सिद्धान्तशिरोमणि: गणिताध्याय (भास्कराचार्य - ग्रहगणित, मध्यमाधिकार व स्पष्टाधिकार सटीक)

502 Verse 2. These estimates being multiplied by the difference of the radius and Śīghrakarṇa and divided by thrice the Śīghra-antyaphalajyā are to be added or sub- tracted from the mean values above given according as the Śīghrakarṇa is less or greater than the radius to give the rectified values. Three minutes of arc are to be construed as one angula in this respect. Comm. When the Śīghrakarṇa equals the radius we know that the planet is situated at the mean distance. The word planet here stands, of course, for the Mandaspaṣṭa- graha which may be roughly taken to be the mean planet, the equation of centre being small. If the Śīghrakarṇa falls short of the radius, then evidently the planet is nearer the earth than the mean position so that disc of the planet appears to be bigger. Otherwise, the planet is further and its disc appears to be smaller. It was noted that approximately there was an increase or decrease ⅓ of the magnitude of the disc by the decrease or increase of the Śīghrakarṇa by the antyaphalajyā. Hence, in between the two positions, rule of three is used to obtain the magni- tudes as follows. "If by a difference of the Śīghrakarṇa and radius equal to the antyaphalajyā, there is a difference of ⅓ of the actual magnitude, what would it be for an arbitrary difference?" The answer is d × ⅓ × 1/a = d/3a where d = Śīghrakarṇa or radius and a the antyaphalajyā. This difference is to be added or subtracted to the mean value, as the case may be. Verse 3 and first half of 4. To obtain the time of the conjunction of two planets, compute the difference of the longitudes of two planets, and divide by the difference of their daily motions. If one of the planets be retrograde, divide by the sum of the daily motions. The result gives the number of days approximately after the moment of conjunction if the slower planet has a longitude falling short of that of the quicker. If one of the planets be

503 retrograde, and if its longitude be the lesser then also the conjunction was past by the number of days computed. In the other cases the conjunction is to take place after the number of days computed. If, however, both the planets be retrograde, then if the slower of them has a longitude less than that of the quicker, then the conjun- ction is ahead, otherwise past by the number of days. Comm. Clear. In the case of one or both the planets being retrograde, the word 'slower planet' means that planet whose retrograde motion is slower and not the planet whose mean motion is slower. Latter half of verse 4 and verse 5. To rectify the moment of conjunction. Having computed the approximate time of conjun- ction, obtain the true motions of the planets pertaining to that day, rectify them for Āyana-Dṛk-Karma and following the process indicated in verse (3) above, again compute the moment of conjunction. (This will be a good approxi- mation). This conjunction will be one on the polar latitudinal circle. If Āyana-Dṛk-Karma be not done, then the conjunction will be on the circle of celestial latitude. Comm. Bhāskara says that the conjunction on the circle of polar latitude is preferred because this could be observed as there is a star at the celestial pole and the movable circle of polar latitude could be moved into a position in which the two planets could be seen situated thereupon. The method of successive approximation is self-explanatory. Bhāskara, however, adds that when the conjunction is on the circle of celestial latitude, the planets will be seen to be closer. Verse 6 and first half of verse 7. To obtain the north- south celestial latitudinal distance between two planets.

