1
Genesis of Calculus
K.Chandra Hari∗
Abstract
Present paper is an attempt to have a fresh look at the controversial thesis on the Indian origin of
calculus in the light of the European developments of 17th century that culminated in the
formulation of calculus by Newton and Leibniz. The profile of European advances in mathematics
beginning from the 12th century is replete with relatively older Indian signatures such as Hindu
numerals, place-value notation, algebra, Barrow’s differential triangle, vanishing chord vis-à-vis
tangent at a point to the curve, maxima and minima, Bhaskara’s differential formula as well as
Madhava’s series expansions. All of these signatures were in blossom in Kerala during the 14th to
17th century when the direct maritime contact was established between Europe and Kerala with the
maiden trip of Vasco da Gama taking place in AD 1497. The art of writing history as before us in
numerous instances calls for no restraint in identifying Kerala as the source of critical information
that ignited the European revolution in mathematics and the formulation of calculus.
Key words: Calculus, Bhaskara II, Madhava, mathematics, Newton, Leibniz.
I. Introduction
The fundamental ideas out of which Calculus had its evolution can be traced back to the great
antiquity of early Greek logicians – as early as in 450 BC we can find Zeno speaking of the
infinitesimal in describing the paradox of tortoise beating the Achilles. Around 370 BC Eudoxux
placed the method of exhaustion1 on a sound scientific footing and was to be later used by
Archimedes (287-212 BC) in the determination of the areas and volumes of many geometrical
figures such as circle, sphere, cone etc. Thus was born what we have come to know in modern
times as integration- the art of creating the whole from the infinitesimal. No further progress is
perceptible in the western world until the century of geniuses where we find an array of original
approaches in further developing the age-old geometrical methods to analytical dimensions. This
new impetus had at its base the works of Johannes Kepler (1615) on celestial mechanics and of
Galileo on the free fall of bodies’ vis-à-vis acceleration due to gravity. Kepler had determined the
area of sectors of an ellipse by invoking the principle of infinitesimal in its crude form and Cavalieri
in 1635 developed the method into a geometry of indivisibles. In 1637 Descartes made his
∗
B-204, Parth Avenue, Chandkheda, Ahmedabad – 380005 [email protected]
1
Computing areas/volumes of curved figures by successive approximations by inscribing or circumscribing
polygons and other shapes whose areas/volumes are already known. For example, finding the area of a
circle using approximations by regular polygons with increasing numbers of sides.
2
appearance with the all-encompassing analytic geometry simultaneously with Fermat’s method for
determining maxima and minima and tangents to curves. Fermat’s work on tangents was
succeeded by that of Isaac Barrow in 1669 and the wave of creativity ultimately found its peak in
Isaac Newton (1642-1727) who carved out calculus mainly based on the works of Barrow and
Fermat. Confining our-selves to the 16th and 17th centuries the cardinal steps in this evolutionary
process can be summarized as:
Kepler’s use of the infinitesimal constituents of the whole in his planetary theory
Marriage of geometry and algebra achieved by Descartes
Method for maxima and minima – Fermat
Finding a tangent to the curve – Fermat and Barrow
Newton and Leibniz (1646-1716) discovering the limit of a sum and the development of calculus
as a mathematical method
Newton had arrived on the scene at the most opportune moment to complete the chain of discrete
thinking that had taken place thus far to formulate the calculi of fluents and fluxions into a most
efficient tool for the description of the physics he had invented.
It was against this marvelous European background that Pandit Bapudeva Sastri 2placed a claim of
Indian origin for calculus in 1858 in the journal of the Asiatic Society of Bengal. Since then
refinement of Sastri’s claims have been made by scholars like Brajendranath Seal3, Sengupta4,
Bag5 and Prof. K.S.Shukla6 to establish the first use of differential calculus in India. Present paper
is an attempt to critically examine the soundness of Indian claims vis-à-vis development of calculus
in India in contrast to the evolution of ideas in the European theatre.
