**Lazarus Fuchs**'s parents were the Jewish teacher Rafael Fuchs and his wife Caecilie Katz. We do not have dates for Lazarus's parents but we know that at the time he wrote his doctoral thesis his mother was deceased but his father was still alive. He was first taught by his father in Moschin but he had a strong desire for learning so, in 1846, he went to Posen (now Poznań in Poland) where he was privately taught classical languages for two years but mainly studied on his own. However, his parents were in financial difficulties and could not give him any support. He did get very minimal support from a distant wealthy relative but he lived in extreme poverty. He found a place to live which was cheap but the conditions were extremely bad. For months his only food was coffee and bread. He often begged his better-off friends to give him the last end of a candle so that he could study during the night and achieve his aim of entering the local Gymnasium. In 1848 he entered the third class at the Friedrich Wilhelm Gymnasium in Posen which, at this time, had Gustav Kiessling as its director. While he was studying at the Gymnasium, Albert Gustav Heydemann (1808-1877) was appointed as director in 1850.

It was at the Gymnasium that his remarkable abilities at mathematics became very clear to his teachers despite the fact that he had a very poor preparation in the topic before beginning his studies at the Gymnasium. Mathematics became the subject which, even at this early stage, Fuchs knew was going to dominate the rest of his life. In addition to his school studies, Fuchs had to earn money by giving private tutoring and his abilities for developing hidden skills and talents in his pupils were quickly appreciated so that he became a much sought-after teacher. One of his friends was Leo Königsberger who was four years younger that Fuchs and was also a pupil at the Friedrich Wilhelm Gymnasium in Posen. After eighteen months, Fuchs had completed the Middle School and, missing out a year in the Upper School, he was able to take his maturity examination at Easter 1853 which he passed with grade "excellent".

Despite qualifying to enter university in 1853, he could not begin his studies since he was completely destitute. It was at this stage that Leo Königsberger's parents came to his rescue. They offered to employ Fuchs as a private tutor to Leo for a year, to give him accommodation in their home, and to pay him a modest fee. He happily accepted and, after tutoring Leo Königsberger for a year, he entered the University of Berlin at Easter 1854. The rector of the university at this time was the famous astronomer Johann Franz Encke (1791-1865) and Fuchs was enrolled into the Faculty of Philosophy whose dean was the philosopher Friedrich Adolf Trendelenburg (1802-1872). From Easter 1857 to Easter 1864 Fuchs and Königsberger shared accommodation in Berlin. They lived in a great number of different houses, forced to live a very simple and modest life-style since, especially in the first few years, they lived off the income that Fuchs made through giving private lessons.

At the University of Berlin, Fuchs attended lectures by a number of famous mathematicians including Eduard Kummer and Karl Weierstrass. He gives the full list of those who taught him during his four years at the University of Berlin in his doctoral thesis. These teachers are: mathematician Peter Friedrich Arndt (1817-1866), classical scholar August Böckh (1785-1867), Carl Borchardt, Rudolf Clausius, Lejeune Dirichlet, physicist and meteorologist Heinrich Wilhelm Dove (1803-1879), astronomer Johann Franz Encke (1791-1865), Eduard Kummer, zoologist Martin Heinrich Carl Lichtenstein (1780-1857), experimentalist in chemistry and physics Heinrich Gustav Magnus (1802-1870), chemist and mineralogist Eilhard Mitscherlich (1794-1863), mathematician Martin Ohm, physicist Johann Christian Poggendorff (1796-1877), historian and political scientist Friedrich Ludwig Georg von Raumer (1781-1873), chemist Franz Leopold Sonnenschein (1817-1879), philosopher Friedrich Adolf Trendelenburg (1802-1872), mineralogist Christian Samuel Weiss (1780-1856), and Karl Weierstrass. He writes:-

As well as attending lectures by the people listed above, Fuchs read Gauss'I am, and will always be, grateful to all these men who perfectly rendered outstanding services to me.

*Disquisitiones arithmeticae*and works by Fourier, Laplace and Cauchy. The professors who had the greatest influence on Fuchs were Karl Weierstrass, who introduced him to function theory, and Eduard Kummer who went on to supervise his doctorate. The examiners for his doctoral dissertation,

*De Superficierum lineis curvaturae*Ⓣ, held on 2 August 1858 were Kummer and Martin Ohm (the brother of Georg Simon Ohm) and his opponents were Julius Weingarten, Leo Königsberger and Eduard Fischer. Fuchs was awarded the degree by the University of Berlin 1858.

