Jyotihshastra: Astral and Mathematical Literature c4.s2: C. Maximum Latitudes

Jyotiḥśāstra: Astral and Mathematical Literature c4.s2

Language: English

C. Maximum Latitudes

c4.s2.1

| | Brāhma | Ārya = Ārdharātrika = Saura |
| Saturn | 2;26° | 2;15° |
| Jupiter | 1;34° | 1;15° |
| Mars | 5;38° | 4;26° |
| Venus | 8° | 7;37° |
| Mercury | 4° | 1;36° |
| Moon | 4;30° | 4;30° |

The earliest *siddhānta* that has come down to us is a Paitāmahasiddhānta²⁷ of the early fifth century. This is preserved because of its incorporation into the Viṣṇudharmottarapurāṇa²⁸ (2, 166–174), though it also (or at least the astronomical section, chapters 168–174) has been copied as an independent treatise.²⁹ Its position as the source of the Brāhmapakṣa is made evident by Brahmagupta’s dependence on it in composing the first ten chapters (Daśādhyāyī) of the BSS, and was explicitly recognized by Kamalākara (Siddhāntatattvaviveka 1, 62). Its chronological position is fixed by the fact that Sassanian astronomers in about 450 knew one of its characteristic parameters;³⁰ this makes it more than likely that Pitāmaha is the Svayambhū referred to by Āryabhaṭa (A Gola 50), whose Kalpa-system is a modification of the Brāhmapakṣa’s (he even accepts the Brāhma’s rotation of the *uccas* and *pātas* [A Daśagītikā 7 and Gola 2] despite the fact that in his system that is unnecessary), whose table of Sines (A Daśagītikā 10 and Gaṇitapāda 12) is derived from the Pait. (3, 12), and who copies various other formulae from the same source.

c4.s2.4

²⁷ CESS A 4.

c4.s2.5

²⁸ Bombay, 1912.

c4.s2.6

²⁹ Ed. by V. P. DVIVEDI in Jyautiṣasiddhāntasaṅgraha, BSS 39, 2, Benares, 1912; trans. by D. PINGREE, “The Paitāmahasiddhānta of the Viṣṇudharmottarapurāṇa,” Brahmavidyā 31–32 (1967–68), 472–510.

c4.s2.7

³⁰ D. PINGREE, “The Persian ‘Observation’ of the Solar Apogee in *ca.* A.D. 450,” JNES 24 (1965), 334–336.

c4.s2.8

The Pait. is cast in the form of a lecture by Brahmā to Bhṛgu. Its first two chapters deal with astrology. Chapter three discusses time-measurements, the parameters of the planetary system, Sines and Versines, the declination of the Sun and latitudes of the planets, gnomon-problems, right and oblique ascensions, a star catalogue, the Greek-letter phenomena of the planets, and the calendar; this confused congeries is also indicative of the age of the Pait. as later *siddhāntas* are arranged far more systematically. Chapter four covers the computation of the *ahargaṇa* (number of days lapsed since epoch), the astrological lords of the years, days, and *parvans*, the computation of the mean and true longitudes of the planets and of their latitudes, and the times of the occurrences of the planets’ first and second stations. Chapter five deals with the earth’s shadow (for lunar eclipses), *saṅkrāntis*, *tithis*, *karaṇas*, *yogas*, and the *vaidhṛta* and *vyatīpāta* *pātas*. Chapter six covers some problems in spherical trigonometry and their solution by means of analemmata. Chapter seven is devoted to the computation of the ascendant and of the time lapsed since sunrise. Chapter eight presents the method of computing the times and longitudes of the first and last visibilities of the planets and fixed stars. And chapter nine gives rules for computing the illumination of the Moon and longitudinal and latitudinal parallax (for solar eclipses). Several of these rules are incomplete or wrong as presented in the two editions, and only the preliminaries for computing lunar and solar eclipses are given. One suspects that the original Pait. was a more competent work than what we now possess of it.

c4.s2.9

Āryabhaṭa³¹ was 23 years old in 3600 Kali = A.D. 499 (A Kālakriyā 10) and wrote his Āryabhaṭīya³² at Kusumapura (A Gaṇitapāda 1), which is identified by Bhāskara in his commentary on this verse with Pāṭaliputra (p. 45). The same Bhāskara, in a series of examples relating to the kuṭṭaka, refers to the Aśmakasya gaṇita (p. 138), or Aśmakīya (pp. 140, 146, 147, and 148), or Aśmaka as an authority on the subject, and he applies to the Āryabhaṭīya the epithet Aśmaka (e.g., in MB 1, 3). Scholars, beginning at least with Nīlakaṇṭha (on A Gaṇitapāda 1), have contended on this basis that Āryabhaṭa was born in the Aśmaka country between the Godāvarī and the Narmadā; but the epithet may only reflect Bhāskara’s own origin and training in Aśmaka.

c4.s2.10

³¹ CESS A 1, 50b–53b; A 2, 15b; A 3, 16b; and A 4.

c4.s2.11

³² Edited with the Bhaṭadīpakā of Parameśvara by H. KERN, Leiden 1875, reprinted (with a Hindi translation by UDAYA NARAYANA SINGH), Madhurapur, Etawah, 1906, and Osnabrück 1973; with the Bhāṣya of Nīlakaṇṭha (omitting the Daśagītikā) by K. SAMBASIVA SASTRI (2 vols., TSS 101 and 110) and S. K. PILLAI (TSS 185), Trivandrum 1930–1957; with Sanskrit and Hindi commentaries by B. MISHRA, Patna [1966]; with an English translation by K. S. SHUKLA and K. V. SARMA, New Delhi 1976; with the commentaries of Bhāskara (to Golapāda 6) and Someśvara (Golapāda 6–50) by K. S. SHUKLA, New Delhi 1976; and with the commentary of Sūryadeva Yajvan by K. V. SARMA, New Delhi 1976. The Gaṇitapāda was published with a German translation and commentary by K. ELFERING, Die Mathematik des Āryabhaṭa I, München 1975. There are English translations of the whole work by P. C. SENGUPTA, JDL/UC 16 (1927), art. 6, and by W. E. CLARK, Chicago 1930.

