Proposition: Prop. 10.065: Square on Side of Sum of two Medial Area applied to Rational Straight Line

(Proposition 65 from Book 10 of Euclid's “Elements”)

The square on the square root of (the sum of) two medial (areas) applied to a rational (straight line) produces as breadth a sixth binomial (straight line).1 * Let $AB$ be the square root of (the sum of) two medial (areas), having been divided at $C$. * 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 sixth binomial (straight line).


Modern Formulation

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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 two medials is a sixth binomial. See [Prop. 10.59] (translator's note).