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.).