Square on Medial Straight Line

Proof
Let $A = \rho \sqrt [4] k$ be a medial straight line.

Let $BC = \sigma$ be a rational straight line.

The square on $A$ is $\rho^2 \sqrt k$.

The breadth $CD$ of the rectangle whose area is $\rho^2 \sqrt k$ and whose side is $BC$ is:

where $m$ and $n$ are integers.

Thus $CD$ can be expressed in the form:
 * $CD = \sqrt {k'} \sigma$

which is commensurable in square only with $\sigma$.

By definition, therefore, $CD$ is incommensurable in length with $BC$.