Yuktibhāṣā

Yuktibhāṣā (Malayalam: യുക്തിഭാഷ, lit. 'Rationale'), also known as Gaṇita-yukti-bhāṣā^: xxi and Gaṇitanyāyasaṅgraha (English: 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. 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. 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, but Yuktibhāṣā demonstrates otherwise.

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 Tantra Samgraha. It is considered an early text to give some ideas of calculus like Taylor and infinity series, predating Newton and Leibniz by two centuries. 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.