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numbers, plane and solid, which are of course analogous to 
the corresponding geometrical figures, and he may have con 
sidered 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 parallelepipedal figures including cubes, 
plinthides (square bricks) and SoKiSes (beams), and scalene 
figures with sides unequal every way (= (Soo/j-lo-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 inter 
polation, it confirms the supposition that Theon included in 
his work only this limited amount of geometry and stereo 
metry. 
Section I is on Arithmetic in the same sense as Nicomachus’s 
Introduction. At the beginning Theon observes that arith 
metic will be followed by music. Of music in its three 
aspects, music in instruments (er opydvoLs), 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 con 
tents 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 num 
bers 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 Ih., pp. 16. 24-17. 11. 
3 Ih., pp. 42. 10-45. 9. Of. vol. i, pp. 91-8,