The Kerala School of astronomy and mathematics was an Indian school of mathematics and astronomy founded by Madhava of Sangamagrama in Kerala, South India, which included among its members several scientists. The school flourished in the 14th-16th centuries. In attempting to solve astronomical problems, the Kerala School independently created a number of important mathematics concepts. In this well documented article, Dennis Francis Almeida and George Gheverghese Joseph reconstruct the mathematics of Kerala School and attempt to show the possible ways of its transmission to modern Europe.
Table of contents
Note of the editor
An earlier version of this paper came out in 2007 online in Philosophy of Mathematics Education Journal (No. 20, June 2007, ISSN 1465-2978, edited by Paul Ernest; see the special issue on Social Justice, part 1; online version here (.doc file, 103kb). We are grateful to the two authors Dennis Francis Almeida and George Gheverghese Joseph for allowing republishing on www.MuslimHeritage.com.
According to the literature, the general methods of the calculus were invented independently by Newton and Leibniz in the late I7th century  after exploiting the works of European pioneers such as Fermat, Roberval, Taylor, Gregory, Pascal, and Bernoulli  in the preceding half century. However, what appears to be less well known is that the fundamental elements of the calculus including numerical integration methods and infinite series derivations for p and for trigonometric functions such as sin x, cos x and tan-1 x (the so-called Gregory series) had already been discovered over 250 years earlier in Kerala. These developments first occurred in the works of the Kerala mathematician Madhava and were subsequently elaborated on by his followers Nilakantha Somayaji, Jyesthadeva, Sankara Variyar and others between the 14th and 16th centuries .
In the latter half of the 20th century, there has been some acknowledgement of these facts outside India. There are several modem European histories of mathematics  which acknowledge the work of the Kerala School. However, it needs to be pointed out that this acknowledgement is not necessarily universal. For example, in the recent past a paper by Fiegenbaum on the history of the calculus makes no acknowledgement of the work of the Kerala School . However, prior to the publication of Fiegenbaum’s paper, several renowned publications detailing the Kerala calculus had already appeared in the West . Such a viewpoint may have its origins in the Eurocentrism that was formulated during the period of colonization by some European nations.
In the early part of the second millennium evaluations of Indian mathematics or, to be precise, astronomy were generally from Arab commentators. They tended to indicate that Indian science and mathematics was independently derived. Some, like ?a’id Al-Andalusi, claimed it to be of a high order:
“[The Indians] have acquired immense information and reached the zenith in their knowledge of the movements of the stars [astronomy] and the secrets of the skies [astrology] as well as other mathematical studies. After all that, they have surpassed all the other peoples in their knowledge of medical science and the strengths of various drugs, the characteristics of compounds, and the peculiarities of substances .”
Others like Al-Biruni were more critical. He asserted:
“I can only compare their mathematical and astronomical literature, as far as I know it, to a mixture of pear shells and sour dates, or of pearls and dung, or of costly crystals and common pebbles .”
Figure 2: Statue of Aryabhata on the grounds of Inter-University Centre for Astronomy and Astrophysics, Pune, India. As there is no known information regarding his appearance, any image of Aryabhata originates from an artist’s conception. (Source).
Nevertheless, a common element in most early evaluations is the emphasis on the uniqueness of Indian mathematics. However, by the 19th century and contemporaneous with the establishment of European colonies in the East, the views of European scholars about the supposed superiority of European knowledge were developing racist overtones. Sédillot  reported that not only was Indian science indebted to Europe but also that the Indian numbers are an ‘abbreviated form’ of Roman numbers, that Sanskrit is ‘muddled’ Greek, and that India had no reliable chronology. Although Sédillot’s assertions were based on imperfect knowledge and understanding of the nature and scope of Indian knowledge, this did not deter him from concluding:
“On one side, there is a perfect language, the language of Homer, approved by many centuries, by all branches of human cultural knowledge, by arts brought to high levels of perfection. On the other side, there is [in India] Tamil (sic) with innumerable dialects and that Brahmanic filth which survived to our day in the environment of the most crude superstitions.”
In a similar vein, Bentley  also cast doubt on the chronology of India by locating Aryabhata and other Indian mathematicians several centuries later than was actually the case. He was of the opinion that Brahmins had actively fabricated evidence to locate Indian mathematicians earlier than they existed:
“We come now to notice another forgery, the Brahma Siddhanta Sphuta, the author of which I know. The object of this forgery was to throw Varaha Mihira, who lived about the time of Akber, back into antiquity… Thus we see how Brahma Gupta, a person who lived long before Aryabhata and Varaha Mihira, is made to quote them, for the purpose of throwing them back into antiquity…. It proves most certainly that the Brahma Siddhanta cited, or at least a part of it, is a complete forgery, probably framed, among many other books, during the last century by a junta of Brahmins, for the purpose of carrying on a regular systematic imposition.”
For the record, the actual dates are Aryabhata (born 476 CE), Varahamihira existed around 500 CE, Brahmagupta composed his famous work in 598 CE, and Akbar lived around 1550. So it is justifiable to suggest that Bentley’s hypothesis was an indication of either ignorance or a fabrication based on a Eurocentric history of science. Nevertheless Bentley’s altered chronology had the effect not only of lessening the achievements of the Indian mathematics but also of making redundant any conjecture of transmission to Europe.
