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| | She is known by the title of Myrtle Shryock. Years ago we moved to North Dakota. The preferred hobby for my kids and me is to play baseball but I haven't produced a dime with it. Hiring is her working day occupation now and she will not alter it anytime quickly.<br><br>my web site ... [http://www.fanproof.com/index.php?do=/profile-115513/info/ fanproof.com] |
| '''''Yuktibhāṣā''''' ({{lang-ml|യുക്തിഭാഷ}}; "Rationale in the Malayalam/Sanskrit language"<ref name=sarma/>) also known as '''''Gaṇitanyāyasaṅgraha''''' ("Compendium of astronomical rationale"),<ref name=sarma/> is a major [[treatise]] on [[Indian mathematics|mathematics]] and [[Hindu astronomy|astronomy]], written by [[India]]n astronomer [[Jyesthadeva]] of the [[Kerala school of astronomy and mathematics|Kerala school of mathematics]] in about AD 1530.<ref name="sarma">{{cite journal| author1=K V Sarma |authorlink=K. V. Sarma | author2=S Hariharan | title=Yuktibhāṣā of Jyeṣṭhadeva: A book on rationales in Indian Mathematics and Astronomy: An analytic appraisal
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| |url=http://www.new.dli.ernet.in/insa/INSA_1/20005ac0_185.pdf
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| | journal=Indian Journal of History of Science
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| | volume=26
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| | issue=2
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| | year=1991
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| | accessdate=2006-07-09
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| |format=PDF |archiveurl = http://web.archive.org/web/20060928203221/http://www.new.dli.ernet.in/insa/INSA_1/20005ac0_185.pdf <!-- Bot retrieved archive --> |archivedate = 2006-09-28}}
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| </ref> The treatise is a consolidation of the discoveries by [[Madhava of Sangamagrama]], [[Nilakantha Somayaji]], [[Parameshvara]], [[Jyeshtadeva]], [[Achyuta Pisharati]] and other astronomer-mathematicians of the Kerala school. ''Yuktibhasa'' is mainly based on Nilakantha's ''[[Tantrasangraha|Tantra Samgraha]]''.<ref name="infinity">{{cite web| publisher=D.P. Agrawal — Infinity Foundation |work=Indian Mathemematics|url=http://www.infinityfoundation.com/mandala/t_es/t_es_agraw_kerala.htm| title=The Kerala School, European Mathematics and Navigation| accessdate=2006-07-09}}
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| </ref> It is considered an early text on some of the foundations of [[calculus]] and predates those of [[Europe]]an mathematicians such as [[James Gregory (mathematician)|James Gregory]] by over a century.<ref name="MAT 314">{{cite web| publisher=Canisius College |work=MAT 314|url=http://www.canisius.edu/topos/rajeev.asp| title=Neither Newton nor Leibniz - The Pre-History of Calculus and Celestial Mechanics in Medieval Kerala| accessdate=2006-07-09}}</ref><ref name="scotlnd">{{cite web| publisher=School of Mathematics and Statistics University of St Andrews, Scotland |work=Indian Maths|url=http://www-history.mcs.st-andrews.ac.uk/HistTopics/Indian_mathematics.html| title=An overview of Indian mathematics| accessdate=2006-07-07}}</ref><ref name="pdffile3">{{cite web
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| | publisher=Prof.C.G.Ramachandran Nair |work=Government of Kerala — Kerala Call, September 2004|url=http://www.kerala.gov.in/keralcallsep04/p22-24.pdf| title=Science and technology in free India| accessdate=2006-07-09
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| |format=PDF}}</ref><ref name="charles">{{Citation
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| | author =[[C.M. Whish|Charles Whish]]
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| | year = 1834
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| | title = On the [[Hindu]] [[Quadrature of the circle]] and the [[infinite series]] of the proportion of the circumference to the diameter exhibited in the four [[Sastra]]s, the [[Tantrasangraha|Tantra Sahgraham]], Yucti Bhasha, [[Karanapaddhati|Carana Padhati]] and [[Sadratnamala]]
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| | journal = Transactions of the Royal Asiatic Society of Great Britain and Ireland
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| | publisher = [[Royal Asiatic Society of Great Britain and Ireland]]
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| | doi=10.1017/S0950473700001221
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| | volume=3
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| | issue=3
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| | pages=509–523
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| | jstor=25581775
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| }}</ref> However, the treatise was largely unnoticed beyond [[Kerala]], as the book was written in the local language of Malayalam. However, some have argued that mathematics from Kerala were transmitted to Europe (see [[Kerala school of astronomy and mathematics#Possibility of transmission of Kerala School results to Europe|Possible transmission of Keralese mathematics to Europe]]).
