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

361 for, as mentioned before, the daily motions of the Moon and the Sun are ready computed for every day. Also the advantage in using this formula is that besides the fact that we need to deal only with small quantities instead of the big numbers of Yojanas, the true value for the day of eclipse is got by using the true daily motions. Using the mean daily motions, however, we have for the mean diameter, ²/₁₅ m₁ — ⁵/₁₂ s₁ = ²/₁₅ × 790′–35″ — ⁵/₁₂ × 59′–8″ = 105 ⅔ — 24 ⅔ = 81 very approximately. Note (3) In obtaining a convergent for (s − e)/Ks , since the values of s and Ks are parameters s/Ks is got alright, where e/Ks is more exaggerated as the value of the hori- zontal solar parallax. By this term the result is increased to an extent of 3′ out of which 1′ is mitigated by the fact that we have to take the earth's atmosphere also into

362 the Moon with respect to the ecliptic where λ is the longi- tude of the Moon with respect to the nearer node and 270' or 4½° is taken to be the inclination of the lunar orbit to the ecliptic or what is the same, the maximum latitude of the Moon. Comm. The formula is evident and similar to that for calculating the declination of the Sun. Thus H sin β = (H sin λ × H sin 4½°) / R . But since β is small and also 4½°, we can take β = (H sin λ × 270') / R . The argument used by Bhāskara, however, is ‘ If by a H sin λ equal to R, we have the maximum latitude of 270', what shall we have for H sin λ?’ The result is as given. Note. The modern value for i the inclination of the lunar orbit to the ecliptic is given to vary between 4°-58' and 5°-18'. Verse 11. The definition of the magnitude of a lunar eclipse. Sthagita or the magnitude of an eclipse is defined as P+r+β where P and r are respectively the radii of the eclipsing and eclipsed bodies and β is the latitude of the Moon. If the Sthagita is greater than 2r, then the eclipse is total. [Diagram: Two intersecting circles representing the Moon (center M, radius r) and Earth's shadow (center C, radius P), along lines labeled "LUNAR ORBIT" and "ECLIPTIC"] Fig. 71

363 Comm. In figure 71, the eclipsing body is just con- tacting the eclipsed body. Taking the case of a lunar eclipse, the latitude then of the Moon is evidently P+r ie. β=P+r holds good at the moment of first con- tact. (C) is the cross-section of the shadow-cone at the lunar orbit and (M) is the Moon. (fig. 72), CB+AM-CM=CB+AB+ BM-CM=AB+(CB+BM)-CM= AB+CM-CM=AB ∴ P+r-β=AB=Sthagita. Thus Sthagita gives the portion of the diameter of the eclipsed body which is shadowed. The eclipsed body is termed the Chādya, the eclipsing body as the Chādaka and P+r as the Manaikya-ardha ie. half the sum ofthe diameters of the eclipsing and eclipsed bodies. Fig. 72 When the Sthagita exceeds the diameter of the eclipsed body the eclipse is evidently total ie., when P+r-β>2r ie. P-r>β. Verse 12. Duration of the eclipse and duration of its totality. Sthiti-Khanda = √(P+r)²-β² × 60 = ½ Duration of —————————————— the eclipse m₁-s₁ Marda-Khanda = √(P-r)²-β² × 60 = ½ Duration of —————————————— totality m₁-s₁ where P is the radius of the shadow-cone, r the radius of the Moon's disc, β its latitude taken to be constant during the eclipse, m₁ and s₁ the daily motions of the Moon and Sun respectively. Comm. (1) The time between the moment of first contact and the middle of the eclipse or the moment of

