A Poem by H H Ravi Shankar on New Year

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Was Jesus born on 25th December?

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The last week of December to the first week of January is a time of festivity. It is when the people worldover are celebrating the Birth of Christ as Christmas and the arrival of a New Year.

Christmas is popularly celebrated on 25th December. The Christian Calendar which is also linked to the birth of Jesus starts from 1st January. The Greek orthodoxy, the Armenians, East Europeans and Russians celebrate Christmas on 6th January. It is intriguing, as to why we celebrate these different days as the birthday of Jesus Christ.

How did these 3 days come about?

To understand this, we need to look at the times of Jesus Christ’s birth.

Herod, The Great

The king who ruled when Jesus was born was ‘Herod, The Great’. Herod died in 4 BCE. This is uniformly accepted by both Jewish and Christian historians.

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Herod, The Great

This implies that Jesus was probably not born in year 1 but was born before 4 BCE, when ‘Herod, the Great’ was alive.

This brings into question not only the date of December 25th, but also the year of birth.

0 year

Was Jesus Christ born in the year 0?

The number 0 was not known to the Romans until 1500 CE. In their calendar, -1 BCE transitioned into +1 CE.

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Anno Domini

Infact, the term Anno Domini (A.D., “in the year of our Lord”) as a universal reference point was coined by Christian historian, Dionysius Exiguous, a Roman monk in the year 532 CE.

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Dionysius Exiguous

Before Dionysius Exiguous, the terms BC and AD did not exist.

Council of Nicaea

Which year and date was Jesus born then?

When we dwell into this, we come across an interesting episode.

A council was formed, now popularly known as the Council of Nicaea, in 325 AD to arrive at Jesus Christ’s date of birth.

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Council of Nicaea

A few dates were put up for consideration at the Council

  1. 25th December was suggested by the Romans
  2. 20th May was given by the Clement of Alexandria
  3. 28th March was given by De Pascha Computes
  4. 6th January was given by the Greeks as it was the birthday of their God Dionysus
  5. Egyptians suggested the birthday of their God Osiris

The Urn Method

To choose a day, as the story goes, they put the names into an urn and picked one of them. The date that was picked is not known, but it is better known that the date was not accepted by all.

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An ancient Urn

This probably led to having multiple days for the birth of Jesus Christ.

January 6th

Since then, the Greek orthodoxy celebrate January 6th as Christmas which is essentially the birthday of Greek God Dionysus who was the most popular divinity of the Greek then.

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Greek God Dionysus

December 25th

The Romans thought it fit to celebrate December 25th as the birthday of Jesus, for, their biggest festival then in Rome was ‘Natalis Solis Invicti’ , ‘Nativity of the unconquered Sun’, to celebrate the return of the sun, the beginning of the northward journey of the sun from the Tropic of Capricorn.

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‘Natalis Solis Invicti’, the then biggest festival

Winter Solstice occurred on 25th December then. Now it occurs on 21st December.

This day, 25th December was a day of feasting for the Romans as the warmth of the sun was set to return. This popular festival day of the Romans was adopted by the Christians for the birthday of Jesus Christ.

Mario Righethi, a Roman Catholic writer, writes in his book,  ‘Manual of Liturgical History’ 1955, Vol 2, p 67,

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‘Manual of Liturgical History’

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The Earliest Christ Mass

The earliest celebration of Christ Mass, which has now come to be called Christmas was first observed on 25th December 336 AD, in the Phliocalion Calendar.

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The Phliocalion Calendar

The first Pope to address the Christ Mass – Christmas was Pope Liberius in the year 354 AD.

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Pope Liberius

New Year on January 1st

If the date of birth of Jesus Christ is a tossup between December 25th and January 6th, then how did January 1st come to be celebrated as the first day of the Christian Calendar.

Should not December 25th have been the first day of the Roman Christian Calendar?

Should not January 6th have been the first day of Greek Christian Calendar?

Was January 1st then a compromise date between the two?

