(related to Proposition: Prop. 9.33: Number whose Half is Odd is Even-Times Odd)

- For let the number $A$ have an odd half.
- I say that $A$ is an even-times-odd (number) only.

- In fact, (it is) clear that ($A$) is an even-times-odd (number).
- For its half, being odd, measures it an even number of times [Def. 7.9] .
- So I also say that (it is an even-times-odd number) only.
- For if $A$ is also an even-times-even (number) then it will be measured by an even (number) according to an even number [Def. 7.8] .
- Hence, its half will also be measured by an even number, (despite) being odd.
- The very thing is absurd.
- Thus, $A$ is an even-times-odd (number) only.
- (Which is) the very thing it was required to show.∎

**Fitzpatrick, Richard**: Euclid's "Elements of Geometry"