**Carl Neumann**was the son of Franz Neumann who has a biography in this archive. His mother, Luise Florentine Hagen (born 1800), was Wilhelm Bessel's sister-in-law. Carl's siblings were Franz Ernst Christian Neumann (1834-1918), Friedrich Julius Neumann (1835-1910), and Luise Neumann (1837-1934). Let us note at this point that Franz Ernst Christian Neumann became a pathologist and director of the Pathological Institute at Königsberg, and Friedrich Julius Neumann became an economist, holding professorships in Königsberg, Basel, Freiburg and Tübingen. Luise Neumann became a painter creating landscapes and portraits, as well as interiors of cultural and historical buildings in Königsberg. She also painted porcelain. Luise Neumann wrote a biography of her father,

*Franz Neumann: Erinnerungsblätter*Ⓣ (Tübingen-Leipzig, 1904), which we reference below [3].

Carl received his school education, both primary and secondary, in the city of Königsberg where his father was the Professor of Physics at the University. His secondary education was at the Altstadt Gymnasium which, when he began his studies, was situated in the street Danziger Keller near Königsberg Castle but, after a period in temporary accommodation, the school moved to a new building in Altstädtischer Kirchplatz in which Carl completed his schooling. His two younger brothers, Friedrich Julius Neumann and Franz Ernst Christian, also attended this school.

Neumann entered the University of Königsberg where he became close friends with two of his teachers, Otto Hesse and Friedrich Julius Richelot (1808-1875). Richelot had been a student of Carl Jacobi and taught at the University of Königsberg from 1831, becoming a full professor in 1844. He published on mathematical analysis but his real strength was as a teacher. After graduating with a qualification to teach mathematics in secondary schools, Neumann continued to study at Königsberg for his doctorate which was awarded in 1855 for his thesis *De problemate quodam mechanico, quod ad primum integralium ultraellipticorum classem revocatur* Ⓣ. In this work he studied applications of hyperelliptic integrals to specific problems in mechanics. He had been advised during his doctoral studies by Richelot.

After receiving his doctorate, Neumann studied for his habilitation advised by Eduard Heine who had been appointed as an ordinary professor at Halle in 1856. Neumann submitted his habilitation thesis *Explicare tentatur, quomodo fiat, ut lucis planem polarisationis per vires electricas vel magneticas declinetur* Ⓣ to the University of Halle in 1858. In this work [18]:-

In that year he received the right to lecture when he became a Privatdozent at Halle. He was promoted to extraordinary professor there in 1863.... he treated mathematically the rotation of the plane of polarization of light by magnetism(the Faraday effect).

Neumann did not remain at Halle for long after his promotion for he was offered a professorship at the University of Basel. Arriving in Basel in 1863 he only spent two years at the university there before being offered a professorship at the University of Tübingen. However, during these two years in Basel he married Hermine Mathilde Elise Klose on 7 April 1864 in St Matthew's Church in Berlin. Mathilde, who was born in 1834 to the registrar Heinrich Theodor Klose, died aged 40 in 1875. They had no children. Neumann's inaugural lecture at Tübingen was *Der gegenwdrtige Standpunct der Mathematischen Physik* Ⓣ and in this he discussed applications of potential theory to electrodynamics, optics and the theory of heat. After a slightly longer time, namely three years, spent in Tübingen, from 1865 to 1868, Neumann was on the move again, this time to a chair at the University of Leipzig. Appointed to Leipzig in the autumn of 1868 he gave his inaugural lecture *Über die Principien der Galilei-Newton'schen Theorie* Ⓣ in 1869. In this lecture he discussed [10]:-

The German text of this lecture is given in [2].... what sort of intelligibility and comprehensibility should be demanded of physical laws, and what sort of comprehension of physical phenomena physics can really offer.

