Definition: Def. 10.16: Sixth Apotome

And if neither (the whole nor the attached straight line is commensurable), a sixth (apotome).

Modern Formulation

The sixth apotome is a straight line whose length is \[\sqrt{\alpha}-\sqrt{\beta},\] where \(\alpha,\beta\) denote positive rational numbers.

Proofs: 1 2 3 4 5 6 7
Propositions: 8 9 10 11

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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",, 2016