Major is Irrational

Proof
From :
 * 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:

and:

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

Therefore $AC^2$ is irrational.

Hence $AC$ is irrational.

Such a straight line is called major.