All the Way with Gauss-Bonnet and the Sociology of Mathematics

I was stimulated to write this story by the discussion in The American Mathematical Monthly between Peter Hilton and Jean Pederson on the one hand and Branko Grunbaum and G. C. Shephard on the other hand [HP] [GS]. The discussion as well as my story involves the Euler-Poincare Number, alias the Euler Characteristic. The discussion centers on whether the Euler-Poincare Number should be discussed in a historical way without mentioning the vast and dramatic generalization and depth of understanding that this most interesting invariant has acquired in this century. My position in this discussion is that Topology should not be viewed as an advanced subject whose theorems and concepts should be avoided until graduate school. Rather it is the study of continuity, and thus underlies the most basic geometric results. In this paper I show how the basic concept of angle leads naturally to the basic topological ideas of degree of mapping and of the EulerPoincare Number. My stozy spans the history of mathematics. It concerns what may be the most widely known non-obvious theorem of mathematics and it contains the same stunning generalization that characterizes the recent history of the Euler-Poincare number. In fact, it concerns one of the most important and earliest of the applications of the Euler-Poincare number. It shows the fickleness of mathematical fame, it shows the unreasonable power of unreasonable points of view, and it shows how easy it is for mathematicians to miss and forget beautiful and important theorems as well as simple and revealing points of view. This is a history of the Gauss-Bonnet theorem as I see it. I am not a mathematical historian. I quote only secondary sources or first hand papers that I quickly scanned, and I did not conduct any thorough interviews. Nonetheless, I am writing this history because I have contributed the last sentence to it (for the moment). I especially want to acknowledge the help of Hans Samelson. His scholarship greatly altered the thrust of earlier versions of this paper. He discovered Satz VI. He informed me of many points in this history; about Gauss' work, Descartes work, and Hopf's work. And he was a student of Hopf who generalized the GaussBonnet theorem himself.