Proof: By Euclid
(related to Proposition: Prop. 9.34: Number neither whose Half is Odd nor Power of Two is both Even-Times Even and Even-Times Odd)
- In fact, (it is) clear that $A$ is an even-times-even (number) [Def. 7.8] .
- For it does not have an odd half.
- So I say that it is also an even-times-odd (number).
- For if we cut $A$ in half, and (then cut) its half in half, and we do this continually, then we will arrive at some odd number which will measure $A$ according to an even number.
- For if not, we will arrive at a dyad, and $A$ will be (one) of the (numbers) doubled from a dyad.
- The very opposite thing (was) assumed.
- Hence, $A$ is an even-times-odd (number) [Def. 7.9] .
- And it was also shown (to be) an even-times-even (number).
- Thus, $A$ is (both) an even-times-even and an even-times-odd (number).
- (Which is) the very thing it was required to show.
Thank you to the contributors under CC BY-SA 4.0!
Adapted from (subject to copyright, with kind permission)
- Fitzpatrick, Richard: Euclid's "Elements of Geometry"