Art. XVIII.—Notes on Hindu Astronomy and the History of our Knowledge of it

James Burgess

doi:10.1017/S0035869X00022553

Art. XVIII.—Notes on Hindu Astronomy and the History of our Knowledge of it

Published online by Cambridge University Press: 15 March 2011

James Burgess

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The following notes are perhaps somewhat miscellancous, but they may help to re-direct attention to an interesting subject, recalling the history of European studies of it with some of the results obtained. And since the scheme arranged by Mr. Whitley Stokes for cataloguing the Sanskrit works in Indian libraries, private and other, and for obtaining copies of the rarer ones has yielded such excellent results in all departments, it will perhaps be possible for Orientalists now to publish and translate some of the more important Siddhântas and Karaṇas hitherto inaccessible, and which would be most useful in tracing the origins and history of this Indian science.

Type: Original Communications
Information: Journal of the Royal Asiatic Society , Volume 25 , Issue 4 , October 1893 , pp. 717 - 761

DOI: https://doi.org/10.1017/S0035869X00022553 [Opens in a new window]
Copyright: Copyright © The Royal Asiatic Society 1893

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References

^{page 717 note 1} Strabo, Geograph. lib. xv. cap. i. § 70 (Casaub. p. 719); conf. Lassen, Rhein. Mus. für Phil. Bd. I. S. 183, and Ind. Alterthumsk. (2nd ed.), Bd. I. S. 1002 n.; Weber, , Hist. Ind. Lit. p. 28Google Scholar.

^{page 718 note 1} See Prof. Weber's paper Über den Veda Kalendar, Namens Jyotisham, (in Abhandlung. d. Königl. Akad. der Wissensch. zu Berlin, 1862) and an important paper by DrThibaut, in Jour. As. Soc. Beng. vol. xlvi. (1877), pt. i. pp. 411–437Google Scholar, cited below, § 34.

^{page 718 note 2} Dr. Rhys Davids has called my attention to the following passage in the Tevijja Sutta, Mahâ-Sîlam, 4: “Or, whereas some Samaṇa-Brâhmans, who live on the food provided by the faithful, continue to gain a livelihood by such low arts and such lying practices as these: that is to say, by predicting— ‘There will be an eclipse of the moon.’ ‘There will be an eclipse of the sun.’ ‘There will be an eclipse of a planet.’ ‘The sun and the moon will be in conjunction,’ ‘The sun and the moon will be in oppositions.’ ‘The planets will be in conjunction.’ ‘The planets will be in opposition.’ ‘There will be falling meteors, and fiery coruscations in the atmosphere,’ etc.… He [the recluse] on the other hand, refrains from seeking a livelihood by such low arts, by such lying practices.”—See the whole passage io Rhys Davids' Buddhist Suttas (Sacred Books of the Hast, vol. xi. pp. 197–8). The work is supposed to be an early one in Buddhist literature and the reference it contains, condemnatory of the practices of astrology, is of interest. It reminds ns of “the dividers of the heavens, the star-gazers, the monthly prognosticators” of ancient Chaldea (Isaiah, xlvii. 13).

^{page 719 note 1} He seems to have considered the measurements made for the earth's circumference as only approximate, and put it at 250,000 or 252,000 stadia. The distance of the sun he made 804,000,000 stadia; but what stadium did he use? If 8 4/7 stadia he taken as equal to an English mile, then the first would be 29,200 miles, and the sun's distance 93,800,000 miles: not very far from the truth.

^{page 721 note 1} The astronomical use of the word ἔξελιγμòς is not given in Liddell and Scott's Lexicon. Ptolem. M. Syntaxis, lib. iv. cap. 2; Geminos, Eisag. eis ta Phainom.

^{page 721 note 2} Censorinus (a.d. 238) has the following passage (de Die Natali, cap. xviii. ed. Nisard, p. 377), to which my attention has been called by Prof. H. Jacobi, of Bonn: “Est præterea annus, quern Aristoteles maximum potius, quam magnum, adpellat: quern solis, lunæ, vagarumque quinque stellarum orbes conficiunt, cum ad idem signum, ubi quondam simul fuerunt, una referuntur, cuius anni hiems summa est κατακλσμòς, quam nostri diluvionem vocant; æstas auteni ⋯κπ⋯ρωσις, quod est mundi incendium. Nam his alternis temporibus mundus turn exignescere, turn exaquascere videtur. Hunc Aristarchus putavit esse aunorum vertentium duum millium CCCCLXXXIV; Aretes Dyrrachinus quinque millium DLII; Heraclitus et Linus decem millium CC∞ (10,800); Dion X.M.CC∞ XXCIV (10,884); Orpheus CMXX (120,000); Cassandrus trecies sexies centum millium (360,000). Alii vero infinitum esse, nee unquam in se reverti existimarunt.”

