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

INTRODUCTION XXVII the author’s intention and become blind to its merits and peculiarities, which otherwise they could have easily seen. Equally off the mark is the emendation of tithi into stvatha by Neugebauer and Pingree (NP for short). See below, Explanatory Notes, for the real reason for this ‘slipping away from the real’.” On this subject might be advantageously referred to the section ‘The place of the Vāsiṣṭha in the his- tory of Hindu astronomy’ in T.S.K. Sastry’s paper ‘The Vāsiṣṭha Sun and Moon’ in the Jl. of Or. Research, Madras, 25 (1955-56) 19-41, reprinted in his Collected Papers on Jyotiṣa (Tirupati, 1989, pp.1-28) Table of Common topics Ahargaṇa Vāsiṣṭha II.8 Paulīśa I.11-13 Paitāmaha XII.5 Romaka I.8-10 Paitāmaha XII.1-2 Vyatīpāta-Vaidhṛti Paulīśa III.20-22 Nakṣatra-Tithi Paitāmaha XII.4 Paulīśa III.16 Candra-grahaṇa Vāsiṣṭha II.7 Vāsiṣṭha-Paulīśa VI.1-14 Ravi-sphuṭa Saura X.1-7 Paulīśa III.1-3 Ravi-grahaṇa Vāsiṣṭha II.1 Paulīśa VII.1-6 Candra-sphuṭa Romaka VIII.1-18 Paulīśa III.4 Saura IX.1-27 Vāsiṣṭha II.2-6 Grahāstodaya Aharmāna Vāsiṣṭha-Paulīśa XVIII.1-60 Paulīśa III.11-12 Saura XVII.11-12 12. Varāhamihira : His Life and Works One of the foremost early Indian astronomer and astrologer, Varāhamihira belongs to the sixth century A.D. In the Pañcasiddhāntikā (I.8) he takes the cut-off date or epoch for computations using the Paulīśa Siddhānta as Śaka 427, which corresponds to A.D. 505. Since the practice in Indian astronomical manuals (Karaṇa-grantha-s) is to take a contemporary date, as near to the composition of work, answering to certain specifications, as the cut-off date, it is reasonable to presume that PS was composed some time after A.D. 505. Regarding his demise there is a statement by Āmarāja in his commentary on Brahmagupta’s Brāhmasphuṭasiddhānta, which reads : navādhika-pañcasamkhyā- śake Varāhamihirācāryo divam gataḥ. “In śaka 509 Varāhamihira attained to the heavens.’ This would mean that VM passed away in A.D. 587. This date is corroborated by VM’s mention in PS XV.10, of Āryabhaṭa who composed his Āryabhaṭīya in A.D. 499, which work should have become well known by the time that VM composed his PS.

