The Significance of the Sacred Thread, Yagnopavitham

An interview by – Siddharth Swamy,
Student of Maths & Economics, Trinity College, Dublin

Today, full moon, in Sravan month is the day when many Indians perform the thread changing ceremony.

We bring to you an interaction of an inquiring youngster with D. K. Hari and D.K. Hema Hari of Bharath Gyan, on the relevance of this practice of wearing a sacred thread.

Yagnopavitham3

What is this thread I have been made to wear? It was put on during my Upanayanam. What is an upanayanam?

Let us go to the etymological meaning of Upanayanam. Upa means ‘near’ or ‘by the side of’ and nayanam means ‘by the eyes’. Therefore upanayanam denotes being by the side or supervision of a teacher.

Upanayanam is typically performed at the age of 7 years, the time when a child is ready to start schooling. At that time, parents conduct this ceremony and then take the child to the Gurukula, school.

This thread ceremony, also called Brahma Upadesham (Brahmopadesham) is to prepare the child to enter school and the schooling phase of life (Brahmacharyam).

I have heard people using the terms Yagnopavitham or Upakarman also for this ceremony.

Yagna, besides worship, sacrifice, also denotes commitment to a focused act. So committed, that one sacrifices everything else to ensure this act is accomplished. Pavitham means to cleanse. Yagnopavitham means to cleanse, purify one’s mind as well as intention behind the Yagna, act of commitment – studying in the case of a student.

Upakarman denotes preparation or that activity that aides the start, execution of any mission, it comes from Upa, beside and karman meaning activity.

Upakarma

A group  Yagnopavitham ceremony at the Art of Living Ashram

I was told that this thread I wear will protect me. How can this thread protect me and from what?

On your Upanayanam, this thread is given to you as a sign of committing you to schooling, education. It is a way by which parents tell you that you are now entering a phase where you have to stay fully committed to learning, avoiding all distractions.

The thread you wear acts as a constant reminder and helps you to make sure that you stay committed to the cause you have taken up and also to avoid all distractions, which may come in your wake.

Also, during the upananayanam you would have received the first lesson, i.e., the Gayatri Mantra from your first guru, who is your father. This is the Brahma Upadesham, counsel on the cosmos, from the father. This Gayatri Mantra is to be recited atleast twice a day, at dawn and dusk, the time windows in a day considered to be most conducive to learning.

This mantra is powerful and the vibrations it causes in the body and in the surroundings rejuvenate the body and mind with positive energy keeping them in good health and thus protected.

Is this protection only for Brahmin boys?

No, in older times every child, irrespective of varna/jati (loosely translated as caste in present times) at the start of schooling underwent this ceremony and went to gurukula to study basic veda and other subjects in line with their family profession or aptitude. So this particular investiture is meant for all, not just Brahmins.

What about girls then? Do they not need such protection during their studying age?

Scriptures show how girls too underwent such a ceremony. Perhaps in the medieval period when India came under onslaughts, girls being physically vulnerable, were kept away from schools to protect them from the invaders. With that maybe the number of girls going to gurukula reduced, thereby reducing the practice.

So if I do already know what this stands for and have a sense of commitment to my studies, then do I still need to wear it?

This thread is like a school uniform and gives you a sense of identity. When you wear a school uniform, your mind automatically gets conditioned and constrained  from indulging in acts that do not behoove school going children such as perhaps going to a bar,  discotheque or movies with the uniform on, this thread also conditions the mind and keeps one focused in their mission.

It is not only school uniform, any uniform be it that of a soldier, a policeman or a nurse, conditions one to conduct oneself in a manner specific to the category / institution they represent, when in that uniform. This thread is like that.

In that vain of thought, this helps us to be centered to our commitment at hand.

Similarly, when one gets married, there is an additional thread that gets added to the previous set. This is to condition one to stay on the path of societal norms of a married man who has to look after his family as well as support the community.

Why should he continue with the previous set of thread, now that he is already married and out of school?

The other set is to remind him that in life one is always a student and has to seek knowledge that can help one journey through the various phases of life with ease and relish.

Why do we change this thread every year?

It is to renew our commitment every year, the way people make resolutions every New Year. Since this thread is made from cotton yarn, it needs to be changed atleast once a year from a point of hygiene.

