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

533 longitudes of the Sun, Moon and Rāhu are respectively 3-2-28-12, 8-19-4-26 and 6-11-43-28 (in Rāśis, degrees etc), taking mean motions into account. When the Moon has the maximum southern declination in the position M₁ of fig. 129 his longitude will be however 8-10-9-35 and then his declination will be 1169. At that moment the decli- nation of the Sun will be 1398. Even here the Moon's declination falls short of that of the Sun. Again after 13⅔ days, we find the declination of the Moon falling short of that of the Sun. Even after 2 months, the Moon cannot have a declination equalling that of the Sun. This phenomenon occurs when Rāhu of the lunar orbit happens to be at Libra, and the Sun is at the summer solstice approximately. Verse 8. The definitions of Vyatīpāta and Vaidhṛti. When the Sun and the Moon are in opposite Ayanas, but in the same Goḷa, and if then their declinations be equal, then that moment is said to constitute vyatīpātayoga. If on the other hand, if both the Sun and the Moon be in the same Ayana and opposite Goḷas, and if then their declinations be equal, that moment is said to constitute Vaidhṛti Yoga. Comm. Explained before. Verse 9 and first half of verse 10. To prognosticate the occurrence of a pāta. When the sum of the tropical longitudes of the Sun and the Moon happens to be 180° or 360°, then the Vyati- pāta and the Vaidhṛti respectively will occur or will have occured. The number of minutes of arc by which the sum of the tropical longitudes falls short of or exceeds 180° or 360° as the case may be, are to be divided by the sum of the true daily motions of the Sun and the Moon, which will give aqproximately the number of days after which or

before which the Yogas will occur or would have occured. Compute the declinations of the Sun and the Moon for that moment from the then true daily motions. Comm. From fig. 124, it is clear that triangle ♈SL and ♎ MN are congruent so that ♈S = ♎ M ∴ SS₁ = MS₁ ∴ ♈S + ♈M = ♎ M + ♈M = 180°. Again triangle ♈SL and ♈M'N' are congruent ∴ ♈S = M'♈ ∴ ♈S + ♈M' = ♈M' + M'♈ = 360°. Thus in the former case which constitutes Vyatīpāta, the sum of the tropical longitudes of the Sun and the Moon is 180°; whereas in the latter case, which constitutes the Vaidhṛtipāta, the sum of those two longitudes is equal to 360°. Hence we are asked to note when the sum of the two longitudes is likely to amount to 180° or 360°. On a particular day suppose the sum is 180 — θ or 360 — θ; so that θ is to be made up by the then velocities of the Sun and the Moon conjointly. Then θ / (u + v) where u, v are their respective velocities gives the number of days or fractiou of a day, by which the sum would be 180° or 360° as the case may be. As the velocities change from moment to moment, the above θ / (u + v) is only approximate. So, compute again the respective positions and respective velocities. Suppose the sum of the longitudes is 180 ± θ' or 360 ± θ' and the velocities u' and v', Then θ' / (u' + v') gives the fraction of a day after or before the moment in question when the pātas occur Vyatīpāta or Vaidhṛti. Verse 10. To know whether the Yoga is past or future. If the declination of the Moon situated in an odd quadrant exceeds that of the Sun or falls short of the

535 Sun's in an even quadraut the moment of the occurence of pāta has elapsed. Otherwise the pāta is to take place shortly after. Comm. This is clear because in an odd quadrant, the declination of the Moon is on the increase. So if it be already greater than the Sun's, in future it will be far greater and as such cannot equal the Sun's declination. In other words the pāta had already taken place. Similarly in an even quadrant, the declination of the Moon is on the decrease. As such if it be greater than the Sun's it will become equal in future. Thus the pāta is to take place. On the other hand if the Moon's decli- nation is less than the Sun's, it will decrease further so that prior to the moment concerned a pāta should have occured. Latter half of verse 11 and verses 12, 13, 14. To compute the time of the occurrence of pāta through a consideration of declinations. Obtain the difference of the declinations if they be of the same direction or their sum if they be of opposite directions in the case of Vyatipāta ; obtain the sum or difference of the declinations according as the declinations are of the same or different directions in the case of Vaidhṛti. Call this sum or difference 'the Ādya'. After a lapse of an arbitrary time or before the moment concerned, obtain the positions of the Sun, Moon and the Rāhu ; also compute their declinations and form their difference or sum as the case may be. Call this Anya. If on both the occasions it is indicated that the pāta has elapsed or is to elapse, obtain the difference of the Ādya and Anya ; otherwise their sum. Divide the arbitrary time taken, by the above sum or difference of the Ādya and Anya and multiply by the Ādya. Take the result in ghatis. Taking this as the arbitrary time, repeat the

