Calculus was discovered in India

— Dr. M Lavanya

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.


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).

Lasting Impact of an Orienting Speech

The speech of Swami Vivekananda at Chicago conference on September 11th, in 1893 was only for 6 minutes consisting of 887 words but what he stirred up with that has been echoing across continents for 120 years and will continue to echo for centuries to come.

Swami Vivekananda

Indians rediscovering India

This short speech recreated an interest about India in the west for the 20th century. It also helped Indians discover Swami Vivekananda and through him Indians were able to discover themselves, the strength of their tradition, culture and the respect for their own spirituality. The awakening of Indians among the American intelligentsia had resonance back in his homeland in India.  This strength later paved the way for the Indian independence movement. In many a ways, 11th September is the day the Indians through Swami Vivekananda rediscovered their own strength.

 Swami Vivekananda at the Parliament of Religions, Chicago, 1893

Harbinger of yoga and spirituality in the west

This speech also gave impetus to Indian yoga and spirituality. Today, yoga is a thriving industry in the west but its seeds were laid by Swami Vivekananda in the west.

Religious Tolerance

People today talk of religious tolerance. The seed of this idea of religious tolerance and the idea of religious tolerance was sown by Swami Vivekananda at this speech.

4 years in America

After this speech, Swami Vivekananda kept getting invitations to speak at different occasions and was compelled to stay on in America for 4 years. This momentous speech catapulted Swami Vivekananda, an unknown swami of the orient to become one of the most powerful thought providers.

 A true prajapathi, a thought leader in every sense!

Introspecting On The Fall Of The Rupee

At the time of Independence of India, in 1947, Rupee was equal in value to the Dollar. While we talk about having progressed, our Rupee has been depreciating every year since then.

5 year plans

From the time of Independence, the government has been pushing through 5 year plans and poverty alleviation programs ‘Nehruvian’ in thought, adopted from the now defunct Soviet model. We are now in the twelfth 5 year plan.

These government policies while they have shown some progress, they seem to have favoured cities more than the villages.  The development schemes have been lopsided to say the least, as is evident to all.

Poverty Continues

At the time of Independence, we had about 50 % of population living in poor conditions.  65 years later, when we look around the land we still see half the population living in appalling conditions. While the government statistics show a picture of progress, the fact is that appalling poverty is still apparent after 65 years of poverty alleviation programs.

 “Isn’t there obviously something fundamentally wrong with this model then?”

Three waves of plunder

At the time of Independence, we got our freedom from the vice grip of the British who had plundered us mercilessly in the name of ruling us.

 Plunder of India

Plunder of India

But it was not just the British who had plundered us.

India had already come under an earlier wave of plunder from the Near West marauders before the British. When we look back at the history of this land, we hear of the prosperity of every kingdom across the land, through the times. It was this prosperity which was eroded by wave after wave of plunder.

The First Wave of Plunder

The near west onslaughts started by Muhamad of Ghauri and Mahmud of Ghazni, around 1000 CE, continued till Nadir Shah around 1700 CE and constituted the first wave of plunder.

First wave of plunderMuhmad of ghazni

 The Second Wave of Plunder

The second wave of plunder started with the invasion by the Portuguese in 1500 CE and intensified with the arrival of British in 1600 CE and continued till 1947. It was plunder by colonization.

EIC Flag

 A Third Wave of Plunder

The British left India in 1947. India should then have started flourishing again without any oppression and become prosperous like before.

But a different destiny awaited India.

After the departure of the British, the plunder did not come to an end. It was now the turn of Indians themselves who took to plundering the land with corrupt practices which have grown into bigger and bigger scams. This can be classified as the third wave of plunder, the Home-Grown Plunder.

Ever wondered, how come this civilization is being plundered repeatedly?

Only if a land is generating wealth continuously, can it be plundered over and over again and still have enough loot left for the succeeding plunderers.  The British have equated the wealth generating capacity of India to that of a natural spring that keeps gushing out water continuously.

Fall from Rajiv Gandhi to Rahul Gandhi

While even after the 1st and 2nd waves of plunder, there was 1:1 parity between Dollar and Rupee at the time of Independence, it is the current wave, the 3rd wave that has been bleeding the land heavily and rapidly, right in our own times.

