Ancient (and recent) Indian Mathematics

Kaushal
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Ancient (and recent) Indian Mathematics

Postby Kaushal » 18 Aug 2002 19:22

I will try to reconstruct this thread from what i have saved,Kaushal

The archived thread on Indic mathematics

http://www.bharat-rakshak.com/cgi-bin/ultimatebb.cgi?ubb=get_topic&f=13&t=000012&p=

posted by Gganesh

I came across this great link that provides an index of Ancient Indian Mathematicians. It basically made me want to read more. I did a google search and came up with a few more useful links. I would greatly appreciate any relevant input.

History Topics: Index of Ancient Indian mathematics
http://www-gap.dcs.st-and.ac.uk/~history/Indexes/Indians.html

History of Mathemacics: India
http://aleph0.clarku.edu/~djoyce/mathhist/india.html

Astronomy and Mathematics in Ancient India
http://www.cerc.utexas.edu/~jay/india_science.html

Indian Mathematicians
http://www.ilovemaths.com/ind_mathe.htm

HOW ADVANCED WERE WE?
http://www.lifepositive.com/mind/culture/indology/ancientindia.asp

Vigyan: A website of Indian Science and Technology
http://www.vigyan.org.in/mathlinks.html

History of Mathematics in India
]http://members.tripod.com/~INDIA_RESOURCE/mathematics.htm]

Equations and Symbols
http://www.gosai.com/chaitanya/saranagati/html/vishnu_mjs/math/math_5.html
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posted 14 August 2002 02:38 PM

http://www.ourkarnataka.com/vedicm/vedicms.htm

http://www1.ics.uci.edu/~rgupta/vedic.html

http://www.astrocommunity.com/VM/sutras.php

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Re: Ancient (and recent) Indian Mathematics

Postby Kaushal » 18 Aug 2002 19:30

Kaushal
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posted 15 August 2002 10:37 AM
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The use of concise mathematical symbolism (<,>,=,*,(), inf., sup.,the integral sign, the derivative sign, the sigma sign etc is a relatively recent revolution in mathematics, When calculus was invented by Newton and Leibniz independently, they used different notations. Newton used dots on top of the quantity and called them fluxions, while Leibniz used the 'd' notation. Unfortunately when all the major advancements in science and mathematics were happening in Europe in the 17th and 18th century, India was stagnant, caught in interminable wars and conquests.

Ironically it was the Indian place value notation that triggered the advance in Europe. Prior to this it was common to express all problems in wordy sentences and verse.

As to the value of PI (cannot be expressed as a fraction) it is impossible to calculate it without the use of some variant of the 'method of exhaustion'. In this particular instance it was a matter of using increasingly larger number of isosceles triangles( forming an n sided polygon) within the circle. This is the germ of the method of series expansions. The ancient Indians were well aware of this technique and used it for several trigonometric calculations. The felicity with which Indians did series expansions extended to Ramanujam. He was the incomparable master and there may be none like him on the face of this earth again.

Again there is no claim that the Indians did everything. For example there is no evidence that the Indians were familiar with the representation of complex numbers and complex variables or advanced topics such as the calculus of variations.

The ancient vedic indians were interested in very practical aspects of mathematics, namely the positions of the stars, developing a panchanga (calendar), ordinary mathematics for everyday use, measurements, such as that of land and weights etc.. There is no evidence that they were familiar with the science of mechanics for instance, which developed in Europe in the 18th century after Newton.

Kaushal

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Re: Ancient (and recent) Indian Mathematics

Postby Kaushal » 18 Aug 2002 19:32

Kaushal
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posted 15 August 2002 11:32 AM
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http://www.udupipages.com/book/hindhu.html

a History of PI (does not give details)

http://www.math.rutgers.edu/~cherlin/History/Papers2000/wilson.html
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vsunder
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posted 16 August 2002 09:07 PM
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From Thangadurai's web page:

http://www.geocities.com/thangadurai_kr/PILLAI.html

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Re: Ancient (and recent) Indian Mathematics

Postby Kaushal » 18 Aug 2002 19:33

http://www.geocities.com/ifihhome/articles/sk003.html

The Beautiful Tree


Subhash Kak


Sulekha Columns, May 22, 2001


As a young boy raised in small towns of Jammu and Kashmir, I often came across people who could not read or write. The school books said that literacy in all of India was low, perhaps 30 percent or so, and this was despite the introduction of the British education system more than 100 years earlier. The books implied that before the arrival of the British the country was practically illiterate. This thought was very depressing. Perhaps I shouldn't have believed the story of India's near total illiteracy in the 18th century so readily. India was rich 250 years ago when the British started knocking at the door for a share of its trade. Paul Kennedy, in his highly regarded book, The Rise and Fall of the Great Powers : Economic Change and Military Conflict from 1500 to 2000 estimates that in 1750 India's share of the world trade was nearly 25 percent.



To understand this figure of 25 percent, consider that this is USA's present share of the world trade, while India's share is now only about half a percent. India was obviously a very prosperous country then, and this wealth must have been mirrored in the state of society, including the literacy of the general population.



Unfortunately, education in medieval India is not a subject that has been well researched. But thanks to the pioneering book, The Beautiful Tree by Dharampal, we now have an idea of it before the coming of the British. The book uses British documents from the early 1800s to make the case that education was fairly universal at that time. Each village had a school attached to its temple and mosque and the children of all communities attended these schools.



W. Adam, writing in 1835, estimated that there were 100,000 schools in Bengal, one school for about 500 boys. He also described the local medical system that included inoculation against small-pox. Sir Thomas Munro (1826), writing about schools in Madras, found similar statistics. The education system in the Punjab during the Ranjit Singh kingdom was equally extensive.



These figures suggest that the literary rate could have approached 50 percent at that time. From that figure to the low teens by the time the British consolidated their power in India must have been a period of continuing disaster.



Amongst Dharampal's documents is a note from a Minute of Dissent by Sir Nair showing how the British education policy led to the illiteratization of India: "Efforts were made by the Government to confine higher education and secondary education, leading to higher education, to boys in affluent circumstances... Rules were made calculated to restrict the diffusion of education generally and among the poorer boys in particular... Fees were raised to a degree, which, considering the circumstances of the classes that resort to schools, were abnormal. When it was objected that minimum fee would be a great hardship to poor students, the answer was such students had no business to receive that kind of education... Primary education for the masses, and higher education for the higher classes are

discouraged for political reasons."



According to Dr Leitner, an English college principal at Lahore, "By the actions of the British the true education of the Punjab was crippled, checked and is nearly destroyed; opportunities for its healthy revival and development were either neglected or perverted."



Dharampal's sources appear unimpeachable and the only conclusion is that 250 years ago the Indian basic education system was functional. Indeed, it may have been more universal than what existed in Europe at that time.



