8/31/2006

Lalu Prasad to teach at IIM-A

Just a follow up on my previous post on Lalu. He has been invited to lecture at IIM-A on his management skills. According to rediff

Ever since Prasad took over as India's Railway Minister, the Railways have become the second-largest PSU profit-earner after the Oil and Natural Gas Corporation. Lalu has surprised many by emerging as one of the top-performing ministers in Prime Minister Manmohan Singh's Cabinet.


Building on his experience milking cows, Lalu has apparently adapted his mantra "If you do not milk the cow fully, it falls sick," quite efficiently to the Indian Railways context. Check out rediff for the complete article.

8/26/2006

India and Freedom of Expression.

The recent campaigns against M. F. Husain and blocking of various internet sites by the Indian government has prompted me to write this note. The outrage which bloggers have expressed against the government decision to block website has been quite impressive. But at the same time, as Kingsley pointed out in his blog title "Free Speech Free Riders", the outrage has been sometimes very ``selfish'' in nature. I mean, many people are okay with government blocking websites as long as its not theirs. The scariest part is that a NDTV poll "demonstrated" that a majority of Indians are okay if government blocks websites which are anti-national.

This brings me to the fundamental point of this article.

"How much of freedom of speech do we have in India ?"

The short answer is that there is no simple answer. A better way to look at the problem will be to compare India with various other countries in the world and see where it stands. If you compare with China or many other African and Middle Eastern countries, free speech in India is a paradise. But if you compare the 1862 Indian Penal Code (with some amendments) with the 1791 First Amendment in USA, the concept of free speech in India becomes really "free speech" with huge quotes around it. It is quite primitive and India has a long long way to go. One of the prime reasons is that sections of Indian Penal Code is sometimes very loosely defined and people and political parties interpret it according to their needs. Let me take the campaign against M. F. Husain by some self appointed, disillusioned protectionist of Hinduism, who think they own this religion. They claim they can use IPC 295A which states

"Whoever, with deliberate and malicious intention of outraging the religious feelings of any class of citizens of India, by words, either spoken or written, or by signs or by visible representations or otherwise, insults or attempts to insult the religion or the religious beliefs of that class, shall be punished with imprisonment of either description for a term which may extend to three years, or with fine, or with both."

Note the use of words like "deliberate" and "malicious intention". These are outright subjective and very difficult to judge and implement, whether the case is genuine or not. 295A also prohibits someone from critisizing someone else's views or beliefs. IPC is filled with such subjective cases which makes its misuse quite rampant. Another famous case of misuse is the loose definition of "cruelty" in IPC 498A, India's anti-dowry law.

Let me now quote the First Amendment from USA Bill of Rights.

"Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the government for a redress of grievances."

I hope I am making myselves clear here. As you can see, there is very little room for interpretation in the First Amendment. You are right if you feel that there should be a limit to First Amendment in the sense that no one should be able to make "derogatory" or false statements about a living person, which could in principle hurt his/her job prospects, political career etc. Slander and libel laws are then there to protect the victims in such cases. But then, even if the statements are inherently "derogatory", First Amendment will allow it in cases where it is deemed necessary. For example, making defamatory statements against corrupt politicians, which can be establised to be true is protected by First Amendment.

Finally, I think the notion that something needs to be banned or censored because it hurts people's beliefs, whether it religious, political or some other belief, is flawed. Having such a law restricts freedom of thoughts and expression and also suppresses creativity and vibrancy. It also gives politicians a sharp and dirty tool to use it to their own purposes. Some examples which come to my mind are

