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

260 BRAHMAGUPTA AS AN ALGEBRAIST by n², then we are justified in saying that if we put nx=u, the equation (i) becomes Mu²±c=y² (ii), and clearly the first root of (i) is equal to the first root of (ii) divided by n. The corresponding second root will be the same for both the equations. FORM a²x²±c=y² : We find Brahmagupta giving the following rule in this connection : This is a solution of a particular form of a varga-prakṛti or Square-nature. If the multiplier be a square, the interpolator divided by an optional number and then increased and decreased by it, is halved. The former (of these results) is the second root; and the other divided by the square-root of the multiplier is the first root.¹ Thus the solutions of the equation a²x²±c=y² are : x = 1/(2a) (±c/m - m) y = 1/2 (±c/m + m) where m is an arbitrary number. Bhāskara II and Nārāyaṇa have also given the same solutions as proposed by Brahmagupta. Rational Geometrical Figures In the days of the Taittirīya Saṁhitā and the Śatapatha Brāhmaṇa, Indian mathematicians got familiarity with the solution of such equations x²+y²=z² and the results were arrived geometrically on the basis of the law of rectangle as propounded by Baudhāyana in the Śulba Sūtras and which goes by his name. The reader is referred to the Chapter on Baudhāyana, the first Geometer in the author's "Founders of Sciences in Ancient India". Baudhāyana (c. 800 B.C.) gave a

1. वर्गे गुणके क्षेपः केनचिदुद्धृतयुतोनितो दलितः । प्रथमोऽन्यन्मूलमन्त्यो गुणकारपदोद्धृतः प्रथमः ॥ —BrSpSi. XVIII. 69