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

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

457 is the centre of the sphere and the Hsine of KP is KN so that it is the Āyanavalanajyā. Hence ON² = OK² — KN² ∴ ON = Yaṣṭi = √R² — Āyanavalanajyā² = Āyanavalanakoṭijyā. From the similarity of the triangles OKN and GRF, OK / ON = GR / GF ∴ GF = (GR × ON) / OK = (H sin MR × Yaṣṭi) / R GR could be taken equal to MR, the latter being small and GF could be taken to be equal to GR′ ie. H sin MR′ and so equal to MR′. ∴ MR′ = (MR × Yaṣṭi) / R Adding MR′ to the declination of M, we get the modern declination of R′ ie. the Sphuṭakrānti of R. Note (5) In fact the spherical triangle MKP is just like the spherical triangle r E ♋ (fig.). In the place of the paramakrānti E ♋, we have the Āyanavalana KP (fig. 108), and in the place of Dyujyā ♋ L (fig. 19) we have Yaṣṭi. Note (6) An alternative is given in the verse for this namely MR′ = (MR × H cos δ′) / R where δ′ is the decli- nation of a point whose longitude is 90+λ. In other words we have to prove that H cos v = H cos δ′ or v = δ′ (of a point whose longitude is 90+λ. Since declination is given by the formula H sin δ = (H sin λ × H sin ω) / R Putting 90+λ for λ, H sin δ′ = (H cos λ × H sin ω) / R I But from triangle MKP, sin v / sin ω = sin (90 — λ) / sin (90 — δ) = cos λ / cos δ 58

458 ∴ sin ν = (sin ω cos λ) / (cos δ) or H sin ν = (H sin ω × H cos λ) / (H cos δ) II Comparing I and II, ν would be equal to δ′ provided H cos δ = R. This is accepted as an approximation, as δ is generally small. This approximate formula is given by Sūryasiddhānta in the context of Āyana Dṛk-karma in verse 10, ch. 7, where ν is assumed to be equal to δ′ defined above. Verses 4 and 6. The process called Āyana Dṛk-karma. The Āyana Dṛk-karma correction measured in minutes is obtained by multiplying Āyanavalana by the unrectified celestial latitude, and divided by H cos δ and then multiplied by 1800 and divided by the rising time of the Rāśi which is occupied by the planet. Or again it is obtained approximately by the product of the Āyana- valana and the unrectified celestial latitude divided by the Yaṣṭi. This Āyana Dṛk-karma correction is negative if the hemisphere and the latitude have the same direction. On symbols, Āyana Dṛk-karma correction = (Āyanavalana × β × 1800) / (H cos δ × T) or approximately equal to (β × Āyanavalana) / (Yaṣṭi (= H cos ν)) . Comm. Let G be the position of the planet when the foot of the planet’s secondary (called Grahasthāna) to the Ecliptic namely A is rising, where rCA is the Ecliptic. Let rML be the celestial Equator. Let P and K be the poles of the celestial Equator and Ecliptic respectively. The arc AC of the Ecliptic intercepted between A and the declination circle of G expressed in minutes represents the आयनकलोः formulated in the verse. To find the magnitude of AC, the first formula mentioned above envisages finding

459 Fig. 109 it through the magnitude of ML the corresponding Equatorial arc which is itself found through finding the magnitude of AB. AB expressed in minutes will be equal to the number of asus taken by the declination circle of G namely PGB to have traversed from the position PDA to PGB which is the time taken by the planet G to be at its position from the moment of its rising at the Equatorial horizon namely PDA, D being then its position. When G was at D, K, the pole of the Ecliptic should have been itself rising on the Equatorial horizon. Thus it is sought to find the time expressed in asus taken by K from its moment of rising on the Equatorial horizon to its present position at K. When Bhāskara mentions in the commen- tary under the verse that ‘‘क्षितिजकदम्बयोरन्तरं तदेवोत्तरमायनं वलनम्’’ by the word क्षितिज, he had at the back of his mind लङ्काक्षितिज ie. Equatorial horizon and not the horizon of the place shown in the figure. Also as was mentioned before in a previous context the time expressed in asus taken by an equatorial arc to rise is equal to the number of minutes in that arc. Since the time taken by the planet to traverse the arc of the diurnal circle namely DG is the same as the time taken by the equatorial arc LM to rise, Bhāskara seeks to find the number of minutes in AB, then compute, the number of Asus taken by LM to rise

