modification of planetary theory by kerala astronomers

Reorinted from Currpnt .f;riPl'wP HISTORICAL COMMENTARY AND NOTF.S

The 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 motion

K. Ramasubramanian, M D. Srinivas w:d M. S. Sriram

We 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 accurate formulation of the equati°1!.of centre for 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-loped in Europe. The Kerala astrono-mers obtained the infinite series for ~sine and cosine functions and also deve-

loped fast convergent approximations tothem'. Here we report that the Keralaschool also made equally significantdiscoveries in astronomy. in particular.planetary theory.

We show that Nilakantha Somasutvan

of Trkkantiyur (1444-1550 AD) carriedout, in his treatise Tantrasangraha(1500 AD). a major revision of theearlier Indian planeta,y model for thei'nterior planets Mercury and Venus.This led Nilakantha to a much betler

784

formulation of the equation of centre forthese planets than was available eitherin the ~lier Indian works or in the Islamic

or European traditions. of astronomytill the work of Kepler. which was tocome more than a hundred years later.

We also note that Nilakantha in his

later works. Golasara. Siddhanta-

darpana and more importantly thecelebrated Aryabhaliyabhashya. ex-plains that the computational schemedeveloped by him implies a geometricalpicture of planetary motion. where thefive planets Mercury. Venus. Mars,Jupiter and Saturn move in eccentricorbits around the mean Sun. which inturn goes around the Earth. Most of theKerata astronomers who succeeded

Nilakantha. such as Jycsthadcva. AcyutaPisarati.PutumanaSomayaji. etc. seemto have adopted this planetary model.

The conventional planetarymodel of Indian astronomy

In the Indian astronomical tradition, atleast from the time of Aryabhata (499AD), the procedure for calculating thegeocentric longitudes of the five planets,Mercury, Venus, Mars. Jupiter andSaturn involves essentially the follow-ing steps'. First, the mean longitude(called the madhyamagraha) is calcu-lated for the desired day by computingthe number of mean civil days elapsedsince the epoch (this number is calledahargana) and multiplying it by themean daily motion of the planet. Thentwo correctionsnamely manda samskaraand sighra samskara are applied to themean planet to obtain the true longitude.

Themandasamskara is equivalenttotaking into account tbe eccentricity

CURRENT SCIENCE,VOL. 66, NO. 10.25 MAY 1994

HISTORICAL COMMENTARY AND NOTES

A

A

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 thetrue heliocentric longitude.

Thesighra sanrskarais applied to thismandosphutagraha to obtain the truelongitude known as splrutagralra. Thesighra correction. as we explain below.is equivalent to convening the hclio.centric longitude into the geoccntriclongitude. The exterior and interiorplanets are treated differently inapplying this correction. and wc takeIbem up one aller the olher.

Exterior planets

For the exterior planets Mars. Jupiterand Saturn~ the mCiln heliocentric

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 referenceto Figure I. Longitudes are always mea-sured in Indian astronomy with respectto a fixed poinl in the Zodiac known asthe Nirayana Meslradi denoted by A inthe figure. E is the Earth and G is themandoJ'plllllagraha at n distance R. S isthe mean Sun referred to as the

sighrocca for an exterior planet. DrawGP = r parallel to ES. Then P corres-ponds to the true planet. We havc~

LI/EG = 9.s= MandasplrulaLilES = 9s = Longitude of siglrrocca

(mean Sun)LAEP = 9 = True geocentric longitude

of the planet

CURRENT SCIENCE. VOL. 66. NO. I U. 25 MAY 1994

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 result

sin(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 true heliocentriclongitude 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 planets

For the interior planets Mercury andVenus.ancientIndianastronomers~atleast from the tim. of Aryabhata. tookthe mean Sun as the madilyamagralraorthe mean planet. For these planets. themean heliocentric period is the period ofrevolution of the planet around the Sun.while the mean geocentric period is thesame as that of the Sun. The ancientastronomersprescribedapplicationofthemandacorrectionortheequationofcentre characteristic of the planet. to themean Sun. instead of the mean helio-centric planet as is done in the currentlyaccepled model of the solar system.However.the ancient Indian astrono.mersintroduceda sighroccafor theseplanetswhoseperiodis thesame as themean heliocentric period of Iheseplanets. Thus the longitude of thissighri>ccawill be the same as the meanheliocentric longitude of the interiorplanet.

