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Interview with George published in Kalakaumudi
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Professor George Gheverghese Joseph was interviewed by V.D. Selvaraj and published in Kalakaumudi, a Malayalam weekly dated February 26, 2012 (No. 1903). It was translated into English by Professor M. Dakshinamurthy.

The Secret of Srichakra Revealed

When we talk of discoveries and inventions, we always look up to the West. It is assumed that everything modern has come from the West. Whether it is the toothbrush, a new star in the sky, the pressure cooker, or the rabies vaccine, the story is the same. The printing press that has made it possible for you to read these lines has also come from the West.

But, what is the foundation of the scientific and technological progress and the consequent economical advancement achieved by the West? Was it purely a Western enterprise? Definitely not, says Professor George Gheverghese Joseph, Emeritus Professor at Manchester University in England.

Calculus, the foundation of modern mathematics, was born in Kerala in India. Calculus is a necessary tool for measuring time, making almanacs and for finding directions at sea. Did the Europeans learn some of the elements of calculus from Kerala? This is a million dollar question. It was the versatility of calculus that helped them to conquer the world. As a result, Europe became financially powerful. Did the Europeans import mathematical books along with spices from India? These are broad questions for which we may never find definitive answers.

When George Gheverghese Joseph says this, he is neither decrying the West nor praising the East to the skies. He is trying to excavate non-European mathematical traditions. This search eventually led him to medieval Kerala, the Kerala of 13th, 14th, 15th, 16th, and 17th centuries. The growth of mathematics in Kerala was truly remarkable. He goes on to describe that phenomenon.

One may wonder why Professor George Gheverghese Joseph refers only to mathematics. When he talks about modernity, didn't physics, chemistry and biology lead to modern discoveries and inventions and help scientific and technological advancement? Yes. But he implicitly accepts the Indian view that mathematics is at the head of all sciences.

Yathaasikhaa mayuuraanaam
Naagaanaam manayo yathaa
Tadvad vedaanga saastraanaam
Ganitam moorddhani sthitam.
(Like the crest of the peacock,
Like the jewel on the head of a snake
Mathematics is at the head of all knowledge)

These lines are from Vedaangajyotisham written around the fifth century before Christ and it provides the title of one of Professor George's book. Those who are not aware of this quotation would remember Tilakan's declaration in the film Sphatikam, "The world is a big zero without mathematics".

Two books written by Professor George as part of his research into the history of mathematics are famous. The first book is The Crest of the Peacock: Non European Roots of Mathematics. This book was first published by Penguin at London in 1991. Five editions of this book have come out, the latest extended third edition in 2011 being brought out by Princeton University Press. The book has been translated into six languages. In total about 30,000 copies of this book have been sold to date... The second book, A Passage to Infinity: Medieval Indian Mathematics from Kerala and its Impact, was published in 2009. This too is now receiving a lot of attention. Professor George is engaged in another great research project. He is trying to trace the path through which medieval Kerala mathematics may have reached Europe, which is part of a wider project trying to trace the shadowy figures behind Newton and Leibniz, usually seen as the founders of calculus. He hopes that this project will involve scholars from Oxford University (England), McMaster University (Canada), University of Singapore apart from his own University of Manchester (England) and Indian scholars.

Professor George talked about his search into the history of mathematics and his conclusions. His words resounded with the heat and excitement of the intellectual life of Keralam.

Let us first acquaint ourselves with Professor George. He was born into a family famous for patriotism, journalism and teaching. The great journalist Pothan Joseph and his elder brother Barrister George Joseph belonged to the Oorayil family of Chengannur.

Barrister George Joseph was the father of Professor George's mother. He got his degree in Philosophy from the University of Edinburgh and then qualified as a Barrister (Middle Temple) from London in 1908. He was the editor of Gandhiji's Young India and Motilal Nehru's Independent. A nationalist and a close associate of Mahatma Gandhi, the Nehrus, Rajagopalacharia, Vallabhbhai Patel, and he was in the forefront of Vaikom Satyagraha. When Gandhiji said that non-Hindus should not participate in the agitation, Barrister Joseph argued publicly with Gandhiji and left the scene. His sister Sarah Abraham became the principal of the famous Sophia College in Mumbai. P. M. Joseph, another brother of the barrister, was awarded Padmashri for his contributions to the field of physical education.

