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

452 (b) In the case of the superior planets, Bhāskara mentions in the Golādhyāya “पातोऽथवा शीघ्रफलं विलोमं कृत्वा स्फुटात् तेन युतात् शरोऽनः ” verse 22 (Golabandhādhikāra). This also clearly indicates that the orbits of the superior planets are also heliocentric, for, otherwise, there is no purpose of subtracting the Sīghraphala from the True position of the planet. The purpose in doing so is to obtain the heliocentric longitude of the planet (vide fig. 106). True geocentric longitude of J is A/EJ = ASJ/. Subtracting the S'ighraphala EJS = JEJ' from ASJ', we have AŜJ the heliocentric longitude. Adding the retrograde longitude of N (तेन युतात् as cited above in verse 22) means adding ASN to AŜJ which gives NŜJ. This is the argument to calculate the latitude of J. (c) Bhāskara alludes to a confusion in the mind of Chaturvedāchārya in this context while commenting upon verses 23, 24 in Golabandhādhikāra (Golādhyāya). Chatur- vedāchārya commenting on Brahmagupta's words (verse 10 Grahayukti-adhikāra ch. 9, Brahma Sphutasiddhānta) exclaims. "The latitude of Mercury and Venus will be the same as what they are at the point of S'īghroccha ; correct- ness of the result alone is proof ; no other reason could be adduced !" Bhāskara clears his misconception in the following words. "The number of sidereal revolutions of the nodes of Mercury and Venus mentioned in the chapter on mean motion, are to be increased by the number of the sidereal revolutions of the S'īghrocchas of Mercury and Venus, as mentioned by Mādhava in his work “ Siddhānta- chudāmani ”. This means that the S'īghra anomaly is to be added to the position of the node obtained by the smaller number of revolutions given in the chapter on mean motion.

453 The misconception in the mind of Chaturvedāchārya as well as the wrong notion in the minds of Mādhava and even Bhāskara in construing that the number of revolu- tions of the nodes are to be increased by the number of revolutions of the S'īghrocchas, are due to the fact that it was overlooked that the position of S'īghrocohas with respect to Mercury and Venus are given by their helio- centric longitudes. In other words as is mentioned by the verse “ ज्ञशुकयोः ग्रहः सूर्यः भवेत्तौ शीघ्रनामकाँ ” the S'īghroc- chas of Mercury and Venus are no other than the helio- centric positions of those planets. Thus the exclamation of Chaturvedācharya cited above arose out of thinking that the S'īghrocchas differ from the planets, whereas they are the same heliocentrically, though they differ geocentri- cally. Heliocentrically the planets longitude is (vide fig. 105) ÂSP which is equal to A/EP' geocentrically. The geocentric longitude of the planet is on the other hand A/EP differing from the above, though both the helio- centric and geocentric longitudes point to the same planet. Bhāskara, no doubt, gave a correct procedure but missed to identify the S'īghroccha and the heliocentric position of the planet. In this context, the reader is referred to the author's ‘ peculiar concept of S'īghroccha in Hindu astronomy published in the journal of Oriental Research of the S. V. University Vol. XIV, Part 2, Dec. 1971. Verse 2. The Hsine of the arc of the planet’s orbit Vimandala intercepted between the nearer node and the planet multiplied by the maximum latitude of the planet cited, and divided by the S'īghrakarṇa gives the latitude of the planet at the given place. Comm. From the formula akin to that which gives the declination of a point on the ecliptic namely sin δ = sin λ sin ω or in the Hindu form H sin δ = (H sin λ H sin ω) / R ,

454 H sin β = (H sin λ H sin i) / R where β is the latitude required, i the maximum latitude and λ the arc of the planetary orbit intercepted between the nearer node and the planet. Since β and i are small H sin β and H sin i could be taken to be β and i. Hence we have β = (i × H sin λ) / R . Since this value is had at the distance of the Śīghrakarṇa, its value at a distance of R should be given by β = (i H sin λ) / R × R / K = (i H sin λ) / K as formulated. Verse 3. √(R² - H sin² v) is called Yaṣṭi where v is Āyanavalana. The latitude β of the planet multiplied by Yaṣṭi and divided by R or multiplied by H cos δ where δ is the declination of the point whose longitude is 90 + λ, λ being that of the planet and divided by R gives the value of the rectified latitude which could be added to the declination of the foot of the latitude to give what we say the modern declination of the planet. [Diagram: Spherical geometry diagram showing points P, P', K, R, R', B, B', M, M', S, L, N, ω, γ] Fig. 107