After his appointment to Leipzig he found notes of Jacobi's lectures on analytical mechanics which had been delivered in Berlin in the winter semester of 1847-1848. The notes had been taken by Wilhelm Scheibner who was now Neumann's colleague in Leipzig. Neumann [18]:-

Neumann held the chair at Leipzig until he retired in 1911 but sadly, as we indicated above, his wife died in 1875. Ernst Richard Julius Neumann (1875-1955) was the son of Neumann's brother, the pathologist Franz Ernst Christian Neumann. Ernst Neumann became a mathematician and studied at Königsberg, Heidelberg and then at the University of Leipzig from 1893 to 1898. In Leipzig he was a student of his Carl Neumann, the subject of this biography. Ernst Neumann went on to become a professor at Breslau and Marburg. He continued to undertake research along the same lines as his uncle on potential theory.... characterized Jacobi's lectures as constituting an exceptionally penetrating critique of the foundations of mechanics. Now, like Jacobi, Neumann took his goal to be to start from some unanalyzable basic assumptions or principles and to deduce general theorems from them, in this way constructing a physical theory of the field under consideration. His goal thus was to establish a physical theory by employing an axiomatic approach like that found in geometry. He stressed, however, what he considered to be a fundamental difference between mathematical axioms and physical principles: He claimed that the latter, which form the basis of a physical theory, can never be described as true or probably true; they always embody some uncertainty or arbitrariness.

Wussing writes in [1]:-

He worked on a wide range of topics in applied mathematics such as mathematical physics, potential theory and electrodynamics. Of course he was not the only one working on theories of electrodynamics and a discussion (in some cases an argument) arose about which was "correct." Assumptions had to be made about various physical properties, such as how electromagnetic forces propagate, in constructing a mathematical model. Experimental verification was an important factor in coming to a decision but other factors also played a role. Neumann made assumptions some of which followed those made by Wilhelm Weber. This is not surprising since Neumann's father, Franz Neumann, had adopted the same assumptions. As early as 1847 Hermann von Helmholtz had claimed that these assumptions violated the law of conservation of energy. Helmholtz published his own theory of electrodynamics inNeumann, who led a quiet life, was a successful university teacher and a productive researcher. More than two generations of future Gymnasium teachers received their basic mathematical education from him.

*Über die Bewegungsgleichungen der Elektricität für ruhende leitende Körper*Ⓣ (1870). This led to a debate between various scientists, in particular between Helmholtz and Neumann. Karl-Heinz Schlote writes in [18]:-

Neumann also made important pure mathematical contributions. He studied the order of connectivity of Riemann surfaces inThe Helmholtz-Neumann debate might seem somewhat surprising, since both used potential theory, and Helmholtz's approach corresponded closely to Neumann's methodological views on the structure of a physical theory. Thus, Helmholtz assumed the validity of conservation of energy and derived a general formula for the electrodynamic potential that covered "the entire experimentally known area of electrodynamics ... with one and the same relatively simple mathematical expression." He noted with satisfaction that Neumann had required many hypotheses to reach a similar result. Neumann stuck to his guns, however, although he acknowledged that conservation of energy was a basic principle of physics, and he admitted that Helmholtz's theory appeared to have a simpler structure. One important reason, in general, that Neumann opposed Helmholtz's approach was that Helmholtz's states of energy and their interaction made the explanation of many electrodynamic phenomena more complicated than explanations based upon Weber's or Maxwell's theories.

*Vorlesungen uber Riemann's Theorie der Abel'schen Integrale*Ⓣ (1st edition 1865, second edition 1884).

During the 1860s Neumann wrote papers on the Dirichlet problem and the 'logarithmic potential', a term he coined. He used the logarithmic potential to solve the Dirichlet problem in the plane in 1861. This problem requires, given the values of a function on the boundary of a region in the plane, finding a function *f* which satisfies Δ*f* = 0 in the region and takes the given values on the boundary. He solved the problem in more generality in *Zur Theorie des Logarithmischen und des Newtonschen Potentials. Erste Mittheilung* Ⓣ (1870) and continued to return to produce further methods and results in this area. In 1890 Émile Picard used Neumann's results to develop his method of successive approximation which he used to give existence proofs for the solutions of partial differential equation. This is discussed in detail in [6].

