KERALA MATHEMATICS AND ITS POSSIBLE TRANSMISSION TO EUROPE
Dennis Francis Almeida
University of Exeter, UK
George Gheverghese Joseph
University of Manchester, UK
Abstract: Mathematical techniques of great importance, involving elements of the calculus, were developed between the 14th and 16th centuries in Kerala, India. In this period Kerala was in continuous contact with the outside world, with China to the East and with Arabia to the West. Also after the pioneering voyage of Vasco da Gama in 1499, there was a direct conduit to Europe. The current state of the literature implies that, despite these communication routes, the Keralese calculus lay confined to Kerala. The paper is based on the findings of an ongoing research project, which examines the epistemology of the calculus of the Kerala school and its conjectured transmission to Europe.
According to the literature the general methods of the calculus were invented independently by Newton and Leibniz in the late 17th century after exploiting the works of European pioneers such as Fermat, Roberval, Taylor, Gregory, Pascal, and Bernoulli in the preceding half century. However, what appears to be less well known is that the fundamental elements of the calculus including numerical integration methods and infinite series derivations for p and for trigonometric functions such as sin x, cos x and tan-1 x (the so-called Gregory series) had already been discovered over 250 years earlier in Kerala. These developments first occurred in the works of the Kerala mathematician Madhava and were subsequently elaborated on by his followers Nilakantha Somayaji, Jyesthadeva, Sankara Variyar and others between the 14th and 16th centuries . In the latter half of the 20th century there has been some acknowledgement of these facts outside India. There are several modern European histories of mathematics which acknowledge the work of the Kerala school. However it needs to be pointed out that this acknowledgement is not necessarily universal. For example, in the recent past a paper by Fiegenbaum on the history of the calculus makes no acknowledgement of the work of the Kerala school. However, prior to the publication of Fiegenbaum's paper, several renowned publications detailing the Keralese calculus had already appeared in the West. Such a viewpoint may have its origins in the Eurocentrism that was formulated during the period of colonisation by some European nations.
 See, for example, Margaret Baron, The Origins of the Infinitesimal Calculus, Oxford, Pergamom, 1969, p 65.
 See, for example, Charles Edwards, The Historical Development of the Calculus, New York, Springer-Verlag, 1979, p189, and Victor Katz, "Ideas of calculus in Islam and India", Mathematics Magazine, Washington, 68 (1995), 3: 163-174, p 163 and p 164.
 See the work of K Venkateswara Sarma, A History of the Kerala School of Hindu Astronomy, Hoshiarpur, Vishveshvaranand Vedic Research Institute, 1972, p 21 and p 22 and the paper by Charles Whish, "On the Hindu quadrature of the circle and the infinite series of the proportion of the circumference to the diameter exhibited in the four Shastras, the Tantrasamgraham, Yukti-Bhasa, Carana Padhati, and Sadratnamala", Transactions of the Royal Asiatic Society of Great Britain and Ireland, London, 3 (1835): 509-523, p 522 and p 523.
 For example, Margaret Baron, Origins of Calculus, op cit, p 62 and p 63; Ronald Calinger, A Contextual History of Mathematics to Euler, New Jersey, Prentice Hall, 1999, p 284
 Leone Fiegenbaum, "Brook Taylor and the Method of Increments", Archive for History of Exact Sciences, Baltimore, 34 (1986): 1-140, p 72
 For example, Charles Whish, "On the Hindu quadrature of the circle and the infinite series of the proportion of the circumference to the diameter exhibited in the four Shastras, the Tantrasamgraham, Yukti-Bhasa, Carana Padhati, and Sadratnamala", Transactions, 3 (1835): 509-523; C T Rajagopal and M S Rangachari, "On an Untapped Source of Medieval Keralese Mathematics", Archive for History of Exact Sciences, Baltimore, 18 (1978): 89-102; C T Rajagopal and T V Vedamurthi, "On the Hindu proof of Gregory's series", Scripta Mathematica, New York, 18(1951): 91-99