# 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.

### 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.

### 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.

### 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.

### 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;

### 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;

### 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.

## 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.

## 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.