opposition or conjunction as the case may be is called Sparśa-Sthiti-Khaṇḍa. (2) The time between the middle of the eclipse and the moment of last contact is called the Mokṣa-Sthiti- Khaṇḍa. (3) The time between the commencement of total eclipse and the middle of the eclipse is called the Sammi- lana-Marda-Khaṇḍa. (4) The time between the middle of the eclipse and the end of total eclipse is called Unmilana-Marda-Khaṇḍa. The suffix Khaṇḍa meaning ‘half’ is generally omit- ted while refering to these phases. In the above verse we are given formulae for Sthiti-Khaṇḍa and Marda- Khaṇḍa only without specifying whether they pertain to Sparśa or Mokṣa. Though the same formulae serve for both the Sparśa phase as well as the Mokṣa phase under the supposition that β does not vary, it will be noted that the Sparśa-Sthiti-Khaṇḍa will not be equal to Mokṣa-Sthiti-Khaṇḍa and that the Sparśa-Manda- Khaṇḍa will not be equal to the Mokṣa-Marda Khaṇḍa in as much as β changes from moment to moment. In fig. 73, let C₁ be the position of the eclipsing body at the moment of first contact and C₂ its position at the moment of last contact. In the figure is shown only one position of the Moon's disc signifying that we may con- sider the motion of the eclipsing body keeping the eclipsed body fixed (or what is the same relative to the position of the eclipsed body). It is evident from the fig. that C₁M = C₂M = P + r so that C₁M C₂ is an Isosceles triangle. Let MN be the ⊥ᵃʳ dropped from M on C₁ C₂. C₁ C₂ is the ecliptic because the centre of the shadow will be moving along the ecliptic, for, in fig. 66, SE the ecliptic passes through D the centre of the cross-section of the shadow- cone, as well as through the vertex V of the shadow-cone.

868 Fig. 73 MN is therefore the latitude of the Moon. Since the lati- tude is not the same at the moment of first contact and that of the last contact, the figure drawn does not represent the true figure but only a figure drawn on the supposition that β remains the same and C₁ Moves relative to M. From the figure C₁N² = (P+r)² − β² = C₂N² I The Sthiti-Khanda defined in this verse is the time taken by C₁ to reach the position N ie. the position at the moment of opposition, and again from the position N to the position C₂. The velocity of C₁ relative to M is no other than the excess of the velocity of, the Moon over that of the Sun. (The velocity of the Earth is the relative velocity of the Sun with respect to the Earth and this is equal to the velocity of the shadow moving along the ecliptic). So, the time taken by C₁ to reach the position of N relative to the Moon is equal to [ 60 × √(P + r² − β²) ] / [ m₁ − s₁ ] Similarly the time taken by the centre of the shadow from N to C₂ ie. from the point of opposition to the moment

386 of last contact has also the same formula where in each case β is the latitude at the moment of opposition. The path taken by the centre of the shadow is called 'ग्राहकमार्ग' ie. the path of the eclipsing body. The actual case when both C and M are both moving and when β is considered as a non-changing quantity is shown in fig. 74. In this Fig. 74 case, three positions are shown, (1) that at the first con- tact (2) that at opposition and (3) that at last contact, where C₁, M₁, C₂, M₂ and C₃, M₃ give the positions of the centre of the shadow and that of the Moon's disc respe- ctively, both the centres being shown as moving. Since the Moon moves faster than C and as such overtakes C, the path of M from M₁ M₃ which synchronizes with the path of C from C₁ to C₃, is shown to be longer. But, one may wonder, how C₁ N₁ and C₃ N₃ represent the Sparśa- Sthiti-Khanda and Mokṣa-Sthiti-Khanda respectively. The distance overtaken by M with respect to C from the point of first contact to the point of opposition is M₁ M₂ — C₁ C₂ = C₁ N₁. Hence we compute C₁ N₁ by the formula C₁ N₁² = (P+r)² — β². Similarly from the point of opposition to the point of last contact M overtakes C by the distance M₂ M₃ — C₂ C₃ = C₃ N₃ = √(P+r)² — β². Fig. 75 shows the situation when β changes as is the actuality. When the opposition takes place after the Moon crosses the node, then β₃ > β₂ > β₁, whereas if

