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

the current month, the total tithis are obtained. These lessened by the number of elided days in the period between the Epoch and the time taken gives the Days from Epoch, for the elided days are the tithis left out of reckoning in one to one correspondence between the tithis and the days. Here the elided days are obtained by the proportion, if for the tithis in the yuga numbering 10,57,500 there are 16,547 elided days, how many elided days are there for the tithis from the Epoch to the taken time; i.e. 10,57,500 : 16,547 :: the intervening tithis : the intervening elided days. So, we have the equation, the intervening tithis × 16,547 ÷ 10,57,500 = the elided days. Here the multiplier for the tithis, viz., the fraction 16,547/10,57,500 can be expressed as a continued fraction to find a suitable smaller fraction for easy work, thus: 1 16547 1057500 63 1 1508 15039 9 1 41 1467 35 1 9 32 3 4 4 5 1 0 1 1 1 1 1 1 1 1 1 1 i.e. 16,547/10,57,500 = ——— ——— ——— ——— ——— ——— ——— ——— ——— 63+ 1+ 9+ 1+ 35+ 1+ 3+ 1+ 4+ The successive convergents obtained from this are: 1/63, 1/64, 10/639, 11/703, 395/25244 etc. Of these the author has taken 11/703 as being simple and, at the same time, sufficiently accurate for the purposes of this work, for even during a period as large as the yuga, the difference in the elided days will be only 10,57,500 (1654/10,57,500 − 11/703) = 1/17, and this is small in comparison with the difference caused by actually using the true tithi in the formula, which we are constrained to use, in the place of the mean tithi which, according to theory we must use. Now, at the time of Epoch there was a fraction of elided day equal to 514/703, and, as this has also to be added, the additive constant 514 is given. As done in the case of the intercalary month, here also we can examine the correctness of the constant, 514, thus: the fraction of elided day is the part of the current tithi gone before the time of beginning of the new day, as in the present case, viz., the Romaka before sunset at Yavanapura. We have seen before that at Epoch there remains 16½ nāḍikās for the mean new moon to end, i.e. about 43 nāḍikās have ended in mean Amāvāsyā tithi. The constant 514 means that 514/703 part of the Amāvāsyā has gone and this is equal to about 43 nāḍikās and thus the constant is practically correct. It is because of the existence of this constant that we have interpreted, caitra-śuklādau as ‘when the first tithi of Caitra was about to begin’. Further, we have seen that at Epoch Amāvāsyā is current and Caturdaśī is gone. But, taking the Amāvāsyā as gone, the tithis to be used in the formula are asked to be reckoned from the first tithi of the month. That is why we gave the instruction to add the tithis from Śukla-Pratipad to the current tithi, though the usual instruction would be to add only the tithis gone. It must be noted that the author’s instruc- tion is simpler and at the same time not incorrect. Also, there is the usual practice of comparing the week day for the obtained Days from Epoch, with the actual week day of the taken time, and adding or subtracting a day from the days got, if necessary, which will take care of everything. Thus the whole thing is explained. The Śaka year is the year of the Śaka era which began at 3179 Kali (elapsed), for the Siddhāntas instruct that 3179 should be added to the Śaka year to get the Kali year. The purpose of mentioning that Caitra Śukla Pratipad occurred near the Epoch is to indicate that the months gone must be

10 PAÑCASIDDHĀNTIKĀ I.10 counted from Caitra and the tithis from Śukla Pratipad. The moment of Epoch is given as mid- sunset at Yavanapura, because the Sun has an angular diameter of about 32', and the time between the beginning and end of its immersion below the horizon is considerable. The practice of beginning the day at sunset was, in those days, prevalent in the countries near Yavanapura, which practice is still followed by Jews and Muslims, as in India certain Siddhāntas like the Sūrya Siddhānta begin the day at midnight, which is used for certain injunctions of the Dharma-śāstras, while certain other works like the Āryabhaṭīya etc. begin the day at sunrise which is used for certain other injunctions of the Dharma-śāstras. Yavanapura is Alexandria in Egypt, the ancient capital of the country, where Ptolemy II, the famous astronomer and author of the Almagest, ruled and which was well known to the astronomers of India. How do we know that it is Alexandria and no other city? In III. 13 the time-difference between Yavanapura and Ujjain due to their difference in longitude is given as seven nāḍīs and twenty vināḍīs and sunset at Yavanapura is later. From this we can see that it must be a well known place 44° west of Ujjain in longitude and its position agrees with that of Alexandria. We have said that the moment of Epoch begins Monday, somadivasādye. This reading is that of Bhaṭṭotpala, quoting the verse in his commentary of the Bṛhatsaṃhitā and we have adopted it as the correct one. It does not matter if we adopt another reading, saumyadivasādye, for we can interpret this as 'the day pertaining to the Moon', i.e. Monday, because the word saumya can be interpreted as 'belonging or pertaining to the Moon'. It cannot mean Wednesday, as it might appear at first sight, (the word saumya being a name for Mercury), for it must be Monday because the Lord of that day as computed from I.20 is the Moon and not Mercury. We shall show how. In I.17 it is instructed that 2227 should be added to the Days from Epoch to get the lords of the year, month, day and horā. Because the Days from Epoch gone is patently zero at the Epoch itself, we have 2227 + 0 = 2227, from which to get the Lord of the day. The instruction is to divide this out by seven, and take the remainder, which gives the Lord of the day gone counting from the Sun, in the order Sun, Moon, Mars etc. Now we want the Lord of the 2228th day, and dividing 2228 by 7, the remainder is 2, i.e. Moon is the Lord of the day and it must be Monday. This can be shown in other ways also but this is enough here. When there is this fact of a Monday and the reading somadivasādye to support it, the interpretation by some as 'at the beginning of Wednesday' has to be discarded. There is another reading, bhaumadivasa which has been accepted by the two scholars, S.B. Dikshit and Bhau Daji, and also by NP, not remembering that the formula has been and can be constructed only on the basis of the mean constants and not of the true constants and not understanding the purpose of the statement caitraśuklādau, as such that reading has also to be discarded. Note also that the Romaka ahargaṇa mentioned in verse 17 below, viz. 2227, works out only to Monday, not Tuesday, since the cycle commences from Sunday. We have given as one interpretation of nāticire Pauliśe 'py evam, 'It can be taken as the Pauliśa rule also, provided the time taken for computation is not very far from the Epoch'. Strictly speaking, in the rule given by a particular Siddhānta, only the synodic month and the tithi of that Siddhānta must be used to get the Days from Epoch. But as given in I.4, the tithi of the Romaka was near that of Pauliśa at the time of Epoch and so the Romaka rule could be used for the Pauliśa for some time, especially because there is the check by comparing the week-days. Another thing to be noted is this: Whatever Siddhānta is used to compute the days from Epoch, the result must be the same. That is why no separate rule has been given either for the Vāsiṣṭha or for the Saura, for we can use days of the Romaka or Pauliśa for these also, mutatis mutandis. TS interpret nāticire Pauliśe' py evam as 'the rule is the same for also the Pauliśa Siddhānta which was

