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

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 ।

70 PAÑCASIDDHĀNTIKĀ III.31 The rule is explained thus: The Moon moves in its own orbit, inclined to the ecliptic at an angle equal to the maximum latitude. Hindu astronomy assumes this motion to be on the ecliptic itself and gives the Moon's longitude, because there is only a maximum difference of 7'. Between the ascending and the descending nodes the orbit is north of the ecliptic and therefore the latitude measured on the great circle perpendicular to the ecliptic and passing through the Moon, is north. Between the descending node and the ascending, the orbit is south and the latitude is south. The distance between the node and the Moon, the latitude and the angle of inclination forming a spherical triangle, we have : sin latitude = sin interval × sin maximum latitude. The maximum being small and the latitude being generally less than this, sin latitude and sin maximum latitude are propor- tional to the latitude and maximum latitude, and we have the formula, lat = max. lat × sin interval. But the Paulīśa takes the latitude proportionate to the interval itself and gives the rule. We have to consider here whether the author intends that the latitude is to be found in propor- tion to the actual degrees of the interval or the sine of the interval in degrees. The triangle being spherical, the correct thing would be to use the sine. But we have reason to think that the degrees themselves are intended to be used in the proportion, for if the sine is to be used it must be men- tioned. Also, this Siddhānta uses only the proportion by degrees in other places also, where propor- tion by sines alone would be correct, as for e.g. in the solar eclipse, in correcting Rāhu and in calculating valana, i.e. transformation of direction (see VI. 2-4, 8). Therefore the original Paulīśa itself has instructed proportion by degrees, as being sufficiently accurate, which the author reiterates here. But in computing the parallax (in time) in the case of the solar eclipse the sine is used either by the Paulīśa itself, to avoid too much inaccuracy, or by the author VM to save the Siddhānta from ridicule. As for TS they say that the author intends here proportion only by sine, that being the proper thing to do. Another thing should be mentioned here. In using proportion by degrees, the maximum error will be in the neighbourhood of the nodes. With the maximum latitude 280', the latitude for 13° interval would be 40½ minutes, which would be incompatible with the formula for eclipses (vide VII. 5-6). Therefore it seems that the Siddhānta, though knowing that proportion by sine is the correct thing for latitudes, gives proportion by degrees for the sake of ease of computation. Or by taking liptāśatatrayeṇāśītim as the correct reading, which would make the maximum 380', the incompatibility can be avoided. It may be argued that the error in the latitude would be great in the neighbourhood of the maximum. But this is only erring one side while some others err on the other. The mean maximum latitude is 309'. If some like Ptolemy give 240', which is less by 70', what is wrong in taking that the Paulīśa gives 380', which is greater by the same amount? Now for the readings: In the fourth foot of verse 30, mu is corrected into su because murāt is meaningless. In the third foot of the 31st verse two syllables are wanting and nā is added for pur- poses of syntax. For the same reason, śī-tama-nupā in the fourth foot is corrected into śī-tyā-nupā. The correction of aśīti into saptati by TS here is unwarranted, because it is not known what it was in the original Paulīśa. If this is done in conformity with the Saura, why not in conformity with the Romaka, which gives 280', (see VIII. 11), and which is nearer to the Paulīśa? Now follow six verses devoted to criticising the views of the Romaka, and an astronomer by name Bhadraviṣṇu, the intention of the author being to create faith in his own work. (This was a custom in those days, vide for instance the Brāhma-Sphuṭa-Siddhānta, Dūṣaṇādhyāya and the Vaṭeśvara Siddhānta, Madhyamā-dhikāra, Chapter X). In several places of the text the readings are not clear and we cannot be sure of what exactly the author intends to say, though the gist is clear, the matter

