Jinamanjari 1996 09 No 14

Author(s): Jinamanjari
Publisher: Canada Bramhi Jain Society Publication

Page 80

He also uses the formula for the sum of a finite geometric progression:
S(n) = a + ar + a2 + ans + ... + ar n-1 = (1 -1)
1- 1
in his computation of the number of the heavenly bodies (Dhavalā on CK 1.4.4; vol. 4, pp. 150-159) and of the areas of the concentric islands and oceans (on CK 1.4.24 25; vol.4, pp. 193 202). It is, therefore, not improbable that the inventor of the fornrula took the limit of n in:
s(n) = ^{(0)" - 1}
Another derivation of the same formula seems to have been equally possible for him. By multiplying both sides of the equation,
S=a+ + +...+70),
by p, he could obtain:
pS = op+{c++-+-+0)} = ap+5.
Hence the above formula. The expression '<O>'in the above equations indicates a 'space point' (āgāsa-padesa). This notion is clearly seen in Vīrasena's expression of the 'thickness' of a plane figure' 'The circumference of its (the cylinder's) top, horizontal circle whose thickness (bāhalla) is one 'space-point is this much 371/113 (on CK 1.3.2; vol. 4, p. 12).
As for the transformation of geometric figures without changing their areas or volumes, I simply point out that it was
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