Proposition: Prop. 10.060: Square on Binomial Straight Line applied to Rational Straight Line

(Proposition 60 from Book 10 of Euclid's “Elements”)

The square on a binomial (straight line) applied to a rational (straight line) produces as breadth a first binomial (straight line).1 * Let $AB$ be a binomial (straight line), having been divided into its (component) terms at $C$, such that $AC$ is the greater term. * And let the rational (straight line) $DE$ be laid down. * And let the (rectangle) $DEFG$, equal to the (square) on $AB$, have been applied to $DE$, producing $DG$ as breadth. * I say that $DG$ is a first binomial (straight line).


Modern Formulation

(not yet contributed)

Proofs: 1

Proofs: 1 2

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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


  1. In other words, the square of a binomial is a first binomial. See [Prop. 10.54] (translator's note)