One might, with hindsight, complain that the curriculum in the pathshalas was not satisfactory. Dharampal's book lists the texts used and they appear to have provided excellent training in mathematics, literature, and philosophy. Perhaps the curriculum could have had more of sciences and history. I think the school curriculum was not all that bad in itself. Judging by the standards of its times, it did a good job of providing basic education.



What was missing was a system of colleges to provide post-school education. After the destruction of ancient universities like Taxila and Nalanda, nothing emerged to fill that role. Without institutions of higher learning, the Indian ruling classes did not possess the tools to deal with the challenges ushered in by rapid scientific and technological growth.



The phrase ‘the beautiful tree’ was used by Mahatma Gandhi in a speech in England to describe traditional Indian education. Gandhi claimed that this tree had been destroyed by the British. Dharampal's book provides the data in support of Gandhi's charge.



The Macaulayite education system, put in place by the British, almost succeeded in erasing the collective Indian memory of vital, progressive scientific, industrial and social processes. But not all records of the earlier history were lost. Dharampal has authored another important book, Indian Science and Technology in Eighteenth Century: Some Contemporary European Accounts which describes the vitality of Indian technology 250 years ago in several areas.



It is not just colonialist ideas that are responsible for the loss of cultural history. The need to pick and choose in today's information age is also leading to an erosion of cultural memory. The scholar and mathematician C. Muses from Canada did his bit to counter it by writing about Ramchundra (born 1821 in Panipat), a brilliant Indian mathematician, whose book on Maxima and Minima was promoted by the prominent mathematician Augustus de Morgan in London in 1859. Muses's work appeared in the respected journal The Mathematical Intelligencer in 1998. Ramchundra had been completely forgotten until Muses chanced across a rare copy of his book.



Muses called me over a year ago, just before he died, to tell me how he got interested in India. He said that he wanted to make sense of why Indians had not developed science, as colonialist and Marxist historians have long alleged. But the deeper he got into the original source materials, he found an outstanding scientific tradition that had been misrepresented by historians who were either biased or plain incompetent.



Although Muses did not so speculate, one might ask if de Morgan's own fundamental work on symbolic logic owed in part to the Indian school of Navya Nyaya. De Morgan, in his introduction to Ramchundra's work, indicates that he knew of the Indian tradition of logic, "There exists in India, under circumstances which prove a very high antiquity, a philosophical language (Sanskrit) which is one of the wonders of the world, and which is a near collateral of the Greek, if not its parent form. From those who wrote in this language we derive our system of arithmetic, and the algebra which is the most powerful instrument of modern analysis. In this language we find a system of logic and metaphysics."



Finally, there is the loss of memory taking place due to the carelessness with which we are preserving our heritage. This is a process of permanent loss, although on a few lucky occasions long-forgotten documents are found. One example of this latter event is the recovery of the lost notebooks of Srinivasa Ramanujan (1887-1920), who may have been the greatest mathematical genius of all time. Ramanujan had been called a second Newton in his own lifetime, yet the full magnitude of his achievements was appreciated only when his [lost] notebooks, full of unpublished results, were discovered in the eighties.



You can read a fine biography of Ramanjuan by Robert Kanigel titled The Man Who Knew Infinity. I also recommend Ramanujan: Letters and Commentary, edited by Bruce Berndt and Robert Rankin.

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Re: Ancient (and recent) Indian Mathematics

Postby Kaushal » 18 Aug 2002 20:30

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Re: Ancient (and recent) Indian Mathematics

Postby Rahul Mehta » 18 Aug 2002 21:19

I had posted an article in that previous thread. It was something as follows:

There are claims that Madhava (an Mathmatcian in in Medivial India) knew about trgonometric series such as arctan(x) = x - x*2/2 + ...

And using this series he obtained value of PI correct upto several places after decimal.

Can someone post the EXACT and ORIGINAL Sanskrit text written by Madhav or his contemperories? I have a 1000 page Sanskrit dictionary and would like to do some translation myself so that I can verify.

Thanks

-Rahul Mehta

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Re: Ancient (and recent) Indian Mathematics

Postby Kaushal » 18 Aug 2002 21:45

More on Ramchundra

http://www.mfo.de/Meetings/Documents/1998/43/Report_41_98.doc

http://www.dcs.warwick.ac.uk/bshm/abstracts/R.html

Raina, Dhruv, ‘Mathematical foundations of a cultural project or Ramchandra’s treatise "Through the unsentimentalised light of mathematics"’, Historia mathematica 19 (1992), 371-384
The 19th century witnessed a number of projects of cultural rapprochement between the knowledge traditions of East and West. In his Treatise on the problems of maxima and minima, the Indian polymath Ramchundra tried to render elementary calculus amenable to an Indian audience in the indigenous mathematical idiom. The "vocation of failure" of the book is discussed within the context of encounter and the pedagogy of mathematics.

http://xerxesbooks.com/cats/alge534.mv

34. Ramchundra. TREATISE ON PROBLEMS OF MAXIMA AND MINIMA SOLVED BY ALGEBRA London 1859 Wm. H. Allen. 8vo., 185pp., 8 plate, original cloth. Owner signed and owner bookplate (Edward Ryley) . Good, cloth faded, cover and spine ends worn, one tiny chip in spine cloth. $235.00

http://mcs.open.ac.uk/puremaths/pmd_research/pmd_resnews8.htm

My own lecture, entitled ‘Multiculturalism in history: voices in 19th century mathematics education east and west’, was on the history of interaction between European and Indian mathematics education in the 19th century, discussing the contributions in particular of Henry Colebrooke, Yesudas Ramchundra, and Mary Everest Boole.

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Re: Ancient (and recent) Indian Mathematics

Postby kautilya » 18 Aug 2002 22:13

Originally posted by Rahul Mehta:
I had posted an article in that previous thread. It was something as follows:

There are claims that Madhava (an Mathmatcian in in Medivial India) knew about trgonometric series such as arctan(x) = x - x*2/2 + ...

And using this series he obtained value of PI correct upto several places after decimal.

Can someone post the EXACT and ORIGINAL Sanskrit text written by Madhav or his contemperories? I have a 1000 page Sanskrit dictionary and would like to do some translation myself so that I can verify.

Thanks

-Rahul Mehta
I have a couple of different dictionaries of Sanskrit, and I am a begining student in sanskrit, besided learning it in my school. Despite, that I cannot even begin to claim to be able to translate the esoteric texts.

If you think you can do it just using a dictionary, there can only be following explainations
a) you are a super genius
b) you don't understand how sanskrit works, and why it is untranslatable through a simple dictionary, especially the esoteric texts, becaue dictionary can only give you a translation of root words. Also, the mathematical texts especially use a notation unfamiliar to a person who is not a scholar in the field.

