**James Pierpont**'s father was Cornelius Pierpont, a well-off businessman living in New Haven. The Pierpont family had lived for many generations in New Haven. One of the early members, also named James Pierpont (1659-1714), was a graduate of Harvard University and a founder of The Collegiate School of Connecticut in 1701 which shortly after was named Yale College. James, the subject of this biography, was one of his parents' four children. He had one older brother, one younger brother and a sister. He attended New Haven High School where he prepared for a college education. After graduating from high school, he entered Worcester Polytechnic Institute, taking courses there which would lead to him becoming a mechanical engineer. However, as he progressed he realised that it was pure mathematics which interested him far more than the practical aspects of engineering. He graduated with a B.S. in 1886 but already he had made plans to study pure mathematics at some of the top European universities.

Of course one has to realise that the college education that Pierpont had received in no way prepared him for doctoral studies so he had to go through the full range of undergraduate and postgraduate studies in Europe. He adopted the standard German experience of moving between various different universities but the majority of his first few years of study were spent at the University of Berlin. There he was taught by the leading mathematicians of the day including Lazarus Fuchs, Karl Weierstrass and Leopold Kronecker. However, it was Kronecker's algebraic school that influenced Pierpont most and this influence could clearly be seen in Pierpont's work for many years. The last few years of Pierpont's European training were spent at the University of Vienna where he undertook research for his doctorate. There he was advised by Gustav Ritter von Escherich and, from 1893 onwards, also by Leopold Gegenbauer who was appointed to Vienna in that year. Von Escherich (1849-1935), an expert on analytical geometry, had been appointed to the University of Vienna in 1884. Pierpont was awarded a doctorate in 1894 for his thesis *Zur Geschichte der Gleichung fünften Grades bis zum Jahre 1858*. He had enjoyed his six years in Europe and made many good friends such as Wilhelm Wirtinger who had habilitated in Vienna in 1890. Øystein Ore writes [3]:-

This period of studies was a very happy one, and in his later years Pierpont liked to reminisce about it, repeating often what a revelation the acquaintance with European mathematics and mathematicians had been to him.

After the award of his doctorate, Pierpont returned to the United States where he was appointed a Lecturer in Mathematics at Yale in 1894. He held this position for a year and then was appointed an Instructor in Mathematics in 1895. The American Mathematical Society held its first Colloquium at Buffalo in September 1896. Two series of lectures were given, one by Maxime Bôcher on *Linear Differential Equations, and their Application* and the other by Pierpont on* Galois's Theory of Equations*. Bôcher's lectures were not published, but those by Pierpont appeared in the *Annals of Mathematics* in 1900. In the autumn of 1896 Pierpont became an Assistant Professor at Yale, then in 1898 he was appointed as a full professor. The title of his Colloquium lectures is a good indication of Pierpont's main interests in the period up to his appointment to a full professorship. He published some nicely written articles, but there is little in them in the way of original results. He also wrote some good historical articles on the theory of equations such as *Lagrange's place in the theory of substitutions *(1894), and* Early history of Galois' theory of equations* (1897).

He attended the summer meeting of the German Mathematical Society at Munich in September 1899. From this time on he became interested in the theory of functions of a real variable and integration, publishing papers such as *On multiple integrals *(1905) and* On improper multiple integrals* (1906). Between these two papers his first book was published, the first volume of the two-volume text *Lectures on the Theory of Functions of Real Variables* (1905). He explains in the Preface what his motivation was in writing the book:-

The student of mathematics, on entering the graduate school of American universities, often has no inconsiderable knowledge of the methods and processes of the calculus. He knows how to differentiate and integrate complicated expressions, to evaluate indeterminate forms, to find maxima and minima, to differentiate a definite integral with respect to a parameter, etc. But no emphasis has been placed on the conditions under which these processes are valid. Great is his surprise to learn that they do not always lead to correct results. ... The problem therefore arises to examine more carefully the conditions under which the theorems and processes of the calculus are correct, and to extend as far as possible or useful the limits of their applicability.

G A Miller writes in a review [2]:-

... one feels that one is being led by a master of his subject and a sympathetic teacher, and these elements combined with the nature of the subject make the present work one of the most significant publications on pure mathematics that have ever appeared in this country.

