◀ ▲ ▶Branches / Geometry / Elements-euclid / Book-10-incommensurable-magnitudes / Proposition: Prop. 10.075: Second Apotome of Medial is Irrational

Proposition: Prop. 10.075: Second Apotome of Medial is Irrational

(Proposition 75 from Book 10 of Euclid's “Elements”)

• For let the medial (straight line) $CB$, which is commensurable in square only with the whole, $AB$, and which contains with the whole, $AB$, the medial (rectangle contained) by $AB$ and $BC$, have been subtracted from the medial (straight line) $AB$ [Prop. 10.28].
• I say that the remainder $AC$ is an irrational (straight line).
• Let it be called a second apotome of a medial (straight line).

Modern Formulation

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

\[\alpha^{1/4}-\frac{\sqrt{\beta}}{\alpha^{1/4}},\]

where $\alpha$ and $\beta$ are positive rational numbers. See also [Prop. 10.38].

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