Truth is Emerging Finally
The following is a main news item appeared in
“A little known school of scholars on the coast of southwest
“Dr George Gheverghese Joseph from the University of Manchester, who did the new study with Dennis Almeida at the University of Exeter, says the '
“There were many reasons why the contribution of the
"For some unfathomable reasons, the standard of evidence required to claim transmission of knowledge from East to West is greater than the standard of evidence required to knowledge from West to East."
In this context, I am pasting below, an article on the Kerala legacy to the World Mathematics.
Kerala Legacy to
- An Overview -
A basic change is taking place among the Indian youth - a change from indifference to eagerness to know our roots and heritage. Many individuals and organisations like IISH are inspiring this awakening. Motivated by the essay contest organised by the IISH, I present here an overview of the contributions of Kerala, within the framework of Indian scientific heritage, to the world of mathematics.
Kerala has a very long history as indicated by the mention of 'Kerala Puthra' in Mahabharata (3417 BC2). Kerala had been a centre of maritime trade, even at the time of the Babylonians:, we had contacts with many countries including Arabia and
A myth propagated by the Euro-centric scholars was that Indian Mathematics was standstill after Bhaskaracharya, until the British introduced modern Mathematics. This is not true. Though, the progress was slowed down in Northern India due to Islamic invasions, in Kerala, this period marked a high point in the development of astronomy and Mathematics.
The remarkable discoveries of Kerala mathematicians include a formula for the ecliptic, the Newton-Gauss interpolation formula; the formula for the sum of an infinite series, etc. Their contributions were also there to the infinite series, convergence, differentiation, and iterative methods for solution of non-linear equations. Yuktibhasa, the first calculus text written by Jyesthadeva of Kerala School explores methods and ideas of calculus repeated only in 17th' century.
Let us now explore the major contributions of some of the leading lights of the
Madhava of Sangamagramma
Madhava of Sangamagramma (c. 1340-1425) was the founder of the
Many of his writings have been lost. His contributions were discovered through the Tantrasamgraha, a major treatise written by Nilakantha, who came 100 years later.
His most significant contribution was :
Moving on from the finite procedures of ancient mathematics to treat their limit passage to infinity, which is the essence of modem classical analysis, and thus he is considered the father of mathematical analysis.
His other discoveries were:
· Trigonometric series for sine, cosine, tangent and arctangent functions.
· Methods of Polynomial expansion.
· Tests of convergence of infinite series.
· Analysis of infinite continued fractions.
· The solution of some transcendental numbers by iteration.
· Value of it to 11 decimal places (3.14159265359), the most accurate value given after a thousand years
· Sine tables to 12 decimal places and cosine tables to 9 decimal places of accuracy, (the best value till 17th century).
· Procedure to determine the lunar positions every 36 minutes.
· Methods to estimate the planetary motions.
· Rules of Integration.
Initiating the development of Calculus. It was then continued by his successors at the
Narayana Pandit (c. 1340-1400), the earliest of the notable Kerala mathematicians, had written two works, an arithmetical treatise called Ganita Kaumudi and an algebraic treatise called Bijaganita Vatamsa. Narayana is also thought to be the author of an elaborate commentary of Bhaskara II's Lilavati, titled Karmapradipika (or Karma-Paddhati).
Narayana's other major works are :
Parameshvara (c. 1370-1460) wrote commentaries on the works of Bhaskara I, Aryabhatta and Bhaskara II. His Lilavati Bhashya is a commentary on Bhaskara II's Lilavati.
The Siddhanta-dipika by Parameshvara is a commentary on the commentary of Govindsvamin on Bhaskara I's Maha-Bhaskariya. It contains:
In Nilakantha Somayaji's (1444-1544) most notable work Tantra Samgraha, he elaborates and extends the contributions of Madhava. He was also the author of Aryabhatiya-bhasa a commentary of the Aryabhatiya. Of great significance in Nilakantha's work includes:
· The presence of inductive mathematical proof.
· Proof of the Madhava-Gregory series of the arctangent.
· Improvements and proofs of other infinite series expansions by Madhava.
· An improved series expansion of π/4 that converges more rapidly.
· The relationship between the power series of π/4 and arctangent.
Chitrabhanu (c. 1530) was also from Kerala who gave integer solutions to 21 types of systems of two simultaneous equations with two unknowns.
