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Reorinted from Currpnt .f;riPl'wP HISTORICAL COMMENTARY AND NOTF.SThe article published below draws attention to the work of the Kerala school of astronomers, particularly Nilakanta(/500 AD) in modelling planetary motion. In the exchanges between authors and referee, it became clear that thisschool did not stop with :opying their predecessors but attempted to wrestle with the problems of the old(geocentric) system. Whether their work constituted a clean break towards a true heliocentric system. as proposedby Srinivas and colleagues. appears to hinge upon some subtle points of interpretation of the original texts. Forexample, did the Kerala astronomers maintain the distinction between the mean and the centre of the epicycle of aninterior planet. even though both move together in the sky? They could be at different distances, as a refereesuggests. In any case. one cannot but note the vitality of this tradition of mathematics and astronomy which evenstudied infinite series some years later. while the rest of the cOl/ntry was going through an academic dark age.

- Editor

Modification of the earlier Indian planetary theory by theKerala astronomers (c. 1500 AD) and the implied heliocentricpicture of planetary motionK. Ramasubramanian, M D.Srinivas w:d M. S. SriramWe report vn a significant contribution made by the Kerala School of Indian astronomers toplanetary theory in thefifteenth century. Nilakantha Somasutvan, the renowned astronomer of theKerala School, carried out a major revision of the older Indian planetary model for the interiorplanets, Mercury and Venus, in his treatise Tantrasangraha (1500 AD),' and for the first time inthe history of astronomy, he arrived at an accurateformulation of the equati1!.of centrefor theseplanets. He also described the implied geometrical picture of planetary motIon, where the fiveplanets - Mercury, Venus, Mars, Jupiter and Saturn - move in ecceniric orbits around the Sun,which in turn goes around the Earth. The later astronomers of the Kerala School seem to have byand large adopted the planetary model developed by Nilakantha.It is now widely recognized that theKerala school of Indian astronomy'.starting with Madhava of Sangama-grama in the fourteenth century. madeimportant contributions to mathematicalanalysis much before this subjoct deve-

formulation of the equation of centre forthese planets than was available eitherin the ~lier Indianworks or in the Islamicor European traditions. of astronomytill the work of Kepler. which was tocome more than a hundred years later.

The conventional planetarymodel of Indian astronomyIn the Indian astronomicaltradition, atleast from the time of Aryabhata (499

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HISTORICAL COMMENTARY AND NOTES

AA

Figure 1. Sighra samskara for an exterior planet.

/", ,,/ ,/ ,I ,,- \/ / \t , \, t I~ l I\ G \ I, I \ I\ I I\ I I\ I I\ ~ /" -'"""

A

A

Figure 2. S'ghra samskara for an interior planet.

of the planet's orbit. DitTerent compu-tational schemes for .the mandasamsknra arc discussed in Indianastronomical literature. Howcvcr~ themanda correction in all these schcmescoincides. to tirst order in eccentricity.with the equation of centre currentlycalculated in astronomy. The manda.corrected mean longitude is call~dmandasphulagraha. As we explainbelow- for exterior plancts. themandasphulagraha is the same as the

sidereal period is identical with themean geocentric sidereal period. Thus,the mean longitude calculated prior tothe manda samskara is the same as themean heliocentric longitude of theplanel as we undcrstand loday. As themanda sams/cara is applied 10 thislongitude to obtain the mandasplrula-graila. Ihe laller will be the true helio-centric longitude of the planet.The sighra samskara for the exteriorplanets can be explained with reference

LGEP = 8- 8.s = Sigilra correction.The ditTerencc between the longitudesof the siglrracca and the mandasplruta,namely.

(\)is called the sigilrakendra (anomaly ofconjunction) in Indian astronomy.DrawPF perpendicular to the extension of theline EG. From the triangle EPF we caneasily obtain the resultsin(8 - 9m.d

rsina- [(R+rcosa)' +r' sin' aJ'" (2)

which is the siglrra correction formulagiven by Indianastronomersto calculatethe geocentric longitude of an exteriorplanet.From the figure it is clear that thesighra samskara transforms the trueheliocentric longitudes into true geo.centric longitudes: for. LASP = LAEGis thc trueheliocentricongitude andone has to add LGEI' to it to get thetrue geoeen"ic longitude. This is Irueonly if rlR is equal to the ratio of theEarth-Sun and Planet-Sun distancesand is indeed very nearly so in theIndian lexls. But equation (2) is still anapproximation as it is based upon the.identification of the mean Sun with thetrue Sun.Interior planetsFor the interior planets Mercury andVenus.ancientIndianastronomers~tleast from the tim. of Aryabhata. tookthe mean Sun as the madilyamagralraorthe mean planet. For these planets. themean heliocentric period is the periodofrevolution of the planet around the Sun.

