Summary
Ancient and medieval India produced a host of eminent scholars who contributed richly to the development of astronomy and mathematics. For about four centuries since the fourteenth century c.e. there was an upsurge in Indian mathematical activities centred at Kerala. The achievements of this period have quite a different flavor in the sense that the barriers of the finite were broken and the highly fertile area of the infinite explored. The discipline attained new heights with the discovery and analysis of the infinite series for the circumference and the power series for half chords. Attempts to increase accuracy and expedite convergence by transforming partial sums using correction terms opened the doors of series approximation and error analysis. En route, heavy knockings on the doors of calculus and infinitesimal analysis can be heard. This article outlines some of these major discoveries and attempts to take a peep into the methodology and motivations. The emphasis laid by Indian mathematicians in general, and Kerala mathematicians in particular, on the notion of proof, the different types of proofs used, the purpose of rationale, the nature, and style of exposition are examined briefly. Commentaries and other expository works are capable of throwing light on the methodology and motivations in Indian mathematics.
V. Madhukar Mallayya’s field of research is the history of mathematics.
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Notes
- 1.
Sastry, Parameśvara: Transcript copy C.1024.E of the manuscript Golasāra Siddhāntadarpaṇamca of Gārgya Kerala Nīlakaṇṭha Mss No. T 846. B, (KUORI and Mss Library, Thiruvananthapuram), iii vs.2. Also ref. K. V. Sarma (ed.): Golasāra, Cr. Ed. with English Translation, VVRI, Hoshiarpur, (p. 14) 1970.
- 2.
Sastri, T. S. Kuppanna (ed.): Mahābhāskarīya of Bhāskarācārya with the Bhāṣya of Govindasvamin and the super commentary Siddhāntadīpika of Parameśvara, Madras, 1957 (iv.22, pp. 201–202); For details ref. R. C. Gupta: Second Order Interpolation in Indian Mathematics up to the Fifteenth Century, IJHS, Vol. 4, (pp. 86–98) 1969.
- 3.
Sastri, T. S. Kuppanna (ed.): Mahābhāskarīya of Bhāskarācārya with the Bhāṣya of Govindasvāmin and the super commentary Siddhāntadīpika of Parameśvara, Madras, 1957 (iv. 22, p. 205, vs. 14–16); For details see R. C. Gupta: An Indian Form of Third Order Taylor Series Approximation for Sine, Historia Mathematica, Vol. 1, (pp. 287–289) 1974.
- 4.
Sastri, T. S. Kuppanna (ed.): Mahābhāskarīya of Bhāskarācārya with the Bhāṣya of Govindasvāmin and the super commentary Siddhāntadīpika of Parameśvara, Madras, 1957 (iv. 22, p. 205, vs. 18–19). For details see R. C. Gupta: A Mean Value Type Formula for Inverse Interpolation of Sine, Mathematics Education, Vol. 10, No. 1 (pp. 17–20) 1976.
- 5.
Sastry, K. Sambasiva: Āryabhaṭīyam with Bhāṣya of Nīlakaṇṭha, Part I, TSS No. 101, p. 55, Thiruvananthapuram (1930).
- 6.
Thampuran, Rama Varma Maru and Akileswara Aiyar (ed): Yuktībhāṣā, Part I, Mangalodayam, Trichur, 1947, (Ch. vii). Also see C. T. Rajagopal and A. Venkataraman: The Sine And Cosine Power Series in Hindu Mathematics, Journal of Royal Asiatic Society of Bengal – Kolkata, Vol. XV. (pp. 1–13) 1949.
- 7.
Golasāra (iii.vs. 6–14).
- 8.
Mallayya, V. Madhukar: An Interesting Algorithm for Computation of Sine Tables From the Golasāra of Nīlakaṇṭha, Gaṇitā Bhārati, Vol. 26, Nos. 1–4 (pp. 40–55) 2004. For details, ref. Mallayya, V.Madhukar: The Golasāra Concept of Jyā: A Study in Modern Perspective, SSVV, Thiruvananthapuram (Chaps.3 and 4), 2004.
- 9.
Sastry, K. Sambasiva: Āryabhaṭīyam with Bhāṣya of Nīlakaṇṭha, Part I, Thiruvananthapuram Sanskrit series No. 101, (p. 41) Thiruvananthapuram (1930).
- 10.
ibid(pp. 41–42).
- 11.
ibid(p. 42).
- 12.
Sarma, K. V.:Līlāvatī of Bhāskarācārya with Kriyākramakarī of Śaṅkara and Nārāyaṇa, V.V.R.I. Hoshiarpur (p. 377), 1975.
- 13.
Sarma, K. V.: Sadratnamala of Śaṅkara Varman, INSA, Delhi, iv. 2 (p. 26), 2001.
- 14.
Thampuran, Maru: Yūktibhāṣā, Part I, Ch. vi, (pp. 84–116).
- 15.
ibid(pp. 116–142).
- 16.
Sarma, K. V.: Kriyākramakarī (p. 385).
- 17.
ibid(pp. 387–391).
- 18.
Thampuran, Maru: Yuktībhāṣā, Part I Trichur (Ch. vi, pp. 120–142), 1947.
- 19.
Hayashi, T., Kusuba, T., and Yano, M.: The Correction of Madhava Series for the Circumference of a Circle, Centaurus, Vol. 33, pp.149–174 (1990).
- 20.
Mallayya, V. Madhukar: Śaṅkara’s Correction Functions for Series Approximations, Recent Trends in Mathematical Analysis, Allied Publishers, Delhi (pp. 176–189) 2003.
- 21.
Sarma, N. Anantakrishna: Transcript copy of the Ms. Parameśvarakṛta Līlāvatīvyākhyā, p. 89. Also see T. A. Sarasvati Amma: Geometry in Ancient and Medieval India, Delhi (pp. 108–109) 1991.
- 22.
Dvivedi, Padmakara (ed): The Gaṇita Kaumudi of Nārāyaṇa Paṇḍita, Part II, Benaras, 1942 (p. 58, vs. 48). For more details see K. V. Sarma: Kriyākramakarī, pp. 348–362 and Maru Thampuran: Yuktībhāṣā, Part I, Ch. vii (pp. 228–237).
- 23.
Vasista, Viharilal: Bījagaṇīta, with commentary, Bījāṅkura of Kṛṣṇa Daivajña, Jammu (p. 16, vs. 13), 1977.
- 24.
Pillai, Suranad Kunjan: Āryabhaṭīya with Bhāṣya of Nīlakaṇṭha, Part III, Golapādā, Trivandrum Sanskrit series No. 185, Thiruvananthapuram (vs. 48), 1957.
- 25.
Amma, T. A. Sarasvati: Geometry in Ancient and Medieval India (p. 3).
- 26.
Apte, V. G.: Līlāvatī with Buddhīvilāsinī of Gaṇeśa Daivajña and LīlāvatīVivaraṇa of Mahidhara, Anandasramom series. No. 107, Poona (Part I)(p.1,vs.4) 1937.
- 27.
Sarma, K. V. and Sastry, V. Kutumba: Science Texts in Sanskrit in the Manuscripts Repositories of Kerala and Tamil Nadu, Rashtriya Sanskrit Sansthan, Delhi (2002).
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Mallayya, V.M. (2009). The Indian Mathematical Tradition with Special Reference to Kerala: Methodology and Motivation. In: Yadav, B., Mohan, M. (eds) Ancient Indian Leaps into Mathematics. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-4695-0_10
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