Kerala School of Astronomy and Mathematics | |
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Location | |
India | |
Information | |
Type | Hindu, astronomy, mathematics, science |
Founder | Madhava of Sangamagrama |
The Kerala School of Astronomy and Mathematics was a school of mathematics and astronomy founded by Madhava of Sangamagrama in Kerala, 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 discovered a number of important mathematical 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]
The Kerala school has made a number of contributions to the fields of infinite series and calculus. These include the following (infinite) geometric series:
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:
for large n.
They applied ideas from (what was to become) differential and integral calculus to obtain (Taylor–Maclaurin) infinite series for , , and . [8] The Tantrasangraha-vakhya gives the series in verse, which when translated to mathematical notation, can be written as: [1]
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.) [1] They also made use of the series expansion of to obtain an infinite series expression (later known as Gregory series) for : [1]
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 : [1]
They used the improved series to derive a rational expression, [1] for correct up to nine decimal places, i.e. . 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, [9] though the notion of a function, or of exponential or logarithmic functions, was not yet formulated.
In 1825 John Warren published a memoir on the division of time in southern India, [10] 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". [11] 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, [12] [13] a commentary on the Yuktibhasa's proof of the sine and cosine series [14] and two papers that provide the Sanskrit verses of the Tantrasangrahavakhya for the series for arctan, sin, and cosine (with English translation and commentary). [15] [16]
In 1952 Otto Neugebauer wrote on Tamil astronomy. [17]
In 1972 K. V. Sarma published his A History of the Kerala School of Hindu Astronomy which described features of the School such as the continuity of knowledge transmission from the 13th to the 17th century: Govinda Bhattathiri to Parameshvara to Damodara to Nilakantha Somayaji to Jyesthadeva to Acyuta Pisarati. Transmission from teacher to pupil conserved knowledge in "a practical, demonstrative discipline like astronomy at a time when there was not a proliferation of printed books and public schools."
In 1994 it was argued that the heliocentric model had been adopted about 1500 A.D. in Kerala. [18]
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. [19] Kerala was in continuous contact with China and Arabia, and Europe. The suggestion of some communication routes and a chronology by some scholars [20] [21] could make such a transmission a possibility; however, there is no direct evidence by way of relevant manuscripts that such a transmission took place. [21] 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". [8] [22] V.J. Katz notes some of the ideas of the Kerala school have similarities to the work of 11th-century Iraqi scholar Ibn al-Haytham, [9] suggesting a possible transmission of ideas from Islamic mathematics to Kerala. [23] [ dubious ]
Both Arab and Indian scholars made discoveries before the 17th century that are now considered a part of calculus. [9] According to V.J. Katz, they were yet 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 have today", like Newton and Leibniz. [9] The intellectual careers of both Newton and Leibniz are well-documented and there is no indication of their work not being their own; [9] 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". [9] 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. [9]
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Mādhava was an Indian mathematician and astronomer from the town believed to be present-day Aloor, Irinjalakuda in Thrissur District, Kerala, India. He is considered the founder of the Kerala school of astronomy and mathematics. One of the greatest mathematician-astronomers of the Middle Ages, Madhava made pioneering contributions to the study of infinite series, calculus, trigonometry, geometry, and algebra. He was the first to use infinite series approximations for a range of trigonometric functions, which has been called the "decisive step onward from the finite procedures of ancient mathematics to treat their limit-passage to infinity".
Jyeṣṭhadeva was an astronomer-mathematician of the Kerala school of astronomy and mathematics founded by Madhava of Sangamagrama. He is best known as the author of Yuktibhāṣā, a commentary in Malayalam of Tantrasamgraha by Nilakantha Somayaji (1444–1544). In Yuktibhāṣā, Jyeṣṭhadeva had given complete proofs and rationale of the statements in Tantrasamgraha. This was unusual for traditional Indian mathematicians of the time. An analysis of the mathematics content of Yuktibhāṣā has prompted some scholars to call it "the first textbook of calculus". Jyeṣṭhadeva also authored Drk-karana a treatise on astronomical observations.
