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.

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1 Al-Biruni , 1030, India, trans. Qeyamuddin Ahmad ( New Delhi , National Book Trust , 1999 ), p. 70 .

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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.

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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 )

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K. V. Sarma , A History of the Kerala School of Hindu Astronomy ( Hoshiarpur , Hoshiarpur Vishveshvaranand Institute , 1972 )).

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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.

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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 ).

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6 See, for example, Cortesao and L. de Albuquerque , Obras completas de D. Joao de Castro, Vol. IV ( Coimbra , University of Coimbra , 1982 ).

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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 )

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and C. B. Boyer , The History of the Calculus ( New York , Dover , 1949 ).

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10 J. F. Scott , The Mathematical Work of John Wallis ( New York , Chelsea , 1981 ), p. 66 .

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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 .

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12 Al-Biruni, 1030, op. cit., pp. 11–12.

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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.

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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.

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18 D. E. Smith , History of Mathematics, 2 Vols ( Boston, MA , Ginn , 1923–25 ; reprinted by New York, Dover, 1958), Vol. 1, p. 435 .

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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 )

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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 .

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23 G. G. Joseph, 2000, op. cit., p. 215.

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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

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28 A selection is M. E. Baron , The Origins of the Infinitesimal Calculus ( Oxford , Pergamon , 1969 )

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29 M. E. Baron, 1969, op. cit., p. 65.

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30 R. Calinger, 1999, op. cit., p. 28.

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31 G. G. Joseph, 2000, op. cit.

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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.

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38 B. L. van der Waerden , ‘ Pells equation in Greek and Hindu mathematics ’, Russian Math Surveys (Vol. 31, 1976 ), pp. 210 – 225 , p. 210 .

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39 B. L. van der Waerden, 1976, op. cit., p. 221.

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40 L. A. Sedillot, 1873, op. cit., p. 463.

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43 See, for example, T. Dantzig , The Bequest of the Greeks ( London , Allen & Unwin , 1955 ).

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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 ).

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46 See, for example, K. V. Sarma, 1972, op. cit.; V. J. Katz, 1992, op. cit.

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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 , ), pp. 33 – 48 , especially pp. 42 – 43 .

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48 See J. F. Scott, 1981, op. cit., p. 43.

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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 , ), pp. 115 – 122 .

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51 V. J. Katz, 1992, op. cit., p. 368.

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55 R. Calinger, 1999, op. cit., p. 282.

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56 C. Brezinski, 1980, op. cit., p. 34.

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57 As discussed in G. G. Joseph, 1995, op. cit.

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