THE THEORY OF PROPORTION AND MEANS 85 
Fabricius. dvaXSycov is not the correct form of the word, but 
the meaning would be ‘proportions’ or ‘proportionals’, and 
the true reading may be either tcov dvaXoyicov (‘ proportions ’), 
or, more probably, tcov dvd Xoyov (‘ proportionals ’); Diels 
reads tcov dvd A oyov, and it would seem that there is now 
general agreement that dXoycov is wrong, and that the theory 
which Proclus meant to attribute to Pythagoras is the theory 
of proportion or proportionals, not of irrationals. 
(a) Arithmetic, geometric, and harmonic means. 
It is true that we have no positive evidence of the use by 
Pythagoras of proportions in geometry, although he must 
have been conversant with similar figures, which imply some 
theory of proportion. But he discovered the dependence of 
musical intervals on numerical ratios, and the theory of means 
was developed very early in his school with reference to 
the theory of music and arithmetic. We are told that in 
Pythagoras's time there were three means, the arithmetic, 
the geometric, and the subcontrary, and that the name of the 
third (‘ subcontrary ’) was changed by Archytas and Hippasus 
to ‘ harmonic b 1 A fragment of Archytas’s work On Music 
actually defines the three; we have the arithmetic mean 
when, of l^hree terms, the first exceeds the second by the 
same amount as the second exceeds the third; the geometric 
mean when, of the three terms, the first is to the second as 
the second is to the third; the ‘ subcontrary, which we call 
harmonic ’, when the three terms are such that ‘ by whatever 
part of itself the first exceeds the second, the second exceeds 
the third by the same part of the third ’. 2 That is, if a, b, c 
are in harmonic progression, and a — b + a , 
n 
b = c + j whence in fact 
n 
a _ a — b 
c b — c ’ 
we must have 
Nicomachus too says that the name ‘ harmonic mean ’ was 
adopted in accordance with the view of Philolaus about the 
‘ geometrical harmony ’, a name applied to the cube because 
it has 12 edges, 8 angles, and 6 faces, and 8 is the mean 
1 Iambi, in Nicom., p. 100. 19-24. 
2 Porph. in Ptol. Harm., p. 267 [Vors. i 3 , p. 834. 17sq.).