I.10          I. INTRODUCTION OF THE WORK          11 written not long ago’. But the time of a work is irrelevant to a manual of the sort the author is writing and he is not interested in giving it. As a result of this interpretation, they have taken that the Pauliśa rule is the same as the Romaka rule, with the result that they have not been able to see that the following verses 11-13 give the rule of the Pauliśa, though they are quite capable of under- standing and interpreting them. NP translate, ‘It is not very different in the Pauliśa’, without explaining nāticire. [पौलिशसिद्धान्तानुसारी अहर्गण:] ‘दि’घ्नाः सा‘ष्टनवरसा’ दिवसा ( ‘एकर्तु’)सप्तनव’भक्ताः | पौलिशमतेऽधिमासाः ‘त्रिकृत’दिनान्यवमसंक्षेपः || ११ || Days from Epoch according to Pauliśa 11. (The formula for Days from Epoch according to the Pauliśa, is as follows:) As in Romaka (I.8-10), deduct 427 from the Śaka year (elapsed). Multiply by 12 and add the months gone from Caitra. Multiply by 30. The ‘Solar days’ (S-days) to the end of the current solar month are got. Multiply the S-days by 10, add 698, and divide by 9761. The quotient are the intercalary months. (Again, as in Romaka), multiply the months got by 30 and add to the S-days, and add also the tithis from śukla-pratipad, inclusive of the current tithi. The sum is the tithis gone from Epoch. Multiply this by 11, add 444 (tri-kṛta) and divide by 703. The quotient are the elided days. Deduct this from the tithis gone. The remainder are the Days from Epoch. Here the word divasāḥ is interpreted as ravi-divasāḥ, i.e. ‘solar days’, because it comes in the place of ‘solar months’ in the formula. The number of ‘solar days’ is equal the number of degrees traversed by the Sun, the time taken for moving one degree being taken as one ‘solar day’ by Indian astronomers. It is not what is meant in modern astronomy, the time interval taken by the Sun for the successive crossing of the meridian. To avoid error of syntax, ‘sāṣṭānavarasa’ is emended into ‘sāṣṭanavarasā’. Following the sense, in the place of kurtu and rutu, the reading ekartu is substituted. NP editorially add before divasāḥ the word saura, which is not necessary, as it can be inferred. NP’s translation gives the number 9761 with an emended reading kṛtusaptanava. Again, the ms. reading tri-kṛta has been changed to tri-ṣaṭ, with the translation, ‘there is an omitted tithi every 63 days’, missing to see that tri-kṛta (444) is the Pauliśa kṣepa in place of the Romaka kṣepa 514 of the previous verse, to be used in the Pauliśa calculation. 11a. A1.2.D. दिग्नाः; C. दिघ्ना          A1.2. कर्तु; B1.2. रुतु; C. क्रतु; D. [कृतु] A1.2.B1.2. साष्टा            c. C. त्रिक्रतु; D. त्रि [षड्] A1.2.C. नवरस; D. नवरसाः [सौर] दि०    d. B2. ०नान्यिवम० b. A1.2. B1.2. om. ए            A1.2. संशेषा

14 PAÑCASIDDHĀNTIKĀ I.13 should be included for greater accuracy and it can be done by an appropriate addition in the S-days, by the proportion: If 10/9761 intercalary month is got for one S-day, by how many S-days is (1 + 1/550)/9761 intercalary month got? Thus we get S-days equal to, (1 + 1/550)/9761 ÷ 10/9761 = (1 + 1/550)/10 = 1/10 + 1/10 × 1/550 . This is for every 107 years, and so, for every 107 years, 1/10 S-day has to be added for greater accuracy in getting the intercalary months and for every 550 such addi- tions one more tenth is to be added, which is the instruction given. (This is the reason for our giving as the correct reading, 'tithidaśamāṁśam where tithi according to the context means S-day). Now we proceed to explain the part of the formula relating to the elided days. We got before that there are 11,40,37,61,190 elided days in a period of 7,28,80,32,70,590 lunar tithis or simply tithis. Cancelling out a factor 30, we have 38,01,25,373 elided days for 24,29,34,42,353 tithis. So, to obtain the elided days for tithis gone we have the proportion, 24,29,34,42,353: 38,01,25,373 :: tithis gone: elided days during the period, i.e. elided days = tithis gone × 38,01,25,373 ÷ 24,29,34,42,353. The multiplying fraction 38,01,25,373/24,29,34,42,353 can be expressed as a continued fraction thus: 1 38,01,25,373 24,29,34,42,353 63 1 3,45,81,519 34,55,43,854 9 2,71,336 3,43,10,183 126 .... .... 38,01,25,373/24,29,34,42,353 = 1/(63+) 1/(1+) 1/(9+) 1/(1+) 1/(126+) ......... The successive convergents are 1/63, 1/64, 10/639, 11/703, 1396/89217 etc. Of these, our author has taken 11/703 (note that this is the same as that of the Romaka) as being enough for a first approx- imation. By taking this, 38,01,25,373/24,29,34,42,353 − 11/703 = 2,71,336/(24,29,34,42,353 × 703) elided day is left out for every tithi. In the period of 245 years, given in the rule, there are, from the constants given before, 7,28,80,32,70,590 × 245 ÷ 1,96,40,88,000 tithis. So in this period the left out elided day is {2,71,336/24,29,34,42,353 × 703} × {72,88,03,70,590 × 245 ÷ 1,96,40,88,000} = 16,61,933/(16,36,740 × 703). This can be included in the formula by making a proportionate change in the tithi thus: To get 11 elided days we have to take 703 tithis, to get the elided days left out in 245 years, we must take tithis equal to 703 × 16,61,933 ÷ (16,36,740 × 703 × 11) = 16,61,933 ÷ (16,36,740 × 11) = (1 + 25,193/16,36,740)/11 = 1/11 + 25,193/(16,36,740 × 11). In this the first term 1/11 is given by the instruction to add an eleventh of a tithi every 245 years. The second term does not agree with the instruction to omit adding one eleventh for every addition of 2,03,279 elevenths. This may be due to several reasons. It may be that the mean motion for 3031 days is given to the nearest minute, and small as this is, it can affect the value of the correction which itself is very very small. Or the Paulīśa Moon is slightly different from the Vāsiṣṭha Moon, which we have assumed for the Paulīśa. Or there is some error in the text here. We must be satisfied with the other and more important items of agreement. It must be remembered here that TS have omitted even the translation of these two verses, as a hopeless task.

I.16 I. INTRODUCTION OF THE WORK 15 Now we proceed to examine the kṣepas used in the formula. At the time of Epoch, the Vāsiṣṭha mean Moon is 11ʳ 25° 6′ (vide II.3). As done before, we assume this for the Pauliśa also. The Pauliśa mean (‘mean’ here is the assumed mean) Sun is 11ʳ 29° 44′ (vide III.1). From these we can see that the mean new moon will occur after 23 nāḍikās. From the kṣepa for elided day given, 444, we can see that the end of the Amāvāsyā occurs, before the beginning of the next day by 444 × 59/703 = 37 nāḍikās, i.e. 23 nāḍikās after the Epoch, and thus there is agreement. (This shows that the reading ‘trikṛtadināny avamasaṅkṣepaḥ’ is correct). We shall examine the kṣepa for the intercalary month. The kṣepa given is 698. Dividing by the given divisor, 9761, we see that at the time of Epoch there is a fraction of 698/9761 intercalary month left. As the fraction of intercalary month is the interval from new moon to the next ending moment of the solar month, we get that 698/9761 synodic month = 2 days and 6½ nāḍikās after new moon, the Sun enters the next rāśi, here Meṣa. We have seen that the mean new moon itself falls 23 nāḍikās after Epoch. Therefore we get that the Sun enters Meṣa 2 days 6½ nāḍikās + 23 nāḍikās = 2 days 29½ nāḍikās after Epoch. The proper mean Sun computed for Epoch is 11ʳ 27° 33′ (vide III. 1-3), i.e. after traversing 2° 27′, i.e. after 2 days 29½ nāḍikās, the Sun will enter Meṣa. This is the same as what we have computed from the kṣepa 698, and thus it is verified. Perhaps the reader has noted here that in the verification of the kṣepa for elided day we have used the assumed mean Sun (written ‘mean’ Sun) at Epoch and of the kṣepa for intercalary month, the proper-mean-Sun at Epoch. Is it proper, he may ask? Logically it is not. But, after all, what we want is to get the Days from Epoch correctly. If, by this shift, the rule is simplified, without sacrificing accuracy, then there is no harm in having recourse to it, thinks the author. We have already said that the mean Sun and Moon can alone be taken in framing the rule here. What we have called above, the ‘proper-mean’ is really the mean and so that part is all right. If here the assumed mean Sun is used, which is practically the true Sun at Epoch, an intercalary Vaiśākha will be falling immediately which will necessitate giving a kṣepa almost equal to the divisor 9761 and cause a lot of trouble. So the author has done what is only proper here. Then why not use the mean Sun to get the elided day kṣepa also? The Pauliśa, in giving its peculiar method, has assumed the beginning of the true Solar year as that of the mean Solar year, so that the true Sun at that point is assumed as the mean Sun. Our author has taken it as it is given and computed the kṣepa for the elided day accordingly, for, as we have already said, there must be the check by comparing the weekday and that will take care of everything. Or, some astronomer, unaware of the illogicality, has handled the kṣepa. While TS omit to translate the verses 11-13, merely stating that the details are obscure (Tr. p.5), NP change several ms. readings, daśamāṃsa to daśāṃsa, pañcakṛtadvisammitāḥ to pañcatanudvid- vimitāḥ, ekīkartum to eka ṛtu, without getting anywhere near the correct sense. [सौर-रोमकयोः रवि-चन्द्रयुगम्] वर्षायुते ‘धृति’घ्ने ‘नववसुगुणरसरसाः’ स्युरधिमासाः | सावित्रे ‘शरनवखेन्द्रियार्णवाशाः’ तिथिप्रलयाः || १४ || रोमकयुगमर्कैन्द्वोर्वर्षाण्या‘काशपञ्चवसुपक्षाः’ | ‘खेन्द्रियदिशो’ऽधिमासाः ‘स्वरकृतविषयाष्टयः’ प्रलयाः || १५ || युगवर्षमासपिण्डं रविमानं साधिमासकं चान्द्रम् | अवमविहीनं सावनमैन्दवमब्दान्वितं त्वाक्षम् || १६ ||

