imported>MPGuy2824: /* Modern editions */ fixed long volume error in ref

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Modern editions: fixed long volume error in ref

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{{short description|Treatise on mathematics and astronomy}}
{{Use dmy dates|date=February 2020}}
{{italic title}}
'''''Yuktibhāṣā''''' ({{lang-ml|യുക്തിഭാഷ|lit=Rationale}}<ref name="sarma" />), also known as '''{{transl|ml|Gaṇitanyāyasaṅgraha}}''' (''Compendium of Astronomical Rationale''),<ref name="sarma" /> is a major [[treatise]] on [[Indian mathematics|mathematics]] and [[Hindu astronomy|astronomy]], written by the [[India]]n astronomer [[Jyesthadeva]] of the [[Kerala school of astronomy and mathematics|Kerala school]] of mathematics around 1530.<ref name="sarma">{{cite journal| author1=K V Sarma |author-link=K. V. Sarma | author2=S Hariharan | title=Yuktibhāṣā of Jyeṣṭhadeva: A book on rationales in Indian Mathematics and Astronomy: An analytic appraisal
|url=http://www.new.dli.ernet.in/insa/INSA_1/20005ac0_185.pdf
| journal=Indian Journal of History of Science
| volume=26
| issue=2
| date=1991
| access-date=9 July 2006
|archive-url = https://web.archive.org/web/20060928203221/http://www.new.dli.ernet.in/insa/INSA_1/20005ac0_185.pdf  |archive-date = 28 September 2006}}
</ref> The treatise, written in Malayalam, is a consolidation of the discoveries by [[Madhava of Sangamagrama]], [[Nilakantha Somayaji]], [[Parameshvara]], [[Jyeshtadeva]], [[Achyuta Pisharati]], and other astronomer-mathematicians of the Kerala school.

The work was unique for its time, since it contained [[Mathematical proof|proof]]s and derivations of the [[theorem]]s that it presented; something unusual for Indian mathematicians of that era.<ref name="jyest">{{cite web|title=Jyesthardeva|url=http://www-gap.dcs.st-and.ac.uk/~history/Biographies/Jyesthadeva.html|work=Biography of Jyesthadeva|publisher=School of Mathematics and Statistics University of St Andrews, Scotland|access-date=7 July 2006}}
</ref> Some of its important topics include the [[infinite series]] expansions of functions; [[power series]], including of [[Pi|π]] and π/4; [[trigonometric series]] of [[sine]], [[cosine]], and [[arctangent]]; [[Taylor series]], including second and third order approximations of [[sine]] and [[cosine]]; radii, diameters and circumferences; and [[Integral test for convergence|tests of convergence]].

{{transl|ml|Yuktibhāṣā}} 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| access-date=9 July 2006}}
</ref> It is considered an early text on the ideas of [[calculus]], predating Newton and Leibniz by centuries.<ref name="raju">
{{cite journal| author1=C. K. Raju | title=Computers, mathematics education, and the alternative epistemology of the calculus in the Yuktibhāṣā
|url=http://ckraju.net/papers/Hawaii.pdf
| journal=Philosophy East & West
| volume=51
| issue=3
| date=2001
| pages=325–362
| doi=10.1353/pew.2001.0045
| access-date=11 February 2020
}}
</ref><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|access-date=9 July 2006|url-status=dead|archive-url=https://web.archive.org/web/20060806040307/http://www.canisius.edu/topos/rajeev.asp|archive-date=6 August 2006}}</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| access-date=7 July 2006}}</ref><ref name="pdffile3">{{cite web|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|access-date=9 July 2006|url-status=dead|archive-url=https://web.archive.org/web/20060821195309/http://www.kerala.gov.in/keralcallsep04/p22-24.pdf|archive-date=21 August 2006}}</ref><ref name="charles">{{Citation
| author =Charles Whish
| author-link =C.M. Whish
| date = 1834
| 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 Sastras, the Tantra Sahgraham, Yucti Bhasha, Carana Padhati and Sadratnamala
| journal = Transactions of the Royal Asiatic Society of Great Britain and Ireland
| doi=10.1017/S0950473700001221
| volume=3
| issue=3
| pages=509–523
| jstor=25581775
| url =https://zenodo.org/record/2223599
}}</ref> The treatise was largely unnoticed outside India, as it was written in the local language of Malayalam. It is often generalized that early Indian scholars in astronomy and computation lacked in proofs, but {{transl|ml|Yuktibhāṣā}} demonstrates otherwise.<ref name=":0">{{Cite journal|last=Divakaran|first=P. P.|date=2007|title=The First Textbook of Calculus: "Yuktibhāṣā"|journal=Journal of Indian Philosophy|volume=35|issue=5/6|pages=417–443|doi=10.1007/s10781-007-9029-1|jstor=23497280|issn=0022-1791}}</ref> In modern times, due to wider international cooperation in mathematics, the wider world has taken notice of the work. For example, both Oxford University and the Royal Society of Great Britain have given attribution to pioneering mathematical theorems of Indian origin that predate their Western counterparts.<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|access-date=9 July 2006|url-status=dead|archive-url=https://web.archive.org/web/20060806040307/http://www.canisius.edu/topos/rajeev.asp|archive-date=6 August 2006}}</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| access-date=7 July 2006}}</ref><ref name="pdffile3">{{cite web|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|access-date=9 July 2006|url-status=dead|archive-url=https://web.archive.org/web/20060821195309/http://www.kerala.gov.in/keralcallsep04/p22-24.pdf|archive-date=21 August 2006}}</ref><ref name="charles">{{Citation
| author =Charles Whish
| author-link =C.M. Whish
| date = 1834
| 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 Sastras, the Tantra Sahgraham, Yucti Bhasha, Carana Padhati and Sadratnamala
| journal = Transactions of the Royal Asiatic Society of Great Britain and Ireland
| doi=10.1017/S0950473700001221
| volume=3
| issue=3
| pages=509–523
| jstor=25581775
| url =https://zenodo.org/record/2223599
}}</ref>

