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

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

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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

495 Let in fig. 119, ST be the Bhuja, MT the Koti and SM the Karṇa formerly defined. As per verse (5) 6B / K = Valana where B = ST, K = SM, and 6 = MG so that GE = 6B / K = Valana, which is in practice drawn as E'G' from the east point e, in the form of a Hsine. CG goes by the name Valanasūtra. Compute the Śukla in angulas after making the prescribed correction in the longitude of the Moon as stated by Bhāskara and dividing the elon- gation of the Moon, thereafter obtained, by 15. Mark off the Śukla in angulas along the Valanasūtra from G. Suppose GD is the Śukla. Draw the diameter AB perpendicular to the Valanasūtra passing through M. Draw the circle circumscribing D, A, B. Its centre lies evidently on the Valanasūtra, say C. The circle drawn is called Parilekha Vṛtta, its radius CA is called Pari- lekhasūtra and the point C Parilekha Vṛtta Madhya. In the triangle CAM, which is right-angled CA is the Karṇa, AM the Bhuja and MC the Koti. CD is equal to the Karṇa CA so that MD = CD – CM = Karṇa – Koti = K – k (say). We have AM² = CA² – CM² = K² – k² = 36 = B²; Hence K + k = B² / (K – k). Here K – k = MD is known because GD the Śukla is known. So B² / (K – k) = K + k is known. Thus knowing K – k and K + k, by using what is called Saṁkramagaṇita ie. by adding K + k and K – k, we have K and by sub- tracting we have k. Here the Koti CM is called Vibhā and the Karṇa CD the Swabhā. The Parilekhasūtra or the radius of the Parilekha Vṛtta being the Karṇa, the Swabhā is thus the Parilekhasūtra. In the wake of this exposition, the translation of verse (7) runs as follows. "Let the compliment of the elongation (corrected as directed in verse (6)) divided by 15 be the denominator; let the numerator be 36; take the

496 result after division and put it in two places; with this and the complement of the elongation divided by 15, using Samkramagaṇita, we have successively Vibhā and Swabhā''. Comm. Here, the meaning of taking the complement of the elongation and dividing by 15 is to obtain the magnitude of MD directly; for, (90 - e) / 15 = 6 - e/15 = MG - e/15 = MG - GD = MD where e is the elongation and e/15 is the Sukla. In other words, finding the Sukla by dividing the elongation by 15 and subtracting it from the radius to get MD is the same as taking the complement of the elongation and dividing it by 15 to obtain MD. Bhāskara quotes in the course of the commentary, the verse from his Leelāvatī, namely "भुजाद्वर्गितात् कोटिकर्णान्त राप्तम् द्विधा कोटिकर्णान्तरेणोनयुक्तम्। तदर्धे क्रमात्कोटिकर्णौ भवेताम्, इदं धीमताऽऽवेद्य सर्वत्र योज्यम्" which means B² / (K - k) = K + k since K² - k² = B². This situation arises when the Bhuja B and the difference of Koti and Karṇa ie. K - k are given and it is required to find K and k separately. Verses (8) and (9). Draw a circle with radius 6 angulas to represent the disc of the Moon. Mark the disc with the Cardinal points signifying the east, west, north and south. Compute the Valana as directed in verse (5) (which represents E' G' in fig. 119) and mark it off as a Hsine from the west point w in the last quarter and from the east point e in the first quarter. From the centre M of the Moon's disc, mark off Vibhā MC (computed as directed) along the join of MG'. With centre C and radius Swabhā (already computed as directed) draw a circle. The spherical sector of the Moon's globe thus demarked by the arc of the circle ADB namely AGBD is found to have the elevated cusp in the opposite direction in which the Valana has been marked.

497 Comm. The directions are clear in the wake of the previous commentary. In marking off the Valana, Bhās- kara says ‘from the west in the last quarter and from the east in the first quarter’. In the first quarter, the Moon will be visible immediately after Sunset in the western sky and the illuminated part of the Moon will be towards the western side of the disc in which direction the Sun is situated. The eastern point and the western point of the disc are easily discernible on the background of the horizon. Drawing first the diameter ‘WE’ of the disc along the vertical circle in case the Moon is due west or else along a small circle parallel to the prime-vertical called Upa-Vṛtta, we have the east point of the disc namely e vertically above the disc in the first quarter. (The figure 119 is shown for the first quarter). Then we are directed to mark off the Valana E′ G′ in the form of a Hsine'. Since the Valana is already computed as directed in verse (5), to mark off this Valana we have to lay the scale perpendicular to WE. Such a point G′ on the circum- ference of the disc is to be marked from which the Hsine equals the Valana computed. It is to be noted here that it is not marking G′ E′ from G′, for, G′ is not known, nor E′ for the matter of that, but locating G′ knowing the magnitude of the Valana. Though we happened to mention before that GE is the Valana, in the light of what Bhāskara mentions in verse (8) we have to under- stand that in the first quarter, the practice is to mark off Valana from the point e marked on the disc before hand. Though it does not matter when we take EG to be the Valana E′ G′, we have to notice the practice followed. The direction given in the course of the commentary “केन्द्राद्वलनोपरिवृत्ताद् बहिरपि खटिकया सूत्रमुच्छाद्यम्” meaning thereby that from the point M, we have to mark off along MG′ where G′ is the point pertaining to Valana, a line with the help of a chalk and a fine thread, namely MC. This practice is still in vogue adopted by masons to draw straight lines. 63

