Lemma: Lem. 10.016: Incommensurability of Sum of Incommensurable Magnitudes

(Lemma to Proposition 16 from Book 10 of Euclid's “Elements”)

If a parallelogram, falling short1 by a square figure, is applied to some straight line then the applied (parallelogram) is equal (in area) to the (rectangle contained) by the pieces of the straight line created via the application (of the parallelogram). * For let the parallelogram $AD$, falling short by the square figure $DB$, have been applied to the straight line $AB$. * I say that $AD$ is equal to the (rectangle contained) by $AC$ and $CB$.


Modern Formulation

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Proofs: 1

Proofs: 1

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Adapted from (subject to copyright, with kind permission)

  1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"


  1. Note that this lemma only applies to rectangular parallelograms (translator's note).