Yuktibhāṣā

Yuktibhāṣā (Malayalam: യുക്തിഭാഷ, lit. 'Rationale'), also known as Gaṇita-yukti-bhāṣā and Gaṇitanyāyasaṅgraha (English: Compendium of Astronomical Rationale), is a treatise on mathematics and astronomy, written by the Indian astronomer Jyeṣṭhadeva 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 Nambudiri, Jyeṣṭhadeva, 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 proofs and derivations of the theorems that it presents. The Yuktibhāṣā demonstrates that at least some early Indian scholars in astronomy and computation had the concept of proofs.

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.

Yuktibhāṣā mainly gives rationale for the results in Nilakantha's Tantrasamgraha. 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. 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. 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.