^{page 136 note *} Mem. de l'Acad. des Sciences, tom. 8. p. 28r. &c.

^{page 137 note *} Traité de l'Astronomie Indienne et Orientale, par M. Bailly. Paris, 1787.Google Scholar

^{page 139 note *} Mem. sur l'Astronomie des Indiens, par M. le Gentil, Hist. de l'Acad des Scien. 1772, II. P. 207. The phrase which we here translate constellations, signifies the places of the moon in the twelve signs.

^{page 140 note *} Mem. Acad. des Scien. 1772, II. P. 189.Google Scholar

^{page 140 note †} Ibid. 209.

^{page 141 note *} Mem. Acad. des Scien. 1772. II. P. 200.Google Scholar The zodiac they call sodi mandolam, or the circle of stars.

^{page 141 note †} Ibid. 194. Ast. Indienne, p. 43, &c.

^{page 142 note *} Mem. Acad. Scien. tom. 8. p. 312. Ast. Indienne, p. 11. § 14.

^{page 143 note *} Ast. Ind. p. 7. ✶ 6.

^{page 143 note †} Mem. Acad. Scien. tom. 8. p. 328.

^{page 143 note ‡} Ast. Ind. p. 124. The tables of Tirvalore make the year 6′′ less.

^{page 144 note *} The equation of the sun, or what they call the chaiaa, is calculated in the Siamese tables only for every 15 of the matteiomme, or mean anomaly. , Cassini, ubi supra, p. 299Google Scholar.

^{page 144 note +} Ast. Ind. p. 9.

^{page 144 note *} The error, however, with respect to the apogee, is less than it appears to be; for the motion of the Indian zodiac, being nearly 4′ swifter than the stars, is but 6′′ slower than the apogee. The velocity of the Indian zodiac is indeed neither the same with that of the stars, nor of the sun's apogee, but nearly a mean between them.

^{page 145 note *} The Indian period is more exact than that of our golden number, by 35′. Ast. Ind. p. 5. The Indians regulate their festivals by this period. Ibid. Disc. Prelim, p. viii.

^{page 145 note +} Ast. Ind. p. 11. & 20.

^{page 145 note †} Ast. Indienne, p. 13. , CassiniMem. Acad. tom. 8. p. 304.Google Scholar

^{page 146 note *} Mem. Acad. Scien. tom. 8. p. 302. & 309.

^{page 146 note +} Ast. Ind. p. 12. It brings us to a meridian 82°, 34′, east of Greenwich. Benares is 83°, 11', east of the same, by Rennel's map.

^{page 146 note ‡} These tables are published by Bailly, M., Ast. Ind. p. 335,Google Scholar &c. See also p. 31, &c.

^{page 147 note *} Ast. Ind. p. 49, &c.

^{page 148 note *} They were explained, or rather decyphered by M. le Gentil in the Memoirs of the Academy of Sciences for 1784, p. 482, &c.; for they were not understood by the missionary who sent them to Europe, nor probably by the Brahmins who instructed him. M. le Gentil thinks that they have the appearance of being copied from inscriptions on stone. The minutes and seconds are ranged in rows under one another, not in vertical columns, and without any title to point out their meaning, or their connection. These tables are published, Mem. Acad. ibid. p. 492, and Ast. Ind. p. 414.

^{page 149 note *} Tirvalore is a small town on the Coromandel coast, about 12 G. miles weft of Negapatnam, in Lat. 10°, 44′, and east Long. from Greenwich, 79°, 42′, by Rennel's map. From the observations of the Brahmins, M. le Gentil makes its. Lat. to be 10°, 42′, 13′. (Mem. Acad. Scien. II. P. 184.) The meridian of Tirvalore nearly touches the west side of Ceylon, and therefore may be supposed to coincide with the first meridian, as laid down by Father du Champ. There is no reduction of Longitude employed in the methods of Tirvalore.

^{page 149 note †} These are Indian hours, &c.

^{page 149 note ‡} Mem. Acad. des Scien. II. P. 187. Ast. Indienne, p. 76, &c.

^{page 149 note ∥} The Indian hours are here reduced to European.

^{page 150 note *} Mem. Acad. des Scien. ibid. p. 229. Ast. Ind. p. 84.

^{page 150 note †} M. le Gentil has given this table, Mem. Acad. ibid. p. 261.

