IAMBLICHUS
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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.
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