Proof: By Euclid
(related to Proposition: Prop. 8.17: Number does not divide Number iff Cube does not divide Cube)
 For let the cube number $A$ not measure the cube number $B$.
 And let $C$ be the side of $A$, and $D$ (the side) of $B$.
 I say that $C$ will not measure $D$.
 For if $C$ measures $D$ then $A$ will also measure $B$ [Prop. 8.15].
 And $A$ does not measure $B$.
 Thus, $C$ does not measure $D$ either.
 And so let $C$ not measure $D$.
 I say that $A$ will not measure $B$ either.
 For if $A$ measures $B$ then $C$ will also measure $D$ [Prop. 8.15].
 And $C$ does not measure $D$.
 Thus, $A$ will not measure $B$ either.
 (Which is) the very thing it was required to show.
∎
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"