c4.s2.12

The Āryabhaṭīya, which is the basic text of the Āryapakṣa, is not, strictly speaking, a siddhānta since it combines, in extreme conciseness, the fundamentals of both astronomy and mathematics. It is divided into four pādas. The Daśagītikā gives the planetary parameters, the divisions of the Kalpa, and a table of the first order differences of the Sines; at one point Āryabhaṭa refers to the diurnal rotation as a rotation of the earth (A Daśagītikā 3; see also Golapāda 9–10) but elsewhere (A Golapāda 16) he mentions the mathematically equivalent theory of the rotation of the fixed stars (bhagola). The theory of the diurnal rotation of the earth was fiercely attacked by many later Indian astronomers, beginning with Varāhamihira (PS 13, 6–7). The Gaṇitapāda, on mathematics, will be briefly discussed in chapter III. The Kālakriyāpāda defines time units, describes two geometrical models of planetary motion (eccentric with epicycle and concentric with two epicycles), and gives rules for computing the true longitudes of the planets. Finally, the Golapāda expounds Āryabhaṭa’s cosmology and geography, explains the apparent motions of the heavenly bodies, and gives solutions for problems in spherical trigonometry and rules for computing eclipses. The Āryapakṣa which evolved from the A was for most of its history popular mainly in South India, though the earliest extant commentaries were composed in Gujarāt—the Āryabhaṭīyabhāṣya of Bhāskara³³ at Valabhī in 629 and that of Someśvara in the eleventh or twelfth century; equally attesting to its popularity in Western India in the eighth and early ninth centuries is its use by Arabic astronomers of the period, who call Āryabhaṭa Arjabhar.³⁴ Al-Bīrūnī, however, while he possessed a copy of the A in the Panjāb in the 1020’s, was confused about its identity.³⁵ The later Indian commentators on the A, almost all of its manuscripts, and most of the texts following the Āryapakṣa³⁶ come from South India. The Sanskrit commentaries in addition to those named above were composed by Sūryadeva Yajvan, who was born in 1191, at Gaṅgāpura in Cola country; by Parameśvara³⁷ in about 1450 at Aśvatthagrāma in Kerala; by Yallaya at Skandasomeśvara in Āndhrapradeśa in 1480; by Nīlakaṇṭha³⁸ in Kerala shortly after 1501; by Raghunātha Rāja at Ahobila in Āndhrapradeśa in 1597; by Bhūtiviṣṇu,³⁹ probably near Kāñcī in Tamilnadu; and by Ghaṭīgopa⁴⁰ in Kerala in about 1800. There are also commentaries in Telugu and Malayālam and a translation into Marāṭhī.

c4.s2.13

³³ CESS A 4.

c4.s2.14

³⁴ D. PINGREE, “The Greek Influence on Early Islamic Mathematical Astronomy,” JAOS 93 (1973), 32—43, esp. 37—38.

c4.s2.15

³⁵ D. PINGREE, “Al-Bīrūnī’s Knowledge of Sanskrit Astronomical Texts,” The Scholar and the Saint, New York 1975, pp. 67—81, esp. 68—69.

c4.s2.16

³⁶ K. V. SARMA, “Tradition of Āryabhaṭīya in Kerala: Revision of Planetary Parameters,” IJHS 12 (1977), 194—199.

c4.s2.17

³⁷ CESS A 4.

c4.s2.18

³⁸ CESS A 3, 175b—177b, and A 4.

c4.s2.19

³⁹ CESS A 4.

c4.s2.20

⁴⁰ CESS A 2, 147a—147b, and A 4.

c4.s2.21

Āryabhaṭa wrote a second work that formed the basis of the Ardharātrikapakṣa, apparently entitled Āryabhaṭasiddhānta. This is now lost, though a number of astronomers have mentioned its views, and some commentators—especially on the Sūryasiddhānta—have quoted its verses.⁴¹ Especially prevalent in these quotations are verses describing Āryabhaṭa’s simple and crude observational instruments (e.g., by Mallikārjuna Sūri⁴² and by Tammayajvan⁴³) and his clepsydras (by Tammayajvan and by Rāmakṛṣṇa). The first treatise to follow the Ārdharātrikapakṣa of the Āryabhaṭasiddhānta was a revised version of the Sūryasiddhānta. This was composed by a pupil of Āryabhaṭa named Lāṭadeva⁴⁴ according to the information available to Al-Bīrūnī; and there is no good reason to doubt this attribution. As summarized by Varāhamihira in the PS it is a karaṇa rather than a siddhānta; its epoch is midnight 20/21 March 505. It is nuclear whether this form is that of Lāṭadeva’s original, or was imposed on it by Varāhamihira.

c4.s2.22

⁴¹ See especially K. S. SHUKLA, “Āryabhaṭa I’s Astronomy with Midnight Day-reckoning,” Gaṇita 18 (1967), 83—105, and “Glimpses from the Āryabhaṭasiddhānta,” IJHS 12 (1977), 181—186.

c4.s2.23

⁴² CESS A 4.

c4.s2.24

⁴³ CESS A 3, 85a—86a.

c4.s2.25

⁴⁴ PS ed. NEUGEBAUER—PINGREE, vol. 1, pp. 12—15; A with the commentary of Bhāskara ed. SHUKLA, pp. lxi—lxiv.

c4.s2.26

The earliest extant *siddhāntas* to adhere to something approaching the classical form belong to the Āryapakṣa. They were composed by that Bhāskara who commented on the A at Valabhī in 629; of these the Mahābhāskariya⁴⁵ was written before the Āryabhaṭīyabhāṣya which quotes from it under the title Karmanibandha (pp. 33, 40, 131, 179, 203, 204, 205, and 225–226), while the Laghubhāskariya⁴⁶ summarizes the MB, and probably postdates the Āryabhaṭīyabhāṣya (and the BSS).

