We would like to thank the unstinting help given by Anne Dinan, librarian at the University of Exeter, St Lukes Campus, in locating and acquiring the material needed to research this paper. We would also like to acknowledge the .nancial support of both the University of Exeter and AHRB in the researching the mathematics of the Kerala School.
2.
1 Al-Biruni , 1030, India, trans. Qeyamuddin Ahmad ( New Delhi, National Book Trust , 1999), p. 70.
3.
2 Madhava began a school that had the following teacher-student lineage: Madhava (fl. 1380–1420) → Parameswara (fl. 1380–1460) → Damodara (fl. 1450) Narayana (fl. 1529) and Sankara → Nilakantha (b. 1444) → Chitrabhanu (fl. 1530) → Variyar (fl. 1556). Also: Damodara → Jyesthadeva (fl. 1500–1575) → Achuta Pisharoti (fl. c.1550, d. 1621). The names italicised are generally recognised as the major .gures of the Kerala School.
4.
3 The mathematical and astronomical works of the Madhava (or Kerala) School were written either in Sanskrit (for example, the Tantrasangraha of Nilakantha) and/or Malayalam (the local language of Kerala, as, for example, the Yuktibhasa of Jyesthadeva). Many of these texts have not yet been studied, but several scholars with the required linguistic skills have studied parts of the key texts such as the Tantrasangraha and the Yuktibhasa and written articles for the bene.t of the Englishspeaking academic community (for example: T. A. Saraswati Amma , Geometry in Ancient and Medieval India ( Varanasi, Motilal Banarsidass , 1963)
5.
C. T. Rajagopal and M. S. Rangachari , ‘ On an untapped source of medieval Keralese mathematics ’, Archive for the History of Exact Sciences (Vol. 18, 1978), pp. 89–102
6.
K. V. Sarma , A History of the Kerala School of Hindu Astronomy ( Hoshiarpur, Hoshiarpur Vishveshvaranand Institute , 1972)).
7.
4 From Plato’s Republic to Proclus’ Neoplatonism, mathematical entities always played the role of intermediaries between the immaterial realities of the highest realm of being and the confusedly complex objects of the sense world.
8.
5 Greek dif.culties with ‘irrational numbers’ (numbers such as the square root of two or the ratio of circumference of a circle to its diameter (Π) whose values cannot be exactly determined) arose from the attempt to establish a close correspondence between geometric and arithmetic quantities, the result being a heavy emphasis on a geometric interpretation of the irrationality of numbers. Because of this geometric bias, the Greeks were not at ease with irrational numbers and consequently operations with numbers were reduced to a narrow geometric realm, robbing them of considerable potency in arithmetic. On the other hand, with the stress in the Indian tradition on operations with numbers rather than the numbers themselves, Indian mathematics steered clear of any problem with incommensurability. See G. G. Joseph , ‘What is a square root? A study of geometrical representation in different mathematical traditions’, in C. Pritchard (ed.), The Changing Shape of Geometry ( Cambridge University Press, Cambridge , 2003).
9.
6 See, for example, Cortesao and L. de Albuquerque , Obras completas de D. Joao de Castro, Vol. IV ( Coimbra, University of Coimbra , 1982).
10.
7 R. Hooykaas , Selected Studies in History of Science ( Coimbra, Por ordem da Universidade Coimbra , 1983), p. 590.
11.
8 C. T. Rajagopal and M. Marar , ‘ On the Hindu quadrature of the circle ’, Journal of the Royal Asiatic Society (Bombay branch) (Vol. 20, 1944), pp. 65–82 .
12.
9 For a discussion about the controversy over in.nitesimals, see D. M. Jesseph , Squaring the Circle: the war between Hobbes and Wallis ( Chicago, IL, University of Chicago Press , 1999)
13.
and C. B. Boyer , The History of the Calculus ( New York, Dover , 1949).
14.
10 J. F. Scott , The Mathematical Work of John Wallis ( New York, Chelsea , 1981), p. 66.
15.
11 S. Al-Andalusi , c.1068, Science in the Medieval World, trans. S. I. Salem and A. Kumar ( University of Texas Press , 1991), pp. 11–12.
16.
12 Al-Biruni, 1030, op. cit., pp. 11–12.
17.
13 L. A. Sedillot , ‘The great autumnal execution’, in Bulletin of the Bibliography and History of Mathematical and Physical Sciences (published by B. Boncompagni, member of Ponti.c Academy , 1873), reprinted in Sources of Science (Vol. 10, 1964), especially pp. 460–2, 467.