Inadequate understanding of Indian mathematics was not confined to run of the mill scholars. More recently Smith , an eminent historian of mathematics of the 20th century, claimed that, without the introduction of western civilization in the 18th and 19th centuries, India would have stagnated mathematically. He went on to say that: “Not since Bhaskara (i.e. Bhaskara II, b. 1114) has she produced a single native genius in this field.”
Figure 3: View of Jantar Mantar Observatory in New Delhi, India, completed in 1734. (Source).
This inclination to ignore advances in and priority of discovery by non-European mathematicians persisted until even very recent times. For example there is no mention of the work of the Kerala School of mathematics and astronomy in Edwards’ text  on the history of the calculus nor in articles on the history of infinite series by historians of mathematics such as Abeles  and Fiegenbaum . A possible reason for such puzzling standards in scholarship may have been the Eurocentrism that accompanied European colonization . With this phenomenon, the assumption of white superiority became dominant over a wide range of activities, including the writing of the history of mathematics. The rise of nationalism in 19th-century Europe and the consequent search for the roots of European civilisation, led to an obsession with Greece and the myth of Greek culture as the cradle of all knowledge and values and Europe becoming heir to Greek learning and values . Rare exceptions to this skewed version of history are provided by earlier writers such as Ebenezer Burgess and George Peacock .They, respectively, wrote:
“Professor Whitney seems to hold the opinion, that the Hindus derived their astronomy and astrology almost bodily from the Greeks… I think he does not give the Hindus the credit due to them, and awards to the Greeks more credit than they are justly entitled to .”
“… (I)t is unnecessary to quote more examples of the names even of distinguished men who have written in favour of a hypothesis [of the Greek origin of numbers and of their transmission to India] so entirely unsupported by facts .”
However, by the latter half of the 20th century, European scholars, perhaps released from the powerful influences induced by colonization, had started to analyse the mathematics of the Kerala School using largely secondary sources such as Rajagopal and his associates . The achievements of the Kerala School and their chronological priority over similar developments in Europe were now being aired in several Western publications . However these evaluations are accompanied by a strong defence of the European claim for the invention of the generalised calculus. For example, Baron  states that: “The fact that the Leibniz-Newton controversy hinged as much on priority in the development of certain infinite series as on the generalisation of the operational processes of integration and differentiation and their expression in terms of a specialised notation does not justify the belief that the [Kerala] development and use for numerical integration establishes a claim to the invention of the infinitesimal calculus.” Calinger  writes: “Kerala mathematicians lacked a facile notation, a concept of function in trigonometry…Did they nonetheless recognise the importance of inverse trigonometric half chords beyond computing astronomical tables and detect connections that Newton and Leibniz saw in creating two early versions of calculus? Apparently not.”
These comparisons appear to be defending the roles of Leibniz and Newton as inventors of the generalised infinitesimal calculus. While we understand the strength of nationalist pride in the evaluation of the achievements of scientists, we do find difficulty in the qualitative comparison between two developments founded on different epistemological bases. It is worthwhile stating here that the initial development of the calculus in 17th century Europe followed the paradigm of Euclidean geometry in which generalisation was important and in which the infinite was a difficult issue . On the other hand, from the 15th century onwards the Kerala mathematicians employed computational mathematics with floating point numbers to understand the notion of the infinitesimal and derive infinite series for certain targeted functions  (Whish, 1835). In our view it is clear that using qualitatively different intellectual tools and in different eras to investigate the similar problems are likely to produce qualitatively different outcomes. Thus the sensible way to understand Kerala mathematics is to understand it within the epistemology in which it was developed. To do otherwise is akin to trying to gain a full appreciation of the literature of Shakespeare by literally translating it into Urdu – the semantic and cultural connotations would undoubtedly be lost.
The basis for establishing the transmission of science may be taken to be direct evidence of translations of the relevant manuscripts. The transmission of Indian mathematics and astronomy since the early centuries in common era via Islamic scholars to Europe has been established by direct evidence. The transmission of Indian computational techniques was in place by at least the early 7th century for by 662 AD it had reached the Euphrates region . A general treatise on the transmission of Indian computational techniques to Europe is given by Benedict . Indian Astronomy was transmitted westwards to Iraq, by a translation into Arabic of the Siddhantas around 760 CE  and into Spain. This transmission was not just westwards for there is documentary evidence of Indian mathematical manuscripts being found and translated in China, Thailand, Indonesia and other south-east Asian regions from the 7th century onwards . In the absence of such direct evidence, the following is considered by some to be sufficient  to establish transmission:
Further, there is van der Waerden’s ‘hypothesis of a common origin’ to establish the transmission of (mainly Greek) knowledge . Neugebauer uses his paradigm to establish his conjecture about the Greek origins of the astronomy contained in the Siddhantas. Similarly van der Warden uses the ‘hypothesis of a common origin’ to claim that Aryabhata’s trigonometry  was borrowed from the Greeks. Van der Waerden makes a similar claim about Bhaskara’s work on Diophantine equations and, whilst offering an argument based on methodological similarities, he is sufficiently convinced about the existence of an unknown Greek manuscript which was available to Bhaskara and his students . Van der Waerden concludes his work on the Greek origins of these works of Aryabhata and Bhaskara work by stating that scientific discoveries are, in general, dependent on earlier prototypical works .
