At the International Conference on Indian History, Civilisation and Geopolitics (ICIH 2009) that was held at New Delhi in January, Dr. C.K. Raju presented a brilliant paper about how calculus, an Indian invention, was picked up by the Jesuit priests from Kerala in the second half of the 16th century and taken to Europe. This is how the Westerners got their calculus. Over time, people forgot this link and the Europeans began to claim calculus as their own invention. This myth still persists despite calculus texts existing in India since thousands of years.

Here is the press release that was issued during ICIH-2009 regarding this issue:

‘Calculus is India’s Gift to Europe’In his speech at ICIH 2009, Professor C.K. Raju revealed that calculus was an Indian invention that was transmitted by Jesuit priests to Europe from Cochin in the second half of 16th century. “Indian infinite series has been known to British scholars since at least 1832, but no scholar tried to establish the connection with the calculus attributed to Newton and Leibnitz,” he said.

Dr. Raju’s 10-year research that included archival work in Kerala and Rome was published in a book “Cultural Foundations of Mathematics.” It established that the Jesuit priests took trigonometric tables and planetary models from the Kerala mathematicians of the Aryabhata school and exported them to Europe starting around 1560 in connection with the European navigational problem.

“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,” Dr. Raju added. Ironically, some British scholars claimed credit for this research despite being warned against plagiarizing Professor Raju’s work.

I produce below an abstract of Dr. Raju’s paper:

The Infinitesimal Calculus:How and Why it Was Imported into EuropeBy 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.

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.

http://www.indianscience.org/essays/31-%20E–Infinitesimal%20Calculus.PDF

About Dr. C.K. RajuC. 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.

Please read this web site and more evidences. Prof CK Raju is right

http://www.sciencedaily.com/releases/2007/08/070813091457.htm

Indians Predated Newton ‘Discovery’ By 250 Years, Scholars Say

ScienceDaily (Aug. 15, 2007) — A little known school of scholars in southwest India discovered one of the founding principles of modern mathematics hundreds of years before Newton — according to new research.

The discovery is currently – and wrongly – attributed in books to Sir Isaac Newton and Gottfried Leibnitz at the end of the seventeenth centuries.

The team from the Universities of Manchester and Exeter reveal the Kerala School also discovered what amounted to the Pi series and used it to calculate Pi correct to 9, 10 and later 17 decimal places.

””””””””””””””””””contin……………………………..

And there is strong circumstantial evidence that the Indians passed on their discoveries to mathematically knowledgeable Jesuit missionaries who visited India during the fifteenth century.

………..contin……………………………………….

Dr George Gheverghese Joseph from The University of Manchester says the ‘Kerala School’ identified the ‘infinite series ‘- one of the basic components of calculus – in about 1350.

Jesuits were founded in 1540, so they could not visit India in fifteenth century. Just a note. ;) However, the appearance of calculus in Europe is likely to be in the manner described here. Newton was too devoted to alchemy and magic to be curious enough for science.

The article says Jesuits began to take manuscripts from Kerala to Europe starting 1560. It doesn’t say 15th century.

http://www.sciencedaily.com/releases/2007/08/070813091457.htm

Dr George Gheverghese Joseph from The University of Manchester says the ‘Kerala School’ identified the ‘infinite series ‘- one of the basic components of calculus – in about 1350.

FYI, there’s more to it.

C.K Raju has accused G.G.Joseph & another chap of plagiarizing his “works”.

Dr George Gheverghese Joseph has been dismissed by the University of Manchester for unethical conduct and plagiarising C.K. Raju’s research. Dr. Raju’s research was shamelessly lifted by these people and circulated in their name. They lost their jobs.

That will be a surprise to both GG Joseph and the University, as they continue to employ him 5 years later!

Use of Calculus in Hindu Mathematics:

1. Differential Calculus

A Controversy

Attention was first drawn to the occurrence of the differential formula

δ (sin θ) = cos θ δ θ

in Bhaskara II’s (1150) Siddhanta Siromani by Pandit BapuDeva Sastri in 1858. The Pandit published a summarised translation of the passages which involve the use of the above formula. His summary was defective in so far as it did not bring into prominence the idea of the infinitesimal increment which underlies Bhaskara’s analysis. Without making clear to his readers, the full significance of Bhaskara’s result, the Pandit made the mistake of asserting — what was plain to him — that Bhaskara was fully acquainted with the principles of the differential calculus.

