Lemma: Lem. 10.041: Side of Sum of Medial Areas is Irrational

(Lemma to Proposition 41 from Book 10 of Euclid's “Elements”)

We will now demonstrate that the aforementioned irrational (straight lines) are uniquely divided into the straight lines of which they are the sum, and which produce the prescribed types, (after) setting forth the following lemma. * Let the straight line $AB$ be laid out, and let the whole (straight line) have been cut into unequal parts at each of the (points) $C$ and $D$. * And let $AC$ be assumed (to be) greater than $DB$. * I say that (the sum of) the (squares) on $AC$ and $CB$ is greater than (the sum of) the (squares) on $AD$ and $DB$.


Modern Formulation

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Proofs: 1

Proofs: 1 2

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Adapted from (subject to copyright, with kind permission)

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