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Mathematical Expositions of Vīrasena in Dhavalā Commentary : (207)
(4)
Mensuration
In initial stages, mathematics was not divided in many branches. Still the geometry was quite in vogue like arithmetic. The geometrical calculations are made on five kinds of shapes. The Jainas assumed their universe as circular, and, accordingly, they developed the geometry of the circle in terms of circumference, area and volume etc. in comparison to Hindus who developed their geometry on the basis of rectangular bricks etc. used for Yajñas. Dhavalā has good number of geometrical topics. Some of which are given here: (a) Value of T.
It is very necessary to learn the inter-relationship between c, d and r of a circle. Their relationship is expressed through a for which four approximations are given:
c= V(10 d?) (2) c/d = 3 or V10 (Vol. 4, p. 209) (3) c=3d + (16/116) d or (355/113) d = 3.14 d (Vol. 4, p. 221-22)
c = 371/119= 3.13 (Vol. 4, p. 40) The texts like TP have generally given two values for 1-3 or V10. The Dhavalā author has given four formulae and has used it = V10 in many places for area and circumference. However, he has quoted a finer value of n through a quotation, which is nearer to 3.14. Now a days it is available up to 22 digits. However, the second value indicates that the Chinese value of it (Yen Tsing and Chih, 5 century) was known and used in India even before China where it got, perhaps, through Buddhist people from India. Vīrasena has also given the formula for area of a circle as A=3 (d/2)2 = 7tr2 (1) or c. d/4 (older form). (b) Volume of the Frustum of Cone
In the fourth volume of Dhavalā, while discussing the shape of the universe, Vīrasena has obtained the volume of the frustum of a cone by an unprecedented infinite process of cutting it into plain boundaries without changing its volume and finding the volume of each section. Applying the geometrical progressional summation, Jain has given the volume as
V = 1/3 th (a? + b2 + 2ab) = (1 + b 2)h.t
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