Yagnopavitham2
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The possible transmission of calculus from India to Europe – Dr. Bhaskar Kamble

The development of the infinitesimal calculus is considered to be a watershed event in the history of science and mathematics. Its importance in the natural sciences cannot be overestimated. Among the people credited for its invention are John Wallis (1616-1703 CE), Isaac Newton (1642-1727 CE), and Gottfried Leibniz (1646-1716 CE).

Very roughly speaking, calculus handles those problems where the rate at which ‘something’ is increasing is itself changing in time. Simple examples can include the case where the speed (the rate at which the distance is increasing) is changing in time, or the case where the acceleration (the rate at which the speed is changing in time) is changing in time. Together with Newton’s three laws of motion, which are physical in content, it offers a powerful tool to mathematically describe physical phenomena.

However, the standard story of calculus being developed in Europe independently by Newton and Leibniz, which is so universally accepted today, may well be in need of a major revision. Just like the concept of zero and the decimal number system originated in India, it is now well known that the concept of calculus also originated in India three centuries before it first appeared in Europe [1,2]. There is strong circumstantial evidence to suggest that these ideas and concepts were systematically appropriated by the church in Europe at the end of the 16th century and subsequently passed off as a European invention.

The earliest notion of calculus, specifically differential calculus, is to be found in the notion of tatkalika gati (Sanskrit: instantaneous velocity), of Bhaskaracharya (1114-1185 CE), in his monumental work Siddhanta Shiromani. In this text, he explicitly demonstrates and makes use of the relation which is a standard result of differential calculus, to determine the instantaneous velocity of a planet. He also states one of the most important results of differential calculus – that the derivative vanishes at the points of minima or maxima, and also states what is today known as the Rolle’s theorem in analysis/calculus [2].

sine_differential

The tradition of mathematics in India has a long and hoary past, with several shining names such as Aryabhata, Bhaskara I, Bhaskaracharya, Brahmagupta, Varahamihira, and so on. The schools started by several of these mathematicians would constantly develop and improve upon the discoveries of the earlier mathematicians, and come up with significant new results in the process. The most sophisticated insights and developments undoubtedly come from the work of what is known as the Kerala school of mathematics, which was extant from 1300-1600 CE. They wrote commentaries on the works of earlier mathematicians such as Aryabhata and Bhaskara, and made important discoveries in what is known today as calculus. That these mathematicians developed calculus 300 hundred years before Newton and Leibniz did is obvious [1,2], but what is more interesting is how and why their work was hauled off to Europe, plagiarized, and passed off as a European invention. In this post I will try to shed light on how this occurred.

Till the 15th – 16th centuries, it is important to remember that the church dominated all spheres of life in Europe. Central to its aims was the establishment of Christianity throughout the world and destroy all ‘pagan’ and ‘heathen’ cultures in the process. The genocide of Red Indians in the Americas, or of aborigines in Australia, or Hindus in India (especially the Goa inquisition in the 16th century), are a direct consequence of these aims, and so are the continued attempts of today by Christian missionaries to convert people of other faiths into Christianity. To further these expansionist policies, it was necessary to go to far-away lands and ‘civilize’ and Christianize the ‘natives’. To do so, however, needed navigational skills which, in turn, needed a good knowledge of astronomy (for example while navigating with the help of the stars) and a good knowledge of trigonometry (for example to calculate the latitude and longitude). In particular, trigonometric tables of the sine and cosine functions are a must for accurately determining the latitude at sea based on the altitude of the pole star.

There was another very important reason why the church needed astronomical knowledge – to carry out the calendar reform. The calendar originally used by the church was the Julian calendar, which had an error of one day in a century. This error was accumulating over the centuries and was causing the date of Easter to drift further and further away from the spring equinox into summer. It was very important for the church to set it right. A good calendar is also essential for good navigation, and thus the problem of navigation and the calendar were closely related. Thus, the issues of navigation and the calendar were high priority programs by the church, and several mathematicians involved with the church were actively involved in finding solutions. Attractive prizes were offered to anyone who could come up with solutions to these problems. The most important member in this regard is Christoph Clavius, who modified the curriculum of the priests in Collegio Romano to teach them mathematics, and himself designed the mathematical content of the curriculum, as well as writing a text book on mathematics to be used by the priests in their education.

