[Pramāṇa](https://indike.org/pramana/) › [Studies](https://indike.org/pramana/ins) › [Science, technology and education](https://indike.org/pramana/ins.science) › [A History of the Kerala School of Hindu Astronomy (in Perspective)](https://indike.org/pramana/ins.science.sarma-1972) › c8.s7 # 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 - **Language:** English ### 11. Some other highlights ### [c8.s7.1](https://indike.org/pramana/ins.science.sarma-1972.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](https://indike.org/pramana/ins.science.sarma-1972.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](https://indike.org/pramana/ins.science.sarma-1972.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](https://indike.org/pramana/ins.science.sarma-1972.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](https://indike.org/pramana/ins.science.sarma-1972.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. ## Linked texts (3) - [Āryabhaṭīya](https://indike.org/pramana/in.astro.siddhanta.early.aryabhatiya) — Āryabhaṭa, Sanskrit - [Govindakṛti](https://indike.org/pramana/in.astro.kerala.tantra.govindakrti) — Govindasvāmin (astronomer), Sanskrit - [Laghubhāskarīya](https://indike.org/pramana/in.astro.siddhanta.early.laghubhaskariya) — Bhāskara I, Sanskrit --- Source: [https://indike.org/pramana/ins.science.sarma-1972.c8.s7](https://indike.org/pramana/ins.science.sarma-1972.c8.s7) · Pramāṇa