“Read Euler, read Euler, he is master of us all,” LaPlace exhorted us. And it is true, Euler writes with unerring grace and ease. He is exceptionally clear thinking and clear speaking. It is a joy and a pleasure to follow him. It is especially so with Ed Sandifer as your guide. Sandifer has been studying Euler for decades and is one of the world’s leading experts on his work. This volume is the second collection of Sandifer’s “How Euler Did It” columns. Each is a jewel of historical and mathematical exposition. The sum total of years of work and study of the most prolific mathematician of history, this volume will leave you marveling at Euler’s clever inventiveness and Sandifer’s wonderful ability to explicate and put it all in context.

**Preface**

**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 $$F_5$$ (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)

**About the Author**

### About the Author

**C. Edward Sandifer** is Professor Emeritus of Mathematics at Western Connecticut State University. He earned his PhD at the Univeristy of Massachusetts under John Fogarty, studying ring theory. He became interested in Euler while attending the Institute for the History of Mathematics and Its Uses in Teaching, IHMT, several summers in Washington DC, under the tutelage of Fred Rickey, Victor Katz, and Ron Calinger. Because of a series of advising mistakes, as an undergraduate Ed studied more foreign languages than he had to, and so now he can read the works of Euler in their original Latin, French, and German. Occasionally he reads Spanish colonial mathematics in its original as well. Ed was the secretary of The Euler Society, and he wrote a monthly online column, “How Euler Did It,” for the MAA—this volume is a collection of some of those columns. He has also written *The Early Mathematics of Leonhard Euler* and *How Euler Did It*, both also published by the MAA, and edited, along with Robert E. Bradley, *Leonhard Euler: Life, Work, and Legacy*. He and his wife Theresa, live in a small town in western Connecticut. Ed used to be an avid runner and he has over 35 Boston Marathons on his shoes.

### MAA Review

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. Continued...