Chapter 16 - Eclipses (grahana)

vedanivistam ravigrahanakaranam 16. 1. 1. 16. grahanam - ECLIPSES yattva surya svarbhanustamasavidhyadasurah | aksetravid yatha mugdho bhuvananyadidhayuh || 5 || svarbhanoradha yadindra maya avo divo vartamana avahan | gulham suryam tamasa'pavratena turiyena brahmana'vindad atrih || ma mamimam tava santam atra irasya dugdho bhiyasa ni garit | tvam feat asi satyaradhastau mehavatam varunasca raja || gravno brahma yuyujanah sarpayan kirina devan namasopasiksan | atrih suryasya divi caksuradhat svarbhanorapa ya adhuksat || 8 || yam vai suryam svarbhanustamasa'vidhyadasurah | atrayastamanvavindan na hyanye asaknuvan || 6 || Cause stated in the Rgveda (Rgveda, 5.40. 5-9) O Sun, when the demon Svarbhanu overspread you with darkness, all the worlds stood as if not knowing where they were. (5) O Indra, you destroy the illusions of Svarbhanu which exist under the sky. Sage Atri got back the Sun who was hidden by darkness, by means of the fourth brahma incantation. (6) O sage Atri, may that malicious demon, desirous of food, not devour me with that dreadful darkness. You are a friend and truth is your wealth. May you and god Varuna protect me. ( 7 ) The sage Atri set the press-stone, propitiated the gods with prayers and salutations and dispelled Svarbhanu and set his eye on Sun's light. (8) ness. Atri and his descendents alone could restore the Sun when the demon had overspread (the Sun ) with darkNone besides them had the power to do so. (9) 16. 1.2. svarbhanurvva asura adityam tamasa''vidhyat | tam deva na vyajanan | te atrimupadhavan | tasya anirbhasena tamo'pahanyat | prathamamapan sa krsnavirabhavat | dvitiyam sa rajata, yat trtiyam sa lohiti yatha varnamabhyatrnat sa suklasit || (Tandya | Pancavimsa Brahmana, 6.6.8) The Demoniac Svarbhanu struck the Sun with darkness; the gods did not discern it (the Sun, hidden as it was by darkness ) : they resorted to Atri; Atri repelled its darkness by the bhasa. The part of the darkness he first repelled became a black sheep, what (he repelled) the second time (became) a silvery (sheep), what (he repelled) the third time (became ) a reddish one, and with what (arrow) he set free its original appearance ( colour), that was a white sheep. 1 ( W. Caland). candragrahanahetuh 16. 2. 1. bhucchayam svagrahane bhaskaramarkagrahe pravisatinduh | pragrahanamatah pascannendorbhanosca purvarddhat || 8 | vrksasya svacchaya yathaikaparsve bhavati dirghacaya | nisi nisi tadvad bhumeravaranavasaddinesasya || 8 || muryatsaptamarasau yadi codagdaksinena natigatah | candrah purvabhimukhaschayamova tada visati || 10 || candro'dhasthah sthagayati ravimambudavatsamagatah pascat | pratidesamatascitram drstivasad bhaskaragrahanam || 11 || avaranam mahadindoh kunthavisanastato'rddhasanchannah | svalpam raveryato'tastiksnavisano ravirbhavati || 12 || evamuparagakaranamuktamidam divyadrgbhiracaryaih | rahurakaranamasminnityuktah sastrasadbhavah || 13 || na kathancidapi nimittairgrahanam vijnayate nimittani | anyasminnapi kale bhavantyathotpatarupani || 16 || pancagrahasamyoganna kila grahanasya sambhavo bhavati | tailam ca jale'stamyam na vicintyamidam vipascidbhih || 17 || (Varaha, Brhat-Samhita, 5.8-17) Cause of the Lunar eclipse At a lunar eclipse the Moon enters the shadow of the Earth, and at a solar eclipse, she enters the Sun's disc. That is the reason why the lunar eclipse does not begin at the western limb, nor the solar one at the eastern limb. (8) Just as the shadow of a tree goes on increasing on one side as a result of the Sun's movement, even so is the case with the shadow of the Earth every night by its hiding the Sun during its revolution. (9) In her course towards the east, if the Moon tenanting the seventh house from the Sun, does not swerve much either to the north or the south (when her declination is very little), she enters the Earth's shadow. (10) 1 The interesting feature of the above passage is the detailed observation of the change of colour in the Sun's disc during the progress of an eclipse.

The Moon situated below and moving from the west, obstructs the solar disc like a cloud. The solar eclipse, therefore, is different in different countries according to the visibility of the eclipsed disc. (11) When the lunar eclipse takes place, the obstructing agency is very large, whereas in the solar eclipse it is small. Hence in semi-lunar and semi-solar eclipses, the luminous horns become blunt and sharp, respectively. ( 12 ) In this manner the ancient seers endowed with divine insight have explained the causes of eclipses. Hence the scientific fact is that Rahu is not at all the cause of eclipses. (13) An eclipse can by no means be ascertained through omens and other indications. For, the portents such as the fall of meteors and earth tremors occur at other times as well. (16) Scholars should not believe the traditional statement to the effect that an eclipse cannot take place except when there is a combination of five planets in the same zodiacal Sign, and that a week before the eclipse, i.e. on the previous 8 th lunar day, its characteristics can be inferred from the behaviour or appearance of a drop of oil poured on the surface of water. (17). (M.R. Bhat) candragrahanagananasiddhantah 16. 3. 1. sardhani sat sahasrani yojanani vivasvatah | viskambho mandalasyendoh sasitistu catussati || 1 || sphutasvabhuktigunitau madhyabhuktya hatau svakau | ravessvabhaganabhyastah sasankabhaganoddhrtah || 2 || sasankakaksyagunito bhajito va'rkakaksyaya | viskambhascandrakaksyayam tithyapto manaliptikah || 3 || sphutendubhuktirbhuvyasagunita madhyayoddhrta | labdham suci mahivyasasphutarkasravanantaram || 4 | madhyenduvyasagunitam madhyarkavyasabhajitam | visodhya labdham sucyastu tamoliptasca purvavat || 5 || bhanorbhardhe mahicchaya tattulye'rkasame'thava | sasankapate grahanam kiyadbhagadhikonake || 6|| tulyau rasyadibhih syatamamavasyantakalikau | suryendu paurnamasyante bhardhe bhagadibhissamau || 7 || gataisyaparvanadinam svaphalenonasamyutau | samalipto bhavetam tau patastatkaliko'nyatha || 8 || (Suryasiddhanta , 4.1-11) Lunar eclipse: Principle of computation The diameter of the Sun's disc is six thousand five hundred yojanas; of the Moon's, four hundred and eighty. (1) 15.* 16. 4. 1 These diameters, each multiplied by the true motion, and divided by the mean motion of its own planet, give the corrected (sphuta) diameters. If that of the Sun be multiplied by the number of the Sun's revolutions in an Age, and divided by that of the Moon's, (2) Or if it be multiplied by the Moon's orbit ( kaksa ), and divided by the Sun's orbit, the result will be its diameter upon the Moon's orbit: all these, divided by fifteen, give the measures of the diameters in minutes. (3) Multiply the Earth's diameter by the truc daily motion of the Moon, and divide by her mean motion: the result is the Earth's corrected diameter (suci). The difference between the Earth's diameter and the corrected diameter of the Sun is to be multiplied by the Moon's mean diameter, and divided by the Sun's mean_diameter: subtract the result from the Earth's corrected diameter (suci ), and the remainder is the diameter of the shadow; which is reduced to minutes as before. (4-5) The Earth's shadow is distant half the Signs from the Sun: when the longitude of the Moon's node is the same with that of the shadow, or with that of the Sun, or when it is a few degrees greater or less, there will be an eclipse. (6) The longitudes of the Sun and the Moon, at the moment of the end of the day of new moon (Amavasya), are equal, in Signs, etc. : at the end of the day of full moon (Paurnamasi) they are equal in degrees, etc., at a distance of half the Signs. (7) When diminished or increased by the proper equation of motion for the time, past or to some, or opposition or conjunction, they are made to agree, to minutes: the place of the node at the same time is treated in the contrary manner. 1 (8). candragrahanagananam -- vasistha-paulisau samakalacandrasuryo 16. 4. 1. naisyastithinadyo'rke deyascandre samenduravivivarat | divasodbhavasca sodhyah sa bhavati tatkalasasiliptah || candragrahanasambhavah rahoh samasatkrtikalam hitvamsam tacchasankavivaramsaih | grahanam trayodasantah pancadasantastamastasya || 2 || grahanasthitikalah viksepakalakrtivarjitasya panconasastivargasya | mulam dvigunam tithivad vibhajya kalah sthiterbhavati || 3 || * For notes and comments, see Su. Si.: Burgess, pp. 143-45.

16. 4. 1 fandsma: INDIAN ASTRONOMY-A SOURCE-BOOK sasitimiravivarabhagah trayodasonah sarahatah ksepyah | sthityam vinadikastah rahavadhike'nyatha hanih || 4 || kintvantaramsahinaih pancabhiruna hata dasa 'krta ' ghnah | tatpadamekasvighnam pancaso'smadvimardakalah || 5 || sparsamoksadisah sthitidalavimardadalayovisesake tamah sakalamattindum | pragrahanamoksasasirahuvivarabhagaisca dig vacya || 6 || viksepaviparyasantariyabhage krte tayodasadha | paridhau prakprabhrtindorgrahanasamse vadet parva || 7 || sasiparidhidalardhaghne khendvantarabhagasamgune cakse | 'khakharupasta ' hrte pragvalanam vamam cyute savyam || 8|| grahanakalam varnam ca ་ tithyante grahamadhyam prakparatah sthitidalena catyantau | raktakapilau ca varnavuccadhassthe pare nitaram || 6 || rahumukhonam cakram 'dhidviyama ' gunam sasankasamyuktam | (juketthage'yamuccah ) kriyadikanyantage nicah || 10 | (Varaha, Pancasiddhantika, 6.1-10) Vasistha-Paulisa systems Sun and Moon of equal longitude Minutes of arc equal to the nadis of the full moon tithi, to go, after sunset, are to be added to the Sun, (which has been computed for sunset). Minutes of arc, equal to the nadis to go from the end of the full-moon or new-moon tithi in the day-time, upto sunset, are to be so added to the Sun. Thus corrected, the Sun becomes equal to the Moon in (degrees and) minutes at the end of the full or new moon tithi, (i..., at full or new moon). (1) Possibility of a lunar eclipse Deduct one degree and thirtysix minutes from Rahu's Head or Tail, (whichever is near the Moon) and find the interval in degrees between that and the Moon (at full moon found above). If it is less than thirteen, a lunar eclipse will occur then. If it is less than fifteen, (and above thirteen) there will be a slight darkening, that is all. (2) Duration of the eclipse Square the Moon's latitude, subtract it from the square of 55, (i.e. from 3025), and find its square root. Double this, and multiplying by 60, divide by the difference of the daily motions of the Sun and Moon, in minutes. The approximate time of the duration of the eclipse is obtained in nadis. (3). If Moon Sun is less than 13°, multiply the degrees by 5. The result is in vinadis. Add these vinadis to the 198 duration if the longitude of Rahu is greater than that of the Moon, and subtract if the Moon is greater than Rahu. Thus the time of duration becomes correct. (4) Total obscuration Deduct the difference of the longitudes between the Moon and Rahu from five degrees. Deduct this from ten degrees, and multiply the remainder by this itself and by four. Find the square root of the result and multiply it by 21. The minutes of arc of total obscuration is obtained. This, divided by the daily motion, gives the time. (5) Direction of the eclipse During the interval from the time of first contact to the beginning of totality, Rahu, (i.e. darkness), swallows the Moon completely. The directions of the points of first and last contacts are to be calculated from the Moon Rahu of those times. (6) Divide the semi-orbit of the Moon situated opposite to the direction of latitude into 13 parts by straight lines parallel to the east-west diameter, at equal distances from one another. At the part of the rim equal to the degrees of Moon Rahu, on the eastern or western part of the orbit, are the points of first and last contacts, from which the direction can be read. (7) Multiply a fourth of the Moon's rim, (in whatever unit taken, as for e.g. minutes or digits) by the latitude, and again by the degrees of the Moon east or west of the meridian. Divide this by 8100. By so many units is the east or west point of contact bent northward or away from the north, respectively, if the Moon is east of the meridian, and bent away from the north and northward, respectively, if the Moon is west of the meridian. (8) Moment of the eclipse and colour moon. The middle of the eclipse is at the moment of new The times of first and last contacts are earlier and later than the middle, by half the time of duration (9 a-b). When the eclipse is total, the colour of the Moon is red or brown as it is farthest or nearest to the earth respectively and mixed, more or less, in between. When the eclipse is near sunset or sunrise, the moon is smoky in colour. When the eclipse is partial, the Moon has the colour of rain-cloud. (9 c-d). Subtract the Head of Rahu from 12 rasis, multiply it by 228, and add the Moon's longitude. If this is between 6 and 12 rasis, the Moon is farther, and if between 0 and 6 rasis, it is nearer.¹ (10). (T. S. Kuppanna Sastry). 1 For the elucidation of the several rationales involved, see Pancasiddhantika: T. S. Kuppanna Sastry: 6.1-10.

199 - saurasiddhantah tamobimbah 16. 5. 1. sthityardha kalah ravikaksya navatiguna 'sadastadasro 'ddhrtendukaksayah | chedah ' sattri ' ghnayah labdhenonasca sadvargah ||| 1|| 'viyadarka ' gune sasikaksyaya hrte karmukam tamovyasah | candratamovyasayutim dvabhyam hrtva tato vargat || 2 || viksepavargahinadasannapade 'viyadvicandra 'ne | suryendubhuktivivaroddhrte sthiternadika labdhah || 3|| pragrahanendoh krtva viksepamato'naya sthitairbhavati | evam bhuyo bhuyah sthityavisesah krto yavat ||| 4 || istakalagrasapramanam vimardakalah 16. ECLIPSES arkendubhuktivivaram vanchitanadihatam tu sastihrtam | sthitiliptastabhyastattatkale dosca viksepat || 5 || krtiyogapadam sodhyam sasirahukalapramanayogadalat | yacchesam tad grastam jneyam tatkalamarkendvoh || 6 || antyadyayorvisesadavanativiksepavargavivarapadam | dvigunam tithivat krtva vimardakalo'rkacandramasoh || 7 || (Varaha, Pancasiddhantika, 10.1-7 ) -Saura Siddhanta Diameter of the shadow Multiply the Moon's true distance in its orbit by 36, and divide by the Sun's true distance multiplied by 90 and divided by 286. Subtract this result from 36, multiply by 120, divide by Moon's true distance and get the arc of the resulting sine. This is the angular diameter of the shadow. (1-2 a) Duration of the eclipse Add the angular diameters of the Moon and the shadow, divide by two, and square it. Subtract the square of the Moon's latitude from this, and find the square root. Multiply this by 120 and divide by the difference of the motions per day of the Sun and the Moon, pertaining to the time of eclipse. The duration of the eclipse is obtained in nadikas. (2 b-3). Find the Moon's latitude at first contact, and using this find a more correct duration. Repeat this till there is no difference between the previous and the next durations. (4) Obscuration at any desired moment Take the nadis before or after full or new moon up to the time for which the amount eclipsed is wanted. Multiply this by the difference of the Sun's and Moon's daily motions, ( mentioned above), and divide by 60. 16. 6. 1 The 'corresponding minutes of arc' are got. Square this, square the Moon's latitude for the momert, add them and extract the square root. Subtract this from the half-sum of the diameters of the eclipsing and the eclipsed bodies. The remainder is the minutes of arc eclipsed, at the moment taken, of the Moon in the case of the lunar eclipse, and of the Sun in the case of the solar eclipse. (5-6) Time of obscuration Take the difference of the angular semi-diameters, instead of the sum. Square it, subtract the square of the parallax-corrected latitude (in the case of the solar eclipse), of the latitude (in the case of the lunar ), find the square root, double it, and treat it as tithi, i.e. multiply by 60 and divide by the difference of the parallax-corrected daily motions for the solar eclipse, cr of the mere daily motions in the case of the lunar. The time of total obscuration is got.1 (7) (T. S. Kuppanna Sastry). -- aryabhatiyam 16. 6. 1. candre jalamarko'gnih mrdbhuschayapi ya tamastaddhi | chadayati sasi surya, sasinam mahati ca bhucchaya || 37 || sphutasasimasante'kam patasanno yada pravisatinduh | bhucchayam paksante tadadhikonam grahanamadhyam || 38 || bhuravivivaram vibhajed bhugunitam tu ravibhuvisesena | bhucchayadirghatvam labdham bhugolaviskambhat || 36 || chayagracamndravivaram bhuviskambhena tat samabhyastam | bhucchayaya vibhaktam vidyat tamasah svaviskambham || 40 || tacchasisamparkardhakrteh sasiviksepavargitam sodhyam | sthityardhamasya mulam jneyam candrarkadinabhogat || 41 || candravyasadhanasya vargitam yattamomayardhasya | viksepakrtivihinam tasmanmulam vimadardham || 42 || tamaso viskambhardham sasiviskambhardhavarjitamapohya | viksepadyacchesam na grhyate tacchasankasya || 43 || viksepavargasahitat sthitimadhyadistavarjitanmulam | samparkardhacchodhyam sesastatkaliko grasah || 44 || (aryabhata I, Aryabhatiya , 4. 37-44) Aryabhatiya Moon, Sun, Earth and Shadow The Moon is water, the Sun is fire, the Earth is earth, and what is called Shadow is darkness (caused by the Earth's Shadow). The Moon eclipses the Sun and the great Shadow of the Earth eclipses the Moon. ( 37 ) 1 For detailed elucidation, rationales and calculations, see Pancasiddhantika : T. S. Kuppanna Sastry, 10. 1-7.

16. 6. 1 Occurrence INDIAN ASTRONOMY-A SOURCE-BOOK When at the end of a lunar month, the Moon, lying near a node (of the Moon), enters the Sun, or, at the end of a lunar fortnight, enters the Earth's Shadow, it is more or less the middle of an eclipse, (solar eclipse in the former case and lunar eclipse in the latter case). (38) Length of Shadow Multiply the distance of the Sun from the Earth, by the diameter of the Earth and divide (the product) by the difference between the diameters of the Sun and of the Earth the result is the length of the Shadow of the Earth (i.e. the distance of the vertex of the Earth's shadow) from the diameter of the Earth (i.e. from the centre of the Earth). (39) Earth's shadow Multiply the difference between the length of the Earth's shadow and the distance of the Moon by the Earth's diameter and divide (the product) by the length of the Earth's shadow: the result is the diameter of the Tamas (i.e., the diameter of the Earth's shadow at the Moon's distance). (40) Half-duration From the square of half the sum of the diameters of that (Tamas) and the Moon, subtract the square of the Moon's latitude, and (then) take the square root of the difference; the result is known as half the duration of the eclipse (in terms of minutes of arc). The corresponding time (in ghatis etc.) is obtained with the help of the daily motions of the Sun and the Moon. (41) Subtract the semi-diameter of the Moon from the semi-diameter of that Tamas and find the square of that difference. Diminish that by the square of the (Moon's) latitude and then take the square root of that: the square root (thus obtained) is half the duration of totality of the eclipse. (42) Non-eclipsed portion Subtract the Moon's semi-diameter from the semidiameter of the Tamas; then subtract whatever is obtained from the Moon's latitude: the result is the part of the Moon not eclipsed (by the Tamas). (43) Measure of the eclipse Subtract the ista from the semi-duration of the eclipse; to (the square of) that (difference) add the square of the Moon's latitude (at the given time); and take the square root of this sum. Subtract that (square root) from the sum of the semi-diameters of the Tamas and the Moon; the remainder (thus obtained) is the measure of the eclipse at the given time. (44) - aryabhatiyardharatrapaksah 16. 7. 1 a. ravicandrau samaliptau tithigatagamyaghatikaphalonayutau | patonacandrajiva viksepo navagunesuhrta || 1 || 200 'bhavadasa ' gunite ravisasigati 'nakhaih ' 'svarajina ' te mane | sastya bhaktam 'tattvasta ' gunitayorantaram tamasah || 2|| viksepam samsodhya pramanayogardhatastamaschannam | sarvagrahanam grahyadadhike khandagrahanamune || 3 | chadyardhena cchadakadalasya yuktonakasya vargabhyam | viksepakrtim pro pade tithivat sthitivimadardhe || 4 || bhuktih sastihrta sthitivimardadalanadikagunarkendvoh | adavrnamante dhanamasakrt tenanyatha pate || 5 || vistasthitidalaviksepaliptikavargayutipadenonam | manaikyardham channam madhye viksepaliptonam || 6 || trijyaptacapabhagairnataksajivavadhadudagyamyaih | purvaparayoh purva vibhayuggrahyayanamsaisca || 7 || (Brahmagupta, Khandakhadyaka, 1.4.1-7) -Aryabhata's Midnight system The degrees, minutes and seconds (omitting the Signs) in the longitudes of the Sun and Moon should be made equal by adding to or subtracting from them their respective motions during the ghatikas, which are to pass till the purnanta (or time of opposition) or that have passed since then, respectively. Subtract the longitude of the pata¹ from that of the Moon. The jya of the remainder, multiplied by 9 and divided by 5, gives the viksepa of the Moon in minutes. (1) The true daily motions of the Sun and Moon multiplied, respectively, by 11 and 10, and divided by 20 and 247, give their angular diameters in minutes. The difference between 8 times the true motion of the Moon and 25 times that of the Sun, when divided by 60, gives the angular diameter of the Earth's shadow in minutes. (2) When the viksepa of the Moon is subtracted from half of the sum of the diameters of the obscuring and the obscured bodies, the remainder is the portion obscured by the shadow. If the obscured portion is greater than the obscured body, there is total eclipse; if less, there is partial eclipse. (3). Find the sum and difference of the semi-diameters of the obscuring and obscured bodies. Subtract the 1 Pata in this chapter is Moon's Node.

