Proposition: Prop. 10.114: Area contained by Apotome and Binomial Straight Line Commensurable with Terms of Apotome and in same Ratio

Euclid's Formulation

If an area is contained by an apotome, and a binomial whose terms are commensurable with, and in the same ratio as, the terms of the apotome then the square root of the area is a rational (straight line). * For let an area, the (rectangle contained) by $AB$ and $CD$, have been contained by the apotome $AB$, and the binomial $CD$, of which let the greater term be $CE$. * And let the terms of the binomial, $CE$ and $ED$, be commensurable with the terms of the apotome, $AF$ and $FB$ (respectively), and in the same ratio. * And let the square root of the (rectangle contained) by $AB$ and $CD$ be $G$. * I say that $G$ is a rational (straight line).


Modern Formulation

(not yet contributed)

Proofs: 1 Corollaries: 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