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

VIII. 11 VIII. ROMAKA-SIDDHĀNTA — SOLAR ECLIPSE 187 Example 4. Compute Rāhu for the moment, 59 days from epoch. (i) Revolution etc. = (59 × 24 + 56,266) ÷ 1,63,111 = rā. 0-4-7-19. (ii) Head of Rāhu = rā. 12-0-0 - rā. 4-7-19 - rā. 7-22-41. From this the tail = rā. 7-22-41 + rā. 6-0-0 = rā. 1-22-41. We have said that the Moon's node is called Rāhu, on account of the connection between the two. Of the two nodes, the first is the Head and the second, situated six signs away, is the Tail of Rāhu. According to the Romaka, there are 24 revolutions of the Moon's nodes in 1,63,111 days. There- fore, multiplying the days by 24 and dividing by 1,63,111 the revolutions are got. As the motion is retrograde, what is obtained has got to be treated as negative, and therefore to be subtracted from 12 signs or full revolutions. At the moment 56,266/24 days before Epoch, the Head of Rāhu was a full revolution, and in order to reckon from that time 56,266 is added to the days multiplied by 24. As for the correctness of the numbers, we cannot verify them since the original is lost. But we can see how nearly correct the Romaka Rāhu here given is, by comparison with that of other systems. At Epoch the Head or Rāhu according to the Romaka = rā. 12-0-0 - (0 × 24 + 56,266) ÷ 1,63,111 = rā. 7-25-49. Actually it is rā. 7-26-0. According to the Paulīśa it is rā. 7-25-59, and according to the Saura, rā. 7-26-6. The time for one tropical revolutions is 1,63,111 ÷ 24 = 6796-17-30 days. The correct time is 6798-21-48. The difference of 2-4-18 days is caused by the wrong constant of precession adopted by the Romaka, of 34" instead of the correct 50". Thus, since the Romaka precession is less by five minutes in the time taken by one revolution, its period of revolution must be about two days less as it is found to be, and the disagreement is small indeed. [लम्बनम्] दिनमध्यमसंप्राप्ता यावत्यो नाडिका व्यतीता वा | ताभ्यः षड्गुणिताभ्यो ज्यात्रिंशां'शस्थितेर्नार्म [:] || ९ || Parallax in longitude 9. (This is the same as VII. 1. and explained completely there. There is no difference in meaning between the readings there and here, dinamadhyama- samprāpyā and dinamadhyamasamprāptā). [दृक्क्षेपः] उदयात् प्रभृति च नाड्यो याः स्युः प्राग्लग्नमानयेत्ताभिः | तस्मात्तु नवसमेतादपक्रमांशान् विनिश्चत्य || १० || लग्नत्र्यगुविवरज्यां द्विगुणां स्व'रसां'शसंयुतामपमात् | जह्याद् दिग्द्व्यत्यासे विक्षेपैक्ये तयोर्योगः || ११ || 9. Quoted by Utpala on BS 5.18 c. B. षड्गुणिता योज्या 9a. B. मध्यसमं प्राप्ता d. A.B. तिथिर्नार्म; C. तिथेनार्म b. B. यायत्या. A. त्यो दिनाधिका 25

188 PAÑCASIDDHĀNTIKĀ VIII. 12 उत्तरमक्षाच्छुद्धं याम्यं साऽक्षं च दक्षिणं विद्यात् । उत्तरमक्षादधिकमुत्तरमेवं विजानीयात् ॥ १२ ॥ Declination of the Nonagesimal 10. At any time (for which the zenith distance of the nonagesimal, ZDN, is desired,) find the orient ecliptic point, OEP. Add nine signs to it. (This point is called the nonagesimal). Find its declination. 11. Subtract the Head of Rāhu from the nonagesimal, find its sine, double it, and add a sixth of the quantity got by doubling, (i.e. find the latitude of the Moon, supposing it to be situated at the nonagesimal). Add this to the declina- tion found above if both are of the same direction, and subtract it from the declination if they are of different directions. (Thus the declination of the nonagesimal is corrected). 