# Side of Rational plus Medial Area is Irrational

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## Theorem

In the words of Euclid:

*If two straight lines incommensurable in square which make the sum of the squares on them medial, but the rectangle contained by them rational, be added together, the whole straight line is irrational; and let it be called the***side of a rational plus a medial area***.*

(*The Elements*: Book $\text{X}$: Proposition $40$)

## Proof

- Let $AB$ and $BC$ be straight lines which are incommensurable in square which make the sum of the squares on them medial but the rectangle contained by them rational.

We have that:

- $AB^2 + BC^2$ is medial

and from Proposition $6$ of Book $\text{X} $: Magnitudes with Rational Ratio are Commensurable:

- $2 \cdot AB \cdot BC$ is rational.

So:

- $AB^2 + BC^2$ is incommensurable with $2 \cdot AB \cdot BC$.

So from Proposition $16$ of Book $\text{X} $: Incommensurability of Sum of Incommensurable Magnitudes:

- $AC^2 = \left({AB + BC}\right)^2 = AB^2 + BC^2 + 2 \cdot AB \cdot BC$ is also incommensurable with $2 \cdot AB \cdot BC$.

But $2 \cdot AB \cdot BC$ is rational.

Therefore $AC^2$ is irrational.

Hence $AC$ is irrational.

Such a straight line is called the **side of a rational plus a medial area**.

$\blacksquare$

## Historical Note

This proof is Proposition $40$ 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