None of the many possible geometries need to refer to Euclidean space, however intuitive Euclidean space may be.

The importance of a rigorous account of any part of mathematics should not be ignored, but the acceptance of non-Euclidean and Riemannian geometries went beyond the presentation of a consistent formalism. It marks the acceptance of the abstract view that geometry is whatever can be described in the Riemannian formalism: one has a very general framework, allowing for a dizzying number of concrete specifications. Thus the door was opened to the view that there are many geometries, each of which mu