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

287 Comm. We computed Dinārdha Śaṅku by the formula. D.S. = (Antyā × U.S.) / Charajyā = (Hṛti × Koṭi of a L.T.) / (Karṇa of L.T.) under verse 36 ; similarly Iṣṭa Śaṅku (I.S.) will be given by Iṣṭa Śaṅku = (Iṣṭāntya × U.S.) / Charajyā = (Iṣṭa Hṛti × Koṭi of a L.T.) / (Karṇa of the L.T.) But Iṣṭāntyā = [R (sin φ sin δ + cos φ cos δ cos h)] / [cos φ cos δ] (as under verse 56) ∴ Iṣṭa Śaṅku = [R (sin φ sin δ + cos φ cos δ cos h)] / [cos φ cos δ] × (R sin δ sin φ) / (R tan φ tan δ) from formulae (13) and (19) = R (sin φ sin δ + cos φ cos δ cos h) = R cos z = H cos z Or again Iṣṭa Hṛti = (R cos z) / (cos φ) from formula (11) ∴ Iṣṭa Śaṅku = (R cos z / cos φ) × (H cos φ / R) = R cos z = H cos z Having got H cos z, using the formula H sin² z = R² — H cos² z, H sin z ie. Dṛk-jyā can be computed. Also K = 12R / (H cos z) and S = (KH sin z) / R give the Chāyā Karṇa and Chāyā. Bhāskara cautions us that H sin z cannot be computed from Hṛti as mentioned in verse 37. because there in that verse, the H sin z computed is that at noon alone. Verse 62. Alternate method of obtaining K. The Chāyākarṇa when the Sun is on the unmaṇḍala multiplied by Kujyā or that when the Sun on the prime- vertical multiplied by Taddhṛti, or again that when the Sun is on the meridian multiplied by Hṛti, divided by Iṣṭa Hṛti, will be equal to the Iṣṭa-Karṇa K.

288 Comm. Equation (23) under verse 41 is (Iṣṭa Śaṅku / Iṣṭa Hṛti) = (D.S. / Hṛti) = (S.S. / Taddhṛti) = (U.S. / Kujyā) = cos φ I and 12 / K = (H cos z) / R (under verse 40) ie. K = 12R / Iṣṭa Śaṅku which means Dinārdha Karṇa = 12 R / D.S. ; Sama Karṇa = 12R / S.S. and unmaṇḍala Karṇa = 12 R / U.S. Substituting for the numerators in I 12R / (K × Iṣṭa Hṛti) = 12R / (DK × Hṛti) = 12R / (S.K × Taddhṛti) = 12R / (U.K. × Kujyā) II ∴ Iṣṭa Karṇa × Iṣṭa Hṛti = S.K. × Taddhṛti = U.K. Kujyā ∴ Iṣṭa Karṇa = (Uumaṇḍala Karṇa × Kujyā) / Iṣṭa Hṛti = (Sama Karṇa × Taddhṛti) / Iṣṭa Hṛti = (Dinārdha Karṇa × Hṛti) / Iṣṭa Hṛti Verse 63. Just a caution. If in any context where the word ūna-yuta has been used, the quantity to be subtracted exceeds the quantity from which it is to be subtracted, it goes without saying that subtraction should be reversely effected and in the place of addition subsequently prescribed subtraction should be done and Vice-versa. Comm. Bhāskara gives three examples to illustrate his point. In verse 54, while defining Sūtra (H sin 90 – h) we are asked to subtract chara from unnata when δ > 0