Barrister George Joseph's daughter Sarah Joseph was Professor George Gheverghese Joseph's mother. She was one of the first Asian principals appointed to a secondary school in Kenya. Professor George's father Adangapuram Gheverghese Joseph Panikkar came from an old and well-established family in Kallooppara. After a short stint in the British Army, he became a secondary school teacher in Kenya. Though Professor George was born in his mother's house at Chengannur, he was brought up and educated outside Kerala. He had his primary education at Madurai. He then went to Kenya and later to England. Professor George says: "I was born in my grandfather Barrister Joseph's ancestral house at Chengannur. Apart from that I have had little connection with Chengannur.

I lived in Madurai till the age of nine. My mother was a teacher at St. Joseph's Convent School, Madurai. My grandfather Barrister George Joseph practiced at Madurai court for a long time after his return from England. He too had left Chengannur during his youth.

"Madurai Meenakshi Temple is part of the treasure trove of my memories. I used to go the temple with my friends and play there. There was an image of Ganapati, which had become smooth over time because everybody used to stroke its stomach respectfully". He remembers listening to memorable musical concerts at the temple. He remembers Chitti Babu's concerts even now. A number of famous musicians used to give concerts at the temple so that they were recipient of Meenakshi's blessings and become better musicians. There was a big tank inside the temple. Professor George used to sit on the steps of the tank with his friends. He and his friends used to wander inside the temple. The temple was so near his house that he could go to the temple by himself unescorted. Later his mother would come there to take him home.

He continues: "My house was near a street full of flower shops leading to the temple. The street filled with the sweet smell of flowers was always clean. There were many astrologers with parrots in cages in front of the temple. When a cage is opened, the parrot would come out and pick up a card. The astrologer would look at the picture in the card and make some calculations. It seemed to a young boy complicated calculations. However, it was those calculations that introduced me to the mystery of mathematics. I became friendly with a couple of these astrologers and they allowed me to sit with them while they performed. I forget what I learnt from them. But I shall never forget the parrots and the cards.

"Though we laugh at parrot-astrologers today, they were part of our rich mathematical heritage. In olden days mathematics and astronomy and astrology were inextricably linked. But that tradition has been lost.

"Just as I am interested in mathematics, I am interested in classical music as well. It was the temple that made me love music. I listened to the veena recitals at the temple. I can now appreciate Western music because I have lived in the West for so long. But what awakens my soul is our classical music and dance. Later I recognized that our music was based on mathematics. No other music or dance has so much mathematical underpinnings. The steps of Bharatanatyam are based on mathematical shapes and structures. There is a mathematical rhythm and pattern in the movements of the dancers. The well-known dancer Shobana Jeyasingh approached me some years ago. She wanted to present a dance opera based on the life and work of Srinivasa Ramanujan. She wanted my help to choreograph the mock-theta function, one of Ramanujan's famous discoveries during the last year of his life. Now, 'mock -theta function' is a difficult topic in pure mathematics. I found it a most rewarding experience. This year we are celebrating Ramanujan's 125th birth anniversary. Mahavira of the seventh century, who wrote the famous book Ganita Sarasamgraha, stated that Mathematics has its role in the art of love, in music, and in many other fields of endeavours'.

"I went to school at the age of four. It was the school where my mother taught. Though it was a school for girls, a few boys including myself were admitted. I was there only for four years. I went to Kenya in 1948, when I was in my ninth year.

"Even after going abroad I have visited Madurai at least four or five times. The gopurams of the temple attract me even today. Its vastness, height, colour, decorations… everything attracts me. I wonder whether the temple is as clean today as it was those days.