c4.s2.27

⁴⁵ Edited with the commentary, Karmadīpikā, of Parameśvara by B. D. APATE, ASS 126, Poona 1945; with the Bhāṣya of Govindasvāmin and the super-commentary, Siddhāntadīpikā, of Parameśvara by T. S. KUPPANNA SASTRI, Madras GOS 130, Madras 1957; and with an English translation and commentary by K. S. SHUKLA, Lucknow 1960.

c4.s2.28

⁴⁶ Edited with the commentary, Pārameśvara, of Parameśvara by B. D. APATE, ASS 128, Poona 1946; with the Vivaraṇa of Śaṅkaranārāyaṇa by S. VENKATA-SUBRAMONIA IYER and S. KOCHUKANJU ASARI, TSS 162, Trivandrum 1949; and with an English translation and commentary by K. S. SHUKLA, Lucknow 1963.

c4.s2.29

An arrangement less rigid than that of the classical *siddhānta* is evident in the contents of the MB’s eight chapters:

c4.s2.30

1. *ahargaṇa*; mean longitudes; planetary *kuṭṭaka*.
2. *deśāntara* correction.
3. spherical trigonometry; catalogue of stars.
4. planetary equations (including the model involving a concentric with equant); *tithis*, *karaṇas*, *nakṣatras*, *pātas*, and *yogas*.
5. solar and lunar eclipses.
6. first visibility and illumination of the Moon; first and last visibilities of the planets; planetary conjunctions.
7. parameters of the Āryapakṣa and of the Ārdharātrikapakṣa.
8. examples.

c4.s2.31

The arrangement of the eight chapters of the LB, which is probably influenced by that of the Daśādhyāyī of the BSS, represents the first stage of the fully developed form:

c4.s2.32

1. *ahargaṇa*; mean longitudes; *deśāntara* correction (*madhyama*).
2. planetary equations; *nakṣatras*, *tithis*, *karaṇas*, and *pātas* (*spaṣṭa*).
3. spherical trigonometry (*tripraśna*).
4. lunar eclipses (*candragrahaṇa*).
5. solar eclipses (*sūryagrahaṇa*).
6. first visibility and illumination of the Moon (*śṛṅgonnati*).
7. first and last visibilities and conjunctions of the planets (*udayāsta* and *grahayuti*).
8. catalogue of stars; conjunctions of planets with stars (*bhagrahayuti*).

c4.s2.33

Both of these *siddhāntas* were, like most Āryapakṣa works, popular only in South India; it is there that most of their manuscripts were copied, and their commentaries written. These commentaries include that composed by Govindasvāmin⁴⁷ (ca. 800–850) in Kerala on the MB, that written by his pupil, Śaṅkaranārāyaṇa, at Kollapurī, Kerala, in 869 on the LB, that composed by Udayadivākara⁴⁸ in 1073 on the LB, and those composed by Parameśvara (ca. 1380–1460) in 1408 on the LB and after 1443 on the MB; Parameśvara also wrote after 1432 a super-commentary on the Mahābhāskarīyabhāṣya of Govindasvāmin.

c4.s2.34

⁴⁷ CESS A 2, 143b–144a, and A 3, 35b. ⁴⁸ CESS A 1, 56b–57a. ⁴⁹ CESS A 4. ⁵⁰ Edited with his own Sanskrit ṭīkā by S. DVIVEDIN, Benares 1902, and with his own Sanskrit and Hindi ṭīkās and excerpts froın those of Pṛthūdakasvāmin and DVIVEDIN by R. SARMAN, 4 vols., New Delhi 1966. A new critical edition with the commentary of Pṛthūdakasvāmin, an English translation, and a commentary is being prepared by D. PINGREE. ⁵¹ Based on it was a Mahāsiddhānta which was translated into Arabic as Al-Zīj al-Sindhind al-kabīr in 771 or 773 by Muḥammad ibn Ibrāhīm al-Fazārī; see D. PINGREE, “The fragments of the Works of al-Fazārī,” JNES 29 (1970), 103–123. This tradition was also followed by al-Fazārī’s contemporary, Ya‘qūb ibn Ṭāriq; see D. PINGREE, “The Fragments of the Ya‘qūb ibn Ṭāriq,” JNES 27 (1968), 97–125. See also D. PINGREE, The Thousands of Abū Ma‘shar, London 1968. ⁵² Concerning the relation of the Sindhind tradition—particularly in the form of the Latin version of al-Khwārizmī’s Al-Zīj al-Sindhind and its commentaries—to the BSS and other Sanskrit works see D. PINGREE, “The Indian and Pseudo-Indian Passages in Greek and Latin Astronomical and Astrological Texts,” Viator 7 (1976), 141–195, esp. 151–169.