367 [चित्र: Fig. 75] Fig. 75 the opposition precedes the Moon’s position at the node β₃ < β₂ < β₁. Also, when β changes, M₁ M₂ does not exceed C₁ C₂ exactly by C₁ N₁. So, on both the counts, the formulae, given in verse 12 are approximate. What is done in practice is that β is computed for the moment of opposition and estimating the Sparsa-Sthiti-Khanda by the formula given above, and subtracting it from the time of opposition the moment of first contact is got. Then β is computed for that time and again the formula is applied to get the Sparsa-Sthiti-Khanda. Repeating the process, we rectify the Sparsa-Sthiti-Khanda. Even then, we do not have the actual value of the Sparsa-Sthiti-Khanda, because M₁ M₂ does not exceed C₁ C₂ exactly by C₁ N₁. A more correct procedure would be to compute the time between the moment of first contact and the moment of opposition and by that time, to compute the length of M₁ M₂ and take [m' (β₁ ~ β₂) / (M₁ M₂)] in the place of m' and use the formula of verse 12. This nicety, however, need not be attended to with respect to the duration of totality, for, it does not make much difference. Another way of obtaining a better value for T, the Sparsa-Sthiti-Khanda is to take average values for β₁ and β₂, m₁ and m₂, s₁ and s₂ where m₁ and m₂ are the values of the Moon’s daily motion and s₁ and s₂ are those of the

368 Sun's at the point of first contact and the moment of con- junction respectively. We may also use calculus to obtain δT, the variation in time for a variation of δβ in β and a variation of δm₁ in m₁ ignoring the small variation in s₁, as follows. T² = ((P + r)² - β²) / (m₁ - s₁) ∴ 2T δT = [(m₁ - s₁) × -2β δβ - ((P + r)² - β²) δm₁] / (m₁ - s₁)² ∴ δT = - β δβ / [T (m₁ - s₁)] - δm₁ ((P + r)² - β²) / [T (m₁ - s₁)²] The first term on the Right hand side gives the variation for δβ and the second for δm₁. Fig. 76 shows the case of totality. [Diagram: Fig. 76] Fig. 76 M₁ M₂ = N₁ C₁ + C₁ C₂ ∴ The Moon has to over- take C from the moment of the beginning of totality to the moment of opposition by the distance C₁ N₁ with a relative velocity of m₁ - s₁. Hence the time of Sammilana- Marda-Khanda is equal to [√(C₁M₁² - β²) × 60] / (m₁ - s₁) = [√((R - r)² - β²) × 60] / (m₁ - s₁) as given, taking β to be constant. Similarly the Un- milana-Marda-Khanda from the position (M₂ C₂) to the position (M₃ C₃) will also be the same, taking β to be constant.

369 Rectification of this time, when β is considered as varying will proceed on the same lines as before. Verse 13. Rectification of the times of Sparsa-Sthiti- Khanda and Mōksha-Sthiti-Khanda. From the position of the Moon and that of the Node obtained for the moment of opposition, have to be com- puted their positions for the moment of first contact and those for the moment of last contact. For this, (T × v) / 60 is to be subtracted and added respectively to the positions at the moment of opposition of the Moon and Node, where T is the time of the Sparsa-Sthiti-Khanda, and v the daily motion (of the Moon or the Node as the case may be). From these positions β has to be computed for the moment of first contact and that of last contact, and from this β the time of Sparsa-Sthiti-Khanda and Mōksha-Sthiti- Khanda have to be rectified by the method of successive approximation. Comm. From fig. 75, C₁ N₁ and C₂ N₂ are the dist- ances gained by the Moon over the centre of the shadow so that to get their correct values β₁ and β₂ are to be used and not β. Hence β₁ and β₂ are to be computed using T the time of Sparsa-Sthiti-Khanda and that of the Mōksha- Sthiti-Khanda which are taken to be equal in the first instance. Since T is the time taken as a first approxi- mation, β₁, β₂ are also approximate in the first instance. From these β₁, β₂ T is to be rectified and in this rectifi- cation, we have T₁ and T₂ differing, as the times of Sparsa- Sthiti-Khanda and Mōksha-Sthiti-Khanda. From these rectified times again β₁, β₂ are further to be rectified and from them again T₁, T₂ are to be further rectified. This procedure is to be continued till constant values are obtained for T₁ and T₂. 47