289 Fig. 56 and add Chara to Unnata when δ < 0 and take the H sine of the result. Let us first consider the case when δ > 0. (Refer fig. 56) when the Sun is at ☉, Iṣṭa Śaṅku is H sin ☉L; and Unmandala Śaṅku is H sin BM. When the Sun is at ☉₁, Iṣṭa Śaṅku = H sin ☉₁N. In the first case Iṣṭayaṣṭi = (H sin ☉L — H sin BM) which will be the orthogonal projection of ☉B on the meridian plane. Unmandala Śaṅku and the Iṣṭa Śaṅku in the position ☉₁ are similarly the orthogonal projections of BM and ☉₁N on the same plane. In the position ☉ Iṣṭa Śaṅku = Unmandala Śaṅku + Iṣṭayaṣṭi, whereas in the position ☉₁, Iṣṭa Śaṅku = Unmandala Śaṅku — Iṣṭayaṣṭi which is now downwards. Thus in the place of addition we have subtraction of Iṣṭayaṣṭi. This reversion has arisen out of 37

290 the fact that in the position ☉, Sūtra is the H sine of (KA — KE) whereas in the position ☉₁ Sūtra is the H sine of EC ie. H sine of (KE — KC) ie. in the former position Sūtra = H sine (Unnata-Chara) and in the lattter Sūtra = H sin (Chara-Unnata). Thus a reversion in subtraction here, effects a reversion of addition of the Iṣṭayaṣṭi. Similar is the case in the other cases cited by Bhāskara. Analytically this happens so because cos h, when h > 90, becomes negative and adding cos h tantamounts to subtracting sin θ where h = 90 + θ. Verse 64. Another point to be observed. Hvers (90 + θ) = R — H cos 90 + θ̅ = R + H sin θ. The Unmandala Sanku is not observable when δ is south in as much as it is below the horizon ; none the less it may be computed for the purposes of effecting proportion. Fig. 57 Hvers (CG) = Hvers (CÔG) = GH Hvers (EG) = Hvers (EÔG) = Hvers (90 + θ) = R + OH' = R + EL = R + H sin ☉ as defined by Bhāskara,

291 Comm. Under verse 58, we had to form Hvers h; a doubt might arise as to what this Hvers h would be if h > 90. Hence Bhāskara defines it and the definition is clear from fig. 57. The Unmandala S'anku has the formula R sin δ sin φ so that either when δ is negative or when φ is negative, it will be negative which means it will be in the opposite direction ie. vertically downwards. Since negative latitudes are not considered by the Hindu astronomers, the other case alone is considered. Even from a figure it is evident that when the Sun is in the south of the Equator, the Unmandala S'anku is vertically downwards. In proportions like I given under verse 62, we can use the magnitude of this Unmandala S'anku also and it does not vitiate the results, when we take its numerical value. Verse 65. The Sun crosses the prime-vertical when his northern declination falls short of the latitude. Then alone there is sense in giving the magnitude of his shadow at that moment. When the Sun does not cross the prime- vertical at all, the Sama S'anku which could be computed out of its formulation, though it does not exist, under the Sun, yet, it could be used in proportions (like I under verse 62) and no blunder is committed. Comm. This is a beautiful example cited by Bhāskara where he intuitively uses the so-called principle of geometrical continuity. We have formulated Sama S'anku as (R sin δ) / (sin φ) ie. H cos z = (R sin δ) / (sin φ) . We have a real value of z when δ > φ, for, then only H cos z < R. When φ > δ, then H cos z > R which is impossible, for, no Hindu sine or Hindu cosine could be greater than R just as no modern sine or cosine could be greater than unity.

292 TERMINOLOGY N.B.—In as much this Tripraśnādhyāya has a good number of technical terms whose understanding is necessary to understand the Hindu methods of solving diurnal problems, we shall collect here all such technical terms under this heading for guidance.

