Yuktibhāṣā

Yuktibhāṣā (Malayalam: യുക്തിഭാഷ; "Rationale in the Malayalam/Sanskrit language") also known as Gaṇitanyāyasaṅgraha ("Compendium of astronomical rationale"), is a major treatise on mathematics and astronomy, written by Indian astronomer Jyesthadeva of the Kerala school of mathematics in about AD 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. Yuktibhasa is mainly based on Nilakantha's Tantra Samgraha. It is considered, possibly the first text, on the foundations of calculus and predates those of European mathematicians such as James Gregory and Newton by many centuries. The treatise was largely unnoticed as the book was written in the local language of Malayalam and it was thought that many Indian ideas in astronomy and computation lacked proofs or foundations. Yuktibhasa however demonstrates founding principles and the development and proofs of theorems. However, both Oxford University and Royal Society of Great Britain have accepted that Calculus and many such pioneering mathematical theorems originated in India.

The work was unique for its time, since it contained proofs and derivations of the theorems that it presented; something that was not usually done by any Indian mathematicians of that era. Some of its important developments in analysis include: the infinite series expansion of a function, the power series, the Taylor series, the trigonometric series for sine, cosine, tangent and arctangent, the second and third order Taylor series approximations of sine and cosine, the power series of π, π/4, θ, the radius, diameter and circumference, and tests of convergence.

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