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

190 BRAHMAGUPTA AS AN ALGEBRAIST A.D.). There can be no doubt regarding the age of these two authors. Bhāskara II completed his great work on the Siddhānta- śiromaṇi in 1072 Śaka, and Karaṇa-kutūhala a practical astro- nomical treatise in 1105 Śaka; these dates are based on the passages given by Bhāskara himself in his works. The Bīja- gaṇita and the Līlāvatī form parts of the great treatise, the Siddhānta-śiromaṇi. The genuineness of the text is established, as Colebrooke says, with no less certainty by numerous commen- tators in Sanskrit, besides a Persian version of it. Those com- mentaries comprise a perpetual gloss, in which every passage of the original is noticed and interpreted : and every word of it is repeated and explained. From comparison and collation of various texts, it appears then that the work of Bhāskara, exhibit- ing the same uniform text which the modern transcripts of it do, was in the hands of both Muhammedans and Hindus, between two or three centuries ago : and numerous copies of it having been diffused throughout India, at an earlier period, as of a performance held in high estimation, It was the subject of study and habitual reference in countries and places so remote from each other as the north and west of India and the Southern Peninsula. This though not marking any extraordinary antiquity, nor approaching to that of the author himself, was a material point to be determined : as there will be in the sequel, so says Cole- brooke, occasion to show, that modes of analysis, and in parti- cular, general methods for the solution of indeterminate prob- lems both of the first and second degrees, are taught in the Bīja-gaṇita, and those for the first degrees repeated in the Līlāvatī, which were unknown to the mathematicians of the West, until invented anew in the last two centuries by algebraists of France and England.¹ Bhāskara who himself flourished more than six hundred and fifty years ago, was in this respect a compiler and took those methods from Indian authors as much more ancient than himself. Regarding the age of the precursors of Bhāskara II, Cole- brooke says : The age of his precursors cannot be determined with equal precision. He then proceeds to examine the evidence as follows :

1. Colebrooke, H. T., Miscellaneous Essays, p. 421.