*Written on the occasion of our Centennial in 1999*

This year the Department of Mathematics at the University of Pannsylvania celebrates its centennial as a separate intellectual discipline. The history of mathematics at Penn goes back to the earliest days of the institution, but in those early days considerable overlap existed with what we now recognize as the separate intellectual disciplines of physics, astronomy, and philosophy.

On November 13, 1749, twenty-four trustees were constituted as governors of the institution that eventually became the University of Pennsylvania. The institution began as an Academy that consisted of three Schools: Latin, English, and Mathematics which provided pre-college training. A separate Charitable School was opened for children of poor citizens.

On December 17, 1750 the trustees appointed the first master in mathematics. The event is recorded in the minutes this way: "Mr. Theophilus Grew having offered himself as a Master in the Academy to teach Writing, Arithmetick, Merchants Accounts, Algebra, Astronomy, Navigation, and all other branches of the Mathematicks; it is ordered that he be received as such at the rate of one hundred and twenty five pounds a year, his service to commence on the seventh day of January next." So in some sense, 1999 is the 249th birthday of mathematics at Penn!

Five years later on June 10, the trustees received a charter creating the College and soon after Theoliphus Grew was appointed Mathematical Professor in the College. From that time on, the highest ranking faculty position in the College was that of professor.

Grew was simultaneously professor in the College and master of the Mathematics School in the Academy. His textbook, *The Uses of Globes*, was the first textbook to be published by any member of the faculty. When Grew died in 1759, the master of the English School, Thomas Pratt, taught mathematics without the title of professor of mathematics until the Reverend Hugh Williamson was appointed professor in 1761. Williamson had been one of the first seven students receiving degrees from the College at the first commencement on May 17, 1757. Williamson did not stay long, asking to be relieved of his duties in late 1763 to serve as one of the delegates from North Carolina to the Constitutional Convention that met in Philadelphia. He later became a member of the first Congress that met in New York in 1787.

After Williamson's departure Thomas Pratt again added mathematics instruction to his duties. Thomas Dugan was appointed master of the Mathematics School on January 21, 1766. Dugan held his position until April 18, 1769. From then until 1773, there is no record of a master of the mathematics school or a professor of mathematics. It appears that instruction in mathematics during this gap was covered by various tutors and masters of the English School. In 1773 James Cannon was appointed professor of mathematics, and from then on there has always been at least one professor of mathematics at Penn. Indeed, the total number of professors of mathematics from 1755 to 1899, a period of 144 years, was ten. In the 100 years since 1899, there have been sixty professors of mathematics at Penn.

Minutes of meetings of the College faculty tell the story of the time. In the early 1800s, many faculty meetings were devoted in part or entirely to disciplinary issues. The institution had not begun to flourish in those days and the total number of students seldom exceeded fifty. Both discipline and attendance were problems. According to minutes of the faculty meetings, infractions included speaking out of turn, stomping when entering class, talking during chapel, throwing snowballs through the College door, leaving the Math Room and not returning, "disturbing the math recitation by speaking when at his seat so as to disturb the exercises," scuffling, "disturbing the exercises immediately after silence had been directed by the Professor," and "indecorous conduct in the Math Room in attempting to cast ridicule upon the direction of the Professor by not recalling his exercises from the Board." In 1828, the Faculty took the remarkable step of appealing to the trustees for assistance in securing discipline. The trustees, on the other hand, regarded the problem to be a lack of devotion and ability on the part of the faculty. Their solution was to adopt a resolution terminating all existing professorships in the Department of Arts at the end of the spring term in 1828!

Against this background of the early University of Pennsylvania and instruction in mathematics there, we have the general background of mathematics in the civilized world. Flourishing in the Renaissance, mathematics became a European discipline during the 17th, 18th, and 19th centuries. The 18th century saw the flowering of the work of the Bernoulli family, of Leonhard Euler--an all time giant of mathematics, of Joseph Lagrange, Adrien M. Legendre, and Pierre S. Laplace. All these people made contributions to mathematics: that are still important, used, and taught today. From their viewpoint, mathematics in the North American colonies of England hardly existed. No important mathematics journal was published in the United States before 1876 and English was not a main scientific language. Nevertheless, the colleges and universities in colonial America always had mathematics teachers on their faculties.

