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

475 in the east, thus rising heliacally. After attaining the maximum elongation in his retrograde motion, his motion will then become direct, so that he again approaches the Sun, going from west to east. Next a little before overtaking the Sun, he sets heliacally in the east, and thereafter, having overtaking the Sun, he rises heliacally in the west. Here, one point is to be mentioned. The time between Mercury's maximum elongation in the west and again the maximum elongation in the east (which are termed maximum eastern elongation and maximum western elongation with respect to the Sun) will be far less than the time between the maximum elongation in the east (ie. maximum western elongation with respect to the Sun) and that in the west (ie. maximum western elongation with respect to the Sun) in as much as when Mercury is retrograde, and the Sun having always direct motion, the relative velocity will be the sum of the retrograde velocity of Mercury and the direct velocity of the Sun. In the case of the superior planets, as mentioned above, the superior planets set heliacally in the west and rise heliacally in the east. They will not rise heliacally in the west and will not set heliacally in the east which happens only if either the superior planet has a greater velocity than the Sun or the Sun has a retrograde motion which is never the case. Verse 5. Speciality with respect to Mercury and Venus. Mercury and Venus rise heliacally in the west in their direct motion, (attain maximum elongation before they) become retrograde, set heliacally there itself, then rise heliacally in the east continuing to be retrograde, (attain maximum elongation there before they) next become direct

476 and gradually set there (to rise again in the west) as before. Comm. Explained above. Verse 6. Kālāmśas or distance in degrees from the Sun within which the planets rise or set heliacally. The Kālāmśas with respect to the Moon, Mars, Mer- cury, Jupiter, Venus and Saturn, or the degrees of distance from the Sun within which they rise or set heliacally are 12, 17, 14, 11, 10, 15 respectively. In the case of Mercury and Venus when they are retrograde the Kālāmśas are 12, and 8 respectively. Comm. The Kālāmśas given above depend upon the luminosity of the respective planets. When Mercury and Venus happen to be retrograde, the Kālāmśas happen to be 2° less in each case because they are then nearest to the earth and as such being most luminous as seen by us will not set heliacally till they are very near the Sun. Verse 7. To compute the moment when a planet rises or sets heliacally. If it is to be known when a planet rises or sets helia- cally the position of the Prāk-Dṛk-graha or the Paschima- Dṛk-graha as the case may be (Prāk-Dṛk-graha in case the rising or setting takes place in the east or the other in the other case) and that of the Sun also are to be computed on a day a little before the day of rising or setting as prognosticated by the Śīghra anomaly. In case the planet rises or sets in the west, to obtain the lagna the position of the Sun is to be increased by 180°. Comm. Clear.

477 First half of verse 8. To obtain what are called Iṣṭa-Kālāṁśas. The time between the rising of the planet or of the Dṛk-graha and that of the Sun measured in ghaṭīs multi- plied by six gives what are called Iṣṭa-Kālāṁśas. Comm. Having found the approximate position of the Dṛk-graha as mentioned above when the planet is likely to rise or set heliacally, let the time in between the rising of this Dṛk-graha and that of the Sun (if it be the case of setting or rising in the west the position of the Sun is to be increased by 180° because the astalagna directed to be found in verse (1) is the point of intersection of the ecliptic with the eastern horizon, which is removed 180° from the setting point of the Sun) be multiplied by six. Since both the Dṛk-graha and the Sun's position are points on the Ecliptic and since we have considered the time in between their rising moments, which is measured on the equator, and again since this time is measured in ghaṭīs, the number of ghaṭīs multiplied by six give the degrees, for, each ghaṭī corresponds to six degrees of the equator (sixty ghaṭīs corresponding to one sidereal day). These degrees are said to be Iṣṭa-Kālāṁśas, which means that it gives the arc in between the feet of the declination circles of the Dṛk-graha and the Sun at the Iṣṭa-kāla ie. that particular time considered. Second half of verse 8 and verses (9) and (10). If the number of degrees so found ie. the Iṣṭa- Kālāṁśas fall short of or exceed the number of Kālāṁśas postulated for the rising or setting of the planet, then the planet's rising has to take place or has already taken place respectively and vice versa in the case of setting. The number of minutes of the difference of the prescribed and Iṣṭa-Kālāṁśas multiplied by 1800 and divided by the rising time of the Rāśi expressed in Kalās and again