785


Table 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 interiorplanets 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.

LA£S - 8ms = Mandasphwa

LAEG = 8.. = Longitude of sighrocca

LAEP = 8 = True geocentric longitudeof the planet

LSEP = 8 - 8ms -Sighra correction.

Again. the sighrakendra (T is definedas the difference between the sighroccaand the mandasphutagraha. Thus.

Let PF be perpendicular to the line £S.From the triangle EP F we get the sameformu la

sin(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 the

geocentric 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. we

give Aryabhata's values for hoth theexterior and interior planets alongwith the modern values based on themean Earth-SW1 and 'Sun-Planetdistances.

Since the manda correction or

equation of centre for an interior planetwas applied to the longitude or the meanSun instead of the mean heliocentric

longitude of the planet. the accuracy ofthe computed longitudes of the interiorplanets according to the old~r Indianplanetary models would not have heenas good as that achieved for the exteriorplanets.

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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 sin (8/1 - 8.).(5)

This is also essentially the rule forcalculatir.g the latitude. as given inIndian texts, at least from the time of

Aryabhata. 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 hasnothing to do with its true heliocentriclongitude.

However, all the older Indian tcxts onastronomy stipulated that for interiorplanets. the latitude is to be calculatedfrom equation (5) with

OH = Os+ manda correction,

the manda-corrected longitude of thesighrocca. Since the longitude of thesighrocca for an interior planet. as we

PLANETARY

ORBIT

N ECLIPTIC

Figure 3. Latitude of a planet.

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explained above. is equal to the rnt.'anheliocentric longitude of the planet.equation (7) leads to the correc<identification. that even for an intcriN

planet. 8/1 in equation (5) has to be thetrue heliocentric longitude.

Thus. we see that the earlier Ind'ar,

astronomical texts did provide a fain;-accurate theory for the planeta"latitudes. But they had to live with twoentirely different rules for calculatilWlatitudes. one for the exterior plancls(equation (6». where the mandasphula-graha appeared and an entirely differer,tone for the interior planets (equatior..(7)). which involved the sighrocca ofthe planet. with the mauda correcUUHincluded.

This peculiarity of the rule forcakulating the, latitude of an intcriNplanet was repeatedly noticed by variou,Indian astronomers. at least trom thl'

time of Bhaskaracharya I (629 AD),who in his Aryabhatiyabhashya drewattention to the fact that the procedurefor calculating the latitude of an interiorplanet is indeed very differeot from thaIadopted for the exterior pla~ots"'.Bhaskaracharya II in his own com;a~!"';tary Vasanabhashya on Siddhanta.':{r(;--mani (1150 AD) quotes the statement 01Chaturveda Prithudakaswamin (860AD) that this peculiar procedure fa, theinterior planets can be justified only onthe ground that this is what has beenfound to lead to drigganilai~ya, G' pre.dictions which are in conformity \','i!~observationss. ~

Planetary model of Nilakanth..Somasutvan (c 1500 AD)

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Nilakantha Somasutvan (1444-1550).the renowned Kcrala astronomc{

appears to have been led to his impo;.tant reformulation of the older Indian

planetary model. mainly by the t~ct In.:ithere obtained two entirely diffcrcn\rules for the calculation of plancmrylalitudes. As he explains in hi,Aryabhatiyabhashya6. the latitude Rrjs~s

CURRENT SCIENCE. VOL. 66. NO. 10.25 MAY 199.'

