Glorious past of INDIA

8/14/2019 Glorious past of INDIA

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Who Invented Calculus

Newton, Leibniz or Indians ?

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Sir Isaac Newton (1642 1727) was an Englishphysicist andmathematician

Gottfried Wilhelm von Leibniz was aGerman mathematician and philosopher.He occupies a prominent place in thehistory of mathematics and the history ofphilosophy.Born : 1646, Died : 1716

Both men publishin the 1680s,

Leibniz in 1684 ifounded journalEruditorum and

In India, CalculuIndian Mathemat

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https://www.google.co.in/search?biw=1360&bih=673&q=gottfried+wilhelm+leibniz+born&sa=X&ei=tr4PUtycEoSPrgeDlYGgDQ&sqi=2&ved=0CKQBEOgTKAEwFAhttps://www.google.co.in/search?biw=1360&bih=673&q=gottfried+wilhelm+leibniz+died&sa=X&ei=tr4PUtycEoSPrgeDlYGgDQ&sqi=2&ved=0CKgBEOgTKAEwFAhttps://www.google.co.in/search?biw=1360&bih=673&q=gottfried+wilhelm+leibniz+died&sa=X&ei=tr4PUtycEoSPrgeDlYGgDQ&sqi=2&ved=0CKgBEOgTKAEwFAhttps://www.google.co.in/search?biw=1360&bih=673&q=gottfried+wilhelm+leibniz+born&sa=X&ei=tr4PUtycEoSPrgeDlYGgDQ&sqi=2&ved=0CKQBEOgTKAEwFA


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Newtons book of calculus

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MARCOPOLO Italian Traveller in INDIA1295

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Marco polo travel route 1250-1295

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Museum of Venice

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Marco polo paintings and idols

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Marcopolo presenta libro indiancalcolo di Sacro Romano ImperEnrico VII

Marcopolo is presenting indian book on roman emperor Henry VII

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Del Sacro Romano ImperFerdinando Iche d il libro di calcolo iBonaventura Francesco C1634

Holy Roman EmperorFerdinand I,giving the book of Indicalculus to BonaventuFrancesco Cavalieri 1634

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Guido Bonatti

Guido Bonatti (died between1296 and 1300) was an Italianastronomer and astrologerfrom Forl. He was the mostcelebrated astrologer inEurope in his century.

He mentioned about thebooks of Indian mathematicsbrought by Marco polo

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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 1556 1560

In his book Nova Scientia he wrote -

a mio avviso il libro indiano di flussioni grande classica in matematica chedimostra altamente intellettuali di Archimede e le opere di Euclide, che ho tradottoin italiano.

in my view the indian book of fluxions is great classical in mathematics which showshighly intellectuals than Archimedes and Euclid's works which I translated in Italian.

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Bonaventura Francesco CavalieriBonaventura Francesco Cavalieri (in Latin, Cavalerius)(1598 1647) was an Italian mathematician. He is knownfor his work on the problems of optics and motion, workon the precursors of infinitesimal calculus, and theintroduction of logarithms to Italy.

Cavalieri's principle in geometry partially anticipatedintegral calculus.

Cavalieris works was studied by Newton, Leibniz, Pascal,Wallis and MacLaurin as one of those who in the 17th and18th centuries "redefine the mathematical object".

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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 tolearn mathematics. Leonardo became an amicable guest of the Emperor Frederick II, whoenjoyed 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


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Fibonacci and his Guru (Brahmagupta ?) see Brahmagupta Fibonacci identity A K TIWARI


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Varahmihir(505 587 CE),

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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 thdifference (of the mean anomalies) and divided by the cheda , added or subtracontrarily (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 (

The formula occurs also in the works of Aryabhata II (950). Bhaskara II (1150), and lawriters.

Bhaskara II indicates the method of obtaining the differential of sine . His method isprobably the same as that employed by his predecessors.

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Attention was first drawn to the occurrence of the differential formu (sin ) = cos

in Bhaskara II's (1150) Siddhanta Siromani shows that Bhaskafully acquainted with the principles of the differential calculus.

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http://www.google.co.in/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&docid=8il4H7BcWPxjlM&tbnid=gA91VyFVwFIVXM:&ved=0CAUQjRw&url=http%3A%2F%2Fahmedabadganitmandal.org%2Fmathem.php&ei=uR4RUomoD4nWrQf3q4DwCg&bvm=bv.50768961,d.bmk&psig=AFQjCNFQe1PcCycvLbrOSXzBa0ncdZQP7Q&ust=1376939958988205http://www.google.co.in/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&docid=emR0HlnK2BmOqM&tbnid=ZPnSCGBntIBjFM:&ved=0CAUQjRw&url=http%3A%2F%2Fwww.nbcindia.com%2Fbooks%2F1-siddhanta-siromani-treatise-astronomy-by-bhaskaracharya-pandit-muralidhara-jha-ed%2F8130712199&ei=fx4RUrK3C8uTrgf3wIDwDA&bvm=bv.50768961,d.bmk&psig=AFQjCNFQe1PcCycvLbrOSXzBa0ncdZQP7Q&ust=1376939958988205


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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 Hinduastronomers 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 anyparticular time and u', v', w' the values of the respective quantities at a subsequent instant; and 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 undconsideration. Aryabhata and Brahmagupta used the sine table to find the value of (sin w'w).

