पञ्चसिद्धान्तिका (आचार्य वराहमिहिर - सूर्य, रोमक, पौलिश, वासिष्ठ एवं पितामह सिद्धान्त)

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

Chapter Nine SAURA-SIDDHĀNTA — SOLAR ECLIPSE ९. नवमोऽध्यायः सौरसिद्धान्तः — रविग्रहणम् Introductory In the first portion of this chapter the Sun, Moon and Rāhu according to the Saura Siddhānta are given, and in the latter portion, the computation of the solar eclipse according to the same. In agreement with the author’s statement in his Introduction to the PS, that the tithi got by the Saura is very accurate, we see that not only the tithi but most other constants as well are wonderfully accu- rate, and approximate closely to the modern values. Among the five Siddhāntas this is the only one that uses epicycles to compute the Equation of the centre of the Sun and the Moon, and later in chapter XVII, the Equation of the centre and equation of conjunction of the ‘star-planets’, followed later by astronomers like Āryabhaṭa. The Ārdharātrika-pakṣa of Āryabhaṭa, expounded by Brahmagupta in his Khaṇḍakhādyaka, follows this Saura-Siddhānta in its constants. Though the computation of Days from Epoch (‘days’) has not been specially given for the Saura, (the rule given in I.13 not being clear whether it is related to the Saura or not), yet from the Yuga-elements of the Saura in I.14, it is possible to formulate rules for ‘Days from Epoch’, following the Saura, as has been shown by us in our Notes under I.14. We have also explained how the ‘days from Epoch’ obtained from the Romaka or Pauliśa rules can be used for the Saura also, provided we bear in mind the variation in time of commencement of the Epoch, as for instance, that the epoch for the mean Sun and Moon, their apogees, and the Moon’s node is for mid-day at Ujjain, and for the star-planets it is mid-night. Now, the author, intending to deal with eclipses, gives first the Sun, Moon and Rāhu on which eclipses depend, beginning with the mean Sun. [रविमध्यम्] द्युगुणोऽर्कोऽष्टशतघ्ने विपक्ष‘वेदार्णवे’ऽर्कसिद्धान्ते । ‘स्वरखाऽश्विद्विनवयमो’द्धृते क्रमाद्दिनदलेऽवन्त्याम् ॥ १ ॥ Mean Sun

1. According to the Saura-Siddhānta, to get the mean Sun in revolutions etc., multiply the days from Epoch by 800, deduct 442, and divide by 2,92,207. This is for Ujjain mean noon.
2. Paraphrased by Utpala or BS 2, p.65. 1a. A.B1. णेर्केष्ट. A2. शतघ्नो c. A.B1.2. ॰खाद्विधिनव०; C. ॰खाद्विद्विनव० d. A. हृते; B1.2. धृते. A2. दिनं. A1. ॰वत्यां; B. ॰वत्या

198 PAÑCASIDDHĀNTIKĀ IX.4 That is, take the days from Epoch got by the Romaka or Pauliśa rule. The mean Sun at Ujjain mean noon, just preceding the Epoch, (i.e. sunset at Yavanapura beginning Monday), = (days × 800 − 442) ÷ 2,92,207, in revolutions etc. Example 1 (a). Find the mean Sun, given days from epoch, 5,28,931. (This is midday, Ujjain, 21-5-1953 A.D.). (b) Get the mean Sun for Ujjain mean noon, just preceding the Epoch, i.e. for zero day. What is it at Epoch, i.e. sunset at Yavanapura? (a) Mean Sun = (5,28,931 × 800 − 442) ÷ 2,92,207 = revol. 1448-1-5-15.7 = rā 1-5-15.7. (b) Mean Sun at Ujjain mean noon preceding Epoch = (0 × 800 − 442) ÷ 2,92,207 = − 32'.7 = rā 11-29-27-3. Since mean sunset at Yavanapura is nā. 7-20 later than that at Ujjain, the mean motion for nā 22-20, about 22', has to be added, and the required mean Sun at Epoch is rā. 11-29-49.3. (According to modern astronomy it is 11-29-37.2, assuming that at that period the vernal equinox coincided with the First point of Meṣa. See how accurate the value is.) The rule is explained thus: We showed under I.14 that in the Saura yuga of 1,80,000 years, i.e. 1,80,000 mean solar revolutions, there are 6,57,46,575 days. Therefore, the revolutions for the given days = days × 1,80,000 ÷ 6,57,46,575 = days × 800 ÷ 2,92,207, reducing the numerator and denominator by the factor 225. Now, according to the Saura, the revolution of the mean Sun was completed 442/800 days after Ujjain mean-noon prior to Epoch. Therefore 442 eight hundredth parts have to be subtracted from the total number of eight-hundredths, and hence the deduction of 442. Though the original Saura is not obtainable now, yet from the Ārdharātrika system of Āryabhaṭa, and the Khaṇḍakhādyaka following it, we can see that 442/800 day after the said Ujjain- mean noon the mean Sun's revolution was completed. The Epoch was near the end of Śaka 427, i.e. 427 + 3179 = 3606, Kali years gone. Kali began with Friday, Ujjain mean mid-night. For 3606 revolutions, the days (from the beginning of Kali) = 3606 × 2,92,207 ÷ 800 = 13,17,123 42/800. Dividing out by 7, we have the remainder 3 42/800, i.e. 42/800 days after midnight ending Sunday, the revolutions was complete. Since Sunday mid-day is half a day or 400/800 day earlier than midnight, it is 42/800

