= 1, the series reduces to the standard power series for these trigonometric functions, for example: and Use of 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 used.) Use of the series expansion of to obtain the Leibniz formula for π: A rational approximation of error for the finite sum of their series of interest. For example, the error, , (for n odd, and i = 1, 2, 3) for the series: Manipulation of error term to derive a faster converging series for : Using the improved series to derive a rational expression, 104348/33215 for π correct up to nine decimal places, i.e. 3.141592653. Use of an intuitive notion of limit to compute these results. A semi-rigorous (see remark on limits above) method of differentiation of some trigonometric functions. However, they did not formulate the notion of a function, or have knowledge of the exponential or logarithmic functions. 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. Rajagopal and his associates. Their work includes commentaries on the proofs of the arctan series in Yuktibhāṣā given in two papers, a commentary on the Yuktibhāṣā'''s proof of the sine and cosine series and two papers that provide the Sanskrit verses of the Tantrasangrahavakhya for the series for arctan, sin, and cosine (with English translation and commentary). Narayana Pandit is a 14th century mathematician who composed two important mathematical works, an arithmetical treatise, Ganita Kaumudi, and an algebraic treatise, Bijganita Vatamsa. Narayana is also thought to be the author of an elaborate commentary of Bhaskara II's Lilavati, titled Karmapradipika (or Karma-Paddhati). Madhava of Sangamagrama (c. 1340–1425) was the founder of the Kerala School. Although it is possible that he wrote Karana Paddhati a work written sometime between 1375 and 1475, all we really know of his work comes from works of later scholars. Parameshvara (c. 1370–1460)