Proposition: Prop. 10.112: Square on Rational Straight Line applied to Binomial Straight Line

(Proposition 112 from Book 10 of Euclid's “Elements”)1

The (square) on a rational (straight line), applied to a binomial (straight line), produces as breadth an apotome whose terms are commensurable (in length) with the terms of the binomial, and, furthermore, in the same ratio. Moreover, the created apotome will have the same order as the binomial.


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. Heiberg considers this proposition, and the succeeding ones, to be relatively early interpolations into the original text (translator's note).