c4.s2.35

Bhāskara’s contemporary rival as an astronomer was Brahmagupta,⁴⁹ who was born in 598 and completed his Brāhmasphuṭasiddhānta⁵⁰ in 628 at Bhillamāla in Rājasthāna during the reign of the Cāpa (Cāvoṭaka) monarch Vyāghramukha. The BSS, which belongs to the Brāhmapakṣa, was enormously influential on later Indian astronomy as well as on Islamic⁵¹ and Western European.⁵² It consists of twenty-four chapters, to which a twenty-fifth—a karaṇa—is appended in many manuscripts. These begin with a Daśādhyāyī (chapters 1–10), which summarizes and expands the teachings of the Pait. The relation to (and possible influence on) the LB will be clear from Table 4. This is followed by a chapter (11) in which Brahmagupta criticizes his predecessors and praises the Brāhmapakṣa (tantraparīkṣā) and another (12) on mathematics (gaṇita). Subject to the tantraparīkṣā’s animadversions are Āryabhaṭa, the Jainas, Pradyumna, Lāṭadeva, Varāhamihira, Vijayanandin, Viṣṇucandra’s Vasiṣṭhasiddhānta, Śrīṣeṇa’s Romakasiddhānta, and Siṃha. Additional information concerning those of these authors whose works are lost can be found in the PS, in the Āryabhaṭīyabhāṣya of Bhāskara, and in several later commentaries (e.g., those of Bhaṭṭotpala on the Bṛhatsaṃhitā and of Pṛthūdakasvāmin on the BSS). The next five chapters of the BSS contain additions and corrections to the Daśādhyāyī: chapter thirteen to chapter one, fourteen to two, fifteen to three, sixteen to four and five, and seventeen to seven. Brahmagupta followed this practice of providing supplementary chapters (uttara) in his Kh also. The last seven chapters of the BSS are more heterogeneous: eighteen is on the kuṭṭaka, nineteen on gnomon problems (śaṅku), twenty on prosody (chandas), twenty-one on cosmology (gola), twenty-two on instruments (yantra), twenty-three on measurements (māna), and twenty-four on the contents of the BSS and on its author (saṃjñā). The BSS was particularly popular in North and West India; the scholar who brought its derivative, the Mahāsiddhānta, to Baghdād in 771 or 773 came from Sind, and the two commentaries known to have been written on it were composed in the North. Balabhadra⁵³ wrote his at Kānyakubja, probably under Yaśovarman (ca. 725–750); it is now lost except for fragments preserved by Pṛthūdakasvāmin, Bhaṭṭotpala, and al-Bīrūnī. Pṛthūdakasvāmin⁵⁴ wrote his monumental Vāsanābhāṣya on the BSS at Sthāneśvara (Sthāṇvīśvara) in Kurukṣetra in the 860's; the commentary on chapters sixteen to twenty, on most of twenty-two, and on twenty-three and twenty-four is lost.

c4.s2.36

⁵³ CESS A 4.

c4.s2.37

⁵⁴ CESS A 4.

c4.s2.38

Earlier another astronomer had apparently lived at Sthāneśvara, where, in the eighth century, someone wrote a new Pauliśasiddhānta.⁵⁵ This astronomer, while accepting the planetary parameters of the Ārdharātrikapakṣa and the main division of the Kalpa according to the Āryapakṣa, compromised with smṛti (and the Brāhmapakṣa) to the extent of accepting the traditional division of a Mahāyuga. In order to achieve a mean conjunction at the beginning of the current Kaliyuga (which he placed 3,888,000 years after the beginning of the current Mahāyuga), he delayed the beginning of celestial motion 648,000 years so that there would remain 3,240,000 years (three of Āryabhaṭa's yugas) before the beginning of the current Kaliyuga. The Pauliśasiddhānta is not extant in its entirety, but many fragments remain in the works of Pṛthūdakasvāmin, Bhaṭṭotpala, al-Bīrūnī, and Āmarāja.⁵⁶

c4.s2.39

⁵⁵ CESS A 4.

c4.s2.40

⁵⁶ D. PINGREE, "The Later Pauliśasiddhānta," Centaurus 14 (1969), 172–241.

c4.s2.41

Probably also in the eighth century, though perhaps in the early ninth, Lalla composed his Śiṣyadhīvṛddhidatantra⁵⁷ following the Āryapakṣa, but with bījas for periods of 250 years subsequent to 498 (the numerators of some of these bījas are borrowed from Haridatta, who wrote in 683). Lalla may have lived in Daśapura in Mālava, which he mentions (SDV 2, 9, 10). His approximate date is indicated by his initial bīja-period, which ends in 748; by his sharing with the Pauliśasiddhānta and the Sūryasiddhānta the Brāhmapakṣa's divisions of a Mahāyuga while retaining Āryabhaṭa's parameters; and by the new organization of the siddhānta that he introduced and that was imitated by Vaṭeśvara, who wrote his VS in 904, and, in part, by Śrīpati⁵⁸ when he wrote his SŚŚ toward the middle of the eleventh century. This new organization of the siddhānta retained essentially the contents of the BSS's Daśādhyāyī in a first section on computation (grahaṇita), while gathering theoretical and cosmological material in a second section (gola). The ŚDV was commented on by Bhāskara⁵⁹ (b. 1114) and by Mallikārjuna Sūri⁶⁰—probably he who commented on the Sūryasiddhānta in 1178 in Sanskrit, and previously in Telugu. Thus Lalla’s work, as is to be expected of one belonging to the Ārya-pakṣa, was popular in the South; but the presence of manuscripts in Western India and the references to Lalla in, for instance, Āmarāja’s commentary on the Kh, show that it was also studied in Gujarāt and Rājasthāna.

c4.s2.42

⁵⁷ Edited by S. DVIVEDIN, Benares 1886.

c4.s2.43

⁵⁸ Śrīpati also modelled his Jyotiṣaratnamālā on Lalla's Jyotiṣaratnakośa.

c4.s2.44

⁵⁹ CESS A 4.

c4.s2.45

⁶⁰ CESS A 4.

