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

401 If a moment in between the Sammīlana and Unmīlana were taken, the then Bhuja and Koṭi could be no doubt computed, but the question of magnitude of the eclipse does not arise as the entire disc has been plunged in the shadow. Second half of verse 31 verse 32 and first half of verse 33. Alternative method of depicting the eclipse geo- metrically. Joining the upper end of the latitude of the middle moment of the eclipse to those of the first and last contacts, we have what are called the Pragrahamārga and Mokṣamārga ie. the path of the centre of the eclipsing body from the first contact to the middle moment of the eclipse and that from the middle moment to the last contact. The lengths of these paths could be computed and they could be drawn before hand. Then with the centre of the Moon as centre and radius equal to p—r, if a circle be drawn, it cuts the paths described above each in one point. With those points as centre and radii equal to p, if circles be drawn, they will touch the Moon’s disc each in one point which are respectively the points of Sammīlana and Unmīlana. Comm. In as much as the latitude of the Moon differs from moment to moment, the Pragrahamārga and the Mokṣamārga are separated to achieve a little more accuracy than could be got by joining the upper extre- mities of the initial and final latitudes. The remaining statement is evident, for, at the moments of Sammīlana and Unmīlana, the distance between the centres of the eclipsing body and the eclipsed will be p—r, so that the points of intersection of the Pragrahamārga and Mokṣa- mārga with the circle whose centre is the centre of the eclipsed body and radius p—r will give the centre of the eclipsing body. 51

402 Latter half of verse 33. To know the magnitude of the eclipse at any given moment during the course of the eclipse. Let the product of the time elapsed from the moment of first contact and the length of the path of the eclipsing body traced from the moment of the first contact to the middle of the eclipse divided by the time between the moment of first contact and the middle of the eclipse, be x. Similarly let the product of the time before the end of last contact and the path of the eclipsing body traced between the middle moment of the eclipse and the moment of last contact divided by the time between the middle moment and the moment of last contact be y. Lay off x and y units of length from the first and last points of the path of the eclipsing body along the path respectively. Then we get the points of the centre of the eclipsing body at the required moments. With these points as centre and radius p, if circles be drawn, they represent the eclipsing body. The length of the diameter of the eclipsed body shaded, gives the magnitude of the eclipse called grāsa. Comm. Here rule of three is applied namely “If during time T₁ or T₂ a path equal to l₁ or l₂ in length is traced what length will be traced in times t₁ or t₂?”, where T₁ and T₂ are the times called Sparsa-Sthiti-Khanda and Mokṣa-Sthiti-Khanda respectively, l₁ and l₂ are the times elapsed from the moment of first contact or before the moment of last contact and t₁, t₂ are the times from the beginning of the eclipse and before the end of the eclipse respectively. Then x and y give the points where the centre of the eclipsing body lies. Verse 35. Given the magnitude of the eclipse at any time to obtain the time elapsed after the first contact. The time taken by the centre of the eclipsing body to move through the segment of the path of the eclipsing

408 body which lies between the position of the eclipsing body at the moment of first contact and the point of intersection with the path of the eclipsing body of the circle drawn with the centre of the eclipsed body as centre and radius equal to the difference of p+r -g where g is the magni- tude of the eclipse (grāsa) at the moment, or similarly the time taken by the centre of the eclipsing body to move through a similar and equal segment of the path of the eclipsing body on the other side, gives the time elapsed after the moment of first contact or the time before the moment of last contact. Comm. This is the converse of the above problem. The method is clear being based on rule of three as above. Both the problems could be algebraically expressed as follows. Let T, t, l, g, and k, stand respectively for the Sthiti-Khanda ie. the time between the moment of first contact to the middle of the eclipse or the time between the middle moment to the moment of last contact ; (2). the time elapsed after the moment of first contact or the time before the moment of last contact, as the case may be ; (3) the length of the Pragrahamārga or Mokṣamārga ; (4) the grāsa which is defined as p+r-k ; (5) the Karṇa whose expression is √(B²+β²), B being the Bhuja defined and β the latitude of the Moon. Then the following working is stipulated (a) If in time T, a path of length l is traced, what will be traced in t? The result is lt/T (b) Then B = l - lt/T (c) B² + β² = K² (d) p+r-k=g. Thus combining all the steps {l (T-t) / T}² + β² = (p+r-g)² ie. l² (T-t)² + β² T² = T² (p+r-g)² I given t, this equation gives g and given g it gives t.

