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

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

II.6 II. VĀSIṢṬHA-SIDDHĀNTA 31 value, and θ is the anomaly. Note that this is the first term of the series for the equation of the centre in modern astronomy, with its sign reversed, and the reversing is necessary because the ano- maly was reckoned by the ancients from the apogee, not perigee. But in the Vāsiṣṭha it is of the form — (665 — 5P) P/63 for the first two quadrants and + (665 — 5P') P'/63 for the last two. These are derivable from the equation for the Moon’s daily true motion given in III.4, (as we shall show there), which assumes the increase or decrease of motion as uniform. Here we shall assume them and derive the two formulae of II.6. As said before, (e) + (ii) is given by the formulae and (ii) is — (665 — 5P) P/63, for the first formula. Therefore (e) + (ii) = 27 209/248 P — (665 — 5P) P/63 = (63 × 27 209/248 — 665 + 5P) P/63 = (1754 — 665 + 5P) P/63 = (1089 + 5P) P/63 = {1094 + 5 (P — 1)} P/63, which is the first formula. For the second formula (ii) is + (665 — 5P') P'/63. ∴ (e) + (ii) = 27 209/248 P' + (665 — 5P') P'/63 = (27 209/248 P' × 63 + 655 — 5P') P'/63 = (1754 + 665 — 5P') P'/63 = (2419 — 5P') P'/63 = {2414 — 5 (P' — 1)} P'/63 which is the second formula. We have already shown how for the half-gati 6ʳ 0° 4' is got instead of the mean motion 6ʳ 1° 32½'. This means that there is in this an equation of the centre equal to — 88½', combined with it. So, when the equation of the centre given by + (665 — 5P') P'/63 = + 88½, then it is actually zero according to this Siddhānta. Solving this equation, we get P' = 9 or P' = 124. As P' is minus-pada, which is the original padas got less 124, we get that the equation of the centre actually becomes zero at original padas, P= 133, and P = 248. As P = 248, is the end of the gati, this is what we expect, as the anomaly has again become zero. Also, by computation or examination we can get from the equation of the cyclic part of the for- mula for the first half-gati, — (665 — 5P) P/63, the numerically greatest value of the negative equa- tion of the centre, which is — 351', for P = 66½. In the same way, from that the formula for the second half-gati, + (665 — 5P') P'/63 we can get the maximum + 351', for P' = 66½; but as there is a residue of — 88½' in the second half-gati, 351' — 88½' = 262½' is the actual maximum. The numerical mean is 307' which, we see, is very nearly equal to that of the other Hindu Siddhāntas. It is not that VM does not know that zero equation of the centre must occur at P = 124, and not at 133, for in his own Romaka and Saura it is so. Nor is it difficult for VM, a master in the science, to give the two formulae so as to have the equation of the centre zero at P = 124, (so that for a half- gati we get the correct 6ʳ 1° 32½'), retaining, at the same time, the equation of the centre desired by him. If he had given the two formulae in the form (1134 + 5P) P/63, and (2374 — 5P') P'/63, he could have secured this. But adherence to the Siddhānta has prevented him from doing this. So closely does he follow the original that he does not even give the two formulae in the more simplified forms, (1089 — 5P) P/63, and (2419 — 5P') P'/63. The following things are to be noted in connection with this Siddhānta. Of both the Sun and the Moon, the mean motion and the equation of the centre is mixed in a peculiar manner and thereby the true motion is given. We shall see that it is the same in the case of the Pauliśa also. The period of 3031 days called ghana here is the same as what is called kālānala in the Vāk- yakaraṇa, which gives for this period, the mean motion, 11ʳ 7° 31', neglecting full revolutions. The number 248 given here is there mentioned as devara.