II. Western Reasoning and Origin of Calculus
Sufficient information and illustrations on the origin and development of Calculus is available with
Internet resources on the history of mathematics. The salient features of the development can be
outlined as follows:
2
Sastri, Bapudeva, ‘Bhaskara’s Knowledge of the Differential Calculus’, Journal of the Asiatic Society of
Bengal, 27, pp.213-216, 1858.
3
Seal Brajendranath, The Positive Sciences of the Ancient Hindus, Motilal Banarssidas, New Delhi, 1991,
pp.77-80.
4
Sengupta, P.C., Journal of the Department of Letters, Vol.XXII, 1931, Calcutta University.
5
Bag, A.K., Mathematics in Ancient and Medieval India, Chaukhamba Orientalia, 1979, Varanasi
6
Shukla Kripa Shankar, ‘Use of Calculus in Hindu Mathematics’ Indian Journal of History of Science, 19 (2),
1984.
3
•
Fermat had investigated the problem of maxima and minima by configuring the tangent to the
curve as parallel to the x-axis i.e., in the same way as the modern method by equating the
derivative of the function to zero. Lagrange as such considered Fermat to be the inventor of the
calculus.
•
Similar conceptions involving the derivative can be found in the works of Hudde and Barrow
with the latter describing the tangent as the limit of a chord across two merging points. Barrow
also considered the problem of motion with variable speed and had been aware of the process
of obtaining velocity as the derivative of distance and the inverse process. Barrow as such had
been on the trail of the fundamental relation between integration and differentiation when
Newton arrived on the scene.
•
Both Leibniz and Newton configured the process in terms of graphs rather than functions.
Newton considered variables changing with time and the objective was probably limited to the
creation of a geometric technique to express his own physical discoveries while Leibniz thought
of variables x and y as consisting of infinitesimally small increments dx and dy and chose to
develop it as an analytical tool with appropriate notations. In fact Leibniz had been on the
search for a lingua generalis as is evident from what he wrote at the age of 20 in his De arte
combinatorial – “a general method in which all truths of the reason would be reduced to a kind
of calculation”.
•
Leibniz used the f(x)dx notation for the first time on 21November 1675. Newton had been using
his method of fluxions to deal with change and motion since 1665 but he published the ideas
only in 1687 – three years after the publication of Leibniz’s paper, “A new method for maxima
and minima as well as tangents…and a curious type of calculus for it”.
•
Newton combined the ‘infinitesimal’ of the Greeks and the graph system of Descartes and
conceived geometrical figures as ‘fluents’ evolving from the continuous motion of a point or line
and the velocity of the moving point or line became the fluxion of the fluent. Thus emerged
probably the earliest conception of a continuous function. Newton then applied the processes
of differentiation and integration in expanding the works of Fermat and Barrow for finding the
maxima and minima, tangents/curvature of curves etc. Two more fundamental operations thus
appeared on the scene – differentiation to find the ‘limit’ (the ratio of changes in two variables
as these changes approaches zero∗) while integration worked on the reverse from an equation
∗
Value of a fraction as the numerator and denominator both shrink towards 0 is called the limit. As two points
on a curve slide together the vertical and horizontal distances between them remain coupled, even as they
fade away, by the relationship of y to x expressed in the original equation of the curve. As they merge the
ratio of their differences approaches a definite limit that can be evaluated in terms of y and x. This limit is
the derivative or the instantaneous slope of the curve at the precise spot where the two points merge- the
derivative y wrt x.
4
of the rate of change to the variables involved and Newton used it to work out the laws of
motion and gravitation.
•
Apart from the deductions obtained by differentiating the Keplers 2nd law, the new operator
yielded a great insight to Newton that ‘the constant factor’ in many processes of nature is the
rate at which a rate of change changes. As for example if we consider the equation of free fall y
= 16t2, ý=32t and ÿ =32.i.e the rate of increase in the speed of a falling body is a constant
=32ft/sec.
•
None could have put forward calculus on a fine morning as a classified collection of rules.
