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

194 ∴ log sin δ₂ = 9.6093 + 9.9375 = 9.5468 ∴ δ₂ = 20° – 30′ ∴ log sin EA₁ = 9.3634 + 9.5758 = 8.9392 ∴ EA₁ = 4° – 59′ = 299′ ∴ rE = 3463 – 299 = 3164 asus ∴ Rising time of Sāyana Vṛṣabha for the locality = 3164 – 1505 = 1659 asus. rA₂ = 5400, sin EA₂ = tan 13 tan δ₂ = tan 13 tan 24°. ∴ log sin EA₂ = 9.3634 + 9.6486 = 9.0120 ∴ EA₂ = 5° – 54′ = 354 asus ∴ rE = 5400 – 354 = 5046 ∴ Rising time of Mithuna is 5046 – 3164 = 1882 asus. Before we proceed to find the rising times of Karka- taka, Simha and Kanyā, we shall cast our previous proce- dure into the Hindu form. In fig. 21, let rS be the Sāyana Mesha. The rising time of rS is measured by rE, because when r is at E, Meṣa is just about to rise and when r is in the position indicated, the extremity of Meṣa namely S is rising. So it means that as rS of the ecliptic has risen, a portion rE of the Equator has risen. As time is measured by the arc of the equator which rises with a uniform speed, we measure the rising time of rS by the arc rE; but rE = rA – AE. rA is the Equatorial rising time of rS, because when A is at E, S will be at B i.e. A and S will then be on the equatorial horizon EB simul- taneously. Hence rA = 1670 as proved before and stated by Bhāskara. EA is the chara for 30°. The chara for one angula or inch (inch is here used technically, and does not mean what it means in ordinary parlour) as has been stated by Bhāskara and proved by us is 10 Vinadis. (Vide page 181)

195 But we have taken 13° as our latitude, so that tan ϕ = .2309. Since s/12 = tan ϕ = .2309 ∴ s = 2.7708; let us take this as 2.8″ so that the charas 10, 8, 3⅓ found for one inch are to be multiplied by 2.8 to give the local charas. They are in Vinādis, 28, 22.4, 9⅓ or in asus 168, 134·4, 56; for convenience let us take 134.4 as 134, so that the charas are 168, 134, 56. Thus the rising time of Sāyana Meṣa at this locality is 1670 − 168 = 1502 asus i.e. 250 Vinādis = 4-10 Nādis. Then let rS now represent 60°, instead of subtracting EA from rA to get the combined rising time of Meṣa and Vṛiṣabha, the Hindu practice is to subtract the chara pertaining to Vṛiṣabha from the equatorial rising of Vrishabha i.e. rE = 1793 − 134 = 1659 as got before. Here it must be noted that EA° is the chara not pertaining to Vṛiṣabha alone but to Meṣa and Vṛiṣabha put together. That is why for ease, the chara to Meṣa, the increase in chara for Vṛiṣabha, and the increase in chara for Mithuna as well as their individual equatorial rising times are given. The increments in the charas are called chara-khandas just as the increments in the H sines are called Jyā-khandas (khands means segments). Simil- arly the rising time of Sāyana Mithuna is equal to 1937 − 56 = 1881 asus = 313.5 Vinadis = 5-14 Nadis. Now with respect to Karkataka, its equatorial rising time is 1793, for, from fig. 25, the equator at the equatorial place being prime Vertical, if rM be Meṣa, its time of rising is given by rE where E is the foot of the declination circle of M. Similarly if MV represents Vṛiṣabha, when V comes to the horizon, N the foot of the declination circle comes to the horizon. Thus the rising time of any arc of the ecliptic at an equatorial place is given by the correspond- ing arc of the equator, which is intercepted between the declination circles of the ends of the arc. So from fig. 26 if r ed ≃ be the equator, rED. ≃ the ecliptio, A the Ayana or Summer solotice, P the pole rE, ED, DA etc. the Sāyana Rasis Meṣa etc. a, b, c etc. the feet of the

