Mathematics in Ancient and Medieval India c4.s3.t13.9

c4.s3.t13.9

The *Līlāvatī* which is based on Brahmagupta’s *Brāhmasphuṭasiddhānta*, Śrīdhara’s *Pāṭīgaṇita* and Āryabhaṭa II’s *Mahāsiddhānta* exhibits a profound system of arithmetic and also contains many useful propositions in geometry and arithmetic. In addition to fundamental operations his rule of three, five, seven, nine, and eleven, permutation and combination, treatment of zero prove his deep knowledge in the subject. The work was translated into Persian by Fyzi in 1587 A.D. by the direction of the emperor Akbar. A curious account of the occasion of writing the *Līlāvatī* is current which is given by Fyzi¹ in the preface of his translation. His *Bījagaṇita* contains problems of unknown quantities, surds, pulverizer (or *kuṭṭaka*), simple and quadratic equations including general rule which went beyond that of Śrīdhara, solutions of several indeterminate equations of second degree and solutions of certain equations of the third and fourth degree. The full solution of the equation and of its more general form ax^2+bx+c=y^2 was given by Bhāskara II. About his ‘Cyclic method’ Hankel² wrote, “It is beyond all praise: it is certainly the finest thing achieved in the theory of numbers before Lagrange.” H.T. Colebrooke³ points out that the method is exactly the same which Lord Brouncker devised to answer a question proposed by way of challenge by Fermat in 1657. In his *Siddhāntaśiromaṇi*, we get evidence of his knowledge of trigonometry including sine table and different relations among the three functions known as *jyā*, *kojyā* and *utkramajyā*. The idea of differentiation of the earlier workers was finally developed by him. He was acquainted with the principle of infinitesimal calculus and is often given credit for originating the idea of integration long before Newton and Leibniz.