Section 4.4 Themes of Lectures
First we discuss the invited lectures delivered by émigré mathematicians who settled in the region. Then we examine the lectures by their mathematical classification: analysis, geometry/topology, algebra/number theory, probability/ statistics, curriculum/pedagogy, and special topics.
Émigrés
Like the rest of the country, the Philadelphia Section benefited enormously from mathematicians forced to flee their homeland. During the period 1933-1941 six invited lectures were given by four different émigré mathematicians, all of whom chose to speak on some aspect of analysis except for Richard Courant, who spoke on applied mathematics.
The preceding chapter discussed the 1930 lecture by one immigrant from the 1920s, James A. Shohat. The Penn professor accepted two more invitations in the 1930s, the first at the 1934 meeting held at Penn. In this lecture Shohat showed that the trapezoidal formula and Simpson’s Rule for approximating a definite integral could be derived by a simple application of Taylor’s Formula.
The 1939 annual meeting held at Lehigh provided the highlight of Shohat’s association with the Philadelphia Section. At the business part of the meeting, held between the morning and afternoon sessions, he was elected chairman of the section for the next year. On the program he presented a talk contrasting Lagrangian and Hermitian interpolations of certain orthogonal polynomials, based on recent results of Bernstein, Faber, and Fejér.
Princeton’s Salomon Bochner was born in Krakow, Poland, in 1899. He was forced to leave his position of lecturer in Munich, Germany, in 1932, and just one year later Oswald Veblen arranged a faculty appointment for Bochner at Princeton. Two years after that, in Bochner’s 1935 talk to the Philadelphia Section, he presented an outline of the theory of periodic and almost-periodic functions as viewed from the theory of expansions of general functions on groups. He emphasized that his approach explained adequately why the exponentials formed a complete system of pure periodic functions of one variable. We do not know the cause of James Shohat’s emigration from Russia in 1923 but there is no doubt why Salomon Bochner left Poland – to avoid the Nazi persecution of Jews. Bochner always remained grateful to the United States for saving his life.
Hans Rademacher, the third émigré to speak to the Philadelphia Section, also left his native Germany to avoid Nazi persecution, but not for religious reasons. Rademacher was a Protestant who became a Quaker when he came to Philadelphia. He was dismissed from his position as full professor at the University of Breslau in 1933 because he belonged to the International League for the Rights of Man and was president of the Breslau chapter of the German Society for Peace.
During his lifetime Hans Rademacher presented four invited lectures at annual meetings of the Philadelphia Section, two during the period under consideration, the first at the 1937 meeting at Haverford College. A distinguished figure in analysis and analytic number theory, he chose to speak about Bernoulli numbers in his initial talk. Here he regarded the general binomial coefficient \(n\choose
j\) as an integral-valued polynomial of the \(j^{th}\) degree, which was then expressed as a linear combination of binomial coefficients by means of integral coefficients \(A_{q j}\) . The resulting formula yielded the \((q + 1)^{th}\) Bernoulli polynomial. The master number theorist used this fact to prove that the Von Staudt theorem followed as a direct consequence of a congruence property of the \(A_{q j}\text{.}\)
Rademacher delivered his second invited lecture to the Section at the 1940 annual meeting at his home institution, the University of Pennsylvania. On this occasion he spoke about one of his favorite topics, Dedekind sums, based on a forthcoming paper written with his first Ph.D. student at Penn, Albert Whitman. Rademacher lectured on an arithmetical property of Dedekind sums called the “theorem of reciprocity”.
The lecture by the remaining émigré, Richard Courant, is noteworthy for several reasons. For one, it was the last lecture delivered during the period under discussion. For another, the subject matter was one dear to the speaker’s heart: applied mathematics. Another émigré, Lipman Bers, commenting on the value of many immigrants’ knowledge of applied mathematics, wrote, “Almost overnight, refugee mathematicians became a boon rather than a burden.” Like Bochner, Courant had been born in Poland but educated in Germany, obtaining his doctorate under David Hilbert at Göttingen in 1910. He was professor of mathematics and Director of the Mathematical Institute at Göttingen from 1920 until his dismissal by the Nazis in 1933. After spending the next academic year at Cambridge University, Courant crossed the ocean to become a professor of mathematics and head of the department at New York University. He held both posts until his retirement in 1958. Curiously, the fascinating biography of Courant by Constance Reid does not mention his lecture to our section.
