HINDU THOUGHTS ON CALCULUS BEFORE NEWTON AND LEIBNIZ

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1 Hindu thoughts on Calculus before Newton and Leibniz 106 we may consider Madhava to have been the founder of mathematical analysis. Some of his discoveries in this field show him to have possessed etraordinary intuition George Gheverghese Joseph CHAPTER - IV HINDU THOUGHTS ON CALCULUS BEFORE NEWTON AND LEIBNIZ 4.1 INTRODUCTION Study of Mathematics in India has a long tradition. Mathematics and mathematicians were deeply respected in India. Hindu numerals, place value notation, algebra, astronomy are remarkable contributions of the Hindus to mathematics. Since the time of Sulbasutras (the oldest tets prior to 800 BC still in eistence) which dealt with mathematics of surds, construction of altars and fireplaces of different size and shapes, method of etracting roots. Differential equations of sine function of Aryabhatta and solution of quadratic indeterminate equation of Jaydeva and Bhaskaracharya (1 th century) laid the landmarks of Indian mathematics. Manjula (or Munjala) composed a treatise on Nakshatra Vidya (astronomy)- Laghumaanasa. He was born at Prakaasa pattana (may be in Gujarat). Laghumaanasa dealt with simple and brief rules involving less calculations. Boudhayana invented two methods of finding quadrature of a circle in addition to so called Pythagoras Theorem. The profile of European mathematics began from 1 th century only. Many researchers challenged the traditional history of calculus in particular and also mathematics in general. They are of the opinion that medieval Indian mathematicians developed some core concepts of calculus which were being neglected or ignored due to many reasons. One of the most important core

2 Hindu thoughts on Calculus before Newton and Leibniz 107 concepts of calculus is Infinite series. Infinite series was first developed in India in 14 th century. George Gheverghese Joseph of University of Manchester and Dennis Almeida of University of Eeter spent three years in India to study Indian tets, commentaries etc. They are of the opinion that calculus was invented in India 50 years before Newton. But calculus was studied to a limited etent and used for specific purposes (astronomical) in India during early and medieval times. So, calculus was used to find favorable and unfavorable times for religious rites and rituals on the basis of the moments of the eclipses, conjunction of the planets and conjunction of the planets and the stars also. 4.. THE KERALA SCHOOL OF ASTRONOMY AND MATHEMATICS Probably the most famous school of mathematics in India that contributed remarkable discoveries to mathematics in general and to calculus in particular is the Kerala School of Astronomy and Mathematics. Here was the Guruparampara or chain of teachers originating with Madhava of Sangamagrama (modern Irinjalakuda near South Malabar region of Kerala) in the late 14 th century and continuing up to the beginning of the 17 th century. Parameswara, Damodara, Neelkantha Somayaji, Jyesthadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikar were active members of the School. Calculus was developed during this period and many books were written. The ancestry of the calculus scholars of Kerala School are as follows: Madhava ( ) Parameswara ( ) Damodara ( ) Neelkantha ( ) Jyesthadeva ( ) Achyuta ( Melpathur ( )

3 Hindu thoughts on Calculus before Newton and Leibniz : Madhava and Yuktibhasa Madhava s most significant contribution was the transition from finite procedures of ancient mathematics for treating their limit passage to infinity which is considered to be of vital importance in analysis. The School made intuitive use of mathematical induction to discover semi-rigorous proof of some results. K.V.Sarma, an authority on Madhava identified Madhava as the author of the following books : 1. Golavada. Madhyanayanaprakara 3. Mahajyananaprakara 4. Venvaroha 5. Lagnaprakararana 6. Sphutacandrapti 7. Candravakyani 8. Aganita-grahacara Jyesthadeva, a student of Madhava wrote Yuktibhasa or vernacular of rationales in Malayalam in about 1530 AD. It is a major treatise in which the discoveries of Madhava, Neelkantha, Parameswara, Jyesthadeva, Achyuta Pisharati and others were included. It contains detailed proofs of mathematical procedures stated in Tantra-Sangraha. Madhava was responsible for the discovery of the following results: r sin r =. ( r(1 cos ) = r r ( + ) r ) r +. ( r. ( + ) r ) r. (4. (4 + 4) r 4) r when r = 1, the results found in modern factorial notation as sin = + 3! 5! [Madhava-Newton power series]

