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

247 Verse 24. (Kujyā × Bhuja) / Karṇa = second segment of Agrajyā (Krāntijyā × Koṭi) / Karṇa = first segment of Agrajyā and Agrā = Sum of the two segments. Comm. The first statement is based on the similarity between the seventh triangle and others and the second on the similarity between the 6th and the others. Verse 25. (First segment of Agra × Bhuja) / Koṭi = Un-mandala-Śaṅku and (Krāntijyā × Bhuja) / Karṇa = Un-mandala-Śaṅku. (19) Comm. Both the statements are based upon the similarity of the sixth triangle and others. Thus un- mandala śaṅku = U.S. = (H sin δ × H sin φ) / R = = R sin δ sin φ. Verse 26. (First segment of Agrā × Koṭi) / Bhūja = Un-mandala-Śaṅku = Kujyā × Koṭi / Karṇa. Comm. The first statement is based on the similarity of the sixth latitudinal triangle and others and second that between the seventh and others. Sama-śaṅku − unmandala śaṅku = upper segment of Sama-śaṅku. Verse 27. (Agrā × Bhuja) / Karṇa = Kujyā Taddhṛti − Kujyā = upper segment of Taddhṛti.

248 Comm. The first statement is based on the similarity between the third triangle and the others and the second statement is evident. Second half of verse 27. Other elements could be derived from what is already known and from what has been obtained. Also by alternando and invertendo we could pass from one element to the other and vice- versa. Verse 28. Karṇa = √(Bhuja² + Koti²) Bhuja = √(Karṇa² − Koti²) Koti = √(Karṇa² − Bhuja²) Thus the third could be had from the other two in all the triangles. Verse. There are sixty-three ways of obtaining H sin ϕ and H sin ϕ. On account of hundreds of ways of obtaining Agrajyā etc., there are an infinite number of ways of obtaining H cos ϕ etc. Comm. Under verse 23 Bhāskara says that there are 98 ways of obtaining Taddhṛti. Taking the third latitude triangle, H sin δ could be obtained in seven ways, from this H sin δ, Kujyā could be obtained in seven ways; hence, according to the principle of association namely that when one thing could be done in m ways and another in n ways, both the operations could be together performed in mn ways, so Kujyā could be obtained in 7 × 7 = 49 ways; similarly the upper segment of Taddhṛti could be had in 49 ways; so that adding the two Taddhṛti could be obtained in 98 ways. Similarly suppose we have to find H sin δ. H cos ϕ could be found in seven ways and from H cos ϕ, H sin δ could be found in seven ways. Thus H sin ϕ could be

249 found in 7 × 7 ways = 49 ways. From R, H sin φ could be found in seven ways by using similarity with the other seven latitudinal triangles except the second. Also obtaining H cos φ in seven ways and using the formula H sin φ = √(R² — H cos² φ) we have seven more ways. Thus in all there are 7 × 7 + 7 + 7 = 63 ways. Similarly H cos φ could be found in 63 ways. Extending this to Agrajyā etc. which could be in as many or more ways themselves finding H sin therefrom means again finding it in 69 × 69 ways and so on. Since there is no end in counting all these ways, it is said that there are infinite ways to find it. The word 'infinite' here connotes only a very large number of ways not exactly what we mean by the word 'infinity'. Verse 30. To find what is known as Koṇa-Śaṅku. As a first approximation take Koṇa-Śaṅku = √(R² — 2A²) where A = Agrajyā and Koṇa- Śaṅku means H cos z when the azimuth is equal to 45°. Then take Agrajyā ± the above Koṇa-Śaṅku × s/12 = Śaṅku- bhuja = b (say) then again Koṇa-Śaṅku = √(R² — 2b²). Then again take Agrajyā ± the above Koṇa-Śaṅku × s/12 as the new bhuja and proceeding thus by the method of successive approximations, we arrive at a constant value which gives the Koṇa-Śaṅku. Comm. Bhāskara gives later the method of obtaining H cos z ie. the Śaṅku pertaining to any zenith-distance. So, he need not have given a separate treatment for this Koṇa-Śaṅku. But in as much as Brahmagupta and other previous writers gave it he has also given the same. He gives here the method of finding the Koṇa-Śaṅku by the method of successive approximations as given by Śrīpati. 32

