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

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.

421 Bhāskara's proof proceeds in two stages from first principles. In the first place when V coincides with z, Lambana is seen to be equal to 4 nādis on the horizon and zero at Z ie. it is zero when H sin Z☉ = 0 and 4 nādis, a maximum when H sin Z☉ = R. So, it is meet that Lambana should be taken to be proportional to H sin Z☉ ie. proportional to natajyā. In this context the lambana termed as Madhyamalambana is entirely along the ecliptic. It is taken to be in the form of Karṇa, because in the position of ☉C also it is in the form of a Karṇa. Then let the Ecliptic be deflected from the zenith (deflected = क्षिप्त). Vitribhalagna then being deflected from the position of Z, occupies the position of V (fig. 92). So ZV is called Dṛk-kshepa since the Ecliptic which was in the form of a Dṛk-mandala is deflected from that posi- tion. Also the circle ZV is called Dṛk-kshepa-mandala because V is deflected along that circle. Now consider the △ whose sides are H sin ZV, H cos ZV and R. H cos ZV equal to R and as such in the form of a Karṇa corresponds to the Madhyamalambana which is also in the form of a Karṇa ; when this Vitribha-Sanku assumed the form H cos ZV, ie. rendered a Kōti from its form of a Karṇa, R, the Sphutalambana is also rendered a Kōti in the form of ☉ D so that Madhyamalambana / R = Sphutalambana / H cos ZV ∴ Sphutalambana = (H cos ZV / R) Madhyamalambana = (H cos ZV / R) × (4 H sin Z☉ / R) . Then Bhāskara says that we could look at this, from another angle in the words “यदेव स्फुटलम्बनस्य कोटिरूपत्व- मुपपन्नम् etc. ”.

422 In fig. 92, H sin Z☉ is in the form of a Karṇa; H sin ZV is in the form of the corresponding Bhuja. This triangle formed by these two as sides may be taken to be similar to the triangle ☉DC, both being called parallax Δs. This plane triangle ☉DC is like the plane triangle which has for its sides H sin λ, H sin δ, where λ and δ are the longitude and declination of a point of the Ecliptic. In the Triprasnādhikara, we had occasion to deal with this triangle and there we had R √(H sin² λ - H sin² δ) --------------------- H sin α where α is the right ascen- H cos δ √(H sin² Z☉ - H sin² ZV) sion of the point. Similarly R ------------------------ H cos ZV = H sin V☉. In other words Dṛk-nati is H sin V☉ projected into a circle of radius H cos ZV from a circle of radius R. We have the proportion ∴ C☉ CD D☉ -------- = ------------- = ------------------------- H sin Z☉ ☉ sin ZV √(H sin² Z☉ - H sin² ZV) D☉ = Sphutalambana = ------------------------- √(H sin² Z☉ - H sin² ZV) The quantity under the radical in the denominator is called Dṛk-nati for the following reasons. When V coincides with Z, V ☉ is the Dṛk-mandala- nata, so that when V is deflected also, we continue to view the Dṛk-nati placed along V☉. Since Madhyama- lambana ☉C is in the form of a Karṇa in the Δ ☉DC, we perceive it to be in the form of a Karṇa even when V coincides with Z. This Madhyamalambana being equal to Sphutalambana when V coincides with Z, Sphuta- lambana is also in the form of a Karṇa then. Now in the position ☉DC, Sphutalambana has assumed the position of a Kōti ie. the Sphutalambana which, in the form of a Karṇa, being placed along Dṛk-mandala natāmsa, is now