Figure 4: 2 pages from Rigveda manuscript of mathematics in Sanskrit on paper, India, early 19th century, 4 vols., 795 folios. The manuscript is in Devanagari script with deletions in yellow, Vedic accents, corrections etc in red. (Source).
What we see from these paradigms is that a case for claiming the transmission of knowledge from one region to another does not necessarily rest on documentary evidence. This is a consequence of the fact that many documents from ancient and medieval times do not now exist, having perished due to variety of reasons. In these circumstances priority, communication routes, and methodological similarities appear to establish a socially acceptable case for transmission from West to East. Despite these elements being in place, the case for the transmission of Kerala mathematics to Europe seems to require stronger evidence. One has merely to survey the literature of the history of mathematics to date to see hardly any credible mention about the possibility of this transmission .
So how can our conjecture of transmission of Kerala mathematics possibly be established? The tradition in renaissance Europe was that mathematicians did not always reveal their sources or give credit to the original source of their ideas. However the activities of the monk Marin Mersenne between the early 1620’s to 1648 suggest some attempt at gathering scientific information from the Orient. Mersenne was akin to being “the secretary of the early republic of science .” Mersenne corresponded with the leading renaissance mathematicians such as Descartes, Pascal, Hobbes, Fermat, and Roberval. Though a Minim monk, Mersenne had had a Jesuit education and maintained ties with the Collegio Romano. Mersenne’s correspondence reveals that he was aware of the importance of Goa and Cochin (in a letter from the astronomer Ismael Boulliaud to Mersenne in Rome) , he also wrote of the knowledge of Brahmins and “Indicos”  and took an active interest in the work of orientalists such as Erpen – regarding Erpen he mentions his “les livres manuscrits Arabics, Syriaques, Persiens, Turcs. Indiens en langue Malaye .”
It is possible that between 1560 and 1650, knowledge of Indian mathematical, astronomical and calendrical techniques accumulated in Rome, and diffused to neighbouring Italian universities like Padua and Pisa, and to wider regions through Cavalieri and Galileo, and through visitors to Padua, like James Gregory. Mersenne may have also had access to knowledge from Kerala acquired by the Jesuits in Rome and, via his well-known correspondence, could have helped this knowledge diffuse throughout Europe. Certainly the way James Gregory acquired his Geometry after his four year sojourn in Padua where Galileo taught suggests this possibility.
All this is circumstantial – to make our case for the transmission of Kerala mathematics we will use stronger criteria. In addition to the Neugebauer criteria of priority, communication routes, and methodological similarities, we propose to test the hypothesis of transmission on the grounds of motivation and evidence of transmission activity by Jesuits missionaries. In the next section all these aspects will be discussed.
The priority of Kerala developments in the calculus over that of Newton and Leibniz is now beyond doubt . Madhava (1340-1425) is credited with the original ideas in the Kerala mathematics. These ideas led to derivation for the infinite series for p which we have illustrated earlier and to infinite series for a range of trigonometric functions. These developments in the calculus, therefore, precede the late 17th century calculus of Newton and Leibniz by at least 250 years. A communication route between the South of India and the Arabian Gulf (via the port of Basrah) had been in existence for centuries . The arrival of the Portuguese Vasco da Gama to the Malabar Coast in 1499 heralded a direct route between Kerala and Europe via Lisbon. Thus, after 1499, despite its geographical location, which prevented easy communication routes with the rest of India, Kerala was linked with the rest of the world and, in particular, directly to Europe.
Whilst the two aspects of priority and communication routes are readily established, the existence of methodological similarities requires further discussion. Firstly there is a similarity in the approach to calculus in the Yuktibhasa and the approach to calculus adopted by Fermat, Pascal, Wallis, and others. In the Yuktibhasa the following key result is proved :
This exact result was adopted by Fermat, Pascal, Wallis, and others in the 17th century to evaluate
The area under the parabolas y=xk, or, equivalently, calculate ò xk dx.
At this point, we remark that our conjecture on the transmission of Kerala mathematics is given credence by the fact that Wallis used reasoning similar to the ones given in the Yuktibhasa. That is, Wallis replaces the term n2 by n(n +1) implicitly implying that, as n tends to ¥, (n + 1) can be replaced by n so that he can use the approximation n(n + 1) ˜ n2 . Methodological similarities between the mathematics of Aryabhata school, upon which Kerala mathematics is based, and the works of the renaissance mathematicians are not infrequent. We give two further examples.
