**Cyrus Colton MacDuffee**'s parents were Cyrus Thompson MacDuffee (1842-1917) and Elisabeth Louise Sewirr (1856-1928). Cyrus Thompson MacDuffee was born in Mcduffietown, Seneca, New York, on 11 February 1842. His parents were Harmon Harrison Mcduffee and Margaret S Warne. His first wife was Kate M Schwartz, his second wife was Anna Hubbard, then he married Elisabeth Louise Sewirr, the mother of Cyrus Colton MacDuffee. Cyrus Thompson MacDuffee died in Oneida, New York, on 14 July 1917.

MacDuffee was educated in New York and went to Colgate University to study for his first degree. This University had been founded in 1820 as the Baptist Education Society of the State of New York, and was known as the Hamilton Literary and Theological Institution from 1823 to 1846. It was then named Madison University until given the name Colgate University in 1890, named after the soap manufacturer William Colgate. MacDuffee graduated from Colgate University with a B.S. degree in 1917 and then went to Chicago University to study for a Master's degree. He was awarded an S.M. by Chicago in 1920, continuing to study there for his doctorate advised by Leonard Eugene Dickson. He submitted his 16-page thesis *Invariantive Characterizations of Linear Algebras with the Associative Law Not Assumed* in 1921 and was awarded a Ph.D. in 1922. He published his thesis in the *Transactions* of the American Mathematical Society in 1922.

On 7 September 1921, MacDuffee married Mary Augusta Bean; they had four children, Robert Colton MacDuffee, Fredric Dearborn MacDuffee, Mary Elizabeth MacDuffee, and Helen Sewirr MacDuffee. MacDuffee was appointed as an instructor at Princeton University in 1922 and taught there for three years. During these years, he was promoted to assistant professor in 1924. While at Princeton he published a number of papers including: *On transformable systems and covariants of algebraic forms* (1923), *On covariants of linear algebras* (1924), *The nullity of a matrix relative to a field* (1925) and *On the complete independence of the functional equations of involution* (1925).

MacDuffee left Princeton in 1925 and was appointed as an assistant professor at Ohio State University. He was promoted to associate professor at Ohio State in 1929 and then to full professor there in 1933. He published his classic book *The Theory of Matrices* in the same year he became full professor.

He continued to publish on rings and algebras with papers such as *A correspondence between matrices and quadratic ideals* (1927), *An introduction to the theory of ideals in linear associative algebras *(1929), *The discriminant matrix of a semi-simple algebra* (1931), and *Matrices with elements in a principal ideal ring* (1933). In 1933, in a joint paper with Claiborne G Latimer, he published *A Correspondence Between Classes of Ideals and Classes of Matrices* in the *Annals of Mathematics*. This paper contains the theorem which today is known as the Latimer-MacDuffee theorem. He made a considerable effort to build the graduate school at Ohio during his years there and during the first half of the 1930s he was thesis advisor for six students who were undertaking research for a Ph.D.

In 1935 MacDuffee moved to the University of Wisconsin where he was appointed as a professor. He spent the rest of his career at Wisconsin becoming a major figure in the American mathematical scene. He gave long service to the Mathematical Association of America. He was vice president in 1942-44, president during 1945-1946 and served on the board of governors as an ex-president from 1947 to 1952. He delivered the address *An Objective in Education* to a joint meeting of the Wisconsin Section of the Mathematical Association of America, the Mathematics Section of the Wisconsin Education Association and the Mathematics Club of Milwaukee at the Milwaukee State Teachers College on 5 May 1945. He gave the address *The Scholar in a Scientific World* as retiring president in January 1948.

When his period on the board of governors of the Mathematical Association of America came to an end he wrote the following letter to the Secretary-Treasurer Harry M Gehman:

*Dear Gehman:*

*Thank you for your nice letter kicking me off the Board of Governors. I had figured that it was about due. As Julius Caesar once remarked, tempus sure does fugit and it doesn't seem long ago that I attended my first meeting of the Board. The members all seemed very ancient to me then, but now I undoubtedly seem ancient to the new members. I trust that you will have many re-elections to the office of Secretary-Treasurer until the newly elected members will say to one another, "Do you see that old man over there with the cane and the long beard? That is Harry Gehman!"*

*Sincerely*,

*Mac*

In 1945 MacDuffee succeeded W D Cairns on the War Policy Committee. He was also a member of the National Research Council and served as secretary of the American Association for the Advancement of Science. He delivered the address *Mathematics curriculum in perspective* to a symposium on mathematics instruction run by the American Association for the Advancement of Science at Indianapolis on 27 December 1957.

In addition, he was also a very active member of the American Mathematical Society serving on its council as well as taking on the role of editor of the *Transactions*. In 1952 he succeeded R E Langer as chairman of the Mathematics department at the University of Wisconsin.

We have already mentioned MacDuffee's 1933 book *The Theory of Matrices*. This was the first of four classic books that MacDuffee wrote, the other three being *An introduction to abstract algebra* (1940), *Vectors and matrices* (1943) and *Theory of equations* (1954).

The extracts from MacDuffee's addresses give a good indication of his views on the teaching of mathematics. Because he was so involved in all aspects of teaching, it seems fitting to end this biography by quoting from [9] part of his description of a course he gave at the University of Wisconsin:-

The course which I have been giving at Wisconsin for the last couple of years is still entitled the Theory of Equations, but might more properly be called the Theory of Polynomials. This approach seems to unify the somewhat scattered topics in the theory of equations, and to give a deeper insight into the subject which is particularly valuable to those who go on in algebra and to those who contemplate teaching algebra.

You may not agree with me in bringing in about a week of the theory of numbers, but I have found it desirable, and after all the only way to check a pedagogical theory is to try it out. Only a little of the theory of numbers is required, the definitions of primes and units, scales of notation, the greatest common divisor algorithm, and the unique factorization theorem. The ideas are here presented in their simplest form free of computational difficulties.

Later the corresponding concepts and theorems must be proved for polynomials. Just as a matter of experience I have found that students have trouble with this topic if it is first presented to them by way of polynomials. The introduction of this bit of number theory seems necessary in order to teach the polynomial theory regardless of its intrinsic interest. After this bit of number theory it is easy to attack the problem of finding the integral solutions of an equation having integral coefficients, and the rational solutions of an equation having rational coefficients.

Let us consider for a moment the theorem that if an equation with integral coefficients has a rational solution, when this solution is expressed in lowest terms the numerator is a divisor of the constant term of the equation. The proof depends upon the theorem in number theory that if a number divides a product and is relatively prime to one of the factors, it must divide the other factor. This students are ordinarily asked to accept as obvious, but I have seen some able students quite disturbed by it. It is proved in our bit of number theory and students having had this week of number theory really seem to understand the proof.

The plan which I have been following is to start with polynomials over the rational field, later take up polynomials over the real field, and finally polynomials over the complex field. It is well to call attention to the properties of a field, but it is not necessary to call them postulates or to introduce the notion of an abstract field. Other fields, if introduced at all, are postponed until late in the course.

The high points in the theory of polynomials are the Euclid algorithm, the unique factorization theorem, the representation of one polynomial as a polynomial in powers of a second, and the properties of the derivative in regard to multiple zeros. All of these results hold for every coefficient field but can first be introduced for polynomials over the rational field. At this point the decomposition of a rational function into a sum of partial fractions can be rigorously established. The students are familiar with the process from integral calculus and some of them are astonished when you point out that the universality of the method had not been proved to them.

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