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