{{short description|Treatise on mathematics and astronomy}} {{Use dmy dates|date=February 2020}} {{italic title}} {{Infobox book | name = Yuktibhasa | title_orig = | translator = | image = Yuktibhasa cover.jpg | caption = Front and back cover of the [[Palm-leaf manuscript]]s of the Yuktibhasa, composed by [[Jyesthadeva]] in 1530 | author = [[Jyesthadeva]] | cover_artist = | country = Modern-day [[Kerala]], [[India]] | language = [[Malayalam]] | series = | subject = | genre = [[Mathematics]] and [[Astronomy]] | publisher = | pub_date = 1530 | english_pub_date = 2008 | media_type = | pages = | isbn = | oclc = | dewey = | congress = | preceded_by = | followed_by = | orig_lang_code = ml | native_wikisource = | wikisource = | image_size = 300px }} '''''Yuktibhāṣā''''' ({{langx|ml|യുക്തിഭാഷ|lit=Rationale}}), also known as '''Gaṇita-yukti-bhāṣā'''{{rp|xxi}} and '''{{Transliteration|ml|Gaṇitanyāyasaṅgraha}}''' ([[English language|English:]] ''Compendium of Astronomical Rationale''), is a treatise on [[Indian mathematics|mathematics]] and [[Hindu astronomy|astronomy]], written by the [[India]]n astronomer [[Jyeṣṭhadeva]] of the [[Kerala school of astronomy and mathematics|Kerala school of mathematics]] around 1530.{{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}} The treatise, written in Malayalam, is a consolidation of the discoveries by [[Madhava of Sangamagrama]], [[Nilakantha Somayaji]], [[Parameshvara Nambudiri]], Jyeṣṭhadeva, [[Acyuta Piṣāraṭi|Achyuta Piṣāraṭi]], and other astronomer-mathematicians of the Kerala school. It also exists in a Sanskrit version, with unclear author and date, composed as a rough translation of the Malayalam original. The work contains [[Mathematical proof|proof]]s and derivations of the [[theorem]]s that it presents. The {{Transliteration|ml|Yuktibhāṣā}} demonstrates that at least some early Indian scholars in astronomy and computation had the concept of proofs.{{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|s2cid=170254981|issn=0022-1791}} Some of its important topics include the [[Series (mathematics)|infinite series]] expansions of functions; [[power series]], including of [[Pi|π]] and π/4; [[trigonometric series]] of [[Sine and cosine|sine, cosine]], and [[Inverse trigonometric functions|arctangent]]; [[Taylor series]], including second and third order approximations of sine and cosine; radii, diameters, and circumferences. {{Transliteration|ml|Yuktibhāṣā}} mainly gives rationale for the results in Nilakantha's ''[[Tantrasamgraha]]''.{{cite book|author=Glen van Brummelen|title=The mathematics of the heavens and the earth: The early history of trigonometry|publisher=[[Princeton University Press]]|date=2009|pages=128–129|isbn=9780691129730|url=http://press.princeton.edu/titles/8956.html}} It is regarded as an early work containing techniques involving infinite series, including series expansions of certain trigonometric functions, predating the works of Newton and Leibniz by approximately two centuries.{{Citation | last1 = Rajagopal | first1 = C. | last2 = Rangachari | first2 = M. S. | year = 1977 | title = On an untapped source of medieval Keralese mathematics | doi = 10.1007/BF00348142 | journal = Archive for History of Exact Sciences | volume = 18 | issue = 2 | pages = 89–102 | s2cid = 51861422 | postscript = . }}{{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 | doi-access =free }}{{Cite book |last=George Gheverghese Joseph |url=http://archive.org/details/crestofpeacockno00jose |title=The crest of the peacock |date=2000 |publisher=Princeton University Press |others=Internet Archive |isbn=978-0-691-00659-8}} However, it did not combine several ideas under the unifying concepts of the [[derivative]] and the [[integral]], show the connection between the two, or turn calculus into the powerful problem-solving tool we have today.{{Cite journal|last=Katz |first=Victor J. |author-link=Victor J. Katz |date=June 1995 |title=Ideas of Calculus in Islam and India |url=https://www.tandfonline.com/doi/full/10.1080/0025570X.1995.11996307 |journal=[[Mathematics Magazine]] |language=en |volume=68 |issue=3 |pages=163–174 |doi=10.1080/0025570X.1995.11996307 |issn=0025-570X |jstor=2691411|url-access=subscription }} The treatise was largely unnoticed outside India, as it was written in the local language of Malayalam. In modern times, due to wider international cooperation in mathematics, the wider world has taken notice of the work. For example, the [[University of Oxford]] and the British [[Royal Society]] have given attribution to pioneering mathematical theorems of Indian origin that predate their Western counterparts.