P.P.S.T. SCIENCE SERIES NO. 1
PANCASIDDHANTIKA OF VARAHAMIHIRA WITH TRANSLATION AND NOTES
BY
T.S. KUPPANNA SASTRY
C R I T I C A L L Y EDITED
WITH INTRODUCTION AND APPENDICES BY
K.V. SARMA Adyar Library and Research Centre, Madras
P.P.S.T. F O U N D A T I O N ADYAR, MADRAS 1993
Published by : P P S T Foundation, P O B 2080, Adyar, Madras 600 020 (India)
ALL RIGHTS RESERVED First Edition 1993
Printed i n India By V i p i n Sachdev, Printers Plates, , Royapettah H i g h Road, Madras 600 014
PREFACE T h e Pancasiddhantika of the sixth century astronomer Varahamihira is a major work on mathematical/astronomy of early India. T h e work is particularly significant for the fact that, besides providing an insight into the level of contemporary development i n the discipline, it forms also a resume, though uneven, of five astronomical schools, to wit, the Vasistha, Paitdmaha, Romaka, Paulisa and Saura, that were i n vogue i n India d u r i n g the early centuries of the ChrisUan era, but whose original texts have not came down to us and are lost on account of improved astronomical systems having been developed in course of time. There have appeared two earlier edidons of this cryptic technical text, the first by G . Thibaut and Sudhakar Dvivedi (Banaras, 1889) and second by O . Neugebauer and D . Pingree (Copenhagen, 1970-71). But both these editions have limitations and imperfections, even as their editors themselves have indicated in their Introductions to the said editions. T h e main difficulty in gaining a proper understanding, let alone producing a correct edition, of the Pancasiddhantika has been confounded on account of all the available manuscripts having descended from a single corrupt archetype. This aspect of the matter cannot be better expressed than i n the words of G . Thibaut i n the Preface (p.v.) to his edition of that work. H e says : " I m p e r f e c t a n d f r a g m e n t a r y as (the) text a n d (the) t r a n s l a t i o n are, we m a y assert, at a n y rate, that i n o u r e n d e a v o u r s to o v e r c o m e the q u i t e u n u s u a l obstacles w h i c h the c o r r u p t a n d b a r e text o f the P a n c a s i d d h a n t i k a o p p o s e s to the i n t e r p r e t e r , we have s p a r e d n o t r o u b l e . T h e t i m e a n d t h o u g h t d e v o t e d to the present v o l u m e w o u l d , I m a y say w i t h o u t e x a g g e r a t i o n , have a m p l y sufficed f o r the e d i t i n g a n d e x p l a i n i n g o f twenty times the a m o u n t o f text p r e s e n t i n g o n l y n o r m a l difficulties." Hence the need for a further attempt for a better edition and interpretation o f this important text on early Indian astronomy. T h e present edition which is based o n all the available manuscripts of the text and also external testimonia and which takes into consideration the work o f interpretation attempted i n the two previous editions, presents as critical and readable a text as is possible on the basis of the above-said material. T o this is added a literal translation with explanatory words added as necessary. This is followed by detailed explanatory notes i n terms inclusive of modern mathematics, and adumbrated with tables and diagrams. Whenever computations are involved, illustrative examples are given and worked out. T h e r e again, most of the chapters are provided with explanatory introductions indicating the general contents and method o f approach of the matter contained i n the chapters.
T h e above work has been a labour of love pursued by the late P r o f T . S . K u p p a n n a Sastry, formerly Professor i n the Sanskrit College, Madras. A scholar i n Sanskrit, a student of modern mathematics and one fully conversant with Jyotissastra, Prof. Sastry was an ideal combination of Indian and Western schools o f astronomy. A n d , as such, he was best fitted for the task of expounding a difficult text on Indian astronomy like the Pancasiddhantika.
VI
PREKAC.E
When Prof. Sastry passed away in 1982, he left behind his handwritten manuscript which was in different states of perfection. While the earlier chapters were in their final form, chapter X I V had been left untouched and so also were verses 35 to 55 of chapter X V I I l . T h e later chapters, portions of which had been issued in the form of articles, were in their rough draft form. T h e above-said material was placed in my hands by Dr. T . K . Balasubramanian, Scientist, Bhabha Atomic Research Centre, Bombay, Prof. Sastry's son, with the suggestion that the same might be duly processed and perfected and made pressworthy and placed before the world of scholars in printed form. As an academic associate of Prof. Sastry for nearly three decades, I accepted the challenge and set to work on it without delay. In this matter I had the cooperation of two eminent scholars in astronomy. Prof. K . S . Shukla of Lu( know and Shri. S. Hariharan of Bangalore. Work on Prof. Sastry's manuscript was twofold. T h e first related to the perfection of the existing Translation and Notes and the supply of the same for the sections which were left out by Prof. Sastry. Prof. Shukla translated C h . X I V with notes and diagrams and Shri. Hariharan supplied the Translation and Notes for the verses left out i n C h . X V I I I . While the above was done at Lucknow and Bangalore, respectively, in Madras, the end chapters were put in proper form. Alongside, the endre manuscript, r u n n i n g to about 500 pages, was duly perfected. T h i s revision work included also such matters as the crosschecking of entries, supply of diacritical marks to Sanskrit expressions, marking off of paragraphs, making the presentation uniform, and several other allied matters. The manuscript, revised as above, had also to be typed out and checked again. The large number of diagrams occurring in the work were also drawn to scale with the use of geometrical instruments and added at appropriate places. T h e second task related to the critical editing of the textual verses. A draft press copy was prepared on the basis of the readings adopted by Prof. Sastry. Copies of all manuscripts of Poncasiddhdntikd available at different repositories were procured and collated with the draft press copy. T h e text in the two printed editions and in the external tcstimonia, which had also been assembled, was collated similarly and the variants recorded. A n d , on the basis of the above, the final press copy of the critical text was prepared. Varied typograph) for the half a dozen different items of the edition was also selected to setoff the same distinctly in print. T h e resultant edition, provided with an Introduction and necessary Appendices including a Subject Index, is now placed before scholars. It is to be hoped that this edition of Pancasiddhantika will contribute, in some measure, to the furtherance of the study and appraisal of early Indian mathematics and astronomy. T h e publication of this volume had been made possible by the generous contribution of friends and patrons of academic studies. T h e Birlas made a gracious donation of Rs. 20,000/- and Dr. T . K . Balasubramanian, of Rs. 5,000/-. T h e Rashtriya Veda Vidya Pratishthan, New Delhi, has extended the major financial assistance in the form of purchasing copies of the book amounting to about rupees one lakh. T h e most profound thanks are due to the above named patrons for their kind gesture. Thanks are due to the Bhandarkar Oriental Research Institute, Poona, Oriental Institute, Baroda, and National Library, Calcutta, for their kind cooperation
\
PREFACE
VII
by supplying copies of the manuscripts of Pancasiddhantika available with them. For the beautiful printing and nice get-up, thanks are due to Ms. Printers Plates, one of the leading presses of Madras. T h e P P S T Foundation, Madras, an organisation set up for the popularisation of Indian sciences, has kindly undertaken the responsibility for the distribution of this publication. Last but not least, my grateful thanks are due to Prof. K . S . Shukla and Shri. S. Hariharan, who, besides making their personal contribution to the volume, had been available for reference and advice at all stages of the work on the present edition of the Pancasiddhantika. Madras, January 4, 1993
K.V. S A R M A
i
CONTENTS
Page
V
PREFACE
IX
CONTENTS
XIII
INTRODUCTION
Introductory - Source Material - Presentation o f the Text - Translation - Notes - Division of Paricasiddhandka - C h . X V I I I of Paiicasiddhantika - T h e Five Siddhantas in PS ; T h e i r distribution - Content analysis of the Pancasiddhantika - Depiction of the Siddhantas in the PS - Comparative study of the Siddhantas - Varahamihira : His life and works. Page Ch.I. I N T R O D U C T I O N OF T H E W O R K Aim of the Work (verses 1-2) 1 The Five Schools of Astronomy (3-4) 4 Contents of the Work (5-7) 5 Days from Epoch : Romaka (8-10) 6 Days from E p o c h : Paulisa (11-13) 11 Yuga of the Sun and the M o o n : Romaka and Saura (14-16) 15 Lordoftheyear(17-18) 18 L o r d of the M o n t h (19) 20 L o r d o f the H o r a etc. (20-25) 20 Names of the 30 days of the Parsi months 24 Ch.II.
53 54 56 57 58 64 65 65 66 67 69 71 71 71
VASISTHA-SIDDHANTA:
PLANETARY COMPUTATIONS
ETC.
Introductory T r u e Sun (1) T r u e M o o n (2-4) NaksatraandTithi(7) Day-dme (8) Gnomonic Shadow (9-10) Lagnafrom shadow and vice versa (11-13) Ch.III.
Page Local Sunset time (15) Naksatra computation (16) Sun's daily motion (17) Karanas (18-19) Vyatlpata and Vaidhrta (20-22) Sadasiti-punyakala (23-24) Solstices (25) Sankranti-kala (26) Tridinasprg-yoga (27) R a h u (Node) (28-29) Moon's latitude (30-31) Defect i n Bhadravisnu (32) Defect i n Padaditya (33) Defect in Romaka (34-37)
25 25 27 33 34 35 37
FAULISA-SIDDHANTA:
PLANETARY COMPUTATION
Introductory T r u e Sun (1-3) T r u e Motion of M o o n (5) Equation of the centre (5-9) Cara or Oblique ascension (10) Day-time (11-12) Desantara (13-14)
ETC,
40 40 45 47 48 50 51
Ch.IV. T H R E E P R O B L E M S : T I M E , PLACE A N D DIRECTION Introductory Table o f R Sines (1-15) Declinadon of the Sun and the M o o n (16-18) Gnomonic shadow (19) Latitude from Shadow (20-21) Sine zenith distance (22) Sine Co-latitude and Day-diameter (23-25) Cara or Oblique Ascension (26) Latitude from Cara (27-28) Rt. ascensional difference (29-30) Rising Signs (31) T i m e to reach the Prime vertical (32-34) Great gnomon (Sama-sartku) and its shadow (35)
76 76 87 90 91 95 96 98 100 102 104 106 110
c:oNrEN'rs
X
Page
Page Astronomer's qualifications (36-37) Gnomonic sfiadow and the prime vertical (38) Agra : Sine amplitude (39) Latitude from Agra (40) Shadow at desirea time (41 -44) Timeafter sunrise (45-47) T i m e for Sunset (48) Shadow from time (49) Moon's shadow (50-51) Directions from shadow (52-54) Sun from shadow (55-56) Ch.V.
112 113 113 114 115 118 121 122 123 126 132
137 137 143 147
PAULISA-
SIDDHANTA: LUNAR
ECLIPSE
Introductory Sun and M o o n of equal longitude (1) Probability o f a n eclipse (2) Duration ofthe eclipse (3-4) Total obscuration (5) Direction of the eclipse (6-8) Moment ofthe eclipse and its colour(8-10) Diagrammatic representation (11-13) L u n a r and Solar eclipses — Differences (14)
152 152 154 156 160 162 165 168 171
Ch.VII. ( P A U L i : § A - S I D D H A N T A ) : SOLAR ECLIPSE
Introductory Parallax of longitude (1) Parallax in latitude (2-4) Eclipse computation (5-6)
172 172 176 178
Ch.VIII. R O M A K A - S I D D H A N T A : SOLAR ECLIPSE
Introductory
189 191 192 192
SOLAR ECLIPSE
Introductory T i m e o f Moon's visibility (1-3) Diagram of the Moon's cusps (4-7) Daily rising and setting of the M o o n (8-10) (VASI§THA-)
181 183 185 186 187 187
Ch.IX. S A U R A - S I D D H A N T A :
PAULISA-SIDDHANTA:
MOON'S CUSPS
Ch,VL
T r u e Sun (I-3) T r u e M o o n (4-6) Daily motion of the Sun and the Moon (7) Rahu (8) Parallax in longitude (9) Declination of the Nonagesimal (10-12) Parallax correction and orbital diameter (13-14) T r u e diameterofthe orbs (15) Moment of the eclipse (16) Eclipse diagram (17-18)
181
Introductory M e a n Sun (1) Mean M o o n (2-4) R a h u : M a x i m u m latitude (5-6) T r u e Sun and M o o n (7-9) Epicyclic theory Bhujantara correction Udayantara correction Desantara correction (10) MeanmotionoftheSunandtheMoon(lI) M o t i o n o f Moon's anomaly (12) T r u e motion of Sun and M o o n (13-14) Kaksa of the Sun and the M o o n (15) Measure of the orbs (16) Sin Zenith Distance of Meridian point(17-18) Drksepa ofthe Sun (19-20) G n o m o n (21) Parallax-corrected New M o o n (22-23) Parallax in latitude (24-25) Duration of the eclipse (26-27)
197 197 198 201 203 206 209 209 209 210 211 211 214 215 216 219 220 221 223 225
Ch.X. S A U R A - S I D D H A N T A : L U N A R ECLIPSE
Introduction 228 Diameter of the Shadow (1-2a) 228 Duration ofthe Eclipse (2b-4) 230 Obscuration at any desired moment (5-6) 232 T i m e oftotal obscuration (7) 235
CONTENTS
Page
Page Ch.XI. ECLIPSE D I A G R A M Introduction Marking the ecliptic etc. (1-3) Markingof'p(jints of contact etc. (4-5) Conversion olminutes into angles (6)
236 236 238 240
Ch.XII. P A I T A M A H A S I D D H A N T A Introducuon Days from Epoch (1-2) T i t h i , Naksatra etc. (3) Vyatipata (4) Duration of a day (5)
241 241 243 244 246
Ch.XIII. S I T U A T I O N O F T H E EARTH: COSMOGONY Situation of the earth (1-4) Rotation of the earth (5-8) Situation of the Gods and Asuras (9-13) SignsandYojanas(14-16) Position of Lartka and L'jjayinI (17) Measures of the earth etc. (18-19) Visibility ofthe Sun (20-29) Astronomical observation (30-34) Moon's visibility (35-38) T h e Planets and their situation (39-41) Lords of the Months, Days and Year (42)
248 249 250 252 253 253 253 258 258 259 260
Ch.XIV. G R A P H I C A L M E T H O D S AND ASTRONOMICAL INSTRUMENTS Introductory Ascensional differences o f the zodiacal signs (1-4) R sine ofthe Sun's zenith distance for the given time (5-6) Right ascensions ofthe signs (7) Gnomon (8) Local latitude (from the equinoctial midday shadow) (9-1 Oa) Sun's longitude (lOb-11) V-Shapedyasti(12) Moon's longitude (13) Cardinal directions (by means of a gnomon (14-16)
XI
T h e Celestial Spfiere( 17-18) Hemispherical B o v l and its use (19-20) H o o p and its use (21-22) ArmiUary Sphere (23-25) Sun's northward and southward journeys (26) Instruments for measuring time (27-28) Local longitude i n terms of time (29-30) TheNadiorGhati(31-32) Conjunction ofthe M o o n w'th a Star (33) Positions of certain JunctionStars (34-37) Digits between the M o o n and a Star in Conjunction and T i m e of Conjunction (38) Heliacal Rising of Canopus (39-41)
270 271 272 273 273 274 274 275 276 277
277 278
Ch.XV. SECRETS O F A S T R O N O M Y Eclipses (1-10) Situationatthe Poles (11-16) Weekday (17-18) Day-reckoning (19-29)
282 286 288 289
Ch.XVI.SAURA SIDDHANTA: MEAN PLANETS Introductory Mean positions of the Star-planets (1-19) V M ' s B I j a corrections (10-11)
292 293 298
Ch.XVII. S A U R A - S I D D H A N T A : T R U E PLANETS 261 261 264 266 267 267 268 269 269 270
Epicycles ofthe planets (1-3) T r u e planets (4-9) Special work for Mercury and Venus (10-11 a) ' Retrograde motion ( l i b ) Heliacal rising (12) Latitudes of planets (13-14)
300 301 307 309 309 310
Ch.XVIII. V A S . - P A U . SIDDH.: RISING A N D S E T T I N G O F P L A N E T S Venus (1-5) Jupiter (6-13) Saturn (14-20) Mars (21-35)
312 316 322 339
XII
CONTENTS
Page Mercury (35-36) Hints (57-60) General (61-63)
341 351 353
SPURIOUS S U P P L E M E N T : T R U E PLANETS Introduction (64-66) T r u e Mars (67-69) T r u e Mercury (70-72) T r u e Jupiter (73-75) T r u e Venus (76-78)
355 357 359 361 362
Page T r u e Saturn (79-81)
363
APPENDICES I. Verse Index II. Index o f P S verses quoted by later astronomers III. Bhutasaiikhya used i n the PS I V . Index o f Places, Persons and Texts quoted in the PS V . Bibliography V I . Subject Index
367 371 372 373 374 377
INTRODUCTION 1. Introductory T h e Pancassiddhdntikd (PS) o f Varahamihira ( V M ) , (6th cent. A . D . ) , occupies an important place in the history of early Indian astronomy, for, herein we have been given certain aspects of five systems of Indian astronomy current d u r i n g the first centuries of the Christian era. T h e work supplies also considerable additional material on the astronomical concepts, computational methods and instruments used d u r i n g the dmes o f the author. V M makes mention ofthe objectives ofthe work towards its commencement: purvdcdryamatebhyo yad yad srestham laghu sphutam bijam \ tat tad ihdvihalam ahum rahasyam udyato vaktum 11 1.1 | Paulisa-Romaka-Vdsistha-Saura-Paitdmahds tuparkasiddhdntdh || 2 | 'Here, I shall state i n full the best of the secret lore o f astronomy extracted from the different schools of the ancient teachers so as to be easy and clear. 'The five siddhdnta-s, of which this work is a compendium, are the Paulisa, the Romaka, the Vasistha, the Saura and the Paitdmaha. Following the above statement, V M specifies also how he is intending to deal with the said five astronomical schools : yat tatparam rahasyam bhavati matiryatra tantrakdrdridm \ tad aham apahdya matsaram asmin vaksye graham bhdnoh \ \ dik-sthiti-virmrda-karna-prarndna-veld grahdgrahdv indoh tdrdgrahasarnyogam desdntarasddhanam cd 'smin 11 sarmmandala-candrodaya-yantra-cchedydni sdhhavacchdyd \ upakaranddy aksajyd-'valambakd-'pakrarriddydni \ \ 1.5-7 | 'I shall tell i n this work, avoiding all jealousy, the computation o f the solar eclipse which is guarded as a great secret and i n which the m i n d 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 difference i n 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.' While the importance of the work i n the reconstruction of early Indian astronomy would be apparent from the above statement, the paucity of reliable manuscripts of the work makes the preparation of a correct edition o f the work a formidable task, affecting, i n its turn, a proper understanding, translation and interpretation of the work. T h e gravity o f the problem could be gauged from what G . Thibaut has stated in the Preface to the first edition ofthe work (TS), issued in 1889. He says: 1. The Pancasiddhantika, the astronomical work of Varaha Mihira. The Text edited with an original commentary in Sanskrit and an English translation, by G . Thibaut and Mm Sudhakara Dvivedi, Leipzig : Varanasi, 1889; Rep. Lahore, 1930; Rep. Varanasi, 1968. (Page references made are to this reprint.)
XIV
PANCASIDDHANTIKA
"There is some reason to fear that the feeling of any one who may examine in detail this edition and translation of Varaha Mihira's astronomical work will, in the first place, be wonder at the boldness of the editors. I am fully conscious that on the imperfect materials at our disposal an edition i n the strict sense of the word cannot be based, and that what we are able to offer at present deserves no other name but that of a first attempt to give a general idea of the contents ofthe Paiicasiddhantika. It would, in these circumstances, possibly have been wiser to delay an edition o f the work u n d l more correct Manuscripts have been discovered. T w o consideradons, however, i n the end, influenced us no longer to keep back the results, however imperfect, of our long continued endeavours to restore and elucidate the text of the Paiicasiddhantika. I n the first place, we were encouraged by the consideration that texts of purely mathematical or astronomical contents may, without great disadvantages, be submitted to a much rougher and bolder treatment than texts of other kinds. What interests us in these works is almost exclusively their matter, not either their general style or the particular words employed, and the peculiar nature of the subject often enables us to restore with nearly absolute certainty the general meaning of passages the single words of which are past trustworthy emendation. A n d , i n the second place, we feel convinced that even from that part of the Paiicasiddhantika which we are able to explain more is to be learned about the early history of Sanskrit Astronomy than from any other work which has come down to our time." (p.v.). About the manuscript material available to h i m and the editorial criteria adopted by him Thibaut says in his Introduction to the edition: " T h e present edition o f the Pancasiddhantika is founded on two Manuscripts, belonging to the Bombay Government. T h e text ofthe better one o f those two Manuscripts is reproduced in the left hand columns of our edition, while the foot notes give all the more important different readings from the other Manuscript. A comparison o f the traditional text with the emended one, as given i n the right hand columns of the edition, will show that the former had, in many cases, to be treated with great liberty. Not unfrequendy, the emended text is merely meant as an equivalent in sense of what we suppose Varaha M i h i r a to have aimed at expressing, while we attach no importance to the words actually employed i n the emendadon." (p.lx).
T h e Pancasiddhantika has again been edited recently by O . Naugebauer and D . Pingree, (NP)^. A n d Pingree too observes, on the state of the manuscript material: "The present edition of the Pancasiddhantika does not solve all the remaining problems connected with this text. We suspect that much will never be understood unless better manuscripts material becomes available." (Vol. I, Intro., p. 19) It, however, so happens that d u r i n g the hundred years that have passed by since the publication of its first edition i n 1889 no 'really' new manuscript of the work has come to light. A few manuscripts that have become available,^ all go back to the two manuscripts used i n the first edition, as shown by the commonality in them of omissions and corruptions which occur in the newly available manuscripts.
2. The Pancasiddhantika of Varahamihira. Pt. I. Text and Translation by D. Pingree; Pt. II. Notes by O. Neugebauer, Cobenhavn, Munksgaard, 1970. 3. For details. See below under 'Manuscript material'.
INTRODUCTION
XV
T h e edition of Pancasiddhantika which is now issued is also based on the manuscripts used for the two above-said editions. T h e question that would naturally arise here would be: W h y then is the need for another edition when no new source material is to be had? T h e answer is threefold: i. First, it was felt that in reconstructing the text from the corrupt manuscripts, which alone are available, both T S and N P have subscribed to an editorial principle voiced by Thibaut when he says: "What, i n the attempt to reconstitute the text of an astronomical or mathematical work, has chiefly to be kept i n view, is of course to arrive at rules which are capable of being proved mathematically. This consideration has, i n more than one place, led us to introduce changes even where such appeared hardly to be required by the external form o f the traditional text." (Introduction, p.lxi). A n d , this they have done to an extent which seems to be hardly justified i n editing a classical text. T h e n again, such emendations are often inserted without specific indication, especially i n the edited text of N P , with the result that the reader takes the emended text as the 'real' manuscript text. T h e translation and interpretations that follow are, primarily, based o n the emended text and not on the 'real' text. In fact, an editorial principle which has to be applied with the utmost caution and i n as limited a manner as possible, seems to have been used rather extensively. In the present edition of Pancasiddhantika, the principle oisthitasya gatis cintaniyd, justification of the extant reading should be thought o f , has been primarily adhered to, alongside the correction of the copyist's errors by visualising the psychology of the scribe who is illiterate with reference both to the language and the subject of the text. 'Real' emendations have been comparatively small and far between. In all cases, however, when changes had to be made to the manuscripts readings, they have been specifically indicated by their being placed within curved brackets i n the case of scribal errors and i n square brackets i n the case of editorial supplementation. A n d , whenever there has been an emendation, the reasons for suggesting the emendation have been given in the Notes that follow each verse. ii. Secondly, i n a number of places, T S nor N P do not seem to have caught the correct import of the text and this has affected their Translation and Notes. A l l these have been attempted to be rectified. In many places, the untenability o f the T S and N P readings, translations and notes have also been pointed out. iii. Thirdly, and what is most important, special effort has been made to digest the textual verses fully, and offer, i n the case of knotty places and apparently vague passages, detailed interpretations and elucidations, adumbrated with tables and illustrations. Moreover, a number of examples have also been worked out to illustrate the rules enunciated by Varahamihira. U n d e r the circumstances, it is to be hoped that the present publication will form another step towards understanding and evaluating the principles and practices o f early Indian astronomy. 2. Source Material A-B. T h e available manuscripts of Pancasiddhantika, of which five have been collated for preparing the present edition, fall into two recensions which have been designated as A and B . C o m m o n corruptions and omissions indicate that even these two recensions go back to a common original which too should have been far from perfect i n the matter of accuracy. T h e technical nature of the work, brizzling with unusual terminology, have made the scribes commit all types of imaginable errors except i n the case of well-known words and expressions. These errors include, as a reference to the footnotes recorded in the edition would show, wrong spellings, etrauc sandhi-s and splitting
XVI
PANCASIDDHANTIKA
of words, omission of vowel signs, verses made to stop short in the middle or to r u n into another, numbering of verses i n the wrong places and so on. In several cases some of these corruptions are common to all the manuscripts, confirming that these errors have to be traced back to a common archetype of both the recensions. T h e said five manuscripts have been designated A l , A 2 and B l , B 2 , B 3 , according to the twc recensions and reladve reliability ofthe manuscripts. A l l the manuscripts are in paper, written in Devanagari script. A , . Ms. N o . 338/1879-80 of the Bhandarkar Oriental Research Institute, Pune, described i n A Catalogue of Collections of Mss. deposited in the Deccan College by S.R. Bhandarkar (Bombay, 143. It has 22 folios with 11 lines per page. It is complete and has been copied at Stambhatirtha (modern Cambay i n Gujarat) i n Sam. 1673,Saka 1538(AD. 1616),bySaAkarasonofGovinda.This manuscript is the 'better of the two manuscripts' used i n the T S . edition o f the PS. T h e manuscript is far from perfect and exhibits numerous scribal errors and some transposidons, but it is definitely better than the B manuscripts. A j . M s . N o . 49, currently preserved i n the National Library, Calcutta, but it originally belonged to the erstwhile Imperial Library, Calcutta. It is i n 24 folios with 9 lines a page. It is incomplete and extends to a pordon of X V I I I . 90 d. T h e writing is very readable but is very much error-ridden. The readings are closely associated to A , . Pingree suspects that it "agrees almost entirely with A (our A , ) of which it is most probably a copy." (see his edition of PS, Introducuon, p. 20). T h i s cannot be a copy of A , for the reason that there occur differences between the two, for which see I. 3c, 7d, 10a, 12a and a number of other contexts. Pingree doubts also that is "perhaps the copy utilized by Thibaut and Dvivedin" (Introduction, p.20). T h i s goes against Thibaut's statement that his "edition of PS is founded on two Manuscripts belonging to the Bombay Government" (T's Introduction, p. L X ) . It is also to be noted that while A j is complete, is incomplete. It is again to be noted that minor over-writings and corrections above the lines occur i n A^ at several places obviously having been added by a modern user of the manuscript. These latter, being modern, have not been noted as variants i n the footnotes to the present edidon.
B, . M s . N o . 37/1874-75 of the Bhandarkar O r i . Res. Inst., Pune, described in A Catalogue of Co lections of Mss. deposited in the Deccan College (Bombay, 1888). C o p i e d i n modern "Universal fo paper, with a dde page i n Devanagari reading "number 37-Satra 1872 [A.D.] Pancasiddhandka, patrani 49-15-1930", it is i n 49 pages, with 15 to 17 lines a page. T h i s is the second of the two manuscripts used by T S , from which they have documented only some of the variants, as recorded i n the footnotes of their edition. Pingree states (Intro., p.20), that i n the edition he has documented only those variants recorded by T S i n their footnotes. I n the present edidon, however, the manuscript has been fully collated and all the variants therein recorded. T h i s manuscript carries, through its entire length, corrections, obviously made by Thibaut. B^. Ms. N o . 64 of the National Library, Calcutta. It contains 108 pages numbered 7 to 114, and is incomplete, commencing only from 1.22, the previous verses having been written on the folios 1-6, now lost. T h e manuscript is shapely and the writing readable, but the matter contained is extremely
INTRODUCTION
XVII
corrupt. T h e numbering of the verses is also erratic. Several verses are broken i n their middle and verse numbers are interposed. A t times the last line of a verse is continued with the beginning letters of the next verse, entailing at times, the beginnings and ends being half-words, (see III. 1 and 2). T h e manuscript seems to be the handiwork of a good-handed scribe from a highly corrupt original. Corrections by a modern hand i n lighter ink is seen at places. Pingree suspects that "this manuscript seems to be a copy of B - perhaps that used by Thibaut and Dvivedin." (Introduction, p.20). B. . Ms. N o . 7165 of the Oriental Institute, Baroda. T h i s manuscript i n 33 folios contains the complete work. A post-colophonic statement says that it was copied i n Sarti,. 1928/saka 1793 ( A . D . 1872) by Uttamarama Durlabharama, a resident of Amadavada (Ahmedabad). T h e wridngis readable but corrupt readings persist. Verse numbers are written for the first chapter, but not for the later chapters. Numbers expressed i n the verses are written also i n digits i n many places. A few more manuscripts of Pafkasiddhdntika are known to exist (or to have existed) but could not be used for the present edition. T h e y are : 1. Ms. N o . 288 of the Bombay University. T h i s is at present missing i n the Library. Pingree has rsed this manuscript. In 32 folios it contains the full text of PS. It was copied i n Saih. 1928, corresDonding to A D . 1871, by Nathurama Parika. 2. Ms. N o . 6288 of the India Office, L o n d o n , (Buhler 268) described i n the A Catalogue of Skt. and °kt. Mss. in the India Office Library, vol. II by A . B . Keith. T h i s is a copy of our Ms. A , , copied i n Sath 1936, Saka 1802 ( A . D . 1879). We have not used it nor has Pingree. About still other manuscripts, Pingree states : "Besides these seven manuscripts, there existed i n 1890 the manuscript belonging to J . B . M o d a k of T h a n a which was copied from B (our B,), and we enow of a manuscript (no. 6674) of the Pancasiddhantika i n the Anandasrama i n Poona. T h e man- . iscripts, recorded as the property of Sjt. Pushpachandra Sarma Daloi o f Helach i n Assam and of he Arsha Library i n Vijayanagara (no. 506), probably contain the Bhdsvati of Satananda, which is lometimes confused with o u r text." (Introduction, p. 21). C. T h e emended text of Thibaut-Sudhakara Dvivedi, as printed i n the right hand columns of heir edition. O n this Thibaut says (Introduction, pp. Ix-lxi): "The present edition o f the Paiicasiddhantika is founded on two manuscripts belonging to the Bombay Government. T h e text of the better one of these two Manuscripts is reproduced in the left hand columns of our edidon, while the foot notes give all the more important different readings from the other Manuscript... the emended one as given i n the right hand columns ofthe edition. "What, in the attempt to reconstruct the text of an astronomical or mathemadcal work, has chiefly to be kept i n view, is of course, to arrive at rules which are capable of being proved mathematically. T h i s consideration has, i n more than one place, led us to introduce changes even where such appeared hardly to be required by the external form of the traditional text." »Jotwithstanding the wild emendations which Thibaut-Sudhakar Dvivedi have made, at times, heir emended readings have been recorded as C i n the footnotes o f the present edition. D. T h e edition, Translation and Notes of TS by O . Neugebauer and D . Pingree, (2 vols., Copenhagen, 1970). Pingree edits and translates the PS, while Neugebauer offers the Notes. A l l
XVIII
PANCASIDDHANTIKA
our manuscripts are used i n this edidon too. H e r e i n occur a number of emendations which are sometimes put within brackets, but sometimes without brackets. Often the emendations are wilder than those of Thibaut-Sudhakar Dvivedi. E. External testimonia. About 125 Pancasiddhantika verses have been identified as quoted in later texts, the largest number thereof, 117 being in the commentary of Utpala of Kashmir {A.D. 966) on the Brhatsamhita of Varahamihira. Other authors who quote from the PS are Prthiidakasvamin ( A . D . 854), Makldbhatta (14th cent.), Parame^vara of Kerala (1360-1460), Nilakantha Somayaji, (b. 1443), 9'^ain of Kerala, and Siiryadevayajvan (b. 1191) of T a m i l n a d u , i n South India. These PS quotations are taken as External Testimonia and the variants found i n such readings have been noticed in the footnotes with ' E ' prefixed to the abbreviations o f authors/works which quote the verses. These sources are : E. Jy. Jyotirmimdmsd of Nilakantha Somayaji. E.M. Makkibhatta's C o m . on the Siddhdntasekhara o f Sripati. E.N. Nilakantha Somayaji's Bhasya on the Aryabhatiya. E.Pa. Paramesvara's com. on the Aryabhatiya. E.Pr. Prthudakasvamin's com. on the Brdhmasphutasiddhdnta of Brahmagupta. E.S. Suryadevayajvan's com. on the Aryabhatiya. E.U. Utpala's com. on the Brhatsamhitd of Varahamihira. Pingree has adopted this method of referring to external testimonia, and we have followed him in the matter. In fact, we have been much benefitted by his identifications and our labour relates only to texts which have not been noticed by h i m . O n the provenance of PS, Pingree states that: "So far there is no indisputable evidence that the Paiicasiddhantika was known outside of an area roughly corresponding to the modern states of Madhya Pradesh, Gujarat, Rajasthan, the Punjab, Kashmir, and West Pakistan. "However, some verses from the text are quoted by fifteenth century Kerala astronomers of the drgganita school i n their commentaries o n the Aryabhatiya. T h u s Paramesvara (c. 1380-1460) cites a verse, and Nilakantha (b. 1443) several others. It is noteworthy that all four verses that they quote are also found i n Utpala's commentary on the Brhatsrtihita, which was known in Kerala; it is not proved, then that they had a copy o f the Pancasiddhantika." (Introduction p. 17).
Pingree's assertion as above is not correct for the reason that Kerala and South Indian astronomers quote not only the said four verses occurring i n Utpala's commentary, but five more PS verses which do not occur i n Utpala's commentary, they being PS, 1.3, 4 by Nilakantha in his Jyotirmimdmsd, the verses safikhyd tu tesdm andydny atahprati again by Nilakantha i n his AryabhatiyaI V . 10 and PS X I I I . 3 6 by Siiryadevayajvan i n his Aryabhattyavydkhyd. (See below A p p . II: Index o PS verses quoted by later astronomers). This would mean that PS should have been prevalent in South India also. Recording of Variant readings Textual variants recorded i n footnotes i n the present edition are restricted to be above-said material listed under A , B , C , D and E . Some o f the PS verses have, indeed been studied by scholars but emendations and variants occurring in these studies have not been recorded here mainly for the
INTRODUCTION
XIX
reason that they have mostly been docitmented by Pingree i n his edition of the PS and so can be referred to therefrom.' 3. Presentotionof theText While the generally accepted conventions of critical editing o f Indie texts are duly followed i n the present edition, attention might be drawn to certain methodologies which are stressed herein, in view of the technical nature o f the text, the defective nature o f the manuscripts and the tentativeness of many of the emendations and textual changes effected i n the two earlier editions. i. O n account o f the inadequacies o f the copyists o f the parent manuscripts a n d also due to deficiencies i n the parent manuscripts themselves, some emendations have to be done i n the edited text. However special care has been taken i n 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 Varaha Mihira's Pancasiddhantika',//. of Asiatic Society of Bengal, 53 (1884) 259-93. ii. S.B. Dikshit, 'The Original Surya-siddhanta', Indian Antiquary, 19 (1890) 45-54. iii. S.B. Dikshit, 'The Romaka Siddhanta', Indian Antiquary, 19 (1890), 133-42. iv. S.B. Dikshit, 'The Paiicasiddhantika', Indian Antiquary, 19 (1890) 439-40. V. J . Burgess, 'The Romaka Siddhanta', Indian Antiquary, 19 (1890) 284-85. vi. J . Burgess, 'The sines and arcs in the Pancasiddhantika', Indian Antiquary, 20 (1891) 228. vii. M.P. Kharegat, 'On the interpretation of certain passages in the Pancha Siddhantika of Varahamihira, an old historical work'//, of the Bombay Branch of the Royal Asiatic Society. 19 (1895-97) 109-41. viii. K.S. Shukla, 'On three stanzas from Pancasiddhantika,' Ganita, 5 (1954) 129-36. T.S. Kuppanna Sastri ix. 'The Vasistha Sun and Moon in Varahamihira's Paiicasiddhantika', Jl. of Or. Research, Madras, 25 (1955-56) 19-41. X. 'Some misinterpretations and omissions in Thibaut and Sudhakara Dvivedi in the PS of V M ' , Vishveshvaranand IndologUal Journal, 11 (1973) 107-18. xi. 'The epoch of the Romaka Siddhanta in the PS and the epoch longitudes of the Sun and Moon the Vasistha Siddhanu', Indian Jl. of Hist, of Science, 13 (1978) 151-58. xii. 'The Vasistha-Paulisa Venus in the PS of V M ' , Collected Papers of T.S. Kuppanna Sastry, Tirupati, 1989, pp. 141-47. xiii. 'The Vasistha-PauliSa Jupiter and Saturn in the PS', Collected Papers, pp. 148-68. xiv. 'The Vasistha-PauliSa Mars in the PS of V M ' , ColUcted Papers, pp. 169-87. XV. 'The epoch-constants ofthe Vasistha-Paulisa star-planets', Collected Papers, pp. 201-5. xvi. 'Saurasiddhanta of PS : Planetary constants and computation', Collected Papers, pp. 206-40. xvii. 'Paiicasiddhantika XVIII. 68-81 : A n interpolation", ColUcted Papers, pp. 241-54.
XX
PANCASIDDHANTIKA
ii. Thibaut and Dvivedi who print their emended text i n the right hand column of their edition do not specifically indicate their emendations and one has to identify the emendations oneself Pingree indicates his emendations in many places by angular brackets but i n many other places prints the emended text without any indication. In the present edition, all their emendations are idendfied and nodced i n the footnotes denoting them by the sigla C and D . W h e n these emendations are accepted i n the present edition also, they are not separately marked as above. This procedure is expected to enable the comparison between the emendations of T S and N P 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 T S and N P , especially of the latter, are often far-fetched, ungrammadcal, offending the metre or failing to give a cogent sense. See, for example, the T S / N P emendations in 1.23; I X . 5 ; X I . 2 , 4, 5; XII.5a, 5d; X I I I . 3 8 d , 4 I d : X V I I . I , 12; X V I I I . 2 d , 3, 19, 24, 25. iii. In order that a discernmg student ofthe text shall have before him, i n full, what occurs in the several manuscripts, an attempt has been made to record all variants, right or wrong. T h i s has been done for three reasons : (a) Correct forms of corrupt passages can be visualized only i f all the readings, as found i n the manuscripts, are before one's eyes; (b) T h e n alone would it be possible for an editor to vindicate the emendations suggested by h i m ; (c) T h e corrupt and apparently meaningless readings i n the manuscripts which the editor could not correct or has wrongly emended can be corrected or better emendations suggested by other scholars i f all the variants are given. However, obvious errors of a purely scribal nature, like separate words written jointly, using anusvdra for anundsika 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 corrected silently and not noticed i n the footnotes. 4. Translation T h e translation provided i n the edition is as literal as possible without sacrificing readability and not going against the English idiom. Elucidatory expressions and words which are understood i n 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 T h e 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. T h e emendations made by T S and N P are also examined and observations offered. A n 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 chapters, towards setting out the significance ofthe contents ofthe respective chapters. 6. Division of Pancasiddhantika T h e colophons ofthe several Sections of P S , as found i n 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 adhydya-s (chapters) and also indicate the numbers thereof. T h e several colophons read :
INIRODUCTION
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.
XXI
Karanavatarah Naksatradicchedah Iti Paulisasiddhantah Id Karanadhyayas caturthah Id Sasidarsanam Candragrahane sasthodhyayah Paulisasiddhante ravigrahanam Id Romakasiddhante 'rkagrahanam astamo 'dhyayah Id Suryasiddhante 'rkagrahanam navamo 'dhyayah Candragrahanam dasamo'dhyayah A(nu)varnanam ekadaso 'dhyayah Id Paitamahasiddhante dvadaso 'dhyayah Trailokyadarsanam nama trayodaso 'dhyayah Iti Chedyakayantrani caturdaso 'dhyayah Jyotisopanisat paiicadaso 'dhyayah Siiryasiddhante madhyagatih Taragrahasphutikaranam sodaso (? saptadaso) 'dhyayah Paulisasiddhante taragrahah
However, whether there be full chapter headings and chapter numbers or there be only the mention of the topics in the colophons, the commencement of a new subject helps to ascertain the beginnings of the chapters. T h e omission of the specification of the chapter headings and numbers has to be ascribed to the imperfections i n the original archetype.
This helps us to correct Pingree's edition where Chs. X V I and X V I I are taken as a single chapter, which he numbers as X V I . N o w , after eleven verses, here, there occurs the colophon 'Suryasiddhante madhyagatih', which is the subject of those eleven verses. After still another fourteen verses occurs i n the colophon 'Tdrdgrahasphutikaranam sodaso 'dhyayah', and an entirely different subject is treated in those fourteen verses. Ignoring the radical difference i n 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 ofthe second set of verses, Pingree combines the two sets of verses, 11 plus 14 = 25, to constitute ch. X V I and takes ch. X V I I I of TS and ofthe present edition as ch. X V I I . 7. Chapter XVIII of Pancasiddhantika T h e theme of C h . X V I I I is the heliacal rising and setting o f the star-planets according to the Vasistha and Paulisa 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, Varahamihira, 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. H a v i n g thus completed his treatise, couched entirely i n the Arya metre, following a convention, adopted by Sanskrit writers, of concluding works by a closing colophonic verse couched i n a different metre, V M breaks into a different metre, the Vasantatilaka, for the purpose : dvantyahah samdsdc chisyahitdrtham sphutdnkasamam cakre Vardhamihiras tdrdgrahakdrikdtantram || 61 | Pradyumna-bhumitanaye jive saure 'tha Vijayanandikrte \ budhe ca bhagnotsahah sphutam idam karanam bhajatdm 11 62 11
PANCASIDDHANTIKA
XXII
drstam Vardhamihirena sukhaprabodham \ • II 63 II T h e manuscripts are defective here, omitting the next three lines of the concluding verse, some exhibidng also a gap. In continuation of the above, the manuscripts commence another short work with a mahgalasloka ('verse of salutation') and then a second verse expressing a pratijnd ('resolve') to compose a 'better work' by 'Varahamihira himself.' prastdve 'pi na dosan jdnann api vakti yah paroksasya \ prathayati gundrhs ca tasmai sujandya namah parahitdya \ astddasabhir baddhdny d tdrdgraham etad drydbhih \ varam iti Vardhamihiro daddti nirmatsarah karanam \ \ These verses end abruptly without any closing colophonic verse, as might be expected.
Not taking into consideration the fact that V M had closed the PS most formally with all attendant paraphernalia, with verse 63, both T S and N P treat this short work, as the concluding part of Pancasiddhantika and as a work of Varahamihira himself, and translate it and explain it as such. It is no noticed by T S and N P 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 i n the computation of Venus and Mercury, which one cannot expect to be committed by VM.
T h e prose colophon 'Paulisasiddhante taragrahah evam' ('Thus the star-planets of the Paulisasiddhanta') does not have any reality behind it since the computations therefrom do not accord with those o f the Paulisasiddhanta elucidated earlier. These points have been discussed in detail in the Notes, below, to these verses. Obviously, these verses are apocryphal and are the handiwork o f a n inferior astronomer who has ascribed them to V M . F o r this reason, in the present edition, Pancasiddhdntikd is formally closed with verse 63 and the verses following given as'-a praksepa (interpoladon) 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 Siddhantas in PS : Their Distribution It had been mentioned earlier that V M ' s treatment of the five siddhantas in the PS are uneven. It is not that each Siddhanta is taken u p one by one and a resume of the same given fully or systematically. Select topics are taken up, apparently arbitrarily, and are dealt with individually or jointly when the enunciations o f two schools are similar. Alongside, chapters are devoted also to general astronomical topics which are applicable to all the schools. T h e assortment of the treatment of the subjects i n the several chapters are as indicated below.
9. Content Analysis of the Pancasiddhantika C h . I . 1-7 8-10 11-13 14-16
General Romaka Paulisa Saura-Romaka
Introducuon Days from epoch (Ahargana) Days from epoch (Ahargana) Yuga o f S u n and Moon, Lords ofthe year, month, and hora.
XXIII
INTRODUCTION
C h . I . 17-25
Romaka
Ch.II
1-13 1 2-6 7 7 8 9-10 11-13
Vasistha
Ch.III 1-37 1-3 4 5 6-9 10 11-12 13-14 15 16 17 18-19 20-22 23-24 25 26 27 28-29 30-31 32 33 34-35 36-37
Paulisa
General
C h . I V 1-58
General
1-15 16-18 19-22 23-25 26-28
//
n It If II 11 ti
11 11 II It It It 11 II II If 11 11 11 11 II II II II 11 II 11
11 11 11
II
11
Lords of the days of the month
T r u e Sun (Ravi-sphuta) T r u e M o o n (Candra-sphuta) Naksatra computation T i t h i computation Day-time (Aharmana) Gnomonic shadow (Saiikucchaya) Lagna
T r u e Sun (Ravi-sphuta) T r u e motion of M o o n (Candragati) Equation of the centre (Mandaphala) Sense obscure Cara Day-time (Aharmana) Desantara Local time (Istadesakala) Naksatra computation Sun's daily modon (Ravi-gati) Karana-s Vyatipata and V a i d h r t i Sadasiti-kala Solsdces (Ayana) Saiikrand-kala Tridinasprk-yoga Rahu (Node) Moon's latitude (Candra-viksepa) Bhadravisnu, defect i n Padaditya, defect i n Romaka, defect i n Astronomy, importance of Problems of Time, Space and DirecUon (Triprasna) Table of R sines (Jyah) Declination of Sun and M o o n (Krand) Gnomonic shadow and derivatives (saAkucchaya) Sine colatitude and Day-diameter (Lambajya and Dinamana) Cara and derivatives
PANCASIDDHANTl K A
XXIV C h . I V . 29-30
"
31-34 35-36 37-38 39-40 41-49
" " " " "
50-51 52-54 55-58
" " "
Rt. ascensional difference (Lartkodaya-rasimana) Rising of signs (Rasyudaya) Gnomonic shadow (Sartkucchaya) Astronomer's qualifications Sine amplitude (Agra) Gnomonic shadow at required time (Istacchaya) Moon's shadow (Candracchaya) Directions from shadow (Diksadhana) Sun from gnomonic shadow (chayatah ravih)
Paulisa
Moon's horns (Candrasrngonnati)
Ch.VIl-14
Vasistha and Paulisa
L u n a r eclipse (Candragrahanam)
C h . V I I 1-6
Paulisa
Solar eclipse (Ravi-grahana)
C h . V I I I 1-18
Romaka
Solar Eclipse (Ravi-grahana)
C h . I X 1-27
Saura
Solar eclipse (Ravi-grahana)
C h . X 1-7
Saura
L u n a r eclipse (Candra-grahana)
C h . V 1-10
C h . X I 1-6
General
C h . X I I 1-6 1-2 3-4 4 5
Paitamaha "
"
Eclipse diagram (Grahanaparilekha)
Days from epoch (Ahargana) T i t h i , Naksatra etc. Vyatipata Day-time (Aharmana)
C h . X I I I 1-42
General
Situadon of ther worlds (Trailokyasathsthanam)
C h . X I V 1-41
General
Astronomical instruments (Chedyaka-yantrani)
C h . X V 1-29
General
Jyotisopanisat
Ch.XVII-II 1-9 10-11
Saura " "
Mean planets Mean planets (Graha-madhya) Bija correction by V M
XXV
INTRODUCTION
C h . X V I I . 1-14 1 -11 a lib 12 13- 14
Saura " " " "
True planets True planets (Graha-sphuta) Retrograde motion (Vakragati) Heliacal rising (Grahastodaya) Planetary latitudes (Graha-viksepa)
C h . X V I I I 1-60
Vasistha-Paulisa
Heliacal rising and setting of planets (Grahastodaya) Venus (Sukra) Jupiter (Guru) Samm(Sani) Mars(Kuja) Mercury (Budha) Hints Conclusion of PS
1-5 6-13 14- 20 21-35 36-56 57-60 61-63 64-81 64-66 67-69 70-72 73-75 76-78 79-81
" " " " " " General Spurious Supplement " " " " " "
True planets Introduction True Mars True Mercury Truejupiter True Venus True Saturn
10. Depiction of the Siddhantas in PS The above Content Analysis of the PS would enable the identification ofthe extent of selective depicuon of the different siddhantas by V M in the work, as shown below.
PAULISA-SIDDHANTA
Ahargana Naksatra Mandaphala Ravigati Raviphuta Candra-sphuta Ayana SaAkranti Tridinasprkyoga Desantara Cara Aharmana Istadesakala Karana Vyatipata Vaidhrti
I.1I-13 H I . 16 III.5 III.17 III.1-3 III.4 III.25 III.26 III.27 in.I3-14 III. 10 III.I1-12 HI.15 III.18-19 III.20-22 III.20-22
Sadasltikala Candra-viksepa Rahu Candra-grahana Ravi-grahana CandrasrAgonnati Grahastodaya
III.23-24 III.30-3I III.28-29 V I . 1-I4 V I I . 1-6 V.1-10 X V I I I . 1-60
ROMAKA-SIDDHANTA Ahargana Yuga Lordoftheyear L o r d ofthe month L o r d ofthe H o r a L o r d ofthe Days Ravi-grahana
1.8-10 1.14-16 1.17-18,21 1.19,21 1.20,21 1.23-25 V I I I . 1-18
XXVI
PANCASIDDHANTIKA
SAURA-SIDDHANTA Yuga L o r d of the year L o r d of the month L o r d of the H o r a Graha-madhya Graha-sphuta Graha-viksepa Candra-grahana Ravi-grahana Grahastodaya
1.14-16 1.17-18,21 1.19,21 1.20,21 X V I . 1-11 XVII.1-11 XVn.l3-I4 X.I-7 I X . 1-27 X V I I . I 1-12
VASI§THA-SIDDHANTA Ravi-sphuta
II. I
II.2-6 II.7 II.7 II.8 II.9-10 ILIl-13 VI.I-I4 X V I I I . 1-60
XII.I-2 XII.3 XII.3 XII.4 XII.5
With regard to the Chapters and verses not included i n the above table, some, like chapters X I to X V , and several verses, are o f a general nature, applicable to all the siddhantas and i n the case of the others it is difficult to identify positively the siddhantas to which they pertain. 11. Comparative Study of the Siddhantas It can be seen from the above Table that several of the topics selected by V M from the different siddhantas for depiction in the PS are common. T h i s should enable a comparative study of the common topics, the more important o f which are noticed i n the Table below.
In this connection it is necessary to point out, an erroneous view current, to the effect, that the Paulisa siddhdnta is accurate, the Romaka is equally accurate, the Saura is still more accurate and tha the Vasistha and Paitdmaha highly inaccurate. T h i s view has been brought into vogue through two unwarranted emendations introduced by T S and N P into the text of the following verse i n the PS: Paulisatithis sphuto 'sau tasydsannas tu Romaka-proktah \ spastatarah sdvitrah parisesau duravibhrastau \ \ PS. 1.4 'The tithi resulting from Paulisa is tolerably accurate and that o f the Romaka approximate to that. T h e tithi of the Saura is very accurate. B u t that of the remaining two (viz., die Vasistha and the Paitamaha) have slipped far away (from the real).' H e r e V M speaks only about the tithi, (lunar day), as computed d u r i n g his time according to the different siddhantas; H e is not making a relative estimate ofthe siddhantas as such or i n respect of the various other computations. As elucidated i n the Note to this verse i n the body of this book (p.5):
"The five Siddhantas are compared here with reference to their tithi alone because that is the chief of the five afigas, viz., tithi, vara, naksatra, yoga and harana; that is most useful not only religous purposes but also for civil purposes; that is independent of the origin o f reference in the ecliptic and can be examined for correctness by observation of eclipses and heliacal rising; and that is used i n finding the days from Epoch, ihesine qua non of all astronomical computation. "This being the case, the change o f tithi into krta by the late D r . 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. D o i n g this, they have condemned the Vdsistha Siddhdnta 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 ( N P for short). See below. Explanatory Notes, for die real reason for this 'slipping away fr®m the real'." O n this subject might be advanUgeiously referred to the section 'The place of the Vasistha i n the history of H i n d u astronomy' i n T . S . K . Sastry's paper T h e Vasistha Sun and M o o n ' i n the/Z. of Or. Research, Madras, 25 (1955-56) 19-41, reprinted i n his Collected Papers on Jyotisa (Tirupati, 1989, pp.I-28)
Table of Common topics Ahargana Paulisa Romaka Paitamaha
I.I1-13 1.8-10 XII.1-2
III.16 II.7
Paulisa Vasistha
ni.1-3 n.i
Candra-sphuta Paulisa Vasistha
III.4 II.2-6
Aharmana Paulina
III.11-12
III.20-22 X1I.4
Candra-grahana Vasistha-Paulisa Saura
Ravi-sphuta
n.8 xn.5
Vyatipata-Vaidhfti Paulisa Paitamaha
Naksatra-Tithi Paulisa Vasistha
Vasistha Paitamaha
VI.1-14 X.I-7
Ravi-grahana Paulisa Romaka Saura
V I I . 1-6 vni.1-18 I X . 1-27
Grahastodaya Vasistha-Paulisa Saura
xvni.i-60 X V I I . I 1-12
12. Varahamihira: His l i f e and Works One ofthe foremost early Indian astronomer and astrologer, Varahamihira belongs to the sixth century A . D . I n the PancasiddhdntUtd (1.8) he takes the cut-off date or epoch for computations using the Paulisa Siddhdnta as Saka 427, which corresponds to A . D . 505. Since the practice i n Indian astronomical manuals (Karatia-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 P S was composed some time after A . D . 505. Regarding his demise there is a statement by Amaraja i n his commentary on Brahmagupta's Brdhmasphutasiddhdnta, which reads : navddhika-pancasarhkJiydsake Vardhamihirdcdryo divamgatah."In saka 509 Varahamihira attained to the heavens.' This would mean that V M passed away i n A . D . 587. T h i s date is corroborated by V M ' s mention i n P S X V . 10, of Aryabhata who composed his Aryabhatiya i n A . D . 499, which work should have become well known by the time that V M composed his P S .
XXVIII
PANCASIDDHANTIKA
Personal details about V M are forthcoming from his own writings as also from those of others. Towards the close of his Brhajjdtaka, V M says: Adityaddsa-tanayas tadavdptabodhah Kdpittakah savitrlabdhavaraprasddah | Avantiko munimatdny avahkya samyag Hordm Vardhamihiro ruciram cakdra \ \ (Br.J. 26.1 (Edn. Triv. Skt. Ser., No.91) T h u s V M was the son of Adityadasa; he learnt the sastra from his own father; his native place was Kapitthaka; he was blessed by L o r d Sun (at Kapitthaka). H e (later) resided at A v a n d (Ujjain) where he composed his v/ork Hord (Brhajjdtaka). O n Kapitthaka V M ' s commentator Utpala says : Kdpitthdkhye grdmeyo 'sau bhagavdn savitd suryah, tasmdt labdhah prdpto varah prasddo yena lage of Kapittha where he received the blessing of G o d Sun'). Kapitthaka, the nadve village of V M , has been idenufied by Ajay M i t r a Shastri (vide his India as seen by Varahamihira, ( M L B D , Delhi, 1969, p. 19) o n the basis ofthe mention thereof by 7th century Chinese traveller Y u a n Chwang with Kapittha populary known as "Sartkasya (modern Sankisa) i n the Farrukhabad district o f Uttar Pradesh...."
Utpala states in his commentary on V M ' s Brhatsarhhitd (1.1) that V M was a 'Magadhadvija' : tad ayam apy Avantyakdcdryo MagadhMvija-Vardhamihiraharkahbdhavaraprasddojyotissdstrasungra (Ed. Sarasvati Bhavana Granthamala, Varanasi, 1968, p.2). Utpala makes such a statement also in his commentary on V M ' s Yogaydtrd. T h i s and the surname 'Mihira' which is borne by many Sakadvipa brahmanas, who are worshippers ofthe Sun, would indicate that V M belonged to this clan of Brahmanas whose forefathers migrated to India from the Maga country in Persia and settled in the village of Kapitthaka whence V M came to the city of Ujjain where he wrote his works. By all accounts, Varahamihira had the Sun as his tutelary deity. T o quote A . M . Shastri (op. cit., pp. 20-21): 'That Varahamihira was a devotee of the Sun admits of no doubt. H i s father's name was Adityadasa, his own name-ending ' M i h i r a ' , derived from 'Mithra', the Iranian Sun-god, his obtaining a boon from the Sun, his obeisance to the Sun i n all his works except the Vivdhapatala, (which, appropriately enough, opens with an invocation to Kama, the Indian god o f love), and his devoting a comparatively larger number o f verses to the description of Surya icons, all indicate that the sun was his family deity. H i s son Prthuyasas also invokes the Sun i n the opening verse o f his SatpancdJikd. A s we have seen, Varahamihira was regarded as an incarnation of the Sun.' T h e fame of V M has given rise to several legends about his birth and incidents i n his life, including his being a courtier o f K i n g V i k r a m a and one of the nine gems (nava-ratnas) i n his court. A l l these have to be considered as more fable and eological, not based on facts. Varahamihira was an astute astronomer and astrologer and wrote extensively on all the three branches of the science, viz., Tantra or mathematical astronomy, Hord (Jdtaka) or horoscopy, and Sarhhitd or mundane or natural astrology. It is interesting that for all his major works. V M has prepared abridged versions also for the benefit o f those who desist from works which are too lengthy, who, as Utpala says, are vistaragrantha-bhiru-s.
INTRODUCTION
XXIX
O n Tantra the major work o f V M is the Pancasiddhdntikd i n eighteen chapters. It would seem from a statement of Utpala towards the beginning of his commentary on VM's Laghujdtaka that V M had prepared on abridgement also o f that work Cf. Vardhamihira jyotissdstrasangraham krtvd tadeva vistara-granthn-bhtrunarh krte' 'sanksiptam ganitasdstram' krtvd hordsdstram vaktukdmah e O n horoscopy V M has produced two works, the Brhajjdtaka, called also Hordsdstra i n 26 chapters, and its abridged version, the Laghujdtaka, called also Svalpa-jdtaka and Sdksmajdtaka, i n thirteen chapters.
O n natural or mundane astrology also V M has two works, the Brhatsamhitd called also Vdrdhisarhhitd in 106 chapters and Samdsasamhitd, known also as Laghu-samhitd and Svalpa-sarhhitd 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 Brhatsamhitd. Vatakanikd, which exists only i n the form of quotations, is a work of V M on omens.
O n military astrology, three works of V M are available; (i) Mahdydtrd, known also as Brhadydtrd, Brhadyogaydtrd, Yaksyesvamedhikdh-yatrd (based i n the commencing expression yaksye 'svamedhe vijitya i n the second verse of the work), (ii) Svalpaydtrd or Tikanikdydtrd, and (iii) Yogaydtrd. O n marital horoscopy, V M has written a work endtled Vivdhapatala, a n d according to Utpala, there is also a Svalpavivdhapatala (vide, his com. on Brhajjdtaka X X . 10). M o r e than 30 more texts are mentioned, i n manuscripts and elsewhere, to have been composed by V M (Cf. A . M . Shastri, op. cit., pp. 29-31) but these lack authenticity i n their ascription. Alongside his wide range of scholarship, V M ' s writings are also characterised by chaste language, brevity and linguistic elegance. H e is a master not only of expression but also of metre. In illustration of his poetic talents one might refer to the figures of speech expressed through verses X I X . 13-15 of the Brhatsarhhitd describing Agastyodaya, the rising of the star Agastya. In the same work, Brhatsamhitd, he utilises the entire chapter 104, containing 64 verses, the Gocarddhydya ('Transits of planets'), to illustrate the metres, including the dandaka-s, alongside depicting the subject proper. It is also instructive that the names ofthe several metres are also deftly incorporated i n the verses by means of slesa or double entandre. Utpala is not, perhaps off the mark when he says, towards the commencement of his commentary on Brhatsamhitd, extolling V M as an incarnation ofthe S u n : Yac cdstram savitd cakdra vipulam skandhatrayair jyautisam tasyocchittibhaydt punah kaliyuge sarhsrjyayo bhdtalam \ bhuyah svalpataram Vardhamihira-vydjena sarvam vyadhdd ittham yam pravadanti moksakusalds tasmai namo bhdsvate 'The science of Jyautisa i n its triple aspects (oi Tantra, Jdtaka, and Sarhhitd) was propounded at length by G o d Sun. Fearing that it would be lost i n the K a l i age, G o d Sun incarnated i n the world in the form of Varahamihira and expounded all (the said three aspects) again i n shorter form. So say about the Sun those who are knowledgeable about salvation. Obeisance to that Sun.' Madras January 4,1993
K.V. SARMA
PANCASIDDHANTIKA
Chapter One INTRODUCTION OF THE WORK
cb
Aim of the Work 1-2. A f t e r s a l u t i n g , at the outset, w i t h great d e v o t i o n , the v a r i o u s great sages like S u r y a , V a s i s t h a , a n d others, a n d m y father a n d teacher w h o t a u g h t m e this sastra, I s h a l l state i n f u l l the best o f the secret l o r e o f a s t r o n o m y extracted f r o m the d i f f e r e n t schools o f the a n c i e n t teachers so as to be easy a n d clear. Mss used A , . B O R I , Poona, Ms N o . 338/1879-80; A ^ . National Library, Calcutta, M s N o . 49. B , . B O R I , M s N o . 37/1874-75; B^. O r . Inst., Baroda, Ms N o . 7165; B , . 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 o f PS by Naugebauer - Pingree, Munksgaard, 1970. External Testimonia (E) Jy. M. N. Pa.
J
PS Quotations in theJyotirmimdmsd of Nilakantha Somayaji Quotations i n Makkibhatta's C o m . o n the Siddhdntasekhara ofSrTpati Quotations in Nilakantha Somayaji's C o m . o n the Aryabhatiya Quotations i n Paramesvara's C o m . o n the Aryabhatiya
Pr. S. U.
Quotations in Prthudaka's C o m . o n the Brdhmasphutasiddhdnta Quotations in Siiryadevayajvan's C o m . on the Aryabhatiya Quotations i n Utpala's C o m . o n the Brhatsamhitd. (BS).
A 1 . 2 . Begin with: ?ft
PANCASIDDHANTIKA
4
1.2
In the second verse, the letter ka has been added to supply the one syllable wanting and in keeping with the sense. T h e Sun being the Atman 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 Siddhantas on astronomy, viz. Surya, Soma, Pitamaha, Vasistha, A t r i , Parasara, Kasyapa, Narada, Gargya, Marici, M a n u , Aiigiras, Romasa, Paurukutsa, Cyavana, Yavana, B h r g u and Saunaka and, by saluting these, the author salutes all ancient authors who follow these Siddhantas.
The Five Schools of Astronomy 3. T h e five Siddhantas, o f w h i c h this w o r k is a c o m p e n d i u m , are the Paulisa, the Romaka, the Vdsistha, the Saura a n d the Paitdmaha. O f these five, the first two, v i z . , the Paulisa a n d the Romaka have b e e n c o m m e n t e d u p o n by Latadeva.
O f the Siddhantas here mentioned, B r a h m a is the author of the Paitdmaha; Vasistha, that o Vdsistha; Paulisa that of Paulisa; Romaka that of Romaka; and Surya, that of Saura. From a dialog between Surya and A r u n a , it can be learnt how these five Siddhantas were given to their respective recipients. According to tradition, at the first instance, Brahma saw this lore of astronomy embedded in the Vedas and extracted it i n the form of the Paitdmaha. H e taught this to his son, Vasistha at the behest of V i s n u and again to Surya who was created with the express purpose of giving T i m e to the Universe. Vasistha gave this lore to his son, Parasara who, in turn gave the Parasara Siddhanta to th sages. O n e sage, Paulisa taught this to the sages Garga etc. and this is the Paulisa Siddhdnta. Surya himself, being born among the Yavanas by the curse of Brahma, taught the science :o Romaka and Duryavana in the city of Romaka, and Romaka propounded it as the Romaka Siddhdnta. T h u s thes five Siddhantas are the most ancient.
It is to be noted here that the five Siddhdntas 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. T h e Paulisa used by the author is different from the Paulisa quoted by Bhattotpala in his commentary on the Brhatsamhitd which latter agrees with the Saura o f our author and disagrees with his Paulisa. T h Romaka and Vdsistha now extant are different from those o f V M , agreeing as they do with the now well-known SUrya Siddhdnta, (called by scholars as the 'Modern' or 'Later' Sdrya Siddhdnta to guish it from the ancient Saura Siddhdnta). T h e author's Saura does not agree with the 'Mode Surya Siddhdnta, though one would expect agreement from the similarity in name, but it agrees wit a work of the ancient Aryabhata, now lost to us, and called by his commentator Bhaskara I as the Ardharatra-Paksa, which again is the basis of the Khanda-khddyaka-karana o f Brahmagupta. As 3. Quoted in the JyodrmimamsaQy) of Nilakantha, p.7.
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1.3
I. I N T R O D U C T I O N O F T H E W O R K
5
the Paitdmaha, there are several works now extant claiming Pitamaha or B r a h m a for their author. One is the Brahma Siddhdnta given by B r a h m a to Narada, which follows the 'Modem' SUrya Siddhdnta in its constants. Another is the Pitamaha Siddhdnta, forming a part o f Visnudharmottara, which has been taken by Brahmagupta as the basis o f his Brdhma-Sphuta Siddhdnta. A third one, now lost, is the basis of the Aryabhatiya. But the Paitdmaha o f our author is different from all these. As for the Latadeva mentioned here, he is the Latacarya referred to i n X V . 18 o f this work, for, there, the author says that this Acarya has taken sunset at Yavanapura as the beginning o f the day and from 1.8. we understand that the Paulisa and Romaka do the same and here it is mentioned that Latadeva is the commentator o f these two Siddhdntas.
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1.7
PANCASIDDHANTl KA
Contents of the Work 5-7. I s h a l l tell i n this w o r k , a v o i d i n g a l l j e a l o u s y , the c o m p u t a t i o n o f the solar eclipse, w h i c h is g u a r d e d as a great secret a n d i n w h i c h the m i n d o f the a s t r o n o m e r reels. I s h a l l also tell the o c c u r r e n c e o r n o n - o c c u r r e n c e o f the l u n a r eclipse, the d i r e c d o n s o f the first a n d last contacts, the d u r a t i o n , the total phase, the 'hypotenuse' at any m o m e n t w i t h related quantity o f obscuration a n d t i m e a n d also the m u t u a l c o n j u n c t i o n s o f the stars a n d the planets a n d the c o m p u t a t i o n o f differences i n l o n g i t u d e as also the p r i m e v e r t i c a l , m o o n r i s e , a s t r o n o m i c a l i n s t r u m e n t s a n d o t h e r requirements, g r a p h i c a l representations, the g n o m o n i c s h a d o w , the sines o f l a t i t u d e , co-latitude a n d d e c l i n a d o n s a n d s u c h o t h e r matters.
T h e textual recording tddavacchdya is emended as sankavacchdyah because (i) tddava is meanin less, and it may be a corruption o f sdnkava meaning relating to the sanku or gnomon which is suggested by the juxtaposition with chdya meaning 'shadow' and (ii) chdya must be chdyah because grammer requires the accusative case of the word. T S , N P take the word as sdnkavacchdyd, without the final visarga. As for the mention of the computation o f the solar eclipse as a 'great secret' it is because o f the diffKulty ofthe 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. T h e technical words that occur here like prime vertical etc. will be explained in their respective contexts.
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1.10
7
I. I N T R O D U C T I O N O F T H E W O R K
Days from E p o c h according to Romaka 8-10. D e d u c t 4 2 7 f r o m the S a k a y e a r (elapsed) o f the t i m e t a k e n . M u l t i p l y the r e m a i n d e r by 12. A d d the m o n t h s g o n e , c o u n d n g f r o m C a i t r a . P u t this result i n two places. I n o n e place, m u l d p l y it by 7, d i v i d e by 228 a n d take the q u o t i e n t w h i c h constitute the i n t e r c a l a r y m o n t h s . A d d this to the result k e p t i n the s e c o n d place. ( T h e total are the s y n o d i c m o n t h s gone.) M u l t i p l y this by 30 a n d a d d the tithis c o u n t e d f r o m sukla-pratipad to the c u r r e n t tithi. P u t the s u m i n two places. I n o n e place m u l d p l y by 11, a d d 5 1 4 , d i v i d e by 703 a n d take the q u o t i e n t , ( w h i c h constitute the e l i d e d days o r avamas). D e d u c t this f r o m the s u m p u t i n the o t h e r place. T h e r e m a i n d e r are the ' D a y s f r o m E p o c h ' (dyugana), the m o m e n t o f E p o c h b e i n g m i d - s u n s e t at Y a v a n a p u r a , b e g i n n i n g M o n d a y w h e n the first tithi o f C a i t r a was a b o u t to b e g i n . T h i s r u l e is a c c o r d i n g to the Romaka. It c a n be t a k e n as the Paulisa r u l e also, p r o v i d e d the t i m e t a k e n f o r c o m p u t a t i o n is n o t v e r y far f r o m the E p o c h , (or the p a r t o f the r u l e f o r avama m a y be u s e d f o r the Paulisa also, p r o v i d e d the t a k e n date is n o t v e r y far f r o m the E p o c h ; or i n the Paulisa too the m o v e m e n t o f E p o c h is m i d - s u n s e t at Y a v a n a p u r a , b e g i n n i n g M o n d a y . )
Example 1 . Find the Days from Epoch for Tuesday the sixth day of the dark fortnight of Asddha, Sa (elapsed). Saka year (elapsed) of date is 499. 499 - 427 = 72 years gone. Months gone = 72 x 12-1- months counted from Caitra upto Asadha = 72 X 12 -I- 3 = 867. 867 X 7 - 228 = 6069 -r- 228 = 26 141 (=Q)+ (=Rem) 228
867 A d d i n g the quotient Synodic months gone
26 893
T h e tithis = 893 X 30 -I- tithis i n the current month = 893 X 30 -I- 21 = 26,811 (26,811 X 1 H - 5 1 4 ) ^ 7 0 3 = 4 2 0 ( = Q ) - H 175/703 (= Rem) Deducting the quotient
26,811 420
Days from Epoch gone
26,391
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PANCASIDDHANTIKA
1.10
Dividing 26,391 out by 7, the remainder got is I, 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.) T h e 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 1.15). 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 1.17). N o w , because the Epoch is 427 Saka (elapsed), by deducting 427 from the Saka 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 X 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. T h e intercalary months d u r i n g this period is obtained by proportion from the solar months and the intercalary months o f theyw^a, viz. 34,200: 1050 :: the solar months gone: the intercalary months during the period. T h u s we have the equation, the intercalary months = the solar months gone X 1050 ^ 34,200. T h e 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 o f the yuga. T h e number o f months gone frotn 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 X 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. T h e rule for 'Days from Epoch' does not mention any constant {ksepa) 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 Y u g a began, wherefrom the fraction required can be obtained, still from the constant for the mean Sun and M o o n in Chapter V I I I 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. F r o m this we learn that 65/150 days, (i.e. 26 nadikas) after Epoch, the mean solar month ends and therefore at Epoch the mean Sun is 1 F 29 ° 34 ' 30 ". Again, from the constants in the fourth verse giving the mean M o o n , we can learn that the mean M o o n at E p o c h is 11' 26° 12'. F r o m these, it can be computed that the mean new moon occurs about 16 '/2 nddikds after Epoch. As the interval from new moon to the end of the solar month is the fraction of intercalary month, we get 26 - 1 6 V2 = 9V2 nadikas, as the fraction. As for one intercalary month consisting of about 29 Vz days there are 228 parts as constant, for 9 '/a nddikds we get 1 as constant. T h i s 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 nddikds. That is why i f an intercalary month has actually fallen i n the current year before the taken time, we take the fraction of the computed intercalary month as whole and add one, and i f no intercalary month has fallen we omit one from the computed months when the fraction left over is small. T o 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
1.10
9
I. I N T R O D U C T I O N O F T H E W O R K
the current month, the total tilhis 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, i f 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 the intervening elided days. So, we have the equation, the intervening m/jM X 16,547 -r 10,57,500 = the elided days. Here the multiplier for the tithh, 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 1 1 1 4 i.e.
16,547/10,57,500 =
J 63+
16547 1508 41 9 4 0 _ i 1+
9+
l
l 1+
J 35+
_ 1+
1057500 15039 1467 32 5 1 l l i 3+
1+
63 9 35 3 1 l 4+
The successive convergents obtained from this are: 1/63,1/64,10/639, 11/703,395/25244 etc. O f these the author has taken 11/703 as being simple and, at the same time, sufficientiy accurate for the purposes of this work, for even d u r i n g a period as large as xheyuga, 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 ofthe 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. W e have seen before that at Epoch there remains 16'/2 nddikds for the mean new moon to end, i.e. about 43 nddikds have ended in mean Amavasya tithi. The constant 514 means that 514/703 part o f the Amavasya has gone and this is equal to about 43 nddikds and thus the constant is practically correct. It is because of the existence of this constant that we have interpreted, caitra-suklddau as 'when the first tithi of Caitra was about to begin'. Further, we have seen that at Epoch Amavasya is current and Caturdasi is gone. But, taking the Amavasya as gone, the tithis to be used i n 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 Sukla-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 instruction 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, i f necessary, which will take care of everything. T h u s the whole thing is explained. The Saka year is the year of the Saka era which began at 3179 Kali (elapsed), for the Siddhdntas instruct that 3179 should be added to the Saka year to get the K a l i year. T h e purpose of mentioning that Caitra Sukla Pratipad occurred near the Epoch is to indicate that the months gone must be
10
PANCASIDDHANTIKA
1.10
counted from Caitra and the tithis from Sukla Pratipad. T h e moment of Epoch is given as midsunset 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. T h e 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 Siddhdntas like the Surya Siddhdnta begin the day at midnight, which is used for certain injunctions of the Dharma-sdstras, while certain other works like the Aryabhatiya etc. begin the day at sunrise which is used for certain other injunctions of the Dharma-sdstras. 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. H o w do we know that it is Alexandria and no other city? In III. 13 the dme-difference between Yavanapura and Ujjain due to their difference in longitude is given as seven nddis and twenty vinddis and sunset at Yavanapura is later. F r o m 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 o f Epoch begins Monday, somadivasddye. T h i s reading is that of Bhattotpala, quoting the verse i n his commentary ofthe Brhatsamhitd and we have adopted it as the correct one. It does not matter i f we adopt another reading, saumyadivasddye, for we can interpret this as 'the day pertaining to the M o o n ' , i.e. Monday, because the word saumya can be interpreted as 'belonging or pertaining to the M o o n ' . 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 L o r d of that day as computed from 1.20 is the M o o n and not Mercury. W e shall show how. I n 1.17 it is instructed that 2227 should be added to the Days from E p o c h to get the lords of the year, month, day andhord. Because the Days from Epoch gone is patendy zero at the Epoch itself, we have 2227 + 0 = 2227, from which to get the L o r d o f the day. T h e instruction is to divide this out by seven, and take the remainder, which gives the L o r d o f the day gone counting from the Sun, in the order Sun, M o o n , Mars etc. Now we want the L o r d ofthe 2228th day, and dividing 2228 by 7, the remainder is 2, i.e. M o o n is the L o r d of the day and it must be Monday. T h i s can be shown in other ways also but this is enough here. W h e n there is this fact of a Monday and the reading somadivasddye to support it, the interpretation by some as 'at the beginning o f Wednesday' has to be discarded. There is another reading, bhaumadivasa which has been accepted by the two scholars, S.B. Dikshit and B h a u Daji, and also by N P , 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 ofthe statement caitrasuklddau, as such that reading has also to be discarded. Note also that the Romaka ahargana mentioned i n verse 17 below, viz. 2227, works out only to Monday, not Tuesday, since the cycle commences from Sunday. We have given as one interpretation oindticire Paulise 'pyevam, 'It can be taken as the Paulisa rule also, provided the time taken for computation is not very far from the Epoch'. Strictly speaking, in the rule given by a particular Siddhdnta, only the synodic month and the tithi of that Siddhdnta must be used to get the Days from Epoch. B u t as given i n 1.4, the tithi o f the Romaka was near that of Paulisa at the time of E p o c h and so the Romaka rule could be used for the Paulisa for some time, especially because there is the check by comparing the week-days. A n o t h e r thing to be noted is this: Whatever Siddhdnta 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 Vdsistha or for the Saura, for we can use days of the Romaka or Paulisa for these also, mutatis mutandis.
T S interpret ndticire Paulise'py evam as 'the rule is the same for also the Paulisa Siddhdnta which w
1.10
1. I N T R O D U C T I O N O F T H E W O R K
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 Paulisa 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 Paulisa, though they are quite capable of understanding and interpredng them. N P translate, 'It is not very different in the Paulisa', without explaining ndticire.
Days from E p o c h according to Paulisa 11. ( T h e f o r m u l a f o r D a y s f r o m E p o c h a c c o r d i n g to the Paulisa, is as follows:) A s i n R o m a k a (1.8-10), d e d u c t 4 2 7 f r o m the S a k a year (elapsed). M u l d p l y by 12 a n d a d d the m o n t h s g o n e f r o m C a i t r a . M u l t i p l y by 3 0 . T h e ' S o l a r days' (S-days) to the e n d o f the c u r r e n t solar m o n t h are got. M u l t i p l y the S-days by 10, a d d 6 9 8 , a n d d i v i d e by 9 7 6 1 . T h e q u o t i e n t are the i n t e r c a l a r y m o n t h s . ( A g a i n , as i n R o m a k a ) , m u l t i p l y the m o n t h s got by 30 a n d a d d to the S-days, a n d a d d also the tithis f r o m s u k l a - p r a t i p a d , inclusive o f the c u r r e n t tithi. T h e s u m is the tithis g o n e f r o m E p o c h . M u l t i p l y this by 11, a d d 4 4 4 {tri-krta) and divide by 703. T h e quotient are the elided days. Deduct this from the tithh gone. T h e remainder are the Days from Epoch. Here the word divasdh is interpreted as ravi-divasdh, i.e. 'solar days', because it comes i n the place of 'solar months' in the formula. T h e 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 i n modern astronomy, the time interval taken by the Sun for the successive crossing of the meridian. T o avoid error of syntax, 'sdstdnavarasa' is emended into 'sdstanavarasd'. Following the sense, in the place o f kurtu and rutu, the reading ekartu is substituted. N P editorially add before divasdh the word saura, which is not necessary, as it can be inferred. NP's translation gives the number 9761 with an emended reading kvrtusaptanava. A g a i n , the ms. reading tri-krta has been changed to tri-sat, with the translation, 'there is an omitted dthi every 63 days', missing to see thattri-krta (444) is the Paulisa ksepa in place of the Romaka ksepa 514 of the previous verse, to be used i n the Paulisa calculation.
lla.
