22 Chapter 2
repeatability that he phrased in musical terms. In this, differences between arithmetic and
geometry, music ’ s sisters in the quadrivium, came forward musically, mathematically, and
astronomically.
Oresme began by showing that the radii of the different celestial spheres cannot be
expected to be commensurable with each other, based on Euclid ’ s propositions about
spheres circumscribing regular solids.^3 He drew on his older contemporary, Johannes de
Muris, best known for theoretical writings on music, though Oresme went much further
in examining the consequences. He concluded that the radii of different spheres are far
more likely to be incommensurable than commensurable with each other, which still
remains possible, though improbable.^4 But if two celestial motions are incommensurable,
then any given initial position of the two bodies will never recur. His argument is simple;
assume hypothetically the contrary, that their motions (say, their velocities) obey a certain
ratio, say, 3:2. Then after six revolutions of the first body, both bodies will have regained
their initial position, the first body having “ lapped ” the second body twice over. If the ratio
were m : n , it would take m · n revolutions; if, on the contrary, their motions are not com-
mensurable, m and n are not any finite numbers, and neither is m · n. Hence, the initial
positions of these two bodies would never recur even after an infinite time. If so, the
cosmos will not return to any given initial configuration, continually assuming different
states from any that came before.
On the other hand, Plato had described a cosmic cycle of 26,000 solar years, the “ Great
Year, ” after which the planets would return to their initial configuration.^5 Oresme con-
cluded that no such recurrence was possible, hence no Great Year. He emphasized that his
result disproves astrology: the impossibility of recurrence disallows the recurrent astral
configurations on which astrological predictions depend. Learning this, he hoped that the
ignorant would abandon astrological determinism and understand their free will.
At many points, Oresme ’ s inclusion of musical matters plays a central role. His judg-
ments reflected his interest in new music and in new ideas, including the speculation that
the Earth might not be the center of the universe. More than century and a half before
Copernicus, Oresme ’ s writings give an invaluable view into the status of geocentric
cosmology. Though he addressed this issue at several points in his earlier writings, his
most extended treatment comes in his final work, Le Livre du ciel et du monde (1377),
a translation and extensive commentary on Aristotle ’ s On the Heavens written for
Charles V ( figure 2.1 ).
Oresme devotes considerable space to the suggestion that the Earth may not be at the
center of the cosmos and might move, rather than remaining at rest, as Aristotle had argued.
What Oresme calls the “ Italian or Pythagorean ” view places “ the sphere of fire ” at the
middle, so that “ the earth is a dark star moving in a circle around the center and ... this
is the cause of our nights and days. ” Explaining the heliocentric view, Oresme compares
the planets moving clockwise around “ the entire circumference of the wheel ... just like
people in a carole , ” a circular dance ( chorea in Latin) that during the twelfth century was