250 We saw before that Agrā = Śaṅku-tala + Śaṅku- bhuja. So, in the first place as a first approximation, Agrā is taken as Śaṅku-bhuja. Since, when z = 45° the perpendiculars from the celestial body on the planes of the prime-vertical as well as meridian are equal, and since the perpendicular on the plane of the prime-vertical is called Śaṅku-bhuja = b (say) 2b² = H sin² z. This is so because H sin² z = Sum of the squares of the perpendiculars on the planes of the meridian and prime-vertical. But H sin² z = R² − H ocs² z. ∴ R² − 2b'² = H cos² z. So, taking Agrā as the bhuja b as a first approximation, √R² − 2b² gives us H cos z. From this using formula III under latitudinal triangles namely H cos z × s/12 = Śaṅku-tala, obtain the approximate Śaṅku-tala, from the approximate H cos z got above. Now using the formula Agrā = Śaṅku-tala + Śaṅku-bhuja obtain Śaṅku-bhuja as Agrā ⩲ Śaṅku-tala, where the +ve sign is taken when the Sun has a southern declination, and the difference sign when the declination is north. Taking this Śaṅku-bhuja, b', Koṇa-Śaṅku is now √R² − 2b'². In the first place we took the Agrā itself as the bhuja; but here we have a better approximation for the Śaṅku-bhuja. From this Koṇa-Śaṅku again, obtain a still better approximation for Śaṅku-bhuja and proceeding thus till a constant value is obtained, we have the required Koṇa-Śaṅku. This is a beautiful example where the method of successive approxi- mation was used by the Hindu Astronomers to a good advantage. It will be noted here that the Śaṅku-tala is always treated as extending south ie. the Śaṅku-tala will be south of the Śaṅku since India's latitudes are all north. Also it is said that when the Sun has southern declination, A + S = B and when northern A − S = B when A = Śaṅku-Agrā or simply Agrā (in contradistinction to Karṇāgrā reduced to a circle of radius K the chayākarṇa) S = Śaṅku-tala and B = Śaṅku-bhuja. This convention

251 of signs is to be correlated with the modern. We have from the formula derived out of PZS, A = S + B. According to modern convention when δ is north, it will be taken to be positive and when a is to the north of the East point it also will be taken to be positive so that, (1) when δ is north and a north, A = S+B ie. B = A-S; this accords with the Hindu convention namely ‘सौम्येऽन्तरम्’ (2) when δ is south and a south, - A = S - B so that B = A + S; this also accords with the Hindu convention, namely याम्ये योग: (3) But, however, when δ is north and a south ie. when the Sun having northern declination comes to the south of the prime-Vertical, A = S - B so that B = S - A. This accords with the Hindu convention if only we take B = | A-S | when δ is north. Bhāskara makes two statements at the end of the commentary under this verse namely that when δ is south and A > 2431, there will be no Koṇa-Śaṅku and that when δ is north and s > 17'' - 5''' there will be four Koṇa-Śaṅkus. We have to verify these statements. The first statement is evident because H sin (Agrā) > H sin 45°

252 as may be seen by taking δ = 20°, φ > 61° - 6'. Thus Bhāskara gave the minimum latitude which could enjoy four Koṇa-Śaṅkus. Fig. 41 Fig. 42

253 From fig. 41, it is clear that there are two Koṇa- Śaṅkus at S₁ and S₂ during the forenoon and similarly two at S₁' and S₂' in the afternoon where S₁' and S₂' are the symmetrical points of S₁ and S₂. From fig. 42, it is clear that if Agrā < H sin 45° when δ is south, there will be one Koṇa-Śaṅku in the forenoon at S₁ and one in the afternoon at the symmetrical S₁'. Verses 31 and 32. H cos z at noon known as Dinārdha-Śaṅku. By 'northern hemisphere' it is meant that the Sun is in the northern hemisphere ie. his Sāyana longitude ie. modern longitude lies between 0° and 180°, and 'the southern hemisphere' means that the Sun's longitude lies between 180° and 360°. The direction of δ may be got from the above convention. The latitude and colatitude are always deemed as south and north respectively. The latitude and colatitude being 'added to subtracted from or being decreased by' as the case may be, the declination, we have the zenith-distance and the altitude of the celestial body at Noon. The zenith-distance and the altitude are mutually complements. Comm. In Hindu Astronomy the words “उत्तरगोले” “दक्षिणगोले” are very often used to connote that the Sun is on the north or the south of the celestial equator respec- tively, so that the declination could be automatically known to be north or south respectively. Regarding the latitude, the peculiarity in Hindu Astronomy is that what we call north latitude in modern astronomy is construed as south in as much as the celestial equator gets depressed south in northern latitudes. The colatitude SQ in fig. 41 on the other hand extends north from the south, so that, it is construed as north. The word 'Saṃskāra' is used in Hindu Astronomy in the meaning given above in the translation. Fo