In 1655, John Wallis stated in his Arithmetica Infinitorum that the convergents of the continued fraction satisfied certain relations. Exactly these recurrence relations were discovered by Bhaskara II some 500 years earlier in his Bijaganita . By itself, this fact may not be much more than a coincidence except that Wallis gives exactly Bhaskara’s proof of ‘Pythagoras‘ theorem in his treatise on angular sections . Brezinski posits that Wallis may have derived his results independently , but does objectively suggest a scenario for the possible transmission of Bhaskara’s work . Another similarity with Bhaskara’s works is the challenge problem that Fermat issued in 1657 “What is for example the smallest square which, multiplied by 61 with unity added, makes a square?” As Struik  notes, Indian mathematicians already had a solution to this problem – the case of A = 61 is given as a solved example in the BeejGanita text of Bhaskara II. This coincidence is not trivial when we consider that the solution [x = 1,766,319,049, y = 226,153,980] involves rather large numbers. A similar problem had earlier been suggested by the 7th-century Brahmagupta, and Bhaskara II provides the general solution with his chakravala method. Thus, this challenge problem suggests a connection of Fermat with Indian mathematics.
Figure 5a-b: Palm leaf manuscript from Kerala. (Source).
The motivation for the import of knowledge from India to Europe arose from the need of greater accuracy in arithmetical computation [as is well attested by historian of mathematics], the calendar, and in astronomy. For example, by the middle of the 16th century there was an error in the calculations that formed the basis of the existing Julian calendar. The true solar year was around 11.25 minutes less than the assumed 365.25 days thus causing a cumulative error which was offsetting the date of Easter appreciably. For example, the vernal equinox was scheduled by the calendar to take place on 21 March but it actually took place on March 11 — thus, without correction Easter would eventually take place in Summer rather than in Spring. The evidence for this may be found in the use of the Indian calendrical term ‘tithi’  in Viète’s critique of the Gregorian calendar reform.
In astronomy we point to the attested remarkable similarities between the planetary model by the Kerala mathematician Nilakantha and the later one by Tycho Brahe and the adoption by Kepler of the 10th century Indian lunar model of the astronomer Munjala .
The arrival of Francis Xavier in Goa in 1540 heralded a continuous presence of the Jesuits in the Malabar till 1670. While the early Jesuits were interested in learning the vernacular languages and conversion work, the latter Jesuits who arrived after 1578 were of a different mould. The famous Matteo Ricci was in the first batch of Jesuits trained in the new mathematics curriculum introduced in the Collegio Romano by Clavius. Ricci was an accomplished mathematician . Ricci also studied cosmography and nautical science in Lisbon prior to his arrival in India in 1578. Ricci’s arrival in Goa was significant in respect of Jesuit acquisition of local knowledge. His specialist knowledge of mathematics, cosmography, astronomy and navigation made him a candidate for discovering the knowledge of the colonies and he had specific instructions to investigate the science of India .
Figure 6: Old photo of the Madayi Mosque in the Malabar Coast, Kerala State, in the period 1919-1929. (Source).
Subsequently several other Jesuit scientists trained both by Clavius or Grienberger (Clavius’ successor as Mathematics Professor at the Collegio Romano) were sent to India. Most notable of these, in terms of their scientific activity in India, were Johann Schreck and Antonio Rubino. The former had studied with the French mathematician Viète, well known for his work in algebra and geometry. At some point in their stay in India these Jesuits went to the Malabar region including the city of Cochin, the epicentre of developments in the infinitesimal calculus . Later, we shall see that all these scholarly Jesuits attempted to acquire local knowledge. For the moment, we point out that the Jesuits had an interest in the calendar that stemmed from the Church’s desire to reform the erroneous dating of Easter and other festivals – Clavius headed the commission that ultimately reformed the Gregorian calendar in 1582. Others have conjectured that these Jesuits were part of an interchange of scientific ideas between Europe on the one hand and India and China on the other .
We have mentioned above that there was a batch of mathematically able Jesuits who arrived in India in the late 15th and early 16th centuries and who had specific objectives to study local knowledge. We should point out that the mathematics of the Kerala school was essentially astronomy which had application to astrological prediction. Consequently, it is our conjecture that the Jesuits had the opportunity and the motivation to transmit the calculus back to Europe bundled in the knowledge of the astronomy and calendrical science. With regard to the earlier Jesuits in the Malabar Coast we observe that several references in the historical works of Wicki  indicate that they were interested in arithmetic, astronomy and timekeeping of the region. They were able to appreciate this knowledge by their learning of the vernacular languages such as Malayalam and Tamil . The rationale for learning the vernacular languages was to aid their work in converting the local populace to Jesuit Catholicism by understanding their science, culture and customs. A prominent early Jesuit de Nobili, for instance, in 1615, wrote a critical paper on the Varamihira’s Vedanga Jyotisa . Indeed, it appears that the Jesuits tried to formalise this policy by including local sciences such as astrology or jyotisa in the curriculum of the Jesuit colleges in the Malabar Coast . The early Jesuits were also active in the transmission of local knowledge back to Europe. Evidence of this knowledge acquisition is contained in the collections Goa 38, 46 and 58 to be found in the Jesuit historical library in Rome (ARSI). The last collection contains the work of Father Diogo Gonsalves on the judicial system, the sciences and the mechanical arts of the Malabar region. This work started from the very outset . The translation of the local science into European languages prior to transmission to Europe was epitomised by Garcia da Orta’s popular Colloquios dos simples e drogas he cousas mediçinas da India published in Goa in 1563 – there may have been other publications of this type which remain obscured because possibly due to linguistic and nationalistic reasons .
