Kerala Legacy to World Mathematics

Truth is Emerging Finally

The following is a main news item appeared in Telegraph, U.K. on 14/08/07.(http://www.telegraph.co.uk/earth/main.jhtml?view=DETAILS&grid=&xml=/earth/2007/08/14/scinewt114.xml)

A little known school of scholars on the coast of southwest India discovered one of the founding principles of modern mathematics hundreds of years before Newton published them, according to a new study.

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“Dr George Gheverghese Joseph from the University of Manchester, who did the new study with Dennis Almeida at the University of Exeter, says the 'KeralaSchool' identified the 'infinite series'- one of the basic components of calculus - some 250 years earlier.

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“There were many reasons why the contribution of the KeralaSchool has not been acknowledged - a prime reason is neglect of scientific ideas emanating from the Non-European world - a legacy of European colonialism and beyond.

"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
World Mathematics

- An Overview -


By VIKAS SARANGADHARA
"The National Geographic has declared Kerala, the South- west coast near the tip of the Indian peninsula, as God's Own Country.  In recent years, Kerala has gained recognition for its role in the reconstruction of medieval Indian Mathematics"1

1.0 Introduction

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 China. Therefore, it was likely that Kerala could have been a. hub in the Knowledge Highway connecting India to the World. Kerala started peaking in fourteenth century. In fact, after Bhaskara-II (twelfth century), the next remarkable contribution had to wait for 200 years until Madhava, who invented the Taylor Series and rigorous mathematical analysis. And Madhava was from Kerala. He was the founder of; now famed Kerala School of Mathematics.

2.0 Kerala School

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 KeralaSchool. It is to be mentioned here that some scholars believe that Aryabhata and Bhaskara, two great Astronomer-Mathematicians were also belonging to the KeralaSchool. However, there is no consensus in this regard3.

Madhava of Sangamagramma

Madhava of Sangamagramma (c. 1340-1425) was the founder of the KeralaSchool and one of the greatest mathematician-astronomers of the Middle Ages.

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.

·         Additional Taylor series approximations of sine and cosine 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 KeralaSchool

Narayana Pandit

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 :

  • A rule to calculate approximate values of square roots.
  • Solutions of indeterminate higher-order equations.
  • Mathematical operations with zero.
  • Discussion of magic squares and similar figures.
  • Contribution related to the cyclic quadrilaterals.

Parameshvara

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:

  • A mean value type formula for inverse interpolation of the sine.
  • It presents a one-point iterative technique for calculating the sine of a given angle.
  • A more efficient approximation that works using a two-point iterative algorithm, which is essentially the same as the modem secant method.
  • Give the radius of circle with inscribed cyclic quadrilateral, an expression that is normally attributed to Lhuilier (1782).

Nilakantha Somayaji

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

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

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 KeralaSchool mathematicians, particularly Madhava's. Similar to the w6rk of Nilakantha, it is almost unique in the history of Indian mathematics, in that it contains:

  • Derivations of rules and series.
  • Proofs of most mathematical theorems and infinite series earlier discovered by Madhava and others
  • Proof of the series expansion of the arctangent function (equivalent to Gregory's proof), and the sine and cosine functions.
  • The earliest statement of Wallis' theorem.
  • Geometric derivations of series.

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 India and outside too. Of course, there are notable exceptions such as the American historian Will Durant, Australian Indologist A. L. Basham and the great Mathematician Laplace. They acknowledge that, mathematics today owes a huge debt to the outstanding contributions made by Indian mathematicians over many centuries. What is unfortunate is our reluctance to recognize this. We still prefer to teach in our schools that discoveries mentioned above as European in origin. Many scholars argue that Kerala Mathematics may have been transmitted to Europe; Kerala was in contact with Europe from around 1500. The methodological similarities, possible communication routes and strong correlation in the chronology of development are hard to dismiss.

3.0 Conclusion

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.

References    

1 The KeralaSchool, European Mathematics and Navigation, By D.P. Agrawal, http://www.infinityfoundation.com/mandala/t es/t es agraw kerala.htm

2 htto://www.ece. Isu.edulkak/Mahabharatall.pdf.

3 Prof. Subhash Kak, India's schoolbook histories, http://www.rediff.com/news/20031aug,'22kak.htm

4 http:i/www.namhoothiri.corn/articles/calendar.htm

For Furthur Reading

  • T.R.N. Rao and S. Kak, Computing Science in Ancient India. USL Press, Lafayette, 1998.)
  • Boyer, C.B.. 1968. A History of Mathematics. New York: John Wiley.
  • Chattopdhayaya, D. 1986. History of Science and Technology in Ancient India: the Beginnings. Calcutta: Firma KLM.
  • Eves, H. 1983. An Introduction to History of Mathematics: A Reader. Philadelphia: Sunders.
  • Joseph, G.G.1994. The Crest of the Peacock: Non-European Roots of Maths. London:Penguin Books. Pp.286-289.
  • Rajagopal, C.T. and M.S. Rangachari. 1986. On Medieval Keralese maths. Archive for Exact Sciences. 35:91-99.
  • Raju, C.K. 2000. Talk given at the international seminar on East-West Transitions, National Institute of Advanced Studies, Bangalore, Dec 2000.
  • Raju, C.K. 2001. Computers, Mathematics Education, and the Alternative Epistemology of the Calculus in the Yuktibhasa. Philosophy East and West, 51(3), 2001, 325—61
  • IISH(Indian Institute of Scientific Heritage) Publications
  •   http://www.iish.org/sanatananews_details.asp?id=83