32 PAÑCASIDDHĀNTIKĀ II.6 We can see that the remark in I.4 about the tithi of the Vāsiṣṭha being very incorrect is appropriate, but it can be shown that it is not due to the error in the Moon, but in the Sun, whose sidereal year is taken as 365-15-0 days. The Moon's motion for 3031 days is in cycles etc. 110-11-7-32 = 110 5063/5400 cycles. In one day the motion is 110 5063/5400 ÷ 3031 = 5,99,063/ (5400 × 3031) cycle. In one day the Sun's motion is 1/365¹/4 = 4/1461 cycle. The relative motion, i.e. their separation per day is 5,99,063/(5400 × 3031) − 4/1461 cycle. The time taken for a separation of one cycle, i.e. the synodic month, is in days, 1/{59,90,63/(5400 × 3031) − 4/1461} = 1461 × 3031 × 5400 ÷ (1461 × 5,99,063 − 4 × 5400 × 3031) = 7,97,09,23,800 ÷ 26,99,20,481 = 29 - 31 - 50 - 17 - 38. But the correct synodic month computed for the time near that of our author is 29 - 31 - 50 - 7 - 47. Therefore in successive synodic months the tithi comes later by days etc. 0-0-0-9-51, according to Vāsiṣṭha. In about 29¹/2 years this will accumulate to one nāḍikā. This is bad indeed, and merits VM's remark in I.4 taditarau dūravibhraṣṭau (i.e., 'The tithis of the other two have slipped far away from the real'.) Now, we have seen that according to this Siddhānta the mean Moon moves revs. 110-11-7-32-0, while the real motion for the period is revs. 110-11-7-32-15. Therefore in 3031 days the Vāsiṣṭha nakṣatra is delayed by a little more than one vināḍi. So a delay of one nāḍi is caused in 440 years only. So the delay of one nāḍi in the tithi per 29¹/2 years mentioned above, must be due almost wholly to the error in the Sun, the result of giving the time per cycle as 365-15-0 days. Again, in every 3031 days, the Vāsiṣṭha Moon lags behind the correct one by 15". A lagging behind by one degree will take place in 3031 × 60 × 60 ÷ 15 days, i.e. in about 2000 years, a very long period indeed. Bearing this in mind, we shall try to answer the question already raised, viz. whether the Vāsiṣṭha Sun and Moon are given for sunrise at Ujjain or sunset at Yavanapura; and, incidentally, we shall show that the kṣepa, 1ʳ 7° 29′ given by śaśi-muninavayāmāśca rāśyādyāḥ and obliterated by TS by their drastic emendation as yama-hṛtāś ca is necessary in II.3. The following is the mean Moon for Epoch (viz. Śaka 427 elapsed, i.e. in A.D. 505), sunset, at Yavanapura, beginning Monday, Caitra Sukla being about to begin. i. Computed for the period according to modern astronomy, assuming the ayanāṁśa to be practically 0 for the time 354° 48′ ii. According to Saura 355° 6′ iii. According to Siddhānta Śiromaṇi 355° 41′ iv. According to Romaka 356° 12′ v. According to Vāsiṣṭha, assuming that the mean Moon is given for Ujjain sunrise 355° 6′ -do- -do- for sunset at Yavanapura 346° 54′ -do- Ujjain sunrise, without Kṣepa 317° 37′ -do- sunset at Yavanapura without Kṣepa 309° 25′ (For use by anyone interested in making the calculations himself, the Kalidina etc. of Epoch is 13,17,122-37-20. Also, the Epoch is 5,09,432-22-40 days before mean sunrise at Ujjain of the first January 1900). An examination of the table will show that the Vāsiṣṭha Moon agrees with that of every other fairly well, taking it as being given for Ujjain sunrise, and taking that the kṣepa is given. If, on the other hand, it is assumed that it is given for sunset at Yavanapura, there is a difference of about 8°, which can happen only in 1600 years, which is very very unlikely; for this to happen the Vāsiṣṭha should