The title of Courant’s lecture was “Problems of stability and instability demonstrated by soap film experiments”. In it he stated that if a dynamical system permits different states of stable equilibrium, corresponding to relative minima of the potential energy, there must exist transitions between these two states leading over an intermediary state of unstable equilibrium. Courant then demonstrated the principle with soap film experiments of a novel type, relating to minimal surfaces and systems of such surfaces. After that Courant discussed how these experiments pointed to a new field within the calculus of variations. He ended his talk by linking the newly formulated problems to classical isoperimetric problems.
Before discussing other invited lectures during 1933-1941, we pause to mention another one on applied mathematics. Given at the 1935 meeting, Penn’s Enos Eby Witmer spoke on “Quantum mechanics”. In his talk Witmer viewed the development of quantum theory as developed by Planck in terms of the history of classical physics. His major theme was that every physical problem should be approached as an eigenvalue problem. He concluded, “and that is what quantum mechanics is.”
Analysis
The section above described lectures on analysis by James Shohat, Salomon Bochner, and Hans Rademacher. From 1933 to 1941 five other speakers chose to speak on a theme from analysis. Altogether 10 of the 38 talks presented to the Philadelphia Section during the nine-year period under discussion were on analysis, which reflects a slight de-emphasis from the period 1928-1932, which featured 8 of 23.
Haverford’s Cletus O. Oakley delivered two invited addresses during the 1930s. In 1934 he lectured on ordinary differential equations. The title of his 1939 talk, “Equations of polygonal configurations,” belies its content, which concerned semilinear equations involving linear forms. By invoking properties of linear operators, Oakley thereby introduced many of the members to topics in modern functional analysis.
A lecture given three years earlier removed all doubt about the intrusion of “soft analysis” into the area. In his 1936 talk, Penn’s James A. Clarkson discussed convexity properties of Banach spaces and their subsets, the first time such spaces were mentioned officially at a meeting of the Section. In his talk Clarkson presented a simple proof of the Radon-Riesz theorem on weak convergence in \(L_p\) spaces for \(p > 1\text{.}\)
Two Bryn Mawr College professors also spoke on analysis. In 1935 Gustav Arnold Hedlund took a rather nontraditional approach to dynamical systems. Hedlund’s lecture did not feature the usual particles. Instead he considered Euclidean and non-Euclidean billiards with a single ball, which maintains its velocity and is reflected from the sides at equal angles. In her 1938 lecture Anna Pell Wheeler presented a study of the correspondence between certain classes of functions analytic in the interior of the unit circle.
P. T. Maker of Rutgers University preceded Courant’s talk at the 1941 meeting held at Penn. Like Pell Wheeler, Maker spoke about analytic functions. In his talk he discussed the contribution of measure theory to the problem of relaxing conditions in order for a function of a complex variable to be analytic, ending by generalizing the Cauchy theorem to functions on closed sets.
Geometry/Topology
Eight lectures at the annual meetings of the Philadelphia Section held from 1933 to 1941 were devoted to topics in topology or geometry. Although the previous five-year period witnessed lectures by Princeton topologists James Alexander and Morris Knebelman, nobody from that famous school of topologists spoke during the period at hand. Nonetheless, the three talks devoted to topology maintained the quality achieved by the earlier speakers.
The earliest address on topology was an expository talk by Raymond L. Wilder, one of the first graduates of the Texas school of topology under R. L. Moore. At the time of his lecture Wilder was at the Institute for Advanced Study at Princeton on leave from Michigan. In his talk he outlined the development of various notions of connectivity in higher dimensional polyhedrals, then he described recent extensions of these ideas to arbitrary topological spaces.
