## How Euler Did ItHow 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 |

Combinatorics | 83 |

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 |

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