Definition:That which produces Medial Whole with Rational Area

Definition

Let $a, b \in \R_{>0}$ be in the forms:

$a = \dfrac \rho {\sqrt {2 \left({1 + k^2}\right)} } \sqrt{\sqrt {1 + k^2} + k}$

$b = \dfrac \rho {\sqrt {2 \left({1 + k^2}\right)} } \sqrt{\sqrt {1 + k^2} - k}$

where:

$\rho$ is a rational number

$k$ is a rational number whose square root is irrational

$\lambda$ is a rational number whose square root is irrational.

Then $a - b$ is that which produces a medial whole with a rational area.

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 $77$)