216 BRAHMAGUPTA AS AN ALGEBRAIST Second Rule: The absolute term multiplied by the coefficient of the square of the unknown is increased by the square of half the coefficient of the unknown; the square-root of the result diminished by half the coefficient of the unknown and divided by the coefficient of the square of the unknown is the unknown.² This when expressed in the modern algebraic notations would be Vac+b/2) (6/2) a Here if the quadratic equation is x= ax²+bx+c=0 the 'absolute term' is c (the one without the unknown x), 'the coefficient of the square of unknown' means the coefficient of x², i.e. a. and the 'coefficient of the unknown' means the coefficient of x. i.e. b. The above two methods of Brahmagupta are exactly the same as were suggested by Aryabhata I. The root of the quadratic equation for the number of terms of an arithmetic progression (A.P.) is given by Brahmagupta according to the first rule 2: 11 Third Rule: Brahmagupta also suggests a Third Rule which is very much the same as is used commonly now. Though it has not been expressedly suggested as a new rule, we find its application in a few. instances. For example this rule has been suggested in connection with the following problem on interest: √8bs+(2a-b)²-(2a-b) 26 A certain sum (p) is lent out for a period (t₁); the interest accrued (x) is lent out again at this 1. वर्गात रूपाणामव्यक्तार्धकृत संयुतानां यत् । पदमव्यक्तार्थोनं तवर्ग विभक्तमव्यक्तः || 2. उत्तरहीनद्विगुग्णादि शेषवर्ग धनोत्तराष्टवघे । प्रक्षिप्य पदं शेषोनं द्विगुणोदरहृतं गच्छः ॥ -BrSpSi. XVIII. 45 -BrSpSi. XII. 18