504 Having computed the number of days by which the celestial latitudinal conjunction was past or is going to take place, let the common celestial longitude of the two planets be obtained by the method of successive approxi- mation for the moment of celestial latitudinal conjunction. Let the celestial latitudes of the two planets be rectified for parallax in latitude as in the case of solar eclipse. The difference of these celestial latitudes in case they are of the same direction or the sum if, of opposite direction, gives the north-south distance of the planets with respect to the ecliptic. Having known the directions of the celestial latitudes with respect to the ecliptic, if the two celestial latitudes happen to be both south or both north, then the planet with lesser celestial latitude is said to be in the opposite direction with respect to the other; that is, suppose both have northern celestial latitudes and suppose p₁ has a smaller northern celestial latitude than p₂, then p₁ is said to be south of p₂. Similar is the case if both the celestial latitudes happen to be south. Comm. Here Bhāskara does not specify whether he is talking of conjunction on a polar latitudinal circle or on a celestial latitudiaal circle, though the previous procedure indicated by him to obtain the moment of con- junction gives preference to polar latitudinal conjunction which is more easily observable. But, here, as he prescribes rectification of the latitudes for parallax in celestial latitude, we have to construe that he is speaking of conjunction with respect to celestial longitudes alone, because, in the context of parallax, no method was indi- cated by him for parallzx in polar latitude. Latter half of verse 7 and verses 8 and 9. Case of occultation. If the north-south celestial latitudinal distance happens to be less than the sum of the angular radii of the two planets, an occultation occurs (what we say 'eclipse'

505 with respect to the Sun and the Moon, holds good with respect to occultation). In this case of occultation, we have to rectify the time of conjunction with respect to parallax in latitude also. For obtaining this parallax in longitude, let the planet which is nearer the earth be taken as the Moon and the other the Sun. But to obtain the longitude of the Vithribha, which is necessary to compute parallaxes in longitude and latitude, the lagna of the moment of conjunction is to be computed from the position of the Sun and not that of the planet assumed to be the Sun as directed above. Having obtained the parallax in longitude, the computed moment of conjun- ction is to be rectified for parallax in longitude, (if necessary by the method of successive approximation) to obtain the actual moment of apparent conjunction ie. occultation here. This procedure is worth-adopting only when the occultation in question takes place above the horizon and is observable. The north-south celestial latitudinal distance in this case of occultation corresponds to the celestial latitude of the Moon in the case of a solar eclipse. The direction of this celestial latitude is to be construed as that of the direction in which the planet near the earth is situated with respect to the other. If the planet which is nearer the earth happens to have a motion lesser than that of the other, or be retrograde, then the planet which is situated at a greater distance from the earth will be over taking the other so that the higher planet gets occulted in the eastern direction of its disc. Thus the first contact is to be known to be in the east and the last contact would be in the west; (If otherwise, the other way). Comm. Self-explanatory. Here ends the Grahayutyadhikāra. 64

506 Bhagrahayuti (Conjunction of a planet with respect to a star) The longitudes of the stars (professed to be polar). The polar longitudes of the stars from Aswini includ- ing Abhijit are as follows. R d m R d m Aswinī 0- 8- 0 Swātī 6-19- 0 Bharaṇī 0-20- 0 Visākhā 7- 2- 5 Kritticā 1- 7-18 Anūrādhā 7-14- 5 Rohiṇī 1-19-18 Jyeṣṭhā 7-19- 5 Mṛgasīrṣa 2- 3- 0 Mūlā 8- 1- 0 Ārdrā 2- 7- 0 Purvāṣādhā 8-14- 0 Punarvasū 3- 3- 0 Uttarāṣādhā 8-20- 0 Puṣyamī 3-16- 0 Abhijit 8-25- 0 Asreṣā 3-18- 0 Sravaṇam 9- 8- 0 Makhā 4- 9- 0 Dhaniṣṭhā 9-20- 0 Purvāphalgunī 4-27- 0 Satabhiṣak 10-20- 0 Uttarāphalgunī 5- 5- 0 Purvābhādrā 10-26- 0 Hasta 5-20- 0 Uttarābhādrā 11- 7- 0 Chitrā 6- 3- 0 Revatī 0- 0- 0 Comm. In the enumeration of these longitudes, Bhāskara makes three statements which we have to note. (1) That these longitudes are rectified for Āyana-Dṛk- Karma ie. that they are polar longitudes, (2) That the longitudes of Kritticā and Rohinī are less by 32′. (The longitudes shown in the table above are those incorporating this specified correction), (3) That the longitudes of Visākhā, Anūrādhā and Jyeṣṭhā are to be increased by 5′. (This correction is also incorporated in the table given above).