II.6 II. VĀSIṢṬHA-SIDDHĀNTA 33 have been written 1600 years earlier. If there is not the kṣepa, the difference is 37°, an impossible thing, not to speak of the assumption that it is for Yavanapura sunset and there is no kṣepa, both together which will make the difference 45° and worse. Hence the Vāsiṣṭha epoch is definitely at sunrise at Ujjain and not at sunset at Yavanapura (Alexandria). We have shown the kṣepa necessary. But TS have emended kalāḥ dviguṇaghanāḥ, śaśi-muni- navayamāś ca rāśyādyāḥ (verse 3) into phalaṁ dviguṇaghanāḥ śaśi-muni-yama-hṛtāś ca rāśyādyāḥ and spoiled the already correct reading and introduced two extra syllables in the last foot, which spoils the āryā metre too. (It must be noted that already there are 16 mātrās in the last foot, i.e. one mātrā extra, which can be explained away or corrected by reading rāśyādyāh as rāśyādi.) One another point: TS have expressed their inability either to interpret II.6 or to explain why 6ʳ 0° 4′ is to be added for a half-gati (vide Com, page 9). But still thinking II.6 gives the pure equation of the centre, the commentary goes on: arthāt vedārkālpa-padeṣu ṛṇam, adhikeṣu dhanam iti buddhimad- bhiḥ svayam eva ūhyam, i.e. “It goes without saying that when the padas are less than 124, the result is subtractive, and when more it is additive”, which is wrong for we have seen that the result of both the formulae are additive. Moreover, the failure, both by TS and NP, to realise that the expression ‘rāśyādyāḥ’ specifically instructs that the digits in śaśimuninavayama are to be taken as ‘beginning from rāśi’, i.e. as 1ʳ 7° 29′ and not as a whole number 2971 (TS) or as “2ʳ 9;7, 1⁰” (NP) have led to incorrect interpretations by them; also, the Notes of NP (vol. II, pp. 16-19) and the deductions made (p.19) have to be revised in the light of all that has been stated above. [नक्षत्र-तिथी] श (श्यर्ध)दलं त्रिकृतिघ्नमृक्षमंश(स्थि)ता मुहूर्ताः स्युः । व्यर्केन्दुदलं विषयाऽऽहतं तिथिस्तद्गदेवोक्तः ॥ ७ ॥ Nakṣatra and Tithi 7. Divide the True Moon by 4 and multiply by 9. What we get in the rāśi column is the nakṣatra. What is got in the degree column are the muhūrtas. Deduct the true Sun from the true Moon, divide the result by 2 and multiply by 5. Tithis are got in the rāśi column and thirtieths of tithis in the degree column. As the 27 nakṣatra-segments are divided into the 12 rāśi - segments, there are 2¼ = 9/4 nakṣatras per rāśi. Hence the instruction to divide the rāśis by 4 and multiply by 9 to get the nakṣatras. As degrees are thirtieths of rāśis, the resulting numbers in the degree column are thirtieths of nakṣatras, called muhūrtas by this Siddhānta. It must be noted that the word muhūrta originally meant the 30th part of a nakṣatra, but later came to be applied to the 30th part of a day as well, because both are practically the same in duration. The interval of longitude between the Sun and the Moon is the tithi, 12° forming one tithi, i.e. there are 2½ = 5/2 tithis per rāśi. Hence the instruction to divide the rāśis by 2 and multiply by 5 to get the tithis. 7a. A. शशादलं; B. शशखदमन्तिर b. A. मंशस्थिता; B. ॰मक्ष (B3. मक्षु) -मंशास्थिता d. A. तिथिस्तंद्॰

