[Pramāṇa](https://indike.org/pramana/) › [Studies](https://indike.org/pramana/ins) › [Science, technology and education](https://indike.org/pramana/ins.science) › [Jyotihshastra: Astral and Mathematical Literature](https://indike.org/pramana/ins.science.pingree-1981) › c4.s5 # Jyotihshastra: Astral and Mathematical Literature c4.s5: South Indian Astronomers Jyotiḥśāstra: Astral and Mathematical Literature c4.s5 - **Language:** English ### South Indian Astronomers ### [c4.s5.1](https://indike.org/pramana/ins.science.pingree-1981.c4.s5.1) ²³⁵ The most recent general treatment of South Indian astronomers, astrologers, and mathematicians is K. V. SARMA, A History of the Kerala School of Hindu Astronomy, VIS 55, Hoshiarpur 1972. ### [c4.s5.2](https://indike.org/pramana/ins.science.pingree-1981.c4.s5.2) The Dravidian speaking areas of the Indian peninsula—Āndhrapradeśa, Karṇāṭaka, Tamilnadu, and Kerala—developed traditions of astronomy (and mathematics) based on the Ārya and Saura pakṣas, but largely independent of the traditions in the rest of India; this insularity was particularly true of Kerala, where some of the most brilliant work ever to be done in India in mathematics was produced in the fourteenth and fifteenth centuries without having any effect whatsoever on the north. In the south in particular developed three interrelated systems: the Parahita and Vākya following the Āryapakṣa, and the Dṛggaṇita following the Saurapakṣa. Further, a genre of literature called tantra or karaṇa evolved in which the author could deal with selected topics in astronomy rather than giving a complete system as a siddhānta or northern karaṇa would. Alongside these texts, which are collections of problems, there were written short essays dealing with single facets of astronomy. To some extent, as we shall see, such essays were also composed in the north, but never in the numbers in which they were written in Kerala. ### [c4.s5.3](https://indike.org/pramana/ins.science.pingree-1981.c4.s5.3) The Parahita system was first embodied in the Grahacāranibandhana²³⁶ of Haridatta. Traditionally the date of the inception of the system was 683; since Haridatta is already quoted as an authority of Govindasvāmin in his Prakaṭārthadīpikā, written in the first half of the ninth century, 683 is not an unreasonable date for Haridatta. The GCN versifies the mean daily motions of the planets according to the Āryapakṣa, the sines of their equations at intervals in the argument of 3;45° (he calls these tables vākyas), the śīghra anomalies necessary for first and last stations, and the mean longitudes of the planets on 20 March 355; all of the numbers are expressed by the kaṭapayādi system. In another work—probably the lost Mahāmārganibandhana—Haridatta gave bīja corrections to the mean yearly motions of the planets, to be applied to the years following 522. ### [c4.s5.4](https://indike.org/pramana/ins.science.pingree-1981.c4.s5.4) ²³⁶ Edited by K. V. SARMA, Madras 1954. ### [c4.s5.5](https://indike.org/pramana/ins.science.pingree-1981.c4.s5.5) Following the Parahita system, and thereby the Āryapakṣa, are the vākyas. These are based on the synodic periods of the five planets and cycles based on those periods, and, in the case of the Moon, on cycles of its anomalistic period; within each synodic period true longitudes at regular time intervals are given, and in the smallest lunar cycle daily longitudes of the Moon are given. ### [c4.s5.6](https://indike.org/pramana/ins.science.pingree-1981.c4.s5.6) The earliest attested epoch of the lunar vākyas is 1184. The