Who are the ten best baseball players of all time? Or the ten best pieces of music? With so many excellent choices, who could ever agree on the answers?
Bill Dunham faced similar questions when he picked Euler's best work from over 35,000 pages, over 800 works. He had experience, though. When he wrote his excellent book, Journey through Genius, he narrowed the field to just twelve great theorems in mathematics, and won awards for his efforts.
In Euler: The Master of Us All, Bill Dunham describes some of Euler's most important contributions in eight mathematical fields, Number Theory, Logarithms, Infinite Series, Analytic Number Theory, Complex Variables, Algebra, Geometry and Combinatorics. These eight chapters, sandwiched between a Biographical Sketch at the beginning, and a few pages of Conclusions, make this a short book and leave the reader hungry for more. Dunham anticipated this in his Preface, where he confesses "I have omitted virtually ALL of Euler's work." He also admits that "Fifty different authors would come up with fifty different books (and I'd be interested in the other forty-nine)."
Most chapters follow a fixed pattern. First there is a Prologue, describing the state of a subject before Euler started his work. Then there is a section titled "Enter Euler," where some of Euler's contribution to the subject are described. Finally, there is an Epilogue, which sometimes describes subsequent results, sometimes provides modern proofs of Euler's results, and sometimes sketches some of Euler's related results.
One of my favorite chapters is the one on Analytic Number Theory, a chapter with a relatively short Prologue, because Euler virtually invented the subject. Dunham focuses on the results from one of Euler's earlier papers, written in 1737, when Euler was only 30. Euler had already solved the "Basel problem," summing the reciprocals of the square integers. Dunham put that result in the chapter on Infinite Series. In this chapter, Dunham shows how Euler used his solution to the Basel problem to sum some unusual series, like the reciprocals "whose denominators are one less than all perfect squares which simultaneously are other powers." The sum turns out to be 7/4 – π2/6, but that result isn't really very interesting. The method, though, is interesting.
Dunham then shows us how Euler uses that same method on any sum of reciprocals of powers to discover the sum-product equality for the zeta function. Of course, it wouldn't be called the zeta function until the next century, but he demonstrates the equality anyway.
As a corollary, Euler gives a new proof that there are infinitely many prime numbers. Knowing that ζ(1) diverges, since that's the harmonic series, he knows that the product representation of ζ(1) also diverges. The factors in the product representation have all the prime numbers in their numerators, and, for a product to diverge, it must be an infinite product. Hence there are infinitely many prime numbers, and the new field of Analytic Number Theory is born.
The chapter on Analytic Number Theory isn't quite typical. The others tend to focus on better known results. The Number Theory chapter tells us about Euler's sigma function, the sum of all the divisors of n, about his work with perfect and amicable numbers and his discovery that the fifth Fermat number is composite. In the Geometry chapter, we learn about the Euler Line. Euler's solution to the Basel problem is in the chapter on Infinite Series.
Of course, Dunham had to leave things out. We won't dwell on those here. We'll leave that to the authors of the other forty-nine books on Euler. The other forty-nine books about Euler's work don't exist yet, at least not in English. There are excellent books on Newton, Cantor, Fermat, Descartes, Hilbert, and several other great mathematicians.
Now there's one about Euler.
Ed Sandifer (email@example.com) is professor of mathematics at Western Connecticut State University and has run the Boston Marathon 26 times.