478 divided by the difference of the daily motions of the planet and the Sun expressed in minutes of arc if the planet is direct in motion or divided by the sum of those daily motions if the planet be retrograde gives the days elapsed after rising or to elapse for the rising to take place. Again, compute the positions of the Dṛk-graha and the Sun for the moment thus obtained and repeat the process till the actual moment is obtained. Comm. Suppose the prescribed Kālāṁśas for rising be x and suppose the Iṣṭa-Kālāṁśas are y such that y<x; then for the planet to rise, the arc of the equator in between the feet of the declination circles of the Sun and the Dṛk-graha has to increase for the planet to rise which means that this takes some more time to happen. Similarly if y>x, the planet has already risen. The question of the arc decreasing does not arise in this case of rising, because the planet's position in the east is behind that of the Sun, and in the case of a superior planet the Sun has to advance further for the planet to rise ie. the arc has to increase and in the case of a retrograde inferior planet also, the arc will be increasing. In the case of an inferior planet being direct, the arc will be decreasing no doubt, but we have to remember that this is a case of an inferior planet setting and not rising which we are considering. In the case of setting, in the east, however, of the inferior planet the moment of setting has already elapsed and thus the condi- tion is reverse to that of rising. The case of setting of a superior planet in the east never happens. In the case of setting of a superior planet in the west, if the Iṣṭa- Kālāṁśas be less than the prescribed Kālāṁśas, ie. y<x, setting must have already taken place, the Sun having approached the planet from behind and already effected heliacal setting. Thus this is also reverse to the condition of rising as stated. In the case of an inferior planet setting in the west if y>x, the planet should have set already since the inferior planet is retrograde while setting in the west.

479 This condition is also reverse to what has been stated in the case of rising. The case of a superior planet rising in the west also never happens, because it is the Sun that overtakes the planet and also the superior planet cannot be retrograde while near the Sun. The case of rising which has already elapsed the Iṣṭa- Kālāṁśas y will be evidently greater than x ie. y > x. To obtain the time by which the rising will take place after the moment in question, ie. the rising of a superior planet which is direct and an inferior planet which is retrograde, we have to take the difference of the prescribed and Iṣṭa- Kālāṁśas and find out the time by rule of three as follows. (1) Since we have to find the corresponding arc of the ecliptic in minutes of arc, from (y - x) × 60' the difference of the Kālāṁśas of the equator converted into minutes ie. asus we have first to use the proportion. "If by the rising time of the Rāśi t in asus in which the planet is situated we have 1800' of the ecliptic, what will we have by (y - x) × 60? The answer is (y - x) × 60 × 1800 / t . Next we have to find the days when this arc of the ecliptic is covered by the proportion. "If the Sun overtakes the planet by (u - v) minutes of the ecliptic per day, how many days are taken to cover the above arc?" The answer is as stated. The answer gives in days together with fraction of a day when the planet is likely to rise. We have to repeat the process because the motions of the planet as well as the Sun differ from moment to moment and the time calcu- lated above by rule of three taking into account their motions for the entire day will be only approximate. The computation in the case of setting of a planet may be similarly considered, the setting of a superior planet in the west or that of a direct inferior planet in the east.