404 Again the following relation holds good between T and l, l² = (p+r)² – β² II and l / (m₁–s₁) = T with the nomenclature already employed which means T = √(p+r)²–β² / (m₁–s₁) III In the above working, the fundamental elements are p, r, β, m₁ and s₁ with which the other elements could be worked out. Replacing the other elements from equation I, we have {(p+r)²–β²} { √(p+r)²–β² / (m₁–s₁) – t }² + β² (p+r²–β²) / (m₁–s₁)² = (p+r)²–β² / (m₁–s₁)² (p+r–g)² ie. {(p+r)²–β²} [√(p+r)²–β² – t (m₁–s₁)]² + β² (p+r²–β²) = (p+r²–β²) (p+r–g)² ie. {√p+r²–β² – t (m₁–s₁)}² + β² = (p+r–g)² IV Putting t=0 in this equation, we have (p+r)² = (p+r–g)² ie. g=0 which means at the moment of first contact, the grāsa is zero. Again putting t = T ie. t = √(p+r)²–β² / (m₁–s₁) ie. t (m₁ – s₁) = √p+r²–β² we have β² = p+r–g² ie. g = p+r–β which gives the grāha at the middle of the eclipse which was defined as the Sthagita. In equation IV which we may take as a funda- mental equation, the two unknowns are t and g one of which being given the other could be got. Verse 36. The colour of the eclipse. When less than half the disc of the Moon is eclipsed, the colour will be what is called Dhumra ie. of the colour

405 of smoke; when the disc is half eclipsed, the colour is black; when more than half is eclipsed, the colour would be a blend of black and red and when the entire disc is eclipsed, the colour will be what is called pisanga or reddish-brown. Comm. Clear. Verse 37. When declare the occurrence of an eclipse. When even one-sixteenth of the diameter of the Moon's disc is shadowed, the eclipse will not be visible in as much as the shadowed portion is covered by the illaminating rays of the disc. In the case of the Sun, when even one-twelfth of the diameter is shadowed, the eclipse will not be visible for the same reason. Hence we shall not declare the occurence of an eclipse upto the shadowing of the discs to the extents stated above. Verses 38 and 39. Examples which disclose the invalidity of construing Valana in terms of Hversine instead of Hsine. When the Sun is in the zenith, the Ecliptic being vertical, the Valana is clearly seen to be the Agra of (☉+90) where ☉ is the longitude of the Sun. If you could show that the Valana will be the same on the basis of Hversine-formula, then I would accept that what Lallāchārya postulated in his work Śiṣya-Dhī-Vṛddhida is correct. Again, in a place of latitude 90—ω, ω being the obli- quity of the Ecliptic (ω is taken to be 24°), when the Sun being situated in Meṣa, Vṛṣabha, Mīna or Kumbha, the Moon contacts him from the south at the moment of a solar eclipse, in as much as the Ecliptic coincides with the horizon. In this circumstance, how could the Valana be equal to R, as made out by the Hversine-formula.

