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

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-

499

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

GRAHAYUTYADHIKĀRA Verse 1. The mean diameters of Mars, Mercury, Jupiter, Venus and Saturn are respectively 4′-45″, 6′-15″, 7′-20″, 9′, 5′-20″. Comm. Bhāskara gives us a method under verse (5) of Chandragrahaṇādhikāra, as to how the angular diameters of celestial bodies were being measured with an instrument that we may call 'protractor', describing it as follows. “यस्मिन् दिने अर्कस्य मध्यतुल्या स्फुटा गतिः स्यात् तस्मिन् दिने उदयकाले चक्रकलाव्यासार्धमितेन यष्टिद्वितयेन मूलमिलितेन तत्रस्थदृष्ट्या तदग्राभ्यां बिम्बप्रान्तौ विध्येत् “या यष्ट्यग्रयोरन्तरकलाः ता रविबिम्बकला भवन्ति मध्यमाः” ie. On the day on which the true motion of the Sun equals the mean, in the morning, observe with an instrument having two equal rods jointed at one end (and the other ends being connected by a flexible protractor marked with minutes and seconds of arc) placing the eye. at the joint of the rods, and the two rods pointing to the extremities of a diameter of the disc. The magnitude of the disc is then read on the protractor, which gives the mean diameter”. This method is alright so far as it goes. ' The measur- ing being done at the time of morning and that too with the naked eye might have been responsible for the exaggerated magnitudes of the diameters of the discs of the planets, as compared with their modern values. This kind of exaggerate estimate is due what is called the phenomenon of irradiation which increases the apparent size of a brilliant body when seen at some distance. Even Tycho Brahe, an accurate and brilliant astronomer prior to the invention of the telescope gave estimates of these angular diameters which are nearly the same as remarked by Burgess in his translation of Sūryasiddhānta under verses 13, 14 ch. VII.

502 Verse 2. These estimates being multiplied by the difference of the radius and Śīghrakarṇa and divided by thrice the Śīghra-antyaphalajyā are to be added or sub- tracted from the mean values above given according as the Śīghrakarṇa is less or greater than the radius to give the rectified values. Three minutes of arc are to be construed as one angula in this respect. Comm. When the Śīghrakarṇa equals the radius we know that the planet is situated at the mean distance. The word planet here stands, of course, for the Mandaspaṣṭa- graha which may be roughly taken to be the mean planet, the equation of centre being small. If the Śīghrakarṇa falls short of the radius, then evidently the planet is nearer the earth than the mean position so that disc of the planet appears to be bigger. Otherwise, the planet is further and its disc appears to be smaller. It was noted that approximately there was an increase or decrease ⅓ of the magnitude of the disc by the decrease or increase of the Śīghrakarṇa by the antyaphalajyā. Hence, in between the two positions, rule of three is used to obtain the magni- tudes as follows. "If by a difference of the Śīghrakarṇa and radius equal to the antyaphalajyā, there is a difference of ⅓ of the actual magnitude, what would it be for an arbitrary difference?" The answer is d × ⅓ × 1/a = d/3a where d = Śīghrakarṇa or radius and a the antyaphalajyā. This difference is to be added or subtracted to the mean value, as the case may be. Verse 3 and first half of 4. To obtain the time of the conjunction of two planets, compute the difference of the longitudes of two planets, and divide by the difference of their daily motions. If one of the planets be retrograde, divide by the sum of the daily motions. The result gives the number of days approximately after the moment of conjunction if the slower planet has a longitude falling short of that of the quicker. If one of the planets be

503 retrograde, and if its longitude be the lesser then also the conjunction was past by the number of days computed. In the other cases the conjunction is to take place after the number of days computed. If, however, both the planets be retrograde, then if the slower of them has a longitude less than that of the quicker, then the conjun- ction is ahead, otherwise past by the number of days. Comm. Clear. In the case of one or both the planets being retrograde, the word 'slower planet' means that planet whose retrograde motion is slower and not the planet whose mean motion is slower. Latter half of verse 4 and verse 5. To rectify the moment of conjunction. Having computed the approximate time of conjun- ction, obtain the true motions of the planets pertaining to that day, rectify them for Āyana-Dṛk-Karma and following the process indicated in verse (3) above, again compute the moment of conjunction. (This will be a good approxi- mation). This conjunction will be one on the polar latitudinal circle. If Āyana-Dṛk-Karma be not done, then the conjunction will be on the circle of celestial latitude. Comm. Bhāskara says that the conjunction on the circle of polar latitude is preferred because this could be observed as there is a star at the celestial pole and the movable circle of polar latitude could be moved into a position in which the two planets could be seen situated thereupon. The method of successive approximation is self-explanatory. Bhāskara, however, adds that when the conjunction is on the circle of celestial latitude, the planets will be seen to be closer. Verse 6 and first half of verse 7. To obtain the north- south celestial latitudinal distance between two planets.

