{"history":[],"article":{"id":"d4e9ff3e-5655-4896-801d-79461373612b","title":"Yuktibhāṣā","slug":"yuktibh","content":":::summary\n![Front and back cover of the Palm-leaf manuscripts of the Yuktibhasa, composed by Jyesthadeva in 1530](https://upload.wikimedia.org/wikipedia/commons/b/b4/Yuktibhasa///cover.jpg)\n**Name:** Yuktibhasa \n**Author:** [Jyesthadeva](https://bliptext.com/articles/jyesthadeva) \n**Country:** Modern-day [Kerala](https://bliptext.com/articles/kerala), [India](https://bliptext.com/articles/india) \n**Language:** [Malayalam](https://bliptext.com/articles/malayalam) \n**Genre:** [Mathematics](https://bliptext.com/articles/mathematics) and [Astronomy](https://bliptext.com/articles/astronomy) \n**Pub Date:** 1530 \n**English Pub Date:** 2008 \n**Orig Lang Code:** ml \n**Image Size:** 300px \n:::\n\n# \n**Yuktibhāṣā**, also known as **Gaṇita-yukti-bhāṣā** and **Gaṇitanyāyasaṅgraha** ([English:](https://bliptext.com/articles/english-language) Compendium of Astronomical Rationale), is a major [treatise](https://bliptext.com/articles/treatise) on [mathematics](https://bliptext.com/articles/indian-mathematics) and [astronomy](https://bliptext.com/articles/hindu-astronomy), written by the [Indian](https://bliptext.com/articles/india) astronomer [Jyesthadeva](https://bliptext.com/articles/jyesthadeva) of the Kerala school of [mathematics](https://bliptext.com/articles/indian-mathematics) around 1530.\nThe treatise, written in Malayalam, is a consolidation of the discoveries by [Madhava of Sangamagrama](https://bliptext.com/articles/madhava-of-sangamagrama), [Nilakantha Somayaji](https://bliptext.com/articles/nilakantha-somayaji), [Parameshvara](https://bliptext.com/articles/parameshvara), [Jyeshtadeva](https://bliptext.com/articles/jyeshtadeva), [Achyuta Pisharati](https://bliptext.com/articles/achyuta-pisharati), and other astronomer-mathematicians of the Kerala school.\nIt also exists in a Sanskrit version, with unclear author and date, composed as a rough translation of the Malayalam original.\nThe work contains [proofs](https://bliptext.com/articles/mathematical-proof) and derivations of the [theorems](https://bliptext.com/articles/theorem) that it presents.\nModern historians used to assert, based on the works of Indian mathematics that first became available, that early Indian scholars in astronomy and computation lacked in proofs, but Yuktibhāṣā demonstrates otherwise.\nSome of its important topics include the [infinite series](https://bliptext.com/articles/infinite-series) expansions of functions; [power series](https://bliptext.com/articles/power-series), including of [π](https://bliptext.com/articles/pi) and [π](https://bliptext.com/articles/pi)/4; [trigonometric series](https://bliptext.com/articles/trigonometric-series) of [sine](https://bliptext.com/articles/[sine](https://bliptext.com/articles/sine)), co[sine](https://bliptext.com/articles/[sine](https://bliptext.com/articles/sine)), and [arctangent](https://bliptext.com/articles/arctangent); [Taylor series](https://bliptext.com/articles/taylor-series), including second and third order approximations of [sine](https://bliptext.com/articles/[sine](https://bliptext.com/articles/sine)) and co[sine](https://bliptext.com/articles/[sine](https://bliptext.com/articles/sine)); radii, diameters and circumferences.\nYuktibhāṣā mainly gives rationale for the results in Nilakantha's [Tantra Samgraha](https://bliptext.com/articles/tantrasangraha).\nIt is considered an early text to give some ideas of [calculus](https://bliptext.com/articles/calculus) like Taylor and infinity series, predating Newton and Leibniz by two centuries.\nThe treatise was largely unnoticed outside India, as it was written in the local language of Malayalam.\nIn modern times, due to wider international cooperation in mathematics, the wider world has taken notice of the work.\nFor 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.\n\n# Contents\nYuktibhāṣā contains most of the developments of the earlier Kerala school, particularly [Madhava](https://bliptext.com/articles/madhava-of-sangamagrama) and [Nilakantha](https://bliptext.com/articles/nilakantha-somayaji).\nThe text is divided into two parts – the former deals with [mathematical analysis](https://bliptext.com/articles/mathematical-analysis) and the latter with astronomy.\nBeyond 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.