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). * For let the two numbers $A$ and $B$ make (the numbers) $D$ and $E$ (respectively, by) multiplying some number $C$. * I say that as $A$ is to $B$, so $D$ (is) to $E$.


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