XXVIII PAÑCASIDDHĀNTIKĀ Personal details about VM are forthcoming from his own writings as also from those of others. Towards the close of his Bṛhajjātaka, VM says: Ādityadāsa-tanayas tadavāptabodhaḥ Kāpittakaḥ savitṛlabdhavaraprasādaḥ | Avantiko munimatāny avalokya samyag Horām Varāhamihiro ruciram cakāra || (Br.J. 26.1 (Edn. Triv. Skt. Ser., No.91) Thus VM was the son of Ādityadāsa; he learnt the śāstra from his own father; his native place was Kāpitthaka; he was blessed by Lord Sun (at Kāpitthaka). He (later) resided at Avanti (Ujjain) where he composed his work Horā (Bṛhajjātaka). On Kapitthaka VM's commentator Utpala says : Kāpitthākhye grāme yo 'sau bhagavān savitā sūryaḥ, tasmāt labdhaḥ prāpto varaḥ prasādo yena ('In the vil- lage of Kāpittha where he received the blessing of God Sun'). Kāpitthaka, the native village of VM, has been identified by Ajay Mitra Shastri (vide his India as seen by Varāhamihira, (MLBD, Delhi, 1969, p.19) on the basis of the mention thereof by 7th century Chinese traveller Yuan Chwang with Kāpittha populary known as "Saṅkāśya (modern Sankisa) in the Farrukhabad district of Uttar Pradesh...." Utpala states in his commentary on VM's Bṛhatsaṁhitā (I.1) that VM was a 'Magadhadvija' : tad ayam apy Āvantyakācāryo Magadhadvija-Varāhamihiraḥ arkalabdhavaraprasādo jyotiśśāstrasaṅgrahakṛt, (Ed. Sarasvati Bhavana Granthamālā, Varanasi, 1968, p.2). Utpala makes such a statement also in his commentary on VM's Yogayātrā. This and the surname 'Mihira' which is borne by many Śāka- dvīpa brāhmaṇas, who are worshippers of the Sun, would indicate that VM belonged to this clan of Brāhmaṇas whose forefathers migrated to India from the Maga country in Persia and settled in the village of Kāpitthaka whence VM came to the city of Ujjain where he wrote his works. By all accounts, Varāhamihira had the Sun as his tutelary deity. To quote A.M. Shastri (op. cit., pp. 20-21) : 'That Varāhamihira was a devotee of the Sun admits of no doubt. His father's name was Ādityadāsa, his own name-ending 'Mihira', derived from 'Mithra', the Iranian Sun-god, his obtaining a boon from the Sun, his obeisance to the Sun in all his works except the Vivāhapaṭala, (which, appropriately enough, opens with an invocation to Kāma, the Indian god of love), and his devoting a comparatively larger number of verses to the description of Sūrya icons, all indicate that the sun was his family deity. His son Pṛthuyaśas also invokes the Sun in the opening verse of his Ṣaṭpañcāśikā. As we have seen, Varāhamihira was regarded as an incarnation of the Sun.' The fame of VM has given rise to several legends about his birth and incidents in his life, includ- ing his being a courtier of King Vikrama and one of the nine gems (nava-ratnas) in his court. All these have to be considered as more fable and eological, not based on facts. Varāhamihira was an astute astronomer and astrologer and wrote extensively on all the three branches of the science, viz., Tantra or mathematical astronomy, Horā (Jātaka) or horoscopy, and Saṁhitā or mundane or natural astrology. It is interesting that for all his major works, VM has prepared abridged versions also for the benefit of those who desist from works which are too lengthy, who, as Utpala says, are vistaragrantha-bhīru-s.

INTRODUCTION <span style="float: right;">XXIX</span> On Tantra the major work of VM is the Pañcasiddhāntikā in eighteen chapters. It would seem from a statement of Utpala towards the beginning of his commentary on VM’s Laghujātaka that VM had prepared an abridgement also of that work Cf. Varāhamihira ..... jyotiśśāstrasaṅgrahaṁ kṛtvā tadeva vistara-grantha-bhīrūṇāṁ kṛte’ ‘saṅkṣiptaṁ gaṇitaśāstraṁ’ kṛtvā horāśāstraṁ vaktukāmaḥ etc. On horoscopy VM has produced two works, the Bṛhajjātaka, called also Horāśāstra in 26 chapters, and its abridged version, the Laghujātaka, called also Svalpa-jātaka and Sūkṣmajātaka, in thirteen chapters. On natural or mundane astrology also VM has two works, the Bṛhatsaṁhitā called also Vārāhī- saṁhitā in 106 chapters and Samāsasaṁhitā, known also as Laghu-saṁhitā and Svalpa-saṁhitā known through quotations. These are works of an encyclopaedic nature, dealing with astrological and many other subjects of human interest, such as architecture and iconography, water divining, omens, cosmetics, horticulture, characteristics of animals, gemmology, weapons, species of men and women and their qualities and the like. A wide range of information on the geography of India and its people is also to be round in the Bṛhatsaṁhitā. Vaṭakaṇikā, which exists only in the form of quotations, is a work of VM on omens. On military astrology, three works of VM are available; (i) Mahāyātrā, known also as Bṛhadyātrā, Bṛhadyogayātrā, Yakṣyeśvamedhikāh-yatrā (based in the commencing expression yakṣye 'svamedhena vijitya in the second verse of the work), (ii) Svalpayātrā or Ṭikaṇikāyātrā, and (iii) Yogayātrā. On marital horoscopy, VM has written a work entitled Vivāhapaṭala, and according to Utpala, there is also a Svalpavivāhapaṭala (vide. his com. on Bṛhajjātaka XX. 10). More than 30 more texts are mentioned, in manuscripts and elsewhere, to have been composed by VM (Cf. A.M. Shastri, op. cit., pp. 29-31) but these lack authenticity in their ascription. Alongside his wide range of scholarship, VM’s writings are also characterised by chaste language, brevity and linguistic elegance. He is a master not only of expression but also of metre. In illustra- tion of his poetic talents one might refer to the figures of speech expressed through verses XIX. 13-15 of the Bṛhatsaṁhitā describing Agastyodaya, the rising of the star Agastya. In the same work, Bṛhatsaṁhitā, he utilises the entire chapter 104, containing 64 verses, the Gocarādhyāya (‘Transits of planets’), to illustrate the metres, including the daṇḍaka-s, alongside depicting the subject proper. It is also instructive that the names of the several metres are also deftly incorporated in the verses by means of śleṣa or double entandre. Utpala is not, perhaps off the mark when he says, towards the commencement of his commentary on Bṛhatsaṁhitā, extolling VM as an incarnation of the Sun : Yac cāstram savitā cakāra vipulam skandhatrayair jyautiṣam tasyocchittibhayāt punaḥ kaliyuge saṁsṛjya yo bhūtalam | bhūyaḥ svalpataram Varāhamihira-vyājena sarvam vyadhād ittham yaṁ pravadanti mokṣakuśalās tasmai namo bhāsvate ‘The science of Jyautiṣa in its triple aspects (of Tantra, Jātaka, and Saṁhitā) was propounded at length by God Sun. Fearing that it would be lost in the Kali age, God Sun incarnated in the world in the form of Varāhamihira and expounded all (the said three aspects) again in shorter form. So say about the Sun those who are knowledgeable about salvation. Obeisance to that Sun.’ Madras January 4, 1993 <span style="float: right;">K.V. SARMA</span>

