GRAHANYAYADEEPIKA OF PARAMESWARA an old astronomy text from Kerala

Grahananyaayadeepika of Parameswara

In this short work Parameswara speaks of the principles of eclipses in detail.The first thing he mentions is the distance between the central point of Sun and moon and earth.
The distance from earth to sun is 459585 yojana.To moon it is 34377.These are respectively multiplied by the karma or radius of the orbits of each ,and then divided by Thrijya (3438)and thus the real distance from earth to sun and moon obtained.This is depending upon the time and position of earth in its orbit.3438 is in minutes.Divide by 60 to get 23 Hora and 58 minutes.(the dinadairghya)and this number is used as the chandravyasa in Indian astronomy right from the beginning.Till 18th century Europeans did not know why such a fixed number is used by Indians in calculations ,yet we teach students that India had borrowed its astronomy from outsiders.Now modern scince of astronomy uses chandravyasa as 34375,while Indians were using 3437.5 as decimal and made it a complete number by making it 3438.
The ratio of distance of earth to sun and moon is 1:13
34377:499585
499585 yojana multiplied by 8.8 gives 302517.6
Divide by 3438 we get 1176.3 and 87.9(which is written as 88 and in decimal as 8.8)
This is to measure the lightray,its distance,time of spread,the gathi of it,and its shadow cast which is all about the knowledge of eclipse.

Ravisasibhoovyasam

Suns vyaasa is 4410,moon’s 315 and earths 1050.To make yojanaas into kala (karma or thrijya being in kala)we have to see the sphutayojanakarna and multiply by thrijya.
For sun 4410/459585X 3438
For moon 315/34377X 3438
Ratio of Moon;earth;sun=315:1050:4410
That is,earth is 3.3 more than moon and sun is 14 times than moon.in the size of the gola.Thus the 3 gola and their distance,and bimba and the ratios between them are first obtained.
Distance earth :moon 1
Earth :sun 13
Bimba moon :earth:sun=1:3.3:14
In grahana or eclipse the gathy and kaala of sunray and moonray are calculated with the same numbers as the modern astronomers now use is significant .To demonstrate that see below the modern units of lightyears (kpc,mpc,kiloparsec,megaparsec)
One parsec=3.1 X 10 13=3.3 yrs
One kpc=3.1 X 10 16 =3.3 thousand years
One mpc=3.1 X 10 19 =3.3 million years
It is in this way the 33 crores of prakaasadevaas or light rays are calculated by Indian scientists.If one have to make a diagrammatic representation of this ratios of earth,sun and moon,one has to show earth as center and sun in the periphery.Figure below.
But if this is being communicated to people who do not know the basic rules of grahana ,it is misunderstood.Ptolemy when he learned from India made this mistake perpetuated by several astronomers for a long time.Since the observer sits in earth ,the dinadairghya or 23 hours 58 minutes is constant for him,and hence 3438 as karma in minutes is constant for all calculations of all structures as far as the observer is concerned.Therefore the bhookendrasthithy of the observer is important for calculations.This is a basic lesson for astronomy which the Ptolemian astronomers of west could not grasp and they said that earth is central ,whereas Indians knew that Sunis central but for calculation one has to take a geocentric view .15 thithi X 24 hours gives 360
Thithy X raasi or 15 X 12 gives half of it
If thithy is multiplied by 23.58 the exact value is 359.5 not 360,and this shows how accurate they were in using Thrijya for calculations.In Panchasidhanthika also we find the same rule.
The bhoochaaya or image of the earth is measured by a sankudeepa.The vyasadala of sun is the same as the height of the deepa.That of earth is the same as the height of sanku .The distance between sanku and deepa gives distance between earth and sun in yojana
The distance between sanku and deepa multiplied by sanku ,and divided by the difference in their heights gives sankuchaaya which is the difference between bhoochaaya and difference between heights of sanku and deepa.