Here we have a fair counterpart of the Hindu theory of Yugas; and, as Prof. Jacobi also points out, so far, at least, as Aristotle is concerned, Usener has shown (Rheinische Museum, Bd. xxviii. Ss. 392 f.) that the statement of Censorinus is correct. The annus maximus of Aristotle is mentioned by Tacitus (Dial. 16, in ed. Nisard, p. 481): “Ut Cicero in Hortensio scribit, is est magnus et verus annus, quo idem positio cœli siderumque, quæ quum maxime est, rursum existet, isque annus horum quos nos vocamus annorum XII M.DCCCCLIV (1·2,954), eomplectitur.” In this period a precession of 50″·023 annually would carry the equinoctial points round just 180°.

^{page 723 note 1} I have revised the times from modern tables, assuming the longitude of Siam at 6h. 42m. E. from Greenwich; Cassini (Mem. de l'Acad. tome viii. p. 311) adopted 6h. 34m. E. from Paris, which is only 1 1/2m. in excess of this.

^{page 723 note 2} Mem. de l'Acad. 1666–1699, tome viii. p. 309.

^{page 723 note 3} Astron. Indienne et Orientate, p. 12.

^{page 723 note 4} Astron. Indienne, pp. viii. 7, 44.

^{page 723 note 5} Mem. de l'Acad. tome viii. p. 304.

^{page 724 note 1} Al-Berûnî's India, Sachau's transl. vol. ii. p. 58; see below § 38.

^{page 724 note 2} Wilkinson's, Siddhânta S'iromaṇi, Golâdhyâya, iv. 12Google Scholar, where it is misprinted 64 1/11 for 64–1/11; conf. also Al-Berûnî's, India, Sachau's tr. vol. ii. pp. 37Google Scholar, 47, 52, and 54.

^{page 724 note 3} Al-Berûnî's, India, vol. ii. p. 18Google Scholar; the exact value with this element is 3231·98752 days, the difference between this and 3232 days amounts only to one day in 80·2 revolutions, or 686 years.

^{page 725 note 1} This portion of the paper was written before Thibaut's Pañchasiddhântikâ, of Varâha Mihira, reached this country. It contains an outline of the Pauliśa Siddhânta.

^{page 725 note 2} Petropoli, 1738.

^{page 725 note 3} Beschi († 1742) also published Tiruchabei Kanidam, a Tamil work on astronomy.

^{page 725 note 4} Histoire de l'Academie Royale des Sciences, 1772, 2nde Partie,—Mémoires, pp. 169–189; suite, pp. 190–214, 221–266.

^{page 725 note 5} Ib. pp. 169, 170; conf. Strabo, Geog. lib. xvi. c: i. § 20, ed. Casaub. p. 805.

^{page 726 note 1} Mém. de l'Acad. pp. 221 ff.

^{page 726 note 2} See Warren's Kala Saṅkalita, p. 118, etc. Probably this was the method of the Pauliśa Siddhânta.

^{page 726 note 3} Kala Saṅkalita, pp. 118 ff. 340, and Tables xxvi. ff.

^{page 726 note 4} Astron. Indienne, pp. 87, 245.

^{page 726 note 5} Mém. de l'Acad. 1772, pt. ii. pp. 187, 188.

^{page 726 note 6} Mém. ut sup. pp. 209 ff.

^{page 727 note 1} Mém. de l'Acad. p. 221.

^{page 727 note 2} The word (except Sittandij, as used by Beschi) seems to have been unknown to Warren; conf. Kala Saṅkalila, pp. 7, 51–56, 83, Le Gentil says it means ‘ancient,’ and Vâkyam means ‘new’; but his meanings and derivations are not to be trusted—Kaliyuga, for example, he says is from Kaly an ‘epoch,’ and ugam ‘misfortune’! Sittandij is probably a Dravidian derivative of Siddhânta, i.e. following the Siddhânta rules; conf. Waltheri, Doctr. temp. Indica, in Bayeri, , Hist. Reg. Grœc. Bactriani, pp. 184Google Scholar, 198.

^{page 727 note 3} Bailly, , Astron. Ind. p. 296Google Scholar:—“L'antiquité des Chaldéens n'auroit pas suffi aux 2500 ans. La plus ancienne date des Chaldéens en Astronomie est de l'an 2234 avant notre ère, 2100 ans. environ avant Hypparque. D'ailleurs j'ai remarqué plus haut que les observations d'éclipses, du moins les observations exactes, ne paroissent pas remonter à Babylone au-delà de Nabonassar; il faut done que ces observations aient été faites ailleurs, et on ne peut guères se refuser à croire qu'elles ont été faites dans l'Inde ou les Chaldéens semblent avoir emprunté les premiers éléments de leur Astronomie.” And, p. 300,—“II semble que ce n'est point sur une suite d'observations d'éclipses qu' Hypparque à établi la période de 126,007j. lh., mais sur les Tables indiennes. II en résulte par consequent que les Astronômes d'Alexandrie tiennent des Indiens les connoissances primitives et fondamentales de la théorie de la lune.” See also pp. 303, 306, etc.

