IAMBLICHUS 
115 
result, and so on. Then, says Iamblichus, the final result 
will he the number 6. E.g, take the numbers 10, 11, 12; the 
sum is 33. Add the digits, and the result is 6. Take 
994, 995, 996 : the sum is 2985 ; the sum of the digits is 24 ; 
and the sum of the digits of 24 is again 6. The truth of the 
general proposition is seen in this way. 1 
Let N = n 0 + 10 n x +10 2 n 2 + ... 
be a number written in the decimal notation. Let JS(JV) 
represent the sum of its digits, ¿C 2 ) (N) the sum of the digits 
of S (jS t } and so on. 
Now N — 8{N) = 9 (n x + lln 2 + 111%+ ...), 
whence i\T EE $(A) (mod. 9). 
Similarly 8(If) = S&N (mod. 9). 
Let iSfC*- 1 ) (iV) = (mod. 9) 
be the last possible relation of this kind; S^JS 7 will be a 
number A T/ S 9. 
Adding the congruences, we obtain 
N = N' (mod. 9), while N' < 9. 
Now, if we have three consecutive numbers the greatest 
of which is divisible by 3, we can put for their sum 
JV = (3^+1) + (3^ + 2) + (3^+3) = 9j?+6, 
and the above congruence becomes 
9p + 6 = N' (mod. 9), 
so that N' = 6 (mod. 9) ; 
and, since N' ^ 9, N' can only be equal to 6. 
This addition of the digits of a number expressed in our 
notation has an important parallel in a passage of the 
Refutation of all Heresies by saint Hippolytus, 2 where there 
is a description of a method of foretelling future events 
called the ‘ Pythagorean calculus ’. Those, he says, who 
claim to predict events by means of calculations with numbers, 
letters and names use the principle of the pythmen or base, 
1 Loria, op. cit., pp. 841-2, 
2 Hippolytus, Refut. iv, c. 14. 
I 2