12. The north declination, being less and therefore deducted from the latitude of the place, the remainder (which is the ZDN) is south. The south declination must be added to the latitude, and the sum (forming the ZDN) is north. The part of the north declination greater than the latitude, (i.e. the remainder after deducting the latitude from the north declination, which forms the ZDN), is north. Lagnatryaguvivara actually means the difference between the OEP and the Head of Rāhu, plus three signs. Clearly this is equal to the difference between the nonagesimal and the Head of Rāhu, as translated above. Therefore, if the reading, lagnāsuravivara is adopted, the word lagna must be taken to mean tribhonalagna or nonagesimal. If the nonagesimal is greater than the Head and less than the Tail, the latitude obtained is north, otherwise south. Why this is so has been explained in connection with finding the Moon’s latitude according to the Pauliśa. Though, in a general way, the nonagesimal latitude is asked to be deducted from the declination if of different directions (instruc- tion contained in verse 11), in the case where the declination is less, the declination is to be deducted from the latitude, the direction of the corrected declination being the direction of the latitude. The instructions contained in verse 12 envisages only places north of the equator, as usual. 10-12, Quoted by Utpala on BS 5.18 10a. A. व for च b. A1.C. सवसांस; B1.2. यासां स; b. B. या: पु: प्रालग्ममानये ताभिः B2. सारसां; D. स्वरसाप्तामपक्रमांशात् | c. A. B1. नवमसेता A1. संयुतममरान्; | B. संपुतयममरान् d. A.C.D. ॰मांशा; B. ॰माशात् c. A.B. जह्या दिग्व्यत्यासौ A.B. विनिश्चत्या (B2. ॰त्यं); C.D. विनिश्चिन्त्याः d. A.B. विज्ञेयैके 11a. A. वन्नासुरविरज्यां; B.C. लग्नासु (B3. लग्नास) रविरत्यां; 12a. B. मक्षाछुढ्ढं D. लग्न्नासुर वि [व] रज्यां b. B. य for च C.U. विन्द्यात् c. B. उत्तमक्षां

VIII.13 VIII. ROMAKA-SIDDHĀNTA — SOLAR ECLIPSE 189 Thus, the following has got to be done: (i) The OEP for the time for which the parallax corrected latitude is required, is found, by using the local ascensional differences. (ii) Nonagesimal = OEP + 9 signs. (iii) Find the declination of the nonagesimal, marking its direction north or south. (iv) Sine (nonagesimal − Head of Rāhu) × 7/3 = latitude pertaining to nonagesimal. This is north if (nonagesimal − Head of Rāhu) is within 6 signs, south otherwise. (v) Corrected declination = declination ± latitude, found in (iv), (the upper sign of same direc- tion, otherwise lower, the direction of the result being that of the greater. (vi) ZDN = Latitude of the place ± corrected declinations, the upper sign if the corrected declina- tion is south, lower sign otherwise. In the latter case, if the latitude is greater, the direction of ZDN is south, if the declination is greater it is north). The work is thus explained: In computing the solar eclipse it has been mentioned under VII.1, that in the place of the Moon's latitude, the same corrected for parallax has got to be used. To get the correction the sine of the ZDN is required. For ease of computation, the Romaka takes the difference between the latitude of the place and the declination of the nonagesimal (the directions being taken into consideration,) as the ZDN, the error being small as can be seen from the figure under VII.1. This is given by verse 12 above. Further, the parallax correction for latitude depending on sine ZDN is on the supposition that the Moon moves on the ecliptic, which is only approximately true. Actually the Moon moves in its orbit, and a small correction has got to be made for this, and the work of verse 11 above is intended for this. Practically, all astronomers before the famous Bhāskarācārya II have given this rule, on the surmise that taking a point on the Moon's orbit, corresponding to the nonagesimal, things will be all right. But the mistake in this has eluded all these ancient astronomers, including the astute Brahmagupta. It was Bhāskarācārya who detected their mistake, showed, by means of an example, how the rule was wrong, and gave the correct rule. (Vide the Vāsanā-Bhāṣya at the end of Sūrya- grahaṇādhikāra, Gaṇitādhyāya, Siddhānta Śiromaṇi). From the rule given by verse 11, it can be inferred that according to this Siddhānta the obliquity of the Moon's