The Kerala school of astronomy and mathematics was a school of mathematics and astronomy founded by Madhava of Sangamagrama in Malabar, Kerala, South India, which included among its members: Parameshvara, Neelakanta Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar. The school 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 independently created a number of important mathematics concepts. Their most important results—series expansion for trigonometric functions—were described in Sanskrit verse in a book by Neelakanta called Tantrasangraha, and again in a commentary on this work, called Tantrasangrahavakhya, 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 Yuktibhasa (c. 1500 – c. 1610), written in Malayalam, by Jyesthadeva, and also in a commentary on Tantrasangraha.^{[1]}
Their work, completed two centuries before the invention of calculus in Europe, provided what is now considered the first example of a power series (apart from geometric series).^{[2]} However, they did not formulate a systematic theory of differentiation and integration, nor is there any direct evidence of their results being transmitted outside Kerala.^{[3]}^{[4]}^{[5]}^{[6]}
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See also : Madhava series
The Kerala school has made a number of contributions to the fields of infinite series and calculus. These include the following (infinite) geometric series:
This formula, however, was already known in the work of the 10th century Iraqi mathematician Alhazen (the Latinized form of the name Ibn alHaytham) (965–1039).^{[8]}
The Kerala school made intuitive use of mathematical induction, though the inductive hypothesis was not yet formulated or employed in proofs.^{[1]} They used this to discover a semirigorous proof of the result:
They applied ideas from (what was to become) differential and integral calculus to obtain (TaylorMaclaurin) infinite series for sinx, cosx, and arctanx.^{[9]} The Tantrasangrahavakhya gives the series in verse, which when translated to mathematical notation, can be written as:^{[1]}
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.)^{[1]} They also made use of the series expansion of arctanx to obtain an infinite series expression (later known as Gregory series) for π:^{[1]}
Their rational approximation of the error for the finite sum of their series are of particular interest. For example, the error, f_{i}(n + 1), (for n odd, and i = 1, 2, 3) for the series:
They manipulated the error term to derive a faster converging series for π:^{[1]}
They used the improved series to derive a rational expression,^{[1]} 104348 / 33215 for π correct up to nine decimal places, i.e. 3.141592653. They made use of an intuitive notion of a limit to compute these results.^{[1]} The Kerala school mathematicians also gave a semirigorous method of differentiation of some trigonometric functions,^{[10]} though the notion of a function, or of exponential or logarithmic functions, was not yet formulated.
The works of the Kerala school were first written up for the Western world by Englishman C.M. Whish in 1835, though there exists some other works, namely Kala Sankalita by J. Warren in 1825^{[11]} which briefly mentions the discovery of infinite series by Kerala astronomers. 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."^{[12]} 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 Yuktibhasa given in two papers,^{[13]}^{[14]} a commentary on the Yuktibhasa's proof of the sine and cosine series^{[15]} and two papers that provide the Sanskrit verses of the Tantrasangrahavakhya for the series for arctan, sin, and cosine (with English translation and commentary).^{[16]}^{[17]}
In the fields of geometry, arithmetic, and algebra, the Kerala school discovered a formula for the ecliptic,^{[citation needed]} Lhuilier's formula for the circumradius of a cyclic quadrilateral by Parameshvara,^{[18]}^{[19]} decimal floating point numbers,^{[20]} the secant method and iterative methods for solution of nonlinear equations by Parameshvara,^{[18]}^{[21]} and the NewtonGauss interpolation formula by Govindaswami.^{[citation needed]}
In astronomy, Madhava discovered a procedure to determine the positions of the Moon every 36 minutes, and methods to estimate the motions of the planets.^{[22]} Late Kerala school astronomers gave a formulation for the equation of the center of the planets,^{[22]}^{[23]} and a heliocentric model of the solar system.^{[22]}
Nilakanthan Somayaji (1444–1544), in his Aryabhatiyabhasya (a commentary on Aryabhata's Aryabhatiya), developed his own computational system for a partially heliocentric planetary model, in which Mercury, Venus, Mars, Jupiter and Saturn orbit the Sun, which in turn orbits the Earth, similar to the Tychonic system later proposed by Tycho Brahe in the late 16th century. Nilakantha's system, however, was mathematically more efficient than the Tychonic system, due to correctly taking into account the equation of the centre and latitudinal motion of Mercury and Venus.^{[24]}^{[25]} Nilakanthan's planetary system also incorporated elliptic orbits^{[26]} and the Earth's rotation on its axis.^{[27]}
In his Tantrasangraha (1500), Nilakantha further revised Aryabhata's model for the interior planets Mercury and Venus. His equation of the centre for these planets was more accurate at predicting their heliocentric orbits than the later Tychonic and Copernican models, and remained the most accurate until the 17th century when Johannes Kepler reformed the computation for the interior planets in much the same way Nilakantha did.^{[28]}^{[29]} Most astronomers of the Kerala school of astronomy and mathematics who followed him accepted his planetary model.^{[24]}^{[25]}
In linguistics, the ayurvedic and poetic traditions of Kerala were founded by this school, and the famous poem, Narayaneeyam, was composed by Narayana Bhattathiri.^{[citation needed]}
Main article : Madhava of Sangamagrama
Madhava of Sangamagrama (c. 1340–1425) was the founder of the Kerala School. Although it is possible that he wrote Karanapaddhati, a work written sometime between 1375 and 1475, all we really know of his work comes from works of later scholars.
