VI.10 VI. VĀS.-PAUL. SIDDHĀNTA – LUNAR ECLIPSE 167 [चित्र: Fig. VI. 6 - Sun, Earth, Moon, Moon's orbit] Fig. VI. 6 In its nature this phenomenon is similar to the Sun apparently rising earlier and setting later, and appearing red at both times. As for the distinction between red and brown, it can be seen from the figure that there is great illumination at a greater distance, and, so, when nearer, there is less red, which gives a brown colour as mentioned by the author. The redness may vary by other causes, like the dust or water vapour in the atmosphere, and it goes to the credit of the ancients that distance was distinguished as one cause, affecting the redness. When the Moon is partially eclipsed, the glare of the illuminated part dims more or less the red- ness of the eclipsed part, so that it looks almost dark like a rain cloud. Near sunset or sunrise, the red glow from the Moon has again to pass a long distance through the atmosphere, and get filtered out, so that the colour becomes smoky. Verse 8-A is given only by Bhaṭṭotpala, the text manuscripts omitting it. The original of VM must have contained this verse, for it supplies several lacunae. The word evam in verse 9 requires a pre- vious verse mentioning colour. Omission of this verse has compelled TS to emend sarvagrāsiny evam into sarvagrāse pītam. The rule in verse 10, giving when the Moon is at ucca and nīca has a purpose only with 8-A which requires the information. If TS had verse 8-A before them, they would not have misunderstood 10, and declared that it is something pertaining to astrology. It also instructs us how to find the times of the first and last contacts, and when the middle occurs, which instruction we assumed before, in working the examples. NP rightly include this verse in brackets and give it the number 9, the further verses being numbered from 10. We have said that verse 10 gives a rule to find when the Moon is far and when it is near. We shall explain how: The rule says that when the result got by the rule is nine rāśis, the Moon is at the greatest distance, i.e. at ucca, and when the result obtained is three rāśis, it is nearest, i.e. at nīca, as we have explained. Therefore if the rule gives about nine rāśis for the Moon at ucca, i.e. when the Moon is replaced by ucca, then it must be correct. We shall show that it is so. The rule is: Moon − 228 R = ucca − 228 R (where R is the Head of Rāhu at any full-moon). At the full moon just preceding the Epoch, by II.3, taking the reading vasumuninava etc. it can be cal- culated that according to Vāsiṣṭha-Pauliśa, the ucca is rā. 8-14-21.4. (The Siddhānta does not give the ucca direct, but we can find it by the relation, Moon − Moon's kendra = ucca). Every synodic month the ucca increases by 3° 17′.3 according to this Siddhānta. Therefore after m synodic months, the ucca is rā. 8-14-21.4 + m × 3° 17′.3. The corrected Rāhu, R, at the full moon before Epoch is rā. 7-26-45.2. It decreases by 1° 33′.88 every synodic month. Therefore, after m synodic months, R = rā. 7-26-45.2 − m × 1° 33′.88 ∴ ucca − 228 R = rā. 8-14-21.4 + m × 3° 17′.3 − 228 × (rā. 7-26-45.2 − m × 1° 33′.88) = rā. 8-14-21.4 − 228 × rā. 7-26-45.2 + m (3° 17′.2 + 228 × 1° 33′.88) = rā. 8-14-21.4 − rā. 11-9-45.6 + m (3° 17′.3 + 356° 43′.9)

168 PAÑCASIDDHĀNTIKĀ VI.13 = rā. 9-4-35.8 + m × 1'.2 = practically nine rāśis, for a long time after or before Epoch. Nineteen years before Epoch, this would have been exactly nine rāśis. A small difference in Rāhu, (if it is 1'.2 more) would make it nine rāśis even at the taken time. It must be noted that one or even two rāśis either way will not matter in our context, of redness at one end and brownness at the other, for the difference between redness and brownness itself is slight. Only after two or three thousand synodic months will there be perceptible difference, and the rule cease to hold good. From the proof of the rule it will be seen that our emendation of triyamadviguṇam into dhīdviyama- guṇam is necessary. triyamadviguṇam had perhaps been wrongly written by some scribe who had the 'saros', (consisting of 223 lunations) in his mind. As antithetical to kriyādikanyāntago nīcaḥ, we have emended the meaningless group of letters, epikleśoyam ucca into jūketthageyam uccaḥ. Neither TS nor NP seem to have understood the significance of this verse. TS observes on it: "A stanza of doubtful import, see the Sanskrit commentary" (Tr., p.45) and the Sanskrit commentary suspects it to be of astrological import (com., p.34). NP gives an incorrect translation and says "The synodic months in an 18 year eclipse cycle is 223, but the role of this number in the present context remains obscure to me" (Pt. II, p.55) [ग्रहणपरिलेखा:] सप्तदशाष्टत्रिंशत् तद्द्वयलिप्ता (युतोने) सूत्रेण । शशि (राहु) स्थितिवृत्तान्येक (स्थानानि चाऽऽलिख्य ॥ ११ ॥ प्रोक्ता (शशाङ्क) लङ्का-पूर्वाऽप (रायाश्च) पार्श्वयोश्चाऽपि । आयामिन्यो रेखास्त्रयोदश समान्तराः कार्याः ॥ १२ ॥ चन्द्रच्छेदकमेतद् व्याख्यागम्यं समासतोऽभिहितम् । ग्रासविमर्दस्थितयः संस्थानेनाऽत्र दृश्यन्ते ॥ १३ ॥ Diagrammatic representation 11. Draw three concentric circles with radii 17, 38 + 17 (= 55), and 38 - 17 (= 21), minutes of arc. These circles relate to the Moon, the duration and the obscuration, respectively. (Drawing the part of the Moon's orbit forming the path of the Moon), mark the points (of first and last contacts) and also those of inversion and emergence if any). 12. Draw the diameter (making an angle equal to the Valana given in verses 7-8), with the ecliptic which, (according to this Siddhānta), is east-west with reference to the equator. (This diameter shows the east-west of the place). (As shown in fig. 5c) draw thirteen equally spaced lines parallel to this east-west diameter. (The directions of the points of contact etc. are given by this figure).