  • Congress blocks blogs because their content "hurts" them.
  • Aubrey Menon's Rama Retold is banned because it "hurts" Hindu sentiments.
  • Seven states in India bans Da Vinci because it "hurts" some Christians.
  • BJP bans movies by Rakesh Sharma on Godhra and Anand Patwardhan on Nuclear Tests.
  • West Bengal bans Tasleema Nasreen's writings because it "hurts" some politicians.
  • The teleserial "Bible Ki Kahaniyan" came under fire because it potrays Abraham, who is considered a prophet according to Koran. Hence, NO ONE should potray him in images (or cartoons for that matter).
  • Congress bans Salman Rushdie's book because it "hurts" muslim sentiments. As a side note, India banned it even before Iran declared its fatwa. During the same Rajiv Gandhi government, The Last Temptation of Christ is also banned.
  • Violent riots erupt in Kerala (with incentives from Churches) following the release of a play scripted by P. M. Antony based on Nikos Kasantasakis’ famous novel The Last Temptation of Christ.
I guess I can keep continuing with tons of such examples. I only hope India comes up with somthing similar to the First Amendment in the near future. Let me end by quoting Gurudev's Chitto jetha bhayshunyo (Where the mind is without fear).

Where the mind is without fear and the head is held high;
Where knowledge is free;
Where the world has not been broken up into fragments by narrow domestic walls;
Where words come out from the depth of truth;
Where tireless striving stretches its arms towards perfection:
Where the clear stream of reason has not lost its way into the dreary desert sand of dead habit;
Where the mind is lead forward by thee into ever-widening thought and action--
Into that heaven of freedom, my Father, let my country awake.
As a quick trivia, let me quote the last two verses from the original Bengali, which is much harsher.
Lord Father, strike {the sleeping} Bharat (India) without mercy, so that it may awaken into such a heaven.



8/22/2006

Perelman wins 2006 Fields Medal (Math Nobel Prize)

Grisha Perelman is one of the winners of the 2006 Fields Medal, though he has declined to accept it. The other three winners are Andrei Okounkov, Terence Tao and Wendelin Werner.

I am glad that I happened to sit for Okounkov's course on Representation Theory while I was in Princeton. He was a very lucid and friendly teacher.

Terence Tao is only 31. Swap the digits in 31, you get 13. The age at which he won a gold medal at the Internation Mathematics Olympiad, a feat which hasent been achieved again yet.

Some kids start thinking about primes while others of their age are still learning to use the potty. What else can you say ?

8/17/2006

Grisha Perelman: The genius who made Poincare Conjecture a Theorem.

Perelman is in news again. After three years of proof checking, mathematicians are convinced that this guy has indeed cracked the conjecture.

Apparently, this unwordly genius is "missing" somewhere in Russia and is not responding to emails. People have also started conjecturing tha he will turn down the Clay Math million dollar prize and the prestigious Fields Medal, which he will most likely receive next tuesday.

NY Times (free subscription maybe required) has a nice article on the problem, its history and Perelman. Click here for a related article by Simon Singh.

For the mathematically inclined, Bruce Kleiner and John Lott have a page with materials related to Perelman's work on Ricci flow.

The list of Clay Math Millenium Prize Problems can be obtained here.

8/14/2006

Amusing interests of Kerala Ministers.

This is straigh off the Kerala state government website. It lists the council of ministers and their hobbies. Most of them list the standard boring stuff like reading, library etc as their hobbies. But the following two definitely are the interesting ones.

A. K. Balan,
Minister for Welfare of Scheduled Castes/Scheduled Tribes and
Backward Classes, Electricity
Hobbies:
Reading, Plamistry, Magic, etc

I am assuming its a typo and actually Palmistry. Even then, palmistry and magic for a communist minister seems like a very useful tool. He can use his palmistry to predict the bright prospects of joblessness in Kerala and use his magic to convince people that joblessness is actually good for them.

N. K. Premachandran, Minister for Irrigation, Command Area Development Authority Ground Water Development, Water Supply and Sanitation
Favorite Pastime and Recreation: Cinema, Traveling, Reading and playing cards

Playing cards, i.e., sheetukali. This guys should be having a lot of free time to engage in the pleasures of playing cards.

8/13/2006

Big Fat Mallu Wedding :)

A very funny, sad, true, sarcastic blog from "Dog's Own Country ?" Must read I should say. You wont have time to stop laughing at this ridicule of marriage institution.

Zen and Hindu beliefs are demonic.