^{page 151 note *} The accuracy of the geography of the Hindoos, is in no proportion to that of their astronomy, and, therefore, it is impossible that the identity of the meridians of their tables can be fully established. All that can be said, with certainty, is, that the difference between the meridians of the tables of Tirvalore and Siam is, at most, but inconsiderable, and may be only apparent, arising from an error in computing the difference of longitude between these places. The tables of Tirvalore are for Long. 79°, 42′; those of Siam for 82°, 34′; the difference is 2°, 52′, not more than may be ascribed to an error purely geographical.

As to the tables of Chrisnabouram, they contain a reduction, by which it appears, that the place where they are now used is 45′ of a degree east of the meridian for which they were originally constructed. This makes the latter meridian agree tolerably with that of Cape Comorin, which is in Long. 77°, 32, 30′, and about half a degree west of Chrisnabouram. But this conclusion is uncertain; because, as M. Bailly has remarked, the tables sent from Chrisnabouram, and understood by Father du Champ to belong to that place, are not adapted to the latitude of it, but to one considerably greater, as appears from their rule for ascertaining the length of the day. (Ast. Ind. p. 33.)

The characters, too, by which the Brahmins distinguish their first meridian, are not perfectly consistent with one another. Sometimes it is described as bisecting Ceylon; and at other times, as touching it on the west side, or even as being as far west as Cape Comorin. Lanka, which is said to-be a point in it, is understood, by Fath. du Champ, to be Ceylon. M. Bailly thinks that it is the lake Lanka, the source of the Gogra, placed by M. Rennel, as well as the middle of Ceylon, in Long. 80°, 42; but, from a Hindoo map, in the Ayeen Akbery, vol. iii. p. 25. Lanka appears to be an island which marks the intersection of the first meridian of the map, nearly that of Cape Comorin, with the equator; and is probably one of the Maldivy islands. See also a note in the Ayeen Akbery, ibid. p. 36.

^{page 153 note *} Mem. Acad. Scien. 1772, II. P. 214.Google Scholar Ast. Ind. p. 129.

^{page 153 note †} Mem. Acad. de Berlin, 1782, p. 287.Google Scholar Ast. Ind. p. 144.

^{page 153 note ‡} Ast. Ind. p. 130.

^{page 154 note *} Ast. Ind. p. 110. The Brahmins, however, actually suppose the epoch to be 6 hours later, or at sunrise, on the same day. Their mistake is discovered, as has been said; by comparing the radical places in the different tables with one another.

^{page 154 note †} Ast. Ind. p. 83.

^{page 155 note *} Ast. Ind. p. 142, &c. The first meridian is supposed to pass through Benares; but even if it be supposed 30 farther west the difference, which is here 37′, will be only increased to 42′.

^{page 156 note *} Ast. Ind. p. 114.

^{page 156 note †} Ibid. p. 115.

^{page 156 note ‡} Ibid. p. 117.

^{page 156 note ∥} Ibid. p. 118.

^{page 157 note *} Mem. Acad. Scien. tom. 8. p. 286.

^{page 157 note †} Ast. Ind. p. 145.

^{page 157 note ‡} Ibid. p. 126.

^{page 158 note *} The reasoning here reserred to is the following: As the mean motions, in all astronomical tables, are determined by the companion of observations made at a great distance of time from one another; if x be the number of centuries between the beginning of the present, and the date of the more ancient observations, from which the moon's mean motion in the tables of Chrisnabouram is deduced; and if y denote the same for the more modern observations: then the quantity by which the moon's motion, during the interval x–y, falls short of Mayer's, for the same interval, is (x ²–y ²)9′.

If, therefore, m be the motion of the moon for a century in the last mentioned tables, m(x–y)–9′ (x ²–y ²) will be the mean motion for the interval x–y in the tables of Chrisnabouram. If, then, a be any other interval, as that of 43·83 centuries, the mean motion assigned to it, in these last tables, by the rule of proportion, will be . Let this motion, actually taken from the tables be = na, then ma–na – 9a(x+y); or , in the present case. It is certain, therefore, that whatever supposition be made with respect to the interval between x and y, their sum mast always be the same, and must amount to 3219 years. But that, that interval may be long enough to give the mean motions with exactness, it can scarcely be supposed less than 2000 years; and, in that case, x = 3609 years, which therefore is its least value. But if 3609 be reckoned back, from 1700, it goes up to 1909 years before Christ, nearly, as has been said.

