INDIAN knowedge system research in phd kera school of matehmctics

its all about the INDIAN knowledge system research in phd kera school of matehmctics

1. Kerala School of Mathematics:Orientation,Concerns and Perspectives C.Rajendran

2. History of Indian mathematics Six periods Harappan (3000-1500BC) Vedic(1500-500BC) Late V edic/ Jaina (500-200BC ) Bakshali Ms Period(200BC-400AD) Classical(400-1200AD) Medieval(1200-1600AD)

3. Some very outstanding contributions ‘Arabic ’ numerals Decimal system?

4. Harappan Mathemaatics:Orientations 1.2 million Kilometers :Indus valley, parts of east Punjab, Uttar Pradesh Northern Rajasthan, Costal Gujarat, Northern area of Persian border. Linked with town planning: brick technology: uniform ratio4:2:1 Mensuration : Weights Series of a standard ‘plumb bob’ with decimal ratio:0.05,0.1,0.2,0.5,1,2,10,20,50,100,200,500 ….. Indus scale :accurate (A shell 66.2millimeter with nine exactly parallel divisions ‘Indus inch’ 1.32 inch

5. “Vedic” Mathematics How ‘ vedic ’ is Swami Bharati Krishna Tirthaji’s Vedic Mathematics? 16 sutras and 13 sub sutras from a parisista of Atharvaveda ??

6. Multiplication: An example 88X96=8448 90X95=8550 Number Deficiency from 100 88 12 96 4 84 48 Number Deficiency 90 10 95 5 85 50

7. Vedic Mathematics:Astronomical and ritualistic orientations Necessity of selecting auspicious time for ritual Construction of altar : Brick technology Vedangas : Jyotisa and Kalpa > sulbasutras as ’chord’ geometry Baudhayana , (800BC) Apastamba ,(600BC) Katyayana (200BC) Home worship and community worship ‘falcon altar ‘ community ritual Sulbas for craftsmen for laying altars?

8. Sulba geometry Pythagoras theorem( Apastamba and Baudhayana ) Merging equal/unequal squares to obtain a third square To transform a rectangle into a square of equal area Squaring a circle/circling a square Evaluating irrational square roots to a high degree of approximation √ 2=1.4142156 …..

9. Jaina Mathematics Decline of Vedic sacrifice Jaina scientific temperament : Anekantavada Delinking from ritual: Mathematics cultivated for its own sake Philosophical preoccupations Number theory Permutations and combinations Binomial theorem Jaina Astronomy

10. Main works Suryaprajnapti Jambudvipaprajnapti Uttaradhyayanasutra Bhagavatisutra Anuyogadvarasutra

11. Features of Jaina Mathematics Enumeration of very large numbers :Measure of time called Sirsa prahelika 75X10x(8400000) 28 days Sequences, progressions possibly influenced by their philosophical theory of cosmological structures: Concentric rings of alternate continents and oceans

12. ‘ Bakshali ’ Period Mathematics becomes more practical and applied to everyday problem Devoted to arithematics and algebra The method becomes more systematic (1) Sutra- statement of rule (2) Udaharana -Statement of example (3)Karana-Operation of the rule

13. Classical Period Revival in the middle of the first millennium Kusumapura , Ujjayini and Mysore emerge as three centers Incorporation of Babylonian and Hellenistic ideas Emergence of Siddhantas containing astronomical methods and practices Zodiac system replaces stellar astronomy Planetary movements and eclipses correctly calculated.

14. Major Astronmers -mathematicians of the Classical period AryabhataI (AD 476) Kusumapura Brahmagupta (AD 598) Ujjayini Mahavira (AD 850)Mysore Sridhara (AD900)Bengal/ S.India ? Bhaskaracharya ( BhaskaraII ) (1114) Ujjayini Narayana Pandita (1370) Delhi

15. Medieval Kerala School of Mathematics Marks the zenith of achievement in Mathematics and astronomy in 14-17 centuries No external influence Probably influenced Western mathematics

16. Antecedents Vararuci : Katapayadi system Haridatta (650-700)introduced Parahita system based on Aryabhatiya Govindasvami (first half of 9 th century ) author of bhasya on Mahabhaskariya Sankaranarayana (825-900) patronized by King Kulasekhara Ravivarma of Mahodayapura

17. Major authors Madhava of Sangamagrama (1340-1425) Paramesvara of Vatasseri (1360-1445)Founder of Drgganita system –watched eclipses for 55 years Alattur village on Nila Bank Also a famous astrologer Nilakantha Somayajin (1443-1543) Kelallur at Trkkantiyur Became a Somayajin after performing a soma sacrifice. Corresponded with Sundararaja of Tamilnadu on astronomical matters;Patronised by Alvancheri Tamprakkal -misunderstandings

18. Sankara Varier of Trikkutaveli ((1500-60) Sankara of Mahisamangalam (1494-1570) Jyesthadeva (1500-100) Yuktibhasa Trikkantiyur Acyuta Pisaroti (1550-1621) Putumana Somayajin (1700-60) Sankaravarman of Katattanadu (1800-38) Sadratnamala

19. Kerala’s contributions: An Overview Gregory and Leibniz series for the inverse tangent The power series for π [usually attributed to Newton]and a number of rational approximations The Newton power series for sine and cosine and approximation for sine and cosine functions[usually attributed to Tylor ]

20. Did Kerala Mathematics migrate to the West ? Views of Raju C.K In 16 th C Jesuit missionaries were translating and transmitting very many Indian texts to Europe Their activities were centered in the vicinity of their Cochin College They were teaching Malayalam to local children Yuktibhasa and other related texts were in common use there for purposes like calendar making Infinite series in these Indian texts started appearing in works of Cavalieri,Pascal,Gregory etc from 1630 onwards These European scholars had access to Jesuit archives at the Cochin College Romano Tylor series expansions are found in mathematics / astronomy/ Jyotisha texts , like those of Madhava , Nilaknatha and Jyesthadeva

21. Circumstances of migration Jesuits required local calendar knowhow Their Calendars off the mark with clumsy Roman numerals making it difficult to handle fractions European Navigators like Numes , Mercator, Stevin, and Clavius desperately required good calendar with trigonometric value of Indian texts European governments required good navigational methods for reliable trade routes to India. Vasco da Gama required an Indian Pilot for navigation in the Indian ocean.

22. European Project European navigation theorist Nunes appointed Professor of Mathematics in 1529 Spanish [1567],Dutch[1636]French[1666]and British[1711] institute prizes French Royal Academy and British Royal Society established mainly to tackle navigational problems Prior to Clock technology of 18 th C, navigational problems had to be tackled on the basis of ‘celestial’ navigation, based on mathematics and astronomy, as found in Kerala texts Latitude problem required a correction of European calendar by 10 days Change in the date of equinoxes and Easter becomes necessary Trained Jesuits sent to India Ricci[1581] ‘Trying to understand Jyotisa from an intelligent Brahmin or an honest moor ‘in the vicinity of Cochin

23. Conclusions: Orientations ,Concerns and Perspectives Indian mathematics Harappan period:practical issues Vedic period:Ritual , linked with astronomy Jaina philosophical inputs Kerala: Astronomy,Astrology,Ritual , Vastuvidya Migration: Navigation

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