The Pandit was adversely criticised by Spotiswoode, who without consulting the original on which the Pandit based his conclusions, remarked (1) that Bapu Deva Sastri had overstated his case in saying that Bhaskaraciirya was fully acquainted with the principles of the differential calculus, (2) that there was no allusion to the most essential feature of the differential calculus, viz. the infinitesimal magnitudes of the intervals of time and space therein employed, and (3) that the approximative character of the result was not realized.

Since the above controversy took place no serious investigation of the subject seems to have been made by any scholar. In order that the reader may be better able to judge the merit of the Hindu claim to the invention of the differential calculus, it is desirable that the problems which required the use of the above differential formula be stated first.

Problems in Astronomy

The calculation of eclipses is one of the most important problems of Astronomy. In ancient days this problem was probably more important than it is now, because the exact time and duration of the eclipses could not be foretold on account of lack of the necessary mathematical equipment on the part of the astronomer. In India, the Hindus observed fast and performed various other religious rites on the occasion of eclipses. Thus their calculation was a matter of national importance. It afforded the Hindu astronomer a means of demonstrating the accuracy of his science and his own ability to the public who patronised him. The problem of the calculation of conjunction of planets and occupation of stars was equally important both from scientific as well as religious view points.

In problems of the above nature it is essential to determine the true instantaneous motion of a planet or star at any particular instant. This instantaneous motion was called by the Hindu astronomers tat-kalika-gati. The formula giving the tat-kalika-gati (instantaneous motion) is given by Aryabhata and Brahmagupta in the following form:

u’- v’ = v’ – v ± e (sin w’ – sin w) (i)

where u, v, w are the true longitude, mean longitude, mean anomaly respectively at any particular time and u’, v’, w’ the values of the respective quantities at a subsequent instant; and e is the eccentricity or the sine of the greatest equation of the orbit. The tat-kalika-gati is the difference u’-u between the true longitudes at the two positions under consideration. Aryabhata and Brahmagupta used the sine table to find the value of (sin w’ — sin w). The sine table used by them was tabulated at intervals of 3° 45′ and thus was entirely unsuited for the purpose. To get the values of sines of angles, not occurring in the table, recourse was taken to interpolation formulae, which were incorrect because the law of variation of the difference was not known.

A Differential Formula

Manjula (932) was the first Hindu astronomer to state that the difference of the sines, sin w’ – sin w = (w’ – w) cos w,

where (w’ – w) is small.

He says:

“True motion in minutes is equal to the cosine (of the mean anomaly) multiplied by the difference (of the mean anomalies) and divided by the cheda, added or subtracted contrarily (to the mean motion).”

Thus according to Manjula formula (i) becomes

u’ – u = v’ – v ± e(w’ – w) cos w, (ii)

which, in the language of the differential calculus, may be written as

δu = δv ± e cos θ δ θ.

We cannot say exactly what was the method employed by Manjula to obtain formula (ii). The formula occurs also in the works of Aryabhata II (950). Bhaskara II (1150), and later writers. Bhaskara II indicates the method of obtaining the differential of sine θ. His method is probably the same as that employed by his predecessors.

Proof of the Differential Formula

Let a point P (See Fig. 1) move on a circle. Let its position at two successive intervals be denoted by P and Q. Now, if P and Q are taken very near each other, the direction of motion in the interval PQ is the same as that of the tangent at P. Let PT be measured along the tangent at P equal to the arc PQ. Then PT would be the motion of the point P if its velocity at P had not changed direction.

Discussing the motion of planets, Bhaskaracarya says: “The difference between the longitudes of a planet found at any time on a certain day and at the same time on the following day is called its (sphuta) gati (true rate of motion) for that interval of time.”