196 Fig. 25 Fig. 26 declination circles of A, B, C etc., since PAE is secondary both to the ecliptic and equator (i.e. perpendicular circle) spherical triangles PAD, PAB are congruent; PBC, PDE are congruent and PC ♎ is congruent with PE ♈. Hence ab = ad i.e. rising times of Karkataka and Mithuna are equal; bc = de i.e. those of Simha and Vriṣabha are

197 equal and similarly those of Meṣa and Kanyā. It will be noted that the equatorial risings alone are equal in the above cases but not at any other place, for in a place with some latitude when a point like B (fig. 26) comes to the horizon, the foot of its declination circle namely b will not be on the horizon and there arises the chara in bet- ween, which has to be taken into account, and be subtract- ed from or added to the equatorial rising time as the case may be. Now regarding the chara-khanda of Karkaṭa it is again 56; why it should be so is not proved by Bhāskara but merely stated, nor any commentator took the pains to prove. It can be proved as follows. Let in fig. 26, δ₁, δ₂, δ₃ be the declinations at the ends of Vṛṣabha, Mithuna and Karkaṭaka respectively. We know δ₁ = δ₃. The chara-segments for Mithuna and Karkaṭaka i.e. when Mithuna and Karkaṭaka are rising are to be proved to be equal, here 56 asus. Their expressions are tan ϕ tan δ₂ − tan ϕ tan δ₁ and tan ϕ tan δ₃ − tan ϕ tan δ₂ i.e. tan ϕ (tan δ₂ − tan δ₁) and tan ϕ (tan δ₃ − tan δ₂). Since δ₁ = δ₃ we perceive that they are equal but of opposite signs. So Bhāskara says rightly “अपचीयमानत्वात् धनम्” i.e. because of negative sign, the chara-segment of Karka- taka while being subtracted will be rendered positive’. Hence the rising times of Karkaṭaka, Simha and Kanya will be respectively 1937+56, 1793+134, 1670+168 asus or 1993, 1927, 1838 asus or 332, 321, 306 Vinadis or 5-32, 5-21 and 5-6 nadis. Thus, in as much as the rising times of Meṣa to Kanyā are 1670−168, 1793−134, 1937−56, 1937+56, 1793+134, 1670+168 their total is 30 nadis as should be expected because the equator bisects the ecliptic between ♈ and ♎ and the equatorial interval between ♈ and ♎ is 30 nadis. It will be noted that while the equatorial rising times of Meṣa to Kanyā are symmetrical as 1670, 1793, 1937, 1937, 1793, 1670, their rising times at any other place are not like that but

198 constitute a different kind of symmetry as 1670—168, 1793—134, 1937—55, 1937＋55, 1793＋134, 1670＋168. We have now to comment upon the statement “तुलादितोऽमी च विलोमसंस्थाः ” which means that the rising times from Tulā to Mīna are in the reverse order i.e. the rising time of Tulā equals that of Kanyā; that of Vris- chika equals that of Simha and so on the rising time of Mīna equalling that of Meṣa. Thus the rising times of Tulā to Mīna being in the reverse order are 1670＋168, 1793＋134, 1937＋55, 1937—55, 1793—134, 1670—168 for the aforesaid locality. Why it should be so can be easily seen from the fact that Kanyā and Tulā are sym- metric with respect to the line ♈ ♎ which bisects the ecliptic (Ref. fig. 27). Or again we can see this in another Fig. 27 way; the chara-segments are successively (tan ϕ tan δ₁ — tan ϕ tan 0), (tan ϕ tan δ₂—tan ϕ tan δ₁), (tan ϕ tan ω— tan ϕ tan δ₂), (tan ϕ tan δ₂—tan ϕ tan ω), tan ϕ tan δ₁— tan ϕ tan δ₂ (tan ϕ tan 0—tan ϕ tan δ₁), (tan ϕ tan δ₁— tan ϕ tan 0), (tan ϕ tan δ₂—tan ϕ tan δ₁), (tan ϕ tan ω— tan ϕ tan δ₂), (tan ϕ tan δ₂—tan ϕ tan ω) (tan ϕ tan δ₁— tan ϕ tan δ₂), (tan ϕ tan 0—tan ϕ tan δ₁) where δ₁ = declination of 30°, and δ₂ that of 60°. These are there as found before 56, 134, 168, —168, —134, —56 upto Kanyā. But the remaining, though apparently are 56, 134, 168, —168, —134, —56 must be taken with a reverse sign because the rising point of the ecliptic will be to the south of the east point and δ will be negative from 180° to 360° longi- tude. Hence the chara Segments are 56, 134, 168, —168, —134, —56, —56, —134, --168, 168, 134, 56 so that from Tulā onwards they are in the reverse order as Bhāskara

