A history of Greek mathematics - Wilbourhall.org

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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.
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