==Contents==
{{transl|ml|Yuktibhāṣā}} contains most of the developments of the earlier Kerala school, particularly [[Madhava of Sangamagrama|Madhava]] and [[Nilakantha Somayaji|Nilakantha]]. The text is divided into two parts – the former deals with [[mathematical analysis]] and the latter with astronomy.<ref name="sarma"/>

===Mathematics===
[[Image:Yuktibhasa.svg|200px|thumb|Explanation of the [[Law of sines|sine rule]] in {{transl|ml|Yuktibhāṣā}}]]
The first four chapters of the {{transl|ml|Yuktibhāṣā}} contain elementary mathematics, such as division, the [[Pythagorean theorem]], [[square root|square roots]], etc.<ref name="pdf2">{{cite web
| publisher=Dr Sarada Rajeev |
work=The Pre-History of Calculus and Celestial Mechanics in Medieval Kerala
|url=http://www.canisius.edu/topos/archives/rajeev2.pdf
| title=The Yuktibhasa Calculus Text
| access-date=9 July 2006
}}
</ref> Novel ideas are not discussed until the sixth chapter on [[circumference]] of a [[circle]]. {{transl|ml|Yuktibhāṣā}} contains a derivation and proof for the [[power series]] of [[Inverse trigonometric function|inverse tangent]], discovered by Madhava.<ref name="infinity"/> In the text, Jyesthadeva describes Madhava's series in the following manner:
{{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.}}
In modern mathematical notation,

:<math> r\theta={r\frac{\sin\theta}{\cos\theta}}
-\frac{r}{3}\frac{\sin^3\theta}{\cos^3\theta}
+\frac{r}{5}\frac{\sin^5\theta}{\cos^5\theta}
-\frac{r}{7}\frac{\sin^7\theta}{\cos^7\theta}
+\cdots</math>

or, expressed in terms of tangents,
:<math>\theta = \tan\theta - \frac13 \tan^3\theta + \frac15 \tan^5\theta - \cdots \ ,</math>

which has been previously attributed to [[James Gregory (astronomer and mathematician)|James Gregory]], who published it in 1667.

The text also contains Madhava's [[series (mathematics)|infinite series]] expansion of [[Pi|π]] which he obtained from the expansion of the arc-tangent function.

:<math>\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots + \frac{(-1)^n}{2n + 1} + \cdots</math>

Using a rational approximation of this series, he gave values of the number [[Pi|π]] as 3.14159265359, correct to 11 decimals, and as 3.1415926535898, correct to 13 decimals.