It must be remembered, that what is here investigated is the limit, or the most modern date possible to be assigned to the observations in question. The supposition that x–y = a, is the most probable of all, and it gives x = 4801, which corresponds to the beginning of the Calyougham.

^{page 159 note *} Mem. Acad. des Scien. 1786, p. 235, &c.Google Scholar

^{page 160 note *} Mem. Acad. des Scien. 1786, p. 260.Google Scholar

^{page 161 note *} Mem. de l'Acad. de Berlin, 1782, p. 170, &c.Google Scholar

^{page 161 note †} Ast. Ind. p. 160, &c.

^{page 161 note ‡} Supra, § 18. and 10.

^{page 162 note *} Mem. Acad. Berlin, 1782. p. 289.Google Scholar

^{page 162 note †} Ast. Ind. p. 160.

^{page 162 note ‡} Ibid. p. 161.

^{page 162 note ∥} He says, “Sans doute il ne peut résulter de ce calcul qu'un apperçu.”

^{page 163 note *} Ast. Ind. p. 163.

^{page 164 note *} Mem. Acad. Berlin, 1782, p. 287.Google Scholar

^{page 164 note †} Ast. Ind. p. 165.

^{page 165 note *} Ast. Ind. p. 173, &c.

^{page 165 note †} Ibid. p. 177.

^{page 166 note *} Ast. Ind. p. 194.

^{page 166 note *} Ibid. p. 199, &c.

^{page 167 note *} Ast. Ind. p. 181.

^{page 167 note †} Ibid. p. 184. § 13.

^{page 168 note *} Mem. Acad. Berlin. 1782, p. 246.Google Scholar Ast. Ind. p. 186.

^{page 168 note *} Ast. Ind. p. 188.

^{page 169 note *} Esprit des Journeaux, Nov. 1787. p. 80.Google Scholar

^{page 169 note †} The inequality of the precession of the equinoxes, (§ 22.); the acceleration of the moon; the length of the solar year; the equation of the sun's centre; the obliquity of the ecliptic; the place of Jupiter's aphelion; the equation of Saturn's centre; and the inequalities in the mean motion of both these planets.

^{page 170 note *} Ast. Ind. p. 355, &c.

^{page 171 note *} Mem. Acad. des Scien. II. P. 175.

^{page 171 note †} To judge of the accuracy of this approximation, suppose O to be the obliquity of the ecliptic, and x the excess of the semidiurnal arch, on the longest day, above an arch of 90°, then sin. x = tan. lat. But if G be the height of a gnomon, and S the length of its shadow on the equinoctial day, Therefore or in minutes of time, reckoned after the Indian manner,

If 0 = 24°, then tan. O = ·4452, and the first term of this formula gives x = , which is the same with the rule of the Brahmins.

For that, reduced into a formula, is .

They have therefore computed the coefficient of with sufficient accuracy; the error produced by the omission of the rest of the terms of the series will not exceed 1′, even at the tropics, but, beyond them, it increases fast, and, in the latitude of 45°, would amount to 8′.

^{page 173 note *} Acad. des Scien. 1772, II. P. 205.Google Scholar

^{page 174 note *} Mem. Acad. des Scien. 1772, II. P. 259.Google Scholar

^{page 174 note *} Ibid. 241.

^{page 175 note *} Hist. Acad. II. P. 109. Ibid. Mem. 253,‒256.

^{page 175 note †} In the language, however, of their rules, we may trace some marks of a fabulous and ignorant age, from which indeed even the astronomy of Europe is not altogether free. The place of the moon's ascending node, is with them the place of the Dragon or the Serpent; the moon's distance from the node, is literally translated by M. le Gentil, la lune offensée du dragon. Whether it be that we have borrowed these absurdities from India, along with astrology, or if the popular theory of eclipses has, at first, been every where the same, the moon's node is also known with us by the name of the cauda draconis. In general, however, the signification of the terms in these rules, so far as we know it, is more rational. In one of them we may remark considerable resinement; ayanang sam, which is the name for the reduction made on the sun's longitude, on account of the precession of the equinoxes, is compounded from ayanam, a course, and angsam, an atom. Mem. Acad. II. P. 251. The equinox is almost the only point not distinguished by a visible object, of which the course or motion is computed in this astronomy.