100 has stated. Bhāskara's statement yet has another mean- ing. The ecliptic from 180° to 360 being in the reverse as can be seen in fig. 27, the time by which a particular Rāsi rises, is equal to the time by which its seventh Rāsi or diametrically opposite Rāsi sets. It is important to note that the rising time of a particular Rāsi is not equal to its setting time as can be gauged from the rising times, for, then the rising times of Meṣa and Tulā must be equal which is not the case. The word विलोमसंस्थाः has this meaning as well, rising and setting being reverse directions. Incidentally there is what is called a वृद्धकारिका handed down i.e. a traditional statement which men- tions "मीनमेषौ चतुर्नड्यः सार्धं चत्वारिगोघटौ etc." This clearly pertains to the local rising times of the Sāyana Rāsis at a place of latitude 17°-45' i.e. in between Rajamundry and Vizianagaram, as in this latitude the chara Segments in asus will be 230, 184, 73 which give rise to such rising times. We next find what are called Nirayana Swōdayas i.e. the rising times for the locality of the Nirayana Rāsis i.e. the Rāsis from the zero-point of the Hindu zodiac. We shall obtain these magnitudes for Rajahmundry whose latitude is 17°-2'. We shall however take it as 17°; also we shall take the Ayanāṁsas to be 21° at present following Varāhamihira. This is one of the contexts where the knowledge of Ayanāṁsas is essential. The method is the same as followed before; only, we have to find the decli- nations, right ascensions, charas and therefrom the rising times of arcs of magnitude 21°, 51°, 81°......351°. Sub- tracting the preceding rising times from the following we have successively the required rising times. We shall give them under the following table.

200

201 In modern terms, the rising times of Rasis could be found by solving the spherical triangle γEA (fig. 28) using the Inner side inner angle formula namely . [Fig. 28] Fig. 28 cos 𝑥 ✕ cos 𝜔 = sin 𝑥 cot 𝜆 + sin 𝜔 tan 𝜙 I sin 𝑥 cot 𝜆 − cos 𝑥 cos 𝜔 = − sin 𝜔 tan 𝜙 putting sin 𝑥 = t, this reduces to t cot 𝜆 − √(1 − t²) cos 𝜔 = − sin 𝜔 tan 𝜙 or (t cot 𝜆 + sin 𝜔 tan 𝜙)² = (1 − t²) cos² 𝜔 ∴ t² (cos² 𝜔 + cot² 𝜆) + 2t ✕ sin 𝜔 tan 𝜙 cot 𝜆 − cos² 𝜔 = 0 ∴ t = −sin 𝜔 tan 𝜙 cos 𝜆 ± √[sin² 𝜔 tan² 𝜙 cot² 𝜆 + cos² 𝜔 (cos² 𝜔 + cos² 𝜆)] ────────────────────────────────────────────────────────────────────────── cos² 𝜔 + cot² 𝜆 Putting successively 𝜆 = 30°, 60°...360° and ignoring the negative sign of the radical we have the sines of rising times of arcs of the ecliptic of 30°, 60° etc. Subtracting 26