Before the award of his doctorate, Fuchs began to worry about whether he should convert from Judaism to Christianity. He knew that if he remained a follower of the Jewish faith he would suffer discrimination and his career prospects would be very limited. Although Jews had been officially discriminated against during the early years of Fuchs' life, a new constitution in Prussia in 1850 gave all citizens equal rights irrespective of their religion. Despite the constitution, equality did not happen and discrimination continued in Prussia as did many restrictions and limitations on Jews. Fuchs spent three years deeply worried about whether he should convert to Christianity. Of course, it was not simply a personal matter since he worried greatly about how his Jewish family would feel if he converted. Both Kummer and Weierstrass were well aware of Fuchs' painful suffering on the issue and encouraged him to take the step. He was also given great support from Julius Müllensiefen (1811-1893), who was a pastor at the church of St Marien in Berlin. In 1860, Fuchs converted to Evangelical-Lutheran Christianity.

Before obtaining his doctorate, Fuchs was appointed to a teaching post at a Gymnasium in Berlin on 19 March 1859. In fact he taught in several different institutions before moving on to a mathematics teaching position at the Friedrich Werder Trade School. During this time he was undertaking research with the aim of becoming a university professor. It was during this period that he did outstanding work on homogeneous linear differential equations with variable coefficients which was published in the important 40-page paper *Zur Theorie der linearen Differentialgleichungen mit veränderlichen Coefficienten* Ⓣ which appeared in Crelle's Journal in 1866. This work is described in more detail below. He began his university teaching career when he was appointed as a Privatdozen at the University of Berlin in August 1865 after submitting his habilitation thesis *Zur Theorie der linearen Differentialgleichungen mit veränderlichen Coeffizienten* Ⓣ. He was promoted to extraordinary professor there on 2 December 1866 and taught at the University until winter semester 1868-69 when he accepted an appointment as a full professor at Greifswald. Fuchs also held a second post in Berlin from 23 May 1867 when he was appointed as professor of mathematics at the Artillery and Engineering School.

In 1868 Fuchs married Marie Anders (23 July 1849 - 10 May 1917). They had four sons and two daughters including Clara Fuchs (1869-1954) and Richard Fuchs who became a mathematician and has a biography in this Archive. Clara married the mathematician Ludwig Schlesinger who also has a biography in this Archive. We note that Richard Fuchs and Ludwig Schlesinger jointly edited Lazarus Fuchs' complete works: *Gesammelte mathematische Werke von L Fuchs* which was published by Mayer & Müller, Berlin in three volumes in 1904, 1906 and 1909.

Fuchs took up the professorship in Greifswald on 3 February 1869, the position becoming vacant since Leo Königsberger, who had taught at Greifswald for five years, had been appointed to a chair of mathematics at Heidelberg. After spending five years in Greifswald, Fuchs moved again, this time to Göttingen where he took up an appointment as an ordinary professor on 23 January 1874. Then in the following year he went to Heidelberg and taught there for nine years. In many ways these years were the most enjoyable period of his life. He loved the natural environment around the city, something that was important to Fuchs who had deep feelings for nature. Also in Heidelberg he had a particularly good relationship with his many outstanding doctoral students, and he got on extremely well with the other members of staff in many different faculties of the university. In fact during nine years at Heidelberg he supervised the doctoral studies of at least eight students who went on to become professors of mathematics. We will say a little below about the correspondence he carried out with Henri Poincaré during his years at Heidelberg. In the summer semester of 1884 he returned to Berlin to fill Kummer's chair when his old teacher retired. Fuchs held this post for the rest of his life. He also undertook important editorial duties in the final ten years of his life when he was the editor of Crelle's journal, the *Journal für die reine und angewandte Mathematik*. Fuchs published many papers in this journal over his career: see the papers at THIS LINK.

Ernest Julius Wilczynski attended Fuchs' lectures in Berlin. He writes in [17] that he:-

Fuchs worked on differential equations and the theory of functions. George Mathews writes [12]:-... remembers the small and crowded lecture-room in the University of Berlin, poorly ventilated, stuffy and hot in the summer days, but so full of meaning and inspiration to the earnest and thoughtful student. Fuchs was not a brilliant lecturer. He spoke in a quiet, undemonstrative manner, but what he said was full of substance. To the student there was the inspiration of seeing a mathematical mind of the highest order full at work. For Fuchs worked when he lectured. He was rarely well prepared, but produced on the spot what he wished to say. Occasionally he would get lost in a complicated computation. Then he would look around at the audience over his glasses with a most winning and child-like smile. He was always certain of the essential points of his argument, but numerical examples gave him a great deal of trouble. He was fully conscious of this failing, and I remember well one occasion when, after a lengthy discussion, he laid considerable emphasis upon the fact that "in this case, two times two is four."