507 It is to be noted that these longitudes are the same given by Brahmagupta originally and copied by S'rīpati as well as the corrections indicated above. But herein a mistake was committed as clarified by Bhāskara later in the end of the chapter namely that the polar longitudes are subject to what is called Āyanavikāra though the celestial longitudes are not. We say that the celestial longitudes are not subject to such an Āyanavikāra ie. that change due to the phenomenon known as the precession of equinoxes, because the Hindu system is Nirayana ie. reckons the longitudes from the first point of Aswinī which is its zero point instead of reckoning from r. If longitudes are measured from r, as r is preceeding, the longitudes of stars will be steadily increasing all at the same annual rate of precession namely about 50-25" per year. These increasing longitudes of the modern system go by the name Sāyana longitudes. It might be thought that since the polar longitudes also are measured from Aswinī and along the ecliptic like celestial longitudes, they also don't vary like the Nirayana celestial longitudes ; but it is not so, because the effect of precession on the right ascension and declination of a star have both their effect upon the polar longitude as well as polar latitude. We cannot say that Bhāskara did not know this but we may say that the effect on the polar longitude and latitudes were construed by him as neglible and would be appreciable only in the long run. Verses 4, 5, 6. Sphutas'aras or rectified latitudes of the stars. Aswinī 10°- 0 north Ārdrā 11 - 0 south Bharaṇī 12°- 0 ,, Punarvasū 6 - 0 north Kṛtticā 4 -30 ,, Puṣyamī 0 - 0 ,, Rohinī 4 -30 south Asreṣā 7 - 0 south Mṛgas'irṣa 10 - 0 ,, Makhā 0 - 0 north

508 Purvāphalguni 12 - 0 north Purvāṣādhā 5 -20 south Uttarāphalguni 13 0 ,, Uttarāṣādhā 5 - 0 ,, Hasta 11 - 0 south Sravaṇam 30 - 0 north Chitrā 1°-45' ,, Abhijit 62 - 0 ,, Swātī 37°- 0 north Dhaniṣṭha 36 - 0 ,, Visākhā 1 -20' south Śatabhiṣak 0 -20 south Anūrādhā 1 -45 ,, Purvābhādrā 24 - 0 north Jyeṣṭhā 3 -30 ,, Uttarābhādrā 26 - 0 ,, Mulā 8 -30 ,, Revatī 0 - 0 ,, (a) These are also what were given by Brahmagupta and copied by Śrīpati, (b) Bhāskara mentions in the Golādhyāya "नक्षत्राणां स्फुटा एव स्थिरत्वात् पठिताः शराः, दृक्कर्मणा- ऽऽयनेनैषां संस्कृताश्च तथा ध्रुवाः" ie. In as much as the stars are fixed we have given their latitudes and longitudes rectified. But here, there is a point to be noted as Bhāskara has put us in a doubt namely that the rectified latitudes which he speaks of elsewhere under verse (3) Grahacchāyādhikāra, Gaṇitādhyaya are not exactly the polar latitudes spoken of here. (Ref. fig. 120). Let ♈MR be the ecliptic and ♈N be the celes- tial equator. Let S be a star, k be the pole of the ecliptic, and p be the celestial pole. Then ♈R is the celestial longitude, SR the celestial latitude which are known as the Dhruvaka and S'ara (Asphuta- s'ara). ♈M is the polar longitude [Fig. 120] and SM the polar latitude. MR is the arc connoting the Āyana- Dṛk-Karma correction, so that celestial longitude ± Āyana- Dṛk-Karma correction = polar longitude (plus or minus according as the longitude lies in the 2nd and 4th quadrants or 1st and 3rd quadrants). If the longitude just equals 0°, 90°, 180° or 270°, the correction of Āyana--Dṛk

