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

INTRODUCTION XIX reason that they have mostly been documented by Pingree in his edition of the PS and so can be referred to therefrom.¹ 3. Presentation of the Text While the generally accepted conventions of critical editing of Indic texts are duly followed in the present edition, attention might be drawn to certain methodologies which are stressed herein, in view of the technical nature of the text, the defective nature of the manuscripts and the tentative- ness of many of the emendations and textual changes effected in the two earlier editions. i. On account of the inadequacies of the copyists of the parent manuscripts and also due to deficiencies in the parent manuscripts themselves, some emendations have to be done in the edited text. However special care has been taken in this edition to indicate such emendations by placing them within curved brackets; square brackets are used to enclose fillings of apparent omissions or newly suggested readings. Doubtful suggestions are marked by an interrogation mark.

1. Studies mentioned in this footnote have mostly been identified by Pingree and on pages 18-19 of his edition of PS and variants. i. G. Thibaut, 'Notes from Varāha Mihira's Pañcasiddhāntikā', Jl. of Asiatic Society of Bengal, 53 (1884) 259-93. ii. S.B. Dikshit, 'The Original Sūrya-siddhānta', Indian Antiquary, 19 (1890) 45-54. iii. S.B. Dikshit, 'The Romaka Siddhānta', Indian Antiquary, 19 (1890), 133-42. iv. S.B. Dikshit, 'The Pañcasiddhāntikā', Indian Antiquary, 19 (1890) 439-40. v. J. Burgess, 'The Romaka Siddhānta', Indian Antiquary, 19 (1890) 284-85. vi. J. Burgess, 'The sines and arcs in the Pañcasiddhāntikā', Indian Antiquary, 20 (1891) 228. vii. M.P. Kharegat, 'On the interpretation of certain passages in the Pañcha Siddhāntikā of Varāhamihira, an old his- torical work' Jl. of the Bombay Branch of the Royal Asiatic Society, 19 (1895-97) 109-41. viii. K.S. Shukla, 'On three stanzas from Pañcasiddhāntikā,' Gaṇita, 5 (1954) 129-36. T.S. Kuppanna Sastri ix. 'The Vāsiṣṭha Sun and Moon in Varāhamihira's Pañcasiddhāntikā', Jl. of Or. Research, Madras, 25 (1955-56) 19-41. x. 'Some misinterpretations and omissions in Thibaut and Sudhakara Dvivedi in the PS of VM', Vishweshvaranand Indological Journal, 11 (1973) 107-18. xi. 'The epoch of the Romaka Siddhānta in the PS and the epoch longitudes of the Sun and Moon the Vāsiṣṭha Siddhānta', Indian Jl. of Hist. of Science, 13 (1978) 151-58. xii. 'The Vāsiṣṭha-Pauliśa Venus in the PS of VM', Collected Papers of T.S. Kuppanna Sastry, Tirupati, 1989, pp. 141-47. xiii. 'The Vāsiṣṭha-Pauliśa Jupiter and Saturn in the PS', Collected Papers, pp. 148-68. xiv. 'The Vāsiṣṭha-Pauliśa Mars in the PS of VM', Collected Papers, pp. 169-87. xv. 'The epoch-constants of the Vāsiṣṭha-Pauliśa star-planets', Collected Papers, pp. 201-5. xvi. 'Saurasiddhānta of PS : Planetary constants and computation', Collected Papers, pp. 206-40. xvii. 'Pañcasiddhāntikā XVIII. 68-81 : An interpolation', Collected Papers, pp. 241-54.

