**William Feller**'s parents were Ida Oemichen and Eugen V Feller. In fact his first name was not William. Rather it was Vilibald which was the name of the Saint whose day falls on his birthday. His mother was a Roman Catholic and so he was christened Vilibald, but was known as Willy (and he even used the name Willy Feller on his papers into the 1940s). William's father was a chemist working in industry. William was educated by private tutors and had no secondary schooling. He entered the University of Zagreb and was awarded his first degree in 1925. His Ph.D. was awarded when he was only twenty years old by the University of Göttingen in 1926, where he studied under Hilbert and Courant. He spent another two years at Göttingen before he accepted an appointment as head of the applied mathematics laboratory at the University of Kiel where he worked until 1933.

Hitler came to power in Germany in 1933 and there was no way that Feller would accept the requirements placed on academics. He already knew Harald Bohr well since both had been in Göttingen together and Bohr now undertook to try to help academics fleeing from Hitler's Nazis to find employment elsewhere. Feller went to Copenhagen where he remained until 1934, then he moved to the University of Stockholm where he joined the probability group headed by Harald Cramér. Being a research associate of Cramér was valuable to Feller, as was the fact that he was able to have useful discussions with Marcel Riesz who was also working in Sweden. In 1938 Feller married Clara Mary Nielsen; they had no children. Feller and his wife emigrated to the United States in 1939 and he became an associate professor of mathematics at Brown University, in Providence Rhode Island.

Founded in 1931 with Neugebauer as the editor, the reviewing journal *Zentralblatt für Matematik* had rapidly become an indispensable tool for mathematicians world wide. After Hitler came to power in Germany in 1933, production of *Zentralblatt* became increasing difficult and Harald Bohr helped Neugebauer to move to the University of Copenhagen in January 1934. Feller was still in Copenhagen at this time. Neugebauer took the editorial office of *Zentralblatt für Matematik* to Copenhagen with him and from 1934 until 1938 *Zentralblatt* continued to flourish from its headquarters there. The struggle to produce the reviewing journal became more difficult throughout this period, however, for the Nazis tried more and more to influence the editorial policy of the journal. Tragically some mathematicians were seduced by the Nazi ideas and mathematicians such as Blaschke attacked the journal. It was clear that the mathematical world required a reviewing journal which was not subjected to political pressures and, when Neugebauer was appointed to Brown University, the American Mathematical Society saw its chance to produce a major reviewing journal. Feller became the first executive editor of *Mathematical Reviews* which started reviewing articles which appeared from July 1939 and the first issue appeared in January 1940. Its successful launch was largely the result of both Neugebauer and Feller's major efforts to get the journal established.

Feller became an American citizen in 1944, and in the following year accepted a professorship at Cornell University where he became a colleague and friend of Mark Kac who had emigrated to the United States in similar circumstances to Feller. He was to work there for five years until he was appointed Eugene Higgins Professor of Mathematics at Princeton in 1950. Gian-Carlo Rota writes:-

From the time he moved from Cornell to Princeton in1950, [Feller's]whole life revolved around a feeling of inferiority. He secretly considered himself to be one of the lowest ranking members of the Princeton mathematics department ... In retrospect, nothing could be farther from the truth. Feller's treatise on probability is one of the great masterpieces of mathematics of all time.

Feller worked on mathematical probability using Kolmogorov's measure theoretic formulation. His approach was pure mathematical but he did study applications of probability, particularly to genetics. He transformed the relation between Markov processes and partial differential equations. Later he put his results in a functional analysis framework. Feller made notable contributions to the mathematical theory of Brownian motion and diffusion processes during the years 1930-1960.

Some of the first papers reviewed by *Mathematical Reviews* were written by Feller himself such as *Completely monotone functions and sequences* (Duke Journal, 1939) and *Die Grundlagen der Volterraschen Theorie des Kampfes ums Dasein in wahrscheinlichkeitstheoretischer Behandlung* (1939). Doob, in a review of the latter paper in *Mathematical Reviews*, writes:-

The author makes a careful study of the development of a population whose number at time t, N(t), is treated as a random function. Thus if it is supposed that the probability that each individual in a time interval of length dt has probability l dt of producing a second individual, the exact value of the probability of having n individuals at time t is found....