: 509 Karma vanishes. It must be noted here that when the longitude is 0° or 180° Āyanavalana is maximum but Āyana-Dṛk-Karma vanishes. There is no ambignity here in this polar longitude because Bhāskara is unequivocal in defining this. But under verse (3) Grahacchāyādhikāra, Bhāskara means by Sphuṭaśara SL and not SM but by Sphuṭaśara here he means SM. It is ridiculous to suppose that Bhāskara did not know that B √(R²—Āyanavalana is less than β. This B √(R²—a²)/R (a=Āyanavalanajyā) he defined as Sphuṭaśara there under verse (3) cited. SR is defined by him as Asphuṭaśara or simply Śara. Now here in the commentary he makes us believe that Sphuṭaśara is SM which is the polar latitude. This confusion created by Bhāskara leads Ramaswarup the editor of Brahmagupta Siddhānta as well as one Mukhopadhyaya the author of the thesis, ‘Hindu Nakshatras’ to suppose that Bhāskara was wrong in supposing that (B/R) √(R²—a²) > β. Bhāskara could not evidently commit such a silly mistake but we must infer that in that context he called SL as the Sphuṭaśara which being added to Rn the Krānti or what is the same LN gives SN the Sphuṭakrānti or the modern declination. Rn is called by him as Asphuṭakrānti. In this context he calls SM as the Sphuṭaśara since it is Dhruvābhimukha ie. directed towards the pole. Of course, Bhāskara should not have called both SL and SM as Sphuṭaśaras but since he was deliberately defining the Sphuṭaśara as SL previously and now as SM, and since he could not commit such a glaring mistake as to construe (B/R) √(R²—a²) as greater than β, we should not rush to pronounce that Bhāskara was wrong. Only we could say that he is inconsistent to that extent. Verse 7. The polar longitudes of Agastya (Canopus) is 87° and his polar latitude is 77 south. The polar

510 longitude of Lubdhaka (Sirius) 86° and its polar latitude is 40° south. Comm. Clear. The star Agastya is considered to be important in Indian literature because the heliacal rising and setting of Agastya are directed to be noted and in fact are being noted in every panchanga even today from times.immemorial. Even Kālidāsa alludes to this heliacal rising of Agastya as inaugurating the Sarat-kāla or autumn which was reiterated by Varaha Mihira in his Bṛhat-Saṁhitā under the verse “भगवति जलधरपक्ष्मक्षपाकरा- केंक्षणे कमलनाभे, उन्मीलयति तुरङ्गमकरिरथनीराजनं कुर्यात्” ie. when Lord Vishnu whose eyes are supposed to be the Sun and the Moon and his eyelids the clouds, opens his eyes ie. on Kārtica Ekādasi the 11th day of the bright half of the lunar month of Kārtica then Kings are directed to perform Nīrājanavidhi for his horses, elephants and chariots (to start on an expedition for war). It is to be noted that this was so long ago in times of yore that Agastya used to rise at the beginning of autum. Now the star is rising about 22nd August long in advance even in the rainy season. This is on account of the effect of precession of equinoxes. Verse 8. The Iṣṭanādis for Agastya are said to be two, for Lubdhaka 2⅙, for other stars which are next in size, 2⅓ and for still smaller ones the Iṣṭanādis are to be taken still more. Comm. Iṣṭanādis ie. the time in nādis (where a nādi is equal to 24′ of time) giving the time in between the rising moments of the star and the Sun, when the star rises heliacally. In other words, let the star Agastya rise at a particular time t. If the Sun rises at t+48′, then it is the time for Agastya to rise heliacally. This again means that if the Sun sets at time T and the star at T+48′ in its diurnal motion, it is time for the star to set heliacally in the west. Similarly for the other stars. It will be noted here that stars set heliacally in the west and rise

