Page:Algebra, with Arithmetic and mensuration, from the Sanscrit.djvu/16

 iv DISSERTATION. 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 occasion to show, that modes of analysis, and, in parti- cular, general methods for the solution of indeterminate problems both of the first and second degrees, are taught in the Vjja-ganita, and those for the first degree repeated in the Lildvatl, which were unknown to the mathematicians of the west until invented anew in the last two centuries by algebraists of France and England. It will be also shown, that Bhascara, 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 aucient than himself That Bha'scaka's text (meaning the metrical rules and examples, apart from the interspersed gloss;) had continued unaltered from the period of the compilation of his work until the age of the commentaries now current, is apparent from the care with which they have noticed its various readings, and the little actual importance of these variations; joined to the considera- tion, that earlier commentaries, including the author's own explanatory annotations of his text, -wcro extant, and lay before them for consultation and reference. Those earlier commentaries ar« occasionally cited by name : particularly the Ganita-caumudi, which is repeatedly quote^i by more than one of the scholiasts.* No doubt then can be reasonably entertained, that we now possess the arithmetic and algebra of Bha'scara, as composed and published by him in the middle of the twelfth century of the Christian era. The age of his pre- cursors cannot be determined with equal precision. Let us proceed, how- ever, to examine the evidence, such as we can at present collect, of their antiquity. Towards the close of his treatise on Algebra,* Bha'scara informs us, that it is compiled and abridged from the more diffuse works on the same subject, bearing the names of Brahme, (meaning no doubt Brahmegupta,) Srid'hara and Padmana'bha; and in the body of his treatise, he has cited a passage of Srid'hara's algebra,' and another of Padmana'bha's.* He repeatedly adverts to preceding writers, and refers to them in general terms,, ' For example, by Su'ryada'sa, under Lildvati, §74; and still more frequently by Ranca- naYha. » Vija-gaiiUa, § 218. ' Ibid. § 131. ♦ Ibid. § 142.