VI.13 VI. VĀS.-PAUL. SIDDHĀNTA – LUNAR ECLIPSE 169 13. The graphical representation of the lunar eclipse has here been described briefly, and can be understood properly only by explanation (followed by demonstration). From this, the total duration, the total obscuration, the magnitude, etc. can be found by inspection. The representation given by the author can do duty for all the figures used by us to explain verses 3, 5, 6 and 7, the centre of the concentric circles being the centre of the Shadow in each. An impor- tant difference is that in the previous illustrations, the orbs, of the Moon and the Shadow were shown separately, while here they are replaced by one circle for each of duration (radius = 38' + 17') and totality, (radius = 38' - 17'), the Moon being reduced to a point coinciding with its centre. As the points of contact etc. showing the direction cannot be marked on the point-Moon, another circle is drawn to represent the Moon, with the same centre. As the directions of the points on the two circles are diametrically opposite to the directions of the same points with reference to the Moon, the points can be marked on the Moon-circle by the intersection of the diameter on its oppo- site half. An examination of fig. 7 will show this, and an example will make everything clear. Example 3. (Note: This example is intended only as an illustration). The latitude of a place is 20°. The full moon occurs 4 nāḍīs after sunset. The Moon at that time is rā. 5-15-0, and the Sun, rā. 11-15-0. The uncorrected Rāhu (Head) at full moon is rā. 5-6-36, from which the latitude is 35'N. The full moon is 11 nāḍīs before mid-night. The daily motion of (Moon - Sun) = 780'. Find the times of first contact etc., and verify by a graphical representation. (i) The corrected Rāhu = rā. 5-6-36 - 1° 36' = rā. 5-5-0. Moon ~ Rāhu = rā. 5-15-0 - rā. 5-5-0 = 10°. This is less than 13°. ∴ there is a lunar eclipse. (ii) Minutes of duration = √(55² - lat²) = √(55² - 35²) = 42, minutes of arc. Now, 2 × 42 × 60 ÷ 780 = nā. 6-28 = Uncorrected duration. Difference between Moon and uncorrected Rāhu = rā. 5-15-0 ~ rā. 5-6-36 = 8° 24. Correction = 8° 24' × 5 = 42 vināḍīs. As Rāhu is less, deducting from uncorrected duration, the correct duration = nā. 5-46. Half this is nā. 2-53. Subtracting from the time of new moon, 4 nāḍīs, the first contact is at nā. 1-7 after sunset. Adding, the last contact is at nā. 6-53. (iii) Moon ~ cor. Rāhu = 10°. As this is greater than 5°, there is no total phase. (iv) The first contact is nā. 15-0 - nā. 1-7 = nā. 13-53 before midnight, i.e. the Moon is 83° east of the meridian, 11a. A.B. दशाष्टा 12a. A. प्रोक्तायां सकलङ्का; B. प्रोक्ता यो सवूलांका (B2. बू) B1.2. त्रिशतद्वय; B3. त्रिंशद्वय D. प्रोक्ताया [मं] शकलका [त्] b. A. मतेन सूत्रेण; B.C.D. मितेन सूत्रेण b. A.B.D. पूर्वापरयोश्च (A om र); C. पूर्वापरायाश्च c. A.B. शशिना बहुस्थिति (A2. नां) d. A. समात्ताराः; B. समां ताराः d. A. नि एक छेना निष्पालेख्य; B. नि एकछत्रो निवासलेख्य; 13a. A. छेदक; B. षेदक; C.D. छेद्यक D. न्येकस्थानि वा संलेख्य d. B1. मंस्थ्यानेनात्र (B2.3. सं०)

170 PAÑCASIDDHĀNTIKĀ VI.13 The Moon's circumference = 17' × 44/7 = 107'. The bending of the direction due to latitude = (107'/4) × 20 × 83 ÷ 8100 = 5½ minutes of arc. As Moon is east of the meridian, the equatorial east point is bent 5½' north of the east point of the place, i.e. the east point of the place is situated south by 5½ minutes-length. At the time of last contact, the Moon is 49° east of the meridian. The bending of the point of last contact at the western limb of the Moon =(107/4) × 20 × 49/8100 = 3.2', southward, as the Moon is east of the meridian, i.e. the west point of the place is 3½ minutes-length north of the equatorial west point. (v) We shall show all these graphically in Fig. 7. The latitude to be used in the figure is the corrected latitude = 10° × 380' ÷ 90 = 42'N. [चित्र: Fig. VI. 7] Moon's orbit, Eclipse circle M', M Moon's counterpart to mark directions West point as seen from place at last contact W, Ecliptic, E Total eclipse circle Position of last contact, Position of first contact East point as seen from place at first contact Fig. VI. 7 M, M', are the points of first and last contacts. MM' = duration in minutes of arc. MM' = 3".6 by measurement, = 3.6 × 20' = 72'. From this, the time of duration = 72 × 60 ÷ 780 = nā. 5-32.

VI. 14 VI. VĀS.-PAUL. SIDDHĀNTA – LUNAR ECLIPSE 171 See how close this is to the time calculated, viz. nā. 5-46. The Moon’s orbit does not touch the circle of totality. Therefore there is no total obscuration, as already found by calculation. The arc in the figure from the east point of place to the first point of contact gives the direction of the point at the beginning, and the arc from the west-point of place to the point of last contact gives the direction at the end. We have amended matena into yutona, nabahu into Rāhu, ekachanoni into ekasthānāni, yāmasabū into śaśāṅka and pūrvāparayośca into pūrvāparāyāśca as necessitated by the context and proximity of the lettering. [रविचन्द्रग्रहणयोर्भेदः] स्वे भूच्छायामिन्दुः स्पृशत्यतः स्पृश्यते न पश्चार्द्धे । भानुग्रहेऽर्कमिन्दुः प्राक् प्रग्रहणं रवेर्नाऽतः ॥ १४ ॥ Lunar and solar eclipses – Differences 14. In the lunar eclipse, the Moon, (moving eastward), contacts the earth’s shadow. Therefore the ‘first contact’ (occurs at the eastern limb of the Moon, and so) does not occur at the Moon’s western limb. In the solar eclipse, the Moon meets the Sun, and therefore, (the Sun being contacted as its western limb), the first contact does not occur at the eastern limb of the Sun. The Moon’s motion being more than thirteen times that of the Sun or the Shadow, (whose motion is the same as that of the Sun), it moves eastwards relative to the Sun or Shadow and con- tacts them at their western limb, and its own eastern part. As the lunar and solar eclipses are with reference to the Moon and the Sun, respectively, the first contacts are at the eastern and western limbs, respectively. It need not be mentioned that the last contacts are, respectively, on the western and eastern limbs. 14. Quoted Utpala on BS. 5.12 C. स्पृशति तथा स्पृशति दृश्यते पश्चात् । 14a. U. स्वं. B 1.2. भूयच्छायां; B3. भूच्छायां. A.B. ॰मिन्दु A. पश्चार्द्धं; B.D. पश्चार्धः b. AB.3. स्पृशतः स्पृ०; B 1. 2̂. स्पृश्यतः स्पृ० c. A.B1. भागनुग्रहे. A.B.C.D.U.मिन्दोः d. A. रवे [इति पञ्चसिद्धान्तिकायां वराहमिहिरविरचितायां चन्द्रग्रहणं नाम षष्ठोऽध्यायः ।]¹ Thus ends Chapter Six entitled ‘(Vāsiṣṭha-) Paulīśa Siddhānta: Lunar Eclipse’ in the Pañcasiddhāntikā composed by Varāhamihira

1. A.B.D. चन्द्रग्रहणं षष्ठोऽध्यायः C. इति चन्द्रग्रहणं नाम षष्ठोऽध्यायः 14

Chapter Seven (PAULIŚA-SIDDHĀNTA) – SOLAR ECLIPSE ७. सप्तमोऽध्यायः पौलिशसिद्धान्तः — रविग्रहणम् Introductory This chapter deals mainly with the solar eclipse according to the Pauliśa. But the last two verses giving the computation of the solar eclipse gives the lunar eclipse also. It is from this that we have to conclude that the lunar eclipse of Chap VI is that of the Vāsiṣṭha. The method of the Pauliśa for correcting the Moon’s latitude for parallax is peculiar. The correction is done on Rāhu, and thence carried to the latitude. Also, this is the earliest siddhānta to deal with the solar eclipse, and thus, with its peculiar method, historically important. [लम्बनम्] दिनमध्यमसंप्रा(प्या) यावत्यो नाडिका व्यतीता वा | ताभ्यः षड्गुणिताभ्यो ज्यात्रिंशांशस्तिथे(नामः) || १ || Parallax of longitude