पञ्चसिद्धान्तिका PAÑCASIDDHĀNTIKĀ

Chapter One INTRODUCTION OF THE WORK १. प्रथमोऽध्यायः करणावतारः [ग्रन्थोद्देशः] दिनकरवसिष्ठपूर्वान् विविधमुनीन्द्रान् प्रणम्य भक्त्यादौ । जनकं गुरुं च शास्त्रे येनास्मिन् नः कृतो बोधः ॥ १ ॥ पूर्वाचार्यमतेभ्यो [यद्] यच्छ्रेष्ठं लघु स्फुटं बीजम् । तत्तदिहावि [क]लमहं रहस्यमभ्युद्यतो वक्तुम् ॥ २ ॥ Aim of the Work 1-2. After saluting, at the outset, with great devotion, the various great sages like Sūrya, Vasiṣṭha, and others, and my father and teacher who taught me this śāstra, I shall state in full the best of the secret lore of astronomy extracted from the different schools of the ancient teachers so as to be easy and clear. Mss used A₁. BORI, Poona, Ms No. 338/1879-80; A₂. National Library, Calcutta, Ms No. 49. B₁. BORI, Ms No. 37/1874-75; B₂. Or. Inst., Baroda, Ms No. 7165; B₃. National Library, Ms No. 64. C. Readings/Emendations in the edn. of PS by Thibaut-Sudhakara Dvivedi, Varanasi, 1889; Rep. 1938, 1968. D. Readings/Emendations in the edition of PS by Neugebauer - Pingree, Munksgaard, 1970. External Testimonia (E) Jy. PS Quotations in the Jyotirmīmāṃsā of Nīlakaṇṭha Somayājī M. Quotations in Makkibhaṭṭa's Com. on the Siddhāntaśekhara of Śrīpati N. Quotations in Nīlakaṇṭha Somayājī's Com. on the Āryabhaṭīya Pa. Quotations in Parameśvara's Com. on the Āryabhaṭīya Pr. Quotations in Pṛthūdaka's Com. on the Brāhmasphuṭasiddhānta S. Quotations in Sūryadevayajvan's Com. on the Āryabhaṭīya U. Quotations in Utpala's Com. on the Bṛhatsaṃhitā. (BS). A1.2. Begin with: श्री रामचाय नमः B1. Begins: पञ्चसिद्धान्तिका । ; B2. Begins: श्रीगणेशाय नमः । अथ श्रीपञ्चसिद्धान्तिका लिख्यते । B3. Beginning lost. C. Begins: [श्रीः । अथ पञ्चसिद्धान्तिका वराहमिहिरकृताऽऽरभ्यते । ] श्री रामच[न्द्र]ाय नमः । D. Begins: [श्रीवराहमिहिरविरचिता पञ्चसिद्धान्तिका प्रारभ्यते । ] श्री रामच[न्द्र]ाय नमः ।