AI.2.D.fe(I:;C.f^ A1.2.B1.2.W
A1.2.f| B l . 2 . ^ c. C.1^Rt3;D.1?r[^]
AI.2.C.^1^; D . ^ T ^ : [^]f^°
d. B2.°Hli^c|H°
b. AI.2.B1.2.om.tT
A1.2.tf^
D . [fg]
12
PANCASIDDHANTIKA
f?rf8I^(W)
1.13
'WTi3|W^:
^ 5 g ^ ^ ' w c ) ^ t d ( ^ ( ^ < a ^ A ' ^ 1 1 ^ ^ II 12-13. ( F o r g r e a t e r a c c u r a c y i n finding the i n t e r c a l a r y m o n t h s by the above f o r m u l a , ) a d d 1/10 S-day to the S-days f o r every 107 years a n d f o r still greater accuracy, a d d a n e x t r a 1/10 S-day f o r every 5 5 0 s u c h 1/10 S-days a d d e d , (i.e., f o r 55 S-days a d d e d ) . I n the same way, i n g e t t i n g the e l i d e d days, f o r every 2 4 5 years a d d 1/11 tithi to the tithis g o n e a n d f o r every 2 , 0 3 , 2 7 9 such a d d i t i o n s omit one addition. What is the purpose of the author in aiming at such accuracy? T h e author intends that the Paulisa should be used by people for a very long time as can be seen by his giving it fully i n this work. (He gives the Saura also fully with the same view.) So he wishes to secure greater accuracy for the Days from Epoch got by the Paulisa. Example 2 . The day taken is Monday, Mdrgasirsa Amdvdsyd, Saka 1821 (elapsed). Find the Epoch, for the moment of the beginning of the day. T h e S-days to the end o f the Solar month in the current synodic month = {(1821 - 427) 12 -I8} X 30 = 5,02,080. T h e tenths of solar days to be added to this for computing the intercalary months accurately = (1821 - 427)-h 1 0 7 = 13. A d d i n g 5,02,080-f 1 3/10 = 5,02,081 3/10. Multiplying by 10, adding 698 and dividing by 9761, the intercalary months got are: (5,02,0813 -I- 698) -^9761 = 514 4357/9761. T a k i n g the full months, multiplying by 30, adding the tithis inclusive of Amavasya and adding to the S-days, the tithis gone are 514 X 30 -I- 30 -f 5,02,080 = 5,17,530. T h e elevenths o f tithis to be added to this for accurately computing the elided days = (1821 - 427) 245 = 5. A d d i n g to the tithis gone, multiplying by 11, adding 444 and dividing by 703, the elided days obtained are {(5,17,530 -I- 5/11) X 11 -I- 444 } -r- 703 = 8098 3857703. Deducting the full elided days from the tithis gone, the Days from Epoch gone = 5,17,530 - 8098 = 5,09,432. Dividing out by 7, the remainder is 0, i.e. Monday is beginning. This agrees with the weekday of the taken time, viz. Monday, and therefore the Days from Epoch are 5,09,432. Now, for the explanation ofthe Paulisa rule. Before proceeding, the following things should be noted: (1) T h e text here is badly vitiated, (2) the original Paulisa has been lost, and no Siddhdnta with I2a. A I . ^ S T ^ " ; A 2 . ^ - g a p J 7 ? ° BI.2.
d. A1.2.f?M^9g:; B1.2.C.D.f?raTfl9g; 13a. B1.3tf«Rra% B2.3rf«wra% b. D . [ c ^ ^ t g O W ] ;
Al.BI.C.-qFEIfai; A 2 . ™ n c-d. A l . f l f f ^ ; A 2 . f § ^ ; B I . 2 . f l ; ^ ;
B1.2.°%l%;^ D . ^ f ^ q i ^ c.
BI.2.°^
13
1. I N T R O D U C U O N O F T H E WORK
1.13
the constants of the original Paulisa is extant so that the constants here given cannot be verified directly; (3) the condensed Paulisa of our author does not give the mean M o o n . But as the Vdsistha and the Paulisa are found mixed together in this work and as some peculiar technical terms are found common to both, we can assume that they are connected in some way and the Vdsistha M o o n is intended for the Paulisa also. N o w we can explain the rule. From l l l . l we learn that there are 43,831 days in 120 Solar years. F r o m 11.3 we learn that in a period of 3031 days, called a Ghana, the mean M o o n moves through 110 full revolutions 11 signs, 7 degrees and 32 minutes. Assuming that no fraction of a minute has been added or left out, the modon for tlie period in revolutions is 110+ 11/12 + 7/12 X 30 + 32/12 X 30 X 60 = 110 5063/5400 = 5,99,063/5400. T h e days for 5,99,063 revolutions are, 5,99,063 X 3031 ^ (5,99,063/5400) = 5400 X 3031 =1,63,67,400. For 1,63,67,400 x 43,831 = 7,17,39,95,09,400 days, the Moon's revolutions are 5,99,063 x 43,831 = 26,25,75,30,353. In the same way, the mean Sun's revolutions in the period of 7,17,39,95,09,400 days are 120 x 1,63,67,400 = 1,96,40,88,000. Therefore, using 1.16, in a period of 7,17,39,95,09,400 days, there are 26,25,75,30,353 revolutions ofthe M o o n , and there are 1,96,40,88,000 revolutions ofthe Sun. Subtracting, there 24,29,34,42,353 synodic months, Multiplying Sun's rev. by 12 there are 23,56,90,56,000 solar months, Subtracting, there are 72,43,86,353 intercalary months. Multiplying synodic months by 30 there are 7,28,80,32,70,590 tithis. Now, the days are, as above 7,17,39,95,09,400. Subtracting the days, there are 11,40,37,61,190 elided days. From this we can see that in a period of 23,56,90,56,000 solar months, i.e. for 7,07,07,16,80,000 'solar days' there are 72,43,86,353 intercalary months. So, by proportion we find the intercalary months for the given solar days by 7,07,07,16,80,000 : 72,43,86,353 :: S-days : intercalary months, i.e. intercalary months = S-days X 72,43,86,353/7.07,07,16,80,000. T h e multiplying fraction here, viz., 72,43,86,353/7,07,07,16,80,000, can be expressed as a continued fraction thus = 10
724386353
707071680000
976
1
18391633
70599472
3
5
2967080
15424553
5
21315
589153
72,43,86,353/7,07,07,16,80,000 = — _ L - L _ L _1 _L 976+ 10+ 3+ 1+ 5+ 5+ T h e successive convergents are 1/976, 10/9761, 31/30,259,41/40,020 etc. O f these, our author has taken 10/9761, as being sufficiently simple and accurate. (Incidentally, this gives the justification for giving the corrected reading ekartusaptanava). By taking this fraction for use, for every S-day, 72,43,86,353/70,70,71,68,0000 - 10/9761 = 1,83,91,633/(7,07,07,16,80,000 x 9761) intercalary month is lost, i.e. in every period of 107 solar years, containing 107 x 360 S-days, the above fraction multiplied by 107 x 360 is lost, which reduces to (1 + 1214/6,54,696)/9761 = (1 + l/550)/9761 very nearly (or exactly, i f possibly 32' given above is a correction to the nearest minute). So this also
14
1.13
PANCASIDDHANTIKA
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 -tl/550)/9761 intercalary month got? T h u s we get S-days equal to, (1 -t- l/550)/9761 10/9761 = (1 + l/550)/10 = | p - f - | ^ x L _ . This is for every 107 years, and so, for every 107 years, 1/lOS-day has to be added for greater accuracy in getting the intercalary months and for every 550 such additions one more tenth is to be added, which is the instrucdon given. (This is the reason for our giving as the correct reading, 'tilhidasamdmsam 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 propordon, 24,29,34,42,353: 38,01,25,373 :: tithis gone; elided days d u r i n g the period, i.e. elided days = tithis gone X 38,01,25,373 ^ 24,29,34,42,353. T h e muldplying fraction 38,01,25,373/24,29,34,42,353 can be expressed as a condnued 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 = ^ 63+
^ 1+
^ 9+
^ 1+
126+
T h e successive convergents are 1/63, 1/64, 10/639, 11/703, 1396/89217 etc. O f these, our author has taken 11/703 (note that this is the same as that of the Romaka) as being enough for a first approximation. By taking this, 38,01,25,373/24,29,34,42,353 - 11/703 = 2,71,336/(24,29,34,42,353 x 703) elided day is left out for every tithi. In the period of 245 years, given i n the rule, there are, from the constants given before, 7,28,80,32,70,590 X 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 x 703} X {72,88,03,70,590 x 245 ^ 1,96,40,88,000} = 16,61,933/(16,36,740 x 703). This can be included i n the formula by making a proportionate change i n the tithi thus: T o get 11 elided days we have to take 703 tithis, to get the elided days left out i n 245 years, we must take tithis equal to 703 x 16,61,933 (16,36,740 x 703 25
193
11)= 1 6 , 6 1 , 9 3 3 ( 1 6 , 3 6 , 7 4 0 x II) = (1 + IQ^Q^J^QV^^ = 1/11 + 25,193/(16,36,740 x 11). In this the first term 1/11 is given by the instruction to add an eleventh oi a tithi every 245 years. T h e 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 modon 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. O r the Paulisa M o o n is slightly different from the Vdsistha M o o n , which we have assumed for the Paulisa. O r there is some error i n the text here. We must be satisfied with the other and more important items of agreement. It must be remembered here that T S have omitted even the translation of these two verses, as a hopeless task. X
1.16
1. I N T R O D U C T I O N O F T H E W O R K
15
Now we proceed to examine the hsepas used in the formula. A t the time of Epoch, the Vdsistha mean Moon is 11' 25° 6' (vide II.3). As done before, we assume this for the PaM&'o also. The Paulisa mean ('mean' here is the assumed mean) Sun is 1 V 29° 44' (vide III. I). F r o m these we can see that the mean new moon will occur after 23 nddikds. F r o m the ksepa for elided day given, 444, we can see that the end of the Amavasya occurs, before the beginning of the next day by 444 X 59/703 = 37 nddikds, i.e. 23 nddikds after the Epoch, and thus there is agreement. (This shows that the reading 'trikrtadindny avamasanksepah' is correct). We shall examine the ksepa for the intercalary month. T h e ksepa 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 6 V2 nddikds after new moon, the Sun enters the next rdsi, here Mesa. We have seen that the mean new moon itself falls 23 nddikds after Epoch. Therefore we get that the Sun enters Mesa 2 days 6 V 2 nddikds + 23 nddikds - 2 days 29 V2 nddikds after Epoch. T h e proper mean Sun computed for Epoch is 1 T 27° 33' (vide III. 1-3), i.e. after traversing 2 ° 27', i.e. after 2 days 29'/2 nat/i^ds, the Sun will enter Mesa. This is the same as what we have computed from the ksepa 698, and thus it is verified.
Perhaps the reader has noted here that in the verification of the ksepa for elided day we have used the assumed mean Sun (written 'mean' Sun) at Epoch and of the ksepa for intercalary month, the proper-mean-Sun a l 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 i n having recourse to it, thinks the author. We have already said that the mean Sun and M o o n can alone be taken i n framing the rule here. What we have called above, the 'proper-mean' is really the mean and so that part is all right. I f here the assumed mean Sun is used, which is practically the true Sun at Epoch, an intercalary Vaisakha will be falling immediately which will necessitate giving a ksepa almost equal to the divisor 9761 and cause a lot of trouble. So the author has done what is only proper here. T h e n why not use the mean Sun to get the elided day ksepa also? T h e Paulisa, 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. O u r author has taken it as it is given and computed the ksepa 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. O r , some astronomer, unaware ofthe illogicality, has handled the ksepa. While T S omit to translate the verses 11-13, merely stating that the details are obscure (Tr. p.5), N P change several ms. readings, dasamdrnsa to dasdrnsa, pancakrtadvisammitdh topaiicatanudvidvimitdh, ekikartum to eka rtu, without getting anywhere near the correct sense.
16
PANCASIDDHANTIKA
1.16
Yuga of the Sun and the Moon (Romaka and Saura) 14. I n the Saura Siddhdnta, a p e r i o d (actually the minor yuga) o f 1,80,000 solar years c o n t a i n s 6 6 , 3 8 9 i n t e r c a l a r y m o n t h s a n d 10,4.5,095 e l i d e d days. 15. T h e l u n i - s o l a r yug^a o f the Romaka Siddhdnta consists o f 2 8 5 0 solar years. In this p e r i o d , t h e r e are 1050 i n t e r c a l a r y m o n t h s a n d 16,547 e l i d e d days. 16. T h e solar years i n xheyuga m u l t i p H e d by 12 gives the solar m o n t h s i n the yuga. T h e solar m o n t h s p l u s the i n t e r c a l a r y m o n t h s are the s y n o d i c m o n t h s i n xheyuga. T h e tithis got by m u l t i p l y i n g the s y n o d i c m o n t h s by 30 r e d u c e d by the e l i d e d days, are the c i v i l days, (i.e. days) i n xheyuga. T h e civil days plus the solar years are the s i d e r e a l days i n the yuga (or the s y n o d i c m o n t h s plus the solar years are the M o o n ' s r e v o l u t i o n s i n Xheyuga).
Example 5. Give the revolutions of the Sun and the Moon, the civil days etc. in a yuga (minor) of the S Siddhdnta. There are 1,80,000 solar years in the Saura m i n o r yuga, and as a solar year is the period of revolution of the Sun, there are 1,80,000 solar revolutions i n xheyuga. M u l t i p l y i n g the solar years by 12, the solar months in a yuga are 12 x 1 , 8 0 , 0 0 0 = 21,60,000. T h e synodic months are solar months plus intercalary months = 21,60,000 + 66,389 = 22,26,389. T h e tithis are 30 x 22,26,389 = 6,67,91,670. T h e (civil) days a r e , / M w - e l i d e d days = 6,67,91,670 - 10,45,095 = 6,57,46,575. T h e sidereal days are, civil days plus solar years = 6,57,46,575 -I- 1,80,000 = 6,59,26,575. T h e lunar revolutions are, synodic months + solar years = 22,26,389 -I- 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 time of revolution of each, etc. Sun's revolutions = solar years = 2850. T h e solar months are, 12 X 2850 = 34,200. T h e synodic months are, 34,200 -I- 105o' = 35,250. T h e tithis are, 30 X 35,250 = 10,57,500. T h e civil days are, 10,57,500 - 16,547 = 10,40,953. T h e lunar revolutions are, 35,250 -I- 2850 = 38,100. Dividing the days i n Xheyuga 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 i n days, etc. 10,40,953 ^ 35,250 = 29-31 -50-5-37. Dividing the days by the lunar revolutions, the dme for one revolution got is, in days etc. 10,40,953 38,100 = 27-19-17-46. T h e following points should be noted. The Romaka Siddhdnta, now extant, agrees with xhe Modem Sdrya Siddhdnta i n its constants like the period of xheyuga, the number of revolutions of the planets in the Yuga etc. But the Romaka Siddhdnta condensed by our author is quite different and seems to 14a.
B 1 . ^ ; B 2 . ^
b. A2.°Tr^''
d. A l . 2 . ^ ; B l . W ^ ; B2.-P?fcl
c. Al.°M<=il^ll:; D . hciipR<4i°] c-d.
B1.2.^!ife!iT-gap?nfer«I
15a. B 2 . ^ f o r ^ B1.2.^Rti=^; b. B 1 . 2 . W ^ ( B 2 . ^ ) W :
B 1 . 2 . i R W ^ : A l . - g ^ ; A.2.'CT5r 16. Q u o t e d b y U l p a t a o n B S 2 , p . 2 9 a.
B1.2.^gT^^lft^
b. A1.2.^!#mra* d.
A1.2.C.D.^^B1.2.W^
1.16
17
I. I N T R O D U C T I O N O F T H E W O R K
be lost. Therefore we cannot determine whether the period of 2850 years mentioned here is the actual yuga ofthe original Siddhdnta or a m i n o r 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 o f solar years and intercalary months in the yuga by the factor, 150, we see that there are 7 intercalary months i n a period of 19 years or 228 solar months, which is the wellknown Metonic cycle. F r o m all this we can conclude that this Siddhdnta is from a Greek source.
In the case o f the Saura, the period o f 1,80,000 years given here is certainly a minor yug^a of the original Saura, for by multiplying this by 24 we get the number o f years in the yuga o f the original, viz., 43,20,000 years. F r o m this we can infer that i n theyuga o f the original there are 1,80,000 X 24 = 43,20,000 solar revolutions, 6,57,46,575 x 24 = 1,57,79,17,800 civil days and 24,06,389 x 24 = 5,77,53,336 lunar revolutions. W e have already mentioned that all these agree with the Ardhardtrapaksa o f Aryabhata given in the Mahdbhaskariya, with the Kharidakhddyaka which is based on Ardhardtrapaksa and with the Paulisa quoted by Bhattotpala i n his commentary on the Brhatsarnh but not with the M o d e r n and well-known Sdrya Siddhdnta.
Now what is the purpose of our author i n giving theyuga-elements of these two Siddhdntas alone? Our author expects that, like the Paulisa, 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 o f the Romaka itself, the accumulation o 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 o f 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 theyuga. T h e synodic month is the interval between two consecutive conjunctions of the Sun and the M o o n . In the Yuga the M o o n makes r revolutions and, therefore, in one day makes rid revolution. In the same way, the Sun makes Rid revolution. In one day they move r —R apart by — g — revolution. W h e n the separation equals one revolution they are in the next conjunction. T h e period o f separation equal to one revolution, i n days = 1/^^—^^ = is the length in days, of the synodic month
f_
, which
(1)
For dlir - R) days, there is one synodic month; for d days (i.e. the days of theyuga) there are di {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: T h e synodic months, Caitra etc. are those that end in the solar months Mesa etc., and there is normally one to one correspondence between the two sets. B u t as the synodic month is shorter than the solar it successively 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 i f the correspondence between the set Caitra etc. with the set Mesa etc. has to be maintained. T h i s is the Adhikamdsa or intercalary month.
18
PANCASIDDHANTIKA
1.18
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 theyuga. Therefore the length of a solar month in days = dll2R. T h e length in days of a synodic month, already derived, = dl(r — R). Therefore i n every solar month the end of the synodic month (i.e. the new moon) occurs earlier hyd/l2R - d/(r — R) = d(r - \SR )/l2R (r R). W h e n 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, i n terms of solar months, d/{r - R)^ {d{r - 13/?)/ 12R(r - R)} = 12R/(r — 13R). Therefore, i n the yuga containing 12R solar months the number of intercalary months i = \2RI {\2RI{r - 13/?)} = r - \m{r - R) - \2R=m -\2R. Therefore 12R + i= m (3), i.e. the solar months + the intercalary months give the synodic months. (iii) W e shall explain the occurrence of elided days and derive their number: T h e length of a tithi is a little less than a day and so every day the tithi occurs earlier and earlier in the day, u n d l the time so accumulated becomes equal to one tithi and gets immersed i n the day, with the result that the correspondence, one tithi to one day, is broken. Such tithis are left out by reckoning and are called 'submerged tithis' or 'elided days'. N o w , as there are i n theyuga d days and t tithis, the duration of one tithi = dit. In one day, the tithi falls earlier by I - dit day. T h i s accumulates to one tithi i n dit (1 — dit) =dl{t - d) days, which is the time for one elided day to happen. Therefore, the number of elided days happening in a Yuga = e = dl{dl(t -d))=t-d. Thereforerf = t-e (4), i.e. deducting the elided days from the tithis we get the days. (iv) W e shall explain the sidereal day and derive the number of sidereal days i n theyuga. T h e 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. A s there are n sidereal days and d days i n theyuga, i n one sidereal day the Sun makes din revolution. Therefore i n one sidereal day he lags behind by \ — din = {n - d)ln, revolution. T h i s lagging behind is due to the Sun's eastward motion in the Sky and its magnitude is the Sun's motion i n terms of revolutions d u r i n g a sidereal day. This is equal to RIn. Therefore, (u — d)ln = RIn. 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.
Lord of die year 17. A d d 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. A d d 1, multiply by 3, deduct 2 and divide out by 7. T h e remainder counted in the order Sun (Ravi), (Moon, Bhauma, Budha, G u r u , Sukra and
I. I N T R O D U C T I O N O F T H E W O R K
1.18
19
M a n d a ) is the L o r d o f the y e a r ( i n w h i c h the t a k e n day falls) (i.e. I f Q is the q u o t i e n t t a k e n , the n u m b e r to be d i v i d e d o u t by 7 is e q u a l to ( Q + 1) x 3 - 2). Example 5. The days from Epoch is 3479. Give the Lord of the year. A d d i n g the ksepa 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. T h e fourth from the Sun, B u d h a is the L o r d of the year. The processes mentioned here are explained thus: A t the moment 2227 days before Epoch, beginning Sunday, the days for calculating the L o r d 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 L o r d was the Sun. T o find these Lords for any dme, 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 o f calculating the L o r d of the Year, the sdvana year comprising 360 days is used by our author and the L o r d o f the first day of the sdvana year is the L o r d of the year. In the same way, to calculate the L o r d of the month, i\\e sdvana month of 30 days is used, the L o r d of the first day of the month being the L o r d 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 i n the same order. Hence the instruction to divide the days out by 2520 and take the remainder alone. T h i s remainder is set in 3 places to find the Lords ofthe year, the month and the day. T a k i n g the remainder of the days, the L o r d of the first year is that o f the first day, the L o r d o f the second year is that of the first day in the next year, i.e. ofthe 361st day, i.e. that o f the (358 + 3)th day, i.e. that o f the day three days after; the L o r d ofthe year next to that is that ofthe day 6 days after that of the first and so on. T h u s , the L o r d of the nth year is that o f (n - 1)3 + I, i.e. that of n X 3 - 2. If Q is the number of years gone, then n = Q + 1, and the L o r d is that of ( Q + 1)3 — 2, which is the rule given. A s the same L o r d is repeated by the addition o f 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 sdvana years and giving the L o r d of the first day ofthe year as the L o r d of the year is peculiar to o u r author. F o r others the L o r d o f the first day oi xhe saura year and for yet others that of Caitra Sukla Pratipad is the L o r d o f 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 C o m mentary). It has been hidden by another mistake made by h i m , viz., adopting the reading 'anghri' (= 2) but using the reading 'abdhi' (= 4) i n the derivation. T h e reading pratirdsca is really pratirdsya. Both N P and T S take the readingpratirdsi and moreover, S gives it the incorrect meaning5e5aw, 'remainder'.
17-18. Quoted by Utpala on BS 2.2, pp. 30-31. 17a.
B2.-3F'^
c. A 1.2.'5»fcRra; B 1 . 2 . ^ 9 ; C.D.3lRraf?I A1.2.cJ?^eT° d. A1.2.^TRnfcT
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4
20
1.20
PANCASIDDHANTIKA
Lord of the Month 19. Take the remainder set apart (as mentioned in verses 17-18), divide by 30 and take the quotient. A d d 1, multiply by 2 and deduct 1. T h e remainder, after dividing out by 7, is the L o r d of the month, counted from the Sun. T h e rule is ( Q + 1)2 — 1, where Q is the quotient taken. Here i n the place ofthe reading 'kdrydh' accepted both by T S and N P , we have adopted the reading vyekdh, given by Bhattotpala in his Br. Sam. commentary, as being the correct one and as necessary here. Also i n the place of prapanna Bhattotpala readspratipada. Whatever be the reading here, we want the meaning T. T h e rule is derived thus: As mentioned before for the L o r d of the year, to get the L o r d of the month the days are divided into sdvana month o f 30 days duration and the L o r d of the first day of the month is the L o r d of the M o n t h . T h u s the L o r d o f the very first day, viz. the Sun is the L o r d of the first month. As the days i n the month, 30, divided out by 7 leaves the remainder 2, the Lords of the successive months are those o f 2, 4, 6 etc. days after that of the first month, i.e. the L o r d of the nth month is given by (n - 1 )2 + 1 = n X 2 - 1. A s n is the current month, it is equal to ( Q + 1). Therefore n x 2 - l = ( Q - f - l ) 2 - 1, which is divided out by 7 gives the L o r d ofthe month. Here too the derivation of M . M . Sudhakara Dwivedi is wrong (vide his commentary on the verse, p. 6). T h e translation of both T S and N P are incorrect for having taken the reading Aarya/^ for vyekdh ('deduct 1'), not realising which N P complain: " T h e text's (1.19) 'increase the (resulting) months by the current one' should be replaced by 'discard the fractional part o f the current (month)' (Pt. II, p. 13, footnote). O n verses 17-19, K . S . Shukla has a detailed note i n his paper, 'The PS o f V M (2)' Ganita, 28(1977) 9 9 f f " Example 6. Far the same day as given in Ex. 5 give the Lord of the month. T h e remainder set apart (in the Ex. 5) is 666. Dividing by 30, the Quotient, Q , obtained is 22. (22 -I- 1)2 - I = 45. Dividing out by 7, the remainder is 3. Hence, the third from the Sun, viz. B h a u m a is the L o r d o f the month.
Cm^) 19. Q u o t e d b y U t p a l a o n B S 2 . p . 3 I .
cbM^^VI: II ?o || c. B I . 2 . ^ ? r # ^
19a. B I . 2 . 3 m ^ : ; U.3lRP?c?ri|
d.
BI.^sj^
1.21
21
I. I N T R O D U C T I O N O F T H E W O R K
Lord of the Hora 20. Take the remainder set apart in verses 17-18. Divide out by 7 and the remainder is the Lord of the Day, counting from the Sun. Take this remainder, multiply by 3, add 1, and add also the number of hords (i.e. the hours) counted from the beginning of the day, (i.e. the previous sunset) inclusive of the hord in which the taken moment falls. Multiply by 5 and divide out by 7. T h e remainder, counted from the Sun, gives the L o r d ofthe H o r a . If the L o r d of the day is dth from the Sun and the time taken falls i n the hth hord, then the number for the L o r d of the H o r a is {M + I + h) x 5. It should be noted here that the hord, h, is counted from sunset, because the time of Epoch is sunset and the day is said to commence there. T h e derivation of the two rules: T h e rule for the L o r d of the Day is obvious for the order of the Lords, Sun M o o n , Bhauma, etc. is meant to be the order of the Lords of the weekdays, Sunday, Monday, etc. T h e rule for the L o r d of the hord is derived thus: F r o m the Sdstra we learn that the L o r d of the hord beginning at sunrise is the same as the L o r d of that day. T h e L o r d ofthe hord beginning Sunday, i.e. ofthe hord']\xst after sunset of Saturday, (i.e. Mandavara), is B u d h a , since the L o r d of the hord after sunrise on Mandavara is M a n d a and the successive Lords of the hords are the fifth after each, i.e. the sixth counting from each, (vide the next verse, 21). B u d h a is the 4th i n order. After this if {n - I) hords are gone, the L o r d of the nth hord is given by (n - 1)5 -I- 4. Let us find the L o r d of the/jora for the A-th/lora of the d-th d a y . T h i s is {(d— 1)24 -I-A}thAorfl. Therefore the L o r d of the hord is, substituting this for n in the above formula, {{d - I )24 -l-h - l } 5 - l - 4 = (24rf -I- h 25)5 + 4 = {2ld + 3d +h + I- 26)5 + 4 = {3d + I +h)5 +4 +5 x2ld-5 X26 = {M +I + h)5-l- 105
Example 7. (a) Who is the Lord of the Day, for the day given in Ex. 5? (b) On the same day, who is the Lo of the Hora, fifth after sunrise? (a) T h e remainder set apart according to verses 17-18 is 666. Dividing out by 7, the remainder left is 1, i.e. the L o r d of the Day is the Sun. (b) In the example, d = l,h = 5 + 12 = 17 (because h is counted from the beginning of the day, i.e. the previous sunset). Substituting, (1 X 3 -I- 1 + 17)5 = 105. Casting out 7, the remainder is 0 or 7 and the 7th from the Sun, M a n d a is the L o r d of the hord.
20.
Quoted by Utpala on BS 2, p.34. BI.2.^T<^
C.
20a, b.
A 1 . 2 . B I . 2 . ' " ^ 5 r a ? t ^ : ; C.D.U.BPfnt
d.
C D . U.^11I?T; Bl.WWrf; B2.^?r
Bl.ft^
AI.2.*M^1\VII:; BI.*W^Tlq:; B2.^R^??ft?T:
22
1.22
PANC:ASIDDHANTIKA
21. T h e f o u r t h c o u n t e d from the L o r d o f any year is the L o r d o f the year next to that. T h e t h i r d from the L o r d o f a n y m o n t h is the L o r d o f the m o n t h next. T h e s i x t h from the L o r d o f any hord is that o f the n e x t hord. T h e L o r d s o f the day c o m e c o n s e c u t i v e l y , i n the o r d e r g i v e n . This the explanation: It has been said that the L o r d of the year is that of the sdvana year of 360 days, coming one after another. T h e L o r d of the first day of the year is the L o r d of the year and the L o r d of the 358th day is the same. T h e L o r d of the next year is that of the 361st day, which is the fourth counting from 358. T h u s the L o r d o f the next year is the fourth counting from that of the previous year. In the same way, the L o r d of the next (sdvana) month is that of the 31st day, counted from the first day o f the previous month. T h e L o r d o f the 29th day is the same as that of the first. T h e 31st day is the 3rd counting from the 29th. Therefore, the L o r d of the 31st day, i.e. the L o r d of the next month, is the third from that o f the previous month. T h e Lords of the hords come in the order, Manda, G u r u , B h a u m a , Ravi, Sukra, B u d h a and Soma, which is the descending order of the distances of their orbits. T h e planet next in this series, who is the L o r d of the next hord, is the 6th in the series given by our author, and hence the statement that the sixth from the previous is the L o r d of the next hord. T h a t the Lords of the day come consecutively is obvious, for the series Ravi, Soma, Bhauma, etc. is given i n the very order o f the Lords o f the day. One thing must be said here. T h e author has taken the Lords i n the arbitrary order Ravi, Soma etc. as it is well known by means of the week-days we are using in o u r day-to-day affairs. But the order o f the Lords of the hord, viz. Manda, G u r u , etc. based on their distances is fundamental and given by the Sdstras, which give the L o r d of the week-day itself as being the same as the L o r d of the first hord after sunrise on that day, taking the L o r d o f the hord as known. T a k i n g this order we can make the following statements: T h e L o r d of the next day is the 4th as counted from that ofthe current day, the L o r d of the next month is the 7th counted from that of the current month (or, which is the same, the one previous to that of the current month) and the L o r d ofthe next year is the third counted from that of the current year.
22. C o n s u l t i n g the w o r k s o f Sages, I s h a l l tell i n m y f u t u r e w o r k f o l l o w i n g the Hord-Tantra, the p r e d i c t i o n s , v i z . w h i c h results w i l l flow d u r i n g the r e i g n o f w h i c h L o r d o f the year o r L o r d o f the m o n t h . There is a gap in this verse in every manuscript, tattadvrttaih being missing. So we have adopted the reading of Bhattotpala in his Br. Sam. commentary which is full. 21. Quoted by Utpala on B S 2, p.35. 21a. A 1 . 2 . B 1 . ^ ^
22a
B 3 . Commences with this verse. B1.2.3.^W
b. A1.2.^lfireT«TFrat; B1.2.^lfcrRT«ir?ml;
b. B 2 . ^ ^
c. B1.2.?faf«r#?i£r. A 1 . 2 . B 1 . 2 . ^
c. A . B . o n . c R T ^ : ; C . [dt1c»t>d^] A1.2.BI.2.^% d B1 and B 2 . # a for#n
d. A l . f ^ « i a ; U.f^^Rni«?:"Fn^
A1.°T#«II^:; B2.cTf^«?A; B. f%«n^
1.25
I. I N T R O D U C T I O N O F T H E W O R K
Rvi.^^
( ^ )
23
(^^)
(^:)
^ :
jj
2 3 - 2 5 . A d d 1 to the D a y s f r o m E p o c h , d i v i d e by 3 6 5 , take the r e m a i n d e r a n d d i v i d e this by 3 0 . T h e q u o t i e n t are the m o n t h s g o n e . T h e r e m a i n d e r gives the L o r d s o f the c u r r e n t d e g r e e i n the c u r r e n t m o n t h . T h e y are, c o r r e s p o n d i n g to each d e g r e e , K a m a l o d b h a v a ( B r a h m a ) , P r a j a p a t i , S v a r g a ( H e a v e n ) , W e a p o n , Tree, A n n a (Food), Residence, K a l a (Time), A g n i , A b h r a (Cloud), S u n , M o o n , I n d r a , C o w s , N i y a t i (Fate), H a r a , B h a v a , G u h a , M a n e s , V a r u n a , B a l a d e v a , V a y u , Y a m a (the r u l e r o f the W o r l d o f the m a n e s ) , V a k ( G o d d e s s o f S p e e c h ) , S r i (the G o d d e s s o f W e a l t h ) , K u b e r a , H e l l , E a r t h , V e d a s a n d the Supreme Being. This matter must have been taken from the ancient Samhitas by o u r author and given here. For the purpose of giving the L o r d of the degrees they must have divided the days into years of 365 days (why not the exact duration o f the solar year, we cannot say) and the years into months o f 30 days as can be inferred from the instruction. B u t then it comes to giving the L o r d of not the degrees of the rasi but that of each o f the savana days i n the sdvana month. For the 5 days left over at the end of the year it must be taken that the first 5 Lords are repeated. Example 8. Give the Lord of the Degree of the rdsi for Daysfrom Epoch 3479. A d d i n g one and dividing by 365, the remainder is 195. Dividing by 30, the remainder is 15. Therefore the fifteenth i n the list, Niyati (Fate) is the L o r d required. Ed. Note: N P have identified verses 23-25 as relating to the Magas, emending the expression mdsds syuh i n verse 23 to magdbddh syuh and have correlated the 30 names enumerated i n the verses with the lords of the 30 days i n the month according to the Magan calendar. K . S . Shukla has studied these three verses i n detail, noted that these names are enumerated also i n the Vatesvara Siddhdnta (ch. I, sn.v, TO. 117 c-d, 118) and has traced the names to their Zoroastrian (Parsi) originals, as per the following Table, i n his paper ' T h e PS o f V M (2)', Ganita, 28 (1977) 99-116. 23a.
A 2 . ^ ; B l - 3 ^
b. A 2 . W ! i i . A I . 2 . ^ ; B I . 3 . « H | ^ . AI.2.'Rrai^:; D . [tPHS^:] ^ : d. AI.2.^f55mt;BL2.^^=?a^ 24a. A L ^ ' W i k d l ; BI-3.'»H
AI.2.^;B3.^:;C.^;D.^
A1.2.^55?1PI; 8 1 . 2 . ^ ^ ° ; B3.*sltlM<> c. A 1 . 2 . B 1 . 2 . C . D . W I R ^ I R R g q : (B2.<*9^:) d. C.D.'llFi^d'M:.
B.m^
25a. A I . 2 . ^ ^ ^ ; B I . ^ ; D . [m^]
B 3 . ^ ;
ftg
b. A I . ^ c l ^ ; A.2.^^el|^ AI.2.'HHlcWufl c. B1.3.3nf ?ft«R^ I A I . 2 . B 1 . 3 . C . D . P n ? l l d. A I . 2 . ^ ; BI.3. C . D . ^ : A . 2 . ^ . A I . 2 . ^ :
PANCASIDDHANTIKA
24
Names of the 30 days of the Parsi months Name i n V M 1. Kamalodbhava (Lotus-born) 2. Prajesa (Protector of creatures) 3. Svarga (Heaven) 4. 5. 6. 7. 8. 9. 10.
Name i n Vatesvara Siddhdnta
Zoroastrian (Parsi) name
Brahma
A h u r m a z d ( L o r d God)
Prajapati (Protector o f creatures) Dyauh (Heaven)
B a h m a n (Protector of creatures. Brahman) Ardibahesht (Holder ofthe keys of heaven) Shahrivar (Lord of pure metal) Spandarmad (Charitable) K h u r d a d (Lord of festivals) Amordad Depadar (Associate of Ahurmazd) A d a r (Fire) A v a n (Waters)
Sastra (Weapon) T a r u (Tree) A n n a (Food) Vasa (Residence) Kala (Yama) A g n i (Fire) K h a (Same as Abhra)
11. 12. 13. 14. 15. 16. 17.
Sastra (Weapon) Druma(Tree) A n n a (Food) Vasa (Residence) Kala (Yama) Anala(Fire) A b h r a (Filled with water, Cloud) Ravi (Sun) Sasi(Moon) Indra (God of rain) G o (Cow) Niyati (Destiny) H a r a (Mihira, Sun) Bhava (Siva)
18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29.
Guha Pitr (Manes) Varuna Baladeva (Balarama) Samlrana(Wind) Yama V a k (Speech) Sri (Righteousness) Dhanada (Kubera) Niraya(Hell) Dhatri (Earth) Veda
Aja ( U n b o r n God) Pitr (Manes) Varuna H a l i (Balarama) V a y u (Wind) Yama V a k (Speech) Sri (Righteousness) Dhanada (Kubera) Niraya(Hell) B h u m i (Earth) Veda
30.
ParahPurusah (Supreme Being)
Parapurusa (Supreme Beipg)
Ravi (Sun) Sasi (Moon) Indra (God of rain) G o (Cow) Niyati (Destiny) Savitr (Sun) G u h a (Son of Siva)
Khurshed (Sun) M a h (Moon) T i r (Distributor of water) Gosh (Cow) Depmehr (Ahurmazd's associate) Meher(Sun) Sarosh (Protector of the living and the dead) Rashna Farwardin (Farohars of the dead) B e h r a m (or Varenes) Ram Govad (Wind) D e p d i n (Ahurmazd's associate) Din Ashisvang (Righteousness) Ashtad (Aingel created by Mazda) A s m a n (Sky) Zamvad (Earth) Marespand (Zarathustrian law and religion) A n e r a n (Endless lights of shining heaven)
1. A.B.C.D.*
Thus ends Chapter One, entitled 'Introduction of the Work', in the Pancasiddhantika composed by Varahamihira
Chapter Two VASISTHA-SIDDHANTA—PLANETARY COMPUTATIONS ETC.
Introductory Now follow the five Siddhdntas. O f these the Vdsistha is given first, as being the most primitive among the Siddhdntas that distinguish between true and mean motions, unlike the Paitdmaha which gives only the mean motion. For a detailed exposition of some of the verses see T . S . K . Sastry, ' T h e Vasistha Sun and M o o n ' , / O P 25 (1955-56) 19-41 and K . S . Shukla, ' T h e PS o f VM(2)', Ganita 28(1977)99-116.
True Sun 1. M u l t i p l y the D a y s f r o m E p o c h by 4 a n d a d d 6. D i v i d e this by 1461 ( a n d take the r e m a i n d e r ) . T a k e f r o m this, successively, the q u a n t i t y 126, r e d u c e d by 1, 0, 0, 0, 2, 4, 7, 9, 9, 8, 6, 5 (i.e., the twelve quantities 125, 126, 126, 126, 124,122, 119, 117,117,118,120, 121). ( T h e S u n ' s rdiw. M e s a etc. are successively got.) T h e direction is: Multiply the days by 4, add 6, divide by 1461 and take the remainder. F r o m this first take off 125, and consider that Mesa is gone. T h e n from what remains deduct 126 and consider Rsabha is gone, and so on. T h e Sun is i n the rdsi corresponding to the number which cannot be deducted on account o f its being less than what is left over. Multiply what is left over by 30 and divide by what cannot be deducted. T h e position o f the S u n i n that rdsi is got, i n degrees. It is to be noted that even 'Days from Epoch' is not mentioned here but we take it as understood because every work of this sort requires it. It is not specifically mentioned that this rule is for computing the true Sun but we can infer it from the quantities here given and the work asked to be done. Even the work is not clearly and completely given. B u t knowing what the author is about, we can see what is wanted to be done. T S have refrained from interpreting this, as an impossible task.
la. A l . f c r p i q i ^ A 2 . w r W W g ; BC.^Knpjra^ a-b. A 3 . ^ ^
b. D.1^1^ c-d.
A.B.C.^IsRfKT
f o r f ^
26
PANCASIDDHANTIKA
T h e text here svarakrta has been changed into krtasvara by interchanging the words, as the nature of the work requires it and as this kind of transposition is somedmes found in manuscripts. It is impossible that the Siddhdnta itself has made this mistake, not noticing the ascending nature of the series i n this part. Next, we are in doubt here about the time o f the day (like sunrise, sunset, noon or midnight) for which the Sun is here given. O n e may think that because no time is mentioned here, not even the instrucdon to take the 'Days from Epoch', one is expected to take the Days of the Romaka or the Paulisa and with its own time of sunset at Yavanapura, i.e. thirty-seven nddis twenty vinddis from sunrise at Ujjain. B u t later, in dealing with the Romaka itself and with the Saura, the author gives different times of day for different computations (vide V I I I . 5 , I X . I, X V I . 1) and hence this doubt. It is likely that the Vdsistha Sun and Moon are given for sunrise at Ujjain, as we shall show while dealing with the M o o n .
Another point to be noted is this. T h e rule gives the ' T r u e ' Sun direcdy, without giving the 'Mean' Sun. This is possible because this Siddhdnta, like the other Siddhdntas of the period like the Aryabhatiya, has taken the apogee ofthe Sun as fixed and, so, for a given day in the solar year there, is a given anomaly with a given equadon ofthe centre, which means a given true Sun. (It is so with the Vdkyakarana also, which follows the Mahdbhaskariya based on the Aryabhatiya, with this diffe ence that here the days for fixed intervals ofthe true Sun is given, while in the Vdkyakaraija the days for the Sun and the Sun for the days, both are given.) T h e rule is explained thus: In this Siddhdnta the solar year consists of 365 '/4 days, (like the Julian year), i.e. o f 1461 quarter-days. For convenience of computation, the Days from Epoch are also converted into quarter-days. According to this Siddhdnta the true solar year began, i.e. the true Sun was at the first point of Mesa, 1 V 2 days, i.e. 6 quarter-days, before Epoch. So 6 is added to the quarterdays from Epoch to give the true Sun from the beginning of Mesa. As after periods of 365 V4 days, i.e. 1461 quarter-days, the Sun returns to the first point of Mesa, we can divide the quarter-days out by 1461 and take the remainder alone to find the Sun, i.e. its position from the beginning of Mesa. N o w this Siddhdnta has found empirically that the true Sun traverses Mesarasi i n 31 ' A days, i.e. 125 quarter-days, Rsabha-rasi i n S P . ^ days, i.e. i n 126 quarter-days and so on. T h u s i n 125 -t126 -I- 126 + 126 + 124 -I- 122 4- 119 -(- 117 + 117 -I- 118 -I- 120 -I- 121 = 1461 quarter-days the Sun traverses all the twelve rdsis and reaches Mesa again. T h a t these numbers add upto 1461, and 1461/4 = 365V4, the days ofthe year, is proof of the correctness of our interpretation ofthe rule. T h u s we see that the solar months Mesa etc. contain each 31 ' / i , 31 V2, 31 '/2,31 ' / 2 , 3 1 , 3 0 ' / 2 , 2 9 % , 29'/4, 29'/4, 29'/2, 30 and 30'/4 days, respectively. It can be seen that these fairly agree with what is given by the other Siddhdntas. T h u s i f 125 quarter-days are left over in the year the Sun has traversed Mesa, i f 125 -I- 126 are left over, it has traversed Mesa, Rsabha etc. It is obvious that its position within a rdsi is to be found by the propordon: If 30° are for the quarter-days of the rdsi, how many degrees for the quarter-days ultimately left over. Example l.(a). Days from Epoch 4246. Find the true Sun. (b) Find the true Sun for zero day. (a) Days converted into quarter-days = 4 X 4246 = 16,984. A d d i n g 6 we get 16,990. Dividing out by 1461, the remainder is 919. Deducting 125, 794 is left over; Mesa is gone. Deducting 126, 668 is left over; Rsabha is gone. Deducting 126 again, 542 is left over; M i t h u n a is gone. Deducting 126 for Karkata, 416 is left over. Deducting 124 for Simha 292 is left over. Deducting 122 for Kanya,
II.4
II. V A S I S T H A - S I D D H A N T A
27
170 is left over. Deducting 119 for T u l a , 51 is left over, in Vrscika, i.e. 5 1 1 1 7 o f Vrscika is gone, i.e. 30 X 51/117 degrees = 13° 5'. Therefore the true S u n = T 13° 5 ' . (b) For 0 day, 0-1-6 = 6, quarter-days. 30° x 6/125 = 1° 26' gone i n Mesa. Therefore the true Sun = 0 M ° 2 6 ' .
• f e ^ ^ q r r a t ' 7 T i ? ! ? n ^ ( ^ ) 8 i v i ^ t H d l : -Bern: StrTJT:
True Moon 2. A d d 1936 to the Days from Epoch, and divide the Sun by 3031. T h e quotient are called ghanas. Multiply the remainder by 9 and divide by 248. T h e quotient are called gatis and the remainder are called padas. 3. Divide the ghanas out by 16 and take remainder alone. Multiply this by 3, divide by 4 and take the result as rdsi etc. Subtract this from 12 rdsi and take the remainder. A d d to this, minutes equal to twice the total ghanas. A d d also r 7° 29'. (The mean M o o n at the end oi the ghanas is got). 4. Multiply the gatis by 185, subtract a tenth o f the gatis and add these also, taken as minutes. (The mean Moon at the end of the gatis is got.) If the number oi padas is less than 124, they are called ph\%-padas. If 124 or more, 124padas are taken and set apart as a hali-gati. T h e remaining/>a
B.^;c!f5»I0I«Rt(B2.°^°)
A(=AI.2.)^';B(BI.2.3.)^
b. B . ^ ^ A.'SRH f o r « R F
d. B.TR
d. A . ^ a ^ ; D . o m ? ^ C . W W 4a. B l . T p f i r a T W " ; B2.3.-nf!lfII'raT
b. Al.^T^lf^"; D.-nRmt''. B1.2."?W5rai: B2.'atTf5i
3a. A l . m i ^ f l ^ ; A2.B2.?R'fe?I^?t^ BI.?^^ b. A.-atspiraf^; B.2.sil^i€Jfei; c.sil^mfei c. C.tll^AI.tl^preRr; A2.«RI;
c. B.^Ra#K°. D.«<2i)MM«S c-d. d.
B1.2.°^^'M4
28
11.4
PANCASIDDHANTIKA
nothing is said, then the emendadon gatikdsthdmsa is the proper one, which we have given. (TS also give this). If, o n the other hand, we take it that the instruction is to subtract 2 minutes per ghana, taking the word projjhya in the previous instruction to be understood here also, then the emendation gatyastdmsa will be the proper one. In this case we would also have to keep the letter ta of the original as it is, without changing it into tha. B u t the addition o f 2 minutes per ghana alone would agree with the correct mean motion for the period of our author which is i n cycles etc. 110-11 -7-3215 for 3031 days, the Vasistha mean morion being 110-11-7-32. NP have emended the word as sasthdmsa which would not give the correct result. Example 2. Find the mean Moon for the end of the gatijust before Days from Epoch, 3,06,131. A d d i n g 1936 to Days from Epoch, the days for computation got is, 1936 + 3,06,131 = 3,08,067. Dividing by 3031, the quotient 101 got zreghanas. T h e remainder is 1936. Multiplying 1936 by 9 and dividing by 248, the quotient 70 are gatis. T h e remainder, 64, are padas. These being less than 124, are plus-jbo
= 8"^ 7° 30' = 0 3 22 = I 7 29 = 7 5 43
T h e mean M o o n at the end o f the last gati
4 24 4
5. If a ha\{-gati has been obtained, for the sake of that hali-gati, add rdsis, etc. 6-0-4. A d d also degrees equal in number to the plus-padas or minm-padas. Using the plus-padas or minus-padas, respectively, in the two following formulae, find the value, which is in minutes and add that also. (The true M o o n is got). 6. Deduct one from the plus-padas or minus-/»a
A.tRqiags^
c. A I . W ^ : ; A 2 . W ? E i m - . d. AI.tTSI; A 2 . B . C . D . ^ A . « R 4 ^ B.«Fl'irfgPra(B2<'^<')
A.Sr^
6a.
B.-^
I
BI.3.°fe^
b. B 3 . ^ . B . f g ^ d. A . B . 1.151^^1^; A.%<1I
[.6
29
II. V A S I S T H A - S I D D H A N T A
T h e two formulae can be written down thus: (i) If P is the number of plus-padas, {1094 -I- 5(P - 1)} P/63. (ii) If P' is the number o f minus-padas, {2414 - 5(P' - 1)} PV63. Example 3. Continue Ex.2 and compute the true Moon for the days given. The mean Moon got in Ex. 2 to the end ofthe gati The padas obtained are 64, plus-padas (P) Adding degrees equal to P = 64, Using formula (i) intended for plus-padas, {1094 -I- 5 (64 - I)} 64/63 = 1431 minutes
=
4^ 24°
+
2
4
0
+
0
23
51
7
21
55
The true M o o n Example 4. The Days from Epoch are 1219. Find the True Moon.
1219 + 1936 = 3155 (= days for computation). Dividing by 3031, g/wna got is 1, remainder 124. M u l d p l y i n g 124 by 9 and dividing by 248, the g-ato got are 4. T h e remainder 124 are padas. T h i s is just one hali-gati and no pada is left over. Ghana 1 x 3 / 4 = 0' 22° 30'. Deducting from 12 rdsis A d d i n g minutes 1 x 2 Ksepa Gatis 4, X 184 9/10 = 740 (minutes) For the hali-gati, add
= -1-f-(-
-1-
True Moon
r 11 0 I 0 6
O 7 0 7 12 0
30 2 29 20 4
6
27
25
Example 5. Find the true Moon for Days from Epoch, 1228. 1228 + 1936 = 3164 (= days for computation). D i v i d i n g by 3031, gAonos got 1, remainder 133. Multiplying by 9 and dividing by 248, the quotient 4 are the gatis got, and the remainder 205 are padas left over. A half-ga
II 0 I 0 6 2
7 0 7 12 0 21
30 2 29 20 4 0
I
13
9
True Moon
11
I
34
T h e following is the explanation of the processes: T h e true M o o n at a given time t is: (i) the mean M o o n at t plus (ii) the equation o f the centre for t. (i) is given here i n five parts. W e shall call them (a), (b), (c), (d), (e) which are to be a d d e d u p to get the total mean M o o n .
4'
30
PANICASIDDHANTIKA
II.6
(a) (Usually called the MiUa-dhruva or Ksepa) is the mean M o o n at a point of dme 1936 days before the E p o c h , when the Moon's apogee a n d the mean M o o n exacdy coincided according to this Siddhdnta. T h i s is given as sasi-muni-navayamds ca rdiyddydh, i.e. V 7° 29'.
(b) is the mean motion d u r i n g whole numbers of cycles of 3031 days from the point of time 1936 before E p o c h , each cycle equal to 110 anomalisdc revoludons of the M o o n . T h i s (b) is found by muldplying the mean m o d o n per cycle (110 revoludons, 11 rdsis, 7 degrees, 32 minutes) by the number of cycles, called ghanas, obtained as quotient, by dividing the Days from E p o c h plus 1936, by 3031. A s full revolutions can be neglected, it is enough i f we m u l d p l y the ghanas by 11 rdsis 7 degrees 32 minutes, which may be done as ghanas X 2' + ghanas X 1 r 7° 30'. Ghanas X 2 is given by dvigunaghandh kaldh (yojydh). Because 16 ghanas X I T 7°30' equals 15 full revolutions, it enough i f we divide out the ghanas by 16 and take the remainder alone for multiplication (for we shall be neglecdng only full revoludons), which we are asked to do by ghanasodasahrta-sesam. A s 1 T 7° 30' is % rdsi less than a full revoludon, we can multiply the remainingg/winas by % rdsi and take this as subtractive, which we are instructed to do by projjhyddhas trigumtam caturbhaktam bhddi (rd T h u s b is disposed of.
(c) is the mean motion d u r i n g the subsequent full anomalistic revolutions called gatis, which form the quotient got by dividing the remaining days by the anomalistic period, 248/9 days, (i.e. multiplying the days left over by 9 and dividing by 248). F o r each gati the mean motion is I revolution and 184 9/10 minutes (which can be obtained by dividing the motion per ghana, viz. 110 rev. 1 T 7° 32' by thenumberofg-a
We shall first explain II.6 by showing how the formulae here combine the residual mean motion, viz. padas X 27 209/248 minutes (= e) with what is identifiable with the equation o f t h e centre (= ii). T h e equation o f the centre o f the Vdsistha is peculiar. Usually i n the Siddhdntas the equation of the centre varies as the sine ofthe anomaly, and therefore is zero at zero degree anomaly, g o i n ^ to a m i n i m u m at 90°, again rising to zero at 180°, then going to a m a x i m u m at 270°, and then falhng to 0 to 360°, i.e. zero°. T h u s it is negative i n the first two quadrants a n d positive i n the third and fourth quadrants and of the form, ' - a sin 0, where ' a ' is the m a x i m u m or m i n i m u m numerical
II.6
II. V A S I S T H A - S I D D H A N T A
31
value, and 0 is the anomaly. Note that this is the first term o f the series for the equation of the centre in modern astronomy, with its sign reversed, and the reversing is necessary because the anomaly was reckoned by the ancients from the apogee, not perigee. B u t i n the Vdsistha it is ofthe form - (665 - 5P) P/63 for the first two quadrants and -I- (665 - 5P') P763 for the last two. These are derivable from the equation for the Moon's daily true motion given i n III.4, (as we shall show there), which assumes the increase or decrease o f motion as uniform. Here we shall assume them and derive the two formulae o f II.6. As said before, (e) -I- (ii) is given by the formulae and (ii) is - (665 - 5P) P/63, for the first formula. Therefore (e) -I- (ii) = 27 209/248 P - (665 - 5P) P/63 = (63 X 27 209/248 - 665 + 5P) P/63 = (1754 - 6 6 5 - f 5 P ) P/63 = (1089+ 5P) P/63 = 11094 + 5(F1)1 P/63. which is the first formula. For the second formula (ii) is 4- (665 - 5P') P763. .-. (e) + (ii) = 27 209/248 P ' -I- (665 - 5P') P763 = (27 209/248 P ' x 63 + 655 - 5P') P763 = (1754 + 665 - 5P') P763 = (2419 - 5P') P763 = {2414 - 5 (P' - I)} P763 which is the second formula. We have already shown how for the half-gati 6"^ 0° 4 ' is got instead of the mean motion 6"^ 1° 32 Va'. This means that there is i n this an equation o f the centre equal to — 88 Va', combined with it. So, when the equation of the centre given by -t- (665 - 5P') P763 = -I- 88 Va, then it is actually zero according to this Siddhdnta. Solving this equation, we get P ' = 9 or P ' = 124. A s P ' is minus-pada, which is the original padas got less 124, we get that the equation ofthe centre actually becomes zero at original padas, P = 133, and P = 248. A s P = 248, is the end of the gati, this is what we expect, as the anomaly has again become zero. Also, by computation or examinadon we can get from the equation o f the cyclic part o f the formula for the first hali-gati, — (665 — 5P) P/63, the numerically greatest value o f the negative equation of the centre, which is - 351', for P = 66 Va. In the same way, from that the formula for the second hali-gati, + (665 - 5P') P'/63 we can get the m a x i m u m -f- 351', for P ' = 66 Va; but as there is a residue of— 88 Va' i n the second hali-gati, 351' —88 Va' = 262 Va' is the actual m a x i m u m . T h e numerical mean is 307' which, we see, is very nearly equal to that of the other H i n d u Siddhdntas. It is not that V M does not know that zero equation o f the centre must occur at P = 124, and not at 133, for in his own Romaka and Saura it is so. N o r is it difficult for V M , a master i n the science, to give the two formulae so as to have the equation of the centre zero at P = 124, (so that for a halfgait we get the correct 6' 1° 32 Va'), retaining, at the same time, the equation o f the centre desired by h i m . If he had given the two formulae i n the form (1134 -I- 5P) P/63, and (2374 - 5P') P'/63, he could have secured this. B u t adherence to the Siddhdnta has prevented h i m from doing this. So closely does he follow the original that he does not even give the two formulae i n the more simplified forms, (1089 - 5P) P/63, and (2419 - 5P') P'/63. T h e following things are to be noted i n connection with this Siddhdnta. O f both the Sun and the M o o n , the mean motion and the equadon o f the centre is mixed i n a peculiar manner and thereby the true motion is given. We shall see that it is the same i n the case o f the Paulisa also. T h e period of 3031 days called ghana here is the same as what is called kdldnala in the Vdkyakarana, which gives for this period, the mean motion, I F 7° 3 1 ' , neglecting full revolutions. T h e number 248 given here is there mentioned as devara.
32
PANCASIDDHANTIKA
II.6
We can see that the remark i n 1.4 about the tithi of the Vdsistha being very incorrect is appropriate but it can be shown that it is not due to the error i n the M o o n , but i n the Sun, whose sidereal year is taken as 365-15-0 days. T h e Moon's motion for 3031 days is i n cycles etc. 110-11-7-32 = 110 5063/5400 cycles. In one day the motion is 110 5063/5400+3031 = 5,99,063/ (5400 x 3031) cycle. In one day the Sun's motion is 1/365 V4 = 4/1461 cycle. T h e relative motion, i.e. their separation per day is 5,99,063/(5400 x 3031) - 4/1461 cycle. T h e dme taken for a separation of one cycle, i.e. the synodic month, is i n days, 1/(59,90,63/(5400 X 3031) - 4/1461} = 1461 x 3031 x 5400 ^ (I46I X 5,99,063 - 4 X 5400 x 3031) = 7,97,09,23,800 H- 26,99,20,481 = 29 - 31 - 50 - 17 - 38. But the correct synodic month computed for the time near that of o u r author is 29 - 31 - 50 - 7 47. Therefore i n successive synodic months the tithi comes later by days etc. 0-0-0-9-51, according to Vdsistha. In about 29 V 2 years this will accumulate to one nddikd. T h i s is bad indeed, and meri V M ' s remark i n 1.4 taditarau duravibhrastau (i.e., ' T h e tithis o f the other two have slipped away from the real'.)
Now, we have seen that according to this Siddhdnta the mean M o o n moves revs. 110-11-7-32-0, while the real motion for the period is revs. 110-11-7-32-15. Therefore i n 3031 days the Vdsistha naksatra is delayed by a little more than one vinddi. So a delay of one nddi is caused in 440 years only So the delay of one nddi in the tithi per 29 V2 years mentioned above, must be due almost wholly to the error in the Sun, the result o f giving the time per cycle as 365-15-0 days.
A g a i n , i n every 3031 days, the Vdsistha M o o n lags behind the correct one by 15". A lagging behind by one degree will take place i n 3031 x 60 X 60 -H 15 days, i.e. in about 2000 years, a very long period indeed. Bearing this i n m i n d , we shall try to answer the question already raised, viz. whether the Vdsistha Sun and M o o n are given for sunrise at Ujjain or sunset at Yavanapura; and, incidentally, we shall show that the ksepa, T 7° 29' given hy sasi-muninavayamdsca rdsyddydh and obliterated b by their drasdc emendation asyanm-hrtds ca is necessary i n II.3.
Thefollowing is the mean Moonfor Epoch (viz. Saka 427 elapsed, i.e. in A.D. 505), sunset, at Yava beginning Monday, Caitra Sukla being about to begin.
i. Computed for the period according to modern astronomy, assuming theayandijisa to be practically 0 for the time 354°48' i i . According to Saura 355° 6' iii. According to Siddhdnta Siromani 355°41' iv. According to Romaka 356°12' V. According to Vasistha, assuming that the mean M o o n is given for Ujjain sunrise 355° 6' 346° 54' -do-do- for sunset at Yavanapura 317° 37' -doUjjain sunrise, without Ksepa -dosunset at Yavanapura without Ksepa 309° 25' (For use by anyone interested i n making the calculations himself, the Kalidina etc. o f Epoch is 13,17,122-37-20. Also, the E p o c h is 5,09,432-22-40 days before mean sunrise at Ujjain ofthe first January 1900).
A n examination of the table will show that the Vdsistha M o o n agrees with that of every other fairly well, taking it as being given for Ujjain sunrise, and taking that the ksepa is given. If, o n the other hand, it is assumed that it is given for sunset at Yavanapura, there is Si difference of about 8°, which can happen only i n 1600 years, which is very very unlikely; for this to happen the Vdsistha should
II.6
II. V A S I S T H A - S I D D H A N T A
33
have been written 1600 years earlier. I f there is not the ksepa, the difference is 37°, an impossible thing, not to speak of the assumption that it is for Yavanapura sunset and there is no ksepa, both together which will make the difference 45° and worse. Hence the Vasistha epoch is definitely at sunrise at Ujjain and not at sunset at Yavanapura (Alexandria).
We have shown the ksepa necessary. B u t T S have emended kalah dvigunaghandh, sasi-muninavayamds ca rdsyddydh (verse 3) into phalarh dvigunaghandh sasi-muni-yama-hrtds ca rdsyddydh a spoiled the already correct reading and introduced two extra syllables i n the last foot, which spoils the arya metre too. (It must be noted that already there are 16 mdtrds i n the last foot, i.e. one mdtrd extra, which can be explained away or corrected by reading rdsyddydh as rdsyddi.)
One another point: T S have expressed their inability either to interpret 11.6 or to explain why 6' 0° 4' is to be added for a half-gati (vide C o m , page 9). B u t sdll thinking 11.6 gives the pure equation of the centre, the commentary goes on: arthdt veddrkdlpa-padesu rnam, adhikesu dhanam iti buddhimad bhih svayam eva dhyam, i.e. "It goes without saying that when the padas are less than 124, the result is subtractive, and when more it is addidve", which is wrong for we have seen that the result of both the formulae are additive. Moreover, the failure, both by T S and N P , to realise that the expression 'rdsyddydh' specifically instructs that the digits i n sas'imuninavayama are to be taken as 'beginning from rdsi', i.e. as F 7° 29' and not as a whole number 2971 (TS) or as " 2 ' 9;7, I°" (NP) have led to incorrect interpretations by them; also, the Notes of N P (vol. II, pp. 16-19) and the deductions made (p. 19) have to be revised i n the light o f all that has been stated above.
Naksatra and Tithi 7. D i v i d e the T r u e M o o n by 4 a n d m u l t i p l y b y 9. W h a t we get i n the rdsi c o l u m n is the naksatra. W h a t is got i n the d e g r e e c o l u m n are the muhurtas. D e d u c t the t r u e S u n f r o m the t r u e M o o n , d i v i d e the result by 2 a n d m u l t i p l y by 5. Tithis are got i n the rdsi c o l u m n a n d thirtieths of tithis i n the degree c o l u m n . As the 27 nafoafra-segments are divided into the 12 rdsi - segments, there are 2 V4 = 9/4 ruiksatras per rdsi. Hence the instruction to divide the rdsis by 4 and multiply by 9 to get the naksatras. As degrees are thirtieths o f rdsis, the resulting numbers i n the degree column are thirtieths of naksatras, called muhurtas by this Siddhdnta. It must be noted that the word muhurta originally meant the 30th part o f a naksatra, but later came to be applied to the 30th part o f a day as well, because both are practically the same i n duration. T h e interval of longitude between the Sun and the M o o n is the tithi, 12° forming one tithi, i.e. there are 2 V2 = 5/2 tithis per rdsi. Hence the instruction to divide the rdsis by 2 and multiply by 5 to get the tithis. 7a.
A . mW;
B.J^IVIWHPH
b. A.^i^lPHJcll; B . ° ^ ^ ( B 3 . ^ ^ - ^ W l f k m d. A.fM«ra5°
34
PANt:ASIDDHANTIKA
II.7
Example 6. The true Sun is 10' 18°, and the true Moon is 5' 22°. Find the naksatra and the tithi. Naksatra: T h e M o o n is 5' 22°. D i v i d i n g by 4, (5^ 22°)/4 = T 13°. M u l d p l y i n g by 9, 9x T 13° = 12 - 27, i.e. twelve naksatras have gone and i n the 13th, (Hasta), 27 muhdrtas have gone. Tithi: M o o n - Sun = 5'22° - 10TS° = Dividing by 2, (7^4°)/2 = 3 ^ 7 ° . Multiplying by 5, 3 M 7 ° X 5 = 1 7 - 2 5 . Seventeen tithis are gone and in the eighteenth (Bahula Trtlya) 25/30 parts have gone.
Day-time 8. W h e n the S u n is i n t h e 3 rdsis, M a k a r a etc., the S u n m e a s u r e d i n rdsis p l u s t h r e e is the d u r a t i o n o f d a y - t i m e i n muhurtas. W h e n it is i n the 3 rdsis. M e s a etc., the S u n p l u s fifteen is the d u r a t i o n o f d a y - d m e . W h e n i n the 6 rasis, K a r k a t a k a etc., the S u n p l u s n i n e is the d u r a t i o n o f t h e n i g h t - t i m e . ( T o get the d u r a t i o n o f the d a y - t i m e , this s h o u l d be s u b t r a c t e d f r o m 30).
T h o u g h no measure o f time is mentioned here, we can infer that it is the muhurta (2 nddis) because by adding the shortest day, 12, and the longest, 18, we get 30 which must be equal to the whole day, i.e. 60 nddikds.
T h u s , for the S u n at the beginning of each rdsi. Mesa etc., the day-dme in muhHrtas is 15, 16, 17 18, 17, 16, 15, 14, 13, 12, 13, 14. T h e longest day is 18 muhurtas when the Sun is at the first point of Karkataka (Cancer) at Summer solstice and the shortest 12 muhdrtas when at the first point of Makara (Capricorn) at W i n t e r solstice. T h e day and night are equal at the first points of Mesa (Aries) and T i i l a (Libra), i.e. at the equinoxes. T h e daily increase or decrease i n day-light is 4 vinddis per day. In essence, the same formula for day-light is found in the VeddngaJyotisa and the Paitdmaha Siddhdnta (PS, XII.5),with th'« difference that here the true Sun is used, but there, because they have no true Sun but only f' le mean Sun, the day which is proportionate to the mean Sun, is used.
Evidentiy not understanding what is given here, T S have made a drastic change in the text, writing bhusvarga-tithimito for mesddau tithiyuto, intending to make this agree with the next verse giving noon-day shadow. But, even within that verse, there is contradiction and this need not have been attempted, at such cost. T o crown all their interpretation with their emendation is full of contradiction within itself, which has been set out i n detail by me in a paper entitled 'Vasistha Sun and Moon' in the Journal of Oriental Research, 25 (1955-56) 19 - 4 1 . T h e uniform increase and decrease i n day-time given here is wrong, of course, and it varies with the position of the S u n , being greatest at the equinoxes and falling to zero at the solstices. T h e m a x i m u m or m i n i m u m day-time itself varies with the latitude of the place (depending on tan.
8b. A.*1
A.B.C.^:. A 2 . f ^ I
c.
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d.
E.mPl
11.10
35
II. V A S I S T H A - S I D D H A N T A
latitude), what is given here being for some place having a N o r t h latitude of 35° 45'. (This matter is dealt with i n the text in III. 10 and IV.26). T h e rules for day-time is explained thus: F r o m the beginning o f Makara to the end o f M i n a the Sun i n rdsis increases from 9 to 12. T h e day-dme also increases following it, from 12 muhdrtas at winter solstice to 15 at Equinox. A s (9 to 12) -1- 3 = (12 to 15), the instruction to add 3 for the Sun i n this quadrant follows. I n the same way, from the beginning o f Mesa to the end of Mithuna, the Sun i n rdsis increases from 0 to 3. T h e day-time increases from 15 at Equinox to 18 at summer solstice. (0 to 3) -I- 15 = (15 to 18), and this explains the addition o f 15. As there is the maximum day of 18 muhdrtas at summer solstice, there is the m i n i m u m night there, of 12 muhdrtas. This increases to m a x i m u m night, 18 muhdrtas for Sun at the beginning of Makara, 6 months after. As a result, as the Sun's rdsi increases from 3 to 9, the night increases from 12 to 18. (3 to 9) -f 9 = (12 to 18) and this explains the addition of 9 (three times three). Example 7. Give the day-time for the Sun at the beginning of: (i) Rsabha, (ii) Kumbha and (iii) Kanyd. (i) T h e Sun is in the quadrant 0 to 3 rdsis. T h e beginning of Rsabha is one rd.si. Therefore 1 -I15 = 16, muhdrtas, is the day-time. (ii) T h e Sun at the beginning of K u m b h a is 10 rdsis. T h i s is in the quadrant 9 to 12. Therefore 10 -I- 3 = 13, muhurtas, is the day-time. (iii) T h e Sun at the beginning of Kanya is 5 rdsis and is i n the 6 rdsis 3 to 9. Therefore 5 + 9 = 14, muhurtas, is the night-time. Deducdng from 30, 30 - 14 = 16, muhurtas, is the day-time.
II
I
Gnomonic Shadow 9. When the Sun is in the six rdsis beginning with Karkataka, the number of rdsis traversed from the beginning of Karkataka, multiplied by 2, is the midday shadow (of the twelve-digit gnomon) in digits. When the Sun is in the six rdsis beginning with Makara, find the distance in rdsis traversed by the Sun from the beginning of Makara, and muldply by 2. Subtract this from 12, to find the mid-day shadow. 10. When the Sun is in its southward-course, (i.e., in the six rdsis from Karkataka), half the mid-day shadow plus three is the longitude ofthe Sun in rdsis. When in the northward course in the six rdsis from Makara, half the noon-shadow subtracted from fifteen is the Sun in rdsis. d. A.B.C.D.i^-^lfW-1. A.^to5cTr#«zil^ 9a. A 1 . ^ F « 5 ; B 2 ° % B . ^
b. A . B l . ^ R e q ^ . A . ' l ^ c. A l . ^ ; A 2 . ^ 5
B.T?5^HII
10a. A l . ^ W n t d. B . W ^
36
PANCASIDDHANTIKA
11.10
Example 8. (a) On a certain day the Sun's longitude is 5 rdsis. (b) On another day it is tdsis 11-15 casesfindthe mid-day shadow.
(a) T h e Sun is 5 rdsis and is within the six rdsis from Karkataka, being 2 rdsis from the beginning of the first point o f Karkataka (i.e. 3 rdsis). So, 2 X 2 = 4 digits is the shadow. (b) T h e Sun's longitude is rdsis 11-15. T h i s is within the six rdsis from the first point of Makara (9 rdsis), the Sun's posidon being 1 n 5 ° - 9'^ 0° = 2' 15° = 2 V 2 rdsis from that point. 12 - (2 V 2 X 2) = 7 digits is the noon-shadow.
Example 9. (a) The Sun is in its southward course and the shadow is 4 digits. Find the longitude oft (b) The Sun is in its northward course and the noon-shadow is 7 digits. Find the Sun. (a) H a l f the shadow = 4/2 = 2. As the course is southward add 3 rdsis; the Sun's longitude is 5 rdsis. (b) H a l f the shadow = 7/2 = 3 V 2 . As the Sun's course is northward, deduct from 15. 15 - 3 V 2 = 1 1 V 2 rdsis. T h i s is the longitude o f the Sun. F r o m the two sets of examples it can be seen that the two formulae are the inverse of each other.
T h e formulae are explained thus: T h i s Siddhdnta assumes that the noon-shadow is zero when, at the end of its northward course, it reaches the first point of Cancer. T h e n as it moves southward, the shadow increases to 12 digits at the end ofthe course, i.e. after 6 months, when the Sun reaches the first point of Capricorn. A s s u m i n g the increase to be uniform, there is an increase of 2 digits per rdsi. As the shadow is zero for the first point o f Cancer, the longitude i n rdsis measured from this point, multiplied by 2 gives the shadow. T h u s , i f c is the Sun in rdsis measured from the first point of Cancer and s is the shadow i n digits, s = 2c for the 6 months till the Sun reaches Capricorn, where the shadow is 6 X 2 = 12 digits. T h e n the shadow decreases at the same rate to zero at the first point of Cancer, i n the course o f 6 months. Therefore i f c is the Sun measured from the first point of Capricorn, when the shadow is 12, and i the shadow, then s = 12 - 2c.
Now for the longitude of the Sun from the noon-shadow. W e have seen that for the six rdsis from Cancer,.? = 2c. Thereforec = 5/2. But c is counted from the first p o i n t o f Cancer, i.e. 3rdsis. Therefore the Sun's longitude in rdsis is 3 -I- c = 3 -(- s/2, which is the same as the instruction to divide the shadow by 2 and add three rdsis. For the six rdsis from Capricorn, we have seen that ,s = 12 — 2c Therefore c = (12 — s)/2. B u t c is counted from the first point of Capricorn, i.e. 9 rdsis. Therefore the Sun = 9-t-c = 9 + ( 1 2 - s)/2 = 15 - s/2, which is the instruction given. It must be noted here that both the formulae are very rough. A t summer solstice, when the Sun is at the first point of Cancer, its north declination is m a x i m u m and given by H i n d u astronomy as 24°. A t that time, the mid-day Sun is at the zenith at places on 24° north latitude (like the region of Ujijain) and so it is only in this region that the shadow is zero at this time. W h e n the Sun reaches its southernmost point at winter solstice, i.e. the first point of Capricorn, its south declination is 24°. Therefore the zenith distance o f t h e noon-Sun as seen from latitude 24° N o r t h at that time must be 48° towards the South, and the shadow at that time must be greater than 13 digits and not 12. (All this will be shown in Chapter IV). If the shadow is to be 12 digits, the Sun's zenith-distance must be 45° and the region where the Sun is seen at this zenith-distance is 21° N o r t h latitude. T h u s there is contradiction even here. In verse 8, we showed that the rule is intended for a region having about 36° N o r t h latitude, neither 24° nor 21°. T h u s , so far as these things are concerned, the Siddhdnta seems to be a hotch-potch.
11.11 [BTSTRft cTO
II. V A S I S T H A - S I D D H A N T A
37
WIT g ]
Lagna from shadow and vice versa 11. A d d 12 to the s h a d o w ( o f the t w e l v e - d i g i t - g n o m o n , m e a s u r e d i n digits) at a n y t i m e o f the d a y , a n d d e d u c t f r o m it the m i d - d a y s h a d o w f o r the d a y . D i v i d e 36 by this a n d take the result. T h i s r e s u l t t a k e n as rasis, p l u s the S u n i n r a i i s is the lagna at the m o m e n t , i f it is f o r e n o o n . I f a f t e r n o o n , d e d u c t this f r o m the S u n p l u s s i x r o i t s a n d t h e lagna is got. T h e formulae (a) for the forenoon and (b) afternoon respectively can be expressed thus: (a) Lagna = S u n -I- 36/(12 + shadow - n o o n shadow), (b) Lagna = S u n + 6 - 36/(12 + shadow noon shadow). What is called lagna is the Orient Eclipdc Point, i.e. the point o f the ecliptic rising o n the eastern horizon.
Example 10. (a) On a certain day, the Sun is 9 rdsis and the mid-day shadow 12. At a time in the mornin the gnomonic shadow is 36. Find the lagna for the moment, (b) On a certain day, the longitude of the Sun is rdsis and the noon-shadow 6 digits. At a time in the evening the gnomonic shadow is 24 digits, find the la for that moment. (a) F r o m f o r m u l a ( a ) , L a g T i a = 9 + 36/(I2 + 3 6 - I2) = 9-(- I = 10, rojw. Hence the first point o f K u m b h a is rising i n the east. (b) F r o m formula (b), Lagna = 6
6 24 6 - 36/(12 -f 2 4 - 6 ) = 1 2 - 1 ^ = IQ^rdsis. Hence
the 25th degree o f K u m b h a is rising i n the east. These rules are r o u g h and there is no question of strictly proving them. B u t we can explain them thus. F r o m sunrise to noon, as the aldtude o f the S u n increases, the lagna goes on increasing and the shadow decreasing, till it is shortest at noon. Therefore the increase i n lagna can be roughly expressed as, ^/(shadow -I- b), where a and b are constants to be determined. N o w , i f the place is supposed to be situated o n the equator, and the ecliptic o n which the Sun moves is supposed to coincide with the celestial equator, then at noon the shadow will be zero. A t that dme the increase in lagna (after sunrise) would be 3 rdsis, as the S u n has reached an altitude o f 90°. Therefore 3 = fl/(0 -I- b). A g a i n , seven and a half nddis after sunrise, the S u n would have risen to an altitude of 45°. So the increase i n lagna now is I rdsis and the shadow is 12 tan 45° = 1 2 digits. Therefore, I Va = a/(I2 + b). Solving these two equations for a and b we get a = 36, * = 12. Therefore the increase in lagna is 36/(shadow -I- 12), o f course, o n the given two assumptions. B u t the place may not be on the equator and the ecliptic does not coincide with the celestial equator and the S u n moving on it has a varying declination, with the result that generally the noon-shadow is not zero. According to the length o f the noon-shadow at other times also there will be an increase i n the shadow over what
l l a . Al.SWf^:; A2.?Kf^:. A . ^ b. B.^R«n#. B.i?ii?n:. Ai.°5W5Rim;
c. A.B.^-J^l4
38
Il.li
PANCASIDDHAMIKA
it would otherwise be, for which the rule has been formulated. As the deduction of the noonshadow for the day of observation from the shadow would, to some extent, remove this extra length of shadow and bring about an approximation to the ideal condition, the noon-shadow is asked to be deducted from the shadow in the formula. Therefore the increase in lagna is given by 36/ (shadow - noon shadow -f 12). As at sunrise the Sun is the lagna, adding the increase to the Sun we get the lagna, i.e. lagna = Sun -I- 36/(12 -I- shadow - noon-shadow). This is for the forenoon. In the afternoon, what happens in the forenoon with reference to the shadow is reversed, and therefore the rule gives the part of the lagna to increase from the dme of observing the shadow to sunset. So, it is less than the lagna at sunset by what is got from the formula. But the lagna at sunset is the Sun plus six rdsis. Therefore the formula for the afternoon becomes: Sun -I- 6 — 36/(12 -lshadow - noon-shadow). Again let it be remembered that the rules are rough.
cw4v^<: 'VH'*^
12-13. D e d u c t the S u n f r o m the lagna a n d c o n v e r t the r e m a i n d e r into m i n u t e s o f arc, i f f o r e n o o n . I f a f t e r n o o n , d e d u c t the m i n u t e s f r o m a h a l f circle, (i.e. f r o m 10,800 m i n u t e s ) , a n d take these as m i n u t e s . D i v i d e 64,800 by the i n i n u t e s got. A d d the result to the n o o n - s h a d o w o f date a n d d e d u c t 12 f r o m this. T h i s is the s h a d o w at the t i m e o f the g i v e n lagna. T h i s is a c c o r d i n g to the succinct Vasistha Siddhdnta. The formula (a) for the forenoon, and (b) for the afternoon, respectively, are: (a) shadow = {64,800 {lagna - Sun, in minutes) + noon-shadow - 12}. (b) shadow = 64,800 4- {10,800 (lagna — Sun, in minutes)} -I- noon-shadow — 12.
Example 11. (a) On a certain day at a time in the forenoon, the Sun is 9 rdsis and the lagna 10 rds the noon-shadow of date is 12 digits. Find the shadow for the time, (b) On another day, for a time in noon, the sun is 6 rdsis and the lagna 10 rdsis 25 degrees and the noon shadow of dale is 6 digits. Find the shadow. a] Using formula (a), shadow = 64,800 - {(10 - 9) X 1800} - 12 + 12 = 36 digits. b] Using formula (b), shadow = 64,800 ^ {10,800 - (10 5/6 - 6) X 1800} - 12 -I- 6 = 64,800 -r(10,800 - 8640) - 12 + 6 = 64,800 + 2160 - 12 + 6 = 30 - 12 + 6 = 24 digits. These formulae (a) and (b) can be derived from the previous formulae (a) and (b) of verse 11, for these are only the inverse ofthe previous operations. 12a.
