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

281 First half of verse 56. The Unmandala S'anku multiplied by the Sūtra and divided by Charajyā is also Iṣṭayaṣṭi. Comm. The Unmandala S'anku and Iṣṭayaṣṭi are the lines in vertical planes corresponding to Charajyā and Sūtra in the Equatorial plane. Hence the proportion. It will be noted that the Unmandala S'anku and Iṣṭayaṣṭi are not in the same vertical plane but parallel vertical planes. None the less the proportionality holds good. Latter half of verse 56 and first half of verse 57. The Sūtra increased or decreased by the Charajyā according as the Sun is in the northern or southern hemi- sphere is what is known as Iṣṭāntyā ; similarly the Kalā increased or decreased by Kujyā is what is known as Iṣṭa- Hṛti. Comm. In fig. 52, Iṣṭāntyā = AN = AB + BN = ME + BN = Charajyā + Sūtra. Similarly in fig. 53, Iṣṭa Hṛti = an = ab + bn = sg + bn = Kujyā + Kalā. ∴ Iṣṭāntyā = H cos h + R tan δ tan φ = R (sin φ sin δ + cos φ cos δ cos h) / (cos φ cos δ) in modern terms (28) Latter half of verse 57. Similarly Iṣṭayaṣṭi increased or decreased by the Unmandala S'anku is Iṣṭa S'anku or H cos z. Comm. Let in fig. 54 which represents the plane of the prime-vertical AA', EW, BB', FF', qq' represent the lines of intersection of this plane with planes parallel to the horizon and passing through A, B, F, q of fig. 21. Then Oa = Unmanda-S'anku, Oβ = Sama S'anku, Or = Dinārdha S'anku. If xx' be the line of intersection of this plane of the prime-vertical with a plane passing through an arbitrary position of the Sun in the diurnal circle and 36

282 Fig. 54 parallel to the horizon. then Ox = Iṣṭa-S'anku = Oα + ax = Un-mandala S'anku + Iṣṭa yaṣṭi. ar = yaṣṭi. In the Southern hemisphere Oα, the Unmandala S'anku will be below the horizon so that Iṣṭayaṣṭi decreased by the Unmandala S'anku will be Iṣṭa-S'anku. Thus we have the method of obtaining the Iṣhta- S'anku from the Unnata Kāla as detailed above. We shall see what this process means in modern terms. Unnatakāla-Charakāla = ⊙P̂A − AP̂E (Fig. 21) = ⊙P̂E where ⊙ is the foot of the declination circle of the Sun e in any arbitrary position in his diurnal path. But ⊙P̂E = QP̂E − QP̂⊙ = 90−h where h is the hour angle of the Sun. Thus Sutra = H sin (90−h) = H cos h I

283 ∴ Kalā = (H cos h × H cos δ) / R II ∴ Iṣṭa-yaṣṭi = [(H cos h × H cos δ) / R] × [(H cos φ) / R] III Now Unmaṇḍala-Śaṅku is derivable from the sixth latitudinal triangle in which Krāntijyā is the Karṇa and Unmaṇḍala-Śaṅku is the Bhuja. Comparing it with the second latitudinal triangle Krāntijyā / R = U. S. / (H sin φ) where U. S. is Unmaṇḍala-Śaṅku. ∴ U. S. = (Krāntijyā × H sin φ) / R = (H sin φ H sin δ) / R IV ∴ Iṣṭa-Śaṅku as per the above formulation is given by Iṣṭa-Śaṅku (I. S.) = (H sin δ H sin δ) / R + (H cos φ H cos δ H cos h) / R² = H cos z V [चित्र: गोलीय त्रिभुज PZS (Spherical triangle PZS) — शीर्ष: Z (zenith), P (celestial pole), S (celestial body / Sun); भुजाएँ: ZP = 90 - φ, ZS = z, PS = 90 - δ; कोण: ∠PZS = 90 - A, ∠ZPS = h, ∠PSZ = η] (Ref. fig. 55) Fig. 55 Formulae for △PZS Z = zenith ; P = celestial pole. S = celestial body, say, the Sun. z = zenith-distance of the celestial body S.