Planet Aryabhaliya Modern value

Mercury 0.361 to 0.387 0.387

Venus 0.712 to 0.737 0.723

Mars 0.637 to 0.662 0.656

Jupiter 0.187 to 0.200 0.192

Saturn 0.114 to 0.162 0.105


from the deflection of the planet (fromthe ecliptic) and not from that of asighrocca, which is different from the

planet. 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 for

all planets. exterior or interior.Nilakantha has presented his improved

planetary 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 mean

heliocentric 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 be

applied to the madhyamagraha asdescribed above to obtain the mallda-

sphutagraha. Thus. in Nilakantha'smodel, the ffl,mdasphutagl'oha willbe equal to the true heliocentric longi-tude for both the interior anti exteriorplanets.

Subsequently. the sp/III/agraha or thegeocentric longitude is to be obtainedby applying the sighra correction.While Nilakanthn's formulation of the

sighra correction is the same nsin theearlier planetary theory for the exteriorplanets. his formulation of the: .righracorrection for the inlerior planets isdifferentandisexplainedbdo\\.

According to Nilnkantha the moan Sunshould be taken as the siglwocca forinterior planets also. just as in the caseof exterior planets. In Figur~ 4. P is themanda.corrccted planet. E is lhe EarthandS thesighroc'xa or the m~an Sun.We have.

CURRENT SCIENCE. VOL. 66. NO. 10. 25 MAY 1994

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 the mandasphutagraha. Thenfrom triangle ESP. we get 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

Figure 2. we notice that they are thesame except for the interchange of thesighrocca and the mandasphutagraha.The manda correction or the equation ofcentre is now associated with P whereasit was associated with S earlier.

In the seventh chapter of Tan/ra-sangraha. Nilakantha gives formula (5)for calculating the latitudes ofplanets'.,and prescribes that for all planets, bothexterior and interior. 8H in equation (5)should be the mandasphutagraha. Thisis as it should be. for in Nilakantha's

model even for an interior planet. themandasphulagraha (the manda.corrcc.ted mean longitude) coincides with thetrue heliocentric longitude, just as in thecase of the exterior planets. Thus

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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 byclubbing together Venus along with theexterior planets and singling out Mer-cury as following a slightly deviantgeometricalmodelof motion12.

Even the celebrated Copernican re-volution brought about no improvementin the planetary theory for the interiorplanets. As is widely known now", theCopernican model was only a refor-mulation of the Ptolemaic model (withsonie modifications borrowed from theMaragha School of Astronomy of Nasirad-Din at-Tusi (1201-74 AD), Ibn ash.Shatir (1304-75) and others) for aheliocentric frame of reference, withoutaltering his computational scheme inany substantial way for the interiorplanets. The same holds true. for the

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787


geocentric 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 17th century.

Geometrical model ofplanetary motion

It 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 observations

and the following sophistication of th"ircomputations of course suggested somegeometrical models. and once in a whilethe Indian astronomers did discuss the

geometrical 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<ciple ofP8l'3mesvara was the teilcher ofNilakantha. Nilakantha otien refers to

Paramesvara as Paramaguru.In his commentary on ArYClbhatiya.

Paramesvara briefly discusses in 12verses". the geometrical model ofmotion as implied by the conventionalplanetary model of Indian astronomy. Inhis super-commentary Siddh"nIadipika(on Govindasvamin's commentary onMahabhaskariya of Bhaskaracharya-I(629 AD). Paramesvara gives a moredetailed exposition of the geometricalmodel of plan clary motion. He noticesthat for an interior planet. the finallongitude thai is calculated (LA£P inFigure 2) is the geocentric longitude ofwhat is called the sighron'u of theplanel (in the conventional planelarymodel). Paramesvara th"refore suggeslsat the end. that what has been called as

the sighrocca of an interior planet inconvenlional plan"tary model should beidentified as the' planet ilself and themean Sun should be taken as Ihe

sighrocca for all the planets. whilecomputing the sighra correction. Thusmanyof the basic ideaswhich wcre used

788

a

b

- Groha-bhromono -vrtta

$'Io-"r"o

Moftdo - wtto

-r- Groho-bhromono...vrtta

Figure 5. B. Geometrical model of planetary motions according to SiddhantadBrpsnaof Nilakantha. illustrated for interior planets. b. Geometrical model 01 planetary motionsaccording to Siddhantadarpsna ot Nilakantha. illustrated for exterior planets.

by Nilakantha in formulating his newmodel were already present in the workof Paramesvara.

