70 
PYTHAGOREAN ARITHMETIC 
mg the Egyptian view The Pythagoreans ‘ made number 
out ot' one’ 1 2 ; some of them called it ‘ a progression of multi 
tude beginning from a unit and a regression ending in it’. 3 
(Stobaeus credits Moderatus, a Neo-Pythagorean of the time 
of Nero, with this definition. 4 ) Eudoxus defined number as 
a ‘ determinate multitude ’ (7rX^doy oopia-pevov). 5 * Nicoma- 
chus has yet another definition, ‘ a flow of quantity made up 
of units ’ (l (TTocror^roy e/c povdSoav crvyKtipevov). Aris 
totle gives a number of definitions equivalent to one or other 
of those just mentioned, ‘limited multitude’, 7 ‘multitude (or 
‘ combination ’) of units ’, 8 ‘ multitude of indivisibles ’, 9 £ several 
ones’ (eVa 7rXeio)), 10 ‘multitude measurable by one’, 11 ‘multi 
tude measured ’, and ‘ multitude of measures ’ 12 (the measure 
being the unit). 
Classification of numbers. 
The distinction between odd (7repiaraos) and even (dprios) 
doubtless goes back to Pythagoras. A Philolaus fragment 
says that ‘ number is of two special kinds, odd and even, with 
a third, even-odd, arising from a mixture of the two; and of 
each kind there are many forms ’. 13 According to Nicomachus, 
the Pythagorean definitions of odd and even were these: 
‘ An even number is that which admits of being divided, by 
one and the same operation, into the greatest and the least 
parts, greatest in size but least in number (i. e. into two halves) 
. . ., while an odd, number is that which cannot be so divided 
but is only divisible into two unequal parts.’ 14 
Nicomachus gives another ancient definition to the effect 
that 
‘ an even number is that which can be divided both into two 
equal parts and into two unequal parts (except the funda 
mental dyad which can only be divided into two equal parts), 
but, however it is divided, must have its two parts of the same 
kind without part in the other kind (i. e, the two parts are 
1 Iambi, in Nicom. ar. introd., p. 10. 8-10. 
2 Arist. Metaph. A. 5, 986 a 20. 3 Theon of Smyrna, p. 18. 3-5. 
4 Stob. Ed. i. pr. 8. 5 Iambi, op. cit., p. 10. 17. 
K Nicom. i. 7. 1. 7 Metaph. A. 13, 1020 a 13. 
8 lb. I. 1, 1053 a 30; Z. 13, 1039 a 12. 
■» lb. M. 9, 1085 b 22. 10 Phys. hi. 7, 207 b 7. 
11 Metaph. I. 6, 1057 a 3. 12 Jb. N. 1, 1088 a 5. 
13 Stob. Ed. i. 21. 7 C (Vors. i 3 , p. 310. 11-14). * 14 Nicom. i. 7. 3.