480 Verses 11 and 12. If the Prāk-Dṛk-graha has a longitude greater than that of the Sun or the Paschima- Dṛk-graha has one less than that of the Sun, the sum of the prescribed Kālāṁśas and Iṣṭa-Kālāṁśas converted into minutes of arc will have to be used to compute the elapsed days or the days after which the respective phenomenon is going to occur. If the Iṣṭa-Kālāṁśas y be greater than the prescribed namely x, the reverse to what has been stated in the second half of verse (8) and in the first line of verse (9) happens ie. a phenomenon which has elapsed when y < x will happen in the future and vice versa. Comm. If the Prāk-Dṛk-graha has a longitude greater than that of the Sun, (since the prescribed Kālāṁśas are between the planet and the Sun, the planet being behind the Sun, and the Iṣṭa-Kālāṁśas are now on the other side of the Sun) the planet has advanced over its position of heliacal setting be it a superior planet or inferior by the sum of the prescribed Kālāṁśas and Iṣṭa-Kālāṁśas. Hence the days that have elapsed after the heliacal setting have to be calculated with the sum of the two Kālāṁśas. If it be also a case of a retrograde inferior Prāk-Dṛk-graha rising, having a longitude greater than that of the Sun, even then the prescribed Kālāṁśas are behind the Sun and the Iṣṭa-Kālāṁśas ahead of the Sun. So from the position of the planet’s heliacal rising, the Sun and the planet have advanced in opposite direction to a distance which is the sum of the prescribed Kālāṁśas and Iṣṭa-Kālāṁśas. Hence computation has to proceed with the sum of the two Kālāṁśas to get the time that has elapsed after the planet’s heliacal rising. Similar is the argument for the Paśchima- Dṛk-graha. We have commented on verses from the latter half of (8) upto (10) where the Iṣṭa-Kālāṁśas y happen to be less

481 than x. If y>x, or again the Prāk-Dṛk-graha has a longi- tude greater that of the Sun, the phenomenon of rising or setting which happened when y<x, will have to be taken as going to happen and that which was going to happen when y<x, must have happened already. It is enough to consider just one case instead of all the four cases since the argument is similar as before. Let us take y>x and it is an inferior planet in the east behind the Sun. According to the latter half verse (8) the rising had to take place y being less than x, whereas, now, y being greater than x, rising had already taken place, in contra- distinction to what happens when y<x. Here ends the Udayāstādhikāra.

SṚṄGONNATYADHIKĀRA Verse 1. Either in the last quarter of a lunation, or in the first quarter, on the day when the elevation of the cusps of the crescent Moon is to be determined, then either at the moment of Moon-rise or Moon-set or (for the matter of that during any part of the night) the Hsine of the altitude of the Moon is to be computed by noting the time from the moment of Moon-rise. Comm. Either in the first quarter or the last quarter of the lunation, ie. when the phase of the Moon according to the definition of phase in modern terms is less than half, the Moon will be a crescent. Also, generally, on the back-ground of the horizon, we notice that one of the cusps is more elevated than the other. This elevation goes by the name Sṛṅgonnati. Even in the middle half of the lunation, Bhāskara mentions that Brahmagupta and some others (meaning Sripati whom he closely follows) attempted at finding the elevation (strictly speaking elevation during the second quarter and depression during the third quarter) of the dark horns. Bhāskara does not appreciate this since, nobody would think about this as it does not appeal to the eye at all. Verse 2. To find the Hsine of the altitude of the Sun. The Hsine of the altitude of the Sun is to be com-puted, assuming the rising Sun to be in the opposite hemisphere, south or north (ie. if he be originally in the northern, assume him to be in the south) and using the formula given in verse 54 of Tripras'nādhyāya "अथोन्नता- दूनयुताच्चरेण" given the time measured in asus that has elapsed after Sunset.