4 Hindu thoughts on Calculus before Newton and Leibniz cos = [ do ]! 4! (i) (ii) 3 5 tan tan = tan ! 5! tan = (i) is attributed to Gregory and (ii) is attributed to Gregory and Leibniz. 4..: Madhava s arc-difference rule To find an unknown arc from known sine and cosine, procedure invented by Madhava was eplained in the verses of Nilakantha s Tantra-sangraha. The formula for the arc is as follows: The divisor [derived] from the sum of the cosines is divided by the difference of the two given sines. The radius multiplied by two is divided by that result. That is the difference of the arcs In other words, if the sines and cosines of known arc θ and unknown arc θ + θ are given, then their difference θ is given as θ R cosθ + cos( θ + θ ) sin( θ + θ ) sinθ [41]. In Yuktibhasa, the series epansion of pi, sinθ, cosθ, arctanθ of Madhava were presented with proof in terms of the Taylor series epansions. In Europe, such series were developed by James Gregory in Obtaining a fast convergent from a slow convergent is a major development of mathematical analysis. This was also discussed in Yuktibhasa in a different way. Madhava s work was outstanding because of his estimate of an error term. In Mahajyanayanaprakara (Methods for the great sines) of Madhava, the infinite epansion of π was given as π 1 = which he obtained from the power series epansion of arctangent function.

5 Hindu thoughts on Calculus before Newton and Leibniz 110 Also, he gave correction term as (i) 1 R n =, or (ii) 4n n R n = (iii) 4 n +1 R n + 1 = n 4n + 5 n The third correction gives us a highly accurate value of π. Mdhava derived the results h h sin( θ + h) = sinθ + cosθ r r h h cos( θ + h) = cosθ sinθ r r sinθ cosθ which are the special cases of Taylor series established in 1700 AD. Yukti-Bhasha generally uses the technique of delaying the step of taking the limit as much as possible. In finding 3 θ sinθ = θ 3! 5 θ + 5!... the word sunya prayam whose literal meaning is being similar to zero was used. Today we would have called it infinitesimal! There is also the usage of the word yathestam whose meaning is as one wishes. For eample, dividing lines into arbitrarily large subdivisions, the word yathestam was used [55]. The surface area and volume of a sphere also obtained in Yukti-Bhasha by integration of the infinitesimal elements. The notation of integrals, termed sankalitam is found in the statement Ekadyekothara pada sankalitam samam padavargathinte pakuti which means the integration of a variable (pada) equals half that variable squared (varga) [50]. For these reasons, many scholars consider Yuktibhasa as the world s first tet book on calculus [17][].

6 Hindu thoughts on Calculus before Newton and Leibniz 111 Madhava also developed the concept of differentiation, term by term integration after finding epansion, iterative methods for solution of non-linear equations and the evaluation of the area under a curve. G.G Joseph states, we may consider Madhava to have been the founder of mathematical analysis. Some of his discoveries in this field show him to have possessed etraordinary intuition. Ian G.Pearce (00) states that Madhava has been called the greatest mathematician-astronomer of medieval India. In addition to those above, they used improved series to find a rational epression /3315 for π. Parameswara, one of Madhava s student discovered Mean Value Theorem possibly derived from Bhaskara s work. The theorem is commonly known as Cauchy s theorem. Parameswara s book of calculus, including Drig-ganita was available even to Arabs also POSSIBLE PROPAGATION Clock technology came into light in 18 th century. Prior to it, navigational theorists and mathematicians like Stevin, Mersenne felt that for the solution of latitude problem, a reformed calendar was necessary. The eisting European calendar was off by 10 days (more than 3 degrees). For this reason, large inaccuracies occurred in calculating latitude of a place from solar altitude at noon. The method of determination of latitude was described in Laghu Bhaskariya of Bhaskara I ( AD). The Jessuits were aware of it. Kerala, a narrow strip of land is located in between West Ghat mountains and the Arabian sea. Because of its peeper production and geographical location, Kerala became a major centre of international trade for many centuries. Many researchers are of the opinion that the European navigators, navigational theorists and mathematicians in the Malabar Coast came in contact with the scholars of Kerala School of Astronomy and Mathematics. The port of Muziris was a major trade centre located near Sangamagrama. Jessuit missionaries were active in that area. Jessuit like Matteo Ricci who was trained in mathematics and astronomy by Clavius, the head of the Gregorian Reform Calendar was sent to India to learn

