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

206 BRAHMAGUPTA AS AN ALGEBRAIST quadratic or of higher powers, one side of it is cleared of the unknowns of all denominations and the other side of it of the absolute terms, so that the equation is ultimately reduced to one of the form ax² + bx = c, where a, b, c may be positive or negative; some of them may even be zero. Thus Brahmagupta observes : From which the square of the unknown and the un- known are cleared, the known quantities (rūpāṇi) are cleared (from the side) below that.¹ On this Pṛthūdaka Svāmī comments as follows : This rule has been introduced for that case in which the two sides of the equation having been formed in accordance with the statement of the problem, there are present the square and other powers of the unknown together with the (simple) unknown. The absolute terms should be cleared off from the side opposite to that from which are cleared the square (and other powers) of the unknown and the (simple) unknown. When perfect clearance (saṁśodhana) has been thus made...² Śrīdhara and Bhāskara II have also given the rules of clearance almost on the same lines. Thus the equation yā va 0 yā 10 rū 8̇ yā va 1 yā 0 rū 1 after perfect clearance having been made will be (according to Pṛthūdaka Svāmī) yā va 1 yā 10 rū 9̇ i.e. the equation 10x - 8 = x² + 1 after clearance would become x² - 10x = -9. Classification of Equations Usually equations are classified as : simple equation : yāvat-tāvat quadratic : varga

1. वर्गाव्यक्ताः शोध्या यस्माद् रूपाणि तदधस्तात् ॥ — BrSpSi. XVIII. 43.
2. BrSpSi. XVIII. 43 (com.)