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.
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). 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.
|Kerala School of Astronomy and Mathematics|
|Type||Hindu, astronomy, mathematics, science|
|Founder||Madhava of Sangamagrama|
The Kerala school made intuitive use of mathematical induction, though the inductive hypothesis was not yet formulated or employed in proofs. 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 . The Tantrasangraha-vakhya gives the series in verse, which when translated to mathematical notation, can be written as:
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.) They also made use of the series expansion of to obtain an infinite series expression (later known as Gregory series) for :
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 :
They used the improved series to derive a rational expression, for correct up to nine decimal places, i.e. . They made use of an intuitive notion of a limit to compute these results. The Kerala school mathematicians also gave a semi-rigorous method of differentiation of some trigonometric functions, 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, 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". 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, 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).
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."
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. Kerala was in continuous contact with China and Arabia, and Europe. The suggestion of some communication routes and a chronology by some scholars could make such a transmission a possibility; however, there is 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. 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. The intellectual careers of both Newton and Leibniz are well-documented and there is no indication of their work not being their own; 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". 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.
A History of the Kerala School of Hindu Astronomy (in perspective) is the first definitive book giving a comprehensive description of the contribution of Kerala to astronomy and mathematics. The book was authored by K. V. Sarma who was a Reader in Sanskrit at Vishveshvaranand Institute of Sanskrit and Indological Studies, Panjab University, Hoshiarpur, at the time of publication of the book (1972). The book, among other things, contains details of the lives and works of about 80 astronomers and mathematicians belonging to the Kerala School. It has also identified 752 works belonging to the Kerala school.
Even though C. M. Whish, an officer of East India Company, had presented a paper on the achievements of the mathematicians of Kerala School as early as 1842, western scholars had hardly taken note of these contributions. Much later in the 1940s, C. T. Rajagopal and his associates made some efforts to study and popularize the discoveries of Whish. Their work was lying scattered in several journals and as parts of books. Even after these efforts by C. T. Rajagopal and others, the view that Bhaskara II was the last significant mathematician pre-modern India had produced had prevailed among scholars, and surprisingly, even among Indian scholars. It was in this context K. V. Sarma published his book as an attempt to present in a succinct form the results of the investigations of C. T. Rajagopal and others and also the findings of his own investigations into the history of the Kerala school of astronomy and mathematics.A Passage to Infinity
A Passage to Infinity : Medieval Indian Mathematics from Kerala and Its Impact is a book by George Gheverghese Joseph chronicling the social and mathematical origins of the Kerala school of astronomy and mathematics. The book discusses the highlights of the achievements of Kerala school and also analyses the hypotheses and conjectures on the possible transmission of Kerala mathematics to Europe.Achyutha Pisharadi
Achyutha Pisharodi (c. 1550 at Trikkandiyur (aka Kundapura), Tirur, Kerala, India – 7 July 1621 in Kerala) was a Sanskrit grammarian, astrologer, astronomer and mathematician who studied under Jyeṣṭhadeva and was a member of Madhava of Sangamagrama's Kerala school of astronomy and mathematics. He is remembered mainly for his part in the composition of his student Melpathur Narayana Bhattathiri's devotional poem, Narayaneeyam.C. M. Whish
Charles Matthew Whish (1794–1833) was an English civil servant in the Madras Establishment of the East India Company. Whish was the first to bring to the notice of the western mathematical scholarship the achievements of the Kerala school of astronomy and mathematics. Whish wrote in his historical paper: Kerala mathematicians had ... laid the foundation for a complete system of fluxions ... and their works ... abound with fluxional forms and series to be found in no work of foreign countries.
Whish was also a linguist and had prepared a grammar and a dictionary of the Malayalam language.C.M. Whish was a collector of palm-leaf manuscripts in Sanskrit and other languages. After his premature death in 1833 at the age of thirty-eight years, Whish's brother, J.L. Whish, who was also employed in the service of East India Company deposited these manuscripts in the Royal Asiatic Society of Great Britain and Ireland in July 1836. A catalogue of these manuscripts list 195 items. Though the manuscripts collected by Whish are not distinguished
by great age, there are many rare and valuable ones among them. Perhaps the most important of all are the
Mahabharata manuscripts which represent a distinct recension of the great epic. These manuscripts were related a wide range of subjects: vedic literature, ancient epic poetry, classical Sanskrit Literature, and technical and scientific literature.