In [1] Jerome Manheim writes:-Fuchs's mathematical papers are very pleasant to read and free from that tendency to heaviness which is apt to belong to memoirs on differential equations. He had the faculty of bringing out clearly the really important points without over-elaborate detail, and he did not disdain to show the power of his methods by applying them to specific and definite problems.

In 1865 Fuchs studied nth order linear ordinary differential equations with complex functions as coefficients. This is described by Bölling in [3]:-Fuchs was a gifted analyst whose works form a bridge between the fundamental researches of Cauchy, Riemann, Abel, and Gauss and the modern theory of differential equations discovered by Poincaré, Painlevé, and Émile Picard.

It was Fuchs' work on this inverse function which led Poincaré to introduce what he called a Fuchsian group, and use this as a fundamental concept in the development of the theory of automorphic functions. The first contact between Poincaré and Fuchs was a letter written by Poincaré on 29 May 1880:-Fuchs enriched the theory of linear differential equations with fundamental results. He discussed problems of the following kind: What conditions must be placed on the coefficients of a differential equation so that all solutions have prescribed properties(e.g. to be regular or algebraic). This led him(1865,1866)to introduce an important class of linear differential equations(and systems)in the complex domain with analytic coefficients, a class which today bears his name(Fuchsian equations, equations of the Fuchsian class). ... He succeeded in characterising those differential equations the solutions of which have no essential singularity in the extended complex plane. Fuchs later also studied non-linear differential equations and moveable singularities.

Fuchs' study(1876with Hermite)of elliptic integrals as a function of a parameter marks an important step towards the theory of modular functions(Klein, Dedekind). In a series of papers(1880-81)Fuchs studied functions obtained by inverting the integrals of solutions to a second-order linear differential equation in a manner generalising Jacobi's inversion problem.

Although we have given the extract from Poincaré's letter in English (following [2]) in fact he wrote in French while Fuchs replied in German. Both seemed happy with this arrangement. Verhulst writes in [2]:-I have read with great interest the remarkable treatise that you had recently included in the last issue of Crelle's Journal with the title "Über die Verallgemeinerung des kehrungsproblems."ⓉI hope you will grant me, dear sir, to request from you certain clarifications on the subject. ... I have to confess, dear sir, that these thoughts have raised with me some doubts about the generality of the results that you have published and I have taken the freedom to approach you about it, hoping that it will not trouble you to clear this up.

On 12 June 1880 Poincaré wrote to Fuchs:-Fuchs, despite being21years Poincaré's senior, consistently maintained a tone of friendship and interest, even when it began to become clear that his young French correspondent was developing an approach that was quite different from his own and more complete.

On 20 March 1881, Poincaré wrote his last letter to Fuchs:-I find in the case that there are two singular points only, that the function you have introduced has very remarkable characteristics, and because I intend to publish the results I have obtained, I am asking your permission to call them Fuchsian functions; for it was you who discovered them.

Fuchs also investigated how to find the matrix connecting two systems of solutions of differential equations near two different points. A survey of Fuchs work appears in [8] where Gray also describes how this work influenced Klein, Jordan, Poincaré and others. In this interesting paper Gray also discusses the relationships between Fuchs' ideas and his mathematical tools, and illustrates how solutions of certain problems led Fuchs to the study of further problems. Gray writes in his introduction:-I have continued with the functions that I named after you and I hope to publish my results shortly. These functions contain as a special case the elliptic functions and also the modular function. With these and other functions that I have called zeta Fuchsian, one can solve:(1)All linear differential equations with rational coefficients that have three singularities only, two finite and one infinite.(2)All second order equations with rational coefficients.(3)A large number of equations of various orders with rational coefficients.

[After discussing the concepts named for Fuchs, Gray writes [8]:-Fuchs']work can profitably be seen as an attempt to impose upon the inchoate world of differential equations the conceptual order of the emerging theory of complex functions. As well as being the architect of the rigorous modern theory of linear equations, he raised many questions which were taken up by his contemporaries and provided an interesting battleground for the schools of invariant theory and transformation group theory.

In [3] Bölling describes Fuchs' character as follows:-But it would be idle to debate the justice of remembering Fuchs by things Fuchsian. Rather, Fuchs' career remains of interest because it shows clearly and dramatically how mathematical ideas are many-sided, and how many new ideas may be needed to solve a problem. There is an ironic truth in the assessment that Fuchs opened up a new province: repeatedly Fuchs pointed to problems in analysis that could best be solved by group theory. Denying himself this tool, which belonged to the younger generation, one might say that Fuchs could only stand like Moses and gaze upon the promised land.

... Fuchs is a representative of both Berlin's classical and its post-classical era. His personality has been described as indecisive, timid, but at the same time humorous and full of kindness.

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

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