**Eduard Study** was the son of Carl Traugott Wilhelm Study, a teacher of Latin, Greek, German and history at the Coburg Gymnasium, and Caroline Therese Henriette von Langsdorff. Eduard's talent for mathematics was probably inherited from his mother for his great grandfather Karl Christian von Langsdorff had been a professor of mathematics at Heidelberg from 1806 to 1834 while his grandfather Wilhelm Gustav von Langsdorff (1803-1847) had taught applied mathematics at St Petersburg and Mannheim. Eduard was christened Christian Hugo Eduard Study on 16 April 1862 but he always used the name Eduard. When he was four years old, his mother died. His father wrote (see [4]):-

Monday,13August1866. At7:55early in the morning my dearly beloved, good, good wife breathed her last breath. She died after a long and terrible suffering with pulmonary tuberculosis. ... So I am left alone with the task of educating the child that she gave me. When will I find my motivation?

Carl Study, Eduard's father, got a lot of support from the rest of his family. In particular, his sister-in-law Elise was a great help to him bringing up Eduard and Carl married Elsie two years after the death of his first wife. Sadly, Elsie also died of tuberculosis when Eduard was eleven years old and after that, alone with his father, he enjoyed a strict upbringing. It was not a religious upbringing; quite the reverse for he was brought up by his father to think for himself and he received no religious education. Although Eduard seems to have inherited more from his mother than from his father, one love which he did inherit from his father was a love of nature. Carl was a member of the Alpine Club and enjoyed trips with botanists. When Eduard was twelve years old his father sent him for a 3-week holiday in the Upper Bavarian forests. Rather remarkably, he sent the twelve year old Eduard on this holiday on his own. Eduard attended the Coburg Gymnasium where his two favourite subjects were mathematics and biology and, after graduating from school in 1880, he entered the University of Jena to study with the biologist Ernst Häckel (1834-1919). Häckel was professor of comparative anatomy at Jena from 1862 to 1909.

After studying at the University of Jena for a year, Study decided that he wanted to move towards mathematics and in order to do this he went to the University of Strasbourg to work with Theodor Reye. Reye had published a two volume work on synthetic geometry *Geometrie der Lage* in 1866 and 1868. Study solved all the problems in the first volume of this book during the academic year 1881-82 when he was at Strasbourg. He spent the winter semester of 1882-83 at the University of Leipzig before returning to Strasbourg. He did not spend long on this second period at Strasbourg for he soon moved to the Ludwig-Maximilians University of Munich.

We should note that Study, almost certainly because of his upbringing, worked in a very solitary manner. He did not attend many lectures preferring instead to work on his own. This, in many ways, was not because he thought this the best method of working. Rather it seems that he would liked to have behaved differently but felt that he had not had the best training in his youth. He wrote in a letter to Friedrich Engel in January 1892 (see [4]):-

I envy anyone who has learned anything decent in his youth. My self-taught patchy knowledge annoys me ...

He sat his final examination at the University of Munich on 21 July 1884. This consisted of three written questions, two on mathematics and one on physics. Despite the fact that Study chose to prove a generalisation of one of the problems that had been set, Philipp von Seidel criticised his solution as clumsy. He then had an oral examination with examiners Philipp von Jolly (1809-1884), Gustav Bauer (1820-1906), the professor of botany Ludwig, Adolph Timotheus Radlkofer (1829-1927), and the dean Adolf von Baeyer (1835-1917), who was a chemist. He was awarded Grade I in mathematics, his major subject, and also in physics and in botany. Naturally, the overall Grade was also I.

He obtained his doctorate from the University of Munich in 1884 for his thesis* Über die Massbestimmung extensiver Grössen*. He had been advised by Philipp von Seidel and Gustav Bauer. He published his thesis in 1885 but this was not his first mathematics publication since *Über Distanzrelationen* (1882),* Elementare Beweise einiger geometrischer Sätze *(1883), and* Geometrische Konstruktion der Abbildung des Kreisringes auf ein Rechteck *(1884) had already appeared. The last two mentioned papers were published in Crelle's Journal. After the award of his doctorate, Study returned to the University of Leipzig where he worked on his habilitation thesis encouraged by Felix Klein to whom he expressed his "very special thanks". In 1885 Study was appointed as an assistant in mathematics at the University of Leipzig where, in addition to Klein, he met David Hilbert who had just arrived there. The fact that he was largely self taught in mathematics clearly showed in his very individual approach. Although his interests were close to those of Hilbert, he found it difficult to work with him. Hilbert wrote in a letter to Adolf Hurwitz (see [5]):-