370 Note. T₂ will be less than or greater than T₁ accord- ing as β₂ ≷ β₁. Verse 14. Rectification of the Sammilana-Marda- Khanda and Unmilana-Marda-Khanda. Proceeding on the same lines as above and obtaining β₃ and β₄ the rectified latitudes of the Moon for the moments of the commencement and end of totality of the eclipse, the Sammilana-Marda-Khanda and Unmilana- Marda-Khanda, T₃ and T₄ are to be rectified. Note. We have the formula sin β = sin λ sin i so that by differentiating. we have cos βδβ = cos λδλ × sin i sin i cos λ Δλ ∴ δβ = —————————————— cos β This formula gives in one stroke the rectified latitudes of the Moon at the respective moments from which the respective rectified times could be got. Verse 15 and the first half of verse 16. The definition of Bhuja and the method of finding it at an intermediate point of time. The word ‘Iṣṭa’ is used to connote ‘At any given time’. The word ‘Spārśika-Iṣṭa’ means ‘At a given time after the moment of first contact’; similarly the word ‘Maukṣika Iṣṭa’ means ‘At a given time before the moment of last contact’. (T–t) (m₁—s₁) where (m₁—s₁) is in degrees (m₁ and s₁ of course being expressed in minutes); T stands for the Sthiti-Khanda (Spārśika or Maukṣika) and t stands for the Iṣṭa (Spārśika or Maukṣika) gives the Bhuja. Similarly with respect to obtaining the Marda-Bhuja. (The former is called Sthiti- Bhuja). Comm. In fig. 77, let C and M be the centres of the Rāhu (cross-section of the shadow-cone at the lunar orbit)

271 Fig. 77 and the Moon respectively; let MN be the perpendicular from M on the Grāhaka-mārga or the path of the eclipsing body (ie. the ecliptic). Then CN is called the Bhuja at the time. At the moment of first contact, the value of CN is √((P+r)²—β²) where β is the latitude of the Moon at that moment. At any subsequent moment, from fig. 77, CN is equal to √((P+r—AB)²—β²) where β is the latitude at the subsequent moment and AB the portion of the radius of the eclipsed body shaded. Hence we could obtain the Bhuja at any subsequent moment, by computing the latitude at that moment and the value of AB. But AB could be computed only by knowledge of CN and β. Hence

342 the necessity for knowing the value of CN at any subse- quent moment arises. β, of course, could be computed, knowing the hourly variation of β, which in its turn could be known, by a knowledge of the hourly variation in λ, the longitude of the Sapātachandra. The magnitude of CN is calculated by the rule of three "If by the Sparsa-Sthiti-Khanda we have initially the initial value of CN, what shall we have for (T –t) ?" The result is [(T–t) × CN] / T where CN is the initial value of CN and T the Sparsa-Sthiti-Khanda. Substituting the values of CN and T from verse 12, where CN = √((R+r)²–β²) and T = [√((R+r)²–β²) × 60] / (m₁–s₁) we have the required Bhuja as [(T–t) (√(R+r)²–β²)] / [60 (√(R+r)²–β²)] × (m₁–s₁) = [(T–t) {m₁–s₁}] / 60 minutes = (T–t) (m₁–s₁) degrees as given. Similarly we could find the Bhuja with respect to 'totality' ie. the 'Marda-Bhuja' as it is called. Note. One might mistake M₁ M₂ of fig. 74 (M₁ per- taining to a subsequent moment) to be the Bhuja defined above, which is the join of the centre of the eclipsing body and the foot of the latitude at the middle of the eclipse. That is why Bhāskara uses the word 'Madhya-Sarāgra- Chihna' in the commentary, meaning thereby not the foot of the actual latitude at the middle of the eclipse but only the point N of fig. 77 which 'signifies' it. Second half of verse 16 and first half of verse 17. Taking the latitude of the Moon at a given time as Koti, and Bhuja as the Bhuja of the moment defined above, we have the Karṇa of the moment as √(Bhuja² + β²); R+r–Karṇa gives the Grāsa at the moment.