^{page 177 note *} Euc. Lib. IV. Prop. 15.

^{page 177 note †} See these tables, Ast. Ind. p. 414.

^{page 178 note *} The formula deduced from this hypothesis, for calculating the equation of the centre from the anomaly of the eccentric, is the following: Let x be the equation of the centre, φ the anomaly of the eccentric, e the eccentricity of the orbit, or the tangent of half the greatest equation; then .

^{page 179 note *} This method of calculation is so nearly exact, that even in the orbit of Mars, the equation calculated from the mean anomaly, rigorously on the principle of his angular motion being uniform, about a point distant from the centre, as described above, will rarely differ a minute from that which is taken out from the Indian tables by this rule. It was remarked, (§ 37.) that it is not easy to explain the rules for finding the argument of the equation of the centre, for the planets. What is said here explains fully one part of that rule, viz. the correction made by half the equation manda; the principle on which the other part proceeds, viz. the correction by half the equation fehigram, is still uncertain.

^{page 183 note *} Almagest. lib. XI. cap. 9. & 10.

^{page 185 note *} Ayeen Akbery, Vol. III. p. 32.

^{page 187 note *} Ast. Ind. p. 307.

^{page 188 note *} Ast. Ind. p. 309. le Gentil, M., Mem. Acad. Scien. 1772. P. II. p. 221.Google Scholar

^{page 191 note *} It should have been remarked before, that M. Bailly has taken notice of the analogy between the Indian method of calculating the places of the planets, and Prolemy's hypothesis of the equant, though on different principles from those that have been followed here, and such as do not lead to the same conclusion. In treating of the question, whether the sun or earth has been supposed the centre of the planetary motions by the authors of this astronomy, he says, “Ils semblent avoir reconnu que les “deux inégalités (liquation du centre et la parallaxe de l'orbe annuel) etoient vues de “deux centres differens; et dans l'impossibilité où ils étoient de déterminer et le lieu et “la distance des deux centres, ils ont imaginé de rapporter les deux inégalités à un point “qui tînt le milieu, c'est-à-dire, à un point également éloigné du soleil, et de la terre. “Ce nouveau centre resemble assez au centre de l'équant de Ptolemee. (Ast. Ind. Disc. Prel. p. 69.) The fictious centre, which M. Bailly compares with the equant of Ptolemy, is therefore a point which bisects the distance between the sun and earth, and which, in some respects, is quite different from that equant; the fictitious centre, which, in the preceding remarks, is compared with the equant of Ptolemy, is a point of which the distance from the earth is bisected by the centre of the orbit, precisely as in the case of that equant. M. Bailly draws his conclusion from the use made of half the equation schigram, as well as half the equation manda, in order to find the argument of this last equation. The conclusion here is establishied, by abstracting altogether from the former, and considering the cases of oppositions and conjunctions, when the latter equation only takes place. If, however, the hypothesis of the equant shall be found of importance in the explanation of the Indian astronomy, it must be allowed that it was first suggested by M. Bailly, though in a sense very different from what it is understood in here, and from what it was understood in by Ptolemy.

For what farther relates to the parts of the astronomy of Chaldea and of Greece, which may be supposed borrowed from that of India, I must refer to the 10th Chap, of the Astronomie Indienne, where that subject is treated with great learning and ingenuity. After all, the silence of the ancients with respect to the Indian astronomy, is not easily accounted for. The first mention that is made of it, is by the Arabian writers; and M. Bailly quotes a very singular passage, where Massoudi, an author of the 12th century, says, that Brama composed a book, entitled, Sind-Hind, that is, Of the Age of Ages, from which was composed the book Maghifti, and from thence the Almagest of Ptolemy. Ast. Ind. Disc. prel. p. 175.

The fabulous air of this passage is, in some measure, removed, by comparing it with one from Abulfaragius, who says, that, under the celebrated Al Maimon, the 7th Khalif of Babylon, (about the year 813 of our era) the astronomer Habash composed three sets of astronomical tables, one of which was ad regulas Sind Hind; that is, as Mr Costard explains it, according to the rules of some Indian treatise of astronomy. (Afiatic Miscel. Vol. I. p. 34.) The Sind-Hind is therefore the name of an astronomical book that existed in India in the time of Habash, and the same, no doubt, which Massoudi says was ascribed to Brama.