The Early Mathematics of Leonhard Euler, edited by C. Edward Sandifer, The Mathematical Association of America, Washington, D. C. 2007. 393 pp., $25.95(hardbound). ISBN 0883855593.
Leonhard Euler is certainly one of the most prolific mathematicians of all time, as Sandifer states in the preface, his exact rank depends on how you define the nature of the publications. There is a brief discussion of the ranking of Erdös versus Euler based on how coauthors are scored. What is not a matter of dispute is who was the dominant mathematician of the middle of the eighteenth century. In that time, Euler had no real peer.
This book is a listing and analysis of the first mathematical papers that Euler had published, they are sequentially segmented by year or a short interval of years. The first timeframe is 1725-1727 and the last is 1741. Approximately 50 papers are examined, each is summarized and annotated with explanations of the content and the historical value it had. A few considered to be the major ones of historical significance are given asterisks for emphasis.
Since these are all his early papers, some are fairly nondescript and even Sandifer says so. What is clear from these papers is a clear development of mathematics, both in content and in the notational representation. One paper contains the first use of now standard f(x) notation for functions and others the development of new techniques used in proof. When he was wrong, and even Euler was wrong, he advanced the field of mathematics. When you embark on even a rudimentary examination of mathematical history you realize how valuable modern notation is compared to the relatively crude forms used several hundred years ago.
Euler was truly a giant in mathematics, in this book you can read about the early emergence of his work. It sounds odd to say it but these fifty pages are just the preliminary material of his output. As singer Carly Simon put it, “Nobody does it better.”
This book was made available for free for review purposes