# Medial is Irrational

## Theorem

In the words of Euclid:

*The rectangle contained by rational straight lines commensurable in square only is irrational, and the side of the square equal to it is irrational. Let the latter be called***medial***.*

(*The Elements*: Book $\text{X}$: Proposition $21$)

## Proof

By definition, a medial is a mean proportional between two rational line segments which are commensurable in square only.

Let $\rho$ and $\rho \sqrt k$ be two rational line segments which are commensurable in square only.

Then:

- $\rho: \rho \sqrt k = \rho^2 : \rho^2 \sqrt k$

From Commensurability of Elements of Proportional Magnitudes, $\rho^2 \sqrt k$ is incommensurable in length with $\rho^2$.

From Book $\text{X}$ Definition $4$: Rational Area, $\rho^2 \sqrt k$ is an irrational area.

Thus $\sqrt {\rho^2 \sqrt k} = \rho \sqrt [4] k$ is by definition an irrational straight line.

By definition of mean proportional, $\rho \sqrt [4] k$ is the medial between $\rho$ and $\rho \sqrt k$

$\blacksquare$

## Historical Note

This proof is Proposition $21$ 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