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HISTORICAL COMMENTARY AND NOTESTable 1. Comparison of r IR (variable) in Aryabhatiya with modernvalues (ratio of the mean values of Earth -Sun and Planet-Sun

distances for exterior planets and the inverse ratio for interior planets)

i.T.he sighra samskara for the interior

planets can be explained with referenceto Figure 2. Here E is the Earth and S isthe mandasphutagraha. Draw SP = rparallel to EG. Then P corresponds tothe true planet. We have.

LAS - 8ms= MandasphwaLAEG = 8.. = Longitude of sighroccaLAEP = 8 = True geocentric longitudeof the planetLSEP = 8 - 8ms -Sighra correction.Again. the sighrakendra (Tis definedas the difference between the sighroccaand the mandasphutagraha. Thus.

Let PF be perpendicular to the line S.From the triangle EPF we get the sameformu lasin(8 -8mS)- rsinc 4- 2. ., 2. II'" ( )(R +rcosO") +r'sin 0"1-

which is the sighra correction .given inthe earlici Indian texts to calculate thegeocentric longitude of an interiorplanet. Both for Mercury and Venus. thevalue specified for rlR is very nearly.equal to the ratio of the Planet-Sun andEarth-Sun distances. In Tahle I. wegive Aryabhata's values for hoth theexterior and interior planets alongwith the modern values based on the

Computation of the planetary'latitudes

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Planetary latitudes (called vikshepa inIndian astronomy) play an importantrole in the prediction of planetaryconjunctions, occultation of stars byplanets. etc. [n Figure 3. P denotes theplanet moving in an orbit inclioed atangle i to the ecliptic. intersecting theecliptic at the point N, the node (calledpala in Indian astronomy). If f3 is thelatitude of the planet. 8" its heliocentriclongitude, and 8.. the heliocentriclongitude of the node, then for small iwe hav~

siD f3 = sini sin(8/1 - 8.) =d s in (8/1 - 8.).(5)

This is also essentially the rule forcalculatir.g the latitude. as given inIndian texts, at least from the time ofAryabhata. For the exterior planets, itwas stipulated that

the mandasphlllagraha. which as wesaw earlier. coincides with the helio-centric longitude of the exterior planet.The same :ule applied for interiorplanets would not have worked, becauseaccording to the earlier Indian planetarymodel, the manda.corrected meanlongitude for the interior planet has

PLANETARYORBIT

N ECLIPTICFigure 3. Latitude of a planet.

(6)

explained above. is equal to the rnt.'anheliocentric longitude of the planet.equation (7) leads to the correc

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HISTORICAL COMMENTARY AND NOTESfrom the deflection of the planet (fromthe ecliptic) and not from that of asighrocca, which is different from theplanet. Therefore. he argues that whatwas thought of as being the sighrocca ofan interior planet should be identifiedwith the mean planet itself and themanda correction is to be applied to thismean planet, and not to the mean Sun.This, Nilakantha argues. renders the rulefor calculation of latitudes the same forall planets. exterior or interior.Nilakantha has presented his improvedplanetary model for the interior planetsin an earHer treatise Tanlrosangrahawhich, according to Nilakantha's pupilSankara Variar, was composed in 1500AD'. We shall describe here. the mainfeatures of Nilakantha's model in so faras they differ from the earlier Indianplanetary model for the interior planets.In the first chapler of Ton/ra-

sangraha, while presenting the meansidereal periods of planets. Nilakanthagives the usual values of 87.966 daysand 224.702 days (which are tradi-tionally ascribed to the sighraccas ofMercury and Venus). but asserts thatthese are' svaparyayas'. Le. the mean.evolution pori ods of the planetsthemselves'. As these are the meanheliocentric periods of these planets. themadhyamagraha as calcu Imed inNilahntha's model will be equal 10 themean heliocentric longitude of the planet,for the case of interior planets also.In lhe second chapter of Tanlra-

sangraha. Nilakamha discusses themanda correction or the equation ofcentre and states. that this should beapplied to the madhyamagraha asdescribed above to obtain the mallda-sphutagraha. Thus. in Nilakantha'smodel, the ffl,mdasphutagl'oha will

Figure 4. True longitude of an interior planet according to Nilakantha.