In the moola or base chaya is equal to bhoovyasa.In the end it is elongated as the tail of a cow (gopucha)since sunrays fall everywhere equally.If sun is a deepasthamba or a dwaja or yupa of 12 feet ,earth is a sankhu or a conch of 12 angula only.One is light and the other is naada.The sanku should be made as a sreeyanthra in thrikona,pyramid shape.
To see the length of chaaya at chandrasthaana
Formula is chayadairghya +distance between moon and earth at center X bhoovyaasa divided by chaayadairghya,multiplied by thrijya and divided by distance from moon in yojana we get chayavyasa in kala.

Suppose the chayavyasa from chayagra is same as bhoovyasa ,how to find its vyasa at chandrasthaana?First see the kala as we saw for the moon.bhooovyasa and thamovyasa are same since thamograha is Rahu or earth itself.The statement that the earths shadow has a gopucha like elongation is noteworthy here.Earth has a tail and with its vyasa the eclipse of moon and sun happen was known to astronomers of India(see the word ulka as used in Dravidadesa and Indonesia described in previous chapter)From the diksoothra of an observer from earth how much time sun and moon stay ,will be the time or duration of an eclipse.How much moon stay in earths shadow will be duration of lunar eclipse.In pournami moon travels through center of chaaya.The yoga of sun and moon will be just before or after the new moon in amavasi.This is because of lambana or parallax.
Madhyahnalagnam or point of ecliptic on meridian is traditionally by Lankodayapramaana natanaazhika ,by adding or subtracting surya rekhansa from lagnakhanda according to Parameswara ,just like Varahamihira and the astronomers before him.That means whether the observer is in Ujjain or in Kerala,whether it is in 5th century AC.E or in 14th century ,Lanka was accepted as the meantime just as we calculate GMT after we became colony to British.Drikshepalagna is praaglagna minus 3 raasi .
See local meantime and lanka rasimaana and natanaadika according to it.The segment of ecliptic or madhyakhanda to it is seen and reduced from suns rekhansa,and in afternoon madhyalagna is added to suryarekhansa just as described by Varahamihira and the earlier astronomers like vasishta and Paithamaha.
Madhyajya is the jya of madhyalagna.It is obtained by adding or subtracting declination of madhyajya and akshamsa.Madhyajya is madhyadrikjya or the highest distance of madhyalagna.Its sanku is called Madhyasanku,(cos madhyajya)
Madhyajya=madhyalagna,akshamsa (either addition or subtraction as the case may be)The sine of the sum or angle thus obtained
Drikshepasanku is the sanku of drikshepalagna or the cos of drikshepalagna
Madyasanku X thrijya/madhyalagna minus praaglagna

Golathrikona being a samakonathrikona the rule of cos and sine is applied here just as Varahamihira did.
Madhyalagnajya minus Harijam is madhyasanku.Then how to find drikshepasanku with thrijya?
Mahaachaya =drikshepajya
It is height of karma and mahachaya of graham .Driggathijya is its bhuja.
The vargamoola of drigjya minus the root of sin drikshepa gives sine driggathy
The position of observer is different because of the change in position of earth .Therefore one has to calculate driggathijya considering change in rekhansa.The difference in akshamsa due to drikbheda is measured with the drikshepajya.It is same as the chaya or sine uchabhedam.Therefore drikshepajya is the same as drikjya.Drikshepalagnajya minus sine zenith distance of nongesimal.
Inkhamadya drigbheda is soonya or zero.In harija and in antharala the the bhoovyasa in its kakshya is found.according to sine of zenith distance.and its ratio.
The picture for this calculation in its poorvaroopa is seen in the Harappa/Mohenjadaro pictographs.Which shows the antiquity of the knowledge of astronomy and of zero in Indian subcontinent.
When the observer is the center point of observation (local time)the direction of drikbheda is upside down.(sloka 23)
Lambana or parallax is according to distance of zenith distance and its sine(difference of mahaachaya)But rekhansalambana can be obtained from driggathijya itself.Drikshepajya or lambanam from akshamsa(parallax in latitude )is called Nathy.
Lambanaamsa of rekhansa is ansa of ecliptic.Nathy is the parallax which is at right angles to ecliptic.Drikshepajya,driggathyjya,are multiplied by half of earths vyasa and divided by thrijya to get akshamsalambana or nathy and rekhansalambana or lambana respectively in yojana in orbits of the planets.
Since drikshepajya and driggathijya are equal to thrijya ,their nathy and lambana are equal to half of vyasa of earth.To see nathy and lambana of any drikshepajya and driggathijya respectively divide by sphutakarna (distance from earth)and multiply by thrijya .The results will be for the particular graham in the same thrijyavritha.moons akshamsalambana minus suns akshansalambanam gives the nathy used for calculation of eclipses .Those of moon have to be found out from those of sun.The lambana and nathy of moon and sun are equal in yojana but different in kala or minutes.Kala will be more for moon than sun.

