turies before Christ-has been supposed to attribute to a philosopher contemporary with Tchou-kong a knowledge of the famous property of the right-angled triangle. In this fragment, the philosopher mentioned concludes, that if the two sides of a right-angled triangle are respectively equal to 3 and 4, the base will be equal to 5. Here we must remark, that the numbers 3 and 4 are not taken at hazard, but selected from some mysterious connexion supposed by Con-fu-tso to exist between these numbers and the universe*. Gaubil concludes, that in the time of Tchou-kong, the Chinese had methods for the resolution of rightangled triangles, though spherical trigonometry was unknown to them till the time of Cocheou-king, twelve centuries after Christ.

The history of Con-fu-tso, which extends from 720 to 481 B. C., records several eclipses, of which a good many have been verified by modern astronomers; others appear to have been marked in the wrong month; from the rough way in which all are given, they can only serve to fix the dates of Chinese chronology, and to show the assiduity with which these phenomena were noticed at so early an age. The most interesting observation of this period regards the position of the winter soltice, which was placed in the beginning of the constellation Nieou, the first star of which was Capricorni. Now, as there can be no doubt that the astronomers who found this position of the solstice, were acquainted with the observations of Tchou-kong, who placed it at 2° from Aquarii, nearly 9° of longitude distant, it seems evident that they must have perceived the apparent retrogradation of the solstices and equinoxes. Such is the conclusion drawn, and apparently with reason, by Gaubil + and La Place; but in this case, it seems strange that the astronomers of the Hans, two centuries before Christ, should have been ignorant of the effects of precession.

Subsequently to this time the Chinese astronomy ceases to possess the same interest for us: as it has always remained in so rude and imperfect a state, that nothing but the antiquity of the determinations to be found in their books could make them worthy of attention. Complete astronomical treatises, as early

Lett. Ed. p. 117. The numbers 3, 4, and 5, multiplied into each other, give 60-the number of years of the Chinese cycle.

+ Lett. Ed. xxvi. p. 247, ed. Paris, ‡ Mécan, Célest. vol. v. p. 246.

as the year 108 B.C., are still in existence; but the reader who wishes to investigate their tables and methods, is referred to the treatises of Gaubil on this subject, in the second and third volumes of the collection of Souciet. We shall only notice here the method of establishing their epochs, followed by all the Chinese astronomers, anterior to Cocheou-king, above mentioned; and we will take, as an example, the treatise of Lieou-hiu, the most ancient of those now extant. The epoch of these tables is a general conjunction of the sun, moon, and planets, the moon being on the ecliptic, about 143127 years, before the year 104 B.C.* This latter seems to be the real epoch of his tables: the other, it is scarcely necessary to observe, is obtained by calculating back from the year 104 B.C., with the mean motions found for the different planets, till a general conjunction was obtained. This, as we have noticed, was the method usually followed by the Chinese; but sometimes, in order to avoid such large numbers as that just given, they would content themselves with a very rough approximation to a general conjunction, and neglect the errors arising on the mean motions, as too small to be noticed; which, if the epoch were at all distant, they would really be. The pretended general conjunction in the reign of Tchuen-hiu, above noticed, is, in all likelihood, an epoch obtained in this way. In the astronomy of Lieouhiu we find the cycle of nineteen years very clearly explained; and, indeed, not only this, but the period of seventy-six years proposed in Greece by Callippus, was known in China before the Christian æra t.

It seems that, about the year 164 after Christ, the Chinese began to have communication with subjects of the Roman empire; and it is worthy of notice, that very shortly afterwards some important reforms were made in their astronomy. They now ascertained the eccentricity of the solar orbit, the principal inequality of the moon, and a more exact value of the solar year; and we now find, for the first time, a distinct account of precession; a phenomenon, however, with which we believe they must have been acquainted long before. That some of the discoveries just mentioned were introduced from the west is only a conjecture; but it is a conjecture which derives considerable force from Souciet, vol, ii. p. 16. † Ibid., p. 21. Ibid., p. 24.

the circumstance, that during the whole interval between the second and thirteenth centuries after Christ, the Chinese, though continuing to observe with assiduity, made little or no progress, and certainly not one discovery. Their greatest improvements did not go beyond some trifling ameliorations in the elements of their tables. Those who feel any curiosity to examine their observations, will find in the works of Gaubil, edited by Souciet, and in the Connoissance des Temps for 1809, a considerable quantity of observations of solstices made with the gnomon, and of solar eclipses; with some notices of occultations, of comets, and of appulses of Jupiter to the fixed stars. Of these the most important are the observations of the gnomon; some of which have been used by La Place to determine the diminution of the obliquity of the ecliptic to the equator.