In spite of this, as is well known, European astronomy and mathematics of those times was hopelessly lacking in the required knowledge [3]. And at the same time, the astronomical and mathematical knowledge of India was much superior to that of Europe. The works of several Indian mathematicians were well known in Europe, thanks to Arabic translations of Sanskrit texts, and the subsequent translations into Latin [4]. Also Fibonacci had introduced the Indian number system to Europe in 1202 CE. The navigational skills of Indian merchants was also something of a legend [5].

To realize just how advanced the Indian mathematics was at this time, we need to look at the achievements of the Kerala school. The tradition of the Kerala school was started by Madhava of Sangamagrama (1340-1425 CE), who was followed by several brilliant mathematicians and astronomers which include Parameshvara (1380-1460 CE), Nilakantha Somayaji (1444-1544 CE), and Jyeshthadeva (1500-1610 CE). Madhava is credited with many of the discoveries of the Kerala school, but verly little of his writings survive. The results obtained by him are further elaborated and developed by later scholars such as Nilakantha Somayaji in his work Tantra Sangraha, and Jyeshthadeva in his work Yukti Bhasha. The Yukti Bhasha is a veritable text book of calculus, and offers detailed explanations of most of the results obtained by the Kerala scholars. The scholars of this school also made several astronomical observations and collected the data in their works, and proposed significant improvements of the then prevailing astronomical models. Among the achievements of the Kerala school are the systematic development of the ‘limit’ procedure, which is so central to calculus, the systematic analysis of inifinte series, infinite series expansions of the sine, cosine and arctan functions, (the so-called Taylor series of today), a plethora of series expansions of pi (including the one known today as the Gregory series, 300 years before Gregory discovered them), important contributions in spherical trigonometry, and the development of much improved astronomical models based on actual observations. A practical application, much sought after by European navigators, was the calculation of sine tables, which had been carried out by Madhava up to an accuracy of eight decimal places [6,7]. An interesting application of this work was the calculation of pi up to 17 decimal places, which is coded beautifully through the kattapayadi system in the Sadratnamala of Shankara Varman.  In fact many of the works of these mathematicians are still subjects of active research by modern mathematicians! And of course, behind this there was a whole body of work by earlier Indian mathematicians such as Aryabhata, Bhaskara I and II, Brahmagupta etc.

It is in the light of this vastly superior Indian mathematics and astronomy, and the tremendous eagerness of the church to possess this knowledge, that the situation in Europe in the 15th – 16th centuries must be viewed. As already mentioned, Christoph Clavius had set up the mathematical syllabus of the Jesuit priests, and in 1578, the first batch of the most capable priests trained by him, which included Matteo Ricci, Johann Schreck, and Antonio Rubino, were dispatched to the Malabar region of Kerala, including Cochin, which was the epicenter of the Kerala mathematics.

Once they were there, they set up a printing press, learnt the local language, and gained the patronage and trust of the local scholars and royal personages. And now began in earnest the task to acquire Indian texts, translate them, and dispatch them back to Europe [7]. However, all this was kept a top secret. Even today, if you make a Google search on Matteo Ricci, you will never find the real reason why he was there, although it will be mentioned that he was in Kerala. And this, in spite of the fact Ricci and Rubino have been recorded in correspondence as answering requests for astronomical information from Kerala sources [8].

However, there is enough circumstantial evidence to prove that the transfer of the calculus from India indeed took place. First, there is little doubt about the real intention behind the trip of the Jesuits to Kerala: before being sent to India in 1578, not only were they trained in mathematics by the leading astronomer of those days, Christoph Clavius,  but also that, soon thereafter in 1582, the Gregorian calendar reform took place [9]! Remember that the calendar reform was one of the pressing concerns of the church and, what is more, the committee that carried out this reform was also headed by Christoph Clavius!

Next, as mentioned already, the Kerala mathematicians had created extensive tables of sines and cosines to a high degree of accuracy. Now, in 1607, Clavius published these tables under his name, without explaining how he carried out the calculations [10]! This again leaves no doubts as to the source of these tables.

