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

50 PAÑCASIDDHĀNTIKĀ III.11 [अहर्मानम्] मेषादि(षु तदु)पचि(तैः) कर्कट[का]द्येषु तदपचयमितैः । दिनवृद्धिरसाध्ये(त) क्षयस्तुलाद्येषु [कर्कटकात्] ॥ ११ ॥ सागरहिमगिरिपरिधौ स्पष्टमिदं चरविनाडिकाकर्म । अन्यत्राऽपि यथैतत् स्पष्टं तच्छेद्वके वक्ष्ये ॥ १२ ॥ Day-time 11-12. To find the day-time in Meṣa, Vṛṣabha and Mithuna, add the cara differences one by one, in the order given, to 30 nāḍikās, and in the next three, subtract in the reverse order. In the next three rāśis, Tulā, etc. subtract from 30 nāḍīs in the given order, and for Makara etc. add in the reverse order. This will give the day-time fairly accurately for places in Northern (whole?) India. I shall give the method to find the day-time accurately in other places (in the IV chapter) when dealing with spherical astronomy. Thus, when the Sun is in the six rāśis, Meṣa, etc. 30 nāḍikās cara-vināḍīs = day-time. When in the six rāśis, Tulā etc. 30 nāḍikās cara-vināḍīs = day-time. The increase over 30 nāḍikās and the decrease to 30 is by Eq. shadow × {20, 16½, 6¾, 6¾, 16½, 20}vināḍīs. This is repeated in the decrease from 30 nāḍīs and the increase to 30, in the six rāśis from Tulā. We shall explain all this in the IV chapter. The cara-vināḍīs computed in the above manner and the day-time got by them will be accurate only in Northern (whole?) India it has been said. The reason for this is as follows: From the expla- nation of the method of computing the cara-vināḍīs, it can be noted that sine cara is proportionate to the Eq. shadow, and the cara-vināḍīs are proportionate to the arc obtained from sine cara. It is well known that when the sines are small they are proportionate to their arcs. Therefore when sine cara is fairly small, i.e. when the Eq. shadow is small, the arcs are proportionate to the Eq. shadow, i.e. the cara-vināḍīs are proportionate to the Eq. shadow. In India the Eq. shadow is fairly small, and therefore the cara-vināḍīs for different places in India can be formed by proportion, using the Eq. shadow. In higher latitudes like places North of the Himalayas, the shadow increasingly becomes greater, and the inaccuracy of using the given method will gradually increase. Example 5. (a) Eq. Shadow 5, Sun rā. 2-10-0. Find the day-time. (b) The Eq. shadow is 7, the Sun is in rāśis 7. Find the day light. (a) The cara-vināḍīs are 5(20 + 16½ + 2¼) = 5 × 38¾ = 194. As the Sun is in the 6 rāśis, Meṣa etc., the day-time is greater than 30 nāḍīs, and, therefore, the day-time is 30 nāḍīs + 194 vināḍīs = 33-14 nāḍīs. 11a. A.B. मेषादिषडुपचितं (B2. ॰ष्ट॰; B2. चितं) 12a. A1. हिमापरि धौ; A2. B1.C. हिमाद्रिपरिधौ; b. A.B. कर्कटाद्येषु. C. ॰षु [च] तद्. A. चयमितै B2.3. हिमद्रिपरिधौ c. A.B. साद्येने; C. स्याद्येन c-d. A2. यथैतस्पष्टं; B2.3. तथैतत् d. B. तुलाद्येषुकु. A.C.D. ॰षु वैषुवतात्; B. ॰षु वैषुवनात् d. A1. तछेद्यके

III.12 III. PAULIŚA-SIDDHĀNTA 51 (b) The cara-vināḍīs are 7 × 20 = 140. The Sun being within the 6 rāśis from Tulā, the day-time is less than 30 nāḍīs. Therefore the day-time is 30 nāḍīs − 140 vināḍīs = nāḍīs 27-40. [देशान्तरम्] यवनान्तरजा नाड्यः सप्ताऽवन्त्यां त्रिभागसंयुक्ताः | वाराणस्यां ‘त्रिकृति’: साधनमन्यत्र वक्ष्यामि || १३ || Deśāntara 13. The correction to the time of the longitude of Yavanapura to get the time of the longitude of Ujjain is seven nāḍikās, 20 vināḍikās and that of Banaras (Vārāṇasī) is nine nāḍikās. How to find the correction for other longitudes will be given (in the next verse). In short, what is here given is the difference in time due to difference in longitude alone from Yavanapura of Ujjain and Banaras, Deśāntara-nāḍīs, being difference in time of the occurrence of any event due to longitude, the occurrence being earlier by this time if the place is East, and later if West. Actually the time difference due to longitude for Ujjain from Yavanapura is nāḍīs 7-38, and for Banaras, nāḍīs 8-50. The Greenwich East Longitude of Yavanapura, Ujjain and Banaras are 30°, 75° 50′ and 83°. From this we can find the actual time difference: (75° 50′ − 30°)/6 nāḍīs = 7-38, (83° − 30°)/6 nāḍīs = 8-50. But considering the difficulty of doing this faced by the ancients for want of facilities, the achievement of the Siddhānta is commendable. ‘त्रिकृति’घ्नात् ‘खवसु’हृताद् योजनपिण्डात् स्वताडिताज्जह्यात् अक्षद्वयविवरकृतिं मूल्याः षट्कोद्धृता नाड्यः || १४ || 14. Take the distance in yojanas between the two places between which the time difference for longitude has to be found. Multiply this by 9 and divide by 80. (The result is their distance in degrees). Square the result. From this deduct the square of the difference in latitude between the two places. Find the square root of the remainder. (This is the East-West difference in degrees). This divided by 6 is the time difference in nāḍikās. The purpose of finding this time difference is ultimately to find the time difference for longitude from Yavanapura, so that it may be used in the reduction of the planet found for Yavanapura mean sunset to the local sunset. If by this rule, the time difference from Ujjain or Banaras is found, then 14a. A.त्रिकृतिघ्ना खवसु 13a. A. यवनान्तरजा; B 1.3. युवनाच्चरनाड्यः b. A.पिडात्; B 1.2.पिण्डाक्षता०; B3.पिण्डक्षता० b. A.B 1.2.वन्त्यास्त्रिभाग A.B.°ता जह्यात् ( B3. जह्यात्) c. B2.वाणारस्यां c. A.B.विवरक्रति d. A1.साधमन्यत्र: A2.साधनमत्पत्र d. A.B.2.मूला:; B 1.3.मूल्या:; C.मूलं A1.B.षट्कोधृता; D.षट्कोद्धृतं 6