So, when you made a claim in one of your previous posts that you had translated a lot of sanskrit text 10 years ago, just based on that dictionary, I rejected it. But, who knows you could be a super genius(though hard for me to believe from the quality of your posts and ideas), and in that case I bow to you, o the learned one.

Otherwise, please give us a break.

P.S. this is a personal question, so, you may ignore it if you want. how old were you 10 years ago?

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Re: Ancient (and recent) Indian Mathematics

Postby Calvin » 19 Aug 2002 04:45

Is there any kind of museum dedicated to Madhava in Cochin?


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Re: Ancient (and recent) Indian Mathematics

Postby A_Gupta » 19 Aug 2002 09:24

In "The Golden Age of Indian Mathematics", by S. Parameswaran, published by the Swadeshi Science Movement, Kerala, the following verse is given by Madhava :

Vibudha-netra-gaja-ahi-hutasana
tri-guna-veda-bha-vaarana-baahavah
nava-nikharva-mite vrtivistare
paradhi-manam idam jagadur budhah
This is said to be in the Bhuta Samkhya system (the other system which remained popular is the Katapayadi system). The Katapayadi system is based on giving the consonants a digit value. The Bhuta Samkhya system uses well-known objects or concepts to denote numbers.

0 is denoted by sunya (void), kha (sky), antariksa (atmosphere), purna (whole), randhra (hole) etc.

1 is denoted by sasi (moon), bhumi(earth), go(cow) etc.

2 is denoted by netra (eyes), bahu (hands), karna (ears), paksa(moon's waxing and waning periods) etc. - each of which has a pair of members.

3. is denoted by kala (time - past, present, future), loka (heaven, earth and hell) etc. - each of which has a trio of components

4 is denoted by veda, yuga, dik or dis(directions) etc.....

5. is denoted by bhuta (elements), pandavas, etc. .....

32 is denoted by danta...

The principle of place value was used but the mode of writing was from right to left; thus netra-kala-yuga and bahu-loka-veda stand for 432 (not 234)....

The same word sometimes stood for two and occasionally for more than two different numbers, e.g., go has been used for 1 as well as 9, paksa for 2 as well as 15; loka for 3 as well as 14; dik for 8 was well as 10...These ambiguities must have created some confusion and also gone against the universal acceptance and prolonged survival of the Bhuta Samkhya system.....
Thus the verse given above is interpreted that the circle of diameter nava-nikharva or 9 * 10**11 is 2,827,433,388,233 units.

(vibudha = 33, netra = 2, gaja = 8, ahi = 8, hutasana = 3, tri = 3, guna = 3, veda = 4, bha = 27, vaarana = 8, baahavah = 2 )

George Gheverghese Joseph, in The Crest of the Peacock gives the following translation

Gods (33), eyes(2), elephants (8), snakes (8), fires (3), three (3), qualities (3), vedas (4), naksatras (27), elephants (8), and arms (2) - the wise say that this is the measure of the circumference when the diameter of a circle is 900,000,000,000.

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Re: Ancient (and recent) Indian Mathematics

Postby Pennathur » 19 Aug 2002 21:03

All said the present trend among Indians and people of Indian Origin (PIO) seems to be to avoid mathematics. There are some incredibly good mathematicians in India in TIFR; JNU; MATSCIENCE; ISI and the IITs - but beyond that there is nothing much of substance happening. My uncle a maths professor in Madras got his daughter to switch to study something more applied - Chemisty - and she is now studying for a Ph.D. in the US - a loss for Mathematics. Another cousin - a genius at IIT-M came to the US on fellowship to study maths. He went back with a Ph.D. and after a great struggle landed a job with TCS. Browse the list of Ph.D. candidates in US university websites - the greats are all from the past. Little that is recent. A cause for great concern.

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Re: Ancient (and recent) Indian Mathematics

Postby Kaushal » 19 Aug 2002 21:25

I have to bring to the attention of the forum that the name Pennathur is associated with a prolific spawning ground for Indian mathematicians, namely the Pennathur ShivaswamiIyer (sp.) High School in Madras. At least 2 of the members of my family have passed thru this school to go on and win the Stewart prize (Gold medal in Mathematics) at the University of madras.

I agree with Pennathur's comments that the creme de la creme are now going on to applied fields. TIFR was a great spawning ground for Pure mathematics, I do not know where it stands today.

Maybe the answer is to institute a all India Mathematics Olympiad along the lines of those in european countries such as Hungary which has been the originator of a large number of Mathematicians.

It is perhaps good for the long run that only those with a genuine passion for the subject will discipline themselves to the rigors of a Mathematical profession.

Kaushal

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Re: Ancient (and recent) Indian Mathematics

Postby Pennathur » 19 Aug 2002 23:11

Kaushal the school you refer to is Pennathur Subramaniam Iyer High School - founded by my great grand uncle (also illustrates the fact how descendants rarely match the brilliance of the illustrious ones after one generation :D ) PS as the school is known has produced a long string of luminaries and some of the first dalit high school graduates in the 1940s. Out here in the US the telent spotting system is widespread and well managed. Math Olympiads happen all the time - right from grad 3 or so. There are many students who veer away from maths when it is time to go to university. even at the IITs it is rare to find students taking up maths after the JEE selection. Nevertheless Indian science and engg. students who come to study in the US are very good at maths despite the lack of formal undergrad qualifications (apart from the ancilliary courses that everyone goes thru). TIFR and MATSCIENCE do a lot of work in Maths unfortunately it is too late by then. ISI does have an excellent undergrad program - the B.Stat. where students get to choose between maths and stats. The course is very very tough and admits a few students (50-60 I believe). Notice that I use the terms maths and stats interchangeably as in the realms of theoretical work - the lines are often blurred - however as a rule stats work tends to be a lot more focused.

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Re: Ancient (and recent) Indian Mathematics

Postby Nate » 19 Aug 2002 23:25

Mostly the veering away (from Math studies at Univ. level)happens due to peer pressure and lack of job opportunities. I know of a person who was doing a Ph.D in number theory at a US university(described by many as the queen of mathematical sciences) who took up C programming to get a job here in the US. Sad is the state of Math studies even in the US. Hope things will change if the armed forces find some good use for Math talent such as in cryptography. I was told that MacDonalds employed a large number of statisticians to predict demographic changes and use it to their advantage.

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Re: Ancient (and recent) Indian Mathematics

Postby Kaushal » 19 Aug 2002 23:32

PS high School has been the alma mater for many members of the previous generations of my family (including my mother). One of her classmates(the son of her math teacher), she tells me, attained a very high position at BARC, later on.She also tells me wryly that she always outscored him in math.

It merely highlights what you are saying that neither of them went on for a career in Mathematics.

Kaushal

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Re: Ancient (and recent) Indian Mathematics

Postby Sunder » 19 Aug 2002 23:33

Thanks for the consolidation Kaushal.. Just a suggestion, that when you copy and paste material from an earlier thread, you lose the A HREF links posted by members.. for example, I had provided 3 links and none of them appear when you cut and paste them :) Like below..