• 400/800 = 442/800 day earlier, than the time of full revolution, as we have taken and used. Incidentally, we also got that it is Sunday mean noon, agreeing with the fact that the Epoch is at sunset at Yavanapura on that day. Further, we get that according to this Siddhānta the length of the year is 2,92,207 ÷ 800, days = 365 - 15 - 31.5 days. [चन्द्रमध्यं उच्चं च] नवशतसहस्रगुणिते ‘स्वरैकपक्षाम्बरस्वरर्तूने’ । ‘षट्शून्येन्द्रियनववसुविषयजिनै’र्भाजिते चन्द्रः ॥ २ ॥ नवशतगुणिते दद्याद् ‘रसविषयगुणाम्बरर्तुयमपक्षान्’ । ‘नववसुसप्ताष्टाम्बरनवाश्वि’ भक्ते शशाङ्कोच्चम् ॥ ३ ॥ ‘शशिविषय’घ्नानीन्दोः ‘(क्र)कर्मि’हतानि मण्डलानि ऋणम् । स्वोच्चे ‘दि(ग्घ्ना)’नि धनं ‘स्वर(न्ध्र)यमो’द्धृते विकलाः ॥ ४ ॥

IX.4 IX. SAURA-SIDDHĀNTA — SOLAR ECLIPSE 199 Mean Moon 2. Multiply the Days by 9,00,000, deduct 6,70,217, and divide by 2,45,89,506. The approximate mean Moon in revolutions etc. is got. 3. Multiply the Days by 900, add 22,60,356, and divide by 29,08,789. The approximate Moon's apogee in revolutions etc. is obtained. 4. Multiply the revolutions of mean Moon by 51, and divide by 3121. The resulting seconds of arc are to be subtracted to get the exact mean Moon. Multiply the revolutions of apogee by 10 and divide by 297. The resulting seconds are to be added to get the exact apogee. The following are the formulae: (i) Mean Moon in revs. etc = (Days × 9,00,000 − 6,70,217) ÷ 2,45,89,506 − number of revolutions ×

200 PAÑCASIDDHĀNTIKĀ IX.4 Example 3. (a) Days 5,28,931. Find the Moon's apogee. (b) Find the Moon's apogee for Ujjain mean noon prior to Epoch, and for Epoch. (a) By (ii) the approx apogee in revs. = (5,28,531 × 900 + 22,60,356) ÷ 29,08,789 = Revs. 164-5-5-33.1 The additive seconds = 164 × 10 ÷ 297 = 6. Adding, the exact apogee = rā. 5-5-33.2. (b) For the said Ujjain mean noon, the apogee = (0 × 900 + 22,60,356) ÷ 29,08,789 = rā. 9-9-45. Adding 2'.5, the mean motion of apogee in 22 1/3 nāḍikās, the apogee at Epoch = rā. 9-9-47.5. (Note: The actual position was rā. 9-9-34. See how close this is.) The explanation for the formula relating to the mean Moon is as follows: It was shown under I.14 that in the Saura yuga consisting of 6,57,46,575 days there are 24,06,389 revolutions of the Moon. For the sake of convenience, the author has first assumed that in whole numbers there are 9,00,000 revolutions in 2,45,89,506 days, intending to give a correction as a second step. Therefore we get that in 6,57,46,575 days there are 6,57,46,575 × 9,00,000 ÷ 2,45,89,506 revolutions = rev. 24,06,389-0-10-55-27. Thus we get 10° 55' 27'' more than what we should get, and this has to be deducted, proportion- ately to the revolutions got. For one revolution the deduction is, 10° 55' 27'' /24,06,389 = 39,327''/ 24,06,389. In the place of this fraction the author gives the approximate but simpler fraction 51''/ 3121, since the error caused will be only plus 4'' in the yuga. The manuscript reading, kharkāgni if read as khārkāgni as done by TS and NP, ( = 51''/3120) will cause an error of minus 8'', which also is negligible but unlikely, since the author then would have given the reduced form, 17''/1040. That is why we have read it as kvarkāgni, 3121. The deduction of 6,70,217 is explained in the manner of the Sun's deduction: We have seen that at the end of Śaka 427, the end of 3,606 solar years from the beginning of Kali fell 42/800 days, i.e. nā. 3-9, after Ujjain mean midnight after Epoch. Under I.14 it was shown that according to the Saura there are 24,06,389 revolutions of the Moon in 180,000 years. Therefore, in 3,606 years the revolutions gone are 48,207.992966̇. At the beginning of Kali, the Moon, like the Sun, began a revolution, according to the Saura. So, .007033̇ revolution remains to be completed now. We have seen that for 2,45,89,506 fractional parts there is one revolution. So, for .007033̇ revolution, the parts to go are 2,45,89,506 × .007033̇ = 1,72,946. These must go after the completion of the solar year to complete the revolution. But the year ends nā. 3-9 + nā. 30 = nā. 33-9 from mean noon. In one day, there are 9,00,000 parts, and for nā. 33-9, the parts to go are 9,00,000 × 33.15 ÷ 60 = 4,97,250. There- fore at mean noon the parts to go for completing the revolution are 1,72,946 + 4,97,250 = 6,70,196. Since these have to go, this number is deducted from the total parts got by multiplying the days by 9,00,000. Here, the author gives 6,70,217 arrived at by using approximate work in the place of 6,70,196, for the difference is small, the error caused being only minus one second in the yuga. Now for the explanation of the rule to get the longitude of apogee: We do this using the element given in Āryabhaṭa's Ārdharātrika system, or which is the same, in the Khaṇḍakhādyaka, since this is not given in I.14, and the original Saura is not available. From them we learn that in the Mahāyuga of

