Proposition: Prop. 10.016: Incommensurability of Sum of Incommensurable Magnitudes
(Proposition 16 from Book 10 of Euclid's “Elements”)
If two incommensurable magnitudes are added together then the whole will also be incommensurable with each of them. And if the whole is incommensurable with one of them then the original magnitudes will also be incommensurable (with one another).
* For let the two incommensurable magnitudes $AB$ and $BC$ be laid down together.
* I say that the whole $AC$ is also incommensurable with each of $AB$ and $BC$.
* And so let $AC$ be incommensurable with one of $AB$ and $BC$.
* I say that $AB$ and $BC$ are also incommensurable.
Modern Formulation
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Table of Contents
Proofs: 1
 Lemma: Lem. 10.016: Incommensurability of Sum of Incommensurable Magnitudes
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Proofs: 1 2 3 4 5 6 7 8 9 10
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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