The Kerala School, European Mathematics and Navigation
By D.P. Agrawal

The National Geographic has declared Kerala, the south-west 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 astronomer-mathematicians 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 well-known 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 (1445-1555) was mainly an astronomer,
but his Aryabhatiya Bhasya and Tantra Samgraha contain work
on infinite-series 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. 1500-75)
and Sankara Variar (c. 1500-1560), entitled Kriyakramakari, also contains
a discussion of Madhava's work. The Karana Paddhati by Putumana Somayaji
(c. 1660-1740) 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 infinite-series 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 third-order 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 circum-radius of
a cyclic quadrilateral,
which goes under the name of l'Huilier's formula;

The use of the Newton-Gauss interpolation formula (to
the second order) by Govindaswami; and

The statement of the mean value theorem of differential
calculus, first recorded by Paramesvara (1360-1455) 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 14th-15th 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 calendar-making 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 well-known 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 large-scale 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 re-interpretation
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" non-Eurocentric
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 present-day university mathematics by shifting away from formal
and spiritual mathematics-as-proof to practical and empirical mathematics-as-calculation.
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 "re-birth"
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 time-keeping (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 Indo-Arabic 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 Indo-Arabic 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: Non-European Roots of Maths.
London: Penguin Books. Pp. 286-289.

Rajagopal, C.T. and M.S. Rangachari. 1986. On Medieval Keralese maths. Archive
for Exact Sciences. 35:91-99.

Raju, C.K. 2000. Talk given at the international seminar on East-West 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.