16 PAÑCASIDDHĀNTIKĀ I.16 Yuga of the Sun and the Moon (Romaka and Saura) 14. In the Saura Siddhānta, a period (actually the minor yuga) of 1,80,000 solar years contains 66,389 intercalary months and 10,45,095 elided days. 15. The luni-solar yuga of the Romaka Siddhānta consists of 2850 solar years. In this period, there are 1050 intercalary months and 16,547 elided days. 16. The solar years in the yuga multiplied by 12 gives the solar months in the yuga. The solar months plus the intercalary months are the synodic months in the yuga. The tithis got by multiplying the synodic months by 30 reduced by the elided days, are the civil days, (i.e. days) in the yuga. The civil days plus the solar years are the sidereal days in the yuga (or the synodic months plus the solar years are the Moon's revolutions in the yuga). Example 3. Give the revolutions of the Sun and the Moon, the civil days etc. in a yuga (minor) of the Saura Siddhānta. There are 1,80,000 solar years in the Saura minor yuga, and as a solar year is the period of revolu- tion of the Sun, there are 1,80,000 solar revolutions in the yuga. Multiplying the solar years by 12, the solar months in a yuga are 12 × 1,80,000 = 21,60,000. The synodic months are solar months plus intercalary months = 21,60,000 + 66,389 = 22,26,389. The tithis are 30 × 22,26,389 = 6,67,91,670. The (civil) days are, tithis − elided days = 6,67,91,670 − 10,45,095 = 6,57,46,575. The sidereal days are, civil days plus solar years = 6,57,46,575 + 1,80,000 = 6,59,26,575. The lunar revolutions are, synodic months + solar years = 22,26,389 + 1,80,000 = 24,06,389. Example 4. Give the revolutions of the Sun and the Moon, the civil days etc. in the Romaka yuga and the time of revolution of each, etc. Sun's revolutions = solar years = 2850. The solar months are, 12 × 2850 = 34,200. The synodic months are, 34,200 + 1050 = 35,250. The tithis are, 30 × 35,250 = 10,57,500. The civil days are, 10,57,500 − 16,547 = 10,40,953. The lunar revolutions are, 35,250 + 2850 = 38,100. Dividing the days in the yuga by the solar revolution, the time taken for the one revolution, i.e. the solar year is, in days etc. 10,40,953 ÷ 2850 = 365-14-48. Dividing the days by the synodic months, the period of synodic revolution (month) got is in days, etc. 10,40,953 ÷ 35,250 = 29-31-50-5-37. Dividing the days by the lunar revolutions, the time for one revolution got is, in days etc. 10,40,953 ÷ 38,100 = 27-19-17-46. The following points should be noted. The Romaka Siddhānta, now extant, agrees with the Modern Sūrya Siddhānta in its constants like the period of the yuga, the number of revolutions of the planets in the Yuga etc. But the Romaka Siddhānta condensed by our author is quite different and seems to 14a. B1. धृतिपे; B2. धृतिधे b. A2. ॰गुणा॰ d. A1.2. स्वकृत; B1. स्यात्कृत; B2. स्वकृत c. A1. ॰न्द्रिर्णवाशाः; D. [नवकेन्द्रिया॰] B1.2. क्रियाष्ट्यः A1. ष्ट्यप्र; A2. ष्ट्या प्र c-d. B1.2. खेन्द्रिया-gap शास्तिथि 16. Quoted by Ulpata on BS 2, p.29 15a. B2. युग्मे for युगे a. B1.2. युगवर्षं सपिण्डं B1.2. मकैन्दो; b. A1.2. साधिभासकं b. B1.2. पक्षयेस्तु (B2. वस्तु) पक्षाः d. A1.2. C.D. चार्क्षम्; B1.2. तार्क्षम्

I.16 I. INTRODUCTION OF THE WORK 17 be lost. Therefore we cannot determine whether the period of 2850 years mentioned here is the actual yuga of the original Siddhānta or a minor yuga (i.e. a fraction of it in whole years) given for convenience. Patently, the solar year given here is tropical and agrees with the value given to it by the ancient Greeks, like Ptolemy II and Herodotus. It is so with the duration of the synodic month also. Reducing the number of solar years and intercalary months in the yuga by the factor, 150, we see that there are 7 intercalary months in a period of 19 years or 228 solar months, which is the wellknown Metonic cycle. From all this we can conclude that this Siddhānta is from a Greek source. In the case of the Saura, the period of 1,80,000 years given here is certainly a minor yuga of the original Saura, for by multiplying this by 24 we get the number of years in the yuga of the original, viz., 43,20,000 years. From this we can infer that in the yuga of the original there are 1,80,000 × 24 = 43,20,000 solar revolutions, 6,57,46,575 × 24 = 1,57,79,17,800 civil days and 24,06,389 × 24 = 5,77,53,336 lunar revolutions. We have already mentioned that all these agree with the Ārdharā- trapakṣa of Āryabhaṭa given in the Mahābhāskarīya, with the Khaṇḍakhādyaka which is based on the Ārdharātrapakṣa and with the Pauliśa quoted by Bhaṭṭotpala in his commentary on the Bṛhatsaṃhitā but not with the Modern and well-known Sūrya Siddhānta. Now what is the purpose of our author in giving the yuga-elements of these two Siddhāntas alone? Our author expects that, like the Pauliśa, the Saura also would be used for a long time. So, if the time taken is far from the Epoch, he expects the reader to make his own rule, taking the elements given here, following the method of the Romaka. In the case of the Romaka itself, the accumulation of error in the rule can be prevented by deducting multiplies of 2850 years from the years gone from Epoch and doing the work with the small number of years left. Also, in the case of both, we can use the elements given here to check the constants given in later work, for mistakes. We shall now explain the rules of verse 16, indicating the Sun’s revolution as R, the Moon’s r, the synodic months m, the intercalary months i, the elided days e, the Tithis t, the civil days d, and the sidereal days n. (i) We shall explain the synodic month and derive the relation between the synodic months and lunar revolutions in the yuga. The synodic month is the interval between two consecutive conjunc- tions of the Sun and the Moon. In the Yuga the Moon makes r revolutions and, therefore, in one day makes r/d revolution. In the same way, the Sun makes R/d revolution. In one day they move apart by (r – R)/d revolution. When the separation equals one revolution they are in the next con- junction. The period of separation equal to one revolution, in days = 1/ [(r – R)/d] = d / (r – R) , which is the length in days, of the synodic month ....... (1) For d/(r – R) days, there is one synodic month; for d days (i.e. the days of the yuga) there are d/ {d/(r – R)} = r – R synodic months, i.e. r – R = m, r = m + R.........(2); i.e. adding the Sun’s revolutions to the synodic months, the lunar revolutions are obtained. (ii) The explanation of the intercalary month and its relation to the synodic month: The synodic months, Caitra etc. are those that end in the solar months Meṣa etc., and there is normally one to one correspondence between the two sets. But as the synodic month is shorter than the solar it succes- sively ends earlier and earlier in the solar and when it happens that the synodic month ends so early in the solar that another synodic month also ends within the same solar, obviously it has to be left out of reckoning if the correspondence between the set Caitra etc. with the set Meṣa etc. has to be maintained. This is the Adhikamāsa or intercalary month.

