Proposition: 7.11: Proportional Numbers have Proportional Differences

(Proposition 11 from Book 7 of Euclid's “Elements”)

If as the whole (of a number) is to the whole (of another), so a (part) taken away (is) to a (part) taken away, then the remainder will also be to the remainder as the whole (is) to the whole.


Modern Formulation

In modern notation, this proposition states that if \[\frac{AB}{CD}=\frac{AE+EB}{CF+FD}=\frac{AE}{CF},\] then \[\frac{EB}{FD}=\frac{AB}{CD}.\]

Proofs: 1

Proofs: 1

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