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
- European perspectives on Indian and Kerala mathematics
- A Discussion on transmission
- The case for transmission: Priority, communication routes and methodological similarities
- The case for the transmission of Kerala mathematics to Europe: Motivation and evidence of transmission activity by Jesuits missionaries
- A conjecture for the mode of acquisition by the Jesuits of the manuscripts containing the Kerala calculus
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:
- the identification of methodological similarities;
- the existence of communication routes;
- a suitable chronology for the 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 Ne