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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 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 ∗ B6-103, ONGC Colony, Gandhinagar, Ahmedabad - 382424 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.

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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:

1. 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.

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.

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2. 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.

3. 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”.

4. 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”.

5. 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 of the rate of change to the variables involved and Newton used it to work out the laws of motion and gravitation.

6. 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.

∗ 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 which can be evaluated interms 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.

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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

“ 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 θθθθ

The Hindu sine table had its origin with the 24 values of a quadrant equally placed at 03045’.

7 Shukla Kripa Shankar, ‘Use of Calculus in Hindu Mathematics’ Indian Journal of History of Science, 19 (2), 1984.

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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} →(3)

Sin{(θ-ε)/2} = {(sin θ + sin ε)2 + (cos θ - cos ε)2}1/2 →(4)

{(Cos θ - Sin θ)2/2}}1/2 = Sin (45-θ) →(5)

1- 2 Sin2 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: 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 θθθθ as well as the Barrow’s differential triangle.

8 Somayaji, D.A., A Critical Study of Ancient Hindu Astronomy, p.9, Karnatak University, 1971

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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. 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.

9 Seal Brajendranath, The Positive Sciences of the Ancient Hindus, Motilal Banarssidas, New Delhi, 1991, p.79. 10 Shukla Kripa Shankar, ‘Use of Calculus in Hindu Mathematics’ Indian Journal of History of Science, 19 (2), 1984.

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(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 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 the 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 ππππ/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)] ,

11 George G. Joseph, The Crest of the Peacock, pp.286-293.

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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 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

12 Bag, A.K., Mathematics in Ancient and Medieval India, Chaukhamba Orientalia, 1979, Varanasi. 13 Datta, B., Singh, A.N., Revised by K.S.Shukla, “Use of Series In India”, IJHS, 28(2), p.121 14 Details about the works of European mathematicians have been taken from Encyclopedia Britannica, Vol. 11, 1971 and Grolier Encyclipedia.

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(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 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 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.

15 Taken from the Internet resources; History of Mathematics

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(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 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.

16 Chandra Hari, K., “Sangamagrama Madhava” , Paper under submission to IJHS.

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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 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 17 George G. Joseph, The Crest of the Peacock, p.287

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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 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 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” – 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.