# Definition:Apotome

## Contents

## 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

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

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

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**.