For unequal magnitudes, the greater (magnitude) has a greater ratio than the lesser to the same (magnitude). And the latter (magnitude) has a greater ratio to the lesser (magnitude) than to the greater. * Let $AB$ and $C$ be unequal magnitudes, and let $AB$ be the greater (of the two), and $D$ another random magnitude. * I say that $AB$ has a greater ratio to $D$ than $C$ (has) to $D$, and (that) $D$ has a greater ratio to $C$ than (it has) to $AB$.
In modern notation, this proposition reads that if \(\alpha > \beta\) then \[\frac\alpha\gamma > \frac\beta\gamma\] and \[\frac\gamma\beta > \frac\gamma\alpha\]
for all positive real numbers \(\alpha,\beta,\gamma\).
see rules of calculation with inequalities (Rules 6 and 11)
Proofs: 1