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

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

522 by the ancient Hindu astronomers, before the time of Bhāskara, for the reason that the Moon does not exactly move in the ecliptic but in his own orbit whose inclination to the ecliptic was taken to be 4½°. The points of intersec- tion of the ecliptic with the lunar orbit, or what are called Rāhu and Ketu, the nodes of the lunar orbit, have themselves a motion backwards on the ecliptic in 18.59575 solar years as per Bhāskara. This makes the lunar orbit oscillate about the mean position of the ecliptic, so that sometimes the lunar orbit lies between the celestial equator and some-times not between them. This phenomenon makes the computation more difficult, to obtain the declination of the Moon. Bhāskara says that even great Ācāryas like Lalla and Brahmagupta went wrong or got confused in computing the moments of the occurence of a Pāta. Bearing upon Bhāskara’s statement that such Ācāryas also got confused, some second-rate astronomer exclaimed in the following interesting manner “त्रिस्कन्धविद्याकुशलैकमल्लो लल्लोऽपि यत्राऽप्रतिभा बभूव, यातेऽपि किञ्चित् गणिताधिकारे पाताधिकारे मम नाऽधिकारः” ie. “when even an unrivalled scholar like Lalla, who was well-versed in the three branches of Jyotiṣa betrayed his ignorance in this context of Pātādhi- kāra, though I am a bit of a mathematician, I could have no pretensions to any knowledge in this difficult chapter”. In Fig. 124, Vyatīpāta occurs when SL=MN, where S and M are the Sun and the Moon situated respectively in Uttarāyaṇa and Dakṣi- ṇāyana but in the same uttaragola. The positions of S and M could be interchanged ie. S may be in the second quadrant of the ecliptic and M in the first quadrant and if their declinations be equal, then Fig. 124 also Vyatīpāta occurs. Of course in this figure 124, we have taken the Moon also situated on the ecliptic. In

528 actual computation, we should not take the Moon as such. The pāta named Vaidhṛti occurs when in the same Fig. 124, SL=M¹N¹. Herein both the Sun and Moon are in the same Ayana namely Uttarāyaṇa but in opposite goḷas. If we assume the Moon to be moving on the ecliptic, Vyatīpāta occurs when the sum of the tropical longitudes of the Sun and Moon equals 180° and Vaidhṛti occurs when the sum of those longitudes is equal to 360. Verse 2. Definitions of Goḷa-Sandhi and Ayana- Sandhi with respect to the Sun. When the tropical longitude of the Sun ie. his longitude measured from ♈ along the ecliptic is equal to 0° or 180°, then he is said to be at his goḷa Sandhi. In other words, when the Sun who moves along the ecliptic comes to the celestial equator, he will be at his Goḷa sandhi. Similarly when his longitude is 90° or 270°, he is said to be at the his Ayana-Sandhi. Comm. This means that when the Sun is about to pass from the Dakṣiṇa goḷa to the uttaragoḷa or from the uttaragoḷa to the Dakṣiṇagoḷa he is said to be at a Goḷa Sandhi. Similarly when he is about to go south or when he is about to go north, he is said to be at Ayana Sandhi. The word 'Sandhi' means junction. Thus the points ♈ and ♎ (Libra) are said to be goḷa Sandhis whereas the points denoting Cancer and Capricorn are Ayana-Sandhis. Since in Bhāskara's time the Ayanamsas were 11°, ie. the point ♈ was behind the zero point of the Hindu Zodiac by 11°, therefore Bhāskara gives the Goḷa Sandhis as the points having longitudes 349° and 169° respectively. Similarly the Ayana Sandhis were the points having longitudes 79° and 259° respectively. Bhāskara gives the method of locating these goḷa Sandhis or Ayana Sandhis as follows. Erect a gnomon

