Mathematics in Ancient and Medieval India c6.s9.92

c6.s9.92

The expression for the volume of a cone occurs first in India in the work of Brahmagupta. He used the term *sūcī*, meaning thereby generally a pyramid with a base of any form. The base may be a circle and hence the term includes a cone as well. According to Brahmagupta¹ (c. 628 A. D.) *volume of a cone (or pyramid* = ⅓ of the base. This formula reappears in the works of Mahāvīra² (c. 850 A. D.), Āryabhaṭa II³ (c. 950 A. D.), Nemichandra⁴ (c. 975 A. D.), Śrīpati⁵ (1039 A. D.) and Bhāskara II⁶ (1150 A. D.). The formula refered to by Kaye⁷ is (Circumference/6)². This formula no doubt occurs in the works of the above writers except Mahāvīra. But in connection with this formula it has been specifically mentioned by all of them that the formula is to be employed only in case of "the measurements of the maunds grain" (*rāśi-vyāvahāra*), not in other cases. It has been further remarked by Brahmagupta, Śrīpati and Bhāskara II that in that case the height of the maund must be assumed to be equal to the circumference of the base divided by 9, 10 or 11 according to calculates two approximate values called *vyavahārika gaṇita* or ‘approximate value’ ( A ) and *antra gaṇita* or ‘gross volume’ ( G ). Mahāvīra calls them respectively *karmāntika-phala* and *auṇḍra-phala*. It is stated : A = ( area from half the sum of the linear dimensions of face and base ) × height, and G = ½ ( area of face + area of base ) × height. The accurate volume ( V ) of the frustum is then given to be V = ⅓ ( G – A ) + A. Now for the frustum of a right circular cone noted before, $A = ((r + R)²)/2 π × h, G = ½ (π r² + π R²) × h$ ∴ V = (π)/3 (r² + R² + rR) × h as stated before.