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plied, from which it appears, that the property of the right angled triangle was very well known to the Hindu mathematicians, and its use in subjecting geometrical figures to algebraic calculation. One of these, to find the sides of a right angled triangle, the sum of the three sides being given = 40, and the rectangle of the two sides about the right angle = 120, is very neatly resolved with a considerable display of geometrical knowledge and some address, in avoiding the introduction of surd quantities. If the sum of the sides be s, and the hypothenuse h, it is inferred from a lemma previously demonstrated, that sh twice the rectangle of the sides, or 240. But it has htwice also been shown that (s + h) ( s − h) = -h, therefore (s + h) (s − h) 240, and since s + h 40, sh 6, so that s 23, and h 17. Therefore 23 is to be divided, so that the rectangle of the parts = 120, which gives an ordinary quadratic equation, from which the sides about the right angle are found to be 15 and 8. The Persian translator here, as in some other places, refers to the propositions in Euclid; but this is a mere interpolation, for which there is no authority in the original.

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On considering the whole of this Fragment, for as such we must regard it, we cannot but be of opinion, that it goes farther to resolve the question, whether the remains of mathę~ matical science in India are derived from Arabia, and so from Greece, or are either the native productions of the country in which they are found, or emanations from a source which is entirely unknown, than any thing that has yet appeared. To judge as to the first of these, we must compare the methods of the Bija with those of Diophantus, and of the Arabian authors. With respect to Diophantus, the comparison is easy; and Mr Strachey has justly observed, in a memoir On the Early History of Algebra, that very little resemblance can be observed between them. The solutions of particular problems given by the Greek geometer, are extremely elegant, and do great credit to his ingenuity. As to general methods, we find none; and, much less, general formule. Accuracy and simpli city are the characters of Diophantus's investigations; but there is little of generalization. His work might be the foundation of such solutions as we find in the Bija Gannita; and it may perhaps be regarded as the foundation of those of Euler and La Grange. But much exercise of inventive genius, many

*The Arabian and Persian geometers quote the 47th of Euclid by the name of the figure of the Bride, and the figure of the Chair's one does not see from what analogy.

efforts made with great skill, must have been interposed between the one point and the other. It is not among the modern Hindus, nor among them such as they have existed for many ages, that these efforts can be supposed to have been made. The alge bra of India, therefore, is not derived from that of Diophantus. The same argument holds with respect to the science of the Arabs. Here traditional, and even historical accounts, are in favour of the Arabians having derived their science from India. Dr Hutton states this curious anecdote, that BOMBELLI, whose name is well known in the early history of Algebra, published a treatise on that science, at Rome, in 1579, in which he says that he had translated part of Diophantus, and that he found the Indian authors often cited in it, from which he 'concludes, that the science of Algebra was known to the In'dians before the Arabians had it." This rests entirely on the testimony of Bombelli; and none of the manuscripts of Diophantus, that are at present known, contain any thing to the effect here mentioned. Whatever weight we allow to the testimony of Bombelli, it is certain that the Muslemans now residing in India, of whom there are some very learned in the mathematical sciences, all consider algebra as having originated in the latter country, as well as the arithmetic now in use. If we proceed farther, and inquire into the state of algebra among the Arabians, we shall find, that among the works of their own countrymen, there is none that can be compared with the Bija Gannita, in what respects the solution of indeterminate problems. Mr Strachey, during his residence in India, took particular pains to inform himself concerning the best treatises of algebra in Arabic; and he was led, on the authority of the most learned of that nation, to study with care the Khulásat-ul-Hisab, the work of Baha-ul-din, an astronomer and mathematician, born at Balbec, in the year 953 of the Hegira, or 1578 of the Christian era. Mr S. made a translation of a great part of it from the viva voce interpretation into Persian, of a learned Musleman, MAULAVI-ROSHEN-ALI, who was perfectly master of the subject of the treatise, as well as of the Persian and Arabic languages. From the abstract given in the memoir above referred to, and inserted likewise in Dr Hutton's History of Algebra, (Tracts, Vol. II. p. 179, &c.) we are led to form a very favourable opinion of this work. first part contains arithmetic, and some practical rules of mensuration; and the second part carries algebra as far as quadratic equations, and on the whole nearly to the length it attained in the hands of the Italian mathematicians, who in the 14th and 15th century imported this science into Europe. Bb

VOL. XXI. NQ. 42.

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From this, however, to the methods contained in the Bija Gannita, the step is prodigiously great, in as much at least as regards indeterminate problems, a subject of difficult discussion, and treated of by the Hindu mathematicians, as we have already observed, in a manner not unworthy of the reputation of EULER and LA GRANGE. The Indian treatise, though so much more ancient, is much more profound than the Arabic, which at this moment is reputed the best in that language; a fact which furnishes a complete answer to the assertion, that the sciences of the former country are borrowed from the latter. We know indeed of no country, in a condition to lend the truths, which the Hindus are alleged to have borrowed, except modern Europe, and that only since the middle of last century.

This argument, in favour of the originality of the Hindu algebra, is quite independent of that which we formerly stated, as grounded on the singular application of the word colour, and the names of the different colours. This is a distinguishing mark, to which nothing similar occurs in the science of Arabia, or of any other country, and has a strong claim to the character of originality.

