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

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

442 = (R+r-g)²-β² / (m₁-s₁)² so that δt = -βδβ / (t (m₁-s₁)²) and T² = (R+r)²-β² / (m₁-s₁)² so that δT = -βδβ / (T (m₁-s₁)²) Thus we see that as β increases both t and T decrease on account of the minus signs in δt and δT. Also if I were to increase, βδβ / (t (m₁-s₁)²) should be greater than βδβ / (T (m₁-s₁)²) ie. 1/t must be greater than 1/T ie. T should be greater than t and this is so. Hence as β increases t and T decrease but T-t increases ie. as T decreases T-t increases ie. I increases. So, roughly I is taken to be inversely proportional to T so that I' = (I × T) / T' If strict inverse proportionality is there, since, t + δt = t' = t - βδβ / (t (m₁-s₁)²) and T + δT = T' = T - βδβ / (T (m₁-s₁)²) and therefore t'/T' = (t/T) · { 1 - βδβ / (t² (m₁ - s₁)²) } / { 1 - βδβ / (T² (m₁ - s₁)²) } = (t/T) · { t² - βδβ / (m₁ - s₁)² } / { T² - βδβ / (m₁ - s₁)² } t² must be roughly equal to T² which means T - t = I must be small. In other words the formula of the verse holds good only for small I's or Iṣṭakālas. In fact this process of rectification of the Bhuja was first given by Brahmagupta in verses 18 and 19 of Sūrya- grahaṇādhikāra and accepted by Śrīpati in verse 14. The verse 19 under Lunar eclipses in the Sūrya- siddhānta also seems to indicate this process but uses the words Madhya-Sthiti-khaṇḍa and Spaṣṭa-Sthiti-khaṇḍa in

443 an ambiguous sense. Ranganātha in his commentary takes the Madhya-Sthiti-khanda to mean that rectified for variation in β, and Spaṣṭa-Sthiti-khanda to mean that rectified for parallax. Some modern traditional commen- tators, for example pandit Sitha Ramajha commenting on the verse of the Sūryasiddhānta and the commentator of the verses of Brahma Sphutasiddhānta published under the editorship of Acharya Rama Swarupa Sharma, have ignored the variation in β and confined themselves only to the effect of parallax in longitude. Yet one more proof could be adduced in this behalf. Let D, be the moment of geocentric conjunction, t the time for a given grāsa in between D and T the moment of first geocentric contact. Then the times D, t and T are to be corrected both for parallax in longitude and that in latitude. Take the geocentric Sthiti-khanda, and geocentric Bhuja as the mean-values T and B and those corrected first for parallax in longitude as the True values T' and B'. Also take the time t corrected for parallax in longitude to be t'. We could write T' - T = δT, t' - t = δt and B' - B = δB. In the above, consider B expressed in time. Then D - T = S the mean Sthiti-khanda, D - t = B the mean Bhuja, D' - T' = S', the true Sthiti-khanda recti- fied for parallax in longitude, D' - t' = B' the Bhuja also rectified for the same. The δB = δD - δt, δI = δD - δT. Then using the proportion "If for D - T we have δD - δT as the variation what shall we have for D - t?" The result is ((D - t) (δD - δT)) / (D - T) = δB ∵ B + δB = B' = D - t + ((D - t) (δD - δT)) / (D - T) = (D - t) (1 + (δD - δT) / (D - T)) = (D - t) ((D + δD) - (T + δT)) / (D - T) = (D - t) (D' - T') / (D - T) = (Mean Bhuja × True Sthiti-khanda) / (Mean Sthiti-khanda)