1. Find the interval between mid-day and the time of new moon, in nāḍīs. Multiply this by 6. Degrees are got. Find its sine. Divide it by 30. The result is the parallax in nāḍīs to be deducted from the time of new moon if new moon is before mid-day, and to be added to the time of new moon, if after mid-day. The new moon corrected for parallax in longitude is obtained. Thus: i. nāḍīs of parallax = sine (interval in nāḍīs between mid-day and new moon × 6) ÷ 30. ii. Parallax corrected new moon = new moon ∓(i), minus for forenoon, and plus for afternoon. The rationale of parallax correction is as follows: A lunar or solar eclipse occurs when the Moon gets so close to the earth’s shadow or the Sun, that it enters the Shadow so as to be darkened by it, or hides the Sun from the observer’s view. Now, the Moon being darkened by the Shadow is practi- cally independent of the position of the observer on the earth. But the Moon hiding the Sun depends upon the observer’s position, owing to parallax. So parallax correction has to be done in the solar eclipse. The critical angular distance is the sum of the semi-diameters. The angular dis- tance between the Sun and the Moon is calculated from the longitudes of both and the latitude of 1a. A.B.C.D. संप्राप्ता c. B. गुणितान्यो b. B1.2. यावन्त्यो A. व्यतीता वत; B. व्यतीता वत्त | d. A.B. ज्यास्त्रिंशांशः A.B. तीथिनाम; C. तिथेर्नाम

VII.1 VII. PAULIŚA-SIDDHĀNTA — SOLAR ECLIPSE 173 the Moon. These being given with reference to the centre of the earth, the angular distance calcu- lated is as seen by an observer at the centre of the earth. But we want the distance as seen by an observer on the surface of the earth, and a correction has got to be made for this. This is correction for parallax or simply parallax. See Fig. 1-a. Z D | / | / | / | / | / S | / / P | / / .---. / / / | \ / M / | X / | | / \ / | | / \ / | C / \ / C Earth's centre \ / / P Position of observer \ / / Z Zenith '---' M Moon S Sun Fig. VII. 1-a. (Note: The figure is only diagrammatic and does not represent actual distances). PD = a line drawn parallel to CM. The observer at C sees the Moon along CM, and for him the Moon’s zenith distance (z.d.) is ZCM (= ZPD). But the observer at P sees the Moon along PM, and its z.d. for him is ZPM. This is equal to ZPD + DPM = ZCM + PMC, and PMC is the parallax correction to ZCM. It can be seen that the parallax correction for the Sun is PSC, and less than that for the Moon. It is actually about 2/27 of the parallax of the Moon, according to the Hindu Siddhāntas, its distance being about 27/2 times that of the Moon, according to them. (It must be noted that actually the Sun’s distance is about 390 times the Moon’s, and accordingly the Sun’s parallax is about 9", and practically negligible). The amount of parallax, PMC can be calculated trigonometrically thus: Sin PMC/PC = sin CPM/CM = sin (PCM + PMC)/CM = sin PCM/CM (∵ PMC is small). Sin PMC = sin CPM × PC/CM. Arc PMC × 120/57.3 = sin CPM × PC/CM (∵ PMC is small). Parallax of Moon in minutes = sin z.d. × (PC/CM) × 60 × 57.3

120 = sin z.d. × 28.65 × Earth’s radius/Moon’s distance. Similarly, the Sun’s parallax in minutes = sin z.d. × 28.65 × Earth’s distance ÷ Sun’s distance.

174 PAÑCASIDDHĀNTIKĀ VII.1 At a solar eclipse, z.d. is practically equal for the Moon and the Sun, and in measuring the angular distance between the Sun and the Moon relative parallax can be applied to the Moon, the Sun being supposed unaffected. We can write: Relative parallax = Moon's parallax −Sun's parallax = sin z.d. × 28.65 × earth's radius × (1/Moon's dis. − 1/Sun's dis.) When the Sun or Moon is at the horizon, sin z.d. is 120, and the relative parallax, (hereafter we shall call it merely parallax), called horizontal parallax, is a maximum, and this Siddhānta takes it as equal to 49'. ∴ Parallax = 49' sin z.d./120. Also, we can see from the fig. 1a that the Moon is depressed, away from the zenith by parallax, along the vertical circle, ZM, increasing the z.d. and that is why it is called lambanam, i.e. 'depression', in Sanskrit. This general parallax has to be resolved into two parts, correction to longitude, (p. long.) and cor- rection to latitude (p. lat.). This is shown in fig. 1b. Z Zenith N Nonagesimal (Lagna − 90°) O Orient ecliptic point (Lagna) A First point of Aries MAQ = ω Fig. VII. 1-b.

VII.4 VII. PAULIŚA-SIDDHĀNTA — SOLAR ECLIPSE 175 ZM = z.d., and MM′ is the general parallax. PM′ is the parallax in latitude, and MP, that in longitude ZQ. = latitude of the place, and ZN is the zenith distance of the nonagesimal (z.d.N.). We shall find an expression for PM′, the p.lat. p.lat = PM′ = MM′ × sin M′ MP/120 = 49′ sin ZM × sin ZM N/120² = 49′ sin ZN/120 = 49′ sin zdN/120, (from rt △ MNZ). Similarly for p.long, i.e. MP, p.long = MP = MM′ cos M′ MP/120 = 49′ sin ZM. cos ZMN/120² = 49′ cos ZN × sin MN/120² (from rt △ MNZ) = 49′ cos z.d.N × cos OM/120², OM being (lagnam − Moon). Clearly, this is positive when the Moon is east of the nonagesimal, and negative when west. It is p.long. that we are concerned with in this verse, and it must be given in terms of the hour- angle (h, or natāṃśa) cos z.d.N × cos OM = (i) cos w. cos Ø. sin h. sec δ [δ being the Moon’s declination] + (ii) [sin w × cos Moon’s longitude × {cos Ø × tan δ × sin (nāḍīs after sunrise or before sunset × 6°)/120²} − sin Ø/120]. The Siddhānta omits (ii) which is small in comparison with (i), sin Ø/120 being small in India. For the same reason, and as w and δ cannot exceed 24°, it takes cos w. cos Ø. sec δ as 120. ∴ p.long = 49′ sin h/120. As said before, this is positive, i.e. it increases the Moon’s longitude when h is east, and negative, i.e. it decreases, when h is west. Therefore the corrected time of new moon is earlier and subtractive in the forenoon, and later and additive in the afternoon. Now, 49′ of p-long, converted into time, using the mean relative daily motion of the Sun and the Moon, (731′.5), = 49′ × 60/731′.5 = 4 nāḍīs nearly. ∴ nāḍīs of parallax = 4 × sin h/120 = sin h/30, as given by the text. If (ii) is not neglected, the Moon being east or west of the nonagesimal will be the criterion for the subtraction and addition of the nāḍīkās. [नतिः] पञ्चघ्नात् ‘(त्रि)घना’ऽऽप्तादक्षान्मुखपु(च्छ)योर्धनर्णे तत् | (सशशि)चरणा(पमगुणा) धनर्णनाड्यो ‘[धृति]’विभक्ता ॥ २ ॥ उदगयने पूर्वार्धे धनमृणं दक्षिणे प्राच्याम् | पश्चाद्धनं तु याम्ये (दि)गुदग्गुणं वामतः पुच्छे ॥ ३ ॥ दिनयातशेषनाड्यश्चन्द्रा(पम)संगुणास्त्वशीतिहताः | (मे)षतुलादि ऋणधनं विपरीतं वामतः पुच्छे ॥ ४ ॥