The next lecture on topology was given by Stuart S. Cairns of Lehigh University, who spoke on the deep subject of simplicial complexes in both point set and algebraic topology, the latter then known as “combinatorial topology”. After presenting his solution to a problem involving regular manifolds, Cairns applied the material to the usual definition of arc length before supplying a proof of the generalized Stokes Theorem.
The third, and final, invited lecture on topology was given by one of the section’s most active members, John C. Oxtoby of Bryn Mawr College, who was elected chairman of the section in 1955. Oxtoby opened the 1940 meeting at Penn by discussing transitive flows, a topic that deals with one-parameter continuous groups of automorphisms of a topological space. As is typical of many aspects of Lie groups, this material was applied to the theory of differential equations.
The five talks on geometry during this period dealt with five entirely different aspects of the subject. At the 1940 meeting John Oxtoby was followed by J. L. Vanderslice of Lehigh University, who gave an expository talk on generalizations of classical differential geometry in which he supplied evidence for the indispensability of tensors and tensor differentiation. Two years earlier R. C. Yates of the University of Maryland presented a lecture titled “Linkages”. After presenting a history of the subject from the common pantograph of Scheiner in 1631 to work by J. J. Sylvester, the speaker exhibited several working models of linkages designed for line motion, trisection, and the description of conics and many higher plane curves. Yates would speak to the section again in 1951, at which time he had moved to the U.S. Military Academy at West Point. In 1934 J. Alfred Benner of Lafayette College spoke about a certain class of curves defined by polar equations.
- Jacob Alfred Benner (1900-1974) is a native of Hopewell, PA. He received a bachelors degree from Penn State in 1922 and two masters degrees, one from Lafayette in 1925 and another from Columbia four years later. J. Alfred Benner spent his entire career at Lafayette, being appointed an instructor in 1922 and eventually promoted to full professor in 1950.
In 1933 Heinrich Brinkmann of Swarthmore College spoke about projective geometry, a topic also discussed by Penn State’s Frederick Owens in 1937. Owens presented an expository paper centered around one theorem: the impossibility of constructing three distinct triangles, real or imaginary, for which each triangle is in six-fold perspective with each of the others. Helen Owens spoke before the section two years later. The fact that the Allegheny Section scheduled annual meetings in the spring enabled mathematicians who had formerly been active in our section to continue their formal ties.
Algebra/Number Theory
During the period 1933-1941 there was only one invited lecture at the Philadelphia Section that truly fits into the classification of algebra. In the last year of the period Heinrich Brinkmann discussed the relations between the discriminant of a cubic polynomial and its roots, concluding with an elementary proof of a theorem due to Voroni. Since this topic deals with the classical algebra of polynomials, it means that the section had still not heard a talk about any part of the abstract algebra that was developed in the late 1920s and throughout the 1930s, even though the acknowledged leader in the field, Emmy Noether, taught at Bryn Mawr College from 1933 until 1935. In fact, the renowned algebraist Nathan Jacobson replaced Noether at Bryn Mawr, but there is no record of his participation in the section either. Edward Kasner of Columbia University gave an invited lecture on a related topic at the annual meeting in 1933. Titled “Polygons and groups,” he extended results from a \(1903\) Monthly article to bi- point transformations connected with a given polygon.
In spite of a dearth of talks on modern algebra, the period 1933-1941 bore witness to five invited lectures on number theory. In the first year Emory P. Starke of Rutgers University discussed binomial congruences on a level accessible to undergraduate students. Starke continued to be active in sectional affairs, serving as chairman for the year 1946-1947. Two of the section’s invited talks on number theory were delivered at the 1937 meeting at Haverford College. The one by Hans Rademacher was described above; the Rutgers mathematician H. S. Grant gave the other, on Farey series. In 1941 Richard P. Bailey of Lafayette College gave a talk whose title suggests geometry, “The problem of the square pyramid,” but instead deals with integral solutions of systems of equations based on the work of the French number theorist, F. E. A. Lucas.
The final lecture on number theory was the opening presentation at the 1939 meeting held at Lehigh University. The title of D. H. Lehmer’s talk has a very modern ring to it, “Mechanical aids to the theory of numbers”. In it he described commercial multiplying machines and their adaptations, punch card equipment, sieve and stencil devices, and his own recently constructed electric sieve.
