Yuktibhāṣā
Yuktibhāṣā (), also known as ("Compendium of Astronomical Rationale"), is a major treatise on mathematics and astronomy, written by the Indian astronomer Jyesthadeva of the Kerala school of mathematics around 1530. 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 proofs and derivations of the theorems that it presented; something unusual for Indian mathematicians of that era. Some of its important topics include the infinite series expansions of functions; power  series, including of 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 tests of convergence. Contents. contains most of the developments of the earlier Kerala school, particularly Madhava and Nilakantha. The text is divided into two parts – the former deals with mathematical analysis and the latter with astronomy. Mathematics. The first four chapters of the contain elementary mathematics, such as division, the Pythagorean theorem, square roots, etc. Novel ideas are not discussed until the sixth chapter on circumference of a circle. contains  a derivation and proof for the power series of inverse tangent, discovered by Madhava. In the text, Jyesthadeva describes Madhava's series in the following manner: In modern mathematical notation, or, expressed in terms of tangents, which has been previously attributed to James Gregory, who published it in 1667. The text also contains Madhava's infinite series expansion of π which he obtained from the expansion of the arc-tangent function. 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. 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 termformula_5 was used in the infinite series expansion of formula_6 to improve the approximation of π to 13 decimal places of accuracy when "n"=76. Apart from these, the contains many elementary and complex mathematical topics, including, Astronomy. Chapters seven  to seventeen deal with subjects of astronomy: planetary orbits, celestial spheres, ascension, declination, directions and shadows, spherical triangles, ellipses, and parallax correction. The planetary theory described in the book is similar to that later adopted by Danish astronomer Tycho Brahe. Modern editions. The importance of 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". 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. For the first time, an edition of the entire Malayalam text, alongside an English translation and detailed explanatory notes, was published by Springer in 2008. A third volume presenting a critical edition of the Sanskrit Ganitayuktibhasa has been published by the Indian Institute of Advanced Study, Shimla in 2009.