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Jyotihshastra: Astral and Mathematical Literature c4.s4: Koṣṭhakas

Jyotiḥśāstra: Astral and Mathematical Literature c4.s4

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English

Koṣṭhakas

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¹⁷¹ On astronomical tables see D. PINGREE, SATIUS, SATE, and "On the Classification of Indian Planetary Tables," JHA 1 (1970), 95—108.

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Closely associated with *karaṇas* in North and West India since at least the twelfth century were astronomical tables (*koṣṭhakas* or *sāraṇīs*) which were intended to facilitate the computation of planetary positions and the cusps of the astrological places for those who cast horoscopes, or that of various calendaric functions such as *tithis*, *nakṣatras*, and *yogas* for the makers of almanacs (*pañcāṅgas*). Tables had much earlier been an integral part of astronomical writings, in which they were versified along with the mathematical formulae. But the composing of tables using numerical symbols arranged in columns was apparently introduced into India as an imitation of Islamic *zījes*. The Indian genius for mathematical computation led them to develop many ingenious ways to present the complex elements of their planetary and calendaric models in tabular forms very different from those in use in Islam, but the ultimate derivation of the idea from the West seems unquestionable.

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In fact, the earliest such *koṣṭhaka* concerning which we have any information is apparently the *zīj* composed by Durlabha¹⁷² at Multān in Sind; its epoch is 932. Our only source of knowledge of this *zīj*, however, is al-Bīrūnī's India; and he quotes from the instructions rather than the tables, which presumably accompanied those instructions if the work was indeed a *zīj*.

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¹⁷² CESS A 3, 116b.

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Planetary tables in India can be classified into three categories. "Mean linear" arrangements tabulate the increments in the mean longitudes of the planets in successively smaller units of time; the *manda* and *śīghra* equations are tabulated separately. This is the form of planetary table employed by Ptolemy and by most of his eastern and western successors. The "true linear" arrangement tabulates the true longitudes of the planets in ideal years at given intervals (*avadhis*). The ideal years assume initial mean longitudes of the planet at equal distances around the zodiac while the initial mean longitude of the Sun is always at Aries 0°; the intervals between the initial mean longitudes that are attested are 3°, 6°, 12°, 13;20° (a *nakṣatra*), and 30°. Mean motion tables in terms of these intervals allow one to locate the planet within this system for any given year. The *avadhis* are normally periods of fourteen days. The "true linear" arrangement seems to be an Indian invention. Finally, the "cyclic" arrangement utilizes the Babylonian goal-year periods (that of Venus is changed from 8 to 227 years). The planet's true longitude for the beginning of each *avadhi* in each year of its goal-year period is entered in the table; after one period the cycle is repeated. The goal-year periods themselves were introduced into Indian astronomy from Islamic in the early seventeenth century;¹⁷³ the "cyclic" tables produced in that century and the succeeding two in West and Northwest India combined these goal-year periods with the native tradition of true linear tables.

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¹⁷³ D. PINGREE in JHAS 2 (1978), 319—320.

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The planetary tables that have been studied heretofore are as follows.¹⁷⁴

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¹⁷⁴ The texts that accompany them are edited by D. PINGREE in a collection entitled Minor Astronomical Texts in Sanskrit, shortly to be published in the GOS.

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1. The Grahajñāna¹⁷⁵ by Āśādhara,¹⁷⁶ belonging to the Brāhmapakṣa. Its epoch is 20 March 1132; it is true linear. Apparently written in Gujarāt. Expanding the instructions for using these tables is the Gaṇitacūḍāmaṇi composed by Harihara in Gujarāt in about 1580.

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¹⁷⁵ SATE 69–72.

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¹⁷⁶ CESS A 1, 54b; A 2, 16a; A 3, 16a; and A 4.

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2. The Laghukhecarasiddhi¹⁷⁷ of Śrīdhara, also belonging to the Brāhmapakṣa. Its epoch is 20 March 1227; it is mean linear. Probably written in Khāndeśa.

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¹⁷⁷ SATE 73–76; and D. PINGREE, Śrīdhara's Laghukhecarasiddhi, Baroda 1976.

