Proposition: Prop. 10.022: Square on Medial Straight Line

(Proposition 22 from Book 10 of Euclid's “Elements”)

The square on a medial (straight line), being applied to a rational (straight line), produces as breadth a (straight line which is) rational, and incommensurable in length with the (straight line) to which it is applied.

Let $A$ be a medial (straight line), and $CB$ a rational (straight line), and let the rectangular area $BD$, equal to the (square) on $A$, have been applied to $BC$, producing $CD$ as breadth.
I say that $CD$ is rational, and incommensurable in length with $CB$.

fig022e