66
PYTHAGOREAN ARITHMETIC
doctrines down to the time of Philolaus to be attributed
to a pledge of secrecy binding the school ; at all events, it
did not apply to their mathematics or their physics ; the
supposed secrecy may even have been invented to explain
the absence of documents. The fact appears to be that oral
communication was the tradition of the school, while their
doctrine would in the main be too abstruse to be understood
by the generality of people outside.
In these circumstances it is difficult to disentangle the
portions of the Pythagorean philosophy which can safely
be attributed to the founder of the school. Aristotle evi
dently felt this difficulty ; it is clear that he knew nothing
for certain of any ethical or physical doctrines going back
to Pythagoras himself ; and when he speaks of the Pytha
gorean system, he always refers it to ‘ the Pythagoreans
sometimes even to ‘ the so-called Pythagoreans ’.
The earliest direct testimony to the eminence of Pythagoras
in mathematical studies seems to be that of Aristotle, who in
his separate book On the Pythagoreans, now lost, wrote that
‘ P3Thagoras, the son of Mnesarchus, first worked at mathe
matics and arithmetic, and afterwards, at one time, condescended
to the wonder-working practised by Pherecydes.’ 1
In the Metaphysics he speaks in similar terms of the
Pythagoreans :
‘ In the time of these philosophers (Leucippus and
Democritus) and before them the so-called Pythagoreans
applied themselves to the study of mathematics, and were
the first to advance that science ; insomuch that, having been
brought up in it, they thought that its principles must be
the principles of all existing things.’ 2
It is certain that the Theory of Numbers originated in
the school of Pythagoras; and, with regard to Pythagoras
himself, we are told by Aristoxenus that he ‘ seems to have
attached supreme importance to the study of arithmetic,
which he advanced and took out of the region of commercial
utility ’. 3
1 Apollonius, Hist, mirabil. 6 (Vors. i 3 , p. 29. 5).
2 Arist. Metaph. A. 5, 985 b 23.
3 Stobaeus, Eel. i. proem. 6 (Vors. i 3 , p. 346. 12).