IX.6 IX. SAURA-SIDDHĀNTA — SOLAR ECLIPSE 201 1,57,79,17,800 days there are 4,88,219 revolutions of the Moon’s apogee. If the approximate rule given as the first part is used, we get that there are, 900 × 1,57,79,17,800 ÷ 29,08,789 revolutions = rev. 4,88,218-11-25-26-48 for the Mahāyuga. But this is 4° 33' 12" less than the correct value, and this latter has got to be added, per Mahāyuga, i.e. for 4,88,219 revolutions. Therefore the addition for the revolutions gone is, revolutions gone × 16,392" ÷ 4,88,219. In the place of this fraction the author gives 10"/297, as the difference is very small, for by using this the error will be only plus 2" in the Saura yuga of 1,80,000 years, which is negligible, especially in the apogee. If, instead of our (as also NP's) emendation, svararandhrayama, we make another emendation vasurandhrayama giving the fraction as 10"/298, then it will be very correct. As for our reading randhra in the place of the author’s dasra, it is necessary since otherwise there will be an error of plus one degree and a half in the Mahāyuga. That is why TS have given the emendation svaranandayama meaning the same as our reading, but randhra fits the letters better than nanda. The correctness of the kṣepa is shown hereunder: 3606 years of Kali ended nā. 3-9 after Ujjain mid- night next to Epoch. The revolutions of apogee for 3606 years = 4,88,219 × 3606 ÷ 43,20,000 = 407.527248611. At the beginning of Kali the longitude of apogee was 0.25 revolutions. Therefore at nā. 3-9 after the said Ujjain midnight, the longitude is 0.25 + 0.527248611 = 0.777248611 rev. The fractional parts (at 900 per day), for 0.77248611 rev. = 29,08,789 × 0.777248611 = 22,60,852. This is the kṣepa to be added at the end of the year. But Ujjain mean moon, for which we want the apogee, is nā. 33-9 earlier, and the parts for this interval = 900 × 33.15 ÷ 60 = 497 has to be deducted. ∴ the kṣepa is 22,60,355. The author gives 22,60,356, which differs by only one unit and causes practically no difference. [राहु:] ‘(त्रि)घन (शत)’घ्ने‘नवैकैकपक्षरामेन्दुदह(नरस)’सहिते | ‘(स्वर)यमवसुभूतार्णव-गु(ण)धृति’भ(क्ते) [क्र]माद् राहो: || ५ || चक्रात् पतितं (वक्त्रं) षड्राशियुतं तु पुच्छाख्यम् | (नव)तिविवरस्य लिप्ता विक्षेप: सप्त(तिर्द्वि)शति || ६ || Rāhu: Maximum latitūde 5. Multiply the days from Epoch by 2700, add 63,13,219 and divide by 1,83,45,827. Revolutions etc. are obtained, to be used in getting Rāhu. 6. This deducted from twelve rāśis is the Rāhu-head (i.e. ascending node of the Moon.) Rāhu-head plus six rāśis is the Rāhu-tail (i.e. descending node). At the (maximum) distance of 90° from Rāhu (the node), the Moon’s latitude is 270 minutes (i.e., this is the maximum latitude.) 5a. A1.B1.2.दिघ्नगजघ्ने (B1.2.°घ्नेन्) C.दशाघ्ने. B.चक्रे- धृतिभूता साद्राहो:; C. धृतिभि: D. °द्राहु: b. A.दहशब्दा:; B.दहनशब्दा:; C.दहशब्द्या: |; D.दहन षट्- 6a. A.B1.2. चक्रं B.प्रहिते (B2.3.°न्ते:) b. A.युतं वसुशाख्यं; D.च for तु c. A.चरयम; B.वरयम; C. om स्वर; D. करयम c. A.सहति; B1.अहति; B2.3.ग्रहति A.वसुघृतार्णव C.सहित; D.तिमिर d. A.गुणधृतभक्तभाद्राहो:; (A2.माद्राहो:) B.गुणा d. A.B1.2.सप्तता दिशती

