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 $D$.
Modern Formulation
In modern notation, this proposition states that if \[\frac AB=\frac CD,\] then \[\frac AC=\frac BD.\]
Table of Contents
Proofs: 1
Mentioned in:
Proofs: 1 2 3 4 5 6 7 8 9 10
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References
Adapted from (subject to copyright, with kind permission)
 Fitzpatrick, Richard: Euclid's "Elements of Geometry"
Adapted from CC BYSA 3.0 Sources:
 Prime.mover and others: "Pr∞fWiki", https://proofwiki.org/wiki/Main_Page, 2016