TheKerala school of astronomy and mathematics or theKerala school was a school ofmathematics andastronomy founded byMadhava of Sangamagrama inTirur,Malappuram,Kerala, India, which included among its members:Parameshvara,Neelakanta Somayaji,Jyeshtadeva,Achyuta Pisharati,Melpathur Narayana Bhattathiri andAchyuta Panikkar. The school flourished between the 14th and 16th centuries and its original discoveries seem to have ended withNarayana Bhattathiri (1559–1632). In attempting to solve astronomical problems, the Kerala school independently discovered a number of important mathematical concepts. Their most important results—series expansion for trigonometric functions—were described inSanskrit verse in a book by Neelakanta calledTantrasangraha (around 1500), and again in a commentary on this work, calledTantrasangraha-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 workYuktibhasa (c. 1530), written inMalayalam, by Jyesthadeva, and also in a commentary onTantrasangraha.[1]
Their work, completed two centuries before the invention ofcalculus in Europe, provided what is now considered the first example of apower series (apart fromgeometric series).[2][3][4]
Islamic scholars nearly developed a generalformula for findingintegrals ofpolynomials by 1000 AD —and evidently could find such a formula for any polynomial in which they were interested. But, it appears, they were not interested in any polynomial of degree higher than four, at least in any of the material that has been found to date.Indian scholars, on the other hand, were by the year 1600 able to use formula similar toIbn al-Haytham's sum formula for arbitrary integral powers in calculating power series for the functions in which they were interested. By the same time, they also knew how to calculate the differentials of these functions. So some of the basic ideas of calculus were known in Egypt andIndia many centuries beforeIsaac Newton. It does not appear, however, that either Islamic or Indian mathematicians saw the necessity of connecting some of the disparate ideas that we include under the name calculus. They were apparently only interested in specific cases in which these ideas were needed.[5][6]
The Kerala school has made a number of contributions to the fields ofinfinite series andcalculus. These include the following infinite geometric series:
The Kerala school made intuitive use ofmathematical induction, though theinductive hypothesis was not yet formulated or employed in proofs.[1] They used this to discover a semi-rigorous proof of the result:
for largen.
They applied ideas from (what was to become)differential andintegralcalculus to obtain (Taylor–Maclaurin) infinite series for,, and.[8] TheTantrasangraha-vakhya gives the series in verse, which when translated to mathematical notation, can be written as:[1]
where, for the series reduce to the standard power series for these trigonometric functions, for example:
and
(The Kerala school did not use the "factorial" symbolism.)
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 to obtain an infinite series expression (later known as Gregory series) for:[1]
Their rational approximation of theerror for the finite sum of their series are of particular interest. For example, the error,, (forn odd, andi = 1, 2, 3) for the series:
where
They manipulated the terms, using the partial fraction expansion of : to obtain a more rapidly converging series for:[1]
They used the improved series to derive a rational expression,[1] for correct up to nine decimal places, i.e.. They made use of an intuitive notion of alimit to compute these results.[1] The Kerala school mathematicians also gave a semi-rigorous method of differentiation of some trigonometric functions,[9] though the notion of a function, or of exponential or logarithmic functions, was not yet formulated.
In 1825 John Warren published a memoir on the division of time in southern India,[10] called theKala Sankalita, which briefly mentions the discovery of infinite series by Kerala astronomers.
The works of the Kerala school were first written up for the Western world by EnglishmanC. M. Whish in 1835. 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".[11] However, Whish's results were almost completely neglected until over a century later, when the discoveries of the Kerala school were investigated again byC. T. Rajagopal and his associates. Their work includes commentaries on the proofs of the arctan series inYuktibhasa given in two papers,[12][13] a commentary on theYuktibhasa's proof of the sine and cosine series[14] and two papers that provide theSanskrit verses of theTantrasangrahavakhya for the series for arctan, sine, and cosine (with English translation and commentary).[15][16]
A. K. Bag suggested in 1979 that knowledge of these results might have been transmitted to Europe through the trade route fromKerala by traders andJesuit missionaries.[18] Kerala was in continuous contact with China andArabia, andEurope. 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 toDavid Bressoud, "there is no evidence that the Indian work of series was known beyond India, or even outside of Kerala, until the nineteenth century".[8][21] V. J. Katz notes some of the ideas of the Kerala school have similarities to the work of 11th-century Iraqi scholarIbn al-Haytham,[9] suggesting a possible transmission of ideas fromIslamic mathematics to Kerala.[22]
Both Indian andArab scholars made discoveries before the 17th century that are now considered a part of calculus.[9] According to Katz, they were yet to "combine many differing ideas under the two unifying themes of thederivative and theintegral, show the connection between the two, and turn calculus into the great problem-solving tool we have today", likeNewton andLeibniz.[9] The intellectual careers of both Newton and Leibniz are well documented and there is no indication of their work not being their own;[9] however, it is not known with certainty whether the immediatepredecessors 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".[9] This is an active area of current research, especially in the manuscript collections of Spain andMaghreb, research that is now being pursued, among other places, at theCentre national de la recherche scientifique inParis.[9]
^abcdefghRoy, Ranjan. 1990. "Discovery of the Series Formula for by Leibniz, Gregory, and Nilakantha."Mathematics Magazine (Mathematical Association of America) 63(5):291–306.
