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.
Table of Contents
Proofs: 1
Mentioned in:
Proofs: 1 2 3 4 5 6 7
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References
Adapted from (subject to copyright, with kind permission)
 Fitzpatrick, Richard: Euclid's "Elements of Geometry"
Adapted from CC BYSA 3.0 Sources:
 Prime.mover and others: "Pr∞fWiki", https://proofwiki.org/wiki/Main_Page, 2016