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.