Proof: By Euclid
(related to Proposition: Prop. 9.17: Last Element of Geometric Progression with Coprime 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.
∎
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References
Adapted from (subject to copyright, with kind permission)
 Fitzpatrick, Richard: Euclid's "Elements of Geometry"