orbit, giving the maximum latitude of the Moon, is 280 minutes, (got from: 120 × 2 (1 + 1/6) = 120 × 7/3 = 280). We shall see that this agrees with the rule given by verse 14, giving the Moon's latitude. But TS have adopted the incorrect reading, kharasāṃśasammitām and dividing the doubled sine by sixty, got the latitude, which they are constrained to consider to be in degrees. NP too, accept the same sense as TS with an emended reading kharasāptām apakramāṃśāt. By this the maximum latitude according to the Romaka would be 4°. It is very strange that they do not see this is too far from the correct value, highly improbable in the Romaka which they themselves praise inordinately, and disagrees with their own (TS's) commentary under verse 14. [नतिः बिम्बमानं च] तज्ज्यार्द्धं शशिभुक्तिं हत्वा ‘धृतिभिः शतैः’ स्मृता [ऽवनतिः] । मध्यममानं त्रिंशद् भानोः शशिनश्चतुस्त्रिंशत् ॥ १३ ॥

190 PAÑCASIDDHĀNTIKĀ VIII.14 Parallax correction and orbital diameter 13. Multiply the true daily motion of the Moon by the sin of ZDN, thus found, and divide by 1800. This is the parallax correction for latitude. The mean angular diameter of the Sun is 30 minutes, and that of the Moon, 34 Minutes (according to the Romaka). Thus: (i) Parallax in latitude = sine corrected ZDN × true daily motion of the Moon ÷ 1800 (Its direc- tion is that of the ZDN). (ii) Mean angular diameter of the Sun = 30′. (iii) Mean angular diameter of the Moon = 34′. (Using (ii) and (iii) the respective true angular diameters are to be found). Under VII.1, it was explained that the parallax correction for latitude, to be used in the solar eclipse, is obtained by multiplying the horizontal parallax of the Moon relative to the Sun, by the sine of the ZDN and dividing by 120, (the max. sine). It was also shown there that the horizontal parallax itself varies inversely as the distance of the Moon from the earth, being greatest when the Moon is nearest. Hindu astronomers take it that the distance is inversely proportionate to the true daily motion, though this is only approximately correct. Therefore it is taken here that the relative parallax is proportionate to the motion, the Sun's parallax being very small compared to that of the Moon. Here, the parallax correction i.e. relative horizontal parallax × sin (corrected) ZDN ÷ 120 = Moon's daily motion × sin (corrected) ZDN ÷ 1800. From this it can be seen that according to this Siddhānta, the relative horizontal parallax is the daily motion divided by 15. Therefore the mean relative horizontal parallax = 790′.5 ÷ 15 = 52.7 minutes, as mentioned already. As for the mean angular diameters that is what the Siddhānta has found them to be, by observation or analysis of eclipses. समलिप्ता (ऽगु)विवरज्या [ऽभ्य]स्ता 'मूर्च्छना' नवहताश्च । अवनत्यायुतविश्लेषिताश्च दिक्साम्यवैलोम्ये ॥ १४ ॥ 14. Twentyone, multiplied by the sine of (Sun or Moon at new moon ~ Rāhu) and divided by nine is the latitude. This, with the parallax correction added is the parallax-corrected latitude, when both are of the same direction. When of different directions, their difference is the corrected latitude. 13. Quoted by Utpala on BS 5.18 13a. A. तज्ज्याघ्री; B1. तज्ज्याघ्नी; B2.3. तज्ज्याघ्री b. B. शनैः. A. B1.2. स्मृता नवभिः; D. स्फुटावनतिः c-d. B. त्रिशन्दानोः d. A. °स्त्रिंशान्न; B1.3. °स्त्रिशान् 14. Quoted by Utpala on BS 5.18 14a. A. लिप्ताद्वविवर; B. लिप्तिताद्व विवर; C.D. U. लिप्ताराहुविवर b. A.B1.2. न्यस्ता. D. हता च c. A. अनवद्या; B. अवनधा B. विशिल d. B. षिवाक्ष; D.U. षिता च. A1.B1. साम्ये; A2. सान्ये

