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.
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