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

XV.9 XV. SECRETS OF ASTRONOMY 285 7. Though the Sun is low near the horizon, near sunrise or sunset, the Moon, being higher up, can hide the Sun like a cloud. VM here answers an objector to his stand (5-6). He is unaware that the case in (7) is similar to that (5-6). TS’s emendations paramocca, which means ‘standing high up in the sky’, will make an eclipse itself impossible. NP’s emendation candroparasamavastha does not give the sense of ‘being under the Moon’ given by them for the expression. अस्माकमुदयसमये येषामल्पास्तगो दिवसनाथः । मध्याह्नो वा येषां तेषामपि न युगपद्ग्रहणम् ॥ ८ ॥ तदतीतमुदयगानां क्षणद्वये (नैष्य)दस्त(देशा)नाम् । मध्याह्नदे(श)गानामनवरतं वर्तमानेन ॥ ९ ॥ 8. For people who have sunset and for people who have mid-day, when we have sunrise, for all of us, the solar eclipse does not occur at the same time. 9. Throughout the time when there is eclipse for the mid-day people, it is past for the sunrise people by four nāḍikās, and will be yet to occur for the sun- set people by four nāḍikās. The context is the solar eclipse and the Moon is near the Sun, So for the sunrise people the apparent longitude of the Moon has increased by parallax, and the circumstances of the eclipse are advanced by more than four nāḍīs. The opposite happens for the sunset people, and the circumstances are delayed by four nāḍīs. The duration itself is shortened by a slow rate of change of parallax for both. Therefore when the mid-day people’s eclipse begins, the morning people’s eclipse has ended, and when it ends, the evening people’s eclipse has not begun. So there is no overlapping of them. The explanation given by me is general, and several other circumstances will have to be taken into account. But VM’s statement is correct in a general way. Hindu astronomers give the maximum parallax converted into time as four nāḍīs. We must contrast this with the lunar eclipse, which begins and ends at the same moments, wherever the Moon is visible on the earth. 7a. A.B. यद्यं (B. द्य) द्युदये; C. ग्रासे ह्युदये 8a. B. Hapl. om of one ये b. A.B. निञ्चस्थो A. ॰मंशुमा भवति b. B. दिवनाथः c. A.B. चन्द्रोपरमवस्थो; C. चन्द्रः परमोच्चस्थो; d. B. तेषां मेयिनमुग्रापत ग्रहणम् । D. चन्द्रो पर[स]मवस्थो 9a. B. तदानीतमु—यश्रंमानां d. A1. ध्नन्द्वानोः; A2. द्यनद्धानोः; b. A.B. नेषदस्तदोषानाम् (B3. सेष्य) B. ग्रनह्रा (B3. द्रा) तो c. A. देशे In B2, there is transposition of folios here. d. B. मनपरत वर्तमान; B. gap for, न ॥ उ (of next verse)

286 PAÑCASIDDHĀNTIKĀ XV.16 (उक्तश्च) संहितायां मया प्रपञ्चोऽ(स्य) राहुचारादौ । ग्रहणस्य यन्निमित्तं विनै[व] राहुं रविहिमांश्वोः ॥ १० ॥ 10. This matter of eclipses has been expatiated upon by me at the beginning of the chapter on Rāhu’s (the Node’s) motion (in the Bṛhatsaṁhitā, 5. 8-11). Also the causes for the eclipse of the Sun and the Moon without the consider- ation of Rāhu has been dilated upon. VM indicates both the nodes as Rāhu (the dragon of mythology), one as the ‘head’ and the other as the ‘tail’. Here, VM takes the correct astronomical position in the matter of the eclipses. In this chapter as also as in his Bṛhajjātaka, VM refers to without inhibition, the incorrect views of the authors of the early astronomical saṁhitās. [ध्रुवस्य स्थानम्] मेरोर्न दिग्विभागो यस्मात् प्राची न भास्करात्तस्मिन् । उदय(ते) याव(द्धि)वं पर्येतीव सुन्दरी तावत् ॥ ११ ॥ अणुमात्रदर्शनात् प्राग्विभाग इति चेत् समार्धमि(त्वा तु) । तस्मिन्नेवाऽस्तमये किं वा प्राची भवेत् त्वपरा ॥ १२ ॥ तेषामपक्रमवशाद्दिवसो न खलु भ्रमाद्यथास्माकम् । षष्टिर्नाड्योऽस्माकं वर्षमहोरात्रममराणाम् ॥ १३ ॥ वर्षे वर्षे द्युनिशं सुरासुराणां विपर्ययेणाह्नः । मासं तु तत् पितॄणां मनुजानां नाडिकाषष्टिः ॥ १४ ॥ यन्मात्रं भूवृत्तात् क्षण[मात्रेणो]न्नतिं व्रजत्यर्कः । तन्मात्रान्तर(चा)रिणममराः पश्यन्ति नो(र्ध्व)म(तः) ॥ १५ ॥ होराधिपतिदिनेश्वरपरम्परा न [घ]टते यथास्माकम् । षष्टिर्नाड्यस्तस्मि[न्] नाहोरात्रो भवति यस्मात् ॥ १६ ॥ Situation at the Poles 11. There is no distinction of direction at the North pole, because East cannot be determined there using the Sun (rising and setting and culminating), for, as long as the Sun stays risen, it goes round and round the sky like a beautiful damsel. The idea is that there is no daily rising and setting of the Sun to determine east and west. In fact, at the North pole all directions are south. 10a. A1. उक्तं च; A2. उक्तं व; B. —क्तं च      11a. B1.2. दिविभागो b. B. संतायामवाप्रपंचोस्य              b. B. भास्करामस्मिन् c. A1. यन्निमितं; B. यनिमितं             c. B. नदयति d. A.B. विनैराहुं (B. हु)              d. A1. यावद्विवं; A2. यावर्द्विवं; B. हुपरपि हिमांश्च                 B. यावद्विपर्येतीव C.D. यावद्दिनपः पर्येति वसुन्धरीं तावत्

