EPADEL:A Semisesquicentennial History, 1926-2000

Table 3.5.1 lists the invited lectures at the annual meetings of the Philadelphia Section from the third meeting, held in \(1928\) at the University of Pennsylvania, to the seventh, at Swarthmore College in \(1932\text{.}\) We analyze the lectures in two different ways, first in terms of the presenters, including affiliations, and second by mathematical classification.

During those five years the section sponsored \(23\) lectures by \(21\) different individuals associated with \(11\) different colleges. Lehigh paved the way with five speakers, followed by Penn, Penn State, and Princeton with three each, and Swarthmore with two. The affiliations are shown in Table 3.5.2. The five schools with one speaker each are Brown (Bennett), Bryn Mawr (Lehr), Columbia (Ritt), Lafayette (Smith), and Ursinus (Clawson). Bennett himself might even be regarded as representing Lehigh since he had left there just one year before.

Table 3.5.1 shows that \(8\) of the \(23\) talks sponsored by the section in this period dealt with topics in the broad area of analysis, with at least one given each year. However, all eight lectures covered topics that can be considered classical analysis, not the emerging area of functional analysis.

Penn State’s Orrin Frink presented talks at the first and last meeting of this five-year period. In 1928 he described a method of obtaining formulas from differential calculus without the use of any limiting process. Frink wrote that his method differed from early writers on the calculus by “being rigorous”; it was, in fact, based on the theory of analytic functions of a hyper-complex variable. One can only wonder what the successful calculus textbook author Lloyd Smail thought of this stinging critique while seated in the audience.

Smail, a charter member of the Philadelphia Section, had spoken at the organizational meeting. Five years later, in 1931, he talked again at the second meeting held at Lehigh. This time it was Smail’s turn to criticize calculus textbook authors, asserting that a definition due to Konrad Knopp was the only satisfactory alternative to the usual textbook definitions of infinite series. Smail proposed that the general principle of convergence be substituted in place of the usual limit definition of the convergence of infinite sequences. He also stressed the importance of the concept of summability.

In 1932, four years after his first talk to the section, Orrin Frink spoke on measure theory. He began his lecture by noting deficiencies in the definitions of measure given by Jordan, Borel, Lebesgue, Caratheodory, and Hausdorff, then introduced a result by von Neumann that enabled a measurable function to be defined for all sets. During Frink’s talk he referred to Banach’s use of bounded linear and planar sets, an indication of the section’s acquaintance with the emerging specialty in functional analysis, but Frink never mentioned Banach’s development of linear operators.

Columbia University’s Joseph Ritt became the section’s first invited speaker from outside the region when he spoke at the 1929 meeting at the University of Pennsylvania. He might have traveled by train, because the commute between New York and Philadelphia was convenient, 30th Street Station lying within easy walking distance of Penn’s Bennett Hall. Ritt described the work of Joseph Liouville on the impossibility of performing certain integrations in finite terms but the possibility of solving certain differential equations in finite terms.

The Program Committee for the 1930 meeting (Tomlinson Fort, J. R, Kline, and John Miller) selected three speakers who presented talks on classical analysis at the meeting held at Penn. The host institution’s James A. Shohat, whose contributions to the Philadelphia Section will be detailed in the next chapter, opened the meeting with a lecture on Tchebycheff polynomials. After reminding the audience of the definition of these polynomials, he proved existence and uniqueness theorems for orthogonal polynomials, and then he proved results about their minimum properties and about the distribution of their roots. Penn State’s I. M. Sheffer spoke about non-analytic functions of a complex variable, referring on several occasions to the work of Edward Kasner. Sheffer ended the talk by introducing matrices associated with the differentiation of non-analytic functions. The day’s final speaker was Lehigh’s Tomlinson Fort, who had also delivered an invited address at the section’s second meeting in 1927. In summarizing Fort’s talk, the section’s secretary, Perry Caris, wrote briefly that Fort “discussed the fundamental notions of the papers by Harald Bohr.” That was no mean feat, as the famous Danish mathematician published his results in three long papers that ran to almost 300 pages in the esteemed journal Acta Mathematica in 1925 and 1926.