Probability/Statistics
During the 1930s the section sponsored talks on two emerging areas, probability and statistics, by one person, Samuel Stanley Wilks, of Princeton. The first talk was given at the 1935 meeting at Penn. Titled “Inverse probability and fiducial inference,” Wilks explained some of the difficulties that arise in attempting to apply Bayes’s Theorem on inverse probability to the problem of drawing inferences about unknown population parameters from observations, and conditions under which such difficulties can be avoided by fiducial arguments.
S. S. Wilks spoke a second time five years later, at the 1940 meeting again at the University of Pennsylvania. The title of his presentation was “Statistics involved in College Entrance Examinations”. In this invited lecture he discussed statistical problems that arise in the construction and reading of examinations, particularly those administered by the College Entrance Examination Board. Wilks stressed that to secure maximum reliability in an examination with a given number of questions, care must be taken to select items of suitable difficulty satisfying the subject-matter requirements such that the correct and incorrect responses on each pair of questions are highly positively correlated.
Curriculum/Pedagogy
Five papers presented at the 1932-1941 annual meetings dealt with some aspect of education, two each at the meetings in 1935 and 1938 and one in 1936. Three of the lectures concerned the college curriculum exclusively, one focused on secondary mathematics, and the remaining one bridged the gap between high school and college.
At the 1938 meeting, the first hosted by Ursinus, Albert W. Tucker described the two one-semester geometry courses he offered at Princeton. One of them, called an “Introduction to Modern Geometry,” was designed for sophomores and dealt with geometries characterized by their groups of transformations. The other, “Elementary Topology,” was a course constructed for juniors. It dealt with topological properties in two and three dimensions, including the classification of 2-dimensional manifolds and examples of 3-dimensional manifolds, culminating in properties of general topological spaces. By examining manifolds in \(\cal R^3\text{,}\) Tucker anticipated a topic that would earn Fields Medals for four mathematicians in the second half of the \(20^{th}\) century: John Milnor in 1962, Steven Smale in 1966, William Thurston in 1982, and Michael Freedman in 1986.
In 1936 W. R. Murray of Franklin and Marshall College described the undergraduate comprehensive examination developed by the faculty at F & M. Murray maintained that although F & M’s system of comprehensive examinations offered decided advantages in coordinating a student’s course of study, it was more difficult to construct in mathematics than in other areas. In spite of observing several desirable gains, “the experiment is still too new and the operation too imperfect to make any enthusiastic claims for the plan.” Apparently the system of comprehensive exams flourished until the 1960s, when they began to disappear across the college. The major hurdle seemed to be the conundrum of the student who, having passed all courses, performed poorly on the comprehensive exams. In mathematics, one of the consequences was to lower the level of the exam to meet the students’ achievement, clearly not the intention of the system. Sometime in the early 1970s Franklin & Marshall eliminated comprehensive examinations altogether.
At the tenth meeting of the section in 1935, the first at Lafayette College, the leadoff speaker was Richard P. Bailey from the host institution. His colleague Will Smith, who was chairman of the Section in 1934-1935 and served three terms on the Program Committee in the nine-year period under discussion, introduced him. The section’s secretary, Perry Caris, described Bailey’s talk in one sentence: “Doctor Bailey presented the results of a survey of the mathematics curricula of the colleges of the Philadelphia Section, calling attention in particular to the most prevalent types of curriculum organization and the major problems to which they give rise.” Unfortunately, no further details were provided. Bailey himself had just come to Lafayette that year as an instructor in mathematics, and he remained at Lafayette until 1944. Certainly Secretary Caris knew about Bailey’s work in mathematics, because Bailey had just completed his doctoral dissertation at Penn under James Shohat a few months before delivering his invited lecture to the section.