VIII.15 VIII. ROMAKA-SIDDHĀNTA — SOLAR ECLIPSE 191 It is stated here that, (i) The Moon’s latitude at new moon = sin (Moon ~ Rāhu) × 7 ÷ 3. (ii) Parallax-corrected latitude = Moon’s latitude ± parallax correction given in verse 13. (The upper sign is to be taken if both are of the same direction, and the lower sign, if of different direc- tions, the resulting direction being that of the greater). The latitude at new moon is the distance of the Moon north or south of the Sun, as seen by an observer at the centre of the earth. For an observer on the surface, there is a difference in this, equal to the parallax in latitude. Therefore they have to be combined, taking the directions into consi- deration, to find the actual distance as observed, i.e. if of the same direction they have to be added, and if of different directions the differences is to be taken, the direction being that of the greater. Though the author wants this to be done at new moon, as the use of the word sama-lipta indicates – perhaps following the instructions of the original Siddhānta – it will be better if it is done at new moon corrected for parallax in longitude, that being generally nearer the circumstances. It is given that the sine of (Moon ~ Rāhu) multiplied by 21 and divided by 9, (it will be easier to multiply by 7, and divide by 3), is the latitude. From this, the maximum latitude according to this Siddhānta = the maximum sine × 7 ÷ 3 = 120′ × 7 ÷ 3 = 280′. This agrees with verse 11 above, as already said. But TS say here that the maximum is 270′, contradicting their statement under verse 11, that it is 4°, i.e. 240′. Without any reason, they assume here that the maximum is 270′, and since the maximum sine multiplied by 21 and divided by 9 does not give 270′, they say that the multiplier and the divisor given are approximate!! The same applies also to NP, vide their derivation (pt.II, p.63) of the result “i ≈ 4° .... (3c)” and “i ≈ 4:30° ... (10), in contrast to (3c)” (pt.II. p.64). Further, TS’s statement, that the parallax due to the Sun has been omitted by the author on account of its smallness, is wrong, for the intention of the author is only to give the relative parallax. The correct statement would be, “The Sun’s parallax has not been separately computed and deducted from the Moon’s, as the difference in effect would be negligible”. [स्फुटबिम्बमानम्] मध्यममानाऽभ्यस्ता स्फुटभुक्तिर्मध्यभुक्तिभक्ता च । भवति कलापरिमाणं तत्कालीनं रविहिमांश्वोः ॥ १५ ॥ True diameter of the orbs 15. The mean angular diameters of the Sun and the Moon, respectively, multiplied by their true daily motions and divided by their mean daily motions, gives the true angular diameters at the time of eclipse. Thus: (i) The angular diameter of the Sun = 30′ × Sun’s true daily motion ÷ 59. (ii) The angular diameter of the Moon = 34′ × Moon’s true daily motion ÷ 791. 15. Quoted by Utpala on BS. 5.18 B1.2. भुक्तिमभुक्ति; B3. स्फुटभक्तिमध्यमभुक्तिमत्ता च 15a. A. ॰मभाना॰. A.B. न्यस्ता c. A.B.कलाप्परि. B2. परिमानं b. A. भुक्तिमध्यमभुक्ति; d. B1.2.हिमांणो:;

192 PAÑCASIDDHĀNTIKĀ VIII.17 It is a matter of experience that an object looks bigger, the nearer it is, smaller the farther away it is, i.e. the angle formed by the object at the eye is inversely proportionate to the distance. We have already mentioned that approximately the daily true motion of the Sun and the Moon is inversely proportionate to the distance. Therefore, the angle at the eye is proportionate to the daily true motion, approximately. Hence, from the proportion, Mean motion: True motion:: Mean angular diameter; True angular diameter, we have, True angular diameter = Mean angular diameter × true motion ÷ mean motion, which is the rule given. [ग्रहणकालः] अवनतिवर्गं जह्याद् रवीन्दुपरिमाणयोगदलवर्गात् । तन्मूलात्तु द्विगुणात् तिथिभुक्तवदादिशेत् कालम् ॥ १६ ॥ Moment of the eclipse 16. Subtract the square of the parallax-corrected latitude from the square of the sum of the semi-diameters. The square root of the remainder, multi- plied by two, is the number of minutes of arc giving the duration. These minutes, multiplied by 60 and divided by the minutes of relative true daily motion gives the time of duration in nāḍikās. The following is to be done: (i) Minutes of arc of duration = 2 √(sum of the semi-diameters)² - (parallax corrected·latitude)². (ii) Time duration in nāḍīs = minutes of arc of duration × 60 ÷ daily relative true motion in minutes of arc. (Half this, subtracted from, and added to the time of new moon corrected for parallax gives the first and last contacts respectively). The rationale of the work has been shown in connection with the Paulīśa (chap. VII). We must add the following: If the latitude as corrected for parallax is found separately, each for the time of first contact and the time of last contact, and used in the work, then each will be more correct. Thus, the corrected latitude and the time are interdependent, each requiring the other for its computa- tion, and therefore the method of successive approximation is indicated here. This is not mentioned by the work, as being easily understood, or the author does not give it because it is not found in the original. [ग्रहणपरिलेखाः] रविशशिमानयुतिदलादवन [ति] हीनाद्वदन्ति या लिप्ताः । तान्यङ्गुलानि विद्याद् भानोश्छन्नानि चन्द्रमसा ॥ १७ ॥ 16. Quoted by Utpala on BS 5.18 b. A. रविन्दु. B. ०द्दलवग्रति 16a. A1.B1.2. वर्गे; A2. वग्र c. A. ०मलात्तु. B. द्विगुणा

VIII.18 VIII. ROMAKA-SIDDHĀNTA — SOLAR ECLIPSE 193 अर्धेनाऽऽलिख्य रविं दत्वाऽवनतिं यथादिशं मध्यात् । अवनत्यन्ताच्चन्द्रं विलिखेद् ग्रासार्थमर्धेन ॥ १८ ॥ Eclipse diagram 17. Subtract the parallax-corrected latitude for the time of parallax-corrected new moon, from the sum of semi-diameters. The remainder in minutes are the digits of obscuration of the Sun by the Moon. 18. To represent the amount of obscuration graphically, draw a circle of radius equal to the semi-diameter of the Sun, measure the parallax-corrected latitude north or south according as where the Moon is situated, and with the point marking its end as centre draw a circle of radius equal to the Moon's semi-diameter, to represent the Moon. (The part common to both the circles is the part obscured, and its measure in digits is its width in minutes of arc.) Obscuration in digits = Sum of the semi-diameters, in minutes − parallax-corrected latitude, in minutes. (This for the time of parallax-corrected new moon). The Fig. to illustrate this is given at the end of Example 5, as part thereof. It can be seen from there that the amount of obscuration, AB = SB − SA = SB − (SM − MA) = SB + MA − SM = radius of the Sun + radius of the Moon − corrected latitude = sum of the semidiameters − corrected latitude. The author has taken it that one minute of arc appears to the eye as one digit, though actually the apparent size varies, ('apparent' because this is an illusion), the heavenly bodies appearing to be bigger the nearer they are to the horizon. Example 5. After 59 days has passed from epoch, on the 60th day, there is a solar eclipse. Compute this for Pudukkottai (in S. India) (lat. 10° 23'; longitude 48° east of Yavanapura, represented by 8 nāḍīs of time). The first things to be found are: The Sun and Moon at new moon, the daily motion etc. In Example 1, we have found that the true Sun for 59 days from epoch is rā. 1-28-24. In Example 2, the true Moon is found to be rā. 1-19-36. In Example 3, the Sun's true daily motion for the 60th day is found to be 57', and the Moon's, 835'. In Example 4, the Head of Rāhu is found to be rā. 7-22-41. (All the three longitudes are for mean sunset at Yavanapura, that being the time of day of epoch.) Sun − Moon = rā. 1-28-24 − rā. 1-19-36 = 8° 48'. The Sun being greater, the new moon is to come. The relative daily motion = the difference of the true motions = 835' − 57' = 778'. 18a. B. अद्धनालिख्य रवि 17a. B1.2. ॰दवनिति. A. भवन्ति b. B.1.2. दंत्रा. A.B1.2. नवति c. A.C.D. U. विंद्यात् c. A. यांतश्चंद्रं; B. यातश्चन्द्रं d. A. भानो छन्नानि. A1. चंद्रममसा; A2. चंद्रमदमस्म d. B. विलिखेत्तु ग्रासार्द्धे

