A history of Greek mathematics - Wilbourhall.org

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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
Theon of Smyrna, ed. Hiller, pp. 111-13.
2 lb., pp. 16. 24-17. 11.
3
lb., pp. 42. 10-45. 9. Cf. vol. i, pp. 91-3.