202 the preceding from the succeeding, we have successively the rising times. Thus the rising time of Mesha is √3 sin² ω tan² ϕ + cos² ω (cos² ω + 3 − √3 sin ω tan ϕ II ──────────────────────────────────────────────────────── 3 + cos² ω That of Meṣa and Vṛṣabha put together, the rising time is . √sin² ω tan² ϕ + 3 cos² ω (cos² ω + ⅓) − sin ω tan ϕ III ─────────────────────────────────────────────────── √3 (cos² ω + ⅓) The combined rising time of the 3 Rasis Meṣa, Vṛṣa- bha and Mithuna is from I using tables cos⁻¹ (tan ω tan ϕ) = 5046 asus which exactly accords with what we have found previously namely 1505+1659+1882 = 5046. Also putting in I λ = 180°, we have sin x = 0 or x = 180° = 10800 asus ; subtracting 5046, we have the combined times of rising of Karkataka, Simha ahd Kavya to be 5754 asus as we have had. Putting again λ = 270, we have cos°⁻¹ (tan ω tan ϕ) = 2π − 5046 whioh signifies that the sum of the rising times of the last three rasis is the same as that of the first three which again means that the sum of the rising times of Tula to Dhanus is equal to the sum of the rising times of Karkaṭaka, Simha and Kanya establish- ing Bhaskara's statement “विलोमसंस्थाः”. Verse 60. Computations of Lagna, Udayāntara and the like from the rising times of big arcs of the ecliptic like Rasis will be approximate, whereas one desirous of greater approximation has to find the same from the rising times of smaller arcs likes Horas and Dṛkkāṇas, so as to be more correct. Comm. Rasis divided into halves are called horas and if divided into one-third parts are called Dṛkkāṇas. The meaning of the verse is that if after having found the rising time of a particular Rāśi say Meṣa, we say that one-third of that rising time is that of one-third of that Rāśi, we will be making only an appro-

203 ximate statement just like saying that ‘Since 12 Rasis rise in the course of a sidereal day, so each rasi rises in 1800 asus’ which is far from truth being based on a crude rule of three. So, Bhāskara says, Acharyas like Aryabhata insisted on finding the rising times of Dṛkkāṇas, in as much, as while computing the Lagna or the point of inter- section of the Ecliptic with the horizon, we will be nearer the truth by using the rising times of smaller arcs like Dṛkkāṇas than broadly using those of Rasis. Verse 61. Bhujāntara correction. The equation of centre of the Sun multiplied by the equatorial rising time of the Rāsi occupied by the Sun, and divided by 1800, and then again multiplied by the true daily motion of a planet and divided by the number of asus in a day, is the correction in the planet positive or negative according as the equation of centre of the Sun is positive or negative. Comm. This Bhujāntara correction arising out of the Sun’s equation of centre is prescribed even for the Sun, as well as for the other planets. The planets are originally computed for the rising time of the Mean Sun, whereas we are interested to know the positions at the True Sun- rise. The position of the Sun also is originally computed for his mean rise and is therefore to be rectified to get his position at his true rise. So even the Sun is not exempt from the correction. It will be noted here that as the equation of centre pertains to the eccentricity of the Sun’s orbit, this correction of Bhujāntara is a correction for the so-called modern ‘Equation of time due to eccentricity’. In other words, the Sun’s equation of centre converted into time is exactly what is called the Equation of time due to eccentricity. The formula prescribed for the correction is as follows. Suppose the Sun is in a particular Rāsi which rises at the equator in x asus. Let the equation of centre of the Sun be