driggathijyaX bhoovyasardha/mean distance of earth to moon X 60/mean gathy of moon in a day gives lambananaadika of the eclipse
Therefore driggathijya /863=lambananaadika
Parallax in longitude in naadika is driggathijya
The errors due to usage of moons madhyamadoora is removed by division with mean daily motion.To remove all the errors (samasthalambanadosha)one can just divide with the samasthagathi of moon for one day.
There are 3 different methods used by astronomers
1.akshamsa of drikshepalagnaja plus highest distance of drikshepalagna to get drikshepalagna in chandrakakshya
2. others add chandralambana
3.adding akshamsa,rekhansalambana respectively
Drikshepajya/863 X difference between moon and suns dinagathy/60=sphutanathy
In the morning Parvaantham minus lambanam
In the afternoon parvaanthum plus lambanam
Because moon is always below relative to sun
Parvaantham is lunisolar yoga or the end of amavasya.To calculate lunisolar yoga ,lambana is to be found by adding lambana if moon is on western side of sun and subtracting if on the eastern side.
Here poorvahna,aparahna are to denote whether the position of sun is to the east or west of the drikshepalagna (if east poorvahna)
To do lambanasamskrithakriya one has to find the lambana for yogavela upto a point when there is no difference between two results (either x-y =0 or x=y )That is one has to see the integer to the nearest.
Thus the samskrithalambana is equalized and called sphutalambana.Sphutalambana does not mean the lambana at yogavela.From the very first lambanavela there is parallax for the graham and it is not sphutalambana.Or it is not parallax but the calculation without lambana or parallax that is called sphutalambana.The madhyagrahana of sun happens at yogavela where there is lambanasamskrithy is done without parallax.At that time the straight line(ray)that pass through center of sun and moon also passes through the observers eye in straight line.The moons akshamsa used for suns eclipse is akshamsa of grahanamadhya added with drikbramsa of that period.If they are both in same direction.If in opposite direction ,minus them.See chandralambana from moons position at akshamsa agra.and from the nathy from there.The direction of madhyajya is direction of drikshepa and of akshamsalambana also.According to the kshepa of a bamboo the thread fastened to its end also changes.Similarly the lambitha of akshamsa also changes with drikshepa.(same is said in Panchasidhanthika of Varahamihira in 5th century)The bimba that is grasped,the bimba that grasps and their touching(sparsa)and at that time the distance between their centers which is the sum of their radii are to be known.Contact circle(samparkamandala)is drawn with this sum as radius.
When sun is in center and moon is within samparkamandala ,it is solar eclipse.In lunar eclipse the the time of eclipse is from the first sparsabindu to the last sparsabindu.Sun is always considered at center of samparkamandala.Moon in grahanamadhya is at equal distance from samparkamandalakendra to his own kendraakshamsa.Moon is in the paridhi during first and last sparsavela.This statement shows that the real suns position as center (heliocentric)has to be known for eclipse prediction .And Indians knew it.

