Proposition: 5.15: Ratio Equals its Multiples
(Proposition 15 from Book 5 of Euclid's “Elements”)
Parts have the same ratio as similar multiples, taken in corresponding order.
 For let $AB$ and $DE$ be equal multiples of $C$ and $F$ (respectively).
 I say that as $C$ is to $F$, so $AB$ (is) to $DE$.
Modern Formulation
In modern notation, this proposition reads that \[\frac\alpha\beta=\frac{m\,\alpha}{m\,\beta},\]
for all positive real numbers \(\alpha\), \(\beta\), and all multiples of aliquot parts \(m > 1\).
Generalized Modern Formulation
since $\frac mm=1$, this follows immediately from the existence and uniqueness of real 1.
Table of Contents
Proofs: 1
Mentioned in:
Proofs: 1 2 3 4 5
Sections: 6
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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