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

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

434 complement of the meridian-zenith-distance may be assumed to be the Vitribha-Sanku approximately. Then the parallaxes in latitude and longitude could be computed respectively from the two as described before. The following figure gives a particular nomenclature that was in the mind of Kamalākara, the author of Siddhāntatattvavivēka. [चित्र: Fig. 99-A] Madhya-Mandala Dṛk-mandala Garbha-chinha Mandala Dṛk-chinha Mandala Para-Mandala Krāntimandala Krāntisadrasamandala Fig. 99-A Latter half of verse 14. The purpose of obtaining the parallax in latitude.

435 The apparent latitude of the Moon is equal to the algebraic sum of his geocentric latitude and the parallax in latitude. From this apparent latitude are to be calcu- lated the Sthiti-khanda and Marda-khanda of the solar eclipse (by the method described in the chapter on lunar eclipse, taking the eclipsing body or grāhaka to be the Moon and the eclipsed or Grāhya to be the Sun). Comm. The geocentric parallax of the Moon has a double effect on the occurence of a solar eclipse as men- tioned before. If the parallax be resolved along the Ecliptic and along a secondary to the Ecliptic, we have respectively the parallax in longitude and that in latitude. The parallax in longitude makes the apparent moment of conjunction at a given place, differ from the moment of the geocentric conjunction, whereas the parallax in latitude makes the magnitude of apparent latitude of the Moon at the place of observation differ from that of the geocentric. The apparent latitude is equal to the sum or difference of the geocentric latitude and the parallax in latitude. Having got the apparent latitude, the computation of the Sthiti and Marda-khandas could be done according to the method described in the chapter on lunar eclipse. For a point C on the surface of the Earth, a solar eclipse occurs if the latitude of the Moon be less than MD Fig. 100

436 (vide fig. 100) where M is the centre of the Moon and MD the latitude of the Moon at the point of first contact MD = MB + BD = m + BÊD where m is the semi- diameter of the Moon's disc. But BÊD = BEA + AÊS = CB̂E − CÂE + AÊS = P − p + s where P and p are the parallaxes of the Moon and the Sun and s the angular semi-diameter of the Sun. Thus in order that a solar eclipse may be possible for some point of the Earth, the latitude of the Moon at the moment of conjunction must be less than P + s + m − p = 57' + 16' + 15' = 89' approximately. The lesser the northern latitude of the Moon at the moment of conjun- ction, more places situated on the surface of the Earth between C and F will have solar eclipse where F is the sub-solar point ie. the point of the Earth which has the Sun in the zenith at the time of conjunction. Similarly, if the southern latitude of the Moon is less than 89' at the moment of conjunction the places situated on the Earth between G and F will have solar eclipse. In particular the sub-solar point F will have solar eclipse if the latitude of the Moon at the moment of conjunction is less than HD ie. less than s+m ie. 33' approximately. The sub- solar point will have no parallax, so that the terms P and p in P+s+m−p vanish. For the other points ie. points between F and C or G parallax will be there and the latitude may be greater than s + m but less than s+m+P−p to have an eclipse. A latitude of 33' corres- ponds to a distance of (33 × 15) / 70 = 99 / 14 = 7 1°/14 of the Sun with respect to a node. Thus if at the moment of conjun- ction, the latitude of the Moon be less than 7°, even the sub-solar point must have an eclipse. Verses 15, 16, 17. To find Spars'akāla, Mokṣakāla, Sammīlanakāla and Unmīlanakāla.

437 First compute the time called Sthiti-khanda (as mentioned in the chapter on lunar eclipses). The ending moment of local Amāvāsyā or what is called the moment of local conjunction is known as the Madhya-Grahakāla or the moment of the middle of the eclipse. Subtract the Sthiti-khanda from the computed time of Geocentric conjunction ; the result will be the approximate Sparśa- kāla. This has to be rectified for parallax in longitude as well as the approximate Madhyagrahakāla of geocentric conjunction to obtain the local Sparśakāla and the local Madhyagrahakāla ; Similarly the Mokṣakāla, the Sammī- lana and the Un-mīlanakālas are to be rectified for parallax in longitude. But while effecting this correction for the parallax in longitude, the Moon's latitude also differs for the corrected time which in turn effects the durations of Sthiti-khanda, Mokṣa-khanda etc. Correcting the first computed Sthiti-khanda. Mokṣa-khanda etc. for this variation in the latitude, and subtracting the Sthiti- khanda from the time of Madhya-graha, we have a better approximation for the Sparśakāla. In as much as parallax in longitude, that in latitude, and the Moon's latitude vary from time to time, and the times of Sparśa, Madhyagraha etc. are effected by them, the process of computation proceeds by the method of successive approximation. Subtracting the rectified Marda-khanda from the rectified Madhyagrahakāla, we have the true Sammīlanakāla ; similarly adding the former to the latter we have the true Un-mīlanakāla. If, as mentioned before in verse 9, the parallax in longitude is found without using the method of successive approximation, the Sparśa-kāla and the Mokṣa-kāla are had at once. But the latitude of the Moon and the parallax in latitude are to be computed using the then longitudes of the Moon and the non-agesimal. Comm. Clear.

