Definition:Side of Rational plus Medial 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 the side of the sum of (two) medial areas.

In the words of Euclid:

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

(The Elements: Book $\text{X}$: Proposition $40$)

Also see

• Construction of Components of Side of Rational plus Medial Area

• Side of Rational plus Medial Area is Irrational