1. Quoted by Utpala on BS 2.2 1a. B1.वशिष्ट; B2.वशिष्ठ b. B1.मुनिन्द्रा; U.मुनीन् भावतः प्रणम्यादौ 2b. A1.2.B.1.3 ०भ्यो यद्दिष्ट लघु c. A1..2. तत्कदिहाविलमहं; B1.2. तत्तदिहाखिलमहं

4 PAÑCASIDDHĀNTIKĀ 1.2 In the second verse, the letter ka has been added to supply the one syllable wanting and in keeping with the sense. The Sun being the Ātman of the Universe and also the chief of the grahas, all the gods and all the grahas are propitiated by His worship. By the expression 'various great sages' the author means the eighteen primary authors of the Siddhāntas on astronomy, viz. Sūrya, Soma, Pitāmaha, Vasiṣṭha, Atri, Parāśara, Kāśyapa, Nārada, Gārgya, Marīci, Manu, Aṅgiras, Romaśa, Paurukutsa, Cyavana, Yavana, Bhṛgu and Śaunaka and, by saluting these, the author salutes all ancient authors who follow these Siddhāntas. [पञ्च सिद्धान्ताः] पौलिश-रोमक-वासिष्ठ-सौर-पैतामहास्तु पञ्च सिद्धान्ताः । पञ्चभ्यो द्वावाद्यौ व्याख्यातौ लाटदेवेन ॥ ३ ॥ The Five Schools of Astronomy 3. The five Siddhāntas, of which this work is a compendium, are the Paulīśa, the Romaka, the Vāsiṣṭha, the Saura and the Paitāmaha. Of these five, the first two, viz., the Paulīśa and the Romaka have been commented upon by Lāṭadeva. Of the Siddhāntas here mentioned, Brahmā is the author of the Paitāmaha; Vāsiṣṭha, that of Vāsiṣṭha; Paulīśa that of Paulīśa; Romaka that of Romaka; and Sūrya, that of Saura. From a dialogue between Sūrya and Aruṇa, it can be learnt how these five Siddhāntas were given to their respective recipients. According to tradition, at the first instance, Brahmā saw this lore of astronomy embedded in the Vedas and extracted it in the form of the Paitāmaha. He taught this to his son, Vasiṣṭha at the behest of Viṣṇu and again to Sūrya who was created with the express purpose of giving Time to the Universe. Vasiṣṭha gave this lore to his son, Parāśara who, in turn gave the Parāśara Siddhānta to the sages. One sage, Paulīśa taught this to the sages Garga etc. and this is the Paulīśa Siddhānta. Sūrya himself, being born among the Yavanas by the curse of Brahmā, taught the science to Romaka and Duryavana in the city of Romaka, and Romaka propounded it as the Romaka Siddhānta. Thus these five Siddhāntas are the most ancient. It is to be noted here that the five Siddhāntas used by the author in his work are all different from works of the same name current at present and it seems they have been lost to us. The Paulīśa used by the author is different from the Paulīśa quoted by Bhaṭṭotpala in his commentary on the Bṛhatsaṁhitā which latter agrees with the Saura of our author and disagrees with his Paulīśa. The Romaka and Vāsiṣṭha now extant are different from those of VM, agreeing as they do with the now well-known Sūrya Siddhānta, (called by scholars as the 'Modern' or 'Later' Sūrya Siddhānta to distin- guish it from the ancient Saura Siddhānta). The author's Saura does not agree with the 'Modern' Sūrya Siddhānta, though one would expect agreement from the similarity in name, but it agrees with a work of the ancient Āryabhaṭa, now lost to us, and called by his commentator Bhāskara I as the Ārdharātra-Pakṣa, which again is the basis of the Khaṇḍa-khādyaka-karaṇa of Brahmagupta. As for 3. Quoted in the Jyotirmīmāṁsā (Jy) | B1.2. वाशिष्ट; A1.2. वासिष्ट of Nīlakaṇṭha, p. 7. | b. A1. पैतामहास्तु a. A1. पौलिश | c. B1.2. पञ्चेभ्यो B1. रोमय्; Jy. रोमश | A1. द्वावाद्यो; A2. द्वावोद्यौ; Jy. द्वावन्त्यौ

