Commission on the History of Mathematics

Report on the ICHM-sponsored Day of Lectures (January 9, 2004)
at the AMS-MAA meeting in Phoenix, Arizona

Karen Parshall

The ICHM co-sponsored (with the American Mathematical Society and the Mathematical Association of America) a day of lectures on Friday, 9 January, 2004 at the Joint Meetings held this year in Phoenix, Arizona. The session—co-organized by Joseph W. Dauben, Lehman College (CUNY), Karen V. H. Parshall, University of Virginia, and David E. Zitarelli, Temple University—drew an audience that varied during the day between 50 and 200. It was comprised of the following talks:

Analytic Geometry: Descartes versus Fermat

Maria Sol de Mora, Plaza Adriano, 7, Sobretico, 08021 Barcelona, Spain.

Abstract: One of the first ideas Descartes tries to develop is the generality of the mathematical method. He sought to resolve problems by analogy with the procedure employed in arithmetic, conceiving of the solution of geometrical problems in terms of the construction of figures, rather than in terms of a satisfactory algebraic solution. For Descartes, algebraic equations are tools for constructing and classifying geometrical problems. In most cases, he carried out his calculations without even writing down the equations of the curve explicitly. He considered the degree of a curve as a measure of its simplicity. As for coordinates, Descartes only considered positive values in the first quadrant. He was aware that his method could be extended to curves and surfaces in three dimensions, but he did not carry out this extension. Fermat did. I believe that the idea of associating equations to the curves is much clearer in Fermat than in Descartes. Fermat had a more modern vision and in exploring various reasons why algebraic geometry is attributed to Descartes, we will try to explain the subsequent "oblivion" of Fermat.

Title: The Brachistochrone Problem and Its Sequels

Rüdiger Thiele, Medizinische Fakultät, Karl-Sudhof-Institut, Augustusplatz 10/11, 04109 Leipzig, Germany

Abstract: To a large extent Hilbert's well-known list of problems (Paris, 1900) steered the course of mathematics in the 20th century. However, posing problems is an old mathematical tradition and there are many famous problems from the 17th century, among them the most influential Brachistochrone Problem (Johann Bernoulli, 1696). As a consequence of this problem mathematical physics (in its actual meaning) got its start by developing essential variational methods that resulted in a new branch of mathematics. Moreover, the concept of an analytic function was formulated (Joh. Bernoulli, 1697) and extended (Euler, since 1727). This lecture gives a comprehensive overview on these cornerstones of mathematics.

Is the English Version of the Theory of Incommensurability Commensurable with Its Chinese Translation?

Yibao Xu, Ph.D. Program in History, Graduate Center, CUNY, 365 Fifth Avenue, New York, NY 10016-4309

Abstract: Henry Billingsley's English version of Euclid's Elements (1570) has recently been identified as the source for the Chinese translation of the last nine books of the Elements by the British missionary Alexander Wylie and the Chinese mathematician Li Shanlan, first published in 1857. This discovery has not only made it possible to reevaluate the quality of the translation as a whole, but provides a means as well to investigate how the theory of incommensurability was introduced to China, where traditional Chinese mathematics lacked the concept of magnitude, greatly increasing the difficulty of interpreting the concept of incommensurability. By focusing on the Chinese translation of Book X of the Elements, it is possible to assess the accuracy of the translation and to evaluate the impact the theory of incommensurability and irrational numbers may have had upon Chinese mathematicians at the time, among other historical issues.

Science in Translation: The Transmission of Probability Theory into Late Imperial Chinese Mathematical Culture (1880-1911)

Andrea Eberhard-Brard, 311 Clinton Street, Brooklyn, NY 11231

Abstract: When the first translation of a treatise on probability theory (Jueyi shuxue, original Thomas Galloway's article in the 7th edition of the Encyclopaedia Britannica) appeared in China in 1896, neither statistical thinking nor algebraic symbolism was common knowledge to Chinese literati. The purpose of this talk is to portray the migration of probabilistic thinking through cultures and time as a linguistic attempt to translate unfamiliar phenomena into an existing taxonomic system. This involved not only nding precise terms for new mathematical concepts or describing the phenomenon by means of paraphrases, but also to interpret Western mathematical notations with the terminological set available from the Chinese algorithmic tradition.

