During his tenure at theUniversity of Göttingen, Klein was able to turn it into a center for mathematical and scientific research through the establishment of new lectures, professorships, and institutes. Hisseminars covered most areas of mathematics then known as well as their applications. Klein also devoted considerable time to mathematical instruction and promoted mathematics education reform at all grade levels in Germany and abroad. He became the first president of theInternational Commission on Mathematical Instruction in 1908 at the Fourth International Congress of Mathematicians in Rome.
Felix Klein was born on 25 April 1849 inDüsseldorf,[2] toPrussian parents. His father, Caspar Klein (1809–1889), was a Prussian government official's secretary stationed in theRhine Province. His mother was Sophie Elise Klein (1819–1890,née Kayser).[3] He attended theGymnasium in Düsseldorf, then studied mathematics and physics at theUniversity of Bonn,[4] 1865–1866, intending to become a physicist. At that time,Julius Plücker had Bonn's professorship of mathematics and experimental physics, but by the time Klein became his assistant, in 1866, Plücker's interest was mainly geometry. Klein received his doctorate, supervised by Plücker, from the University of Bonn in 1868.
Plücker died in 1868, leaving his book concerning the basis ofline geometry incomplete. Klein was the obvious person to complete the second part of Plücker'sNeue Geometrie des Raumes, and thus became acquainted withAlfred Clebsch, who had relocated to Göttingen in 1868. Klein visited Clebsch the next year, along with visits toBerlin and Paris. In July 1870, at the beginning of theFranco-Prussian War, he was in Paris and had to leave the country. For a brief time he served as a medical orderly in thePrussian army before being appointedPrivatdozent (lecturer) at Göttingen in early 1871.
After spending five years at the Technische Hochschule, Klein was appointed to a chair ofgeometry atLeipzig University. His colleagues includedWalther von Dyck, Rohn,Eduard Study andFriedrich Engel. Klein's years at Leipzig, 1880 to 1886, fundamentally changed his life. In 1882, his health collapsed and he battled with depression for the next two years.[7] Nevertheless, his research continued; his seminal work on hyperelliptic sigma functions, published between 1886 and 1888, dates from around this period.
Klein accepted a professorship at theUniversity of Göttingen in 1886. From then on, until his 1913 retirement, he sought to re-establish Göttingen as the world's prime center for mathematics research. However, he never managed to transfer from Leipzig to Göttingen his own leading role as developer ofgeometry. He taught a variety of courses at Göttingen, mainly concerning the interface between mathematics and physics, in particular,mechanics andpotential theory.
The research facility Klein established at Göttingen served as model for the best such facilities throughout the world. He introduced weekly discussion meetings, and created a mathematical reading room and library. In 1895, Klein recruitedDavid Hilbert from theUniversity of Königsberg. This appointment proved of great importance; Hilbert continued to enhance Göttingen's primacy in mathematics until his own retirement in 1932. Klein and Hilbert jointly invitedEmmy Noether to Göttingen in 1915 where she introduced Einstein to group theory and the relationship between symmetries and conservation principles.[8]
In 1893, Klein was a major speaker at the International Mathematical Congress held in Chicago as part of theWorld's Columbian Exposition.[9] Due partly to Klein's efforts, Göttingen began admitting women in 1893. He supervised the first Ph.D. thesis in mathematics written at Göttingen by a woman, byGrace Chisholm Young, an English student ofArthur Cayley's, whom Klein admired. In 1897, Klein became a foreign member of theRoyal Netherlands Academy of Arts and Sciences.[10]
Around 1900, Klein began to become interested in mathematical instruction in schools. In 1905, he was instrumental in formulating a plan recommending thatanalytic geometry, the rudiments of differential and integralcalculus, and thefunction concept be taught in secondary schools.[11][12] This recommendation was gradually implemented in many countries around the world. In 1908, Klein was elected president of theInternational Commission on Mathematical Instruction at the RomeInternational Congress of Mathematicians.[13] Under his guidance, the German part of the Commission published many volumes on the teaching of mathematics at all levels in Germany. In 1892 theManchester Literary and Philosophical Society awarded Klein Honorary membership of the Society.[14]TheLondon Mathematical Society awarded Klein itsDe Morgan Medal in 1893. He was elected a member of theRoyal Society in 1885, and was awarded itsCopley Medal in 1912. He retired the following year due to ill health, but continued to teach mathematics at his home for several further years.
