From the Beginning to Plato

256 FROM THE BEGINNING TO PLATO
general. For rationals and irrationals he formulated definitions and
differentiae, determined also many orders of the irrationals, and brought to
light whatever of definiteness is to be found in them.
([8.60], I.1)
Later Pappus writes:
Those who have written concerning these things declare that the Athenian
Theaetetus assumed two lines commensurable in square [only] and proved
that if he took between them a line in ratio according to geometric
proportion, then the line named the medial was produced, but that if he
took the line according to arithmetic proportion, then the binomial was
produced, and if he took the line according to harmonic proportion, then
the apotome was produced.
([8.60], II.17)
These assertions require some explication. I begin with the notions of geometric,
arithmetic and harmonic proportion, and with a fragment of Archytas’ On Music.
There are three musical means, the first arithmetic, the second geometric,
the third subcontrary (hupenantios), which is also called harmonic. There
is an arithmetic mean when there are three terms in proportion with respect
to the same excess: the second term exceeds the third term by as much as
the first does the second…. There is a geometric mean when the second
term is to the third as the first is to the second… There is a
subcontrary mean (which we call harmonic) when the first term exceeds
the second by the same part of the first as the middle exceeds the third by a
part of the third.
(Porphyry [8.73], 93.6–15, DK 47 B 2)
Here Archytas speaks of three types of means rather than three types of
proportions, although the vocabulary of proportions also slips into what he says.
In the present context I shall speak of the three types of mean and use the word
‘proportion’ only for expressions of the form ‘x:y :: z:w’. The geometric mean
is, of course, the middle of three terms standing in a standard proportion. The
arithmetic mean is simply the arithmetic average of two terms x and z, that is
. The harmonic mean is usually given a more general definition which
we find in Nicomachus ([8.55], II.25.1; cf. Theon of Smyrna [8.92], 114.14–17),
according to which:
y is the harmonic mean between x and z if and only if