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 \[\frac AC=\frac BD.\]

Proofs: 1

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)

  1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"

Adapted from CC BY-SA 3.0 Sources:

  1. Prime.mover and others: "Pr∞fWiki",, 2016