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| The work was unique for its time, since it contained [[Mathematical proof|proof]]s and derivations of the [[theorem]]s that it presented; something that was not usually done by any Indian [[mathematicians]] of that era.<ref name="jyest">{{cite web | |
| | publisher=School of Mathematics and Statistics University of St Andrews, Scotland |
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| work=Biography of Jyesthadeva
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| |url=http://www-gap.dcs.st-and.ac.uk/~history/Biographies/Jyesthadeva.html
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| | title=Jyesthadeva
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| | accessdate=2006-07-07
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| }}
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| </ref> Some of its important developments in analysis include: the [[infinite series]] expansion of a function, the [[power series]], the [[Taylor series]], the [[trigonometric series]] for [[sine]], [[cosine]], [[tangent (trigonometric function)|tangent]] and [[arctangent]], the second and third order Taylor series approximations of [[sine]] and [[cosine]], the power series of [[π]], π/4, [[θ]], the radius, diameter and circumference, and [[Integral test for convergence|tests of convergence]].
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| ==Contents==
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| ''Yuktibhasa'' contains most of the developments of earlier Kerala School mathematicians, particularly [[Madhava of Sangamagrama|Madhava]] and [[Nilakantha Somayaji|Nilakantha]]. The text is divided into two parts — the former deals with [[mathematical analysis]] of [[arithmetic]], [[algebra]], [[trigonometry]] and [[geometry]], [[Logistic function|logistic]]s, [[algebraic equation|algebraic problems]], [[Fraction (mathematics)|fraction]]s, [[Rule of three (mathematics)|Rule of three]], ''Kuttakaram'', [[circle]] and disquisition on R-Sine; and the latter about astronomy.<ref name="sarma"/>
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| ===Mathematics===
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| [[Image:Yuktibhasa.gif|200px|thumb|Explanation of the [[Law of sines|sine rule]] in ''Yuktibhasa'']]
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| As per the old Indian tradition of starting off new chapters with elementary content, the first four chapters of the ''Yuktibhasa'' contain elementary mathematics, such as division, proof of [[Pythagorean theorem]], [[square root]] determination, etc.<ref name="pdf2">{{cite web
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| | publisher=Dr Sarada Rajeev |
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| work=The Pre-History of Calculus and Celestial Mechanics in Medieval Kerala
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| |url=http://www.canisius.edu/topos/archives/rajeev2.pdf
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| | title=The Yuktibhasa Calculus Text
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| | accessdate=2006-07-09
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| |format=PDF}}
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| </ref> The radical ideas are not discussed until the sixth chapter on [[circumference]] of a [[circle]].
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| ''Yuktibhasa'' contains the derivation and proof of the [[power series]] for [[Inverse trigonometric function|inverse tangent]], discovered by Madhava.<ref name="infinity"/> In the text, Jyesthadeva describes Madhava's series in the following manner:
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| {{cquote|The first term is the product of the given sine and radius of the desired arc divided by the cosine of the arc. The succeeding terms are obtained by a process of iteration when the first term is repeatedly multiplied by the square of the sine and divided by the square of the cosine. All the terms are then divided by the odd numbers 1, 3, 5, .... The arc is obtained by adding and subtracting respectively the terms of odd rank and those of even rank. It is laid down that the sine of the arc or that of its complement whichever is the smaller should be taken here as the given sine. Otherwise the terms obtained by this above iteration will not tend to the vanishing magnitude.}}
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| This yields
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| : <math> r\theta={\frac {r \sin \theta }{\cos \theta
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| }}-(1/3)\,r\,{\frac { \left(\sin \theta \right) ^
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| {3}}{ \left(\cos \theta \right) ^{3}}}+(1/5)\,r\,{\frac {
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| \left(\sin \theta \right) ^{5}}{ \left(\cos
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| \theta \right) ^{5}}}-(1/7)\,r\,{\frac { \left(\sin \theta
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| \right) ^{7}}{ \left(\cos \theta \right) ^{
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| 7}}} + \cdots</math>
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| which further yields the theorem
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| :<math>\theta = \tan \theta - (1/3) \tan^3 \theta + (1/5) \tan^5 \theta - \cdots</math>
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| attributed to [[James Gregory (astronomer and mathematician)|James Gregory]], who discovered it three centuries after Madhava.
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| The text also elucidates Madhava's [[series (mathematics)|infinite series]] expansion of [[π]]:
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| :<math>\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots + \frac{(-1)^n}{2n + 1} + \cdots</math>
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| which he obtained from the power series expansion of the arc-tangent function.
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| Using a rational approximation of this series, he gave values of the number [[π]] as 3.14159265359 - correct to 11 decimals; and as 3.1415926535898 - correct to 13 decimals. These were the most accurate approximations of π after almost a thousand years.{{Citation needed|date=February 2007}}
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| The text describes that he gave two methods for computing the value of π.
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| *One of these methods is to obtain a rapidly converging series by transforming the original infinite series of π. By doing so, he obtained the infinite series
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| :<math>\pi = \sqrt{12}\left(1-{1\over 3\cdot3}+{1\over5\cdot 3^2}-{1\over7\cdot 3^3}+\cdots\right)</math>
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| and used the first 21 terms to compute an approximation of π correct to 11 decimal places as 3.14159265359.
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| *The other method was to add a remainder term to the original series of π. The remainder term was used
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| :<math>\frac{n^2 + 1}{4n^3 + 5n}</math>
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| in the infinite series expansion of <math>\frac{\pi}{4}</math> to improve the approximation of π to 13 decimal places of accuracy when n = 76.