"The reason for our migration to Kenya was my father's job. He was in the British Army. When the Second World War ended, Britain rewarded the soldiers by providing them with money or land and allowed them to settle in Kenya which was then a British colony.

"When we joined my father he had found a teaching job in a coastal town called Mombasa. I went to school there. When I completed plus two, I got a scholarship from the Government of Kenya. It enabled me to do my degree course in Britain.

"I was seventeen when I left my parents and went to England. I joined Leicester University for a three year degree course with mathematics as the main subject. When I completed my degree course, I was offered a studentship to continue my education at Leicester. But I decided to return to Kenya because I could only continue my postgraduate education if the Kenyan Government gave me permission to do so. If I did not return to Kenya, my father would be responsible for paying back the scholarship amount.

"On my return, I was treated as if I was an Englishman with an honours degree and paid a good salary with a number of privileges. I worked as Education Officer in Kenya for six years. "It was the beginning of the sixties. By the middle of the sixties Kenya had become independent. Though I was not born in Kenya, I had spent my childhood, adolescence and youth there. I had African friends. Their mother tongue Swahili was virtually my mother-tongue too. Even today I feel more at home with Swahili and English than Malayalam.

"When Africanisation movement grew stronger, I wondered whether I should continue living in Kenya. I returned to England in 1967 and joined Manchester University for higher education. After I got my M.Sc. degree, I was appointed on the staff of the same department. There I did my Ph. D. I have spent more than forty years at Manchester University. Though I retired in 2000, I have continued to be associated with the University and taken the opportunity to spend short periods at other Universities in Papua Guinea, Canada, New Zealand and India. I have continued to give lectures and undertake research in India, having been granted Royal Society Visiting Fellowships twice and having been awarded research grants by a number of bodies in the United Kingdom. More recently, I have become interested in philosophy since mathematics cannot be separated from philosophy as the ancient Greeks discovered.

"When I think of philosophy, Indian philosophy comes first to mind. The Indian proof tradition is independent and quite different from that of Europe. The European proof tradition is derived from ancient Greeks. In fact, as some one once said, Western philosophy generally since the Greeks is little more than footnotes on the works of Plato and Aristotle. The Indian proof tradition owes little to other sources. The thought process that lies behind different ways of knowing determines the proof traditions.

"My interest in Indian philosophy led me to ancient Indian works, including Indian mathematical and astronomical works. Till then I had no idea that the land of my birth, the southernmost part of India, had contributed greatly to the growth of modern mathematics. The mathematicians who are now called the Kerala School of Mathematics lived in Kerala between A.D. 1300 and 1700."

Would you please explain the origin and development of the Kerala School of Mathematics?

Professor George Gheverghese Joseph: "The mathematical tradition that grew and flourished in Keralam during the 300 years from the 14th to the 17th centuries is now known as the Kerala School of Mathematics.

"The mathematician who laid the foundation for this great age was Madhavan (1340-1425) of Sangamagramam. He was followed by Parameswaran, Damodaran, Jyeshtadevan, Achyuta Pisharodi, Neelakantan, Chitrabhanu, Narayanan, Sankara Warriar and Putumana Somayaji. Through them the school grew to great heights. They were all mathematicians as well as astronomers. During those days mathematics and astronomy went hand in hand.

"Madhavan was a member of an Embraantiri family of Sangamagramam in Central Keralam. The name of the village comes from the name of the temple in the village, Sangameswara Temple. It is believed that Sangamagramam is the present day Iringaalakkuda in Thrissur district. Madhavan's known main works are Venvaroham, Chandravakyapaddhati and Muhorttaratnam. There were equations for determining the position of the moon in Vararuchi's Chandravaakyapadhati. Madhavan modified them and made them more exact. Vararuchi, who lived in A. D. fourth century, is considered to be the founder of Kerala astronomy (and astrology).