176 PAÑCASIDDHĀNTIKĀ VII.4 Parallax in latitude 2. Multiply the degrees of latitude by 5 and divide by 27. Add or subtract the resulting degrees, respectively, to Rāhu's head or from Rāhu's tail, where the Moon is situated. (i) 3. Add three rāśis to the Moon, and find its declination in degrees. This multiplied by the nāḍīs of parallax (given by verse 1) and divided by 18, are to be added to the Head if it is forenoon and Uttarāyaṇa (i.e. the Sun is in its northward course), or afternoon and Dakṣiṇāyana. The degrees are to be sub- tracted from the Head if it is forenoon and Dakṣiṇāyana or afternoon and Uttarāyaṇa. For the Tail, the addition and subtraction should be inter- changed. (ii) 4. Take nāḍīs from sunrise to new moon if forenoon, the nāḍīs from new moon to sunset if afternoon. Multiply these by the degrees of the Moon's declination and divide by 80. The resulting degrees are to be added to the Head if the Moon's longitude is between 6 and 12 rāśis, and subtracted if bet- ween 0 and 6 rāśis. For the Tail, interchange the addition and subtraction. (iii) The corrections are: i. Ø° × 5 ÷ 27 ii. 'Degrees' × the nāḍīs of verse 1 ÷ 18, where 'Degrees' are to be got from sin 'degrees' = sin w sin (Moon + 90°)/120 iii. Degrees of Moon's declination × the time in nāḍīs from sunrise to (parallax corrected) new moon, or to sunset from parallax corrected new moon ÷ 80. The addition or subtraction is as instructed in the translation above. These rules follow from the formula derived already for parallax correction in latitude: p-lat = 49' sin z.d. N/120. 49' sin z.d. N/120 = sin w. cos Ø sin (AQ)/120³ − cos w. sin Ø/120², (from the two rt. △ s. Z NA and Z QA) = (i) − 49' cos w, sin Ø/120² + (ii) − 49' sin w cos Ø sin (Moon + 90°) × sin h. sec δ/120³ + (iii)

• 49' cos w. cos Ø tan δ. cos h/120³, (h being the hour angle of the Moon at parallax-corrected new moon). 2a. A.B1. पञ्चघ्नलि०. D. यमाप्ता b. B1. दक्षन्; D. [क्षेपे] क्षे मुख . A. पुच्छयोः; B1.2. पृष्ठयोः c. A. यक्षाधनं A.B. धनर्णं तन्न |; D. धनर्णः d. A. om दिग्; B1. दृग्गुणं; B3. दृगुड्गुणं; C.D. om दिग् c. A.B.C. राशिचरणा०; D. तद्राशिचरणा०; A. पुछे; B. पृष्ठे A.B.C.D. ०यनगुणं 4a. B. शेषं d. A.B.C. धनमृणनाड्यो (B1.3. ०ताड्यो). b. A.C.D. चन्द्रायन; B. चन्द्रानयन A. त्रशीति; A.B.C.D. दिक्० B. श्काशीति B. विभक्ता B1. ऋताः; B2.3. क्षताः 3a. B. धनं मृणं c. A.B. शेष. D. तुलाघ्णं घनं b. A. दिणे; B. दक्षिणं d. B. विपरीतं. A.B. पुछे

VII.4 VII. PAULIŚA-SIDDHĀNTA — SOLAR ECLIPSE 177 To secure correction 49' in latitude, a correction = 90° × 49/380 = 11°.6, must be applied to Moon ~ Rāhu which can be done by applying it to Rāhu, as the Siddhānta does. The purpose of this replacement is to extend the method of computing the lunar eclipse using Moon ~ Rāhu to the solar eclipse also. So 49' has to be replaced by 11°.6 in the rules, (i), (ii) and (iii). We shall take the rules one by one and derive the Siddhānta rules. (i) = – 11°.6 cos ω sin Ø/120² = – 11°.6 × 109.6 × (degrees of latitude × 120/57.3)/120², (∵ when latitude is not much, its sine ∝ degrees) = – 5 × degrees of latitude ÷ 27, as given. As the Siddhānta has only the north latitudes in view, this part of p.lat is always negative. There- fore Moon's north-latitude must become less, and south latitude more, by the correction. This can be done by increasing Rāhu-Head, i.e. by adding the correction to the Head, and by decreasing Rāhu's Tail, i.e. by subtracting from the Tail, as instructed. (ii) = – 11°.6 sin ω cos. Ø sin (Moon + 90°) sin h. sec δ /120³ = – 11°.6 sin ω . sin (Moon + 90°) × nāḍīs of correction to new moon ÷ (cos ω × 120 × 4), (∵ sin h = 120² × nāḍīs of correction × cos δ ÷ (cos ω × cos Ø × 4), in verse 1). = – 11°.6 × sin declination of point (= Moon + 3 rāśis) × nāḍīs of correction ÷ (109.6 × 4) = – 11°.6 × {Degrees of declination of point (= Moon + 3 rāśis) × 120/57.3} × nāḍīs of correc- tion ÷ (109.6 × 4) = – 'Degrees' of declination of point (= Moon + 3 rāśis) × nāḍīs of correction to new moon ÷ 18. The 'Degrees' are north, i.e. positive for Moon's Uttarāyaṇa, and negative for Dakṣiṇāyana. The nāḍīs stand for positive p-long in the forenoon, and negative p-long in the afternoon as already shown. Thus, for Uttarāyaṇa and forenoon, the p-lat is negative, and so the result of (ii) is to be added to Head, and subtracted from Tail. Clearly, it is the same for Dakṣiṇāyana and afternoon, as then also p-lat is negative. If Uttarāyaṇa and afternoon, or Dakṣiṇāyana and forenoon, p-lat. becomes positive, and so the correction is subtractive to Head, and additive to Tail, as instructed. (iii) = + 11°.6 cos ω. cos Ø. tan δ. cos h/120³. = + 11°.6 × 109.6 × cos Ø × sin δ × cos h ÷ (cos δ × 120³) = + 11°.6 × 109.6 × degrees of Moon's declination × cos h ÷ (120 × 57.3), (∵ sin δ = degrees of declination × 120 ÷ 57.3, nearly, and taking cos Ø/cos δ as equal to unity, δ being not great, and Ø being not great in India) = + 11°.6 × 109.6 × δ × nāḍīs after sunrise or before sunset ÷ (120 × 57.3 × 15), ∵ the Siddhānta, takes cos h as equal to nāḍīs after sunrise or before sunset ÷ 15, being satisfied with approximate values, i.e. Ø not being great, the time from sunrise or sunset to noon is taken as 15 nāḍīs always. Secondly, the angle is taken as ∝ to sine, as often done before. ∴ cos h = sin (90° – h) = (90° – h)/90° = (15 nāḍīs – nāḍīs to meridian)/15 = nāḍīs after sunrise or before sunset ÷ 15 = + δ × nāḍīs after sunrise or before sunset ÷ 80) δ is north, i.e. positive for Moon, 0 to 6 rāśis, and negative for Moon, 6 rāśis to 12 rāśis. The original of the nāḍīs cos h, is positive both forenoon and afternoon. ∴ p.lat is positive for Moon between 0 to 6 rāśis, and so the result is deducted from Head and

