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

Proof: By Euclid

(related to Proposition: Prop. 9.26: Odd Number minus Odd Number is Even)

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

• For since $AB$ is odd, let the unit $BD$ have been subtracted (from it).
• Thus, the remainder $AD$ is even [Def. 7.7] .
• So, for the same (reasons), $CD$ is also even.
• And hence the remainder $CA$ is even [Prop. 9.24].
• (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"