XV.16 XV. SECRETS OF ASTRONOMY 287 12. If it is argued that from the point where the Sun just appears above the horizon east is determined, as the Sun sets at the same point after half a year, can this east become west also? What VM says in these two verses is essentially true. But his statement that it sets at the same point after half a year is not correct. It is not exactly half a year, and the Sun will disappear at any point proportionate to the fraction remaining over the full sidereal days gone between rising and setting. 13. For the gods at the North Pole, the day is determined by the Sun's decli- nation, (north declination being day-time, and south declination night), not like ours, depending on the daily rotation. It is 60 nāḍikās for us, and one year for the gods. 14. Every year the day and the night of the gods and of the asuras (demons, at the South Pole) is opposite, (i.e. when it is day-time for the gods it is night- time for the asuras, and vice versa); for the pitṛs (on the Moon), the day-night is one synodic month; and for men, it is sixty nāḍikās. 15. To the extent the Sun rises above the horizontal by two muhūrtas, (i.e. 24° above the horizon) to that extent the gods at the pole see the Sun rising above the horizon, and not more than that. The Sun spirals round and round after rising, with its altitude increasing with its north declina- tion. As the maximum declination is 24° (according to Hindu astronomy, and fairly correct at VM's time), the altitude never exceeds 24°. After that is spirals down. 16. The series of the Lords of the Horās and Lords of the days do not fit there as it does for us, because the sixty-nāḍikā-day-night does not obtain there. This statement fits not only the pole, but also all places in the arctic zone, when the Sun seen above the horizon exceeds 24 hours. The horā is one hour's time, and the Lords of the horās are successively Saturn, Jupiter, Mars, Sun, Venus, Mercury and Moon, and again Saturn etc. The lord of the first horā after sunrise is the lord of the day. It can be seen that the lord of the 25th horā is the lord of the day next in the day series. So, if the day is more than 24 hours, the two series cannot fit. 12a. A.B. अनुमात्र b. A.B. समार्धमिवा नु ( A2. ॰मिचा नु; B. ॰मित्वात्) c. B. तस्मिन्वास्तमये d. B. प्राचि भवेत् परा 13a. A. तेषामपक्रम; B. तेवाम ( B1.2. om म) पक्रमवशादिवसो ( B2. दि) c. B. षष्टिनोर्द्धोस्माकं d. B3. gap after णाम्; ( B1.2. no gap) 14a. B. वर्षे वर्षेन्दुर्निशं c. B. मास तु A.B. पितॄणां d. B. मनुज्ञानां B. नाडिकषष्टिः 15. Quoted by Utpala on BS 17.4-5. 15a. B1. भ्रमक्ता; B2. भ्रमवत्ता; B3. भ्रमवता for भूवृत्तात् b. A.C.D.U. क्षणद्वयेनोन्नतिं; B. क्षणवृयेणोन्नति ब्र ( B3. ०ण्णदाये) c. A.B.वारिण ( B. om ण) d. B. पश्यन्ति A. नोर्धमधः; B. नोर्धमंधः 16a. A. नटते ( A. नद्यते) यथास्माकं a-b. C. परम्परा तत्र नो यथास्माकम् | D. परम्परा न स्यात्तु यथास्माकत् c. A. तस्मिन्नाहो; B. षष्ठिनाड्यस्तस्मिन्नहो 38

288 PAÑCASIDDHĀNTIKĀ XV.21 [वारज्ञानम्] दिनवारप्रतिपत्तिर्न समा सर्वत्र कारणं कथितम् । (नहोऽपि) भवति यस्मात् विप्रवदन्तेऽत्र दैवज्ञाः ॥ १७ ॥ द्युगणाद्दिनवाराप्ति (र्धु) गणोऽपि हि देशकालसम्ब (न्धात्) । ला(टा)चार्येणोक्तो यवनपुरे (ऽर्धा) स्तगे सूर्ये ॥ १८ ॥ Weekday 17. The determination of the weekday is not the same everywhere. As no reason is given in this matter too, even astrologers disagree among there- selves. The commencement of the day is fixed arbitrarily, as a matter of convention, like midnight, sun- rise, sunset, etc, following the custom of different peoples. 18. The weekday is obtained from the total days commencing from a stated point of time, of a particular day at a particular place. Ācārya Lāṭadeva has said that the day begins at the exact (mean) sunset at Yavanapura. This convention is that of the Romaka and the Pauliśa Siddhāntas, mentioned by VM in PS I 8-10, Lāṭadeva is said to have redacted these two siddhāntas. Yavanapura is Alexandria in Egypt, as can be fixed from the longitude correction for Ujjain in PS III. 13. see also I. 8 and note on p. 10 above. [दिनगणना] रव्युदये लङ्कायां सिंहाचार्येण दिनगणोऽभिहितः । यवनानां निशि दशभिर्मुहूर्तैश्च तद्गुरुणा ॥ १९ ॥ लङ्कार्धरात्रसमये दिनप्रवृत्तिं जगाद चार्यभटः । भूयः स एव सूर्योदयात्प्रभृत्याह लङ्कायाम् ॥२० ॥ देशान्तरसंशुद्धिं कृत्वा चेन्न घटते तथा तस्मिन् । कालस्याऽस्मिन् साम्यं (तै)रेवोक्तं यथाशास्त्रम् ॥ २१ ॥ 17-20. Quoted by Makkibhaṭṭa on 18-29. Quoted by Utpala on BS 2, pp. Si. Śekhara, 2.10. 31-32 17a. A. वारप्रतिपत्ति न ; B. वारप्रति पति न 18a. B. द्युगणांदिन b. M. कारणे कथिता B1.2. ०प्तिद्विगुणोपि; (B3. ०प्तिर्द्वि०) c. A.B.C.D.M. नेहापि b. A1. संवधा; A2. संबन्धा; B.D. M.U. सम्बन्ध: c. A.B1.2. लाजाचार्ये d. A.B. पुरेवा (B. या) स्तगे; M. चास्तगे

XV.27 XV. SECRETS OF ASTRONOMY 289 मध्याह्नं भ(द्राश्वेऽ)स्तमयं कुरुष्व्[त्तरेषु] केतुमालानाम् । कुरुतेऽर्धरात्रमुद्य(न् भा)रतवर्षे युगपदर्कः ॥२२॥ उदयो यो लङ्कायां सोऽस्तमयः सवितुरेव सिद्धपुरे । मध्याह्नो यमकोट्यां रोमकविषयेऽर्धरात्रः सः ॥ २३ ॥ अधिमासकोनरात्रग्रहदिनतिथिविसमेषचन्द्रार्काः । अयन(र्त्वा)र्क्षगतिनिशाः समं प्रवृत्ता युगस्यादौ ॥ २४ ॥ अन्यद् रोमकविषयाद् देशान्तरमन्यदेव यवनपुरात् । लङ्कार्धरात्रसमयादन्यत् सूर्योदयाच्चैव ॥ २५ ॥ सूर्यस्यार्धास्तमयात् प्रतिदिवसं यदि दिनाधिपं ब्रूमः । तत्राऽपि नाऽऽप्तवाक्यं न (वा) युक्तिः काचिदन्याऽस्ति ॥ २६ ॥ सन्ध्या क्वचित् क्वचिदहः क्वचिन्निंशा [दिवसपतेः] क्वचित् क्वचित् । स्वल्पे स्वल्पे स्था(ने) व्याकुलमेवं दिनपतित्वम् ॥ २७ ॥ Day-reckoning 19. Siṃhācārya has declared that reckoning day-total commences at a sun- rise in Laṅkā. The preceptor of the Yavanas has said that the day commences for the Yavanas ten muhūrtas, or twenty nāḍikās, in the night, (i.e. after sun- set). Hindu siddhāntas suppose Laṅkā to be on the equator, at the junction of the Ujjain meridian. Siṃhācārya's view is that of many later siddhāntas. The preceptor of the Yavanas mentioned is probably the Yavanācārya of the Yāvanajātaka, the well-known astrological work. There is a section devoted to astronomy also in that work. If the people in Greece are meant by Yavanas here, Yavanācārya perhaps tries to fit a Greek astronomical work into serviceability in Ujjain, for 20 nāḍīs after sun-set in Greece is the moment of sun-rise at Ujjain, assuming a rough longitude correction 10 nāḍīs. 20. Āryabhaṭa has said that the day commences at mid-night at Laṅkā. He himself again has said, the day commences from sunrise at Laṅkā. Āryabhaṭa has written two works. One is the wellknown Āryabhaṭīya. He has written another work, not extant now. It is referred to by others as the Midnight School and commences the day at midnight. Bhāskara I has given its system in chap VII of his Karmanibandha, better known as Mahābhāskarīya. The system given in this is the same as that of the Saurasiddhānta of the PS. Brahmagupta professes to follow this in his Khaṇḍakhāyaka. 20a. Jy, N. समयात् 19c. B. यवनानांशिनिशिभिर्गतैर्मु॰ 20. Quoted by Nīlakaṇṭha in his b. B1.3. प्रवृत्तिञ्-गाद; B2. प्रवृत्तिः जगाद A.D.U. ॰द्शभिर्गतैर्मु॰; Jyotirmīmāṃsā, p.8, as also on A. चार्यमट्टः; B1.3. चार्यभटः M. यवना निशीह दश॰ ABh. Kāla. 16. c. M, N, U. चार्कोदयात्