455 Comm. Let rM (fig. 107) be the ecliptic and rN the celestial equator whose poles are K and P respectively. Let R be a celestial body whose latitude is β and whose modern declination is RL. Let M be the foot of the latitude circle and let δ be the declination of M. The word Krānti in Hindu astronomy is applied to connote the declination of a point on the ecliptic alone and not of any other point like R. RL is called Sphuta Kranti which is equal to R'N = R'M + MN. RM is called Vikṣepa and R'M Sphuta Vikṣēpa. K̂MP is called the Āyanavalana at the point M of the ecliptic. Produce MP to P' where PP' = δ so that MK = MP = 90°. Hence P̂' is a right angle and KP' = K̂MP' = v̂ = Āyanavalana. From the spherical triangle KPP', cos ω = cos v cos δ. Draw perpendicular RR' on MP so that MR' = β' = Sphuta Vikṣēpa = β cos v = (β H cos v) / R. But H cos v = √(R² - H sin² v) = Yaṣti ∴ β' = (Yaṣti × β) / R which is to be added to δ to obtain the Sphutakrānti R'N or RL, the modern declination. Note (1) One Mukhopādhyāya, in his thesis ‘The Hindu nakṣatras’ submitted to the Calcutta University, mistook RM' to be the Sphutavikṣēpa instead of R'M and hastily remarked that Bhāskara was wrong in making RM' less than RM. Note (2) v̂ shown in the fig. 107, is called Sthanīya- valana or valana at the point M which is considered to be place of the planet ‘Sthāna’ on the ecliptic. K̂RA, on the other hand is called Bimbīya-valana or valana at the bimba or disc of the planet. Note (3) v̂ could be obtained from the spherical triangle KMP where KM = 90°, PM = 90-δ, using the

456 formula cos ω = sin (90-δ) cos v or in the Hindu form H cos v = (R H cos ω) / (H cos δ) . In this case β' = (β H cos ω) / (H cos δ) . Or again noting that M K̂ P = 90-λ where λ is the longitude of R, we could use 'Inner side Inner angle formula' with respect to the triangle KMP, which gives 0 = sin 90 cot δ - sin (90 -λ) cot v or cot v = cot ω / cos λ or tan v = cos λ tan ω. But this formula implies the tangent functions which were not used by the Hindu astronomers. Similarly using the elements 90-λ, 90°, v, and 90-δ of the same triangle KMP, another formula could be got for v. Or again noting that K P̂ M = 90+α, we could yet get more formulae where α is the Right ascention of M. Note (4) Thus far we have used modern formulae. Let us now see as to how Bhāskara derives his formula. He takes the triangle MKP' (fig. 107) wherein MK = 90°, KP' = v and P̂' = 90. From fig. 108, the Hsine of MK is Fig. 108 KO equal to R where 'O' is the centre of the sphere and the Hsine of KP is KN so that it is the Āyanavalanajyā, Hence

457 is the centre of the sphere and the Hsine of KP is KN so that it is the Āyanavalanajyā. Hence ON² = OK² — KN² ∴ ON = Yaṣṭi = √R² — Āyanavalanajyā² = Āyanavalanakoṭijyā. From the similarity of the triangles OKN and GRF, OK / ON = GR / GF ∴ GF = (GR × ON) / OK = (H sin MR × Yaṣṭi) / R GR could be taken equal to MR, the latter being small and GF could be taken to be equal to GR′ ie. H sin MR′ and so equal to MR′. ∴ MR′ = (MR × Yaṣṭi) / R Adding MR′ to the declination of M, we get the modern declination of R′ ie. the Sphuṭakrānti of R. Note (5) In fact the spherical triangle MKP is just like the spherical triangle r E ♋ (fig.). In the place of the paramakrānti E ♋, we have the Āyanavalana KP (fig. 108), and in the place of Dyujyā ♋ L (fig. 19) we have Yaṣṭi. Note (6) An alternative is given in the verse for this namely MR′ = (MR × H cos δ′) / R where δ′ is the decli- nation of a point whose longitude is 90+λ. In other words we have to prove that H cos v = H cos δ′ or v = δ′ (of a point whose longitude is 90+λ. Since declination is given by the formula H sin δ = (H sin λ × H sin ω) / R Putting 90+λ for λ, H sin δ′ = (H cos λ × H sin ω) / R I But from triangle MKP, sin v / sin ω = sin (90 — λ) / sin (90 — δ) = cos λ / cos δ 58