“This is indeed rough motion (sthulagati). I now describe the fine (suksma) instantaneous (tat-kalika) motion. The tatkalika-gati (instantaneous motion) of a planet is the motion which it would have, had its velocity during any given interval of time remained uniform.”

During the course of the above statement, Bhaskara II observes that the tat-kalika-gati is suksma (“fine” as opposed lo rough), and for that the interval must be taken to be very small, so that the motion would be very small. This small interval of time has been said to be equivalenttoa ksana which according to the Hindus is an infinitesimal interval of time (immeasurably small). It will be apparent from the above that Bhaskara did really employ the notion of the infinitesimal in his definition of Tat-kalika-gati.

But in actual practice, the intervals that are considered are not infinitesimal. How are we, then, to apply the notion of tat-kalika-gati to actual problems ? The answer to the above question is given by Bhaskara II as follows:

In equation (i) we have to consider the sine-difference (sin w’ – sin w). Let an arc of 90° be divided into n parts each equal to A, and let us consider the sine differences R(sin A – sin O), R (sin 2A – sin A), R (sin 3A – sin 2A)… These differences are termed Bhogya khanda. Bhaskara-II says: “These are not equal to each other but gradually decrease, and consequently while the increase of the arc is uniform, the increment of the sine varies — on account of deflection of the arc.”

In the figure given above, let the arc PQ = A. Then

R (sin BOQ – sin BOP) = QN – PM = Qn

which is the Bhogya Khanda. Bhaskara introduces the notion of Tat-kalika Bhogya Khanda (instantaneous sine difference) in order to find the variation of the sine at P. According to him if the arc BP instead of being deflected towards Q, be increased in the direction of the tangent, so that PT = PQ = A, then TS – PM = Tr is the Tat-kalika Bhogya Khanda of the sine PT, i.e. the “instantaneous sine difference”. By having recourse to this artifice Bhaskara II avoids the use of the infinitesimal in his analysis. It should be borne in mind that the “instantaneous sine difference” for a finite arc PQ, is a purely artificial quantity created with a special end in view, and is different from the actual “sine difference” R (sin BOQ – sin BOP).

Now from the similar triangles PTr and PMO, we at once derive the proportion

R : PT : : R cos w : Tr ……………. (iii)

Tr = PT cos w.

But Tr = R(sin w’ – sin w) and PT = R(w’ – w)

(sin w’ – sin w) = (w’ – w) cos w.

Thus the Tat-kalika Bhogya Khanda (the instantaneous sine difference) in modern notation is

δ (sin θ) = cos θ δ θ.

This formula has been used by Bhaskara to calculate the ayana-valana (“angle of position”).

If the above were the only result occurring in Bhaskara II’s work, one would be justified in not accepting the conclusions of Pandit Bapu Deva Sastri. There is however other evidence in Bhaskara II’s work to show that he did actually know the principles of the differential calculus. This evidence consists partly in the occurrence of the two most important results of the differential calculus:

(i) He has shown that when a variable attains the maximum value its differential vanishes.

(ii) He shows that when a planet is either in apogee or in perigee the equation of the centre vanishes, hence he concludes that for some intermediate position the increment of the equation of centre (i.e. the differential) also vanishes.

The second of the above results is the celebrated Rolle’s Theorem, the mean value theorem of the differential calculus.

Remarks

The use of a formula involving differentials in the works of ancient Hindu mathematicians has been established beyond the possibility of any doubt. That the motions of instantaneous variation and that of motion entered into the Hindu idea of differentials as found in works of Manjula, Aryabhata II and Bhaskara II is apparent from the epithet Tat-kalika (instantaneous) gati (motion) to denote these differentials. The main contribution of Bhaskara II to the theory of these differentials, which were already worked out by his predecessors, seems to be his proof of the formula by the rule of proportion without actually using the infinitesimal or varying quantities. He has, however, made it quite clear that the differentials give true results only when very small variations are concerned.

Nilakarttha’s Result

Nilakantha (c. 1500) in his commentary on the Aryabhatiya has given proofs, on the theory of proportion (similar triangles) of the following results.

