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.

Proofs: 1

Proofs: 1 2 3 4 5
Sections: 6

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