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

RULE OF DISSIMILAR OPERATIONS 223 is divided by the difference of them, and this (latter) is added to and subtracted from the quotient and then divided by two ; (the results are) the residues whence the number of elapsed days (can be found).¹ This viṣama-karma or dissimilar operation has been descri- bed by other Indian algebraists also, as Āryabhaṭa II (Mahāsi- ddhānta, XVII, 22); Śrīpati (Siddhānta-śekhara, XIV. 13) ; Bhā- skara II (Līlāvatī) and Nārāyaṇa (Gaṇita-kaumudī, I, 32). Indeterminate Equations of the First Degree Āryabhaṭa I should be given the credit of giving for the first time a treatment of the indeterminate equation of the first degree. In his Āryabhaṭīya, we find a method for obtaining the general solution in positive integers of the simple indeterminate equation : by – ax = c for integral values of a,b,c, and further indicated how to extend it to get positive integral solutions of simultaneous indeter- minate equations of the first degree. His disciple Bhāskara I (522) showed that the same method might be applied to solve : by – ax = – c and further that the solution of this equation would follow that of by – ax = – 1. These methods of Āryabhaṭa I and Bhā- skara I have also been adopted by Brahmagupta, and in certain cases, the improvement were suggested by Āryabhaṭa II in the middle of the tenth century A.D. The problems which were treated by ancient Indian algeb- raists and which led them to the investigation of the simple inde- terminate equation of the first degree may be classified under three heads : Class I : To find a number N which being divided by two given numbers (a,b) will leave two given remain- ders (R₁, R₂). Thus we have : N = ax + R₁ = by + R₂

1. तद्वर्गान्तरमाधे तदन्तरं चान्तरोद्धृतयुतोनम् । वर्गान्तरं विभक्तं द्वाभ्यां शेषे ततोद्युगणः । BrSpSi, XVIII, 97