vākyas themselves, called Candravākyāni²³⁷ and attributed to Vararuci, use the relation: 9 anomalistic months = 248 days. The larger cycles of the system are: 110 anomalistic months = 3031 days, and 449 anomalistic months = 12,372 days. The two smaller cycles had already been given by the Vasiṣṭhasiddhānta known to Varāhamihira (PS 3, 8–9); the first is ultimately Babylonian. Several Sanskrit poems composed in Kerala contain, embedded in their verses by means of the kaṭapayādi system, Vararuci’s lunar vākyas; the most notable of these is the Haricarita²³⁸ composed by Parameśvara²³⁹ in the fifteenth century, in which the life of Kṛṣṇa provides the surface meaning for the versified astronomical tables. These vākyas were still in use in south India in modern times. They were described by John Warren in 1825²⁴⁰ as part of a discussion of the pañcāṅga-computations; in this form, and misnamed “Tamil,” they generated some interest among non-Indian historians of science in the 1950’s and 1960’s.²⁴¹ ### [c4.s5.7](https://indike.org/pramana/ins.science.pingree-1981.c4.s5.7) ²³⁷ Edited by C. KUNHAN RAJA, Madras 1948. ### [c4.s5.8](https://indike.org/pramana/ins.science.pingree-1981.c4.s5.8) ²³⁸ Edited by V. KRISHNAMACHARYA, ALS 63, Madras 1948. ### [c4.s5.9](https://indike.org/pramana/ins.science.pingree-1981.c4.s5.9) ²³⁹ CESS A 4. ### [c4.s5.10](https://indike.org/pramana/ins.science.pingree-1981.c4.s5.10) ²⁴⁰ J. WARREN, Kala Sankalita, Madras 1825, pp. 118–146. ### [c4.s5.11](https://indike.org/pramana/ins.science.pingree-1981.c4.s5.11) ²⁴¹ O. NEUGEBAUER, “Tamil Astronomy,” Osiris 10 (1952), 252–276; I. V. M. KRISHNA RAV, “The Motion of the Moon in Tamil Astronomy,” Centaurus 4 (1956), 198–220; B. L. VAN DER WAERDEN, “Tamil Astronomy,” Centaurus 4 (1956), 221–234; and G. J. TOOMER, “A Note on Tamil Astronomical Tables,” Centaurus 9 (1963–1964), 11–15, and “A Further Note on Tamil Astronomical Tables,” Centaurus 9 (1963–1964), 254–256. ### [c4.s5.12](https://indike.org/pramana/ins.science.pingree-1981.c4.s5.12) The planetary vākyas are given in an anonymous text entitled Vākyakaraṇa,²⁴² in which the epoch of the lunar vākyas is 22 May 1282; the text seems to have been composed in about 1300, probably near Kāñcī. There is a commentary composed by Sundararāja, who was a native of Viprasadgrāma (Aṇḍaṇa-nal-lur near Trichinopoly), in the late fifteenth or early sixteenth century; Sundararāja studied under Padmagarbha during the reign of (Gopendra) Tipparāja,²⁴³ who ruled Koṇḍavīḍu in the Guntur District for the Vijayanagara king, Kṛṣṇadeva Rāya, from about 1515 till 1520, and he corresponded with the well-known astronomer, Gārgyakerala Nīlakaṇṭha Somayājin, whom we shall presently discuss in more detail. ### [c4.s5.13](https://indike.org/pramana/ins.science.pingree-1981.c4.s5.13) ²⁴² Edited with the commentary, Laghuprakāśikā, of Sundararāja by T. S. KUPPANNA SASTRI and K. V. SARMA, Madras 1962. ### [c4.s5.14](https://indike.org/pramana/ins.science.pingree-1981.c4.s5.14) ²⁴³ CESS A 2, 133 b. ### [c4.s5.15](https://indike.org/pramana/ins.science.pingree-1981.c4.s5.15) Before turning to the most important school of Kerala astronomers, I should mention the first text in a genre, the tantra, cultivated exclusively in south India. A tantra is like a karaṇa in its emphasis on pragmatic solutions to particular astronomical problems—sometimes only to those that were regarded as needing a special treatment. But since it follows the Āryapakṣa, in which there is a mean conjunction of all the planets at the beginning of the current Kaliyuga (6 A.M. at Laṅkā on 18 February—3101), a tantra uses that date as its epoch rather than a date closer in time to the author’s. ### [c4.s5.16](https://indike.org/pramana/ins.science.pingree-1981.c4.s5.16) The earliest such tantra is the Vārṣikatantra²⁴⁴ written by Viḍḍaṇa or Viddaṇa. His own name and that of his father, Mallaya, indicate a Dravidian origin, probably in Mysore whence most of the manuscripts of his work come. The Vārṣikatantra was written before 1370 since it is quoted by Ekanātha in his commentary on Bhāskara’s KK. It consists of eleven adhyāyas in a somewhat unusual order: dhruvamadhya (= madhyama), grahagatinirṇaya (= sphuṭa), chāyā (= tripraśna), candragrahaṇa, sūryagrahaṇa, valilekhana (= parilekha), mahāpāta, udayāstamaya, grahayuti, bhagrahayuti, and śṛṅgonnati. A manuscript was acquired in the Karṇāṭaka and brought to Bikāner in Rājasthān by the Mahārāja Anūpasimha²⁴⁵ in about 1675; there the work was revised by the court astronomer, Virasimha, in 1678, and a set of koṣṭhakas calculated for it. ### [c4.s5.17](https://indike.org/pramana/ins.science.pingree-1981.c4.s5.17) ²⁴⁴ I have used the manuscript Harvard 1113 (Poleman 4391), in which II 5–III 10 are missing. ### [c4.s5.18](https://indike.org/pramana/ins.science.pingree-1981.c4.s5.18) ²⁴⁵ CESS A 1, 43b–44a. ### [c4.s5.19](https://indike.org/pramana/ins.science.pingree-1981.c4.s5.19) But the most important school of astronomers from south India was founded by its most brilliant member, Mādhava²⁴⁶ of Saṅgamagrāma (near Cochin in Kerala), in the late fourteenth century. His most ingenious work was done in the investigation of the series that express the value of π and of other trigonometric functions used in astronomy; this achievement is discussed more fully in the chapter on mathematics. The surviving astronomical works of Mādhava are for the most part concerned with refinements of the vākya system. They include, on the lunar vākyas, the Sphuṭacandrāpti;²⁴⁷ its elaboration, the Veṇvāroha,²⁴⁸ whose epoch is 1403; and a new set of Candravākyāni²⁴⁹ giving daily longitudes of the Moon correct to the second sexagesimal place instead of to the first as in the Candravākyāni of Vararuci. The idea of the Sphuṭacandrāpti and the Veṇvāroha is to utilize the cyclic nature of the lunar vākyas, in which nine anomalistic months equal 248 days, to determine easily the lunar longitude at nine equally distant times in one day. A Malayalam commentary was written on the Veṇvāroha by Acyuta Piṣāraṭi²⁵⁰ in the early seventeenth century. On the computation of the longitudes of the planets Mādhava wrote a Madhyamānayanaprakāra and an Agaṇitagrahacāra, the latter apparently in 1418. He also composed a Lagnaprakaraṇa on computing the ascendant. None of these has yet been published. ### [c4.s5.20](https://indike.org/pramana/ins.science.pingree-1981.c4.s5.20) ²⁴⁶ CESS A 4. A colection of short texts produced by Mādhava’s school was edited by K. V. Sarma under the litle Gaṇitayuktayaḥ as PUIS 24, Hoshiarpur 1979. ### [c4.s5.21](https://indike.org/pramana/ins.science.pingree-1981.c4.s5.21) ²⁴⁷ Edited by K. V. SARMA, VIS 62, Hoshiarpur 1973. ### [c4.s5.22](https://indike.org/pramana/ins.science.pingree-1981.c4.s5.22) ²⁴⁸ Edited with the Malayalam ṭīkā of Acyuta Piṣāraṭi by K. V. SARMA, RSG 7, Tripunithura 1956. ### [c4.s5.23](https://indike.org/pramana/ins.science.pingree-1981.c4.s5.23) ²⁴⁹ Edited by K. V. SARMA as appendices to his editions of the Sphuṭacandrāpti and to the Veṇvāroha. ### [c4.s5.24](https://indike.org/pramana/ins.science.pingree-1981.c4.s5.24) ²⁵⁰ CESS A 1, 36b–38b; A 2, 11a; and A 4. ### [c4.s5.25](https://indike.org/pramana/ins.science.pingree-1981.c4.s5.25) Mādhava’s most distinguished pupil was that Parameśvara of Aśvatthagrāma in Kerala whom we have already encountered as a commentator on Āryabhaṭa’s A; on Bhāskara’s MB and LB (the latter in 1408); on the SS in 1432; on Govindasvāmin’s Mahābhāskariyabhāṣya in 1432; and on Muñjāla’s LM in 1409. He made a series of observations of eclipses of the Sun and the Moon between 1393 and 1432.