Everything about The Kerala School Of Astronomy And Mathematics totally explained
The
Kerala school of astronomy and mathematics was a school of
mathematics and
astronomy founded by
Madhava of Sangamagrama in
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.
Their discovery of these three important series expansions of calculus—several centuries before calculus was developed in Europe by
Leibniz and
Newton—was a landmark achievement in mathematics. However, the Kerala School can't be said to have invented
calculus,
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).
The Kerala school made intuitive use of
mathematical induction, though the
inductive hypothesis wasn't yet formulated or employed in proofs. The
Tantrasangraha-vakhya gives the series in verse, which when translated to mathematical notation, can be written as: though the notion of a function, or of exponential or logarithmic functions, wasn't yet formulated.
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
Yuktibhasa given in two papers, a commentary on the Yuktibhasa'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).
Geometry, Arithmetic, and Algebra
In the fields of
geometry,
arithmetic, and
algebra, the Kerala school discovered a formula for the
ecliptic,
Lhuilier's formula for the circumradius of a
cyclic quadrilateral by
Parameshvara, decimal
floating point numbers, the
secant method and
iterative methods for solution of
non-linear equations by
Parameshvara, and the
Newton-
Gauss interpolation formula by Govindaswami.
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. Late Kerala school astronomers gave a formulation for the equation of the center of the planets, and a
heliocentric model of the solar system. the second-order Taylor series approximations of the sine and cosine functions and the third-order Taylor series approximation of the sine function, the power series of
π (usually attributed to
Leibniz), the solution of
transcendental equations by
iteration, and the approximation of
transcendental numbers by
continued fractions. He also extended some results found in earlier works, including those of
Bhaskara.
Although the
Karmapradipika contains little original work, it contains seven different methods for squaring numbers, a contribution that's wholly original to the author, as well as contributions to algebra and
magic squares.
Nilakantha was also the author of
Aryabhatiya-bhashya, 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 Madhava-Gregory 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.
»
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.
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. such as communication routes and a suitable chronology certainly make such a transmission a possibility. However, there's no direct evidence by way of relevant manuscripts that such a transmission took place. 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."
Both
Arab and Indian scholars made discoveries before the 17th century that are now considered a part of calculus.
[ 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 problem-solving tool we've today."][ The intellectual careers of both Newton and Leibniz are well-documented and there's no indication of their work not being their own;][ however, it isn't 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're not now aware."][ This is an active area of current research, especially in the manuscript collections of Spain and Maghreb, research that's now being pursued, among other places, at the Centre national de la recherche scientifique in Paris.][Further Information]
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