Hyperbolic geometry: The first 150 years

Abstract

This will be a description of a few highlights in the early history of non-euclidean geometry, and a few miscellaneous recent developments.An Appendix describes some explicit formulas concerning volume in hyperbolic 3-space.The mathematical literature on non-euclidean geometry begins in 1829 with publications by N. Lobachevsky in an obscure Russian journal.The infant subject grew very rapidly.Lobachevsky was a fanatically hard worker, who progressed quickly from student to professor to rector at his university of Kazan, on the Volga.Already in 1829, Lobachevsky showed that there is a natural unit of distance in non-euclidean geometry, which can be characterized as follows.In the right triangle of Figure 1 with fixed edge a, as the opposite vertex A moves infinitely far away, the angle 9 will increase to a limit 9 0 which is assumed to be strictly less than 7r/2.He showed that a = -log tan(0 o /2) if the unit of distance is suitably chosen.In particular, a « (TT/2) -0 O if a is very small.(In the interpretation introduced by Beltrami forty years later, this unit of distance is chosen so that curvature = -1.)FIGURE 1.A right triangle in hyperbolic space By early 1830, Lobachevsky was testing his "imaginary geometry" as a possible model for the real world.If the universe is non-euclidean in Lobachevsky's sense, then he showed that our solar system must be extremely small, in terms of this natural unit of distance.More precisely, taking the vertex A in Figure 1 to be the star Sirius and taking the edge a to be a suitably chosen radius of the Earth's orbit, he used the (unfortunately incorrect) estimate 7T -20 ss 1.24 seconds of arc s6x 10~6 radians

Locations

• Project Euclid (Cornell University) - View - PDF
• Bulletin of the American Mathematical Society - View - PDF