Kitap 9
33
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.
33
Therefore A is even-times odd only. Q. E. D.
PROPOSITION 34.
34
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.
34
Now that A is even-times even is manifest; for it has not its half odd. [VII. Def. 8]
34
I say next that it is also even-times odd.
34
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.
34
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.
34
Thus A is even-times odd.
34
But it was also proved even-times even.
34
Therefore A is both even-times even and even-times odd. Q. E. D.
PROPOSITION 35.
35
If as many numbers as we please be in continued proportion, and there be subtracted from the second and the last numbers equal to the first, then, as the excess of the second is to the first, so will the excess of the last be to all those before it.
