Overview

Ever since the time of Euclid, mathematicians have felt that Euclid's fifth postulate, which lets only one straight line be drawn through a given point parallel to a given line, was a somewhat unnatural addition to the other, more intuitively appealing, postulates. Eighteenth-century mathematicians attempted to remove the problem either by deriving the postulate from the others, thus making it a theorem, or by replacing it with a simpler statement. Nineteenth-century mathematicians would change the postulate to generate logically consistent non-Euclidean geometries, which twentieth-century physicists would in turn propose as the true geometry of space and time.

Background

In his Elements of Geometry, the great Greek mathematician Euclid (335-270 B.C.) was forced to adopt a rather awkwardly worded fifth and final postulate:

If a straight line falling on two straight lines makes the interior...