So says Christian Evengelist Albert Mohler, President of Southern Baptist Theological Seminary. You can view the video clip on youtube. Its an old clip, but I just came across it. Quoting Mohler,

I would have to say that as a Christian that I believe that any belief system, any worldview, whether it’s Zen Buddhism or Hinduism or dialectical materialism for that matter, Marxism, that keeps persons captive and keeps them from coming to faith in the Lord Jesus Christ ... is a demonstration of satanic power.

Without any offence to Mohler's belief or questioning his right to express it, I believe that he is stupid and dumb to make such a statement in the media. Its very very different from cracking a joke (about what you don't believe) with your friends over a weekend beer party. I hope the Southern Baptists choose a more diplomatic leader next time.

I see a future takeover for Pat Robertson.

The Argumentative Indian

I just finished reading Amartya Sen's Argumentatvie India . Hats off to Prof. Sen for this passionate and patriotic account of India's past, its culture, argumentative tradition, identity and tips for India's future.

Growing up in Mumbai (previously Bombay), I always wondered why we Indians argue over anything and everything. (I still vividly remember an incident when I was travelling in a Mumbai local train when two middle aged men were passionately arguing over the potency of Salman Khan). Whether it be religion, national and international politics, cricket, movies, you name it and we can immediately jump into a heated argument. I am sure most of you would have thought of this yourselves. One of the recent polls in my school email group asked:

India/Pakistan, Hindu/Muslim why are these debates never ending ?


One of the answer choices (rightly) was

"Because we are Indians"

To add fuel to the fire, some people also argue that India's development is slow because people keep arguing with themselves. Looks like arguement is kind of a double edged sword. But wait. Sen argues that arguments and debates are good for a society in the long run. He argues his point by comparing India with China. India for example, after having adopted democracy hasent had a major drought as opposed to China where dissent it not very much tolerated.

He also puts forth the argument that arguing is not a recent phenomena but somthing which has been running in an Indian's blood since time immemorable. The creation song from Vedas ends with the following skeptical questions:


" Who really knows? Who will here proclaim it? Whence was it produced? Whence is this creation? The gods came afterwards, with the creation of this universe. Who then knows whence it has arisen?

Whence this creation has arisen – perhaps it formed itself, or perhaps it did not – the one who looks down on it, in the highest heaven, only he knows or perhaps he does not know."

The books starts off with an overview of heterodoxy in Indian thoughts, epics, sciences and a very elaborate discussion on various kinds of secularism. It points out that India's past diversity is very essential to understand its present and also to remove the biased Western perceptions which are still alive. Debates and exchanges between various schools of thoughts are recorded in Indian texts. Rituals and religiosity were both defended by its defenders and questioned by skeptics. Over some time, even the atheistic school became influential. One can see very clearly different forms of agnosticism developing here. Vedic ritualism, Buddhism, Jainism, agnosticism and atheism were all debating with each other during this "philosophical golden age". Infact, the first chapter in Madhavacarya's Sarvadarsanasamgraha is devoted to the defense of Carvaka's skeptic atheistic school, though the later chapters denounce this school of thought with counterarguments. (That was the style adopted in text in those times). Atheism flourished for quite some time and also had representatives in Akbar's multi-religious court as chronicled in Abul Fazl's writings.

Without going into too many details, let me end by just pointing out some prime highlights as follows

  • The extreme diversity and flexibility in past Indian thought is crucial not only to understand India's present but also to frame policies for a better tomorrow.

  • Because of its past diversity, its unfair to categorize India in religious terms. The form of inclusive secularism which India adopted (very different from prohibitive secularism in France for example) after independence, inspired by Ashoka and Akbar (definitely two prime characters in the book) should continue. Infact, the broader term Hindu had a very different meaning till recent times. The Arabs used Hindu to refer to people in the geographical area arond current India. For example, muslims in India were called "Hindu Muslims" and christians were called "Hindu Christians".

