which case, the product of the four remainders, obtained as above, is the square of the area. BHASCARA, in the Lilavati, remarks the error of BRAHMEGUPTA, taking notice, very justly, that the four sides of a quadrilateral do not determine its area. He does not, however, take notice of the case where the quadrilateral is subjected to the condition of being inscribable in a circle, and in which case the sides do determine the area.

There is a like inattention to the due limitations of two other theorems concerning quadrilaterals. It is stated very rightly by BRAHMEGUPTA, § 27 (p. 209), that the product of the two sides of a triangle, divided by twice the perpendicular (on the third side), is equal to the radius of the circle described about the triangle. This proposition is true of triangles; but the Hindu geometer applies it also to quadrilateral figures, without taking notice, that all such figures are not capable of being inscribed in circles. When they are, the proposition is applicable to them also; and is, that the rectangle under two contiguous sides of the quadrilateral, divided by twice the perpendicular on the diagonal that joins those sides, is equal to the radius of the circle described about the quadrilateral. This indeed is the same with the former proposition.

Another theorem, by no means very easy to be demonstrated, is enunciated as true of all quadrilaterals; though, like the former, it holds only of those that can be inscribed in a circle. It is, that the sum of the products of the opposite sides of the quadrilateral is equal to the product of the diagonals. This proposition is remarkable, as having been demonstrated by PPOLEMY in his Syntaxis, and made the foundation of the construction of his Trigonometrical Tables. The rule for constructing the Indian Table of Sines, as MR DAVIS has given it in the Asiatic Researches, may have been deduced from the same theorem. However that be, it is certain that the knowledge of this and the propositions formerly mentioned, argues a very extensive knowledge of elementary geometry, and such as is by no means easily acquired. Unfortunately, we have not the original demonstrations.

To this we must add, that these Geometers knew the theorems from which the area of the circle is computed, and also the superficies and solidity of the cone and sphere. That the area of a circle is equal to the rectangle under the radius, and half the circumference of the circle, is demonstrated in a very ingenious and palpable manner, not altogether according to the rigour of the Greek geometry, but abundantly satisfactory to those who are pleased with an argument when it is sound, though it be not

dressed exactly in the costume of science. The circle is supposed to be divided into two semicircles; and the circumference of each of these semicircles to be divided into a great number of equal parts. From the points of division in one of the semicircles, straight lines are then drawn to the centre, so as to cut the area into as many equal parts as there are divisions in the semicircumference. Then that semicircumference being straightened, or drawn out into a straight line, the triangular spaces into which the area is divided, will stand out from the circumference in the shape of small isosceles triangles, the whole figure resembling the teeth of a saw. The other semicircle, treated in the same manner, will afford the same result; and, if the two be made to approach, so that the one set of teeth shall fall into the intervals of the other, they will form a rectangular area, of which the length is the semicircumference, and the breadth the radius of the circle. Therefore, the areas of the two semicircles, or the whole area of the circle, is equal to a rectangle of which the length is half the circumference, and the breadth the radius. There is something very ingenious and simple in this reasoning, and such as might be readily admitted in a system of geometrical demonstration that was not very refined, or very scrupulous about introducing mechanical considerations.

The way of demonstrating that the superficies of the sphere is equal to four great circles of the sphere, is not pointed out. That the solidity is equal to the superficies multiplied into a third part of the radius, is derived from supposing the sphere to be made up of a great number of small pyramids, having their bases in the superficies, and their common vertex in the centre of the sphere. Such, we doubt not, has been, in all countries, the aspect under which this truth first appeared, and the original form in which it entered the mind of ARCHIMEDES as well as of BRAHMEgupta.

Among many subjects of wonder which the study of these ancient fragments cannot fail to suggest, it is not one of the least, that algebra has existed in India, and has been diligently cultivated, for more than twelve hundred years, without any signal improvement, or the addition of any material discovery. The works of the ancient teachers of science have been commented on, elucidated, and explained, with skill and learning; but no new methods have been invented, nor any new principle introduced. The methods of resolving indeterminate problems, that constitute the highest merit of their analytical science, were known to BRAHMEGUPTA, hardly less accurately than to BhasCara;

and they appear to have been understood even by Arya Bhatta, more ancient, by several centuries, than either. A long series of Scholiasts display, in their annotations, great acuteness, intelligence, and judgment; but they never pass far beyond the line drawn by their predecessors, which probably seemed, even to those learned and intelligent men, as the barrier within which the science was for ever to be confined. In India, indeed, every thing seems equally immoveable; and truth and error are equally assured of permanence in the situations they have once occupied. The politics, the laws, the religion, the science, the manners, seem all nearly the same, at this moment, as at the remotest period to which history extends. Is it because the power which brought about a certain degree of civilization, and advanced science to a certain height, has either ceased to act, or has met with such a resistance as it is barely able to balance, but not sufficient to overcome?. Or, is it because the discoveries which the Hindus are in possession of, are an inheritance from some more inventive and more ancient people, of whom no memorial remains, but some of their attainments in science?

