Madan Mohan Malviya

Madan Mohan Malviya is one of those notable leaders that this country has seen. His role in the Indian Freedom struggle and his contributions towards education can scarcely be missed as his legacy stands tall even today after 71 years of his leaving the mortal coil.

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Madan Mohan Malviya

Great Educationist

The Banaras University, the largest residential university in Asia is one of those great legacies that Malviya left behind in his role as an educationist. The university now provides higher education to more than 12000 students across various streams like science, arts and technology.

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Banaras Hindu University

Freedom Fighter

As the president of the Indian National Congress for four times, Malviya played a vital role in the freedom struggle. He was an important in the non-cooperation movement of Mahatma Gandhi.

On Par with Gandhi

Malviya is perhaps the only freedom fighter who has been compared with ‘the Mahatma’. Mrs Sarojini Naidu described Malviya’s courtesy as being far greater and sweeter than ‘the Mahatma’.

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Sarojini Naidu

Mahatma Gandhi himself lauded Malviya’s efforts in saving innocent lives of Indians after the Jallianwala Bagh tragedy.

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Madan Mohan Malviya and Mahatma Gandhi

A committee was formed under the presidency of Malviya in 1919, soon after the tragedy to build a memorial for the martyrs who died in the attack.

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Jallainwala Bagh Memorial

Prince among Beggars

Mahatma Gandhi called Malviya, the ‘Prince among Beggars’ for his capacity to repeatedly collect funds as huge as 1 crore rupees for public cause. The Banaras University was formed as result of Malviya’s ability to collect funds.

Other Initiatives and Roles

Malviya was also one of the founders of the Indian scouts, and also founded the newspaper, ‘The Leader’ published in the year 1909. He also served as the chairman of Hindustan Times newspaper from 1924 to 1946.

Great Orator

Malviya was also a great orator which earned him the title “Silver tongued orator”, due to his great command in English.

Mahamana’

For the varied roles in many fields, he was conferred the title, “Mahamana’, by the people, meaning ‘the Great One’, towards the end of his life, as the whole nation recognized the contributions of this great freedom fighter, politician, educationist and orator.

Madan Mohan Malviya passed away on November 12th, 1946 at Varanasi.

He was bestowed with the Bharat Ratna in 2015.

Swami Vivekananda’s Yatra to Kanyakumari

Swami Vivekananda travelled mostly on foot all over India from the Himalayan peaks, through the land, to Kanyakumari, the southeren tip of Indian peninsula, reaching there on 24th December, 1892.

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Swami Vivekananda

Wandering Monk

It is for this ceaseless travel that Swami Vivekananda has been respectfully referred to as the ‘Wandering Monk’. He swam across the waters to a large rock, just off the tip of the land, sat there in meditation for 3 days, 24th, 25th and 26th, realized the reasons as to why his mother land, Bharatha Desha which had its days of glory, fallen in his times.

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Swami Vivekananda, the Wandering Monk

The answer that came to him in meditative state, after his physically observing the length and breadth of the land during his travels, are poignant indeed. Some of the thoughts that came to him which he has expressed in his speeches and works can be assimilated here for our understanding.

Bhogya – Self-Indulgence

This land has Bhog, food prosperity, whereas people had become Bogyam, interested only themselves, their immediate fame and their self-aggrandizement, not giving a thought to the consequences of what it would lead to and who was feeding them this bhogya, and for what purpose.

If somebody is offering you bhogya, food and wealth and other minor worldly pleasures, they are doing so to please you, for, they expect from you to do something to you in return, what you normally would not do. It is selling of one self’s momentary bodily pleasures, pleasures of flesh.

These thoughts that passed through his mind, was the clarity that occurred to Swami Vivekananda.

Halasyam – Laziness

This was leading to apparent stupidity among people because of their craving. This in turn led to laziness, halasyam. The key people who had the role to be vibrant were put into a stupor.

Nirbhayatvam – To be fearless

To do any work successfully against the existing odds, one has to be fearless. This land and its people had the kshatriya bhava, to do things fearlessly. The word kshatriya does not denote only caste. It denotes a state of mind which exhibits fearlessness, a sense of capacity, capability, in trying to achieve ones goal that one has set out to achieve. What Swami Vivekananda saw was that, this innate quality of nirbhayatvam had gone missing from the masses.

In its place, there was a mentality of paying obeisance to the new colonial masters. This nirbhayatvam or kshatriya bhavam is not only to be exhibited in the battlefield, but is that which must exist in our attitude, for, every day is a battle. The land as a whole had slid into divisions.

