# Dropping parallel postulate and infinitude of straight line

I was reading an article about back ground of Killing's work by Thomas Hawkins from Historia mathematica $1980$ (doi: 10.1016/0315-0860(80)90027-0). It was written that Killing stated that if one drop assumptions infinitude of straight line and parallel postulates four possibilities arise, which we can judge. I got only one which is very simple. Lines which has a transverse need not intersect at a point on the side where sum of interior angles is less than two right angles.

Is it correct? Then what are the other possibilities Killing suggests?

## 1 Answer

The cited statement appears on page 301 of the article. I believe the answer to your question is given on page 311, where the following appears: "Just as previous generations of mathematicians had offered proofs of the parallel postulate, those of the second half of the 19th century provided many proofs that there are only three, or in some cases, four geometries compatible with experience: Euclidean and Lobachevskian geometry, the spherical geometry of Riemann, and elliptic geometry." This statement is in reference to the prevailing late 19th century opinion that, pace Riemann, geometry meant manifolds of constant curvature.

Lobachevsky's geometry is the geometry of constant negative curvature, which has the property you mention. The sphere has constant positive curvature. There are no parallel lines, and lines perpendicular to a given line will meet at two antipodal points. (As a consequence, two points do not always uniquely determine a line.) Elliptic geometry is the geometry of the sphere with antipodal points identified, which has the topology of the projective plane. Like spherical geometry, this has constant positive curvature, but lines perpendicular to a given line meet at only one point, and the property that two points uniquely determine a line is restored.

Some of the historical background for the above is given in the pages preceding page 301 of the article. Page 298 explains how Riemann's suggestion of a surface of constant positive curvature was initially widely understood to be spherical geometry, until the independent discovery of elliptic geometry by Klein and Newcomb had time to sink in.