Proposition: 7.14: Proportion of Numbers is Transitive

(Proposition 14 from Book 7 of Euclid's “Elements”)

If there are any multitude of numbers whatsoever, and (some) other (numbers) of equal multitude to them, (which are) also in the same ratio taken two by two, then they will also be in the same ratio via equality.


Modern Formulation

In modern notation, this proposition states that if \[\frac AB=\frac DE\wedge \frac BC=\frac EF,\] then \[\frac AC=\frac DF.\]

Proofs: 1

Proofs: 1 2 3 4 5 6 7 8

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