Necessity is the mother of invention and new techniques emerge on the call of situations based
on first principles. This is the usual sequence that we see in the history of science. When
Newton and Leibniz had arrived on the scene all mathematical analysis was leading up to the
ideas and methods of the infinitesimal calculus. Calculus had begun to cast its shadows even
in the writings of Napier, Kepler, Cavalieri, Pascal, Fermat, Wallis, and Barrow and for Newton
formulation of calculus was only a minor test of his ingenuity.
III. Indian Claims on the Origin of Calculus
The different stages of development of calculus in India can be gleaned from the works of past
authors and can be summarized as follows:
(a) Just as Kepler used the ‘infinitesimal’ in his astronomical theory to compute the area of
segments of an ellipse swept out by the radius vector, in India also calculus had its beginning in
astronomy. But the Indian development precedes Kepler by almost thousand years – in the 6th and
7th centuries Aryabhata and Brahmagupta had expressed the notion of instantaneous motion
(tatkalika-gati) of a planet as:
δL(true) = δL(mean) ± e (sin m1 – sin m2),
where L stands for the longitude, m for mean anomaly and e for eccentricity or the sine of the
greatest equation of the orbit.
The Hindu sine table having values tabulated at intervals of 03045’ and interpolation did not give
the correct velocity tallying with the observations and hence the above equation was fated to
undergo further modification in future. In AD 932, Manjula transformed it into a differential equation
in his work Laghumanasa II.7:7
7
Shukla Kripa Shankar, ‘Use of Calculus in Hindu Mathematics’ Indian Journal of History of Science, 19 (2),
1984.
5
“ 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 ( = 1/e = 360/Perphery of the epicycle), added or
subtracted contrarily (to the mean motion)”.
i.e., in modern parlance, δL(true) = δL(mean) ± e ( m1 – m2) cos m2 or
δL(true) = δL(mean) ± e cos m δm
In contrast to the European situation where we have got detailed documentation of the evolutionary
stages of thinking, we have got no information as to how Manjula had arrived at the above
differential formula. But it can be inferred from the works of later astronomers like Aryabhata II
(AD950) and Bhaskara II (1150 AD) that a tradition of using the differential expression had begun
in India as early as AD 932, which surely and certainly was impossible in the absence of clear
reasoning as well as proof.
(b) Bhaskara’s method for the differential of Sin θ
The Hindu sine table had its origin with the 24 values of a quadrant equally placed at 03045’.
According to Somayaji, Bhaskara had improvised the technique to bring down the interval of 03045’
to that of a degree in the following steps:8
1) Tradition had the sines of 300, 450, 600, 180 and 360 by inscribing regular polygons in a circle.
From these five basic values the 24 sines equally placed at 03045’ were obtained using the
formulae: Sin2θ + Cos2θ =1 →(1) and Sin θ/2 = ½sqrt(sin2θ+tan2θ) = sqrt Tan θ →(2).
2) Bhaskara by the use of trigonometrical expressions could divide the quadrant further upto 30
and 90 in finding out the sine values as is evident from verses 12 to 20 of Goladhyaya of
Siddhantasiromani. These expressions are:
Sin {(90±θ)/2} = {(1± Sin θ)/2}
2
→(3)
2 1/2
Sin{(θ-ε)/2} = {(sin θ + sin ε) + (cos θ - cos ε) }
→(4)
{(Cos θ - Sin θ)2/2}}1/2 = Sin (45-θ)
→(5)
2
1- 2 Sin θ = Sin (90- 2θ)
→(6)
Sin (θ ± 1) = Sin θ [1 –(1/6569)] ± (10/573) Cos θ
→(7)
Last of these expressions in fact led him to the differential formula δ (Sin θ) = Cos θ δθ in the
following manner:
8
Somayaji, D.A., A Critical Study of Ancient Hindu Astronomy, p.9, Karnatak University, 1971
6
Sin (θ ± 1)0 = Sin θ0 Cos 10 + Cos θ0 Sin 10 . As Cos 10 is nearly equal to 1, we can write:
Sin (θ ± 1)0 - Sin θ0 = Cos θ0 . 60’ , by taking Sin θ = θ , when θ is small.