"Madhavan's disciple Vadasseri Parameswaran (1360-1460) was born in Vadasseri Mana (mana=house), famous for astronomy. Vadasseri Mana is in Aalattiyoor village in Ponnani Taluk in Malappuram District. Goladeepika and Drkganitam are his important mathematical works. Parameswaran argues that the laws of science should be verified through observation. He refers to the calculation of the earth's perimeter and the revolution of planets in his Drkganitam. Being interested in astrology, he has written a commentary on Madhavan's Muhorrtaratnam. "Neelakantan (1443-1560), the third important name in the Kerala School, has prophesied the movements of planets and the method of finding their centre. He lived more than a century before Johannas Kepler (1571-1630), and has given a formula to find the centre of the planets Mercury and Venus in his book Tantrasangraham. Neelakantan's formula is more accurate than that of Aryabhata.

"Neelakantan, who was born in the Somayaji family of Trikkantiyor desam in Ponnani Taaluk and was reputed to have lived over hundred years. He was famous throughout South India. His work Sundararaja Prasnottaram is a proof of his fame. This work is a record of the discussion that took place between Neelakantan and the famous Tamil astronomer Sundararaajan. Sundararaajan in his Vaakyakaranam calls Neelakantan shad darsanapaarangatan (Great Scholar and Master of Six systems of Philosophy).

"Chitrabhaanu Nampootiri (1475-1559), who was born at Chovvara in Thrissur District, was Neelakantan's disciple. He wrote Karanamritam in 1530 and Ekavimsati Prasnottaram a few weeks later. Both these are important works of medieval astronomy. Though Karanamritam is a commentary of Parameswaran's Drkganitam, it gives all the details required for making an almanac. Chitrabhaanu's disciple Naryanan (1500-1575) belonged to Trikkaattiri family of Ottappaalam. Narayanan's Kriyakramakari, a commentary on Bhaskaracharya's Leelavati, is a work that deserves special mention. As a matter of fact, it was Sankara Wariyar (1500-1560) who began the work Kriyakramakari. Sankara Wariya lived under the patronage of Aazhwancheri Tamprakkal. As directed by the Tampraakkal, Sankara Wariyar wrote a commentary on Neelakantan's Tantrasangraham too. It is called Laghuvivritti. It is supposed to have been written in 1556. Sankara Wariyar had earlier written two commentaries on Tantrasangraham, called Yuktidepika and Kriyakalpam.

"Jyeshtadevan (1500-1610), a member of Aalattiyor Parakottu family, and a contemporary of Sankara Wariyar, wrote Yuktibhasa, the most important work of the Kerala School of Mathematics. It is written in old Malayalam. In this book he explains and comments on all the principles and formulae used by mathematicians, astronomers and astrologers of the day. Jyeshtadevan's book was based on Neelakantan's Tantrasangraham.

"Achyuta Pishaarodi (1550-1621) was born at Trikkantiyoor in Ponnaani Taaluk. He was well versed in astronomy, medicine and literature. He was Jyeshtadevan's disciple. He begins his work Uparagakriyaakarmam, written in 1592, by remembering his guru. He observes the planets in detail in his work Sphutanirnayam. Achyuta Pisharodi made these observations at the same time when Tycho Brahe (1501-1546) introduced his formula 'Reduction to the Ecliptic'.

"Karanapaddhati is a notable mathematical work produced in Keralam in the seventeenth century. It was written by Somayaaji of Putumana Illam of Chovvara in Thrissur District. Charles Whish, an Englishman, who first brought to the notice of the West the importance of the Kerala School of Mathematics, refers to Somayaaji's work. Moreover he says that when he wrote the article in 1830, Somayaaji's son was seventy.

"The known last link in the Kerala School of Mathematics was Sankara Varma. He wrote a mathematical work called Sadratnamala in 1823".

What made you go in search of these mathematicians?

"It all began with a footnote in an article by Professor Whiteside of Cambridge University. There is a reference to the work of the Kerala School of Mathematicians in that footnote. Whiteside was an authority on Newton's works. He had brought out the collected works of Newton. When I met Whiteside, I referred to that article. He asked me: 'You are from that region, aren't you? The southernmost part of India… I forget the name of the place. It's there in the footnote. Did you read it?'

