Proof: By Euclid
(related to Proposition: 7.18: Ratios of Multiples of Numbers)
- For since $A$ has made $D$ (by) multiplying $C$, $C$ has thus also made $D$ (by) multiplying $A$ [Prop. 7.16].
- So, for the same (reasons), $C$ has also made $E$ (by) multiplying $B$.
- So the number $C$ has made $D$ and $E$ (by) multiplying the two numbers $A$ and $B$ (respectively).
- Thus, as $A$ is to $B$, so $D$ (is) to $E$ [Prop. 7.17].
- (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"