290 PAÑCASIDDHĀNTIKĀ XV.27 21. If it is argued that the different times for commencing the day can be accounted for by correction for longitude, it does not agree with what they themselves have said in this matter, according to the śāstras, (which is as follows). 22. 'The sun rising in Bhārata-varṣa, makes at that very moment, mid-day in Bhadrāśva-varṣa, sun-set in Uttara-kuru-varṣa, and mid-night in the Ketumāla-varṣa. 23. What is sun-rise at Laṅkā, that same moment is sun-set at Siddhapura, noon at Yamakoṭi, and mid-night in the Romaka-pura. In the above two verses the early siddhāntic conception of a world geography is given briefly. The equator is the Jambūdvīpa, with the North Pole at its centre. Laṅkā is the point where the Ujjain meridian cuts the equator. The point 90° east of Laṅkā is Yamakoṭi, also called Yavakoṭi. Here seems to be a vague concept of Java, called Yavadvīpa, whose exact distance was not realised. Ninety degrees west of Laṅkā is Romaka-pura, answering to Rome, whose exact position was not realised. The antipode point of Laṅka is called Siddhapura. A vague notion of the Mayan and Aztec civilisa- tion brought in by early exporters sailing the seas might have given rise to the idea. The astronomical idea of sun-rise, moon etc. is correct according to the conception. The four varṣas mentioned, Bhārata, Bhadrāśva, Kuru, and Ketumāla are supposed to be situated round the North Pole, at its south, east, beyond the pole and west, from our stand point, in Bhāratavarṣa. The Purāṇas give seven divisions of the Jambūdvīpa, and these are the principal four. The purāṇic concept is that of an earlier period of a flat earth, with the mountain Meru at the centre with Jambūdvīpa arranged all round, transmitted by tradition. The Siddhāntas tried to fit whatever is possible of the Purāṇic geography, into the conception of the spherical earth, refuting the rest outright or explaining them away. 24. At the beginning of the yuga, the intercalary months, the omitted days, the planetary days, the lunar days, the first point of Meṣa, the Moon, the Sun, half-years, ṛtus, and the sidereal days, begin together (and can be reckoned anew). 25. The longitude correction reckoned from the Romaka region is different from that from Yavanapura. Reckoning time from mid-night at Laṅkā is different from that from sun-rise. 21a-b. B. द्धिंत्तचन्न (B3. ०न०) | 23a. A. Hap. om of one यो; c. B. ०स्मात् साम्यं (B3. साम्यां) | B. दनम्रो यो लङ्कायां d. A. नैर्वोक्तं | c. A.B1. यमकोद्यां U. यवकोट्यां मध्याह्नं B1.2.3. gap indicated for यथाशास्त्रं | d. B. रोमवियेर्द्ध U. रात्रं च and part of the next line up to कुरुषू | 24a. B. 'अ' lost. B. रात्रिग्रह 22a. A. भद्रेभ्रस्त; C.D.U. भद्राश्वेष्वस्तमयं | b. B. दिवसमयूष A. मेषं b. A.C.D.U. om उत्तरेषु and read | c. A. अयन-र्क्ष; B. अयनत्वर्क्ष; D. अयनत्वृक्षगति कुरुषु केतुमालानाम; | A1. युगस्पादौ B. .....तरेषु कालेंतुलानानाम् c. A. ०मृद्यद्धरात | 25b. B. विषयादेशा d. B2. युगपदर्कः | d. A. दन्यसूर्यो; B. दन्यः सूर्यो

XV.29 XV. SECRETS OF ASTRONOMY 291

26. If we determine the Day-lord from the half-setting of the Sun every day, there is neither traditional authority nor reasoning to support this.

27. Even in quite adjacent places, in one place there is sun-rise or sun-set, and not in the other, day-time in one place and night in the other, and vice versa. Thus, there is confusion among people in the matter of the Lord of the day.

होरावार्ता(प्ये)वं यस्माद् होरा दिनाधिपस्याद्या | तस्याऽपरिनि(ष्ठा)ने होराधिपतिः कथं भवति || २८ ||

अविचार्यैवं प्रायो दिनवा(रे) जनपदः प्रवृत्तोऽयम् | स्फुटतिथिवि(च्छे)दसमं युक्तमिदं प्राहुराचार्याः || २९ ||

28. The matter of determining the Horā-lord also is in the same mess. When the Day-lord is not determined, how can the Horā-lord be determined?

29. Without giving a thought to all these difficulties, people generally use the name of the Day-lord in their daily routine, (and get on with their work). Learned authorities say that the best thing would be to use the true tithi, (lunar day) and its parts for daily intercourse (as for fixing a definite point of time etc.)

What is meant is as follows:- Sun-rise etc. may vary from place to place, according to the local time. But the lunar day is the same for every place on the earth. So this can fix a point of time without any ambiguity. We learn that the ancient Babylonians used the lunar day as the unit of time, just as we use the solar day.

In the above verses VM indulges in a lot of discussion about Horā-lord, Day-lord, etc. But these are only matters of convention, and astrologers and governments can agree upon some convention to avoid difficulties, e.g. do we not have the standard mean time for our daily dealings.

26a. B. ॰र्धस्तमयात् 27b. A.B. निशा दिनपतिः 28a. A. वार्त्ताध्येवं; B. वत्ताप्येवं (B3. ॰र्त्ता॰) b. B. ॰दिवस यदि B. दिनाधिपत्यं A2. ब्रूमः A2. क्वचिक्वचित् b. B. ॰धिपश्वाद्याः c. B. नाप्तं वाक्यं c. A.B. स्थानं c. A. निष्ठाने; B. निष्ठाने d. A. नवयुक्तिः; B.C.D. न च युक्तिः d. D. व्याकुलमेव 29a. U. अविदित्वैवं B. ॰दन्यास्तः; U. ॰दप्यस्ति b. A. वारौ; B1.2. वारै; B3. वारैः c. B. स्फुरतिथि A.B1.3. विच्छेद d. B1. राचार्योः

[इति पञ्चसिद्धान्तिकायां वराहमिहिरविरचितायां ज्योतिषोपनिषन्नाम पञ्चदशोऽध्यायः ||]¹ Thus ends Chapter Fifteen on ‘Secrets of Astronomy’ in the Pañcasiddhāntikā composed by Varāhamihira