483 Fig. 112 Comm. During the early part of the night or during the latter part thereof, when the Sun is below the horizon, the Sun will be occupying symmetrical positions with respect to the horizon at times which are equally removed from Sunset and the next Sun-rise. This is clear from the figure 112. Let A and B be two such symmetrical positions where AM and BN are the Hsines of the altitudes. Evidently AM = BN, C P̂ A = C P̂ B. Since the rising eastern hour-angle of the Sun equals he setting western hour- angle, and since C P̂ A = C P̂ B, the time elapsed after Sunset when the Sun is at B, will be equal to the time before Sun-rise in the position A. Hence by congruence AM = BN = Hsine of the altitudes in the two symmetric positions. Using the modern formula from the triangle PZS, cos z = sin ϕ sin δ + cos ϕ cos δ cos h, when C P̂ A = C P̂ B = h cos z will be the same in the two positions A and B. Putting z = 90 + θ, cos z = - sin θ will be the same ie. H sin θ will be the same in the two positions ie. the Śaṅkus will be the same. In the formula

484 cos z = sin ϕ sin δ + cos ϕ cos δ cos h Put z = 90 + θ, h = 90 + H, δ = δ in the positions A, B and z = 90−θ, h=90−H. δ=−δ in the positions A', B' We have then − sin θ = sin ϕ sin δ − cos ϕ cos δ sin H and sin θ = − sin ϕ sin δ + cos ϕ cos δ sin H which are identical. This means that the altitude θ below the horizon with +δ in the positions A, B is computable with −δ in the positions A', B'. This accounts for the statement made ‘गोलविपर्ययेण’. The second statement of the latter half of verse (2) says Śankutala = Śanku × ˢ/₁₂ which we have proved in Tripraśnādhyāya. Verse (3) and first half of (4). To obtain the Bhuja of the Sun. The Śankutala of the inverse altitude or the altitude below the horizon, is north (in contradistinction to what it is above the horizon). The sum or difference of the Agrā and Śankutala according as they are of the same direction or of opposite directions, is the Bhuja. The sum or difference of the Bhujas of the Sun and Moon, according as they are of opposite or the same directions is what is called the Spaṣṭa Bhuja whose direction is to be construed as that of the Moon. If the Bhuja of the Moon falls short of that of the Sun, then the direction of the Spaṣṭa Bhuja is that opposite to that of the Moon. Comm. In fig. 113, we have shown five different positions of the Sun S₁ to S₅ the feet of the Śankus being B₁ to B. From these feet of the Śankus draw perpendiculars on the Udayāstasutras as well as on the East-west line their points of intersection being respectively A₁ to A₅ and C₁ to C₅. The perpendiculars from the feet of the Śankus on the East-west line go by the name Bhujas, the perpendi-

485 Fig. 113 cular distances between the Udayāstasutras and the East- west line are called Agrās and the perpendiculars from the feet of the S'ankus on the Udayāstasutras are called S'ankutalas which were all defined in the course of the Tripras'nādhyāya. We have from the spherical triangle PZS, sin δ = sin ϕ cos z + cos ϕ sin z sin a so that sin δ / cos ϕ = tan ϕ cos z + sin z sin a which was shown as A=S+B in the Tripras'nādhyāya ie. Agrā = Sankutala + Bhuja. Changing δ into —δ and a into —a we have different formulae, the standard form being A=S+B. We don't propose to change ϕ into —ϕ, because in India this case does not arise. Thus our latitude being north, the S'anku- talas defined above as the distances from the feet of the Sankus from the Udayāstasutras are always deemed as south. It need not be reiterated that the Udayāstasutras

