Felix Klein

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Felix Klein
Grave site in Göttingen

Felix Christian Klein (born April 25, 1849 in Düsseldorf , † June 22, 1925 in Göttingen ) was a German mathematician .

Felix Klein achieved significant results in geometry in the 19th century . In addition, he has made an outstanding contribution to the application of mathematics and teaching. Klein, who was also an important science organizer, played a key role in making Göttingen a center of mathematics.

Life, studies and professional career

Felix Klein was born on April 25, 1849 in Düsseldorf. Klein's father, Caspar Klein (1809–1889), was of old Prussian Protestant character and came from Ennepetal in southern Westphalia . He was land rent master of the government's main treasury in Düsseldorf , while Klein's mother came from circles of the Aachen industry. After first lessons from his mother Sophie Elise, née Kayser (1819–1890), Felix Klein, with previous knowledge of reading, writing and arithmetic, entered a private elementary school in Düsseldorf at the age of six , and then in autumn 1857 the Catholic humanistic school Transfer to Royal High School Düsseldorf . Despite this purely philological upbringing, his early interest in natural science found inspiration in the pharmacy of the father of his friend and classmate Wilhelm Ruer, as well as in the small observatory in the city of Düsseldorf with Karl Theodor Robert Luther , director of the small planets, who explored the small planets . In addition, his father gave him a few factory tours.

In the autumn of 1865, Felix Klein began studying mathematics and natural sciences at the University of Bonn . Klein studied with Rudolf Lipschitz and Julius Plücker , whose assistant he became. After Plücker's death, Alfred Clebsch took over the editing of his unfinished work and transferred this work to the gifted Klein. Klein received his doctorate from Plücker in 1868 with a topic from geometry applied to mechanics .

In 1869 he went to the Humboldt University in Berlin , where he heard a lecture by Leopold Kronecker on square shapes. He took part in the mathematical seminars of Ernst Eduard Kummer and Karl Weierstrass , where he also met Sophus Lie , with whom he went to Paris for a study visit in 1870 and was friends. Due to the Franco-German War , he returned to Germany. He completed his habilitation with Clebsch in Göttingen in 1871 and stayed in Göttingen as a private lecturer in 1871/72. At the instigation of Clebsch, he was offered a professorship in Erlangen in 1872 . His further professional path led him to the Technical University of Munich in 1875 .

Also in 1875 he married Anna Hegel, the daughter of the historian Karl Hegel and granddaughter of the philosopher Georg Wilhelm Friedrich Hegel .

In 1880 he became a corresponding member of the Bavarian Academy of Sciences . In the same year, Klein followed the call to Leipzig as professor of geometry. During this time in Leipzig fell his most fruitful scientific creative period. He corresponded with Henri Poincaré and at the same time devoted himself intensively to the organization of teaching. This double exposure ultimately led to a physical breakdown in 1882, followed by depression in 1883–1884.

In 1886 he accepted a call to Göttingen, where he stayed until his death. His daughter Elisabeth was born here on May 27, 1888, and it was here that he mainly devoted himself to scientific organizational tasks, while David Hilbert, who was called to Göttingen for his work in 1895, further expanded its reputation as one of the world centers of mathematics at that time. Also in 1886 he was elected a member of the Leopoldina .

In 1897 he became a corresponding member of the Académie des Sciences in Paris; In 1915 membership was canceled. In 1898 he was awarded the Bavarian Maximilian Order for Science and Art and elected to the National Academy of Sciences , in 1904 to the American Academy of Arts and Sciences . From 1908 he represented the University of Göttingen in the Prussian manor house . In 1912 he was awarded the Copley Medal . From 1913 he was a corresponding member of the Prussian Academy of Sciences . In 1914 he received the Ackermann-Teubner Memorial Prize . In 1924 Klein became an honorary member of the German Mathematicians Association , of which he was president in 1897, 1903 and 1908.

