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

Proof: By Euclid

(related to Proposition: Prop. 9.16: Two Co-prime Integers have no Third Integer Proportional)

• For let the two numbers $A$ and $B$ be prime to one another.
• I say that as $A$ is to $B$, so $B$ is not to some other (number).

• For, if possible, let it be that as $A$ (is) to $B$, (so) $B$ (is) to $C$.
• And $A$ and $B$ (are) prime (to one another).
• And (numbers) prime (to one another are) also the least (of those numbers having the same ratio as them) [Prop. 7.21].
• And the least numbers measure those (numbers) having the same ratio (as them) an equal number of times, the leading (measuring) the leading, and the following the following [Prop. 7.20].
• Thus, $A$ measures $B$, as the leading (measuring) the leading.
• And ($A$) also measures itself.
• Thus, $A$ measures $A$ and $B$, which are prime to one another.
• The very thing (is) absurd.
• Thus, as $A$ (is) to $B$, so $B$ cannot be to $C$.
• (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"