504 Having computed the number of days by which the celestial latitudinal conjunction was past or is going to take place, let the common celestial longitude of the two planets be obtained by the method of successive approxi- mation for the moment of celestial latitudinal conjunction. Let the celestial latitudes of the two planets be rectified for parallax in latitude as in the case of solar eclipse. The difference of these celestial latitudes in case they are of the same direction or the sum if, of opposite direction, gives the north-south distance of the planets with respect to the ecliptic. Having known the directions of the celestial latitudes with respect to the ecliptic, if the two celestial latitudes happen to be both south or both north, then the planet with lesser celestial latitude is said to be in the opposite direction with respect to the other; that is, suppose both have northern celestial latitudes and suppose p₁ has a smaller northern celestial latitude than p₂, then p₁ is said to be south of p₂. Similar is the case if both the celestial latitudes happen to be south. Comm. Here Bhāskara does not specify whether he is talking of conjunction on a polar latitudinal circle or on a celestial latitudiaal circle, though the previous procedure indicated by him to obtain the moment of con- junction gives preference to polar latitudinal conjunction which is more easily observable. But, here, as he prescribes rectification of the latitudes for parallax in celestial latitude, we have to construe that he is speaking of conjunction with respect to celestial longitudes alone, because, in the context of parallax, no method was indi- cated by him for parallzx in polar latitude. Latter half of verse 7 and verses 8 and 9. Case of occultation. If the north-south celestial latitudinal distance happens to be less than the sum of the angular radii of the two planets, an occultation occurs (what we say 'eclipse'

505 with respect to the Sun and the Moon, holds good with respect to occultation). In this case of occultation, we have to rectify the time of conjunction with respect to parallax in latitude also. For obtaining this parallax in longitude, let the planet which is nearer the earth be taken as the Moon and the other the Sun. But to obtain the longitude of the Vithribha, which is necessary to compute parallaxes in longitude and latitude, the lagna of the moment of conjunction is to be computed from the position of the Sun and not that of the planet assumed to be the Sun as directed above. Having obtained the parallax in longitude, the computed moment of conjun- ction is to be rectified for parallax in longitude, (if necessary by the method of successive approximation) to obtain the actual moment of apparent conjunction ie. occultation here. This procedure is worth-adopting only when the occultation in question takes place above the horizon and is observable. The north-south celestial latitudinal distance in this case of occultation corresponds to the celestial latitude of the Moon in the case of a solar eclipse. The direction of this celestial latitude is to be construed as that of the direction in which the planet near the earth is situated with respect to the other. If the planet which is nearer the earth happens to have a motion lesser than that of the other, or be retrograde, then the planet which is situated at a greater distance from the earth will be over taking the other so that the higher planet gets occulted in the eastern direction of its disc. Thus the first contact is to be known to be in the east and the last contact would be in the west; (If otherwise, the other way). Comm. Self-explanatory. Here ends the Grahayutyadhikāra. 64

506 Bhagrahayuti (Conjunction of a planet with respect to a star) The longitudes of the stars (professed to be polar). The polar longitudes of the stars from Aswini includ- ing Abhijit are as follows. R d m R d m Aswinī 0- 8- 0 Swātī 6-19- 0 Bharaṇī 0-20- 0 Visākhā 7- 2- 5 Kritticā 1- 7-18 Anūrādhā 7-14- 5 Rohiṇī 1-19-18 Jyeṣṭhā 7-19- 5 Mṛgasīrṣa 2- 3- 0 Mūlā 8- 1- 0 Ārdrā 2- 7- 0 Purvāṣādhā 8-14- 0 Punarvasū 3- 3- 0 Uttarāṣādhā 8-20- 0 Puṣyamī 3-16- 0 Abhijit 8-25- 0 Asreṣā 3-18- 0 Sravaṇam 9- 8- 0 Makhā 4- 9- 0 Dhaniṣṭhā 9-20- 0 Purvāphalgunī 4-27- 0 Satabhiṣak 10-20- 0 Uttarāphalgunī 5- 5- 0 Purvābhādrā 10-26- 0 Hasta 5-20- 0 Uttarābhādrā 11- 7- 0 Chitrā 6- 3- 0 Revatī 0- 0- 0 Comm. In the enumeration of these longitudes, Bhāskara makes three statements which we have to note. (1) That these longitudes are rectified for Āyana-Dṛk- Karma ie. that they are polar longitudes, (2) That the longitudes of Kritticā and Rohinī are less by 32′. (The longitudes shown in the table above are those incorporating this specified correction), (3) That the longitudes of Visākhā, Anūrādhā and Jyeṣṭhā are to be increased by 5′. (This correction is also incorporated in the table given above).