  • The book also addresses concerns over labelling of India as a Hindu ( using the current notion of Hinduism) nation by certain political parties in recent times. Not only is such a categorization not unique (i.e., a nation can be categorized using numerous concepts, each of which has its own properties and shortcomings. Think of the blindmen elephant story), but it completely ignores contributions made by other sects like Buddhists, Christians, Jains, Sikhs, Muslims, Parsees, agnostics, atheist and many other communities. Amartya gives the following examples which are equally good attributes of the majority Indians

    • category of low- or middle-income people
    • class of non-owners of much capital
    • rural Indians
    • people who do not work in the organized industrial sector
    • Indians who are against religious persecution


    • As can be clearly seen from the above list, trying to redefine India as a Hindu nation because majority of Indians happen to be Hindus is a very narrow minded Indianview.


  • The book offers a very detailed account as to why the West looked at India with a skewed mystic perspective.

  • Gives a very strong defense for globalization of trade and thoughts. Infact, as Amartya points out, globalization is not a new phenomena at all. Adoption of Indian mathematics by Arabs and Greeks, import of Buddhism by China, adoption of chinamsuka (chinese silk), chinake (camphor), paper, chinaka (fennel), chinapista (vermilion), gunpowder by India, the appointment of Gautama Siddhaartha (Qutan Xida, in Chinese) as president of Board of Astronomy in China in 8th century, Xuanzang's stay in Nalanda University, etc are old examples of trade and exchanges on a global scale.

Finally, let me end this by quoting how Sen ends his book

" Jamsetji Tata had the pride not of a Parsee who happened to be Indian, but of an Indian who happened to be a Parsee".


Truly an excellent read to grasp the mystery and diversity that is India.

8/06/2006

Kalaripayattu: Ancient Martial Art from Kerala, India

Known to be one of the most ancient forms of martial arts in the world, Kalaripayittu was developed in Kerala no later than 1200 according to some historians. There are two styles for this art, the northern style practised in Malabar region and the southern style practised in Travancore. It should be noted that the term Malabar comes from the Portuguese malabaristas, meaning acrobats.

Though known very little outside Kerala, it is finally getting noticed in the international community thanks to the Jackie Chan movie "The Myth". There are also reasonable evidences that Kalaripayittu was taken to China by Bodidharma and that Shaolin Kung Fu and Japanese martial arts were derived from Kalaripayittu. These assertions are disputed though. Kalaripayittu shows up in lots of movies from Kerala, but outside Kerala, it has shown up in only one movie. Kamal Hassan's Indian.

Click here for a National Geographic documentary on Kalaripayittu and hope you enjoy it. Towards the end, there is a fight scene between two men with one of them blindfolded. Dont miss it.

Here are some relevant links on Kalaripayittu.

Kalaripayattu at Wikipedia
Disputed History of Kalaripayattu at Wikipedia
Jackie Chan and the art of Kalaripayattu

8/04/2006

Ancient Indian Mathematics

"I have never found one among them who could deny that a single shelf of a good European library was worth the whole native literature of India and Arabia. "



India was usually known to outsiders until very recently as the land of naked sadhus, snake charmers, overtly religious mystical land which taught the world how to make love. Such a perspective definitely helped the colonisers in assigning an inferior status for the colonised and justify their acts of colonisation and "civilisation". Macaulay's remarks in his Minute on Indian Education is a good read to understand the British perspective then about India. Many influential political leaders (Gandhi for example) who had renounced the more superstitious religious beliefs, in favour of less superstitious beliefs also contributed to the mainstream western perspective on India remaining a status quo. This is surprising considering the fact that even during Gandhi's time, there were huge difference of opinion between Gandhi and other more rational minded people like Tagore and Ambedkar. Gandhi in his essay Hind Swaraj, strongly condemds technology and industrialization, whereas Tagore believed in science, industry and rationality. Tagore was particularly disgusted by Gandhi remark that the 1934 Bihar earthquake was a divine retribution for ill treatment of dalits. Tagore, who was as much against caste exploitation as Gandhi was, however could not stand such an irrational explanation for the earthquake suggested by Gandhi. Their debates on this and many other issues (like Gandhi's belief that everyong should use the carka for atleast 30 mins a day) in the form of letters make a very good read [1]. Another non ignorable figure who had serious issues with Gandhi's irrationality was Ambedkar. The latter firmly believed in the evils of caste system, where as Gandhi who was against the opression against dalits, nevertheless believed in the sanctitty of the age old caste system and even justified its working as a means to achieving social stability. You can read Ambedkars response to Gandhi's views on caste in his essay "Annihilation of Caste". It looks like Tagore and Ambedkar were not at all influential in changing outsiders outlook towards India. Gandhi definitely takes the spotlight.