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3. The Mahādevī¹⁷⁸ of Mahādeva,¹⁷⁹ again belonging to the Brāhmapakṣa. Its epoch is 28 March 1316; it is true linear. Written on the banks of the Godāvarī, most probably in Mahārāṣṭra.

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¹⁷⁸ SATIUS 37a–39a; SATE 82; and O. NEUGEBAUER and D. PINGREE, "The Astronomical Tables of Mahādeva," PAPhS 111 (1967), 69–92.

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¹⁷⁹ CESS A 4.

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4. The Makaranda¹⁸⁰ of Makaranda,¹⁸¹ belonging to the Saurapakṣa and including tithi, nakṣatra, yoga, and eclipse as well as planetary tables. Its epoch is 27 March 1478; it is mean linear, and the only set of mean motion tables that is entirely sexagesimal. Written at Kāśī.

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¹⁸⁰ SATIUS 39b–46b, and SATE 92.

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¹⁸¹ CESS A 4.

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5. The Kheṭamuktāvalī¹⁸² of Nṛsiṃha,¹⁸³ belonging to the Gaṇeśapakṣa. Its epoch is 31 March 1566; it is true linear. Written at Nandigrāma in Gujarāt.

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¹⁸² D. PINGREE, "The Kheṭamuktāvalī of Nṛsiṃha," Sanskrit and Indian Studies, Essays in Honour of D. H. H. Ingalls, Dordrecht 1980, pp. 143–157.

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¹⁸³ CESS A 3, 202b–204a, and A 4.

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6. The Candrārkī¹⁸⁴ and the Kheṭasiddhi¹⁸⁵ of Dinakara,¹⁸⁶ belonging to the Brāhmapakṣa. The former deals with the Sun and Moon, the latter with the other planets. Their epoch is 31 March 1578; they are true linear. Written at Bārejya in Gujarāt.

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¹⁸⁴ SATIUS 51b–53a, and SATE 101.

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¹⁸⁵ SATE 101–112.

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¹⁸⁶ CESS A 3, 102b–104b, and A 4.

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7. The Grahakaumudī¹⁸⁷ of Nṛsiṃha, the author of the Kheṭamuktavalī to whose Gaṇeśapakṣa the Grahakaumudī also adheres. Its epochs are 31 March 1588 and 31 March 1603; it is also true linear. Written at Nandigrāma in Gujarāt.

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¹⁸⁷ SATE 118–123 and D. PINGREE in Sanskrit and Indian Studies, pp. 149–156.

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8. The Rāmavinoda¹⁸⁸ of Rāma, who wrote the Saura karaṇa of the same name. Its epoch is 11 March 1590; it is true linear. Written at the Mughal court.

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¹⁸⁸ SATE 114–118.

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9. The Ravisiddhāntamañjarī¹⁸⁹ of Mathurānātha,¹⁹⁰ belonging to the Saurapakṣa. Its epoch is 29 March 1609; it is mean linear. It includes parallax-tables for use in computing solar eclipses. Probably composed in Bengal.

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¹⁸⁹ Edited by VIŚVAMBHARA JYOTIṢṆARṆAVA, BI 198, Calcutta 1911: see SATE 128–134.

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¹⁹⁰ CESS A 4.

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10. The Grahaprabodha¹⁹¹ of Nāgeśa,¹⁹² which follows the Gaṇeśapakṣa. Its epoch is 5 March 1619, the beginning of dhruvāṅka 9; it is mean linear. Written in Gujarāt. There is an udāharaṇa by Yādava, in which the example is for 9 December 1663.

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¹⁹¹ SATIUS 63b–64b.

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¹⁹² CESS A 3, 145b–146b, and A 4.

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11. The Khecaraśīghrasiddhi or Grahasāraṇī¹⁹³ of Gaṅgādhara,¹⁹⁴ also belonging to the Gaṇeśapakṣa. Its epoch is 1 March 1630, the beginning of dhruvāṅka 10; it is mean linear. It includes tables for computing eclipses. Written at Kāśī.