Study is a strange person, almost at the opposite pole from my nature and, as I think I can judge, from yours too. Dr Study approves, or rather he knows, only one field of mathematics and that is the theory of invariants, very exclusively the symbolic theory of invariants. Everything else is unmethodical 'fooling around' ... He condemns for this reason all other mathematicians; even in his own field he considers himself to be the only authority, at times attacking all the other mathematicians of the symbolic theory of invariants in the most aggressive fashion. He is one who condemns everything he doesn't know whereas, for example, my nature is such that I am most impressed by just that which I don't yet know.

Klein suggested that both Hilbert and Study should visit Erlangen and discuss their research with Paul Gordan who was the leading expert on invariant theory. However, the visit did not take place at that time. Klein then told both Study and Hilbert that they should visit Paris. They both went in early 1886, Study arriving first. Klein had given them instructions as to which of the Paris mathematicians they should visit and they did as he told them, alternately writing to Klein about their experiences. One of the first mathematicians they visited was Henri Poincaré who returned their visit a few days later. The two young visitors read their letters to Klein out loud to each other so that they would not both tell him the same things. He replied to each in turn, making clear that he was treating them equally. In a letter written to Study, Klein continued to insist that he must have personal contact with Gordan and Noether. In Paris, Camille Jordan gave a dinner for Study and Hilbert to which George-Henri Halphen, Amédée Mannheim and Gaston Darboux were invited. On this occasion the French mathematicians all spoke German out of politeness to their German guests who complained to Klein afterwards that the mathematical conversation had been very superficial. They were also disappointed with their meeting with Pierre Bonnet who they felt was too old for mathematical discussions. The mathematician with whom they seemed to get on best was Charles Hermite. Although they considered him very old (he was 64), he was "extraordinarily friendly and hospitable" and discussed the big problems of invariant theory which interested Study. Since they had found their visit especially useful, they returned to Hermite's home for a second visit a few days later. Study returned to Germany and reported in person to Klein about his Paris visit. However, Klein was disappointed that Study did not speak as much about mathematics as he had expected.

Back at Leipzig, Study habilitated and gave various lecture courses: *Einleitung in die analytische Geometrie der Ebene und des Raumes* (Summer semester 1886); *Einführung in die Theorie der Wärmeleitung, in Verbindung mit der Theorie der Fourier'schen Reihen und Fourier'schen Integrale* (Winter semester 1886); *Einleitung in die neuere Algebra und deren geometrische Anwendungen* (Summer semester 1887); *Einleitung in die Ausdehnungslehre und Quaternionentheorie* (Summer semester 1887); *Neuere Algebra* (Winter semester 1887); *Mechanische Theorie der Wärme* (Winter semester 1887); and *Principien der Mathematik (Zahlbegriff und geometrische Axiome) für Fortgeschrittenere* (Summer semester 1888). Study spent a month, from 15 January 1887 to 15 February 1887, at Erlangen where he had discussions with Gordan. This was at Klein's suggestion but this time Study was keen to go. He wanted to get to know Gordan personally and at Erlangen he had many useful discussions with him. Study, who seems to have got on with Gordan extremely well, wrote in letters to Klein and to Friedrich Engel:-

Of course, I first visited Professor Gordan, who showed me the greatest kindness and devotion. I am with him for several hours every day(probably this is possible due to the small number of students here). He talks to me on mathematical matters which I find very stimulating and helpful. I am fortunate that I have learned sufficient to be able to follow him. His kindness goes so far as to hold lectures on advanced topics just for me. I have learnt much, even things that I can use in my current research. ... He is one of the cleverest people I have ever met - he is deep, thorough and clear like nobody else I have ever met. ... He is a very amiable man who treated me in the kindest possible way and I am most grateful to him.

Study had first met Engel in July 1885 just after he had returned from a ten month visit to Sophus Lie in Oslo. From that time on, Study corresponded with both Engel and Lie concerning his mathematical results. Study's father died in 1888 after a long illness. This meant that he inherited enough money to allow him to marry. Soon after his father's death he married his cousin Lina von Langsdorff; they had only one child, a daughter Trude (born 26 June 1889).