In Europe, the 19th century marked an explosion of mathematics as a subject. This was the age when physics came to prominence with the gradual understanding of electrical and magnetic phenomena, subjects impossible to understand without more advanced knowledge than simply calculus. The spur of the applied mathematicians in England (Green and Stokes), the burgeoning experimental data accumulated on the continent and in England (Ampere, Oerstead, Ohm, Faraday), and the natural growth of abstraction at the hands of Hamilton, Jacobi, Dirichlet, Kummer, Weirstrass, and the two giants Gauss and Riemann, provided a spurt of achievement such as had not occurred for almost two millennia. The German universities had instituted the notions of graduate school (in the Arts and Sciences) and the Ph.D. degree. Advanced training was available only in Europe, and all who taught in America had spent some time studying and obtaining advanced degrees there.

This situation began to change when, in 1876, Johns Hopkins University founded the first Graduate School of Arts and Sciences and instituted the granting of Ph.D. degrees for the first time in America. They imported the famous British mathematician J.J. Sylvester to head their mathematics graduate program. During his roughly ten years in the United States, Sylvester founded the American Journal of Mathematics (still an active and high quality journal), put Hopkins on the intellectual map, and provided the impetus for advanced graduate mathematical training to be established at universities in the USA.

Back at the University of Pennsylvania, the trustees, in March of 1881, approved the formation of a Faculty of Philosophy for instruction leading to the degree of Doctor of Philosophy. In that month there also occurred one of the most significant events in the history of the department. Thomas A. Scott, who had become President of the Pennsylvania Railroad Company, wrote to the trustees. "My Dear Sir: I want to present to the University of Pennsylvania fifty thousand dollars of 6% Bonds to endow a chair of Mathematics in the Arts Department as I understand help is needed for a chair of this character. May you please deliver the Bonds to the proper authorities of the University to be used as indicated. If you will call or send to this office, Mr. Barclay will hand the bonds to you or your representative. Trusting my action in this premises may be of use to the coming young men of the University, I am Very truly yours, Thomas A. Scott."

This remarkable gift for mathematics and instruction in mathematics established the fourth chair in the University and only the second in the Arts and Sciences. On June 7, 1881, just a few days after Scott's death, the trustees designated Scott's gift as the Thomas A. Scott Professorship in Mathematics. Ezra Otis Kendall, LLD, the then current Professor of Mathematics, became the first holder of the Scott Chair. Subsequent holders of this distinguished professorship in mathematics with their dates of appointment are Edwin S. Crawley (1899), George H. Hallett (1933), John R. Kline (1941), Hans A. Rademacher (1956), Eugenio Calabi (1967), Shmuel Weinberger (1994), and Herbert Wilf (1998).

The first appointments were made in November of 1882 to the Faculty of Philosophy. Thirteen professors representing thirteen subject areas were named, with E. Otis Kendall, LLD, and Scott Professor of Mathematics, as Dean. The first meeting of that faculty on December 8, 1882 is regarded as the beginning of the Graduate School at Penn. The first Ph.D. in Mathematics went to Edwin S. Crawley in 1892. In that same year, Kendall was also named to the newly created Flower Chair of Astronomy but resigned from the Flower Chair three years later, so that it could be awarded to Professor Doolittle who became Flower Professor of Astronomy and professor of mathematics. In 1896, Kendall retired, but he retained the title of the Scott Professor of Mathematics until his death in 1899. His death set in motion several changes which we now recognize as the beginning of the Department of Mathematics at Penn as a separate intellectual discipline. Crawley, our first Ph. D., was appointed to the Scott Chair and became chairman of the department. Professor Doolittle dropped the title of professor of mathematics, retaining the Flower Professorship in Astronomy. At this point, mathematics had achieved final independence from other disciplines such as astronomy, physics, and moral philosophy.