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¹⁹³ SATE 134–141.

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¹⁹⁴ CESS A 2, 82b–85a, and A 4.

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12. The Grahavidyādhara¹⁹⁵ of Vidyādhara. The tables themselves are not preserved, but Vidyādhara states that they belonged to the Brāhmapakṣa. Its epoch is about 29 March 1638; it is mean linear. Written at RāJakoṭa in Saurāṣṭra.

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¹⁹⁵ I have used IO 2083c.

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13. The Jagadbhūṣaṇa¹⁹⁶ of Haridatta, belonging to the Brāhmapakṣa. Its epoch is 31 March 1638; it is the first set of cyclic planetary tables. Written in Mewar, Rājasthān.

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¹⁹⁶ SATIUS 55b–59b, and SATE 141–142.

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14. The Khecaradīpikā¹⁹⁷ of Kalyāṇa,¹⁹⁸ also following the Brāhmapakṣa. Its epoch is 31 March 1649; it is true linear, being an adaptation of the Mahādevī. The fact that two of its verses are derived from the Candrārkī of Dinakara support the impression that it was composed in Gujarāt or Rājasthān.

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¹⁹⁷ SATIUS 61b–62b.

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¹⁹⁸ CESS A 2, 25b.

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15. The Grahaprakāśa¹⁹⁹ of Devadatta,²⁰⁰ belonging to the Adjusted Saurapakṣa. Its epoch is 28 March 1662; it is mean linear. As his nephew Murāri traveled to Kāśī in order to bathe at Maṇikarṇikā, Devadatta probably did not live in that city; but nothing positive is known concerning his locale.

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¹⁹⁹ SATE 142–149.

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²⁰⁰ CESS A 3, 118b–119a.

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16. The Kheṭataraṅgiṇī²⁰¹ of Āpadeva,²⁰² belonging to the Gaṇeśapakṣa. Its epoch is 18 March 1702; it is mean linear. Written at Janasthāna in Mahārāṣṭra.

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²⁰¹ I have used Baroda 3095.

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²⁰² CESS A 1, 49b–50a; A 3, 15b; and A 4.

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17. The Grahasiddhi and Bhramaṇasāraṇī of Trivikrama,²⁰³ the tables of which are probably those of the Anonymous of 1704,²⁰⁴ which belong to the Brāhmapakṣa. Its epoch is 1 April 1704; it is cyclic. Composed at Nalinapura, presumably a locality in Rājasthān.

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²⁰³ CESS A 3, 92b–93b, and A 4.

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²⁰⁴ SATIUS 64b–66a.

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18. The Gaṇitarāja²⁰⁵ of Kevalarāma Pañcānana²⁰⁶ contains eclipse tables and tithi, nakṣatra, and yoga tables as well as planetary tables; it follows the Adjusted Saurapakṣa. Its epoch is 30 March 1728; it is mean linear. The equations are normed so as to be always positive; this feature was probably borrowed from Islamic tables. Written in Navadvīpa, Bengal.

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²⁰⁵ SATE 158–168.

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²⁰⁶ CESS A 2, 63a, and A 4.

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19. The Grahāgama²⁰⁷ of Govindasūnu,²⁰⁸ belonging to the Gaṇeśapakṣa. Its epoch is 6 February 1773 Julian, which is the beginning of dhruvāṅka 23; it is mean linear. Composed at Śipośi, which is presumably a locality in Rājasthān.

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²⁰⁷ SATE 168–169.

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²⁰⁸ CESS A 2, 143b, and A 4.

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A second category of koṣṭhakas that were popular in India provides tables for computing the beginnings (and ends) of tithis (periods during which the elongation of the Moon from the Sun increases by 12°), nakṣatras (periods during which the longitude of the Moon increases by 13; 20°), and yogas (periods during which the combined motion of the Sun and the Moon equals 13; 20°). The computations of the true instances of the beginnings of tithis and yogas is made complex by the fact that both of the luminaries have varying velocities, the Sun's dependent on the distance of its mean longitude from a more or less fixed sidereal point (its mandocca) and the Moon's on the distance of its mean longitude from a constantly moving point (its mandocca). The solutions adopted by various Indian koṣṭhakakāras are not yet understood in their totality by the present writer. The tithis, nakṣatras, and yogas themselves needed to be tabulated for the convenience of astrologers computing muhūrtas and of the constructors of pañcāṅgas which informed the populace of the details of the calendar and the proper times for festivals and the performance of saṃskāras.

