Definition:Apotome of Medial/First Apotome

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Let $a, b \in \set {x \in \R_{>0} : x^2 \in \Q}$ be two rationally expressible numbers such that $a > b$ be in the forms:

$a = k^{1/4} \rho$
$b = k^{3/4} \rho$


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

Then $a - b$ is a first apotome of a medial.

In the words of Euclid:

If from a medial straight line there be subtracted a medial straight line commensurable with the whole in square only, and which contains with the whole a rational rectangle, the remainder is irrational. And let it be called a first apotome of a medial straight line.

(The Elements: Book $\text{X}$: Proposition $74$)

Also see