Klein died in Göttingen on June 22, 1925. He found his final resting place there in the city ​​cemetery . His estate is kept by the Central Archives of German Mathematicians' bequests at the Lower Saxony State and University Library in Göttingen .

Scientific achievements

Geometry and Erlangen program

When Klein was appointed to Erlangen in the winter semester of 1872, he was already one of the most important representatives of 19th century geometry. B. worked on projective geometry, Plücker's line geometry and non-Euclidean geometry. His scientific program writing became known as the so-called Erlanger program . It is based on the considerations of Klein and Lie and represents a systematization of the different geometries known at the time. With this, the Euclidean and the non-Euclidean geometries were placed in a common context with the help of projective geometry . Groups of transformations of the plane or of the space considered small. He assigned a geometry to each group of transformations, under which certain geometric properties (such as orthogonality, parallelism) remain invariant. In this way he created an organizing system for the geometries known up to then.

Theory of the icosahedron and equations of the fifth degree

Klein recognized a connection between algebraic equations and the invariant theory of linear substitutions. The regular polyhedra play a special role in these relationships . Klein has particularly studied the icosahedron . He realized that the icosahedral equation is a Galois equation and that its Galois group is isomorphic with the group of icosahedral rotations. In his book on the icosahedron, Klein shows the interplay between function theory, the theory of algebraic equations and group theory. A short outline can be found in his "Elementary Mathematics from a Higher Viewpoint".

Function theory

Klein dealt with elliptic functions and hyperelliptic functions (later referred to as Abelian functions), and also with Riemann's function theory and the theory of automorphic functions. He came to an essential result in the theory of uniformity, in a friendly competition with Henri Poincaré, although he suffered a physical and mental breakdown due to his overload and Poincaré had to "win". In Klein's own opinion, his career as an active researcher was then over. The connection with Poincaré began with the fact that Klein was annoyed about the naming of some of "his" groups by Poincaré, who was not very knowledgeable in literature, after Lazarus Fuchs (which did not get any better when Poincaré named other groups after Klein to compensate for this, because this was the case in Klein's eyes was again unjustified), but then ended in an intensive correspondence. Strangely enough, it was Poincaré and not Klein who discovered non-Euclidean geometry in the work of the module group in the complex upper half-plane.

The Klein model of the non-Euclidean (hyperbolic) plane consists of the inner points of the unit circle E as points and the chords (without their end points) of E as straight lines.

The points of Poincaré's model are the points of the open upper half-plane (in the real number plane), and the straight lines are the circles that intersect the real axis perpendicularly (as far as they are in the upper half-plane), including the "improper circles" (i.e. straight lines) belong.


Klein was also very interested in the applications of mathematics, which have a large space in the encyclopedia. He dealt with trusses and other applications of geometry in mechanics and worked with his student Arnold Sommerfeld on the theory of the gyroscope , about which they wrote an extensive, four-volume standard work.

At the age of almost seventy, Klein was still familiarizing himself with Albert Einstein's general theory of relativity . Much of the second part of his "Lectures on the Development of Mathematics in the 19th Century" testifies to this interest. In addition, there and in the “ Encyclopedia of Mathematical Sciences ”, the penetration of physics with vector and tensor calculus that began at the time of publication of the encyclopedia becomes clear. He was particularly fascinated by the discovery of his Göttingen colleague Hermann Minkowski that behind the special theory of relativity was nothing other than non-Euclidean geometry, one of Klein's favorite subjects. He was also enthusiastic about the emerging application of group theory in physics, especially through a sentence by Emmy Noether about the connection between symmetries and conservation laws, the Noether theorem . Klein also published on this topic.