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| Apart from these, the ''Yukthibhasa'' contains many [[elementary mathematics|elementary]], and complex mathematics, including,
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| * Proof for the expansion of the [[sine]] and [[cosine]] functions.
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| * Integer solutions of systems of first degree equations (solved using a system known as ''kuttakaram'')
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| * Rules for finding the sines and the cosines of the sum and difference of two [[angles]].
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| * The earliest statement of and the [[Taylor series]] (only some for some functions).
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| * Geometric derivations of series.
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| * Tests of [[Convergent series|convergence]] (often attributed to [[Augustin Louis Cauchy|Cauchy]])
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| * Fundamentals of calculus:<ref name="pdffile3"/> [[derivative|differentiation]], term by term [[Integral|integration]], [[iterative method]]s for solutions of [[Nonlinearity|non-linear]] equations, and the theory that the area under a curve is its integral.
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| Most of these results were long before their European counterparts, to whom credit was traditionally attributed.
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| ===Astronomy===
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| Chapters seven to seventeen of the text deals essentially with subjects of astronomy, viz. [[Planetary orbit]], [[Celestial sphere]], [[Right ascension|ascension]], [[declination]], directions and shadows, [[spherical trigonometry|spherical triangle]]s, [[ellipse]]s and [[parallax]] correction. The planetary theory described in the book is similar to that later adopted by [[Danish people|Danish]] astronomer [[Tycho Brahe]].<ref name="brahe">{{cite web
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| | publisher=India Resources |
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| work=South Asian history
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| |url=http://india_resource.tripod.com/mathematics.htm
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| | title=Science and Mathematics in India
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| | accessdate=2006-07-09
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| }}
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| </ref>
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| ==Modern edition of Yuktibhasa==
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| The importance of Yuktibhasa was brought to the attention of modern scholarship by [[C.M. Whish]] in 1834 through a paper published in the ''Transactions of the Royal Asiatic Society of Great Britain and Ireland''. However, an edition of the mathematics part of the text (along with notes in Malayalam) was published only in 1948 by Ramavarma Thampuran and Akhileswara Aiyar. For the first time, a critical edition of the entire Malayalam text, along with English translation and detailed explanatory notes, has been brought out in two volumes by [[Springer Science+Business Media|Springer]]
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| in 2008.<ref>{{cite book|last=Sarma|first=K.V.|authorlink=K. V. Sarma|coauthors=Ramasubramanian, K., Srinivas, M.D., Sriram, M.S.|title=Ganita-Yukti-Bhasa (Rationales in Mathematical Astronomy) of Jyesthadeva|publisher=Springer (jointly with Hindustan Book Agency, New Delhi)|year=2008|edition=1|series=Sources and Studies in the History of Mathematics and Physical Sciences |volume=Volume I: Mathematics Volume II: Astronomy|pages=LXVIII, 1084|isbn=978-1-84882-072-2|url=http://www.springer.com/math/history+of+mathematics/book/978-1-84882-072-2|accessdate=17 December 2009}}</ref>
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| A third volume presenting a critical edition of the Sanskrit Ganitayuktibhasa has been brought out by the Indian Institute of Advanced Study, Shimla in 2009.<ref>{{cite book|last=Sarma|first=K.V. |title=Ganita Yuktibhasa|publisher=Indian Institute of Advanced Study, Shimla, India|year=2009|volume=Volume III|isbn=81-7986-052-3|url=http://www.iias.org/p_ganita_yuktibhasa_volume-III.html|accessdate=16 December 2009|language=Malayalam and English}}</ref>
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| ==See also==
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| * [[Ganita-yukti-bhasa]]
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| * [[Indian mathematics]]
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| * [[Kerala school of astronomy and mathematics|Kerala School]]
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| * [[Kerala school of astronomy and mathematics#Possibility of transmission of Kerala School results to Europe|Possible transmission of Kerala mathematics to Europe]]
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| ==Notes==
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| <references/>
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| ==References==
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| <div style="font-size: 90%">
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| * {{cite book
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| | author =K V Sharma & S Hariharan
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| | year = 1990
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| | title = Yuktibhasa of Jyesthadeva — A book on rationales in Indian Mathematics and Astronomy - an analytic appraisal
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| | publisher = Indian Journal of History of Science
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| }}
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| </div>
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| ==External links==
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| * [http://www-gap.dcs.st-and.ac.uk/~history/Biographies/Jyesthadeva.html Biography of Jyesthadeva — School of Mathematics and Statistics University of St Andrews, Scotland]
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| {{Indian mathematics}}
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| {{Use dmy dates|date=October 2010}}
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| {{DEFAULTSORT:Yuktibhasa}}
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| [[Category:Astronomy books]]
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| [[Category:Astrological texts]]
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| [[Category:Indian mathematics]]
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| [[Category:Hindu astronomy]]
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| [[Category:Hindu astrology]]
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| [[Category:History of mathematics]]
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| [[Category:Kerala school]]
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| [[Category:Mathematics manuscripts]]
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