That is, for an increment of say δθ in θ the sine value had a variation of δθ Cos θ, or we can
write: Sin (θ+ δθ) – Sin θ = Cos θ δθ. Or, in modern notation, δ (Sin θ) = Cos θ δθ.
Prof. Shukla has given a detailed account of the geometrical reasoning of Bhaskara in IJHS, 19(2)
and it is interesting to note that Bhaskara had used a tangential triangle in evaluating what he
referred as a “tatkalika bhogyakhanda” – the infinitesimal increment of y against the infinitesimal
increment in x when the arc increases by δθ (δy = δx. tanθ). It is doubtlessly clear that Bhaskara
intercepted the differential formula while successively moving from a division of the quadrant into
24 equal parts to 90 equal parts and then ultimately of the “infinitesimal arcs” that constituted the
quadrant and in arriving at his result he made use of the concept of a tangential triangle at every
point of the quadrant in a manner analogous to the early developments in Europe more than five
hundred years later. The quadrant of Bhaskara as well as the tangential triangle he made use
of can rightly be considered as the precursors of the Cartesian x-y axes, the curve y = r sin
θ as well as the Barrow’s differential triangle.
The role of the ‘infinitesimal’ is well evident from Bhaskara’s description of the tatkalika –gati in the
Gatisphutiprakarana, Ganitadhyaya of Siddhantasiromani:9
“…iyam kila sthoola gatih; atha sukshma tatkaliki kathyate…yada asannasthityantastada tatkalika
gatya tithisadhanam kartum yujyate…yatascandragatih mahattvat pratiksanam sama n bhavati
atastadartham viseshobhihitah…”
The emphasis added by the present author may kindly be noted, which speaks of nothing other
than the instantaneous velocity of the moon – the differential of its longitude. It must be noted here
that till the advent of the Cartesian frame and the coordinate notation (x-y), longitude and latitude
were the terms with which the coordinate axes were referred to in Europe as is evident from the
historical records of AD 1350. Bhaskara’s notion is deficient only in terms of the conception of
calculus as a tool for mathematical reasoning at the hands of Newton and Leibniz as a result of the
avalanche of creative developments in mathematics that preceded them in the works of Napier,
Kepler, Cavalieri, Pascal, Fermat, Wallis, and Barrow.
9
Seal Brajendranath, The Positive Sciences of the Ancient Hindus, Motilal Banarssidas, New Delhi, 1991,
p.79.
7
Bhaskara’s claim to be regarded as the originator of calculus receives further reinforcement in a
number of other results he had to which a mention is available in the paper of Prof. K.S. Shukla. To
quote:10
“If the above [δ (Sin θ) = Cos θ δθ] 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’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 perigee the equation of the center vanishes;
hence he concludes that for some intermediate position the increment of the equation of center
(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”.
In these results, we can find a reflection/anticipation of the results of Fermat on maxima and
minima of functions and that of Leibniz, who first reported his results under the title-“A new method
for maxima and minima as well as tangents…and a curious type of calculus for it”. When we note
that it was Fermat’s work on the maxima and minima that inspired Lagrange to credit the invention
of calculus to him, it is nothing but fair to credit the same with Bhaskara-II on the basis of the
evidence reviewed above.
(c) Sangama-grama Madhava
In the works Analysis with infinite series written in 1669 and Method of fluxions and infinite series
written in 1671 Newton had given the series expansion for sin θand cos θ, which are now called the
Taylor or Maclaurin series. But more than three hundred years before around 1350 AD, the sine
and cosine series had its origin in Kerala – the tract of land belonging to the south-west corner of
India – at the hands of Sangamagrama Madhava. Madhava came three hundred years after
Bhaskara II and the available evidence makes him the patriarch of the medieval astronomical
tradition of Kerala. Even though the original sources of Madhava’s mathematical discoveries
remain untraced the Gurukula tradition of which he was the Patriarch has preserved the critical
10
Shukla Kripa Shankar, ‘Use of Calculus in Hindu Mathematics’ Indian Journal of History of Science, 19 (2),
1984.