(1) The sine-difference sin (θ + δ θ) – sin θ varies as the cosine and decreases as θ increases.

(2) The cosine-difference cos (θ + δ θ) – cos θ varies as the sine negatively and numerically increases as θ increases.

He has obtained the following formulae:

The above results are true for all values of δθ whether big or small. There is nothing new in the above results. They are simply expressions as products of sine and cosine differences.

But what is important in Nilakantha’s work is his study of the second differences. These are studied geometrically by the help of the property of the circle and of similar triangles. Denoting by Δ2 (sin θ) and Δ2 (cos θ), the second differences of these functions, Nilakarttha’s results may be stated as follows:

(1) The difference of the sine-difference varys as the sine negatively and increases (numerically) with the angle.

(2) The difference of the cosine-difference varys as the cosine negatively and decreases (numerically) with the angle.

For Δ2 (sin θ), Nilakantha has obtained the following formula

Besides the above, Nilakantha, has made use of a result involving the differential of an inverse sine function. This result, expressed in modern notation, is

In the writings of Acyuta (1550-1621 A.D.) we find use of the differential of a quotient also

2. Method of infinitesimal-integration

Surface of the Sphere

For calculating the area of the surface of a sphere Bhaskara II (1150) describes two methods which are almost the same as we usually employ now for the same purpose.

First Method: “Make a spherical ball of clay or of wood. On it take a (vertical) circumference circle and divide this into 21600 parts. Mark a point on the top of it. With that point as the centre and with the radius equal to the 96th part of the circumference, i.e. to 225′, describe a circle. Again with same point as the centre with twice that arc as radius describe another circle; with thrice that another circle; and so on up to 24 times. Thus there will be 24 circles in all. The radii of these circles will be the jya 225′ (= R sin 225′), etc. From them the lengths of the circles can be determined by proportion. Now the length of the extreme circle is 21600′ and its radius is 3438′. Multiplying the Rsines (of 225′, 450′, etc.) by 21600 and dividing by 3438, we shall obtain the lengths of the circles. Between two and two of these circles there lies annular strips and there are altogether 24 such. They will be many more in case of many R-sines being taken into consideration (bahujya-pakse-vahuni syuh). In each annulus considering the larger circle at the lower end as the base and the smaller circle at the top as the face and 225’as the altitude (of the trapezium), find its area by means of the rule ‘half the sum of the base and the face multiplied by the altitude etc. Similarly the areas of all the annular figures severally can be found. The sum of all these areas is equal to the area of the surface of half of the sphere. So twice that is the area of the surface of the whole sphere. And that is equal to the product of the diameter and the circumference.”

In other words, if Tn denotes the nth jya (or Rsine), Cn the circumference of the corresponding circle, An the area of the nth annulus and S the area of the surface of the sphere, then we shall have

Therefore,

the summation being taken so as to include all the Rsines in a quadrant of the circle. Since there are ordinarily 24 Rsines in a Hindu trigonometrical table, we have

Hence approximately

S = 21600 x 2 x 3437

Bhaskara II states:

Area of the surface = circumference x diameter.

Second Method: “Suppose the (horizontal) circumference-circle on the surface of the sphere to be divided into parts as many as four times the number of Rsines (in a quadrant). As the surface of an emblic myrobalan is seen divided into vapras (i.e. lunes) by lines passing through its face (or top) and bottom, so the surface of the sphere should be divided into lunes by vertical circles as many as the parts of the above mentioned (horizontal) circumference-circle. Then the area of each lune should be determined by (breaking it up into) parts. And this area of a lune is equal to the sum of all the Rsines diminished by half the radius and divided by the semi-radius. Since that is again equal to the diameter of the sphere, so it has been said that the area of the surface of a sphere is equal to the product of its circumference and diameter.”