Gladly enough, thanks to the IT age, outsourcing and the recent economic boom, this image is rapidly changing. Such a categorization, not only misrepresented India, but also ignored completely many other aspects like agnosticism (Buddhism), atheism (Carvaka) and science and mathematics which were very much being developed in ancient India. Buddhism, which originated in India and spread as a rebel against caste system and rituals, was in full glory for about a millenium and was one of the prime exports of India during those days to China and other eastern countries. So much that China refered to India as the Kingdom of Buddhism. Rationalism and skepticism too, was a school of thought which cannot be ignored during these times. Hinduism, Buddhism, Jainism, Atheism and many other schools were competing at a time for acceptance as a mainstream philosophy. Amartya Sen's book The Argumentatvie India forms an excellent read on this subject.

As far as ancient Indian sciences are concerned, it was developed and transmitted to regions outside India by Chinese and Arabs. Many mathematics and science texts were translated by Al-Beruni[2], Al-Khwarizmi (who is responsible for the temrs algebra and algorithm) into Arabic and found its way to Europe there on. One of the prime motivations for doing mathematics those days was astronomy. Approximations for trigonometric values, pi, circumference of circle, etc were developed. Bhaskara's approximation for sin(x) as a rational function of x, Brahmagupta's calculations of eclipses, Aryabhatta I's approximation of pi, Aryabhatta I's method to solve linear Diophantine equations are just some examples amongst a huge literature.

Broadly, ancient Indian mathematics can be categorized into the following based on their period of development.

  • Pre Vedic Indus mathematics dating back to as early as 3000BC. Development of mathematics were highly influenced by practical applications like measuring scales, calculating brick ratios etc.
  • Vedic or Sulbasutras, which contained rules to construct altars for various rites and rituals. Various constructions based on Pythagoras theorem are listed, approximation to square root of 2, approximation to pi are some of the achievements. This period probably lasted till around 500BC.
  • Jaina mathematics, from 600BC to 500AD. Prime achievements are various notions of infinities, Pascal's triangle, form of set theory, operations with roots of order larger than 2 etc
  • A set of manuscripts were found around 1880s, called the Bakhshali manuscript. The dates of these manuscripts are assumed to be around 400AD. This book contains sets of problems and solutions in linear equations, fractions, square roots etc
  • Golden age of Indian mathematics set off by Aryabhatta I. Numbers start becoming more abstract and makes it possible to consider zero and negative numbers. Brahmagupta, Mahavira and Bhaskara lists down rules for multiplying, subtracting and (wrongly) dividing by zero. All of them missed that dividing by zero does not make sense :) It was during this period that Al-Khwarizmi, who lived from 790 to 840, wrote Al'Khwarizmi on the Hindu Art of Reckoning and passed on Indian achievements to Islamic and Arabic countries. Ibn Ezra, who lived from 1092 to 1167 in Spain also helped transmit Indian number system to Europe.
  • Keralese mathematics: The golden age started to decline after Bhaskara's works around 1200. There is some consensus that political instabilities are partly to be blamed for this decline. Despite all these, mathematics continued to develop in Kerala. The time period between 1400 and 1600 is considered to be the peak period of this development. In addition to building upon and extending previous works by Bhaskara, Brahmagupta, Aryabhatta etc, one of the significant achievements of this period was mathematical inductive proof in the works of Nilakantha, Jyeshtadeva. The prominent figure during this time period is definitely Madhava of Sangamagramma. He built tools essential for modern analysis. Though not fully certain, he is supposed to have independently derived taylor series expansion for arctan, sin, cos and many other function. A list of 13 different expansions attributed to Madhava is listed here. Madhava also managed to calculate the value of pi to 17 decimal places (3.14155265358979324), much ahead of his contemporaries.

