THE earliest treatise on Algebra, which has come down to the present times, is that of Diophantus of Alexandria, in Egypt, who flourished about the middle of the fourth century after Christ, and wrote a work on this subject, in the Greek language, which, according to the testimony given by himself, in his introductory address to Dionysius, consisted, originally, of thirteen books, though, unfortunately for the interests of science, only the first six, and an imperfect tract on triangular and polygonal numbers, are now extant; which were first printed, in a Latin translation, by Xylander, in 1575, and afterwards, in Greek and Latin, with a comment, in 1621 and 1670, by Bachet and Fermat (a).

Other works on this subject, however, of a more easy and elementary kind, must, evidently, have existed long before the time of Diophantus; since he no where treats, in the abovementioned per

(a) This writer had a number of interpreters among the ancients, whose works are now lost of which we have more particularly to regret the commentary of the learned Hypatia, the daughter of Theon the philosopher (anno Domini 410); whose talents, virtues and misfortunes, will ever entitle her to the homage of posterity. See, for a more particular account of this illustrious victim of fanaticisin, Bossut's History of Mathematics, English edition, 8vo. 1803.

formance, of the first principles and leading rules of the science, as a writer in the infancy of the art would have done, but proceeds, at once, to the resolution of a particular class of indeterminate problems, relating chiefly to the finding of square and cube numbers, which, even at present, are generally considered as forming one of the most dif'ficult branches of pure analysis.

But whether we are indebted for this admirable invention to the genius of the Greeks, as has, hitherto, been commonly thought, or to that of some other ancient nation, cannot, at this distance of time, be easily ascertained; though, from the information which, for more than a century past, has been gradually obtained through our intercourse with the East, there are strong reasons for believing that algebra, as well as our common system of arithmetic, originated among the Hindoos, or natives of India; who are known to possess some very valuable works on this subject, which contain rules and principles that do not appear to have been derived from any foreign source.

The chief of these, and indeed the only performances of the kind, that have come to our knowledge, are the two celebrated works called 'the Beja Ganita, and the Lilavati; the first of which treats wholly on algebra and its applications, and the latter on arithmetic, algebra and mensuration, or practical geometry; and were both of them written in Sanscrit, about the end of the twelfth, or beginning of the thirteenth century of

the Christian era, by Bhascara Acharya, a famous Hindoo astronomer and mathematician of the city of Biddur, in the country of the Deccan.

It is chiefly, however, from the Persian translation of the Beja Ganita, which was made in 1634, by Ata Allah Rusheedee, at Agra or Delhi, and that of the Lilavati, by the celebrated Fyzee, in 1587, that we have obtained any certain knowledge of the contents of these curious performances; for which we are principally indebted to Mr. Edward Strachey, of the East India Company's Bengal civil establishment, who has lately favoured the public with a very interesting account of the former of these works, and some parts of the latter, taken from the source above mentioned (b).

From this publication it appears, that besides a knowledge of most of the common rules of algebra, at present in use, exclusively of cubic and biquadratic equations, the Hindoos were acquainted with some branches of the science that were not known in Europe till about the middle of the eighteenth century; it being here shown, by a va

(b) The late Mr. Reuben Burrow, who resided several years in India, collected, in that country, a number of Oriental manuscripts on mathematical subjects; some of which he left to his friend Mr. Dalby, Professor of Mathematics in the Royal Military College, Wycombe; particularly fair copies, in Persian, of the Beja Ganita and Lilavati, with the English of each word, written above the Persian, which Mr. Strachey mentions as having been of some service to him in composing his present performance.

riety of examples, extracted from the Persian translation of the Beja Ganita, that they were able to resolve all the possible forms of indeterminate equations, both of the first and second degree, by methods that were successively invented by Bachet de Mezeriac, Fermat, Euler and Lagrange, at a much later period; and as the questions here given, as well as the manner of treating them, are different from those of Diophantus, it is scarcely to be doubted that these people had among them a number of propositions relating to this science, which were of their own invention.

The Beja Ganita, it is true, as well as the Lilavati, are works of comparatively modern date; but from Bhascara's own account, who assumes no other character than that of a compiler, it appears that they were extracted from other performances of the same kind, which existed in his time; so that it is highly probable there are much older Indian treatises on algebra, in that country, than are yet known to us (c); and it is to be hoped,

(c) Mr. Davis, the well known author of two papers on Indian astronomy, in the third volume of the Asiatic Researches, who has furnished Mr. Strachey with a number of extracts and translations from the original Sanscrit Beja Ganita, which are given by him at the end of his present work, observes, "that almost any trouble and expense would be compensated by the possession of the three copious treatises on algebra from which Bhascara declares he extracted his Beja Ganita; and which, in this part of India, are supposed to be entirely lost."