The successive approximation method of Parameswara and of Varahamihira shows how scientific Indian astronomers had been in their approach to problems of the cosmos and its laws.
Valanam and akshavalanam are described in detail by Varahamihira and by Parameswara.Valana is the deviation in direction of the ray of light which is now called geodesic .Akshamsavalana is is the difference we feel due to akshamsa difference
Sine of latitude X sine of colatitude divided by 3438=akshavalanamThis is curved to north in morning and to south in evening.In the center there is no northsouth difference .Maximum difference is at Harija or horizon (according to Bhaskara 2 in grahanavaasana this nyaya is wrong)
The fact remains that Miletus of Thales in Greece who predicted a eclipse for the first time according to textbooks believed that earth is flat as a mat ,and such a man could never have predicted a eclipse (only if earth is shherical or gola it can cast a shadow on other gola like moon and sun is basic knowledge).Varahamihira and Parameswara lived 1000 years apart.Yet we find them using the same traditional Indian methods used by astronomers before them.Both have discussed the laws in detail and emphasized the importance of constant observation.It was during the time of Varahamihira 2 the Romance of Alexander was written in Greece and an Armenian copy for it came in 14th century.In this bookthere is a picture showing Alexander was born on a eclipse day.In it a triangle like a sanku and saptharshimandala with Arundhathy and 8 planets with lunisolar yoga of earth are shown.

Parameswara or Kunnathur Naagalassery Vadassery Parameswara called Drigganithaparameswara lived in 13th-14th century in Kerala and he did Drigganitha and parahithaganitha and wrote many books on eclipses and their laws.His observations were from the banks of Nila in thirunaava /Alathhiyur .Vadssery is the vadasreni of south Malabar(sundararajas vakyakaranaavyakhyana ed T.S.Kuppannasasthry ,K.V.Sharma Madras 1962 pp 8,23,93)He was born in Brighuvansa ,Aswalaayanagothra.Cheri ,sreni ,etc were names given to a group or samooha.The northern banks of river Nila ,where it joins Arabian sea we find Vadassery .It was a branch of kudallur mana.Parameswaran lived in Alathiyur.He has said that from the samarekha 18 yojana west ,in 647 akshamsa (10.51 N)in sakavarsha 1360 Goladeepika was written by him.Smarekha is here Lanka ,not Ujjain as many people have misinterpreted.9 yojana(correctly 8.8 -8.9 yojana)is one degree in gola.18 yojana west of Lanka thus means 2 degree west of Lanka or with 8 minutes time difference ,and it is a wellknown golaniyama.The guru of Parameswara was Rudran and also Golavith samgramagama Madhavan ,Narayanan .Damodaran was his son and disciple and Neelakandasomayaji was another disciple.Parameswara did 55 years of observation from an observatory in South Malabar continuously and after that in 1353 saka(CE 1431)wrote Drigganitha.In saka 1365(CE 1443)Goladeepika 2 came.In CE 1445 (kali 4536)he wrote a major work according to Neelakandasomayaji.That is from 1431 backwards 55 years ,when he started his observation he must be just a child (in 1375)Somayaji was born in 1442.Parameswara lived from CE 1360-1455.
His works were
1.drigganitha a karanagrantha
2Goladeepika 1
3 Goladeepika 2 Both these are on spherical geometry
4Grahanamandana
5grahananyayadeepika
6 Grahanashtaka (these 3 are on eclipses)
7 vakyakarana(astronomical tables)
8 bhatadeepika a commentary to Aryabhata
9.Parameswara Laghubhaskareeyabhashya
10.sidhanthadeepika mahabhaskareeyabhashya .These 2 are commentaries to Bhaskara 1
11sidhanthadeepika is a commentary to the commentary of Govindaswamy for Mahabhaskareeyabhashya
12 laghumaanasabhashya of Munjala
13.vivarana suryasidhanthabhashya
14vivarana leelavathy commentary to Bhaskara2
15vrithy vyatheepaatha ashtaka ,laatavaidhritha
16.vrithy bhashya to goladeepika 2
17.Acharasamgraha
18 jaathakapadhathy
19 muhurtharathnabhashya
20 bhashya to sreepathys jathakaadesa
21commentary to prasnasathpanchasikha of prithuyasa
(from 17 to 21 are astrology books)
22muhurthashtakadeepika
23 vakyadeepika
24 bhaadeepika
(22-24 are not yet obtained)
In Aryabhateeyabhashya of Neelakandasomayaji it is said that Paramesara had yanthra for his observations .Varahamihira and Parameswara used yanthra and both used beejaganitha for sphuta of calculations.Both gives the importance of correction by observations.Both call the sanskaara as Kaalaantharasamskaara though they lived 1000 years apart and one in Ujjain and the other in Kerala.