At that same meeting Albert G. Rau, the Dean at Moravian College, presented his research on the history of the curriculum at Moravian secondary schools and at Moravian College. He indicated that teachers in the Moravian schools were acquainted with algebra and geometry; these subjects became part of the curriculum after 1780. This background enabled Moravian College to add conic sections into the curriculum after 1820. Analytic geometry and calculus were added after 1860 but were discontinued from about 1870 until 1900, perhaps due to the growing tradition at some Lehigh Valley colleges of students burning their calculus books at the end of the year. Rau concluded his paper by stating, “The general question of mathematics in other Pennsylvania German schools can not be examined for lack of evidence and is postponed for a later paper.” We do not know if Dean Rau ever pursued his research further in this direction, but we pause to introduce this active member of our section.
- Albert George Rau (1868-1942) is the only person to speak to our section who was a full-time administrator his whole professional life. Born on August 7, 1868, in Bethlehem, PA, Rau obtained his bachelors degree from Lehigh University in 1888. In spite of graduating shortly before his 20th birthday, Rau assumed the position of Superintendent of the Moravian Preparatory School that September. He remained in that post until 1909, when he was appointed Dean of the Moravian College and Theological Seminary. Rau attended school in the evening, resulting in a masters degree from Lehigh in 1900 and a Ph.D. from Moravian in 1910. His publications show a wide breadth of interest, a trait reflected in his professional memberships. Rau not only belonged to the MAA 1 and the AMS 2 , but to the American Society for Political and Social Science, perhaps being the only member in the intersection of these three sets. Except for a one-year leave as lecturer on rural sociology at the Teachers College of Columbia University in 1927, Rau retained his deanship until his death on February 23, 1942.
Being the second speaker at the 1938 meeting, Albert W. Tucker discussed his two topology courses before the business meeting, conducted right before lunch. After the meal the audience was treated to another lecture dealing with education, but this one at the secondary level. The speaker, W. D. Carpenter, a mathematics teacher at the private Germantown Academy in Philadelphia, centered his talk about two issues. The first was the growing influence of the College Board Examinations. Carpenter felt that the inclusion of elementary calculus and analytic geometry would cause a lack of thoroughness in the other subjects taught, a theme repeated by many college teachers in the 1980s and 1990s. Next the speaker addressed criticisms of secondary mathematics, again anticipating basic issues that would arise 50 years later. He put the blame for the poor preparation of high-school teachers squarely on the shoulders of the college faculty themselves, and then he laid out a plan of improved methods in the training of future teachers.
Special Topics
Four of the talks delivered to the Philadelphia Section in the 1930s do not fit into any of the categories above but are too important to overlook. In 1934 Charles N. Moore presented an invited lecture in which he compared mathematics and poetry on the basis of certain common aesthetic elements. Moore spent the 1934-35 academic year at the Institute for Advanced Study in Princeton on leave from his usual post at the University of Cincinnati. He had served as vice-president of the MAA 3 in 1931.
In 1937 Albert Harry Wheeler, a high-school teacher from Worcester, Massachusetts, presented a talk titled “Stellated polyhedra, illustrated by models”. In his presentation Wheeler showed by means of paper models some unusual transformations of solids. In addition to paper models he showed photographs of many forms of stellated polyhedra, probably passing the photos among the 57 listeners in the room because overhead projectors were not yet in use.
The annual meeting at Lehigh in 1939 included two invited lectures on special topics. One was by Dr. Elmer Johnson from the Schwenkfelder Library in Pennsburg, PA, who exhibited numerous old mathematical books and surveying instruments in the library’s collections. He also discussed the history of some of those books and instruments, including facts concerning their acquisition.
The other lecture was by Penn State’s Helen B. Owens, whose title was “Mathematics clubs, old and new”. We have already discussed Helen and her husband, Frederick, during the time Penn State was aligned with the Philadelphia Section, and even after the formation of the Allegheny Section in 1933. Here Owens suggested various activities that could be used to stimulate interest in mathematics among undergraduate students, including the use of problem contests, joint club meetings, mathematical exhibits (like the one described by Albert Wheeler above), public lectures, and student mathematical publications. The section’s Deborah A. Frantz, the indefatigable Kutztown University professor who served as EPADEL’s Student Chapter Coordinator throughout the 1990s, carried out every one of these activities in that decade.
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