# Square on Medial Straight Line

## Theorem

In the words of Euclid:

The square on a medial straight line, if applied to a rational straight line, produces as breadth a straight line rational and incommensurable in length with that to which it is applied.

### Lemma

In the words of Euclid:

If there be two straight lines, then, as the first is to the second, so is the square on the first to the rectangle contained by the two straight lines.

## 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:

 $\ds CD$ $=$ $\ds \frac {\rho^2 \sqrt k} \sigma$ $\ds$ $=$ $\ds \dfrac {\rho^2 \sqrt k} {\sigma^2} \sigma$ $\ds$ $=$ $\ds \sqrt k \dfrac m n \sigma$

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 $BC$.

$\blacksquare$

## Historical Note

This proof is Proposition $22$ of Book $\text{X}$ of Euclid's The Elements.