The above two circumstances are quite strong to come to the conclusion that the Europeans surreptitiously used the Kerala texts, but there is more. At the end of the 16th century, the Danish astronomer Tycho Brahe came up with his ‘Tychonic model’ of planetary motion, wherein Mercury, Venus, Mars, Jupiter and Saturn revolve around the sun, but the sun is revolving around the earth. What is interesting to note here is that this is exactly the model proposed by Nilakantha in his Tantra Sangraha some 300 years earlier [11]! What a ‘coincidence’! Remember that Tycho Brahe in the capacity of the Royal astronomer of the Holy Roman Empire had easy access to all the Kerala texts sent by missionaries such as Ricci. He was also known to be extremely secretive and jealous about the astronomical observations and other documents in his possession [11]. The only explanation and conclusion is that Brahe was in possession of the work of the Kerala school of mathematics which he used to come up with his ‘Tychonic model’.

We must also mention that Jyeshthadeva’s Yuktibhasha gives a formula involving a passage to infinity to calculate the area under a parabola. The same formula was used by Fermat, Pascal, and Wallis [8]. Wallis is also given partial credit for the development of calculus. It is thus quite safe to conclude that the Kerala texts fell into the hands of these mathematicians, based on whose work Newton and Leibniz came up with the ideas of calculus. The possibility that Newton and Leibniz had direct access to these texts cannot be ruled out.

Finally the question may be raised as to why the church kept all this activity so secret. The answer is obvious: the church could not possibly carry out its noble mission of ‘civilizing pagan cultures’ and at the same time accept that these cultures had a much advanced scientific culture upon which it (the church) was so dependent! This only makes sense since it is difficult for a ‘superior’ race to cope with the fact that an ‘inferior’ race can have a civilization and culture much more advanced than theirs. This is the reason why the Aryan race theory was created by the European imperialists when the antiquity and culture of the Hindu civilization was discovered [12]. Moreover, in the case of the church, anyone who professed to be using ‘pagan’ sources of knowledge ran the certain risk of being a heretic and being burnt at the stake for ‘devil-worship’. This certainly was a good enough incentive for anyone to conceal the true sources of knowledge! In this context, it is instructive to read the following quote from [13]:

“There is nothing ‘natural’ or universal in hiding what one has learnt from others: the Arabs, for instance, did not mind learning from others, and they openly acknowledged it. This is another feature unique to the church: the idea that learning from others is something so shameful that, if it had to be done, the fact ought to be hidden. Therefore, though the church sought knowledge about the calendar, specifically from India, and profusely imported astronomical texts … this import of knowledge remained hidden.”

Since the modern world is fortunately not governed by ecclesiastical restrictions anymore, and since it is good scientific practice to give credit where it is due, it is time that we revise the standard story of calculus and honor and remember its original inventors from Bharat.

References:

[1] ‘On the Hindu quadrature of the circle, and the infinite series of the proportion of the circumference to the diameter exhibited in the four Sastras, the Tantra Sangraham, Yucti Bhasha, Carana Padhati, and Sadratnamala’, by C. M. Whish, published in the  Transactions of the Royal Asiatic Society of Great Britain and Ireland, Vol.  3, No. 3, pp. 509–523.

[2] Encyclopedia of the history of science, technology and medicine in non-western cultures (two volumes), ed. Helaine Selin, Springer.

[3] To understand the status of European navigation in the 16th century, look up Navigation, Maths and Astronomy: the Pagan Knowledge, by D. P. Agrawal ( http://www.indianscience.org/essays/15-%20E–Navigation%20&%20Math.pdf  ).

[4] In this context we note how the modern names for the trigonometric functions ‘sine’ and ‘cosine’ originated: “When Arabic writers translated his (Aryabhata’s) works from Sanskrit into Arabic, they referred it as jiba. However, in Arabic writings, vowels are omitted, and it was abbreviated as jb. Later writers substituted it with jaib, meaning “pocket” or “fold (in a garment)”. (In Arabic, jiba is a meaningless word.) Later in the 12th century, when Gherardo of Cremona translated these writings from Arabic into Latin, he replaced the Arabic jaib with its Latin counterpart, sinus, which means “cove” or “bay”; thence comes the English sine” (http://en.wikipedia.org/wiki/Aryabhata#Trigonometry).

[5] One of the best kept secrets of Western history is that Vasco da Gama and Columbus were no good navigators at all. It is commonly assumed that Vasco da Gama ‘discovered’ India- he did nothing of the sort. In fact he was safely escorted to India by an Indian merchant from Gujarat, named Kanha, from the African coast.