square of the viksepa of the Moon from the square of each of the results. Find the square roots of the remainders; and thus calculate, respectively, the half durations of the eclipse and of the total obscuration in the same way as in the case of tithis. (4) Multiply the true daily motion of the Sun or the Moon by the number of ghatikas, etc., in the half duration of the eclipse or of total obscuration. Divide each product by 60. Add the result to or subtract from the respective longitude of the Sun or of the Moon. The first gives the longitude at the end of the eclipse, or the total obscuration, as the case may be; and the second gives the longitude at the beginning of the eclipse or the total obscuration. In the case of the pata, the process must be reversed. The corrections should be applied repeatedly. (5) From the half duration of the eclipse, whether at the beginning or at the end, subtract the given time, after which or before which, respectively, the obscured portion is wanted. From that time, calculate the arc in minutes gained by the Moon and also its viksepa. Find the square root of the sum of the squares of these two quantities. Subtract it from half the sum of the diameters of the obscuring and obscured bodies. The remainder is the obscured portion. At the time of the madhyagrahana, the obscured portion is obtained by subtracting the number of minutes in the Moon's viksepa from half the sum of the diameters of the obscuring and obscured bodies. (6) Multiply the natajya (samamandaliya-natansa-jya) of the obscured body by the aksajya and divide by the trijya. Considering the result as the jya, find the number of degrees in the corresponding arc. This arc is to the north or south according as the obscured body is in the eastern or the western half of the celestial sphere. Take the sum of difference of the number of degrees in this arc and that in the ayanavalana, that is, the declination calculated from the longitude of the obscured body increased by 90°, according as they are of the same or of opposite denominations. The result is the variation of the eastward direction of the ecliptic from the eastward direction of the disc of the obscured body.1 (7) (Bina Chatterjee) parvajnanam 16.7.1 b. patonaraverbhardhaccakranconadhikah kala bhaktah | tadgatiyutyaptadinairasanne'rkasya masante || 16 | parvendoh paksante pragadhikona yutirbhavati pascat | tanmadhye na grahanam yadi bhanoh 'pancajinabharasah || 20 || 1 For the rationale and formulae involved, see Khandakhadyaka: Bina Chatterjee, I. 117-22. 16. 8. 1 | graffqqun faunt faqinar'feafaqur'zaGHARA | 'vyomatidhrtidviyugani ' ' rasasarasca ' candrapatasya || 21 || kham nanda dviyama khabdhayo grhadyastathestaparvagunah | ksepyah parvanyesyati sodhyah pate'nyathatite || 22 || grahane yatha ravindvoh spastikaranadyamuktavat krtva | evam parvajnanam grahanajnanam sphutam ganitat || 23 || (Brahmagupta, Khandakhadyaka, 2.4. 19-23) Syzygy computation Find the mean longitudes of the Sun and Moon's pata on a given day. Subtract the longitude of the pata from that of the Sun. Subtract the remainder completely from 6 and 12 Signs and express the difference in minutes. Divide this difference by the sum of the mean daily motions of the Sun and the pata. The result is in days etc. If the difference in the longitudes of the Sun and the pata is greater than 6 or 12 Signs the difference will be equal to it before the time in the result; if less, the difference will be equal to it after the time in the result. If the time is Amavasya, there is the possibility of a solar eclipse; if near Purnima, there is the possibility of a lunar eclipse. If there is no eclipse at present, solar or lunar, (it should then be examined whether there was one, 12, 18, 24, etc., months before or there will be one after the same intervals. For this purpose, the longitudes of the Sun, Moon its ucca and pata before or after these intervals must be found). The half-yearly motions of the Sun, Moon, its ucca and pata are respectively 5 signs 24° 27' 6", 5 signs 22° 12' 53", 19° 42' 56" and 9° 22' 40". Multiply these motions by the number of half years before or after which the possibility of an eclipse is to be determined. The products resulting from the motions of the Sun, Moon and its ucca should be added to the respective longitudes, if the time is after the given day; and deducted if the time is before the given day. The product resulting from the pata's motion should be (The applied to its longitude in the reverse manner. final results are the mean longitudes at the time when the possibility of an eclipse is being determined. (19-23). (Bina Chatterjee) " -- aryabhatopari brahmaguptakrtah sodhah 16. 8. 1. svaphalamrnam cakrardhadune kendre'dhike dhanam madhye | tattithinatakendrajyavadho raveh sasinavendugunah || 1 || indornavanavavedaistrijyakrtilabdhavikalikonah prak | pascadadhiko'rko'sakrdrne'nyathendurdhane hinah || 2 || ksayadhanahanidhanani prak pascadanyatha kherindoh | pragvat pascat svagatau dhanaksayaksayadhanani prak || 3 || dinagatasesalpayutam svapadayuktam dinam dinardhahrtam | arsyafaan fagdzialatesti qeft: 118 11 angulalipta vitusairyavodarairanagulam sadbhih

grahyagrahakadalatatsamasaviksepaliptikacchedah anagulalipta trijyavalanajyanam bhavedistah || 5 || (Brahmagupta, Khandakhadyaka, 2.4.1-5) -Emendation by Brahmagupta If the mandakendra of the Sun or Moon is less than 6 Signs, the mandaphala of each should be subtracted from its mean longitude; if the mandakendra is greater than 6 Signs, the mandaphala should be added. The result in each case will be the corrected longitude. From these longitudes calculate the time of opposition or conjunction, according as it is lunar or solar eclipse. Find the natakala at that time. Multiply the jya of the Sun's mandakendra by the jya of the natakala. Multiply the product again by 191 and divide by the square of the trijya. Add or subtract the result in seconds, to or from the Sun's corrected longitude, according as it is in the western or eastern half of the sky. (The result gives the correct true longitude.) Multiply the jya of the Moon's mandakendra by the jya of its natakala. Multiply the product again by 499 and divide by the square of the trijya. The result is in seconds. If the mandaphala of the Moon is subtractive, add or subtract the result to or from its corrected longitude, according as it is in the eastern or western half of the sky. If the mandaphala is additive subtract the result from the corrected longitude of the Moon, whether it is in the eastern or western half of the sky. (The result is its corrected true longitude.) The process should be repeated till the longitudes are fixed. (1-2) If the Sun is in the eastern half of the sky, the correction to its motion is subtractive, additive, subtractive and additive, according as its mandakendra is in the first, second, third and fourth quadrants, respectively. If the Sun is in the western half, the process is reversed. In the case of the Moon, when it is in the western half, the correction to its motion is subtractive, additive, subtractive and additive, according as its mandakendra is in the first, second, third and fourth quadrants, respectively. If the Moon is in the eastern half, the correction is additive, subtractive, subtractive and additive, according as its mandakendra is in the first, second, third and fourth quadrants respectively. (3) In the case of a solar eclipse, add the length of the day (on which the eclipse occurs) to its one-quarter. Add to the sum the dinagata or the dinasesa, whichever is less, and divide by half the length of the day. The result is the number of minutes in an angula on that day. (In a lunar eclipse the same method must be followed using the length of the day of the Moon.) The breadth of six grains of barley without the husk is equivalent to one angula. (4) The semi-diameters of the obscuring and the obscured bodies, their sum and the viksepa of the Moon are expresssed in angulas, when the number of minutes in these lengths is divided by the number of minutes in an angula. The trijya and the valanajya are expressed in angulas by dividing each by any number. (It is 6 according to ancient astronomers.) 1 (5). (Bina Chatterjee) -MTEHT: 9 16. 9. 1 a parvanadyo ravau deyastah salipta nisakare | evam pratipadah sodhyah samaliptadidrksuna || 1|| 'pancavasvisurandhresusagara ' stigmatejasah | karnah 'parvatasailagnivedarama ' nisakrtah || 2 || avisesakalakarnataditau trijyaya hrtau | sphutayojanakarnau tau tayoreva yathakramam || 3 | 'qfsakrunnraarent' za'feafuforait'-gui: vyaso vasundharayasca 'vyomabhutadisah ' smrtah || 4|| yojanavyasasamksunnam viskambhardham vibhajayet | sphutayojanakarnabhyam liptavyasau sphutau tayoh || 5 || karnah ksunnah sahasramsormedinivyasayojanaih | medinyarkavisesena bhucchayadairghyamapyate || 6 | candrakarnavihine'smin bhumivyasena tadite | chayadairghyahrte vyasascandravattamasah kalah || 7 || (Bhaskara I, Laghubhaskariya 4. 1-7) -Bhaskara I Longitudes of the Sun and the Moon . One who wants to obtain (the longitudes of the Sun and the Moon when there is) equality in minutes of arc² should add as many minutes of arc as there are parvanadis, to the Sun's longitude (at sunrise) and the same together with the minutes of arc (of the difference between the longitude of the Sun as increased by 6 signs, and the longitude of the Moon in the case of opposition, or of the difference between the longitudes of the Sun and the Moon in the case of conjunction) to the Moon's longitude (when opposition or conjunction of the Sun and Moon is to occur); similarly, (when opposition or conjunction of the Sun and Moon has occurred) one 1 For the explanation of and proof, see Khandakhadyaka:Bina Chatterjee, I. pp. 149-51. * When the Sun and the Moon are in opposition, their longitudes differ by six signs; when they are in conjunction, their longitudes are the same. The minutes, however, are the same. The equality in minutes of arc refers here to the time of opposition or conjunction.

should subtract the pratipad-nadis (etc. from the longitude of the Sun and the Moon). (1) Mean distances of the Sun and the Moon 4,59,585 is (in yojanas) the (mean) distance of the Sun and 34,377 that of the Moon. (2) True distances of the Sun and the Moon These (above-mentioned mean distances of the Sun and the Moon) multiplied by their true distances in minutes obtained by the method of successive approximations and divided by the radius (i.e., by 3438') give their true distances in yojanas. (3) Diameters of the Sun, the Moon and the Earth The diameter of the Sun is 4410 ( yojanas ) ; of the Moon, 315 (yojanas); and of the Earth, 1050 (yojanas). (4) Angular diameters of Sun and Moon Multiply the radius (i.e., 3438') (separately) by their diameters in yojanas and divide by their true distances in yojanas: then are obtained their true (i.e., angular). diameters in minutes of arc. (5) Length of Earth's shadow Multiply the Sun's (true) distance (in yojanas) by the diameter of the Earth in yojanas and divide by the differrence between (the diameters of) the Sun and the Earth. Then is obtained (in yojanas) the length of the Earth's shadow. (6) Diameter of the Earth's shadow This (length of the Earth's shadow) diminished by the (true) distance of the Moon and multiplied by the diameter of the Earth and (then) divided by the length of the Earth's shadow gives (in yojanas) the diameter of the Earth's shadow (at the point where the Moon crosses it). This should be reduced to minutes of arc like (the diameter of) the Moon. ( 7 ) . ( Kripa Shankar Shukla) candraviksepah 16. 9. 1 b. patonasamaliptendorjiva 'khatighana ' hata | karnena hriyate labdha t viksepah saumyadaksinah || 8 || induhinatamovyasadalaliptavivarjitah | viksepasya na grhyante tamasa sasalaksmanah || 6 || viksepavargahinayah samparkardhakrteh padam | gatyantarahrtam hatva sastya sthityardhanadikah || 10 || sphutabhuktihata nadyah sastya nityam samuddhrtah | labdhaliptah ksayascandre ksepasca sparsamoksayoh || 11 || viksepascandratastasmannadika liptikah sasi | avrtya karmana tena sthityardhamavisesayet || 12 || sthityardhenavisistena hinayukta tithih sphuta | sparsamoksau tu tau syatam parvamadhyam grahasya tat || 13 || grahyagrahakavislesadalaviksepavargayoh | 16.9.1 b vislesasya padam pragvad vimardardhasya nadikah || 14 || tithimadhyantaralanamasunamutkramajyaya | visuvajjya hata bhajya trimorvya labdhadikkramah ||15 || prakkapale tu bimbasya purvapascimabhagayoh | udagdaksinato'ksasya valanam pascime'nyatha || 16 || tatkalendvarkayoh kotyorutkramajyapamo gunah | ayanadvimbapurvardhe pascardhe vyatyayena dik || 17 || yogastaddhanusoh samye disorbhede viparyayah | samparkardhahata tajjya trijyaptam valanam hi tat || 18 || ekadikkam ksipet ksepe vidikkam tadvisodhayet | valanam tat sphutam jneyam suryacandramasorgrahe || 16 || samparkardhadhikam taddhi sankhyaya yatra labhyate | samparkat sakaladdhitva valanam tatra sisyate || 20 || asamyuktamavislistam sparsavat kevalam sphutam | viksipya grahamadhyasya tasya syad vyastadikkramah || 21 || bhaskarendutamovyasaviksepavalanodbhavah | anagulanyadhita liptasta eva harijasthite || 22 || (Bhaskara I, Laghubhaskariya , 4.8-22) Moon's latitude Multiply the R sine of the difference between the longitudes of the Moon, when in opposition with the Sun, and its ascending node by 270 and divide (the product) by the true distance of the Moon, in minutes: the result is the Moon's (true) latitude north or south. ( 8 ) Moon's diameter unobscured by the shadow Diminishing the (minutes of arc of the ) Moon's latitude (obtained above) by half of the minutes of arc resulting on diminishing the diameter of the shadow by that of the Moon are obtained those of (the diameter of) the Moon which remain unobscured by the shadow. (9) Sparsa and moksa sthityardhas Diminish the square of half the sum of the diameters of the Moon and the shadow (samparkardha) by the square of the (Moon's) latitude (for the time of opposition of the Sun and Moon) and then take the square root (of that). That divided by the difference between the (true) daily motions (of the Sun and Moon) and multiplied by 60 gives, in nadis, the (first approximation to the sparsa or moksa) sthityardha. (Then) multiply those nadis by the true daily motion ( of the Moon) and always divide by 60. The resulting minutes should then be severally subtracted from, and added to the longitude of the Moon (calculated for the time of opposition) to get the longitudes of the Moon for the times of the first and last contacts.

16. 9. lb INDIAN ASTRONOMY-A SOURCE-BOOK From the Moon's longitude (for the first contact as also for the last contact) calculate the Moon's latitude; and from that successively determine the (corresponding (sthityardha in terms of) nadis, the corresponding minutes of arc (of the Moon's motion), and the longitude of the Moon (for the first contact as also for the last contact). Repeating this process again and again, find the nearest approximations to the (sparsa and moksa) sthityardhas. (10-12). First and the last contacts Diminish and increase the true time of opposition by the (sparsa and moksa) sthityardhas, obtained by the method of successive approximations, (respectively): then are obtained the times of the first and the last contacts. The time of the middle of the eclipse is the same as that (of opposition of the Sun and the Moon).1 (13) Sparsa and moksa vimardardhas The square root of the difference between the squares of the Moon's latitude and half the difference between (the diameters of) the eclipsed and eclipsing bodies leads, * This is how the exact times of the beginning and end of a lunar eclipse are determined. In practice, however, the exact beginning and end of an eclipse are not perceived with the unaided eye. A lunar eclipse is seen to begin after a portion of the Moon's disc is already obscured by the shadow. Sankaranarayana tells us how to find the times when a lunar eclipse is actually seen to begin and end, He says: "At the beginning, having diminished the sixteenth part of the Moon's diameter from half the sum of the diameters of the Moon and the shadow, (then) having squared it and subtracted from it the square of the Moon's latitude, one should obtain half the (apparent) duration of the lunar eclipse by the method of successive approximations. Or, one should multiply the sixteenth portion of that (semi-duration) in minutes by 60 and divide by the difference between the daily motions of the Sun and the Moon, and then reduce that to vighatis etc. Having thus ascertained the corresponding time (in vighatis) the apparent instant of the first contact should be declared by adding that to the instant of the first contact. After that, in order to determine the instant of the last contact, the moksa-sthityardha obtained by the method of successive approximations should be added to the instant of opposition and the result taken, as before, as the instant of the last contact. There also the (apparent) time should be announced after diminishing it by onesixteenth (of the time corresponding to the moksa-sthityardha). Then adding the two sthityardhas (i.e., the sparsa and moksa sthityardhas), the sum should be declared, in ghatis etc., to be the duration of the eclipse." In support of his statement, Sankaranarayana quotes the following verse of Acarya Bhatta Govinda: sasidehastyamsonam samparkadalam yada nater adhikam | bhavati tadendugrahanam na bhavatyalpe 'rdhasamparke || i.e., When half the sum of the diameters of the Moon and the shadow diminished by the sixteenth portion of the Moon's diameter is greater than the Moon's latitude (for the time of opposition), then does a lunar eclipse occur (i.e., is observed). When half the sum of the diameters of the Moon and the shadow (diminished by the sixteenth part of the Moon's diameter) is smaller, a lunar eclipse does not seem to occur (i.e., is not observed). The statement that the time of the middle of the eclipse is the same as that of opposition of the Sun and Moon is only approximately true. An accutare expression for the difference between the two instants was first given by Ganesa Daivajna (1520). (Kripa Shankar Shukla) 204 as before, to the determination of the (nearest approximation in) nadis of the (sparsa vimardardha as also of the moksa vimardardha. (14) Aksa-valana and Ayana-valana Multiply the R sine of the (local) latitude by the R versed-sine of the asus between the times of (the beginning, middle, or end of) the eclipse and the middle of the night or day,¹ and divide by the radius (i.e., 3438'): (the result is the R. sine of the aksa-valana). The direction of the result (i.e., aksa-valana) is (determined) in the following manner : (If the eclipsed body, at the time of the first or last contact, is) in the eastern half of the celestial sphere, the directions of the aksa-valana for the eastern and western halves of the disc (of the eclipsed body) (i.e., of the sparsa and moksa valanas in the case of the Moon and vice versa in the case of the Sun) are north and south, (respectively); (if the eclipsed body is) in the western half of the celestial sphere, (they are to be taken) reversely. (15-16). Magnitude and direction of the ayana-valana The R sine of the declination calculated from the R versed sine of the koti of the tropical (sayana) longitude of the Sun or Moon² for that time (i.e., for the beginning, middle, or end of the eclipse) (is the R sine of the ayanavalana). In the eastern half of the disc (of the Sun or the Moon), the direction (of the ayana-valana) is the same as that of the ayana³ (of the Sun or the Moon). In the western half, the direction is contrary to that of the ayana. (17) Resultant valana Take the sum of their arcs (i.e., of the aksa-valana and ayana-valana (when they are of like directions) and the difference when they are of unlike directions. Multiply the R sine of that (sum of difference) by the sum of the semi-diameters of the eclipsed and eclipsing bodies and divide by the radius: this result is the valana. (18) Corrected valana: sphuta-valana If the valana (obtained above) is of the same direction (as the Moon's latitude) add it to the Moon's latitude; if it is of the contrary direction, subtract it (from the 1 Night when the eclipse is lunar, and day when the eclipse is solar. * The Sun is taken when the eclipse is solar, and the Moon is taken when the eclipse is lunar. 3 Ayana means "the northerly or southerly course (of a planet)". The course (ayana) is north or south according as the planet lies in the half orbit beginning with the tropical Sign Capricorn or in that beginning with the tropical sign Cancer.

16/10.1 205 16. ECLIPSES Moon's latitude). The ( sum or difference thus obtained is known as the corrected valana (sphuta-valana) in the case of solar and lunar eclipses. In case that (corrected valana) is found to be greater than the sum of the semi-diameters of the eclipsed and eclipsing bodies, it should be subtracted from the entire sum of the semi-diameters of the eclipsed and eclipsing bodies and the remainder (thus obtained) should be taken as the (corrected ) valana. (19-20 ) Valana for the middle of the eclipse The (resultant) valana for the middle of the eclipse obtained in the same way as for the first contact without any further addition or subtraction of the Moon's latitude tude is the corrected (valana for the middle of the eclipse). The direction of that (Moon's latitude) is to be taken reversely (in the projection of a lunar eclipse ). ( 21 ) Converting minutes of arc into angulas The minutes of arc of the diameters of the Sun, the Moon, and the shadow and those of the (Moon's) latitude and the (corrected) valana when divided by two are reduced to angulas. (But when the Sun and Moon are) on the horizon, they (i.e., minutes of arc) are the same (as argulas). ( 22 ). ( Kripa Shankar Shukla) - lallah grahanakalah 16. 10. 1. akasakaksya grahamuktiliptah dinakarastasamaye savidhuntudo ravividha vidadhita parisphuti | prathamapaksajapancadase tithau sasadharagrahanavagamecchya || 1 || dasagunam gunayet 'khakhasadnghanam ' yugabhavairbhaganaih sisiradyuteh | bhavati yojanamanamahah pate- rdyutiyujo nabhasah parigheridam || 2 || 'khakhanakhadri ' hrtam bhayujo bhaved grahayujo nijaparyamyahrt prthak | 'sarayamanga 'hata 'bhanavagni ' hrd grahavrteh sravanah phalamucyate || 3 || ravicandrayoh bhumadhyantaram 'visayanagasaranakasarabdhayom ' dinakrtah khalu yojanaja srutih | 'turagasailahutasakrtagnayah ' sasadharasya kumadhyatadantaram || 4 || nijamrdusravanena hate sruti vibhagunena hrte bhavatah sphute | sravanamadhyamabhuktihato'thava nijanijasphutabhogavibhajitah || 5 || tithigunam sasiyojanamandalam dinakrto gaganendukrtabdhayah | dinakrtah sravanah sarasanaguno nrpahrto bhavatiha mahiprabha || 6 || sphutasasisravanena vivarjita gaganapancakakubgunita kubha | apahrta ca taya prabhaya bhuvo bhavati yojanabimbamagoh phalam || 7 || tribhagunena hatani ravisruti- smarasuhrcchravanopahrtanyatah | phaladhanumsi vapumsi phalani va himagu - tigmamayukha - sasidvisam || 8 || sivahata dvibhahrcchasino gati- stanukalah syurinasya nakhoddhrtah | gunitayorbhujagaih saralocanaih 'kharasa ' hrd vivaram tamaso'thava || 6 || timiramavaranam himadidhite- dinakarasya nisakaramandalam | bhavati mandalakhandayutistayo- stadabhidhavaranavaraniyayoh || 10 || ravisamanakalasya kalavato vitamaso gunitam bhujasinjinim | tithibhirindunavendubhirhared vyagunisakaragolavasaccharah || 11 || cyutasaravaranavaraniyayo- dalayutih sthagitasya mitirbhavet | samadhikavaraniyatanoryada sakalameva tadavrtamadiset || 12 || vivaramavaranavaraniyayo- rapanayedathava dalitam sarat | tadavasesamitih prakata bhave- dyadi na sesamasesatanurgrahah || 13 || dalitamavaranam yutamunitam prthagathavaraniyadalena ca | svagunitam saravargavivarjitam krtapadam gaganangahatam haret || 14 || gativiyogakalabhirato bhavet sthitidalam ghatikadi samardalam | ravisasankatamogatisagunam 'kharasa ' hrt svaphalani prthak prthak || 15 ||

ravisasaphale samaliptayoh ksayadhane bhavatah prathamantyayoh | dhanamrnam tamasah svaphalam kalah sthitivimardadaladasakrt tatah || 16 || tithyantam sthitidalavarjitam yutam ca praggrasam kramasa usanti moksakalam | samyogam sthitidalayoh sthitesca kalam mardardhadvayayutimindvadarsanasya || 17 || madhyagrasah syat tithescavasane mardardhenantyena yuktah sa uktah | kalastajjnairnunamunmilanasya hinascadyenendusammilanasya || 18 || istonitasthitidalena viyogalipta-: bhuktyorhata gaganasatkahrta bhujah syat | tatkalikodupasaram kathayanti koti doh kotivargayutimulamabhistakarnah || 16 || evam vimadardhahate ca gatyoh syadantare doh svasarasca kotih | nimilanonmilanakarnasiddhayai grasastu manaikyadalam vikarnam || 20 || vistagrase manayorardhayoge svaghne mulam ksepavargonitam yat | tat sastighnam bhuktivislesabhaktam svasthityardhacchuddhamistastu kalah || 21 || sparsadyato grahyamanasya khande drste sese mucyamanasya sesah | tatkalendoh ksepamaniya samyak kuryat tavat karma yavat sthiram syat || 22 | sparsadikalajanatotkramasinjinibhih ksunnaksabha palabhava sravanena bhakta | capani purvanatapascimayoh phalani saumyetarani samavehi prthak kramena || || 23 || ahardaladravidalavasanam yavat kapalam kathayanti purvam | tato dinardhantamapurvamindo- rbhanorbhavetam grahane'nyatha te || 24 || grahyat sarasivitayad bhujajya vyasta tatah pragvadapakramajya | tasya dhanuh satrigrhendudik syat ksepo vipatasya vidhodisi syat || 25 || apakramaksepapalodbhavanam yutih kramadekadisam kalanam | karya viyogo'nyadisam tato jya grahya bhavat sa valanasya jiva || 26 | - Lalla anagulani valanasya ca jiva 'khesvarairvasugunabdhihutasah | unnato nijadinardhavibhaktah sardhayugmayugathanagulaliptah || 27 || manaikyardham chadakacchadyayosca ksepaschannam karnadoh kotayasca | bhajyah sarve'pyanagulanam kalabhi Time of the eclipse jayante te vangulani kramena || 28 || 206 (Lalla, SiDhVz., 5.1-28) If one wants to ascertain (the time of) a lunar eclipse, one must find the true longitudes of the Sun, the Moon, and its ascending node, on the fifteenth tithi (i.e., full moon day) in the light half of the lunar month, at sunset. (1) Circumference of the sky Multiply 21,600 by 10 and then by the revolutions of the Moon in a yuga. The result is the yojanas in the circumference of the sky up to which the rays of the Sun reach. (2) Distance of the planet from the Earth (The circumference of the sky in yojanas) divided by 72,000 gives the circumference of the orbit of the asterisms. Again, (the circumference of the sky in yojanas) divided by the number of revolutions of each planet in a yuga, gives the circumference of the planet's orbit (in yojanas). When this circumference is multiplied by 625 and divided by 3,927, the result is the distance of the planet from the earth (in yojanas). (3) Distance of the Sun and the Moon from the Earth The mean distance of the Sun from the Earth's centre is 4,59,585 yojanas and that of the Moon is 34,377 yojanas. (4) When their mean distances are multiplied by their respective mandasphuta hypotenuse and divided by the radius, the results are their correct distances. Or, the mean distance multiplied by the mean motion and divided by the true motion, gives the correct distance ( of the Sun or Moon) from the earth. (5) The diameter of the Moon is 315 yojanas and that of the Sun is 4410. The Sun's distance from the earth multiplied by 5 and divided by 16 gives the height of the cone of the Earth's shadow. (6) Diameter of the Earth's shadow in yojanas The length of the earth's shadow (as found above) diminished by the correct distance of the Moon from the

