How Euler Did It

With the recent publication of five 'outstanding' books on the life and work of Leonhard Euler, the MAA has contributed magnificently to celebrating the 300th anniversary of Euler’s birth. Amongst that group of five is this book by Ed Sandifer, but the MAA set the ball rolling eight years ago, when it published William Dunham’s acclaimed book Euler: The Master of Us All  [1]. Before that event, and prior to these more recent MAA publications, it would have been necessary to browse a variety of books on the history of mathematics in order to extract examples of ‘how Euler did it’. From now on, a very pleasant alternative to that circuitous process is provided by this range of highly readable books, collectively providing us with compact and insightful overviews of Euler’s life and work.

How Euler Did It is a collection of 40 columns about the mathematical and scientific work of this great 18^th century Swiss mathematician. These columns appeared monthly on MAA Online between November 2003 and February 2007. They now form the 40 chapters that make up this book, and each is a self-contained account of a particular aspect of Euler’s work. The book is organised into four parts, with six chapters on Geometry, six chapters on Number Theory, five chapters on Combinatorics and twenty-two chapters on Analysis. For those wishing to examine the original sources, used by the author for the writing of the online columns, each chapter concludes with specific references to their location within the Euler Archive, at http://www.eulerarchive.org .

Each chapter commences with a discursive introduction, providing a wide thematic context for discussion of the ensuing mathematics. These are analysed with great clarity by the author, whose style of writing encourages the reader to pick up pen and paper and experiment with Euler’s ideas. Typical of this format is the introduction to chapter 13, where Euler’s approach to experimentation and proof are compared to those of Lebniz and Newton. Specifically, we learn how his approach to mathematics in early life was Leibnizian and Cartesian, but it later became more Newtonian. This leads to discussion of the role of proof in Euler’s work and takes place alongside analysis of the way in which he examined properties of numbers of the form 2aa + bb, where a and b are relatively prime. And this reminds me that very little is lost in Sandifer’s transciption of Euler’s ideas, because much of the original notation is retained and the book contains numerous diagrams taken from Euler’s publications.

Due to the sheer size of Euler’s mathematical output, Ed Sandifer’s task has been made easier and yet more difficult. ‘Easier’ because there is so much material from which to choose, and yet ‘difficult’ due to the process of ensuring that the selection is a fair representation of Euler’s achievements. After all, it’s quite impossible, in a single book, to cover the total work of a man whose mathematical and scientific output occupies 640 volumes pertaining to number theory, analysis, geometry, astronomy, graph theory, differential equations, logic etc. But whether or not the 40 themes covered in this book are truly representative (and I feel they are), Ed Sandifer’s exposition is nonetheless both informative and entertaining.

The first chapter (Euler’s Greatest Hits) summarises the results of an informal poll, conducted within a group of about 30 mathematicians. This was playfully designed to gauge the ten most popular of results due to Euler. Voted first was the Basel problem, i.e., determining the sum of the series of reciprocals of the squares. Second was his polyhedral formula and third was the famous equation e^πi =  –1; but information as to the remaining six can be ascertained only by purchasing the book!

In the chapter on ‘Piecewise Functions’, the narrative focuses upon Euler’s consolidation function concept and the manner in which he planted it firmly at the heart of analysis. Strangely, such conceptual developments rarely receive the popular acclaim normally accorded to the invention of theorems and important equations. And the same comment applies to advances made in the use of mathematical notations (Cajori [2] lists Euler, alongside Leibniz, as being one of foremost providers of modern notation).

Ensuing chapters reveal some of the vagaries in Euler’s thinking. For example, the first two of the six chapters of geometry are concerned with Euler formula V – E + F = 2, and the author points to the errors of Euler’s ways in his supposed vindication of this result (which was first discovered, in a slightly different form, by Descartes). Much later, chapter 34 reveals Euler’s ‘deliberate vagueness’ about his ideas on sums and products, for which his thinking was based upon the spurious principles of continuity and continuation formulated by Leibniz. On the other hand, chapter 4 (‘Cramer’s Paradox) shows the full power of Euler in top gear, where his incisive logic resolved one of the early conundrums of algebraic geometry. In other parts of the book, it is explained how he would gradually refine his proofs by producing several different versions for the same theorem.

Speaking of geometry, I was struck by the very first comment of chapter 6 that discusses ‘The Euler-Pythagoras Theorem’. Here it is said that ‘Euler didn’t do a lot of geometry’. Well, if Euler didn’t do a lot of geometry, who did? He certainly produced far more than Descartes, and his results on triangle geometry alone would have ensured him lasting fame. In fact, Coolidge [3] regarded him as one of the key figures in the history of algebraic and differential geometry. For instance, Euler was the first to make use of ideas that are central to intrinsic geometry and his work on developable surfaces inspired Monge make further advances in that area. He produced results on centres of similitude and devised formulae for rigid motions in space that enabled him to reduce equations of second order surfaces to canonical form, and Eule’s lemma for homogeneous polynomials appears in most introductions to algebraic geometry. Of course, as a proportion of his overall output, Euler’s work on geometry may have been relatively small; but, in mathematics, the power of ideas always outweighs prolificacy.

Anyway, the largest part of this book (Analysis) consists of the twenty-two chapters covering a surprisingly diverse range of topics, many of which are of an obviously analytic nature (‘Foundations of Calculus’, ‘Mixed Partial Derivatives’, ‘Divergent Series’ etc). But many other topics seem to have no ostensible analytic connotation. For example, chapter 22 deals with a problem concerning number theory, taken from ‘Calculi integralis’. For his solution of this, Euler invoked the used partial fractions — a topic that he had previously covered in his Introductio in Analysin Infinitorum . Yet it seems that he included this problem to illustrate the use of partial fractions in topics other than integration by parts. A similar comment applies to the previous chapter that deals with the solution of equations by recursion. In fact, it was only at this stage that I discovered any disparity between chapter contents and Euler’s orginal writing. Anyway, those minor errors will have been already detected by vigilant readers, when this chapter appeared online in June 2005.