What is the use of going right over the old track again?There is an adder in the path which your own feet have worn. You must make tracks into the Unknown.—Henry David Thoreau1

In order to place Euler’s formula in a modern context, we must discuss a mathematical field called graph theory. This is not the study of graphs of functions that we encountered in high school precalculus (y = mx + b is a line, y = x2 is a parabola, and so on.). It is the study of graphs such as those shown in figure 11.1. They are made of points, called vertices, and lines joining these points, called edges.*

In 1736, during his first stay in St. Petersburg, Euler tackled the now famous problem of the seven bridges of Königsberg. ...

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