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Proposition: Prop. 11.23: Sum of Plane Angles Used to Construct a Solid Angle is Less Than Four Right Angles

(Proposition 23 from Book 11 of Euclid's “Elements”)

To construct a solid angle from three (given) rectilinear angles, (the sum of) two of which is greater than the remaining (one, the angles) being taken up in any (possible way). So, it is necessary for the (sum of the) three (angles) to be less than four right angles [Prop. 11.21]. * Let $ABC$, $DEF$, and $GHK$ be the three given rectilinear angles, of which let (the sum of) two be greater than the remaining (one, the angles) being taken up in any (possible way), and, further, (let) the (sum of the) three (be) less than four right angles. * So, it is necessary to construct a solid angle from (rectilinear angles) equal to $ABC$, $DEF$, and $GHK$.

Modern Formulation

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Table of Contents

Proofs: 1

1. Lemma: Lem. 11.23: Making a Square Area Equal to the Difference Of Areas of Two Other Incongruent Squares

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Proofs: 1

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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