M. Bailly's attempt in behalf of the originality of the Hindu astronomy has found almost a parallel in the Uranographie Chinoise of M. Gustave Schlegel (La Haye, 1875), in which the author attempts to prove that the early astronomy is originally Chinese, and has been imported by Chaldeans, Greeks, Indians, etc., from China.

^{page 728 note 1} Astron. Ind. pp. iii. xi. 49 and 391.

^{page 728 note 2} Ib. pp. 31, 32 ff. 317 ff. 319 n. There is a small village of the name in long. 77° 40′ E. lat. 14° 30′ N. about twelve miles south of Anantapur; but there are several other Kṛishṇapurams, one in N. Arkaḍ, long. 78° 28 1/2′ E., lat. 12° 53′ N.; another on the Kṛishṇâ, long. 79° 16′ E., lat. 16° 21′ N.; a fourth in Trichinapalli, long. 78° 51′ E., lat. 11° 23′ N.; a fifth on the Kâverî, long. 77° 1′ E., lat. 12° 13′ N.; a sixth in Tinnevelli, long. 77° 51′ E., lat. 8° 41′ N.; a seventh in Travankoḍ, long. 76° 35′ E., lat. 9° 9′ N. It is very unlikely that the tables of P. DuChamp came from the first, as Bailly assumes, simply because it is the only Kṛishṇapuram on D'Anville's map.

^{page 728 note 3} Ib. p. 32.

^{page 728 note 4} Ib. pp. 336, 337.

^{page 728 note 5} Here, again, we have no definite locality, for there are several towns of the name of Narsâpur, and others named Narsipur. Narsâpur in long. 73° 28′ E., lat. 18° 59′ N.; another in long. 78° 19′ E., lat. 19° 2′ N.; a third in long. 83° 41′ E., lat. 18° 35′ N.; and a fourth in long. 81° 44′ E., lat. 16° 26′ N.; and a fifth in long. 79° 1′ E., lat. 15° 4′ N., which is perhaps meant bv Bailly. Narsipur in long. 76° 18′ E., lat. 12° 47′, a pretty large town on the Hemâvatî in Maisur; another in long. 78° 4′ E., lat. 13° 8 1/2′ N.; a third on the Kâverî in long. 76° 58′ E., lat. 12° 12′ N.; a fourth in long. 81° 50′ E., lat. 16° 21′N., etc.

^{page 729 note 1} The longitude of Benares is 83° E. from Greenwich: which Narasiṁhapuram he means is uncertain.

^{page 729 note 2} Astron. Indienne, pp. 49, 55, 60. The Graha Lâghava, according to Warren (Kala Sankalita, p. 365), was written about 1556 a.d., but Whitney says it was the composition of Ganes'a, and dated S'ake 1442 (a.d. 1520). The Siddhânta Sundara of Jñânarâja also belongs to the beginning of the sixteenth century. The Siddhânta Rahasya was written in S'. 1513 (A.D. 1591); Ranganâtha completed his commentary on the Sûrya-Siddhânta in S'. 1525 (A.D., 1603); and his son Munis'vara wrote the Siddhânta-Sârvabhauma. and a commentary on the Siddhânta-S'iromaṇi of Bhâskara-Âchârya. The Graha Taraṅgini was written in 1618, the Siddhânta Manjari in 1619, and Kamalákara wrote the Siddhânta Tattva-Viveka about 1620 (Jour. Amer. Or. Soc. vol. vi. p. 422). It thus appears that during the century 1520–1620, after intercourse with Europe had been established, there was considerable activity in the compilation of new astronomical text-books.

^{page 729 note 3} Astron. Ind. pp. 204 f.; Burgess' Sûrya-Siddhânta, xii. 80–90: Al-Berûnî's, India (Sachau's tr.), vol. ii. pp. 67–73Google Scholar; Gladwin's, Âyin-Akbari (8vo. ed.), vol. ii. p. 306Google Scholar; also S'âstri, Bâpu Deva in Trans. Benares Institute, 1865, pp. 18–27Google Scholar.

^{page 729 note 4} Ptol. Syntaxis, lib. v. cap. xv. and Arkhai (ed. Halma, Hypoth, etc.), p. 61.

^{page 730 note 1} Esprit des Journeaux, Nov. 1787, p. 80.

^{page 730 note 2} Trans. R. Soc. Edinb. vol. ii. pp. 135–192.

^{page 730 note 3} Asiat. Res. vol. ii. pp. 225–287.

^{page 730 note 4} Asiat. Res. vol. ii. p. 238.

^{page 730 note 5} Astron. Ind. pp. xli. xlii. 165, 166.

