By CK Raju

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 (

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. 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. focussed 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 (

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

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 of understand local methods of timekeeping (

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

*C. K. Raju holds a Ph.D. from the Indian Statistical Institute. He taught mathematics for several years before playing a lead role in the C-DAC team which built Param: India’s first parallel supercomputer. His earlier book ‘Time: Towards a Consistent Theory’ (Kluwer Academic, 1994) set out a new physics with a tilt in the arrow of time. He has been a Fellow of the Indian Institute of Advanced Study and is a Professor of Computer Applications.**This paper questioning the origin of calculus was proposed by him at the Indian Mathematical Congress held at Delhi in January 2009*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. 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. focussed 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 Bhâskarîya*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 CalendarReform 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 of understand local methods of timekeeping (

*jyotisa*), from both Brahmins and Moors, 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 Bhâskarîya*, 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 17th c. This led to the development of the chronometer, an appliance that could be mechanically used without application of the mind.
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