He joined the service of East India Company in 1812 as Register of Zillah Court in South Malabar and rose up the judicial ladder to become finally a Criminal Judge at Cuddapah. Cuddapah Town Cemetery had a tomb in the name of C.M. Whish with the inscription "Sacred to the memory of C.M. Whish, Esquire of the Civil Service, who departed this life on the 14th April 1833, aged 38 years".Damodara
Kottessori Damodara Sottaidiri was an astronomer-mathematician of the Kerala school of astronomy and mathematics who flourished during the fifteenth century CE. He was a son of Kottessori Paramesvara (1360–1425) who developed the drigganita system of astronomical computations. The family home of Paramesvara was Vatasseri (sometimes called Vatasreni) in the village of Alathiyur, Tirur in Kerala.Damodara was a teacher of Nilakantha Somayaji. As a teacher he initiated Nilakantha into the science of astronomy and taught him the basic principles in mathematical computations.Ganita-yukti-bhasa
Ganita-yukti-bhasa (also written as Ganita Yuktibhasa) is either the title or a part of the title of three different books:
Ganita-yukti-bhasa (Rationales in Mathematical Astronomy) of Jyesthadeva, published by Springer, is the first critical edition with an English translation of Yuktibhasa, a seminal treatise in the Malayalam language composed in c.1530 CE by Jyeshthadeva. one of the most significant personalities of the Kerala school of astronomy and mathematics. The book is published in two volumes as Volume I and Volume II. Jyeshthadeva's Yuktibhasa discusses various topics in mathematics and astronomy and it is purported to be an enunciation of the rationales underlying the various mathematical assertions and the astronomical concepts and computations in the great treatise Tantrasamgraha of Nilakantha Somayaji (1444–1544). (An edition of the mathematical portion of Yuktibhasa with some explanatory notes in Malayalam had appeared in 1948.)
In addition to Jyshthadeva's Yuktibhasa, Kerala school of astronomy and mathematics had produced another work called Ganita Yuktibhasa. In contrast to Jyeshthadeva's Yuktibhasa, this work is in Sanskrit. The date of composition and the name of the author of this work in Sanskrit have not been determined with certainty. The book appears to be of a later period and seems to have been composed as a somewhat rough Sanskrit translation of the Malayalam original Yuktibhasa.
Ganita Yuktibhasa (Volume III) is the title of a book published by Indian Institute of Advanced Study (IIAS), Shimla, and it is a critical edition of the Sanskrit Ganita Yuktibhasa. This is intended as the third volume of a series on Yuktibhasa, the first two volumes being Volume I and Volume II of Ganita-yukti-bhasa (Rationales in Mathematical Astronomy) of Jyeshthadeva published by Springer.Govinda Bhattathiri
Govinda Bhaṭṭathiri (also known as Govinda Bhattathiri of Thalakkulam or Thalkkulathur) (c. 1237 – 1295) was an Indian astrologer and astronomer who flourished in Kerala during the thirteenth century CE. His major work was Dasadhyayi, a commentary on the first ten chapters of the astrological text Brihat Jataka composed by Varāhamihira (505 – 587 CE). This is considered to be the most important of the 70 known commentaries on this text. Bhaṭṭathiri had also authored another important work in astrology titled Muhūrttaratnaṃ. Paramesvara (ca.1380–1460), an astronomer of the Kerala school of astronomy and mathematics known for the introduction of the Dṛggaṇita system of astronomical computations, had composed an extensive commentary on this work. In this commentary Paramesvara had indicated that he was a grandson of a disciple of the author of Muhūrttaratnaṃ.Govinda Bhaṭṭatiri was born in the Nambudiri family known by the name Thalakkulathur in the village of Alathiyur, Tirur in Kerala. He was traditionally considered to be the progenitor of the Pazhur Kaniyar family of astrologers. He is a legendary figure in the Kerala astrological traditions.Indian astronomy
Indian astronomy has a long history stretching from pre-historic to modern times. Some of the earliest roots of Indian astronomy can be dated to the period of Indus Valley Civilization or earlier. Astronomy later developed as a discipline of Vedanga or one of the "auxiliary disciplines" associated with the study of the Vedas, dating 1500 BCE or older. The oldest known text is the Vedanga Jyotisha, dated to 1400–1200 BCE (with the extant form possibly from 700–600 BCE).Greek astronomy was influenced by Indian astronomy and vice versa beginning in the 4th century BCE and through the early centuries of the Common Era, for example by the Yavanajataka and the Romaka Siddhanta, a Sanskrit translation of a Greek text disseminated from the 2nd century.Indian astronomy flowered in the 5th–6th century, with Aryabhata, whose Aryabhatiya represented the pinnacle of astronomical knowledge at the time. Later the Indian astronomy significantly influenced Muslim astronomy, Chinese astronomy, European astronomy, and others. Other astronomers of the classical era who further elaborated on Aryabhata's work include Brahmagupta, Varahamihira and Lalla.
