The story of Calculus is intriguing and reveals that our Indian mathematical system was not only far advanced and ahead by centuries, in concepts, calculations and precisions, from the rest of the world, but was also subjected to unlimited appropriations by westerners to claim it as their own. When in fact, all they did is mostly application and compilation of the existing Indian results. The fact of the research and matter is that Calculus was neither a European invention nor anything close was the subject of their thoughts until the dire needs of the Church to travel abroad and spread Christianity had them stumbled upon the vast Indian mathematical knowledge-body.
But sadly, the history of (mathematical and other) science as commonly taught to us in India is mostly and ‘falsely’ Eurocentric. Not only this, nearly all the fundamental knowledge in Astronomy[1], navigation, agriculture and mathematics that European countries claim to have mastered was either taken as it is from India to Europe or was translated and secretly transported to Europe in a ‘mission’ mode. This all started in 16th[2] century from Greece and with the help of Arabs who retransmitted almost everything what they had learnt from India to Europe[3].This happened via either trade, missions of Jesuit priests, in the name of European scholarly pursuits and scholarly translations of existing Indian mathematical and astronomical systems.
What is known in the world today as modern calculus of the 17th century due to Newton and Leibniz was the direct result of the use of all the fundamentals (infinitesimals, infinite series[4], since cosine series etc.) that were developed independently in India.
To substantiate our claims, we first need to look at the major developments in two other parts of the world at that time, besides Europe and India: Greece (or ancient Greek) and Arabs – the two main countries where the Scientific literature from India was consumed and used for transportation to Europe.
Mathematical Story of Greece
Mathematics in Greece predates that of Europe by several centuries but Greek language and culture spread across these areas to a great extent and hence mathematics published in England, America, German (even Moritz Cantor) thus have only dealt with Greek mathematics largely in their works. (But Literature of Greece itself was all Indian underneath with Greece polishing).
To know the chronology of all the inventions and discoveries that happened in Greek, the ages can be laid down as follows:
From Greek Dark Ages (c. 1100 – c. 800 BC) to classical antiquity (8th –7th century BC – times of Homer), no evidence of development related to calculus or geometry can be found. Then followed the ages of classical Greece, Hellenistic period (323-146 BC), end of “Ancient Greece” (30 BC), emergence of Christianity and the decline of the Roman Empire (5th century AD), end of Antiquity culminating in the Early Middle Ages (600–1000 AD).
Beginning of elementary geometry is credited to Thales (ca. 624–548 BC) who gave Thales theorem and Intercept theorem using deductive proofs. Contributions of Pythagoras (c. 570 – c. 495 BC) are still debatable – so as of Aristotle (384–322 BC) for his influential and aesthetic philosophy over his mathematical abilities. Nevertheless, geometry was founding its roots with his theorems. Now, the only precursor to calculus was the method of exhaustion used by Antiphon (5th century BC) – and later by Archimedes (c. 287 – c. 212 BC) and Exodus (c. 390– c. 337 BC), student of Plato, who proved areas of circles and squares. Archimedes was able to use some infinitesimals in a way that is similar to modern integral calculus of today. However, he had no contribution to Calculus. Some of his best inventions included screw pump, pulleys, and defensive war machines to protect his native Syracuse from invasion. Later on, drawing on these concepts, Euclid (300 BCE) published ‘Elements’ in 13 books with several results (including above) and came to be known as the ‘father of geometry’. Largely, Greece had not yet developed any concepts of calculus on their own.
The story of Original Calculus by India
Calculus begins with two simple questions that cannot be solved using only algebra and geometry.
Question 1: How to calculate an instantaneous speed (or rate of change) of something rather than just an average speed (integral calculus).
Now this problem was not new to Hindu astronomers Aryabhata (476–550 CE) and Brahmagupta. They called this instantaneous motion as tat-kalika (Sanskrit for instantaneous velocity) and gave the formula for it long back[5][6].
And question 2: How to calculate the area under a curve rather than just the area of “regular” objects like squares, triangles, circles.
All these details were known to the scholars and mathematicians of what was known as Kerala School (1300-1600 CE), a well-known center of mathematics and astronomy in the 15th and 16th centuries long before Europeans were familiar with these concepts. As such, for many concepts such as development of Infinitesimal series and its applications which were central to calculus and transcend many other branches of Modern mathematics today, the origin was neither Greece nor Europe (definitely not China). Mathematicians who belonged to this school developed comprehensive theories and codified the science in palm leaf bundles (granthas) equivalent to (but far superior yet) modern day books[7].Those included Bhaskaracharya, Brahmagupta, Varahamihira, and so on.