Issac Asimov, the famous scientist in his book, ‘Book of Facts’ says,

“In 534 AD, the first man who calculated the year of Jesus’ birth made a mistake and we’ve been stuck with it forever.”

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Issac Asimov and his ‘Book of Facts’

All in all, we now have 3 dates, December 25th, January 1st and January 6th, celebrated in different parts of the world as the birth date of Jesus Christ.

Not arrived at from Bethlehem

It is interesting to note that all these 3 dates have not been arrived at in Bethlehem where Jesus Christ was born, but have been arrived at in other places  from Niceae, to Rome to Greece.

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Birth of Jesus at Bethlehem

What does modern science say?

Various scholars have tried to come out with a correct date for the birth of Jesus Christ using modern scientific tools.

Archaeo-Astronomy

One of these tools is Archaeo-Astronomy.

An Australian Astronomer, Dave Reneke, using sky charts, arrived at 17th June, 2 BCE, as the date of Jesus Christ’s birth.

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Dave Reneke

Based on the sky chart, the researcher opined that there could have been “a beacon of light” visible across the eastern sky at dawn, as the planets Jupiter and Venus moved across the Leo constellation.

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Leo constellation

“While these two are planets, they could have been called the Star of Bethlehem” says Dave Reneke in support of his date.

Further researches however cast doubts on this date.

There have been other researchers like this who have looked at the date of birth of Jesus Christ in 5 BCE.

As science evolves, it would be wonderful, if we could find the exact date of Jesus Christ’s birth. Until then, let us celebrate all 3 days as the birthday of Jesus Christ.

Raja Bhoja – the Polymath Monarch

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An eBook – Raja Bhoja – The Polymath Monarch

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An eBook – Raja Bhoja – The Polymath Monarch

This eBook is a peep into some of the Gizmos that Raja Bhoja had invented to obtain efficiency / effectiveness / enchantment in some of the day to day activities of himself and his staff.

These are the ones as described in his treatise, Samarangana Sutradhara, as translated by Shri.Prabhakar Pandurang Apte, Samskrt scholar from Pune and as visualized and animated by D.K.Hari and D.K.Hema Hari, Founders of Bharath Gyan.

The objective is to showcase through these innovations, Raja Bhoja’s all-round innovative spirit besides his other skills in order to highlight how a scientific and enquiring temperament, along with an innovative spirit is not new or alien to India. How, they have flowered in this land across millennia time and again and this century is yet another such milestone in history that Indians have to take advantage of.

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

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).

Narmada River

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Narmada is one of the 3 main rivers in India that flows westwards into the Arabian Sea, the other two being Tapti and Mahi. It is the fifth largest river in the Indian sub-continent and the third largest of the rivers flowing entirely within India.

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Narmada River

Narmada has its source at the Narmada Kund of the Amarkantak Plateau of the Bilaspur district of Chhattisgarh, and flows through Madhya Pradesh, Maharashtra and Gujarat. The river flows for 1300 kms, before draining into Arabian Sea.

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Narmada Kund, Amarkantak

One of the specialties of this river is that it flows through a rift valley, between the Vindhya and Satpura range. It is also one of those major rivers that doesn’t form any delta.

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Narmada flowing through Rift valley

Narmada –names, legends and importance

Narmada means “that which gives pleasure”. It is also known as Rewa, meaning, “swift”, due to the swiftness of its water currents.

Narmada and Adi Shankara

Adi Shankara once calmed the raging waters of Narmada River, using his kamandalu, to save his Guru Govinda Bhagavatpada, who was immersed in dhyana, meditation at a cave nearby.

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Adi Shankara calming the waters of Narmada

Narmada Ashtakam

Adi Shankara glorifies Narmada in the Narmada Ashtakam, which he composed in the glory of Narmada Devi. The opening verse of this hymn reads,

Sa-Bindu-Sindhu-Suskhalat-Tarangga-Bhangga-Ran.jitam
Dvissatsu Paapa-Jaata-Jaata-Kaari-Vaari-Samyutam |
Krtaanta-Duuta-Kaala-Bhuuta-Bhiiti-Haari-Varma-De
Tvadiiya-Paada-Pangkajam Namaami Devi Narmade ||1||

English Meaning:

Salutations to Devi Narmada whose River-body illumined with Sacred drops of Water, flows with mischievous playfulness, bending with Waves.

