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

THEON OF SMYRNA 243
to the seven heavenly bodies and the sphere of the fixed stars.
The whole of this passage (chaps. 15 to 16, pp. 138-47) is no
doubt intended as the promised account of the ' harmony in
the universe ', although at the very end of the work Theon
implies that this has still to be explained on the basis of
Thrasyllus's exposition combined with what he has already
given himself.
The next chapters deal with the forward movements, the
stationary points, and the retrogradations, as they respectively
appear to us, of the five planets, and the saving of the phenomena
by the alternative hypotheses of eccentric circles '
'
and
epicycles (chaps. 17-30, pp. 147-78). These hypotheses are
explained, and the identity of the motion produced by the
two is shown by Adrastus in the case of the sun (chaps. 26, 27,
pp. 166-72). The proof is introduced with the interesting
remark that ' Hipparchus says it is worthy of investigation
by mathematicians why, on two hypotheses so different from
one another, that of eccentric circles and that of concentric
circles with epicycles, the same results appear to follow '. It
is not to be supposed that the proof of the identity could be
other than easy to a mathematician like Hipparchus ; the
remark perhaps merely suggests that the two hypotheses were
discovered quite independently, and it was not till later that
the effect was discovered to be the same, when of course the
fact would seem to be curious and a mathematical proof would
immediately be sought. Another passage (p. 188) says that
Hipparchus preferred the hypothesis of the epicycle, as being
his own. If this means that Hipparchus claimed to have
discovered the epicycle-hypothesis, it must be a misapprehension
; for Apollonius already understood the theory of epicycles
in all its generality. According to Theon, the epicyclehypothesis
is more according to nature ' ' ; but it was presumably
preferred because it was applicable to all the planets,
whereas the eccentric-hypothesis, when originally suggested,
applied only to the three superior planets ; in order to make
it apply to the inferior planets it is necessary to suppose the
circle described by the centre of the eccentric to be greater
than the eccentric circle itself, which extension of the hypothesis,
though known to Hipparchus, does not seem to have
occurred to Apollonius.
R 2
244 SOME HANDBOOKS
We next have (chap. 31, p. 178) an allusion to the systems
of Eudoxus, Callippus and Aristotle, and a description
(p. 180 sq.) of a system in which the 'carrying' spheres
(called hollow ' '
') have between them solid spheres which by
their own motion will roll (dveXigovcri) the carrying spheres in
the opposite direction, being in contact with them \ These
'
solid ' spheres (which carry the planet fixed at a point on
their surface) act in practically the same way as epicycles.
In connexion with this description Theon (i.e. Adrastus)
speaks (chap. 33, pp. 186-7) of two alternative hypotheses in
which, by comparison with Chalcidius, 1 we recognize (after
eliminating epicycles erroneously imported into both systems)
the hypotheses of Plato and Heraclides respectively. It is
this passage which enables us to conclude for certain that
Heraclides made Venus and Mercury revolve in circles about
the sun, like satellites, while the sun in its turn revolves in
a circle about the earth as centre. Theon (p. 187) gives the
maximum arcs separating Mercury and Venus respectively
from the sun as 20° and 50°, these figures being the same as
those given by Cleomedes.
The last chapters (chaps. 37-40), quoted from Adrastus, deal
with conjunctions, transits, occultations and eclipses. The
book concludes with a considerable extract from DercyHides,
a Platonist with Pythagorean leanings, who wrote (before the
time of Tiberius and perhaps even before Varro) a book on
Plato's philosophy. It is here (p. 198. 14) that we have the
passage so often quoted from Eudemus
'
Eudemus relates in his Astronomy that it was Oenopides
who first discovered the girdling of the zodiac and the revolution
(or cycle) of the Great Year, that Thales was the first to
discover the eclipse of the sun and the fact that the sun's
period with respect to the solstices is not always the same,
that Anaximander discovered that the earth is (suspended) on
high and lies (substituting Keircu for the reading of the manuscripts,
KLveirai, moves) about the centre of the universe, and
that Anaximenes said that the moon has its light from the
sun and (explained) how its eclipses come about' (Anaximenes
is here apparently a mistake for Anaxagoras).
1
Chalcidius, Comm. on Timaens, c. 110. Cf. Aristarchus of Samos,
pp. 256-8.