B3.f^
b. A . ^ W T A.?5lf«M^ B.iJtrsraig Bl.^^gJT^:; B 3 . ^ « ? i ^ I C. A.B.^Rra#^: ( B 3 . ^ : ^ : )
13. A l . c ^ ; A 2 . c ^ . B3.t?^ b. A . B . ^ l i S m . B.yni^Tbl d.
A2.^;B1.2.^lfOT;B3.gif?TS
II.13
II. V A S I S T H A - S I D D H A N T A
39
The previous (a), is: lagna = Sun + 36/(12 + shadow — noon-shadow). Therefore, lagna — Sun = 36/ (12-1- shadow - noon-shadow). Therefore S6/{lagna — Sun) =12-1- shadow - noon-shadow. Therefore, shadow = 36/{lagna — Sun) -f noon-shadow — 12, where it is to be noted that (lagna — Sun) is i n rdsis. If it is to be expressed i n minutes, we have, shadow = S6/{lagna — Sun) X 1800 + 1800 + noon-shadow - 12 = 36 x 1800/{lagna — Sun) in minutes + noon-shadow - 1 2 = 64800/{lagna - Sun) in minutes -I- noon - shadow - 12, which is (a) here. The previous (b) is: lagna = Sun + 6 — 36/(shadow -I- 12 — noon-shadow). Therefore 36/(shadow + 12 — noon-shadow)= Sun -I- 6 — lagna = 6 — (lagna — Sun). Therefore 36/{6 — (lagna — Sun)} = shadow -f- 12 - noon-shadow. Therefore shadow = 36/(6 - (lagna - Sun)} + noon-shadow - 12, where (lagna — Sun) is in rdsis. If it is to be expressed in minutes, we have, shadow = 36/6 — { 6 — (lagna - Sun) X 1800 ^ 1800} + noon-shadow - 12 = 36 x 1800/{6 X 1800 - (lagna - Sun) in minutes} -I- noon-shadow — 12 = 64,800/(10,800 — (lagna — Sun) i n minutes} -I- noon-shadow — 12, which is (b) here. The concluding words, Vdsistha-samdsa-siddhdnte, though forming a part of the sentence giving the rules of verses 12-13, may be detached from it and taken to refer to the whole chapter II and mean ' A l l this is given as i n the succinct Vdsistha Siddhdnta'. The word, naksatrddicchedah is found at the conclusion i n the manuscripts. Perhaps it is, naksatrddhicchedah, meaning 'Naksatra section' and the title is appropriate because this is the chief thing given here and other things depend on it. T h u s i n this chapter the true Sun and M o o n are given according to the Vdsistha and also other matters depending on them like the naksatra, the tithi, day-time shadow and lagna. (The star planets i.e. the regular planets, of this Siddhdnta will be given i n X V I I I . ) T S have professedly not understood verses 1, 5 and 6 and gone wrong in verses 3 (and 8), which means they have practically not understood the Siddhdnta at all. Thibaut even thinks that verse 1 may be dealing with the M o o n . But this professed ignorance d i d not prevent him from making the unwarranted remark i n the Introduction (vide p. X X X V I I I ) "the methods are so crude and so completely omit to distinguish between mean and true astronomical quantities, that the Vasistha Siddhanta can hardly be included within Scientific H i n d u Astronomy."
A . B . have as C o l o p h o n , C D . ^^^l(<^<: |
Thus ends Chapter Two entitled 'Vasistha-Siddhanta: Planetary Computations etc' in the Pancasiddhantika composed by Varahamihira
Chapter Three PAULI^A-SIDDHANTA — PLANTERY COMPUTATIONS ETC.
Introductory
This chapter is a compendium o f the part o f the Paulisa Siddhdnta dealing with the Sun, M o o n and Rahu. It has already been mendoned that the original Paulisa is now lost, perhaps for ever. This and the Saura are the only Siddhdntas dealt with by the author i n full, the others being scrappy. For some reason not known to us, at present the Paulisa is mixed up with the Vdsistha, for, the mean M o o n , together with its peculiar technical terms pada, gati and ghana are used here, without mentioning how they are got. T h e formula for the Moon's daily true motion, which patently belongs to Vdsistha has strayed into this chapter, another evidence of their being mixed up. In this chapter the Sun, the M o o n and Rahu, the methods of computing the dzily naksatra, tithi dndkararm, the two yogas vyatipdta and vaidhrti, the day-light in any given place in India, and certain holy days necessary for religious observations are dealt with.
True Sun 1. Multiply the days from Epoch by 120, deduct 33 and divide by 43,831. T h e mean Sun in revolutions, rdsis etc. is obtained. A d d 2 0 ° to this mean Sun. What is called kendram is got. 2-3. For the first six rdsis of kendra there are the following six quantities: 11, 48, 69, 70, 54 and 26, all deductive and in minutes. For the next six rdsis are the following: 10, 48, 70, 71, 54 and 25, all additive and in minutes. (If these are taken one after another according to rdsis of the kendra gone and) applied to the mean Sun, it becomes true Sun.
III.3
III. P A U U S A - S I D D H A N T A
41
In short, these twelve quantities are intervals of the equation of the centre for every rdsi of the kendra, the word being used i n a peculiar sense here, and not the usual one o f mean anomaly. T h e faulty readintrs, munyahti, mupta.krjta is corrected as mutkrti, meaning 26, and thereby the excess of one syllable in the foot also gets corrected. T S and N P have emended it as maksakrti, meaning 25, which, by its form, seems to be less likely to be the original, and which keeps the defect of the excess of one syllable. Example 1. (a) Find the true Sun for days from Epoch 690. (b) Compute the true Sun at Epoch. (a) Multiplying the days 690 by 120 and deducting .33, we have, 690 x 120 - 33 = 82,767. Dividing by 43,831, the revoludon got is 1 and remainder 38,936. Multiplying this by 12 and dividing by 43,831, the rdsis got is 10. T h e remainder is 28 9/22, which multiplied by 30 and divided by the same divisor, gives degrees 19. T h e remainder 34,871 multiplied by 60 and divided by the same divisor gives 48 minutes. T h u s the mean Sun is, omitting revolutions, ra. 10-19-48. Kendra = Mean Sun -I20° = rd. 10-19-48 -f- 20° = rd. 11-9-48. F o r ' l l full rdsis oi kendra and 9° 48 ' of the 12th, the minutes to be applied are, - 1 1 . - 4 8 , - 6 9 , - 7 0 , - 5 4 , -26,-1-10, -f48, -1-70, -1-71, -•-54, -t-25 X 9° 48730°, which added together is - 1 7 . A p p l y i n g this to the mean Sun, the true Sun is 10' 19° 48' - 1 7 ' = 10M9°31'. (b) A t Epoch the days are zero. Therefore 0 X 120 - 33' = - 3 3 . Mean Sun = -33/43,831 revolutions = - 3 3 X 12 X 30 X 60/43,831 minutes = - 1 6 ' = 1 T 29° 44', (since cycles of 12 rdsu can be added or omitted). Kendram = 1V 29° 44' -f 20° = 0' 19° 44'. A s no full ran orkendram is gone and there are 19° 44' in the first rdsi, the minutes to be applied are: - 1 1 ' x 19° 44'/30° = - 7 ' . T h e true Sun = 1 r 2 9 ° 4 4 ' - 7' = 1 T 29° 37'. It is to be noted that the word kendram here is not used i n its usual sense i n later astronomical works, as the mean anomaly (counted from the H i g h e r Apsis) which is obtained by deducting the longitude o f the higher apsis from the mean planet. I n fact, the use of the expression in modern western astronomy itself is different i n the sense that the anomaly is obtained by subtracting the lower apsis or perigee from the mean planet. T h e Surya Siddhdnta instructs that the kendram is to be obtained by deducting the mean planet from the higher apsis which is equal to the kendram given by others, subtracted from 12 rdiis. T h u s the common characterisdc of these different kendras is that it is used as the argument for the equation o f the centre and in this sense its use is appropriate here also. So we can take it that the mean Sun itself is used as the argument i n the table, the values being given for the intervals 340° - 1 0 ° , 10° - 40°, 40° - 70°, 70° - 100°etc. instead of 0° - 30°, 30° - 60°, 60° - 9 0 ° , 90° - 120° etc. (vide Table)
3. Quoted by Utpala on B S 2, p.40. l a . A.fcTrm b. A.'Tsn^. A l . ^ l c i m c. A . f c ^ ; B 1 . 2 . f ^ ; 63.1^1. A.*Hir
b. B . - ^ ' T ^ ^ c. Al.-g=^?fRTa; A 2 . g : ^ a . ; B . ^ 5 < T f c i a ; C.D.^^Jf^a d. A . W 3a. A l . W . A l . D . - ? ^ ^ : c. B . haplographical omission of«"
42
III.3
PASICASIDDHANTIKA
Kendra Mean Sun
0°
30°
60°
90°
120°
150°
340°
10°
40°
70°
100°
130°
Values for intervals Real Anomaly
-11' 264°
Values taking 140' as maximum
-69'
-48' 294°
-12.9'
-70'
324°
-45.6'
-54' 24°
354°
-67.7'
-71.6'
-26' 54°
-56.3'
-26.0'
180°
210°
240°
270°
300°
330°
360°
160°
190°
220°
250°
280°
310°
340°
+ 10' 84°
+ 48' 114°
+ 12.9'
+ 70' 144°
+ 45.6'
+ 71' 174°
+ 67.7'
+ 71.6'
+ 25'
+ 54' 234°
204° + 56.3'
264°
+ 26.0'
T h e procedure is explained thus: A c c o r d i n g to the Paulisa Siddhdnta there are 120 solar revolutions i n 43,831 days. So,in any desired number of days, the Sun's mean motion is days X 120 43,831 revolutions, which multiplied by 12,30 and 60 successively gives rdsis, degrees and minutes. T h e mean solar year begins 16V^ nddikds after Epoch. Therefore to reckon days from the beginning ofthe year, 16 V 2 nddikds or 33/120 days must be subtracted from the days from Epoch. A s the days have already been multiplied by 120 and converted into one h u n d r e d and twentieth parts, we have to deduct 33 parts in order to deduct 33/120 days, which is the instruction. But what is called mean Sun here is not the real mean Sun. It is the real mean Sun plus the equation o f the centre for the beginning o f the year (this makes it the true Sun) plus 7 minutes of arc. T h a t it is so can be seen thus: A t the beginning o f the year the so-called mean S u n is zero, the S u n having made full revolutions, starting from the zero point 16 V 2 nddikds from E p o c h . T h e Kendram at that time is 0° + 20° = 20°. F o r this we have the difference or interval o f equation o f the centre, - 1 1 X 2 0 ° / 3 0 ° = - 6 ' . 6 7 . D e d u c t i n g t h i s from r a 0-0-0, we have die true Sun ra. II-29-53.Deducting the equadon ofthe centre from this, the real mean S u n is got, for the true S u n is obtained by adding the equation ofthe centre to the real mean Sun. T h u s the so-called mean S u n = the real mean S u n + the equation o f the centre + 7 minutes = the real mean S u n -I- 142' (135'.8 -I- 6'.67), the equation of the centre at this point being 135 '.8 (which we shall show presently). Thus, the so
III.3
III. P A U L I S A - S I D D H A N T A
43
to be 140', if the equation of the centre is computed for these anomalies, and the intervals tabulated, we get - 12.9, - 45.6, - 67.7, - 71.6, - 56.3, - 26.0, + 12.9, + 45.6, +67.7, + 71.6, + 56.3, + 26.0, i n minutes. As these are not much different from the series - 11, - 48, - 69 etc., we see that our assumption about the apsis is correct and the series — 11, —48, are derivable from it. I n the two series, the constants are the same i n some places, differ by 1' i n some, by 2' i n some, the difference being 3' only i n one place. Even this small difference may be due to the constants ofthe given series being empirical, the values having been discovered by repeated observations only. I n this connection it may be mendoned that the tabular values ofthe Romaka differ far more than this does from the same values obtained from formula. O r it may be that the formula used for the derived series may not be full, some terms being omitted. Actually, there are such additional terms, omitted by the ancients, and works like the Surya Siddhdnta give a certain correction to the epicycles which can give the equivalent of such additional terms. T h u s even the small difference is explained. Now for the additive or subtractive nature of the tabular values. T h e y are the equivalents of what is obtained by computing the equation o f the centre for the anomalies 294°, 324°, 354°, etc. and deducting the previous from the next successively. A t the beginning o f the year the anomaly is 284° = rd.9-14-0. T h e n the anomaly increases to 12 rdsis, when the Sun reaches the higher apsis, ra.2-16-0. As d u r i n g this interval the anomaly is i n the fourth quadrant, the positive equation ofthe centre is decreasing. Therefore the tabular values obtained by subtracting the previous from the next are negative, and given as - 11, - 48, - 69. W h e n the Sun has crossed the higher apsis and moves 90° farther, the anomaly is i n the first quadrant. H e r e the negative equation o f the centre increases. Therefore the tabular values are again negative, and we have - 7 0 , - 5 4 , - 2 6 . T h e n the anomaly moves i n the second quadrant, where the negative equation ofthe centre decreases. So the tabular differences are positive and given as -I- 10,-1- 48, -I- 70. After this the anomaly moves i n the third quadrant, where the positive equation o f the centre increases. So the intervals are again positive and given as + 71, + 54, + 25. T h u s , by giving the true Sun increased by 7' as the mean and applying the two series, the first negative and the next positive, i n the place ofthe equation of the centre, the true Sun is computed by this Siddhdnta.
We shall now show that the equation o f the centre for the beginning of the year is 135.8 minutes. T h e mean anomaly for the beginning o f the year is 0 ° - 76° = 284°. Its tabular sine (see Chapter IV) 116' 25", multiplied by the m a x i m u m 140' and divided by the radius 120', gives 135.8'. Itis this we used to show that the mean Sun mentioned i n this Siddhdnta is the real mean Sun plus 142'. W e shall show that this is^so i n another way. W e have seen that the mean Sun, when equal to 76°, is at the higher apsis. As the equatior of the centre must be zero here, the mean anomaly being zero, the true Sun must be equal to the real mean S u n . T h e tabular values to be applied are, - I I ' (for mean Sun interval 340° to 10°, - 48° (for interval 10° to 40°), - 69° (for 40° to 70°) and - 7 0 x 6°/ 30° (for 6° i n the next internal) = - 142'. Deducting this from the mean Sun 76° we get true Sun = 73° 38'. which is also the real mean Sun. T h u s we see the real mean S u n is indeed 142' less than the so called mean Sun. But a doubt arises: T h e real mean Sun ofthe Paulisa at E p o c h is ra. 11-27-22. B u t this does not agree with those of other Siddhdntas, as for e.g., Saura rd. 11-29-49, Romaka rd. 11-29-34, Siddhdnta Siromani, rd 0-0-42. H o w is this difference o f more than two degrees to be explained? This is the answer: Even though the Siddhdntas are generally agreed that longitudes are to be reckoned from the beginning o f the stellar segment Asvini (called the first point), they differ with regard to the actual point where the segment begins. So, reckoned from different points, the longitudes
44
PANCASrODHANTIKA
III.3
naturally differ. Secondly, even i f all Siddhdntas had started reckoning from the same point originally, the difference in the duration of their solar years would, in course of time, cause differences in longitude when computed for a particular moment of time, causing apparently a shift of the first point. T h u s as the first point of the Paulisa is about 2° east of that ofthe Saura or Romaka. the mean Sun is less, as in the case of the Siddhdnta Siromani, which is one degree more because its first point is one degree west of that of the Saura or Romnka. It is also well-known that there is a difference of three degrees between the first point of the Caitra and Raivata Paksas. Now we shall explain why 20° is added to the mean Sun to form a peculiar type of Kmdram to give the values. W h y have the values not been given directly for intervals of mean Sun. 0° to 30°, 30° to 60° etc. T h e reason must be that the author has taken these values from the original Pan.lisa. or the original Paulisa itself from its source, where they would have been given for intervals of mean Sun, 0° to 30°, 30° to 60° etc. But the source or original Paulisa's first point might have been 10° east of that o f the Paulisa here given, and it might have been shifted west in course of time to the present first point adopted, (there is evidence i n the Vedas o f this kind of shift being made, as evident, for instance, i n the case of the beginning of the year from the Sun at Mrgasira to the Sun at Krttika and so on) with the result that what was originally 0° had become 10°, what had been 30° had become 40° and so on reckoned from the new point. So the values are as i f they are given for 340° to 10° etc. and to make them full rdsis for convenience, 20° are asked to be added and the name kendram given to it. T h u s everything is properly explained. T S have not understood the method given here. So far as the explanation o f the part referring to the mean Sun goes, what they say is correct. B u t after that what they say is all wrong. There is the express instrucdon after giving the first series 11, 48 etc. that the values should be subtracted from the mean Sun. T h e second series 10,48 etc. come after that in a separate sentence and a separate verse, with the instruction that the values should be added. B u t somehow T S have understood the two instructions to mean that the two series should first be added one to one and then the resulting new series, which they think is the equation of the centre it.self, should be applied to the mean Sun. If this is done, the instruction where to add and where to subtract is lost, which they have not noted. They have failed to see that the Siddhdnta gives differences of values for every 30° intervals of kendram. T h e y have never considered why i f the equation o f the centre itself is given, it is given in two series which are almost identical. A g a i n , the new series of theirs only appears to be the equadon of the centre, which is because the differences of a sine-function-series is a cosine function series, which is again a sine-function-series with a lateral shift of 90° in the argument. In verifying the series they have formed by comparing it with the actual, which they have computed, they have found a difference of 6' and 7' in two terms, but waived them aside as negligible. B u t 6' or 7' are too large to be negligible. Further, they have failed to see that the word kendram is used here in a peculiar sense as we have already said. T h e y have taken it to mean the regular anomaly. B u t the mean anomaly can be found only i f the longitude ofthe higher apsis is given, which is nowhere to be found. T h e y explain this by saying that the longitude o f the apsis was well known and therefore not given! Different SiddMntas give different values for the longitudes o f the apsis; the Romaka gives 75°, the Saura gives 80°, a n d the Aryabhatiya and Surya Siddhdnta give 78°. Which of these are we to take? Certain things and operations, we can understand from the context and nature of the work, but this is not a thing which can be learnt without being told, as also the instruction when to add o r when to subtract the values, which, according to them, has also to be learnt otherwise. If they had only tried to work out some examples, as for instance, taking the mean S u n as 80°, using their interpretation ofthe rules, then they would have discovered their mistake.
II.4
III. P A U L I S A - S I D D H A N T A
45
True Motion of Moon 4. If the padas obtained (by II.2) are phis-padas, (i.e., in the first half-gaft) subtract 9 from the padas, multiply by 10 and divide by 7. A d d the result to 702. T h e Moon's daily true motion in minutes is got. In the second half-g-ai!i, (i.e., if the padas are m'mus-padas), deduct 9, muldply by 10 and divide by 7 and subtract the result from 879. T h e resulting minutes are the daily true motion of the Moon. Example 2. Find the Moon's daily true motion for plus padas 9,18, 27, 36, 63, 71, 72, 81,117,124 mmus-padas 9, 63, 71, 72,117, 124. P = 9 P = 18 P = 27 P = 36 P = 63 P = 71 P = 72 P = 81 P=117 P = 124 P = 133 P' = 9 P ' = 63 P ' = 71 P ' = 72 P'=117 P'=I24
daily motion " " " " " " " " " " " " " " " "
= = = = = = = = = = = = = = = = =
and
10 (9 - 9)/7 + 702 = 702 minutes 10(18 - 9)/7 + 702 = 702 + 12 6/7 = 714 6/7 minutes 10 (27 - 9)/7 + 702 = 702 + 2 X 12 6/7 = 727 5/7 minutes 10 (36 - 9)/7 + 702 = 702 + 3 X 12 6/7 = 740 4/7 minutes 10 (63 - 9)/7 + 702 = 702 + 6 X 12 6/7 = 779 1/7 minutes 10(71 - 9)/7 + 702 = 702 + 88 4/7 = 790 4/7 minutes 10 (72 - 9)/7 + 702 = 702 + 7 x 12 6/7 = 792 minutes 10 (81 - 9)/7 + 702 = 702 + 8 x 12 6/7 = 804 6/7 minutes 10 (117 - 9)/7 + 702 = 702 + 12 X 12 6/7 = 856 2/7 minutes 10(124 - 9)/7 + 702 = 702 + 164 2/7 = 866 2/7 minutes 10 (133 - 9)/7 + 702 = 702 + 177 1/7 = 879 1/7 minutes 879 - 10 (9 - 9)/7 = 879 - 0 = 879 minutes 879 - 10 (63 - 9)/7 = 879 - 6 X 12 6/7 = 801 6/7 minutes 879 - 10(71 - 9)/7 = 879 - 88 4/7 = 790 3/7 minutes 879 - 10(72 - 9)/7 = 879 - 7 X 12 6/7 = 789 minutes 879 - 10 (117 - 9)/7 = 8 7 9 - 12 X 12 6/7 = 724 5/7 minutes 8 7 9 - 10 (124 - 9)/7 = 879 - 164 2/7 = 714.5/7 minutes
T h e daily modon when the pada is less than 9 cannot be found from the formula as it is, but we can frame a rule by considering the nature o f the variation o f motion. M'mus-padas 1 to 8 are the successive padas after p\us-padas 124 and previous to mmus-padas 9. Therefore the m o d o n must lie between 866 2/7 and 879, increasing from 866 2/7 (vide example). Therefore, mmus-padas 1 to 8, muldplied by 10 and divided by 7, added to 866 2/7 will give the motion. In the same way, pluspadas 1 to 8 are successive padas after minus-padas 124, and before p\us-padas 9. Therefore the modon lies between ll^bll and 702, decreasing gradually. Therefore p\us-padas 1 to 8, muldplied by 10 and divided by 7 must be deducted from 714 5/7 to get the motion. 4a., A.B.C.H'II^M^I^. B1.3.^?Trara b. A . B . B f w M ; C . ^ « [ W ^ ] B1.3.^I^:; B2.^7^^:
46
PANCASIDDHANTIKA
II1.4
T h e two rules for the true daily modon can be shown to be connected with the two rules for finding the true motion in II.6, by deriving these from those, and the exact derivation itself is a proof of the correctness of the rules. T h e rule for plus-padas (i.e. applicable to the first hali-gati) is, {1094 + 5(P - 1)} P/63. If m is the mean motion at the end of the last iullgati, then the true M o o n = m. + P° + {1094 + 5(P - I)} P/63 minutes. T h e daily motion of the current day is got by subtracting the true M o o n at the end of the previous day from that at the end of the current day and if the pada is P at the end of the current day, it is P - 9 at the end of the previous day. Therefore the motion for the current day = [m + 9° + {1094 + 5(P - I)} X P/63] - [m + (P - 9)° + {1094 + 5 (P - 9 - 1)} (P - 9)/ 63]' = (m - m) + P° - (P - 9°) + {1089 + 5P)} P/63 minutes - {1089 + 5(P - 9)} (P - 9)/63 minutes = i n minutes, 540 + 1089 P/63 + 5 P % 3 - 1089 (P - 9)/63 - 5(P - 9)%3 = 540 + 1089 x 9/63 + 5 X 9V63 + 90 (P - 9)/63 = 540 + 1134/7 + 10 (P - 9)/7 = 540 + 162 + 10 (P - 9)/7 = 702 + 10 (P — 9)/7. which is the rule here given. In the same way, taking the rule for the true M o o n i n the second half-ga/«, i.e. for minus-padas, we have, the daily motion = [m + rd. 6-0-4 -I- P° -)- {2414 - 5 (P' - 1)} P'/63 minutes] - [m + rd. 6-0-4 + (P' - 9)° + {2414 - 5 (P' - 9 - 1)} (P' - l)/63 minutes] = in minutes, 504 + {2419 - 5P'} P'/63 - {2419 - 5(P' - 9)} (P' - 9)/63 = 540 -t- 2419 P'/63 - 5 P ' % 3 - 2419 (P' - 9)/63 5 (P' - 9)^/63 = 540 -h 2419/7 - 45/7 - 10(P' - 9)/7 = 540 + 339 1/7 - 10 (P' - 9)/7 = 879 1/7 - 10 (P' — 9)/7, which is the rule for daily motion i n the second hali-gati, omitting the small fraction of minutes 1/7. (Note that in the example we actually got this 879 1/7 as the maximum).
T r o m the relationship between these two sets of rules shown here, we can understand that the rules for daily motion, though they have strayed into the chapter dealing with the Paulisa, actually belongs to the Vdsistha. If they are to be used for the Paulisa also, it is because they are interconnected and mixed up, as the use of the same technical terms, and the absence of the method to find the mean M o o n , show. W e can even say that the author does not intend this for the Paulisa because another set of rules is given for this i n III.9.
Because the daily increase in padas is 9, the daily increase or decrease in the true motion is 10 X 9/7 = 12 6/7 minutes. By integrating the motions and deducting the mean motion d u r i n g the days for which the integration is done, we can find the equation of the centre implied in the rules: For convenience let us take padas 9, 18, etc. and work out for plus-padasfirst.T h e total true motion in minutes for P/9 days, i.e. to the end oiPpadas is:702 + 10 (9 - 9)/7 -I- 702 -I- 10 (18 - 9)/7 -I- 702 -I- 1 0 ( 2 7 - 9)/7 + 702-^ 10 (P - 9)/7 = 702 x P/9 + 9 x 10 {1 2 + 3-I(P - 9)/9}/7 = 702 P/9-1-9 X 10 {i/2(P - 9)/9} P/(9 X 7) = 702 x P / 9 + 5 P ( P - 9)/63 = 702 x 7P/63-h5PV63 - 45P/63 = 4869P/63-f-5PV63. T h e mean motion per day can be found from 11.2-4 to be 790' 35" and for P/9 days, the mean motion is 790' 35" x P/9 = 5534 P/63 minutes. Subtracting this from the true motion found, the equation ofthe centre obtained is 4869 P/63 + 5 P % 3 - 5534 P/63 = 5P%3 - 665 P/63 = (5P - 665) P/63. In the same way, we can find the equation o f the centre connected with the second half-gaft, i.e. minuspadas. The total true motion = 879 1/7 x P79 - { 5 P ' % 3 - 45 P'/63}= 6154 P'/63+45 P'/63 5 P ' % 3 . Deducting the total mean motion, the equation ofthe centre = (6154 -t- 45 - 5534) P'/63 - 5 P ' % 3 = 665 P'/63 - 5 P ' % 3 = P(665 - 5P)/63. It is these two rules for the equation o f the centre that we used in II.6, to derive the rules there. From inspection we see that P(5P - 665)/63 is negative for all values of P, as it ought to be i n the first half-gati. P ' (665 - 5P')/63 is positive, as it ought to be i n the second hali-gati for all the values of
111.4
III. P A U L I S A - S I D D H A N T A
47
P'. T h e numerical m a x i m u m is for P o r P' = 66'/2. For P = 124, the value is — 88 V 2 minutes which is cancelled by + 88'/2 minutes for P ' = 124, withtheresultthatattheendoftheg-a
Equation of the centre 5. Reduce the plus or rmnns padas by one and multiply by 40. Subtract this from 5261. Multiply the result by the padas and divide by 729. T h e resulting minutes are equation of the centre. T h e formula is: T h e equation ofthe centre = {5261 -40 pada, plus or minus, without distinction.
{pada - I)] padan29, where pada is any
Example 3. (a) Pada = 63. Find the equation of the centre, (b) Pada = 9. Find the equation ofthe centre. (a) . (5261 - 40 (63 - 1)} 63/729 = (5261 - 2480) 63/729 = 2781 x 7/81 = 240 1/3 minutes. (b) . {5261 - 40 (9 -- 1)} 9/729 = (5261 - 320) 9/729 = 4941 x 1/81 = 61 minutes.
5a. B.M
c. A . ^ t ^ ; B1.3.-!?l'2Tr; B 2 . ' ^ d. B 2 . ^
48
PANCASIDDHANTIKA
III.10
6-9. (Translation and Commentary later).
[^:] ' 1 ^ : 9 1 % ' - ' T f g : W ^ : ' * q # T T : W ' TMM>ie
Cara or Oblique ascension 10. Multiply the constants, 20,16 Va and 6% by the equinoctial shadow. T h e results are oblique ascensional differences (carakhandas) in vinddis, first in the given order, then in the reverse order for the first six months (solar) and again the given and reverse orders for the second half of the ecliptic, the second six months. T h e differences are for the solar months Mesa etc. i n vinddis: 20, 16 Va, 6 % , 6 % , 16 Va, 20, 20, 16 Va, 6 ¥ 4 , 63/4, 16 Va, 20) x equinoctial shadow. T h e formula for Cara is given i n IV.26. thus: Sine Cara = sin latitude X sin Sun's declination X the diameter -r- sin (90° — latitude) x the day-diameter. T h e sines used here are tabular sines, the
6a.
BS-yrf^*
b. B.^lfeTtarr C.^:^:"?T^ft?n^. B.ft?P3; d.
A . T I c q ^ ; B.^IcJTS^
8a. Al.°f«I*^ A . ^ J T ^ : ; B.^!R3fW
c. C.TOf^
7b. A1.M«W Bl.^?RlfiT-fiTa^. Al.B2.3.^rd*w< c. A.^R^cff:; C."^^:. A . B . ^ ^ ; C.<=hld'q%
9a. C . ^ a-b. C . ^ c f T I ^ a [ ^ ] ^ ^ % i ^ I b-c. A . % q t : ; B.T^1cHHM^'ld
d. C . T l f e ^ . A . C . W : ; B.^CcT: Bi.s.^^rarg; B2.^^irai^ 10. Quoted by Utpala on BS 2, p.60. 10a. A . f R l f ^ : . C . W a - b . U . ^ * n ^ : b. B.4l
c. Bl.-jl&RTr d. A . ^ ° ; B.a?ira:. A . C . ^ ^ ^ ; B . 1 | : T P % §
111.10
III. 1 ' A U L I ! ^ A - S I D D H A N T A
49
unit or radius being 120', and the diameter 240'. T h e day-diameter is twice the tabular cosine of the Sun's declination. T h u s this formula reduces to the modern form: sin cara — k tan latitude x tan declination. T h e degrees of cara got from the formula, converted into minutes and divided by 3 gives the whole cara. fhe cara-kharidas or differences are got deducting the whole cara of the beginning of a rdsi from that of the end of the rd.ii. We shall not dciivc this foimula now. We shall restrict ourselves to deriving the given carakhandas from the l o n u u l a . The diameter', as we have said, is 240' (vide IV. 1). The tabular sines of declination at the ends ofthe three rdsis Mesa, etc. is 24' 24", 42' 15", 48' 48" (from 1V.24). T h e daydiameters for the same are 235', 224' 40", 219' 15" (from IV.25). Sin lauiude ^ sin (90° - latitude) = equinoctial shadow -H12. (This will be shown when dealing with IV). Therefore; successively, sine cara = equinoctial shadow x {(240 X 24 2/5) (235 X 12), (240 x 4 2 ' A ) H- (224 2/3 x 12), (240 x 48 4/5) -^(219'/4 X 12)} = E q . shadow X (2'.08,3'.78,4'.45). T h e arcs of these cannot be found in terms of the Eq. Shadow unless it is known. But as the author intends this rule only for North India, where the eq. shadow may be taken as 6 digits oh the average, we shall frame the rule for 6 and then use it for other places by the rule of proportion. So, multiplying the numbers 2'.08,3'.78, 4'.45 by 6, we get sine cara = 12' 29", 22' 41", 26' 42". Using the tabular sines, the arcs o f these = 5° 58'.3, 10° 53'.6, 12° 51 '.6 = 358'.3,653'.6, 771 '.6. Dividing by 3, we get the whole cara invinddis, 119.4, 217.9, 257.2. Dividing these by 6 and multiplying by E q . shadow, we get the whole cara vinddis for any E q . shadow, viz., E q . shadow X (19.9, 36.3, 42.9). Deducting the next from the previous, and because the cara is zero for the beginning of Mesa, as declination is zero, the cara differences (khandas) got are, E q . shadow X (19.9, 16.4, 6.6). T h i s is practically the same as E q . shadow X (20, 16'/a, 6%) given by the author. In the Vdkyakarana, Mahdbhdskariya and Siddhdnta Siromarii, the same method is given. As we have said, the cara-vmdclis are zero for the beginning of Mesa when day-light is 30 nddis. T h e n at the end of each rd.si, they increase by a quantity equal to the differences, reaching a maximum at the end of Mithuna, the day-light then being a m a x i m u m also, as the declination has reached a maximum. After that the declination decreases in the same manner in which it has increased, and the day-light decreases from m a x i m u m and the cara also decreases. A t the end of Kanya the declination becomes zero again, the day-light equals 30 nddikds again, and the cara becomes zero again. So the differences have to be used in the reverse order for Karkataka, Siinha, and Kanya. After this the South declination increases and decreases just like the N o r t h declination and corresponding to this the night-time increases from 30 nddikds to a m a x i m u m and decreases again to 30 nddikds at the end of the next six months; and the cara also increases from zero to a maximum and then decreases to zero, repeating what it is for North declination. Hence the instruction to repeat for the next six months ofthe solar year. If the cara is required for any day within the month, it is to be got by interpolation, which is well known and therefore not mentioned. Example 4. (a). In a place the Eij. shadow is 5 digits. Find the cara-vinddis when the Sun is rd.2-10-0. (b) The Eq. shadow is 7. the Sun is at the end of Tula, find the cara-vinddis. (a) T h e constant for Mesa is 20, for Vrsabha i O V a a n d f o r 10° of Mithuna, 6 % X 10°/30°= 2V4. Adding, the U)tal cara is 20+ 16'/i; + 2'/4 = 3 8 % . M u l d p l i e d by Eq. shadow, the wmac^w are 5 X 3 8 % = 193%. (b) T h e Sun is m.7-0-0 and therefore 1 rdsi has gone in the second half of the zodiac. Therefore liw cara-vinddis = 7 X 20 = 140.
50
III.l
PANCASIDDHANTIKA
Day-time 11-12. T o find the day-time in Mesa, Vrsabha and Mithuna, add the cara differences one by one, in the order given, to 30 nddikds, and in the next three, subtract in the reverse order. In the next three rdsis, T u l a , etc. subtract from 30 nddis in the given order, and for Makara etc. add in the reverse order. This will give the day-time fairly accurately for places in Northern (whole?) India. I shall give the method to find the day-time accurately in other places (in the IV chapter) when dealing with spherical astronomy.
Thus, when the Sun is i n the six rdsis, Mesa, etc. 30 nddikds cara-vinddis = day-time. W h e n in t six rasis, T u l a etc. 30 nddikds cara-vinddis = day time. T h e increase over 30 nddikds and the decrea to 30 is by E q . shadow X {20, I 6 V 2 , 6 % , 6 V 4 , 1 6 V 2 , 20}vinddis. This is repeated in the decrease from 30 nddis and the increase to 30, i n the six rdsis from T u l a . W e shall explain all this in the I V chapter.
T h e cara-vinddis computed i n the above manner and the day-time got by them will be accurate only in Northern (whole?) India it has been said. T h e reason for this is as follows: F r o m the explanation ofthe method o f computing the cara-vinddis, it can be noted that sine cara is proportionate to the E q . shadow, and the cara-vinddis are proportionate to the arc obtained from sine cara. It is well known that when the sines are small they are proportionate to their arcs. Therefore when sine cara is fairly small, i.e. when the E q . shadow is small, the arcs are proportionate to the Eq. shadow, i.e. the cara-vinddis are proportionate to the E q . shadow. In India the Eq. shadow is fairly small, and therefore the cara-vinddis for different places in India can be formed by proportion, using the E q . shadow. In higher latitudes like places N o r t h of the Himalayas, the shadow increasingly becomes greater, and the inaccuracy of using the given method will gradually increase.
Example 5. (a) Eq. Shadow 5, Sun rd. 2-10-0. Find the day-time, (b) The Eq. shadow is 7, the S 7. Find the day light.
(a) T h e cara-vinddis are 5(20 -I- I 6 1 / 2 -I- 2 ' A ) = 5 X 3 8 % = 194. As the Sun is i n the 6rdsis, Mesa etc., the day-time is greater than 30 nddis, and, therefore, the day-time is 30 nddis + 194 vinddis = 33-14 nddis.
l l a . A.B.iimRtl^Ml^d ( B 2 . ° ^ ° ; B 2 . M ) b. A . B . ^ ^ f e ^ . C . ° ^ [ ^ ] ^ . A . ' g ^ l f t ^ c. A.B.^Hlt^;
C.V^
d. B.gcTI%f. A . C . D . ° 3 % ^ ; B . ° g % ^
12a. Al.f|Hrqft«ft; A 2 . B l . C . f | i n f 5 ; q M ; B2.3.f|Rf5^^ c-d. A2.^^cnnt; B2.3.W0cl^ d. Al.rld?J%
III.12
51
III, P A U L I S A - S I D D H A N T A
(b) T h e cara-vinddis are 7 X 20 = 140. T h e Sun being within the 6 rdsis from T u l a , the day-time is less than 30 nddis. Therefore the day-time is 30 nddis — 140 vinddis = nddis 27-40.
Desantara 13. T h e c o r r e c t i o n to the t i m e o f the l o n g i t u d e o f Y a v a n a p u r a to get the t i m e o f the l o n g i t u d e o f U j j a i n is seven nddikds, 2 0 vinddikds a n d that o f B a n a r a s ( V a r a n a s i ) is n i n e nddikds. H o w to find the c o r r e c t i o n f o r o t h e r l o n g i t u d e s w i l l be g i v e n ( i n the n e x t verse). In short, what is here given is the difference i n time due to difference i n longitude alone from Yavanapura of Ujjain and Banaras, Desdntara-nddis, being difference i n dme o f the occurrence of any event due to longitude, the occurrence being earlier by this dme i f the place is East, and later if West. Actually the time difference due to longitude for Ujjain from Yavanapura is nddis 7-38, and for Banaras, nddis 8-50. T h e Greenwich East Longitude of Yavanapura, Ujjain and Banaras are 30°, 75° 50' and 83°. F r o m this we can find the actual dme difference: (75° 50' - 30°)/6 nddis = 7-38, (83° - 30°)/6 nddis = 8-50. B u t considering the difficulty o f d o i n g this faced by the ancients for want of facilities, the achievement of the Siddhdnta is commendable.
3T^g.^(^c«(d ipTT; N^chic^dl ^ T l ^ ; ||
\
14. T a k e the distance myojanas b e t w e e n the t w o places b e t w e e n w h i c h the t i m e d i f f e r e n c e f o r l o n g i t u d e has to be f o u n d . M u l t i p l y this by 9 a n d d i v i d e by 8Q. ( T h e result is t h e i r distance i n degrees). S q u a r e the result. F r o m this d e d u c t the s q u a r e o f the d i f f e r e n c e i n l a t i t u d e b e t w e e n the t w o places. F i n d the s q u a r e r o o t o f the r e m a i n d e r . ( T h i s is the E a s t - W e s t d i f f e r e n c e i n degrees). T h i s d i v i d e d by 6 is the t i m e d i f f e r e n c e i n nddikds. T h e purpose of finding this time difference is uldmately to find the time difference for longitude from Yavanapura, so that it may be used i n the reduction of the planet found for Yavanapura mean sunset to the local sunset. If by this rule, the time difference from Ujjain or Banaras is found, then I4a. A.ft'+ldWHslcl^ b. A.ftSig; B1.2.ft'i5rac!I°; B3.ft«5«cr° 13a. A.^1<^HW
6
A.B.°^^5ip^(B3.^3llI^)
b. A.B1.2.<=lTA|lfel'*M
c. A.B.f^=<^*rd
c. B2.=ll"IK«i d. Al.^n«FRJJf: A2.'?ntHHW
d.
A.B.2.^:;B1.3.^:;C.-^ A I . B . ^ « P ; D.^^f5|rr
52
PANCASIDDHANTIKA
III.14
by adding or subtracdng this, according as the place is East or West, to or from nadis 7-20 (for Ujjain) or 9-0 (for Banaras), respectively, the time difference for longitude from Yavanapura can be got.
Example 6. The latitude of a place west of Ujjain is 27° and the distance between them is 44 4l9y Assuming the latitude of Ujjain to be 24°,findthe time differencefor longitude of the placefrom Yavanap T h e distance in degrees = 9 X 44 4/9 = 5°. T h e difference in latitude in degrees is 27° - 24° = 3°. V 5'"^ - 3^ = 4 = the east-west difference in degrees. 4/6 nddikds = 40 vinddikds is the time difference. A s the place is West of Ujjain, deducting this from 7-20, the time difference from Yavanapura is nddikds 6-40.
T h e rule is explained thus: It is well known that i n a plane right-angled triangle the square ofthe hypotenuse is equal to the sum ofthe squares ofthe sides containing the right angle. Now, let the places be P, and Pj. Let the point where the line of longitude passing through one of the places, say P,, cuts the latitude passing through the other say P2, be C . T h e n C is practically a right angle of which C P , is one arm and C P ^ the other, and PjPj is the hypotenuse of the right angled triangle P j C P j . If P1P2 is not too great, then this triangle, though on the surface of a sphere, may be treated as practically a plane triangle. P | C is the difference i n latitude ofthe places. P j C is the difference in longitude, which is wanted. P i P j is the distance between them. We are going to find the difference i n longitude P j C i n degrees. T h e difference i n latitude P . C is also in degrees. So P i P j also must be found i n degrees. So the distance myojanas is converted into distance i n degrees by multiplying Xheyojanas by 9 and dividing by 80, because according to this Siddhdnta there are 9° for 80yojanas on the Earth, i.e. the circumference ofthe Earth (360° i f given in degrees) is 3200yo7owa,$. T h u s we have the difference in longitude in degrees =CP2^ = V PiP2^ — CP,^. T h e degrees are converted into dme by the proportion, i f there are 60 nddikds for 360° of longitude, how much for the degrees got. Therefore degrees got x 60/360 = degrees got/6, are the nddikds of difference in longitude.
In view of the right angled triangle not being exactly plane, a better result will be got if the nddikds obtained are muldplied by the circumference of the earth and divided by the circumference of the line of latitude midway between the places. But the author has not mentioned this because this method is intended for India and in India the two circumference do not differ much and therefore the difference between the two methods will be negligible. Sdrya Siddhdnta etc. give the correct method. T h e author's method is given by the Mahdbhdskariya with the name Adhvd (Tath') and by the Vatesvara Siddhdnta with the name 'Adhvavaha' ('Marching on the Path'), only to be condemned as inaccurate. In the Vdkyakarana, which is also satisfied with rough results, the distance along the latitude (CPg in the explanation) is taken as found and a rule given. T h e method of determining the latitude of a place is given i n IV. 20-21 and the author expects us to get it by using that method and use it in the formula. Another thing must be mentioned here. If the two places are distant from each other or intervened by a sea or mountain or some such obstacle, as for instance Yavanapura and Ujjain, Yavanapura and Banaras, or Ujjain and Banaras, then by observation, from the two places, of celestial phenomena that are visible everywhere at the same moment, like the circumstances of a lunar eclipse, the time difference for longitude can be obtained. (All Siddhdntas give methods based on this principle, and the Mahdbhdskariya i n Chapter II, which is, devoted exclusively to Desdntara, gives two methods.) T h e following is the method: Let us assume that the meridian of Ujjain as the prime meridian (generally given i n all Siddhdntas do) and by computation it has been found that at 5 nddis after midnight the total obscuration o f the moon begins. (The beginning or end of the
III.14
HI. P A U L I S A - S I D D H A N T A
53
total phase can be observed well aind therefore specially chosen by the Surya Siddhdnta). In anotner place, say in Banaras, the beginning o f the total phase is observed to be at nadikas 6-12, (of course by its own time, i.e. local time). T h e time difference for longitude must be patently the difference between the two times, i.e. nddis 6-12 minus nddis 5, i.e. na. 1-12. A s the local dme must increase as we go east, we can also say that Banaras is east o f Ujjain.
[^a^VIIWchlH:]
Local Sunset time 15. If the Sun is in the sixrdsis Mesa etc., subtract half the carat^ma& from the time difference for longitude. If the Sun is in the six rdsis T u l a etc. add half the cara-vinddis to the time difference for longitude. Find the motion of the planet (the Sun or Moon, in this context) during this dme. Subtract this from the longitude of the planet computed. (The planet for the beginning of the local day, i.e. for local sunset, is obtained).
Example 7. The time differencefor longitude at a place (in India) with reference to Yavanapura is 10 The Sun is rd. 9-0-0 and the cara for that day at the place is 4 nddis. The Moon computed is rd. 4-8-0 and it daily motion 840'. Find the Moon at the beginning of the local day, i.e. at local sunset of the place. Longitude time difference = 10 nddis. H a l f cara = 4/2 = 2, nddis. A s the Sun is within the six rdsis from T u l a , adding 10 and 2, we get 12 nddis. T h e motion per day is 840'. T h e motion for 12 nddis is, 840' X 12/60 = 168' - 2° 48'. Subtracdng from the computed M o o n , the M o o n at local sunset is, ra. 4-8-0 - 2" 48' = ra. 4-5-12. T h e explanation o f the procedure is as follows: A s the days from E p o c h are from mean sunset at Yavanapura, the planets computed for the days from E p o c h are for mean sunset at Yavanapura and i f they are required for any time i n the day, they have to be found by adding the motion d u r i n g the time. Therefore i f the planets are required for local sunset at any other place, the time interval between the local sunset and Yavanapura mean sunset has to be found first, and the motion for the interval applied to the planet computed. T h i s motion is subtractive i f local sunset is earlier, and additive i f later. Now, the author intends the procedure for India alone, and at places in India the local sunset is always earlier than Yavanapura mean sunset. This is because nowhere in India (including Afghanistan) is the difference for longitude from Yavanapura less than 4 Vs nddis, and the half cara greater than this. Because India is east o f Yavanapura i n the local mean sunset is earlier by the time difference for longitude. T h e actual sunset is earlier than the mean sunset or later by half the cara-nddis. It is later i f the S u n is i n the six rdsis. Mesa etc., because the day-time is longer. It is earlier i f the Sun is in the six rdsis, T u l a etc. Therefore the local actual sunset is earlier than Yavanapura mean sunset by the time difference for longitude minus half the nddis o f cara when the daytime is greater. B u t 15a. A . ^ : ^ ! ^ ; B.'RlStRT b. C.^^ni5I«f^°. B . ' ^ f ^
c. B.(S|*WI-I d. B . ^ . B.'kRpT; D.«(FF?f^
54
PANCASIDDHANTIKA
III. 15
as the half cara is always less, there is always a remainder when this subtraction is made by which time, therefore, local sunset is always earlier. W h e n the Sun is i n the 6 rasis, T u l a etc., the local sunset is earlier by the time difference for longitude and still earlier by the half cara. Therefore it is earlier by the sum of the two. T h u s , i n all cases, with regard to places in India, the sunset is earlier than the mean sunset at Yavanapura and therefore the motion for the dme is always to be subtracted; and, that is the instruction. If the place is north of India or west, it may happen that the half cara is greater than the dme difference for longitude. T h e n when the Sun is in the six rasis from Mesa, the sunset may be later and the motion for the later time will have to be added to the planet. Further, there are two other corrections, Bhujantara (correction for the Equation o f the centre) and Udayantara (Reduction to the Equator), the equivalent o f the equation of time which have got to be made, but not given by this Siddhdnta either because these are very small or because this Siddhdnta is not aware of its existence. T h e Vdkyakarana omits to give the Udaydntara alone because it is not found even in its source, Bhdskariya. T h e Sdrya Siddhdnta omits the cara and Udaydntara corrections, the former because begins the day at midnight which is not affected by cara and the latter because it is not aware of its existence. Sripati is the first to give the Udayantara. T S are unaware that the two corrections mentioned above are given here, which can be seen from the commentary pp. 12-13, and English notes pp. 16-17. They also seem to think that the instrucdon to subtract or add is for places i n the northern and southern hemispheres, respectively, (cf. Skt. com., p. 13). N P , too, have not got the sense fully, for they observe on this verse; " A fragmentary passage which i n the present form makes no sense, e.g., because one cannot add longitudinal differences and ascensional differences." (Ch. II, p. 31).
Naksatra computation 16. For every 800 minutes of arc in the Moon's longitude there is one naksatra (asterismal segment). Deduct the Sun's longitude from the Moon's. For every twelve degrees of the remainder there is one tithi. T h e time of the ending moment of the naksatra should be found by proportion using the Moon's daily motion. T h e time ofthe ending moment ofthe tithi should be found by proportion, using the difference in the daily motions of the Sun and the Moon. T h e idea is this: Compute the Moon's longitude for the end of the day on which the naksatra is to be found, and also the motion for the day. Convert the longitude into minutes and divide by 800. T h e quotient are the naksatras gone, and the last o f them is the naksatra ending i n the day, before the incomplete one began. Multiply the remainder by 60 and divide by the daily motion i n minutes. 16a. B l . 3 . ? ^ ; B2.?%. A . f ^ s # ; B.f^RllTrat b. A.tcTfsrfS"; B1.2.°^s^=5[fM«Tf5°
c. Bl.S.^JtJMT d. B.^n#5
III.16
III. P A U L I S A - S I D D H A N T A
55
T h e result are nddis of the incomplete naksatra gone i n that day. Deduct it from 60. T h e ending moment of the naksatra gone or last gone on that day is got in nddis from the beginning ofthe day. T o get the tithi, subtract the Sun's longitude from the Moon's and convert it into minutes. Divide by 720. T h e quotient are full tithis gone after new moon. T h e last tithi gone is the tithi ending before the incomplete one begins. M u l d p l y the remainder by 60 and divide by the difference of the Sun's and Moon's motions i n minutes, for that day. T h e result are nndis ofthe incomplete tithi on that day. Deduct the nddis from 60. T h e remaining nddis are the ending moment of the tithi ending or last ending on that day. If the total tithis got are more than 15, count again from one, i.e. Prathamd. Example 8. At the end of a certain day the Sun is rd. 2-15-10 and its daily motion 57'. The moon is rd. 10-18-30 and its daily motion 827'. Find the naksatras etc. for the day. T h e M o o n = rd. 10-18-30 = 318° 30' = 19,110'. Dividing this by 800 the quotient, i.e. full naksatras gone is 23, and the remainder 710' has gone i n the 24th. Multiplying by 60 and dividing by the daily motion, 710 x 60 827 = 51-31, nddis, belong to the 24th. Therefore the 23rd, i.e. Sravistha ends 60.0 - 51.31 = 8.29 nddis, after the beginning of the day. T h e Tithi: M o o n - Sun = rd. 10-18-30 - rd. 2-15-10 = rd. 8-3-20 = 243° 20' = 14,600'. Dividing by 720, the full tithis gone are 20, and the remainder 200' has gone i n the nexttithi. Multiplying this by 60 and dividing by the difference of the daily motions, the nddis got are 200 X 60 (827 - 57) = 200 X 60/770 = 15-35, which is the time occupied by the 2lst tithi. Therefore the 20th tithi, i.e. Bahula PancamI ends at 60 nddikds-15.35 nddikds, i.e. 44.25 nddikds after the beginning of the day. T h e length of a naksatra is 800' and the Moon's mean daily motion 791', is not much different from it. T h e length of a tithi is 720' and its mean daily passage 732 is not very much different from it. Therefore generally there is one naksatra or one tithi ending i n a day. B u t it may happen that the remainder is so small and the motion or passage for the day so great, that, remainder -I- length of a naksatra or tithi < the daily motion or passage. In this case, two naksatras or two tithis end on the same day. In the case of the tithi this is called avama, the second ending tithi being immersed i n the day and not counted for reckoning days. O n the other hand, the remainder may be greater than the motion or passage for the day, with the result that no naksatra or tithi ends on that day, i.e. they begin at close ofthe previous day, extend throughout the day and end at the beginning of the next day. W h e n this happens in the case of a dthi it is called Tridina-sprk, literally 'touching three days'. T h e explanation o f t h e rules is as follows: T h e Zodiac consists of 12 rdsis, i.e. 12 X 30 X 60 = 21,600 minutes. It is divided into 27 equal segments called naksatras and so there are 21600 ^ 27 = 800 minutes for each segment. What is called naksatra i n the verse and sought to be found, is the segment in which the M o o n is situated. Therefore the Moon's longitude i n minutes is divided by 800 and the quotient are the segments passed. T h e remainder is the position of the M o o n i n the next segment. T h e time taken by the M o o n to pass that portion is found by the proportion; If the daily motion takes 60 nadikas to pass, how long will the remainder take? Therefore the time when the M o o n has been at the end of the segment just passed, i.e. the end of the naksatra passed, falls before the end of the day by the obtained nddikds. Now for Tithi: W h e n after new moon the M o o n leaves the Sun behind for every 12° one tithi is gone. Therefore by subtracting the Sun from the M o o n , the total degrees left behind is found, and dividing this by 12° or 720' the tithis gone is found. Everyday, i.e. i n every 60 nddikds, the Sun is left behind by the difference of their motions. Therefore the time taken for leaving behind the remainder is: remainder X 60 difference o f t h e motions. A s the remainder is of the incomplete tithi, the completed tithi ends before the end o f the day, by a time equal to the nddikds found.
56
III.17
PANCASIDDHANTIKA
' Iml - f ? l f e - T p T T - 5 f j r - W ^ ' - * " f ^ ^
^cBT
^ - C F T '
Sun's daily motion 17. T h e d a i l y m o d o n o f the S u n i n m i n u t e s d u r i n g each o f the twelve m o n t h s , M e s a etc. is 58, 57, 57, 57, 58, 5 9 , 6 1 , 6 1 , 6 1 , 6 1 , 60, 5 9 . T h e daily motion for the month is found thus: D u r i n g every solar month, the Sun moves one rdsi, i.e. 1800'. Dividing this by the days of the month the daily motion is got. O r , according to the length of the month, take 31,30 or 29 days of the month, almost covering it. Find the modon for these days and divide by the number of days taken, the daily motion for the month is got. Because the daily motion thus found is very near sixty minutes, the author enumerates their difference from sixty, for the sake of convenience. T h e mean daily motion is 59' 8", which can be got dividing the minutes in 12 rdsis, i.e. 21,600, by the days i n the solar year. Because the higher apsis, i.e. the apogee of the Sun, is about the middle of M i t h u n a , the Sun's motion in Vrshabha, M i t h u n a and Karkataka is very slow, and the daily motion, 57' given for these is proper. Because the lower apsis, i.e. the perigee, is about the middle of Dhanus, the Sun's motion i n Vrscika, Dhanus and Makara is very quick, and the daily motion, 61' given for these is proper. We shall h^re derive the motion for Mesa and T u l a . Let us begin with the moment on the first day of Mesa, when the mean Sun is zero. B y III. 1, the kendram is 20° and the correction for making the Sun true is - 11' x 20°/30° = - 7'. T h e true Sun is 0 ° - 7 ' = rd. 1 1 - 2 9 - 5 3 . 31 days after this moment, the mean Sun is, 31 X 120 + 43831 X 360° = 30° 33'. T h e kendram is 30° 33' + 20° = 50° 33'. T h e true Sun is 30° 33' - 11' - 48' x 20° 33'/30 = ra, 0-29-49. T h e motion ofthe Sun for 31 days is i?a. 0 - 2 9 - 4 9 - r a . 11-29-53 = ra.0-29-56. T h e motion per dayis 29° 56'/31 = 1736'/31 = 58'. Therefore 58' is the correct motion for Mesa and not what is mentioned i n the given verse, and that is why we have suggested the emendation yama for guna in the text. We shall examine the daily motion for the T u l a . We shall begin work from the first day of T u l a when the mean Sun is 185°. The kendram for that is 185° + 20° = 205°. T h e true Sun is 185° - 11' - 48' - 69' - 7 0 ' - 54' - 25' + 10' x 25°/30° = 180° 3 1 ' . N o w we shall take a dme 30 days later. T h e mean Sun then is 185° -t- 30 X 120 x 360°/43831 = 214° 34'. T h e kendram is 214° 34' + 20° = 234° 34' . T h e true Sun is 214° 34' - 11! - 48' - 69' - 70' - 54' - 48' - 25' + 10' + 48' X 24° 34'/30° = 210° 4 6 ' . T h e motion for 30 days i s 2 1 0 ° 4 6 ' - 1 8 0 ° 3 r = 30° 1 5 ' . T h e motion per day is 30° 15' 30 = 6 O V 2 ' . T h e author gives this as a whole number, 6 1 ' ; so it is all right.
17a. A . B . C . D . ^
d. C.SJIl^
b. B . f ^ A.C.^
c. C.«TFIT
for^
III.19
57
III. P A U L I S A - S I D D H A N T A
[ch
Karanas 18. In the bright fortnight take the M o o n minus Sun and subtract from it 6°. For the dark fortnight (take the M o o n minus Sun from the beginning of the dark fortnight, i.e.) take the M o o n minus Sun with 6 rdsis subtracted from it, and add 6°. Convert it into minutes and divide by 360'. What are obtained are the (Cara) kararms (Bava etc. coming one after another repeatedly). (Take the remainder and treat it as) the remainder in calculating the tithi, (i.e., multiply by 60, and divide by the difference of the daily motions of the Sun and Moon in minutes etc.) (and thus get the ending moment of the last kararia. In each tithi the first half is one kararia and the second half another). 19. From the middle of the fourteenth tithi of the dark fortnight (are the four half tithis, viz., the second half of Bahula-Caturdasi, the two halves of Amavasya, and the first half of Sukla-pratipad, which) are the Sthira-kararms, Sakuni, Catus-pdda, Ndga and Kimstughna, respectively. (The other Kararias are) movable. A kararia is half a tithi. T h e n from the remaining half o f sukla-pratipad the Cara-karanas come i n the order Bava, Balava, Kaulava, T a i d l a , Gara, Vanijya (Vanija) a n d Visti (Bhadra), (repeating eight times). As said before, there are two karanas i n a tithi, the first ending at the middle o f the tithi, and the second ending with the tithi. Therefore, the ending moment o f the second need not be computed separately. Even that of the first is not computed by almanac-makers, the m i d - p o i r « o f the tithi being taken for this. A n o t h e r thing is to be mentioned: Just as two tithis can end o n the same day, three karanas can end o n the same day. Example 9. Calculate the kararia from the data supplied in Example 8. M o o n - S u n got there is rd. 8-3-20, a n d the difference o f daily motion 770'. As it is Bahula-paksa, deducting 6 rdsis and adding 6°, rd. 8-3-20 - rd. 6-0-0 -t- 6° = rd. 2-9-20 = 4160'. D i v i d i n g by 360', the quotient got is 11, and remainder 200'. A s i n the case ofthe tithi, the ending moment is : 200 X 60 -r 770 = na. 15-35 before the end o f the day, i.e. the karana ends nd. 44-25 after the beginning 18a. A.(«de|«ld^:; Bl.fefSFJI^:; B2.3.l^d=)jc^gl: b. A.B.MHllfa<(^. A.%TPI, 19a-B. B1.3.
4f*l "=llf"I;
B2.^i|5T^^
b. Al."?^™?^;
B.°1^-^J1«M<
c. A . ° w i P # ; B.°?nftm c-d, A . B . C . W ^ « f ; D . ^ g ^ r ^ d. A.B.^K"IIlPlcf^'5I^#%; C . *<"I1IH
ilc|
D . ^B^fcTsr:3ra#%
58
111.19
PANCASIDDHANTIKA
of the day, and the karana that has ended, being the 11 th, is Taitila. (Note that this is also the ending moment of the tithi). In the place of vajjaladhoh (in verse 18) the reading bahulayoh is suggested, following the meaning and keeping to the letters. O r , it can be corrected as kajjalayoh which will give the same meaning, 'dark fortnight'. Secondly, to avoid splitting the word ardhe between the third and fourth feet (in verse 19), cardni and karandni have been interchanged, as possibly the scribe has interchanged them by the similarity of letters. T h e word, ca is interposed in the third foot, to make up the deficiency of one syllable, as an original ca cardni might possibly have been written as cardni. T h e extra syllable in the fourth foot can be explained in the manner we did earlier.
3 i ^ ^ 4 ) J N ^ % ? T ^ ^ y r ^ H f | ^ (^) ^ r r ( ^ )
oqfdMidl^TJ^
(^^^:)
||
|
Vyatipata and Vaidhf ta 20. W h e n the sum of the true longitudes ofthe Sun and the M o o n equals one complete revolution, (i.e., twelve rdsis) there is theyog-a called Vaidhrta. When this sum plus ten naksatras (i.e., rdsis 4-13-20) equals a complete revoludon, (i.e., twelve rdsis or twenty-four raiis), then is the yoga called Vyatipdta. Their dme is to be found by using the sum of the daily motions of the Sun and the Moon. T h e yogas are found thus: T h e true Sun and M o o n are computed for the ending moment of every day. W h e n the sum of these is a little over 12 rdsis, d u r i n g that day the end ofthe Vaidhrta will occur. ('A litde over' means, not more than the sum of their daily modons). W h e n the sum plus ra. 4-13-20 is, in the same way, a litde over 12 rdsis or 24 rdsis, the ending moment ofthe Vyatipdta will occur d u r i n g that day. T h e ending moments must be found like the ending moments of naksatras mentioned already, using, i n the place of the daily motion of the M o o n , the sum of the daily motions of the Sun and the M o o n .
Example 10. (a) At the ending-moment of a day the true Sun is 2' 20° and the true Moon 9' 15°. Their da motions for the day are 57° and 783' respectively. Show that Vaidhrta unit occur on that day, andfind it moment. T h e sum of the true longitudes is 12*^5°. T h i s is a little, i.e. 5°, over 12 rasis, (5° being less than 57' + 783'). Therefore Vaidhrta will end on that date. T h e ending moment is when the sum is exactly 12 rdsis, i.e. the sum is less by 5° or 300'. Therefore, the ending moment is 300 X 60 + (57' -I- 783') = nd. 21-26 earlier than the end ofthe day, i.e. nd. 38-34 from the beginning ofthe day.
(b) At the ending moment ofa day, the true Sun and Moon are: 9' 15° 20' and Iff 7° 20', respectively, a their daily motions 61' and 749'. Show that Vyatipdta ends on that day andfind the moment. T h e sum ofthe true Sun and M o o n = 19"^ 22" 40'. T h e sum plus 4^ 13° 20' = 24' 6° 0'. This is more than a full revoludon by 6° which is less than the sum o f the daily modons, 6 1 ' -I- 749' = 13° 30'. Therefore Vyatipdta ends on the day. T h e ending moment is: 360' X 60 ^ (61' -I- 749') = nd. 26-40, before the end o f the day, i.e. nd. 33-20 from the beginning o f the day. 20a. C.4|J|H<^ b. A . ^ ^ . A . B . C . ^ ^ l f l ^ D.'^lflf!^
c. A l . ^ ; B . C . D . ^ . D.°^dlmdl d. A . B . f ^ l q ^ : ; C ^ ' ^ ' l ^ : ; D . i f j l ^ q ^ :
III.20
III. P A U L I S A - S I D D H A N T A
59
The author here gives the computation of two of the twenty-seven yogas, Viskambha, etc., of which Vyatipdta is the seventeenth and Vaidhrta is the twenty-seventh. These have to be known because offerings are made to the manes and other deeds of merit are performed at these times, as at Visuva, Ayana, Sartkrama, etc. (which also are going to be given), these being two well-known days among the ninety-six Sraddha-days. A l l astronomers know that the twenty-seven yogas are computed like the twenty-seven naksatras, using the sum of the true Sun and M o o n i n the place of the true M o o n and the sum of the daily motions i n the place of the Moon's daily motion, because the yoga and the naksatra have equal extent, viz. 800 minutes. Vaidhrta, the twenty-seventh of the Viskambha series, and theVaidhrta here given are identical because the one ends at twenty-seven naksatra segments, i.e. one full revolution, and the other also ends at a full revolution, the duration of both being the same. In the same way, the Vyatipdta given by the author is the same as the seventeenth yoga of the same name i n the Viskambha series because, true S u n -I- true M o o n -I- 10 naksatralengths = one revolution = 27 naksatra-lengths. Therefore true Sun -I- true M o o n = 27 - 10 = 17 naksatra-lengths, given for Vyatipdta of the Viskambha series. For the sake of syntax sahitesu has been corrected as sahite tu. For the sake of grammar cakrah is corrected as cakram, for the neuter gender alone means a cycle or revolution, viz. 12 rdsis. O r let it be the masculine cakrah itself, meaning collection which ultimately can yield the idea of a collection of 12 rdsis. yitair bhdgaih is corrected as yutair bhdgaih, i.e. 'the sum o f the daily motions', which is necessary. Gatairbhdgaih or sthitair bhdgaih can satisfy the context, but will not be sufficient, for division by the sum ofthe daily motions cannot ordinarily be understood without being told. B u t the correction, of cakre, which is quite all right, as satke by T S is unwarranted and due to ignorance of what is wanted here. Another thing must be mentioned. If the twenty-sevenyogas, Viskambha etc., are computed, as i n the later-day works like the Surya Siddhdnta, then there would be no need to take the trouble of computing these two, viz. the seventeenth and the twenty-seventh, alone separately. I f we are instructed to do these two separately, it is because the Paulisa d i d not have the twenty-seven yogas. We have reason to believe that even d u r i n g the time of the V M these d i d not exist, for i n the Brhatsarnhitd, while naksatra, tithi and karana are taken up for astrological predictionsyoga is not so taken. T h e following is the history of the yoga. T h e yoga is not mentioned i n the Vedas, and the Veddfiga Jyotisa does not give it. F r o m the Paitdmaha condensed by our author we can infer that the original Paitdmaha Siddhdnta gave the Vyatipdta for the first time, for this condensed Paitdmaha gives the rule for Vyatipdta: "Multiply the days from Epoch, (this E p o c h is different), by 12, and divide by 305" (XII. 8). In the B a u d d h a and J a i n astronomical works, like Sdryaprajnapti and Kdlalokaprakdsa, too, the Vyatipdta alone is mentioned. Because the author gives both Vyatipdta and Vaidhrta here, we can guess that the original Paulisa had Vaidhrta also. Aryabhata, a contemporary of V M , mentions Vyatipdta alone i n the sdtra, "The Sun's cycles plus the Moon's cycles are the number o f Vyatlpdtas" {ABh, Kdla. 3,), but commentators take h i m to mean Vaidhrta also by implication, for, the Mahdbhdskariya, which is practically a commentary on the Aryabhatiya says, "When the Sun plus the M o o n equals six signs it is the Vyatipdta, when it is twelve signs it is Vaidhrta, and when it is equal to the distance of A n u r a d h a it is the yoga Sdrpamastaka" (IV. 35). H e r e the sum being equal to twelve signs, gives the Vaidhrta mentioned i n the context, which is patent. T h e distance o f A n u r a d h a being equal to seventeen naksatra segments, Sdrpamastaka is to be identified with the Vyatipdta of the Paulisa, which, as we have shown, is the seventeenth o f the Viskambha series. Govindasvami too, in his Mahdbhdskariya-Bhdsya on this verse quotes the original Aryahhatiya-Sutra and explains that Bhaskara here gives both Vyatipdta and Vaidhrta. (As for the Vyatipdta given by the sum equal to six signs, that
60
PANCASIDDHANTIKA
III.2C
is the Mahdvyatipdta, distinct from the seventeenth of the Viskambha series, which is not what we are talking about here.)
Prabhakara, generally mendoned as a disciple oi Aryabhata, has mentioned sevenyog-os, which he called Mahddosah ('the great Inauspicious'). This information we have from two slokas quoted by Sankarandrdyana i n his commentary on the Laghubhdskariya as Prabhakara's. T h e slokas say, "F Sun plus M o o n , in terms of naksatra-segments. W h e n they are equal to twenty-seven (i.e. a full revoludon), when 14, 8, 12, 5, 17, 18 and 10 are added, there are the Mahddosas Nirodha, Parigha Vajra, Danda, Ganda Sula and Vyatipdta, respectively. In this group, all excepting Danda, can identified i n the Viskambha series. Prabhakara has not included Vaidhrta i n the group, perhaps because he does not consider it as zMahddosa. Because these are computed individually, by a special rule, we can conclude that the twenty-seven yogas, Viskambha etc., were not i n vogue in the days of Prabhakara. We have mentioned that i n the days of Bhaskara a senior contemporary of Brahmagupta, also the twenty-seven yogas d i d not exist. T h o u g h it may be supposed that the twenty-seven yogas had come into vogue by Brahmagupta's days from the statement, "The minutes of the sum o f the longitudes of the Sun and the M o o n , divided by 800 are the yogas", {Br. SpSi., Spasta. 63) and on the strength o f this we ourselves have written that Brahmagupta knew the twenty-seven yogas, i n o u r Introduction to the Mahdbhdskariya, it is now learnt that the statement is an interpolation because this is not taken up and commented upon by Prthudakasvdmi in \v\sBhdsya of the Brdhmasphutasiddhdnta and also because in giving the computation oipunyakdlas at the ends of tithis, naksatras, etc. according to custom, like Vatesvara and Sripati, Brahmagupta omits yoga while the others includeyoga as well. In the Sdrya-Siddhdnta etc. which are later, the Viskambha serie find a place. T h u s of the five angas, the yoga was the last to develop.
We said that the Vyatipdta was the first yoga born and next Vaidhrta. W e shall consider thei nature and how they arose. T h e Vedic priests and astronomers were i n the habit of observing the sky looking for celestial occurrences like the rising and setting o f the Sun and the M o o n , because of the need of this kind of knowledge for the performance oiyajnas and out of thirst for knowledge. It is said that the Gavdm-ayana Satra was designed for this very purpose. T h e following facts were observed by them. A t one time the Sun rises farthest south o f the East point, that is the end of Daksindyana and beginning of Uttardyana. (This is the winter solstice). After that, the Sun rises more and more north every day and, at the end o f six months, rises farthest north. T h e n is the end of Uttardyana and the beginning of Daksindyana. (This is the summer solstice). F r o m that time it begins to rise more and more to the south, until after six months again it is farthest south. This is the end of Daksindyana and the beginning o f Uttardyana again. T h u s i n a year there are the two courses of the Suri, northward and southward. In a given place, the exact point north or south where the Sun rises depends o n its declination north or south. Like the Sun, the M o o n too, according to its declination, rises north or south o f the east-point and has its Uttardyana i n about fourteen days and its Daksindyana in about the same period, the total taking a little more than twenty-seven days. Now, the day o n which the Sun and the M o o n rise almost at the point, one moving south-ward, and the other moving north-ward, coming to meet each other as it were, that day is the Vyatipdta. Because they cross each other moving i n different directions, the phenomenon is called Vyatipdta or Vyatipdt Now, how can the time of the phenomenon be computed? Because they must rise nearly at the same point, their declinations must be nearly equal. That they must be moving in opposite directions, i.e. their respective ayanas should be different, has been mentioned. These two conditions can be approximately secured i f the position of one is as far away on one side of the junction o f Uttardyana and Daksindyana, as that of the other is on the other side of the junction. Let us take it that the
III.20
III.
PAULISA-SIDDHANTA
61
longitudes are reckoned from the starting point of the Uttarayana (winter solstice), as i n the Veddnga Jyotisa and the Paitdmaha from Sravisthd. T h e two being at equal distances on both sides of the zero point means that the sum of their longitudes is equal to one full revolution, i.e. twelve rdsis. It is this that the Paitdmaha gives by its rule, 'Multiply the days by twelve and divide by 305.' But, because the true declination of the M o o n will generally differ from that of the Sun at this dme, on account of its latitude, the time given is only approximate and the Paitdmaha intends that the actual time should be found by observation. If we reckon the longitude not from Sravisthd as the zero point but from Asvini, then the longitudes will each be five naksatras less, because Asvini is five naksatras forward, and the sum will be ten naksatras less. Therefore, i f ten naksatras are added to the sum o f the longitudes (as the author asks us to do) we have the condition fulfilled, and therefore the Vyatipdta. But in course o f time, on account of the precession of the equinoxes, the winter solstice had moved to the beginning o f Makara at the dme of the author, and now still more backward so that conformity to definition is growing less and less. But on account of respect for the o l d Sdstras, the 17th continued and still continues to be the Vyatipdta, just as we continue to observe Uttarayana rites still when the Sun enters Makara because Uttardyana was once there, though now it has come down into M u l a . A new type of Vyatipdta called the Mahdvyatipdta came into existence to satisfy the definition. T h i s is mentioned by the author i n the next two verses. T h e memory of a sacred day at the sum being a full revolution resulted i n the creation of a new sacred day, even when reckoned from Asvini, and it was called Vaidhrta, because the old Vyatipdta was 'sustained' (dhrta), as it were, by this. Because it has grown in the place o f the Vyatipdta, this Vaidhrta itself is sometimes called Vyatipdta. F o r e.g. the Sary a-Siddhdnta says, "This is another well known Vyatipdta, called by die different name of Vaidhrti" (XI. 8) and " T h e three fearsome Vyatipdtas" (XI. 22). Govindasvami also says this: "When the sun plus M o o n is equal to six signs, there is Vyatipdta. When it is equal to twelve signs it is Vaidhrta and this is also called Vyatipdta; for it is said ' T h e sum of the revolutions of the Sun and those of the M o o n are the Vyatipdtas (in the yuga)' (ABh. Kala, 3)." How does this mean that? T h i s is how: T h e sUtra primarily gives only the Vaidhrtas that come at the end of full revolutions, which are called Vyatipdtas because both have the same characterisdcs. T h e effect of both being the same, Vaidhrta is called Vyatipdta. So the vyatipdtas characterised by full revolutions and half revolutions are both given by the sutra. (Govindasvami's Bhdsya, Mahabhaskariya IV.35). Sankaranarayaiia too, by saying "Aryabhata mennons the two types oi vyatipdtas", i n his commentary on Laghubhdskariya, II. 29, understands Vaidhrta also by the word Vyatipdta.
21. W h e n the S u n b e g a n to t u r n s o u t h , i.e. w h e n the s u m m e r solstice was at the m i d d l e o f the asterism, A s l e s a , the r e q u i r e m e n t o f the d e f i n i t i o n that the S u n a n d the M o o n s h o u l d be i n d i f f e r e n t ayanas was satisfied. B u t n o w the t u r n i n g s o u t h takes place at t h r e e q u a r t e r s o f P u n a r v a s u . T h e r e f o r e the d e f i n i t i o n has b e c o m e faulty. F r o m this we can infer that the author knew the precession of the equinoxes. In the Brhatsamhitd also he says the same thing, "Certainly at one time, the summer and winter solsdces were at the middle of Aslesd and the beginning of Dhanisthd, respectively, because such has been mentioned in
62
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III.21
the ancient lore. But now the summer solstice is at the beginning of Cancer and the other one at the beginning of Capricorn. If at any time this is not conformed to, then there is a further change, which can be seen and measured by observation and examination." (fir. Sam. III. 1-2). It is from this that we have interpreted Punarvasu as "the point at three quarters of Punarvasu". T h e ancient lore mentioned here includes Vedanga-Jyotisa and Paitdmaha Siddhdnta. T h e Ydjusa-Jyotisa says, " A t the beginning of ^ravistha the Sun and the M o o n turn northward and at the middle of Aslesa they turn southward, with the Sun always i n the Magha and Sravana months, respectively" (verse 7). "When the Sun and the M o o n rise i n the sky together, with Sravistha with them, the Yuga begins then as also the month o f Magha, the seasonal month Tapas, the bright fortnight, and the turning northward " (verse 6). As the Paitdmaha too counts the naksatras of longitudes from Sravisthd and says that it is Vyatipdtd when the sum o f their longitudes is a whole revolution, we can infer that the turning northward is at Sravisthd.
22. W i t h the M o o n a p p r o a c h i n g to m e e t the S u n , m o v i n g i n a d i r e c t i o n opposite to that o f the S u n , v^'hen its true d e c l i n a t i o n (i.e. the m e a n declination p l u s its latitude) b e c o m e s e q u a l to the S u n ' s a n d w h e n the s u m o f t h e i r l o n g i tudes is n e a r l y six signs, t h e n is the Vyatipdta c o n f o r m i n g to the d e f i n i t i o n , (i.e. the Mahdvyatipdta). T h e m i n i m u m and sufficient conditions for the Mahdvyatipata are that the Sun and M o o n should have different southward or northward courses and that their true declinations must be equal, both being north or both being south. T h e second part o f the second condition, though not mentioned by the verse, is implied i n the requirement that the sum should be nearly six signs. Because the northward or southward courses and the declinations depend on the tropical longitudes, we can understand that the sum also is of the tropical longitudes (i.e. the sdyana longitudes) of the Sun and the M o o n . I f this is not stated it is because d u r i n g the time o f the author the Ayandihsa, i.e. the difference between the tropical and sidereal longitudes, was pracdcally zero and the author intended the work as a karana not to be used for a very long time when the ayandmsa would become considerable. We have interpreted "half revolution as approximately six signs" because when the M o o n has a ladtude as generally it would have, the equality i n declination will happen not exacdy at the sum being six signs. O n l y the mean declination o f the M o o n will be equal to that of the Sun when the sum is exactly six signs, as Bhaskara I says i n his commentary o n the Aryabhatiya, (Kdla, 3), "Vyatipdta
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III.22
III.
PAULISA-SIDDHANTA
63
occurs when the declinations are the same and the courses are different. T h e expression halfrevolution i n that connection is only meant to be approximate, because by the latitude of the M o o n it may be a little more or less." Therefore we should examine whether a Vyatipdta would occur at the neighbourhood of the sum being six signs, because it can occur only there. But sometimes it may not occur at all, because the definition is not satisfied ( A l l this is expounded clearly i n works like the Siddhdnta Siromani and we stop with this). One may think that we are making contradictory statements by saying in the history of the origin of the Vyatipdta, that it occurs at the sum being full revolutions and here that it occurs at half revolutions. There is no contradiction because the origin from which the longitudes are measured is different in the two cases. In the former the winter solstice was taken as the origin, and, i n the latter, the spring equinox. T h e r e is a difference o f three signs between the origins, which causes the same difference in each of the two longitudes, with the result that there is a difference of six signs i n the sum. That they are the same can be shown thus: T h e Sun measured from winter solstice, (say, a) = the Sun measured from spring equinox (say, 6) + 3 signs. T h e M o o n measured from winter solstice, (say, a) = the M o o n measured from spring equinox (say, b) 3 signs. Therefore a -I- a' = b -t- b' -I- 6 signs. If a -I- a' = full revolution, b -I- b' -I- 6 signs = full revolution, therefore b 4- b' = full revolution — 6 signs = half revolution, which proves the sameness. Spring equinox is not mentioned because at the author's time it was situated at the beginning of Asvini and longitudes are reckoned from there. Example 11. The Sun and the Maori, at the end of the day are rd. 1-10-0 and rd. 4-23-30, and their daily motion 57' and 783'. Taking the spring equinox to be at the beginning of Asvini, i.e. the winter solstice at the beginning of Capricorn, examine the possibility of Vyatipdta, in both ways. Because the longitudes are from zero Asvini, they are the same as reckoned from spring equinox also, both points being the same i n the problem. Therefore sum of longitudes = ra. 1-10-0- -I- ra. 4-23-30 = rd. 6-3-30. T h i s is 3° 30', i.e. 210', over a half revolution. T h e sum of the daily motions = 57' -I- 783' = 840'. Therefore at 210 x 60 840 = 15, nddis before the end of the day, the sum is equal to a half revoludon or 6 signs, and so Vyatipdta may occur i n its neighbourhood. Otherwise, i f the longitudes as measured from winter solstice, the Sun = rd. 1-10-0 - rd. 9-0-0 = rd. 4-10-0. T h e M o o n = rd. 4-23-30 - rd. 9-0-0 = rd. 7-23-30. T h e i r sum = rd. 4-10-0 + rd. 7-2330 = rd. 12-3-30, and this is 210' over a full revolution. Therefore 210' X 60 H- 840 = 15, nd4is before the end of the day. T h e sum is a full revolution and the Vyatipdta may occur as its neighbourhood. (Note that worked in both ways, the time is the same). Now for the readings. In the place ofpdto we have tskenydto because the scribe may easily mistake pd ioryd. But the correction bhdgo o f T S does not agree with the second case i n arkakasthdm and deserves to be rejected. T h e wrong reading, sasi-saviksepah has been corrected by us into sasi saviksepah, by a simple lengthening of. B u t T S and N P have made itsasiraviksepah which is incorrect and also does not agree with kdsthdm. T h e meaning which they have taken for this verse itself is wrong. T h e i r interpretation of kdstha into 'maximum dechnation' i.e. 24° (or 23° 20') is not proper, for, in his work (see C h a p . IV), kdsthdnta is used for m a x i m u m declinadon and kasthd is taken to mean only declinauon. Let us concede it is m a x i m u m declinadon and therefore means 24°. Even this does not agree with the meaning given by them because they want and imply 23° 20' only there. If 24° is given roughly for 23° 20', why not 23° which is nearer. T h e y do not seem to have understood at all what is sought to be conveyed by the author.
