The Genius of Euler: Reflections on His Life and Work

Leonhard Euler and William Dunham make for a potent combination. Dunham’s 1999 book Euler: The Master of Us All is still perhaps the best general introduction to the mathematical work of Euler available. Dunham also devotes chapters to Euler in his classic Journey Through Genius and in The Calculus Gallery. Euler lived from 1707 to 1783 and it’s a great pleasure that Dunham has given us this new book to mark the tercentenary.

In The Genius of Euler, Dunham plays the role of editor rather than author. To be sure, his chapter “Euler and the Fundamental Theorem of Algebra” and his introductory material are integral pieces in the collection, but much of the appeal of this book is the broad range of perspectives on Euler’s work. The chapters, collected from the various MAA journals and a few other sources, feature the work of more than thirty authors. About a quarter of the papers first appeared in a memorial issue of the Mathematics Magazine from 1983, the bicentenary of Euler’s death, but there is so much more here. The articles were originally published between 1872 and 2006. Among the authors there are many well-known mathematicians and historians of mathematics, including Boyer, Cajori, Erdös, Morris Kline, Pólya, W. W. Rouse Ball and André Weil.

Dunham divides book into two sections. Articles of a biographical nature and those which provide historical background to Euler’s life and work make up Part I. The larger second part of the book is devoted to articles that describe Euler’s mathematical achievements and the subsequent related work of other mathematicians.

Because Part I includes a number of biographies, there is some repetition in this section. Therefore, you may not wish to take the entire first section in order on the first reading. Still, it’s worth beginning with the first two pieces. The collection starts with a brief and very readable mathematical biography by Benjamin Finkel, who founded the American Mathematical Monthly in 1894 and included his article “Leonhard Euler” in its December 1897 issue. Also not to be missed on the first reading is Clifford Truesdell’s 30-page biography “Leonhard Euler, Supreme Geometer,” which follows immediately on Finkel’s piece. Truesdell (1919-2000) was one of the great Euler scholars of 20^th century. He edited three of the volumes of Euler’s Opera Omnia and also founded the journal Archive for the History of Exact Sciences. This entertaining and informative article originally appeared in a 1972 book and is slightly better known from the author’s 1984 collection an Idiot's Fugitive Essays on Science. Dunham’s inclusion of this wonderful but obscure essay is an indication that he really did his homework when compiling this volume.

Not all the pieces in Part I are biographies. Gerald Alexanderson, for example, gives an appreciation of Euler as expositor and Carl Boyer writes about Euler’s Introductio, “The Foremost Textbook of Modern Times.” Even among the biographies, J. J. Bruckhardt’s article brings something to the collection that is lacking elsewhere: a survey of Euler’s work in mechanics and astronomy. Whereas all other articles in this collection deal more or less exclusively with Euler the mathematician, it’s important to remember that Euler’s contributions to physics were as significant as his contributions to mathematics. Euler’s 1734 Mechanica set classical mechanics on its modern course, as a systematic study of differential equations rather an ad hoc collection of geometrical problems. Also, Euler’s first lunar theory made him a player in the longitude problem, and the recipient of ₤300 in prize money from the Longitude Commission.

Part I ends playfully with three poems, one of them by Dunham and Charlie Marion, arguing that Euler deserves a place in the “Mount Rushmore” of mathematicians, alongside with the usual trio of Archimedes, Newton and Gauss.

Part II of The Genius of Euler consists of 18 articles about Euler’s mathematical discoveries and achievements. It’s certainly fitting that the majority of these concern Euler’s work in analysis. This includes the oldest article in the collection, a paper on Euler’s constant by J. W. L. Glaisher from the 1872 volume of the British journal The Messenger of Mathematics. The 1974 article by Raymond Ayoub on the zeta function, which won the MAA ’s Lester R. Ford award for expository excellence, straddles the disciplinary boundaries between analysis and the second most prevalent topic of this section, number theory.  

Articles devoted entirely to number theory include one on pentagonal numbers by George E. Andrews, another on quadratic reciprocity by Harold M. Edwards, and a broader survey of Euler’s number theory by Paul Erdös and Underwood Dudley. Much as I enjoyed the Erdös/Dudley article, I have to take exception with their claim that Euler merely “almost proved” the case n=3 in Fermat’s Last Theorem, and their contention that Johann Bernoulli corresponded with Leibniz in 1669, when he was only 2 years old (both claims seem to come from uncritical reading of L. E. Dickson’s History of the Theory of Numbers). More seriously, I don’t understand why the authors harp on at such length about what Euler didn’t do: he didn’t attempt to give an asymptotic formula for the number of partitions of n as a function of n as Rademacher did in 1937, he didn’t find an integer solution to the equation a⁴ = b⁴ + c⁴ + d⁴ + e⁴, as Norrie did in 1911, and he didn’t prove the prime number theorem by the methods employed by Hadamard and de la Valle Poussin in 1896. On the last point, they seem unaware of how close Euler came to the prime number theorem in one of his 1737 papers (see Ed Sandifer’s March 2006 How Euler Did It column at MAA Online).

Part II ends on a high note with three great articles about Euler’s combinatorial and topological achievements. Pólya’s article on “Guessing and Proving” examines Euler’s tetrahedral formula as a case study in conjecture and proof. Brian Hopkins and Robin Wilson give us “The Truth about Königsberg,” contrasting the modern exposition of the famous problem of the Seven Bridges with Euler’s actual solution of 1736. The most recently published article in the collection is the piece by Dominic Klyve and Lee Stemkoski on Graeco-Latin squares, which appeared in the College Mathematics Journal in 2006. The authors, who are responsible for the magnificent Euler Archive (at http://www.eulerarchive.org ), were graduate students when they wrote this paper about these special magic squares, which some claim to be the precursor of Sudoku puzzles. Klyve and Stemkoski examine Euler’s work on Graeco-Latin squares and the 20^th century resolution of his conjecture that no such squares of order 4k + 2 exist.

The thing that appeals most to my inner geek is the Glossary, which first appeared in the 1983 memorial issue of the Mathematics Magazine. Compiled by St. Olaf College students Karl Anderson and Jeff Ondich, this is a 13-page dictionary of the many terms, formulas, equations, techniques and functions that bear Euler’s name.

The articles in The Genius of Euler are written for a general mathematical audience. They are accessible to faculty and graduate students and, in many cases, to undergraduates. The book does not assume any particular background in the history of mathematics. Indeed, historians will find it tiresome to read so many authors denigrating Euler’s “inattention to issues of convergence,” a classic case of imposing a modern point of view on a historical figure, rather than taking his measure relative the context of his own time.

Read this book for the sheer fun of it. It’s very entertaining and well-written — as an indication of the excellent expository writing, three of the articles won the MAA ’s Pólya Award and another won the Ford Prize. It’s also a very topical book, having appeared early in Euler’s tercentennial year. It seems a must for any undergraduate mathematics library and an excellent source of supplementary material for a course in the history of mathematics. I can also imagine it being used with great success as the primary textbook for a capstone seminar course in mathematics, centered on the life and achievements of Leonhard Euler.

Rob Bradley is professor of mathematics at Adelphi University. His main research interest is the history of mathematics. He is currently the president of the Euler Society and past president of the Canadian Society for the History and Philosophy of Mathematics.