The conquest of China by Gent-Chiskhan, who brought with him men well versed in the astronomy of Ptolemy and the Arabs, gave a fresh impulse to the languid state of the Chinese astronomy; but the ameliorations then introduced belong rather to the history of the middle ages. It is time now to turn to a people whose astronomical reputation is greater, though perhaps less deserved, than that of the Chinese.

CHAPTER II.

The Indians.

SOME learned men have been disposed to attribute an extraordinary antiquity to the cultivation of astronomy in India. This opinion, which is founded upon the elements of astronomical tables brought from India, has been supported at great length and with much ingenuity by M. Bailly, in a work professedly on this subject; which, though it may contain erroneous conclusions, must always be considered as a model of elegance in scientific composition. The tables in which the elements of the Indian astronomy are to be found have been brought into Europe at various times: the earliest known were those imported from Siam by M. de la Loubère, the French envoy, on his return from a mission to that country about the year 1687.

These

tables have been analysed and explained by D. Cassini (in the Mémoires de l'Académie des Sciences, tom. viii.), and his explanation has been adopted with some immaterial alterations by Bailly*. It

Astron, Indienne, chap. I.

appears that the epoch of these tables is the 21st of March, 638 A.D., at the moment that the sun entered the beginning of the zodiac; for the Siamese, like all the Indians, had a zodiac of twentyseven signs or constellations, the position of which was entirely determined by the fixed stars, and which had, from the effects of precession, a progressive motion in longitude, successively occupying different situations with regard to the equinox. They had also a division of the zodiac into twelve signs, but this seems to have been merely an abstract mathematical division for purposes of calculation; and these twelve signs were by no means identified with any of the constellations. The epoch once determined, the Siamese calculate the mean motions of the sun and moon by means of two periods; the first of 800 years comprising 292207 days; the second, of 19 years corresponding to 235 lunar revolutions. The first gives us a sidereal year of 365d 6h 12m 30s, about 3m 24$ greater than the real value: in the second they appear to have taken the tropical year as equal to 365 days; the lunar revolution being supposed equal to 29d 12h 44m 3.

The tables of Chrisnabouram* offer little remarkable; we do not find in them the Siamese period of nineteen years, but a method of intercalation, which has the same object. The solar apogee, which the Siamese consider fixed, is here supposed moveable, though its motion is slower than it ought to be, according to our observations. The mean motions differ very considerably from those of the Europeans; but it is remarkable that the error for the sun and for the moon, in a given interval, is the same, so that the calculation of the time of an eclipse, which seems to be the principal object of all the Hindoo astronomers, is little affected by it. The epoch of these tables is fixed at sun-rise, on the 10th of March, 1491 A.D. The tables of Narsapur resemble a good deal those of Siam: they have the same period of 800 years containing 292207 days; but, instead of the second period of the Siamese Tables, they calculate the moon's motion directly by supposing that she makes 800 revolutions in 21857 days. This gives a sidereal revolution

Sent from India by P. Duchamp. Chrisnabouram is a town of the Carnatic: Narsapur is in the vicinity of Massulipatnem: Tirvalore is near Pondicherry. The original translation of tables of

Chrisnabouram into French, from Sanscrit, by Duchamp, is to be found at the end of the Astronomie Indienne of Bailly,

of 274 7h 42m 36 considerably too great; and, indeed, the Brahmins seem to have perceived the necessity of correcting the mean motions here assigned, which they do by renewing their epoch every eighty-seven years: at least in these tables of Narsapur there are two epochs, the one in 1569, the other in

1656.

The most curious of all the Indian tables, are those brought from Tirvalore by M. Le Gentil, and analysed at length by Bailly in his Indian Astronomy. The epoch of these tables is the year 3102 before Christ: at which time, they suppose a general conjunction on the ecliptic of the sun, moon, and planets. The object of the construction of these tables, like all those of the Indians, seems to be the calculation of eclipses; their methods have been explained at length by Bailly and Le Gentil: and the latter of these astronomers has applied their methods to an eclipse really observed by himself in India; the error he found to amount to about twenty-two minutes of time.

From a comparison of the four tables, just mentioned, Bailly deduces that they have all one common origin: and on this head his arguments are pretty conclusive. He remarks-I. That the tables of Siam contain a reduction for the difference of meridians, which show them to have been borrowed from a place, which has about the same longitude as Benares, a town which has been always looked upon by the Hindoos with especial veneration, and which seems to have been the residence of their most learned men. The tables of Tirvalore, Chrisnabouram, and Narsapur, contain no reduction of this kind, as all these towns lie nearly on what appears to have been assumed as the first meridian.

