![]() To enter a land area and then leave it requires that there be two lines (bridges) connecting each dot. When the problem is recast as a network, it is evident that each node is connected to every other node by an odd number of lines – three or five. Map of Königsberg converted to a network with nodes, demonstrating Euler’s argument () And you can move the nodes around all you want, as long as you keep the connections intact. So you can reduce the areas to points, or nodes, and treat the bridges as lines that connect the nodes. The shape of the areas does not matter, nor the routes between the bridges. Euler solved the problem, and to do so he had to invent something called graph theory and found a discipline called topology.Įuler pointed out that everything about the problem is irrelevant except the seven bridges and the areas they connect. The best mathematicians of the day, including all of the Bernoullis, were unable to say whether it is possible or impossible, much less come up with a proof. Someone before Euler had posed a problem, as mathematicians liked to do in those days, which seems very simple: Is it possible to set out from one part of the city and cross each of the 7 bridges once and once only? A variant asks the same question but requires that the bridge-crosser return to the starting point. ![]() A late 16th-century map shows the situation ( second image). In Euler's time, there were seven bridges that connected the two islands and the two parts of the city on both sides of the river. The first problem is called "The Seven Bridges of Königsberg." Königsberg, now Kaliningrad, sits on the Pregel River near the Baltic Sea the city also includes two large islands in the river. Six bridges are visible, the seventh is upstream from lower right () Map of Königsberg, hand-colored woodcut, late 16th-centeury. However, there are two topics that we can understand, one of them investigated by Euler himself, the other more recently but named after Euler, and I thought we could look at these and use them to better appreciate Euler's genius. One can say that Euler brought new understanding to the nature of integral and differential calculus, and that he did important work on power series, and proved significant theorems about prime numbers, but that means very little to most of us who are not serious mathematicians. Mathematics being what it is, it is hard for ordinary folk to appreciate the nature of Euler's accomplishments. Later, he lost sight in his left eye as well, but it didn't seem to slow him down. Petersburg, and by the time he went to Berlin, he was completely blind in his right eye. Euler began losing the sight in his right eye during his first stint in St. He essentially spent his entire life being paid by enlightened sovereigns to prove abstract mathematical theorems, and all his patrons seem to have been quite happy to do so. Petersburg, which he did in 1766, and he lived out the rest of his life there, dying in 1783. Russia was meanwhile going through some turbulent years, but when Catherine the Great came to power, she persuaded Euler to return to St. After 14 productive years in Russia, Euler was invited by Frederick the Great of Prussia to move to the Academy in Berlin, where he was even more productive for the next 25 years. Petersburg, he persuaded them to invite young Euler as well. Born in Basel, he was tutored by Johann Bernoulli of the prominent Bernoulli family, all excellent mathematicians themselves, and when son Daniel Bernoulli was invited to join the Russian Academy in St. In a century filled with great mathematicians, Euler was, by general agreement, the greatest of them all. Leonhard Euler, a Swiss mathematician, was born Apr.
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