Definition:Bimedial

Definition

First Bimedial

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

$a = k^{1/4} \rho$

$b = k^{3/4} \rho$

where:

$\rho$ is a rational number

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

Then $a + b$ is a first bimedial.

In the words of Euclid:

If two medial straight lines commensurable in square only and containing a rational rectangle be added together, the whole is irrational; and let it be called a first bimedial straight line.

(The Elements: Book $\text{X}$: Proposition $37$)

Second Bimedial

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

$a = k^{1/4} \rho$

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

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 a second bimedial.

In the words of Euclid:

If two medial straight lines commensurable in square only and containing a medial rectangle be added together, the whole is irrational; and let it be called a second bimedial straight line.

(The Elements: Book $\text{X}$: Proposition $38$)

Order of Bimedial

The order of $a + b$ is the name of its classification into one of the two categories: first or second.