100 
PYTHAGOREAN ARITHMETIC 
composite a product of prime factors (excluding 2, which is 
even and not regarded as prime), and (3) ‘ that which is in itself 
secondary and composite but in relation to another is prime and 
incomposite’, e.g. 9 in relation to 25, which again is a sort of 
intermediate class between the two others (cc. 11-13); the 
defects of this classification have already been noted (pp. 73-4). 
In c. 13 we have these different classes of odd numbers ex 
hibited in a description of Eratosthenes’s ‘ sieve ’ (koctklvov), an 
appropriately named device for finding prime numbers. The 
method is this. We set out the series of odd numbers begin 
ning from 3. 
3, 6, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 
Now 3 is a prime number, but multiples of 3 are not; these 
multiples, 9, 15... are got by passing over two numbers at 
a time beginning from 3 ; we therefore strike out these num 
bers as not being prime. Similarly 5 is a prime number, but 
by passing over four numbers at a time, beginning from 5, we 
get multiples of 5, namely 15, 25 ...; we accordingly strike 
out all these multiples of 5. In general, if n be a prime num 
ber, its multiples appearing in the series are found by passing 
over n — 1 terms at a time, beginning from n; and we can 
strike out all these multiples. When we have gone far enough 
with this process, the numbers which are still left will be 
primes. Clearly, however, in order to make sure that the 
odd number 2 n + 1 in the series is prime, we should have to 
try all the prime divisors between 3 and </(2 71+1); it is 
obvious, therefore, that this primitive empirical method would 
be hopeless as a practical means of obtaining prime numbers 
of any considerable size. 
The same c. 13 contains the rule for finding whether two 
given numbers are prime to one another; it is the method of 
Eucl. VII. 1, equivalent to our rule for finding the greatest 
common measure, but Nicomachus expresses the whole thing 
in words, making no use of any straight lines or symbols to 
represent the numbers. If there is a common measure greater 
than unity, th% process gives it; if there is none, i. e. if 1 is 
left as the last remainder, the numbers are prime to one 
another. 
The next chapters (cc. 14-16) are on over-perfect {vTrepTeXijs),