The Kerala School
The Kerala School, European Mathematics and Navigation
By D.P. Agrawal
The National Geographic has declared Kerala, the southwest coast near the tip of the Indian peninsula, as God’s Own Country. It has been a centre of maritime trade, with its rich variety of spices greatly in demand, even as early as the time of the Babylonians. Famous travellers and explorers such as Ibn Battuta and Vasco da Gama came from across the Arabian Sea. In recent years, Kerala has gained recognition for its role in the reconstruction of medieval Indian mathematics.
Joseph (1994) has very emphatically brought out the significance of the Kerala school of Maths in his The Crest of the Peacock, though the Eurocentric scholars have severely criticized it. C.K. Raju, the well known mathematician and historian of science, has also written a good deal not only on the famous work, Yuktibhasa by Jyesthadeva, but also on the export of Maths from India to Europe. Until recently there was a misconception that mathematics, in India made no progress after Bhaskaracharya, that later scholars seemed ‘content to chew the cud, writing endless commentaries on the works of the venerated mathematicians who preceded them, until they were introduced to modern mathematics by the British. Though the picture about the rest of India is not clear, in Kerala, the period between the fourteenth and seventeenth centuries marked a high point in the indigenous development of astronomy and mathematics. The quality of the mathematics available from the texts that have been studied is of such a high level, compared with the earlier period that it is impossible to bridge the gap between the two periods. Nor can one invoke a ‘convenient’ external agency, like Greece or Babylonia to explain the Kerala phenomenon. There were later discoveries in European mathematics, which were anticipated by Kerala astronomermathematicians two hundred to three hundred years earlier. And this leads us to ask whether the developments in Kerala had any influence on European mathematics. The only scholar who has dealt with this issue to my mind is C.K. Raju, whose views would also be discussed in this essay.
Joseph informs that in 1835, Charles Whish published an article in which he referred to four works – Nilakantha’s Tantra Samgraha, Jyesthadeva’s Yuktibhasa, Putumana Somayaji’s Karana Paddhati and Sankara Varman’ s Sadratnamala – as being among the main astronomical and mathematical texts of the Kerala school. While there were some doubts about Whish’s views on the dating and authorship of these works, his main conclusions are still broadly valid. Writing about Tantra Samgraha, he claimed that this work laid the foundation for a complete system of fluxions [‘Fluxion’ was the term used by Isaac Newton for the rate of change (derivative) of a continuously varying quantity, or function, which he called a ‘fluent’]. The Sadratnamala, a summary of a number of earlier works, he says ‘abounds with fluxional forms and series to be found in no work of foreign countries’. The Kerala discoveries include the Gregory and Leibniz series for the inverse tangent, the Leibniz power series for p, and the Newton power series for the sine and cosine, as well as certain remarkable rational approximations of trigonometric functions, including the wellknown Taylor series approximations for the sine and cosine functions. And these results had apparently been obtained without the use of infinitesimal calculus.
In the 1940s it was Rajagopal and his collaborators who highlighted the contributions of Kerala mathematics, though none of their results has as yet percolated into the standard Western histories of mathematics. For example, Boyer (1968, p. 244) writes that ‘Bhaskara was the last significant medieval mathematician from India, and his work represents the culmination of earlier Hindu contributions.’ And according to Eves (1983, p. 164), ‘Hindu mathematics after Bhaskara made only spotty progress until modem times.’
Madhava’s work on power series for p and for sine and cosine functions is referred to by a number of the later writers, although the original sources remain undiscovered or unstudied. Nilakantha (14451555) was mainly an astronomer, but his Aryabhatiya Bhasya and Tantra Samgraha contain work on infiniteseries expansions, problems of algebra and spherical geometry. Jyesthadeva (c. 1550) wrote, in a regional language rather than in Sanskrit,Yuktibhasa, one of those rare texts in Indian mathematics or astronomy that gives detailed derivations of many theorems and formulae in use at the time.This work is mainly based on the Tantra Samgraha of Nilakantha. A joint commentary on Bhaskaracharya’s Lilavati by Narayana (c. 150075) and Sankara Variar (c. 15001560), entitled Kriyakramakari, also contains a discussion of Madhava’s work. The Karana Paddhati by Putumana Somayaji (c. 16601740) provides a detailed discussion of the various trigonometric series. Finally there is Sankara Varman, the author of Sadratnamala, who lived at the beginning of the nineteenth century and may be said to have been the last of the notable names in Kerala mathematics. His work in five chapters contains, appropriately, a summary of most of the results of the Kerala school, without any proofs though.
