1. Who Invented Calculus ? Newton, Leibniz or Indians ? A K TIWARI 1

2. Sir Isaac Newton (1642 1727) was an English physicist and mathematician Gottfried Wilhelm von Leibniz was a German mathematician and philosopher. He occupies a prominent place in the history of mathematics and the history of philosophy. Born: 1646, Died: 1716 Both men published their researches in the 1680s, Leibniz in 1684 in the recently founded journal Acta Eruditorum and Newton in 1687 In India, Calculus is being used by Indian Mathematicians since 932 AD A K TIWARI 2 3. Newtons book of calculus A K TIWARI 3 4. A K TIWARI 4 5. MARCOPOLO Italian Traveller in INDIA 1295 A K TIWARI 5 6. Marco polo travel route 1250-1295 A K TIWARI 6 7. Museum of Venice A K TIWARI 7 8. A K TIWARI 8 9. Marco polo paintings and idols A K TIWARI 9 10. Marcopolo presenta libro indiano sul calcolo di Sacro Romano Imperatore Enrico VII Marcopolo is presenting indian book on calculus to holy roman emperor Henry VII A K TIWARI 10 11. Del Sacro Romano Impero Ferdinando I che d il libro di calcolo indiano a Bonaventura Francesco Cavalieri 1634 Holy Roman Emperor Ferdinand I, giving the book of Indian calculus to Bonaventura Francesco Cavalieri 1634 A K TIWARI 11 12. Guido Bonatti Guido Bonatti (died between 1296 and 1300) was an Italian astronomer and astrologer from Forl. He was the most celebrated astrologer in Europe in his century. He mentioned about the books of Indian mathematics brought by Marco polo A K TIWARI 12 13. Italian mathematicians Scipione del Ferro (1465 1526) Gerolamo Cardano (1501 1576) Practica arithmetice et mensurandi singularis Milan, 1577 (on mathematics). Niccol Fontana Tartaglia (1499/1500, Brescia 1557, Venice) his treatise General Trattato di numeri, et misure published in Venice 15561560 In his book Nova Scientia he wrote- a mio avviso il libro indiano di flussioni grande classica in matematica che dimostra altamente intellettuali di Archimede e le opere di Euclide, che ho tradotto in italiano. in my view the indian book of fluxions is great classical in mathematics which shows highly intellectuals than Archimedes and Euclid's works which I translated inA K TIWARI 13 14. Bonaventura Francesco Cavalieri Bonaventura Francesco Cavalieri (in Latin, Cavalerius) (1598 1647) was an Italian mathematician. He is known for his work on the problems of optics and motion, work on the precursors of infinitesimal calculus, and the introduction of logarithms to Italy. Cavalieri's principle in geometry partially anticipated integral calculus. Cavalieris works was studied by Newton, Leibniz, Pascal, Wallis and MacLaurin as one of those who in the 17th and 18th centuries "redefine the mathematical object". A K TIWARI 14 15. Leonardo Pisano Bigollo (c. 1170 c. 1250) known as Fibonacci, and also Leonardo of Pisa, Leonardo Pisano, Leonardo Bonacci, Leonardo Fibonacci was an Italian mathematician, considered by some "the most talented western mathematician . He traveled Arab and India to learn mathematics. Leonardo became an amicable guest of the Emperor Frederick II, who enjoyed mathematics and science. In 1240 the Republic of Pisa honored Leonardo, referred to as Leonardo Bigollo,[6] by granting him a salary A K TIWARI 15 16. There, following my introduction, as a consequence of marvelous instruction in the art, to the nine digits of the Hindus. I use method of the Hindus. (Modus Indorum) -- In the Liber Abaci , book by Fibonacci Fibonacci and his Guru (Brahmagupta ?) see Brahmagupta Fibonacci identity A K TIWARI 16 17. A K TIWARI 17 18. Varahmihir (505587 CE), A K TIWARI 18 19. Manjula (932) Manjula (932) was the first Hindu astronomer to state that the difference of the sines, sin w' - sin w = (w' - w) cos w, (i) 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 u = v e cos . 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 . His method is probably the same as that employed by his predecessors. A K TIWARI 19 20. Attention was first drawn to the occurrence of the differential formula (sin ) = cos in Bhaskara II's (1150) Siddhanta Siromani shows that Bhaskara was fully acquainted with the principles of the differential calculus. A K TIWARI 20 21. Problems in Astronomy 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 K TIWARI 21 22. The chapter which describe computation of tatkalik gati (of planets : time and motion) named Kaal Kolash 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. A K TIWARI 22 23. Method of infinitesimal-integration 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. 