Kerala school of astronomy and mathematics

Kerala school of astronomy and mathematics
Kerala school of astronomy and mathematics
	Chain of teachers of the Kerala school
Location
	Central and Northern Kerala, India
Information
Type	Astronomy, Mathematics, Science
Founder	Madhava of Sangamagrama

The Kerala school of astronomy and mathematics or the Kerala school was a school of mathematics and astronomy founded by Madhava of Sangamagrama in Tirur, Malappuram, 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 its original discoveries seem to have ended with Narayana 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 in Sanskrit verse in a book by Neelakanta called Tantrasangraha (around 1500), 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. 1530), 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]^[3]^[4]

Background

Islamic scholars nearly developed a general formula for finding integrals of polynomials 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 to 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 Isaac 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]

Contributions

Pages from the Yuktibhasa c.1530

Infinite series and calculus

The Kerala school has made a number of contributions to the fields of infinite series and calculus. These include the following infinite geometric series:

{\frac {1}{1-x}}=1+x+x^{2}+x^{3}+\cdots {\text{ for }}|x|<1

^[7]

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:

1^{p}+2^{p}+\cdots +n^{p}\approx {\frac {n^{p+1}}{p+1}}

for large n.

They applied ideas from (what was to become) differential and integral calculus to obtain (Taylor–Maclaurin) infinite series for $\sin x$ , $\cos x$ , and $\arctan x$ .^[8] The Tantrasangraha-vakhya gives the series in verse, which when translated to mathematical notation, can be written as:^[1]

r\arctan \left({\frac {y}{x}}\right)={\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 ,{\text{ where }}{\frac {y}{x}}\leq 1.

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}}}-\cdots

r\left(1-\cos {\frac {x}{r}}\right)=r{\frac {x^{2}}{(2^{2}-2)r^{2}}}-r{\frac {x^{2}}{(2^{2}-2)r^{2}}}\cdot {\frac {x^{2}}{(4^{2}-4)r^{2}}}+\cdots

where, for $r=1,$ the series reduce to the standard power series for these trigonometric functions, for example:

\sin x=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots

and

\cos x=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots

(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 $\arctan x$ to obtain an infinite series expression (later known as Gregory series) for $\pi$ :^[1]

{\frac {\pi }{4}}=1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots

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:

{\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)

where

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}}.

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]

{\frac {\pi }{4}}={\frac {3}{4}}+{\frac {1}{3^{3}-3}}-{\frac {1}{5^{3}-5}}+{\frac {1}{7^{3}-7}}-\cdots

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,^[9] though the notion of a function, or of exponential or logarithmic functions, was not yet formulated.

Recognition

In 1825 John Warren published a memoir on the division of time in southern India,^[10] called the Kala 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 Englishman C. 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 by C. T. Rajagopal and his associates. Their work includes commentaries on the proofs of the arctan series in Yuktibhasa given in two papers,^[12]^[13] a commentary on the Yuktibhasa's proof of the sine and cosine series^[14] and two papers that provide the Sanskrit verses of the Tantrasangrahavakhya for the series for arctan, sin, and cosine (with English translation and commentary).^[15]^[16]

In 1972 K. V. Sarma published his A History of the Kerala School of Hindu Astronomy which described features of the School such as the continuity of knowledge transmission from the 13th to the 17th century: Govinda Bhattathiri to Parameshvara to Damodara to Nilakantha Somayaji to Jyesthadeva to Acyuta Pisarati. Transmission from teacher to pupil conserved knowledge in "a practical, demonstrative discipline like astronomy at a time when there was not a proliferation of printed books and public schools."

In 1994 it was argued that the heliocentric model had been adopted about 1500 A.D. in Kerala.^[17]

Possible transmission of Kerala school results to Europe

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".^[8]^[21] V. J. Katz notes some of the ideas of the Kerala school have similarities to the work of 11th-century Iraqi scholar Ibn al-Haytham,^[9] suggesting a possible transmission of ideas from Islamic mathematics to Kerala.^[22]

Both Indian and Arab 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 the derivative and the integral, show the connection between the two, and turn calculus into the great problem-solving tool we have today", like Newton and Leibniz.^[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 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".^[9] 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.^[9]

Notes

^ ^a ^b ^c ^d ^e ^f ^g ^h Roy, Ranjan. 1990. "Discovery of the Series Formula for $\pi$ by Leibniz, Gregory, and Nilakantha." Mathematics Magazine (Mathematical Association of America) 63(5):291–306.
^ (Stillwell 2004, p. 173)
^ (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 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."
^ 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."
^ Singh, A. N. (1936). "On the Use of Series in Hindu Mathematics". Osiris. 1: 606–628. doi:10.1086/368443. S2CID 144760421.
^ ^a ^b Bressoud, David. 2002. "Was Calculus Invented in India?" The College Mathematics Journal (Mathematical Association of America). 33(1):2–13.
^ ^a ^b ^c ^d ^e ^f ^g Katz, V. J. 1995. "Ideas of Calculus in Islam and India." (pdf) Mathematics Magazine (Mathematical Association of America), 68(3):163-174.
^ John Warren (1825) A Collection of Memoirs on Various Modes According to which Nations of the Southern Part of India Divide Time from Google Books
^ Whish, Charles M. (1835). "XXXIII. On the Hindú Quadrature of the Circle, and the infinite Series of the proportion of the circumference to the diameter exhibited in the four S'ástras, the Tantra Sangraham, the Yucti Bháshá, Carana Padhati, and Sadratnamáka". Transactions of the Royal Asiatic Society. 3: 509–523.
^ 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. S2CID 51861422.
^ Rajagopal, C.; Rangachari, M. S. (1986). "On Medieval Kerala Mathematics". Archive for History of Exact Sciences. 35 (2): 91–99. doi:10.1007/BF00357622. S2CID 121678430.
^ K. Ramasubramanian, M. D. Srinivas & M. S. Sriram (1994) Modification of the earlier Indian planetary theory by the Kerala astronomers (c. 1500 A.D.) and the implied heliocentric picture of planetary motion, Current Science 66 (10): 784–90
^ 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. S2CID 170341845.
^ ^a ^b Almeida, 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.
^ Katz 1995, p. 174.

