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

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-

381 Natāṁśa (defined before) and H sin z may be taken to be roughly equal to 90h / (D/2) (as depicted before). In the East the Ākṣa Valana is north, for, in fig. 80 ĜMI which gives the East of the Equator with respect to the the East of the prime vertical, is north; whereas in the West ĤMJ is south. (The definition of the direction of the Valana is given as a directive to add the two kinds of Valanas if they be of the same direction otherwise to take the difference; in the fig. 80, the Āyana Valana ie. ÎMA is also north, so that adding ĜMI + ÎMA = ĜMA is the Sphuta Valana or the actual Valana). Hence Sphuta Valana measured by GMA is had by the sum or difference of the two angles ĜMI and ÎMA which define respectively the Āyana and the Ākṣa Valanas. Similarly, at the point of intersection of the Ecliptic and the prime-vertical, the Sphuta Valana is a maximum which is the sum or difference of the Valanas as the case may be. At points removed 90° on either side, from the point of intersection of the Ecliptic and the prime vertical, in as much as the north-south directions with respect to the Ecliptic and the prime-vertical coincide, the Sphuta Valana is zero. If (as Lallāchārya said) the Valana varies as the Hversine at those points which are removed by 90° from the points of intersection of the Ecliptic and the prime- vertical, the Sphuta Valana will not be zero (which is against common sense). Hence the Valanajya varies as Hsine and not as Hversine. We shall look at the subject from another point of view for the sake of clarity.........Fix a circle on the sphere with the celestial pole as centre and ω as the angular

382 radius. This circle is called Kadamba-Bhrama-Vritta or the circle in which the pole of the Ecliptic revolves round P (due to diurnal revolution of the Earth). In that circle Hsine of an angle will be H sin δ............Or again draw the great circle with the planet's position as the pole, called the horizon of the planet. The arc intercepted on this circle between the Ecliptic and the celestial Equator will be Āyana Valana and that intercepted between the celestial Equator and the horizon is the Ākṣa Valana; and the arc intercepted between the Ecliptic and the horizon is the Sphuṭa Valana. Or again draw a circle with K as centre and radius ω = 24°. This circle is called the Jina-Vritta where the word Jina means 24. Let a secondary to the Ecliptic passing through K and K' the poles of the Ecliptic revolve with KK' as fixed. When this revolving circle passes through Cancer (Sāyana) it will be passing through P. The angle turned through by this circle from Cancer, will be equal to the angle turned through from P. The Hsine of that angle in the Jina Vritta will be H sin δ of a longitude equal to that angle. This is the Āyana Valana and it arises at the end of Dyujyā, since the north-polar distance of the planet is (90-δ) whose Hsine is Dyujya ie. H cos δ. The corresponding Āyana Valana in a circle of radius R is got by multiplying by R and divided by H cos δ. Let us clarify Bhāskara's mind. (Ref. fig. 81) Let PBD be the Jina Vritta drawn on the sphere with K, the pole of the Ecliptic as centre and ω=24° as radius. Let a revolving secondary to the Ecliptic coincide initially with KC where C is Cancer. Let it occupy subsequently the position KM where M is the centre of the Moon's disc taken to be on the Ecliptic as is the case very approxi- mately at the moment of an eclipse. Now the Āyana Valana is the angle KMP. Let MA be the declination of M. Produce MP to L such that MK=ML=90°. Hence

383 Fig. 81 PL = δ since PA = 90° and LM = 90°. The Āyana Valana K̂MP is measured by the arc ML where ML is an arc of the Grahakṣitija or the horizon of the planet M (ie. the circle with M as centre and 90° as radius drawn on the sphere or what is the same the great circle whose pole is M). PB is an arc of the small circle parallel to KL which is an arc of a great circle. Then in the Jina Vritta sin PB = sin PK × sin P̂KB = sin ω sin (90 - λ) = sin ω cos λ ∴ sin KL = sin ω cos λ/cos PL = (sin ω cos λ) / (cos δ) = sin PMK.