Nilakantha describes the geometricalpicture associated with his model otpla~etary mOlion in his works Go/asara.Siddhamadarpana (with his own comm-entary), and in much greater detail in hisAryabhaliyabhashya. There is also aIract of his. on planetary latitudes.Grahasphulanayane Vikshepavasanals.which deals wilh Ihis topic.

.In his Aryabhaliyabhashya, Nila-kantha explains that the orbits of theplanets. i.e. the geometrical model ofplanetary motion is to be inferred fromIhe computational scheme for calculat-ing the sphulagraha (geocentriclongitude) and vilcshepa (latitude of theplanels)'.. The geometrical model validfor both exterior and interior planets as

presented by him in verses 19-21 ofChapter I of Siddhantadarpanal7 is asfollows:

~.At~M<'I~7t.A oI'"C.'t''b''''4..,.fI ~,~ "..;~~Cct>1..."'UIo"lfi'''I.tl. II~.."'I

~ rsf. """" .~,.nrT ¥" :"1&.1,,11'\~ ~V~ol'~ ,ri;Tt, ~ ,..,.,m; ~,I~'i~"- ""'" rfi'I'ft.r ~ "\'

"The [eccentric) oroits on which planelsmove (graha-bhramanavrlla) them-selves move at the same rate as theapsides (ucca-gali) on manda-vrlla, [orthe manda epicycle drawn with itscentre coinciding with the centre of themanda concentric). In the case of theSun and the Moon. the centre of the

CURRENT SCIENCE. VOL. 66. NO.1 0, 2S MAY 1994


Earth 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~ oftheir motion given by Nilakantha in hisAryabhaliyabhashya2.: ;

;nrt: ~ ,"::.. ;-f: ~,..~~oi":I"~;>:I~~~...~ -, """ ,...,1,Q"""'.mr...J<~'f1IN'" "':',':~,,<oj"'"~~~ &a t'7n.Pt"im:f r..zA ~ ~ ~.wTX; T'III'XI

~The eanh is not circumscribed by their[i.e. the interior planets, Mercury andVenus] orbits. The Earth is always

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 ineccentric 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 thatthere is 8 vast literature on astronomy(including mathematics) both inSanskrit and Malayalam, produced bythe Kerala school. during the period14th-19th century. Only a small fractionof it has been published and so far onlya few studies of these texts have

appeared. Whal seems to emerge clearlyfrom the source-works already publi-shed is Ihat by the later part of the 15thcentury, if not earlier. Kerala astrono-mers had arrived at many of thediscoveries in mathematical analysis andastronomy which are generally hailed as

CURRENT SCIENCE, VOL. 66, NO. 10, 25 MAY 1994

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 decadesprior to 1835 is indicatedbyJ. Warrenin his book Kala Sanlcalita (Madras.1825, pp. 93, 309-3iO). Curiously, thewhole issue gOt' totally ignored after1835 till 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/cara I and Somesvara (cd. Shukla.K. S.), INSA. New Delhi. 1976, pp. 32.247.

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

6. Aryabhatiyam with the Bhashya of Nild-kantha Somasutvan:. Golapada (cd.Pillai, S. K.), Trivandrum SanskritSeries, No. 185, 1957, p. 8.

7. Tanlrasangrahaof Nilakantha Soma-sutvan with the commentaryLaghuvivritli of Sankara Variar (cd.Pillai. S. K.). Trivandrum SanskritSeries. No. 188,1958, p. 2.

8. Ref. 7. p. 8. Nilakantha's modificationof traditional planetary model byidentifying what were referredto as thesighroccas of Mercury and Venus withthe planetS themselves, seems to havegone unnoticed so far. even in thosestudies where allusions arc made to

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Nilakaotha'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 oj

Astronomy 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.) Sphwanirnayufanlro

of 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 of

last 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

modification of planetary theory by kerala astronomers

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