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and dated about the year 4383 of the Cali-yug, or 516 years ago; and the mean annual motions of the planets given in that work, were, on the principles of the Hindoo aftronomy, calculated to give the pofitions of the planets in the heavens, at that time, as near, at least, as the author could determine by obfervation. However, in order to do away thefe delufions, I fhall, before 1 proceed to the investigation of the antiquity of the Suryá Siddhánta, explain, in as fimple a manner as poffible, the principles upon which the Hindoo fyftems are founded, and the manner in which they are formed.'

The charge here brought against M. Bailly is, without doubt, a very heavy one, and affects his character deeply as an astronomer and a man of fcience. To have had a fet of aftronomical tables put into his hands, and not to have been able to discover their principles, or the fuppofitions on which they were calculated, might indeed involve no reproach at all. Their form might be fo enigmatical, they might be fo imperfect, and of fo little extent, as not to afford data for the required determination. But if fuch were the cafe, the astronomer must at least be fenfible of thefe defects. He must know whether he understood the matter before him or not. This is what a man, not to fay of science, but of common sense, could not but perceive; and if M. Bailly has really written a quarto volume on a fubject which he did not understand,-if he has treated of it at fo much length, and deduced from it so many confequences, it will be very difficult to reconcile his conduct with the ability and modefty by which he is usually thought to have been diftinguifhed. We shall beg leave to confider, therefore, how far this charge is well founded, and whether thofe refults which Mr Bentley intends to do away, are really the delufions which he fuppofes them to be.

The mean motion of any of the planets, or the angle which, at a medium, it defcribes in a given portion of time, is deduced from two determinations of its place, feparated by a confiderable interval of time from one another. The more accurate the obfervations, and the greater the length of time between them, the more exact will be the mean motion derived from this comparison. The length of the interval, even if the obfervations are not very exact, may fo far compenfate their inaccuracy, as to give great precifion to the refult. If, for example, we were to determine the length of the folar year, and if the obfervations compared were made at the interval of 2000 years, then, though the error in these observations fhould amount to fix hours, or a quarter of a day, the determination of the length of the year would neverthelefs be exact to the 2000dth part of fix hours, or to ten feconds nearly. It is thus that time adds

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to the accuracy of aftronomical determinations, and is capable of doing fo indefinitely; on which account, fuch determinations might continually approximate to the truth, in as much at least as regards the mean motions, even were no improvement to take place in the inftruments or methods of obfervation. The improvements in these last have no other effect than to render the approximation more rapid.

It is chiefly to this effect of time, in giving a value to ob fervations, that we are to ascribe the progreffive accuracy in the tables of the planetary motions. Thus, Ptolemy was enabled to give those motions more accurately than Hipparchus; the Arabs more accurately than Ptolemy; Tycho than the Arabs; and the modern aftronomers much more exactly than any of their predeceffors. With regard to the latter, it is true that great improvements in aftronomical inftruments have taken place; but, even independently of the fuperior accuracy derived from this fource, the mere lapfe of time would have produced a near approach to the fame refults.

This is the natural progrefs of aftronomical improvement, and is the infeparable concomitant of the antiquity of fcience. In the Indian aftronomy, there appears to be a contrivance calculated somewhat to retard and derange this natural progress; and it is on this contrivance, and the effect of it, that Mr. Bentley lays fo much stress, in the account of what he calls the artificial systems of the Indian aftronomy.

The contrivance referred to is this ;-The Indian aftronomers, having first determined the mean motion of the fun or any of the planets, from two or more observations made and compared as fuppofed above, have from thence gone back by calculation to fome fictitious epocha connected with their mythological fyftem, which, in all their future calculations, they choofe to affume as an observation actually made, and as the ftandard with which other observations are afterwards to be compared. The effect of this fiction must be, to prevent the knowledge of the mean motions from improving and becoming more perfect in the progrefs of time, in fo confiderable a degree as it has done in the aitronomy which has defcended from the Greeks, through the Arabs, to the nations of modern Europe.

This retardation of improvement, and the continuance of the science nearly in the same state for a succession of ages, are the only possible effects that could result from the practice here referred to. This may be made evident by a very simple instance. Suppose that the motion of the sun were reckoned at 360° in 365 days 6 hours, as in the Julian calendar, and that the instant in a certain

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tain year, when the sun was in the vernal equinox, had also been observed. Suppose, likewise, that at the end of ten times 365 days and 6 hours, or after 3652 days and a half, the sun's place is again observed, and is found to be advanced beyond the vernal equinox by a small arch, such as the sun-passes over in 1. 50. then it is evident that the sun's annual motion has been supposed too slow by the tenth part of the small arch of excess just mentioned, or every year two long by the tenth part of 1h. 50m, that is by 11 minutes, so that the true length of the year is 365d. 5h. 49m.

But now, let us suppose that, after the first observation was made, the astronomer had counted back 10 years or 3652 days and a half, and at the commencement of that period had concluded the sun to be in the vernal equinox; then if he compares his second observation, not with the first, which was only 10 years distant from it, but with the fictitious observation, which is 20 years distant from it, he will conclude that the year has been reckoned too long only by the 20th part of 1h. 50m. instead of the 10th, and therefore the correction which he applies in order to obtain the true length of the year will be only the half of what he ought to have applied. If, however, preserving the fictitious epocha, other observations at the distance of 30, 40, 50 years, &c. be compared with it, the corrections found will approximate to the true corrections as the fractions, 4, 4, do to,

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; so that the errors will decrease in the proportion of the differences of the above, or as the fractions, T, 25, &c. The only effect here is to retard the progress of improvement, but by no means to give the system any greater accuracy than its principles naturally involve.

