1 The Birth of Science - MSRI

78 3. Other Hellenistic Scientific Theories
the sphere of the fixed stars loses its function. It is not an accident that
both Aristotle and Ptolemy, who believed in the earth’s immobility, also
believed in a rigid sidereal sphere, and that the first person to challenge
the notion of that sphere was apparently the same who first asserted that
the earth rotates — Heraclides of Pontus, who maintained that the universe
is infinite and that every celestial body is a world in itself (and even
has its own atmosphere). 118 An interesting argument in favor of an infinite
universe is reported by Lucretius. 119
Aristarchan heliocentrism brought to bear a new argument that increased page 115
enormously the traditional dimensions of the cosmos. The supporters of
heliocentrism, indeed, had to explain why ever our motion around the
sun causes no observable parallax effects — that is, why the appearance
of the constellations does not change as our vantage point moves relative
to them throughout the year. According to Archimedes, Aristarchus overcame
that objection by assuming that the radius of the earth’s orbit is to the
radius of the sphere of fixed stars as a sphere’s center is to its radius. This
wording, which appears more or less unchanged in Geminus, Cleomedes,
Ptolemy and other authors, 120 is criticized by Archimedes, who says that
the ratio between two lengths is necessarily nonzero. 121
This issue calls perhaps for a mathematical parenthesis. The aim of the
Arenarius may have been precisely to defend the “Archimedean postulate”
(see page 45 in Section 2.7), by showing that one can assign a finite,
nonzero ratio to any two nonzero lengths (or other homogeneous magnitudes).
To accomplish this it was necessary to work out a numbering
system able to express even the largest imaginable ratio between homogeneous
magnitudes, such as the ratio between the volume of the sidereal
sphere and that of a grain of sand; this is what Archimedes does in his
tract. The triumph of Archimedes’ views on commensurability took away
the rationale for the task undertaken in the Arenarius and rendered the
work hard to understand (it has always been felt to be strange). Obviously,
what Aristarchus (and the other authors mentioned) intended in
saying that two lengths are in the same ratio as a point is to a circumfer-
118 [DG], 328b, 4–6; 343, 15. The same opinions were attributed by Aetius also to “Pythagoreans”;
these may have been the same Pythagoreans, such as Hicetas and Ecphantus (page 74), who
asserted that the earth moves.
119 Lucretius states that if the universe were finite, all masses would already be concentrated in its
center. Thus he supposes that gravity affects all bodies, not just “heavy” ones (Lucretius, De rerum
natura, I, 984–997).
120 For example, Geminus, Eisagoge eis ta phainomena, XVII, 16; Cleomedes, Caelestia, I, 8, 1–5 (ed.
Todd); Ptolemy, Almagest, V, xi, 401, 22–402, 1. Aristarchus likewise postulates that the ratio between
the earth and the orbit of the moon is equal to the ratio between a point and a circumference
(On the sizes and distances of the sun and moon, hypothesis 2 = [Heath: Aristarchus], p. 352).
121 Arenarius, 135, 19–22 (ed. Mugler, vol. II).
Revision: 1.13 Date: 2002/10/16 19:04:00