Calculus, India's gift to Europe
The Jesuits took the trignometric tables and
planetary models from the Kerala School of Astronomy and Maths and exported
it to Europe starting around 1560 in connection with the European
navigational problem, says Dr Raju.
Dr C K Raju was a professor Mathematics and played a leading role in the C-DAC
team which built Param: India’s first parallel supercomputer. His ten year
research included archival work in Kerala and Rome and was published in a
book called " The Cultural Foundations of Mathematics". He has
been a Fellow of the Indian Institute of Advanced Study and is a Professor
of Computer Applications.
“When the Europeans received the Indian calculus, they couldn’t
understand it properly because the Indian philosophy of mathematics is
different from the Western philosophy of mathematics. It took them about 300
years to fully comprehend its working. The calculus was used by Newton to
develop his laws of physics,” opines Dr Raju.
The Infinitesimal Calculus: How and Why it Was Imported into Europe
By Dr C.K. Raju
It is well known that the “Taylor-series” expansion, that is at the
heart of calculus, existed in India in widely distributed mathematics /
astronomy / timekeeping (“jyotisa”) texts which preceded Newton and
Leibniz by centuries.
Why were these texts imported into Europe? These texts, and the accompanying
precise sine values computed using the series expansions, were useful for
the science that was at that time most critical to Europe: navigation. The
‘jyotisa’ texts were specifically needed by Europeans for the problem of
determining the three “ells”: latitude, loxodrome, and longitude.
How were these Indian texts imported into Europe? Jesuit records show that
they sought out these texts as inputs to the Gregorian calendar reform. This
reform was needed to solve the ‘latitude problem’ of European
navigation. The Jesuits were equipped with the knowledge of local languages
as well as mathematics and astronomy that were required to understand these
Indian texts.
The Jesuits also needed these texts to understand the local customs and how
the dates of traditional festivals were fixed by Indians using the local
calendar (“panchânga”). How the mathematics given in these Indian
ancient texts subsequently diffused into Europe (e.g. through clearing
houses like Mersenne and the works of Cavalieri, Fermat, Pascal, Wallis,
Gregory, etc.) is yet another story.
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 in 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 connection 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. 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, attempts to solve the
European navigational problem in the 16th and 17th centuries focused on
mathematics and astronomy. These were (correctly) believed to hold the key
to celestial navigation. It was widely (and correctly) held by navigational
theorists and mathematicians (e.g. by Stevin and Mersenne) that this
knowledge was to be found in the ancient mathematical, astronomical and
time-keeping (jyotisa) texts of the East.
Though the longitude problem has recently been highlighted, this was
preceded by the latitude problem and the problem of loxodromes. The solution
of the latitude problem required a reformed calendar. The European calendar
was off by ten days. This led to large inaccuracies (more than 3 degrees) in
calculating latitude from the measurement of solar altitude at noon using,
for example, the method described in the Laghu Bhâskarîya of Bhaskara I.
However, reforming the European calendar required a change in the dates of
the equinoxes and hence a change in the date of Easter. 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. Clavius subsequently reformed the Jesuit
mathematical syllabus at the Collegio Romano. He also headed the committee
which authored the Gregorian Calendar Reform of 1582 and remained in
correspondence with his teacher Nunes during this period.
Jesuits such as Matteo Ricci who trained in mathematics and astronomy under
Clavius’ new syllabus were sent to India. In a 1581 letter, Ricci
explicitly acknowledged that he was trying to understand the local methods
of time-keeping (‘jyotisa’) from the Brahmins and Moors in the vicinity
of Cochin.
Cochin was then the key centre for mathematics and astronomy since the
Vijaynagar Empire had sheltered it from the continuous onslaughts of Islamic
raiders from the north. Language was not a problem for the Jesuits since
they had established a substantial presence in India. They had a college in
Cochin and had even established printing presses in local languages like
Malayalam and Tamil by the 1570’s.
In addition to the latitude problem (that was settled by the Gregorian
Calendar Reform), there remained the question of loxodromes. These were the
focus of efforts of navigational theorists like Nunes and Mercator.
The problem of calculating loxodromes is exactly the problem of the
fundamental theorem of calculus. Loxodromes were calculated using sine
tables. 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.
Madhava's sine series
sin x = x - x^3/3! + x^5/5! - x^7/7!+......
The Europeans encountered difficulties in using these precise sine values
for determining longitude, as in the Indo-Arabic navigational techniques or
in the Laghu Bhâskarîya. This is because this technique of longitude
determination also required an accurate estimate of the size of the earth.
Columbus had underestimated the size of the earth to facilitate funding for
his project of sailing to the West. His incorrect estimate was corrected in
Europe only towards the end of the 17th century CE.
Even so, the Indo-Arabic navigational technique required calculations while
the Europeans lacked the ability to calculate. This is because 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 century and as Swift (of
‘Gulliver’s Travels’ fame) had satirized in the 17th century. This led
to the development of the chronometer, an appliance that could be
mechanically used without any application of the mind.