384 Here sin ω cos λ is called Sa-thribha-graha-ja-kranti or the declination of a point whose longitude is 90+λ where λ is the longitude of M. As we have the formula sin δ = sin ω sin λ, sine of the declination of such a point is equal to sin ω sin (90+λ)=sin ω cos λ. When δ is very small sin PMK may be taken to be sin ω cos λ or what is the same Sa-thribha-graha-ja-krānti as is formulated by Sūrya- siddhānta. It may be doubted how sin PB=sin PK sin 90-λ. (Ref. fig. 82). Let K′ be the centre of the circle PBD, Fig. 82 K′ being in the plane of PBD. K′P and K′B are radii of this circle. Since the arc PB stands for 90-λ PK′B= 90-λ. Draw the H sine of arc PB, which is PB′. Now K′P=H sin ω, as PK′ is ⊥ar drawn on OK ∴ PB′=PK′ sin PK′B =H sin ω ✕ sin PK′B = (H sin ω ✕ H sin PK′B) / R = (H sin ω H cos λ) / R ∴ H sin KL = [(H sin ω H cos λ) / R] ✕ H cos δ = (H sin ω H cos λ) / (H cos δ) as given.

385 [चित्र: Fig. 83 — Equator, Ecliptic, Prime vertical] Fig. 83 Note 1. Bhāskara says PB' (fig. 82) is Krānti-Sinjanī; so it is because PB' = PK' sin (90 - λ) = (H sin ω H cos λ) / R = H sine of the declination of a point whose longitude is 90 + λ = Satribha-grahaja-krānti as is mentioned by Bhāskara and Sūrya-siddhānta. Note 2. If M be the centre of the Moon's disc and ABC its horizon defined above, the arc AB intercepted between the Ecliptic and the Equator is Āyana Valana, the arc BC intercepted between the Equator and the prime- vertical is Ākṣa Valana and the arc AC intercepted between the Ecliptic and the prime-vertical is Sphuta- Valana. Note 3. The analysis of Ākṣa Valana proceeds on similar lines, only we have P and N in the place of K and P. Note 4. The mistake of Lallāchārya alluded to by Bhāskara is as follows. The Āyana Valana, we have seen is zero at the Ayanas ie. the solstices and maximum at ♈ and ♎ ie. the equincotial points removed by 90° from the Ayanas. Now Hversine = R — H cosine so that when 90 — λ = 0 ie. λ = 90°, Hversine (90 — λ) = R — H cos (90 — λ) = R — H sin 90° = R — R = 0 and when 90 — λ = 90° ie. 49

386 λ = 0 Hversin (90 — λ) = R — H cos (90 — λ) = R — H sin λ = R — H sin 0° = R — 0 = R. Hence Lallācharya took by mistake that the Āyana Valana varies as Hvers (90 — λ) instead of H sin (90 — λ) since both Hversine and H sine of 90 — λ̄ are zero at the Ayanas and maximum at r and ♎. The same mistake was committed by Lallācharya in the context of the Moon’s phase also as criticised by Bhāskara as we shall see later. In fact, this latter criticism is not so justified as the former, as will be shown in that context. Note 5. If instead of taking the Āyana Valana to vary as H sin (90 — λ) we happen to take according to Lallācharya that it varies as Hversine, then in places (Ref. verses 38, 39 Valana Vāsanā, Golādhyāya) removed by 90° from the points of intersection of the Ecliptic and the prime-vertical, where there should be no Sphuta- Valana, we do get that there is some Sphuta Valana there, since the value of Hversine differs from Hsine, though these two functions happen to be zero simultaneously and maximum simultaneously. Bhāskara continues in verses 66-68 (Ibid) “ I shall now depict Ākṣa Valana by means of the hour-angle. Take the sum or difference of S'anku-Agrā and S'anku-tala according as they are of the same direction or not ; compute √(R² — B²) where B is the result ; then H sin ϕ H sin h ─────────────── is equal to H sin ξ where ξ is the Ākṣa Valana ”. √(R² — B²) Comm. We saw before in the Tripraśnādhyāya that A = S + B where A = S'anku-Agrā, S = S'anku-tala, and B = S'anku-bhuja = H sin µ where µ is represented in fig. 79. Hence √(R² — B²) = H cos µ so that the above formula gives H sin ϕ × H sin h H sin ξ = ───────────────────── which is the same as got by H cos µ the modern formula in Equation II.

