Rudolf Friedrich Alfred Clebsch was a German mathematician. Together with Carl Neumann at Göttingen, he founded the mathematical research journal Mathematische Annalen in 1868.
Background
Clebsch was born on January 19, 1833, in Königsberg, Prussia (now Kaliningrad, Russian Federation), the son of Ernst Friedrich Leopold Clebsch and Pauline Ramberg.
Ernst Clebsch's father, the paternal grandfather of Alfred Clebsch, was Johannn Friedrich Leberecht Clebsch (1759-1847), a surgeon in the state of Colberg where he worked at the military hospital.
Education
In 1850 Clebsch entered the University of Königsberg, where the school of mathematics founded by Jacobi was then flourishing. His teachers included the mathematical physicist Franz Neumann and the mathematicians Friedrich Richelot and Ludwig Otto Hesse, both pupils of Jacobi.
After graduation, in 1854, Clebsch went to Berlin, where he was taught under the direction of Karl Schellbach at various schools. His academic career began in 1858, when he became Privatdozent at the University of Berlin. Soon afterward he moved to Karlsruhe, where he was a professor at the Technische Hochschule from 1858 to 1863. From 1863 to 1868 he was professor at the University of Giessen, collaborating with Paul Gordan. From 1868 until his death, he was professor at the University of Göttingen and in the forefront of contemporary German mathematics. In 1868 he and his friend Carl Neumann, son of Franz Neumann, founded the Mathematische Annalen.
Clebsch’s doctoral dissertation at the University of Königsberg concerned a problem of hydrodynamics, and the main problems considered in the first period of his scientific career were in mathematical physics, especially hydrodynamics and the theory of elasticity. His book on elasticity (1862) may be regarded as marking the end of this period. In it he treated and extended problems of elastic vibrations of rods and plates. His interests concerned more the mathematical than the experimental side of the physical problems. He soon moved on to pure mathematics, where he achieved a dominant place.
Although in his analytical papers Clebsch already proved himself to be highly skilled in calculus, his fame as a leader of contemporary scientists was first gained through his contributions to the theory of projective invariants and algebraic geometry. In the nineteenth century these fields were called modern geometry and modern algebra. The name “Modern algebra” was applied to the algebra of invariants, founded by the English mathematicians Cayley, Salmon, and Sylvester. One of the first German contributors to this discipline was Aronhold. It was especially the papers of Aronhold that incited Clebsch to his own researches in the theory of invariants, or “algebra of quantics,” as it was called by the English. The results in the theory of invariants are to be interpreted by geometric properties of algebraic curves, surfaces, and so on. This connection between algebra and geometry attracted Clebsch in a special way. He was soon a master of the difficult calculations with forms and determinants occurring in the theory of invariants. In this he surpassed his teacher Hesse, whose ability and elegance in analytical geometry were praised at the time.
Clebsch completed the symbolic calculus for forms and invariants created by Aronhold, and henceforth one spoke of the Clebsch-Aronhold symbolic notation. The general interest in the theory of invariants began to abate somewhat in 1890, when Hilbert succeeded in proving that the system of invariants for a given set of forms has a finite basis. In 1868 Gordan had already proved the precursory theorem on the finiteness of binary invariants. The theory of binary invariants thus being more complete, Clebsch published a book on this part of the theory (1872), giving a summary of the results obtained.
In the last year of his life Clebsch planned the publication of his lectures on geometry, perhaps to include those on n dimensions, results by no means as self-evident then as now. After Clebsch’s death his pupil Karl Lindemann published two volumes of these lectures (1876–1891), completed with his additions but confined to plane and three-dimensional geometry. Between 1906 and 1932 a second edition of volume I, part I of this work appeared under the name of both Clebsch and Lindemann. The first volume contained almost all of the known material on plane algebraic curves and on Abelian integrals and invariants connected with them.
As successors to Clebsch there arose the German school of algebraic geometry, led by Brill and M. Noether, both regarded as his pupils. At the end of the nineteenth century, algebraic geometry moved to Italy, where particular attention was paid to the difficult theory of algebraic surfaces. But the beginnings of the theory of algebraic surfaces go back to Cayley, Clebsch, and Noether, for Clebsch described the plane representations of various rational surfaces, especially that of the general cubic surface. Clebsch must also be credited with the first birational invariant of an algebraic surface, the geometric genus that he introduced as the maximal number of double integrals of the first kind existing on it.
Clebsch’s first researches in pure mathematics were suggested by Jacobi’s papers concerning problems in the theory of variation and of partial differential equations. He had not known Jacobi personally but collaborated in the edition of his Gesammelte Werke. For general problems in the calculus of variations, Clebsch calculated the second variation and promoted the integration theory of Pfaffian systems, surpassing results that Jacobi had obtained in these fields.
Connections
Clebsch's wife was Dorothe Charlote Mathilde Heinel (1838-1866), the daughter of the Priest Heinel in Marienburg. Alfred and Dorothe Clebsch had four sons.