34 Chapter 2
blocks her advocacy of the alternative possibility “ that the celestial velocities are propor-
tioned by numbers. ”^29
In rebuttal, Geometry notes that no one really supposes “ that any celestial motions are
related as any one of the principal concords, ” which seems explicitly to recognize the
problem that Arithmetic had ignored. Despite this, Geometry does not abandon the concept
of celestial harmony in her alternative: “ However, should the celestial spheres produce
some concord while moving, this ought not be measured in terms of the velocities of the
motions, but rather by the volumes of the spheres, or the quantities of the orbs. ”^30 Both
these statements accord with Oresme ’ s own stated positions earlier in the work, thus giving
further evidence that Geometry finally wins the debate, even though we never get to hear
Apollo ’ s verdict. In the wake of his Proposition 11, Oresme had offered the same response
to the problem he disclosed that now appears in the mouth of Geometry: the harmonies
relate volumes and their correlate “ quantities of the orbs ” (probably meaning their masses,
proportional to their volumes). Neither Oresme nor Geometry abandons the concept of
celestial harmony, even in the face of its inaudibility and the incompatibility of the simplest
ancient versions based on the principal concords.
Geometry ’ s resolution also comes with an important corollary: to resolve Oresme ’ s
problem, celestial harmonies must involve some aspect of incommensurability. To that
end, Geometry suggests harmonizing celestial volumes, rather than velocities. The logic
of her statement bears close examination: by using volumes as a way to introduce incom-
mensurability, she seems to presume that the volumes of spheres have some kind
of incommensurability. This in turn seems to reflect Oresme ’ s knowledge of the work of
Archimedes, who had proved that the volumes of spheres are proportional to the cube
of their radii. As a consequence, commensurable spherical volumes have incommensu -
rable radii. 31
Oresme ’ s arguments about the irreversible and never-repeating course of the universe
are inextricably joined with arguments about mathematical incommensurability and with
the “ new song ” that results in ever-novel cosmic harmonies. In this intricate tapestry of
ideas, music plays an essential role mediating between astronomical, arithmetical, and
geometric ideas. Oresme celebrates this “ new song ” in his climactic image that “ the
heavens are like a man who sings a melody and at the same time dances, thus making
music in both ways — cantu et gestu — in song and in action. ”^32 Pure song is the melodic
and harmonic ideal, incarnating mathematical relationships in the visible dance of stars
and planets. The combination of new music with ancient results about incommensurability
led Oresme to reject the simplest versions of the music of the spheres even as he indicated
new possibilities for celestial harmony. Two centuries later, Music moved her sisters to
reconsider their relation entirely.