After completing his first degrees (for he had two!), McShane continued to study at Tulane for his Master's degree in mathematics while teaching mathematics as an instructor. It is noted in  that he:-
... once remarked that he had never regretted his decision to become a mathematician rather than an engineer.With the award of a Master's degree from Tulane in 1927, McShane went to the University of Chicago to undertake studies for his doctorate. However, financial difficulties caused him to interrupt his studies and take out the session 1928-29 when he taught at the University of Wichita to earn some money. Returning to Chicago, where his studies were supervised by Bliss, he completed work on his thesis and he was awarded his PhD in 1930.
McShane spent part of the following two years at Princeton, part at Ohio State, part at Harvard and part at Chicago. During this period he married Virginia Haun on 10 September 1931. They would have two daughters, Jennifer and Ginger, and one son, Neill, all of whom were highly talented in both music and mathematics.
Now 1932 was not a good year to be trying to find a university post in the United States due to the effects of the Depression. McShane and his wife spent the session 1932-33 at Göttingen, but if 1932 was a bad year in the United States, 1933 was certainly a bad year in Germany with Hitler coming to power and bringing in his first anti-Jewish legislation. Having seen the frightening start of the Nazi rise to power, the McShane family returned to the United States where McShane spent the next two years at Princeton.
In 1935 McShane was appointed to a professorship at the University of Virginia at Charlottesville, where, except for temporary appointments elsewhere, he was to remain for the rest of his life. One of these periods away was when he went to head the Ballistics Research Laboratory at the Aberdeen Proving Ground, Maryland in 1942-45 during the Second World War. His joint treatise Exterior Ballistics written during this period, and published in 1953, was considered the leading work on ballistics at the time.
In  a nice story is told regarding McShane during the McCarthy era. In the early 1950s United States senator Joseph R McCarthy whipped up strong feelings against communism. McShane had been asked to complete a questionnaire. One question asked:-
... whether he had ever been involved with organisations that had at any time advocated the violent overthrow of the U.S. government.It was quite a brave move for McShane to reply "yes", because he was an employee of the State of Virginia! At the University of Virginia this sense of humour added to his popularity with both staff and graduate students who liked :-
... his clear lectures, his amusing anecdotes, and his willingness to think about their problems.McShane is famous for his work in the calculus of variations, Moore-Smith theory of limits, the theory of the integral, stochastic differential equations, and ballistics. In fact he wrote three important books on integration, the first being Integration written in 1944 to provide a readable and clear introduction to Lebesgue integration for students. In 1953 he wrote Order preserving maps and integration processes which was :-
... an outgrowth of his search for a mathematically correct setting in which to treat the divergent integrals in quantum physics.In 1974, the year he retired and was made Professor Emeritus at Virginia, McShane published Stochastic calculus and stochastic models which again reflected his work on the mathematical setting for quantum mechanics. Then in 1983 he published his third book on integration, Unified integration which provides :-
... a theory of integrals with applications to physics.McShane was much involved with both the American Mathematical Society and with the Mathematical Association of America. He was American Mathematical Society Colloquium Lecturer in 1943, and later served the Society as President in 1959-60. He also served as President of the Mathematical Association of America in 1953-54. Among the honours which he received was the Award for Distinguished Service to Mathematics from the Mathematical Association of America in 1964. Another honour is the fact that the paper  belongs to a special volume of the Society for Industrial and Applied Mathematics Journal on Control and Optimization dedicated to him.
He is described in  as:-
... a deeply cultured man, with a flair for languages and a great variety of other interests. He was widely known for his warmth, generosity, and modesty, and for his fund of humorous and interesting anecdotes.
Article by: J J O'Connor and E F Robertson
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