Euler: The Mathematics of Musical Sadness 149
Euler thus discovered not just the first important insights that later grew into the field
of topology but also, more deeply, indexing as a crucial (and novel) tool of what became
the topological approach itself. Music was a particularly appropriate first venue for this
new topological thinking because musical intervals do not have the kind of spatial structure
that seems to govern elementary geometry. The lack of visible evidence — and his judgment
of the insufficiency of the traditional criterion of “ simplicity ” of ratio — opened the door
to his definition of degree, tied to the sensual criteria of suavitas. After Euler took this
initial step away from the traditional givens of mathematics, such as pure ratio, it was
probably easier to think in essentially the same way when he came to the bridge problem
and then to polyhedra. To be sure, the concept of degree was already familiar in the realm
of algebra, such as the degree of a polynomial equation. But the example of music required
a still bolder application of this general concept of degree in a case where it has no obvious
prior meaning, unlike the algebraic degree of an equation that is manifest in the highest
power of the unknown.^45 In the cases of music, bridges, and polyhedra, Euler had to devise
a degree for each that would have a decisive, invariant significance; this required him to
discern from a number of surface details those that could constitute the kind of parameter
that would answer his questions. For Euler, musical questions opened the way to a new
mathematics.