BRAHMAGUPTA AS AN ALGEBRAIST A.D.). There can be no doubt regarding the age of these two authors. Bhaskara II completed his great work on the Siddhanta- firomani in 1072 Saka, and Karana-kutuhala a practical astro- nomical treatise in 1105 Saka; these dates are based on the passages given by Bhaskara himself in his works. The Bija- ganita and the Lalavatı form parts of the great treatise, the Siddhanta siromani. 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 Bhaskara, 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. 190 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 Bija-ganita, and those for the first degrees repeated in the Lalavati, which were unknown to the mathematicians of the West, until invented anew in the last two centuries by algebraists of France and England. Bhaskara 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 Bhaskara 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.