1. Col.: A.B.D. ज्यो (A1. ज्यौ) तिषोपनिषत् पञ्चदशो (B. दशमो) ध्यायः | C. इति ज्योतिषोपनिषन्नाम पञ्चदशोऽध्यायः ||

Chapter Sixteen

SAURA SIDDHĀNTA : MEAN PLANETS

१६. षोडशोऽध्यायः

सौरसिद्धान्तः — ग्रहमध्यमानयनम्

Introductory Chapter XVI of the Pañcasiddhāntikā deals with the computation of the mean star-planets, Mars etc., according to the Saura Siddhānta, and chapter XVII of their true motions, with their heliacal risings and latitudes. The mean planets are made true by employing the method of epicycles, as in the case of the Sun and the Moon, in chapters XI and X. Of the five siddhāntas condensed by Varāhamihira the Saura alone uses epicycles, and there is no evidence of its use in any other. So, in the originals also, only the Saura must have used epicycles, since VM follows the originals as far as necessary. Thus the Saura is the most mature, and may be considered to begin the highest developed stage of Hindu astronomy, represented by the Āryabhaṭīya, the Brāhma-sphuṭa-siddhānta, the Later Sūrya Siddhānta etc. Though VM's Saura, being a karaṇa, does not use yuga-cycles for the planets, the original must have had them and they can be reconstructed from the epoch-constants given, as we have done in the case of the Sun, Moon, Moon's apogee and nodes. These can be seen to agree with the corres- ponding parameters of the Pauliśa quoted by Bhaṭṭotpala in his commentary on the Bṛhatsaṁhitā, and with the Ārdharātrika-pakṣa of Āryabhaṭa, a work now lost, but reconstructible from its descrip- tion given in the Mahābhāskarīya, chapt. VII, 21-35, and from the Khaṇḍakhādyaka of Brahmagupta, which latter expressly follows the Ārdharātrika-pakṣa. Not only the yuga-cycles, but also the yuga days, and epicycles and apogee positions and nodes agree in these. Strangely enough, the 'New' Sūrya Siddhānta does not agree with the 'Old' in many things. In the matter of computing the latitudes of the star-planets, the Saura gives the same method as the Ārdharātrika-pakṣa combining two types of latitudes, but the Khaṇḍakhādyaka follows the Āryabhaṭīya itself exactly as propounded in the Mahābhāskarīya, VI. 52-55. As for agreement of VM's Saura with the other siddhāntas of the period, a perusal of the table given under XVII. 11 will show this. But it must be noted that the agreement in mere number of cycles is not real agreement, because, the yuga days being different, there will be difference in the calculated mean values. But at the period we are considering, viz. c. 500 A.D., the mean positions fairly agree with one another, and also with what would be got by modern astronomy, showing thereby the accuracy of their observations. For example, for PS's epoch, all except the later Sūrya Siddhānta give nearly 236° for Rāhu, including the moderns. This is seen only in the Rāhu of the Later Sūrya Siddhānta. (Evidently, there is error of reading here. In I.33 of the Later Sur. Sid., the original should have been vasvaśviyamāśviśikhidasrakāḥ instead of vasvagni etc. This mis-reading must have occurred before the commentator Raṅganātha, for he gives aṣṭarāmākṛtirāmadvimitāḥ. If what I suggest is correct, 3° will be added to the 232° 29' got according to the wrong reading making Rāhu = 235° 29', giving fair agreement). There is agreement in the degrees for heliacal rising and setting and the method of computing the star-planets between the Saura of the PS and the Later Sūrya Siddhānta, though the epicycles differ in many ways.

XVI. 1 XVI. SAURA : MEAN PLANETS 293 Another important matter should be mentioned. In XVI. 10-11, and XVII. 10-11a, VM gives corrections, which are his own, to secure agreement with observation to make the Saura fit for correct almanac-making, which naturally will be demanded by the literate. Thus, in XVI.10-11, certain bījas are given to correct the means of Mars, Jupiter and Saturn and the śīghra of Mercury and Venus. The corrections amount, in terms of yuga-cycles, to: Mars, + 57; Mercury, + 400; Jupiter, – 33¹/₂; Venus, – 150; and Saturn, + 25. These corrections are similar and approximately equal to the famous Vāgbhāvona correction on the Āryabhaṭīya, propounded by his successors in his school, to correct his cycles to agree with their observation. I do not suggest that VM was aware of the Vāgbhāva correction in that form, but the tendency to correct the earlier results with bījas based on observations is found everywhere, whether north or south, a healthy sign of the growth of the science. One might refer also to XVII.10-11a, where VM attempts to correct Mercury and Venus to secure agreement with observation. Another thing is to be noted. In (1) the Āryabhaṭīya, in (2) the Ārdharātrika-pakṣa (which means ipso facto the Khaṇḍakhādyaka), VM's Saura and Bhaṭṭotpala-quoted Paulisa, and in (3) the Later Sūryasiddhānta, the yuga cycles are such that the mean planets are all zero at the beginning of Kali, the Moon's apogee is 90°, and the Moon's node 180°. Now the Āryabhaṭīya had equal yuga-pādas, Kṛta, Tretā, Dvāpara and Kali, i.e., they are equal in length. The other siddhāntas have unequal yuga divisions, Kṛta being 4 parts, Tretā 3 parts, Dvāpara 2 parts and Kali 1 part. If the other siddhāntas also postulate, like the Āryabhaṭīya, that the planets were created and began to move from the beginning of the Kalpa from a zero position, then the cycles should be divisible by 20. But they are not so divisible in all. This necessity is avoided by postulating a time later than the beginning of the Kalpa called 'the time of creation of planets' by the Later Sūrya-Siddhānta, as started in the verse, graharkṣadevadaityādi sṛjato'sya carācaram kṛtābdhivedā divyābdāḥ śataghnā vedhaso gatāḥ || I.24 || and by having both the number of cycles and yuga-cycles divisible by four. In the case of the Moon's apogee, the cycles should be odd, and in the case of Rāhu the cycles should be even, but not divisible by 4. These necessary conditions are indeed found in the Later Sūrya Siddhānta and its kind. Thus, if there is any observed difference in the mean planets, Moon's apogee and nodes, they must be due to the 3600 years elapsed after Kali, for the period *c.*499 A.D. But the observed differences should be only small, and due to error of observation. The cycles must have been, and have been, constructed with an eye to this also. In fact, the number of cycles have been determined by observa- tion, and by using the Diophantine equation (kuṭṭaka). The difference of just 300 days in the length of the yuga, (it does not matter much if it is 328 days, as in the Later Sūrya Siddhānta) to secure equal- ity at *c.*499 A.D., between the Ārdharātrika-pakṣa and the Āryabhaṭīya, which is called, for the sake of distinction, the Audayika-pakṣa, meaning the type beginning the day from mean sunrise at Ujjain, provided the number of cycles are the same. (See tables under XVII.11). There is a difference of just a quarter of a day accumulated from zero Kali to *c.*499 A.D. and the difference is made zero at this point of time. [ताराग्रहाणां मध्यमानयनम्] एष निशार्धेऽवन्त्यां ताराग्रहनि(र्णयोऽ)र्कसिद्धान्ते । तत्रेन्दुपूत्रशुकौ तुल्यगतौ म(ध्य)मार्केण ॥ १ ॥

