86
PYTHAGOREAN ARITHMETIC
between 12 and 6 according to the theory of harmonics (Kara
tt)v apfxovLKrjv)}
lamblichus, 2 after Nicomachus, 3 mentions a special ‘ most
perfect proportion ’ consisting of four terms and called
‘musical’, which, according to tradition, was discovered by
the Babylonians and was first introduced into Greece by
Pythagoras. It was used, he says, by many Pythagoreans,
e. g. (among others) Aristaeus of Croton, Timaeus of Locri,
Philolaus and Archytas of Tarentum, and finally by Plato
in the Timaeus, where we are told that the double and triple
intervals were filled up by two means, one of which exceeds
and is exceeded by the same part of the extremes (the
harmonic mean), and the other exceeds and is exceeded by
the same numerical magnitude (the arithmetic mean). 4 The
proportion is
a + h 2 ah 7
a: —— = —: b,
2 a + h
an example being 12:9 = 8:6.
(/3) Seven other means distinguished.
The theory of means was further developed in the school
by the gradual addition of seven others to the first three,
making ten in all. The accounts of the discovery of the
fourth, fifth, and sixth are not quite consistent. In one place
lamblichus says they were added by Eudoxus 5 ; in other
places he says they were in use by the successors of Plato
down to Eratosthenes, but that Archytas and Hippasus made
a beginning with their discovery, 0 or that they were part of
the Archytas and Hippasus tradition. 7 The remaining four
means (the seventh to the tenth) are said to have been added
by two later Pythagoreans, Myonides and Euphranor. 8 From
a remark of Porphyry it would appear that one of the first
seven means was discovered by Simus of Posidonia, but
that the jealousy of other Pythagoreans would have robbed
him of the credit. 9 The ten means are described by
1 Nicom. ii. 26. 2. 2 Iambi, in Nicom., p. 118. 19 sq.
3 Nicom. ii. 29. 4 Plato, Timaeus, 86 A.
5 Iambi, in Nicom., p. 101. 1-5. 6 lb., p. 116. 1-4.
7 lb., p. 113, 16-18. 8 lb., p. 116. 4-6.
9 Porphyry, Vit. Pyth. 3; Vors. i 3 , p. 343. 12—15 and note.