[6] See for example http://en.wikipedia.org/wiki/Madhava’s_sine_table

[7] C.K. Raju (2007). Cultural foundations of mathematics: The nature of mathematical proof and the transmission of calculus from India to Europe in the 16 thc. CE. History of Philosophy, Science and Culture in Indian Civilization. X Part 4. Delhi: Centre for Studies in Civilizations. pp. 114–123.

[8] D. F. Almeida and G. G. Joseph, Eurocentrism in the history of mathematics: the case of the Kerala school, Race and Class, Vol. 45(4): 45-59 (2004).

[9] Resulting in the so-called Gregorian calendar, which is the one used today.

[10] Christophori Clavii Bambergensis, Tabulae Sinuum, Tangentium et Secantium ad partes radij 10,000,000 (Ioannis Albini, 1607), as quoted in C. K. Raju, Teaching mathematics with a different philosophy, Part 2: Calculus without Limits, Science and Culture 77(7-8) (2011) pp. 280-285.

[11] C. K. Raju, Ending Academic Imperialism: a Beginning. Available online at http://www.ckraju.net/papers/Academic-imperialism-final.pdf

[12] An excellent account of the Aryan race theory is given in Breaking India, by Rajiv Malhotra and Aravindan Neelakandan.

[13] D. P. Agrawal, Navigation, Maths and Astronomy: the Pagan Knowledge. The article can be accessed at http://www.indianscience.org/essays/15-%20E–Navigation%20&%20Math.pdf

Teacher’s Day vs Guru Utsav

– Sandeep Singh, Business Consultant, Writer, Friend of Bharath Gyan

The syllable ‘gu’ means shadows
The syllable ‘ru’, means he who disperses them.
Because of the power to disperse darkness the guru is thus named.

Aaradvayatka Upanishad 14—18, verse 5

There is an unfortunate artificial darkness created around Teachers Day and Guru Utsav. I tried to understand the darkness and learned quit a bit about Teacher, Guru, Teachers Day, World Teachers Day, Dr Sarvepalli Radhakrishnan, and Guru Purnima. I am sharing the same.

Sanskrit has got four words Sikshak, Adhyapak, Acharya and Guru which are often used as synonym. Each of these words can be broadly understood as:

  • Teacher being equivalent to Sikshak
  • Professor being equivalent to Adhyapak
  • Principle being equivalent to Acharya
  • Guru doesn’t have an equivalent word in English. Hence it is used as Guru in all the languages.

Teacher gives the basic education.

Guru is a word much bigger than Teacher. As a noun the word means the imparter of knowledge. As an adjective, it means ‘heavy,’ or ‘weighty,’ in the sense of “heavy with spiritual wisdom”. Guru is one’s spiritual guide on earth. One is considered ‘orphan’ without a Guru.

In fact a Tamil saying describes the word “Guru” beautifully:
Guru illaakru vidhaiyum illai, mudhal illaarku labamum illai” i.e. This saying in Tamil means The Person who has no guru has no skill; just like a business without principle makes no profit.

Teachers’ day is celebrated in many countries but date varies from country to country.
World Teachers’ Day is distinct from Teachers’ days, and is officially celebrated across the world on October 5.

Sarvepalli Radhakrishnan’s birthday, is celebrated as Teachers Day on 5th September from 1962 in India. Sarvepalli Radhakrishnan was born in a Telugu family at a village near Thiruttani , in Tamil Nadu near the border of Andhra Pradesh. His thesis for the M.A. degree was “The Ethics of the Vedanta and its Metaphysical Presuppositions”. His philosophy professor, Dr. Alfred George Hogg commended that Radhakrishnan has done most excellent work. Radhakrishnan’s thesis was published when he was only 20.

According to Radhakrishnan himself, the criticism by Hogg and other Christian teachers of Indian culture “disturbed my faith and shook the traditional props on which I leaned.” Radhakrishnan himself describes how, as a student, “The challenge of Christian critics impelled me to make a study of Hinduism and find out what is living and what is dead in it. My pride as a Hindu, roused by the enterprise and eloquence of Swami Vivekananda, was deeply hurt by the treatment accorded to Hinduism in missionary institutions.” This led him to his critical study of Indian philosophy and religion, and a lifelong defence of Hinduism against “uninformed Western criticism”.

