ThatsMaths

Carl Friedrich Gauss is generally regarded as the greatest mathematician of all time. The profundity and scope of his work is remarkable. So, it is amazing that, while he studied non-Euclidian geometry and defined the curvature of surfaces in space, he overlooked a key connection between curvature and geometry.  As a consequence, decades passed before a model demonstrating the consistency of hyperbolic geometry emerged.

The Parallel Postulate

Given a line L and a point P not on it, there is exactly one line through P parallel to L.

Or is it two? Or perhaps none at all? The statement above is Playfair’s version of the fifth postulate of Euclid, the Parallel Postulate.

The Parallel Postulate was a source of frustration and dissatisfaction for centuries. While Euclid’s first four postulates are simple, unequivocal and intuitive, the fifth is cumbersome in form and far from self-evident. Concerted efforts were made to prove the parallel property as a theorem based on the other four postulates. But all attempts failed.

In the early nineteenth century two mathematicians, one in Hungary and one in Russia, showed that a meaningful version of geometry can be derived in which more than one line through a point P and parallel to L exists. They were Janos Bolyai and Nikolai Ivanovich Lobachevskii.

Farkas (Wolfgang) Bolyai, father of Janos, who had studied with Gauss, sent a description of his son’s work to the man later described as the Prince of Mathematicians. But Gauss responded that he had obtained similar results decades earlier although he had not published them. This had a devastating effect on poor Janos.

Gauss had begun to investigate the Parallel Postulate around 1796 (when he was only nineteen) and had arrived at a system that he later called non-Euclidian geometry. But it is clear that he did not progress as far as Bolyai or Lobachevskii, so his response to Bolyai’s father seems rather harsh.

The True Geometry

If there was more than one geometry, which was the correct one? Which was a true model of the physical world? Indeed, was the new geometry, which Felix Klein latter called hyperbolic geometry, meaningful at all?

Around 1868, the Italian mathematician Eugenio Beltrami showed that hyperbolic geometry could be modelled using a pseudo-sphere. That is, the axioms and theorems of the new geometry are true for geodesic curves on this surface. The pseudo-sphere is a surface of constant negative curvature.

The pseudo-sphere, a surface of constant negative curvature [ Image from http://virtualmathmuseum.org ]

Differential Geometry

Gauss studied the properties of curved surfaces embedded in space, profoundly transforming and extending the subject of differential geometry. He defined the curvature of a surface, relating it to the curvature of the intersections of the surface with planes containing the normal to it.

Gauss discovered an amazing result that he called the Theorema Egregium or remarkable theorem. This showed that curvature was intrinsic to the surface: it could be calculated within the surface itself, without reference to the embedding space.

Gauss studied surfaces having constant curvature, positive like the sphere or ellipsoid, zero like the plane or cylinder and negative like the pseudo-sphere or hyperbolic paraboloid. But he failed to make the link with non-Euclidian geometry. Had he done so, he could have constructed a model of the hyperbolic geometry of Bolyai and Lobachevskii.

Because of this oversight, Gauss never realized that a demonstration of the consistency of the new geometry was there at his finger-tips.

Sources

Gray, Jeremy, 2007: Worlds Out of Nothing. Springer. 376pp. ISBN 978-1-84628-632-2

Sossinsky, A. B. 2012: Geometries. Amer. Math. Soc. 301pp. ISBN 978-0-8218-7571-1