For (magnitudes) having a ratio to the same (magnitude), that > For (magnitudes) having a ratio to the same (magnitude), that (magnitude which) has the greater ratio is (the) greater. And that (magnitude) to which the latter (magnitude) has a greater ratio is (the) lesser. * For let $A$ have a greater ratio to $C$ than $B$ (has) to $C$. * I say that $A$ is greater than $B$.
In modern notation, this proposition reads that if \[\frac\alpha\gamma > \frac\beta\gamma,\] then \[\alpha > \beta,\]
for all positive real numbers \(\alpha,\beta,\gamma\).
see rules of calculation with inequalities (Rules 6)
Proofs: 1