Astronomy provided the main motive for the study of infiniteseries expansions of p and rational approximations for different trigonometric functions. For astronomical work, it was necessary to have both an accurate value for p and highly detailed trigonometric tables. In this area Kerala mathematicians made the following discoveries:

The power series for the inverse tangent, usually attributed to Gregory and
Leibniz; 
The power series for p, usually attributed to Leibniz, and a number of rational
approximations to p; and 
The power series for sine and cosine, usually attributed to Newton, and approximations for sine and cosine functions (to the second order of small quantities), usually attributed to Taylor; this work was extended to a thirdorder series approximation of the sine function, usually attributed to Gregory.
Apart from the work on infinite series, there were extensions of earlier work notably of Bhaskaracharya:

The discovery of the formula for the circumradius of a cyclic quadrilateral,
which goes under the name of l’Huilier’s formula; 
The use of the NewtonGauss interpolation formula (to the second order) by Govindaswami; and

The statement of the mean value theorem of differential calculus, first recorded by Paramesvara (13601455) in his commentary on Bhaskaracharya’s Lilavati.
Here it may be relevant to note some points of the debate that CK Raju has been carrying out with the West in general, and with Whiteside (the famous historian of Maths) in particular, about the export of Maths to Europe.
Raju’s Encounter with Eurocentric scholars
Raju (personal communication) explains that Whiteside, while conceding Madhava’s priority for the development of infinite series, distorts the dates of both Madhava and the Yuktibhasa, by about a century in each case. (Madhava was 14th15th c. CE,not 13th, while the Tantrasangraha [1501 CE] and Yuktibhasa [ca. 1530 CE] are both 16th c. CE texts, not 17th.) In fact, in the 16th c. CE Jesuits were busy translating and transmitting very many Indian texts to Europe; during the 16th c. CE, their activities were especially concentrated in the vicinity of their Cochin College, where they were teaching Malayalam to the local children (especially Syrian Christians) whose mother tongue it was, and where copies of the Yuktibhasa and several other related texts were and still are in common use, for calendarmaking for example.
After the trigonometric values in the 16th and early 17th c. CE, exactly the infinite series in these Indian texts started appearing in the works, from 1630 onwards, of Cavalieri, Fermat, Pascal, Gregory etc. who had access in various ways to the Jesuit archives at the Collegio Romano. Since Whiteside has a copy of the printed commentary on the Yuktibhasa, he could hardly have failed to notice this similarity with the European works with which he seeks to make the
Yuktibhasa contemporaneous!
Raju has no doubt that in the course of “the fabrication of ancient Greece” (in Martin Bernal’s words), some Western historians acquired ample familiarity with this technique of juggling the dates of key texts. Having anticipated this, the evidence for the transmission of the calculus from India to Europe is far more robust than the sort of evidence on which “Greek” history is built – it cannot be upset by quibbling about the exact date of a single wellknown manuscript like the Yuktibhasa.
While the case for the origin of the calculus in India, and its transmission to Europe is otherwise clear, there remains the important question of epistemology (“Was it really the calculus that Indians discovered?”). For, while European mathematicians accepted the practical value of the Indian infinite series as a technique of calculation, many of them did not, even then, accept the accompanying methods of proof. Hence, like the algorismus which took some five centuries
to be assimilated in Europe, the calculus took some three centuries to be assimilated within the European frame of mathematics.Raju has discussed this question in depth, in relation to formalist mathematical epistemology from Plato to Hilbert, in an article “Computers, Mathematics Education, and the Alternative Epistemology of the Calculus in the Yuktibhasa“.