\n# [Pages from the Yuktibhasa | upload.wikimedia.org/wikipedia/commons/b/bc/Pages///from///Yuktibhasa.jpg]\n\n## Mathematics\nThis subjects treated in the mathematics part of the Yuktibhāṣā can be divided into seven chapters:\nThe first four chapters of the contain elementary mathematics, such as division, the [Pythagorean theorem](https://bliptext.com/articles/pythagorean-theorem), [square roots](https://bliptext.com/articles/square-root), etc. Novel ideas are not discussed until the sixth chapter on [circumference](https://bliptext.com/articles/circumference) of a [circle](https://bliptext.com/articles/circle).\nYuktibhāṣā contains a derivation and proof for the [power series](https://bliptext.com/articles/power-series) of [inverse tangent](https://bliptext.com/articles/inverse-trigonometric-function), discovered by Madhava.\nIn the text, Jyesthadeva describes Madhava's series in the following manner:\nThe 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.\nAll 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.\nIt 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.\nOtherwise the terms obtained by this above iteration will not tend to the vanishing magnitude.\nIn modern mathematical notation,\nor, expressed in terms of tangents,\nwhich in Europe was conventionally called [Gregory's series](https://bliptext.com/articles/gregory-s-series) after [James Gregory](https://bliptext.com/articles/james-gregory-astronomer-and-mathematician), who rediscovered it in 1671.\nThe text also contains Madhava's [infinite series](https://bliptext.com/articles/series-mathematics) expansion of [π](https://bliptext.com/articles/pi) which he obtained from the expansion of the arc-tangent function.\nwhich in Europe was conventionally called [Leibniz's series](https://bliptext.com/articles/leibniz-formula-for), after [Gottfried Leibniz](https://bliptext.com/articles/gottfried-leibniz) who rediscovered it in 1673.\nUsing a rational approximation of this series, he gave values of the number [π](https://bliptext.com/articles/pi) as 3.14159265359, correct to 11 decimals, and as 3.1415926535898, correct to 13 decimals.\nThe text describes two methods for computing the value of π.\nFirst, obtain a rapidly converging series by transforming the original infinite series of π.\nBy doing so, the first 21 terms of the infinite series\nwas used to compute the approximation to 11 decimal places.\nThe other method was to add a remainder term to the original series of π.\nThe remainder term was used in the infinite series expansion of to improve the approximation of π to 13 decimal places of accuracy when n=76.\nApart from these, the Yuktibhāṣā contains many [elementary](https://bliptext.com/articles/elementary-mathematics) and complex mathematical topics, including,\n# [Explanation of the sine rule in Yuktibhāṣā | upload.wikimedia.org/wikipedia/commons/3/3e/Yuktibhasa.svg]\n\n## Astronomy\nChapters eight to seventeen deal with subjects of astronomy: [planetary orbits](https://bliptext.com/articles/planetary-orbit), [celestial spheres](https://bliptext.com/articles/celestial-sphere), [ascension](https://bliptext.com/articles/right-ascension), [declination](https://bliptext.com/articles/declination), directions and shadows, [spherical triangles](https://bliptext.com/articles/spherical-trigonometry), [ellipses](https://bliptext.com/articles/ellipse), and [parallax](https://bliptext.com/articles/parallax) correction.\nThe planetary theory described in the book is similar to that later adopted by [Danish](https://bliptext.com/articles/danish-people) astronomer [Tycho Brahe](https://bliptext.com/articles/tycho-brahe).\nThe 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, [Vyatipata](https://bliptext.com/articles/vyatip-ta) (when the sun and moon have the same declination), visibility correction for planets and phases of the moon.\nSpecifically,\n

