Definition: Def. 10.12: Second Apotome

(Definition 12 from Book 10 of Euclid's “Elements”)

And if 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 second apotome.

Modern Formulation

The second apotome is a straight line whose length is \[\frac{\alpha}{\sqrt{1-\beta^{\,2}}}-\alpha,\]

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

Corollaries: 1
Proofs: 2 3 4 5 6 7 8 9 10 11 12
Propositions: 13 14 15 16 17 18 19 20 21 22

Thank you to the contributors under CC BY-SA 4.0!



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