It is necessary to attend particularly to the import of the term originality, as employed here. We do not pretend, by it, to say of what country this algebra is the original production, but only that it is not derived from any known source, or any system of science with which the Western world has yet been made acquainted. It may either be an indigenous production of India; or it may be, as indeed there is reason to think it is, a fragment of a system that is lost; a remain of a great body of science which enlightened the world at some very remote period, when the Sanscrit was a spoken language, or when some parent language, still more antient, sent forth those roots which have struck with more or less firmness into the dialects of so many and such remote nations, both of the East and of the West.

Or it may be a fragment of antediluvian science, that has escaped the ruin produced by one of those great catastrophes which have shaken or overwhelmed the earth, and brought destruction on so many of its inhabitants.

But whatever opinion is formed at present, it can be considered only as probable, and provisionary, till such time as all the evidence can be examined. It would contribute essentially to this object, to have both the books of Bhascara, the Lilavati and the Bija Gannita, accurately translated from the Sanscrit originals, accompanied with such notes or commentaries as the translator should judge proper, but in such a form, that

these last should be quite separated from the text. It is not too much to hope for this from the Asiatic Society, to which we are already indebted for so much information concerning the antiquities of India. Perhaps it might be reckoned an undertaking not unworthy of the protection of the Company itself, whose very liberal and disinterested exertions in behalf of science, we have more than once had occasion to remark.

It would also be necessary, to render the data complete, to have fuller information concerning the Arabic works in algebra. Though MR STRACHEY has given a very full and satisfactory account of one of the best of those treatises, and has done, indeed, all that an individual could be expected to perform, there may be other treatises on the same subject deserving of notice, and concerning which it were desirable that farther inquiry should be made. If Arabia have any claim to be considered as the instructor of India, it is in this way only that such claim can be established. Indeed we have very little doubt that the truth, to a certain length at least, would be ascertained on good evidence, if the question were discussed with perfect fairness and impartiality, without any love of paradox, on the one hand, or any desire to prove, at all hazards, the great antiquity of Indian science; and, on the other hand, without any fear of discovering proofs of such antiquity, or any desire of reducing it within limits previously determined. To one who is not quite aware how much prejudice warps all our opinions, it would seem very unnecessary to exhort men to impartiality on an inquiry into questions of the most remote antiquity, and that can affect, one should think, the personal interest of no one individual at present on the surface of the earth. Yet every one who has attended to what has already passed on the subject of the astronomy of India, must know, that such cautions as we are now presuming to offer, are by no means unnecessary. We cannot dismiss this subject without again reminding our readers how much they are obliged to Mr STRACHEY, for supplying a document so important in this question, as that of which we have been giving an account. He has entered on the research with candour and ability; has pursued it with great industry, and at great expense of time, in a situation where his time was probably of great value. He may, at least, have fairly the satisfaction to think, that he has done a service to all who are interested in the history of knowledge; and that nothing which has yet been produced, has thrown so much light on the science of the East, as that which he has laid before the public.

ART. VII. A Tour through Italy, exhibiting a View of its Scenery, its Antiquities, and its Monuments; particularly as they are Objects of Classical Interest and Elucidation: With an Account of the present State of its Cities aud Towns, and occasional Observations on the recent Spoliations of the French. By the Rev. John Chetwode Eustace. 2 vol. 4to. pp. 1342. London. Mawman. 1813.

THIS is one of the best books of travels that have appeared since we began our labours; and, consistently with our high sense of its value, we cannot delay bringing it fully before our readers. Of the subject, little needs be said. It is, perhaps, the most interesting to which a traveller could devote himself. In the design, we may have occasion to regret certain omissions, and to wish that Mr Eustace had taken a somewhat wider range in his inquiries and observations. Of the execution, we must speak more in detail as we proceed.

Mr Eustace is a Roman Catholic clergyman, who travelled with an amiable young gentleman of the name of Roche, since deceased; and having, during the year 1801, fallen in company with Lord Brownlow and Mr Rushbrooke at Vienna, they all resolved to undertake together the tour of Italy, which they accomplished the following year. He does ample justice to the good qualities of his companions; and in particular, expresses his obligations to Lord Brownlow, for much valuable assistance in the course of his work. A good Catholic travelling in Italy cannot fail to find frequent opportunities of reminding his readers, that their religious creeds differ; yet we must say for Mr Eustace, that he is by no means narrow-minded or uncharitable in his observations. There is no doubt, however, that he is considerably tinctured with enthusiasm, religious as well as classical. He plainly feels inspired as much with the modern as the ancient recollections, excited by the scenes which he visits: But there is little or no bigotry mixed up with his enthusiasm; and we know not that his book is the worse for this peculiarity in his faith. It certainly lends animation and interest to many parts, which in former travellers were somewhat tame; and this may serve to make amends for the excesses of description into which it leads him, when he gets among churches and ceremonies. At all events, the frank and manly avowal contained in the following passage of his Preface, must be allowed to give the reader full warning upon this topic; and our Protestant alarmists have themselves to blame, if they run the risk of seduction, by entering the scarlet gentlewoman's dwelling, after reading so plain an inscription over the door-way.

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