Who Invented Calculus ?
Newton, Leibniz or Indians ?
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Sir Isaac Newton (1642 –
1727) was an English
Gottfried Wilhelm von Leibniz was a
German mathematician and philosopher.
He occupies a prominent place in the
history of mathematics and the history of
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
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Marcopolo presenta libro indiano sul
calcolo di Sacro Romano Imperatore
Marcopolo is presenting indian book on calculus to holy
roman emperor Henry VII
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Del Sacro Romano Impero
che dà il libro di calcolo indiano a
Bonaventura Francesco Cavalieri
Holy Roman Emperor
giving the book of Indian
calculus to Bonaventura
Francesco Cavalieri 1634
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• Guido Bonatti (died between
1296 and 1300) was an Italian
astronomer and astrologer
from Forlì. He was the most
celebrated astrologer in
Europe in his century.
• He mentioned about the
books of Indian mathematics
brought by Marco polo
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• Scipione del Ferro (1465 –1526)
• Gerolamo Cardano (1501 –1576)
• “Practica arithmetice et mensurandi singularis” Milan, 1577
• Niccolò Fontana Tartaglia (1499/1500, Brescia –1557,
Venice) his treatise General Trattato di numeri, et misure published in Venice
In his book ‘Nova Scientia’ he wrote-
a mio avviso il libro indiano di flussioni è 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
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
Cavalieri’s 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".
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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 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, by granting him a salary A K TIWARI 15
instruction in the
art, to the nine
digits of the
Hindus. I use
method of the
Indorum)” -- In
the Liber Abaci ,
Fibonacci and his Guru (Brahmagupta ?) see Brahmagupta –Fibonacci identity
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• 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
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
• Bhaskara II indicates the method of obtaining the differential of sine θ. His method is
probably the same as that employed by his predecessors.
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• Attention was first drawn to the occurrence of the differential
δ (sin θ) = cos θ δ θ
in Bhaskara II's (1150) Siddhanta Siromani shows that Bhaskara was
fully acquainted with the principles of the differential calculus.
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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 Hindu
• 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
• 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
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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.
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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
• 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
ज्या का फ़लाक कोज्या होता है.
Means d sinx/dx = cos x
िकसी राशी की िघाद्विघात का फ़लाक उस
राशी के िघाद्विगुण होता है.
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
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Kerala school of astronomy and
• 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 (1559–1632).
• In attempting to solve astronomical problems, the Kerala school independently created a number of important mathematics
• Their most important results—series expansion for trigonometric functions—were described in Sanskrit verse in a book by
Neelakanta called Tantrasangraha, and again in a commentary on this work, called Tantrasangraha-vakhya, of unknown
• 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
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• Nilakantha (c. 1500) in his commentary on the Aryabhatiya has given
proofs, on the theory of proportion (similar triangles) of the following
(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.
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• 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
(2) The difference of the cosine-difference varys as the cosine negatively and decreases (numerically) with
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
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• 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
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• 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.
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• The French or Catalan Dominican missionary Jordanus
Catalani was the first European to start conversion in
• 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
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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 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.
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• 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
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