III.24
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64
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l^a^asiti-pu^yakala 23-24. A t t h e first p o i n t o f M e s a (Aries) a n d T u l a ( L i b r a ) a r e the s p r i n g a n d a u t u m n a l e q u i n o x e s ( a n d the sacred days t h e r e o f are w h e n t h e S u n is there.) T h e c o m m e n c e m e n t s o f t h e sacred days c a l l e d Sadasttis are at p e r i o d s o f 86 solar degrees c o m m e n c i n g w i t h Tula-zero p o i n t . T h e days i n t h e solar m o n t h s after t h e respective c o m m e n c e m e n t o f t h e Sadasttis a r e sacred as c o n n e c t e d w i t h t h e manes. T h e c o m m e n c e m e n t o f t h e Sadasttis a r e after 14 degrees o f K a n y a , ( V i r g o ) , after 18 degrees o f M i t h u n a ( G e m i n i ) , after 2 2 degrees o f M i n a , (Pisces) a n d after 2 6 degrees o f D h a n u s (Sagittarius). The main purpose o f the author i n giving the equinoxes here is to indicate the sacred days connected with them as can be gathered from the context. T h e equinoxes, i.e. the points of intersection between the ecliptic and the celestial equator, though moving westward slowly along the ecliptic, (this is the precession of the equinoxes), were at the first points o f Mesa and T u l a only at the period of the author. A t the present day the equinoxes have moved far into Uttara-Bhadrapada and Uttara-PhalgunI, but the sacred days are still observed with the Sun entering Mesa and T u l a by blind routine. T h e time taken by the Sun to move one degree is a 'solar day' according to H i n d u astronomers. (We have put it within inverted commas, because i n English it means the ordinary day caused by the Sun and therefore quite different). So i n a solar year there are 360 'solar days', and i n each solar month 30 'solar days'. As for counting from zero-Tula, this is enjoined by the Dharma-sdstras. T h e commencements of the Sadasltimukha-s are, 1 X 86° = 86°, 2 X 86° = 172°, 3 x 86° = 258° and 4 X 86° = 344°. from zero-Tula, i.e. from rd. 6-0-0. Therefore they are rd. 6 -I- 86°, rd. 6 -I- 172°, rd. 6 -f- 258° and ra. 6 -I- 344°, and these are, respectively, 26 degrees of Sagittarius, 22 degrees of Pisces, 18 degrees o f G e m i n i and 14 degrees o f Virgo. These sacred days are not observed i n these days and it would be interesting to know when a n d how they went out of vogue. W h e n the Sun enters Sagittarius, Pisces, Gemini and V i r g o , we observe the sacred day, calling it Sadastti; and in the place of the last sixteen 'solar days' of V i r g o , (these seem to have secured importance at the time of SUrya Siddhdnta) the dark fortnight o f Bhadrapada is dedicated to the Manes, with the name of Mahdlayapaksa. T h e dark fortnight o f Asvina also is observed as a secondary Mahdlaya-paksa and it is the belief that the Manes are sent back to their world on Naraka-Caturdasi. Now, what is the speciality about 86 solar days, it may be asked. T h i s period is three synodic months less one day. It may be
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III.24
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65
that a section of people observed a sacred day for the manes once i n three synodic months, and then this came i n its place.
Solstices 25. T h e Sun's t u r n i n g n o r t h w a r d is w h e n it reaches the zero-point o f M a k a r a , ( C a p r i c o r n ) , i.e. at w i n t e r solstice, a n d its t u r n i n g s o u t h w a r d is at the z e r o p o i n t o f K a r k a t a k a ( C a n c e r ) i.e. at s u m m e r solstice, w i t h the a t t e n d a n t sacred days. T h e seasons ^isira etc. c o m m e n c e w i t h the w i n t e r solstice a n d e a c h season lasts two t r o p i c a l solar m o n t h s . T h e precession of the equinoxes implies the precession o f the solstices as well and therefore the solsdces at the zero-points of Karkataka and Makara is true only for the period of the author. If the sacred days are observed still when the Sun enters these signs, it is again blind custom. As the seasons depend upon the position o f the mid-day Sun in the sky and the length o f day dme, and these depend on the Sun's declination depending on tropical (sdyana) longitude o f the Sun, the seasonal months are different from either the solar sidereal months Mesa etc. or the synodic months Caitra etc., and these cannot correctly represent the seasons. That is why the Vedas give a new set of months, (actually tropical months) for the seasons: M a d h u and Madhava are the months constituting the Vasanta (spring) season, Sukra and Suci are the months constituting the Grisma (summer season), Nabha and Nabhasya are the months constituting the Varsa (rainy) season, Isa and Urja are the months constituting the Sarad (post-rainy season); Sahas and Sahasya constituting the Hemanta (pre-winter) season; and Tapas and Tapasya constituting the Sisira (winter) season. (Suklayajurveda, 13.25). Even i n the Veddnga Jyotisa we have the information that the Sisira season begins with the Uttardyana (winter solstice). T h e Ydjusa-Jyotisa (verse 6) says, "When the Sun and the M o o n rise together, with Sravistha, from then commence the yuga, the month o f Magha, the seasonal month Tapas, the light fortnight of the month, and Uttardyana". As Tapas is the first month o f Sis'ira we understand Sisira begins with Uttardyana. B y mentioning Magha and Tapas distinctly, we understand that the Vedas wish us not to confuse the two. But confusion there has been, and still continues, with the result that people call Mesa and even Vrsabha spring months, though patently we have summer then, K u m b h a and M i n a being practically the spring months now. T h i s confusion has resulted i n M a d h u , Madhava etc. and Caitra, Vaisakha etc. as synonyms. People who know are amused, when i n the sankalpa recited for H i n d u rituals the month of Vrsabha, which is advanced summer, is mentioned as spring.
25. Quoted by Utpala on BS 2, p.23.
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P A N C A S I D D H A N T I K A
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III.27
SaAkranti-kala 26. T h e a n g u l a r d i a m e t e r o f the S u n i n m i n u t e s , m u l t i p l i e d by sixty a n d d i v i d e d by the d a i l y m o t i o n o f the S u n , are total sacred nddis o f Safikranti (literally 'crossing'). H a l f this t i m e before a n d after the S u n e n t e r i n g a rdsi, is sacred. T h e Paulisa does not give the angular diameter of the Sun, so it must be the intention of the author to use the angular diameter given by the Romaka or the Saura.
Example 12. The angular diameter of Sun is 31' and its daily motion 57'. The Safikramana is 19 nadis afte sunrise. Find the sacred nadis. A n g u l a r diameter X 60 ^ daily motion = 3 1 ' X 60 57' =raa.32-38. H a l f this is 32-38/2 = nd. 16-19. Therefore nd. 19-0 - nd. 16-19 = nd. 2-41 to nd. 19-0 -h 16 - 19 = n a . 35-19 is the sacred period. T h e rule is proved thus: T h e time of the centre of the Sun's orb crossing to the next sign is the dme of Safikramana. A t this time half the orb is i n the previous sign and half in the next. T h e period when parts o f the orb are i n both signs is the sacred period. So it begins when the east point of the orb just enters the next sign and ends when the west point just leaves the previous sign. So, d u r i n g the interval the S u n moves a distance equal to its own diameter. This time is got by the propordon: daily motion: angular diameter :: 60 nddikds: the required dme. Therefore aAg. diameter X 60 -H 60 is the time i n nddikds. As half this time is required for the mid-point to reach the junction of the signs, half this period placed on either side o f the time o f the mid-point crossing over gives the beginning and end of sacred period. It must be noted that if the angular diameter is computed according to the old H i n d u astronomical works and used, the sacred period would be constant whatever be the daily motion, and the sacred period can easily be given as so imny nddikds before and a.ix.er safikramana. How? Let x be the mean angular diameter in minutes. According to H i n d u astronomy the angular diameter is propordonate to the daily modon, (because the m o d o n is taken inversely proportionate to the distance and the angular diameter also is inversely propordonate to the distance) (See V I I I . 15, I X 14-16). Therefore the angular diameter = x multiplied by daily motion mean daily motion. T h e period = angular diameter x 60 -H daily motion = xX daily motion X 60 - i - (daily motion X mean daily motion) = x X 60 - i - mean daily motion which is constant. I f to avoid this we assume that the mean diameter is intended to be used i n the rule, then the rule is unreasonable. O r we have to accept it on the injunction of the Dharmasdstras, throwing the burden o n them. W e said, "according to the old H i n d u astronomical works", because actually the angular diameter is not exactly proportional to the daily motion.
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111.27
III.
HAUI.ISA-SIDDHANTA
67
Tridinaspfg-yoga 27. W h e n a tithi e x t e n d s t h r o u g h o u t a day, c o i n c i d i n g w i t h a part o f the p r e v i o u s day a n d the next day, the o c c u r r e n c e is c a l l e d Tridinasprg, (literally 'touch o f three days'). (If, besides a w h o l e tithi, parts o f the previous a n d next tithis fall o n the same day, the occurrence is called avatm, literally, ' u n c o u n t e d tithi'). T h e only thing we have done to the reading in the first half of the verse is to change sd into sd, which is quite warranted. But T S have changed de into di and introduced a new word, anya. Still their reading of the text cannot yield the meaning. T o agree with tithitraya-sparsanat, we corrected ahnah into avamah, because avama alone results by contact with three tithis. T h e word ahnah is necessary also, but can be understood from the context, though not mentioned, but not so, avamah. If this part is left uncorrected as T S have left, the expression would be non-sensical like Sudhakara's meaning: "Because the day touches three tithis, it is called 'Three-day touching'. B u t Thibaut has grasped the idea here, though calling it "the conjunction touching three Tithis". N P too, have caught the idea, but since the relevant emendation to avama d i d not strike them, they merely say '(there is a yoga)'." [TT|:]
Rahu (Node) 28. M u l t i p l y the days f r o m E p o c h by 8 a n d d i v i d e by 151. R a h u ' s m o t i o n is got i n degrees etc. A d d m i n u t e s e q u a l to r e v o l u t i o n s . ( T h e m o t i o n becomes exact.) 29. D e d u c t the m o t i o n f r o m 7' 25° 5 9 ' . T h e r e m a i n d e r is R a h u ' s H e a d (what is c a l l e d D r a g o n ' s H e a d , a p o p u l a r n a m e f o r the A s c e n d i n g N o d e ) . A d d 6 rdsis to R a h u ' s H e a d , ( D r a g o n ' s T a i l o r D e s c e n d i n g N o d e ) , is got. Example 13. (a) Days from Epoch, 75,500; find Rdhu's Head and Tail, (b) Find the Head of Rdhu at Epoch, i.e. for Zero day. 27b.
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I'ANC:ASIODHANTIKA
111.29
(a). Rahu's motion for 75,500 days = 8 x 75,500 divided by 151, degrees = rev. 11-1-10-0. T h e exact motion = rev. 11-1-10-0 + 11 minutes = rev. 11-1-10-11 = rd. 1-10-11, omitdng full revolutions. Rahu's Head = rd. 7-25-59 - ra. 1-10-11 = ra. 6-15-48. T a i l = rd. 6-15-48 - rd. 6-0-0 = rd. 0-15-48. As the m o d o n of R a h u is zero, the H e a d of R a h u is the constant itself, viz. rd. 7-25-59. As the motion is 8° in 151 days, according to this Siddhdnta, to move 360°, i.e. one revoludon, it takes 360 X 151 H- 8 = 6795 days. But d u r i n g this period it moves one minute more, i.e. the exact motion is 360° 1' i n 6795 days, i.e. one revolution takes 360° X 6795 360° 1' i.e. 6794 days, 19 nddis. We have seen that the H e a d of R a h u for the Epoch, according to the Paulisa is the constant itself, viz. rd. 7-25-59. According to the Saura (condensed by the author) it is rd. 7-26-6. According to the Vdkyakarana it is ra. 7-26-11. According to modern astronomy, taking the ayandmsa as being zero for the period it is rd. 7-26-0. According to the Siddhdnta Siromani it is ra. 7-27.13. We see that all except the value of Siromani agree closely, verifying the Paulisa value for the Epoch. T h e disagreement of the Siromani value is only apparent, for the zero point of the Siromani zodiac is about one degree behind that of the rest, (as may be seen by comparing with its co-ordinates of the stars or from its Sun being one degree more than that of the rest) and i f the same point is taken as the origin, the Siromarii too gives about rd. 7-26-13.
T h u s the value at epoch is necessary to get the R a h u at any moment, and it is this that is given by Vrscikabhdgd Rdhoh etc. B u t T S have not understood this need (as they did not understand the need for the ksepa i n the case of the M o o n (see II.3) and gave a wrong interpretation of sasimuninava-yamdsca rdsyddydh) and give the following laughable explanation: "The measurement of the Umbs of R a h u having the form o f a scorpion is 25 minutes. Deducting this from the motion of Rahu obtained from (28), the head or face o f R a h u is to be found. This plus six rdsis is the tail. We have to rely only on the words of the ancients to know that the scorpion-like limbs of Rahu measure 25 minutes, there is no other reason." N o w we ask: Let it be that they have not understood the need for the ksepa. H o w d i d it not occur to them that R a h u can be got only by deducUng the motion from something, whether it is a cycle or some other constant, because the motion is retrograde (as they themselves have said i n other places: "Rahu deducted from a full revolution is the Head, this plus six rasis is the tail", I X . 6, "deducted from the end of Pisces is the head", V I I I . 8). It also appears here that Thibaut is not sadsfied with Sudhakara's explanation. Further, how did it not occur to them that ekaUptikdluptdhsadxnmsativrsciakabhdgdh, means rd. 7-25-59, when they have correctly interpreted simhasya vasuyamdrnsdh as ( X V I . a) rd. 4-2S-0, sdrdhdhpancdlino ( X V I I I . 1) asra. 7-5-30, nai^a sdrdhdhkanydrnsdh ( X V I I I . 11) asra. 5-9-SO,sodmavrsabhasydmsdhnavaliptikdvarjitdh ( X V I I I . 18) as rd. 1-15-51? It is really astounding what tricks the m i n d can play! Incidentally, the following should be mentioned here for general information. Following the nomenclature of the ancient Samhitds, the author calls the ascending node Rahu's head, and the descending node 'Rahu's T a i l ' , both being Rahu, though generally i n later astronomical works the word Pdta is used. In recent times, somehow the term Ketu has come to be applied to the descending node, Rahu being retained for the ascending node, though there is no authority in astronomical works or Purdnas to bring in Ketu here. T h e ancient Samhitds use the term Ketu for the DhQmaketus or comets and, as deities they are generally referred to i n the plural. T h e y are also characterised by unpredictable motions and in the Brhatsamhitd too the author says so. (This is the view of the ancients though we now know that a number of them are periodic, and their positions can be predicted with tolerable accuracy). T h e y are worshipped i n the collective, as seated on doves.
III.29
III.
P A U U S A - S I D D H A N T A
69
with the expression, "Salutation to the Ketus." F r o m the singular i n the mantra of their invocation, Ketum krnvannaketave, one may think that Ketu is referred to i n the singular also. It is not so. Here the word does not mean the 'Comet' Ketu at all. T h e mantra itself is i n praise o f the Sun arid Ketu here means the activity caused by the Sun i n the sleeping inactive world. H o w then is this mantra used to invoke the Ketus? T h e utterance of the word Ketu here is sufficient, as the utterance of the m a n y i a sounds i n the Ma.ntr?i sarnrwdevth etc. (Rgveda 10.9.4) is sufficient to propitiate Sanaiscara. though the mantra itself refers to the water-deities, or the utterance o f the word mayura i n the mantra, dmandrair intra haribhih etc. (Rgveda 3.45.1) is sufficient to propitiate Subrahmanya, though the mantra refers to Indra, as also words like, Om, atha kalyana etc. cause auspiciousness by their mere utterance. So, according to the Sdstras, Ketu refers only to the Dhumaketus, and Rahu is both the nodes, the recent application of the term Ketu to the descending node being unwarranted. Therefore, when the Dharmasdstras enjoin the eclipses caused by Rahu as sacred periods, they take in both the ascending and the descending nodes, and the suggestion by some that they do not take i n the descending node on the score of some un-informed people calling it Ketu is wrong because what they call Ketu is really R a h u .
Moon's latitude 30. I f the M o o n lies b e t w e e n the H e a d a n d the T a i l it is n o r t h o f the ecliptic, (i.e. its latitude is n o r t h ) . I f it lies b e t w e e n the T a i l a n d the H e a d it is s o u t h o f the ecliptic, (i.e., l a t i t u d e is south). 3 1 . T h e latitude is a m a x i m u m e q u a l to 2 8 0 m i n u t e s w h e n the M o o n is 90 degrees distant f r o m e i t h e r H e a d o r T a i l . T h e latitude is to be f o u n d by p r o p o r t i o n , at o t h e r places, u s i n g the distance i n degrees f r o m H e a d o r T a i l , w h i c h e v e r is n e a r e r . Example 14. The true Moon is rd. 2-7-0. Rdhu's Head is rd. 5-3-0. Find the latitude of the Moon. The T a i l is H e a d rd. 6-0-0- = rd. 11-3-0. T h e M o o n is between T a i l and Head. Therefore the latitude is south. T h e distance o f the M o o n from the nearer limb. H e a d is rd. 5-3-0 - rd. 2-7-0 = rd. 2-26-0 - 86 degrees. For 90° the latitude is 280'. For 86° latitude is 86° x 280' -r 90° = 267 V 2 ' , South, as seen. 30a.
B.^g^JKl^
b. A . B l . ' J ^ i l ^ . B . ° ^
31b. B.°^tfwf?T c-d. A.%^q?ftfeqyiral^ 1;
c. A i . B i . ^ ; C . ^ ^ ^ : ^ « i ^
B.?[^cq#af^qral^ I;
d. A . B . o f t % r a # ; C.°f^Sgcrai%;
C.^t
70
P A N C A S I D D H A N T I K A
111.31
T h e rule is explained thus: T h e M o o n moves i n its own orbit, inclined to the ecliptic at an angle equal to the m a x i m u m latitude. H i n d u astronomy assumes this motion to be on the ecliptic itself and gives the Moon's longitude, because there is only a m a x i m u m difference of 7'. Between the ascending and the descending nodes the orbit is north of the ecliptic and therefore the ladtude measured on the great circle perpendicular to the ecliptic and passing through the M o o n , is north. Between the descending node and the ascending, the orbit is south and the latitude is south. T h e distance between the node and the M o o n , the latitude and the angle of inclination forming a spherical triangle, we have : sin latitude = sin interval X sin m a x i m u m latitude. T h e m a x i m u m being small and the latitude being generally less than this, sin latitude and sin m a x i m u m latitude are proportional to the latitude and m a x i m u m latitude, and we have the formula, lat = max. lat X sin interval. But the Paulisa takes the latitude proportionate to the interval itself and gives the rule. We have to consider here whether the author intends that the latitude is to be found in proportion to the actual degrees of the interval or the sine of the interval i n degrees. T h e triangle being spherical, the correct thing would be to use the sine. B u t we have reason to think that the degrees themselves are intended to be used i n the proportion, for i f the sine is to be used it must be mentioned. Also, this Siddhdnta uses only the proportion by degrees i n other places also, where proportion by sines alone would be correct, as for e.g. in the solar eclipse, in correcting Rahu and in calculating valana, i.e. transformation of direction (see V I . 2-4, 8). Therefore the original Paulisa itself has instructed proportion by degrees, as being sufficiendy accurate, which the author reiterates here. B u t i n computing the parallax (in time) i n the case of the solar eclipse the sine is used either by the Paulisa itself, to avoid too much inaccuracy, or by the author V M to save the Siddhdnta from ridicule. As for T S they say that the author intends here proportion only by sine, that being the proper thing to do. Another thing should be mentioned here. In using proportion by degrees, the m a x i m u m error will be i n the neighbourhood of the nodes. W i t h the m a x i m u m latitude 280', the latitude for 13° interval would be 40 V 2 minutes, which would be incompatible with the formula for eclipses (vide V I I . 5-6). Therefore it seems that the Siddhdnta, though knowing that proportion by sine is the correct thing for latitudes, gives proportion by degrees for the sake o f ease of computation. O r by taking liptdsatatrayendsitiih as the correct reading, which would make the m a x i m u m 380', the incompatibility can be avoided. It may be argued that the error i n the latitude would be great in the neighbourhood of the m a x i m u m . B u t this is only erring one side while some others err on the other. T h e mean m a x i m u m latitude is 309'. If some like Ptolemy give 240', which is less by 70', what is wrong i n taking that the Paulisa gives 380', which is greater by the same amount? Now for the readings: In the fourth foot of verse 30, mu is corrected into su because murdt is meaningless. In the third foot of the 31st verse two syllables are wanting and nd is added for purposes of syntax. For the same reason, st-tama-nupd i n the fourth foot is corrected into si-tyd-nupd. T h e correction of asiti into saptati by T S here is unwarranted, because it is not known what it was in the original Paulisa. If this is done i n conformity with the Saura, why not i n conformity with the Romaka, which gives 280', (see V I I I . 11), and which is nearer to the Paulisa? Now follow six verses devoted to criticising the views of the Romaka, and an astronomer by name Bhadravisnu, the intention o f the author being to create faith in his own work. (This was a custom in those days, vide for instance the Brahma-Sphuta-Siddhdnta, Dusanddhydya and the Vatesvara Siddhdnta, Madhyamd-dhikdra, Chapter X ) . In several places o f the text the readings are not clear and we cannot be sure of what exactly the author intends to say, though the gist is clear, the matter
111.31
111.
71
P A U U S A - S I D D H A N T A
not being scientific and amenable to intelligent guess. Still, as there is much to say here too, we are dealing with these, unlike T S who have refrained from doing so.
Defect in Bhadravi^pu 32. I f the tithis a n d naksatras as seen f r o m o b s e r v a t i o n o f the sky agree w i t h those c o m p u t e d a c c o r d i n g to the Nostra, t h e n the Nostra is c o r r e c t a n d fit to be a c c e p t e d . It is n o t so i n the case o f B h a d r a v i s n u ' s w o r k ; still p e o p l e d o n o t t u r n away f r o m that a n d f o l l o w the c o r r e c t Sdstras. (This remark about the nature o f people is true even today). For the sake of syntax the long dd has been shortened by us. Consistent with the idea intended, the negative particle na has been added i n the fourth foot.
Defect in Padaditya 33. S u n r i s e o r sunset is n o t at the same m o m e n t i n a l l places o n the e a r t h ; (so the place m u s t be m e n t i o n e d w h o s e s u n r i s e o r sunset is t a k e n as the e p o c h f o r finding the days a n d d o i n g the c o m p u t a t i o n ) . B u t P a d a d i t y a , w h o has p l a c e d the e p o c h at sunset, has n o t m e n t i o n e d the sunset o f w h i c h place he refers to. (So his w o r k is faulty.) There seems to be some error i n the last foot and we are not sure whether Padaditya is a person as we have interpreted, or something else or whether the word is the same at all and not an nth incarnation of the original.
32a.
A.B.C.D.^
b. A . B l . T ^ ' ^ P l ^ ; B 2 . ' S l f c N i ^ . B . ^ c. B.»^s^i^s»^l d.
A.B.C.D.om.
33a.
A.B.«ng:
c. D.°5WPT: d.
B.MKIMH;
D.wf^-T
A.^feiq^; B.^rtrfift^:; C . ^ r p f t ^
III.34
P A N C A S I D D H A N T I K A
72 Defect in Romaka
34. T h i s ganita w o r k h a d d e v i a t e d f r o m the r i g h t p a t h h a n d e d d o w n by a h i e r a r c h y o f g o o d teachers a n d the day o f its e x p o s u r e is n o t far distant. W i t n e s s its d o w n f a l l i n 6 8 , 5 5 0 years! T o agree with the fifth case in mdrgdt we have corrected upeta into apeta. It may appear strange that the author calls 68,550 years as a not far distant date. B u t that depends on the outlook of people, and the H i n d u mind, especially the old H i n d u mind may consider even this as a comparatively short period. O r this verse may belong to the criticism of the Romaka following immediately, strayed to this place by the mistake of the scribe. In that case the first letter kha i n khavisayabhutdstarasaih should be corrected as sva and the expression taken to mean "6855 years, by its own measure", in which case the following is the meaning:" T h i s R o m a k a has not c o m e d o w n t h r o u g h a h i e r a r c h y o f g o o d teachers because i t follows the T r o p i c a l y e a r i n s t e a d o f the t r a d i t i o n a l sidereal year. It w i l l be e x p o s e d i n a p e r i o d o f 6 8 5 5 o f its o w n t r o p i c a l years, a n d p e o p l e w i l l a b a n d o n it." H o w this will happen and how the number 6855 can be arrived at almost exactly, will be shown in explaining the next verse.
35. I f we a d o p t the days f r o m e p o c h r e s u l t i n g f r o m the t r o p i c a l year as a d o p t e d by the Romaka a n d the S u n o r M o o n r e s u l t i n g t h e r e f r o m , we m u s t accept P u n a r v a s u as the naksatra o f the f u l l m o o n o f the m o n t h o f C a i t r a , i n s t e a d o f the e x p e c t e d H a s t a o r C i t r a , P u n a r v a s u w h i c h is the naksatra o f Caitra-Sukla-navami. T h i s connection has been strongly established i n people's m i n d by the observance on Caifranavami as the birthday of L o r d Rama, hero of the Rdmdyana, well known as born i n the asterism Punarvasu. This is how this will happen: T h e months Caitra, Vaiksakha etc. are so called because the M o o n at new moon i n these months is i n the vicinity of Citra, Visakha etc. T h u s , in a given month, the full moon, i.e. the M o o n of the 15th tithi, is i n a given naksatra or nearby, so that the other tithis also are 34a. b.
A.B.C.D.^RFrf^
a.b. A . B 1 . ° ^ ^ ^ ° ; B2.3. C . D . ° ^ W ^ * ° A.o^citW; B . ° ^ W :
B.^^31^^
D. °AIdl
c. A . ^ f ^ j B . o m ^ c.
35a.
A.B.C.D.Ttq^
B2.MVWI^1
d. D . ™
C.°^WT;
III.35
III.
P./VULISA-SIDDHANTA
73
connected with particular wafoa/ra^. For e.g. as the M o o n of the fifteenth WA? i n Caitra is near Hasta or Citra, the M o o n of navami, six days before, is near Punarvasu or Pusya, because the tithi, the day and the naksatra, have approximately the same duration. In the same way, as the M o o n of the 15th tithi of Sravana is near Sravana, the M o o n at Sravana Bahula Astami, eight days after that, is near Rohini, which is also a thing well known. Now, when the months (synodic) are 'tied' to the naksatras as mendoned, there will be this conformity. B u t the months are kept tied to the naksatras if the solar year is sidereal, and not tropical like that of the Romaka, i.e. i f the solar year begins as in all other Siddhdntas with a fixed point on the ecliptic like the first point of Mesa, and not the movable vernal equinox, the so called First Point o f Aries, as i n the Romaka or the new Indian Rashtriya Panchang. As the difference between the two points (i.e. the ayandmsa) increases, the above mentioned conformity will gradually decrease and when the ayandrnsa accumulates to 30° the full moon of the first month will fall i n the Phalgunis instead o f near Citra and though called Caitra the first month will really be 'Phalguna'. This non-conformity has already happened i n the Rashtriya Panchang, with the ayandmsa more than twenty degrees now. If it accumulates to 6 naksatras, (i.e. 80°) the full moon of the first month, sdll called Caitra, will be i n the naksatras Punarvasu or Pusya, though the month will really be Pausha. T h u s Punarvasu connected with the N a v a m i of the real Caitra will occur at the full moon of the so called Caitra of the Romaka (or the Rashtriya Panchang). T h e Navami of the new Caitra, occurring 6 days before the full moon day, will occur in Apabharani, 6 naksatras earlier. When a situation thus arises contradictory to their belief, people will realise that the sidereal year is the proper thing and discard the Romaka. Incidentally we may mention that the same confusion will arise i n following the Rashtriya Panchang also. O n e thing we want to say here. W e do not deny that the tropical year best suits civil purposes, but a luni-solar calendar based on the sidereal year also will suit our religious purposes best. Therefore they have to be kept apart. (A civil calendar based on the tropical year, we already have i n the Christian Calendar we have been following, which is pracdcally worldwide. As for the defects in it, the 'Calendar Reform' will take care of it, while i n making this reform we have taken a step in isolating ourselves.) O n e thing could have prevented the confusion. If they had adopted the seasonal-month-names like M a d h u , Madhava etc. preserved for us in the Vedas (vide III. 25) instead of trying to fit in the sidereal luni-solar-calendar months, Caitra, Vaisakha, etc., this confusion would have been avoided and they would not have simply added one more to the one hundred contradictory Panchangs already extant. T o continue: W e shall compute after how many years the exposure of the Romaka, as mentioned by the author, would happen. T h e ayandrnsa was zero at the author's time, as we have shown on several occasions. We must calculate when it accumulates to 6 naksatras. T h e Siddhdntas of the author's time use a sidereal year of days, 365-15-31 nearly, and the tropical year of the Romaka is days 365-14-48 (See V I I I . 1). Therefore, the Romaka year begins earlier by 43 vinddis, every year, and this is equivalent to the rate of ayandmsa per annum, 42", nearly. l( ayandmsa to become 42" takes one year, to become 6 naksatras it will take, 6 X 800 X 60 divided by 42 = 6857 years nearly. It is this that is given by the author as 6855 in the previous verse explained.
36. A l l the i n j u n c t i o n s o f the V e d a s a n d Smrtis are based on the proper dme, and by not performing the rites at those times the performer, especially a twice-born.
74
111.36
P A N C A S I D D H A N T I K A
acquires sin which is to be expiated. Therefore, a study o f this Romaka itself is to be expiated. How is that sin may accrue by not performing a rite at the dme enjoined by the Sdstras. But by simply studying the Romaka, we cannot say one would also perform the rite at the improper time. T h e Romaka may give the dme wrongly, but one may study it not for the sake o f using its time, but for other purposes, as for instance to understand where it goes wrong, and expose its weakness to others and save them, for which the man who studies even deserves merit. Let us understand that these things, sin and merit, are subtle and cannot be known without a deep study of the Sdstras, and if the author steeped in the Dharma Sdstras says a thing, let us accept it. T h e Vedas promise svarga not only to the performer o f the yajna but also to one who knows how to perform it properly. We frequently meet i n the Vedas the expression ya u cainam evam veda. T h e Veddnga Jyotisa says that people who know astronomy know as it were the correct performance o f the sacrifices themselves, yojyotisam veda sa veda yajndn and they go to svarga after establishing a long line o f progeny i n this world. Now, it stands to reason that i f the mere study o f a good thing gives merit, the mere study of a bad thing brings sin. It is said that even association with bad characters and sinners bring sin, as also doing sinful things even i n dreams. A s for the argument that the person even deserves merit for intending to keep off people from the i m p r o p e r times, he does deserve it and will get it. B u t that does not mean expiation is not called for. Contact with craftsmen may be necessary to keep the temple idols i n form, but that does not mean that purificatory ceremonies need not be performed for the idols o n that account. W e can say this m u c h that i n these cases the expiadon is light, like the utterances of the Lord's N a m e , like ' K r s n a K r s n a , Siva Siva! W e should also take into consideration the spirit i f the times i n which these statements were made.
37. T h e p e r s o n h a v i n g c o r r e c t k n o w l e d g e o f the S u n , M o o n , etc. gets Dharma, w h i c h w i l l take care o f his f u t u r e w o r l d , Artha w h i c h w i l l e n s u r e his p r o s p e r i t y i n this w o r l d a n d f a m e , w h i c h w i l l p e r p e t u a t e his m e m o r y . B u t the b a d a s t r o n o m e r w h o m i s l e a d s p e o p l e by his w r i t i n g s w i l l c e r t a i n l y h a v e to go to h e l l a n d d w e l l t h e r e .
36a.
A . t ^
b. B.?ildl Wl°
c-d.
A . $*<«irc|
% ^ * l T ^ ^ < i ^*<"I*K:
(A2.^:)
c. B 1 . 2 . ' S I R M . Al.^Tsrat C. ^
In A and B , ' ^ etc. occurs as the
^1fcf^H<^^de|iyi: I
second half o f the verse. It is put here as
D . ^ ^ i ^ " ! ! ^ fesfl ^ ^>*pr?R52 [IT]
^
the first half to suit the sense, a. D.'cTSSciT
^
111.37
III.
7.5
PAULIJ^A-SIDDHANTA
It must be noted here that when even the person with correct knowledge gets so much, the writer will get more. It must also be noted that only the writer of bad astronomy goes to hell, not the reader, whose sin is small. In this verse there is a j u m b l i n g of words and phrases and induction into the text extraneous words intended as commentary. T h e words, sphutaganitavid etc. seem to be the first half of the verse because i n the first foot there are twelve and i n the second eighteen syllables. Therefore what comes before that is the second half. In that, there are may syllables more than the required twentyseven. Selecting the required words alone, we have reconstructed the third and fourth foot. For the observations o f K S. Shukla on 32-37 vis-a-vis N P , see his paper ' T h e PS o f V M 9(1974) 62-76.
(lyjIHS
1. A.B.MlPeli^lRi^lTl:; C.D.?f^i
Thus ends Chapter Three entitled Taulisa-Siddhanta: Planetary Computations etc' in the Paiicasiddhantika composed by Varahamihira
Chapter Four THREE PROBLEMS — TIME, PLACE AND DIRECTION
Introductory Problems on T i m e , Place and Direction, involving spherical trigonometry, are dealt with in this chapter. T h e first fifteen verses are devoted to the construction o f a table of sines. As this kind of matter does not involve constants specific to any siddhdnta and is commfjnly found in all siddhdntas, we cannot say which Siddhdnta this belongs to, Paulisa or Saura, the only two siddhdntas meant to be expounded in detail by the author. Probably it is the author's own, meant for both, or taken from both. T w o things point to this conclusion: In the part of the work dealing with the Saura, xiiz., chs. I X , X , X I , X I I I , X I V , X V , X V I , and X V I I , no space is given to the sine tables, though required, and to the problems dealt with here, and therefore i f these are not meant for Saura, it would be imperfect though almost full. O n the other hand, certain redundant and crude rules point to this chapter's connection with the Paulisa, as also its position i n the chapter distribution in the PS text.
Table of R Sines 1. T a k e the c i r c u m f e r e n c e as m e a s u r e d i n 360 units, square it, take the tenth p a r t o f the square, a n d f i n d its square root. T h e result is the d i a m e t e r o f the circle i n the units t a k e n . W e assume the d i a m e t e r to be 4 ° , (i.e., 2 4 0 ' ) a n d h e r e u n d e r give the t a b u l a r sines o f angles f o r 3° 4 5 ' i n t e r v a l . T h e rule is: diameter = Vcircumference^/10. It comes to this: d = c/V 10. T h e formula, d = c/n is w e l l k n o w n , a n d t h e a u t h o r h a s t a k e n V 10 as an approximation for n which is incommensurable and usuallyrepresented by the approximate values, 22/7, 355/133, 3.1416 etc. Vhe Surya Siddhdnta toogivesVlO as the value of n in its instruction to find the circumference of the earth from its diameter (1.59.): "The earth's diameter is \&00yojanas. Square this, multiply by 10, and find the square root. This is the earth's circumference." By thus taking VTO for n, an error of about 0.0067% results, and for a circumference of 21,600', we get the radius 3415', instead of the well-known 34-38'. B u t it must be mentioned here that this error does not affect the computation of the sines la-b.
A.B.^^Pf
b. A.l5r^«T:
c. A . B . dR^Wliag^ ( B . °'^) d. A.'H«*
IV.2
IV, T H R E E P R O B L E M S
77
mentioned in the succeeding verses, because it can be shown that the author derives the sines from a correct formula, (not dependent on this wrong ratio of the diameter to the circumference), based on 120' as the radius of the circle. If he had depended on the wrong value, the first tabular sine, i.e. sin 3° 45' would be 7' 54", (being the 96th part of the circumference, where the sine is indistinguishable from the arc), and not the correct 7' 51" as given by the author. Taking the diameter as 4°, and thereby the m a x i m u m sine (i.e. the radius) as 120', is arbitrary. In general, the Siddhdntas give the m a x i m u m sine, 3438', as arrived at from taking the circumference as 360° or 21600'. T h e Vdkyakarana makes it 43°. In actual work, the sines enter only as a ratio to the maximum sine, and therefore no harm, will result by taking these different m a x i m u m sines. T S and N P have not understood the meaning of the second half of the verse, and mis-interpret armacatuskam as quadrant.
gjc|ch
|| ^
2. T h e square o f the r a d i u s , (i.e. 14,400), is called dhruva (karani), (literally, T i x e d I r r a t i o n a l ' ) . T h e f o u r t h p a r t o f it, (i.e. 3600), is the karani (Irrational) related to the first s i g n , (or 3 0 ° ) . Dhruvakaranl minus the karani o f M e s a , (i.e. 14,400 - 3 6 0 0 = 10,800), is the kararii o f t w o signs, (or 60°). T h e square r o o t o f a karani is the t a b u l a r sine. Being square of tabular sines given in minutes, the karanis are squares of minutes, which is their peculiarity as given by the author, though this is not mentioned explicitly. T h e other well-known characteristic of a karani, viz. irrationality, is found i n all karanis except 14,400 and 3600, though the author calls these also karanis i n a general way. In modern terminology the word sine used i n connection with the angle is defined thus:
Fig. IV. l-a
In the right angled triangle, (fig. l-a), sine Z. B = A C / A B , o r s i n e Z_ A = B C / A B , i.e. as the ratio of the opposite side to the hypotenuse. In tabulating the sines, the hypotenuse is taken as unity, and the ratio expressed as a decimal fraction. 2a. A . B . f ^ ^ o ; C . D . f f i T ^ " b. A.B.offcn. A . B . f ? l t W : m . A.B.^?l?tTO
c. B 1 . 3 . % * n ; B 3 . ^ d.
A . ^ ^ B . ^ 1
78
PANCASIDDHANTIKA
IV.2
T h e ancients however expressed the sines in minutes-length or, more accurately, in minutes and seconds-lengths, the m a x i m u m sine called Trijya (meaning 'the sine of three signs', i.e. 90°), occurring separately in the work to make up the ratio. T h i s is the way in which they conceived the sine (meaning 'bow-string' from its Sanskrit equivalent imjmj, synonymous with731a). In Fig. 1-b. A3 E F3 D is the circumference of the circle, centre B . A part of the circumference like A D F , A , D F , , etc. is called dhanus (literally, 'bow') or arc. D
Fig, IV. I-b
T h e straight lines A C F , A , C | F , , etc. forming the 'bow-strings' o f the respective 'bows' are thejyds or full sines. B u t i n actual practice, the halves of the full sines A C , A | C | , etc. above are used with the name of 'sines', with respect to the half-bows or arcs, A D , A , D , , etc. Because the arcs A D etc. are as the angles A B D etc., the sines A C etc. are spoken of with respect to the angles A B D (= A B C ) etc. also. Thus, A C is the sine of L A B D o r a r c A D , AiC, is the sine of L A , B D o r a r c A | D | a n d s o o n . l t is this connection of the sine with the arc that has given it the nature of a length, which is expressed in minutes and seconds on account of the connection of the arc with the angle at the centre. It may be mentioned here that C D , C , D , CgD etc., appearing like the arrows on the respective bow-strings, are called iara (meaning 'arrow'). If A j B D is a right angle, i.e. three signs, then, obviously, A3B is the sign of this angle, i.e. it is the sine o f three signs, and therefore called trijya. Its length is clearly half the diameter A 3 B F 3 , i.e. the radius, equal to 120'. Now, let the angle A B D be equal to one sign, i.e. 30°. A B D = D B F = 30°..-. L A B F = 60°. A B = B E , being radii..-. L B A F = L B F A = 60°. Thus A B F is an equilateral triangle, and AF = A B = 120'..-. A C = A F / 2 = 60'. T h u s sine 30° = 60'. Its, karani is its square, viz. (60')^ = 3600, the fozranf of Mesa as mentioned by the text. T h e n , let L AgBD be equal two signs, or 60°. A , B D = DBF., = 60°. .-. C^ is a right angle. So the karani of 2 signs = A ^ C / = AjB^ - B C / = A.,B' - A C ^ (•.• A A^B Cj = A B A G ) , = 120' - 6 0 ' = 10,800, A j B being the radius. T h i s also agrees with what the text says. (The square root of 10800 minutes, i.e. 103' 55", is the sine of 2 signs, which agrees with the value given in the table.)
IV. T H R E E
79
P R O B L E M S
Incidentally, we shall derive the karani and sine of one and a half signs, i.e. 45°, mentioned in verse 4. Let A , B D be equal to 45°. A = 45°, and Z_ C is a right angle. .-. A , C = C , B . But, A , B ' = A , C / + C | B - = 2 A , C / . . - . A | C r ' = I20'/2 = 14400/2 = 7200 = they^aram of one and a half signs as mentioned in the text. Its root, 84'51", is sine 45°, agreeing with what is given in the tables.
3-5. T h e other tabular sines, (i.e. sine 3° 45', sin 7° 30' etc. other than the four mentioned of the total 24) are formed successively in the following manner: Let the angle or arc for which the sine is required be 0. I. Sin'0 = iA[sin'20 + (120 - sin(90° - 20)}'] II. Sin'0 = 60 X {120' - sin (90° - 20)}, where the sines are i n minutes etc. O f the 24 sines, the^aranf o f the nth sine = 14400 - the Aaranf o f the (24 - n)th sine. 7200 is the karani o f one and a half signs, i.e. 45°.
Thus, as karanis 8, 12, and 16 are known, those o f their halves etc. and (24 - halves) etc. can be found successively. T h u s all the sines from 1 to 24 can be found. O f the two formulae, the first is suited to geometrical representation, and the second to computadon. Example 1. Given the 8th kararii (i.e. of 30°) 3600 and its sine 60', the 16th karani (i.e. of 60°) 10800, and its sine 103' 53".33,find the 4th and 20th karanis and sines, using each of the two formulae. The
3a.
desired sine is the 4th, i.e. o f 4 X 3° 4 5 ' = 15°. 20 = 30°, 90° - 20 = 60°.
A.«?3ll. A.B.C.D.°^gDm°
4a.
A.C.D.^!pr#n; B.^uil^lwi
b. A.cI5^. B . ° 1 ^ d.
C D . [t5r5qTci^^i5PTf] f l 5a. d.
A.TO^; B . ^ ^
B.^; CD.
A.B.C.D.°fW3qi
b. A . M
A.m^;
b.
A.-^;B1.2.^?^;B3.^W C . D . ^ a | l ^ 3 ^ 3 . A.farf^jgrP: A.C^^iJ^lfe'JuH ( C . ° ^ ) B 3 . ^ A.B.^RF5!n
c. A.^^nrnft; B . ^ ^ C f e ' j U M I ' f e y ' l l ^ : ; D.fl^CT^S] ^55[oft
d.
B . m A.B.°^RTiT7tTO
80
IV. 5
PANCASIDDHANTIKA
I. T h e 4th karani = = = = =
V4 [sin' 30° + (120 - sin 60°)'] 1/4 [60' + (120 - 103' 55".33)'] i/4[3600 + (I6'4".67)'] 1/4 (3600 + 258 97/144) 1/4 (3858 97/144) = 964 385/576
.-.the fourth sine, i.e. sine 15° = V964 385/576 = 3 1 ' 4" II. T h e 4th karani = 60 (120 - sin 60°) = 6 0 ' ( 1 2 0 ' - 103'55".33) = 60' X 16' 4".67 = 964 385/576 F r o m this, sin 15° = V964 385/576 as before = 3 1 ' 4" We shall prove the first formula geometrically, and derive the second from the first. Fig. IV. 2
In Fig. 2, F B D = 0, and D F is its arc of which the sine wanted is D E . D E ' = the wanted karani (i.e. sine' 0). D F A = 2 D F . .-. D E = V 2 D A . . - . s i n ' 0 = D E ' = 1 / 4 D A ' = V 4 ( A C ' + C D ' ) . Now, •.• A C = sine 2 arc F D = sin 20, A C ' = s i n ' 20; and •.• C D ' = ( B D - B C ) ' = ( B D - A G ) ' = [120' - sin (90° - 20)', sin'0 = V4 ( A C ' + C D ' ) = 1/4 [sin' 20 + {120' - sin (90° - 20)}'] F r o m this, sin0 = Vsin'0. F r o m I, we can derive II thus: 1/4 [sin' 20 + {120' - sin (90° - 20)}'] = 1/4 {sin' 20 + 120' + s i n ' (90° - 20) - 2 X 120 X sin (90° - 20)} = 1/4 {120' + 120' - 2 X 120 X sin (90° - 20)} (•.• s i n ' 20 + s i n ' (90° - 20) = s i n ' 20 + C o s ' 20 = radius') 9 V 1 90'
= ^
{120' - sin (90° - 20)}
= 60' {120' - sin (90° - 20)} Now, of T S , Thibaut alone proves the formula, while Sud. Dvivedi uses it. T h e form of the first formula given by them differs from that given by us, and is as foUows:sin'0 = (1/2 sin'20) + [1/2 {120' - sin (90° - 20)}'] T h o u g h their formula is correct, it entails more work, and to give the formula in this form they have made many unwarranted changes i n the already correct readings. We have made only one correction, and that grammatical, for the sake of syntax, viz. dhanurdvigunapaddt into dhanurdvigunam paddt, which entails the occurrence o f an extra syllable, which can be explained, as before, by rules of prosody. O r , let the reading hesese tviste dhanurdvi-gunam paddt projjhya etc. Now follow six verses giving the sines, computed by the author himself.
IV. 9
IV.
I H R E E
' w r g * '
PROBLEMS
81
' w ^ ' f ^ '
6-7. T h e o t h e r sines are the f o l l o w i n g : I n the first s i g n , the m i n u t e s parts are successively 7, 15, 20 + 3, 20 + 11, 20 + 18, 5 X 9,50 + 3, a n d 60. T h e seconds, respectively, are: 50 + 1, 5 X 8, 25, 4, 30 + 4, 56, 5 a n d 0. T h u s we have for the first sign —
1
2
3
4
5
6
7
8
Arc or angle
3° 45'
7° 30'
iri5'
15°0'
18° 15'
22°30'
26°15'
30° 0'
Sine
7'51"
15'40"
23'25"
31'4"
38'34"
45'56"
53'5"
60' 0"
Sine no.
^'^-M^^c)l(1?riT)TTf?ffir'R4fldchl
II 6
8-9. Of the sines i n the s e c o n d s i g n , the m i n u t e s parts t a k i n g the i n c r e m e n t s i n the c u r r e n t s i g n a l o n e , are, successively, 6, 13, 20 — 1, 3 X 8, 30 -I- 0, 30 -I5, 30 -I- 9, a n d 30 + 13. T h e respective seconds are: 40, 3, 7, 50 + 1, 13, 12, 60 - 14, a n d 60 - 5. 6a. A . B . ? 1 ^ ; C . D . ^ ^ ^ : . B . ^ H ^ b. A.°f^att?lRT:; B.of^ait^:. B.^afl?TT:
c. A.!^i<^N< c-d.
A . B . ^ ^ C B . H ) WTfPT: ( A . o m f ^ : ) °
c. A.B.-?Rnt 7a. B . ^ ^ ^
b. B.c.D.^raiwra c. B.-^cjI^i^'*!
d. B.f?R1TSBT(B3.ft) 9a. B 2 . ^ ^ . B1.2.°?W b. A l . g r i M . A.B.'^+filid 'ifrl; C.'S**lld^'ldl;
d. A.^HsrarosTT: 8a. B.^lS3Rl. A.B.°^?ter1^°; c.D.?;;^Nfil^° b. B . f ^ . C . ' J I S ^ S ^ : . A . t ^
c. A.B.C.^ffetTI; C.ts#lK?rafe d. A . B . ^ ;
C.^i^Hlt:
82
PANCASIDDHANTIKA
IV.
11
A d d i n g the minutes and seconds, and 60' for the end o f the first sign, the sines are:-
Sine no.
9
10
11
12
13
14
15
16
Arc or angle
33°45'
37° 30'
41° 15'
45° 0'
48°45'
52° 30'
56° 15'
60° Of
Sine
66' 40"
It.
73'3"
79' 7"
84' 5:1"
90' 13"
95'12"
99' 46"
103'55"
10-11. O f the sine i n c r e m e n t s g o n e i n the t h i r d sign, above the second, the m i n u t e s are: 3, 6 , 9 , 10 + 2, 10 + 3 , 1 0 + 5, 10 -1-5, a n d 10 -I- 6. T h e respective seconds are: 6 0 - 18, 6 0 - 3, 6 0 - 1 8 , 0 , 5 0 - 3 , 4 , 5 0 - 1, a n d 5. N e x t f o l l o w the sine i n t e r v a l s . A d d i n g the given minutes and seconds to 103' 55" the sine o f 2 signs, we have:
Sine no.
17
18
19
20
21
22
23
24
A r c or angle
63°45'
67° 30'
7ri5'
75° 0'
78°45'
82°30'
86°15'
90° 0'
Sine
107'37"
110'52"
113'37"
115'55"
117'42"
118-59"
119'44"
120'0"
In one or two places we have corrected the corrupt readings, having in view what exactly should be the number as found by computation. B u t T S have made corrections that give wrong values for the already correct values. For e.g. the fourteenth sine, 95' 12" given by the text is correct, while they make it 95' 13" by an unwarranted change, giving it an unlikely form. T h e 16th sine, 103' 55" given by the text is correct, but they make it 103' 56" so that i n every sine of the third sign, 17-24,
10a.
A.B.C.D.iJ,"H
a-b. A . B . ° ^ a 1 | ; C . o ^ ^ l | { ^ ° . D . o ^ ^ r i
c. D.^lR^kll:. A . B . D . f t ^ s q ; C . f t ' ^ d. B.ilSflWdl; C.
b. A.B.^35P35ni<^^d
d. B l . 2 . ^ ; B 3 . ^ . A . ^ ^ :
IV.
IV. T H R E E P R O B L E M S
14
83
there is one second more. B y this mistake the 24th, i.e. the radius, has become 120' 1", and even this obvious mistake they have failed to note. T h e promised sine-mtervals are here given:
(ir)ravi(dfe(c)4;
II
'^jlliu|
||
12. O f the intervals the minute parts are, in the first sign, 7, 7, 7, 7, 7, 7, 7,6; in the second sign: 6 , 6 , 6 , 5 , 5 , 4 , 4 , 4 ; and in the third sign, 3 , 3 , 2 , 2 , 1 , 1,0,0. 13-14. T h e seconds in the first sign are, 50 + 1, 50 - 1, 50 - 5, 50 - 11, 30, 22, 9, (and here is a break resulting in the loss of the 4th foot of the 13th verse and the first two feet with 4 mdtrds of the 3rd foot of the 14th verse. From the values of the sines we can compute that the seconds in the 8th interval must be 55, which must have been given in the missing part. By examining the remaining part we can construct the meaning of this verse thus). T h e seconds of the intervals in the second sign are, respectively 10 x (4, 2, 0, 4, 2, 5, 3, 0) + (0, 3, 4, 4, 2, 9, 4, 9), added each to each. T S have not been able to see that i n what is left o f the 14th verse, the digits i n the unit places o f the eight numbers giving the seconds of the intervals o f the second sign are given. This is because they have made mistakes in the 14th and 16th sines resulting in an error o f -f- 1" i n the 14th interval, — 1" i n the 15th and -1-1" again i n the 16th. This has prevented them from finding the correct values by comparison, so that they make many changes i n the readings here, saying that the text is very corrupt, here. It may be seen that not a single word is wrong here. 12a. A.-i^H^lv^; B.-^rql^ b. A.^PT^; B . W r a ^ ; . C . ^ i^:) D.faifrisif^ c. A . f t l f e q f c l ^ : ;
to 14c. D suggests for 13d [trai^^qq:^^?pn] W
us. D suggests for 14: B.WR^
d. C.^lfsPr^. A - B . ^ | ; C . ^ D . ^ [ 1 ] 13a. A.rc|-»dl|vid° b. B l . - ^ ; B 2 . ^ ; Al.BI.'fe^^ c. A l . B 2 . f l l ^ 13d. A . B . U n i n d i c a t e d g a p o f ISd.upto^^JT of 14c. t
C. indicates the gap by dots as done by ^ W ] ! F [<5f^=] f?pfe [f|[] ^TO^r [1^]
84
P A N C A S I D D H A N T I K A
IV. 15
15. T h e seconds of the intervals in the third sign, are, 3 x 14, 3 x 5, 3 x 15, 3 X 6,5 X 8 + 7, 17,9 X 5 and 16. T h e intervals can be got by deducting the previous sines from the succeeding ones, and compared with the author's concern for correctness, they are correct. It is to be noted that the intervals of T S also come correct i n the third sign, because there is a uniform error of 1" in all sines. T h e intervals are as tabulated:
No.
0
Int.
1
2
3
4
5
6
7
7'51"
7'49"
7'45"
7'39"
7' 30"
7' 22"
7'9"
No.
8
9
10
11
12
13
14
15
Int.
6' 55"
6' 40"
6' 23"
6'4"
5' 44"
5'22"
4' 59"
4' 34"
No.
16
17
18
19
20
21
22
23
Int.
4'9"
3'42"
3' 15"
2'45"
2' 18"
I'47"
1'17"
0' 45"
N o . 24 Int.O' 16" These intervals are useful for interpolation. T h e method of interpolation has not been given by the author, as being obvious. T h e intervals being increments in the series for successive increments in the arcs (or angles) o f 3° 45', we can find the value for what is left over after taking the tabular value, by proportion, and adding it to the tabular value find the value wanted, whether it is arc for sine, or sine for arc. Further, the author has given the sines o f arcs upto 3 signs, as usually given in tables. B u t the method to compute the sines of arcs greater than three signs, has not been mendoned by him. We shall find a method. T h e circle is divided into four quadrants (Fig. 3), A O B , B O C , C O D and D O A , each quadrant being three signs. Arcs A E = H C = C K = O A , from which their sines, E F = H J = J K = G F . But, since the sines increase i n the first quadrant from O at A to B O (= 120') and then decrease i n the second quadrant from 120' to O at C and again increase i n the third quadrant to D O (= 120') and then again decrease in the fourth quadrant to O at A , sine A E = sine A H = sine
15a.
B.ylH(=(N<4. A.om'?5
b. B.f'I'j'JII M^na"*
c. B.°<5)HM^+^ d. A .
(i.e. on 5)
85
IV. T H R E E P R O B L E M S
o\
D
H
Fig. IV. 3
A K = sine A G (neglecting the first sign) because, E F , H J , J K , G F , are all equal, as already mendoned. Therefore, for an arc or angle in the second quadrant, (3 to 6 signs), deduct it from six signs and get the sine of the remainder. For that i n the third quadrant, (6 to 9 signs), deduct six signs and find the sine of the remainder. F o r that i n the fourth quadrant, deduct it from twelve signs and find the sine of the remainder. Example 2 (a). Find the sine of the arc or angle equal to 4 signs. It is in the second quadrant. Therefore, sine 4 signs = sine 6 signs - 4 signs = sine 2 signs = 1 0 3 ' 55". Example 2 (b). Find the sine of signs 7-11-15. This is in the third quadrant. Therefore sine of sign 7-11-15 = sine (7-11-15 - 6-0-0) = sine 1-1115 - 79' 7". Example 2 (c). Find the sine of signs 9-15-0. This is in the 4th quadrant. Therefore sine of signs 9-15-0 = sine (12 signs — sign 9-15-0) = sine 2-15-0= 115'55". Declination From here to the end of the chapter, problems based on the solution of spherical triangles are dealt with, being problems involving position, time and direction. A s a preliminary, the declination of the Sun and the M o o n are required, which are given first. Stellar sphere (Bhagola) T h e ancient astronomers speak of three spheres, the Terrestrial sphere or Earth sphere on which we live, the Stellar sphere or the Sphere o f the stars, and the Sky sphere or the Sphere of the sky. We shall describe the terrestrial sphere i n connection with the Saura chap. X I I I - X V . O f the other two, we shall now describe the stellar sphere, a knowledge of which is immediately required. It is the apparent sphere on which the stars appear to be fixed, and form a frame of reference for
86
PANCASIDDHANTIKA
I V . 15
the motion o f bodies like the Sun, M o o n , planets etc. Actually the stars are at widely different distances from us and are moving i n various directions at speeds of several miles per hour. But with all this they appear to be, and can be represented, as being fixed on a sphere, because of their enormous distances from us, so much so that we are pracdcally viewing the same sphere as people several generations ago did. But the ancients believed that the stars were luminous bodies fixed on the under-surface of a sphere of radius only sixty times that o f the orbit of the Sun (actually the earth) r o u n d the earth, with the centre o f the sphere at the earth's centre. T h i s sphere seems to be rotating about once a day, by the actual rotation of the earth, about once a day.
NP
/
A
S
o
E
SP Fig. IV. 4
O n this sphere (Fig. 4) there is an important great-circle called the ecliptic {Krdnti-vrtta) (EC), marked by the twenty-seven asterisms, Asvini etc., on which the Sun (S) moves, (actually appears to move on account o f the motion o f the earth), completing a revolution once a year. T h e M o o n and the planets move in orbits inclined to the ecliptic at small angles. T h e i r longitudes are reckoned along the eclipdc from a fixed point called the first point of Mesa or A s v i n i . Latitudes are measured on secondaries to the great circle, meeting at a point called the pole of the ecliptic {Kadamba). Another great circle called the Celestial Equator ( A O B ) cuts the ecliptic at r, called the First point of Aries or Vernal Equinox point, which, instead of being fixed, has a slow westward motion on the ecliptic. A t the period of the authors of the first Siddhdntas, this point coincided with the first point of Mesa, this being the reason why that particular point was taken by them for reckoning from. T h e angle between the two great circles (SrR) is called the Obliquity of the echptic. It was about 24° at the time of Varahamihira, and now it is about 23° 27'. T h e declination (SR,) {Krdnti) of a body like the Sun, is measured along the secondary passing through the body, (NPSRSP,) called the Declination circle, all the secondaries meeting at the celestial poles, the poles o f the stellar sphere, (SPNP), on the axis j o i n i n g which the sphere apparently rotates. T h e declination is found by solving the spherical right angled triangle SrR. Spherical triangles were in general solved by the H i n d u astronomers using the properties of plane right angled triangles formed by the sections o f the sphere along the great circle arcs forming the spherical triangle (The
IV. 16
IV. T H R E E P R O B L E M S
87
Greeks solved them by an extension o f Manelau's T h e o r e m to figures on the sphere.) W e shall content ourselves with giving the formulae for solution and refer to them whenever necessary by way of proof. Let A B C (fig. 5) be a spherical triangle, right angled at C , and R the radius of the sphere. I. II. III. IV. V.
Cos A B = Cos A C X Cos B C ^ R sin B C = sin A B x sin A R Cos A = R X Cos A B . sin A C / s i n A B . Cos A C Cos A = Cos B C X sin B ^ R C o s A B = R. Cos A X C o s B / ( s i n A X s i n B )
These, together with the general identides, cos0 = sin (90° - 0), s i n ' 0 + C o s ' 0 = 1 , will suffice for explaining the formulae occurring i n the text. ^
Fig. IV. 5
Declination of the Sun and the M o o n 16. T h e sine o f S u n ' s d e c l i n a t i o n is f o u n d b y m u l t i p l y i n g the sine o f its l o n g i t u d e b y 61 a n d d i v i d i n g by 150. Its arc is the d e c l i n a t i o n . T h e d e c l i n a t i o n o f the M o o n f o u n d t h u s is the m e a n d e c l i n a t i o n . Its t r u e d e c l i n a t i o n is the m e a n d e c l i n a t i o n plus l a t i t u d e . T h e intervals o f d e c l i n a t i o n s f o r intervals o f q u a r t e r signs are g i v e n ( i n the n e x t t w o verses, 17-18). T h i s is the formula: sin dec. = sin sayana long, x 61/150. F r o m this the arc forming the declinadon is found by using the tables, and then the true declination o f the M o o n , using this. It must be noted that when the longitude reckoned from r (i.e. sayana long.) is within 6 signs, the declination is N o r t h , and when more than 6 signs it is South. I n the case of the M o o n , if the declination and the latitude are o f the same direction they should be added to get the true declination. If they are of different directions, their difference is the true declinadon, its direction being that o f the greater. T h e author has not mentioned this because it is obvious. Example 3 (a). Find the maximum declination of the Sun. Obviously, the m a x i m u m sine o f the Sun's longitude (i.e. when the S u n is 3 signs or 9 signs) will give the m a x i m u m declinadon. It is therefore given by sin dec. = 120' X 61 ^ 150 =48' 48". Its arc, 24° is the m a x i m u m declination.
I6a. A.^ft^IHJl^o; B . # ^ a n - ^ ; C . ^ a n s s i ^ B.Hms^.
B.^mm;
D . [yi^rdkll]. A.B.^5rara:; C.^frora:;
A.c.D.^ramr.-
b. A . ^ l ^ : ^ ; B . ^ ^ ^ ; C . ^ i ^ i f e ;
d. A . B . C . D . d
88
PANCASIDDHANTIKA
IV. 17
Example 3 (b). Sdyana Sun is rdsi 4-7-30. Find its declination. Sin 4' 7° 30' = Sin (6^-0-0 - 4^-7-30) = sin r-22-30 = 95' 12". Sin Declination = 95' 12" X (60 + 1) -r 150 = 95' 12" X (2/5 + 2/5 x 60) = 38' 5" + 38" = 38' 43". Its arc, 18° 50', is the declination. Since the sdyana Sun is within 6 rdsis, the declination is north. Example 3 (c). The sdyana Moon is 9'0°0'. Its latitude is 4°N. Find its declination. Thesdyana longitude being 9'^0° 0', the mean declination is the m a x i m u m , south, i.e. 24°S. Its lat. is 4°N, i.e. of opposite direction..-. the true declination is 24° - 4° = 20° S. South because South is greater. The author uses the word kdsthd to signify declination, which is uncommon. Sometimes this word itself is used to mean sine declination. The rule for sin declination is explained thus: Our siddhdntas take the m a x i m u m declination to be 24°. As the m a x i m u n declination occurs when thesdyana longitude is 3 signs, the angle between the ecliptic and the celestial equator (i.e. the obliquity o f the ecliptic) also is 24°. In Fig. 5, take A B and A C as parts of the ecliptic and celestial equator. T h e n , A = 24° and A B is thesdyana longitude. B C is the declination wanted. B y formula II under the present verse, sin dec. = long, x sin 24° -H 120'. B u t sin 24° -f- 120' = 48' 4 8 " 120' = 61/150. Hence,sindec = sin long X (60 ^ 1)/150, which is the given rule. Now for the direction: See F i g . 4. A t r the Sun moving along the ecliptic crosses the celesdal equator, and passes from South to N o r t h . A s great circles bisect one another, till the longitude is 6 sings it moves north o f the celestial equator, for which declinations are reckoned, and then moves south o f it. Therefore for longitude 0 to 6 rdsis, the declination is N o r t h , and for longitude 6 to 12 rdsis, it is South. As the M o o n and other planets move i n their own orbits inclined to the ecliptic, the declinadons computed from their longitudes reckoned along the ecliptic are only approximate. T o get correct declinations, their distances north or south from the ecliptic points, called their 'latitudes', should be combined i n the proper manner as instructed. But the result by thus adding or subtracting will be only approximate, because the latitudes are directed towards the pole o f the eclipdc (Kadamba), while the declinations are directed towards the Celestial Pole. T h e m a x i m u m error that can occur thus is about 24'. C o m b i n e d with the error i n latitude due to other factors like proportion by degrees of the argument o f latitude (advocated by the Paulisa) instead of the sine etc., the error will be considerable. Now, for the readings. F o r syntactical purposes, and getting the proper meaning, satdrmsdssaikd has been corrected as satdmsaghnaikd, sastidinesa as sastirdinesa, kdsthdnta as kdsthd jyd, apakramardsi as apakramo rdsi andpddenyah as pddebhyah. As for T S , they have not touched this verse and the next two, saying, i n so many words, that they cannot interpret them. NP's interpretation of the three verses has also been affected by the highly corrupt text.
I V . 18
89
IV. T H R E E P R O B L E M S
17-18. T h e ( p r o m i s e d ) intervals o f d e c l i n a t i o n s i n m i n u t e s for intervals o f quarter-signs, are, i n S a y a n a M e s a : 180 -I- 3, 180 + 0, 180 - 5, 180 - 14, i n S a y a n a V r s a b h a , 100 + 4 x 14, 100 + 4 x 11, 100 -I- 4 X 7, 100 + 4 x l , a n d i n S a y a n a M i t h u n a , 9 0 , 6 3 , 4 0 a n d 11. ( T h u s the intervals are 183, 180, 175, 166, 156, 144, 128, 1 0 4 , 9 0 , 6 3 , 4 0 , 11.) T h e s e are to be a d d e d successively to get the d e c l i n a t i o n s f r o m M e s a (Aries) to M i t h u n a ( G e m i n i ) . T h e n f r o m K a r kataka ( C a n c e r ) to K a n y a ( V i r g o ) , these s h o u l d be d e d u c t e d i n the reverse o r d e r , u n t i l at the e n d o f K a n y a , the d e c l i n a t i o n is z e r o . T h e s e d e c l i n a t i o n s f r o m M e s a to the e n d o f K a n y a are n o r t h . T h e n f r o m T u l a ( L i b r a ) to the e n d o f D h a n u s (Sagittarius), the s o u t h d e c l i n a t i o n s increase i n the g i v e n o r d e r , a n d f r o m M a k a r a ( C a p r i c o r n ) to M i n a (Pisces) the s o u t h d e c l i n a t i o n s decrease i n the reverse o r d e r , u n t i l at the e n d o f M i n a the d e c l i n a t i o n is z e r o again.) Example 4. Find the declination of the ecliptic point ending (Sdyana) Capricorn. T h e end of Capricorn is rdsi lO-O-O. T h i s falls between rdsis 6 and 12..-. T h e declinauon is south. T h e declination ending Sagittarius (i.e. beginning Capricorn) is 24° S. T h e dechnation at the end of Capricorn is 24° - 11' - 40' - 63' - 90' = 24° - 3° 24' = 20° 36'S. T h e author has perhaps computed the declination by applying the formula of verses 16 and got the intervals by deducting the previous from the next. O r , these verses are taken in toto from the original Paulisa and given here, for there are small differences from the computed values. O r , the differences are scribal errors. Both are given hereunder for comparison: Degrees
0
Declinations by 0 formula
7V2
15
22'/^
30
371/2
45
183
363
537
704
860
1003
Intervals
183
180
174
167
156
143
127
Given intervals
183
180
175
166
156
144
128
Declinations
0
183
363
538
704
860
1004
17a. A.B.C.-?l?W#a;D.°TTOtf^ b.
A.^?lf^W^f)ftfSPmfli; B . <;vifeiJ^iiy*i(iik-!i ( B l . T ) C. ^feraH?Pfqf^?TR^ I D . <5»lBl«
18a. B . ^ b. B . H a p l o g r a p h i c a l o m . ^ a [om. t d ' ^ a i n verse 19]
c. A.B.c."nf^^q3*ragpr-; D . - n f ^ n i g q ^
c. D.*^'
d. A . B . C . ^ ^ a ^ : ; D . ^ M 1^:. B . ^
d. c.g^nf^q^
90
IV. 19
PANCASIDDHANTIKA
Degrees
52>/2
Declinations by 1130 formula
60
67'/2
75
821/2
90
1237
1324
1388
1427
1440
Intervals
107
87
64
39
13
Given intervals
104
90
63
40
11
Declinations
1132
1236
1326
1389
1429
1440
Bearing the need for agreement in mind, the syllables ka, tu have been inserted in verse 17 to make up for deficiency i n syllables; dsit ta has been corrected into asityd for the sake of sense, as also desastrisa into mese trikha, manundm into manunam, and gavise into gavi; and ntare has been corrected into nte to delete one syllable, and also make the word sensible. In the manuscripts, aher catvdrimsacchivdsca in the 18th verse, the end of the 19th, ydmyottare kdrye and the beginning of the 20th, Visuvaddina(? va)samadhye have strayed. O n l y after these is found the end of the 18th, mithundntare(? nte). T h e portion from here, upto na divdnisi in V . 9 , is missing in one set of manuscripts.
Gnomonic shadow 19. (Plant a g n o m o n at the c e n t r e of) a circle h a v i n g a d i a m e t e r e q u a l to f o u r times the g n o m o n . M a r k the two p o i n t s w h e r e the s h a d o w o f the g n o m o n enters the circle a n d e m e r g e s f r o m it. T h e l i n e j o i n i n g the p o i n t s is the eastwest l i n e . T h e l i n e d r a w n p e r p e n d i c u l a r to this by m e a n s o f e q u a l i n t e r s e c t i n g circles, is the n o r t h - s o u t h l i n e . T h o u g h the east-west line is first asked to be drawn and the north-south next, as perpendicular bisector to the east-west, it will be better i f the north-south line is first drawn by means of equal intersecdng circles with the two points as centres. T h e n using the points of intersection of the north-south line and the original circle as centres, by the same means, the perpendicular bisector forming the east-west line can be drawn, which will pass through the centre as required. O n the other hand, i f the east-west line is first drawn by j o i n i n g the first two points, another line parallel to it and passing through the centre is to be drawn as the desired east-west line. I9a. A.%g<^R<4dl>; C. S^J^-ft^-'Jclfa^ft c. A . 3 i q ^ ; c.D.aqqHl d.
A.^ajC.D.^g
• B . A f t e r ^ one leaf missing uptof^^
in V . 9c. as noted by the scribes. Thus, B I addsinthemargin,^'?^'?^; B 2 . adds,3lfimt3^, and B 3 . adds^lfift TlfftJ^. Andalladdsnil'Rlfef I
IV. 21
91
IV. T H R E E P R O B L E M S
The gnomon should be twelve units in length, not necessarily digits; no harm will result, provided all measurements are given i n the same units. T h e circle also can be of any desired diameter, not exactly four gnomons in length. We are not sure whether the word for 'four' occurs at all i n the text, it is so corrupt in that part. W e can only say it cannot be sahkvangula as corrected by T S ; the letters are so different. Finding the directions in the manner described is explained thus: T h e N o r t h is directed towards the north pole of the earth. Corresponding to this is the celestial north pole, (from which we can find the north, i f we can only observe it correctly, and therefrom the other directions). A t mid-day the Sun is o n the meridian, and at equal times before and after, its altitudes and directions are equal, provided its declination does not change. As the Sun's position is thus symmetrical, before and after noon, with the meridian as the line of symmetry, the gnomonic shadow is symmetrical with the north-south Hue (which corresponds to t h g ^ e r i d i a n ) as the line of symmetry. Therefore if two gnomonic shadows, one in the m o r n i n g and one i n the evening, of equal lengths, are marked on a horizontal surface, the bisector of the angle between the two shadows is the line of symmetry, and therefore the north-south line. F r o m this the east-west line, which is its perpendicular bisector, is drawn. T h e circle, asked to be drawn, serves the purpose of marking the equal shadows. By the same symmetry, the ends of the shadows are at equal distances from the east-west line, and so the line drawn between them is also east-west, being parallel to the east-west line drawn. Therefore the author asks us to draw the east-west line formed by j o i n i n g the two points first, and proceed. W e have said that the Sun's declination must be the same, i.e. does not change during the interval. B u t actually it changes. So if we do the work at a dme o f the year when the change in declinadon is very litde, then the direcdons found will be nearly accurate. T h i s happens near the solstices, and therefore the work should be done when the sun is near the solstices. Methods to find the directions accurately even when the declinations are changing rapidly, are given by writers like Vatesvara, Paramesvara etc., and also explained by Govindasvamin i n his commentary on the Mahabhaskariya, III.l.
Tg^TRTlg?!
fa^elx:^NI^d
fo^ldl
II
II
Latitude from Shadow 20. Measure the mid-day shadow on the day when the Sun is at the equinoxes (the equinoctial shadow), square it, add 144, and find the square root. By this divide the product of the shadow into 120. 21. T h e result is the sine of the latitude of the place, called Visuvajjtvd (or Visuvajyd). Its arc is the latitude. O r , do this work on any day and get the arc. If the Sun is in the six signs from sdyana-Mesa, i.e. if the Sun's declination is north, add the declination to the arc, the latitude is got. If the Sun is in the six
92
IV. 21
PANCASIDDHANTIKA
signs from sdyana-Tula, i.e. if the declination is south, subtract the declination from the arc, the latitude is got. That is: i. Sine latitude = 120' X equinoctial s h a d o w V 1 4 4 + equinoctial shadow'. T h e arc from this is the latitude. ii. Sine south zenith distance of the Sun, (SZD) = 120' X mid-day shadow T h e arc of this is the SZD.
V144 -I- midday-shadow'.
U s i n g SZD, Latitude = S Z D ± d e c l i n a t i o n ('plus' should be used i f the declinadon is north, and minus i f south.) Example 5 (a). At a certain place the equinoctial shadow is 5 units. Find the latitude of the place. Sin. lat. = 5 X 120' ^ V5' + 144 = 600' - M 3 = 46' 9". A r c of 46' 9" ^ 22° 37'. T h e ladtude is 22° 37'. Example 5(b). At a place when the Sun is at the end ofsdyana Tula, the mid-day shadow isfound to be 9 units. Find the latitude of the place. F r o m the formula, (themid-day-Sun's) sin SZD = 9 X 120'
V 9 ' -f- 144 = 1080'/15 = 72'.
SZD = a r c o f 7 2 ' = 3 6 ° 5 3 ' . T h e declination o f the Sun at the end o f L i b r a is 11° 44'S (from 16-18). T a k i n g the minus sign, since the declinadon is south, the latitude = 36° 53' - 11° 44' = 25° 9'. Note: Rule (i) can be used everywhere, while rule (ii) should be used only i f the midday sun is south o f the zenith. I f it is north, having north zenith distance, (NZD), declination = N Z D = ladtude. B u t the work being a Karana, the author intends it to be used only i n N o r t h India, where the midday zenith distance is always south, and hence this has not been mentioned by him. Further the author envisages only north latitudes by his formulae. Sky-sphere (Khagola) H e r e onwards, explanations require a knowledge o f the sky-sphere (khagola) with the stellar sphere imposed o n it. Therefore we shall describe the sky sphere. H i n d u astronomers describe it as the 'Casket Boundary of our universe' (Brahmdnda-katdha-sampula), and marked by the penetration of sunlight. Beyond that there is no sunlight. T h e measure of a great circle on the sky-sphere is said to be the number ofyojanas the M o o n , or the Sun or any planet moves i n a kalpa (yuga according to the followers of Aryabhata), — though, really, the sphere is only illusory and supposed to have an indefinite radius. A s the stellar sphere also is enormous, we can take the surfaces of the two spheres sliding o n each other, and forming spherical triangles by arcs o n each intersecting those on the other. T h e problems will entail the solution o f these triangles. (The formulae for solution have already been given). This will be understood by examining Fig. 6.
20-21. Quoted by Utpala onBS 2. pp. 59-60. 20a. A . W T ^ ; C.I^'RSJIfJ b. A . om^^'im-om. c. A . - ? R r . A . D . ^
d. A . C . D . ^ 12b. A.^^FT^^. A.C.D.°8rai'T^B^ d. A.^fS:
I V . 21
IV. T H R E E P R O B L E M S
93
N
S Fig. IV. 6
N E S W Z is the sky-sphere. It is the sky as seen by an observer on the earth who fancies himself stationary, though taking part i n the rotation o f the earth, and thinks that the stellar sphere is rotating on the axis j o i n i n g the celestial poles, N P , SP. N E S W is the horizon marked by the cardinal points. N o r t h , East, South, West. (Note that the east and west points are interchanged so as to appear as we see them when looking up at the sky.) Z is the zenith, corresponding i n the sky to the observer's position on the earth, and is the point where the line from the centre of the earth, through the observer, joins the sky-sphere. N N P Z O S is the meridian. E Z W is called the prime vertical. E N P is part o f what is called unmandalam. Z S H is part of the vertical circle from the zenith to the horizon passing through the sun, moon, etc. (Note that the prime vertical is the vertical circle passing through the east and west points, and that the meridian itself can be considered as a vertical circle passing through the N o r t h and South points.) ZS is the zenith distance of S, and H S is the altitude. SjSSj is the diurnal circle, the apparent path o f S daily, due to earth's rotation. T h e stellar sphere (Fig.4) can be recognised here by the celestial equator E r R O W , by the ecliptic EcrSC, by the north celestial pole N P , by the position of the Sun S, etc. and by the declination circle. N P S R , of which SR is the arc of declination. Because o f the position of the observer o n the earth with reference to the terrestrial N o r t h pole, the celestial N o r t h pole (NP) seems lifted up along the meridian from the north-point (N) so that its altitude is equal to the latitude of the place, and by this the celestial equator is depressed southward by the same amount from the prime vertical.
94
I V . 21
PANCASIDDHANTIKA
Therefore the latitude of the place = N N P = Z O . (What we have said is for places in the northern hemisphere, i.e. north latitudes. In the southern hemisphere, i.e. at places of south latitudes, SP is lifted up from S, and the celestial equator is depressed northward by the same amount.) T h e complement of Z O , O S , is called the co-latitude (Lamba). T h u s in triangles formed by great-circlearcs of the stellar sphere and the sky-sphere, the latitude is involved directly or indirectly. T h e formulae for the solution o f these triangles have been already given. Now, the two formulae for ladtude can be proved by using the meridian, thus: see Fig.7 O b : observer N : north point S: south point NS.^ZSiOSoS,: T h e meridian Z: zenith O : point of intersecdon of meridian and celestial equator. S = So, S,, S^, S3,: four positions of the mid-day sun. O S : Sun's declination Fig. IV. 7
As already described, O Z = ladtude. O n the equinoctial days at mid-day the Sun, S^ is at O . .-. ZS,, (the south zenith distance of the Sun) = Z O = ladtude (first formula). O n other days, the Sun may be (i) south of O , (S3), or (ii) north of O but south of Z, (S,), (iii) north of O and north of Z, (S^). i. Here, the latitude = O Z = S3Z - S3O = the south zenith distance of the Sun — the declination, (second part o f second formula). ii. Here the ladtude = O Z = ZS, + S , 0 = the south zenith distance of the sun -I- the declination (first part of the second formula). UI. Here the latitude = Z O = S2O - S^Z = the declination - the north zenith distance of the Sun. (This case is not given by the author). T h e zenith distance o f the midday Sun used i n the formulae is to be found thus: see Fig.8. E G = gnomon o f 12 units E T = T h e midday shadow, T G = T h e shadow hypotenuse, ZGS = the zenith distance = angle T G E . Sin zenith distance (ZD) = sin Z G S = sin T G E = T E X 120'
TG
= Shadow X 120' Shadow hypotenuse = Shadow X 120' Vshadow' -f- gnomon^ = Shadow X 120' -^ Vshadow' + 144., (where shadow is i n the units taken). F r o m sin Z D , arc Z D is found. Fig. IV. 8
I V . 22
IV. T H R E E P R O B L E M S
9.5
Sine zenith distance 22. S u b t r a c t the S u n ' s d e c l i n a t i o n f r o m the l a t i t u d e (of the place), i f the d e c l i n a d o n is n o r t h , a n d a d d i f it is s o u t h . T h e m i d d a y S u n ' s Z . D . is got. F i n d its sine a n d m u l t i p l y by twelve. D i v i d e this b y the r o o t o f the d i f f e r e n c e o f the squares o f the r a d i u s a n d sine Z D . T h e m i d - d a y s h a d o w is o b t a i n e d i n afigulas. T h e following are the rules: i . Degrees of zenith distance = Latitude + Sun's declination (the upper sign being used for north declination and the lower for the south.) ii. Mid-day shadow = 12 X sin Z D -f- V I 2 0 ' - s i n ' Z D (where 120 is written for radius). It must be noted that the incompleteness mentioned i n connection with the second formula if the previous work is found here too. Example 6. The latitude of a place is 25° 9'. The Sun's declination is 11° 44', south, (the Sun being in the part of the ecliptic beginningfrom Libra). Find the mid-day shadow. T h e declinadon being south, Z D = 25° 9' -I- 11° 44' = 36° 52'. Sin Z D = 72'. .-. Middayshadow = 12 X 72 - V 1 2 0 ' - 7 2 ' = 12 x 72 ^ 96 = 9angulas. T h e rules are explained thus: F r o m the previous rule. Latitude = zenith distance ± declination, (-1- for north declination, and - for south declinadon), we have, zenith distance = latitude + declination, (for north and south declinations, respectively). From this sine zenith distance is got. U s i n g this, the shadow is obtained from the previous rule, sin Z D = shadow X 120 H- V shadow' -f 144. Squaring both sides, s i n ' Z D = shadow' X 120' H- (shadow' -I- 144). sin' Z D X (shadow' -t- 144) = shadow' X 120' sin' Z D X shadow' 4- 144 s i n ' Z D = shadow' X 120' 120'. shadow' - s i n ' Z D . shadow' = 144 s i n ' Z D shadow' = 144 s i n ' Z D ( 1 2 0 ' - s i n ' ZD) shadow = 12 sin Z D ^ V 1 2 0 ' - s i n ' Z D .
