◀ ▲ ▶Branches / Geometry / Elements-euclid / Book--9-number-theory-applications / Proof: By Euclid

Proof: By Euclid

(related to Proposition: Prop. 9.24: Even Number minus Even Number is Even)

• For let the even (number) $BC$ have been subtracted from the even number $AB$.
• I say that the remainder $CA$ is even.

• For since $AB$ is even, it has a half part [Def. 7.6] .
• So, for the same (reasons), $BC$ also has a half part.
• And hence the remainder [$CA$ has a half part].
• Thus, $AC$ is even.
• (Which is) the very thing it was required to show.
  ∎

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References

Adapted from (subject to copyright, with kind permission)

1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"