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.
 Let $AB$ and $C$ be two unequal magnitudes, of which (let) $AB$ (be) the greater.
 I say that if (a part) greater than half is subtracted from $AB$, and (if a part) greater than half (is subtracted) from the remainder, and (if) this happens continually, then some magnitude will (eventually) be left which will be less than the magnitude $C$.
Modern Formulation
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Table of Contents
Proofs: 1
Mentioned in:
Proofs: 1 2 3 4 5 6 7
Sections: 8
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References
Adapted from (subject to copyright, with kind permission)
 Fitzpatrick, Richard: Euclid's "Elements of Geometry"
Adapted from CC BYSA 3.0 Sources:
 Prime.mover and others: "Pr∞fWiki", https://proofwiki.org/wiki/Main_Page, 2016
Footnotes