22. Quoted by Utpala o n B S 2, p.61. 22a. A . D . 3 1 ^ . A . C . D . U . ° ^ b. A.c#35fiT; C D . U.dfci^ltPcl. A . ' ^
c. A.C.D.i|&mi d. A.'RISOII;^
96
PANCASIDDHANTIKA
IV. 23
Sine Co-latitude and Day-diameter 23. Square the sine of latitude and deduct from the square of the radius. Its square root is the 'sine of co-latitude', (its arc being the 'co-latitude'). Square the sine of declination, deduct from the square of the radius and find its root. Twice the result is the 'day diameter'. Now, we have (i) sine co-latitude = V radius' — s i n ' latitude (ii) Day-diameter = 2 x V radius' - s i n ' declination Example 7 (a), sin lat. = 72. Find sin co-lat, and its arc, viz. the co-lat. sin co-lat. = V 1 2 0 ' - 7 2 ' = 96'. A r c 96' = 53° 8' = co-latitude. Example 7 (b). The Sun is at the end of the sign Aries. Find the day-diameter. T h e Sun's longitude = rdsi. I-O-O. Sineraii. 1-0-0 = 60'. .-. sin declination = 60' (60-H)/I50 = 24' 24". Theday-diameter = 2 X V 1 2 0 ' - 24' 24"^ = 2 x 1 1 7 ' 30" = 235'. In the right angled triangle having the radius as the hypotenuse and the sine of ladtude as the base, the sine of the co-latitude stands as the perpendicular or lamba. Therefore it is called lambajyd. By the analogy with co-sine for sine, co-tangent for tangent, and co-secant for secant, the term colatitude for latitude, has been invented for 9 0 ' - l a t i t u d e , for convenience of expression. Therefore: (since base' + perpendicular' = hypotenuse'), sin'lat -I- s i n ' co-lat = radius'. F r o m this, s i n ' co-lat = radius' — s i n ' lat. Np .•. sin co-lat = Vradius' — s i n ' lat. ^—"-"I As for the day-diameter, by the diurnal rotation of the earth on its axis, the Sun apparently moves r o u n d the earth every day i n a circular path, at a distance from the celestial equator equal to the latitude, with the axis of the earth perpendicular to the plane of the circle. T h i s circle is called the diurnal circle or day-circle and its diameter, the daydiameter. (See this shown i n Fig.6.) T h e diameter can be measured thus: see Fig.9.
c
Q
Fig. IV. 9
23. Quoted by Utpala on 2, p. 60. 23a. A . ^ ^ ^ S R W R q i ^ b. A . T J c R F f ^ : ; C . i J e T ^ ^ :
SP
c-d. A.°rf>irTl'^lRl'^ls^Wt1<1<; C-D.°tc^Tl
I V . 25
IV. T H R E E P R O B L E M S
97
N P C S P Q i s the stellar sphere, with centre E , C Q i s the celestial equator, SC is the declination of the Sun S, S E C = degrees of declination, SB is the diurnal circle, with the straight line SB as its diameter, and S A as its radius. Suppose the sphere is cut into equal halves, with the cross section N P C SP Q E exposed to view and the axis N P E SP forming a diameter. SE is the radius, and SD (= A E ) = sin declination. T h e n , S A = V S E ' - S D ' . But S A = half day-diameter. .-. day-diameter = 2 S A = 2 V radius' - s i n ' declination.
^'1?rftr' ^ l ^ ( l ^ ) i 5 R F c n 5 « r f « R F T [ : ] x r f ^ :
II
24. T h e sines o f d e c l i n a t i o n s o f the p o i n t s o f the ecliptic e n d i n g A r i e s , T a u r u s a n d G e m i n i are 24' 24", 42' 15", a n d 48' 48". We shall show these to be correct by computing them. T h e sine declination of the end of Aries, i.e. rd. 1-0-0 has been derived i n example 7(b) to be 24' 24". T h e sine of declination of the end o f Gemini, i.e. rdsi 3-0-0, has been shown to be 48' 48", (the maximum) in the example above. So we shall derive here only sine declination o f the end o f Taurus, i.e. rdsi 2-0-0. Sine rdsi. 2-0-0 = 103' 55" (from tables). T h e sine o f its declination by I V . 16 is, 103' 55" X (60 -l-I)/150 = 4 1 ' 34" -f- 41" 34"' = 42' 15"34"'. Here, though 34"' is greater than half a second, the author has omitted it and given 42' 15", to the nearest quarter minute.
25. T h e respective d a y - d i a m e t e r s are, i n the m i n u t e s parts: 200 -i- 35, 200 -I24, a n d 200 -1- 19, w i t h 40" a n d 15" a d d e d to the s e c o n d a n d t h i r d , (i.e. the dayd i a m e t e r s are, 235', 204' 40" a n d 219' 15"). O f these, the day-diameter o f the end o f Aries has been worked out i n Example 7 (b). W e shall derive the other two. T h e day-diameter for the Sun at the end of Taurus = 2 \^I20' — sin' declinadon of the end of Taurus, = 2 V 1 2 0 ' - 4 2 ' 15"- = 224' 38".
24b. A.C.D.^I^SPg
25a. A . omMafel^
c. A.q<<»m<.; D . q ^ [ ^ ] t 5 t ^
b. A . o ^ a w . C . D . t l w l
d. A.°f«J^TT
d. A . ° f ^ . A . ^ R R ^
98
I V . 26
PANCASIDDHANTIKA
But the author gives 224' 40" as being more convenient to use. T h e day-diameter at the end of G e m i n i = 2 V 1 2 0 ' - 48' 48"' = 219' 15", which is the same as given by the author. T h e missing part of the text, (pancatrirhsat), has been found out by computation, (tisamyu) has been guessed as being necessary to supply the meaning, which is clear.
Cara 26. Multiply the sine of latitude by 240' and by the sine of declination. Divide by the sine of co-latitude and by the day-diameter. Find the arc of the sine obtained in minutes — (This arc is called half-cara)-and divide by 3. T h e result are the accurate minutes of cara, (which might be called 'day-difference'). From the cara we can obtain the cara-intervats, (or cara differences). This is the formula: (i) Sine half-cara = 240' X sine latitude X sine declination
(sine co-latitude x day-diameter)
F r o m this the half-cara arc is got. T h e n , (ii) Cara, i.e. day-difference i n vinddis = minutes o f half-cara ^ 3. In III. 12 the author gave a rule for the cara-vinddis to be used i n North-India and its neighbourhood and said that he would give the general rule later i n the Chedyaka section. T h i s is it. Further, in the rule o f III. 10, the interval o f the vinddis were given for long intervals i n degrees, like whole signs, and the value obtained can only be rough. T h i s rule can give accurate values. T h e reading perhaps is 'cara-pinda' for which the scribe has written 'cara-khanda' by mistake. Example 8. The sine of latitude of a place is 72', and the sine of co-latitude 96'. The Sun is at the end of Mithuna, with the sine of its declination 48' 48". The day-diameter for the day is 219' 15". Find the caravinddis. By the formula, sine half-cara = 240' X 72' x 48' 48" ^ (96' x 219' 15") =40' 4". A r c 40' 4" = half-cara = 19° 3 1 ' = 19 X 60'-H 3 1 ' = 1171'. Cara-vinddis = 1171/3 = 390, i.e. nddis 6-30. T h e work is thus explained: (See fig. 10)
26a. A.aTRT:a#3 b. A.^sqra^
I V . 26
IV. T H R E E P R O B L E M S
99
NP
Fig. IV. 10
In the stellar sphere C E C is the celestial equator, N P and SP being the north and south poles. N E S P is the Unmandala or horizon of a place on the equator. Z is the zenith o f the place, N and S being the north and south points and E is the east point. DsD is the day-diameter of the Sun, (s), i n the northern hemisphere, making the declination sd. D|S,D, is the day-diameter of the Sun, (s,), i n the southern hemisphere, making the declination sd. D S D is the day-diameter of the Sun (s,), in the southern hemisphere, making the declination s,d,. s and s, are the rising points of the Sun as seen from the place, NsEs,S being its horizon. T h e altitude o f the N o r t h Pole. N N P = angle N E N P , is the latitude, which is equal to SE SP, from which it is seen that for places in the northern hemisphere, the Unmandala is raised from the horizon by this angle i n the north, and depressed by this angle i n the south. A s the Sun, in its diurnal circuit, takes exactly half a day to move from the eastern Unmandala to the western, the day-time is longer when the Sun's declinadon is north, for it has to travel, after rising, an arc i n the d i u r n a l circle (equivalent to the great circle arc dE) to reach the Unmandala and an equal time while setting. T h e time is less when the declination is south, because before rising it has to travel less by an arc equivalent to E d , to reach the horizon from the unmandala (and an equal time less while setdng). d E and E d , are the arcs o f half-cara. Therefore when the declinadon is north, the time corresponding to 2 D E i n the day-difference, (the day time being greater than 30 nddikds by this amount,) and when it is south the dme equivalent of 2 E d , is the day-difference, (the day-time being less than 30 nddikds by this amount). So we have to calculate d E , and E d , . In A dEs, right angled at d, by fundamental formula III, sin d E = Radius X sin S d X Cos sEd -r(Cos sd X sin sEd). But, sd is the declination and sEd = 90° - N E N P = 90° - ladtude. .-. sin half-cara = 120' x sin dec X cos ( 9 0 ° - lat.) -^{(Cos dec X sin (90° - lat)}] 9
IV. 28
P A N C A S I D D H A N T I K A
100
= 120' sin dec x sin lat ^ ((Cos dec X sin (90° - lat)} = 120' sin dec X sin lat (day-radius X sin co-ladtude) = 240' sin dec X sin lat H- (day-diameter x sin co-latitude). F r o m this the arc d E is got. E d , for south declination is got i n the same way, from A s,E d,. F r o m d E or E d , , the cara-vinddis are got thus: F o r the whole circle of 360° or 21600 minutes o f arc, there are 60 X 60 = 3600 vinddis..-. F o r the arc of half cara i n minutes there are 3600 X arc of half-cora 21600 = arc of half cara/6 vinddis. T h e whole cara-vinddis are twice this, and equal to 2 X minutes (rf'half-cara/6 = minutes of half-cara/3. As we have said, these are added to 30 nddikds to find the day-time, when the declination is north, i.e. when the Sun is i n six signs from Aries. Those vinddis are subtracted when the Sun is in the south, i.e. i n the six signs L i b r a etc. T h e part of the formula, sin declination x sin-ladtude sin co-ladtude, is called 'Earth sine', (ksitijyd), i n H i n d u astronomical works, which is required to be multiplied by the radius and divided by the day-diameter to get sin half-cara. In certain works the half-cara itself is called cara.
Latitude from Cara 27. Divide the vinddis of cara by tv\renty and find the sine of the resulting degrees. Multiply the day-diameter by this, and divide by 240. Put the result in two places. In one place square it and add the square of the sine of declination and find its root. 28. Multiply the result kept in the other place by the radius, and divide by this root. T h e result is the sine of latitude. Its arc is the latitude. 90' minus latitude is the co-latitude, and its sine, sine co-latitude. T h e following is the work to be done: i. T h e vinddis oicara -i- 20 = degree o f half-cara. F i n d its sine. ii. Sine half-cara X day-diameter ^ 240 = sine x. (This is earth-sine or ksitijyd).
27-28. Quoted by Utpala onBS, 2, p.60. 27b. A.°JT5^° C. A.o4l<^fe<+>C<:||; C . \ ^ V l ^ C c | l c l ^ d. A . ^
28a. A.Mf3«IT; C.D.r^lfcl^. c.
A.^^1^^1^"1HI
A . ^ ^
I V . 28
IV, T H R E E P R O B L E M S
iii. Earth-sine x 120
101
Vsin' earth-sine + s i n ' dec = sin lat. F r o m this the latitude is found
iv. 90° — latitude = co-latitude. Its sine, co-lat. Example 9. At a certain place on a certain day, the vinddis of cara are 390 113. The day-diameter is 219' 15". Find the latitude of the place, and sine co-latitude. T h e sine ofdeclination required for the formulae is, by (IV.23),V 120' - (219' 1572)' = 48' 48". i . Degree o f half-cara = 390 1/3
20 = 1171/(3 X 20) = 19° 3 1 ' . F r o m this, sine half-cara = 40'
4". i i . (Earth)-sme = 40' 4" X 219' 15"
240' = 36' 36".
iii. Sinlat.= 36' 36" x 120' ^ V36'36'"-I-48'48"' = 120 ^ V I -1-16/9 = 120' x 3/5 = 72'. F r o m this, lat = 36° 52'. iv. Co-latitude = 90° - 36° 52' = 53° 8'. F r o m this sine co-latitude = 96'. T h e rules are thus derived: a. F r o m the rule, vinddikds of cara = minutes o f half-cara -H'3. B y transposing, we have: Minutes of half-cara = vinddikds of cara X 3. Degrees of half-cara = vinddikds of cara X 3/60 = vinddikds o f cara/20, which is (i). b. From the rule, sine half-cara = sin lat.X 240 X sin dec -r- (sin co-lat x day-diarneter), we get; Sin dec = sin half-cara X sin co-lat . X day-diameter -H (240 X sin lat) = sin co-lat X earth-sine -r- sin lat. Using this i n (iii) above, we have: Sin lat = earth-sine X 120'
Vearth-sine' + s i n ' co-lat X earth-sine' ^ sin'lat.
120' H- Vsin'lat -I- sin'co-lat -^ sin'lat. = earth-sine X 120' ^ (earth-sine) V I -H s i n ' co-lat = 120' H- Vsin' lat + s i n ' co-lat ^ sin'lat. = 120' ^ Vl/sin'lat = V sin'lat = sin lat, thus proving (iii). From this the latitude is got. T h e n , -.• ladtude + co-latitude = 90°, Co-ladtude = 90° - latitude. It should be noted that o f the sin declination and the day-diameter required i n the rules, one is sufficient, because the other can be got from that. A s for the word kharida, meaning 'interval' or 'difference', we have already said that it is pinda ('the whole') we get first, and thence the khanda. As for the reading, we have corrected, carathanakapaksdrma, into carakhandakhapaksdmsa, making ka into kha, because 'twenty' is required here as the divisor. T h i s is the only correction we have made. B u t T S , followed by N P , have made several corrections, not realising that i f Bhattotpala's reading is adopted no other correction would be required.
102
I V . 30
PANCASIDDHANTIKA
'g^gpR^' 'HT^^RfsRT 'l5rf53Jra^' [STfl] (^)
Rt. ascensional difference 29. Square the sine of the longitude of a point on the ecliptic, and deduct from it the square of the sine of the declination of the point. Find its root, multiply it by the diameter and divide by the day-diameter. Find the arc of the resulting sine in degrees. Multiply the degrees by 10. T h e Right ascension of the point is obtained in vinddis. deducting the right ascension of the next rdsi from that of the previous, the right ascentional difference of the rdsis are obtained. 30. T h e vinddis of right ascentional difference for the three signs from Mesa are 278, 299 and 323. In the next quadrant they are the same in the reversed order, viz. 323, 299 and 278. In the half of echptic beginning from Libra, the difference are those of the first half, taken in the reverse order. T h e formula is: Sin Right ascension = 240' x Vsin'longitude - s i n ' dec
day-diameter.
T h e degrees of right ascension multiplied by 10, are the vinddis o f right ascension. T h e differences as calculated, are, for Aries etc. 2 7 8 , 2 9 9 , 3 2 3 , 3 2 3 , 2 9 9 , 2 7 8 , 2 7 8 , 2 9 9 , 3 2 3 , 3 2 3 , 299, 278. Now, what is the meaning of saying that i n the second half the differences are in the reverse order of those in the first half, when reversing the order does not make any difference? T r u e . But the author must have meant this statement for ascensional difference i n general, for, then, owing to the subtraction and addition of half day-differences (cardrdha) in the first and second quadrants, the reverse order becomes different. Further, the vinddis mentioned here are sidereal and not mean solar, because the vinddis per degree are obtained by dividing the time of a full revolution by 360, and the time of a full revolution of the stellar sphere is a sidereal day, and not a mean solar day which is the time o f the diurnal revolution of the mean Sun. 30. Quoted by Utpala on B S 2, p.61. 29a. A . W W s q i ; C.il«
I V . 30
103
IV. T H R E E P R O B L E M S
Example 10. Find the right ascensions of the points of the ecliptic ending Aries, Taurus, and Gemini, i.e. longitudes 30°, 60° and 90°. From them find their respective differences. Sin 30° = 60', sin 60° = 103'55" and sin 90° = 120'. Sin dec. of the points ending Aries etc. are, respecdvely, 24'24", 42' 15" and 48'48". T h e respective day-diameters are 235', 2244'38", and 219'15". (a) For the point 30°, sin Rt. asc = V 6 0 " - 24' 24"' X 240 H- 235 = 54' 49" X 240 Its arc = 27° 49'. M u l d p l y i n g by 10, the vinddis o f Rt. asc. are 27° 49' x 10 = 278.
235 = 55' 59".
(b) For the point 60°, sin Rt. asc. = \ ^ 3 ' 55"' - 42' 15"' X 240 - 224' 38" = 94' 57" x 240' ^ 224' 38"= 101' 26". Its arc = 57° 42'. T h e wna^is of Rt. asc. = 57° 42' x 10 = 577. (c) For the point 90°, sin Rt. asc. = V 1 2 0 " - 48' 48"' X 240' ^ 219' 15" = 109' 37".5 x 240' -4219' 15" = 120'. Its arc = 90°. T h e vinddis of Rt. asc. arc. 90° X 10 = 900. T h e difference for G e m i n i = Rt. asc. for 90° - Rt. asc for 60° = 900 - 577 = 323 T h e difference for Taurus = Rt. asc. for 60° - Rt. asc. for 30° = 577 - 278 = 299. As the Rt. asc. o f the first point of Aries is zero, the difference for Aries = Rt. asc. for 30° - Rt. asc. for 0° = 278 - 0 = 278. A l l these are the same as given by the author. This is how the formula is arrived at: T h e dme taken by each sign of the ecliptic, beginning from Aries, to rise above the eastern horizon, for an observer on the equator, is i n vinddis 278, 299, etc., and their total is the time taken by any point to rise, after the rising o f the First point of Aries. T h i s is represented by the arc o f the celestial equator (called the Rt. asc.) measured from the First point of Aries, and we have to find this arc. In Fig. 11, r is the First point o f Aries. P is the point on the ecliptic of which the time o f rising is required, and P d is the declination o f the point, equal to the arc of the horizon from the east point to the rising point, d r is the arc o n the celestial equator, called the Right-ascension of the point P, which is required to be found. F r o m the fundamental formula iv. Sin Rt. asc. = sin dr = sin Pd X cos P r d X Radius = sin Pr X cos P r d = sin P d X radius
(Cos P d X sin Prd)
cos P d (•. • by the fundamental formula i i , sin P r d sin Pr.)
= sin Pr X V Radius' - Radius^ sin'Pd -H sin'' Pr ^ cos P d = sin Pr X Radius Vsin' Pr - s i n ' P d ^ (sin Pr X cos Pd) = Radius X Vsin' Pr - sin'' P d
COS P d
= 120' X Vsin' long. - sin'dec. -^ Vz day-diameter = 240' X Vsin' long. - sin"' dec.
day-diameter.
T h e arc of this is the Rt. asc. As there are 3600 vinddis for a Rt. asc. of 360°, for the Rt. asc. got, the time is, Rt. asc. x 3600 ^ 360 = Rt. asc. x 10. T h e n by subtracting the vinddis pertaining to the Rt. asc. of the beginning o f the sign from that o f the end o f the sign, the differences are got.
104
P A N C A S I D D H A N T I K A
IV.31
Because the sine of the longitude and the sine of the declination (which itself varies as the sine of the longitude) decrease in the second quadrant i n the reverse order of the increase in the first, and this increase and decrease are repeated i n the third and fourth quadrants, the differences of vinddis follow the same course.
Rising Signs 3 1 . T a k e the differences o f R t . asc. o f three signs at a t i m e . F r o m the first t r i p l e t subtract the differences o f half-cara,s, o n e by o n e , t a k e n i n the g i v e n o r d e r . A d d the half-cara differences o n e by o n e , t a k e n i n the reverse o r d e r , to the s e c o n d triplets. T o the t h i r d t r i p l e t a d d the half-cara differences t a k e n i n the g i v e n o r d e r . F r o m the f o u r t h t r i p l e t subtract the half-cara differences o n e by o n e , i n the reverse o r d e r . T h e vinddis o f the r i s i n g signs, called the ascens i o n a l differences, as seen f r o m any place, are o b t a i n e d . T h e seventh f r o m the r i s i n g signs set d u r i n g the same t i m e as the signs themselves rise. T h e ascensional differences for the several signs are as follows: Aries Taurus Gemini Cancer Leo Virgo Libra Scorpio Sagittarius Capricorn Aquarius Pisces
278 299 323 323 299 278 278 299 323 323 299 278
+ + + + + + -
half-cara half-cara half-cara half-cara half-cara half-cara half-cara half-mra half-rara half-cara half-cara half-cara
difference difference difference difference difference difference difference difference difference difference difference difference
for for for for for for for for for for for for
Aries Taurus Gemini Gemini Taurus Aries Aries Taurus Gemini Gemini Taurus Aries
It can be noted that the ascensional differences for the six signs. Libra etc., are those of the six signs Aries etc. taken in the reverse order, as mentioned by us earlier. It should also be noted that signs Aries etc. mentioned here are sdyana. For nirayana mesa etc. (reckoned from the first point of Asvini) the differences, obviously, will be different, and there will not be this symmetry about the first point of Mesa or T u l a . Also, we have already said that the vinddis are sidereal. Note also, that for places on the equator, the ascensional differences are those given in IV.30 itself because the
31. Q u o t e d b y U t p a l a B S , 2 , p . 6 1 . 31a. A.C.D.r«+M
b. A . ' s m i ^ d. A.^Plf%. B.°^«re!I^
IV. 31
IV. T H R E E
105
PROBLEMS
cara is zero there, the day-time being always 30 nddis there. T h e Sanskrit name 'Lankodaya itself suggests this, Lanka representing a place on the equator. Exaynple 11. At a certain place the equinoctial shadow of a twelve-unit gnomon is 5 units. Find the ascensional differences of the twelve dsis. (sdyana). By 111.10 the cara-wr7ac?w —differences for the place, pertaining to Aries, Taurus and G e m i n i , are 5 X (20, 16'/2, 6%) = 100, 82'/2, 3 3 % . T h e half-cara differences are, respecdvely, 50, 41, 17 vinddis. in the southern hemisphere it is the other way. It is called Unmandala because it is raised in one's Aries Taurus Gemini Cancer Leo Virgo
: 278 - 50 : 299-41 : 3 2 3 - 17 :323-M7 :299-f-41 :278 + 50
= 228 = 258 = 306 = 340 = 340 = 328
Libra : 278 Scorpio : 299 Sagittarius: 323 Capricorn : 323 Aquarius : 299 Pisces : 278
+ -f -1-
50 41 17 17 41 50
= = = = = =
328 340 340 306 258 228
The procedure is thus explained: T h e horizon o f a place on the etjuator (i.e. zero latitude) appears raised towards the north pole to a person in the northern hemisphere on account of the elevation of the pole as we go north and submerged towards the submerged south-pole. T o a person in the southern hemisphere it is the other way. It is czWedUnmandala because it is raised in one's own hemisphere. T h e Right ascensional differences having reference to the horizon o f zero latitude, i.e. the unmandala. B u t what we want are the ascensions, i.e. risings from the horizon o f the place. Therefore the risings are earlier when the declination o f the rdsi is north, (for places in the northern hemisphere), by the time the Sun takes to move from the horizon to the unmandala along the diurnal circle, and later by the same time when the declination is south. It has been explained that this time is equal to the half-cara vinddis. So, with reference to the points o f the triplet Aries, Taurus and G e m i n i , whose declination is north, the half-cara has to be deducted. A s the declination increases, rdsi by rdsi, the differences o f half-cara have to be subtracted one by one, until the maximum half-rara is reached. There the declination decreases as it has increased, still being north, and the half-cara which has to be deducted decreases in the same manner. So the differences are added in the reverse order in the second triplet, i.e. Cancer, L e o and Virgo. In the next triplet, viz. Libra, Scorpio and Sagittarius, the south declination increases, i.e. the additive half-ca?a increases, and to the half-cara differences are again added, in the regular order, because in the third triplet the south declinadon increases in the same manner as the north declination in the first triplet, r h e n in the fourth triplet, i.e. Capricorn, Aquarius and Pisces, the south declination decreases, i.e. the additive cara decreases, and so the differences have to be deducted. (All this can be seen clearly on a globe). F r o m the explanation it can be seen that for places in the southern hemisphere, the risings of the rdsis are those of their seventh in the northern hemisphere. As great circles intersect one another, the part o f the ecliptic above the horizon is always half a great circle, and therefore the distance between the rising point and the setting point o f the eclipdc is always six signs, as also that o f the celestial equator. Therefore the change in the Rt. asc. o f the setting point o f the ecliptic is equal to that o f the rising point, with the result that the time of the setting of a sign seventh from the rising point is that of the rising point.
106
P A N C A S I D D H A N T I K A
IV. 33
[dsldeblH:]
T i m e to reach the Prime vertical 32. W h e n the S u n is w i t h i n 6 signs f r o m A r i e s , (i.e. w h e n the Sun's d e c l i n a t i o n is n o r t h ) , m u l d p l y the sine o f the d e c l i n a t i o n b y 120' a n d d i v i d e by the sine o f the latitude, (the place b e i n g p r e s u m e d to be n o r t h o f the e q u a t o r also). T h e sine o f the Sun's a l t i t u d e at P r i m e v e r d c a l , (sama-sanku), is got. F i n d its arc. T r e a t this arc as p a r t o f the ecliptic, a n d find its R t . ascension i n vinddis.
33. T h i s is the t i m e t a k e n by the S u n to r e a c h the P r i m e v e r t i c a l i n the foren o o n after c r o s s i n g the wnmandc/fl,and the t i m e r e m a i n i n g to r e a c h it after r e a c h i n g the P r i m e v e r d c a l , i n the a f t e r n o o n . T h e S u n does not t o u c h the P r i m e v e r t i c a l w h e n it is i n the six signs b e g i n n i n g f r o m L i b r a , (i.e. w h e n the d e c l i n a t i o n is south), (as seen f r o m places i n the n o r t h e r n h e m i s p h e r e ) . T h e following is the work asked to be done: (i) Sin altitude at Prime vertical = 120' x sin dec (ii) Sin rt. asc. = Vsin' alt - sin" dec x 240' Find the arc of this.
sin lat.
day - diameter.
(iii) A r c in degrees X 10 = time in vinddis to reach the prime vertical from the unmandala (or vice versa in the afternoon) (iv) A d d the total half-cara vinddis if the time from sunrise, (or to set, if afternoon) is wanted. Here, the author has not mentioned the work of ii-iv explicitly, intending to give it subsequently. But it is clear that he is giving the time connected with the prime vertical, and that too, not the time before noon or afternoon, but the time from sunrise or to sunset. But it is not mentioned whether the rising or setting is with reference to the horizon o f the place or to the unmandala. But as the rt. ascension in the manner of computing the Lankodaya is clearly meant, rising or setting with reference to the unmandala alone seems to be in the author's mind, for the time with reference to that alone can be got. So to get the dme from actual sunrise or sunset, the half-cara has got to be added, (secdon iv of the work), though this is not mentioned by the author. T h e half-cara has already been given, and need not be computed afresh. 32-33 Quoted by Utpala o n B S . 2,p.41. b. A . ^ . A.cI^«Rcli; D . ^ l l ^ M W i c. A.fcIT^. A . # ^ d. C . ^ q ^ ^ :
33b. A . W n . Al.t^RI?!^; A 2 . i ^ ^ U . 1 ^ ^ d.
Al.^vl
IV. 33
107
IV. T H R E E P R O B L E M S
It may be mentioned in this connection that T S understand here only the work upto finding the sine of altitude at Prime vertical. As for the time, they say it is equal to the time taken by the Sun to reach the altitude found out, when the question is how to find this very time. It should also be noted that the work upto finding the sine of rt. ascension mentioned in (i) and (ii-) can be done easily, thus: Work (iv) presupposes the knowledge of sin half-cara. Using that, sin rt. ascension mentioned in (ii) = sin half-cara X s i n ' colat -f- s i n ' lat. = sin half-cara X 144 -r- square of equinoctial shadow. If the sin rt. ascension obtained is greater than 120', then, even when the Sun's declination is north, the Sun does not touch the prime vertical. We shall explain this later. Example 12. On a certain day, the longitude of the Sun is rdsi 1-15. The latitude of the place (north of equator) is 30°. (The equinoctial shadow is 6 angulas 55.7 vyangulas). When, after sunrise, does the Sun cross the prime vertical at that place, on that day. We require the sine of declination and sine half-cara for the given time and place. Sin dec = sin T 15° X 61/150 = sin 45° X 61/150 = = 84' 51" X 61/150 = 34' 30".3. Theday-diameter = 2 X V 120' - 34' 30".3' = 229' 51".4 Sin half-cara = 240' X sin lat X sin dec -H (sin co.lat. X day-diameter) = 240' X 60' X 34' 30".3 ^ (103' 55" x 2 2 9 ' 51".4) = 20' 48". Half-cara = arc of 20' 48" = 9° 59'. Half-cara vinddis arc 9° 59' x 10 = 100 = na.1-40. A l l this is supposed to be known already. Now for the computation of the time: (i) sin altitude = 34' 30".3 x 120' - 60' = 69' 1". (ii) sinrt. asc = V69' 1 " ' - 3 4 ' 3 0 " . 3 ' X 240' -r- 229' 51".4 = 62' 24". Its arc is 31° 2 1 ' . (iii) T h e corresponding time = 31° 2 1 ' x 10 = SlSvinddis
= nd. 5-13.
(iv) T h e time of crossing the prime vertical after sunrise = nd. 5-13 4- nd. 1-40 = nd. 6-53. This is for the forenoon. For the afternoon, deducting this time from the time o f sunset, nd. 3320, the time of crossing is nd. 33-20 - nd. 6-53 = nd. 26-27. Now, according to the short-cut in the place of (i) and (ii). Sin rt. asc. = sin half-cara x s i n ' co.lat -i- s i n ' lat. = 20' 48" X 103' 55"' ^ 6 0 " = 20' 48" x 3. = 62' 24". (See this obtained by the regular rule). O r , sin rt. asc. = sin half-cara X 144 equinoctial shadow = 20' 48" X 144 -i-iGang. 55.7 vyang.f = 20' 48" X 3 62' 24", as already obtained. T h e rules are explained as follows, supposing the place to be north of the equator. (For places south of the equator also the same can be used, interchanging the directions north and south, wherever they occur.) See Fig. 12.
108
PANCASIDDHANTIKA
I V . 33
N
Fig. IV. 12
S
In this figure of the sky-sphere, Z is the zenith, and N P is the north pole. D , D , , D D , etc. are four diurnal circles, o n which four positions of the sun, S,, S, etc are indicated. D2M2D2 is a part of the unmandala, visible. In all the diurnal circles, the Sun S, etc. rising at D , etc. moving westward, moves a little south, little by little, until it reaches the meridian point M , etc., where the 'southing' is equal to the latitude, N N P , and then proceeds to move westward, moving north little by little, setting i n the west at a point having the same amplitude as the rising point, (assuming that the declination does not change). O n the two equinoxes, the Sun rises due east (D2) and sets due west (Dj) southing on the meridian by Z M j (= N N P = ladtude), and thus is always south o f the prime vertical. So, when the declination is south, the diurnal circle (DjDj) is always south of the prime vertical and so the Sun (S3) never touches the prime vertical. Even when the declination 8,82 (= M|M2)is greater than the ladtude (ZMj) then the Sun is always north of the prime vertical, the diurnal circle D , S ; M | D | being north of it. It is this that was referred to by us as the case not mentioned by the author, viz. the case of the declination being north, but still not crossing the prime vertical, the case that is possible in the southern part o f India. T h e r e is only one case left, that of the Sun's dechnation being north, but less than the latitude, (e.g. the Sun moving on the diurnal circle D U S M D ) , in which alone the Sun crosses the prime verdcal as at S. T h e time by which the S u n rising at D describes the part of the diurnal circle, DS, is to be found. H e r e there are two parts, the time from D to U which is the half-cara, and the time from U to S, i.e. the dme after crossing the unmandala, which alone, we have said, has been mendoned explicidy by the author, and for which alone the rules o f computation have been given by h i m . T h a t is why we have said that the two dmes should be combined to get the dme after sunrise. O f these, the method for computing the half-cara has been explained already. Therefore we shall explain the second part alone. T h e time to move from U to S i n the d i u r n a l circle is clearly the time to move from D2 to S2 on the celestial equator, and given by the arc Da Sj which is to be got by solving the spherical triangle
I V . 34
IV. T H R E E PROBLEMS
109
SS2D2, right angled at Sj. SSg is the declination. A n g l e S2D2S = ZM2 = ladtude. Therefore, from the fundamental formula I V , sin D2S2 = Cos S2D2S X sin SS2 X radius 4- (sin S2D2S.COS SSj)
= Cos lat X sin dec X radius -r- (sin lat.X cos dec.) = sin colat X sin dec X radius H- (sin latx day-diameter/2) = sin colat X sin dec x 240' X (sin lat.x day-diameter) (From sin DgS, arc D2S2is found and converted into time at 10 vinddis per degree, as mentioned before.) We shall prove the author's method by showing that his formula is equal to this. T h e author's formula is: Sin D2S2 = V s i n ' alt, at prime-vertical - s i n ' dec X 240 ^ day-diameter = Vsin' dec X 120' -r- s i n ' lat - s i n ' dec x 240 ^ day-diameter (•.• sin D2S = sin SS; sin S2D2S, by fundamental formula II) = V (sin' dec (120' -- s i n ' lat) ^ s i n ' lat X 240 ^ day-diameter. = V sin' dec. s i n ' colat -r- s i n ' lat X 240 -4- day-diameter). = sin dec x si 1 colat X 240 (sin lat x day-diameter) This is identical with the formula derived by us. (The author himself will be giving this form i n the next verse.) We shall now show how the formula for the condensed work is got. T h e formula for half-cara is: Sin half-cara = sin dec X sin lat X 240'
(sin colat X day-diameter)
Multiplying the numerator and the denominator of the formula arrived at by (sin lat X sin colat), we have. Sin Rt. asc. = sin declination X sin lat X s i n ' colat X 240 -r- (sin' lat X day-diameter X sin colat) — sin half-cara X s i n ' colat -r- sin'lat, given by us. Again, s i n ' colat -j- s i n ' lat. = 120' X 12 -i- equinoctial hypotenuse' -r- (120' X equinoctial shadow -H equinocdal hypotenuse)' = 12'/equinoctial shadow' = 144 square o f equinoctial shadow. So this can be substituted for s i n ' colat s i n ' lat. It must be noted that i f the declination is greater than ladtude, i.e. i f sin dec > sin lat, then sin colat > day-diameter. Therefore sin rt. asc. > 120', for which there is no arc, which means that at no aldtude, or at no time does the Sun cross the prime verdcal. T h i s is what was referred to earlier and here shown mathematically.
34. Multiply sine declination by 240 and again by sin co-latitude and divide by the
product of the
sine of latitude and day-diameter. F i n d its arc in
degrees and divide by six. (The time in nadis, taken by the Sun to move from
110
PANC;ASIDDHA.\TIkA
IV. 35
the unmandala to the p r i m e v e r t i c a l is got.) A d d to it the t i m e o f half-cara. T h i s is the t i m e f r o m sunrise for the S u n to r e a c h the p r i m e v e r t i c a l . T h e following is the work: (i) Sin (arc corresponding to time from unmandala to prime vertical) = 240 ' X sin dec X colat (sin lat X day-diameter) (ii) T h e arc in degrees of (i) is to be got. Dividing by 6, the time in nddis is got. (iii) T h e time got by (ii) -I- the half-cara is the time after sunrise, for the Sun to cross the prime vertical. Note that the formula here given is what we arrived at earlier, as what the author's formula reduces to in verses 32-33. T h e n , why is this repetition? In the previous two verses, the work was not given clearly and fully. Here it is clear and full.
Now for the reading: F r o m the words khajinaghni krdntijyd lambaghni, it is clear that the produc of two sines must be the divisor. Therefore, we have corrected dhruvaguna dyudairghyahrtd into dhnivaguna-dyudairghya-hatd, which is otherwise also a better reading. Other small corrections have been made according to the idea intended to be expressed, and according'to syntax. T h u s it is clearly seen that in the work sin colat appears as part of the numerator, and sin lat. as part of the denominator, from which it can be seen clearly that the formula is concerned with finding the dme of the Sun's rise from unmandala to the prime vertical, and not the half-cara. T h e mention of the half-cara here is just to say that it should be added to find the whole dme. However, both T S and N P have been misled by the mention of the expression 'half-cara' into thinking that the formula itself is to find the half-cara, with the result that they take the numerator as the denominator, and the denominator as the numerator, not realising that by their interpretation the rule for half-cara would be a repedtion, because in I V . 26 also the same has been given, and in the same form, which N P , too, have, noticed and observe: "This in fact, is only a repetition of I V . 26. It is here out of place." (pt.II, p.43). But it may be asked whether the work according to our interpretation is not a repetition of the work of the previous two verses. We say the work as given here is clear, succinct and full. B u t when what is the use of the two previous verses? T h e work there given is easy to explain on the basis of the rule for the Rt. ascension o f the ecliptic point, gone before. O r , that method perhaps is that of the Paulisa, the author giving the same in a better form here. T h e example on this has already been worked out i n Example 12.
34a. A l . ^ % r a t ; A 2 . ^ i ^
d. C . D . ^ ? 1 ^
b. C D . (HM^dl ^'^''W. A.ffcng^
A.f^^l^^HI^:; C f ^ ^ ^ ^ ^ ^ K ^ : ;
c. A.d^lMi^l
D.f^[fsr]^^^:
I V . 35
IV. IHRF.K I'ROBI.K.MS
111
Great gnomon (Sama-saftku) and its shadow 3 5 . W h e n the S u n is i n the n o r t h e r n h e m i s p h e r e , (i.e. i n the six signs, A r i e s etc.), m u l t i p l y the sine o f the l o n g i t u d e o f the S u n by the sine o f the m a x i m u m d e c l i n a t i o n , (i.e. by 4 8 ' 48"), a n d d i v i d e by the sine o f l a t i t u d e . T h e m i n u t e s so o b t a i n e d are c a l l e d the m i n u t e s o f the ' G r e a t g n o m o n ' o r $anku, (i.e. sine o f a l d t u d e ) , ( a n d i n this case, the sine o f P r i m e v e r d c a l altitude). F r o m this the s h a d o w o f the S u n o n the p r i m e v e r t i c a l m u s t be c a l c u l a t e d . (i) Sin prime vertical altitude = sin Sun's long x 48' 48"-^ sin latitude. T h i s is the Great gnomon, and the radius is the Great hypotenuse. T h e square root of the square of the hypotenuse lessened by the square o f the gnomon is the shadow. Therefore the Great shadow = Vradius' — sin ' prime vertical alt. Therefore, by the similarity between the Great shadow and the shadow triangles, we have the proportion. Great gnomon: Great shadow :: Twelve unit gnomon: shadow. F r o m this, the required, (ii) Shadow = 12 x V 120' - s i n ' prime vertical alt.
sin prime vertical aldtude.
Example 13. The longitude of the Sun is rdsi 1-0. The latitude is 30°. Find the Great gnomon of the Sun at prime vertical, and thereby the gnomonic shadow at that time. (i) T h e Great gnomon = sin prime vertical altitude = Sin Sun's longitude X 48' 48" sin ladtude = 60' X 48' 48" 4- 60' = 48' 48". (ii) Shadow = 12 X V120' - 48' 48"' ^ 48' 48" = 12 X 109' 38" H - 4 8 ' 4 8 " = 12 X 109 19/30 4- (61/L50) = 1644 - 30 ^ 61 = 26 units and 58 parts, angulas and vyangulas T h e equation (i) can be written as. Sin prime vertical alt. = sin Sun's l o n g . X sin max. dec. sin lat. = sin Sun's long. X sin max. dec x radius ^ (sin lat x radius) = (sin Sun's long. X sin max. dec -H radius) x (radius X sin lat) Here, it can be shown that sin Sun's long x sin max. dec
radius =
' *^ Kig. IV. Vi
d e c , thus:
Sin Sun's l o n g . X sin max. dec radius = sin Sun's l o n g . X 48' 48" ^ 120' = sin Sun's long^x 61/150 = sindc. (by I V . 16). O r , from Fig. 13, thus: In the triangle right-angled at R, rS is the Sun's long, and SR is the declination o f the Sun. SrR is the m a x i m u m declination. B y fundamental formula II, sin rS x sin SrR 4- radius = sin SR. .-.sin Sun's long x sin max. dec ^ radius = sin dec. 35. Quotedby Utpala o n B S 2, p.42 35b. A.+ISlTli'Juil
d. A.*liy
112
I V . 37
PANCASIDDHANTIKA
N o w we shall show that sin prime vertical alt = sin dec X radius ^ sin lat. In Fig. 12, SZD., is the prime vertical, and the part DgS is the altitude of the Sun S, and the sine of the altitude is to be found. But, sin D2S = Cos SZ, since DgZ = 90', and SZ = 90° - D2S. Observe the triangle S Z N P , right angled at Z. Here, Z N P is the co-latitude. S N P = 90° - SS,,, (•.• N P S, = 90°). Now, from fundamental formula i , cos SZ = cos S N P X radius ^ cos Z N P . .-.sin Pv alt = cos (90° - dec) X radius ^ cos co-lat = sin dec X radius sin (90° - colat) = sin dec x radius sin lat. As stated earUer, the Sun crossing the prime vertical can occur, i f at all, only when it is in the northern hemisphere, (of course for north-latitudes) and this is mentioned in the verse by uttaragole. It should be noted that it is this sin pv. alt that is asked to be derived i n I V . 32 by the statement: istottaragoldpakramamsakajydm khabhaskarabhystam hrtvdksajivaya, which can be seen by comparing the work. Only, the namd Sama-sanku (i.e. sin pv.alt.) is not mendoned there. So, the arrangement would have been better i f the author had first given the formula for sin pv.alt, and then given the dme o f crossing the prime vertical by either I V . 34 or I V . 32-33, and, last of all, the shadow of the Sun on prime verdcal. B u t the great transcend all restricdon! O r , there is plenty of all sorts of errors committed by scribes in this part of the text, as we have reason to think.
Astronomer's qualifications 36. Only he is fit to be called an expert astronomer knowing the problems dealing with the Sun, who can compute the time of the Sun crossing the prime vertical, and prove his method mathematically and graphically. 37. Even a person with very little knowledge can, by using pieces of potsherds, and strokes tackle (by means of computation) problems like finding the Sun's motion in a desired number of days, given the motion is twelve rdsis per year. T h e idea is that anybody can tackle problems depending on mere proportion. O n l y an expert can understand how to solve difficult problems Hke computing the time of the Sun's crossing the prime vertical, and prove the soundness o f his method by means of graphical representations. 36. Quoted by Utpala on i3S 2, p.42 a. A . e R I #
b.
A.%?II;
37b. A.cRt
P. U . M . A.*
I V . 39
IV. T H R E E
P R O B L E M S
113
Gnomonic shadow and the prime vertical 38. O n a circle w i t h the east-west l i n e d r a w n , a n d the d i r e c t i o n s m a r k e d , ( a c c o r d i n g to I V . 19), the t i m e w h e n the g n o m o n i c s h a d o w perfectly c o i n cides w i t h the east-west l i n e is the t i m e o f the S u n c r o s s i n g the p r i m e v e r t i c a l . T h e idea is that i f this time is found by measuring instruments, compared with the computed time and the agreement shown, people will acquire faith in the method. It can be shown that when the Sun is on the prime vertical, the gnomonic shadow must be along the east-west line. T h e prime vertical is the vertical great circle of the sky-sphere, passing through the east-west points and the zenith, and therefore the east-west line forms the intersecdon of this vertical plane and the plane of the horizon which is horizontal. As the gnomon standing vertical and also the Sun on the prime vertical lie i n the vertical plane, the shadow (intercepted by the horizontal plane) must also lie on the vertical plane, and therefore must fall on the east-west line, which is the intersection of the two planes.
Agra : Sine amplitude 39. M u l t i p l y the S u n ' s d e c l i n a t i o n by the r a d i u s a n d d i v i d e by the sine o f col a d t u d e , a n d find the sine (of the a m p H t u d e o f the r i s i n g o r setting p o i n t , c a l l e d Agra). A t a p o i n t distant by this a m o u n t f r o m the east-west l i n e (accordi n g to the d e c l i n a t i o n , n o r t h o r south) the S u n rises o r sets. Agra, (i.e. sine amplitude) = sin dec X radius sin colat. F i n d the arc of this sine. By an angle equal to this from the east to west point does the Sun rise or set on the horizon. Example 14. The latitude of a place is 60°. The longitude of the Sun is rdsi 4-0. Find the direction of rising or setting of the Sun. First, sin declination is to be found. As the Sun is in the second quadrant. Sin rdsi 4-0 = Sin rdsi 2-0 = 103' 55". Sin dec = 103' 55" X 61/150 = 42' 15", and this declination is north. Sin amplitude = 42' 15" x 120' -f- sin (90° - 60°) = 42' 15" X 120' X 60' = 84' 30". = 84' 30". 38. Quoted by Utpala o n B S 2, p.41. 38a. A . f f c l f ^ l ^ c. A . ? l f :
39a. A.^^qM; D . ^ b. A . ^ T O ^ ; D.5ziimw(^) el^*ixt)*i»iii^j,: d. Al.'fralcqi; A2.'f#T^correctedto^THl^ I
114
PANCASIDDHANTIKA
IV. 40
T h e arc o f 84' 30" is 44° 4 6 ' . Therefore the Sun rises at a point 44° 46' north o f the east point, and sets at a point 44° 46" north of the west point, (assuming that the declinauon has not changed). T h e formula for amplitude is got thus: See Fig. 12. T a k e S, as the Sun on the diurnal circle north of equator. T h e n SjSg is the declination o f the Sun, D,D2 is the amplitude o f sunrise. F r o m the figure it can be seen, D i D j = 90° — N D , Therefore sin D j D j = Cos N D , . F r o m the right angled triangle N D N P i n the figure, Cos N D can be got thus: By the fundamental formula I, Cos N D , = 120' X Cos D , N P Cos N N P . But, Cos D , N P = Cos S, N P = Cos (90° - 8,8^) = sin S,S2 = sin 8,82 Cos N N P = sin (90° - N N P ) = sin colat. .-. Sin amplitude = 120' X sin dec ^ sin colat. W h e n the Sun is south o f the celestial equator, (e.g. S3 i n the figure), D , is the rising point, and D2D3 is the amplitude. Its sine is got thus, from the triangle, DzDjE, right-angled at E. By the fundamental formula II, Sin D2D3 = sin ED3 x radius 4- sin angle D j D j E , Here, ED3 is the declination. DjDsE = 90° - ED2Z = 90° - lat. .-.sin amplitude = sin declination X radius 4- sin (90° — lat) = sin dec X radius ^ sin co-lat., which is the formula given.
'iSl^'Ht thlV^^l d M c h 1 ^ [ ^ ] (m) Wl I ^ ^ c l f d f ^ ^ H I ('^%^) % 5 ^ W r n T : ^ : \\>io II
^ItTT
Latitude from A g r a 40. Multiply sine declination by 120 and divide by the sine of amplitude. T h e sine of co-latitude is got. Find its arc in degrees. Deduct the degrees from 90. T h e remainder are the degrees of latitude. Now, sin co-lat = 120' X sin dec. -r- sin amplitude. F r o m this, the arc, co-lat is got. 90° — colatitude = latitude, as already stated i n (IV. 28). F r o m the formula o f the previous verse. Sin amp = radius X sin dec sin co-lat. Sin colat = radius X sin dec -j- sin amp. = 120' X sin dec -^ sin amp. F r o m the amplitude of the setting Sun also, the latitude can thus be found. T h e amplitude can be marked o n a circle with the directions already marked by the observation o f sunrise or sunset. 40a. A.Wb. A 1 . ^
A.m^ ( A 2 . ^ corrected to ^ )
c-d. A . # 1 T ^ ^ W T I :
IV. 43
IV. T H R E E P R O B L E M S
115
The sine o f amplitude to be used i n the formula can be got by measuring the arc of amplitude, or the distance o f the point, from the east-west line. Example 15. Sine declination is 42' 15". Sine amplitude is 84' 3ff'. Find the latitude. From the formula, sin. colat = 120' x 4 2 ' 15" H- 84' 30" = 6 0 ' . Coladtude arc o f this, i.e. 30°, ladtude = 90° - 30° = 60°.
Shadow at desired time 41. T o find the gnomonic shadow caused by the Sun at any time: Take the cara in vinddis and divide by 20. Degrees of half-cara are obtained. Place the degrees in two places. Convert the time from sunrise in nddis into degrees by multiplying by 6. From these degrees, deduct or add the half-cara degrees according as the sun is in the six signs beginning with Mesa or in the six signs beginning with T u l a , respectively. 42. Find the sine of the resulting degrees, and add or subtract this from the sine of the half-cara kept apart in the second place, according as the Sun is in the 6 signs Mesa etc., or in the six signs T u l a etc. (The result is a sine. If the half-cara degrees cannot be deducted from the time converted into degrees, then simply find the sine of the degrees of sine, and take it for further work.) 43. Multiply this sine by the sine of colatitude and the day-diameter and divide by 28,800. T h e result is sine altitude of the Sun.
41-44. Quoted by Utpala o n B S 2, p.61. 41b. c. 42a.
43a.
A.^El^?it?ltfePR
b. A.^°!l)t^H|c|;
A.m^ A.
fl^i
d. 10
U.^T^
A . Haplographical omission of
b. A . ° 4 l ( ^ d l ^ ^ C.D.4irHdNj
A.*--^! ^-^1
U.°%^^
A.MWWIAM+T^OMI ( A 2 . ^ f ^ )
43 c-d, 44 and 45a-b: [... ci»l'i] RJaiq^ Hence they are added here from Utpala's quotaton thereof.
116
PANCASIDDHANTIKA
I V . 44
44. S q u a r e this a n d d e d u c t f r o m 14,400. T a k e its square r o o t , m u l d p l y this by twelve, a n d d i v i d e by sine a l t i t u d e . T h e result is the l e n g t h o f the shadow o f the t w e l v e - d i g i t g n o m o n . T h e following are the steps i n the work: (i) Sine altitude = {sine (degrees of the + degrees of half-cara) ± sine half-cara} X sin colat X day-diameter -r 28,800. (Here, of + or ± , the upper sign should be taken for the 6 signs Mesa ^ t c , and the lower for the 6 signs T u l a etc.) (ii) T h e shadow = 12 X V 14,400 - s i n ' a l t i t u d e
sin altitude
Example 16(a). At a certain place where the sine of the co-latitude (i.e. cos. lat.) is 103' 55", when the Sun is in the 6 signs from Mesa on a particular day, the cara is 200 vinddis, and the day-diameter is 229' 51'. Find the length of the shadow at 8 nddis from Sunrise. (i) Degrees of half-cara = 200 ^ 20 = 10°. Degrees of time = 8 x 6 = 48°. As the Sun is i n the six signs from Mesa, deducting 10° from 48°, we get 38°. Sine 38° = 73' 35". The sine o f the half-cara, i.e. sin 10° = 20' 50". A d d i n g the two signs, (since the Sun is from Mesa), 73' 35" + 20' 50" = 94' 25". Sine altitude = 94' 25" x 229' 51" x 103' 55" ^ 28,800 = 78' 19" (ii) T h e shadow = 12 x V 1 4 , 4 0 0 - 78' 19"^
78' 19" = 13 ang 56 vya^.
Example 16 (b). At the same place, on the same day, find the shadow at one nddi after sunrise. (i) T h e degrees o f half-cara (already found) = 10°. T h e degrees of time = 1 x 6 = 6°. T h e halfcara degrees have to be deducted, but cannot be deducted, being greater. Therefore, taking the sine of the 6° alone, we have 12' 32". Sin altitude = 12' 32" X 229' 51" X 103' 55" ^ 22,800 = 10' 24". (ii) shadow X 12 x V 1 4 , 4 0 0 - 10' 24"' ^ 10' 24" = 137 angulas 57 vyangulas. But it should be mentioned here, that the author's instruction for the case when the degrees of half-cara cannot be deducted from the degrees of time, will give only a rough result. This will not matter m u c h i n places where the degrees of half-cara is small, as i n India, and therefore given by the author. For correctness, the following instruction is to be followed. If the degrees of half-cara cannot be deducted from the degrees of time, deduct the degrees of time from the degrees of half-cara, find its sine, and deduct this from the sine o f half-cara. T h i s sine should be multiplied by sin colat. etc. and sine altitude is to be got. Because this will not produce m u c h differences in our country, the author has not given this detail. (Even if the cara is 5 nddis the difference in sin alt. will be only 15'.) Further, the measurement of long shadows cannot be accurate, and any inaccuracy caused by the author's rough work will be submerged i n the inaccuracy of measurement. 44a.
C.D.ft^TRlf^
117
IV. T H R E E P R O B L E M S
I V . 44
We have mentioned that the author's rough procedure is indicated only when the S u n is i n the six signs from Mesa, because only then have we to deduct the degrees o f half-cara, and the question, what is to be done when the half-cora is greater, arises. A s for the subtraction o f sine half-caro i n the six signs from T u l a , that will always be less, and the question cannot arise. T S have not understood the author here, and say something unconnected and useless. (See their commentary p.25, and English Translation, pp.34-35). T h e rules, (for the S u n i n the northern hemisphere) can be derived thus: (see fig. 14.) N
Z = Zenith P = N o r t h pole E = East point S = Sun D D ' = D i u r n a l Circle A S = Altitude o f the S u n ZS = zenith distance of the Sun. E P = Unmandalam Sin A S = Sin aldtude of the S u n = Sankuliptds (or Mahd Safiku or Great gnomon)/120
E
By the well-known formula o f the spherical triangle. Sin altitude of the S u n = sin A S = cos ZS = cos S P. cos Z P
-I-
sin S P. sin Z P. cos PZ. S
Here, using the tabular sines,
j,.^
sin S P = dyujyd/120 = diameter of the diurnal circle/240, sin Z P = sin co-latitude = lambajyd/120. cos S P = sin (90° - SP) = sin dechnation of the Sun = krdntijyd/120. cos Z P = sin latitude = aksajyd/120. cos S PZ = sin S P E = sin (D PS - D PE) = sin (degrees of the taken time - degrees o f half-cara) .-. Sankuliptds (i.e. Great gnomon) = sin declination X sin latitude 120 + day-diameter X sin co-latitude X sin (degrees o f taken dme - degrees of half-cara) 28,800. = day-diameter X sin colatitude X {sin (degrees of taken d m e - d e g r e e s of half-cara)-I- sin declination X sin latitude x 240 (day-diameter X sin colatitude)} 28,800 = day-diameter X sin colat. {sin (degrees o f taken time - degrees o f half-cara) -I- sin half-cara} 4- 28,800 = rule (i) applied to S u n i n the northern hemisphere. In the same manner, the rule can be proved for the S u n i n the southern hemisphere, but here the degrees o f half-cara is first to be added (instead o f being subtracted) to the degrees of dme, and sin half-cara is to be subtracted instead o f being added, because here, S P E = D PS -I- D P E , and these changes have to be made accordingly.
P A N C A S I D D H A N T I K A
118
Rule (ii) is derived from the Great gnomon thus: T h e radius itself being the Great hypotenuse, and the Great gnomon and the Great shadow (this is 120 cos altitude or 120 sin zenith distance) are the sides o f the right angled triangle, we have: Great shadow = V 120' — Great g n o m o n ' T h e n the shadow o f the 12 digit gnomon is found by the proportion: Great gnomon: Great shadow:: 12 digit gnomon : shadow, and we get the rule (ii), shadow = 12 X V 14,400 - Great gnomon' ^ Great gnomon. In this connection, it may be noted that later authors like Bhaskaracarya II give different terms to different sections o f the work. F o r instance they call sin (degrees o f taken time + degrees of halfcara) as Sutram. Sutram ± half-cara is called by them Istantyd. T h e y call Istdntyd x day-diameter -r 240 as Istahrti. T h e n from htahriti the Sartkulipta is obtained by the proportion: 120: sin colat:: Istahriti: Sankuliptd, by the similarity o f the tksdksetras.
' y i 9 i c « w ( % ) f f t ( ^ * ) (i^r) « R i c T 3 ^ w i ^
||>JM
T i m e after sunrise 4 5 . S q u a r e the s h a d o w m e a s u r e d i n d i g i t s , a d d 144, a n d get its square root. M u l t i p l y this by the sine o f c o - l a t i t u d e a n d by this p r o d u c t d i v i d e 1,72,800. T h e q u o t i e n t is c a l l e d the T i r s t sine'. 46. N o w , m u l t i p l y the sine o f d e c l i n a t i o n o f the S u n by the sine o f latitude a n d d i v i d e b y the sine o f c o - l a t i t u d e . ( L e t us call this by its actual n a m e , the E a r t h - s i n e . ) P l a c e this E a r t h - s i n e i n two places. I n one place, subtract this f r o m o r a d d this to the 'first sine', a c c o r d i n g as the S u n is i n the n o r t h e r n o r southern hemisphere. 47. T h i s result, a n d the E a r t h - s i n e , are e a c h to be m u l t i p l i e d by 2 4 0 a n d d i v i d e d by the d a y - d i a m e t e r . T h e s e are t w o sines. F i n d the arcs o f e a c h o f these. W h e n the S u n is i n the n o r t h e r n h e m i s p h e r e a d d the two arcs. O t h e r wise subtract o n e f r o m the o t h e r . D i v i d e t h e r e s u l t by 6. T h e r e s u l t is the t i m e i n nadis after s u n r i s e . 45c.
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IV.47
IV. T H R E E
PROBLEMS
119
The following are the steps in the work to be done: (i) T h e Tirst sine' = 1,72,800 ^ (sin coladtude x V144 + shadow' (ii) T h e Earth-sine (which is to be placed i n two places) = sin ladtude X sin declination 4- sin colatitude. (iii) sine I = (Tirst sine' + Earth-sine) X 240 (iv) sine II = Earth-sine X 240
day-diameter
day-diameter.
(v) Find arc I and arc II of sin I and sin II The desired dme in nddis = arc 1/6 ± arc II/6. In (iii) and (v) the upper sign is to be taken for the Sun in the six signs from Aries, i.e. for the Sun in the northern hemisphere; otherwise the lower sign is to be taken. It must be added here, ui accordance witn what was said in the same context i n getdng the shadow from the time, that i f the Tirst sine' is less than ihe Earth-sine and therefore the Earth-sine cannot be deducted in (iii), the T i r s t sine' is to be deducted from the Earth-sine, and the result, i.e. sine I, is to taken as negative. T h e n in (v) the nddis got from this, viz. arc 1/6, are also negative, and therefore deducted from arc II/6, to get the dme. H e r e too, i f the ladtude o f the place is not too high, the reverse of the author's method i n the context can be used without any appreciable error, though this has not been mentioned here by the author. T h i s is the work to be done: Here the Earth-sine is greater than the T i r s t sine'; omit the Earth-sine and do (iii) and (v) with the Tirst sine' alone, i.e. muldply the Tirst siue' by 240, divide by the day-diameter, get the arc of this, and divide by 6 and thus to get the nddis after sunrise. The following points must be noted here. In the work of computing the nddis from the shadow, as the exact time is not known, the exact Sun and therefrom the exact declination cannot be known, and we have to use the declination of the S u n at sunrise or sunset. T h e r e may be a small error o n account of this. This can be avoided by repeating the work using the declination of the Sun for the computed time. It has not been specifically mentioned by the author because this can be inferred by the computer. Secondly, the author has given all this for places i n the northern hemisphere in the forenoon. For places i n the southern hemisphere and the afternoon, changes have to be made in the work, which have not been given by the author. It must also be noted that the ancients considered the computation of the time from the shadow or the shadow from the time as very important because this was the best means available to them of knowing the times of births and muhurtas. Example 17 (a) For a place (in the northern hemisphere) sin lat. is 60', and therefrom sin colat is 103' 55". On a particular day the sin declination is 34' 30", (the sun being in the 6 signs from Aries) and therefore the day-diameter is 229' 51". Find the time from sunrise if the shadow of the 12 digit gnomon is 13 ang 50 vyangulus. (i) T i r s t s i n e ' = 1,72,800 ^ ( 1 0 3 ' 5 5 " X VIS 14/15' -I- 144 = 1,72,800 H - ( 1 0 3 ' 5 5 " x 18.389) = 90' 26". (ii) Earth-sine = 60' X 34' 30" ^ 103' 55" = 19' 55' (iii) Sine I = (90' 26" - 19' 55") 240 ^ 229' 51" = 73' 38' (iv) Sine II = 19' 55" X 240 ^ 229' 51" = 20' 48'
120
PANCASIDDHANTIKA
IV.47
38° 3' 9° 59' (v) A r c I = 38° 3'. A r c II = 9° 59'. T h e time from sunrise in nddis = — g — + — g — = 8 nddis. (Note that this work is the inverse o f example 15 (a). There, 8 nddis were given, and the shadow 13 ang- 56 vyafig was computed. H e r e , for the shadow 13 ang 56 vyafig, the nddis amounting to 8 have been computed). Example 17 (b) For the same place, on the same day, find the time when the shadow is 137 ang 57 vyang. (i) T i r s t s i n e ' = 1,72,800 H- (103' 55" X V 144 + 137 57/60' = 12' 1". (ii) Earth-sine = 60' x 34' 30" - 103' 55" = 19' 55". (iii) Sine I = (12' 1" - 19' 55") X 240 ^ 229' 51" = - 8' 15" (iv) Sine II = 19' 55" x 240 ^ 229' 51" = 20' 48". (v) A r c I = - 3° 57', A r c II = 9° 59'. T h e time from sunrise = - 3° 57'/6 -I- 9° 59'/6 = 1 nddi. (Note that is the inverse o f Example 15 (b). T h e r e the shadow 137 ang 57 vyang was computed for one nddi from sunrise. Here for the same shadow the time one nddi is computed.) We shall do the same by the inverse operation o f the work previously given by the author: T h e T i r s t sine', computed is 12' T ' . T h e Earth-sine computed is 19'55",and greater than the T i r s t sine'. Therefore taking the T i r s t sine' alone, 12' 1" X 240 229' 51" = 12' 33". T h e arc o f this = 6° 2'. Dividing by 6, the time obtained is one nddi and 1/3 vinddi, and neglecting the negligible 1/3 vinddi, we see the same time is got. F o r proof o f the rules here given, we shall derive these from the rules for the shadow given the time, as the operation is practically the inverse o f the operation given there. In the previous work, rule (ii) gives: 12 X V 14,400 - s i n ' altitude sin altitude = shadow. .-. 144 X (14,400 - s i n ' alt.) = s i n ' alt. = shadow'. .-. 144 X 14,400 = s i n ' alt. X shadow' ± 144 s i n ' alt. = s i n ' alt. (shadow' + 144). .-. 12 X 120 = s i n a l t . x V s h a d o w ' - I - 12'. .-. 12 X 1 2 0 ^ V shadow'-H 12' _ _ _ _ _ _ = sin alt. = 12 X 120 X 120 x sin colat. -^ (120 X sin colat. X V shadow' + 12' = T i r s t sine' X sin colat. -i- 120, (because, 12 X 120 x 120 4- (sin colat x V shadow''-I- 12==) = 1,71,800 (sincolat x Vshadow^' -I- 12') = T i r s t sine' as given). Similarly, i n the previous rule (i), sin alt. = {sine (degrees of time + degrees of half-cara) ± sin half-cara} x sin colat. X day-diameter -;- 28,800, = T i r s t sine' x sin colat. 120. .-. Tirst sine' X 240 ^ day-diameter = {sin (degrees o f time + degrees o f half-cara) ± sin halfcara}. .-. Tirst sine' X 240 -r- day-diameter + sin half-caro = sin (degrees o f time + degrees o f half-cara). .-. T i r s t sine' x 240 H- day-diameter if Earth-sine x 240 4- day-diameter = sin (degrees o f time + degrees o f half-cora). .-. (Tirst sine' + earth-sine) X 240 day-diameter = sin (degrees o f time + degrees o f half-cara) = sin (degrees o f time after the S u n has touched the unmaridala)
IV. T H R E E P R O B L E M S
I V . 48
121
From the sin degrees of time, and thence by dividing by 6, the time in nddis after the Sun has touched the unmandala is obtained. T h e addition or subtraction o f the half-cara to this gives the time from sunrise, to obtain which sin half-cara is got from the Earth-sine, and then its arc, viz the degrees of half-cara.
T i m e for sunset 4 8 . O r r o u g h l y , m u l t i p l y the d u r a t i o n o f d a y t i m e i n nddis b y 6, a n d d i v i d e b y the s h a d o w i n c r e a s e d b y 12 a n d d e c r e a s e d b y the m i d d a y s h a d o w o f date. T h e t i m e f r o m s u n r i s e is got i n the f o r e n o o n , a n d the t i m e to elapse f o r sunset is o b t a i n e d i n the a f t e r n o o i i . T h e shadows mentioned here are those of the twelve-digit gnomon and not the shadows of a person measured by his foot. T h e rule is the time in nddis = 6 x daytime i n nddis
(shadow -f- 12 — mid-day shadow).
Example 18. Given the duration of daytime, nddis 33-20, and mid-day shadow, 2 ang 50 vyang. Find the time when the gnomonic shadow is 13 ang 56 vyang. T h e time = 6 X 33 1/3 ^ (13 14/15 -f- 12 - 2 5/6) = 2 0 0 - ^ 2 3 1/10 = n a & 8-37. T h e data given in the example are for the place and day i n Example 16 (a), and we must get nddis 8, as the time. But we get nddis 8-37. F r o m this we can have an idea of the roughness of this method. Evidently V M wants us to use this rule i f we feel that this accuracy is sufficient, for, this is easy to use, provided the daytime and the midday shadow are tabulated beforehand and kept ready. T h e rule may be explained i n the manner we explained the similar rule with Vdsistha Siddhdnta. Let us assume, time = x X day-time -j- (shadow — mid-day shadow -I- y), where x and y are two constants to be determined. (The daytime occurs as a multiplier i n the rule because, other things being equal, the time must vary with the daytime. For the deduction o f the mid-day shadow from the shadow, see the explanation i n the Vdsistha.) A t noon the shadow is equal to the mid-day shadow of date, and the time is daytime/2. Therefore we have: X X day time 4- (mid-day shadow - mid-day shadow + y) = daytime/2. .-. 2x X daytime = daydme x y. .-. 2x = y. Therefore, whatever be the multiplier for the daytime, twice that is the constant additive to the shadow, as in the author's rule here, 6 and 12, respectively. O n l y so far can we go i n the explanation 48. Quotedby Utpala o n B S 2, p.62 48a. A . q < ^ « < c t l C D . ^ H \ ^ b. A . o ^ f t i T H n t
c. AI.TSRsn; A2.7ransl d. A . ^ : . A.lTS^
122
PANCASIDDHANTIKA
I V . 49
whether actually the constants are 6 and 12, as here or 5 and 10, or some other number and double that, depends upon the accuracy of the result we get. For the matter of that there is another rule, very popular and attributed to our author himself in the following form: dme = 5 X daydme (shadow - mid-day shadow +10), given by the popular verse: chdyd nijestd dinamadhyabhdgacchdyonitd diksahitd taydpte \ dine saraghne gatagamyanddih srlmdn Vardho vadati syayuktyd || Here too the shadow is that of the 12 digit gnomon. Note that the muldplier here is 5, and the additive constant double that, viz. 10. Actually, different constants for different places, and for different dmes, even i f the place is the same, may have to be used i f sufficient accuracy is to be secured. So the average for a particular region may be used for that region i n the rough rule. Let us now compute the constants using the data of Example 16 (a), and examine the degree of accuracy o f the constants 5 and 10 used i n the above verse. In the example we find that the time is 8 nMis for shadow ang 13-56. T h e daytime for the day is nd. 33-20 and mid-day shadow, ang. 2-50, as we have already given i n Example 17. U s i n g the assumed form, X X 33 1/3 - (13 14/15 -f- 2x - 2 5/6) = 8. X X 33 1/3 = 8 (2 X 11 1/10) = 16 X -t- 88 4/5. 17 I / 3 x = 88 4/5. X = 88 4/5 ^ 17 1/3 = 444 x 3 - (5 x 52) = 5 8/65. As 8/65 is small, x, the multiplier, may be taken as 5, and y (i.e. 2x) may be taken as 10, with tolerable accuracy, as V M himself seems to have done i n the popular verse. Let us examine the accuracy given by this by working Example 17 using this. T h e time = 5 x 33 1/3 ^ (13 14/15 + 1 0 - 2 5/6) = 500 x 10 -^ (3 x 211) =nd. 7-54. Note how near this is to the correct 8 nddis, and contrast with the result o f the rule given by the text, nd. 8-37. Let us once again examine the relative accuracy by computing the time sought i n the example under I V . 41-44, from the shadow caused by the S u n o n the prime vertical, at the place and time of Example 16 (a). T h e prime vertical shadow was given as ang. 17-4. T h e time got there was nd. 6-53. Using the rule of the text, dme = 6 X 33 1/3 H- (17 1/15 - 2 5/6-1- 12) = nd. 7-37, which is far from the correct nd. 6-53. U s i n g the popular verse, dme = 5 X 33 1/3 (17 1/15 - 2 5 / 6 + 10) = nd. 6-53, agreeing exactly with the correct dme. What are we to conclude from this?
Shadow from time 4 9 . R o u g h l y a g a i n , the s h a d o w c a n be got t h u s f r o m the t i m e : M u l t i p l y the d a y t i m e by 6, a n d d i v i d e by the t i m e f o r w h i c h the s h a d o w is s o u g h t . A d d the
IV. 50
123
IV. T H R E E P R O B L E M S
mid-day shadow to the result and deduct 12. T h e shadow of the gnomon, caused by the Sun, is got. This means: Shadow at any time =6 X daytime 4- the time taken + midday shadow — 12. Example 19. Given daytime = nd. 33-20, mid-day shadow = ang. 2-50, find the shadow at na. 8-0 from sunrise. Shadow = 6 X 33 1/3 ^ 8 -H 2 5/6 - 12 = 25 + 2 5/6 - 12 = ang. 15-50. (Actually the shadow is ang. 13-56, which can be seen from the previous examples). But i f the constants i n the popular verse, 5 a n d 10, are used, then the rule becomes. Shadow = 5 X daytime -r- taken time -I- mid-day shadow — 10. Using this, the shadow = 5 X 33 1/3
8 +2 5/6 - 10 = 20 5/6 + 2 5 / 6 - 10 = ang. 13-40.
See how close this is to the correct, 13-56. Being the inverse of the operation of finding the time from the shadow, this rule can be derived from the previous rule, viz, 6 x daytime -r- (shadow — mid-day shadow + 12) = time in nddis. .-. 6 x daytime
time = shadow — mid-day shadow + 1 2 .
.-. shadow = 6 x daytime -r- time + mid-day shadow — 12, which is the present rule.
Moon's shadow 50. T o compute the Moon's shadow at any time in the night, the time after sunset is to be added to the nddis from moonrise to sunset if the M o o n rises in the day. If the M o o n rises after sunset, the time of moonrise after sunset is to be subtracted from the taken time. This is to be used as the time taken for computation, and work done as in the case of the Sun to get the Moon's shadow. The work is to be done thus: U p t o the desired time after sunset, the time after moonrise is to be found, and this time is to take the place o f the time after sunrise i n the work of finding the shadow as in I V . 41-44. So, for the desired time the Moon's true declination and day-diameter have to be found and these are to be used i n the place of the Sun's declination and day-diameter. T h e required cara etc. are to be found using these. A s the nddis pertain to the solar day, they should be made lunar and used. T h e two examples given hereunder will make the work clear. T h e author's intention is 49. 49a.
Quoted by Utpala on iJS 2, p.62
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A.SRIT
124
PANCASIDDHANTIKA
IV. 50
to convey that the inverse process o f finding the dme from the Moon's shadow is also to be done as from the Sun's. T h e time o f moonrise required i n this work will be given by the author in V . 8-10. T h e Moon's true declination has been given already i n I V . 16. T h e proof o f the work is similar to that o f the Sun's. It must be remembered that i n getting the time from the Moon's shadow, successive approximation has to be done, as i n the case o f the sun, for the same reason. T h e following should also be noted. If the desired dme after sunset for which the shadow is sought is less than the dme of moonrise after sunset, the work need not be done. O r i f the moon sets i n the night before the desired dme, the work need not be done. Obviously, these should be examined before commencing the work. M u c h has to be said here, for which the reader is referred to works like the Siddhdnta Siromani.
Example 20. The sine of latitude of a place is 45' 56", and thence the sine of colatitude 110' 52". There, on a particular day the daytime is nd. 32-24. The moonrise is at nd. 27-18 after sunrise. At that time the Moon's true declination is 15° south, (i.e. the Moon is in the southern hemisphere). Since Moon's declination is 31' 4", and thence the day-diameter 231' 50". The cara-vinddis from these for the day is 132. The lunar day, i.e. the duration ofmoonrise to moonrise is 62 nddis. Compute the shadow caused by the Moon at nd, 4-8 after sunset T h e time to be taken for computation = the time from moonrise to the given time = the time from moonrise to sunset -t-the given time (after sunset) = nd. 32-24 - nd. 27-18 4-8 = nd. 9-14. T h e Moon's cara-vinddis = 1 3 2 , given. Both should be converted to the lunar measure. For 62 nddis there is one lunar day, i.e. 60 lunar nddis; so for nd. 9-14, there are 9-14 X 60/62 = 8-56 lunar nddis. Converdng into degrees, we have (8-56) X 6 = 53° 36'. Similarly, the cara-vinddis made lunar = 132 X 60/62 = 128. Converted into degrees, 128/20 = 6° 24'. Now, using the rules o f verses 40-44, (i) Sin altitude = {sin (53° 36' -t- 6° 24') - sin 6° 24')} X 231' 50" x 110' 52" - 28,800 (the upper sign is taken because the M o o n is i n the southern hemisphere). = = = =
(sin 60° - sin 6° 24') x 231' 50" x 110' 52" -f- 28,800 (103' 55" - 13' 23") 231' 50" x 110' 52" 28,800 9 0 ' 32" X 231' 50" x 110' 52" 4- 28,800 80' 49".
(ii) gnomonic shadow caused by the M o o n = 12 V 14,400 - 8 0 ' 49"' ^ 8 0 ' 49" = ang. 13, vyang II.
Example 21. For the same place and the same time of Example 20, find the time, given the shadow caused by the Moon is ang. 13-11, extending the method of verse 45-47 to the Moon. T h e required elements already given i n Example 20 are: sin lat. 45' 56", sin colat. 110' 52", sin Moon's declination 3 1 ' 4", sin Moon's day-diameter 231' 50", time o f moonrise nd. 27-18 after sunrise, duration o f the day nd. 32-24, a n d the duration o f the lunar day = 62 nddis.
I V . 51
IV. T H R E E P R O B L E M S
125
(i) 'Firstsine'= 1,72,800 ^ ( 1 1 0 ' 5 2 " x V 13 11/60 + 12') = 87' 26". (ii) Earth-sine = 45' 56" X 3 1 ' 4" - M 1 0 ' 52" = 12' 52" (iii) Sine I = (87' 26" + 12' 52") x 240 -f- 231' 50" = 103' 55", (since the M o o n is i n the southern hemisphere). (iv) Sine II = 12' 52" x 240' ^ 231' 50" = 13' 23". (v) A r c sine I = 60°. A r c sin II = 6° 24'. T h e time of shadow after moonrise = (60° - 6° 24')/6 = 53° 36'/6 = na. 8-56, (Moon being i n the southern hemisphere). This time pertains to the lunar sdvana day, and converted into ordinary (i.e. solar) sdvana, the time after moonrise = 8-56 X 62 60 = nd. 9-14. T h e time from sunrise = nd. 27-18 + nd. 9-14 = nd. 36-32. T h e time from sunset = nd. 36-32 - nd. 32-24 = nd. 4-8. T h e result is correct, because in Example 20, we took this same time and got the shadow ang. 13-11, which we have used i n this example.
5 1 . F o r the others, (i.e. for the l u m i n a r i e s o t h e r t h a n the S u n a n d the M o o n , viz. the star-planets) also, d e t e r m i n i n g the c o r r e s p o n d i n g o p e r a t i o n s , a n d u s i n g t h e i r respective latitude a n d d a y - d i a m e t e r , a n d g e t t i n g the cara-nddis etc. (in terms o f t h e i r respective sdvana days), (not o n l y the w o r k o f finding the s h a d o w for the g i v e n t i m e a n d t i m e f o r the g i v e n s h a d o w as above, b u t also) t h e i r daily risings a n d settings a n d r e d u c t i o n to d i f f e r e n t localities s h o u l d be thought out a n d done. T h e following is the idea. T h e computation o f the rising and setting o f the Sun has been given already in this chapter. T h e Moon's rising and setting will be given below, in chapter V . Understanding the nature of the operation from these and taking the star-planets corrected to the different longitudes and computing their respective sdvana days and cara-vinddis, using their latitudes to get their true declinations and day-diameters, everything done i n connection with the Sun and the M o o n should be done i n connection with the star-planets also. It is from this that we understand that i n the work of computing the Moon's shadow we have to use the true declination, day-diameter, and time measured i n the Moon's sdvana day, as we have done already. Therefore this verse may also be taken as an extension of the previous verse. Here T S and N P have done a lot of emendations that are unnecessary for, without those emendations we get the same idea as they have given, at such pains.
51a.
