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

THEON OF SMYRNA 239
edited by E. Hiller (Teubner, 1878) and finally, with a French
translation, by J. Dupuis (Paris, 1892).
Theon's date is approximately fixed by two considerations.
He is clearly the person whom Theon of Alexandria called
'the old Theon', top' iraXaibv Qecova} and there is no reason
to doubt that he is the ' Theon the mathematician ' (6 fiaOr)-
fianKos) who is credited by Ptolemy with four observations
of the planets Mercury and Venus made in a.d. 127, 129, 130
and 132. 2 The latest writers whom Theon himself mentions
are Thrasyllus, who lived in the reign of Tiberius, and
Adrastus the Peripatetic, who belongs to the middle of the
second century a.d.
Theon's work itself is a curious medley,
valuable, not intrinsically, but for the numerous historical
notices which it contains. The title, which claims that the
book contains things useful for the study of Plato, must not
be taken too seriously. It was no doubt an elementary
introduction or vade-mecum for students of philosophy, but
there is little in it which has special reference to the mathematical
questions raised in Plato. The connexion consists
mostly in the long proem quoting the views of Plato on the
paramount importance of mathematics in the training of
the philosopher, and the mutual relation of the five different
branches, arithmetic, geometry, stereometry, astronomy and
music. The want of care shown by Theon in the quotations
from particular dialogues of Plato prepares us for the patchwork
character of the whole book.
In the first chapter he promises to give the mathematical
theorems most necessary for the student of Plato to know,
in arithmetic, music, and geometry, with its application to
stereometry and astronomy. 3 But the promise is by no means
kept as regards geometry and stereometry : indeed, in a
later passage Theon seems to excuse himself from including
theoretical geometry in his plan, on the ground that all those
who are likely to read his work or the writings of Plato may
be assumed to have gone through an elementary course of
theoretical geometry. 4 But he writes at length on figured
1
Theon of Alexandria, Comm. on Ptolemy s Syntaxis, Basel edition,
pp. 390, 395, 396.
2
Ptolemy, Syntaxis, ix. 9, x. 1, 2.
3
Theon of Smyrna, ed. Hiller, p. 1. 10-17.
4
lb., p. 16. 17-20.
240 SOME HANDBOOKS
numbers, plane and solid, which are of course analogous to
the corresponding geometrical figures, and he may have considered
that he was in this way sufficiently fulfilling his
promise with regard to geometry and stereometry.
Certain
geometrical definitions, of point, line, straight line, the three
dimensions, rectilinear plane and solid figures, especially
parallelograms and parallelepiped al figures including cubes,
plinthides (square bricks) and SoKiSes (beams), and scalene
figures with sides unequal every way (= ^(o/ilo-kol in the
classification of solid numbers), are dragged in later (chaps.
53-5 of the section on music) 1 in the middle of the discussion
of proportions and means ; if this passage is not an interpolation,
it confirms the supposition that Theon included in
his work only this limited amount of geometry and stereometry.
Section I is on Arithmetic in the same sense as Nicomachus's
Introduction. At the beginning Theon observes that arithmetic
will be followed by music. Of music in its three
aspects, music in instruments (ev opyavois), music in numbers,
i.e. musical intervals expressed in numbers or pure theoretical
music, and the music or harmony in the universe, the first
kind (instrumental music) is not exactly essential, but the other
two must be discussed immediately after arithmetic. 2 The contents
of the arithmetical section have been sufficiently indicated
in the chapter on Pythagorean arithmetic (vol. i, pp. 112-13)
it deals with the classification of numbers, odd, even, and
their subdivisions, prime numbers, composite numbers with
equal or unequal factors, plane numbers subdivided into
square, oblong, triangular and polygonal numbers, with their
respective ' gnomons ' and their properties as the sum of
successive terms of arithmetical progressions beginning with
1 as the first term, circular and spherical numbers, solid numbers
with three factors, pyramidal numbers and truncated
pyramidal numbers, perfect numbers with their correlatives,
the over-perfect and the deficient; this is practically what
we find in Nicomachus. But the special value of Theon's
exposition lies in the fact that it contains an account of the
famous side- ' '
' and diameter- ' numbers of the Pythagoreans. 3
1 2
Theon of Smyrna, ed. Hiller, pp. 111-13. lb., pp. 16. 24-17. 11.
3
lb., pp. 42. 10-45. 9. Cf. vol. i, pp. 91-3.