The Early Mathematics of Leonhard Euler

C. Edward Sandifer's book on  The Early Mathematics of Leonhard Euler gives readers the feeling of having studied alongside Euler himself. This book does more than just dazzle us with Euler's brilliance; it provides readers with a more comprehensive appreciation of Euler's work than has previously been attainable from a single volume. Here we learn about Euler's mathematics in breadth and in depth, we gain insight into the development of his ideas and his personal style, and we find historical information that places Euler's work in the context of his life and times. Sandifer's skilled exposition makes this book accessible to a general mathematical audience while including all of the essential technical details.

There have been many excellent books written about specific mathematical contributions of Leonhard Euler, such as Havil's Gamma: Exploring Euler's Constant, Nahin's Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills (which discusses exp(i x) = cos(x) + i sin(x)), and Varadarajan's Euler Through Time: A New Look at Old Themes (which mostly discusses Euler's contributions to Number Theory). Dunham's wonderful work,Euler: The Master of Us All, includes many of Euler's greatest theorems across many branches of mathematics. In the book The Early Mathematics of Leonhard Euler, Sandifer takes the next great step in writing about Euler by providing a complete portrait of Euler's early mathematical work — not just the important or the famous results, but all of them.

The style and content of Sandifer's book are unique among current books on Euler in a number of ways. First, as the title indicates, the book focuses on Euler's early mathematics: the body of his work written during his first stay at the Academy of Sciences at St. Petersburg (1725 to 1741), prior to his departure to work at the Berlin Academy. This restriction to specific dates excludes a number of popular topics from consideration in this volume, such as Euler's polyhedral formula. Second, this book is a systematic presentation of Euler's work. There are forty-nine sections in this book, one for each of the mathematical works written by Euler during this period. While many of these papers contain now-famous results, some are less than exciting; Sandifer presents the fundamental alongside the forgettable, in order to provide comprehensive coverage. Third, the sections are not arranged by subject matter but rather by date of writing. These dates were deduced and indexed by Gustav Eneström in a bibliography published in 1913; Sandifer stays true to this standard historical notation. For the reader particularly interested in reading Euler's "important" works, Sandifer identifies such works using a rating system (zero to three "stars") in the table of contents.

The topics that Euler investigated during his St. Petersburg years are as varied as they are important; truly, there is something for everyone in this book. There are far too many topics to be listed here; nevertheless, we shall attempt to provide an idea of the great scope of material. Readers with a preference towards analytic topics can enjoy Euler's discovery of the equality of mixed partial derivatives, development of integrating factors, solutions of differential equations, use of the Euler-Maclaurin summation formula, and creation of the calculus of variations. Geometric topics include approximations for π, methods for calculating (to a high degree of precision) the sine and tangent of angles, computations of arc lengths, and isoperimetric problems. Those interested in combinatorics will appreciate the sections about the invention of generating functions, solutions of partition problems, and analysis of the Königsberg bridges problem. Topics involving number theory appear in abundance; they include continued fractions, Fermat numbers, Fermat's Little Theorem, Pell's equation, the Chinese remainder theorem, Fermat's Last Theorem in the case n = 4, the Gamma and Zeta functions, and the Basel problem (determining the sum of reciprocals of square integers). This collection also includes the first use of function notation, f (x). The book concludes with an essay by Euler, On the Utility of Higher Mathematics, translated into English for the first time by Sandifer and Robert Bradley — a most fitting way to end this book.

While the primary focus of this book is on forty-nine of Euler's works, readers will gain a much more comprehensive overview of Euler's achievement than this. There are numerous references throughout this book to Euler's other works, both the non-mathematical works written during this period and the mathematical works written after 1741. For example: Euler's first ten works (those with Eneström index numbers E1 through E10) contain five that may be considered "mathematical", and each of these has an entire section devoted to it. While reading the book, however, we see that each of the other five of Euler's first ten works are also mentioned. As one further example: One of Euler's early mathematical papers, E54, includes Euler's first proof of Fermat's Little Theorem (and his first use in print of the technique of mathematical induction). In the section devoted to E54, Sandifer also briefly discusses future papers of Euler, even though they occur after 1741, including E134 (a similar inductive proof of Fermat's Little Theorem, written in 1747), E262 (a completely different proof of Fermat's Little Theorem, written in 1758), and E271 (where Euler proves the full Euler-Fermat theorem, written in 1760).

The presentation of each of Euler's papers is a pleasure to read. Sandifer's exposition has the feel of an experienced tour guide; he shows us the way through each paper and writes about each as if it were an old friend. He is able to point out the marvelous ideas contained in each article, even those whose ultimate worth Euler himself did not recognize. Rather than just point to the important parts of each publication and move on, however, Sandifer elucidates and helps the reader to work through Euler's derivations; thus, the reader gains as much appreciation for the method and approach as for the end results.

Sandifer stays true to Euler's notation, using xx instead of x²; denoting sequences as A, B, C, D,... rather than a₁, a₂, a₃, a₄, ...; even copies of diagrams from original publications are included. Readers gain a real sense of Euler's style, implicitly by working through proofs and explicitly from Sandifer's commentary on the proofs.

As readers progress through the sections, they have the opportunity to watch Euler's style evolve over the sixteen year period detailed in this book. At the very beginning of this book, for example, we learn the characteristics of an early work of Euler. We later learn how Euler chooses his problems, from whom Euler receives inspiration (e.g., a discussion of his correspondence with Christian Goldbach), and where Euler publishes his results and why. We develop an appreciation for the way Euler structures many papers: proceeding from specific examples to general cases. The rare mistake of Euler is gently pointed out, and later corrections discussed. There is considerable cross-referencing; each article is placed in the context of related articles from Euler's past and future work, providing a truly unified treatment.

At regular intervals throughout the book, Sandifer presents "interludes": brief descriptions of events taking place throughout the world, of scientific academies and political turmoil, of Euler's scientific colleagues and other contemporaries, of events taking place in Euler's life, and of Euler's mathematical and nonmathematical work. This also serves as a unifying force, helping us to gain a better understanding of the life and times of Leonhard Euler, while placing his work in this greater context.

There is an old saying: you never really know a man until you walk a mile in his shoes. To paraphrase: you never really know a mathematician until you work through his theorems. And so readers of The Early Mathematics of Leonhard Euler will be rewarded with more than just an appreciation of Euler's mathematics — they will feel closer to Euler himself.

Lee Stemkoski is Assistant Professor of Mathematics at Adelphi University; with Dominic Klyve, he created the Euler Archive, an online repository of Euler's work.