18.
14 L. A. Sedillot, 1873, op. cit., p. 460.
19.
15 J. Bentley , A Historical View of the Hindu Astronomy ( Calcutta, Baptist Mission Press , 1823), p. 151.
20.
16 G. G. Joseph , The Crest of the Peacock: non-European roots of mathematics ( Princeton, NJ, Princeton University Press , 2000).
21.
17 By a Eurocentric history of science, we mean any account of modern science that appeals solely to causes and ideas within Europe and simultaneously marginalises the growth of modern scientific ways of thinking outside of Europe.
22.
18 D. E. Smith , History of Mathematics, 2 Vols ( Boston, MA, Ginn , 1923–25; reprinted by New York, Dover, 1958), Vol. 1, p. 435.
23.
19 The works in question are J. Warren , A Collection of Memoirs on the Various Modes According to which the Nations of the Southern Parts of India Divide Time ( Madras , 1825)
24.
and C. M. Whish , ‘ On the Hindu quadrature of the circle and the in.nite series of the proportion of the circumference to the diameter exhibited in the four Shastras, the Tantrasamgraham, Yukti-Bhasa, Carana Padhati, and Sadratnamala ’, Transactions Royal Asiatic Society of Great Britain and Ireland (Vol. 3, 1835), pp. 509–523 .
25.
20 C. H. Edwards , The Historical Development of the Calculus ( New York, Springer-Verlag , 1979).
26.
21 F. F. Abeles , ‘ Charles L. Dodgson’s geometric approach to arctangent relations for Pi ’, Historia Mathematica (Vol. 20, 1993), pp. 151–159
27.
L. Fiegenbaum , ‘ Brook Taylor and the method of increments ’, Archive for the History of Exact Sciences (Vol. 34, no. 1, 1986), pp. 1–140 .
28.
22 G. G. Joseph , ‘ Cognitive encounters in India during the age of imperialism ’, Race & Class (Vol. 36, no. 3, 1995), pp. 39–56 .
29.
23 G. G. Joseph, 2000, op. cit., p. 215.
30.
24 M. Bernal , Black Athena ( London, Free Association Books , 1987).
31.
25 E. Burgess , The Surya Siddhanta: a text-book of Hindu astronomy (1860; reprinted by New Delhi, Motilal Banarsidass, 1997), p. 387.
32.
26 G. Peacock , ‘Arithmetic – including a history of the science’, Encyclopedia Metropolitana; Or, Universal Dictionary of Knowledge, Part 6, First Division ( London, J. J. Grif.n , 1849), p. 420.
33.
27 For example, C. T. Rajagopal and T. V. Vedamurthi , ‘ On the Hindu proof of Gregory’s series ’, Scripta Mathematica (Vol. 18, 1952), pp. 65–74
34.
C. T. Rajagopal and M. S. Rangachari , ‘ On Medieval Keralese mathematics ’, Archive for the History of Exact Sciences (Vol. 35, 1986), pp. 91–99 ; T. A. Saraswati Amma, 1963, op. cit.
35.
28 A selection is M. E. Baron , The Origins of the Infinitesimal Calculus ( Oxford, Pergamon , 1969)
36.
V. J. Katz , A History of Mathematics: an introduction ( New York, HarperCollins , 1992)
37.
V. J. Katz , ‘ Ideas of calculus in Islam and India ’, Mathematics Magazine (Vol. 68, no. 3, 1995), pp. 163–174
38.
R. Calinger , A Contextual History of Mathematics to Euler ( Englewood Cliffs, NJ, Prentice Hall , 1999).
39.
29 M. E. Baron, 1969, op. cit., p. 65.
40.
30 R. Calinger, 1999, op. cit., p. 28.
41.
31 G. G. Joseph, 2000, op. cit.
42.
32 S. R. Benedict, A Comparative Study of the Early Treatises Introducing into Europe the Hindu Art of Reckoning, PhD thesis, University of Michigan (Rumford Press, 1914).
43.
33 These transmission facts are dealt with in G. Saliba , A History of Arabic Astronomy ( New York, New York University Press , 1994); and G. G. Joseph, 2000, op. cit.
44.
34 B. V. Subbarayappa and K. V. Sarma , Indian Astronomy: a source-book ( Bombay, Nehru Centre , 1985).
45.
35 O. Neugebauer , The Exact Sciences in Antiquity ( New York, HarperCollins , 1962).
46.