Göttingen - Center of Mathematics

Klein's appointment to Göttingen at the Georg August University was carried out by the Ministerial Director Friedrich Althoff of the Prussian Ministry of Culture. In the years that followed, Althoff and Klein systematically expanded Göttingen University into the world's most important mathematical center, which was to remain so until the Nazis expelled many German mathematicians. In addition to important mathematicians such as David Hilbert , Richard Courant , Hermann Minkowski , Hermann Weyl , Emmy Noether and others, important physicists such as Walther Nernst , Max Born , James Franck and Peter Debye were later brought to the university. The mathematics and physics faculties in Göttingen thus became the model for many international research institutions.

In 1893 Klein traveled to the United States of America for the first time (to the Evanston Colloquium at Northwestern University ), further trips followed and ensured him a great influence with American mathematicians, many of whom came to Göttingen to study. In the USA he received suggestions for a stronger orientation of mathematics at the university towards application. Klein implemented in Göttingen what he had seen in the USA. He sought contact with engineers in the Association of German Engineers and found partners in Carl Linde and Henry Böttinger who brokered contacts with industry. This made it possible to set up a department for technical physics with the help of industrial funding. In 1898 the Göttingen Association for the Promotion of Applied Physics was founded, the first organization in Germany to combine industry and the university. As a result of further activities, further institutes were founded in Göttingen in the following years, which were dedicated to the application of physics or mathematics. Other important scientists came to Göttingen as a result, B. the hydrodynamicist Ludwig Prandtl and the applied mathematician Carl Runge .

For an understanding of Klein's role in German mathematics at the end of the 19th century, the contrast between the schools of Berlin mathematicians (especially Karl Weierstrass , Leopold Kronecker , Ernst Eduard Kummer ), with their emphasis on mathematical rigor, and the school of Felix Klein (and his teachers , Alfred Clebsch and Julius Plücker), who preferred geometrical and physical investigations, were of importance. This went as far as open hostilities, for example in the judgments of Weierstrass (Klein would rather snack and would be a blender ) and Lazarus Fuchs (who saw Klein's icosahedron book as a compilation in the features style of his own work and that of Schwarz ) when Felix Klein was shortlisted for the successor to Weierstrass (the Berliners put Hermann Amandus Schwarz through). The contrast between Berlin and Göttingen mathematicians as the dominant schools in Germany lasted until the first half of the 20th century.


Felix Klein was also very committed to mathematics didactics . Already in his time in Erlangen he began to deal with teaching and within the framework of his professorships he was always concerned with the organization of teaching and didactics. He not only studied the history of mathematics teaching, but also found out about current international developments. As early as 1894, Klein joined the association for the promotion of mathematics and science teaching . It was not until 1900 that an imperial decree finally made mathematical and natural science subjects formally equal to humanistic subjects in German schools, not least through Klein's work. Klein continued to commit himself to the implementation and developed reform proposals that became known as Klein's reform and were officially included in the Merano proposals of 1905. He called for the strengthening of spatial perception, an education in the habit of functional thinking and the introduction of the calculus as compulsory teaching subject (this last point was not implemented until 1925).

In 1908 the German Committee for Mathematics and Science Education (DAMNU) was founded. Klein took over the chairmanship of the committee for teacher training. In the same year, the International Mathematical Teaching Commission (IMUK) was founded, which Klein also took over as chairman and held it until 1916. His three-volume “Elementary Mathematics from the Higher Point of View”, which is aimed at teachers and in which many remarks on mathematics education can also be found in international comparison, testify to his educational efforts.

In 1897 he gave a plenary lecture at the first International Congress of Mathematicians in Zurich (on the question of higher mathematics teaching).

It was Althoff who commissioned Klein to promote women's studies in Göttingen. At the time, women were only able to study with great difficulty, if at all. Sofja Kowalewskaja , who studied with Karl Weierstrass in Berlin, could not do her doctorate in Berlin; so she came to the Göttingen University. The first woman to do her doctorate with Klein in Göttingen was Grace Chisholm .


