 In the middle of the 19th century, alternative conceptions of geometrical space were developed, so-called non-Euclidean geometries. Gauss, Bolyai, Lobachevski and Riemann, showed that one can develop geometries that falsify Euclid’s parallel axiom while preserving all the other Euclidean axioms. The axiom states (in a later but equivalent version to the one given by Euclid) that for any line and a point external to that line, there is one and only one parallel to the given line passing through the point. The statement contains a claim of existence and a claim of uniqueness. It can be thus falsified by denying existence or by accepting the existence but denying the uniqueness. Both alternatives have been developed, with some of the earliest interpretations using surfaces. The first alternative, where no parallels exist to any given line that pass through a given point external to the given line, is known as elliptic geometry. An instance of elliptic geometry is spherical geometry, so called because it can be modeled on the surface of a sphere. The second alternative is known as hyperbolic geometry and in it for every line in the model and any point outside of the line there are infinitely many parallels to that line passing through the point. A portion of the surface of a horse saddle can be used to model hyperbolic geometry. (The pictures below are based on those in Luminet 2008: 49.) 