# How Euler Did It

MAA, Aug 30, 2007 - Mathematics - 237 pages
How Euler Did It is a collection of 40 monthly columns that appeared on MAA Online between November 2003 and February 2007 about the mathematical and scientific work of the great 18th century Swiss mathematician Leonhard Euler. Almost every column is self-contained and gives the context, significance and some of the details of a particular facet of his work. First we find interesting stories about Euler's work in geometry. In a discussion of the Euler polyhedral formula the author speculates about whether Descartes had a role in Euler's discovery and analyzes the flaw in Euler's proof. We also learn of Euler's solution to Cramer's paradox and its role in the early days of linear algebra. Number theory is well-represented. We see Euler's first proof of Fermat's little theorem for which he used mathematical induction, as well as his discovery of over a hundred pairs of amicable numbers, and his work on odd perfect numbers, about which little is known even today. Elsewhere in the book we learn of the development of what we now call Venn diagrams, what Euler knew about orthogonal matrices, Euler's ideas on the foundations of calculus (before the days of limits, epsilons and deltas), and his proof that mixed partial derivatives are equal.Professor Sandifer based his columns on Euler's own words in the original language in which they were written. In this way, the author was able to uncover many details that are not found in other sources. For example, we see how Euler used differential equations and continued fractions to prove that the constant e is irrational, several years before Lambert, who is usually credited with this discovery. Euler also made an observation equivalent to saying that the number of primes less than a number x is approximately x/Inx, an observation usually attributed to Gauss some 15 years after Euler died.The collection ends with a somewhat playful, but factual, account of Euler's role in the discovery on America.

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### Contents

 Eulers Greatest Hits 1 Analysis 3 19th Century Triangle Geometry May 2006 19 The EulerPythagoras Theorem January 2005 33 Amicable Numbers November 2005 49 Euler and Pell April 2005 63 Philip Naudés Problem October 2005 85 Derangements September 2004 103
 Arc Length of an Ellipse October 2004 157 Mixed Partial Derivatives May 2004 163 Goldbachs Series February 2005 167 Bernoulli Numbers September 2005 171 Divergent Series June 2006 177 Who Proved e is Irrational? February 2006 185 Infinitely Many Primes March 2006 191 Formal Sums and Products July 2006 197

 Piecewise Functions January 2007 115 Finding Logarithms by Hand July 2005 121 Roots by Recursion June 2005 127 Theorema Arithmeticum March 2005 133 A Mystery about the Law of Cosines December 2004 139 A Memorable Example of False Induction August 2005 143 Foundations of Calculus September 2006 147 Walliss Formula November 2004 153
 Estimating the Basel Problem December 2003 205 Basel Problem with Integrals March 2004 209 Cannonball Curves December 2006 213 Propulsion of Ships February 2004 219 How Euler Discovered America October 2006 223 The Euler Society May 2005 227 Index 233 Copyright

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Ed Sandifer is Professor of Mathematics of Western Connecticut State University. He earned his PhD at the University 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 he studied more foreign languages than he had to, 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. Now he is secretary of the Euler Society, and he writes a monthly on-line column, How Euler Did It, for the MAA. he has also written The Early Mathematics of Leonhard Euler, also published by the MAA, and edited along with Robert E. Bradley, Leonhard Euler: Life, Work and Legacy. He and hi wife Theresa, live in a small town in western Connecticut, and he has run the Boston Marathon every year since 1973.