Describe the work of Gauss, Bolyai and Lobachevsky on non-Euclidean geometry, including mathematical details of some of their results. What impact, if any, did the rise of non-Euclidean geometry have on subsequent developments in mathematics?
Word Count: 1912
Euclidean geometry is the everyday “flat” or parabolic geometry which uses the axioms from Euclid’s book The Elements. Non-Euclidean geometry includes both hyperbolic and elliptical geometry [W5] and is a construction of shapes using a curved surface rather than an n-dimensional Euclidean space. The main difference between Euclidean and non-Euclidean geometry is the nature of parallel lines. There has been much investigation into the first five of Euclid’s postulates; mainly into proving the formulation of the fifth one, the parallel postulate, is totally independent of the previous four. The parallel postulate states “that, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.” [W1] Many mathematicians have carried out extensive work into proving the parallel postulate and into the development of non-Euclidean geometry and the first to do so were the mathematicians Saccheri and Lambert. Lambert based most of his developments on previous results and conclusions by Saccheri. Saccheri looked at the three possibilities of the sum of the angles in a triangle. He found the first to be <180°, one being =180° and the final being >180°. He showed that the first, that the angles total <180°, was viable but only through the use of a new geometry system, non-Euclidean geometry therefore beginning the investigation into this. Although both Lambert and Saccheri contributed to this field of mathematics, the most famous mathematicians linked to non-Euclidean geometry are Carl Friedrich Gauss, Janos Bolyai and Nikolai Lobachevsky.
Carl Friedrich Gauss, a German mathematician, was once referred to as the “Prince of Mathematics” [W2] and was the first mathematician to really understand the problem of the parallels [W3]. He did mathematical work in the fields of prime and complex numbers, shape construction, statistics and non-Euclidean geometry. By 1817 Gauss had determined that the fifth of Euclid’s postulates, the parallel postulate, was independent of the other four. He began to work out the consequences of a geometry in which more than one line could be drawn through a given point parallel to a given line [W3]. Although he did not publish any of these findings, Gauss was seen as the initiator for further work by Bolyai and Lobachevsky due to the time he spent surveying for the Royal House of Hanover where he questioned Euclidean geometry. Gauss started to investigate the shape of the Earth and the idea that the world was not flat but curved. He also became interested in differential geometry and curvature, the study of curves and surfaces in a 3-D Euclidean space.
Gauss showed many of his theorems to other mathematicians via letters. Some of his findings included him calculating the upper bound of the area of the hyperbolic triangle, the circumference of the hyperbolic geometric circle and the formula for the area of the hyperbolic triangle [B1 Pg 27]. Gauss said in a discussion with Taurinus in November 1824 about triangles in non-Euclidean space, "the assumption that the angle sum is less than 180° leads to a geometry quite different from Euclid's, logically coherent, and one that I am entirely satisfied with. It depends on a constant, which is not given a priori. The larger the constant, the closer the geometry to Euclid's and when the constant is infinite they agree.”[A, Pg63] There is not much evidence of Gauss’ contributions to trigonometry in non-Euclidean geometry, but in a letter to Schumacher in July 1831 Gauss gives 12πk(erk-e-rk), where k is a large constant, or infinite in the Euclidean case, as the formula for the circumference of a semi-circle [A, Pg63]. Gauss also had discussions with a friend of his, Farkas Bolyai, the father of Janos Bolyai, about his theory of the parallels, who himself tried to create a faultless proof. Janos followed on from his father’s research and continued to work on other parallel postulate proofs.
Janos Bolyai published work on the parallel postulate in 1831 and Gauss, although being amazed at Bolyai’s intelligence and his remarkable breakthrough, also disappointed Bolyai by telling him that he had already done an extensive amount of work into these problems. In 1829, around the same time as the publication of Bolyai’s work, a Russian mathematician Nikolai Lobachevsky had also published a non-Euclidean geometry text, On the Principles of Geometry. Neither Bolyai nor Gauss knew of Lobachevsky, showing similar problems with Euclidean mathematics were being explored simultaneously around the world.

Lobachevsky created a substitute for the parallel postulate after much work on it. Lobachevsky’s definition was that “All straight lines which in a plane go out from a point can, with reference to a given straight line in the same plane, be divided into two classes – cutting and not-cutting. The boundary lines of the one and other class of those lines will be parallel to the given line.”[B2 Pg 13]
Lobachevsky constructed the diagram below to examine parallel lines:

According to Katz [B3 Pg 775] Lobachevsky stated that if BC is a line, A is a point not on that line and AD is a perpendicular from A to BC. You can draw AE, a perpendicular to the line AD, which does not meet BC. Lobachevsky then assumed that there may be other lines drawn through A, e.g. AG that do not meet BC. There are two different types of line that can be drawn, AF which is considered a cutting line as it crosses the line BC, and AG which is a not-cutting line. In between the lines AF and AG there must be a line AH that is the boundary between these two sets. Lobachevsky sees AH as parallel to BC as it does not intersect it, and the angle HAD between AH and the perpendicular AD as dependent on the length p of AB. This angle HAD is called the angle of parallelism and is written as ∏(p). The Euclidean case occurs if ∏(p) = 90° therefore there is only one line through A parallel to BC. If ∏(p) < 90° there will be a corresponding line AK on the other side of AD opposite to AH that also makes the same angle ∏(p) with AD. This means there will be two parallels, one on each side of the line AD and this is always the non-Euclidean case to distinguish two different sides in parallelism. Under the non-Euclidean assumption it can be seen that there are indefinitely many lines that can be drawn through A that do not meet BC on each side of AD.

Bolyai explored the concept of parallels in a similar way to Lobachevsky. This is the diagram he uses which is from Pg5 in his book The Science Absolute of Space.