People have started recognizing and acknowledging role of Indians, but awareness still seems to be a big issue. I think it will take some time before terms like Madhava-Gregory series, Leibniz-Gregory-Madhava constant, Hemachandra-Fibonacci series, Bhaskara-Brouncker Algorithm, Aryabhatta algorithm, Aryabhatta Remainder Theorem, Aryabhatta Kuttaka etc become commonplace.You might want to check the following links to dig deeper into the history of Indian mathematics.

    1. Indian Mathematics: Redressing the balance
    2. C. T. Rajagopal and M. S. Rangachari. 'On an untapped source of medieval Keralese mathematics', Archive for History of Exact Sciences 18 (pages 89-102). 1978.
    3. Keralese Mathematics
    4. Possible transmission of Keralese mathematics to Europe.


    Here is a list of prominent Indian mathematicians in chronological order. The names and brief description are obtained from various websites including but not limited to University of St. Andrews.



    Baudhayana 800BC - 740BC

    Baudhayana was the author of one of the earliest Sulbasutras: documents containing some of the earliest Indian mathematics.



    Apastamba 600BC - 540BC

    Apastamba was the author of one of the most interesting of the Indian Sulbasutras from a mathematical point of view.



    Panini 520BC - 460BC

    Panini was a Sanskrit grammarian who gave a comprehensive and scientific theory of phonetics, phonology, and morphology.



    Katyayana
    200BC - 140BC

    Katyayana was the author of one of the Sulbasutras: documents containing some of the earliest Indian mathematics.



    Yavanesvara
    120 - 180

    Yavanesvara was an Indian astrologer who translated an important Greek text on astrology.



    Aryabhata the Elder 476 - 550

    Aryabhata I was an Indian mathematician who wrote the Aryabhatiya which summarises Hindu mathematics up to that 6th Century.



    Yativrsabha 500 - 570

    Yativrsabha was a Jaina mathematician who gave a description of the universe which is of historical importance in understanding Jaina science and mathematics.



    Varahamihira 505 - 587

    Varahamihira was an Indian astrologer whose main work was a treatise on mathematical astronomy which summarised earlier astronomical treatises. He discovered a version of Pascal's triangle and worked on magic squares.



    Brahmagupta 598 - 670

    Brahmagupta was the foremost Indian mathematician of his time. He made advances in astronomy and most importantly in number systems including algorithms for square roots and the solution of quadratic equations.



    Bhaskara I 600 - 680

    Bhaskara I was an Indian mathematicians who wrote commentaries on the work of Aryabhata I.



    Lalla 720 - 790

    Lalla was an Indian mathematician who wrote mainly on the application of mathematics to astronomy.



    Govindasvami 800 - 860

    Govindasvami was an Indian mathematical astronomer whose most famous treatise was a
    commentary on work of Bhaskara I.




    Mahavira 800 - 870

    Mahavira Mahavira was an Indian mathematician who extended the mathematics of Brahmagupta.


    Prthudakasvami 830 - 890

    Prthudakasvami was an Indian mathematician best known for his work on solving equations.



    Sankara Narayana
    840 - 900

    Sankara Narayana was an Indian astronomer and mathematician. He wrote a commentary on the work of Bhaskara I.


    Sridhara 870 - 930

    Sridhara was an Indian mathematician who wrote on practical applications of algebra and was one of the first to give a formula for solving quadratic equations.


    Aryabhata II 920 - 1000

    Aryabhata II was an Indian mathematician who wrote about astronomy as well as geometry. He constructed tables of sines accurate up to about 5 figures.



    Vijayanandi 940 - 1010

    Vijayanandi was an Indian mathematician and astronomer who made some contributions to trigonometry.


    Sripati 1019 - 1066

    Sripati was an Indian who wrote works on astronomy and arithmetic.


    Brahmadeva 1060 - 1130

    Brahmadeva was an Indian mathematician who wrote a commentary on the work of Aryabhata I.