The method has been further elucidated by him in his gloss thus:

“As many as are Rsines in the table of any particular work selected, take four times that number, and suppose the (horizontal) circumference-circle on the sphere is to be divided into, as many parts. Like the natural lines seen on the surface of a round emblic myrobalan passing through its face and base and thus dividing it into lunes, draw circles on the surface of the given sphere, passing through its top and bottom and thereby dividing it into lunes as many as the number of parts into which the (horizontal) circumference-circle is divided. Next the area of each lune has to be determined. It can be done thus: For instance in the Dhivrddhida, there are 24 Rsines. So suppose the (horizontal) circumference-circle measures 96 cubits. On drawing the vertical circles through every cubit, there will be as many lunes. Then the upper half of any one lune ondrawing the transverse arcs at distances of every cubit, will be divided into portions equal to the number of Rsines, that is, 24. The lengths of these transverse lines will be obtained by dividing the Rsines severally by the radius. Of these the lowest line measures one cubit; but the upper and upper ones are a little smaller and smaller according to the Rsines. But the altitude is all along one cubit in length. Now by finding the area of each portion in accordance with the rule, “half the sum of the top and the base multiplied by the altitude etc: they should be added together. This sum gives the area of half a lune; twice that is the area of a lune. For the determination of that the rule is, “the sum of all the Rsines minus half the radius etc.” Now the sum of all the Rsines, 225 etc., is 54233. This diminished by the semi-radius becomes 52514. Dividing the result by the semi-radius we get the area of each lune as 30;33. Now 30;33 is equal to the diameter of a circle whose circumference measures 96. And as the number of lunes is equal to the number of portions of the circumference it is consequently proved that the area of the surface of a sphere is equal to the product of its circumference and diameter”.

If ln denotes the length of the nth transverse arc, we have

Therefore,

the summation being taken so as to include all the Rsines. Hence

Hence the area of a lune is numerically equal to the diameter of the sphere. As the number of limes is equal to the number of parts of the circumference of the sphere, we get

Area of the surface = circumference x diameter.

Volume of the Sphere

To find the volume of a sphere Bhaskara II states the following method:

“Consider on the surface of the sphere pyramidal excavations, each of a base of an unit area having unit sides and of a depth equal to the radius, as many as the number of units of area in the surface. The apices of these pyramids meet at the centre of the sphere. The sum of the volumes of the pyramids is equal to the volume of the sphere. So it is proved (that the volume of a sphere is equal to the sixth part of the product of the surface area and diameter).

The above results are the nearest approach to the method of the integral calculus in Hindu Mathematics. It will be observed that the modern idea of the “limit of a sum” is not present. This idea, however, is of comparatively recent origin so that credit must be given to Bhaskara II for having used the same method as that of the integral calculus, although in a crude form.

Two British researchers challenged the conventional history of mathematics in June when they reported having evidence that the infinite series, one of the core concepts of calculus, was first developed by Indian mathematicians in the 14th century. They also believe they can show how the advancement may have been passed along to Isaac Newton and Gottfried Wilhelm Leibniz, who are credited with independently developing the concept some 250 years later.

“The notation is quite different, but it’s very easy to recognize the series as we understand it today,” says historian of mathematics George Gheverghese Joseph of the University of Manchester, who conducted the research with Dennis Almeida of the University of Exeter. “It was expressed verbally in the form of instructions for how to construct a mathematical equation.”

Historians have long known about the work of the Keralese mathematician Madhava and his followers, but Joseph says that no one has yet firmly established how the work of Indian scholars concerning the infinite series might have directly influenced mathematicians like Newton and Leibniz.

Unfortunately – for the western mind that has hard time comprehend anything but material gain, plagiarising comes easily. In India sharing is the foundation. At present times – India has been misled by the western tought and has lost its way. India must take responsibility.

Hopefully, they get back and real good work and distribute it free to the world. From public to public domain…this private enterprise is over-rated and at least as corrupt as the public enterprise. The evidence is for all to see.

US needs to lead by example – but have gained it by wrongful means, it must dissipate it through the same. An opportunity lost.

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Excellent News.

Dr George Gheverghese Joseph from The University of Manchester says the ‘Kerala School’ identified the ‘infinite series ’- one of the basic components of calculus – in about 1350.

According to Dr George Gheverghese Joseph of Great Britain’s University of Manchester, the discovery is currently – and wrongly – attributed in books to Sir Isaac Newton and Gottfried Leibnitz at the end of the seventeenth centuries.