204 E minutes of arc. Then the equatorial rising time of E is (E × x) / 1800 asus because there are 1800' in a Rasi. We have to provide a correction in the planetary position including that of the Sun for this time. If the planet goes Y' during 21659 asus of a day, what arc is covered by the planet or the Sun in (E × x) / 1800 asus? The result is (E × x) / 1800 × y / 21659 minutes of arc. This correction is positive if the equation of centre of the Sun is positive, for, we want the planetary position for a latter time than Mean Sunrise, since the positive equation of centre advances the True Sun over the Mean. This correction will be appreciable only in the case of the Moon having a rapid motion. Verses 62, 63. The correction known as Udayāntara. The difference in minutes of arc in the longitude of the Sāyana mean Sun and the asus in his Right ascension, multiplied by the daily motion of the planet and divided by 21659 is the result to be added to or subtracted from the planet's longitude according as the asus in the Sun's Right ascension are greater or less than the minutes of arc of his longitude. This is what is called Udayāntara correction in the planetary position. Comm. (1) We have seen that the Bhujāntara is a correction in the mean planetary position due to the Equation of time in Eccentricity. (2) This Udayāntara is a correction in the same due to the Equation of time in obliquity. (3) Some have misconstrued that this Udayāntara correction is the Equation of time in the obliquity itself, whereas it is a correction to be effected in the planetary position due to the Equation of time due to obliquity.

205 (4) The maximum equation of centre in the Sun has a magnitude = 13 2°/3 × 1/2π = 41/3 × 7/44 = 287/132 = 2° — 10' Converting this into time at the rate of 15° per hour (since in one hour diurnal rotation of the earth is equal to 15°) we have 13/6 × 4 minutes = 26/3 = 8'—40''. The maxi- mum equation of centre according to modern astronomy is 2e expressed in radiaus where e = 1/60 (.0167339) = 2 × 1/60 radiaus = 2 × 1/60 × (180 × 7)/22 degrees = 21°/11 = 1°—54' — 33'. As such the max. Equation of time due to eccentricity is 21/11 × 4' = 84'/11 = 7'—38''. The small difference in the two values arises out of the difference in the max. equ- ations of centre. Any way it is clear that, in as much as the Bhujāntara correction is necessitated on account of the equation of centre in the Sun, to obtain the planetary positions computed for the mean Sunrise at the time of True Sunrise, the Bhujāntara correction is a correction in the planetary position, on account of the equation of time due to eccentricity. The max. correction to be effected in even the quick moving Moon amounts to (8 × 790)/(24 × 60) = 79/18 = 4'—23.3''. (5) We have said in the translation of the verse. ‘The asus in the Right ascention of the mean Sun’, where what exactly is stated by Bhāskara is, the time of rising of the small arc covered by the Sun in the particular Sāyana Rasi in which the Mean Sun is, (भुक्तासवः) together with the rising times of the previous Sāyana Rasis covered by the Sun. The meaning is therefore the rising time of an arc of the ecliptic equal to the Sāyana longitude of the ecliptic which is measured by his Right ascension at the

rate of 15° per hour or 6° per nadi or 10 Vinādis per degree or 60 asus per 60 minutes of arc or as many asus as there are minutes of arc in the right ascension of the mean Sun. So, what is stated by Bhāskara is the difference of the minutes of arc in the mean longitude of the Sun and the minutes of arc in his right ascension ie. (l—a) expressed in minutes. We know that the total equation of time arising out of unequal motion in the true longitude of the Sun ie. ☉ in comparision with the equal motion in his right ascension ie. a is measured by ☉—a which may be expressed as (☉—l) + (l—a) where l is his mean longi- tude, ☉—l = Equation of centre and so the time expressed by ☉—l is the equation of time due to obliquity. The difference l—a arises out of the obliquity of the ecliptic and so the time expressed by l—a is the equation of time due to obliquity. We have the modern formula Cos ω = tan a / tan l so that (1 — cos ω) / (1 + cos ω) = (tan a — tan l) / (tan a + tan l) = (sin a — l) / (sin a + l) ∴ Sin (a — l) = tan² ω/2 sin (a + l). As a is very nearly equal to l, we could write, when expressed in radiaus a—l = tan² ω/2 sin 2l. Thus the maximum difference between a and l arises when 2l = 90° ie. l = 45 degrees ie. at the mid-point of the first quadrant; the mini- mum difference is when 2l = 270 ie. l = 135 ie. at the middle point of the 2nd quadrant. Also the numerical magnitudes of the max. as well as the minimum value are each tan² ω/2 ie. they are equal. Since this is expressed in radiaus, converting into time the numerical value of the max. and minimum equation of time due to obliquity is 9.87′. Thus we can write l—a = 9.87′ sin 2l. Again where l = 225, 2l = 450 so that l—a will have a positive max.; and again when l = 315, 2l = 630 so that l—a will have a negative maximum value. Also when l has values 0, 90, 180, 270 it is zero. Thus the equation of time due to obliquity is zero at ♈, ie. the vernal equinox; +ve in the first quadrant