Recall that F. M. Weida spoke at the annual AMS¹-MAA² meeting at Penn in December 1926. His invited lecture to open the 1928 annual meeting of the Philadelphia Section was on numerical analysis, the only time this topic was addressed in the five-year period. He began the lecture by declaring, “The theory of errors is a branch of mathematics which belongs to practical analysis in applied mathematics.” We pause to profile Weida before analyzing other lectures.

During the five-year period under discussion the section sponsored three lectures on standard topics from applied mathematics: astronomy, mechanics, and potential theory. In 1928 Swarthmore’s John Miller spoke about his first love, mathematical astronomy. He suggested a way of explaining solar coronas by considering particles ejected from the solar surface at successive intervals. In his lecture Miller proposed a theory based on conic sections and orthogonal projections of planes. He concluded, “This theory has been applied to a great many solar coronas that have been photographed in the past twenty years with long focus telescopes ... Of course this does not prove that the coronas are produced in this way but it does offer one explanation of their peculiarities.” At the 1929 meeting, one year after Miller’s talk, charter member Kenneth Lamson of Lehigh University presented an invited lecture on experiments that led to the use of quantum theory. He described the work of the Austrian physicist Erwin Schrödinger leading to the conclusion that energy in mechanics corresponds to frequency in optics. Here, the Philadelphia Section was being kept abreast of current research being conducted in Europe. Another Lehigh faculty member, George E. Raynor, spoke about potential theory in his 1932 talk to open the meeting at Swarthmore College. In the first part he described recent progress on the Dirichlet problem. In the second he supplied necessary and sufficient conditions for the Dirichlet-Neumann problem to have a solution for a sphere with a singular point at the center.

Only two invited speakers lectured on topics in algebra. One was the 1929 talk on group characters by Howard Mitchell, his second talk to the section he helped found three years earlier. Mitchell concluded his presentation by using group representations to prove a result of William Burnside that no group whose order is divisible by just two different primes can be simple. The only other lecture on algebra during this five-year period was the opening presentation at the 1931 meeting by Penn State’s C. A. Rupp, who used notions from linear algebra to “draw some geometric consequences of the linear dependence of flat spaces in a space of n dimensions.”

The section certainly provided the opportunity for mathematicians in the area to keep abreast of progress in differential geometry and topology. With Princeton housing arguably the most prominent set of topologists in the world, it is no wonder that the section would consider the latest developments. James Alexander delivered the first such talk at the section’s third meeting, held at Penn in 1928 in front of a record crowd of 75 people. (That record lasted until the 1954 meeting at Princeton, when 115 attended.) Simply titled “Knots,” his address examined the unsolved problem of finding sufficient invariants to determine completely the knot type of an arbitrary simple closed curve in three-dimensional space. This topic has a modern ring to it even now.

The following year Luther Eisenhart, a graduate of Gettysburg College, presented an invited lecture on his specialty, differential geometry. His talk was based on a paper that had appeared in the Annals of Mathematics just one month before.

The third Princeton faculty member to speak on topology or differential geometry was one of the section’s charter members, Morris Knebelman, who had been at Lehigh University when his colleague J. B. Reynolds first broached the idea of an MAA⁴ section in 1925. Knebelman’s 1931 talk in Packard Laboratory at Lehigh came three years after he received his Princeton Ph.D. In his lecture Knebelman extended the concept of the curvature of a surface – as studied by Gauss, Rodrigues, and Riemann – to topological properties of a space. The section’s secretary, Perry A. Caris, wrote, “it is only within the last two or three years that the question of curvature has undergone a closer scrutiny.” Knebelman’s lecture considered some of the newer results. It ended up being the only lecture he ever delivered to the section, for shortly thereafter he moved to Washington State University. Although he returned to Bucknell in 1964, there is no record of any further participation in the Philadelphia Section.

