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HISTORY OF MATHEMATICS: ON LEONHARD EULER (1707-1783)

The following points are made by Albert-Laszlo Barabasi (citation below):

1) On September 18, 1783, in St. Petersburg, Leonhard Euler (1707-1783) started the day as usual. He gave a mathematics lesson to one of his grandchildren and took up some calculations on the flight of balloons. Just three months earlier, south of Lyon, the Montgolfier brothers had launched an enormous balloon that rose 6500 feet into the air and landed safely about a mile away. Euler was working out the mechanics of the balloon's motion as the Montgolfier brothers were preparing to launch a sheep into the air in front of King Louis XVI in Paris, a flight that took place the next day, on September 19. Euler never heard about the event, however. After lunch, working with his assistants, he made some calculations on the orbit of the recently discovered planet Uranus. The equations introduced by him, capturing the planet's peculiar orbit, would lead decades later to the discovery of the planet Pluto. Euler did not live to witness that discovery either. About five o'clock in the afternoon, he suffered a brain hemorrhage and uttered, "I am dying," before losing consciousness. He died that evening, ending the most prolific career in mathematics of all time.

2) Euler, a Swiss born mathematician who spent his career in Berlin and St. Petersburg, had an extraordinary influence on all areas of mathematics, physics, and engineering. Not only was the importance of his discoveries unparalleled, their sheer quantity is also overwhelming. Opera Omnia, the still incomplete record of Euler's collected works, currently runs to over 73 volumes, 600 pages each. The last 17 years of Euler's life, between his return to St. Petersburg in 1766 and his death at the age of 76, were rather tumultuous. Yet, despite many personal tragedies, about half of his works were written during these years. These include a 775-page treatise on the motion of the moon, an influential algebra textbook, and a three-volume discussion of integral calculus, completed while he continued to publish an average of one mathematics paper per week in the journal of the St. Petersburg Academy. The amazing thing is that he barely wrote or read a single line during this time. Having partially lost his sight soon after returning to St. Petersburg in 1766, Euler was left completely blind after a failed cataract operation in 1771. The thousands of pages of theorems were all dictated from memory.

3) Three decades earlier, his sight intact, Euler had written a short paper addressing an amusing problem that originated in Konigsberg, a town not too far from Euler's home in St. Petersburg. Konigsberg, a flowering city in eastern Prussia, did not suspect in the early 18th century the sad and war-torn fate that awaited it as host for one of the fiercest battles of the Second World War. Contemporary etchings show a thriving city on the banks of the Pregel, where a busy fleet of ships and their trade offered a comfortable life to the local merchants and their families. The healthy economy allowed city officials to build not fewer than seven bridges across the river. Most of these connected the elegant island Kneiphof, which was caught between the two branches of the Pregel, with other parts of the city. Two additional bridges crossed the two branches of the river. The people of Konigsberg, enjoying a time of peace and prosperity, amused themselves with mind puzzles, one of which was: "Can one walk across the seven bridges and never cross the same one twice?" No one was to find such a path until a new bridge was built in 1875.

4) Almost 150 years before the new bridge, in 1736, Euler offered a rigorous mathematical proof stating that with the seven bridges such a path does not exist. He not only solved the Konigsberg problem but in his brief paper inadvertently started an immense branch of mathematics known as graph theory. Today graph theory is the basis for our thinking about networks. During the centuries after Euler it grew into a mature field, to which most great mathematicians contributed.

5) Euler's proof is simple and elegant, easily understood even by those not trained in mathematics. Nevertheless, it is not the proof that made history but rather the intermediate step that he took to solve the problem. Euler's great insight lay in viewing Konigsberg's bridges as a graph, a collection of nodes connected by links. For this he used nodes to represent each of the four land areas separated by the river, distinguishing them with letters A, B, C, and D. Next he called the bridges the links and connected with lines those pieces of land that had a bridge between them. He thus obtained a graph whose nodes were pieces of land and links were bridges.

6) Euler's proof that in Konigsberg there is no path crossing all seven bridges only once was based on a simple observation. Nodes with an odd number of links must be either the starting or the end point of the journey. A continuous path that goes through all bridges can have only one starting and one end point. Thus, such a path cannot exist on a graph that has more than two nodes with an odd number of links. As the Konigsberg graph had four such nodes, one could not find the desired path.

For our purpose the most important aspect of Euler's proof is that the existence of the path does not depend on our ingenuity to find it. Rather, it is a property of the graph. Given the layout of the Konigsberg bridges, no matter how smart we are, we will never succeed at finding the desired path. The people of Konigsberg finally agreed with Euler, gave up their fruitless search, and in 1875 built a new bridge between B and C, increasing the number of links of these two nodes to four. Now only two nodes (A and D) with an odd number of links remained. It was then rather straightforward to find the desired path. Perhaps the creation of this path was the hidden rationale behind building the bridge?

In retrospect, Euler's unintended message is very simple: Graphs or networks have properties, hidden in their construction, that limit or enhance our ability to do things with them. For more than two centuries the layout of Konigsberg's graph limited its citizens' ability to solve their coffee house problem. But a change in the layout, the addition of only one extra link, suddenly removed this constraint.

In many ways Euler's result symbolizes an important message of this book: The construction and structure of graphs or networks is the key to understanding the complex world around us. Small changes in the topology, affecting only a few of the nodes or links, can open up hidden doors, allowing new possibilities to emerge.

7) Graph theory boomed after Euler with contributions made by mathematical giants such as Augustin Cauchy (1789-1857), William Hamilton (1805-1865), Arthur Cayley (1821-1895), Gustav Kirchhoff (1824-1887), and George Polya (1887-1985). They uncovered just about everything that is known about large but ordered graphs, such as the lattice formed by atoms in a crystal or the hexagonal lattice made by bees in a beehive. Until the mid-20th century the goal of graph theory was simple: It aimed to discover and catalogue the properties of the various graphs. Famous problems included finding a way to escape from a maze or labyrinth, first solved in 1873, or finding a sequence of moves with a knight on a chess board such that each square is visited only once and the knight returns to its starting point. Some of the more difficult problems have gone unsolved for centuries.

Adapted from: Albert-Laszlo Barabasi: Linked: The New Science of Networks. Perseus Publishing 2002, p.9. More information at: http://www.amazon.com/exec/obidos/ASIN/0738206679/scienceweek

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