Mathematics in Ancient and Medieval India c6.s9.115

c6.s9.115

This expression is of course easily reducible to the second Hindu form. But the first Hindu form i. e. the Brahmagupta-Mahāvīra form is considerably different from the Chinese form. Putting a = b, a' = b' in (3), we get the formula for the volume of a frustum of a pyramid with a square base as ⅓ ( a² + a'² + aa' ) h. This particular formula occurs in the *Chiu Chang Suan-Shu*³, *Chang Ch'iu-Chien Suan-Ching*⁴ and Heron's *Stereometry*.⁵ It was also known to the ancient Egyptians.⁶ Cantor⁷ followed by Tropfke⁸ and Smith⁹ thinks that Brahmagupta's rule was meant for such a particular case only. But we do not think so.¹⁰ For there is nothing in Brahmagupta's definition of his rule to suggest such a limitation. These writers were probably misled by an example of a well with a square face (a case of a truncated square pyramid turned up-side down) given by the commentator Pṛthudakasvāmī in illustration of Brahmagupta’s rule. Mahāvīra’s rule is absolutely free from any kind of limitation. For an illustration of it, he has given examples relating to a truncated pyramid with a triangular, square and rectangular base, a frustum of a right circular cone and also a truncated wedge.¹