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

Proof: By Euclid

(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.
  ∎

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References

Adapted from (subject to copyright, with kind permission)

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