# Algebra, with Arithmetic and Mensuration/Dissertation

​The history of sciences, if it want the prepossessing attractions of political history and narration of events, is nevertheless not wholly devoid of interest and instruction. A laudable curiosity prompts to inquire the sources of knowledge ; and a review of its progress furnishes suggestions tending to promote the same or some kindred study. We would know the people and the names at least of the individuals, to whom we owe particular discoveries and successive steps in the advancement of knowledge. If no more be obtained by the research, still the inquiry has not been wasted, which points aright the gratitude of mankind.
In the history of mathematical science, it has long been a question to whom the invention of Algebraic analysis is due? among what people, in what region, was it devised? by whom was it cultivated and promoted? or by whose labours was it reduced to form and system? and finally from what quarter did the diffusion of its knowledge proceed? No doubt indeed is entertained of the source from which it was received immediately by modern Europe; though the channel have been a matter of question. We are well assured, that the Arabs were mediately or immediately our instructors in this study. But the Arabs themselves scarcely pretend to the discovery of Algebra. They were not in general inventors but scholars, during the short period of their successful culture of the sciences: and the germ at least of the Algebraic analysis is to be found among the Greeks in an age not precisely determined, but more than probably anterior to the earliest dawn of civilization among the Arabs; and this science in a more advanced state subsisted among the Hindus prior to the earliest disclosure of it by the Arabians to modern Europe.
The object of the present publication is to exhibit the science in the state in which the Hindus possessed it, by an exact version of the most approved ​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​He completed his great work, the Sidd'hanta-sírómańi, as he himself informs us in a passage of it, in the year 1072 Saca. This information receives cor- roboration, if any be wanted, from the date of another of his works, the Caj-ana-cutuhala, a practical astronomical treatise, the epoch of which is 1105 Saca; 33 years subsequent to the completion of the systematic treatise. The date of the Sidd' hánta-śirómańi, of which the Vija-ganita and Lilávati are parts, is fixt then with the utmost exactness, on the most satis- factory grounds, at the middle of the twelfth century of the Christian era, A.D. 1150.
The genuineness of the text is established with no less certainty by nume- rous commentaries in Sanscrit, 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. A comparison of them authenticates the text where they agree ; and would serve, where they did not, to detect any alterations of it that might have taken place, or variations, if any had crept in, subsequent to the composition of the earliest of them. A careful collation of several commentaries, and of three copies of the original work, has bppn made , and it will be seen in the notes to the translation how unimportant are the discrepancies. From comparison and collation, it appears then, that the work of Bháscara, exhibiting the same uniform text, which the modem transcripts of it do, was in the hands of both Mahommedans and Hindus between two and three centuries ago : and, numerous copies of it having been diffused through- out 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 : or, to speak with the utmost precision, Jambusara in the west, Agra in North Hindustan, and Párthapúra, Gólagràma, Amarávati, and Nandigráma, in the south. ​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. ​DISSERTATION. v where his commentators understand him to allude to Arya-bhat't'a, to Brahmegupta, to the latter's scholiast Chaturve'da Prit'hudaca Swa'miV and to the other writers above mentioned. Most, if not all, of the treatises, to which he thus alludes, must have been extant, and in the hands of his commentators, when they wrote; as appears from their quotations of them; more especially those of Brahmegupta and Arya-bhat't'a, who are cited, and particularly the first mentioned, in several instances." A long and diligent research in various parts of India, has, how- ever, failed of recovering any part of the Padmanabha v'lja, (or Algebra of Padmana'bha,) and of the Algebraic and other works of Arya-bh atta.' But the translator has been more fortunate in regard to the works of Suii)'HARA and Brahmegupta, having in his collection Srid'hara's compendium of arithmetic, and a copy, incomplete however, of the text and scholia of Brah- megupta's Brahma-sidd'hanta, comprising among other no less interesting matter, a chapter treating of arithmetic and mensuration ; and another, the subject of which is algebra : both of them fortunately complete.* The commentary is a perpetual one ; successively quoting at length each A'erse of the text ; proceeding to the interpretation of it, word by word ; and subjoining elucidations and remarks : and its colophon, at the close of each chapter, gives the title of the work and name of the author.' Now the name, which is there given, Chaturve'da Prit'hudaca Swa'mi, is that of a cele- brated scholiast of Brahmegupta, frequently cited as such by the commen- tators of Bha'scara and by other astronomical writers: and the title of the work, Brahma-siddhunta, or sometimes Brahma sphuta-sidd' hdnta, corre- sponds, in the shorter form, to the known title of Brahmegupta's treatise in the usual references to it by Bha'scara's commentators;* and answers, in the longer form, to the designation of it, as indicated in an introductory couplet which is quoted from Brahmegupta by Lacshmida'sa, a scholiast of Bha'scara.^ Remarking this coincidence, the translator proceeded to collate, with the ' ^V'-g"'*- Ch. 5. note of Su'rtada'sa. Also V'tj.'gaii. § 174 ; and Lil. § 246 ad finem.
' Vitand-bhdshya by Chatuuve'da Piut'uu'daca Swa'mi, son of Mad'uu'su'dana, on the Brahma-sidiThdnta ; (or sometimes Brahma-sphuta-sidd'/tunta.)
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* ↑ Góládháya ; or lecture on the sphere, c. 11. § 56, As. Res. vol. 12. p. 314.
* ↑ As. Res. ibid.
* ↑ Though the matter be introductory, the preliminary treatises on arithmetic and algebra may have been added subsequently, as is hinted by one of the commentators of the astronomical part. (Vártic.) The order there intimated places them after the computation of planets, but before the treatise on spherics; which contains the date.
* ↑ Note A. 1