406 Comm. Lallāchārya gave the Valana in terms of the following verses “स्पर्शादिकालजनितोत्क्रमशिञ्जिनीभिः, क्षुण्णाक्षभा पलभवश्रवणेन भक्ता, चापानि पूर्वनतपश्चिमयोः फलानि, सौम्येतराणि समवेहि पृथक् क्रमेण; ग्राह्यात् सराशित्रितयाद् भुजज्या व्यस्ता ततः प्राग्वदप- क्रमज्या...” Verses 23, 25 Chandragrahaṇādhikāra, wherein he formulated the Valana in terms of Hversine in the place of Hsine. The reason for his slip, we have already explained. Now Bhāskara gives two glaring examples to substantiate his formula and to show up the flaw in Lallāchārya's formulation. In the first example, where the Ecliptic takes the form of a Vertical, the Sun being in the zenith, the Spaṣṭa Valana which is the angle between the Ecliptic and the prime-vertical is the same as the arc between the East point and the intersection of the Ecliptic with the horizon known as Lagna. Since the Sun is then in the zenith, the longitude of the Lagna is (90 + ☉) so that the said arc is the Agra of the point whose longitude is 90 + ☉ as stated. Hence Spaṣṭa Valanajyā = sin A = sin δ/cos ϕ where A is the agrā (using Napier's rule from triangle PNL where L is the Lagna N the north-point and P the celestial pole). In the Hindu form, this is given by H sin V = H sin A = (R H sin δ) / (H cos ϕ) where δ is the declination of a point of the Ecliptic whose longitude is (90 + ☉). But Lallāchārya's formula gives the Valanajyā as Hvers δ, δ being the declination of a point of the Ecliptic whose longitude is (90 + λ), λ being the longitude of the Eclipsed body ignoring the latitude. In other words, in the case of the lunar Eclipse when the Moon is in the zenith his Valanajyā = Hvers δ (δ having the above value) the Ākṣa Valanajyā here being zero. Since (R H sin δ) / (H cos ϕ) ≮ Hvers δ, the mistake committed by Lallāchārya is evident even supposing H cos ϕ = R when we ignore the latitude ie. take ϕ to be zero.

407 In the second example cited by Bhāskara the Ecliptic coincides with the horizon, the pole of the Ecliptic being in the zenith. Then in a Solar Eclipse the Moon eclipses the Sun from the south showing that H sin V = R. That H sin V = R is also evident from the fact that the Ecliptic makes 90° with the prime-vertical, having coincided with the horizon. But here according to Lallāchārya’s formula, H sin ξ = Hvers 90° = R and Āyana Valanajyā is Hvers δ, where δ is the declination of a point whose longitude is 90° more than ☉. If ☉ = 30°, 60° H sin θ = R/2 sin ω/R or (√3/2) (R sin ω)/R ie. (sin ω)/2 or √3/2 sin ω. Evidently the sum of the two Valanas Āyana and Ākṣa cannot be 90° as is also vouchsafed from geometry. So, here also, the flaw is evident. Note 1. Śrīpatyāchārya also followed Lallāchārya vide verses 18, 19, 20 Chandragrahaṇādhyāya, Siddhānta Śekhara. It will be noted that the commentator of Siddhānta Śekhara, while reiterating Bhāskara’s stand as the correct one, himself commits a mistake in saying “सममण्डलीय नतांशज्यास्थाने नतकाकोत्क्रमज्या गृहीता” In fact sin ξ = (sin ϕ sin h) / cos μ = (sin ϕ sin z) / cos δ as proved by us before. The commentator cited above overlooked that sin ξ could be also equal to (sin ϕ sin h) / cos μ , wherein natakāla also is implied. Note 2. It will be noted that even Pṛthūdakāchārya, while commenting on Brahmasphuṭa Siddhānta, ignored Brahmagupta and followed Lallāchārya blindly. Note 3. The formula given by Lallāchārya and followed by Pṛthūdaka as well as by Śrīpati is very rough besides containing the flaw cited, in as much as both μ and δ are taken to be zero, which are not so,

SURYAGRAHANĀDHIKĀRA Verse 1. In as much as the observer situated on the surface of the Earth and as such elevated by the radius of the Earth from the centre there of, perceives not the Sun and the Moon having the same longitude at the moment of conjunction, to be in the same line of sight, heyt being depressed unequally having different orbits, so I proceed to elucidate what are called Lambana and Nati ie. parallax in longitude and latitude, on which account they are not in the same line of sight. Fig. 91 Comm. (Refer fig. 91) Let E be the centre of the Earth, M and S the centres of the discs of the Moon and

