I 
116 
PYTHAGOREAN ARITHMETIC 
that is, what we call a digit of a number expressed in our 
decimal notation ; for the Greeks, in the case of any number 
above 9, the pythmen was the same number of units as the 
alphabetical numeral contains tens, hundreds, thousands, &c. 
Thus the pythmen of 700 (\fr in Greek) is 7 (£); that of 
(6000) is <y (6), and so on. The method then proceeded 
to find the pythmen of a certain name, say Ayayeyvcov. 
Taking the pythmenes of all the letters and adding them, 
we have 
1 + 34-1+4 + 5 + 4 + 5 + 8 + 5 = 36. 
Take the pythmenes of 36, namely 3 and 6, and their sum is 
9. The pythmen of Ayayeyrcov is therefore 9. Next take 
the name "Ektcop; the pythmenes are 5, 2, 3, 8, 1, the sum of 
which is 19 ; the 'pythmenes of 19 are 1, 9 ; the sum of 1 and 
9 is 10, the pythmen of which is 1. The pythmen of "Ektcop 
is therefore f, ‘ It is easier ’, says Hippolytus, ‘ to proceed 
thus. Finding the pythmenes of the letters, we obtain, in the 
case of "Ektcop, 19 as their sum. Divide this by 9 and note 
the remainder: thus, if I divide 19 by 9, the remainder is 1, 
for nine times 2 is 18, and 1 is left, which will accordingly 
be the pythmen of the name "Ektcop.’ Again, take the name 
TldrpoKXos. The sum of the pythmenes is 
8 + l + 3 + l + 7 + 2 + 3+ ^ + 2 = 34 ; 
and 3 + 4 = 7, so that 7 is the pythmen of Udrpo/cXoy. 
‘ Those then who calculate by the rule of nine take one-ninth 
of the sum of the pythmenes and then determine the sum of 
the pyythmenes in the remainder. Those on the other hand 
who follow the “ rule of seven ” divide by 7. Thus the sum 
of the pythmenes in Harpo/cXcy was found to be 34. This, 
divided by 7, gives 4, and since 7 times 4 is 28, the remainder 
is 6. . . .’ ‘ It is necessary to observe that, if the division 
gives an integral quotient (without remainder), . . . the 
pythmen is the number 9 itself ’ (that is, if the rule of nine is 
followed). And so on. 
Two things emerge from this fragment. (1) The use of the 
pythmen was not appearing for the first time when Apollonius 
framed his system for expressing and multiplying large 
numbers; it originated much earlier, with the Pythagoreans.