Yuktibhāṣā, also known as Gaṇitanyāyasaṅgraha, 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, tangent and arctangent; Taylor series, including second and third order approximations of sine and cosine; radii, diameters and circumferences; and tests of convergence. Yuktibhāṣā is mainly based on Nilakantha's Tantra Samgraha. It is considered an early text on the ideas of calculus, predating Newton and Leibniz by centuries. 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 Yuktibhāṣā demonstrates otherwise. 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.
Contents
Yuktibhāṣā 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 Yuktibhāṣā 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. Yuktibhāṣā 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 term was used in the infinite series expansion of to improve the approximation of π to 13 decimal places of accuracy when n=76. Apart from these, the Yuktibhāṣā contains many elementary and complex mathematical topics, including,
Proofs for the expansion of the sine and cosine functions
The importance of 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. However, the mathematics part of the text, along with notes in Malayalam, was first published only in 1948 by Rama Varma Maru 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.
