VI.57 Christian Felix Klein
b. Düsseldorf, Germany, 1849; d. Göttingen, Germany, 1925
Higher geometry; function theory; theory of algebraic equations;
pedagogy
Klein had originally intended to be a physicist but during the course of his studies with Julius Plücker in Bonn, with whom he studied both mathematics and physics, he turned to mathematics, receiving his doctorate for a thesis on line geometry in 1868. After Plücker’s death in 1868 he went to Göttingen to study with Alfred Clebsch, where he worked exclusively on mathematics. In 1869-70 he spent some months in Berlin studying with WEIERSTRASS [VI.44] and KUMMER [VI.40] before joining LIE [VI.53] for a trip to Paris to see HERMITS [VI.47]. After passing his habilitation in Göttingen in 1871, he took positions successively at Erlangen, Munich, and Leipzig, returning to Göttingen in 1886, where he remained until he retired (because of poor health) in 1913. In 1875 he married Anna Hegel, a granddaughter of the philosopher Georg Wilhelm Friedrich Hegel.
In 1872 Klein published his celebrated “Erlanger Programm,” a creative and unified conception of geometry. Building on a paper of CAYLEY [VI.46] of 1859 in which Cayley had shown how to deduce EUCLIDEAN GEOMETRY [I.3 §6.2] from PROJECTIVE GEOMETRY [I.3 §6.7], Klein applied his knowledge of group theory (learned from JORDAN [VI.52] in Paris) to create a hierarchy of all geometries. He had recognized that each geometry could be characterized by a group of transformations and classified accordingly (see SOME FUNDAMENTAL MATHEMATICAL DEFINITIONS [I.3 §6.1]). The classification showed, as Klein had anticipated, that of all the geometries, projective is the most basic and that the others, e.g., affine, hyperbolic, Euclidean, etc., are subsumed at some level beneath it. Moreover, it was clear from his construction that a contradiction in NON-EUCLIDEAN GEOMETRY [II.2 §§6-10] would simultaneously involve a contradiction in Euclidean geometry.
Klein regarded his work in function theory as his greatest achievement. As his career progressed, he moved more and more from Plücker’s and Clebsch’s strictly geometric viewpoint toward the wider outlook embraced by RIEMANN [VI.49], who had regarded analytic functions as given by conformal mappings between given domains. In his “Riemanns Theorie der algebraischen Funktionen und ihrer Integrale” (1882), Klein gave a geometric treatment of function theory in which he fused Riemann’s ideas with the rigorous power-series methods of Weierstrass.
In 1882, when he was at the height of his powers, Klein’s health broke down. His attempt to keep up with POINCARÉ [VI.61] in the race to develop the theory of automorphic functions (which are generalizations of periodic functions such as trigonometric functions, ELLIPTIC FUNCTIONS [V.31], etc.), during which he had proved his famous Grenzkreis (boundary circle) Theorem, had left him exhausted, and he was never again able to work with such intensity and at such a high level.
After his breakdown Klein’s interest shifted progressively from research toward pedagogy. In his efforts to modernize mathematical education he developed outstanding organizational skills and initiated important and far-reaching editorial projects ranging from the preparation of lecture notes to coediting the twenty-four-volume Encyklopädie der mathematischen Wissenschaften (1896-1935). He was an editor of the Mathematische Annalen for almost fifty years, and was among the founding members of the Deutsche Mathematiker-Vereinigung (1890). He also played an active role in establishing mathematical applications in science and engineering, as well as promoting the better understanding of mathematics by engineers.
Among Klein’s other achievements were important results in the theory of algebraic equations (through a consideration of the icosahedron he obtained a complete theory of the general fifth-degree equation (1884)) and in mechanics, in which, jointly with Arnold Sommerfeld, he developed the theory of the gyroscope (1897-1910). He also worked on ideas involving the application of group theory to the theory of relativity, producing papers on the LORENTZ GROUP [IV.13 § 1] (1910) and gravitation (1918). Klein was an international figure who traveled widely, including to the United States and the United Kingdom, and he played a significant role in the first International Congresses of Mathematicians. His many foreign students included several from the United States, e.g., Maxime Bôcher and William Fogg Osgood, and a number of women, notably Grace Chisholm Young and Mary Winston.
Klein’s achievements made Göttingen the scientific center of Germany and one of the mathematical centers of the world. He possessed an outstanding ability to “see” the truth in mathematical statements and to bring mathematical fields together without feeling the necessity for detailed calculations and justification (which he left to his students and others). He believed strongly in the unity of mathematics.
Further Reading
Frei, G.1984. Felix Klein (1849-1925), a biographical sketch. In Jahrbuch Überblicke Mathematik, pp. 229-54. Mannheim: Bibliographisches Institut.
Klein, F. 1921-23. Gesammelte mathematische Abhandlungen, three volumes. Berlin: Springer. (Reprinted, 1973. Volume 3 contains lists of Klein’s publications, lectures, and dissertations directed by him.)
———. 1979. Development of Mathematics in the 19th Century, translated by M. Ackerman. Brookline, MA: MathSci-Press.