# How Euler Did Even More

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*C. Edward Sandifer*

MAA Press: An Imprint of the American Mathematical Society

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.

#### Reviews & Endorsements

… 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. … A third way to read this book would seem to summarize a great deal of Sandifer's writing on Euler. … If you haven't yet dipped into these books, I'd encourage you to do so.

-- Joel Haack, MAA Reviews

… On the whole, this collection is notable for the clarity of its exhibition, the wide range of subjects, and the sophistication of its mathematical treatment. One comes away with a renewed appreciation for the genius of Euler, as well as an improved understanding of what mathematical practice in the eighteenth century really looked like. Anyone with an interest in Euler or the development of mathematics in the eighteenth century will find a wealth of important material here.

-- Douglas M. Jesseph, Mathematical Reviews Clippings

… Like the other books in the series this one is also very readable and gives once more insights into the works of Leonhard Euler.

-- Zentrallblatt

# Table of Contents

## How Euler Did Even More

- Cover cov11
- Half Title i2
- Copyright Page ii3
- Title Page iii4
- Series Page iv5
- Contents ix10
- Preface xi12
- Part I: Geometry 116
- Part II: Number Theory 3954
- Part III: Combinatorics 5974
- Part IV: Analysis 8196
- 12 e, π and i: Why is “Euler” in the Euler Identity (August 2007) 8398
- 13 Multi-zeta Functions (January 2008) 88103
- 14 Sums of Powers (June 2009) 95110
- 15 A Theorem of Newton (April 2008) 101116
- 16 Estimating π (February 2009) 109124
- 17 Nearly a Cosine Series (May 2009) 119134
- 18 A Series of Trigonometric Powers (June 2008) 124139
- 19 Gamma the Function (September 2007) 132147
- 20 Gamma the Constant (October 2007) 136151
- 21 Partial Fractions (June 2007) 141156
- 22 Inexplicable Functions (November 2007) 146161
- 23 A False Logarithm Series (December 2007) 154169
- 24 Introduction to Complex Variables (May 2007) 160175
- 25 The Moon and the Differential (October 2009—A Guest Column by Rob Bradley) 166181

- Part V: Applied Mathematics 173188
- Part VI: Euleriana 207222
- About the Author 237252