511 heliacally in the east. This phenomenon has been long in the notice of even the most illiterate people of India from times immemorial, as they were used to get up from beds round about the time when a brilliant star rose heliacally and stood in the eastern horizon, shining for a good length of time before Sun-rise. Each star of first magnitude thus played the part of a morning star for some time, though perhaps the illiterate folk mistake them to be the same star. Especially the brightest stars of the zodiac thus play the part of morning and evening stars. The case with respect to Agastya and Lubdhaka is different in that even though they are far away from the zodiac, yet they were noticed to be morning and evening stars by virtue of their being of the first magnitude at a spot of the sky in the vicinity of which no other such brilliant stars are there. This is the reason why panchangā—Computers have been in the habit of recording in the panchangā even to-date the heliacal rising and setting of Agastya, also because the heliacal rising of this star synchronized with the setting in of S'arat-kāla or autumn. Since in India the lunar Kārtica Ekādasi, ie. the 11th day of the bright half of the lunar month roughly synchronized with 15th Nov., when the Sun rose far in the south of the horizon, this Agastya, though it be in the far south, happens to play the part of a morning star and about the day when it rose heliacally, there were no more rains and waters of the rivers stood crystal-clear as described by many a Sanskrit poet like Kālidāsa (vide the famous verse of Kālidāsa प्रसादोदयादम्भः कुम्भयोने र्महौजसः 4th canto Raghu- vams'a). The star Lubdhaka or Sirius, the dog-star as it is called in English parlance also was conspicuous as a morning and evening star, whose heliacal rising was noticed and recorded by English poets like Shakespeare and Milton. Since one nādi corresponds to 6°, two nādis correspond to 12°, degrees, which are spoken as Kālāṁs'as for the

512 heliacal rising of Agastya. They are so termed, because they indicate the Kāla or the time in between the rising of the star and the Sun which signifies the moment of its heliacal rising. Indirectly therefore these Kālāmsas indi- cate which star is of which magnitude. A star which has 12° as Kālāmsas is therefore of first magnitude ; and as the Kālāmsas increase, the magnitude also increases. It will be rembered that the higher the magnitude of a star, the fainter it will be and not the brighter as is likely to be misconstrued by lay people. Verse 9. To compute the moment of conjunction of a planet and a star. The Āyana-Dṛk-Karma is to be done as mentioned before (with respect to the planet) and the Sphutasara is to be computed to know the time of (polar latitudinal) conjunction. Comm. We are directed to use the polar longitude and polar latitude with respect to the planet because this kind of conjunction will be more conspicuous than a celestial latitudinal conjunction because the Ecliptic is far more inclined than the Equator with respect to the horizon. Verses 10, 11. The difference in the longitudes of the planet and the star, divided by the daily motion of the planet, gives the number of days approximately after or before the moment of conjunction. If the planet be retrograde, the conjunction past or future will be in the reverse ie. future or past. Comm. Let x and y be the polar longitudes of the planet and the star and let x<y. Since y is constant as the star has no motion, x has to increase to the extent of

513 y to be in a polar latitudinal conjunction. Hence (y—x) is to be covered by the planet as per its daily motion say m. So the number of days that is to elapse for conjun- ction is (y—x)/m. If x>y, then the conjunction was past by x—y/m. If, the planet be retrograde and x<y, the con- junction was past by (y—x)/m days and if x>y, the planet being retrograde, the conjunction is to take place in (x—y)/m. Note. Though Bhāskara does not mention here Asakṛt-Karma ie. method of successive approximation, it is implied because the motion of the planet differs from moment to moment as well as its Sphutas'ara. Hence having obtained the approximate moment of conjunction, compute again the true motion of the planet at that instant, as well as its Sphutas'ara and Āyana-Dṛk-Karma. The latter ie. Āyana-Dṛk-Karma is to find the polar longitude from the celestial and the Sphutas'ara is the polar latitude ie. SM of fig. 120. The Sphutas'ara is required for the purpose of finding the distance in between the planet and the star on a polar latitudinal circle and not to compute the moment of conjunction. Verses 12, 13, 14. To compute the moments of heliacal rising and heliacal setting of a star. Compute the Udayalagna and the Astalagna of Agastya and Lubdhaka doing Ākṣa-Dṛk-Karma alone. Assuming the Udayalagna to be the Sun, compute the lagna for the Iṣṭa-kāla nāḍīs (ie. after a lapse of Iṣṭanādīs after the moment) given before (namely 2 nāḍīs for Agastya and 2⅙ for Lubdhaka) which will be the longitude of the Sun, when the star (Agastya or Lubdhaka or whatever it be) rises heliacally. Thus the Udayārka is a point of the ecliptic which rises when the star rises heliacally. 65