"That was the inspiration. My search for the mathematical tradition of Keralam began with that footnote. In that footnote Whiteside referred to an article on Kerala School of Mathematics by C.T. Rajagopal. When I read the article by Rajagopal, I came to know that it was Charles Whish, an Englishman, who first wrote about the Kerala School of Mathematics.

"It was in the 1830's that Charles Whish wrote about the mathematical tradition of Keralam. He says that Neelakantan's Tantrasangraham, Jyeshtadevan's Yuktibhaasha, Putumana Somayaaji's Karanapaddhati and Sankara Varma's Sadratnamala were the earliest works on calculus. But nobody seems to have taken note of it. Charles Whish was Assistant Collector at Koodallur in Tamil Nadu. He knew Malayalam and Sanskrit. He was genuinely interested in mathematics. It was a century after the publication of his article that Rajagopal and his collaborators wrote about the Kerala School. Till then nobody took notice of that article. Thus the Kerala School remained unknown to the world. Usually we take note of a thing only when Westerners point it out. Unfortunately, though Charles Whish wrote about the Kerala work in the 1830's, we did not take note of it.

"C T Rajagopal, was a mathematics professor at Madras University. Like Rajagopal, K.V. Sarma, a Sanskrit scholar, had also studied the mathematical tradition of Keralam in depth. He knew old Malayalam well, and it was he who brought out the works of Neelakantan.

"Thus I came to know that Indians had contributed greatly to the growth of mathematics even after Bhaskaracharya . I decided to study it in depth.

"Even though I say this confidently, I must confess my trepidations in writing a book. I had only a basic knowledge of Sanskrit. Most of the old works are in Sanskrit. It was K. V. Sarma who helped me to overcome this diffidence. He gave me his English translation of Jyeshtadevan's Yuktibhasa. The original work is in old Malayalam. I do not know how it happened to be so. Those days books were mostly written in Sanskrit. Yuktibhasa is one of the most valuable books introducing us to Kerala mathematics.

You have been talking about the prominent mathematicians who were part of the Kerala School of Mathematics which lasted three centuries. Are there not others who are still unknown? Are there not other works that remain unknown even today?

"Only a few of the available palm leaf manuscripts have been studied. There are literary works as well as mathematical works that have not seen the light of day because they lie bundled up in various archives and libraries. For example, a copy of Yuktibhasa was obtained from Baroda Archives. "K V Sarma says that palm leaf manuscripts of 3473 works in Sanskrit by Kerala mathematicians and astrologers have been found. Moreover 12,244 copies of these texts are kept in various collections in Keralam and Tamil Nadu.

"How many of these have been classified? How many books have been subjected to careful study? Nobody knows. What treasure may lie hidden in such a vast ocean!"

In what way is the mathematical tradition of Keralam related to that of India?

"The classical age of Indian mathematics and astronomy extends from AD 450 to 1150. Aryabhata was the first and foremost of the mathematicians of this age. Aryabhata, who was born in A.D. 476 wrote Aryabhateyam in A.D. 499 when he was twenty three. It is believed that he was born at Kusumapuram near Patna in Bihar. It may or may not be true. But it is certain that he wrote Aryabhateyam at Kusumapuram. Besides this Aryabhata, there were probably two other Aryabhatas. Aryabhata II worte Maha Aryasiddhantam in A.D. 950. Al Biruni in his book History written in 1036 states that there was another Aryabhata before the Aryabhata of Kusumapuram. But today there is one other Aryabhata, and that is Aryabhata II, who wrote a commentary on Aryabhateyam called Maha Aaryasiddhantam.

