1849-1925

German Mathematician

Felix Klein's work had a profound effect on mathematical thought. He unified Euclidean geometry with the non-Euclidean geometries of Nikolai Lobachevsky (1792-1856) and Georg Riemann (1826-1866) by showing that they all could be derived as special cases of a larger system called projective geometry.

Projective geometry is more fundamental than Euclidean geometry because it deals with properties such as when points lie on the same line and when a set of lines meet in one point. These properties are invariant under a larger group of transformations, or mathematical manipulations, than the congruence, or equality of lengths, angles, and areas with which Euclidean geometry mainly concerns itself. "Projective geometry has opened up for us with the greatest facility new territories in our science," Klein wrote, "and has rightly been called a royal road to its own particular field of knowledge."

Klein, whose full given...