Proof: By Euclid
(related to Proposition: Prop. 8.27: Similar Solid Numbers have Same Ratio as between Two Cubes)
 For since $A$ and $B$ are similar solid (numbers) , two numbers thus fall (between) $A$ and $B$ in mean proportion [Prop. 8.19].
 Let $C$ and $D$ have (so) fallen.
 And let the least numbers, $E$, $F$, $G$, $H$, having the same ratio as $A$, $C$, $D$, $B$, (and) equal in multitude to them, have been taken [Prop. 8.2].
 Thus, the outermost of them, $E$ and $H$, are cube [Prop. 8.2 corr.] .
 And as $E$ is to $H$, so $A$ (is) to $B$.
 And thus $A$ has to $B$ the ratio which (some) cube number (has) to a(nother) cube 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"