Definition:Apotome/Fourth Apotome

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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 $4$)


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


\(\ds \frac {\sqrt {a^2 - b^2} } a\) \(=\) \(\ds \frac {\sqrt {9 - 2} } 3\)
\(\ds \) \(=\) \(\ds \frac {\sqrt 7} 3\) \(\ds \notin \Q\)

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