486

are the straight lines joining the rising and setting points of the celestial body. As such, these Udayāstasutras are all parallel to the East-west line. In fig. 114, in the horizontal plane containing the points A's, B's and C's in the respective cases, according to the Hindu convention, we talk of Agrā and Bhuja as being Uttara Agrā, Dakṣiṇāgrā, Uttara Bhuja and Dakṣiṇa Bhuja. Śaṅkutalam is always taken to be south except in the fifth case shown in figs. 113 and 114 when the altitude happens to be below the horizon. In this case the Śaṅkutala is spoken as north. Thus the general formula A=S+B assumes the following various forms, in the respective cases. (1) Uttarāgrā—Dakṣiṇa Śaṅkutala = Uttara Bhuja. This corresponds to A, B, C of fig. 114 where AC, is Uttarāgrā, BA, Dakshina Śaṅkutala and BC, Uttara Bhuja. (2) In the case of A₃ B₃ C₃, Śaṅkutala—Agrā = Dakṣiṇa Bhuja since B₃ A₃—C₃ A₃=B₃ C₃. This occurs after the Sun's diurnal path has crossed the prime-vertical and the Sun has a position on the south of the prime- vertical, having a northern declination. In the case of A₄ B₄ C₄, B₄ C₄ = B₄ A₄ + A₄ C₄ ie. Dakṣiṇa Bhuja = Dakṣiṇa Śaṅkutala + Dakṣiṇāgrā. In the fifth case when the Sun's altitude is below the horizon, B₅ C₅ = B₅ A₅ + C₅ A₅ ie. Uttara Bhuja = Uttara Śaṅkutala + Uttarāgrā. In the sixth case when the Sun has a southern declination and an altitude below the horizon, B₆ C₆ = A₆ C₆ + A₆ B₆ ie. Dakṣiṇa Bhuja = Dakṣiṇāgrā minus Uttara Śaṅkutala. In the case of A₂ B₂ C₂, Uttarāgrā =

487 Dakṣina Sankutala + Uttara Bhuja, since A₂ C₂ = B₂ A₂ + B₂ C₂. In the absence of a unifying convention, all these formulae are loose and apt to create confu- sion. So we shall unify all these formulae into the standard form A=S+B which holds good universally, if we have the conventions. (1) Agrā shall be deemed positive if it be north and negative if it be south ; (2) Similarly with respect to the Bhuja. But, with respect to the Śankutala, we shall deem it positive if it be south and negative if north. These conventions compre- hend all the cases and unify them into the standard form A=S+B. The corresponding conventions with respect to δ, a are that +δ corresponds to Uttarāgrā and +a corresponds to Uttara Bhuja. Having the above conventions, and finding the Bhujas of the Moon above the horizon and the Sun below the horizon, the Spaṣṭa Bhuja required in finding the Śṛṅgon- nati in the present chapter, is defined as the difference of the Bhujas of the Sun and the Moon and the sum thereof according as they are of the same direction north or south or of different directions. Also this Spaṣṭa Bhuja is by convention said to have the same direction as that of the Moon. If, however, the Bhuja of the Moon falls short of that of the Sun, then the Spaṣṭa Bhuja is said to have the direction opposite to that of the Moon. Verse (4). Definition of Koti. I deem that the Koti should be taken as the sum of the Śankus of the Sun and the Moon, the one being below the horizon and the other above respectively. Comm. We have defined above the Spaṣṭa Bhuja as the north-south distance between the feet of the Śankus

488 of the Sun and the Moon. Herein Bhāskara defines the Koti as the sum of the Saṅkus of the Sun and the Moon which is the vertical distance between the points on the armillary sphere which represent the Sun and the Moon. Bhāskara says 'I deem' to signify that he differs from Brahmagupta in this, who defines the chord joining the Sun and Moon on the armillary sphere which is equal to 2 Hsine of the SM on the sphere as the Karṇa. Brahma- gupta defines the Bhuja and Karṇa from which he deduces the Koti as √(Karṇa² — Bhuja²). Bhāskara argues that since the Karṇa defined by Brahmagupta is not in the vertical plane, since the Sun and the Moon are not in the same vertical plane, so, the Koti defined by him throught he Karṇa will not be in the vertical plane. The Karṇa defined by Bhāskara is not on the other hand the chord joining the Sun and the Moon but lies in the vertical plane in which the perpendicular from the Moon's position on the armil- lary sphere, on the horizontal plane containing the Sun's position on that sphere. Also the Bhuja defined both by Brahmagupta and Bhāskara is the projection on the north- south line of the join of the Sun's position on the sphere and the foot of the perpendicular from the Moon's position on the horizontal plane through the Sun's position and it is not actually the above join. Bhāskara is evidently guided by the right angled triangle SMN' which is not strictly a spherical triangle as per its modern definition as the arc SN is not that of a great circle. This figure guided him to take the Bhuja horizontal and the Koti vertical so that his Karṇa is also in a Fig. 115 vertical plane. Bhāskara's