18 PAÑCASIDDHĀNTIKĀ I.18 Now, in one solar year there are 12 R solar months. As there are R years in the yuga, there are 12 R solar months in the yuga. Therefore the length of a solar month in days = d/12R. The length in days of a synodic month, already derived, = d/(r - R). Therefore in every solar month the end of the synodic month (i.e. the new moon) occurs earlier by d/12R - d/(r - R) = d(r - 13R )/12R (r - R). When this is equal to one synodic month and gets immersed in the solar, then one intercalary month happens, and the time for this to happen is, in terms of solar months, d/(r - R) ÷ {d(r - 13R)/ 12R(r - R)} = 12R/(r - 13R). Therefore, in the yuga containing 12R solar months the number of intercalary months i = 12R/ {12R/(r - 13R)} = r - 13R (r - R) - 12R = m - 12R. Therefore 12R

• i = m...... (3), i.e. the solar months + the intercalary months give the synodic months. (iii) We shall explain the occurrence of elided days and derive their number: The length of a tithi is a little less than a day and so every day the tithi occurs earlier and earlier in the day, until the time so accumulated becomes equal to one tithi and gets immersed in the day, with the result that the correspondence, one tithi to one day, is broken. Such tithis are left out by reckoning and are cal- led 'submerged tithis' or 'elided days'. Now, as there are in the yuga d days and t tithis, the duration of one tithi = d/t. In one day, the tithi falls earlier by 1 - d/t day. This accumulates to one tithi in d/t ÷ (1 - d/t) = d/(t - d) days, which is the time for one elided day to happen. Therefore, the number of elided days happening in a Yuga = e = d/{d/(t - d)} = t - d. Therefore d = t - e ..... (4), i.e. deducting the elided days from the tithis we get the days. (iv) We shall explain the sidereal day and derive the number of sidereal days in the yuga. The time taken by the stellar sphere to move (apparently) one round, is the sidereal day. But the day, i.e. the civil day, is related to the apparent diurnal movement of the Sun, from sunset to sunset, from sunrise to sunrise, from midnight to midnight etc. As there are n sidereal days and d days in the yuga, in one sidereal day the Sun makes d/n revolution. Therefore in one sidereal day he lags behind by 1 - d/n = (n - d)/n, revolution. This lagging behind is due to the Sun's eastward motion in the Sky and its magnitude is the Sun's motion in terms of revolutions during a sidereal day. This is equal to R/n. Therefore, (n - d)/n = R/n. Therefore, (n - d) = R. Therefore n = R + d .....(5), i.e. adding the solar years to the days, we get the sidereal days. Thus all the rules of verse 16 have been explained. [वर्षाधिपः] 'मुनियमयमद्वि'युक्ते द्युगणे 'शून्यद्विपञ्चयम' भक्ते । प्रति (राश्य) 'खर्तुदहनै' लब्धं वर्षाणि यातानि ॥ १७ ॥ तानि प्रपन्नसहिता'न्यग्नि'गुणा'न्यङ्घ्रि'वर्जितानि हरेत् । सप्तभिरेवं शेषो वर्षाधिपतिः क्रमात् सूर्यात् ॥ १८ ॥ Lord of the year

17. Add 2227 to the days from Epoch, divide out by 2520 and take the remainder. Set this in 3 places. In one place divide the remainder by 360 and take the quotient.
18. Add 1, multiply by 3, deduct 2 and divide out by 7. The remainder counted in the order Sun (Ravi), (Moon, Bhauma, Budha, Guru, Śukra and

I.18 I. INTRODUCTION OF THE WORK 19 Manda) is the Lord of the year (in which the taken day falls) (i.e. If Q is the quotient taken, the number to be divided out by 7 is equal to (Q + 1) × 3 – 2). Example 5. The days from Epoch is 3479. Give the Lord of the year. Adding the kṣepa to the days given, 3479 + 2227 = 5706. Dividing out by 2520, the remainder is 666. Dividing this by 360, the quotient obtained is 1. (1 + 1)3 – 2 = 4. The fourth from the Sun, Budha is the Lord of the year. The processes mentioned here are explained thus: At the moment 2227 days before Epoch, beginning Sunday, the days for calculating the Lord of the year etc. began and, as for the first day from that point of time, for the first month and the first year also beginning from that moment, the Lord was the Sun. To find these Lords for any time, the days from this point must be found and as the Epoch is 2227 days from this point, the days required are got by adding 2227 to the days from Epoch. For the purpose of calculating the Lord of the Year, the sāvana year comprising 360 days is used by our author and the Lord of the first day of the sāvana year is the Lord of the year. In the same way, to calculate the Lord of the month, the sāvana month of 30 days is used, the Lord of the first day of the month being the Lord of the month also. Now, as 2520 is the least common multiple of 360, 30 and 7, after each period of 2520 days, these Lords are repeated in the same order. Hence the instruction to divide the days out by 2520 and take the remainder alone. This remainder is set in 3 places to find the Lords of the year, the month and the day. Taking the remainder of the days, the Lord of the first year is that of the first day, the Lord of the second year is that of the first day in the next year, i.e. of the 361st day, i.e. that of the (358 + 3)th day, i.e. that of the day three days after; the Lord of the year next to that is that of the day 6 days after that of the first and so on. Thus, the Lord of the nth year is that of (n – 1)3 + 1, i.e. that of n × 3 – 2. If Q is the number of years gone, then n = Q + 1, and the Lord is that of (Q + 1) 3 – 2, which is the rule given. As the same Lord is repeated by the addition of multiples of 7, by casting out 7 we get the same and hence the instruction to cast out seven and take the remainder alone. Dividing the days into sāvana years and giving the Lord of the first day of the year as the Lord of the year is peculiar to our author. For others the Lord of the first day of the saura year and for yet others that of Caitra Śukla Pratipad is the Lord of the year. Some give two Lords. There is a flaw in the derivation of this rule by M.M. Sudhakara Dwivedi (vide page 6 of his Com- mentary). It has been hidden by another mistake made by him, viz., adopting the reading ‘aṅghri’ (= 2) but using the reading ‘abdhi’ (= 4) in the derivation. The reading pratirāśca is really pratirāśya. Both NP and TS take the reading pratirāśi and moreover, S gives it the incorrect meaning śeṣam, ‘re- mainder’. 17-18. Quoted by Utpala on BS 2.2, pp. 30-31. 18b. A1.2. गुणान्यब्धि; B1. गुणान्याघ्रि; B2. गुणान्यङ्घ्रि; 17a. B2. गुनियम D.U. गुणान्यंश्चि c. A1.2. प्रतिराश्च; B1.2. गतिराश्च; C.D. प्रतिराशि A1.2. वर्जिता हरेत् A1.2. दहनै ल° c. e. शेषं d. A1.2. पाताति d. A1. वपाधिपतिः; A2. वषाधिपतिः; B1.2. वर्षाधिपति क्र°