In this paper, Raju proposes a new understanding of mathematics. He argues that formal deductive proof does not incorporate certainty, since the underlying logic is arbitrary, and the theorems that can be derived from a particular set of axioms would change if one were to use Buddhist logic, or, say, Jain logic.
Raju further states, “Indeed, I should point out that my interest in all this is not to establish priority, as Western historians have unceasingly sought to do, but to understand the historical development of mathematics and its epistemology. The development of the infinite series and more precise computations of the circumference of the circle, by Aryabhata’s school, over several hundred
years, is readily understood as a natural consequence of Aryabhata’s work, which first introduced the trigonometric functions and methods of calculating their approximate numerical values. The transmission of the calculus to Europe is also readily understood as a natural consequence of the European need to learn about navigation, the calendar, and the circumference of the earth. The centuries of difficulty in accepting the calculus in Europe is more naturally understood in
analogy with the centuries of difficulty in accepting the algorismus, due, in both cases, to the difficulty in assimilating an imported epistemology. Though such an understanding of the past varies strikingly from the usual “heroic” picture that has been propagated by Western historians, it is far more real, hence more futuristically oriented, for it also helps us to understand e.g. how to tackle
the epistemological challenge posed today in interpreting the validity of the results of largescale numerical computation, and hence to decide, e.g., how mathematics education must today be conducted.
‘I would not like to go further here into the difficult question of epistemology, and the interaction between history and philosophy of mathematics, except to link it to Whiteside’s use of the phrase “Hindu matmatics” [sic]. Am I to understand that Whiteside now implicitly accepts also the possible influence of Newton’s theology on his mathematics, and is alluding, albeit indirectly, to some subtle new changes brought about by Newton in the prevailing atmosphere of, shall we say, “Christian
mathematics”? Probably not. I presume instead that, despite his protestations to the contrary, Whiteside is really referring to the Eurocentric belief that there is only one “mainstream” mathematics, and everything else needs to be qualified as “Hindu mathematics”,”Islamic mathematics” etc.
‘Now it is true that I have commented on formalist mathematical epistemology from the perspective of Buddhist, Jain, Nyaya,and Lokayata notions of proof (pramana),in my earlier cited paper and book. I have also commented elsewhere, from the perspective of Nagarjuna’s sunyavada, on the reinterpretation of sunya as zero in formal arithmetic, and the difficulties that this created in the European understanding of both algorismus and calculus, difficulties that persist to
this day in e.g. the current way of handling division by zero in the Java computing language. Nevertheless, having also scanned the OED for the meaning of “Hindu”, I still don’t quite know what this term “Hindu” means, especially in Whiteside’s “ruggedly individualistic” nonEurocentric sense, and especially when it is linked with mathematics! Given the fundamental differences between the four schools listed above, it is very hard for me to dump them all, like Whiteside, into a single category of “Hindu”; on the other hand, if we exclude some, which counts as “Hindu” and which not, and why? And exactly how does that relate to mathematics?
‘A key element of the Project of History of Indian Science, Philosophy, and Culture, as I stated earlier, is to get rid of this sort of conceptual clutter ,authoritatively sought to be imposed by colonialists (and their victims/collaborators), and to rewrite history from a fresh, pluralistic
perspective. In my case, it is part of this fresh perspective to redefine the nature of presentday university mathematics by shifting away from formal and spiritual mathematicsasproof to practical and empirical mathematicsascalculation. Since my objective is truth and understanding, I am ever willing to correct myself, and I remain open to all legitimate criticism, but I do not recognize dramatic poses, assertions of authority, abuse, cavil, misleading circumlocutions, etc. as any part of such legitimate criticism.
‘There are numerous other points in Whiteside’s prolix response, to which it would be inappropriate to provide detailed corrections here. [E.g., I do not share the historical view needed to speak of the “rebirth” of European mathematics in the 16th and 17th c., which view Whiteside freely attributes to me, though I would accept that direct trade with India in spices also created a direct route for Indian mathematics, bypassing the earlier Arab route.] For the record, I deny as similarly inaccurate all the interpolations and distortions he has introduced into what I have said.