294 PAÑCASIDDHĀNTIKĀ XVI.11 Mean positions of the star-planets

1. The following is the determined position of the star-planets at midnight at Ujjain according to the Saura Siddhānta. For their computation, the mean Sun should be taken as the mean Mercury and Venus. Note: I follow TS's emendations. Example: Find the mean Venus at 1,20,553 days after Epoch for the star-planets, viz. 427 śaka elapsed midnight at Ujjain. This is the mean Sun at 1,20,553.5 days from midday of the Saura epoch, (vide expl. under IX. 1). Therefore the mean Venus = the mean Sun = 1,20,553.5 × 800 - 442) ÷ 2,92,207 = 17° 18′ 27″. जीवस्य 'शता'भ्यस्तं 'द्वित्रियमाग्नित्रिसागरै' (र्वि) भजेत् । द्युगणं कुजस्य चन्द्राऽऽहतं तु 'सप्ताष्टषड्' भक्तम् ॥ २ ॥ सौरस्य 'सहस्र' गुणा (द्) 'ऋतुरस (श्) न्यर्तुषट्कमुनिखैकैः' ॥ यल्लब्धं ते भगणाः शेषा म(ध्य) ग्रहाः क्रमेणैव ॥ ३ ॥ दश दश भगणे भगणे संशोध्यास्तत्पराः सुरेज्यस्य | 'मनवः' कुजस्य देयाः शनेश्च 'बाणा' विशोध्या (स्तु) ॥ ४ ॥ राशिचतुष्टयमंशद्वयं कलाविंशतिर्वसु'समेताः | 'नववेदा'श्च विलिप्ताः शनेर्ध(नं) मध्य (मस्यै) व ॥ ५ ॥ अष्टौ भा (गा) लि(प्ता) '(ऋत)वः' 'ख(पक्षौ)' गुरौ विलिप्ताश्च | क्षेपः कुजस्य '(य)मतिथि-पञ्चत्रिंश'च्च राश्याद्याः ॥ ६ ॥ शतगुणिते बुधशीघ्रं 'स्वरनवसप्ताष्ट'भाजिते क्रमशः | अत्रार्धपञ्चमास्तत्पराश्च भगणाह(ताः) क्षेपः ॥ ७ ॥ सितशीघ्रं दशगुणिते द्युगुणे भक्ते 'स्वरार्णवाश्विव्यमैः' | अर्धैकादश देया विलिप्तिका भगणसंगुणिताः ॥ ८ ॥ सिंहस्य 'वसुयमां'शाः 'स्व(रेन्दु)वो' लिप्तिका ज्ञशीघ्रधनम् | शो [ध्याः] सितस्य विकलाः 'शशिरसनवप [क्ष] गुणदहनाः' ॥ ९ ॥ [वराहमिहिरकृतः शोधः ] क्षेप्याः 'स्वरेन्दु'विकलाः प्रतिव (र्षं) मध्यमक्षिति (जे) | दश दश गुरोर्विशोध्याः शनैश्चरे सार्धसप्तयुताः ॥ १० ॥ 'पञ्चा (ब्ध) यो' विशोध्याः सिते बुधे 'खाश्विचन्द्र'युताः | 'खखवेदेन्दु' विकलिकाः शोध्याः [स्युः ] सुरपूजितस्य मध्याः स्युः ॥ ११ ॥

XVI.11 XVI. SAURA : MEAN PLANETS 295

2-9 To get mean Jupiter, multiply the days from epoch by 100, and divide by 4,33,232. Revolutions etc. are got. Deduct 10‴ per revolution. Add 8ʳ 6° 20″, the mean at epoch (This is called kṣepa.) (A bīja correction is given by VM, to this, for which see verses 10-11, below.) To get Mean Mars, divide the days by 687. Revolutions etc. are got. Add 14‴ per revolution. Add 2ʳ 15° 35′ 0″, the mean at epoch. (See verses 10-11, below, for bīja correction.) To get mean Saturn, multiply the days by 1000 and divide by 1,07,66,066. Revolutions etc. are got. Deduct 5‴ per revolution. Add 4ʳ 2° 28′ 49″, the mean at epoch. (See verses 10-11, below, for bīja correction). To get the Śīghra of Mercury, multiply the days by 100 and divide by 8797. Revolutions etc. are got. Add 4½‴ per revolution. Add 4ʳ 28° 17′ 0″, the Śīghra at epoch. (See verses 10-11, below, for bīja correction.) To get the Śīghra of Venus, multiply the days by 10 and divide by 2247. Revolu- tions etc. are got. Add 10½″ per revolution. Add 8ʳ 27° 30′ 39″, the śīghra at epoch. (See verses, 10-11, below, for bīja correction).

[Apparatus Criticus — Column 1]

1a A. ॰र्धेवत्पां; B. धैवत्यां b. A. निर्णेर्किसिधांते; B. गणकसिद्धान्ते c. A. महमार्केण; B. मध्यमाकेसा 2a. B. जिवस्य B2. शताभ्यासं b. A1.C. ॰यमाग्निचिसागरैर्विभजेत्; B. ॰विभजेन d. A1. हतं A1. ॰ष्ट्र्द्धक्तं 3a. B. repeats words from previous verse: सौम्यस्य सप्ताभ्यस्तं द्वित्रियमाग्निसामरैः सहस्रगुणा a-b. D. गुणमृत् b. A. दतुरससून्युर्तु; B. रुतु॰ B. खैकः d. B1.2.D. शेषा मध्या 4a. B1.3. दशांश भगणे and one भगणे om by haplography b. B. ॰ध्यास्तसराः c. B. नमवः कुकुक्षु देया d. B1.3. शनैश्च B3. विशोध्य A. ॰ख्रु; B. स्युः 5a. B. ॰मंशं b. B3. ॰शतिवसु C.D. समेता c. B. ॰वेदाक्षलिप्ताः d. A. ॰धनेर्मध्यमास्येव; B. शने मध्यमस्त्वेयम् 6a. A. भामा लिप्त; B. मागाः लिप्त b. A.C.D. र्तवः; B. तवः; A. खमक्षोगरौ (A2. ॰क्षौ॰); B. तवः शेषसौ गरु विः

[Apparatus Criticus — Column 2]

C. खमक्षो गुरौ; D. खपछो गुरोः c. B. क्षेवः A. जमतिथि; B. यमतितिथि d. B. त्रिशद्य 7a. B. गुणितं c. B. ॰र्धपंचमौस्त॰ (B3. ॰स्तस. d. A. हतः; B. हतक्षिपा; C.D. क्षेप्याः 8a. B. गुणीते b. A. ॰वाश्वियमैः c. A1. अर्कैका॰; A2. 'अर्कैका d. A. विलिप्ता 9a. A. सिंहस्य; B. सिंहेस्य b. A. खरेन्वो; B. खरे देवो विलिप्तिका c. A. शोसितस्य; B3. शोषितस्य B. विकला d. A.B. पक्षा गुणा दहनाः (A. ताः) 10a. A.B. क्षेप्या B. विकला b. A.B. वर्षमाध्यम A.B. क्षितिजो 11a. A.B 1.2 पञ्चद्वयो; B3. ॰द्वयोः b. A.B. ॰स्ताश्वि B. चन्द्रयुक्ताः c. A.B. विकालिकाः d. A.B.C.D. om स्युः B. सुर —— प्जतिस्य (B3. पू). A.B.C.D. मध्यात् In D. ch. XVII of the Mss. and of C is continued as part of ch. XVII. with verse numbers duly altered.

296 PAÑCASIDDHĀNTIKĀ XVI. 11 Note 1. I follow TS's emendations, except in verse 6, where I have read khamakṣau as khapakṣau instead of their khamakṣo, makṣo being meaningless. But their meaning, 20, is all right. In 7, the word kṣepa can stand, and need not be emended as done by them. Note 2. The word madhya with reference to Mars, Jupiter and Saturn is mean planet in modern parlance, and śīghra with reference to Mercury and Venus, is mean planet according to modern terminology. Note 3. How to get the days from epoch has already been explained, and it should only to be brought to the mid-night following to be used here. Example 1. Find the mean Mars at 1,20,553 days from the midnight following the Romaka epoch, which is the epoch given for star-planets. 1,20,553 ÷ 687 = 175 revolutions and = 5ʳ 21° 52' 40" The revolution correction = 175 × 14" = + 41" Kṣepa or mean at epoch = 2ʳ 15° 35' 0"

Mean Mars at required date = 8ʳ 7° 28' 21" Example 2. Find the Śīghra Venus at 1,20,553 days for epoch. 1,20,553 × 10 ÷ 2247 = 536 revolutions and 6ʳ 2° 19' 23" Revolution Correction: 536 × 10½" = + 1° 33' 48" Śīghra at epoch = 8ʳ 27° 30' 39"

