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

236 BRAHMAGUPTA AS AN ALGEBRAIST where z and w denote the ahargaṇa and the revolutions per- formed by Mars respectively. The general solution of this equation is z = 131493125 s + 118076020 w = 191402 s + 171872 where s = 0, 1, 2, 3......When s = 0, we have the least solution. Brahmagupta’s Rules Concerning Indeterminate Analysis of the First Degree For the solution of Āryabhaṭa’s problem, Brahmagupta gives the following rule : What remains when the divisor corresponding to the greater remainder is divided by the divisor correspond- ing to the smaller remainder—that (and the latter divisor) are mutually divided and the quotients are severally set down one below the other. The last residue (of the reciprocal division after an even number of quo- tients has been obtained) is multiplied by such an optio- nal integer that the product being added with the differ- ence of the (given) remainders will be exactly divisible (by the divisor corresponding to that residue). That optional multiplier and then the (new) quotient just obtained should be set down (underneath the listed quotients). Now, proceeding from the lower-most number (in the column), the penultimate is multiplied by the number just above it and then added by the number just below it. The final value thus obtained (by repeating the above process) is divided by the divisor corresponding to the smaller remainder. The residue being multiplied by the divisor corresponding to the greater remainder and added to the greater remainder will be the number in view.¹

1. अधिकाग्रभागहारादूनाग्रच्छेद भाजिताच्छेषम् । यत् तत् परस्परहृतं लब्धमधोधः पृथक् स्थाप्यम् ॥ शेषं तदिष्टगुणितं यथाऽनयोरन्तरेण संयुक्तम् । शुध्यति गुणकः स्थाप्यो लब्धं चान्त्यादुपान्त्यगुणः ॥ स्वोर्ध्वोऽन्त्य युतोऽग्रान्तो हीनाग्रच्छेदभाजितः शेषम् । अधिकाग्रच्छेदहत मग्रिकाप्रयुतं भवत्यग्रम् ॥ —BrSpSi. XVIII, 3-5