8
information necessary to glean the genius of Madhava. Madhava is credited with the following
discoveries by the subsequent authors such as Jyestadeva (cf. Yuktibhasa), Narayana (cf.
Kriyakramakari) and Nilkantha/Sankara Varrier (cf. Tantrasamgraha):11
1. Sine and Cosine series
Sin θ = θ - θ3/3! + θ5/5! - … and Cos θ = 1- θ2/2! + θ4/4! - θ6/6! + …
Madhava’s sine values accurate up to 8th or 9th decimal places suggest the use of these
expressions in deriving those values.
Also Madhava is known to have derived:
Sin (θ+ h) = sin θ + (h/r) cos θ - (h/2r2) sin θ and
Cos (θ+h) = cos θ - (h/r) sin θ - (h/2r2) cos θ, which are special cases of Taylor series (1700 AD)
2. Infinite series for the arc of a circle in terms of sine and cosine functions
Detailed discussion of the method is available in reference (). Madhava’s series in modern notation
is:
θ = tan θ - [(tan3θ)/3] + [(tan5θ)/5] - … This is equivalent to the inverse tangent series
discovered by James Gregory in 1667 and Leibniz in 1671.
3. Euler’s series for π/4
When θ = 450, the series reduces to Euler’s series: [π/4] = 1 – 1/3 + 1/5 – 1/7… and by putting θ =
300,Madhava obtained an approximation for π as: π = ⊕12 [1-1/32 + 1/32.5 – 1/33.7+…].
Nilkantha has credited Madhava also with the expression:
πd = 4d – [4d/3 + 4d/5] -…± [4dn/((2n)2 +1)] , where d is the diameter of the circle and n the
number of terms. By replacing the last term with {4d(n2+1)/n[(n+1)2+1]}, Madhava could estimate
the value of π correct up to 11 decimal places, i.e., π = 3.14159265359.
Bag12 has discussed the geometrical proof available in Yuktibhasa in detail. As is the case with
Bhaskara Jyestadeva has considered the division of the quadrant in to n infinitesimal parts with the
same logic as that of the Barrow’s differential triangle. The proof obviously may have its origin with
Madhava and might have survived time through his disciples like Paramesvara of Drgganita fame.
On the face of the available evidence the only fact apparent is that the derivation of the formulae
11
12
George G. Joseph, The Crest of the Peacock, pp.286-293
Bag, A.K., Mathematics in Ancient and Medieval India, Chaukhamba Orientalia, 1979, Varanasi
9
must have been through geometrical methods and differential calculus may not have played any
significant role in the process. But it must be noted that at the base of the geometrical proof we can
find the differential formula of Bhaskara and therefore it is very difficult to comprehend that
Madhava was unaware of what we may call as the “differential connection” of the result.
In this connection we may note that Govindaswami (AD 800) and Bhaskara were in possession of
what has come to known as the Newton – Gauss interpolation formula up to the 2nd order, which is
expressed as:
f(a+xh) = f(a) + x δf(a)+ (1/2)(x)(x-1)[δf(a) - δf(a-h)] where x = δθ/h.
f(a+δθ) = f(a)+ δθ.δf(a)+( ½!)(δθ)2. δ(δf(a))+…(as in Taylor series)
Comparing this with Bhaskara’s result referred earlier Sin (θ+δθ) = Sin θ.Cos δθ + Sin δθ.Cos θ =
Sin θ+ δθ. Cos θ, a genius like Madhava could have easily realized that it is an infinite series
truncated under the approximation of (δθ)2 τ 0. Obviously his next step would have been to explore
the infinite series expression for sin θ in terms of the “differentials”. It is quite unlikely that a
complex geometrical construction would have unfolded in his mind to yield the sine and cosine or
some other series. On the contrary the geometrical construction would have been laboriously
worked out to prove the intuitive deductions he might have made using the differentiation process.