423 rendered a Kōti and is placed along V ☉, So the quantity √(H sin² Z☉ - H sin² ZV) which is the Kōti of the △ formed by H sin Z☉ and H sin ZV, corresponds to the Kōti of Sphuta-lambana ☉D. So we call √(H sin² Z☉ - H sin² ZV) as Dṛk-nati, in as much as the Sphutalambana being placed along V☉ in the form of a Karṇa when V☉ is Dṛk-mandala-nata, continues to be placed along V☉ in the deflected position also and becomes a Kōti corresponding to the Kōti of the triangle formed by H sin Z☉ and H sin ZV ie. corresponding to the quantity √(H sin² Z☉ - H sin² ZV). The Sphuta- lambana should be construed as being associated with Dṛk-mandala-nata which term is now abbreviated to the term Dṛk-nati. At A of fig. 92, the Dṛk-nati = √(R² - H sin² ZV) = H cos ZV = H cos ZV = Vitribha-lagna-Śanku. Hence the proportion proceeds in accordance with this Dṛk-nati. Verse 6 (latter half) and first half of verse 7. Alternative method of obtaining parallax in longitude. √(((H cos ZV) / (R/4))² - ((H cos Z☉) / (R/4))²) or √(((H sin Z☉) / (R/4))² - ((H sin ZV) / (R/4))²) gives the parallax in longitude expressed in nādīs. Comm. These formulae just constitude another mode of expressing the parallax in longitude and the equivalence of the formulae with the formula (4/R) Dṛk-nati is evident. Latter half of verse 7. Use of the parallax in longitude. The time of the ending moment of New Moon ie. the moment of geocentric conjunction is to be rectified by this

424 parallax in longitude to get the moment of apparent conjunction by the method of successive approximation. Comm. In as much as the moment of apparent conjunction for an observer situated somewhere on the surface of the Earth precedes or follows the moment of geocentric conjunction being preponed or belated by the parallax in longitude, we have got to take this parallax in longitude into account and compute the moment of apparent conjunction. This computation has to proceed according to the method of successive approximation since the hourly motions of the Sun and the Moon vary as well as the parallax in longitude. When the Sun is in advance of V, the Sphutalambana advances the Moon more than the Sun so that the moment of apparent conjunction is past. Hence the correction is negative and vice versa. Verses 8 and 9. Computation of the parallax in longitude without an appeal to the method of successive approximations. Let Para = (13 / 32) H cos ZV ; {Para ~ H sin ☉ L}² + H cos² ☉ L = K² H sin⁻¹ { (H cos ☉ L × Para) / K } = parallax in longitude. Ref. fig. 95, E₁ E₂ is taken to be what is termed Para equal to (4 / R) H cos ZV, H cos ZV being the Vitribha-Sanku. Since 4 / R could be written as (H sin 24 / R), since 4 ghatis = (4 × 360) / 60 = 24°, 60 ghatis being equivalent to 360°, (4 / R) H cos ZV = (H sin 24 / R) × H cos ZV. Imagining for a moment H cos ZV has come in the place of H sin d

425 pertaining to the formula H sin δ = (H sin λ × H sin 24) / R , (4/R) H cos ZV = Para = the H sine of the declination of that point whose longitude is equal to Vitribha-Sanku. In other words Para is termed as the Vitribha-Sanku- Rūpa-Krānti-Vṛttiya-Bhujajyājanita-Krāntijyā. Now take E₁ E₂ = Para defined above. Draw circles of equal radii with E₁ and E₂ as centres. Call (E₁) and (E₂) as the Chandra-Kakshāmandala and Ravi-Kaksha Mandala. Para by its formulation as (4/R) H cos ZV, is equal to the maximum parallax in longitude for a given H cos ZV ie. for a given position of V with respect to Z. This being so, the parallax in longitude for an arbitrary position of ☉ with respect to V will be [Para × H sin (☉ - v)] / R according to the previous formu- lation thereof. This form of the formula by its similarity with the formula (a/R) H sin m, pertaining to the eccentric- circle-theory, suggested to Bhāskara that the parallax in longitude could be derived from the theory of the eccen- trics or Prati-Vṛtta-Bhangi. In fig. 95, it will be noted that E₁, E₂ are not the centre of the Earth and the position of the observer on the surface of the Earth but such points as E₁ E₂ is made equal to (4/R) H cos ZV or (a/d) H cos ZV of the modern figure 4/R being equal to a/d, so that E₁ E₂ is of a variable magnitude varying with H cos ZV. Comm. When H cos ZV = R ie. when V coincides with Z, we have the maximum parallax. What then will 54

