THEORY OF NUMBERS
319
Xenocrates of Chalcedon. (396-314 b. c.), who succeeded
Speusippus as head of the school, having been elected by
a majority of only a few votes over Heraclides, is also said
to have written a book O n Numbers and a Theory of Numbers,
besides books on geometry. 1 These' books have not survived,
but we learn that Xenocrates upheld the Platonic tradition in
requiring of those who would enter the school a knowledge of
music, geometry and astronomy; to one who was not pro
ficient in these things he said ‘ Go thy way, for thou hast not
the means of getting a grip of philosophy’. Plutarch says
that he put at 1,002,000,000,000 the number of syllables which
could be formed out of the letters of the alphabet. 1 2 If the
story is true, it represents the first attempt on record to solve
a difficult problem in permutations and combinations. Xeno
crates was a supporter of ‘ indivisible lines ’(and magnitudes)
by which he thought to get over the paradoxical arguments
of Zeno. 3
The Elements. Proclus’s summary [continued).
In geometry we have more names mentioned in the sum
mary of Proclus. 4
‘ Younger than Leodamas were Neoclides and his pupil Leon,
who added many things to what was known before their
time, so that Leon was actually able to make a collection
of the elements more carefully designed in respect both of
the number of propositions proved and of their utility, besides
which he invented diorismi (the object of which is to deter
mine) when the problem under investigation is possible of
solution and when impossible,’
Of Neoclides and Leon we know nothing more than what
is here stated; but the definite recognition of the Siopuryos,
that is, of the necessity of finding, as a preliminary to the
solution of a problem, the conditions for the possibility of
a solution, represents an advance in the philosophy and
technology of mathematics. Not that* the thing itself had
not been met with before: there is, as we have seen, a
1 Diog. L. iv. 13, 14.
2 Plutarch, Quaest. Conviv. viii. 9. 18, 733 A.
3 Simpl. in Phys., p. 138. 3, &c.
4 Proclus on Eucl. I, p. 66. 18-67. 1.
