Non–Euclidean Geometry

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Non-Euclidean Geometry

non-Euclidean geometry refers to certain types of geometry that differ from plane geometry and solid geometry, which dominated the realm of mathematics for several centuries. There are other types of geometry that do not assume all of Euclid’s postulates such as hyperbolic geometry, elliptic geometry, spherical geometry, descriptive geometry, differential geometry, geometric algebra, and multidimensional geometry. These geometries deal with more complex components of curves in space rather than the simple plane or solids used as the foundation for the geometry invented by Euclid of Alexandria (c. 325–c. 265 BC), what is called Euclid’s geometry or Euclidean geometry. The first five postulates of Euclidean geometry will be listed in order to better understand the changes that are made to make it non-Euclidean.

A straight line can be drawn from any point to any point.
A finite straight line can be produced continuously in a straight line.
A circle may be described with any point as center and any distance as a radius.
All right angles are equal to one another.
If a transversal falls on two lines in such a way that the interior angles on one side of the transversal are less than two right angles, then the lines meet on the side on which the angles are less than two right angles.

A consistent logical system for which one of these postulates is modified in an essential way is non-Euclidean geometry. Although there are different types of non-Euclidean geometry which do not use all of the postulates or make alterations of one or more of the postulates of Euclidean geometry, hyperbolic and elliptic are usually most closely associated with the term non-Euclidean geometry.

Hyperbolic geometry is based on changing Euclid’s parallelpostulate, which is also referred to as Euclid’s fifth postulate, the last of the five postulates of Euclidian Geometry. Euclid’s parallel postulate may also be stated as one and only one parallel to a given line goes through a given point not on the line.

Elliptic geometry uses a modification of Postulate II. Postulate II allows for lines of infinite length, which are denied in Elliptic geometry, where only finite lines are assumed.

The history of non-Euclidean geometry

Euclid was thought to have instructed in Alexandria after Alexander the Great established centers of learning in the city around 300 BC. Euclid was the mathematician who collected all of the definitions, postulates, and theorems that were available at that time, along with some of his insights and developments, and placed them in a logical order and completed what mathematicians now know as Euclid’s Elements.

The influence of Greek geometry on the mathematics communities of the world was profound for in Greek geometry was contained the ideals of deductive thinking with its definitions, corollaries, and theorems which could establish beyond any reasonable doubt the truth or falseness of propositions. For an estimated 22 centuries, Euclidean geometry held its weight.

Despite the general acceptance of Euclidean geometry, there appeared to be a problem with the parallel postulate as to whether or not it really was a postulate or that it could be deduced from other definitions, propositions, or axioms. The history of these attempts to prove the parallel postulate lasted for nearly 20 centuries, and after numerous failures, gave rise to the establishment of non-Euclidean geometry and the independence of the parallel postulate.

Several Greek scientists and mathematicians considered the parallel postulate after the appearance of Euclid’s Elements, around 300 BC. Aristotle’s treatment of the parallel postulate was lost. However, it was Arab scholars who appeared to have obtained some information on the last text and reported that Aristotle’s treatment was different from that of Euclid since his definition depended on the distance between parallel lines. Proclus and Ptolemy also published some attempts to prove the parallel postulate.

Persian mathematician and astronomer Omar Khayyam (1048–1131) provided extensive coverage on the proof of the parallel postulate or theory of parallels in his discussions on the difficulties of making valid proofs from Euclid’s definitions and theorems. During the thirteenth century, Husam al–Din al–Salar wrote a text on the parallel postulate in an attempt to improve on the development by Omar Khayyam.

The eighteenth century produced more sophisticated proofs and although not correct, produced developments that were later used in non-Euclidean geometry. Italian mathematician Girolamo Girolamo Saccheri (1667–1733) in one of his proofs considered non-Euclidian concepts by making use of the acute–angle hypothesis on the intersection of two straight lines.

The attempt to solve this problem was made also by Farkas Bolyai, the father of Johann Bolyai, one of the founders of non-Euclidean geometry but his proof was also invalid. It is interesting to note that Johann’s father cautioned his son not to get involved with the proof of the parallel postulate because of its complexity.

The founders of non-Euclidean geometry

The writings of German mathematician Carl Fredrich Gauss (1777–1855) showed that he, too, first considered the usual attempts at trying to prove the parallel postulate. However, a few decades later, 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. 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, Hungarian mathematician Janos Bolyai (1802–1860) and Russian mathematician Nicolai 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 PanGeometry 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 to 1822. In 1823, Bolyai discovered the formula for the transformation that 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. He 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.) and (5) Problems that can be solved by elementary methods. (Squaring the circle, on the hypothesis that the fifth postulate is false.)

Elliptic non-Euclidean geometry

A later development following that of Bolyai’s and Lobachevsky’s hyperbolic non-Euclidean geometry was that of elliptic non-Euclidian geometry. German mathematician Georg Friedrich Bernhard Riemann (1826–1866) developed the rudiments of elliptic non-Euclidean geometry. His introduction to his foundations of spherical geometry apparently was used as the basis for his elliptic geometry that made use of the postulate that the sum of the angles of a triangle

in space are greater than 180^°. Based on the foundations that Riemann had introduced, Klein was able to further develop elliptic non-Euclidean geometry and was actually the mathematician who defined this new field as elliptic non-Euclidian geometry. Klein’s Erlanger Program made a significant contribution in providing major distinguishing features among parabolic (Euclidean geometry), hyperbolic, and elliptic geometries.