Proposition: 7.18: Ratios of Multiples of Numbers

(Proposition 18 from Book 7 of Euclid's “Elements”)

If two numbers multiplying some number make some (other numbers) then the (numbers) generated from them will have the same ratio as the multiplying (numbers).


Modern Formulation

In modern notation, this propositions states that if \[a\,c = d\quad\text{ and }\quad b\,c=e,\] then \[\frac ab=\frac de,\] where all symbols denote numbers.

Proofs: 1

Proofs: 1 2 3 4 5 6 7

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