²⁵¹ These should have influenced his Grahaṇamaṇḍana,²⁵² whose epoch is 15 July 1411; his Grahaṇanyāyadīpikā;²⁵³ and his Grahaṇaṣṭaka.²⁵⁴ Such an influence, however, is not clearly discernible. Parameśvara’s main importance in planetary astronomy in south India was through his dṛggaṇita system, expounded in the Dṛggaṇita²⁵⁵ of 1431. In this the mean motions (with slight modifications in the cases of the lunar mandocca and node) were taken from the Saurapakṣa, though the mean planets were assumed not all to be at Aries 0° at the beginning of the Kaliyuga; the longitudes of the mandoccas (except for Saturn’s) are derived from Muñjāla’s LM; and the dimensions of the manda and śīghra epicycles are new. Parameśvara also wrote two Goladīpikās in which he discusses general astronomical and geographical theories; the first²⁵⁶ of these was written in 1443 and is accompanied by a commentary composed by the author himself, the second²⁵⁷ was composed toward the end of his life, probably in the 1450’s. ### [c4.s5.26](https://indike.org/pramana/ins.science.pingree-1981.c4.s5.26) ²⁵¹ At the end of adhyāya 5 of his Siddhāntadīpikā; see D. PINGREE in JAOS 87 (1967), 337–339. ### [c4.s5.27](https://indike.org/pramana/ins.science.pingree-1981.c4.s5.27) ²⁵² The two recensions were edited by K. V. SARMA, VIS 34, Hoshiarpur 1965. ### [c4.s5.28](https://indike.org/pramana/ins.science.pingree-1981.c4.s5.28) ²⁵³ Edited by K. V. SARMA, VIS 35, Hoshiarpur 1966. ²⁵⁴ K. V. SARMA, “The Grahaṇaṣṭaka of Parameśvara. A Short Manual on Eclipses: Edition and Translation,” JOR Madras 28 (1958–59), 47–60. ²⁵⁵ Edited by K. V. SARMA, VIS 30, Hoshiarpur 1963. ²⁵⁶ Edited with Parameśvara’s own vivṛti by K. V. SARMA, ALPS 32, Madras 1957. ²⁵⁷ Edited by T. GANAPATI SĀSTRĪ, TSS 49, Trivandrum 1916. ²⁵⁸ Edited with the Laghuvṛtti of Śaṅkara by S. K. PILLAI, TSS 188, Trivandrum 1958; and with the Yuktidīpikā as well as Śaṅkara’s Laghuvṛtti by K. V. SARMA, Hoshiarpur 1977. ²⁵⁹ Edited by K. V. SARMA, VIS 47, Hoshiarpur 1970. ²⁶⁰ Edited by K. V. SARMA, ALPS 30, Madras 1956; and re-edited with Nīlakaṇṭha’s own ṭīkā by the same scholar, PUIS 7, Hoshiarpur 1976. ²⁶¹ Edited with Nīlakaṇṭha’s own ṭīkā by K. V. SARMA, PUIS 6, Hoshiarpur 1976. ²⁶² Edited by K. V. SARMA, PUIS 11, Hoshiarpur 1977. ### [c4.s5.29](https://indike.org/pramana/ins.science.pingree-1981.c4.s5.29) A student of Parameśvara’s son and student, Dāmodara, was Gārgyakerala Nīlakaṇṭha Somayājin, one of the commentators on Āryabhaṭa’s A. Nīlakaṇṭha was born on about 14 June 1444 at Kuṇḍapura near Tirur in Kerala; in the Āryabhaṭīyabhāṣya (on Kālakriyā 12–15) he refers to his observation of solar eclipses in 1467 and 1501. Nīlakaṇṭha made several efforts to establish new parameters for the mean motions of the planets (most of his parameters for the other elements of the planetary models are taken from the Āryapakṣa or the Saurapakṣa). The first set of new parameters is found in the Tantrasaṅgraha²⁵⁸ composed in 1501, and in abbreviated form in the Golasāra;²⁵⁹ the commentaries on the Tantrasaṅgraha—the Yuktidīpikā