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

184 BRAHMAGUPTA AND ARITHMETIC The above method of working Rule of Three is found among Arabs, although it does not seem to have been used in India after Brahmagupta. Problem Containing Quadratic Equation Perhaps Āryabhaṭa I is the first man in the history of mathematics to give a solution of a quadratic equation (499 A.D.). In his Āryabhaṭīya, he gives a rule for the solution of the following problem (I am reproducing it as described by Datta and Singh) : The principal sum p (=100) is lent for one month (interest unknown = x). This unknown interest is then lent out for t(=six) months. After this period, the original interest (x) plus the interest on this interest amounts to A(=16). The rate-interest (x) on the principal (p) is required. This problem requires the solution of the quadratic equation :— tx² + px - AP = 0 which gives x = (-p/2 ± √((p/2)² + Apt)) / t The negative value of the radical does not give a solution of the problem; so that the result is x = (√(Apt + (p/2)²) - p/2) / t This solution is stated by Āryabhaṭa I in the following words : Multiply the sum of the interest on the principal and the interest (A) by the time (t) and by the principal (p). Add to this result the square of half the principal (p/2)². Take square-root this. Subtract half the principal (p/2) and divide the remainder by the time (t). The result will be the (unknown) interest (x) on the principal.¹ Here the Sanskrit terms are mūla for principal and phala for interest.

1. मूलफलं सफलं कालमूलगुणमर्धमूलकृतियुक्तम् । मूलं मूलार्धोनं कालहृतं स्यात्स्वमूलफलम् ॥ Ārya. II. 25,