Definition:Apotome/First Apotome
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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 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 $1$)
Example
Let $a = 9$ and $b = \sqrt {17}$.
Then:
\(\ds \frac {\sqrt {a^2 - b^2} } a\) | \(=\) | \(\ds \frac {\sqrt {81 - 17} } 9\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\sqrt {64} } 9\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 8 9\) | \(\ds \in \Q\) |
Therefore $9 - \sqrt {17}$ is a first apotome.
Also see
- Definition:Second Apotome
- Definition:Third Apotome
- Definition:Fourth Apotome
- Definition:Fifth Apotome
- Definition:Sixth 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.