centre of the earth, then multiplied by 1050 and divided by itself, gives, as result, the diameter of the earth's shadow in yojanas (in the Moon's orbit). (7) Angular diameter of the Sun etc. The diameters of the Sun, the Moon and the shadow, (each expressed in yojanas), and multiplied by the radius. and divided, respectively, by the Sun's distance from the earth, the Moon's distance and the Moon's distance, (in yojanas), give the respective angular diameters in minutes, whence the arcs corresponding to the quotients as R sines are found. The results can also be treated approximately as diameters (without finding the arcs). (8) Or, the true motion of the Moon multiplied by 11 and divided by 272 gives its angular diameter in minutes. That of the Sun is obtained by multiplying its true motion by 11 and dividing by 20. The difference between 8 times the true motion of the Moon and 25 times that of the Sun, divided by 60, gives the angular diameter of the shadow (in minutes). (9) The eclipser and the eclipsed The shadow is the Moon's obscuring body and the Moon is the Sun's obscuring body. There are total and partial eclipses of the obscured body caused by the obscuring body. (The eclipse) is named after the (eclipsed portion of the) obscured body.1 (10) (Bina Chatterjee) Latitude of the Moon When the Moon is equal to the Sun in respect of minutes etc., subtract from its longitude, the longitude of its ascending node. Find the R sine of the remaining arc, multiply it by 15 and divide by 191. The result is the latitude of the Moon, and its direction is according to the hemisphere in which the Moon happens to be diminished by its node. (11) Obscured portion at mid-eclipse When this latitude is subtracted from the sum of the semi-diameters of the obscuring and the obscured bodies, the remainder is the portion obscured (at the time of mid-eclipse). When the portion is greater than the obscured body, the latter is said to be completely obscured. (12) Or, subtract half the difference of the diameters of the obscuring and the obscured bodies from the latitude of the Moon. The remainder is the portion not obscured. If there is no remainder, the obscured body is completely obscured. (13) 1 For rationales and demonstrations, see Si.Dh.Vr: Bina Chatterjee II.112-13. Half-duration of the eclipse 16. 10.1 Take the sum or difference in the semi-diameters of the obscuring and the obscured bodies. Square it and subtract from it the square of the Moon's latitude. Find the square root of the remainder. Multiply it by 60 and divide by the difference of the true motions in minutes of the two bodies. The results are, respectively, the approximate half durations of the eclipse and the total eclipse in ghatikas. When these times are severally multiplied by the true motions of the Sun, Moon and its node and each product divided by 60, the results are, respectively, the motions of the Sun, Moon and its node during these times. (14-15) The Sun's and Moon's motions in minutes should, respectively, be subtracted from their longitudes and the node's motion added to its longitude, if the first half of duration of the eclipse or of the total eclipse is required; the reverse process is to be followed if the second half is required. From these longitudes, again calculate the half. duration of the eclipse and of the total eclipse. Repeat the process till the times are fixed. (16) When the first half of the duration of an eclipse is subtracted from the time of the full moon, the remainder gives the time when the eclipse began. When the second half of the duration of an eclipse is added to the time of the full moon, the sum gives the time when the eclipse ends. So the wise say. The sum of the first and second half of the duration of the eclipse is its total duration. The sum of the first and second half of the duration of the total eclipse is the duration of the complete disappearance of the Moon. (17) The end of the full moon is the time of mid-eclipse. That time, diminished by the first half of the duration of the total eclipse, gives the time when the Moon is completely obscured. Again, that time added to the second half of the duration of the total eclipse, gives the time when the Moon begins to reappear. So say those who know. (18) Obscured portion at any time. When the given time (during an eclipse) is subtracted from the first or second half of the duration of an eclipse, and the remainder is multiplied by the difference in minutes of the motions (of the obscuring and the obscured bodies) and divided by 60, the result is called bhuja or base. The latitude of the Moon at that time is called koti or perpendicular. The square root of the sum of the squares of the base and the perpendicular is the hypotenuse at that time. (19)

In the same way, when the first or second half of a total eclipse is multiplied by the difference of the true motions (of the obscuring and the obscured bodies, and the product is divided by 60, the result is the base). The latitude of the Moon at these times is the perpendicular. (The square root of the sum of the squares of the base and the perpendicular is the hypotenuse at the beginning of the total eclipse (if the first half is taken), and is the hypotenuse for the end of the total eclipse (if the second half is taken). (In both cases), when the hypotenuse is subtracted from the sum of the semi-diameters of the two bodies, the remainder is the obscured portion. (20) Time from the obscured portion Subtract the given obscured portion from the sum of the semi-diameters of the two bodies. Square the remainder. Subtract from it the square of the Moon's latitude (at the time of mid-eclipse). Find the square root of the remainder. Multiply it by 60 and divide by the difference of the true motions of the two bodies. Subtract the result from the first half of the duration of the eclipse, if the observed portion is between the beginning of the eclipse and the mid-eclipse. But if it is between the mid-eclipse and the end, subtract the quotient from the second half of the duration of the eclipse. The result is approximately the time when the given portion is obscured. Find the Moon's latitude at this time and repeat the process till the time is fixed. (This then is the correct time.) (22) Valana: Deflection due to latitude and declination Multiply the equinoctial midday shadow by the R versed sine or the utkramajya of the hour-angle at the beginning of the eclipse, etc. and divide by the hypotenuse of the equinoctial shadow. Remember that the arc corresponding to this quotient as the R. sine, (which is called aksavalana), is to be taken as north or south according as the obscured body is in the eastern or western hemisphere. (23) In a lunar eclipse, the Moon, (the obscured body), is said to be in the eastern hemisphere from midday till midnight. From midnight till midday it is said to be in the western hemisphere. In a solar eclipse, (for the Sun, which is the obscured body), the contrary is the case. (24) Increase the number of degrees in the longitude of the obscured body by 3 Signs and then find its R. versed sine or utkramajya. Hence find, as before, the R. sine of the declination. The corresponding arc (ayanavalana): has the same denomination as that of the Moon increased 208 by 3 signs. The latitude of the Moon has the same denomination as that of the Moon diminished by its node. (25) The aksavalana, the ayanavalana and the Moon's latitude, all in minutes, should be added together if they are of the same denomination but their difference must be taken when of different denominations. Then find the R sine of the sum or difference, as the case may be. The result is the R sine of the valana or variation of the eastward direction of the ecliptic from the eastward direction of the disc of the obscured body. (26) Divide the R. sine of the valana and 3438, the radius, by 110, to convert them into angulas. When the altitude in time (unnatakala) on any day at any given time, is divided by half the duration of the day and 2 added to the result, the sum is the number of minutes equivalent to one angula. (27) Conversion of angular deflection to linear deflection The sum of the radii of the obscuring and the obscured bodies, the latitude of the Moon, the obscured portion, the hypotenuse, the base and the perpendicular, (expressed in minutes) should be divided by the number of minutes in one angula.1 (28). (Bina Chatterjee) -bhaskarah 2 grahanasambhavah 16. 11. 1 a. kalergatabda 'ravi ' bhirvinighna- scaitradimasaih sahitah prthaksthah | dvighnah sva 'nagankagajam ' sahinah \ 'qsang'Haal: quanfaat: 24: 11 9:11 masah prthak te dviguna 'stripurna atun'fant can'qqYI'AYFAT: | tribhirvibhaktah phalamamsapurvam masaughatulyaisca grhairyutam syat || 2 || sapatasuryo'sya bhujamsaka yada 'manu ' nakah syad grahanasya sambhavah | grhardhayuktasya sapatabhasvato bhujamsakan goladiso'vagamya ca || 3 || jneyo'rko ravisamkramad gatadinairdarsantanadinata- 'dvedam ' sena grhadinonasahitah prakpascime'syapamah | aksamsaih khalu samskrto 'rasa ' lavenasyatha te samskrtah patadhyarkabhujamsaka yadi 'nago ' nah syustadarkagrahah || 'rupam ' 'viyat ' 'purnakrtan ' sapadan ksiptva sapate pratimasamarke | * For the rationale and exposition, see Sisyadhivrddhida, : Bina Chatterjee, II. 113-28.

b tatsambhavam pragavalokya dhiman - Bhaskara II : grahan grahartham vidadhita tatra || 5 || Possibility of an eclipse (Bhaskara II, SS., 1. 4. 1-5) Multiply the number of years that have elapsed from the beginning of the Kaliyuga by twelve and add the number of months elapsed from the beginning of the luni-solar year. Let the result be x. Then add 2 x (1 - 8bai 6 ) 65 to x. Let the result be y. Then the longitude of what is called Sapata-Surya or the longitude of the Sun with respect to a node will be x rasis+ (2 y+503) (1+130) 3 x 30 rasis. If this longitude be less than 14°, then a lunar eclipse is likely to occur. (1-3 a) Special note for the solar eclipse Add half a rasi to the longitude previously obtained; find out on which side the Sun lies, north or south; compute the longitude of the Sun from the number of days elapsed after the Sankramana day (i.e. the day on which the Sun has left one rasi and entered another. (3 b) Obtain the hour-angle in nadis of the Sun at the ending moment of the Amavasya, i.e. at the moment of new moon; add or subtract one-fourth thereof in rasis from the position of the Sun according as the Sun is in the western or eastern hemisphere; then finding the declination of that point and from the sum or difference of the declination and latitude of the place, obtain the zenith-distance of the culminating point of the ecliptic; taking that point to be roughly the Vitribha, i.e. the point of the ecliptic which is 90° behind the Sun on the ecliptic, find one-sixth of the zenith-distance; taking the sum or difference of the result and the longitude of the Sun with respect to the node (obtained in the beginning by adding half a rasi to its position at full moon) if the result happens to fall short of 7°, then we could expect a solar eclipse. (4) If there be no eclipse at the current new moon, then go on adding one rasi-0°-40'-15" to the longitude of the Sun with respect of the Node (which will be its longitude for the moment of the next new moon) and repeating the procedure indicated, the occurrence of an eclipse or otherwise could be known. If such an occurrence be indicated, then compute the actual positions of the Sun, Moon and Rahu and following the procedure to be indicated in the chapter on Solar Eclipses, the moment of the occurrence of the eclipse and other relevant details could be computed. 1 (5). (Arka Somayaji) 1 For an exposition, see Siddhantasiromani: Arka Somayaji, pp. 331-45. 16 arkendroh kaksavyasardhah 16. 11. 1 b. 'naganagagninavastarasa ' rakhe prasapramanam 'rasarasesumahisu ' mita vidhoh | nigaditavanimadhyata ucchritih srutiriyam kila yojanasamkhyaya || 3 || mandasrutirdvaksrutivat prasadhya taya vibhajya dviguna vihina | trijyakrtih sesahrta sphuta sya- lliptasrutistigmarucervidhosca || 4 || liptasrutighnastrigunena bhaktah spasto bhavedyojanakarna evam | • bimbam 'khedvisarartu ' samkhya- nindoh 'khanagambudhi ' yojanani || 5 || bhuvyasahinam ravibimbamindu- karnahatam bhaskarakarnabhaktam | bhuvistrtirlabdhaphalena hina bhavet kubhavistrtirindumarge || 6 || suryendubhubhatanuyojanani trijyahatanyarkasasindukarnaih | bhaktani tatkarmukaliptikasta- stesam kramanmana kala bhavanti || 7 || sapatatatkalikacandradoya 'khabhaih ' hata vyasadalena bhakta | sapatasitadyutigoladik sya- dviksepa indoh sa ca banasamjnah || 10 | yacchadyasamchadakamandalaikya- khandam saronam sthagitapramanam | tacchadyabimbadadhikam yada sya Preliminaries Orbital radius jjeyam ca sarvagrahanam tadanim || 11 || (Bhaskara II, Siddhantasiromani , 1.5.3-7,10-11) The distances of the centres of the globes of the Sun and the Moon from the centre of the Earth in yojanas are respectively 689,377 and 51,566. (3) Hypotenuse The radius vector is to be computed even in the case of the Equation of centre as we did in the case of sighraR 2 phala. If it be 'K', 2 R- K will be what is kalakarna both in the case of the Sun, as well as the Moon. (4) The above kalakarna multiplied by the karna given in yojanas and divided by the Radius gives the rectified yajanakarna. (5)

16. 11. 1 b Spherical radii of Sun and the Moon The spherical diameters of the Sun and the Moon are respectively, 6522 and 480 yojanas. (5 b) (S-E) Km Ks 210 INDIAN ASTRONOMY-A SOURCE-BOOK Duration of the eclipse √ (P+r)² -ss³*60 Sthiti-khanda= duration of m₁-$1 the eclipse √ (P_r) 2 ss2*60 Marda-khandaduration of mi -$1 totality =2 a (where e-Earth's diameter, s= Sun's diameter; Km=Moon's distance from the Earth's centre; Ks=Sun's distance from the Earth's centre, and a radius of the Earth's shadow cone at the lunar orbit). Angular measure (6) The diameters of the Sun, the Moon, and Rahu in yojanas multiplied by R=3438', and divided, respectively, by Ks, Km and Km, give their angular measures. (7) Latitude of the Moon Viksepa or Sara, as it is also called, i.e. the latitude R. sin l * 270 of the Moon, is obtained by the formula = R and it will have the same direction as the Moon with respect to the ecliptic, where A is the longitude of the Moon with respect to the nearer node and 270' or 4° is taken to be the inclination of the lunar orbit to the ecliptic or, what is the same, the maximum latitude of the Moon. (10) Magnitude of a lunar eclipse Sthagita or the magnitude of an eclipse is defined as P+r+ (where P and r are, respectively, the radii of the eclipsing and eclipsed bodies and SS is the latitude of the Moon). If the Sthagita is greater than 2 r, then the eclipse is total.1 (11). (Arka Somayaji) sthityardhah 16. 11. 1 c. manardhayogantarayoh krtibhyam sarasya vargena vivarjitabhyam | mule 'khasat ' samgunite vibhakte bhuktyantarena sthitimardakhande || 12 || feucusanatufuran eryfa: sastya hrta tadrahitau yutau ca | krtvendupatavasakrcchrabhyam sthityardhamadyam sphutamantimam ca || 13 || evam vimadardhaphalonayukta - sapatacandrodbhavasayakabhyam | prthak prthak purvavadeva siddhe zye za ananmufan&que 11 98 11 (Bhaskara II, Sisi., 1.5. 12-14) 1 For an exposition and the rationale of the several processes involved, see Siddhantasiromani : Arka Somayaji, pp. 347-63. - where P is the radius of the shadow-cone, r the radius of the Moon's disc, SS its latitude taken to be constant during the eclipse, m₁ and s₁ the daily motions of the Moon and Sun respectively. (12) Rectification of the times From the position of the Moon and that of the Node obtained for the moment of opposition, have to be computed their position for the moment of first contact and those for the moment of last contact. (13) Proceeding on the same lines as above and obtaining B₂ and SS the rectified latitudes of the Moon for the moments of the commencement and end of totality of the eclipse, the Sammilana-marda-khanda and Unmilanamarda-khanda, T 3 and T 4, are to be rectified. 1 (14). (Arka Somayaji) sparsadivyavastha 16. 11. 1 d madhyagrahah parvaviramakale prak pragraho'smat paratasca muktih | sthityardhanadisvatha mardajasu sammilanonmilanake tathaiva || 16 || 'khanka ' hatam svadyudalena bhaktam sparsadikalotthanatam lavah syuh | tesam kramajya palasinjinighni bhakta maurvya yadavaptacapam || 20 || prajayate pragapare nate krama- dudagyamasam valanam palodbhavam | First contact etc. (Bhaskara II, Siddhantasiromani , 1.5.19-21 b) The 'Middle of the eclipse' (or, strictly speaking) the moment when the portion eclipsed is a maximum) occurs at the moment of opposition. Sparsa or pragraha is at the moment of first contact and moksa is at the moment of last contact, separated from the moment of the middle of the eclipse by times equal to sparsa-sthitikhanda and moksa-sthiti-khanda respectively before and after. Similarly, sammilana and unmilana or the moment of the commencement of totality and the end thereof occur before and after the moment of 'the middle of the eclipse' by times equal to sammilana-marda-khanda and unmilana-marda-khanda, respectively. (19) 1 For the rationale involved, see Siddhantasiromani: Arka Somayaji, pp. 363-70.

211 Valana or Deflection 16. ECLIPSES The hour angle of the eclipsed body expressed in nadis, multiplied by 90 and divided by half the duration of night (if it be lunar eclipse) or half the duration of day (if it be solar ), as the case may be, will give the degrees of an angle, whose R sine being multiplied by the R sine of the latitude and divided by ( R cos 8), (where 8 is the declination of the eclipsed body), gives the R sine of what is called Aksa-valana which is north when the hour angle is east, and south otherwise. (20-21 ab). (Arka Somayaji) -karanaratnam bimbavyasah 16. 12. 1. candraviksepah sparsamoksau grahanasambhavah grahanadesyata vimarvakalah 'dasa ' gunitendorbhuktih 'sasisarayama ' bhajita nijam bimbam | 'navanava 'bhih sva raho- staya raveh sva 'puranena || 2 || phanirahitasamakalendorjya svadasamsonita'tra viksepah | tatkrtirahite phanisasibimbasamasardhavarge syat || 3 || mulam sthityardhakala ravisasibhuktyantaroddhrta nadyah | parvani draha syat pragrahanam tatra samyute moksah || 4 || pratipadi viparitamidam tvavisistam jayate grahanamadhyam | suryagrahane'pyevam rahusthane tu sasibimbam || 5 || samparkadalpe viksepe grahanamasti nastyadhike | rahoryutisca drstiryada raveh syat tada bhavati || 6 || dvadasabhagadunam grahanam taiksnyadraveranadesyam | sodasabhagadindoh svacchatvadadhikamadesyam || 7 || viksepakrti tyaktva phanisasiviskambhavivaradalavargat | mulam bhuktyantarahrtamatha ghatikah syurvimadardhe || 8 || sthitidalaghatikasadrsi samkhya visame pade svaviksepe | sparse sodhya ksepya same'nyatha moksakale syat || 6 || krtva'visesamevam yavad dvitulyarupata bhavati | sthitidalaviksepau tau punah punah tavadaniyat || 10 || tenanitasthitidalaghatikakalau tu tau sphutau jneyau | (Deva, Karanaratna, 2.2 - 11 b ) -Karanaratna Angular diameters of Sun, Moon and Rahu Ten times the Moon's daily motion when divided by 251 gives the Moon's own diameter and when divided by 99, gives the diameter of Rahu (i.e., Shadow); and 16.12.1 10 times the Sun's daily motion when divided by 18 gives the diameter of the Sun. (2) Moon's latitude From the longitude of the Moon at full moon subtract the longitude of the Moon's ascending node. The R sine of that diminished by one-tenth of itself is the Moon's latitude ( at full moon ) . ( 3 ) Times of first and last contacts Subtract the square of that (Moon's latitude) from the square of half the sum of the diameters of the Moon and Shadow. The square root of that is half the duration of the eclipse in terms of minutes. This divided by the motion-difference of the Sun and Moon (in terms of degrees) gives the nadis (of half the duration of the eclipse) . ( When the time is measured from sunrise ) on the full moon tithi, these nadis being subtracted from the time of opposition (of the Sun and the Moon), the result is the time of the first contact; and the same number of nadis being added to the time of opposition, the result is the time of the last contact. (When the time is measured from sunrise) on the next tithi (called Pratipad), the process is just the reverse. (That is, the time of the last contact is obtained by subtracting the above ghatis from the time of opposition and the time of the first contact is obtained by adding those ghatis to the time of opposition). The time of the middle of the eclipse is obtained by iterating the above process. In the case of a solar eclipse, the process is the same, except for that in the place of Rahu (Shadow) one has to use the Moon's disc. (3 cd-5). Possibility of a Lunar eclipse When the Moon's latitude (for the time of opposition ) is less than half the sum of diameters of the eclipsed and eclipsing bodies, an eclipse (of the Moon) is possible; when greater (or equal), it is not possible. The conjunction of Rahu (Shadow) with the Moon occurs when the Sun sees the Moon (i.e., when the Moon is diametrically opposite to the Sun ) . ( 6 ) Prediction of an eclipse A solar eclipse should not be predicted when it amounts to less than one-twelfth of the Sun's diameter (as it might not be visible to the naked eye) on account of the brilliancy of the Sun. But a lunar eclipse must be declared whenever it amounts to more than onesixteenth of the Moon's diameter, as it will be visible (to the naked eye) on account of the transparency of the Moon. (7) Duration of totality Subtract the square of the Moon's latitude (for the time of opposition) from the square of half the difference

between the diameters of the eclipsed and eclipsing bodies and take the square root thereof, and then divide (that square root) by the motion-difference (of the Sun and Moon) (in terms of degrees): the quotient gives the ghatikas of half the duration of total eclipse. (8) Moon's latitude for first or last contact (In order to obtain the Moon's latitude) for the first contact, subtract as many minutes from the Moon's latitude (for the time of opposition) as there are ghatis in half the duration of the eclipse if the eclipse occurs in an odd nodal quadrant, and add the same number of minutes if the eclipse occurs in an even nodal quadrant. (In order to find the Moon's latitude) for the last contact, proceed reversely. (9) Semi-duration of eclipse by iteration Having done this, apply the process of iteration in the following way: Calculate the semi-durations of the eclipse and the Moon's latitude (for the first and last contacts) again and again until the successive values are the same. The times, in ghatis, of the semi-durations of the eclipse calculated from them (i.e., from the Moon's latitudes for the first and last contacts, obtained by iteration) are the true values of the two (semi-durations), 1 (10-11 a). (Kripa Shankar Shukla). _ettafa: candramasanayamam 16. 13. 1 a. 'candranganando ' nasako ''rka ' nighna- _Sripati caitradimasairyugadho 'dvi 'nighnah | 'panco ' nitah sviya ' nrpanka ' bhaga- hinah 'saranga ' ptaphalena yuktah || 2 || (Sripati, Dhikoti, 1. 2) Lunar months since Saka 961, the epoch Diminish the (current) year of the Saka era by 961, (then) multiply (the remainder) by 12, (then) add (to the resulting product) the number of months elapsed since the beginning of Caitra; (then set down the resulting sum in two places one below the other.) Multiply the sum in the lower place by 2, (then) diminish (the product) by 5, and (then) diminish (the remainder obtained) by 961 th of itself. Divide whatever is obtained (as the remainder) by 65 an add (the quotient to the sum standing in the upper place. 2 (2) 1 For rationale, see: Karanaratna: Kripa Shankar Shukla, pp. 42-47. * This rule assumes, following Aryabhata I, that one solar month is equivalent to (1 + 2 (1-1) 1) 916 lunar months approx. 650 The subtractive 5 in the text is meant to account for the intercalary excess at the beginning of the Saka year 961. darsanta - purnimanta- madhuvah grahanasambhavasca 16. 13. 1 b. vidha 'karabhyam 'dhrti ' bhi ' rbhuva ' ca tam masavrndam gunayet tato'ntyah | nija ' bhranetram sayuga ' bhrasadbhi '- bhaktah phaladhayastvatha madhyarasih || 3 || sa ' sasti ' bhaktah phalamurdhvarasau datva tato 'bhai ' vibhajecca sesam | darsantako bhadhruvakastu sah syat 'bhuva ' 'bdhivedai ' sca 'rasaih ' sametah || 4 || darsantakah syacchasina ' thanande- 'rana ' yuktah sa ca purnimantah | 'cakrardha ' cakrantarake dhruvasya 'purna ' yadodhvam grahanam vicintyam || 5 || 212 (Sripati, Dhikoti, 1. 3-5) Darsanta-bhadhruva and Purnimanta-bhadhruva Set down the resulting "aggregate of months" in three places (one below the other) and multiply (the numbers in the upper-most, the middle and the lowest places) by 2, 18, and 1, respectively. Then increase the last result (which is in the lowest place) by one-twentieth of itself. Then divide that by 60, (retain the remainder) and add the quotient to the number in the middle place. (Then) divide that (i.e., the number now in the middle place) by 60, (retain the remainder) and add the quotient to the number in the uppermost place. Then divide the number now in the uppermost place) by 27 (discard the quotient and retain the remainder). (To the remainders standing in the uppermost, the middle and the lowest places denoting naksatra, ghiti and pala in order) add 1, 44 and 6, respectively. Thus is obtained the Darsanta-bhadhruva.¹ (3-5 a) 1 The Darsanta-bhadhruva being increased by 1, 9 and 0 (in the uppermost, the middle and the lowest places denoting naksatra, ghati and pala, respectively) gives the Purnimanta-bhadhruva. The Darsanta-bhadruva denotes the longitudinal distance of the Mean Sun from the Moon's ascending node at the time of conjunction of the Sun and the Moon in terms of naksatra, ghati and pala. The naksatra, ghati and pala here are the divisions of the circle like the degrees, minutes and seconds. The whole circumference of the circle is divided into 27 equal parts called naksatras, each naksatra is subdivided into 60 equal parts called ghatis, and each ghati is further subdivided into 60 equal parts called palas. The Purnimanta bhadhruva denotes the longitudinal distance of the mean Sun from the Moon's ascending node at the time of opposition of the Sun and the Moon, in terms of naksatra, ghati and pala. The above rule assumes, following the Siddhantasekhara, that the rate of separation of the Sun from the Moon's node is equivalent to 2 naksatras 18 ghatis and 1 1/20 palas per month. In a fortnight, likewise, this separation to 1 naksatra 9 ghatis O pala. 31 naksatra 44 ghatis and 6 palas, used as an additive in the above rule, is the bhadhruva for the beginning of Caitra Saka 961, the starting point of our calculation.

