employed in proofs. They used this to discover a semi-rigorous proof of the result: for large n. They applied ideas from (what was to become) differential and integral calculus to obtain (Taylor–Maclaurin) infinite series for , , and . The Tantrasangraha-vakhya gives the series in verse, which when translated to mathematical notation, can be written as: where, for the series reduce to the standard power series for these trigonometric functions, for example: (The Kerala school did not use the "factorial" symbolism.) The Kerala school made use of the rectification (computation of length) of the arc of a circle to give a proof of these results. (The later method of Leibniz, using quadrature (i.e. computation of area under the arc of the circle), was not yet developed.) They also made use of the series expansion of to obtain an infinite series expression (later known as Gregory series) for : Their rational approximation of the error for the finite sum of their series are of particular interest. For example, the error, , (for n odd, and i = 1, 2, 3) for the series: They manipulated the terms, using the partial fraction expansion of : to obtain a more rapidly converging series for : They used the improved series to derive a rational expression, for correct up to nine decimal places, i.e. . They made use of an intuitive notion of a limit to compute these results. The Kerala school mathematicians also gave a semi-rigorous method of differentiation of some trigonometric functions, though the notion of a function, or of exponential or logarithmic functions, was not yet formulated. Recognition In 1825 John Warren published a memoir on the division of time in southern India, called the Kala Sankalita, which briefly mentions the discovery of infinite series by Kerala astronomers. The works of the Kerala school were first written up for the Western world by Englishman C. M. Whish in 1835. According to Whish, the Kerala mathematicians had "laid the foundation for a complete system of fluxions" and these works abounded "with fluxional forms and series to be found in no work of foreign countries". However, Whish's results were almost completely neglected, until over a century later, when the discoveries of the Kerala school were investigated again by C. T. Rajagopal and his associates. Their work includes commentaries on the proofs of the arctan series in Yuktibhasa given in two papers, a commentary