52 PAÑCASIDDHĀNTIKĀ III. 14 by adding or subtracting this, according as the place is East or West, to or from nāḍīs 7-20 (for Ujjain) or 9-0 (for Banaras), respectively, the time difference for longitude from Yavanapura can be got. Example 6. The latitude of a place west of Ujjain is 27° and the distance between them is 44 4/9 yojanas. Assuming the latitude of Ujjain to be 24°, find the time difference for longitude of the place from Yavanapura. The distance in degrees = 9 × 44 4/9 = 5°. The difference in latitude in degrees is 27° − 24° = 3°. √(5² − 3²) = 4 = the east-west difference in degrees. 4/6 nāḍikās = 40 vināḍikās is the time differ- ence. As the place is West of Ujjain, deducting this from 7-20, the time difference from Yavanapura is nāḍikās 6-40. The rule is explained thus: It is well known that in a plane right-angled triangle the square of the hypotenuse is equal to the sum of the squares of the sides containing the right angle. Now, let the places be P₁ and P₂. Let the point where the line of longitude passing through one of the places, say P₁, cuts the latitude passing through the other say P₂, be C. Then C is practically a right angle of which CP₁ is one arm and CP₂ the other, and P₁P₂ is the hypotenuse of the right angled triangle P₁CP₂. If P₁P₂ is not too great, then this triangle, though on the surface of a sphere, may be treated as practically a plane triangle. P₁C is the difference in latitude of the places. P₂C is the difference in longitude, which is wanted. P₁P₂ is the distance between them. We are going to find the differ- ence in longitude P₂C in degrees. The difference in latitude P₁C is also in degrees. So P₁P₂ also must be found in degrees. So the distance in yojanas is converted into distance in degrees by multiplying the yojanas by 9 and dividing by 80, because according to this Siddhānta there are 9° for 80 yojanas on the Earth, i.e. the circumference of the Earth (360° if given in degrees) is 3200 yojanas. Thus we have the difference in longitude in degrees = CP₂² = √(P₁P₂² − CP₁²). The degrees are converted into time by the proportion, if there are 60 nāḍikās for 360° of longitude, how much for the degrees got. Therefore degrees got × 60/360 = degrees got/6, are the nāḍikās of difference in longitude. In view of the right angled triangle not being exactly plane, a better result will be got if the nāḍikās obtained are multiplied by the circumference of the earth and divided by the circumference of the line of latitude midway between the places. But the author has not mentioned this because this method is intended for India and in India the two circumference do not differ much and therefore the difference between the two methods will be negligible. Sūrya Siddhānta etc. give the correct method. The author’s method is given by the Mahābhāskarīya with the name Adhvā (‘Path’) and by the Vaṭeśvara Siddhānta with the name ‘Adhvavaha’ (‘Marching on the Path’), only to be condemned as inaccurate. In the Vākyakaraṇa, which is also satisfied with rough results, the distance along the latitude (CP₂ in the explanation) is taken as found and a rule given. The method of determining the latitude of a place is given in IV.20-21 and the author expects us to get it by using that method and use it in the formula. Another thing must be mentioned here. If the two places are distant from each other or inter- vened by a sea or mountain or some such obstacle, as for instance Yavanapura and Ujjain, Yavanapura and Banaras, or Ujjain and Banaras, then by observation, from the two places, of celes- tial phenomena that are visible everywhere at the same moment, like the circumstances of a lunar eclipse, the time difference for longitude can be obtained. (All Siddhāntas give methods based on this principle, and the Mahābhāskarīya in Chapter II, which is, devoted exclusively to Deśāntara, gives two methods.) The following is the method: Let us assume that the meridian of Ujjain as the prime meridian (generally given in all Siddhāntas do) and by computation it has been found that at 5 nāḍīs after midnight the total obscuration of the moon begins. (The beginning or end of the

III. 14 III. PAULIŚA-SIDDHĀNTA 53 total phase can be observed well and therefore specially chosen by the Sūrya Siddhānta). In another place, say in Banaras, the beginning of the total phase is observed to be at nāḍikās 6-12, (of course by its own time, i.e. local time). The time difference for longitude must be patently the difference between the two times, i.e. nāḍīs 6-12 minus nāḍīs 5, i.e. nā. 1-12. As the local time must increase as we go east, we can also say that Banaras is east of Ujjain. [इष्टदेशास्तकालः] देशान्तरनाडीभ्य-श्चरनाड्यर्धं क्षयस्तु पूर्वार्द्धे । चक्रस्यार्धे चान्ये वृद्धिस्तद्भोगमपि जह्यात् ॥ १५ ॥ Local Sunset time 15. If the Sun is in the six rāśis Meṣa etc., subtract half the caravināḍis from the time difference for longitude. If the Sun is in the six rāśis Tulā etc. add half the cara-vināḍis to the time difference for longitude. Find the motion of the planet (the Sun or Moon, in this context) during this time. Subtract this from the longitude of the planet computed. (The planet for the beginning of the local day, i.e. for local sunset, is obtained). Example 7. The time difference for longitude at a place (in India) with reference to Yavanapura is 10 nāḍīs. The Sun is rā. 9-0-0 and the cara for that day at the place is 4 nāḍīs. The Moon computed is rā. 4-8-0 and its daily motion 840'. Find the Moon at the beginning of the local day, i.e. at local sunset of the place. Longitude time difference = 10 nāḍīs. Half cara = 4/2 = 2, nāḍīs. As the Sun is within the six rāśis from Tulā, adding 10 and 2, we get 12 nāḍīs. The motion per day is 840'. The motion for 12 nāḍīs is, 840' × 12/60 = 168' = 2° 48'. Subtracting from the computed Moon, the Moon at local sunset is, rā. 4-8-0 − 2° 48' = rā. 4-5-12. The explanation of the procedure is as follows: As the days from Epoch are from mean sunset at Yavanapura, the planets computed for the days from Epoch are for mean sunset at Yavanapura and if they are required for any time in the day, they have to be found by adding the motion during the time. Therefore if the planets are required for local sunset at any other place, the time interval between the local sunset and Yavanapura mean sunset has to be found first, and the motion for the interval applied to the planet computed. This motion is subtractive if local sunset is earlier, and additive if later. Now, the author intends the procedure for India alone, and at places in India the local sunset is always earlier than Yavanapura mean sunset. This is because nowhere in India (including Afghanistan) is the difference for longitude from Yavanapura less than 4¹/² nāḍis, and the half cara greater than this. Because India is east of Yavanapura in the local mean sunset is earlier by the time difference for longitude. The actual sunset is earlier than the mean sunset or later by half the cara-nāḍīs. It is later if the Sun is in the six rāśis, Meṣa etc., because the day-time is longer. It is earlier if the Sun is in the six rāśis, Tulā etc. Therefore the local actual sunset is earlier than Yavanapura mean sunset by the time difference for longitude minus half the nāḍīs of cara when the daytime is greater. But 15a. A. नाडीभ्य; B. नाडीम्य c. B. विक्रस्याद्वै b. C. नाड्यर्थक्षय०. B. पूर्वार्द्धम् d. B. वृक्षि. B. भोरामपि; D. भागमपि

54 PAÑCASIDDHĀNTIKĀ III.15 as the half cara is always less, there is always a remainder when this subtraction is made by which time, therefore, local sunset is always earlier. When the Sun is in the 6 rāśis, Tulā etc., the local sunset is earlier by the time difference for longitude and still earlier by the half cara. Therefore it is earlier by the sum of the two. Thus, in all cases, with regard to places in India, the sunset is earlier than the mean sunset at Yavanapura and therefore the motion for the time is always to be subtracted; and, that is the instruction. If the place is north of India or west, it may happen that the half cara is greater than the time difference for longitude. Then when the Sun is in the six rāśis from Meṣa, the sunset may be later and the motion for the later time will have to be added to the planet. Further, there are two other corrections, Bhujāntara (correction for the Equation of the centre) and Udayāntara (Reduction to the Equator), the equivalent of the equation of time which have got to be made, but not given by this Siddhānta either because these are very small or because this Siddhānta is not aware of its existence. The Vākyakaraṇa omits to give the Udayāntara alone because it is not found even in its source, Bhāskarīya. The Sūrya Siddhānta omits the cara and Udayāntara corrections, the former because it begins the day at midnight which is not affected by cara and the latter because it is not aware of its existence. Śrīpati is the first to give the Udayāntara. TS are unaware that the two corrections mentioned above are given here, which can be seen from the commentary pp. 12-13, and English notes pp. 16-17. They also seem to think that the instruction [L2