c4.s2.46

The Saurapakṣa was initiated in about 800 by the unknown author of the “modern” Sūryasiddhānta,⁶¹ who has modified many of the planetary parameters of the Ārdharātrikapakṣa (and thereby of Lāṭadeva’s Sūryasiddhānta) while retaining others along with the midnight epoch, and who has taken over from the Brāhmapakṣa the traditional divisions of both the Kalpa and the Mahāyuga, but delayed the beginning of the motions of the planets by 17,064,000 years so that the rotations of each within the remainder of the Kalpa are divisible by four. Other elements in the SS indicate its author’s knowledge of Brahmagupta’s BSS. The popularity of the work was enormous in all parts of India, but especially in the South and Northeast. Virtually every commentator, however, has rearranged the text, adding and subtracting verses ad libitum; the manuscript traditions of these many recensions, especially the earlier ones, will have to be fully investigated before a clear idea of the exact contents of the original can emerge. The most important of the extant Sanskrit commentators are: Mallikārjuna Sūri,⁶² who wrote a commentary in Telugu before composing his Sanskrit Sūryasiddhāntaṭīkā in 1178; Caṇḍeśvara,⁶³ a Maithila Brāhmaṇa whose Sūryasiddhāntabhāṣya was written in 1185; Madanapāla,⁶⁴ a Mahārājādhirāja of the Ṭāka family, who wrote his Vāsanārṇava between 1375 and 1400; Parameśvara,⁶⁵ who composed his Sūryasiddhāntavivaraṇa in Kerala in 1432; Yallaya, who wrote his Kalpavallī in Āndhrapradeśa in 1472; Rāmakṛṣṇa Ārādhya, whose Subodhinī was also composed in 1472; Bhūdhara,⁶⁶ who wrote his Sūryasiddhāntavivaraṇa at Kāmpilya in 1572; Tamma Yajvan,⁶⁷ who composed his Kāmadogdhrī at Paragīpurī in 1599; Raṅganātha, whose Gūḍhārthaprakāśaka was completed in Kāśī in 1603; Nṛsiṃha,⁶⁸ who wrote his immense Saurabhāṣya in the same city in 1611; Viśvanātha, also of Kāśī, who composed his Gahanārthaprakāśa in about 1628; Kamalākara,⁶⁹ another astronomer of Kāśī, whose Sauravāsanā on SS 1—10 was written after 1658; and Dādābhāī,⁷⁰ a Cittapāvana Brāhmaṇa, who composed his Kiraṇāvalī in 1719. The dates of three other commentators, all of whom lived in South India, are unknown; they are Kāmabhaṭṭa⁷¹ (Sūryasiddhāntaṭīkā), Cola Vipaścit⁷² (Gaṇakopakāriṇī), and Bhūtiviṣṇu⁷³ (Gurukaṭākṣa).

c4.s2.47

⁶¹ Edited with the commentary of Raṅganātha by F. HALL and BĀPŪ DEVA ŚĀSTRIN, BI 25, Calcutta 1859, repr. Amsterdam [1974]; edited with the same commentary by JĪVĀNANDA VIDYĀSĀGARA, 2nd ed., Calcutta 1891; edited with a Hindī translation by UDAYA NARAIN SINGH, Biddoopur, Meerut, 1903; edited with his own commentary, Sudhāvarṣiṇī, by SUDHĀKARA DVIVEDIN, BI 173, Calcutta 1911, 2nd ed. Calcutta 1925; edited with his own bhāṣya, Tattvāmṛta, by KAPILEŚVARA ŚĀSTRIN, KSS 144, Banārasa 1936; edited with his own Malayālam commentary by PULIYOOR PURUSHOTTAMAN NAMPUTIRI, Trivandrum 1950; edited with the commentary of Raṅganātha and a Hindī translation by B. P. MISRA, Bombay 1956; and edited with the commentary of Parameśvara by K. S. SHUKLA, Lucknow 1957. The madhyama was edited with his own ṭīkā, Sudhādhavalā, by UMĀDATTA JOŚI, Hādiyāvāda [ND]. The Sūryasiddhānta was translated into English by E. BURGESS with the assistance of W. D. WHITNEY, JAOS 6, 2 (1860), 141--498, reprinted New Haven 1860 and Calcutta 1935, and by BĀPŪ DEVA ŚĀSTRIN, BI 32, Calcutta 1861, reprinted Amsterdam [1974].

c4.s2.48

⁶² CESS A 4.

c4.s2.49

⁶³ CESS A 3, 40b—41a.

c4.s2.50

⁶⁴ CESS A 4.

c4.s2.51

⁶⁵ CESS A 4.

c4.s2.52

⁶⁶ CESS A 4.

c4.s2.53

⁶⁷ CESS A 3, 85a—86a.

c4.s2.54

⁶⁸ CESS A 3, 204a—205a.

c4.s2.55

⁶⁹ CESS A 2, 23a.

c4.s2.56

⁷⁰ CESS A 3, 97a—97b.

c4.s2.57

⁷¹ CESS A 2, 31b.

c4.s2.58

⁷² CESS A 3, 52b—53a.

c4.s2.59

⁷³ CESS A 4; a fragment was edited by K. M. K. Sarma, “The Sūryasiddhāntavyākhyā of Bhūtiviṣṇu,” Brahmavidyā 5 (1941), 77—83.

c4.s2.60

A siddhānta in which the influence of both the Āryapakṣa and the Saurapakṣa as well as that of Brahmagupta’s BSS are apparent is the Vaṭeśvarasiddhānta⁷⁴ composed by Vaṭeśvara at Ānandapura in Gujarāt in 904; the arrangement of the chapters follows that of Lalla’s ŚDV. One of the more interesting sections is that (VS 1, 1, 10) in which Vaṭeśvara criticizes Brahmagupta as Brahmagupta had criticized his predecessors, and especially Āryabhaṭa (BSS 11).

c4.s2.61

⁷⁴ Adhikāras 1—3 are edited in an unsatisfactory manner by R. S. Sharma and M. Mishra, New Delhi 1962; see also T. S. Kuppanna Shastri, “The System of the Vaṭeśvara Siddhānta,” IJHS 4 (1969), 135—143, and K. S. Shukla, “Hindu Astronomer Vaṭeśvara and His Works,” Gaṇita 23 (1972), 65—74.

c4.s2.62

In about 950—1000 a second Āryabhaṭa⁷⁵ wrote a Mahāsiddhānta⁷⁶ in large part based on the Brāhmapakṣa, but beginning the planets’ rotations 3,024,000 years after the beginning of the Kalpa in the fashion of the Saurapakṣa and substituting a new set of planetary parameters. He also reports another set of new parameters which he found in a treatise ascribed to Parāśara (MS 1, 2). And, while following Lalla in dividing his work into a grahagaṇita and a gola section, he follows Brahmagupta’s BSS in including chapters on arithmetic and geometry (2, 2 = 15) and on the kuṭṭaka (2, 5 = 18) in his siddhānta.