III.31 III. PAULIŚA-SIDDHĀNTA 71 not being scientific and amenable to intelligent guess. Still, as there is much to say here too, we are dealing with these, unlike TS who have refrained from doing so. [भद्रविष्णुमते दोषः] तिथिनक्षत्रच्छे(द)प्रतिपत्तिर्यदि तथा ततः साधुः | न तथा च भद्रविष्णोस्तथाऽपि [न] विनिवर्तते लोकः || ३२ || Defect in Bhadraviṣṇu 32. If the tithis and nakṣatras as seen from observation of the sky agree with those computed according to the Śāstra, then the Śāstra is correct and fit to be accepted. It is not so in the case of Bhadraviṣṇu's work; still people do not turn away from that and follow the correct Śāstras. (This remark about the nature of people is true even today). For the sake of syntax the long dā has been shortened by us. Consistent with the idea intended, the negative particle na has been added in the fourth foot. [पादादित्यमते दोषः] न युगपदुदयो भानोरस्तमयो वाऽपि भवति सर्वत्र | कस्मिन् देशेऽस्तमये पादादित्येन (नोक्त)मिदम् || ३३ || Defect in Pādāditya 33. Sunrise or sunset is not at the same moment in all places on the earth; (so the place must be mentioned whose sunrise or sunset is taken as the epoch for finding the days and doing the computation). But Pādāditya, who has placed the epoch at sunset, has not mentioned the sunset of which place he refers to. (So his work is faulty.) There seems to be some error in the last foot and we are not sure-whether Pādāditya is a person as we have interpreted, or something else or whether the word is the same at all and not an nth incarnation of the original. [रोमकमते दोषः] मार्गा(द)पेतमेतत् काले लघुता न तावदतिदूरे | 'खविषयभूताष्टरसै'रब्दैः पश्याऽस्य विनिपातम् || ३४ || 32a. A.B.C.D. छेदा 33a. A.B. भानुः b. A.B1. प्रतिपत्तियदि; B2. प्रतिपतियदि. c. D. ॰ऽस्तमयः B. साधु d. B. पादादित्येन; D. पादाद्विनेन c. B. भद्रविष्टमो A. भुक्तिमिदं; B. भक्तिमिन्दुः; C. भुक्तमिदम्; d. A.B.C.D. om. [न] D. भुक्तं विदुः

72 PAÑCASIDDHĀNTIKĀ III.34 Defect in Romaka 34. This gaṇita work had deviated from the right path handed down by a hierarchy of good teachers and the day of its exposure is not far distant. Witness its downfall in 68,550 years! To agree with the fifth case in mārgāt we have corrected upeta into apeta. It may appear strange that the author calls 68,550 years as a not far distant date. But that depends on the outlook of people, and the Hindu mind, especially the old Hindu mind may con- sider even this as a comparatively short period. Or this verse may belong to the criticism of the Romaka following immediately, strayed to this place by the mistake of the scribe. In that case the first letter kha in khaviṣayabhūtāṣṭarasaiḥ should be corrected as sva and the expression taken to mean "6855 years, by its own measure", in which case the following is the meaning:- "This Romaka has not come down through a hierarchy of good teachers because it follows the Tropical year instead of the traditional sidereal year. It will be exposed in a period of 6855 of its own tropical years, and people will abandon it." How this will happen and how the number 6855 can be arrived at almost exactly, will be shown in explaining the next verse. (रौ)मकमहर्गणं (वा) (तद)र्कमिन्दुं च गणयतां ग्रा(ह्यम्) चैत्रस्य पौर्णमास्यां नवमी नक्षत्रमादित्यम् ॥ ३५ ॥ 35. If we adopt the days from epoch resulting from the tropical year as adopted by the Romaka and the Sun or Moon resulting therefrom, we must accept Punarvasu as the nakṣatra of the full moon of the month of Caitra, instead of the expected Hasta or Citrā, Punarvasu which is the nakṣatra of Caitra-Śukla-navamī. This connection has been strongly established in people's mind by the observance on Caitra- navamī as the birthday of Lord Rāma, hero of the Rāmāyaṇa, well known as born in the asterism Punarvasu. This is how this will happen: The months Caitra, Vaikśākha etc. are so called because the Moon at new moon in these months is in the vicinity of Citrā, Viśākhā etc. Thus, in a given month, the full moon, i.e. the Moon of the 15th tithi, is in a given nakṣatra or nearby, so that the other tithis also are 34a. A.B.C.D. मार्गादुपेत a-b. A.B1. ॰णं पादं मर्क॰; B2.3. C.D. ॰णं पादमर्क॰ b. B. न तावदति दूरो A. ॰यतां तां ग्राह्या; B. ॰यतां ग्राह्याः C. ॰यता ग्राह्या; c. A. षविषय; B. om ख D. ॰यतां ग्राह्य c. B2. पौर्णिमास्यां 35a. A.B.C.D. रोमक d. D. नवम्यां