458 ∴ sin ν = (sin ω cos λ) / (cos δ) or H sin ν = (H sin ω × H cos λ) / (H cos δ) II Comparing I and II, ν would be equal to δ′ provided H cos δ = R. This is accepted as an approximation, as δ is generally small. This approximate formula is given by Sūryasiddhānta in the context of Āyana Dṛk-karma in verse 10, ch. 7, where ν is assumed to be equal to δ′ defined above. Verses 4 and 6. The process called Āyana Dṛk-karma. The Āyana Dṛk-karma correction measured in minutes is obtained by multiplying Āyanavalana by the unrectified celestial latitude, and divided by H cos δ and then multiplied by 1800 and divided by the rising time of the Rāśi which is occupied by the planet. Or again it is obtained approximately by the product of the Āyana- valana and the unrectified celestial latitude divided by the Yaṣṭi. This Āyana Dṛk-karma correction is negative if the hemisphere and the latitude have the same direction. On symbols, Āyana Dṛk-karma correction = (Āyanavalana × β × 1800) / (H cos δ × T) or approximately equal to (β × Āyanavalana) / (Yaṣṭi (= H cos ν)) . Comm. Let G be the position of the planet when the foot of the planet’s secondary (called Grahasthāna) to the Ecliptic namely A is rising, where rCA is the Ecliptic. Let rML be the celestial Equator. Let P and K be the poles of the celestial Equator and Ecliptic respectively. The arc AC of the Ecliptic intercepted between A and the declination circle of G expressed in minutes represents the आयनकलोः formulated in the verse. To find the magnitude of AC, the first formula mentioned above envisages finding

459 Fig. 109 it through the magnitude of ML the corresponding Equatorial arc which is itself found through finding the magnitude of AB. AB expressed in minutes will be equal to the number of asus taken by the declination circle of G namely PGB to have traversed from the position PDA to PGB which is the time taken by the planet G to be at its position from the moment of its rising at the Equatorial horizon namely PDA, D being then its position. When G was at D, K, the pole of the Ecliptic should have been itself rising on the Equatorial horizon. Thus it is sought to find the time expressed in asus taken by K from its moment of rising on the Equatorial horizon to its present position at K. When Bhāskara mentions in the commen- tary under the verse that ‘‘क्षितिजकदम्बयोरन्तरं तदेवोत्तरमायनं वलनम्’’ by the word क्षितिज, he had at the back of his mind लङ्काक्षितिज ie. Equatorial horizon and not the horizon of the place shown in the figure. Also as was mentioned before in a previous context the time expressed in asus taken by an equatorial arc to rise is equal to the number of minutes in that arc. Since the time taken by the planet to traverse the arc of the diurnal circle namely DG is the same as the time taken by the equatorial arc LM to rise, Bhāskara seeks to find the number of minutes in AB, then compute, the number of Asus taken by LM to rise

460 through which the number of minutes in AC could be computed. AB = AG sin AĜB = (AG × H sin AĜB) / R = (β × Āyanavalanajyā) / R where AG = β. In the formula given the word Āyanavalana is used for Āyanavalanajyā ie. the Hindu sine of Āyanavalana for brevity. Also, since generally the angle AĜB defined as Āyanavalana happens to be small, its Hsine will be almost equal to it. In this sense also the term Valana is used where Valanajyā is to be used. From AB we pass on to the magnitude of LM. From Hindu spherical Trigonometry LM = (AB × R) / (H cos AL) = (AB × R) / (H cos δ) = (β × Āyanavalanajyā) / R × R / (H cos δ) = (β × Āyanavalanajyā) / (H cos δ) . (H cos δ is called Dyujyā which is connoted by the term Dyu-guna in the verse, the words Jyā and Guna being synonymous). Thus LM = (β × Āyanavalanajyā) / Dyu-guna From LM we pass on to the corresponding arc of the Ecliptic namely AC, which does not imply the latitude of the place. So we have to use what are called Niraksha- Udayas of Rāśis ie. the rising times of the Rāśis at Equatorial places. Rule of three is used here to find AC from LM. Locating the particular Rāśi in which the planet is situated, and using the proportion. “If a Rāśi of 30° ie. 1800′ of the Ecliptic rises at an Equatorial place in say x asus, what arc in minutes corresponds to the number of minutes or what is the same the number of Asus

