# Definition:Medial

Jump to navigation
Jump to search

## Definition

A **medial** is a strictly positive real number which is the mean proportional between two rational line segments which are commensurable in square only.

Thus a magnitude $a \in \R_{>0}$ is **medial** if and only if $a$ is of the form:

- $a = \rho \sqrt [4] k$

where:

- $\rho$ is a rational number
- $k$ is a rational number whose square root is irrational.

In the words of Euclid:

*The rectangle contained by rational straight lines commensurable in square only is irrational, and the side of the square equal to it is irrational. Let the latter be called***medial***.*

(*The Elements*: Book $\text{X}$: Proposition $21$)