Summary
The author starts with the role of King Aśoka the Great in the expansion of Buddhism to the whole of Central and South Asia and with that, in addition, not only of the religion but also of Indian culture and civilization. Then he concentrates on China and describes how Indian Buddhist monks carried with them astronomy and mathematics to ancient China. He gives concrete examples to show how Indian and Chinese mathematicians worked on similar problems. Finally, he describes in detail the works of the great Chinese Astronomer, I-Hsing, to show India’s contribution to Chinese mathematics.
Reprinted from phGaṇita Bhāratī, Vol. 11: 38–49, 1989, with a change of title.
R. C. Gupta has been a professor of Mathematics at the Birla Institute of Technology, Mesra Ranchi, India, and the editor of Gaṇita Bhāratī, the Bulletin of the Indian Society for History of Mathematics. His areas of interest include the History of mathematics and mathematics education.
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Notes
- 1.
Bapat, P. V. (general editor): 2500Years of Buddhism. Publications Division, Delhi, p.53 (1964).
- 2.
Ibid., p.397.
- 3.
Ibid., pp.59 and 110.
- 4.
Mukherjee, P. K.: Indian Literature Abroad (China). Calcutta Oriental Press, Calcutta, p.1 (1928).
- 5.
Ibid., pp.2–3.
- 6.
Chou Hsiang-Kuang: The History of Chinese Culture. Central Book Depot, Allahabad, p.106 (1958).
- 7.
Bapat (ref. 2), p.239, and Mukherjee (ref. 5), pp.78–79.
- 8.
See K. S. Shukla, Āryabhaṭa (booklet), New Delhi, p.5 (1976).
- 9.
Bapat (ref. 2), pp.163–164.
- 10.
Yabuuti, K.: Indian and Arabian Astronomy in China. In: The Silver Jubilee volume of the Zinbun-Kagaku-Kenkyusyo, Kyoto, pp.585–603 (1954).
- 11.
Mukhopadhyay, S. K. (ed.): The Śārdūlakarāvadāna. Visvabharati, Santiniketan, pp.46–53 and p.104 (1954).
- 12.
Vaidya, P. L. (ed.): Lalitavistara. Darbhanga, p.103 (1958). The last number in the final count will be equal to \(1{0}^{7+9\times 46}\ =\ 1{0}^{421}.\)
- 13.
Abhidharmakośa edited by Dvārikadas Sastri, 2 Volumes, Varanasi, III, 45–60 (1981) (Vol. I, pp.506–518).
- 14.
Ibid., p.544.
- 15.
Bapat (ref. 2), p.115.
- 16.
Chin Keh-mu, “India and China: Scientific Exchange” in D. Chattopadhyaya (ed.): Studies in History of Science in India. Vol. II, pp.776–790, (1982) (Solar month \(30\frac{1} {2}\) days (year = 366 d.), \(P = 27\frac{21} {60}\) (cf. \(27\frac{21} {67}\)), and \(S = 29\frac{30} {62}\) (cf. \(29\frac{32} {62}\)).
- 17.
Mukherjee (ref. 6), p. 38.
- 18.
Needham, J.: Science and Civilization on China. Vol. III, Cambridge, UK, p.88 (1959).
- 19.
See Bapat (ref. 2), p.214; Mukherjee (ref. 5), p.34; and Needham (ref. 19), p.707, where the Chinese title of the second work appears as Li Shih A-Pi-Than Lun (Philosophical Treatise on the Preservation of the World).
- 20.
Needham, J. (ref. 19), p.716, and Chin Keh-mu (ref. 17), p.784.
- 21.
Gupta, R. C.: Indian Astronomy in China During Ancient Times. Vishveshvaranand Indological Journal, XIX, 266–276, p.270 (1981).
- 22.
Ibid., pp.271–273.
- 23.
The work has been fully translated with notes by Kiyori Yabuuti in his paper “Researches on the Chiu-Chih Li Indian Astronomy under the Thang Dynasty” Acta Asiatica, Vol. 36, pp.7–48 (1979).
- 24.
Waerden, B. L. van der:Geometry and Algebra in Ancient Civilization. Springer–Verlag, Berlin, p.53 (1983).
- 25.
Shukla, K. S. (ed.);Āryabhatīya with the commentary of Bhāskara I andSomeśvara, INSA, New Delhi, India, pp.99–100 (1976).
- 26.
Waerden, B. L. van der (ref. 25), pp.50–51.
- 27.
