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

542 zero ?" The answer is t (a + b) / Adya - Anya which gives the time after which Vaidhṛti is likely to occur. Suppose a and b the declinations observed at a particular moment, are one north and and the other south. Compute their difference a - b; after a few ghatis or days again from the difference of the declinations c and d ie. c - d. If c - d < a - b then Vaidhṛti is going to occur, otherwise it has elapsed. This is so because the difference has to vanish for Vaidhṛti to occur. Viewing the situation algebraically, ie. construing northern declination to be positive and southern negative, we could have laid down only one criterion, instead of separating the issues and stipulating separate criteria. Proceeding on this basis, suppose the the Sun and the Moon are one in the first quadrant and the other in the second. Here both declinations are therefore positive. Compute their difference a - b; compute again the difference of c and d which are again northern declinations observed after a few ghatis or days. If c - d < a - b then Vyatīpāta is going to occur. Now take a case when one of a and b is north and the other south ; so also c and d. We are talking of a, c to be in the first quadrant and b, d in the fourth quadrant. Even now if c - d < a - b Vyatīpāta is going to occur. Thus there is no need to stipulate "Sum or difference". Difference alone counts now, because difference of one positive and the other negative quantities means sum only. Thus in the case of Vyatīpāta difference should tend to zero, whether both the declinations are north, or one north and the other south. Similarly in the case of Vaidhṛti the sum should tend to zero because the algebraic sum of one positive northern declination and the other negative southern declination

543 shall be zero, if the declinations were to be equal and of opposite directions. Here ends the Pātādhyāya. Note Bhāskara gives two examples in the course of the commentary, one, to show the fallacy in asserting that there would be a pāta when the Sun is in an even quadrant, the Moon in an odd quadrant and the Moon's declination is less than that of the Sun; and the other to show the fallacy in asserting that the pāta has elapsed when it is still to take place, and that it is still to occur when it has already elapsed. He quotes the example given in his Golādhyāya wherein the longitudes of the Sun, Moon and Rāhu are 120°, 60° and 180° respectively, there being no Ayanāṁśas. (Since at the time of Lalla, there were no Ayanāṁśas, Bhāskara gives such an example), Since the Sun is in an even quadrant and the Moon in an odd quadrant, Bhāskara says, that as per the verse of Lalla ‘सूर्यापमात् ओजपदोद्भवात् etc.’ there should be a pāta. But since the longitude of Rāhu is 180°, Ketu is at the equinoctial point so that the lunar orbit lies between the ecliptic and the equator. Hence the declination of the Moon remains smaller than that of the Sun for a good length of time never equalling the declination of the Sun. This, Bhāskara says, is a fallacy regarding ‘भावाभावत्व’ ie. ‘Is there a pāta or not’ because when there is actually no pāta, Lalla's criterion leads to assert that there is one. It may be reiterated here, that Lalla gave a general criterion ignoring the latitude of the Moon ie. supposing the Moon to be moving on the ecliptic. Bhāskara takes the latitude also into consideration and proves that there is no pāta. In the second example, which Bhāskara gives, he tries to show the fallacy with respect to “गतेष्यत्व” ie. asserting that a pāta has elapsed when it is still to occur, and that it is to occur when it has already elapsed.

544 In this example the declinations of the Sun and Moon are respectively 1416′, 1324′, when their tropical longitudes are 80° and 100°. The Moon is in an even quadrant, and his declination is less than that of the Sun. As per the criterion of Lalla given in the verse “सूर्यापमात् etc.” the pāta is not there at all. But Bhāskara computes and shows that the pāta has occured 70 nādīs before (Rāhu having a reverse tropical longitude of 100°). We shall show here, how using differential calculus we could cut short the process. We have sin δ = sin λ sin ω ; differentiating cos δ Δδ = cos λ δλ sin ω or Δδ = (sin ω cos λ / cos δ) Δλ = Āyanavalana × Variation in Bhuja. This could be seen graphically also from fig. 129. Let P and Q be two contiguous positions of the Sun (say) on the ecliptic when arc pQ = Δλ. Let the increment in his Fig. 129 declination in going from P to Q be NQ equal to Δδ. Taking PNQ to be a plane Δ Δδ = Δλ sin v where v̂ = QP̂N. We have named this angle to be v, because it is no other than the Āyanavalana which was formulated to be equal to (sin ω cos λ / cos δ) . Thus Δδ = Δλ × Āyana- valanajyā.