With scanty evidence it’s of course very difficult to recreate the thinking of Madhava after almost
650 years and after the extinction of the tradition sprung from him. Such intuitive or some kind of
analytical deduction is not altogether impossible when we note that the binomial expansion for
(a+b)n for integer values of n and the Pascal triangle have been known in India13 since very early
times. It is true that as has happened in the west after 300 years, the technique did not develop in
to a fundamental operation and the works of his disciples remained centered over epicyclic
astronomy. In view of the fact that Madhava’s works on mathematics remains untraced, we lack the
confirmatory evidence and therefore it can be conceded that Madhava did not formulate calculus in
the Newtonian fashion. But ironically it is Newton - coming after 300 years - who offers the best
circumstantial evidence in support of Madhava’s knowledge of the Newtonian methods of calculus
or at the least all its precursors.
(d) Comparable profiles of the genesis of calculus with Newton and Madhava14
(i)
Earliest documentary evidence of the invention of calculus is a manuscript dated May28,
1665, written at the age of 23 and according to A Short Account of the History of
13
Datta, B., Singh, A.N., Revised by K.S.Shukla, “Use of Series In India”, IJHS, 28(2), p.121
Details about the works of European mathematicians have been taken from Encyclopedia Britannica, Vol.
11, 1971 and Grolier Encyclopedia.
14
10
Mathematics (4th edition, 1908) by W. W. Rouse Ball it was about the same time that he
discovered the binomial theorem.
(ii)
Newton’s work on Analysis with infinite series was written in 1669 and his Method of fluxions
and infinite series was written in 1671. In these two works the series expansion for sin x and
cos x appeared for the first time. These works were published respectively in 1711 and 1736.
(iii)
The inordinate delay in the publication of these tracts can be suspected as due to Newton’s
lack of confidence in the method of fluxions. In the Principia Newton therefore has presented
the topic by the method of limits. Further, though he had derived many of his results of
astronomy and mechanics by the method of fluxions, he has attempted rigorous geometrical
proofs for the same.
We need to take note of this fact specially because analogous may be the case with Madhava,
whose results meet only with geometrical proofs in the traditional records. Had Newton been not
doubtful of the method of fluxions/limts he had no reason to attempt cumbersome geometrical
proofs. At the least he was not confident that the new technique was superior to the then prevalent
methods of geometry. This is not surprising when we note that several mathematicians like
Huygens had opposed the method of calculus from the very beginning. Skepticism on calculus
vanished only in 1823 when Cauchy published his treatise on differential calculus.
It is quite likely that similar factors might have been on play at the time of Madhava also, which
prevented the recognition of new methods as genuine mathematical techniques. In the European
theatre situation was far more satisfactory as Descartes had already brought into existence the
power of reasoning and a new paradigm of mathematical thinking with the publication of his work
La géométrie.
(iv)
Development of calculus in fact required as prerequisites only the binomial theorem and the
differential/integral formula and the rest were simply details emerging out of reasoning. In the
words of Hegel:15
“The whole method of the differential calculus is complete in the proposition that d(x)n = nx(n - 1)dx,
or (f(x + i) - fx)/i = P, that is, is equal to the coefficient of the first term of the binomial x + d, or x + 1,
developed according to the powers of dx or i. There is no need to learn anything further: the
development of the next forms,of the differential of a product, of an exponential magnitude and so
on, follows mechanically; in little time, in half anhour perhaps — for with the finding of the
15
Taken from the Internet resources; History of Mathematics
11
differential the converse the finding of the original function from the differential, or integration, is
also given — one can be in possession of the whole theory”.
It becomes therefore apparent that Madhava had both the pre-requisites as well as the end results
such as the sine and cosine series in his possession. Whether he had the method or not to move
from the pre-requisites to the series expansions of Newton is a very silly question, because without
the correct method he could not have obtained the correct results.
