##
Proposition 34

If a number neither is one of those which is continually doubled from a dyad, nor has its half odd, then it is both even-times even and even-times odd.
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. If we bisect A, then bisect its half, and do this continually, we shall come upon some odd number which measures A according to an even number. 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. Thus A is even-times odd. But it was also proved even-times even. Therefore A is both even-times even and even-times odd.
Therefore, if a number neither is one of those which is continually doubled from a dyad, nor has its half odd, then it is both even-times even and even-times odd.
Q.E.D.

###

(Forthcoming)

Book IX Introduction - Proposition IX.33 - Proposition IX.35.

© 1996
D.E.Joyce
Clark University