148 Chapter 9
a “ vertex ” by Legendre (about 1794). If a solid polyhedron (closed but not necessarily
regular) is bounded by planar faces, Euler argued that “ the sum of the number of solid
angles plus the number of faces exceeds the number of edges by 2, ” or V + F – E = 2 (to
state it algebraically, which he does not do), now widely called “ Euler ’ s formula. ” Though
his arguments are somewhat flawed, the truth and depth of this proposition make it one
of his most celebrated results. It represents a three-dimensional generalization of the
K ö nigsberg bridge problem, in which the requirement of closure for the solid polyhedron
corresponds to the connectedness of an Euler walk, its return to its starting point.^42 By
identifying and tabulating V , F , and E , Euler was able to calculate the index V + F – E =
2 that also characterizes such polyhedra.
The structure of this relation between vertices, edges, and faces is strikingly similar to
the structure of the degree of agreeableness of musical intervals, s – n + 1. Without intend-
ing any direct connection between polyhedra and Euler ’ s hierarchy of musical intervals, as
such, both these relations ( V + F – E = 2 and s – n + 1) give the kind of general categoriza-
tion we now think of as topological and which Euler thought of in terms of geometria situs.
To be sure, these relations are very different, and not just in the objects they describe. Euler ’ s
formula is an equation describing a necessary and sufficient condition for closed, convex
polyhedra; his formula for musical degree defines a hierarchy between different intervals.
They both pose a general schematization that categorizes a vast domain, of polyhedra or of
musical intervals, respectively, subsuming many different individuals under a larger genus.
Thus, polyhedra of many different shapes and numbers of sides fall under Euler ’ s formula,
which (as modern topology phrases it) describes polyhedral surfaces of genus 0 , those
having no “ holes ” or “ handles. ” Later topologists generalized Euler ’ s formula to manifolds
of higher genus than zero (such as a doughnut whose hole gives it genus 1, for instance) by
defining the Euler characteristic χ = V + F – E. In this way, χ gives a “ degree ” of such
surfaces that is analogous to the musical degree d = s – n + 1, which gives the “ topology ”
of musical intervals, their general grades of classification.^43
Thus, Euler ’ s early work classifying musical intervals grouped different intervals under
a single degree, expressing a higher commonality among them, despite their differences.
He did so without any earlier precedent in mathematics, for the traditional hierarchy of
musical intervals was based on fairly arbitrary numerological criteria of “ simplicity. ”^44
Euler ’ s degrees group together intervals by cutting across these traditional classes; his
criterion for setting up his degrees is freely chosen according to his notions of what would
be more “ intelligible ” and hence more “ agreeable ” ( suavis ). In his musical work, Euler
first devised the general classificatory strategy that he then applied to the bridge problem
and later to polyhedra. To use a later mathematical term, his approaches in these cases
were isomorphic , that is, they had the same essential structure. Because the musical
example came first, it arguably was the arena in which he first found and applied the kind
of approach that he later (and perhaps without realizing it) then found appropriate to
bridges and polyhedra.