II. The epochs of these tables are so connected by the mean motions, that from one of them all the others may be found by employing the mean motions of the tables of Chrisnabouram. This would show that the Indians have in reality only one epoch, from which the others have been deduced by calculation; and, as one of these epochs may be obtained from the other without any correction for the difference of meridians, it follows that all these tables have been originally formed for the primitive meridian.

III. There is an extraordinary re† Astron, Ind, chap, v.

• Chap. iv.

semblance in all the elements of the solar orbit, as given in these tables, though those of the lunar orbit differ. We find the same mean motion of the sun in all, the same length of the year, the same equation of the centre*. The coincidence is too striking to be fortuitous, and proves, that the tables of the sun have been taken from the same source, while those of the moon have undergone various alterations.

IV. All the Brahmins versed in astronomy agree in considering these tables as borrowed from ancient works, and principally from one called the Surya Siddhantar: a work that, in the time of Bailly, was supposed to be guarded with great jealousy, and to be entirely shut up from all but a few learned men. This, however, turns out not to be the case: the work in question is rare, but that and many other treatises on the same subjects have been procured by Englishmen resident in India; and an interesting analysis of the Surya Siddhanta, though not so complete as might be wished, is to be found in the Asiatic Researches. It is evident from an inspection of the extracts given by Mr. Davis, that the solar tables of the Carnatic, which we have been discussing, are essentially borrowed from the Surya Siddhanta; in this latter the sidereal year is given at 365d 6h 12m 36: the greatest equation to the centre at 2° 10' 32", the obliquity of the ecliptic at 24°. An acquaintance with the Surya Siddhanta, and the other systems of astronomy existing in Sanscrit, puts beyond a doubt, that it was the custom of the Hindoos, as we have seen it was of the Chinese, to take for epoch a fictitious

The term Siddhanta is applied by the Hindoos to the newer. to their older systems of astronomy, in opposition There are several Siddhantas; this of Surya (a Hindoo God, representing the Sun) is

supposed to have been received by divine revela

tion, 2,164,930 years ago. The original, which exists in Sanscrit, (now a dead language in India,) has never been translated into any European langiven of it, and by the extracts from it, in the seguage: it is known principally by the account

veral memoirs in the Asiatic Researches quoted in this chapter. The same may be said of other ancient astronomical systems, particularly that of Bramagupta.

Vol, ii, p. 225. London, 4to.

general conjunction of the planets obtained by calculating backwards, with the respective mean motions attributed to the several planets by the authors of the system. No doubt can remain upon this head after a perusal of the memoirs of Messrs. Bentley and Davis in the Asiatic Researches, which are founded on an examination of the original documents.

However, Bailly has devoted the greater part of his work to establishing the point that the epoch of the Indian tables in the year 3102 before Christ was not imaginary, but founded upon actual observation. The talent and research with which he has argued the question, make it worth while to give a rapid summary of the proofs he has alleged, and some remarks upon their insufficiency. His argument may be divided into two parts, I., as it regards the epochs of the tables; II., as it regards the mean motions.

I. The nature of his argument with regard to the epochs is this. The positions of the sun, moon, and planets, as well as the position of the solsticial and equinoxial colures are so accurately determined by the Indians for the time of the Calyougam*, that these positions must have been actually observed, and could not have been merely the results of calculation in later times, as the Hindoos have never possessed that high degree of science, necessary to make these calculations with any precision. But have they in reality given us these positions for the Calyougam with so much accuracy? This is a point which it is necessary to examine. Now in the first place, the Indian tables give, at the epoch, a general conjunction of the sun, moon, and planets; calculating by our modern tables, we find such a general conjunction to have been impossible. It is true that for times so distant the errors of our tables will cause an uncertainty of five or six degrees, but the calculations of Bailly himself show that the planet Venus never could have been in conjunction or indeed near it at the time specified, making every allowance for this uncertainty; and even with regard to the other planets, he is obliged to content himself with an approximation. The impossibility of this pretended conjunction appears to La Place in itself a sufficient reason to reject the