Śīghra of Venus at 1,20,553 days = 3ʳ 1° 23' 50" Note 4. The rules give to find the mean planets etc. depend on the fact that there are approxi- mately 100 revolutions of Jupiter in 4,33,232 days, one revolution of Mars in 687 days, 1000 revolutions of Saturn in 1,07,66,066 days, 100 śīghra (truly mean) revolutions of Mercury in 8,797 days and 10 of Venus in 2247 days. The revolution corrections make these exact. The epoch constants are the means at epoch. Note 5. From the rules given we can reconstruct the yuga cycles of the original Saura-siddhānta of which the Saura of the PS is a Karaṇa, and from these the epoch constants. These we shall do now. The yuga days of the original Saura are 1,57,79,17,800, as computed from the short Saura yuga given in I.14, from which it can be computed that in 1,80,000 years there are 6,57,46,575 days, since the yuga is 43,20,000 years, being 24 times the short yuga. We might now verify by calculation, the yuga revolutions (yuga-paryaya) and epoch constants (kṣepa) of the several planets. Jupiter: Yuga revolutions 1,57,79,17,800 × 100 ÷ 4,33,232 = 3,64,220, rev. 0ʳ 17° 25' 1" Revolution correction = 3,64,220 × 10''' = – 16° 51' 43"

∴ The number of rev. etc. in the yuga = 3,64,220 rev., 0ʳ 0° 31' 18"

XVI. 11 XVI. SAURA : MEAN PLANETS 297 The error in the karaṇā method is 31' 18" in 43,20,000 years, which is negligible when we consider that the rule is given in a karaṇa, which is not intended to be used for such a long period. The yuga revolutions 3,64,220, is indeed that given in the original, as seen from the Ārdharātrika-pakṣa and Bhaṭṭotpala’s Pauliśa and Khaṇḍakhādyaka. Epoch Constant (kṣepa) for Jupiter The epoch is 427 Śaka i.e. 427 + 3179 = 3606 years from zero Kali, i.e. midnight, − 3 nāḍīs, 9 vināḍis. For 3606 years, the motion is 3,64,220 × (1/1200 + 1/(1200 × 600)) = 304 rev., 0ʳ 8° 6' 36" Subtracting the motion for 3 nāḍīs, 9 vināḍīs, − 16" Jupiter’s epoch constant got = 0ʳ 8° 6' 20" This is exactly what is given above in verse 6. Saturn: Yuga revolutions 1,57,79,17,800 × 1000 ÷ 1,07,66,066 = 1,46,564 rev., 0ʳ 3° 26' 12" The cycle correction = 1,46,564 × 5''' = − 3° 23' 34" ∴ The Yuga cycles got = 1,46,564 rev., 0ʳ 0° 2' 38" This is indeed the yuga cycles given in the Ārdharātrika-pakṣa etc. neglecting the small error of 2' 38" accumulating in 43,20,000 years, due to the karaṇa roughness. Epoch constant for Saturn 1,46,564 (1/1200 + 1/(600 × 1200)) = 122 rev., 4ʳ 2° 28' 5" Deducting for 3 nāḍīs, 9 vināḍis − 6" The epoch constant got = 4ʳ 2° 28' 49" Mars: Yuga revolutions 1,57,79,17,800 ÷ 687 = 22,96,823 rev. 6ʳ 29° 4' 59" Rev. Correction = 22,96,823 × 14''' = + 4ʳ 28° 52' 5" ∴ Yuga-cycles = 22,96,823 rev. 11ʳ 27° 57' 4" = 22,96,824, in round numbers, being short only by 2° 3', negligible in the long period. We see agreement with the original. Epoch constant for Mars The epoch constant is 22,96,824 (1/1200 + 1/(600 × 1200)) = 1917 rev., 2ʳ 15° 36' 43".2

298 PAÑCASIDDHĀNTIKĀ XVI.11 Deduction for 3 nāḍīs, 9 vināḍīs − 1′ 39.5″ Less 1917‴ ÷ 5 − 6″ The epoch constant = 2ʳ 15° 34′ 58″ There is agreement. Mercury: Yuga revolutions 1,57,79,17,800 × 100 ÷ 8997 = 1,79,36,998 rev. 11ʳ 21° 41′ 33″ Rev. Correction = 1,79,36,99 × 4½‴ = + 1ʳ 13° 41′ 15″ The Yuga cycles = 1,79,37,000 Rev. 0ʳ 5° 22′ 48″ There is fair agreement with the Ārdharātrika-pakṣa etc. with an excess of 5° 22′ 48″ in the yuga, which need not be considered great in a karaṇa rule. 4 7/16 instead of 4½‴ would have taken this difference also into account. Epoch constant for Mercury 1,79,37,000 ( 1/1200 + 1/(600 × 1200) ) = 14,972 rev., 4ʳ 28° 30′ 0″ Subtracting for the excess 3 nāḍīs, 9 vināḍīs − 23′ 53″ For 1/16 repeat the correction + 16″ Epoch constant = 4ʳ 28° 17′ 23″ Here the constant seems to have been given to nearest minute. Venus: Yuga revolutions 1,57,79,17,800 × 10 ÷ 2247 70,22,331 rev. 1ʳ 8° 55′ 54″ Revolution correction = 70,22,331 × 10½ = 56 Rev. 10ʳ 21° 47′ 52″ Yuga cycles = 70,22,388 0ʳ 0° 43′ 46″ There is a small error of 43′ 46″, negligible in the long period of yuga, owing to the karaṇa rule. 10 85/178 would have been very correct. Epoch constant for Venus Epoch constant, 70,22,388 ( 1/1200 + 1/(600 × 1200) ) = 5861 rev. 8ʳ 27° 35′ 38″.4 Less for 3 nāḍīs, 9 vināḍīs − 5′ 2″ Extra in the correction + 2′ 4″ Epoch correction (in full agreement) = 8ʳ 27° 30′ 39″ In the Sun, Moon, Rāhu, and Moon’s apogee too we see much exact agreement with Ārdharāt- rika-pakṣa, Khaṇḍakhādyaka and Bhaṭṭotpala-quoted Paulīśa, from which we can conclude that the source of VM’s Saura is the Old Saura-siddhānta.

XVI.11 XVI. SAURA : MEAN PLANETS 299 VM’s Bīja corrections 10-11. Add 17" per year to mean Mars. Deduct 10" per year from mean Jup- iter. Add 7¹/₂" per year to mean Saturn. Add 120'' per year to the śīghra (‘mean’ according to modern parlance) of Mercury. Subtract 45" per year from the śīghra (modern ‘mean’) Venus. In addition, subtract 1400" or 23' 20", constant from Jupiter’s mean. Note. 1 I follow TS’s corrections. Note 2. These corrections are obviously VM’s own, to secure agreement with observation, because VM sees the Saura used widely for almanac making, (besides himself being its follower) and uses these bīja corrections to the Saura. Being VM’s own, we cannot verify the numbers used, but we can compare these corrections with those given by the followers of the Āryabhaṭīya belonging nearly to his time. Note how close they are, and commend the tendency to observe and correct, instead of blindly following the masters. The Kerala school following the Āryabhaṭīya gives the well-known vāgbhāva correction: vāgbhāvōnāc chakābdād dhanaśatalayāhān mandavailakṣyarāgaiḥ prāptābhir liptikābhir virahitatanavaś candratattuṅgapātāḥ | śobhānīrūḍhasamvidgaṇakanarahatān māgarāptāḥ kujādyāḥ saṁyukytā jñārasaurāḥ suragurubhṛgujau vajitau bhānuvarjam || (Kaṭapayādi notation is used here.) According to this the corrections per annum are for Mars