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126
I V . 54
PANCASIDDHANTIKA
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&t|cMlcii^j,(uwi (wm:)
I I \-R
(^T^^)lot^mlTT(lrr:) g5t(Sn) ' ^ ^ ^ \\ Directions from shadow 5 2 . T w e l v e times the r a d i u s (i.e. 1440) is to be d i v i d e d by the ' S h a d o w h y p o t e n u s e ' , i.e. the r o o t o f the s u m o f the squares o f the s h a d o w a n d 12. T h i s m u l t i p l i e d by sine l a t i t u d e a n d d i v i d e d by sin co-latitude a n d d i v i d e d by sin c o - l a t i t u d e is c a l l e d Suryagra (otherwise w e l l - k n o w n as Sankvagram o r iankutalam). 5 3 . F r o m this Suryagra, the sine o f the Sun's d e c l i n a t i o n d i v i d e d by the sine o f co-latitude ( w h i c h is o t h e r w i s e c a l l e d Agra) s h o u l d be d e d u c t e d o r a d d e d , a c c o r d i n g as the S u n is i n the six signs b e g i n n i n g w i t h A r i e s , o r the six signs b e g i n n i n g w i t h L i b r a , (i.e. a c c o r d i n g as the d e c l i n a t i o n is n o r t h o r south). T h e result is to be m u l t i p l i e d by the ' S h a d o w - h y p o t e n u s e ' a n d d i v i d e d by the r a d i u s , (i.e. b y 120). 54. W h a t is o b t a i n e d are t e r m e d Koti, (or ' P e r p e n d i c u l a r ' ) , m e a s u r e d i n digits. T h e r o o t o f the square o f the Koti d e d u c t e d f r o m the square o f the s h a d o w is c a l l e d Baku (or 'Base'.) a n d the Koti is to be so c o n s t r u c t e d as to be p e r p e n d i c u l a r to the 'Base', (whose e x t e n s i o n b o t h ways is the p r i m e vertical). T h u s the d i r e c t i o n s are got. T h e following are the steps i n the work: (i) Shadow hypotenuse = V shadow' + 144 (ii) Suryagra 12 X 120 X sin latitude -r- (shadow hypotenuse X sin colat.) (iii) Agra or A m p l i t u d e = sin max. dec. of Sun x sin Sun's long, -f- sin colat. = sin dec. of Sun X 120 H- sin colat. (The declination is north if the Sun is in the six signs from Aries, and south otherwise) 52a-b. A . C . D . ^ ( C D . ^ T I ^ ) ' ^ i g ^ W * b. D . # I ( P r a i ] t ^ | d. A.-?IfPW)I 53a. A.+l^ill4>;C.D.+l8l^dl4. b. A.R
c. A.°HI^leIT^ 54a. A . ^ b. A.clSRI. A.om-J?^^ c. A L ^ I ^ H ^ ^ t ; A 2 . ^ ^ l l ? ^ d. A.ftld^^lti^^Jj"!. C.D.^OTPff
IV. T H R E E P R O B L E M S
I V . 54
127
(iv) {Suryagra + Agra X shadow hyp. 120 = 'Perpendicular' (of the upper sign is for north declination, and the lower for south. If the 'Perpendicular' got is positive then it is north, if negative, south.) (v) V shadow' - Perpendicular'* = Base Here, steps (ii), (iii) and (iv) can be simplified and put in the form: 'Perpendicular' = (12 X sine latitude + shadow hypotenuse x sin dechnation) -^ sin colat, (of + , the upper sign is for north declination and the lower for south. As already said, the Perpendicular obtained is north i f positive and south i f negative. If, when the declination is north. Shadow hypotenuse X sin declination > 12 x sin latitude, then deduct the less from the greater and take it as negative, i.e. take the resulting Perpendicular as south.) G
Base Fig. IV. 15
Example 22. The latitude of a place is 3(f, whence sin lat = 60', and sin colat = 103' 55". The Sun at the time of taking the shadow = rdsi 1-15, whence sin Sun's long = 84' 51", sin declination = 48' 48" X 84' 51", -r 120 == 34' 30", (north, as the Sun is in the first 6 signs). For this place and time if the shadow is 5 digits, find the direction of the shadow. (i) shadow hypotenuse = V 5 ' -I- 144 = 13. (ii) Silrydgrd = 12 X 120 X 60
(13 X 103' 55") = 63' 57".2
(iii) Agra = 48' 48" X 84' 51" ^ 103' 55" = 34' 30" x 120 ^ 103' 55" = 39' 51".4 (iv) 'Perpendicular' = (63' 57" - 39' 51") X 13 - i - 120 = ang. 2-36.6 (The 'Perpendicular' is north, as the result is positive) (v) T h e 'Base' = V 5 ' - (2 - 36.6)' = ang. 4-16. Or, using the simplified form, the Shadow-hypotenuse, 13 ang, being known, 'Perpendicular' = (12 X 60' - 13 X 34' 30") ^ 103' 55" = 271' 30" 103' 55" = ang. 2-36.6. T h e n the 'Base' is calculated as done above. Using the 'Base' and the 'Perpendicular', the direction o f the shadow is found thus graphically, (see Fig. 15).
128
PANCASIDDHANTIKA
IV. 54
Here A B is the 'Base' which, extended on both sides, is the prime verdcal. A C is the 'Perpendicular', extending northwards from A B that lies east-west. Angle C A B = 90". B C is the shadow, and angle A B C is the angle made by the shadow with the east-west line. T h e direction o f the sun is the line C B extended backwards, making the same angle with A B extended.
Example 23. For the same place and the same day, find the Sun's direction, when the shadow is ang. 27-30 (i) Shadow hypotenuse = V 2 7 V F T T 4 4 = ang. 30 (ii) Suryagra = 12 X 120 X 60
(30 X 103' 55") = 27' 42".8
(iii) Agra = 39' 51".4, found in example 22. (iv) 'Perpendicular' = (27' 42".8 - 39' 51".4) X 30-^120 = ang. - 3-2, i.e. ang. 3-2 southward. (Or, which is the same, deducting 27' 42".8 from 39' 5r'.4, and doing the work with the remainder 12' 8".6, the perpendicular obtained is ang. 3-2, negative and.-. southward). (v) 'Base' = V 2 7 V 2 ' - 3 1/30' = ang. 27-20. O r , by the simplified formula, 'Perpendicular' = (12 x 60' - 30 X 34' 30") ^ 103' 55" = (720' - 1035') ^ 103' 55" =-315' ^ 103'55" = ang. 3-2 southward. F r o m the 'Perpendicular' the 'Base' is found as already done. T h e direction o f the shadow is found graphically tltus:
Here, A B is the 'Base', which extended both ways, is the prime vertical. A C is the Perpendicular, directed southwards. B C is the shadow. Angle A B C is the direction o f shadow. A t an equal angle to the east-west o n the opposite side is the Sun.
Example 24. Sin lat of place = 72', whence sin colat = 96'. The longitude of the Sun = rdsi. 11-0, from which sin longitude of the Sun = 60', and thence sin decl = 24' 24", south, since the Sun is within the six sig from Libra. Find the direction when the shadow is ang. 27-30. (i) Shadow-hypotenuse = V l 4 4 -I- 27V2' = ang. 30. (ii) Silrydgrd = 12 X 120 X 72'
(96 x 30) = 3 6 ' .
IV.
54
IV. T H R E E
PROBLEMS
]
29
(iii) Agra = 48' 48" X 60 ^ 96' = 30' 30". (iv) T h e Sun being in the six signs from Libra, Terpendicular' = (36' + 30' 30") X 30 ang. 16-37.5 north.
120' =
(v) 'Base' = V271/2' - 16-V8' = ang. 21-54. O r by the simplified formula, Perpendicular = (12 x 72' + 30 X 24' 24")/96 = ang. 16-37.5. (+ is taken, as the declination is south). From this the 'Base' is calculated to be ang. 21-54 as before. The direction is graphically represented thus:
Fig. IV. 17
Here too, the angle of shadow is A B C , and the direction of the Sun is opposite to the shadow, making the same angle. We shall now prove the steps, taking them one by one: (i) Shadow-Hypotenuse: In the right angled triangle having the shadow as base and the twelve digit gnomon as perpendicular, the shadow-hypotenuse is the hypotenuse. Hence by the well-known formula, base' + perpendicular' = hypotenuse', Vshadow' + g n o m o n ' = shadowhypotenuse. As the gnomon is 12 angulas and the shadow too is measured in angulas, the shadowhypotenuse measuredmangulas = Vshadow' -I- 12'. (ii) Suryagra: This is the distance between the linejoining the rising and setting points and the diurnal circle (see Fig. 18). This is called sankvagra by the earlier Bhaskara I and his followers and sankutalam by the later Bhaskara II and others. It has been mentioned that, as seen from places on the earth other than the equator, since the circles on the stellar sphere are bent southwards (this is from the point of view of people i n the northern hemisphere) the diurnal circles following these are also bent southwards. Therefore by the intersection of the arcs on the stellar sphere and the celestial sphere several right angled triangles are formed by their sine lengths, which triangles are called 'Latitude-caused triangles' {Aksaksetras). F r o m the similarity of these triangles, when the length elements of one are known the corresponding length elements of another may be calculated by the rule of proportion. A m o n g these, two similar triangles answer to our need, in one, which is well-known, sin lat is the base, sin colat is the perpendicular, and
130
PANCASIDDHANTIKA
I V . 54
the radius is the hypotenuse; and in the other Suryagra (i.e. sankuialam) is the base, the Great gnomon is the perpendicular and what is called Taddhrti is the hypotenuse (Vide Sid. Siromani, Gola, Triprasna 49). Therefore, when sin lat, sin colat, and the Great gnomon are known SHrydgrd can be calculated by the proportion: Sin colat: sin l a t : : Great gnomon: Surydgrd. Surydgrd = Great gnomon X sin lat H- sin colat. T h e Great g n o m o n can be found from the similarity of the two triangles, i n one of which the shadow is the base, the twelve-digit gnomon is the perpendicular and the shadow-hypotenuse is the hypotenuse, and i n the other sin zenith distance is the base, the Great gnomon is the perpendicular, and the radius is the hypotenuse. Therefore by the proportion: shadow-hypotenuse : 12 :: radius : Great gnomon, the Great gnom o n = 12 x I 2 0 ' H- shadow-hypotenuse. Hence by substituting we get, Surydgrd = 12 X 120' x sin lat -J- (shadow-hypotenuse x sin colat). Since the celestial sphere is bent southward, Surydgrd is really south, permanently, (from the point of view o f a m a n i n the northern hemisphere, as we have already said). But here, as we are dealing not with the Sun but with the shadow, which is opposite to the Sun, we have taken the Suryagra as always north. W e shall illustrate these things by Fig. 18. We have mendoned that for observers i n the northern hemisphere the diurnal circles bend southward, resulting i n the 'southing' o f the celestial bodies, because o f the southward bend of the stellar sphere. As the shadow moves i n the direction opposite to the Sun, the tip of the shadow moves i n circles bent northwards, like I, II, III, i n the Fig. Also, it should be remembered, as we are depicting the shadows i n the Fig, the direction of Agrd and Surydgrd are reversed.
Fig. IV. 18.
I V . 54
IV. T H R E E P R O B L E M S
131
I: T h e circle on which the tip of the shadow moves on a day when the Sun is in the southern hemisphere. II: T h e circle on which the tip of the shadow moves on a day when the Sun is on the equator. I l l : T h e circle on which the dp of the shadow moves on a day when the Sun is in the northern hemisphere. A , B = rising and setdng points of the Sun, on the day related to I. C, D = rising and setting points o f the Sun on the day related to II, and E , F, related to III. A B , C D , and E F are the lines j o i n i n g the respeaive rising and setdng points and are parallel to one another. With reference to I, (i.e. for a day when the Sun is i n the southern hemisphere), G A = H J = Agrd (directed northward), J K = Surydgrd (directed northward) and H K = Agrd + Surydgrd, from which it is obvious that the Perpendicular is also directed northward. With reference to II, (i.e. for a day when the Sun is on the equator), the Sun rises at C itself and sets at D itself, and therefore the Agrd is zero. L M is the Surydgrd (directed northward) and the T e r pendicular' = Surydgrd +Agrd, is also L M . With reference to III, (i.e. for a day when the Sun is i n the northern hemisphere), Agrd = Q P = N O (directed southward) and P R or O S is the Surydgrd (directed north). A t a time sufficiently near sunrise or sunset, for which O S is the Surydgrd, the Perpendicular is N S (directed southward). This is the case where Agrd is deductive but numerically greater than the Surydgrd. At a time sufficiently near noon, for which P R is the Surydgrd, the Perpendicular is Q R got by P R - P Q , Q P being numerically less than PR. (iii) Agra: This is the amplitude, and forms the distance between the parallel lines constituting the prime vertical and the linejoining the rising and setdng points. This is also the sine of the angles of the rising and setting points made from the East or West points, respectively. T h e author has given the formula for this in V . 39, without mentioning its name Agrd, as also here without mendoning its name. T h e derivation o f the formula has been given by us there. W h e n the Sun is in the northern hemisphere, this is north, and when in the southern, it is south. But here, as we are dealing with the shadow, we have reversed the directions. One thing must be mentioned in this connection: T S and N P interpret the word Surydgrd as Agrd or 'Sine of the amplitude of the Sun', evidently assuming the derivation suryasya agrd = Suryagra, i.e. Agrd itself, because the context is the Sun here. As for Surydgrd itself, they simply call it 'a sine'. They have failed to nodce that i f taken thus, the formula for getdng them would become wrong. Even i f somehow, by changing the order of words i n the sentence, we make the formulae agree in this work, in the next work of getdng the sun from the direcdon of the shadow, it would be impossible to secure agreement between the words there. But we must mendon here that i n the Mahdbhdskariya, Agrd is termed Arkdgrd', evidently by the derivation, arkasya agrd arkdgrd. Surydgrd is there called Sankvagra, as we have already said. (Vide Mahdbhdskariya, III. 53-54). B u t here we have no choice except to go by the text. (iv) Perpendicular: F r o m what we have already said, and from the Fig. 18, it can readily be seen that (Surydgrd + Agrd) is the distance between the Prime vertical and the tip o f the Great shadow. This is called 'Bhuja' by other authors. T h e Bhuja corresponding to the shadow is got from this by the proportion. 11
132
IV. 56
PANCASIDDHANTIKA
Radius: 'Bhuja':: shadow-hypotenuse : shadow-Mw/a, So we have, (Surydgrd ± Agra) X shadow-hypotenuse -r- 120 = shadow Bhuja. O u r author calls this Koti or 'Perpendicular', as we have already said. B u t this does not matter, for in a right angled triangle, with the hypotenuse given (as here the shadow), the other two sides are perpendicular to each other, and any one may be taken as the base, and the other as the perpendicular. (v) Base: W h e n the 'Perpendicular' is got from the well-known formula of the right angled triangle. Base' -t- Perpendicular ' = hypotenuse', (the shadow being the hypotenuse here,) we have,'Base' = Vshadow' - Perpendicular'. Since the Perpendicular is north-south, the 'Base' is east-west, and is a part of the east-west line, as the foot o f the shadow is on the east-west line. Since the east-west line is known, we can lay the 'Base' on it, lay the 'Perpendicular', at right angle, and draw the shadow. Clearly, if initially we have the shadow marked on the ground, we can get the directions by using this method. T h e angle between the shadow and the base gives the direcdon of the shadow. Obviously the direcdon of the Sun is given by the equal angle vertically opposite. What has been said here for the shadow may be said for the sun without reversing the direction as we have done for the sake of the shadow, and the Sun's direction can be got. F r o m that the direction of the shadow may be got as being vertically opposite. B u t it is clear that the author says everything here for the shadow, and not for the Sun.
l^fgRT H1!T(^) rTT^^fq ^R^:
|m^ |
Sun from Shadow 5 5 . (Explanatory translation): By a process reverse to the previous one, the longitude of the Sun can be computed from the shadow, thus: Take the distance of the tip of the shadow in angulas, from the east-west line, multiply it by 120' , and divide by the angulas of the shadow hypotenuse (mentioned in the previous work). This is 'the sine'. (It may be seen that this is the Silrydgrd + Agra, of the previous work). If the shadow is north of the east-west line then 'the sine' also is north. If the shadow is south, 'the sine' is south. Compute the surydgrd as given already in the previous work. This is to be taken as always north (as already mentioned). If'the sine' and SHrydgrd are of different directions, then 'the sine' plus Silrydgrd is Agrd. (It must be remembered that they will be of different directions only when the Sun is in the northern hemisphere, i.e. within the six signs from Aries). If they are of the same direction, then the Agrd is one deducted from the other. (If'the sine' is greater, then the Sun is in the southern hemisphere, i.e. within the six signs from Libra. If
IV. 56
IV. T H R E E P R O B L E M S
133
Surydgrd is greater, t h e n the S u n is i n the n o r t h e r n h e m i s p h e r e , i.e. i n the six signs f r o m A r i e s ) . 56. T h e Agrd t h u s got m u h i p l i e d by the sine o f c o l a t i t u d e , a n d d i v i d e d by 4 8 ' 48". is the sine o f the S u n ' s l o n g i t u d e a n d f r o m that the s u n is o b t a i n e d . ( F r o m this sine, first the d e g r e e s o f Bhuja, D , is got. I f the S u n is i n the n o r t h e r n h e m i s p h e r e , t h e n the S u n ' s l o n g i t u d e is D , o r (six signs — D ) . I f the S u n is i n s o u t h e r n h e m i s p h e r e , the S u n ' s l o n g i t u d e is six signs + D , o r (twelve signs — D ) . W h a t e x a c d y it is o f the d i a d m u s t be d e t e r m i n e d by the S u n ' s ayana). ( F o l l o w i n g the m e t h o d f o r the S u n , the o t h e r grahas also c a n be got). The following are the steps i n the work:(i) As already seen, shadow-hypotenuse = Vshadow' + 1 4 4 . (ii) As already seen, Surydgrd = 12 x 120 X sin lat ^ (shadow-hypotenuse x sin colat). (iii) ' T h e sine' = the distance in angulas from the east-west line to the tip of the shadow X 120' -i- shadow-hypotenuse. (If the shadow is north of the east-west hne, 'the sine' is north, if the shadow is south o f the east-west line, 'the sine' is south). (iv) (a) If'the sine' is north, and greater than the Surydgrd, Agrd = 'the sine' — Suryagrd, and the Sun is i n the southern hemisphere. (b) If 'the sine' is north and less than the Surydgrd, Agrd = Suryagra — 'the sine', and the Sun is in the northern hemisphere. (c) If 'the sine' is south, Agrd = Suryagra + 'the sine', and the S u n is i n the northern hemisphere. (v) Sine longitude of the S u n = Agrd x sin colat
48' 48" = Agrd X sin colat X 5
244.
(vi) From the sine of longitude, the Bhuja degrees D , and using that the longitude of the Sun by examination, are to be obtained. As i n the previous work, (iii), (iv) and (v) can be simpUfied thus: Sine sun's longitude = (12 Xsin lat i sin colat X the distance in angulas between the tip of the shadow and the east-west line) X 150 H- (61 X shadow hypotenuse). In + if the shadow is south of the east-west line then the upper sign is to be taken, and the Sun then is i n the northern hemisphere. If the shadow is north, the lower sign is to be taken. In this case, if 12 X sin lat is greater, the Sun is i n the northern hemisphere, and if sin colat X distance in angulas, is greater, the sun is i n the southern hemisphere. Example 25. Of a certain place, sin lat = 60', sin colat = 103' 55". There, on a day during Uttardyana, when the length of the shadow is 5 angulas, the distance of the shadow tipfrom the east-west line is measured to be ang. 1-36.6, north of the east-west line. Find the longitude of the Sun. (i) Shadow-hypotenuse = V 5 ' -I- 144 = 13ang.
55a.
D.-^-rlR^iail d. A . - g ^ . A 2 . ^ 56a.
A.D.'ffl^
Al.o^RiclKll; A2.°^t
A.
b. A . W .
A.TRt*:; D . l ^ :
c.
A2.^-%T
d.
A.WSrat
134
PANCASIDDHANTIKA
IV.
56
(ii) Suryagra = 12 X 120' X 60' ^ (13 X 103' 55") = 63' 57".2 (iii) ' T h e Sine' = ang. 2-36.6 x 120 - 13 ang- = 24' 6". (This is north as shadow is north). (iv) A s 'the sine' is north, the lower sign is to be used, i.e. the difference is to be found. There, as Suryagra is greater, Agrd = 63' 57".2 - 24' 6" = 39' 51" (The Sun is i n the northern hemisphere). (v) T h e sine of Sun's longitude = 39' 51" X 103' 55" x 5
244 = 84' 51".
(vi) T h e Bhuja degrees D = A r c of 84' 51" = rdsi. 1-15. As the sun is in the northern hemisphere, the longitude isrdsi 1-15, or rdsi 6-0 — rdsi 1-15, i.e. rdsi 4-15. A s the S u n is i n Uttardyana, the longitude of the sun is rdsi 1-15. Using the simplified method, and taking the lower sign since the distance is north, sin Sun's long = (12ang-X 60'-ang-2-36.6 X 103'55") x 150-^(61 x 13a?ig.) = (720' - 271' 30") x 150
(61 x 13)
= 84' 51". (As 12 Xsin lat is greater, the sun is i n the northern hemisphere). T h e rest of the work is the same. Example 26. Of a certain place, sin lat = 60', sin colat = 103' 55". On a Daksiridyana day, when the shadow is ang. 27-30, its tip is found to be ang 3-2.15 south of the east-west line. Find the Sun. (i) Shadow-hypotenuse = V T 4 4 T 2 7 V ? = ang. 30. (ii) surydgrd = 12 X 120' x 60' -r- (30 X 103' 55") = 27' 42".8. (iii) ' T h e sine' = ang. 3-2.15 X 120
ang. 30 = 12' 8".6 (south, as the shadow is south).
(iv) A s the sine is south, the upper sign is to be taken, and the Sun is in the northern hemisphere, and therefore, Agrd = 27' 42".8 + 12' 8".6 = 39' 51". (v) Sin longitude of S u n = 39' 51" X 103' 55" X 5 ^ 244' = 84' 51". (vi) T h e degrees of Bhuja = A r c 84' 51" = rdsi 1-15 As the Sun is in the northern hemisphere, the longitude is rdsi 1-15 or rdsi 4-15. As it is Daksiridyana, the Sun is roij 4-15. A p p l y i n g the simplified method for this, as the upper sign is to be taken, since the shadow tip lies south of the east-west line, sin long = (12 X 60 + 103' 55" x 3.2) x 150 ^ (61 x 30) = 84' 51", and the Sun must be i n the northern hemisphere. T h e rest of the work is the same as done already. Example 27. Of a certain place sin lat = 72', and sin colat = 96'. There, on a certain day in Uttardyana, when the shadow is ang. 27-30, the distance of its tip from the east-west line is ang. 16-37.5 north. Find the Sun. (i) Shadow hypotenuse = Vl44 -f 27 V 2 ' = 30. (ii) Silrydgrd = 12 X 120' X 72' H- (30 x 96) = 36'. (iii) ' T h e sine' = ang. 16-37.5 X 120
ang. 30 = 66' 30", (north, as the distance is north).
I V . 56
IV. T H R E E P R O B L E M S
135
(iv) As 'the sine' is north, the difference is to be taken. A s 'the sine' is greater, Agrd = 66' 30" 36' = 30' 30", (and the Sun is i n the southern hemisphere). (v) Sine ladtude of Sun = 30' 30" x 96 x 5
244 = 60'.
(vi) T h e degrees oi Bhuja = rdsi 1 -0. A s the Sun is i n the southern hemisphere, the Sun is rdsi 6-0 -I- rdsi 1-0, i.e. rdsi 7, or rdsi 12-0 — rosi 1-0, i.e. rdsi 11. A s it is Uttardyana, the Sun's longitude must bera.Il. A p p l y i n g the simpHfled method, since the lower sign is to taken as the distance is north, sin long = (12 X 72 ~ 96 X 16-37.5) x 150 (61 x 30). Here since distance x sin colat is greater. Sin long = (96 X 1 6 - 3 7 . 5 - 12 x 72) x 1 5 0 ( 6 1 x 30) = 60', and the Sun must be i n the southern hemisphere. T h e rest o f the work is the same. T h e proof o f the above rules is as follows: I n the previous work, the 'Perpendicular', i.e. the distance of the tip of the shadow from the east-west line, was calculated, given the Sun and the shadow, and from that the 'Base' and the direction were calculated. Here, given the distance and the 'Perpendicular', the Sun is computed. Therefore this is the converse o f the previous work, and can be derived from that. Steps (i) and (ii) are the same as steps (i) and (ii) of the previous work, and have been derived there. W e shall therefore derive (iii), (iv) and (v) from (iii), (iv) and (v) there. In the previous work i n (iv), 'Perpendicular' = (Surydgrd + Agra) X shadow hypotenuse 120. .-. 'The sine = (Surydgrd +Agrd) = Perpendicular X 120 -j- Shadow hypotenuse, as i n (iii) here. Since, 'the sine' = (Surydgrd + Agrd), when the Sun is in the northern and southern hemispheres, respectively, Agrd — Surydgrd ~ 'the sine'. It has been mentioned that Surydgrd is always north, 'the sine' is either south or north according to the line to the tip of the shadow from the east-west line, and Agrd is south i f the Sun is i n the northern hemisphere and vice versa. Therefore, when Agrd is north, (i.e. when the Sun in the southern hemisphere,) 'sine' is north, and greater than Surydgrd. Therefore, in using ('the sine' — Surydgrd), we get that the Sun is i n the southern hemisphere. If Agra is south, and therefore to be got negative by the addition of Surydgrd, (i.e. when the Sun is in the northern hemisphere), and 'the sine' is north, Surydgrd is greater than 'the sine'. Here we have to use (Surydgrd — 'the sine'), and we get that when the Sun is in the northern hemisphere. If Agrd is south again, (i.e. the Sun is in the northern hemisphere, again), and 'the sine' is also south, then we have the case, Agrd = Surydgrd + 'the sine', in which case also the Sun is in the northern hemisphere. From the Agra, the sine o f Sun's longitude is got thus: In step (iii) of the previous work, Agra = M a x i m u m declination X sine longitude o f the Sun -r- sin colatitude. .-.sin long, of the Sun = Agra X sine colatitude max. dec. = Agra sin colat 48' 48", as we get here i n step (v). T h e explanation of getting the Sun's longitude from its sine has already been given i n connection with getting the sines for degrees (IV. 1-15). Another point to be noted i n this connecdon is this: I n what the author gives, there is nothing to say about the addition of 'the sine' and Surydgrd when they are of different direcdons, and therefore
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IV. 56
about the S u n being i n the northern hemisphere. B u t when they are of the same direction, and one is to be deducted from the other, the author mentions only the case where Surydgrd is to be deducted from the Sun (thereby assuming 'the sine' to be greater) the case i n which the Sun is taken to be i n the southern hemisphere. We have seen that when 'the sme' and Suryagrd are of the same direction, the Sun is to be taken as i n the northern hemisphere i n the case (iv) i n which Suryagrd is greater, and 'the sine' is deducted from it. T h e author has omitted to mention this case. Has he forgotten it? We think not. H e hopes that the reader himself will infer the changes to be made i n this condngency, viz, that 'the sine' is to be deducted from Suryagrd, and as the result is to be considered negative, and as the Agrd thus got is negative it is to be taken as south, and as south Agrd is for the Sun i n the northern hemisphere, the Sun i n the northern hemisphere will be inferred. As for T S and N P , here too they interpret Surydgrd as Agrd. T h e y are not aware of the error that would be caused by this i n the situadon of the Sun, the hemispheres being reversed. For the matter of that they do not refer to the Sun's situation at all, nor to the contingency of the reversal the subtractor and the subtrahend.
Thus ends Chapter Four entitled *Three Problems: Time, Place and Direction in the Paiicasiddhantika composed by Varahamihira
1. C o l . A.C.D.^^ROII«Mgg«?:
Chapter Five PAULISA-SIDDHANTA — MOON'S CUSPS
Introductory In this chapter the Moon's visibihty after or before hehacal setting, the appearance of its horns at the time of visibility wfith its geometrical representation, and the daily rising and setting o f the Moon with its time of reaching the meridian are dealt with. W e can surmise that this chapter is a part of the Paulisa Siddhdnta because the things required for the computations like the declination of the Sun and the M o o n with the latitude o f the M o o n , are available to us only from the Paulisa, the Romaka and the Saura having not been dealt with as yet, and because the methods are too rough to be attributed to the Saura.
T i m e of Moon's visibility 1. F i n d the d i f f e r e n c e i n l o n g i t u d e o f the S u n a n d the M o o n , as also the d i f f e r e n c e o f t h e i r decHnations, (the m e a n d e c l i n a t i o n o f the M o o n b e i n g u s e d f o r this p u r p o s e ) . M u l t i p l y the s u m o f these t w o differences by t h e i r d i f f e r e n c e ancl find the square r o o t o f the p r o d u c t . B y this 'square r o o t ' d i v i d e the p r o d u c t o f the M o o n ' s l a t i t u d e a n d the d i f f e r e n c e o f d e c l i n a t i o n a l r e a d y found. 2. T h e 'result' is to be subtracted f r o m the difference i n l o n g i t u d e , i f visibility i n the west is i n q u e s t i o n a n d the l a t i t u d e a n d ayana (i.e. c o u r s e n o r t h w a r d o r s o u t h w a r d ) o f the M o o n are o f the same d i r e c t i o n , o r a d d e d to the difference i n l o n g i t u d e i f o f o p p o s i t e d i r e c t i o n s . I f visibility i n the east is i n q u e s t i o n , reverse the s u b t r a c t i o n a n d a d d i t i o n . 1-3. Q u o t e d b y U t p a l a o n B S 4 . 1 5 . l a . A.D.31
2a. A . liiciRiiM^) b. A l . # s r a ^ ° ; A 2 . ? 5 f « ^ ^ ^ ° c. U . ^ « 2 W W I ^
138
PANCASIDDHANTIKA
V. 3
3. I n the case o f the visibility p e r t a i n i n g to the west, i f a segment e q u a l to the c o r r e c t e d d i f f e r e n c e i n l o n g i t u d e takes at least two nddis to rise i n the east as r e c k o n e d by u s i n g the ascensional d i f f e r e n c e (for the place) o f the seventh rdsi f r o m the S u n , t h e n the M o o n w i l l be visible, p r o v i d e d the sky is clear. I n the case o f visibility i n the east, use the ascensional d i f f e r e n c e o f the Sun's rdsi itself. The following are the steps in the operation :i. F i n d the difference i n the longitude of the Sun and the M o o n . i i . F i n d the difference of the Sun's declination, and the Moon's mean declination. iii. T h e square root = V( diff. in long. + diff. in dec.) X (diff. i n long. — diff. i n dec.) iv. Result = diff. in dec. X Moon's lat -4- the square root. V. Corrected diff. i n long = diff. in long + result. (Of + the upper sign is to be used i f visibility pertains to the west, and the latitude and the course of the M o o n are of the same direction, or if the visibility pertains to the east and the latitude and course of the M o o n are o f different directions. The lower sign is to be used otherwise. vi. If the visibihty pertains to the west, find the time of rising of an ecliptic segment equal to (v), by using the ascensional difference (for the place) of the seventh sign from the Sun and M o o n . If it pertains to the east, use the ascensional difference of the Sun's rdsi itself. If the time so found is greater than two nddis, the M o o n is visible; otherwise it is not visible. The time that we find i n (vi) is the time of Moonset after sunset i n the west, and the time of moonrise before sunrise in the east. T h e sun. M o o n , declinations and latitude of these times should be used and the work repeated for greater accuracy. Other siddhdntas mention this, though the author here has not done so specifically. O r , even before beginning the work, we can know the approximate times of moonset and moonrise, and do the work using the elements of these times. Near the time of new moon the M o o n is invisible because the lighted up part is very small, and the sky itself is bright by the nearness of the Sun below the horizon. It has been fixed by the ancient authors by observation, that i f the M o o n sets within two nddis after sunset, or i f the sun rises within two nddis after moonrise, the m o o n is not visible. (In places near the equator this criterion will be satisfied i f the elongadon o f the M o o n is i n the neighbourhood of twelve degrees.) It is this we are finding by the computation, and it is obvious that the nearer the time o f the elements used to the result, the better will be the result itself. Therefore is the need for successive approximation. If it is only for the sheer beauty o f its appearance i n the sky which has been described by poets like Kalidasa, the first digit o f the M o o n is fit to be sought. B u t it is necessary for religious purposes 3b. A . ° ^ l t ^ . c . A . c l f ^ d. A.C.D.U.dl+HWird I
V. P A U L I S A MOON'S GUSPS
139
also. T h e Baudhayanas have to avoid Isti being performed on the day of the first appearance of the M o o n , and do it on the previous day, and the offerings to the manes have to be done on the day previous to the Isti. T h e Dharmasastras describe the seeing of the first digit of the M o o n as meritorious. The Muslims consider their months ending with the first appearance o f the M o o n , and so this is important to them for calendrical purposes. T h e observance of the last digit of the M o o n was necessary i n ancient times, for from that they had to determine whether the same day or the next one would be the new moon day, so necessary for their religious rites. T h e importance can be guessed from the special names they had for the days at new moon, Sinivdli and Kvhu i n which the streak o f the M o o n will be visible and invisible, respectively. Example l.Ata certain place having lat. 30°N. examine the visibility of the Moon in the evening, given, the Sun at sunset = rdsi 1-0, the Moon at sunset = rdsi 1-15, and the Moon's latitude = 240' south. From the S u n and the M o o n , their respective declinations are 7 0 4 ' N and 1004'N (mean). F r o m the latitude 30°N, and Sun's declination the vinddis of ascension at the place, of Scorpio, the seventh rdsi from Sun and M o o n , can be calculated to be 355. F r o m these, i. Diff. i n long = ra. 1-15 - ra. 1-0 = ra. 0-15 = 15°. ii. Diff. i n dec. = 1004' - 704' = 300' = 5° iii. T h e square root = V(15° + 5°) X (15° - 5°) = 14° 8'.4 iv. T h e Result = 5° X 4°
14° 8'.4 = 85'.
V. Corrected diff. i n long = 15° -t- 85' = 16° 25', (the lower sign, because the work pertains to the west (evening) and the Moon's latitude is south, while its ayana is north), vi. A s the work pertains to the west, the seventh rdsi measure is to be used, which we have found to be 355 vinddis. U s i n g this, the time taken for 16° 25' to rise is 16° 25' X 355 30° = 194 vinddis. This is more than 2 nddis and so the M o o n will be visible. A s the time found is far above the requirement, we need not repeat the work using the elements of the time of moonset. Example 2. At a certain place (north of the equator) on a particular day in ttie evening the Sun rd 6-0. The Moon is rd. 6-15. The Moon's latitude is 4° 40'. The equinoctial shadow of the place is 4 digits. Examine for Moon's visibility. From the Sun, its declination is 0', and from the M o o n its mean declination is 363' S. F r o m the equinoctial shadow and the Sun's declination, the measure of the ascension of Aries, (seventh from Sun and M o o n , since the computation pertains to the west) can be calculated to be 228 vinddis. Using these, i. diff. i n long. = ra 6-15 - ra 6-0 = 15° = 900' i i . diff. i n dec. = 363' - 0' = 363'. iii. Thesquareroot = V(900' + 363') X (900' - 363') = 823'.5 iv. T h e resuh = 363' X 280' ^ 823.5 = 123'.4 V. T h e corrected diff. i n long. = 900' - 123'.4 = 12° 56'.6 (The upper sign because, the work pertains to the west, and the a^iana and latitude of the M o o n are of the same direction.) vi. A s the work refers to the west, using the measure o f Aries, the seventh rdsi from Sun and 12
140
PANCASIDDHANTIKA
V.3
M o o n , the time for a segment equal to 12° 56'.6 to rise is, 228 x 12° 56'.6 H- 30 = 99 vinddis. This is less than 2 nddis and so the M o o n will not be visible that day. A s the time got is far less than the requirement, repetition of the work is unnecessary. T h e steps are explained thus:
Here, W D is the celestial equator. S M ' C is the ecliptic and L M ' is the diurnal circle o f the M o o n projected on the ecliptic. S is the Sun, M is the M o o n and M ' is the same projected o n the echptic. M M ' is the Moon's latitude. S M ' is the difference i n longitude which is found i n step (i). W S is the Sun's declination, and D M ' is the Moon's mean declination. .-. S L is the difference of the declinations, found i n step (ii). Assuming the triangles as plane triangles, i n the right angled triangle S L M ' , L M " = S M " - S L ' = ( S M ' + SL) ( S M ' - SL) .-. L M ' = V ( S M ' + S L ) ( S M ' - SL), and^ L M ' being the square root, it is equal to V (diff. i n long. 4- diff. i n dec.) (diff. i n long - diff. i n dec.).... (step iii) M ' C is the result and it is found thus: As M M ' is perpendicular to SC, triangle M M ' C is right angled at M ' . .-. angle S M ' L = angle C M M ' Therefore the two triangles are similar. .-. M ' C / M M ' = S L / L M .
v.3
V. P A U L I S A M O O N S CUSPS
141
.-. M ' C = S L x M M ' ^ L M ' , i.e. 'the result' = difference i n declination X moon's ladtude H- 'the square root', (which is step iv). Now for the additive or subtractive nature of'the result': If the Moon's ayana is northward, i.e. i f the echptic is inclined northwards (as i n fig. 1), the M o o n having south latitude, being at the end o f a perpendicular to it, is lifted up. Therefore the M o o n projected at M ' is projected at C , as it were, and the difference i n longitude which is the distance between S and M ' , is increased. So, i n this case, 'the result' M ' C is to be added. Now consider the case, when the ayana does not change, but the latitude also is north, like the ayana, as i n Fig. 2.
South
Now, clearly the M o o n M at the end of M ' M is bent downwards, with the result that M ' C is deductive i n this case, as the instruction says. Let us next consider the case when the Moon's ayana is southward as i n Fig. 3.
Clearly i n this case the M o o n having north ladtude is lifted up, and 'the result', C M ' , is addidve, and the M o o n having south latitude is depressed, and C M ' is subtractive. T h u s we have, for ayana and latitude having identical direction, 'the result' is subtractive and having different directions it is additive. This is for visibility i n the west.
142
PANCASIDDHANTIKA
Now, for the visibility in the east: we are now looking eastward and successive points on the eclipdc are lower and lower towards the horizon. Therefore i n figs. 1,2 and 3, other things being the same, the ecliptic alone is to be represented as being directed downwards, as in Fig. 4.
Fig. V. 4
Therefore, i n each case taken up for consideration above, the direction of the ayana being changed, we see that for the ayana and latitude having different directions, 'the result' is subtracdve, and having the same direction it is additive. T h u s step (v) is explained. Now for step (vi). W e have already said that the M o o n will be visible i f it does not set within two nddis after sunset, or i f it rises before 2 nadis before sunrise. (As visibility depends actually on other factors like the keenness of the eyesight o f the observer, we have only to take the authority of the Sdstras i n this matter). So i n the evening we have to find the time by which the M o o n will set after sunset, i.e. the segment constituting the corrected difference i n longitude will set. As the distance between the rising and setting points i n always i n 6 rdsis, this time is equal to that of the rising of an equal segment i n the east, which can be calculated by using the vinadis of the ascensional difference of the rising sign, which being six rdsis away, is the seventh from the Sun (or Moon). I f this dme is greater than 2 nddis, the M o o n would not have set, and therefore be visible. In the matter of visibility in the east, the same explanation holds, except that now the time of rising of the segment i n the east is wanted, using the ascensional difference of the rising sign i n which the Sun (or Moon) itself is situated, and hence the instruction to use that sign. This instrucdon to use the ascensional difference of the same sign as the Sun i n the case of visibility in the east is implied by the use of the word vd, though not explicitly stated, and can also be inferred from the nature of the explanation. T S - N P do not seem to have noted the difference in the methods to be pursued i n the operation. A n o t h e r mistake they have made is that they have discarded the correct reading, ayandnukdlaviksipte (v. 2) and chosen the incorrect readingapamdnukdlaviksipte and accordingly, have given the condition for additiveness or subtractiveness, " I f the moon's latitude is
v.5
V.-PAULISA MOON'S
143
CUSPS
of the same direction as the difference i n declination etc." Declinauon had direcdon, but what direction can be attributed to the difference in declinadon as given i n the text? O r how can the word for declination mean difference in declination? Whatever the ladtude, 'the result' is zero at the junction of the ayanas, which means its sign, i.e. its additiveness or subtractiveness changes there, and therefore the ayana should be a criterion for additiveness or subtractiveness. T h e very name of this correction, Ayanadrkkarma (this name is not mentioned here by the author, but it is this) will suggest that the ayana of the M o o n must form part of the criterion. Another thing must be mentioned: T h e work given here is very rough, because spherical triangles are taken as plane triangles, and another correction called Aksadrkharma which is to be done for the sake of the latitude of the observer has been omitted. Therefore the reader should refer to works like the Mahabhaskariya and Siddhdnta Siromani for greater accuracy.
Diagram of the Moon's cusps 4. M u l t i p l y the l a t i t u d e o f the place i n degrees by t w o a n d d i v i d e by fifteen. B y the r e s u l t i n g n u m b e r o f angulas o r digits ( m e a s u r e d a l o n g the r i m ) , the n o r t h e r n t i p o f the h o r n o f the M o o n s h o u l d be r a i s e d u p w a r d s (as c a u s e d by the latitude at the t i m e o f first visibihty). T h i s r a i s i n g s h o u l d be d i r e c t e d u p w a r d s l i k e the 'Bhuja' w h i c h we are g o i n g to m e n t i o n . T h e n u m b e r o f digits o f i l l u m i n a t i o n o f the M o o n ' s o r b , (usually c a l l e d m e r e l y digits), is the twelfth part o f the d i f f e r e n c e i n l o n g i t u d e i n degrees, last f o u n d , a n d s h o u l d be d i r e c t e d like ' H y p o t e n u s e ' , w h i c h we are g o i n g to m e n t i o n . 5. T h e d i f f e r e n c e i n d e c h n a t i o n last f o u n d s h o u l d be a d d e d to the M o o n ' s latitude o r subtracted f r o m it, as the d i r e c t i o n s o f the M o o n ' s ayana and its l a t i t u d e be the same o r d i f f e r e n t . ( T h i s refers to the visibility i n the west in the e v e n i n g . W i t h reference to the visibility i n the east i n the m o r n i n g , the addit i o n a n d s u b t r a c t i o n , is d o n e vice versa). T h e result is c a l l e d 'Koti'. T h e difference i n l o n g i t u d e is c a l l e d 'the H y p o t e n u s e ' . T h e 'Bhuja' is the s q u a r e root o f the d i f f e r e n c e o f the squares o f the ' H y p o t e n u s e ' a n d the 'Koti'. 5a. A.STITO; C . D . s m P i R . A l . ^ ^ ; 4-7. Quoted by Utpala on £ 5 4.15. 4a. A.fe^pl^; C.U.f^H'i")^!; D.fggoiT^ b. A.-5JTg?f^Tjonf«rqf%: d. A.'+ufe^cbiSfl: D.=b"?fet(e<+>ivi:
A2.f«I^ b. A l . 2 . ^ ^ ^ ;
A 2 . ^ ^ U . ^ g ^
A.'MldKHdl; C.-^tPrai c. A.T%f5rat
144
V.7
PANCASIDDHANTIKA
6. T h e 'Koti' is to be drawn on that side of the M o o n towards the Sun, north or south, which is got in computing it, using the scale, one angula = one degree of 'Koti'. T h e Bhuja and the Hypotenuse also should be drawn to the same scale. 7. Thus, first there is the Hypotenuse from the centre of the M o o n to that of the Sun. F r o m the centre of the Sun the 'Koti' is laid in the direction computed for it. T h e n from its termination the 'Bhuja' is laid towards the Moon's centre. O n the rim of the M o o n represented by a circle of fifteen angulas, the raising of the horn in atigulas due to the latitude of the place is to be done. At the centre of the two ends of the horn the illumination in digits is to be represented on the diameter. There the arc (forming the upper boundary of the illumination) is to be drawn (by making the arc pass through the two ends of the horn and the point in the middle to which the illuminadon extends). T h o u g h it is plain that these four verses give instructions for the graphical representation of the M o o n at the times of visibility, (specifically its first visibility i n the evening i n the west), yet on account of possible incorrect copyings, and because we are not sure of the degree of roughness of the result intended by the Siddhdnta, we encounter a lot of difficulty i n ordering the words and interpretting them. T h e author has not given the diameter of the M o o n i n angulas, which is necessary to draw the orb, and represent i n it the illuminadon and the uplifting of the horn. B u t we can infer the diameter to be fifteen angulas thus: O n Astami, at the middle of either fortnight, when the hypotenuse is 90°, according to the rule for getting the illumination, we have 90/12 = 7^/^ afigulas of illuminadon. W e know that half the M o o n is illuminated then, and therefore the whole M o o n should have a diameter o f fifteen angulas, as we have stated. This agrees also with the 'elevadon of the h o r n ' due to the latitude, which can be shown thus: T h e line j o i n i n g the tips o f the h o r n seen horizontal by a person o n the equator, is seen verdcal by a person at the pole, i.e. at 90° latitude, because the celestial equator is inclined by 90° there, so as to be coincident with the horizon. As the hypotenuse at the dme for which the elevation is required is small, we can take it that the elevation o f the hord is proportionate to the degrees of latitude. According to the rule for elevation given by the author, it is for 90° and 9 0 x 2 - ^ 1 5 = 1 2 angulas, along the r i m of the quadrant, from the horizontal to the vertical. Therefore the whole r i m , i.e. the circumference, is 4 X 12 = angulas and this shows that the diameter must be 48 X 7/22 = fifteen angulas very nearly. This agreement i n the diameter, as calculated by the two rules, itself is a criterion for the correctness o f the rules. 7c. A . o ^ ^ P I : ; C . D . U . " ^ ^ d. D.J^fi<^M«qRro 6b. A.5R|3qiqfig#qa^:
A . c R ^ ^ ; D.^Rg [^]
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V. PAULISA M O O N S CUSPS
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Now, we shall show why this elevation is always on the northern limb. As mentioned several dmes before, when latitude is used i n the rules given, it is always north latitude that the author means. As seen from north latitudes, the circles o n the stellar sphere are all bent towards the south above the horizon. Therefore the hypotenuse also is inclined south, the angle of inclination being equal to the latitude, the hypotenuse being small and taken as a straight line. By this inclination south, the linejoining the tips of the horns, which is perpendicular to the Hypotenuse is elevated in the north and depressed i n the south, the angle of elevation being equal to the latitude. T h i s elevation, measured on the r i m in angulas is, as we have shown, twice the latitude divided by fifteen. In the matter of the addition or subtraction o f the difference o f declination and the Moon's latitude, we have said that the author has i n view only the visibility i n the west i n the evening, for then alone is the statement correct. Perhaps the author thinks that this is enough, because the elevation of the horn at evening appearance alone is observed anxiously by people, as an omen of good or evil. O r the author thinks that the readers themselves will understand the reversal of addition and subtraction for the m o r n i n g appearance, by analogy with what was done before i n the case of visibility. It must also be noted that the object here is only to represent the orb of the M o o n as it appears, and the Hypotenuse, Bhuja and Koti are given to serve this end. Therefore it would not matter i f these are represented on a different scale from that on which the M o o n is given, as for instance an angula per degree here. (On this scale the M o o n will have to be represented by a diameter of a \\a\i-angula.) There is a view that the elevation of the horns should be observed when the orb of the M o o n is on the horizon. In that case, the Sun will be below the horizon, and the question of the difference in scale will not arise at all. So, these are the steps in the work:i. T h e elevation of the horn due to latitude in angulas = latitude in degrees x 2
15.
ii. Illumination or digit of illumination in angulas = the difference in longitude in degrees -^12. iii. Koti in angulas = d i f f in declination i n degrees ± latitude in degrees. (For evening i n the west, if the Moon's latitude and ayana are of like direction, addition, and i f o f different directions, subtraction. F o r m o r n i n g i n the east, reverse the addition and subtraction). iv. Hypotenuse = angulas equal i n number to difference i n longitude. V. Bhuja in angulas = V Hypotenuse' — K o t i ' . vi. See fig. 5, below. O n the surface on which the phenomenon is to be represented draw a horizontal Hne and mark the north and south sides o n both ends. M a r k the point S on it to represent the Sun. M a r k a point A on the horizontal line on the side i n which the M o o n is situated, (this is known when finding the Koti) such that S A = Koti. F r o m A draw a perpendicular upwards equal to the Bhuja and at the end mark M , the centre o f the M o o n . M S is the Hypotenuse. W i t h M as centre draw the orb o f the M o o n having a diameter o f 15 angulas. A t M draw a diameter B C perpendicular to the Hypotenuse. F r o m the northern end the diameter, say C , measure the angulas of elevation due to the latitude of the place, along the r i m , and mark the point D . Draw the diameter D M E . D and E are the tips of the horns. O n the lower semicircle caused by D E , mark its mid-point, F. Draw the radius F M . O n this mark a point G , such that F G = the angulas of illumination. Draw the arc D G E by the well-known method o f making a circle pass through 3 points. T h i s is the upper limit of the illumination. T h e figure o f the M o o n is now as it will be seen i n the sky. T h e horizon is between the Sun and the M o o n , parallel to the original horizontal line. It must be remembered that
146
V.7
PANCASIDDHANTIKA
what the Siddhdnta gives is only approximate, though easy to do, and for greater accuracy, we have to do a lot o f work like calculating the Great gnomons of the M o o n and the Sun etc. Example 3. Represent graphically the Moon ofexample 1. There, we are given, latitude ofthe place = 30°N, the Moon's ayana is northward, and its latitude 4°S, and we get the difference in longitude = 13°, and the difference in declination = 3°.
S
A Fig. V. 5
F r o m the data given above: i. T h e angulas of elevation due to the latitude of the plane = 30 X 2 i i . Angulas o f illumination = 15
15 = 4.
1 2 = IV4.
iii. Koti = 5° — 4° = 1°, and .-. 1 afig., the Sun being to the south, (because it is evening observation, and Moon's ayana and lat. are of diff. direction). iv. Hypotenuse = 15 angulas. V. Bhuja = V 225-1 — nearly 15 angulas. vi. Representation: Fig. 5: (Scale 1' unit = dafig.) It should be remembered that the fig. is intended only for the appearance of the M o o n , with the illumination, and elevation of the horns represented on it, and none else. T h e line D G E F D is the part illuminated, D and E being the tips of the horns. Actually the Sun is down, on the line M F .
v.9
147
V. PAULINA M O O N ' S CUSPS
Now for the readings: A s the elevation due to the latitude of the place is considerable, it cannot be neglected and must be represented; therefore we have corrected dviguneche tithyamsa into dvigune'kse tithyamsa, by changingc/ia intoksa. B u t T S have adopted the readingdvigunecchdtithyamsa and considering it a combination of dvigunecchd and atithyamsa thinking that the subject matter is astrological, (which is obviously unlikely). We have corrected paridhavaksondmnh into paridhdvaksonndmah, for the instruction to apply the elevation due to the latitude must be given. But T S and N P take the reading as it is, and say that something on the r i m o f the M o o n is named aksa, which is purposeless. T h e i r readings themselves in these two cases are from their own edition of Bhattotplala's commentary on the Brhatsarmhitd, and to say that their (TS's) readings agree with those of the Bhattotpala may be improper, for probably they have themselves put the readings there.
d^^^lchlc^H i ^ g r ^ ^ y i V I I ^ < 4 )
W3l:
Daily rising and setting of the M o o n 8. Multiply the Moon's latitude in degrees by the equinoctial shadow and divide by twelve. A d d the resulting degrees to the longitude of the Moon, or subtract from it, according as the Moon's latitude is south or north, if the times of daily moonrise is to be computed. If the times of daily moonset is to be found, reverse the addition and subtraction, i.e. subtract and add, respectively. 9. Subtract the longitude of the Sun from that of the M o o n corrected thus. Find the time for this segment of the ecliptic to rise, after sunrise. By so much dme after sunrise, the M o o n will rise. If this segment is less than sixrdsis, then the moonrise will fall in the day-time, if greater, the moon will rise at night.
8-10. Quoted by Utpala o n B S 4.15. 8b. A.°il3^^5qM^ c.fls^cqrai^
9a-b. A.-< h. C . i s t S ^ : ; D . t % n . A . ^ s f ^ ^ c.
A.-^^ B1.2.3. Commence again from ° ^ f ^ after the big gap which
A.
commenced at IV.20. d. A . M . B3.i^li
148
PANCASIDDHANTIKA
V . 10
10. I n the m a n n e r g i v e n ( i n verse 8), correct the M o o n f o r moonset, d e d u c t the S u n f r o m this c o r r e c t e d M o o n , a n d d e d u c t 6 rdsis f r o m the r e m a i n d e r . F i n d the t i m e by w h i c h the r e m a i n i n g segment w i l l rise, after sunrise. T h i s is the t i m e f r o m sunrise w h e n the M o o n w i l l set. A t the t i m e exactly m i d w a y betw e e n m o o n r i s e a n d m o o n s e t , the M o o n w i l l r e a c h the m e r i d i a n , (i.e. w i l l be at upper culmination). The following is the work to be done: i. T h e correction (for latitude) = the Moon's latitude X the equinoctial shadow known as Aksadrkkarma).
12 (This is
ii. This correction is to be applied to the true Moon. Corrected Moon = True Moon ± Correction. (If the dme of moonrise is to be found, then the correction is subtractive if the Moon's latitude is north, and additive if it is south. If moonset is wanted, if the Moon's ladtude is south, the correction is subtracdve, i f north it is addidve). iii. T h e dme of moonrise is found thus: T h e corrected M o o n - T r u e Sun = elongation. T h e dme of rise of the segment of elongation from sunrise is the time of moonrise. (In other words, the corrected Moon's position on the ecliptic being known, the time when that point rises is the time of moonrise). W h e n the elongation is less than 6 rdsis, moonrise is i n the day-dme, otherwise at night. iv. T h e dme of moonset is found thus: Corrected M o o n - Sun = elongation. T h e time of rise of (the segment of elongation — 6 rdsis) from sunrise, is the time o f moonset. (In other words, the time of rising of the point diametrically opposite to corrected M o o n is the dme o f moonset). Here, if the elongation is less than 6 rdsis, then the moonset is i n the night, and i f greater, it is i n the daytime. V. Moonrise to moonset is the moon-day-time. It is obvious that at the middle of its day time it is on the meridian. It is obvious that the times o f rising and setting will be correct i f the longitudes and Moon's ladtude of those times are used. B u t as the computadon as done here is only approximate, we can guess the approximate dmes of moonrise and moonset for the day from the tithi of the day, and use the elements o f those dmes, to get tolerably accurate times. Sample 4. The equinoctial shadowfor a certain place (in the northern hemisphere)is4 angulas. The ascensional differences for the place are for Aries 236 vinddis, Taurus 265, Gemini 309, Cancer 337, Leo 333, lOa.
A.S^;
E.Tf^.
A . B . ^
b. A . B . C . D . U.5!I*; ( B 2 . 5 # ) ^ f t W l e T
d. B 3 . ^ and C . D . t ¥ ^ for'?lfiFr A.B.U.I^'TOI^. D.Re|^6^
c.
A . B . T l f W Z l : ; C . m\\
V . 10
V. PAULISA M O O N S CUSPS
149
Virgo 320, Libra 320, Scorpio 330, Sagittarius 337, Capricorn 309, Aquarius 265, and Pisces 236. There, on a certain day, the true longitude of the Moon at sunrise isra, 1-18, the true Sunisrd. 10-3, the Sun's daily motion is 60', the Moon's daily motion is 840', the latitude of the Moon is 272' S, and its motion per day 8' S. Find the moonrise, moonset and upper culmination. T h e distance of the M o o n from the Sun = rd. 1-18 - ra. 10-3 = ra. 3-15, (equal to 8 % From this the approximate dme of moonrise is, 8 % x 2 = 17 '/2 nddis. Therefore the dme o f moonset is approximately, 17 '/2 -H 31 = 4 8 V2 nddis. T h e Sun at approx. moonrise is ra. 10-3-18, the moon rd. 1-22-5, and its latitude 274' S. A t approximate moonset, the sun is ra. 10-3-49, the M o o n rd. 1-29-19, audits latitude 278' S. Using each set, the computation is as follows: i. T h e correction for moonrise = 274' X 4
12 = 9 1 ' .
T h e correction for moonset = 2 7 8 ' x 4 - ^ 12 = 93'. ii. T h e corrected M o o n for moonrise = ra. 1-22-5 -(- 9 1 ' = ra. 1-23-36. T h e corrected moon for moonset = ra. 1-29-19 - 93' = ra. 1-27-46. iii. C o m p u t i n g Moonrise: Elongation = Corrected M o o n - Sun = ra. 1-23-36 - ra. 10-3-18 = ra. 3-20-18. As this is less than six rdsis, the moonrise is in the day-time. T h e time for the segment, ra. 3-20-18, to rise after sunrise is found thus: For the rest of Aquarius, which is the sign occupied by the Sun, to rise, the time taken is 265 x 1602' 1800' = 236 vinddis. For Pisces to rise, 236, for Aries 236, for the corrected M o o n to rise i n Taurus, 265 X 1416' 1800' = 208 vinddis. So the total time taken is, 236 -I- 236 -I- 236 + 208 = nddis 15-16. This is the time of moonrise. iv. Moonset: Elongation = Corrected M o o n — Sun = rd. 1-27-46 - rd. 10-3-49 = ra. 3-23-57. Deducting six rdsis from this, we have rd. 9-23-57. T h e time for the rise of this much segment is found thus: For the rest of Aquarius to rise, the time taken is 265 x 1571' 1800' = 231 vinddis. For Pisces 236, Aries 236, Taurus 265, Gemini 309, Cancer 337, Leo 333, V i r g o 320 and Libra 320. For Scorpio to rise upto the point diametrically opposite to the corrected M o o n , 333 X 1666' 1800' = 308. A d d i n g up, the time of moonset is na. 48-15. This agrees with what we can infer from elongation, for the elongation found is less than 6 rdsis, and the moonset must be in the night. v. T h e duration of the lunar day is na. 48-15 — nd. 15-16 = nd. 32-59. H a l f this is nd. 16-30. A d d i n g this to moonrise, mid-moon-day, the time of upper culmination of the M o o n is na. 15-16 -I- nd. 16-30 = na 31-46, after sunrise. T h e instruction is thus explained: T h e problem is to find the time of rising or setting of the M o o n , which is i n its orbit, at a distance equal to its latitude from the ecliptic. If it can be projected
150
PANCASIDDHANTIKA
V . 10
on the ecliptic in such a way that its projected position rises and sets at the same time as it itself rises and sets, then the time can be found like lagma, by the method given in Chap. I V for that purpose. In order to effect the said projection two corrections have to be applied to the M o o n , one for the inclination of the ecliptic called Ayana-drkkarma (we d i d this for visibility), and the other for the latitude of the observer called Aksa-drkkarma. T h e Siddhdnta gives the correction for latitude alone here, which is got by multiplying the latitude of the M o o n by the equinoctial shadow and dividing by 12. T h e following is its rationals: T h e latitude is measured on the great circle perpendicular to the ecliptic and directed towards the pole of the ecliptic. If the latitude is projected on secondaries to the pole, and we get the true declination of the M o o n by adding this to the mean declination, we can find its time of rising and setting directly, as we find the rising and setting of the Sun, by computing its cara etc. and getting its own ascensional differences. But if the latitude is small, it can roughly be taken as the correction for the Moon's mean declination, given by its longitudes. So the correction to the vinddis of true cara can be found by the latitude taken as part of the declination, by proportion from the cara-vinddis for the mean declination already used in finding the ascensional differences. Therefore, as in getting the cara-vinddis, here too we have to multiply by the equinoctial shadow and divide by 12. But the division by the diurnal radius is not done, because here we are not finding actually the vinddis of cara, but an element of the ecliptic corresponding to the cara, for the sake of which we have to multiply again by the radius of the diurnal circle, and the two cancel out. T h u s the correction will permit us to consider the M o o n to be on the ecliptic. We shall now consider when it is additive, and when subtractive. In the northern hemisphere, the Unmandala is elevated above the horizon, the elevation increasing towards the north. Therefore if the Moon is a litde to the south of the ecliptic on account of its south latitude, it rises later. As successive points on the ecliptic rise later and later, the correction got from the Moon's south latitude is equivalent to an increase in the Moon's longitude, and so the correction to the longitude is additive. From this we can see why the correction is subtractive if the Moon's latitude is north. As for the time of the setting of the M o o n , the further north a body is, the later it sets, and therefore the correction is additive i f the latitude is north, from which we see it is subtractive if the latitude is south. T h e Siddhdnta has in view only observers in the northern hemisphere, as we have already said. We ha\e already drawn the attention of the reader to the omission of the correction due to the inclination of the ecliptic (Ayana-drkkarma). It may be that the author expects us to make this correction also, taking the hint from the computation of the heliacal rising of the M o o n (visibility). T h e part of the instruction to find the time when the corrected M o o n rises is explained thus: T h e corrected M o o n minus Sun is the segment of the ecliptic between them, and the time taken for its rise after sunrise is the time of the rise of the corrected M o o n itself. If this segment is less than six rdsis, the M o o n must rise in the day-time, because just after the rise of six rdsis from sunrise the sun sets, but this segment is less. Clearly, i f it is more than six rdsis the Sun has set, and it is night when the M o o n rises. As for moonset, the point of the corrected Moonplus (or minus) six rdsis rises at that time. So, the time of its rise, or which is the same, the time taken by the corrected M o o n mimis Sun ± six rdsis, after sunrise is the time. O f ± , the author has chosen minus, because the effect of both is the same. By analogy with mid-day Sun, the M o o n is on the meridian at the middle of its day-time, provided its motion and change of declination is tolerably uniform. We must add here that it would have been sufficient i f the author had said, 'Treat the corrected moon as Lagna, and its time of rise is the time of moonrise. Treat the corrected Moon plus (or minus) six rd.sis as Lagna, and its time of rise is the time of moonset'.
V . 10 •
V. P A U L I S A M O O N S
CUSPS
151
Now for the readings: In verse ten if the meaning is taken as it is, then we shall be getting the time of sunset after moonrise, which serves no purpose and cannot be the intention of the author to get, and it is incompatible with the tiine of the meridian M o o n sought to be found in the fourth foot. This is the middle of the Moon's day-time and for this, moonset has to be found. (TS and N P too interpret this verse as giving moonset). Therefore we have corrected vyarkam candram visodhya cakrdrdhdt into vyarkdt candrdt visodhya cakrdrdham, by interchanging the case endings, and thus got the time of moonset, required for meridian M o o n . T h e reading mesodayakdla ior sesodayakdla has been discarded as being unconnected with the problem. T S and N P have given the impossible correction, nisi divase'stam sasi ydti for sasidivasdrdhe sashnadhya found in the manuscripts. T h e i r aim, viz. to get the time of moonset, is all right, but their intei pretadon of the stanza to get this is wrong, and also self-contradictory. See the Sanskrit commentary for the said interpretation: 'Deduct the Moonminus Sun from 6 raiw; the time of the rising of this is the time of moonset, reckoned from sunrise. Here, if the M o o n sets in the day-time then Moon-minus-Sun must be deducted from 6 rdsis. If in the night, the M o o n itself is to be deducted from 6 rdsis. This order of procedure should be understood.' If their instruction in the first sentence is followed, the dme of sunset after moonrise will be got, as we have already said, but not the time of moonset. It is to avoid this that we interchanged the case endings. As for the instruction in the second sentence, the first part of it disagrees with the second part. We shall illustrate these defects found in their interpretation, by applying them to two examples. (a) T h e Sun is rd. 0-15, M o o n is rd. 2-0, the Moon's latitude is zero, i.e. there is no corrections. In this case, the moonset according to T S is to be found thus: M o o n — Sun = rd. 2-0 — rd. 0-15 = rd. 1-15. Deducting this from 6rasw, the remainder is rd. 4-15. T h e y say, the time taken for rd. 4-15 to rise, after sunrise, is the time of moonset after sunrise. T h e absurdity of this can be seen by finding the time of moonrise, which is the time taken by the Moon-Sun to rise, i.e. for rd. 1-15 to rise; i.e. the interval between moonrise and moonset is the rising time of 3 rdsis. O r , by the instrucdon in the second sentence, as the M o o n does not set in the day, it sets in the night, and therefore deducting the M o o n itself from 6 rdsis, we getrd. 4-0, and they say by the time of rise of rd. 4-0 from sunrise the Moon sets. Does it occur in the night at all? Perhaps they meant sunset. T h e n , let us take another case. (b) T h e Sun is rd. 0-15, the M o o n is rd. 8-0, and Moon's latitude is 0, again. T h e n , M o o n — Sun = rd. 7-15. The moonset is in the day-time, clearly. Therefore deducting this from 6 rdsis, we have rd. 6-0 — rd. 7-15 = rd. 10-15. According to them the M o o n sets by theriseof this segment after sunrise. Clearly according to this the moonset will fall in the night, and not in the day-time as required. Assuming sunrise is a mistake for sunset, reckoned from sunset also it will be wrong, for then the moonset will be rd. 4-15 from sunrise, which is wrong, for it is correctly the time of rising of rd. 1-15 from sunrise. This demonstration shows their interpretadon to be wrong, and at the same time justifies our interchanging the case-endings, by which alone the dme of moonset can be got correctly.
Thus ends Chapter Five entitled Paulisa-Siddhanta: Moon's Cusps
Chapter Six (VASISTHA-) PAULISA-SIDDHANTA: LUNAR ECLIPSE
Introductory This chapter deals with the lunar eclipse. Nothing is given in the colophon at the end of the chapter about the Siddhdnta to which this belongs. T h i s cannot belong to the Saura for the lunar eclipse of the Saura is dealt with in Chapter X . T h e Sun, M o o n and Rahu of the Romaka are given i n Chapter V I I I , and in the same chapter the solar eclipse according to that Siddhdnta occurs, and its lunar eclipse cannot be given here, earlier. Also, the method here does not have the refinement of the Romaka solar ecUpse. So this chapter cannot belong to the Romaka. That it may belong to the Paitdmaka is out o f question, since only the mean Sun and the M o o n , and that very crudely, being given by the Paitdmaha, and R a h u is not given. Also, the Siddhdnta occupies a later chapter, the twelfth. T h i s leaves the Vdsistha and the Paulisa for consideradon. Perhaps it belongs to both combined, as we have observed i n the case o f their M o o n and its daily motion. It cannot belong to the Vdsistha separately for the Vasistha does not give R a h u , which we have to get from the Paulisa. Also, it cannot belong to Paulisa separately, for then at least part of the computation, Hke the duration of the eclipse will become redundant, because the duration of the lunar eclipse with its limits occurs in chapter V I I also, together with the computation of the solar eclipse, which from the colophon and from the nature of the method given, must belong to the Paulisa. Also, details usually given i n connecdon with ecHpses, Hke the direction o f contacts, colour etc. are found only here i n the V I chapter. Therefore we can conclude that chapters V I and V I I belong both to the Vdsistha and the Paulisa, and that the solar eclipse i n the V I I chapter belongs to the Paulisa.
Sun and M o o n of equal longitude 1. Minutes of arc equal to the na4is of the full moon-tithi to go, after sunset, are to be added to the Sun, (which has been computed for sunset). Minutes of arc equal to the nddis to go from the end of the full moon or new moon-tithi in the day-time upto sunset are to be so added to the Sun. Thus corrected, the Sun becomes equal to the M o o n in (degrees and) minutes at the end of the full or new moon-tithi, (i.e. at full or new moon). T h e idea is that by thus finding the Sun, we can, without any trouble, get the M o o n , for, i f new moon, the Sun thus got is the M o o n and, i f full moon, the Sun plus 6 rdsis is the M o o n .
VI. 1
VI. VAS.-PAUL. S I D D H A N T A - L U N A R ECLIPSE
153
T h e following is the rationale of the work: T h e lunar edipse occurs at the end of full moon-tithi. A t that time, the Sun and the M o o n are separated from each other exactly by 6 rdsis. Therefore the degrees and minutes or, which is the same, the total minutes left over after finding the rdsis at 1800 minutes a rdsi, are the same for both. Therefore they are called sama-liptas, i.e. 'having equal minutes'. T h e solar eclipse is at the end of the new moon-tithi, at which time they are the same even in rdsi, not to speak of the degrees and minutes, and therefore samaliptas. So, if we know the Sun at these times, we know the M o o n , for, if new moon, they are the same, and i f full moon, different by 6 rdsis. Now, in the Paulisa the 'days from epoch' are found for sunset, and from them the Sun and the M o o n are found for sunset first. (We have already drawn the attention of the reader to this, while commenting on III. 15.) T h e n , the ending moment of the tithi is calculated by using the difference of their motions. T h e Sun's motion is roughly one minute of arc per nddi. Therefore i f one minute per nddi of the time from sunset to full moon (we take only the full moon because with new moon at night there will be no solar eclipse) is added to the sunset Sun, the Sun at full moon is got, and the Moon is got from it by adding 6 rdsis. Thus the M o o n is easily got, for otherwise we must calculate the Moon's motion d u r i n g the interval by propordon from its daily motion, add this to the sunset M o o n , and get the M o o n . In the case when new or full moon-tithi ends i n the day-dme, it is obvious that the minutes of arc accrued d u r i n g the interval up to sunset should be deducted from the sunset Sun, to get the M o o n , as a preliminary to computing either the solar eclipse or the lunar eclipse. T h e Siddhdnta is content with thus getting the Moon roughly, for that will be sufficient considering the crudeness of its method of computation. If greater accuracy is desired, we must multiply the difference of the Sun's daily motion from 60 minutes o f arc by the dme to go or dme gone, and, taking the product as seconds of arc, subtract or add them, respectively, to the M o o n i f the daily motion is less, and add or subtract respectively if greater. T h e daily motion required for this is given in III. 17, which we have already seen. T S have understood that the M o o n at new or full moon is found here, but not the manner in which it is done so simply, for they interpret the instruction to mean that the M o o n is to be got from its daily motion by proportion. T o obtain this meaning, they make wild emendations of the words. But we have kept the words mostly as they are, and we can see that they are sufficient to give the correct idea. For example, in all the three readings, naisydh, naisndh and vaisndh (snaisndh) there is 'nai' which therefore must have been in the original word. Therefore, by changing 5 to .v, we get naisydh, meaning 'belonging to the night', which so well agrees with the idea. We have changed candram into cdndra for the sake of syntax and agreement with the idea. Between vi and va, we have introduced vi, thus reading ravi-vivardt, which is a likely haplographical omission, and get a word that fits so well with the idea. T a k i n g the meaningless reading, nrvasudbhavdcca, and keeping as far as possible to the letters there, we have reconstructed the form as divasodbhavdcca, fitting in with the
la. A . B 2 . D . V l I : ; B l . 3 . ^ : ; B 3 . ^ : ; C . ^ ^ i l ^ A . B . C . D . Hl*il<+;f b. A . ^ ^ ; B . ^ g ^ ; (B2.°a=j^) C . ^ ^ d W c ^ d l ^ : ^ ^ ? ! : I D.<^M*IA . ^ % d ^ ; B.-Mi)-^
c. A.^g^P^-?Ttsqi; B.'5S3-^cna6I:-?tl«zn:; C . See above; D.WS^gf?Th2ir: d. A . dc+MJ^lRflftHyi^ ^\M^^: I D.?mlf^:
154
PANCASIDDHANTIKA
VI.2
idea. N P insert vi to get the reading ravi-vivardt but leave the other errors untouched or making unwarranted emendations.
Probability of an eclipse 2. D e d u c t o n e degree a n d thirty-six m i n u t e s f r o m R a h u ' s H e a d o r T a i l ( w h i c h e v e r is n e a r the M o o n ) a n d find the i n t e r v a l i n degrees between that a n d the M o o n (at f u l l m o o n f o u n d above). I f it is less t h a n t h i r t e e n , a l u n a r eclipse w i l l o c c u r t h e n . I f it is less t h a n fifteen, ( a n d above thirteen), there w i l l o n l y be a slight d a r k e n i n g . The following is the explanation: A t the moment when the distance between the centres of the M o o n and the Shadow circle is equal to the sum of their semi-diameters there is the first contact or the last contact of the eclipse, because the rims just touch each other then, see Fig. l a , below.
Fig, VI. l-a
The centre of the Shadow is always six rdsis distant from the Sun. A t full moon (i.e. the end of the tithi) the M o o n projected on the ecliptic (i.e. the longitude of the Moon) is 6 rdsis distant from the Sun, as we have already said. Therefore the centre of the Shadow also is there. But the actual M o o n
2a. B . ' e ^ S ^ l ^ b. A 2 . f H ^ ( A l . ^ ^ ) ; B.IBmi^ld8i!i!l*
(B2.sim, B 3 . ^ ° ) d. A.B.°
VI.2
VI. V A S . - P A U L . S I D D H A N T A - L U N A R ECLIPSE
155
is on its orbit, at a distance equal to its ladtude. Therefore only when the latitude is equal to the sum of the semi-diameters, is there at least a grazing of the rims'. (See Fig. lb)
As according to the Paulisa the sum of the semi-diameters is always 55', (this will be shown later), and as this much latitude can be got only i f M o o n ~ R a h u is 13°, and not m o r e ' there can be no eclipse, i f Moon ~ Rahu is greater than 13°. As for the little darkening from 13° to 15°, it is due to the M o o n entering the penumbra alone and getdng out, instead o f entering the umbra. It is well-known that i f the source o f light is not a point, there is a region not so dark round the shadow, which is darker and .darker as the shadow is approached, and becomes sufficient to be seen. T h i s Siddhdnta has taken this region to be about 8', round the shadow. Therefore, the Moon's orb will touch this region at lull moon i f the ladtude is 63', and for this its distance from R a h u must be 15', as given here. As for deducUng r 3 6 ' from Rahu, the author has found this is necessary by observaUon, and we have to accept it, as agreement with observation is necessary, otherwise people will lose faith in the Sdstra. O r the Paulisa Siddhdnta itself gives this correction for agreement with observation, for, in the phenomena intended to be seen, such correction is the practice of the writers of this Sdstra. But it may be asked how this need for correction arises at all. This implies that either the longitude of Rahu or that of the M o o n is incorrect. We showed in ch.III that at epoch Rahu-head was 235° 59' 1. What we have said here is a little inexact and taken as such by most of the ancient authors. Actually, since the Moon's orbit is inclined 'o the ecliptic by about 5°, the minimum distance between the Moon and the Shadow, given by SM', the perpendicular on the orbit from S, is a little less than SM, and it is only when SM' is equal to the semi-diameters that the grazing occurs. Therefore even if the latitude at full moon is a little greater, an eclipse can occur, but this has been neglected as being very small, actually less than a quarter of a minute of arc. 2. As explained by us under III.31, taking 380' as the maximum latitude (i.e. for 90° distance), and taking the latitude as proportionate to the Bhuja of Moon-Rahu, as given by the siddhdnta there, we gel 380' X 13° 90° = 55', for Moon ~ Rahu equal to 13°. This agreement here is the proof of the correctness of what we said above in the explanation that the latitude is proportionate to the degrees of Bhuja. If we take the maximum latitude to be 280', as given by the reading of the text there, or to be 270' as TS have taken there without assigning any reason, neither taking the latitude as proportionate to the degrees of Moon ~ Rahu, nor correctly as proportionate to sin (Moon — Rahu) will give 55'. This is the reason why TS themselves have, in this section, in verse 5, abandoned both 280' and 270', and taken 240' as the maximum latitude, (vide their Sanskrit and English explanations under VI.5.). U
156
PANCASIDDHANTIKA
VI.4
according to the Paulisa and this agrees beautifully with its position than according to modern astronomy, 236°, and tolerably well with those of other Siddhdntas. Therefore the incorrectness must be i n the M o o n , and as much error i n the tithi is unlikely, nor i n the Sun as well. O n examinadon we find it is indeed so; we find that at the period of the author the Sun and the M o o n of Paulisa were less by about a degree and a half, than those of other Siddhaiitas. By this error in the M o o n , the value of M o o n minus R a h u will be less by about a degree and a half, (1° 36') and instead of correcting the error by adding it to the M o o n , we subtract 1° 36' from Rahu, which is the same. We do not add it to the M o o n , because i f we do, we must add the same quantity to the Sun to keep the tithi intact, and this will affect the Samkramanas, and thus cause a lot of disturbance. If added to Rahu nobody will even notice it. It may be asked whether it is not wrong to use the latitude calculated in III.31 from uncorrected R a h u i n our work here, as we are going to do. Indeed it will be wrong, and that is why the author gives a correcdon below, in stanze 4, to set it right.' T S do not understand the nature of this correction, not even its amount and its connecdon with stanza 4. T h e i r ignorance i n the matter of the computation of Rahu, which we exposed in III. 2829, they exhibit here also, (see their Sanskrit C o m m . page 40).
[i<^uife
R < m f e c h i w i T m r ^ f ^ ^ s ^ g n ^ : ||>i |
Duration of the eclipse 3. Square the Moon's latitude, subtract it from the square of 55, (i.e. from 3025), and find its square root. Double this, and multiplying by 60, divide by the difference of the daily motions of the Sun and Moon, in minutes. T h e approximate time of the duration of the eclipse is got in nddis. 4. If M o o n ~ Sun is less than 13°, multiply the degrees by 5. T h e result are vinddis. A d d these vinddis to the duration if the longitude of Rahu is greater than that of the Moon, and subtract if the Moon is greater than Rahu. Thus the time of duration becomes correct. The following are the steps i n the work: i. U s i n g III.31, find the Moon's ladtude, (using uncorrected Rahu). ii. Uncorrected time of duration in nddis = V 3025 — (latitude in minutes)' x 1 2 0 d i f f e r e n c e of daily motions of Sun and M o o n i n minutes. 3a. A.-?F^ c. d.
A.B.f^ A.B.^nelf^
4a.
A.B.C.D.'^:
c. A.B.C.D.fw??T
157
VI. V A S . - P A U L . S I D D H A N T A - . L U N A R ECLIPSE
VI.4
iii. Corrected time of duration = uncorrected time ± 5 X degrees of M o o n ~ Rahu in vinadis (If Rahu is greater use the upper sign, i f less use the lower sign. M o o n ~ Rahu is what we get in verse 2, in this section). Example I. On a day, at sunset, the longitude of the Sun is rd. 10-10-12, longitude of the Moon rd. 4-8-57, Sun's daily motion 60' and Moon's daily motion 810'. (From these the end of the full moon tithi falls at 6 nddis after sunset). The Tail of Rdhu at full moon w rd. 4-7-42. Examine whether a lunar eclipse will occur, and if so, find the duration. As a preliminary, the M o o n at full moon should be found by verse 1, thus: T h e tirne to elapse after sunset, for full moon, is 6 nddis. A d d i n g 6' to the Sun at sunset, the Sun at full moon is rd. 1010-12 + 6' = rd. 10-10-18. Therefore the M o o n is rd. 4-10-18. Now, examine whether there will be an eclipse. Rahu at full moon = rd. 4-7-42 (given). Corrected Rahu = rd. 4-7-42 - 1° 36' = rd. 4-6-6. M o o n ~ corrected Rahu = 4° 12'. As this is less than 13°, there is eclipse. Now for the work of getting the duration: i. T h e Moon's latitude (supposed known already) = 380' X (rd. 4-10-18—rd. 4-7-42 in degrees) ^ 9 0 ° = 11'. ii. Uncorrected duration in nddis = V 3 0 2 5 - 11' X 120 - (810 - 60) = V 2904 X 120 8-37.
750 = nd.
iii. Corrected duration = nd. 8-37 — 5 X 4.2 vinddis = nd. 8-16. (subtraction because Rahu is less than the Moon). The following is the rationale of the procedure: We have said that, according to the Siddhahta, when the distance between the centres of the Shadow and the M o o n is 55', the eclipse begins or ends, because at these times the distances are equal to the sum of the semi-diameters. Moon's orbit
Ml
M
M2
Ecliptic
A
S
B
Fig. VI.2
See S M , and SM., in Fig 2, where S is the Shadow and M , , M ^ are the M o o n . A t full moon, the latitude is the distance between the centres, at that time. (SM), because at full moon the centre of the shadow is at S, because it is always 6 rdsis distant from the Sun. Since the inclination of the orbit to the ecliptic is siuall, the siddhdnta takes it that the orbit and the eclipdc are practically parallel, and that the latitudes of the M o o n at first contact ( A M , ) at full moon (SM), and at last contact (BM^) are
158
P A N C A S I D D H A N T I K A
VI.4
equal. As the three lines of latitude are perpendicular to the ecliptic, A M , A S . and A M,^ BS, are equalandright angled at A and B . Therefore A S = SB = V(sum of semi-diameters)- — latitude' = V ' 5 5 ' — lat'-, where A S or S B are the difference of the Moon's longitude at first or last contacts from its longitude at new moon, and measured in minutes of arc. As the motion of S is the same as that of the Sun, the time taken by the longitude to move from A to S or from S to B = the minutes of difference ^ the difference of the motions of the Sun and the M o o n i n minutes = V 5 5 ' — lat' X 60 -^ difference of daily motion in minutes, in nddis. Therefore the total time in nddis horn A to B (this is the uncorrected whole duration) = 2 X 60 x V 5 5 ' — lat' H- the difference of daily modons in minutes. But actually the latitudes at the first and last contacts differ enough from that of the full moon to justify the use of their exact value. So each should be used separately, and the half-duration before full moon, and that after full moon should be found. Using the latitudes of these times again, if necessary, they should be found again. Certain Karanas (manuals) like the Vdkyakarana apply a certain coi rection in the place of this repetition of work. This correction depends upon the M o o n ~ Rahu at full moon, like the correction in stanza 4, given by the author for correcting the duration, and therefore it is possible that the correction for the difference in latitude has been included in that correction. But as the correction given is rough, we cannot analyse it and find out whether the author has done so or not. Let us now consider the rationale of the correction in stanza 4. We have already hinted that this is to compensate for using the latitude got in III.31 from uncorrected Rahu, instead of that from corrected Rahu, which is to be used i n our work here.