426 [चित्र: Fig. 95] Fig. 95 be had for an arbitrary H cos ZV ? The result is (H cos ZV / 3438) × H sin 24 since 4 nādīs correspond to 24°, 60 nādīs corresponding to 360°. Hence the result is (H cos ZV × 1397) / 3438 . Converting 3438 / 1397 into a continued fraction we have 2 + 1/(2+) 1/(5+) 1/(1+) 1/9 ......of which the con- vergents are 2/1, 5/2, 27/11, 32/13 and 32/13 is a very good convergent preceding a large quotient namely 9. So the result may be written as (13 H cos ZV) / 32 which is symbolized as Para. Now parallax in longitude = (Para × H siu ( ☉ − v )) / R . When H sin ( ☉ − v ) = R, the parallax will be equal to

427 Para. This formula by its similarity with the formula pertaining to the eccentric theory led Bhāskara to use the method of eccentric circles to obtain the parallax. It is indeed ingenious on his part to have conceived the appli- cability of that method. Further it is rather curious that 4 nādīs of the maximum parallax should correspond to 24°. This also led Bhāskara to conceive similarity between the formulae H sin δ = (H sin λ × H sin 24°) / R (the formula used to obtain the declination δ given the longitude λ of a point of the Ecliptic) and the formula (H cos ZV × H sin 24) / R = Para. So from an arbitrary H sin λ equal to Vitribha- Sanku, Para is derivable as H sin δ. In other words Para is called Vitribha-S'anku-Rūpa-Krānti-Vṛttiya-Bhujajyā- Janita-Krāntijyā. Now the doubt arises, namely that when the formula longitudinal parallax = (Para × H sin (☉ — v)) / R resembles the formula a/R H sin m which pertains to the Equation of centre, why does Bhāskara suggest that the parallax is derivable without the application of the method of succes- sive approximations, by appealing to the method S'īghra- phala. The doubt is here two fold (1) where is the necessity for the method of successive approximation to obtain the parallax, though it be called for, to obtain the moment of conjunction ? (2) why does Bhāskara appeal to S'īghrakarma and not Mandaphala, when the formula suggests the latter, by the presence of R and there is no K at all ? The answer is as follows. In the first place, even in the modern formula for parallax namely a/d sin z, Z is

428 the zenith-distance pertaining to the observer and not the geocentric zenith-distance, which are respectively called prṣṭhīya and garbhīya natāṁsas. Also the parallax is the angle between the geocentric direction of the Moon and that of the observer. (Vide fig. 91 where parallax = E M̂ A). In deriving this parallax, we are using the apparent zenith-distance of the Moon and not the geocentric zenith- distance of the Moon. In fig. 92 the position of ⊙ corres- ponds to the geocentric position, whereas D corresponds to the position of the observer on the surface of the Earth. So, as we use the apparent zenith-distance as argument to obtain parallax along the vertical, so we have to use, VD as the argument to derive the parallax in longitude and not V ⊙. So, the method of suceessive approximations is called for as ⊙ D is first computed from the argument V ⊙ and VD is to be made the argument thereafter. This means that V ⊙ may be construed as Madhyakēndra and VD as Sphuṭakēndra. Now applying this idea to fig. 95, V₂ E₂ ⊙ may be construed as Sphuṭakēndra whereas V₂ E₁ ⊙ may be construed as Madhyakēndra. From the similarity of the triangles ⊙ NM, and E₂ LM, ⊙N / LM = ⊙M / E₂M ∴ ⊙N = (⊙M / E₂M) × LM = (Para / K) × ⊙K = (Para / K) × H sin K Ê₂ ⊙ where E₂M is termed the Karṇa and K Ê₂ ⊙, the Sphuta- kēndra is made the argument. Thus parallax in longitude which was originally formulated as (Para × H sin (⊙ - v)) / R (in which case, the method of successive approximation was called for), is now formulated as (Para / K) × H sin (KE₂ ⊙)