^{page 731 note 1} As. Res. vol. ii. p. 262. Pandit Bâpu Deva S'âstri gives the distance of 51566 yojanas as equal to 468,780 miles: Trans. Benares Institute, 1865, p. 21.

^{page 731 note 2} Ib. pp. 266, 270; Burgess' Sûrya Siddhânta, iii. 9–12 and notes.

^{page 731 note 3} Ib. vol. ii. pp. 289–306.

^{page 731 note 4} Ib. pp. 389–403.

^{page 731 note 5} The Hindu astronomers teach “that the vernal equinox oscillates from the third of Mîna to the twenty-seventh of Mesha and back again in 7200 years; which they divide into four pâdas, and consequently that it moves, in the two intermediate pâdas, from the first to the twenty-seventh of Mesha, and back again, in 3600 years; the colure cutting their ecliptic in the first of Mesha, which coincides with the first of Aświnî, at the beginning of every such oscillatory period.” Ib. p. 392, also pp. 394 and 398.

^{page 732 note 1} Phil. Trans, vol. lxxx. pt. ii. (1790), pp. 660–684Google Scholar.

^{page 732 note 2} His principal authorities seem to have been Beschi's, Tamil Grammar (1738)Google Scholar; Roger's, AbrahamMœurs des Brames (1670)Google Scholar; and Bailly's, Astron. Indienne, p. 326Google Scholar.

^{page 732 note 3} Astron. Ind. pp. xxviii. 184, etc.

^{page 732 note 4} Phil. Trans, vol. lxxx. pt. ii. p. 583.

^{page 732 note 5} Asiat. Res. vol. iii. pp. 209–227.

^{page 732 note 6} Asiat. Res. vol. iii. pp. 215, 219; Varâha Mihira makes the fraction , the equivalent of 8° 42′·72 of Jupiter's motion, which takes place in 104·840987 days (J.R.A.S. N.S. Vol. V. p. 48). Varâha's is the only rule known by Al-Berûnî, (a.d. 1030); India, (ed. Sachau, ), vol. ii. p. 123Google Scholar.

^{page 733 note 1} Delambre says the Hindus knew nothing of heliocentric longitudes, Hist. Astron. Anc. tome i. p. 481. This is true scientifically, but the mean motion of a superior planet is its equivalent, conf. As. Res. vol. iii. p. 212.

^{page 733 note 2} Warren, , Kalasanhalita, p. 203Google Scholar, has made a mistake in converting the fraction on the supposition that Jupiter moves through 30° only in 360 saura days. Both Mr. Davis and Col. Warren have rather complicated their operations by the introduction of saura time, which is quite unnecessary; the heliocentric longitudes saving confusion. The simple nature of the fractions will readily appear when we take the cycle year of the Jyotistattva, or Ârya Siddhânta, of 361·02268 days; of this is 4·23600 days; and the sum of these numbers is 365·25868 days, or exactly the solar year. For the mean motion of Jupiter, also, we have 30° in one year of the cycle; of 30° is 21′·12; and the sum 30°21′·12 is the mean motion of the planet in one solar year.

^{page 734 note 1} To bring out the exact values of the fractions in this and the other rules, we must assume that the solar and chakra reckoning commence from the same point, and not at 2·14757 days apart, during which Jupiter's motion would be 10′ 42″-45. The rules immediately following, however, show that the Hindu writers were not particular about even larger discrepancies in the position of the planet.

^{page 734 note 2} If we compute by the formula with K=3179 complete, instead of K— 1, we get signs = 3^s 9° 0′ 28″·8, or 3 years of the cycle and 108·4031 days expired; or a whole cycle year and 4·236 days, that is exactly one solar year, too great.

^{page 735 note 1} These, for the Jyotistattva rule, would become .

^{page 735 note 2} Conf. Warren's, Kalasankalita, pp. 202–204Google Scholar; Ind. Ant. vol. xviii. pp. 198–201 and 380 f. The differences in the mean places of Jupiter for different dates may be tabulated thus (the remainder on dividing the expired cycle year by 12, giving the sign completed):—

The S′. year 573, by the Jyotistattva, thus began with the 42nd year of the cycle; the formula of Varâha Mihira would have given 5^s 0° 3′·36 elapsed, making the saṁvatsara begin 0·67d. before the Saṁkrânti.

^{page 737 note 1} The S′aka years, in which expunged years of the Bṛihaspati chakra oocur, according to the Jyotistattva rule, are given by the formula—

n being any suitable integer. Thus putting n = 12, we have—

an expunged year occurred in S′ 1083, by the Jyotistattva rule.

Similarly, for the rules applicable to the Sûrya Siddhânta, we obtain

for Kaliyuga dates when expunged cycle years occur, (1) according to the text, and (2) with the bîja.

^{page 737 note 2} In Southern India the Saṁvatsara is made to coincide with the year beginning with Mesha-saṁkrânti, and is eleven in advance of the northern reckoning. Hence they must have coincided before the Kshaya saṁvatsara Which occurred in S′aka 827, when, probably they began to diverge. The formula for the South Indian reckoning is (K + 12) 60, which gives the elapsed cycles and years.