III. 16 III. PAULIŚA-SIDDHĀNTA 55 The result are nāḍīs of the incomplete nakṣatra gone in that day. Deduct it from 60. The ending moment of the nakṣatra gone or last gone on that day is got in nāḍīs from the beginning of the day. To get the tithi, subtract the Sun's longitude from the Moon's and convert it into minutes. Divide by 720. The quotient are full tithis gone after new moon. The last tithi gone is the tithi ending before the incomplete one begins. Multiply the remainder by 60 and divide by the difference of the Sun's and Moon's motions in minutes, for that day. The result are nāḍīs of the incomplete tithi on that day. Deduct the nāḍis from 60. The remaining nāḍīs are the ending moment of the tithi ending or last ending on that day. If the total tithis got are more than 15, count again from one, i.e. Prathamā. Example 8. At the end of a certain day the Sun is rā. 2-15-10 and its daily motion 57'. The moon is rā. 10-18-30 and its daily motion 827'. Find the nakṣatras etc. for the day. The Moon = rā. 10-18-30 = 318° 30' = 19,110'. Dividing this by 800 the quotient, i.e. full nakṣatras gone is 23, and the remainder 710' has gone in the 24th. Multiplying by 60 and dividing by the daily motion, 710 × 60 ÷ 827 = 51-31, nāḍis, belong to the 24th. Therefore the 23rd, i.e. Śraviṣṭhā ends 60.0 - 51.31 = 8.29 nāḍis, after the beginning of the day. The Tithi: Moon - Sun = rā. 10-18-30 - rā. 2-15-10 = rā. 8-3-20 = 243° 20' = 14,600'. Dividing by 720, the full tithis gone are 20, and the remainder 200' has gone in the next tithi. Multiplying this by 60 and dividing by the difference of the daily motions, the nāḍis got are 200 × 60 ÷ (827 - 57) = 200 × 60/770 = 15-35, which is the time occupied by the 21st tithi. Therefore the 20th tithi, i.e. Bahula Pañcamī ends at 60 nāḍikās - 15.35 nāḍikās, i.e. 44.25 nāḍikās after the beginning of the day. The length of a nakṣatra is 800' and the Moon's mean daily motion 791', is not much different from it. The length of a tithi is 720' and its mean daily passage 732 is not very much different from it. Therefore generally there is one nakṣatra or one tithi ending in a day. But it may happen that the remainder is so small and the motion or passage for the day so great, that, remainder + length of a nakṣatra or tithi < the daily motion or passage. In this case, two nakṣatras or two tithis end on the same day. In the case of the tithi this is called avama, the second ending tithi being immersed in the day and not counted for reckoning days. On the other hand, the remainder may be greater than the motion or passage for the day, with the result that no nakṣatra or tithi ends on that day, i.e. they begin at close of the previous day, extend throughout the day and end at the beginning of the next day. When this happens in the case of a tithi it is called Tridina-spṛk, literally 'touching three days'. The explanation of the rules is as follows: The Zodiac consists of 12 rāśis, i.e. 12 × 30 × 60 = 21,600 minutes. It is divided into 27 equal segments called nakṣatras and so there are 21600 ÷ 27 = 800 minutes for each segment. What is called nakṣatra in the verse and sought to be found, is the segment in which the Moon is situated. Therefore the Moon's longitude in minutes is divided by 800 and the quotient are the segments passed. The remainder is the position of the Moon in the next segment. The time taken by the Moon to pass that portion is found by the proportion; If the daily motion takes 60 nāḍikās to pass, how long will the remainder take? Therefore the time when the Moon has been at the end of the segment just passed, i.e. the end of the nakṣatra passed, falls before the end of the day by the obtained nāḍikās. Now for Tithi: When after new moon the Moon leaves the Sun behind for every 12° one tithi is gone. Therefore by subtracting the Sun from the Moon, the total degrees left behind is found, and dividing this by 12° or 720' the tithis gone is found. Everyday, i.e. in every 60 nāḍikās, the Sun is left behind by the difference of their motions. Therefore the time taken for leaving behind the remainder is: remainder × 60 ÷ difference of the motions. As the remainder is of the incomplete tithi, the completed tithi ends before the end of the day, by a time equal to the nāḍikās found.

56 PAÑCASIDDHĀNTIKĀ III. 17 [रविमुक्तिः] ‘[यम]-शिखि-गुणा-ऽग्नि-यम-शशि’-‘वियुता सैका सरूप रूपै-का’ ‘खै’-‘क’वियुता च भानोः षष्टिर्भुक्तिः क्रमादेवम् ॥ १७ ॥ Sun’s daily motion 17. The daily motion of the Sun in minutes during each of the twelve months, Meṣa etc. is 58, 57, 57, 57, 58, 59, 61, 61, 61, 61, 60, 59. The daily motion for the month is found thus: During every solar month, the Sun moves one rāśi, i.e. 1800′. Dividing this by the days of the month the daily motion is got. Or, according to the length of the month, take 31, 30 or 29 days of the month, almost covering it. Find the motion for these days and divide by the number of days taken, the daily motion for the month is got. Because the daily motion thus found is very near sixty minutes, the author enumerates their difference from sixty, for the sake of convenience. The mean daily motion is 59′ 8″, which can be got dividing the minutes in 12 rāśis, i.e. 21,600, by the days in the solar year. Because the higher apsis, i.e. the apogee of the Sun, is about the middle of Mithuna, the Sun’s motion in Vṛṣhabha, Mithuna and Karkaṭaka is very slow, and the daily motion, 57′ given for these is proper. Because the lower apsis, i.e. the perigee, is about the middle of Dhanus, the Sun’s motion in Vṛścika, Dhanus and Makara is very quick, and the daily motion, 61′ given for these is proper. We shall here derive the motion for Meṣa and Tulā. Let us begin with the moment on the first day of Meṣa, when the mean Sun is zero. By III. 1, the kendram is 20° and the correction for making the Sun true is − 11′ × 20°/30° = − 7′. The true Sun is 0°−7′ = rā. 11−29−53. 31 days after this moment, the mean Sun is, 31 × 120 ÷ 43831 × 360° = 30° 33′. The kendram is 30° 33′ + 20° = 50° 33′. The true Sun is 30° 33′ − 11′ − 48′ × 20° 33′/30 = rā. 0-29-49. The motion of the Sun for 31 days is Rā. 0-29-49 − rā. 11-29-53 = rā. 0-29-56. The motion per day is 29° 56′/31 = 1736′/31 = 58′. Therefore 58′ is the correct motion for Meṣa and not what is mentioned in the given verse, and that is why we have suggested the emendation yama for guṇa in the text. We shall examine the daily motion for the Tulā. We shall begin work from the first day of Tulā when the mean Sun is 185°. The kendram for that is 185° + 20° = 205°. The true Sun is 185° − 11′ − 48′ − 69′ −70′ − 54′ − 25′ + 10′ × 25°/30° = 180° 31′ . Now we shall take a time 30 days later. The mean Sun then is 185° + 30 × 120 × 360°/43831 = 214° 34′. The kendram is 214° 34′ + 20° = 234° 34′ . The true Sun is 214° 34′ − 11′ − 48′ − 69′ − 70′ − 54′ − 48′ − 25′ + 10′ + 48′ × 24° 34′/30° = 210° 46′. The motion for 30 days is 210° 46′ − 180° 31′ = 30° 15′. The motion per day is 30° 15′ ÷ 30 = 60½′. The author gives this as a whole number, 61′; so it is all right. 17a. A.B.C.D. गुण c. C. भानां b. B. वियुक्ता d. C. क्रमाद् [भानोः] A.C. खैकै for सैका

III.19 | III. PAULIŚA-SIDDHĀNTA | 57 [करणानि] सित (बहुल्योः) क्षयधनं षड्भागाशीतगो रविभोगात् | लिप्ताः 'खर्तुहुताशैर्लब्धं करणं तिथिवदन्यत् || १८ || बहुलचतुर्दश्यर्धाद् ध्रुवाणि शकुनिश्चतुष्पदं नागः | किंस्तुघ्नमिति [च] करणान्यर्धे (चराणि) प्रवर्तन्ते || १९ || Karaṇas 18. In the bright fortnight take the Moon minus Sun and subtract from it 6°. For the dark fortnight (take the Moon minus Sun from the beginning of the dark fortnight, i.e.) take the Moon minus Sun with 6 rāśis subtracted from it, and add 6°. Convert it into minutes and divide by 360'. What are obtained are the (Cara) karaṇas (Bava etc. coming one after another repeatedly). (Take the remainder and treat it as) the remainder in calculating the tithi, (i.e., multiply by 60, and divide by the difference of the daily motions of the Sun and Moon in minutes etc.) (and thus get the ending moment of the last karaṇa. In each tithi the first half is one karaṇa and the second half another). 19. From the middle of the fourteenth tithi of the dark fortnight (are the four half tithis, viz., the second half of Bahula-Caturdaśī, the two halves of Amāvāsyā, and the first half of Śukla-pratipad, which) are the Sthira-karaṇas, Śakuni, Catuṣ-pāda, Nāga and Kiṁstughna, respectively. (The other Karaṇas are) movable. A karaṇa is half a tithi. Then from the remaining half of śukla-pratipad the Cara-karaṇas come in the order Bava, Bālava, Kaulava, Taitila, Gara, Vaṇijya (Vaṇija) and Viṣṭi (Bhadra), (repeating eight times). As said before, there are two karaṇas in a tithi, the first ending at the middle of the tithi, and the second ending with the tithi. Therefore, the ending moment of the second need not be computed separately. Even that of the first is not computed by almanac-makers, the mid-point of the tithi being taken for this. Another thing is to be mentioned: Just as two tithis can end on the same day, three karaṇas can end on the same day. Example 9. Calculate the karaṇa from the data supplied in Example 8. Moon – Sun got there is rā. 8-3-20, and the difference of daily motion 770'. As it is Bahula-pakṣa, deducting 6 rāśis and adding 6°, rā. 8-3-20 – rā. 6-0-0 + 6° = rā. 2-9-20 = 4160'. Dividing by 360', the quotient got is 11, and remainder 200'. As in the case of the tithi, the ending moment is : 200 × 60 ÷ 770 = nā. 15-35 before the end of the day, i.e. the karaṇa ends nā. 44-25 after the beginning 18a. A.सितवज्जलधोः; B1.सितवजलधेः; c. A. ॰घ्नामिति; B. ॰घ्नासित B2.3.सितवनुलधेः c-d. A.B.C. चराण्यर्धे; D. चराण्यर्धं b. A.B. भागोविरवि. A. भोगान् d. A.B.करणानिवत् प्रवर्तन्ते; 19a-B. B1.3. बहुलयतु दृश्यं द्धधि वाणि; C. करणानि प्रवर्तन्ते; B2. बहुलचतु दृश्यं तां द्वयध्रुवाणि D. करणं तिथेः प्रवर्तन्ते b. A1. ॰श्र्वष्पदं; B. ॰नि चतुष्पदं