489 Karṇa therefore has nothing to do with the arc SM but only has the virtue of being in a vertical plane. Sripati adopted Brahmagupta's method. Kamalākara neither follows Brahmagupta nor Bhāskara but follows a method of his own. In fact the methods adopted by these ancient Hindu astronomers are not mathematically correct because the question of determining the cusps as well as phase of the Moon is concerned with the actual positions of the Sun, Moon and the earth in space and not as seen on the sphere. Hence M. M. Sudhakara Dwivedi has written a small book by the name Vāstava-Sṛṅgonnati following modern methods. The modern method which gives the truth of the matter is depicted in standar modern texts. Verse 5. The hypotenuse or Karṇa is the square-root of the sum of the squares of the Bhuja and Koti. The Bhuja multiplied by 6 and divided by the Karṇa gives what is known as the Dik-valana of the Moon. The dire- ction of the Valana has the same direction as the Spasṭa Bhuja defined before. Comm. The idea is that taking the radius of the Moon's disc to be six angulas or units, representing the Karṇa, the magnitude of the Bhuja on the same scale gives a measure of what is defined as Valana. Thus Valana = 6B/K where B and K stand for the Bhuja and Karṇa defined before. The idea of this Valana will be clarified in the ensuing verses. Verse 6. The Hsine of the elongation of the Moon is to be multiplied by the radius vector of the Moon measured in Yojanas, and divided by the radius vector of the Sun, also measured in Yojanas ; the arc of the Hsine so obtained is to be added to the longitude of the Moon in the bright half of the lunation and is to be subtracted from the same in the dark half. 62

490 Comm. This is a correction to be made in the longi- tude of the Moon to depict graphically the phase of the Moon. Bhāskara says that many of the prior astronomers took that the phase of the Moon was in direct proportion to the elongation of the Moon. Taking the radius of the Moon's disc to be six angulas or units, and assuming that when the elongation is 180°, the entire disc being illumi- nated, it was thought that 12 units of the diameter correspond to 180° of elongation so that the Śukla of the disc measured by the central width of the illuminated disc increases at the rate of 1 unit for 15° of elongation. (The word Śukla may be taken to correspond to the modern word phase. Śukla is expressed in angulas, taking the diameter of the disc to be 12 angulas. The measure of the portion of the diameter of the Moon's disc perpendicular to the diameter which is the join of the extremities of the cusps, covered by the illuminated part of the disc, (which may be defined as the maximum width of the illuminated part of the disc) measured in angulas is said to be the Śukla. The word 'phase' is used to signify the ratio of the above width to the diameter. Śukla is expressed in angulas, whereas phase is expressed as a ratio. The Śukla is equal to twelve times phase). Bhāskara rightly argues that this method of measuring the Śukla is approximate because he says that six angulas of Śukla is had when the elongation is not 90°, but only 85°-45'. This may be substantiated as follows from fig. 116. Let E, M, S stand for the earth, Moon and the Sun. Let ^ EMS = 90° so that it is a moment of dichotomy ie. the moment when the phase is half and the Śukla 6 angulas. Let θ be then the elongation of the Moon, so that cos θ = EM / ES . Taking the average values given for EM and ES by Bhāskara, cos θ = 51566 / 689377 = 19 / 254 (obtaining a convergent) = .0748.

