Proposition: Prop. 10.015: Commensurability of Sum of Commensurable Magnitudes

(Proposition 15 from Book 10 of Euclid's “Elements”)

If two commensurable magnitudes are added together then the whole will also be commensurable with each of them. And if the whole is commensurable with one of them then the original magnitudes will also be commensurable (with one another). * For let the two commensurable magnitudes $AB$ and $BC$ be laid down together. * I say that the whole $AC$ is also commensurable with each of $AB$ and $BC$. * And so let $AC$ be commensurable with $AB$. * I say that $AB$ and $BC$ are also commensurable.


Modern Formulation

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Proofs: 1

Proofs: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

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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