Mathematics in Ancient and Medieval India c7.s5.157

c7.s5.157

From these an unlimited number of integral so'utions can be readily obtained by principle of composition (*samāsabhāvanā*) discovered by Brahmagupta (*Brāhmasphuṭasiddhānta*, ch. 18, p. 64–55) and applied by later mathematicians.¹ In algebraic symbo's the principle of *samāsabhāvanā* may be expressed as fol'ows : For conveniently chosen values of c₁ and c₂, if (a₁, b₁) and (a₂, b₂) be a set of solutions of Nx² + c₁ = y² and Nx² + c₂ = y² respectively, then x = a₁b₂ ± a₂b₁ and y = b₁b₂ ± Na₁a₂ wi'l satisfy the equation Nx² + c₁c₂ = y². The aim of the scholars like Brahmagupta and Śrīpati was to obtain solutions of Nx² + 1 = y² in positive integers. They however discovered that an integral solution of Nx² + k = y² can always be found if k = ± 1, ± 2 or ± 4. Having got the solution, an infinite number of solution can be obtained by repeated application of *samāsabhāvanā*. This is undoubtedly a remarkable feat when we realise that this was done in 628 A. D. The method would have been perfect if only a method other than trial be avilable to Brahmagupta for the solution. Bhāskara II did it by method known as *cakravāla* as following.