A look at some of the notable achievements of this school and what they are known by in the world today gives a fair idea of how much successful Europeans are today in their attempt to claim Indian Math as their own.
a] The concept such as ‘Limit’, infinite series was developed in this School in around 1350 (300 hundred years before Newton and Leibniz)[8],
b] The text Yukti Bhasha, written by the Indian astronomer Jyesthadeva of this school was a veritable text book of original calculus, and offers detailed explanations of most of the results used today. All of them are now being named after European mathematicians. It contained extensive trigonometric tables developed by Madhava (c. 1340 – c. 1425)[9] along with his seminal contributions to the study of Infinitesimal series expansions for sine, cosines, arctangents, asymptotic expansions, value of Pi (earlier cited in the Mahajyānayana prakāra), that were never introduced anywhere else in the world, with an astounding accuracy. All of these tables was published by Clavius, in 1607, under his name[10] without any proof of any calculations leaving no doubts as from where he took them. In Europe, the first such series were developed by James Gregory in 1667 three centuries after Madhava. (Today, it is referred to as the Madhava-Gregory-Leibniz series).
While the use of arithmetic and geometric series appeared in Vedic literature in 2000 B.C, but Madhava laid the foundations for the development of modern calculus[11].
c] Series expansion for trigonometric functions were described by Neelakanta in Sanskrit verses in an astronomical treatise called Tantrasangraha. The same expansions of sine, cosine and arctan functions became Taylor series of today [12][13][14]), and the series expansions of pi became Gregory series (developed by him 300 years before Gregory discovered them),
The Kerala mathematicians had “laid the foundation for a complete system of fluxions” and these works abounded “with fluxional forms and series to be found in no work of foreign countries” – C. M. Whis, noted Western Englishman who first wrote up the works of Kerala School in 1835
d] Neelakanta in the same book proposed models that became ‘Tychonic model’ of planetary motion (published by Tycho Brahe in 1583), centuries later.
e] Jyeshthadeva’s Yuktibhasha formula involving a passage to infinity to calculate the area under a parabola was used by Fermat, Pascal, and Wallis[15]
f] Bhaskaracharya’s (1114-1185 CE) monumental works such as Karaṇakutūhala
(“Calculation of Astronomical Wonders”) and Siddhāntaśiromaṇi (“Head Jewel of Accuracy”) that later became Rolle’s theorem[16] continued to challenge French mathematicians for centuries[17]
g] In Jyeṣṭhadeva we find the notion of integration, termed sankalitam, (lit. collection), as in the statement:
ekadyekothara pada sankalitam samam padavargathinte pakuti,
And even after all of this, the above account is only an ‘infinitesimal’ attempt to put light on the origin of these great mathematical discoveries. But an important question to be asked here is why this ‘intellectual loot’ was so easy for Europeans. Did we know it was happening back then, did we not care enough or were we weak to defend?
Several reasons exist to explain this. Notably, the First of it lies in the way Hindu Philosophy works. We believe in doing things not for the sake of prestige or award, but for the sake of exploration and knowledge and ultimate realization. For a Hindu, search for truth has always been an end in itself and not an end to achieve anything more. Be it Math or science, literature or philosophy. Ramanujan once said “an equation has no meaning for me, unless it expresses a thought of God”. No other philosophy in the world, let alone Europeans believes in this.
The other reason was Jesuits.
The Society of Jesus of the Catholic Church had its members called by the name Jesuits. The Church in its prejudiced colonialism mind believed that discoveries and inventions should be at the disposal of only and only Europeans and anything that bear any other name should be suppressed. (Jesuits mission was as given in the reference point [2]).
Kerala at that time was in continuous contact with China, Arabia, and Europe. The port of Muziris, near Sangamagrama, was a major center for maritime trade[18], and a number of Jesuit missionaries and traders were active in this region. While in Kerala, use of Astronomy, mathematics, calendar etc. aided many socio cultural practices of that time such as weather forecasts, determining positions and movements of celestial bodies, in Europe that was needed to make explorations to meet the objective of the Church.
Enamored by the vastness of the Indian literature-body, Europeans took the task of translating the Indian work (due to little knowledge of the medieval form of the local language of Kerala, Malayalam and Sanskrit) and transmitting them to Europe by Jesuits missionaries and traders whose Presence in Kerala from the middle of the 16th century was well documented in many historical records belonging to that period[19][20].
