CLASSIFICATION OF NUMBERS
73
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of it, excluding the number itsfelf, is 1); Theon of Smyrna
gives euthymetric and linear as alternative terms, 1 and the
latter (ypayyiKos) also occurs in the fragment of Speusippus.
Strictly speaking, the prime number should have been called
that which is rectilinear or linear only. As we have seen,
2 was not originally regarded as a prime number, or even as
a number at all. But Aristotle speaks of the dyad as ‘ the
only even number which is prime/ 2 showing that this diver
gence from early Pythagorean doctrine took place before
Euclid’s time. Euclid defined a prime number as ‘ that which
is measured by a unit alone ’, 3 a composite number as 4 that
which is measured by some number’, 4 while he adds defini
tions of numbers 4 prime to one another ’ (‘ those which are
measured by a unit alone as a common measure ’) and of
numbers 4 composite to one another ’ (‘ those which are mea
sured by some number as a common measure’). 5 Euclid then,
as well as Aristotle, includes 2 among prime numbers. Theon
of Smyrna says that even numbers are not measured by the
unit alone, except 2, which therefore is odd-like without being
prime. 6 The Neo-Pythagoreans, Nicomachus and lamblichus,
not only exclude 2 from prime numbers, but define composite
numbers, numbers prime to one another, and numbers com
posite to one another as excluding all even numbers; they
make all these categories subdivisions of odd? Their object
is to divide odd into three classes parallel to the three »subdivi
sions of even, namely even-even = 2 n , even-odd = 2 (2m + 1)
and the quasi-intermediate odd-even =2 ,l+1 (2mpl); accord
ingly they divide odd numbers into (a) the prime and
incomposite, which are Euclid’s primes excluding 2, (h) the
secondary and composite, the factors of which must all be not
only odd but prime numbers, (c) those which are 4 secondary and
composite in themselves but prime and incomposite to another
number/ e.g. 9 and 25, which are both secondary and com
posite but have no common measure except 1. The incon
venience of the restriction in (h) is obvious, and there is the
1 Theon of Smyrna, p. 23. 12.
2 Arist. Topics, 0. 2, 157 a 39.
3 End. VII. Def. 11. 4 lb. Def, 13.
5 lb. Defs. 12, 14.
fi Theon of Smyrna, p. 24. 7.
7 Nicom. i, cc. 11-13 ; Iambi, in Nicoin., pp. 26-8.