Proof: By Euclid
(related to Proposition: 7.12: Ratios of Numbers is Distributive over Addition)
- For since as $A$ is to $B$, so $C$ (is) to $D$, thus which(ever) part, or parts, $A$ is of $B$, $C$ is also the same part, or parts, of $D$ [Def. 7.20] .
- Thus, the sum $A$, $C$ is also the same part, or the same parts, of the sum $B$, $D$ that $A$ (is) of $B$ [Prop. 7.5], [Prop. 7.6].
- Thus, as $A$ is to $B$, so $A$, $C$ (is) to $B$, $D$ [Def. 7.20] .
- (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"