when one of Gauss’s students, Bernhard Riemann, showed that hyperbolic geometry was not the only non-Euclidean geometry possible. In a brilliant lecture delivered in Göttingen on June 10, 1854 (figure 45 shows the first page of the published lecture), Riemann presented his views “On the Hypotheses That Lie at the Foundations of Geometry.” He started by saying that “geometry presupposes the concept of space, as well as assuming the basic principles for constructions in space. It gives only nominal definitions of these things, while their essential specifications appear in the form of axioms.” However, he noted, “The relationship between these presuppositions is left in the dark; we do not see whether, or to what extent, any connection between them is necessary, or a priori whether any connection between them is even possible.” Among the possible geometrical theories Riemann discussed elliptic geometry, of the type that one would encounter on the surface of a sphere (figure 41c). Note that in such a geometry the shortest distance between two points is not a straight line, it is rather a segment of a great circle, whose center coincides with the center of the sphere. Airlines take advantage of this fact—flights from the United States to Europe do not follow what would appear as a straight line on the map, but rather a great circle that initially bears northward. You can easily check that any two great circles meet at two diametrically opposite points. For instance, two meridians on Earth, which appear to be parallel at the Equator, meet at the two poles. Consequently, unlike in Euclidean geometry, where there is exactly one parallel line through an external point, and hyperbolic geometry, in which there are at least two parallels, there are no parallel lines at all in the elliptic geometry on a sphere. Riemann took the non-Euclidean concepts one step further and introduced geometries in curved spaces in three, four, and even more dimensions. One of the key concepts expanded upon by Riemann was that of the curvature—the rate at which a curve or a surface curves. For instance, the surface of an eggshell curves more gently around its girth than along a curve passing through one of its pointy edges. Riemann proceeded to give a precise mathematical definition of curvature in spaces of any number of dimensions. In doing so he solidified the marriage between algebra and geometry that had been initiated by Descartes. In Riemann’s work equations in any number of variables found their geometrical counterparts, and new concepts from the advanced geometries became partners of equations.
Figure 45
Euclidean geometry’s eminence was not the only victim of the new horizons that the nineteenth century opened for geometry. Kant’s ideas of space did not survive much longer. Recall that Kant asserted that information from our senses is organized exclusively along Euclidean templates before it is recorded in our consciousness. Geometers of the nineteenth century quickly developed intuition in the non-Euclidean geometries and learned to experience the world along those lines. The Euclidean perception of space turned out to be learned after all, rather than intuitive. All of these dramatic developments led the great French mathematician Henri Poincaré (1854–1912) to conclude that the axioms of geometry are “neither synthetic a priori intuitions nor experimental facts. They are conventions [emphasis added]. Our choice among all possible conventions is guided by experimental facts, but it remains free.” In other words, Poincaré regarded the axioms only as “definitions in disguise.”
Poincaré’s views were inspired not just by the non-Euclidean geometries described so far, but also by the proliferation of other new geometries, which before the end of the nineteenth century seemed to be almost getting out of hand. In projective geometry (such as that obtained when an image on celluloid film is projected onto a screen), for instance, one could literally interchange the roles of points and lines, so that theorems about points and lines (in this order)