{{Cite web |title=Indian mathematics |url=https://mathshistory.st-andrews.ac.uk/HistTopics/Indian_mathematics/ |access-date=2026-03-15 |website=Maths History |language=en}} ==Contents== {{Transliteration|ml|Yuktibhāṣā}} contains most of the developments of the earlier Kerala school, particularly those of [[Madhava of Sangamagrama|Madhava]] and [[Nilakantha Somayaji]]. The text is divided into two parts – the former deals with [[mathematical analysis]] and the latter with astronomy. Beyond this, the continuous text does not have any further division into subjects or topics, so published editions divide the work into chapters based on editorial judgment.{{rp|xxxvii}} [[File:Pages from Yuktibhasa.jpg|center|thumb|420x420px|Pages from the ''Yuktibhasa'']] ===Mathematics=== [[Image:Yuktibhasa.svg|thumb|Explanation of the [[Law of sines|sine rule]] in {{Transliteration|ml|Yuktibhāṣā}}]] The subjects treated in the mathematics part of the {{Transliteration|ml|Yuktibhāṣā}} can be divided into seven chapters:{{rp|xxxvii}} # ''parikarma'': logistics (the eight mathematical operations) # ''daśapraśna'': ten problems involving logistics # ''bhinnagaṇita'': arithmetic of fractions # ''trairāśika'': rule of three # ''[[Kuṭṭaka|kuṭṭakāra]]'': pulverisation (linear indeterminate equations) # ''paridhi-vyāsa'': relation between circumference and diameter: infinite series and approximations for [[Pi|π]] # ''jyānayana'': derivation of Rsines{{Clarify|reason=Rsines?|date=March 2026}}: infinite series and approximations for sines.For more details on contents see Kinokuniya DataBase: {{cite web|url=http://bookwebpro.kinokuniya.co.jp/booksea.cgi?ISBN=1848820720http%3A%2F%2Fwww.buscalibros.cl%2Flibro.php%3Flibro%3D2104208|title=Ganita-yukti-bhasa (Rationales in Mathematical Astronomy) of Jyesthadeva|access-date=1 May 2010|archive-date=20 July 2011|archive-url=https://archive.today/20110720103656/http://bookwebpro.kinokuniya.co.jp/booksea.cgi?ISBN=1848820720http://www.buscalibros.cl/libro.php%3Flibro=2104208|url-status=dead}} The first four chapters of the section contain elementary mathematics, such as division, the [[Pythagorean theorem]], [[square root]]s, etc.{{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 | archive-date=8 August 2006 | archive-url=https://web.archive.org/web/20060808185740/http://www.canisius.edu/topos/archives/rajeev2.pdf | url-status=dead }} Novel ideas are not discussed until the sixth chapter on the [[circumference]] of a [[circle]]. {{Transliteration|ml|Yuktibhāṣā}} contains a derivation and proof for the [[power series]] of [[Inverse trigonometric functions|inverse tangent]] discovered by Madhava.{{Cite journal | last = Bressoud | first = David | author-link = David Bressoud | title = Was Calculus Invented in India? | journal = College Mathematics Journal | volume = 33 | issue = 1 | pages = 2–13 | year = 2002 | doi=10.2307/1558972| jstor = 1558972 }} In the text, Jyeṣṭhadeva 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, : 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 or, expressed in terms of tangents, :\theta = \tan\theta - \frac13 \tan^3\theta + \frac15 \tan^5\theta - \cdots \ , which in Europe was conventionally called ''[[Gregory's series]]'' after [[James Gregory (astronomer and mathematician)|James Gregory]], who independently discovered it in 1671. The text also contains Madhava's infinite series expansion of π which he obtained from the expansion of the arc-tangent function. :\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots + \frac{(-1)^n}{2n + 1} + \cdots \ , which in Europe was conventionally called ''[[Leibniz formula for π|Leibniz's series]]'', after [[Gottfried Leibniz]] who independently discovered it in 1673. Using a rational approximation of this series, Jyeṣṭhadeva gave values of 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 :\pi = \sqrt{12}\left(1-{1\over 3\cdot3}+{1\over5\cdot 3^2}-{1\over7\cdot 3^3}+\cdots\right) 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\frac{n^2 + 1}{4n^3 + 5n} was used in the infinite series expansion of \frac{\pi}{4} to improve the approximation of π to 13 decimal places of accuracy when ''n''=76.{{Cite web |title=Madhava - Biography |url=https://mathshistory.st-andrews.ac.uk/Biographies/Madhava/ |access-date=2025-02-18 |website=Maths History |language=en}} Apart from these, the {{Transliteration|ml|Yuktibhāṣā}} contains many [[elementary mathematics|elementary]] and complex mathematical topics, including,{{citation needed|date=October 2023}} * Proofs for the expansion of the sine and cosine functions *The [[List of trigonometric identities|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 * Statements of Taylor series for some functions ===Astronomy=== Chapters eight to seventeen deal with subjects of astronomy: [[planetary orbit]]s, [[celestial sphere]]s, [[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]].{{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}} The topics covered in the eight chapters are computation of mean and true longitudes of planets, Earth and celestial spheres, fifteen problems relating to ascension, declination, longitude, etc., determination of time, place, direction, etc., from gnomonic shadow, eclipses, [[Vyatipāta|Vyatipata]] (when the sun and moon have the same declination), visibility correction for planets and phases of the moon. Specifically,{{rp|xxxviii}}
