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Proposition: Prop. 10.074: First Apotome of Medial is Irrational

(Proposition 74 from Book 10 of Euclid's “Elements”)

• For let the medial (straight line) $BC$, which is commensurable in square only with $AB$, and which makes with $AB$ the rational (rectangle contained) by $AB$ and $BC$, have been subtracted from the medial (straight line) $AB$ [Prop. 10.27].
• I say that the remainder $AC$ is an irrational (straight line).
• Let it be called the first apotome of a medial (straight line).

Modern Formulation

A first apotome of a medial (straight line) is a straight line whose length is expressible as

\[\delta^{1/4} - \delta^{3/4},\]

for some positive rational number \(\delta\). See also [Prop. 10.37].

Table of Contents

Proofs: 1

Mentioned in:

Proofs: 1 2 3 4
Propositions: 5 6

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References

Adapted from (subject to copyright, with kind permission)

1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"

Adapted from CC BY-SA 3.0 Sources:

1. Prime.mover and others: "Pr∞fWiki", https://proofwiki.org/wiki/Main_Page, 2016