^{page 737 note 3} Asiat. Res. vol. iv. pp. 159–163.

^{page 737 note 4} Sir Robt. Barker had given an account of the observatory at Benares in Phil. Trans, vol. lxvii. pp. 598 ff.: see also Bernoulli's ed. of Tieffenthaler's Desc. de l'Inde, tome i. pp. 316 f. and 347 f. for those at Jaypur and Ujjain; conf. also As. Res. vol. v. pp. 190–211. But little has been done since to describe oriental instruments: see Jour. As. Soc. Beng. vol. viii. pp. 831–838; on a Persian astrolabe, Ib. vol. x. pp. 759–765, and vol. xi. pp. 720–722; also conf. Ib. vol. ii. pp. 251 ff. Paṇḍit Bâpû Deva Sâstri described the Mânmandra observatory at Benares, , in the Trans. Benares Institute, 1860, pp. 191–196Google Scholar.

^{page 738 note 1} As. Res. vol. iv. p. 163. Mr. Davis was afterwards a Director of the Hon. E. India Co., and was the father of the late Sir John Francis Davis, Bart.

^{page 738 note 2} Trans. S. Soc. Edin. vol. iv. pp. 83–106.

^{page 739 note 1} Connaissance des Tems, 1808, pp. 447–453; and Phil. Mag. vol. xxviii. (1807) pp. 18–25Google Scholar. Delambre computed a table of the sines for every 3° 45′ of the quadrant with 233·5 as the divisor, which agrees practically with the Siddhânta table; four of the sines only differing by more than half a minute from the Hindu values. Had the author of the Siddhânta, however, known the property used by Briggs, he would have seen that as the second differences have a constant relation to the sines, the sum of any number of sines of equidistant arcs divided by the sum of their second differences must give the constant divisor.

^{page 739 note 2} Tattva stands for 25, and aśvina for 2, and all such numbers are written down from right to left. That the divisor should be equal to the first sine, the arcs would require to have been multiples of 3° 47′ 48″-48—values which would have been of no use in a table, even had the Hindus possessed the means of computing it. Again, the divisor 225 is correct only for multiples of 3° 49′ 13″·54 (or with the correct value of π, 3° 49′ 14″·22), which are equally unsuitable. The Hindu sines are expressed in minutes, the radius being made equal to 3438′, which gives 3·14136 for the value of π, or 57° 18′ for radius. How this value was arrived at we know not. Archimedes, about 250 B.C., had determined the ratio of the diameter to the circumference to lie between 1 to and 1 to . These give respectively 57° and 57° ; and, rejecting the fraction, the latter might readily be adopted as lying between the limits, though very near the second.

^{page 740 note 1} Thus in the Hindu table the sum of the sines of 23 arcs is 50795, and of the second differences 218; and dividing the first by the second we have 233 as the approximate value. Had the Bines been calculated to decimals we should have 50791·01 217·495 = 233·527; and for the 24 arcs 54229 232·217 = 233·527—both correct to the third place of decimals. By modern tables we get the true value thus:

2(1—cos. 3° 45′) = 4sine² = 0042821523 = ; and sine 3° 45′ = 0·065403129, multiplied by R′ = 3437′·746770785, gives 224′·8393961, instead of 225′—the Hindu value. Log. 233·5273583 =2·3683377665.

It is evident that the Sûrya Siddhânta rule waa founded on inspection of the first few sines of the Table, and not the table on the rule. Ranganâtha, in his commentary, makes a similar deduction from a false conclusion. He states that the last second difference is 15′ 16″ 48‴—which is evidently found by dividing R = 3438′ by 225; then he makes the proportion: As radius to any other sine, so is this difference to the second difference at that sine. This gives a roughly approximate value in a table already formed, but which could not be constructed with this divisor. Even with R = 3438′ the second difference at 90° is 14′ 43″·322—the correct value being 14′ 43″·257567.

^{page 740 note 2} Asiatic Researches, vol. vi. pp. 537–588. See Colebrooke's Essays, vol. ii. pp. 389, 390 (or Cowell's ed. p. 341); Weber's, Sanskrit Literature, p. 261Google Scholar.