202 PAÑCASIDDHĀNTIKĀ IX.6 The Head of Rāhu in revolutions etc. = - (Days × 2700 + 63,13,219) ÷ 1,83,45,827. The tail of Rāhu = the above + 6 rāśis. As for Moon's latitude, for a maximum moon ~Rāhu, equal to 90°, there is the maximum latitude, 270'. For other differences, lat = 270' sin (Moon ~ Rāhu) ÷ 120, as given in verse 25, which reduces to, lat = 9 sin (Moon ~ Rāhu)/4. This is given by the Saura, and followed by all later Siddhāntas. Example 4. Compute Rāhu (a) for Ujjain mean noon prior to Epoch, and (b) for Epoch. (a) In this case, days from Epoch is zero. ∴ Head of Rāhu in revs. = - (0 × 2700 + 63,13,219) ÷ 1,83,45,827 = - rā.4-3-53-3 = rā.7-26-6-57. (b) Since the Epoch is nā. 22-20 later, the motion for this interval, rev. 67/180 × 2,700 ÷ 1,83,45,827 = 1' 11'' has to be deducted. Rāhu-head according to the Saura for the time of Epoch, viz. mean sunset at Yavanapura, is rā. 7-26-5-46. Actually it is rā. 7-26-0, and the difference is within 6'. The calculation of the latitude will be explained in the context of the computation of eclipses. The rule of Rāhu: Like that for the apogee, this rule must be derived from the constants given in the two works that follow Saura since the original Saura is lost. In the Mahāyuga consisting of 1,57,79,17,800 days, there are 2,32,226 revolutions of Rāhu, (i.e. Moon's nodes). Using the rule for Rāhu here, we get, 2700 × 1,57,79,17,800 ÷ 1,83,45,827 = rev. 2,32,226-0-0-46-2 of Rāhu per yuga. This is 46' 2'' more than what we should get, but neglected by author as being small especially in a karaṇa intended to be used for a comparatively short period, considering the fact that even in 10,000 years the error is only 6'', which will not affect the result. That is why the second step of correction is not given, unlike in the case of the mean Moon and apogee. If a correction is wanted here also, multiply the revolutions by 10, divide by 848, and add the resulting seconds to Rāhu. Or, instead of using 1,83,45,827 as divisor use 1,83,45,827.2 i.e., in the rule, take the multiplier to be 27,000, kṣepa 6,31,32,190, and the divisor 18,34,58,272. At the end of 427 Śāka or Kali years 3606, the revolutions to get Rāhu = the longitude at the commencement + the revolutions in 3606 years. = 1/2 + 3606 × 2,32,226 ÷ 43,20,000 = 194 + 1,23,913/3,60,000. Omitting the full revolutions, the parts for the fraction remaining are the kṣepa, for the end of 3606 years Kali. Since there are 1,83,45,827 parts for a revolution, the parts of kṣepa = 1,83,45,827 × 1,23,913/ 3,60,000 = 63,14,684. Since we want the kṣepa for Ujjain mean noon, nā. 33-9 earlier, we have to subtract the parts for this time. Since there are 2700 parts in a day, for nā. 33-9 we have nā. 33-9 × 2700 ÷ nā. 60 = 1492 parts. ∴ the kṣepa for mean noon is 63,14,684 - 1492 = 63,13,192.

IX.9 IX. SAURA-SIDDHĀNTA — SOLAR ECLIPSE 203 The author gives 63,13,219, the difference, 27 parts, giving a difference of 2" in longitude being very small; for by neglecting a small fraction equal 1/7 in the divisor to make it a whole number, can give this difference. The readings here are extremely corrupt: Our explanation itself will show that the emendations we have made are necessary. We have read dvighanagaja as trighanaśata while TS give the correction trighanadaśa. The textual reading, carayamavasubhūtārṇavaguṇādhṛtibhakta is corrected by us as svarayamavasubhūtārṇavaguṇādhṛti-bhakte. But TS give the correction yamavasubhūtārṇavagu- ṇadhṛtibhiḥ. Here it is improper on their part to omit two letters cara though they require this omis- sion since in trighanagaja they have given daśafor gaja, instead of śata given by us. Nothing is gained by reading daśa instead of śata for gaja. Further, by omitting cara which is a corruption for svara, the number 7 in the unit's place is omitted by them, with the result that in the yuga an error of plus 30° and more is caused in Rāhu, while it is actually 46' 2" according to our correction. NP make the cor- rect emendation trighanaśataghne but emend cara to kara. We have corrected dahanaśabdāḥ as dahanarasa which fits the rule as shown. But TS content themselves with remarking that here the numbers of the kṣepa cannot be determined owing to the extreme corruption of the text. NP have made the emendation dahanaṣaṭ here, which too will serve the purpose. That the Head of Rāhu obtained by deducting what is got from 12 rāśis has been explained in dealing with the Paulīśa. We read sahati in the text as navati, since the difference of 90° between moon and Rāhu gives the maximum latitude, which is 270' according to the Saura, as also in all later Hindu Siddhāntas like the Āryabhaṭīya. Or we may read it as mahati, since the greatest difference, viz. 90° will give the greatest latitude, viz. 270'. But TS read it as sahita, and give something farfetched and unacceptable. NP emend sahati as timira, which neither accords with the lettering of the manuscript nor give the sense 90° required here. That the latitude is proportionate to the sine of (Moon ~ Rāhu) has already been explained in the context of the Romaka, and will also be shown below, in verse 25 of this chapter. [स्फुटरविचन्द्रौ] अंशाऽशी(त्या ही)नोऽर्कः केन्द्रं स्वोच्चवर्जितश्चन्द्रः | (तज्ज्या)ऽर्कस्य 'मनु'घ्नी 'रूपाऽग्नि'गुणा शशाङ्कस्य || ७ || 'व्योमरसाऽ नल' भक्ते तच्चा(पं) द्विस्थितं (स्वकेन्द्र)वशात् प्रथमे चक्रस्यार्धे क्षयश्रयः पश्चिमे भागे || ८ || सौर्यं स्थापितचापं तद्भुक्तिघ्नं 'खखा(ष्टि)यम'भक्तम् | प्रथमवदर्के कार्यं चन्द्रे च दिवाकरवशेन || ९ || (True Sun and Moon) 7. The mean longitude of the Sun minus 80° is called the Sun's (mean) anomaly. The mean Moon minus its apogee is its (mean) anomaly. Multiply the sine of the anomaly of the Sun by 14, and that of the Moon by 31. 8. Divide each by 360, and find their arcs. Put the Sun's arc in two places, for subsequent use. The arc of each is to be deducted from its mean longitude if