254 example in the equation A = S + B, we say that the Bhuja is had by a Samskāra between A and S. The meaning of Samskāra given by Bhāskara is “समदिशोर्योगः भिन्नदिशोरन्तरम् संस्कारः” Latitude being regarded as southern, if the Sun's declination is 12° north and the latitude 20°, then as they are of opposite direction, effecting the Samskāra as directed 20 - 12 = 8 = zenith-distance (South) = Nata as it is called similarly 70 + 12 = 82 = Altitude = Unnata; here we have added because, both lamba and declination are north. Similarly when δ = 24° north, and φ = 20° as before (south) 24 - 20 = 4° = zenith-distance (north) = Nata 70 + 24 = 94 = unnata (north). But, we take 180 - 94 = 86°. In the above working in the first case we found φ - δ, whereas in the second we found δ - φ. This difference in treatment is not taken objection to, since, the word Antara is used to take the positive value of the difference alone and so in the first instance the nata is pronounced as south, whereas in the second it is pro- nounced north. In modern astronomy, however, we have the formula z + δ = φ, considering z as positive if south, δ and φ positive if north. Here 8° + 12° = 20° (first case cited above) and (- 4°) + 24° = 20° (2nd case, z being negative, for, it is north. In the Hindu symbolism we have to pronounce separately when z is south or north, whereas in modern symbolism the sign alone informs its direction. Similarly in the equation A = S + B, we have to pro- nounce ‘north bhuja’ or ‘south bhuja’ as the case may be, whereas having a convention that δ is + ve when north, and also the Hindu azimuth (measured from the East point) the sign of bhuja indicates its direction. In other words we differentiate the two cases A - S and S - A giving them signs and deducing the direction of the bhuja

255 from the sign itself without an appeal to a picture or without ascertaining whether the northern Agrā prevails over the Southern Saṅku-tala or the Southern Saṅku-tala prevails over the northern Agrā. Thus the Dinārdha Saṅku in symbolism = H cos (φ ± d) (20). Verse 33. Here at noon, Dṛg-jyā is the H sine of nata and the Saṅku is H sine of unnata. Second half of 33 and first half of Verse 34. The product of R and the unmandala-Saṅku divided by Charajyā is called Yaṣṭi. The Yaṣṭi increased by Un-mandala-Saṅku gives H cos z according as the Sun is north or south of the equator. Comm. Unmandala-Saṅku is H cos z when the Sun is on the unmandala. From the sixth latitudinal triangle, wherein Unmandala-Saṅku is Bhuja and Krāntijyā Karṇa, so by comparing with the second latitudinal triangle (or rather operating with the second triangle to signify the Hindu method). (Krāntijyā × Bhuja) / Karṇa = Unmandala-Saṅku = (H sin δ × H sin φ) / R (already derived under (19)). We saw before Charajyā = R tan φ tan δ. Hence as directed in the verse (R × H sin φ H sin δ) / (R × R tan φ tan δ) = Yaṣṭi = (H cos φ H cos δ) / R (21). ∴ H cos z (at Noon) = (H cos φ H cos δ) / R ± (H sin φ H sin δ) / R according as the Sun is on the north or south of the equator.

256 Hence H cos z (at Noon) = Natajyā = (H cos φ H cos δ ± H sin φ H sin δ) / R or in modern symbolism cos (φ ∓ δ) already derived under (20). We shall now show how the formulation is done by the simple rule of three (Ref. fig. 39). If a parallel through o₄d is drawn to cut Aa at a₁, then Aa₁ is called the Yaṣṭi, which is vertical. The triangles Aoa₁ and Do₄d are similar so that Aa₁ / Ao = Dd / Do₄ ∴ Aa₁ = (DD × Ao) / Do₄ . But Ao / Do₄ = R / Charajyā for, all the lines of the diurnal circle and the equator stand in the ratio (H cos δ) / R = Ao / R = Do₄ / Charajyā = Kujyā / Charajyā ∴ Ao / Do₄ = R / Charajyā ∴ Aa₁ = (Unmandala Śaṅku × R) / Charajyā = Yaṣṭi as formulated. Now Dinardha Śaṅku = Aa = Aa₁ + a₁ a = Aa₁ + Dd = Yaṣṭi + Unmandala Śaṅku. It is evident from fig. 21 why in the northern sphere the sum is to be taken whereas in the southern, the difference is to be taken. Latter half of verse 34. Definition of Hṛti and Antyā. The sum or difference of Dyujyā and Kujya will be similarly Hṛti, whereas the sum or difference of Charajyā and radius will be Antyā. Comm. We defined formerly Hṛti and Antyā under our commentary on the latitudinal triangles. From fig. 39 Hṛti = o₁A = o₁o + oA = o₄D + oA = Kujya + H cos δ (Dyujyā) (22).