If the early and late Jesuits were involved in learning the local sciences then, given the academic credentials of the Jesuits such as Ricci, Schreck and Rubino of the middle period, it is a plausible conjecture that this work continued and with greater intensity. There is some evidence that this did happen. It is also known that Ricci made enquiries about Indian calendrical science – in a letter to Maffei he states that he requires the assistance of an “intelligent Brahmin or an honest Moor” to help him understand the local ways of recording and measuring time or jyotisa . Then there is de Menses who, writing from Kollam in 1580, reports that, on the basis of local knowledge, the European maps have inaccuracies . There were other later Jesuits who report of scientific findings on such diverse things as calendrical sciences and inaccuracies in the European maps and mathematical tables. Antonio Rubino wrote, in 1610, similarly about inaccuracies in European mathematical tables for determining time . Then there is the letter from Schreck, in 1618, of astronomical observations intended for the benefit of Kepler  – the latter had requested the eminent Jesuit mathematician Paul Guldin to help him to acquire these observations from India to support his theories .
Whilst this does not establish the fact that these Jesuits obtained manuscripts containing the Kerala mathematics it does establish that their scientific investigations about the local astronomy and calendrical sciences would have lead them to an awareness of this knowledge. There are some reports  that the Brahmins were secretive and unwilling to share their knowledge. However this was not an experience shared by many others. For example in the mid-17th century Fr. Diogo Gonsalves, who learnt the local language Malayalam well, was able to write a book about the administration of justice, sciences and mechanical arts of the Malabar. This book is to be found in the MS Goa 58 in the Jesuit Historical library (ARSI), Rome. And a Brahmin spent eight years translating Sanskrit works for Fr. Frois during the same time . The information gathering and transmission activities of the Jesuit missionaries are thus not in doubt. In addition after the 1580 annexation of Portugal by Spain and subsequent loss of funding from Lisbon, the rationale for transmission acquired another dimension, that of profit . Whatever the nature of the profit, intellectual or material, the motivation may have been sufficient for the learned Jesuits to have acquired the relevant manuscripts containing the Kerala mathematics.
Even if the conjecture of the transmission of the Kerala calculus by the Jesuit missionaries is accepted it leaves open the question as to how might the Jesuits have obtained key manuscripts of Indian astronomy such as the Tantrasangraha and the Yuktibhasa? Such manuscripts would require the Jesuits being in close contact with scholarly Brahmins or Kshatriyas. We have already mentioned above that at least one scholarly Brahmin was working for the Jesuits. In addition, as we shall now show, the Jesuits were in communication with the Kshatriya kings of Cochin whose scholarship enabled them to be in very close proximity to the Kerala mathematics.
The kings of Cochin came from the scholarly Kshatriya Varma ‘Tampuran’ family who were knowledgeable about the mathematical and astronomical works of medieval Kerala. This is attested by  and Srinivasiengar  who state that the author of the Sadratnamala is Sankara Varma, the younger brother of Raja of Cadattanada near Tellicherry and further states that the Raja is a very acute mathematician. Srinivasiengar  further refers to the Malyalam History of Sanskrit Literature in Kerala which identifies the King of Cochin, Raja Vanna, as being aware of the chronology of the Karana-Paddhati. Rama Varma Tampuran who, in 1948 (together with A.R. Akhileswara Iyer) had published an exposition in Malayalam on the Yuktibhasa was one of the princes of Cochin.
Rajagopal and Rangachari  state that Rama Vanna Tampuran supplied them with the manuscript material relating to Kerala mathematics. Sanna  identifies the valuable contribution to the analysis of Kerala astronomy by Rama Varma Maru Tampuran. Mukunda Marar (who is the eldest son of the last king of Cochin) stated in a personal communication to the Aryabhata Forum that the kings of Cochin would have, at the least, been aware both of the astronomical methods for astrological prediction and of the manuscripts that contained these methods. Several were scholarly enough to publish commentaries of the mathematical and astronomical works. Mukunda Marar, himself, worked with Rajagopal and published a work . Moreover, various authors, from Charles Whish in the 19th century to Rajagopal and Rangachari in the 20th century have acknowledged that members of the royal household were helpful in supplying these manuscripts in their possession.
This suggests that the former royal family in Cochin, which was in possession of a large number of MSS, had not only a scholarly tradition, but also a tradition of helping other scholars. Thus, the royal family could itself have been a possible source of knowledge for the Jesuits. Indeed the Jesuits working on the Malabar Coast had close relations with the kings of Cochin . Furthermore, around 1670, they were granted special privileges by King Rama Varma  who, despite his misgivings about the Jesuit work in conversion, permitted members of his household to be converted to Christianity . The close relationship between the King of Cochin and the foreigners from Portugal was cemented by King Rama Varma’s appointment of a Portuguese as his tax collector . Given this close relationship with the Kings of Cochin, the Jesuit desire to know about local knowledge, and the royal family’s contiguity to the works on Indian astronomy, it is quite possible that the Jesuits may have acquired the key manuscripts via the royal household.