• 11.5", for Mercury + 105", for Jupiter – 12", for Venus – 39", and for Saturn + 5". See that these compare well with VM’s. Example: Give the bīja corrections for Mars and Venus at 1,20,553 days from epoch. This is 330 years. The correction for Mars = 330 × 17" (positive) = + 1° 33' 30". The correction for Venus = 330 × 45 (negative) = – 4° 7' 30". Thus bīja-corrected mean Mars of date is 8ʳ 9° 1' 51", and bīja-corrected Venus, 2ʳ 27° 16' 20". [इति पञ्चसिद्धान्तिकायां वराहमिहिरविरचितायां सौरसिद्धान्ते मध्यमानयनं नाम षोडशोऽध्यायः ||] Thus ends Chapter Sixteen entitled ‘Saura-Siddhānta – Mean Planets’ in the Pañcasiddhāntikā composed by Varāhamihira

1. Col.: A.B.D. सूर्यसिद्धान्ते मध्यगतिः (B. शनि for गति) (D. gives this as a section colophon). C. इति सूर्यसिद्धान्ते मध्यगतिर्नाम षोडशोध्यायः

Chapter Seventeen SAURA-SIDDHĀNTA — TRUE PLANETS १७. सप्तदशोऽध्यायः सौरसिद्धान्तः — ताराग्रहाणां स्फुटीकरणम् As stated earlier, the computation of the Saura star-planets is continued in ch. XVII, the topics treated being, the True planets, their heliacal risings and their latitudes. [स्फुटकर्म] शीघ्राख्योऽर्कोऽन्येषां भौमादीनां तु परिधयो द्विगुणाः । 'पक्षस्वरा' (श्च खं षड्यमाः)' 'खकृता' श्चकुजादीनाम् ॥ ३ ॥ Epicycles of the planets

1. For the other planets (i.e. other than Mercury and Venus, viz., for Mars, Jupiter and Saturn), the Sun is their Śīghra. The epicycles of equation of the apsis of Mars etc. are twice, 35°, 14°, 16°, 7°, and 30°, (i.e., of Mars 70°, of Mercury 28°, of Jupiter 32°, of Venus 14° and of Saturn 60°.) Note 1. I follow TS's emendation in pañcatriṁśanmanavaḥ. But I read surāḥ as svarāḥ and not śarāḥ, like TS because svarāḥ is nearer the given reading surāḥ, and also 14° is the epicycle given in the Ārdharātrika-pakṣa etc. Five mātrās are wanting in the last foot, and it must be supplied with some such words as bhāgāḥ, as all numbers are already given. But TS make it ṣaḍyutās-triṁśāḥ and, strangely enough, translate it as 24, confusing the addition mentioned by themselves for subtrac- tion. (However, on page XXIII of the Introduction Thibaut gives the correct 30° × 2 = 60°.) Note 2. The first foot is to be read with verses 7 and 8 of chap. XVI where the śīghra of Mercury and Venus have already been given. Properly speaking, the matter in the foot should have been given in chap. XVI. 'रस-भव-वसु-वेदा-र्का' विंशतिगुणिताः कुजस्य दशको (ना) : । मन्दगति नाम (?) भागाः कुजबुधगुरुशुक(सौ)राणाम् ॥ २ ॥
2. 6, 11, 8, 4, 12 multiplied by 20, Mars's being less by 10°, (i.e. 110°, 220°, 160°, 80°, and 240°) are the apogee positions of Mars, Mercury, Jupiter, Venus and Saturn. 1a. B. र्को d. A. सुरास्त्रिंशाः || B. ष्ट्यः सुरास्त्रिंशा ||; C. ष्ट्यः b. ०नां नु परि; D. तु [मन्द] परिधयः [स्युः] | शराः षड्युतास्त्रिंशाः ||; c. D. द्विगुणाःपञ्च० A.त्रिंशत्सनवो D. ष्ट्यः स्वस्त्रिंशा [श्च] ||

XVI.6 XVII. SAURA : TRUE PLANETS 301 Note 1. I follow TS's emendations. Mandagatināmabhāgāḥ does not make any sense. But the meaning is obvious, it must mean apogee positions. Some drastic emendation of the word can be made to give this meaning, but I am against such as emendation. Note 2. These positions agree with those given in the Ārdharātrika-pakṣa etc., as also in the Āryabhaṭīya. The correct positions according to modern astronomy are 128°, 234°, 170°, 290°, and 244°, respectively. Note 3. The apogee 80° for Venus and the epicycle 14° are the same as given for Sun. The apogee position 80° given is near the perigee position of modern astronomy, so far away. We shall explain this under verses 10-11a, below. शीघ्रपरिधावंशाः ‘कृतगुणप(क्ष)’-‘द्विवह्निशीतकराः’ । ‘पक्षस्वरा’ (श्व खंषड्य माः)’ ‘खकृता’ श्वकुजादीनाम् ॥ ३ ॥ 3. The degrees of epicycles of conjunction of Mars is 234, of Mercury 132, of Jupiter 72, of Venus 260, and of Saturn 40. Note 1. I have generally adopted TS's corrections. But the text is corrupt in the third foot, and TS's correction itself wants one mātrā. I would read the third and fourth feet thus: pakṣasvarāś ca kham ṣaḍyamāḥ khakṛtāś ca kujādīnām. This would follow the original work. Note 2. The values agree with the Ārdharātrika-pakṣa, Khaṇḍakhādyaka, and Bhaṭṭotpala-quoted- Pauliśa group, as to be expected. [स्फुटग्रहाः] शीघ्रान्मध्यमहीनाद् (रा)शित्रितये गतैष्यदं(श)ज्ये । भुजकोटी तत्परतः षड्(भिः) प(ति)ते स एव विधिः ॥ ४ ॥ स्वपरिधिगुणिते भाज्ये ‘खर्तुगुणै’ [स्ते] विप(रिण)ते तच्च । कोटिफलं व्यासार्धे मृगकर्क्यादौ चयापच(यम्) ॥ ५ ॥ तद्भुज[कृति]योगपदैर्भा(ज्यं भुजजं ‘ख)सूर्य’घ्नम् । तच्चापार्धं मन्दे हानिधनं शीघ्रकेन्द्रवशात् ॥ ६ ॥ 2a. B.रससंवत्सुवेदाकर्कों (B2.रसंस०) 3a. B.परिधा यथाशा b. B.गुणिता A.क्रजस्य A.दशकोणाः; B.दशकोणस्पणाः b. B.ततदगुण० A.B.C.D.पक्षा द्वि. B1.2.वाहि (B2.°ष्यणाः; B2.स्पदणाः) c. B.om पक्षस्वराः A.स्वराऽऽऽऽ खंवद्य B.खषद्य c. D.गतीनां भागाः B.नाम लाधवं ||कुज d. A.B.मा ख (B3.माःख) C.D.कृताःस्युःकुजादीनाम् d. A.शुक्रसोराणां०