524 vertically with the help of a plumb-line. Draw a circle on the horizontal plane having the foot of the gnomon as the centre and any arbitrary radius. Draw the East-West and North - South diameters of the circle. The longitudes of the Sun when the shadow of the gnomon lies along the East - West diameter give the gola Sandhis. Note the direction of the shadow daily after the day when his shadow is along the East - West line. If the Sun rises thereafter a little towards North of the East point, then he is said to be in the uttara gola; his shadow will be a little south of the East - West line. The point at which the Sun thus begins to rise north of the east point, is the Meṣa gola Sandhi or the vernal equinox. Gradually the Sun goes on rising at points which are farther and farther away from the East point. When he has reached the extreme north point, ie when the gnomon's shadow is extreme south, he is at the Cancer. Similarly when he rises at the extreme South point on the horizon, he is at Capricorn. Bhāskara actually noted these four points and the longi- tudes cited above were given by him as the Gola Sandhis and Ayana Sandhis. Indirectly, he could know that the Ayanāṁśas at the time of his writing the book, were 11°. Verses 3 to 6. Speciality with respect to the Moon. Let the Hsine and Hcosine of the tropical longitude of the pāta (Rāhu the ascending node of the lunar orbit) as per the smaller table of Hsines when the radius is taken to be 120', be respectively multiplied by 123 and 7 and divided respectively by 4 and 12. The results are known as the Bāhuphala and Koṭiphala respectively. According as the tropical longitude of Rāhu ie λ be such that as 270° < λ < 90° or 90° < λ < 270°, the Bāhuphala is to be divided by 362 ± Kotiphala. Subtract the result from the Gola Sandhis and Ayana Sandhis of the Sun, to get those of the Moon.

525 Comm. from the Hindu Astronomical point of view this is an intricate procedure which made that particular half-learned astronomer declare "पाताधिकारे मम नाऽधिकारः" ie. "I have no pretensions to have understood the chapter known as Pātādhikārā", We perceive herein Bhāskara's mathematical understanding of the problem. His procedure is approximate because he uses (1) the smaller crude table of Hsines taking the radius to be 120' instead of 3438. (2) also because he gives the Sun's declinations for longitudes 15°, 30° etc. as well as Moon's celestial latitudes for his longitudes of 15°, 30° etc. relative to the node in his orbit. Of course, his mathema- tical procedure was correct. He exemplifies his mathematical procedure by solving a problem given in the prasna-Adhyāya of his book Golādhyāya. The problem set by him is as follows. युक्तायनांशोऽशशतं शशी चेत् अशीतिरर्को द्विशती विपातः चन्द्रः, तदानीं वद पातमाशु धीवृद्धिदं त्वं यदि बोबुधीषि । ie. If the tropical longitude of the Moon is 100°, that of the Sun 80°, and the longitude of the Moon measured in his orbit from the Node Rāhu is 200°, compute the moment of the occurence of the pāta, if you know what was said by Lalla in his work Siṣya dhīvṛddhida. We shall first understand Bhāskara's mind and subsequently we shall give a modern procedure.

526 Fig. 125 Refer to fig. 125. Gr NK is the celestial equator. RrL is the ecliptic and RGAM the Moon's orbit where R is the Rāhu or ascending node of the lunar orbit. Let the obliquity of the ecliptic be ω (omega) and the inclination of the lunar orbit to the ecliptic be i. ω was taken to be 24° and i 4½° by Bhāskara. Let the lunar orbit RGAM cut the celestial equator in G which is called the Moon's Gola Sandhi. r is the Sun's Gola Sandhi. Bhāskara first wants us to locate G ie. to find rQ which gives its longitude. Let A be the position of the Moon when his celestial longitude is zero ie rA is perpendicular to the ecliptic. Let M be the position of the Moon when his celestial longitude is 15° (Here the figure is not drawn to scale but a little exaggerated for the sake of clarity). Let ML be the celestial latitude of the Moon in its position. M. If LK be drawn perpendicular to the equator, LK is called the Asphuṭa - krānti of the Moon. ML is called the Asphuṭa-Vikṣepa of the Moon. MN drawn perpendicular to the celestial equator ie. the declination of the Moon is called Sphuṭakrānti of the Moon. Draw perpendicular LS on MN. Then MS is called the Sphuṭa Vikṣepa of the Moon. Since MN = Ms + LK Sphuṭakrānti = sphuṭa Vikṣepa + Asphuṭakrānti. Let Ap be the declination of the Moon when his longitude is zero.