34 PAÑCASIDDHĀNTIKĀ II.7 Example 6. The true Sun is 10ʳ 18°, and the true Moon is 5ʳ 22°. Find the nakṣatra and the tithi. Nakṣatra: The Moon is 5ʳ 22°. Dividing by 4, (5ʳ 22°)/4 = 1ʳ 13°. Multiplying by 9, 9x 1ʳ 13° = 12 – 27, i.e. twelve nakṣatras have gone and in the 13th, (Hasta), 27 muhūrtas have gone. Tithi: Moon – Sun = 5ʳ 22° – 10ʳ 18° = 7ʳ 4°. Dividing by 2, (7ʳ 4°)/2 = 3ʳ 17°. Multiplying by 5, 3ʳ 17° × 5 = 17 – 25. Seventeen tithis are gone and in the eighteenth (Bahula Tṛtīyā) 25/30 parts have gone. [अहर्मानम्] मकरादौ 'गुण'युक्तो मेषादौ 'तिथि'युतो र(वि)दिवसः । कर्कटकादिषु सत्सु 'त्रयस्त्रिकाः' शर्वरीमानम् ॥ ८ ॥ Day-time 8. When the Sun is in the 3 rāśis, Makara etc., the Sun measured in rāśis plus three is the duration of day-time in muhūrtas. When it is in the 3 rāśis, Meṣa etc., the Sun plus fifteen is the duration of day-time. When in the 6 rāśis, Karkaṭaka etc., the Sun plus nine is the duration of the night-time. (To get the duration of the day-time, this should be subtracted from 30). Though no measure of time is mentioned here, we can infer that it is the muhūrta (2 nāḍīs) because by adding the shortest day, 12, and the longest, 18, we get 30 which must be equal to the whole day, i.e. 60 nāḍikās. Thus, for the Sun at the beginning of each rāśi, Meṣa etc., the day-time in muhūrtas is 15, 16, 17, 18, 17, 16, 15, 14, 13, 12, 13, 14. The longest day is 18 muhūrtas when the Sun is at the first point of Karkaṭaka (Cancer) at Summer solstice and the shortest 12 muhūrtas when at the first point of Makara (Capricorn) at Winter solstice. The day and night are equal at the first points of Meṣa (Aries) and Tūla (Libra), i.e. at the equinoxes. The daily increase or decrease in day-light is 4 vināḍīs per day. In essence, the same formula for day-light is found in the Vedāṅga Jyotiṣa and the Paitāmaha Siddhānta (PS, XII.5),with this difference that here the true Sun is used, but there, because they have no true Sun but only the mean Sun, the day which is proportionate to the mean Sun, is used. Evidently not understanding what is given here, TS have made a drastic change in the text, writing bhūsvarga-tithimito for meṣādau tithiyuto, intending to make this agree with the next verse giving the noon-day shadow. But, even within that verse, there is contradiction and this need not have been attempted, at such cost. To crown all their interpretation with their emendation is full of contradiction within itself, which has been set out in detail by me in a paper entitled 'Vāsiṣṭha Sun and Moon' in the Journal of Oriental Research, 25 (1955-56) 19 – 41. The uniform increase and decrease in day-time given here is wrong, of course, and it varies with the position of the Sun, being greatest at the equinoxes and falling to zero at the solstices. The maximum or minimum day-time itself varies with the latitude of the place (depending on tan. 8b. A. मेषादौ. C. भूर्स्वर्गातित्थिमितो रखेर्दिवसः c. B2. सत्सु; C.D. षट्सु A.B.C. रखेः. A2. दिक्कम् । d. B. मानाम्

II. 10 II. VĀSIṢṬHA-SIDDHĀNTA 35 latitude), what is given here being for some place having a North latitude of 35° 45′. (This matter is dealt with in the text in III.10 and IV.26). The rules for day-time is explained thus: From the beginning of Makara to the end of Mīna the Sun in rāśis increases from 9 to 12. The day-time also increases following it, from 12 muhūrtas at winter solstice to 15 at Equinox. As (9 to 12) + 3 = (12 to 15), the instruction to add 3 for the Sun in this quadrant follows. In the same way, from the beginning of Meṣa to the end of Mithuna, the Sun in rāśis increases from 0 to 3. The day-time increases from 15 at Equinox to 18 at summer solstice. (0 to 3) + 15 = (15 to 18), and this explains the addition of 15. As there is the maximum day of 18 muhūrtas at summer solstice, there is the minimum night there, of 12 muhūrtas. This increases to maximum night, 18 muhūrtas for Sun at the beginning of Makara, 6 months after. As a result, as the Sun's rāśi increases from 3 to 