The extent of this plunder is apparent from the statements of Rajiv Gandhi and Rahul Gandhi, just one generation apart.

100 paisa to 17 paisa to 2 paisa

In 1990, the dynamic, late Prime Minister Rajiv Gandhi had lamented, Only 17 paise of every rupee reaches the villages”.

  Rs1 to 17 p

In 2008, Rahul Gandhi, his son, as a General Secretary of All India Congress Committee, had this to say – “Only 2 paise of every rupee spent reaches the people in the villages.”

 1Rs to 2 p

These two statements are indicators of the state of affairs and the depth of malice in the 3rd wave of plunder.

Today, Rupee has dipped to an all time low of 65 Rupees to a Dollar. In 65 years, we have economically eroded ourselves 65 times. This is not to say that the American dollar is strengthening but that the poverty alleviation programs and the delivery mechanism of the government have failed the people.

If these models are so wrong what should the right model be?

If the Rupee had parity with the Dollar at the time of Independence in 1947, it means that something was still right, even then.

Far Sighted At Same Time Near Sighted

Two things stand out clear when we analyze the way this civilization had organized itself and functioned to create such untold amount of wealth and prosperity.

  1. Far Sighted Goal – the people and governance were geared for prosperity creation and not poverty alleviation. Creating widespread reach of prosperity automatically meant alleviation of poverty.

Plunder and Poverty are two sides of the same coin. Where one goes, the other follows.

Poverty and Prosperity are also like the two sides of a coin. When you see one, the other disappears.

  1. Near Sighted Vision – the structure of governance was so organized that power of managing the land, the resources, the people’s productivity and the yield, vested at the village level, with the Panchayat. It was not far removed and centralized with the King alone.

The people therefore could prioritize money collection and spend according to their local needs and that made the spend effective and productive. It also cut out all unnecessary layers and thus scope for embezzlement / corruption as well as administrative expenses in money collection and disbursement from a central point.

A Prosperity Generation Model

Instead of talking about poverty alleviation, if we focus on prosperity generation, then the shift from poverty to prosperity can happen in present day India too.

While at the policy level, this needs to be the mantra, at implementation level, the tantra should be to move away from the centralized system of governance that was imposed by the British on India.

Centralization in India was imposed by the British as the intent of British administrators was to collect taxes. Which is why even today, we have the chief administrator of a district being called “a collector” – a colonial hangover.

This land was prosperous historically, because of a decentralized system of administration where every village panchayat was responsible for its prosperity. Kings and Kingdoms came and went but prosperity continued village after village across the land through the ages.

This was because the ethos of the land was driven towards prosperity generation through a decentralized system of self-governance. This made this civilization as a whole, an economic giant. It is time now to revert back to this time tested model that has given prosperity to this land.

Mindset change

It is possible to bring back parity of Rupee and Dollar within the next few decades itself.

For this, minor tinkering of economic policies is NOT the way.

It needs a mindset change – a change to recognize the ethos of this land, to recognize that the prosperity of this land emanates from a decentralized model.

That would give prosperity to the land at the grass root, administrative unit level – the panchayats.

While we achieved Swarajya, self governance 65 years ago, this would help us achieve, Svatantrata now, the practice of indigenous models.

A simple but effective prosperity generating model!

Giving Dhana

The word dhana has a root syllable, dha which means to give. It is interesting to note that the word donate also comes from the same root. To give, to donate is an intrinsic part of human nature exhibited in every civilization that has flourished. The Indian civilization on its part had been a prosperous one. This prosperity here forth is manifest in the munificent practice of donation, dhana, an act where one shares the prosperity with an open heart.

Dhana box

This word dhana can be seen not only in India but also in South East Asian languages. In every place of worship, in Indonesia, Bali, Thailand and other places, there is a dhana box to offer donations, much like the hundi that one sees in temples and other religious places of India.

Donation box in Indonesia

Donation box at a temple in Indonesia

Mother’s offering every day

Every day, when a mother makes food, she ensures that the first handful is kept aside for the birds, street dogs and other domestic animals. This has been the practice from Punjab in North India to Tamil Nadu in deep south.

feeding dogs


The land of India has been vast from time immemorial. People have been walking to and fro, all over this land, to various Punya Tirtha, the holy spots. Chatram were built as a tradition of the land in almost every village, where these pilgrim travelers could rest in clean surroundings, partake free food offered by the local community and continue with their pilgrimage.

