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Summary: A fascinating synopsis of the most prolific mathematician
Review: To say that Leonhard Euler possessed mathematical insight is like saying that the last year has been an interesting one in American politics. One would also have to add many superlative qualifiers in front of it if you were to use the word prolific to describe his published mathematical output. The St. Petersburg Academy of Sciences was still publishing his material a quarter century after his death. As the author points out, nothing short of a forklift will allow you to carry a set of his published works around and the ongoing project to publish them was started in 1911 and is still incomplete! With that as a backdrop, it is a challenging task to "squeeze" the flavor of Euler's work into a mere 180 pages that can be balanced on a finger. However, Dunham proves equal to the challenge, providing a tantalizing whiff of most of the areas that Euler contributed to.
Many of the proofs presented here take avenues that lead to the solution but leave you puzzled. It is a tribute to his genius that Euler could see that a seemingly irrelevant and complicating operation is in fact a disguised giant step towards the solution. To think nothing of the fact that fathering thirteen children, moving from St. Petersburg to Berlin and then back again and losing his sight in one eye and then the other did not slow him down at all. It is also ironic that he wrote a popular science book that was a bestseller in his time. The book, Letters to a German Princess, was translated into many languages and was even published in the United States in 1833.
No doubt there are people who will look at this book and express regrets concerning what was not included. Please be gentle with them. This is a book that could only provide brief glimpses into the works of the most prolific mathematician of all time. And in that sense, Dunham is very successful. Each proof will dazzle you with the combination of simplicity and style.
Published in Journal of Recreational Mathematics, reprinted with permission
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Summary: William Dunham has done it again!
Review: With the publication of this, his third book, Dunham has once more shown himself to be a master himself of mathematical explanation. Unlike his previous two books, The Mathematical Universe and Journey Through Genius, which covered results by a variety of mathematicians, this book focuses on selected results that sprang from the remarkable mind of Leonard Euler, one of the most prolific and important mathematicians of all time. What sets Euler apart is not only the vast quantity of his output (the publication of his collected works, the Opera Omnia, spans six dozen volumes, or over 25,000 pages in all!), but also the breadth and originality of his work. Not only did Euler contribute to a wide array of mathematical fields -- from number theory to complex analysis to geometry -- but in many cases, he was the founder of those fields. For example, Euler invented the field of analytical number theory, and he was the first mathematician to recognize the importance of and to discover the important properties of complex numbers. This book in many ways resembles Dunham's Journey Through Genius. As in that book, Dunham has selected 15 or so theorems to present in detail, and he makes an effort to keep the proofs similar in spirit to the original proofs. Although the proofs are complete and the book is full of equations, they are accessible to anyone with a high school level of mathematics education. But in addition to the proofs, Dunham also provides historical context, as well as commentary on how later mathematicians used and improved upon Euler's work. For example, we learn that Euler began to loose the sight in his right eye at the age of 32, and that despite his virtual blindness by the age of 65, he continued his prolific rate of output until his death at age 84. The book's title is taken from a quote by Laplace, who said, ``Read Euler, read Euler. He is the master of us all.'' Indeed, if you have any interest in mathematics, you will almost certainly find yourself in complete agreement with Laplace's sentiments by the time you finish reading this wonderful book. ...
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