APPENDIX LIST OF TECHNICAL TERMS

1. Adhikamāsa : A month gained by the lunar reckoning over the solar. This is located in that lunar month which does not contain a Saṁkramaṇa,
2. Agrajyā : The Hindu sine of the arc of the horizon in between the rising point of the Sun and the east point.
3. Akṣa (or) Pala : Latitude (Terrestrial)
4. Ākṣa Dṛkkarma : The arc of the ecliptic between the point of intersection of the ecliptic with a secondary through the star to the prime vertical and the point of intersection of the ecliptic with the star's declination circle.
5. Akṣakarṇa : The hypotenuse of the gnomonic triangle when its shadow is equal to what is called Viṣuvat-chāyā
6. Akṣa-Kṣetra : A right-angled spherical triangle one of whose sides may be the arc of a small circle namely the diurnal circle but one of whose angles is a right angle and another equal to the terrestrial latitude.
7. Akṣavalanam : The angle at the point of the star in between the declination circle of the star and a secondary to the prime vertical through the star.
8. Antyā : The Hindu sine of an arc of the celestial equator corresponding to Hṛti. 69

546 9 Asta : Setting or heliacal setting. 10. Ayanabindu : Solstice. 11. Ayana-Dṛkkarma: The arc of the ecliptic intercepted between its point of intersection with the star's declination circle and the secondary to the ecliptic through the star. 12. Ayanāṁśam : The arc of the ecliptic in between the vernal equinoctial point and the Hindu zero of the ecliptic ie the first point of the Zodiacal sign called Aśvinī. 13. Ayanavalanam : The angle at the point of a star, between its declination circle and the secondary to the ecliptic through the star. 14. Bārhaspatyamāna: The time taken by Jupiter to reside in a Rāśi, on the average, is called a jovian year. This falls short of a solar year. 15. Bhāga : A degree 16. Cāpa or : Arc. Kārmuka 17. Carajyā . : The Hindu sine of the arc intercepted between the east point and the declination circle of a rising star or planet or the Sun. 18. Cāndra-māsa : The time between two consecutive full- moons or New moons. 19. Chāyā or Bhā : Shadow cast by the gnomon. 20. Chāyābhuja : The projection of the shadow on the east- west line. 21. Chāyākarṇa or : The hypotenuse of the gnomonic triangle Bhākarṇa whose two sides are the gnomon and its shadow.

547 22. Chāyākoṭi : The perpendicular from the extremity of a shadow on the east-west line. 23. Dhruva : The star near the celestial pole or the celestial pole itself. 24. Dhruvaka : The celestial longitude. 25. Dhruva- : The declination circle. protavṛttam 26. Digjyā : The Hindu sine of the azimuth measured by the angle between the prime vertical and the vertical of a star or a planet. 27. Dorjyā or : Hindu sine of celestial longitude. Bhujajyā 28. Dṛgjyā : The Hindu sine of the Zenith distance. 29. Dṛg-lambana : Total parallax. 30. Dvāparayuga : Twice the period of a kaliyuga. 31. Dyujyā : The Hindu cosine of declination or the radius of the diurnal circle taking the radius of the celestial equator to be R equal to 3438 units. 32. Dyujyā-vṛtta or : The diurnal circle of a star or a planet. Ahorātra-vṛtta 33. Ghaṭi or Nāḍi : An interval of time equal to 24.' (minutes) 34. Grahaṇa : Eclipse. 35. Hṛti or Iṣṭahṛti : The Hindu sine of the arc of the diurnal circle from a point of the same upto the plane of the horizon. 36. Kadamba : Pole of the ecliptic. 37. Kadamba- : A secondary to the ecliptic through a star protavṛtta or planet.

548 38. Kakṣāmaṇḍala : The deferent of a planet or the circle with the earth as centre and radius equal to 3438 units. 39. Kalā : The Hindu sine in the diurnal circle corresponding to the Sūtra (given below). 40. Kalā or Liptā : A minute of angle. 41. Kaliyuga : The period consisting of 4,32,000 mean solar years. 42. Kalpa : Dvāparayuga is twice Kaliyuga; Tretāyuga thrice and Kṛta four times. All these put together constitute a Mahāyuga. 71 Mahā- yugas make one Manvantara. Fourteen Manvantaras with what are called Sandhi periods on either side equal to a Kṛtayuga or thousand Mahāyugas make a Kalpa. The creation is supposed to last one Kalpa, which is said to constitute the day time of Brahma, the Creator. His night also is of the same duration when there is no creation. It is held that we are now in what is called Śveta-Varāha Kalpa, wherein the seventh Manvantara called Vaivasvata manvantara is current. In this Manvantara, twenty seven Mahāyugas are supposed to have elapsed and in the twenty-eighth Mahāyuga, kṛta Treta and Dvāparayugas have elapsed and that we are now in the Kaliyuga. In this Kaliyuga which began on the day when Lord Kṛṣṇa gave up his mortal coil, 5080 mean solar years have elapsed by about 21 March, 1979. 43. Kramajyā or : Hindu sine of an angle. simply jyā or Jīvā or guṇa 44. Karaṇa : Half of the duration of a tithi.

549 45. Karṇāgrajyā or : The Hindu sine Agrajyā multiplied by K, simply Karṇāgrā and divided by R where K is the hypotenuse of the gnomonic triangle whose two sides are the gnomon and its shadow and R the radius of the celestial sphere taken to be 3438 units. 46. Kendra : The centre of a circle. 47. Ketu : The diametrically opposite point of Rāhu. Rāhu also means the circular section of the earth’s shadow at the Moon. 48. Kha-Svastika : The Zenith at a place. 49. Koṭijyā : Hindu Cosine of an angle. 50. Krānti or Apama: The declination of a point on the ecliptic. 51. Krānti-Vṛttam : Ecliptic. 52. Kṛtayuga : Four times the period of a Kaliyuga. 53. Kṣayamāsa : That lunar month in which there are two Saṁkramaṇas. 54. Kṣitija : The Horizon at a place. 55. Kujyā : The Hindu sine measured in the diurnal circle corresponding to the Carajyā defined above. 56. Lagna : The Rāśi which rises at any moment or the rising point of the ecliptic. 57. Lambajyā : The Hindu Cosine of the latitude. 58. Lambana : Parallax in longitude. 59. Mahāyuga : The Sum of the four yugas mentioned above. 60. Manvantara : A period equal to 71 Mahāyugas.