"But there is a dispute regarding Aryabhata's birthplace. Some say that he was born at Asmaki, and not at Kusumapuram. Asmaki is referred to in the commentary on Aryabhateeyam written by Neelakantan in A.D. 1500. It is said that Asmaki is in southern Keralam, But there is no evidence to prove this. Whatever it may be, Aryabhata spent many years at Kusumapuram after his education at Nalanda: Kusumapuram and Ujjain were important centres of mathematics in ancient India. Those days Pataliputra was the gateway to knowledge in India. Knowledge from abroad came into India through Patilaputra , and knowledge from India reached foreign countries through Patilaputra. Patilaputra was the capital of the Gupta Empire. Aryabhata's discoveries reached Patilaputra from Kusumapuram and from there they reached Arabia and Europe.

"It was in Keralam that most of the commentaries on Aryabhateyam were written.

"By discovering a method to determine the value of pi, Aryabhata triggered off the solution of many problems in mathematics. The value of pi was found to be 3.1416 in the fifth century A.D. Later the value of pi was determined to be up to fourteen decimal places.

"The Sulbasutras gave a method of constructing a square that is equal in area to a circle. Of course this is not possible in an exact sense. However, the method implies a value of pi is given as 3.088. The earliest of the Sulbasutras was written in 800 B.C., 1300 years before Aryabhata l"

What is the greatest contribution to mathematics made by Keralam?

"The work on infinite series is the greatest contribution of Keralam to mathematics. From this contribution calculus developed. It is generally believed that Newton and Leibniz discovered calculus in the seventeenth century. The truth is that one of the essential strands of calculus had been developed in Keralam 250 years before that. The infinite series relating to sine, cosine, and arc-tan (from which the series for pi) were all derived from the Kerala tradition of mathematics.

"There is some circumstantial evidence to prove that Europe got the idea of calculus from Keralam. The method of calculus involves differentiation and integration. Differentiation makes it possible to measure a thing by dividing it into small bits. For example, if we want to find the area of a circle, we should first divide the circle into hundreds of small triangles by drawing hundreds of radii from the centre to the circumference. Then the circle consists of hundreds of extremely small triangles. If we find the area of one triangle, we can multiply it by the number of triangles and thus get the area of the circle. This is integration, putting small things together to form a whole. This is a simplistic way to describe differentiation and integration. The early foundation of calculus has been attributed to Madhavan by his disciples.

"After the publication of the Crest of the Peacock, the Newton sine and cosine series were renamed Madhava-Newton Series. And similarly the Gregory Series became the Madhava-Gregory Series. What this indicates is that there is a gradual recognition of the contribution of Kerala to the development of mathematics. I am proud of the fact that I have played a small part in changing this perception. "When we talk of scientific revolution, we begin with Copernicus. Scientific revolution was not the monopoly of Europe. India, China and Persia had contributed greatly to the scientific revolution. Arun Bala's book Dialogue Among Civilizations deals with this. He says that modern science has grown out of the interaction between various civilizations. This is a new idea. I have pointed this out to indicate that the history of modern science requires major corrections."

What was the context that made it necessary for Europe to depend on Keralam in the field of modern mathematics?

"There were many mistakes in the old European calendar (Julian calendar) of the sixteenth century. The date of Easter could not be determined accurately. If one went according to the time indicated by the calendar, there could be a mistake of three degrees in the direction taken by the ship. A difference of three degrees at sea may lead the ship to another continent. European countries offered huge rewards to those who could rectify the mistakes in the calendar. Spain was the first country to announce a reward in 1567. The prize money was increased in 1598. The Dutch announced a reward in 1636 and the French in 1666. The British Emperor announced a similar reward in 1711. Europeans wanted to reach India without depending on Persians. It was their social need. So they were determined to find a way to reach India by sea at any cost.

"A committee was formed in Europe in 1582 to revise the Julian calendar. The committee contained the famous Jesuit mathematician Christopher Clavius. Three of his students, Mateo Ricci, Johan Schreck and Antonio Rubino, who were Jesuit priests, reached Keralam and immersed themselves in mathematical studies".

How did Kerala mathematics reach Europe through these Jesuit priests, who were disciples of Clavius?