the Sun. Let A be the position of an observer on the surface of the Earth, elevated by the radius EA from E. Let M and S be in the same line of sight as seen from E. But as seen from A, AS and AM are respectively the lines of sight to the Sun and the Moon. Evidently these lines of sight differ the Moon being depressed more than the Sun. If a line AS' be drawn which is parallel to the central line of sight namely EMS, we find that the Sun is depressed by the angle S'AS whereas the Moon is depressed by the angle S'AM'. These angles differ because the orbits of the Sun and Moon differ. Here the angle S'AS will be very very small, its magnitude being in truth just about 8" only. But the angle S'AM' will be sufficiently large since the Moon is very near the Earth compared with the Sun. Taking ES and AS to be almost parallel due to the largeness of the Sun's distance, the angle SÂM will be almost equal to AM̂S so that we could consider that the Moon is depressed from AS the line of sight to the Sun by the angle SÂM' = AM̂E. This angle AME is called the geocentric parallax of the Moon and the angle AŜE that of the Sun M'ÂS = angle of depression of the Moon over and above that of the Sun = M'AS' - M'AS = EM̂A - EŜA = geocentric parallax of the Moon minus geocentric parallax of the Sun. Verse 2. The presence or absence as well as the positiveness and negativeness of the parallax in longitude. Compute the Lagna at the moment of conjunction of the Sun and the Moon. There will be no parallax in longitude when the Sun is situated at the point called Vitribha or the point whose longitude is ≡ L - 90°, L being the longitude of the Lagna ie. the ascendant which is the point of intersection of the Ecliptic with the 52

410 horizon. If the Sun's longitude falls short of the longitude of the Vitribha or exceeds it, there will be parallax in longitude which will be positive in the former case and negative in the latter. Fig. 92 Comm. (Ref. fig. 92) Let SN be the horizon, Z the zenith and VA the Ecliptic. A is the ascendant or Lagna. Let V be the point called Vitribha which is 90° behind A. Strictly speaking V is called Vitribhalagna or lagna from which three Rāśis or 90° are subtracted (Bha = Rāśi. त्रिभिर्विरहितम् वित्रिभम्; वित्रिभम् च तत् लग्नम् च वित्रिभलग्नम् ie. a point whose longitude is got by subtracting three Rāśis from that of the Lagna). Let ZV be the vertical of V so that ZV̂A = 90°. It will be seen that AV = 90° as follows. Let A' be the point where the Ecliptic intersects the horizon on the west. One will construe that the Ecliptic is bisected by the meridian; but it is not so. Spherical triangles AVZ and A'VZ being right-angled at

411 V are congruent because AZ = A'Z and ZV is common ∴ AV = VA'. But AV + VA' = 180° because the Ecliptic and the horizon being two great circles, they bisect each other. Hence AV = 90°. Then a celestial body situated at V will be depressed along ZV the vertical, say, to a point B. Let ☉ be any arbitrary position of the Sun; then ☉ will be depressed along the vertical Z☉, say, to a point C. Draw CD perpendicular on the Ecliptic. Then ☉D is the component of the parallax ☉C along the Ecliptic where as DC is its component perpendicular to the Ecliptic. Thus ☉D is the parallax in longitude and DC is the parallax in latitude. The word ‘Lambana’ means etymologically लम्बते अनेनेति लम्बनम् ie. that amount by which the celestial body is depressed (along the Ecliptic). In Hindu Astronomy the word Lambana is applied to parallax in longitude alone whereas the word Nati is applied to parallax in latitude. Hence to translate Lambana as parallax alone is not correct. The word Drik-lambana is applied to mean parallax along the vertical, and the word Sphutalambana is occasionally used to connote parallax in longitude. As Bhāskara rapidly comments on the verses in this Gaṇitādhyāya, he having dealt with the subject of parallax elaborately under the caption, Grahaṇa Vāsanā, in the Golādhyāya, to catch up his thought, we have to treat the subject first from the modern view point and then elucidate what he has said in the Golādhyāya, much matter of which is reiterated by him under the commentary here in the Gaṇitādhyāya. (Ref. Fig. 91) From the △EAM, (sin EMA) / a = (sin EÂM) / d = (sin ZÂM) / d where a is the radius of the Earth and d the distance of the celestial body (here the Moon)

