The Madhava Trignometric Series

The Madhava Trignometric series is one one of a series in a collection of infinite series expressions discovered by Madhava of Sangramagrama ( 1350-1425 ACE ), the founder of the Kerala School of Astronomy and Mathematics. These are the infinite series expansions of the Sine, Cosine and the ArcTangent functions and Pi. The power series expansions of sine and cosine functions are called the Madhava sine series and the Madhava cosine series.

The power series expansion of the arctangent function is called the Madhava- Gregory series.

The power series are collectively called as Madhava Taylor series. The formula for Pi is called the Madhava Newton series.

One of his disciples, Sankara Variar had translated his verse in his Yuktideepika commentary on Tantrasamgraha-vyakhya, in verses 2.440 and 2.441


Multiply the arc by the square of the arc, and take the result of repeating that (any number of times). Divide (each of the above numerators) by the squares of the successive even numbers increased by that number and multiplied by the square of the radius. Place the arc and the successive results so obtained one below the other, and subtract each from the one above. These together give the jiva, as collected together in the verse beginning with "vidvan" etc.


Rendering in modern notations

Let r denote the radius of the circle and s the arc-length.

The following numerators are formed first:

s.s^2,
s.s^2.s^2
s.s^2.s^2.s^2

These are then divided by quantities specified in the verse.

1)s.s^2/(2^2+2)r^2,
2)s. s^2/(2^2+2)r^2. s^2/4^2+4)r^2
3)s.s^2/(2^2+2)r^2.s^2/(4^2+4)r^2. s^2/(6^2+6)r^2

Place the arc and the successive results so obtained one below the other, and subtract each from the one above to get jiva:


Jiva = s-(1-2-3)

When we transform it to the current notation

If x is the angle subtended by the arc s at the center of the Circle, then s = rx and jiva = r sin x.


Sin x = x - x^3/3! + x^5/5! - x^7/7!...., which is the infinite power series of the sine function.


By courtesy www.wikipedia.org and we thank Wikipedia for publishing this on their site.

Adverse Jupiter does it for India !

We had warned against the adverse transits of Jupiter, adverse for both Dhoni and India.

While Jove in the 10th indicates " loss of position and wandering about" for India, for Dhoni Jove transits the hostile 8th, indicating fall !

India whitewashed and humiliated, we can now understand. Sachin losing his 100th century also.

Just shows how powerful the Nine Revolving Heavens are and how the physique and the psyche of the natives are affected.

Now India will have to resurrect herself from the ruins!

Good luck, India. Well done, England !

Of Natural Strength, Naisargika Bala

Natural Strength, Naisargika Bala, is the inherent property of a celestial object, which possesses the following properties


1) This force is constant for a celestial object, not varying in time.

2) This force is proportional to the size of the diameter of the planets.

3) This force is inversely proportional to the distance, r, from the Sun.

4) It increases in the order from the farthest planet to the nearest planet to the Sun. From Saturn,Jupiter, Mars, Venus, Mercury, Moon and Sun.

5) This Force is a major factor when planets are involved in Planetary War ( Graha Yuddha ), when their longitudes are more or less identical in the Ecliptic.

Let F1 and F2 be the Naisargika Bala of planets 1 and 2 situated in the same distance, r, from earth. Then we have

F1 = F(D1)/r
F2 = F(D2)/r

The ratio of the planetary Naisargika Bala is

F1/F2 = F(D1)/F(D2)

The forces are given by ( according to Newtonian modern theories)

F1= ( M1M/r^2)
F2= ( M2M/r^2)

The ratio of the gravitational forces are

F1/F2 = M1/M2


M1 = v1 d1
M2 = v2 d2 ( v volume d density )


If d1 = d2

then

F1/F2 = V1/V2 = F(D1)/F(D2)


Therefore the ratio of the Naisargika Balas of two planets at the same identical position in the Zodiac, as defined by the Indian astronomers, is almost identical to the ratios of the modern gravitational forces of these planets if their mass densities are identical.

India becomes a laughing stock !

It is to the credit of Gooch that England owes its batting renaissance. Gooch gave tips to the English batsmen and you can see the result.

Gooch scored 333 against India in one Test and he knows the methodology to counter Indian swing and spin. Mentally and physically he prepared the English batsmen to score massive scores.

England did score heavily in Australia, where they trounced the Kangaroos in Tests. Now it is their turn to defeat India.

India is now a laughing stock, despite Dravid's 35th ton, an unconquered 146 not out. India most probably will lose this Test and it is time to do some rethinking !

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.

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.

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.