Definition: Def. 10.08: Fourth Binomial

So, again, if the square on the greater term is larger than (the square on) [the lesser] by the (square) on (some straight line) incommensurable in length with (the greater) then, if the greater term is commensurable in length with the rational (straight line previously) laid out, let (the whole straight line) be called a fourth binomial (straight line).

Modern Formulation

The fourth binomial 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 8
Propositions: 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