378 Comm. The word 'Grāsa' at the moment stands for AB of fig. 77, Karṇa for CM, where CN is the Bhuja and MN is the Kōti. The 'Grāsa' at the moment of opposition has the special name Sthagita. Second half of verse 17 and verse 18. To obtain the time after the moment of first contact, knowing 'Grāsa' at the moment. T — √((P+r —Grāsa)²—β²) / (m₁—s₁) = t; this 't' is to be rectified by obtaining the β of the moment and again finding t and repeating the process till an invariable magnitude is got. Comm. This is the converse of finding the Grāsa given the time. The method of rectification is also evident. In the above equation considering β and t as variables, and differenting, δt = (1 × —βδβ) / (2 (m₁—s₁) √((P+r—g)²—β²)) = —βδβ / ((T—t) (m₁—s₁)²) Knowing δβ, δt could be got without taking recourse to the method of successive approximations. Verse 19. Certain definitions. The 'Middle of the eclipse' (or strictly speaking the moment when the portion eclipsed is a maximum) occurs at the moment of opposition. Sparsa or Pragraha is at the moment of first contact and Mokṣa is at the moment of last contact, separated from the moment of the middle of the eclipse by times equal to Sparsa-Sthiti-Khanda and Mokṣa-Sthiti-Khanda respectively before and after. Similarly Sammilana and Unmilana or the moment of the commencement of totality and the end thereof occur before and after the moment of 'the middle of the eclipse' by times equal to Sammilana-Marda-Khanda and Un- milana-Marda-Khanda respectively.

374 Comm. Clear. Verse 20. To get what is called the Valana. The hour-angle of the eclipsed body expressed in nādīs, multiplied by 90 and divided by half the duration of night (if it be lunar eclipse) or half the duration of day (if it be solar) as the case may be will give the degrees of an angle, whose H sine being multiplied by the H sine of the latitude and divided by (H cos δ, (where δ is the decli- nation of the eclipsed body), gives the H sine of what is called Ākṣavalana which is north when the hour angle is east, and south otherwise. Comm. This subject of Valana requires a detailed treatment as is given in the Golādhyāya by Bhāskara. Here only a practical formula is given to proceed with the computation. For an understanding of this formula we have to necessarily draw upon the treatment in Golādhyāya. The word ‘Valana’ means ‘deflection’. The problem posed is at what point of the disc of the eclipsed body does the eclipse begin and at what points it ends. Since an observer sees the disc of the eclipsed body on the back- ground of the spherical surface of the sky, the specification of the point of first contact must necessarily be made with respect to east, west, north and south. These directions could be specified with respect great circles drawn second- ary to the prime-vertical. But the Earth's shadow moves along the ecliptic and the Moon is also very nearly moving on the ecliptic at the moment of an eclipse. Thus ‘Valana’ should give the angle between the ecliptic and the prime-vertical; rather it should be described by two diameters of the Moon's disc, one a secondary to the prime-vertical and one a secondary to the ecliptic. In other words we have to get the angle subtended at the centre of the Moon's disc between those diameters.