About the Author

The author of the above essay is my son and he received ‘Shastra Ratna’ award from IISH (Indian Institute of Scientific Heritage) for this essay. on 25th December 2006. It was a part of an Essay Competition on Indian Science Heritage organized by IISH for Indian students from across the World. Vikas won the award for coming top of his category (Junior College Students). He received the prize on 25/12/06 at Guruvayoor.                                                                                                           -Dr.P.E.S.Kartha
 

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.

http://kartha-pes.sulekha.com/blog/post/2007/08/kerala-legacy-to-world-mathematics/comment/785406.htm

http://kartha-pes.sulekha.com/blog/post/2007/08/kerala-legacy-to-world-mathematics/comment/781666.htm
 
There is a link to an excellent PPT presentation provided in the above. A direct link to that is given below.

With regards,

Kartha

 


Dr. P. E. Sarangadhara Kartha

dear karigarji,
thank you so much for your great interest and bringing up a lot more info than in the blog itself.
thank you.
kartha


karigar / / 7 yrs ago
karigar

 
the story of the math genius srinivasa ramanujam never ceases to fascinate ..excerpts from interview of an author who wrote fiction based on his life...
 
http://ia.rediff.com/news/2007/sep/03inter.htm
the rediff interview/
author david leavitt
god, math, and ramanujan's fascinating story
september 03, 2007
...........
 
 

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.


of course, ramanujan was from a very different world, the world of tamil nadu, a very high brahmin but [from a] poor family.

i must also say i have been fascinated by indian writers; among them, r k narayan.  narayan's world, in a way, is very similar to that in which ramanjun grew up. when i visited kumbakonam, the place ramanujan grew up, in many ways it reminded me of malgudi [narayan's fictional town].  

what else interested you about the relationship between the two men?

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. 


how did hardy take it?

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.


you have also said that you have been fascinated for a long time with artistic imagination.
true. i have always been interested in the artistic imagination and the creative process but i was getting a little bit tired of writing about painters or writers or musicians, and the idea of mathematician as a kind of artist was something that fascinated me. the more i learned about hardy and ramanujan the more they seemed very much to be artists in terms of their approach to their lives and approach to the world around them. and i just began to envision a novel, a sort of novel with a fairly big scope.
 
 


karigar / / 7 yrs ago
karigar

 
thanks for the links , dr k..
 
being too lazy to write a blog on this myself, i'm treating your son's blog as a repository for the "new" info i've gathered...
 
hope the knowledge spreads....
 
 


Dr. P. E. Sarangadhara Kartha

dear riverine,
thanks. i have provided a direct link to karigarji's comments in the main body.
thank you once more for the suggestion.
kartha


riverine / / 7 yrs ago
riverine

 
i saw the valuable add-ons in the comment section by karigar. thanks a lot.
 
dr. kartha, may i request you to please add these to the main section of the blog as a ps?


karigar / / 7 yrs ago
karigar

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....

mathematical proofs made poetic

the

ganita 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

comments

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.

posted by:
aditya | oct 26, 2006 11:46:22 pm
 


Dr. P. E. Sarangadhara Kartha

dear ksn,
sorry for replying this late. i am sorry.
thank you for your onam greetings and your appreciation of my son's article and the blessings you have showered on him.
thank you again and again.
 
dear karigarji,
sorry for being late.
yes, the inventory of our contributions to all branches of science and specifically to maths is very long. thinking  in big terms vis-a-vis space and time had been part of the vedic philosophy and hence come to us naturally. that is what induced carl sagan make a  remark that meant approximately, " when the west were not able to think beyond a few thousand years (look at the genesis stories), indians had established a complete spectrum of number system that spread from zeo to infinity. apart from conceiving, they could also use these numbers in real life." 
look at our number names. we have names (existing since many thousands of years) for a very very small fraction to a number as gigantic as  1053 .
all these are fine. but how many of us are aware of our great contributions and proud ? thinking in that line shows us how pathetic our edn. system really is. i am getting a strong feeling that starting from the primary level, if our education system is allowed to grow and get nourished from our own roots, our intellectual centre of gravity may go up much much higher than now.
kartha 
                          


karigar / / 7 yrs ago
karigar

 
interesting piece quoted below (don't know the credentials of authors, but seems well written, especially extracts from lilavati explained...)
 
