Sunday, April 15, 2007

Leonhard Euler 300

Today is the 300th birthday of Leonhard Euler, the most creative and productive mathematician ever. He grew up in Basel, Switzerland, where he learned the nuts and bolts of the mathematics of his time with the Bernoullis, but left his home town at the age of 20 to never come back, and spent all of his later life in Berlin and Petersburg.

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, 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.



  1. Interesting! I didn't know Euler himself introduced the notation 'i'. A brilliant man. Though I can't say he is particularly good looking ;-) Best,


  2. Dear Bee,

    I didn't know Euler himself introduced the notation 'i'.

    I didn't know either until today. You can learn something when writing blog posts ;-)

    Though I can't say he is particularly good looking ;-)

    He had some kind of infectious desease when he was 31, which caused the loss of his right eye. Later, in 1771, he became even totally blind, and his house in Petersburg burned down. But in spite of all these camalities, he continued working and producing his amazing results.

    It seems that he had a quite happy family life. He was married, and the couple had 13 children - five of them survived childhood.

    And then all these hundreds of papers he wrote, that's completely crazy...

    Best, stefan

  3. I enjoy the history lessons.

    Navigating Celestial Currents by Erica Klarreich of Science News

    In the 18th century, European mathematicians Leonhard Euler and Joseph-Louis Lagrange discovered that in this rotating frame there are five gravitational sweet spots, now called Lagrange points. At these equilibrium points, the competing pulls on the third body balance each other, and the body remains motionless.

    See Genesis Spacecraft uses Tubes as Freeways

    When you look at the cosmos in this way it takes on a new frontier of gravitational inclinations.

    Graduating to cosmology is a string theorist's evolution? One cannot help looking at the universe with regards to gravity.zs

  4. A diagram of the Königsberg bridges

    I was interested in Topologies early history.

    "i" as a "imaginary number" in Dirac's matrices? I always thought it was Dirac who introduced the "i"

  5. Great post, Stefan. Thanks.

    With all the hullaballoo about the centenary of Einstein's Annus Mirabilus, I think this should be declared as the 'Year of Matheamtics'.

  6. The year 2000 was declared to be the "world mathematical year", keeping in tune with the anniversaries of more contemporary results.

  7. Euler's Equation unites algebra with analytic geometry. If you do the math you've done the world. Quantum mechanics has no corresponding connection to reality.

    Massive resources have been invested to fold gravitation into QFT. Wouldn't it be just like the universe to have success move in the opposite direction... after we get gravitation correct? Organic chemists call this umpolung, and it works.

  8. What's with the rag on his head? ;-) (that is not in any way some epithet, so no one should take it that way, thanks, I'm just curious as to this strange headwear)

    And the reason he could write so many papers and still have 13 kids is because his poor wife did all the work. :-)

  9. Euler was from another planet, the guy was so smart; and, what a pretty equation!

    Thanks for the post, changcho.

  10. Hi stefan, another amazing post, so much we have to be thankful for, there are so many Mathematical_genius's, again great thread, best pv.

  11. Arguably the greatest genius to ever live (right there with Newton, Einstein and Gauss)

    I think his eulogy sums it up best:

    "On the 7th of September 1783, after amusing himself with calculating on a slate the laws of the ascending motion of air balloons, the recent discovery of which was then making a noise all over Europe, he dined with Mr Lexell and his family, talked of Herschel's planet (Uranus), and of the calculations which determine its orbit. A little after, he called his grandchild, and fell a playing with him as he drank tea, when suddenly the pipe, which he held in his hand, dropped from it, and he ceased to calculate and to breathe. The great Euler was no more"


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