XX PAÑCASIDDHĀNTIKĀ ii. Thibaut and Dvivedi who print their emended text in the right hand column of their edition do not specifically indicate their emendations and one has to identify the emendations oneself. Pin- gree indicates his emendations in many places by angular brackets but in many other places prints the emended text without any indication. In the present edition, all their emendations are iden- tified and noticed in the footnotes denoting them by the sigla C and D. When these emendations are accepted in the present edition also, they are not separately marked as above. This procedure is expected to enable the comparison between the emendations of TS and NP and those made in this edition and evaluate the merit and appropriateness between the two. In fact, it is felt that many of the emendations of TS and NP, especially of the latter, are often far-fetched, ungrammatical, offending the metre or failing to give a cogent sense. See, for example, the TS/NP emendations in I.23; IX.5; XI.2, 4, 5; XII.5a, 5d; XIII.38d, 41d: XVII.1, 12; XVIII.2d, 3, 19, 24, 25. iii. In order that a discerning student of the text shall have before him, in full, what occurs in the several manuscripts, an attempt has been made to record all variants, right or wrong. This has been done for three reasons : (a) Correct forms of corrupt passages can be visualized only if all the rea- dings, as found in the manuscripts, are before one’s eyes; (b) Then alone would it be possible for an editor to vindicate the emendations suggested by him; (c) The corrupt and apparently meaning- less readings in the manuscripts which the editor could not correct or has wrongly emended can be corrected or better emendations suggested by other scholars if all the variants are given. However, obvious errors of a purely scribal nature, like separate words written jointly, using anusvāra for anunāsika and vice versa, using double consonants for single consonants and vice versa, giving the benefit of doubt for a letter that could be read rightly or wrongly, have been cor- rected silently and not noticed in the footnotes. 4. Translation The translation provided in the edition is as literal as possible without sacrificing readability and not going against the English idiom. Elucidatory expressions and words which are understood in the context are added within brackets, the ultimate aim being to make the matter dealt with clear and fully understood. Topical headings have been provided to verses or groups of verses with the same objective, again, towards the above-said objective. 5. Notes The notes added are generally detailed and self-contained. There again, they seek to elucidate the verses and the underlying ideas, primarily from the Indian standpoint. Tables and geometrical diagrams are provided wherever warranted. Quite often, the need for the emendations suggested in the text is explained. The emendations made by TS and NP are also examined and observations offered. An aspect of the Notes which deserves special mention is the addition of self-suggested mathematical or astronomical problems and working them out according to methods enunciated in the verses, and also by employing modern methods. Introductions are prefixed for several chap- ters, towards setting out the significance of the contents of the respective chapters. 6. Division of Pañcasiddhāntikā The colophons of the several Sections of PS, as found in the manuscripts, which all go back to a defective archetype, are uneven. Some of the colophons merely mention the topic treated in the respective sections but some others designate the sections as adhyāya-s (chapters) and also indicate the numbers thereof. The several colophons read :

INTRODUCTION XXI

1. Karaṇāvatāraḥ
2. Nakṣatrādicchedaḥ
3. Iti Pauliśasiddhāntaḥ
4. Iti Karaṇādhyāyaś caturthaḥ
5. Iti Śaśidarśanam
6. Candragrahaṇe ṣaṣṭhodhyāyaḥ
7. Pauliśasiddhānte ravigrahaṇam
8. Iti Romakasiddhānte 'rkagrahaṇam aṣṭamo 'dhyāyaḥ
9. Iti Sūryasiddhānte 'rkagrahaṇam navamo 'dhyāyaḥ
10. Candragrahaṇam daśamo 'dhyāyaḥ
11. A(nu)varṇanam ekādaśo 'dhyāyaḥ
12. Iti Paitāmahasiddhānte dvādaśo 'dhyāyaḥ
13. Trailokyadarśanam nāma trayodaśo 'dhyāyaḥ
14. Iti Chedyakayantrāṇi caturdaśo 'dhyāyaḥ
15. Jyotiṣopaniṣat pañcadaśo 'dhyāyaḥ
16. Sūryasiddhānte madhyagatiḥ
17. Tārāgrahasphuṭīkaraṇam ṣoḍaśo (? saptadaśo) 'dhyāyaḥ
18. Pauliśasiddhānte tārāgrahāḥ However, whether there be full chapter headings and chapter numbers or there be only the men- tion of the topics in the colophons, the commencement of a new subject helps to ascertain the beginnings of the chapters. The omission of the specification of the chapter headings and numbers has to be ascribed to the imperfections in the original archetype. This helps us to correct Pingree's edition where Chs. XVI and XVII are taken as a single chapter, which he numbers as XVI. Now, after eleven verses, here, there occurs the colophon 'Sūryasid- dhānte madhyagatiḥ', which is the subject of those eleven verses. After still another fourteen verses occurs in the colophon 'Tārāgrahasphuṭīkaraṇam ṣoḍaśo 'dhyāyaḥ', and an entirely different subject is treated in those fourteen verses. Ignoring the radical difference in the subjects dealt with in the two sets of verses and the colophon after the first set of verses and guided merely by the colophon at the end of the second set of verses, Pingree combines the two sets of verses, 11 plus 14 = 25, to con- stitute ch. XVI and takes ch. XVIII of TS and of the present edition as ch. XVII. 7. Chapter XVIII of Pañcasiddhāntikā The theme of Ch. XVIII is the heliacal rising and setting of the star-planets according to the Vāsiṣṭha and Pauliśa schools. Venus, Jupiter, Saturn, Mars and Mercury are treated, in that order, in verses 1-56, and certain allied matters in verses 57-60. In verse 61 the author states that he, Var- āhamihira, hailing from Avanti, has composed the PS for the benefit of students. In the next verse, 62, he asks astronomers dissatisfied with Pradyumna and Vijayanandi, to resort to his work. Having thus completed his treatise, couched entirely in the Āryā metre, following a convention, adopted by Sanskrit writers, of concluding works by a closing colophonic verse couched in a different metre, VM breaks into a different metre, the Vasantatilakā, for the purpose : āvantyakaḥ samāsāc chiṣyahitārtham sphuṭāṅkasamam | cakre Varāhamihiras tārāgrahakārikātantram || 61 || Pradyumna-bhūmitanaye jīve saure 'tha Vijayanandikṛte | budhe ca bhagnotsāhaḥ sphuṭam idam karaṇam bhajatām || 62 ||

