114 
PYTHAGOREAN ARITHMETIC 
as a race-course (SictvXo?) 1 formed of successive numb ers 
from 1 (as start, vanXgg) up to n, the side of the square, 
which is the turning-point (Kap-mgp), and then back again 
through {n — 1), (77 — 2), &c., to 1 (the goal, vvacra), thus: 
1+2 + 3 + 4... (tt,-1) + 
n 
1 + 2 + 3 + 4...(t7/-2) + (77-1) + • 
This is of course equivalent to the proposition that n 2 is the 
sum of the two triangular numbers ^n{n+\) and %(n—l)n 
with sides n and n — 1 respectively. Similarly Iamblichus 
points out 2 that 'the oblong number 
71 (n — l) = (1 + 2 + 3 + ... + 77/) -f- (77/ — 2 + 71 — 3 + ... -|-3 + 2). 
He observes that it was on this principle that, after 10, 
which was called the unit of the second course (Sevrepco- 
Sovpevr] povds), the Pythagoreans regarded 100 = 10.10 as 
the unit of the third course (rpiooSovptvr] povds), 1000 = 10 3 
as the unit of the fourth course (rerpcoSoopevr] pouds), and 
so on, 3 since 
1+24-3 + ... + 10 + 9 + 8-1-...+2 + 1 — 10.10, 
10 + 20 + 30 + ... + 100 + 90 + 80 + ...+ 20 + 10 = 10 3 , 
100+ 200+ 300+ ... + 1000+ 900+ ... + 200+ 100 = 10 4 , 
and so on. Iamblichus sees herein the special virtue of 10 : 
but of course the same formulae would hold in any scale 
of notation as well as the decimal. 
In connexion with this Pythagorean decimal terminology 
Iamblichus gives a proposition of the greatest interest. 4 
Suppose we have any three consecutive numbers the greatest 
of which is divisible by 3, Take the sum of the three 
numbers; this will consist of a certain number of units, 
a certain number of tens, a certain number of hundreds, and 
so on. Now take the units in the said sum as they are, then 
as many units as there are tens in the sum, as many units as 
there are hundreds, and so on, and add all the units so 
obtained together (i.e, add the digits of the sum expressed 
in our decimal notation). Apply the same procedure to the 
1 Iambi, in Nlcom., p. 75. 25-77. 4. 2 lb., pp. 77. 4-80. 9. 
3 Ih, pp. 88. 15-90. 2. 4 Ih., pp. 103. 10-104. 13. ’