213 Possibility of an eclipse 16. ECLIPSES. 'rasadribana ' pramitaccharonat When the difference of the (Darsanta or Purnimanta) bhadhruva from half a circle (i.e. 13 naksatras 30 ghatis), or a full circle (i.e. 27 naksatras) yields zero in the uppermost place (denoting naksatras ), an eclipse (of the Sun or the Moon) is to be considered (possible ). 1 (5 b). (Kripa Shankar Shukla) ravikanadyah 16. 13. 1 c. 'vahnaya 'sta ' dig ' 'rudra ' 'gaja ' gni ' samkhyah karkadike'rke ravika rnakhyah | 'sat ' 'dik ' 'bhava''sa ' 'rasa ' 'sunya ' samkhya mrgadike'rke dhanasamjnanadyah || 6 || Difference between true and mean Sun (Sripati, Dhikoti, 1. 6) When the Sun is in the six Signs beginning with Cancer (i.e. in Cancer, Leo, Virgo, Libra, Scorpio, and Sagittarius), the Ravika Nadis (ie ghatis of the Sun's correction), which are then negative in sign, amount to 3, 8, 10, 11, 8 and 3 ( respectively ); when the Sun is in the six Signs beginning with Capricorn (i.e. in Capricorn, Aquarius, Pisces, Aries, Taurus and Gemini) the (corresponding Ravika Nadis, which are now positive in sign, amount to 6, 10, 11, 5, 6 and 0 ( respectively). 2 (6) grahanagananam 16. 13. 1 d. samskrtya purvam ravikabhirasya cakrardhacakrantarake'tha sesam | liptadikam madhyasarah sa yamyo bhardhadhike, nyunatare'tha saumyah || 7 || madhyesuhina 'rasabana ' lipta- schannam ca tasmin sasimandalone | syatkhandaparvabhyadhike'tha purnam 'karagni ' liptamitamindumanam || 8 || 'sadagnicandragni ' mitardhayukta- vargattu viksepakrti visodhya | mule'tha sastya ' gunite ''bhrarama- nagaisca bhakte sthitikhandanadyah || 8 || 1 This means that an eclipse of the Sun or the Moon should be considered possibe if at the time of conjunction or opposition of the Sun and the Moon the distance of the Sun from the nearest node is less than one naksatras, i.e. 13° 20'. *The Ravika-nadis above give the difference between the true and mean positions of the Sun, i.e. the Sun's correction. This correction can be easily identified with the Sun's equation of the centre. It is to be noted that the correction stated above is zero in Gemini, the sign occupied by the Sun's apogee, and is again zero somewhere in Sagittarius, the sign occupied by the Sun's perigee. In the other signs its variation behaves like that of the Sun's equation of the centre. Moreover, its maximum value is 11 nadis, i.e. 2° 26' 40", which roughly corresponds to the Indian value of It may be added that 11 nadis the Sun's equation of the centre. is the approximate maximum value of the correction in round figures. 16-* pragvatkrte madhyavimardanadyah | sparsasthitih syat sthitikhandahine 16. 13. 1 d moksasthitih syat sthitikhandayukte || 10 || mardonamadhye tu nimilanam sya- dunmilanam madhyavimardayukte | madhyagrahah parvatitheh samaptau saradike tu vihatenaguladi || 11 || purvasritah kimcidivesadiksthah sparsasca vayodisi moksa uktah | syaduttarasyam disi madhyakhandam saumyah sasamkasya yada sarah syat || 12 || sparsastathagnau nirrtau ca moksah khandam ca yamye yadi yamyabanah | nimilanam pascimadigvibhage unmilanam purvadisastathendoh || 13 || Eclipse computation (Sripati, Dhikoti, 1. 7-13) Having first applied these Ravika Nadis (as a negative or positive correction) to the Bhadhruva, one should find the difference of the (corrected) Bhadhruva from half a circle or a full circle (as the case may be ). The difference (in ghatis etc. thus obtained) gives the Madhyasara (i.e. the Moon's latitude for the middle of the eclipse) in terms of minutes etc. When the (corrected) Bhadhruva is greater than half a circle, the Madhyasara is south; when less it is north. 1 (7) Moon's diameter. Subtract the Madhyasara (i.e. the Moon's latitude for the middle of a lunar eclipse) from 56 minutes (denoting the sum of the semi-diameters of the Moon and the Shadow): then is obtained the measure of the lunar eclipse (in minutes ). When it is less than the Moon's diameter, the eclipse is partial; when greater, the eclipse is total, the measure of the Moon's diameter is 32 minutes. 2 (8) 1 The (corrected) Bhadhruva denotes the longitudinal distance of the apparent (or true) Sun from the Moon's ascending node, in terms of naksatra, ghati and pala. The difference between the corrected Bhadhruva and half a circle or a full circle (as the case may be) gives the distance of the apparent Sun from the Moon's nearer node. For demonstration see Dhikoti: Kripa Shankar Shukla, pp. 15-16. * The following table gives the mean (angular) diameters of the Sun, the Moon, and the Shadow as used by Sripati in the present work and also the corresponding modern values. Sripati's value Sun's diameter Moon's diameter Diameter of shadow 33' 32' 80' Modern value 32' 4" 317!! 82' approx.

16. 13. Id Duration of the eclipse INDIAN ASTRONOMY A SOURCE-BOOK Subtract the square of the Moon's latitude from 3136, which is the value of the square of the sum of the semidiameters of the Moon and the Shadow (in minutes). Then multiply the square root of that (difference) by 60 and divide by 730: the quotient gives the duration of the (lunar) eclipse in nadis. (9) Duration of totality for a total lunar eclipse Subtracting (the square of) the Moon's latitude from 576 (i.e. the square of the difference of the semi-diameters of the Moon and the Shadow in minutes)1 and proceeding as before, are obtained the nadis of half the totality (for a total lunar eclipse). (10 a) First and last contacts and immersion and emersion (The time of the middle of the lunar eclipse) being diminished by half the duration of the lunar eclipse gives the time of first contact: the same being increased by half the duration of the lunar eclipse gives the time of separation of the Moon from the Shadow (i.e. the time of the last contact.) (The time of the middle of the lunar eclipse) being diminished by half of the duration of totality gives the time of immersion; and the same increased by half the duration of totality gives the time of emersion. The middle of the eclipse happens to be (approximately) at the end of the Parva-tithi (i.e. at the time of opposition of the Sun and the Moon). (The minutes of) the Moon's latitude etc. being divided by 3 are reduced to angulas. (10 b-11) When the Moon's latitude is north, the first contact occurs in the east slightly deviated towards the northeast; the separation occurs in the north-west; and the middle of the eclipse in the north. (12) When the (Moon's) latitude is south, the first contact occurs in the south-east; the separation in the southwest; and the middle of the eclipse, in the south. The immersion of the Moon takes place in the west and emersion, in the east. (13). (Kripa Shankar Shukla) - vakyakaranam grahanopakaranam 16. 14. 1 a pratipatparvaghatikah kala rnadhanam ravau || 5 || Mit'alar'arcarear: famen: gaag zati 'nica ' nnyune tu bhoge syurvyatyayat sa samo ravih || 6 || candrena paurnamasyante cakrardhena samanvitah | bimbam ravergatikala 'mana 'ghna 'dhana ' bhajitah || 7 || 1 The semi-diameter of the shadow has been taken to be 40 minutes. 214 indoh ' sastra ' hrta bhuktih bimbam tanmana ' taditam | zreffha azifara a'uzi' diggi faat: || 5 || patonacandrabahujya 'stana 'ghna ksepacapakam | svasva 'putram 'sasahitam dhanarne jukamesatah || 6 || (Vakyakarana, 4. 5 b-9) -Vakyakarana The Ecliptic Elements Deduct from the Sun's longitude at sunrise as many minutes as the nadis gone in Prathama, or add as many minutes as the nadis to go for the end of the Parva. Deduct or add seconds equal to the product of the nadis and the excess in minutes of the daily motion over 60'. If the daily motion is less than 60' use the defect, and add or deduct, respectively. The longitude of the Sun at the end of the Parva is got. This will be equal to the longitude of the Moon, if the Parva is Amavasya. If it is Purnima, this will be equal to the Moon plus 6 rasis. (5 b-7 a) Multiply the Sun's daily motion in minutes by 5 and divide by 9. The angular diameter of the Sun is in minutes is got. Divide the daily motion of the Moon in minutes by 25. The Moon's angular diameter is obtained. The Moon's angular diameter multiplied by 5 divided by 2 and increased by 1, is the diameter in minutes of the shadow which hides the Moon (in the Lunar eclipse. (7 b-8) Deduct Rahu from the Moon find its bhuja and the sine of the bhuja. Multiply this by 6 and add a 21 st part of the result. This is the latitude of the Moon in minutes. This is positive (i.e. South) if Moon minus Rahu is from 6 to 12 rasis and negative (i.e., North) if from 0 to 6 rasis. (9).1 (T. S. Kuppanna Sastry-K. V. Sarma ) grahanakarma 16. 14. 1 b. grahyagrahakasamyogadalam vikseparvajitam | sistam grasangulam, grahyadadhike sakalagrahah || 10 || grahyagrahakasamyogadalavargad grahyagrahaka samyogadalavargad vivarjitat | viksepavargena padam hrtam gatyantaramsakaih || 11 || sthityardhanadikastadvad vimardardham tadantarat | sthityardhenonitam parva sparsakalah prakirtitah || 12 || samyukto moksakalah syanmadhyakalah svayam bhavet | milanakhyo bhavetkalo vimadardhena varjitah || 13 || unmilanakhyah samyuktah sarvagrase tu te ubhe | fa¿iutaifaqisifufaadimfuofaat 11 98 11 sparsamoksa, yutau yugme milanonmilane api | natyarkagrahane, ksepe natisuddhe padantaram || 15|| (Vakyakarana, 4. 10-15) 1 For worked out examples see Vakyakarana: T. S. Kuppanna Sastry-K. V. Sarma , pp. 278-279.

The circumstances of the Eclipse: General Add the angular diameters of the eclipsing and the eclipsed bodies, and divide by 2. Deduct the latitude of the Moon. The remainder is the Magnitude in minutes or angulas. (If the latitude is greater, there is no eclipse). If the remainder is greater than the eclipsed body, the eclipse is total. (10) Add the eclipsed and the eclipsing and divide by 2. Square it. Deduct the square of the latitude from this, and find the square root. Divide this by the difference of the daily motions of the Sun and the Moon, in degrees. The result are nadikas of half-duration of the eclipse. If, instead of adding the eclipsed and the eclipsing, we subtract one from the other and do the calculation, we get the half-duration of the total phase. (11-12 a) The end of the parva is the Middle of the eclipse. Deducting the half-duration of the eclipse from this, the aproximate time of the First contact or the beginning of the eclipse is got. Adding, the approximate time of the Last Contact or the ending of the eclipse is got. (12 b-13 a ) Deducting and adding the half-duration of the total phase from the end of the parva, the approximate times of Immersion and Emergence are got. (13 b-14 a) Take vinadis equal to half the latitude of the Moon. in minutes, and deduct from it 1 / 6 th of itself. Deduct these vinadis from the times of the First contact and Last contact and from the times of Immersion and Emergence, if any, if (Moon minus Rahu ) is in the odd quadrants; add if in even quadrants. The respective correct times are got. In the case of the solar eclipse, use of the Moon's latitude corrected for parallax (PCL) here. If it happens, in this case, that the latitude has changed sign by the parallax-correction, add for odd quadrants and subtract for even quadrants. 1 (14 b-15) -grahalaghavam 16. 15. 1. gatagamyadinahatadyubhakteh 'kharasa ' ptamsaviyugyuti grahah syat | tatkalabhavastatha ghatighnyah 'kharasai 'rlabdhakalonasamyutah syat || 1 || evam parvante virahnarkabaho- 'rindra 'lpamsah sambhavasced grahasya | tem'sa nighnah 'samkaraih ' 'saila ' bhakta . vyagvarkasah syat prsatkom'guladih || 2 || ' vyasusara 'gati ' svam ' so 'dig 'yugbhavedvapurusnago- ratha ferval bimbam bhukti ' yugacala ' bhajita | 1 For worked out examples, see Vakyakarana : T. S. Kuppanna Sastry-K. V. Sarma , pp. 280-81. tadapi himagobimbam trighnam ni 'jesa lavanvitam vi 'vasu ' bhavati ksmabhabimbam kilamgulapurvakam || 3 || chadayatyarkaminduvidham bhumibha chadakacchadyamanaikyakhandam kuru | taccharonam bhavecchannametadyada grahyahinavasistam tu khacchannakam || 4 || manaikyakhanda ' misu 'na sahitam dasaghnam channahatam padamatah sva 'rasa ' sahinam | globimbahrt sthitiriyam ghatikadika sya- nmarda tatha tanudalantarakhagrahabhyam || 5 || yugmahatairvyagubhujamsasamaih palaih sa dvistha sthitivirahita sahita'rkasadabhat | une vyagavitaratha'bhyadhike sthiti stah sparsantime kramagate ca tathaiva marde || 6 || tithiviratirayam grahasya madhyah sa ca rahitah sahito nijasthitibhyam | grahanamukhaviramayostu kala- viti pihitapihite svamardakabhyam || 7 || pihitahatestam sthitivihrtam tat | sacarana 'bhu ' yug grasanamabhistam || 8 || tribhayutonaravih svavidhugrahe- 'yanalavadhya itascaravahalaih | 'nagasarendu ' mitairvalanam bhavet svaravidik tvatha madhyanatacca yat || 6 || visayalabdhagrhadita uktavad valana ' maksa ' hrtam palabhahrtam | udagapagiha purvapare kramad 'rasa ' hrtobhayasamskrtiranachrayah || 10 || manaikyardhahrtat 'khapad ' ghnapihitanmulam tadasamghrayah khacchannam sadalaikayuk ca gaditah khacchannajasamghrayah | savyasavyamapagudagvalanajasamghrin pradadyacchara- sayah syad grahamadhyamanyadisi khagraso'thavasesakam || madhyacchannasamdhibhih prak ca pasca- dindorvyastam tusnagoh sparsamoksau | khagrastat khacchannapadaih pare prag- dattairindormilanonmilane stah || 12 || - Grahalaghava (Ganesa, GL, 5. 1-12) Multiply the interval that has elapsed or yet to elapse (from sunrise or any standard time) by the daily motion of the planet. Divide by 60. The result in degrees is to be added to the position of the planet, if the previous interval is yet to elapse, it should be subtracted otherwise. The result gives the position of the planet.

If the interval (gone or to elapse) is in ghatis, multiply the same by the daily motion of the planet and divide by 60. The quotient is to be added to subtracted from the position of the planet in minutes or seconds. After this correction, the true position of the planet is obtained. (1) At the end point of full or new moon, subtract the position of Rahu from that of the Sun. Find the value of the bhuja of the Sun. If it is less than 14 degrees, an eclipse will occur. In the case of the occurrence the eclipse, multiply the bhuja thus got by 11 and divide by 7. The quotient in angulas give the sara. Its direction is the same as that of the Sun minus Rahu. (2) Subtract 55 minutes from the true daily motion of the Sun (x). Take (x/5+10). This in angulas gives the diameter of the Sun. 1/74 of the daily motion of Moon gives diameter of Moon in angulas, (y). Find (3 y+3 y/11-8). This gives the diameter of Earth's shadow. (3) The Moon eclipses the Sun, and the Earth's shadow eclipses the Moon. Find the sum of the radii of the Sun and the Moon (during the solar eclipse) and that of the Moon and the Earth's shadow (during the lunar eclipse). This is termed manaikya-ardha (half-sum of the diameters). Subtract from it the sara obtained earlier. The result gives in angulas the position of the eclipsed body (x). Subtract the angular diameter of the eclipsed body from x. The result is called (kha-cchannaka or sarvagrasa or (occurring at a total eclipse). (4) Find the sum of the sara and the sum of the radii of the Sun and the Moon. Multiply the result by 10. Let it be x. Multiply x by the grasa, the position that is eclipsed. And take its square root, (y). Subtract 3/6 from it. Divide (y-y/6) by the angular diameter of Moon. The quotient gives the duration of eclipse in ghatikas. Add the sara to the difference of the radii of the Sun and Moon. Multiply by 10. Find the square of its product by khagrasa (y). Divide (y-y/6) by the angular diameter of the Moon. The result gives marda in ghatis. (This is for total eclipse). (5) The first and last points of contact When the position of the Sun with Rahu (Node) is less than 12 rasis or 6 rasis, multiply the above bhuja by 2, (x). Take the number of palas equivalent to the number of degrees in x. Subtract this value from the time, ghatis of the middle of the eclipse to get the time of first contact; by adding the time of last contact is to be had. The process should be reversed if Rahu plus Sun is more than 12 rasis. 216 The process is the same for finding the commencement and end of the eclipse. (6) The end of a tithi is the middle of the eclipse. The time of first point of contact is obtained by subtracting the sparsa-tithi from this; the time of the last point is got by adding its sthiti. The instants of the commencement and the end are obtained in the same manner by taking the nimilana and unmilana marda instead of the sthiti. (7) Eclipsed body at any time Multiply the desired time in ghatikas by the value of grasa, and divide it by the duration (stithi). Add 1° 25' to the quotient. The result in angulas gives the position of the eclipse at the desired time. (8) Valana: deflection Ayana-valana. In the case of the solar eclipse add 3 rasis to that of the Sun; in respect of the lunar eclipse subtract 3 rasis. Add the ayanarsa. Following the process to find cara, find the result by using 7, 5 and 1 as cara-khandas. That equals ayana-valana. Its direction is the same as that of the Sun plus 3 rasis (solar eclipse) or the Sun minus 3 rasis (lunar eclipse). (9) Aksa-valana Divide by 5 the hour angle of the middle of the eclipse (madhya-kala-nata) and the result is in rasis etc. As in verse 9, take 7, 5, 1 as carakhandas and repeat the process to find cara (x). Multiply x by the equinoctical shadow and divide by 5. The result gives aksa-valana. In the case of eastern hour angle it is northern and for western hour angle it is southern. The sphuta-valana is the algebraic sum of these two. Onesixth of sphuta-valana is called sphuta-valananghri. (10) The grasa multiplied by 60 is to be divided by the sum of the radii. Take the square root of the quotient. That equals grasanghri. In the case of total eclipse, replace the sum of the radii by their difference. Then kha-grasanghri (kha-channanghri) is obtained. Draw a circle with any centre, and radius equal to (the disc) of the eclipsed body. If the valana is southwards, mark a point equal to valananghri from the southern tip and to the right of the sara; reverse the process for northern valanas, i.e. mark the point to the left of the northern tip of the sara. The middle point indicates the middle of the eclipse. Total eclipse is in a direction opposite to that of the middle. In case it is not a total eclipse, the remainder after the middle is in the same direction. (11)

217 The direction of first and last points of contacts 16. ^ ECLIPSES In the case of lunar eclipse: On the eastern and western sides of the point denoting the middle eclipse, mark the points equal to the asanghri. They denote respectively the first and last points. In the case of the solar eclipse reverse the process. In the case of a total lunar eclipse: On the western and eastern sides the point denoting khagrasa, mark the points equal to the khagrasanghri. Then denote respectively the immersion and emersion of the total eclipse. (12) (V. S. Narasimban) candragrahanalekhanam - vasistha-paulisau 16. 16. 1 a. saptadasastatrimsattadddvayaliptayutonasutrena | sasinavarahusthitivrttanyekasthanani calikhya ||11|| proktasamsakalankapurvaparayasca parsvayoscapi | ayaminyo rekhastrayodasa samantarah karyah || 12 || candracchedakametad vyakhyagamyam samasato'bhihitam | grasavimardasthitayah samsthanenatra drsyante || 13 || (Varaha, Pancasiddhantika, 6. 11-13) Lunar eclipse diagram Draw three concentric circles with radii 17, 38 +17 (=55), and 38-17 (= 21), minutes of arc. These circles relate to the Moon, the duration and obscuration, respectively. (Drawing the parts of the Moon's orbit forming the path of the Moon), mark the points of first and last contacts, and also those of immersion and emergence if any. (11) Draw the diameter making an angle equal to the valana given in verses 7-8 with the ecliptic to which, (according to this siddhanta) is east-west with reference to the equator. This diameter shows the east-west of the place. Draw thirteen equally spaced lines parallel to the east-west diameter. (12) Here the graphical representation of the lunar eclipse has been described briefly, and can be understood properly only by explanation (followed by demonstration.) From this, the total duration the total obscuration, the magnitude etc., can be found by inspection. J (13). (T. S. Kuppanna Sastry) ravicandragrahanayovisesah 16. 16. 1 b. sve bhucchayaminduh sprsatyatah sprsyate na pascardhe | bhanugrahe'rkaminduh prak pragrahanam ravernatah | | 14 || (Varaha, Pancasiddhantika, 6. 14) 1 For the rationale, see Pancasiddhantika : T. S. Kuppanna Sastry : 6. 11-13. 16. 17. 1 Difference between solar and lunar eclipses In the lunar eclipse, the Moon, ( moving eastward), contacts the Earth's shadow. Therefore the 'first' contact' (occurs at the eastern limb of the Moon and so) does not occur at the Moon's western limb. In the solar eclipse, the moon meets the Sun, and therefore, (the Sun being contacted at its western limb), the first contact does not occur at the eastern limb of the Sun. (14). (T. S. Kuppanna Sastry) - aryabhatardharatrapaksah 16. 17. 1. grahyasamasavyasardhangulatulyena karkatena bhuvi | vrttatritayam krtva tasmin diksadhanam kuryat || 6 || prakprabhrtindoh pascadarkasya disah svavalanajivabhih | viksepa viparitascandrasya yathadisam savituh || 7 || suryasya pragrahane mokse sasino viparyayadvalana | deya sasino grahane mokse suryasyanulomat || 8 || vyasardhe valanajyam datva jyavat samapyate yatra | tasmanmadhyam yavadrekham nitva taya ca yatra || 6 || samyogastasmadapi manaikyardhe prthak svaviksepah | grahyo viksepagrat parilekhyo grahakamrdhena || 10 || tiksnakiranasya madhye disam daksinottaram jnatva | viksepavasattasyam valana deya viparyayacchasinah ||11|| pragrahamoksanugatam tadagrato vrttamadhyagam rekham | datva viksepamitam sutram nihsarayet pratipam tat || 12 || madhye krtva grahyam parilikhya grahakapramanena | pragrahamoksa sa digbhuparilekhe bhavatyevam || 13 || pascat pragrahane pranamokse ravibimbamadhyato bahuh | svavalanasiddhayam disi viparitah sitakaramadhyat || 14 || bhanumato bahvagradyathadisam kotiranyatha sasinah | ravisasimadhyat karnastiryakkarnagrakotiyuteh || 15 || parilekham grahyasya grahakamanena purvavat krtva | tatkalikasamsthanam nimilanonmilane caivam || 16 || (Brahmagupta, Khandakhadyaka 2.4.6-16) -Aryabhatiya Midnight System Draw by means of karkata (a kind of compass); on the ground, three concentric circles, whose radii are, respectively, equal to the radius of the obscured body, the sum of the radii of the obscuring and the obscured bodies and the trijya. ( These circles are, respectively, called grahayavrtta, samasavrtta and trijyavrtta). Then mark the directions north, south etc., in these circles. (6) In a lunar eclipse, for contact, the valanajya should be marked along the trijyavrtta, from the east point in its own direction; and for separation, from the west point in the opposite direction. In a solar eclipse, for contact,

