72 
PYTHAGOREAN ARITHMETIC 
Trepirra/ciS') and ‘even-times odd’ (7repirra dpTLctKis), which 
are evidently used ip the simple sense of the products of even 
lind even, odd and odd, odd and even, and even and odd 
factors respectively. 1 Euclid’s classification does not go much 
beyond this; he does not attempt to make the four defini 
tions mutually exclusive. 2 An ‘ odd-times odd J number is of 
course any odd number which is not prime; but ‘ even-times 
even ’ (‘ a number measured by an even number according to 
an even number ’) does not exclude ‘ even-times odd ’ (‘ a 
number measured by an even number according to an odd 
number’); e.g. 24, which is 6 times 4, or 4 times 6, is also 
8 times 3. Euclid did not apparently distinguish, any more 
than Plato, between ‘ even-times odd ’ and ‘ odd-times even ’ 
(the definition of the latter in the texts of Euclid was pro 
bably interpolated). The Neo-Pythagoreans improved the 
classification thus. With them the ‘even-times even ’ number 
is that which has its halves even, the halves of the halves 
even, and so on till unity is reached ’ 3 ; in short, it is a number 
of the form 2 n . The ‘ even-odd ’ number (dpTioirepirros in one 
word) is such a number as, when once halved, leaves as quo 
tient an odd number, 4 i.e. a number of the form 2(2771+1). 
The ‘odd-even’ number (nepicrcrdpTLos) is a number such that 
it can be halved twice or more times successively, but the 
quotient left when it can no longer be halved is an odd num 
ber not unity, 5 i.e. it is a number of the form > 2 ,,+1 (2771+1). 
The ‘ odd-times odd ’ number is not defined as such by , 
Nicomachus and Iamblichus, but Theon of Smyrna quotes 
a curious use of the term ; he says that it was one of the 
names applied to prime numbers (excluding of course 2), for 
these have two odd factors, namely 1 and the number itself. 0 
Prime or incomposite numbers (npcoros kul davvOeTos) and 
secondary or composite numbers (Sevrepos kul avrderos) are 
distinguished in a fragment of Speusippus based upon works 
of Philolaus. 7 We are told 8 that Thymaridas called a prime 
number rectilinear {evOvypappLKos), the ground being that it 
can only be set out in one dimension 9 (since the only measure 
1 Plato, Parmenides, 143 e. 2 See Eucl. VII. Defs. 8-10. 
3 Nicom. i. 8. 4. 4 Ih. i. 9. 1. 5 Ih. i. 10. 1. 
0 Theon of Smyrna, p. 23. 14-23. 
7 Theol. Ar. (Ast), p. 62 (Vors. i 3 , p. 304. 5). 
3 Iambi, in Nicom., p. 27. 4. lJ Cf. Arist. Metaph. A. 13, 1020 b 3, 4.