For his services to education he was knighted by George V in 1931. He stopped use of the title after India became independent. He preferred to use his academic title of ‘Doctor’. In 1939 Pt. Madan Mohan Malaviya invited him to succeed him as the Vice-Chancellor of Banaras Hindu University (BHU). He served as its Vice-Chancellor till January 1948. His political career started after BHU.

Radhakrishnan did not have a background in the Congress Party. His motivation lay in his pride of Hindu culture, and according to Brown, “He had always defended Hindu culture against uninformed Western criticism and had symbolized the pride of Indians in their own intellectual traditions.”

Gurupurnima falls on the day of, Purnima (full moon), in the month of Ashadh (June–July) of the Shaka Samvat (Gregorian calendar). Gurupurnima is as old as civilisation and is celebrated by all spiritual religions of India. Indian from all the fields, ranging from music to dance, academic to sports etc. celebrate this day by thanking their teachers as well as remembering past teachers and scholars.

Looking at above facts, it makes perfect sense to observe Radhakrishnan’s birthday as Guruutsav rather than as Teachers Day. Radhakrishnan was beyond “direction or language” division. Infact efforts should be made to celebrate his birthday as International Guruutsav Day. The world will be happy to accept it.

Paul Artur Schillp has said “….nor would it be possible to find a more excellent example of a living “bridge” between the East and the West than Professor Radhakrishnan.” While Michael Hawley said “Radhakrishnan’s concern for experience and his extensive knowledge of the Western philosophical and literary traditions has earned him the reputation of being a bridge-builder between India and the West.”

Last but not the least, Modern English, which includes the works of William Shakespeare and the King James Version of the Bible, is generally dated from about 1550. And only after the United Kingdom became a colonial power, English spread outside England. To top it more than 65% of English words are actually taken from other languages including from India. It will also be important to mention that the word Guru is more English than the modern day apologists of English and as old as English itself. The word Guru was first used in English in the year 1613.

Calculus was discovered in India

— Dr. M Lavanya
Physicist

Knowledge and Indian civilization

 Most of the amazing science and technology knowledge systems of the modern world are credited to have started around the time of the Renaissance movement in Europe in ~ the 15th century. These knowledge systems are generally traced back to roots in the civilization of Ancient Greece, and occasionally, that of Ancient Egypt. Hence, most of the heroes we are taught about in school and college are European, or Greek.

 As for India, or even China, it would appear that they have played a minimal role in this magical story. Hence, many (western) accounts of the “Ascent of Man” do not devote even a single line to India’s contributions.

 The trouble of course is that few of us know what exactly the Indian contributions are. This is due to the utter neglect of organized, extensive, detailed, and scholarly studies of these in modern India. Incidentally, this is in contrast to the attitude in almost any other country – people elsewhere have a keen interest and fierce pride and celebrate their own contributions to world knowledge and heritage. Several countries also make a living out of their past through tourism!

 However, there does exist, thanks in part to valiant individual efforts, some kind of a background awareness that the Indian civilization is in fact one of the most ancient and glorious, and that India has contributed enormously, perhaps even predominantly, to the growth of world civilization and knowledge in practically every field, ranging from the mundane and practical to the unworldly and spiritual.

 Some well-known early Indic contributions to Mathematics

 In the sciences, seminal contributions have in fact been made by Ancient India to mathematics, astronomy, chemistry, metallurgy, the list is long.

 Some Indian contributions to mathematics are well known (at least in India) : the zero, the decimal place value system and the commonly used numerals, the so-called “Indo-Arabic” numerals (called Arabic numerals in the West) were discovered in Ancient India. In fact, the importance of these is such that without these, mathematics (and science, commerce, etc.) as we know it would not have even existed!

 Further, few are aware that there has been a continuous unbroken tradition of mathematics in India from at least a thousand BCE (and perhaps even several thousand BCE) to ~ 200 years ago, and then again in the modern era.

  The discovery of the Kerala School of Mathematics

 A relatively recently discovered field is what goes by the name of the “Kerala School of Mathematics” which flourished in a tiny corner of present-day Kerala during ~ 1300-1600 CE.  Many details about the work of this school and the story of the mathematicians who contributed to it are only now being researched. This despite the fact that this work was brought to the attention of western scientists almost 200 years ago.  In 1834,  an Englishman named Charles  M. Whish  published an article  entitled “On the Hindu quadrature of the circle and the infinite series of the proportion of the circumference to the diameter exhibited in the four sastras, the Tantrasangraham, Yukti-Bhasha, Caruna-Padhati and Sadratnamala” in a journal called the ‘Transactions of the Royal Asiatic Society’ of Great Britain and Ireland. But the article was long ignored.

