386 12. MODERN GEOMETRIES
was available. The word topology first appeared in the title of his 1848 book Vorstu-
dien zur Topologie (Prolegomena to Topology). Like geometry itself, topology has
bifurcated several times, so that one can now distinguish combinatorial, algebraic,
differential, and point-set topology.
4.1. Early combinatorial topology. The earliest result that deals with the com-
binatorial properties of figures is now known as the Euler characteristic, although
Descartes is entitled to some of the credit.^42 In a work on polyhedra that he never
published, Descartes defined the solid angle at a vertex of a closed polyhedron to
be the difference between 2ð and the sum of the angles at that vertex. He asserted
that the sum of the solid angles in any closed polyhedron was exactly eight right
angles. (In our terms, that number is Air, the area of a sphere of unit radius.)
Descartes' work was found among his effects after he died. By chance Leibniz saw
it a few decades later and made a copy of it. When it was found among Leibniz'
papers, it was finally published. In the eighteenth century, Euler discovered this
same theorem in the form that the sum of the angles at the vertices of a closed
polyhedron was 4n - 8 right angles, where ç is the number of vertices. Euler noted
the equivalent fact that the number of faces and vertices exceeded the number of
edges by 2. That is the formula now generally called Euler's formula:
V-E + F = 2.
Somewhat peripheral to the general subject of topology was Euler's analysis
of the famous problem of the seven bridges of Konigsberg in 1736. In Euler's
day there were two islands in the middle of the River Pregel, which flows through
Kongisberg (now Kaliningrad, Russia). These islands were connected to each other
by a bridge, and one of them was connected by two bridges to each shore, the other
by one bridge to each shore. The problem was to go for a walk and cross each
bridge exactly once, returning, if possible to the starting point. In fact, as one can
easily see, it is impossible even to cross each bridge exactly once without boating
or swimming across the river. Returning to the starting point merely adds another
condition to a condition that is already impossible to fulfill. Euler proved this fact
by labeling the two shores and the two islands A, B, C, and D, and representing
a stroll as a "word," such as ABCBD, in which the bridges are "between" the
letters. He showed that any such path as required would have to be represented
by an 8-letter word containing three of the letters twice and the other letter three
times, which is obviously impossible. This topic belongs to what is now called graph
theory; it is an example of the problem of unicursal tracing.
4.2. Riemann. The study of analytic functions of a complex variable turned out
to require some concepts from topology. These issues were touched on in Riemann's
1851 doctoral dissertation at Gottingen, "Grundlagen fur eine allgemeine Theorie
der Functionen einer veranderlichen complexen Grosse" ("Foundations for a gen-
eral theory of functions of a complex variable"). Although all analytic functions of
a complex variable, both algebraic and transcendental, were encompassed in Rie-
mann's ideas, he was particularly interested in algebraic functions, that is functions
w = f(z) that satisfy a nontrivial polynomial equation p(z, w) = 0. Algebraic func-
tions are essentially and unavoidably multivalued. To take the simplest example,(^42) Much of the information in this paragraph is based on the following website:
http: //www.math.sunyeb.edu/ tony/vhatsnew/column/descartes-0899/cle3cartes2.html.