ब्राह्मस्फुटसिद्धान्त (ब्रह्मगुप्त - शून्य, कुट्टक, बीजगणित एवं सम्पूर्ण २१ अध्याय सान्वय सटीक)

258 BRAHMAGUPTA AS AN ALGEBRAIST this type was a problem, write Datta and Singh, which could not be solved completely nor satisfactorily by Brahmagupta. In fact, Brahmagupta had to depend on trial. Success in this direc- tion was, however, remarkably attained by Bhāskara II. He evol- ved a simple and elegant method which assisted in deriving an auxiliary equation having the required interpolator ±1, ±2, or ± 4, simultaneously with its two integral roots, from another auxi- liary equation empirically formed with any simple integral value of the interpolator, positive or negative. This method has been technically known as Cakravāla or the cyclic method. This is so called because it proceeds as in a circle, the same set of opera- tions being applied again and again in a continuous round. For the details of this method, our reader is requested to consult the Algebra of Bhāskara II and the narrative on this method as given by Datta and Singh under the title "Cyclic Method" in their History of Hindu Mathematics: Algebra, 1962 Edition, pp. 161-72. Solution of Indeterminate Quadratic Equation It is remarkable to see that Brahmagupta was the first algebraist in the history of mathematics to find a general solu- tion of the indeterminate quadratic equation Nx²±c=y² in positive integers. We have the following verse in the Brāhmasphuṭasiddhānta in this connection: From two roots (of a Square-nature or varga-prakṛti) with any given additive or subtractive, by making (combination) with the roots for the additive unity other first and second roots (of the equation having) the given additive or subtractive (can be found).¹ Let us take the following two equations: a₁k=an+b; and b₁k=bn+Na From them we get : by eliminating n a₁b−ab₁=1

1. रूप प्रक्षेपपदे पृथगिष्टक्षेपशोध्यमूलाभ्याम् । कृत्वाऽऽन्त्याथपदे ये प्रक्षेपे तेने ॥ —BrSpSi. XVIII. 66