550 61. Nakṣatra : A star. Also the time which elapses as the longitude of the Moon increases by 13⅓ degrees starting from the zero point of Aśvini. 62. Nākṣatra-Dina : The time that elapses between two consecutive risings of a star. 63. Nākṣatra-māsa : The time taken by the Moon to go from Aśvini again to Aśvini. 64. Natakāla : Hour angle measured in ghaṭis. 65. Natāṁśajyā : Hindu sine of the Zenith distance. 66. Nati : Parallax in latitude. 67. Nīcoccavṛtta : Epicycle. 68. Parama-Antyā : The Antyā when the celestial body is at the point of intersection of the celestial equator and the meridian or what is the same at the point of culmination. 69. Paramakrānti : Obliquity of the ecliptic taken by the Hindu astronomers to be 24°. 70. Pāta : The point of time when the declinations of the Sun and the Moon are equal and of the same sign or opposite sign. Also it means the point of intersection of two great circles. 71. Prācī : East point. 72. Prācyaparā : East-west line. 73. Prativṛtta : The circle in which the planet moves and whose centre is at a distance from the centre of the Kakṣāmaṇḍala or the orbit of the mean planet. This is the same as the orbit of the the true planet supposed to be a circle.

551 74. Rāhu : The point of intersection of the Moon's path with the ecliptic (ascending point of the Moon's path). Also it means the circular section of the earth's shadow at the Moon. 75. Rāśi : An arc equal to 30° (on the ecliptic). 76. Samabindu or : North point. Udagbindu 77. Samamaṇḍala : Prime vertical. 78. Sama-Śaṅku : The Hindu cosine of the altitude when the Sun is on the Prime Vertical. 79. Saṁkramaṇa : The point of time when the Sun enters from one Rāśi to another of the twelve Rāśis, Meṣa etc. measured from the zero- point of the Hindu Zodiac. 80. Śaṅku : Gnomon. 81. Śaṅku : The Hindu sine of the Altitude. (Also means) 82. Śaṅku-cchāyā : The shadow cast by the gnomon. 83. Saura-māsa : The time when the Sun occupies one Rāśi. 84. Śara : The Hindu versed-sine of the hour-angle. Also it means celestial latitude. 85. Sāvanāha : The time between two consecutive Sun-rises. 86. Sūtra : The Hindu sine of the complement of the hour-angle. 87. Taddhṛti : The Hṛti of a celestial body when it is on the prime vertical. 88. Tithi : The time taken by the elongation of the moon to increase by 12° starting from zero.

552 89. Tithi-kṣaya : Since a Tithi falls short of a Sāvanāha or civil day, in course of time, it so happens, that a tithi begins after sun-rise and does not last upto the next sun-rise. Such a tithi is said to be lost and goes by the name Tithi kṣaya. One such tithi kṣaya occurs out of 64 tithis approximately or eleven out of 703 more approximately. 90. Tretāyuga : Thrice the period of a Kaliyuga. 91. Trijyā : The Hindu sine of three Rasis or 90° equal to R or 3438 units. 92. Udaya : Rising or heliacal rising. 93. Unmaṇḍala or : The great circle through the celestial pole Udvṛtta and the east point. 94. Unmaṇḍala : The Hindu cosine of the altitude of a Śaṅku celestial body situated on the unmaṇḍala. 95. Upavṛtta : A small circle parallel to the prime vertical through a star or a planet. 96. Vighaṭi or Vināḍi: One sixtieth of a ghaṭī. 97. Vikṣepa or śara : Celestial latitude. or Viśikha 98. Viṣuvatbindu : Equinoctial point. 99. Viṣuvat-chāyā : The shadow cast by the gnomon when the Sun is at the point of intersection of the celestial equator and the meridian on an equinoctial day. 100. Viṣuvat-Vṛtta : Celestial equator. 101. Vṛtta or Maṇḍala: A circle.

553 102. Yāmyottara- : The Hindu sine of the Altitude of a saṁku or Celestial body situated on the meridian. Dinārdha-Śaṅku 103. Yāmyottaravṛtta : The meridian at a place. 104. Yaṣṭi : R² - Ayanavalanajā² Yaṣṭi has also another meaning namely the length of the perpendicular from a point on the diurnal circle on the plane parallel to the plane of the horizon through the point of intersection of the diurnal circle with the unmaṇḍala. 105. Yoga : The time which elapses when the sum of the longitudes of the Sun and the Moon to increase by 13¹⁄₃º starting from zero. 106. Yuti : Conjunction.