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298 | Cara Cāpa | Arc of the celestial Equator bet. the East point and foot of the declination circle | | | | | Carajyā | H sine of the above or the corresponding line of Kujya in the plane of the celestial Equator | 13 | 13-17 | (R · H sin φ · H sin δ) / (H cos φ · H cos δ) | R tan φ tan δ | | Natakāla | Hour angle h | | | | | | Unnata | Time elapsed after rise | | | | | | Śara | Corresponding line of phala in the Equatorial plane | 29 | 58 | H vers (h) | R (1 — cos h) | | Phala | ⊥ᵃʳ from the culminating point on a line through the celestial body parallel to Udayāstastūtra | 30 | | (H vers h · H cos δ) / R | R cos δ (1 — cos h) | | Ūrdhwa | Orthogonal projection of phala on the plane of the meridian | 31 | 59 | (H vers h · H cos φ · H cos δ) / R² | R cos φ cos δ (1-cosh) |

299 Yet (R sin δ) / (sin φ) will have a value greater than R ie. even though the Sama-S'anku is never born so to say, it has a magnitude. ‘तत्कथमिदं वन्ध्यासुतवत्’ Bhāskara exclaims with respect to the magnitudes of Sama-S'anku and Taddhṛti as well, both of which are not there, yet, both of which have magnitudes greater than R. So he says “those magnitudes of the Sama-S'anku and Taddhṛti are like the sons of a barren lady”. Then he says ‘तदपि प्रदर्श्यते’ ie. ‘We shall show how they arise.’ Here he uses his intuition of the principle of geo- metrical continuity. Even when the diurnal circle does not cut the prime-vertical, their planes intersect, out- side the sphere and the perpendicular dropped from the point of intersection on the plane of the horizon is the Sama S'anku which has a magnitude greater than R. Similarly the Taddhṛti could be seen what it is now. These magnitudes can enter into a proportion like the I in verse 22, and do help us to get the other real magnitudes like the Unmandala S'anku etc. Verses 66, 67 and first half of 68. To obtain the time from the shadow. Iṣṭāntyakā = (U.K. × Carajyā) / I.K. = (D.K. × Antyā) / I.K. = (k × R²) / (R cos δ × I.K.) ; Rvers⁻¹ (Antyā − I. A.) = h Dinārdha − h = Unnatakāla where K = Karṇa, k = Vishuvat Karṇa, I.A. = Iṣṭāntya. Comm. We had under verse 62. Iṣṭa-Karṇa × Iṣṭa-Hṛti = D. K. × Hṛti = S. K. × Taddhṛti = U. K. × Kujyā multiplying throughout by R / (R cos δ)

300 Iṣṭa-Karṇa × I. A. = D. K. × Antyā = U. K. × Carajyā so that I.A. = (D.K. × Antyā) / I.K. – (U.K. × Carajyā) / I.K. I But U.K. = 12 R / U.S. (verse 40) ∴ U.K. × Carajyā = (12R × Carajyā) / U.S. = (12R × Kujyā × R) / (U.S. × H cos δ) since Carajyā = (Kujyā × R) / (H cos δ) But Kujyā and U.S. are the Karṇa and Koṭi of the seventh latitudinal triangle so that Kujyā / U.S. = k / 12 Comparing with the first fundamental latitudinal triangle. ∴ U.K. × Carajyā = (12R² × k) / (H cos δ 12) = kR² / (H cos δ) Hence substituting in I I.A. = (U.K. × Carajyā) / I.K. = kR² / (H cos δ × I.K.) Thus we have proved the first part of the statement. Having obtained I.A., from fig. 52 we have Antyā – I.A. =(FQ–AN)=CQ. The Utkrama Cāpa of CQ = NQ = h and Dinārdha – h = Unnatakāla. Let us see what this procedure means in practice. Since on any day at any place, φ and the declination of the Sun are known we can compute all the magnitudes given in the verse or more easily H cos δ so that from the formula Iṣṭāntyā = 12R² / (H cos δ × I.K.) where K = √(S² + 12), the shadow being observed Iṣṭāntya could be computed in no time. Also the Antyā of the day R+(H tan φ tan δ) can be computed so that the segment CQ can be got. The inverse Hversine of this is h. The arc CQ above was symbolized as Sara.

