Proposition: Prop. 10.001: Existence of Fraction of Number Smaller than Given Number

Euclid's Formulation

If, from the greater of two unequal magnitudes (which are) laid out, (a part) greater than half is subtracted, and (if from) the remainder (a part) greater than half (is subtracted), and (if) this happens continually, then some magnitude will (eventually) be left which will be less than the lesser laid out magnitude.


Modern Formulation

(not yet contributed)

Proofs: 1

Proofs: 1 2 3 4 5 6 7
Sections: 8

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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. This theorem is the basis of the so-called {\em method of exhaustion}, and is generally attributed to Eudoxus of Cnidus (translator's note).