Euler at 300: An Appreciation

An explosion of Euler books.

Recent years have seen a marked increase in the attention paid to Leonhard Euler (1707-1783). The book under review is in fact the tenth review of a book about Euler in MAA Reviews , bringing the current list to

The collection [2] consists of articles written by many luminaries thoughout the last century and [4] consists of articles written for a 1983 conference. The other eight books are thoroughly new contributions. I found the MAA Online reviews of the first nine to be very useful in giving me a quick overview of all this material.

Why the sudden increase in attention? The obvious reason is that 2007 is Euler's tercentenary. In fact, books [1]–[5] above together form the official MAA Tercentenary Euler Celebration. It seems to me that there is also a second reason, namely a general sentiment that the very high level of esteem in which Euler is generally held is in fact not quite high enough. One of the lighter parts of [2] adds Euler to Eric Temple Bell's famous top three of Archimedes, Newton, and Gauss, combining four familiar images into a mathematical Mount Rushmore. That this would be a more just configuration seems an undercurrent throughout the books.

Contents of Euler at 300.

So what is in the tenth of these Euler books? There are twenty-one articles. Two are written by pairs of then-students. The rest are all single authored, in an echo of eighteenth century practice. Almost all the papers are written versions of talks given at meetings in the last seven years. The papers quite uniformly reflect their origins: they are aimed at a fairly wide audience and have a somewhat informal feel. The balance is good. In particular, the fact that Euler's work in applied mathematics is at least as extensive as his work in pure mathematics is duly reflected. I'll discuss four articles as representatives of the collection.

Dominic Klyve and Lee Stemkoski's paper The Euler Archive: Giving Euler to the World is aptly titled. Volumes of Euler's Opera Omnia have been appearing for a century and the collection is now at seventy-six large volumes with six more volumes to come. Unfortunately, as Klyve and Stemkoski point out, the currently published volumes cost $17,000. Klyve and Stemkoski, with student assistants and volunteer contributors, have created a free alternative, the online Euler Archive. This archive now contains almost all of Euler's work. Equally importantly, the process of translating Euler's work into English is well underway. The article describes how the Euler archive was created.

David Pengelley's paper Dances between continuous and discrete: Euler's summation formula illustrates why Euler was called "analysis incarnate." Divergent series appear, and one of Euler's correct principles is that these series can be used "until they begin to diverge." A typical application is Euler's quick calculation that 1000! ≈ 4.023872 × 10²⁵⁶⁷. This paper represents Pengelley's position that historical aspects of mathematics should be thoroughly incorporated into the mathematics curriculum. Arguments for this position are available on Pengelley's home page.

Carolyn Lathrop and Lee Stemkoski's paper Parallels in the Work of Leonhard Euler and Thomas Clausen focuses on three interesting parallels. One is via the "Quadrature of Lunes." A second is via Graeco-Latin squares; Euler conjectured around 1776 that there are no 6 × 6 such squares and Clausen announced a proof in a 1842 letter to Gauss. A third is via the Fermat numbers F_n = 2²ⁿ+1; Euler factored F₅ = 641 × 6,700,417 in 1732 while Clausen factored F₆ = 274,177 × 67,280,421,310,721 in a 1855 letter to Gauss. There is an air of mystery about Clausen. The authors report that he had a "brain inflammation" and his activities in 1834–1840 are unknown. Whether he actually had a non-existence proof for 6 × 6 Graeco-Latin squares is unknown, but the authors argue that he did and that the first published proof in 1900 followed the same lines. How Clausen factored F₆ is likewise also the subject of speculation, and the first published factorization was in 1880.

Ed Sandifer's paper Euler Rows The Boat explains how Euler's contribution to Naval Science is substantial: twelve papers and two books spread through his career. The focus of Sandifer's paper is Euler's work on propelling ships without sails. The ideas put forth by Jakob Bernoulli and then Euler range from "goofy" to "prescient" and are all illustrated by figures from original texts. This article is a longer version of the February 2004 installment of Sandifer's monthly column How Euler Did It. These columns constitute short independent excursions into Euler's work; many are collected in [3].

Euler in the classroom?

Questions of relative rank aside, it seems to me that Euler occupies a singular position in the history of mathematics from the point of view of undergraduate-level pedagogy. Material earlier than Euler tends to be more history than mathematics, and is often very foreign to undergraduates. Material after Euler is often too advanced for undergraduates. In contrast, translations of many of Euler's original works can be made directly meaningful to students and Euler's work has many points of contact with the undergraduate curriculum. The fact that Euler himself wrote many texts helps here too.

As a bonus, Euler's life story seems almost designed to fit into the classroom and attract student interest. His life is divided neatly into four parts — Basel, St. Petersburg, Berlin, St. Petersburg. The compelling narrative of Euler's life gives sense to his complicated and varying surroundings. For example, learning about Euler makes it easy to place Peter, Frederick, and Catherine the Greats in their proper historical contexts. Students might even be able to distinguish the eight mathematical Bernoullis without memorizing. Even the less-easily impressed could only regard Euler's life as heroic: serenity and a stable family life through deaths and other tragedies; sustained prodigious output through loss of sight in one eye and then the other.

The material on Euler accessible to undergraduates has vastly increased in the past ten years. It would now be easily feasible to insert a multiweek student-driven unit on Euler into a standard history of math course. We can thank many people for opening up this option, including the contributors to Euler at 300.

David Roberts is an associate professor of mathematics at the University of Minnesota, Morris.