# Rationality of Rectangle Contained by Medial Straight Lines Commensurable in Square

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## 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