7 Hindu thoughts on Calculus before Newton and Leibniz 11 local methods of time-keeping (Jyotisa). In a 1581 letter, Ricci acknowledged that he was trying to understand local methods of time-keeping from an intelligent Brahmin. G.G.Joseph is of the view that the writings of KSAM may have been transmitted to Europe via Jessuit missionaries and traders during 15 th to 16 th century. In this contet, we would like to refer to the writings of C.K.Raju. Raju writes, Jesuit records show that they sought these tets as inputs to the Gregorian Calendar Reform, which, I point out, was needed to solve the latitude problem of navigation. The Jesuits were equipped with knowledge of both the local language and the mathematics and astronomy needed to understand local customs, and how dates of traditional festivals were fied using the local calendar (panchanga) [0]. This way, the difficulties with the calendar reform were settled. Yet, the problem of eact sine values which were the objectives of study of the navigational theorists like Nunes, Mercator remained unsolved. Aryabhatta s sine tables were etended by Madhava using series epansion of sine functions. These were the most accurate sine tables. It is mentioned in Jyotisa tets, including the Karanapaddhati. Nunes, Stevin, Clavius were all aware of it. As a result, European development in the field of analysis and calculus might have been influenced by the Indian contributions ON THE ORIGIN OF CALCULUS Mathematical astronomy or what is known as the Siddhanta tradition has been the dominant and continuing mathematical tradition in India, flourished almost continuously for over seven centuries, starting with Aryabhatta ( ) etending to Bhaskara II ( ) and beyond. Madhava s works were referred to the works of later mathematicians of KSAM as the source for several infinite series epansions representing the first step from the traditional finite processes of algebra to considerations of the infinite, with its implications for the future development of calculus and mathematical analysis.

8 Hindu thoughts on Calculus before Newton and Leibniz 113 Madhava s outstanding works remained unnoticed for several decades due to some reasons: (i) Since Madhava s mathematical works were relevant to astronomical calculations, his insight and techniques were covered by the huge astronomical calculations. (ii) His works were written in local language Malayalam and no attempt has been made to translate those works up to (iii) The investigators like R. C. Gupta, C.T. Rajagopal and M.S.Gopalachari made remarkable investigations on the works of the Kerala School of Astronomy and Mathematics in 1960 and later published in Indian Journals only. Many of these journals were not to be found in Europe and America in those days. Because of this, the outstanding discoveries of Indian scholars remained unnoticed in Europe and America. Pandit Bapudeva Sastri placed a claim that calculus had its origin in India. His paper was published in the Journal of the Asiatic Society of Bengal in Later Brajendranath Seal, Achyut Kumar Bag and K.S.Sukla also tried to establish the fact that differential calculus was first used in India. On the basis of the works of these authors, we can cite several points. (i) Aryabhatta and Brahmagupta in 6 th and 7 th century epressed the notion of instantaneous motion (tatkalika-gati) of a planet which was given by the formula u u = v v ± e(sin w sin w) where u, v, w are the true longitude, mean longitude, mean anomaly respectively at any particular time and u, v, w are the values of the respective quantities at a subsequent instant and e is the eccentricity of the orbit. Manjula (93 AD) modified the formula into a differential equation in his book Laghumaanasa as u u = v v ± e( w w) cos w

9 Hindu thoughts on Calculus before Newton and Leibniz 114 In modern notation of differential calculus when L L δ = e wcos w ( true) δ ± δ ( mean) w w is small The formula was found in use in the works of Aryabhatta II (950AD). In 1150, Bhaskara II ( AD) gave the proof of the formula. He states the tatkalikagati (instantaneous motion) of a planet is the motion which it would have, had its velocity during any time interval of time remained uniform. Then he noticed that for tatkalikagati, the time interval must be very small which according to the Hindus is an infinitesimal interval. This equation provided his familiarity not only with the notion of differentials, but also the knowledge of the epression d (sin α) = cosα. dα (ii) By the use of trigonometric epression, Bhaskara divided the quadrant upto 30 and 90 divisions to find sine values which is found in verse 1 to 0 of Goladhyaya of Siddhantasiromani. In one epression, we find 1 10 Sin ( θ + 1 ) = sin θ[1-( )] + ( ) cosθ This led him to differential equation as follows: sin( θ 1) sin cos1 + cos sin = θ θ, since 1 1 cos 0 and sin θ = θ when θ is small. Then 0 0 sin( θ + 1) sinθ = cosθ 60 0 i.e for an increment δθ in θ sin( θ + δθ) sinθ = cosθδθ i.e δ (sin θ) = cosθδθ This differential formula occurred in Siddhanta Siromani of Bhaskara II ( ) and was used by Bhaskara II to calculate ayana-valana (angle of position).