9, the night increases from 12 to 18. (3 to 9) + 9 = (12 to 18) and this explains the addition of 9 (three times three). Example 7. Give the day-time for the Sun at the beginning of: (i) Ṛṣabha, (ii) Kumbha and (iii) Kanyā. (i) The Sun is in the quadrant 0 to 3 rāśis. The beginning of Ṛṣabha is one rāśi. Therefore 1 + 15 = 16, muhūrtas, is the day-time. (ii) The Sun at the beginning of Kumbha is 10 rāśis. This is in the quadrant 9 to 12. Therefore 10 + 3 = 13, muhūrtas, is the day-time. (iii) The Sun at the beginning of Kanyā is 5 rāśis and is in the 6 rāśis 3 to 9. Therefore 5 + 9 = 14, muhūrtas, is the night-time. Deducting from 30, 30 − 14 = 16, muhūrtas, is the day-time. [शङ्कुच्छाया] कर्कटकादिषु भुक्तं द्विगुणं माध्यन्दिनी भवेच्छाया । मकरादिषु चाप्येवं, (किंत्वस्मिन्) मण्डलाच्छोध्यम् ॥ ९ ॥ मध्याह्नच्छायार्धं सत्रिभमर्कोऽयने भवेद्याम्ये । उदगयने संशोध्यं पञ्चदशभ्यो रविर्भवति ॥ १० ॥ Gnomonic Shadow 9. When the Sun is in the six rāśis beginning with Karkaṭaka, the number of rāśis traversed from the beginning of Karkaṭaka, multiplied by 2, is the mid- day shadow (of the twelve-digit gnomon) in digits. When the Sun is in the six rāśis beginning with Makara, find the distance in rāśis traversed by the Sun from the beginning of Makara, and multiply by 2. Subtract this from 12, to find the mid-day shadow. 10. When the Sun is in its southward-course, (i.e., in the six rāśis from Karkaṭaka), half the mid-day shadow plus three is the longitude of the Sun in rāśis. When in the northward course in the six rāśis from Makara, half the noon-shadow subtracted from fifteen is the Sun in rāśis. d. A.B.C.D.किं चास्मिन्. A. मंडलात्छोध्याम्; 9a. A1.कर्कटादिषु; B2॰दिवु. B. भक्तं B. मदलात्सोध्या b. A.B1.मध्यंदिनी. A. भवेछाया 10a. A1.॰ह्नछायार्द्धं c. A1.चाप्येव; A2.वाप्येवं d. B. पञ्चदशेभ्यो 5

36 PAÑCASIDDHĀNTIKĀ II.10 Example 8. (a) On a certain day the Sun’s longitude is 5 rāśis. (b) On another day it is rāśis 11-15. In both cases find the mid-day shadow. (a) The Sun is 5 rāśis and is within the six rāśis from Karkaṭaka, being 2 rāśis from the beginning of the first point of Karkaṭaka (i.e. 3 rāśis). So, 2 × 2 = 4 digits is the shadow. (b) The Sun’s longitude is rāśis 11-15. This is within the six rāśis from the first point of Makara (9 rāśis), the Sun’s position being 11ʳ 15° − 9ʳ 0° = 2ʳ 15° = 2½ rāśis from that point. 12 − (2½ × 2) = 7 digits is the noon-shadow, Example 9. (a) The Sun is in its southward course and the shadow is 4 digits. Find the longitude of the Sun. (b) The Sun is in its northward course and the noon-shadow is 7 digits. Find the Sun. (a) Half the shadow = 4/2 = 2. As the course is southward add 3 rāśis; the Sun’s longitude is 5 rāśis. (b) Half the shadow = 7/2 = 3½. As the Sun’s course is northward, deduct from 15. 15 − 3½ = 11½ rāśis. This is the longitude of the Sun. From the two sets of examples it can be seen that the two formulae are the inverse of each other. The formulae are explained thus: This Siddhānta assumes that the noon-shadow is zero when, at the end of its northward course, it reaches the first point of Cancer. Then as it moves southward, the shadow increases to 12 digits at the end of the course, i.e. after 6 months, when the Sun reaches the first point of Capricorn. Assuming the increase to be uniform, there is an increase of 2 digits per rāśi. As the shadow is zero for the first point of Cancer, the longitude in rāśis measured from this point, multiplied by 2 gives the shadow. Thus, if c is the Sun in rāśis measured from the first point of Cancer and s is the shadow in