**Oskar Bolza**'s parents were Luise Koenig and Moritz Bolza. The family moved around in southern Germany until 1873 when they settled in Freiburg im Breisgau. Oskar's father manufactured printing presses and it was the family's intention that Oskar should follow in his father's trade. Oskar began his secondary schooling in Neuchâtel but when the family moved to Freiburg he continued his secondary schooling there. He took his university entrance examinations in the summer of 1875 entering the University of Berlin in the October of that year. Initially he was interested in languages and physics so he took two courses, a liberal education course at the university and at the same time a technical course at the Berlin Gewerbeakademie. This proved to be too demanding for Bolza so he decided to concentrate on physics at the University. He studied under two leading physicist, namely Helmholtz and Kirchhoff, from 1876 to 1878. This went well but when he spent the summer of 1878 working in August Kundt's physics laboratory in Strassburg he found that the experimental work was not to his liking. He changed to study pure mathematics from the autumn of 1878 and worked with on this topic for two years. From 1878 until 1881 Bolza studied mathematics under Christoffel and Reye at Strasbourg, under Schwarz at Göttingen and under Weierstrass at Berlin. He attended Weierstrass's 1879 lecture course on the calculus of variations which was to have a lasting effect on the direction that Bolza's mathematical interests would take.

Despite working under such an eminent collection of mathematicians, Bolza did not find a research topic. Deciding that he did not have the talent to do original mathematics, he took the examinations to qualify to become a secondary school teacher in 1882. After teaching at a gymnasium in Freiburg for a year to qualify as a teacher, he found that he could not give up the idea of research so easily and he began working on hyperelliptic functions. He was well on his way to a doctorate when he learnt that Goursat had discovered the results he had obtained before him, and even worse was the fact that Goursat's methods were more elegant than Bolza's. Given his earlier doubts one might think that this would be the final straw which would make Bolza give up completely the notion of research. However, it had the opposite effect for he now realised that he did have the necessary talent so he was able to press ahead. He returned to the University of Berlin where he worked with Kronecker and Fuchs but, after corresponding with Klein about the results he was obtaining, he decided to make one final change in his long route to a doctorate and went to Göttingen to be supervised by Klein. He completed his doctoral studies after eight years of study and many changes of direction, and received his doctorate in 1886 from the Georg- August- Universität Göttingen for his thesis *Über die Reduction hyperelliptischer Integrale erster Ordnung und erster Gattung auf elliptische, insbesondere über die Reduction durch eine Transformation vierten Grades* Ⓣ.

Bolza had become a close friend of Maschke's while the two studied together at Berlin. They were now together again at Göttingen where they both worked with Klein as postdoctoral research assistants. Maschke found working with Klein in his home in the evenings very rewarding, but once again Bolza suffered self doubts, feeling that he was not good enough to get an academic post in Germany. Bolza wrote that he felt:-

... an immense distance between Klein's brilliant genius, supported by a wonderful capacity for geometric visualisation ...[and my]... purely analytic gifts, deficient of all fantasy and lying in an entirely different direction.

When one of his friends moved to the United States, Bolza was tempted to follow his example. He spent the winter of 1887-88 in England improving his English, then emigrated to the United State in 1888. He travelled to Baltimore hoping to get an appointment at Johns Hopkins University, carrying with him a strong recommendation from Klein. There he met Newcomb who told Bolza that had he known that he was intending to emigrate to the United States he would have strongly advised against it. Newcomb then wrote to Klein on 23 April 1888 in an attempt to stop further German mathematicians coming to the United States:-

I never advise a foreign scientific investigator to come to this country, but always tell him that the difficulties in the way of immediate success are the same that a foreigner would encounter in any country. ... We have indeed several hundred so-called colleges but I doubt that if one half of the professors of mathematics in them could tell what a determinant is. All they want in their professors is an elementary knowledge of the branches they teach and the practical ability to manage a class of boys, among whom many will be unruly. Considerations of religion, personal influence, and former connection with the college also come into play.

