Proof: By Euclid
(related to Proposition: Prop. 9.17: Last Element of Geometric Progression with Co-prime Extremes has no Integer Proportional as First to Second)
- For, if possible, let it be that as $A$ (is) to $B$, so $D$ (is) to $E$.
- Thus, alternately, as $A$ is to $D$, (so) $B$ (is) to $E$ [Prop. 7.13].
- And $A$ and $D$ 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 $B$.
- And as $A$ is to $B$, (so) $B$ (is) to $C$.
- Thus, $B$ also measures $C$.
- And hence $A$ measures $C$ [Def. 7.20] .
- And since as $B$ is to $C$, (so) $C$ (is) to $D$, and $B$ measures $C$, $C$ thus also measures $D$ [Def. 7.20] .
- But, $A$ was (found to be) measuring $C$.
- And hence $A$ also measures $D$.
- And ($A$) also measures itself.
- Thus, $A$ measures $A$ and $D$, which are prime to one another.
- The very thing is impossible.
- Thus, as $A$ (is) to $B$, so $D$ cannot be to some other (number).
- (Which is) the very thing it was required to show.
Thank you to the contributors under CC BY-SA 4.0!
Adapted from (subject to copyright, with kind permission)
- Fitzpatrick, Richard: Euclid's "Elements of Geometry"