— 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 15^{th} 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 wellknown 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 socalled “IndoArabic” 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 presentday Kerala during ~ 13001600 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, YuktiBhasha, CarunaPadhati 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 predates 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 Āryabhatiyabhāsya :
“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 templevillages around a river called Nila in the ancient days (and currently called river Bharatha, the second longest river in Kerala) during ~ 13001600 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 presentday 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 Sangamagrama of which the pioneer, Mādhava (~13401420) 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 Āryabhatiyabhāsya.

Jyesthadeva (c. 1530)
Author of the celebrated Ganita 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, “GanitaYuktiBhasha (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).
I saw an other good book in the U.S.:
“Geometry in Ancient and Medieval India”, by T. A. Sarasvati Amma
It covers a method for the cosine infinite series. But, I could not figure it out all the pieces. The method seems different than what we learn today in math. Very Geometric. I am not steeped in the methods used then. So it will require more work. Concepts of first and second derivative were used and the concept of the second derivative of cosine is minus cosine was exploited.
You could refer to the works of Late Dr K. V. Sarma of Hoshiyarpur University who has done extensive work in the field of Kerala Mathematics. His work is now being carried forward by Dr. M. D. Srinivas of K. V. Sarma Research Foundation, Chennai.
You mentioned commerce.
Lall Nigam wrote an article about Indian Double Entry bookkeeping likely to be the root of Double Entry bookkeeping. He notes in the article fiar Luca Pacioli wrote one of the first math, titled “Summa de arithmetica”, books shortly after the Gutenberg movable type press. The last chapter was on double entry bookkeeping. He also brought Indian numerals to accounting and math in Europe, about 250 years after FIbbonaci.
Might some math books from India be in firar Luca Pacioli’s in his travels? HIs book was published in Venice which has an old library.
An article I found very interesting is:
Lall Nigam, “BahiKhata: The PrePacioli Indian Doubleentry System of Bookkeeping, ABACUS, Volume 22, Issue 2, September 1986, Pages 148–161
Corrections:
He notes in the article fiar Luca Pacioli wrote one of the first European printed math book, titled “Summa de arithmetica”, shortly after the Gutenberg movable type press. That book is in Italian. Venice became a book publishing center.
Might some math books from India be in firar Luca Pacioli’s in his travels? HIs book was published in Venice which has several old libraries. And he was well traveled in Italy.
Thanks for these interesting additional inputs.