Definition: Def. 10.07: Third Binomial

And if neither of the terms is commensurable in length with the rational (straight line previously) laid out then let (the whole straight line) be called a third binomial (straight line).

Modern Formulation

The third binomial 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 7
Propositions: 8 9 10

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