Slide of Indian civilization

While, the division of labour and specialization of family skills and helped in the productivity of land through millennia, its stratification and hardening that happened from time to time in the mindset, had created vicissitudes in the very body of the population. This had happened like never before. This aided the divide and rule policy that the colonial masters were quick to seize up for their benefit. These were the 3 key point that Swami felt were the reasons for slide of Indian civilization from its hoary heights. These were the stark point that came to his minds when he swam back on 27th, from the rock in the sea to the mainland, where the Kanyakumari temple stands at the land’s end.

Meditating on Bharath Mata

Through times, different sanyasis had meditated on different divinities for boons and or for wellbeing of society. In contrast, Swami Vivekananda meditated on a rock in the sea of Kanyakumari at the foot of the land mass of Bharatha and meditated on the divinity of Bharat Mata, trying to find out the reasons for our slide, how the civilization can once again be rejuvenated for the single-minded effort of his. Swami Vivekananda is commonly referred to as Patriotic Saint. This is not to say that the other saints are not patriotic, but is to empathize that, Swami Vivekananda’s call of patriotism, an analysis stood out in the hour at the time of dire needs.

After the successful visit of Swami Vivekananda to USA, in particular to the Chicago address at the world religions conference on 11th September in 1893, Swami Vivekananda further analyzed the reasons for the decline of the Indian civilization.

In comparison to the American civilization doing well then, India lacked of organizing ability, to perform in a united way.

In stark contrast, he could see this organizing ability and standing united among Americans during his nearly 4 year stay in USA. The other stark aspect that came forth to his mind was that the Indians did not know the greatness of their land.

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).

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

Srinivasa Ramanujan

Srinivasa Ramanujan, the genius mathematician was born on 22nd December 1887. In December 2011, Ramanujan’s birthday was declared as ‘National Mathematics Day’, in recognition of his contributions to the field of mathematics.

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Srinivasa Ramanujan

A person who lived for a little over 32 years, Ramanujan was born in Kumbhakonam, the famous temple town in the Cauvery River delta.

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Kumbakonam

Kumbakonam, the rice bowl of Tamil Nadu has been famous for many things, from temples to rice and now for the aromatic Kumbakonam Degree Coffee.

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Kumbakonam Rice Fields

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 Kumbakonam Degree Coffee

 But, the greatest son of Kumbakonam is the mathematician, Srinivasa Ramanujan.

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Srinivasa Ramanujan’s house

In his Dreams

Ramanujan attributed the mathematical formulae that he came up with, to Namagiri Thayar, the Goddess of Namakkal temple.

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                         Namakkal Temple                                                               Goddess Namagiri

He often mentioned that it was Goddess Namagiri who came to him in his dreams and gave answers to his mathematical problems.

From Wife

The wife of Ramanujan, Janakiammal has an interesting input about her husband.

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Janakiammal

Ramanujan used to feverishly do all his basic calculations on a black slate. This was the norm of every student in India then.

She says, “Ramanujan did his calculations on a hand held slate, then transferred the final results to his note books, erasing the slate.”

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Ramanujan did his calculations on a slate

Thus we have few clues as to how he arrived at these equations, and there is no doubt that they are true.

This is expressed by the mathematics historian George Gheverghese Joseph in his book ‘The Crest of the Peacock’, Page 11.

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The Crest of Peacock Book

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George Gheverghese

His work notes and formulae that he arrived at are available in his now famous notebooks.

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Ramanujan’s notebook

Mathematicians till to date are trying to understand and use them.

To Cambridge University

When Ramanujan was working as a Clerk in Madras Port Trust, he sent some of his mathematical workings to Prof. G H Hardy of Cambridge University.

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Prof. G H Hardy

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Cambridge University

Ongoing through the notes, Prof G H Hardy felt that here was an absolute genius at work.

Prof Hardy invited Ramanujan to the Cambridge University.

Ramanujan spent 6 to 7 years in Cambridge. The work that Ramanujan did then along with Hardy has now become a part of the legend of Mathematics.

The mathematical formula that Ramanujan came up has been used as algorithms in modern computer systems.

Unfortunately, due to severe cold weather of England, Ramanujan who was more used to the tropical climate of Kumbhakonam, could not acclimatize and picked up an illness. The illness grew from bad to worse and he sailed back to India.

A sick and sad Ramanujan returned to Madras on April 2nd 1919.  He passed away on 26th April, 1920 at Chetpet in Madras.

Ganitham

Srinivasa Ramanujan belonged to an illustrious lineage of mathematicians that India has offered to the world starting from Boudhayana, Apastambha, Aryabhatta, Varahamihira, Brahma Gupta, Bhaskaracharya, Madhava and a galaxy of others.

All these illustrious people through the ages specialized in this field of Ganitham, the Indian term for mathematics.