c4.s2.63

⁷⁵ CESS A 1, 53b—54a; A 2, 15b—16a; and A 4.

c4.s2.64

⁷⁶ Edited with his own Sanskrit commentary by Sudhākara Dvivedin, BSS 36, Benares 1910; the grahagaṇita was edited with an English translation by S. R. Sarma, 2 vols., Marburg 1966.

c4.s2.65

Śrīpati, who wrote various works at Rohiṇīkhaṇḍa (about 150 miles south of Ujjayinī) between 1039 and 1056, followed the Brāhmapakṣa when he composed the Siddhāntaśekhara,⁷⁷ though he adds the bījas to the planetary mean motions that characterize the later history of this school. He also adds to Brahmagupta’s astronomy rules for computing the second component (udayāntara) of the equation of time (SŚŚ 3, 46) and the evection of the Moon (sphuṭacaraphala) (SŚŚ 11, 2–4); the latter rule, which has ultimately an Islamic Ptolemaic origin, had previously been given by Muñjāla. Śrīpati arranged the twenty chapters of the SŚŚ in much the same order as Lalla’s ŚDV, though without the latter’s division into two sections and with the addition of two chapters (13 and 14) on arithmetic and geometry (vyakta-gaṇita) and on algebra (avyaktagaṇita) respectively. The commentary, Gaṇitabhūṣaṇa, composed by Makkibhaṭṭa⁷⁸ in 1377, is preserved only up to 4, 75.

c4.s2.66

⁷⁷ Edited with the commentary on 1–4 of Makkibhaṭṭa and with his own commentary on 5–20 by BABUĀJĪ MIŚRA, 2 vols., Calcutta 1932–1947.

c4.s2.67

⁷⁸ CESS A 4.

c4.s2.68

But the most notable siddhānta produced in India is the Siddhāntaśiromaṇi⁷⁹ written by Bhāskara⁸⁰ at Vijjaḍaviḍa (?) probably in his native Sahyādri north of Devagiri in 1150. Modelled in form on Lalla’s ŚDV (though with the addition of a chapter on the trigonometrical functions, jyotpatti and of another describing the seasons) the SŚB is the foremost text of the Brāhmapakṣa after the BSS. Bhāskara changes a few parameters minutely, repeats some of the improvements introduced by earlier astronomy such as the equation of time, and presents for the first time the formula for finding Sin (α + β),⁸¹ but the most impressive quality of the SŚB is its comprehensiveness. Commentaries on the SŚB were composed by, among others, Bhāskara himself (Mitākṣarā); by Lakṣmīdāsa in 1501 (Gaṇitatattvacintāmaṇi); by Gaṇeśa⁸² at Nandipura in Gujarāt in the early seventeenth century (Śiromaṇiprakāśa); by Nṛsiṃha⁸³ at Kāśī in 1621 (Vāsanāvārttika) and by Munīśvara⁸⁴ at Kāśī before 1638 (Marici).

c4.s2.69

⁷⁹ The gaṇitādhyāya with Bhāskara’s own Mitākṣarā was edited by L. WILKINSON, Calcutta 1842, reprinted Calcutta 1855; the golādhyāya with the Mitākṣarā by L. WILKINSON, Calcutta 1842, reprinted Calcutta 1856; edited with the Mitākṣarā by BĀPŪDEVA ŚĀSTRIN, Benares 1866, revised ed. by CANDRADEVA, Benares 1891, revised ed. by GAṆAPATIDEVA ŚĀSTRIN, KSS 72, Banarasa 1929; the golādhyāya with the Mitākṣarā by JĪVĀNANDA VIDYĀSĀGARA, 2nd ed. Calcutta 1899; the gaṇitādhyāya with the Mitākṣarā edited by JĪVĀNANDA VIDYĀSĀGARA, Calcutta 1881, revised ed. by AŚUBODHA VIDYĀBHŪṢAṆA and NITYABODHA VIDYĀRATNA, Kalikātā 1915; the golādhyāya with the Mitākṣarā and a Bengālī translation edited by RASIKAMOHANA CHAṬṬOPĀDHYĀYA, Calcutta 1887; the golādhyāya with the Mitākṣarā and a Bengālī translation published in Aruṇodaya 1 (1890), pt. 6; the golādhyāya published in Bṛhaspati 1 (1896/97); edited with the Mitākṣarā and a Hindī translation by UDAI NARAIN SINGH, Bombay 1905; adhyāyas 1–8 of the golādhyāya with Marāṭhī translation and commentary edited by V. P. KHĀNĀPURKAR, Mumbaī 1911; the golādhyāya with the Mitākṣarā and a Hindī commentary edited by GIRIJĀPRASĀDA DVIVEDIN, Lucknow 1911, reprinted Ahmadabad 1936; the gaṇitādhyāya with Marāṭhī translation and commentary edited by V. P. KHĀNĀPURKAR, Puṇem 1913; adhyāya 1 of the gaṇitādhyāya with the Mitākṣarā and the commentaries of Nṛsiṃha and Munīśvara edited by MURALĪDHARA JHĀ, Benares 1917; the golādhyāya with the Mitākṣarā and a Bengālī translation edited by RĀDHĀVALLABHA, Calcutta 1921; the gaṇitādhyāya with the Mitākṣarā and the commentary of Gaṇeśa edited by DATTĀTREYA APAṬE, ASS 110, 2 vols., Poona 1939–41; the golādhyāya with the Mitākṣarā and the commentary of Munīśvara edited by DATTĀTREYA APAṬE, ASS 122, 2 vols., Poona 1943–52; adhyāyas 1–2 of the gaṇitādhyāya with the Mitākṣarā edited by MURALĪDHARA THAKKURA, KSS 149, Banārasa 1950; and the gaṇitādhyāya with the Mitākṣarā and the commentary of Munīśvara (the latter not on adhyāya 1) edited by KEDĀRADATTA JOŚĪ, 3 vols., Vārāṇasī 1961–64. The gaṇitādhyāya was translated into Latin by E. ROER, JAS Bengal 13 (1844), 53–66; the golādhyāya was translated into English by L. WILKINSON, revised by BĀPŪDEVA ŚĀSTRIN, BI 32, Calcutta 1861.