438 Verses 18, 19. To obtain the true values of the Bhuja and the Iṣṭakāla. The remaining work proceeds on the lines indicated in the chapter on ‘lunar eclipses’ (ie. the computation of the Bimbavalana, Bhuja, Koti and the like is to be done as indicated there). The Bhuja will be rectified by multi- plying it by the Sthiti-khanda obtained by adopting the latitude of the Moon effected by parallax in latitude and divided by the Sthiti-khanda rectified for parallax in longitude. Similarly given the grāsa ie. the magnitude of the eclipse, the result found before by verse 15 in the chapter of lunar eclipses, is to be multiplied by the Sthiti- khanda rectified for parallax in longitude and divided by that obtained adopting the latitude of the Moon effected by parallax in latitude, and the result so obtained being subtracted from the Sthiti-khanda, we get the Iṣṭa-kāla. Comm.

439 parallax in latitude, so that it is a variable. We are to make a correction for this variability, both in the com- putation of Sthiti-khanda, Bhuja and Iṣṭa-kāla. The formula for the Sthiti-khanda is √(R+r)² - β² / (m₁ - s₁) where β is the latitude of the Moon at the moment of conjunction. The value of MN at the moment of conjunction will not be equal to its value at any intermediate point because parallax in latitude differs from position to position of the Moon. In other words β is variable. The formulae given for the rectification of the Bhuja B or the Iṣṭakāla I are B' = (B × T') / T and I' = T' - √(R+r - g)² - β² / (m₁ - s₁) × T / T' where T' is the Sthiti-khanda rectified for the variability of β, B' is the Bhuja rectified for the same whereas T and B are the values of the Sthiti-khanda and Bhuja computed taking the effect of parallax in longitude above. The effect of parallax in longitude is to prepone or postpone the moment of first contact as well as that of conjunction. The verse under commentary uses two terms Sphuta-Sthiti-khanda and Sphuteshuja-Sthiti-khanda. The former is the Sthiti-khanda rectified for parallax in longitude whereas the latter is that rectified for parallax in latitude ie. by adopting β' instead of β in the formula √(R+r² - β²) / (m₁ - s₁) where β' = β ± effect of parallax in latitude. Suppose on account of parallax in longitude the moment of first contact t₁ becomes t₁ + δt₁, and let the moment of conjunction t₂ become t₂ + δt₂. Then the Sthitikhanda unrectified for parallax in longitude will be (t₂ - t₁) whereas that rectified for parallax will be (t₂ - t₁)

• (δt₂ - δt₁). This rectified Sthitikhanda is called Sphuta-Sthiti-khanda. Now the verse under commentary gives a procedure to rectify the Bhuja, and Iṣṭakāla in the

440 wake of β being effected by parallax in longitude. The formula for Bhuja is √(R+r -g² — β²) / (m₁ — s₁) wherein all quantities except β may be taken to be constant. Suppose β effected by parallax be comes β'. If β' > β, Bhuja will decrease ; also the Sthiti-khanda whose formula is √(R+r)² — β² / (m₁ — s₁) decreases if β' > β. Hence if T' be the new value of T the Sthiti-khanda, T' < T. In other words when β' > β, B' < B and T' < T. Also if β' < β, both B' and T' will be greater than B and T respectively. Hence as a rough measure B and T are taken to vary together positively or negatively and as such proportionally. Though both increase or both decrease together, strictly speaking the concept of proportionality is there ; but roughly speaking they are taken to vary proportionally which means B' = (B × T') / T . The fact that proportionality is not there could be seen in two ways. B = (T — I) (m₁ — s₁) (1) with usual notation so that taking B and T to vary on account of the variation in β, δB = δT (m₁ — s₁), I not varying so that B + δB = B' = (T + δT) (m₁ — s₁) — I (m₁ — s₁) = T' (m₁ — s₁) — I (m₁ — s₁) = (T' — I) (m₁ — s₁) (2). Dividing (1) by (2) B/B' = (T — I) / (T' — I) which will be approximately equal to T/T' provided I is very small compared with T. Assuming so, B/B, could be taken to be equal to T/T', which means B' = (B × T') / T as mentioned in the verse. Or again, considering the formulae for B and T and differentiating them with respect to β and getting B' and T', we shall have the following working. B² = ((R+r—g)² — β²) / (m₁ — s₁) so that 2 BδB = (-2 βδβ) / (m₁ — s₁)

441 or δB = -βδβ / B (m₁ - s₁) ; similarly T² = (R + r)² - β² / m₁ - s₁ so that 2T δT = -2βδβ / m₁ - s₁ so that δT = -βδβ / T (m₁ - s₁) ∴ B¹ = B + δB = B - βδβ / B (m₁ - s₁) and T¹ = T + δT = T - βδβ / T (m₁ - s₁) ∴ B¹ / T¹ = (B² - βδβ / B (m₁ - s₁)) × (T (m₁ - s₁) / T² - βδβ) = (B² - βδβ) T / B (T² - βδβ) = (B - βδβ / B) / (T - βδβ / T) . Since βδβ / B ± βδβ / T B¹ / T¹ ± B / T . Regarding the finding of Iṣṭakāla when g the grāsa is given, we have the formula T - I = B / m₁ - s₁ = √(R+r-g)² - β² / m₁ - s₁ so that putting T - I = t t = √R+r-g² - β² / m₁ - s₁ ∴ As β increases t decreases so that T - t increases ie. I increases. Whereas as β increases T decreases. This means in a way that as T decreases. I increases so that I is taken to be inversely proportional to T ie. I' is taken to be (I × T) / T' as given. Here also, it could be seen that the inverse propor- tionality is not strictly there ; for β increasing both t and T decrease so that T - t ie. I will increase only if the decrease in t is greater that in T. But t² 56