XXII PAÑCASIDDHĀNTIKĀ dṛṣṭam Varāhamihireṇa sukhaprabodham | ... ... ... ... ... ... ... ... ... ... || 63 || The manuscripts are defective here, omitting the next three lines of the concluding verse, some exhibiting also a gap. In continuation of the above, the manuscripts commence another short work with a maṅgala- śloka ('verse of salutation') and then a second verse expressing a pratijñā ('resolve') to compose a 'better work' by 'Varāhamihira himself.' prastāve 'pi na doṣān jānann api vakti yaḥ parokṣasya | prathayati guṇāṁś ca tasmai sujanāya namaḥ parahitāya || aṣṭādaśabhir baddhāny ā tārāgraham etad āryābhiḥ | varam iti Varāhamihiro dadāti nirmatsaraḥ karaṇam || These verses end abruptly without any closing colophonic verse, as might be expected. Not taking into consideration the fact that VM had closed the PS most formally with all attendant paraphernalia, with verse 63, both TS and NP treat this short work, as the concluding part of Pañ- casiddhāntikā and as a work of Varāhamihira himself, and translate it and explain it as such. It is not noticed by TS and NP that computation according to these verses can give only rough results since the equation of the centre has been dispensed with, which makes them valueless. Further, there are mistakes in the computation of Venus and Mercury, which one cannot expect to be committed by VM. The prose colophon 'Pauliśasiddhānte tārāgrahāḥ evam' ('Thus the star-planets of the Pauliśasiddhānta') does not have any reality behind it since the computations therefrom do not accord with those of the Pauliśasiddhānta elucidated earlier. These points have been discussed in detail in the Notes, below, to these verses. Obviously, these verses are apocryphal and are the handi- work of an inferior astronomer who has ascribed them to VM. For this reason, in the present edi- tion, Pañcasiddhāntikā is formally closed with verse 63 and the verses following given as a prakṣepa (interpolation) by someone else who has moreover blacked out in the original archetype the three lines in verse 63, being the concluding verse of PS. 8. The Five Siddhāntas in PS : Their Distribution It had been mentioned earlier that VM's treatment of the five siddhāntās in the PS are uneven. It is not that each Siddhāntā is taken up one by one and a résumé of the same given fully or sys- tematically. Select topics are taken up, apparently arbitrarily, and are dealt with individually or jointly when the enunciations of two schools are similar. Alongside, chapters are devoted also to general astronomical topics which are applicable to all the schools. The assortment of the treatment of the subjects in the several chapters are as indicated below. 9. Content Analysis of the Pañcasiddhāntikā Ch.I. 1-7 General Introduction 8-10 Romaka Days from epoch (Ahargaṇa) 11-13 Pauliśa Days from epoch (Ahargaṇa) 14-16 Saura-Romaka Yuga of Sun and Moon, Lords of the year, month, and horā.