If the fictitious epocha were taken at a very great distance from the date of the real observations, then the progress of the science might be in some measure stopped; the retardation of improvement would become so great, that the science would continue in the same state for a vast number of ages. This is in a great measure the case with the Indian systems, into which, for reasons that we do not know, but probably connected with religion, such fictitious epochas have been introduced. In consequence of this fiction, these systems remain more stationary than they would do otherwise; and it seems as if the astronomers of the East had been led, by a kind of instinct, to a device that was to give the same inactivity and inertness to their science, that pervades among them the whole of the moral and intellectual world.

Now, admitting all this to be just, and to be a true defcription of the Indian tables of aftronomy, let us fee how it will

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affect the argument concerning their antiquity. It is evident, that this aftronomy, in as far as relates to the mean motions, has little fufceptibility of improvement, and cannot have acquired by fubfequent obfervation much more exactness than at the time when the tables were firft conftructed, and the artificial fyftem introduced. We muft therefore confider whatever exactnefs we find in such a system, as going back to its commencement, when the mean motions must have been determined by accurate or diftant observations. Now this is precifely the ground of M. Bailly's argument, which, therefore, is either not at all affected, or only a little ftrengthened by the confideration fuppofed by Mr Bentley to be fo entirely fubverfive of it. Indeed, it is on the accuracy of the mean motions, as they are actually fet down, that the proofs of the antiquity of the tables must depend, and every other condition may be fafely fet afide. The circumftance on which Mr Bentley lays fuch stress, is really extraneous to the conftruction of the tables; it amounts to nothing but a fecurity that, in the fubfequent edi- . tions, they have received but little improvement; and M. Bailly, had he fuppofed the fact to be as has been ftated, could not have drawn any other conclufion than that which he has actually done. It muft till have been by the accuracy of the mean motions, as contained in the tables, that their merit, and their claim to antiquity was decided. The mean motions of the heavenly bodies can be discovered in one way only, viz. by the comparison of obfervations made at a great distance, in time, from one another; and the principles on which this is done must be every where the fame. A man cannot fet about making a fyftem of aftronomy by the mere force of his fancy or his genius, as he may write a romance or an epic poem. It is not by invention, but by obfervation and discovery, that his task is to be performed. The principles on which he muft proceed, if he would attain accuracy, must be every where the fame, in whatever age or country he is placed,-whether he has gone to work on the banks of the Ganges or the fhores of the Atlantic,--has lived in the antediluvian ages, or in the nineteenth century,-has been inftructed by the philofophy of Newton, or amufed by the fictions of Varáha.

The author of the paper on the antiquity of the SURYA SIDDHANTA fhould therefore have thought well before he hazarded an affèrtion that was to charge with ignorance or prefumption fuch men as Caffini and Bailly, who had explained the aftronomy of the East; it was not likely that an amateur, however distinguished, fhould convict thefe aftronomers of grofs ignorance, or find it fo eafy to do away their opinions, in a matter that concerned their own profeffion, -a fcience which, day and night, had been for many years the fubject of their study.

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Let us next confider the criterion which Mr Bentley himself propofes for determining the age of a fyftem of aftronomical tables, from the confideration of the tables themselves, independently of teftimony, tradition, or any external evidence. Such a criterion is precifely the thing wanted on the present occasion, but we can by no means approve of that particular one which he endeavours to establish. It is founded on this maxim, that the time of the conftruction of any fet of tables must be that at which they agree bet with the heavens. Hence, when fuch tables are given, and we wish to determine their antiquity, we have only to compute from them, the places of the fun and moon, &c. for different times confiderably distant from one another: to compare these places with thofe given by the best modern tables, and the time when they approach the nearest to one another, is to be taken for. the time when the tables were constructed. As it must be an object, in all astronomical tables, to reprefent the ftate of the heavens tolerably near the truth at the time when they are compofed, it must be allowed that this rule is not deftitute of plaufibility. On examination, however, it will be found very fallacious, and fuch as might lead into great mistakes.

Astronomical tables are liable to errors of two different kinds, that may fometimes be in the fame, fometimes in oppofite directions. One of them concerns the radical places at the epoch from which the motions are counted; the other concerns the mean motions themselves, that is to fay, the mean rate or angular velocity of the planet. Of these the first remains fixed, and its effect at all times is the fame; the fecond again is variable, and its effect increafes proportionally to the time If, therefore, they are oppofite, the one in excefs, and the other in defect, they muit partly deftroy one another; and the one increafing continually, will at length become equal to the other, when there will, of confequence, be no error at all; after which the error will fall on the oppofite fide, and will increase continually. Here, the moment of no error, or that when the tables are perfectly correct, is evidently disfant from the time of the construction of the tables, and may be very long, either before, or after that period. Suppose, for example, that, in conftructing tables of the fun's motion, we are to fet off from the beginning of the prefent century, and that we make the fun's place for the beginning of the year 1801 more advanced by half a degree than it was in reality. Suppofe, alfo, that the mean motion fet down in our tables is erroneous in a way oppofite to the former, and is lefs than the truth by 1" in a year. The place of the fun then as affigned from the tables for every year, fubfequent to 1800, will, from the first of the above

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