^(Bressoud 2002, p. 12) Quote: "There is no evidence that the Indian work on series was known beyond India, or even outside Kerala, until the nineteenth century. Gold and Pingree assert [4] that by the time these series were rediscovered in Europe, they had, for all practical purposes, been lost to India. The expansions of the sine, cosine, and arc tangent had been passed down through several generations of disciples, but they remained sterile observations for which no one could find much use."
^Plofker 2001, p. 293 Quote: "It is not unusual to encounter in discussions of Indian mathematics such assertions as that "the concept of differentiation was understood [in India] from the time of Manjula (... in the 10th century)" [Joseph 1991, 300], or that "we may consider Madhava to have been the founder of mathematical analysis" (Joseph 1991, 293), or that Bhaskara II may claim to be "the precursor of Newton and Leibniz in the discovery of the principle of the differential calculus" (Bag 1979, 294). ... The points of resemblance, particularly between early European calculus and the Keralese work on power series, have even inspired suggestions of a possible transmission of mathematical ideas from the Malabar coast in or after the 15th century to the Latin scholarly world (e.g., in (Bag 1979, 285)). ... It should be borne in mind, however, that such an emphasis on the similarity of Sanskrit (or Malayalam) and Latin mathematics risks diminishing our ability fully to see and comprehend the former. To speak of the Indian "discovery of the principle of the differential calculus" somewhat obscures the fact that Indian techniques for expressing changes in the Sine by means of the Cosine or vice versa, as in the examples we have seen, remained within that specific trigonometric context. The differential "principle" was not generalized to arbitrary functions—in fact, the explicit notion of an arbitrary function, not to mention that of its derivative or an algorithm for taking the derivative, is irrelevant here"
^Pingree 1992, p. 562 Quote: "One example I can give you relates to the Indian Mādhava's demonstration, in about 1400 A.D., of the infinite power series of trigonometrical functions using geometrical and algebraic arguments. When this was first described in English by Charles Whish, in the 1830s, it was heralded as the Indians' discovery of the calculus. This claim and Mādhava's achievements were ignored by Western historians, presumably at first because they could not admit that an Indian discovered the calculus, but later because no one read anymore theTransactions of the Royal Asiatic Society, in which Whish's article was published. The matter resurfaced in the 1950s, and now we have the Sanskrit texts properly edited, and we understand the clever way that Mādhava derived the serieswithout the calculus; but many historians still find it impossible to conceive of the problem and its solution in terms of anything other than the calculus and proclaim that the calculus is what Mādhava found. In this case the elegance and brilliance of Mādhava's mathematics are being distorted as they are buried under the current mathematical solution to a problem to which he discovered an alternate and powerful solution."
^Katz 1995, pp. 173–174 Quote: "How close did Islamic and Indian scholars come to inventing the calculus? Islamic scholars nearly developed a general formula for finding integrals of polynomials by A.D. 1000—and evidently could find such a formula for any polynomial in which they were interested. But, it appears, they were not interested in any polynomial of degree higher than four, at least in any of the material that has come down to us. Indian scholars, on the other hand, were by 1600 able to use ibn al-Haytham's sum formula for arbitrary integral powers in calculating power series for the functions in which they were interested. By the same time, they also knew how to calculate the differentials of these functions. So some of the basic ideas of calculus were known in Egypt and India many centuries before Newton. It does not appear, however, that either Islamic or Indian mathematicians saw the necessity of connecting some of the disparate ideas that we include under the name calculus. They were apparently only interested in specific cases in which these ideas were needed. There is no danger, therefore, that we will have to rewrite the history texts to remove the statement that Newton and Leibniz invented the calculus. They were certainly the ones who were able to combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between them, and turn the calculus into the great problem-solving tool we have today."