401 in the arc LM?" We get the magnitude of the arc AC in minutes. The computation of this arc AC is intended to know the difference between the times of rising of the point A and the point G the former being the point of the ecliptic signifying the position of the planet and the latter being the planet itself. This difference of the times of rising is itself required to know the actual time of rising of the planet by a knowledge of the time of rising of the point which signifies the planet’s position on the Ecliptic by its longitude. In verse 5, Bhāskara gives an alternate and easy method of obtaining the magnitude of AC construing AGC to be roughly a plane triangle which does not involve any appreciable error. AC = AG tan AĜC = (β × H sin AĜC) / (H cos AĜC) = (β × Āyanavalanajyā) / Yaṣṭi since Yaṣṭi is defined as H cos AGC in verse 3. Note. The point C is called the Kṛta-Āyana-Dṛk- Karma Sthāna ie. the point of the Ecliptic signifying the planetary position rectified for the Samskāra or correction called Āyana-Dṛk-karma. The longitude of C thus got is called the polar longitude of the planet which is defined as the longitude of the planet as measured on the Ecliptic upto the point of intersection of the planet’s declination circle with the Ecliptic. Bhāskara mentions elsewhere, “ नक्षत्राणां स्फुटा एव स्थिरत्वात् पठिताः शराः दृक्कर्मणाऽऽयनेनैषां संस्कृताश्च तथा ध्रुवाः ” ie. in the case of stars which are fixed, the Sphuta-saras or polar latitudes (given by GC in the above case) are

462 given and longitudes rectified for Āyana Dṛk-karma are given ie. polar longitudes. Bhāskara made this as an approximate statement, for, we know, even in the case of stars, though they be fixed, their polar latitudes also do change by the precession of equinoxes be it just a little; whereas their polar longitudes do not change by the same constant quantity as their celestial longitudes. Verses 6, 7 and 8. Obtain the Carakhandas or ascentional differences of the Sphuta and Asphuta Krāntis. If they be of the same direction their difference is to be taken, if of opposite direction, their sum is to be taken. The result in asus gives the Ākṣa Dṛk-karma correction if the celestial latitude is of appreciable magnitude. If it be not appreciably large, it (the celestial latitude) is to be multiplied by the Ākṣavalana, then divid by H cos ϕ or what is the same, multiplied by the equinoctial shadow and divided by 12. The result is to be multiplied by the radius and divided by H cos δ. Then we have the Ākṣa Dṛk-karma correction in asus. Assuming the planet’s position corrected for Āyana Dṛk-karma to be the Sun, obtain the lagna using the asus of the Ākṣa-Dṛk-karma. If the planet has a southern latitude let the lagna be found in the positive direction; otherwise in the negative dire- ction. Then we have the rising lagna of the planet or what is the same the longitude of the rising point of the planet. Again assuming the planet’s position increased by 180° to be the Sun, and using the asus of Ākṣa Dṛk- karma, obtain the lagna in the positive direction with respect to a northern latitude of the planet or else in the negative direction. The result gives the longitude of the setting planet or what is called the Asta-lagna. Comm. (Refer to fig. 110) Let p be the planet whose Graha-sthāna or foot of the latitude is p′. Let q be what is called the Kṛta-Āyana-Dṛk-Karmaka-Sthāna or the planet’s position corrected for the correction of Āyana