507 It is to be noted that these longitudes are the same given by Brahmagupta originally and copied by S'rīpati as well as the corrections indicated above. But herein a mistake was committed as clarified by Bhāskara later in the end of the chapter namely that the polar longitudes are subject to what is called Āyanavikāra though the celestial longitudes are not. We say that the celestial longitudes are not subject to such an Āyanavikāra ie. that change due to the phenomenon known as the precession of equinoxes, because the Hindu system is Nirayana ie. reckons the longitudes from the first point of Aswinī which is its zero point instead of reckoning from r. If longitudes are measured from r, as r is preceeding, the longitudes of stars will be steadily increasing all at the same annual rate of precession namely about 50-25" per year. These increasing longitudes of the modern system go by the name Sāyana longitudes. It might be thought that since the polar longitudes also are measured from Aswinī and along the ecliptic like celestial longitudes, they also don't vary like the Nirayana celestial longitudes ; but it is not so, because the effect of precession on the right ascension and declination of a star have both their effect upon the polar longitude as well as polar latitude. We cannot say that Bhāskara did not know this but we may say that the effect on the polar longitude and latitudes were construed by him as neglible and would be appreciable only in the long run. Verses 4, 5, 6. Sphutas'aras or rectified latitudes of the stars. Aswinī 10°- 0 north Ārdrā 11 - 0 south Bharaṇī 12°- 0 ,, Punarvasū 6 - 0 north Kṛtticā 4 -30 ,, Puṣyamī 0 - 0 ,, Rohinī 4 -30 south Asreṣā 7 - 0 south Mṛgas'irṣa 10 - 0 ,, Makhā 0 - 0 north

508 Purvāphalguni 12 - 0 north Purvāṣādhā 5 -20 south Uttarāphalguni 13 0 ,, Uttarāṣādhā 5 - 0 ,, Hasta 11 - 0 south Sravaṇam 30 - 0 north Chitrā 1°-45' ,, Abhijit 62 - 0 ,, Swātī 37°- 0 north Dhaniṣṭha 36 - 0 ,, Visākhā 1 -20' south Śatabhiṣak 0 -20 south Anūrādhā 1 -45 ,, Purvābhādrā 24 - 0 north Jyeṣṭhā 3 -30 ,, Uttarābhādrā 26 - 0 ,, Mulā 8 -30 ,, Revatī 0 - 0 ,, (a) These are also what were given by Brahmagupta and copied by Śrīpati, (b) Bhāskara mentions in the Golādhyāya "नक्षत्राणां स्फुटा एव स्थिरत्वात् पठिताः शराः, दृक्कर्मणा- ऽऽयनेनैषां संस्कृताश्च तथा ध्रुवाः" ie. In as much as the stars are fixed we have given their latitudes and longitudes rectified. But here, there is a point to be noted as Bhāskara has put us in a doubt namely that the rectified latitudes which he speaks of elsewhere under verse (3) Grahacchāyādhikāra, Gaṇitādhyaya are not exactly the polar latitudes spoken of here. (Ref. fig. 120). Let ♈MR be the ecliptic and ♈N be the celes- tial equator. Let S be a star, k be the pole of the ecliptic, and p be the celestial pole. Then ♈R is the celestial longitude, SR the celestial latitude which are known as the Dhruvaka and S'ara (Asphuta- s'ara). ♈M is the polar longitude [Fig. 120] and SM the polar latitude. MR is the arc connoting the Āyana- Dṛk-Karma correction, so that celestial longitude ± Āyana- Dṛk-Karma correction = polar longitude (plus or minus according as the longitude lies in the 2nd and 4th quadrants or 1st and 3rd quadrants). If the longitude just equals 0°, 90°, 180° or 270°, the correction of Āyana--Dṛk