digits, s = 2c for the 6 months till the Sun reaches Capricorn, where the shadow is 6 × 2 = 12 digits. Then the shadow decreases at the same rate to zero at the first point of Cancer, in the course of 6 months. Therefore if c is the Sun measured from the first point of Capricorn, when the shadow is 12, and s the shadow, then s = 12 − 2c. Now for the longitude of the Sun from the noon-shadow. We have seen that for the six rāśis from Cancer, s = 2c. Therefore c = s/2. But c is counted from the first point of Cancer, i.e. 3 rāśis. There- fore the Sun’s longitude in rāśis is 3 + c = 3 + s/2, which is the same as the instruction to divide the shadow by 2 and add three rāśis. For the six rāśis from Capricorn, we have seen that s = 12 − 2c. Therefore c = (12 − s)/2. But c is counted from the first point of Capricorn. i.e. 9 rāśis. Therefore the Sun = 9 + c = 9 +(12 − s)/2 = 15 − s/2, which is the instruction given. It must be noted here that both the formulae are very rough. At summer solstice, when the Sun is at the first point of Cancer, its north declination is maximum and given by Hindu astronomy as 24°. At that time, the mid-day Sun is at the zenith at places on 24° north latitude (like the region of Ujjain) and so it is only in this region that the shadow is zero at this time. When the Sun reaches its southernmost point at winter solstice, i.e. the first point of Capricorn, its south declination is 24°. Therefore the zenith distance of the noon-Sun as seen from latitude 24° North at that time must be 48° towards the South, and the shadow at that time must be greater than 13 digits and not 12. (All this will be shown in Chapter IV). If the shadow is to be 12 digits, the Sun’s zenith-distance must be 45° and the region where the Sun is seen at this zenith-distance is 21° North latitude. Thus there is contradiction even here. In verse 8, we showed that the rule is intended for a region having about 36° North latitude, neither 24° nor 21°. Thus, so far as these things are concerned, the Siddhānta seems to be a hotch-potch.

II.11 II. VĀSIṢṬHA-SIDDHĀNTA 37 [छायातो लग्नं लग्नतः छाया च] द्वादशभिः सच्छायैर्मध्याह्नोनेर्भजें'द्रसहुताशम्' | अपराह्ने चक्रार्धाद्विशोध्य सार्कं भवति लग्नम् || ११ || Lagna from shadow and vice versa 11. Add 12 to the shadow (of the twelve-digit-gnomon, measured in digits) at any time of the day, and deduct from it the mid-day shadow for the day. Divide 36 by this and take the result. This result taken as rāśis, plus the Sun in rāśis is the lagna at the moment, if it is forenoon. If afternoon, deduct this from the Sun plus six rāśis and the lagna is got. The formulae (a) for the forenoon and (b) afternoon respectively can be expressed thus: (a) Lagna = Sun + 36/(12 + shadow – noon shadow). (b) Lagna = Sun + 6 – 36/(12 + shadow – noon shadow). What is called lagna is the Orient Ecliptic Point, i.e. the point of the ecliptic rising on the eastern horizon. Example 10. (a) On a certain day, the Sun is 9 rāśis and the mid-day shadow 12. At a time in the morning the gnomonic shadow is 36. Find the lagna for the moment. (b) On a certain day, the longitude of the Sun is 6 rāśis and the noon-shadow 6 digits. At a time in the evening the gnomonic shadow is 24 digits, find the lagna for that moment. (a) From formula (a), Lagna = 9 + 36/(12 + 36 – 12) = 9 + 1 = 10, rāśis. Hence the first point of Kumbha is rising in the east. (b) From formula (b), Lagna = 6 + 6 – 36/(12 + 24 – 6) = 12 – 1 6/30 = 10 24/30 rāśis. Hence the 25th degree of Kumbha is rising in the east. These rules are rough