The sine table used by them was tabulated at intervals of 3 45' and thus was entirely unsuited the purpose.

To get the values of sines of angles, not occurring in the table, recourse was taken to interpolatiformulae, which were incorrect because the law of variation of the difference was not known.

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The chapter which describe computation oftatkalik gati (of planets : time and motion)

named K a al K o l as h 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 sametime on the following day is called its ( sphuta ) gati (true rate of motion) interval of time."

"This is indeed rough motion ( sthulagati ). I now describe the fine ( sukinstantaneous ( tat-kalika ) motion. The tatkalika-gati (instantaneous moplanet is the motion which it would have, had its velocity during any giveninterval of time remained uniform."

During the course of the above statement, Bhaskara II observes that thekalika-gat i is suksma ("fine" as opposed lo rough), and for that the interval mbe taken to be very small, so that the motion would be very small. This smallinterval of time has been said to be equivalenttoa ksana which accordinHindus is an infinitesimal interval of time (immeasurably small). It will beapparent from the above that Bhaskara did really employ the notion of theinfinitesimal in his definition of Tat-kalika-gati .

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Method of infinitesimal-integration For calculating the area of the surface of a

sphere Bhaskara II (1150) describes twomethods which are almost the same as weusually employ now for the same purpose.results are the nearest approach to themethod of the integral calculus in HinduMathematics.

It will be observed that the modern idea ofthe "limit of a sum" is not present.

This idea, however, is of comparatively recentorigin so that credit must be given toBhaskara II for having used the same methodas that of the integral calculus, although in acrude form.

Means d sinx/dx =

.

Means d x 2/dx =

Indian mathematicword FALAX in pderivative 1150 A

Newton used the win place of derivat

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Kerala school of astronomy andmathematics (1300-1632)

The Kerala school of astronomy and mathematics was a school of mathematics and astronomy founded by Madhava ofSangamagrama in Kerala, India, which included among its members: Parameshvara, Neelakanta Somayaji, Jyeshtadeva, AchyuPisharati, 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 withNarayana Bhattathiri (1559 1632).

In attempting to solve astronomical problems, the Kerala school independently created a number of important mathematicsconcepts.

Their most important results series expansion for trigonometric functions were described in Sanskrit verse in a bNeelakanta called Tantrasangraha, and again in a commentary on this work, called Tantrasangraha-vakhya, of unknown authors

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 Tantrasangra 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 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

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Nilakantha (c. 1500) in his commentary on the Aryabhatiya has givproofs, on the theory of proportion (similar triangles) of the followiresults.

(1) The sine- difference sin ( + ) - sin varies as the cosinedecreases as increases.

(2) The cosine- difference cos ( + ) - cos varies as the sin

negatively and numerically increases as increases.

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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 studiedgeometrically by the help of the property of the circle and of similar triangles. Denoting by 2 (si(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 theangle.

(2) The difference of the cosine-difference varys as the cosine negatively and decreases (numerically) withthe angle.For 2 (sin ), Nilakantha has obtained the following formula

Besides the above, Nilakantha, has made use of a result involving the differential oan 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 aquotient also

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Two British researchers challeng ed the co nventional history ofmathematics in June when they reported having evidence that theinfinite series, one of the core concepts of calculus, was first developed byIndian mathematicians in the 14th century. They also believe they canshow how the advancement may have been passed along to IsaacNewton and Gottfried Wilhelm Leibniz, who are credited withindependently developing the concept some 250 years later.

The notation is quite different, but its very easy to recognize the seriesas we understand it today, says historian of mathematics GeorgeGheverghese Joseph of the University of Manchester, who conducted theresearch with Dennis Almeida of the University of Exeter. It wasexpressed verbally in the form of instructions for how to construct amathematical equation.

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http://www.manchester.ac.uk/aboutus/news/archive/list/item/?year=2007&month=august&id=121685http://www.manchester.ac.uk/aboutus/news/archive/list/item/?year=2007&month=august&id=121685http://www.manchester.ac.uk/aboutus/news/archive/list/item/?year=2007&month=august&id=121685


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Historians have long known about the work of the Keralesemathematician Madhava and his followers, but Joseph says that one has yet firmly established how the work of Indian scholarsconcerning the infinite series and calculus might have directlyinfluenced mathematicians like Newton and Leibniz.

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http://en.wikipedia.org/wiki/Madhava_of_Sangamagramahttp://en.wikipedia.org/wiki/Madhava_of_Sangamagrama


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The French or Catalan Dominican missionary Jordanus

Catalani was the first European to start conversion inIndia. 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 inCalicut, Mangalore, Thane and Broach (north of Thane).

He translated many Sanskrit and Malayalam books onMathematics , Science, metallurgy, Construction andvetnary

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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 trigon

calculus , astronomical calculation and a they says the universe is not only billions, but trillions of years in age and that we areeternal 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 vstages 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 usingextremely accurate astronomical, heliocentric calculations for both Earth and celestial motions, indicating an understanding thathe Sun is at the center of the solar system and that the Earth is round. Elliptical orbits were also calculated for all moving celesbodies. 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 cyc

25,920 years .

The cosmology of India describes our universe as having fourteen parallel realities on multiple levels, all existing and intersewithin the material realm in which we are currently living.

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Glorious past of INDIA