# Definition:Bimedial

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## 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.