    Acharya Hemchandra 1089 - 1173

    Hemachandra was a Jaina scholar who presented what is now called the Fibonacci sequence around 1150, about 50 years before Fibonacci (1202). He was considering the number of cadences of length n, and showed that these could be formed by adding a short syllable to a cadence of length (n−1), or a long syllable to one of (n−2). This recursion relation F(n) = F(n−1) + F(n−2) is what defines the fibonacci sequence.
    Bhaskara 1114 - 1185

    Bhaskara II or Bhaskaracharya was an Indian mathematician and astronomer who extended Brahmagupta's work on number systems.


    Narayana Pandit 1340-1400

    Narayana Pandit is the author of an arithmetical treatise called Ganita Kaumudi and an algebraic treatise called Bijganita Vatamsa. Narayana is also thought to be the author of an elaborate commentary of Bhaskara II's Lilavati, titled Karmapradipika (or Karma-Paddhati).


    Madhava of Sangamagrama 1340-1425

    Madhava of Sangamagrama was the founder of the Kerala School and considered to be one of the greatest mathematician-astronomers of the Middle Ages. It is vaguely possible that he may have written Karana Paddhati a work written sometime between 1375 and 1475 but all that is known of Madhava comes from works of later scholars.


    Parameshvara 1370-1460

    Parameshvara, the founder of the Drgganita system of astronomy, was a prolific author of several important works. He belonged to the Alathur village situated on the bank of Bharathappuzha. He is stated to have made direct astronomical observations for fifty-five years before writing his famous work, Drgganita. He also wrote commentaries on the works of Bhaskara I, Aryabhata and Bhaskara II. His Lilavati Bhasya, a commentary on Bhaskara II's Lilavati, contains one of his most important discoveries:


    Nilakantha Somayaji 1444 - 1544

    Nilakantha was a mathematician and astronomer from South India who wrote texts on both astronomy and infinite series.


    Jyesthadeva 1500 - 1575

    Jyesthadeva was a mathematician from South India who wrote an important work on mathematics and astronomy which summarises the work of the Kerala school.


    Citrabhanu c. 1530

    Citrabhanu was a 16th century mathematician from Kerala who gave integer solutions to 21 types of systems of two simultaneous Diophantine equations in two unknowns. These types are all the possible pairs of equations of the following seven forms:


    Kamalakara 1616 - 1700

    Kamalakara was an Indian astronomer and mathematician who combined traditional Indian astronomy with Aristotelian physics and Ptolemaic astronomy as presented by Islamic scientists.


    Jagannatha Samrat 1690 - 1750

    Jagannatha was an Indian mathematician who is important as a translator of important Greek works into Sanskrit.


    Sankara Varman 1800-1838

    Sankara Varman
    There remains a final Kerala work worthy of a brief mention, Sadratnamala an astronomical treatise written by Sankara Varman serves as a summary of most of the results of the Kerala School. What is of most interest is that it was composed in the early 19th century and the author stands as the last notable name in Keralese mathematics. A notable contribution was his compution of π correct to 17 decimal places.



    [1]

    The Mahatma and the Poet : Letters and Debates between Gandhi and Tagore 1915-1941, Compiled and Edited by Sabyasachi Bhattacharya, National Book Trust, India


    [2]

    Alberuni's India, Edward C. Sachau, Trobner & Co., London, 1888, Rupa & Co., 2002. You can read a review of this book at hindu.com or obtain a pdf file of his book from infinityfoundation.com

  • 8/02/2006

    Carvaka on rituals, Aswamedha and self contradictions in Vedas

    Here is some Carvaka stuff from Sarvadarsanasamgraha by Madhavacarya. (Please note that this is the monist Madhavacarya and not the dualist Madhvacarya.) In the text below, Carvaka very strongly critises rituals as well as the obscenety bestiality/necrophilia associated with Aswamedha [*].