the valanajya should be marked along the trijyavrtta from the west point in a direction opposite to its own; and for separation, from the east point in its own direction. In a lunar eclipse, the viksepa is marked along the samasavrtta in a direction opposite to its own, both for contact and separation. In a solar eclipse, the viksepa is marked along the samasavrtta in its own direction both for contact and separation. Following the above rules, mark off on the circumference of the trijyavrtta a length equal to the valanajya beginning from the east or west point, as the case may be. Join the point thus marked and the centre of the concentic circles by a straight line. (This line is called. valanasutra). Mark the point where this line cuts the samasavrtta. From this point along the circumference of the samasavrtta cut off a length equal to the viksepa (according to the rules given above). (The point thus marked is the centre of the obscuring body. With this as centre and the radius of the obscuring body as radius, describe a circle, representing the obscuring body. (The respective diagrams give the positions during contact and separation.) (7-10) For a diagram at the madhyagrahanakala, that is, at purnanta or darsanta, first mark the north and the south points in the concentric circles already drawn. In a solar eclipse, the valanajya should be marked along the trijyavrtta from the north point, if the Moon's viksepa is south. The valanajya should be marked eastward, if its direction is opposite to that of the viksepa, and westward if same. In a lunar eclipse the reverse process must be followed. Join the point thus marked and the centre of the concentric circles by a straight line. From the centre along this line cut off the length of the viksepa, in its own direction in the case of a solar eclipse, and in an opposite direction in the case of a lunar eclipse. Mark this point which is the centre of the obscuring body. With this point as centre, and the radius of the obscuring body as radius describe a circle. This represents the obscuring body. Thus one should draw the diagrams on the ground to represent contact, separation and the middle of an eclipse. (11-13) In a solar eclipse, from the centre of the Sun or grahyavrtta mark along the valanasutra a length equal to a bhuja. If the given time is between the madhyagrahanakala and the beginning of the eclipse, the length must be marked to the west; and to the east, if the time is between the madhyagrahanakala and the end of the eclipse. At the point thus marked, draw a line perpendicular to the bhuja and equal to the length of the koti, in the same direction as that of the Sun's koti. The straight line __ 218 joining the centre of the Sun to the end of the koti is called karna. With that point of intersection of the koti and the karna as centre, and with the radius of the Moon or obscuring body as radius, draw a circle. Thus is found the obscured portion of the obscured body at a given time. In the same manner the diagrams for immersion and emergence may be drawn. In a lunar eclipse, from the centre of the Moon or grahyavrtta mark along the valanasutra a length equal to the bhuja. If the given time is between the beginning of the eclipse and the madhyagrahanakala, the length must be marked to the east; and to the west, if the time is between the madhyagrahanakala and the end of the eclipse. At the point thus marked, draw a line perpendicular to the bhuja, and equal to the length of the Moon's koti, in a direction opposite to its own. The straight line joining the centre of the moon to the end of the koti is called karna. (The remaining construction is the same as that in a solar eclipse). 14-16. (Bina Chatterjee) -ATEHT: 9 16. 18. 1. grahyanagulardhavistrtya vrttam sutrena likhyate | grahyagrahakasamparkadalasanakhyena caparam || 23 || purvaparayatam sutram tanmatsyat saumyadaksinam | krtva yathadisam kendradvalanam niyate sphutam || 24 || vinyastamatsyamadhyena sutram purvapare disau | nitva tu bahyavrttantam tatah kendram samanayet || 25 || grahyamandalatadyogoh vyaktam yatropalaksyate | qurrugatat zaka Zy famiga: 11 2E 11 tulyadigvalanaksiptyorvalanam varunim nayet | anyathaindrim ravervyastam sutram tanmatsyato bahih || 27 || viksepasya vasat kendramanayet tat yathadisam | viksepam kendrato nitva bindum tatra prakalpayet || 28 || grahakanagulaviskambhadalasanakhyena khandayet | grahyabimbam tatha madhye grahakasyavatisthate || 26 || pragrasamadhyamoksanam bindunam mastakanugam | matsyadvayotthavrttam yad vartma syat grahakasya tat || 30 || sthityardhenestahinena hatva gatyantaram haret | sastya labdhakrtim yuktva viksepasya krteh padam || 31 || tannayet kendrato vartma yatra samyak tayoryutih | tattrestakalajo graso grahakardhena likhyate || 32 | (Bhaskara 1, Laghubhaskariya , 4. 23-32) -Bhaskara I Draw a circle with a thread equal in length to half the angulas of the diameter of the eclipsed body (as radius) and another (concentric circle) with a thread equal in length to half the sum of the diameters of the eclipsed and eclipsing bodies.

( Then ) having drawn (through the common centre ) the east-west line and with the help of a fish-figure the north-south line, lay off from the centre (of the circle ) the corrected valana ( for the first or last contact) accord ing to its directions. About that point draw a fish-figure (in the east-west direction). (Then) passs a thread through the middle of that fish-figure and produce it towards the east or west (as the case may be) to meet the outer and from there carry it to the centre. : The point where the junction of the circle of the eclipsed body and that (thread ) is clearly seen (in the figure) is the place where the Moon is eclipsed or is separated (from the shadow ). When the valana and the Moon's latitude (for the middle of the eclipse) are alike in direction, the valana should be laid off towards the west (from the centre); otherwise, towards the east. In the case (of the eclipse) of the Sun, it should be done reversely. (Then) through the fish-figure drawn (along the north-south direction) about that point, pass a thread and extend it beyond the fish-figure (towards the north or south), according to (the direction of) the Moon's latitude to meet the outer circle, and from there carry the thread to the centre. Then from the centre along that thread lay off the Moon's latitude in the proper direction and put there a point. (With that point as centre and) with the angulas of the semi-diameter of the eclipsing body (as radius), draw a circle cutting the disc of the eclipsed body. The portion of the eclipsed body thus cut off lies submerged in the eclipsing body. The circle which is drawn through the points (i.e., the centres of the eclipsing body) corresponding to the beginning, middle, and end of the eclipse, with the help of two fish-figures, is the path of the eclipsing body. (23-30) Phase of the eclipse for given time Multiply the difference between the (true) daily motions (of the Sun and Moon) by the sthiyardha minus the given time and divide that (product ) by 60. Then adding the square of that to the square of the Moon's latitude (for the given time ), take the square root (of that sum). (The square root thus obtained is the distance between the centres of the eclipsed and eclipsing bodies at the given time.) Lay that off from the centre so as to meet the path of (the centre of) the eclipsing body. With the meetimg point as centre and half the diameter of the eclipsing : 16. 19. 1 body and radius, draw the eclipsed portion for the given time. ( 31-32). (Kripa Shankar Shukla) - lallah 16 19 1 purvasayam pragrahah sitarasmeh - Lalla pascanmoksastigmagoranyatha to | ksepah sarve vyatyayena syurindo- ryadvad bhanoragatastadvadeva || 26 || grahyam vrttam manayogardhavrttam trijyavrttam calikhet sadhitasam | trijyavrtte sitagoh purvabhage jyavad dadyad valananyanagulani || 30 || pascadbhage tigmagoscandrabhanvoh pascad vyastanyantajanyadijani | yamyat saumyan madhyamani pradesa- danyaikasanyanayet prakpraticyoh || 31 || ksepam jnatva tigmagoranyathendo- stebhyah sutranyanayet kendrabhanji | adyadantyat sutramanaikyayoga- jjyavat ksepau sarayedadyamoksau || 32 | madhyaksepam madhyato madhyasutre ksepagrebhyo grahakardhena tebhyah | vijnayante khandite tu kramena || grahye sparso madhyamagrasamoksau || 33 || ksepagrattrayamandalaistimiyugrasyasyasthitasaktayo rajjvoryogabhuvah saratrayasirah prapyalikhenmandalam | tat syat grahakavartma kendravisrtam yuktam sruti tatsprsam krtva grahakamalikhedabhimatagrasadisamsiddhaye || 34 || mucyamanamudupe paranmukhim chadyamanamiha sakradinmukhim | samprasarya vidhivacchruti tato viddhayabhistaditamanyatha ravau || 35 || (Lalla, Sisyadhivrddhida , 5. 29-35) In a lunar eclipse, the contact takes place in the eastern portion of the disc of the Moon and separation in the western portion. The contrary is the case in a solar eclipse. The latitudes of the Moon should always be drawn in a direction contrary to their own (in the projection) of a lunar eclipse. (But in the projection) of a solar eclipse, the latitudes should be drawn in their own direction. (29) Draw three (concentric) circles, with radii respectively equal to the radius of the obscured body, the sum of the radii of the obscuring and the obscured bodies, and

the radius (trijya). Mark the directions (north, etc.) in these circles. (In the projection) of a lunar eclipse, for contact, the valana or deflection should be marked along the third circle, from the east point, in its own direction. For separation, it should be marked from the west point in a direction opposite to its own. In both the cases it should be expressed as an R sine. In a solar eclipse, for contact, the valana should be marked from the west point in a direction opposite to its own; and for separation, it should be marked from the east point in the same direction as its own. Here again, it should be expressed as an R sine. In the projection of a solar eclipse, at the time of mid-eclipse, if the Moon's latitude is north, the valana or deflection should be marked from the north point eastward, if its own direction is opposite to that of the latitude, and westward, if its own direction is the same. Again, if the Moon's latitude is south, the valana or deflection should be marked from the south point eastward, if its own direction is opposite to that of the latitude, and westwards, if its own direction is the same. In the projection of a lunar eclipse, the contrary is the case. (In each case, from the end of the valana or deflection thus marked), draw a straight line passing through the centre of the concentric circle. (This is called valanasutra). Mark the point where it cuts the circle with radius equal to the sum of the radii of the obscuring and the obscured bodies. From this point mark along the same circle the latitude for contact and separation, each expressed as an R sine. In a solar eclipse the latitudes should be drawn in their own direction and in a lunar eclipse in the opposite direction. At mid-eclipse, the latitude should be marked along the valanasutra from the centre of the (concentric) circles, (towards the valana). In each case, with the extremity of the latitude as the centre and the radius of the obscuring body as radius, describe a circle (cutting the obscured body). Thus are known the points of contact and separation and also the obscured part at the mid-eclipse. (30-33) (Mark) the three extremities of the latitudes, (at the beginnning, middle and end of the eclipse). (Draw) two fish-figures, (one passing through the first two points and the other through the last two points). Draw two lines passing through the mouth and tail of each fish-figure. (With the point of intersection of these two lines as centre) draw a circle passing through the three extremities of the latitudes. This is the path of the obscuring body. Then place the hypotenuse from the centre (of the concentric circles) just touching the path. 220 (In the projection of a lunar eclipse, when the obscured portion is increasing, that is between the beginning and middle of the eclipse), the hypotenuse must be drawn eastward. But when the obscured portion is decreasing, (that is, between the middle and end of the eclipse), the hypotenuse must be drawn westward. In the projection of a solar eclipse, the contrary is the case. With this point of intersection as the centre, draw the obscuring body. Thus is found the obscured portion at any time. (34-35). (Bina Chatterjee) - suryasiddhantah 16. 20. 1. na chedyakamute yasmatksepa grahanayoh sphutah | jnayante tatpravaksyami chedyakajnanamuttamam || 1 || susadhitayamavanau bindum datva tato likhet | saptavargangulenadau mandalam valanasritam || 2 || grahyagrahakayogardhasammitena dvitiyakam | mandalam tatsamasakhyam grahyardhena trtiyakam || 3 || yamyottara pracyampara sadhanam purvavad disam | anfuratugui quan-atentschen fauciana 11*|| yathadisam praggrahanam valanam himadidhiteh | mauksikam tu viparyastam viparitamidam raveh || 5 || valanagrannayenmadhyam sutram tadyatra samsprset | acanta aat sut faciat untfernt 11 & 11 viksepaprat punassutram madhyabindum pravesayet | tadgrahyavrttasamsparse grasamoksau vinirdiset || 7 || nityaso'rkasya viksepah parilekhe yathadisam | viparitam sasankasya tadvasadatha madhyamam || 8 || valanam pranmukham neyam tadviksepaikata yadi | | bhede pascanmukham neyam indorbhanoviparyayat || 6|| valanagrat punah sutram madhyabindum pravesayet | madhyat sutrena viksepam valanabhimukham nayet || 10|| viksepagrallikhedvrttam grahakardhena tena yat | grahyavrttam samakrantam tadgrastam tamasa bhavet || 11 || - Suryasiddhanta Since, without a projection (chedyaka), the precise (sphuta) differences of the two eclipses are not understood, I shall proceed to explain the exalted doctrine of the projection. (1) Having fixed, upon a well prepared surface, a point, describe from it, in the first place, with a radius of fortynine digits (angula), a circle for the deflection (valana). (2) Then a second circle, with a radius equal to half the sum of the eclipsed and eclipsing bodies; this is called

the aggregate-circle (samasa ) ; then a third, with a radius equal to half the eclipsed body. (3) The determination of the directions, north, south, east, and west, is as formerly. In a lunar eclipse, contact (grahana) takes place on the east, and separation (moksa) on the west; in a solar eclipse, the contrary. (4) In a lunar eclipse, the deflection (valana) for the contact is to be laid off in its own proper direction, but that for separation in reverse; in an eclipse of the Sun, the contrary is the case. (5) From the extremity of either deflection draw a line to the centre: from the point where that cuts the aggregate-circle (samasa ) are to be laid off the latitudes of contact and of separation. (6) From the extremity of the latitude, again, draw a line to the central point: in either case, where that touches the eclipsed body, there point out the contact and separation. (7) Always, in a solar eclipse, the latitudes are to be drawn in the figure (parilekha) in their proper direction; in a lunar eclipse, in the opposite direction. In accordance with this, then, for the middle of the eclipse, the deflection is to be laid off eastward, when it and the latitude are of the same direction; when they are of different directions, it is to be laid off westward: this is for a lunar eclipse; in a solar, the contrary is the (8-9) case. From the end of the deflection, again, draw a line to the central point, and upon this line of the middle lay off the latitude, in the direction of the deflection. (10) From the extremity of the latitude describe a circle with a radius equal to half the measure of the eclipsing body: whatever of the disc of the eclipsed body is enclosed within that circle, so much is swallowed up by the darkness (tamas).1 (11). (Burgess) - bhaskarah 2 16. 21. 1. grahyasurdhatrena vidhaya vrttam manaikyakhandena ca sadhitasam | bahye'tra vrtte valanam jyakavat prakcihnatah sparsabhavam himamsoh || 26 || savyapasavyam khalu yasyasaumyam mauksam tada pascimatasca deyam | ravigrahe pascimapurvataste viksepadicihnata eva sadhyam || 27 || 1 For elucidation, see Su.Si: Burgess, pp. 178-83. istaprasah sutrani kendradvalanagrasakta- nyankayanyatah sparsavimuktibanau | jyavannijabhyam valanagrakabhyam art yathasavatha madhyabanah || 28 || kendrat pradeyo valanasya sutre tebhyah prthaggrahakakhandakena | | vrttaih krtaih sparsavimuktimadhya- grasah kramenaivamihavagamyah || 26 || kendrad bhujam sve valanasya sutre saram bhujagracchravanam ca kendrat | prasarya kotisrutiyogacihnad vrtte krte grahakakhandakena || 30 || sammilanonmilanakestakala- grasasca vedya yadi vanyathami | ye sparsamuktyorvisikhagracihne tabhyam prthanamadhyasaragrayate || 31 || rekhe kila pragrahamoksamargom tayosca mane viganayya vedye | bimbantarardhena vidhaya vrttam kendre'tha tanmargayutidvaye'pi || 32 || bhubhardhasutrena vidhaya vrtte sammilanonmilanake ca vedye | margangulaghnam sthitikhandabhakta- mistam syuristangulasamjnakani || 33 || istangulanistavasat svamarge dattva ca grahakakhandavrttam | krtvestakhandam yadi vavagamyam - Bhaskara II sthulah sukhartham parilekha evam || 34 || 16. 21.1 (Bhaskara II, Siddhantasiromani , 1.5. 226-34) Draw a circle with radius equal to that of the radius of the disc of the eclipsed body and also a circle of radius equal to r+p, the sum of the radii of the eclipsed and eclipsing bodies; let directions (east etc.) be marked in the figure. In the outer circle, draw the valanajya or the R sine of the sphutavalana. In the case of the Moon, the valanajya pertaining to the moment of first contact should be marked from the east point and that pertaining to the moment of last contact should be marked from the west point. In the case of the Sun the reverse is to be done. If the valana is south, it should be marked in the clockwise direction, otherwise anticlockwise. (26-27) Having marked the valanajya in the form of an R sine, draw the line joining the centre to the top of the valanajya, i.e. to the point of intersection of the R sine with the outer

16. 21.1 circle. INDIAN ASTRONOMY - A SOURCE-BOOK The celestial latitude of the Moon is to be drawn from this top of the valanajya in the form of an R. sine again. If the latitude pertains to the moment of first contact, it should be drawn from the top of the valanajya pertaining to that moment, and if it pertains to the moment of last contact, it should be laid off from the top of the valanajya pertaining to the moment of last contact. (28) The celestial latitude pertaining to the middle of the eclipse should be drawn from the centre along the line of valanasutra or the line joining the centre to the top of the valanajya. Taking the extremities of these latitudes, circles are to be drawn with the radius of the eclipsing body to depict the eclipse at the respective moments. (29) First contact etc. The bhuja is to be laid from the centre of the Moon along its valanasutra or the line indicating the direction of the ecliptic; the latitude is to be drawn from the end of the bhuja and perpendicular to the bhuja. The hypotenuse is to be drawn from the centre of the Moon. Taking the point of intersection of the latitude (koti) and the hypotenuse, as centre, and radius equal to that of the eclipsing body, if circles be drawn, from these circles. could be known the points where totality begins and ends as well as the magnitude of the eclipse at any given moment. Or, these could be found in another way as follows. (30-31 a) Joining the upper end of the latitude of the middle moment of the eclipse to those of the first and last contacts, we have what are called the pragrahamarga and moksamarga, i.e. the path of the centre of the eclipsing body from the first contact to the middle moment to the last contact. The lengths of these paths could be computed and they could be drawn beforehand. Then, with the centre of the Moon as centre and radius equal to (pr) if a circle be drawn, it cuts the paths described above each in one point. With these points as centre and radii equal to p, if circles be drawn, they will touch the Moon's disc each in one point which are respectively the points of sammilana and unmilana. (31 b-33 a) Eclipse at any moment Let the product of the time elapsed from the moment of first contact and the length of the path of the eclipsing body traced from the moment of the first contact to the middle of the eclipse divided by the time between the moment of first contact and the middle of the eclipse, be x. Similarly, let the product of the time before the end of last contact and the path of the eclipsing body traced between the middle moment of the eclipse and the moment of last contact divided by the time between the middle moment and the moment of last contact be y. Lay off x and y units of length from the first and last points of the path of the eclipsing body along the 222 path, respectively. Then we get the points of the centre of the eclipsing body at the required moments. With these points as centre and radius p, if circles be drawn, they represent the eclipsing body. The length of the diameter of the eclipsed body shaded gives the magnitude of the eclipse called grasa. 1 ( 33 b-34) . (Arka Somayaji) --karanaratnam 16. 22. 1. sasiravivrttam lekhyam sphutabimbadalena dikcatustayavat || pragrahanamuktivalane pragaparam tannayet paridhau | anyadisindorarkagrahane parapurvayoh samanadisi || 12 || tatra ravindvorvacya sparsavimoksapradesau tau | tadvipradesamadhyadarabhyendvarkaparidhimadhyagata || 13 || rekha cottaradaksinasamadivastha samvidheyendoh | tattranyaddisi nyasenmadhyadviksepamatmadisi bhanoh | |14 krtva'trankam bhramayed grahakabimbardhasadrsasutrena | channapatite ravindvoryatha yatha cchedite pradrsyete || 15 || nabhasi ca tatha tatha te bhavatah samlaksite grahane | bimbadvayayutidalasamasutram pragaparato nayenmadhyat || 16 || pragrahanamoksabindu, sparsatvankau tadagrasthau | ankatrayadvimatsyanmukhapucchasprksutrasangame nyasya || 17|| sutnat tritayankasprgrekha ya grahako margah | sahitabimbardhasammitasutragram madhyabindutah pragvat ||18 || grahakamarga yatra sprsati tamastatra parilekhyam | parilekhyamanametacchasiparidhi yatra samsprsettatra || 16 || samcchaditau pradesau pascadevam pradrsyete | istaprasah istaghatikavihinam sthitidalamarkendubhuktivivarena || 20 || samgunya 'kharasa ' labdham tatkrtiviksepavargayutam | yattatra bhavati mulam tenonam bimbamanayogadalam || 21 || yacchesam tadgrasam viksepakalavivarjitam madhye | tanmulasadrsasutram madhyat prakpascimam nayedindoh | vyastam kheriha, patham sprsati yathastham tamo vilikhet || istena sthityardhe madhyagrasangulani sanagunya | hrtva sthitidalakalairistagrasangulam bhavati || 23 || (Deva, Karanaratna, 2. 11 b-23) - Karanaratna With half the diameter of the Moon (in the case of a Junar eclipse) or with half the diameter of the Sun (in the case of a solar eclipse), draw a circle, and furnish it with the four cardinal points. (11 b) In the case of a lunar eclipse, lay off the resultant valanas for the first and last contacts towards the east and 1 For explanation, see Siddhantasiromani : Arka Somayaji, pp. 396-402.