194 PAÑCASIDDHĀNTIKĀ VIII.18 The nāḍis of new moon from mean sunset at Yavanapura, = 60 × 8° 48′ ÷ 778′ = nā. 40-43. The time of new moon from mean sunset at Pudukkottai = nā. 40-43 + nā. 8-0 = nā. 48-43. i.e. on the 60th day, after mean sunrise at Pudukkottai, the new moon is at nā. 48-43 − nā. 30-0 = nā. 18-43. Given the half-cara for the day, 39 vināḍis, the new moon is at nā. 18-43 + vi. 39 = nā. 19.22. (No correction is made for equation of time since the Siddhānta does not give it.) At new moon, the Sun = the Moon = rā. 1-28-24 + 48′ = rā. 1-29-12. Head of Rāhu at new moon = rā. 7-22-41 − 2′ = rā. 7-22-39. Correction of new moon for parallax (by verse 9.): Half day-time is nā. 15-0 + vi. 39 = nā. 15-39. The time elapsed after noon = nā. 19-22 − nā. 15-39 = nā. 3-43. Corresponding to this, there are 22° 18′. Sine 22° 18′ = 45′ 32″. The parallax bending of the new moon, (later) = 45′ 32″/30 = nā. 1-31. Parallax-corrected new moon = nā. 19-22 + nā. 1-31 = nā. 20-53. The OEP at new moon: The required ascensional difference for every Drekkāṇa for Pudukkottai in vināḍis are 90, 94, 98 for Taurus; 102, 105, 108 for Gemini; 109, 110, 109 for Cancer; 108, 105, 103 for Leo; 101, 101, 99 for Virgo. The new moon is 1162 vināḍis from sunrise. 8 vināḍis after sunrise Taurus ends, 315 from this Gemini ends, 328 from this Cancer ends, and 316 from this Leo ends. For the remaining 195 in Virgo, the part risen is 10° + 9° 18′ = 19° 18′. ∴ OEP = rā. 5-19-18. Nonagesimal = OEP + rā. 9-0-0 = rā. 2-19-18. sin declination of nonagesimal = 48′ 48″ × sine (rā. 2-19-18) ÷ 120′ = 47′ 58″. Declination = 23° 33′, North. Corrected declination of the Nonagesimal (verses 10-11) Nonagesimal ~ Head of Rāhu = rā. 2-19-18 ~ rā. 7-22-39 = rā. 6-26-39. Sine of this = sin rā. 0-26-39 = 53′ 49″. 53′ 49″ × 2 (1 + 1/6) = 126′, south, (since the Nonagesimal is more than 6 rāśis distant from Head of Rāhu). Being of different directions, 23° 33′ − 126′ = 21° 27′, North is the corrected declination. Z̄ D N (by verse 12) : corrected declination − latitude = 23° 27′ − 10° 23′ = 11° 4′, north (being north declination and greater than latitude). Parallax in latitude (by verse 13) = Sin Z D N × Moon’s true daily motion in minutes ÷ 1800 = 22′ 50″ × 835 ÷ 1800 = 10′.6, north, (same direction as Z D N). The uncorrected latitude at new moon = (verse 14): Moon − Head of Rāhu = rā. 1-29-12 − rā. 7-22-39

VIII.18 VIII. ROMAKA-SIDDHĀNTA — SOLAR ECLIPSE 195 = rā. 6-6-33. Sine rā. 6-6-33 = Sin 6° 33′ = 13′ 41″. The latitude = 13′ 41″ × 7/3 = 31′.9, south, (the Moon being more than 6 rāśis distant from Head of Rāhu). Parallax-corrected latitude = (by verse 14), 31′.9 − 10′.6 = 21′.3, south. Sum of true semi-diameters (by verse 15) : True diameter of Sun = 30′ × 57 ÷ 59 = 29′. True diameter of Moon = 34′ × 835 ÷ 791 = 35′.9. Sum of semi-diameters = (29′ + 35′.9)/2 = 32′.4. Duration (by verse 16): Minutes of arc of duration = 2 × √(32.4² − 21.3²) = 2 × 24′.4 = 48′.8. Time of duration = 48′.8 × 60 ÷ 778′ = nā. 3-46. Half duration = nā. 1-53. Subtracting this from parallax-corrected new moon, first contact is, nā. 20-53 − nā. 19-0, after sunrise. Adding to parallax-corrected new moon, last contact is, nā. 20-53 + nā. 1-53 = nā. 22-46, after sunrise. Part obscured in digits (by verse 17): sum of semi-diameters − parallax-corrected latitude = 32.4 − 21.3 = 11.1. Graphical representation of obscuration S = centre of the Sun M = centre of the Moon SM = parallax-corrected latitude AB = the measure of the obscuration = 1′′.11 = 11.1 digits. Fig. VIII. 1 In this work, I.8-10 give the ‘days from epoch’ according to the Romaka; I.15, gives the elements concerning the Sun and Moon in the Romaka-yuga; VIII.1-8 give the true Sun, Moon and Rāhu; and VIII.9-18 give the solar eclipse according to the Romaka. It is the ‘days of