^abBressoud, David. 2002. "Was Calculus Invented in India?"The College Mathematics Journal (Mathematical Association of America). 33(1):2–13.
^abcdefgKatz, V. J. 1995. "Ideas of Calculus in Islam and India." (pdfArchived 8 December 2023 at theWayback Machine)Mathematics Magazine (Mathematical Association of America), 68(3):163-174.
^Rajagopal, C.; Rangachari, M. S. (1949). "A Neglected Chapter of Hindu Mathematics".Scripta Mathematica.15:201–209.
^Rajagopal, C.; Rangachari, M. S. (1951). "On the Hindu proof of Gregory's series".Scripta Mathematica.17:65–74.
^Rajagopal, C.; Venkataraman, A. (1949). "The sine and cosine power series in Hindu mathematics".Journal of the Royal Asiatic Society of Bengal (Science).15:1–13.
^Rajagopal, C.; Rangachari, M. S. (1977). "On an untapped source of medieval Keralese mathematics".Archive for History of Exact Sciences.18 (2):89–102.doi:10.1007/BF00348142.S2CID51861422.
^Rajagopal, C.; Rangachari, M. S. (1986). "On Medieval Kerala Mathematics".Archive for History of Exact Sciences.35 (2):91–99.doi:10.1007/BF00357622.S2CID121678430.
^A. K. Bag (1979)Mathematics in ancient and medieval India. Varanasi/Delhi: Chaukhambha Orientalia. page 285.
^Raju, C. K. (2001). "Computers, Mathematics Education, and the Alternative Epistemology of the Calculus in the Yuktibhasa".Philosophy East and West.51 (3):325–362.doi:10.1353/pew.2001.0045.S2CID170341845.
^abAlmeida, D. F.; John, J. K.; Zadorozhnyy, A. (2001). "Keralese Mathematics: Its Possible Transmission to Europe and the Consequential Educational Implications".Journal of Natural Geometry.20:77–104.
^Gold, D.; Pingree, D. (1991). "A hitherto unknown Sanskrit work concerning Madhava's derivation of the power series for sine and cosine".Historia Scientiarum.42:49–65.
Bressoud, David (2002), "Was Calculus Invented in India?",The College Mathematics Journal,33 (1):2–13,doi:10.2307/1558972,JSTOR1558972.
Gupta, R. C. (1969) "Second Order of Interpolation of Indian Mathematics",Indian Journal of History of Science 4: 92-94
Hayashi, Takao (2003), "Indian Mathematics", in Grattan-Guinness, Ivor (ed.),Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, vol. 1, pp. 118–130, Baltimore, MD: The Johns Hopkins University Press, 976 pages,ISBN0-8018-7396-7.
Plofker, Kim (1996), "An Example of the Secant Method of Iterative Approximation in a Fifteenth-Century Sanskrit Text",Historia Mathematica,23 (3):246–256,doi:10.1006/hmat.1996.0026.
Plofker, Kim (2001), "The "Error" in the Indian "Taylor Series Approximation" to the Sine",Historia Mathematica,28 (4):283–295,doi:10.1006/hmat.2001.2331.
Plofker, K. (20 July 2007), "Mathematics of India", in Katz, Victor J. (ed.),The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, Princeton, NJ: Princeton University Press, 685 pages (published 2007), pp. 385–514,ISBN978-0-691-11485-9.
C. K. Raju. 'Computers, mathematics education, and the alternative epistemology of the calculus in the Yuktibhâsâ',Philosophy East and West51, University of Hawaii Press, 2001.
Roy, Ranjan (1990), "Discovery of the Series Formula for by Leibniz, Gregory, and Nilakantha",Mathematics Magazine,63 (5):291–306,doi:10.2307/2690896,JSTOR2690896.
Sarma, K. V.; Hariharan, S. (1991). "Yuktibhasa of Jyesthadeva : a book of rationales in Indian mathematics and astronomy – an analytical appraisal".Indian J. Hist. Sci.26 (2):185–207.
