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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:

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


$\rho$ 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 $21$)

Also see