‘There is, however, one issue, which remains puzzling, even from a purely Eurocentric perspective. In what sense did Newton invent the calculus? Clearly, the calculus as a method of calculation preceded Newton, even in Europe. Clearly, also, the calculus/analysis as something epistemologically secure, within the formalist frame of _mathematics as proof_, postdates Dedekind and the formalist approach to real numbers. While Newton did apply the calculus to physics, that would no more make him the inventor of the calculus than the application of the computer to a difficult problem of genetics, and possible adaptations to its design, would today make someone the inventor of the computer. Doubtless Newton’s authority conferred a certain social respectability on the calculus. The credit that Newton gets for the calculus depends also upon his quarrel with Leibniz, and the rather dubious methods of “debate” he used in the process. But none of this convincingly establishes the credit for calculus given to Newton, even within the Eurocentric (as distinct from Anglocentric) frame. So what basis is there to give credit to Newton for originating the calculus, while denying it, for example, to Cavalieri, Fermat, Pascal, and Leibniz?
Navigation and Calculus
In his recent talk (2000) Raju emphasised that the calculus has played a key role in the development of the sciences, starting from the “Newtonian Revolution”. According to the “standard” story, the calculus was invented independently by Leibniz and Newton. This story of indigenous development, ab initio, is now beginning to totter, like the story of the “Copernican Revolution”. The English speaking world has known for over one and a half centuries that “Taylor” series expansions for sine, cosine and arctangent functions were found in Indian mathematics/astronomy/timekeeping (jyotisa) texts, and specifically the works of Madhava, Neelkantha, Jyeshtadeva etc. No one else, however, has so far studied the connection of these Indian developments to European mathematics.
The relation is provided by the requirements of the European navigational problem, the foremost problem of the time in Europe. Columbus and Vasco da Gama used dead reckoning and were ignorant of celestial navigation. Navigation, however, was both strategically and economically the key to the prosperity of Europe of that time. Accordingly, various European governments acknowledged their ignorance of navigation, while announcing huge rewards to anyone who developed an appropriate technique of navigation.
The Jesuits, of course, needed to understand how the local calendar was made, especially since their own calendar was then so miserably off the mark, partly because the clumsy Roman numerals had made it difficult to handle fractions. Moreover, European navigational theorists like Nunes, Mercator, Stevin, and Clavius were then well aware of the acute need not only for a good calendar, but also for precise trigonometric values, at a level of precision then found only in these Indian texts. This knowledge was needed to improve European navigational techniques, as European governments desperately sought to develop reliable trade routes to India, for direct trade with India was then the big European dream of getting rich. At the start of this period, Vasco da Gama, lacking knowledge of celestial navigation, could not navigate the Indian ocean, and needed an Indian pilot to guide him across the sea from Melinde in Africa, to Calicut in India.
These rewards spread over time from the appointment of Nunes as Professor of Mathematics in 1529, to the Spanish government’s prize of 1567 through its revised prize of 1598, the Dutch prize of 1636, Mazarin’s prize to Morin of 1645, the French offer (through Colbert) of 1666, and the British prize legislated in 1711. Many key scientists of the time (Huygens, Galileo, etc.) were involved in these efforts: the navigational problem was the specific objective of the French Royal Academy, and a key concern for starting the British Royal Society.
Prior to the clock technology of the 18th century, attacks on the navigational problem in the 16th and 17th c. focused on mathematics and astronomy, which were (correctly) believed to hold the key to celestial navigation, and it was widely (and correctly) believed by navigational theorists and mathematicians (e.g. by Stevin and Mersenne) that this knowledge was to be found in ancient mathematical and astronomical or timekeeping (jyotisa) texts of the east. Though the longitude problem has recently been highlighted, this was preceded by a latitude problem, and the problem of loxodromes.
