KERALA SCHOOL OF MATHEMATICS:ORIENTATION,CO

NCERNS AND PERSPECTIVESC.Rajendran

History of Indian mathematicsSix periods

Harappan(3000-1500BC) Vedic(1500-500BC) Late Vedic/Jaina(500-200BC) Bakshali Ms Period(200BC-400AD) Classical(400-1200AD) Medieval(1200-1600AD)

Some very outstanding contributions

‘Arabic’ numerals Decimal system?

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

“Vedic” Mathematics How ‘vedic’ is Swami Bharati Krishna

Tirthaji’s Vedic Mathematics? 16 sutras and 13 sub sutras from a

parisista of Atharvaveda??

Multiplication: An example 88X96=8448

90X95=8550

Number Deficiency from 10088 1296 484 48

Number Deficiency90 1095 585 50

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?

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…..

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

Main works Suryaprajnapti Jambudvipaprajnapti Uttaradhyayanasutra Bhagavatisutra Anuyogadvarasutra

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

‘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

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.

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

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

Antecedents Vararuci :Katapayadi system Haridatta (650-700)introduced Parahita

system based on Aryabhatiya Govindasvami (first half of 9th century )

author of bhasya on Mahabhaskariya Sankaranarayana (825-900) patronized

by King Kulasekhara Ravivarma of Mahodayapura

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

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

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]

Did Kerala Mathematics migrate to the West ? Views of Raju C.K

In 16th 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

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.

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 18th 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

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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