The formula v - e + f = 2 has a lon

The formula v - e + f = 2 has a long and complicated history. It first appears in a letter from Euler to Goldbach dated 1750. However, Euler placed no restrictions on his polyhedra and his reasoning can only be applied in the convex case. It took sixty years before Lhuilier drew attention (in 1813) to the problems raised by polyhedra such as those shown in our Figs 1.2 and 1.3. The precise statement oftheorem (1.1), and the proof outlined below, are due to von Staudt and were published in 1847. Outline proo.(. A connected set of vertices and edges of P will be called a graph: connected simply means that any two vertices can be joined by a chain of edges in the graph. More generally, we shall use the word graph for any finite con-nected set of line segments in 3-space which fit together nicely as in Fig. 1.4. (If two segments intersect they are required to do so in a common vertex.) A graph which does not contain any loops is called a tree. Notice that for a tree, the number ofvertices