Proposition: Prop. 10.063: Square on Major Straight Line applied to Rational Straight Line

(Proposition 63 from Book 10 of Euclid's “Elements”)

The square on a major (straight line) applied to a rational (straight line) produces as breadth a fourth binomial (straight line).1 * Let $AB$ be a major (straight line) having been divided at $C$, such that $AC$ is greater than $CB$, and (let) $DE$ (be) a rational (straight line). * And let the parallelogram $DF$, equal to the (square) on $AB$, have been applied to $DE$, producing $DG$ as breadth. * I say that $DG$ is a fourth binomial (straight line).


Modern Formulation

(not yet contributed)

Proofs: 1

Proofs: 1

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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 major is a fourth binomial. See [Prop. 10.57] (translator's note).