Women and minorities were welcomed into the Ph.D. program at Penn from its earliest stages. The first woman Ph.D. was Roxana Hayward Vivian in 1901. The first African-American mathematician to receive a Ph.D. at Penn -- and only the second in the US -- was Dudley Weldon Woodard in 1928. William Waldron Schieffelin Claytor (1933) was the second African-American mathematician to receive a Ph.D. at Penn and the third in the US. The Department's Ph.D. program has been very active over the years. The list of Ph.D.'s now totals 320.

The notion of an undergraduate major was not introduced until 1914. In September of that year the College adopted "The New Curriculum," which was quite different from "The Old Curriculum." One was that students in the new curriculum would complete nine units of work in one of fifteen "Major Subjects" including mathematics. In the old curriculum, students completed nine units of "Group Work" in two or three of twenty-three subjects including mathematics. Another substantial change was in the degree awarded. In the old curriculum, students who presented an appropriate study of Latin and Greek received a Bachelor of Arts degree, whereas those who completed their required foreign language in two languages from Latin, French, German, Italian, and Spanish received the degree of Bachelor of Science. In the New Curriculum of the College, the emphasis on Greek was dropped and all students received the Bachelor of Arts degree.

On February 21, 1896, the Class of 1880 presented the University with $1000 to endow a prize for the best paper presented by "a candidate for admission to the course in arts and sciences" in a special examination in mathematics. The examination was described in the gift documents as covering algebra through quadratics and plane geometry. Over the years the examination was opened to all freshmen undergraduates in any school of the university -- indeed, in some years it was called the Freshman Entrance Prize. The content has evolved to keep pace with the knowledge expected of an incoming student. The initial exam prize of $50 -- all of the annual income from the endowment -- was awarded in 1896. Tuition for a full academic year of 37 weeks was $160 at that time so this was a very substantial prize. There is a spirited competition for this prize each year and we have a long list of 1880 Exam prize winners, many of whom have gone on to very distinguished careers in many different disciplines.

On the world stage, mathematics continued at an accelerating pace both in fecundity of thought and in its applications to the study of natural phenomena. This was the time of Henri Poincaré, who not only founded the subject of topology but did research of extreme profundity in celestial mechanics. It was also the time of David Hilbert, who made so many different contributions to the whole landscape of mathematics that one can not escape his name today anywhere in mathematics. Added to this was the gradual mathematization of physics (the Maxwell equations are example enough), of chemistry, and the crisis of abstract thought caused by special relativity. Mathematics assumed its current essential role in Western thought by the beginning years of the 20th century.

In the United States, three institutions took a commanding lead in the establishment and cultivation of mathematics as a living intellectual entity: the University of Chicago, Harvard, and Princeton. They established their dominance by importing European mathematicians of stature to be on and lead their faculties, by searching out and educating a cadre of coming leaders in mathematics, and by constantly promulgating original work in mathematics. Most other institutions (even those of prominence) did not insist on this mode of excellence until the 1920s.

The decade between 1930 and 1940 was tumultuous for science, mathematics, and intellectual life in general in America. The political climate in Europe forced out a substantial portion of the scientific and artistic elite, and with their emigration to America the center of intellectual life shifted to the United States. Almost overnight, the United States became the leading country in practically every discipline and certainly so in mathematics.

Hans Rademacher, one of the European émigrés, was attracted to Penn. He left Germany in 1934 and came to Penn in part because of its situation in Quaker Philadelphia. A distinguished mathematician famous for his work in analytic number theory, the theory of functions of a real variable, quantum theory, and mathematical genetics, he was also a kind and charming person. In the mid-1960s, a group of mathematicians and scientists "wishing to honor Dr. Rademacher in his lifetime by associating his name creatively with the future of mathematics" provided seed funding for the creation of The Hans A. Rademacher Instructorships at the University. The University also honored Rademacher at his retirement in 1962 by conferring on him an honorary Doctor of Science degree. The citation noted not only his scientific contributions, but also the "great charm and winsome tolerance of human frailties that endeared him to his students."

