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A History of the Kerala School of Hindu Astronomy (in Perspective) c8.s7: 11. Some other highlights

A History of the Kerala School of Hindu Astronomy (in Perspective) c8.s7

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11. Some other highlights

c8.s7.1

In the context of the continuity of astronomical tradition and the diligence with which its traditional exponents cultivated its study, a reference might be made here to certain other significant methodologies evolved and discoveries made by Kerala astronomers and mathematicians. The introduction of the *Kaṭapayādi* notation and the expression of astronomical data through mnemonic tables and verses composed with the help of this facile notation has already been noticed. In his table of sine-differences enumerated in the verse *makhi-bhakhi* etc. (*Gītika* 10), *Āryabhaṭa* gives only their rough values, correct to the minute. It is interesting that Govindasvāmin (*c.* 800-50) developed their evaluation still more accurately, correct to the second order of sexagesimal fractions, viz., seconds and thirds, which he gives in his *Bhāṣya* on *Mahabhāskarīya*. He also gives a formula for the derivation of the value of the last sine-difference when the tabular interval is halved. Developing the topic further, he enunciates also a rule for computing, from a given last sine-difference, the other sine-differences lying between 60° and 90°.¹

c8.s7.2

¹ See *Mahabhāskarīya* with Govindasvāmin's *Bhāṣya*, *op. cit.*, ch 4, verse 22, pp. 201-2. For an exposition of these rules and their rationale, see R.C. Gupta, 'Fractional parts of Āryabhaṭa's sines and certain rules found in Govindasvāmin's *Bhāṣya* on the *Mahabhāskarīya*', Indian JI. Hist. of Sc. 6 (1971) 51-59.

c8.s7.3

The *Kuṭṭakāra-gaṇita*, which is a complex type of calculation involving the Rule of three, Indeterminate equations and Continued fractions, has been developed by Kerala mathematicians to a high degree. It enabled them to solve various problems which require the application of higher algebra according to modern working. *Śaṅkaranārāyaṇa* (*c.* 825-900) quotes in his commentary on the *Lahgubhāskarīya* (*op. cit.*, pp. 95-97) a short disquisition on the subject from *Govindakṛti*, a lost work of Govindasvāmin, while the later works like *Yuktibhāṣā* (16th cent.) give full accounts of the processes involved.¹ Nīlakaṇṭha Somayājī (1444-1545) has made significant contributions including logical proofs and demonstrations to geometrical theorems, a formula for the sum of an infinite, geometrically progressing convergent series, and the rationale for the formula for the sum of a geometrical progression. He has also offered geometrical demonstrations for the sum of an arithmetic progression, the sum of triangular numbers and the sums of squares and cubes of natural numbers.²

c8.s7.4

¹ See *Yuktibhāṣā*, ch. 5, op. cit., pp. 54-71. For the rationale of *Kuṭṭakāra* according to modern mathematical terms, see *ibid.*, Appendix, pp. xli-lxi.

c8.s7.5

² See for details, T.C. Saraswathi, 'The development of mathematical series in India', *Bul. of the National Sciences of India*, No. 21 (1962) 320-43.

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