Proposition: 5.13: Relative Sizes of Proportional Magnitudes

(Proposition 13 from Book 5 of Euclid's “Elements”)

If a first (magnitude) has the same ratio to a second that a third (has) to a fourth, and the third (magnitude) has a greater ratio to the fourth than a fifth (has) to a sixth, then the first (magnitude) will also have a greater ratio to the second than the fifth (has) to the sixth.

For let a first (magnitude) $A$ have the same ratio to a second $B$ that a third $C$ (has) to a fourth $D$, and let the third (magnitude) $C$ have a greater ratio to the fourth $D$ than a fifth $E$ (has) to a sixth $F$.
I say that the first (magnitude) $A$ will also have a greater ratio to the second $B$ than the fifth $E$ (has) to the sixth $F$.

fig13e

Modern Formulation

In modern notation, this proposition reads that if \[\frac\alpha\beta=\frac\gamma\delta\text{ and }\frac\gamma\delta > \frac\epsilon\zeta\] then \[\frac\alpha\beta > \frac\epsilon\zeta\]

for all positive real numbers $\alpha,\beta,\gamma,\delta,\epsilon,\zeta$.