444 In other words to obtain the Bhuja rectified for parallax in longitude, we have to multiply the mean Bhuja by the Sthiti-khanda rectified for parallax and divide by the Mean Sthiti-khanda. Similarly to obtain the Sthiti- khanda rectified for the variation in latitude also, take the Sthiti khanda rectified for parallax as the mean and that rectified for variation in β as the True and then by the same procedure, Bhuja rectified for parallax in latitude will be equal to Sthiti-khanda rectified for parallax in latitude multiplied by the Bhuja rectified for parallax in longitude divided by the Sthiti-khanda rectified for parallax in longitude Bhāskara gives the names Sphuta Sthiti- khanda and Sphuta Saraja Sthiti-khanda to the Sthiti- khanda rectified for parallax in longitude and that rectified further for parallax in latitude. The proof adduced above accords with the statements of Brahmagupta, Śrīpati and Bhāskara ; Ranganātha bearing in mind Bhāskara's version puts a correct interpretation on verse 19, Lunar eclipses of Sūryasiddhānta. But the usage of the words Madhya and Sphuta in that verse, misled the modern traditional scholars including Burgess. If is to be men- tioned here that the word Koti translated as perpendicular by Burgess is misinterpreted by him (see his commentary under verse 19, lunar eclipses) as ‘ The perpendicular is furnished us in time and the rule supposes it to be stated in the form of the interval between the given moment and that of contact or separation’. This translation goes against the definition of Koti contained in verse 18 just above, since the Koti of Sūryasiddhānta is the Bhuja of Bhāskara and vice versa. We shall now see why there arose confusion in the minds of many modern commentators including Burgess. The problem mooted by the Sūryasiddhānta in calling for a rectification of the Koti, is to be noted as that when the grāsa g is given and not the Iṣtakāla I. The two formulae

445 for Koti are √(R+r - g²- β²) / (u - v) and (T - I) (u - v). When I is given we have to use the latter formula and take T" in the place of T where T, T' and T" are respectively the Sthityardhas (1) Mean, (2) Mean rectified for parallax in latitude and (3) Mean rectified for parallax in longitude. Of course here to arrive at T", method of successive approximation is to be used as β goes on changing from time to time and there is an inter-play between the simultaneous effects of parallax in longitude and that in latitude. On the other hand when g is given we have to use the former formula for the Koti and the Koti thus obtained is to be rectified for the variation in β. So Sūryasiddhānta proposes the formula √(R+r - g)² - β² / (u - v) × T' / T" meaning thereby that the correction for the variation in β is more important because the formula is in terms of β and not the other formula. This formulation is approximate but adopted for the sake of ease. Otherwise from the grāsa, I is to be obtained and the other formula could be used which method is more laborious. Bhāskara's Correction of Brahmagupta's Statement Verses 1, 2 & 3. The statement of Brahmagupta namely that the arc of the Moon's Dṛk-kṣepa will be obtained by the sum or difference of that of the Sun with the latitude of the Vitribha, I (Bhāskara) do not accept. I shall give the reason why. In a place where the latitude is 24°, when the longitudes of the Sun, the Moon and the Node are all 180°, at the time of Sun-rise, the ecliptic occupies the position of the prime-vertical. The Moon will not leave the ecliptic even though depressed by parallax in longitude. Thus there is no parallax in

446 latitude. The Vitribha then being in the zenith, and its latitude being 4½°, the Dṛk-kṣepa of the Moon according to Brahmagupta's formula will be 4½° and therefore the parallax in latitude obtained by the Dṛk kṣepa will be (790'-35 / 15) × (H sin 4½° / 3438) = (52'-42'' / 3438) 270 = 4'-8" which is not the case actually. Comm. Having thus shown the flaw in Brahma- gupta's approximate formula, Bhāskara proceeds to show how that formula could be justified in a particular way. Brahmagupta assumed the Moon's orbit to be the ecliptic because at the moment of an eclipse, the latitude of the Moon is very small so that he might be taken to be on the ecliptic. In figure 101 let the Ecliptic coincide with the prime-vertical ZE. Let EV be the Moon's orbit where V is the Vitribha of the Moon's orbit. The arc of the Sun's Dṛk-kṣepa is here zero and that of the Moon's Dṛk-kṣepa is ZV which is the sum of the arc of the Sun's Dṛk kṣepa namely zero and the latitude of the Moon's Vitribha namely ZV, since the pole of the ecliptic now coincides with the south point S, which means that ZV is the latitude of V. Let VV' be the nati which is equal to AB, since V'A is the so-called Vikṣepa Sadṛśamaṇḍala or the deflected position of VE on account of parallax in latitude. The four minutes of parallax in latitude is now VV'=AB. This parallax is obtained because, Bhāskara argues, the nati obtained by Brahmagupta is there because he took VE to be the Ecliptic and so obtained that nati with respect to VE. Now, Bhāskara says, this is to be corrected by the difference of the Moon's latitudes the original one and that obtained after the Moon is deflected by the parallax in longitude. This difference is 0-AB=BA. Correcting AB with BA, the result is AB+BA=0 so that ultimately