Sunder
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posted 14 August 2002 02:38 PM
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Vedic Mathematics: Sulabha Sutra.
Nuggets from Vedic Mathematics

Another Excellent link..

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Re: Ancient (and recent) Indian Mathematics

Postby Kaushal » 19 Aug 2002 23:44

Sunder, thanks for pointing that out. I corrected that.

Kaushal

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Re: Ancient (and recent) Indian Mathematics

Postby Ponniyin Selvan » 20 Aug 2002 11:37

Originally posted by pennathur:
Kaushal the school you refer to is Pennathur Subramaniam Iyer High School - founded by my great grand uncle (also illustrates the fact how descendants rarely match the brilliance of the illustrious ones after one generation :D )
Oh come now, don't sell yourself short :-)

Anyway, there are actually two schools I believe. PS High School - the old school which most people know as it is right next to Mylapore tank. It is said that you can't throw a stone in a music academy kutcheri without hitting a P.S. alum :-) The old school is not so "elite" anymore .. it is
affiliated to the Matriculation system. The new school is "P.S. Senior" and is affiliated to the CBSE system .. a younger school without the reputation of the PSBB/Vidya Mandir/DAV but with a pretty reasonable performance. I have lots of friends from PS Senior .. an interesting innovation in the new school is the "Insurance" program in plus-2 .. the brain child of the late correspondent/founder. I think it was a misguided program (career limiting) but nevertheless an interesting idea.

Oh btw, before I forget, PS Senior is something like 99% TamBrahm :-) Make of it, what you will. One of my friends who went there from a govt. school in another state found it an extremely unsettling atmosphere although he was a TamBrahm as well. A bit too homogeneous ..

Pennathur, most people in the IITs study engineering. The IITs are really equivalent to a "School of Engineering" in the US universities. Although they offer other ocurses, they are primarily engineering schools. The same applies to the RECs, Roorkee, BIT etc. IISc is one exception to this, although it is only devoted to graduate study.

The IITs are great training grounds, but it's only in the last 5-10 years that really good quality research is starting to come out on a fairly regular basis (cf. PRIMES in P, IIT-B success in VLDB-02 conference acceptances etc).

I won't call this a problem, but a big part of the issue is the pressure to study engineering, esp. in the south. Most people do this as a matter of course .. the herd mentality aka "kootathode govinda".

Even among the IIT grads, it's not all that many who want to go into academia .. however you must remember that in the sciences, especially in something like Math, academia is pretty much the only viable career option. So it requires a different kind of committment .. in fact, much more maturity than you'd expect in the average undergrad.

Personally, I have looked at the output of TIFR, Matscience and Chennai Math Inst (used to be SSF). I think they are doing wonderful work in theoretical comp. sci. (can't speak about Math, because .. er I'm weak in Math :-)

In any case, there is nothing wrong per se in doing good applied work. Besides, there are plenty of Indian theoreticians in the US as well .. except that they have a lower profile in general than the money making KB Chandrasekharans of this world.

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Re: Ancient (and recent) Indian Mathematics

Postby Harry Van » 20 Aug 2002 13:48

Hey you guys are right about the P.S.School.There was a guy who joined us from P.S. school during +2 and he was very good at maths.He was a wizard.He always stood first in the class and we would go to him with our doubts.

How come members from a particualr school ahve particular strengths.Does that school admit only those students who are mathematically brilliant ?

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Re: Ancient (and recent) Indian Mathematics

Postby Rahul Mehta » 22 Aug 2002 20:26

Originally posted by kautilaya:
I have a couple of different dictionaries of Sanskrit, and I am a begining student in sanskrit, besided learning it in my school. Despite, that I cannot even begin to claim to be able to translate the esoteric texts.

If you think you can do it just using a dictionary, there can only be following explainations
a) you are a super genius...
I used to be extremely interested in Sanskrit till Xth. When I was in Xth, I used to read the Sanskrit books of FYBA/SYBA and had purchased several books to get grip on Vocab as well as Grammar (perticularly Sandhi/Samaas/rup etc). Even in whole of my after-Xth vacation, I did a lot of Sanskrit readings even though Sanskrit was not be a subject in XIth. So I had grip over Sandhi, Samaas, other aspects of Sanskrit Grammar (all 50-60 kind of rups, like akaaraant pulling, aakaaraant pulling, ikaaraant pulling etc and corresponding strillling, and various dhaatu with several tenses, all having their ekvachan, dvichan and baahuvachan and so forth).

With that much knowledge of sandhi/samaasa/rup/dhaatu, tranlation was more than easy. I could not see any Maths etc. Then I showed it to a Sanskrit professor, who was a PhD, only if there were OTHER possible translations than the simple ones I obtained. None of us could see any Maths/Physics there.

Now Sandhi Vichcched, Samaas Vichcched and Translation were the ONLY three tools we used. Now if there was something over and over these, I would not know.

Now can we come to the point?

Can anyone provide me the EXACT Sanskrit shlokas which gives us value of trigonometricl series, value of PI etc? If I cant translate it, I will take it to some PhDs in Sanskrit as well. Please no translations, I want the RAW text i.e. the ORIGINAL shlokas as Madhava or their conteperory wrote them. My knowledge of Sanskrit is an irrelevant issue.

Or is that we have to accept the claims WITHOUT asking for any text as evidences?

-Rahul Mehta

Later addition:

Originally posted by Arun_Gupta:
In "The Golden Age of Indian Mathematics", by S. Parameswaran, published by the Swadeshi Science Movement, Kerala, the following verse is given by Madhava :
Vibudha-netra-gaja-ahi-hutasana
tri-guna-veda-bha-vaarana-baahavah
nava-nikharva-mite vrtivistare
paradhi-manam idam jagadur budhah
....
Btw, which book is this? Just curious if I can get ENTIRE Sanskrit version of this book? Has any publisher publsihed it?

-Rahul Mehta

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Re: Ancient (and recent) Indian Mathematics

Postby Kaushal » 22 Aug 2002 20:56

Rahul, you have to go to a reputed University library (maybe IIM, Ahmedabad has one i dont know) and look for magazines (such as Ganita )and books. I have been told IIT, Bombay has an outstanding library

The original Sanskrit texts have been published i believe by Munshiram Manoharlal & Motilal Banarsidas. Here are a list of publishers

http://dir.indiamart.com/indianexporters/book_art.html

It is very difficult to find original sanskrit texts in the US and we have to order from Delhi, so i would be grateful if you could post any sources which you have found useful in your search. I have been looking for a book on the Apastamba Sulava Sutra with original verses.

I will be happy to help you search , in case you have a slow internet connection.

