History of Mathematics: Euler

Leonhard Euler was born on the 15th of April in 1707 and was in my opinion the best mathematician of all time. In fact, he is so “awesome” that he is the only mathematician to have two numbers named after him. The ever so important Euler’s number used in calculus which is approximately equal to 2.71828, as well as the Euler-Mascheroni Constant otherwise known as Euler’s constant approximately equal to 0.57721. Euler didn’t focus his time on a single part of mathematics, instead he dipped his brilliance into almost all areas including geometry, infinitesimal calculus, trigonometry, algebra, graph theory, lunar theory, and continuum physics. If all of his work would have been printed, it would take up between 60 and 80 quarto volumes which are 8 page books front and back. Out of all of this work, my favorite is a problem Euler introduced named The Seven Bridges of Königsberg.

The Seven Bridges of Königsberg is such an important problem in the history of mathematics because the negative resolution essentially laid the foundations of graph theory and prefigured the idea of topology. The solution to this problem was considered to be the first theorem of graph theory. And, when Euler recognized that the important information was the number of bridges and the list of their endpoints rather than the exact location and layout of each bridge this was the beginning development of topology. The idea of topology is not concerned with rigid shape of objects.

This problem was based in the city of Königsberg in Prussia along the Pregel River. The city had two large islands within which were connected by seven bridges (Hence the name of the problem.) The problem Euler proposed was to find a walk through the city that would cross each bridge exactly one time. The islands couldn’t be reached by any route other than the bridges, and every bridge must have been crossed completely every time, in other words, you couldn’t walk half way across a bridge and turn around then cross the other half at a later time. A final criterion was that the walked didn’t have to start and end at the same spot. Euler had eventually proved that the problem didn’t have a solution, but the question was why?

Euler noticed that whenever someone enters a island, or vertex, by a bridge they will then have to leave by a bridge (an edge) except for the end points of the walk. Therefore, the number of times someone enters an island equals the number of times they leave it. From this observation Euler was able to conclude that the number of bridges touching a specific land mass must be even (even degree) in order for this walk to be possible or have a zero or two odd degrees. Each of the islands had an odd degree thus providing a contradiction to the proposition of crossing each bridge only once.

Such a walk as this is now referred to an Eulerian path which was later proven by mathematician Carl Hierholzer. The lack of an Eulerian circuit, where you begin and start at the same point is another reason this problem had no solution.

To put more of a stamp on the start of graph theory, Euler also discovered the formula V – E + F = 2 which relates the number of vertices, edges, and faces of a convex polyhedron and now known as the Euler characteristic for the graph.

The Seven Bridges of Königsberg and its impact on graph theory and topology is just one of the many impacts Euler made on the history of mathematics and is why I find him so amazing. Without his works, who knows where our math would be today. Euler is definitely a Math Avenger in my book.