. 207 increasing from zero to 45 degrees and then decreasing from 45° to 90°; assuming the value zero at 90°, then negative in the second quadrant negatively increasing from zero to a maximum as l increases from 90° to 135°, and then negatively decreasing from the maximum value to zero at the end of the second quadrant ; again behaving in the 3rd quadrant as in the first and in the fourth as in the second. (6) It was stated by Mr. Mazumdar in his introduction to the Siddhānta S'ekhara of S'ripati published by the Calcutta University as well as by pandit Babuaji Misra, the editor thereof that this Udayāntara correction was first mentioned by S'ripati, and it meant equation of time due to obliquity. In fact S'ripati states (Verse I ch. eleven) “ अन्त्यभ्रमेण गुणिता रविबाहुजीवाऽभीष्टभ्रमेण विहृता, फलकार्मुकेण, बाहोः कलासु रहितास्ववशेषकं ते, यातासवो युग्मयुजोः पदयोः धनर्णम् ” l expressed in minutes — H sin⁻¹ (H sin ☉ H cos ω / H cos δ) express- ed in asus = what are called elapsed asus and are +ve, +ve, +ve and —ve in the successive quadrants. We saw before that H sin α = (H sin ☉ H cos ω / H cos δ) so that S'ripati meant l - α, the former expressed in minutes of arc and the latter in asus, or what is the same (l—α) both expressed in minutes or asus. These give the gain of l over α. Immediately after this verse S'ripati goes to a different topic, and never mentions (as understood from the printed text) any further details as to what is to be done with these Yātāsus. Since Bhāskara says explicitly that for these Yātāsus, the planets are to be corrected, we may surmise that there should have been in S'ripati's text also another verse detailing the usage of those asus. (7) We shall now attend to what Bhāskara gives by way of explanation of the verses in question, The Ahar-

208 gaṇa found by us is what is called Madhyama Sāvana- ahargaṇa or number of days according to mean solar reckoning, a reckoning made on the basis of taking uniform motion of the Sun in right ascension. The planets computed for the Sun-rise on this basis, are to be corrected for the true Sun-rise which goes according to the measure of ☉ ie. the true longitude of the Sun. But in the name of Bhujāntara we have-effected the correction for ☉ - l ; so what remains is to effect correction for l - a. This correction is known as Udayāntara. In other words, since we could not compute the Sphuṭa Sāvana-ahargaṇa, when we deem that x days have elapsed according to the mean solar reckoning from, the epoch upto the mean Sun-rise x ± f might have elapsed according to Sphuṭa Sāvana-ahargaṇa where f is a fraction. A correction is to be effected for this fraction f of a day and it is of the form ☉ - a = ☉ - l + l - a. In the beginning Bhāskara said that mean planets are had being computed out of the mean solar ahargaṇa, at the time when the mean Sun is about the eastern horizon of the equator. Why did he say ‘about the horizon’? It is because the mean solar ahargaṇa indicates a Sun-rise on the basis of equal motion in right ascension whereas on the basis of equal motion in l, he would be about the horizon. Had we been able to compute Sphuṭa-Sāvana-ahargaṇa, we could have got direct the planetary position when the true Sun is on the horizon. The correction for the difference in ☉ and l having been attended to through Bhujāntara, we are now to attend to the correction for the difference l - a. Verse 64. Another way of looking at the same. Had we obtained the Ahargaṇa in terms of the local risings of the Rasis in the place of the equatorial, and computed the planetary positions for the local true Sun- rise obtained that way, we would have done the three