Kaushal

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Re: Ancient (and recent) Indian Mathematics

Postby Kaushal » 22 Aug 2002 21:23

A good place to start are some of these articles;

http://www-gap.dcs.st-and.ac.uk/~history/References/Bhaskara_I.html

References for Bhaskara I

--------------------------------------------------------------------------------

Biography in Dictionary of Scientific Biography (New York 1970-1990).
Books:

K Shankar Shukla, Bhaskara I, Bhaskara I and his works II. Maha-Bhaskariya (Sanskrit) (Lucknow, 1960).
K Shankar Shukla, Bhaskara I, Bhaskara I and his works III. Laghu-Bhaskariya (Sanskrit) (Lucknow, 1963).
Articles:

R C Gupta, Bhaskara I's approximation to sine, Indian J. History Sci. 2 (1967), 121-136.
R C Gupta, On derivation of Bhaskara I's formula for the sine, Ganita Bharati 8 (1-4) (1986), 39-41.
T Hayashi, A note on Bhaskara I's rational approximation to sine, Historia Sci. No. 42 (1991), 45-48.
P K Majumdar, A rationale of Bhaskara I's method for solving ax ± c = by, Indian J. Hist. Sci. 13 (1) (1978), 11-17.
P K Majumdar, A rationale of Bhatta Govinda's method for solving the equation ax - c = by and a comparative study of the determination of "Mati" as given by Bhaskara I and Bhatta Govinda, Indian J. Hist. Sci. 18 (2) (1983), 200-205.
A Mukhopadhyay and M R Adhikari, A step towards incommensurability of and Bhaskara I : An episode of the sixth century AD, Indian J. Hist. Sci. 33 (2) (1998), 119-129.
A Mukhopadhyay and M R Adhikari, The concept of cyclic quadrilaterals: its origin and development in India (from the age of Sulba Sutras to Bhaskara I, Indian J. Hist. Sci. 32 (1) (1997), 53-68.
K S Shukla, Hindu mathematics in the seventh century as found in Bhaskara I's commentary on the Aryabhatiya, Ganita 22 (1) (1971), 115-130.
K S Shukla, Hindu mathematics in the seventh century as found in Bhaskara I's commentary on the Aryabhatiya II, Ganita 22 (2) (1971), 61-78.
K S Shukla, Hindu mathematics in the seventh century as found in Bhaskara I's commentary on the Aryabhatiya III, Ganita 23 (1) (1972), 57-79
K S Shukla, Hindu mathematics in the seventh century as found in Bhaskara I's commentary on the Aryabhatiya IV, Ganita 23 (2) (1972), 41-50.
I I Zaidullina, Bhaskara I and his work (Russian), Istor. Metodol. Estestv. Nauk No. 36 (1989), 45-49.

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Re: Ancient (and recent) Indian Mathematics

Postby Kaushal » 22 Aug 2002 21:26

http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/References/Indian_sulbasutras.html

References for The Indian Sulbasutras
Books

B Datta, The science of the Sulba (Calcutta, 1932).
G G Joseph, The crest of the peacock (London, 1991).
Articles

R C Gupta, New Indian values of from the Manava sulba sutra, Centaurus 31 (2) (1988), 114-125.
R C Gupta, Baudhayana's value of 2, Math. Education 6 (1972), B77-B79.
S C Kak, Three old Indian values of , Indian J. Hist. Sci. 32 (4) (1997), 307-314.
R P Kulkarni, The value of known to Sulbasutrakaras, Indian J. Hist. Sci. 13 (1) (1978), 32-41.
G Kumari, Some significant results of algebra of pre-Aryabhata era, Math. Ed. (Siwan) 14 (1) (1980), B5-B13.
A Mukhopadhyay and M R Adhikari, The concept of cyclic quadrilaterals : its origin and development in India (from the age of Sulba Sutras to Bhaskara I, Indian J. Hist. Sci. 32 (1) (1997), 53-68.
A E Raik and V N Ilin, A reconstruction of the solution of certain problems from the Apastamba Sulbasutra of Apastamba (Russian), in A P Juskevic, S S Demidov, F A Medvedev and E I Slavutin, Studies in the history of mathematics 19 "Nauka" (Moscow, 1974), 220-222; 302.

--------------------------------------------------------------------------------

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Re: Ancient (and recent) Indian Mathematics

Postby Kaushal » 22 Aug 2002 21:30

Dharmapal's Bha_rata: eCourse




DHARAMPALs INDIA

AN INTRO-ORIENTATION COURSE ON THE DHARAMPALJIs WORK

Background

For the un-initiated, Dharampal is a Gandhian - Historian who has
researched and brought to the notice of the public at large many new
insights and levels of thinking that were neither professed nor
popularised by the mainstream historians. His books on Eighteenth
Century Indian Science and Technology, Pre-British Indian Education
System, Panchayat System in the pre-Colonial era and others all have
brought about a new method of looking at the past of this country
and at the archival material available with the various archives,
museums and libraries on India.

It has been his quest to understand the meaning and life of ordinary
people of this country, their methods of organising their lives and
the tools and means they adopt in doing so, their customs and
culture and their aspirations, he has travelled a vast research
journey mostly alone and documenting scraps of material which
incidentally has opened a huge body of knowledge for the rest of the
world. He maintains, that it is the need to convince himself of the
meaning behind the events in history and ascertain and validate the
facts that has driven him in his work. That some of them have been
published and the publications has inspired many individuals and
launched institutions is a consequence he probably did not predict
or prepare for.

eCourse

The eCourse - DHARAMPAL'S INDIA would highlight his views, work and
interpretation of historical events. Though by no means this would
claim to be comprehensive, we have designed this course to be an
introduction and orientation to his larger works.

SAMANVAYA has had the privilege of working with Dharampalji for over
2 years now.

The course will be limited to few people who are inclined towards
further learning / work based on this understanding. The course
would consist of short capsule of materials selected by people who
have worked with Dharampalji; these would be mailed across followed
by a discussion on the same.

If you are interested to register, please mail samanvaya@vsnl.com
with a note on yourself and why you are interested. The course will
be for a duration of 3 weeks on-line starting late September.

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Re: Ancient (and recent) Indian Mathematics

Postby Pennathur » 22 Aug 2002 22:23

Ponniyin Selvan,

I am aware of the old and the new PS schools. The new PS school like many other hard-schools in other parts of India goes for basics and virtuosity in the hard sciences. Competition being what it is you must be able to get into the highest percentile. However other schools are not lagging behind. It may be distasteful to get into issues of FC/OC/BC etc., but illuminating all the same. In TN the cut-off marks in engg. and medical entrance tests for open competition (which is where "unreserved" students go) and backward class students is now about 1% (97-96). There is disproportionate clustering at the top end these days. It is also an indication of the raw talent that is available through India's underfunded and privately run primary schools.