III.35 III. PAULIŚA-SIDDHĀNTA 73 connected with particular nakṣatras. For e.g. as the Moon of the fifteenth tithi in Caitra is near Hasta or Citrā, the Moon of navamī, six days before, is near Punarvasu or Puṣya, because the tithi, the day and the nakṣatra, have approximately the same duration. In the same way, as the Moon of the 15th tithi of Śrāvaṇa is near Śravaṇa, the Moon at Śravaṇa Bahula Aṣṭamī, eight days after that, is near Rohiṇī, which is also a thing well known. Now, when the months (synodic) are 'tied' to the nakṣatras as mentioned, there will be this conformity. But the months are kept tied to the nakṣatras if the solar year is sidereal, and not tropical like that of the Romaka, i.e. if the solar year begins as in all other Siddhāntas with a fixed point on the ecliptic like the first point of Meṣa, and not the movable vernal equinox, the so called First Point of Aries, as in the Romaka or the new Indian Rashtriya Panchang. As the difference between the two points (i.e. the ayanāṃśa) increases, the above mentioned con- formity will gradually decrease and when the ayanāṃśa accumulates to 30° the full moon of the first month will fall in the Phalgunis instead of near Citrā and though called Caitra the first month will really be ‘Phālguna’. This non-conformity has already happened in the Rashtriya Panchang, with the ayanāṃśa more than twenty degrees now. If it accumulates to 6 nakṣatras, (i.e. 80°) the full moon of the first month, still called Caitra, will be in the nakṣatras Punarvasu or Puṣya, though the month will really be Pausha. Thus Punarvasu connected with the Navamī of the real Caitra will occur at the full moon of the so called Caitra of the Romaka (or the Rashtriya Panchang). The Navamī of the new Caitra, occurring 6 days before the full moon day, will occur in Apabharaṇī, 6 nakṣatras earlier. When a situation thus arises contradictory to their belief, people will realise that the sidereal year is the proper thing and discard the Romaka. Incidentally we may mention that the same confusion will arise in following the Rashtriya Panchang also. One thing we want to say here. We do not deny that the tropical year best suits civil purposes, but a luni-solar calendar based on the sidereal year also will suit our religious purposes best. Therefore they have to be kept apart. (A civil calendar based on the tropical year, we already have in the Christian Calendar we have been following, which is practically worldwide. As for the defects in it, the ‘Calendar Reform’ will take care of it, while in making this reform we have taken a step in isolating ourselves.) One thing could have prevented the confusion. If they had adopted the seasonal-month-names like Madhu, Mādhava etc. preserved for us in the Vedas (vide III. 25) instead of trying to fit in the sidereal luni-solar-calendar months, Caitra, Vaiśākha, etc., this confusion would have been avoided and they would not have simply added one more to the one hundred contradictory Panchangs already extant. To continue: We shall compute after how many years the exposure of the Romaka, as mentioned by the author, would happen. The ayanāṃśa was zero at the author's time, as we have shown on several occasions. We must calculate when it accumulates to 6 nakṣatras. The Siddhāntas of the author's time use a sidereal year of days, 365-15-31 nearly, and the tropical year of the Romaka is days 365-14-48 (See VIII. 1). Therefore, the Romaka year begins earlier by 43 vināḍis, every year, and this is equivalent to the rate of ayanāṃśa per annum, 42", nearly. If ayanāṃśa to become 42" takes one year, to become 6 nakṣatras it will take, 6 × 800 × 60 divided by 42 = 6857 years nearly. It is this that is given by the author as 6855 in the previous verse explained. कालाऽपेक्षा विधयः श्रौताः स्मार्ताश्च तदपचारेण । प्रायश्चित्ती भवति द्विजो यतोऽतोऽधिगम्येदम् ॥ ३६ ॥ 36. All the injunctions of the Vedas and Smṛtis are based on the proper time, and by not performing the rites at those times the performer, especially a twice-born,