429 where that method of successive approximation is circumvented and where by the presence of K in the place of R, analogy is with the eccentric method of formulation of Śīghraphala and not that of Mandaphala. Also K² = E₂M² = E₂L² + ML² = (E₂K–LK)² + ⊙K² = (E₂K–M⊙)² + ⊙K² = (H cos KÊ₂ ⊙ –Para)² + H sin² KÊ₂ ⊙. But if L be the lagna of the moment L⊙ = 90 – V⊙ so that H cos KÊ₂ ⊙ = H sin ⊙L and H sin KÊ₂ ⊙ = H cos ⊙L ∴ K² = (H sin ⊙L–Para)² + H cos² ⊙L as formulated in the verse. Fig. 95 is in the plane of the Ecliptic. The parallax in the vertical circle is projected on to the plane of the Ecliptic by taking 4/R H cos ZV as the Para, and deriving the parallax in longitude from this Para. Now, the doubt arises as to why the Śīghrocca is not taken to coincide with the Vitribha but is taken as removed 180° therefrom. Verse 10. H sin ZV (of fig. 92) is called the Dṛk- kṣepa of the Sun, which is considered to be north in case the northern declination of the Vitribha is greater than ϕ the latitude, otherwise south. Comm. Let in fig. 96, AV be the Ecliptic whereof A is the ascendant or Lagna and V the Vitribhalagna. Let EQR be the celestial Equator. Let δ be the decli- nation of the Vitribhalagna. Then if δ > ϕ. then ZV, the arc of the the Dṛk-kṣepa, (H sin ZV being defined as the Dṛk-kṣepa) as well as H sin ZV are considered to be north. Thus in fig. 96, it is north whereas in fig. 97 it is south. (In fig. 97, r is shown outside the celestial sphere, signi-

430 fying that r is in the western hemisphere and is brought into view for clarity). Fig. 96 Fig. 97 Verse 11 and first half of verse 12. Then the sum of ZV and the latitude of V assuming V to be the Moon, or the difference of the above two, as the case may be, according as both of them are north or of opposite directions, gives the arc whose Hsine is the Dṛk-kṣepa of the Moon, The Dṛk-kṣepas of the Sun and the Moon multiplied respectively by 1/15th of their daily motions and divided by the radius R (equal to 3438′) are the parallaxes of the Sun and the Moon in latitude. The sum or difference of these parallaxes according as they are of opposite or the same direction, is the true parallax in latitude in the context of a solar eclipse. Comm. The true parallax in latitude sought above is the relative parallax of the Sun and the Moon in latitude. Suppose in fig. 92, VB is the parallax in latitude pertain- ing to the Sun and VB′ that pertaining to the Moon; then BB′ is the relative parallax, the difference being taken in this case because both are of the same direction. Parallax in latitude namely CD in fig. 92, we saw equal to VB which is equal to (4H / R) sin ZV, In other words

431 the parallax in latitude either of the Sun or the Moon is equal to 4/R H sin of the zenith-distance of the respective Vitribhalagna wherever the Sun or the Moon be situated in their orbits namely the Ecliptic or the Vimandala. Let H sin ZV be the Dṛk-kṣepa of the Sun, V being the Vitribhalagna pertaining to the Sun and let H sin Zv be the Dṛk-kṣepa of the Moon where v is the Vitribha- lagna of the Moon. (Ref. figures 98 and 99) Let K' be the pole of the Vimandala and vv₂ the latitude of v. Since v and V are in the proximo, the latitude of v may be taken to be very nearly equal to the latitude of V so that ZV ± latitude of V is very nearly equal to Zv. In fig. 98, ZV—latitude of V is very nearly equal to Zv because both ZV and latitude of V are of the same direction. In fig. 99 ZV+latitude of V is very namely equal to Zv because both arc of opposite direction. Thus Zv = ZV ± latitude of V approximately and H sin ZV and H sin Zv are the Dṛk- kṣepas of the Sun and the Moon respectively. Having got these Dṛk-kṣepas 4/R × Dṛk-kṣepa gives the nati in each case ie. the parallax in latitude and the sum or difference of these natis as mentioned in the beginning of the commentary of this verse gives the relative parallax of the Moon with respect to the Sun which is called the Fig. 98 Fig. 99

