The Early Mathematics of Leonhard Euler.

The Early Mathematics of Leonhard Euler

Edward Sandifer Published by The Mathematical Association of America, 2007 400 pp., hard cover ISBN 0-88385-558-5 US$46.95

When you pick up this volume, you cannot help thinking: what a magnificent book! The high quality glossy coloured cover imparts a sense of value which is not dispelled when one opens the book. This is Volume I of the MAA Tercentenary Euler Celebration. It covers the period of Euler's life from 1725 to 1742--the period over which Euler wrote his first 49 mathematical papers.

The structure of the book is as follows. For each year (mostly) there is a short 'Interlude', which covers world events, happenings in Euler's life, a summary of Euler's other non-mathematical work, and a brief description of Euler's mathematical articles. This is followed by individual chapters allocated to the mathematical articles. These chapters are beautifully written, with excerpts from Euler's work interspersed with helpful commentary by the author about the context and the direction of the argument. Diagrams, which look to be scans of the originals, are included. There are also several scanned pages of the original printed Latin text, which give a feel of this early work. There is a detailed list of contents, an index, and a list of articles placed in topical groups, rather than chronological order. The quality throughout is first class.

Now, who would use this book? One would have to say it is tertiary rather than secondary in its level. It is a wonderful addition to mathematical history, and an invaluable resource for students of Euler. On the other hand, it is a book to be dipped into, rather than read right through. Also, one has to ask about the value of reproducing some of these articles. The (book) author gives his own, admittedly subjective, rating of the articles. He gives four articles ***, four **, six *, and the remaining thirty-five no stars at all! The highest rated articles deal with the Konigsberg bridges problem, the Basel problem (to find the sum of 1 + 1/4 + 1/9 + 1/16 + ...), an essay on continued fractions, and some observations about infinite series.

This book is a good resource for seeing the way a great mathematician of an earlier era tackled the problems of the day, and of the notation he used. It is also interesting reading how the mathematics of the time arose from contexts quite different from ours.

Scott Paul