Proposition: Prop. 10.064: Square on Side of Rational plus Medial Area applied to Rational Straight Line

(Proposition 64 from Book 10 of Euclid's “Elements”)

The square on the square root of a rational plus a medial (area) applied to a rational (straight line) produces as breadth a fifth binomial (straight line).1 * Let $AB$ be the square root of a rational plus a medial (area) having been divided into its (component) straight lines at $C$, such that $AC$ is greater. * And let the rational (straight line) $DE$ be laid down. * 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 fifth 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 the square root of a rational plus medial is a fifth binomial. See [Prop. 10.58] (translator's note).