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| | Greetings! I am Marvella and I feel comfortable when people use the full name. For many years he's been working as a meter reader and it's something he truly appreciate. South Dakota is her beginning place but she requirements to transfer simply because of her family members. To gather coins is what his family and him appreciate.<br><br>Here is my blog ... [http://chatmast.com/index.php?do=/BookerSipessk/info/ at home std test] |
| {{for|other uses of this name|Kerala school (disambiguation)}}
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| The '''Kerala school of astronomy and mathematics''' was a school of [[Indian mathematics|mathematics]] and [[Indian astronomy|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 [[Melpathur Narayana Bhattathiri|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''.<ref name=roy>Roy, Ranjan. 1990. "Discovery of the Series Formula for <math> \pi </math> by Leibniz, Gregory, and Nilakantha." ''Mathematics Magazine'' (Mathematical Association of America) 63(5):291-306.</ref>
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| 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).<ref>{{Harv|Stillwell|2004|p=173}}</ref> However, they did not formulate a systematic theory of [[derivative|differentiation]] and [[integral|integration]], nor is there any direct evidence of their results being transmitted outside [[Kerala]].<ref>{{Harv|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."</ref><ref>{{Harvnb|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
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| derivative or an algorithm for taking the derivative, is irrelevant here"</ref><ref>{{Harvnb|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 the ''Transactions 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 series ''without'' 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."</ref><ref>{{Harvnb|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 calculus. Thy 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."</ref>
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| ==Contributions==
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| ===Infinite Series and Calculus===
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| The Kerala school has made a number of contributions to the fields of [[Series (mathematics)|infinite series]] and [[calculus]]. These include the following (infinite) geometric series:
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| :<math> \frac{1}{1-x} = 1 + x + x^2 + x^3 + \dots </math> for <math>|x|<1 </math><ref name =singh>{{cite journal | last1 = Singh | first1 = A. N. | year = 1936 | title = On the Use of Series in Hindu Mathematics | url = | journal = Osiris | volume = 1 | issue = | pages = 606–628 |doi = 10.1086/368443 }}</ref>
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| This formula, however, was already known in the work of the 10th century [[Iraq]]i [[Islamic mathematics|mathematician]] [[Ibn al-Haytham|Alhazen]] (the [[Latin]]ized form of the name Ibn al-Haytham) (965-1039).<ref>Edwards, C. H., Jr. 1979. ''The Historical Development of the Calculus''. New York: Springer-Verlag.</ref>
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| The Kerala school made intuitive use of [[mathematical induction]], though the [[Inductive hypothesis#Formal description|inductive hypothesis]] was not yet formulated or employed in proofs.<ref name=roy/> They used this to discover a semi-rigorous proof of the result:
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| :<math>1^p+ 2^p + \cdots + n^p \approx \frac{n^{p+1}}{p+1}</math> for large ''n''. This result was also known to Alhazen.<ref name=roy/>
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| They applied ideas from (what was to become) [[Derivative|differential]] and [[integral]] [[calculus]] to obtain ([[Taylor series|Taylor-Maclaurin]]) infinite series for <math>\sin x</math>, <math>\cos x</math>, and <math> \arctan x</math>.<ref name=bressoud>Bressoud, David. 2002. "Was Calculus Invented in India?" ''The College Mathematics Journal'' (Mathematical Association of America). 33(1):2-13.</ref> The ''Tantrasangraha-vakhya'' gives the series in verse, which when translated to mathematical notation, can be written as:<ref name=roy/>
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| :<math>r\arctan(\frac{y}{x}) = \frac{1}{1}\cdot\frac{ry}{x} -\frac{1}{3}\cdot\frac{ry^3}{x^3} + \frac{1}{5}\cdot\frac{ry^5}{x^5} - \cdots , </math> where <math>y/x \leq 1. </math>
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| :<math>r\sin \frac{x}{r} = x - x\cdot\frac{x^2}{(2^2+2)r^2} + x\cdot \frac{x^2}{(2^2+2)r^2}\cdot\frac{x^2}{(4^2+4)r^2} - \cdot </math>
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| :<math> r(1 - \cos \frac{x}{r}) = r\cdot \frac{x^2}{(2^2-2)r^2} - r\cdot \frac{x^2}{(2^2-2)r^2}\cdot \frac{x^2}{(4^2-4)r^2} + \cdots , </math> where, for <math> r = 1 </math>, the series reduce to the standard power series for these trigonometric functions, for example:
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| ::<math>\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots </math> and
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| ::<math>\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots </math> (The Kerala school themselves did not use the "factorial" symbolism.)