I.3 I. INTRODUCTION OF THE WORK 5 the Paitāmaha, there are several works now extant claiming Pitāmaha or Brahmā for their author. One is the Brahma Siddhānta given by Brahmā to Nārada, which follows the ‘Modern’ Sūrya Siddhānta in its constants. Another is the Pitāmaha Siddhānta, forming a part of Viṣṇudharmottara, which has been taken by Brahmagupta as the basis of his Brāhma-Sphuṭa Siddhānta. A third one, now lost, is the basis of the Āryabhaṭīya. But the Paitāmaha of our author is different from all these. As for the Lāṭadeva mentioned here, he is the Lāṭācārya referred to in XV.18 of this work, for, there, the author says that this Ācārya has taken sunset at Yavanapura as the beginning of the day and from I.8. we understand that the Paulīśa and Romaka do the same and here it is mentioned that Lāṭadeva is the commentator of these two Siddhāntas. पौलिशतिथिः स्फुटोऽसौ तस्यासन्नस्तु रोमकप्रोक्तः । स्पष्टतरः सावित्रः परिशेषौ दूरविभ्रष्टौ ॥ ४ ॥ 4. The tithi resulting from the Paulīśa is tolerably accurate and that of the Romaka approximate to that. The tithi of the Saura is very accurate. But that of the remaining two (viz. the Vāsiṣṭha and the Paitāmaha) have slipped far away (from the real). The five Siddhāntas are compared here with reference to their tithi alone because that is the chief of the five aṅgas, viz. tithi, vāra, nakṣatra, yoga and karaṇa, that is most useful not only for religious but also civil purposes, that is independent of the origin of reference in the ecliptic and can be examined for correctness by observation of eclipses and heliacal rising and that is used in finding the days from Epoch, the sine qua non of all astronomical computation. This being the case, the change of ‘tithi’ into ‘kṛta’ by the late Dr. G. Thibaut and M.M. Sudhakara Dvivedi (TS for short), especially when the manuscripts read only tithi or tithaḥ, is unwarranted, to say the least. Doing this, they have condemned the Vāsiṣṭha Siddhānta beyond the author's intention and become blind to its merits and peculiarities, which otherwise they could easily have seen. Equally off the mark is the emendation of tithi into stvatha by Neugebauer and Pingree (NP, for short). See below, explanatory Notes, for the real reason for this ‘slipping far away from the real’. [प्रतिपाद्यवस्तु] यत्तत्परं रहस्यं भ्रमति मतिर्यत्र तंत्रकाराणाम् । तदहमपहाय मत्सरमस्मिन् वक्ष्ये ग्रहं भानोः ॥ ५ ॥ दिक्स्थितिविमर्दकर्णप्रमाणवेला ग्रहाग्रहाविन्दोः । ताराग्रहसंयोगं देशान्तरसाधनं चास्मिन् ॥ ६ ॥ 4. Quoted in the Jyotirmīmāṁsā (Jy) P.7. b. A1. ०सन्नस्तु a. A1.2. पौलिशतिथि स्फु०; B1.2. पौलिशतिथः; A1.2. रोमकः; Jy. रोमशः C. पौलिश [कृत]; c. B1.2. ०तरस्सावित्रः; D. ०तरः सवित्रः D. पौलिश [स्त्वथ]; Jy. पौलिश इति d. A1. दूरं A.2. स्फुटोद्यौ A1.2. ०विभ्रष्टौ

