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How Euler Did Even More

C. Edward Sandifer
Mathematical Association of America
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MAA Spectrum
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The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Joel Haack
, on

C. Edward Sandifer’s How Euler Did Even More is the second collection of his monthly columns from MAA Online, “How Euler Did It.” The first collection, also titled How Euler Did It, appeared in 2007 as part of the five-volume set published by the MAA in recognition of the tercentenary of Euler’s birth. It contained Sandifer’s columns from November 2003 through February 2007. This second collection contains his columns from March 2007 through February 2010, with the addition of two guest columns by Rob Bradley and one by Dominic Klyve. (Bradley assisted Sandifer with the details of the publication of this collection.)

There are several ways to read this book. First, one may choose simply to open it at random to read Sandifer’s discussion of how Euler attacked and thought about certain problems. Sandifer places Euler’s work into context of the mathematics of his time, then describes what Euler did and how he did it and why it mattered, keeping in mind the advice of John Fauvel that Sandifer references in How Euler Did It: “Content, Context and Significance.” An alternative would be to read the columns for particular topics that Euler considered; the columns are organized into sections on geometry, number theory, combinatorics, analysis, applied mathematics, and Euleriana. This last section includes two columns reflecting on Euler as teacher, two on light-hearted topics (Euler and the hollow earth and Euler and pirates), and one discussing of Euler’s fallibility.

A third way to read this book would seem to summarize a great deal of Sandifer’s writing on Euler. That is, one could use individual sections as invitations and guides to read Euler’s texts in their original languages or in translations. The background that Sandifer provides in each column, along with the sense of “here’s what Euler’s doing” will make reading Euler much more accessible. (As an aside, I am particularly appreciative of the way Sandifer consults what Euler actually wrote, rather than relying on secondary sources, in his discussions of what Euler did.)

In the first column of How Euler Did It, Sandifer reported about Euler’s greatest hits, as selected by a poll of the participants in the MAA 2007 short course at the Joint Mathematics Meeting. He addressed many of the top ten in How Euler Did It (the numbers in parentheses give the ranking of each of these): (2) \(V-E+F=2\), (5) the Euler product formula, (7) the density of primes, and (8) generating functions and the partition problem. He covered (half of 4) the Knight’s tour and half-covered (tied for 9) the Euler-Fermat theorem; he provided some coverage of (1) the Basel Problem. In How Euler Did Even More, we find discussions of (3) \(e^{\pi i}+1=0\) and (tied for 9) the Gamma function. Adding another of Sandifer’s 2007 MAA publications, The Early Mathematics of Leonhard Euler to the mix, we find that here Sandifer addresses (1) the Basel problem, (the other half of 4) the Königsberg bridge problem, (5) the Euler product, the beginnings of (6) the Euler-Lagrange necessary condition in the calculus of variations, (7) the density of primes, (8) generating functions and the partition problem, the beginnings of (tied for 9) the Euler-Fermat theorem, and (tied for 9) the Gamma function. In short, if a reader is interested in a top-ten result of Euler’s, consulting these three MAA volumes of Sandifer’s discussion of Euler’s work will provide an entree to the topic.

If you already have How Euler Did It, I can’t imagine that you’d not also enjoy How Euler Did Even More. If you haven’t yet dipped into these books, I’d encourage you to do so.

Joel Haack is Professor of Mathematics at the University of Northern Iowa.

Part I: Geometry
1. The Euler Line (January 2009)
2. A Forgotten Fermat Problem (December 2008)
3. A Product of Secants (May 2008)
4. Curves and Paradox (October 2008)
5. Did Euler Prove Cramer’s Rule? (November 2009–A Guest Column by Rob Bradley)
Part II: Number Theory
6. Factoring F5 (March 2007)
7. Rational Trigonometry (March 2008)
8. Sums (and Differences) that are Squares (March 2009)
Part III: Combinatorics
9. St. Petersburg Paradox (July 2007)
10. Life and Death–Part 1 (July 2008)
11. Life and Death–Part 2 (August 2008)
Part IV: Analysis
12. e, π and i: Why is “Euler” in the Euler Identity (August 2007)
13. Multi-zeta Functions (January 2008)
14. Sums of Powers (June 2009)
15. A Theorem of Newton (April 2008)
16. Estimating π (February 2009)
17. Nearly a Cosine Series (May 2009)
18. A Series of Trigonometric Powers (June 2008)
19. Gamma the Function (September 2007)
20. Gamma the Constant (October 2007)
21. Partial Fractions (June 2007)
22. Inexplicable Functions (November 2007)
23. A False Logarithm Series (December 2007)
24. Introduction to Complex Variables (May 2007)
25. The Moon and the Differential (October 2009–A Guest Column by Rob Bradley)
Part V: Applied Mathematics
26. Density of Air (July 2009)
27. Bending Light (August 2009)
28. Saws and Modeling (November 2008)
29. PDEs of Fluids (September 2008)
30. Euler and Gravity (December 2009–A Guest Column by Dominic Klyve)
Part VI: Euleriana
31. Euler and the Hollow Earth: Fact or Fiction? (April 2007)
32. Fallible Euler (February 2008)
33. Euler and the Pirates (April 2009)
34. Euler as a Teacher–Part 1 (January 2010)
35. Euler as a Teacher–Part 2 (February 2010)

Comments's picture

Euler generates excitement in everyone who reads his work, and no one captures this better than Ed Sandifer, so the new volume is very welcome. Maybe MAA could bring back the first How Euler Did It in a similar paperback format? The book collector in me would like to have a matched pair.