and there is no question of strictly proving them. But we can explain them thus. From sunrise to noon, as the altitude of the Sun increases, the lagna goes on increasing and the shadow decreasing, till it is shortest at noon. Therefore the increase in lagna can be roughly expressed as, a/(shadow + b), where a and b are constants to be determined. Now, if the place is supposed to be situated on the equator, and the ecliptic on which the Sun moves is supposed to coincide with the celestial equator, then at noon the shadow will be zero. At that time the increase in lagna (after sunrise) would be 3 rāśis, as the Sun has reached an altitude of 90°. Therefore 3 = a/(0 + b). Again, seven and a half nāḍīs after sunrise, the Sun would have risen to an altitude of 45°. So the increase in lagna now is 1 1/2 rāśis and the shadow is 12 tan 45° = 12 digits. Therefore, 1 1/2 = a/(12 + b). Solving these two equations for a and b we get a = 36, b = 12. Therefore the increase in lagna is 36/(shadow + 12), of course, on the given two assumptions. But the place may not be on the equator and the ecliptic does not coincide with the celestial equator and the Sun moving on it has a varying declination, with the result that generally the noon-shadow is not zero. According to the length of the noon-shadow at other times also there will be an increase in the shadow over what 11a. A1. द्वादशभिः; A2. द्वादभिः. A. सछायै A2. द्रसंजताशं b. B. मध्याह्नाने. B. हृतांशः. A1. ०द्रसजताशं; c. A.B. चक्रार्द्धाद्

38 PAÑCASIDDHĀNTIKĀ II.11 it would otherwise be, for which the rule has been formulated. As the deduction of the noon- shadow for the day of observation from the shadow would, to some extent, remove this extra length of shadow and bring about an approximation to the ideal condition, the noon-shadow is asked to be deducted from the shadow in the formula. Therefore the increase in lagna is given by 36/ (shadow − noon shadow + 12). As at sunrise the Sun is the lagna, adding the increase to the Sun we get the lagna, i.e. lagna = Sun + 36/(12 + shadow − noon-shadow). This is for the forenoon. In the afternoon, what happens in the forenoon with reference to the shadow is reversed, and therefore the rule gives the part of the lagna to increase from the time of observing the shadow to sunset. So, it is less than the lagna at sunset by what is got from the formula. But the lagna at sunset is the Sun plus six rāśis. Therefore the formula for the afternoon becomes: Sun + 6 − 36/(12 + shadow − noon-shadow). Again let it be remembered that the rules are rough. व्यर्के लग्ने लिप्ताः प्राक्पश्चाच्छोधितास्तु चक्रार्धात् । कार्यश्छेदः 'शून्याम्बराष्टलवणोदषट्कानाम्' ॥ १२ ॥ लब्धं द्वादशहीनं मध्याह्नच्छायया समायुक्तम् । सा विज्ञेया छाया वासिष्ठसमाससिद्धान्ते ॥१३ ॥ 12-13. Deduct the Sun from the lagna and convert the remainder into minutes of arc, if forenoon. If afternoon, deduct the minutes from a half circle, (i.e. from 10,800 minutes), and take these as minutes. Divide 64,800 by the minutes got. Add the result to the noon-shadow of date and deduct 12 from this. This is the shadow at the time of the given lagna. This is according to the succinct Vāsiṣṭha Siddhānta. The formula (a) for the forenoon, and (b) for the afternoon, respectively, are: (a) shadow = {64,800 ÷ (lagna − Sun, in minutes) + noon-shadow − 12}. (b) shadow = 64,800 ÷ {10,800 − (lagna − Sun, in minutes)} + noon-shadow − 12. Example 11. (a) On a certain day at a time in the forenoon, the Sun is 9 rāśis and the lagna 10 rāśis and the noon-shadow of date is 12 digits. Find the