## Tuesday, January 13, 2009

### Euler

Leonhard Euler was one of the great mathematicians of all time, and arguably the most prolific. (Only Paul Erdős has published more papers, but Euler published more pages.) Euler's greatest contributions were to number theory, but he also did important work in algebra, calculus, geometry, and probability, as well as in the applied fields of acoustics, artillery, astronomy, finance, mechanics, navigation, and optics.

Euler's influence on the history of mathematics can best be illustrated by a listing of all of the topics in math that bear his name. This article would take me years to write if I were to try to explain all of these topics, and I could hardly do a good job of it. Instead I'd like to write about a few of the things that Euler considered "distractions." As well as being a prolific mathematician, Euler was a prolific puzzle solver and writer (is it any wonder that Project Euler bears his name?). A few of his distractions have had a deeper impact on mathematics than even Euler himself may have predicted they would.

Thirty-six Officers Problem

Arrange 36 officers in a 6x6 square so that one from each of six regiments appears in each row, and one from each of six ranks appears in each column. The problem is equivalent to finding two mutually orthogonal Latin squares of order six.

Euler conjectured that the Thirty-six Officers Problem had no solution when he first proposed it between 1779 and 1782 (accounts vary), but his conjecture wasn't proven until 1901, by Gaston Tarry. The more than century-long search for a proof led to important developments in combinatorics.

Seven Bridges of Königsberg

Perhaps the best known of the puzzles analyzed and solved by Euler, the Seven Bridges of Königsberg Problem was first posed by the curious citizens of Königsberg (now Kaliningrad, Russia). The town of Königsberg featured a pair of islands in the river Pregel. Seven bridges crossed the Pregel, connecting different parts of the town (see the map below).

The problem posed by the citizens of Königsberg was whether or not it was possible to walk across all seven bridges without crossing any bridge more than once. No one had been able to find a path that fit both conditions, but was such a path possible?

In 1736, Euler published a paper titled Solutio problematis ad geometriam situs pertinentis (The solution of a problem relating to the geometry of position) in which he gave a solution to the Seven Bridges of Königsberg. In order for a path traversing each bridge exactly once to exist, Euler said, either one of two conditions must be met. If the path were to begin and end at the same point, each land mass would need to have an even number of bridges connected to it. If the path were allowed to begin at one land mass and end at another, those two land masses could have an odd number of connecting bridges, while all the other land masses must have an even number of connecting bridges. The two paths described in Euler's paper would eventually take the names Eulerian circuit and Eulerian path, respectively.

Since the Königsberg bridges did not meet the criteria outlined by Euler, a walk that involved exactly one crossing of each bridge was impossible. Euler's paper was important not because it satisfied the curiosity of the townfolk of Königsberg, but because it laid the groundwork for the branches of mathematics now known as graph theory and topology.

Publications in Euler's later life, and after his death

When he lost the use of his right eye in 1738, Euler said "Now I will have less distraction." This proved prophetic, as his rate of publication subsequently increased. This productivity gain wasn't an isolated incident, as his rate of publication increased again after he lost sight in his left eye in 1766.

Euler died only moments after calculating the orbit of Uranus on September 18, 1783, at the age of 76. It took another 47 years after his death for all of the math that Leonhard Euler wrote during his lifetime to finally be published.

Notes
For a deeper look into the life of one of history's most fascinating mathematicians, see Euler: The Master of Us All.

This is a programming blog, so I'm assuming if you've read this far you've previously heard of Project Euler. If you haven't, I'm sure you'll have fun there.

I'd like to thank Jeff Moser for influencing my decision to write about Euler in a comment he made to my post on Gauss.

Jeff Moser said...

Nice post.

I picked up the "Euler: The Master of us All" at a math conference when I was in college. It's a fun book, sometimes a bit hard to digest in parts.

Between Gauss and Euler, I'd have to say that Gauss was probably a tad more pragmatic while Euler seemed to have more beautiful/wizard like results.

Regardless, both were brilliant men.

Who's next? How about an amateur like Fermat? :-)

Bill the Lizard said...

Jeff,

I agree that Euler had a few very magical results. I wrote and rewrote a section on Euler's formula about half-a-dozen times before I finally gave up and admitted to myself that I just don't fully understand the damned thing. I had also read some accounts of the formula being featured in the encounter between Euler and Diderot, but that story apparently has been retold so many times that it's now impossible to distinguish what the truth behind the story really is.

Fermat, interesting... :)

Chris Rathman said...

Seeing your reference here and a mention in a previous comment, I started looking at Project Euler. Fun stuff, though there goes my free time. The questions are expressed quite nicely and I like the way it measures progress. I can't say I like the forums (and the one's I've solved are locked).

I've managed to work through 30 of the problems this week using the Oz programming language. I've posted the Oz solutions, though I gather there are some who dislike such postings. The only thing I wish the site had was a better collection of the solutions in various languages.

Anyhow, thanks for pointing me in the direction of Project Euler.

Bill the Lizard said...

Chris,

I am generally opposed to posting solutions to problems that are from sites other than your own. I prefer hints and useful background information rather than answers and full code listings. I'm not sure why the forums are locked to new solutions, since that really is the right place to post them. Regardless, thanks for posting a spoiler warning with your code, and for not posting the actual answers.

Mitch Wheat said...

ah, a fellow Eulerphile!

Edu said...

Hi,

This is my firs time walking around your blog and seems good, so I will try to keep reading it. Talking about Mathematicians, I have to say that I really like Riemann. Specially the Riemann hypothesis, I have been always attraced by prime numbers and the Riemman hypothesis seems important in its pattern. Also Riemann introduce the widely know Integral.

In addition talking about Euler, prime numbers, and Riemann there is a relation betweent them that I really like the "Proof of the Euler product formula for the Riemann zeta function". I consider that proof somehow magical. Thanks for the post.

Bill the Lizard said...

Edu,
I share your fascination with Bernhard Riemann and his work. If you haven't read it already, I highly recommend John Derbyshire's Prime Obsession, all about Riemann and the Riemann Hypothesis.