(v)
We have already seen above that the Indian tradition had all such things as Barrow’s
differential triangle, vanishing chord vis-à-vis tangent at a point to the curve as well as
maxima and minima in the work of Bhaskara. Madhava, in fact, is the astronomer who has
made a most efficient use of the maxima and minima aspect involved in planetary motion in
the computation of true longitudes of planets. Madhava’s famous works, Venvaroha and
Aganita, computes true positions of the moon and planets using anomalistic revolutions
banking on the fact that the equation of center vanishes when the moon/planet is at its
apogee (perigee also). Detailed theory of the method can be found elsewhere and is beyond
the scope of the present paper. But the astronomical technique employed by Madhava offers
testimony for the fact that mathematics had given him enough conviction of the significance
of the anomalistic revolutions in the computation of true positions. In the history of astronomy
none had so much reliance on anomalistic revolutions as Madhava had and perhaps the
accuracy of Madhava’s moon stood unbeaten till the advent of Brown’s theory16.
(vi)
Revolutions of physics and astronomy as well as the publication of Discours de la methode
by Descartes in 1637 in fact paved way for the development of calculus as a technique. All
the great names of physics and astronomy were involved in the game. In 1655 John Wallis
had published Arithmetica Infinitorum – infinitesimal algebra, which brought forth infinite
series straight from algebraic grounds. By this time all the ingredients such as analytic
geometry, infinitesimal methods, study of areas and tangents were ready and the theory of
dynamics necessarily required (dx/dt) and (d2x/dt2) in the creation of acceleration and force. It
was calculus that made acceleration conceivable and inspired the conception of force. While
Madhava and his successors were groping with the quadrant in developing the different
series expressions dynamics and Galileo had given Newton the path function x = x (t) in
terms of the universal independent variable t. It was mechanics that (had become stand still
since the days of Archimedes) shaped calculus in the mind of Newton while the Indian
16
Chandra Hari, K., “Sangamagrama Madhava”, Paper under submission to IJHS, INSA, New Delhi-2
12
theatre of astronomy and mathematics provided no such intellectual stimuli to those who
succeeded Bhaskara and Madhava.
IV. New Impetus to Mathematics in Europe – 16th Century on wards
When we reflect upon factors that might have inspired the new impetus to mathematics in Europe
from 16th century onwards, the first thing that strikes our attention is the marriage of ‘Islamic’
algebra with Greek geometry achieved by Descartes in his analytic geometry. This cardinal step in
the evolution of modern science was preceded by:
Islamic scholars had brought in the wisdom of both the Hindus and the Greeks to the west
and in AD 1202 Leonardo Pisano introduced the ‘Arabic’ numerals and place-valued
decimal system in Europe.
Early in the 16th century great progress was made in algebra - another Arabic subscription
from India transmitted to Europe – and Niccolo Tartaglia discovered the general solution
for cubic equations.
In the late 16th century Francois Viete demonstrated the value of symbols by using (+)/(-)
signs for operations and letters to represent unknowns.