Klein's way of working was such that he carried out his ideas in lectures and students selected by him then worked them out. This resulted in a series of books and lecture notes that were widely distributed in Leipzig and Göttingen. Examples are his "Lectures on Non-Euclidean Geometry" (1928), which appeared in the basic teaching series published by Springer Verlag. He has also written with Robert Fricke and the extensive monographs "Lectures on the theory of elliptic modular functions" band first 2 ( BG Teubner 1890, 1892) and "Lectures on the theory of automorphic functions" (BG Teubner 1902, two volumes) and an extensive multi-volume work on the theory of the gyro with Arnold Sommerfeld . His three-volume “Elementary Mathematics from a Higher Viewpoint” and his “Lectures on the Icosahedron”, in which he combines Galois equation theory with function theory and geometric applications of group theory, have been reissued even more recently. A collection of his lectures is in the Mathematical Institute of the University of Göttingen.

Klein not only authored numerous writings and textbooks, but was also active as an editor: the journal Mathematische Annalen , founded by Alfred Clebsch and Carl Gottfried Neumann in 1868, was published by him from 1876 onwards.

Over the years he worked closely with the publishing house BG Teubner in Leipzig. Another major work was the publication (with Franz Meyer ) of the Encyclopedia of Mathematical Sciences, including its applications . Thanks to his extensive contacts, Klein was able to engage the best authors. The publication of his complete works as well as two volumes of lectures on the development of mathematics in the 19th century , Julius Springer Verlag, which appeared in 1926 and 1927 from the estate (he gave the lectures during the First World War) should be named as late works . He was also one of the initiators and editors of the works of Carl Friedrich Gauß .

The most important works are:

  • Comparative considerations on recent geometrical research. Deichert, Erlangen 1872 ( digitized and full text in the German text archive )
  • About Riemann's theory of algebraic functions and their integrals , Teubner, Leipzig 1882 ( digitized and full text in the German text archive )
  • Lectures on the icosahedron and the solution of the equations of the fifth degree , BG Teubner, Leipzig 1884
  • Non-Euclidean geometry (2 parts), BG Teubner, Leipzig 1890
  • with Robert Fricke: Lectures on the theory of elliptical module functions (2 volumes), BG Teubner, Leipzig 1890 and 1892
  • The Evanston Colloquium. Lectures on Mathematics delivered from August 28 to September 9, 1893 before members of the congress of mathematics held in connection with the world's fair in Chicago, New York: Macmillan 1894
  • Lectures on selected questions of elementary geometry, Leipzig: Teubner 1895 (English translation: Famous problems of elementary geometry, Ginn and Company, 1897)
  • With Arnold Sommerfeld: About the theory of the gyro (4 issues), BG Teubner, Leipzig 1897–1910
  • The mathematical theory of the top , Scribners 1897 (Princeton Lectures)
  • with Robert Fricke: Lectures on the theory of automorphic functions (2 volumes in 4 deliveries), BG Teubner, Leipzig 1897, 1901, 1911, 1912
  • Elementary mathematics from a higher point of view (3 volumes), BG Teubner, Leipzig 1908, 1909, Springer Berlin 1928
  • Collected mathematical treatises (3 volumes), Julius Springer Verlag, Berlin 1921, 1922 and 1923
  • Lectures on the development of mathematics in the 19th century (2 volumes), Julius Springer Verlag, Berlin 1926 and 1927
  • Lectures on higher geometry , Springer Verlag, Grundlehren der Mathematischen Wissenschaften 1926
  • Lectures on non-Euclidean geometry , basic mathematical science, Springer Verlag 1928
  • Lectures on the hypergeometric function , Springer 1933 (basic teachings of the mathematical sciences)
Collections of articles
  • Collected mathematical treatises. Edited by Robert Fricke , A. Ostrowski , Hermann Vermeil, Erich Bessel-Hagen . Volume 1-3. Berlin: Springer; Reprint of the Berlin 1922 edition (Springer Collection Works in Mathematics.)
1. Line geometry - basic geometry for the Erlangen program . 1922.
2. Descriptive geometry - substitution groups and equation theory - on mathematical physics .
3. Elliptic functions, especially module functions - hyperelliptic and Abelian functions - Riemannian function theory and automorphic functions .


