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Proposition: Prop. 10.073: Apotome is Irrational

Euclid's Formulation

If a rational (straight line), which is commensurable in square only with the whole, is subtracted from a(nother) rational (straight line) then the remainder is an irrational (straight line). Let it be called an apotome.

• For let the rational (straight line) $BC$, which commensurable in square only with the whole, have been subtracted from the rational (straight line) $AB$.
• I say that the remainder $AC$ is that irrational (straight line) called an apotome.

Modern Formulation

An apotome is a straight line whose length is expressible as

\[1 -\sqrt{\delta},\]

for some positive rational number \(\delta\). See also [Prop. 10.36].

Table of Contents

Proofs: 1

Mentioned in:

Corollaries: 1
Definitions: 2
Proofs: 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39
Propositions: 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58

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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