and the Laghuvṛtti, both written by Nīlakaṇṭha’s student, Śaṅkara, in about 1550—are important for the transmission of Mādhava’s work on power series. The second set is presented in the Siddhāntadarpaṇa,²⁶⁰ on which Nīlakaṇṭha composed his own commentary. Nīlakaṇṭha also wrote a small work, the Candracchāyāgaṇita,²⁶¹ on computing the Moon’s shadow; it is also accompanied by a commentary from the author’s own hand. But his most fascinating work, and one that is apparently unique in the history of Indian astronomy, is the Jyotirmīmāṃsā²⁶² that he wrote in 1504. In this treatise Nīlakaṇṭha vigorously defends the necessity constantly to correct astronomical parameters on the basis of observation, especially with regard to eclipses, but also with regard to the planets other than the Sun and the Moon; this defense is clearly aimed at the efforts in this direction made not only by Nīlakaṇṭha himself, but also by his paramaguru, Parameśvara. ### [c4.s5.30](https://indike.org/pramana/ins.science.pingree-1981.c4.s5.30) Another pupil of Parameśvara’s son, Dāmodara, was the famous author of the Malayālam Yuktibhāṣā, Jyeṣṭhadeva;²⁶³ and among his pupils was Acyuta Piṣāraṭi of Kuṇḍapura in Kerala, the author of the Malayālam commentary on Mādhava’s Veṇvāroha whom we have previously mentioned. Acyuta, who died on 7 July 1621 at the age of about seventy, was a prolific author on astronomy. On planetary theory, eclipses, and the pātas of the Sun and the Moon he wrote before 1596 a Karaṇottama with a vivaraṇa.²⁶⁴ His Sphaṭanirṇayatantra²⁶⁵ deals not only with the computation of the positions of the planets, but also with the constellations and with the celestial and terrestrial spheres; on this also Acyuta wrote his own commentary. On eclipses he composed in 1593 an Uparāgakriyākrama (a work of the same title was written by a contemporary Kerala astronomer, Nārāyaṇa,²⁶⁶) and an Uparāgaviṃśati,²⁶⁷ and on the Moon’s shadow a Chāyāṣṭaka,²⁶⁸ which attempts to compress the material expressed by Nīlakaṇṭha in thirty-two verses in the Candracchāyāgaṇita into the compass of a poem of a quarter of that length. But Acyuta’s most unusual work is the Rāśigolasphuṭānīti²⁶⁹ in which he expounds a method for reducing the Moon’s longitude in its orbit to an ecliptic longitude in accordance with an Islamic tradition initiated by Yaḥyā ibn Abī Manṣūr in the 820’s.²⁷⁰ ### [c4.s5.31](https://indike.org/pramana/ins.science.pingree-1981.c4.s5.31) ²⁶³ CESS A 3, 76b–77a, and A 4. ### [c4.s5.32](https://indike.org/pramana/ins.science.pingree-1981.c4.s5.32) ²⁶⁴ Both edited by K. R. PILLAI, TSS 213, Trivandrum 1964. ### [c4.s5.33](https://indike.org/pramana/ins.science.pingree-1981.c4.s5.33) ²⁶⁵ Edited with the author’s vivaraṇa by K. V. SARMA, PUIS 3, Hoshiarpur 1974. ### [c4.s5.34](https://indike.org/pramana/ins.science.pingree-1981.c4.s5.34) ²⁶⁶ CESS A 3, 150b–151a, and A 4. ### [c4.s5.35](https://indike.org/pramana/ins.science.pingree-1981.c4.s5.35) ²⁶⁷ Edited with a Malayālam commentary by RAMA VARMA MARU TAMPURĀN, RSG 5, Tripunithura \[1954\]. ### [c4.s5.36](https://indike.org/pramana/ins.science.pingree-1981.c4.s5.36) ²⁶⁸ Edited by K. V. SARMA in his edition of the Sphaṭanirṇayatantra, pp. 69–71. ### [c4.s5.37](https://indike.org/pramana/ins.science.pingree-1981.c4.s5.37) ²⁶⁹ Edited by K. V. SARMA, “The Rāśigolasphuṭānīti of Acyuta,” Brahmavidyā 18 (1954), 306–335; rev. ed. as PUIS 8, Hoshiarpur 1977. ### [c4.s5.38](https://indike.org/pramana/ins.science.pingree-1981.c4.s5.38) ²⁷⁰ D. PINGREE in JHAS 2 (1978), 