209 corrections namely Bhujāntara, Chara and the Udayāntara as well. Comm. The correction of chara arises out of the difference between the equatorial risings and local risings of the mean Sun. The local mean Sun pertaing again to uniform motion along the celestial equator Udayāntara has to be effected for l—a, ie. for the local mean Sun on the ecliptic. Thus we have obtained the planetary posi- tions for the local mean Sun-rise and to obtain them for the True Sun-rise, Bhujāntara is to be effected. Verse 65. An alternative method of effecting the Udayāntara correction. Double the H sine of the Sāyana longitude of the Sun derived out of the smaller H sin-table, being multiplied by the daily motion of the planet and divided by 270 and the result in seconds of arc is to be corrected in the plane- tary position positive or negative according as the Sun is in the even or odd quadrants. Comm. (1) In symbols the correction indicated is (m H sin 2l) / 270 where m is the planet's mean daily motion in minutes of arc, l the longitude of the mean Sun, and H sine is taken where the radius = 120. In the commen- tary under the verse, Bhāskara adds 'Each quadrant of the eclipse rises (at the equator) in ¼th of a day but each Rāśi does not rise in 1/12 of a day. Since this Udayāntara correction vanishes when the Sun is at the ends of quadrants it must be construed that this correction increases positively or negatively from the beginning of the quadrant to the middle and decreases from the middle to the end of a quadrant. Saying that a particular Rāśi rises at the equator in n nādikās is only an approximate statement since the Rāśi does not rise uniformly. That is why astronomers like Aryabhata 27

stipulated finding the risings of smaller arcs like horas and Dṛkkāṇas. Find (H sin l × H cos ω) / (H cos δ) ie. H sin α. Take the asus in the arc of this H sine ie. find α in minutes. Then l - α gives the number of asus for which the correct- ion in the planetary position is to be effected, for, by these asus the True Sun-rise is accelerated or belated. In the middle of a quadrant these asus will be a little above 26 Vinadis. To obtain them at any point of the quadrant, take H sin 2l as the argument so that the maximum correct- ion will be had at the middle of the quadrant by this argument. Then the rule of three is ‘ If by 120 as radius, we have 26 Vinadis, what shall we have for H sin 2l ?’ The result is (H sin 2l × 26) / 120 = (H sin 2l) / 4½ approximately. Then another rule of three. “ If by 60 Vinadis we have m' of the planetary motion, where the daily motion of the planet is m°, what shall we have for (H sin 2l) / 4½ ?” The result is (H sin 2l × m) / (60 × 4½) minutes = (H sin 2l × m) / 270 . Then the sign of the correction is clear. (2) We shall now prove Bhāskara’s statement that at the middle of the quadrant, the correction is 26 Vinadis. We saw above that l - α = tan² ω/2 H sin 2l. It is really creditable on the part of Bhāskara to have seen by intuit- ion that the argument is H sin 2l. The maximum correction is therefore tan² ω/2 expressed in asus or minu- tes of arc. Let x = tan² ω/2 log x = 2 log tan ω/2. Take ω = 24 Then log x = 2 × 1̄.3275 = 2̄.6550 ∴ x = ·04519 radian = 155 minutes of arc ie. 155 asus = 155 / 6 = 26 apply.

