# Cochell: The Early History of the Cornell Mathematics Department

6. CORNELL MATHEMATICS

AT THE END OF THE CENTURY

At the end of the nineteenth century, the Cornell Mathematics Department still counted some of the old guard among its faculty. Professor and Chair Lucien Wait celebrated his thirtieth year in the Department, and Professor George Jones his twenty-fourth. Two-thirds of the team of Oliver, Wait and Jones still guided the Department as it approached the twentieth century. A new Department was taking shape, however. In 1894, two new instructors were hired. John Irwin Hutchinson (1866-1935), a student working under the direction of Oskar Bolza (1857-1942) at the University of Chicago, finished his Chicago Ph.D in 1896, while Daniel Murray (1862-1934) came to Cornell with his Hopkins Ph.D. in hand. As mentioned above, Snyder returned fresh from his studies in Göttingen to take yet another new instructorship in 1895. The following year found Tanner back in Ithaca with a promotion to an assistant professorship, and 1897 brought George A. Miller (1863-1952) to an instructorship following a two-year study tour abroad. These "young lions" solidified the graduate program at Cornell as the nineteenth century ended and directed it as the twentieth century began.

The biggest star in this group of young faculty members was Virgil Snyder. In more than forty years at Cornell he published over eighty articles and books, while directing the Ph.D.'s of thirty-nine students, thirteen of whom were women. He also had national visibility, serving as Vice-President (1916) and President (1927-28) of the American Mathematical Society [20].

Relative to research, Snyder also excelled, although his approach was clearly grounded in the nineteenth century even as others moved beyond those techniques in the early decades of the new century. As a student, Arthur Coble characterized it, Snyder's early research began

. . . at a time when geometers were exploring the superstructures of their subject, particularly in space and hyperspace. By adding the radius of a sphere to its coefficients, Lie had defined a sphere by six homogeneous coordinates subject to a non-singular quadratic relation. This situation also occurs with the Plücker line-coordinates so that the parallel between line geometry in three-space and Lie's "Kugelgeometrie" was apparent. Snyder's doctoral dissertation (Göttingen, 1895) was concerned with linear complexes of spheres. Of twenty-one papers he published in the next ten years, twelve were concerned with the metric side of this parallel and dealt with annular, tubular, and developable surfaces, their asymptotic lines, and lines of curvature, or with the development of collateral algebra. [20,468]

In the middle period of his research, Snyder published a series of articles on surfaces invariant under infinite discontinuous groups of birational transformations. Many of these articles were done in collaboration with his student and colleague, Francis Robert Sharpe (1870-1948). Snyder's culminating work appeared - under the title Selected Topics in Algebraic Geometry - in the Bulletins No. 63 (1928) and 96 (1934) of the National Research Council of which he was chair. Coble described these accomplishments this way: "Of this digest of journal articles up to the date of publication he personally wrote about one quarter of the text and he took on with energy and enthusiasm the entire responsibility for editing and publishing the volume" [20,470]. The supplementary volume was "almost entirely his own work" [20,470]. Parshall and Rowe summarized Snyder's overall research status within the American mathematical research community: "Up until the 1920s, Snyder's prolific output and his talents as a teacher made him, together with Frank Morley of Johns Hopkins, one of the most influential algebraic geometers in the nation. Together with Henry White, in fact, Snyder emerged as a principal heir to Klein's geometric legacy" [46,218].

In addition to his extensive body of research in algebraic geometry, Snyder also actively engaged in the education of undergraduates. In the spirit of the Oliver-Wait-Jones department Snyder coauthored a Treatise on Differential Calculus (1898) with McMahon, Differential and Integral Calculus (1902) with Hutchinson, and Plane and Solid Geometry (1911) with Tanner. All three of these books were used in the lower level courses at Cornell. and elsewhere.

Other members of the Cornell Mathematics Department also contributed to the newly emerging American mathematical scene. John Tanner was active in both the American Association for the Advancement of Science (AAAS) and the American Mathematical Society (AMS), becoming treasurer of the later in 1907. He also received recognition in American Men (and Women) of Science when he earned a star for his research, a distinction reserved for the top people in each field.

John Hutchinson extended the work of his advisor, Bolza, first in his thesis "On the Reduction of a Hyperelliptic Function to Elliptic Functions by a Transformation of the Second Degree" and later in continued work in elliptic and hyperelliptic function theory [46,393-394]. At Cornell, Hutchinson produced Ph.D. students in this and related areas in the early years of the twentieth-century. During his career he also published numerous articles and gave many talks at meetings of the AAAS and the AMS, in addition to serving as Associate Editor of the Transactions of the American Mathematical Society and as Vice-President of the AMS in 1910 [28,186]. Like Snyder, he earned a star in American Men (and Women) of Science.

