Definition:Apotome
Definition
Let $a, b \in \set {x \in \R_{>0} : x^2 \in \Q}$ be two rationally expressible numbers such that $a > b$.
Then $a - b$ is an apotome if and only if:
- $(1): \quad \dfrac a b \notin \Q$
- $(2): \quad \paren {\dfrac a b}^2 \in \Q$
where $\Q$ denotes the set of rational numbers.
In the words of Euclid:
- If from a rational straight line there be subtracted a rational straight line commensurable with the whole in square only, the remainder is irrational; and let it be called an apotome.
(The Elements: Book $\text{X}$: Proposition $73$)
First Apotome
Let $a, b \in \set {x \in \R_{>0} : x^2 \in \Q}$ be two rationally expressible numbers such that $a - b$ is an apotome.
Then $a - b$ is a first apotome if and only if:
- $(1): \quad a \in \Q$
- $(2): \quad \dfrac {\sqrt {a^2 - b^2}} a \in \Q$
where $\Q$ denotes the set of rational numbers.
In the words of Euclid:
- Given a rational straight line and an apotome, if the square on the whole be greater than the square on the annex by the square on a straight line commensurable in length with the whole, and the whole be commensurable in length with the rational straight line set out, let the apotome be called a first apotome.
(The Elements: Book $\text{X (III)}$: Definition $1$)
Second Apotome
Let $a, b \in \set {x \in \R_{>0} : x^2 \in \Q}$ be two rationally expressible numbers such that $a - b$ is an apotome.
Then $a - b$ is a second apotome if and only if:
- $(1): \quad b \in \Q$
- $(2): \quad \dfrac {\sqrt {a^2 - b^2}} a \in \Q$
where $\Q$ denotes the set of rational numbers.
In the words of Euclid:
- But if the annex be commensurable in length with the rational straight line set out, and the square on the whole be greater than that on the annex by the square on a straight line commensurable in length with the whole, let the apotome be called a second apotome.
(The Elements: Book $\text{X (III)}$: Definition $2$)
Third Apotome
Let $a, b \in \set {x \in \R_{>0} : x^2 \in \Q}$ be two rationally expressible numbers such that $a - b$ is an apotome.
Then $a - b$ is a third apotome if and only if:
- $(1): \quad a \notin \Q$
- $(2): \quad b \notin \Q$
- $(3): \quad \dfrac {\sqrt {a^2 - b^2}} a \in \Q$
where $\Q$ denotes the set of rational numbers.
In the words of Euclid:
- But if neither be commensurable in length with the rational straight line set out, and the square on the whole be greater than the square on the annex by the square on a straight line commensurable with the whole, let the apotome be called a third apotome.
(The Elements: Book $\text{X (III)}$: Definition $3$)
Fourth Apotome
Let $a, b \in \set {x \in \R_{>0} : x^2 \in \Q}$ be two rationally expressible numbers such that $a - b$ is an apotome.
Then $a - b$ is a fourth apotome if and only if:
- $(1): \quad a \in \Q$
- $(2): \quad \dfrac {\sqrt {a^2 - b^2}} a \notin \Q$
where $\Q$ denotes the set of rational numbers.
In the words of Euclid:
- Again, if the square on the whole be greater than the square on the annex by the square on a straight line incommensurable with the whole, then, if the whole be commensurable in length with the rational straight line set out, let the apotome be called a fourth apotome;
(The Elements: Book $\text{X (III)}$: Definition $4$)
Fifth Apotome
Let $a, b \in \set {x \in \R_{>0} : x^2 \in \Q}$ be two rationally expressible numbers such that $a - b$ is an apotome.
Then $a - b$ is a fifth apotome if and only if:
- $(1): \quad b \in \Q$
- $(2): \quad \dfrac {\sqrt {a^2 - b^2}} a \notin \Q$
where $\Q$ denotes the set of rational numbers.
In the words of Euclid:
- if the annex be so commensurable, a fifth;
(The Elements: Book $\text{X (III)}$: Definition $5$)
Sixth Apotome
Let $a, b \in \set {x \in \R_{>0} : x^2 \in \Q}$ be two rationally expressible numbers such that $a - b$ is an apotome.
Then $a - b$ is a sixth apotome if and only if:
- $(1): \quad a \notin \Q$
- $(2): \quad b \notin \Q$
- $(3): \quad \dfrac {\sqrt {a^2 - b^2}} a \notin \Q$
where $\Q$ denotes the set of rational numbers.
In the words of Euclid:
- and if neither, a sixth.
(The Elements: Book $\text{X (III)}$: Definition $6$)
Order
The order of $a - b$ is the name of its classification into one of the six categories: first, second, third, fourth, fifth or sixth.
Terms of Apotome
The terms of $a - b$ are the elements $a$ and $b$.
Whole
The real number $a$ is called the whole of the apotome.
Annex
The real number $b$ is called the annex of the apotome.
Warning
It is a mistake to suggest that $a$ and $b$ may be any real numbers such that $\dfrac a b \notin \Q$ and $\paren {\dfrac a b}^2 \in \Q$.
For example, let $a = \pi \sqrt 2$ and $b = \pi$.
Then we see that:
- $(1): \quad \dfrac a b = \sqrt 2 \notin \Q$
- $(2): \quad \paren {\dfrac a b}^2 = 2 \in \Q$
But $a - b = \pi \paren {\sqrt 2 - 1}$ is not an apotome.
Also see
Linguistic Note
The term apotome is archaic, and is rarely used nowadays.
It is pronounced a-POT-o-mee, just as "epitome" is pronounced e-PIT-o-mee.
It is transliterated directly from the Ancient Greek word ἀποτομή, which is the noun form of ἀποτέμνω, from ἀπο- (away) and τέμνω (to cut), meaning roughly to cut away.
Therefore, ἀποτομή means roughly (the portion) cut off.