In Fig. 3, R is the uncorrected position of Rahu, and R' is its corrected position. T h e distance between them is 1° 36', given in stanza 2. W h e n the M o o n is greater than Riihu, ( M , M ' ) then its uncorrected latitude is M B , and the corrected latitude is M ' B , greater by M ' M . This is case I. When the M o o n is less than Rahu ( M , , M , ' ) , then the uncorrected latitude is M , A , and the corrected latitude is M , ' A , less by M , ' M , . T h i s is case II. In case I, if the uncorrected latitude, which is less, is used in the work,v55' - lat' will be greater than what it should be, and the duration should be lessened by a correction. Therefore it is said that the correction is subtractive when the M o o n is greater than Rahu. In case II, since the uncorrected latitude is greater, V5^5' — lat' will be less than what it should be, and so it is said that the correction is additive i f the M o o n is less than Rahu, i.e. if Rahu is greater. T h e Fig. is for Rahu-Head. It can be seen that at Rahu-Tail too the same holds.
VI.4
VI. V A S . - P A U L . S I D D H A N T A - L U N A R ECLIPSE
159
We can show this by theoretical considerations as well, thus: R a h u lessened by 1° 36', is equivalent to M o o n increased by 1° 36', in its effect on M o o n —Rahu. T h e latitude is proportionate to the distance o f the M o o n from R a h u in each quadrant. In the first and third quadrants, the M o o n is greater than Rahu, and the increase i n the M o o n by the correction increases the latitude. In the second and fourth quadrants, the M o o n is less than Rahu, and the increase in the M o o n by the correction lessens its distance from Rahu, with the result that the corrected ladtude is less. T h u s for M o o n greater than Rahu the correct latitude is greater, and for M o o n less, it is less. T h e rest is as we have shown already. Now we shall find the quandty of correcdon: As the angles at R and 7?' are equal, the corrected and the original orbits are parallel, (see the Fig.) Therefore the differences in the latitudes at any position Hke M ' , M , , M ' M , and C R are equal. But C R, being the latitude caused by 1° 36' of longitude, is equal to 380' X 1° 36' 90° = 6 % minutes of arc. Therefore the difference in any position is 6 % ' . We shall first see how much difference this will produce i n the half duration measured i n minutes of arc. Clearly it is = V 5 5 ' - (lat ± 6 3 / 4 ) ' - V 5 5 ' - lat' = V 5 5 ' - l a t ' + 131/2 lat - V 5 5 ' - lat', (if 6 % ' is neglected, being small i n comparison with 55'.) = V55' - lat' V 1 + 131/2 lat/(55' - lat') - ^ 5 ' - lat' = V55'-lat'{l are neglected.
+131/2
lat/2(55' - lat')} - V55' - lat' (if higher powers of
131/2
lat/(5.5' - lat')
= T 131/2 laty2 V55' - lat', and the author has neglected lat' and taken this as + 131/2 lat/2 x 55. Using the difference of the mean daily modons of the Sun and the M o o n , because this will not matter m the already rough result, and doubling for the whole duradon, the viyiadis of correcdon are, I3i/^ lat X 2 X 60 X 60 + (2 x 55 x 720) = 131/2 X lat X 5/55 = 131/2 X {(Moon ~ Rf x 55/13°} x 5/55 (since lat = M o o n ~ Rahu)° X 55/13°,) = 5 X (Moon ~ Rahu), roughly, as given here. The greater the ladtude, the greater the roughness, but this wiU be submerged in the roughness caused by several other things like the incorrect semi-diameters etc., but the method has the advantage of being easy to apply. We shall consider the readings now. In verse 3, the need for correcting mulah into mulam and kdlasthiteh into kdlah sthiteh will be clear, as also for sthityd into sthitydm i n verse 4. As for correcting bhdgaih into bhdgdh, this is justified by what we have shown in the explanation, viz. that it is M o o n ~ Rahu that is to be multiplied by 5 to give the vinddis of correction. If the word bhdgaih, is taken as it is, the instruction should be taken to mean "five multiplied by the difference of M o o n ~ Rahu and 13°. By this, the correction, instead o f being zero, as it should be for zero ladtude, is the maximum of 65 vinddis. Instead of being the m a x i m u m for m a x i m u m latitude, the correction becomes zero. Further, on both sides of zero latitude, where there is a transition from the M o o n being greater, to R a h u being greater, there is a j u m p from - 65 vinddis to + 65 vinddis, which itself is an indication that the formula is incorrect. B u t it may be objected that i f our correction into bhdgdh is accepted the word trayodasondh serves no purpose, for the computation will be begun only if the difference is less than 13°, and therefore this need not be mentioned. T h e answer is this: F r o m our explanation of the formula for correction it may be seen that it is applicable if the difference is
P A N C A S I D D H A N T I K A
160
VI.5
13°, and even a litde more. T h e author instructs that the correction should be applied only if the difference is less than 13°. But T S (also NP), have taken the word bhdgaih as it is and given the interpretation, because they do not know the nature or the rationale of the correction. Here, Thibaut alone says (vide page 43 of E n g . Translation): " T o the duration so found stanza 4 directs us to apply a correction whose rationale we are however unable to assign", and thus accepts ignorance. But S. Dvivedi in his Sanskrit Commentary says that the Moon's motion varies from time to time, and this correction is to rectify the error due to the variation. H e is unable to see that the variation of the Moon's motion has nothing to do with the correction here, and cannot be related to it.
Total obscuration 5. D e d u c t the, difference o f the l o n g i t u d e s between the M o o n a n d R a h u f r o m five degrees. D e d u c t this f r o m ten degrees, a n d m u l t i p l y the r e m a i n d e r by this itself a n d by f o u r . F i n d the square root o f the result a n d m u l t i p l y it by 2 1 . T h e m i n u t e s o f arc o f total o b s c u r a t i o n is got. T h i s d i v i d e n d by the daily m o t i o n gives the t i m e . T h e rule given is as follows: Minutes of obscuration = 21 x V { 5 - (Moon ~ Rahu)} [10 - {5 (Moon ~ Rahu)}] x 4/5. This can be simplified as: Minutes of obscuration = 2 x 2 1 V 5 ' - (Moon ~ Rahu)'/5. = 2 X 2 1 X V25 - (Moon ~ Rahu)'/5. This multiplied by 60 and divided by the daily motion gives the duration of obscuration in nddikds. Here, 21 x (Moon ~ Rahu)/5 is the Moon's latitude at full moon. T h e latitude according to the Paulisa has been shown to be (Moon ~ Rahu) X 380790, where M o o n ~ Rahu is in degrees. As 380/ 90 is very nearly equal to 21/5, we can say: Latitude in minutes = 21 X (Moon ~ Rahu)/5. It must be noted that we use here the corrected Rahu to get M o o n ~ Rahu. So, the latitude obtained is the correct latitude. Therefore, no correction is necessary here corresponding to that of verse 4, above. Now, the rule is explained thus: In this Siddhdnta, the difference between the semi-diameters of 5a. B.^'aRRI?!?)^: (B3.i%'5cPT^°) b. B l . w f t . B . W ^ o m ^ . A.^^rai: c. B.omW d. A . B . W R T l
VI.5
VI. V A S . - P A U L . S I D D H A N T A - L U N A R ECLIPSE
161
the M o o n and the Shadow is 21 minutes of arc. Therefore, when the difference between their centres is 2 1 ' , the total obscuration begins or ends, as at M , o r i n Fig. 4, below.
Fig. VI. 4
Here M is the M o o n at new-moon, and S is the centre of the Shadow. M , S = MgS = 2 1 ' = the difference of the semi-diameters, constant according to this Siddhdnta. S M is the latitude at full moon. Therefore, Minutes o f obscuration = M , M g = 2 ( M , ^ or MM,,) = 2 VSM^^="SNP = 2 V2 1' - ladtude^ = 2 V 2 F ^ {21 X (moon ~ Rahu)/5}' = 2 X 21 V5' - (moon ~ Rahu)'/5 = 2 X 21 X V 25 - (moon ~ Rahu)'', the simplified rule, from which by inverse operation, we get the original rule, 21 x \^5 - (moon ~ Rahu)} x [10 - {5 - (moon ~ Rahu)}] 4/5. T h e conversion of the minutes of arc of obscuration into time is, as already given, by the proportion. Minutes of daily motion of (Moon — Sun): Minutes of obscuration :: 60 nddikds:nddikds of obscuradon. F r o m the simplified rule it will be readily seen that when M o o n ~ R a h u is 5°, the minutes of obscuration, and thence the time, is zero. Therefore only when the difference is less than 5°, there is obscuration, not when greater, i.e. i f the correct latitude at new moon is greater than 2 1 ' , there is no total ecHpse. Example 2. The Moon at new moon is rd. 8-13-24. Rdhu (Tail) is rd. 8-12-0. The daily motion of (Moon — Sun) = 750'. Find the minutes of obscuration and the time. T h e corrected R a h u = rd. 8-12-0 - 1° 36' = rd. 8-10-24 M o o n ~ Rahu = rd. 8-13-24 - rd. 8-10-24 = 3°. By the simplified rule, the minutes of obscuration = 2 X V 2 5 - 3 ' X 21/5 = 2 x 4 X 21/5, minutes. T h e duration of obscuration = 2 x 4 x 21 X 60/(5 X 750) = nd. 2-41.
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PANCASIDDHANTIKA
VI.8
F r o m verse 3, giving the general duration, we see that the sum o f the semi-diameters of Shadow and M o o n is 55'. Here we see that their difference is 2 1 ' . Hence, (55' + 21') = 76', is the diameter of the Shadow according to this Siddhdnta, and 55' - 21' = 34', is the diameter of the M o o n , giving the semi-diameters as 38' and 17' respectively. By a strange confusion of ideas T S and N P have concluded here that when M o o n ~ Rahu is less than 10°, there must be a total eclipse, (vide the Sanskrit com.p.33, and English translation, p.44, N P , Pt. II, p.53). We have shown that for a total eclipse to occur the difference must be less than 5°. It is easy to see which is correct. If the difference is greater than 5°, the number under the radix becomes negative, and no real root can be obtained. For e.g. if we take 7°, as the difference, according to T S there must be total obscuration. But using it in the formula, we get, 21 X V(5 - 7){10 - (5 - 7)} 4/5 = 21 V'^=^/5 = 21 x 4 \ ^ ^ W 5 , which does not give a real value. T h e i r error is due to their confusing in their work, the 21 minutes, sine of 10°, as 21 minutes, latitude to be got for 10° difference. Further the postulation by T S o f a m a x i m u m latitude of 240' in this context is unwarranted. T r u e , from the above a latitude of 55' for 13° difference, and 2 1 ' for 5° difference will follow, if the correct formula, with the sine of (Moon ~ Rahu) is used, instead of the degree of difference. But nowhere i n H i n d u astronomy is 240' given, and the Vdsistha does not give the latitude at all. So when the Vdsistha wants the latitude to be used in V . 3, we have got to use only the Paulisa formula, and the reading gives 280', which T S have translated tacitly as 270'. 280', if taken, will give 63' and 24' for differences 13° and 5°, and 270' will give 6 1 ' and 23', in,stead of 55' and 21', both of which are unsupported by the context. This is the reason why we changed the reading to mean 380', instead of 280', and, following the instructions strictly, gave the rule, latitude = 380' X difference in degrees/90, getting 55' and 21' for differences of 13° and 5°. This is also in keeping with the practice of the Paulisa, which usually uses for proportion the degrees i n the place of sines.
Direction of the eclipse 6. During the interval from the time of first contact to the beginning of totality, Rahu (i.e. darkness), swallows the M o o n completely. T h e direcdons of the points of first and last contacts are to be calculated from the Moon ~ Rahu of those times. 7. Divide the semi-orb of the M o o n situated opposite to the direction of ladtude into 13 parts, by straight lines parallel to the east-west diameter, at
VI. V A S . - P A U L . S I D D H A N T A - L U N A R ECLIPSE
VI.8
163
e q u a l distances f r o m o n e a n o t h e r . A t the p a r t o f the r i m e q u a l to the degrees o f M o o n ~ R a h u , o n the e a s t e r n o r w e s t e r n p a r t o f the o r b , are the p o i n t s o f first a n d last contacts, f r o m w h i c h the d i r e c t i o n s c a n be r e a d . 8. M u l t i p l y a f o u r t h o f the M o o n ' s r i m , (in w h a t e v e r u n i t t a k e n , as f o r e.g. m i n u t e s o r digits) by the l a t i t u d e , a n d a g a i n by the degrees o f the M o o n east or west o f the m e r i d i a n . D i v i d e this by 8 1 0 0 . B y so m a n y u n i t s is the east o r west p o i n t o f contact b e n t n o r t h w a r d o r away f r o m the n o r t h respectively i f the M o o n is east o f the m e r i d i a n , a n d b e n t away f r o m the n o r t h a n d n o r t h w a r d respectively, i f t h e M o o n is west o f the m e r i d i a n . The instructions to obtain the directions of the points of contact have been explained by figures 5 a, b, c.
Fig. VI. 5-a.
Ecliptic
6a.
A.feR;eT
b. A.fe^t^l^^; B . f ^ ^ ) ^ ^ ; C.f%#f%cW:;
c. Al.^Tft^; A2.^*3f>
d. A . J1^>J|IWI^1; B . W ^ " ^ ; D.^T^^imi (c1^) A . «<*d1ildlTlT^ B. •H^dHdl-^: ( B 3 . ° ^ ) ; 8a. c. A . W ^ l ^ ; B I . 3 . W n ^ ; C.D.-sra?itl% 7a. C.(^M4WI; D . f t q % :
b. c-D.-gOWr
A.B.°c;ciifstl
b.
A.B.^^
c.
B.^^W^
d. A.B.e)«rHNW^ ( B . ^ ^ ) ; D . ^ ^ ' ^ i s p j I
164
P A N C A S I D D H A N T I K A
VI.8
E
In all the three figures, M , M , , M^,, M ' represent the centres of the M o o n , and S, S,, S,^, the centres of the Shadow. S' is the point of contact. R is Rahu. In fig. 5a, R M., is part of the Moon's orbit, and RS^ is part of the ecliptic. In position Sg which is the Hmit for the occurrence of an eclipse, MS^ = 1.3°, and M.^S.j is the ladtude, equal to 55', = the sum of the semi-diameters, i.e., M.^S' + S'S.,. S', the point of contact, is seen 90° from the east point, directed towards the north from the ecliptic, i.e. at the north point of the Shadow, but at the south point with reference to the M o o n . In position S, the .Moon is at the node, Rahu, and (Moon ~ Rahu) is 0°. Clearly, S', the first point of contact, is at the east point. In position S, between the above two, it is seen that S', the first point of contact, makes an angle P M , S, with the east point, on the south, with reference to the Vloon. It may be seen that the sine of the angle is proportionate to PS,, the latitude, which itself is proportionate to (Moon ~ Rahu) as we have shown. T h u s , at any intermediate position, the point of contact makes an angle with the east, whose sine is propordonate to M o o n ~ Rahu. Hence the rule to divide the Moon's half opposite to the direction of the ladtude into 13 parts by parallel lines at equal intervals, and take the point o f contact of that Une which corresponds to the degrees. M o o n ~ Rahu. T h i s is shown clearly in fig. 5c. H e r e P' S', the sine of the angle P ' M S ' , which is the direction, is seen propordonate to PS, the latitude, which is propordonate to M o o n ~ Riihu. T h e figure is for M o o n ~ Rahu equal to 7°. Fig. 5b is intended to show both the first and last points of contact, and because of the increase (or decrease) of the ladtude d u r i n g the interval, there is an increase (or decrease) in the angle. In the figure, the angle of first contact, P'MS', corresponds to the latitude M A and is smaller; the angle of last contact, P'M'S' corresponds to the greater latitude M ' B , and is greater. It is also to be noted that the last contact is at the western part of the M o o n , the M o o n now being east of the Shadow. The directions mentioned above are with reference to the ecliptic, taking it as east-west, (neglecting the angle of inclination of the Moon's orbit). B u t the directions have to be given as seen by the observer. For this, two corrections have to be applied, one to convert it with reference to the eastwest of the equator, called Ayana-vaiana, and the other to correct it for the east-west of the place, depending on the latitude of the place, called the Aksavalana. Both these have been mentioned and explained in connection with the observation of the first appearance of the moon given in C h a p V . T h e author here gives the Aksavalana alone following the original Siddhdnta, neglecting the other one, though that is not negHgible. Even in this, he takes into consideration only the northern hemisphere. T h e r e the celestial equator is inclined south. A n observer facing east looking at a body on the celestial equator sees the east point bent northward, and the west point bent southward. Similarly, an observer facing a body west, sees the east point bent south, and the west point bent north.
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165
T h e directions are changed accordingly. (For illustration see fig. 7, below given under example 3). A l l this has been explained in chap. V . T h e Siddhdnta takes it that the bending is proportionate to the latitude, being zero at the equator and 90° at the pole. So, in terms of the length of the circumference, the bending = (ladtude/90°) x circumference/4. But this amount of bending is only at the horizon. O n the meridian there is no bending. In between, the Siddhdnta takes it as proportionate to the angle of the M o o n from the meridian. Thus, Bending = (latitude/90°) X (circumference/4) x (degrees of the M o o n from the meridian/90°) = quarter circumference X latitude X degrees of the M o o n from the meridian/8100, as given. As for the M o o n being "devoured", it is a figurative expression, the Shadow being identified with the demon, Rahu, in the Puranas. In saying that M o o n ~ Rahu should be done once for finding the point of first contact, and again for the last contact, the author recognises that the difference may be considerable, and thereby indicates that it will be good if the times also are computed separately, using the different latitudes. But T S and N P by their eme.nAation, pragrahamokse, have shut out all this suggestion. T S do not seem to understand why the division into thirteen parts is instructed to be made, for Dr. 1 hibaut says, "We do not know the reason for the direcdon, given in stanza 7, to divide each quarter of the circumference into thirteen parts." (p.45). It is not each quarter, and it is not the quarter-circumference that is to be equally divided. We have emended grahandsvdm.se into graharidsdvise, whereby we understand that the point of contact is at the point where the parallel line corresponding to M o o n ~ Rahu meets the circumference. T h e emendation grahandsd 'tad' vadet by N P is not warranted. In the matter of the directions of Aksavalana, Thibaut says the opposite of what Sudhakara Dvivedi says, and neither gives the correct direction, (vide. C o m . and Translation)
Moment of the eclipse and its colour 8 A-9. T h e m i d d l e o f the eclipse is at the m o m e n t o f new m o o n . T h e times o f first a n d last contacts are e a r l i e r a n d later t h a n the m i d d l e , by h a l f the t i m e o f d u r a d o n . W h e n the eclipse is total, the c o l o u r o f the M o o n is r e d o r b r o w n as it is farthest o r nearest to the e a r t h , respectively, a n d m i x e d , m o r e o r less, i n between. W h e n the eclipse is n e a r sunset o r sunrise, the M o o n is s m o k y i n c o l o u r . W h e n the eclipse is p a r t i a l , the M o o n has the c o l o u r o f r a i n c l o u d .
166
PANCASIDDHANTIKA
VI.10
10. Subtract the H e a d o f R a h u f r o m 12 rdsis, m u l t i p l y it by 2 2 8 , a n d a d d the M o o n ' s l o n g i t u d e . I f this is b e t w e e n 6 a n d 12 rdsis, the M o o n is farther, a n d i f b e t w e e n 0 a n d 6 rdsis, it is nearer. ( T h e i d e a is, that the n e a r e r this s u m is to 9 rdsis, the f a r t h e r is the M o o n a n d its c o l o u r at total eclipse is n e a r e r to r e d . T h e n e a r e r this s u m is to 3 rdsis the n e a r e r is the M o o n , a n d its c o l o u r is n e a r e r to b r o w n ) . Since this Siddhdnta uses the latitude, (or M o o n ~ Rahu) at full moon to find the duration, the part of the duradon before full moon is equal to that after full moon, and the middle is at full moon. But other siddhdntas repeat the work, using the latitudes at first and last contacts separately, so that the two parts are not equal, and the middle does not occur at full moon. Still all siddhdntas technically call the moment of full moon as the middle, since at that time the eclipse is practically the maximum. As for the colour of the M o o n at ecHpse, it is based on observation, and given sHghdy differentiy by different Siddhdntas. Some take the fraction of the M o o n ecHpsed as the criterion for the colour, others the time of the ecHpse and its nearness to sunset or sunrise, etc. Here, this Siddhdnta uses, i n addition, a new criterion, not given by any other Siddhdnta, viz. the distance of the M o o n from the observer, and there is truth i n what the Siddhdnta says. Here, it may be asked how at all is it possible for the M o o n to have any colour at eclipse. It is an opaque body, and what illumination it has comes from the Sun's rays falling o n it. W h e n it is immersed i n the Sun's umbra, i.e. full shadow, (we consider the M o o n in umbra alone as eclipsed, and not i n pen-umbra), the Sun's rays cannot fall on it. It cannot be the earth-shine falling on the M o o n and dimly illuminating it, as i n the crescent M o o n , giving rise to the popular belief of "the old M o o n i n the arms o f the new". A t times of new moon, when the lunar eclipse occurs, there is no earth-shine opposite the M o o n to illuminate it. This is the answer: T h o u g h the M o o n is i n the earth's shadow geometrically speaking, the Sun's rays, refracted by the earth's atmosphere, fall on the M o o n and illuminate it with a red or brown glow, red light alone being able to reach the M o o n after passing through the long section o f the earth's atmosphere undispersed, on account of its greater wave-length. (See fig. 6).
8A.9. Quotedby Utpala o n B S 5.18 8 A . O m both in A and B , but included in this edition on account of its essentiality i n this context and its being quoted as a verse of V M by Utpala i n continuity with verse 8. While C o m its this verse since it is not available i n the text mss. D adds it. (as no. 9, and the further verses numbered as 10 etc.) since Utpala has it.
verses 10-14 as 11-15. c. A . B . 3 ^ q n c F I W ^ ( b . ^ ) ; C.3^^TR^raRT^ d.
A.B.C.°^^^5#lefn^(B.om^)
lOa. D.g#T3|* b. A . B . C . D . B l ^ * / ! ; D.?Tf?I c. A . B . ^ p r a ^ ? T l ^ a ( B . ^ ) , ( A 2 ^ « ) ;
9a. A . M441liy-^<^; C . ^ ^ D . numbers the verse as 10 and the
d. D.Uvqi^:. B.TJt*?:; D.->Tt^:
VI.10
VI. VAS.-PAUL. S I D D H A N T A - L U N A R
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167
Eig. V I . 6 In its nature this phenomenon is similar to the Sun apparently rising earlier and setting later, and appearing red at both times. As for the distinction between red and brown, it can be seen from the figure that there is great illumination at a greater distance, and, so, when nearer, there is less red, which gives a brown colour as mentioned by the author. T h e redness may vary by other causes, like the dust or water vapour in the atmosphere, and it goes to the credit of the ancients that distance was distinguished as one cause, affecting the redness. When the M o o n is partially eclipsed, the glare of the illuminated part dims more or less the redness of the eclipsed part, so that it looks almost dark like a rain cloud. Near sunset or sunrise, the red glow from the M o o n has again to pass a long distance through the atmosphere, and get filtered out, so that the colour becomes smoky. Verse 8-A is given only by Bhattotpala, the text manuscripts omitting it. T h e original of V M must have contained this verse, for it supplies several lacunae. T h e word evam i n verse 9 requires a previous verse mendoning colour. Omission of this verse has compelled T S to emend sarvagrdsiny evam into sarvagrdse pitam. T h e rule in verse 10, giving when the M o o n is at ucca and nica has a purpose only with 8-A which requires the informauon. If T S had verse 8-A before them, they would not have misunderstood 10, and declared that it is something pertaining to astrology. It also instructs us how to find the times of the first and last contacts, and when the middle occurs, which instruction we assumed before, in working the examples. N P rightly include this verse in brackets and give it the number 9, the further verses being numbered from 10. We have said that verse 10 gives a rule to find when the M o o n is far and when it is near. We shall explain how: T h e rule says that when the result got by the rule is nine rdsis, the M o o n is at the greatest distance, i.e. at ucca, and when the result obtained is three rdsis, it is nearest, i.e. at nica, as we have explained. Therefore i f the rule gives about nine rdsis for the M o o n at ucca, i.e. when the M o o n is replaced by ucca, then it must be correct. We shall show that it is so. The rule is: M o o n - 228 R = ucca - 228 R (where R is the H e a d of Rahu at any full-moon). A t the full moon just preceding the Epoch, by II.3, taking the reading vasumuninava etc. it can be calculated that according to Vdsistha-Paulisa, the u£ca is rd. 8-14-21.4. (The Siddhdnta does not give the ucca direct, but we can find it by the relation. M o o n - Moon's kendra = ucca). Every synodic month the ucca increases by 3° 17'.3 according to this Siddhdnta. Therefore after m synodic months, the ucca is rd. 8-14-21.4 + m x y 17'.3. T h e corrected Rahu, R, at the full moon before Epoch is rd. 7-26-45.2. It decreases by 1° 33'.88 every synodic month. Therefore, after wi synodic months, R = rd. 7-26-45.2 - w X r 3 3 ' . 8 8 .-. ucca -228R = rd. 8-14-21.4 +TOx 3° 17'.3 - 228 X (rd. 7-26-45.2 - m x 1° 33'.88) = rd. 8-14-21.4 - 228 x rd. 7-26-45.2 +TO(3° 17'.2 + 228 x 1° 33'.88) = rd. 8-14-21.4 - rd. 11-9-45.6 +TO(3° 17'.3 + 356° 43'.9)
168
PANCASIDDHANTIK-A
VI.13
= ra. 9-4-35.8 + m X 1'.2 = practically nine rdsis, for a long time after or before Epoch. Nineteen years before Epoch, this would have been exacdy nine rdsis. A small difference in Rahu, (if it is r . 2 more) would make it nine rdsis even at the taken time. It must be noted that one or even two rdsis either way will not matter in our context, of redness at one end and brownness at the other, for the difference between redness and brownness itself is slight. O n l y after two or three thousand synodic months will there be perceptible difference, and the rule cease to hold good. From the proof of the rule it will be seen that our emendation oitriyamadvigunam into dhidviyamaguTiam is necessary, triyamadviguriam had perhaps been wrongly written by some scribe who had the 'saros', (consisting of 223 lunations) in his m i n d . A s antithetical to kriyddikanydntago nicah, we have emended the meaningless group of letters, epiklesoyam ucca into juketthageyam uccah. Neither T S nor N P seem to have understood the significance of this verse. T S observes on it: " A stanza of doubtful import, see the Sanskrit commentary" (1 r., p.45) and the Sanskrit commentary suspects it to be of astrological import (com., p.34). N P gives an incorrect translation and says " T h e synodic months in an 18 year eclipse cycle is 223, but the role of this number in the present context remains obscure to me" (Pt. II, p.55)
Diagrammatic representation 11. Draw three concentric circles with radii 17, 38 + 17 (= .55), and 3 8 - 1 7 (= 21), minutes of arc. These circles relate to the Moon, the duration and the obscuration, respectively. (Drawing the part of the Moon's orbit forming the path of the Moon), mark the points (of first and last contacts) and also those of inversion and emergence if any). 12. Draw the diameter (making an angle equal to the Valana given in verses 7-8), with the ecliptic which, (according to this Siddhdnta), is east-west with reference to the equator. (This diameter shows the east-west of the place). (As shown in fig. 5c) draw thirteen equally spaced lines parallel to this east-west diameter. (The directions of the points of contact etc. are given by this figure).
VI.13
VI. VAS.-PAUL. S I D D H A N T A - L U N A R
ECLIPSE
169
13. T h e graphical representation o f the l u n a r eclipse has here been described b r i e f l y , a n d c a n be u n d e r s t o o d p r o p e r l y o n l y by e x p l a n a t i o n ( f o l l o w e d by d e m o n s t r a t i o n ) . F r o m this, the total d u r a t i o n , the total o b s c u r a t i o n , the m a g n i t u d e , etc. c a n be f o u n d by i n s p e c t i o n . T h e representation given by the author can do duty for all the figures used by us to explain verses 3, 5, 6 and 7, the centre of the concentric circles being the centre of the Shadow i n each. A n important difference is that in the previous illustrations, the orbs, of the M o o n and the Shadow were shown separately, while here they are replaced by one circle for each of duration (radius = 38' -I17') and totality, (radius = 38' - 17'), the M o o n being reduced to a point coinciding with its centre. As the points of contact etc. showing the direction cannot be marked on the point-Moon, another circle is drawn to represent the M o o n , with the same centre. As the directions of the points on the two circles are diametrically opposite to the directions of the same points with reference to the M o o n , the points can be marked on the Moon-circle by the intersection of the diameter on its opposite half. A n examination of fig. 7 will show this, and an example will make everything clear. Example 3. (Note: This example is intended only as an illustration). The latitude of a place is 20°. The full moon occurs 4 nddis after sunset. The Moon at that time is rd. 5-15-0, and the Sun, rd. 11-15-0. The uncorrected Rdhu (Head) atfull moon is rd. 5-6-36, from which the latitude is 35'N. Thefull moon is 11 nddis before mid-night. The daily motion of (Moon — Sun) = 780'. Find the times cffirst contact etc., and verify by a graphical representation. (i) T h e corrected Rahu = rd. 5-6-36 - 1° 36' = rd. 5-5-0. M o o n ~ Rahu = rd. 5-15-0 - rd. 5-5-0 = 10°. This is less than 13°..-. there is a lunar ecHpse. (ii) Minutes of duration = V55' - lat' = V55' - 3 5 ' = 42, minutes of arc. Now, 2 X 42 X 60
780 = nd. 6-28 = Uncorrected duration.
Difference between M o o n and uncorrected Rahu = rd. 5-15-0 ~ rd. 5-6-36 = 8° 24. Correction = 8° 24' X 5 = 42 vinddis. As Rahu is less, deducting from uncorrected duradon, the correct duration = nd. 5-46. H a l f this is wd. 2-53. Subtracting from the time of new moon, 4 nddis, the first contact is at nd. 1-7 after sunset. A d d i n g , the last contact is at nd. 6-53. (Hi) M o o n ~ cor. R a h u = 10°. As this is greater than 5°, there is no total phase. (iv) T h e first contact is nd. 15-0 - nd. 1-7 = nd. 13-53 before midnight, i.e. the M o o n is 83° east of the meridian, I l a . A.B.^?II5I B1.2.tl?ra??T; B 3 . i [ ^
12a. A.MlxtiWi y * d | ; l ; B . ' t o ^ ^ l ^ ^ ( B 2 . D D.'atrOTlL't] !^l*d*l [^]
b. A.H^H^>I; B.C.D.PHtlH^^"!
b. A . B . D . ^ ^ r f ^ S ( A o m T ) ; C . ^ ^ i w n S
C. A.B.J^lRfHH^lMr ( A 2 . ^ )
d. A.'HHWRI:; B.'HHidKI:
D . % J ^ ^ ^
^_ B I . i i ^ ( B 2 . 3 . ^ ° )
170
PANCASIDDHANTIKA
VI.13
T h e Moon's circumference = 17' x 44/7 = 107'. T h e bending o f the direction due to ladtude = (I07'/4) X 20 X 83 + 8100 = 5 Vs minutes o f arc. As M o o n is east o f the meridian, the equatorial east point is bent 5V2' north o f the east point of the place, i.e. the east point o f the place is situated south by 5 Vs minutes-length. A t the time o f last contact, the M o o n is 49° east o f the meridian. T h e bending o f the point o f last contact at the western limb o f the M o o n =(107/4) x 20 X 49/8100 = 3.2', southward, as the M o o n is east o f the meridian, i.e. the west point of the place is 3 V2 minutes-length north of the equatorial west point. (v) W e shall show all these graphically i n F i g . 7. T h e latitude to be used i n the figure is the corrected latitude = 10° X 380' -=- 90 = 4 2 ' N .
Fig. VI. 7
M , M ' , are the points o f first and last contacts. M M ' = duradon i n minutes o f arc. M M ' = 3".6 by measurement, = 3.6 X 20' = 72'. From this, the dme o f duration = 72 X 60 + 780 = nd. 5-32.
V I . 14
VI. V A S . - P A U L . S I D D H A N T A - L U N A R ECLIPSE
171
See how close this is to the time calculated, viz. nd. 5-46. T h e Moon's orbit does not touch the circle of totality. Therefore there is no total obscuration, as already found by calculation. T h e arc i n the figure from the east point of place to the first point of contact gives the direcdon of the point at the beginning, and the arc from the west-point of place to the point of last contact gives the direction at the end. We have amended malena mioyutona, nabahu into Rdhu, ekachanoni into ekasthdndni, ydmasabu into sasdhka and purvdparayosra into purvdpardydsca as necessitated by the context and proximity of the lettering.
Lunar and solar eclipses - Differences 14. In the lunar eclipse, the Moon, (moving eastward), contacts the earth's shadow. Therefore the Tirst contact' (occurs at the eastern limb of the Moon, and so) does not occur at the Moon's western limb. In the solar eclipse, the M o o n meets the Sun, and therefore, (the Sun being contacted as its western limb), the first contact does not occur at the eastern limb of the Sun. T h e Moon's motion being more than thirteen times that o f the Sun or the Shadow, (whose motion is the same as that of the Sun), it moves eastwards relative to the Sun or Shadow and contacts them at their western limb, and its own eastern part. As the lunar and solar eclipses are with reference to the M o o n and the Sun, respectively, the first contacts are at the eastern and western limbs, respectively. It need not be mentioned that the last contacts are, respectively, on the western and eastern limbs.
14. Quoted Utpala o n B S . 5.12 12a. U . ^ . B1.2.'f?Srat; B3.''PFn. A . B . ° f t ^ b. A B . 3 . ^ : ^ ° ; B 1 . 2 . ^ 5 ^ : ^ °
C.^?^^«lTl^^^at^ | A.^gi^; B . D . W * c. A . B l . ^ F T J J l l . A.B.C.D.U.fiT^-. d. A . T ^
Thus ends Chapter Six entitled '(Vasi§tha-) Paulisa Siddhanta: Lunar Eclipse' in the Paiicasiddhantika composed by Varahamihira u
1. A.B.D.-^-s^Jl^u'l ^ 5 « q r a :
C.?f^^^=5[3^^^5HFT:
Chapter Seven (PAULISA-SIDDHANTA) - SOLAR ECLIPSE
Introductory This chapter deals mainly with the solar eclipse according to the Paulisa. But the last two verses giving the computation of the solar eclipse gives the lunar eclipse also. It is from this that we have to conclude that the lunar eclipse of C h a p V I is that of the Vdsistha. T h e method of the Paulisa for correcting the Moon's latitude for parallax is peculiar. T h e correction is done on Rahu, and thence carried to the latitude. Also, this is the edirWest. siddhdnta to deal with the solar eclipse, and thus, with its peculiar method, historically important.
Parallax of longitude I. F i n d the i n t e r v a l b e t w e e n m i d - d a y a n d the t i m e o f new m o o n , i n nddis. M u l t i p l y this by 6. D e g r e e s are got. F i n d its sine. D i v i d e it by 3 0 . T h e result is the p a r a l l a x i n nddis to be d e d u c t e d f r o m the t i m e o f n e w m o o n i f new m o o n is before m i d - d a y , a n d to be a d d e d to the t i m e o f n e w m o o n , i f after m i d - d a y . T h e n e w m o o n c o r r e c t e d f o r p a r a l l a x i n l o n g i t u d e is o b t a i n e d . T h u s : i . nddis of parallax = sine (interval i n nddis between mid-day and new moon X 6) ii. Parallax corrected new moon = new moon +(i),
30.
for forenoon, and/^/wi for afternoon.
T h e rationale of parallax correction is as follows: A lunar or solar eclipse occurs when the M o o n gets so close to the earth's shadow or the Sun, that it enters the Shadow so as to be darkened by it, or hides the Sun from the observer's view. N o w , the M o o n being darkened by the Shadow is pracdcally independent of the position of the observer on the earth. But the M o o n hiding the Sun depends upon the observer's position, owing to parallax. So parallax correction has to be done in the solar eclipse. T h e critical angular distance is the sum of the semi-diameters. T h e angular distance between the Sun and the M o o n is calculated from the longitudes of both and the ladtude of
la. A.B.C.D.W
c. B.|ifiJ|clMl d. A.B.^lRN^liSfl: A.B.#r«rrfq; C.fiT«t*
VII.1
173
VII. P A U L I S A - S I D D H A N T A — S O L A - R E C L I P S E
the M o o n . These being given with reference to the centre of the earth, the angular distance calculated is as seen by an observer at the centre of the earth. B u t we want the distance as seen by an observer on the surface of the earth, and a correction has got to be made for this. This is correction for parallax or simply parallax. See Fig. l-a.
Fig. VII. l-a.
{Note: T h e figure is only diagrammatic and does not represent actual distances). PD = a line drawn parallel to C M . The observer at C sees the M o o n along C M , and for h i m the Moon's zenith distance (z.d.) is Z C M (= ZPD). B u t the observer at P sees the M o o n along P M , and its z.d. for h i m is Z P M . This is equal to Z P D + D P M = Z C M -t- P M C , and P M C is the parallax correcdon to Z C M . It can be seen that the parallax correction for the Sun is PSC, and less than that for the M o o n . It is actually about 2/27 of the parallax of the M o o n , according to the H i n d u Siddhdntas, its distance being about 27/2 dmes that of the M o o n , according to them. (It must be noted that actually the Sun's distance is about 390 times the Moon's, and accordingly the Sun's parallax is about 9", and pracdcally negligible). The amount of parallax, P M C can be calculated trigonometrically thus: Sin P M C / P C = sin C P M / C M = sin ( P C M -f- P M C ) / C M = sin P C M / C M (•.• P M C is small). Sin P M C = sin C P M X P C / C M . Arc P M C X 120/57.3 = sin C P M X P C / C M (•.• P M C is small). Parallax of M o o n in minutes = sin z.d. X ( P C / C M ) X 60 X 57.3 120 = sin z.d. X 28.65 x Earth's radius/Moon's distance. Similarly, the Sun's parallax i n minutes = sin z.d. x 28.65 X Earth's distance
Sun's distance.
174
PANCASIDDHANTIKA
VII.1
At a solar eclipse, z.d. is practically equal for the M o o n and the Sun, and in measuring the angular distance between the Sun and the M o o n reladve parallax can be applied to the M o o n , the Sun being supposed unaffected. W e can write: Relative parallax = Moon's parallax —Sun's parallax = sin z.d. X 28.65 X earth's radius X (1/Moon's dis. - 1/Sun's dis.) When the Sun or M o o n is at the horizon, sin z.d. is 120, and the relative parallax, (hereafter we shall call it merely parallax), called horizontal parallax, is a m a x i m u m , and this Siddhdnta takes it as equal to 49'. .-. Parallax = 49' sin z.d./I20. Also, we can see from the fig. l a that the M o o n is depressed, awav from the zenith by parallax, along the vertical circle, Z M , increasing the z.d. and that is why it is called lambanam, i.e. 'depression', in Sanskrit. This general parallax has to be resolved into two parts, correcdon to longitude, (p. long.) and correction to ladtude (p. lat.). This is shown in fig. l b .
Fig. VII. I-b.
VII. P A U L I S A - S I D D H A N T A — S O L A R E C L I P S E
VII.4
175
Z M = z.d., and M M ' is the general parallax. P M ' is the parallax in ladtude, and M P , that in longitude Z Q . = ladtude of the place, and Z N is the zenith distance of the nonagesimal (z.d.N.). We shall find an expression for P M ' , the p.lat. p.lat = P M ' = M M ' Xsin M ' MP/120 = 49' sin Z M x sin Z M N / 1 2 0 ' = 4 9 ' s i n ZN/120 = 49' sin zdN/120, (from rt A M N Z ) . Similarly for p.long, i.e. M P , p.long = M P = M M ' cos M ' MP/120 = 49' sin Z M . cos Z M N / 1 2 0 ' = 49' cos Z N X sin M N / 1 2 0 ' (from rt A M N Z ) = 49' cos z.d.N X cos O M / 1 2 0 ' , O M being (lagnam - Moon). Clearly, this is positive when the M o o n is east of the nonagesimal, and negative when west. It is p.long. that we are concerned with i n this verse, and it must be given in terms of the hourangle (h, or natdmsa) cos z.d.N X cos O M = (i) cos w. cos 0. sin h. sec < / [ V b e i n g the Moon's declinauon] + (ii) [sin w X cos Moon's longitude X {cos 0 x tan ( / X sin {nadis after sunrise or before sunset X 6°)/120'} sin 0/120]. T h e Siddhdnta omits (ii) which is small in comparison with (i), sin 0/120 being small in India. For the same reason, and as w and (f cannot exceed 24°, it takes cos w. cos 0. sec c/'as 120. .-. p.long = 4 9 ' s i n h/120. As said before, this is positive, i.e. it increases the Moon's longitude when h is east, and negative, i.e. it decreases, when h is west. Therefore the corrected time of new moon is earlier and subtractive in the forenoon, and later and additive i n the afternoon. Now, 49' of p-long, converted into time, using the mean relative daily motion of the Sun and the M o o n , (731'.5), = 49' X 60/731'.5 = 4 nddis nearly. .-. nddis of parallax = 4 X sin h/120 = sin h/30, as given by the text. If (ii) is not neglected, the M o o n being east or west o f the nonagesimal will be the criterion for the subtraction and addition of the nddikds.
4c|fiif SRlJOT
3IT5EITi^ I
176
VII.4
PANCASIDDHANTIKA
Parallax i n latitude 2. M u l t i p l y the degrees o f l a t i t u d e by 5 a n d d i v i d e by 27. A d d o r subtract the r e s u l t i n g degrees, respectively, to R a h u ' s h e a d o r f r o m R a h u ' s tail, w h e r e the M o o n is situated, (i) 3. A d d t h r e e rdsis to the M o o n , a n d f i n d its d e c l i n a t i o n i n degrees. T h i s m u l t i p U e d by the nddis o f p a r a l l a x ( g i v e n by verse 1) a n d d i v i d e d by 18, are to be a d d e d to the H e a d i f it is f o r e n o o n a n d Uttardyana (i.e. the S u n is i n its n o r t h w a r d course), o r a f t e r n o o n a n d Daksindyana. T h e degrees are to be subtracted f r o m the H e a d i f it is f o r e n o o n a n d Daksindyana o r a f t e r n o o n a n d Uttardyana. F o r the T a i l , the a d d i t i o n a n d s u b t r a c t i o n s h o u l d be interc h a n g e d , (ii) 4. T a k e nadis f r o m s u n r i s e to n e w m o o n i f f o r e n o o n , the nddis f r o m new m o o n to sunset i f a f t e r n o o n . M u l t i p l y these by the degrees o f the M o o n ' s d e c h n a t i o n a n d d i v i d e by 80. T h e r e s u l t i n g degrees are to be a d d e d to the H e a d i f the M o o n ' s l o n g i t u d e is b e t w e e n 6 a n d 12 rdsis, a n d subtracted i f betw e e n 0 a n d 6 rdsis. F o r the T a i l , i n t e r c h a n g e the a d d i t i o n a n d s u b t r a c t i o n , (iii) T h e corrections are: i. 0° X 5
27
ii. 'Degrees' X the nddis of verse 1-^18, where 'Degrees' are to be got from sin 'degrees' = sin w sin (Moon -I- 90°)/120 iii. Degrees of Moon's declination X the time i n nddis from sunrise to (parallax corrected) new moon, or to sunset from parallax corrected new m o o n + 80. T h e addition or subtraction is as instructed i n the translation above. These rules follow from the formula derived already for parallax correction in latitude: p-lat = 49' sinz.d.N/120. 49' sin z.d. N/120 = sin w. cos 0 sin (AQ)/120^ - cos w. sin 0/120', (from the two rt. A s. Z N A and ZQA) = (i) - 49' cos w, sin 0/120' + (ii) - 49' s i n u r cos 0 sin (Moon -I- 90°) x sin h. sec cr/120-' -I- (iii) -I- 49' cos ur.cosO tan c/'-cos h/120^, (h being the hour angle of the M o o n at parallax-corrected new moon). 2a. A.BI.M:llwfel°. D . W 3 T b. B I . ^ D . [ ^ ] % g ! I . A . ^ : ; B I . 2 . ^ : A.B.SFMcIJr|;D.SR'^:
c. A.^«I«R d. A . o m % B I . ^ ; B 3 . ^ ^ ; C . D . o m t ^
c. A.B.C.
A.^; B . ^ 4a.
B.i^
b. A . C . D . ^ ^ ^ ; B . W F W B.SRTIRT
B l . W : ; B2.3.8!raT: c. A.B.-?tW. D . ^ f T ^ d. B . f ^ ^ . A . B . ^
A . ^ ;
VII.4
VII. P A U L I S A - S I D D H A N T A — S O L A R E C L I P S E
177
T o secure correction 49' in latitude, a correction = 90° X 49/380 = 1 T.O, must be applied to M o o n ~ R a h u which can be done by applying it to R a h u , as the Siddhdnta does. T h e purpose of this replacement is to extend the method of computing the lunar eclipse using M o o n ~ R a h u to the solar eclipse also. So 49' has to be replaced by 1 T.b in the rules, (i), (ii) and (iii). We shall take the rules one by one and derive the Siddhanta rules. (i) = - l l ° . 6 c o s
sin 0/120'
= - i r . 6 X 109.6 X (degrees of latitude x 120/57.3)/120', (•.• when latitude is n o t m u c h , its sine degrees) = - 5 X degrees of latitude + 27, as given. As the Siddhdnta has only the north latitudes i n view, this part of p.lat is always negative. Therefore Moon's north-latitude must become less, and south latitude more, by the correction. T h i s can be done by increasing R a h u - H e a d , i.e. by adding the correction to the H e a d , and by decreasing Rahu's T a i l , i.e. by subtracting from the T a i l , as instructed. (ii) = - 1 l ° . 6 s i n u r cos. 0 sin (Moon + 90°) sin h. sec d" /120' = - I T . O s i n u r ' . sin (Moon + 90°) X nddis of correction to new moon (cos ur X 120 X 4), (•.• sin h = 120' X nddis of correction X cos (f -r- (cos «A»'X cos 0 X 4), in verse I). = — 11°.6 X sin declination of point (= M o o n -I- 3 rdsis) X nddis of correction (109.6 x 4) = — 11°.6 X {Degrees of dechnation o f point (= M o o n -I- 3 rdsis) X 120/57.3} X nddis of correction + (109.6 X 4) = - 'Degrees' ofdeclination of point (= M o o n + 3 rddis) X nddis of correction to new moon 18. T h e 'Degrees' are north, i.e. positive for Moon's Uttardyana, and negative for Daksindyana. T h e nddis stand for positive p-long i n the forenoon, and negative p-long i n the afternoon as already shown. T h u s , for Uttardyana and forenoon, the p-lat is negative, and so the result of (ii) is to be added to H e a d , and subtracted from T a i l . Clearly, it is the same for Daksindyana and afternoon, as then also p-lat' is negative. If Uttardyana and afternoon, or Daksindyana and forenoon, p-lat. becomes positive, and so the correction is subtractive to Head, and additive to T a i l , as instructed. (iii) = 4- I r . 6 cos uT. cos 0. t a n ^ f - cos h/120'. = -I- I l ° . 6 X 109.6 X cos 0 X s i n ^ x cos h + {cos (fx 120') = -I- I r . 6 X 109.6 X degrees of Moon's declination x c o s h -f- (120 x 57.3), (•.• s i n / = degrees of declination x 120 + 57.3, nearly, and taking cos 0/cos (T as equal to unity, c f b e i n g not great, and 0 being not great in India) =-I- l l ° . 6 x 109.6 X cT X ndfifis after sunrise or before sunset-j-(120 X 57.3 X 15), •.• the Siddhdnta, takes cos h as equal to nddis after sunrise or before sunset 15, being satisfied with approximate values, i.e. 0 not being great, the time from sunrise or sunset to noon is taken as \5 nddis always. Secondly, the angle is taken as
178
VII.6
PANCASIDDHANTIKA
added to T a i l i f M o o n is between 0 and 6 rdsis. F o r M o o n between 6 rdsis and 12 rdsis cf is negative and p.lat. is negative, and therefore the result is to be added to Head and subtracted from Tail. Thus, by making the three corrections to Rahu, p.lat. is secured, though approximately. T S have not translated or explained these three verses, not having understood the exact form of the rules or their derivation. T h e y merely surmise that it is some work done on Rahu to correct the latitude for parallax. N P too have not understood these verses correctly for they say i n the notes to verse 2. "This verse is corrupt" etc., i n the notes to verse 3, "The text as it stands seems to have confused the different cases" and in the notes to verse 4, " T h e VeyXseems to instruct us" etc. (pt. II, p.57). In accordance with the correct form of the rules, we have i n verse 2, emended dvika into dhrti, rdsicarandyana into sardsicarandpama and candrdyana into candrdpama.
[ii^U|c»>4]
Eclipse computation 5. Deduct 1° 36' from Rahu and find M o o n ~ Rahu, in the case of the lunar ecHpse. Deduct 1° 36' from Rahu corrected (by verses 2-4) and find M o o n ~ Rahu, in the case of the solar ecUpse. If the difference is less than 13° there is a lunar ecUpse. If the difference is less than 8°, there is a solar eclipse, (otherwise not). 6. For the lunar eclipse, deduct the square of the difference from 169, find its square root and take three fourths of it. This is the total duration in nddis. For the solar ecHpse, deduct the square of the difference from 64, find its square root, and take three fourths of it. This is the total duration in nddis.
T h u s , nddis of total duration = V169 - (Moon ~ Rahu)', or 3/4 V64 (Moon ~ Rahu)', respectively. H a l f this subtracted or added to the full moon, or parallax corrected new moon, gives the times of first and last contacts.
5a. A.feelt; B . ^ b. A l . t ^ ^ ; A 2 . t ^ ^ ; B . t ^ U A . B I . 2 . ^ d. B.Wnn^: 6a. AB.I.2^*H^-<1 h. A.B.^^(B.^)^^m^mm'^
( B . ^ ) I C.D.^^:#T(D.f
VII.6
VII. P A U L I S A - S I D D H A N T A — S O L A R E C L I P S E
179
T h e formulae here are similar to that of V I . 5 as reduced by us, giving the total duration in the lunar eclipse, according to Vdsistha. Therefore, the explanadon is similar. A s i n the Vasistha, in the Paulisa too, the limit of the lunar eclipse is seen to be 13°, giving 55' latitude. Therefore, in the Paulisa too, the sine of the semi-diameters of the Shadow and the M o o n is 55', wherefrom their respecdve semi-diameters may be taken as the same, i.e. 38' and 17'. T h e limit of the solar eclipse is seen to be 8°, at which the latitude is 8 X 55'/13 = 33'.8. Therefore the sum of the semi-diameters of the Sun and the M o o n is 33'.8, from which the semi-diameter of the Sun is found to be 16'.8. If actually the Moon's semi-diameter is a little more or less in the Paulisa, to that extent that of the Sun must be less or more. But we have no means of knowing it exactly, since the original Paulisa is not extant, the Paulisa quoted by Bhattotpala i n the Brhat Samhitd being different, as already mentioned. F r o m the formulae, the minutes ofarc pertaining to duration is, 55' X 2 V169 — (Moon ~ Rahu)'/ 13 i n the case of the lunar eclipse. T o be correct, the dme must be found from this by dividing by the true relative motion, which is not done here. If the mean relative motion is used, we get: {2 X 55 V 169 - 0/13} x 60 + 731.5 = 9, being the na
180
PANCASIDDHANTIKA
Next, correction to Rdhu: (i) 5 X 0/27 = 5 X 10.4/27 = 1° 56'. As the M o o n is near the T a i l , this is to be subtracted. . - . T a i l - 1° 5 6 ' = r a 2-1-0 - 1° 5 6 ' = r d . 1-29-4. (ii) Correction to new moon = 2 nddis. (Moon -t- 3 rdsis) = 5 rdsis. The declination of this point is 11° 44'. T h e correction = 2 X 1 1 ° 4 4 ' / 1 8 = 1° 18'. As new moon is afternoon, Uttardyana, and T a i l , this is addidve. A d d i n g to corrected Tail we get, rd. 1-29-4 + V 18' = rd. 2-0-22. (iii) T h e nddis of corrected new moon before sunset = Sunset ~ cor. new moon = 60 nddis — nd. 51-20 = na. 8-40. T h e Moon's dechnadon, from its longitude, is 20° 36'. T h e correction, = 8 2/3 X 20° 36'
80 = 2° 14'.
As the M o o n is between 0 and 6 rdsis, and it is T a i l , this is additive. A d d i n g to corrected T a i l , we get, rd. 2-0-22 -I- 2° 14' = rd. 2-2-36. Subtracting 1° 36' from the corrected T a i l , we have rd. 2-2-36 - 1° 36' = rd. 2-1-0, as corrected Rahu to be used in the formula: M o o n ~ Rahu = rd. 2-1-0 - rd. 2-0-0 = 1° As this is less than 8°, there is a solar eclipse. Duradon = 3 X V61 - 174 = 5-57 nddis. H a l f this isnd. 2-59. Subtracting and adding this to the corrected new moon, we have: T i m e of first contact = Tid. 51-20 - nd. 2-59 = nd. 48-21 T i m e of last contact = nd. 51-20 -I- nd. 2-59 = nd. 54-19 We have already said that ...e results will be very rough.
1. C o l . A . B . C . D . ^ ( A . D . omifcf)
'^ftf^Rlfe^^f^^IPT^
( A . B . D . o m ^ ) W^SHjpiT:
Thus ends Chapter Seven entitled 'Paulisa-Siddhanta - Solar Eclipse' in the Paiicasiddhantika composed by Varahamihira
Chapter Eight ROMAKA-SIDDHANTA — SOLAR ECLIPSE
Introductory In this chapter the Sun, M o o n and R a h u according to the Romaka Siddhdnta are given, as also the solar eclipse, dependent on these. B u t the lunar eclipse is not dealt with. We have already mendoned that this Romaka is different from the Romaka extant now.
1. A c c o r d i n g to the Romaka, the m e a n S u n i n r e v o l u t i o n s etc. is o b t a i n e d by m u l t i p l y i n g the Days f r o m E p o c h by 150, d e d u c t i n g 65 f r o m the p r o d u c t , a n d d i v i d i n g by 5 4 , 7 8 7 . 2. B o t h the S u n a n d the M o o n are to be m a d e t r u e by intervals o f the e q u a tion o f the centre for half-signs o f the respective m e a n a n o m a l i e s g i v e n for the first three signs. F o r the n e x t t h r e e signs they are to be t a k e n i n the reverse o r d e r . T h i s is r e p e a t e d for the next six signs. I n the case o f the S u n , the a n o m a l y is got by d e d u c t i n g nd. 2-15-0 f r o m the m e a n S u n . 3. T h e minutes o f intervals for the S u n , are 20 + 1 5 , 2 0 + 1 4 , 2 0 + 1 0 , 2 0 + 4, 2 0 - 6 a n d 20 - 14, f r o m w h i c h seconds 18 a n d 5, are to be subtracted, a n d 2, 10, 16 a n d 18 are to be a d d e d , i n the g i v e n o r d e r . Thus, (i) Mean Sun = (Days from epoch x 150 - 65)
54,787.
(ii) T h e mean anomaly of Sun = Mean Sun - rdsi 2-15-0.
182
PANCASIDDHANTIKA
VIII.3
(iii) T h e intervals of equation of the centre are 34'42", 3 3 ' 5 5 " , 3 0 ' 2 " , 24' 10", 14'16" and 6' 18". These are subtracdve in the given order i n the first quadrant of anomaly, additive in the reverse order in the second quadrant, additive in the given order in the third quadrant, and subtracdve in the reverse order in the fourth quadrant. (iv) T r u e S u n = (i) + (iii) Example 1. Compute the true Sun, for the moment, 59 days from Epoch. (i) M e a n S u n = (59 X 150 - 65) ^ 54,787 = rasi 1-27-44. (ii) M e a n anomaly = rdsi 1-27-44 - rdsi 2-15-0 = rdsi 11-12-44. (iii) T h e equation of the centre = - 3 4 ' 42" - 3 3 ' 55" - 3 0 ' 2" - 2 4 ' 10" - 1 4 ' 16" - 6 ' 18" -f 6' 18" -1-14' 16"+24' 10"+30'2"-l-33'55"-l-34'42"-h34'42"-l-33'55"+30' 2"-1-24' 10"-I-14' 16"-f6' 18" - 6 ' 1 8 " - 1 4 ' 1 6 " - 2 4 ' 1 0 " - 3 0 ' 2 " - 3 3 ' 5 5 " x 12° 4 4 ' ^ 15° (= - 2 8 ' 4 7 " ) = +39' 50". ( A n examination o f work (iii) will suggest how to get the total easily). (iv) A d d i n g to the mean S u n , T r u e S u n = rdsi 1-27-44 +40' = rdsi 1-28-24. It must be noted that this is at mean sunset at Yavanapura. T h e rule for the mean S u n can be derived from the constants given o r derivable from 1.15. T h e r e it was shown i n the Notes that i n the Romaka yuga consisting 2850 solar years, there are 10,40,953 civil days. T h u s , as the solar year is the period of revolution of the Sun, the number of revolutions, say inx days, is = x x 2850 H- 10,40,953 = x X 150 -r 54,787, as given. As for th^ deducdve constant, 65, we infer that according to the Romaka, 65/150 days after epoch the mean S u n was a full revolution, but we cannot verify this, the original Romaka being lost. B u t it must agree with the relevant constant i n 1.10, and we have shown i n the Notes there that it indeed does. W e have also shown there that this mean S u n is tropical, and not sidereal. T h i s is peculiar to the Romaka. T h e Sun's apogee given as rd. 2-15-0 is what the Romaka must have found by observation and computation, and we have to take it as it is. Actually, the apogee was at rd. 2-17-19 at epoch. As for the intervals of the equation o f the centre, the Siddhdnta is right i n giving them in accordance with the anomaly. B u t the values are slightly different from what they will be i f the correct terms, a sin 0 + b sin 2 0, has been used. Therefore, either the Romaka gives only empirical values obtained from observation, like the PaulUa, or the Romaka like the Surya Siddhdnta etc., apply an equation on the epicycle itself. A n y deviation from this may be due to scribal errors. T a k i n g the sum 1. Quoted by U t p a l a o n B S 2 . p . 4 0 la.
Al."^;A2.'g^
b. A.Bl.C.D.fM«raig; B 2 . 3 . W ^ . A.B.^Rl^mjT
c. A . B . C . D . ^ 1 ^ d. A . B . ^ ; C . ^ . B . o ^ 3b. A.B.#1I»1|R5(I(CI^THI (B.Rvild^HI)
c. B . o f i ^ H M d ^ d. A . B . ° ^ (B.^)d!-i
c.
A1.^;A2.^
d. B.^fil5° A.B.cWlfeifocll ( B 2 . o ^ ^ )
VIII.6
183
VIII. R O M A K A - S I D D H A N T A — S O L A R E C L I P S E
of the intervals as the maximum equation of the centre, and neglecting the correction of the epicycle, we give hereunder the given and computed values for comparison. Anomaly Computed intervals Given intervals
15° 37'6" 34'42"
30° 34'36" 33'55"
45° 29'42" 30'2"
60° 22'46" 24'10"
75° 14'20" 14'16"
90° 4'53" 6'18"
T h e sum forming the m a x i m u m is 143' 23", and very near that of the Paulisa, and very much more than the actual. T h i s excessive roughness itself is an indication that the Siddhdnta is not indigenous. T h e author has not clearly mentioned where the intervals are to be taken in the given order and where in the reverse order, as also where they are additive and where subtracdve. O r , adopting the reading mithunadalat, we can understand that the equation of the centre is subtractive in the six signs of the mean Sun beginning from the middle o f G e m i n i , and therefore additive in the other six. F r o m the corrected reading, vyutkramasasca, we understand that after taking the intervals i n the given order, we take them i n the reverse order. F r o m these everything else is inferred.
True Moon 4. T h e m e a n M o o n i n r e v o l u t i o n s etc. is got by m u l t i p l y i n g the 'day' by 3 8 1 0 0 , s u b t r a c t i n g 10,984, a n d d i v i d i n g by 10,40,953. 5. T h e m e a n a n o m a l y i n r e v o l u t i o n s etc. is o b t a i n e d b y m u l t i p l y i n g the days by 110, a d d i n g 6 0 9 , a n d d i v i d i n g b y 3 0 3 1 , the result b e i n g for sunset at Ujjain. 6. F o r the half-signs o f a n o m a l y the intervals o f e q u a t i o n o f the c e n t r e are: 1° + 14' + 25", I ° + 1 1 ' + 48", r + 2 ' - 9 " , 4 8 ' - 15", 4 8 ' - 18" - 0", a n d 4 8 ' - 18' - 2 0 ' - 1" (i.e. (1) r 14' 25", (2) 1° IV 48", (3) 1° 1' 51", (4) 4 7 ' 45", (5) 3 0 ' 0", (6) 9 ' 59"). 4a. B . ^ ^ ^ ? ^ ° . A.B.^nrs??!^ b. A . B . a K T I E A . B . C . D . ^ T ^ A . ^ l l % ^ B . ^ f i % c. A . B . ^ I ^ ^ ^ ° ; . D.^^iq^^afcW d. D . ' ^ W ^ . A.B.#lRTl: 5a. A.B.<*5FR^ b.
B.W^
d. A . < * i q ^ B l . q 5 | ^ B 2 . 3 . q q ^ 6a. A . B . ^ ? l i c ^
b.
C.D.#^A2.^A.B.C.D.^ A.B.«*If^^; C.D.^^frm
c. A.3f5i?rA.B.TeW^(B.^); D . ^ ^ . d. A.B.'^Pii^ (A2.-wlclB
184
VIII.6
PANCASIDDHANTIKA
T h e true M o o n is got thus:(i) mean M o o n in revolutions = (days X 38,100 - 10,984) ^ 10,40,953. (ii) mean anomaly in revolutions etc. = (days X 110 + 609)
3031.
T h i s is for sunset at Ujjain. If required for sunset at Yavanapura, 622'/2 should be used in the place o f 609, we shall explain how, later. (iii) T h e intervals o f equation o f the centre for the 6 half-signs in a quadrant are, 1° 14' 25", 1° 11' 48", r 1' 51", 47' 45", 30' 0" and 9' 59". In the first quadrant these are to be deducted in the given order, in the second they are to be added in the reverse order, in the third they are to be added in the given order, and in the fourth they are to be subtracted in the reverse order. (iv) T r u e M o o n = (i) + (iii). Example 2. For days 59, (from sunset at Yavanapura), compute the true Moon. (i) T h e mean M o o n = (59 X 38,100 - 10,984) ^ 10,40,953 = Rev. 2-1-23-36-30 = rd. 1-23-36-30. (ii) M e a n anomaly = (59 X 110 -I- 622 V2) H- 3031 = rd. 4-4-46. (iii) T h e equation o f the centre = - 1° 14'25" - 1 ° 1 1 ' 4 8 " - 1° 1'51" - 4 7 ' 4 5 " - 3 0 ' 0 " - 9 ' 5 9 " -I- 9' 59" + 30' 0" + 4° 46' X 47' 45" H- 15° = - 4 ° 0' 39". (iv) T r u e M o o n = (i) -I- (iii) = rd. 1-23-36-30 - 4° 0' 39" = rd. 1-19-36. (Note: This is for sunset at Yavanapura).
T h e rules are explained as in the case o f the Sun thus: In 1.15, it has been mentioned that in the Romaka yuga o f 2850 solar years, there are 1050 intercalary months and 16,547 supprec^'^d tithis Therefrom it has been shown, that i n the yuga there are 2850 x 12 = 34,200 solar months, 34,200 + 1050 = 35,250 synodic months, 35,250 + 2850 = 38,100 lunar revolutions, and 35,250 x 30 16,547 = 10,40,953 mean solar or civil days. So, from the proportion: If there are 38,100 lunar revolutions in 10,40,953 days, how many are there in the days from epoch, we have, the number of revolutions = days X 38,100 10,40,953.
T h e mean M o o n at epoch should be added to the mean M o o n or the time by which the M o o n completes the current revolution should be omitted from the days. According to the Romaka, by 10,984 38,100 days after epoch, the mean M o o n is a full revolution, though we cannot verify this, as the original Romaka is lost. Therefore, we have to deduct from the product o f days from epoch, (10,984 38,100) X 38,100 = 10,984, as instructed. W i t h the given deductive constant we get that the inornate mean M o o n in revolutions at epoch = (0 X 38,100 - 10,984) ^ 38,100 =ra. 11-26-12. See how close this is to the actual, rd. 11-24-48, to the Saura, rd. 11-25-6, and the Siddhdnta $iromani's rd. 11-25-49. T h i s is why we corrected the reading, krtdstanavakaikd (1984) int krtdstanavakhaika (10,984). If the reading is taken as it is as done by T S and N P then the mean M o o n at epoch would become rd. 11-29-19, which is improbable, being too far from the actual. We have also shown under I. 8-10 that the mean M o o n o f the corrected reading alone would agree with the constants there.
In the Romaka, as in the Vdsistha-Paulisa there are 110 anomalistic revolutions of the Moon in 303 days. Therefore multiplying the days by 110 and dividing by 3031, the mean anomaly of the M o o n in revolutions etc. is got. A s , according to the Romaka 609/110 days before sunset at Ujjain, it was a full revolution, we have the addidve constant 609. W e cannot understand why the anomaly alone is given for sunset at Ujjain, while it could also be given for Epoch, i.e. for sunset at Yavanapura by
VIII.7
185
VIII. R O M A K A - S I D D H A N T A — S O L A R E C L I P S E
making the additive constant 622 V2. That is why i n our rules for computation we have given this constant. Perhaps the author wanted to avoid the fraction i n the constant. T h e anomaly computed for Epoch i n revolution etc. = (0 X 110 -1- 622 V2) ^ 3031 = rd. 9-12-16. Compare this with the actual, rd. 9-9-34, Saura's rd. 9-9-47, and Siddhdnta ^iromarii's rd. 9-11-23. The intervals of the equation of the centre are given in minutes and seconds as in the case of the Sun, with the special mention of degrees where there are full degrees. B u t the text here is so corrupt that we are not certain about the numbers, since the original Siddhdnta is lost. So we have to depend much on guessing. A d d i n g the intervals we understand that in this Siddhdnta the Moon's maximum equation of the centre is 4° 55' 48". Using this, and not doing the correction to the epicycle, since it is not known, we have computed the intervals and given them hereunder, for comparison with the given values: Anomaly Computed Values Given Values
15° ri6'34"
ri4'25"
30° Til'20" Til'48"
45° 1°1'15"
60° 47'1"
75° 29'33"
90° 10'5"
ri'51"
47'45"
30'0"
9'59"
In the matter o f order o f taking the intervals and o f adding or subtracting them our remarks under the sun hold here too. r
Daily motion of the Sun and the Moon 7. T h e daily m o t i o n o f the m e a n M o o n is 7 9 0 ' , a n d that o f the m e a n anomaly, 7 8 4 ' . F o r w o r k r e l a t i n g to the d a y - t i m e t h e true d a i l y m o t i o n is the d i f f e r e n c e between the t r u e M o o n s o f t h e t a k e n d a y a n d t h e p r e v i o u s day. F o r w o r k r e l a t i n g to the n i g h t - t i m e the t r u e d a i l y m o t i o n is t h e difference b e t w e e n t h e true M o o n ' s o f the t a k e n d a y a n d the next day. T h e true daily motion, given in the second half, pertains both to the Sun and the M o o n . T h e daily mean motions of the M o o n and its anomaly alone is given because in the case o f the Sun both are the same, practically, equal to 59' 8", and well known. It would have been better if the Moon's mean daily motion had been given as 791'. T h u s , the following is intended: (i) T o get the true daily motion o f the Sun, take the last interval used i n obtaining the true Sun, divide it by 15, and apply it to 59' 8" as the quantity got from the last interval has been applied to the mean Sun. This can be taken as true daily motion for both the day-time and the night-time as there is not much difference. (ii) T o get the true daily motion of the M o o n : (a) for the day-time work, find from the intervals the equation o f the centre for the last 784' of the anomaly, and apply it to 790' as the last part of the interval itself is applied. 7a. A l . B l . W I T b. A.a>cl^ B.craW". B . m ^ i ^
c. B . ^ : ^ ; CD.<4ldl ^ 1 d. A.B1.2.^H^snTTlpT
PANCASIDDHANTIKA
186
V111.8
(b) for the night-time work, find from the intervals the equation of the centre for the 784' following, i n the anomaly, and apply it to 790' as that itself would be applied.
Example 3. The daysfrom epoch is 59, (given in the previous two examples). Find the Sun's and true daily motion, for the day gone and the day to come. (i) I n example 1, the last interval used is - 3 3 ' 55". T h e 15th part of this is - 2 ' 16". A p p l y i n g this to 59' 8", the Sun's true daily motion for both days is 59' 8" - 2' 16" = 57' (in full minutes). (ii) In example 2, the Moon's mean anomaly used is rd. 4-4-46. (a) For the day previous, the last 784' o f this begins from ra. 3-21-42. T h e equation of the centre pertaining to this part o f the anomaly = +30' X 8° 1 8 ' 1 5 ° - f - 4 7 ' 4 5 " X 4° 4 6 ' - 15° = + 16'36" + 15' 10"= + 3 r 4 6 " , (say-I- 32'). A p p l y i n g to the mean motion, 790', the true motion for the previous day = 790 + 32 = 822'. (b) For the next day, we have to find the equation o f the centre for anomaly from rd. 4-4-46 to rd. 4-17-50. T h i s is equal to, + 47' 45" X 10° 14' ^ 15° + 1° 1' 51" X 2° 50' ^ 15° = + 32' 35" -f U ' 41" = -I- 44' 16". A p p l y i n g to the mean motion, the daily motion for the day following = 791' -I- 44" = 835'. T h e instruction is easy to understand, for, clearly the difference in the longitudes of two consecutive days is the motion for the day. As, i n the Romaka, the day begins at sunset for which the longitude is computed, the day-time before sunset falls i n the day previous, and the night following sunset falls i n the day next. Hence for work i n each, respectively, the motion for the previous day and the next day has to be taken. T o avoid computing the longitudes of both days, we have given an easy method, which should have been intended by the author also, for, otherwise, he need not have given the mean motions of the M o o n and its anomaly.
Rahu 8. M u l t i p l y the days f r o m e p o c h b y 24, a d d 5 6 , 2 6 6 a n d d i v i d e by 1,63, 111. S u b t r a c t the r e v o l u t i o n s etc. o b t a i n e d , f r o m the e n d o f Pisces, (i.e. f r o m any w h o l e n u m b e r o f r e v o l u t i o n ) . T h e H e a d o f R a h u is o b t a i n e d . T h e following is instructed to be done: (i) Revolutions etc = (days X 24 -I- 56,266) 1,63,111 (ii) H e a d of R a h u = rd. 12-0-0 — Revolutions etc, omitting the full revolutions. 8a. B.'^KIlI"!^
d. A.shHI^yTllotltl; B.^HI^^sli-dl-^j^
b. A . ^ :
(B2.aiqig^°); C.*Hlcj,^t
c. Ai.?;qFPjfg
D.a5qig^?mg;#5qc»
V I I I . 11
VIII. R O M A K A - S I D D H A N T A — S O L A R E C L I P S E
187
Example 4. Compute Rahu for the moment, 59 days from epoch. (i) Revolution etc. = (59 x 24 -I- 56,266)
1,63,111 = rd. 0-4-7-19.
(ii) Head of R a h u = rd. 12-0-0 - rd. 4-7-19 - rd. 7-22-41. From this the tail = rd. 7-22-41 -I- rd. 6-0-0 = rd. 1-22-41. We have said that the Moon's node is called Rahu, on account of the connection between the two. O f the two nodes, the first is the H e a d and the second, situated six signs away, is the T a i l of R a h u . According to the Romaka, there are 24 revolutions of the Moon's nodes i n 1,63,111 days. Therefore, multiplying the days by 24 and dividing by 1,63,111 the revolutions are got. A s the motion is retrograde, what is obtained has got to be treated as negative, and therefore to be subtracted from 12 signs or full revolutions. A t the moment 56,266/24 days before Epoch, the H e a d oiRdhu was a full revolution, and i n order to reckon from that time 56,266 is added to the days multiplied by 24. As for the correctness of the numbers, we cannot verify them since the original is lost. B u t we can see how nearly correct the Romaka R a h u here given is, by comparison with that of other systems. A t Epoch the H e a d or R a h u according to the Homa/ta = ra. 1 2 - 0 - 0 - (0 X 24 -t- 56,266) -^1,63,111 = rd. 7-25-49. Actually it is rd. 7-26-0. According to the Paulisa it is rd. 7-25-59, and according to the Saura, rd. 7-26-6. T h e time for one tropical revolutions is 1,63,111 24 = 6796-17-30 days. T h e correct time is 6798-21 -48. T h e difference of 2-4-18 days is caused by the wrong constant of precession adopted by the Romaka, of 34" instead of the correct 50". T h u s , since the Romaka precession is less by five minutes i n the time taken by one revolution, its period of revolution must be about two days less as it is found to be, and the disagreement is small indeed.
(^mimn^MiMi ^ng?it mfechi otidldi -m
Parallax in longitude 9. ( T h i s is the same as VII. 1. a n d e x p l a i n e d c o m p l e t e l y t h e r e . T h e r e is n o d i f f e r e n c e i n m e a n i n g b e t w e e n the r e a d i n g s t h e r e a n d h e r e , dinamadhyamasamprdpyd a n d dinamadhyamasamprdptd).
9. Q u o t e d b y U t p a l a o n 5 S 5 . 1 8 9a. B.^T^j^HH UMI b. B.^qp;?n. A.cqtRnirti^ii IB
c. B.t|i'j|u|dl 4l«
188
PANCASIDDHANTIKA
V111.12
Declination of the Nonagesimai 10. A t any t i m e (for w h i c h the z e n i t h distance o f the n o n a g e s i m a i , Z D N , is desired,) f i n d the o r i e n t ecHptic p o i n t , O E P . A d d n i n e signs to it. ( T h i s p o i n t is c a l l e d the n o n a g e s i m a i ) . F i n d its d e c l i n a t i o n . 11. S u b t r a c t the H e a d o f R a h u f r o m the n o n a g e s i m a i , f i n d its sine, d o u b l e it, a n d a d d a s i x t h o f the q u a n t i t y got by d o u b U n g , (i.e. f i n d the latitude o f the M o o n , s u p p o s i n g it to be situated at the n o n a g e s i m a i ) . A d d this to the d e c l i n a t i o n f o u n d above i f b o t h are o f the same d i r e c t i o n , a n d subtract it f r o m the d e c l i n a t i o n i f they are o f d i f f e r e n t d i r e c t i o n s . ( T h u s the d e c l i n a t i o n o f the n o n a g e s i m a i is c o r r e c t e d ) . 12. T h e n o r t h d e c l i n a t i o n , b e i n g less a n d t h e r e f o r e d e d u c t e d f r o m the l a t i t u d e o f the place, the r e m a i n d e r ( w h i c h is the Z D N ) is s o u t h . T h e s o u t h d e c l i n a t i o n m u s t be a d d e d to the l a t i t u d e , a n d the s u m ( f o r m i n g the Z D N ) is n o r t h . T h e p a r t o f the n o r t h d e c l i n a t i o n g r e a t e r t h a n the l a t i t u d e , (i.e. the r e m a i n d e r after d e d u c t i n g the l a t i t u d e f r o m the n o r t h d e c l i n a t i o n , w h i c h f o r m s the Z D N ) , is n o r t h . Lagnatryaguvivara actually means the difference between the O E P and the H e a d o f Rahu, plus three signs. Clearly this is equal to the difference between the nonagesimai and the Head of Rahu, as translated above. Therefore, i f the reading, lagndsuravivara is adopted, the word lagna must be taken to mean tribhonalagna o r nonagesimai. If the nonagesimai is greater than the H e a d and less than the T a i l , the latitude obtained is north, otherwise south. W h y this is so has been explained in connection with finding the Moon's latitude according to the Paulisa. T h o u g h , i n a general way, the nonagesimai latitude is asked to be deducted from the declination if of different directions (instruction contained i n verse 11), i n the case where the declination is less, the declination is to be deducted from the latitude, the direction o f the corrected declination being the direction o f the latitude. T h e instructions contained i n verse 12 envisages only places north of the equator, as usual.
10-12, Quoted by Utpala o n BS 5.18
b. A 1 . C . ^ 1 ^ ^ ; B 1 . 2 . W ^ ; B2.^?(roaT; D.WyMIHMsti^lJiill^ |
10a.
A.^for^
b. B.^:^:MI
Ai.'H^dHmi-j; I B . % i w r o ^ c. A.B.^I^^'^^R'IRft d.
A . B . f ^
12a. B.^^nf^i b.
B.^for^ C.U.f^^ c. B.3TPT^
VIII.13
Vlll. ROMAKA-SIDDHANTA — SOLAR ECLIPSE
189
Thus, the following has got to be done: (i) T h e O E P for the ume for which the parallax corrected latitude is required, is found, by using the local ascensional differences. (ii) Nonagesimai = O E P -I- 9 signs. (iii) Find the declination of the nonagesimai, marking its direction north or south. (iv) Sine (nonagesimai — H e a d of Rahu) x 7/3 = latitude pertaining to nonagesimai. T h i s is north i f (nonagesimai — H e a d of Rahu) is within 6 signs, south otherwise. (v) Corrected declination = declination +, latitude, found i n (iv), (the upper sign of same direction, otherwise lower, the direction of the result being that of the greater. (vi) Z D N = Latitude of the place + corrected declinations, the upper sign i f the corrected declination is south, lower sign otherwise. In the latter case, if the latitude is greater, the direction of Z D N is south, if the declination is greater it is north). T h e work is thus explained: In computing the solar eclipse it has been mentioned under V I I . 1, that i n the place of the Moon's latitude, the same corrected for parallax has got to be used. T o get the correction the sine of the Z D N is required. F o r ease o f computation, the Romaka takes the difference between the latitude o f the place and the declination o f the nonagesimai (the directions being taken into consideration,) as the Z D N , the error being small as can be seen from the figure under V I I . 1. T h i s is given by verse 12 above. Further, the parallax correction for latitude depending on sine Z D N is o n the supposition that the M o o n moves on the ecliptic, which is only approximately true. Actually the M o o n moves i n its orbit, and a small correction has got to be made for this, and the work of verse 11 above is intended for this. Practically, all astronomers before the famous Bhaskaracarya II have given this rule, on the surmise that taking a point on the Moon's orbit, corresponding to the nonagesimai, things will be all right. But the mistake i n this has eluded all these ancient astronomers, including the astute Brahmagupta. It was Bhaskaracarya who detected their mistake, showed, by means of an example, how the rule was wrong, and gave the correct rule. (Vide the Vdsdnd-Bhdsya at the end of Suryagrahariddhikdra, Ganitddhydya, Siddhdnta ^iromani). From the rule given by verse 11, it can be inferred that according to this Siddhdnta the obliquity of the Moon's orbit, giving the m a x i m u m latitude of the M o o n , is 280 minutes, (got from: 120 x 2 (1 -f- 1/6) = 120 x 7/3 = 280). We shall see that this agrees with the rule given by verse 14, giving the Moon's latitude. But T S have adopted the incorrect reading, kharasdmsasamrnitdm and dividing the doubled sine by sixty, got the latitude, which they are constrained to consider to be in degrees. N P too, accept the same sense as T S with an emended reading kharasdptdm apakramdmsdt. B y this the m a x i m u m latitude according to the Romaka would be 4°. It is very strange that they do not see this is too far from the correct value, highly improbable i n the Romaka which they them«elves praise inordinately, and disagrees with their own (TS's) commentary under verse 14.