10 Hindu thoughts on Calculus before Newton and Leibniz 115 Also, Bhaskara showed that (a) when a variable attains the maimum value, its differential vanishes. (b) when a planet is either in apogee or perigee the equation of the centre vanishes, hence he concludes that for some intermediate position the equation of centre also vanishes. Today, this result is known as Rolle s Theorem, the Mean Value Theorem of differential calculus. Bhaskara II made it clear that the differentials give true results only when very small variations were occurred. (iii) Neelakantha s result: In his commentary on the Aryabhatiya, Neelkantha gave proofs on the theory of proportion (similar triangles) of the following results: (i) The sine-difference as θ increases sin( θ + δθ ) sinθ varies as the cosine and decreases (ii) The cosine-difference cos( θ + δθ ) cosθ varies as the sine negatively and numerically increases as θ increases in the following manner (a) (b) δθ δθ sin( θ + δθ ) sinθ = sin.cos( θ + ) δθ δθ cos( θ + δθ ) cosθ = sin.sin( θ + ) But his studis on nd differences bear significance. Neelakantha studied the nd differences geometrically by the help of the properties of circle and similar triangles. He obtained δθ sinθ = sinθ[sin( ) ] i.e. the difference of the sine-difference varies as the sine negatively and increases numerically with the angle. Besides, he used the formula epressed in modern notation δ (sin 1 esin w) = ecosw 1 e sin δw w

11 Hindu thoughts on Calculus before Newton and Leibniz 116 In order to determine accurate motion of a planet at a particular moment, Bhaskaracharya divided a day into a very large number of intervals of time unit scalled truti equals to 1/33750 of a second. He also suggested that the differential vanishes when the time interval is diminished to the absolute minimum [1]. In the writings of Achyuta ( ), we find the differential of a quotient also. 4.5: REMARKS The use of formula involving differentials has been established beyond doubt that the Hindu scholars used calculus. The concept of instantaneous motion (tatkalikagati) found in works of Manjula, Bhaskara II established that these scholars contributed to the theory of differential calculus. They made it clear that the differentials gave true results only when very small variations are taken into account. After Bhaskara II, India had eperienced foreign attacks and rule for a long time, and could not produce any mathematicians of his caliber during that period. 4.6: HINDU IDEAS ON INFINITESIMALS (1) For the calculation of surface area of a sphere, the Hindu mathematicians used circular strips of very small width. It is shown in the following : Bhaskara II found that Fig. 4.1 Area of the surface of the sphere = circumference diameter

12 Hindu thoughts on Calculus before Newton and Leibniz 117 () For the calculation of volume of a sphere: Bhaskara II states the following method for determining volume of a sphere: considering pyramidal ecavations on the surface of the sphere, each of a base of unit area having unit side and of a depth equal to the radius of the sphere. The apices of these pyramids meet at the centre of the sphere. Then It proves that Fig. 4. Volume of the sphere = sum of the volumes of the pyramids 1 Volume of the sphere = 6 surface area diameter The results are the nearest approach to the method of integral calculus and the method is analogous to the concept of limit of a sum []. 4.7: CONCLUSION This chapter is an attempt to have a look at the glorious past of the Hindu tradition of calculus. The Hindu tradition of calculus is much more older than that of European. Yet, the contributions of Kerala School of Astronomy and Mathematics have not been acknowledged properly. Gheverghese said, A prime reason is neglect of scientific ideas emanating from the Non-European world, a legacy of European colonialism and beyond. It should carefully be investigated and the nomenclature of formulas, theorems, results etc. should be renamed so

13 Hindu thoughts on Calculus before Newton and Leibniz 118 that the original contributors may get due recognition from the international community of mathematics. Indian researchers are still now reluctant to do historical research. So the hidden treasures of Indian mathematics and science, in particular, are yet to be analyzed to the fullest etent.