Recently MM Joshi has pushed a proposal that has converted all RECs into National Institutes of Technology. Henceforth the NITs will receive more funding for undergrad programs while IITs, IISc will focus on PG and research. There are also plans to move more in line with the admission process in the US although the numbers involved in India's case are humongous.

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Re: Ancient (and recent) Indian Mathematics

Postby Narayan_L » 22 Aug 2002 23:03

Pennathur, Ponniyin Selvan:

Thanks for the interesting bit of trivia about the PS Institutions. Didn't the PS family hold a majority stake in Enfield India (or Royal Enfield as it used to be called). Apart from cranking out those fabulous OHV 350ccs, looks like they also did a lot for education.

No offense to any one from their neighbor and fierce rival, Vidya Mandir. I had friends from both schools, but the latter is best remembered for its cricket players than mathematicians (I think). L. Sivaramakrishnan, Kris Srikant, V. Sivaramakrishnan and one of my favorite TN players from yester-years, T.E. Srinivasan.

All this talk about Mylapore, Luz, PS, Vidya Mandir, etc makes me quite nostalgic. Remember, Shanti Vihar, the fine eatery not too far from both these schools? Well, those were the days. Since this is a math thread, let me take my reminiscing elsewhere.

Selvan:

Thalaivare! , you rock! Just read your comments on the "Sue the Brits" thread. Accurate and hilarious. I follow your posts with great interest. Vazhga Vazhamudan, Ungal Pugazh Paravattum. (May you live long and prosper. May your fame spread wide.)

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Re: Ancient (and recent) Indian Mathematics

Postby karana » 23 Aug 2002 23:46

to complete the trilogy of PS,vidya Mandir we should not forget the invaluable contribution of rosary matric ;)

but seriously PS i guess had excellent standards and infact the AIR-1 in the year i took JEE was from PS.

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Re: Ancient (and recent) Indian Mathematics

Postby Narayan_L » 23 Aug 2002 23:52

karana>>"we should not forget the invaluable contribution of rosary matric"

karana:

Thanks for reminding us! My heart-felt salutations to their "contributions", many many times. :D

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Re: Ancient (and recent) Indian Mathematics

Postby member_4595 » 27 Aug 2002 02:51

OK, so we all don't get totally lost in the murky past, and since this topic is also about recent Indian
Mathematics- see the link below:

http://www.iitk.ac.in/infocell/announce/algorithm/

Thankth...

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Re: Ancient (and recent) Indian Mathematics

Postby Kaushal » 27 Aug 2002 20:07

http://www.infinityfoundation.com/sourcebook.htm

ECIT Sourcebook on Indic Contributions in Math and Science

This sourcebook will consist primarily of reprinted articles on Indic contributions in math and science, as well as several new essays to contextualize these works. It will bring together the works of top scholars which are currently scattered thoughout disparate journals, and will thus make them far more accessible to the average reader.

There are two main reasons why this sourcebook is being assembled. First, it is our hope that by highlighting the work of ancient and medieval Indian scientists we might challenge the stereotype that Indian thought is "mystical" and "irrational". Secondly, by pointing out the numerous achievements of Indian scientists, we hope to show that India had a scientific "renaissance" that was at least as important as the European renaissance which followed it, and which, indeed, is deeply indebted to it.

Currently, the following table of contents is proposed for this volume:

1. Editors' Introduction (Subhash Kak)

Section 1: Mathematics

2. D. Gray, 2000. Indic Mathematics etc.

3. Joseph, George Ghevarughese. 1987. "Foundations of Eurocentrism in Mathematics". In Race & Class 28.3, pp. 13-28.

4. A. Seidenberg, 1978. The origin of Mathematics. Archive for History of Exact Sciences 18.4, pp. 301-42.

5. Frits Staal, 1965. Euclid and Panini. Philosophy East and West 15.2, pp. 99-116.

6. Subhash Kak, 2000. Indian binary numbers and the Katapayadi notation. ABORI, 81.

7. Subhash Kak, 1990. The sign for zero. Mankind Quarterly, 30, pp. 199-204.

8. C.-O. Selenius, 1975. Rationale of the chakravala process of Jayadeva and Bhaskara II. Historia Mathematica, 2, pp. 167-184.

9. K.V. Sarma, 1972. Anticipation of modern mathematical discoveries by Kerala astronomers. In A History of the Kerala School of Hindu Astronomy. Hoshiarpur: Vishveshvaranand Institute.

Section 2: Science, General

10. Staal, Frits. 1995. "The Sanskrit of Science". In Journal of Indian Philosophy 23, pp. 73-127.

11. Subbarayappa, B. V. 1970. "India's Contributions to the History of Science". In Lokesh Chandra, et al., eds. India's Contribution to World Thought and Culture. Madras: Vivekananda Rock Memorial Committee, pp. 47-66.

12. Saroja Bhate and Subhash Kak, 1993. Panini's grammar and computer science. ABORI, 72, pp. 79-94.

Section 3: Astronomy

13. Subhash Kak, 1992. The astronomy of the Vedic altars and the Rgveda. Mankind Quarterly, 33, pp. 43-55.

14. Subhash Kak, 1995. The astronomy of the age of geometric altars. Quarterly Journal of the Royal Astronomical Society, 36, pp. 385-395.

15. Subhash Kak, 1996. Knowledge of planets in the third millennium BC. Quarterly Journal of the Royal Astronomical Society, 37, pp. 709-715.

16. Subhash Kak, 1998. Early theories on the distance to the sun. Indian Journal of History of Science, 33, pp. 93-100.

17. B.N. Narahari Achar, 1998. Enigma of the five-year yuga of Vedanga Jyotisa, Indian Journal of History of Science, 33, pp. 101-109.

18. B.N. Narahari Achar, 2000. On the astronomical basis of the date of Satapatha Brahmana, Indian Journal of History of Science, 35, pp. 1-19.

19. B.L. van der Waerden, 1980. Two treatises on Indian astronomy, Journal for History of Astronomy 11, pp.50-58.

20. K. Ramasubramanian, M.D. Srinivas, M.S. Sriram, 1994. Modification of the earlier Indian planetary theory by the Kerala astronomers (c. 1500 AD) and the implied heliocentric picture of planetary motion. Current Science, 66, pp. 784-790.

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Re: Ancient (and recent) Indian Mathematics

Postby Kaushal » 27 Aug 2002 20:14

http://www.infinityfoundation.com/kumargrant.htm

A Brief Discussion on the Contributions of India
"The first nation (to have cultivated science) is India. This is a powerful nation having a large population, and a rich kingdom (possession). India is known for the wisdom of its people. Over many centuries, all the kings of the past have recognized the ability of the Indians in all branches of knowledge," wrote Sa`id al?Andalusi, a leading natural philosopher of the eleventh century Muslim Spain (Salem and Kumar, 1991, p. 11). The emphasis in the above quotation is not on India being "the first nation to cultivate science." It is on the fact that the European scholars, as late as the eleventh-century, thought India as a leader in science and technology. This is in contrast with the modern common perception about India in the Western minds or with the colonial period.

