Indeterminate higher-order polynomial equations. Geometry: Gave a proof of the Pythagorean theorem. Calculus: Conceived of differential calculus. Discovered the derivative. Discovered the differential coefficient. Developed differentiation. Stated Rolle's theorem, a special case of the mean value theorem (one of the most important theorems of calculus and analysis). Derived the differential of the sine function. Computed π, correct to five decimal places. Calculated the length of the Earth's revolution around the Sun to 9 decimal places. Trigonometry: Developments of spherical trigonometry The trigonometric formulas: Kerala mathematics (1300–1600) The Kerala school of astronomy and mathematics was founded by Madhava of Sangamagrama in Kerala, South India and included among its members: Parameshvara, Neelakanta Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar. It flourished between the 14th and 16th centuries and the original discoveries of the school seems to have ended with Narayana Bhattathiri (1559–1632). In attempting to solve astronomical problems, the Kerala school astronomers independently created a number of important mathematics concepts. The most important results, series expansion for trigonometric functions, were given in Sanskrit verse in a book by Neelakanta called Tantrasangraha and a commentary on this work called Tantrasangraha-vakhya of unknown authorship. The theorems were stated without proof, but proofs for the series for sine, cosine, and inverse tangent were provided a century later in the work Yuktibhāṣā (c.1500–c.1610), written in Malayalam, by Jyesthadeva. Their discovery of these three important series expansions of calculus—several centuries before calculus was developed in Europe by Isaac Newton and Gottfried Leibniz—was an achievement. However, the Kerala School did not invent calculus, because, while they were able to develop Taylor series expansions for the important trigonometric functions, differentiation, term by term integration, convergence tests, iterative methods for solutions of non-linear equations, and the theory that the area under a curve is its integral, they developed neither a theory of differentiation or integration, nor the fundamental theorem of calculus. The results obtained by the Kerala school include: The (infinite) geometric series: A semi-rigorous proof (see "induction" remark below) of the result: for large n. Intuitive use of mathematical induction, however, the inductive hypothesis was not formulated or employed in proofs. Applications of ideas from (what was to become) differential and integral calculus to obtain (Taylor–Maclaurin) infinite series for sin x, cos x, and arctan x. The Tantrasangraha-vakhya gives the series in verse, which when translated to mathematical notation, can be written as: where, for r