The team from the Universities of Manchester and Exeter reveal the Kerala School also discovered what amounted to the Pi series and used it to calculate Pi correct to 9, 10 and later 17 decimal places.

And there is strong circumstantial evidence that the Indians passed on their discoveries to Jesuit missionaries who visited India during the fifteenth century. That knowledge, the team argues, may have eventually been passed on to Newton himself.

Dr Joseph made the revelations while browsing through obscure Indian papers for a yet to be published third edition of his best selling book ‘The Crest of the Peacock: the Non-European Roots of Mathematics’ by Princeton University Press.

He said: “The beginnings of modern maths is usually seen as a European achievement but the discoveries in medieval India between the fourteenth and sixteenth centuries have been ignored or forgotten.

“The brilliance of Newton’s work at the end of the seventeenth century stands undiminished – especially when it came to the algorithms of calculus. But other names from the Kerala School, notably Madhava and Nilakantha, should stand shoulder to shoulder with him as they discovered the other great component of calculus- infinite series.

“There were many reasons why the contribution of the Kerala school has not been acknowledged – a prime reason is neglect of scientific ideas emanating from the Non-European world – a legacy of European colonialism and beyond.

“But there is also little knowledge of the medieval form of the local language of Kerala, Malayalam, in which some of most seminal texts, such as the Yuktibhasa, from much of the documentation of this remarkable mathematics is written.

He added: “For some unfathomable reasons, the standard of evidence required to claim transmission of knowledge from East to West is greater than the standard of evidence required to knowledge from West to East.

“Certainly it’s hard to imagine that the West would abandon a 500-year-old tradition of importing knowledge and books from India and the Islamic world. But we’ve found evidence which goes far beyond that: for example, there was plenty of opportunity to collect the information as European Jesuits were present in the area at that time.

“They were learned with a strong background in maths and were well versed in the local languages.

“And there was strong motivation: Pope Gregory XIII set up a committee to look into modernizing the Julian calendar. On the committee was the German Jesuit astronomer/mathematician Clavius who repeatedly requested information on how people constructed calendars in other parts of the world. The Kerala School was undoubtedly a leading light in this area.

“Similarly there was a rising need for better navigational methods including keeping accurate time on voyages of exploration and large prizes were offered to mathematicians who specialized in astronomy.

“Again, there were many such requests for information across the world from leading Jesuit researchers in Europe. Kerala mathematicians were hugely skilled in this area.”

Aryabhatta-siddhanta (AD 499), one of the greatest Indian astronomer-mathematician, was the first to discover that the Moon and the planets reflect the light from the Sun, and that the planets follow an elliptical orbit around the Sun.

all are excellent but madhava did not invent calculus instead it was bhakaracharya in 1150 ad. but madhava developed it in 1350 ad.

You speak of two core concepts, which although important, are hardly the entirety of calculus. Inspired or not, whether certain material was borrowed or not, it indeed was Newton and Leibniz that developed calculus into the ordered mathematical system that exists today. It is disingenuous to claim otherwise.

Moreover, you don’t mention algebra at all. Modern algebra didn’t come about till the 1600s, and you can’t really do anything of major scientific or mathematical significance unless you use algebra. Specifically, the rate of change of functions and how that can be applied (differential calculus as we know it today), and the area under a curve of a function and how that can be applied to other sciences and mathematics (integral calculus).

So yes, credit should be given where credit is due and Madhava should be remembered when speaking about the history of calculus, but to say that the Jesuits “took calculus from India” is a bit of a stretch.

There’s always a cococunt wandering about, Good job, chaiwala. Go fetch for your masters.

Ok now that I have gotten that out of the way, with out those two core concepts, how does Newton and Leibnis create calculus?

Oh well done. Responding to a cogent argument with bigotry and racism. How exactly are you refuting his argument? Oh wait, you can’t even understand it.

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Couldn’t understand calculus? The country that gifted decimal place value, trigonometry and algebra to the world? Read a book, dollltroll.