58 PAÑCASIDDHĀNTIKĀ III. 19 of the day, and the karaṇa that has ended, being the 11th, is Taitila. (Note that this is also the ending moment of the tithi). In the place of vajjaladhoḥ (in verse 18) the reading bahulayoḥ is suggested, following the meaning and keeping to the letters. Or, it can be corrected as kajjalayoḥ which will give the same meaning, 'dark fortnight'. Secondly, to avoid splitting the word ardhe between the third and fourth feet (in verse 19), carāṇi and karaṇāni have been interchanged, as possibly the scribe has interchanged them by the similarity of letters. The word, ca is interposed in the third foot, to make up the deficiency of one syllable, as an original ca carāṇi might possibly have been written as carāṇi. The extra syllable in the fourth foot can be explained in the manner we did earlier. [व्यतीपातवैधृती] अर्केन्दुयोगचक्रे वैधृतमुक्तं दर्शक्षसहिते

III.20 III. PAULIŚA-SIDDHĀNTA 59 The author here gives the computation of two of the twenty-seven yogas, Viṣkambha, etc., of which Vyatīpāta is the seventeenth and Vaidhṛta is the twenty-seventh. These have to be known because offerings are made to the manes and other deeds of merit are performed at these times, as at Viṣuva, Ayana, Saṅkrama, etc. (which also are going to be given), these being two well-known days among the ninety-six Śrāddha-days. All astronomers know that the twenty-seven yogas are com- puted like the twenty-seven nakṣatras, using the sum of the true Sun and Moon in the place of the true Moon and the sum of the daily motions in the place of the Moon's daily motion, because the yoga and the nakṣatra have equal extent, viz. 800 minutes. Vaidhṛta, the twenty-seventh of the Viṣkambha series, and theVaidhṛta here given are identical because the one ends at twenty-seven nakṣatra segments, i.e. one full revolution, and the other also ends at a full revolution, the duration of both being the same. In the same way, the Vyatīpāta given by the author is the same as the seven- teenth yoga of the same name in the Viṣkambha series because, true Sun + true Moon + 10 nakṣatra- lengths = one revolution = 27 nakṣatra-lengths. Therefore true Sun + true Moon = 27 - 10 = 17 nakṣatra-lengths, given for Vyatīpāta of the Viṣkambha series. For the sake of syntax sahiteṣu has been corrected as sahite tu. For the sake of grammar cakraḥ is corrected as cakram, for the neuter gender alone means a cycle or revolution, viz. 12 rāśis. Or let it be the masculine cakraḥ itself, meaning collection which ultimately can yield the idea of a collection of 12 rāśis. yitair bhāgaiḥ is corrected as yutair bhāgaiḥ, i.e. 'the sum of the daily motions', which is necessary. Gatairbhāgaiḥ or sthitair bhāgaiḥ can satisfy the context, but will not be sufficient, for divi- sion by the sum of the daily motions cannot ordinarily be understood without being told. But the correction, of cakre, which is quite all right, as ṣaṭke by TS is unwarranted and due to ignorance of what is wanted here. Another thing must be mentioned. If the twenty-seven yogas, Viṣkambha etc., are computed, as in the later-day works like the Sūrya Siddhānta, then there would be no need to take the trouble of com- puting these two, viz. the seventeenth and the twenty-seventh, alone separately. If we are instructed to do these two separately, it is because the Pauliśa did not have the twenty-seven yogas. We have reason to believe that even during the time of the VM these did not exist, for in the Bṛhat- saṁhitā, while nakṣatra, tithi and karaṇa are taken up for astrological predictions yoga is not so taken. The following is the history of the yoga. The yoga is not mentioned in the Vedas, and the Vedāṅga Jyotiṣa does not give it. From the Paitāmaha condensed by our author we can infer that the original Paitāmaha Siddhānta gave the Vyatīpāta for the first time, for this condensed Paitāmaha gives the rule for Vyatīpāta: "Multiply the days from Epoch, (this Epoch is different), by 12, and divide by 305" (XII. 8). In the Bauddha and Jain astronomical works, like Sūryaprajñapti and Kālalokaprakāśa, too, the Vyatīpāta alone is mentioned. Because the author gives both Vyatīpāta and Vaidhṛta here, we can guess that the original Pauliśa had Vaidhṛta also. Āryabhaṭa, a contemporary of VM, mentions Vyatīpāta alone in the sūtra, "The Sun's cycles plus the Moon's cycles are the number of Vyatīpātas" (ĀBh, Kāla. 3,), but commentators take him to mean Vaidhṛta also by implication, for, the Mahābhās- karīya, which is practically a commentary on the Āryabhaṭīya says, "When the Sun plus the Moon equals six signs it is the Vyatīpāta, when it is twelve signs it is Vaidhṛta, and when it is equal to the distance of Anūrādhā it is the yoga Sārpamastaka" (IV. 35). Here the sum being equal to twelve signs, gives the Vaidhṛta mentioned in the context, which is patent. The distance of Anūrādhā being equal to seventeen nakṣatra segments, Sārpamastaka is to be identified with the Vyatīpāta of the Pauliśa, which, as we have shown, is the seventeenth of the Viṣkambha series. Govindasvāmī too, in his Mahābhāskarīya-Bhāṣya on this verse quotes the original Āryabhaṭīya-Sūtra and explains that Bhāskara here gives both Vyatīpāta and Vaidhṛta. (As for the Vyatīpāta given by the sum equal to six signs, that

60 PAÑCASIDDHĀNTIKĀ III.20 is the Mahāvyatīpāta, distinct from the seventeenth of the Viṣkambha series, which is not what we are talking about here.) Prabhākara, generally mentioned as a disciple of Āryabhaṭa, has mentioned seven yogas, which he called Mahādoṣaḥ ('the great Inauspicious'). This information we have from two ślokas quoted by Śaṅkaranārāyaṇa in his commentary on the Laghubhāskarīya as Prabhakāra's. The ślokas say, "Find Sun plus Moon, in terms of nakṣatra-segments. When they are equal to twenty-seven (i.e. a full revolution), when 14, 8, 12, 5, 17, 18 and 10 are added, there are the Mahādoṣas Nirodha, Parigha, Vajra, Daṇḍa, Gaṇḍa Śūla and Vyatīpāla, respectively. In this group, all excepting Daṇḍa, can be identified in the Viṣkambha series. Prabhākara has not included Vaidhṛta in the group, perhaps because he does not consider it as a Mahādoṣa. Because these are computed individually, by a special rule, we can conclude that the twenty-seven yogas, Viṣkambha etc., were not in vogue in the days of Prabhākara. We have mentioned that in the days of Bhāskara a senior contemporary of Brahmagupta, also the twenty-seven yogas did not exist. Though it may be supposed that the twenty-seven yogas had come into vogue by Brahmagupta's days from the statement, "The minutes of the sum of the longitudes of the Sun and the Moon, divided by 800 are the yogas", (Br. SpSi., Spaṣṭa. 63) and on the strength of this we ourselves have written that Brahmagupta knew the twenty-seven yogas, in our Introduction to the Mahābhāskarīya, it is now learnt that the statement is an interpolation because this is not taken up and commented upon by Pṛthūdakasvāmi in his Bhāṣya of the Brāhmasphuṭa-Siddhānta and also because in giving the computation of puṇyakālas at the ends of tithis, nakṣatras, etc. according to custom, like Vaṭeśvara and Śrīpati, Brahmagupta omits yoga while the others include yoga as well. In the Sūrya-Siddhānta etc. which are later, the Viṣkambha series find a place. Thus of the five aṅgas, the yoga was the last to develop. We said that the Vyatīpāta was the first yoga born and next Vaidhṛta. We shall consider their nature and how they arose. The Vedic priests and astronomers were in the habit of observing the sky looking for celestial occurrences like the rising and setting of the Sun and the Moon, because of the need of this kind of knowledge for the performance of yajñas and out of thirst for knowledge. It is said that the Gavām-ayana Satra was designed for this very purpose. The following facts were observed by them. At one time the Sun rises farthest south of the East point, that is the end of Dakṣiṇāyana and beginning of Uttarāyaṇa. (This is the winter solstice). After that, the Sun rises more and more north every day and, at the end of six months, rises farthest north. Then is the end of Uttarāyaṇa and the beginning of Dakṣiṇāyana. (This is the summer solstice). From that time it begins to rise more and more to the south, until after six months again it is farthest south. This is the end of Dakṣiṇāyana and the beginning of Uttarāyaṇa again. Thus in a year there are the two courses of the Sun, northward and southward. In a given place, the exact point north or south where the Sun rises depends on its declination north or south. Like the Sun, the Moon too, according to its declination, rises north or south of the east-point and has its Uttarāyaṇa in about fourteen days and its Dakṣiṇāyana in about the same period, the total taking a little more than twenty-seven days. Now, the day on which the Sun and the Moon rise almost at the point, one moving south-ward, and the other moving north-ward, coming to meet each other as it were, that day is the Vyatīpāta. Because they cross each other moving in different directions, the phenomenon is called Vyatīpāta or Vyatipāta. Now, how can the time of the phenomenon be computed? Because they must rise nearly at the same point, their declinations must be nearly equal. That they must be moving in opposite directions, i.e. their respective ayanas should be different, has been mentioned. These two conditions can be approximately secured if the position of one is as far away on one side of the junction of Uttarāyaṇa and Dakṣiṇāyana, as that of the other is on the other side of the junction. Let us take it that the

