30 Chapter 2
Though Geometry does not really address the troubling sensual displeasure produced
by irrational proportions, she implicitly embraces it as the price of really interesting cosmic
music that will not repeat itself ad infinitum. Where Arithmetic had spoken only indirectly
of “ the architect ” (notably restrained in her reference to revelation), Geometry explicitly
describes God the creator and his artistic alternatives. In contrast to the pleasures offered
by Arithmetic, Geometry considers her own ideal student to be “ a subtle man [who] per-
ceives the beauty in much diversity, while an ignorant man, who fails to consider the whole,
thinks that the sequence in this diversity is confused, just as he who does not realize that
what we call an irrational ratio is part of our order and plan. And yet the infinite plan of
God distinctly realizes this diversity which, put in its proper place, is pleasing to the divine
sight and makes the celestial revolutions more beautiful. ”^14
Geometry then reinterprets Arithmetic ’ s biblical reference to “ number, order, and
measure ” to mean that measured magnitudes are just as necessary as pure numbers. Geom-
etry also discerns a wider horizon of mathematical possibilities for harmony, not just in
planetary velocities or periods (as Arithmetic had implied) but also in the “ magnitude ” of
the spheres, their weight or size, and hence by implication their spatial dimensions. Con-
sistent with her specific subject matter, Geometry emphasizes the full spatial reality of the
celestial spheres as structures with determined radii, not just a mathematical model (as
Ptolemy had argued) but a measurable geometric edifice.
The musical implications of these questions lead us back to the issue of audibility.
Arithmetic had treated celestial music as audible, hence excluding perceptibly inharmoni-
ous irrationals; Geometry, in contrast, cites the ancient authorities who treat cosmic
“ harmony ” as inaudible, in which case sensory disharmony would be an irrelevant crite-
rion. Even so, she keeps using musical language to describe the cosmic harmonies. Oresme
seems to want to hang on to the imaginative and artistic possibilities of musical discourse
even as he questions its sensory basis.
The ending of this singular debate leaves us hanging. After Geometry finishes speaking,
“ Apollo, believing himself adequately informed, ordered silence. ” But Oresme feels
“ astonished and confounded by the novelty of so many things, ” especially by the manifest
contradictions between the arguments of Arithmetic and Geometry. Perceiving this, Apollo
reassures him not to believe “ that there is genuine disagreement between these most illus-
trious mothers of evident truth. For they amuse themselves and mock the stylistic mode
of an inferior science. ” Apollo announces that he will now announce the truth in the form
of his judgment, so that “ with the most ardent desire did I await his determination, but,
alas, the dream vanishes, the conclusion is left in doubt, and I am ignorant of what Apollo,
the judge, has decreed on this matter. ”^15
This enigmatic interruption could be interpreted as a wry expression that we cannot,
after all, know the truth of such exalted matters, on the lines of Oresme ’ s skeptical account
of geocentrism. But a number of clues allow us to conclude that Oresme ’ s own opinion
lies finally with Geometry. We know, from several of his writings (including the Livre du