which the corresponding side has to the corresponding side, that is, of the ratio which the number C has to F, D to G, and also E to H. Q. E. D.
PROPOSITION 20.
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If one mean proportional number fall between two numbers, the numbers will be similar plane numbers.