301 Bhāskara’s proof of I.A. = kR² / (H cos δ × I.K.) proceeds from first principles as follows :—(i) If by k we have 12 as the Koṭi what have we for R? The result is H cos z, Mahāśaṅku. ∴ Mahā Śaṅku = 12 R / I.K. From Mahā Śaṅku we pass on to Iṣṭa Hṛti with which it forms a latitudinal triangle. If by the gnomon of 12 units we have k the Viṣuvat Karṇa, what have we by Mahā Śaṅku? The result is (12 R / I.K.) × (k / 12) = kR / I.K. Again from the Iṣṭa-Hṛti we pass on to I.A. by multiplying by R / (H cos δ) so that I.A. = kR² / (H cos δ × I.K.) as given. Second half of verse 68. The inverse Hversine if a quantity x greater than R, is 5400 + H sin ⁻¹ (θ) when x - R = θ. Comm. Since Hvers (90 + θ) = R + H sin θ = x (say) 90 + θ = Hvers ⁻¹ (R + H sin θ) = But 90° are equal to 5400 asus and θ = H sin ⁻¹ (H sin θ) = H sin ⁻¹ (x - R) = त्रिज्याधिकभागस्यक्रमचापम् ∴ 5400 + त्रिज्याधिकभागक्रमचापम् = Utkrama Cāpa of a त्रिज्यादिक quantity. Verse 69. Alternate method of obtaining the time that has elapsed after Sunrise noting the shadow S. Subtract Charajyā from or add it to Iṣṭāntyā according as δ is north or south. Obtain inverse H sine of the remainder and add the Caracāpa to this inverse H sine. Then we have the unnatakāla by converting the result into time.

302 Comm. Ref. fig. 52. I.A. = AN. I.A. - Carajyā = B.N. Hvers⁻¹ (BN) = arc EN. Arc EN + Caracāpa = EN + EM = MN. This converted into time is evidently the Unnatakāla because the arc MN of the equator is the arc intercepted between the feet of the declination circles at rising and at the time concerned M being the foot of the rising declination circle and N the foot of that at the time in question. The convention of signs is clear. Verse 70 and first half of 71. To obtain the Sun's longitude from the shadow S. The gnomonic shadow at noon, being multiplied by R and divided by K, the inverse H sine of the result gives the meridian zenith-distance. This being decreased or increased by the latitude gives the Sun's declination according as the extremity of the shadow is north or south. From the declination, we have the Sun's longitude by the formula H sin δ = (H sin λ H sin ω) / R . Comm. We have from the triangle formed by the gnomon and the shadow S, S / K = sin z or SR / K = H sin z ∴ H sin⁻¹ (SR / K) = z. Since we are directed to take the mid-day shadow, we have the meridian zenith-distance and from the formula z + δ = φ we have δ. If the extremity of the shadow be north, the Sun is south of the zenith, and then φ ~ z = δ. The word वियुक्ताः is used to signify difference which is positive. If z > φ then the declination is south and vice-versa. If the extremity of the shadow is on the south, the Sun is on the north of the zenith. in which case φ + z = δ. Second half of verse 71. To obtain φ from δ.

308 If the zenith-distance and the declination are of the same direction, their difference, otherwise their sum will be the latitude. [Fig. 58] Comm. Suppose the zenith- distance is north and decli- nation also north, then clearly from fig. 58, in this position S₁ of the Sun QS₁ — ZS₁ = ϕ = δ — Z (1) Again in the position S₃ of the Sun, zenith-distance is south and declination is also south ; so, here also difference gives ϕ ie. ZS₃ — QS₃ = Z — δ = ϕ. (3) In the position S₂, how- ever, Z is south and δ is north, so that their sum is equal to ϕ. It will be noted that the Hindu convention of signs does not contemplate negative declination and also it uses the word 'difference' to signify the positive difference, as for example, in the first two cases δ ~ Z is taken as ϕ. The modern formula Z + δ = ϕ applies universally with the convention that δ is +ve if north, ϕ is +ve if south. Verses 72, 73. To obtain the Bhuja from the shadow. Karṇa Vrittāgrā = (A × K) / R where A = Agrā and K the Chāyākarṇa. This Karnāgrā is to be taken as belonging to the opposite hemisphere to the Sun. Calling this Karnāgra as a and the equinoctial shadow as s, a ⨦ s according as δ is south or north gives the Chāyābhuja b. Thus a ⨦ s = b. If the extremity of the shadow be on the north and δ be north, then b + a = s ; if δ be south b ~ a = s. If b be north, b ~ s = A, otherwise ie. if south b + s = a. (R × a) / K = Agrā and (Agrā × Koṭi of latitudinal triangle) / (Karṇa of the latitudinal triangle) = H sin δ.

