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

214 BRAHMAGUPTA AS AN ALGEBRAIST This problem would today be expressed in terms of the following equation : S(t+x)=x {s+((x-1)/2)b}, where x is the number of days after which the first overtakes the second. We may write this equation as bx²-{2(S-s)+b}x=2tS whence the value x would be after solving the quadra- tic : x= (√{2(S-s)+b}²+8bts+{2(S-s)+b}) / 2b The Bakhasālī Manuscript gives this solution as follows : The daily travel (S) diminished by the march of the first day (s) is doubled; this is increased by the common increment (b). That (sum) multiplied by itself is designated (as the kṣepa quantity). The product of the daily travel and the start (t) being multiplied by eight times the common increment, the kṣepa quantity is added. The square-root of this (is increased by the kṣepā quantity; the sum divided by twice the common increment will give the required number of days). (BMS. Folio 5, recto) Āryabhaṭa I (499 A.D.) is regarded as the founder of algebra, since he gives the solutions of a few quandratic problems. For example, to find the number of terms of an arithmetical pro- gression (A.P.), he gives the following rule : The sum of the series multiplied by eight times the common difference is added by the square of the dif- ference between twice the first term and the common difference: the square-root (of the result) is diminished by twice the first term and (then) divided by the com- mon difference : half of this quotient plus unity is the number of terms.¹ In the modern notations of algebra, the solution would be expressed as follows :

1. गच्छोऽष्टोत्तर गुणिताद् द्विगुणाद्युत्तर विशेषवर्गयुतात् । मूलं द्विगुणाद्यूनं स्वोत्तर भजितं सरूपार्थं ॥ —Ārya. II, 20