Your Sacred Water has the divine power to transform those who are prone to Hatred, the Hatred born of Sins,

You put an end to the fear of the messenger of Death by giving Your protective Armour (of Refuge),
O Devi Narmada, I Bow down to Your Lotus Feet, Please give me Your Refuge.

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Narmada Devi

Daughter of Rishi Mekla

In Purana, Narmada is mentioned as the daughter of Rishi Mekla, who lived and meditated at the foothills of Vindhya Mountains. Hence Narmada also has the name Mekalaa and Mekalakanya. The are other legends which point to Mekala being the mountain from where Narmada rises.

Life Line of Madhya Pradesh

The river is today known as the “life line of Madhya Pradesh” on account of its major contribution to the state.

One of the 7 holy rivers

In the Indian tradition, Narmada is of the 7 holy rivers, the others being Ganga, Yamuna, Sindhu, Kaveri, Sarasvati and Godavari. The ancient Indian texts like the Ramayana, Mahabharata and the Purana talk about this river. Like the Ganga, river Narmada is worshipped as a deity – Narmada Devi. The Vayu and Skanda Purana speak about the origin of this river in detail.

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The connect with the Trinity

As per one legend, Narmada has her origin from the sweat of Lord Shiva, and is therefore also known as Shankari. Another legend states that the river was born from the tear drop of Lord Brahma. These legends also state that Narmada is older than the Ganga.

The Omkareshwar Jyothirlinga is located on the banks of Narmada River, at the Khandava district of Madhya Pradesh.

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  Narmada at Omkareshwar                                      Omkareshwar Jyothirlinga

The resting place of Lord Shiva

Padma Purana states that Lord Shiva rested on the banks of River Narmada, before proceeding on his mission of vanquishing the Tripuras, the three aerial cities of the Asuras. The pebbles on the banks of Narmada are thus regarded to be highly sacred and are worshipped as lingam. These pebbled are known as Banalinga and are sought after for worship.

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Natural Narmada Banalinga

One of the biggest of these Banalinga has been installed in the Brihadeeshvara temple, at Tanjavur in Tamil Nadu.

The battleground of Indra and Vrtra

The Bhagavata Purana states that the battle between Indra and Vrtra, happened on the banks of Narmada River.

In Ramayana

In the Ramayana, it is mentioned that King Kartivirya Arjuna once picnicked with his wives on the banks of Narmada. Ravana also comes here at the same time, and in a battle between Ravana and Kartivirya, the former is humbled.

In the search for Sita, Sugreeva asks his Vanara army to conduct a search amongst the Vindhya mountains, where the Narmada river flows.

A Shiva temple with Narmada as Shakti

A Shiva temple exists on the banks of River Narmada, with Narmada as consort. Devi Narmada is worshipped as Shakti in this temple.

Suprabhat from Narmada Temple on the banks of Narmada at Maheshwar – a Shiva temple with an image of Narmada as Shakti

Pushkaram – The traditional festival

A festival, Narmada Pushakaram is held every 12 years here, in worship of River Narmada, and lasts for 12 days.

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Narmada Pushakaram

Narmada Basin

The Narmada basin covers a large area and is located between Vindhya and Satpura ranges, in the states of Madhya Pradesh, Gujarat, Chhattisgarh and Maharashtra and Telangana. It has one of the oldest teak hardwood forest in India. The Narmada eco region is home to 76 species of mammals and 276 species of birds.

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Narmada Basin

Bhimbetka rocks

The Bhimbetka rock shelters in the Narmada valley, in Madhya Pradesh contain many ancient paintings, that are 30000 years old. These 243 rock shelters at Bhimbetka have been declared as World Heritage Site by the UNESCO.