shadow for the time. (b) On another day, for a time in the after- noon, the sun is 6 rāśis and the lagna 10 rāśis 25 degrees and the noon shadow of date is 6 digits. Find the shadow. a] Using formula (a), shadow = 64,800 ÷ {(10 − 9) × 1800} − 12 + 12 = 36 digits. b] Using formula (b), shadow = 64,800 ÷ {10,800 − (10 5/6 − 6) × 1800} − 12 + 6 = 64,800 ÷ (10,800 − 8640) − 12 + 6 = 64,800 ÷ 2160 − 12 + 6 = 30 − 12 + 6 = 24 digits. These formulae (a) and (b) can be derived from the previous formulae (a) and (b) of verse 11, for these are only the inverse of the previous operations. 12a. B3.लिप्ता b. A.प्राक्पक्षा A.छोधितास्तु; B.छोधितासु 13. A1.लब्धं; A2.लब्ध. B3.हिनं B1.चक्राद्धति:; B3.चक्रार्धात् । द्वोतिः b. A.B.°ह्रच्छायया. B.समायुक्ता c. A.B.कायछेदः (B3.कायः छेदः) d. A2.वासिष्ट; B1.2.वाशिष्ट; B3.वाशिष्ठ

II.13 II. VĀSIṢṬHA-SIDDHĀNTA 39 The previous (a), is: lagna = Sun + 36/(12 + shadow − noon-shadow). Therefore, lagna − Sun = 36/ (12 + shadow − noon-shadow). Therefore 36/(lagna − Sun) = 12 + shadow − noon-shadow. There- fore, shadow = 36/(lagna − Sun) + noon-shadow − 12, where it is to be noted that (lagna − Sun) is in rāśis. If it is to be expressed in minutes, we have, shadow = 36/(lagna − Sun) × 1800 ÷ 1800

• noon-shadow − 12 = 36 × 1800/(lagna − Sun) in minutes + noon-shadow − 12 = 64800/(lagna − Sun) in minutes + noon − shadow − 12, which is (a) here. The previous (b) is: lagna = Sun + 6 − 36/(shadow + 12 − noon-shadow). Therefore 36/(shadow
• 12 − noon-shadow) = Sun + 6 − lagna = 6 − (lagna − Sun). Therefore 36/{6 − (lagna − Sun)} = shadow + 12 − noon-shadow. Therefore shadow = 36/{6 − (lagna − Sun)} + noon-shadow − 12, where (lagna − Sun) is in rāśis. If it is to be expressed in minutes, we have, shadow = 36/6 − { 6 − (lagna − Sun) × 1800 ÷ 1800} + noon-shadow − 12 = 36 × 1800/{6 × 1800 − (lagna − Sun) in minutes}
• noon-shadow − 12 = 64,800/{10,800 − (lagna − Sun) in minutes} + noon-shadow − 12, which is (b) here. The concluding words, Vāsiṣṭha-samāsa-siddhānte, though forming a part of the sentence giving the rules of verses 12-13, may be detached from it and taken to refer to the whole chapter II and mean ‘All this is given as in the succinct Vāsiṣṭha Siddhānta’. The word, nakṣatrādicchedaḥ is found at the conclusion in the manuscripts. Perhaps it is, nakṣatrā- dhicchedaḥ, meaning ‘Naksatra section’ and the title is appropriate because this is the chief thing given here and other things depend on it. Thus in this chapter the true Sun and Moon are given according to the Vāsiṣṭha and also other matters depending on them like the nakṣatra, the tithi, day-time shadow and lagna. (The star planets i.e. the regular planets, of this Siddhānta will be given in XVIII.) TS have professedly not understood verses 1, 5 and 6 and gone wrong in verses 3 (and 8), which means they have practically not understood the Siddhānta at all. Thibaut even thinks that verse 1 may be dealing with the Moon. But this professed ignorance did not prevent him from making the unwarranted remark in the Introduction (vide p. XXXVIII)..... “the methods are so crude and so completely omit to distinguish between mean and true astronomical quantities, that the Vasistha Siddhanta can hardly be included within Scientific Hindu Astronomy.” [इति पञ्चसिद्धान्तिकायां वराहमिहिरविरचितायां वासिष्ठसिद्धान्ते ग्रहादिगणितं नाम द्वितीयोऽध्यायः ॥] A.B. have as Colophon, नक्षत्रादि छेदः; C.D. नक्षत्रादिच्छेदः | Thus ends Chapter Two entitled ‘Vāsiṣṭha-Siddhānta: Planetary Computations etc.’ in the Pañcasiddhāntikā composed by Varāhamihira