375 This angle between the two diameters mentioned, is, for convenience divided into two parts namely K̂MP and P̂MN, where K, P, and N are the poles of the ecliptic, celestial equator and the prime-vertical and M is the centre of the Moon's disc. K̂MP is called Āyana Valana, so called because it depends upon the obliquity of the ecliptic to the Equator (अयनयोः वलनं आयनं वलनम् ie. the deflection due to the deflection of the solsticial points from the equator) whereas the angle P̂MN is called Ākṣa Valana ie. defle- ction of a secondary to the prime-vertical namely NM with respect to a secondary to the celestial equator namely PM which is due to Ākṣa or latitude of the place. We shall first treat the subject on modern lines and then depict Bhāskara's treatment. Let θ, ξ, η stand respectively for the Āyana, Āṣka and total Valanas respectively, where by 'total Valana' we mean K̂MN which is the algebraic sum of K̂MP and P̂MN. From the spherical triangle KMP fig. 78. Fig. 78

376 Sin 90 ∓ α / Sin (90 - β) = sin θ / sin ω = sin (90 - λ) / sin (90 - δ) ∴ Sin θ = (sin ω cos λ) / cos δ or (sin ω cos α) / cos β I [Fig. 79] Similarly from fig. 79, where P = celestial pole, N = North- point, M = centre of the Moon's disc, Q = latitude of the place, h = hour-angle of the Moon, ξ = Āksha Valana and z = Arc of the prime-vertical inter- cepted between the zenith and the foot of the secondary to the prime-vertical drawn through M, which arc goes by the name Sama-Vṛitta-Natāṃ, sa or zenith-distance measured along the prime-vertical- μ = distance of M from the prime-vertical measured along the above secondary, Sin ξ / Sin φ = sin (180 - h) / sin (90 - μ) = sin z / sin (90 - δ) Sin ξ = (sin φ sin h) / cos μ = (sin φ sin z) / cos δ II In fig. 80, where (M) is the Moon's disc, AB, the diameter of the disc extending along the ecliptic (assuming the Moon's centre almost on the ecliptic, which is the case at the time of an eclipse), K, P, N respectively the pole of the ecliptic, the celestial pole and the north point and θ, ξ the Āyana and Ākṣa Valanas defined above, the eclipse starts at A, the eastern side of AB, called the Krānti-Vṛitta-Prāchī, AB being perpendicular to EF a diameter of the disc secondary to the ecliptic. An observer with his physical eye construes the diameter CD, which is secondary to the prime-vertical as indicating north and south. Naturally therefore, it is required to specify the

377 Fig. 80 location of A, the point of first contact, with respect to the diameters GH and CD, which are respectively East- West and North-South. Suppose θ+ξ̅ = 45° = GMA, then we say that the eclipse begins at the north-point of the disc and so on. For this purpose, the concept of Valana arose. We have said above that the angle KM̂N = GMA is to be got, and that it is the algebraic sum of θ and ξ, meaning thereby that when K comes in between P and N, or below N, which is also likely for places of latitude less 48

378 than ω, the obliquity of the ecliptic, KM̂N will be equal to ξ - θ and θ - ξ respectively. The formula given in the present verse is ξ = H sin⁻¹ (H sin 90h / (D/2) × H sin ϕ / H cos δ) or H sin ξ = H sin (90h / (D/2)) × H sin ϕ / H cos δ where h and D/2 are measured in nādīs, h being the hour angle of the Moon and D/2 half-the duration of the Moon's stay above the horizon. Evidently the formula is intended as an approximate one, for all practical purposes considered equivalent to formula II given above namely sin ξ = sin ϕ sin z / cos δ or H sin ξ = H sin ϕ H sin z / H cos δ . Thus in the place of z we are given 90h / (D/2) which means " when z = 90°, D/2 is the hour angle measured in nādīs, what is z when the hour angle is h?". The answer is h × 90 / (D/2) . This formula is approximate because h and z are not strictly in proportion though h increases or decreases along with z. Nonetheless, the formula serves for practical purposes very approximately and the beauty lies in the concept of Sama-Vritta-Natāṁśa, which means measuring hour- angle in terms of the arc of the prime-vertical instead of an arc of the celestial equator. The error, it will be noted will not be much in low latitudes. So far with respect to the commentary on the present verse. Now we shall see how Bhāskara tackles the pro- blem rigorously in Golādhyāya under the caption Valana Vāsanā' ie. 'concept of Valana'.