20 PAÑCASIDDHĀNTIKĀ I.20 [मासाधिपः] त्रिंशद्भक्ते मासाः प्रपन्नसहिता द्विसंगुणा [व्येकाः] | सप्तोद्धृतावशेषे मासाधिपतिस्तथैवार्कात् || १९ || Lord of the Month 19. Take the remainder set apart (as mentioned in verses 17-18), divide by 30 and take the quotient. Add 1, multiply by 2 and deduct 1. The remainder, after dividing out by 7, is the Lord of the month, counted from the Sun. The rule is (Q + 1)2 − 1, where Q is the quotient taken. Here in the place of the reading 'kāryāḥ' accepted both by TS and NP, we have adopted the read- ing vyekāḥ, given by Bhaṭṭotpala in his Br. Sam. commentary, as being the correct one and as neces- sary here. Also in the place of prapanna Bhaṭṭotpala reads pratipada. Whatever be the reading here, we want the meaning '1'. The rule is derived thus: As mentioned before for the Lord of the year, to get the Lord of the month the days are divided into sāvana month of 30 days duration and the Lord of the first day of the month is the Lord of the Month. Thus the Lord of the very first day, viz. the Sun is the Lord of the first month. As the days in the month, 30, divided out by 7 leaves the remainder 2, the Lords of the successive months are those of 2, 4, 6 etc. days after that of the first month, i.e. the Lord of the nth month is given by (n − 1)2 + 1 = n × 2 − 1. As n is the current month, it is equal to (Q + 1). Therefore n × 2 − 1 = (Q + 1)2 − 1, which is divided out by 7 gives the Lord of the month. Here too the derivation of M.M. Sudhakara Dwivedi is wrong (vide his commentary on the verse. p. 6). The translation of both TS and NP are incorrect for having taken the reading kāryāḥ for vyekāḥ ('deduct 1'), not realising which NP complain: “The text’s (I.19) ‘increase the (resulting) months by the current one’ should be replaced by ‘discard the fractional part of the current (month)’ (Pt. II, p. 13, footnote). On verses 17-19, K.S. Shukla has a detailed note in his paper, ‘The PS of VM (2)’ Gaṇita, 28 (1977) 99ff.” Example 6. For the same day as given in Ex. 5 give the Lord of the month. The remainder set apart (in the Ex. 5) is 666. Dividing by 30, the Quotient, Q, obtained is 22. (22

• 1)2 − 1 = 45. Dividing out by 7, the remainder is 3. Hence, the third from the Sun, viz. Bhauma is the Lord of the month. [होराधिपः] सप्तोद्धृते दिनेशः त्रिगुणेऽ(ध्येके) [युते च] होराभिः | (पञ्चघ्ने) सप्तहते विज्ञेयः कालहोरेशः || २० ||

19. Quoted by Utpala on BS 2. p.31. 19a. B1.2. प्रभवसहिताः; U. प्रतिपत्सहिताः b. A1.2. B1.2. C.D. कार्याः for व्येकाः c. B1.2. सप्तोधृता U. शेषे d. B1. वार्ध्यात्

I.21 I. INTRODUCTION OF THE WORK 21 Lord of the Horā 20. Take the remainder set apart in verses 17-18. Divide out by 7 and the remainder is the Lord of the Day, counting from the Sun. Take this remainder, multiply by 3, add 1, and add also the number of horās (i.e. the hours) counted from the beginning of the day, (i.e. the previous sunset) inclusive of the horā in which the taken moment falls. Multiply by 5 and divide out by 7. The remainder, counted from the Sun, gives the Lord of the Horā. If the Lord of the day is dth from the Sun and the time taken falls in the hth horā, then the number for the Lord of the Hora is (3d + 1 + h) × 5. It should be noted here that the horā, h, is counted from sunset, because the time of Epoch is sun- set and the day is said to commence there. The derivation of the two rules: The rule for the Lord of the Day is obvious for the order of the Lords, Sun Moon, Bhauma, etc. is meant to be the order of the Lords of the weekdays, Sunday, Monday, etc. The rule for the Lord of the horā is derived thus: From the Śāstra we learn that the Lord of the horā beginning at sunrise is the same as the Lord of that day. The Lord of the horā begin- ning Sunday, i.e. of the horā just after sunset of Saturday, (i.e. Mandavāra), is Budha, since the Lord of the horā after sunrise on Mandavāra is Manda and the successive Lords of the horās are the fifth after each, i.e. the sixth counting from each. (vide the next verse, 21). Budha is the 4th in order. After this if (n − 1) horās are gone, the Lord of the nth horā is given by (n − 1)5 + 4. Let us find the Lord of the horā for the h-th horā of the d-th day. This is {(d − 1)24 + h}th horā. Therefore the Lord of the horā is, substituting this for n in the above formula, {(d − 1)24 + h − 1} 5 + 4 = (24d + h − 25) 5 + 4 = (21d + 3d + h + 1 − 26) 5 + 4 = (3d + 1 + h)5 + 4 + 5 × 21d − 5 × 26 = (3d + 1 + h) 5 + 105d − 126 = (3d + 1 + h)5 + 15d × 7 − 18 × 7. As no change in the Lord happens by adding or deleting multiples of 7, this reduces to (3d + 1 + h)5, which is the rule given. (Here too the deri- vation of M.M. Sudh. is wrong. Let the readers examine his commentary.) The acceptance of the expression vyeka in place of the ms. reading 'dhyeka both by TS and NP has rendered their trans- lations incorrect. Example 7. (a) Who is the Lord of the Day, for the day given in Ex. 5? (b) On the same day, who is the Lord of the Hora, fifth after sunrise? (a) The remainder set apart according to verses 17-18 is 666. Dividing out by 7, the remainder left is 1, i.e. the Lord of the Day is the Sun. (b) In the example, d = 1, h = 5 + 12 = 17 (because h is counted from the beginning of the day, i.e. the previous sunset). Substituting, (1 × 3 + 1 + 17)5 = 105. Casting out 7, the remainder is 0 or 7 and the 7th from the Sun, Manda is the Lord of the horā. वर्षाधिपश्चतुर्थो मासाधिपतिस्ततो योऽन्यः । होराधिपश्च षष्ठो निरन्तरं दिवसनाथश्च ॥ २१ ॥ 20. Quoted by Utpala on BS 2, p.34. c. C.D. U. पञ्चम; 20a. B1.2. सप्तोद्धृते B1. सप्तहृते; B2. सप्तहृते; C.D. U. सप्तहतो b. A1.2. B1.2. ०ध्येकशहोरादिः; C.D. U. त्रिगुणो d. A1.2. विज्ञेया; B1. विज्ञेय व्येको युतश्च होराभिः A1.2. कालहोरेशाः; B1. कायहोरेशः; B2. कायहोरेशः

