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

pages! The page itself is in English, with English text. We should transcribe the text as it is (English, transliterated Sanskrit, numbers, symbols).

Let's look at: ^ ^ ^ ^ vs the angles Â, B̂, Ĉ, D̂, the angles between the vertical Wait, what if we do: Wait, let's look at: If an OCR line detector is used, it detects: ^ ^ ^ ^ Wait, does Tesseract detect ^ ^ ^ ^? Usually Tesseract might detect it as ^ ^ ^ ^ or ^ ^ ^ ^. Wait, but if it's considered a line: Let's see what happens if someone DOES consider it a line: If we omit it, we have 2 fewer lines! Wait, let's count total lines: With carets as separate lines: 241 on the line parallel to the East-West line and north of it ... If we look at the image, those carets are very distinct, printed with substantial space: ^ ^ ^ ^ Wait, let's look at the first caret line: ^ ^ ^ ^ It is indented to match A, B, C, D. And the second caret line: ` ^ ^

242

Bb = Sama-S'anku, and Dd unmandala S'anku. O₂b is not actually the Agrajyā but parallel and equal to it, since Agrajyā is the H sine of SE of fig. 21, which is the per- pendicular from S on Eω. Similarly do₄ is not the second segment of Agrajyā but a parallel and equal segment. Thus Bb is the perpendicular distance between EE' and FF' ie. Eω and a parallel through F to Eω (fig. 21.) Dd is the perpendicular distance between DD' and BB'; bo₂ is the perpendicular distance between Eω and SS'; do₄ the perpendicular distance between DD' and SS' (fig. 21) ; Bo₂ the perpendicular distance between FF' and SS' ; Do₄ the perpendicular distance between BB', SS'. In this fig. 39, O₁A is called Hṛti, Co₃ = Ishta-Hṛti or any arbitrary hṛti. Taddhṛti Hṛti and Kujyā are special cases of Ishtahṛti. Hṛti is the maximum of Ishtahṛti. Calling Aa, Bb, Cc, Dd S'ankus in general ao₁, bo₂, co₃, do₄ are called S'ankutalas. Aa is called Dinārdha-S'anku or the S'anku of the mid-day ; whereas Dd is the unmandala- S'anku and Bb Sama-S'anku. Cc is called Ishta-S'anku. Perpendiculars from A, B, C, D on the plane of the prime vertical are called S'anku-bhujas. Since B is on the prime-vertical itself, the S'anku-bhuja is zero and at this point BO₂ is Agrajyā. In the arbitrary case at C, the perpendicular from C on the plane of the prime- vertical being S'anku-bhuja, which is equal to the per- pendicular from C on Eω, and Co₃ being the S'anku-tala, and since the perpendicular distance between ML and Eω is the Agrajyā, which is equal to the sum of O₃ C and the S'anku-bhuja S'anku-tala + S'anku-bhuja = Agrajyā. (10) which is a different expression of (5). We shall present this analytically. Putting S'anku = H cos z, and using the latitudinality of Cco₃ (H cos z) / 12 = Co₃ / s = Co₃ / K applying similarity with the first fundamental latitudinal triangle where s = equinoctial shadow, and K the Viṣuvat-Karṇa. Hence we have

243 Co₃ = Ishta-hṛti = K/12 H cos z = (H cos z)/(H cos φ) = (cos z)/(cos φ) (11) Co₃ = Śanku-tala = s/12 H cos z = H cos z tan φ = (H cos z H sin φ)/(H cos φ) = R cos z tan φ (12). From the triangle PZS, we have the formula sin δ = sin φ cos z + cos φ sin z sin a written under verse 8 ∴ (sin δ)/(cos φ) = cos z tan φ + sin z sin a or in the Hindu form (R sin δ)/(cos φ) = H cos z tan φ + (H sin z H sin a)/R . I. We saw under verse 8 that (R sin δ)/(H cos φ) = (RH sin δ)/(H cos φ) = Agrajyā (formula 5). Also we have from (12) above H cos z tan φ = Śanku-tala. From fig. 40, if SM be drawn secondary to the prime-vertical, the right-angled spherical triangle SMZ gives [L14