The oldest member of the new generation of Cornell mathematicians, James McMahon was, like his colleagues, active in the AAAS and the AMS, holding the AAAS posts of Secretary (1897), General Secretary (1898), and Vice-President (1901). Not as prolific as Snyder, he nevertheless, published, presented his work personally at meetings [28,186], and was starred in American Men (and Women) of Science.

Finally, George Miller (1863-1951) worked actively in the theory of groups and directed the studies of two Ph.D. students during his four-year tenure at Cornell. MIller had received his Ph.D from Cumberland University in Tennessee in 1893, and while the quality of that degree may be open to given the state of graduate education in America at the time, Miller augmented his education by traveling to Europe in 1895-97. He spent the 1895-96 year at the University of Leipzig, where he may have heard the lectures of Sophus Lie (1842-1899), and the 1896-97 year at the University of Paris, where Camille Jordan (1838-1922) was lecturing on group theory. Miller brought this mathematics to the faculty and students at Cornell and used it to further his own research agenda. In 1900, in fact, he received the International Prize in Mathematics from the Cracow Academy of Sciences for his work on group theory. This prize carried a cash award of $260, and in Miller's words ". . . would appear to be the first prize in pure mathematics awarded by a foreign academy to an American" [52]. Miller's work also earned him one of the coveted stars of American Men (and Women) of Science.

Miller left Cornell for Standord in 1901 and eventually moved to the University of Illinois. In all, he published well over one hundred articles and books during his career. Miller was also active in 1) the AAAS, serving as its Secretary (1907-1912); 2) the AMS, assuming leadership roles in both the San Francisco and Chicago Sections; and 3) the Mathematical Association of America where he held the vice-presidency in 1916.

This young and energetic faculty worked hard to maintain and improve the graduate program at Cornell. On paper, at least, the graduate courses they offered looked much like those of the American leader, the University of Chicago. Consider for example, Chicago's listing of advanced courses for the academic year 1898-1899 as published in the Bulletin of the AMS:

The University of Chicago. By Professor Moore: Seminar devoted to research work, especially in groups, algebra, and arithmetic; Transfinite totalities; Elliptic modular functions; Abstract groups; Projective geometry. By Professor Bolza: Elliptic functions; Hyperelliptic functions; Advanced integral calculus. By Associate Professor Maschke: Seminardevoted to research work, especially in linear homogeneous substitution groups; Theory of invariants; Functions of a complex variable; Modern analytic geometry; Higher plane curves. By Assistant Professor Young: Mathematical pedagogy; Theory of equations. By Dr. Boyd: Differential equations and applications. By Dr. Hancock: Calculus of variations; Theory of equations. By Dr. Slaught: Advanced integral calculus; Solid analytics. By Dr. Laves: Analytical mechanics. By Dr. Miller (of Cornell University): Seminar in permutation groups . [16,490]

Compare this to the listing given for Cornell in that same year:

Cornell University. By Professor Wait: Advanced analytic geometry; Advanced differential calculus. By Professor Jones: Higher algebra and trigonometry; Probabilities, etc. By Professor McMahon: Higher plane curves; Quaternions; Potential, and spherical harmonics; Mathematical theory of sound. By Professor Tanner: Binary quantics; Theoretical mechanics; German readings. By Dr. Murray: Differential equations; Finite differences; Astronomy. By Dr. Hutchinson: Advanced integral calculus; Elliptic functions; Surface and twisted curves. By Dr. Snyder: Projective geometry; General function theory; Line geometries. By Dr. Miller: Substitution groups; Continuous groups; Theory of numbers. [16,557]

Were these advanced courses at Cornell "modern" by the standards of the day? The course descriptions would make them seem so [26,135-138]. For example, Miller described the content of his course, Theory of Groups of a Finite Order, this way: "Some of the recent literature on these subjects is examined and the difficulties of problems that await solution are pointed out. The later part of the course is devoted to applications, the Galois theory of equations receiving the most attention" [26,136]. HIs complementary course on "Theory of Groups of an Infinite Order" involved "a study of the theory of Lie's continuous groups and their application to the theory of differential equations" [26,137], making it fairly up-to-date for 1900. Hutchinson underscored his awareness of recent work in the descriptions of both his course on the Calculus of Variations, where he recommended Adolph Kneser's recent "work for reference and collateral reading" [26,137], and in his course on the theory of functions, where he included "Applications to geometry, particularly to hyperelliptic surfaces and the generalized Kummer surface" [26,138]. Similar indications may be found in courses given by Tanner and Snyder as well. At the turn of the twentieth century, the Cornell Mathematics Department was doing a conscientious job of exposing its students to the standard-setting research being done abroad.

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