  What was the main contribution of the Kerala school?

 The Kerala school of mathematicians drew inspiration from much earlier texts, mainly Āryabhata’s Āryabhatiya (499 CE). The Āryabhatiya had in fact been a very influential text all over the country, and also, through its translations, in the Arab world and in Europe.

 The Kerala mathematicians, starting with Mādhava, developed some amazing mathematics – in particular, the branch of mathematics that is known today as Calculus, one of the foundation stones of modern science which developed from Europe.

 We have all been taught in school that Calculus was discovered by (Leibniz and) Newton. But Newton’s Magnum Opus, the Principia Mathematica, in which he discusses the Calculus essential for his Laws of Motion, was written around ~1700 CE. Thus, even orthodox historians and scientists now agree that the Kerala Calculus pre-dates that of Newton by at least a clear 200 years.

   A little more on some of the contributions of the Kerala school

 Calculus is the mathematical study of change, and its essence is the use of  infinitesimals / limits  (and, one of the passages to “limit” is by summing an infinite series).

 The concept of limit as given by Nīlakantha in Āryabhatiya-bhāsya :

“k+.TMa :pua:naH ta.a:va:de:va va:DRa:tea ta.a:va:dõ Ra:tea .ca ?”

How is it that [the sum of the series] increases only up to that [limiting value] and that certainly increases up to that [limiting value]?

  • Infinite series expansions for trigonometric functions (e.g., sine, cosine, arctan, ..) (now attributed to Newton), and finite series approximations to them.

  • Estimation of correction terms and their use in the generation of faster convergent series.

  • Extrapolations for sin Ө and cos Ө for nearby Ө’ values to the second and third order of (Ө- Ө’).

  • Binomial series expansion.

  • Taylor series expansion.

  • Infinite series expansion of  π (now known as the “Gregory – Leibniz series”).

  • Discussion of irrationality of  π.

  • Sum of natural numbers

  • Summation of series (Sankalita in Sanskrit) (i.e., Integration ).

  • Instantaneous velocity (of planets) and derivatives.

Besides arriving at the infinite series, that several forms of rapidly convergent series could be obtained is remarkable. Further, many equations that we use in Calculus which are attributed to western mathematicians were clearly known to the Indian mathematicians. They laid the foundations of Calculus, which is recognized as one of the foundations of modern science, and which has applications in many fields including engineering and economics.

These mathematicians also made important contributions to astronomy, but those will be the subject of a separate article. In fact,  much of this work seems to have arisen from an interest in predicting planetary positions, sunrise, sunset etc. to a very high accuracy for the  conduct of worldly affairs.

 Who were these people ? – some historical details

Most of these developments took place in temple-villages around a river called Nila in the ancient days (and currently called river Bharatha, the second longest river in Kerala) during ~ 1300-1600 CE. In fact, the area over which this work was carried out was so localized, that some scholars suggest that the school is more appropriately named the “Nila School of Mathematics”. One of the key villages was Sangamagrāma, which was possibly the present-day village of Irinhalakkuta (about 50 km to the south of Nila). (However, there are a few other possible candidates for Sangamagrāma , such as Kudalur and Tirunavaya). What is more certain was the existence of a remarkable lineage of mathematicians in and around Sangama-grama of which the pioneer,  Mādhava (~1340-1420)   seems to be the one who discovered many of the basic ideas of Calculus.

The Kerala school was a culmination of the school of Āryabhata and seems to have been the last bastion of mathematics in India till the modern era. The school seems to have died out soon after the arrival of the Portuguese in Kerala for obvious historical reasons.

The Lineage

  • Mādhava (c.1340–1420) of Sangamagrāma

 Pioneer of the Kerala School, discovered many of the basic ideas of Calculus.

 The only works of his which seem to be extant are Venvāroha and Sphutacandrāpati.

  • Parameśvara (c. 1380–1460) of Vatasseri

Mādhava’s disciple, great observer and prolific writer.

