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.