Leonhard Euler (1707 - 1783) in a 1753 portrait by Emanuel Handmann
(Öffentliche Kunstsammlung Basel, via MacTutor)
Euler has left his imprint in all branches of mathematics, and created some new ones, such as graph theory. He also contributed to physics, astronomy, and engineering - Wikipedia has a list of about 50 topics that are named after him. Ed Sandifer, in the February 2007 issue of his monthly column "How Euler did it", presents Euler's Greatest Hits in mathematics. Wow - so many stuff you have probably heard about: the polyhedral formula, the Königsberg bridge problem, the product formula, the Euler-Lagrange equations... Lubos has a nice post on the relevance of many of Euler's discoveries for string theory.
And, of course, there is the famous formula
about which whole books have been written. When I was thinking about something a tiny bit original to post about Euler, I thought I might try to trace the origin of this formula in Euler's writings. Mathworld gives as source of the full Euler identity
page 104 of Introductio in Analysin Infinitorum, Vol. 1. Lausanne, 1748. Now, this can be searched for in the Euler Archive, and we find it as entry E101: the Introduction to the Analysis of the Infinite, volume 1, where "Euler lays the foundations of modern mathematical analysis". The original text is available online from gallica.fr, and here is what we read on page 104:
There it is: From this can be seen how imaginary exponential quantities are reduced to sines and cosines of real arguments. It is
It's written in a now old-fashioned notation - in fact, the notation "i" was introduced only in 1777 by, guess whom, Euler - and in an even more old-fashioned language - but what it says is timeless!
There are several web sites commemorating Euler's birthday, for example at theMathematical Association of America, the Euler Society, and the Leonhard Euler Tercentenary - Basel 2007.
The Euler biographies at Wikipedia and MacTutor have much information and many interesting links. If you want to read a book about Euler's life and time, Leonhard Euler by Emil A. Fellmann has not too much maths and no technical details at all - but it is a very readable biography, has a good choice of illustrations, and conveys a lively picture of a 18th century life in science and mathematics. And there is even a comic about Euler.
TAGS: Leonhard Euler, Imaginary Numbers