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later on, many mathematicians like aryabhatta, bhaskaracharya, shridhar, etc. were seen in the country. of them bhaskara-charya wrote siddhanth shiromani in 1150. this great book has four parts: (1) leelavati, (2) algebra, (3) goladhyaya, and (4) graha ganit.

in his book bhaskaracharya, shri gunakar muley writes that bhaskaracharya has acknowledged the basic eight works of mathematics:

1. addition
2. subtraction
3. multiplication
4. division
5. square
6. square root
7. cube
8. cube root.

all these mathematical calculations were prevalent in india for thousands of years. however, bhaskaracharya tells leelavati a strange thing, “at the root of all these calculations there are only two basic calculations—rise and fall or increase and decrease. addition is increasing and subtraction is decreasing. the entire mathematics permeates from these two basic acts.”

these days, the computer solves the biggest and the most difficult calculations in a short time. all calculations are made with only two signs of addition and subtraction (+ and -). these are turned into electric signals i.e., the positive flow for addition and the reverse flow for subtraction. with this, calculations can be made at the speed of lightening. we understand increase, decrease, one and zero today but bhaskaracharya had the basic knowledge at that time.

these days, mathematics is considered a dry subject. but bhaskaracharya’s leelavati is an example of how it can be taught with fun by intermixing it with entertainment, curiosity, etc. let us see an example from leelavati:

from a bunch of pure lotuses’ 1/3’1/5’ and 1/6’ parts were used for the worship of shiv, vishnu and durga respectively; 1/4 was used to worship parvati and the 6 that were left, were used for the worship of the guru’s feet. now, leelavati, quickly tell me how many lotus flowers were there in the bunch?” the answer is 120 flowers.

explaining the square and the cube, bhaskaracharya says, “leelavati, a square shape and its area are called the square. the multiplication of two equal numbers is also called a square. similarly, the multiplication of three equal numbers is called cube and a solid with 12 compartments and equal arms is also called a cube.”

mool’ or root in sanskrit, meant the root of a tree or plant or, in a more expansive form, it means cause of something or origin also. hence, in ancient mathematics, square root meant the reason for the square or origin that is one arm of a square. likewise, we can understand the meaning of cube root in the same way. a number of ways were prevalent to find out the cube root and the square root.

in the same way, bhaskaracharya mentions the trairashik. it had trinomial sums hence its name. for example, if one gets ‘pr’ (as in pramaan) in ‘ph’ (as in phal), then what will we find in ‘i’ (as in ichchha, that is desire)?

in trairaashik[trinomial] questions, the phal number should be multiplied by the ichchha number and the result should be divided by the pramaan number. what is thus acquired is the itccha phal (desired fruit). about 2000 years ago, the trairaashik [trinomial] rule was discovered in india. it reached the arab countries in the 8th century ad. the arab mathematicians called the trairaashik ‘free raashikaat al hind’. later, it spread to europe where it was given the title of the golden rule.

the ancient mathematicians had knowledge not only of the trairaashik but also of the pancharaashik, saptaraashik and the navaraashik. (penta-or quinqnomial, hepta or septunomial and nonomial)

algebra
india is the birthplace of algebra. it was called indistinct or cryptic mathematics. the arab scholar moosa-al-khawarizmi came to india in the 9th century to learn this and wrote a book called alijeb oyal muquabila. thence, this knowledge went to europe.

in ancient times, mathematicians such as aapastamba, bodhaayan, katyaayan and later, brahmagupta and bhaskaracharya worked on algebra.

bhaskaracharya says that algebra means unexpressed mathematics but the initial reason is expressed. hence, in leelavati, arithmetic, which has expressed mathematics, was discussed at the start.

in algebra, bhaskaracharya talks about zero and infinity.

vadha au viyat khan khenadhaate, khaharo bhavet khen bhaktashch raashih

this means that if zero is divided by any number or multiplied by any number, the result is zero. if any number is divided by zero, the result is infinity.

zero and infinity are the two precious jewels of mathematics. life can continue without jewels but mathematics is nothing without zero and infinity.

zero and infinity have no place or name in the physical world and are only the creations of the human mind. yet, through the medium of mathematics and science, they clarify even the most difficult mysteries of the world.

brahmagupta discovered various equations. he gave them the names of ek varna, anek varna, madhyamaharan and mapit. there is one unknown number in an ek varna equation and many unknown numbers in anek varna.

(this book is available with ocean books (p) ltd, 4/19 asaf ali road, new delhi-110 002.)
 


KSN / / 7 yrs ago
KSN

dr kartha  ...congrats for your sons outstanding essay. whish you a happy onam too.
 
this is a outstanding blog on indian mathematics.  the work of jesuit monks in transmiting valuable knowledge to their home country is well know. we do have very well dated research on astronomy and mathematics.
the deriviation of pi and pythogorous theorem is now being attributed to india rather then the greek scholars.
what is more painful is the fact the it is we indians who fail to acknowledge our heritage and celebrate them. we do have a very horrible tendency to seek approval and knowledge from rest of the world. guess it is the years of servitude that has repressed our self-belief.


Dr. P. E. Sarangadhara Kartha

dear melody,
thanks a lot for recommending it as a blogger's pick.
kartha