west respectively along the circumference in the opposite direction¹ (i.e., towards the north or south according as the valana is of south or north direction); and in the case of a solar eclipse, towards the west and east respectively, in its own direction (north or south). (And set down points there). (12) These points should be declared as the points of the first and last contacts of the Moon (in the case of a lunar eclipse) or of the Sun (in the case of a solar eclipse). Then draw lines proceeding from these two points and reaching the centre of the circle representing the Moon or Sun. (13) Also draw another line joining the north and south cardinal points. Starting from the centre, lay off along this line the Moon's latitude (for the middle of the eclipse) in the contrary direction in the case of the Moon, and in its own direction in the case of the Sun. (14) Put down a point there. Taking it as centre and the semi-diameter of the eclipsing body as radius draw a circle by revolving the compass. As is a portion of the Sun or Moon seen intercepted by the eclipsing body in the diagram, just so is the (actual) Sun or Moon seen eclipsed in the sky during the eclipse. (15-16 a) Path of the eclipsing body From the centre (of the circle) draw two lines, each equal to half the sum of the diameters of the eclipsed and eclipsing bodies, towards the east and west, one towards the point of the first contact and the other towards the point of the last contact. The extermities of these lines are the points (denoting the positions of the centre of the eclipsing body at the times) of the first and last contacts. (The point at the extremity of the Moon's latitude for the middle of the eclipse is the third point). (16 b-17 a). Now, with the help of these three points construct two fish-figures, and keeping one end of a thread at the intersection of the head and tail lines of the two fishfigures, draw a circular arc (lit. line) through the above three points: this is the path of the eclipsing body. (17 b-18 a) Now take a thread of length equal to half the sum of the diameters of the eclipsed and eclipsing bodies, and stretch it from the centre (of the circle towards the east and west), as before. Where the other extremity of this thread meets the path of the eclipsing body (towards the east or west), taking that as centre draw a circle with radius equal to that of the Shadow. The point where this circle touches the circumference of the Moon, 1 In fact, only the viksepa-valana should be of the opposite direction; the other two valanas should be of their own direction. 16. 23. 1 This is there lies the point of the first or last contact. how the points of the first and last contacts are seen afterwards (in the sky). (16 b-20 a) Obscuration at the given time (Ista-grasa) Diminish (the ghatis of) half the duration of the eclipse by the given ghatis, then multiply by the motiondifference of the Sun and Moon, and then divide (the product) by 60. Add the square of that to the square of the Moon's latitude, and take the square root (of that sum). By that (square root) diminish half the sum of the diameters of the eclipsed and eclipsing bodies. The remainder is the measure of eclipse at the given time. (The minutes of half the sum of the diameters of the eclipsed and eclipsing bodies) diminished by the minutes of the Moon's latitude give the measure of eclipse at the time of the middle of the eclipse. (20 b-22 a) Graphical representation of Ista-grasa Stretch a thread of length equal to the square root (obtained in the previous rule) from the centre towards the east or west of the Moon, or towards the west or east of the Sun (according as the given time relates to the first or second half of the eclipse), so as to meet the path of the eclipsing body as it stands. At that point draw the Shadow. (22) Method for Ista-grasa Multiply the angulas of the measure of eclipse at the time of the middle of the eclipse by the given time (elapsed since the first contact or to elapse before the last contact) in ghatis and divide (the product) by (the ghatis of) half the duration of the eclipse: the quotient gives the measure of eclipse at the given time, in terms of angulas. (23). (Kripa Shankar Shukla) surya grahanagananam alfanfagra: lambananadyah 16. 23. 1. dinamadhyamasampraptya yavatya nadika vyatita va | art: qeyforattan sunfamiaftasata 19 afcreiente: grahanakarma pancaghnat trighanaptadaksanmukhapucchyordhanarne tat | sasasicaranapamaguna dhanarnanadyo dhrtibhaktah || 2 || udagayane purvardhe dhanamrnam daksine pracyam | qrangi z uzu fayayoi atka: god 11 3:11 tu dinayatasesanadyascandrapamasamgunastvasitihrtah | Hagonfa AEuari fauridi anna: god 11 8 || agt: auchfanai fecaizi advangfaazis: 1 ugui autconta: afazi maleaasta: 11 % ||

tadvargamapasyendornavarturupacchrutirasacca | tanmulam padonam sthitikalascandrabhanvosca || 6 || (Varaha, Pancasiddhantika 7.1-6) Solar eclipse computation: -Paulisa Siddhanta Parallax of longitude Find the interval between midday and the sine of new moon, in nadis. Multiply this by 6. Degrees are got. Find its sine. Divide it by 30. The result is the parallax in nadis, to be deducted from the time of new moon is before midday, and to be added to the time of new moon, if after midday. The new moon corrected for parallax in longitude is obtained. (1) Parallax in latitude (i) Multiply the degrees of latitude by 5 and divide by 27. Add or subtract the resulting degrees, respectively, to Rahu's head or from Rahu's tail, where the moon is situated. (ii) Add three rasis to the Moon, and find its declination in degrees. This multiplied by the nadis of parallax (given by verse 1) and divided by 18, are to be added to the Head if it is forenoon, and Uttarayana (i.e., the Sun is in his northward course), or afternoon and Daksinayana. The degrees are to be subtracted from the Head, if it is forenoon and Daksi- nayana or afternoon and Uttarayana. For the Tail, the addition and subtraction should interchanged. (2-3) if (iii) Take the nadis from sunrise to new moon, forenoon, the nadis from new moon to sunset if afternoon. Multiply these by the degrees of the moon's declination, and divide by 80. The resulting degrees are to be added to the Head if the moon's longitude is between 6 and 12 rasis, and subtracted if between 0 and 6 rasis. For the Tail, interchange the addition and subtraction. (4) Computation Deduct 1° 36' from Rahu, and find the Moon Rahu, in the case of the lunar eclipse. Deduct 1° 36' from Rahu corrected (by verses 2-4), and find Moon Rahu, in the case of the solar eclipse. If the difference is less than 13° there is a lunar eclipse. If the difference is less than 8°, there is a solar eclipse; (otherwise not). (5) For the lunar eclipse, deduct the square of the difference from 169, find its square root, and take three fourths of it. This is the total duration in nadis. For the solar eclipse, deduct the square of the difference from 64, find its square root, and take three fourths of it. This is the total duration in nadis.¹ (6). (T. S. Kuppanna Sastry) 1 i.e., nadis of total duration=√169-(moon Rahu)* or √64-(moon Rahu)* respectively. Half this subtracted or added to the full moon, or parallax corrected new moon, gives the times of first and last contacts. For details see Pancasiddhantika: T. S. Kuppanna Sastry, 7. 1-6. - romaka siddhantah aftafafa: 16 24 1. dinamadhyamasamprapta yavatyo nadika vyatita va | tabhyah sadgunitabhyo jyatrimsamsastithernama || 6 || vrksepajyacapam 224 udayat prabhrti ca nadyo yah syuh praglagnamanayettabhih | tasmattu navasametadapakramamsan viniscitya || 10 || lagnavyaguvivarajya dvigunam sva ' rasam 'sasamyutamapamat | jahyad digvyatyase viksepaikye tayoryogah ||| 11|| uttaramaksacchuddham yamyam saksam ca daksinam vidyat | uttaramaksadyadadhikamuttaramevam vijaniyat || 12 || avanatih, jimbamanam ca tajjyaghni sasibhukti hrtva 'dhrtibhissataih ' smrtavanatih | madhyamamanam trimsad bhanoh sasinascatustrimsat || 13 || avanati samskrtaviksepah sphutabimbamanam grahanakalah samalipta rahuvivarajyabhyasta 'murcchana ' navahrtasca | avanatyayutavislesitasca diksamyavailomye || 14 || madhyamamanabhyasta sphutabhuktirmadhyabhuktibhakta ca | vafa thenufzuri achciti zfafguizat: 11.92 || avanativargam jahyad ravinduparimanabhogadalavargat | tanmulattu dvigunat tiyibhuktavadadiset kalam || 16 || (Varaha, Pancasiddhantika, 8. 9-16) -Romakasiddhanta : Parallax in longitude Find the interval between mid-day and the time of new moon, in nadis. Mulitply this by 6. Degrees are got. Find its sine. Divide it by 30. The result is the parallax in nadis to be deducted from the time of new moon if new moon is before mid-day, and to be added to the time of new moon if after mid-day. The new moon corrected for parallax in longitude is obtained. (9) Declination of the Nonagesimal At any time (for which the zenith distance of the nonagesimal, ZDN, is derived), find the orient ecliptic point (OEP). Add nine Signs to it. (This point is called the nonagesimal). Find its declination. (10) Subtract the Head of Rahu from the nonagesimal, find its sine, double it, and add a sixth of the quantity got by doubling, (i.e., find the latitude of the Moon supposing it to be situated at the nonagesimal). Add this to the declination found above if both are of the same direction, and subtract it from the declination if they are of different directions. (Thus the declination of the nonagesimal is corrected.) (11)

The north declination, being less, and therefore deducted from the latitude of the place, the remainder (which is the ZDN) is south. The south declination .must be added to the latitude, and the sum (forming the ZDN) is north. The part of the north declination greater than the latitude, (i.e., the remainder after deducting the latitude from the north declination, which forms the ZDN), is north. (12) Correction to the parallax and diameter of the orbit Multiply the true daily motion of the Moon by the sine of the ZDN thus found, and divide by 1800. This is the parallax correction for latitude. The mean angular diameter of the Sun is 30 minutes, and that of the Moon, 34 minutes, (according to the Romaka ) . ( 13 ) Twentyone, multiplied by the sine of (Sun or Moon at new moon Rahu) and divided by nine is the latitude. This, with the parallax correction added is the parallaxcorrected latitude, when both are of the same direction. When of different directions, their difference is the corrected latitude. (14) True diameter of the orbits The mean angular diameters of the Sun and the Moon, respectively, multiplied by their true daily motions and divided by their mean daily motions, the true angular diameters at the time of eclipse.1 (15) Moment of the eclipse Subtract the square of the parallax-corrected latitude from the square of the sum of the semi-diameters. The square root of the remainder, multiplied by two, is the number of minutes of arc giving the duration. These minutes multiplied by 60 and divided by the minutes of the relative true daily motion gives the time of duration in nadikas.2 (16) . ( T. S. Kuppanna Sastry) -saurasiddhantah ravicandrakakse 16. 25 1. bimbamanam madhyajya munikrtagunendriya 'ghnah sphutakarnah 'khakrta 'bhajito'rkasya | kakseti candrakarno 'dig ghnah kaksa sasankasya || 15 || 'svaravasumunindravisaya ' bhanoh 'khakrtartuvasugunah ' sasinah | tatkalikamanartham sphutakaksabhyam prthag vibhajet || 16 || madhyarkalambitatithe ranaksarasyudgamaih pratipamsah | prak samalipta hanih kramena pascaddhanam karyam || 17 || 1 I.e., (i) The angular diameter of the Sun = 30' X Sun's true daily motion :59. (ii) The angular diameter of the Moon=34' X Moon's true daily motion :- 791. 2 For the rationales and the working, see Pancasiddhantika : T. S. Kuppanna Sastry, 8.9-16. 17 raverdrkksepah sanakuh lambitaparvantah natih 16. 25. 1 tanmadhyavilagnakhyam tasmaccapakramamsakah kramasah | tairaksaviyutayuktairya jya madhyabhidhana sa || 18 || tithyantavilagnajya kasthantajyahata svalambahrta | madhyajyaghni vyasardhabhajita vargita sa ca || 16 || madhyajyakrtivislesitam prthak sthapya mulamekasyah | saviturdrkksepakhyam samskrtyartham prthak sthapyam || 20 | drkksopakrti jahyat trijyavargat tato'sya yanmulam | lagnarkavivaramaurvya gunitam trijyoddhrtam sakuh || 21 || sanakvangulakhyavimsatisatakrtyorantarena vislesat | sthitavarganmulam dvinavakahatam tadvibhajya kaksyabhyam || 22 bhagavisesattithivattithyantanama punah punastat syat | evam mrgyah kalastutpanno yavadavisesah || 23 || avisesad drkksepam 'vasveka ' ghnam vibhajya kaksabhyam | labdhantaracapamsa madhyajyadigvasena natih || 24 || jyavidhina viksepam tatkalam prapya tena sahitona | spasta natih pramanaih svaih svairgrasam sthitam ca vadet || 25 || avanativargam jahyad ravinduparimanabhogadalavargat | tanmulattu dvigunat tithibhuktavadadiset kalam || 26 || tithyavanamo grahanadinamavislesito yutasthityam | golanyatve deyastvavanamauksikasyaivam || 27 || - Saurasiddhanta Kaksa of the Sun and the Moon (Varaha, Pancasiddhantika, 9. 15-27) The Sun's radius vector multiplied by 5347 and divided by 40 is called its kaksa. The Moon's radius vector multiplied by 10 is its kaksa. (15) Measure of the orbit Divide 5, 14, 787 by the Sun's kaksa, and 38,640 by the Moon's, to get the respective angular diameters in minutes at the time. ( 16 ) Find the interval between midday and the moment of new moon. If the Sun is east of the meridian, (i.e., if new moon falls in the forenoon), find the degrees of right ascension corresponding to this time using the ascensional differences of zero latitude, (Lankodayathe Sun. mana), backwards from Subtract these degrees from the Sun (=Moon) of the moment of If the Sun is west of the meridian, (i.e., new moon. if new moon is in the afternoon), find the degrees corresponding to the interval counting forward from the Sun and add to the Sun ( = Moon). ( 17 ).

16. 25. 1 > INDIAN ASTRONOMY-A SOURCE-BOOK The merdian point of the ecliptic (madhya-lagna) is obtained. Find its declination, north or south. If north, find the difference between the declination and the latitude of the place. If south, add them. The sine of the result is called madhyajya, (i.e., sine zenith distance) of the point. (18) Drkksepa of the Sun Find the sine of the longitude of the Orient Ecliptic point (OEP) at new moon, multiply by the sine of maxium declination, 48' 48", and divide by the sine of the colatitude. (This is sine amplitude of OEP, called udayajya.) Multiply this by the sine of the zenith distance, (ZD) of MEP, already found, and divide by 120'. Square the result and subtract from the square of the sine ZD, of the MEP. (19) Set the remainder in two places. In one place, find its square root. This is the sine of the zenith-distance of the nonagesimal (ZD of N) called the Sun's drkksepa. Keep this aside for future work. (20) Gnomon Subtract from 14,400 the square of sin ZD of N, (kept unused in the other place in the previous work,) and find its square root. Multiply this by the sine of the distance between the Sun and the OEP, and divide by 120'. The result, which is the sine of the Sun's altitude, is called sanku, i.e., the Sun's sanku. (21) Subtract the square of the Sun's sanku obtained above from 14,400. From the remainder subtract the square of the Sun's drk-ksepa kept apart in the previous work and find its square root, technically called drggati. Multiply this by 18 and divide by each of the kaksas of the Sun and the Moon. (22) Find the respective arcs (in minutes) and get their difference. Treat this as the minutes of tithi and find the tithi-nadikas for this. Subtract the nadikas from the time of new moon if forenoon, and add, if afternoon. The parallax-corrected new moon (PCN) is determined. Repeat the operation of finding the PCN, till there is no difference (in time) in two successive operations. This is the PCN (to be used in the subsequent work.). (23) Parallax in latitude Take the sine ZD of N last obtained in the successive approximation, multiply by 18, and divide by the respective kaksas. The respective sine parallax in latitude is got. The arc of their difference is the relative parallax in latitude and its direction is that of sine ZD of MEP (i.e., of M from Z). (24) The Moon's latitude at the time taken is to be got by using the sine (of Moon Rahu), and this is to be 226 added to or subtracted from the parallax correction in latitude, (according to their direction). This is the parallax-corrected latitude. This is to be determined separately for each of the times separately, and from them the times of total obscuration and total duration are to be got. (25) Subtract the square of the parallax-corrected latitude from the square of the sum of the semi-diameters of the Sun and the Moon and find the square root. Double this, and find the time for it, treating it as the motion of tithi. (The duration of the eclipse is got.) (26) Find the nadis of parallax for the time of the beginning. If the time of beginning and the new moon are both in the forenoon or both in the afternoon, find the difference of the nadis of parallax and add it to the half duration to get the correct half duration, (to be subtracted from the time of the corrected new moon). If one is before noon and the other afternoon, add the nadis of parallax, and add it to the half-duration, to get the correct half duration (to be subtracted from the time of parallax-corrected new moon). Do the same for the sine of the end of the eclipse (to find the correct half duration to be added to the parallaxcorrected new moon, to get the correct last contact).1 (27). (T. S. Kuppanna Sastry) -aryabhatiyardharatrapaksah 16. 26. 1. vitribhalagnapakramaviksepaksamsayutivisesonat | bhavitayajjya chedastrijyardhakrteh phalena hrta || 1 || fafavarami-roitar afenifa atari go | rnamadhike dhanamune vitribhalagnattithavasakrt || 2 || ye yutivisesabhagastajjyavanatirguna trayodasabhih | 'khabdhi ' hrta viksepam krtva tatkalikasasanakat || samyogantaramavanatisasankaviksepayoh samanyadisoh | sphutaviksepah sasivat sthityardhavimardadalanadyah || 4 || pragvallambanamasakrt tithyantat sthitidalena hinayutat | tanmadhyantarayuktam sthitidalamevam vimadardham || 5 || adhike'dhikantarajyalambanamevam tadrnadhanaikatve | hine hinam bhede tadaikyayutamuktavatte ca || 6 || (Brahmagupta, Khandakhadyaka, 1. 5. 1-6) -Aryabhata's Midnight system Find the sum or difference of the kranti and viksepa of the vitribha-lagna and the latitude of the place. Subtract the result from 90° and find the jya of the remainder. Divide the square of half the trijya by this jya. Divide the jya of the difference of the longitudes of the Sun and vitribha-lagna by this result. Thus is obtained the 1 For detailed elucidation and rationale involved, see Pancasiddhantika:T. S. Kuppanna Sastry, 9. 15-27.

lb lambana in terms of ghatikas, etc. This time should be added to or subtracted from the instant of conjunction, according as the Sun is less or greater than the vitribhalagna. This process should be repeated (till the time is fixed). (1-2) The jya of the degrees, etc., in respect of the Sun or difference of the kranti and the viksepa of the vitribhalagna and the latitude of the place, multiplied by 13 and divided by 40, gives the avanati. Find the viksepa of the Moon from its longitude at the instant of conjunction. The sum or difference of the avanati and the viksepa, according as they are in the · same or different directions, gives the sphutaviksepa of the Moon. This should be used to calculate, as in the case of the lunar eclipse, the half duration in ghatikas of the solar eclipse or of the total obscuration. (3-4) As before, the lambana should be calculated by repeated process from the instant of apparent conjunction of the Sun and Moon, decreased or increased by the duration of the first or second half of the eclipse, respectively, till it is fixed. When the lambanas for the beginning and middle of the eclipse, that is the sparsalambana and the madhyalambana, are both subtractive, and the former is greater than the latter, and when both are additive, and the former is less than the latter, then their difference, when added to the duration of the first half of the eclipse, gives its corrected duration. When the sparsalambana and the madhyalambana are both subtractive, and the former is less than the latter, and when both are additive, and the former is greater than the latter, then their difference, when subtracted from the duration of the first half of the eclipse, gives its corrected duration. Again, when the sparsalambana and the madhyalambana are of different denominations, then their sum, when added to the duration of the first half of the eclipse, gives its corrected duration. In the same manner, the correct duration of the first half of the total eclipse is calculated. Similarly, one can find the correct duration of the second half of the eclipse or of the total obscurations. (5-6).1 (Bina Chatterjee) -HIGHT: 9 16. 27. 1 a. lambakabhihata trijya paramakrantisamhrta | labdham svadesasambhuto vyavacchedah prakirtitah || 1 || lanakodayanupataptanavagamya rakherasun | tithimadhyantarasubhyo hitva sodhyam gatam tatah || 2 || sese'pi yavatam santi vyutkramat tavatastyajet | bhaga liptasca purvahne madhyalagnamudahrtam || 3 || 1 For formulae involved and the rationale, see Khandakhadyaka:Bina Chatterjee, I.123-28. aparahne cayah karyo gantavyadevivasvatah | qidgtarad: howi faretu: attualetur: || 8 || (Bhaskara I, Laghubhaskariya , 5. 1-4) -Bhaskara I Specialities Multiply the radius by the R sine of the colatitude and divide by the R. sine of the (Sun's) greatest declination: the result is called the local divisor. (1) Having calculated the asus (of the right ascension) of the traversed portion of the Sun's Sign, by proportion with the right ascensions of the Sun's Sign, and (then) having subtracted them from the asus between the times of geocentric conjunction of the Sun and the Moon and midday, subtract the traversed portion of the Sun's Sign from the Sun's longitude. From the remainder, also subtract in the reverse order, as many Signs as have their right ascensions included (in the remaining asus) (as also)) the degrees and minutes (of the fraction) of a Sign, if any. The result (thus obtained) is known as the (tropical) longitude of the meridian ecliptic point in the forenoon. (When the geocentric conjunction of the Sun and the Moon occurs) in the afternoon, addition should be made of the untraversed portion of the Sun's Sign, etc. (2-4 a) From that (tropical longitude of the meridian ecliptic point) diminished by the longitude of the Moon's ascending node, calculate the celestial latitude north or south, (as the case of the Moon). (4 b). (Kripa Shankar Shukla) drkksepah 16. 27. 1 b. madhyalagnapamaksepapalajyadhanusam yutih | duggatijya lambanam grufaaca fafaanini faraqznafarania 11 QUAL madhyajiva taya ksunnam pragvilagnabhujam haret | vyavacchedena yallabdham vargikrtya visodhayet || 6 || madhyajyavargatah seso vargo drkksepasambhavah | tatkalasankuvargena yuktva tam pravisodhayet || 7 || viskambhardhakrtermulam 'ruparandhranisakaraih ' | hrtva labdhasya bhuyo'so vijneyo yo'rdhapancamaih || 8 || lambanakhyo bhavetkalo nadikadyo ravergrahe | parvanah sodhyate prahne diyate madhyato'pare || 6 || evam krtena bhuyo'pi parvana karma kalpyate | kalasya lambanakhyasya niscalatvam didrksuna || 10 || (Bhaskara I, Laghubhaskariya , 5. 5-10)

16. 27.1 b Drkksepa INDIAN ASTRONOMY-A SOURCE-BOOK Take the sum of the declination of the meridian ecliptic point and the celestial latitude (calculated from the tropical longitude of the merdian ecliptic point ), and of the (local) latitude when they are of like direction and the difference when they are of unlike directions, the directions of the remainder (in the latter case) being that of the minuend. (The R sine of the sum or difference is) the madhyajya. By that multiply the R sine of the bhuja of the tropical longitude of the rising point of the ecliptic and divide (the product ) by the (local) divisor ( defined in stanza 1 ) . Square whatever is thus obtained and subtract that from the square of the madhyajya. The remainder is the square of the R sine of the drkksepa. (5-7 a) Drggatijya Having added that (square of the drkksepajya) to the square of the R. sine of the instantaneous altitude (of the Sun), subtract that from the square of the radius: (the result is the square of the drggatijya). (7 b-8 a) Lambana To Having divided the square root thereof by 191, further divide the quotient by 4 and a half; the result in nadis is the time known as lambana in the case of a solar eclipse. It is subtracted from the time of (geocentric) conjunction if the latter occurs in the forenoon and is added to that if that occurs in the afternoon. get the nearest approximation for the lambana, (i.e., the lambana for the time of apparent conjunction of the Sun and Moon), one should similarly perform the above. operation again and again with the help of the time. of (geocentric) conjunction. (8-10). (Kripa Shankar Shukla) natih 16. 27. 1 c. drkksepajyamavislistam gatyantarahatam haret | 'khasvaresvekabhutakhyai ' rlabdhasta liptikadayah ||| 11|| tatkalasasiviksepasamyuktastulyadiggatah | bhinnadikka visesyante raveravanatih sphuta || 12 || (Bhaskara I, Laghubhaskariya , 5. 11-12) Nati Multiply the R sine of the drkksepa obtained by the method of successive approximations, 1 (i.e., multiply the R sine of the drkksepa for the time of apparent conjunction) by the difference between the daily motions. ( of the Sun and Moon) and divide by 51,570 : the result is (the nati) in minutes of arc, etc. ( 11 ) 1 While finding the nearest approximation to the lambana for the time of apparent conjunction by the method of successive approximations, the R sines of the dykkesepa and the drggati were calculated at every stage. By the R sine of the dykkesepa obtained by the method of successive approximation is here meant the value of the R sine of the dykksepa calculated at the last stage, which corresponds to the time of apparent conjunction. 228 (The nati) and the Moon's latitude for that instant. should be added if they are of like directions and subtracted if they are of unlike directions: thus is obtained the true latitude (of the Moon) in the case of a solar eclipse. ( 12 ). ( Kripa Shankar Shukla) -lallah ravigrahanasadhanani 16. 28. 1. madhyalagnam atha ravi grahanavagamodyami tadudaye vidadhita parisphutan | dinakarendunisakaravidvisascaramapaksajapancadase tithau || 1 || tithyantam rajanidalena sahitam krtva sabhardham ravi purvahne kriyate yaduktavidhina lagnam niraksodayaih | madhyahnat parato dinardharahitam krtvavasanam titheh suryam cavikrtam vidhaya sudhiyastanmadhyalagnam jaguh || 2 || madhyalagnasanakuh udayajya drkksepah lambanasuddhaparva tatkrantikasthasahitah svadiso'ksabhaga madhyahvayah syuratha bhinnadisorviyuktah | tajjya bhavettadabhidha vibhamadhyabhaga- vislesamorvyapi ca madhyavilagnasankuh || 3 || inodayattithyavasanalagnam yat svodayaistasya bhujamsamaurvi | jinamsamorvya gunita vibhakta lambajyaya syadudayabhidha jya || 4 || tam madhyajivagunitam vibhaktam vibhajyaya bahumudaharanti | tanmadhyajivabhavavargayosca drkksepamahurvivarasya mulam || 5 || tadvargatithyantajadrstijiva- vargantaram drggativargamuktam | tanmulanighnan saradrstibanan dvisthan bhajet suryasasisrutibhyam || 6 || phalantare sastigune vibhakte bhuktyantarenendusahasrarasmyoh | 'vilambanam syad ghatikadi krtva nate tithau svam vidadhita bhuyah || 7 || tithernatasya kramasinjini hata svamadhyalagnaprabhavena sankuna | 'ksamasadankabdhisarankanetra ' hrd bilambane syad ghatikadi va phalam || 8 || hrtathava drstigatih 'khasadgajai '- vilambanam tat svamrnam kramad bhavet |