There existed a third reason too. The intellectual careers of both Newton and Leibniz are well-documented and there is no indication of their work not being their own. But unfortunately this is not the case with Indian Literature (as we have followed the oral traditions mainly – which although is far more authentic). The Earliest known/recovered Indian work is a mutilated copy of Bhakshali Manuscript (containing work as old as of 300 AD). Other such works are Aryabhatiya (499 AD), Trisatika (750 AD)[21][22].
A very good example of this fallacy arising out of incomplete documentation can also be found in the paper located here (http://users.uoa.gr/~apgiannop/Sources/Roy-pi.pdf) that states that ‘the details of the circumstances and ideas leading to the discovery of the series by Leibniz and Gregory are known but the Indian proof and contribution ascribed to Nilakantha is not.
It is evident and clear today that Calculus was indeed an Indian product.
And the same arguments and facts holds true for other discoveries such as the Laws of Motion, uncertainty principle, to modern day concepts of zero, the decimal number, microwave communication, genetics and the list is exhaustive.
So Europeans did nothing then?
Isaac Newton and Gottfried Leibniz, supposedly independently (but not so surprisingly since they already had the foundation of infinitesimals), pioneered the infinitesimal methods and developed strong algorithmic-compendiums that came to be known as Calculus or ‘Infinitesimal Calculus’ or Modern Calculus[23]. Still, both never arrived to any genuineness and proof of their results and their only point of argument (and defense against Jesuits, Bishop George etc. [23]) was that since the results they were producing were correct, the methods have to be fundamentally correct too. Their arguments continued for two more centuries until a French mathematician Cauchy[24] proved theorems of calculus in his book of infinitesimal calculus titled ‘Cours d’Analyse’ published in 1821. Europeans just developed new methods of doing the same thing, albeit in a more articulated fashion.
But all these arguments need one more point to reason why it happened with India and not with any other country. A good point in fact from those who will still not believe what is being presented above with facts i.e. why present day India was so ahead millennia ago, that Europeans could only dream of even today?
One of the most important reasons why our science was so advanced was that the results were not based on inference and testimony alone using material objects of limited precision and human errors. That these are incomplete tools to explore any scientific discipline in question to its fullest realization, we knew well. Take a look at this verse from Aitareya Upanishad..
तत्प्रज्ञानेत्रम् प्रज्ञाने प्रतिष्ठितं प्रज्ञानेत्रो लोकः प्रज्ञानं ब्रह्म [25]
‘Prajna’ above, as also hinted in Vedas and elaborated in Upanishads, refers to the highest and purest form of wisdom, intelligence and understanding higher than any knowledge obtained by reasoning and inference[26]. Rational intellect, that Europeans today so boast of, is where logic concludes in shame without reaching even the periphery of such (cosmic) intelligence (‘aulokika pratakshya’) and deeper meaning. (In terms of an equation, think of it as – using a pre-developed equation or proving an equation in a different way which is already been proved is not same as actually devising a completely new equation. This was not possible for Newton). While with Prajna, the enlightenment and understanding that dawns is beyond what one has heard about, or deduced from external sources.
In a more profound level, anything which is outside of one’s own consciousness can never reveal the true nature of a thing – be it mathematical formulae – or a discovery. Nothing else can explain why even ordinary ‘Rsis’ (scientists seers) and ‘kavis’ of Bharath could easily know the advanced concepts of today like twin star Arundhati-Vashishtha, the atomic state of matter, concept of time, ‘shunya’ and ‘eternity’ etc. It was the direct result of harnessing that power which is the source of structures of all human experience itself [27].
To them, these were ordinary things, what Europe and the world at large are discovering and exploring even today after hundreds of years of research. Such a state of ‘Rtambhara Prajnya’ was also experienced nominally by Ramanujan.
What is sad today is not the fact that Bharatvarsha have been robbed of its uniqueness and identity but that no one in power, for whatsoever reason, cared to revolt and take it back. Many ancient techniques we lost to no one’s use and many concepts and scientific knowledge lost to the outside world like Calculus. Even hundreds of Indian medicinal herbs formulae have been lost to foreign US patentees who were too quick to know of their value and now we cannot manufacture them.
