Definition: Def. 10.14: Fourth Apotome

Again, if the square on the whole is greater than (the square on) the attached (straight line) by the (square) on (some straight line) incommensurable [in length] with (the whole), and the whole is commensurable in length with the (previously) laid down rational (straight line), then let the (apotome) be called a fourth apotome.

Modern Formulation

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

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

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

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