# Definition:Medial

## 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.