*By Karen V. H. Parshall*

This set of lectures, delivered by an international group of historians of mathematics, once again formed the first day of a two-day Special Session on the History of Mathematics at the Joint Mathematics Meetings held this year in Washington, D.C.

The session was organized by Joseph W. Dauben (The Graduate Center of the City University of New York), Patti W. Hunter (Westmont College), Deborah Kent (Hillsdale College), and Karen V. H. Parshall (University of Virginia)

and was comprised of—in the order of presentation—the following speakers and talks:

*Toke Lindegaard Knudsen, SUNY Oneonta, USA*

Among the unpublished and unstudied mathematical works of medieval India is the *Bijagaita* of Jñanaraja, who flourished in what is now the state of Maharashtra around 1500 C.E. While the treatise is the third and last part of the Siddhāntasundara, a larger work on astronomy and cosmology, it treats mathematics as a topic in its own right. The presentation focused on mathematical results from medieval India based on a study of the *Bijagaita*.

*Glen R. Van Brummelen, Quest University, Canada*

As odd as it sounds, interpolation was a branch of trigonometry during the medieval period in India and Islam. It led a peculiar life on the fringes of mathematical astronomy, a mathematical helpmate, but at least in Islam, an over-shadowed sister to geometry. Some of the best scientists (al-Biruni and al-Kashi among them) made serious mistakes when trying to generate second-order schemes, but others (including Brahmagupta and Ibn Yunus) constructed conceptual mechanisms that led to equivalents of modern formulas. This talk examined how several authors conceived of their methods, and what they thought of them in relation to the entire discipline.

*Kim Plofker, Union College, USA*

Rules on various kinds of proportions, from the familiar "Rule of Three" on up as far as the "Rule of Eleven," formed an important part of Sanskrit arithmetic texts in medieval India. The Central Asian mathematician al-Biruni, who studied Sanskrit mathematics and astronomy in northern India in the early eleventh century, was sufficiently intrigued by these rules to write an Arabic treatise comparing them to Euclidean notions of ratio: the Maqāla f&ibar; rāashikāt al-Hind, or "Treatise on Proportions of the Indians." This presentation examined the way al-Biruni treated his subject, why he thought it was important, and what its consequences were for the interaction of Indian and Islamic mathematics.

*Robin Wilson, The Open University, UK*

The Gresham Chair of Geometry is the oldest mathematics professorship in England, being founded in 1596, after Sir Thomas Gresham left instructions in his will for the founding of a college at which free lectures in seven subjects would be given to interested members of the general public. Four hundred years on, this is still the case. This talk outlined the first one hundred years of the college, and described the geometry professors during this period, including Henry Briggs (co-inventor of logarithms), Isaac Barrow (first Lucasian professor in Cambridge), and Robert Hooke. It also outlined the founding of the Royal Society, which was intimately associated with Gresham College, and was based there, for the its first fifty years.

*Maria Sol de Mora, University of the Basque Country UPV/EHU, Spain*

This talk explored the encounters and disencounters (mainly the second) of Leibniz and the renowned Chevalier de Méré, especially during Leibniz's stay in Paris and in the years immediately after, when Leibniz was interested in mathematical aspects of the theory of probability. The talk examined the contacts of Leibniz with Pascal's work in the field of probability theory, and the resolution of the two problems proposed by Méré to Pascal, and referred to by Pascal in his letter to Fermat of July 1654. The concern of Leibniz with the questions of contingency is well known and goes back to his youth. In 1665, he submitted the "Disputatio juridica de conditionibus," in which he used numbers to represent what he called "degrees of probability"; he was only 19. Faced with a contingent set of circumstances, he came to a decision not totally justified by the "Art of Demonstration or of Judgment," but pertaining to the "Art of Conjecturing."

*Craig Fraser, University of Toronto, Canada*

The Cauchy-Riemann equations connect the real and imaginary parts of an analytic function. f(z) = p(x; y) + iq(x; y) is an analytic function on a given domain if and only if the Cauchy-Riemann equations are satisfied there. Although the equations are named after Augustin Cauchy (1789–1857) and Bernhard Riemann (1826–1866), they first appeared in 1752 in a book by Jean d'Alembert (1717–1783) on fluid dynamics. The paper traced the emergence of the equations and their subsequent development in the period before Cauchy. This history requires examination of the early theory of partial differential equations, a subject pioneered by Leonhard Euler (1707–1783) and d'Alembert from a few principles and methods of solution. The prehistory of the Cauchy-Riemann equations provides insight into the core technical ideas that would later develop into complex analysis.

*George E. Smith, Tufts University, USA*

Did Newton employ the calculus in developing his *Principia*? This frequently asked question is unfortunate not merely because it has no simple "yes-no" answer, but even more so because it draws attention away from the extraordinary range of mathematical techniques that he did employ in that book. The talk first surveyed the range of techniques he did employ, and in the process explained why the question whether he employed the calculus has no "yes-no" answer. It then considered a couple of the many problems peculiar to the physics of the *Principia* that forced Newton to devise novel mathematical techniques that reach far beyond any of those people usually have in mind when they ask whether he did or did not employ the calculus in his physics.

*Kathryn James, Yale University, USA*

Textbooks, student notebooks, and household recipe and account books are among the many different types of sources showing how mathematics was taught and used in popular culture in 17th- and 18th-century England. Fewer explicit sources exist to show how women were taught mathematics in the early modern period. This paper looked at two late 17th-century student notebooks (one in the collections of the Folger Shakespeare Library, and one in those of Yale's Beinecke Library) kept by two young girls who studied arithmetic and calligraphy with their teacher, Mrs. Elizabeth Bean. The notebooks allow insight into two areas: first, how mathematics was approached as one facet of a more general education and, second, how arithmetic was situated within a broader commercial and popular cultural context. A comparison of these notebooks with other student notebooks and printed textbooks of the period allows us to begin to characterize how mathematics was taught in the early modern period, and how it might have been inflected for different audiences.