Philip Jourdain, however, does have some criticisms in his review. For example [1]:-

I am here doubting the advisability of any mode of exposition which leads to the remark of Prof Pierpont: "It is too early to make the reader see the necessity of this step. ..." If it is ever advisable to teach mathematics in this way, it is by introducing the student by a path which can never be used for the discovery of any important truths.

However, on the whole, Jourdain too heaps praise on the book [1]:-

Its chief excellencies seem to be:

(1)The introduction of various current inaccurate statements as examples in criticism ...

(2)analytical representations of some apparently 'lawless' functions. ...

(3)examples of continuous functions lacking differential quotients at isolated points, -- functions often ignored by teachers ...

(4)the careful distinction between "any" and "every," which is often neglected.

The second volume of *Lectures on the Theory of Functions of Real Variables* was published in 1912. Pierpont writes in the Preface:-

The present volume has been written in the same spirit that animated the first. The author has not intended to write a treatise or a manual; he has aimed rather to reproduce his university lectures with necessary modifications, hoping that the freedom in the choice of subjects and in the manner of presentation allowable in a lecture room may prove helpful and stimulating to a larger audience.

Pierpont's next textbook was *Functions of a complex variable* (1914). He explains in the Preface what drove his choice of approach and topics:-

The present volume has arisen from lectures on the 'Theory of Functions of a Complex Variable' which the author has been accustomed to give to juniors, seniors, and graduate students at Yale University during the last twenty years. As these students often do not intend to specialise in mathematics, many topics which might properly find a place in a first course in the function theory have not been treated for example Riemann's surfaces. On the other hand the author, having in mind the needs of the students of applied mathematics, has dwelt at some length on the theory of linear differential equations, especially as regards the functions of Legendre, Laplace, Bessel, and Lamé. As a splendid application of the principles of the function theory and also on account of their intrinsic value, three chapters have been devoted to the elliptic functions.

These books were based on courses Pierpont taught at Yale, so it is clear that he derived pleasure from lecturing. Ore gives this description [3]:-

He enjoyed teaching and contact with the students, and he usually succeeded in transmitting some of his own enthusiasm to them. His lectures were forceful and clear and showed the impression of his wide knowledge of mathematics. At times this knowledge would throw him off on a tangent so that his lecture would end in fields far from the scheduled subject, to the delight of the enthusiastic students and to the grief of those whose interests were chained to the text of the day. He spent considerable time with the students, discussed and gave advice and helped them both mathematically and otherwise.

In addition to the American Mathematical Society Colloquium Lectures that he gave in Buffalo in 1896, Pierpont address the International Congress of Arts and Science in St Louis in September 1904 on the *History of Mathematics in the Nineteenth Century*, he addressed the American Mathematical Society summer meeting at Wellesley in 1921 on *Some mathematical aspects of the theory of relativity*, he gave the Gibbs Lecture in Kansas City in 1925 on *Some modern views of space*, he addressed the annual meeting at Nashville in 1927 on* Mathematical rigor, past and present*, he addressed the annual meeting at New York in 1928* On the motion of a rigid body in a space of constant curvature, *and the annual meeting at Berkeley in 1929 on *Non-Euclidean geometry, a retrospect*.

Among the service he gave to the American Mathematical Society, we note that he served on the Council three times: 1899-1901, 1927-1929, and 1931-1933. He was honoured by being elected Vice-President of the Society in 1905. He was awarded two honorary degrees: Yale University in 1899 and Clark University in 1909. In 1923 he was named Erastus L DeForest Professor of Mathematics at Yale.

To get a feeling for Pierpont as a man let us quote Ore again [3]:-

Pierpont enjoyed friends, company, and conversation. Walking was his only exercise. He loved a good discussion and when one came to see him it usually would not be long before one was drawn into some argument. Even if one at times would disagree with him, it was impossible not to admire his wide knowledge and memory. Personally he was always very modest about his own achievements. At times he could show a burst of temper, mostly in cases where matters of principle were involved, sometimes in meetings of the department or faculty.

Finally let us record some of Pierpont's more amusing aspects. For many years he held the record for borrowing the largest number of books from the Yale University library. The range of topics that interested him was enormous; science, literature, geography, biography, travel and languages to name just a few. He could write equally well with his left and right hand and, even more surprisingly, he could write upside-down - a useful skill when teaching students.

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