Little is known about Madhava, who lived at Irinjalakuda,at that time known as Iringattikudal in Thrissur district between the years 1340 and 1425. Sanskrit scholars used to call the town as Sangamagramam, taking into consideration of the meaning of Kudal apprearing in Iringattikudal, which has the meaning Sangamam in Sanskrit.
Nilkantha attributes the series for sine to him. It is not known if Madhava discovered the other series as well, or whether they were discovered later by others in the Kerala school.
Madhava's discoveries include the Taylor series for the sine,^{[10]} cosine, tangent and arctangent functions;^{[30]} the secondorder Taylor series approximations of the sine and cosine functions and the thirdorder Taylor series approximation of the sine function; the power series of π, usually attributed to Leibniz^{[31]} but now known as the MadhavaLeibniz series;^{[32]}^{[33]} the solution of transcendental equations by iteration;^{[citation needed]} and the approximation of transcendental numbers by continued fractions.^{[31]} Madhava correctly computed the value of π to 9 decimal places^{[1]} and 13 decimal places,^{[31]} and produced sine and cosine tables to 9 decimal places of accuracy^{[34]} (see also Madhava's sine table). He also extended some results found in earlier works, including those of Bhaskara.^{[31]}
Narayana Pandit (1340–1400), had written two works, an arithmetical treatise called Ganita Kaumudi and an algebraic treatise called Bijganita Vatamsa. Narayanan is also thought to be the author of an elaborate commentary of Bhaskara II's Lilavathi, titled Karmapradipika (or KarmaPaddhati).^{[35]}
Although the Karmapradipika contains little original work, it contains seven different methods for squaring numbers, a contribution that is wholly original to the author, as well as contributions to algebra and magic squares.^{[35]}
Narayanan's other major works contain a variety of mathematical developments, including a rule to calculate approximate values of square roots, investigations into the second order indeterminate equation nq^{2} + 1 = p^{2} (Pell's equation), solutions of indeterminate higherorder equations, mathematical operations with zero, several geometrical rules, and a discussion of magic squares and similar figures.^{[35]} Evidence also exists that Narayana made minor contributions to the ideas of differential calculus found in Bhaskara II's work. Narayana has also made contributions to the topic of cyclic quadrilaterals.^{[19]}
Parameshvara (1370–1460), the founder of the Drigganita system of Astronomy, was a prolific author of several important works. He belonged to the Alathur village situated on the bank of Bharathappuzha.He is stated to have made direct astronomical observations for fiftyfive years before writing his famous work, Drigganita. He also wrote commentaries on the works of Bhaskara I, Aryabhata and Bhaskara II. His Lilavathi Bhasya, a commentary on Bhaskara II's Lilavathi, contains one of his most important discoveries: an early version of the mean value theorem.^{[18]} This is considered one of the most important results in differential calculus and one of the most important theorems in mathematical analysis, and was later essential in proving the fundamental theorem of calculus.
