Proof: By Euclid
(related to Proposition: Prop. 9.25: Even Number minus Odd Number is Odd)
- For let the odd (number) $BC$ have been subtracted from the even number $AB$.
- I say that the remainder $CA$ is odd.
- For let the unit $CD$ have been subtracted from $BC$.
- $DB$ is thus even [Def. 7.7] .
- And $AB$ is also even.
- And thus the remainder $AD$ is even [Prop. 9.24].
- And $CD$ is a unit.
- Thus, $CA$ is odd [Def. 7.7] .
- (Which is) the very thing it was required to show.
Adapted from (subject to copyright, with kind permission)
- Fitzpatrick, Richard: Euclid's "Elements of Geometry"