Proof: By Euclid
(related to Proposition: Prop. 10.008: Magnitudes with Irrational Ratio are Incommensurable)
 For let the two magnitudes $A$ and $B$ not have to one another the ratio which (some) number (has) to (some) number.

I say that the magnitudes $A$ and $B$ are incommensurable.

For if they are commensurable, $A$ will have to $B$ the ratio which (some) number (has) to (some) number [Prop. 10.5].
 But it does not have (such a ratio).
 Thus, the magnitudes $A$ and $B$ are incommensurable.
 Thus, if two magnitudes ... to one another, and so on ....
∎
Thank you to the contributors under CC BYSA 4.0!
 Github:

 nonGithub:
 @Fitzpatrick
References
Adapted from (subject to copyright, with kind permission)
 Fitzpatrick, Richard: Euclid's "Elements of Geometry"