146 Chapter 9
position alone, and investigates the properties of position; it does not take magnitudes
into consideration, nor does it involve calculation with quantities. ”^39 Euler ’ s letter of
1736, though, shows his puzzlement as to what this geometria situs really means: “ You
have assigned this problem to the geometry of position, but I am ignorant as to what
this new discipline involves, and as to what types of problems Leibniz and Wolff
expected to see expressed in this way. ” Euler ’ s paper was first presented in 1735; the
field became known as analysis situs and is now called topology, of which this paper
is one of its first great results.
Euler immediately generalized the K ö nigsberg problem to “ any configuration of the
river and the branches into which it may divide, as well as any number of bridges, to
determine whether or not it is possible to cross each bridge exactly once, ” which has come
to be called an Euler walk.^40 He reduced topography to alphabetic symbolism and derived
simple rules, though without defining a numerical index that would “ involve calculation
with quantities, ” as he put it.
Euler did devise such an index when he returned to the “ geometry of position ” in his
“ Elements of the Doctrines of Solids ” (1752), the first of two papers in which he studied
the relations between the number of vertices ( V ), edges ( E ), and faces ( F ) of a polyhedron
( figure 9.5 ).^41 Euler ’ s crucial innovation was defining the edge ( acies ) of a polyhedron,
which, curiously enough, had never before been stated. Euler also identified the polyhe-
dron ’ s faces ( facies ) and its angulus solidus , by which he means not “ solid angle ” (as a
subtended, finite angle) but the point from which such an angle emerges, only calledFigure 9.4
Euler ’ s diagram of the city of K ö nigsberg, the Kneiphof island (A) and the seven bridges over the River Pregel,
a , b , ... , g.