The SiddhantaDeepika by Paramesvara is a commentary on the commentary of Govindsvamin on Bhaskara I's Mahabhaskareeya. This work contains some of his eclipse observations, including one made at Navakshethra in 1422 and two made at Gokarna in 1425 and 1430. It also presents a mean value type formula for inverse interpolation of the sine function, a onepoint iterative technique for calculating the sine of a given angle, and a more efficient approximation that works using a twopoint iterative algorithm, which is essentially the same as the modern secant method.^{[18]}
Parameshvaran was also the first mathematician to give the radius of a circle with an inscribed cyclic quadrilateral, an expression that is normally attributed to L'Huilier (1782).^{[18]}
Main articles : Nilakantha Somayaji, Tantrasamgraha
Nilakantha (1444–1544) was a disciple of Govinda, son of Parameshvara. He was a brahmin from Trkkantiyur in Ponnani taluk. His younger brother Sankara was also a scholar in astronomy. Nilakantha's most notable work Tantrasamgraha (which 'spawned' a later anonymous commentary Tantrasangrahavyakhya and a further commentary by the name Yukthideepika, written in 1501) he elaborates and extends the contributions of Madhava.^{[36]}
Nilakantha was also the author of Aryabhatiyabhashya, a commentary of the Aryabhatiya. Of great significance in Nilakantha's work includes the presence of inductive mathematical proofs, a derivation and proof of the MadhavaGregory series of the arctangent trigonometric function, improvements and proofs of other infinite series expansions by Madhava, an improved series expansion of π that converges more rapidly, and the relationship between the power series of π and arctangent.^{[36]} He also gave sophisticated explanations of the irrationality of π, the correct formulation for the equation of the center of the planets, and a heliocentric model of the solar system.^{[22]}
Nilakanta also authored a work titled Jyotirmimamsa stressing the necessity and importance of astronomical observations to obtain correct parameters for computations and to develop more and more accurate theories.
Citrabhanu (c. 1530) was a 16th century mathematician from Kerala who gave integer solutions to 21 types of systems of two simultaneous Diophantine equations in two unknowns. These types are all the possible pairs of equations of the following seven forms:^{[37]}
For each case, Chitrabhanu gave an explanation and justification of his rule as well as an example. Some of his explanations are algebraic, while others are geometric.^{[37]}
Main articles : Jyesthadeva, Yuktibhasa
Jyesthadeva (c. 1500–1600) was another member of the Kerala School. His key work was the Yuktibhasa (written in Malayalam, a regional language of the Indian state of Kerala), the world's first Calculus text. It contained most of the developments of earlier Kerala School mathematicians, particularly from Madhava. Similar to the work of Nilakantha, it is unique in the history of Indian mathematics, in that it contains proofs of theorems, derivations of rules and series, a derivation and proof of the MadhavaGregory series of the arctangent function, proofs of most mathematical theorems and infinite series earlier discovered by Madhava and other mathematicians of the Kerala School. It also contains a proof of the series expansion of the arctangent function (equivalent to Gregory's proof), and the sine and cosine functions.^{[36]}
He also studied various topics found in many previous Indian works, including integer solutions of systems of first degree equations solved using kuttaka method, and rules of finding the sines and the cosines of the sum and difference of two angles.^{[36]} Jyesthadevan also gave the earliest statement of Wallis' theorem, and geometrical derivations of infinite series.
Main articles : Sankara Varman, Sadratnamala
There remains a final Kerala work worthy of a brief mention, Sadratnamala an astronomical treatise written by Sankara Varman (1774–1839) that serves as a summary of most of the results of the Kerala School. What is of most interest is that it was composed in the early 19th century and the author stands out as the last notable name in Kerala mathematics. A remarkable contribution was his compution of π correct to 17 decimal places.^{[31]}
A. K. Bag suggested in 1979 that knowledge of these results might have been transmitted to Europe through the trade route from Kerala by traders and Jesuit missionaries.^{[38]} Kerala was in continuous contact with China and Arabia, and Europe. The suggestion of some communication routes and a chronology by some scholars^{[39]}^{[40]} could make such a transmission a possibility; however, there is no direct evidence by way of relevant manuscripts that such a transmission took place.^{[40]} According to David Bressoud, "there is no evidence that the Indian work of series was known beyond India, or even outside of Kerala, until the nineteenth century."^{[9]}^{[41]}
Both Arab and Indian scholars made discoveries before the 17th century that are now considered a part of calculus.^{[10]} However, they were not able to, as Newton and Leibniz were, to "combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between the two, and turn calculus into the great problemsolving tool we have today."^{[10]} The intellectual careers of both Newton and Leibniz are welldocumented and there is no indication of their work not being their own;^{[10]} however, it is not known with certainty whether the immediate predecessors of Newton and Leibniz, "including, in particular, Fermat and Roberval, learned of some of the ideas of the Islamic and Indian mathematicians through sources of which we are not now aware."^{[10]} This is an active area of current research, especially in the manuscript collections of Spain and Maghreb, research that is now being pursued, among other places, at the Centre national de la recherche scientifique in Paris.^{[10]}