^{page 741 note 1} Thus the positions of Mercury, Venus, Jupiter, Saturn, and the moon's apogee yielded dates at which they agreed with Lalande'a tables, varying between 887 and 945 A.D., and dividing the sum of the errors at any fixed date by the sum of the errors of annual motion we obtain 924 a.d. as the approximate date at which the Siddhânta elements gave generally correct results for these planets. But for Mars the result would be about 1458 a.d., which is suggestive of a much later epoch, or a revision of the text, It is with the moon's motions, however, that Hindu astronomy is most concerned, and we might fairly suppose that its elements would form the hest test of the age of a Siddhânta. Taking from the Sûrya Siddhânta the positions relative to the sun, we have:

In the last case it will he seen how little the smaller annual error affects the result, and if the first of the three be omitted the average is 1463 A.D. The mean longitudes of Venus, Mars, and Jupiter, give respectively A.D. 1509, 1455, and 1575, when correct, as computed with the bîja, and the mean brought out as above is 1516. As an error of 20′ is perhaps not too much to allow in any observation taken by the Hindus, the Sûrya Siddhânta may fairly be ascribed to the thirteenth century, and the bîja corrections to the latter half of the fifteenth or even to the sixteenth century A.D.; but the observations on which each edition is based were most probably taken at various dates and never reduced to one epoch.

^{page 742 note 1} Asiatic Researches, vol. viii. pp. 195–244.

^{page 743 note 1} As. Res. vol. ix. (1807), pp. 323–376Google Scholar; reprinted in Colebrooke's Essays, vol. ii. pp. 321–373.

^{page 744 note 1} As. Res. vol. xii. (1816), pp. 209–250Google Scholar; also in Colebrooke's, Essays, vol. ii. pp. 374–416Google Scholar.

^{page 744 note 2} London, 1817; reprinted in Essays, vol. ii. pp. 417–531.

^{page 744 note 3} Essays, vol. ii. p. 429; Weber's, Sanskrit Literature, p. 257 n.Google Scholar; see below, §31.

^{page 744 note 4} Professor Playfair must have felt the severity of Delambre's remarks: see his paper “On the Algebra and Arithmetic of the Hindus,” Edinburgh Review, vol. xxix. (11 1817), pp. 162Google Scholar, 163. In his review (vol. x. p. 456) Playfair apparently implies that he also wrote the first notice in vol. i. pp. 42, 43.

^{page 746 note 1} These tables were reprinted in the Appendix to Rothman's History of Astronomy (1834).

^{page 746 note 2} Part of this paper was reproduced in the new edition of Colebrooke's, Essays, edited by his son, vol. ii. pp. 366–374Google Scholar.

^{page 746 note 3} Histoire des Mathématiques (1758), tome i. pp. 402–404.

^{page 746 note 4} As. Res. vol. ii. p. 292.

^{page 746 note 5} Whish (Trans. Lit. Soc. Madras, p. 67) gives a variant reading of this pâda, viz.:— Taukshika Âkokero Hṛidogaś chesthusi kramasaḥ|,.—giving Isthusi for Ἰχθ⋯ς. Kulîra, though closely resembling κóλουροι, is not connected with it, but, like Karkaṭa also, is a Sanskrit word: Varâha, quoting Yavanes'vara, has —Karkî kulîrakṛitiramba samstho, etc. Whish also mentions that it is found in the Horâ S′ûstra, xi. 9.

^{page 747 note 1} Colebrooke, , Essays, vol. ii. pp. 364Google Scholar, 526 f. or new ed pp. 320, 476 f.; Muir, J., Journ. As. Soc. Seng. vol. xiv. pp. 810 f.Google Scholar; Weber, , Ind. Stud. vol. ii. pp. 254Google Scholar, 261; Jacobi, , De Astrologiae Indicae, etc., pp. 8Google Scholar, 11, 33, 35; Kern, , Bṛihat Saṁhitû, int. p. 28Google Scholar. Whish's paper was translate by Lassen, , Zeitsch.f. Kunde d. Morgenl. vol. iv. p. 302Google Scholar.

^{page 748 note 1} Maternus, Firmicus, Math. ii. 4Google Scholar; Salmasii, Plinianœ Exercitationes, pp. 460 f.Google Scholar; Colebrooke, , Essays, vol. ii. pp. 364–370Google Scholar.

^{page 748 note 2} Sûrya Siddh. xii. 79.

^{page 748 note 3} The ordinary dictionaries, Greek or Sanskrit, do not explain the precise meaning of these terms in astrology or astronomy.

^{page 748 note 4} Conf. Sir G. Lewis, Survey of the Astronomy of the Ancients, on the early history of astrology.

^{page 749 note 1} In 1832 was published an Account of British India, prepared by Dr. Hugh Murray and others, in three volumes, in the last of which was a chapter (2nd ed. vol. iii. pp. 288–323) on Hindu Astronomy and Mathematics, written by Prof. W. Wallace, of Edinburgh University, giving a good popular account of previous researches.

In 1834 MrRothman, , in his History of Astronomy (pp. 116–128)Google Scholar, gave a brief outline of the subject with Mr. Bentley's tables of the planets. Mountstuart Elphinstone, in his History of India (1839), in the first chapter of book iii., also gave a brief summary of what was known up to that time.

^{page 749 note 2} Jour. As. Soc. Beng. vol. iii. pp. 504–519.

^{page 749 note 3} Indian Metrology, pp. 174–259.

