**Carl Johannes Thomae** was a German mathematician who worked in function theory.

- When he was five years old Thomae entered the primary school in Laucha.
- His teacher at the gymnasium, Moritz Hülsen, saw what an outstanding pupil he had in Thomae and coached him with additional work beyond the usual course of study.
- Thomae's work in mathematics was quite outstanding and it won for him a scholarship which he was awarded from October 1860.
- Thomae entered the University of Halle in 1861, which was essentially his local university.
- There he attended lectures by Carl Neumann, who was a privatdozent at the time, on applied mathematics and also by Eduard Heine who was an ordinary professor.
- It was Heine who had the greatest influence on Thomae, giving him a love for function theory which was to set the course of his research for the whole of his career.
- In 1862 Thomae, following the usual tradition of German students of his time, moved to study at another university.
- His lectures were taken over by Schering and Thomae went on to undertake his doctoral studies supervised by Schering.
- After being awarded his doctorate in 1864, Thomae went to Berlin where he studied elliptic functions with Weierstrass for two terms.
- Thomae took part in the campaign in Bohemia, where the main Prussian armies met the main Austrian forces.
- Thomae took part in three battles in this short campaign, the most decisive being the Battle of Königgrätz on 3 July.
- After the Seven Weeks' War (as this short war is called) Thomae returned to Göttingen and gave a lectures on determinants and on the differential and integral calculus.
- In 1867 Thomae was appointed as a privatdozent at Halle where he became a colleague of Heine and Cantor.
- Thomae was promoted to extraordinary professor at Halle in 1872 and two years later he moved to the University of Freiburg where he was appointed as ordinary professor.
- After spending five years at Freiburg, Thomae moved to Jena in 1879 when he accepted the chair there.
- Both Thomae and Frege spent the rest of their careers at Freiburg.
- The relationship between Thomae and Frege is an interesting one.
- Thomae was Dean of the Philosophy Faculty in 1884, 1891, 1898, and for a fourth time in 1905.
- Thomae retired in 1914 but continued to publish papers up to 1919.
- The approach of both Riemann and Weierstrass strongly influenced Thomae's study of function theory.
- One of Thomae's first papers is Die allgemeine Transformation der Thetafunktionen mit beliebig vielen Variablen Ⓣ(The general transformation of theta functions with any number of variables) (1864).
- An important formula, which is still often used today, is Thomae's formula which expresses branch points of hyperelliptic curves in terms of hyperelliptic theta constants.
- In the second of the papers Thomae also showed that the roots of a polynomial can be expressed in terms of hyperelliptic theta functions.
- Thomae was the first to attempt to introduce "trans-Archimedean numbers" but Cantor argued that these were unworthy of the name of magnitude or quantity.
- He is also famed for the introduction of the Thomae θ-gamma function.
- Thomae was the first to attempt a general proof of the invariance of dimension but it was not satisfactory since the necessary topological tools had not been developed at this time.
- Thomae's proof, published in August 1878, was criticized at the time because of its unwarranted assumption of a decomposition property.
- Thomae's textbook Elementare Theorie der analytischen Funktionen einer komplexen Veränderlichen Ⓣ(Elementary theory of analytic functions of a complex variable) published in 1880 contained an introduction to arithmetic that fitted together and extended many earlier ideas.
- Frege, Thomae's colleague, strongly opposed these remarks.
- He held the opposite view to Thomae, and tried to set up arithmetic on a purely logical basis.
- In the second edition of his textbook published in 1890, Thomae tried to address the concern's of his colleague while still believing in his type of approach.
- Frege did not calm down, however, and became even more vehemently opposed to Thomae's point of view.
- Thomae's final four papers are Die Liebmannsche Formel für das Ponceletsche Dreieck Ⓣ(The Lie formula for the Poncelet triangle) (1918), Über die harmonischen Kovarianten zweier Kegelschnitte Ⓣ(On the harmonic covariates of two conics) (1918), Die harmonische Kovariante zweiter Art für zwei Kegelschnitte mit vier reellen Schnittpunkten Ⓣ(The harmonious covariate of the second kind of two conic sections with four real intersections) (1919), and Über die Cassinischen Kurven Ⓣ(On the Cassini curves) (1919).

Born 11 December 1840, Laucha (Unstrut), Germany. Died 1 April 1921, Jena, Germany.

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Analysis, Origin Germany

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