III.20 III. PAULIŚA-SIDDHĀNTA 61 longitudes are reckoned from the starting point of the Uttarāyaṇa (winter solstice), as in the Vedāṅga Jyotiṣa and the Paitāmaha from Śraviṣṭhā. The two being at equal distances on both sides of the zero point means that the sum of their longitudes is equal to one full revolution, i.e. twelve rāśis. It is this that the Paitāmaha gives by its rule, ‘Multiply the days by twelve and divide by 305.’ But, because the true declination of the Moon will generally differ from that of the Sun at this time, on account of its latitude, the time given is only approximate and the Paitāmaha intends that the actual time should be found by observation. If we reckon the longitude not from Śraviṣṭhā as the zero point but from Aśvinī, then the longitudes will each be five nakṣatras less, because Aśvinī is five nakṣatras forward, and the sum will be ten nakṣatras less. Therefore, if ten nakṣatras are added to the sum of the longitudes (as the author asks us to do) we have the condition fulfilled, and therefore the Vyatīpāta. But in course of time, on account of the precession of the equinoxes, the winter solstice had moved to the beginning of Makara at the time of the author, and now still more backward so that conformity to definition is growing less and less. But on account of respect for the old Śāstras, the 17th continued and still continues to be the Vyatīpāta, just as we continue to observe Uttarāyaṇa rites still when the Sun enters Makara because Uttarāyaṇa was once there, though now it has come down into Mūla. A new type of Vyatīpāta called the Mahāvyatīpāta came into existence to satisfy the definition. This is mentioned by the author in the next two verses. The memory of a sacred day at the sum being a full revolution resulted in the creation of a new sacred day, even when reckoned from Aśvinī, and it was called Vaidhṛta, because the old Vyatīpāta was ‘sustained’ (dhṛta), as it were, by this. Because it has grown in the place of the Vyatīpāta, this Vaidhṛta itself is sometimes called Vyatīpāta. For e.g. the Sūrya-Siddhānta says, “This is another well known Vyatīpāta, called by the different name of Vaidhṛti” (XI. 8) and “The three fearsome Vyatīpātas” (XI. 22). Govindasvāmi also says this: “When the sun plus Moon is equal to six signs, there is Vyatīpāta. When it is equal to twelve signs it is Vaidhṛta and this is also called Vyatīpāta; for it is said ‘The sum of the revolutions of the Sun and those of the Moon are the Vyatīpātas (in the yuga)’ (ABh. Kāla, 3).” How does this mean that? This is how: The sūtra primarily gives only the Vaidhṛtas that come at the end of full revolutions, which are called Vyatīpātas because both have the same characteristics. The effect of both being the same, Vaidhṛta is called Vyatīpāta. So the vyatīpātas characterised by full revolutions and half revolutions are both given by the sūtra. (Govindasvāmi’s Bhāṣya, Mahābhās- karīya IV.35). Śaṅkaranārāyaṇa too, by saying “Āryabhaṭa mentions the two types of vyatīpātas”, in his commentary on Laghubhāskarīya, II. 29, understands Vaidhṛta also by the word Vyatīpāta. आश्लेषार्धादासीद् यदा निवृत्तिः किलोष्णकिरणस्य । युक्तमयनं तदाऽऽसीत् सांप्रतमयनं पुनर्वसुतः ॥ २१ ॥ 21. When the Sun began to turn south, i.e. when the summer solstice was at the middle of the asterism, Āśleṣā, the requirement of the definition that the Sun and the Moon should be in different ayanas was satisfied. But now the turning south takes place at three quarters of Punarvasu. Therefore the definition has become faulty. From this we can infer that the author knew the precession of the equinoxes. In the Bṛhatsaṃhitā also he says the same thing, “Certainly at one time, the summer and winter solstices were at the middle of Āśleṣā and the beginning of Dhaniṣṭhā, respectively, because such has been mentioned in

62 PAÑCASIDDHĀNTIKĀ III.21 the ancient lore. But now the summer solstice is at the beginning of Cancer and the other one at the beginning of Capricorn. If at any time this is not conformed to, then there is a further change, which can be seen and measured by observation and examination." (Br. Sam. III. 1-2). It is from this that we have interpreted Punarvasu as "the point at three quarters of Punarvasu". The ancient lore mentioned here includes Vedāṅga-Jyotiṣa and Paitāmaha Siddhānta. The Yājuṣa-Jyotiṣa says, "At the beginning of Śraviṣṭhā the Sun and the Moon turn northward and at the middle of Āśleṣā they turn southward, with the Sun always in the Māgha and Śrāvaṇa months, respectively" (verse 7). "When the Sun and the Moon rise in the sky together, with Śraviṣṭhā with them, the Yuga begins then as also the month of Māgha, the seasonal month Tapas, the bright fortnight, and the turning northward " (verse 6). As the Paitāmaha too counts the nakṣatras of longitudes from Śraviṣṭhā and says that it is Vyatīpātā when the sum of their longitudes is a whole revolution, we can infer that the turning northward is at Śraviṣṭhā. विपरीतायन(यां)तो यदार्ककाष्ठां श(शी) सविक्षेपः । भवति तदा व्यतिपातो दिनकृच्छशियोगचक्रार्धे ॥२२॥ 22. With the Moon approaching to meet the Sun, moving in a direction opposite to that of the Sun, when its true declination (i.e. the mean declination plus its latitude) becomes equal to the Sun's and when the sum of their longi- tudes is nearly six signs, then is the Vyatīpāta conforming to the definition, (i.e. the Mahāvyatīpāta). The minimum and sufficient conditions for the Mahāvyatīpāta are that the Sun and Moon should have different southward or northward courses and that their true declinations must be equal, both being north or both being south. The second part of the second condition, though not mentioned by the verse, is implied in the requirement that the sum should be nearly six signs. Because the northward or southward courses and the declinations depend on the tropical longitudes, we can understand that the sum also is of the tropical longitudes (i.e. the sāyana longitudes) of the Sun and the Moon. If this is not stated it is because during the time of the author the Ayanāṁśa, i.e. the difference between the tropical and sidereal longitudes, was practically zero and the author intended the work as a karaṇa not to be used for a very long time when the ayanāṁśa would become considerable. We have interpreted "half revolution as approximately six signs" because when the Moon has a latitude as generally it would have, the equality in declination will happen not exactly at the sum being six signs. Only the mean declination of the Moon will be equal to that of the Sun when the sum is exactly six signs, as Bhāskara I says in his commentary on the Āryabhaṭīya, (Kāla, 3), "Vyatīpāta 21. Quoted by Utpala on BS 2, p.40. 22a. A.B.D. ॰यनपातो C. ॰यनभागो 21a. A1.B1.2. अश्लेषार्धा॰ b. B. पदार्क A.B. शशिसविक्षेपः b. B. किलोकृकिरणस्य B. काष्टांशशि; C.D. काष्ठांश [श] शिरविक्षेपः c. A. युक्तमथनं c. B. भवेति d. A. ॰तमथनं d. A. दिनक्रछशि