178 PAÑCASIDDHĀNTIKĀ VII.6 added to Tail if Moon is between 0 and 6 rāśis. For Moon between 6 rāśis and 12 rāśis ☊ is negative and p.lat. is negative, and therefore the result is to be added to Head and subtracted from Tail. Thus, by making the three corrections to Rāhu, p.lat. is secured, though approximately. TS have not translated or explained these three verses, not having understood the exact form of the rules or their derivation. They merely surmise that it is some work done on Rāhu to correct the latitude for parallax. NP too have not understood these verses correctly for they say in the notes to verse 2. “This verse is corrupt” etc., in the notes to verse 3, “The text as it stands seems to have con- fused the different cases” and in the notes to verse 4, “The text seems to instruct us” etc. (pt. II, p.57). In accordance with the correct form of the rules, we have in verse 2, emended dvika into dhṛti, rāśicaraṇāyana into sarāśicaraṇāpama and candrāyana into candrāpama. [ग्रहणकर्म] राहोः स'षट्कृति'कलां हि(त्वांशां) त(च्छ)शाङ्कविवरांशैः । ग्रहणं त्रयोदशान्तः शशिनो भानोस्तथाष्टान्तः ॥ ५ ॥ तद्वर्गं(मपास्ये)न्दो'(र्न)र्तुरूपात्' ('श्रुतिरस्सा'च्च) । तन्मूलं पादोनं स्थितिकालश्चन्द्रभान्वोश्च ॥ ६ ॥ Eclipse computation 5. Deduct 1° 36′ from Rāhu and find Moon ~ Rāhu, in the case of the lunar eclipse. Deduct 1° 36′ from Rāhu corrected (by verses 2-4) and find Moon ~ Rāhu, in the case of the solar eclipse. If the difference is less than 13° there is a lunar eclipse. If the difference is less than 8°, there is a solar eclipse, (other- wise not). 6. For the lunar eclipse, deduct the square of the difference from 169, find its square root and take three fourths of it. This is the total duration in nāḍis. For the solar eclipse, deduct the square of the difference from 64, find its square root, and take three fourths of it. This is the total duration in nāḍis. Thus, nāḍis of total duration = ¾ √(169 – (Moon ~ Rāhu)²), or ¾ √(64 – (Moon ~ Rāhu)², respectively. Half this subtracted or added to the full moon, or parallax corrected new moon, gives the times of first and last contacts. 5a. A. किलां; B. कला b. A1. हित्वा सं; A2. हिन्चा सं; B. हिच स. A.B1.2. तछ (B. सश्रुत) । d. B. तथाष्टींतः C.D. रूपाद्रवेः श्रुति (D. कृत) रसाच्च 6a. AB.1.2 समासेन्दो c. A. पादोन b. A.B. न वर्तृ (B. चर्त्तु) रूपात्रेचेष्टरसाश्च d. A.B. भानोश्च

VII.6 VII. PAULIŚA-SIDDHĀNTA — SOLAR ECLIPSE 179 The formulae here are similar to that of VI. 5 as reduced by us, giving the total duration in the lunar eclipse, according to Vāsiṣṭha. Therefore, the explanation is similar. As in the Vasiṣṭha, in the Pauliśa too, the limit of the lunar eclipse is seen to be 13°, giving 55′ latitude. Therefore, in the Pauliśa too, the sine of the semi-diameters of the Shadow and the Moon is 55′, wherefrom their respective semi-diameters may be taken as the same, i.e. 38′ and 17′. The limit of the solar eclipse is seen to be 8°, at which the latitude is 8 × 55′/13 = 33′.8. Therefore the sum of the semi-diameters of the Sun and the Moon is 33′.8, from which the semi-diameter of the Sun is found to be 16′.8. If actually the Moon’s semi-diameter is a little more or less in the Pauliśa, to that extent that of the Sun must be less or more. But we have no means of knowing it exactly, since the original Pauliśa is not extant, the Pauliśa quoted by Bhaṭṭotpala in the Bṛhat Saṁhitā being different, as already mentioned. From the formulae, the minutes of arc pertaining to duration is, 55′ × 2 √(169 − (Moon ~ Rāhu)²)/ 13 in the case of the lunar eclipse. To be correct, the time must be found from this by dividing by the true relative motion, which is not done here. If the mean relative motion is used, we get: {2 × 55 √(169 − 0)/13} × 60 ÷ 731.5 = 9, being the nāḍis for the maximum. But, by the formula, the maximum is, 3√(169 − 0)/4 = 9¾ nāḍis. But this is nearer the correct value. For the solar eclipse the maximum by the formula is, 3 √(64 − 0)/4 = 6 nāḍis. But actually, in the solar eclipse the maximum differs according to the time of new moon, being about 5 nāḍis at sunrise or sunset, and about 10 nāḍis near noon. TS, in their explanation here, take the maximum lat. to be 270′, instead of their taking 240′ for the Vāsiṣṭha. But, by this, the respective sums of semi-diameters must be got as 61′ and 38′. But they give 58′ and 35′, which is quite wrong. They seem to have taken these wrong values deliberately, with a view to deriving the formulae using the mean relative motion. NP too have failed to get the correct meanings and so close their notes on verse 7 with th.e statement “We cannot explain the origin of this discrepancy” (pt. II, p.59), the apparent discrepancy being in the formula for total duration of the eclipse as calculated by them. We shall illustrate the whole thing with two examples. Example (a). In chap. VI, example 3, the Moon at full-moon was given as rā. 5-15-0, and Rāhu at that time as rā. 5-6-36. Compute the lunar eclipse:- The Rāhu corrected for eclipse = rā. 5-6-36 − 1° 36′ = rā. 5-5-0 Moon ~ Rāhu = rā. 5-15-0 − rā. 5-5-0 = 10° Total duration = 3√(169 − 10²)/4 = 3 × 8.307/4 = nā. 6-14. (Compare this with the nāḍis got by Vāsiṣṭha, nā. 5-46). Example (b). The latitude of Pudukkottai in S.India is 10° 24′ N. On a certain day, there, sunrise is nā. 28- 40, after sunset, midday is after nā 44-20, and new moon is after nā. 49-20 The Sun = Moon = rā. 2-0-0, at new moon and Rāhu (Tail) is rā. 2-1-0, Compute the solar eclipse, if any, at Pudukkottai. First, Parallax-corrected new moon:- Time of new moon ~ time of midday = nā. 49-20 − nā. 44-20 = 5 nāḍikās, west. = 5 × 6 = 30°, degrees from meridian west. Correction for new moon = sin 30°/30 = 60/30 = 2, nāḍikās. Degrees being west, adding to new moon, the parallax corrected new moon = nā. 49-20 + nā. 2-0 = nā. 51-20.