22 PAÑCASIDDHĀNTIKĀ I.22 21. The fourth counted from the Lord of any year is the Lord of the year next to that. The third from the Lord of any month is the Lord of the month next. The sixth from the Lord of any horā is that of the next horā. The Lords of the day come consecutively, in the order given. This the explanation: It has been said that the Lord of the year is that of the sāvana year of 360 days, coming one after another. The Lord of the first day of the year is the Lord of the year and the Lord of the 358th day is the same. The Lord of the next year is that of the 361st day, which is the fourth counting from 358. Thus the Lord of the next year is the fourth counting from that of the previous year. In the same way, the Lord of the next (sāvana) month is that of the 31st day, counted from the first day of the previous month. The Lord of the 29th day is the same as that of the first. The 31st day is the 3rd counting from the 29th. Therefore, the Lord of the 31st day, i.e. the Lord of the next month, is the third from that of the previous month. The Lords of the horās come in the order, Manda, Guru, Bhauma, Ravi, Śukra, Budha and Soma, which is the descending order of the distances of their orbits. The planet next in this series, who is the Lord of the next horā, is the 6th in the series given by our author, and hence the statement that the sixth from the previous is the Lord of the next horā. That the Lords of the day come consecutively is obvious, for the series Ravi, Soma, Bhauma, etc. is given in the very order of the Lords of the day. One thing must be said here. The author has taken the Lords in the arbitrary order Ravi, Soma etc. as it is well known by means of the week-days we are using in our day-to-day affairs. But the order of the Lords of the horā, viz. Manda, Guru, etc. based on their distances is fundamental and given by the Śāstras, which give the Lord of the week-day itself as being the same as the Lord of the first horā after sunrise on that day, taking the Lord of the horā as known. Taking this order we can make the following statements: The Lord of the next day is the 4th as counted from that of the cur- rent day, the Lord of the next month is the 7th counted from that of the current month (or, which is the same, the one previous to that of the current month) and the Lord of the next year is the third counted from that of the current year. वर्षे यद्यस्य फलं मासे च मुनिप्रणीतमालोक्य । [तत्तद्वृत्तै]र्वक्ष्ये होरातंत्रोत्तरविधाने ॥ २२ ॥ 22. Consulting the works of Sages, I shall tell in my future work following the Horā-Tantra, the predictions, viz. which results will flow during the reign of which Lord of the year or Lord of the month. There is a gap in this verse in every manuscript, tattadvṛttaiḥ being missing. So we have adopted the reading of Bhaṭṭotpala in his Bṛ. Sam. commentary which is full. 21. Quoted by Utpala on BS 2, p.35.      22a. B3. Commences with this verse. 21a. A1.2. B1. चतुर्थे                B1.2.3. वर्ष यस्य फलं b. A1.2. पतितस्थानतो; B1.2. पतितस्तथा ततो;    b. B2. मासे वा C.D. पतितस्था                c. A.B. om. तत्त द्वृत्तैः; C. [तत्तत्फलं च] [तृतीयोऽन्यः]                  A1.2. B1.2. वक्षे c. B1.2. होराधिपतिश्च. A1.2. B1.2. षष्टो       d. B1. होम and B2. होरां for होरा d. A1.दिवनाथश्च; U.दिवसनाथः स्यात्         A1. °त्तविधानैः; B2. तविधाने;                          B.विधानै

I.25 I. INTRODUCTION OF THE WORK 23 द्युगणे 'रूपा'भ्यधिके 'पञ्चर्तुगुणो'द्धृतेऽथ मासाः स्युः । त्रिंशद्भक्ते शेषं ज्ञेयं राश्यंशकेन्द्राणाम् ॥ २३ ॥ कमलोद्भवप्रजेशौ स्वर्गः शस्त्रं (द्रुमान्नवासांसि) । (कालानलाभ्ररवयः) शशीन्द्रगोनियतयः क्रमशः ॥२४ ॥ हरभवगुहपितृवरुणा बलदेवसमीरणौ यमश्चैव । (वाक्) श्रीधनदौ (निरयो) धात्री (वेदाः) परः पुरुषः ॥ २५ 23-25. Add 1 to the Days from Epoch, divide by 365, take the remainder and divide this by 30. The quotient are the months gone. The remainder gives the Lords of the current degree in the current month. They are, corresponding to each degree, Kamalodbhava (Brahmā), Prajāpati, Svarga (Heaven), Weapon, Tree, Anna (Food), Residence, Kāla (Time), Agni, Abhra (Cloud), Sun, Moon, Indra, Cows, Niyati (Fate), Hara, Bhava, Guha, Manes, Varuṇa, Baladeva, Vāyu, Yama (the ruler of the World of the manes), Vāk (Goddess of Speech), Śrī (the Goddess of Wealth), Kubera, Hell, Earth, Vedas and the Supreme Being. This matter must have been taken from the ancient Saṃhitās by our author and given here. For the purpose of giving the Lord of the degrees they must have divided the days into years of 365 days (why not the exact duration of the solar year, we cannot say) and the years into months of 30 days as can be inferred from the instruction. But then it comes to giving the Lord of not the degrees of the rāśi but that of each of the sāvana days in the sāvana month. For the 5 days left over at the end of the year it must be taken that the first 5 Lords are repeated. Example 8. Give the Lord of the Degree of the rāśi for Days from Epoch 3479. Adding one and dividing by 365, the remainder is 195. Dividing by 30, the remainder is 15. Therefore the fifteenth in the list, Niyati (Fate) is the Lord required. Ed. Note: NP have identified verses 23-25 as relating to the Magas, emending the expression māsās syuḥ in verse 23 to magābdāḥ syuḥ and have correlated the 30 names enumerated in the verses with the lords of the 30 days in the month according to the Magan calendar. K.S. Shukla has studied these three verses in detail, noted that these names are enumerated also in the Vaṭeśvara Siddhānta (ch. I, sn.v, vv. 117 c-d, 118) and has traced the names to their Zoroastrian (Parsi) originals, as per the following Table, in his paper 'The PS of VM (2)', Gaṇita, 28 (1977) 99-116. 23a. A2.द्युगणे; B1-3कगणे A1.2.रुद्रमान्य; B1.2.रुद्रभान्य०; B3.रुद्रन्तान्य० b. A2.पञ्चर्शूं. A1.2.धृतेथ; B1.3.ध्वजेथ. c. A1.2. B1.2. C.D.कमलानलान्तरवयः (B2.०ख्खं यः) A1.2.मासा स्युः; D. [मगाब्दाः] स्युः d. C.D.गोनिर्वृतयः. B.हरस्रव d. A1.2.केन्द्राणां; B1.2.चन्द्राणां 25a. A1.2.चरुणा; B1.वरुण; B3.वरुणां; 24a. A1.2.कमलोद्भव्वा; B1-3.कमलोद्भवं; C.D.कमलोद्भवः D. [भवगुरु] पितृ A1.2.प्रजेसा; B1-3. C.D.प्रजेशः b. A1.वलदेव; A2.वलहेव A1.2.समीकरणौ b. A1.2.स्वर्ग्यं; B3.स्वर्यः; C.स्वर्गी; D.स्वर्गे c. B1.3.प्राक् श्रीधनवौ । A1.2. B1.3. C.D.गिरयो C.०शश्चन्द्रमान्यवासांसि; D. [श] शास्तृरुद्रमन्युवसवः d. A1.2.वेधा; B1.3. C.D.वेघाः A2.पुरुः. A1.2.पूरूषः

24 PAÑCASIDDHĀNTIKĀ Names of the 30 days of the Parsi months

1. A.B.C.D. करणावतारः Thus ends Chapter One, entitled 'Introduction of the Work', in the Pañcasiddhāntikā composed by Varāhamihira

Chapter Two VĀSIṢṬHA-SIDDHĀNTA — PLANETARY COMPUTATIONS ETC. २. द्वितीयोऽध्यायः वासिष्ठसिद्धान्तः — ग्रहादिगणितम् Introductory Now follow the five Siddhāntas. Of these the Vāsiṣṭha is given first, as being the most primitive among the Siddhāntas that distinguish between true and mean motions, unlike the Paitāmaha which gives only the mean motion. For a detailed exposition of some of the verses see T.S.K. Sastry, 'The Vāsiṣṭha Sun and Moon', JOR 25 (1955-56) 19-41 and K.S. Shukla, 'The PS of VM(2)', Gaṇita 28 (1977) 99-116. [स्फुटरविः] 'कृत'-'गुणमृतु'-युत'मेकर्तुमनु'हतं 'षड्यमेन्दु'भिर्विभजेत् । 'शशि-ख-ख-ख-यम-कृत-स्वर-नव-नव-वसु-षट्क-विषयो'नैः ॥ १ ॥ True Sun