In conclusion, it is worth noting that we have focused so far on evidence of direct transmis¬sions of Kerala ideas to Europe. But, as pointed out by Bala, the transmission of the discoveries of Kerala mathematics could have been as ‘know-how’ and computation tech¬niques through the channel of craftsmen and technicians. This may explain the absence of documentary evidence in Jesuit communications. Even if there is documentary evidence in 16th century European manuals used for navigation, map-making and calendar construc¬tion of the use of approximate series derived from the discoveries of the Kerala School, it would hardly have been directly communicated to European mathematicians. After all craftsmen oriented to practical rather than theoretical concerns are not likely to write to leading mathematical figures in Europe or be taken seriously if they did so. For more a detailed discussion of this interesting conjecture, see the aforementioned article by Bala.
The authors would like to acknowledge the Arts and Humanities Research Board (AHRB) support for a project on the investigation of Kerala mathematics during the period 2002-2005. Also acknowledged is the input from C. K. Raju during the period preceding the AHRB project. Parts of this paper were incorporated in a presentation at the XXI International conference on the history of science in September 2001, Mexico City, and in a published article by the authors: ‘Eurocentrism in the History of Mathematics: The case of the Kerala School’, Race and Class, 2004, 45(4), 45-59.
 See, for example, Margaret Baron, The Origins of the Infinitesimal Calculus, Oxford, Pergamon, 1969, p 65.
 See for example, Charles Edwards, The Historical Development of the Calculus, New York, Springer-Verlag, 1979, p. 189, and Victor Katz, “Ideas of calculus in Islam and India”, Mathematics Magazine, Washington, 68 (1995)3: 163-174; pp. 163-164.
 See the work of K Venkateswara Sarma, A History of the Kerala School of Hindu Astronomy, Hoshiarpur. Vishveshvaranand Vedic Research Institute. 1972, pp. 21-22; and the paper by Charles Whish, “On the Hindu quadrature of the circle and the infinite series of the proportion of the circumference to the diameter exhibited in the four Shastras, the Tantrasamgraham, Yukti¬-Bhasa, Carana Padhati, and Sadratnamala,” Transactions of the Royal Asiatic Society of Great Britain and Ireland, London. 3 (1835): 509-523, pp. 522-23.
 For example, Margaret Baron, Origins of Calculus, op. cit., pp. 62-63; Ronald Calinger, A Contextual History of Mathematics to Euler, New Jersey, Prentice Hall, 1999, p. 284.
 Leone Fiegenbaum, “Brook Taylor and the Method of Increments”, Archive for the History of Exact Sciences, Baltimore, 34 (1986): 1-I40, p. 72.
 For example, Charles Whish, “On the Hindu quadrature of the circle and the infinite series of the proportion of the circumference to the diameter exhibited in the four Shastras. The TantrasamgrahamYukti¬-bhasa Padhati, and Sadratnamala”, Transactions, 3 (1835): 509-523; C T Rajagopal and M S Rangachari, “On an Untapped Source of Medieval Kerala Mathematics”, Archive for the History of Exact Sciences, Baltimore, 18 (1978): 89-102: C T Rajagopal and T V Vedamurthi, “On the Hindu proof of Gregory’s series”, Scripta Mathematica, New York, I8(1951): 91-99.
 S. Al-Andalusi, c. 1068, Science in the Medieval World, translated by S. I. Salem and A. Kumar. University of Texas Press, 1991, pp. 11-12.
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 C. H. Edwards, 1979, The Historical Development of the Calculus, Springer-Verlag, New York.
 F.F. Abeles, 1993. “Charles Dodgson’s geometric approach to arctangent relations for p”, Historia Mathematica, 20 (2),151-159.
 L. Fiegenbaum, 1986. “Brook Taylor and the Method of Increments”, Archive for the History of Exact Sciences, 34(1): 1-140.
 G G.Joseph, 1995 . “Cognitive Encounters in India during the Age of Imperialism”, Race and Class, 36 (3): 39-56.
 G. G. Joseph. 2000. The Crest of the Peacock: non-European Roots of Mathematics. Princeton University Press, p. 215. An expanded third edition was published by Princeton in 2010.
 E. Burgess 1860. The Surya Siddhanta: A Text-Book Of Hindu Astronomy, Reprinted 1997. Motilal Banarsidass Publishers Private Limited, New Delhi.
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 For example, C. T. Rajagopal and T. V. Vedamurthi 1952: “On the Hindu proof of Gregory’s series,” Scripta Mathematica, 18, 65-74, C. T. Rajagopal and M. S. Rangachari 1978: “On an Untapped Source of Medieval Kerala Mathematics,” Archive for the History of Exact Sciences, 18, 89-102.
 See for example V. J. Katz 1992. A History. of Mathematics: An Introduction, Harper Collins, New York; V. J. Katz 1995. “Ideas of calculus in Islam and India”, Mathematics Magazine. Washington, 68, 3: 163-174; R. Calinger 1999, A Contextual History of Mathematics to Euler, Prentice Hall, New Jersey.
