# Square on Medial Straight Line

Jump to navigation
Jump to search

## 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.*

(*The Elements*: Book $\text{X}$: Proposition $22$)

### 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.*

(*The Elements*: Book $\text{X}$: Proposition $22$ : Lemma)

## 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*.

## Sources

- 1926: Sir Thomas L. Heath:
*Euclid: The Thirteen Books of The Elements: Volume 3*(2nd ed.) ... (previous) ... (next): Book $\text{X}$. Propositions