INTRODUCTION XXIII Ch.I. 17-25 Romaka Lords of the days of the month Ch.II 1-13 Vāsiṣṭha 1 " True Sun (Ravi-sphuṭa) 2-6 " True Moon (Candra-sphuṭa) 7 " Nakṣatra computation 7 " Tithi computation 8 " Day-time (Aharmāna) 9-10 " Gnomonic shadow (Śaṅkucchāyā) 11-13 " Lagna Ch.III 1-37 Pauliśa 1-3 " True Sun (Ravi-sphuṭa) 4 " True motion of Moon (Candragati) 5 " Equation of the centre (Mandaphala) 6-9 " Sense obscure 10 " Cāra 11-12 " Day-time (Aharmāna) 13-14 " Deśāntara 15 " Local time (Iṣṭadeśakāla) 16 " Nakṣatra computation 17 " Sun's daily motion (Ravi-gati) 18-19 " Karaṇa-s 20-22 " Vyatīpāta and Vaidhṛti 23-24 " Ṣaḍaśīti-kāla 25 " Solstices (Ayana) 26 " Saṅkrānti-kāla 27 " Tridinaspṛk-yoga 28-29 " Rāhu (Node) 30-31 " Moon's latitude (Candra-vikṣepa) 32 " Bhadraviṣṇu, defect in 33 " Pādāditya, defect in 34-35 " Romaka, defect in 36-37 General Astronomy, importance of Ch.IV 1-58 General Problems of Time, Space and Direction (Tripraśna) 1-15 " Table of R sines (Jyāḥ) 16-18 " Declination of Sun and Moon (Krānti) 19-22 " Gnomonic shadow and derivatives (śaṅkucchāyā) 23-25 " Sine colatitude and Day-diameter (Lambajyā and Dinamāna) 26-28 " Cara and derivatives

XXIV PAÑCASIDDHĀNTIKĀ Ch. IV. 29-30 " Rt. ascensional difference (Laṅkodaya-rāśimāna) 31-34 " Rising of signs (Rāśyudaya) 35-36 " Gnomonic shadow (Śaṅkucchāyā) 37-38 " Astronomer's qualifications 39-40 " Sine amplitude (Agrā) 41-49 " Gnomonic shadow at required time (Iṣṭacchāyā) 50-51 " Moon's shadow (Candracchāyā) 52-54 " Directions from shadow (Diksādhana) 55-58 " Sun from gnomonic shadow (chāyātaḥ raviḥ) Ch. V 1-10 Paulīśa Moon's horns (Candraśṛṅgonnati) Ch. VI 1-14 Vāsiṣṭha and Lunar eclipse (Candragrahaṇam) Paulīśa Ch. VII 1-6 Paulīśa Solar eclipse (Ravi-grahaṇa) Ch. VIII 1-18 Romaka Solar Eclipse (Ravi-grahaṇa) Ch. IX 1-27 Saura Solar eclipse (Ravi-grahaṇa) Ch. X 1-7 Saura Lunar eclipse (Candra-grahaṇa) Ch. XI 1-6 General Eclipse diagram (Grahaṇaparilekhā) Ch. XII 1-6 Paitāmaha 1-2 " Days from epoch (Ahargaṇa) 3-4 " Tithi, Nakṣatra etc. 4 " Vyatīpāta 5 " Day-time (Aharmāna) Ch. XIII 1-42 General Situation of ther worlds (Trailokyasaṁsthānam) Ch. XIV 1-41 General Astronomical instruments (Chedyaka-yantrāṇi) Ch. XV 1-29 General Jyotiṣopaniṣat Ch. XVI 1-11 Saura Mean planets 1-9 " Mean planets (Graha-madhya) 10-11 " Bīja correction by VM

