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The Kerala school of astronomy and mathematics was a school of mathematics and astronomy founded by Madhava of Sangamagrama in Kerala, India, which included among its members: Parameshvara, Neelakanta Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar. The school flourished between the 14th and 16th centuries and the original discoveries of the school seems to have ended with Narayana Bhattathiri (1559–1632). In attempting to solve astronomical problems, the Kerala school independently created a number of important mathematics concepts. Their most important results—series expansion for trigonometric functions—were described in Sanskrit verse in a book by Neelakanta called Tantrasangraha, and again in a commentary on this work, called Tantrasangraha-vakhya, of unknown authorship. The theorems were stated without proof, but proofs for the series for sine, cosine, and inverse tangent were provided a century later in the work Yuktibhasa (c. 1500 – c. 1610), written in Malayalam, by Jyesthadeva, and also in a commentary on Tantrasangraha.^{[1]}
Their work, completed two centuries before the invention of calculus in Europe, provided what is now considered the first example of a power series (apart from geometric series).^{[2]} However, they did not formulate a systematic theory of differentiation and integration, nor is there any direct evidence of their results being transmitted outside Kerala.^{[3]}^{[4]}^{[5]}^{[6]}
The Kerala school has made a number of contributions to the fields of infinite series and calculus. These include the following (infinite) geometric series:
This formula, however, was already known in the work of the 10th century Iraqi mathematician Alhazen (the Latinized form of the name Ibn al-Haytham) (965-1039).^{[8]}
The Kerala school made intuitive use of mathematical induction, though the inductive hypothesis was not yet formulated or employed in proofs.^{[1]} They used this to discover a semi-rigorous proof of the result:
They applied ideas from (what was to become) differential and integral calculus to obtain (Taylor-Maclaurin) infinite series for \sin x, \cos x, and \arctan x.^{[9]} The Tantrasangraha-vakhya gives the series in verse, which when translated to mathematical notation, can be written as:^{[1]}
The Kerala school made use of the rectification (computation of length) of the arc of a circle to give a proof of these results. (The later method of Leibniz, using quadrature (i.e. computation of area under the arc of the circle), was not yet developed.)^{[1]} They also made use of the series expansion of \arctan x to obtain an infinite series expression (later known as Gregory series) for \pi:^{[1]}
Their rational approximation of the error for the finite sum of their series are of particular interest. For example, the error, f_i(n+1), (for n odd, and i = 1, 2, 3) for the series:
They manipulated the terms, using the partial fraction expansion of :\frac{1}{n^3-n} to obtain a more rapidly converging series for \pi:^{[1]}
They used the improved series to derive a rational expression,^{[1]} 104348/33215 for \pi correct up to nine decimal places, i.e. 3.141592653 . They made use of an intuitive notion of a limit to compute these results.^{[1]} The Kerala school mathematicians also gave a semi-rigorous method of differentiation of some trigonometric functions,^{[10]} though the notion of a function, or of exponential or logarithmic functions, was not yet formulated.
The works of the Kerala school were first written up for the Western world by Englishman C. M. Whish in 1835, though there exists another work, namely Kala Sankalita by J. Warren from 1825^{[11]} which briefly mentions the discovery of infinite series by Kerala astronomers. According to Whish, the Kerala mathematicians had "laid the foundation for a complete system of fluxions" and these works abounded "with fluxional forms and series to be found in no work of foreign countries."^{[12]} However, Whish's results were almost completely neglected, until over a century later, when the discoveries of the Kerala school were investigated again by C. Rajagopal and his associates. Their work includes commentaries on the proofs of the arctan series in Yuktibhasa given in two papers,^{[13]}^{[14]} a commentary on the Yuktibhasa's proof of the sine and cosine series^{[15]} and two papers that provide the Sanskrit verses of the Tantrasangrahavakhya for the series for arctan, sin, and cosine (with English translation and commentary).^{[16]}^{[17]}
A. K. Bag suggested in 1979 that knowledge of these results might have been transmitted to Europe through the trade route from Kerala by traders and Jesuit missionaries.^{[18]} Kerala was in continuous contact with China and Arabia, and Europe. The suggestion of some communication routes and a chronology by some scholars^{[19]}^{[20]} could make such a transmission a possibility; however, there is no direct evidence by way of relevant manuscripts that such a transmission took place.^{[20]} According to David Bressoud, "there is no evidence that the Indian work of series was known beyond India, or even outside of Kerala, until the nineteenth century."^{[9]}^{[21]}
Both Arab and Indian scholars made discoveries before the 17th century that are now considered a part of calculus.^{[10]} However, they were not able, as Newton and Leibniz were, to "combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between the two, and turn calculus into the great problem-solving tool we have today."^{[10]} The intellectual careers of both Newton and Leibniz are well-documented and there is no indication of their work not being their own;^{[10]} however, it is not known with certainty whether the immediate predecessors of Newton and Leibniz, "including, in particular, Fermat and Roberval, learned of some of the ideas of the Islamic and Indian mathematicians through sources of which we are not now aware."^{[10]} This is an active area of current research, especially in the manuscript collections of Spain and Maghreb, research that is now being pursued, among other places, at the Centre national de la recherche scientifique in Paris.^{[10]}
Calculus, University of Cambridge, Trinity College, Cambridge, Lincolnshire, Physics
Indian Mathematics, Kerala school of astronomy and mathematics, Kerala, Mathematics, Astronomy
Lilāvati, Circle, Sankara Variar, Sanskrit, Astronomy
English language, Yuktibhasa, Sanskrit, Springer Science Business Media, Malayalam