That which produces Medial Whole with Rational Area is Irrational

Theorem

In the words of Euclid:

If from a straight line there be subtracted a straight line which is incommensurable in square with the whole, and which with the whole makes the sum of the squares on them medial, but twice the rectangle contained by them rational, the remainder is irrational; and let it be called that which produces with a rational area a medial whole.

(The Elements: Book $\text{X}$: Proposition $77$)

Proof

Let $AB$ be a straight line.

Let a straight line $BC$ such that:

$BC$ is incommensurable in square with $AB$

$AB^2 + BC^2$ is medial

the rectangle contained by $AB$ and $BC$ is rational

be cut off from $AB$.

We have that:

$AB^2 + BC^2$ is medial

while:

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

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

From:

Proposition $7$ of Book $\text{II} $: Square of Difference

and:

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

it follows that:

$2 \cdot AB \cdot BC$ is incommensurable with $AC^2$.

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

Therefore $AC^2$ is irrational.

Therefore $AC$ is irrational.

Such a straight line is known as that which produces with a rational area a medial whole.

$\blacksquare$

Historical Note

This proof is Proposition $77$ 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