Noneuclidean Geometry

Abstract

Until the nineteenth century, Euclid’s geometry enjoyed absolute authority, both as an axiomatic system and as a description of physical space. Euclid’s proofs were regarded as models of logical rigor, and his axioms were accepted as correct statements about physical space. Even today, euclidean geometry is the simplest type of geometry, and it furnishes the simplest description of physical space for everyday purposes. Beyond the everyday world, however, lies a vast universe that can be understood only with the help of an expanded geometry. The expansion of geometric concepts initially grew from dissatisfaction with one of Euclid’s axioms, the parallel axiom.