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Several of the planetary tables contain also tables of tithis, nakṣatras, and yogas; these are the Makaranda of Makaranda, the Candrārkī of Dinakara, and the Gaṇitarāja of Kevalarāma Pañcānana discussed above. Independent koṣṭhakas on this subject include the following:

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1. The Śīghrasiddhi²⁰⁹ of Lakṣmīdhara, which tabulates functions according to both the Āryapakṣa and the Brāhmapakṣa. Its epoch is 1278, and it was probably composed in the territory of the Yādavas of Devagiri. The Śīghrasiddhi was expanded in 1618–1628 by Janārdana²¹⁰ at Saṅgamanera in Mahārāṣṭra; it was also used by Vidyādhara in 1643 at Rājakota in Saurāṣṭra.

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²⁰⁹ SATE 76–82.

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²¹⁰ CESS A 3, 59a, and A 4.

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2. The Tithikāmadhenu²¹¹ of Mahādeva,²¹² following the Āryapakṣa. Its epoch is 1357, and it was composed at Tryambaka in Mahārāṣṭra.

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²¹¹ SATE 82–89.

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²¹² CESS A 4.

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3. The Daivajñavallabha²¹³ of Sumiśra, belonging to the Ārdharātrikapakṣa (the Khaṇḍakhādyaka). Its epoch is 1447, and it was written in Nepāla.

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²¹³ I have used IO 3283e, the apparently unique manuscript.

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4. The Tithicintāmaṇi²¹⁴ of Gaṇeśa, the author of the Grahalāghava; naturally it follows his Gaṇeśapakṣa. Its epoch is 1525, and it was written at Nandigrāma in Gujarāt. There are commentaries by Nṛsiṃha, who wrote the Saurabhāṣya on the SS in 1611 and the Vāsanāvārttika on the SŚB in 1621, both in Kāśī; by Viśvanātha, who wrote in Kāśī in 1634; by Vyeṅkaṭa alias Bāpū Caṇḍika, who also wrote in Kāśī; and by Yajñeśvara.

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²¹⁴ SATIUS 47b—50b and SATE 100—101. The Tithicintāmaṇi was edited with his own Hindi ṭīkā, Vijayalakṣmī, by MĀTRPRASĀDA PĀṆḌEYA, HSS 76, Benares 1938; and with the udāharaṇa of Viśvanātha by DATTĀTREYA ĀPAṬE, ASS 120, Poona 1942, pt. 1.

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5. The Bṛhattithicintāmaṇi,²¹⁵ also by Gaṇeśa and belonging to the Gaṇeśapakṣa. Its epoch is 1552, and it also was composed at Nandigrāma in Gujarāt. There is a commentary, Subodhinī, composed by Viṣṇu, the uncle of the Nṛsiṃha who commented on the Tithicintāmaṇi, in about 1575.

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²¹⁵ SATIUS 50b—51a and SATE 101. The Bṛhattithicintāmaṇi with the ṭīkā, Subodhinī, of Viṣṇu, but without the tables, was edited by DATTĀTREYA ĀPAṬE, ASS 120, Poona 1942, pt. 2.

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6. The Tithisāraṇi²¹⁶ of Dinakara, the author of the Candrārkī and the Kheṭasiddhi; this work belongs to the Brāhmapakṣa. Its epoch is 31 March 1583; and Dinakara wrote it at Bārejya in Gujarāt.

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²¹⁶ SATE 112—114.

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7. The Tithyādicintāmaṇi²¹⁷ of Dinakara.²¹⁸ Its epoch is 1586, and it was composed at Unnatadurga (Junāgaḍh in Saurāṣṭra).

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²¹⁷ SATIUS 51a—51b.