319. ### [c4.s5.39](https://indike.org/pramana/ins.science.pingree-1981.c4.s5.39) More than a century after Acyuta’s death (if the interpretation of his ahargaṇa as ca. 22 March 1733 is correct) a Somayājin belonging to the Putumana family (Nūtanagṛha) of Śivapura in Kerala wrote a Karaṇapaddhati,²⁷¹ which discusses the basis for many astronomical and mathematical parameters and computations, including Mādhava’s power series and their developments. This Putumana wrote several other astronomical works also, of which the most important is the Nyāyaratna; none of these has been published. This tradition of astronomy in Kerala continued well into the nineteenth century, but the author of the Karaṇapaddhati is the last major figure. ### [c4.s5.40](https://indike.org/pramana/ins.science.pingree-1981.c4.s5.40) ²⁷¹ Edited by K. SAMBASIVA SASTRI, TSS 126, Trivandrum 1937; by P. K. KORU, Cherp 1953; and, with two Malayālam ṭīkās, by S. K. NAYAR, Madras GOS 98, Madras 1956. ## Linked texts (22) - [Tantrasaṅgraha](https://indike.org/pramana/in.astro.kerala.tantra.tantrasangraha) — Nīlakaṇṭha Somayājī 1500, Sanskrit - [Sūryasiddhānta](https://indike.org/pramana/in.astro.siddhanta.early.suryasiddhanta) — Maya, Sanskrit - [Āryabhaṭīya](https://indike.org/pramana/in.astro.siddhanta.early.aryabhatiya) — Āryabhaṭa, Sanskrit - [Goladīpikā](https://indike.org/pramana/in.astro.kerala.golayukti.goladipika) — Parameśvara of Vaṭasseri, Sanskrit - [Vāsiṣṭhasiddhānta](https://indike.org/pramana/in.astro.siddhanta.early.vasistha) — Sanskrit - [Siddhāntadarpaṇa](https://indike.org/pramana/in.astro.kerala.tantra.siddhantadarpana) — Nīlakaṇṭha Somayājī, Sanskrit - [Mahābhāskarīya](https://indike.org/pramana/in.astro.siddhanta.early.mahabhaskariya) — Bhāskara I, Sanskrit - [Yuktibhāṣā (Gaṇitayuktibhāṣā)](https://indike.org/pramana/in.astro.kerala.golayukti.yuktibhasa) — Jyeṣṭhadeva 1520, Malayalam - [Laghubhāskarīya](https://indike.org/pramana/in.astro.siddhanta.early.laghubhaskariya) — Bhāskara I, Sanskrit - [Vākyakaraṇa](https://indike.org/pramana/in.astro.kerala.sphuta.vakyakarana) — Sanskrit - [Sphuṭacandrāpti](https://indike.org/pramana/in.astro.kerala.sphuta.sphutacandrapti) — Mādhava of Saṅgamagrāma, Sanskrit - [Karaṇapaddhati](https://indike.org/pramana/in.astro.kerala.tantra.karanapaddhati) — Putumana Somayājī 1660, Sanskrit - [Haricarita](https://indike.org/pramana/in.astro.kerala.sphuta.haricarita) — Parameśvara, Sanskrit - [Jyotirmīmāṃsā](https://indike.org/pramana/in.astro.kerala.tantra.jyotirmimamsa) — Nīlakaṇṭha Somayājī 1500, Sanskrit - [Dṛggaṇita](https://indike.org/pramana/in.astro.kerala.tantra.drgganita) — Parameśvara of Vaṭasseri, Sanskrit - [Karaṇakutūhala](https://indike.org/pramana/in.astro.siddhanta.karana.karanakutuhala) — Bhāskara II, Sanskrit - [Grahaṇamaṇḍana](https://indike.org/pramana/in.astro.kerala.grahana.grahanamandana) — Parameśvara of Vaṭasseri, Sanskrit - [Karaṇottama](https://indike.org/pramana/in.astro.kerala.tantra.karanottama) — Acyuta Piṣāraṭi 1593, Sanskrit - [Golasāra](https://indike.org/pramana/in.astro.kerala.golayukti.golasara) — Nīlakaṇṭha Somayājī, Sanskrit - [Laghumānasa](https://indike.org/pramana/in.astro.siddhanta.karana.laghumanasa) — Muñjāla, Sanskrit - [Candracchāyāgaṇita](https://indike.org/pramana/in.astro.kerala.candracchayaganita) — Nīlakaṇṭha Somayājī, Sanskrit - [Candravākyāni (Vararuci-vākyas)](https://indike.org/pramana/in.astro.kerala.sphuta.candravakya) — Vararuci (Kerala), Sanskrit --- Source: [https://indike.org/pramana/ins.science.pingree-1981.c4.s5](https://indike.org/pramana/ins.science.pingree-1981.c4.s5) · Pramāṇa