ii DISSERTATION.
treatise on it in the ancient language of India, with one of the earlier treatises (the only extant one) from which it was compiled. The design of this pre- liminary dissertation is to deduce from these and from the evidence which will be here offered, the degree of advancement to which the science had arrived in a remote age. Observations will be added, tending to a compa- rison of the Indian, with the Arabian, the Grecian, and the modern Algebra: and the subject will be left to the consideration of the learned, for a con- clusion to be drawn by them from the internal, no less than the external proof, on the question who can best vindicate a claim to the merit of having originally invented or first improved the methods of computation and analysis, which are the groundwork of both the simple and abstruser parts of Mathe- matics; that is, Arithmetic and Algebra: so far at least as the ancient inven- tions are affected; and also in particular points, where recent discoveries are concerned.
In the actual advanced condition of the analytic art, it is not hoped, that this version of ancient Sansorit treatises on Algebra, Arithmetic, and Mensu— ration, will add to the resources of the art, and throw new light on mathe-
matical science, in any other respect, than as concerns its history. Yet the remark may not seem inapposite, that had an'oarlicr version of these treatises
been completed, had they been translated and given to the public, when the notice of mathematicians was first drawn to the attainments of the Hindus in astronomy and in sciences connected with it, some addition would have been then made to the means and resources of Algebra for the general solu~ tion of problems by methods which have been re-invented, or have been per- fected, in the last age. '
The treatises in question, which occupy the present volume, are the Vija- gmiita and Lilévati of BHA’SCARA A'CHA'RYA and the Gan'itéd‘haya and Cut't'acéd‘hya‘ya of BBAHMEGUPTA. The two first mentioned constitute the preliminary portion of BIIA'SCA RA's Course of Astronomy, entitled Sidd'hénta— .éirémmii. The two last are the twelfth and eighteenth chapters of a similar course of astronomy, by BRAHMEGUPTA, entitled Brahma-Sidd’hénta.
The questions to be first examined in relation to these works are their authenticity and their age. To the consideration of those points we now proceed. I
The period when BnA'scmm, the latest of the authors now named, flou- rished, and the time when he wrote, are ascertained with unusual precision-
