Commentary to Book III of Euclid’s Elements

Abstract

We provide a commentary on the propositions of Book III of Euclid’s Elements, accompanying the Polish translation included in this volume. Focusing on the circle and its relation to triangles, we analyze Proposition III.32 and show that, within absolute geometry, if some right-angled triangle admits a circumscribed circle, then all triangles in the plane are Euclidean (with sum of angles equal two right angles). In this context, we also revisit Saccheri’s theory, in which the existence of a Euclidean right-angled triangle implies the Parallel Postulate, and discuss Lobachevsky’s analysis of an angle of parallelism.