  • Nīlakantha Somayājī (c. 1444–1550) of Kundāgrama

Monumental works: Tantrasangraha and Āryabhatiya-bhāsya.

  • Jyesthadeva (c. 1530)

Author of the celebrated Ganita Yuktibhāsā (in Malayalam prose).

  • Śankara Vāriyar (c.1500–1560) of Tr.ikkutaveli

Author of two major commentaries.

  • Acyuta Pisārati (c. 1550–1621)

Disciple of Jyesthadeva,  a polymath

  • Pudumana Somayaji

Work : Karana Paddhati

  • Rājā Śankaravarman  (c.1830) of Kadattanadu

Work : Sadratnamala.

These (and other ancient) texts were written on (dried) palm leaves, which last for ~ 400 years. The language used was mostly Sanskrit and the mathematics was given in verse! in sutras.

Did Calculus travel from Kerala to Europe?

The big question now is: did the Europeans know of the Kerala Calculus? Circumstantial evidence indicates that they did, as many texts from Kerala were translated and transmitted to Europe during this period by the Jesuit priests who had learnt the local languages. Further, it is well known that there have been strong links through trade from times immemorial between Kerala and the West.

However, scholars suggest that more direct evidence is required that the knowledge of the Kerala mathematics was indeed transferred to the West. For instance, can we find translations of the Kerala texts, dating to around 1600 CE, from Sanskrit and Malayalam to English or any of the European languages? An extensive search needs to be carried out in both Kerala and European libraries. Unfortunately, some important libraries have been lost : in 1663, the Dutch  burned down the Jesuit library of Cochin which contained many volumes in local and European languages; and in 1775, almost all the archives and libraries in Lisbon, Portugal (including those which housed their colonial records), were destroyed by an earthquake.

Summary

As we have mentioned earlier, the essence of Calculus is the use of limits. We end this brief article with the following quotes, the first by Charles Seife in “Zero:The Biographyof a Dangerous Idea” (Viking, 2000; Rupa & Co. 2008):

 “The Greeks could not do this neat little mathematical trick. They didn’t have the concept of a limit because they didn’t believe in zero. The terms in the infinite series didn’t have a limit or a destination; they seemed to get smaller and smaller without any particular end in sight. As a result the Greeks couldn’t handle the infinite. They  pondered the concept of void but rejected zero as a number, and they toyed with the concept of infinite but refused to allow infinity – numbers that are infinitely small and infinitely large – anywhere near the realm of numbers. This is the biggest failure in the Greek Mathematics, and it is the only thing that kept them from discovering Calculus.

Unlike Greece, India never had the fear of the infinite or of the void. Indeed, it embraced them. Indian mathematicians did more than simply accept zero. They transformed it changing its role from mere placeholder to number. The reincarnation was what gave zero its power. The roots of Indian mathematics are hidden by time. Our numbers (the current system) evolved from the symbols that the Indians used; by rights they should be called Indian numerals rather than Arabic ones. Unlike the Greeks the Indians did not see the squares in the square numbers or the areas of rectangles when they multiplied two different values. Instead, they saw the interplay of numerals—numbers stripped of their geometric significance. This was the birth of what we now know of as algebra.”

And finally, a quote by the famous mathematician John von Neumann:

“The calculus was the first achievement of modern mathematics and it is difficult to overestimate its importance. I think it defines more unequivocally than anything else the inception of modern mathematics, and the system of mathematical analysis, which is its logical development, still constitutes the greatest technical advance in exact thinking.”

Further reading

Interested readers can find mathematical and historical  details in the following articles (and references therein):

1) K. V. Sarma, K. Ramasubramanian, M. D. Srinivas and M. S. Sriram, “Ganita-Yukti-Bhasha (Rationales in MathematicalAstronomy) of Jyeshthadeva”, Springer (2008).

2) K. Ramasubramanian and M. D. Srinivas, “Studies in the History of Indian Mathematics”  Ed. by C. S. Seshadri, Hindustan Book Agency, New Delhi, pgs. 201 – 286 (2010).

 3)T. Padmanabhan, “Dawn of Science : Calculus is  developed in Kerala”, Resonance pgs. 106 -115 (Feb 2012).

4) “Science and Technology in Ancient India”, Ed. Editorial Board, Vijnan Bharati, Mumbai (2006).