**Paul Gordan** worked with Clebsch on invariant theory and algebraic geometry. He also gave simplified proofs of the transcendence of $e$ and $\pi$.

- Paul was educated in Breslau where he attended the Gymnasium, going on to study at the business school.
- At this stage Gordan was not heading for an academic career and he worked for several years in banks.
- He had, however, attended lectures by Kummer on number theory at the University of Berlin in 1855 and his interest in mathematics was strongly encouraged by N H Schellbach who acted as a private tutor to Gordan.
- Moving to Königsberg, Gordan studied under Jacobi, then he moved to Berlin where he began to become interested in problems concerning algebraic equations.
- As soon as Gordan had completed his dissertation he went to visit Riemann at Göttingen.
- However, Riemann had caught a heavy cold which turned to tuberculosis so Gordan's visit was cut short.
- In 1863 Clebsch invited Gordan to come to Giessen.
- The first work which Gordan and Clebsch worked on in Giessen was the theory of abelian functions.
- The Clebsch-Gordan coefficients used in spherical harmonics were introduced by them as a result of this cooperation.
- The topic for which Gordan is most famous is invariant theory and Clebsch introduced him to this topic in 1868.
- For the rest of his career, although Gordan did not work exclusively on this topic, it would be fair to say that invariant theory dominated his mathematical research.
- For the next twenty years Gordan tried to prove the finite basis theorem conjecture for nnn-ary forms.
- Gordan did not undertake the bulk of this work at Giessen, however, for he moved to Erlangen in 1874 to become professor of mathematics at the university.
- When Gordan was appointed Klein held the chair of mathematics at Erlangen but he moved in the following year to the Technische Hochschule at Munich.
- In the year 1874-75 when Gordan and Klein were together at Erlangen they undertook a joint research project examining groups of substitutions of algebraic equations.
- Hilbert submitted his results to Mathematische Annalen and, since Gordan was the leading world expert on invariant theory, he was asked his opinion of the work.
- Gordan found Hilbert's revolutionary approach difficult to appreciate and not at all consistent with his ideas of constructive mathematics.
- Gordan was recognised as the leading world expert on invariant theory and he was also a close friend of Klein's.
- Gordan also worked on algebraic geometry and he gave simplified proofs of the transcendence of e and π.
- This was rather an unfortunate episode since it resulted in Gordan, who had enjoyed a fine reputation, losing respect from his colleagues.

Born 27 April 1837, Breslau, Prussia (now Wrocław, Poland). Died 21 December 1912, Erlangen, Germany.

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Algebra, Origin Poland

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive