Book 10
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Let it have the ratio which F has to G.
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And, given any number of ratios we please, namely the ratio which D has to E and that which F has to G, let the numbers H, K, L be taken continuously in the given ratios; [cf. VIII. 4] so that, as D is to E, so is H to K, and, as F is to G, so is K to L.
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Since, then, as A is to C, so is D to E, while, as D is to E, so is H to K, therefore also, as A is to C, so is H to K. [V. 11]
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Again, since, as C is to B, so is F to G, while, as F is to G, so is K to L, therefore also, as C is to B, so is K to L. [V. 11]
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But also, as A is to C, so is H to K; therefore, ex aequali, as A is to B, so is H to L. [V. 22]
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Therefore A has to B the ratio which a number has to a number; therefore A is commensurable with B. [X. 6]
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Therefore etc. Q. E. D.
PROPOSITION 13.
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If two magnitudes be commensurable, and the one of them be incommensurable with any magnitude, the remaining one will also be incommensurable with the same.
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Let A, B be two commensurable magnitudes, and let one of them, A, be incommensurable with any other magnitude C; I say that the remaining one, B, will also be incommensurable with C.
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For, if B is commensurable with C, while A is also commensurable with B, A is also commensurable with C. [X. 12]
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But it is also incommensurable with it: which is impossible.
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Therefore B is not commensurable with C; therefore it is incommensurable with it.
13
Therefore etc.
