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Ideas of Physical Forces and Differential Calculus in Ancient India

We have studied the context and development of the ideas of physical forces and differential calculus in ancient India by studying relevant literature related to both astrology and astronomy since pre-Greek periods. The concept of Naisargika Bala (natural force) discussed in Hora texts from India is defined to be proportional to planetary size and inversely related to planetary distance. This idea developed several centuries prior to Isaac Newton resembles fundamental physical forces in nature especially gravity. We show that the studies on retrograde motion and Chesta Bala of planets like Mars in the context of astrology lead to development of differential calculus and planetary dynamics in ancient India. The idea of instantaneous velocity was first developed during the 1st millennium BC and Indians could solve first order differential equations as early as 6th cent AD. Indian contributions to astrophysics and calculus during European dark ages can be considered as a land mark in the pre-renaissance history of physical sciences. Key words: physical forces, differential calculus, history of science, planetary dynamics, ancient India


Introduction
It is generally believed that astrophysics and advanced mathematics is an offspring of renaissance period Europe.
Several studies [1] suggest that non-European contributions especially prior to the renaissance has also played a significant role in the development of basic sciences. Indian civilization is one of the oldest of that kind in the world which has made original marks in fields like mathematics,philosophy,meteorology,metallurgy and surgery even prior to the Greek invasion period [2][3][4][5][6] Many Indian contributions in astronomical sciences prior to Christian era is perhaps lost which can be partly recovered from other forms of classical literature like astrology and later commentaries [7] Development of differential calculus [8] and ideas fundamental physical forces [9] were considered to be triumphs of the European renaissance through dedicated works of Galileo,Kepler,Newton etc. In this paper In this paper we have studied the following problems (i) The ideas of physical forces developed in ancient India and its relation with concepts in modern physics. (ii) The development of the ideas of differential calculus and planetary dynamics in association with the ancient Indian contributions to astrophysics.
It will be shown that ancient Indian astronomers could define a planetary force which is proportional to the planetary size and inversely relatedto the planetary distance. The variation of force of interaction of planetary objects during its relative orbital motion is also studied by them using quantitative models of planetary motion during the1st millennium AD.We will also discuss the development of ideas of differential calculus in India since 1 st millennium BC. This paper summarises our earlier results based on the studies of Indian historical contributions to Physics and mathematics as a part of astronomy. [10][11][12][13].

Ideas of Physical Forces in ancient India
Astronomy and astrology was practiced in ancient India as an integrated discipline called Jyotisha [ 14,15]. This subject has three important branches viz.
(i) Ganita which includes theoretical and experimental astronomy with branches like karana ( computational procedures) and siddhanta ( treatises on general astronomy) (ii) Hora Sastra Mainly principles and practice of astrology with branches like Jataka,Prasna, Muhurta,Nadi etc (iii) Samhita which is equivalent to a general and scientific encyclopedia including astronomy, applied sciences and astrology.
Ancient Indian astronomers like Parasara,Garga and Varahamihira were known to have written texts in all the three branches of Jyotisha. Physical relations between celestial objects and terrestrial phenomena which comes under the studies on modern astrophysics is studied as a part of astrology during ancient times. Old scientific works on astrology from India will be shown to contain important scientific information related to both astrophysics and mathematics in this paper.
Parasara Hora a treatise in sanskrit written during 1 st millennium BC [16,16a] defines six types of physical forces ( shadbalas) by which Sun,moon and visible planets will physically interact with earth or among themselves. Their details are shown in Table 1 and its relation to modern physics. The scientific ideas in Naisargika and Chesta bala will further receive special attention in this paper. The original sanskrit verses used to establish our results and their sources is given in Appendix I of this paper.

Naisargika Bala
Naisargika Bala or natural force is an inherent property of a celestial object which possesses the following physical properties as defined in astrological texts from ancient and medieval India :-1. This force is a constant for a given object ( eg a planet ) which does not vary with position or time [17] 2. Naisargika Bala increases in the order Saturn,Mercury,Mars,Jupiter,Venus Moon and Sun whose magnitudes are given units of Shastiamsa ( 0.5 degrees) as in Table 2 [18] 3. This force ( Fn) is defined to be proportional to the apparent size or diameter of the planetary objects concerned [18.19] 4. This force varies inversely with distance ( r) of the object from the observer ( for eg in earth) [19] 5. When we want to compare force due to more than one planetary object occupying similar celestial longitudes relative to earth ( this occurs during close planetary conjunctions which is referred as grahayudha or planetary war in ancient India) Naisargika Bala becomes an important criteria to judge to find the physically most influential planet [20][21][22].