6 PAÑCASIDDHĀNTIKĀ I.7 सममण्डलचन्द्रोदययंत्रच्छेद्यानि (शाङ्कवच्छायाः) | उपकरणाद्यक्षज्यावलम्बकापक्रमाद्यानि ॥ ७ ॥ Contents of the Work 5-7. I shall tell in this work, avoiding all jealousy, the computation of the solar eclipse, which is guarded as a great secret and in which the mind of the astronomer reels. I shall also tell the occurrence or non-occurrence of the lunar eclipse, the directions of the first and last contacts, the duration, the total phase, the 'hypotenuse' at any moment with related quantity of obscuration and time and also the mutual conjunctions of the stars and the planets and the computation of differences in longitude as also the prime vertical, moonrise, astronomical instruments and other requirements, graphical representations, the gnomonic shadow, the sines of latitude, co-latitude and declinations and such other matters. The textual recording tāḍavacchāyā is emended as śāṅkavacchāyāḥ because (i) tāḍava is meaning- less, and it may be a corruption of śāṅkava meaning relating to the śaṅku or gnomon which is suggested by the juxtaposition with chāyā meaning 'shadow' and (ii) chāyā must be chāyāḥ because grammer requires the accusative case of the word. TS, NP take the word as śāṅkavacchāyā, without the final visarga. As for the mention of the computation of the solar eclipse as a 'great secret' it is because of the difficulty of the computation which, therefore, would bring honour to a person who can do it and for that reason not given to all. From the contents we can see the importance of the work for religious purposes. The technical words that occur here like prime vertical etc. will be explained in their respective contexts. [रोमकसिद्धान्तानुसारी अहर्गणः] 'सप्ताश्विवेद'संख्यं शककालमपास्य चैत्रशुक्लादौ | अर्धास्तमिते भानौ यवनपुरे सोमदिवसाद्यः || ८ || मासीकृते समासे (द्विष्ठे) सप्ताहतेऽष्टयम (पक्षैः' ) | लब्धैर्युतोऽधिमासैस्त्रिंशद्घ्नस्तिथियुतो द्विष्ठः || ९ || 5a. B1. यत्तत्त्वैः; B2. यतस्त्वै 7b. B1. यन्त्र०; B2. कर्षप्र० c. B1. मक्षर० A2. छेद्यानि d. A1. वक्षे; B1.2. वह्नये A1. तावच्छाया; A2. B2. ताडवच्छाया; 6a. B1. दिक् संस्थिति; B2. दिक् मस्थि B1. ताष्टवच्छाया; C.D. शाङ्कवच्छाया c. A2. ताराग्नं योगं e. B1.2. उपकरणा० d. A1.2. सावनं D. ०ण्यान्यक्षन्या d. B1.2. ०वलम्बपक्रमा०

I.10 I. INTRODUCTION OF THE WORK 7 ‘रुद्र’-घ्नः स‘मनुशरो’ लब्धोन‘गुणखसप्त’भिर्द्युगणः | रोमकसिद्धान्तेऽयं नातिचिरे पौलिशेऽप्येवम् || १० || Days from Epoch according to Romaka 8-10. Deduct 427 from the Śaka year (elapsed) of the time taken. Multiply the remainder by 12. Add the months gone, counting from Caitra. Put this result in two places. In one place, multiply it by 7, divide by 228 and take the quotient which constitute the intercalary months. Add this to the result kept in the second place. (The total are the synodic months gone.) Multiply this by 30 and add the tithis counted from śukla-pratipad to the current tithi. Put the sum in two places. In one place multiply by 11, add 514, divide by 703 and take the quotient, (which constitute the elided days or avamas). Deduct this from the sum put in the other place. The remainder are the ‘Days from Epoch’ (dyugaṇa), the moment of Epoch being mid-sunset at Yavanapura, beginning Monday when the first tithi of Caitra was about to begin. This rule is according to the Romaka. It can be taken as the Pauliśa rule also, provided the time taken for computation is not very far from the Epoch. (or the part of the rule for avama may be used for the Pauliśa also, provided the taken date is not very far from the Epoch; or in the Pauliśa too the movement of Epoch is mid-sunset at Yavanapura, beginning Monday.) Example 1 . Find the Days from Epoch for Tuesday the sixth day of the dark fortnight of Āṣāḍha, Śaka 499 (elapsed). Śaka year (elapsed) of date is 499. 499 − 427 = 72 years gone. Months gone = 72 × 12 + months counted from Caitra upto Āṣāḍha = 72 × 12 + 3 = 867. 867 × 7 ÷ 228 = 6069 ÷ 228 = 26 867 141 (= Q) + ───── (= Rem) 26 228 Adding the quotient ──── Synodic months gone 893 The tithis = 893 × 30 + tithis in the current month = 893 × 30 + 21 = 26,811 (26,811 × 11 + 514) ÷ 703 = 420 (= Q) + 175/703 (= Rem) 26,811 Deducting the quotient 420 ───── Days from Epoch gone 26,391 8-10. Quoted by Utpala on BS 2. (p.30) c. B1. युतो त्रिमासैः; B2. युतो द्विमासैः 8c. A2. अद्धस्त d. A1.2. द्विष्ठः; B1.2. द्विस्थः; U. धःस्थः d. A1.2. सौम्य; B1.2. भौम्य; D. भौम A1. ॰शघ्र॰ A1.2. C. D. U. दिवसाद्ये 10b. B1.2. लब्ध्योनो B1. यो नाति 9a. A2. माग्नीकृते A2. समाग्ने c. A1.2. सिद्धांतोयं A2. संदाभि b. A1.2. द्विष्ठे; B1.2. द्विस्थे A1.2. पक्षै; B1.3. पक्ष्यैः