Klein was one of ninety-three signatories of theManifesto of the Ninety-Three, a document penned in support of the German invasion of Belgium in the early stages ofWorld War I.
Construction of a Klein Bottle from two Möbius strips
Klein's dissertation, online geometry and its applications tomechanics, classified second degree line complexes usingWeierstrass's theory of elementary divisors.
Klein's first important mathematical discoveries were made in 1870. In collaboration withSophus Lie, he discovered the fundamental properties of the asymptotic lines on theKummer surface. They later investigatedW-curves, curves invariant under a group ofprojective transformations. It was Lie who introduced Klein to the concept of group, which was to have a major role in his later work. Klein also learned about groups fromCamille Jordan.[15]
Klein devised the "Klein bottle" named after him, a one-sided closed surface which cannot be embedded in three-dimensionalEuclidean space, but it may be immersed as a cylinder looped back through itself to join with its other end from the "inside". It may be embedded in the Euclidean space of dimensions 4 and higher. The concept of a Klein Bottle was devised as a 3-DimensionalMöbius strip, with one method of construction being the attachment of the edges of twoMöbius strips.[16]
Non-Euclidean geometry models proposed by Klein (left) and Poincaré (right)
In 1871, while at Göttingen, Klein made major discoveries in geometry. He published two papersOn the So-called Non-Euclidean Geometry showing that Euclidean and non-Euclidean geometries could be consideredmetric spaces determined by aCayley–Klein metric. This insight had the corollary thatnon-Euclidean geometry was consistent if and only ifEuclidean geometry was, giving the same status to geometries Euclidean and non-Euclidean, and ending all controversy about non-Euclidean geometry.Arthur Cayley never accepted Klein's argument, believing it to be circular.
Klein's synthesis ofgeometry as the study of the properties of a space that is invariant under a givengroup of transformations, known as theErlangen program (1872), profoundly influenced the evolution of mathematics. This program was initiated by Klein's inaugural lecture as professor at Erlangen, although it was not the actual speech he gave on the occasion. The program proposed a unified system of geometry that has become the accepted modern method. Klein showed how the essential properties of a given geometry could be represented by the group oftransformations that preserve those properties. Thus the program's definition of geometry encompassed both Euclidean and non-Euclidean geometry.
Currently, the significance of Klein's contributions to geometry is evident. They have become so much part of mathematical thinking that it is difficult to appreciate their novelty when first presented, and understand the fact that they were not immediately accepted by all his contemporaries.
Klein considered equations of degree > 4, and was especially interested in using transcendental methods to solve the general equation of the fifth degree. Building on methods ofCharles Hermite andLeopold Kronecker, he produced similar results to those of Brioschi and later completely solved the problem by means of theicosahedral group. This work enabled him to write a series of papers onelliptic modular functions.
In his 1884 book on theicosahedron, Klein established a theory ofautomorphic functions, associating algebra and geometry.Poincaré had published an outline of his theory of automorphic functions in 1881, which resulted in a friendly rivalry between the two men. Both sought to state and prove a granduniformization theorem that would establish the new theory more completely. Klein succeeded in formulating such a theorem and in describing a strategy for proving it. He came up with his proof during anasthma attack at 2:30 A.M. on 23 March 1882.[19]
1884:Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom 5ten Grade
English translation by G. G. Morrice (1888)Lectures on the Ikosahedron; and the Solution of Equations of the Fifth Degree viaInternet Archive
1886:Über hyperelliptische Sigmafunktionen. Erster Aufsatz, pp. 323–356,Mathematische Annalen Bd. 27,
1887:"The arithmetizing of mathematics" in Ewald, William B., ed., 1996.From Kant to Hilbert: A Source Book in the Foundations of Mathematics, 2 vols. Oxford Uni. Press: 965–71.
Rowe, David "Felix Klein, David Hilbert, and the Göttingen Mathematical Tradition", in Science in Germany: The Intersection of Institutional and Intellectual Issues,Kathryn Olesko, ed., Osiris, 5 (1989), 186–213.
