A Spherical Pythagorean Theorem

Abstract

for a spherical right triangle with hypotenuse c and legs a and b, is generally presented as the ‘spherical Pythagorean theorem’. Still, it has to be remarked that this formula does not have an immediate meaning in terms of areas of simple geometrical figures, as the Pythagorean theorem does. There is no diagram that can be drawn on the surface of the sphere to illustrate the statement in the spirit of ancient Greek geometry. In this note I reconsider the issue of extending the geometrical Pythagorean theorem to non-Euclidean geometries (with emphasis on the more intuitive spherical geometry). In apparent contradiction with the statement that the Pythagorean proposition is equivalent to Euclid’s parallel postulate, I show that such an extension not only exists, but also yields a deeper insight into the classical theorem. The subject matter being familiar, I can dispense with preliminaries and start right in with Euclid’s Elements [1].