parallel postulate self-consistent non-Euclidean geometries Albert Einstein general relativity Euclidean space gravitational field proved Euclid axioms theorems geometry
For more than two thousand years, the adjective "Euclidean" was unnecessary because
Euclid's axioms seemed so intuitively obvious (with the possible exception of the parallel postulate) that theorems proved from them were deemed absolutely true, and thus no other sorts of geometry were possible. Today, however, many other self-consistent non-Euclidean geometries are known, the first ones having been discovered in the early 19th century. An implication of Albert Einstein's theory of general relativity is that physical space itself is not Euclidean, and Euclidean space is a good approximation for it only over short distances (relative to the strength of the  gravitational field).