463 Fig. 110 Dṛk-karma. (This correction is given by the arc p'q). The difference of the rising times of the planet p and q expressed in asus is what is sought here. The planet p rose at a and has covered the arc ap of the diurnal circle after rising. So, we have to compute ap and therefrom the corresponding arc of the Equator namely AC. If the planet has no latitude ie. is situated on the ecliptic at p', the arc p'q or Āyana Dṛk-karma vanishes, for, the Āyana Dṛk-karma is no other than the projection of the latitude of the planet on the ecliptic taking P or celestial pole as the vertex of projection. Also if there were no Akṣa ie. if the place be equatorial, p and q rise simultaneously ie. the planet will rise along with the point called Kṛta- Āyana-Dṛk-Karmaka-Sthāna q, mentioned above. In fact the Dṛk-karma corrections Āyana and Ākṣa are contem- plated to obtain the difference in the rising times of p and p' ie. the planet and its position on the ecliptic. This time is resolved into two parts (i) the difference of the rising times of p' and q and (ii) that of the rising times of q and p. The former is given by the equatorial arc BC which goes by the name Āyana-Dṛk-karma and the latter by the arc AC which goes by the name Ākṣa-Dṛk- karma. The algebraic sum of these two corrections gives the difference of the rising time of p and p' ie. the time given by the arc AB. Here AB=AC-BC, In verses

464 4, 5 we have seen how BC is computed. Here we are seeking to compute AC through the corresponding arc ap of the diurnal circle. The verse uses the word Sphuta-Krānti, Asphuta- Krānti, and their Cara-khandas. Here in the figure pC is the Sphuta-Krānti and p'B is the Asphuta-Krānti or simply Krānti. Also pp' is called Asphuta-Vikṣēpa and bp' Sphuta-Vikṣēpa. Sphuta-Krānti = pc = bB = bp' + p'B = Sphuta-Vikṣēpa + Asphuta-Krānti. In other words the declination of p, the planet, is called Sphuta-Krānti and that of its longitudinal position on the ecliptic namely p' is called Asphuta-Krānti or simply Krānti. We have obtained Sphuta-Krānti from Krānti by adding to the latter the Sphuta-Vikṣēpa. The method of obtaining Sphuta-Vikṣēpa from the Vikṣēpa, otherwise called Vikṣēpa Sphuta-karama was dealt with in verse 3 above. Now we have to define the Cara-khandas pertaining to the Sphuta-Krānti pC and the Asphuta-Krānti p'B. The former is defined as AE and the latter by CE very approximately. We say very approximately because when p' comes to the horizon, the Cara-khanda will not be exactly EC but a little more or less than EC since Bp' ≠ Cq. In other words, the arc of the equator between the declination circle of q while rising and the equatorial horizon ie. AE is defined as the Cara-khanda of pC ; similarly CE is that of p'B. Their difference is AE—CE = AC. If we use the modern notation, we have to take their algebraical difference. If that be done, the alternative case of adding the Cara-khandas when the Sphuta and the Asphuta-Krānti are of opposite directions, will be automatically implied. The asus pertaining to this arc AC will be the correction of Ākṣa-Dṛk-karma as mentioned in verse (6) when the latitude of the planet is appreciable. These asus will be equal to the time taken by p the planet to cover the arc ap or what is the same, the

465 time in between the moment of rising of p and that of q, the position of the Kṛta-Āyana-Dṛk-Karmakagraha. When the latitude is very small, an approximate estimate of this correction is given in asus as (β' × H sin ξ) / (H cos ϕ) × R / (H cos δ) or (β' × H sin ϕ) / (H cos ϕ) × R / (H cos δ) . That this formula is (β'=qp) only approximate is seen from the fact that H sin ξ ≉ H sin ϕ. This second formula can be proved as follows. Ākṣavalana in asus = (β' × H sin ϕ) / (H cos ϕ) × R / (H cos δ) where β' is the Sphuṭa-Vikṣēpa. Taking qpa as a plane triangle pa = qp tan p̂qa = qp × tan P̂En approximately = Sphuṭa-Vikṣēpa × (H sin ϕ) / (H cos ϕ) . Hence AC = (β' × H sin ϕ) / (H cos ϕ) × R / (H cos δ) . With respect to the first formula ie. Akṣavalana in asus = (β' × H sin ξ) / (H cos ϕ) × R / (H cos δ) ' pa = qp tan p̂qa = (qp × H sin p̂qa) / (H cos p̂qa) . Here instead H cos p̂qa which is Yaṣṭi previously defined, H cos PEn ie. H cos ϕ, an approximate value is substituted, because the latitude is small. Having obtained the value of pa, it is then reduced to the Equator by multiplying by R and dividing by H cos δ. The convention with respect to the sign is clear. The idea of Kramalagna and Vilōma- lagna may be elucidated as follows. In fig. 110, q, is on the horizon rising whereas p had already arisen. So to · 59

