Kitap 9
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Since then as many numbers as we please beginning from an unit are in continued proportion, and the number A after the unit is prime, therefore D, the greatest of the numbers A, B, C, D, will not be measured by any other number except A, B, C. [IX. 13]
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And each of the numbers A, B, C is even; therefore D is even-times even only. [VII. Def. 8]
32
Similarly we can prove that each of the numbers B, C is even-times even only. Q. E. D.
PROPOSITION 33.
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If a number have its half odd, it is even-times odd only.
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For let the number A have its half odd; I say that A is even-times odd only.
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Now that it is even-times odd is manifest; for the half of it, being odd, measures it an even number of times. [VII. Def. 9]
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I say next that it is also even-times odd only.
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For, if A is even-times even also, it will be measured by an even number according to an even number; [VII. Def. 8] so that the half of it will also be measured by an even number though it is odd: which is absurd.
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Therefore A is even-times odd only. Q. E. D.
PROPOSITION 34.
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If a number neither be one of those which are continually doubled from a dyad, nor have its half odd, it is both eventimes even and even-times odd.
34
For let the number A neither be one of those doubled from a dyad, nor have its half odd; I say that A is both even-times even and even-times odd.
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Now that A is even-times even is manifest; for it has not its half odd. [VII. Def. 8]
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I say next that it is also even-times odd.
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For, if we bisect A, then bisect its half, and do this continually, we shall come upon some odd number which will measure A according to an even number.
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For, if not, we shall come upon a dyad, and A will be among those which are doubled from a dyad: which is contrary to the hypothesis.
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Thus A is even-times odd.
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But it was also proved even-times even.
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Therefore A is both even-times even and even-times odd. Q. E. D.
PROPOSITION 35.