"Within seven years after the death of St. Francis, the Portuguese sent home translations of all the eighteen puranas. A record in the Jesuit Library says that a Brahmin translated all the works of Vyaasa in eight years for the Jesuits. These translations are now kept in the Roman Archives of the Society of Jesus.

"See what the Jesuit historian Henry Bernard says of the arrival of Clavius's disciple Matteo Ricci in Kerala. Matteo Ricci was either in Goa or in Kochi from September 3, 1578 to April 15, 1582. He spent most of this time in Kochi. Kochi was in the forefront of Kerala mathematics at that time. Some members of the Kochi royal family were well versed in mathematics and astronomy. Rubino, another Jesuit mathematician-astronomer, has written that he was learning from a Brahmin the method of determining time accurately. Rubino stated in his letter written in 1610 that the European method of determining time was old-fashioned, and that Kerala mathematicians were far ahead in this field. He stated that Kerala mathematicians could forecast accurately when an eclipse would begin and when it would end. Their forecasts were accurate to the minute. But Rubino could not learn the method because the Brahmins who knew this method would teach it only to a few of their compatriots.

"Similarly, Schreck, in his letter written in 1618, said that he was sending astronomical observations from India, as required by Johannas Kepler. Kepler said that the astronomical observations required for proving his theories were available in India. He requested the famous Jesuit mathematician Paul Gird to send the required astronomical data. It was in response to this request that Schreck sent the astronomical observations.

"Possible transmissions of Kerala mathematics and astronomy to Europe needs further investigation. A colleague and I started this investigation in a research project some years ago. The findings are contained in my book, A Passage to Infinity. However, we did not find any direct evidence of such a transmission through the Jesuits. The circumstantial evidence still remains strong."

How did Vasco da Gama reach Kozhikode without knowing the way? How did Calculus benefit Europe?

"Do you know how Vasco da Gama reached India? He rounded the Cape of Good Hope and reached Malindi in East Africa. He did not know how to proceed further. He met a Gujarati merchant at Malindi. The Gujarati was also a navigator. It was with his help that Gama reached Kozhikode.

"Europeans had little experience of sailing in the open sea at that time. They always sailed close to the shore. It was a sense of adventure that made them sail to nearby places in Europe. And it was a combined sense of adventure and a search for new routes to spices and other luxuries from the East that led Vasco da Gama to set out on his long journey from Portugal to India hugging the coast of Africa for part of the journey. And in that manner Vasco da Gama reached Malindi in East Africa on his way to Calicut (Kozhikode) through the Indian Ocean? With hindsight we can see how knowledge of calculus could have helped navigation. Today nobody remembers the Gujarati who showed Vasco da Gama and his party the way. "Navigation was very crude in the days of Gama. But in Europe there was this spirit of enterprise and adventure which drove their desire to discover new lands and outlets. They sought new ways of navigation. Navigation requires an extremely accurate calendar. A calendar is absolutely necessary to find the place, time and direction."

"Europe may have got the calculus from Keralam. It was through calculus that Europe's stellar navigation technology developed. It is by using calculus that we can find the direction and time during a voyage. From the stars we can find the angle of elevation. We can find the direction accurately from this angle by using sine angle calculations of calculus.


What is the role of India in inventing the digits zero to nine?

Professor George: "The only thing the whole world has accepted across linguistic boundaries the system of ten digits and their symbols (0 1 2 3 4 5 6 7 8 9). The fact that any number can be represented by these ten symbols is not a source of wonder today. But these digits invented in India, have become the magnificient edifice on which modern mathematics rests. Other systems of numeration have been developed. But only the Indian numerals constitute a place value system with zero.

"Indians were fascinated by large numbers. As early as 800 BC they gave 1062 a name. (1062 is 1 followed by sixtytwo zeroes). The names of lesser powers of ten are interesting. Satam=100, sahasram=1000, ayutam=10000, niyutam=100000, prayutam=106, arbudam=107(ten crore), nyarbudam=108 (ten thousand crore), samudram=109, maddhyam=1010, anta (1011) paraardham=1012……". 60,000 is read as shashtim sahasri in Sanskrit.