There are a great many mathematical terms named after 'Neumann' but it is such a common name that often it is difficult to determine which are named after Carl Gottfried Neumann. For example, MathSciNet lists 7357 papers with 'Neumann' in the title. Several terms are easy to establish as being named after Carl Neumann, however, such as the Neumann-Poincaré Operator, the Neumann boundary value problem, the Neumann boundary condition, the Neumann series, and the Neumann problem.

In addition to his research and teaching, Neumann made another important contribution to mathematics as a founder and editor of *Mathematische Annalen*. Together with Alfred Clebsch he founded the journal in 1868. The first volume, published in 1869, contains papers by leading mathematicians such as Heinrich Weber, Jacob Lüroth, Arthur Cayley, Alfred Clebsch, Paul Gordan, Camille Jordan, Leo Königsberger, Alexander von Brill, Hieronymus Georg Zeuthen, Theodor Reye, Hermann Hankel, and Eugenio Beltrami. The volume, in four parts, contained five papers by Neumann, namely: *Geometrische Untersuchung über die Bewegung eines starren Körpers* Ⓣ; *Zur Theorie der Functionaldeterminanten* Ⓣ; *Notizen zu einer kürzlich erschienenen Schrift über die Principien der Elektrodynamik* Ⓣ; *Über die Aetherbewegung in Krystallen* Ⓣ; and *Notiz über das cykloidische Pendel* Ⓣ. His contribution to the Annalen is given in [8]:-

He was honoured with membership of several academies and societies, including the Berlin Academy and the societies in Göttingen, Munich and Leipzig. These underwent various changes in name during his time as a member which we now indicate. He was elected a corresponding member of the Königlichen Societät der Wissenschaften of Göttingen in 1864 and, in 1868 became a foreign member of that Society. In 1893 this became the Königlichen Gesellschaft der Wissenschaften of Göttingen and in 1919 it became the Akademie der Wissenschaften zu Göttingen. His membership continued through these changes until his death in 1925. He was elected a corresponding member of the Königlich-Preussischen Akademie der Wissenschaften of Berlin in 1893 and continued his membership when it became the Preussischen Akademie der Wissenschaften in 1919. He was elected a corresponding member of the Königlich-Bayerischen Akademie der Wissenschaften of Munich in 1895 and continued his membership when it became the Bayerischen Akademie der Wissenschaften in 1919. He was elected a full member of the Mathematical-Physical Class of the Sächsischen Akademie der Wissenschaften of Leipzig in 1919.With Carl Neumann's death, we remember, above all, that he was one of the founders of the Annalen. Among the younger generation of German mathematicians, resistance to Crelle's Journal and its sluggish management had arisen in the sixties of the last century[the1860s]. For this reason, Neumann himself preferred publication in the form of monographs and textbooks. The idea of founding a new journal was considered, but it was Neumann who gave it life by working with the right publisher in the company Teubner, with whom he had been in business for some time, recognised the ideal editor in Clebsch, a friend he studied with in his youth, and brought them together. This happened in a letter to Teubner of10June1868, which has been printed several times, in which Neumann summarizes the duties of a good editor in three sentences of classical correctness and conciseness. How much credit he gave himself for his share in the founding of the Annalen, is seen by the heading which he gave after the death of Clebsch: "Founded by Rudolf Friedrich Alfred Clebsch in conjunction with Carl Neumann". A later period has changed this modest formulation, by placing the two names Clebsch and Neumann side by side. Neumann was probably far from the actual editorial activity. Nevertheless, after Clebsch's death, for four years(1873-1876, vol.6-9), he conducted the main editorial work, but at the same time divided the editorial duties on a broader basis by assuring the cooperation of four subsidiary editors. In this way he has given the editors of the Annalen their still existing form. Since the tenth volume, his share of the editorial work of the Annalen has declined rapidly, and his numerous publications in the Annalen gradually became more and more sparse.

**Article by:** *J J O'Connor* and *E F Robertson*

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