447 there is no parallax in latitude ie. no nati at all, as should be the case. Fig. 101 Fig. 102 Or, Bhāskara's argument could be better illustrated from figure 102. Let VS be the ecliptic where V is the Sun's Vitribha and S, the Sun. Let V'M be the Moon's orbit called Vikṣepamandala where V' is taken to be the Moon's Vitribha and M the Moon. Let S', M' be the deflected positions of the Sun and the Moon on account of parallax. ZV' = ZV + VV' = the arc of the Sun's Dṛk- kṣepa + the latitude at the point V called Vitribhalagna- Bāna, as formulated by Brahmagupta that ZV' is roughly equal to the Moon's Dṛk-kṣepa-Dhanus. (Strictly speaking ZV' ought to be perpendicular to the Moon's orbit V'M, if V' were to be the Vitribha of the Moon; but, as the latitude is small, the error is negligible). Let LS' and L'M' be the so-called Krānti-Sadṛśa-mandala and Vikṣepa Sadṛśamandala or parallel drawn to the ecliptic and the Moon's orbit through the deflected positions S' and M' of the Sun and the Moon. S'n is the Sun's parallax in latitude ie. Nati. Similarly Brahmagupta took M'n' to be the nati of the Moon as stated by Bhāskara. The error com- mitted will be therefore M'n'— S'n = (M'x' + x'n') — (S'x + xn) = M'x' — xn) cancelling x'n' and S'n which are roughly equal. Also xn could be roughly taken to be equal

448 to MS. Hence the relative parallax is equal to M'x'—MS = Difference of the latitudes of the Moon in his original and deflected positions respectively. Hence S'n = M'n' — (M'x' — MS) = Brahmagupta's nati+correction of the difference of the latitudes reversely effected, as stated by Bhāskara. In fact, Bhāskara has misread Brahmagupta's correct procedure, since the latter sought the relative parallax of the Sun and the Moon. Instead of adding the latitude at V to the zenith- distance of V (the arc of the Sun's Dṛk-kṣepa) which implies additional computation of that latitude, as an approximate procedure, MS is added to ZV to get ZV' as an alternate procedure as mentioned by Bhāskara in the course of the commentary. Whereas Brahmagupta was seeking relative parallax, Bhāskara misread that Brahmagupta took M'n' as the nati and did not effect the correction of (M'x—MS) to obtain S'n, which Bhāskara took to be the Sphuta-nati. Not effecting the above correction is interpreted as neglecting it since the Moon's latitude during the course of an eclipse is small.

GRAHACCHĀYĀDHIKĀRA Verse 1. The orbital inclinations of the planets. The inclinations of the orbits of Mars, Mercury, Jupiter, Venus and Saturn to the ecliptic are respectively 110, 152, 76, 136 and 130 minutes of arc. The nodes of Venus and Mercury get rectified by adding their respective Śīghra anomalies to their values obtained originally. Comm. The values given above are said to be the mean values. Those given for the superior planets namely Mars, Jupiter and Saturn approximately accord with their modern values. Bhāskara says that these values pertain to that moment of observation, when the Śīghra anomaly is equal to 90 + ½ H sin⁻¹ a where a is Hsine of the maximum Śīghraphala. This is quite in order because when the Śīghra anomaly assumes the said value, the true planet is at the point of intersection of the deferent and the eccentric, which means that the planet is equidistant from E₁ as well as E₂ (vide fig. 103). Identifying E₁ to be the earth's centre and E₂ to be that of the Sun in the case of the superior planets, the mean latitude of the planet observed will be the same as that observed either from the earth or from the Sun. Hence in the case of the superior planets, the maximum latitudes of the planets observed accord with the geocentric as well as heliocentric observation. These are taken to be the mean values of the maximum latitudes. In the case of the inferior planets, it will be clear why the modern values of 7° and 3°-24' for Mercury and Venus are far higher than the Hindu values namely 152' and 136'. Since the mean planet in this case is taken to be the Sun, the linear values of the latitude observed from E and S, the centres of the Earth and the Sun respectively will be in the ratio SP/ES (vide fig. 104). 57

450 Fig. 103 Fig. 104 In the case of Mercury this ratio is 4/10 and that in the case of Venus is 7/10. Hence the values (420 × 4)/10 and (204 × 7)/10 ie. 168′ and 142′ roughly accord with the Hindu values. In other words the Hindu values are geocentric and the modern heliocentric. Bhāskara adds that the nodes of Mercury and Venus as computed previously are to be increased by the Śīghra anomaly to obtain the actual longitude of the node from which the latitude is to be computed. This directive is a beautiful example to show implied heliocentric motion. Let the convex angle ASN be the heliocentric longitude of the node measured negatively as the node has a negative motion along the ecliptic. Adding the heliocentric Śīghra anomaly to this longitude of the node means convex angle ASN + ÛSP = 360° + N̂SP − ÂSU = N̂SP − ÂSU. Now add the longitude of the planet ie. here ÂSU (= ÂES) to obtain the argument from which the latitude of P the planet is to be computed. The result is N̂SP as should be.

