Rationality of Rectangle Contained by Medial Straight Lines Commensurable in Square

Theorem

In the words of Euclid:

The rectangle contained by medial straight lines commensurable in square only is either rational or medial.

(The Elements: Book $\text{X}$: Proposition $25$)

Proof

Let $A$ and $B$ be two medial straight lines.

Let $A$ and $B$ be commensurable in square only.

Then their lengths are of the form:

$l \left({A}\right) = \rho \sqrt [4] k$

$l \left({B}\right) = \rho \sqrt [4] k \sqrt \lambda$

where:

$\rho$ is a rational number

$k$ and $\lambda$ are rational numbers whose square root is irrational.

The rectangle contained by $A$ and $B$ has area $R$ given by:

$R = \rho^2 \sqrt k \sqrt \lambda$

In general, $R$ is medial.

However, let $\sqrt \lambda = k' \sqrt k$ where $k$ is rational.

Then:

$R = \rho^2 \sqrt k k' \sqrt k = \rho^2 k k'$

which is rational.

$\blacksquare$

Historical Note

This proof is Proposition $25$ 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