If F1 and F2 are Naisargika Bala due to planets 1 and 2 situated at the similar distance r from earth ( say the planets occupy nearly the same geocentric longitude or they are in conjunction) then conceptually we have F1 = F( D1)/r ; F2 = F( D2)/r according to properties 3 and 4 defined above for the Naisrgika Bala. Here D1 and D2 are the apparent diameters of the planets 1 and 2 respectively.
We can find that the ratio of the planetary Nairgika Bala as …… ( 1) According to modern theory of Newtonian gravitation the forces due to the above planetary objects are given by F1= ( M1M) /r^2 and F2 = ( M2M/r^2) The ratio of the gravitational forces are given by We have M1= V1d1 and M2 = V1d2 here V is the volume and d is the mass density of these planets If we assume d1=d2 then , So the ratio of Naisargika Balas of two planets situated at identical distances from earth as defined by ancient astronomers from India is almost identical to the ratios of the modern gravitational forces of these planets if they possess also similar mass densities [23].

Chesta Bala and emergence of differential calculus in ancient India
Ancient Indian astronomers observed that the apparent sizes and brightness of the planets showed temporal variations. This is particularly evident during occasions of retrograde motions of planets like Mars,Jupiter etc as seen from earth . It was also noticed that the relative speed of the planets also showed notable variations during periods of retrograde motion. We can find that these astronomical observations and subsequent studies in the context of astrology lead to the emergence of differential calculus in ancient India.
When planets like mars are in the same side of earth in their orbital motion and when it approaches closer to earth or sun then due to the differences in the eccentricity of their orbits there will be relative motion between earth and the planet. This will appear with repect to fixed stellar constellations ( zodiac signs) change from direct ( say clockwise) to retrograde motion ( say anticlockwise) of the planet from earth and then back to direct motion again. Between direct and retrograde motions there is a turning point where the planet will appear as stationary. As the planet begins the retrograde motion gradually its relative speed increases and it reaches a maxima at the centre of the retrograde loop [24,25]. See Fig 1 for the relative orbital speed changes of Mars in its closed approach to earth during the year 2003 while undergoing apparent retrograde motion.
Along with the speed the geocentric distance of the planet from earth decreases and it reaches a minima at the centre of the retrograde loop. This will appear as increase in apparent size and brightness of the planet which reaches again a maxima at the centre of the retrograde loop. See When the geocentric longitude (λ) of the planet is found to increase with time then we observe direct motion where (dλ/dt) >0. Retrograde motion is observed when λ decreses with time or during (dλ/dt) <0. Between these two types of motion we can find turning points of reversal from direct to retrograde motion or vice versa. At these turning points ( called stationary points in astronomy) we have (dλ/dt) =0. where planets like Mars will appear to be stationary for an observer in Earth.
Ancient Indian astronomers who studied retrograde motion of planets noticed the visible physical properties of these planets ( size or brightness) change in conjunction with the apparent speed changes. Variations in apparent size of the planet during retrograde motion is defined as changes in Chesta bala or dynamic force of the planet in ancient astrological works from India . In Parasara Hora ( 6 th Cent BC or before) eight types of dynamical situations are defined in relation with retrograde motion of planets like Mars as seen from earth [18].The nature of the instantaneous speed and the corresponding apparent size ( Chesta bala) of the planet defined in this Hora text is given in Table 3.
Stationary points are defined as Vikala situation in ancient Hora texts ( Vikala means time variations in planetary longitude is less than one kala or one minute of arc per day) These results suggests that ancient Indian astronomers were clearly aware of the concept of instantaneous velocity as early as 6 th cent BC. This can be considered to be the beginning of the development of differential calculus in India. From the time of Brahmagupta ( celebrated Indian astronomer and mathematician) i.e from 6 th cent AD onwards Indians could solve simple differential equations of the type dλ/dt =0 to find the geocentric longitudes of stationary points [26,27] These values obtained from different astronomical works from ancient/medieval India is comparable with the values found by the Greek astronomer Ptolemy [28].