III.22 III. PAULIŚA-SIDDHĀNTA 63 occurs when the declinations are the same and the courses are different. The expression half- revolution in that connection is only meant to be approximate, because by the latitude of the Moon it may be a little more or less." Therefore we should examine whether a Vyatīpāta would occur at the neighbourhood of the sum being six signs, because it can occur only there. But sometimes it may not occur at all, because the definition is not satisfied (All this is expounded clearly in works like the Siddhānta Śiromaṇi and we stop with this). One may think that we are making contradictory statements by saying in the history of the origin of the Vyatīpāta, that it occurs at the sum being full revolutions and here that it occurs at half revolu- tions. There is no contradiction because the origin from which the longitudes are measured is different in the two cases. In the former the winter solstice was taken as the origin, and, in the latter, the spring equinox. There is a difference of three signs between the origins, which causes the same difference in each of the two longitudes, with the result that there is a difference of six signs in the sum. That they are the same can be shown thus: The Sun measured from winter solstice, (say, a) = the Sun measured from spring equinox (say, b) + 3 signs. The Moon measured from winter solstice, (say, á) = the Moon measured from spring equinox (say, b́) + 3 signs. Therefore a + a' = b + b' + 6 signs. If a + a' = full revolution, b + b' + 6 signs = full revolution, therefore b + b' = full revolution - 6 signs = half revolution, which proves the sameness. Spring equinox is not men- tioned because at the author's time it was situated at the beginning of Aśvinī and longitudes are reckoned from there. Example 11. The Sun and the Moon at the end of the day are rā. 1-10-0 and rā. 4-23-30, and their daily motion 57' and 783'. Taking the spring equinox to be at the beginning of Aśvinī, i.e. the winter solstice at the beginning of Capricorn, examine the possibility of Vyatīpāta, in both ways. Because the longitudes are from zero Aśvinī, they are the same as reckoned from spring equinox also, both points being the same in the problem. Therefore sum of longitudes = rā. 1-10-0- + rā. 4-23-30 = rā. 6-3-30. This is 3° 30', i.e. 210', over a half revolution. The sum of the daily motions = 57' + 783' = 840'. Therefore at 210 × 60 ÷ 840 = 15, nāḍīs before the end of the day, the sum is equal to a half revolution or 6 signs, and so Vyatīpāta may occur in its neighbourhood. Otherwise, if the longitudes as measured from winter solstice, the Sun = rā. 1-10-0 - rā. 9-0-0 = rā. 4-10-0. The Moon = rā. 4-23-30 - rā. 9-0-0 = rā. 7-23-30. Their sum = rā. 4-10-0 + rā. 7-23- 30 = rā. 12-3-30, and this is 210' over a full revolution. Therefore 210' × 60 ÷ 840 = 15, nāḍīs before the end of the day. The sum is a full revolution and the Vyatīpāta may occur as its neighbour- hood. (Note that worked in both ways, the time is the same). Now for the readings. In the place of pāto we have taken yāto because the scribe may easily mistake pā for yā. But the correction bhāgo of TS does not agree with the second case in arkakāṣṭhām and deserves to be rejected. The wrong reading, śaśi-savikṣepaḥ has been corrected by us into śaśī savikṣepaḥ, by a simple lengthening of. But TS and NP have made it śaśiravikṣepaḥ which is incorrect and also does not agree with kāṣṭhām. The meaning which they have taken for this verse itself is wrong. Their interpretation of kāṣṭha into 'maximum declination' i.e. 24° (or 23° 20') is not proper, for, in his work (see Chap. IV), kāṣṭhānta is used for maximum declination and kaṣṭhā is taken to mean only declination. Let us concede it is maximum declination and therefore means 24°. Even this does not agree with the meaning given by them because they want and imply 23° 20' only there. If 24° is given roughly for 23° 20', why not 23° which is nearer. They do not seem to have under- stood at all what is sought to be conveyed by the author.

64 PAÑCASIDDHĀNTIKĀ III.24 [षडशीतिपुण्यकालः] मेषतुलादौ विषुवं षडशीतिमुखं तुलादिभागेषु । षडशीतिमुखेषु रवेः पितृदिवसा ये (ऽव) शेषाः स्युः ॥ २३ ॥ षडशीतिमुखं कन्याचतुर्दशेऽष्टादशे च मिथुनस्य । मीनस्य द्वाविंशे षड्विंशे कार्मुकस्यांशे ॥२४ ॥ Ṣaḍaśīti-puṇyakāla 23-24. At the first point of Meṣa (Aries) and Tulā (Libra) are the spring and autumnal equinoxes (and the sacred days thereof are when the Sun is there.) The commencements of the sacred days called Ṣaḍaśītis are at periods of 86 solar degrees commencing with Tulā-zero point. The days in the solar months after the respective commencement of the Ṣaḍaśītis are sacred as connected with the manes. The commencement of the Ṣaḍaśītis are after 14 degrees of Kanyā, (Virgo), after 18 degrees of Mithuna (Gemini), after 22 degrees of Mīna, (Pisces) and after 26 degrees of Dhanus (Sagittarius). The main purpose of the author in giving the equinoxes here is to indicate the sacred days con- nected with them as can be gathered from the context. The equinoxes, i.e. the points of intersection between the ecliptic and the celestial equator, though moving westward slowly along the ecliptic, (this is the precession of the equinoxes), were at the first points of Meṣa and Tulā only at the period of the author. At the present day the equinoxes have moved far into Uttara-Bhādrapada and Uttara-Phalgunī, but the sacred days are still observed with the Sun entering Meṣa and Tulā by blind routine. The time taken by the Sun to move one degree is a ‘solar day’ according to Hindu astronomers. (We have put it within inverted commas, because in English it means the ordinary day caused by the Sun and therefore quite different). So in a solar year there are 360 ‘solar days’, and in each solar month 30 ‘solar days’. As for counting from zero-Tulā, this is enjoined by the Dharma-śāstras. The commencements of the Ṣaḍaśītimukha-s are, 1 × 86° = 86°, 2 × 86° = 172°, 3 × 86° = 258° and 4 × 86° = 344°. from zero-Tulā, i.e. from rā. 6-0-0. Therefore they are rā. 6 + 86°, rā. 6 + 172°, rā. 6

• 258° and rā. 6 + 344°, and these are, respectively, 26 degrees of Sagittarius, 22 degrees of Pisces, 18 degrees of Gemini and 14 degrees of Virgo. These sacred days are not observed in these days and it would be interesting to know when and how they went out of vogue. When the Sun enters Sagittarius, Pisces, Gemini and Virgo, we observe the sacred day, calling it Ṣaḍaśīti; and in the place of the last sixteen ‘solar days’ of Virgo, (these seem to have secured importance at the time of Sūrya Siddhānta) the dark fortnight of Bhādrapada is dedicated to the Manes, with the name of Mahālaya- pakṣa. The dark fortnight of Āśvina also is observed as a secondary Mahālaya-pakṣa and it is the belief that the Manes are sent back to their world on Naraka-Caturdaśī. Now, what is the speciality about 86 solar days, it may be asked. This period is three synodic months less one day. It may be 23a. A. मेखतुलादौ A.विषुव; B.1.2. C.D.विषुवत्; B3.दिषु b. A. षडसीति d. A.B.विशेषा स्युः c. A.B.दिवसाद्ये 24b. A. °ष्टादर्शे

