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

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

532 when the lunar orbit comes in between the celestial Equator and the ecliptic. This situation, in its turn, depends upon the position of Rāhu. Since Rāhu's sidereal period amounts to nearly 18½ years, the lunar orbit happens to take its position in between the celestial Equator and the Ecliptic for sufficiently a long period as shown below. That Bhāskara could visualize this, reflects credit to his genius. This is why he formulated Δδ ± Δβ stressing upon the plus or minus sign. Fig. 128 Bhāskara gives an example under the above verse in the commentary to justify his finding given above. Suppose Rāhu is at the autumnal equinox and the longi- tudes of both the Sun and the Moon are equal to 79°. Adding the Ayanāṁs'as 11°, their longitudes are each 90° so that both of them are at the summer solstice ie. an Āyana Sandhi. (Of course the Moon is not exactly at the position of the Sun, but he being in his own orbit, his longitude measured along the Ecliptic is 90°). Then evidently the lunar orbit lies in between the celestial equator and tho Ecliptic. Then the declination of the Moon is 24°–4½°＝19½°＝1170′ and the declination of the Sun is 24°＝1440′. Thus the Moon's declination when it is maximum falls short of that of the Sun. After half of the sidereal period of the Moon namely 13⅔ days, the

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

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

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

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

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

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

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

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

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

542 zero ?" The answer is t (a + b) / Adya - Anya which gives the time after which Vaidhṛti is likely to occur. Suppose a and b the declinations observed at a particular moment, are one north and and the other south. Compute their difference a - b; after a few ghatis or days again from the difference of the declinations c and d ie. c - d. If c - d < a - b then Vaidhṛti is going to occur, otherwise it has elapsed. This is so because the difference has to vanish for Vaidhṛti to occur. Viewing the situation algebraically, ie. construing northern declination to be positive and southern negative, we could have laid down only one criterion, instead of separating the issues and stipulating separate criteria. Proceeding on this basis, suppose the the Sun and the Moon are one in the first quadrant and the other in the second. Here both declinations are therefore positive. Compute their difference a - b; compute again the difference of c and d which are again northern declinations observed after a few ghatis or days. If c - d < a - b then Vyatīpāta is going to occur. Now take a case when one of a and b is north and the other south ; so also c and d. We are talking of a, c to be in the first quadrant and b, d in the fourth quadrant. Even now if c - d < a - b Vyatīpāta is going to occur. Thus there is no need to stipulate "Sum or difference". Difference alone counts now, because difference of one positive and the other negative quantities means sum only. Thus in the case of Vyatīpāta difference should tend to zero, whether both the declinations are north, or one north and the other south. Similarly in the case of Vaidhṛti the sum should tend to zero because the algebraic sum of one positive northern declination and the other negative southern declination

543 shall be zero, if the declinations were to be equal and of opposite directions. Here ends the Pātādhyāya. Note Bhāskara gives two examples in the course of the commentary, one, to show the fallacy in asserting that there would be a pāta when the Sun is in an even quadrant, the Moon in an odd quadrant and the Moon's declination is less than that of the Sun; and the other to show the fallacy in asserting that the pāta has elapsed when it is still to take place, and that it is still to occur when it has already elapsed. He quotes the example given in his Golādhyāya wherein the longitudes of the Sun, Moon and Rāhu are 120°, 60° and 180° respectively, there being no Ayanāṁśas. (Since at the time of Lalla, there were no Ayanāṁśas, Bhāskara gives such an example), Since the Sun is in an even quadrant and the Moon in an odd quadrant, Bhāskara says, that as per the verse of Lalla ‘सूर्यापमात् ओजपदोद्भवात् etc.’ there should be a pāta. But since the longitude of Rāhu is 180°, Ketu is at the equinoctial point so that the lunar orbit lies between the ecliptic and the equator. Hence the declination of the Moon remains smaller than that of the Sun for a good length of time never equalling the declination of the Sun. This, Bhāskara says, is a fallacy regarding ‘भावाभावत्व’ ie. ‘Is there a pāta or not’ because when there is actually no pāta, Lalla's criterion leads to assert that there is one. It may be reiterated here, that Lalla gave a general criterion ignoring the latitude of the Moon ie. supposing the Moon to be moving on the ecliptic. Bhāskara takes the latitude also into consideration and proves that there is no pāta. In the second example, which Bhāskara gives, he tries to show the fallacy with respect to “गतेष्यत्व” ie. asserting that a pāta has elapsed when it is still to occur, and that it is to occur when it has already elapsed.

544 In this example the declinations of the Sun and Moon are respectively 1416′, 1324′, when their tropical longitudes are 80° and 100°. The Moon is in an even quadrant, and his declination is less than that of the Sun. As per the criterion of Lalla given in the verse “सूर्यापमात् etc.” the pāta is not there at all. But Bhāskara computes and shows that the pāta has occured 70 nādīs before (Rāhu having a reverse tropical longitude of 100°). We shall show here, how using differential calculus we could cut short the process. We have sin δ = sin λ sin ω ; differentiating cos δ Δδ = cos λ δλ sin ω or Δδ = (sin ω cos λ / cos δ) Δλ = Āyanavalana × Variation in Bhuja. This could be seen graphically also from fig. 129. Let P and Q be two contiguous positions of the Sun (say) on the ecliptic when arc pQ = Δλ. Let the increment in his Fig. 129 declination in going from P to Q be NQ equal to Δδ. Taking PNQ to be a plane Δ Δδ = Δλ sin v where v̂ = QP̂N. We have named this angle to be v, because it is no other than the Āyanavalana which was formulated to be equal to (sin ω cos λ / cos δ) . Thus Δδ = Δλ × Āyana- valanajyā.