460 through which the number of minutes in AC could be computed. AB = AG sin AĜB = (AG × H sin AĜB) / R = (β × Āyanavalanajyā) / R where AG = β. In the formula given the word Āyanavalana is used for Āyanavalanajyā ie. the Hindu sine of Āyanavalana for brevity. Also, since generally the angle AĜB defined as Āyanavalana happens to be small, its Hsine will be almost equal to it. In this sense also the term Valana is used where Valanajyā is to be used. From AB we pass on to the magnitude of LM. From Hindu spherical Trigonometry LM = (AB × R) / (H cos AL) = (AB × R) / (H cos δ) = (β × Āyanavalanajyā) / R × R / (H cos δ) = (β × Āyanavalanajyā) / (H cos δ) . (H cos δ is called Dyujyā which is connoted by the term Dyu-guna in the verse, the words Jyā and Guna being synonymous). Thus LM = (β × Āyanavalanajyā) / Dyu-guna From LM we pass on to the corresponding arc of the Ecliptic namely AC, which does not imply the latitude of the place. So we have to use what are called Niraksha- Udayas of Rāśis ie. the rising times of the Rāśis at Equatorial places. Rule of three is used here to find AC from LM. Locating the particular Rāśi in which the planet is situated, and using the proportion. “If a Rāśi of 30° ie. 1800′ of the Ecliptic rises at an Equatorial place in say x asus, what arc in minutes corresponds to the number of minutes or what is the same the number of Asus

401 in the arc LM?" We get the magnitude of the arc AC in minutes. The computation of this arc AC is intended to know the difference between the times of rising of the point A and the point G the former being the point of the ecliptic signifying the position of the planet and the latter being the planet itself. This difference of the times of rising is itself required to know the actual time of rising of the planet by a knowledge of the time of rising of the point which signifies the planet’s position on the Ecliptic by its longitude. In verse 5, Bhāskara gives an alternate and easy method of obtaining the magnitude of AC construing AGC to be roughly a plane triangle which does not involve any appreciable error. AC = AG tan AĜC = (β × H sin AĜC) / (H cos AĜC) = (β × Āyanavalanajyā) / Yaṣṭi since Yaṣṭi is defined as H cos AGC in verse 3. Note. The point C is called the Kṛta-Āyana-Dṛk- Karma Sthāna ie. the point of the Ecliptic signifying the planetary position rectified for the Samskāra or correction called Āyana-Dṛk-karma. The longitude of C thus got is called the polar longitude of the planet which is defined as the longitude of the planet as measured on the Ecliptic upto the point of intersection of the planet’s declination circle with the Ecliptic. Bhāskara mentions elsewhere, “ नक्षत्राणां स्फुटा एव स्थिरत्वात् पठिताः शराः दृक्कर्मणाऽऽयनेनैषां संस्कृताश्च तथा ध्रुवाः ” ie. in the case of stars which are fixed, the Sphuta-saras or polar latitudes (given by GC in the above case) are

462 given and longitudes rectified for Āyana Dṛk-karma are given ie. polar longitudes. Bhāskara made this as an approximate statement, for, we know, even in the case of stars, though they be fixed, their polar latitudes also do change by the precession of equinoxes be it just a little; whereas their polar longitudes do not change by the same constant quantity as their celestial longitudes. Verses 6, 7 and 8. Obtain the Carakhandas or ascentional differences of the Sphuta and Asphuta Krāntis. If they be of the same direction their difference is to be taken, if of opposite direction, their sum is to be taken. The result in asus gives the Ākṣa Dṛk-karma correction if the celestial latitude is of appreciable magnitude. If it be not appreciably large, it (the celestial latitude) is to be multiplied by the Ākṣavalana, then divid by H cos ϕ or what is the same, multiplied by the equinoctial shadow and divided by 12. The result is to be multiplied by the radius and divided by H cos δ. Then we have the Ākṣa Dṛk-karma correction in asus. Assuming the planet’s position corrected for Āyana Dṛk-karma to be the Sun, obtain the lagna using the asus of the Ākṣa-Dṛk-karma. If the planet has a southern latitude let the lagna be found in the positive direction; otherwise in the negative dire- ction. Then we have the rising lagna of the planet or what is the same the longitude of the rising point of the planet. Again assuming the planet’s position increased by 180° to be the Sun, and using the asus of Ākṣa Dṛk- karma, obtain the lagna in the positive direction with respect to a northern latitude of the planet or else in the negative direction. The result gives the longitude of the setting planet or what is called the Asta-lagna. Comm. (Refer to fig. 110) Let p be the planet whose Graha-sthāna or foot of the latitude is p′. Let q be what is called the Kṛta-Āyana-Dṛk-Karmaka-Sthāna or the planet’s position corrected for the correction of Āyana

