Non-Euclidean Geometrie

Drama of the Discovery

Four names - C. F. Gauss (1777-1855), N. Lobachevsky (1792-1856), J. Bolyai (1802-1860), and B. Riemann (1826-1866) - are traditionally associated with the discovery of non-Euclidean geometries. In non-Euclidean geometries, the fifth postulate is replaced with one of its negations: through a point not on a line, either there is none (B) or more than 1 (C) line parallel to the given one. Carl Friedrich Gauss was apparently the first to arrive at the conclusion that no contradiction may be obtained this way. In a private letter of 1824 Gauss wrote:

The assumption that (in a triangle) the sum of the three angles is less than 180° leads to a curious geometry, quite different from ours, but thoroughly consistent, which I have developed to my entire satisfaction.

From another letter of 1829, it appears that Gauss was hesitant to publish his research because he suspected the mediocre mathematical community would not be able to accept a revolutionary denial of Euclid's geometry. Gauss invented the term "Non-Euclidean Geometry" but never published anything on the subject. On the other hand, he introduced the idea of surface curvature on the basis of which Riemann later developed Differential Geometry that served as a foundation for Einstein's General Theory of Relativity.

Both Bolyai and Lobachevsky first tried to prove the fifth postulate but, in time, felt the task impossible: Bolyai in 1823, Lobachevsky in 1826. Bolyai's father, Farkas - a friend of Gauss and a mathematician himself - was involved with the postulate in his youth. When he learned of his son's interest, Farkas made a dramatic attempt to dissuade his son from taking up the problem that so many others tried but failed to solve. However, when he realized that Janos' work had born fruit, Farkas urged his son to publish it at the earliest opportunity and, eventually, included it as a 26 page Appendix to his book that appeared in 1832.

Gauss, in a letter to F.Bolyai, approved of his son's work but claimed to have developed the same ideas some 30 years earlier. He even provided an elegant proof for one of Janos' theorems. An opinion exists that Janos was crushed by Gauss' letter and suspected Gauss of plagiarism. However it might be, he never published anything on the subject afterwards.

There is no doubt that Gauss, J.Boyai, and N.Lobachevsky were unaware of each other's work. However, Lobachevsky was first to publish a paper on the new geometry. His article appeared in Kazan Messenger in Russian in 1829 and, naturally, passed unnoticed. Trying to reach a broader audience, he published in French in 1837, then in German in 1840, and then again in French in 1855. As a rector of Kazan University, Lobachevsky was awarded a diamond ring by Tsar Nicholas, he also received several medals and citations for other services. However, not until years after his death, his name was associated with the discovery of non-Euclidean geometry. His last government citation Lobachevsky received just a few months before his death. He was cited for the discovery of a new way of processing wool.

Lobachevsky and Bolyai built their geometries on the assumption (C): through a point not on the line there exist more than 1 parallel to the line. This is equivalent to Gauss' assumption that the sum of angles in a triangle is less than 180°. The other possibility, viz., that no two lines are parallel was discharged by Saccheri as contradicting the second postulate. Saccheri, as all the others, assumed (probably, correctly) that Euclid had in mind exactly this interpretation: straight lines may be extended so as to have infinite length.

B.Riemann (1854) was the first to notice that, although meant by Euclid, this interpretation does not necessarily follow from the postulate: A piece of straight line may be extended indefinitely. Riemann wrote:

... we must distinguish between unboundedness and infinite extent ... The unboundedness of space possesses ... a greater empirical certainty than any external experience. But its infinite extent by no means follows from this.

Circles can be extended indefinitely since they have no ends. ("Going in circles" means exactly this: doing something repeatedly without appearing to achieve a certain end or with no end in sight.) However, circles are of finite extent.

Implicit in the first postulate - A straight line may be drawn between any two points - was the assumption that such a line is unique. So Riemann modified Euclid's Postulates 1, 2, and 5 to

  1. Two distinct points determine at least one straight line.
  2. A straight line is boundless.
  1. Any two straight lines in a plane intersect.

It is very easy to envisage the objects (points, lines, and the plane) of a geometry that satisfies the five modified postulates. Plane is a sphere, lines are the great circles, i.e. circles whose plane passes through the center of the sphere, points are regular points (but on the sphere, of course.) As we already saw, in this geometry angles in a triangle sum up to more than 180°.

Logically speaking, the assumption of a single parallel line is equivalent to the fact that angles in a triangle sum to 180°. Furthermore, if this is true of a single triangle, this is also true of all possible triangles. In the geometry of Gauss, Lobachevsky, and Bolyai, parallels are not unique. This is equivalent to the sum of angles in a triangle being less than 180°. And again, this condition holds for all triangles provided it is true for any one of them. Riemann's spherical geometry completes a triad: no parallels and the angle sum is (always) more than 180°.



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