257 In the parallel great circle, the Equator we have therefore Antyā = Charajyā + R (already derived). Verse 35. Antyā = (Hṛti × R) / (H cos δ) = (Hṛti × Charajyā) / Kujyā and ∴ Hṛti = (Antyā × H cos δ) / R = (Antyā × Kujyā) / Charajyā by what is called Guṇa-ccheda-Viparyaya ie. alternando. Comm. Evident. Verse 36. To obtain Dinārdha-Śaṅku from Antyā and Hṛti (Antyā × un-maṇḍala Śaṅku) / Charajyā = (Hṛti × 12) / K = Dinārdha- Śaṅku. Comm. The second formula is derived from the similarity of △ Ao₄a (fig.) with the first latitudinal triangle. From the similarity of Aao₄, Ddo₄ fig. 39, Ao₄/Do₄ = Aa / Dd ∴ Aa = (Hṛti × unmaṇḍala-Śaṅku) / Kujyā . But Aa = Dinārdha-Śaṅku ∴ Dinārdha Śaṅku = (Hṛti × U.S.) / Kujyā (U. S. = Un- maṇḍala-Śaṅku.) But Hṛti / Kujyā = Antyā / Charajyā ∴ D.S. (Dinārdha-Śaṅku) = Charajyā / Antyā × U.S. = (Hṛti × 12) / K VIII. Verse 37. Meridian Zenith distance. Agrā ± (Hṛti × bhuja of a lat. triangle) / (Karṇa of a lat. triangle) = H sin z where z is the meridian zenith distance. 33

258 Comm. This formula is a special case of the formula A = S + B since the H sine of the meridian zenith distance is the Śaṅku-Bhuja at noon. From fig. 39 (Hṛti × bhuja of a latitudinal triangle) / (Karṇa of a latitudinal triangle) = O₁a = Dinārdha Śaṅkutala. The operation of sign has been already explained. Verse 38. An alternative method. The meridian zenith distance of the Sun can be had also by the formula (Hṛti ± Taddhṛti) B.L.T. / K.L.T. where B.L.T. and K.L.T. are the bhuja and karṇa of any latitudinal triangle. Comm. (Ref. figures 43 and 21). Let E₁ be the centre of the armillary sphere so that QE₁R is the diameter of the celestial equator which is on the median plane. Let S₁ F₁ S be the diameter of the diurnal circle of the Sun, which is also on the meridian plane so that S₁ F₁ is the Taddhṛti, S₁ S is the Hṛti and S the position of the Sun on the meridian. H sin z = SM = SF₁ sin F̂₁ = SF₁ sin ϕ = (Hṛti − Taddhṛti) × s/K where s/K can be replaced by B.L.T. / K.L.T. (s=equinoctical shadow and k the Viṣuvat-Karṇa). In the Southern sphere, H sin z = S′N = S′F₁′ sin ϕ = (S′S₂ + S₂F₁′) sin ϕ = (Hṛti + Taddhṛti) × s/k. Verse 39. Still another way of obtaining the m. z. d. (meridian-zenith-distance). R − H versin (altitude) = H sin z

259 Fig. 43 This formula gives not only the meridian zenith distance but H sine of the zenith-distances of the Sun when he is on the Koṇa-Vṛtta or prime-vertical or unmaṇḍala. Comm. From fig. 43, H sin z = SM = LE₁ = E₁s − sL = R − H versine (Ss) = R − H versine (altitude) as given. Since R − H versine (altitude) = R − {R − H cos (90 − z)} = R − (R − R sin z) = H sin z, so this formula applies wherever the Sun be. This is almost begging the question as H sine of z is being sought through H versine of (90 − z).