302 PAÑCASIDDHĀNTIKĀ XVII.7 True planets The first step 4. Deduct the mean from the śīghra. If the remainder (called śīghra-kendra) is within 90°, sin. śīghra-kendra is called bhuja, and sin (90° – śīghra-kendra) is called koṭi. If śīghra-kendra is more than 90° and less than 180°, subtract it from 180°. (Taking this as the śīghra-kendra, sin. śīghra-kendra is bhuja and sin (180° – śīghra-kendra) is koṭi. If śīghra-kendra is more than 180° and less than 270°, deduct 180° from it and take this as śīghra-kendra. Sin. śīghra-kendra is bhuja and sin (90° – śīghra-kendra) is koṭi. If śīghra-kendra is from 270° to 360°, deduct it from 360° and take its sine as the bhuja and sin 90° – śīghra-kendra is the koṭi. (The bhuja of manda-kendra is to be found in the same way using manda-kendra in the place of śīghra-kendra)¹ 5-6. The bhuja and koṭi must be multiplied by the planet’s epicycle of conjunc- tion and divided by 360. Thus transformed, they are called bhuja-result and koṭi-result pertaining to the equation of conjunction. If the śīghra-madhya is from 270° to 90°, the koṭi-result is to be added to 120 (the R. of the PS). If śīghra-madhya is from 90° to 270°, the koṭi-result is to be subtracted from 120. Square this and add it to the square of the bhuja-result. Find its square root, and by this divide 120 × bhuja-result. Find arc-sine of this. Subtract half this from the longitude of apsis if the śīghra-kendra is from 0° to 180°. Add if from 180° to 360°. स्फुटयित्वैवं मन्दं मध्याच्च विशोध्य तस्य भुजम् । परिणाम्य कार्मुकार्धं तन्मन्देनैव धनहानी ॥ ७ ॥ Second step 7. Half rectifying the apogee position thus, deduct it from the mean. The result is to be used as the anomaly of the apsis in the second step. As we find the bhuja of the anomaly of conjunction (śīghra-kendra) so find the bhuja of the anomaly of apsis. Multiply the bhuja by the manda epicycle and divide by 360. and get the transformed bhuja-result of the apsis. (This is sine equation of the 4.5. Quoted by Utpala on BS, 2. pp. 44-45 d. A1. षड्भ्याः; B.C.D.U. षड्भ्यः A.B. पतते 4a. U. मध्यविहीनाद् 5b. A. गुणैर्विपगते तच्च; B. गुणे वियुगतक्ष; b. A. वाशित्रितये A.B. गतैष्व (B2.3. ष्य) C.D. गुणैर्विप [रि] णते तच्च (D. णते ते ततश्च) A.B. ॰दंशे ज्ये; U. ॰दंशज्या d. B. कर्कादौ A.B. चयापचयाः; U. चयापचयः c. B. कोटि ———————————————————————————————————————————————————————————————— ¹ In modern usage, for all the above we can simply say sin. śīghra-kendra is the bhuja and cos. śīghra-kendra is the koṭi, without taking into account the sign + or −.)

XVII.9 XVII. SAURA: TRUE PLANETS 303 centre). Find its arc-sine. Add half this arc to the half rectified longitude of apogee if the anomaly of apsis is from 0° to 180° and subtract if 180° to 360°. Thus the apogee is rectified completely. मध्यात् पु(न)र्विशोध्य (त)स्माद्बा[हुर्न] तस्य यच्चापम् । तन्मध्यमे क्षयधनं कर्तव्यं मन्दकेन्द्रवशात् ॥ ८ ॥ Third step 8. Substract this rectified apogee from the mean and thus get the anomaly of apsis. Find its bhuja and multiply it by the epicycle of the apsis and divide by 360°. The bhuja-result, (this is the equation of the centre), is got. Find the arc- sine of this, and subtract the whole of this arc from the mean if the anomaly of apsis is from 0° to 180°, and add it from 180° to 360°. The result is rectified mean. एवं स्फुटमध्याख्यं शीघ्रात् संशोध्य पूर्वविधिनैव । आदिवदा(प्तं) चापं स्फुटमध्या(ख्ये) चयापच(यम्) ॥ ९ ॥ Fourth step 9. Deduct the rectified mean from the śīghra. The anomaly of conjunction is got. Find the bhuja and koṭi of this in the same manner as we did in the first step. Multiply the bhuja by the epicycle of conjunction and divide by 360°. Sine anomaly of conj. is got. Multiply the koṭi, i.e., cos. anomaly of conjunction, by the epicycle of conj. and divide by 360°. The related cosine is got. Add this to 120 if the anomaly is from 270° to 90° and subtract from 120 if from 90° to 270°. Square this, add the square of the bhuja (i.e. equation of conjunction) and find the square root. Divide the equation of conj. × 120 by this square root. The arc sine of this is the result. Add this result to the rectified mean if the anomaly of conj. is from 0° to 180°. Subtract otherwise. The geocentric true planet is got. 6a. A1. तद्भुज; B. तद्भुजयौग             d. B. धनहानिः; C.D. धनहानि b. A. भाजयेन्नभू (A2. भु) जखं;             B 1.2.3. repeat the verse twice and   B. भाजयेन्नभुजखं; C. विभजेद् भुजं फलं खं;       give them two consecutive verse   D. भाजयेत्ततो भुजं ख               numbers.   B. सूर्यघ्न्यः (B2. घ्नः)              8a. B. मध्यासुरो A. पुरो विशोध्य; C.D. पुनर्विशोध्यः c. A.B. तज्झापा (B. या) र्धं             b. A.B.C.D. तस्माद् बाहुं न d. B. शीघ्रं B. वशातात्               9a. a. मध्याख्यां; C. मध्याख्यान् 7a. B. स्फुटयि (B1. पि, B3. यी) त्वैव मन्द          b. D. संशोध्यं b. C. विशोधितस्य; D. विशोध्यं तस्य           c. A.B.D. आदिवदाप्ते B. चाल्पं c. B2. परिणाम B3. कामुकार्धं             d. A.B. मध्याख्योप (B. च) चयापचयः (B. पापचयः)

304 PAÑCASIDDHĀNTIKĀ XVII.9 Note 1. In verse 4, I follow TS's reading, except that I have emended ṣaḍbhyaḥ into ṣaḍbhiḥ, instead TS's ṣaḍbhyaḥ, because my reading allows us subtraction or addition, as is wanted. In verse 5, I follow TS's except in the second foot, where I give the Bṛhatsaṃhitā reading. Either reading gives the same sense. In verse 6, I follow TS's except in the second foot, where I have given bhājyam for vibhajet, as being more likely. But the meaning is the same. In verse 7, the text required no emending, and TS's dhanabāni is unnecessary. In verse 8, like TS I have corrected puro into punaḥ but I have also corrected bāhum into bāhur which is required by grammar. In verse 9, I have corrected madhyākhyām into madhyākhyam, which is the reading of some of the manuscripts. Otherwise I follow TS. Note 2. VM, here, as elsewhere in the PS, uses his tabular sine where R is 120', as given in chap IV. So we must use his tabular values to get the R sines and R cosines. Of course, we may use the modern table, or the Siddhāntic table with R = 3438. But then the R, 1 for the modern tables, and 3438 for the Siddhāntic tables is to be used instead of 120' which is instructed here. (VM uses bhuja to mean sine, and koṭi to mean cosine, instead using the word jyā). Note 3. The method is the same as what is found in the Later Sūrya Siddhānta, with some changes for convenience. But in the matter of the number or order of the steps, the Āryabhaṭīya and the Siddhānta-Śiromaṇi differ. This is because, correctly speaking, the first two steps are useless, and the last two steps alone are necessary. In essence, the third serves to get the true heliocentric position, and the fourth to convert the heliocentric position into geocentric. The earlier steps are in the fond hope of getting correct positions agreeing with observation, while the real trouble is in the inexact parameters followed by the Siddhāntas. Note 4. The second and third steps are merely akin to finding the equation of the centre and applying to the mean. The first and fourth steps are conversion of heliocentric to geocentric posi- tions, neglecting the latitude, which is small and does not affect the result much. The work can be illustrated thus: Epicycle of Conjunction Planet Bhuja Anomaly of Conjunction Q Koti Sun East <— 120' Chapter XVII. Fig. 1