432 true parallax in latitude. This true parallax in latitude increases or decreases the latitude of the Moon at the moment of conjunction as is going to be mentioned in the latter half of verse 14. In deriving the parallax in latitude from the respe- ctive Dṛk-kṣepas, instead of using the formula 4/R Hsine (Dṛk-kṣepa) which is an expression in time, it is sought to express the same in arc because the latitude of the Moon is expressed in arc and we have to take the sum or differ- ence of the latitude and the parallax in latitude to obtain the apparent latitude of the Moon at the moment of conjunction. In the case of the parallax in longitude we sought to express the same in time because the moment of apparent conjunction was sought therefrom. Latter half of verse 12 and first half of verse 13. An approximate method of obtaining the relative parallax in latitude of the Moon with respect to the Sun. The Hsine of the zenith-distance of the nonagesimal pertaining to the Moon or what is called the Moon's Dṛk-kṣepa multiplied by 2 and divided by 141, gives the relative parallax in latitude of the Moon with respect to the Sun ; or working with the smaller table of Hsines (where R is taken to be 120) the Moon's Dṛk-kṣepa being multiplied by 2 and divided by 5 and the result being increased by 1/60th of itself gives approximately the relative parallax in latitude. Comm. Herein, the Vitribha or the nonagesimal of the Sun is taken to coincide with that of the Moon. In other words the Dṛk-kṣepas (the Hsines of the zenith- distances of the nonagesimals, of both the Sun and the Moon are taken to be identical. Then using the following proportion "If by a Dṛk-kṣepa equal to R, the relative parallax in latitude is equal to 1/15th of the difference of the

433 daily motions namely 48′-46″, what will it be for an arbitrary Dṛk-kṣepa ?" We have (D × 48′-46″) / 3438 . Converting 48¾ / 3438 ie. 195 / 13752 ie. 65 / 4584 into a continued fraction, this will be equal to 1/(70+) 1/(1+) 1/(1+) 1/(10+) 1/3 of which a very approximate convergent is 1/71 as taken by Bhāskara. If the radius be taken to be 120, the coefficient of D will be 48¾ / 120 = 195 / 480 = 13 / 32 = 1/(2+) 1/(2+) 1/6 = 2/5 very approximately. Latter half of verse 13 and first half of verse 14. An easy method to compute the parallax in longitude and latitude. Taking the Dṛk-ṣepa of the Moon as well as the Sun to be the Hsine of the meridian zenith-distance of the Vitribha and the H cosine of its meridian zenith-distance as the Vitribha-Sanku, the parallaxes in latitude and longitude could be got from them respectively. Comm. Parallax in longitude is computed from the Vitribha-Sanku, whereas parallax in latitude is computed from the Dṛk-kṣepa or the Hsine of the zenith-distance of the Vitribha. Thus for both the purpose the Vitribha's position is important, whose zenith-distance and altitude give respectively the parallax in latitude and longitude. Since in practice it is a little cumbrous to obtain the Vitribha's altitude and zenith-distance, an approximate procedure is suggested. Obtaining the declination or the Sphuṭa-krānti of the Moon taking him to coincide with the Vitribha by the method described in verse 3 of the Graha-cchāyādhikāra, and using the formula z+δ=ϕ, the meridian zenith-distance of the Vitribha can be got. This may be assumed to be the Dṛk-ṣepa approximately. The 55