In honor of Klein, the Felix Klein Prize is awarded by the European Mathematical Society and the Fraunhofer ITWM (Fraunhofer Institute for Industrial Mathematics), Kaiserslautern, and the Felix Klein Medal (for lifetime achievement in mathematical education) by the International Commission for Mathematical Instruction. Felix Klein is also the namesake for the Felix Klein Center for Mathematics , an institutional alliance between Fraunhofer ITWM and the Department of Mathematics at the Technical University of Kaiserslautern , for the Felix Klein Gymnasium in Göttingen, and for the Felix Klein lecture hall of the University of Leipzig .

Klein's daughter Sophie was the wife of the lawyer Eberhard Hagemann , who was President of the Province of Hanover and President of the District Court in Verden.


  • Paul Kirchberger: Memories of Felix Klein. In: Vossische Zeitung . June 27, 1925, evening edition, p. 2.
  • Günther Frei : Felix Klein (1849–1925): A biographical sketch. In: Yearbook overviews of mathematics. 1984, pp. 229-254, ISSN  0172-8512 .
  • Isaak Moissejewitsch Jaglom : Felix Klein and Sophus Lie - the evolution of the idea of ​​symmetry in the 19th century. Birkhäuser, 1985, 1988, ISBN 3-7643-3316-2 .
  • Fritz König: Felix Klein. In: Herbert Beckert , Horst Schumann (Ed.) 100 Years of Mathematical Seminar at the Karl Marx University in Leipzig. German Science Publishers, Berlin 1981.
  • Reinhold Remmert : Felix Klein and the Riemann legacy. In: Communications of the German Mathematicians Association, No. 1, 2001, p. 22 f., ISSN  0942-5977 .
  • David E. Rowe : The correspondence between Sophus Lie and Felix Klein, an insight into their personal and scientific relationships. In: NTM. Journal of the History of Science, Technology and Medicine. 25, 1988, pp. 37-47, ISSN  0036-6978 .
  • David E. Rowe: Felix Klein, David Hilbert, and the Göttingen Mathematical Tradition. In: Kathryn M. Olesko (Ed.): Science in Germany. The intersection of institutional and intellectual issues. Department of History and Sociology of Science, University of Pennsylvania, Philadelphia PA 1989, pp. 186-213, ISBN 0-934235-12-0 ( Osiris. Ser. 2, Vol. 5).
  • David E. Rowe: Felix Klein as science politician. In: Umberto Bottazzini, Amy Dahan (Eds.): Changing Images in Mathematics: From the French Revolution to the New Millennium. London 2001, pp. 69-92.
  • David E. Rowe: Klein, Lie, and the Geometric Background of the Erlangen Program. In: David E. Rowe et al. (Ed.): The History of Modern Mathematics. Proceedings of the Symposium on the History of Modern Mathematics, Vassar College, Poughkeepsie, New York, 20. – 24. June 1989. Volume 1: Ideas and their Reception. Academic Press, Boston MA et al. 1989, pp. 209-273, ISBN 0-12-599661-6 .
  • David E. Rowe: Klein, Mittag-Leffler, and the Klein-Poincaré Correspondence of 1881–1882. In: Sergei S. Demidov (ed.): Amphora. Festschrift for Hans Wussing on his 65th birthday (= Festschrift for Hans Wussing on the Occasion of his 65th Birthday. ) Birkhäuser, Basel et al. 1992, pp. 598–618, ISBN 3-7643-2815-0 .
  • Nikolai Stuloff:  Klein, Felix. In: New German Biography (NDB). Volume 11, Duncker & Humblot, Berlin 1977, ISBN 3-428-00192-3 , p. 736 f. ( Digitized version ).
  • Rüdiger Thiele : Felix Klein in Leipzig 1880–1886. In: Annual report of the German Mathematicians Association, Vol. 102, Issue 2, 2000, pp. 69–93, ISSN  0012-0456 .
  • Rüdiger Thiele: Felix Klein in Leipzig. With F. Klein's inaugural speech, Leipzig 1880. Edition at Gutenbergplatz Leipzig, Leipzig 2011, ISBN 978-3-937219-47-9 (EAGLE 047, online ).
  • Renate Tobies : Felix Klein. Teubner, Leipzig 1981 ( biographies of outstanding natural scientists, technicians and medical professionals . 50, ISSN  0232-3516 ).
  • Renate Tobies: Felix Klein. Visions for mathematics, applications and teaching , Springer 2019
  • Renate Tobies, David E. Rowe (eds.): Correspondence Felix Klein - Adolph Mayer. Selection from the years 1871–1907. Teubner, Leipzig 1990, ISBN 3-211-95847-9 ( Teubner Archive for Mathematics. 14).
  • Renate Tobies: Felix Klein in Erlangen and Munich. In: Sergei S. Demidov (ed.): Amphora. Festschrift for Hans Wussing on his 65th birthday (= Festschrift for Hans Wussing on the Occasion of his 65th Birthday. ) Birkhäuser, Basel et al. 1992, pp. 751–772, ISBN 3-7643-2815-0 .
  • Renate Tobies: Mathematics as a program. For Felix Klein's 150th birthday. In: Communications of the German Mathematicians Association, Issue 2, 1999, pp. 15-21, ISSN  0942-5977 .
  • Felix Klein: About the relationship between modern mathematics and applications. Leipzig inaugural lecture 1880. In: Herbert Beckert, Walter Purkert : Leipziger mathematische inaugural lectures. Selection from the years 1869–1922. Teubner, Leipzig 1987 (with biography).
  • Jürgen Weiß: Successful old 68ers. Mathematical annals - messages BG Teubner - Alfred Clebsch - Felix Klein - Carl Neumann. Preface: Jürgen Jost, Leipzig. EAGLE 101st Edition at Gutenbergplatz, Leipzig 2018, ISBN 978-3-95922-101-6 .