8 PAÑCASIDDHĀNTIKĀ I. 10 Dividing 26,391 out by 7, the remainder got is 1, i.e. Monday has gone and Tuesday has begun. (This agrees with the data given and therefore 26,391 are the required days from Epoch.) The rule is thus explained: According to the Romaka, in a yuga containing 2850 solar years, there are 1050 intercalary months and 16,547 elided days (vide I.15). From this we can compute that in the yuga there are 34,200 solar months, 35,250 synodic months (i.e. months), 10,57,500 tithis and 10,40,953 civil days (vide I. 17). Now, because the Epoch is 427 Śaka (elapsed), by deducting 427 from the Śaka year (elapsed) of the time taken, the years gone at the taken time from the Epoch is got. As there are 12 solar months in a year, the years gone × 12 + the months gone upto the time taken = the solar months gone from Epoch to the end of the solar month falling in the current month. The intercalary months during this period is obtained by proportion from the solar months and the intercalary months of the yuga, viz. 34,200: 1050 :: the solar months gone: the intercalary months during the period. Thus we have the equation, the intercalary months = the solar months gone × 1050 ÷ 34,200. The fraction 1050/34,200 reduces to 7/228 which represents the author's instruction to multiply by 7 and divide by 228 to get the intercalary months. It should be noted that we are finding the intercalary months not upto the taken time but upto the end of the solar months falling in the current month, for, logically, the third member of the proportion should be solar months as the first member is the solar months of the yuga. The number of months gone from Caitra upto the time taken is the same as the solar months ending in or before the current month, and therefore, we use it for adding to years gone × 12, to get the solar months gone. From this we can understand that in counting the months from Caitra we should not reckon any intercalary month that has fallen. Note also that the fraction of intercalary month obtained from the proportion is the part of the current synodic month from pratipad upto the end of the solar month and by omitting it, we have found the intercalary months gone before the taken time which is the thing wanted. The rule for 'Days from Epoch' does not mention any constant (kṣepa) to be added to the intercalary month obtained because at the time of Epoch there is practically no fraction of intercalary month. We shall now show how it is practically zero. Even though we do not know the time when the Romaka Yuga began, wherefrom the fraction required can be obtained, still from the constant for the mean Sun and Moon in Chapter VIII we can obtain this, in the following manner. There, in the first verse giving the rule for the mean Sun, 150 is mentioned as the multiplier for the Days from Epoch, and 65 is given as the subtractive constant. From this we learn that 65/150 days, (i.e. 26 nāḍikās) after Epoch, the mean solar month ends and therefore at Epoch the mean Sun is 11ʳ 29° 34' 30". Again, from the constants in the fourth verse giving the mean Moon, we can learn that the mean Moon at Epoch is 11ʳ 26° 12'. From these, it can be computed that the mean new moon occurs about 16½ nāḍikās after Epoch. As the interval from new moon to the end of the solar month is the fraction of intercalary month, we get 26 – 16½ = 9½ nāḍikās, as the fraction. As for one intercalary month consisting of about 29½ days there are 228 parts as constant, for 9½ nāḍikās we get 1 as constant. This is omitted as being negligible, because, after all, we are going to use in the rule not the mean Caitra, etc. but the true Caitra etc. which can differ from the mean upto 36 nāḍikās. That is why if an intercalary month has actually fallen in the current year before the taken time, we take the fraction of the computed intercalary month as whole and add one, and if no inter- calary month has fallen we omit one from the computed months when the fraction left over is small. To continue, adding the intercalary months to the solar, the synodic months gone are got, for the intercalary months are the synodic months omitted in the one to one correspondence of the synodic months with the solar. Multiplying the total synodic months by 30 and adding the tithis in

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

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

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

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

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

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

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

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

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

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

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

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

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

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