204 PAÑCASIDDHĀNTIKĀ IX.9 its anomaly is less than six rāśis, and added if more than six rāśis. (The true Sun and Moon at Ujjain mean noon is got.) 9. Multiply the Sun's arc, kept aside in one place, by the Sun's true daily motion, (in minutes), and that kept in the other place by the Moon's true daily motion (in minutes). Divide each by 21,600. Add or subtract the resulting minutes in the respective true longitude found, according as the Sun's arc was first added or subtracted. (The true Sun and Moon at Ujjain true noon is obtained.) The following are the formulae: (a) To get the true Sun: (i) Mean Sun − 80° = Sun’s anomaly. (ii) Sine Sun's anomaly × 14 ÷ 360 = sin Sun's equation of the centre. Its arc is the equation of the centre. (Eq.C). (iii) Mean Sun ∓ Sun’s equation of the centre = true Sun at Ujjain mean noon. (The upper sign, if the anomaly is less than 6 rāśis, lower if more.) (iv) iii ∓ Sun’s equation of the centre × Sun's daily motion in minutes ÷ 21,600 = True Sun at true mean noon. (Addition or subtraction as in iii.) (b) To get true Moon: (i) Mean Moon − Moon’s apogee = Moon’s anomaly. (ii) Sine Moon’s anomaly × 31 ÷ 360 = sine Moon's equation of the centre. Its arc is the equation of the centre. (iii) Mean Moon ∓ Moon’s equation of the centre = true moon, at Ujjain mean noon. (The upper sign, if the Moon’s anomaly is less than 6 rāśis, lower if more.) (iv) iii ∓ Sun’s equation of the centre × Moon's true daily motion in minutes ÷ 21,600 = True Moon at true noon. (Addition or subtraction as in (a) iii). Example 5. Days = 5,28,931. Find the true Sun at true noon, Ujjain. From example 1 (a) the mean Sun = rā. 1-5-15.7. The longitude of Sun’s apogee is 80°. From these two: (a) (i) The Sun’s mean anomaly = rā. 1-5-15.7 − 80° = rā. 10-15-15.7. (ii) Since anomaly = sine rā. 10-15-15.7 = Sin rā. 1-14-44.3 = 84' 27'', ∴ Sine equation of the centre = 84' 27'' × 14 ÷ 360 = 3' 17''. ∴ Equation of the centre = arc 3' 17'' = 1° 34'.1. (iii) True Sun at mean noon = rā. 1-5-15.7 + 1° 34'.1 = rā. 1-6-49.8, (addition because the anomaly is greater than 6 rāśis). (iv) True Sun at true noon = rā. 1-6-49.8 + 1° 34'.1 × 57.4 ÷ 21,600 = rā. 1-6-49.8 + 0'.3 = rā. 1-6-50.1. (That the daily motion of the Sun is 57'.4 will be given under verse 13, below.)

8a. B. रयानल A. तच्चाप C. द्विःस्थितं; B. दिस्थित 7a. A. ॰शीत्योद्विनो; B. अशात्योद्विनो A.B.C. शशाङ्कवशात्; D. शशाङ्करवौ b. A. B1.2. केन्द्रक्षः A1. वर्जित; A2. वर्जित; B. चज्जित c. B. om प्रथमे c. A. B1. ॰तज्यार्कस्य 9b. B. भक्तिघ्नं. A. B. खरखाब्धि d. A1. गुणिता; A2. गुणता d. A2. चंद्रेव

