Non-Euclidean Geometry

The writings of Gauss showed that he too, first considered the usual attempts at trying to prove the parallel postulate. However, a few decades later, in his unpublished reports in his correspondence with fellow mathematicians such as W. Bolyai, Olbers, Schumacker, Gerling, Tartinus, and Bessel showed that Gauss was working on the rudiments of non-Euclidian geometry, the name he attributed to his mathematics of parallels. Gauss shared his thoughts on this topic and asked them not to disclose this information but Gauss never published them. It has been proposed by historians that Gauss was concerned that these concepts were too radical for acceptance by mathematicians at that time. And if this was the case, it probably was correct since the two founders of non-Euclidean geometry, Bolyai and Lobechevsky, received very little acceptance until after their deaths.

It was at the University of Kazan, in the Russian province of Kazakhstan, that Nicolai Ivanovitch Lobachevsky made his contributions in Non-Euclidean geometry. In his early days at the university, he did try to find a proof of the parallel postulate, but later changed direction. As early as 1826, he made use of the hypothesis of the acute angle already developed by Saccheri and Lambert in his lecture noting that two parallels to a given point can be drawn from a point where the sum of the angles of the triangle is less than two right angles. His works On the Imaginary Geometry, New Principles of Geometry, With a Complete Theory of Parallels, Applications of the Imaginary Geometry to Certain Integrals, and Geometrical Researches are on the theory of parallel lines. He later completed his work in one French and two German publications. Lobachevsky developed his Pan-Geometry on the 28 propositions of Euclidean geometry and the negation of the parallel postulate. He developed the concepts for non-Euclidean geometry by introducing two new figures—the horocycle and the horoscope. Using these two concepts and some transformation formulas, he developed his new geometry.

Although Lobachevsky continued throughout his career improving the development of non-Euclidean geometry, Johann Bolyai, the other mathematician given credit for its development apparently only spent slightly over a decade in his mathematical considerations. As indicated previously, Johann's father suggested that he not waste his time working on the complex problems of the parallel postulate. However, Johann and his friend Carl Szasz worked on the theory of parallels while students at the Royal College for Engineers at Vienna from 1817-1822. In 1823 Bolyai discovered the formula for the transformation which connected the angle of parallelism to the corresponding line. He continued with his development and sent his manuscript to his father who published it in 1832. The article was entitled "The Science of Absolute Space" in the Appendix of his father's book. Prior to its publication, Johann's father had sent the paper to Gauss for his consideration. It is reported that the paper originally sent in 1831 to Gauss was lost. Three months after the publication, the article was sent again to Gauss and in 1832 his father received his reply. Gauss indicated that he was impressed by the work but noted that he had been working on the same problem with similar results and was pleasantly surprised to have the development completed by his friend's son. Johann was deeply suspicious of this reply and apparently suspected Gauss of trying to take credit for his work. However, in this instance there was no problem, since Gauss had no publications on the topic and could not claim priority but Johann continued to be suspicious. After the publication of his work, Johann did very little significant mathematical research. And even though he was interested in having his work published before Lobachevsky when he heard of Lobachevsky's contributions, he never completed the necessary research to report to the mathematical journals.

The most important conclusions of Bolyai's research in non-Euclidean geometry were the following: (1) The definition of parallels and their properties independent of the Euclidian postulate. (2) The circle and the sphere of infinite radius. The geometry of the sphere of infinite radius is identical with ordinary plane geometry. (3) Spherical trigonometry is independent of Euclid's postulate. Direct demonstration of the formula. (4) Plane trigonometry in non-Euclidean geometry. Applications to the calculation of areas and volumes. (5) Problems which can be solved by elementary methods. Squaring the circle, on the hypothesis that the fifth postulate is false.