A History of the Kerala School of Hindu Astronomy (in Perspective) c9.s5: 5. Veṇvāroha method for the Moon
A History of the Kerala School of Hindu Astronomy (in Perspective) c9.s5
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5. Veṇvāroha method for the Moon
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Of all heavenly bodies, the Moon has not only the fastest motion, but also the maximum changes in its velocity during the same day. Therefore, the computation of the Moon at some intermediate time during the day, if calculated on the basis of its true position and motion at sunrise or at sunset, as advocated in some *Pañcabodha* texts, would not give accurate results. An ingenious but facile method for the determination of the Moon at any time of the day with high accuracy has been evolved by the 14th century Kerala astronomer Mādhava of Saṅgamagrāma in his work called *Veṇvāroha*. He has also enunciated accurate moon-mnemonics, correct to the seconds, for being used in the *Veṇvāroha* method. His method seems to have caught the fancy also of later astronomers who have extended the method further to yield still more accurate results. At least four later works of this genre have been identified, they being *Candrasphuṭāpti, Veṇvārohakriya and Dṛg-Veṇvārohakriya, all anonymous, and Veṇvārohaṣṭaka of Puṭumana Somayāji.
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6. Computation of Eclipses
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A branch of astronomy in which intense investigations have been made by Kerala astronomers refers to the computation of eclipses. The phenomenon of eclipses seems to have been exploited fully towards verifying calculations, instituting corrections and devising new methodologies. A reference has already been made to Parameśvara of Dṛggaṇita fame who has recorded the details about a large number of eclipses which he had computed, observed and also suggested corrections to correlate the two. Apart from minor tracts, a number of texts on the computation of eclipses have come down. These include three works having the common title Uparāgakriyā-krama, the first by Nārāyaṇa with commentary, the second by Acyuta Piṣāraṭi, with two commentaries, and a third, which is anonymous; an Uparāgaviṃśati with commentary; Grahaṇamaṇḍana by Parameśvara; six anonymous works with the common title Grahaṇa-gaṇita; four works under the title Grahaṇaṣṭaka, one of which is by Parameśvara and the others anonymous; and the three works, Candra-graha-gaṇita, Candra-sūryoparāga-gaṇita and Sūrya-grahaṇa-gaṇita, all anonymous. Two works on eclipses which deserve special mention are the Grahaṇa-nyāya-dīpikā of Parameśvara, which discusses the rationale of eclipse-computation, and the Grahaṇa-granthādi by Nīlakaṇṭha Somayāji, which is an elaborate disquisition on the tradition and rationale of computing eclipses etc.
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7. Computation of the Shadow
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Computation of the positions of the Sun and the Moon from the Shadow (Chaya) caused by them and vice versa had practical utility in the social and religious life of the people of mediaeval times. Besides calculations made on the basis of the Śaṅku (gnomon), intrinsic calculations involving the Mahācchāyā (Great Shadow or Sine Zenith distance) and the Mahāśaṅku (Great gnomon or Sine Altitude) had been developed in this behalf. Some of the Pañcabodhas and other karaṇa texts mentioned above have sections devoted to Shadow computation. Besides these, several independent works devoted solely to this subject have also been produced. Among the works of this category may be mentioned five different works, all entitled Candracchāyā-gaṇita, one of these being by Parameśvara, another by Nilakaṇṭha Somayāji and the others anonymous; two of these works have also commentaries. Other works on the subject include the Chāyāgaṇita, Chāyāgraha-gaṇita, Chāyādigaṇita and two different works having the common title Sūryacchāyādi-gaṇita. Acyuta Piṣāraṭi, too, has written a succinct work on the subject, entitled Chāyāṣṭakam.
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8. Long-range almanacs
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In the matter of preparing annual almanacs, while the general trend in Kerala, as elsewhere, has been to prepare them for one year at a time, efforts had been made to prepare long-range almanacs covering long periods. It has been recorded that Mādhava of Saṅgamagrāma compiled an Agaṇita-pañcāṅga¹ which is, probably, the same as referred to in the Karaṇapaddhati (IV. 14, 16, 18) and preserved in the Kerala University Oriental Research Institute under the title Agaṇitagrahacāra. Śaṅkaran Nampūtiri of Mahiṣamaṅgalam is credited with a ‘Register of Muhūrtas’² for 1000 years. The Agaṇita of Nalleppaḷḷi Subrahmaṇya Śāstri enunciates methods for the facile computation of the planets for a 1000 years.³
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¹ This reference occurs in an astronomical document preserved in the Oriental Institute, Baroda, Cf. K. V. Sarma, ‘Direct lines of astronomical tradition in Kerala’, Pt. Charudeva Shastri Felicitation Volume, Delhi, 1972.
