Geometric Formulas in the Dhavala of Virasena (780 C.E.)

This article highlights the mathematical sophistication of Virasena, a 9th-century Jain scholar, in his commentary Dhavala on the Chakkhandagama (CK). Virasena introduced and applied numerous geometrical and mathematical formulas for cosmological calculations, especially in Jain cosmography. Key Contributions 1) Early Use of π (Pi): Virasena was the first in India to use 355/113 (~3.14159292) as a value for π, predating its known use elsewhere in India. 2) Infinite Geometric Progression: He used a formula for the sum of an infinite geometric series, another first in Indian mathematical tradition. 3) Volumes and Areas of Geometrical Figures: Calculated volumes of trapezoidal prisms, hexagonal prisms, and combinations thereof to model the Jain cosmological structure (e.g., Upper, Middle, and Lower Worlds). 4) Approximations for Irregular Solids: i) Approximated volumes of animals (e.g., bee, conch-shell, fish) and celestial beings using geometric shapes like semi-cylinders, rectangular solids, and cones. ii) Included dimensional conversions between amgulas, ussehas, and joyanas. 5) World Volume in Jain Cosmography: i) Provided adapted formulas for computing the volume of the Jain Universe, both in the Digambara (rectilinear) and Svetambara (truncated cone) traditions. ii) For Svetambara cosmography, applied transformations of figures and infinite series to calculate volume with precision."

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