Monday, June 27, 2011

Influence of Kerala School of Mathematics on European mathematics.

Hello people,

I recently found a book titled  'Kerala Mathematics - History and its possible transmission to Europe'. I am a staunch propagator of such topics. So I thought why not add to the  arsenal for use while debating with friends who rave about Uncle Sam and Euros. I started going through it and it basically is a congregation of Papers presented at a conference on the same topic. Its a bit too technical for a reader like me who chose to read that just for a brief idea. It uses a lot of mathematical terms and things. The main 'conjecture' on which the whole book is based is that 'Mathematics from medieval India, particularly from the southern state of kerala may have had an impact on European mathematics of 16th and 17th centuries'.

I got into the content and George Vargheese Joesph, who edited the book, makes a point about if indeed the above conjecture was true, who could be the possible conduits to Europe and if that is found , then it would be a major discorey as it would make Leibniz and Newton and Fermat and co share the credit with little known Indian mathematicians. He then makes a guess based on some links that perhaps Jesuits were the conduits. They had to impress the people in India and China with their intellectual capability rather than monetary means. Then its about some evidences and counter evidences and about mostly the high class - the Brahmins and Nairs being involved in mathematics. This the book says may be because of the system that makes the first son responsible for looking after the whole family and the younger ones being discharged of any duties are the ones who take up intellectual work.

Now the interesting part. A paper by George Joseph says that Jyeshtadeva a sixteenth century mathematician in his book writes about arc-tan series with

x = tanx - tan^3(x)/3 +tan^5(x)/5...

Joesph says this series is now know as Gregory series after Scottish mathematician James Gregory who did the same a 100 years later in 1671! In addition to this he writes that Kerala school based on great mathematician from 14th century, Madhava, was able to extent from Aryabhatta a 5th century scholar, the series for π, . But no the same series is known as Leibniz series! He also adds that another scholar Nilakantha[1400s] in his book 'Tantrasangraha' about the irrational nature of π, which is an achievement at that time. There is also mention about Madhava finding the value of π correct to 11 decimal places and then later a scholar named Shankara Varier making it upto 17 places. George Joseph says that the general perception about the Indian mathematics being practical and western mathematics being research oriented is wrong. Indian mathematicians founders of numerals treated them as numerals and not as special and the research was result oriented. He then talks about the origins of the Kerala School and Madhava.He also talks a bit about some of the astrologers who could predict things accurately. I felt this should have been left out from the book as it should have been kept purely mathematical.

Neutral View
Then there is a paper from M. Vijayalekshmi who just writes briefly the history of Kerala school with out referring to the Europeans or the mathematical aspects.

Euro Centric View
Another paper is from Madhukar Mallya who writes about the irrationality of π being pointed out by Nilakantha in 'Aryabhateeyabhashya'. He also elaborates an earlier comment by George Joesph that Indian mathematics was result oriented by saying that Kerala school knew about the Greek way of finding the area inside a semi circle by dividing it into equal arcs, but discarded it because of complex square root calculation involved in each step. They instead chose division into unequal parts and then reaching what is now called an integral.

There is an interesting paper by Arun Bala, where he puts forwards a point that the knowledge might not have been transfered through intellectuals and might have been through locals who use use the maths for navigation and other practical things. This must have been used by their European counterparts. This must also have contributed to the belief that Indian maths is not research based.

Then he writes about circumstantial evidence of transmission. He cites Fermat  challenging European mathematicians to solve x^2-Ay^2 = , where A is any non square integer. The point gets interesting when Fermat asks 'what is the smallest square when multiplied by 61 and added to unity gives another square. The taking of A = 61 Fermat is the snatcher here. Brahmagupta in AD 600s set the same indeterminate equation. This was solved by Bhaskara - II in 1100s and he does this by taking A=61 as an example. This cannot be coincidental as the solutions involved are 10 and 9 digit numbers. This indicates Fermat had access to 'Bijaganita' by Bhaskara II where he explains his general method. Another evidence is when Fermat developes basics of calculus 13 years before the birth of Newton. But 150 years ago Jyeshtadeva devlopes a method for finding the sine series which can also be found in 'Tantrasangraha' by Nilakantha, his predecessor. Fermat uses the same to calculate area under the parabola y= xK. But it is also a general formula to solve some problems in calculus. The same has been used by Pascal, Roberval etc. Bala adds that it is significant that relatively quickly after coming to contact with India, European mathematicians start using the methods known to Indians. Another circumstantial evidense is the use of Indian system of using day numbers in scientific specifications of dates.  This was done since Aryabhata in AD 500s. But this was introduced to Europe by Julius Scaliger in 1582 and is known as Julian day number system.  It differs from the Indian day number system- the Ahargana - in the fact that Indian version starts from the day the Kaliyuga started - 17th Feb 3102 CE while the European version starts from Biblical date creation - 1 Jan 4713 CE. He also credits Indians to discover, the number system, negative numbers, trigonometry, infinite series representation of irrational numbers, logaritms, series representations of circular functions etc. Then there are papers which talk about transmission to Arab world and European Mathematics taken alone.

This book was not a nice read, in the sense it was not riveting as any fiction or some other book you may find, but a pure mathematical one with some published papers stacked together. But the book could have been written better so that common readers could also understand and read it, without losing the maths. An example of such a book is a Book 'Fermat's Last Theorem' by Simon Singh. Its an excellent book which has continuity and keeps you gripped even though it has maths in it.

But its indeed good to know that research is being done in this area after a huge euro centric teaching of science in India. I would recommend this book as there is little else avilable to read in this area. A common reader may not find it interesting and may struggle to read it through. But even a brief skip may help in spreading the idea.


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