514 Having computed the Asta-lagna of the star, taking it to be the Sun, compute the lagna in the reverse direction ie. the lagna which preceeds it by the Iṣṭa-kāla given. If this lagna be decreased by 180°, it will give the longitude of the setting Sun at the time of the heliacal setting of the star. Or again find the longitude of the point of the ecliptic which is ahead of the Astalagna which takes (60— Iṣṭanādis) to rise after the Astalagna; this longitude decreased by 180° gives the longitude of the setting Sun at the time of the heliacal setting of the star. The heliacal rising or setting takes place when the longitude of the Sun equals the longitude of the point of the ecliptic which is technically called the Udayārka or the Astārka respectively. The difference between the longitude of the Sun and that of the Udayārka or the Astārka divided by the daily motion of the Sun gives approximately the number of days that have elapsed or are to elapse for the heliacal rising or setting as the case may be, Comm. Here we are to carefully differentiate between the technical words (1) Udayalagna of the star, (2) Asta- lagna of the star, (3) Udayārka and (4) Astārka of the star. The word Udayalagna means that point of the ecliptic which rises simultaneously with the star. The word Astalagna means that point of the ecliptic which is rising when the star is setting. The word Udayārka means that point of the ecliptic which rises when the star rises heliacally. The word Astārka means that point of the ecliptic which is setting while the star sets heliacally. Thus the four points are only points of the ecliptic. Let A be the Udayalagna of a star on the circle CAB which is the ecliptic. We know that the position of the star should be above the eastern horizon, to rise heliacally, by such a distance that the time in between the rising of the Udayalagna and that point of the ecliptic which will be rising when the star rises heliacally must be the Iṣṭa- kāla nādis. Let B be a point ahead of A on the ecliptic

515 [चित्रम् : Fig. 121] such that the time in bet- ween the rising of B and the rising of A is equal to the Iṣṭa-nādis. By the time B rises, A will have gone up a little above the eastern horizon, as well as the star that has risen along with A and now the position of the star is such that the time in between its rising and the rising of B is equal to the Iṣṭa-nādis. Hence B, which is a point of the ecliptic is termed Udayārka signifying thereby that when the Sun coincides with B, the star rises heliacally. Thus the Udayārka B is ahead of the Udayalagna A and the time in between their risings is equal to Iṣṭa-nādis. Let now A' be the Astalagna ie. the point which rises when the star sets. This point will not be, generally, diametrically opposite to A because, the time between the rising and setting of a star which is given by double the rising hour-angle will be far more than half a sidereal day when the star has a large northern latitude and in the case of one having a large southern latitude, the time will be far less than half a sidereal day. From A' find C' which is behind A' such that the time in between the rising of A' and C' is equal to Iṣṭa-nādis. Then the point C diametrically opposite to C' namely C is called Astārka ie. when the Sun is at C the star sets helia- cally in the west. It is so because, when the Sun is on the western horizon setting at C, C' will be the lagna and when the star is setting A' is the lagna and as these two lagnas .differ by Iṣṭa-nādis, the time between the Sun’s setting and the star’s setting is also equal to Iṣṭa-nādis.