According to Fauvel and Gray [B4 Pg 528-529] Bolyai says that the ray AM is not cut by the ray BN which is in the same plane, but it is cut by all rays like BP that are drawn within the angle ABN. Bolyai therefore says that BN is parallel to AM, written as BN|| AM. Now Bolyai says there is only one ray BN, which passes through any point B, not on AM where the sum of both the angles BAM and ABN cannot exceed 180°. For in moving BC around B until BAM+ABC= 180°, somewhere ray BC first does not cut ray AM, and it is then BC||AM. It is clear that BN||EM, wherever the point E be taken on the straight AM (supposing AM>AE). As C goes to infinity on AM, CD = CB and so CBD = (CBD < NBC) but NBC = 0 and ABD = 0 so CBD = 0.
Bolyai also considered absolute geometry, which was a collection of theorems that were true independent of the parallel postulate. He proved that “in any rectilinear triangle, the [circumference of the] circles with radii equal to its sides are as the sines of the opposite angles.” [B3 Pg 778] This theorem Bolyai discusses with respect to the non-Euclidean plane and defines the circumference of the circle as 2πKsinhrK. His theorem, using this circumference formula, translates into, sinhaK:sinhbK:sinhcK=sinA:sinB:sinC. This is very similar to Lobachevsky’s formula sinAcot(b)=sinBcota, where cot(b)=sinh(b) and cot(a)=sinh(a).
Lobachevsky explored the non-Euclidean plane and the trigonometry within it. Through an intricate argument containing both triangles in the non-Euclidean plane and spherical triangles, he was able to conclude the function (x) but in the form, tan⁡(12)x=e-x. From this formula he could then further derive a relationship between the sides a, b and c and the opposite angles A, B and C of an arbitrary non-Euclidean triangle. Lobachevsky derived the formula sinAcot(b)=sinBcota which gives a similar result to the work by Bolyai above. Lobachevsky also looked into the non-Euclidean plane and it was found that the Lobachevskian plane was a projection of a sphere onto the circle u2+v2=a2 and that any lines drawn within this plane were actually chords within the circle [B3 Pg 784]. This idea of the projective plane was further developed by Henri Poincare in 1882 with the creation of the Poincare disc model where parallel lines were seen as non-intersecting or not-cutting lines within the plane as shown below.

These two lines are parallel to one another as they do not intersect.
[B3 Pg 785]

The ideas on the parallel postulate and non-Euclidean geometry presented by both Bolyai and Lobachevsky were initially rejected, but after a few years a new differential geometry started to be used. The mathematicians involved in this move forwards were Riemann and Beltrami who both used the non-Euclidean space previously explored by Bolyai and Lobachevsky. They managed to measure lengths within different spaces, whether the space in which they were working was Euclidean or not and this was the first rigorous account of the non-Euclidean geometry of Bolyai and Lobachevsky. There were not many other advances for at least another generation due to the lack of trust in the work done by Bolyai and Lobachevsky and the lack of belief in the fact that there were three types of geometry, rather than just Euclidean geometry. Felix Klein in 1871proposed the recasting of Riemann’s work using the language of projective geometry and then showing the existence of three geometries this way rather than by just introducing people to the idea.
The progression throughout the eras started with use of classical geometry by Lambert and Saccheri. This then progressed to Bolyai and Lobachevsky using analysis, and resulted in Riemann and Beltrami using differential geometry. Following on from the discovery of non-Euclidean geometry, Riemann did work in the field of geometry on a surface of negative curvature and also Legendre set to proving that the angle sum of a triangle could not be greater than two right angles. Legendre also demonstrated that if in one triangle the angle sum is two right angles, then the same is true in every triangle [B4 Pg 535]. Unfortunately he did not prove this rigorously enough as he did not show that the sum could not be less than two right angles. These developments of mathematics were spurred on by the work by Gauss, Bolyai and Lobachevsky and without them carrying out work on the parallel postulate; further, more complicated geometries would not have been discovered. Modern geometry is made up of three different geometries; parabolic, hyperbolic and elliptical and without the discovery of non-Euclidean geometry we would only use one type of geometry, the Euclidean geometry.

Bibliography
[A] Jeremy Gray. Gauss and Non-Euclidean Geometry. Centre for Mathematical Sciences, Open University. http://bilder.buecher.de/zusatz/20/20768/20768551_lese_1.pdf.
[B1] András Prékopa, Emil Molnár, Magyar Tudományos Akadémia. Non-Euclidean geometries: Janos Bolyai memorial volume. Birkhäuser publishings. 2006.
[B2] Lobachevsky. Geometrical Investigations on the Theory of Parallel Lines. Translated be G. B. Halsted in Bonola. 1840.
[B3] Victor. J. Katz. A History of Mathematics. 2nd Edition. Addison-Wesley Education Publishers, Inc. 1998.
[B4] J. Fauvel and J.Gray. The History of Mathematics: A Reader. Edited version. Open University. Palgrave Macmillan Publishings. 1987.
[W1]D.E. Joyce. http://aleph0.clarku.edu/~djoyce/java/elements/bookI/bookI.html. 1998.
[W2] Luke Mastin. http://www.storyofmathematics.com/19th_gauss.html. Last accessed 15th Jan 2011.
[W3] J.J. O’Connor, E.F. Robertson. http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Non-Euclidean_geometry.html. Feb 1996.
[W4] Net Industries and its Licensors. http://science.jrank.org/pages/4703/Non-Euclidean-Geometry-founders-non-Euclidean-geometry.html. 2011. [W5] Weisstein, Eric W. "Non-Euclidean Geometry." http://mathworld.wolfram.com/Non-EuclideanGeometry.html. MathWorld, A Wolfram Web Resource. 1999-2010. |