466 obtain the rising time of the planet p, which occured before that of q, Vilōmalagna or an anterior point of time is to be computed, and this is done by subtracting the asus of Akṣa-Dṛk-karma from the rising time of q. The rest follows on these lines of elucidation. Verse 9. The planet rises at the time known as its Udayalagna (which is to be computed as aforesaid) and it sets at the time known as its Astalagna (which is also to be computed as mentioned before). Comm. Clear. Verse. 10. During the night, at a given moment, the computed Udayalagna of the planet is less than the particular lagna of the moment, ie, if the longitude of the Udayalagna is less than that of the current lagna, and also if the Astalagna of the planet computed is greater than that of the then current lagna ie. if the longitude of the Astalagna is greater than that of the current lagna, the planet is visible ie. above the horizon. In the case of the Moon, however, if he is not eclipsed by the rays of the Sun, he may be visible even shortly after Sunrise or a little before Sunset in contradistinction to a planet which could not be seen at all during day time (except perhaps Venus). When a planet is visible, his gnomonic shadow could be computed. (This does not mean that the gnomon casts a shadow of the planet but means that its zenith- distance could be computed according to the methods described in Tripraśnādhikāra. Comm. Clear. Verse 11. To obtain the shadow, the time that has elapsed after the rise of the planet is to be known. If it be required to find the gnomonic shadow of the planet, then the current lagna and the Udayalagna of the

467 planet at the moment are to be computed. The time in between the two lagnas, which will be in Sāvana measure pertaining to the planet gives the time that has elapsed after the rise of the planet. Comm. According to computation, the difference of times of the current lagna and the Udayalagna as com- puted taking the present position of the planet will be the time measured along the diurnal path of the planet and as such is in Sānava measure. Bhāskara mentions here a very subtle point. In the analysis the latitude of the planet is taken into account while finding the Udayalagna. Time measured in this case is called Kṣetrātmika and not Kālātmika for the following reason. Suppose A is the Udayalagna of the planet and B the current lagna both points being on the ecliptic. Let us say that the current lagna is posterior to the Udayalagna (there is no loss of generality in supposing this). By the time of the current lagna, the planet is no more at A but will have moved a little towards B. Let the present position of the planet on the ecliptic be A'. The corresponding arc of AB on the equator gives the sidereal measure of the time that has elapsed after the rise of the planet whereas the corresponding arc of A'B on the equator gives the Sāvana measure of the same time. Here this Sāvana is that pertaining to the planet and not that pertaining to the Sun which is Saura Sāvana. In other words it is the Sāvana pertaining to the particular planet, because the arc AA' is traversed by the planet in question as per its own velocity. The stress is here on the word Tat-Kāla-graha, ie. the longitude of the planet at the moment at which the shadow or the zenith-distance of the planet is sought. Taking this as the position of the planet and taking the latitude of the planet also into account compute the Udayalagna. This will be A' cited above. To obtain the

468 sidereal measure of the same time we have to take A and not A'. Verse 12. Sāvana measure alone is to be employed while finding the gnomonic shadow ie. while the zenith-distance of the planet is to be computed, because the arc of the diurnal circle of the planet indicates only Sāvana measure. Suppose the Udayalagna falls short of the current lagna, then the Bhōgyakāla ie. the remaining rising time of the Rāśi in which the planet is situated added to the elapsed time of the Rāśi of the current lagna together with the sum of the rising times of the Rāśis in between gives the difference of the Udayalagna of the planet and the current lagna. Comm. (Ref. fig. 111). Let p be the planet's place on the Ecliptic at the moment in question when the lagna is L (or the foot of the latitude of the planet in case it is not on the ecliptic). Let PA be the remainder of the Rāśi in which the planet is situated. Then the time of rising of the arc PA is here termed Bhōgyakāla. Let DL be the arc of the Rāśi in which the current lagna L is situated, which has arisen. The time of rising of this arc DL is called Bhuktakāla. The times of risings of the Rāśis in between namely AB, BC, CD are called Madhyōdayās. The sum of the rising times of PA, AB, BC, CD, DL gives the time in between the rise of P and that of L which is the time that has elapsed after the planet has arisen upto the current Lagna ie. the rising time of L. Verse 13. The method of computing the gnomonic shadow of the planet or what is the same the zenith- distance of the planet, as per the method of computing that of the Sun (extended to the case of the planet) after