III.24 III. PAULIŚA-SIDDHĀNTA 65 that a section of people observed a sacred day for the manes once in three synodic months, and then this came in its place. [अयनम्] उद्गयनं मकरादावृतवः शिशिरादयश्च सूर्यवशात् । द्विभवनकालसमानं दक्षिणमयनं च कर्कटकात् ॥ २५ ॥ Solstices 25. The Sun’s turning northward is when it reaches the zero-point of Makara, (Capricorn), i.e. at winter solstice, and its turning southward is at the zero point of Karkaṭaka (Cancer) i.e. at summer solstice, with the attendant sacred days. The seasons Śiśira etc. commence with the winter solstice and each season lasts two tropical solar months. The precession of the equinoxes implies the precession of the solstices as well and therefore the solstices at the zero-points of Karkaṭaka and Makara is true only for the period of the author. If the sacred days are observed still when the Sun enters these signs, it is again blind custom. As the seasons depend upon the position of the mid-day Sun in the sky and the length of day time, and these depend on the Sun’s declination depending on tropical (sāyana) longitude of the Sun, the seasonal months are different from either the solar sidereal months Meṣa etc. or the synodic months Caitra etc., and these cannot correctly represent the seasons. That is why the Vedas give a new set of months, (actually tropical months) for the seasons: Madhu and Mādhava are the months constituting the Vasanta (spring) season, Śukra and Śuci are the months constituting the Grīṣma (summer season), Nabha and Nabhasya are the months constituting the Varṣa (rainy) season, Iṣa and Ūrja are the months constituting the Śarad (post-rainy season); Sahas and Sahasya constituting the Hemanta (pre-winter) season; and Tapas and Tapasya constituting the Śiśira (winter) season. (Śuklayajurveda, 13.25). Even in the Vedāṅga Jyotiṣa we have the information that the Śiśira season begins with the Uttarāyaṇa (winter solstice). The Yājuṣa-Jyotiṣa (verse 6) says, “When the Sun and the Moon rise together, with Śraviṣṭhā, from then commence the yuga, the month of Māgha, the seasonal month Tapas, the light fortnight of the month, and Uttarāyaṇa”. As Tapas is the first month of Śiśira we understand Śiśira begins with Uttarāyaṇa. By mentioning Māgha and Tapas distinctly, we understand that the Vedas wish us not to confuse the two. But con- fusion there has been, and still continues, with the result that people call Meṣa and even Vṛṣabha spring months, though patently we have summer then, Kumbha and Mīna being practically the spring months now. This confusion has resulted in Madhu, Mādhava etc. and Caitra, Vaiśākha etc. as synonyms. People who know are amused, when in the saṅkalpa recited for Hindu rituals the month of Vṛṣabha, which is advanced summer, is mentioned as spring. [संक्रान्तिकालः] षष्टिघ्ना भुक्तिहृता रविबिम्बकला भवन्ति नाड्यस्ताः । संक्रान्तीनां कालः पुण्योऽतोऽर्धेन चाद्यन्तात् ॥ २६ ॥ 25. Quoted by Utpala on BS 2, p.23. b. A. वृत्तकशिशि; B1.2. वृवृंतकशशि 25a. A.B1. मकरादौ c. U. समाना

66 PAÑCASIDDHĀNTIKĀ III.27 Saṅkrānti-kāla 26. The angular diameter of the Sun in minutes, multiplied by sixty and divided by the daily motion of the Sun, are total sacred nāḍis of Saṅkrānti (literally ‘crossing’). Half this time before and after the Sun entering a rāśi, is sacred. The Pauliśa does not give the angular diameter of the Sun, so it must be the intention of the author to use the angular diameter given by the Romaka or the Saura. Example 12. The angular diameter of Sun is 31' and its daily motion 57'. The Saṅkramaṇa is 19 nāḍīs after sunrise. Find the sacred nāḍis. Angular diameter × 60 ÷ daily motion = 31' × 60 ÷ 57' = nā. 32-38. Half this is 32-38/2 = nā. 16-19. Therefore nā. 19-0 - nā. 16-19 = nā. 2-41 to nā. 19-0 + 16 - 19 = nā. 35-19 is the sacred period. The rule is proved thus: The time of the centre of the Sun’s orb crossing to the next sign is the time of Saṅkramaṇa. At this time half the orb is in the previous sign and half in the next. The period when parts of the orb are in both signs is the sacred period. So it begins when the east point of the orb just enters the next sign and ends when the west point just leaves the previous sign. So, during the interval the Sun moves a distance equal to its own diameter. This time is got by the proportion: daily motion: angular diameter :: 60 nāḍikās: the required time. Therefore aṅg. diameter × 60 ÷ 60 is the time in nāḍikās. As half this time is required for the mid-point to reach the junction of the signs, half this period placed on either side of the time of the mid-point crossing over gives the beginning and end of sacred period. It must be noted that if the angular diameter is computed according to the old Hindu astronomical works and used, the sacred period would be constant whatever be the daily motion, and the sacred period can easily be given as so many nāḍikās before and after saṅkramaṇa. How? Let x be the mean angular diameter in minutes. According to Hindu astronomy the angular diameter is proportionate to the daily motion, (because the motion is taken inversely proportionate to the distance and the angular diameter also is inversely proportionate to the distance) (See VIII. 15, IX 14-16). Therefore the angular diameter = x multiplied by daily motion ÷ mean daily motion. The period = angular diameter × 60 ÷ daily motion = x × daily motion × 60 ÷ (daily motion × mean daily motion) = x × 60 ÷ mean daily motion which is constant. If to avoid this we assume that the mean diameter is intended to be used in the rule, then the rule is unreasonable. Or we have to accept it on the injunc- tion of the Dharmaśāstras, throwing the burden on them. We said, “according to the old Hindu astronomical works”, because actually the angular diameter is not exactly proportional to the daily motion. [त्रिदिनस्पृग्योगः] तिथ्यन्तं यदि सूर्यः स्पृशन्नुदेत्ये (षा) वासरं चाऽपि । योगस्तदा त्र्यहस्पृक् तिथित्रयस्पर्शनाद(वमः) ॥ २७ ॥ 26a. B1. भुक्षिहताः B2. भुक्षिहता; B3. भुक्षिहता d. B1.2.पुण्यतोद्धेन b. A1.रबिम्ब०; B.बिम्बककला For चाद्यन्तात्, B1.2. वार्धता कृतिः; B2. न नार्धतात्

III.27 III. PAULIŚA-SIDDHĀNTA 67 Tridinaspṛg-yoga 27. When a tithi extends throughout a day, coinciding with a part of the previous day and the next day, the occurrence is called Tridinaspṛg, (literally 'touch of three days'). (If, besides a whole tithi, parts of the previous and next tithis fall on the same day, the occurrence is called avama, literally, 'uncounted tithi'). The only thing we have done to the reading in the first half of the verse is to change śā into ṣā, which is quite warranted. But TS have changed de into di and introduced a new word, anya. Still their reading of the text cannot yield the meaning. To agree with tithitraya-sparśañat, we corrected ahnaḥ into avamaḥ, because avama alone results by contact with three tithis. The word ahnaḥ is neces- sary also, but can be understood from the context, though not mentioned, but not so, avamaḥ. If this part is left uncorrected as TS have left, the expression would be non-sensical like Sudhākara's meaning: "Because the day touches three tithis, it is called 'Three-day touching'. But Thibaut has grasped the idea here, though calling it "the conjunction touching three Tithis". NP too, have caught the idea, but since the relevant emendation to avama did not strike them, they merely say '(there is a yoga)'." [राहुः] अष्टगुणे दिनराशौ 'रूपेन्द्रियशीतरश्मि'भिर्भक्ते । लब्धा राहोरंशा भगणसमाश्च क्षिपेल्लिप्ताः ॥ २८ ॥ वृश्चिकभागा राहोः षड्विंशतिरेकलिप्तिकालुप्ताः । आदितरः प्रोह्य मुखं षड्राशियुतं तु पुच्छाख्यम् ॥ २९ ॥ Rāhu (Node) 28. Multiply the days from Epoch by 8 and divide by 151. Rāhu's motion is got in degrees etc. Add minutes equal to revolutions. (The motion becomes exact.) 29. Deduct the motion from 7ʳ 25° 59ʹ. The remainder is Rāhu's Head (what is called Dragon's Head, a popular name for the Ascending Node). Add 6 rāśis to Rāhu's Head, (Dragon's Tail or Descending Node), is got. Example 13. (a) Days from Epoch, 75,500; find Rāhu's Head and Tail. (b) Find the Head of Rāhu at Epoch, i.e. for Zero day. 27b. A. स्पृश्यनु; B. स्पृशेत्तु॰ A. ॰देतोशावासनं; B. ॰देत्येशावासंरः; D. ॰नुदेत्येष्यं C. ॰दितोन्यवासनं c. A2. लब्ध्वा. B. शहोरंशा c. B. ॰स्तदन्यहः B. स्पृक d. A2. क्षिपेल्लिप्ताः; B1. क्षिपेछिन्त्रप्ताः d. A.B.C.D. नादहः || 29b. A. विंशति. C.D. लुप्ता 28a. A2.B. गुणो. B. गुणाशशौ c. B. आदियरत. B1.2. प्रोज्य; D. प्रोज्झ्य. B. मुख b. B. ॰भिव्यक्ते d. A2. युतं नु. B2. पुच्छाक्ष्यं

