Definition:That which produces Medial Whole with Medial Area

Definition

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

$a = \dfrac {\rho \lambda^{1/4} } {\sqrt 2} \sqrt {1 + \dfrac k {\sqrt {1 + k^2} } }$

$b = \dfrac {\rho \lambda^{1/4} } {\sqrt 2} \sqrt {1 - \dfrac k {\sqrt {1 + k^2} } }$

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 medial 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, twice the rectangle contained by them medial, and further the squares on them incommensurable with twice the rectangle contained by them, the remainder is irrational; and let it be called that which produces with a medial area a medial whole.

(The Elements: Book $\text{X}$: Proposition $78$)