# Definition:Apotome/Second Apotome

(Redirected from Definition:Second Apotome)

## Definition

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.

## Example

Let $a = 2 \sqrt {3}$ and $b = 3$.

Then:

 $\ds \frac {\sqrt {a^2 - b^2} } a$ $=$ $\ds \frac {\sqrt {12 - 9} } {2 \sqrt 3}$ $\ds$ $=$ $\ds \frac {\sqrt 3} {2 \sqrt 3}$ $\ds$ $=$ $\ds \frac 1 2$ $\ds \in \Q$

Therefore $2 \sqrt 3 - 3$ is a second 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.