Definition:Apotome of Medial

Definition

First Apotome of Medial

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:

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

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

where:

$\rho$ 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 $74$)

Second Apotome of Medial

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 $75$)

Order of Apotome of Medial

The order of $a - b$ is the name of its classification into one of the two categories: first or second.

Terms of Apotome of Medial

The terms of $a - b$ are the elements $a$ and $b$.

Whole

The real number $a$ is called the whole of the apotome of a medial.

Annex

The real number $b$ is called the annex of the apotome of a medial.