  1. ''grahagati'': planetary motion, ''bhagola'': sphere of the zodiac, ''madhyagraha'': mean planets, ''sūryasphuṭa'': true sun, ''grahasphuṭa'': true planets
  2. ''bhū-vāyu-bhagola'': spheres of the earth, atmosphere, and asterisms, ''ayanacalana'': [[precession of the equinoxes]]
  3. ''pañcadaśa-praśna'': fifteen problems relating to [[spherical triangles]]
  4. ''dig-jñāna'': orientation, ''chāyā-gaṇita'': shadow computations, ''lagna'': rising point of the [[ecliptic]], ''nati-lambana'': parallaxes of latitude and longitude
  5. ''grahaṇa'': eclipse
  6. ''[[Vyatipāta|vyatīpāta]]''
  7. visibility correction of planets
  8. moon's cusps and phases of the moon
==Modern editions== [[File:Ganita yukti bhasha verse.jpg|thumb|The first verse from Yukti bhasha in Malayalam language]]The importance of the {{Transliteration|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''. The mathematical part of the text, along with notes in [[Malayalam]], was first published in 1948 by Rama Varma Thampuran and Akhileswara Aiyar.''Yuktibhâsâ, Part I'' (ed) with notes by Ramavarma (Maru) Thampuran and A. R. Akhileswara Aiyer, Magalodayam Ltd., [[Trichur]], [[Kerala]], 1123 [[Malayalam Era]], 1948 [[BCE|CE]]. The first critical edition of the entire Malayalam text, alongside an English translation and detailed explanatory notes, was published in two volumes by [[Springer Science+Business Media|Springer]] in 2008.{{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}}See publishers's ([[Springer Science+Business Media|Springer's]]) [[web page]] on the book: {{cite book|url=https://www.springer.com/mathematics/history+of+mathematics/book/978-1-84882-072-2|title=Ganita-Yukti-Bhasa (Rationales in Mathematical Astronomy) of Jyesthadeva|isbn=9781848820722|access-date=29 April 2010}} A third volume, containing a critical edition of the [[Sanskrit]] Ganitayuktibhasa, was published by the [[Indian Institute of Advanced Study]], Shimla in 2009.{{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}}{{cite book|last=K.V. Sarma|title=Ganita Yuktibhasa (Volume III)|year=2004|publisher=[[Indian Institute of Advanced Study]]|location=[[Shimla]]|isbn=81-7986-052-3}}Publisher's ([[Indian Institute of Advanced Study]]) [[web page]] on the book:{{cite web|url=http://www.iias.org/p_ganita_yuktibhasa_volume-III.html|title=Ganita Yuktibhasa by K.V. Sarma|access-date=1 May 2010|archive-url=https://web.archive.org/web/20100317202603/http://www.iias.org/p_ganita_yuktibhasa_volume-III.html|archive-date=17 March 2010|url-status=dead|df=dmy-all}}For a review of ''Ganita-yukti-bhasa (Rationales in Mathematical Astronomy) of Jyesthadeva'' by [[Mathematical Association of America]] see : {{cite web|url=https://old.maa.org/press/maa-reviews/ganita-yukti-bh-s-rationales-in-mathematical-astronomy-of-jyesthadeva|title=Ganita-Yukti-Bhāsā (Rationales in Mathematical Astronomy) of Jyesthadeva|last=Homer S. White|date=2009-07-17|publisher=The Mathematical Association of America|access-date=28 May 2022}} This edition of the ''{{Transliteration|ml|Yuktibhāṣā}}'' has been divided into two volumes: Volume I deals with mathematics and Volume II treats astronomy. Each volume is divided into three parts: the first part is an English translation of the relevant Malayalam part of the {{Transliteration|ml|Yuktibhāṣā}}, the second part contains detailed explanatory notes on the translation, and in the third part the text in the Malayalam original is reproduced. The English translation is by [[K. V. Sarma]], and the explanatory notes are provided by K. Ramasubramanian, M. D. Srinivas, and M. S. Sriram. An [[open access]] edition of the ''{{Transliteration|ml|Yuktibhāṣā}}'' was published by the Sayahna Foundation in 2020.{{Cite web |last=Sayahna Foundation |date=2020-11-20 |title=Yukthibhasha digital edition |url=https://books.sayahna.org/ml/pdf/yukthibhasha.pdf }} ==See also== * [[Indian mathematics]] * [[Kerala school of astronomy and mathematics|Kerala School]] * [[Madhava's correction term]] ==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] {{Kerala School}} {{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]]