c4.s2.70

⁸⁰ CESS A 4.

c4.s2.71

⁸¹ R. C. GUPTA, “Addition and Subtraction Theorems for the Sine and Cosine in Medieval India,” IJHS 9 (1974), 164—177.

c4.s2.72

⁸² CESS A 2, 106b—107a.

c4.s2.73

⁸³ CESS A 3, 205a—206a.

c4.s2.74

⁸⁴ CESS A 4.

c4.s2.75

For approximately 350 years after the composition of the SŚB no important siddhānta was written; the learning of astronomers was rather concentrated in commentaries, their ingenuity in koṣṭhakas. It was, however, probably during this period that a group of minor siddhāntas belonging to the Saurapakṣa, but attributed to deities or sages, were written. These are the Somasiddhānta,⁸⁵ on which a commentary, Gūḍhārthadīpikā, was composed by Nṛsiṃha⁸⁶ at Koṇḍavīḍu in Āndhrapradeśa in about 1400; a Vasiṣṭhasiddhānta⁸⁷ which claims to be based on that of Viṣṇucandra; a Vṛddhavasiṣṭhasiddhānta,⁸⁸ which is influenced by the Brāhmapakṣa in its planetary equations; and the Brahmasiddhānta⁸⁹ alleged to be part of a Śākalyasaṃhitā.

c4.s2.76

⁸⁵ Edited V. P. DVIVEDIN, BSS 39, 1, Benares 1912, pp. 1—36.

c4.s2.77

⁸⁶ CESS A 3, 201a.

c4.s2.78

⁸⁷ Edited V. P. DVIVEDIN, Benares 1907.

c4.s2.79

⁸⁸ Edited V. P. DVIVEDIN, BSS 39, 2, Benares 1912, pp. 25—78.

c4.s2.80

⁸⁹ Edited V. P. DVIVEDIN, BSS 39, 1, Benares 1912, pt. 2. See CESS A 4.

c4.s2.81

c4.s2.82

¹ Divided into khagolabandha, bhagolabandha, grahagolabandha, and sāmānyagolabandha.

c4.s2.83

(Table 4 cont.)

c4.s2.84

c4.s2.85

¹ Adhikāras 1–11 are sometimes regarded as the pūrvārdha, 12–14 as the aparārdha.

c4.s2.86

² What follows this is unclear.

c4.s2.87

(Table 4 cont.)

c4.s2.88

c4.s2.89

c4.s2.90

¹ The jyotpatti is variously placed in various editions.

c4.s2.91

(Table 4 cont.)

c4.s2.92

c4.s2.93

Brahmā, Sūrya, Soma, Vasiṣṭha, and Pulastya are mentioned as agreeing with his Saurapakṣa parameters in the Siddhāntasundara⁹⁰ (1, 1, 25) by Jñānarāja,⁹¹ who wrote this work at Pārthapura on the Godāvarī in 1503; that by Brahmā he means the *siddhānta* belonging to the Śākalyasaṃhitā is clear from SSJ 2, 1, 3. This work basically follows Lalla’s arrangement, but like the SŚB it adds an *ṛtuvarṇana* (SSJ 2, 6). A commentary, Grahaṇita-cintāmaṇi, was written by Jñānarāja’s son, Cintāmaṇi,⁹² in about 1530.

c4.s2.94

⁹⁰ Unpublished. I have consulted the manuscripts IO 2002; BM Add. 14, 365p; and Oxford CS d. 805 (5).

c4.s2.95

⁹¹ CESS A 3, 75a—76a, and A 4.

c4.s2.96

⁹² CESS A 3, 49b.

c4.s2.97

The influence of Islamic Ptolemaic theory upon Indian astronomy can be traced back to Muñjāla in the tenth century, but the translation of Persian astronomical treatises in some number into Sanskrit began only in the sixteenth and seventeenth centuries. The works selected for translation were for the most part the products of the School of Marāgha and its descendant at Samarqand; among the treatises so translated were the Zij of Ulugh Beg⁹³ (Jīca Ulugbegī) and al-Qūshjī’s⁹⁴ Risālah dar hay’at (Hayatagrantha⁹⁵). The earliest Sanskrit *siddhāntas* to reflect this Islamic influence were composed at the Mughal court at Delhi by Nityānanda⁹⁶ during the reign of Shāh Jahān; he wrote the Siddhāntasindhu for Āsaf Khān in 1628, and the Siddhāntarāja⁹⁷ in 1639. While accepting Islamic parameters and models, Nityānanda presents his material in traditional form and puts forth the claim (copied from the Jñānabhāskara) that this astronomy had originally been revealed to the Yavana (Muslim) by the Sun, who was temporarily incarnate as Romaka (a Roman, or perhaps Qāḍī Zādah al-Rūmī). This Indianized presentation failed to convey to Sanskrit scholars the real superiority of Islamic over Indian astronomy, that is its methodology involving both a reliance on carefully planned and executed observations and a concern with the cinematics of the planetary models. Nityānanda’s efforts produced no discernible effects.

c4.s2.98

⁹³ CESS A 4.

c4.s2.99

⁹⁴ CESS A 4.

c4.s2.100

⁹⁵ Edited by V. BHAṬṬĀCĀRYA, SBG 96, Vārāṇasī 1967; see PINGREE in JHAS 2 (1978), 326—328.

c4.s2.101

⁹⁶ CESS A 3, 173b—174a, and A 4.

c4.s2.102

⁹⁷ PINGREE in JHAS 2 (1978), 323—326.