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| 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.)<ref name=roy/> They also made use of the series expansion of <math>\arctan x</math> to obtain an infinite series expression (later known as Gregory series) for <math>\pi</math>:<ref name=roy/>
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| :<math>\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \ldots </math>
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| Their rational approximation of the ''error'' for the finite sum of their series are of particular interest. For example, the error, <math>f_i(n+1)</math>, (for ''n'' odd, and ''i = 1, 2, 3'') for the series:
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| :<math>\frac{\pi}{4} \approx 1 - \frac{1}{3}+ \frac{1}{5} - \cdots (-1)^{(n-1)/2}\frac{1}{n} + (-1)^{(n+1)/2}f_i(n+1)</math>
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| ::where <math>f_1(n) = \frac{1}{2n}, \ f_2(n) = \frac{n/2}{n^2+1}, \ f_3(n) = \frac{(n/2)^2+1}{(n^2+5)n/2}.</math>
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| They manipulated the terms, using the partial fraction expansion of :<math>\frac{1}{n^3-n}</math> to obtain a more rapidly converging series for <math>\pi</math>:<ref name=roy/>
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| :<math>\frac{\pi}{4} = \frac{3}{4} + \frac{1}{3^3-3} - \frac{1}{5^3-5} + \frac{1}{7^3-7} - \cdots </math>
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| They used the improved series to derive a rational expression,<ref name=roy/> <math>104348/33215</math> for <math>\pi</math> correct up to nine decimal places, ''i.e.'' <math>3.141592653 </math>. They made use of an intuitive notion of a [[Limit (mathematics)|limit]] to compute these results.<ref name=roy/> The Kerala school mathematicians also gave a semi-rigorous method of differentiation of some trigonometric functions,<ref name=katz>Katz, V. J. 1995. "Ideas of Calculus in Islam and India." ''Mathematics Magazine'' (Mathematical Association of America), 68(3):163-174.</ref> though the notion of a function, or of exponential or logarithmic functions, was not yet formulated.
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| 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<ref>[http://www.physics.iitm.ac.in/~labs/amp/kerala-astronomy.pdf Current Science],</ref> 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.''"<ref name="charles">{{Citation
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| | author =Charles Whish
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| | year = 1835
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| | title = Transactions of the Royal Asiatic Society of Great Britain and Ireland
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| | publisher =
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| }}
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| </ref>
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| 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,<ref>{{cite journal | last1 = Rajagopal | first1 = C. | last2 = Rangachari | first2 = M. S. | year = 1949 | title = A Neglected Chapter of Hindu Mathematics | url = | journal = Scripta Mathematica | volume = 15 | issue = | pages = 201–209 }}</ref><ref>{{cite journal | last1 = Rajagopal | first1 = C. | last2 = Rangachari | first2 = M. S. | year = 1951 | title = On the Hindu proof of Gregory's series | url = | journal = Scripta Mathematica | volume = 17 | issue = | pages = 65–74 }}</ref> a commentary on the ''Yuktibhasa'''s proof of the sine and cosine series<ref>{{cite journal | last1 = Rajagopal | first1 = C. | last2 = Venkataraman | first2 = A. | year = 1949 | title = The sine and cosine power series in Hindu mathematics | url = | journal = Journal of the Royal Asiatic Society of Bengal (Science) | volume = 15 | issue = | pages = 1–13 }}</ref> and two papers that provide the [[Sanskrit]] verses of the ''Tantrasangrahavakhya'' for the series for arctan, sin, and cosine (with English translation and commentary).<ref>{{cite journal | last1 = Rajagopal | first1 = C. | last2 = Rangachari | first2 = M. S. | year = 1977 | title = On an untapped source of medieval Keralese mathematics | url = | journal = Archive for the History of Exact Sciences | volume = 18 | issue = | pages = 89–102 }}</ref><ref>{{cite journal | last1 = Rajagopal | first1 = C. | last2 = Rangachari | first2 = M. S. | year = 1986 | title = On Medieval Kerala Mathematics | url = | journal = Archive for the History of Exact Sciences | volume = 35 | issue = | pages = 91–99 | doi=10.1007/BF00357622}}</ref>
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| ==Possibility of transmission of Kerala School results to Europe==
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| 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.<ref>A. K. Bag (1979) ''Mathematics in ancient and medieval India''. Varanasi/Delhi: Chaukhambha Orientalia. page 285.</ref> Kerala was in continuous contact with China and [[Arabia]], and [[Europe]]. The suggestion of some communication routes and a chronology by some scholars<ref>{{cite journal | last1 = Raju | first1 = C. K. | year = 2001 | title = Computers, Mathematics Education, and the Alternative Epistemology of the Calculus in the Yuktibhasa | url = | journal = Philosophy East and West | volume = 51 | issue = 3| pages = 325–362 }}</ref><ref name=almeida/> could make such a transmission a possibility, however, there is no direct evidence by way of relevant manuscripts that such a transmission took place.<ref name=almeida>{{cite journal | last1 = Almeida | first1 = D. F. | last2 = John | first2 = J. K. | last3 = Zadorozhnyy | first3 = A. | year = 2001 | title = Keralese Mathematics: Its Possible Transmission to Europe and the Consequential Educational Implications | url = | journal = Journal of Natural Geometry | volume = 20 | issue = | pages = 77–104 }}</ref> In fact, 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."<ref name=bressoud/><ref name=gold>{{cite journal | last1 = Gold | first1 = D. | last2 = Pingree | first2 = D. | year = 1991 | title = A hitherto unknown Sanskrit work concerning Madhava's derivation of the power series for sine and cosine | url = | journal = Historia Scientiarum | volume = 42 | issue = | pages = 49–65 }}</ref>
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| Both [[Islamic mathematics|Arab]] and Indian scholars made discoveries before the 17th century that are now considered a part of calculus.<ref name=katz/> However, they were not able, as [[Isaac Newton|Newton]] and [[Gottfried Leibniz|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."<ref name=katz/> The intellectual careers of both Newton and Leibniz are well-documented and there is no indication of their work not being their own;<ref name=katz/> 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."<ref name=katz/> 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]].<ref name=katz/>
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| ==See also==
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| *[[Indian astronomy]]
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| *[[Indian mathematics]]
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| *[[Indian mathematicians]]
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| *[[History of mathematics]]
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| ==Notes==
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| {{reflist|2}}
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| ==References==
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| <!--<div style="font-size: 90%">-->
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| *{{Citation
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| | title=Was Calculus Invented in India?