412 ∴ Sin EMA = a/d sin ẐAM = ÊMA expressed in radian measure since ÊMA is very small ∴ ÊMA (expressed in radian measure) = a/d sin z I where z is the apparent zenith-distance of the Moon ie. zenith- distance as seen by the observer (in contradistinction to the geocentric zenith-distance of the Moon namely ẐEM). In particular, when z = 90°, ÊMA = a/d which is the maximum parallax known as the horizontal parallax ie. the parallax when the Moon is situated on the horizon of the observer. Also the parallax is zero when M is situated at Z as is seen from formula I and as is rightly remarked by Bhāskara in the words 'खमध्ये नास्ति लम्बनम्'. In fig. 91, ÊMA is the angle by which the line of sight of the observer namely AM is depressed from the geocentric line of sight EM. Since the plane of the paper represents a vertical through the Sun and the Moon, the depression of either the Sun or the Moon or the excess of the depression of the Moon over the Sun are all in the vertical plane. This depression is called Drik-lambana because it is a lambana or depression in the Drik-maṇḍala or vertical. This Drik-lambana varies as sin z as is seen from formula I where a, and d may be taken to be constants. (Both a and d vary slightly a varying slightly from place to place on the Earth, the Earth being an oblate spheroid, and d varying from position to position of the Moon). The maximum horizontal parallax is given by a/d in radian measure which is equal to, according to Bhāskara's

418 estimate 1581/2 × 1/51566 × 3438 = 53′ approximately. Its modern value is about 57′ so that the Hindu estimate is not far from truth. Here 1581 and 51566 are the values of a and d in Yojanas according to Bhāskara. The Hindu astronomers do not, however, proceed exactly as we have done in the para above to obtain the maximum horizontal parallax. Their treatment is a little different and is as follows. Whereas according to Modern astronomy E M̂′ A (fig. 93) is viewed as the horizontal [चित्र: Fig. 93] Fig. 93 parallax, in Hindu Astronomy MÊM′ or the angular measure of the Moon’s path equal to the radius of the Earth is taken to be the horizontal parallax. Both, of course, mean the same as is seen from the figure. In as much as the Hindu astronomers knew very well what they term कलीकरण or converting linear distances into angular measure, converting a linear magnitude equal to the radius of the Earth namely MM′ at the lunar orbit into angular measure, they got MM′/EM × 3438 = 1581/2 × 1/51566 radians = 52′-42″ as the maximum horizontal parallax.

414 Having got this estimate, they reckoned this angular measure in time as the time taken by the Moon to traverse the distance MM' equal to the radius of the Earth as follows. The Moon traverses 790′-35″, which is exactly 15 times 52′-42″. Hence they said that the maximum horizontal parallax is 1/15th of the Moon's daily motion in arc and expressing it in terms of time, that the maximum horizontal parallax is 1/15 of a day or 1/15th of 60 nādis or 4 nādis. That this horizontal parallax is 4 nādis as a maximum, would have been also verified at the time of a solar eclipse when the Sun was situated on the horizon at the time of conjunction, by the fact that the eclipse occurred four nādis in advance of the moment of geocentric conjunction (which could be calculated very accurately by the Hindu astronomers, as could be seen by the very correct estimate of a lunation in Hindu Astronomy. In fact, the length of a lunation must have been estimated correctly by noting the time-interval between two solar eclipses or lunar and by dividing that time by the integral number of lunations elapsed in between the two eclipses). The question then arises as to how the Hindu Astro- nomers could know the distance of the Moon. From the estimate of the horizontal parallax by actual observation, and from the geometry of fig. 93, a correct estimate of the distance of the Moon must have been arrived at. Having thus known that the Moon traverses a distance equal to the radius of the Earth in 4 nādis, his daily linear motion was estimated to be 15 times the radius of the Earth ie. (15 × 1581 1/6) / 2 = 11858 3/4 Yojanas. The daily motion of the Moon having thus been estimated almost correctly, an act of inexpedience on the part of the Hindu astronomers was that they should have

415 presumed that all the other planets including the Sun would be traversing the same linear distance during the course of a day. This led to a wrong estimate of the Sun’s distance as well as his spherical radius. Also, they supposed wrongly that the parallax of the Sun also would be equal to 1/15th of his daily arcual motion. Their estimate of the spherical diameter of the Moon was, however, very near the truth, for, they argued, that if 790′-35″ angular motion per day corresponded to 11858¾ Yojanas in linear measure, to what linear measure did the angular diameter of the Moon namely 32′-0″-9′′′ corres- pond? The answer was 11858¾ × 32′ 1/400 ------------------ = 480 Yojanas. 790′-35″ [चित्र: Fig. 94] Fig. 94 It may be here pointed out that there is a relation between the angular radii of two celestial bodies as seen from each other and their mutual horizontal parallaxes (Fig. 94). Let E and M bet he centres of the Earth and the Moon respectively. E M̂ A = horizontal parallax of the Moon = angular radius of the Earth as seen from the Moon and B Ê M = Angular radius of the Moon = Hori- zontal parallax of the Earth as seen from the Moon. Thus, we see that the Earth will be seen from the Moon, as a Moon with an angular radius equal to 57′. In other words our Earth will be a Moon to our Moon, having nearly 16 times the area of our Moon’s disc.

