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

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 come down to us. 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

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:

${\displaystyle {\frac {1}{1-x}}=1+x+x^{2}+x^{3}+\cdots {\text{ for }}|x|<1}$^[7]

The Kerala school made intuitive use of

They used this to discover a semi-rigorous proof of the result:

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

for large n.

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

${\displaystyle r\arctan \left({\frac {y}{x}}\right)={\frac {1}{1}}\cdot {\frac {ry}{x}}-{\frac {1}{3}}\cdot {\frac {ry^{3}}{x^{3}}}+{\frac {1}{5}}\cdot {\frac {ry^{5}}{x^{5}}}-\cdots ,{\text{ where }}{\frac {y}{x}}\leq 1.}$

${\displaystyle r\sin {\frac {x}{r}}=x-x\cdot {\frac {x^{2}}{(2^{2}+2)r^{2}}}+x\cdot {\frac {x^{2}}{(2^{2}+2)r^{2}}}\cdot {\frac {x^{2}}{(4^{2}+4)r^{2}}}-\cdots }$

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

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

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

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

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

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

Their rational approximation of the error for the finite sum of their series are of particular interest. For example, the error, ${\displaystyle f_{i}(n+1)}$, (for n odd, and i = 1, 2, 3) for the series:

${\displaystyle {\frac {\pi }{4}}\approx 1-{\frac {1}{3}}+{\frac {1}{5}}-\cdots (-1)^{(n-1)/2}{\frac {1}{n}}+(-1)^{(n+1)/2}f_{i}(n+1)}$

where ${\displaystyle f_{1}(n)={\frac {1}{2n}},\ f_{2}(n)={\frac {n/2}{n^{2}+1}},\ f_{3}(n)={\frac {(n/2)^{2}+1}{(n^{2}+5)n/2}}.}$

They manipulated the terms, using the partial fraction expansion of :${\displaystyle {\frac {1}{n^{3}-n}}}$ to obtain a more rapidly converging series for ${\displaystyle \pi }$:^[1]

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

They used the improved series to derive a rational expression,^[1] ${\displaystyle 104348/33215}$ for ${\displaystyle \pi }$ correct up to nine decimal places, i.e. ${\displaystyle 3.141592653}$. 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

• Indian astronomy
• Indian mathematics
• Indian mathematicians
• History of mathematics
• A History of the Kerala School of Hindu Astronomy
• List of astronomers and mathematicians of the Kerala school

Notes

References

• Bressoud, David (2002), "Was Calculus Invented in India?", The College Mathematics Journal, 33 (1): 2–13,
  JSTOR 1558972
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• 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
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• Joseph, G. G. (2000), The Crest of the Peacock: The Non-European Roots of Mathematics, Princeton, NJ:
  ISBN 0-691-00659-8
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• Katz, Victor J. (1995), "Ideas of Calculus in Islam and India", Mathematics Magazine, 68 (3): 163–174,
  JSTOR 2691411
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• Parameswaran, S. (1992) "Whish's showroom revisited",
  Mathematical Gazette
  76, no. 475 pages 28–36
• 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
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• 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
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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.
• Roy, Ranjan (1990), "Discovery of the Series Formula for ${\displaystyle \pi }$ by Leibniz, Gregory, and Nilakantha", Mathematics Magazine, 63 (5): 291–306,
  JSTOR 2690896
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• 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,
  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

Wikimedia Commons has media related to Kerala school of astronomy and mathematics.

• 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