^{page 749 note 4} Jour. As. Soc. Beng. vol. xiii. pt. i. pp. 53–66.

^{page 750 note 1} It was also issued as a separate volume. Bâpu Deva S′âstri's version was published under the supervision of Archdeacon Pratt, in the Bibliotheca Indica in 1860. An English version of the early part of this Siddhânta had also appeared in the Asiatic Journal, May, June, 1817; and part of the first and the eighth chapter, with a French translation, in Abbé Guerin's, J. M. F.Astronomie Indienne, 1847Google Scholar. The text, with Rañganâtha's commentary, was edited by DrHall, Fitzedward, in the Bibliotheca Indica, 1859Google Scholar.

^{page 750 note 2} A good exposition of the principal steps in the calculation of eclipses is to be found in a paper by the late William Spottiswoode in J.R.A.S. Vol. XX. (1863), pp. 345–370Google Scholar.

^{page 750 note 3} DrHall, Fitzedward, in Jour. Am. Or. Soc. (1860), vol. vi. pp. 556–559Google Scholar, and Dr. Whitney in an additional note to the paper (ib. pp. 560–564), had discussed Âryabhaṭa and his writings; and DrKern, , in J.R.A.S. Vol. XX. pp. 371–387Google Scholar, collected the extracts from Âryabhaṭa found in Bhaṭṭa Utpala's commentary on the Bṛihat Samhitâ.

^{page 750 note 4} Dr. Kern has edited the text of these works—Ârya-bhaṭîya, with the commentary Bhaṭadîpikâ of Paramâdîsvara (Leiden, 1874). A notice of this work by Prof. A. Weber appeared in the Liter. Central-Blatt, 1875, No. 7, and was reprinted in his Indische Streifen, Bd. iii. pp. 300–302.

^{page 751 note 1} Sachau's, Al-Berûnî, transl. vol. ii. p. 7Google Scholar.

^{page 751 note 2} He has been confounded with VitteśVara (A.D. 899), son of Bhadatta, of Al-Berûnî (vol. i. p. 156), author of the Karaṇasâra; conf. Weber's, Sansk. Liter, p. 262Google Scholar. Mallikârjunada, a southern astronomer, is supposed by Warren to have written about a.d. 1178, and used the meridian of Râmeśvaram (79° 22′·1 E. of Greenwich), as did also Balâdityakalu, a Telugu astronomer, who wrote in 1456 A.D. Vâvilâla Kuchchinna, another Telugu astronomer, is said to have written in 1298 a.d. Warren's, Kala Sankalita, pp. 171, 356, 389–90Google Scholar. Notices, in Sanskrit, of a number of astronomical writers, have of late appeared in The Paṇḍit.

^{page 751 note 3} Matériaux pour servir à l'Histoire comparée des Sciences Mathématiques chez les Grecs et les Orientaux, pp. 467–549.

^{page 751 note 4} Die Vedischen Nachrichten von den Naxatra (Berlin, 1860 and 1862)Google Scholar.

^{page 752 note 1} Jour. As. Soc. Beng. vol. xxix. (1860), p. 200Google Scholar.

^{page 752 note 2} Conf. Biot, , Jour, des Savants, 1860Google Scholar; or E'tudes sur l'Astronomie Indienne et Chinoise (1862); Sédillot, , Courtes Observations sur quelques Points de l'Hist. de l'Astronomie (1863)Google Scholar; Whitney, , Jour. Amer. Or. Soc. vol. vi. pp. 325–350Google Scholar, etc.; vol. viii. pp. 1–94, 383–398, and Proc. p. lxxxiii.; Weber, , Ind. Str. vol. ii. pp. 172, 173Google Scholar; Ind. Stud. vol. ix. pp. 424 ff.; vol. x. pp. 213 ff.; Burgess, E., Jour. Am. Or. Soc. vol. viii. pp. 309–334, and Proc. pp. lxvii.–lxviiiGoogle Scholar.; Pratt, Archdeacon, Jour. As. Soc. Beng. vol. xxxi. pp. 49 f.Google Scholar; Whitney, , J.R.A.S. Vol. I. (N.S. 1864), pp. 316–331Google Scholar; Sir E. Colebrooke, ib. pp. 332–338; and Whitney, , Orient, and Linguistic Studies (1874), vol. iiGoogle Scholar.

^{page 754 note 1} Vedische Nachrichten von den Naxatra, 2nd. Th. p. 390.

^{page 754 note 2} Al-Berûnî's, India (Sachau's tr.), vol. ii. p. 87Google Scholar; conf. Ind. Ant. vol. xiv. p. 43; Biot, , E'tudes sur l'Astronomie Indienne (Ex. du Jour. d. Sav.), pp. 81Google Scholar, 82. According to Brahmagupta the moon's sidereal motion in one civil day is 13° 10′ 34″·88, whence the Nakshatra portions or arcs are:—

In Al-Berûnî's account, by neglecting a small fraction in the average daily motion, the value left for Abhijit comes out 4° 14′ 18″·60.