Studies of planetary dynamics in ancient India
Observations related to spatial and temporal changes in the physical properties of planetary objects from earth played an important role in the development of planetary dynamical models in ancient India. Some of these observations are discussed in Hora texts and some are in Siddhanta texts. A synthesis of different ideas given in Indian Jyotisha works can provide more insight in to the history of evolution of this knowledge ( see Table. 4 ) As a planetary object moves from one zodiac sign to another as observed from earth we can observe changes in its apparent size, brightness and orbital speed. The apparent brightness of celestial object is defined in Indian astrological works as rasmi bala ( power of rays) whose values obtained from selected Indian Hora texts is given in Table 5. The geocentric longitude of observation of maxima in apparent size or brightness of a planet during its orbital motion is called Ucha and minima in the same is called Neecha .in ancient Indian astrology. Generally Ucha and Neecha points are separated by 6 signs or 180 degrees indicating the near circular nature of the planetary orbits.The planetary force on earth is maximum when the planet is situated in uccha and the force is minimum when the planet is situated in neecha.In ancient Indian Hora texts methods to interpolate planetary force ( or equivalently apparent brightness or size of the planet) at any arbitrary longitude between Ucha and Neecha is also described. Since the period of Aryabhata ( 5 th cent AD) we can find the definition of mandocca ( the geocentric longitude of minimum orbital speed) equivalent to modern apogee where the apparent size is minima. Similarly an antipodal point called nichocha is defined where the size is maxima or equivalent to modern perigee [29][30][31][32].
For an observer in earth the epoch of observation of maximum apparent size/brightness of a planetary object is during oppositions with Sun for superior planets ( Mars,Jupiter and Saturn) and during superior conjunctions for inferior planets ( Venus and Mercury). This happens during the retrograde motions of these planets and where the geocentric longitudinal separation of the planet from sun is also important. In this phenomena we have to consider the variations in the relative positions of three celestial objects viz.Sun,Earth and the planet undergoing retrograde motion ..
Since time of Aryabhata ( 5 th Cent AD) we can find epicyclic theories of planetary motion described in Indian astronomical ( Siddantic) texts. Astrological calculations, especially estimates of the magnitudes of planetary forces ( shadbalas) improved in India due to the application of epicylic theories. Methods to calculate variations in apparent diameters of planetary objects is descried in Suryasiddhanta and Siddantasiromani. In Sripatipaddati ( a hand book of astrological calculations employing advanced mathematical models of planetary motion )methods to calculate variations in Chesta Bala [32]of planets is at any arbitrary point in its orbital motion described. This is defined proportional to the Sighra anomaly which requires use epicycle of planetary conjunction for its calculation. Cheta Bala = 0.33 ( Sigrocca -average geocentric longitude of the planet) The quantity in bracket is the Sighra anomaly equivalent to the angular separation of the planet from the apparent Sun.This anomaly or Chesta Bala is maximum (separation is 180) at at the centre of the retrograde loop orduring oppositions and the same is minimum during conjunctions with sun ( separation is 0). See also Fig.3. The variations in the apparent diameter of planetary objects during its apparent orbital motion and its dependence on the angular separation from the sun( during retrograde motion) is clearly mentioned in Siddanta siromani of Bhaskaracharya [ 30].  [24,[33][34][35]. In this context the ancient Indian contribution to astrophysics is perhaps the development of ideas of planetary physical forces discussed in this paper several centuries prior to the European renaissance.The concept of Naisargika bala is similar to Newtonian gravitation interaction as shown in section 2.1 of this paper ; an anticipation of the idea of fundamental forces in modern physics several centuries prior to the European renaissance.
Naisargika bala or natural force of planets is defined to be proportional to the size of the celestial objects and inversely related to the distance from the observer. This is first clearly mentioned in the astrological treatise Horasara by Prithuyasas during the 6 th Cent AD.There are atleast three other independent references to the planetary size dependence of the Naisargika Bala from medieval India. This idea is discussed in Kitab-ul-Hind ( 11 th cent AD) by the medieval Muslim astronomer Alberuni in his short note on the principles of Indian astrology following Laghujataka ( an abridged form of Brihatataka written by celebrated Indian astronomer Varahamihira during the 6 th cent AD) [ 36].As mentioned in Section 2 Naisargika Bala is used to compare planetary physical forces due to two or more planets occupying almost identical position in the Zodiac at a given instant of time. This phenomenon is referred as Grahayuddha or planetary war in ancient stronomical/astrological literature which happens during close planetary conjunctions in a given zodiac sign. In the medeval Indian astronomical handbook called Karanaratna written by Devacharya during the 8 th Cent AD it is explained that the planet with larger diameter (implying Naisragika bala ) will be the victor during a close planetary conjunction or Grahayuddha. A similar idea is also found in Sripatipadati.Finally Dasadyayi a well known Sanskrit commentary of the Brihatjataka of Varhamihira by the Kerala astronomer Govinda Bhattathiripad ( 13 th cent AD) also provides independent evidence of the planetary size dependence of the Naisargika Bala One can find a definition in Suryasiddanta [30,31,37] which states that : the dynamics or quantity of motion produced by the action of a fixed force to different planetary objects is inversely related to the quantity of matter in these objects. This definition is equivalent to the statement of Newton's second law of motion a = F/M It can be considered as an indirect perception of the idea of planetary mass by ancient Indian astronomers.