416 The periphery of the Moon’s orbit was arrived at as follows. “If 790′-35″ of the Moon’s angular motion corresponds to 11858¾ Yojanas to what periphery must 360 × 60′ correspond ?” The answer is (11858¾ × 360 × 60) / (790′-35″) = 324000 Yojanas. Reverting to the subject of parallax on hand, the Drik-lambana or the parallax along the vertical has the formula (4 H sin z) / R I nādīs in Hindu Astronomy where 4 nādīs is the maximum parallax obtained when H sin z = R. From fig. 92, C☉² = CD² + D☉² ie. Drik-lambana² = Nati² + Sphuṭalambana² II CD = ☉C sin CÔD = (4 H sin z / R) × sin CÔD = 4 sin z sin VÔZ = 4 sin ☉Z sin VÔZ = 4 sin ZV = (4 H sin ZV) / R = VB III Thus, the parallax in latitude at any point of the Ecliptic is that at the Vitribha which is conveyed by Bhāskara in the words “कक्षयोरन्तरं यत् स्यात् वित्रीभे सर्वतोऽपि तत्”. Also ☉D = C☉ cos CÔD = 4 sin ☉Z cos Z☉V = 4 cos ZV sin V☉ = (4 H cos ZV H sin V☉) / R² IV = Maximum parallax × Vitribha-Sanku × H sine of the arc V☉. In the above working we proceeded in a modern way. It is worth-hearing Bhāskara as to how these results were arrived at elegantly and ingeneously from first principles.

417 In fig. 91, EMS is called Garbha-Sūtra whereas AS is called Dṛṣṭi-Sūtra. दृक्सूत्रात् लम्बितश्चन्द्रः तेन तल्लम्बनं स्मृतम् ie. In as much as the Moon is depressed from the Dṛk-Sūtra, so this phenomenon goes by the name Lambana. It will be noted that in Hindu Astronomy geocentric parallax will not be treated separately for the Moon and the Sun but dealt with simultaneously as it is called for, in the context of a solar eclipse. They were interested in knowing the relative depression of the Moon with respect to the Sun rather than knowing the separate magnitudes with respect to the Moon and Sun, for which they had no application. 'दृग्गर्भसूत्रयोरैक्यात् खमध्ये नास्ति लम्बनम्' ie. In as much as the Garbha-Sūtra and Dṛk-Sūtra are identical in the direction EAZ (fig. 91) there is no parallax at the zenith. Now consider the plane through ZV of fig. 92. Suppose the EMS of fig. 91 is in the direction EV. Then both the Sun and the Moon may be considered to have the same Vitribha at that moment of conjunction. Both the Sun and the Moon being then depressed along ZV, to V' and V'' respectively the Ecliptic will then be a circle parallel to VA (fig. 92) through V' and the orbit of the Moon will be another circle parallel to VA through V'', V'' being below V'. If we neglect, for a moment, the depres- sion of the Sun, and consider VD to be the Ecliptic on which the Sun is situated undeflected, and BC to be the deflected orbit of the Moon relative to the Sun, then VB is the Nati of the Moon, which will be the same distance between VD and BC, ie. the orbits of the Sun and the Moon. This fact was proved by us analytically in the modern way showing that CD = BV. This Nati it is that influences the latitude of the Moon, which may cause apparent conjunction when there is no geo- 53

