Major is Irrational

Theorem

In the words of Euclid:

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

(The Elements: Book $\text{X}$: Proposition $39$)

Proof

From Proposition $33$ of Book $\text{X} $: Construction of Components of Major:

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

We have that the rectangle contained by $AB$ and $BC$ is medial.

From:

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

and:

Porism to Proposition $23$ of Book $\text{X} $: Straight Line Commensurable with Medial Straight Line is Medial:

it follows that:

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

But $AB^2 + BC^2$ is rational.

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

So by 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 $AB^2 + BC^2$.

Therefore $AC^2$ is irrational.

Hence $AC$ is irrational.

Such a straight line is called major.

$\blacksquare$

Historical Note

This proof is Proposition $39$ 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