LAES - 8.,= Sighroeea (mean Sun).LASP = 8ms= Mandasphu/a,LAEP = 8 = True geocentric longitude.of the planet,LSEP = 8 - 8s ~ Sighra correction.The sighrakendra is defined in theusual way by

as the difference between the sigh-roeca and themandasphutagraha. Thenfrom triangle ESP. weget the relation:sin(8 -8s)

I'sincr=-[(R +r~os(1)' +r2sin'(1]"2

which is the sighra correction given byNilakantha for calculating the geo-centric longitude 8 of the planet. Com-paring equations (8) and (9) withequations (3) and (4), and Figure 4 with

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Nilakantha, by his modification of tradi-tional Indian planetary theory, solvedthe long-standing problem in Indianastronomy, of there being two differentrules for calculating the planetary lati-tudes.Nilakantha, by 1500 AD, had thusarrived at a consistent formulation ofthe equation of centre and a reasonableplanetary model which is applicablealso to the interior planets, perhaps forthe first time in the history of astro-nomy. Just as was the case with theearlier Indian planetary model, theancient Greek planetary model ofPtolemy and t~ planetary models deve>-loped in the Islamic tradition during the8th-15th centuries postulated that theequation of centre for an interior planetshould be appiied to the mean Sunrather than to the mean heliocentriclongitude of the planet, as we under.stand today". In fact. Ptolemy seems tohave ""mpounded the confusion by

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HISTORICAL COMMENTARY AND NOTESgeocentric reformulation of theCopernican system due to Tycho Brahe.Indeed, it appears that the correct rulefor applying the equalion of centre foran interior planet, to the meanheliocentric planel (as opposed to themean Sun) was first enunciated inEuropean astronomical tradilion only byKepler in the early 17thcentury.Geometrical model ofplanetary motionIt is well known Ihat the 1nd ian astro-nomers were mainly interestcd in thesuccessful computalions of thelongitudes and latitudes of the Sun.Moon and the planets. and were notmuch worried about proposing modelsof the univers". Detailed observationsand the following sophistication of th"ircomputations of course suggested somegeometrical models. and once in a whilethe Indian astronomers did discuss thegeometrical model implied by theircomputations.The renowned Kerala astronomerParamesvara of Vatasseri (I3KO-1460)has discussed in detail the geometricalmodel implied in the earlier Indianplanet8r'J theory. In the Kerala tradition.Paramesvara has alsn a greai reputationas an observational astronomer.Darnodara the son and di

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HISTORICAL COMMENTARY AND NOTESEarth is the centre of this manda-vrlla:(Verse 19)'For the others [namely the planetsMercury, Venus, Mars, Jupiter andSaturn] the centre of the manda-vrttamoves at the same rate as the mean Sun(madhyarkagati) on the sighra-vrlla [orthe sighra epicycle drawn with itscentre coinciding with the centre of ihesighra concentric]. The sighra-vrlla forthese planets is not inclined with respectto the ecliptic and has the centre of the.celestial sphere as its centre.' (Verse 20)'In the case of Mercury and Venlls,thedimension of the sighra-vrlla is taken tobe that of the concentric and thedimensions [of the epicycles] mentionedare of their own orbits. Further, here themanda-vrlla [and hence 'the manilaepicycle of all the planets] undergoesincrease and decrease in size in thes:une way as the kama [or thehypotenuse or the distance of the planetfrom the centre of the. mandaconcentric]": (Verse 21)The geometrical picture described.byNilakantha is shown in Figures.s a andb. Like the above verses of Siddhanta-

darpana, there are several other graphicdescriptions of this geometrical picturein Nilakantha's works. For the exteriorplanets, he explains in his tract onplanetary latitudes tl.at":