379 We defined above that the angle KM̂P is Āyana Valana and the angle PM̂N as the Ākṣa-Valana. These are respectively called Bimbīya-Āyana Valana and Bimbīya-Ākṣa-Valana being subtended at M, the centre of the Bimba ie. the disc of the Moon. If in fig. 78, T be the foot of the celestial latitude of the Moon, then the respective angles KT̂P and PT̂N are called the Sthānīya- Āyana-Valana and the Sthānīya-Ākṣa-Valana ie. the angles subtended at the Sthāna or the construed position of the Moon on the ecliptic. The Āyana-Valana is zero and a minimum when M or T lies at the solstices, and a maximum equal to ω when those points lie at r or ♎. Similarly the Ākṣa- Valana is a minimum equal to zero when M or T lies on the meridian and a maximum equal to ϕ when those points lie at the east or west points. In other words the Āyana-Valana increases from zero to ω as M or T moves along the ecliptic from a solstice to an equinoctial point; and the Ākṣa-Valana increases from zero to the maxi- mum value of ϕ as M or T moves along zE or zω from z the zenith to E or ω along the prime-vertical. Hence Āyana Valana is perceived to be proportional to H sin (90+λ) where λ is the longitude of M or T, since when λ = 90, H sin (90 + 90) = 0 and when λ = 0, H sin (90+90)=0 and when λ=0, H sin (90+0)=R, a maximum; similarly the Ākṣa-Valana is perceived to be proportional to H sin z where z is the Sama-Vritta- Natāṁśa defined before, since, when z=0, M or T is at the zenith and the Ākṣa Valana is zero and when M or T is at the East or West point, z=90° and H sin z=R, a maximum. It is worth-hearing Bhāskara, at this juncture (Ref. verses 30-74 under the caption Valana Vāsanā pages 305- 306. Ānandāśrama edition of Golādhyāya Vol. 2. Poona).

380 “The north and the south with respect to the Equator and Ecliptic (ie. the north-pole and south-pole) are differ- ent at the points ♈ and ♎, being at a distance of ω from each other. Hence the Āyana Valanajyā at those points is equal to H sin 24° (ω taken to be equal to 24°). But at the solstices, the north and south will be the same (mean- ing thereby that the angle subtended by PK at the solstices is zero, or what is the same, the directions to the respective poles (of the Equator and the Ecliptic) at the solstitial points are the same so that the East will be the same for both the circles at those points. Thus there is no Valana at the solstices ie. P ⌒ K = 0 where ♋ = cancer. In between ♈ and ♋, the Valana is found in proportion to H sin (90 + λ) where λ is the longitude of the point and in inverse proportion to H cos δ, where δ is the decli- nation. Hence H sin θ = [H sin (90 + λ) / H cos δ] × H sin ω, where θ is the Āyana Valana. Similarly at the points of inter- section of the Equator and prime-vertical namely E and W, the Unmandala (the Equatorial horizon) decides the north-south direction with respect to the Equator, whereas the horizon decides the same with respect to the prime- vertical. These north-south directions with respect to those two great circles namely the Equator and the prime- vertical differ by the angle between the Unmandala and the horizon which is equal PN = ϕ, the latitude of the place. Hence at the East and West points the Ākṣa Valanajya or the H sine of Ākṣa-Valana is equal to H sin ϕ. But at the zenith, the north-south directions of the Equator and prime-vertical coincide so that there is no Ākṣa Valana at the zenith. Thus H sine of the Valana is proportional to H sin ϕ in between the points on the prime-vertical between the zenith and the East and West points. (Roughly speaking) H sin ξ = (H sin ϕ H sin z) / (H cos δ) where ξ = Ākṣa Valana, z = Sama-Vritta-