1. Multiply the Days from Epoch by 4 and add 6. Divide this by 1461 (and take the remainder). Take from this, successively, the quantity 126, reduced by 1, 0, 0, 0, 2, 4, 7, 9, 9, 8, 6, 5 (i.e., the twelve quantities 125, 126, 126, 126, 124, 122, 119, 117, 117, 118, 120, 121). (The Sun's rāśis, Meṣa etc. are succes- sively got.) The direction is: Multiply the days by 4, add 6, divide by 1461 and take the remainder. From this first take off 125, and consider that Meṣa is gone. Then from what remains deduct 126 and consider Ṛṣabha is gone, and so on. The Sun is in the rāśi corresponding to the number which cannot be deducted on account of its being less than what is left over. Multiply what is left over by 30 and divide by what cannot be deducted. The position of the Sun in that rāśi is got, in degrees. It is to be noted that even 'Days from Epoch' is not mentioned here but we take it as understood because every work of this sort requires it. It is not specifically mentioned that this rule is for com- puting the true Sun but we can infer it from the quantities here given and the work asked to be done. Even the work is not clearly and completely given. But knowing what the author is about, we can see what is wanted to be done. TS have refrained from interpreting this, as an impossible task. 1a. A1. कृतगुणषषमृतु; A2. कृतगुणपयमृतु; BC. कृतगुणषड्ऋतु b. D. वियुजेत् a-b. A3. मैकर्तु c-d. A.B.C. खरकृत for कृतस्वर

26 PAÑCASIDDHĀNTIKĀ II.1 The text here svarakṛta has been changed into kṛtasvara by interchanging the words, as the nature of the work requires it and as this kind of transposition is sometimes found in manuscripts. It is impossible that the Siddhānta itself has made this mistake, not noticing the ascending nature of the series in this part. Next, we are in doubt here about the time of the day (like sunrise, sunset, noon or midnight) for which the Sun is here given. One may think that because no time is mentioned here, not even the instruction to take the ‘Days from Epoch’, one is expected to take the Days of the Romaka or the Pauliśa and with its own time of sunset at Yavanapura, i.e. thirty-seven nāḍīs twenty vināḍīs from sunrise at Ujjain. But later, in dealing with the Romaka itself and with the Saura, the author gives different times of day for different computations (vide VIII.5, IX.1, XVI.1) and hence this doubt. It is likely that the Vāsiṣṭha Sun and Moon are given for sunrise at Ujjain, as we shall show while dealing with the Moon. Another point to be noted is this. The rule gives the ‘True’ Sun directly, without giving the ‘Mean’ Sun. This is possible because this Siddhānta, like the other Siddhāntas of the period like the Āryabhaṭīya, has taken the apogee of the Sun as fixed and, so, for a given day in the solar year there, is a given anomaly with a given equation of the centre, which means a given true Sun. (It is so with the Vākyakaraṇa also, which follows the Mahābhāskarīya based on the Āryabhaṭīya, with this differ- ence that here the days for fixed intervals of the true Sun is given, while in the Vākyakaraṇa the days for the Sun and the Sun for the days, both are given.) The rule is explained thus: In this Siddhānta the solar year consists of 365 ¼ days, (like the Julian year), i.e. of 1461 quarter-days. For convenience of computation, the Days from Epoch are also converted into quarter-days. According to this Siddhānta the true solar year began, i.e. the true Sun was at the first point of Meṣa, 1 ½ days, i.e. 6 quarter-days, before Epoch. So 6 is added to the quarter- days from Epoch to give the true Sun from the beginning of Meṣa. As after periods of 365 ¼ days, i.e. 1461 quarter-days, the Sun returns to the first point of Meṣa, we can divide the quarter-days out by 1461 and take the remainder alone to find the Sun, i.e. its position from the beginning of Meṣa. Now this Siddhānta has found empirically that the true Sun traverses Meṣarāśi in 31 ¼ days, i.e. 125 quarter-days, Ṛṣabha-rāśi in 31 ½ days, i.e. in 126 quarter-days and so on. Thus in 125 + 126 + 126 + 126 + 124 + 122 + 119 + 117 + 117 + 118 + 120 + 121 = 1461 quarter-days the Sun traverses all the twelve rāśis and reaches Meṣa again. That these numbers add upto 1461, and 1461/4 = 365 ¼, the days of the year, is proof of the correctness of our interpretation of the rule. Thus we see that the solar months Meṣa etc. contain each 31 ¼, 31 ½, 31 ½, 31 ½, 31, 30 ½, 29 ¾, 29 ¼, 29 ¼, 29 ½, 30 and 30 ¼ days, respectively. It can be seen that these fairly agree with what is given by the other Siddhāntas. Thus if 125 quarter-days are left over in the year the Sun has traversed Meṣa, if 125 + 126 are left over, it has traversed Meṣa, Ṛṣabha etc. It is obvious that its position within a rāśi is to be found by the proportion: If 30° are for the quarter-days of the rāśi, how many degrees for the quarter-days ultimately left over. Example 1.(a). Days from Epoch 4246. Find the true Sun. (b) Find the true Sun for zero day. (a) Days converted into quarter-days = 4 × 4246 = 16,984. Adding 6 we get 16,990. Dividing out by 1461, the remainder is 919. Deducting 125, 794 is left over; Meṣa is gone. Deducting 126, 668 is left over; Ṛṣabha is gone. Deducting 126 again, 542 is left over; Mithuna is gone. Deducting 126 for Karkaṭa, 416 is left over. Deducting 124 for Siṃha 292 is left over. Deducting 122 for Kanyā,

II.4 II. VĀSIṢṬHA-SIDDHĀNTA 27 170 is left over. Deducting 119 for Tulā, 51 is left over, in Vṛścika, i.e. 51 117 of Vṛścika is gone, i.e. 30 × 51/117 degrees = 13° 5'. Therefore the true Sun = 7ʳ 13° 5'. (b) For 0 day, 0 + 6 = 6, quarter-days. 30° × 6/125 = 1° 26' gone in Meṣa. Therefore the true Sun = 0ʳ 1° 26'. [चन्द्रस्फुट:] ‘रसगुणनवेन्दु’युक्ते ‘शशिगुणखगुणो’द्धृते घना द्युगणे । शेषे नवभिर्गुणिते गतयो‘ष्टजिनैः’ पदं शेषम् ॥ २ ॥ घनषोडशहतशेषं प्रोज्झ्याऽधस्त्रिगुणितं चतुर्भक्तम् । भादि कला द्विगुणघनाः ‘शशिमुनिनवयमा’श्च राश्याद्याः ॥ ३ ॥ ‘विषयधृतयो’ गतिघ्ना गति(का)ष्ठांशोनिताः कलाः प्रोक्ताः ‘वेदार्का’ : पदसंख्यागत्यर्थं धनमृणं परतः ॥ ४ ॥ True Moon 2. Add 1936 to the Days from Epoch, and divide the Sun by 3031. The quo- tient are called ghanas. Multiply the remainder by 9 and divide by 248. The quotient are called gatis and the remainder are called padas. 3. Divide the ghanas out by 16 and take remainder alone. Multiply this by 3, divide by 4 and take the result as rāśi etc. Subtract this from 12 rāśi and take the remainder. Add to this, minutes equal to twice the total ghanas. Add also 1ʳ 7° 29'. (The mean Moon at the end of the ghanas is got). 4. Multiply the gatis by 185, subtract a tenth of the gatis and add these also, taken as minutes. (The mean Moon at the end of the gatis is got.) If the number of padas is less than 124, they are called plus-padas. If 124 or more, 124 padas are taken and set apart as a half-gati. The remaining padas are called minus-padas. (The three technical terms here, half-gati, plus-pada and minus- pada are for use in verses 5 and

