Proposition: 7.13: Proportional Numbers are Proportional Alternately
(Proposition 13 from Book 7 of Euclid's “Elements”)
If four numbers are proportional then they will also be proportional alternately.
* Let the four numbers $A$, $B$, $C$, and $D$ be proportional, (such that) as $A$ (is) to $B$, so $C$ (is) to $D$.
* I say that they will also be proportional alternately, (such that) as $A$ (is) to $C$, so $B$ (is) to $D$.
In modern notation, this proposition states that if \[\frac AB=\frac CD,\] then \[\frac AC=\frac BD.\]
Table of Contents
Proofs: 1 2 3 4 5 6 7 8 9 10
Thank you to the contributors under CC BY-SA 4.0!
Adapted from (subject to copyright, with kind permission)
- Fitzpatrick, Richard: Euclid's "Elements of Geometry"
Adapted from CC BY-SA 3.0 Sources:
- Prime.mover and others: "Pr∞fWiki", https://proofwiki.org/wiki/Main_Page, 2016