"Everybody Makes Errors": The Intersection of De Morgan's Logic and Probability, 1837-1847

Adrian Rice, Department of Mathematics, Randolph-Macon College, Ashland, VA 23005

Abstract: The work of Augustus De Morgan on symbolic logic in the mid-nineteenth century is familiar to historians of logic and mathematics alike. What is less well known is his work on probability and, more specifically, the use of probabilistic ideas and methods in his logic. The majority of De Morgan's work on probability was undertaken around 1837-1838, with his earliest publications on logic appearing from 1839, a period which culminated with the publication of his Formal Logic in 1847. This paper examines the overlap between his work on probability theory and logic during the earliest period of his interest in both.

Conceptions of General Analysis in Nineteenth-Century Mathematics: Preliminary Report

Craig G Fraser, Institute for the History and Philosophy of Science and Technology, Victoria College, University of Toronto, Toronto, Ontario M5S1K7, Canada

Abstract: The paper concentrates on the decade from 1850 to 1860 and examines parts of analysis related to the calculus of variations. Mathematicians such as Mikhail Ostrogradsky (1801-1862), Otto Hesse (1811-1874) and Alfred Clebsch (1833-1872) formulated the results of their investigation at a greater level of generality than either expository considerations or scientific applications would seem to have warranted. They seemed to view their analytical formulations as instances of a much more general and over-arching theory. The paper describes some examples and explores conceptions of generality that guided research in variational analysis at the middle of the nineteenth century, comparing the outlook at this time with the perspectives of earlier and later researchers.

Creating a Mathematical Area: The Case of Arthur Cayley, J. J. Sylvester, and Invariant Theory

Karen V. H. Parshall Departments of History and Mathematics, University of Virginia, P.O. Box 400137, Charlottesville, VA 22904-4137

Abstract: From 1845 through 1855, J. J. Sylvester earned his living as an actuary at the Equity and Law Life Assurance Company in London. In order to qualify himself for advancement within the firm, he entered the Inner Temple in 1846 to prepare for the Bar. By 1847, he had met Arthur Cayley, who was also preparing for the Bar but at nearby Lincoln's Inn. A personal and mathematical friendship soon developed between the two that led to their creation of the British approach to invariant theory. This talk will explore the dynamic as well as the intellectual path that led to this creation.

Jules Tannery and the Mathematical Research Community in France, 1870-1914: Preliminary Report

W. Thomas Archibald, Department of Mathematics and Statistics, Acadia University, Wolfville, NS B4P 2R6, Canada

Abstract: Jules Tannery (1848-1910) occupied an important position in the development of mathematics in late nineteenth and early twentieth century France. His early research on linear differential equations stands at the origin of much research in this area in France, undertaken by such authors as G. Floquet, E. Picard, and E. Goursat. His textbook writings, particularly the Eléments de la théorie des fonctions elliptiques, jointly written with J. Molk, were fundamental to the French university curriculum for many years. Most important, though, was his work as the assistant director in charge of science at the École normale supérieure, a post he took up in 1884. In this capacity, he exerted a strong influence on the careers of many normaliens. In this talk, I shall survey these three aspects of Tannery's career.

"Some Unsolved Problems of Theoretical Dynamics": An Unpublished Paper by George Birkhoff: Preliminary Report

June E. Barrow-Green, Faculty of Mathematics and Computing, Walton Hall, MK7 6AA Milton Keynes, England

Abstract: In 1941 George Birkhoff, Perkins Professor of Mathematics at Harvard University, presented a paper at a fiftieth anniversary symposium of the University of Chicago. The paper was entitled "Some unsolved problems of theoretical dynamics". It was a preliminary exposition and a summary was published in 'Science' soon afterwards. Two years later Birkhoff agreed to contribute the completed version of the paper to the "Recueil Mathmatique de Moscow." But Birkhoff died in 1944 and the paper was never submitted. In this talk I shall discuss the unpublished paper and give some background to Birkhoff's mathematical career.

Who Invented Steiner Triple Systems and Why?

Robin J. Wilson, Department of Pure Mathematics, Open University, MK7 6AA Milton Keynes, England

Abstract: It is well known that Jakob Steiner's involvement with so-called 'Steiner triple systems' was minimal, and that the credit should largely go to the Revd Thomas Penyngton Kirkman. But what did Kirkman do? How did he become interested in the topic? What previous work did he have to build on? And what is the connection between Kirkman's schoolgirls and the three sons of Noah?