References

Bressoud, David (2002), "Was Calculus Invented in India?", The College Mathematics Journal, 33 (1): 2–13, doi:10.2307/1558972, JSTOR 1558972.
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, ISBN 0-8018-7396-7.
Joseph, G. G. (2000), The Crest of the Peacock: The Non-European Roots of Mathematics, Princeton, NJ: Princeton University Press, ISBN 0-691-00659-8.
Katz, Victor J. (1995), "Ideas of Calculus in Islam and India", Mathematics Magazine, 68 (3): 163–174, doi:10.2307/2691411, JSTOR 2691411.
Parameswaran, S. (1992) "Whish's showroom revisited", Mathematical Gazette 76, no. 475 pages 28–36
Pingree, David (1992), "Hellenophilia versus the History of Science", Isis, 83 (4): 554–563, Bibcode:1992Isis...83..554P, doi:10.1086/356288, JSTOR 234257, S2CID 68570164
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, ISBN 978-0-691-11485-9.
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.
Roy, Ranjan (1990), "Discovery of the Series Formula for $\pi$ by Leibniz, Gregory, and Nilakantha", Mathematics Magazine, 63 (5): 291–306, doi:10.2307/2690896, JSTOR 2690896.
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.
Singh, A. N. (1936), "On the Use of Series in Hindu Mathematics", Osiris, 1: 606–628, doi:10.1086/368443, JSTOR 301627, S2CID 144760421
Stillwell, John (2004), Mathematics and its History (2 ed.), Berlin and New York: Springer, 568 pages, ISBN 0-387-95336-1.
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.

External links

An overview of Indian mathematics, MacTutor History of Mathematics archive, 2002.
Indian Mathematics: Redressing the balance, MacTutor History of Mathematics archive, 2002.
Keralese mathematics, MacTutor History of Mathematics archive, 2002.
Possible transmission of Keralese mathematics to Europe, MacTutor History of Mathematics archive, 2002.
"Indians predated Newton 'discovery' by 250 years" phys.org, 2007

[roy-1] ^ ^a ^b ^c ^d ^e ^f ^g ^h Roy, Ranjan. 1990. "Discovery of the Series Formula for $\pi$ by Leibniz, Gregory, and Nilakantha." Mathematics Magazine (Mathematical Association of America) 63(5):291–306.

[2] (Stillwell 2004, p. 173)

[3] (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."

[4] 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"

[5] 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."

[6] 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."

[singh-7] Singh, A. N. (1936). "On the Use of Series in Hindu Mathematics". Osiris. 1: 606–628. doi:10.1086/368443. S2CID 144760421.

[bressoud-8] Bressoud, David. 2002. "Was Calculus Invented in India?" The College Mathematics Journal (Mathematical Association of America). 33(1):2–13.

[katz-9] ^ ^a ^b ^c ^d ^e ^f ^g Katz, V. J. 1995. "Ideas of Calculus in Islam and India." (pdf) Mathematics Magazine (Mathematical Association of America), 68(3):163-174.

[10] John Warren (1825) A Collection of Memoirs on Various Modes According to which Nations of the Southern Part of India Divide Time from Google Books

[whish-11] Whish, Charles M. (1835). "XXXIII. On the Hindú Quadrature of the Circle, and the infinite Series of the proportion of the circumference to the diameter exhibited in the four S'ástras, the Tantra Sangraham, the Yucti Bháshá, Carana Padhati, and Sadratnamáka". Transactions of the Royal Asiatic Society. 3: 509–523.

[12] Rajagopal, C.; Rangachari, M. S. (1949). "A Neglected Chapter of Hindu Mathematics". Scripta Mathematica. 15: 201–209.

[13] Rajagopal, C.; Rangachari, M. S. (1951). "On the Hindu proof of Gregory's series". Scripta Mathematica. 17: 65–74.

[14] 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.

[15] 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. S2CID 51861422.

[16] Rajagopal, C.; Rangachari, M. S. (1986). "On Medieval Kerala Mathematics". Archive for History of Exact Sciences. 35 (2): 91–99. doi:10.1007/BF00357622. S2CID 121678430.

[17] K. Ramasubramanian, M. D. Srinivas & M. S. Sriram (1994) Modification of the earlier Indian planetary theory by the Kerala astronomers (c. 1500 A.D.) and the implied heliocentric picture of planetary motion, Current Science 66 (10): 784–90

[18] A. K. Bag (1979) Mathematics in ancient and medieval India. Varanasi/Delhi: Chaukhambha Orientalia. page 285.

[19] 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. S2CID 170341845.

[almeida-20] Almeida, 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-21] 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.

[22] Katz 1995, p. 174.

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