Definition:Apotome/Third 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 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 $3$)


Let $a = \sqrt {11}$ and $b = \sqrt {\frac {143} {49} }$.


\(\ds \frac {\sqrt {a^2 - b^2} } a\) \(=\) \(\ds \frac {\sqrt {11 - \frac {143} {49} } } {\sqrt {11} }\)
\(\ds \) \(=\) \(\ds \frac {\sqrt {\frac {396} {49} } } {\sqrt {11} }\)
\(\ds \) \(=\) \(\ds \sqrt {\frac {36} {49} }\)
\(\ds \) \(=\) \(\ds \frac 6 7\) \(\ds \in \Q\)

Therefore $\sqrt {11} - \sqrt {\dfrac {143} {49}}$ is a third 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.