451 Computing the latitude as indicated in the next verse, by the formula (H sin NSP × β × R) / (R × K) where β is the maximum latitude cited above in the respective cases, multiplied by R and divided by K indicates that the linear magnitude of the latitude will be increased or decreased according as K is smaller or greater than R. Fig. 105 Fig. 106 (a) In this context, we are to throw light on some moot points. Bhāskara says “मन्दस्फुटो ग्रहः स्वशीघ्र प्रतिमण्डले भ्रमति, तत्र च तस्य पातोऽपि”, which tantamounts to saying that the node is situated in the heliocentric orbit “स्वशीघ्र- तिमण्डल”. The Vimandala or the planet's orbit, taking for example the inner orbit of P in fig. 105, does not exactly lie in the plane of the ecliptic as shown in that figure but is in an inclined position cutting the coplanar inner orbit in N and N', as shown by the ellipse NPN' where NP = NP'. The argument to calculate the latitude is NP' and the formula is (H sin NP' / R) × β to give the helio- centric linear latitude. To obtain its geocentric linear value, we have to multiply by R / K.

452 (b) In the case of the superior planets, Bhāskara mentions in the Golādhyāya “पातोऽथवा शीघ्रफलं विलोमं कृत्वा स्फुटात् तेन युतात् शरोऽनः ” verse 22 (Golabandhādhikāra). This also clearly indicates that the orbits of the superior planets are also heliocentric, for, otherwise, there is no purpose of subtracting the Sīghraphala from the True position of the planet. The purpose in doing so is to obtain the heliocentric longitude of the planet (vide fig. 106). True geocentric longitude of J is A/EJ = ASJ/. Subtracting the S'ighraphala EJS = JEJ' from ASJ', we have AŜJ the heliocentric longitude. Adding the retrograde longitude of N (तेन युतात् as cited above in verse 22) means adding ASN to AŜJ which gives NŜJ. This is the argument to calculate the latitude of J. (c) Bhāskara alludes to a confusion in the mind of Chaturvedāchārya in this context while commenting upon verses 23, 24 in Golabandhādhikāra (Golādhyāya). Chatur- vedāchārya commenting on Brahmagupta's words (verse 10 Grahayukti-adhikāra ch. 9, Brahma Sphutasiddhānta) exclaims. "The latitude of Mercury and Venus will be the same as what they are at the point of S'īghroccha ; correct- ness of the result alone is proof ; no other reason could be adduced !" Bhāskara clears his misconception in the following words. "The number of sidereal revolutions of the nodes of Mercury and Venus mentioned in the chapter on mean motion, are to be increased by the number of the sidereal revolutions of the S'īghrocchas of Mercury and Venus, as mentioned by Mādhava in his work “ Siddhānta- chudāmani ”. This means that the S'īghra anomaly is to be added to the position of the node obtained by the smaller number of revolutions given in the chapter on mean motion.

453 The misconception in the mind of Chaturvedāchārya as well as the wrong notion in the minds of Mādhava and even Bhāskara in construing that the number of revolu- tions of the nodes are to be increased by the number of revolutions of the S'īghrocchas, are due to the fact that it was overlooked that the position of S'īghrocohas with respect to Mercury and Venus are given by their helio- centric longitudes. In other words as is mentioned by the verse “ ज्ञशुकयोः ग्रहः सूर्यः भवेत्तौ शीघ्रनामकाँ ” the S'īghroc- chas of Mercury and Venus are no other than the helio- centric positions of those planets. Thus the exclamation of Chaturvedācharya cited above arose out of thinking that the S'īghrocchas differ from the planets, whereas they are the same heliocentrically, though they differ geocentri- cally. Heliocentrically the planets longitude is (vide fig. 105) ÂSP which is equal to A/EP' geocentrically. The geocentric longitude of the planet is on the other hand A/EP differing from the above, though both the helio- centric and geocentric longitudes point to the same planet. Bhāskara, no doubt, gave a correct procedure but missed to identify the S'īghroccha and the heliocentric position of the planet. In this context, the reader is referred to the author's ‘ peculiar concept of S'īghroccha in Hindu astronomy published in the journal of Oriental Research of the S. V. University Vol. XIV, Part 2, Dec. 1971. Verse 2. The Hsine of the arc of the planet’s orbit Vimandala intercepted between the nearer node and the planet multiplied by the maximum latitude of the planet cited, and divided by the S'īghrakarṇa gives the latitude of the planet at the given place. Comm. From the formula akin to that which gives the declination of a point on the ecliptic namely sin δ = sin λ sin ω or in the Hindu form H sin δ = (H sin λ H sin ω) / R ,