180 PAÑCASIDDHĀNTIKĀ Next, correction to Rāhu: (i) 5 × Ø/27 = 5 × 10.4/27 = 1° 56′. As the Moon is near the Tail, this is to be subtracted. ∴ Tail − 1° 56′ = rā 2-1-0 − 1° 56′ = rā. 1-29-4. (ii) Correction to new moon = 2 nāḍis. (Moon + 3 rāśis) = 5 rāśis. The declination of this point is 11° 44′. The correction = 2 × 11° 44′/18 = 1° 18′. As new moon is afternoon, Uttarāyaṇa, and Tail, this is additive. Adding to corrected Tail we get, rā. 1-29-4 + 1° 18′ = rā. 2-0-22. (iii) The nāḍis of corrected new moon before sunset = Sunset ~ cor. new moon = 60 nāḍis − nā. 51-20 = nā. 8-40. The Moon’s declination, from its longitude, is 20° 36′. The correction, = 8 2/3 × 20° 36′ ÷ 80 = 2° 14′. As the Moon is between 0 and 6 rāśis, and it is Tail, this is additive. Adding to corrected Tail, we get, rā. 2-0-22 + 2° 14′ = rā. 2-2-36. Subtracting 1° 36′ from the corrected Tail, we have rā. 2-2-36 − 1° 36′ = rā. 2-1-0, as corrected Rāhu to be used in the formula: Moon ~ Rāhu = rā. 2-1-0 − rā. 2-0-0 = 1° As this is less than 8°, there is a solar eclipse. Duration = 3 × √(61 − 1²/4) = 5-57 nāḍis. Half this is nā. 2-59. Subtracting and adding this to the corrected new moon, we have: Time of first contact = nā. 51-20 − nā. 2-59 = nā. 48-21 Time of last contact = nā. 51-20 + nā. 2-59 = nā. 54-19 We have already said that the results will be very rough. [इति पञ्चसिद्धान्तिकायां वराहमिहिरविरचितायां रविग्रहणं नाम सप्तमोऽध्यायः ॥]¹

1. Col. A.B.C.D. इति (A.D. om इति) पौलिशसिद्धान्ते रविग्रहणं नाम (A.B.D. om नाम) सप्तमोऽध्यायः Thus ends Chapter Seven entitled ‘Pauliśa-Siddhānta – Solar Eclipse’ in the Pañcasiddhāntikā composed by Varāhamihira

Chapter Eight

ROMAKA-SIDDHĀNTA — SOLAR ECLIPSE ८. अष्टमोऽध्यायः रोमकसिद्धान्तः — रविग्रहणम्ः Introductory In this chapter the Sun, Moon and Rāhu according to the Romaka Siddhānta are given, as also the solar eclipse, dependent on these. But the lunar eclipse is not dealt with. We have already men- tioned that this Romaka is different from the Romaka extant now. [स्फुटरविः] रोमकसूर्यो द्युगुणात् 'खतिथि' (घ्नाः) 'पञ्चकर्तृ'परिहीणात् । 'सप्ताष्टकसप्तकृतेन्द्रियो'द्धृतानमध्यमः [क्रमशः] ॥ १ ॥ रविशशिनोः स्फुटकरणं स्वके (न्द्र) भवनार्धसंमितैः खण्डैः । (व्यु) त्क्रमशश्च पुनस्तैर्मिथुनद (लं) शोध्यतेऽर्कस्य ॥ २ ॥ 'तिथि-मनु-दश-कृत'सहिता 'रस-मनु'हीना (भिश्च) 'विंशति'र्हीना 'धृति-विषयो'ना 'द्वि-दशा-ष्टि-धृति'षु वृद्धिः कलाविकलाः ॥ ३ ॥ True Sun

1. According to the Romaka, the mean Sun in revolutions etc. is obtained by multiplying the Days from Epoch by 150, deducting 65 from the product, and dividing by 54,787.
2. Both the Sun and the Moon are to be made true by intervals of the equa- tion of the centre for half-signs of the respective mean anomalies given for the first three signs. For the next three signs they are to be taken in the reverse order. This is repeated for the next six signs. In the case of the Sun, the anomaly is got by deducting nā. 2-15-0 from the mean Sun.
3. The minutes of intervals for the Sun, are 20 + 15, 20 + 14, 20 + 10, 20 + 4, 20 − 6 and 20 − 14, from which seconds 18 and 5, are to be subtracted, and 2, 10, 16 and 18 are to be added, in the given order. Thus, (i) Mean Sun = (Days from epoch × 150 − 65) ÷ 54,787. (ii) The mean anomaly of Sun = Mean Sun − rāśi 2-15-0.

182 PAÑCASIDDHĀNTIKĀ VIII.3 (iii) The intervals of equation of the centre are 34' 42", 33' 55", 30' 2", 24' 10", 14' 16" and 6' 18". These are subtractive in the given order in the first quadrant of anomaly, additive in the reverse order in the second quadrant, additive in the given order in the third quadrant, and subtractive in the reverse order in the fourth quadrant. (iv) True Sun = (i) + (iii) Example 1. Compute the true Sun, for the moment, 59 days from Epoch. (i) Mean Sun = (59 × 150 - 65) ÷ 54,787 = rāśi 1-27-44. (ii) Mean anomaly = rāśi 1-27-44 - rāśi 2-15-0 = rāśi 11-12-44. (iii) The equation of the centre = -34' 42" -33' 55" -30' 2" -24' 10" - 14' 16" -6' 18" +6' 18" +14' 16" +24' 10" +30' 2"+33' 55" +34' 42" +34' 42" +33' 55" +30' 2" +24' 10" +14' 16" +6' 18" -6' 18" -14' 16" -24' 10" -30' 2" -33' 55" × 12° 44' ÷ 15° (= -28' 47") = +39' 50". (An examination of work (iii) will suggest how to get the total easily). (iv) Adding to the mean Sun, True Sun = rāśi 1-27-44 +40' = rāśi 1-28-24. It must be noted that this is at mean sunset at Yavanapura. The rule for the mean Sun can be derived from the constants given or derivable from I. 15. There it was shown in the Notes that in the Romaka yuga consisting 2850 solar years, there are 10,40,953 civil days. Thus, as the solar year is the period of revolution of the Sun, the number of revolutions, say in x days, is = x × 2850 ÷ 10,40,953 = x × 150 ÷ 54,787, as given. As for thè deductive constant, 65, we infer that according to the Romaka, 65/150 days after epoch the mean Sun was a full revolu- tion, but we cannot verify this, the original Romaka being lost. But it must agree with the relevant constant in I.10, and we have shown in the Notes there that it indeed does. We have also shown there that this mean Sun is tropical, and not sidereal. This is peculiar to the Romaka. The Sun's apogee given as rā. 2-15-0 is what the Romaka must have found by observation and computation, and we have to take it as it is. Actually, the apogee was at rā. 2-17-19 at epoch. As for the intervals of the equation of the centre, the Siddhānta is right in giving them in accor- dance with the anomaly. But the values are slightly different from what they will be if the correct terms, a sin θ + b sin 2 θ, has been used. Therefore, either the Romaka gives only empirical values obtained from observation, like the Pauliśa, or the Romaka like the Sūrya Siddhānta etc., apply an equation on the epicycle itself. Any deviation from this may be due to scribal errors. Taking the sum

