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Proposition: Prop. 10.002: Incommensurable Magnitudes do not Terminate in Euclidean Algorithm

(Proposition 2 from Book 10 of Euclid's “Elements”)

If the remainder of two unequal magnitudes (which are) [laid out] never measures the (magnitude) before it, (when) the lesser (magnitude is) continually subtracted in turn from the greater, then the (original) magnitudes will be incommensurable.

• For, $AB$ and $CD$ being two unequal magnitudes, and $AB$ (being) the lesser, let the remainder never measure the (magnitude) before it, (when) the lesser (magnitude is) continually subtracted in turn from the greater.
• I say that the magnitudes $AB$ and $CD$ are incommensurable.

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References

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", https://proofwiki.org/wiki/Main_Page, 2016