In addition to the Princeton trio, Penn’s J. R. Kline lectured on a topic that straddles topology and differential geometry at the 1932 meeting at Swarthmore College. This was the only time that Kline spoke before the Philadelphia Section. It is not entirely unexpected that a student of R. L. Moore would subject the foundations of a subject to a painstaking analysis. With this in mind, Kline analyzed various definitions of a curve, noting shortcomings of earlier attempts to define the concept. During his lecture he mentioned the work of two of his own students, Leo Zippin and Norman Rutt. (The official report from the meeting incorrectly lists Rutt’s surname as Ruth.) Kline ended his lecture by showing interrelations among three types of curves: regular, perfect continuous, and those that are the sum of a countable number of arcs.

Four of the talks given to the Philadelphia Section during 1928-1932 were concerned with geometry, one per year except 1929. The first was delivered in Penn’s Bennett Hall by section founder A. A. Bennett, who proposed that triangles be studied using the theory of binary forms instead of the usual approaches to geometry using synthetic methods, analytic geometry, projective methods, or inversive geometry. The second invited lecture was given by Ursinus College’s John W. Clawson, who described a polar reciprocation of a complete quadrilateral and some of its related points and lines with respect to a circle having the focal point (Steiner, Miquel) of the quadrilateral for center. The following year William M. Smith of Lafayette College outlined the facilities available for graduate study in Rome, where he had spent a sabbatical year studying under Enrico Bompiani. Smith also reviewed contributions made by Italian geometers to projective geometry and Riemannian geometry.

In Chapter 2 we pointed out that the 1925 dissertation of Bryn Mawr’s Marguerite Lehr dealt with algebraic geometry. Her 1931 talk at Swarthmore College, titled “On curves with assigned singularities,” evokes a common theme. In it Lehr presented her solution to the following problem: given a set of nonnegative integers satisfying a plane algebraic curve defined by Plücker equations, does a plane algebraic curve exist having these equations as its Plücker characteristic? Given the content of Smith’s talk the year before, it was appropriate for Lehr to make use of a theorem due to the famous Russian born, Italian educated, geometer Oscar Zariski.

The two remaining lectures delivered to the Philadelphia Section during 1928- 1932 dealt with curriculum and history/number theory. We already noted that Arnold Dresden had been hired by Swarthmore College in 1927 to fashion that school’s honors course in mathematics for juniors and seniors. At the 1931 meeting at Lehigh he discussed the results of his endeavor. Minutes from that meeting record only that Dresden gave “an account of the way in which this plan [for honors work] is realized, particularly in mathematics and the natural sciences.” Fortunately a note from the May 1927 issue of the Monthly supplies more details about the program:

The honors program that Dresden designed required students to complete four seminars in mathematics and two seminars in each of two minors. That was the student’s whole course load during the last two years. External examiners conducted examinations in honors. That is still the case today, although parts of the system have been drastically revised; external examiners are still an integral part of the program, however.

One of the founders of the Philadelphia Section, Howard Mitchell, spoke to the section on three separate occasions. We have already described his talks at the organizational meeting in 1926 and the 1929 meeting at his home institution, Penn. He also presented the very last talk in the five-year period under discussion, “The life and work of Ramanujan”. As Perry Caris wrote in his official report from the 1932 meeting at Swarthmore College, “The title ... sufficiently indicates the nature of the paper.” This was not the last time our section would be treated to a talk about the fascinating story of Ramanujan and his phenomenal discoveries in number theory, as Nathan Fine lectured on this same subject in 1952 based on material unavailable to Mitchell at the time. In 1990 George Andrews also spoke about Ramanujan’s mathematics at a student conference held at Moravian College. All three Ramanujan speakers – Mitchell, Fine, and Andrews – were associated with the University of Pennsylvania, the first two as faculty members and the last as a doctoral student. Both Fine and Andrews ended up at Penn State.

The discussion in the preceding paragraph raises the question, “Who has delivered the most talks to the section?” Of those who accepted invitations to lecture during the period of establishment, Marguerite Lehr delivered the most, one per decade: 1931, 1944, 1954, and 1963. James Shohat comes next, having delivered all three of his talks in one decade: 1930, 1934, and 1939. Overall, Lehigh University’s Albert Wilansky gave the most lectures, five. In addition to Lehr, four other mathematicians spoke to the section on four different occasions: Cletus Oakley from Haverford College, and Hans Rademacher, Isaac Schoenberg, and Herb Wilf from the University of Pennsylvania.