516 In other words the star is within the Iṣṭa-nāḍi distance from the Sun and as C is behind A by Iṣṭa-nāḍis, the setting Sun at C will be behind (ie. has a lesser longitude) the setting star by Iṣṭa-nāḍis. Hence the star sets then heliacally. Thus we see that the Udayārka and Astārka given by the points B and C are respectively in advance (ie. has a greater longitude) and behind (ie. has a lesser longitude) of the Udayalagna and Astalagna removed by an Iṣṭa-nāḍī distance. It will be noted that the arcs AC and AB are not equal though the equatorial arcs corres- ponding to them are equal. Instead of finding C' from A' by the Vilomalagna method and taking its diametrically opposite point, Bhās- kara gives an alternative namely to find C from A' by the Kramalagna method the time being (60-Iṣṭa-nāḍikas). This is evident. Bhāskara prescribes only Ākṣa-Dṛk-Karma to be per- formed here in finding the Udayalagna and Astalagna of the star because the star's polar longitude is already one in which the Āyana-Dṛk-Karma is contained. Thus from fig. 121, when the Sun-rises at A, the star rises, when the Suns rises at B the star rises heliacally and when the Sun sets at C, the star sets heliacally. Verse 15. If in the case of a star, the Astārka happens to have a longitude greater than the Udayārka on account of a very big northern latitude, that star does not set heliacally (and so the question of heliacal rising does not arise). Comm. In Fig. 121, the Astārka C happens to have a longitude less than that of B (because CAB is the dire- ction of increasing longitude ie. positive direction). At times it so happens that C lies towards the positive direction of B ie. it has a longitude greater than that of B.

517 This happens when the star has a long northern latitude and therefore the Ākṣa-Dṛk-Karma correction will be sufficiently large and the star has a small north polar distance. This may be substantiated as follows. Fig. 122 We have the formula cos h = — tan ϕ tan δ which gives the rising hour-angle of the star. When δ is nearly equal to 90—ϕ, then cos h will be nearly equal to ‘—1’ which means that the rising hour angle is nearly equal to 180°, ie. the duration of the star’s stay below the horizon will be very small. This means A' approaches A very nearly (fig. 121), for example when it is in the position A₁'. Then let C₁' be the point which is behind A' by Iṣṭa-nāḍīs so that C₁' the diametrically opposite point C₁ is far ahead of B in stead of preceeding it. This means that the star which should first set heliacally and then after a few days rise heliacally, is now in such a position that it is to rise heliacally even before setting heliacally. This is an un- tenable position. That this is not tenable may be seen otherwise. The diametrically opposite point of Astalagna ie. the point C of the ecliptic which should set along with the planet should have a longitude less than that of the

518 Astārka C₁′; but C₁′ will have now a longitude less than C which is incongruous. In such a situation, Bhāskara says, there is no question of the star setting heliacally at all. This is indeed, an ingenious mathematical presentation of a physical pheno- menon, which reflects credit to Bhāskara's genius. Verse 16. Circumpolar stars or Sadōdita stars. If a star has a northern declination greater than 90–ϕ (ie. ϕ > 90–δ) it will be always above the horizon ; also if the southern declination is greater than 90–ϕ such a star will never be seen in a northern latitude, be it Lubdhaka or Agastya or even a planet for the matter of that. Comm. (Vide fig. 122) Let S₁ be a star such that PS₁′ < PN ie. 90–δ < ϕ ie. δ > 90–ϕ. It is clear from the figure that its diurnal path is entirely above the horizon. It is called a Sadōdita star or circumpolar. Take the case of S₂. Its southern declination ie. QS₂′ is greater than the lamba (90–ϕ) ie. QS. It is evident from the figure that its diurnal path is entirely below the horizon. Bhāskara gives two examples here namely (1) where the latitude is greater than 37°, there Agastya will not be visible (having a great north-polar distance), (2) where the latitude is greater than 52°, there Abhijit is always above the horizon (having a small north-polar distance). Bhāskara adds, ‘even a planet’. This will be true, for example, in a high latitude say 89°. Suppose the southern declination of a planet is greater than 1°. On that day and for some more days also, the planet’s diurnal paths will be below the horizon as will be clear from a figure.