244 Hṛti = O₁A = OO₁ + OA = Kujyā + H cos δ or Dhujyā VI since OO₁ = DO₄ = Kujyā. The line in the Equatorial plane cor- responding to Kujyā is Charajyā which is equal to R tan φ tan δ (13) so that we can write Kujyā = R sin δ tan φ = H sin δ tan φ (13'). Also from VI [Fig. 40] Antyā = R + Charajyā = R + R tan φ tan δ (14) which gives us the duration of half the day where R gives 6 hrs and Charajyā the increment in day due to φ and δ. [Fig. 40] Verse 18. To obtain the magnitudes of the various chords or the elements of the latitudinal triangles. The elements or the sides of all these latitudinal triangles are mutually derivable from similarity. Comm. Easy. Second half of Verse 18. The radius multiplied by the Kotis or Bhujas and divided by the Karnas gives H cos φ and H sin φ. Comm. The seven triangles except the second, of the eight latitudinal triangles are compared here with the second. Remembering Bhujas are the sides of the triangles opposite to φ and Kotis opposite to 90 − φ,

245 Bhuja / Karṇa in any triangle = Bhuja / Karṇa in the second latitudinal triangle = (H sin φ) / R ∴ (R × Bhuja) / Karṇa = H sin φ Akshajyā or Palajyā (15) Similarly Koti / Karṇa in any triangle is equal to the Koti / Karṇa in the second latitudinal triangle = (H cos φ) / R so that (R × Koti) / Karṇa = H cos φ = lambajyā (16). Verse. The arcs of H sin φ and H cos φ are respec- tively the Akshamsas and lambamsas as they are called ie. latitude and colatitude. H sin φ and H cos φ are also obtainable thus √(R² - H sin² φ) = H cos φ and H sin φ = √(R² - H cos² φ) Or again (H sin φ × Koti) / Bhuja = cos φ and (H cos φ × Bhuja) / Koti = H sin φ where the Bhuja and Koti may belong to any latitudinal triangle. Verse 20. Agrajyā can be had by multiplying Krāntijyā by Karṇa of any lat. triangle and divided by its Koti. Also Sama-Sanku = Karṇa / Bhuja × Krāntijyā and (Sama-Sanku × Karṇa) / Koti = Taddhṛti. Comm. The first of these statements pertains to the similarity of the third triangles to the others. The second of the statements pertains to the similarity of the fifth latitudinal triangle to others whereas the third pertains to that between the fourth and the others.

246 Thus Sama-Śaṅku or S.S. = (H sin δ × R) / (H sin φ) or (R sin δ) / (sin φ) (17). Taddhṛti = R × (H sin δ) / (H sin φ) × R / (H cos φ) = (R sin δ) / (sin φ cos φ) (18). Verse 21. Taddhṛti = (Karṇa × Agrajyā) / Bhuja Comm. This pertains to the similarity between the fourth latitudinal triangle and the others. Latter half of Verse 21 and first half of Verse 22. Sama-Śaṅku = (Taddhṛti × Koṭi) / Karṇa = (Agrajyā × Koṭi) / Bhuja (Sama-Śaṅku × Bhuja) / Koṭi = Agrajyā. Comm. The first statement is made out of the similarity between the fourth lat. triangle and the others whereas the second statement and the third as well are made out of the similarity between the third and others. II half of Verse 22. Sama - Śaṅku = Upper segment of Taddhṛti × Karṇa / Koṭi Comm. The similarity is between the fifth triangle and others. Verse 23. Kujyā = Krāntijyā × Bhuja/Koṭi Upper segment of Taddhṛti = (Krāntijyā × Koṭi) / Bhuja and Kujyā + Upper segment of Taddhṛti. Comm. The first statement is through the similarity of the third triangle with others, whereas the second is through the similarity of the fifth with the others. The third statement is clear from Fig. 21.

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