International 
Commission on the History of Mathematics

Report on the ICHM Co-Sponsored Session on the History of Mathematics

Saturday, 14 January, 2006, 2:00-6:00 pm

by Karen V.H. Parshall

This session of ten talks was co-organized by Joseph W. Dauben (City University of New York), Patti W. Hunter (Westmont College), and Karen V. H. Parshall (University of Virginia and took place at the Joint Meetings of the American Mathematical Society held in San Antonio, Texas, USA. The talks (titles, speakers, and abstracts below), each of which lasted a half-hour, drew audiences of between 75 and 150.

Reconstructing Early Developments of Determinants in China: Evidence from the"Nine Chapters of Mathematical Methods" (Jiu zhang suan shu) and Later Commentaries.

Roger Hart, University of Texas at Austin

I argue that the early history of the development of determinants should be extended back 1500 years earlier than previously recognized to the Nine Chapters of Mathematical Methods (c. 150 BCE). I focus on problem 13 from chapter 8 of the Nine Chapters, together with solutions preserved in later commentaries. First, I show that among these solutions is found the earliest extant record of a calculation of a determinant (c. 1025 c.e.), and the earliest extant record of a determinantal solution (1661 c.e.). I then present mathematical and textual evidence to reconstruct determinantal solutions to problems in the Nine Chapters and argue that these were known at the time of its compilation.

Western Concepts of Infinity Introduced to China in the Seventeenth Century.

Yibao Xu, Borough of Manhattan Community College of the City University of New York

This preliminary report will focus on Chinese translations of Book V of Euclid's Elements, Aristotle's De Caelo, and Galileo's Le Opperexioni del Compasso Geometrico et Militare. It will consider how the concepts of infinity contained in these books were transmitted to China, and their possible influence on Chinese mathematicians.

Giacomo Rho (1592-1638) and his mathematical work in Beijing.

Jose A. Cervera, ITESM, Campus Monterrey, Dep. Estudios Humanisticos

As Joseph Needham points out, it is necessary to consider social conditions to understand the development of Chinese science. Astronomy, in particular, was very important for the Chinese empire. However, in the Ming dynasty, after centuries of research, astronomy was no longer carefully studied and predictions were not as accurate as in previous times. This is the reason why the Jesuits considered astronomy as the key to establishing a strong position in China. In the 1620s, several Jesuits who were highly qualified in science were sent to China. Among them were Adam Schall von Bell (1592-1666), Johannes Schreck (1576-1630), and Giacomo Rho (1590-1638). After Schreck's premature death, Schall and Rho worked together in order to reform the Chinese calendar. The result was the Chong Zhen Li Shu, a huge mathematical and astronomical work that included more than one hundred small books.
One of the treatises included in the Chong Zhen Li Shu is the Chou Suan by Giacomo Rho. This is a translation of the Rabdology by John Napier (1550-1617). Rho also wrote mathematical books on very specific astronomical topics, such as the movement of the planets. In fact, these books may be considered as the real means by which European astronomy was introduced into the Chinese world.

Descartes, the Princess Elisabeth, and the Problem of Apollonius: The dangers of Underestimating a Geometrical Problem.

Henk J.M. Bos, Utrecht University, The Netherlands

In 1643 René Descartes met the Princess Elisabeth of Bohemia. Elisabeth showed interest in his new method of using algebra in solving geometrical problems. She had been taught some algebra and Descartes suggested that she should try to solve Apollonius' problem. That problem is: Given (the centres and the radii of) three circles in the plane; Find (the centre and the radius of) a circle which touches each of the three given circles. She tried hard and gave up. So Descartes had to show her how it could be done. That was not easy.

A War of Words in Pictures: the Dispute between Montmort and De Moivre over the Probability Calculus.