74 PAÑCASIDDHĀNTIKĀ III.36 acquires sin which is to be expiated. Therefore, a study of this Romaka itself is to be expiated. How is that sin may accrue by not performing a rite at the time enjoined by the Śāstras. But by simply studying the Romaka, we cannot say one would also perform the rite at the improper time. The Romaka may give the time wrongly, but one may study it not for the sake of using its time, but for other purposes, as for instance to understand where it goes wrong, and expose its weakness to others and save them, for which the man who studies even deserves merit. Let us understand that these things, sin and merit, are subtle and cannot be known without a deep study of the Śāstras, and if the author steeped in the Dharma Śāstras says a thing, let us accept it. The Vedas promise svarga not only to the performer of the yajña but also to one who knows how to perform it properly. We frequently meet in the Vedas the expression ya u cainam evam veda. The Vedāṅga Jyotiṣa says that people who know astronomy know as it were the correct performance of the sacrifices themselves, yo jyotiṣaṃ veda sa veda yajñān and they go to svarga after establishing a long line of progeny in this world. Now, it stands to reason that if the mere study of a good thing gives merit, the mere study of a bad thing brings sin. It is said that even association with bad characters and sinners bring sin, as also doing sinful things even in dreams. As for the argument that the person even deserves merit for intending to keep off people from the improper times, he does deserve it and will get it. But that does not mean expiation is not called for. Contact with craftsmen may be necessary to keep the temple idols in form, but that does not mean that purificatory ceremonies need not be performed for the idols on that account. We can say this much that in these cases the expiation is light, like the utterances of the Lord's Name, like 'Kṛṣṇa Kṛṣṇa, Śiva Śiva! We should also take into consideration the spirit if the times in which these statements were made. स्फुटगणितविदिह लब्धा धर्माऽर्थयशांसि दिनकरादीनाम् । (कुकरणकारस्सत्यं सहते नरके कृताऽऽवासा: ॥ ३७ ॥) 37. The person having correct knowledge of the Sun, Moon, etc. gets Dharma, which will take care of his future world, Artha which will ensure his prosperity in this world and fame, which will perpetuate his memory. But the bad astronomer who misleads people by his writings will certainly have to go to hell and dwell there. 36a. A. विधय b. B. श्रौता स्मा० c. B1.2. प्रायश्चिती. A1. भवती c-d. A. कुकरणविदो द्विन्यो ये कथयन्त्यस्फुटं कुकरणकारः (A2. करः) B. अकरणविदो द्वित्यो ये कथयसत्यं अकरणकरः सहते नरके कृतावासाः । C. कुकरणविदोद्विन्यो ये कथयन्त्यस्फुटं कुकरणकरः सहते नूनं नरके कृतवासाः । D. कुकरणविदो द्विजा ये कथयन्त्यस्फुट [म] सत्यं [स गणितम्] । कुकरणकारसहि [ताच्च] ते क्षणं नरके कृतवासाः ॥ 37. In A and B, स्फुट etc. occurs as the second half of the verse. It is put here as the first half to suit the sense. a. D. लब्ध्वा

III.37 III. PAULIŚA-SIDDHĀNTA 75 It must be noted here that when even the person with correct knowledge gets so much, the writer will get more. It must also be noted that only the writer of bad astronomy goes to hell, not the reader, whose sin is small. In this verse there is a jumbling of words and phrases and induction into the text extraneous words intended as commentary. The words, sphuṭagaṇitavid etc. seem to be the first half of the verse because in the first foot there are twelve and in the second eighteen syllables. Therefore what comes before that is the second half. In that, there are may syllables more than the required twenty- seven. Selecting the required words alone, we have reconstructed the third and fourth foot. For the observations of K S. Shukla on 32-37 vis-a-vis NP, see his paper ‘The PS of VM (1)’, JIHS 9 (1974) 62-76. [इति पञ्चसिद्धान्तिकायां वराहमिहिरविरचितायां पौलिशसिद्धान्तो नाम तृतीयोऽध्यायः ||]¹