536 process till an invariable quantity is obtained in ghatis. This is the time by which the moment of the pāta has elapsed or after which it is going to occur. Comm. (1) It might be asked “why bother finding the declinations at all, when it is defined that Vyatīpāta happens when the sum of the longitudes of the Sun and the Moon is equal to 180° and Vaidhṛti is defined when the sum is equal to 360, since easily we could know when this happens without taking recourse to declinations?” The problem is not that simple. The above definition in terms of the sum of the longitudes is an approximate statement, because, the original and correct definition is that their declinations must be equal in magnitude. If they be equal in magnitude but opposite in direction, they constitute Vaidhṛti Yoga. If they be both equal in magnitude and direction, the situation constitutes Vyatī- pātayoga. In other words, when the diurnal paths of the Sun and the Moon coincide, the moment will be Vyatīpāta. If on the other hand their diurnal paths are of equal dimentions but one to the north of the celestial equator and the other to the south, then the moment is Vaidhṛti. Though the sum of the longitudes happens to be 180°, the diurnal paths may not coincide because the Moon does not actually move on the ecliptic. So the situation is more complicated. Further the calculation of the declination of the Moon is to be done by calculating the Asphuṭakrānti ie. the declination of the point of the ecliptic which indicates the longitudinal position of the Moon and his celestial latitude ie. the Asphuṭavikṣepa from the known position of the Node, and then by rectifying the Asphuṭavikṣepa to obtain the Sphuṭavikṣepa and then adding the Asphuṭa- krānti to the Sphuṭavikṣepa to obtain the Sphuṭakrānti. (2) The latter half of verse (11) and the first half of verse (12).

537 Let us consider the case of Vyatipāta. Suppose at the moment concerned, the declinations are of opposite direction. Find the sum of their numerical magnitudes. Suppose they are of the same direction; find their differ- ence. This sum or difference of the declinations gives the distance between the planes of the diurnal paths of the Sun and the Moon. This distance is to vanish in order that the diurnal paths may coincide. So we are asked first to find the above distance. Suppose we understand that the Vyatipātha has elapsed by noting the deolinations. This may be known easily by the criteria given previ- ously. Supposing 𝑥 and 𝑦 to be the declinatious of the Moon and the Sun and supposing that 𝑥 is on the decrease and approaching 𝑦, then the Vyatipātha is to occur. But suppose 𝑥 < 𝑦 and 𝑥 is on the decrease, then the Vyatipāta has taken place. Thus knowing whether the Vyatipātha has elapsed or is to occur, after an arbitrary time 𝑡, com- pute the declinations of the Sun and Moon and form their difference or sum as the case may be which gives the distance between the diurnal paths. Let the first distance found be called Ādya and the second distance the Anya. Then the Anya will be less than the Ādya because we have taken a time towards the occurrence of Vyatipāta when the distance is to get nullified. Find the difference of the Ādya and Anya. Then by the proportion “ If in time ‘𝑡’ taken in between the moments of the Ādya and Anya, the distance between the diurnal paths is reduced by Ādya — Anya, what time will be taken for the distance Ādya to vanish ?” we have the result T = (Ādya × 𝑡) / (Ādya — Anya) which gives approximately the time that has to elapse for the occurrence of the pāta. This will be approximate. After a lapse of time T from the Ādya moment concerned, again compute the declinations and repeat the procsss. We arrive at a particular point of time, which gives the moment of occurrence of the pāta. 68

538 In the course of the commentary under these verses Bhāskara solves two problems and points out that when there is a pāta according to his exposition, the statements made by Lalla, Brahmagupta Śrīpati and Mādhava all indicate that there would not occur a pāta. But we feel that Bhāskara read too much into those statements on account of the following. Lalla states सूर्योपमादोजपदोद्गवाच्चेद्युग्मादिजः चन्द्रमसो लघीयान्, अपक्रमःस्यान्न तदास्तिपातः तदन्यथात्वेऽपमयोः समत्वम् । Brahmagupta states त्रिनवगृहेन्दुकान्तिः मेषतुलादौ दिवाकरक्रान्तेः । ऊना यावदभावः तावत् भावोऽन्यथा चेति ॥ and Śrīpati says त्रिनवभवनजाता क्रान्तिरिन्दोर्यदाऽल्पा दिनकृदपमतः स्यान्मेषजूकादिजातात् । न हि भवति तदा च क्रान्तिसाम्यं रवीन्द्रोः नियतमितरथात्वे जायते सम्भवोऽस्य ॥ In fact all these three statements mean one and the same and are intended to be approximate statements, in the first instance, having ignored the latitude of the Moon. They are just statements like that of Bhāskara himself when he says that there will be a pāta when the sum of the longitudes will be 180° or 360°. The statements pur- port to say that if the declination of the Moon when it is on the decrease happens to be less than that of the Sun which is on the increase, there could be no pāta. These statements so far as they go, ignoring the latitude of the Moon are perfectly in order. Bhāskara brings in a detailed analysis of two critical examples, to prove that there is a pāta, but which is negated by the rough statements of the

