A History of the Kerala School of Hindu Astronomy (in Perspective) c7.s7: 7. Basic features of Kerala astronomy

A History of the Kerala School of Hindu Astronomy (in Perspective) c7.s7

Language: English

7. Basic features of Kerala astronomy

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(a) *Adherence to the Āryabhaṭan system.* From at least the 7th century, if not earlier, Kerala had been the bastion of the Āryabhaṭan school of astronomy. While the view expressed in certain quarters that Āryabhaṭa hailed from Kerala² has yet to be substantiated, there is no doubt about the extensive popularity of this system in the land. The later Kerala schools are all based on the Āryabhaṭan system. Most of the known commentaries on the *Āryabhaṭīya* have been written by Kerala mathematicians. Well-nigh, every one of the scores of astronomical works produced in Kerala follow the *Āryabhaṭīya* basically. And, the efforts of Kerala mathematicians have generally been directed towards the revision, supplementation and correction of Āryabhaṭan astronomy and mathematics with a view to deriving more accurate results.

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² For the latest essay in this direction, see K Damodaran Nambiar, ‘Āryabhaṭan’, *Mathrubhumi* (Malayalam Weekly, Kozhikode, Kerala), Oct. 18, 1970, pp. 15-16, 41.

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(b) *Kaṭapayādi notation.* An extremely convenient method of expressing numbers through letters, known as the *Ka-ṭa-pa-yādi* notation, has been extensively used in Kerala from very early times. A legendary author named Vararuci is credited with this innovation and the authorship of the popular collection of *Candra-vākyas* or *Vararuci-vākyas* (‘Moon computation sentences’), composed in this notation. The *Grahcāranibandhana* of Haridatta (A.D. 683) also uses this notation extensively. In this notation, each of the four series of consonants beginning with *k, ṭ, p,* and *y* in the Sanskrit alphabet stands for the digits 1 to 9. In conjunct letters, the value only of the final consonants is to be taken into account. Vowels following the consonants have no value. *n* and *ñ* and the pure vowels stand for zero.

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The letter *l*, which is peculiar to Dravidian denotes 9.¹ The versatility of the notation rests in the fact that even long and intricate numbers can be expressed through apparently meaningful expressions and verse-bits, facilitating their easy verification and memorisation.² The numerous sine and other mnemonic tables which form a characteristic feature of the Kerala School of astronomy are couched in this notation.

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¹ The simple rule for this notation is given by the verse :

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² See *e.g.*, the chronogram 'ācāryavag abhedya' which is kali day 1434160, on which Śaṅkarācārya introduced certain reforms; and the Moon-sentence, 'gir naḥ śreyaḥ' denoting 0°-12'-3".

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(c) *Basic principles.* In their mathematical work, Kerala authors and, for that matter, Indian authors in general, based themselves on the fundamentals of arithmetic, algebra, geometry and trigonometry. They were aware also of properties of numbers, to some extent. Calculus and higher mathematics, as developed in the West during the last three hundred years, were not known, but strict application of geometry, trigonometry and algebra, coupled with clear analytical thought and intuition, have enabled Kerala astronomers to arrive at important results and enunciate theorems which were later rediscovered in the West, some after several centuries.³

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³ For a resume of the more important of these anticipations, see below, Ch. II.

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8. Parahita system of astronomy

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A significant event in the annals of Kerala astronomy was the promulgation, in A. D. 683, of the *Parahita* system by Haridatta (*c.* 650-700) through his *Grahacāranibandhana*⁴ and *Mahāmārga-*

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⁴ Cr. ed with Intro. by K.V Sarma, K.S. Research Inst., Madras, 1954. On the discovery and identification of this thitherto unknown basic text of the *Parahita* system see K. V. Sarma, 'Parahita-gaṇitattinte mūlagrantham', *Mathrubhumi* (Mal. Weekly, Kozhikode), Oct. 10. 1954.

