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
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.
Now that A
is even-times even is manifest; for it has not its half odd. [VII. Def. 8
I say next that it is also even-times odd.
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.
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.
is even-times odd.
But it was also proved even-times even.
is both even-times even and even-times odd. Q. E. D.