IX.9 IX. SAURA-SIDDHĀNTA — SOLAR ECLIPSE 205 Example 6. Find the true Moon at true noon at Ujjain, the Days being 5,28,931. From example 2 (a), the mean Moon for 5,28,931 days gone is rā. 4-11-25.7. From ex. 3 (a) the Moon’s apogee for the given days gone is, rā. 5-5-33.2. From these, (b) (i) The Moon’s anomaly = rā. 4-11-25.7 – rā. 5-5-33.2 = rā. 11-5-52.5. (ii) Sine anomaly = sine rā. 11-5-52.5 = sine rā. 0-24-7.5 = 49' 2". Sine equation of the centre = 49' 2" × 31 ÷ 360 = 4' 13".3. Equation of the centre = arc 49' 2" = 2° 1'. (iii) True Moon at mean noon = rā. 4-11-25.7 + 2° 1' = rā. 4-13-26.7. (iv) True Moon at true noon = rā. 4-13-26.7 + 1° 34'.1 × 729.1 ÷ 21,600 = rā. 4-13-26.7 + 3'.2 = rā. 4-13-29.9 (That the Moon’s daily motion is 729'.1 will be seen from example under verse 13 below. The addition is as the Sun’s Eq.C.) It should be noted here that the apogee of the Sun, given as 80° is too far from the correct apogee for the time of the work viz. 77° 19'. There is no doubt about the reading here, since the Ārdharātrika and the Khaṇḍakhādyaka too give 80°. So much error is unbelievable in the Saura, and must be explained thus: At first the practice might have been to get the mean longitude of the Sun for the days from the commencement of the true solar year and 80° deducted to get the anomaly, for this would be equivalent to deducting about 77° 50', (since the Eq.C at this time is about 2° 10'), from the correct mean Sun, not much different from the correct 77° 19' to be deducted. Later, by some mistake, the deduction of 80° was instructed to be done from the correct mean Sun itself. The apogee for the time computed by the Modern Sūrya Siddhānta is 77° 15'. From the instruction to multiply the sine of the Sun and Moon’s anomalies by 14 and 31, respec- tively, and divide by 360, to get the sine of the respective equation of the centre, we see that this Siddhānta actually uses epicycles like the Āryabhaṭīya etc, though not mentioning the word, and we can say that epicycles appear in the Hindu Siddhāntas for the first time in the Saura, and the others following using epicycles and excentries. The Modern Sūrya Siddhānta gives the same degrees of epicycle for the Sun, but 32° for the Moon instead of 31°. Further, in the Saura, the epicycle is uniform, while in many Siddhāntas like the Āryabhaṭīya there is difference between the degrees at the ends of odd and even quadrants. For instance, the degree of epicycle mentioned above for the Sun and the Moon in the Sūrya Siddhānta is for odd quadrants, being less by 20 minutes at even quadrants. The computations mentioned above can be simplified, since the multiplier and the divisor are constants and small arcs are proportionate to the sines. Thus, we can get the Sun’s Eq.C. in minutes by multiplying its sine anomaly by 1.114. In the example, multiplying 84' 27" by 1.114 we get 94' 6'', the equation of the centre. We can get the Moon’s Eq.C. by multiplying its sine anomaly by 2.467, and if the result is in excess of 225 minutes, adding 1/235 of the excess to the result. In the example, multiplying 49' 2" by 2.467, we get the equation of the centre, 121' 3". In the same way, we find the Sun’s maximum equation of the centre to be , 120' × 1.114 = 133'.7. The correct maximum for the period of our author is 119'.5. The large difference is due to the Moon’s Annual Equation being wrongly applied to the Sun with its sign changed, in Hindu astronomy, as already alluded to, since by doing so the tithi is not affected, the constants having been derived by the analysis of the syzygies, which are, in essence, ends of particular tithis. Adding the maximum Annual equation to the correct equation of the centre of the period, we have 131'.5. See how close this is to the value, 133.7 of the Saura, and how far from the 140' of the Pauliśa, and the 143' of the Romaka and of Ptolemy.

206 PAÑCASIDDHĀNTIKĀ IX.9 In the same way, the maximum of the Moon’s equation of the centre is 120' × 2.467 + (120' × 2.467 − 250) ÷ 235 = 296' + .3' = 296' .3. This too was determined by analysis of syzygies at the occurrence of eclipses. According to modern astronomy, the mean of the maximum equation of the centre of the Moon at true syzygies is 297'.3, a difference of only one minute! (At mean syzygies it is 303'.5). Epicyclic theory We shall now proceed to explain the epicyclic theory of planetary motion, used by this Siddhānta, and show how it works, by relating it to the modern theory, which latter is as follows: The earth and the other planets like Mercury etc. move round the Sun in eclipses, with the Sun at one of the two foci. The point nearest to the Sun on the ellipse is the perihelion, and the most distant, aphelion, which, from the point of view of the earth, are called the perigee and apogee, respectively. In the same manner, the Moon moves in an eclipse round the earth at one focus. This fact relating to the planets was first discovered by the European astronomer Kepler and is called Kepler’s First Law of planetary motion. The line joining the Sun and the planet (or the earth and the Moon), called the radius vector, sweeps equal areas in equal time. This is Kepler’s Second Law of planetary motion. From this it can be readily inferred that the motion of the planet is swiftest at perihelion (or perigee for the Moon) and lowest at aphelion. Kepler’s Third Law, that the square of the periodic time of the planets round the Sun is proportionate to the cube of the distance, is not wanted for our purpose here. The celebrated astronomer, Newton, showed that all the three laws follow from his Theory of Universal Gravitation, that all bodies attract one another with a force proportionate to their masses, and inversely proportionate to the square of the distance between them. But ancient Indian astronomers held the view that the earth is the centre round which the Moon, Sun and planets move. All the motion is in circles, and uniform. To explain the non-uniformity of the apparent motion caused by the equation of the centre, it was assumed that these bodies moved in circles called epicycles, (manda-vṛttas), the centres of which moved in circles round the earth as centre. In the case of the star-planets, another set of epicycles called epicycles of conjunction or śīghra-vṛttas were assumed, the effect of which is to convert helio-centric positions into geocentric. As the observers are on the earth, it is geocentric positions that are wanted, and therefore given, whether by Hindu astronomy or by modern western astronomy. The inaccuracy in the positions given by the former is due to unawareness of the elliptic motion and the inability to observe accu- rately for want of adequate instruments, with the result that small errors in the constants accumu- lated in course or time, to give large errors. It has been said that whether the heliocentric theory is adopted or the geocentric theory, the result in so far as this goes, is the same. Then, it may be asked, is it not better to adopt the geocentric theory which agrees with our perception? No. There is a clinching proof for the motion of the earth round the Sun, in the phenomenon of aberration which makes the Sun and star planets appear to be a little in advance of their real positions, the quantity being so small that very accurate measure- ment is required to find it, capable only by modern instruments. The heliocentric theory is also simpler, and satisfies the requirement of least assumption. Adhering to only circular motion, so satisfying to their minds, the ancients had to get the equa- tion of the centre, caused by the motion on the ellipse. They sought to achieve it in two ways, by using epicycles, as indicated already, or excentric circles, or both. How the ex-centric is used for the purpose is explained as follows:

IX.9 IX. SAURA-SIDDHĀNTA — SOLAR ECLIPSE 207

Fig. IX. 1-a.