David R Bellhouse, Department of Statistical and Actuarial Sciences, University of Western Ontario

In 1708 Pierre Rémond de Montmort published his book Essay d'analyse sur les jeux de hazard, an analysis of contemporary games of chance using combinatorial methods in probability theory. Three years later Abraham De Moivre published his treatise De Mensura Sortis in which various probability problems were solved including many that Montmort had considered. Montmort felt that De Moivre had plagiarized his work. In 1718 De Moivre published an expanded version of his original Latin treatise under the name The Doctrine of Chances. Montmort remained unhappy with parts of this new work. The dispute is described from surviving publications and letters. Both the Essay d'analyse and The Doctrine of Chances contain allegorical engravings that describe in pictures the nature and importance of their work. These pictures are analyzed in the context of the dispute.

On a Result of Lagrange in the Theory of Infinite Series.

Craig G. Fraser, University of Toronto

In the paper "Sur une nouvelle espèce de calcul relatif à la différentiation et à l'intégration des quantités variables" (1774) Joseph-Louis Lagrange proved the following result. Let u=u(x) be a function of x. Then the derivative of u is given in terms of the finite differences ∆ u , ∆ 2u = ∆ (∆ u), ∆ 3u = ∆ (∆ 2u), ..... by the formula du/dx   ∆ x = ∆ u − ∆2u / 2 + ∆3u/ 3 − ... Lagrange derived this result using analogical reasoning applied to a power series in which the place of the variable was taken by an operation The paper discusses Lagrange's derivation, his extension of power series to operations, and his use of analogical reasoning. His treatment is compared to some earlier work of Leonhard Euler from the 1750s on infinite series with operations, and to a derivation given by Sylvestre Lacroix in 1806 of the same result.

Early Theories of Vectors.

Sandro Caparrini, Dibner Institute for the History of Science, Cambridge MS

According to a commonly-accepted picture of the development of mathematics, vector calculus arose as a consequence of the discovery of the geometric representation of complex numbers, at the beginning of the nineteenth century. This is not entirely true. In fact, there were some very important early influences from geometry and mechanics, which can be traced back to the works of many mathematicians, notably Euler, Carnot, Poinsot and Poisson. The decisive step in the application of these new results to the establishment of a primitive form of vector calculus was taken by Italian mathematician Gaetano Giorgini (1795-1874). His Teoria analitica delle projezioni (Analytic Theory of Projections, 1820) consists almost entirely of formulae which are identical with those of modern vector algebra. Giorgini sought to generalize to non-orthogonal axes the results already known on the projections of directed line segments and plane surfaces. By considering the algebraic sums of the projections, he introduced the compositions of line segments (equivalent to the parallelogram law) and of plane surfaces. While the Teoria analitica is still in the manner of Monge and Hachette, it is difficult to deny that it resembles strongly in aim and content a modern expositions of vector algebra.

Cayley's counting problems.

Anthony J. Crilly, Middlesex University

The mathematical reputation of Arthur Cayley (1821-95) rests primarily on his contributions to group theory, matrix algebra, invariant theory, geometry, and dynamics. Another facet of his extensive production is the attention he paid to combinatorial questions in pure mathematics. His interest in "finite analysis" and the sure-footed intuition he displayed in these researches frequently featured in his mathematical travels, and it was an aspect of his work he shared with his friends, especially James Joseph Sylvester and Thomas P. Kirkman. In this paper, instances where Cayley focused on questions of a combinatorial nature will be examined and the place "Tactic" played in his mathematics generally.

Mathematical crystallography after Hilbert's 18th problem.

Jeanine Daems, Leiden University

In 1910 Bieberbach solved the first part of Hilbert's 18th problem by showing that in each dimension there is only a finite number of crystallographic groups. In this talk, I will briefly discuss his proof and compare it to the standard modern proof of the same theorem. Also, I will go into some of the later developments in 20th century mathematical crystallography.

Euler's Amicable Numbers.

William Dunham, Muhlenberg College

Two whole numbers are amicable if each is the sum of the proper divisors of the other. The Greeks knew the amicable pair 220 and 284, and by the end of the 17th century only two other pairs had been discovered. It is thus remarkable that Leonhard Euler single-handedly found five dozen new ones. In a 1750 paper, he explained his method, one that used properties of what we now call the Euler-sigma function (i.e., the sum of all whole number divisors) to reach the desired end. This talk examines the argument by which Euler increased the world's supply of amicable numbers twenty-fold.