Corollary: Cor. 10.006: Magnitudes with Rational Ratio are Commensurable

(Corollary to Proposition 6 from Book 10 of Euclid's “Elements”)

So it is clear, from this, that if there are two numbers, like $D$ and $E$, and a straight line, like $A$, then it is possible to contrive that as the number $D$ (is) to the number $E$, so the straight line (is) to (another) straight line (i.e., $F$). And if the mean proportion3, (say) $B$, is taken of $A$ and $F$, then as $A$ is to $F$, so the (square) on $A$ (will be) to the (square) on $B$. That is to say, as the first (is) to the third, so the (figure) on the first (is) to the similar, and similarly described, (figure) on the second [Prop. 6.19 corr.] 4. But, as $A$ (is) to $F$, so the number $D$ is to the number $E$. Thus, it has also been contrived that as the number $D$ (is) to the number $E$, so the (figure) on the straight line $A$ (is) to the (similar figure) on the straight line $B$. (Which is) the very thing it was required to show.