454 H sin β = (H sin λ H sin i) / R where β is the latitude required, i the maximum latitude and λ the arc of the planetary orbit intercepted between the nearer node and the planet. Since β and i are small H sin β and H sin i could be taken to be β and i. Hence we have β = (i × H sin λ) / R . Since this value is had at the distance of the Śīghrakarṇa, its value at a distance of R should be given by β = (i H sin λ) / R × R / K = (i H sin λ) / K as formulated. Verse 3. √(R² - H sin² v) is called Yaṣṭi where v is Āyanavalana. The latitude β of the planet multiplied by Yaṣṭi and divided by R or multiplied by H cos δ where δ is the declination of the point whose longitude is 90 + λ, λ being that of the planet and divided by R gives the value of the rectified latitude which could be added to the declination of the foot of the latitude to give what we say the modern declination of the planet. [Diagram: Spherical geometry diagram showing points P, P', K, R, R', B, B', M, M', S, L, N, ω, γ] Fig. 107

455 Comm. Let rM (fig. 107) be the ecliptic and rN the celestial equator whose poles are K and P respectively. Let R be a celestial body whose latitude is β and whose modern declination is RL. Let M be the foot of the latitude circle and let δ be the declination of M. The word Krānti in Hindu astronomy is applied to connote the declination of a point on the ecliptic alone and not of any other point like R. RL is called Sphuta Kranti which is equal to R'N = R'M + MN. RM is called Vikṣepa and R'M Sphuta Vikṣēpa. K̂MP is called the Āyanavalana at the point M of the ecliptic. Produce MP to P' where PP' = δ so that MK = MP = 90°. Hence P̂' is a right angle and KP' = K̂MP' = v̂ = Āyanavalana. From the spherical triangle KPP', cos ω = cos v cos δ. Draw perpendicular RR' on MP so that MR' = β' = Sphuta Vikṣēpa = β cos v = (β H cos v) / R. But H cos v = √(R² - H sin² v) = Yaṣti ∴ β' = (Yaṣti × β) / R which is to be added to δ to obtain the Sphutakrānti R'N or RL, the modern declination. Note (1) One Mukhopādhyāya, in his thesis ‘The Hindu nakṣatras’ submitted to the Calcutta University, mistook RM' to be the Sphutavikṣēpa instead of R'M and hastily remarked that Bhāskara was wrong in making RM' less than RM. Note (2) v̂ shown in the fig. 107, is called Sthanīya- valana or valana at the point M which is considered to be place of the planet ‘Sthāna’ on the ecliptic. K̂RA, on the other hand is called Bimbīya-valana or valana at the bimba or disc of the planet. Note (3) v̂ could be obtained from the spherical triangle KMP where KM = 90°, PM = 90-δ, using the

456 formula cos ω = sin (90-δ) cos v or in the Hindu form H cos v = (R H cos ω) / (H cos δ) . In this case β' = (β H cos ω) / (H cos δ) . Or again noting that M K̂ P = 90-λ where λ is the longitude of R, we could use 'Inner side Inner angle formula' with respect to the triangle KMP, which gives 0 = sin 90 cot δ - sin (90 -λ) cot v or cot v = cot ω / cos λ or tan v = cos λ tan ω. But this formula implies the tangent functions which were not used by the Hindu astronomers. Similarly using the elements 90-λ, 90°, v, and 90-δ of the same triangle KMP, another formula could be got for v. Or again noting that K P̂ M = 90+α, we could yet get more formulae where α is the Right ascention of M. Note (4) Thus far we have used modern formulae. Let us now see as to how Bhāskara derives his formula. He takes the triangle MKP' (fig. 107) wherein MK = 90°, KP' = v and P̂' = 90. From fig. 108, the Hsine of MK is Fig. 108 KO equal to R where 'O' is the centre of the sphere and the Hsine of KP is KN so that it is the Āyanavalanajyā, Hence