Proposition: Prop. 10.067: Straight Line Commensurable with Bimedial Straight Line is Bimedial and of Same Order

(Proposition 67 from Book 10 of Euclid's “Elements”)

A (straight line) commensurable in length with a bimedial (straight line) is itself also bimedial, and the same in order. * Let $AB$ be a bimedial (straight line), and let $CD$ be commensurable in length with $AB$. * I say that $CD$ is bimedial, and the same in order1 as $AB$.


Modern Formulation

(not yet contributed)

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. Euclid's expression "(not) being the same in order" means that the resulting irrational number is "(not) of the same kind" as that irrational number, with which it is commensurable.