From the end of World War II to the mid-1950s, many universities took advantage of the general ferment in that period to strengthen their science faculties. In mathematics, Penn was slow to react. Nonetheless, a number of excellent Ph.D.'s were produced at Penn in that period. With the arrival of M. Gerstenhaber (1953), I.N. Herstein (1953), and C.T. Yang (1956), things began to change. For example, the Department had practically no resources for visiting speakers; so, Gerstenhaber and Herstein became contestants on a local radio quiz show. The $25 they won per show established a small visitors' fund. Moreover, in 1954, Gerstenhaber and Herstein were awarded the first NSF grant in mathematics at the University. Gerstenhaber remains on the faculty to this day, while the late Herstein eventually wound up at Chicago.

Realizing that Penn's Department had fallen behind, Gerstenhaber and Yang pushed hard for a modernization of outlook in the Department. They had small success until they found an ally in 1960 in the new Provost, David Goddard. Goddard was a distinguished botanist who felt that his own department, biology, and mathematics were important departments that needed modernizing and up-grading. At the behest of Gerstenhaber and Yang, he appointed the late Oscar Goldman as chairman of the department in 1962. Goldman's charge was deceptively simple: to bring mathematics at Penn to modern and high quality status. Goldman had wide powers; but, as a check on quality and to guard against mistakes, Goddard consulted regularly with two of his colleagues from the National Academy of Sciences. These were Saunders MacLane of the University of Chicago and Donald C. Spencer of Princeton -- both internationally recognized mathematicians.

Goldman's idea was to use the same method he had successfully used at Brandeis: to build in the three main areas of current mathematics (algebra, analysis, and geometry/topology) by attracting distinguished senior people in the full power of their careers and by appointing a cadre of talented younger people who would grow into leadership in the subject. Algebra was represented by Goldman and Gerstenhaber while Yang was a topologist, so Goldman recognized that his first priorities at the senior level were in analysis and geometry. He attracted R.V. Kadison from Columbia (an analyst) and E. Calabi (a geometer) from Minnesota. Calabi was appointed to the Scott chair, while Provost Goddard resigned his own chair -- the Gustave C. Kuemmerle Professorship -- in favor of Kadison. Calabi was elected to the National Academy of Sciences in 1982. Kadison, still an active member of the Department, was elected Foreign Member of the Royal Danish Academy of Sciences and Letters (1974), Member of the Norwegian Academy of Sciences and Letters (1986), and Member of the National Academy of Sciences (1996).

With the appointments of Calabi and Kadison, Penn began its contemporary period of mathematics. The record has been one of growth in strength, stature, and power as a fully modern mathematics department. Penn's contemporary mathematics department ranks very high on the national scene. In 1985, for example, Penn was the lead institution in the creation of the East Regional Geometry Festival, now a major annual event rotating among Penn, Courant Institute, Duke, University of Maryland, University of North Carolina, and SUNY, Stony Brook. In December, 1988, the department sponsored the first major US-USSR mathematics conference to be held in the United States in modern times. And during one recent six-year period, the number of Sloan Foundation Research Fellowships held by junior faculty in the Department was the largest of any university in the country.

There is great activity, much research, and an attention to detail and quality in the important tasks of training and educating undergraduate and graduate students. Over one hundred mathematicians per year from all over the world visit and speak in the department. Students from every part of the University, at all levels, take courses offered by the department and many earn dual degrees, or have double majors, with their second program in Engineering, the Wharton School, or the other fields in arts and sciences. In the last 35 years, 187 Ph.D.'s were awarded by the department compared with 133 Ph.D.'s between 1892 through 1964.

As befits an international discipline, the department and its student body form an internationally oriented, heterogeneous (by country of origin), interactive group. All are excited about the enterprise in which they are engaged and they engage it with gusto. With the continued support of Penn's administration, the Department of Mathematics will continue to gain in strength and stature as is appropriate for a distinguished university.