Poetry in Science: As early as 2600 B.C., the Egyptian monarch Khufu erected a great "house of writings," thus inaugurating a tradition that, by the time of Rameses II (ca. 1300 B.C.), yielded a collection numbering at least 20,000 papyrus scrolls, each protected in a cloth or leather cover. During the 300 year period, between the sack of Thebes by the Assyrians and the invasion of Alexander the Great's armies in 332 B.C., practically all of Egypt's libraries were turned to ash and dust. We know, of course, that this great loss was soon compensated-and afterward repeated-by the great library at Alexandria. This was not typical of only Egypt. Similar decimation of the written words occurred elsewhere in China, Europe, and the Middle East. The Qin dynasty in China, the Reformation in Europe, and Kublai Khan invasion of the Middle East remind us clearly and concretely: the written words are fragile.

While the textual riches of Alexandria, China, and Rome were being put to the flame, a wholly different tradition of scientific expression was brought to a peak in India, in a manner that would prove enormously more resilient to the vicissitudes of time and adversity. This was the oral, poetic tradition of Indian thought, whose greatest purveyor in astronomy and mathematics was Aryabhata I (b. 476 A.D.). }Aryabhata I, by almost any account, was the equal in Indian astronomy to what Cladius Ptolemy became to the tradition of Greek science in Islam and late medieval Europe.

Aryabhata I composed a most remarkable work in Sanskrit known as the Aryabhatiya. Its text consists of a mere 121 stanzas, each with several lines in varying metre and rhyme scheme. The whole is divided into a brief introduction (ten verses), followed by three major parts, one on mathematics (Ganita, 33 verses), one on time reckoning and planetary models (Kalakriya, 25 verses), and one on the mathematics of the sphere and its applicability to astronomical calculation (Gola, 50 verses).

There are no numbers anywhere in the composition. Nor are there figures, drawings, or equations. The Aryabhatiya expresses the highly sophisticated mathematics of sine functions, volumetric determinations, calculation of celestial latitudes and motions, and much more, in the form of a poetic code. This code, invented by Aryabhata I (though others like it had existed earlier on) and delineated in the introductory section, uses an alphabetical system of numerical notation. Specific values are assigned to specific letters and letter combinations, such that any value can be expressed in words and recited through the given metric scheme.

Use of a poetic code did mean, however, that transmission of this most valuable text could be achieved orally. This means through memorization and spoken transfer, or more accurately, by this particular text becoming a part of living memory among succeeding generations of scholars. This occurred during much of the medieval era, when many famous libraries and teaching centers were destroyed in India, as elsewhere, by foreign conquest and civil strife. However, many books were reproduced later on from the oral tradition

The image brought to us by the history of Aryabhata I's seminal work is indeed striking: at the very moment when Ptolemy's, Hipparchus', Eudoxus' and Euclid's books darkened the Alexandrian sky with textual ash, the Aryabhatiya was being passed and preserved through the most transient yet durable medium, the spoken and memorized word in the hearts of the Indians.

Mathematics: The present place?value notation system (base 10), used to represent numbers, is Indian in origin. The Mayas (base 20) and the Babylonians (base 60) used place?value systems as well, although their systems were different from that of the Indians. The role of place-value notation is quite important in mathematics and computers. For example, the extreme difficulty of performing mathematical operations in the absence of a place?value notation system caused Greek mathematics and astronomy to suffer. Nicholas Copernicus (A.D. 1473?1543) was forced to use the Hindu numerals for their ingenuity (Rosen, 1978, p. 27): "This [Hindu] numeral notation certainly surpasses every other, whether Greek or Latin, in lending itself to computations with exceptional speed. For this reason I have accepted [it]." Copernicus, in his book, On the Revolutions, used Hindu?numerals for his computations while advocating the heliocentric model of the solar system.

Trigonometry deals with specific functions of angles and their applications to calculations in geometry. It unites the disciplines of arithmetic, algebra, geometry, and astronomy. The Alexandrian Hipparchus (ca. 150 B.C.) and Ptolemy (ca. A.D. 100--170) helped to lay the foundations of trigonometry. With the work of Aryabhata I (ca. A.D. 500) in India, trigonometry began to assume its modern form. For example, consider a circle of unit radius, so that the length of an arc of the circle is a measure of the angle it subtends at the center of the circle. In order to facilitate calculations in geometry, the Greeks tabulated values of the chords of different arcs of a circle. This method was replaced by Indian mathematicians with another system that used the half chord of an arc, known today as the "sine" of an angle.

The origin of the word "sine" in trigonometry can be traced to the Sanskrit language. The Sanskrit word for the chord of an arc comes from an analogy with a "bow string," which is called "samasta?jya." The half?chord of the arc was called "jya?ardha," later shortened to "jya." The Arabs borrowed the concept from India and chose a similarly word "jaib" for the mathematical operation, which linguistically means a "lace fold or pocket" used in clothes.

This Arabic word was literally translated into Latin and called "sinus," meaning curved surface or fold. This Latin word was later metamorphosed into "sine" or "sin" in the English language. (Joseph, 1991; Kumar, 1994) Such etymological knowledge can be found in some dictionaries and encyclopedias but, ironically, only rarely in trigonometry books.

When the Arabs borrowed a concept from the Indians, they usually attributed the source correctly, especially in their early medieval literature. They often kept the same or similar pronunciation for many words, as shown by the previous example. In other instances, they translated the Sanskrit word into Arabic.

For example, the Sanskrit word for zero (a Hindu concept) is sunya, which translates literally into the Arabic word sifr. When the Europeans learned of it, they called it cipher or zephirum in Latin. This was transformed eventually as zero in English.

Natural Sciences: Regarding the earth's motion in his heliocentric model, Aryabhata I, some thousand years before Copernicus, suggested that the earth might be in axial rotation, with the heavens at rest, so that the apparent motion of the stars would be an illusion. In order to explain the apparent motion of the sun, Aryabhata I used the elegant analogy of a boat in a river: "As a man in a boat going forward sees a stationary object moving backward, just so at Lanka (Sri?Lanka) a man sees the stationary asterisms (stars) moving backward in a straight line." (Clark, 1930, Gola 9) The interpretation is that a person standing on the equator of the earth (Sri?Lanka) that rotates toward the east would see the stars (asterisms) moving in a westward direction. Aryabhata I's hypothesis of the earth's rotational motion is clearly explained by the analogy of the boatman as given above. He incorporated the concept of relative motion, many centuries before its more formal discussion by the noted Parisian scholar Nicholas Oresme in the fourteenth century. (Toulmin and Goodfield, 1961).