The kalidinasamkhya of eclipses observed by Parameswara are given .The first was observed from Gokarna and Nilaathada,the second from Nilaathada and Samgamagraama,the 3rd from Naavakshethra and all others from Nilaathada.In 12262 days he saw 14 eclipses of sun.


Varahamihira has spoken about the lunar and solar eclipses as given by the ancient five astronomers in his work Panchasidhanthika.In chapter 7 we get the solar eclipse in Poulisasidhantha.The rule for calculating parallax in longitude ,(Rekhansadikbramsa)is to ascertain the hour angle multiplying the given naadikaas by 6(six degrees going to each naadika)and establishing the proportion
Radius:greatest parallax = sine hour angle:desired parallax in longitude (the bending of the thithi is the corresponding sine ‘s thirtieth part).The greatest parallax being assumed as equal to the mean motion during four naadikaas ,we have
Desired parallax=4 X sine hour angle/120=sine hour angle/30
(since the mean movement per day is 24 degree in 4 nazhika,for every naazhika 6 degree)

What is the science of calculating the nathajya or the sine of hour?
Consider a unending(anantha)journey through cosmos where the two points between the journey is 864 000000 Kilometers (compare with Dwaaparayugasamkhya)and the speed is 240000 KM /second the time taken by the traveler to travel that distance is one hour.It is in cosmic scale ,not in our usual day to day understanding ,that the cosmic time of travel of a lightray in an eclipse is calculated.And this was known to Indians is the point I want to emphasise here. .That is why the nathajya is important from a cosmic point of view.This was found out only by Einstein in modern astronomy…
Sloka 2-6 are the samskaravisesha of Raahu in sthithisaadhana according to Hindi commentator Sudhakaradwivedi and he explains them.But the English translator Thibaut says sloka 2,3,4 are unintelligible and he describes only the rest.
Sloka 2 and 3 specifically says to reduce 26 minutes from the longitude of Raahu and find the degrees between Raahu and moon.If it is within 13 degree a lunar eclipse and if it is within 8 degree a solar eclipse will happen.(2nd sloka of 6th ch and 5th sloka of 7th chapter are comparable)For moon the 169 or the varga of 13 and for sun 64 or the varga of 8 is subtracted .1/4 of it reduced from the squareroot of the balance gives the duration of the eclipse as the case may be.
For the moons eclipse,
If the greatest latitude is 270,not 240 minutes,we obtain
Latitude=270 X 21/10 X degrees of interval divided by 120=
270 X 2 X degrees divided by 120=
9 X degrees divided by 2 appproximate.
Having found the approximate latitude,assume half the sum of diameter of moon and the shadow to be equal to 58 minutes ,we find
Measure of half duration of eclipse =√58 to the power of 2 minus latitude square=
9/2√13 square minus degrees square approximately.
From this to find the whole duration in Naadika multiply by 2 X 60 and divide by difference of the daily motion of sun and moon and thus we derive at
9 X 6 /73√169-degrees 2=
3/4√169-degrees 2