304 Comm. These verses are very important and the contents have been already elucidated under verses 13–17. We shall elucidate the convention of signs in more detail both from the modern point as well as from the Hindu traditional point. First we shall discuss the modern. We have the formula A = S + B where A = Agrā, S = Sankutala and B, Sanku-bhuja. Let us confine ourselves to north latitudes alone, for, south latitudes did not concern the Hindu astronomers. Then treating north declination as positive and also north Hindu azimuth as positive the above formula holds universally. (Ref. fig. 59) Let the figure represent the meridian plane. Let S₁, S₂, S₃ be the projections of the Sun's positions in their diurnal circles on to the meridian plane. Let A, B. be the Fig, 59

305 projections of the rising points of the Sun on the same plane. Let perpendiculars be dropped from S₁, S₂, S₃ on the plane of the horizon. Let o be the centre of the sphere. Let ns be the north-south line. In position S₁, S₁C = Śanku-Bhujā, KA = Śankutala, oA = Agrā, so that A = S + B (1) In the position S₂. S₂D = Śanku-Bhuja, LA = Śanku-tala, oA = Agrā, so that B + A = S; but here B is negative, a the azimuth being south so that writ- ing - B for B, - B + A = S ∴ A = S + B again. In position S₃, OB = A, MB = S, OM = B so that A + S = B; but here A is +ve, δ being south and B is negative a being south; hence writing - A and - B for A and B A + S = - B or A = S + B again. This shows that with the convention cited above A = S + B holds good universally. Now let us consider the situation with respect to the Karnāgrā. Each of the three quantities a, b, s have now opposite directions. If δ be north, the Sun will be on the north of Equator, but the extremity of the shadow will now be on the south of the Equinoctial shadow line (E.S.L.) ie. the line which is parallel to the East-west line at a distance of the equinoctial shadow s, and which is tbe locus of the extremity of the shadow on the equinoctial day; thus when the Agrā is considered to be positive being on the north of the East point, the Karnāgrā, though it is on fhe south will have to be considered positive. Similarly when the azimuth of the Sun a is on the north of the prime-vertical and is so considered to be positive, the extremity of the shadow being on the south of the East-west line and has a negative azimuth, the bhuja is still to be considered positive. Again the Śankutala being always south of the Udayāsta Sūtra being considered south and positive, the corresponding quantity into which it gets converted on the horizontal dial namely the equinoctial shadow s will be north of the East-west line and will be considered 39

306 positive. In other words in the equation a = b + s, a is positive when δ is north, b is +ve when the azimuth of the Sun is north of the East point, and s is always positive. Since in north latitudes, Śaṅkutala will be always south of the Udayāstasūtra and considered positive, the E.S.L. will be on the north of the East-west line so that s is considered positive. We should have had to consider s negative in southern latitudes, as per the above convention but as the Hindu astronomers did not have to concern themselves with south latitudes, the question of sign for s did not arise except taking it as always posi- tive. Hence the equation a = b + s will hold universally with the same conventions of sign which we stipulated with respect to the equation A = B + S. The foregoing analysis is on the modern lines. [चित्र: वृत्त के अंतर्गत W-E व्यास, E·S·L समानांतर रेखा, तथा बिन्दु O, C, P, R, M, L, A, N, B, K युक्त ज्यामितीय आरेख] Fig. 60