**Adolf Hurwitz** was born into a Jewish family. His father, Salomon Hurwitz, was in the manufacturing business but was not particularly well off. Sadly, Adolf's mother Elise Wertheimer died when he was only three years old. Hurwitz entered the Realgymnasium Andreanum in Hildesheim in 1868. He was taught mathematics there by Schubert [2]:-

Schubert gave up part of every Sunday to working at geometry with the schoolboy Hurwitz, and the first of the latter's papers, written when he was still at the Andreanum, was a joint paper. It was also Schubert who persuaded Hurwitz's father to allow him to go to university and who sent him with warm recommendations to Klein at Munich.

We note that this first paper by Hurwitz, written jointly with Schubert, was on Chasles's theorem. Salomon Hurwitz could not afford to send his son to university but his friend, Mr Edwards, agreed to help out financially, so making a university career for Hurwitz possible. He entered the University of Munich in 1877, before he was eighteen years old, and spent a year there attending lectures by Klein. Although he was greatly influenced by Klein and had already begun to undertake advanced work with him, he went for academic year 1877-78 to continue his studies at the University of Berlin where he attended classes by Kummer, Weierstrass and Kronecker. In particular he attended the one semester course by Weierstrass *Introduction to the theory of analytic functions* and the notes taken by Hurwitz at this time are reproduced as the book [2]. The lectures contained Weierstrass's version of the arithmetisation of analysis including his "construction" of the real numbers, the *ε*, *δ* approach to analysis and his theory of complex functions based on power series.

While at Berlin Hurwitz continued to keep in contact with Klein and assisted him with a paper on elliptic modular functions which he was writing. After three semesters at the University of Berlin, Hurwitz returned to the University of Munich in 1879 to continue working with Klein, so when Klein moved to the University of Leipzig in October 1880, Hurwitz went with him. His Ph. D. was supervised by Klein and he received the degree in 1881 for his dissertation on elliptic modular functions *Grundlagen einer independenten Theorie der elliptischen Modulfunktionen und Theorie der Multiplikatorgleichungen 1. Stufe.*

It would have been natural for Hurwitz to become a Privatdozent at the University of Leipzig since he was a student of Klein, the professor of mathematics there. However there was a difficulty -- Hurwitz did not have sufficient knowledge of Greek to satisfy the Faculty requirements! Luckily Göttingen had no such requirement and Hurwitz became a Privatdozent at the University of Göttingen after submitting his habilitation thesis there in 1882. Hurwitz had not been at Munich during 1881-82, rather he had returned to Berlin where he attended further courses of lectures by Weierstrass and Kronecker.

In 1884 Hurwitz accepted an invitation from Lindemann to become an extraordinary Professor at Königsberg and he was to remain there for eight years. Here he taught Hilbert and Minkowski, becoming a life long friend of Hilbert. Even after Minkowski left the University of Königsberg and went to Bonn, he still returned to Königsberg for every vacation and joined Hurwitz and Hilbert in their almost daily walks [4]:-

During these walks, continued over the whole of the eight year period of Hurwitz's residence in Königsberg, well nigh every corner of the then known mathematical world was explored.

In Königsberg Hurwitz met Ida Samuel, the daughter a professor in the faculty of medicine, and they married; the marriage produced three children. In 1892 Frobenius left his chair at Eidgenössische Polytechnikum Zürich to return to Berlin and Hurwitz was appointed to the vacant chair at Zürich. Hurwitz remained at Zürich for the rest of his life, unfortunately continually suffering from ill health. His health problems had begun when he contracted typhoid in Munich when he was a student there. The disease was widely spread though the city at that time. In fact he twice contracted typhoid and from then on suffered badly from migraine headaches.

Although, as we have pointed out, Hurwitz remained at Zürich for the rest of his life, that was not because he had not been offered in chair in Germany. Schwarz, who was professor at Göttingen, succeeded Weierstrass by accepting his professorship in Berlin in 1892. Göttingen approached Hurwitz and offered him the vacant chair only weeks after he had accepted the Zürich chair, but he turned down the offer. This must have been a remarkably hard decision for Hurwitz since at that time a chair at a leading German university such as Göttingen would have been much more prestigious to any German than a chair in Switzerland. However Hurwitz was an extremely loyal person, and having given his word that he would accept the Zürich position he would not renege on his promise.

