Proposition: Prop. 10.103: Straight Line Commensurable with Apotome

Euclid's Formulation

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