“The power of the Indian number system is based on the concept of zero. The concept of soonyam in Sanskrit became zero in Indian mathematics. Buddha’s soonyaavastha in which the mind is free from all emotion and thought is a good example of a philosophical dimension to zero. In a seventh century commentary on Patanjali’s Yogasootram, zero is given a special place. ‘It is the same digit 1 that is used to represent hundred in the place of hundred, and ten in the place of ten and one in the place of one just as woman is seen as mother, sister and daughter.’

“The oldest inscription using the Indian place number system is found in a copper plate at Barooch in Gujarat which can be dated to AD 594.

“The Indian numerals traveled to Persia and the Islamic world and Indian mathematics reached Europe through Spain and Sicily from Persia. Both Spain and Sicily were under Muslim rule during that time.”

What is the importance of the number zero?

"When I talk of zero, I think of a multifaceted object. Zero acts as a direction separator. When we think of the space above sea level and that below sea level, zero is the boundary. Zero is in the middle. The measure above zero is height. The measure below zero is depth.

"When you write the number 101, the zero signifies that there are no 'tens'. Zero is known as a place holder. We express 101 verbally as one hundred, zero ten, and one one. Zero as a place-holder existed both in the ancient Babylonian civilization and Chinese civilization. Zero is a digit in India. We invented zero as a number. That is the seminal contribution of India to mathematics. We can now travel from zero to infinity.

There are Indian mathematical concepts in our homakundam, talisman, etc. What is the relationship between the famous Srichakram and mathematics?

"Srichakram has an unusual mathematical artefact. Analogies are sometimes drawn between the triangles in Srichakram and great pyramids of Egypt. Inside the circle of Srichakram there are nine interwoven isosceles triangles cutting one another. There are four triangles that point upward, and five that point downward. All the three vertices of the two largest triangles touch the circle. The base angle of the each of these triangles is 51.5o. It is intriguing that the slope of the face to the base (or the angle of inclination) of the Great Egyptian pyramid is 51o 50' 35". "The mathematical interest of the Srichakram lies in its construction. 43 small triangles are formed by the combination of the nine triangles. Look at the smallest triangle in the middle. There is a small circle inside it. It is called bindu and is supposed to represent both zero and infinity.

"The four triangles that point upward are believed to represent Sakti, the primordial female essence of dynamic energy. The root of the energy of the movement of the universe is feminine essence. The five triangles that point downward represent Siva. Siva is the male essence of primal consciousness. Siva and Sakti are united here. Sankaracharya says that Siva cannot even move without Sakti.

"The small triangles in Srichakram are the seats of various gods. Some Sreeyantrams specify the names of these gods and goddesses. According to Tantrikavidya (mystical theology) Sreeyantram can be used for meditation in two ways. When Srichakram is carved on a sheet of gold or silver and consecrated, it becomes Sreeyantram. The first method of meditation is to concentrate on the bindu in the triangle at the centre and then to move outward. The second method is to concentrate on the outer circle and then to move inward.

"The first method of meditation is based on the philosophy of evolution of the universe. When we start from the bindu and move outward through the triangles and circles, we reach the outer square, which is the boundary of the universe. When we go beyond the four gates of the square, our meditation reaches anakaara pindam (the formless mass).

"In the second method where we move from the circles to the bindu, our attention is turned inwards. Taantriks (those who follow the path of tantrika vidya) call this method of meditation Samharatantram.

"Kalams and chakrams are essential part of Indian scientific tradition. Boudhayana who lived around 800 B.C. is believed to have laid the mathematical foundation of kalams and chakrams. His Sulbasutra gives details of the kalams required for homam. He describes how a square kalam can be changed into a circular kalam of the same area. And that constitutes the beginnings of geometry and mathematics in ancient India. That is another story which I have elaborated on in my book, The Crest of the Peacock: Non-European Roots of Mathematics. It is a source of pride that we are the inheritors of a mathematical tradition which is at least three thousand years old.

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