The text describes two methods for computing the value of π. First, obtain a rapidly converging series by transforming the original infinite series of π. By doing so, the first 21 terms of the infinite series

:<math>\pi = \sqrt{12}\left(1-{1\over 3\cdot3}+{1\over5\cdot 3^2}-{1\over7\cdot 3^3}+\cdots\right)</math>

was used to compute the approximation to 11 decimal places. The other method was to add a remainder term to the original series of π. The remainder term<math display="inline">\frac{n^2 + 1}{4n^3 + 5n}</math> was used 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.

Apart from these, the {{transl|ml|Yuktibhāṣā}} contains many [[elementary mathematics|elementary]] and complex mathematical topics, including,

* Proofs for the expansion of the [[sine]] and [[cosine]] functions
*The [[Sum and difference formula (trigonometry)|sum and difference formulae]] for sine and cosine
* Integer solutions of [[System of linear equations|systems of linear equations]] (solved using a system known as ''kuttakaram'')
* Geometric derivations of series
* Early statements of [[Taylor series]] for some functions
* Tests of [[Convergent series|convergence]] for sums
*[[derivative|Differentiation]], [[Integral|integration]], [[iterative method]]s for solutions of [[Nonlinearity|non-linear]] equations, and the theory that the area under a curve is its integral.<ref name="pdffile3" />

===Astronomy===
Chapters seven to seventeen deal with subjects of astronomy: [[Planetary orbit|planetary orbits]], [[Celestial sphere|celestial spheres]], [[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|title=Science and Mathematics in India|url=http://india_resource.tripod.com/mathematics.htm|work=South Asian History|publisher=India Resources|url-status=dead|archive-url=https://web.archive.org/web/20121017083322/http://india_resource.tripod.com/mathematics.htm|archive-date=17 October 2012|access-date=6 May 2020}}
</ref>

==Modern editions==

The importance of {{transl|ml|Yuktibhāṣā}} was brought to the attention of modern scholarship by [[C. M. Whish]] in 1832 through a paper published in the ''Transactions of the Royal Asiatic Society of Great Britain and Ireland''.<ref name=":0" /> However, the mathematics part of the text, along with notes in Malayalam, was first published only in 1948 by Rama Varma Thampuran and Akhileswara Aiyar.<ref name=sarma/>

For the first time, an edition of the entire Malayalam text, alongside an English translation and detailed explanatory notes, was published by [[Springer Science+Business Media|Springer]] in 2008.<ref>{{cite book|last=Sarma|first=K.V.|author-link=K. V. Sarma|author2=Ramasubramanian, K. |author3=Srinivas, M.D. |author4=Sriram, M.S. |title=Ganita-Yukti-Bhasa (Rationales in Mathematical Astronomy) of Jyesthadeva|publisher=Springer (jointly with Hindustan Book Agency, New Delhi)|date=2008|edition=1st|series=Sources and Studies in the History of Mathematics and Physical Sciences |volume=I-II|pages=LXVIII, 1084|isbn=978-1-84882-072-2|url=https://www.springer.com/math/history+of+mathematics/book/978-1-84882-072-2|access-date=17 December 2009|bibcode=2008rma..book.....S}}</ref>

A third volume presenting a critical edition of the Sanskrit Ganitayuktibhasa has been published 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|date=2009|volume=III|isbn=978-81-7986-052-6|url=http://www.iias.org/p_ganita_yuktibhasa_volume-III.html|access-date=16 December 2009|language=ml, en|url-status=dead|archive-url=https://web.archive.org/web/20100317202603/http://www.iias.org/p_ganita_yuktibhasa_volume-III.html|archive-date=17 March 2010}}</ref>

==See also==
* ''[[Ganita-yukti-bhasa]]''
* [[Indian mathematics]]
* [[Kerala school of astronomy and mathematics|Kerala School]]

==References==
{{Reflist}}

==External links==
* [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]

{{Indian mathematics}}
{{Malayalam Literature |state=collapsed}}

{{DEFAULTSORT:Yuktibhasa}}
[[Category:Astronomy books]]
[[Category:Indian mathematics]]
[[Category:Hindu astronomy]]
[[Category:Hindu astrological texts]]
[[Category:History of mathematics]]
[[Category:Kerala school of astronomy and mathematics]]
[[Category:Mathematics manuscripts]]
[[Category:Indian astronomy texts]]

Yuktibhāṣā - Revision history

imported>MPGuy2824: /* Modern editions */ fixed long volume error in ref