The earth E, is the centre of the Orbit-circle, of 'radius' equal to the sine of Three rāśis (120' in this work). O indicates the first point of Meṣa, in the direction EO. EA is the direction of apogee, A, O EA being the longitude of apogee (P is the perigee). X is the centre of the ex-centric circle on which the Sun, Moon or star-planets move uniformly according to the mean motion. X is on EA, at a distance, towards A, equal to the sine of the maxi-minimum Eq. C. S is the position of the body, angle SXO' being the mean longitude of the body. XO' also is directed to the first point of Mesha. EO and XO being practically parallel. ASX = mean longitude – longitude of apogee = mean ano- maly.

SEO is the true longitude, which has got to be found. Since XO' and EO are parallel, SEO = SXO' – XSE, where XSE is the Eq. C. Since XSE changes sign on the right hand side of PA, SEO = SXO' + XSE on that side. Thus we have that in the first case, when the body is from apogee to perigee, i.e. when the anomaly is less than six rāśis, the equation of the centre is subtractive. In the second case, where the mean anomaly is more than six rāśis, is it additive.

Next, for the equation of the centre. If the maximum Eq.C, represented by EX, is small, as in general, then taking SE and SX to be practically equal, sine Eq. C = sine XSE = XE. sine SXE ÷ SE = XE. sine SXA ÷ SX = max Eq.C × sine mean anomaly ÷ 120' (120' being the radius).

In this computation, the astronomers belonging to the school of Āryabhaṭa find the Eq.C using the actual radius vector, SE, in accordance with the geometric representation. But Bhāskaracārya in his Siddhānta Śiromaṇi does not use it, and gives reasons for not using it. Now, if degrees of epicycle are given, as in this Siddhānta, instead of sine maximum Eq. C, these degrees are multiplied by 120' and divided by 360° to get sine maximum Eq. C. (i.e. EX). Therefore, sine Eq. C = (degrees of epi- cycle × 120' ÷ 360°) × sine anomaly ÷ 120'

208 PAÑCASIDDHĀNTIKĀ IX.9 = degrees of epicycle × sine anomaly ÷ 360°, as in the text. We shall now see how the use of the epicycle gives the Eq.C. [Diagram showing epicycle model with centre E (Earth), deferent circle, epicycle centred at C, body S, apogee A, A', direction to first point of Meṣa, and point O] Fig. IX. 1-b. Here too, E, the centre of the earth is the centre of the orbit circle of radius 120'. On the orbit circle, the centre C of the epicycle on which the body S is situated, moves according to the mean motion of the body. The radius of the epicycle is the degrees of epicycle given in the text × 120' ÷ 360°, which is the maximum Eq.C, as already seen. At the two points of intersection of the epicycle with the line of apogee, EA, arc A the apogee, and P the perigee. The body moves on the circumference of the epicycle, with its mean motion in the direction opposite to the motion of C. Angle ACS is the anomaly. Now, draw EA' parallel to CS. Then angle AEA' also is the anomaly. Since CEO is the mean body, A'EO is equal to the longitude of apogee. Since SEO is the true longitude and CEO is the mean longitude, angle CES is the Eq.C. Therefore, mean longitude minus Eq.C. = true long, (the anomaly being less than 6 rāśis in the figure. If the anomaly is more than 6 rāśis, S is greater than P, and the Eq. C. becomes additive). The Eq. C. is got in the same way as in the excentric method. We have now to show that in either of the methods, the Eq. C. is the same, i.e. angle XSE in fig 1a = angle CES in fig 1b. In the two triangles XSE and CES in the respective figures, it has been