284 PZ = colatitude ; PS = north-polar-distance or co-decli- nation. η is called the parallactic angle ; h = hour-angle of S. (90 — a) = The complement of the Hindu azimuth a being measured from the East point. cos (90 — δ) = cos (90 — φ) cos z + sin (90 — φ) sin z cos (90 — a) sin δ = sin φ + cos z + cos φ sin z sin a (1) cos z = cos (90 — φ) cos (90 — δ) + sin (90 — φ) × sin (90 — δ) cos h = sin φ sin δ + cos φ cos δ cos h (2) cos (90 — φ) cos (90 — a) = sin (90 — φ) cot z — sin (90 — a) cot h ie. sin φ sin a = cos φ cot z — cos a cot h (3) cos (90 — φ) cos h = sin (90 — φ) cot (90 — δ) — sin h × cot (90 — a) ie. sin φ cos h = cos φ tan δ — sin h tan a (4) sin z sin (90 — δ) ─────── = ───────────── ie. sin z cos a = sin h cos δ (5) sin h sin (90 — a) This means in modern terms cos z = sin φ sin δ + cos φ cos δ cos h V which we derive from the triangle PZS. Verse 58. To get H cos z from h the hour-angle or nata Kāla. The H. vers (Nata) is called Sara (CQ of fig. 52) (29) Antyā — Śara = Iṣṭāntyā ie. FQ — CQ = FC = AN (fig. 51) H cos δ Śara × Kujyā Śara × ───────── = ───────────── = phala (CQ) (fig. 53) R Charajyā (30)

285 Hṛti — phala = Iṣṭa-Hṛti ie. fq — cq = fc = an (fig. 53) Comm. (Ref. fig. 52). Nata = arc QN ∴ H. vers (Nata) = QC. (Called Śara) = H. vers (h) Antyā — Śara = FQ — CQ = FC = AN = Iṣṭāntya II Śara × (H cos δ / R) = phala = cq (fig. 53) = (Hvers h × H cos δ) / R Hṛti — phala = fq — cq = fc = an = Iṣṭa-hṛti IV Verse 59. (Phala × Koti of a latitudinal triangle) / Karṇa = Ūrdhwa V = βr (of fig. 54) Comm. (Ref. fig. 54) βr is the corresponding line in the plane of the meridian corresponding to phala in the plane of the diurnal circle. As the angle between these two planes is the latitude itself, by the principle of ortho- gonal projection namely. Magnitude of a projected segment = cosine of the dihedral angle × the magnitude of the segment projected, since Ūrdhwa is the orthogonally projected segment of phala, so, Ūrdhwa = phala × cos φ = (phala × H cos φ) / R VI (31) = (phala × Koti of a latitudinal triangle) / (Karna of the latitude triangle) as stated Thus Ūrdhwa = βr (of fig. 54). Verse 60. Ūrdhwa is also given by Ūrdhwa = [U.S. (Unmandala Śanku) × Śara] / Charajyā Dinārdha-Śanku (D.S.)—Ūrdhwa = Iṣṭa-Śanku (I.S.) = H cos z.

286 Comm. Since U.S. is the projected segment of Charajyā on a vertical plane and since the diahedral angle between the planes is φ the latitude U.S. / Charajyā = Cos φ and so Śara × cos φ = (Śara × H cos φ) / R = Ūrdhwa. From fig. 54, Dinārdha-Śanku — Ūrdhwa = o'q — βr (fig. 54) = o'β' = yX = Iṣṭa-Śanku. In modern times, this means, Śara = H. vers (h) = (R — cos h) = 1 — cos h in modern terms phala = (Śara × H cos δ) / R = (1 — cos h) × cos δ ,, ,, phala × (H cos φ) / R = Ūrdhwa = (1 — cos h) cos δ × cos φ ,, ,, Dinārdha Śanku — Ūrdhwa = Iṣṭa Śanku H cos z = cos z (in modern terms) = H cos (φ — δ) (taking northern declination and following Hindu convention with respect to signs) = Dinārdha·Śanku = cos (φ — δ) in modern terms. ∴ cos (φ — δ) — (1 — cos h) cos φ cos δ = cos z ie. cos φ cos δ + sin φ sin δ — cos φ cos δ + cos φ cos δ cos h = cos z ie. cos z = sin φ sin δ + cos φ cos δ cos h as before. Verse 61. Computation of H cos z (Mahā Śanku) through Antyā and Hṛti. Let the Dinārdha Śanku (D.S.) be computed through Iṣṭāntyā and Iṣṭa Hṛti and therefrom Iṣṭa Śanku. From the Śanku, Drik·jyā ie. H sin z and the shadow (KH sin z) / R could be computed —H sin z should not be computed from Hṛti.

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.