\ngrahagati: planetary motion, bhagola: sphere of the zodiac, madhyagraha: mean planets, sūryasphuṭa: true sun, grahasphuṭa: true planets\nbhū-vāyu-bhagola: spheres of the earth, atmosphere, and asterisms, ayanacalana: [precession of the equinoxes](https://bliptext.com/articles/precession-of-the-equinoxes)\npañcadaśa-praśna: fifteen problems relating to [spherical triangles](https://bliptext.com/articles/spherical-triangles)\ndig-jñāna: orientation, chāyā-gaṇita: shadow computations, lagna: rising point of the [ecliptic](https://bliptext.com/articles/ecliptic), nati-lambana: parallaxes of latitude and longitude\ngrahaṇa: eclipse\n[vyatīpāta](https://bliptext.com/articles/vyatip-ta)\nvisibility correction of planets\nmoon's cusps and phases of the moon\n\n\n# Modern editions\nThe importance of Yuktibhāṣā was brought to the attention of modern scholarship by [C. M. Whish](https://bliptext.com/articles/c-m-whish) in 1832 through a paper published in the Transactions of the Royal Asiatic Society of Great Britain and Ireland.\nThe mathematics part of the text, along with notes in Malayalam, was first published in 1948 by Rama Varma Thampuran and Akhileswara Aiyar.\nThe first critical edition of the entire Malayalam text, alongside an English translation and detailed explanatory notes, was published in two volumes by [Springer](https://bliptext.com/articles/springer-science-business-media) in 2008.\nA third volume, containing a critical edition of the Sanskrit Ganitayuktibhasa, was published by the [Indian Institute of Advanced Study](https://bliptext.com/articles/indian-institute-of-advanced-study), Shimla in 2009.\nThis edition of [Yuktibhasa](https://bliptext.com/articles/yuktibhasa) has been divided into two volumes: Volume I deals with mathematics and Volume II treats astronomy.\nEach volume is divided into three parts: First part is an [English](https://bliptext.com/articles/english-language) translation of the relevant [Malayalam](https://bliptext.com/articles/malayalam) part of [Yuktibhasa](https://bliptext.com/articles/yuktibhasa), second part contains detailed explanatory notes on the translation, and in the third part the text in the [Malayalam](https://bliptext.com/articles/malayalam) original is reproduced.\nThe [English](https://bliptext.com/articles/english-language) translation is by [K.V. Sarma](https://bliptext.com/articles/k-v-sarma) and the explanatory notes are provided by K. Ramasubramanian, M. D. Srinivas, and M. S. Sriram.\nAn [open access](https://bliptext.com/articles/open-access) edition of [Yuktibhasa](https://bliptext.com/articles/yuktibhasa) is published by Sayahna Foundation in 2020.\n# [The first verse from Yukti bhasha in Malayalam language | upload.wikimedia.org/wikipedia/commons/e/ed/Ganita///yukti///bhasha///verse.jpg]","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-26T08:01:34.295Z","updated_at":"2025-01-26T08:01:34.295Z","search_vector":"'yuktibhāṣā':1","search_content":" \n \n**Name:** Yuktibhasa \n**Author:** [Jyesthadeva](https://bliptext.com/articles/jyesthadeva) \n**Country:** Modern-day [Kerala](https://bliptext.com/articles/kerala), [India](https://bliptext.com/articles/india) \n**Language:** [Malayalam](https://bliptext.com/articles/malayalam) \n**Genre:** [Mathematics](https://bliptext.com/articles/mathematics) and [Astronomy](https://bliptext.com/articles/astronomy) \n**Pub Date:** 1530 \n**English Pub Date:** 2008 \n**Orig Lang Code:** ml \n**Image Size:** 300px \n \n\n# \n**Yuktibhāṣā**, also known as **Gaṇita-yukti-bhāṣā** and **Gaṇitanyāyasaṅgraha** ([English:](https://bliptext.com/articles/english-language) Compendium of Astronomical Rationale), is a major [treatise](https://bliptext.com/articles/treatise) on [mathematics](https://bliptext.com/articles/indian-mathematics) and [astronomy](https://bliptext.com/articles/hindu-astronomy), written by the [Indian](https://bliptext.com/articles/india) astronomer [Jyesthadeva](https://bliptext.com/articles/jyesthadeva) of the Kerala school of [mathematics](https://bliptext.com/articles/indian-mathematics) around 1530.\nThe treatise, written in Malayalam, is a consolidation of the discoveries by [Madhava of Sangamagrama](https://bliptext.com/articles/madhava-of-sangamagrama), [Nilakantha Somayaji](https://bliptext.com/articles/nilakantha-somayaji), [Parameshvara](https://bliptext.com/articles/parameshvara), [Jyeshtadeva](https://bliptext.com/articles/jyeshtadeva), [Achyuta Pisharati](https://bliptext.com/articles/achyuta-pisharati), and other astronomer-mathematicians of the Kerala school.\nIt also exists in a Sanskrit version, with unclear author and date, composed as a rough translation of the Malayalam original.\nThe work contains [proofs](https://bliptext.com/articles/mathematical-proof) and derivations of the [theorems](https://bliptext.com/articles/theorem) that it presents.\nModern historians used to assert, based on the works of Indian mathematics that first became available, that early Indian scholars in astronomy and computation lacked in proofs, but Yuktibhāṣā demonstrates otherwise.