This research quickly yielded Euler’s original latin publication in PDF form, as well as an informative a wikipedia entry on the Seven Bridges of Königsberg problem. Perusing the Latin publication by Euler is an interesting experience – particularly seeing Euler’s original drawings of the problem. Euler employs the universal point labeling system in his proof – A, B, C, D…which apparently transcends time and language. It also left me wondering if Euler himself ever drew a figure of nodes and edges to represent the problem, or if the jump from 1736 to 20th century logic came later. Euler's original paper is interesting to ponder for a few moments before reading the Wikipedia entry.
Wikipedia additionally features a nice entry on Euler –a renowned Swiss mathematician and physicist who is famous for Euler's formula in calculus, among other discoveries. For those who are unaware (or need refreshing), the Seven Bridge of Königsberg was a problem involving seven bridges crossing the Pregel River, Prussia, via two large islands in the middle of the river. The object was to determine if it was possible to go for a walk, crossing each bridge exactly once, beginning and ending in the same location. Königsberg was the capital of Eastern Prussia until WW II, when it was occupied by the Soviet Army and renamed Kaliningrad. Kaliningrad still exists today – although it’s unclear if all the bridges survived the Soviet Army. The following, however, is a nice picture of one of The Islands from the Königsberg bridge
problem.
(source: Wikipedia)
Coincidentally, Königsberg was also the name of a German cruiser in WW I. It’s unclear how or why Königsberg the city and Königsberg German warship were related (or maybe it’s just a small world?), but the warship faced a similar fate as the city – it was sunk by a British plane in 1915). Thus, neither the city nor the warship Königsberg remain.
The end of this story, which many are familiar with, is that Euler solved the Bridges of Königsberg problem. He did so by reducing the system of bridges to what we would now call a network: a system of nodes and edges. Ultimately, he argued that the feasibility of such a walk depended entirely on the degree of the nodes in the system. The walk existed only if there were 2 or 0 nodes of an odd degree. This discovery has lead to the existence of the so-called “Euler path” or Eulerian trail, as well as the Eularian circuit, and the beginning of networks theory. Euler and his paths and circuits remain entirely alive and relevant in modern science. A search for Euler circuits on Google scholar brings back 27,700 articles.
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3 comments:
The Latin manuscript does indeed make interesting reading. I studied Latin for six years in grade school and high school. Nevertheless, the paper is mostly unintelligible to me. I remember some, but not enough to do much with it. I'm really glad that I don't have to write my papers in Latin.
Anyway, the Latin stuff reminds me of an incident from grad school. There was a student in my department who had become fascinated with a somewhat odd problem in mathematical physics. Mark was spending more and more time trying to figure this out, and was spending less and less time on his dissertation project. He was several years ahead of me, and I think at this point he probably already should have graduated.
In fairness to Mark, the problem he was infatuated with was pretty interesting, in a quirky sort of way. And he was doing good work on it. As he was toiling away and working on tracking down past work, it occurred to him that it was possible that the problem he was working on had already been solved. This is pretty typical when doing research. What's not typical is that the person who Mark thought might have solved the problem was Jakob Bernoulli in the late 1600's.
A word about the Bernoullis. There's a whole family of them, many of whom have made important and enduring contributions to physics. There's a math joke that all differential equations are solvable in the limit that the number of Bernoulli brothers goes to infinity.
Anyway, Mark proceeded to track down the complete works of Jakob Bernoulli. They were several volumes and as soon as they arrived Mark opened them and discovered they were written in Latin. Anguished, he started roaming the department looking for someone who knew Latin. Somebody told him that I had studied Latin, so he sought me out.
I remember that he appeared in my office kinda late one evening, the complete works of Jakob Bernoulli in his arms. He asked with a tone of desperation if I could help him out. I felt bad for him; Mark was really nervous that he had been scooped by someone from the 1600's. I tried to explain that I really didn't remember enough Latin. I also really didn't have the time to go through multiple Latin volumes trying to find the phrase "hanging chain," which is what Mark was looking for. I suggested that he focus instead on looking for formulas that looked familiar.
Eventually Mark left my office, looking pretty sad. It gets worse. It turns out that Bernoulli did solve essentially the same sort of problem that Dan was working.
Nevertheless, there's a happy ending to the tale. Mark was nevertheless able to publish a nice paper on the topic in the American Journal of Physics. He subsequently found more time to work on his dissertation, graduate shortly thereafter, and as I recall got a decent job in industry.
To answer Iris's question about why the city and the German cruiser shared the same name: the Königsberg wasn't just the name applied to a single ship, but to an entire class of light cruisers. The majority of the cruiser classes in the WW1 era German navy were named after cities, especially industrial centers. (Battleships tended to be named after prominent people.) What is unique about the Königsberg class of light cruisers is that all four of its members were specifically named after cruisers that had already been lost in the Great War -- and this new design intended as an upgrade of sorts. Its members were: Königsberg, Emden, Karlsuhe, and Nürnberg.
Have fun everyone.
Andrew
"(or maybe it’s just a small world?)" I am SO TICKLED I get the math joke. I'm also grateful for Andrew Campbell's thorough answer, although I'm likely to forget it as much as Dave has forgotten his Latin.
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