469 finding the Sphutakrānti (ie. modern declination) of the planet. The Krānti of the planet or the declination of the foot of the latitude of the planet added to the latitude rectified called Sphuṭasāra, gives what is called Spaṣṭakrānti of the planet and its Hsine is called Spaṣṭa-Krāntijyā. From this H sin δ, H cos δ etc. is to be computed (as mentioned in Triprasnādhikāra in the context of finding the zenith- distance of the Sun). From the time that has elapsed after the rise of the planet called the Unnata, the shadow is to be computed as in the case of the Sun’s shadow. Having thus computed the shadow or what is the same the zenith-distance of the Moon or that of the stars, the instru- ment called Nalaka could be pointed to the spot where that celestial body is situated. Comm. The words Krānti, Spaṣṭakrānti were explained before. Also the method of calculating the zenith-distance from a knowledge of the declination was described before in Triprasnādhikāra. Verses 14, 15. Consideration of horizontal parallax in the case of visibility of the Moon or planets. H cos z or what is called the Mahāśaṅku of the Moon or planet is to be reduced by 1/15th of the respective daily motion gives the visible Sanku when the radius is taken to be 3438. If the radius is taken as 120, 1/450th of the daily motion is to be subtracted. If H cos z is less than 1/15th (or 1/450th) of the daily motion, then the Moon is not visible. This applies to the other planets as well, but as this is a neglible matter, [the earlier Āchāryas did not suggest this. Comm. In the context of the subject of parallax, we mentioned that in the case of the Moon, the horizontal parallax happens to be 1/15 of the Moon’s daily motion

470 approximately. The same is extended to the case of the planets as well. To get the proportion in the case of taking the radius to be 120 only instead of 3438, rule of three is applied as follows. “If the radius be 3438, 𝑣/15 is parallax, what will it be if the radius be 120?” The answer is 𝑣/15 × 120/3438 = 𝑣 / (3438/8) = 𝑣/430 very approximately as stated. If H cos z be less than this 1/15 th or 1/430 th of the daily motion, the parallax does not allow the Moon to be seen. Bhāskara specifically talks about the Moon only because, the earlier Āchāryas did not apply the correction of parallax to the case of the planets, because it is not appreciable. Verse 16. If an operation is neglected because its effect is not appreciable, or is not of much use, or because it is apparent, or if it implies great labour, or again if it implies a lot of exposition that would make the text unduly voluminous, ignoring the necessity of that oper- ation should not be treated as wrong. Comm. Here Bhāskara upholds the earlier Āchāryas not stipulating parallax in the case of planets, because it is not appreciable. Here ends Graha-cchāyādhikāra.

GRAHŌDAYĀSTĀDHIKĀRA Verse 1 and first half of verse 2. The Udayalagna of a planet is termed Prāk-Dṛk-graha and the Astalagna is termed the Paschima-Dṛk-graha. If the Prāk-Dṛk-graha happens to be less than the current lagna the planet had already risen. If it be greater, the planet is still to rise. Similarly if the Paschima-Dṛk-graha is less than the current lagna, the planet had already set ; otherwise is yet to set. Comm. The words Prāk-Dṛk-graha and Paschima- Dṛk-graha are coined to indicate the points of intersection of the ecliptic with the eastern and western horizon respectively when the planet is rising or setting, when the planet has latitude. When the planet has no latitude Prāk-Dṛk-graha coincides with the rising planet and the Paschima-Dṛk-graha with the setting planet. When the planet has a latitude, the foot of the latitude, which signifies the planet’s position on the ecliptic differs from the two Dṛk-grahas. In this case ie. when the planet has a latitude the Prāk-Dṛk-graha’s longitude will be less than that of the planet’s longitude whereas the Paschima-Dṛk- graha’s longitude will be greater than that of the planet. If x and y be the differences of the longitudes of the planet and those of the Dṛk-grahas respectively, it is clear from a figure that tan x = (tan β) / (tan θ) and tan y = (tan β) / (tan ϕ) where β is the latitude of the planet and θ and ϕ the angles which the ecliptic makes with the horizon at the planet’s rising and setting respectively. Since θ and ϕ are then respectively the zenith- distances of the pole of the ecliptic in each case, and since