The word Ganitham has in it the phrase Gana, meaning weighty, heavy.  The field of mathematics has always been weighty and heavy.

The Lord of Mathematics in Indian tradition is Ganesha, Ganapathy. The term Gana also means numbers.

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Lord Ganesha, the lord of Mathematics

An illustrious lineage

India has had an illustrious lineage of people who excelled in Ganitham.

Srinivasa Ramanujan is one among this illustrious lineage.

Today in our midst, we have another illustrious mathematician of Indian origin settled in USA, Prof Srinivasa Vardhan who is an Abel Laureate.

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Prof Srinivasa Vardhan

Abel Laureate

It is to be noted that in mathematics there is no Nobel Prize as Alfred Nobel did not like Maths.

The same Norwegian Academy which confers the Nobel Prize year after year has instituted an award for mathematics, equal to novel prize in the name of their Norweigian mathematician, Niels Henrik Abel.

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Niels Henrik Abel

It would be nice if the Indian government could institute an international award in the name of Srinivasa Ramanujan for the lineage that India has given to world in the field of Ganitham, mathematics.

Srinivasa Ramanujan Centre

A centre in the memory of Srinivasa Ramanujan was established in the year 2003 at Kumbakonam known as Srinivasa Ramanujan Centre. A museum on Ramanujan and his work is also housed therein which is the house of Ramanujan. This museum is maintained by Sastra University. This centre and museum was dedicated to the nation by President Abdul Kalam in 2003.

International Award

An international award of 10000 US dollars per annum has also been instituted for a mathematician who has done research on works of Ramanujan. Every year an International Conference is organized by Sastra on 22nd December, the birthday of Ramanujan, where the awards are given away to the selected recipients.

Mathematics – Crest of Peacock

Mathematics among the sciences is given a high place in India, like the crest of a peacock among its colored plum, in its ancient treatises. Vedanta Jyothisa, an ancient treatise on mathematics and astronomy mentions this.

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We discuss in detail on India’s contributions in the field of Mathematics, in our book Brand Bharat – Roots In India, which include zero, infinity, numerals, metrics, algebra, algorithm, geometry, 360 degrees, Pi, trigonometry and calculus.

Image result for brand bharat roots in india

The Man who saw Infinity

In the last decade or so, there has been a spurt of interest on Srinivasa Ramanujan. Books are being written and films are being made on this great man who saw infinity.

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The Man who knew Infinity

We need to sustain this interest to encourage more Indians to take up pure mathematics.

Maha Shivaratri and Winter Solstice

Maha Shivaratri is an auspicious period observed once a year for uniting oneself with the Shiva Tattva, the principle of Shiva in the cosmos.

A detailed understanding of why this festival of Maha Shivaratri is observed, is discussed in our book, “Understanding Shiva”, which is part of the Bharath Gyan Series.

In today’s times, Maha Sivaratri festival is celebrated between mid February and mid March every year.

There is a text called Kaushitaki Brahmana, an accompanying text to the Veda, which has been authored by Rishi Kahola Kaushitaki.

Here, he mentions that he lived about 4 generations after the time of the Mahabharata characters. Assuming an average of 25 years for a new generation to be born, we can thus take his time period to be about 100 years after the Mahabharata times.

Rishi Kahola Kaushitaki should have then lived around 3000 BCE. In his text, the Kaushitaki Brahmana, he mentions that the Maha Shivarathri festival day occurred on a winter solstice day.

In present times, the winter solstice occurs on December 21st. On a winter solstice day, the sun is at the southernmost point in its journey between the northern and southern hemispheres. It is the longest night of the year in the northern hemisphere.

Maha Shivaratri is observed in the lunar month of Magha, on the night preceeding the New Mooon night, Krishna Paksha Chaturdasi.

From the text of Kaushitaki Brahmana we can understand that Maha Shivaratri, Magha Krishna Paksha Chaturdasi and winter solstice occurred on the same day, that year, around 3000 BCE.

The month of Magha occurred around December – January in 3066 BCE as we have seen when we dated Bheeshma’s Nirvana.

In 2018 CE, this month Magha falls around February – March, indicating a gap of close to 70 days.

Can we account for this gap?

Both, the ancient Indian texts and modern astronomy, mention about the precession of equinox. This precession occurs at the rate of approximately one day over 72 years. Because of this precession of equinox, seasons keep slipping by one day in every 72 years. Over a large span of time, what was summer once would probably become autumn or even winter.

Using this precession of equinox giving rise to the slippage of one day in every 72 years and multiplying it by 70 days, which is the difference between the Maha Shivaratri day celebrated now or Magha month now and the Magha month or Maha Shivaratri day celebrated during the time of Mahabharata.