 M. E. Baron 1969. The Origins of the Infinitesimal Calculus. Pergamon, Oxford, p. 65.
 R. Calinger, 1999, A Contextual History of Mathematics to Euler, Prentice Hall, New Jersey, p. 28.
 See for example V.J. Katz 1992, A History of Mathematics: An introduction, Harper Collins, New York.
 Charles A. M. Whish 1835. “On the Hindu quadrature of the circle and the infinite series of the proportion of the circumference to the diameter exhibited in the four Shastras, the Tantrasamgraham, Yukti-Bhasa, Carana Padhati, and Sadratnamala.” Tr. Royal Asiatic Society of Gr Britain and Ireland, 3, 509-523.
 John Berggren 1986. Episodes in the Mathematics of Medieval Islam. New York, Springer Verlag, p. 30.
 Susan Benedict 1914. A Comparative Study of the Early Treatises introducing into Europe the Hindu Art of Reckoning. Ph.D. Thesis, University of Michigan, Rumford Press,
 B V Subbarayappa and K Venkateswara Sarma 1985. Indian Astronomy: A Source-Book. Bombay, Nehru Centre Publications, p. XXXVIII.
 B V Subbarayappa and K Venkateswara Sarma 1985. Indian Astronomy, op. cit., p. XXXVII.
 Otto Neugebauer 1962. The Exact Sciences in Antiquity. New York, Harper, pp. 166-167.
 B van der Waerden 1983. Geometry and Algebra In Ancient Civilizations, Berlin, Springer-Verlag, p. 211.
 B. van der Waerden. Geometry and Algebra, op. cit., p. 133.
 B. van der Waerden 1976. “Pell’s equation in Greek and Hindu mathematics,” Russian Mathematical Surveys 31: 210-225, p. 210. Here he states that “the original common source of the Hindu authors was a Greek treatise in which the whole method was explained.”
 B. van der Waerden, “Pell’s equation in Greek and Hindu mathematics”, Russ. Math Surveys, op. cit., p. 221: “… in the history of science independent inventions are exceptions: the general rule is dependence.”
 A few notable exceptions are: Victor Katz, “Ideas of calculus in Islam and India,” Math Magazine, 68 (1995), 3: 163-174, pp. 173-174; Ronald Calinger, Contextual History of. Mathematics, op. cit., p. 282; George Joseph, The Crest of the Peacock:: Non-European Roots of Mathematics, Princeton, Princeton University Press, 2000, pp. 354-356.
 Ronald Calinger, Contextual History of Mathematics, op. cit., p. 475.
 Marin Mersenne, Correspondance du P. Marin Mersenne, 18 vols., Paris, Presses Universitaires de France, 1945-, vol. XIII, p. 267.
 Marin Mersenne, Correspondance, op. cit., vol. XIII, pp. 518-521.
 Marin Mersenne, Correspondance, op. cit., vol II, pp. 103, 115.
 Margaret Baron 1995. Origins of Calculus. op cit. p 63: Victor Katz, “Ideas of calculus in Islam and India”, Math Magazine, 68 (3): 163-174, pp. 171-174.
 B V Subbarayappa and K Venkateswara Sarma, Indian Astronomy, op. cit., p. XXXVIII: Victor Katz, “Ideas of calculus in Islam and India”, Math Magazine, op. cit., p. 174.
 Victor Katz,”Ideas of calculus in Islam and India”, Math Magazine, op. cit., p. 169.
 Joseph Scott 1981. The Mathematical Work of John Wallis. New York, Chelsea, p. 30.
 Claude Brezinski, History of Continued Fractions, op. cit., p. 43.
 Henry Colebrooke, Miscellaneous Essays, op. cit., vol. 2, p. 395.
 Claude Brezinski, History of Continued Fractions, op. cit., p. 32. He states that “the recurrence relationship of continued fractions (of Bhaskara II) will only be discovered in Europe by John Wallis in 1655, which is 500 years later! Thus it is important to notice that the first English translation of Bhascara II’s work appears in 1816.”
 Claude Brezinski, History of Continued Fractions, op. cit., p. 34. Here he posits that “it is highly probable that Muslim mathematicians were in close contact with their Indian colleagues during this period, and that they brought back their writings to western countries. Since Arabic mathematicians were translated much earlier into European languages, it is possible that the Indian mathematical writings were known in Europe before their translation.”
 Dirk Struik 1969. A Source Book in Mathematics, 1200-1800. Cambridge, Mass., Harvard University Press, pp. 29-30.
 R Bein 2007. “Viète’s Controversy with Clavius Over the Truly Gregorian Calendar,” Archive for History of Exact Sciences 61(1): 39-66.
 Dennis Duke 2007. ‘The Second Lunar Anomaly in Ancient Indian Astronomy”, Archive for the History of Exact Sciences, 61, 147-157.
 Vincent Cronin 1984. The Wise Man From the West. London, Collins, p. 22.
