During 1882-83 Miller attended Franklin Academy which was a part of the College of Lancaster. From there he went to Allentown, Pennsylvania, where he studied at Muhlenberg College for his baccalaureate which he received in 1887. Continuing his studies he was awarded a degree of Master of Arts by Muhlenberg College in 1890. By now Miller was 27 years of age, much older than one might expect, but this was just a consequence of having to support himself financially through his education. In fact he had been the principal of schools in Greeley, Kansas, in 1887-88 and professor of mathematics at Eureka College in Illinois from 1888 while he worked for his Master's Degree. When teaching was finished at Eureka College, Miller went to Johns Hopkins University and the University of Michigan to spend the summers of 1889 and 1890.
Cumberland University in Lebanon, Tennessee, offered a doctorate as a correspondence course. This did not require an original thesis, in fact it was not examined by thesis at all but was awarded on the basis of examinations taken on advanced courses. These courses could be studied as correspondence courses and this is exactly what Miller did. What is perhaps more surprising, he offered himself the same courses for a doctorate to students at Eureka College where he was professor of mathematics. He graduated with a doctorate from Cumberland University in 1892 and in the following year he left Eureka College to take up the position of instructor at the University of Michigan.
When he arrived at the University of Michigan, Miller was offered accommodation in Cole's home. He lived there for two years and this was perhaps the most significant event for his mathematical development for Cole was interested in the theory of groups and he soon had Miller totally fascinated by this topic. Miller spent the years from 1895 to 1897 in Europe attending lectures on group theory by Lie in Leipzig and Jordan in Paris. On his return to the United States Miller was appointed assistant professor at Cornell University. He held this position from 1897 until 1901 when he was appointed to Stanford University. In 1906 he moved from Stanford to the University of Illinois at Urbana-Champaign where he remained for the rest of his career. In 1909 he married Casandra Boggs from Urbana-Champaign but they had no children.
Miller worked mostly on group theory but he was also interested in the history of mathematics. Although interesting because it was done at an early stage, his work fails to show much depth. He wrote more than 800 articles over a period of 40 years about half at research level, the others aimed at school teachers. His collected works appear in five volumes: the first contains 62 papers which Miller published before 1900; the second contains 107 of the 147 papers he published during the years from 1900 to 1907; the third includes 89 of the 180 papers he published during the years 1908 to 1915; the fourth contains 98 of the 232 papers he published during the period 1916 to 1929.
Many of Miller's group theory papers enumerate the possible finite groups which satisfy given conditions such as: the prime factors which divide the order, the orders of two generating permutations and their product; the types of subgroups; or the degree of a representation as a permutation group. Several papers investigate groups generated by two elements satisfying given conditions. For example he considered groups generated by two elements of order three whose product is of order four or three or six. He also considered permutation groups of small degree, groups having a small number of conjugacy classes, multiply transitive groups, and characteristic subgroups of finite groups. He found the list of all possible groups of order 1909 to 1919 inclusive. Miller did not introduce new techniques to attack these group theory questions and one is tempted to say that he should have applied his undoubted skills to produce fewer yet more significant results.
His best historical papers are those which look at the history of group theory. He also wrote papers such as The founder of group theory, Primary facts in the history of mathematics, and Some thoughts on modern mathematical research. In addition Miller wrote a number of books: Determinants (1892); Historical introduction to mathematical literature (1916) and he co-authored Theory and application of finite groups (1916) with Blichfeldt and Dickson.
Miller became a member of the New York Mathematical Society in 1891, three years after it was founded. In 1894 the Society became the American Mathematical Society and Miller helped to organise the San Francisco West Coast section which was set up in May 1902. He was a member of the London Mathematical Society, the German Mathematical Society, and an honorary member of the Indian Mathematical Society.
He was elected to the American Academy of Arts and Sciences in 1919 and to the National Academy of Sciences (United States) in 1921. He was honoured with the award of a prize by the Cracow Academy of Sciences. He was an editor of the American Mathematical Monthly from 1909 to 1915 before it became associated with the Mathematical Association of America. He was the sixth President of the Mathematical Association of America from 1921-22 and gave his presidential address in 1922 calling attention to contradictions and inaccuracies in the mathematical literature. This was characteristic of him in his historical writings. In  this is presented in a positive light:-
He became a severe critic of historical methodology in mathematics and was zealous in rooting out error in conjecture or assumed fact. His letters in the David Eugene Smith collection at Columbia University offer ample evidence of the missionary zeal.In  the author criticises this aspect of Miller's work:-
His historical writings outside the theory of groups often depended on secondary sources and reflected an attitude overly concerned with pointing out error in published accounts.A bequest of one million dollars made to the University of Illinois (where he had taught for 25 years) on his death showed the skill he had in making financial investments. His wife had died in 1949 and he had no close relatives. He explained his motives (see :-
Everything I have I received from the university, and I want to repay my obligation.It is difficult to assess Miller's contributions to mathematics. Although these contributions do not seem today to have been particularly important, his contemporaries rated him highly as is evident from the fact that he was ranked tenth in a list of top American mathematicians for J M Cattell's American Men of Science.
Article by: J J O'Connor and E F Robertson
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