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.
Jyesthadeva’s (c. 1500-1575) key work was the Yukti-bhasa (written in Malayalam), the world's first calculus text. It contained most of the developments of earlier
2.1 Kerala Role in the Growth of Euro Maths in the context of Euro\-centrism
The Euro centric approach that discounts all non-European contributions is still predominant in
Kerala's contribution goes beyond Mathematics. Astronomy/astrology, arts and Sanskrit were also benefited from Kerala's indelible imprints. Recent researches bring out more information about the contribution of Kerala to the world of knowledge. As an example, it may be worth to refer to an article I have read recently. It establishes the Kerala Namboothiri connection in the modification of Gregorian calendar in the sixteenth century4.
2 htto://www.ece. Isu.edulkak/Mahabharatall.pdf.
Prof. Subhash Kak,
For Furthur Reading
About the Author
P.S. There is a couple of scholarly comments - rather excellent additional information on the subject - from my great friend and and highly versatile Shri Karigarji. I am providing a direct link to them from here. Please do not miss them.
what about ramanujan interested you?
ramanujan interested me because i have always been fascinated by india. my brother john leavitt is an anthropologist who has been for quite a while in india and has done most of his research and work in uttar pradesh.
i think another reason why hardy and ramanujan fascinated me was about the whole religious question. hardy was a devout atheist -- the oxymoron! -- whose attitude to religion was extremely hostile, rather like that of christopher hitchens [author of the current bestseller god is not great]. and ramanujan said his mathematical formulae came to him as visions provided by a goddess. this way of conceptualising mathematics was anathema to hardy's code.
hardy had trouble in accepting that possibility, even though ramanujan actually perceived a mystic connection between his mathematics and his religion. in other words, hardy persisted in arguing that ramanujan just claimed to be devout to please his family but in fact he was a rationalist and much more like hardy. but that is something i think very few people believed except for hardy.
found this in my records (source unknown)...please see the pdf (which is a great powerpoint presentation on kerala math ) rajeev5.pdf......by rochester univ's sarada rajeev....
the posted by robin varghese at 04:43 pm | most traditional sanskrit didactic treatises are composed in some metrical form or another (cf. amara-kosa). i doubt it has anything to do with jyestha being a keralavasa, but rather is in line with by that time a millenia-old tradition of memorizing verse. posted by: i find chandan's suggestion that verse was used in the presentation of mathematics as a convenient mnemonic device to be both interesting and plausible. posted by: j. f. staal's 'the sanskrit of science' is a good overview of those parts of the sanskrit corpus that most recall those forms of folk idology and their systematisation that we call natural philosophy in the western tradition. journal of indian philosophy. the genres of sanskrit commentary and discourse originally represented an area of research for me, though when lee siegel's love in a dead language came out, i found i was glad that i hadn't pursued it beyond the m.a.
mathematical proofs made poetic
theganita yuktibhasa (1530) by jyesthadeva of the kerala school of mathematics is thought to be the first text on calculus , summarizing developments in mathematics in india from the 5th century onward, including infinitesimals, infinite series, power series, taylor series and integration. i was reading on the work, including this presentation by sarada rajeev over at rochester's physics department, when i noticed that the school presented mathematical proofs and results in the form of poems. it seems alien to me (malayalee, though i am), but it does paint a picture of a beautiful genre.
posted by robin varghese at 04:43 pm |permalink
most traditional sanskrit didactic treatises are composed in some metrical form or another (cf. amara-kosa). i doubt it has anything to do with jyestha being a keralavasa, but rather is in line with by that time a millenia-old tradition of memorizing verse.
posted by:chandan | oct 26, 2006 5:38:21 pm
i find chandan's suggestion that verse was used in the presentation of mathematics as a convenient mnemonic device to be both interesting and plausible.
posted by:abbas raza | oct 26, 2006 5:56:04 pm
j. f. staal's 'the sanskrit of science' is a good overview of those parts of the sanskrit corpus that most recall those forms of folk idology and their systematisation that we call natural philosophy in the western tradition. journal of indian philosophy.
the genres of sanskrit commentary and discourse originally represented an area of research for me, though when lee siegel's love in a dead language came out, i found i was glad that i hadn't pursued it beyond the m.a.