: 509 Karma vanishes. It must be noted here that when the longitude is 0° or 180° Āyanavalana is maximum but Āyana-Dṛk-Karma vanishes. There is no ambignity here in this polar longitude because Bhāskara is unequivocal in defining this. But under verse (3) Grahacchāyādhikāra, Bhāskara means by Sphuṭaśara SL and not SM but by Sphuṭaśara here he means SM. It is ridiculous to suppose that Bhāskara did not know that B √(R²—Āyanavalana is less than β. This B √(R²—a²)/R (a=Āyanavalanajyā) he defined as Sphuṭaśara there under verse (3) cited. SR is defined by him as Asphuṭaśara or simply Śara. Now here in the commentary he makes us believe that Sphuṭaśara is SM which is the polar latitude. This confusion created by Bhāskara leads Ramaswarup the editor of Brahmagupta Siddhānta as well as one Mukhopadhyaya the author of the thesis, ‘Hindu Nakshatras’ to suppose that Bhāskara was wrong in supposing that (B/R) √(R²—a²) > β. Bhāskara could not evidently commit such a silly mistake but we must infer that in that context he called SL as the Sphuṭaśara which being added to Rn the Krānti or what is the same LN gives SN the Sphuṭakrānti or the modern declination. Rn is called by him as Asphuṭakrānti. In this context he calls SM as the Sphuṭaśara since it is Dhruvābhimukha ie. directed towards the pole. Of course, Bhāskara should not have called both SL and SM as Sphuṭaśaras but since he was deliberately defining the Sphuṭaśara as SL previously and now as SM, and since he could not commit such a glaring mistake as to construe (B/R) √(R²—a²) as greater than β, we should not rush to pronounce that Bhāskara was wrong. Only we could say that he is inconsistent to that extent. Verse 7. The polar longitudes of Agastya (Canopus) is 87° and his polar latitude is 77 south. The polar

510 longitude of Lubdhaka (Sirius) 86° and its polar latitude is 40° south. Comm. Clear. The star Agastya is considered to be important in Indian literature because the heliacal rising and setting of Agastya are directed to be noted and in fact are being noted in every panchanga even today from times.immemorial. Even Kālidāsa alludes to this heliacal rising of Agastya as inaugurating the Sarat-kāla or autumn which was reiterated by Varaha Mihira in his Bṛhat-Saṁhitā under the verse “भगवति जलधरपक्ष्मक्षपाकरा- केंक्षणे कमलनाभे, उन्मीलयति तुरङ्गमकरिरथनीराजनं कुर्यात्” ie. when Lord Vishnu whose eyes are supposed to be the Sun and the Moon and his eyelids the clouds, opens his eyes ie. on Kārtica Ekādasi the 11th day of the bright half of the lunar month of Kārtica then Kings are directed to perform Nīrājanavidhi for his horses, elephants and chariots (to start on an expedition for war). It is to be noted that this was so long ago in times of yore that Agastya used to rise at the beginning of autum. Now the star is rising about 22nd August long in advance even in the rainy season. This is on account of the effect of precession of equinoxes. Verse 8. The Iṣṭanādis for Agastya are said to be two, for Lubdhaka 2⅙, for other stars which are next in size, 2⅓ and for still smaller ones the Iṣṭanādis are to be taken still more. Comm. Iṣṭanādis ie. the time in nādis (where a nādi is equal to 24′ of time) giving the time in between the rising moments of the star and the Sun, when the star rises heliacally. In other words, let the star Agastya rise at a particular time t. If the Sun rises at t+48′, then it is the time for Agastya to rise heliacally. This again means that if the Sun sets at time T and the star at T+48′ in its diurnal motion, it is time for the star to set heliacally in the west. Similarly for the other stars. It will be noted here that stars set heliacally in the west and rise

511 heliacally in the east. This phenomenon has been long in the notice of even the most illiterate people of India from times immemorial, as they were used to get up from beds round about the time when a brilliant star rose heliacally and stood in the eastern horizon, shining for a good length of time before Sun-rise. Each star of first magnitude thus played the part of a morning star for some time, though perhaps the illiterate folk mistake them to be the same star. Especially the brightest stars of the zodiac thus play the part of morning and evening stars. The case with respect to Agastya and Lubdhaka is different in that even though they are far away from the zodiac, yet they were noticed to be morning and evening stars by virtue of their being of the first magnitude at a spot of the sky in the vicinity of which no other such brilliant stars are there. This is the reason why panchangā—Computers have been in the habit of recording in the panchangā even to-date the heliacal rising and setting of Agastya, also because the heliacal rising of this star synchronized with the setting in of S'arat-kāla or autumn. Since in India the lunar Kārtica Ekādasi, ie. the 11th day of the bright half of the lunar month roughly synchronized with 15th Nov., when the Sun rose far in the south of the horizon, this Agastya, though it be in the far south, happens to play the part of a morning star and about the day when it rose heliacally, there were no more rains and waters of the rivers stood crystal-clear as described by many a Sanskrit poet like Kālidāsa (vide the famous verse of Kālidāsa प्रसादोदयादम्भः कुम्भयोने र्महौजसः 4th canto Raghu- vams'a). The star Lubdhaka or Sirius, the dog-star as it is called in English parlance also was conspicuous as a morning and evening star, whose heliacal rising was noticed and recorded by English poets like Shakespeare and Milton. Since one nādi corresponds to 6°, two nādis correspond to 12°, degrees, which are spoken as Kālāṁs'as for the

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