463 Fig. 110 Dṛk-karma. (This correction is given by the arc p'q). The difference of the rising times of the planet p and q expressed in asus is what is sought here. The planet p rose at a and has covered the arc ap of the diurnal circle after rising. So, we have to compute ap and therefrom the corresponding arc of the Equator namely AC. If the planet has no latitude ie. is situated on the ecliptic at p', the arc p'q or Āyana Dṛk-karma vanishes, for, the Āyana Dṛk-karma is no other than the projection of the latitude of the planet on the ecliptic taking P or celestial pole as the vertex of projection. Also if there were no Akṣa ie. if the place be equatorial, p and q rise simultaneously ie. the planet will rise along with the point called Kṛta- Āyana-Dṛk-Karmaka-Sthāna q, mentioned above. In fact the Dṛk-karma corrections Āyana and Ākṣa are contem- plated to obtain the difference in the rising times of p and p' ie. the planet and its position on the ecliptic. This time is resolved into two parts (i) the difference of the rising times of p' and q and (ii) that of the rising times of q and p. The former is given by the equatorial arc BC which goes by the name Āyana-Dṛk-karma and the latter by the arc AC which goes by the name Ākṣa-Dṛk- karma. The algebraic sum of these two corrections gives the difference of the rising time of p and p' ie. the time given by the arc AB. Here AB=AC-BC, In verses

464 4, 5 we have seen how BC is computed. Here we are seeking to compute AC through the corresponding arc ap of the diurnal circle. The verse uses the word Sphuta-Krānti, Asphuta- Krānti, and their Cara-khandas. Here in the figure pC is the Sphuta-Krānti and p'B is the Asphuta-Krānti or simply Krānti. Also pp' is called Asphuta-Vikṣēpa and bp' Sphuta-Vikṣēpa. Sphuta-Krānti = pc = bB = bp' + p'B = Sphuta-Vikṣēpa + Asphuta-Krānti. In other words the declination of p, the planet, is called Sphuta-Krānti and that of its longitudinal position on the ecliptic namely p' is called Asphuta-Krānti or simply Krānti. We have obtained Sphuta-Krānti from Krānti by adding to the latter the Sphuta-Vikṣēpa. The method of obtaining Sphuta-Vikṣēpa from the Vikṣēpa, otherwise called Vikṣēpa Sphuta-karama was dealt with in verse 3 above. Now we have to define the Cara-khandas pertaining to the Sphuta-Krānti pC and the Asphuta-Krānti p'B. The former is defined as AE and the latter by CE very approximately. We say very approximately because when p' comes to the horizon, the Cara-khanda will not be exactly EC but a little more or less than EC since Bp' ≠ Cq. In other words, the arc of the equator between the declination circle of q while rising and the equatorial horizon ie. AE is defined as the Cara-khanda of pC ; similarly CE is that of p'B. Their difference is AE—CE = AC. If we use the modern notation, we have to take their algebraical difference. If that be done, the alternative case of adding the Cara-khandas when the Sphuta and the Asphuta-Krānti are of opposite directions, will be automatically implied. The asus pertaining to this arc AC will be the correction of Ākṣa-Dṛk-karma as mentioned in verse (6) when the latitude of the planet is appreciable. These asus will be equal to the time taken by p the planet to cover the arc ap or what is the same, the

465 time in between the moment of rising of p and that of q, the position of the Kṛta-Āyana-Dṛk-Karmakagraha. When the latitude is very small, an approximate estimate of this correction is given in asus as (β' × H sin ξ) / (H cos ϕ) × R / (H cos δ) or (β' × H sin ϕ) / (H cos ϕ) × R / (H cos δ) . That this formula is (β'=qp) only approximate is seen from the fact that H sin ξ ≉ H sin ϕ. This second formula can be proved as follows. Ākṣavalana in asus = (β' × H sin ϕ) / (H cos ϕ) × R / (H cos δ) where β' is the Sphuṭa-Vikṣēpa. Taking qpa as a plane triangle pa = qp tan p̂qa = qp × tan P̂En approximately = Sphuṭa-Vikṣēpa × (H sin ϕ) / (H cos ϕ) . Hence AC = (β' × H sin ϕ) / (H cos ϕ) × R / (H cos δ) . With respect to the first formula ie. Akṣavalana in asus = (β' × H sin ξ) / (H cos ϕ) × R / (H cos δ) ' pa = qp tan p̂qa = (qp × H sin p̂qa) / (H cos p̂qa) . Here instead H cos p̂qa which is Yaṣṭi previously defined, H cos PEn ie. H cos ϕ, an approximate value is substituted, because the latitude is small. Having obtained the value of pa, it is then reduced to the Equator by multiplying by R and dividing by H cos δ. The convention with respect to the sign is clear. The idea of Kramalagna and Vilōma- lagna may be elucidated as follows. In fig. 110, q, is on the horizon rising whereas p had already arisen. So to · 59