Returning to the United States, Halsted was appointed a tutor at Princeton, a post he held between 1879 and 1881, then from 1881 he was an instructor in postgraduate mathematics at Princeton, continuing until 1884. He communicated his enthusiasm, particularly for non-euclidean geometry, to a small group of undergraduates at Princeton and in particular to Fine, who had a natural bent for questions of a logical order. In 1884 he accepted a professorship at the University of Texas at Austin. The state constitution of 1876 had provided an endowment and, five years later, the state legislature established a campus at Austin, Texas. The University of Texas at Austin opened in 1883, and Halsted was appointed to the Department of Pure and Applied Mathematics there in only the second year of operation of the University. He taught there for nineteen years, and during that time influenced a number of students who were to play major roles in the development of American mathematics. L E Dickson was an undergraduate and worked for his M.S. (awarded in 1894) under Halsted's supervision. Robert Moore entered the University of Texas in 1898 and there he took courses by Halsted who turned him into an enthusiast for mathematical research, even in his undergraduate course. His teaching methods, which we say more about below, were always directed towards students learning by guided research.
There are a number of anecdotes showing Halsted's eccentric nature given in . However, despite being somewhat eccentric, yet he was an inspiring professor :-
Student reminiscences picture Halsted as one of the more colourful professors in the university's early history and a popular speaker on and off campus whether talking about his travels to Germany, Mexico, Japan, and Hungary or about religion or science or about geometry.In 1886 Halsted married Margaret Swearingen in Austin; they had three children. His main interests were the foundations of geometry and he introduced non-euclidean geometry into the United States, both through his own research and writings as well as by his many important translations. Halsted gave commentaries on the work of Lobachevsky, Bolyai, Saccheri and Poincaré and made translations of their works into English. His work on the foundations of geometry led him to publish Demonstration of Descartes's theorem and Euler's theorem in the Annals of Mathematics in 1885, the year after he arrived at Austin, and then, in the same journal, Klein's Evanston lectures in 1893.
Lewis describes his teaching methods in :-
In a classroom of freshmen one of his main purposes was to challenge what he regarded as the ill founded notions that pervaded the teaching of geometry. he could, for example, always count on a student in such a class giving him the common but not mathematically satisfying definition of a straight line as the shortest distance between two points. His criticism of such a definition could be the starting point for a discussion of the fundamentals of geometry. his teaching method may have been one of the inspirations for the Moore method of teaching that his student R L Moore later developed with great success.Halsted was always outspoken, even in situations where it was clear that what he said would land him in trouble. He could criticise the students he taught with little fear of come back, and his colleagues came in for similar criticism which one would have to say was usually deserved. However when it comes to criticising university administrations one should be more careful - Halsted was not. The University of Texas, like all state universities, was governed by a board of regents. When Halsted criticised these men as being political appointments he had gone too far. He was dismissed from his post in 1902.
From 1903 he held posts at St John's College, Annapolis, Maryland (1903), then Kenyon College, Gambier, Ohio (1903-1906). One might have thought that his experience at Texas would have made Halsted a bit more careful of his actions but this seems not to be so. At Kenyon College he was appalled at the student initiations which took place there, with bullying being an accepted way of life for the students. He spoke out strongly against these practices which made him unpopular in all quarters. These sorts of things were a way of life in certain establishments and people did not take kindly to an outspoken person like Halsted criticising them. Finally he went to Colorado State College of Education, Greeley in 1906 (this is now the University of Northern Colorado at Greeley). Again Halsted was not prepared to turn a blind eye to things he thought were wrong and again he was very outspoken, What did he find to attack this time? Well this time his charge was certainly a serious one for he stated very publicly that the college was being mismanaged :-
This last controversy led to the college administration being investigated by a grand jury and by two committees appointed by the state governor. Halsted spent most of the last years of his life continuing with his mathematical writing while working as an electrician in the family-run electricity supply store that he had established in Greeley.When Halsted completed the Princeton University Biographical Questionnaire he wrote:-
I am working as an electrician as there is nothing in cultivating vacant lots.He certainly felt badly treated by the mathematical community.
Writing in , H S Tropp says of Halsted:-
In the period when American mathematics had few distinguished names, the eccentric and sometimes spectacular Halsted established himself as an internationally known scholar, creative teacher and promoter and popularizer of mathematics.His other main interest was in mathematical education and, as a mathematics educator, he criticised the careless way that mathematics was presented in the textbooks of the time. He contributed over ninety article to the American Mathematical Monthly and wrote many biographies of mathematicians such as Lambert, Farkas Bolyai, Lobachevsky, De Morgan, Sylvester, Chebyshev, Cayley, Hoüel and Klein.
Article by: J J O'Connor and E F Robertson