The ancient Indians realized the common connections between plants, animals and humans. They also realized the role of plants in medicine and the role of animals in the preservation of nature. In view of the importance of trees for humans, unnecessary cutting of a tree was against social and religious codes in India. Asoka (2nd century B.C.), a king in India known for his Maurya?empire and propagation of Buddhism, defined a strict law against the demolition of trees. Similarly, protection of animals led to the movement of vegetarianism. The Hindus went to the extent of worshiping both life forms; Animals and plants are revered along with gods. It is a curious fact that the Hindu concept of transmigration of souls was a common belief among the Greek philosophers and not the Greek society. Some connections between the Greek philosophers and Indian philosophy and the possibility of such interactions will also be explored.

Kanada wrote a book, Vaisesika?Sutra, in which he proposed that matter is made up of small particles, called parmanu (atoms). Kanada's description is based on the impossibility of infinite division of matter: "parmanu is not visible; the non?perception of atoms disappears when they mass together to form a combination of three double?atoms, a combination which does assume visibility." Cyril Bailey compared the Indian description of atoms with the Greek description, and wrote: "It is interesting to realize that an early date Indian philosophers had arrived at an atomic explanation of the universe. The doctrines of this school were expounded by Vaicesika Sutra [VaiÑesika?Sãtra] and interpreted by the aphorisms of Kanada . . . .Kanada works out the idea of their combinations in a detailed system, which reminds us at once of the Pythagoreans and in some respect of modern science, holding that two atoms combined in a binary compound and three of these binaries in a triad which would be a size to be perceptible to the sense." (Bailey 1928, p. 64)

As this section amply demonstrates that India made substantial contributions to science and technology in the ancient and medieval period. However, such contributions are not in the mainstream knowledge in the absence of its inclusion in the academic curricula. This project is a modest effort to bridge such gaps.

Salient Features
Most writings related to Indian scientific contributions are written in specialized or obscure journals on Indology. Most articles are written in isolation with little or no connection/comparison with the western science or with other contemporary cultures. In the absence of a cross-cultural study, it is unlikely that such knowledge will ever be assimilated into the mainstream knowledge of science. For this reason, I plan to write my articles/modules with a cross-cultural emphasis. Such a study will not be in conflict with the mainstream science; it will fill the gaps in our understanding of science. I also plan to publish my articles in the mainstream journals/magazines (examples: Science as Culture, The Physics Teacher, Journal of College Science Teaching, or American Educational Research Journal)

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Re: Ancient (and recent) Indian Mathematics

Postby Kaushal » 03 Sep 2002 02:23

Kerala School of mathematics

For more on the Kerala School of mathematics, see 'Science and Technology in Ancient india',published by Vijnan Bharati, Mumbai, pp.56.Prominent among the Kerala School were;

Madhava (1340-1425 ce)
Neelakantha Somayaji (1445-1545 ce)
Jyeshtadeva (1520 ce)
Putumana Somayaji (circa 1730)
Raja Sankara Varma

Cross posted from Indiancivilizations

I would like to add contribution of Kerala to Calculus-
On the page 111 of the research paper "The Sanskrit of Science" by Dr.
Frits Stal (Journal of Indian Philosophy, vol. 23, p.73-127, March 1995),
it was mentioned, " The fact remains that Leibnitz, Madhava , Newton,
Nilakantha and others made the same discoveries, the Indians even
earlier,------"
Madhava and Nilakantha were from Kerala.
N. R. Joshi.

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Re: Ancient (and recent) Indian Mathematics

Postby Calvin » 03 Sep 2002 02:54

Kaushal:

This has been a most interesting thread for me, especially since my father was a mathematician. If you had to put your finger on why Indian mathematics did not become the engine of engineering and then the economy that their european counterparts became, what would be the reason.

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Re: Ancient (and recent) Indian Mathematics

Postby kautilya » 03 Sep 2002 03:02

Calvin,
A part of the answer is in this article --
http://members.tripod.com/~INDIA_RESOURCE/mathematics.htm


Although it appears that original work in mathematics ceased in much of Northern India after the Islamic conquests, Benaras survived as a center for mathematical study, and an important school of mathematics blossomed in Kerala.
Also, if you read the section on Applied mathematics, you will see that it also ends at around 12th C, i.e. after the Islamic conquests. The major centers of learning and patronage were destroyed during that time.

Also, the article says --

Not only did India provide the financial capital for the industrial revolution (see the essay on colonization) India also provided vital elements of the scientific foundation without which humanity could not have entered this modern age of science and high technology.

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Re: Ancient (and recent) Indian Mathematics

Postby kautilya » 03 Sep 2002 03:10

Calvin,
Though this is not strictly mathematics, it talks about "India's rational age, from 1000 B.C. to 4th A.D., when treatises in astronomy, mathematics, logic, medicine and linguistics were produced."

http://members.tripod.com/~INDIA_RESOURCE/scienceh.htm

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Re: Ancient (and recent) Indian Mathematics

Postby Kaushal » 03 Sep 2002 20:20

Prime Numbers

More on the Primality Algorithm developed at IITK,crossposted from the News thread.

From Here to Infinity: Obsessing With the Magic of Primes
http://www.nytimes.com/2002/09/03/science/03ESSA.html

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Re: Ancient (and recent) Indian Mathematics

Postby venkat_r » 03 Sep 2002 20:29

I came know about a person who has proved Format's theorm and is now trying to get it published for sometime now.

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Re: Ancient (and recent) Indian Mathematics

Postby Kaushal » 03 Sep 2002 20:36

If you had to put your finger on why Indian mathematics did not become the engine of engineering and then the economy that their european counterparts became, what would be the reason.

In order for any academic field of endeavor to flourish at an advanced level, there must be peace and relative prosperity in the land and such endeavors must be supported by the state. A good example is the Indian Institute of Science and its successor institutes the IIT's which are churning out good work at a steady rate today. It is AlBiruni who remarked that Ghazni's marauding raids had left all of northwest India in ruin and bereft of the presence of savants in the sciences and mathematics. He specifically makes mention of the fact that many have fled to states beyond the reach of his master. It is no coincidence that Ujjain which was a center of scientific activity during pre-Islamic times ceased to be a center after the medieval era. In 1304 CE Alla ud din in one of his many jihads completely destroyed Ujjain, the center of Indian scholarship. North India never recovered from this as from the earlier destruction of Nalanda by Bakhtyar Khalji in 1198 CE, which destroyed Buddhist scholarship.

The center of gravity of Indian scholarship slowly shifted southwards until the 18th century when the British delivered the coup de grace of English based education, which inflicted the double whammy of lack of continuity as well as cutting of the vast majority of Indians from access to any educational institution.

But the spirit of Indian Mathematics could not be extinguished. It came back with a roar in the person of Ramanujam, the greatest number theorist in all of human history.

Kaushal


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