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. . Means d sinx/dx = cos x . Means d x2 /dx = 2x Indian mathematician used the word FALAX in place of derivative 1150 AD Newton used the word FLUXION in place of derivative 1680 AD A K TIWARI 23 24. Kerala school of astronomy and mathematics (1300-1632) The Kerala school of astronomy and mathematics was a school of mathematics and astronomy founded by Madhava of Sangamagrama in Kerala, India, which included among its members: Parameshvara, Neelakanta Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar. The school flourished between the 14th and 16th centuries and the original discoveries of the school seems to have ended with Narayana Bhattathiri (15591632). In attempting to solve astronomical problems, the Kerala school independently created a number of important mathematics concepts. Their most important resultsseries expansion for trigonometric functionswere described in Sanskrit verse in a book by Neelakanta called Tantrasangraha, and again in a commentary on this work, called Tantrasangraha-vakhya, of unknown authorship. The theorems were stated without proof, but proofs for the series for sine, cosine, and inverse tangent were provided a century later in the work Yuktibhasa (c.1500-c.1610), written in Malayalam, by Jyesthadeva, and also in a commentary on Tantrasangraha. Their work, completed two centuries before the invention of calculus in Europe, provided what is now considered the first example of a power series , differential and integral calculus According to Whish, the Kerala mathematicians had "laid the foundation for a complete system of fluxions" and these works abounded "with fluxional forms and series to be found in no work of foreign countries Possibility of transmission of Kerala School results to Europe Source :http://en.wikipedia.org/wiki/Kerala_school_of_astronomy_and_mathematics A K TIWARI 24 25. 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 ( + ) - sin varies as the cosine and decreases as increases. (2) The cosine-difference cos ( + ) - cos varies as the sine negatively and numerically increases as increases. A K TIWARI 25 26. He has obtained the following formulae: The above results are true for all values of 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 2 (sin ) and 2 (cos ), 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 2 (sin ), 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 A K TIWARI 26 27. 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 its 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. A K TIWARI 27 28. 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 and calculus might have directly influenced mathematicians like Newton and Leibniz. A K TIWARI 28 29. The French or Catalan Dominican missionary Jordanus Catalani was the first European to start conversion in India. He arrived in Surat in 1320. After his ministry in Gujarat he reached Quilon in 1323. He not only revived Christianity but also brought thousands to the Christian fold. He brought a message of good will from the Pope to the local rulers. As the first bishop in India, he was also entrusted with the spiritual nourishment of the Christian community in Calicut, Mangalore, Thane and Broach (north of Thane). He translated many Sanskrit and Malayalam books on Mathematics , Science, metallurgy, Construction and vetnary A K TIWARI 29 30. Father Jordanus Catalani wrote in 1325 Pundits calculated the age of the universe in trillions of years. they use decimals numerals and zero as do trigonometry and calculus, astronomical calculation and a they says the universe is not only billions, but trillions of years in age and that we are eternal beings who are simply visiting the material world to have the experience of being here. So, the point is, India holds a massive cosmological view of us and that humans have existed for trillions of years, in varying stages of existence. And further, over time humans will continue to populate the many universes again and again. There is a lot of evidence that ancient Indian civilization was global and as I mentioned many were seafaring and using extremely accurate astronomical, heliocentric calculations for both Earth and celestial motions, indicating an understanding that the Sun is at the center of the solar system and that the Earth is round. Elliptical orbits were also calculated for all moving celestial bodies. The findings are remarkable. What India calculated thousands of years ago, for example the wobble of the Earth's axis, which creates the movement called precession of the equinoxes the slowly changing motion that completes one cycle every 25,920 years . The cosmology of India describes our universe as having fourteen parallel realities on multiple levels, all existing and intersecting within the material realm in which we are currently living. A K TIWARI 30 31. Conclusio n Indian mathematicians were using calculus since 932 AD Kerala School of mathematics in1300 AD has fully developed differential and integral calculus called it METHOD OF FALAX Many Indian books on mathematics were translated in Italian and transmitted to Italy by Marco polo in 1395 Through Italian mathematicians the calculus reaches to Newton and Leibniz in 1680 A K TIWARI 31