260 First half of the verse 40. To obtain the shadow S and K the Chāyakarṇa of any shadow (H sin z × 12) / (H cos z) = S and (R × 12) / (H cos z) = K. Comm. (Ref. fig. 44). (12. H sin z) / (H cos z) = 12 tan z = S. Also 12 / K = cos z = (H cos z) / R so that (12 R) / (H cos z) = K. The Hindu method of looking at this through the similarity of ΔS OM☉ and Ogn the gnomonic triangle. is as follows. ☉M is called Mahā-Śaṅku ie. H cos z ; ☉L is Dṛkjya or H sin z = OM. 12 / (H cos z) = S / (H sin z) so that S = 12 H sin z/H cos z. Also, On / O☉ = 12 / (H cos z) ie. K / R = 12 / (H cos z) ∴ K = (12 R) / (H cos z) . It will be noted that fig. 44 pertains to any vertical plane. Second half of Verse 40. The Dinārdha-Karṇa is equal to (R × k) / Hṛti where k is the Viṣuvat-Karṇa. Comm. The formula is derived through twice apply- ing the rule of three or what is the same, through the similarities of two sets of triangles From fig. 39, O₁A / Aa = Hṛti / Dinārdha-Śaṅku = k / 12 (a) and from fig. 44 On / O☉ = 12 / Dinārdha-Śaṅku = K / R (b) where K is the required Chāyākarṇa. Dividing (a) by (b) Hṛti / 12 = k / 12 × R / K ∴ K = kR / Hṛti

261 Fig. 44 Verse 41. Alternate method of obtaining K 101530/H sin λ = para (say) where λ is the Sāyana longitude of the Sun ; then, (Para × k) / s = K where K is un-mandala-Karṇa First half of Verse 42. To obtain K when the Sun is on the prime-vertical— Para × s/k = Samavṛttakarṇa. Comm. From fig. 19, from the similarity of triangles BD ☉ and CMA, B ☉ / CA = ☉ D / AM ie. ☉ D = (B ☉ × AM) / CA ie. H sin δ = (H sin λ H sin ω) / R (a) Then consider the similarity of the first and the sixth latitudinal triangles ; then Unmandala Sanku / Krāntijyā = s / k (b) where s is the equinoctial shadow and k the Viṣuvat- Karṇa. Again taking that ☉ the Sun lies on the unman-

262 dala in figure 44, (H cos z) / 12 = R / K = (Unmandala Śaṅku) / 12 (c) Eliminating Krāntijyā and Unmandala Śaṅku from (a), (b) and (c) (Unmandala Śaṅku) / (H sin λ H nis ω/R) = s / k ∴ (12R / K) / (H sin λ H sin ω / R) = s / k ie. 12R² / (KH sin λ H sin ω) = s / k ∴ K = (12R² × k) / (s H sin λ H sin ω). Here 12R² / (H sin ω) = (12 × 3438²) / 1397 = 101531 ; but Bhāskara has taken 101530 taking a more correct value of R. Then 101530 / (H sin λ) is symbolized as para so that para × k / s = K = Unmandala Karṇa. Regard- ing the Samavṛttakarṇa, in the place of (b) above we have (Sama-Śaṅku) / Krāntijyā = k / s (b') by the similarity between the first and the fifth latitudinal triangles. Equation (c) holds good with respect to any H cos z and the corres- ponding K since 12 R = K × Śaṅku and 12 R is a cons- tant. Noting therefore R / K' = (Sama-Śaṅku) / 12 (c') eliminating Krāntijyā and Sama-Śaṅku among (a), (b'), (c'), we shall have K = (12 R² s) / (k H sin λ H sin ω) = para × s / k as stated. Second half of Verse 42. To obtain the Dinārdha- karṇa from the Unmandalakarṇa. (Un-mandalakarṇa × Charajyā) / Antyā = Dinārdhakarṇa. Comm. We have equation (c) above stating 12 R = K × Śaṅku. (c) But