For solar eclipse an analogous process ,assuming the half sum of diameters of sun and moon to be equal to 35 minutes.
Measure of half the eclipse=√35 square -81/4 X degrees 2
=9/2√(35 X 2 /9)2-degrees 2
=9/2√64-degree 2 approximately
For duration of eclipse in naadikaas
2 X 9 X 60/730 X 2√64-Degrees 2
=3/4√64-degrees 2 approximately
This shows Poulisasidhantha knew the shape of the earth,the principle of eclipse as well as the parallax laws and the geodesic of the lightray and how to calculate it approximately .
Chapter 8 of Panchasidhanthika gives solar eclipse according to Romakasidhantha.Multiply ahargana by 150,deduct 65,divide by 54787 to get the mean longitude of sun in due succession (revolutions ,signs etc)Here the yuga fraction of 2850/1040953 as the lunisolar yuga of Romaka its reduced fraction 150/54787 is employed.65 is the kshepa (quantity allowing the calculations to start from epoch chosen.)True place of sun and moon is found measuring half signs of anomaly of sun and moon and arranged in direct as well as inverse order.Of the sun the mean longitude has to be reduced from half of Gemini ,that is two and a half signs from Aries first degree.Romakasidhantha calculates the equation of center of sun and moon for each half sign ,that is progressing from 5-15 degrees.The 6 quantities thus obtained for the first quarter of the circle are employed for the second quarter in inverse order .To get the Kendra that is the anomaly of the sun we have to take the mean longitude and the longitude of suns apogee which is estimated as 2 and a half signs or 75 degrees.
Anomaly and equations thus given by Varahamihira,according to Romakasidhantha is as below.
Anomaly in degree 15 30 45 60 75 90
Equations in mts ,sec 34’ 42’’ 68’37’’ 98’39’’ 122’49’’ 137’5’’ 143’23’’

Sloka 9 of the chapter gives the same rule as sloka 1 of 7th chapter just described in Poulisasidhantha that as many naadikaas elapsed till midday are to be multiplied by 6 and the 30th part corresponding sine is the displacement of the thithi (parallax in longitude).A process preliminary to calculation of parallax in latitude is given here.For that in Hindu astronomy we require the zenith distance of that point of the ecliptic which has the greatest altitude (which point is called vithribha or thribhona)the sine of which distance is technically called the drikkshepa,TZ in figure.AC is ecliptic,ac is moons orbit,P’ZTD the projection of a great circle passing through pole of ecliptic ,the zenith and the thribhona T ,and cutting the moons orbit in t.See below the diagram given by Thibaut in page 51 of his translation.(which is actually the figure given by the Hindi translator for slokaas 19-23 in suryasidhantha ,in the next chapter of Panchasidhanthika.(both figures given below for comparison.
Romakasidhantha makes calculation of drikshepa on the sine of tZ,not on the parallax of latitude.For this it first calculates tT ,after ascertaining dT (interval between thribhona and moons node,).The following proportion is obtained,
Rad:sine greatest latitude of moo=sine dT :sine tT
Greatest latitude is assumed equal to 240’,
Sine tT=240 X sine dT divided by 120=2 sine dT(It is interesting to note that in the commentary of 10th sloka ,Sudhakaradwivedi points out that the result after multiplication with 6 ,is the jya and its thrimsaamsa is the thithy called Lambanam and that from the praaglagna bindu,the last bindu of the 9th raasi of kraanthivritha called apakramakraanthyansa is obtained and it is known to all ganakaas of India.Then he says how Lagna is obtained in astronomy/astrology by lambanasamskaara or lambanaayanavidhi.
Nathaghatika X 6=jya
Thrijya =120=ParamalambanLambana=4 X nathajya /120=Nathajya /30 is the law.
Lagna -3=lagna -3 +12=lagna +9 is the law given in Paithamahasidhantha as a simplification of it.And Paithamahasidhantha ,the oldest sidhantha even knows the basic rules of Grahana /cosmic gathi of light etc.)
Resulting minutes turned into degrees dividing by 60.
To tT found,TZ has to be added to find zenith distance of t.But instead of TZ .LZ which is known from the declination of L and terrestrial latitude is taken.
Multiply daily movement of moon by the sine of that(zenith distance)and divide by 1800.The result is parallax in latitude.The mean measure of sun is 30’and that of moon 34’(4 mts difference).Multiply sine of distance of moon which at the time of yoga/conjunction has the same longitude as the sun,from the node by 21 and divide by 9.Take sum of the result and the parallax in latitude ,if the direction is same.If in opposite direction,difference is taken.
Rules for finding the parallax in latitude and the moons true latitude.