These Chatram dotted  the land, from Peshawar in northwest to Kanyakumari in the south. They are also known as Sarai in North India. From this word, we get the words, Caravan Sarai and Mugal Sarai.

These Sarai were not only on main road but were also along the rivers. The best example is the Taj Mahal. We know it as the wonder of the world and a mausoleum, but what is little known is that Taj Mahal was also a Sarai on the banks of Yamuna for people going on boats up and down the Yamuna River.

Taj Mahal

These Sarai were not only for the poor pilgrims. It was the custom of e land even for the kings to partake the food in Sarai, Chatram during their travel across the land. That was the high quality of accommodation and food that was provided as dhana to the travelers.  These are precursors to the travellers inn and present day hotels.

Traveller's inn

The key difference is that while the Chatram and Sarai were free for one and all, the travelers in hotels have to pay for its use.


 At the time of this important astronomical event, eclipse, there is a popular saying, “de dhan,  chute grhan”, meaning after the event of eclipse, it would be prudent to give dhana.

Dhana, the very ethos of this land

Like this, for every occasion like marriage, festivals, graha pravesham – house warming, after death ceremonies, the giving of dhana has been an integral part of the rituals. The giving of dhana has been interwoven into the tradition of the land, like Anna Dhana, Go Dhana, Bhu Dhana, Vastra Dhana and so on, including Shrama Dhana, volunteering.

This civilization has had this practice of dhana, charity for every occasion throughout the year. A beautiful concept of sharing the prosperity that the land has to offer and living in harmony with one another and society!

Teachers’ Day

After independence, those who gave a shape to modern India, marked different days as days of national importance. One such day of national importance is the Teachers’ Day, which falls on 5th September to honour teachers.

Why did they choose this day to honour teachers?

Dr. Sarvapalli Radhakrishnan was a teacher by profession, who rose to be a leader of modern India.


He started his career as a teacher in the Madras University. He distinguished himself in his chosen field of Philosophy and in his times, was one of the foremost thinkers in this field. His specialisation was Hindu philosophy. We often hear the phrase, “Hinduism is a way of life” as a  way to explain Hinduism. This phrase was coined by Dr. Sarvapalli Radhakrishnan.

From being a teacher, Sarvapalli Radhakrishnan soon rose to the high post of becoming the Vice President of India and later occupied the highest post of the land as the 13th President of India.

One particular instance of his brilliance is when he addressed the august gathering of Master Minds in United Kingdom. One of the norms after a person addresses a Master Mind is for the presiding officer of the function to give his comments. After Dr. Sarvapalli Radhakrishnan’s address at the Master Minds, the presiding officer’s only comment was, “a mastermind on Master Minds.” Such was his erudition indeed!

After his lecture there, one of the students present asked him, whether he possessed the whole of the knowledge, for which Dr. Sarvapalli Radhakrishnan replied that, “he was still searching for grains, in the sands of knowledge”.

Dr. Sarvapalli Radhakrishnan’s humble repartee to the student seems to have been inspired by the famous quotation of the ancient Tamil poetess Avvaiyaru – “What you have learnt is equal to just a handful of soil, what you have yet to learn, is as wide as this Earth”.

True knowledge is that which produced humility and a sense of balance in the individual.

In memory of the scholarship of this teacher and leader, the Government of India declared his birthday as Teacher’s Day.

A good teacher is known by the students he creates.

While the birthday of Dr. Sarvapalli Radhakrishna was chosen as the date to honour teachers, we cannot afford to forget his teacher. For, it was his teacher, Ashutosh Mukherjee who had guided Dr. Sarvapalli Radhakrishna in rising to those great heights.

In 1921, it was Ashutosh Mukherjee who convinced the budding philosopher, Dr. Sarvapalli Radhakrishnan to join the Calcutta University. Ashutosh Mukherjee also inspired and groomed other famous personalities like C. V. Raman, Srinivasa Ramanujam and Subhash Chandra Bose.

This teacher’s day, let us cherish memories of not only the great teacher Dr. Sarvapalli Radhakrishnan but also the great teacher’s teacher, Ashutosh Mukherjee.