IX. 10 IX. SAURA-SIDDHĀNTA — SOLAR ECLIPSE 209 said that EX is equal to CS. XS is equal to CE, both being equal to 120'. Angles SXE and SCE are also equal, since they are 180° minus the equal mean anomalies, SXA and SCA. Therefore, the two triangles are congruent, and so angles XSE and CES are equal, as required to be shown. In the matter of the correctness of the degrees of epicycle we have to take the authority of the work. That it agrees with the Original Saura can be seen, the same being found also in the Ārdharātrika system and Khaṇḍakhādyaka. Bhujāntara correction We shall now show why the correction called Bhujāntara is done. The Days used in the formulae are mean solar days. (That is why the mean longitudes are taken to be proportional to them.) There- fore the true longitudes got are for mean noon. But we want the longitudes for true noon. So we have to apply a correction which is the motion during the interval between mean and true noons. If the Sun's Eq.C. is positive it is east of its mean position, and reaches the meridian later than mean noon by a certain number of prāṇas (prāṇa = 4 seconds of time, one sixth of a vināḍī) equal to the number of minutes of arc. Being later, the Sun and Moon's motion during the interval has to be added. If the Eq.C. is negative, then the true Sun is west of its mean position and true noon is earlier. Therefore the motion is to be subtracted. The motion in the interval is found by the proportion: (Since there are 21600 prāṇas in a day) 21600 : daily motion :: Minutes of Sun's equation of the centre: motion during the interval. From this it can be seen that this correction has got to be done not only to the Sun, but also to the Moon and the star-planets as well. Udayāntara correction We must add that this correction for equation of the centre is not sufficient. Another correction has got to be made for what is called Udayāntara or reduction to the equator, i.e. reducing the motion on the ecliptic to motion on the celestial equator which is the circle on which time has to be measured. Both these corrections form the equation of time, being the interval between true and mean noons. But Hindu astronomers prior Śrīpati were unaware of this correction. [देशान्तरसंस्कारः] पञ्चाशता त्रिभिस् त्र्यंशसंयुतैर्योजनैश्च नाड्येका । समपूर्वपश्चिमस्थैर्नित्यं शोध्या च देया च ॥ १० ॥ Deśāntara correction 10. One nāḍī for every 53 1/3 yojanas has to be deducted or added (to Ujjain noon) by people in places east and west, respectively, of the Ujjain meridian, (to get their own noon.) Ujjain was the Greenwich of the Hindus, and the line of longitude passing through Laṅkā, Ujjain and the North pole was taken as the prime longitude. It is well-known that noon occurs earlier and earlier as the longitude of a place is more and more east, and vice versa. The author says that for every 53 1/3 yojanas of distance east or west, there is one nāḍī earlier or later. The idea is that there- fore the daily motion of the body should be multiplied by the nāḍīs got, divided by 60, and the resulting minutes of arc should be subtracted or added to the true longitude, according as the place is east or west, to get the longitude for local noon. 10a. A. पञ्चांशताः

210 PAÑCASIDDHĀNTIKĀ IX.11 Example 7. Days, 52931. Benaras is east of Ujjain longitude by 68 yojanas. Find the true Moon at noon at that place. From example 6, the true Moon at Ujjain noon is rā.4-13-29.9. The difference in time for Benaras = nāḍikās (68 ÷ 53 1/3), earlier. From this, the correction for local noon = 729'.1 × 68 ÷ (53 1/3 × 60) = 15'.5, subtractive. (The Moon's daily motion for the day will be shown to be 729'.1 under verse 13.) ∴ The true Moon required = rā.4-13-29.9 − 15′.5 = rā.4-13-14.4. The instruction is explained thus: The author takes it that for the region of Ujjain the length of the latitude circle is 3200 yojanas. The Sun, in its apparent diurnal motion westward goes once round the circle in 60 nāḍikās, crossing all the meridians on the earth. There are thus 3200 ÷ 60 = 53 1/3 yojanas for one nāḍī, and a place east by this distance has its meridian crossed by the Sun, i.e. its noon earlier by one nāḍī, and west, later. The motion for this time is calculated by the proportion, 60 nāḍīs: daily motion :: the nāḍīs got: the motion for the same. That the motion is deductive if the noon is earlier, and vice versa is plain. (This is for direct motion, the Sun and the Moon alone being considered here. If the daily motion is retrograde, as is possible in the case of the star-planets, it is obvious that the subtraction and addition have to be reversed.) But it is to be noted that in the Ārdharātrika of Āryabhaṭa and in the Khaṇḍakhādyaka, the diameter of the earth is given as 1600 yojanas from which the equatorial circumference got is 5027 yojanas. Therefore the Original Saura must have given the same values. The Modern Sūrya Siddhānta, and the Siddhāntas that follow it also give the same. From this the latitude circle at or near Ujjain should be given according to them as 5027 cos 24° = 4600 yojanas. According to the Āryabhaṭīya, which uses a yojana measure one and a half times that of Saura etc., the equatorial circumference would be 3300 yojanas. From this, it is 14° latitude circle that would be 3200 yojanas, and not Ujjain latitude circle. Why should the author use the yojana measure of the Āryabhaṭīya instead of that of the Saura, and in that why should he use the yojanas of the 14° latitude circle instead of the Ujjain (24°) latitude circle seems inexplicable. [रविशशिनोर्मध्यभुक्तिः] नव(तिः) सप्तशतीन्दोः सचतुस्त्रिंशद्विलिप्तिका भुक्तिः । षष्ट्यैका विकलाऽष्टकं च मध्या सहस्त्रांशोः ॥ ११ ॥ Mean motion of the Sun and the Moon 11. The mean daily motion of the Moon is 790′ 34″, and that of the Sun is 59′ 8″. These can be derived from the mean motions of the Sun and the Moon given in the first and second verses. In the former it is said that in 2,92,207 days there are 800 revolutions of the Sun. ∴ 800 revolutions ÷ 2,92,207 = 59′ 8″, the motion for one day. In the latter, we have that there are 9,00,000 revolutions of the Moon in 2,45,89,506 days. ∴ the motion per day = 9,00,000 revolu- tions ÷ 2,45,89,506 = 790′ 34″. The correction in verse 4 is too small per day to consider. 11a. A.B.नव (A.नच) सप्तसर्तींदोः b. B.त्रिंशद्विलिप्तिकासु भुक्तिः d. B.मध्याग्रहस्त्रांशोः