\nSome of its important topics include the [infinite series](https://bliptext.com/articles/infinite-series) expansions of functions; [power series](https://bliptext.com/articles/power-series), including of [π](https://bliptext.com/articles/pi) and [π](https://bliptext.com/articles/pi)/4; [trigonometric series](https://bliptext.com/articles/trigonometric-series) of [[sine](https://bliptext.com/articles/sine)](https://bliptext.com/articles/[sine](https://bliptext.com/articles/sine)), co[[sine](https://bliptext.com/articles/sine)](https://bliptext.com/articles/[sine](https://bliptext.com/articles/sine)), and [arctangent](https://bliptext.com/articles/arctangent); [Taylor series](https://bliptext.com/articles/taylor-series), including second and third order approximations of [[sine](https://bliptext.com/articles/sine)](https://bliptext.com/articles/[sine](https://bliptext.com/articles/sine)) and co[[sine](https://bliptext.com/articles/sine)](https://bliptext.com/articles/[sine](https://bliptext.com/articles/sine)); radii, diameters and circumferences.\nYuktibhāṣā mainly gives rationale for the results in Nilakantha's [Tantra Samgraha](https://bliptext.com/articles/tantrasangraha).\nIt is considered an early text to give some ideas of [calculus](https://bliptext.com/articles/calculus) like Taylor and infinity series, predating Newton and Leibniz by two centuries.\nThe treatise was largely unnoticed outside India, as it was written in the local language of Malayalam.\nIn modern times, due to wider international cooperation in mathematics, the wider world has taken notice of the work.\nFor 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.\n\n# Contents\nYuktibhāṣā contains most of the developments of the earlier Kerala school, particularly [Madhava](https://bliptext.com/articles/madhava-of-sangamagrama) and [Nilakantha](https://bliptext.com/articles/nilakantha-somayaji).\nThe text is divided into two parts – the former deals with [mathematical analysis](https://bliptext.com/articles/mathematical-analysis) and the latter with astronomy.\nBeyond 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.\n# [Pages from the Yuktibhasa | upload.wikimedia.org/wikipedia/commons/b/bc/Pages///from///Yuktibhasa.jpg]\n\n## Mathematics\nThis subjects treated in the mathematics part of the Yuktibhāṣā can be divided into seven chapters:\nThe first four chapters of the contain elementary mathematics, such as division, the [Pythagorean theorem](https://bliptext.com/articles/pythagorean-theorem), [square roots](https://bliptext.com/articles/square-root), etc. Novel ideas are not discussed until the sixth chapter on [circumference](https://bliptext.com/articles/circumference) of a [circle](https://bliptext.com/articles/circle).\nYuktibhāṣā contains a derivation and proof for the [power series](https://bliptext.com/articles/power-series) of [inverse tangent](https://bliptext.com/articles/inverse-trigonometric-function), discovered by Madhava.\nIn the text, Jyesthadeva describes Madhava's series in the following manner:\nThe 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.\nAll 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.\nIt 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.\nOtherwise the terms obtained by this above iteration will not tend to the vanishing magnitude.\nIn modern mathematical notation,\nor, expressed in terms of tangents,\nwhich in Europe was conventionally called [Gregory's series](https://bliptext.com/articles/gregory-s-series) after [James Gregory](https://bliptext.com/articles/james-gregory-astronomer-and-mathematician), who rediscovered it in 1671.\nThe text also contains Madhava's [infinite series](https://bliptext.com/articles/series-mathematics) expansion of [π](https://bliptext.com/articles/pi) which he obtained from the expansion of the arc-tangent function.\nwhich in Europe was conventionally called [Leibniz's series](https://bliptext.com/articles/leibniz-formula-for), after [Gottfried Leibniz](https://bliptext.com/articles/gottfried-leibniz) who rediscovered it in 1673.\nUsing a rational approximation of this series, he gave values of the number [π](https://bliptext.com/articles/pi) as 3.14159265359, correct to 11 decimals, and as 3.1415926535898, correct to 13 decimals.\nThe text describes two methods for computing the value of π.\nFirst, obtain a rapidly converging series by transforming the original infinite series of π.\nBy doing so, the first 21 terms of the infinite series\nwas used to compute the approximation to 11 decimal places.