The review of Feller's article *On the logistic law of growth and its empirical verifications in biology* (1940) provides some background:-

The logistic curve exploited by Pearl and others is often accepted as expressing a fundamental law of biological growth and this view(frequently plausible in special cases)seems to find support in the success that has attended the effort at fitting a logistic curve to various data on growth of experimental populations. Instead of being content with achieving a graphical vehicle providing satisfactorily close fit, some writers have sought to infer the necessary working of an autocatalytic reaction and hence to assume that the observed and tabulated growth is controlled by some internal factor resident in the organism, rather than being largely governed by external factors. The author considers the problem as to whether there is evidence to support the universal application of any such reasoning as to the operative causes...

Other papers written by Feller while still at Brown University include: *On the time distribution of so-called random events* (1940), *On the integral equation of renewal theory* (1941), *On A C Aitken's method of interpolation* (1943), *The fundamental limit theorems in probability* (1945) and *Note on the law of large numbers and "fair" games* (1945). The second last of these papers on the limit theorems is on a topic that Feller kept returning to over many years.

Feller's most important work was *Introduction to Probability Theory and its Applications* (1950-61), a two volume work which he frequently revised and improved with new approaches, new examples and new applications. The first volume, first published in 1950, was produced in a second edition in 1957 with substantial additions:-

... the exposition is mathematically rigorous and at the same time elegant and lucid. This fascinating book will remain a standard textbook of mathematical probability for many years to come.

A third edition appeared in 1968, and J C Wendel writes:-

All probabilists will welcome the latest edition of this classic book. While preserving the unique flavour of the former editions the author has improved the treatment of many topics.

Feller was invited to address the International Congress of Mathematicians in Cambridge, Massachusetts, in 1950. He chose to speak on the theory of diffusion which he applied to biology. Feller said that his address *Some recent trends in the mathematical theory of diffusion*:-

... outlines some new results and open problems concerning diffusion theory where we find an intimate interplay between differential equations and measure theory in function space.

Work on the theory of diffusion was a major part of Feller's research at Princeton. He also was given the permanent title of visiting professor by Rockefeller University in 1966 and he spent several years there working both with geneticists and mathematicians while on leave from Princeton.

Rota describes in a colourful fashion his experiences attending Feller's lectures:-

His lectures were loud and entertaining. He wrote very large on the blackboard, in a beautiful Italianate handwriting with lots of whirls. Sometimes only one huge formula appeared on the blackboard during the entire period; the rest was handwaving. ... He took umbrage when someone interrupted his lecturing by pointing out some glaring mistake. He became red in the face and raised his voice, often to full shouting range. It was reported that on occasion he had asked the objector to leave the classroom. The expression "proof by intimidation" was coined after Feller's lectures(by Mark Kac). ... I learned more from his rambling lectures than from those of anyone else at Princeton. I remember the first lecture of his I ever attended. It was also the first mathematics course I took at Princeton(a course in sophomore differential equations). The first impression he gave was one of exuberance, of great zest for living, as he rapidly wrote one formula after another on the blackboard while his white mane floated in the air.

J L Doob wrote the following tribute to Feller:-

Those who knew him personally remember Feller best for his gusto, the pleasure with which he met life, and the excitement with which he drew on his endless fund of anecdotes about life and its absurdities, particularly the absurdities involving mathematics and mathematicians. To listen to him lecture was a unique experience, for no one else could lecture with such intense excitement.

Feller received many honours. He was president of the Institute of Mathematical Statistics and he was a member of the Royal Statistical Society in the UK. He was elected to the National Academy of Sciences (United States) and was also a member of the American Academy of Arts and Sciences. He was awarded the 1969 National Medal for Science but died shortly before the presentation was to be made. His wife received the medal on his behalf.

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

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