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See also

Web links

Commons : Felix Klein  - Collection of images, videos and audio files
Wikisource: Felix Klein  - Sources and full texts

Individual evidence

  1. Felix Klein: autobiography from the communications of the Universitätsbund Göttingen, 5th year, issue 1, 1923, reprint in the report of the Felix Klein Oberschule in Göttingen for the year 1952/53, pp. 32-48.
  2. ^ C. Felix (Christian) Klein in the Mathematics Genealogy Project (English)Template: MathGenealogyProject / Maintenance / id used Template: MathGenealogyProject / Maintenance / name used
  3. ^ Member entry by Felix Klein (with picture) at the Bavarian Academy of Sciences , accessed on June 9, 2016.
  4. ^ Member entry by Felix Klein at the German Academy of Sciences Leopoldina , accessed on June 9, 2016.
  5. ^ List of members since 1666: Letter K. Académie des sciences, accessed on January 5, 2020 (French).
  6. Hans Körner: The Bavarian Maximilian Order for Science and Art and its members. In: Journal for Bavarian State History. Vol. 47, 1984, p. 382 BSB ZBLG
  7. Felix Klein (with picture). Members of the predecessor academies. Berlin-Brandenburg Academy of Sciences , accessed on June 9, 2016 .
  8. Klein: History of Mathematics in the 19th Century , with a chapter on Poincaré.
  9. ^ Karl-Eugen Kurrer : The History of the Theory of Structures. Searching for Equilibrium . Berlin: Ernst & Sohn 2018, p. 515f., P. 556, p. 792, p. 814, p. 847, p. 851 and p. 890ff., ISBN 978-3-433-03229-9 .
  10. For example: Renate Tobies, Mathematics as a program. On the 150th birthday of Felix Klein , Mitt.DMV, 1999, issue 2, p. 15f