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² See Ulloor, Kerala Sāhitya Charitram, Vol. II, (Trivandrum, 1954), p. 272.
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³ Ref. to by Ulloor, op.cit., Vol. IV, (Trivandrum, 1957), p. 291.
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9. Astronomical rationale
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An extremely significant class of astronomical treatises from Kerala refers to disquisitions on the rationale of astronomical formulae and procedures. Short expositions of this nature occur in commentarial literature, especially in the commentaries of Parameśvara, Nilakaṇṭha Somayāji and Nārāyaṇa, author of Kriyākramakarī on the Līlāvatī. Single or groups of verses and short tracts pertaining to the exposition of specific points or procedures are often found inscribed on end-leaves of astronomical codices, without indication of source or author. These go mostly unnoticed or ignored by cataloguers of manuscripts. It would, indeed, be highly worthwhile to record, study and analyse these passages with a view to see if some of them contain anything important. In this connection, the discovery of some full-fledged treatises devoted to the exposition of astronomical theories and processes ought to be highly interesting to scholars and historians of mathematics. These texts, which, generally, have the words *yukti* or *nyāya* ‘rationale’ attached to their titles, explain, interpret and discuss the formulae used and procedures followed in the topics which the texts deal with. Among works of this nature may be mentioned the *Grahaṇa-nyāya-dīpikā* on the computation of eclipses and *Vākyakaraṇa* on the formulation of astronomical mnemonics, both by Parameśvara of Vaṭaśreṇi, the *Yukti-bhāṣā* and *Gaṇita-yukti-bhāṣā*, the former and, possibly the latter too, by Jyeṣṭhadeva, the *Nyāya-ratna* by Puṭumana Somayājī, the *Gaṇita-yuktayaḥ* by an anonymous ‘Bhāradvāja brāhman’ and the *Rāśigola-sphuṭānīti* on ‘Reduction to the ecliptic’ by Acyuta Piṣāraṭi. The *Sundararāja-praśnottara* by Nīlakaṇṭha Somayājī, in which he has explained certain astronomical theories and practices in answer to queries raised by a contemporary astronomer Sundararāja of the Tamil country, also belongs to this category of works. It may also be noted that works solely devoted to the demonstration of procedures call themselves *-kriyā* or *-kriyākrama*, e. g., *Saṅgrahasādhana-‘kriyā’* and *Uparāga-‘kriyākrama’*.
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10. Practical astronomy
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Astronomy being a practical science, progress in that discipline is incumbant no less on practical observation than on computational skill. Several works including the *Golaḍīpikā* I of Parameśvara and the *Bhadradīpa* of Iṭakramañceri Nampūtiri contain sections on the construction and use of the armillary sphere. The commentaries of Parameśvara and Nīlakaṇṭha Somayājī carry references to their lifelong astronomical observations. By far the most interesting treatise in this line is the *Grahaparīkṣākrama* of Nīlakaṇṭha Somayājī, in about two hundred verses. There is also a detailed Malayalam commentary on this work. An equally interesting, though possibly, very late, work is *Nazhikamaṇiyuṭe yukti*, in Malayalam, which is a detailed exposition of the working of the pendulum clock.
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11. Revision of astronomical constants
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As indicated earlier, the composers of *karaṇa* texts generally started their works with contemporary epochs. Sometimes they also started with corrections to the astronomical constants, and to the sine tables which they incorporated in their works. A work mainly devoted to the enunciation of corrected constants is the *Siddhāntadarpaṇa* of Nīlakaṇṭha Somayājī, wherein he has set out the number of revolutions of the planets, and their higher apses, ascending nodes, maximum latitudes, velocities etc., as investigated and verified by him. He has also written an elaborate commentary on this work. Some short tracts call themselves *Khaṇḍavum dhruvavum* and merely give the epoch and the corresponding zero-corrections as determined by their respective astronomer-authors. Then again, there exist a large number of manuscripts which form collections of sine tables and mnemonics and are catalogued in libraries under generic titles like *Jyāḥ, Jyotiṣa-vākyāni* etc. on account of their not being full-fledged and independent works. These collections are potential sources of additions and novelties to the well-known mnemonics and, so, deserve to be examined from this point of view.
Linked texts (1)
- Karaṇapaddhati — Putumana Somayājī 1660, Sanskrit