491 From tables, we find θ = 85°-43' which Bhāskara takes to be 85°-45'. Fig. 116 In the wake of this, Bhāskara tries to make amends in the approximate formula prescribed to obtain the Sukla. He prescribes addition of 4°-15' to the longitude of the Moon in the bright half, and subtraction in the dark. In between the moment of conjunction and the moment of dichotomy, he derives the following formula (vide fig. 117). Fig. 117 The deficiency of 4°-15' is had in the form of the angle S₂ E S₁. When the Sun is at S₁ the Moon being at M₁, it

492 is the moment of conjunction. When the Sun has moved from the point S₁ to S₃, S₁ Ê S₃ being 90°, there is a deficiency of magnitude S₁ E S₂. In other words, when S₂ N has assumed the position S₃ E there is a deficiency of 4°-15′ ie. for an increase of 90° of elongation, there is a deficiency of 4°-15′ in the longitude of the Moon. Hence, for the Hsine to become the radius, there corres- ponds a portion E M₂ or S₂ N, which is the Hsine of 4°-15′, so that the following rule of three is adopted. 'If the Sun's distance E S₃ corresponds to the radius, what does E M₂ the distance of the Moon correspond to ?' The result is (m / s) × R, m and s being the respective distances. Then the following proportion is used "If by H sin ξ equal to R, ξ being the Moon's elongation, we have mR/s, what shall we have for an arbitrary H sin ξ ?" Thus the answer is (H sin ξ × mRs) / R = H sin ξ · (m / s) . H sin⁻¹ ((m / s) × H sin ξ) where m and s are the distances of the Moon and the Sun, is to be added to the longitude of the Moon or to be subtracted as the case may be, to have the rectified longitude of ths Moon from which the phase is to be calculated according to Bhāskara. Here we are to offer the following remarks. No doubt, Bhāskara was correct in estimating the moment of dichotomy to be that when ξ the Moon's elongation is not 90° but 85°-45. But the amended formula is not the correct mathematical form. The modern formula to find the phase is (1+ cos EMS) / 2 aud since from fig. 116, SM is nearly equal to SE, so EMS is very nearly equal to

493 180-MES, so that (1+ cos EMS) / 2 = (1 —cos MES) / 2 which may be taken to be an approximate truth. This ‘formula was indeed given by Lallācārya in the following verse, long before Bhāskara. “रविशीतकरान्तरांशजीवा विपरीता शशिखण्ड- ताडिता च, विहृता त्रिमजीवया सितं स्यात् शशलक्ष्माङ्कवदङ्गुलानि तस्मिन् ” verse 12 (Candra Sṛṅgonnatyadhikāra). Here विपरीता रविशीतकरान्तरांशजीवा mean Hvers ξ = |R—H cos ξ. शशिखण्डता- डिता = multiplied by the radius of the disc of the Moon. विहृता त्रिमजीवया = divided by R. Thus the formula given by Lallācārya amounts to r (R—H cos ξ) / R = d/2 (1—cos ξ) where r gives the angulas in the radius of the Moon's disc and d the diameter, and ξ = elongation of the Moon. Since the definition of phase in modern terms is a ratio, namely the ratio of the maximum width of the crescent to the diameter, phase × d = Sukla of the Hindu astronomers ∴ (1 — cos (elongation of the Moon)) / 2 × d = Sukla = r (1 —cos MES) as given by Lallācārya. So Lallā- cārya's formula is quite correct. We shall trace Lallā- cārya's steps in obtaining such an intricate correct formula which was overlooked by Bhāskara. (Refer fig. 118) Let E be the earth, S the Sun and M₁, M₂, M₃ etc. the positions of the Moon as the elongation gradually increases. No doubt the Sukla increases with elongation as known to all Hindu astronomers. But, the question is, does it increase with the sine or versine? We know both the sine and versine increase with the angle. Lallācārya noticed that Sm₁, Sm₂, Sm₃ as the Moon occupied positions M₁, M₂, M₃ etc., are the Hindu versines which are increasing. So he postulated that the Sukla increases with the Hindu versine. His formulation was thus correct. But why Bhāskara overlooked this correct formula was traceable to his getting prejudiced

494 Fig. 118 against Lallācārya's some other formulae which used the Hversine where he ought to have used the Hsine. For example in the case of the Valana Lallācārya's mistake is traceable to his confusion as to whether he was to choose the Hsine or Hversine when both increase as the angle increases. Verse 7. Graphical depiction of the cusps. Fig. 119