The truth is that all nations, from East to West, are nothing but schooled by ancient Bharat in some way or the other, in art, literature, philosophy, music, science, astronomy, economics and mathematics to an unprecedented accent. And it should be a matter of realization, as equally by them, as it is by us.
Additional Notes
[1] Remember that even the world’s largest stone sundial built in the first half of the 18th century using the advanced principles of astronomy and instrument design of ancient Hindu Sanskrit texts is in Rajasthan)
[2] This is the time when the Church along with European imperialist attitude [4] aggressively took the task of spreading Christianity across the world and destroy or convert all other sects and religions and beliefs existing. To travel abroad and spread their mission further, they needed certain navigational skills akin to mathematics/trigonometry which they had none. Even their Julian calendars were getting outdated.
[3] –[6] Refer References.
[7] Major Teacher -> Students relations of the Kerala School was as follows.
Madhava (1340-1425) – > Parameshevara (1360-1460) – > Damodara (1410-1510) – > Nilakantha (1443-1560), Jyesthadeva (1500-1610){ -> Pisarati (1550-1621)} – > Citrabhanu (1475-1550) – > Narayana (1500-1575) – > Shankara Variyaar (1500-1560)
[8]-[10] Refer References.
[11] Madhava was himself influenced by ideas prevailing in older Educational traditions including that from the present day North India and hence it was a mere continuation of the Indian Mathematical systems. His major influence included Kǖţalur Kizhār (2nd century),Vararuci (4th century), and Sankaranarayana (866 AD). Ended with Narayana Bhattathiri (1559–1632).
[12]-[27]- Refer References.
[28] A further Study with respect to the section ‘Mathematical Story of Greece’ can be referred here ‘A history of Greek mathematics, Thomas Little Heath, Oxford, 1921’
References
[1] [2] Refer Notes
[3] Arabic and Latin translations of (Aryabhata’s) works and several Sanskrit texts
http://en.wikipedia.org/wiki/Aryabhata#Trigonometry
[4] http://www.cbc.ca/news/technology/calculuscreated-in-india-250-years-before-newton-study-1.632433
[5] Toward a Global Science: Mining Civilizational Knowledge By Susantha Goonatilake, Page 134
[6] https://www.math10.com/en/maths-history/math-history-in-india/hindu-maths/hindumaths.html
[7] A Passage to Infinity: Medieval Indian Mathematics from Kerala and Its Impact, By George Gheverghese Joseph (Page 8)
[8] Paper available at – www.jstor.org/stable/25581775. ‘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.
[9] 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.
[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] Refer Notes.
[12] Encyclopedia of the history of science, technology and medicine in non-western cultures (two volumes), ed. Helaine Selin, Springer.
[14] Bressoud, David. 2002. “Was Calculus Invented in India?” The College Mathematics Journal (Mathematical Association of America). 33(1):2-13.
[15] 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).
Aryan race theory was created by the European imperialists when the antiquity and culture of the Hindu civilization was discovered
[16] https://en.wikipedia.org/wiki/Rolle%27s_theorem
[17] https://www.britannica.com/biography/Bhaskara-II#ref98080
[18] https://en.wikipedia.org/wiki/Madhava_of_Sangamagrama
[19] A. K. Bag (1979) Mathematics in ancient and medieval India. Varanasi/Delhi: Chaukhambha Orientalia. page 285
[20] C. T. Rajagopal, M. S. Rangachari, On an untapped source of medieval Keralese.
[21] http://www.journals.uchicago.edu/doi/pdfplus/10.1086/368443
[22] https://en.wikipedia.org/wiki/Bakhshali_manuscript
mathematics (https://link.springer.com/article/10.1007%2FBF00348142)
[24] https://en.wikipedia.org/wiki/Augustin-Louis_Cauchy
[25] Aitareya Upanishad (III.i.3). Also refer to verse no 49 (Chapter 1) from The Yoga Sutras of Patanjali.
[26] https://en.wikipedia.org/wiki/Praj%C3%B1%C4%81_(Hinduism)#cite_note-5
[27] R.V 9.83.3. Also refer ‘Meditations Through the Rig Veda: Four-Dimensional Man’ By Antonio T. De Nicolás
Excellent Analysis, Mr. Amit.
We are so delighted to read the truth of Calculus. I am glad and grateful that someone like you took the pains to point out the truth.
I think the time has come – that truths must be bluntly told. Many more such truths await a careful analysis and exposition.
Many a thanks! Please continue doing your good work of speaking the truth.