Jain Journal 1995 10

Author(s): Jain Bhawan Publication
Publisher: Jain Bhawan Publication

Page 13

PADMAVATHAMMA: ASCETIC MOTHER VISUDDHAMATI
As a bird's eye view of the mathematical contents of the Tiloyapanṇatti, we find Geometry, Arithmetic and Algebra. Geometry is used to find the surface area of circular forms of the Jambudvipa. After deforming topologically figures into cubes, their volumes are found out.
In the calculation of numbers 0 is used. For example, in verse number 312, the time named Acalatma is exhibited through a symbol. Its value is (84)31 x (10)90 Pramāņa Varṣa.
We also find many rules to obtain the sum of arithmetic as well as geometric progressions. Interestingly the sums of many new series have been derived.
(1) For the sum of 7 series the following rule is given.
N
S =
-[(N + 7) D
2
- (7 +1)D + 2A] =
N 12A
2
39
[2A + (N-1) DJ
where A,D,N are respectively the first term, common difference and the number of terms.
Volume 1, chapter 2, verse no. 70
caya-hada-miṭṭhādhiya-padamekkādhiya-iṭṭha-gunida
caya-hinam/
dugunida-vadanena judam, pada-dala-gunidammi
hodi samkalidam//70//
Translation: From the number of terms increased by any chosen number and multiplied by the common difference is to be subtracted the product of the chosen number increased by one and the common difference. To the remainder is added twice the first term and is then multiplied by half of the number of terms. This gives rise to the sum (of the chosen number of series).
In the example, 7 is the chosen number.
(2) The sum S,, of first n terms of a geometric progression is given by the formula:
Sal(r.r....upto n terms) - 1] + (r - 1)
where r is the common ratio and a is the first term. Equivalently,
Sa(r" - 1)/(r- 1)
Volume 1, chapter 3, verse no. 79
gaccha same guṇayāre, paropparam guniya-rūva-parihine/ ekkona-guna-vihatte, gunide vayaṇena guna-ganidam//79//