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

THEON OF SMYRNA 241
In the Section on Music Theon says he will first speak of
the two kinds of music, the audible or instrumental, and the
intelligible or theoretical subsisting in numbers, after which
he promises to deal lastly with ratio as predicable of mathematical
entities in general and the ratio constituting the
harmony in the universe, not scrupling to set out once again
'
the things discovered by our predecessors, just as we have
given the things handed down in former times by the Pythagoreans,
with a view to making them better known, without
ourselves claiming to have discovered any of them '. 1 Then
follows a discussion of audible music, the intervals which
give harmonies, &c, including substantial quotations from
Thrasyllus and Adrastus, and references to views of Aristoxenus,
Hippasus, Archytas, Eudoxus and Plato. With
chap. 17 (p. 72) begins the account of the 'harmony in
numbers', which turns into a general discussion of ratios,
proportions and means, with more quotations from Plato,
Eratosthenes and Thrasyllus, followed by Thrasyllus's divisio
canonis, chaps. 35, 36 (pp. 87-93). After a promise to apply
the latter division to the sphere of the universe, Theon
purports to return to the subject of proportion and means.
This, however, does not occur till chap. 50 (p. 106), the
intervening chapters being taken up with a discussion of
the SeKois and rerpa/cTt/? (with eleven applications of the
latter) and the mystic or curious properties of the numbers
from 2 to 10; here we have a part of the theologumena of
arithmetic. The discussion of proportions and the different
kinds of means after Eratosthenes and Adrastus is again
interrupted by the insertion of the geometrical definitions
already referred to (chaps. 53-5, pp. 111-13), after which
Theon resumes the question of means for ' more precise
treatment.
The Section on Astronomy begins on p. 120 of Killer's
edition. Here again Theon is mainly dependent upon
Adrastus, from whom he makes long quotations. Thus, on
the sphericity of the earth, he says that for the necessary
conspectus of the arguments it will be sufficient to
refer to the grounds stated summarily by Adrastus. In
explaining (p. 124) that the unevennesses in the surface of
1
Theon of Smyrna, ed. Hiller, pp. 46. 20-47. 14.
1523.2 R
242 SOME HANDBOOKS
the earth, represented e.g. by mountains, are negligible in
comparison with the size of the whole, he quotes Eratosthenes
and Dicaearchus as claiming to have discovered that the
perpendicular height of the highest mountain above the normal
level of the land is no more than 1 stades ; and to obtain the
diameter of the earth he uses Eratosthenes's figure of approximately
252,000 stades for" the circumference of the earth,
which, with the Archimedean value of - 2 2
T
- for 7r, gives a
diameter of about 80,182 stades. The principal astronomical
circles in the heaven are next described (chaps. 5-12, pp.
129-35) ; then (chap. 12) the assumed maximum deviations in
latitude are given, that of the sun being put at 1°, that of the
moon and Venus at 12°, and those of the planets Mercury,
Mars, Jupiter and Saturn at 8°, 5°, 5° and 3° respectively; the
obliquity of the ecliptic is given as the side of a regular polygon
of 15 sides described in a circle, i.e. as 24° (chap. 23, p. 151).
Next the order of the orbits of the sun, moon and planets is explained
(the system is of course geocentric) ; we are told (p. 138)
that ' some of the Pythagoreans ' made the order (reckoningoutwards
from the earth) to be moon, Mercury, Venus, sun,
Mars, Jupiter, Saturn, whereas (p. 142) Eratosthenes put the
sun next to the moon, and the mathematicians, agreeing with
Eratosthenes in this, differed only in the order in which they
placed Venus and Mercury after the sun, some putting Mercury
next and some Venus (p. 143). The order adopted by ' some
of the Pythagoreans ' is the Chaldaean order, which was not
followed by any Greek before Diogenes of Babylon (second
century B.C.);
'some of the Pythagoreans' are therefore the
later Pythagoreans (of whom Nicomachus was one) ; the other
order, moon, sun, Venus, Mercury, Mars, Jupiter, Saturn, was
that of Plato and the early Pythagoreans. In chap. 15
(p. 138 sq.) Theon quotes verses of Alexander 'the Aetolian'
(not really the ' Aetolian ', but Alexander of Ephesus, a contemporary
of Cicero, or possibly Alexander of Miletus, as
Chalcidius calls him) assigning to each of the planets (including
the earth, though stationary) with the sun and moon and
the sphere of" the fixed stars one note, the intervals between
the notes being so arranged as to bring the nine into an
octave, whereas with Eratosthenes and Plato the earth was
excluded, and the eight notes of the octachord were assigned