Whatever opinion be adopted on these points, we are persuaded, that the light in which the analytical science of the East has been placed by the researches of Mr COLEBROOKE, must materially affect the conclusions to be formed concerning the originality and antiquity of the astronomy of those countries. On this subject, opinion scems at present to be considerably divided. When the Astronomical Tables of India first became known in Europe, the extraordinary light which they appeared to cast on the history and antiquity of the East made everywhere a great impression; and men engaged with eagerness in a study, promising that mixture of historical and scientific research, which is, of all others, the most attractive. The ardour with which they entered on this pursuit, the novelty of the objects, and the surprise excited, may have led them further, in some instances, than the nature of the evidence, when scrupulously examined, authorized them to proceed. Among those who were perhaps in a certain degree under the influence of such fascination, was the illustrious Historian of Astronomy, whom his talents, his virtues, and his misfortunes, have all combined to immortalize. BAILLY, who, in his Inquiries into the Origin of Astronomy in the West, had constantly found himself stopped, and unable to proceed, on account of the impenetrable obscurity that involves the antiquities of that quarter of the world, was willing to indulge a hope, that the light which seemed now rising in the East, was to dispel the obscurity he had so often complained of, and to discover the secrets contained in the antient history of

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the most ancient of the sciences. He therefore entered with great ardour on the study of the Eastern astronomy; on the exposition of its principles, and on the examination and defence of its accuracy; displaying, in all this, the usual resources of his ingenuity, his knowledge, and his eloquence.

A more minute examination, however, instituted by our countrymen on the spot, led them to doubt of the pretensions to high antiquity that they found in the Astronomical Books of the Hindus, and enabled them to detect errors into which the French astronomer had been betrayed, sometimes from the want of local knowledge, oftener from too much confidence in his informers, and occasionally, no doubt, from that spirit of system from which the men of greatest ardour and genius find it most difficult to defend themselves. The tide of opinion now began to set the contrary way; the recentness, and the inaccuracy of the Indian tables, were maintained no less keenly, and by much more objectionable reasonings than their antiquity and correctness had formerly been.

Among those who have lately taken up this argument, one of the most learned and skilful astronomers in Europe, M. DELAMBRE, is particularly distinguished. In a work just published, he has made an elaborate attack on the facts, the reasonings, and the calculations of the Astronomie Orientale, and has treated the author with a severity and harshness to which, from a brother academician, the memory of BAILLY should hardly have been exposed. His main argument is drawn from this fact, that the Data are nowhere quoted, from which the Indian tables were computed, and that there is no record, nor even any tradition of regular astronomical observations having been made by the Hindus. The truth of this assertion, as far as our present information goes, cannot be denied, and is certainly not very easy to be reconciled with the supposition that the Indian astronomy is as original and as antient as it pretends to be. Yet, as to the originality, there is still something to be said; and it has the more weight, from the originality of the Indian Algebra being rendered so very evident by the facts that we have been considering. This analysis, from all the light that history affords, could not be derived from Greece; at least it can have received from thence

*

Histoire de l'Astronomie Ancienne, Tom. I. p. 400, &c.

* Mr COLEBROOKE, after demonstrating the excellence of this algebra, and comparing its more perfect algorithm and its superior advancement with the Greek algebra, as explained in the work of Diophantus, seems nevertheless willing to admit, that some commu

none of the most improved and refined methods which it contains. In the earliest stage in which we discover it, it was already in possession of very high attainments, such as were not exceeded till very late in the history of European discovery. India itself, in the lapse of more than a thousand years, has added but very little to the perfection of this analysis. From whom then did that analysis derive its origin? If it be not an indigenous production of India, nothing remains but to conclude, agreeably to what we suggested, when much more imperfectly informed concerning the history of the Indian science, that what we now see is a fragment, or a derivation from a system that is lost-the remains of light once more widely dif fused, at the period when the Sanscrit was a living language, or when some parent language, still more ancient, 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. + If this conclusion, to which we are almost unavoidably led, be admitted, it will serve to explain the history of the Eastern Astronomy and its existence, as a wreck which has survived the memory of its authors-of those who made the observations on which it is founded, and supplied perhaps by diligence and length of time, the imperfection of the instruments they employed.

Those who, like DELAMBRE, are disposed to think lightly of the Indian Algebra and Arithmetic, will not admit the probability of this result. That mathematician, however, when he treated this subject, knew only the Lilavati; and probably, after seeing the Vija Ganita, and the treatises of BRAHMEGUPTA in Mr COLEBROOKE's translation, he will think of the matter somewhat differently, and will acknowledge, that India possesses a large portion of mathematical science, which it has neither derived from Arabia nor Greece.

We cannot conclude these remarks, without again adverting

nication about the time of the last mentioned author, may have come from Greece to India, on the subject of the Algebraic Analysis. Of this we are inclined to doubt; for this simple reason, that the Greeks had nothing to give on that subject which it was worth the while of the Indians to receive. Mr COLEBROOKE seems inclined to this concession, by the strength of a philological argument, of the force of which we are perhaps not sufficiently sensible. It seems however certain, that the facts in the history of Algebraic Analysis, aken by themselves, give no countenance to the supposition.

+ Edinburgh Review, vol. xxi. p. 376.