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| | issue=1
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| | jstor=1558972
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| *Gupta, R. C. (1969) "Second Order of Interpolation of Indian Mathematics", Ind, J.of Hist. of Sc. 4 92-94
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| | editor1-first=Ivor
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| | title=Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences
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| | place=Baltimore, MD
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| }}.
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| *Parameswaran, S., ‘Whish’s showroom revisited’, Mathematical gazette 76, no. 475 (1992) 28-36
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| *{{Citation | last = Pingree | first = David | authorlink = David Pingree | title = Hellenophilia versus the History of Science | year = 1992 | journal = Isis | volume = 83 | issue = 4 | pages = 554–563 | jstor = 234257 | doi = 10.1086/356288}}
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| *{{Citation
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| | journal=Historia Mathematica
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| | volume=23
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| | issue=3
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| | year=1996
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| }}.
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| *{{Citation
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| | volume=28
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| | date=20 July 2007
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| | editor1-last=Katz
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| | editor1-first=Victor J.
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| | title=The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook
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| | volume=
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| }}.
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| *C. K. Raju. 'Computers, mathematics education, and the alternative epistemology of the calculus in the Yuktibhâsâ', ''Philosophy East and West'' '''51''', University of Hawaii Press, 2001.<!--website didn't have paper; removed website-->
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| *{{Citation
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| | first=Ranjan
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| | title=Discovery of the Series Formula for <math> \pi </math> by Leibniz, Gregory, and Nilakantha
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| | journal=Mathematics Magazine (Math. Assoc. Amer.)
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| | volume=63
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| | issue=5
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| | year=1990
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| }}.
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| *Sarma, K. V. and S. Hariharan: ''Yuktibhasa of Jyesthadeva : a book of rationales in Indian mathematics and astronomy - an analytical appraisal'', Indian J. Hist. Sci. 26 (2) (1991), 185-207
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| *{{Citation
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| | first=A. N.
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| | title=On the Use of Series in Hindu Mathematics
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| | journal=Osiris
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| | volume=1
| |
| | issue=
| |
| | year=1936
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| | pages=606–628
| |
| | jstor=301627
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| | doi=10.1086/368443
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| }}
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| *{{Citation
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| | year=2004
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| | edition=2
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| | title=Mathematics and its History
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| *Tacchi Venturi. 'Letter by Matteo Ricci to Petri Maffei on 1 Dec 1581', ''Matteo Ricci S.I., Le Lettre Dalla Cina 1580–1610'', vol. 2, Macerata, 1613.
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| </div>
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| ==External links==
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| * ''[http://www.infinityfoundation.com/mandala/t_es/t_es_agraw_kerala.htm The Kerala School, European Mathematics and Navigation]'', 2001.
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| *[http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Indian_mathematics.html An overview of Indian mathematics], ''[[MacTutor History of Mathematics archive]]'', 2002.
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| *[http://www-history.mcs.st-and.ac.uk/history/Projects/Pearce/index.html Indian Mathematics: Redressing the balance], ''MacTutor History of Mathematics archive'', 2002.
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| *[http://www-history.mcs.st-andrews.ac.uk/history/Projects/Pearce/Chapters/Ch9_1.html Keralese mathematics], ''MacTutor History of Mathematics archive'', 2002.
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| *[http://www-history.mcs.st-andrews.ac.uk/history/Projects/Pearce/Chapters/Ch9_4.html Possible transmission of Keralese mathematics to Europe], ''MacTutor History of Mathematics archive'', 2002.
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| *[http://www.physorg.com/news106238636.html "Indians predated Newton 'discovery' by 250 years"] ''phys.org,'' 2007
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| {{Indian mathematics}}
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| {{Ancient Dharmic centres of Higher Learning}}
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| [[Category:Hindu astronomy]]
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| [[Category:History of mathematics]]
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| [[Category:History of astronomy]]
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| [[Category:Kerala school|*]]
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| [[Category:History of Kerala]]
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| [[Category:Medieval Kerala]]
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