The solution of the latitude problem required a reformed calendar: the European calendar was off by 10 days, and this led to large inaccuracies (more than 3 degrees) in calculating latitude from measurement of solar altitude at noon, using e.g. the method described in the Laghu Bhaskariya of Bhaskara I. However, reforming the calendar required a change in the dates of the equinoxes, hence a change in the date of Easter, and this was authorised by the Council of Trent in 1545. This period saw the rise of the Jesuits. Clavius studied in Coimbra under the mathematician, astronomer and navigational theorist Pedro Nunes, and Clavius subsequently reformed the Jesuit mathematical syllabus at the Collegio Romano. Clavius also headed the committee which authored the Gregorian Calendar Reform of 1582, and remained in correspondence with his teacher Nunes
during this period.
Jesuits, like Matteo Ricci, who trained in mathematics and astronomy, under Clavius’ new syllabus [Matteo Ricci also visited Coimbra and learnt navigation], were sent to India. In a 1581 letter, Ricci explicitly acknowledged that he was trying to understand local methods of timekeeping (jyotisa), from “an intelligent Brahmin or an honest Moor”, in the vicinity of Cochin, which was, then, the key centre for mathematics and astronomy, since the Vijaynagar empire had sheltered it from the continuous onslaughts of raiders from the north. Language was not a problem, since the Jesuits had established a substantial presence in India, had a college in Cochin, and had even started printing presses in local languages, like Malayalam and Tamil by the 1570’s.
In addition to the latitude problem, settled by the Gregorian Calendar Reform, there remained the question of loxodromes, which were the focus of efforts of navigational theorists like Nunes, Mercator etc. The problem of calculating loxodromes is exactly the problem of the fundamental theorem of calculus. Loxodromes were calculated using sine tables, and Nunes, Stevin, Clavius etc. were greatly concerned with accurate sine values for this purpose, and each of them published lengthy sine tables. Madhava’s sine tables, using the series expansion of the sine function were then the most accurate way to calculate sine values.
Europeans encountered difficulties in using these precise sine value for determining longitude, as in IndoArabic navigational techniques or in the Laghu Bhaskariya, because this technique of longitude determination also required an accurate estimate of the size of the earth, and Columbus had underestimated the size of the earth to facilitate funding for his project of sailing West. Columbus’ incorrect estimate was corrected, in Europe, only towards the end of the 17th c. CE. Even so, the IndoArabic navigational technique required calculation, while Europeans lacked the ability to calculate, since algorismus texts had only recently triumphed over abacus texts, and the European tradition of mathematics was “spiritual” and “formal” rather than practical, as Clavius had acknowledged in the 16th c. and as Swift (Gulliver’s Travels) had satirized in the 18th c. This led to the development of the chronometer, an appliance that could be mechanically used without application of the mind.
Thus we see that the great Kerala School of Maths needs a fuller treatment in the history of Indian science than has been given so far. We should all be thankful to both G.G. Joseph and C.K. Raju for their valuable contributions in this regard.
Bibliography
Boyer, C.B.. 1968. A History of Mathematics. New York: John Wiley.
Chattopdhayaya, D. 1986. History of Science and Technology in Ancient India: the Beginnings. Calcutta: Firma KLM.
Eves, H. 1983. An Introduction to History of Mathematics: A Reader. Philadelphia: Sunders.
Joseph, G.G. 1994. The Crest of the Peacock: NonEuropean Roots of Maths. London: Penguin Books. Pp. 286289.
Rajagopal, C.T. and M.S. Rangachari. 1986. On Medieval Keralese maths. Archive for Exact Sciences. 35:9199.
Raju, C.K. 2000. Talk given at the international seminar on EastWest Transitions, National Institute of Advanced Studies, Bangalore, Dec 2000.
Raju, C.K. 2001. Computers, Mathematics Education, and the Alternative Epistemology of the Calculus in the Yuktibhasa. Philosophy East and West, 51(3), 2001, 325–61.
Raju, C.K. In press. Cultural Foundations of Mathematics. Delhi: PHISPC/Oxford University Press.