.For Mars and other exterior planets(Kujadi). the centre of their manda-kakshya [which is also the centre oftheir manda deferent circle], is the meanSun (madhyarka) which lies on the orbitof the Sun on the ecliptic'.For the case of interior planets. thefollowing is a graphic descriptio~ of

outside their orbit. Since their orbit isalways confined to one side of the geo-centric celestial sphere, in completingone revolution they do not go aroundthe twelve signs (rasis). For them olsoreally the mean Sun is Ihe sighrocca. Itis only their own revolutions which arestated to be the revolutions of thesighrocca [in ancient texts such as theAryabhaliya]. It is only due to therevolution of the Sun [around the Earth]that they' [i.e. the interior planets.Mercury and Venus] complete theirmovement around the twelve rasis [andcomplete their revolution of the Earth)'.Thus, in Nilakantha's planetarymodel, Mercury, Venus. Mars. Jupiterand Saturn. are assumed to move in

eccentric orbits around the sighrocca.which is the mean Sun going around theEarlh. The planetary orbits are tiltedwith respect to the orbit of the Sun orthe ecliptic and hence cause the motionin latitude.Nilakantha's modification of theconventional planetary model of Indianastronomy seems to have been adoptedby must of the laler astronomers of theKerala schouI. This is not only true ofNilakantha's pupils and contemporariessuch as Sankara Variyar (1500-1560).Chitrabhanu (1530). Jyeshtadeva(1500), who is the author of the cele-brated Yuktibhasha. but also of laterastronomers such as Acyuta Pisarati(1550-1621). Putumana Somayaji(1660-1740) and others. They not onlyadopt Nilakantha's planetary model. butalso seem 10 discuss further improve-ments. For instance, Acyuta Pisarati 10his Sphutanirnayatantra and Rasigola.sphulanilpi discusses in detail thecoricction to planetary longitudesduetolatitudinal effects by the method ofreduction to the ecliptic - a point whichhas been earlier briefly noted byNilakantha in his Aryabhaliya-bhashya22 .In conclusion it may be noted that

the signal achievements of Ihe scientificrenaissance in Europe during the 16thand 17th centuries. Only more detailedinvestigations can lead to a correctappreciation and assessment of the workof the Kerala astronomers during the14-16th centuries and their consequentdevelopments" .I. See for instance (a) Indian Astronomy:A Source Book, (eds. Subbarayappa.B. V. and Sanna. K. V.), Bombay, 1985:(b) A Bibliography of Kerala andKeTola-based Astronomy and ASlrology.Sanna. K. V., Hoshiarpur, 1966.2. See for example (a) Whish, C. M.,Trans. R. Asiatic Soc., 1830,3,509: (b)Mukunda Marar, K. and Rajagopal,C. T.. J. Bombay Br. R. Assoc. Soc.

(NS), 1944, 20, 65: (c) Rajagopal. C.T.. Stud. Math.. 1949, IS, 201: (d)Rajagopal. C. T. and Rangachari. M. S..Archiv Hist. Ex. Sc.. 1976, 18, 89: (e)Takao Hayashi, Centauras. 1990, 33,149: (f) Balagangadharan. K., inScientific Heritage of India: Malhe-maUes (cd. Poulosc. K. G. Tripunithura)1991, p. 29; We may note that C. M.Whish's paper which appeared in 1835is not the first dicussion on thediscovery of infinite series by the Kcralaastronomers.That there was quite somediscussion on the subject in the decadespriorto 1835 is indicatedbyJ. Warrenin his book Kala Sanlcalita (Madras.1825, pp. 93, 309-3iO). Curiously, thewhole issue gOt' totally ignored after1835 til l Prof. C. T. Raj3j;opal and hiscolieagues resurrected the subjectaround 1945.

3. For a general review of Indian astro-nomy, see (a) A Critical Study ofAnc;em Hindu Astronomy. Somayaji.D. A.. Kamatak University. Dharwar,1972: (b) A History of IndianAstronomy (eds. Shukla, K. S. and Sen,S. N.) INSA. New Delhi, 1985.4. Aryabhatiya With the Commentary ofBhas/caraI and Somesvara (cd. Shukla.K. S.), INSA. New Delhi. 1976, pp. 32.247.