1. A.B. पौलिशसिद्धान्तः; C.D. इति पौलिशसिद्धान्तः || Thus ends Chapter Three entitled ‘Pauliśa-Siddhānta: Planetary Computations etc.’ in the Pañcasiddhāntikā composed by Varāhamihira

Chapter Four THREE PROBLEMS — TIME, PLACE AND DIRECTION ४. चतुर्थोऽध्यायः त्रिप्रश्नाधिकारः Introductory Problems on Time, Place and Direction, involving spherical trigonometry, are dealt with in this chapter. The first fifteen verses are devoted to the construction of a table of sines. As this kind of matter does not involve constants specific to any siddhānta and is commonly found in all siddhāntas, we cannot say which Siddhānta this belongs to, Pauliśa or Saura, the only two siddhāntas meant to be expounded in detail by the author. Probably it is the author’s own, meant for both, or taken from both. Two things point to this conclusion: In the part of the work dealing with the Saura, viz., chs. IX, X, XI, XIII, XIV, XV, XVI, and XVII, no space is given to the sine tables, though required, and to the problems dealt with here, and therefore if these are not meant for Saura, it would be imperfect though almost full. On the other hand, certain redundant and crude rules point to this chapter’s connection with the Pauliśa, as also its position in the chapter distribution in the PS text. [ज्यानयनम्] षष्टिशतत्रयपरिधेर्वर्गदशांशात् पदं स विष्कम्भः | तदिहां(शच)तुष्कं संप्रकल्प्य रा(श्य)ष्टभागज्या || १ || Table of R Sines

1. Take the circumference as measured in 360 units, square it, take the tenth part of the square, and find its square root. The result is the diameter of the circle in the units taken. We assume the diameter to be 4°, (i.e., 240′) and hereunder give the tabular sines of angles for 3° 45′ interval. The rule is: diameter = √circumference²/10. It comes to this: d = c/√10. The formula, d = c/π is well known, and the author has taken √10 as an approximation for π which is incommensurable and usually represented by the approximate values, 22/7, 355/133, 3.1416 etc. The Sūrya Siddhānta too gives √10 as the value of π in its instruction to find the circumference of the earth from its diameter (I.59.): “The earth’s diameter is 1600 yojanas. Square this, multiply by 10, and find the square root. This is the earth’s circumference.” By thus taking √10 for π, an error of about 0.0067% results, and for a circumference of 21,600′, we get the radius 3415′, instead of the well-known 3438′. But it must be mentioned here that this error does not affect the computation of the sines 1a-b. A.B. परिधे वर्ग c. A.B. तदिहांशाश्चतुष्कं (B. ॰ष्क) b. A. विष्कुम्भः d. A. संप्रकल्प्य; B. प्रकल्प्य. A.B. राश्याष्ट०