1. Quoted by Utpala on BS 2.p.40 1a. A1. सूया; A2. सूर्यो c. A.B.C.D. तत्क्रम b. A.B1.C.D. तिथ्यन्तात्; B2.3. द्वांच. d. A.B. दल; C. दलात्. B. ॰तेकस्य A.B. परीहीणान्न 3b. A.B. हीना भाविंशतिहीना (B. विंशतिहीना) c. B. ॰ष्टकमापत ते C. ॰न्मुभिर्विंशतिहीना:. D. ॰न्मु[वि] हीना च विंशतिर्हिता d. A.B. ॰द्ध (B. धृ) तान्मध्येमा: || U. मध्यमः सूर्यः c. A1. धृत; A2. धृति 2a. A. स्फूटकरणं; B. स्फुरठकरणं d. B. धृतिष्ठ॰ b. A.B. स्वकेन्दु. B. भवनाद्धम् - सं॰ A.B. कलाद्विरकिला (B2. ॰रकला)

VIII.6 VIII. ROMAKA-SIDDHĀNTA — SOLAR ECLIPSE 183 of the intervals as the maximum equation of the centre, and neglecting the correction of the epicycle, we give hereunder the given and computed values for comparison. Anomaly 15° 30° 45° 60° 75° 90° Computed intervals 37′ 6″ 34′ 36″ 29′ 42″ 22′ 46″ 14′ 20″ 4′ 53″ Given intervals 34′ 42″ 33′ 55″ 30′ 2″ 24′ 10″ 14′ 16″ 6′ 18″ The sum forming the maximum is 143′ 23″, and very near that of the Pauliśa, and very much more than the actual. This excessive roughness itself is an indication that the Siddhānta is not indigenous. The author has not clearly mentioned where the intervals are to be taken in the given order and where in the reverse order, as also where they are additive and where subtractive. Or, adopting the reading mithunadalāt, we can understand that the equation of the centre is subtractive in the six signs of the mean Sun beginning from the middle of Gemini, and therefore additive in the other six. From the corrected reading, vyutkramaśasca, we understand that after taking the inter- vals in the given order, we take them in the reverse order. From these everything else is inferred. [स्फुटचन्द्रः] ‘खखरूपाष्टगुण’घ्नात् ‘कृताष्टन(खै)क’वर्जिताद् द्युगणात् । ‘त्रिविषयनवखकृताशा’परिशुद्धान्मध्यशीतां(शु)ः ॥ ४ ॥ ‘शून्यैकैका’(थ्य)स्ता‘न्नवशून्यरस’न्विताद्दिनसमूहात् । ‘रूपत्रिखगुण’भक्तात् केन्द्रं शशिनोऽस्तगमव(न्याम्) ॥ ५ ॥ ‘मनु-भव-यम’सहि(तोंऽशो) ‘वसुहोता’वर्जी(ते) ‘धृति-कृ(ती)’च । ‘विषयकृति-रष्टषटकं’ ‘नव-तिथि’(रहि)तौ न(‘ख’-‘चन्द्रे’ण) ॥ ६ ॥ True Moon 4. The mean Moon in revolutions etc. is got by multiplying the 'day' by 38100, subtracting 10,984, and dividing by 10,40,953. 5. The mean anomaly in revolutions etc. is obtained by multiplying the days by 110, adding 609, and dividing by 3031, the result being for sunset at Ujjain. 6. For the half-signs of anomaly the intervals of equation of the centre are: 1° + 14′ + 25″, 1° + 11′ + 48″, 1° + 2′ − 9″, 48′ − 15″, 48′ − 18″ − 0″, and 48′ − 18′ − 20′ −·1″ (i.e. (1) 1° 14′ 25″, (2) 1° 11′ 48″, (3) 1° 1′ 51″, (4) 47′ 45″, (5) 30′ 0″, (6) 9′ 59″). 4a. B. खस्वरूपा०. A.B. गुणाष्टघ्नात् b. C.D. होत्रा A2. वस्तु A.B.C.D. वर्जितो b. A.B. क्रताष्ट A.B.C.D. नक्कैक A. वर्जिता द्यु; B. वर्जिद्यु A.B. धृतिकृतौ च; C.D. धृतिकृतश्च c. A.B. त्रिविषये च ख०; D. विषयाङ्कखकृताशा c. A. क्रति A.B. रष्टवष्टकं (B. ट्क); D. रष्टिष्टकं. d. D. परिलब्धान्मध्य. A.B. शीतांशोः C. विषयऋतुष्टष्टट्को- 5a. A.B. ०व्यान्यस्ता d. A.B. नवति (A2. नवतिहितौ) C. ना षष्टिस्रौ च; b. B. समूहन् D. नवतिहीनं [हि] तं A. चन्द्रेना; B. चन्द्रेन; C. चन्द्रेनौ d. A. ०मवद्याम्; B1. मवद्गाम्; B2.3. मवद्‌गाम् 6a. A.B. सहितांशौ

184 PAÑCASIDDHĀNTIKĀ VIII.6 The true Moon is got thus:- (i) mean Moon in revolutions = (days × 38,100 - 10,984) ÷ 10,40,953. (ii) mean anomaly in revolutions etc. = (days × 110 + 609) ÷ 3031. This is for sunset at Ujjain. If required for sunset at Yavanapura, 622½ should be used in the place of 609, we shall explain how, later. (iii) The intervals of equation of the centre for the 6 half-signs in a quadrant are, 1° 14' 25", 1° 11' 48", 1° 1' 51", 47' 45", 30' 0" and 9' 59". In the first quadrant these are to be deducted in the given order, in the second they are to be added in the reverse order, in the third they are to be added in the given order, and in the fourth they are to be subtracted in the reverse order. (iv) True Moon = (i) + (iii). Example 2. For days 59, (from sunset at Yavanapura), compute the true Moon. (i) The mean Moon = (59 × 38,100 - 10,984) ÷ 10,40,953 = Rev. 2-1-23-36-30 = rā. 1-23-36-30. (ii) Mean anomaly = (59 × 110 + 622½) ÷ 3031 = rā. 4-4-46. (iii) The equation of the centre = - 1° 14' 25" - 1° 11' 48" - 1° 1' 51" - 47' 45" - 30' 0" - 9' 59"

