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.

Modern Formulation

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
Propositions: 5

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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"