How Euler Did It

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MAA, Aug 30, 2007 - Mathematics - 237 pages
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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

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 Naudes Problem October 2005
85
Derangements September 2004
103
Analysis
113
Piecewise Functions January 2007
115
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
Estimating the Basel Problem December 2003
205
Basel Problem with Integrals March 2004
209

Finding Logarithms by Hand July 2005
121
Roots by Recursion June 2005 127
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
Arc Length of an Ellipse October 2004
157
Mixed Partial Derivatives May 2004
163
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
About the Author
237
Copyright

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References to this book

Euler as Physicist
Dieter Suisky
Limited preview - 2008

About the author (2007)

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.