An identifiable native Indian astronomical tradition remained active throughout the medieval period and into the 16th or 17th century, especially within the Kerala school of astronomy and mathematics.Jyeṣṭhadeva
Jyeṣṭhadeva (Malayalam: ജ്യേഷ്ഠദേവന്) (c. 1500 – c. 1575) was an astronomer-mathematician of the Kerala school of astronomy and mathematics founded by Madhava of Sangamagrama (c. 1350 – c. 1425). 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.Jyotirmimamsa
In Hindu astronomy, Jyotirmimamsa (analysis of astronomy) is a treatise on the methodology of astronomical studies authored by Nilakantha Somayaji (1444–1544) in around 1504 CE. Nilakantha somayaji was an important astronomer-mathematician of the Kerala school of astronomy and mathematics and was the author of the much celebrated astronomical work titled Tantrasamgraha. This book stresses the necessity and importance of astronomical observations to obtain correct parameters for computations and to develop more and more accurate theories. It even discounts the role of revealed wisdom and divine intuitions in studying astronomical phenomena. Jyotirmimamsa is sometimes cited as proof to establish that modern methodologies of scientific investigations are not unknown to ancient and medieval Indians.The nature of the astronomical and mathematical work, the divine intuition, the experimental details of the science, corrections to the planetary parameters, reasons for the corrections for the planetary revolutions, Vedic authority for inference in astronomy, relative accuracy of different systems, and correction through eclipses, true motion, position, etc., of planets are some of the topics discussed in Jyotirmimamsa.Kriyakramakari
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 ('Operational Techniques'), 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.Sankara Variar (c. 1500 - 1560), the first author of Kriyakramakari, was a pupil of Nilakantha Somayaji and a temple-assistant by profession. He was a prominent member of the Kerala school of astronomy and mathematics. His works include Yukti-dipika an extensive commentary on Tantrasangraha by Nilakantha Somayaji. Narayana (c. 1540-1610), the second author, was a Namputiri Brahmin belonging to the Mahishamangalam family in Puruvanagrama (Peruvanam in modern-day Thrissur District in Kerala).
Sankara Variar wrote his commentary of Lilavati up to stanza 199. Variar completed this by about 1540 when he stopped writing due to other preoccupations. Sometimes after his death, Narayana completed the commentary on the remaining stanzas in Lilavati.Madhava of Sangamagrama
Mādhava (c. 1340 – c. 1425) 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".Some scholars have also suggested that Madhava's work, through the writings of the Kerala school, may have been transmitted to Europe via Jesuit missionaries and traders who were active around the ancient port of Muziris at the time. As a result, it may have had an influence on later European developments in analysis and calculus.Parameshvara
Kottessori Parameshvara Kundisori (c. 1380–1460) was a major Indian mathematician and astronomer of the Kerala school of astronomy and mathematics founded by Madhava of Sangamagrama. He was also an astrologer. Parameshvara was a proponent of observational astronomy in medieval India and he himself had made a series of eclipse observations to verify the accuracy of the computational methods then in use. Based on his eclipse observations, Parameshvara proposed several corrections to the astronomical parameters which had been in use since the times of Aryabhata. The computational scheme based on the revised set of parameters has come to be known as the Drgganita or Drig system. Parameshvara was also a prolific writer on matters relating to astronomy. At least 25 manuscripts have been identified as being authored by Parameshvara.Sangamagrama
Sangamagrama is a town in medieval Kerala believed to be the Brahminical Grama of Irinjalakuda which includes parts of Irinjalakuda Municipality, Aloor, Muriyad and Velookara Panchayaths, Thrissur District. It is associated with the noted mathematician Madhava of Sangamagrama, founder of the Kerala school of astronomy and mathematics.The town is known for the Koodalmanikyam temple.Sankara Variar
Shankara Variyar (IAST: Śaṅkara Vāriyar; c. 1500 – c. 1560) was an astronomer-mathematician of the Kerala school of astronomy and mathematics. His family were employed as temple-assistants in the temple at Tṛkkuṭaveli near modern Ottapalam.Sankara Varman
Sankara Varman (1774–1839) was an astronomer-mathematician belonging to the Kerala school of astronomy and mathematics. He is best known as the author of Sadratnamala, a treatise on astronomy and mathematics, composed in 1819. Sankara Varman is considered as the last notable figure in the long line of illustrious astronomers and mathematicians in the Kerala school of astronomy and mathematics beginning with Madhava of Sangamagrama. Sadratnamala was composed in the traditional style followed by members of the Kerala school at a time when India had been introduced to the western style of mathematics and of writing books in mathematics. One of Varman's contribution to mathematics was his computation of the value of the mathematical constant π correct to 17 decimal places.Tantrasamgraha
Tantrasamgraha, or Tantrasangraha, (literally, A Compilation of the System) is an important astronomical treatise written by Nilakantha Somayaji, an astronomer/mathematician belonging to the Kerala school of astronomy and mathematics.