229 -Lalla tithervirame parapurvabhagayo- muhustadutthamsakalah sasinayoh || 6 || khamadhyage tigmakare yada bhaved vilambanam svam vidadhita tattada | tithau raverdaksinage nisakare 16. ECLIPSES nisakarad daksinage ravavrnam || 10 || drksepe sarayugmabanagunite dvihsthe sasankenayoh karnabhyam vihrte phalantarakala madhyamsadik sa natih | drkksepah sphutabhuktijantarahatah 'khadrisurupesu ' hrd va natyendusarah samanya kakubhoryukto viyuktah sphutah || candragrahoktavidhina sthitimardakhande samsadhya tatsthitidalonayutat prthaksthat | spastat titherasakrdeva vilambanadi- naniya tatsthitile sakrdeva sadhye || 12 || pralambanam samadhikam yadi madhyamat sya- dunam ca mauksamrnasamjnitayostayosca | praggrasamunamadhikam yadi vapi mauksam syanmadhyamaddhanagayosca tadantarena || 13|| yuktam nijam sthitidalam sphutamanyathonam yogena varnadhanalambanayoryutam syat | drgksepadrggunasamatvavilambanam syat samyojayet sthitidale'sti vilambanam yat || 14 || evam vimardadalayorapi samsthitih syat tenonitadatha yutat sphutaparvano'ntat | sadhyau ca madhyasaravat prathamantyabanau grasam tathestamavagantumabhistakotih || 15 || bahuh spastasarodbhavah sthitidalaksunnah sphuto jayate sthityardhena hrtah sphutena sasivacchesasya karyo vidhih | grasat purvavadagatasca samayah ksunnah sphutenasakrt sthityardhena hrtah sphutesujanitenonah sthiteh svad dalat || (Lalla, Sisyadhivrddhida , 6. 1-16) Data for computing the solar eclipse One who intends to ascertain a solar eclipse, must first find the true longitudes of the Sun, the Moon and its Ascending Node, on the fifteenth day of the dark half of the lunar month at sunrise. ( 1 ) Meridian ecliptic point (To find the meridian ecliptic point or madhyalagna), the time when the Amavasya or new moon ends, if before midday, should be added to half the duration of the night and the true longitude of the Sun should be increased by 6 Signs. The lagna calculated from these by means of the times of rising of the Signs of the zodiac at Lanka, according to the methods given above, is 17. 16. 28. 1 called meridian ecliptic point by the wise. If the time when the Amavasya ends is after midday, it should be diminished by half the duration of the day and the true longitude of the Sun should be considered without any change and the lagna calculated. ( This, again, would be the meridian ecliptic point ) . (2) R sine altitude at the Meridian ecliptic point The latitude of the observer's station, expressed in degrees, increased or diminished by the declination corresponding to the longitude of the meridian ecliptic point, according as they are in the same or opposite direction, is called madhya. Its R sine is called madhyajya. The R sine of 90° minus the madhya is called the sanku or R sine of the altitude of the madhya. (3) R sine amplitude of the rising point of the ecliptic Calculate the lagna for the time between sunrise and the end of the amavasya, using the local times of rising of the Signs of the zodiac. Multiply the R sine of the longitude of this lagna by the R sine of 24° and divide by the R sine of the colatitude. The result is called udayajya (or R. sine of the amplitude of the rising point of the ecliptic ) . (4) Ecliptic zenith distance When the udayajya is multiplied by the madhyajya and divided by the radius, the result is called bahu or base. The square root of the difference of the squares of the madhyajya and the bahu is called drkksepajya (or R sine of the ecliptic zenith distance ). ( 5 ) Syzygy corrected for parallax in longitude The difference between the squares of the drkksepajya and that of the R sine of the Sun's zenith distance at the end of the Amavasya, is called the square of the drggatijya. Find its square root and multiply it by 525. Divide the product severally by the distances of the Sun and the Moon from the Earth. Find the difference of the two quotients. Multiply it by 60 and divide by the difference of the motions of the Sun and the Moon. The result is the parallax in longitude in ghatikas or lambana (at the mid-eclipse). It should be applied positively, (or negatively as the case may be), to the calculated time when the Amavasya ends. The parallax should be repeatedly calculated (and applied till the time is fixed). (6-7) Or, the R sine of the hour-angle at the Amavasya multiplied by the R sine altitude of the meridian ecliptic point and divided by 29,54,961, gives the parallax in ghatikas at the mid-eclipse. (8) Or, the drggatijya divided by 860 gives the parallax in ghatikas. It should be added to the calculated time

when the Amavasya ends, if the Sun is in the western hemisphere, and subtracted, if in the eastern hemisphere. (The result is the time once corrected.) The longitudes of the Sun and the Moon must be found for this corrected time by adding or subtracting the minutes resulting (from the motions according as the parallax is additive or subtractive) and hence again the parallax. This process must be repeated (till the parallax and the time. are fixed). (9) When the Sun is on the meridian, the parallax, if any, should be added to the (calculated time) when the Amavasya ends, provided that the Moon is to the south of the Sun. If the Sun is to the south of the Moon, the parallax should be subtracted. (10) Parallax in latitude Multiply the drkksepajya by 525 and divide severally by the distances of the Sun and Moon from the Earth. The difference of the results in minutes is called nati or parallax in latitude. Its direction is the same as that of the madhyajya. Or, the drkksepajya multiplied by the differences in the true motions of the Sun and Moon and divided by 51,570 gives the parallax in latitude. The latitude of the Moon increased or diminished by this parallax according as they are in the same or opposite directions, is the corrected latitude. (11) Application of the parallaxes Calculate the first and second half of the duration of the eclipse and of the total eclipse, following the method given for the lunar eclipse. From the corrected time when the Amavasya ends subtract the first and add to it the second half of the duration of the eclipse. (The results are approximately the times when the eclipse begins and ends, respectively.) Then, from these times calculate the parallax at the beginning and end of the eclipse, and apply them (in the manner given in the next two verses) to the approximately calculated half durations. Repeat the process till these times are fixed, (which are then the apparent or sphuta durations of the first and second halves of the eclipse). (12) If the parallax for the beginning of the eclipse is greater than that for the middle of the eclipse and both are subtractive, and if the parallax for the end of the eclipse is less than that for the middle of the eclipse and both are subtractive, add their differences to the approximately calculated first and second half of the duration of the eclipse. If the parallax for the beginning of the eclipse is less than that for the middle of the eclipse and both are 230 additive, and if the parallax for the end of the eclipse is greater than that for the middle of the eclipse and both are additive, then also add their differences respectively to the approximately calculated first and second half of the duration of the eclipse. If the parallax for the beginning of the eclipse is less than that for the middle of the eclipse and both are subtractive, and if the parallax for the end of the eclipse is greater than for the middle of the eclipse and both are subtractive, then, subtract their differences respectively from the first and second half of the duration of the eclipse. If the parallax for the beginning of the eclipse is greater than that for the middle of the eclipse and both are additive, and if the parallax for the end of the eclipse is less than that of the middle of the eclipse and both are additive, then also subtract their differences respectively from the first and second half of the duration of the eclipse. If the parallax for the beginning of the eclipse is different in denomination from that for the middle of the eclipse or if the parallax for the end of the eclipse is different in denomination from that for the middle of the eclipse, then always add their sums to the first or second half of the duration of the eclipse (as the case may be). The result in each case is the apparent (sphuta) half duration of the eclipse. If there is parallax when drkksepa and drggati are equal it should be added to the half duration of the eclipse. (13-14) Computation of the eclipse The same rule is applicable for calculating the first and second half of the duration of the total eclipse. When the apparent duration for the first half and the second half of the eclipse are respectively subtracted or added to the apparent time when the Amavasya ends, (the results are the apparent times for the beginning and end of the eclipse, respectively). Then calculate the Moon's correct latitude at these times in the same manner as the latitude at the mideclipse is calculated. When the obscured portion at any given time is required, calculate the Moon's correct latitude for that time, and this is the koti or perpendicular. (15) Then calculate the bahu or base (as above). Multiply it by the (approximately calculated first or second) half of the duration of the eclipse, as the case may be, using the corrected latitude, and divide by the apparent duration of the first or second half. The result is the

corrected base. The remaining process (to find the obscured portion at any given time) is the same as that in the case of the Moon. Again when the obscured portion is given, and the corresponding time is required, follow the process as given above. Then multiply this time by the apparent duration of the first or second half of the eclipse, as the case may be; and, using the corrected latitude, divide by the approximately calculated duration. The result is the more correct time. This should be subtracted from the half duration. Repeat the process till the time is fixed. 1 ( 16 ) . (Bina Chatterjee) -- mahasiddhantah ayanadrkkarma 16. 29. 1 a. prak srngonnatimukhye karmani suryagrahavinodayajau | krtva candradinam banah sadhyo'stajau pascat || 1 || dattayanajavyastajyonam gajyam sarena samgunayet | kladhajai ca hared gajyavargenayanamidam kaladiphalam || kotijyesuvadho va jadhamamabhakto'yanesu diksamye | sodhyam kharge tvasamye yojyam syadayanah khetah || 3 || (Aryabhatiya II, Mahasiddhanta , 7. 1-3) -Mahasiddhanta Parallax of longitude In the operation, (in which the projection of) the elevation of the lunar cusps is the main (part), determine the latitude of the Moon etc., after finding out (the longitudes of) the Sun and the planet, in east at sunrise and in west at sunset. (1) Diminish the radius by the versed sine (vyastajya) (of the planet in a ) given progress. Multiply (the remainder) by the latitude (sare) and by 1398, and divide by the square of the radius: the result in minutes etc. is the correction for ecliptic deviation (ayana). 2 (2). ( S.R. Sarma) Multiply the perpendicular-sine by the latitude (isu) and divide by 8455. (The quotient) is to be subtracted from (the longitude of) the planet, when the correction and the latitude are of the same direction, and added when of opposite (direction; the result) is the planet's longitude, as corrected for ecliptic deviation (ayanakheta).3 (3) . (S.R. Sarma) 4 For_elucidation and demonstration, see, Sisyadhivrddhida: Bina Chatterjee , II. 131-42 * For the rationale, see Mahasiddhanta: S.R. Sarma, II. 130-33. 3 From the previous verse, the correction for ecliptic deviation in minutes = sin gr. decl. x ( R - versed sine ) x latitude R 1398 x perpendicular-sine x latitude 1,18,19,844 perpendicular-sine x latitude 8455 aksa drkkarma 16. 29. 1 b. 16.30. la | visuvadbhasaraghatam prahrtam khete ksipencare saumye | pascad yamye jahyad vyastam pragaksakarmetat || 4 || || 4|| (Aryabhatiya II, Mahasiddhanta, 7.4) Parallax of latitude Multiply the equinoctial shadow (visuvadbha) by the latitude, and divide by 12; (the result, when the latitude is north, is to be added to the ( longitude) of) the planet ; (when the latitude is) south, is to be subtracted in the western (hemisphere. The operation is) reverse in the eastern (hemisphere). This is the operation for latitude (aksakarman) 1 (4) . (S.R. Sarma) drksamskarah 16. 29. 1 c diksamye visleso'rkendukantyorasamya aikyam tat | vyarkendujyajyahateh gamaurvyaptayamyamsah || 5 || samskrtya bhajed vyarkendujyatamsena candrabimbaghnam | paura bhaktam valanam samskaravasena dig jneya || 6 || (Aryabhatiya II., Mahasiddhanta , 7. 5-6) Application of parallax Of (the sines of) the declination of the Sun and of the Moon (take) the difference, ( when both are) of the same (direction) and the sum, (when they are) of opposite (direction) (A). Multiply the sine of the difference (in longitudes) of the Moon and the Sun by the sine of latitude and divide by the radius. By the quotient in degrees, which are south, correct (the above difference or sum, A) and divide by one sixth of the sine of the difference (in longitude of) the Moon and the Sun. (The quotient), multiplied by the diameter of the Moon and divided by 12, is the deflection ( valana ), ( whose) direction is to be known as the same as the correction above. (5-6 ). ( S.R. Sarma) - bhaskarah 2 lambananati 16. 30. 1 a. darsantakale'pi samau ravindu drasta natau yena vibhinnakaksau | kvardhocchritah pasyati naikasute tallambanam tena nati ca vacmi || 1 || darsantalagnam prathamam vidhaya nalambanam vitribhalagnatulye | ravau tadune'bhyadhike ca tat sya- devam dhanarnam kramatasca vedyam || 2 || tribhonalagnam tarani prakalpya tallagnayoryah samayo'ntare'sau | tribhonalagnasya bhaved dyuyatah sankvadyatastasya carantyakadyaih || 3 || approximately. (3) 1 For the rationale, see Mahasiddhanta: S.R. Sarma, II. 137-46. For the rationale, see Mahasiddhanta:S.R. Sarma, II. 130-36.

16. 30. la INDIAN ASTRONOMY-A SOURCE-BOOK 232 afa: tribhonalagnarkavisesasinjini 'krta ' hata vyasadalena bhajita | hatat phaladvitribhalagnasankuna trijivayaptam ghatikadi lambanam || 4 || tatsamskrtah parvavirama evam sphuto'sakrt sa grahamadhyakalah || 7 c-d || drgjyaiva ya vivibhalagnasanakoh sa eva drkksepa inasya tavat | saumye'me vivibhaje'dhike'ksat ziratsuent afetu ga aa: 11 90 11 capikrtasyasya tu samskrtasya tribhonalagnotthasarena jiva | drkksepa indornijamadhyabhukti faezinfaent fayunggal at 11 99 11 nati ravindvoh samabhinnadiktve aerazki z afa: tysta 11 92 a-b 11 tu zqczisa mun afacicHATSEATT pragvat prasadhye sthitimardakhandam | 14 c-d || (Bhaskara II, Siddhantasiromani, 1.6. 1-4, 7 c-d, 10-12 b, 14 c-d) -Bhaskara II Parallax in longitude and latitude Inasmuch as the observer who is situated on the surface of the Earth and as such elevated by the radius of the Earth from the centre thereof, perceives not the Sun and the Moon having the same longitude at the moment of conjunction, to be in the same line of sight, their height being depressed unequally having different orbits, so I proceed to elucidate what are called lambana and nati, i.e. parallax in longitude and latitude, on which account they are not in the same line of sight. (1) Compute the lagna at the moment of conjunction of the Sun and the Moon. There will be no parallax in longitude when the Sun is situated at the point called. vitribha or the point whose longitude is =L-90°, (L being the longitude of the lagna, i.e. the ascendant. which is the point of intersection of the ecliptic with the horizon). If the Sun's longitude falls short of the longitude of the vitribha or exceeds it, there will be parallax in longitude which will be positive in the former case and negative in the latter. (2) Compute the R cosine of ZV, by calculating the rising time of Atharvaveda, the kujya, dyujya, and antya pertaining to V, (as was formulated in the Triprasnadhikara), then R sin V., multiplied by 4 and divided by R, and again multiplied by R. cos ZV and divided by R again gives the parallax in longitude. (3-4) The time of the ending moment of new moon, i.e. the moment of geocentric conjunction, is to be rectified by this parallax in longitude, to get the moment of apparent conjunction by the method of successive approximation. (7 c-d) Parallax of latitude R sine ZV is called the Drkksepa of the Sun, which is considered to be north in case the northern declination of the vitribha is greater then o, the latitude, otherwise south. (10) Then the sum of ZV and the latitude of V, assuming V to be the Moon or the difference of the above two, as the case may be, according as both of them are north. or of opposite directions, gives the arc whose R sine is the drk-ksepa of the Moon. The drk-ksepas of the Sun and the Moon multiplied, respectively, by 1/15 part of their daily motions and divided by the radius R. (equal to 3438') are the parallaxes of the Sun and the Moon in latitude. The sum or difference of these parallaxes according as they are of opposite or the same direction, is the true parallex in latitude in the context of a solar eclipse. (11 b-12 b) The apparent latitude of the Moon is equal to the algebraic sum of its geocentric latitude and the parallax in latitude. From this apparent latitude are to be ealculated the sthiti-khanda and marda-khanda of the solar eclipse (by the method described in the chapter on lunar eclipse, taking the eclipsing body or grahaka to be the Moon and the eclipsed or grahya to be the Sun).1 (14 c-d). (Arka Somayaji) sparsamuktisammilanonmilanakalah 16. 30.1 b. tithyantad ganitagatat sthitidale nonadhikallambanam tatkalotthanatipusamskrtibhavasthityardhahinadhike | darsante ganitagate dhanamrnam va tadvidhayasakrj- jneyau pragrahamoksasamjnasamayavevam kramat prasphutau || 15 || tanmadhyakalantarayoh samane zyoo vaai frafaquea, a 1 darsantato mardadalonayuktat sammilanonmilanakala evam || 16 || (Bhaskara II, Siddhantasiromani , 1.6. 15-16) Sparsakala, Moksakala, Sammilanakala and Unmilanakala First compute the time called sthiti-khanda (as mentioned in the chapter on lunar eclipses). The ending moment of local Amavasya or what is called the moment of local conjunction is known as the madhyagrahakala or the moment of the middle of the eclipse. 1 For a detailed exposition and rationale of processess, see Sisi: Arka Somayaji, pp. 408-36. See also fig. 92 there.

Subtract the sthiti-khanda from the computed time of geocentric conjunction; the result will be the approximate sparsakala. This has to be rectified for parallax in longitude as well as the approximate madhyagrahakala of geocentric conjunction to obtain the local sparsakala and the local madhyagrahakala. Similarly, the mokaskala, the sammilana and the unmilanakalas are to be rectified for parallax in longitude. (15) But while effecting this correction for the parallax in longitude, the Moon's latitude also differs for the corrected time which, in turn, affects the durations of sthiti-khanda, moksa-khanda etc. Correcting the first computed sthiti-khanda, moksa-khanda etc. for this variation in the latitude, and subtracting the sthiti-khanda from the time of madhya-graha, we have better approximation for the sparsa-kala. Inasmuch as parallax in longitude, that in latitude, and the Moon's latitude vary from time to time, and the times of sparsa, madhyagraha etc. are affected by them, the process of computation proceeds by the method of successive approximation. Subtracting the rectified marda-khanda from the rectified madhyagrahakala, we have the true sammilanakala; similarly adding the former to the latter we have the true unmilanakala. (16). (Arka Somayaji) grahanakarma 16. 30. 1 c. sesam sasankankagrahanoktamatra sphutesujena sthitikhandakena | hato'tha tenaiva hrtah sphutena bahuh sphutah syad grahane'tra bhanoh || 18 | grasacca kalanayane phalam yat sphutena nighnam sthitikhandakena | sphutesujenasakrduddhrtam tat sthityardhasuddham bhavatistakalah || 16 || (Bhaskara II, SS., 1.6. 18-19) Computation of the eclipse The remaining work proceeds on the lines indicated in the chapter on 'Lunar eclipses' (i.e. the computation of the bimba-valana, bhuja, koti and the like is to be done as indicated there). The bhuja will be rectified by multiplying it by the sthiti-khanda obtained by adopting the latitude of the Moon effected by parallax in latitude and divided by the sthiti-khanda rectified for parallax in longitude. (18) Similarly, given the grasa, i.e. the magnitude of the eclipse, the result found before by verse 15 in the chapter on 'Lunar eclipses', is to be multiplied by the sthiti-khanda rectified for parallax in longitude and divided by that obtained adopting the latitude of the Moon effected by parallax in latitude, and the result so obtained being 16. 31. 1 subtracted from the sthiti-khanda, we get the ista-kala. 1 (19). (Arka Somayaji) balanam 16- 30. Id. yutayanamsodupakotisinjini 'jinam ' samaurvya gunita vibhajita || 21 || jivaya labdhaphalasya karmukam bhavecchasankayanadikkamayanam | tayoh palotthayanayoh samasayo- yuviyuktestu vibhinnakasthayoh || 22 || ya sinjini manadalaikyanighni trijyoddhrta tadvalanam sphutam syat | yairutkramajyavidhinaitaduktam samyana na te golagatim vidanti || 23 || (Bhaskara II, Siddhantasiromani , 1.5. 21 c-23) R cos x R sin w R. cos & R sin 8, where 8 is the ayanavalana. The direction of this valana is that of the hemisphere north or south in which the Moon lies. 2 (21-c 22 a) The R sine of the sum of difference of the two valanas according as they are of the same or opposite directions, multiplied by the sum of the angular radii of the Moon and Rahu, and divided by the radius gives the R sine of the Sphuta-valana. Those who said that the valana is proportional to the R. versine, do not know spherical geometry properly.3 (22 b-23). (Arka Somayaji) : ---karanaratnam 16. 31. 1. parvahardala vivarajanadyastvavisesalambanapadani | madhyalagnam trimsacchistani tada pancadasabhyo'dhikastu yada || 1 || manu-rastasvi- khavedah khasara - navapanca rasasade - kamunih | saragiri-vasumuni-navamunya- sitiratha pancasu trigunah || 2 || purvakapale hinam yutamapare lambanena parva syat | tadvisayamsena tatha candrastasmattu viksepah || 3 || visuvacchayavyangulapindam svatyamsabhagasamyutaya | saptatya hrtalabdham visuvannadyah sada yamya || 4|| dinadalaparvavisese sadgunitem'sa ravau visodhyaste | purvakapale pascad deyah tanmadhyalagnam syat || 5 || madhyavilagna jiva svasaramsona vinadikapurvaih | samadisi yuta visodhya bhinnayamadhikadig grahya || 6 || 1 For the rationale, see Siddhantasiromani :Arka Somayaji, pp. 438-45. 2 For a detailed demonstration and rationale, see Siddhantasiromani: Arka Somayaji, pp. 375-91. 3 For an exposition, see Siddhantasiromani: Arka Somayaji, pp. 391-92.