One of important motivations behind the development of differential calculus by different cultures seem to be scientific modelling of observed astronomical phenomena in general and retrograde motion of planets in particular. Babylonian astronomers collected several centuries of valuable observations related to retrograde motion of visible planets like Mars. The earliest scientific model in this context was developed by Indian astronomers of the Mahabharatha period like Parasara and Garga ( on or before 6 th cent BC) through the definition of Chesta bala corresponding to eight dynamical situations during the retrograde motion of a superior planet.Ancient Indian astronomers had the idea of instantaneous velocity which is known to be equivalent to the idea of derivative in modern calculus.
Greek astronomers contributed towards the mathematical models of retrograde motion of planets. Following Applonius , Ptolemy could solve first order differential equations using numerical techniques like linear interpolation to find stationary points of planetary motion during 1 st cent AD. Several Indian astronomers contributed towards the mathematical techniques of differential calculus since 5 th cent AD [ 38] Aryabhata used linear interpolation techniques to solve difference equations to prepare a Table of Sines. Brahmagupta could find derivatives of complex functions through quadratic interpolation techniques who also solved for the first time in India , differential equations to find stationary points of retrograde motion of planets . Bhaskarachrya had expertise in both physical and mathematical aspects of differential calculus which he applied to planetary dynamic models. During early renaissance period ( 14-16centuries AD) differential and integral calculus was further developed by Kerala astronomers (of now international repute) whose principal contributions is the discovery of infinite series in calculus prior to the European contributions by Gregory,Taylor and Lebniez.
We have shown in section 3 that astrophysical observations was the principal motivation behind the developments of mathematical models of planetary motion in ancient India. The physical properties of an Keplerian elliptical orbit ( ie apogee and Perigee) of a planet could be anticipated by them through their correlation with the extremas in physical properties of the planet ( maxima or minima) such as apparent size,brightness and orbital speed. The connection between astrological studies and planetary dynamics is thus a new paradigm in the history of astrophysics [39] Indian knowledge of planetary dynamics and higher mathematics including calculus is fairly advanced prior to the beginning of the European renaissance period in the 14-15 th century AD. After the decline of Greek and Roman Empires by 4 th cent AD the centre of scientific activity at the global level became India where the socio-political situations were stable and favorable for new knowledge creation in diversified fields.Studies in planetary dynamics in both physical and mathematical perspective by Indian astronomers in the context of astronomy as well as astrology can be considered to be important in the history sciences during the European dark ages . There are independent evidences which suggest that Arab astronomers during the medieval period ( 11 th to 15 th cent AD) were aware of Indian contributions planetary physical forces [40]. There is every possibility that the new knowledge created in India in areas like astrophysics and differential calculus was made available the European scientists during renaissance period directly or indirectly through Arab and Christian missionary routes of transmission .It may be worth remembering that both Isaac Newton and Kepler were aware of astrology too at the advanced level [41][42].     Here E is earth, P is planet and S is sun.
Text: S£ryasiddh°nta [21, 21a]  Depending on the location of ucca the meaning and physical consequence of the given location of the planet in the zodaic will vary. In Saravali it is said that when the planet is situated in ucca it is very bright. When the planet is situated in ucca it is at greatest distance from earth and hence its visible disc (apparent diameter) appears large. When it is situated in neecha it is at minimum or least distance from Earth and therefore its visible disc appears small. We can also observe an inverse of Variation of the apparent size of the planet with distance from the Sun (during retrograde inchain of planets like Mars). The quantity of motion Mhar produced in Sun due to the action of an attractive force is very small compared to Moon because of its large physical size or dimensions compared to Moon.
Since the physical size of visible planets in the solar system like Mars is very small compared to the Sun and Moon the effect of the forces on them will be very large. Thus the displacements (positive or negative from reference points) produced in them will be also of a larger magnitude due to the result of action of these forces.