What Else Did Euler Do?

CHAPTER 3 WHAT ELSE DID EULER DO? Bridges THE PROBLEM On Sundays after church residents of the city of Königsberg in Prussia (Kaliningrad, Russia) in the 18th-century promenade around the city and greet friends as they cross paths. At that time, the city was set on both sides of the Pregel River (Pregolya River) and included two large islands which were connected to each other and the mainland by seven bridges. As the river flowed around Kneiphof, literally meaning pub yard and another island, it divided the city into four distinct regions. The seven bridges were called Blacksmith’s Bridge, Connecting Bridge, Green Bridge, Merchant’s Bridge, Wooden Bridge, High Bridge, and Honey Bridge.1 To amuse themselves, citizens attempted to find a way to take a walk around the city such that they would cross each bridge once and only once. A harmless amusement, but it would lead to a new branch of mathematical topology, which would have applications far beyond their imagination as they strolled around Königsberg. Over time, the town, bridges, and river have changed names, buildings have risen and been demolished, and the town’s found itself in different countries (Prussia, Germany, Russia); but the river’s always flowed just the same.2 Until the Second World War, seven bridges connected the landmasses separated by the River, shown here as they were in the 18th century.3 Figure 3.1. Euler’s drawing of Konigsberg Bridges in his 1736 paper, Solvtio Problematis Ad Geometriam Sitvs Pertinentis. Avctore 35 Chapter 3 Is it possible to take a walk around the town, starting at any point, crossing each of the seven bridges once and once only in any direction? We doubt that any hardy soul attempted all the possible promenades over the bridges – but if someone did so, it wouldn’t be of any special importance. A problem’s solution that will have significant benefits is one that does not apply to just one possible example of like problems, but to all of them. Here are some things that must be considered in seeking a solution to the problem. Where are you going to start and in which direction are you going to travel? Can you simplify the task by eliminating any unnecessary elements so you can focus what is essential? For instance, does it matter how deep the River Pregel is at any point? How about the dimensions of the bridges are the area of the regions connected by the bridges? Well, what is essential? Most important – can one explain why a particular solution does or doesn’t work? Here Is the Problem Solver St Petersburg, March 13, 1736. Leonhard Euler, a 28-year-old mathematician, is sitting at a desk in his office at the recently-founded Russian Academy of Sciences, busily writing a letter to a fellow mathematician, about a problem he has been studying. Euler squints slightly looking out of the window across the river Neva, then looks back at his desk and rubs his eyes. He’s recently recovered from a fever that almost cost him his life, and probably cost him the sight in his right eye. Carl Leonhard Gottlieb Ehler, mayor of Danzig, asked Euler for a solution to the problem of the seven bridges of Königsberg. In his reply to the mayor, Euler attempted to beg off, saying: Thus, you see, most noble Sir, how this type of solution bears little relationship to mathematics, and I do not understand why you expect a mathematician to produce it, rather than anyone else, for the solution is based on reason alone, and its discovery does not depend on any mathematical principle. Because of this, I do not know why even questions which bear so little relationship to mathematics are solved more quickly by mathematicians than by others.4 Nevertheless, Euler was intrigued, when he was told that no-one had been able to demonstrate whether or not it was possible to solve the problem. In his letter, he draws he draws a rendition of the seven bridges of Konigsberg (Figure 3.1). He wondered as he wrote: There are seven bridges. If the problem could be reduced to numbers, why couldn’t I find a mathematical approach to solving it? It’s nothing to do with mathematics – it’s a purely logical problem, but that’s what intrigued me about it.5 36.

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