epoch’ of Romaka that is intended to be used everywhere in the work; since it is the distance between two points of time and therefore the same by whatever siddhānta it is computed. The difference caused by the time of the day like ‘Sunset of Ujjain’, ‘Noon at Ujjain’ etc. will, of course, be there, and must be taken into account. The agreement between I.8-10, I.15, and VIII.1-7, each to each, has been shown in the proper places. We have also shown that the Sun, Moon and Rāhu of the Romaka are tropical, though the author has not mentioned this specifically. The work being a manual, intended to be used not for a long period, the difference caused by precession is neglected, no reference being made to it. The periods being tropical, itself indicates that this Siddhānta is foreign. The Sun’s

196 PAÑCASIDDHĀNTIKĀ maximum equation of the centre, given as 143', also is an indicator, agreeing as it does with Ptolemy's. Though the Moon's maximum equation of the centre given is 296', and Ptolemy's is 301', and thus there appears to be a difference, we are not sure that the given quantity is 296', on account of the extremely corrupt nature of the text in the concerned part. There are also lacunae in the computations intended by the author, which are to be supplied from the siddhāntas dealt with already or known otherwise. The method of computing the true Sun and Moon given here is an improvement on the Pauliśa. Only the solar eclipse is dealt with here. The lunar eclipse is omitted probably because it is not different from that of either the Pauliśa and Vāsiṣṭha given, or the Saura to be given. In contrast with the primitive method of the Pauliśa, the Romaka method of computation of the solar eclipse is far advanced, and almost the same as that of the later siddhāntas like the Āryabhaṭīya or the Saura. For instance, the parallax in latitude is correctly sought to be computed by using sine ZDN, though the ZDN itself is approximate, being got by combining the latitude of the place and the declination of the nonagesimal. Only the method given for correcting the declination for the nonagesimal to compensate for the Moon being situated on its own orbit instead of the ecliptic, is wrong, as commonly seen in works of authors prior to Bhāskarācārya II. Making the parallax in latitude and the Moon's true angular diameter depend on the Moon's true motion, and the Sun's true angular diameter on the Sun's true motion, is in accordance with the later siddhāntas, though giving the respective mean diameters as 34' and 30' is very rough. The first contact, middle, and last contact, as also the directions of the points of contact, are intended to be taken from the Vāsiṣṭha-Pauliśa, not being given here. The omission of the total or annular phases does not matter, since they cannot be got correctly by the rough methods given. Further, let us not mind the omission of the successive approximation to be done in the computation of the circumstances, though necessary as shown. (This may be because it is not found in the original or easily understood to be necessary). But it will certainly be better to use in the computation the parallax-corrected latitude of the new moon corrected for parallax, instead of that of the uncorrected new moon as given by the text, the former being generally nearer the time of the thing computed. We do not know why the author has not said so. Inspite of all this, the Romaka is interesting as being comparatively more ancient, and forming a link between the earlier and the later Siddhāntas. [इति पञ्चसिद्धान्तिकायां वराहमिहिरविरचितायां रोमकसिद्धान्तेऽर्कग्रहणमष्टमोऽध्यायः ||]¹

1. Col. A. रोमकसिद्धान्तेऽर्कग्रहणमष्टमष्टमोध्यायः; B.C.D. इति रोमकसिद्धान्तेऽर्कग्रहणमष्टमोऽध्यायः Thus ends Chapter Eight entitled ‘Romaka-Siddhānta: Solar Eclipse’ in the Pañcasiddhāntikā composed by Varāhamihira