527 [Fig. 126] Fig. 126 Now refer to fig. 126. Let NS be the ecliptic and NM the celestial equator. The quadrant NS equal to 90° is divided into six equal parts at S₁, S₂ etc. The successive declinations of S₁, S₂ etc. are given by Brahmagupta as 362, 703, 1002, 1238, 1388, 1440 where the radius is given to be 3438'. In other words 1440 = H Sin ω = H Sin 24°. These can be easily verified from the formula H Sin δ = (H sin λ × H sin ω) / R putting λ = 15°, 30° etc. and R = 3438'. [Fig. 127] Fig. 127 Now refer to fig. 127. R M is the lunar orbit, R being Rāhu. RL is the ecliptic. Using the formula.

528 H sin β = (H Sin λ × H Sin i) / R or β = (H Sin λ × i) / 120 since β and i are small and R is taken to be 120′, we have the successive values of M₁L₁, M₂L₂ etc. of the celestial latitudes of the Moon for arcs RM₁, RM₂ etc. successively equal to 15°, 30° etc. given by 70, 135, 191, 234, 261, 270. In other words the maximum celestial latitude is 270′ = 4½°. Since when λ = 15°, taking R = 120 β = 70′, the first Sarakhanda ie the celestial latitude for λ = 15°, is 70′ (H cos λ × 70) / 120 Bhāskara Calls Koṭiphala. It will be noted here that the first Sarakhanda is taken to mean the increase in the celestial latitude from zero to 70′ when the longitude increases from 0° to 15°. The argument adduced by Bhāskara in such a context is as follows: “If for a H Cosine λ equal to R equal to 120′ (when λ = 0) we have the first Sarakhanda namely 70′, what shall we have for an arbitrary H Cosine λ”. The result is (H cos λ × 70) / 120 = ⁷⁄₁₂ H cos λ. Since the word Cosine means Koṭijyā, Bhāskara calls this Koṭiphala. Now in fig. 125, let rL = 15°, then LK = 362′ as given by Brahmagupta. Bhāskara takes this 362 as Δ (δ) where δ is the declination of the foot of the celestial latitude of the Moon. Then if β be the celestial latitude of the Moon, Bhāskara construes that Δ β is given by the formula ⁷⁄₁₂ H Cos λ, which he calls Koṭiphala. Taking Δ δ ± Δ β as the joint variation of δ and β which is roughly equated with the variation in the modern declination MN of the Moon, Bhāskara's argument is “If for a longitude r M = 15° of the Moon corresponds Δ δ ± Δ β what should be the longitude corresponding to the declination Ap of the Moon

529 when his longitude is equal to zero?" The result is (Ap × 15) / (Δδ ± Δβ) = II where rQ is looked upon as the longitude of the Moon at G called the Goḷa Sandhi of the Moon. But AP = the Sphuṭa Vikṣepa of the Moon when his longitude is equal to zero where rA is the Asphuṭa Vikṣepa at that point. Using his method of calculating AP from Ar, Ap = (Ar × H cos δ) / R where H sin δ = (H sin ω × H sin (90 + λ)) / R . Here AP pertains to the longitude λ equal to zero, so that H sin δ = (H sin 90° × H sin ω) / R = H sin ω. ∴ H cos δ = H cos ω = H cos 24° = 110 when R = 120' Hence Ap = (Ar × 110) / 120 = (11 / 12) Ar. But Ar is the celestial latitude of the Moon to be calculated from RA taken to be λ ∴ rA = (H sin λ × 270) / 120 = (9 / 4) H sin λ (where 270' = 4½° = i) taking R = 120 ∴ Ap = (9 / 4) H sin λ × (11 / 12). Hence rQ from I = (9 / 4) H sin λ × (11 / 12) × 15 / (Δδ ± Δβ) But (H sin λ / 4) × (135 × 11) / 12 = (123¾ / 4) H sin λ. This has to be divided by Δδ ± Δβ. In the problem given. λ = 100° and Δδ is taken as 362. To obtain Δβ Bhāskara adduces the argument "If at λ = 0, the first Śarakhaṇḍa of 70 corresponds to H cos λ = 120' what amount of Śara- khaṇḍa corresponds to an arbitray H cos λ?" The result is (H cos λ × 70) / 120 = (7 / 12) H cos λ. This he calls Kotiphala because it is based upon H cos λ which is the Kotijyā. 67

