Definition:Apotome of Medial/Second Apotome

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Definition

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:

$a = k^{1/4} \rho$
$b = \dfrac {\lambda^{1/2} \rho} {k^{1/4} }$

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 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 medial rectangle, the remainder is irrational; and let it be called a second apotome of a medial straight line.

(The Elements: Book $\text{X}$: Proposition $75$)


Also see