263 Iṣṭa Śaṅku / Iṣṭa Hṛti = cos φ = constant = Dinārdha Śaṅku / Hṛti = Sama-Śaṅku / Taddhṛti = Unmaṇḍala Śaṅku / Kujyā (d) (23) Again by virtue of the proportionality of Iṣṭa Hṛti / Iṣṭāntyā = Hṛti / Antyā = Kujyā / Charajyā (e) (24) We have Ishta-Śaṅku / Ishtāntyā = Dinārdha Śaṅku / Antyā = Unmaṇḍala Śaṅku / Charajyā ∴ Iṣṭa Karṇa × Iṣṭāntyā = Dinārdha Karṇa × Antyā = Unmaṇḍala Karṇa × Charajyā (f) (25) ∴ Dinārdha Karṇa = (Unmaṇḍala Karṇa × Charajyā) / Antyā as stated in the verse. Verse 43. (Unmaṇḍala Karṇa × Kṣitijyā) / Hṛti = (Sama Vṛtta Karṇa × Taddhṛti) / Hṛti = Dinārdha Karṇa. Comm. From (c) and (d) above Dinārdha Karṇa × Hṛti = Sama Karṇa × Taddhṛti = Unmaṇḍala Karṇa × Kujyā (g). Khitijyā is the same as Kujyā. From this the statement follows : Verse 44. The ancient Achāryas found the gnomonic shadows when the Sun is on the meridian, prime-vertical and the Kona-Vṛtta (ie, Vertical when the northern or southern Hindu azimuths are 45°) by different methods. I consider him to be the very Sun illuminating the lotus- faces of aitronomers, if anybody could give a method to find the shadow in any required direction, which holds good in all cases universally.

264 Comm. Evident. Verse 45. Definition of Dikjyā H sin (azimuth). The angle between any vertical and the Prime-Vertical measured on the horizon is what is called Digamsa and its H́ sine is known as Dik-jyā either in the Eastern hemi- sphere or the Western. Comm. In modern astronomy azimuth is measured along the horizon from the north point towards the east point round the horizon. In Hindu Astronomy however, the azimuth is measured from the East point on either side and from the West point also on either side specifying whether it is north or south. Verse 46 and first half of 47. To obtain the gnomonic shadow in any arbitrary direction. Assume Rs / (H sin a) as the equinoctial shadow and obtain the H sine of the corresponding latitude L. Then the product of that H sin L and H sin δ divided by H sin ϕ will give H sin D where D is a hypothetical decli- nation. With the new L and this D, as the hypothetical latitude and declination, obtain the meridian zenith distance by the formula Z + D = ϕ, and through this m.z d. obtain the shadow, which will be the shadow in the required direction namely 12 tan (ϕ ± D). Comm. Let gL be the gnomonic shadow on the equinoctial day in a given direction given by a° Digamsa (the Hindu azimuth) and let gN be the shadow in the same direction on any day. (fig. 45) We know that the extremity of the gnomonic shadow on the equinoctial day traces a straight line parallel to the East-West line Eω at a distance of the equinoctial shadow s because the Equatorial plane passing through the foot of the gnomon

265 Fig. 45 and that passing through the top of the gnomon being parallel planes cut the horizontal plane in parallel straight lines. (This will be also proved analytically subsequently). Hence LM = s. Now from the figure LM / gL = AB / gA = H sin a / R ∴ gL = H sin a / Rs I This gL is spoken of as Iṣṭa-Drikmandala palabhā because it is the shadow on the equinoctial day in any vertical. LN is the increment in the shadow on account of declination and we have to compute this and correlate gL and LN. For this refer to figs. 46 and 47. In fig. 46, QRT is the equator, so that when the Sun is on the equator on the equinoctial day in the direction given by ZS, ZT is the zenith-distance. Let ZS be the zenith- distance of the Sun in the same direction on any day From the analogy of finding H sin δ from H sin λ, from this figure 34

266 H sin SR = (H sin ST × H sin T̂) / R II and H sin φ = (H sin ZT × H sin T̂) / R III so that (H sin SR) / (H sin φ) = (H sin ST) / (H sin ZT) ∴ H sin ST = (H sin SR / H sin φ) × H sin ZT. Noting that SR = δ and putting ST = D H sin D = (H sin δ / H sin ϕ) × H sin ZT. [Diagrams: Fig. 46 and Fig. 47 showing spherical triangles with vertices P, Q, N, R, S, T, D, δ] Fig. 46 Fig. 47 The same formulae are derivable from fig. 47 also; only in fig. 46 while there is a decrement in the shadow of the day as compared with the shadow on the equinoctial day, in fig. 47, there is an increment. This is seen from the decrease and increase of ST in the zenith-distance ZT of the equinoctial day in the given direction. Now corre- lating fig. 45 with figures 46 and 47, the shadow gL pertains to the zenith-distance ZT on the equinoctial day whereas the shadows gN pertains to the zenith-distance on the day concerned in the same direction. We have, S / √(12² + S²) = (H sin z) / R so that RS / √(12² + S²) = H sin z where S is the shadow at any instant when the zenith-distance is z. The process indicated by saying ‘Obtain H sin φ