How Euler Did it (Spectrum)

"How Euler Did It" is a collection of the monthly columns that appeared on MAA Online between November 2003 and February 2007 about the life and mathematical and scientific work of the great 18th century mathematician Leonhard Euler. Almost every column is self-contained, and gives the context, significance and some of the details of a particular facet of Euler's work. Interesting anecdotes about Euler's work in geometry are described In a discussion of Euler's polyhedral formula the author speculates as to whether Descartes had a role in Euler's discovery and analyzes the flaw in Euler's proof. Euler's solution to Cramer s paradox and its role in the early days of linear algebra is also discussed. Some of Euler's work in number theory is described. Euler's first proof using mathematical induction to prove Fermat s little theorem is explored, as well as his discovery of over a hundred pairs of amicable numbers, and his proof for odd perfect numbers. 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 MAA Online 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. He also made an observation equivalent to saying that the number of prime number less than a number x is approximately x/ln(x), 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 of America.