Proof: By Euclid
(related to Proposition: Prop. 8.16: Number does not divide Number iff Square does not divide Square)
 Let $A$ and $B$ be square numbers, and let $C$ and $D$ be their sides (respectively).
 And let $A$ not measure $B$.
 I say that $C$ does not measure $D$ either.
 For if $C$ measures $D$ then $A$ will also measure $B$ [Prop. 8.14].
 And $A$ does not measure $B$.
 Thus, $C$ will not measure $D$ either.
 So, again, 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. 8.14].
 And $C$ does not measure $D$.
 Thus, $A$ will not measure $B$ either.
 (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"