887 Note. A small circle parallel to the prime-vertical is called Upa-Vṛtta. Also secondaries drawn to the Ecliptic, Equator and the prime-vertical are called Kadamba-Sūtra, Dhruva-Sūtra and Sama-Sūtra. They are also called occasionally as Kadamba-prota-Vṛtta, Dhruva- prota-Vṛtta, and Sama-prota-Vṛtta. Bhāskara proceeds to find the Āyana Valana in a very ingenious way in verses 69-74. We shall first give it a modern treatment so that we may better appreciate his genius. Let (S) be the Sun's disc. (It does not matter whether we take the Sun or the Moon). EQ is its diameter along the diurnal circle, and CL along the Ecliptic. LM is the difference of the declination of L and S. Let SL = Δ λ and LM = Δ δ. We have sin δ = sin λ sin ω ; differentiating cos δ Δ δ = sin ω cos λ Δ λ ∴ Δ δ = (sin ω cos λ / cos δ) × Δ λ ; put Δ λ = b, the angular radius of the disc ∴ Δ δ = (b × sin ω cos λ / cos δ) = (b H sin ω × H cos λ / R × H cos δ) . This gives the Valana in the disc of radius b. If that be so, what will it be on the sphere of radius R? The result is (b H sin ω × H cos λ / R × H cos δ) × R / b = (H sin ω × H cos λ / H cos δ) as got before. Let us hear Bhāskara, " put the disc of the Sun at the point of intersection of the Ecliptic and the diurnal circle. The Valana (LM of fig. 84) at the periphery of the disc is the difference of the declinations of L and S. To get the value of this let us first get the value. SL in terms of λ, the longitude of S. It is (b × B / 225) ; so that LM

388 Fig. 84 will be equal to (b × B) / 225 × (H sin ω) / R where B is the Bhogya- khanda of λ. To obtain the value of the above for a circle of radius R from a circle of radius b, we have to multiply by R/b. So, the result is (b × B) / 225 × (H sin ω) / R × R / b = (B H sin ω) / 225 . But the value of B is got as follows. 'If for H cos λ equal to R we have the first Bhogyakhanda equal to 225, what shall we have for H cos λ?' The result is (225 × H cos λ) / R . Substituting for B, we have (H sin ω) / 225 × (225 × H cos λ) / R = (H sin ω × H cos λ) / R Now, on account of declination, the Sun's disc is inclined like an umbrella. So LM of fig. 84 will take a position like L'M as shown in fig. 85 where the triangle MLL' is similar to SMO, S being the centre of the Sun's disc, O the centre of the sphere. Hence (L'M) / LM = R / (H cos δ) ∴ L'M = R / (H cos δ) × (H sin ω H cos λ) / R = (H sin ω × H cos λ) / (H cos δ) as got before '.

389 Comm. Bhāskara terms SL as the Dorjyāntara ' or the variation in H sin λ, which he knows to (H cos λ Δλ) / R. [Fig. 85: Triangle L'LM] [Fig. 86: Triangle SMO, Hypotenuse = R, SM = H Cos δ, MO = H Sin δ] Fig. 85 Fig. 86 But proceeding from first principles, as he always does, he asks us to consider the Bhogyakhanda at λ namely B. If this be for an interval of 225', what will be it be for ‘b’? The result is (b × B) / 225 . Then to rectify B, the proportion used is as used above. Bhāskara says many a time that the variation in Hsine is proportional to Hcosine. This concept he might have derived by looking at the Hsine table of 90 Hsines. Hence the argument advanced by him to rectify B is ‘If for Hcosine equal to R (at zero-value of the argument) the initial Bhogyakhanda is 225, what will it be for an arbitary H cos λ? The result is (H cos λ) / R × 225. Substituting this for B in the above, we have (b H cos λ) / 225 × 225 = (b H cos λ) / R . This expression we perceive as no other than (H cos λ Δλ) / R as equal to Δ (H sin λ), for b is to be taken as Δλ. This (H cos λ × b) / R is called by Bhāskara as Dorjyāntara meaning thereby Δ (H sin λ).