IV.2 IV. THREE PROBLEMS 77 mentioned in the succeeding verses, because it can be shown that the author derives the sines from a correct formula, (not dependent on this wrong ratio of the diameter to the circumference), based on 120' as the radius of the circle. If he had depended on the wrong value, the first tabular sine, i.e. sin 3° 45' would be 7' 54", (being the 96th part of the circumference, where the sine is indistin- guishable from the arc), and not the correct 7' 51" as given by the author. Taking the diameter as 4°, and thereby the maximum sine (i.e. the radius) as 120', is arbitrary. In general, the Siddhāntas give the maximum sine, 3438', as arrived at from taking the circumfer- ence as 360° or 21600'. The Vākyakaraṇa makes it 43°. In actual work, the sines enter only as a ratio to the maximum sine, and therefore no harm, will result by taking these different maximum sines. TS and NP have not understood the meaning of the second half of the verse, and mis-interpret aṁśacatuṣkam as quadrant. व्यासार्ध[स्य] कृतिर्ध्रुवसंज्ञिका कृतांशस्ततः स मेषस्य | ध्रुवकरणी मेषोना द्वयोस्तु राश्योः पदं ज्याः स्युः ॥ २ ॥ 2. The square of the radius, (i.e. 14,400), is called dhruva (karaṇī), (literally, ‘Fixed Irrational’). The fourth part of it, (i.e. 3600), is the karaṇī (Irrational) related to the first sign, (or 30°). Dhruvakaraṇī minus the karaṇī of Meṣa, (i.e. 14,400 − 3600 = 10,800), is the karaṇī of two signs, (or 60°). The square root of a karaṇī is the tabular sine. Being square of tabular sines given in minutes, the karaṇīs are squares of minutes, which is their peculiarity as given by the author, though this is not mentioned explicitly. The other well-known characteristic of a karaṇī, viz. irrationality, is found in all karaṇīs except 14,400 and 3600, though the author calls these also karaṇīs in a general way. In modern terminology the word sine used in connection with the angle is defined thus: [Figure: Right-angled triangle ABC with right angle at C and angle marked at B] Fig. IV. 1-a In the right angled triangle, (fig. 1-a), sine ∠ B = AC/AB, or sine ∠ A = BC/AB, i.e. as the ratio of the opposite side to the hypotenuse. In tabulating the sines, the hypotenuse is taken as unity, and the ratio expressed as a decimal fraction. 2a. A.B. कृते ध्रुव०; C.D. कृतिध्रुव० c. B1.3. ये योना; B3. येषोना b. A.B. ०ज्ञिता. A.B. कृताशाःस्ततः. A.B. सशेषस्य d. A. दयोस्तु; B. दयो सु

78 PAÑCASIDDHĀNTIKĀ IV. 2 The ancients however expressed the sines in minutes-length or, more accurately, in minutes and seconds-lengths, the maximum sine called Trijyā (meaning ‘the sine of three signs’, i.e. 90°), occur- ring separately in the work to make up the ratio. This is the way in which they conceived the sine (meaning ‘bow-string’ from its Sanskrit equivalent śiñjinī, synonymous with jyā). In Fig. 1-b. A₃ E F₃ D is the circumference of the circle, centre B. A part of the circumference like ADF, A₁D F₁, etc. is called dhanus (literally, ‘bow’) or arc. Fig. IV. 1-b The straight lines ACF, A₁C₁F₁, etc. forming the ‘bow-strings’ of the respective ‘bows’ are the jyās or full sines. But in actual practice, the halves of the full sines AC, A₁C₁, etc. above are used with the name of ‘sines’, with respect to the half-bows or arcs, AD, A₁D₁, etc. Because the arcs AD etc. are as the angles ABD etc., the sines AC etc. are spoken of with respect to the angles ABD (= ABC) etc. also. Thus, AC is the sine of ∠ ABD or arc AD, A₁C₁ is the sine of ∠ A₁BD or arc A₁D₁ and so on. It is this connection of the sine with the arc that has given it the nature of a length, which is expressed in minutes and seconds on account of the connection of the arc with the angle at the centre. It may be mentioned here that CD, C₁D, C₂D etc., appearing like the arrows on the respective bow-strings, are called śara (meaning ‘arrow’). If A₃BD is a right angle, i.e. three signs, then, obviously, A₃B is the sign of this angle, i.e. it is the sine of three signs, and therefore called trijyā. Its length is clearly half the diameter A₃BF₃, i.e. the radius, equal to 120′. Now, let the angle ABD be equal to one sign, i.e. 30°. ABD = DBF = 30°. ∴ ∠ ABF = 60°. AB = BF, being radii. ∴ ∠ BAF = ∠ BFA = 60°. Thus ABF is an equilateral triangle, and AF = AB = 120′. ∴ AC = AF/2 = 60′. Thus sine 30° = 60′. Its karaṇī is its square, viz. (60′)² = 3600, the karaṇī of Meṣa as mentioned by the text. Then, let ∠ A₂BD be equal two signs, or 60°. A₂BD = DBF₂ = 60°. ∴ C₂ is a right angle. So the karaṇī of 2 signs = A₂C₂² = A₂B² − BC₂² = A₂B² − AC², (∵ △ A₂B C₂ ≡ △ BAC), = 120² − 60² = 10,800, A₂B being the radius. This also agrees with what the text says. (The square root of 10800 minutes, i.e. 103′ 55″, is the sine of 2 signs, which agrees with the value given in the table.)

