Proof: By Euclid
(related to Proposition: 7.13: Proportional Numbers are Proportional Alternately)
- 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 the same parts, of $D$ [Def. 7.20] .
- Thus, alterately, which(ever) part, or parts, $A$ is of $C$, $B$ is also the same part, or the same parts, of $D$ [Prop. 7.9], [Prop. 7.10].
- Thus, $A,C,B,D$ are proportional, i.e. as $A$ is to $C$, so $B$ (is) to $D$ [Def. 7.20] .
- (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"
