Proposition: 5.08: Relative Sizes of Ratios on Unequal Magnitudes

(Proposition 8 from Book 5 of Euclid's “Elements”)

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 $AB$.

fig08e

Modern Formulation

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 $\alpha,\beta,\gamma$.