28 PAÑCASIDDHĀNTIKĀ II.4 nothing is said, then the emendation gatikāṣṭhāṃśa is the proper one, which we have given. (TS also give this). If, on the other hand, we take it that the instruction is to subtract 2 minutes per ghana, taking the word projjhya in the previous instruction to be understood here also, then the emenda- tion gatyaṣṭāṃsa will be the proper one. In this case we would also have to keep the letter ṭa of the original as it is, without changing it into ṭha. But the addition of 2 minutes per ghana alone would agree with the correct mean motion for the period of our author which is in cycles etc. 110-11-7-32- 15 for 3031 days, the Vāsiṣṭha mean motion being 110-11-7-32. NP have emended the word as ṣaṣṭhāṃsa which would not give the correct result. *Example 2. Find the mean Moon for the end of the gati just before Days from Epoch, 3,0

II.6 II. VĀSIṢṬHA-SIDDHĀNTA 29 The two formulae can be written down thus: (i) If P is the number of plus-padas, {1094 + 5(P − 1)} P/63. (ii) If P' is the number of minus-padas, {2414 − 5(P' − 1)} P'/63. Example 3. Continue Ex.2 and compute the true Moon for the days given. The mean Moon got in Ex. 2 to the end of the gati = 4ʳ 24° 4' The padas obtained are 64, plus-padas (P) Adding degrees equal to P = 64, + 2 4 0 Using formula (i) intended for plus-padas, {1094 + 5 (64 − 1)} 64/63 = 1431 minutes + 0 23 51 —————————————————————————————————————————————————————————— The true Moon 7 21 55 Example 4. The Days from Epoch are 1219. Find the True Moon. 1219 + 1936 = 3155 (= days for computation). Dividing by 3031, ghana got is 1, remainder 124. Multiplying 124 by 9 and dividing by 248, the gatis got are 4. The remainder 124 are padas. This is just one half-gati and no pada is left over. r ° ' Ghana 1 × ¾ = 0ʳ 22° 30'. Deducting from 12 rāśis = 11 7 30 Adding minutes 1 × 2 + 0 0 2 Kṣepa + 1 7 29 Gatis 4, × 184 9/10 = 740 (minutes) + 0 12 20 For the half-gati, add + 6 0 4 —————————————————————————————————————————————————————————— True Moon 6 27 25 Example 5. Find the true Moon for Days from Epoch, 1228. 1228 + 1936 = 3164 (= days for computation). Dividing by 3031, ghanas got 1, remainder 133. Multiplying by 9 and dividing by 248, the quotient 4 are the gatis got, and the remainder 205 are padas left over. A half-gati (= 124 padas) can be taken from this, and the remaining 81 are minus- padas. r ° ' ghana 1 × ¾ʳ = 0ʳ 22° 30'. Deducting from 12 rāśis 11 7 30 Adding 1 × 2 minutes + 0 0 2 Adding kṣepa + 1 7 29 Gatis, 4, × 184 9/10 = 740 (minutes) + 0 12 20 For the half-gati + 6 0 4 Degrees equal to P' = 81' + 2 21 0 Using formula (ii) (as the left over are minus-padas = P'), 2414 − 5 (81 − 1) 81/63 = 2589 mts. + 1 13 9 —————————————————————————————————————————————————————————— True Moon 11 1 34 The following is the explanation of the processes: The true Moon at a given time t is: (i) the mean Moon at t plus (ii) the equation of the centre for t. (i) is given here in five parts. We shall call them (a), (b), (c), (d), (e) which are to be added up to get the total mean Moon.

30 PAÑCASIDDHĀNTIKĀ II.6 (a) (Usually called the Mūla-dhruva or Kṣepa) is the mean Moon at a point of time 1936 days before the Epoch, when the Moon's apogee and the mean Moon exactly coincided according to this Siddhānta. This is given as śaśi-muni-navayamāś ca rāśyādyāḥ, i.e. 1ʳ 7° 29ʹ. (b) is the mean motion during whole numbers of cycles of 3031 days from the point of time 1936 before Epoch, each cycle equal to 110 anomalistic revolutions of the Moon. This (b) is found by multiplying the mean motion per cycle (110 revolutions, 11 rāśis, 7 degrees, 32 minutes) by the number of cycles, called ghanas, obtained as quotient, by dividing the Days from Epoch plus 1936, by 3031. As full revolutions can be neglected, it is enough if we multiply the ghanas by 11 rāśis 7 degrees 32 minutes, which may be done as ghanas × 2ʹ + ghanas × 11ʳ 7° 30ʹ. Ghanas × 2 is given by dviguṇaghanāḥ kalāḥ (yojyāḥ). Because 16 ghanas × 11ʳ 7°30ʹ equals 15 full revolutions, it is enough if we divide out the ghanas by 16 and take the remainder alone for multiplication (for we shall be neglecting only full revolutions), which we are asked to do by ghanaṣoḍaśāhṛta-śeṣam. As 11ʳ 7° 30ʹ is ¾ rāśi less than a full revolution, we can multiply the remaining ghanas by ¾ rāśi and take this as subtractive, which we are instructed to do by projjhyādhas triguṇitaṁ caturbhaktam bhādi (rāśyādi.) Thus b is disposed of. (c) is the mean motion during the subsequent full anomalistic revolutions called gatis, which form the quotient got by dividing the remaining days by the anomalistic period, 248/9 days, (i.e. multi- plying the days left over by 9 and dividing by 248). For each gati the mean motion is 1 revolution and 184 9/10 minutes (which can be obtained by dividing the motion per ghana, viz. 110 rev. 11ʳ 7° 32ʹ by the number of gatis in a ghana, viz. 3031 × 9/248). Hence the rule to multiply the gatis by 185ʹ and deduct minutes equal to 1/10 of the gatis. This is given by viṣayadhṛtayo gatighnā gatīkāṣṭhām- śonitāḥ kalāḥ yojyāḥ. (d) What are now left of the days are ninths of days called padas (and these obviously would be less than 248). The mean motion per pada is 1 degree 27 209/248 minutes, and so padas × 1° 27 209ʹ/248 should be added to complete the mean motion till t. Of this, the Siddhānta asks us to add 1° per pada first, which is given by śeṣapadasamāṁśāmśāḥ (yojyāḥ). This forms d. (e) The residue 27 209/248 minutes per pada, forming (e), is combined with the equation of the centre (ii) and given by the two formulae of II.6. If the padas contain a half-gati (i.e. 124 padas) the value of (d) + (e) + (ii) for the half-gati part is combined together and given as 180° 4ʹ. This is got as follows. As the half-gati is equal to 124 padas, d = 124°. (e) + (ii) given by the first formula of II.6 is: {1094 + 5(124 − 1)} 124/63 = 3364ʹ = 56° 4ʹ; 124° + 56° 4ʹ = 180° 4ʹ = 6 rāśis 4 minutes, which is given by gatyardhe bhagaṇārdham deyam liptācatuṣkasaṁyuktam and which instruction has so much puzzled TS. But, of course, this is incorrect and the defect lies in the equation of the centre- part of the formulae in II.6, which give zero-value for the equation of the centre not at 124 padas, but at 133 padas, as we shall show presently. We shall first explain II.6 by showing how the formulae here combine the residual mean motion, viz. padas × 27 209/248 minutes (= e) with what is identifiable with the equation of the centre (= ii). The equation of the centre of the Vāsiṣṭha is peculiar. Usually in the Siddhāntas the equation of the centre varies as the sine of the anomaly, and therefore is zero at zero degree anomaly, going to a minimum at 90°, again rising to zero at 180°, then going to a maximum at 270°, and then falling to 0 to 360°, i.e. zero°. Thus it is negative in the first two quadrants and positive in the third and fourth quadrants and of the form, '−a sin θ, where 'a' is the maximum or minimum numerical