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Yuktibhāṣā also known as Gaṇitanyāyasaṅgraha, is a major treatise on mathematics and astronomy, written by Indian astronomer Jyesthadeva of the Kerala school of mathematics in about AD 1530. The treatise, written in Malayalam, is a consolidation of the discoveries by Madhava of Sangamagrama, Nilakantha Somayaji, Parameshvara, Jyeshtadeva, Achyuta Pisharati and other astronomer-mathematicians of the Kerala school. Yuktibhasa is mainly based on Nilakantha's Tantra Samgraha. It is considered, possibly the first text, on the foundations of calculus and predates those of European mathematicians such as James Gregory and Newton by many centuries. The treatise was largely unnoticed as the book was written in the local language of Malayalam and it was thought that many Indian ideas in astronomy and computation lacked proofs or foundations. Yuktibhasa however demonstrates founding principles and the development and proofs of theorems. However, both Oxford University and Royal Society of Great Britain have accepted that Calculus and many such pioneering mathematical theorems originated in India.
This is a summary of differentiation rules, that is, rules for computing the derivative of a function in calculus.
A timeline of calculus and mathematical analysis.
Gregory's series, also known as the Madhava–Gregory series or Leibniz's series, is an infinite Taylor series expansion of the inverse tangent function. It was discovered by the Indian mathematician Madhava of Sangamagrama (1350–1410) and rediscovered in 1668 by James Gregory. It was re-rediscovered a few years later by Gottfried Leibniz, who reobtained the Leibniz formula for π as the special case x = 1 of the Gregory series.
Madhava's sine table is the table of trigonometric sines of various angles constructed by the 14th century Kerala mathematician-astronomer Madhava of Sangamagrama. The table lists the trigonometric sines of the twenty-four angles 3.75°, 7.50°, 11.25°, ..., and 90.00°. The table is encoded in the letters of Devanagari using the Katapayadi system. This gives the entries in the table an appearance of the verses of a poem in Sanskrit.
Karanapaddhati is an astronomical treatise in Sanskrit attributed to Puthumana Somayaji, an astronomer-mathematician of the Kerala school of astronomy and mathematics. The period of composition of the work is uncertain. C.M. Whish, a civil servant of the East India Company, brought this work to the attention of European scholars for the first time in a paper published in 1834. The book is divided into ten chapters and is in the form of verses in Sanskrit. The sixth chapter contains series expansions for the value of the mathematical constant π, and expansions for the trigonometric sine, cosine and inverse tangent functions.
In mathematics, a Madhava series or Leibniz series is any one of the series in a collection of infinite series expressions all of which are believed to have been discovered by Madhava of Sangamagrama, the founder of the Kerala school of astronomy and mathematics and later by Gottfried Wilhelm Leibniz, among others. These expressions are the Maclaurin series expansions of the trigonometric sine, cosine and arctangent functions, and the special case of the power series expansion of the arctangent function yielding a formula for computing π. The power series expansions of sine and cosine functions are respectively called Madhava's sine series and Madhava's cosine series. The power series expansion of the arctangent function is sometimes called Madhava–Gregory series or Gregory–Madhava series. These power series are also collectively called Taylor–Madhava series. The formula for π is referred to as Madhava–Newton series or Madhava–Leibniz series or Leibniz formula for pi or Leibnitz–Gregory–Madhava series. These further names for the various series are reflective of the names of the Western discoverers or popularizers of the respective series.
Kriyakramakari (Kriyā-kramakarī) is an elaborate commentary in Sanskrit written by Sankara Variar and Narayana, two astronomer-mathematicians belonging to the Kerala school of astronomy and mathematics, on Bhaskara II's well-known textbook on mathematics Lilavati. Kriyakramakari, along with Yuktibhasa of Jyeshthadeva, is one of the main sources of information about the work and contributions of Sangamagrama Madhava, the founder of Kerala school of astronomy and mathematics. Also the quotations given in this treatise throw much light on the contributions of several mathematicians and astronomers who had flourished in an earlier era. There are several quotations ascribed to Govindasvami a 9th-century astronomer from Kerala.
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