 Henri Bernard 1973. Matteo Ricci’s Scientific Contribution to China, Westport, Conn., Hyperion Press, p. 38: “Ricci had resided in the cities of Goa and of Cochin for more than three years and a half (September 13, 1578-April 15, 1582). He had been requested to apply himself to the scientific study of this new and imperfectly know country, in order to document his illustrious contemporary, Father Maffei, the ‘Titus Livius’ of Portuguese explorations.”
 See for example Isaia Iannaccone 1998. Johann Schreck Terrentius, Napoli, Instituto Universitario Orientale, pp 50-58 and Ugo Baldini 1992. Studi su filosofia e scienza dei gesuiti in Italia 1540-1632, Firenze, Bulzoni Editore, pp. 214-215.
 Ugo Baldini, Studi su filosofia, op. cit., p. 70: “It can he recalled that many the best Jesuit students of Clavius and Geienberger (beginning with Ricci and continuing with Spinola, Aleni, Rubino, Ursis, Schreck, and Rho) became missionaries in Oriental Indies. This made them protagonists of an interchange between the European tradition and those Indian and Chinese, particularly in mathematics and astronomy, which was a phenomenon of great historical meaning.”
 Josef Wicki 1948-. Documenta Indica, 16 volumes. Rome, Monumenta Historica Societate Iesu, vol. IV, p. 293 and vol. VIII, p. 458.
 Josef Wicki, Documenta Indica, op. cit., vol. XIV, p. 425 and vol. XV, p. 34.
 Josef Wicki, Documenta Indica, op. cit., vol. III, p. 307.
 Domenico Ferroli 1939. The Jesuits in Malabar, 2 vols. Bangalore. Bangalore Press, vol 2, p. 402: “In Portuguese India hardly seven years after the death of St. Francis Xavier the fathers obtained the translation of a great part of the 18 Puranas and sent it to Europe. A Brahmin spent eight years in translating the works of Veaso (Vyasa) several Hindu books were got from Brahmin houses, and brought to the Library of the Jesuit college. These translations are now preserved in the Roman Archives of the Society of Jesus. (Goa 46).”
 George Sarton 1955. The Appreciation of Ancient and Medieval Science during the Renaissance (1450-1600) , Philadelphia, University of Pennsylvania, p. 102.
 Josef Wicki, Documenta Indica, op. cit., vol XII, p. 474.
 Josef Wicki, Documenta Indica, op. cit., vol. XI, p. 185: “I have sent Valignano a description of the whole world by many selected astrologers and pilots, and others in India, which had no errors in the latitudes, for the benefit of the astrologers and pilots that every day come to these lands, because the maps theirs are all wrong in the indicated latitudes, as I clearly saw.”
 Ugo Baldini, Studi su filosofia, op. cit., p. 214: “Comparing the real local times with those inferable from the ephemeridis [tables] of Magini, he [Rubino] found great inaccuracies and, therefore, requested other ephemeridis.”
 Isaia Iannccone, Johann Schreck, op. cit., p. 58.
 Carola Baumgardt 1951. Johannes Kepler: Life and Letters, New York, Philosophical Library, p. 153.
 Pascual D’Elia 1960. Galileo In China: Relations through the Roman College between Galileo and the Jesuit Scientist-Missionaries (1610-1640) . Translated by Rufus Suter and Matthew Sciascia. Cambridge, Mass., Harvard University Press, p. 15.
 Domenico Ferroli, The Jesuits in Malabar, op. cit., p. 402.
 Domenico Ferroli, The Jesuits in Malabar, op. cit., vol. 2, p. 93: “Most of the Jesuit missionaries set to work to master the vernaculars… some of their number studied Indian books and Indian philosophy, not merely with the idea of refuting it, but with the desire of profiting by it.”
 Charles Whish, “On the Hindu quadrature of the circle and the infinite series of the proportion of the circumference to the diameter exhibited in the four Shastras, the Tantrasamgraham, Yukti-Bhasa. Carana Padhati, and Sadratnamala”, Transactions, 3 (1835): 509-523, p. 521.
 C N Srinivasiengar 1967. The History of Ancient Indian Mathematics. Calcutta, World Press, p. 146.
 C N Srinivasiengar, The History of Ancient Indian Mathematics, op. cit., p. 145.
 C T Rajagopal and M S Rangachari 1978. “On an Untapped Source of Medieval Kerala Mathematics”, Archive for History, 18: 89-102, p. 102.
 K Venkateswara Sarma, A History of the Kerala School, op. cit., p. 12.
 C T Rajagopal and Mukunda Marar 1944. “On the Hindu Quadrature of the Circle”, Journal of the Royal Asiatic Society (Bombay branch) . 20: 65-82.
 See for example Josef Wicki, Documenta Indica, op. cit., vol. X, pp. 239, 834, 835, 838, 845.
 Josef Wicki, Documenta Indica, op. cit., vol. XV, p. 224.
 Josef Wicki, Documenta Indica, op. cit., vol XV, p. 7.
 Josef Wicki, Documenta Indica, op. cit., vol. XV, p. 667.
 ABala: ‘Establishing Transmissions: Some Methodological Issues’ in G G Joseph (ed), Kerala Mathematics History and its possible Transmission to Europe, Delhi, B.R Publishing Corporation, 2009, 155-179.