 Front and back cover of thePalm-leaf manuscripts of the Yuktibhasa, composed byJyesthadeva in 1530 | |
| Author | Jyesthadeva |
|---|---|
| Language | Malayalam |
| Genre | Mathematics andAstronomy |
Publication date | 1530 |
| Publication place | Modern-dayKerala,India |
Published in English | 2008 |
Yuktibhāṣā (Malayalam:യുക്തിഭാഷ,lit. 'Rationale'), also known asGaṇita-yukti-bhāṣā[1]: xxi andGaṇitanyāyasaṅgraha (English:Compendium of Astronomical Rationale), is a majortreatise onmathematics andastronomy, written by theIndian astronomerJyesthadeva of theKerala school of mathematics around 1530.[2] The treatise, written in Malayalam, is a consolidation of the discoveries byMadhava of Sangamagrama,Nilakantha Somayaji,Parameshvara,Jyeshtadeva,Achyuta Pisharati, and other astronomer-mathematicians of the Kerala school.[2] It also exists in a Sanskrit version, with unclear author and date, composed as a rough translation of the Malayalam original.[1]
The work containsproofs and derivations of thetheorems that it presents. Modern 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,[3] butYuktibhāṣā demonstrates otherwise.[4]
Some of its important topics include theinfinite series expansions of functions;power series, including ofπ and π/4;trigonometric series ofsine,cosine, andarctangent;Taylor series, including second and third order approximations ofsine andcosine; radii, diameters and circumferences.
Yuktibhāṣā mainly gives rationale for the results in Nilakantha'sTantra Samgraha.[5] It is considered an early text to give some ideas related tocalculus like Taylor and infinite series of some trigonometric functions, predating Newton and Leibniz by two centuries.[6][7][8][9] however they did not combine many differing ideas under the two unifying themes of thederivative and theintegral, show the connection between the two, and turn calculus into the powerful problem-solving tool we have today.[10] 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, 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.[7][8][9]
Yuktibhāṣā contains most of the developments of the earlier Kerala school, particularlyMadhava andNilakantha. The text is divided into two parts – the former deals withmathematical analysis and the latter with astronomy.[2] 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.[1]: xxxvii
This subjects treated in the mathematics part of theYuktibhāṣā can be divided into seven chapters:[1]: xxxvii
The first four chapters of the contain elementary mathematics, such as division, thePythagorean theorem,square roots, etc.[12] Novel ideas are not discussed until the sixth chapter oncircumference of acircle.Yuktibhāṣā contains a derivation and proof for thepower series ofinverse tangent, discovered by Madhava.[5] In the text, Jyesthadeva describes Madhava's series in the following manner:
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,
or, expressed in terms of tangents,
which in Europe was conventionally calledGregory's series afterJames Gregory, who rediscovered it in 1671.
The text also contains Madhava'sinfinite series expansion ofπ which he obtained from the expansion of the arc-tangent function.
which in Europe was conventionally calledLeibniz's series, afterGottfried Leibniz who rediscovered it in 1673.
Using a rational approximation of this series, he gave values of the numberπ 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
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 was used in the infinite series expansion of to improve the approximation of π to 13 decimal places of accuracy whenn=76.[13]
Apart from these, theYuktibhāṣā contains manyelementary and complex mathematical topics, including,[citation needed]
Chapters eight to seventeen deal with subjects of astronomy:planetary orbits,celestial spheres,ascension,declination, directions and shadows,spherical triangles,ellipses, andparallax correction. The planetary theory described in the book is similar to that later adopted byDanish astronomerTycho Brahe.[14]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,Vyatipata (when the sun and moon have the same declination), visibility correction for planets and phases of the moon.[11]
Specifically,[1]: xxxviii
The importance ofYuktibhāṣā was brought to the attention of modern scholarship byC. M. Whish in 1832 through a paper published in theTransactions of the Royal Asiatic Society of Great Britain and Ireland.[4] The mathematics part of the text, along with notes in Malayalam, was first published in 1948 by Rama Varma Thampuran and Akhileswara Aiyar.[2][15]
The first critical edition of the entire Malayalam text, alongside an English translation and detailed explanatory notes, was published in two volumes bySpringer[16] in 2008.[1] A third volume, containing a critical edition of the Sanskrit Ganitayuktibhasa, was published by theIndian Institute of Advanced Study, Shimla in 2009.[17][18][19][20]
This edition ofYuktibhasa has been divided into two volumes: Volume I deals with mathematics and Volume II treats astronomy. Each volume is divided into three parts: First part is anEnglish translation of the relevantMalayalam part ofYuktibhasa, second part contains detailed explanatory notes on the translation, and in the third part the text in theMalayalam original is reproduced. TheEnglish translation is byK.V. Sarma and the explanatory notes are provided by K. Ramasubramanian, M. D. Srinivas, and M. S. Sriram.[1]
Anopen access edition ofYuktibhasa is published by Sayahna Foundation in 2020.[21]
