**Edward Van Vleck**'s mother was Ellen Maria Burr and his father was John Monroe Van Vleck. John Van Vleck was a teacher of mathematics and astronomy at the Wesleyan University from 1853 until his death in 1912. There he taught H S White and F S Woods and was well known nationally, in particular he was one of the two vice-presidents of the American Mathematical Society in 1904, the other being Bolza. Edward was known as Ned by his father and friends.

Edward Van Vleck attended Middletown High School, then Wilbraham Academy before entering the Wesleyan University in 1880. Here he was taught mathematics by his father and lived at his parents' home during his years of study at the university. At this stage his interests were in mathematics, physics and astronomy and after graduating with an A.B. in 1884 he spent a year as an assistant in the physics laboratory at the Wesleyan University.

In 1885 Van Vleck became a graduate student at Johns Hopkins University where his interests still ranged through mathematics, physics and astronomy. He studied mathematics there under Craig, Newcomb and Story, and physics under Henry Rowland. After two years he became convinced that mathematics was the topic for him. He was then advised by his father that he should travel to Göttingen in Germany to continue his studies.

At the University of Göttingen, Van Vleck attended lectures by Klein, Burkhardt, Fricke, Schur, Schwarz, Woldemar, Voigt, Weber and others. His doctorate was awarded by Göttingen in 1893 for the thesis entitled *Zur Kettenbruchentwicklung Laméscher und ähnlicher Integrale* written under Klein's supervision. The thesis studied hyperelliptic and related integrals in continued fractions. After returning to the United States in 1893 Van Vleck was appointed as an instructor at the University of Wisconsin. In the same year he married Hester Laurence Raymond; they had one son, John Hasbrouck Van Vleck, who was born on 13 March 1899.

We should record here that John Hasbrouck Van Vleck was educated at the University of Wisconsin, Madison, and then at Harvard University where he received his doctorate in 1922. He worked at the University of Minnesota, the University of Wisconsin, then Harvard University where he was Hollis professor of mathematics and natural philosophy from 1951 to 1969. He produced the first quantum mechanical theory of magnetism. He was awarded the Nobel Prize for physics in 1977 for his work on the behaviour of electrons in magnetic, noncrystalline solid materials.

Returning to describe the career of Edward Burr Van Vleck, after being an instructor in mathematics at the University of Wisconsin from 1893 to 1895, he returned to the Wesleyan University as an associate professor. In 1898 he was promoted to professor, again moving to the University of Wisconsin in 1906 where he remained as a professor until he retired in 1929.

Almost all Van Vleck's research papers were in the fields of function theory and differential equations. For example he published *On the determination of a series of Sturm's functions by the calculation of a single determinant* (1899), *On linear criteria for the determination of the radius of convergence of a power series* (1900), *On the convergence of continued fractions with complex elements* (1901), *A determination of the number of real and imaginary roots of the hypergeometric series* (1902), *On an extension of the *1894* memoir of Stieltjes* (1903), and *On the extension of a theorem of Poincaré for difference-equations* (1912). In [6] the authors discuss one of Van Vleck's papers. They write:-

A discussion is given of a1908paper by the American E Van Vleck. It is argued that Van Vleck proved the first zero-one law, anticipating the zero-one law of Borel and, more strikingly, that of Kolmogorov. A brief description of the evolution of the link between measure theory and probability theory is given. By following Van Vleck's own steps in deriving consequences of his zero-one law, a result("the extended Van Vleck theorem")is given which is directly comparable to Borel's law of normal numbers. Finally, it is shown that the Van Vleck zero-one law, which in generality falls between that of Borel and that of Kolmogorov, is further distinguished in that it provides the key step in establishing what may be the earliest example in ergodic theory of a metrically transitive transformation.

Perhaps surprisingly, for a man of his talents, his output of just over thirty mathematical articles might seem rather modest. This is explained by Birkhoff in [2] where he writes:-

... as the years went on, the self-imposed demand for elegance and simplicity exercised a inhibitive influence upon Van Vleck's production.

Langer and Ingraham in [4] express similar sentiments. Van Vleck considered a:-

... mathematical result was not something to be transmitted haphazardly to the public. It should be part of a cultural structure and, as such, it should be expressed with precision and elegance.

Van Vleck was American Mathematical Society Colloquium lecturer in 1903 giving six lectures on divergent series and continued fractions. He published these lectures in the first volume of the series *American Mathematical Society Colloquium Publications.* The American Mathematical Society was fortunate to have Van Vleck's support as well as that of his father. Van Vleck was an editor of the *Transactions of the American Mathematical Society* from 1905 to 1910, vice-president in 1909, and president from 1913 to 1914. He gave his retiring presidential address on *The role of the point-set theory in geometry and dynamics.* Perhaps his greatest contribution to the Society came in 1911 when his actions were to see the Society remain as a single national Society instead of breaking into smaller local societies. The crisis arose over where Bôcher would give his presidential address. John Hasbrouck Van Vleck, writing in [7], recalled the events:-

At that time there was bitter feeling, especially by the Chicago group, that no national, rather than sectional, meetings were held west of the Alleghenies. ... The Harvard group was particularly adamant: Osgood, I remember father said, claimed he suffered from insomnia if he left Cambridge. ... Cole, the Secretary of the Society, was apparently willing to settle for a plan to more or less Balkanise it, whereas my father believed that the organisation should be national in scope.

On 27 February 1911 Bôcher wrote to Van Vleck (see [7]):-

I wrote this morning to Cole ... saying I would give my presidential address in Chicago. In doing this I yield my own judgment, which tells me that no useful purpose will be accomplished, to yours. You are on the ground and ought to know.

Van Vleck believed that his actions had [7]:-

... averted an open break in the American Mathematical Society.

In 1916 the University of Chicago celebrated its first 25 years and as part of the Quarter-Centennial Celebration they awarded Van Vleck an honorary doctorate on 6 June 1916. The citation reads:-

Of the American Mathematical Society sometime president, and editor of the Transactions; always wise counsellor and leader; creative mathematician and successful investigator in the theory of functions, and in the theories of differential and difference equations and of functional equations; for these eminent services in mathematics, and especially for your important researches concerning functional equations and analytic continued fractions.

On this occasion Van Vleck delivered an address on *Current tendencies of mathematical research.*

His interests are described in [1] as:-

Travelling and collecting Japanese art... His collection of these prints, numbering thousands of items, is very remarkable, and one of the major private collections.

His contributions are summed up in [3] as follows:-

Van Vleck was remembered by his peers as a cultured and distinguished scholar who did much to enrich the lives and careers of those who knew him.

We should note that the University of Wisconsin named a Hall for him on 13 May 1963, twenty years after his death.

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

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