In this course we traced the development of hyperbolic (non-Euclidean) geometry from ancient Greece up to the turn of the century. This was accomplished by focusing chronologically on those mathematicians who made the most significant contributions to the subject.
We began with an exposition of Euclidean geometry, first from Euclid's perspective (as given in his Elements) and then from a modern perspective due to Hilbert (in his Foundations of Geometry). Almost all criticisms of Euclid up to the 19th century were centered on his fifth postulate, the so-called Parallel Postulate.The first half of the course dealt with various attempts by ancient, medieval, and (relatively) modern mathematicians to prove this postulate from Euclid's others. Some of the most noteworthy efforts were by the Roman mathematician Proclus, the Islamic mathematicians Omar Khayyam and Nasir al-Din al-Tusi, the Jesuit priest Girolamo Sacchieri, the Englishman John Wallis, and the Frenchmen Lambert and Legendre. Each one gave a flawed proof of the parallel postulate, containing some hidden assumption equivalent to that postulate. In this way properties of hyperbolic geometry were discovered, even though no one believed such a geometry to be possible.
The second half of the course covered the discoveries of the 19th century. Gauss, Schweikart, and Taurinus were the first ones to consider the possibility that a non-Euclidean may be self-consistent, but it was Janos Bolyai and Lobachevskii who provided thorough descriptions of hyperbolic geometry, and it is they who are considered the founders of the subject. The analytic nature of these works contrasted sharply with the synthetic arguments of the earlier mathematicians. Riemann changed the subject even more dramatically with his introduction of differential means of describing these geometries. He expanded the class of non-Euclidean geometries to include elliptic geometry (which was then called Riemannian geometry) and also geometries whose properties may vary from point to point (which is now what is meant by Riemannian geometry). This new approach facilitated the discovery of various models of hyperbolic geometry due to Beltrami, Cayley, Poincare, and Klein. These models, the most important of which include the disk models of Beltrami and Poincare and Poincare's half-plane, were then used to prove the logical self-consistency of hyperbolic geometry, thus setting it on equal footing with Euclidean geometry, and to make the connections with complex analysis and algebra that form the points of departure for the modern treatment of the subject. The course concluded with an introduction to Klein's projective geometry, which gave the subject of hyperbolic geometry its name.
References:
Four general references were used throughout this course: Bonola's Non-Euclidean Geometry, Jeremy Gray's Ideas of Space, Greenberg's Euclidean and Non-Euclidean Geometries, and McCleary's Geometry from a Differential Viewpoint. In addition, original works of these mathematicians were used whenever possible, as well as biographies of them. These books included Euclid's Elements, Hilbert's Foundations of Geometry, Proclus's A Commentary on the First Book of Euclid's Elements, Saccheri's Euclid Vindicated, Bolyai's Science of Absolute Space, Lobachevskii's Geometrical Researches in the Theory of Parallels, and Riemann's "On the Hypotheses Which Lie at the Foundations of Geometry," among others.