Much of Hurwitz's mathematics can be seen as being strongly influenced by Klein (and also by Riemann whose ideas where transmitted to Hurwitz via Klein). In fact Hurwitz and Klein complemented each other extremely well for the reasons that Young indicates in [4]:-

Klein's strength ... was sometimes regarded as consisting still more in the fertility and the[genius]of his ideas than in the power of developing them.

Here then was Hurwitz's strength -- in developing Klein's ideas [1]:-

Klein's new view on modular functions, uniting geometrical aspects such as the fundamental domain with group theory tools such as the congruence subgroups and with topological notions such as the genus of the Riemann surface, was fully exploited by Hurwitz.

Hurwitz studied the genus of the Riemann surface. He worked on how to derive class number relations from modular equations. He investigated the automorphic groups of algebraic Riemann surfaces of genus greater than 1, showing that they were finite. He also studied invariant integrals for *SO*(*n*, **R**) and *SL*(*n*, **R**) and Slodowy describes in [11] how this work, together with Schur's work on orthogonality relations and the character formula for the orthogonal groups, led to Weyl's papers on the representation theory of semisimple Lie groups.

Further topics studied by Hurwitz include complex function theory, the roots of Bessel functions, and difference equations. He also wrote several papers on Fourier series. Soon after he went to Zürich he was asked a question by Aurel Stodola, one of his colleagues, concerning when an *n*th-degree polynomial with real coefficients

f(x) =a_{0}x^{n}+a_{1}x^{n-1}+ ... +a_{n}

with positive leading coefficient *a*_{0} > 0 has only roots with negative real parts. Hurwitz solved this problem completely showing that the condition held if and only if a certain sequence of determinants are all positive. He published this in 1895 in the paper *Über die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Theilen besitzt* which appeared in *Mathematische Annalen* in 1895. This remarkably influential paper was reprinted 100 years later in the proceedings of the Hurwitz Symposium on Stability theory in Ascona in 1995. The excellent review [6] appears in the proceedings of the same symposium, and in the paper [5] the genesis of Hurwitz's version of the well-known stability criterion is described in detail.

Hurwitz did excellent work in algebraic number theory. For example he published a paper on a factorisation theory for integer quaternions in 1896 and applied it to the problem of representing an integer as the sum of four squares. A full proof of Hurwitz's ideas appears in a booklet published in the year of his death. This involves studying the ring of integer quaternions in which there are 24 units. He shows that one-sided ideals are principal and introduces prime and primary quaternions.

The paper [9] by Lindström shows another aspect of Hurwitz's work. Here is part of Lindström's summary:-

In1893the Swedish actuary and mathematics historian Gustaf Eneström published a theorem on the complex roots of certain polynomials with real coefficients in a paper on pension insurance(in Swedish). This result is now often called the Eneström-Kakeya theorem, since S Kakeya published a similar result in1912-1913. But Kakeya's theorem contained a mistake, which was corrected by A Hurwitz in1913. Hurwitz informed E Landau about Kakeya's result(corrected); Landau needed the result in a proof of a theorem on infinite power series. ... We mention a generalization of Eneström's theorem and give an application to a similar result by Hurwitz.

Migraine was not the extent of Hurwitz's health problems which became increasingly severe. His kidneys became diseased and he had one removed in 1905. With only one kidney, and that one not functioning properly, the quality of his life was very poor. Young [2] writes that:-

... his life[was]one long struggle with a wasting disease. That this struggle was waged with comparative success for so many years appears almost incredible, and can only be accounted for by the constant care and devotion of[his]wife.

In [1] Hilbert's comments on Hurwitz as a person are recorded:-

Hilbert depicted him as a harmonious spirit; a wise philosopher; a modest, unambitious man; a lover of music and amateur pianist; a friendly unassuming man whose vivid eyes revealed his spirit.

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

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