Definition: Def. 10.13: Third Apotome

And if neither of (the whole or the attached straight line) is commensurable in length with the (previously) laid down rational (straight line), and the square on the whole is greater than (the square on) the attached (straight line) by the (square) on (some straight line) commensurable (in length) with (the whole), then let the (apotome) be called a third apotome.

Modern Formulation

The third apotome is a straight line whose length is \[\sqrt{\alpha}\,\left(1-\sqrt{1-\beta^{\,2}}\right),\]

where \(\alpha,\beta\) denote positive rational numbers.

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

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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