The History of Mathematics: A Brief Course

5. NON-EUCLIDEAN GEOMETRY 347

FIGURE 12. Projection of the Lobachevskii-Bolyai plane onto the

interior of a Euclidean disk.

FIGURE 13. The pseudosphere. Observe that it has no definable

curvature at its cusp. Elsewhere its curvature is constant and

negative.

there were any contradiction in the new geometry, there would be a corresponding

contradiction in Euclidean geometry itself.

A variant of this model was later provided by Henri Poincare (1854-1912), who

showed that the diameters and the circular arcs in a disk that meet the boundary

in a right angle can be interpreted as lines, and in that case angles can be measured

in the ordinary way.

Beltrami also provided a model of a portion of the Lobachevskii-Bolyai plane

that could be embedded in three-dimensional Euclidean space: the pseudosphere

obtained by revolving a tractrix about its asymptote, as shown in Fig. 13.

In 1871 Felix Klein gave a discussion of the three kinds of plane geometry in his

article "Uber die sogennante nicht-Euklidische Geometrie" ("On the so-called non-

Euclidean geometry"), published in the Mathematische Annalen. In that article he

gave the classification of them that now stands, saying that the points at infinity on

a line were distinct in hyperbolic geometry, imaginary in spherical geometry, and

coincident in parabolic (Euclidean) geometry.

The pseudosphere is not a model of the entire Lobachevskii-Bolyai plane, since

its curvature has a very prominent discontinuity. The problem of finding a surface in