Proposition: Prop. 10.011: Commensurability of Elements of Proportional Magnitudes

Euclid's Formulation

If four magnitudes are proportional, and the first is commensurable with the second, then the third will also be commensurable with the fourth. And if the first is incommensurable with the second, then the third will also be incommensurable with the fourth.

Let $A$, $B$, $C$, $D$ be four proportional magnitudes, (such that) as $A$ (is) to $B$, so $C$ (is) to $D$.
And let $A$ be commensurable with $B$.
I say that $C$ will also be commensurable with $D$.
And so let $A$ be incommensurable with $B$.
I say that $C$ will also be incommensurable with $D$.

fig011e