\nThe other method was to add a remainder term to the original series of π.\nThe remainder term was used in the infinite series expansion of to improve the approximation of π to 13 decimal places of accuracy when n=76.\nApart from these, the Yuktibhāṣā contains many [elementary](https://bliptext.com/articles/elementary-mathematics) and complex mathematical topics, including,\n# [Explanation of the sine rule in Yuktibhāṣā | upload.wikimedia.org/wikipedia/commons/3/3e/Yuktibhasa.svg]\n\n## Astronomy\nChapters eight to seventeen deal with subjects of astronomy: [planetary orbits](https://bliptext.com/articles/planetary-orbit), [celestial spheres](https://bliptext.com/articles/celestial-sphere), [ascension](https://bliptext.com/articles/right-ascension), [declination](https://bliptext.com/articles/declination), directions and shadows, [spherical triangles](https://bliptext.com/articles/spherical-trigonometry), [ellipses](https://bliptext.com/articles/ellipse), and [parallax](https://bliptext.com/articles/parallax) correction.\nThe planetary theory described in the book is similar to that later adopted by [Danish](https://bliptext.com/articles/danish-people) astronomer [Tycho Brahe](https://bliptext.com/articles/tycho-brahe).\nThe 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, [Vyatipata](https://bliptext.com/articles/vyatip-ta) (when the sun and moon have the same declination), visibility correction for planets and phases of the moon.\nSpecifically,\n \ngrahagati: planetary motion, bhagola: sphere of the zodiac, madhyagraha: mean planets, sūryasphuṭa: true sun, grahasphuṭa: true planets \nbhū-vāyu-bhagola: spheres of the earth, atmosphere, and asterisms, ayanacalana: [precession of the equinoxes](https://bliptext.com/articles/precession-of-the-equinoxes) \npañcadaśa-praśna: fifteen problems relating to [spherical triangles](https://bliptext.com/articles/spherical-triangles) \ndig-jñāna: orientation, chāyā-gaṇita: shadow computations, lagna: rising point of the [ecliptic](https://bliptext.com/articles/ecliptic), nati-lambana: parallaxes of latitude and longitude \ngrahaṇa: eclipse \n[vyatīpāta](https://bliptext.com/articles/vyatip-ta) \nvisibility correction of planets \nmoon's cusps and phases of the moon \n \n\n# Modern editions\nThe importance of Yuktibhāṣā was brought to the attention of modern scholarship by [C. M. Whish](https://bliptext.com/articles/c-m-whish) in 1832 through a paper published in the Transactions of the Royal Asiatic Society of Great Britain and Ireland.\nThe mathematics part of the text, along with notes in Malayalam, was first published in 1948 by Rama Varma Thampuran and Akhileswara Aiyar.\nThe first critical edition of the entire Malayalam text, alongside an English translation and detailed explanatory notes, was published in two volumes by [Springer](https://bliptext.com/articles/springer-science-business-media) in 2008.\nA third volume, containing a critical edition of the Sanskrit Ganitayuktibhasa, was published by the [Indian Institute of Advanced Study](https://bliptext.com/articles/indian-institute-of-advanced-study), Shimla in 2009.\nThis edition of [Yuktibhasa](https://bliptext.com/articles/yuktibhasa) has been divided into two volumes: Volume I deals with mathematics and Volume II treats astronomy.\nEach volume is divided into three parts: First part is an [English](https://bliptext.com/articles/english-language) translation of the relevant [[Malayalam](https://bliptext.com/articles/malayalam)](https://bliptext.com/articles/malayalam) part of [Yuktibhasa](https://bliptext.com/articles/yuktibhasa), second part contains detailed explanatory notes on the translation, and in the third part the text in the [[Malayalam](https://bliptext.com/articles/malayalam)](https://bliptext.com/articles/malayalam) original is reproduced.\nThe [English](https://bliptext.com/articles/english-language) translation is by [K.V. Sarma](https://bliptext.com/articles/k-v-sarma) and the explanatory notes are provided by K. Ramasubramanian, M. D. Srinivas, and M. S. Sriram.\nAn [open access](https://bliptext.com/articles/open-access) edition of [Yuktibhasa](https://bliptext.com/articles/yuktibhasa) is published by Sayahna Foundation in 2020.\n# [The first verse from Yukti bhasha in Malayalam language | upload.wikimedia.org/wikipedia/commons/e/ed/Ganita///yukti///bhasha///verse.jpg]","revisionCount":0},"hasMore":false,"nextCursor":null}

History