________________
According to Digambara table of measures -
1 utsedha-yojana3 = 15363 pramana amgulas3
Progression:
Virasena also uses the formula for the sum of a finite geometric progression:S(n) = a + ar+ ar2 + ar3 +
a (-1)
r-1
Above formula is used in his computation of the number of the heavenly bodies and the areas of the cocentric islands and oceans.
40
Another derivation of the same formula is -
S
PS
a a
= a + -+p p
=ap +
Jain Education International
a a
-
a + + p p
+ arn-1 =
---
+ <0>
= ap + S
In the above expression <0> indicates a 'space point (agasa-pradesa). This notation is clearly seen in Virasena's expression of the 'thickness' of a plane figure.
+<0
In the previous lines we have given some descriptions of the mathematical excellence of Virasena. Of course all these are available in the Dhavata commentary, not only it, but a lot of more mathematical material is available in Dhavala. On the basis of it, I can say that Dhavala is a book of first rate importance of the historian of mathematics and more systamatic & deeper study is essential from the mathematical point of view.
References:
1. 'Mathematics in Dhavala', Dhavala-IV, p. 4, Sholapur, 1984.
2. (a) Anupam Jain, Katipaya Ajñāta Jaina Ganita Grantha', Ganita Bhārati, 4(3-4), (Delhi), 1981, pp. 61-71.
(b) Jaina Ganitiya Sahitya'. Arhat Vacana (indore), 1(1), September, 1988.
3. See ref. 1.
4. Ganila sara-samgraha' by Mahāvīra, 1963 with a Hindi translation by L.C. Jain, Solapur, Ch. 1, 63-68.
5. Dhavala-III, p. 98, quoted verse 51, cf. Gommata-sara Jivakända, pp. 633. [Satakhandagama. Dhavala Tika by Acarya Virasena, Ed.-Dr. Hiralal Jain, Part 1-16, Amravati, Solapur etc. 1939-59).
6. Ibid, p. 99, quoted verse 52.
7. Ibid, p. 99, quoted verse 53. 8. Ibid, p. 56.
Arhat Vacana, 14 (2-3), 2002
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