This is the name given by the Indians to their epoch 3102 B.C,

pretended antiquity of the Indian tables*. Another consideration urged by the author of the Indian Astronomy is drawn from the accuracy with which he affirms the places of the sun and moon to have been determined for the epoch given. Very little stress, however, can be laid upon this argument, for, as has just been stated, we cannot be certain of the places calculated from our tables for the time of the Calyougam, to nearer than five or six degrees, and this is about the quantity by which the positions given by the Indians differ from ours. Besides, there is an uncertainty, whether the assigned longitude of the sun is the mean or true longitude. The most specious argument brought forward as to the epoch, is that founded on the pretended position of the colures. According to Bailly, the position of the equinox given in the Indian tables for the Calyougam is such, that the star called Aldebaran was in 359° 20′ of longitude: now calculating its place from the modern formulæ of precession, its longitude at the Calyougam would be 18': an agreement which, though it does not at first sight seem very great, was the more remarkable, because the Indians, who supposed the precession to be 54" annually, could not have obtained it by calculating back from a modern epoch. But it turns out that this posi tion of the colures for the Calyougam is merely a calculation of Bailly and Le Gentil, the Indian tables only giving us the longitude of the equinox 3600 years after the Calyougam; whence the astronomers just mentioned have deduced its position for the year 3102 B.C. The reader will see at once that this invalidates the whole of the argument urged by Bailly. Indeed the consideration of the epochs alone seems quite decisive against the pretended_observations of the Calyougam. The equation of the centre of the sun, given in these tables, is much too great for the epoch in question; La Place thinks that this may be accounted for by the circumstance that in eclipses the moon's annual equation increases the sun's apparent equation of the centre, by a quantity which is very nearly equal to the difference between the equation in question as given by the Indians, and that which may be deduced from the modern tables for the period of the Calyougam. This is an ingenious ex

*V. Syst. du Monde, liv. v.
Bailly, chap. v. § 30.
# V. Asiat. Research., vol. ii.

planation; but even should it be admitted as satisfactory, the arguments on the other side are too numerous and weighty to be much affected by it: and La Place himself, as we have before mentioned, does not hesitate to reject as fictitious the epoch of 3102 B.C. The equation of the centre of the moon presents a suspicious resemblance to that of Hipparchus: as to the elements of the orbits of the planets, Bailly himself is obliged to confess that it is only in some of them that we are to look for any precision. While the equation of the centre of Saturn agrees pretty well with theory, on the other hand the disagreement presented by other planets, and particularly by Mars and Jupiter, is very striking. But the limits of this treatise do not allow us to follow Bailly into all these details, particularly as the results are in general of a very unsatisfactory nature. In a discussion of this kind one consideration is obvious; if the Indians really observed the positions and motions of the heavenly bodies somewhere about 3000 years B.C., and if we are to take for a proof of these observations, the accordance of the elements of the sun, moon, and planets, with the values assigned for that period by modern astronomers, then we must expect to find the same degree of accordance nearly for one of these bodies as for another-allowing for the difficulty of the respective observations. No conclusion can be drawn from the agreement of some of these elements; that agreement must be general before any argument can be founded upon it.

II. The second class of arguments given by Bailly are those founded on the mean motions. But here it is necessary to make some preliminary observations, which show that little dependence can be placed on considerations of this kind in determining the age of the Indian or any other tables. The major axes and periodic times of the several primary planets are not subject to any secular inequalities; and it is very evident that they cannot in that case be used for the purpose in question. For the mean motions of the Indian tables must be well or badly determined: in the latter case it is clear that nothing can be deduced from them: in the former all that can be said is, that they were well known at the date of the formation of the tables; but as they suit (any time they cannot prove the tables to belong really to one period more than another.

Thus, for example, we may suppose the Indians to have framed these tables in the year 1491 A.D., and if we were to find the mean motions ever so accurate, might imagine them to have been determined by a thousand previous years of observation, which, while it would fully account for their perfection, would make the origin of astronomy in India considerably posterior to the Christian era. The planets Jupiter and Saturn are subject to a long periodic inequality of about 929 years: but it is clear that this cannot be of much assistance to us, as many of these periods are comprised between the Calyougam and the present time; and, according to La Place, the mean motions of these planets, as given in the tables now under discussion, would suit equally well the Calyougam, and the year 1491*. But the case is different with regard to the moon: the motion of this satellite is subject to a small secular acceleration, which only becomes perceptible when we compare her places at very distant periods of time. This, then, may be used as a test for trying the antiquity of Tables; but we find dif ferent and inconsistent results in the case before us, according to the parti cular tables we make use of. Those of Tirvalore do not give directly the moon's sidereal revolution, but Bailly has de duced it from the motions given with regard to the apogee. Now there are, in these, several periods given for the anomalistic revolution of the moon, from each of which different results may be deduced: the tables of Chrisnabouram differ again from all of these, as do likewise those of Narsapur. Nor do the two last mentioned agree with each other. We cannot then, in this case, deduce any conclusions from the moon's mean motion, which certainly is the only one of the mean motions that affords us in general a criterion.

The theory of Bailly on this subject appears, then, totally untenable; but it may be asked, what is the real age of the Surya Siddhanta, and the Indian astronomy? and have the Brahmins