498 It is interesting to note Bhāskara waxing poetic in the course of the commentary under these verses describing how the illumination of the Moon's disc by the rays of the Sun is effected. The description is quite apt, inter- esting and scientific reminding us of Varāhamihirācārya's description in the verse 'सलिलमये शशिनि etc.' in Bṛhat- Saṁhitā. Of course, following Varāhamihira Bhāskara also takes the Moon's globe to contain water within its bosom which Modern Science has yet to confirm. Verses 10, 11, 12. The Koti and Karṇa defined by Brahmagupta, do not lead us to accordance between computation and observation to locate the cusps. I request good mathematicians to verify this carefully. In a place where the latitude is (90—ω) = (90—24) = 66°, when the ecliptic coincides with the horizon, and when the Sun is in the beginning of Meṣa which is then rising in the east, and the Moon in the beginning of Makara, then the Moon is dichotomized by the meridian and the illuminated part of the Moon's disc is towards the east. This does not hold good according to Brahmagupta's definition of Koti, because then the Bhuja as well as Koti according to his definition is equal to R. When the Bhuja is zero, the cusps will be horizontal, and when the Koti is zero, they will be vertical. Brahma- gupta's Bhuja and Koti both being equal to R, the cusps therefore cannot be vertical which is against truth as stated above. Why should I make this statement? May my homage be to those great. Comm. When the latitude of a place be 90—ω, and when the ecliptic coincides with the horizon, it is clear that in whatever positions be the Sun and the Moon (assuming that the Moon is also approximately on the horizon) the cusps of the Moon are always vertical, there being Bhuja only and no Koti. But according to Brahma-

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gupta's definition of Karṇa, it will be in this particular case. √(R² + R²) = R√2 so that the Koṭi will be √(2 R² — R²) = √R² = R, where the Bhuja also is R (projection of the line joining the Sun and the Moon on the north-south line. Though in the verse (11) above one particular position of the Sun and one of the Moon, are contemplated, the same argument holds good, says Bhāskara in the course of the commentary, whatever positions are occupied by the Sun and the Moon on the ecliptic. In this case there is only Bhuja existing and no Koṭi, so that the cusps will be vertical. But in all these cases, there is Koṭi aecording to Brahmagupta's definition, so that the ousps will not be vertical according to him, which is clearly against truth. Here we have to note that in the example cited by Bhāskara the Bhuja equal to R is along the north-south direction, and even though the Koṭi according to Bhās- karajs definition is conceived to be vertical, the Karṇa according to Brahmagupta's conception being ES where E and S are the east and south points, his Koṭi will be horizontal coinciding with OE, O being the centre of the horizontal ecliptic. Further according to Brahmagupta, the join of the ousps will be perpendioular to the Karṇa whioh does not therefore go against truth. Similarly in all the cases wherever be the Sun and the Moon on the horizon, Brahmagupta's Karṇa being a line joining the centres of the disos of the Sun and the Moon, it will be a horizontal line and so the ousps could be vertical. It is not clear whether or not Bhāskara recognized this namely that the Karṇa and Koṭi of Brahmagupta in the cases cited above are all horizontal lines so that the cusps could be vertical even according to Brahmagupta. That is why he pays homage to Brahmagupta in the last line of (12). Probably Bhāskara expected that Brahmagupta's Koṭi also should have been vertical as his. The fact that he recognized that Brahmagupta's Koṭi will not be vertical

500 but inclined, is justified here since the Kotis in all the cases cited are all horizontal lines. In fact, as we stated before, even Bhāskara's analysis is not sound, for which reason, M. M. Sudhākara Dwivedi, wrote the booklet called Vāstava Sṛngonnati based on modern lines. As this topic could be found in books of modern astronomy, we need not go into the treatment of Sudhākara Dwivedi here. Here ends the Sṛngonnatyadhikāra