were theories proposing the motion of the Earth (P, H). According
to P (Platonic Questions, 1006C, and The face on the Moon, 923A, cf. pseudo-Plutarch,
De placita phil. 891A), Aristarkhos merely hypothesized with demonstrations that the Earth
revolved around the Sun on an inclined circle and also rotated so that the sphere of the
fixed stars would not move. S S later maintained this view. We may
surmise that the account included other basic elements of a heliocentric theory.
Unfortunately, no extant testimony provides any motivation for his heliocentric theory or
even details. Indeed, our other principal source, A’ Sand-Reckoner, 1.4–7, only
mentions that such a universe must be larger than a geocentric one. Even here, we must
infer a motivation for this claim, that otherwise we would expect an observable stellar
parallax, i.e. a change in the angle between stars in different seasons. Indeed, it is easier to
speculate on why the theory was not accepted than why it was proposed. Does
K’ accusation of impiety represent a common judgment among ancient
scientists?
The treatise on the measurement of the
distances and sizes of the Sun and Moon
was profoundly influential in the ancient
world. Interestingly it assumes a geocentric
universe (cf. prop. 6). Using four basic phe-
nomena in addition to the equal apparent
or angular size of the Moon and Sun,
Aristarkhos constructed a valid geo-
metrical analysis of the relative distances
of the Sun and Moon, based on the insight
that the triangle formed by the Sun,
Moon, and observer must form a right tri-
angle at half moon, with the right angle at
the Moon. One phenomenon is that the
shadow of the Earth on the Moon during
a lunar eclipse is two lunar diameters: the
actual ratio varies and is larger (about 2.6).
Aristarkhos also treated the distances of
the Moon and Sun as constant, as would
most contemporary astronomers, and
ignored parallax. Only these assumptions
affect the geometry of his argument.
Although Archimedes ascribes to him the
correct angular size of the sun as ½ ̊, here
it is 2 ̊ (by inference from eclipses and his
value for the moon as^1 / 45 right angle).
These three observational errors facilitate
the geometry and calculation but do not
affect the calculation seriously. Quite different is his assumption that the angle of the Sun-
Earth-Moon at half moon is^1 / 30 less than a right angle (87 ̊ in a later system). The actual
angle, ca 89 ̊ 51 ’, is not observable without sophisticated instruments. At 87 ̊ the Moon is ca
47.5% illuminated, about 4 to 5 hours before true half Moon. So his approximation makes
the Sun much nearer. The principal results are that the distance of the Sun’s distance from
Aristarkhos of Samos: distances and sizes of Sun and Moon © Mendell
ARISTARKHOS OF SAMOS