• 9' 59" + 30' 0" + 4° 46' × 47' 45" ÷ 15° = -4° 0' 39". (iv) True Moon = (i) + (iii) = rā. 1-23-36-30 - 4° 0' 39" = rā. 1-19-36. (Note: This is for sunset at Yavanapura). The rules are explained as in the case of the Sun thus: In I.15, it has been mentioned that in the Romaka yuga of 2850 solar years, there are 1050 intercalary months and 16,547 suppressed tithis. Therefrom it has been shown, that in the yuga there are 2850 × 12 = 34,200 solar months, 34,200
• 1050 = 35,250 synodic months, 35,250 + 2850 = 38,100 lunar revolutions, and 35,250 × 30 - 16,547 = 10,40,953 mean solar or civil days. So, from the proportion: If there are 38,100 lunar revolutions in 10,40,953 days, how many are there in the days from epoch, we have, the number of revolutions = days × 38,100 ÷ 10,40,953. The mean Moon at epoch should be added to the mean Moon or the time by which the Moon completes the current revolution should be omitted from the days. According to the Romaka, by 10,984 ÷ 38,100 days after epoch, the mean Moon is a full revolution, though we cannot verify this, as the original Romaka is lost. Therefore, we have to deduct from the product of days from epoch, (10,984 ÷ 38,100) × 38,100 = 10,984, as instructed. With the given deductive constant we get that the Romaka mean Moon in revolutions at epoch = (0 × 38,100 - 10,984) ÷ 38,100 = rā. 11-26-12. See how close this is to the actual, rā. 11-24-48, to the Saura, rā. 11-25-6, and the Siddhānta Śiro- maṇi's rā. 11-25-49. This is why we corrected the reading, kṛtāṣṭanavakaikā (1984) into kṛtāṣṭanavakhaika (10,984). If the reading is taken as it is as done by TS and NP then the mean Moon at epoch would become rā. 11-29-19, which is improbable, being too far from the actual. We have also shown under I. 8-10 that the mean Moon of the corrected reading alone would agree with the constants there. In the Romaka, as in the Vāsiṣṭha-Pauliśa there are 110 anomalistic revolutions of the Moon in 3031 days. Therefore multiplying the days by 110 and dividing by 3031, the mean anomaly of the Moon in revolutions etc. is got. As, according to the Romaka 609/110 days before sunset at Ujjain, it was a full revolution, we have the additive constant 609. We cannot understand why the anomaly alone is given for sunset at Ujjain, while it could also be given for Epoch, i.e. for sunset at Yavanapura by

VIII.7 VIII. ROMAKA-SIDDHĀNTA — SOLAR ECLIPSE 185

making the additive constant 622½. That is why in our rules for computation we have given this constant. Perhaps the author wanted to avoid the fraction in the constant. The anomaly computed for Epoch in revolution etc. = (0 × 110 + 622½) ÷ 3031 = rā. 9-12-16. Compare this with the actual, rā. 9-9-34, Saura's rā. 9-9-47, and Siddhānta Śiromaṇi's rā. 9-11-23.

The intervals of the equation of the centre are given in minutes and seconds as in the case of the Sun, with the special mention of degrees where there are full degrees. But the text here is so corrupt that we are not certain about the numbers, since the original Siddhānta is lost. So we have to depend much on guessing. Adding the intervals we understand that in this Siddhānta the Moon’s maximum equation of the centre is 4° 55' 48". Using this, and not doing the correction to the epicycle, since it is not known, we have computed the intervals and given them hereunder, for comparison with the given values:

Anomaly 15° 30° 45° 60° 75° 90° Computed Values 1° 16' 34" 1° 11' 20" 1° 1' 15" 47' 1" 29' 33" 10' 5" Given Values 1° 14' 25" 1° 11' 48" 1° 1' 51" 47' 45" 30' 0" 9' 59"

In the matter of order of taking the intervals and of adding or subtracting them our remarks under the sun hold here too.

[रवि-चन्द्र-भुक्तिः] 'खनवनगाः' शशिभुक्तिः ('कृ)तवसुमुनयः' शशाङ्ककेन्द्रस्य | यातस्फुटान्तरे दिवसभुक्तिरागामिकी नैशी || ७ ||

Daily motion of the Sun and the Moon 7. The daily motion of the mean Moon is 790', and that of the mean anomaly, 784'. For work relating to the day-time the true daily motion is the difference between the true Moons of the taken day and the previous day. For work relating to the night-time the true daily motion is the difference between the true Moon’s of the taken day and the next day.

The true daily motion, given in the second half, pertains both to the Sun and the Moon. The daily mean motions of the Moon and its anomaly alone is given because in the case of the Sun both are the same, practically, equal to 59' 8", and well known. It would have been better if the Moon’s mean daily motion had been given as 791'. Thus, the following is intended:

(i) To get the true daily motion of the Sun, take the last interval used in obtaining the true Sun, divide it by 15, and apply it to 59' 8" as the quantity got from the last interval has been applied to the mean Sun. This can be taken as true daily motion for both the day-time and the night-time as there is not much difference.

(ii) To get the true daily motion of the Moon: (a) for the day-time work, find from the intervals the equation of the centre for the last 784' of the anomaly, and apply it to 790' as the last part of the interval itself is applied.

7a. A1.B1.वनगा c. B.यातः स्फुटा; CD. याता स्फुटा b. A.क्रतव B.तत्तव०. B.शशङ्केन्द्रस्य d. A.B1.2.सभुक्ति आगामि

186 PAÑCASIDDHĀNTIKĀ VIII.8 (b) for the night-time work, find from the intervals the equation of the centre for the 784' following, in the anomaly, and apply it to 790' as that itself would be applied. Example 3. The days from epoch is 59, (given in the previous two examples). Find the Sun's and the Moon's true daily motion, for the day gone and the day to come. (i) In example 1, the last interval used is -33' 55". The 15th part of this is -2' 16". Applying this to 59' 8", the Sun's true daily motion for both days is 59' 8" - 2' 16" = 57' (in full minutes). (ii) In example 2, the Moon's mean anomaly used is rā. 4-4-46. (a) For the day previous, the last 784' of this begins from rā. 3-21-42. The equation of the centre pertaining to this part of the anomaly = + 30' × 8° 18' ÷ 15° + 47' 45" × 4° 46' ÷ 15° = + 16' 36"

• 15' 10" = + 31' 46", (say + 32'). Applying to the mean motion, 790', the true motion for the pre- vious day = 790 + 32 = 822'. (b) For the next day, we have to find the equation of the centre for anomaly from rā. 4-4-46 to rā. 4-17-50. This is equal to, + 47' 45" × 10° 14' ÷ 15° + 1° 1' 51" × 2° 50' ÷ 15° = + 32' 35"
• 11' 41" = + 44' 16". Applying to the mean motion, the daily motion for the day following = 791'
• 44" = 835'. The instruction is easy to understand, for, clearly the difference in the longitudes of two consecutive days is the motion for the day. As, in the Romaka, the day begins at sunset for which the longitude is computed, the day-time before sunset falls in the day previous, and the night following sunset falls in the day next. Hence for work in each, respectively, the motion for the previous day and the next day has to be taken. To avoid computing the longitudes of both days, we have given an easy method, which should have been intended by the author also, for, otherwise, he need not have given the mean motions of the Moon and its anomaly. [राहुः] 'त्र्यष्टक'गुणिते दद्याद् 'रसर्तुयमषट्कपञ्चकान्' राहोः । 'भवरूपाग्न्यष्टि'हृते क्रमात् झषान्तो (च्यु)ते वक्त्रम् ॥ ८ ॥ Rāhu

8. Multiply the days from epoch by 24, add 56, 266 and divide by 1,63,111. Subtract the revolutions etc. obtained, from the end of Pisces, (i.e. from any whole number of revolution). The Head of Rāhu is obtained. The following is instructed to be done: (i) Revolutions etc = (days × 24 + 56,266) ÷ 1,63,111 (ii) Head of Rāhu = rā. 12-0-0 - Revolutions etc, omitting the full revolutions. 8a. B. त्र्यष्टगुणिते d. A. क्रमाझखांतोव्यते; B. क्रमादुखान्तोच्चते b. A. नाहो: (B2. क्रमातु दु०) ; C. क्रमात् झषान्तोत्क्रमात् वक्रमू c. A1. रूपानन्यष्ठि D. क्रमात् झषात् सोच्यते