It is well evident from the above that the Islamic contact has played a crucial role in the new
impetus to mathematics in medieval Europe. On the other hand for the Islamic world the repository
of all wisdom and knowledge was of course the Hindus and was thus the Hindu numerals and
algebra reached Europe in the 13th century. May be it’s accidental that the European renaissance
in mathematics accompanied the direct European contact with India, especially Kerala which had
been a center of maritime trade since time immemorial – Vasco da Gama sailed from Lisbon on 8th
July 1497 and had set his foot at Calicut on May 20, 1498 inspired by the European ambition to
outsmart the Muslim traders. In this context the following words of George G. Joseph are
noteworthy:17
“…in Kerala the period between the fourteenth and seventeenth centuries marked a high point in
the indigenous development of astronomy and mathematics. The quality of the mathematics
available from the texts that have been studied is of such a high level compared with what was
produced in the classical period that it seems impossible for the one to have sprung from the other
– there must be ‘missing links’ to bridge the gap between the two periods. There’s no ‘convenient’
external agency, a Greece or Babylonia that we can invoke to explain the Kerala phenomenon. In
17
George G. Joseph, The Crest of the Peacock, p.287
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deed the only point of comparison is with later discoveries in European mathematics, which were
anticipated by Kerala astronomer-mathematicians two hundred to three hundred years earlier. And
this leads us to ask whether the developments in Kerala had any influence on European
mathematics. To answer this question, there is a need for a careful examination of the nature of
the contacts between this most accessible of areas and the Europeans who came here in the wake
of Vasco da Gama. There is some evidence, mentioned by Lach(1965), of a transfer of technology
and products from Kerala to Europe. A lot more research on archival material from maritime,
commercial and religious sources is required before the matter can be satisfactorily resolved”.
If we intend to follow the western tradition and style of writing history by crediting every piece of
early astronomical and mathematical knowledge to Babylon / Greece, I see no reason for a
restraint in identifying Kerala as the source of origin of medieval European revolution in
mathematics. As mentioned earlier all the basic constructs that the Europeans have used in
creating calculus viz., Barrow’s differential triangle, vanishing chord vis-à-vis tangent at a point to
the curve as well as maxima and minima, had been popular in India and especially in Kerala at
least 300 years before the European tryst with them.
None can deny the European maritime contact with Kerala
None can deny the existence of Gregory, Leibniz, Taylor etc., series in Kerala at least in
their primitive form more than 200 years before their discovery by Europeans.
How can Europeans claim originality for their findings of say, sine and cosine series, when they
themselves (scholars like Pingree) have credited Arybhatan astronomy to Babylonian sources on
the ground of similarities?
Also Europeans have credited the Hindu numerals and the sexagesimal system also to Babylonian
sources. So it’s European conclusion that no two places can have the claim of originality of
invention of the same concepts and if it’s so apparently one must be a copying of the other. So
there exists an a priori conclusion that it was Kerala mathematics that ignited the European genius
in the 17th century and hence it is quite logical to place Bhaskara II and Madhava at a stature
equivalent to those of men like Galileo and Kepler if not Newton and Leibniz.
V. Conclusions
A comparative study of the evolution profile of calculus and power series in India at the hands of
Bhaskara II and Madhava with that of the 17th centuries geniuses of Europe like Fermat, Barrow
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and Newton suggests that even in Europe calculus had its emergence through the essentially
Indian constructs such as the quadrant and the tangential triangle at a point, idea of the
infinitesimal, instantaneous velocity, maxima and minima etc. If Fermat’s work on maxima and
minima could inspire Lagrange to credit him with the discovery of differential calculus, nothing
should deter us from ascribing the invention of calculus to Manjula and Bhaskara II – especially the
latter, who has bequeathed to us a proof for the differential formula viz., δsinθ = cosθ. δθ, in terms
of the quadrant (curve), tangential triangle, instantaneous velocity and the notion of infinitesimal.
Similar is the case with Madhava when we consider the power series expansion of trigonometric
functions as well as π. In fact precursors of all the constructs over which the medieval European
revolution in mathematics was founded are essentially Indian and the concepts and techniques
were in full blossom in Kerala, when the Europeans established direct maritime trade contact with
India and Kerala in 1497 AD – in the century preceding the revolution in mathematics. As observed
by George G. Joseph - “There’s no ‘convenient’ external agency, a Greece or Babylonia that we
can invoke to explain the Kerala phenomenon. In deed the only point of comparison is with later
discoveries in European mathematics, which were anticipated by Kerala astronomermathematicians two hundred to three hundred years earlier. And this leads us to ask whether the
developments in Kerala had any influence on European mathematics” – In the light of this
observation if we follow the European tradition of creating history no restraint is called for in
ascribing Madhava and Bhaskara II as the sources of European revolution in mathematics in the
17th century.
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