INTRODUCTION XXV Ch. XVII. 1-14 Saura True planets 1-11a " True planets (Graha-sphuṭa) 11b " Retrograde motion (Vakragati) 12 " Heliacal rising (Grahāstodaya) 13-14 " Planetary latitudes (Graha-vikṣepa) Ch.XVIII 1-60 Vāsiṣṭha-Pauliśa Heliacal rising and setting of planets (Grahāstodaya) 1-5 " Venus (Śukra) 6-13 " Jupiter (Guru) 14-20 " Saturn (Śani) 21-35 " Mars (Kuja) 36-56 " Mercury (Budha) 57-60 " Hints 61-63 General Conclusion of PS 64-81 Spurious Supplement True planets 64-66 " Introduction 67-69 " True Mars 70-72 " True Mercury 73-75 " True Jupiter 76-78 " True Venus 79-81 " True Saturn 10. Depiction of the Siddhāntas in PS The above Content Analysis of the PS would enable the identification of the extent of selective depiction of the different siddhāntas by VM in the work, as shown below. PAULIŚA-SIDDHĀNTA Ṣaḍaśītikāla III.23-24 Candra-vikṣepa III.30-31 Ahargaṇa I.11-13 Rāhu III.28-29 Nakṣatra III.16 Candra-grahaṇa VI.1-14 Mandaphala III.5 Ravi-grahaṇa VII.1-6 Ravigati III.17 Candraśr̥ṅgonnati V.1-10 Raviphuṭa III.1-3 Grahāstodaya XVIII.1-60 Candra-sphuṭa III.4 Ayana III.25 Saṅkrānti III.26 ROMAKA-SIDDHĀNTA Tridinaspr̥kyoga III.27 Deśāntara III.13-14 Ahargaṇa I.8-10 Cara III.10 Yuga I.14-16 Aharmāna III.11-12 Lord of the year I.17-18, 21 Iṣṭadeśakāla III.15 Lord of the month I.19, 21 Karaṇa III.18-19 Lord of the Horā I.20, 21 Vyatīpāta III.20-22 Lord of the Days I.23-25 Vaidhṛti III.20-22 Ravi-grahaṇa VIII.1-18

XXVI PAÑCASIDDHĀNTIKĀ SAURA-SIDDHĀNTA Candra-sphuṭa II.2-6 Nakṣatra II.7 Yuga I.14-16 Tithi II.7 Lord of the year I.17-18, 21 Aharmāna II.8 Lord of the month I.19, 21 Śaṅkucchāyā II.9-10 Lord of the Horā I.20, 21 Lagna II.11-13 Graha-madhya XVI.1-11 Candra-grahaṇa VI.1-14 Graha-sphuṭa XVII.1-11 Grahāstodaya XVIII.1-60 Graha-vikṣepa XVII.13-14 Candra-grahaṇa X.1-7 PAITĀMAHA-SIDDHĀNTA Ravi-grahaṇa IX.1-27 Ahargaṇa XII.1-2 Grahāstodaya XVII.11-12 Nakṣatra XII.3 Tithi XII.3 VĀSIṢṬHA-SIDDHĀNTA Vyatīpāta XII.4 Ravi-sphuṭa II.1 Aharmāna XII.5 With regard to the Chapters and verses not included in the above table, some, like chapters XI to XV, and several verses, are of a general nature, applicable to all the siddhāntas and in the case of the others it is difficult to identify positively the siddhāntas to which they pertain. 11. Comparative Study of the Siddhāntas It can be seen from the above Table that several of the topics selected by VM from the different siddhāntas for depiction in the PS are common. This should enable a comparative study of the common topics, the more important of which are noticed in the Table below. In this connection it is necessary to point out, an erroneous view current, to the effect, that the Pauliśa siddhānta is accurate, the Romaka is equally accurate, the Saura is still more accurate and that the Vāsiṣṭha and Paitāmaha highly inaccurate. This view has been brought into vogue through two unwarranted emendations introduced by TS and NP into the text of the following verse in the PS: Pauliśatithis sphuṭo 'sau tasyāsannas tu Romaka-proktaḥ | spaṣṭataraḥ sāvitraḥ pariśeṣau dūravibhraṣṭau || PS. I.4 ‘The tithi resulting from Pauliśa 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).’ Here VM speaks only about the tithi, (lunar day), as computed during his time according to the different siddhāntas; He is not making a relative estimate of the siddhāntas as such or in respect of the various other computations. As elucidated in the Note to this verse in the body of this book (p.5) : “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 religous purposes but also for 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 titha, is unwarranted, to say the least. Doing this, they have condemned the Vāsiṣṭha Siddhānta beyond

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