International 
Commission on the History of Mathematics

Report on the ICHM-sponsored Euler Session at the Joint Meeting
of the Canadian Society for History and Philosophy of Mathematics(CSHPM)
and the British Society for the History of Mathematics (BSHM)

27-29 July, 2007

by Craig Fraser

The joint CSHPM/BSHM meeting was held at Concordia University in Montreal. The Euler session was organized by Robert Bradley. The keynote address was the

CSHPM's annual Kenneth O. May Lecture: "Five Pearls of Euler."

C. Edward Sandifer, Western Connecticut University

The lecture outlined some of Euler's greatest and most beautiful results and commented on modern criticisms of his presentation of these results. The results were the Basel problem, the polyhedral formula, the Euler identity, the Köonigsberg bridge problem and the Euler product formula. On the one hand, all are beautiful results, products of a creative genius of the highest order. On the other hand, Euler's presentations of each of these results has some sort of flaw that, on close examination, might make the modern reader uncomfortable. Professor Sandifer considered this discomfort, noting that what is a pearl to us is a great irritation to the oyster, and considered the nature of portioning out credit for mathematical discovery.

The speakers, titles and abstracts of the other papers in this session were:

Euler and the Enlightenment Mathematicians: A Scottish Perspective

Amy Ackerberg-Hastings, University of Maryland University College

The professor of mathematics and natural philosophy at Edinburgh University, John Playfair (1748-1819) used expository writing, reviews, and historical accounts to shape British conceptions of mathematics and science. Specifically, he evaluated the contributions made to eighteenth-century mathematics by Leonhard Euler, Jean D'Alembert, Pierre-Simon Laplace, and other Continental mathematicians. His articles appeared in the Transactions of the Royal Society of Edinburgh, the Edinburgh Review, and the Encyclopaedia Britannica. The talk explored the portraits Playfair developed in these writings and considered his body of work's own merit as propaganda and as primary source material for the history of mathematics.

Euler's Continued Fractions

Chris Baltus, State University of New York, Oswego

When Euler first worked with continued fractions, by 1730, the subject consisted of a few formulas, largely from Wallis, and a few particular continued fractions. Euler established ties to differential equations and infinite series, and studied a variety of special forms. When he finished, continued fractions constituted a field within mathematics. His continued fraction work illustrates, or, better, exemplifies, his general approach: the brilliant exploitations of examples to arrive at general forms, the intense interest in computation, the discovery of connections between apparently distant ideas. Euler's lesser interest in theory limited his achievement in the case of the Pell Equation, where the young Lagrange quickly surpassed him.

Euler's Summation of a Divergent Series Involving the Pentagonal Numbers

Jordan Bell, Carleton University

Euler's pentagonal number theorem gives the series expansion of the infinite product (1-x)(1-x2)(1-x3)….. It is called the pentagonal number theorem because the exponents in the series expansion are the pentagonal numbers n(3n ± 1)/2. The pentagonal number theorem was used by Euler to prove recurrence relations for the partition and sum of divisors functions. In "De mirabilibus proprietatibus numerorum pentagonalium" (E542), Euler uses the pentagonal number theorem to sum a divergent series involving the pentagonal numbers. This talk gave an explanation of this argument, and also discussed several other of Euler's uses of infinite products to sum series.

Euler's Resolution of Cramer's Paradox

Rob Bradley, Adelphi University

In a September 1744 letter, Gabriel Cramer introduced Leonhard Euler to a problem in the theory of cubic curves, the generalization of which has become known as Cramer's Paradox. In a 1750 paper (E147), Euler eventually proposed a resolution of Cramer's Paradox by introducing a notion related to linear independence. In Euler's October 1744 reply to Cramer's letter, which has only recently come to light, he correctly identifies the direction in which the paradox ought to be resolved, arguing by analogy in the case of conic sections. This talk examined Euler's arguments in both the 1744 letter and the 1750 article.

A Birthday Gift for Euler

Munibur Rahman Chowdhury, University of Bangladesh

This talk gave an account of Euler's seminal contribution to the theory of residues (1761) culminating with Euler's theorem a φ(n) ≡ 1   (mod n) for every integer a coprime to n, and Euler's formula φ(n)= n Πp/n(1-  1/p). This work of Euler's is one of the sources of group theory. Group theory was used in the talk to prove these results, assuming only a budding acquaintance with the group concept. Everything else was developed ab initio, although a rudimentary knowledge of elementary number theory was advantageous. On the occasion of the tercentenary of Euler's birth, it is proposed that the multiplicative group of the prime residue classes modulo n be called the Euler group modulo n, and be denoted by En.

How Euler Built the Britannia Bridge

Lawrence D'Antonio, Ramapo College

It is a remarkable but little known fact that Leonhard Euler built the Britannia Bridge connecting Wales and the Isle of Anglesey. The bridge, considered a marvel of engineering for its time, was constructed in 1850. This talk consisted primarily of an explication of the contradictions contained in the previous sentences.

Euler's Use of Divergent Series

Craig Fraser, University of Toronto

This talk looked at some of Euler's papers on divergent series, situating his analysis and understanding of the subject with respect to the outlook of late nineteenth-century theory of summability. Euler's definition of the sum of a divergent series is considered and compared to the definitions of sum given by such later analysts as Hölder and Cesàro. In addition to its intrinsic interest, this subject offers general perspectives on the historical development of mathematical ideas.

What is the 'Birthday' of Elliptic Functions?

Adrian Rice, Randolph-Macon College

On December 23, 1751, Euler received a copy of a paper by Count Giulio Carlo de' Toschi di Fagnano on the lemniscate, which directly inspired the creation of Euler's general addition theorems for elliptic integrals. After his major contributions to the subject and the subsequent development and systematization of the theory by Legendre, elliptic functions became one of the dominant areas of mathematical research during the 19th century, leading Jacobi to call December 23, 1751 'the birth day of the theory of elliptic functions.' But to what extent can the subject be said to have been born with Euler in 1751? After all, several other mathematicians, including Jacobi himself, are often credited with laying the foundations of what was to become the theory of elliptic functions, in which case its 'birthday' could be anywhere from 1694 to 1829. By looking at the contributions of Euler, together with those of four other mathematicians, this talk examined whether the theory of elliptic functions really did begin in 1751, or whether there is another date that could more accurately be described as 'the birth day of the theory of elliptic functions.'

How Did Euler Change Mathematics?

Ruediger Thiele, University of Leipzig

On the one hand, Euler best represented the natural sciences in the middle of the 18th century; on the other hand he was known as analysis incarnate. Indeed, it was mathematics—especially the rising analysis—which served for Euler as the ground on which he started his investigations. From this viewpoint it is interesting to see the way in which he changed mathematics and our view of nature. This lecture briefly discussed some essential points of the transition.