    " If a beast slain in the Jyotistoma rite will itself go to heaven, Why then does not the sacrificer forthwith offer his own father? If the Sraddha produces gratification to beings who are dead, Then here, to, in the case of travelers when they start, it is needless to give provisions for the journey. If beings in heaven are gratified by our offering the sraddha here, Then why not give the food down below to those who are standing on the housetop? While life remains let a man live happily, let him feed on ghee even though he runs in debt; When once the body becomes ashes, how can it ever return again? How is it that he comes not back again, restless for love of his kindred? Hence it is only as a means of livelihood that Brahmins have established here. All the ceremonies for the dead, there is no fruit anywhere. The three authors of the Vedas were buffoons, knaves, and demons All the well-known formulas of the pandits, jarphari, turphari, &c. And all the obscene rites of the queen commanded in the Asvamedha, These were invented by buffoons, and so all the various kinds of presents to the priests, While the eating of flesh was similarly commanded by night prowling demons. "


    In the following para, Carvaka derides the Agnihotra as merely a waste of money and useful only to keep the brahmins employed. It goes on to explain how the vedas are full of untruth, self contradictions and tautulogies.

    If you object that if there be no such thing as happiness in a future world, then how should men of experienced wisdom engage in the Agnihotra and other sacrifices, which can only be performed with great expenditure of money and bodily fatigue, your objection cannot be accepted as any proof to the contrary, since the Agnihotra are only useful as a means of livelihood, for the Veda is tainted by the 3 faults of untruth, self contradiction, and tautology; then again the imposters who call themselves Vedic pandits are mutually destructive, as the authority of the jnana-kanda (section on knowledge) is overthrown by those who maintain that of the karma-kanda (section on action), while those who maintain the authority of the jnana-kanda reject that of the karma-kanda; and lastly, the three Vedas themselves are only the incoherent rhapsodies of knaves, and to this effect runs the popular saying:

    "the Agnihotra, the three Vedas, the ascetic's three staves, and smearing oneself with ashes, - Brhaspati says these are but means of livelihood for those who have no manliness nor sense. "




    [*]

    I was shocked myselves initially about what all was involved in Aswamedha. Its a ceremony performed by a king for expanding territory, beget sons and for the well being of his country in general. When its performed to beget sons, a horse is slained and the chief queen is supposed to then "unite" with the dead horse. Apparently, this is what happens in the Bala Kanda of Ramayana when King Dasharatha performs the Aswamedha to beget sons. Chief Queen Kausalya is supposed to have "united" with the horse after the sacrifice. Valmiki's Ramayana mentions this as follows:

    Kausalya walked reverently all around the horse and then with the greatest joy cut it with three knives. Her mind unwavering in her desire for righteousness, Kausalya passed one night with the horse. The priests-- the hotri, the adhvaryu and the udgatri-- saw to it that the second and the junior most of the king's wives, as well as his chief queen, were united with the horse. Then the officiating priest, who was extremely adept and held his senses in check, removed the fat of the horse and cooked it in the manner prescribed in the ritual texts.


    Source: The Ramayana of Valmiki: An Epic of Ancient India, Volume 1: Balakanda

    Alternate Source: www.valmikiramayan.net . Go to verse 33 on the link provided.

    Valmiki does not give any more specific details about the Aswamedha in Ramayana. The complete details about this yagna is found in the 13th Kanda of Satapatha Brahmana (White Yajur Veda). There are many translations of this book available, most popular being one translated by Julius Eggeling. If you can read and understand Sanskrit, you can access this book online here. Use the right frame to jump to Book 13, Chapter 5, Paragraph 2. A very direct and gory translation of the verses pertaining to this ritual is given here.


    B. R. Ambedkar, in his essay titled "Riddles in Hinduism", has also used this practise (among many other things) to criticise Vedas.


    8/01/2006

    Thackeray threatens lawyers

    Maybe its in his blood to make Senaspeaks. Raj, president of MNS (offshoot of Shiv Sena) has threatened to make it difficult for lawyers to "move around Mumbai" if they defend the accused in Mumbai blasts. Clearly infringing the democratic rights of lawyers as well as the rights of accused to a frail trial, Raj's statement has been strongly been condemned by various lawyer's association.

    Maybe this Hitler admirer does not consider democracy to belong to Maharashtra. The "Mee" in
    "Mee Maharashtracha, Maharashtra Maaza" has no place for it. May Shivaji bless them.