519 Verses 17 to 21. Ancient astronomers happened to give a list of polar longitudes and polar latitudes at a time when there were no Ayanāṁsas ie. when the zero-point of the ecliptic as taken by the Hindu Astronomers namely Aświnī coincided with the modern zero-point namely r. In fact these polar longitudes and latitudes do change if there be Ayanāṁsas. Here in this case from the polar latitudes the celestial latitudes are to be computed in a reverse process; with the half of these celestial latitudes, the Āyana-Dṛk-Karma is to be effected in the reverse process to obtain the celestial longitudes. After having obtained the correct celestial longitudes and celestial latitudes, now, bringing the Ayanāṁsas into the picture, compute the correct Dṛk-Karma and also rectify the celestial longitudes, to obtain the correct polar longitudes and polar latitudes to compute the moment of polar latitudinal conjunction. This case may be taken when the Ayanāṁsas are large, otherwise, a small difference there will be, (which does not matter). Comm. Bhāskara·gives the process in the course of the commentary. We have the formula Asphuṭa Vikṣepa × Yaṣṭi Sphuṭa Vikṣepa = ——————————————————————— Radius Here we know Sphuṭa Vikṣepa. At once, we could not Sphuṭa Vikṣepa × Radius say Asphuṭa Vikṣepa = ——————————————————————— Yaṣṭi computing Yaṣṭi from the modern Ayanāṁsas. Yaṣṭi is a function of declination too, because the Āyanavalana is a function of declination. We know, the declinations change in the wake of precession of the equinoxes. So, there is no point in taking the value of the present Yaṣṭi. We should compute it for the Āyanaśūnyakāla or for the time when Aświnī coincided with r. Then the formula could be applied. Then with this celestial latitude obtained, Āyana-Dṛk-Karma is to be effected to obtain the present

520 polar longitude, using the modern Ayanāṁsas. Since in this process, we have no definite knowledge of the then celestial longitude ie. of the Āyanasūnyakāla, the method of successive approximation is appealed to. But this method of Bhāskara may be modified to an easier pro- cess as follows. Let rL, LS be the polar longitude and polar lati- tude of a star as given by Acharyas in whose time the Hindu first point of the zodiac coincided with r. It is required to find rN and NS the celestial longitude and latitude which hold good even today becaus rN and NS are the same as AN, AS, A being the first point of Aświni and a celestial longitude measured from A along the ecliptic is not subject to change on account of precession of the equinoxes as well as the celestial lati- tude. From the spherical triangle rLM, cot rLM = cos rL tan ω (1) and from the triangle SNL, tan LN = cos SL̂N × tan SL = cos rLM tan SL (4) and sin SN = sin SL × sin SLN = sin SL × sin rLM (5). From (1) the angle rL̂M is obtained ; substituting this value in (2) and (3) LN and SN are obtained. Adding LN to rL we have rN. Thus the celestial longitude and latitude are found far more easily and more accurately than from the laborious method indicated which gives only approximate results. Here ends the Bhagrahayutyadhikāra.

PĀTĀDHYĀYA Verse 1. Even scholars get confused while computing the occurence of a Pāta, unable to know whether it has already occurred or is going to occur. So, I seek to clarify the method of computing the moment of occurrence of a pāta. Comm. (1) There are what are called pātās, two in number, called Vyatipāta and Vaidhṛti. The first is defined as occurring at that moment when the Sun and Moon have equal declinations, being situated in opposite Ayanas but the same goḷa. The words Uttaragoḷa and Dakṣiṇagoḷa are used by Hindu Astronomers as the northern and southern halves of the celestial sphere on either side of the celestial equator, respectively; where as the words Uttara-Ayana and Dakṣiṇa-Ayana are used to connote the times when the Sun or Moon have tropical longitudes (measured from r instead of Aswini, the zero point of the Hindu Zodiac) one lying between Capricorn and Cancer, and the other lying between Cancer and Capricorn. Thus Vyatipāta occurs when the Sun and Moon each lies in one of the first or second quadrants of the ecliptic measured from r or each lies in one of the third or fourth quadrants of the ecliptic; (both should not lie in the same quadrant) and have equal declinations. The second Pāta Vaidhṛti occurs when the Sun and Moon have equal declinations, they being situated in the same Ayana but opposite goḷas. These two moments are considered to have malefic effects on humans and the world of life. Though the moments are to be computed by a knowledge of spherical astronomy, they have only an astrological significance. A chapter, usually the last, has been devoted to this subject in every book of Hindu Astronomy. The method of computation was felt difficult 66