Proposition: Prop. 10.066: Straight Line Commensurable with Binomial Straight Line is Binomial and of Same Order

(Proposition 66 from Book 10 of Euclid's “Elements”)

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