^{page 755 note 1} Dr. Hoernle has discussed a portion of the MS. of Pushkarasârin's work in the Jour. As. Soc. Bengal, vol. lxii. (1893) part. i. pp. 9–18Google Scholar, See The Academy, Aug. 12th, 1893, p. 136.

^{page 755 note 2} Vedische Nachrichten von den Naxatra, 2nd Th. p. 377.

^{page 755 note 3} See Thibaut, , in Ind. Ant. vol. xiv, p. 43Google Scholar. Where two numbers are given in the following table, the second is from Pushkarasârin's fragment, published by DrHoernle, in Jour. As. Soc. Beng., vol. lxiiGoogle Scholar. part i.; see also Colebrooke's, Essays, vol. ii. p. 322Google Scholar, and table.

^{page 755 note 4} Burgess's, Sûrya Siddhânta, pp. 175–220Google Scholar, or Jour. Am. Or. Soc. vol. vi. pp. 319–364.

^{page 756 note 1} Alcyone is not given in Ptolemy's Star List, and by Ulagh Beg it is mentioned only as “a small star”: Atlas (27, Tauri) or Merope (23, T.) is probably intended in the Hindu works.

^{page 756 note 2} The MS. of Pushkarasârin's work, found in Central Asia, ascribes 8 (or 7) muhûrtas to Abhijit, which gives 27·267 (or 27·233) days for the moon's sidereal revolution, which is only a rough approximation. Brahmagupta's value is equivalent to muhûrtas, giving 27·32167 days for the sidereal month as derived from the elements he adopts.

^{page 757 note 1} This tract (48 pp.) was reviewed by Prof. A. Weber in the Liter. Central-Blatt, 1873, No. 25, pp. 786–88, reprinted in Indische Streifen, Bd. iii. pp. 165–68; also by Prof. Kern, H. in The Academy, 1873Google Scholar.

^{page 757 note 2} March, 18th, A.D. 505, at 33^gh 9P after noon.

^{page 758 note 1} This is otherwise put as 900,000 revolutions in 24589506 = 24589506·3100775 days, and the rule for the moon's mean place is to multiply the ahargaṇa by 900,000, deduct 670,217, and divide by 24,589,506, and correcting for the fraction by taking of the elapsed revolutions as seconds to be subtracted.

^{page 758 note 2} The almost perfect coincidence of these values with those of Ptolemy, Math. Syn. lib. iii. c. 2, has been pointed out by me, Ind. Ant. vol. xix. p. 284. The Romaka Siddhânta accepts, without modification, the Metonic cycle of 19 years combined with Hipparchus's length of the Tropical year; and this may account for the very slight differences. If, instead of 126007 days plus one hour which Ptolemy uses for the lunar equations, we substituted 126007 days minus one hour (or 1 1/2h) we should get the Romaka Siddhânta values.

^{page 759 note 1} Warren's, Kâla Sankalita, pp. 118 ff.Google Scholar.

^{page 759 note 2} See this in Ind Ant. vol. xx. p. 228.

^{page 759 note 3} Wallis, Hist, and Prac. Algeb. c. 7.

^{page 760 note 1} Pañchasiddhântikâ, Introd. p. xlvii.

^{page 761 note 1} The greatest error is only 3′·4 in the case of Mars, for which the fraction would have given the more accurate value of 768°·691; and for Mercury, the multiplier 2170 and divisor 19, would have heen even closer than 3312 and 29. To convert these arcs into civil time, using Varâha Mihira's value for the sidereal year, 365d. 15gh. 33p., we have to multiply by 1·01460764 or : the results are—Mars, 779·92 days; Jupiter, 398·87; Saturn, 378·08; Mercury, 115·88; and Venus, 583·90 days.

^{page 761 note 2} The principal of these papers are:—Jacobi, , Methods and Tables for verifying Hindu Dates, Titles, etc. in Ind. Ant. vol. xvii. pp. 145–181Google Scholar; The Computation of Hindu Dates in Inscriptions, etc., with Tables in Epigraphica Indica, vol. i. pp. 403–460; Tables for Calculating Hindu Dates in True Local Time, in ibid. vol. ii.; Schram, K., Hilfstafeln für Chronologie, in Denkschriften d. Kais. Acad. d. Wissensch. mat. nat. Cl. Wien, vol. xlv. pp. 289–358Google Scholar; and Ind. Ant. vol. xviii. pp. 290–300; Kielhorn, , The Sixty year Cycle of Jupiter, in Ind. Ant. vol. xviii. pp. 193–209Google Scholar, and 380–386, and Abhand. d. K. Gesellsch. d. Wissensch. z. Göttingen, 1889.

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Art. XVIII.—Notes on Hindu Astronomy and the History of our Knowledge of it

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