36 B. L. van der Waerden , Geometry and Algebra in Ancient Civilizations ( Berlin, Springer-Verlag , 1983).
47.
37 Any equation, usually in several unknowns, that is studied in a problem whose solution may be required to be whole numbers (or integers) or more generally rational numbers. Named after Diophantus of Alexandria (c.250 AD) who investigated such problems in his book Arithmetica. An example of such an equation in two unknowns (x and y) is 3x + 4y =50 which has a number of positive integer solutions for (x, y). For example, x = 14, y = 2 satisfies the equation, as do the solution sets (10,5), (6,8) and (2,11).
48.
38 B. L. van der Waerden , ‘ Pells equation in Greek and Hindu mathematics ’, Russian Math Surveys (Vol. 31, 1976), pp. 210–225, p. 210 .
49.
39 B. L. van der Waerden, 1976, op. cit., p. 221.
50.
40 L. A. Sedillot, 1873, op. cit., p. 463.
51.
41 L. A. Sedillot , Sciences mathématiques chez les Grecs et les Orienteaux, Vol. 2 ( Paris, Libraire de Firmin Didot Frères , 1845–49), p. 465.
52.
42 D. L. O’Leary , How Greek Science Passed to the Arabs ( London, Routledge & Kegan Paul , 1948), p. 109.
53.
43 See, for example, T. Dantzig , The Bequest of the Greeks ( London, Allen & Unwin , 1955).
54.
44 The Kamal was an Indian navigation instrument made available to the Portuguese since the voyage of Vasco da Gama to the Malabar in 1499. Luis Albuquerque discusses this in his book Curso de Historia da Nautica ( Coimbra, University of Coimbra , 1972).
55.
45 D. Potache , ‘ The commercial relations between Basrah and Goa in the sixteenth century ’, STUDIA (Lisbon, Vol. 48, 1989).
56.
46 See, for example, K. V. Sarma, 1972, op. cit.; V. J. Katz, 1992, op. cit.
57.
47 The Aryabhata Group , 2002, ‘ Transmission of the calculus from Kerala to Europe ’, in Proceedings of the International Seminar and Colloquium on 1500 Years of Aryabhateeyam (Kochi, Kerala Sastra Sahitya Parishad, 2002), pp. 33–48, especially pp. 42–43 .
58.
48 See J. F. Scott, 1981, op. cit., p. 43.
59.
49 See K. Ramasubramaniam , ‘ Aryabhateeyam: in the light of Aryabhateeyambhashya by Nilakantha Somayaji ’, in Proceedings of the International Seminar and Colloquium on 1500 Years of Aryabhateeyam (Kochi, Kerala Sastra Sahitya Parishad, 2002), pp. 115–122 .
60.
50 C. Brezinski , History of Continued Fractions and Pade Approximations ( London, Springer-Verlag , 1980).
61.
51 V. J. Katz, 1992, op. cit., p. 368.
62.
52 U. Baldini , Saggi sulla cultura della Compagnia di Gesu (secoli XVI–XVIII) ( Padova, CLEUP Editrice , 2000).
63.
53 For the movements of the Jesuit missionaries in India, see U. Baldini , Studi sufilosofia e scienza dei gesuiti in Italia 1540–1632 ( Bulzoni Editore , 1992)
64.
and I. Iannaccone , Johann Schreck Terrentius ( Napoli, Instituto Universitario Orientale , 1992).
65.
54 With reference to Rubino, see U. Baldini, 1992, op. cit., p. 214; to Ricci, see J. Wicki , Documenta Indica, 16 Vols ( Rome, Monumenta Historica Societa Iesu , 1948), Vol. 12, p. 474.
66.
For additional information and evidence of Jesuit activity to acquire local Kerala knowledge and transmit it to Europe, see D. F. Almeida , J. K. John and A. Zadorozhnyy , ‘ Keralese mathematics: its possible transmission to Europe and the consequential educational implications ’, Journal of Natural Geometry (Vol. 20, no. 1, 2000), pp. 77–104 .
67.
55 R. Calinger, 1999, op. cit., p. 282.
68.
56 C. Brezinski, 1980, op. cit., p. 34.
69.
57 As discussed in G. G. Joseph, 1995, op. cit.
70.
58 G. G. Joseph , ‘Different ways of knowing: contrasting styles of argument in Indian and Greek mathematical traditions’, in P. Ernest (ed.) Mathematics, Education and Philosophy ( London, Falmer Press , 1994).