Swetz, Frank: The AmazingChiu Chang Suan Shu. Math. Teacher, 65, 423–430, p. 429. Translation kindly supplied by D. B. Wagner.
- 28.
Shukla (ed.), op. cit. (ref. 26), pp.100–102. Shukla’s remark (p.299) that the Chinese and Hindu solutions are “quite different” is not justified, since both are ultimately based on the Pythagorean property. The relation\(BC = y + x = {z}^{2}/e\) follows from the property of chords (which itself is based on the Pythagorean property) or from\({y}^{2} - {x}^{2} = {z}^{2}\) and\(y - x = e\). The slight difference in methods is not significant.
- 29.
Mikami, Y.:The Development of Mathematics in China and Japan, reprinted by Chelsea, New York, p.14 (1961).
- 30.
Shukla (ref. 26), p.61.
- 31.
Jain, L.C. (ed.):Gaṇitasārasaṅgraha (with Hindi translation), Sholapur, III, 28, p. 259 (1963).
- 32.
Wagner, D.B.: “Liu Hui and Tsu Keng-chih on the Volume of a Sphere,” Chinese Science, No. 3, 59–79, p.60 (1978).
- 33.
Gupta, R. C.: “Volume of a Sphere in Ancient India,” paper presented at the Seminar on Astronomy and Mathematics in Ancient India, Calcutta, May 19–21, 1987, has details.
- 34.
Mikami (ref. 30), p.43. On p.39 he says that the work “probably belongs to latter half of the sixth century.”
- 35.
Ibid., p.44.
- 36.
Hayashi, Takao: The Bakshali Manuscript, Ph.D. thesis, Brown University, p.649 (1985). He places the work in the seventh century, which is somewhere in the middle of the early (fourth century) and late (tenth century) dates assigned to it.
- 37.
Ibid., p.650; and David Singmaster,Sources in Recreational Mathematics, 3rd Preliminary Edition, p.139, June 1988.
- 38.
Shukla, K. S. (ed.): The Patiganita of Sridharacarya, Lucknow, pp. 80–83 (1959)(text) and 50–51 (transl.), Jain (ref. 30), p.131.
- 39.
Shukla (ref. 39) has given all the 16 solutions. Also see Hayashi (ref. 37), p.650, for more references.
- 40.
Singmaster, op. cit. (ref. 38), pp.139–144.
- 41.
Gupta, R. C.: The Process of Averaging in Ancient and Medieval Mathematics.Gaṇita Bhāratī, III, 32–42 (1981).
- 42.
Needham (ref. 19), p.149.
- 43.
Mikami (ref. 30), p.26.
- 44.
There are similarities in many other mathematical works that we have not discussed here. Some of these are treated by B. Datta in his paper “On the Supposed Indebtedness of Brahmagupta toChiu Chang Suan Shu,”Bulletin of the Calcutta Math. Soc., Vol. XXII, pp.39–51 (1930). Datta does not mention Bhāskara I. Also see van der Waerden (ref. 23), pp.196–208, for π=3.1416, and L.C. Jain, “Jaina School of Mathematics (A Study in Chinese Influences and Transmissions),”in Contribution of Jainism to Indian Culture (ed. by R.C. Dwivedi), Delhi, India, 206–220 (1975).
- 45.
Shukla (ref. 26), p.311.
- 46.
Cullen, C.: “An Eighth Century Chinese Table of Tangents,”Chinese Science, No. 5, 1–33, p.32 (1982).
- 47.
Beer, A., et al.: An Eighth Century Meridian Line: I-Hsing’s Chain of Gnomons. Vistas in Astronomy, Vol. 4, 3–28, p.14 (1961).
- 48.
Cullen (ref. 47), p.32.
- 49.
See Cullen’s paper (ref. 47) andHistoria Mathematica, Vol. 11, pp.45–46 (1984), where it is stated thatLiu Ch′uo (about 600c.e.) knew the formula for interpolation for equal intervals andLi Ch′un-feng (665c.e.) had studied finite differences up to the second order, and interpolation for equal as well as for unequal intervals. See R. C. Gupta.: Second Order Interpolation in Indian Mathematics, etc.Indian J. Hist. Sci., IV, 86–98 (1969).
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Gupta, R.C. (2009). India’s Contributions to Chinese Mathematics Through the Eighth Century C.E.. In: Yadav, B., Mohan, M. (eds) Ancient Indian Leaps into Mathematics. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-4695-0_2
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