William was educated by private tutors as well as attending secondary school at the First Realgymnasium, Zagreb. His private mathematics tutor was Stanko Vlögel, the professor of mathematics and engineering mechanics at the University of Zagreb. William graduated from the Realgymnasium in June 1923 and entered the University of Zagreb in October of that year. At Zagreb University he studied ( or ):-
Mathematics I-IV (with Marije Kiseljak), Differential and Integral Calculus (with Vladimir Varicak), Infinite Series (with Stjepan Bohnicek), Number Theory 1,2 (with Stjepan Bohnicek), Theory of Real Functions (with Vladimir Vranic), Calculus of Variations (with Vladimir Varicak), and two mathematical seminars (with Vladimir Varicak), along with a panoply of experimental and theoretical physics, some chemistry, and a sample of courses in pedagogy and psychology.A fuller list of courses he took is given in . He was also taught by Stanko Hondl (1873-1971), the professor of physics, who had attended Max Planck's lectures. However, Vladimir Varicak was the lecturer who influenced Feller most. He was :-
... one of the leading European experts on Einstein's theory of relativity at that time and author of the often quoted book 'Darststellung der Reltivitätstheorie im dreidimenzionalen Lobatschefskijschen Raume' (Zagreb, 1924).Feller was awarded his first degree in 1925 (after 2 years instead of the usual 4 years) and then went to the University of Göttingen to undertake research. His Ph.D. was awarded, suma cum laude, by the University of Göttingen for his thesis Über algebraisch rektifizierbare transzendente Kurven when he was only twenty years old. His thesis advisor was Richard Courant. He completed work on his thesis in 1926 and his oral examination was held on 3 November of that year with examiners Richard Courant, Gustav Herglotz and James Franck. He was awarded his doctorate by the Faculty of Mathematics and Natural Sciences on 18 July 1927. The delay was a result of the requirement that the thesis be accepted for publication before the doctorate was awarded. At Göttingen, in addition to Courant, Feller was also strongly influenced by David Hilbert. After completing work on his thesis, he spent another two years at Göttingen as an assistant of Richard Courant before he accepted an appointment as head of the applied mathematics laboratory at the University of Kiel where he worked until 1933. This was the year that Feller published his first work on probability theory which was his review of Andrei Nikolaevich Kolmogorov's monograph on probability theory Grundbegriffe der Wahrscheinlichkeitsrechnung . This monograph, published in 1933, built up probability theory in a rigorous way from fundamental axioms in a way comparable to Euclid's Elements. Feller wrote in his review:-
The calculus of probabilities is constructed axiomatically, with no gaps and in the greatest generality, and for the first time systematically integrated, fully and naturally, with abstract measure theory. The axiom system is certainly the simplest imaginable. ... The great generality is noteworthy; probabilities in infinite dimensional spaces of arbitrary cardinality are dealt with. ... The presentation is very precise, but rather terse, directed to the reader who is not unfamiliar with the material. Measure theory is assumed.Hitler came to power in Germany in 1933 and there was no way that Feller would accept the requirements placed on academics to sign a Nazi oath. 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 in the Institute for Insurance Mathematics and Mathematical Statistics 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 who had been his student in Kiel; 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 Otto 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 Wilhelm 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 in 1950, [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). Joseph 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 (1950-71), 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.The first edition was published in 1950 but Feller began writing it in 1941. Gian-Carlo Rota explains why this classic book took so long to appear in print :-
While he was writing the first volume of his book he would cross out entire chapters in response to the slightest critical remark. Later, while reading galleys, he would not hesitate to rewrite long passages several times, each time using different proofs; some students of his claim that the entire volume was rewritten in galleys, and that some beautiful chapters were left out for fear of criticism. The treatment of recurrent events was the one he rewrote most, and it is still, strictly speaking, wrong. Nevertheless, it is perhaps his greatest piece of work. We are by now so used to Feller's ideas that we tend to forget how much mathematics today goes back to his "recurrent events"; the theory of formal grammars is one outlandish example.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.This third edition was translated into Russian and Andrei Nikolaevich Kolmogorov wrote an introduction to this work:-
The first edition of Feller's book has already been met with a lot of approval in the USSR. We now bring to the attention to our readers the translation of the second English edition, improved and revised by the author with a lot of details. Such choice of material gives Feller's book a special place among the reference books on the probability theory ... By his choice of problems Feller exposes their solutions by 'direct' and especially probabilistic methods. This tendency to see probabilistic sense behind their analytical transformations represents the most valuable feature of Feller's book. The author's effort to clearly show character and effects of probabilistic laws on carefully chosen examples deserves our attention. In many cases the author succeeds in introducing the reader into quite interesting questions on the comparison between statistical data and probabilistic theory of events. No other book on Probability theory can be compared to this one - in terms of mathematical strictness, excellency of proof and the number of the examples observed. By explaining the most complex mathematical issues he never loses sight of reality where a developed theory can be applied. The character of the book is such that for a long time it will not become out-of-date.The second volume of Introduction to Probability Theory and its Applications appeared in two editions in 1966 and 1971.
In addition to opinions of the reviewers, we quote from Joseph Doob :-
No other book even remotely resembles it in its combination of the purest mathematics together with a dazzling virtuosity of techniques and applications, all written in a style which displays the enthusiasm of the author. This style has made the book unexpectedly popular with non-specialists, just as its elegance and breadth, not to mention its originality, has made it an inspiration for specialists.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.Feller delivered the one hour plenary talk Some new connections between probability and classical analysis at the International Congress of Mathematicians held in Edinburgh, Scotland, in 1958.
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 hand waving. His proof - insofar as one can speak of proofs - were often deficient. Nonetheless, they were convincing, and the results became unforgettably clear after he had explained them. The main idea was never wrong. 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). During a Feller lecture, the hearer was made to feel privy to some wondrous secret, one that often vanished by magic as he walked out of the classroom at the end of the period. Like many great teachers, Feller was a bit of a con man. 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 elected to the Croatian Academy of Sciences and Arts (1937), to the Royal Danish Academy of Sciences, and to the Royal Statistical Society in London, and he was president of the Institute of Mathematical Statistics. He was elected to the National Academy of Sciences (United States) in 1960 and was also elected a member of the American Academy of Arts and Sciences in 1958. 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 in a presentation by president Richard Nixon in the White House. The medal was conferred:-
... for his important contributions to pure and applied mathematics, for his efforts to make probability theory accessible to broad audience, and for his pioneering work in establishing Mathematical Reviews.Feller had been nominated for the National Medal for Science by Oscar Zariski and the case for the award was written by Joseph Doob, Mark Kac and Jerzy Neyman.
There was a difficult period when Feller was told that he had terminal cancer. In  Mark Kac is quoted as saying:-
Having accepted the verdict himself he tried to make it easy for all of us to accept it too. He behaved so naturally and he took such interest in things around him that he made us almost forget from time to time that he was mortally ill.Let us quote Charles Ashbacher's assessment of Feller :-
Feller was one of the giants in the development of the modern ideas of probability models and analysis. He was creatively active his entire life, a counterexample to the myth that mathematicians have exhausted their creative energies by the age of 50. Feller is also another example of the incredible value that can be derived from immigration. He was one of the first very talented people to leave Germany due to the rise of the Nazis, eventually settling in the eastern United States.Let us end this biography by quoting from Joseph Doob in :-
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 his lecture was a unique experience, for no one else could lecture with such intense excitement. No one could generate in himself as well as in his auditors so much intense excitement. In losing him, the world of mathematics has lost one of its strongest personalities as well as one of its strongest researchers.
Article by: J J O'Connor and E F Robertson