The rule for finding parallax in latitude is founded on supposition that greatest parallax is equal to 15th part of the moons daily movement.
Therefore the proportion,
Radius=daily motion /15=sin zenith distance of thribhona :parallax
Therefore parallax=daily motion X sin zen.distance/15X 120
=daily motion X sin zenith distance /1800












In order to get latitude of moon ,
Radius :sine of greatest latitude (which is 270’)
Moons distance from node :desired latitude
We therefore have,
Latitude=270’ X sin distance /3 X 4=
21 X sin distance /12 X 21 /27=
21 X sin distance/3 X 3 approximately.By increasing or decreasing the latitude thus found,by the parallax we get the true latitude.
Multiply the true motin of sun and moon by their mean measure ,and divide by mean motion.The result is true measure.(in minutes)of the two bodies at a given time.
Deduct the square of the avanathi(latitude of moon as corrected for parallax of latitude)from square of half the sum of the measure of sun and moon..From the double square root of the remainder determine the time (of eclipse)as in the case (of calculation)of the elapsed portion of a thithi.

Rule for determining duration of an eclipse is then given.Take the right angle triangle of which half the sum of diameter of sun and moon form the hypotenuse.The moons corrected latitude is the perpendicular.Base of the triangle represents half the time of the duration ,expressed in minutes of arc.Thus establish the proportion,
Minutes of difference of suns and moons gathy or motion:60 naadika=minutes of arc of duration of eclipse ;naadikaas of duration .As many minutes as there remain on the correted latitude being deducted from half the sum of the measures of sun and moon,so many angulis of the sun know to be obscured by the moon.Describe the circle of the sun with half its diameter and mark off from its center the true latitude ,in its appropriate direction.From the end of that latitude describe the circle of the moon with half her diameter ,and thereby show the amountof obscuration.



Here ,the Hindi translator Sudhakaradwivedi says with the helpof the above figure ,that Ra is the center of sun,and cha that of moon.Racha is the true latitude of moon.The obscured or overlapped amount(ta ka) is therefore equal to suns radius minus ra rata.It is equal to true latitude minus radius of moon.Hence obscured amount equals to sum of two radii minus the latitude.The rule for delineating the eclipse .
This shows that what Parameswara in later centuries was trying to establish was experimental proof for his ancestral lands knowledge,just as Varahamihira was doing by comparison and experimentation and observations .The law was there since antiquity in India and the Rahu of Indian astronomers had always been the Ahorathra (day and night on earth)and the middle letters taken in inverse order (Hora as hour and Raho as earth)as Varahamihira points out in his Horasasthra,and they very well knew the rules of eclipse,of geodesics,bending of lightrays and observation of relativistic laws during a solar eclipse well before Europeans had any idea of it.The above picture is seen in some of the IVC/Harappan relics showing that this knowledge was there from at least 3000-3500 BC onwards in India.(see pp 206 Parpola for the intersecting circles and its relation to Rohini asterism,reaching vernal equinox, during 3054 BC .This was the early Hrappan period and according to Mahabhagavatha and Mahabharatha ,the period of Sreekrishna,born on a Rohini star corresponds to this period).It is noteworthy that the same picture is seen as an adaptation in Euclids text book.

