1 The Birth of Science - MSRI

3.7 From the Closed World to the Infinite Universe 79
ence was to translate in mathematical terms an assumption that the ratio
is too small to be measured or estimated from observed data. To admit the
Aristarchan statement by assumption is not simply equivalent to saying
that one length is negligible with respect to the other in calculations; it constitutes
rather an attempt to construct a model in which lengths form what
we would call a non-Archimedean set. In particular, to think that the stars
lie on a “sphere” whose radius is incommensurably greater than any observable
length is but a step away from introducing a mathematical model
where the “celestial sphere” is a conventional and useful way to represent
the set of directions. This step was effectively taken, as seen from the fact
that Geminus, in his compilation dating probably from around 50 B.C., in- page 116
troduces the “so-called sphere of fixed stars”, explaining its conventional
nature and warning the reader not to suppose it to have a physical existence,
since the stars are at different distances from us. 122
Archimedes was utterly victorious in his assertion that all lengths have
nonzero ratio. But if the history of mathematics for two millennia followed
the path shaped by this view, it is not to be concluded that older formulations
like that of Aristarchus were necessarily erroneous or lacked all
possibility of coherent development. Indeed, the goal of constructing geometries
that admit points “at infinity” came to be achieved in modern
projective geometry.
To go back to astronomy: Not surprisingly, another known proponent
of heliocentrism, Seleucus, 123 likewise did without the sidereal sphere and
believed in an infinite universe. 124
Because the further something is the slower it appears to move, the new
distances suggested for the stars also left room for the possibility that the
stars were not in fact fixed. Thus it is no wonder that Hipparchus also
conjectured that the apparently fixed stars were in fact mobile. According
to Pliny, Hipparchus compiled his catalog of stars precisely so that later
generations might deduce from it (besides the possible appearance of novae)
the displacements of stars. 125 Clearly, Hipparchus too did not believe
in a material sphere in which the stars are set. His catalog achieved its
aim in full: the stellar coordinates listed therein were incorporated into
122 Geminus, Eisagoge eis ta phainomena ( = Elementa astronomiae), I, 23 (a good recent edition being
[Geminus/Aujac]). The exact same approach is adopted today by, for instance, the Encyclopaedia
Britannica (15th edition, Micropaedia, sub “celestial sphere”).
123 Plutarch, Platonicae quaestiones, 1006C.
124 The opinion of Seleucus is reported by Aetius together with that of Heraclides of Pontus; see
[DG], 328b, 4–6.
125 Pliny, Natural history, II, 95.
Revision: 1.13 Date: 2002/10/16 19:04:00