IV. THREE PROBLEMS 79 Incidentally, we shall derive the karaṇī and sine of one and a half signs, i.e. 45°, mentioned in verse 4. Let A₁BD be equal to 45°. ∠ A₁ = 45°, and ∠ C is a right angle. ∴ A₁C = C₁B. But, A₁B² = A₁C₁² + C₁B² = 2 A₁C₁². ∴ A₁C₁² = 120²/2 = 14400/2 = 7200 = the karaṇī of one and a half signs as mentioned in the text. Its root, 84'51", is sine 45°, agreeing with what is given in the tables. शेषेष्विष्टेषु धनु-र्द्विगु(णं) पदात् प्रोज्झ्य शेषगुणहीना [त्] । [व्यासस्याऽर्धार्द्धर्गं] द्विगुणकरण्यां समायोज्यम् ॥ ३ ॥ तत्पादोऽभिमता [स्याद्] ध्रुवा तदूनाऽवशेषपिण्डस्य । ध्रुवकरणीदलमध्यर्धसंज्ञमन्योऽत्र विधिरुक्तः ॥ ४ ॥ इष्टांशद्विगुणोनत्रिभज्ययोना त्रयस्य चापज्या । षष्टिगुणा सा करणी तया ध्रुवोनाऽवशेषस्य ॥ ५ ॥ 3-5. The other tabular sines, (i.e. sine 3° 45′, sin 7° 30′ etc. other than the four mentioned of the total 24) are formed successively in the following manner: Let the angle or arc for which the sine is required be θ. I. Sin²θ = ¼[sin²2θ + {120 − sin(90° − 2θ)}²] II. Sin²θ = 60 × {120′ − sin (90° − 2θ)}, where the sines are in minutes etc. Of the 24 sines, the karaṇī of the nth sine = 14400 − the karaṇī of the (24 − n)th sine. 7200 is the karaṇī of one and a half signs, i.e. 45°. Thus, as karaṇīs 8, 12, and 16 are known, those of their halves etc. and (24 − halves) etc. can be found successively. Thus all the sines from 1 to 24 can be found. Of the two formulae, the first is suited to geometrical representation, and the second to computation. Example 1. Given the 8th karaṇī (i.e. of 30°) 3600 and its sine 60′, the 16th karaṇī (i.e. of 60°) 10800, and its sine 103′ 55″.33, find the 4th and 20th karaṇīs and sines, using each of the two formulae. The desired sine is the 4th, i.e. of 4 × 3° 45′ = 15°. 2θ = 30°, 90° − 2θ = 60°. 3a. A. धनुर्द्वि. A.B.C.D. ॰गुणपदा॰ 4a. A. तपदो; B. पदो; C.D. त[स्य] पदो. b. A. पदायोज्य; B. पदायोज्य्; C.D. पदायॊग॰ A.B.C.D. ॰भिमतज्या A.C.D. गुणहीना; B. गुणाहीना b. A. तदुना. B. ॰विशेषे c. A.B. तृव्यासपादार्द्धाद्धर्गं; (B. om तु; B2. ०र्गं); d. A. र्द्धसंज्ञा; B1.2. र्द्धसंज्ञां; B3. र्द्धं संज्ञा C.D. [त्रिज्या तदर्धवर्गौ] द्वि C.D. संज्ञकोऽन्योऽत्र. A. विधिनुक्तः d. A. कारथो; B. कारयो 5a. A.C. इच्छांशद्विगुणेन (C. ॰णोन) A. समायोज्यं; B. समाप्रोज्यन्त; b. B3. त्रयंस A.B. वायज्या C. द्विगुणज्यार्धस्य संयोज्यः; D. द्विगुण [र्ध] करणी c. A. स कारणी; B. स करणी समायोज्यः d. B. षपा. A.B. ॰नामशेषस्य