539 four Ācāryas. Mādhava's statement in his Siddhānta Cūdāmaṇi is as follows which is also in similar terms as those of the other three. रवेरोजपदक्रान्तेः चन्द्रयुग्मपदोद्भवा स्वल्पाचेन्नतयोःक्रान्त्योः साम्यं स्यादन्यथा भवेत् Verses 15 and 16. To obtain the duration of the pāta. The semi-sum of the diameters of the Sun and the Moon or what is the same the sum of their angular radii being multiplied by the Spaṣṭaghaṭis (obtained in the estimate as per the verses 11-14) and divided by the Ādya in that context, gives the beginning and end of the pāta from the moment of the computed time. The process being repeated according to the method of successive approximations we have the correct estimate of the duration of the pāta. Comm. This is a convention stipulated with respect to the duration of the Pāta. Strictly speaking, when the diurnal paths of the centres of the discs coincide in the case of Vyatīpāta, that will be the middle moment of the Vyatīpāta. The pāta is said to last for such a time as the distance between the Centres of the discs (north-south distance) is less than r + p. This will be clear from a figure. The situation is akin to that of an eclipse. The time obtained for the occurrence of the pāta under verses (11) to (14) indicates the middle of the duration of the pāta. The duration of the pāta is defined as the time that lasts as long as the declination of the highest point of one disc becomes equal to that of the lowest point of the other beginning from the moment at which the declination of the lowest point of the one becomes equal to that of the highest point of the second. In other words, just like in an eclipse, so long as the distance between the diurnal paths traced by the centres

540 of the discs is less than the sum of their angular radii, the pāta lasts. Extending the meaning of this to Vaidhṛti also, so long as the numerical difference of the declinations of the Sun and the Moon ignoring their direction happens to fall short of the sum of the angular radii, the pāta lasts. To obtain this duration of the pāta the argument is “If the sum or difference of the declinations according as they are of opposite or the same direction (which was taken to be Ādya under verses (11) to (14) ) was reduced to zero in the time computed that time being known as Spaṣṭaghaṭis, what time will be taken for a difference of declinations equal to the sum of the angular radii?” The result is ((r + p) × T) / Ādya where r and p are the angular radii of the Sun and the Moon, T the time calculated formerly known as Spaṣṭaghaṭis and Ādya is as defined above. The above result gives the duration of the pāta. Note. An approximate estimate of this duration could be obtained using differentiation. We have sin δ = sin λ sin ω so that cos δ Δδ = sin ω cos λ Δλ Let Δλ be the motion in longitude of the Moon with respect to the Sun per nāḍi which will be on the average 12′ approximately. ∴ Δδ = (sin ω cos λ × 12) / cos δ If in one nāḍi, there be a variation in the declination equal to the above, what time will be taken for 16′ + 15′ the sum of the angular radii approximately ? The result is (31 cos δ) / (12 sin ω cos λ)

541 When λ approaches 90°, the duration will be very much since cos λ → 0. This is true because the variation in the declination when λ = 90°, is very slow even for the Moon. The above formula holds good so long as λ does not approach 90°. Verse 17. It must be deemed that the declinations will be equal so long as the difference in the declinations is less than r + p numerically. In the case of Vaidhṛti, we are asked to take the sum instead of difference and difference instead of sum in contradistinction. Why? We shall see. We know that Vaidhṛti occurs when the declinotions of the Sun and the Moon are of equal magnitude but of opposite directions subject to the condition that the longitudes are such that their sum is approximately 360°. This condition is stipulated because when the Sun is in the first quadrant and the Moon is in the 3rd quadrant and at the same time their diclinations are equal, the moment does not constitute Vaidhṛti, because the sum of the longitudes is then less than 90° + 270° ie. less than 360°. Hence, for Vaidhṛti one of the two celestial bodies must be in the first quadrant and the other in the fourth quadrant, so that the sum of the longitudes could equal 360° and at the same time the declinations could be equal though of opposite sign. Now compute the declinations of the Sun and the Moon at a particular moment. Suppose they are a and b and both are north. After a few ghatis or days compute again their declinations, say c and d, both again being north. Suppose then c + d < a + b, then Vaidhṛti is going to occur. Construing northern declination to be positive and the southern negative, for Vaidhṛti to occur a + b must be reduced to zero ie. a and b are equal and of opposite direction. Now the argument adduced is "If in time t ghatis or days a̅+̅b̅ has become c̅+̅d̅ (calling a̅+̅b̅ as Ādya and c̅+̅d̅ as Anya) there is a decrease of Ādya - Anya, what time should lapse for a̅+̅b̅ or Adya to reduce to

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.