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Bhimbetka rock shelters

Archaeologist have found evidence of Harappa settlements on the banks of Narmada. One of the excavated sites is located at Navadatoli on the south bank of the river, which has remnants of the earlier civilization. Another one was excavated at Mehtakhedi village, in Narmada Valley, Madhya Pradesh.

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An ancient archaeological remains discovered at Narmada Valley

Bharuch

Bharuch is a sacred city located on the mouth of Narmada, and its name is derived from the great Rishi Brigu, the city’s original name being Brigukaccha. Rishi Brigu’s ashram was located on the banks of river Narmada.

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Bharuch, location

As per the Purana, Rishi Brigu is one of the ten sons of Lord Brahma. Many Rishi like Markandeya, Shukracharya, Jamadagni belonged to the lineage of Rishi Brigu. Lord Parasurama was born in the 7th generation this Rishi.

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Rishi Brigu

As per the Skanda Purana, 55 tirtha Sthal, are located along the Narmada River. Bharuch is also a Jain tirtha Sthal.

Today, just like other rivers, pollution has affected Narmada. On December 11th, 2016, the Madhya Pradesh government launched the Narmada Seva Yatra to turn the river pollution free. It sought to create awareness about the conservation of the river.

Narmada is one of the major rivers in this country that has shaped the culture and tradition of this civilization, apart from support life for many a millennia. We need to preserve it, so that it continues to sanctify us for many more millennia.

Music

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The month Margazhi also called Mrigashirsha is that time of the year when at the time of sunset in the west, the stars that rise in the east are the Orion constellation called Mrigashiras or head of a deer.

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Orion Constellation

This month is named after this constellation which is the most prominent in the sky through the night.

This month is also the music season in Chennai, the place having the culture of celebrating music in this month for over 80 years.

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Music Academy, Chennai

This music season has become so well entrenched that it is compared with the music festival of Vienna, Austria, where the world’s largest music festival takes place.

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Open air music festival, Vienna

The culture of the music festival in Vienna came about since Vienna is the home town of the European Classical musician Amadeus Mozart.

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Amadeus Mozart

In the Indian tradition, carnatic music has its primacy of place and is best exhibited in Margazhi Mahotsav – The music festival.

The word Carnatic in “Carnatic music” comes from Karnatakam, meaning traditional. The 2nd meaning being “ear”. Music is heard, relished and passed on from generation to generation. i.e, Carnatic music is the one that has its roots in the tradition of the land and that which is passed on, year after year, from ear to ear.

Among musical instruments, the lyre is considered to be one of the older instruments. This lyre is called the “Yazh”in the old Tamizh language. This Yazh instrument was very famous in the Eastern town of Sri Lanka from which this region got its name Yazh Paanam. The English way of calling the place now is Jaffna, which has no correlation with the ancient musical instrument Lyre, yazh.

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Lyre

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Yazh

While lyre, yazh refers to very fine metal string, which means the metallurgy should have been developed enough then, to have these metal strings.

There are instruments even prior to metallurgy. One among those pre-ancient music instruments is the wind instrument – flute. When the wind blows through a bamboo groove, a natural music whiz is created. Bamboo is, but a variety of grass.

Early man enjoyed this music of Nature and tried to create his own music with a piece of bamboo by making suitable holes for blowing air into it and by tuning the air flow with his finger to get the desired lilting music.

So, obviously flute is one of the oldest musical instruments known to man. Different cultures over thousands of years have made their own variants of the flute. There have also been great flute players.

Of all these flutists, the greatest name that readily comes to mind is Lord Krishna who was born 5000 years back.

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Krishna – The Divine Flutist

In Brindavan, his flute mesmerized the cows, peacocks as well as the gopis. When played normally, the natural raga that comes from the flute is Yaman when played in the North Indian style and HariKhamboji when played in the South Indian style. Krishna was able to captivate everyone with his flute.

While one needs to be gifted to produce melodious music, it does not take much to dissolve into, melt into, unite or become laya with the divine harmony in music. India with its wide repertoire of classical, semi-classical, folk, film and fusion music has a lot to offer.