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*na-ñāv acaś ca śūnyāni, saṅkhyāḥ ka-ṭa-pa-yādayaḥ |*
*miśre tūpāntahalsaṅkhyā, na ca cintyo halaḥ svaraḥ ||*

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*Sadratnamālā* of Śaṅkara Varman nibandhana.¹ The event is recorded in several later works. For instance, the *Sadratnamālā* of Śaṅkaravarman says :

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ācaryāryabhaṭapraṇītagaṇitam prāyaḥ sphuṭam tat kalau
‘gotrottuṅga’-(3623)-mitābdake vyabhicaran brahmādisiddhāntake |
... ... ...
dṛgvaiṣamyavaśāt ‘mahāsthala’-(3785)-mite kalyabdake niścitaḥ
saṃskāro vibudhair yataḥ parahitatvam teṣu vīneṣv ayam || (6.¹, 3)

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¹ This seems to be the full-fledged theoretical treatise of the *Parahita* system while the *Grahacāranibandhana* is only its working manual. Manuscripts of this work have not yet come to light; the work is known through its reference by Haridatta himself in the *Grahacāranibandhana*, 3. 44.

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‘The astronomical treatise composed in Kali 3623 (A.D. 522) by Āryabhaṭa gave well-nigh accurate results, whereas the *siddhāntas* ascribed to Brahmā etc. had (by that time) tended to be inaccurate. (When its results too tended to be inaccurate, as exemplified by their) not tallying with observation, the correction called *Parahita* was adopted by the wise in Kali 3785 (A.D. 683) for the planets other than the Sun’.²

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² For other references to this event and the identity of Haridatta and his works, see K.V. Sarma, *Grahacāranibandhana*, op. cit., Intro., pp. v-ix.

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Tradition says that the inauguration of the system took place on the occasion of the twelve-yearly *Māmāṅkam* (*Mahā-magha*) festival held at Tirunāvāy on the Malabar coast.³

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³ See ibid., pp. vi-vii; Ulloor S. Parameswara Aiyar, *Kerala Sahitya Charitram*, vol. I, (Trivandrum, 1953), p. 165.

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Haridatta based his system on the *Āryabhaṭīya*, but improved upon it in several ways. He made computation easier by specifying simple multipliers and divisors for the various calculations, including the derivation of the positions of the planets. His use of the *Kaṭapayādi* notation made his composition elegant. He introduced the unique system of enunciating graded tables of the Sines of the arcs of Anamoly and of Commutation of the different planets (*Manda-jyā* and *Śīghra-jyā*) at intervals of 3° 45’, to facilitate the computation of the true positions of the planets. His *Grahacāra-nibandhana* being only the computation manual of the system, deals, mainly, with the calculation of the Kali days elapsed, *tithi* and *nakṣatra* of any day, the mean and retrograde motion of the planets and their Mean and True positions. One of the corrections introduced by Haridatta to make Āryabhaṭan results more accurate, is called *Bhaṭa-saṃskāra* or *Śakābda-saṃskāra* (on account of its being calculated for the years beginning from the ‘*śakābda*’ of Āryabhaṭa, viz., 444), and is particularly significant. This correction specifies that for every completed year after Śaka 444, a correction in minutes (*kalā*) of −9/85, −65/134, −13/32, +45/235, +420/235, −47/235, −153/235 and +20/235 should be made to the Mean positions of the Moon, Moon’s apsis, Moon’s node, Mars, Mercury, Jupiter, Venus and Saturn, respectively, no correction being necessary in the case of the Sun.¹

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¹ 2. Ed. K. V. Sarma, as Appendix to the edn of *Grahacāranibandhana*, Madras, 1954.

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A large number of astronomical manuals based on the *Parahita* system, including a *Grahacāranibandhana-saṅgraha*,² have come to be composed during later times, both in Sanskrit and in Malayalam, each taking off from a convenient contemporary epoch (*khaṇḍa-dina*) for which the True positions of the planets would have been fixed after elaborate calculations, these positions being used as zero-corrections for computation according to these manuals. Some of these incorporate also further corrections.

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² 3. Cr. ed. by K V. Sarma, with Intro., V.V.R. Inst., Hoshiarpur, 1963.

Linked texts (1)

Āryabhaṭīya — Āryabhaṭa, Sanskrit