68 PAÑCASIDDHĀNTIKĀ III.29 (a). Rāhu's motion for 75,500 days = 8 × 75,500 divided by 151, degrees = rev. 11-1-10-0. The exact motion = rev. 11-1-10-0 + 11 minutes = rev. 11-1-10-11 = rā. 1-10-11, omitting full revolu- tions. Rāhu's Head = rā. 7-25-59 – rā. 1-10-11 = rā. 6-15-48. Tail = rā. 6-15-48 – rā. 6-0-0 = rā. 0–15–48. As the motion of Rāhu is zero, the Head of Rāhu is the constant itself, viz. rā. 7-25-59. As the motion is 8° in 151 days, according to this Siddhānta, to move 360°, i.e. one revolution, it takes 360 × 151 ÷ 8 = 6795 days. But during this period it moves one minute more, i.e. the exact motion is 360° 1' in 6795 days, i.e. one revolution takes 360° × 6795 ÷ 360° 1' i.e. 6794 days, 19 nāḍīs. We have seen that the Head of Rāhu for the Epoch, according to the Pauliśa is the constant itself, viz. rā. 7-25-59. According to the Saura (condensed by the author) it is rā. 7-26-6. According to the Vākyakaraṇa it is rā. 7-26-11. According to modern astronomy, taking the ayanāṃśa as being zero for the period it is rā. 7-26-0. According to the Siddhānta Śiromaṇi it is rā. 7-27.13. We see that all except the value of Śiromaṇi agree closely, verifying the Pauliśa value for the Epoch. The disagreement of the Śiromaṇi value is only apparent, for the zero point of the Śiromaṇi zodiac is about one degree behind that of the rest, (as may be seen by comparing with its co-ordinates of the stars or from its Sun being one degree more than that of the rest) and if the same point is taken as the origin, the Śiromaṇi too gives about rā. 7-26-13. Thus the value at epoch is necessary to get the Rāhu at any moment, and it is this that is given by Vṛścikabhāgā Rāhoh etc. But TS have not understood this need (as they did not understand the need for the kṣepa in the case of the Moon (see II.3) and gave a wrong interpretation of śaśimuninava-yamāśca rāśyādyāḥ) and give the following laughable explanation: "The measurement of the limbs of Rāhu having the form of a scorpion is 25 minutes. Deducting this from the motion of Rāhu obtained from (28), the head or face of Rāhu is to be found. This plus six rāśis is the tail. We have to rely only on the words of the ancients to know that the scorpion-like limbs of Rāhu mea- sure 25 minutes, there is no other reason." Now we ask: Let it be that they have not understood the need for the kṣepa. How did it not occur to them that Rāhu can be got only by deducting the motion from something, whether it is a cycle or some other constant, because the motion is retrograde (as they themselves have said in other places: "Rāhu deducted from a full revolution is the Head, this plus six rāśis is the tail", IX. 6, "deducted from the end of Pisces is the head", VIII. 8). It also appears here that Thibaut is not satisfied with Sudhakara's explanation. Further, how did it not occur to them that ekaliptikāluptāḥ ṣaḍviṃśativṛściakabhāgāḥ, means rā. 7-25-59, when they have correctly inter- preted siṃhasya vasuyamāṃśāḥ as (XVI. a) rā. 4-28-0, sārdhāḥ pañcālino (XVIII. 1) as rā. 7-5-30, nava sārdhāḥ kanyāṃśāḥ (XVIII. 11) as rā. 5-9-30, ṣoḍaśa vṛṣabhasyāṃśāḥ nava liptikāvarjitāḥ (XVIII. 18) as rā. 1-15-51? It is really astounding what tricks the mind can play! Incidentally, the following should be mentioned here for general information. Following the nomenclature of the ancient Saṃhitās, the author calls the ascending node Rāhu's head, and the descending node 'Rāhu's Tail', both being Rāhu, though generally in later astronomical works the word Pāta is used. In recent times, somehow the term Ketu has come to be applied to the descending node, Rāhu being retained for the ascending node, though there is no authority in astronomical works or Purāṇas to bring in Ketu here. The ancient Saṃhitās use the term Ketu for the Dhūmaketūs or comets and, as deities they are generally referred to in the plural. They are also characterised by unpredictable motions and in the Bṛhatsaṃhitā too the author says so. (This is the view of the ancients though we now know that a number of them are periodic, and their positions can be predicted with tolerable accuracy). They are worshipped in the collective, as seated on doves,

III.29 III. PAULIŚA-SIDDHĀNTA 69 with the expression, "Salutation to the Ketus." From the singular in the mantra of their invocation, Ketum kṛṇvannaketave, one may think that Ketu is referred to in the singular also. It is not so. Here the word does not mean the 'Comet' Ketu at all. The mantra itself is in praise of the Sun and Ketu here means the activity caused by the Sun in the sleeping inactive world. How then is this mantra used to invoke the Ketus? The utterance of the word Ketu here is sufficient, as the utterance of the many śa sounds in the Mantra śaṃ no devīḥ etc. (Ṛgveda 10.9.4) is sufficient to propitiate Śanaiścara. though the mantra itself refers to the water-deities, or the utterance of the word mayūra in the mantra, āmandravir intra haribhiḥ etc. (Ṛgveda 3.45.1) is sufficient to propitiate Subrahmaṇya, though the mantra refers to Indra, as also words like, Om, atha kalyāṇa etc. cause auspiciousness by their mere utterance. So, according to the Śāstras, Ketu refers only to the Dhūmaketus, and Rāhu is both the nodes, the recent application of the term Ketu to the descending node being un- warranted. Therefore, when the Dharmaśāstras enjoin the eclipses caused by Rāhu as sacred periods, they take in both the ascending and the descending nodes, and the suggestion by some that they do not take in the descending node on the score of some un-informed people calling it Ketu is wrong because what they call Ketu is really Rāhu. [चन्द्रविक्षेपः] वक्त्रादधिकश्चन्द्रो हीनः पुच्छाच्च याति भगणोदक् । हीनो वदने पुच्छेऽधिकेऽ(सु) राधाति दक्षिणतः ॥ ३० ॥ भागनवत्या राहोश्चन्द्रोऽन्तरितोऽतिमहति विक्षेपे । लिप्ताशतद्वये [ना] त्यशी(त्याऽ)नुपातोऽतोऽन्यत्र ॥ ३१ ॥ Moon's latitude 30. If the Moon lies between the Head and the Tail it is north of the ecliptic, (i.e. its latitude is north). If it lies between the Tail and the Head it is south of the ecliptic, (i.e., latitude is south). 31. The latitude is a maximum equal to 280 minutes when the Moon is 90 degrees distant from either Head or Tail. The latitude is to be found by pro- portion, at other places, using the distance in degrees from Head or Tail, whichever is nearer. Example 14. The true Moon is rā. 2-7-0. Rāhu's Head is rā. 5-3-0. Find the latitude of the Moon. The Tail is Head + rā. 6-0-0- = rā. 11-3-0. The Moon is between Tail and Head. Therefore the latitude is south. The distance of the Moon from the nearer limb, Head is rā. 5-3-0 − rā. 2-7-0 = rā. 2-26-0 − 86 degrees. For 90° the latitude is 280'. For 86° latitude is 86° × 280' ÷ 90° = 267½', South, as seen. 30a. B. चक्रादधिक b. A.B1.पृच्छाच्च. B.°पोदृक् c. A1.B1.पुंछे; C.वदनात् पुच्छाधिको d. A.B.°धिकोमुराघाति; C.°धिकोऽसुराद्याति; D.धिकोऽमकाद्याति 31b. B.°तोभिमहति c-d. A. येत्यशीतिमनुपातोन्पत्र ।; B. द्वयेत्यशीतमनुपातोन्पत्र ।; C. द्वयाधिकसप्ततिरनुपाततोन्पत्र ।; D. द्वय[मे] त्यशीतिमनपातोतोन्पत्र ।