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

222 BRAHMAGUPTA AS AN ALGEBRAIST These equations have also been treated by Mahāvīra, Bhāskara II and Nārāyaṇa. Nārāyaṇa has attempted two other forms of quadratic equations : (v) x² + y² = c } (vi) x² - y² = m } x - y = d } xy = b } For their solutions, see Datta and Singh, Algebra, P. 84. Rule of Dissimilar Operations : Datta and Singh say that the process of solving the follow- ing two particular cases of simultaneous quadratic equations was distinguished by most Indian mathematicians by the special designation viṣama-karma or dissimilar operation : (i) x² - y² = m } (ii) x² - y² = m } x - y = n } x + y = p } These equations have been regarded by these mathematicians as if of fundamental importance. They have given the following solutions (expressed in modern algebraic symbols) : For the combination (i) : x = ½ (m/n + n), y = ½ (m/n - n), For the combination (ii) : x = ½ (p + m/p), y = ½ (p - m/p). We shall express these solutions as follows in the words of Brahmagupta : The difference of the squares (of the unknowns) is divi- ded by the difference of the unknowns and the quotient is increased and diminished by the difference and divided by two ; (the results will be the two unknown quantities) ; (this is) dissimilar operation. The same rule is restated by him on a different occasion in the course of solving a problem. If then the difference of their squares, also the differe- 'nce of them (are given) ; the difference of the squares

1. योगोऽन्तरयुग्हीनो द्विहतः संक्रमणमन्तर विभक्तं वा । वर्गान्तरमन्तर-युतहीनं द्विहतं विषमकर्म । BrSpSi. XVIII. 36