A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

OTHER LOST WORKS 195
Astronomy.
We are told by Ptolemaeus Chennus * that Apollonius was
famed for his astronomy, and was called e (Epsilon) because
the form of that letter is associated with that of the moon, to
which his accurate researches principally related. Hippolytus
says he made the distance of the moon's circle from the surface
of the earth to be 500 myriads of stades. 2 This figure
can hardly be right, for, the diameter of the earth being,
according to Eratosthenes's evaluation, about eight myriads of
stades, this would make the distance of the moon from the
earth about 125 times the earth's radius.
This is an unlikely
figure, seeing that Aristarchus had given limits for the ratios
between the distance of the moon and its diameter, and
between the diameters of the moon and the earth, which lead
to about 1 9 as the ratio of the moon's distance to the earth's
radius.
Tannery suggests that perhaps Hippolytus made a
mistake in copying from his source and took the figure of
5,000,000 stades to be the length of the radius instead of the
diameter of the moon's orbit.
But we have better evidence of the achievements of Apollonius
in astronomy. In Ptolemy's Syntaxis 3 he appears as
an authority upon the hypotheses of epicycles and eccentrics
designed to account for the apparent motions of the planets.
The propositions of Apollonius quoted by Ptolemy contain
exact statements of
the alternative hypotheses, and from this
fact it was at one time concluded that Apollonius invented
the two hypotheses. This, however, is not the case. The
hypothesis of epicycles was already involved, though with
restricted application, in the theory of Heraclides of Pontus
that the two inferior planets, Mercury and Venus, revolve in
circles like satellites round the sun, while the sun itself
revolves in a circle round the earth ; that is, the two planets
describe epicycles about the material sun as moving centre.
In order to explain the motions of the superior planets by
means of epicycles it was necessary to conceive of an epicycle
about a point as moving centre which is
not a material but
a mathematical point. It was some time before this extension
of the theory of epicycles took place, and in the meantime
196 APOLLONIUS OF PERGA
another hypothesis, that of eccentrics, was invented to account
for the movements of the superior planets only. We are at this
stage when we come to Apollonius. His enunciations show
that he understood the theory of epicycles in all its generality,
but he states specifically that the theory of eccentrics can only
be applied to the three planets which can be at any distance
from the sun. The reason why he says that the eccentric
hypothesis will not serve for the inferior planets is that, in
order to make it
serve, we should have to suppose the circle
described by the centre of the eccentric circle to be greater
than the eccentric circle itself. (Even this generalization was
made later, at or before the time of Hipparchus.) Apollonius
further says in his enunciation about the eccentric that ' the
centre of the eccentric circle moves about the centre of the
zodiac in the direct order of the signs and at a s r peed equal to
that of the sun, while the star moves on the eccentric about
its centre in the inverse order of the signs and at a speed
equal to the anomaly \ It is clear from this that the theory
of eccentrics was invented for the specific purpose of explaining
the movements of Mars, Jupiter, and Saturn about the
sun and for that purpose alone. This explanation, combined
with the use of epicycles about the sun as centre to account
for the motions of A r enus and Mercury, amounted to the
system of Tycho Brahe that system
;
was therefore anticipated
by some one intermediate in date between Heraclides and
Apollonius and probably nearer to the latter, or it may
have been Apollonius himself who took this important step.
If it was, then Apollonius, coming after Aristarchus of
Samos, would be exactly the Tycho Brahe of the Copernicus
of antiquity. The actual propositions quoted by Ptolemy as
proved by Apollonius among others show mathematically at
what points, under each of the two hypotheses, the apparent
forward motion changes into apparent retrogradation and
vice versa, or the planet appears to be stationary.
1
apud Photium, Cod. cxc, p. 151 b 18, ed. Bekker.
2
Hippol. Refut. iv. 8, p. 66, ed. Duncker.
3
Ptolemy, Syntaxis, xii. 1.
o 2