211 (3) Taking the case of the Moon, the max correct- ion amounts to (790 × 120) / 270 = 5' — 51''. Verses 66, 67. The computation of Tithi, Nakṣatra and Yoga. The elongation of the Moon ie. the excess of the Moon's longitude over that of the Sun in degrees. (In case the former is smaller, add 360° to it and subtract Sun's longitude) being divided by 12 and 6, the quotients represent the elapsed tithis and Karaṇas. If the elapsed Karanas are K, count K — 1 beginning from Bava to get the current Karaṇa and count from Śakuni to get the current one beginning from the mid-moment of Kṛṣṇachaturdasī. Take again the planetary position or that of the Moon in particular as well as the sum of the longitudes of the Sun and the Moon both expressed in minutes of arc and divided by 800. The first quotient gives the elapsed stars ie. the Stars covered by the planet or the Moon; whereas the second quotient gives the elapsed Karaṇas. Then take the remainders in seconds and divide by the respective daily motions in minutes i.e. in the case of the planets or the Moon divide by their respective motions and in the case of yogas divide by the sum of the motions of the Sun and the Moon. Then the results give the times in nādīs as to how much the next nakṣatra or yoga have elapsed. If it be required to find as to how long the next nakṣatra or yoga will last, substract the remainder from 800, and divide by the daily motions in minutes as mentioned above. The results give as to how many nādīs beginning from the morning concerned, the next nakṣatra or yoga will last. Comm. (1) A lunation i.e. the time from the moment of New Moon to the next New Moon is divided into 30 parts called Tithis. Their names are pratipat, Dwitiyā etc. upto the 15th purnima or full Moon and again

212 pratipat, Dwitīyā etc. upto the 30th i.e. Amāvāsyā or New Moon. Thus pratipat starts when ☾ = ☉ i.e. when the longitude of the Moon is equal to that of the Sun i.e. from the moment of New Moon when the Moon is in conjunction with the Sun. अमा सह = वसतः सूर्य्याचन्द्रमसावस्या मित्यमावस्या i.e. Amāvāsyā is that point of time when the Sun and the Moon are together. Pratipat lasts till ☾ = ☉ + 12°; then Dwitīyā begins and lasts till ☾ = ☉ + 24; Thus purnima begins from the moment when ☾ = ☉ + 168 and lasts till ☾ = ☉ + 180° and Amāvasyā begins when ☾ = ☉ + 348 and lasts till ☾ is again equal to ☉ Thus a tithi is the time that is taken by the Moon to overcome the Sun by 12° beginning from the moment of New Moon. In other words the tithi is a measure of the phase of the Moon with a particular convention. (2) In modern astronomy the phase of the Moon or that of a planet is measured by the formula (1 + cos EPS) / 2 where P is the planet or the Moon. Since ES is almost Fig. 29 parallel to PS (Fig. 29) ÊPS may be taken to be nearly equal to P̂ES which is the elogation of the planet or the Moon so that phase = (1 - cos EPS) / 2 approximately. So

213 in modern astronomy also the phase is measured through the elongation. But the maximum phase in modern astronomy is taken to be unity as could be seen from the formula, for, when EPŜ = 180°, the phase = 1. Thus the phase multiplied by 15 gives approximately the tithi. We shall have occasion to deal with this topic later. (3) Suppose ☾ − ☉ = E°. The number of the tithis elapsed is equal to the quotient in E / 12. Take the remainder r°. It means during the current tithi which has a duration of 12°, r° have elapsed. Let the daily motions of the Moon and the Sun be m and s so that the Moon overtakes the Sun by m − s (expressed in minutes for convenience) during 60 nādīs. The elapsed time of the current tithi in nādīs is given by (r × 60 × 60) / (m − s). But r × 60 × 60 = Seconds of arc of the remainder so that it is said “गतैष्यविलिप्तिकाः”. If it be required to find how many nādīs the next tithi lasts, (12−r)° or (12−r) × 60′ is to be gained by the Moon over the Sun. So the time in Nādīs = ((12 − r) × 60 × 60) / (m − s) = एष्यविलिप्तिकाः / (m − s) . (4) A lunation is again divided into 60 Karaṇas and so 6° of increase in elongation correspond to a Karaṇa. These Karaṇas are eleven in number, out of which 4 are fixed to occur during the latter half of Kṛṣṇa Chautùrdasī, during the two halves of Amāvāsyā and the first half of the Sukla pratipat. Their names are Sakuni, Chatuṣpāt, Nāgava, and Kimstughna. So there remain 56 halves of tithis during the lunation during which the remaining seven Karaṇas, Known as Bava, Bālava etc. rotate eight times. Thus the first half of the duration of any tithi is covered by one Karaṇa and the latter by another. The computations of Karaṇas proceeds as with respect to tithis