*Steven H. Weintraub, Lehigh University, USA*

The Lehigh University library has a collection of 40 letters written from Arthur Cayley to Robert Harley between 1859 and 1863, and an unpublished manuscript, "A Memoir on the Quintic Equation," that Cayley was working on at the time of his death in 1895. Examination of this material, which is available online as part of the library's digital archive at http://digital.lib.lehigh.edu/remain/con/cayley.html, gives insights into the working relation between Cayley and Harley as they attacked the quintic from an invariant-theoretic point of view, and into Cayley's lifelong interest in the quintic.

*Renaud G. Chorlay, Université Denis Diderot (Paris 7), France*

The comparison between the theory of complex functions à la Riemann and à la Weierstrass has been a standard topic since the end of the 19th century. However, what exactly is at stake remains, to some extent, unclear. This talk argued that an answer to this question calls for both epistemological and historical work. The talk first stressed elements which are common to both mathematicians, such as the rejection of brute calculation, the conception of regular functions, and the use of singularities. It drew on the example of Poincaré's work to show that, long before Weyl's "Idea of a Riemann surface," some mathematicians had successfully devised a mixed approach. This should help pinpoint more precisely where the differences lie. The presentation also documented the ways in which these differences were described by late 19th-century mathematicians: discovery vs. proof, intuition vs. rigor, geometry vs. arithmetic, transcendental vs. algebraic, global vs. local. Analyzing the meaning and use of these pairs can contribute to the historical epistemology (in the sense of Daston) of some standard categories in the mathematical discourse. From a more philosophical viewpoint, it can provide non-standard case studies for the ongoing debates on issues such as purity of methods, choice of "proper" setting, and geometric thinking.

*Peggy Aldrich Kidwell, National Museum of American History, Smithsonian Institution, USA*

In 1899, Albert Sexton, a Philadelphia mechanical engineer, received the John Scott Medal of the Franklin Institute for his invention of the omnimetre, a circular slide rule designed to carry out all the common operations of arithmetic and trigonometry, except addition and subtraction. At the suggestion of a Belgian-born user of the instrument, it was inscribed: "NUMERI MUNDUM REGUNT." In part because of instruments like the omnimetre, numbers increasingly ruled the practical world of the late nineteenth- and early twentieth-century United States. This changed not only engineering, but mathematics education and mathematics itself.

*Thomas Drucker, University of Wisconsin--Whitewater, USA*

Siobhan Roberts's biography of H.S.M. Coxeter has in its subtitle the phrase 'The Man Who Saved Geometry.' The author suggests that Bourbaki was the threat from which geometry was most in need of saving. In particular, negative statements from Coxeter about Bourbaki are used to suggest that he played a heroic role in bringing down the Bourbaki monster. This talk made two claims. First, Coxeter was taking aim at something of a travesty of Bourbaki, although it was not a travesty entirely without foundation. Secondly, the failure of Bourbaki, as Leo Corry suggests in his history of algebra and mathematical structures, can be attributed to internal diffculties with the Bourbakist program. Coxeter's accomplishments do not require being set against Bourbaki in order to be admired.

*Roger L. Cooke, University of Vermont, USA*

The life of Joseph Perott (1854–1924), one of the minor mathematicians of the late nineteenth century, helps to fill in a few mathematical and historical details from this period in Europe and the United States. Born in Saint Petersburg of a Polish father and Russian mother, in later life he told people that he had a French father and a Polish mother. In the 1880s, he interacted with Sof'ya Kovalevskaya (1850–1891) in both mathematical and literary activities. At the end of that decade, he emigrated to the United States and in 1890 became a docent at Clark University, where he remained for the second half of his life.

*Donald G. Babbitt, UCLA, USA and Judith R Goodstein, California Institute of Technology, USA*

Guido Castelnuovo and Francesco Severi were two of most important figures in the history of Italian algebraic geometry who both carried on a fascinating correspondence with their more junior colleague, Beniamino Segre. The correspondence to Segre has fortunately been preserved and offers an interesting socio-mathematical aperçla;u into Italian algebraic geometry. Among this correspondence, there is a 1932 letter from Severi to Segre that offers the former's opinion of who was overrated and who was underrated among the major figures, including Castelnuovo, in (not just Italian) algebraic geometry from 1850 up to the early 20th century. He also gives an unsurprisingly immodest appraisal of his own contributions. There is also a 1938 letter from Castelnuovo to Segre that contains a paragraph assessing Severi's contributions to algebraic geometry focusing mainly on the first, probably most prolific, years of his research career. This talk discussed these two letters together with some background on the temperament and politics of these two personalities, who could not have been more different.

*Colin McLarty, Case Western Reserve University, USA*

One small puzzling proof by David Hilbert in 1888 became the paradigm of modern axiomatic mathematics, and Hilbert knew its importance. With time the affair grew into an origin myth, a titanomachy where new gods defeat the old, and specifically Hilbert defeats one Professor Paul Gordan who is known today for rejecting Hilbert's proof and calling it "not Mathematics but Theology!" In fact, Hilbert found his proof in conversation with this very Prof. Gordan, and Gordan supported it from the start. The talk traces the growth of the legend over 60 years from the original proof until the story was canonized in no less than three versions by Eric Temple Bell.

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