1 Contributions
2 Prominent mathematicians
3 Possibility of transmission of Kerala School results to Europe
4 Notes
5 References
6 External links
7 See also

Kerala school of astronomy and mathematics

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"Kerala School" redirects here. For public school, see Kerala School, Delhi.

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 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 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]

Contributions

Infinite Series and Calculus

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:

$\frac{1}{1-x} = 1 + x + x^2 + x^3 + \dots$ for

| x | < 1

[7]

This formula, however, was already known in the work of the 10th century Iraqi mathematician Alhazen (the Latinized form of the name Ibn al-Haytham) (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 semi-rigorous proof of the result:

$1^p+ 2^p + \cdots + n^p \approx \frac{n^{p+1}}{p+1}$ for large n. This result was also known to Alhazen.[1]

They applied ideas from (what was to become) differential and integral calculus to obtain (Taylor-Maclaurin) infinite series for $sin x$ , $cos x$ , and $arctan x$ .[9] The Tantrasangraha-vakhya gives the series in verse, which when translated to mathematical notation, can be written as:[1]

$r\arctan(\frac{y}{x}) = \frac{1}{1}\cdot\frac{ry}{x} -\frac{1}{3}\cdot\frac{ry^3}{x^3} + \frac{1}{5}\cdot\frac{ry^5}{x^t} - \cdots ,$ where $y/x \leq 1.$ $r\sin \frac{x}{r} = x - x\cdot\frac{x^2}{(2^2+2)r^2} + x\cdot \frac{x^2}{(2^2+2)r^2}\cdot\frac{x^2}{(4^2+4)r^2} - \cdot$ $r - \cos x = r\cdot \frac{x^2}{(2^2-2)r^2} - r\cdot \frac{x^2}{(2^2-2)r^2}\cdot \frac{x^2}{(4^2-4)r^2} + \cdots ,$ where, for

r = 1

, the series reduce to the standard power series for these trigonometric functions, for example: $\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots$ and $\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots$ (The Kerala school themselves 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.)[1] They also made use of the series expansion of $arctan x$ to obtain an infinite series expression (later known as Gregory series) for $π$ :[1]

$\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \ldots$

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:

$\frac{\pi}{4} \approx 1 - \frac{1}{3}+ \frac{1}{5} - \cdots (-1)^{(n-1)/2}\frac{1}{n} + (-1)^{(n+1)/2}f_i(n+1)$ where $f_1(n) = \frac{1}{2n}, \ f_2(n) = \frac{n/2}{n^2+1}, \ f_3(n) = \frac{(n/2)^2+1}{(n^2+5)n/2}.$

They manipulated the error term to derive a faster converging series for $π$ :[1]

$\frac{\pi}{4} = \frac{3}{4} + \frac{1}{3^3-3} - \frac{1}{5^3-5} + \frac{1}{7^3-7} - \cdots$

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 semi-rigorous 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]

Geometry, Arithmetic, and Algebra

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 non-linear equations by Parameshvara,[18][21] and the Newton-Gauss interpolation formula by Govindaswami.^{[citation needed]}

Astronomy

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]

Linguistics

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]}

Prominent mathematicians

Madhavan of Sangamagrama

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.

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