During 1889 Bolza worked at Johns Hopkins University where Newcomb gave him a temporary short-term appointment, then he obtained a position at Clark University. Clark University, which opened in Worcester, Massachusetts, in 1887, was established by Jonas Gilman Clark, a Worcester native and successful merchant, and G Stanley Hall, a psychologist and first president of the university. Clark University started as a graduate institution, the first undergraduates entering only in 1902. The University hired some excellent mathematicians, and there Bolza became a colleague of White and Story. While at Clark, Bolza published the important paper *On the theory of substitution groups and its application to algebraic equations* in the *American Journal of Mathematics*. However a serious political situation arose at Clark University and a vote of no confidence was passed in the president G Stanley Hall. Nine of the eleven members of faculty left Clark including both White and Bolza, who decided to return to Germany.

In 1892 the University of Chicago opened and the head of the mathematics department, Eliakim Moore, began building up a strong unit. He persuaded Bolza not to return to Germany but instead to come to work at Chicago. Bolza joined the University of Chicago in 1892 and then he persuaded Eliakim Moore to bring his friend Maschke to Chicago. The three were highly influential in building up a strong mathematics research school in Chicago. R C Archibald writes:-

These three men supplemented one another remarkably. Moore was a fiery enthusiast, brilliant, and keenly interested in the popular mathematical research movements of the day; Bolza, a product of the meticulous German school of analysis led by Weierstrass, was an able, and widely read research scholar; Maschke was more deliberate than the other two, sagacious, brilliant in research, and a most delightful lecturer in geometry. During the period1892-1908the University of Chicago was unsurpassed in America as an institution for the study of higher mathematics.

Between 1892 and 1910 the mathematics department was outstandingly successful with thirty-nine students graduating with doctorates (nine of them students of Bolza). These included Leonard Dickson, who was the first to be awarded a Ph.D. in mathematics by the University of Chicago, Gilbert Bliss, Oswald Veblen, Robert Moore, George D Birkhoff and T H Hildebrandt.

Bolza published *The elliptic s-functions considered as a special case of the hyperelliptic s-functions* in 1900 which related to work he had been studying for his doctorate under Klein. However, he worked on the calculus of variations from 1901. Papers which appeared in the *Transactions of the American Mathematical Society* over the next few years were: *New proof of a theorem of Osgood's in the calculus of variations* (1901); *Proof of the sufficiency of Jacobi's condition for a permanent sign of the second variation in the so-called isoperimetric problems* (1902); *Weierstrass' theorem and Kneser's theorem on transversals for the most general case of an extremum of a simple definite integral* (1906); and *Existence proof for a field of extremals tangent to a given curve* (1907). His text *Lectures on the Calculus of Variations* published by the University of Chicago Press in 1904, became a classic in its field and was republished in 1961. After the death of his friend Maschke in 1908, Bolza became unhappy in the United States and, in 1910, he and his wife returned to Freiburg in Germany where he was appointed as an honorary professor. Chicago gave him the title of 'non-resident professor of mathematics" which he retained for the rest of his life.

Immediately after his return to Germany Bolza continued teaching and research, in particular on function theory, integral equations and the calculus of variations. Two papers of 1913 and 1914 are particularly important. The first *Problem mit gemischten Bedingungen und variablen Endpunkten* Ⓣ formulated a new type of variational problem now called 'the problem of Bolza' and the second studied variations for an integral problem involving inequalities. This latter work was to become important in control theory. Bolza returned to Chicago for part of 1913 giving lecturers during the summer on function theory and integral equations.

World War I greatly affected Bolza and, after 1914, he undertook no further research in mathematics. He became interested in religious psychology, languages (particularly Sanskrit), and Indian religions. He published *Glaubenlose Religion* Ⓣ in 1930 although it did not appear under his own name, rather under the pseudonym F H Marneck. He returned to mathematics, however, lecturing at Freiburg from 1929 to 1933 when he retired.

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

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