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