c4.s2.103

His contemporaries in Kāśī, however, did not ignore Islamic astronomy, though some were more receptive than others. The two *siddhāntas* written there in the middle of the century follow the Saurapakṣa with some influence from Bhāskara’s SŚB. Munīśvara,⁹⁸ who had earlier composed the gigantic Marīci on the SŚB, completed his Siddhāntasārvabhauma⁹⁹ in 1646. While willingly accepting some Islamic trigonometry and rather indifferently referring to certain doctrines of the Pārasikas (Persians), Munīśvara vigorously attacks the theory of precession,¹⁰⁰ thereby beginning a heated controversy carried on by Raṅganātha, the brother of his rival, Kamalākara, in his Lohalakhaṇḍana, to which Munīśvara’s cousin, Gadādhara,¹⁰¹ replied in his Lohagolasamarthana.¹⁰² Munīśvara wrote his own commentary, the Āśayaprakāśinī, on the SSBM.

c4.s2.104

⁹⁸ CESS A 4.

c4.s2.105

⁹⁹ I 1, 1—3, 133 with Munīśvara’s own commentary were edited by GOPĪNĀTHA KAVIRĀJA, PSWWT 41, 2 vols., Benares 1932—35.

c4.s2.106

¹⁰⁰ PINGREE in JHAS 2 (1978), 321–322. ¹⁰¹ CESS A 2, 115a. ¹⁰² Both works were edited by M. H. OJHA, LG 15, Benares 1961. ¹⁰³ CESS A 2, 21a–23a, A 3, 18a; and A 4. ¹⁰⁴ Edited with Kamalākara’s Śeṣavāsanā by S. DVIVEDIN, BSS 1, Benares 1885; 2nd ed. by MURALIDHARA JHĀ, Benares 1924–35; also edited by GAṄGĀDHARA MIŚRA, Lucknow 1929. ¹⁰⁵ PINGREE in JHAS 2 (1978), 322–323. ¹⁰⁶ CESS A 3, 56a–58a, and A 4. ¹⁰⁷ CESS A 3, 63a–64b, and A 4. ¹⁰⁸ Edited by R. S. ŚARMAN, 3 vols., New Delhi 1967–69. ¹⁰⁹ PINGREE in JHAS 2 (1978), 328–329. ¹¹⁰ N. V. ATHALEY, “Kalandikā-Prakāśa of Somanātha Vyāsa,” A Volume of Studies in Indology Presented to Prof. P. V. KANE, POS 75, Poona 1941, pp. 39–48. ¹¹¹ Cf. CP, Kielhorn XXIII 104: the Bhūbhramavādakhaṇḍananirāsa of the Sihoragrāmasthasabhā, a manuscript dated Śaka 1761 = A.D. 1839. ¹¹² CESS A 4. ¹¹³ Published at Mumbai in 1896.

c4.s2.107

Kamalākara,¹⁰³ who wrote the Sauravāsanā on the SS and completed the Siddhāntatattvaviveka¹⁰⁴ in 1658, while accepting the Saurapakṣa’s planetary parameters, was far more receptive to Muslim astronomy as represented by Ulugh Beg and the School of Samarqand.¹⁰⁵ He was, in particular, prone to agree with Muslim (Ptolemaic) notions of the structure of the planetary system, and he presents what is, apparently, the only Sanskrit treatise on geometrical optics (STV 1, 5).

c4.s2.108

This process of transmitting Islamic astronomy to India was carried much further by the translation of Naṣīr al-Dīn’s Arabic version of Ptolemy’s Σύνταξις μαθηματική into Sanskrit by Jagannātha¹⁰⁶ at Jayapura in 1732 under the patronage of Savāī Jayasiṃha.¹⁰⁷ The Siddhāntasamrāṭ¹⁰⁸ continues after the thirteenth book of Ptolemy with supplements describing the instruments that Jayasiṃha had set up in his observatories in imitation of those installed by Ulugh Beg at Samarqand, and an exposition of Ulugh Beg’s and al-Kāshī’s derivations of sines, followed by remarks on the traditional aspects of Indian astronomy that were not touched upon by Ptolemy.¹⁰⁹ This last effort to expound Ptolemaic astronomy in Sanskrit was also doomed to fail; a century later, modern European astronomy was being taught in Sanskrit at Sihore by Somanātha Vyāsa¹¹⁰ and other paṇḍits of the Sihore High School¹¹¹ in the 1830’s and 1840’s under the influence of Englishmen such as Lancelot Wilkinson. The process of Westernization was practically complete by the time Premavallabha¹¹² wrote his Paramasiddhānta¹¹³ at Jayapura in 1882.

Linked texts (21)

Sūryasiddhānta — Maya, Sanskrit
Bṛhatsaṃhitā — Varāhamihira, Sanskrit
Vaṭeśvarasiddhānta — Vaṭeśvara, Sanskrit
Viṣṇudharmottarapurāṇa — Sanskrit
Brāhmasphuṭasiddhānta — Brahmagupta, Sanskrit
Āryabhaṭīya — Āryabhaṭa, Sanskrit
Paitāmahasiddhānta (lost early version) — Sanskrit
Vāsiṣṭhasiddhānta — Sanskrit
Pauliśasiddhānta — Sanskrit
Āryabhaṭasiddhānta (ārdharātrika system) — Āryabhaṭa, Sanskrit
Siddhāntatattvaviveka — Kamalākarabhaṭṭa 1633, Sanskrit
Śiṣyadhīvṛddhidatantra — Lalla, Sanskrit
Samrāṭsiddhānta — Jagannātha Samrāj 1707, Sanskrit
Mahābhāskarīya — Bhāskara I, Sanskrit
Siddhāntaśekhara — Śrīpati, Sanskrit
Gūḍhārthadīpikā — Vāmadeva, Sanskrit
Pañcasiddhāntikā — Varāhamihira, Sanskrit
Mahāsiddhānta (Āryasiddhānta) — Āryabhaṭa II, Sanskrit
Laghubhāskarīya — Bhāskara I, Sanskrit
Khaṇḍakhādyaka — Brahmagupta, Sanskrit
Siddhāntaśiromaṇi — Bhāskara II, Sanskrit