530 Thus in the given problem where l = 100°, H cos 100° = 7/12 × 21 (since H cos 100 = - H sin 10 = 120 × .1735 = - 21 approximately where R = 120') = - 12′-15″ ∴ Δδ ± Δβ = 362 - 12′-15 = 349-45. Now 123/4 H sin λ to be divided above = (123 × 118) / 4 = 3628′-30″, so that 123/4 H sin λ × 1 / (Δδ ± Δβ) = (3628′-30″) / (349-45) = 10°-22′-28″ = = rQ. Now r's longitude from the zero point of the Hindu zodiac is 11 Rāśis 19° so that Q's longitude = 11 R-19° minus 10°-22′-28″ = 11 Rāśis 8°-37′-32″. This gives us the longitude of G which is called the Moon's Goḷa Sandhi. Adding 3, 6, 9 Rāśis, we get successively the first Āyana Sandhi, the other Goḷa Sandhi and the second Āyana Sandhi of the Moon. (1) Bhāskara overlooked a crudeness in his proce- dure namely that Δβ is perpendicular to the ecliptic whereas Δδ is perpendicular to the celestial Equator, but, since Δβ is small, he overlooked the nicety. Strictly speaking Δβ should have been corrected for Āyanavalana. (2) There is also crudeness in computing Δδ and Δβ for arcs of 15°, whereas they should have been done for increase of every degree in the longitude. He could have done that, because at the end of the Goḷādhyāya he gave the method or computing the H sines for every degree under the caption प्रतिभागज्यकाविधि. We shall now give a modern method of computing the value of rQ. From the spherical triangle RrA, where ⁀rR = 100°, R̂ = i = 4½°, we have sin 100 = (tan rA) / (tan 4½)

531 ∴ log tan rA = log cos 10 + log tan 4½° = 9·9934 + 8·8960 = 8·8894 ∴ rA = 4°-26ʹ. From the spherical triangle rAG, cos 90—ω = tan rA / tau Gr log tan Gr = log tan rA — log sin ω = 8·8894 — 9·6093 = 9·2801. Again from the spherical triangle rQG, cos ω = tan rQ / tan Gr ∴ log tan rQ = log tan Gr + log cos ω = 9·2801 + 9·9697 = 9·2498 ∴ rQ = 10°-5ʹ whereas Bhāskara got 10°-22ʹ-28″. This shows how Bhāskara was correct mathematically. The small difference there is, due to his taking Hsines for arcs of 15° instead of for smaller arcs. Note. In the commentary called Sikhā of one Sri Kedāra Datta Joṣī (page 357) we find a mistake committed namely that he subtracted 3 R-10° the longitude of the pāta which is in Rāśis and degrees from the Sphuṭakrānti 349ʹ-45″. What Bhāskara meant was, that since the longitude 100° exceeds 90°, the cosine will be negative which therefore entails △β to be subtracted from △δ. Verse 7. How to know the occurrence of a pāta. If the Sphuṭakrānti of the Moon, when it is maximum be less than that of the Sun, then there could be no occasion for their declinations to become equal in the near future. Comm. The situation in which the maximum decli- nation of the Moon falls short of that of the Sun, arises