In July 1888, Study left Leipzig and moved to the University of Marburg where he became a privatdocent. His move to Marburg was an attempt to push his career forward more quickly, believing that he would have better opportunities at a Prussian university. He published his first book *Methoden zur Theorie der ternären Formen* in September 1889. Gian-Carlo Rota wrote in his introduction to the 1982 reprint of Study's 1889 original book:-

... the flamboyant German style and the scattered remarks, often sparkling and original, make the reading enjoyable, and the deciphering of the two examples of irrational invariants will be a test of the reader's understanding of the vagaries of the invariant-theoretic mind. ... The reader will miss in Study's book Elliott's philatelic detail, Capelli's combinatorial skill, Hilbert's and Alfred Young's steam-rolling genius, but will find instead a breadth of conceptual view and philosophical insight that displays Continental mathematical thought in its finest hour.

Study became more and more unhappy at Marburg. He only had two or three students attend his lectures and his colleagues did not fare any better. This led to a degree of rivalry between Study and the other lecturers and he came to dislike the lecturing side of his job. The only colleague with whom he got on well was Heinrich Weber but in 1892 Weber left Marburg when he was appointed to the University of Göttingen. Around this time, however, Study began work on writing a book on spherical trigonometry having already published the paper *Über die sphärische Trigonometrie *in 1891. The book *Sphärische Trigonometrie, orthogonale Substitutionen und elliptische Functionen. Eine analytisch-geometrische Untersuchung* was published in 1893. In July 1893 he visited the United States where he attended the mathematical conference at the Columbian Exposition in Chicago in August 1893 and then went to Evanston to attend Klein's series of lectures on contemporary mathematical research held from 28 August to 9 September 1893. After this he went to Johns Hopkins University where Fabian Franklin arranged for him to receive $250 for giving three lectures on invariant theory. While in the United States he published a paper in English, namely *On the addition theorems of Jacobi and Weierstrass* in the *American Journal of Mathematics*. While he was in America he received a letter from Engel telling him that a professorship in Bonn had been advertised. Study was interested but not confident that he would be a strong applicant. He wrote:-

It will probably be a sham, I have no more hope.

Despite his pessimism, two months later he would be appointed to the professorship at Bonn. He left America on 25 April 1894 having first visited J Willard Gibbs at New Haven, Connecticut, Simon Newcomb in Washington, and Henry Fine at Princeton. On his return to Germany in April 1894 he was appointed extraordinary professor of mathematics at Bonn. However, at Bonn he found that he was not receiving much more in salary than he had in Marburg but found Bonn a much more expensive place to live. Again Study was to move after three years, this time to a full professorship at Greifswald. In 1904 he made his final move when he accepted the chair at the University of Bonn which had been left vacant on the death of Rudolf Lipschitz in October 1903. Study held the chair at Bonn until he retired in 1927.

Study became a leader in the geometry of complex numbers. He reformulated, independently of Francesco Severi, the fundamental principles of enumerative geometry due to Hermann Schubert. He also worked on invariant theory helping to develop a symbolic notation. In 1923 he proved important theorems on real and complex algebras of low dimension publishing these results. Study's contribution is summarised by W Burau in [1] as follows:-

... Study demonstrated what he considered to be a thorough treatment of a problem. ... With Corrado Segre, Study was one of the leading pioneers in the geometry of complex numbers. ... Adept in the methods of invariant theory ... Study, employing the identities of the theory, sought to demonstrate that geometric theorems are independent of coordinates. ... Study was the first to investigate systematically all algebras possessing up to four generators overRandC.

Other areas which Study worked on were straight lines in elliptic space, with his student at Bonn J L Coolidge, and he simplified the method of differential operators. In 1903 he published *Geometrie der Dynamen* which considered euclidean kinematics and the mechanics of rigid bodies. In [1] the impact of *Geometrie der Dynamen* is described:-

Unfortunately, because of its awkward style and surfeit of new concepts, this work has never found the public it merits.

Study remained in Bonn after his retirement at the end of the summer semester of 1927, being made professor emeritus at this time. He died of stomach cancer three years later. He was cremated on 9 January 1930 and the urn with his ashes was buried in the Poppelsdorf cemetery in Bonn.

One final fact about Study is of interest. He had always been interested in biology from his student days and one of the ways that he continued this interest through his life was by having an impressive collection of butterflies.

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

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