Definition: 5.07: Having a Greater Ratio

(Definition 7 from Book 5 of Euclid's “Elements”)

And when for equal multiples (as in Def. 5), the multiple of the first (magnitude) exceeds the multiple of the second, and the multiple of the third (magnitude) does not exceed the multiple of the fourth, then the first (magnitude) is said to have a greater ratio to the second than the third (magnitude has) to the fourth.

Modern Formulation

For all positive real numbers \(\alpha,\beta,\gamma,\delta\), if there is natural number $n>0$ such that $n\alpha > \beta$ and $n\gamma\le \delta$ then \[\frac\alpha\beta > \frac\gamma\delta.\]

Proofs: 1 2 3 4 5 6
Propositions: 7 8 9

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  1. Health, T.L.: "The Thirteen Books of Euclid's Elements - With Introduction and Commentary by T. L. Health", Cambridge at the University Press, 1968, Vol 1, 2, 3

Adapted from (subject to copyright, with kind permission)

  1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"