5. MuraJidharaChaturveda (ed.), SidJhanta-siromani. Varanasi. 1981. p.402.

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HISTORICAL COMMENTARY AND NOTESNilakaotha's work. For instance.Pingree. in his review anicle on Indianastronomy presents the mean rates ofmotion of Mercury and Venus given inTantrosongraha as the rates of motionof their sighroccas (Pingree, D.. Historyof Mathematical Astronomy in India. inDictionary of Scientific Biography. NewYork. 1978. vol. XV. p. 622).9. Ref. 7. pp. 44-46.10. Ref. 7. p. 139.-II. See for example (a) A History ojAstronomy from Tholes 10 Kepler.Dreyer. J. L. E.. Dover. New York.1953: (b) Mathematical Astronomy inCopernicus' De Revolut;onibus.Swerdlow, N. M- and Neugebauer. 0..Springer. New York. 1984.2 vols.12. See for example The Almagest byPlolemy. Tr. by Taliaferro R. C.. inGreat Boaks oj the Western World (ed.Hutchins, R. M.). Chicago. vol. 16.1952. For the exterior planclS. theancient Indian planetary model and themodel described by Plolemy are verysimilar except that. white the Indilmastronomers use a vuiable radiusepicycle. Ptolemy introduces the notionof an equant. Ptolemy adopts the samemodel for Venus also. and presents aslightly different model for Mercury. Inboth cases the equation of centre isapplied to the mean Sun. While theancient Indian astronomers successfullyused the notion of the siglrrocca toarrive at a satisfactory theory of thelatitudes of the interior planets. the

Ptolemaic model is totally off the markwhen it comes to the question oflatitudes of these planets. This difficultywith the computation of latitudespersisted till around the time of Kepler.13. Kern. B. (ed.). Aryabhatiyam withVyakhya of Paramesvara. Leyden. 1885.pp. 68-69.14. Kuppanna Sastri. T. S. (ed.). Mahabhas.kari)'am with Govindasvamin's Vyakh,.oand Siddhanladipiko of Paramesvara.Madras Govt. Oriental Series. No. 130.1957. pp. 233-238.15. Sarma. K. V. (ed.). GrohasphulonayaneVilcslrepovasana of Nilakantha Soma-sutvan in GoniloYlIlcloyah. Hoshiarpur.1979. pp. 61-64.16. Sambasiva Sastri. K. (ed.).Aryobhaliyam with the Bhashya ofNilakantha Somasutvan: Kalakriya-pada. Trivandrum Sanskrit Series. No.110.1931. p. 53.17. Sarma. K. V. (editor and translator)Siddhontadarpana. Hoshiarpur. 1976.pp. 18-19.18. This reflects the feature of Indianplanetary models. that the mandacorrection or the equation of centre iscalculated from a variable radiusepicycle model. See for instance Maha-bhaskoriyo of Bhaskaracharya. I.. cd.and translated by Shukla. K. S.,Lucknow. 1960. pp. 136-146.19. Ref. 15.p . 63.20. Ref. 6. p. 9.21. Sarma. K. V. (cd.) Sphwanirnayufanlroof Acyuta Pisarati. Hoshiarpur. 1974:

Ras;golasphutaniti of Acyuta Pisarati.cd. and translated by Sarma. K. V..Hoshiarpur. 1977.22. Ref. 6, p. 6.23. The well known Orissa astronomer oflast century. Chandrasekhara Samanta.who was trained solely in traditionalIndian astronomy. seems to have alsodiscussed a model of planetary motionwhere the five planets go around theSun. in his work Siddhantadarpana.(Siddhanladarpana of Mahamaho.padhyaya Samanta Sri ChandrasekharaSimha. Calcutta 1897. V. 36. See alsothe review by W.E.P. in Nature, 1532.59.437. 1899). We are grateful to Dr P.Nayak for bringing this important factto our attention.

ACKNOWLEDGEMENTS. We thank Prof.K. V. Sarma. Dr S. Balachandra Rao. and DrJ. K. 8ajaj for many helpful discussions. Wealso gratefully acknowledge financialsupport provided by the Department ofScience and Technology. New Delhi. underthe Nationally Coordinated Project onFoundations and Methodology of TheoreticalSciences in the Indian Tradition.

The authors are in the Department ofTheoretical Physics, University ofMadras, Guindy Campus, Madras600 025. India