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²¹⁸ CESS A 3, 104b—105a.

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8. The Tithikalpadruma²¹⁹ of Kalyāṇa,²²⁰ also following the Brāhmapakṣa. Its epoch is 31 March 1605, and it was composed at Maṅgalapura in Saurāṣṭra.

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²¹⁹ SATE 123—128.

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²²⁰ CESS A 2, 24b—25a, and A 4.

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9. The Camatkārasiddhi²²¹ of Virasiṃha,²²² one of the astronomers working for Anūpasimha,²²³ the Mahārāja of Bikāner from 1674 to 1698. Its epoch is 1627.

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²²¹ I have used Oxford CS d. 791 (9).

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²²² See K. M. K. SARMA, “Virasiṃhagaṇaka,” Brahmavidyā 9 (1945), 7—12.

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²²³ CESS A 1, 43b—44a.

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10. The Pañcāṅgavidyādhari²²⁴ by Vidyādhara, the author of the Grahavidyādhara; like Lakṣmīdhara’s Śīghrasiddhi, much of which it copies, it tabulates functions according to the parameters of both the Āryapakṣa and the Brāhmapakṣa. Its epoch is 1643; it was composed for Vīrabhadra, the rājā of Rājakota in Saurāṣṭra.

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²²⁴ SATIUS 60b—61b and SATE 142.

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11. The Candrārkī²²⁵ of Acalajit,²²⁶ which transforms the tables of Dinakara from the Brāhmapakṣa to the Saurapakṣa. Its epoch is 1655; Acalajit wrote at Muraripupura in Gujarāt.

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²²⁵ I have used Baroda 3120 and 3121.

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²²⁶ CESS A 4.

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12. and 13. The Tithidarpaṇa²²⁷ and Laghutithidarpaṇa²²⁸ of Murāri.²²⁹ The former cannot be dated, but the latter, based on the Makaranda of Makaranda, a text belonging to the Saurapakṣa, has as its epoch 1665, and was composed in Kāśī.

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²²⁷ SATE 149–150.

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²²⁸ SATE 151–153.

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²²⁹ CESS A 4.

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14. The Jayavinodasāraṇī²³⁰ ascribed to Savāī Jayasiṃha, the Mahārāja of Jayapura who sponsored Jagannātha’s Siddhāntasamrāṭ in 1732. Its epoch is 1735.

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²³⁰ SATIUS 66b–67a.

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15. The Pattraprakāśa²³¹ of Viśrāmaśukla, belonging to the Adjusted Saurapakṣa. Its epoch is 1750, but it seems to have been composed in 1777, in Kāśī. It appends true-linear planetary tables to those for tithis, nakṣatras, and yogas.

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²³¹ SATE 170–175.

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A final class of astronomical tables includes those designed for use in computing lunar and solar eclipses. Such tables are embedded in the Grahasāraṇī of Gaṅgādhara and the Gaṇitarāja of Kevalarāma, which have already been described. The only set of tables devoted exclusively to the subject of eclipses would seem to be the Karaṇakesarī²³² of Bhāskara.²³³ Its epoch is 1681, and it was composed at Saudamika, evidently a locality in Gujarāt. However, specialized treatises on eclipse computations—e.g., the Parvadvayasādhana of Mallāri—frequently include versified tables; such texts are described more fully below.

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²³² SATIUS 70b–72b; for the text I have referred to Baroda 11268.

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²³³ CESS A 4.

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The koṣṭhakas described in the preceding pages come mostly from western India with a few representatives of a tradition in Kāśī and Bengal. This bias reflects a real situation in the history of Indian astronomy to which we have previously referred—that is, the influence of the Islamic zīj on the Indian koṣṭhaka. However, other parts of India were not without such tables, though those known to me tend to be late.²³⁴ Moreover, the next section of this chapter will describe texts containing versified tables used in South India in imitation of those found in siddhāntas and karaṇas. It was generally in this form that astronomical functions were tabulated in those parts of India not subject to a direct and profound Islamic influence.

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²³⁴ E.g., SATE 178–182, and CESS A 1, 38b.

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