The treatise was completed in 1501 CE. It consists of 432 verses in Sanskrit divided into eight chapters. Tantrasamgraha had spawned a few commentaries: Tantrasamgraha-vyakhya of anonymous authorship and Yuktibhāṣā authored by Jyeshtadeva in about 1550 CE.
Tantrasangraha, together with its commentaries, bring forth the depths of the mathematical accomplishments the Kerala school of astronomy and mathematics, in particular the achievements of the remarkable mathematician of the school Sangamagrama Madhava.
In his Tantrasangraha, Nilakantha revised Aryabhata's model for the planets Mercury and Venus. His equation of the centre for these planets remained the most accurate until the time of Johannes Kepler in the 17th century.It was C.M. Whish, a civil servant of East India Company, who brought to the attention of the western scholarship the existence of Tantrasamgraha through a paper published in 1835. The other books mentioned by C.M. Whish in his paper were Yuktibhāṣā of Jyeshtadeva, Karanapaddhati of Puthumana Somayaji and Sadratnamala of Sankara Varman.Venvaroha
Veṇvāroha is a work in Sanskrit composed by Mādhava (c.1350 – c.1425) of Sangamagrāma the founder of the Kerala school of astronomy and mathematics. It is a work in 74 verses describing methods for the computation of the true positions of the Moon at intervals of about half an hour for various days in an anomalistic cycle. This work is an elaboration of an earlier and shorter work of Mādhava himself titled Sphutacandrāpti. Veṇvāroha is the most popular astronomical work of Mādhava. It is dated 1403 CE. Acyuta Piṣārati (1550–1621), another prominent mathematician/astronomer of the Kerala school, has composed a Malayalam commentary on Veṇvāroha. This astronomical treatise is of a type generally described as Karaṇa texts in India. Such works are characterized by the fact that they are compilations of computational methods of practical astronomy. The title Veṇvāroha literally means Bamboo Climbing and it is indicative of the computational procedure expounded in the text. The computational scheme is like climbing a bamboo tree, going up and up step by step at measured equal heights.
The novelty and ingenuity of the method attracted the attention of several of the followers of Mādhava and they composed similar texts thereby creating a genre of works in Indian mathematical tradition collectively referred to as ‘veṇvāroha texts’. These include Drik-veṇvārohakriya of unknown authorship of epoch 1695 and Veṇvārohastaka of Putuman Somāyaji.In the technical terminology of astronomy, the ingenuity introduced by Mādhava in Veṇvāroha can be explained thus: Mādhava has endeavored to compute the true longitude of the Moon by making use of the true motions rather than the epicyclic astronomy of the Aryabhata tradition. He made use of the anomalistic revolutions for computing the true positions of the Moon using the successive true daily velocity specified in Candravākyas (Table of Moon-mnemonics) for easy memorization and use.Veṇvāroha has been studied from a modern perspective and the process is explained using the properties of periodic functions.Whish
Whish is a surname, and may refer to:
C. M. Whish (1794–1833), English civil servant of the East India Company and author of the first western paper on the Kerala school of astronomy and mathematics
J. L. Whish, his brother and also an English civil servant of the East India Company
David Whish-Wilson (born 1966), Australian author
Claudius Buchanan Whish (1827-1890), Australian sugar-planter
Peter Whish-Wilson (born 1968), Australian politician
Ancient Dharmic centres of higher learning
|Early centres of learning|
|Further centres of learning|