Definition: 5.04: Having a Ratio
(Those) magnitudes are said to have a ratio with respect to one another which, being multiplied, are capable of exceeding one another.
Two positive real numbers \(\alpha,\beta\) have a ratio if there is a natural number $n$ such that $n \alpha > \beta$.
See also Archimedean axiom.
Proofs: 1 2 3 4
- 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)
- Fitzpatrick, Richard: Euclid's "Elements of Geometry"