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 16^{th} 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].

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 15^{th} – 16^{th} 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 16^{th} 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 15^{th} – 16^{th} 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 16^{th} 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 16^{th} 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

Amazing facts!!. I think government should promote research on these things…these info should reach young minds who can take this further

“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.”

And the errors in the tables Clavius published are probably in the originals. Note 10 seems to indicate the circle divided 10 million times. There has to be errors as they were probably computed by hand and the typeset errors would not be in the originals.

I think that would give the direction of travel. Also, there must be copies of the same data in India rather than recomputing the table. A sine or cosine table can be scanned and a computer can pick out the errors.

Of course the theory behind the trigonometric formulas is much more interesting evidence.

Yes, more data on this is available in our book Brand Bharat_Vol2_Roots in India, in the chapter Calculus – Its Origins.