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 BY-SA 4.0! 
- Github:
-
- non-Github:
- @Fitzpatrick
References
Adapted from (subject to copyright, with kind permission)
- Fitzpatrick, Richard: Euclid's "Elements of Geometry"