centric conjunction and which does not show an apparent conjunction when there is a geocentric conjunction. In other words parallax in latitude plays a very important part in solar eclipses. Also ☉ D being the parallax in longitude, the moment of apparent conjunction might be preceded or followed by a geocentric conjunction according as the Sun lies along VA or AV. Thus having determined the exact moment of apparent conjunction using the magnitude of ☉ D, then we have to rectify the latitude using the magnitude of VB. If that rectified latitude of β falls short of R+r where R is the angular radius of the Sun and r that of the Moon, then there will be a solar eclipse. It will be noted that when the Sun coincides with V at the moment of conjunction, there is no parallax in longitude ☉ D being zero (Fig. 92) in that position. Also there will be no parallax in latitude when the Ecliptic assumes the position of a vertical circle passing through the zenith, the Drik-lambana then being entirely along the Ecliptic. In this case the Vitribhalagna V will coincide with Z and (4 H sin V ☉) / R which is termed the Madhyamalambana is now entirely along the Ecliptic and as such it is the Sphutalambana in this case. We have said above that when V coincides with Z, the Madhyamalambana is zero at Z, and that the maximum is equal to 4 nādīs on the horizon. In between Z and the horizon it has the formula (4 H sin V ☉) / R . Noting further that in this case when the H cosine of the zenith-distance of V ie. the Śanku of V is R, the entire lambana is along the Ecliptic, the nāti being zero, and the Madhyamalambana is itself the Sphuta- lambana, and again when V does not coincide with Z, H cos ZV is no longer R but has assumed Kōti-Rūpa ie. the form of a H cosine, as well as the Sphuta-lambana also, which assumes Kōti-Rūpa ie. of the form ☉ D of Fig. 92, where ☉ C is the Madhyamalambana, ☉ D is the

419 Kōti = Sphuta-lambana, and DC = Bhuja = Nāti, it is argued that the Sphuta-lambana is proportional to H cos ZV, assuming a maximum value when V coincides with Z (☉ not being at Z). Verses 3 and 4. Parallax in longitude based on two proportions. Compute the H cosine of ZV, by calculating the rising time of AV, the Kujyā, Dyujyā and Antyā pertain- ing to V, as was formulated in the Tripraśnādhikāra, then H sin V☉, multiplied by 4 and divided by R, and again multiplied by H cos ZV and divided by R again gives the parallax in longitude. Comm. As per the above formula, parallax in longi- tude equal to ☉D of Fig. 92 is equal to (4 H sin V☉ × H cos ZV) / R² . This is evidently derived out of two proportions that the parallax in longitude is proportional to H sin V☉ as well as H cos ZV. This we have already derived through modern methods as formula IV. Under verse 2. The two proportions are (1) V coin- ciding with Z, if by H sin V☉ equal to R, we have 4 nādis as the maximum lambana on the horizon, what shall we have by an arbitrary H sin V☉? The result is (4 H sin V☉) / R and (2) V not coinciding with Z, if by H cos ZV equal to R we have (4 H sin V☉) / R as the Madhyamalambana, what shall we have for an arbitrary H cos ZV? The result is (4 H sin V☉) / R × (H cos ZV) / R as formulated. First half of verse 5. Alternate method of rectifying lambana. The Madhyamalambana multiplied by 12 and

420 divided by the Chāyākarṇa of the Vitribha will also give the Sphuta-lambana. Comm. From Tripras'nādhikāra, we have 12 / K = (H cos Z) / R so that in the formula cited above instead of (H cos ZV) / R we are asked to use 12/K. Latter half of verse 5 and first half of verse 6. Dṛk-nati² = H cos² ZV − H cos² Z☉ = H sin² Z☉ − H sin² ZV (fig. 92) (4 Dṛk-nati) / R = Sphutalambana. Comm. We shall first prove this on modern lines. Cos Z☉ = cos ZV cos Z☉ ∴ cos² ZV − cos² Z☉ = cos² ZV (1 − cos² V☉) = cos² ZV sin² V☉ = (H cos² ZV H sin² V☉) / R⁴ Also cos² ZV − cos² Z☉ = sin² Z☉ − sin² ZV = (H cos² ZV − H cos² Z☉) / R² = (H sin² Z☉ − H sin² ZV) / R² ∴ H cos² ZV − H cos² Z☉ = H sin² Z☉ − H sin² ZV = (H cos² ZV H sin² V☉) / R⁴ ∴ Dṛk-nati defined above = √(H cos² ZV − H cos² Z☉) = √(H sin² Z☉ − H sin² ZV) = (H cos ZV H siu V☉) / R ∴ (4 Dṛk-nati) / R = (4 H cos ZV H sin V☉) / R² = Sphutalambana.