Definition: Def. 10.05: First Binomial

Given a rational (straight line), and a binomial (straight line) which has been divided into its (component) terms, of which the square on the greater term is larger than (the square on) the lesser by the (square) on (some straight line) commensurable 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 first binomial (straight line).

Modern Formulation

The first binomial is a straight line whose length is \[\alpha+\alpha\sqrt{1-\beta^{\,2}},\] where \(\alpha,\beta\) denote positive rational numbers.

Proofs: 1 2 3 4 5 6 7 8 9
Propositions: 10 11 12

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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