Solar eclipse in Suryasidhantha.(ch 9 of Panchasidhanthika)

Mean place of sun is found successively (in revolutions,signs etc)by multiplying the ahargana by 800,442 and dividing by 292207.The place is thus found for midday at Avanthi by Varahamihira.
180000 revolutions of sun in 65746575 days according to suryasidhantha.Therefore in one ahargana,180000 X ahargana /65746575 which fraction is reduced by 225 to 800 X ahargana /292207.The kshepa quantity -442 introduced to enable us to start calculation from epoch of Panchasidhanthika ,427 saaka.
Multiply ahargana by 900000,deduct 670217,divide by 24589506.Result is mean place of moon.

(The figure he uses to describe the laws till 12th sloka of this chapter is the same as that Thibaut adopts as that of Romakasidhantha).In sloka 15 the sphutabimba of sun and moon,kaalasphuta and bimbakalaaparimaana are explained.When sun is ucha the bimba is alpa(small)and when neecha it is vipula according to this sloka which shows the relation of distance,mass and velocity which now we attribute to Newton.
If a system completes 61 revolutions in 60 yrs,and thus neutralize the difference of 6 degrees due to its movement ,(360 degree in 366 days according to Paithamahasidhantha),one can divide the kranthibhedanabindu of 6796 of Romaka(adopted by Ptolemy also)with 366 paithamhadina ,
6796/366=4531/61=74 17/61 or 74.3.
This figure with observation of Dhoomakethu (+ or – 2)which appears in every 76 yrs as observational sphuta,61 bramana or revolution,7 times happen in 427.(60X 7=420 yrs ,61 X 7 =427 )This is the Saka 427 selected by Varahamihira for his calculation.
From Aslesha to Dhanishta (yugadhi of Varaha to Paithamaha)427 X 27 /2 =11529/2=5764 yrs.This is the time of the Romakayuga of Americas and it is 343 yrs after Paithamahayuga and hence (6107 yrs before Varahamihira was the time of Paithamaha.)The yuga of Rome on the other hand is 850 yrs before saka 427(varahamihira)and it corresponds to BC 343.This way,Varahamihira is actually pointing out the timescale of various sidhanthaas including the Yavana(Romaka of the Americaas)and Yavana (Romaka of Rome)and its relation to the old method of calculation of the Indian astronomers .
The number of revolution of the kraanthibhedanabindu /paithamahadinasamkhya=6976/366=19.22 which is closest to the squareroot of time (19 X 19 being 361)and this multiplied by 4 gives the 76.88 yrs ,as the appearance of the comet .Indians noticed that though by calculation it should be 76 ,it is not so and it varies .(see the list of observation from BC 240 to AD 1986 TO VERIFY THIS STATEMENT)
The dhoomakethu of 1986 ,for the era which started in BC 6080,is the 79th time of appearance of it being watched by humanity.In 1986 this was visualized in the last point of Krithika and first point of Rohini,and this shows the ayana has become as before .It has come a full cycle for the Indian astronomers.76 is the chaayaamaanam of Raahu in Indian astronomy.One sakavarsha has 1673 samvatsara and 7 such has 11711.It is the same as that of Paithamaha and poulisa because they take 22 X 76 =1672 .And + I is the 1673 of saaka.1672 X 7 is only 11704 which is taken by Poulisa and the difference of 11711 and 11704 is 7 .This type of difference is seen in the calculation of Kaliyuga of India and that of the Maayan calendar also showing a common way of understanding the cosmos.(Yukathan has a calendar starting from BC 3114 August 11th.The Indian Kaliyuga starts in BC 3104 August 11 when the sun is in the asterism of Aslesham or Ayilyam star .)The way of determining the movement ,the diameter of the planet,the orbit,its eclipse and the geodesics of light etc and the concept of light as wave and paramaanu makes Indian thought equivalent to modern astrophysics in many respects.Only difference is in the language(Sanskrit instead of English) and the lack of evidence of whether they were using the modern technological devices.