16. 31. 1 avanatih INDIAN ASTRONOMY-A SOURCE-BOOK dasabhakta tajjiva ravisasinorbhuktivivarasamgunita | hrtva 'vidyesukrte 'rlabdha'vanatih susuksmatara || 7 || viksepasyavanateh prayutirviyutih samanyadisoh | evam sphutaviksepo drkksepajyam vina'pi dhiya || 8 || samparkardhakalayastulyayam va'thava'dhikayam va | sphutaviksepavanatau sasimandalam raverna runaddhi || 6 || sthityardhasya saramsam sparse lambanavisuddhacandramasi | hitva datva mokse sasiviksepastatah karyah || 10 || samadisi valanatritayam samyojyam bhinnadisi tu vislesyam | grahaka indugrahane . rahuh, suryagrahe candrah || 11|| pragrahanamoksakalikaviksepadanayedvalanan | yutabimbardha - pragrahamoksasthityardhaliptikavivaram || 12 || vargikrtam ca sagram nijaviksepasya krtisahitam | mulam grahyatanughnam grahyagrahakasametabimbahrtam || 13 || viksepavalanametadviksepasama digasya syat | tribhavanarahitaccandrat pragrahane taih samanvitanmokse || 14 || bahujyam krtavedairhrtva''gatamayanavalanam syat | kalavinadyo dala vibhagabhaktastu rasayo jneyah ||15 rasivitayam hitva sparse mokse ca datva dik | tadvahujyam visuvacchayotthavinadikagunam krtva || 16 || 'rasakrtamunigaganendu ' bhiraptam syadaksavalanam tat | lambanantarasamyukte sthityardhe vinirdiset sphute || 17 || tabimbardhaviksepavisleso grasa ucyate | 'paksagni ' gunito grahyabimbabhaktah sphutah smrtah || 18 || grasaih saptabhirastamam 'dviku ' lavairbhagam caturtham vaded 'vedekai 'stu trtiya ' mangasasibhiscardham grhitam raveh | sarvam 'lokayamai 'stribhagarahitam padona 'mangasvibhi '- hanam svastamabhagato 'navayamai 'rindoh tathaikanvitam || (Deva: Karanaratna 3.1-19) Karanaratna The iterated lambana The nadis lying between midday and the end of the new moon tithi (parva) are the padas of the iterated lambana. When they exceed 15, they are subtracted from 30. (1) (The vinadis of the iterated lambana for 1, 2, ..., 15 padas are) : 14, 28, 40, 50, 59, 66, 71, 75, 78, 79, 80, 80, 80, 80, and 80 each multiplied by 3. (2). Application of Iterated lambana In the eastern hemisphere (i.e., in the forenoon), the parva (i.e., the time of geocentric conjunction of the Sun and Moon) should be diminished by the (vinadis of the iterated) lambana and in the western hemisphere 234 (i.e., in the afternoon ), the parva should be increased by the (vinadis of the iterated) lambana (Thus is obtained the time of apparent conjunction of the Sun and the Moon). The longitude of the Moon (for the time of conjunction should be diminished or increased in the same way by 1 / 5 of that (lambana in ghatis ). From the resulting longitude (of the Moon) should be calculated the Moon's latitude. (3) Local latitude Divide the equinoctial midday shadow (of the gnomon), in terms of vyangulas, by 70 (1+1/3); the quotient gives the (local) latitude in terms of nadis. It is always south. (4) Meridian ecliptic point The time (in ghatis) between midday and the parva (i.e., the time of conjunction of the Sun and the Moon) should be converted into degrees by multiplying it by 6. These degrees should be subtracted from the longitude of the Sun for the time of conjunction (if the Sun is) in the eastern hemisphere and added to that (if the Sun is) in the western hemisphere. The result is (the longitude of) the meridian ecliptic point. (5) Zenith distance of meridian ecliptic point Diminish the R sine of the longitude of the meridian ecliptic point by one-fifth of itself: (the result is the declination of the meridian ecliptic point, in terms of vinadis.) Take the sum or difference of this (declination) and the (local latitude in ) vinadis ( already obtained in vs. 4), according as they are of like or unlike directions: (the result is the zenith distance of the meridian ecliptic point in terms of vinadis). Its direction is that of the greater of the two. (6) Parallax in latitude Divide the result (obtained last ) by 10, and then find the R sine of the quotient. Multiply the R sine obtained by the motion-difference of the Sun and Moon and divide by 4518; the quotient is the more accurate value of the avanti. (7) Moon's true latitude Take the sum or difference of the Moon's latitude and avanati, according as they are of like or unlike directions. This is how the Moon's true latitude is obtained without using the drkksepajya, by the application of intellect. (8) Impossibility of a solar eclipse When the Moon's true latitude equals or exceeds half the sum of (the diameters of) the eclipsed and eclipsing bodies, the Moon's disc does not hide the Sun's disc. (9)

235 Moon's latitude for first and last contacts 16. ECLIPSES (To obtain the Moon's latitude for the first or last contact:) First subtract one-fifth of the semi-duration of the eclipse (in terms of vinadis) from the Moon's longitude (for the time of conjunction) corrected for lambana, in the former case, and add the same in the latter case, and then find the Moon's latitude. (10) The three valanas One should take the sum or difference of the three valanas (taking two at a time) according as they are of like or unlike directions. Rahu (i.e., Shadow) is the eclipser in the lunar eclipse and Moon in the solar eclipse. (11) Calculate the valanas (for the first and last contacts) with the help of the Moon's latitude for those times, as follows: (12 ab) Viksepa-valana Find the difference of (i) half the sum of the eclipsed and eclipsing bodies, and (ii) half the duration of eclipse towards the first or last contact, in terms of minutes. Square it and then increase it by the square of the Moon's own latitude (for that time). Multiply the square rcot of that (sum) by the diameter of the eclipsed body and divide by the sum of the diameters of the eclipsed and eclipsing bodies. This is the viksepa-valana and its direction is the same as that of the Moon's latitude. (12 cd-14 ab) Ayana-valana In the case of the first contact, subtract three Signs from the Moon's longitude, and in the case of the last contact, add three Signs to the Moon's longitude. Find the R sine of the bahu thereof and divide that by 44. What is thus obtained is the ayana-valana. (14 cd-15 ab) Aksa-valana Divide the vinadis of the hour angle by (the vinadis of) one-third of half the duration of the day: the result is (the hour angle) in terms of Signs. In the case of the first contact, subtract 3 Signs from that and in the case of the last contact, add three Signs to that. Multiply the R sine of the bahu of that by the vinadis (of the local latitude) arising from the equinoctial midday shadow and divide by 10,746: the result is the aksa-valana. (15 cd-17 ab) True semi-durations in a solar eclipse Half the semi-duration of the eclipse (towards the first contact) should be increased by the difference between the lambanas for the first contact and the middle of the eclipse; and half the semi-duration of the eclipse 16. 32. 1 (towards the last contact) should be increased by the difference between the lambanas for the middle of the eclipse and the last contact. The results thus obtained should be declared as the true values of the two semidurations of the eclipse. (17 cd) Measure of eclipse (Grasa) The difference of (i) half the sum of the diameters of the eclipsed and eclipsing bodies and (ii) the Moon's latitude (both for the time of conjunction and in terms of minutes of arc) is called the measure of eclipse. That multiplied by 32 and divided by the diameter of the eclipsed body is called the true value thereof. (18) The eight phases of a solar eclipse When the eclipsed portion (of the Sun's diameter) amounts to 7', 1/8 (of the Sun's diameter) should be declared as eclipsed; when 12, 1/4; when 14', 1/3; when 16', 1/2; when 23', 1-1/3 (=2/3); when 26', 1-1/4 (=3/4); when 29', 1-1/8 (=7/8); when 32', 1.1 (19). (Kripa Shankar Shukla) -sripatih - situfa: 16. 32. 1. ravestu parvanyatha parvakalah spasto bhavellambanasamskrtasca | sardham ghatinam tritayam trayam ca dvayam tathaika ghatika kramena ||1|| adye dvitiye ca trtiyaturye yamardhake lambanakam rnam syat | dhanam tathaika yugalam trayam ca sardhatrayam pancamakat kramena ||2|| mine'tha mese natira 'nkarama ' mita 'bdhirama ' gavi catha kumbhe | yugme mrge 'saptakara ' 'dhrti ' sca karke'tha cape natiratra yamya || 3 || simhe tatha'lau dasakam kalanam kalascatastrastulakanyayosca | saravanatyoryutirekadikke- 'nyatve'ntaram spastatarah sarah syat || 4 || 'virama ' liptapramita sarona channam ravermandalamindumanam | 'navastakhendu ' pramito'tra yukta au: ngihugad feafara 11.2 11 suryagrahe'pi dyumanevimardah syatsambhavascandramiterbahutvat | tithyantato lambanasamskrtacca fradena acutut fafarcer: 11 & 11 sparsasthitau candramasom'rka muktih sparsobjamuktaviti suryaparva | For the rationale of the several processes, see Karanaratna:Kripa Shankar Shukla, pp. 52-62.

kotidam satkaranam prasiddham srisripatih sarataram cakara || 7 | 236 (with the Moon) is the same as that of the Moon's separation. (from the Shadow). (Sripati, Dhikoti, 7.1-7) This is the situation in a solar eclipse. Sripati Time of apparent conjunction In the case of an eclipse of the Sun, the time of conjunction of the Sun and the Moon becomes apparent when lambana (i.e. correction for parallax in longitude) is applied to it. (1) In the first, second, third and fourth yamardhas, the lambana is negative and its values are 3 1/2 ghatis, 3 ghatis, 2 ghatis, and 1 ghati respectively. In the four subsequent yamardhas, beginning with the fifth, the Lambana is positive and its values are 1 ghati, 2 ghatis, 3 ghatis and 32 ghatis respectively.1 (2) Moon's true latitude When the lagna, i.e. the point of ecliptic lying on the eastern horizon, is) in Pisces or Aries, the value of nati (i.e. correction for parallax in latitude) is 39'; in Taurus or Aquarius it is 34' ; in Gemini or Capricorn, it is 27'; in Cancer or Sagittarius, it is 18' ; in Leo or Scorpio, it is 10'; and in Libra or Virgo it is 4'. The direction of nati here (i.e. in this work ) is always south. When the Moon's latitude and nati are of like directions, they are to be added together; when of unlike directions, their difference is to be taken. Whatever is thus obtained is the Moon's true latitude. (4) First and last contacts etc. 33 minutes minus the Moon's (true) latitude is the measure of the eclipse of the Sun. The Sun's diameter is (roughly) equivalent to the Moon's diameter (already stated). 1089 is here the value of the square of the sum of the diameters of the eclipsed and eclipsing bodies. The duration of the eclipse is obtained as in the case of a lunar eclipse. (5) The measure (of the diameter) of the Moon being (at times) greater ( than that of the Sun ), in the case of a solar eclipse too, total obscuration of the Sun is possible. The times (of first contact and separation, or of immersion and emersion) should be obtained with the help of the time of conjunction as corrected for lambana and the duration (of the eclipse or totality). (6) The position of the Sun's separation (from the Moon is the same as that of the Moon's (first ) contact (with the Shadow); and the position of ( the Sun's first ) contact 1 Tamardha is a unit of time which is equivalent to one-eighth of the day measured from sunrise to sunset. The first yamardha begins at sunrise and the eighth yamardha ends at sunset. This excellent Karana (i.e. calendaric work), which is entitled Dhikoti and is highly condensed is composed by Sri S 1 ipati. (7). (K.SS) -- grahalaghavam lambanasamskarah 16. 33. 1. natih, sarasca grahanakalah varnah istagrasah lagnam darsante tribhonam prthakstham tat krantyamsaih samskrto'kso natamsah | tadvidvayamso vargitasced dvikova sunset nah khanditastadyutah sah || 1 || dvayunah tribhonodayarka- sarko harah syat vislesam 'sa ' samsahinaghnasakrah | haraptah syallambanam nadikadyam tithyam svarnam vinibhe'rkadhikone || 2 || 'triku ' nighnavilambanam kalastat sahitonastithivad vyaguh saro'tah | atha sadgunalambanam lavastai- yugavitribhatah punarnatamsah || 3 || dasahrtanatabhagonahata 'stendava ' stad- rahitasadhrtiliptaih sadbhiraptasta eva | svadigiti natiretatsamskrtah so'nguladih sphuta isuramuto'tra syat sthiticchannapurvam || 4 || sthiti 'rasa ' hatiramsa vitribham taih prthakstham rahitasahitamabhyam lambane ye tu tabhyam | sthitivirahitayuktah samskrto madhyadarsah kramasa iti bhavetam sparsamuktyostu kalau || 5 || mardadevam milanonmilane sto graso nadesyo'ngulalpo ravindroh | dhumrah krsnah pingalo'lpardhasarva- grastascandro'rkastu krsnah sadaiva || 6 || istam dvighnam channaksunnam sparsantyantarnadibhaktam | rupardhenopetam vidyadiste kale'rkasya grasam || 7 || 1 _Grahalaghava Parallax in Longitude and Latitude ( Ganesa, GL, 6.1-7) Find the lagna at the end of new moon and subtract from it 90°. Keep it separate. Find the declination 1 For elacidation and rationale see GL:RCP, II, pp. 1-15.

and correct it for latitude; that gives nati in degrees (say x). (If kranti and aksa are in the same direction, take their sum; if of different signs take their difference). (1) Find the square of x/22. If this is greater than 2, subtract 2 from it. Add half of it to the square. 12. This becomes the 'divisor'. Add If (x/22) is less than 2, add 12 to it. That is the 'divisor'. Find the difference in degrees between the lagna of the Nonagesimal and the Sun. Find 1/10 th of this (x). Subtract x from 14. Multiply the remainder by x, i.e. find (14-x) x. Divide this by the 'divisor'. The result in nadikas gives the lambana. Add the lambana to the end of tithi in ghatis. If the lagna of the nonagesimal is less than that of the Sun, the lambana is to be subtracted from the time in ghatis of the end of the tithi. (2) Correction for the lambana for Sun minus Node Multiply the lambana by 13. The result is in minutes etc. Carry out the correction of this product just as in the case of the tithi, (i.e. if the lambana is positive add, if negative subtract), to the position of the Sun minus node. Convert this corrected value into sara. Multiply the lambana by 6, and add this to the lagna of the nonagesimal if the lambana is positive, (subtract otherwise). The declination (kranti) is to be calculated from this value. From the values of the kranti and aksa (declination and latitude) the corrected value of natamsa can be had. (3) Nati (parallax) and sara from natamsa Divide natamsa by 10. Let it be x. Find (18-x). x. Subtract the same from 6° 18', (y). Divide y by (18-x). x. The result in angulas gives the parallax, its direction being the same as that of natamsa. For the sara already found out, carry out the parallax correction; i.e. take the sum if the directions are the same and take the difference otherwise. Only from the sara thus corrected, is the portion of the body eclipsed and the duration of eclipse are to be calculated. (4) Times of first and last contacts Multiply the duration of the eclipse (already found) in ghatis by 6. It is thus converted into degrees. Take the lagna of the nonagesimal at the end of the tithi. Let it be x. The lagna of the nonagesimal at the first and last contacts are to be had by respectively substracting and adding the duration from and to this value x. The parallax is to be calculated separately from the two found above. 16. 34. 1 The time of first contact is found by correcting the middle of the eclipse minus the duration with the lambana of first contact, i.e. if lambana is positive, add them; if it is negative, take the difference. For the last point of contact take the middle of the eclipse plus the duration. Carry out the correction for lambana as mentioned above. (5) Immersion and Emergence When marda is multiplied by six, degrees are obtained. Repeat the same process for determining the time of immersion or emergence. If the portion eclipsed is less than one angula it is not necessary to find grasamana for both lunar and solar eclipses. (5 b) Colour of the eclipse If Moon is eclipsed partially it is smoky in colour; if half is eclipsed it is black and a full eclipse is reddish brown. The solar eclipse is always balck in colour. (6 b) Portion eclipsed at any time Double the desired time in ghatis. Multiply this by the grasamana. (x) Find the difference between the times of last and first contacts (y). Find x/y. Add to this half-angula (or 30 vyangulas), to get the portion of the body eclipsed at any desired time. (7). (V. S. Narasimban) suryagrahanale khanam - romakasiddhantah 16. 34. 1. ravisasimanadala davana tihinad bhavanti ya liptah | tanyangulani vidyad bhanoschannani candramasa || 17 || ardhenalikhya ravim datvavanati yathadisam madhyat | avanatyantaccandram vilikhed grasarthamardhena || 18 || (Varaha, Pancasiddhantika, 8. 17-18) -Romaka Siddhanta Subtract the parallax corrected-latitude for the time of parallax-corrected new moon, from the sum of the semi-diameters. The remainder in minutes are the digits of obscuration of the sun by the moon. (17) To represent the amount of obscuration graphically, draw a circle of radius equal to the semi-diameter of the Sun, measure the parallax-corrected latitude, north or south according as where the Moon is situated, and with the point marking its end as centre, draw a circle of radius equal to the Moon's semi-diameter, to represent the Moon. (The part common to both the circles is the part obscured, and its measure in digits is its width in minutes of arc). 1 (18). (T. S. Kuppanna Sastry) 1 For elucidation and worked out examples, see Pancasiddhantika: T. S. Kuppanna Sastry, 8.17-18

-saurasiddhantah apamandaladyankanam 16. 35. 1. yastya viddhangulaya vrttam parilikhya samprasarya disam | antyadyadalaikyenatha yadaparamardhena cadyasya || 1 || candrambarantaramsotkramajyaya jyam nihatya vaisuvatim | 'khaka 'samsanudayastamayodagyamyato dadyat || 2|| satrigrhasya himamsorapakramamsan yathadisam kuryat | pragaparasiddhirevam cakrad yamyottare jneye || 3 | sparsamoksabindvankanam digvyatyayena sasino viksepantad digantakam sutram | sparso dvitiyavrtte tasmadanyat sprsenmadhyam || 4|| tatsampate sparso mokso'pyevam viparyayat sadhyah | tatkalikat svabuddhaya mokso dik samvidhatavya || 5 || kalanamanagulikaranam liptayena harije trayena mesurane'ngulam bhavati | anupato'ntarasamsthe kartavyo drstiyuktartham || 6 || (Varahamihira, Pancasiddhantika, 11.1-6) -Saurasiddhanta Diagram for ecliptic etc. Using the stick-instrument with notch-marks of digits, draw the circle called the 'sum-circle', having for its radius the half-sum of the diameters converted into digits. Mark the east-west and north-south lines. Similarly, using the semi-diameter of the eclipsed body converted into digits as radius, draw the 'eclipsed body circle' concentric with the sum-circle. (1) Find the versine of the hour-angle (of the Moon at mid-eclipse), and multiply this by the tabular sine of the latitude of the observer and divide by 120. Find the arc in degrees of the resulting sine. If the hour-angle is east, lay the degrees north of the east-point, if west, south of the east-point. The east-point with reference to the equator is thus obtained. (2) Add three rasis to the Moon's longitude and find the degrees of declination of this point. If the declination is north, lay the degrees north of E', if south, south of E'. This is the east-point with respect to the ecliptic . Draw the straight line through the centre, E"OW'. E"-W" is the ecliptic east-west. By means of circles, (i.e., by drawing the perpendicular bisector) get the ecliptic north-south, viz., N"-S". (3) Marking points of contact release etc. In the case of the lunar eclipses, mark on the 'eclipsed body circle' the direction points in reverse of the points on the sum-circle. (4) On the N S line, mark the Moon's latitude at first contact (converted into digits) according to its direction, and take it (westward) to the sum-circle. Join this point on the sum-circle and the centre with a straight line. (5) Conversion of minutes into angulas In order that the graphical representation may appear as the eclipse is seen actually, the minutes of arc are to be converted into digits, at 2' per digit when the Moon is near the horizon, and at 3' per digit when it is on the tenth sign, i.e. meridian, and proportionately in between.1 (6). (T. S. Kuppanna Sastry) grahanavarnah 16. 36. 1. pragrahanante dhumrah khandagrahane sasi bhavati krsnah | sarvagrase kapilah sakrsnatamrastamomadhye || 46 || (aryabhata I, Aryabhatiya , 4.46) Colour of the eclipse At the beginning and end of its eclipse, the Moon (i.e., the obscured part of the Moon) is smoky; when half obscured, it is black; when ( just ) totally obscured, (i.e. at immersion), it is tawny; when far inside the Shadow, it is copper-coloured with blackish tinge. (46). (Kripa Shankar Shukla) 16. 36. 2. adyantayoh sa dhumrah krsnah khandagrahe'rdhato'bhyadhike | grase sa krsnatasrah sarvagrahane kapilanarnah || 17 || (Brahmagupta, Khandakhadyaka, 2.4.17) Both at the beginning and end of the eclipse, the Moon is dusky; it is dark, when the obscured portion is less than half and is of dark copper colour, when the obscured portion is greater then half; it is tawny, when it is completely obscured. (17). (Bina Chatterjee) 16. 36. 3. adyantayorbahuladhumralavanukari khandagrahe niyatamanjanapunjavarnah | grase dalat samadhike'runakrsnavarnah sarvagrahe bhavati sitakarah pisangah || 36 || (Lalla, SiDh Vr., 5.36) At the beginning and end of an eclipse, the Moon is of dense smoky colour. In a partial eclipse, it is always dark as a mass of collyrium. When the obscured portion is greater than half, it is dark red. When it is completely obscured, it is tawny. (36). (Bina Chatterjee) 16. 36. 4. svalpe channe dhumravarnah sudhamso- rardhe krsnah krsnarakto'dhike'rdhat | sarvacchanne varna uktah pisango bhanocha sarvada krsna eva || 36 || (Bhaskara II, Siddhantasiromani , 1. 5.36) 1 For the rationale and the diagram, see Pancasiddhantika : T. S. Kuppanna Sastry, 11. 1-6.

239 Colour 16. ECLIPSES When less than half the disc of the Moon is eclipsed, the colour will be what is called dhumra, i.e. of the colour of smoke; when the disc is half eclipsed, the colour is black; when more than half is eclipsed, the colour would be a blend of black and red, and, when the entire disc is eclipsed, the colour will be what is called pisanga or reddish-brown. (36). (Arka Somayaji) adarsanataya anadesyam grahanam 16. 37.1. suryenduparidhiyoge'rkastamabhago bhavatyanadesyah | bhanorbhasurabhavat svacchatanutvacca sasiparidheh || (Aryabhata I, Aryabhatiya , 4.47) Eclipses: Conditions when not to be predicted When the discs of the Sun and the Moon come into contact, a solar eclipse should not be predicted when it amounts to one-eighth of the Sun's diameter (or less) (as it may not be visible to the naked eye) on account of the brilliancy of the Sun and the transparency of the Moon. (47). (Kripa Shankar Shukla) 16. 37. 2. dvadasabhagadunam grahanam taiksnyadraveranadesyam | sodasabhagadindoh svacchatvadadhikamadesyam || 18 || (Brahmagupta, Khandakhadyaka, 2.4. 18) If the obscured portion of the Sun is less than its twelfth part, the eclipse is ignored, because the obscured portion is so small that it cannot be seen owing to the brightness of the Sun. If the obscured portion of the Moon is greater than its sixteenth part, the eclipse is considered because though the portion is small, it is visible owing to the clearness of the Moon. (18). (Bina Chatterjee) tamralekhadisu grahananirdesah -- candragrahanam - kalacurisamvat 880 16. 38. 1 tenasityadhikastavatsare sake jate dine gih pateh kartikyamatha rohinibhasamaye ratesca yamatraye | srimad- ratnanaresvarasya sadasi jyotirvidamagratah sarvagrasamanusnagoh pravadata tirna pratijnanadi || (Sarkho plates of Ratnadeva II, Kalachuri year 880: A.D. 1128, lines 23-24, Corpus Ins. Indicarum, IV, p. 427.) Inscriptional references-Lunar eclipse -Kalachuri 880 : A.D 1128 He (Padmanabha) declaring in the assembly of the illustrious Ratnadeva, in the presence of all astronomers, that when the year eight hundred increased by eighty had passed, on the day of the Lord of Speech (i.e. Thursday), on the full moon day of Karttika, during the third quarter of the night when (the Moon would be in) the constellation Rohini, there would be a complete 16. 39. 2 eclipse of the Moon, crossed the river of assertion (i.e. vindicated his prediction). ---sakasamvat 111 e 16. 38. 2. margasirsa paurnamasyam sanaiscaravare somagrahane . (Gadag stone inscription of Vira-Ballala II, Saka 1119: A.D. 1197, lines 43-44, --Saka 1119 : A.D. 1197 Indian Antiquary, 2 (1873) 298 ff.) On the occasion of a lunar eclipse on Saturday, the day of the full moon in the month of Margasirsa . . . --sakasamvat 1151 16. 38. 3. margasirsa - paurnamasyam guruvare candroparage .... | (Nagari copper plate inscription of Anangabhima III, Saka 1151 and 1152: A.D. 1230-31, line 135, Epigraphica Indica, XXVIII, pp. 235 ff.) -Saka 1151 : A.D. 1230-31 On the occasion of a lunar eclipse on Thursday, the bright half of the month of Margasirsa . . tamralekhadisu grahananirdesah -- ravigrahanam M - kalacurisamvat 322 16.39 1 etesam brahmananam • utsarpanartham asadha samvatsare caitramavasyayam jahnavimadhye catakavatasamsthitena grahoparage .. 1 (Nagardhan plates of Svamiraja, Kalachuri year 322: A.D. 580, line 14, Ep. Ind. XXVIII, pp. 1 ff). Inscriptional references: Solar eclipse -Kalachuri year 322: A.D. 580 And, to these (same) Brahmanas, while staying at the Catuka-vata village on the Ganga, on the occasion of the eclipse on the new moon day of Caitra in the year Asadha (donated, with a libation of water, according to the maximum of uncultivated land, the village Amkollika). - kalacurisamvat 404 16. 39.2. mahabaladhikrta srivasavasamadesat likhitamidam devadine- neti | sam 404 de asadha va amavasya suryagrahoparage 1 + • (Kasare plates of Allasakti, Kalachuri year 404: A.D. 662, line 30, Corpus Ins. Indicarum, IV. i, p. 110). -Kalachuri year 404 A.D. 662 This charter is written by Devadinna by the order of the mahabaladhikrta Vasava in the year four hundred and four, on the new moon day in the dark (fortnight) of Asadha, on the occasion of a solar eclipse.

16.39.3 ---sakasamvat 1113 INDIAN ASTRONOMY-A SOURCE-BOOK 16. 39. 3. sakanrpakalatitasamvatsarasatesu trayodasadhikesvekadasasu vartamana- virodhikrt samvatsarantargata jyesthamasamavasyayam aditya- vare suryagrahane . (Gadag stone inscription of Bhillana V, Saka 1113: A.D. 1192, Ep. Ind. III, pp. 219 ff.) -Saka 1113 : A.D. 1192 On the occasion of the solar eclipse on Sunday, the new moon tithi of the month of Jyestha of the year · Virodhikrt in Saka 1113 ..... - sakasamvat 1151 16. 39. 4. karkatakamavasyayam suryoparage . 240 (Nagari copper plate inscription of Anangabhika III, Saka 1151 and 1152, A.D. 1230-31), line 142, Ep. Ind., XXVIII, pp. 235 ff). -Saka 1151: A.D. 1230-31 On the occasion of a solar eclipse on the Karkataka amavasya (i.e. new moon day when the moon's in the zodiac Karkataka).