190
PANCASIDDHANTIKA
Vlll.14
Parallax correction and orbital diameter 13. M u l t i p l y the t r u e d a i l y m o t i o n o f the M o o n by the sin o f Z D N , thus f o u n d , a n d d i v i d e by 1800. T h i s is the "parallax c o r r e c t i o n for latitude. T h e m e a n a n g u l a r d i a m e t e r o f the S u n is 30 m i n u t e s , a n d that o f the M o o n , 34 M i n u t e s ( a c c o r d i n g to the R o m a k a ) . Thus: (i) Parallax i n latitude = sine corrected Z D N x true daily motion of the M o o n tion is that of the Z D N ) .
1800 (Its direc-
(ii) Mean angular diameter of the Sun = 30(iii) Mean angular diameter of the M o o n = 34'. (Using (ii) and (iii) the respective true angular diameters are to be found). U n d e r V I I . 1, it was explained that the parallax correction for latitude, to be used i n the solar eclipse, is obtained by multiplying the horizontal parallax of the M o o n relative to the Sun, by the sine o f the Z D N and dividing by 120, (the max. sine). It was also shown there that the horizontal parallax itself varies inversely as the distance of the M o o n from the earth, being greatest when the M o o n is nearest. H i n d u astronomers take it that the distance is inversely proportionate to the true daily motion, though this is only approximately correct. Therefore it is taken here that the relative parallax is proportionate to the modon, the Sun's parallax being very small compared to that of the M o o n . Here, the parallax correction i.e. relative horizontal parallax X sin (corrected) Z D N 120 = Moon's daily motion X sin (corrected) Z D N H- 1800. F r o m this it can be seen that according to this Siddhdnta, the relative horizontal parallax is the daily motion divided by 15. Therefore the mean relative horizontal parallax = 790'.5 15 = 52.7 minutes, as mentioned already. As for the mean angular diameters that is what the Siddhdnta has found them to be, by observation or analysis of eclipses.
14. T w e n t y o n e , m u l t i p l i e d b y the sine o f ( S u n o r M o o n at n e w m o o n ~ R a h u ) a n d d i v i d e d by n i n e is the latitude. T h i s , w i t h the p a r a l l a x c o r r e c t i o n a d d e d is the p a r a l l a x - c o r r e c t e d l a t i t u d e , w h e n b o t h are o f the same d i r e c t i o n . W h e n o f d i f f e r e n t d i r e c t i o n s , t h e i r d i f f e r e n c e is the c o r r e c t e d latitude.
14. Quoted by Utpala o n f i S 5.18 14a. A . f W < n ^ n ^ ; B.idfk1dlilfte«; 13. Quotedby Utpala on B 5 5.18 13a. A.cfsqraf; Bl.ri^qrat; B2.3.^r5qrat b. B.?1^:. A.B1.2.'MjdH°ll^:; D . ^ W P d : c-d. B . % ? F ^ : d. A.°1Wi?r; B l . 3 . ° i W 3 ;
C D . U.(d'
VIII.15
VIII. R O M A K A - S I D D H A N T A — " S O L A R E C L I P S E
191
It is stated here that, (i) T h e Moon's latitude at new moon = sin (Moon ~ Rahu) X 7 H- 3. (ii) Parallax-corrected latitude = Moon's latitude i p a r a l l a x correction given in verse 13. (The upper sign is to be taken if both are of the same direction, and the lower sign, if of different directions, the resulting direction being that of the greater). r h e latitude at new moon is the distance of the M o o n north or south of the Sun, as seen by an observer at the centre of the earth. For an observer on the surface, there is a difference in this, equal to the parallax in latitude. Therefore they have to be combined, taking the directions into consideration, to find the actual distance as observed, i.e. if of the same direction they have to be added, and i f of different directions the differences is to be taken, the direction being that of the greater. T h o u g h the author wants this to be done at new moon, as the use of the word sama-lipta indicates — perhaps following the instructions of the original Siddhdnta — it will be better if it is done at new moon corrected for parallax in longitude, that being generally nearer the circumstances. It is given that the sine of (Moon ~ Rahu) multiplied by 21 and divided by 9, (it will be easier to multiply by 7, and divide by 3), is the latitude. F r o m this, the m a x i m u m latitude according to this Siddhdnta = the m a x i m u m sine X 7 3 = 120' x 7 ^ 3 = 280'. This agrees with verse 11 above, as already said. But T S say here that the m a x i m u m is 270', contradicUng their statement under verse 11, that it is 4°, i.e. 240'. Without any reason, they assume here that the m a x i m u m is 270', and since the maximum sine multiplied by 21 and divided by 9 does not give 270', they say that the multiplier and the divisor given are approximate!! T h e same applies also to N P , vide their derivation (pt.II, p.63) of the result " i = 4° .... (3c)" and " i = 4:30° ... (10), i n contrast to (3c)" ( p t . l l . p.64). Further, TS's statement, that the parallax due to the Sun has been omitted by the author on account of its smallness, is wrong, for the intention of the author is only to give the relative parallax. The correct statement would be, ' T h e Sun's parallax has not been separately computed and deducted from the Moon's, as the difference in effect would be negligible".
True diameter of the orbs 1.5. T h e m e a n a n g u l a r d i a m e t e r s o f the S u n a n d the M o o n , respectively, m u l t i p l i e d by t h e i r t r u e d a i l y m o t i o n s a n d d i v i d e d by t h e i r m e a n daily m o t i o n s , gives the t r u e a n g u l a r d i a m e t e r s at the t i m e o f eclipse. Thus: (i) T h e angular diameter of the Sun = 30' x Sun's true daily motion ^ 59.
192
Vlll.17
PANCASIDDHANTIKA
It is a matter of experience that an object looks bigger, the nearer it is, smaller the farther away it is, i.e. the angle formed by the object at the eye is inversely proportionate to the distance. We have already mentioned that approximately the daily true motion of the Sun and the M o o n is inversely proportionate to the distance. Therefore, the angle at the eye is proportionate to the daily true motion, approximately. Hence, from the proportion, Mean motion: T r u e motion:: Mean angular diameter; T r u e angular diameter, we have, T r u e angular diameter = Mean angular diameter x true modon
mean motion,
which is the rule given. [i<^U|chlH:]
Moment of the eclipse 16. S u b t r a c t the s q u a r e o f the p a r a l l a x - c o r r e c t e d l a t i t u d e f r o m the square o f the s u m o f the s e m i - d i a m e t e r s . T h e s q u a r e r o o t o f the r e m a i n d e r , m u l t i p l i e d by two, is the n u m b e r o f m i n u t e s o f arc g i v i n g the d u r a t i o n . T h e s e m i n u t e s , m u l t i p l i e d by 6 0 a n d d i v i d e d by the m i n u t e s o f relative t r u e daily m o t i o n gives the t i m e o f d u r a t i o n i n nadikds. T h e following is to be done: (i) Minutes of arc of duration = 2 V (sum of the semi-diameters)^ - (parallax corrected latitude)^. (ii) T i m e duration in nadis — minutes of arc of duration x 60 ^ daily relative true motion in minutes of arc. (Half this, subtracted from, and added to the dme of new moon corrected for parallax gives the first and last contacts respecUvely). T h e rationale of the work has been shown in connection with the Paulisa (chap. V I I ) . We must add the following: If the latitude as corrected for parallax is found separately, each for the time of first contact and the time of last contact, and used in the work, then each will be more correct. Thus, the corrected ladtude and the time a^e interdependent, each requiring the other for its computation, and therefore the method of successive approximation is indicated here. This is not mentioned by the work, as being easily understood, or the author does not give it because it is not found in the original.
[i»ljUIMr<<^
16. Quoted by Utpala on 5 S 5.18
b. A . ^ r f ^ . B.°
16a. A 1 . B 1 . 2 . ^ : A 2 . ^
c. A . ° ^ ^ . B . t O ^
VIII.18
VIII. R O M A K A - S I D D H A N T A — S O L A R E C L I P S E
193
Eclipse diagram 17. Subtract the parallax-corrected latitude for the time o f parallax-corrected new m o o n , f r o m the s u m o f semi-diameters. T h e r e m a i n d e r i n m i n u t e s are the digits o f o b s c u r a t i o n o f the S u n by the M o o n . 18. T o r e p r e s e n t the a m o u n t o f o b s c u r a t i o n g r a p h i c a l l y , d r a w a circle o f r a d i u s e q u a l to the s e m i - d i a m e t e r o f the S u n , m e a s u r e the p a r a l l a x - c o r r e c t e d latitude n o r t h o r s o u t h a c c o r d i n g as w h e r e the M o o n is situated, a n d w i t h the p o i n t m a r k i n g its e n d as c e n t r e d r a w a circle o f r a d i u s e q u a l to the M o o n ' s s e m i - d i a m e t e r , to r e p r e s e n t the M o o n . ( T h e p a r t c o m m o n to b o t h the circles is the part o b s c u r e d , a n d its m e a s u r e i n digits is its w i d t h i n m i n u t e s o f arc.) Obscuration in digits = S u m of the semi-diameters, in minutes — parallax-corrected latitude, in minutes. (This for the time of parallax-corrected new moon). T h e Fig. to illustrate this is given at the end of Example 5, as part thereof. It can be seen from there that the amount of obscuration, A B = SB - Sy\ = SB - (SM - M A ) = SB
-I-
M A - SM
= radius of the Sun + radius of the M o o n — corrected latitude = sum of the semidiameters — corrected latitude. T h e author has taken it that one minute of arc appears to the eye as one digit, though actually the apparent size varies, ('apparent' because this is an illusion), the heavenly bodies appearing to be bigger the nearer they are to the horizon.
Example 5. After 59 days has passed from, epoch, on the 60th day, there is a solar eclipse. Comput for Pudukkottai (in S. India) (lat. 10° 23'; longitude 48° east of Yavanapura, represented by 8 nddis o time). The first things to he found are: T h e S i m and M o o n at new moon, the daily motion etc. In Example 1, we have found that the true Sun for 59 days from epoch is ra. 1-28-24. In Example 2, the true M o o n is found to be r«. 1-19-36. In Example 3, the Sun's true daily motion for the 60th day is found to be 57', and the Moon's, 835'. In Example 4, the H e a d of Rahu is found to be ra. 7-22-41. (All the three longitudes are for mean sunset at Yavanapura, that being the time o f day of epoch.) S i m — M o o n = rd. 1-28-24 — ra. 1-19-36 = 8° 48'. T h e Sun being greater, the new moon is to come. T h e relative daily motion = the difference of the true motions = 835' - 57' = 778'. 18a. B.3ISHl(dte!lTf% 17a. B1.2.°?;#lf^. A.«isif%
b. B . 1 . 2 . ^ . A . B l . 2 . ^
c. A . C . D . U . ^ ? n ^
c.
A.*aK;B.^a^
d. A.«n%^5nPr. A l . W P M ; A2.^9R?qc?T9T
d.
B.ftfe#§HM^
194
PANCASIDDHANTIKA
V l l l . 18
The nddis of new moon from mean sunset at Yavanapura, = 60 x 8° 48' H- 778' = na.40-43. T h e time of new moon from mean sunset at Pudukkottai =na.40-43 + wa.8-0 = na.48-43. i.e. on the 60th day, after mean sunrise at Pudukkottai, the new moon is at na.48-43 — na.30-0 = na. 18-43. Given the half-cara for the day, 39 vinddis, the new moon is at na. 18-43 -I- vi.39 = nd. 19.22. (No correction is made for equation of time since the Siddhanta does not give it.) At new moon, the Sun = the M o o n =rd. 1-28-24 -I- 48' = ra. 1-29-12. Head of Rahu at new moon = ra.7-22-41 - 2' = ra.7-22-39. Correction of new moon for parallax (by verse 9.): Halfday-dme is na. 15-0 + OT.39 = na. 15-39. T h e dme elapsed after noon = nd. 19-22 - na. 15-39 = na.3-43. Corresponding to this, there are 22° 18'. Sine 22° 1 8 ' = 45'32". T h e parallax bending of the new moon, (later) = 45' 32"/30 = wa. 1-31. Parallax-corrected new moon = nd. 19-22 + na. 1-31 = na. 20-53. The OEP at new moon: T h e required ascensional difference for every Drekkdria for Pudukkottai in vmadw are 90, 94, 98 for Taurus; 102, 105. 108 for G e m i n i ; 109, 110, 109 for Cancer; 108, 105, 103 for Leo; 101, 101, 99 for Virgo. T h e new moon is 1162 vinddis from sunrise. 8 vinddis after sunrise Taurus ends, 315 from this G e m i n i ends, 328 from this Cancer ends, and 316 from this Leo ends. For the remaining 195 in Virgo, the part risen is 10° + 9° 18' = 19° 18'. . • . 0 £ P = ra. 5-19-18. Nonagesimai = O E P -l-ra. 9-0-0 = ra. 2-19-18. .nn declination of nonagesimai = 48' 48" X sine (ra.2-19-18) ^ 120' = 47' 58". Declination = 23° 33', N o r t h .
Corrected declination of the Nonagesimai (verses 10-11) Nonagesimai ~ Head of Rahu = ra. 2-19-18 ~ ra.7-22-39 = ra. 6-26-39. Sine of this = sin ra.0-26-39 = 53' 49". 53' 49" X 2 (1 + 1/6) = 126', south, (since the Nonagesimai is more than 6 rdsis distant from Head of Rahu). Being of different directions, 23° 33' - 126' = 21° 27', N o r t h is the corrected declination. ZDN {hy verse 12) : corrected declination - ladtude = 23° 27' - 10° 23' = 11° 4', north (being north declination and greater than latitude). Parallax in latitude (by verse 13) = Sin Z D N X Moon's true daily modon in minutes ^ 1800 = 22' 50" X 835 1800 = 10'.6, north, (same direction as Z D N ) . The uncorrected latitude at new moon = (verse 14): M o o n - H e a d of Rahu = ra. 1-29-12 - ra. 7-22-.S9
VIII.18
195
VIII. ROMAKA-SIDDHANTA —SOLAR ECLIPSE
= ra. 6-6-33. Sine ra.6-6-33 = Sin 6° 3 3 ' = 13'41". T h e latitude = 13'41" x 7/3 = 3 r . 9 , south, (the M o o n being more than 6 rdsis distant from H e a d of Rahu). Parallax-corrected latitude = (by verse 14), 31'.9 - 10'.6 = 21'.3, south. Sum of true semi-diameters (by verse 15) : T r u e diameter o f Sun = 30' x 5 7
59 = 29'.
T r u e diameter of M o o n = 34' X 835 ^ 791 = 35'.9. Sum of semi-diameters = (29' -I- 35'.9)/2 = 32'.4. Duration (by verse 16): Minutes of arc of duration = 2 x V 3 2 . 4 ^ - 2 1 . 3 - = 2 x 24'.4 = 48'.8. T i m e of duration = 48'.8 x 60 ^ 778' = na.3-46. H a l f duration = nd. 1-53. Subtracting this from parallax-corrected new moon, first contact is, no.20-53 - nd. 19-0, after sunrise.
Sun's orb
A d d i n g to parallax-corrected new moon, last contact is, na.20-53 -I- na. 1-53 = na.22-46, after sunrise. Part obscured in digits (by verse 17): sum of semi-diameters - parallax-corrected ladtude = 32.4 - 21.3 = 11.1. Graphical representation of obscuration S = centre of the Sun M = centre of the M o o n S M = parallax-corrected latitude Moon's orb
A B = the measure of the obscuration = l " . l l = 11.1 digits Fig. VIII. 1
In this work, 1.8-10 give the 'days from epoch' according to the Romaka; 1.15, gives the elements concerning the Sun and Moon in the Rormka-yuga; V I I I . 1-8 give the true Sun, M o o n and Rahu; and V I I L 9 - 1 8 give the solar eclipse according to the Romaka. It is the 'days of epoch' of Romaka that is intended to be used everywhere in the work; since it is the distance between two points of time and therefore the same by v/hatever siddhdnta it is computed. T h e difference caused by the time of the day like 'Sunset of Ujyain', ' N o o n at Ujjain' etc. will, of course, be there, and must be taken into account. T h e agreement between 1.8-10, 1.15, and VIII.1-7, each to each, has been shov/n in the proper places. We have also shown that the Sun, M o o n and R a h u of the Romaka are tropical, though the author has not mentioned this specifically. T h e work being a manual, intended to be used not for a long period, the difference caused by precession is neglected, no reference being made to it. T h e periods being tropical, itself indicates that this Siddhdnta is foreign. T h e Sun's
196
PANCASIDDHANTIKA
maximum equation of the centre, given as 143', also in an indicator, agreeing as it does with Ptolemy's. T h o u g h the Moon's maximum equation of the centre given is 296', and Ptolemy's is 301', and thus there appears to be a difference, we are not sure that the given quantity is 296', on account of the extremely corrupt nature of the text in the concerned part. T h e r e are also lacunae in the computations intended by the author, which are to be supplied from the siddhdntas dealt with already or known otherwise. T h e method of computing the true Sun and M o o n given here is an improvement on the Paulisa.
Only the solar eclipse is dealt with here. T h e lunar eclipse is omitted probably because it is not different from that of either the Paulisa and Vdsistha given, or the Sura to be gix>en. In contrast w primitive method of the Paulisa, the Romaka method of computation of the solar eclipse is far advanced, and almost the same as that of the later siddhdntas like the Aryabhatiya or the Saura. F instance, the parallax in latitude is correctly sought to be computed by using sine Z D N , though the Z D N itself is approximate, being got by combining the latitude of the place and the declination of the nonagesimai. Only the method given for correcting the declination for the nonagesimai to compensate for the M o o n being situated on its own orbit instead of the ecliptic, is wrong, as commonly seen i n works of authors prior to Bhaskaracarya II. M a k i n g the parallax in latitude and the Moon's true angular diameter depend on the Moon's true motion, and the Sun's true angular diameter on the Sun's true motion, is in accordance with the later siddhdntas, though giving the respective mean diameters as 34' and 30' is very rough. T h e first contact, middle, and last contact, as also the directions of the points of contact, are intended to be taken from the Vdsistha-Paulisa, not being given here. T h e omission of tfie total or annular phases does not matter, since they cannot be got correctly by the rougfi methods given. Further, let us not m i n d the omission of the successive approximation to be done in the computation of the circumstances, though necessary as shown. (This may be because it is not found in the original or easily understood to be necessary). But it will certainly be better to use in the computation the parallax-corrected latitude of the new moon corrected for parallax, insteadof thatof the uncorrected new moon as given by the text, the former being generally nearer the time of the thing computed. We do not know why the author has not said so. Inspiteof all this, the Romaka is interesdng as being comparatively more ancient, and forming a link between the earlier and the later Siddhdntas.
1. C o l . A . T t i F F i w ^ s ^ m w i O T g ' i ^ : ; B.C.D.lf^-
Thus ends Chapter Eight entitled 'Romaka-Siddhanta: Solar Eclipse' in the Paiicasiddhantika composed by Varahamihira
Chapter Nine SAURA-SIDDHANTA
SOLAR ECLIPSE
Introductory In the first portion of this chapter the Sun, M o o n and Rahu according to the Saura Siddhdnta are given, and in the latter portion, the computation of the solar eclipse according to the same. In agreement with the author's statement in his Introducdon to the PS, that the tithi got by the Saura is very accurate, we see that not only the tithi but most other constants as well are wonderfully accurate, and approximate closely to the modern values. A m o n g the five Siddhdntas this is the only one that uses epicycles to compute the Equation o f the centre of the Sun and the M o o n , and later i n chapter X V I I , the Equation of the centre and equation of conjunction of the 'star-planets', followed later by astronomers like Aryabhata. T h e Ardhardtrika-paksa of Aryabhata, expounded by Brahmagupta in his Khandnkhddyaka, follows this Saura-Siddhdnta i n its constants. T h o u g h the computation o f Days from Epoch ('days') has not been specially given for the Saura, (the rule given in 1.13 not being clear whether it is related to the Saura or not), yet from the Fw^a-elements of the Saura in 1.14, it is possible to formulate rules for 'Days from Epoch', following the Saura, as has been shown by us in our Notes under 1.14. We have also explained how the 'days from Epoch' obtained from the Romaka or Paulisa rules can be used for the Saura also, provided we bear in mind the variation in time of commencement of the Epoch, as for instance, that the epoch for the mean Sun and M o o n , their apogees, and the Moon's node is for mid-day at Ujjain, and for the star-planets it is mid-night. Now, the author, intending to deal with eclipses, gives first the Sun, M o o n and R a h u on which eclipses depend, beginning with the mean Sun.
Mean Sun I. A c c o r d i n g to the Saura-Siddhdnta, to get the m e a n S u n i n r e v o l u t i o n s etc., m u l t i p l y the days f r o m E p o c h by 8 0 0 , d e d u c t 4 4 2 , a n d d i v i d e by 2,92,207. T h i s is f o r U j j a i n m e a n n o o n . I. Paraphrased by Utpala or BS 2, p.65. la. A . B l . ^ . A2.?ra^
c. A . B 1.2. °^rflf«R^9; C . d. A . ^ ; B l , 2 . ^ . A 2 . f ^ . A l . ° ^ ; B . o ^
198
IX.4
PANCASIDDHANTIKA
That is, take the days from E p o c h got by the Romaka or Paulisa rule. T h e mean Sun at Ujjain mean noon, just preceding the Epoch, (i.e. sunset at Yavanapura beginning Monday), = (days X 800 - 442) ^ 2,92,207, in revolutions etc.
Example 1 (a). Find the mean Sun, given days from epoch, 5,28,931. (This is midday, Ujjain, 21A.D.). (b) Get the mean Sun for Ujjain mean noon, just preceding the Epoch, i.e. for zero day. What Epoch, i.e. sunset at Yavanapura? (a) Mean Sun = (5,28,931 x 800 - 442)
2,92,207 = revol. 1448-1-5-15.7 = rd 1-5-15.7.
(b) M e a n Sun at Ujjain mean noon preceding Epoch = (0 X 800 - 442) ra 11-29-27-3.
2,92,207 = - 32'.7 =
Since mean sunset at Yavanapura is nd. 7-20 later than that at Ujjain, the mean motion for nd 22-20, about 22', has to be added, and the required mean Sun at Epoch is rd. 11-29-49.3. (According to modern astronomy it is 11-29-37.2, assuming that at that period the vernal equinox coincided with the First point of Mesa. See how accurate the value is.)
The rule is explained thus: We showed under 1.14 that i n the Saura yuga of 1,80,000 years, i.e 1,80,000 mean solar revolutions, there are 6,57,46,575 days. Therefore, the revolutions for the given days = days X 1,80,000 H- 6,57,46,575 = days X 800 ^ 2,92,207, reducing the numerator and denominator by the factor 225. N o w , according to the Saura, the revolution of the mean Sun was completed 442/800 days after Ujjain mean-noon prior to Epoch. Therefore 442 eight hundredth parts have to be subtracted from the total number o f eight-hundredths, and hence the deduction of 442. T h o u g h the original Saura is not obtainable now, yet from the Ardhardtrika system of Aryabhata, and the Khandakhddyaka following it, we can see that 442/800 day after the said Ujjainmean noon the mean Sun's revolution was completed. T h e Epoch was near the end of ^aha 427, i.e. 427 + 3179 = 3606, Kali years gone. Kali began with Friday, Ujjain mean mid-night. For 3606 revolutions, the days (from the beginning o f Kali) = 3606 X 2,92,207 H- 800 = 13,17,123 42/800. Dividing out by 7, we have the remainder 3 42/800, i.e. 42/800 days after midnight ending Sunday, the revolutions was complete. Since Sunday mid-day is half a day or 400/800 day earlier than midnight, it is 42/800 -f- 400/800 = 442/800 day earlier, than the time o f full revolution, as we have taken and used. Incidentally, we also got that it is Sunday mean noon, agreeing with the fact that the Epoch is at sunset at Yavanapura on that day. Further, we get that according to this Siddhdnta the length of the year is 2,92,207 ^ 800, days = 3 6 5 - 1 5 - 3 1 . 5 days.
IX.4
IX. . S A U R A - S I D D H A N T A — S O L A R E C L I P S E
199
Mean Moon 2. M u l t i p l y the D a y s by 9 , 0 0 , 0 0 0 , d e d u c t 6,70,217, a n d d i v i d e by 2,45,89,506. T h e a p p r o x i m a t e m e a n M o o n i n r e v o l u t i o n s etc. is got. 3. M u l t i p l y the D a y s by 9 0 0 , a d d 2 2 , 6 0 , 3 5 6 , a n d d i v i d e by 2 9 , 0 8 , 7 8 9 . T h e a p p r o x i m a t e M o o n ' s a p o g e e i n r e v o l u t i o n s etc. is o b t a i n e d . 4. M u l t i p l y the r e v o l u t i o n s o f m e a n M o o n by 5 1 , a n d d i v i d e by 3 1 2 1 . T h e r e s u l t i n g seconds o f arc are to be subtracted to get the exact m e a n M o o n . M u l t i p l y the r e v o l u t i o n s o f a p o g e e by 10 a n d d i v i d e by 2 9 7 . T h e r e s u l t i n g seconds are to be a d d e d to get the exact a p o g e e . The following are the formulae:
X
(i) Mean Moon in revs, etc = (Days x 9,00,000 - 6,70,217) 5 1 " - 3121.
2,45,89,506 - number of revolutions
(ii) Moon's apogee in revs. etc. = (Days x 900 + 22,60,356) ^ 29,08,789 + number of revolutions got X 10" 297. Example 2. (a) Days from Epoch = 5,28,931: Eind the mean Moon, (b) Find the mean Moon for Ujjain mean noon, immediately prior to Epoch, and for Epoch. (a) By formula (i), the approximate mean M o o n = (5,28,931 x 9,00,000 - 6,70,217) ^ 2,45,89,506 = revs.f 19,359 - 4-11-31. = T h e subtractive seconds = 19,359 X 51 ^ 3,121 = 316. .-. Exact Mean M o o n = ra.4-11-31 - 316" =
ra.4-11-25-7.
(b) By formula (i), for the said Ujjain mean noon the approx. mean M o o n = (0 X 9,00,000 — 6,70,217) - 2,45,89,506 = - ra.0-9-48.8 = ra. 11-20-11.2 A d d i n g the mean modon of apogee for 22 1/3 nadikds, 4° 54'.3, the mean M o o n at epoch = rd. 11-25-5.5. (Note: T h e actual apogee got by modern constants is rd. 11-24-47.6, and we see the difference is only 18 minutes.)
2b. A . ^ ^ A.m^; c.
B 1.2.^0^: B 3 . W ^
B 3 . t | ^ ^ i | ; B1.3.^f?Af%\q; D.^<<4l*^Ps^'M
4a. B 1 . 2 . 1 ^ 9 1 ° ; B 3 . ^ ^ ^ b.
D.
3a. A . B 2 . 1 I N b. A.'juil=l
c.
A . ^ .
c.
A.«kliei«<
d.
A . B . ^ 5 ^ ^ ^ ^ R ^ (B.'iPRt'ft);
d.
A.^T^lfe^'?!
A.t^;B1.2.C.D.^Sraffjf;
B 2 . i ^
200
PANCASIDDHANTIKA
IX.4
Example 3. (a) Days 5,28,931. Find the Moon's apogee, (b) Find the Moon's apogee for Ujjain me prior to Epoch, and for Epoch. (a)
By (ii) the approx apogee in revs. = (5,28,531 X 900 + 22,60,356) ^ 29,08,789
= Revs. 164-5-5-33.1 T h e additive seconds = 164 X 10 -4- 297 = 6. A d d i n g , the exact apogee = rd. 5-5-33.2. (b) For the said Ujjain mean noon, the apogee = (0 X 900 + 22,60,356) ^ 29,08,789 = rd. 9-9-45. A d d i n g 2'.5, the mean motion of apogee in 22 1/3 nddikds, the apogee at Epoch = rd. 9-9-47.5. (Note: The actimlposition was rd. 9-9-34. See how close this is.)
The explanation for the formula relating to the mean Moon is as follows: It was shown under 1.14 t in the Saurayuga consisdng of 6,57,46,575 days there are 24,06,389 revolutions of the M o o n . For the sake o f convenience, the author has first assumed that in whole numbers there are 9,00,000 revolutions in 2,45,89,506 days, intending to give a correction as a second step. Therefore we get that i n 6,57,46,575 days there are 6,57,46,575 x 9,00,000 H- 2,45,89,506 revolutions = rev. 24,06,389-0-10-55-27. T h u s we get 10° 55' 2 7 " more than what we should get, and this has to be deducted, proportionately to the revolutions got. For one revolution the deduction is, 10° 55' 2 7 " /24,06,389 = 39,327"/ 24,06,389. In the place of this fraction the author gives the approximate but simpler fraction 5 1 " / 3121, since the error caused will be only plus 4 " in the yuga. T h e manuscript reading, kharkdgni if read as khdrkdgni as done by T S and N P , ( = 51 "/3120) will cause an error of minus 8", which also is negligible but unlikely, since the author then would have given the reduced form, 17"/1040. That is why we have read it as kvarkdgni, 3121.
The deduction of 6,70,217 is explained in the manner of the Sun's deduction: We have seen that a end of Saka 427, the end of 3,606 solar years from the beginning of Kali fell 42/800 days, i.e. nd. 3-9, after Ujjain mean midnight after Epoch. U n d e r 1.14 it was shown that according to the Saura there are 24,06,389 revolutions of the M o o n in 180,000 years. Therefore, i n 3,606 years the revolutions gone are 48,207.992966. A t the beginning of Kali, the M o o n , like the Sun, began a revolution, according to the Saura. So, .007033 revoludon remains to be completed now. We have seen that for 2,45,89,506 fractional parts there is one revolution. So, for .007033 revolution, the parts to go are 2,45,89,506 x .007033 = 1,72,946. These must go after the completion of the solar year to complete the revolution. But the year ends nd. 3-9 -f nd. 30 = nd. 33-9 from mean noon. In one day, there are 9,00,000 parts, and forna. 33-9, the parts to go are 9,00,000 X 33.15 60 = 4,97,250. Therefore at mean noon the parts to go for completing the revolution are 1,72,946 -I- 4,97,250 = 6,70,196. Since these have to go, this number is deducted from the total parts got by multiplying the days by 9,00,000. Here, the author gives 6,70,217 arrived at by using approximate work in the place o f 6,70,196, for the difference is small, the error caused being only minus one second in the yuga. Now for the explanation of the rule to get the longitude of apogee: We do this using the element giv in Aryabhata's Ardhardtrika system, or which is the same, in the Khandakhddyaka, since this is not given in 1.14, and the original Saura is not available. F r o m them we learn that in the Mahayuga of
IX.6
201
IX. SAURA-SIDDHANTA — SOLAR ECLIPSE
1,57.79,17,800 days there are 4,88,219 revolutions of the Moon's apogee. If the approximate rule given as the first part is used, we get that there are, 900 X 1,57,79,17,800 H- 29,08,789 revolutions = rev. 4,88,218-11-25-26-48 for the Mahayuga. But this is 4° 33' 12" less than the correct value, and this latter has got to be added, per Mahayuga, i.e. for 4,88,219 revolutions. Therefore the addition for the revolutions gone is, revolutions gone x 16,392" 4,88,219. In the place o f this fraction the author gives 10"/297, as the difference is very small, for by using this the error will be only plus 2" in the Saurayuga of 1,80,000 years, which is negligible, especially i n the apogee. If, instead of our (as also NP's) emendation, svararandhrayama, we make another emendation vasurandhrayama giving the fraction as 10"/298, then it will be very correct. As for our reading randhra i n the place of the author's t ^ i r a , it is necessary since otherwise there will be an error of plus one degree and a half in the Mahayuga. T h a t is why T S have given the emendation svaranandayama meaning the same as our reading, but randhrafitsthe letters better than nanda. The correctness of the ksepa is shown hereunder: 3606 years of Kali ended nd. 3-9 after Ujjain midnight next to Epoch. T h e revolutions o f apogee for 3606 years = 4,88,219 X 3606 ^ 43,20,000 = 407.527248611. A t the beginning of K a l i the longitude of apogee was 0.25 revolutions. Therefore at nd. 3-9 after the said Ujjain midnight, the longitude is 0.25 -(- 0.527248611 = 0.777248611 rev. The fractional parts (af 900 per day), for 0.77248611 rev. = 29,08,789 x 0.777248611 = 22,60,852. This is the ksepa to be added at the end o f the year. B u t Ujjain mean moon, for which we want the apogee, is nd. 33-9 earlier, and the parts for this interval = 900 X 33.15 60 = 497 has to be deducted..-. the ksepa is 22,60,355. T h e author gives 22,60,356, which differs by only one unit and causes practically no difference.
'(^)^^^c^^^r^|u?c^-T|(TlT)q^?T''q(%>) M ^ T I ^ T I ^ :
( ^ ) ( d f a c « < ^ # P r t T l g [ ^ : W ( M ) ? T f ^ || ^
Rahu: Maximum latitude 5. M u l t i p l y the days f r o m E p o c h by 2 7 0 0 , a d d 6 3 , 1 3 , 2 1 9 a n d d i v i d e by 1,83,45,827. R e v o l u t i o n s etc. are o b t a i n e d , to be u s e d i n g e t t i n g R a h u . 6. T h i s d e d u c t e d f r o m twelve rdsis is the R a h u - h e a d (i.e. a s c e n d i n g n o d e o f the M o o n . ) R a h u - h e a d p l u s soLtdsis is the R a h u - t a i l (i.e. d e s c e n d i n g n o d e ) . A t the ( m a x i m u m ) distance o f 9 0 ° f r o m R a h u (the n o d e ) , the M o o n ' s l a t i t u d e is 2 7 0 m i n u t e s (i.e., this is the m a x i m u m latitude.) 5a. A l . B 1 . 2 . f e W ^ ( B 1 . 2 . ° ? l ^ ) C . ^ . B . ^ b. A.?S?II^:; B.<^H!fKI:; C.^S?1sej:|; D . c ^ ^ B.3,%^ (B2.3.o%:)
C. ^Iflf^: D . ° s ^ : ABl2 ^ K ^ ^ ^
^ D . ^ fori
c. A . ^ ; B . ^ ; C . o m ^ ; D . ^
^ A . ^ ; B l . 3 ^ ; B2.3.T??f!T
A.^^3I^ d. A.(i«iRl^Tt>HKH5l:; ( A 2 . w r : ) B.l]pw
C . ^ ; D.fdPR d. A . B 1 . 2 . W c n ^ W ^
202
PANCASIDDHANTIKA
T h e H e a d of R a h u i n revolutions etc. = - (Days X 2700 + 63,13,219)
IX.( 1,83,45,827.
T h e tail of R a h u = the above + 6 rdsis.
As for Moon's ladtude, for a m a x i m u m moon —Rahu, equal to 90°, there is the maximum latitude, 270'. For other differences, lat = 270' sin (Moon ~ Rahu) 120, as given i n verse 25, which reduces to, lat = 9 sin (Moon ~ Rahu)/4. T h i s is given by the Saura, and followed by all later Siddhdntas. Example 4. Compute Rahu (a) for Ujjain mean noon prim to Epoch, and (b) for Epoch. (a) In this case, days from Epoch is zero. .-. Head ofRahu i n revs. = - (0 x 2700 + 63,13,219)
1,83,45,827 = - ra.4-3-53-3 = ra.7-26-6-57
(b) Since the Epoch is nd. 22-20 later, the motion for this interval, rev. 67/180 X 2,700 ^ 1,83,45,827 = r I I " has to be deducted.
Rahu-head according to the Saura for the time of Epoch, viz. mean sunset at Yavanapura, is rd. 7-26-5-46. Actually it is rd. 7-26-0, and the difference is within 6'. T h e calculation of the latitude explained in the context of the computation of eclipses.
The rule of Rahu: Like that for the apogee, this rule must be derived from the constants given in th two works that follow Saura since the original Saura is lost. In the Mahayuga consisting of 1,57,79,17,8 days, there are 2,32,226 revolutions o f R a h u , (i.e. Moon's nodes). U s i n g the rule for Rahu here, we get, 2700 X 1,57,79,17,800 H- 1,83,45,827 = rev. 2,32,226-0-0-46-2 o f R a h u peryuga. This is 46' 2' more than what we should get, but neglected by author as being small especially i n a karana intended to be used for a comparatively short period, considering the fact that even in 10,000 years the error is only 6", which will not affect the result. That is why the second step of correction is not given, unlike i n the case of the mean M o o n and apogee. If a correction is wanted here also, multiply the revolutions by 10, divide by 848, and add the resulting seconds to Rahu. O r , instead of using 1,83,45,827 as divisor use 1,83,45,827.2 i.e., i n the rule, take the multiplier to be 27,000, ksepa 6,31,32,190, and the divisor 18,34,58,272. A t the end of 427 Saka or K a l i years 3606, the revolutions to get R a h u = the longitude at the commencement + the revolutions i n 3606 years. = 1/2 + 3606 X 2,32,226 + 43,20,000 = 1 9 4 + 1,23,913/3,60,000.
Omitting the full revolutions, the parts for the fraction remaining are the ksepa, for the end o 3606 years Kali.
Since there are 1,83,45,827 parts for a revolution, the parts of ksepa =1,83,45,827 X 1,23,913 3,60,000 = 63,14,684.
Since we want the ksepa for Ujjain mean noon, nd. 33-9 earlier, we have to subtract the parts fo this time. Since there are 2700 parts i n a day, forna. 33-9 we have nd. 33-9 X 2700 ^ na. 60 = 1492 parts. .-. the ibepa for mean noon is 63,14,684 - 1492 = 63,13,192.
IX. SAURA-SIDDHANTA — SOLAR ECLIPSE
IX.9
203
The author gives 63,13,219, the difference, 27 parts, giving a difference o f 2" i n longitude being vary small; for by neglecting a small fraction equal 1/7 i n the divisor to make it a whole number, can give this difference. The readings here are extremely corrupt: O u r explanation itself will show that the emendations we have made are necessary. We have read dvighanagaja as trighanasala while T S give the correction trighanadasa. T h e textual reading, carayamavasubhutdrnavaguvMhrtibhakta is corrected by us as svarayamavasubhutdrnavagunadhrti-bhakte. B u t T S give the correction yamavasubhutdrnavagunadhrtibhih. Here it is improper o n their part to omit two letters cara though they require this omission since in trighanagaja they have given dasaiorgaja, instead of sata given by us. N o t h i n g is gained by reading dasa instead ofsata for gaja. Further, by omitting cara which is a corruption iorsvara, the number 7 in the unit's place is omitted by them, with the result that i n theyuga an error of plus 30° and more is caused in R a h u , while it is actually 46' 2" according to our correction. N P make the correct emendation trighanaiataghne but emend cara to kara. We have corrected dahanasabdah as daharuarasa which fits the rule as shown. But T S content themselves with remarking that here the numbers o f the ksepa cannot be determined owing to the extreme corruption o f the text. N P have made the emendation dahanasat here, which too will serve the purpose. That the H e a d o f Rahu obtained by deducting what is got from 12 rdsis has been explained i n dealing with the Paulisa. We read sahati in the text as navati, since the difference o f 90° between moon and Rahu gives the m a x i m u m ladtude, which is 270' according to the Saura, as also i n all later H i n d u Siddhantas like the Aryabhatiya. O r we may read it as mahati, since the greatest difference, viz. 90° will give the greatest latitude, viz. 270'. B u t T S read it as sahita, and give something farfetched and unacceptable. N P emend sahati as timira, which neither accords with the lettering of the manuscript nor give the sense 90° required here. T h a t the latitude is proportionate to the sine o f (Moon ~ Rahu) has already been explained in the context o f the Romaka, and will also be shown below, in verse 25 o f this chapter.
3 r 8 i ^ T:I*WIVJ
II6 II
(True Sun and Moon) 7. T h e m e a n l o n g i t u d e o f the S u n minus 8 0 ° is c a l l e d the S u n ' s (mean) a n o m a l y . T h e m e a n M o o n minus its a p o g e e is its (mean) a n o m a l y . M u l t i p l y the sine o f the a n o m a l y o f the S u n b y 14, a n d that o f the M o o n b y 3 1 . 8. D i v i d e e a c h b y 3 6 0 , a n d find t h e i r arcs. P u t the S u n ' s arc i n t w o places, f o r s u b s e q u e n t use. T h e arc o f e a c h is to be d e d u c t e d f r o m its m e a n l o n g i t u d e i f
204
IX.9
PANCASIDDHANTIKA
its a n o m a l y is less t h a n sixrasis, a n d a d d e d i f m o r e t h a n sixrasis. ( T h e true S u n a n d M o o n at U j j a i n m e a n n o o n is got.) 9. M u l t i p l y the Sun's arc, k e p t aside i n o n e place, by the Sun's true daily m o t i o n , (in m i n u t e s ) , a n d that k e p t i n the o t h e r place by the M o o n ' s true d a i l y m o t i o n (in m i n u t e s ) . D i v i d e each by 2 1 , 6 0 0 . A d d o r subtract the r e s u l t i n g m i n u t e s i n the respective t r u e l o n g i t u d e f o u n d , a c c o r d i n g as the Sun's arc was first a d d e d o r subtracted. ( T h e t r u e S u n a n d M o o n at U j j a i n true n o o n is obtained.) The following are the formulae: (a) To get the true Sun: (i) Mean Sun — 80° = Sun's anomaly. (ii) Sine Sun's anomaly x 14 360 = sin Sun's equation of the centre. Its arc is the equation of the centre. (Eq.C). (iii) M e a n Sun + Sun's equation of the centre = true Sun at Ujjain mean noon. (The upper sign, if the anomaly is less than 6 rdsis, lower i f more.) (iv) iii + Sun's equation of the centre X Sun's daily motion i n minutes -^ 21,600 = T r u e Sun at true mean noon. (Addition or subtraction as in iii.) (b) To get true Moon: (i) Mean M o o n — Moon's apogee = Moon's anomaly. (ii) Sine Moon's anomaly X 31 360 = sine Moon's equation of the centre. Its arc is the equation of the centre. (iii) M e a n M o o n + Moon's equation of the centre = true moon, at Ujjain mean noon. (The upper sign, i f the Moon's anomaly is less than 6 rdsis, lower i f more.) (iv) i i i + Sun's equation of the centre X Moon's true daily motion i n minutes 21,600 = T r u e M o o n at true noon. (Addition or subtraction as in (a) iii). Example 5. Days = 5,28,931. Eind the true Sun at true noon, Ujjain. From example I (a) the mean Sun = rd. I-5-I5.7. T h e longitude of Sun's apogee is 80°. From these two: (a) (i) T h e Sun's mean anomaly = ra. 1-5-15.7 — 80° = ra. 10-15-15.7. (ii) Since anomaly = sine rd. 10-15-15.7 = Sin rd. 1-14-44.3 = 84' 27", .-. Sine equation of the centre = 84' 2 7 " x 14 ^ 360 = 3' 17". .-. Equation of the centre = arc 3' 1 7 ' = r 3 4 ' . l . (iii) T r u e Sun at mean noon = rd. 1-5-15.7 -I- 1° 34'. 1 = rd. 1-6-49.8, (addition because the anomaly is greater than 6 rdsis). (iv) T r u e Sun at true noon = ra. 1-6-49.8-1-1° 34'. I X 57.4 ^- 21,600 = ra. 1-6-49.8-1-0'.3 = ra. 1-6-50.1. (That the daily motion of the Sun is 57'.4 will be given under verse 13, below.) 8a. B.TJ?R?f A.cTS^C.fefelW; B . f S J ^ A.B.C.TRT^^; D.^Wlf^ b. A . B1.2.^?5^: A l . ^ ; A 2 . ^ ^ ; B.'gfee c. A.Bl.°c1^3qi^ d. A l . ^ Q t a ; A2.1ipraT
c. 9b.
B.omWt B.'Ti^iW. A.B.^Q^gifa?
d. A 2 . ^
IX.9
IX. S A U R A - S I D D H A N T A — S O L A R E C L I P S E
205
Example 6. Find the true Moon at true noon at Ujjain, the Days being 5,28,931. From example 2 (a), the mean M o o n for 5,28,931 days gone is rd. 4-11-25.7. From ex. 3 (a) the Moon's apogee for the given days gone is, rd. 5-5-33.2. From these, (b) (i) T h e Moon's anomaly = rd. 4-11-25.7 - rd. 5-5-33.2 = rd. 11-5-52.5. (ii) Sine anomaly = sine rd. 11-5-52.5 = sine rd. 0-24-7.5 = 49' 2". Sine equation of the centre = 49' 2" X 31 - 360 = 4' 13".3. Equation of the centre = arc 49' 2" = 2° 1'. (iii) T r u e M o o n at mean noon = rd. 4-11-25.7 + 2° V = rd. 4-13-26.7. (iv) T r u e M o o n at true noon = ra. 4-13-26.7 + l°.34'.l x 729.1 ^ 21,600 = ra.4-13-26.7 -I- 3'.2 = ra.4-13-29.9 (That the Moon's daily motion is 729'. 1 will be seen from example under verse 13 below. T h e addition is as the Sun's Eq.C.) It should be noted here that the apogee of the Sun, given as 80° is too far from the correct apogee for the time of the work viz. 77° 19'. T h e r e is no doubt about the reading here, since the Ardhardtrika and the Khandakhddyaka too give 80°. So much error is unbelievable in the Saura, and must be explained thus: A t first the practice might have been to get the mean longitude of the Sun for the days from the commencement of the true solar year and 80° deducted to get the anomaly, for this would be equivalent to deducting about 77° 50', (since the E q . C at this time is about 2° 10'), from the correct mean Sun, not much different from the correct 77° 19' to be deducted. Later, by some mistake, the deduction of 80° was instructed to be done from the correct mean Sun itself. T h e apogee for the time computed by the Modern Surya Siddhdnta is 77° 15'. From the instruction to muldply the sine of the Sun and Moon's anomalies by 14 and 31, respectively, and divide by 360, to get the sine of the respective equation of the centre, we see that this Siddhanta actually uses epicycles like the Aryabhatiya etc, though not mentioning the word, and we can say that epicycles appear in the H i n d u Siddhantas for the first time in the Saura, and the others following using epicycles and excentries. T h e Modern Surya Siddhdnta gives the same degrees of epicycle for the Sun, but 32° for the M o o n instead of 31°. Further, in the Saura, the epicycle is uniform, while in many Siddhantas like the Aryabhatiya there is difference between the degrees at the ends of odd and even quadrants. For instance, the degree of epicycle mentioned above for the Sun and the M o o n i n the Surya Siddhdnta is for odd quadrants, being less by 20 minutes at even quadrants. T h e computations mentioned above can be simplified, since the multiplier and the divisor are constants and small arcs are proportionate to the sines. T h u s , we can get the Sun's E q . C . i n minutes by multiplying its sine anomaly by 1.114. In the example, multiplying 84' 27" by 1.114 we get 94' 6", the equation of the centre. We can get the Moon's E q . C . by multiplying its sine anomaly by 2.467, and i f the result is i n excess of 225 minutes, adding 1/235 of the excess to the result. In the example, multiplying 4 9 ' 2" by 2.467, we get the equation o f the centre, 121' 3". In the same way, we find the Sun's m a x i m u m equation of the centre to be , 120' X 1.114 = 133'.7. T h e correct maximum for the period of our author is 119'.5. T h e large difference is due to the Moon's A n n u a l Equation being wrongly applied to the Sun with its sign changed, in H i n d u astronomy, as already alluded to, since by doing so the tithi is not affected, the constants having been derived by the analysis o f the syzygies, which are, in essence, ends o f particular tithis. A d d i n g the m a x i m u m A n n u a l equation to the correct equation of the centre of the period, we have 131 '.5. See how close this is to the value, 133.7 of the Saura, and how far from the 140' o f the Paulisa, and the 143' of the Romaka and of Ptolemy.
206
PANCASIDDHANTIKA
IX.9
In the same way, the m a x i m u m of the Moon's equation of the centre is 120' X 2.467 + (120' X 2.467 - 250) ^ 235 = 296' + .3' = 296' .3. This too was determined by analysis of syzygies at the occurrence of eclipses. According to modern astronomy, the mean of the maximum equation of the centre of the M o o n at true syzygies is 297'.3, a difference of only one minute! (At mean syzygies it is303'.5).
Epicyclic theory
We shall now proceed to explain the epicyclic theory of planetary motion, used by this Siddhdnta, and show how it works, by relating it to the modern theory, which latter is as follows: T h e earth and the other planets like Mercury etc. move r o u n d the Sun in eclipses, with the Sun at one of the two foci. T h e point nearest to the Sun on the ellipse is the perihelion, and the most distant, aphelion, which, from the point of view of the earth, are called the perigee and apogee, respectively. In the same manner, the M o o n moves in an ecHpse r o u n d the earth at one focus. T h i s fact relating to the planets was first discovered by the European astronomer Kepler and is called Kepler's First Law of planetary modon. T h e line j o i n i n g the Sun and the planet (or the earth and the Moon), called the radius vector, sweeps equal areas i n equal time. This is Kepler's Second Law of planetary motion. F r o m this it can be readily inferred that the motion of the planet is swiftest at perihelion (or perigee for the Moon) and lowest at aphelion. Kepler's Third Law, that the square of the periodic dme of the planets round the Sun is proportionate to the cube o f the distance, is not wanted for our purpose here. T h e celebrated astronomer, Newton, showed that all the three laws follow from his Theory of Universal Gravitation, that all bodies attract one another with a force proportionate to their masses, and inversely proportionate to the square of the distance between them. But ancient Indian astronomers held the view that the earth is the centre r o u n d which the M o o n , Sun and planets move. A l l the motion is i n circles, and uniform. T o explain the non-uniformity of the apparent motion caused by the equation o f the centre, it was assumed that these bodies moved in circles called epicycles, (manda-vrttas), the centres o f which moved i n circles r o u n d the earth as centre. I n the case o f the star-planets, another set o f epicycles called epicycles of conjunction or sighra-vrttas were assumed, the effect of which is to convert helio-centric positions into geocentric. As the observers are on the earth, it is geocentric positions that are wanted, and therefore given, whether by H i n d u astronomy or by modern western astronomy. T h e inaccuracy in the positions given by the former is due to unawareness of the elliptic motion and the inability to observe accurately for want of adequate instruments, with the result that small errors i n the constants accumulated in course or time, to give large errors. It has been said that whether the heliocentric theory is adopted or the geocentric theory, the result in so far as this goes, is the same. T h e n , it may be asked, is it not better to adopt the geocentric theory which agrees with our perception? N o . T h e r e is a clinching proof for the motion of the earth r o u n d the Sun, i n the phenomenon of aberration which makes the Sun and star planets appear to be a litde i n advance o f their real positions, the quantity being so small that very accurate measurement is required to find it, capable only by modern instruments. T h e heliocentric theory is also simpler, and satisfies the requirement of least assumption. A d h e r i n g to only circular motion, so satisfying to their minds, the ancients had to get the equation of the centre, caused by the motion on the ellipse. T h e y sought to achieve it in two ways, by using epicycles, as indicated already, or excentric circles, or both. H o w the ex-centric is used for the purpose is explained as follows:
IX.9
IX. S A U R A - S I D D H A N T A — S O L A R ECl IPSE
207
Fig. IX. 1-a.
T h e earth E , is the centre of the Orbit-circle, of'radius' equal to the sine of T h r e e raiw (120' in this work). O indicates the first point of Mesa, in the direction E O . E A is the direction of apogee. A , O E A being the longitude of apogee (P is the perigee). X is the centre of the ex-centric circle on which the Sun, M o o n or star-planets move uniformly according to the mean motion. X is on E A , at a distance, towards A , equal to the sine of the maxi-minimum Eq. C . S is the position of the body, angle S X O ' being the mean longitude of the body. X O ' also is directed to the first point of Mesha. E O and X O being practically parallel. A S X = mean longitude - longitude of apogee = mean anomaly. S E O is the true longitude, which has got to be found. Since X O ' and E O are parallel, S E O = S X O ' - X S E , where X S E is the E q . C . Since X S E changes sign on the right hand side of P A , S E O = S X O ' -l-XSE on that side. T h u s we have that in the first case, when the body is from apogee to perigee, i.e. when the anomaly is less than six rdsis, the equation of the centre is subtractive. In the second case, where the mean anomaly is more than six rdsis, is it additive. Next, for the equation of the centre. If the m a x i m u m E q . C , represented by E X , is small, as in general, then taking S E and SX to be practically equal, sine E q . C = sine X S E = X E . sine S X E SE = X E . sine S X A ^ S X = max Eq.C X sine mean anomaly -J- 120' (120' being the radius). In this computation, the astronomers belonging to the school of Aryabhala find the E q . C using the actual radius vector, S E , in accordance with the geometric representation. But Bhaskaracarya in his Siddhdnta Siromarii does not use i i , and gives reasons for not using it. Now, if degrees of epicycle are given, as in this Siddhdnta, instead of sine maximum Eq. C , these degrees are multiplied by 120' and divided by 360° to gel sine maximum E q . C. (i.e. E X ) . Therefore, sine E q . C = (degrees of epicycle X 120' H- 360°) X sine anomaly 120' 17
208
PANCASIDDHANTIKA
I^-^
= degrees of epicycle X sine anomaly H- 360°, as i n the text. We shall now see how the use of the epicycle gives the E q . C .
Fig. IX. 1-b.
Here too, E , the centre of the earth is the centre of the orbit circle of radius 120'. O n the orbit circle, the centre C of the epicycle on which the body S is situated, moves according to the mean motion o f the body. T h e radius of the epicycle is the degrees of epicycle given in the text X 120' 360°, which is the m a x i m u m E q . C , as already seen. A t the two points of intersection of the epicycle with the line of apogee, E A , arc A the apogee, and P the perigee. The body moves on the circumference of the epicycle, with its mean motion in the direction opposite to the motion of C . Angle A C S is the anomaly. Now, draw E A ' parallel to C S . T h e n angle A E A ' also is the anomaly. Since C E O is the mean body, A ' E O is equal to the longitude of apogee. Since S E O is the true longitude and C E O is the mean longitude, angle C E S is the E q . C . Therefore, mean longitude minus E q . C . = true long, (the anomaly being less than 6 rdsis in the figure. If the anomaly is more than 6 rdsis, S is greater than P, and the E q . C . becomes additive). T h e E q . C . is got in the same way as in the excentric method. We have now to show that i n either of the methods, the E q . C . is the same, i.e. angle X S E in fig l a = angle C E S in fig 1 b. In the two triangles X S E and C E S in the respecdve figures, it has been
IX.IO
IX. S A U R A - S I D D H A N T A — S O L A R E C L I P S E
209
said that E X is equal to CS. X S is equal to C E , both being equal to 120'. Angles S X E and S C E are also equal, since they are 180° miniis the equal mean anomalies, S X A and S C A . Therefore, the two triangles are congruent, and so angles X S E and C E S are equal, as required to be shown. In the matter o f the correctness of the degrees of epicycle we have to take the authority of the work. That it agrees with the Original Saura can be seen, the same being found also in the Ardhardtrika system and Khandakhddyaka.
Bhujantara correction We shall now show why the correction called Bhujantara is done. T h e Days used i n the formulae are mean solar days. (That is why the mean longitudes are taken to be proportional to them.) Therefore the true longitudes got are for mean noon. But we want the longitudes for true noon. So we have to apply a correction which is the motion d u r i n g the interval between mean and true noons. If the Sun's Eq.C. is positive it is east of its mean position, and reaches the meridian later than mean noon by a certain number of prdrias (prdria = 4 seconds of time, one sixth of a vinddi) equal to the number of minutes of arc. Being later, the Sun and Moon's motion d u r i n g the interval has to be added. If the Eq.C. is negative, then the true Sun is west of its mean position and true moon is earlier. Therefore the motion is to be subtracted. T h e motion in the interval is found by the proportion: (Since there are 21600 pranas i n a day) 21600: daily motion :: Minutes of Sun's equation of the centre: motion d u r i n g the interval. F r o m this it can be seen that this correction has got to be done not only to the Sun, but also to the M o o n and the star-planets as well.
Udayantara correction We must add that this correction for equation o f the centre is not sufficient. Another correction has got to be made for what is called Udaydntara or reduction to the equator, i.e. reducing the motion on the ecliptic to motion on the celestial equator which is the circle on which time has to be measured. Both these corrections form the equation o f time, being the interval between true and mean noons. But H i n d u astronomers prior Sripati were unaware o f this correction.
DeSantara correction 10. O n e nddi f o r every 53 1/3 yojanas has to be d e d u c t e d o r a d d e d (to U j j a i n n o o n ) by p e o p l e i n places east a n d west, respectively, o f the U j j a i n m e r i d i a n , (to get t h e i r o w n n o o n . ) Ujjain was the Greenwich of the H i n d u s , and the line of longitude passing through Lanka, Ujjain and the N o r t h pole was taken as the prime longitude. It is well-known that noon occurs earlier and earlier as the longitude of a place is more and more east, and vice versa. T h e author says that for every 53 1/3 yojanas of distance east or west, there is one nddi earlier or later. T h e idea is that therefore the daily motion of the body should be multiplied by the nd4is got, divided by 60, and the resulting minutes of arc should be subtracted or added to the true longitude, according as the place is east or, west, to get the longitude for local noon. 10a.
A.
mmu
210
PANCASIDDHANTIKA
I X . 11
Example 7. Days, 52931. Benaras is east of Ujjain longitude by 68 yojanas. Find the true Moon that place. From example 6, the true M o o n at Ujjain noon is ra.4-13-29.9. T h e difference i n time for Benaras = nddikds (68 53 1/3), earlier. F r o m this, the correction for local noon = 729'. 1 X 68 -r(53 1/3 X 60) = 15'.5, subtracdve. (The Moon's daily modon for the day will be shown to be 729'. 1 under verse 13.) .-. T h e true M o o n required = ra.4-13-29.9 - 15'.5 =
ra.4-13-14.4.
T h e instrucdon is explained thus: T h e author takes it that for the region of Ujjain the length of the latitude circle is 3200 yojanas. T h e Sun, in its apparent diurnal motion westward goes once r o u n d the circle i n 60 nddikds, crossing all the meridians on the earth. There are thus 3200 -H 60 = 53 1/3 yojanas for one nddi, and a place east by this distance has its meridian crossed by the Sun, i.e. its noon earlier by one nddi, and west, later. The modon for this time is calculated by the proportion, 60 nddis: daily motion :: the nddis got: the motion for the same. That the motion is deductive if the noon is earlier, and vice versa is plain. (This is for direct motion, the Sun and the Moon alone being considered here. If the daily motion is retrograde, as is possible i n the case of the star-planets, it is obvious that the subtraction and addition have to be reversed.)
But it is to be noted that in the Ardhardtrika of Aryabhata and in the Khandakhddyaka, the diameter of the earth is given as 1600yo;ana5 from which the equatorial circumference got is 5027 yojanas. Therefore the Original Saura must have given the same values. T h e Modem Sdrya Siddhdnta, and th Siddhdntas that follow it also give the same. F r o m this the latitude circle at or near Ujjain should be given according to them as 5027 cos 24° = 4600 yojanas. According to the Aryabhatiya, which uses a yojana measure one and a half times that of Saura etc., the equatorial circumference would be 3300 yojanas. F r o m this, it is 14° latitude circle that would be 3200yojanas, and not Ujjain latitude circle. Why should the author use the yojana measure of the Aryabhatiya instead of that of the Saura, and in that why should he use the yojanas o f the 14° latitude circle instead of the Ujjain (24°) latitude circle seems inexplicable. [Tfe9#nW«zr^:]
Mean motion of the Sun and the Moon 11. T h e m e a n d a i l y m o t i o n o f the M o o n is 7 9 0 ' 34", a n d that o f the S u n is 5 9 ' 8". These can be derived from the mean motions o f the Sun and the M o o n given in the first and second verses. In the former it is said that in 2,92,207 days there are 800 revolutions of the Sun. .-. 800 revolutions 2,92,207 = 59' 8", the motion for one day. In the latter, we have that there are 9,00,000 revolutions of the M o o n i n 2,45,89,506 days. .-. the motion per day = 9,00,000 revolutions -r- 2,45,89,506 = 790' 3 4 " . T h e correction in verse 4 is too small per day to consider. 11a. A . B . ^ ( A . ^ ) ^kl^dT
d. B.^m^^Hlvi):
I X . 14
IX. S A U R A - S I D D H A N T A — S O L A R E C L I P S E
211
Motion of Moon's anomaly 12. T h e d a i l y m o t i o n o f the M o o n ' s a p o g e e is 6 2/3 m i n u t e s . T h e M o o n ' s m e a n daily m o t i o n less the m o t i o n o f the a p o g e e is the d a i l y m o t i o n o f the M o o n ' s (mean) a n o m a l y . T h e t r u e daily m o t i o n is to be f o u n d u s i n g this motion of anomaly. We get that the daily modon of the Moon's anomaly is 790' .34" - 6'. 40" = 783'.54". T h e Sun's mean daily motion itself is the motion of its anomaly, since its apogee has no motion according to this Saura, as we have already said. Even i f motion is taken into account, it is so small that it is practically nothing per day. T h e rule to get the daily motion of Moon's anomaly is explained thus: F r o m verse 3 above, we see that there are 900 revolutions of Moon's apogee i n 29,08,789 days..-. i n one day, the motion is 900 revolutions -r- 29,08,789 = 6' 41". T h e correction per day is practically nothing and so left out. T h e author gives it as 6' 40", for convenience of expression, since the 1' left out will not affect the result materially. Since anomaly is mean longitude minus longitude o f apogee, the motion of anomaly is mean motion minus motion of apogee, for it is the daily motions that add up to form the longitude.
True motion of Sun and Moon 13. T h e d a i l y m o t i o n o f a n o m a l y s h o u l d be m u l t i p l i e d by the c u r r e n t sinei n t e r v a l a n d d i v i d e d by 2 2 5 . T h i s s h o u l d be r e d u c e d to the e p i c y c l e , i.e. m u l t i p l i e d by the degrees o f epicycle a n d d i v i d e d by 3 6 0 ° . T h e c h a n g e i n sine E q . C , is got. Its arc s h o u l d be subtracted f r o m the m e a n d a i l y m o t i o n , i f the a n o m a l y falls w i t h i n rdsis 9 to 3, a n d a d d e d i f it falls w i t h i n rdsis 3 to 9. 14. T h i s is the t r u e m o t i o n p e r day, for the m o m e n t (for w h i c h the a n o m a l y is taken.) T h e t r u e d a i l y m o t i o n i n the case o f the M o o n is got by s u b t r a c t i n g 12a. B.RlRj^iilll
13a. C.^-i;^H<'j("ldl:; D.°^l|iluidl
b. A.^ST%n
b. B . ^ 3 o i t ^ A.qIWRT:; B . C . D . * i I F ! T
d. B . ' ^ J S ^ a r ^ ^ ; A . s m ^
d.
Bl.3.'^'
212
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PANCASIDDHANTIKA
the p r e v i o u s day's t r u e M o o n f r o m the g i v e n day's t r u e m o o n . T h e d a i l y m e a n m o t i o n , m u l t i p l i e d by 120' a n d d i v i d e d by the m o m e n t a r y m o t i o n p e r day is the r a d i u s vector at the m o m e n t . The following is given here:(A) (i) T h e daily change i n sine E q . C . for short interval = the interval i n the tabular sine of anomaly current X daily motion of mean anomaly x degrees of epicycle ^ (225 X 360). (ii) T h e daily change i n E q . C . i n minutes of arc = (i) X 3438 there is n o difference between sine and arc).
120. (since the sine is small and
It should be noted here that it is possible to simplify the above work, because the daily anomaly and the degrees of epicycle are fixed for each body, and the rest are constants. Only the interval in the tabular sine of anomaly current varies. F o r instance, the Moon's daily motion of anomaly is 783'.9, and epicycle 31°. Therefore, the daily change i n E q . C . i n minutes o f arc = the interval i n the tabular sine of anomaly current X 783.9 X 31 X 3438 ^ (225 X 360 X 120) = described interval X 8.6. For the S u n it is, interval i n the tabular sine of anomaly current x 59.1 x 14 x 3438 360 X 120) = interval etc X .29.
(225 X
(ii) T h e daily rate of true motion for the moment = mean daily motion ± (ii) (additive if anomaly is from rasis 3 to 9 and subtractive i f from 9 to 3.) (B) T h e true daily motion = the given day's true longitude — the previous day's true longitude. (C) T h e radius vector at the moment = 120 X mean daily m o d o n for the moment.
the daily rate of true motion
The sine interval of anomaly used in computing the true M o o n or Sun for noon can easily be used to find the daily rate for the noon i n question. It is this that should b^ used for finding the motion during the interval between mean and true noons, as we have done already, and in correcting for longitude. In eclipses also this true daily rate with radius vector, for the moment of syzygies should be used, because this will give the circumstances accurately. T h i s seems to be the author's idea i n giving these here. As for the Sun, there is no distinction either i n the rate or radius vector between those for the day or for the moment. This is indicated by making the distinction i n the case of the M o o n alone, using the expression sasivisesdt.
Example 8. For the Ujjain true noon of examples 5 and 6,findthe true motion (a) for the Moon (b) Sun. (a) In example 6, the Bhuja, i.e. sin of Moon's anomaly got is ra.0-24-7.5. Sines being given for every 3 3/4 degrees, the current sine interval is the seventh, equal to 7' 9". .-. B y (A ii), the daily change i n E q . C . = 7' 9" X 8.6 = 61 '.5 By (A iii), the true motion for the said noon = 790'.6 - 61'.5 = 729'. 1, (subtraction since anomaly is between rd. 9 to 3). 14a-b. A . B . C . D . ? ! ^ b. A . B 2 . ' ^ ; B l . 2 . * ^ ; C . D . T l I M
d. B1.2.«lfTf) A . B 1 . 2 f ^ : ; B3.l5rII:
IX.15
IX. S A U R A - S I D D H A N T A — S O L A R E C L I P S E
213
(b) T h e Bhuja or Sun's anomaly in example 5 is rd. 1-14-44.3. T h e current interval, 12th. is 5' 44". .-. by (A ii), the daily change i n E q . C . = .5' 44" X .29 = 1' .7. B y (A iii), the true motion = 59'. 1 1 '.7 = 57'.4, (subtraction since anomaly is between rd. 9 and 3). Example 9. For the noon of example 8, find the Moon's radius vector. By (C), the radius vector required = 120' x 790'.6-f- 729'.1 = 130'.1. T h e following is the explanation of the rules: T h e true modon d u r i n g any interval between two moments is the difference between the true longitudes of the moments. T h e shorter this interval, the more accurate is the motion. In the rule, all factors excepting the sine of anomaly are constants. Therefore the accuracy of the motion depends on the sine of anomaly only. In the case of the M o o n , the motion of the anomaly being rapid, there is significant difference i n the sine from time to time even within the day, for four or even five sine intervals pass i n a day, with the result that the motion is got differently for different times. So the motion has got to be found for shorter periods like theydma i n the day, and this is the motion for the time being. For every 225' of anomaly there is one sine interval. So, d u r i n g the time for which the anomaly interval is current, the change i n sine Eq.C. is caused by the corresponding sine interval current. Therefore, the change in sine E q . C . is got by multiplying the current sine interval by the degrees of epicycle, and dividing by 360, for the period covered by the corresponding anomaly interval of 225'. F r o m the change i n sine E q . C . the change in Eq.C. is got in minutes (by multiplying by 3438 and dividing by 120, as we have done). This change is for the time to which the current 225' interval of anomaly corresponds. It is converted into the change per day by multiplying by the daily motion of anomaly i n minutes and dividing by 225. This is applied to the mean daily motion to get the daily rate.
Now, we know that the E q . C . is zero when the anomaly is zero, that it is negative and increases numerically in the first quadrant of anomaly, i.e. upto 3 rdsis, then decreases numerically, still being negative, to the end of the second quadrant, i.e. upto 6 rdsis where the value becomes zero again, then in the third quadrant it is positive and increases to a m a x i m u m at 9 rdsis and then decreases i n the fourth quadrant to zero at the end o f 12 rdsis or zero. F r o m this it can be seen that the E q . C . goes o n decreasing as the anomaly passes from 9 rdsit to 3 rdsis i.e. the change is negative or subtractive, and goes on increasing as the anomaly passes from 3 rdsis to 9 rdsis, i.e. the change is positive or additive, as instructed by the text. N o harm will ensue from the instruction to multiply and divide the sine interval first and then reduce it to the epicycle, since the sine is small. It is to be noted well that the motion per day found is not actually the motion i n the day, but only the rate d u r i n g the short interval or moment taken i n the day. T h a t is why the motion for the day is given by a separate rule, for the M o o n . In the case of the Sun there is no distinction between the two, since the daily motion of the anomaly is small. T h e rule for the radius vector has been explained already when dealing with the Romaka.
214
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PANCASIDDHANTIKA
Kak$a of the Sun and the Moon 15. T h e Sun's r a d i u s vector m u l t i p H e d by 5 3 4 7 a n d d i v i d e d by 4 0 is called its kaksd. T h e M o o n ' s r a d i u s v e c t o r m u l t i p l i e d b y 10 is its kaksd. It is to be noted that the kaksa obtained here, depending as it does on the radius vector is also for the moment taken and its neighbourhood. W e can also derive it directly from the daily rate of motion obtained from verses 3-14, above. T h u s : (a) The Sun's kaksd = Sun'si radius vector x 5347/40 = (120 x 59.13 ^ Sun's daily rate of motion) X 5347/40 = 9,48,558 Sun's daily rate of modon. (b) T h e Moon's kaksd = Moon's radius vector x 10 = (120 x 790.56 motion) X 10 = 9,48,680 H- Moon's daily rate of motion.
Moon's daily rate of
Example 10. Pudukkottai (Lat.10° 24'), on a particular day, new moon falls at nd.20-40 after sunris that moment, the longitude of the sun — the longitude of the Moon = rd.2-0-0. The Rdhu-head, at tha rd. 7-29-24. The Sun's rate of motion for the time is 57' per day and the Moon's 810'. The daytime is nd 20. Compute the solar eclipse occurring. For this, the kaksd is found first: (a) T h e Sun's Kaksd for the time = 9,48,558 H- 57 = 16,641 (b) T h e Moon's for the time = 9,48,680 H- 810 = 1171.2 T h e author does not use the word kaksd here i n its usual sense of orbit, but for the actual distance reduced by some factor, for the orbit is constant, while what we get here is a quantity varying with the rate of motion. Since only the proportion o f the distances of the Sun and the M o o n from the earth is significant, the reduction will not cause any error. T h a t is why, the wordyo;a«a giving the measure o f distance, is not used here. Now, the mean kaksd, derived trom the mean radius vector, 120', is for the Sun, 120 X 5347 H- 40 = 16,041. For the M o o n it is 120 X 10 = 1200. These obviously are the respective reduced mean distances from the earth. I n the Original Saura the Sun's orbit is given as 6,89,358 yojanas, and the Moon's 51,566 yojanas, as we learn from the Ardhardtrika system etc. Since the mean distances are proportionate to the orbits etc, i f the Moon's orbit, 51,566, is reduced to 1200 as here, the Sun's orbit, by the same factor, must be reduced to, 6,89,358 X 1200 -f- 51566 = 16,042. T h i s agrees very closely with 16,041 got above. T h e difference of one may be due to giving the multiplier correct to the nearest whole number, as 5347. Further, the orbits, which is the same for our present purpose as saying distances, are inversely proportionate to the yuga cycles given i n the ^astras. Therefore, from the Saura cycles of Sun and M o o n i n 1.14, by the proportion 1,80,000:24,06,389:: 1200:x, we get 16,042 for X , the Sun's reduced mean distance, when the Moon's is 1200. This agreement is the justificadon for our correcting the reading drighna into digghna. B u t T S correct it into gnighna. Also, they correct khar into khdrkn though there is the alternate reading khakrta fitting correcdy i n the rule and adopted by us. B y their corrections the Sun's mean kaksd will be 5347 and the Moon's 360. T h u s the Sun's kaksd becomes, 5347/360 ( = 14.85) times the 15a. A . ^ B 3 . Has an unnecessary gap after B 1.2. do not have it.
b. A . •IslJlj'Hll^d); C.M^n^IcTt c. B . ^ ^ . A.^OTt; B . ^ d. A.B.fSJT:; C . t W :
IX.16
215
IX. S A U R A - S I D D H A N T A — S O L A R E C L I P S E
Moon's, against the fact that it is only 13.37 times from the Sastras, and what they give is equal to saying that there are 14.85 revolutions of the M o o n in the year for the kaksds vary directly as the periods of revolution, i.e. inversely as the number of cycles i n a given period. We shall also see what havoc their wrong corrections play i n the angular diameters following.
Measure of the orbs 16. D i v i d e 5,14,787 by the Sun's kaksd, a n d 3 8 , 6 4 0 by the M o o n ' s to get the respective a n g u l a r d i a m e t e r s i n m i n u t e s at the t i m e . It is to be noted that the angular diameters (of the orbs of the Sun and Moon) are always given in minutes by our Sastras. Thus, (a) T h e Sun's angular diameter in minutes = 5,14,787/Sun's kaksd. (b) T h e Moon's angular diameter in minutes = 38,640/Moon's kaksd.
Example 11. To continue the problem of example 10, find the angular diameters of the Sun and Moon fo the time given. (a) T h e Sim's angular diameter = 5,14,787' - 16,641 = 3 r . O . (b) T h e Moon's angular diameter = 38,640' -f- 1171.2 = 33'.0. T h e derivation of the rules for the angular diameters is as explained below. T h e angle formed at the eye by the diameter of the orbs of the Sun and the M o o n is their angular diameter and given in minutes. We all know from experience that the nearer the orbs, i.e. the lesser the radius vector (given myojanas), the greater is the angle, and the farther away is the orb, i.e. the greater the radius vector, the lesser is the angle. T h u s the angle and the radius vector are in inverse ratio, as also the kaksd which is directly proportionate to the radius vector. So we have: Sine angle at the eye = 120 X diameter i n yojanas the radius vector in yojanas.
X
T h e angular diameter in minutes = 3438 X 120 X diameter in yq/awas 120.)
(radius vector inyq;anas
Here, the author has reduced the diameter i n yojanas by the same factor used i n verse 15 to reduce the radius vector i n yojanas to the kaksd, and multiplying by 3438, as explained already, to convert the sine into minutes, he has given 5,14,787 for the Sun, and 38,640 for the M o o n . T h e mean kaksd of the Sun derived by us i n the explanations is 16,042. D i v i d i n g 5,14,787 by 16,042, we have the Sun's mean angular diameter, 32'. 1, and the Moon's is 38,640 ^ 1200 = 32'.2. 16a. A.B1.2.^Q^!M1«5°; C.D.^aapgjig^; (D.^) b. B . ^ a ^ ° ; A.B.C.om^f; C.1!7FII:
c. B 2 . d ^ l l c l * ; B1.2. B 2 . Unnecessary gap after of?lf?H:
216
PANCASIDDHANTIKA
IX.18
T h e angular diameters derived from the Original Saura (and the modern SUrya Siddhdnta) are 32'.3 and 32'.0, respecdvely. T h o u g h the difference is small, we must investigate why there is a difference at all. T h e error is about 160th part of itself in each case. Either the author has taken values slightly different from those of the Original to derive the numbers here, or there are some errors i n the readings here.
N o w for the readings. T h e first foot o f the verse is i n excess by one mdtrd. There are eight digits in the number, khakhavasu-khamumndravisaydh, though there should be only six digits. So we ha corrected khakhavasukha into svaravasu following the form of the letters also. T S correct it khavasukha etc., giving seven places i n the number, as 5147080. By this, the dividend has become ten times what it actually is. Since they give the divisor, viz. the Sun's kaksd, as one third of the actual, (as seen already), they have made the angular diameter in minutes, thirty times the actual. Unaware of the mistakes they have made they wonder why the angular diameter comes thirty times the actual, and make the following curious comment: T h e correct angular diameters can be got only by dividing what we get here by 30. B u t the author does not say anything about dividing by thirty. Perhaps i n his days there was the well-known understanding that what is got is to be divided by 30 to get the angular diameter. T h a t is why, we surmise, the author has not instructed the division by 30 !! (vide page 50 of the Sanskrit commentary.) A l l this is the result of the errors i n their correction of the readings. N P too have sensed the error here and emend the expressions as khavasukha munlnduvisaydh (5,17,080), with the result that "the radius of the Sun is about four times the radius of the earth, the radius o f the M o o n about one third" (pt. 11, p.73). In the same way, there is one mdtrd less i n the second foot. Therefore, supplying the lost letter we have read khakrtartusugandh as khakrtartuvasuguridh. N P too, do the same. B y this we get the places required i n the dividend. B u t T S read it as khakrtartusuraguridh, thus making the dividen nine times what it is. T h e y have already made the divisor, the moon's kaksa, three tenths of what the author has said it is. B y this the angular diameter o f the M o o n also has been made thirty times the real value, by them, again rousing their wonder in the manner mentioned before.
Sin Zenith Distance of Meridian pt. 17. F i n d the i n t e r v a l b e t w e e n m i d d a y a n d the m o m e n t o f new m o o n . I f the S u n is east o f the m e r i d i a n (i.e. i f n e w m o o n falls i n the f o r e n o o n ) , find the degrees o f r i g h t a s c e n s i o n c o r r e s p o n d i n g to this t i m e u s i n g the ascensional differences o f z e r o l a t i t u d e , (lankodayamdna.), b a c k w a r d s f r o m the S u n . S u b t r a c t these degrees f r o m the S u n ( = M o o n ) o f the m o m e n t o f n e w m o o n . I f the S u n is west o f the m e r i d i a n , (i.e. i f n e w m o o n is i n the afternoon), find the degrees c o r r e s p o n d i n g to the i n t e r v a l c o u n t i n g f o r w a r d f r o m the S u n , a n d a d d to S u n ( = M o o n ) .
IX.18
IX. S A U R A - S I D D H A N T A — S O L A R E C L I P S E
217
18. T h e m e r i d i a n p o i n t o f the e c l i p t i c (madhyalagnam) is got. F i n d its d e c l i n a t i o n , n o r t h o r s o u t h . I f n o r t h , find the d i f f e r e n c e b e t w e e n the d e c l i n a t i o n a n d the latitude o f the place. I f s o u t h , a d d t h e m . T h e sine o f the result is c a l l e d madhyajyd, i.e. s i n z e n i t h distance o f the p o i n t . T h e madhyajyd is got thus: (i) Interval between noon and new moon = time of noon ~ time of new moon. (ii) T h e degrees of rise of the ecliptic, backward or forward from the Sun, using the corresponding ascensional differences of zero latitude, for the interval in (i) in the forenoon or afternoon respectively is to be found. (iii) (Longitude of) meridian ecliptic point = T h e Sun (or Moon) at new moon qp (ii), (— for forenoon, + for afternoon). (iv) T h e declination north or south o f (iii) is to be found. (v) Sine zenith distance o f m.e.p. i n (iii) = sine (declination found i n (iv) + latitude), (-I- i f the declination is south, and the sine found is directed south, ~ i f the declination is north, and the sine is directed north or south according as the declination or the latitude is greater.) Example 12. To continue example 10 using the times given there. (i) Interval between noon and new m o o n = na.20-40 - na. 15-40 = 5 na., afternoon. (ii) Given, Sun = M o o n = rd. 2-0-0, at new moon. Forward counting is to be done (because afternoon) from the first point of Gemini where the Sun is. F o r successive 10° of rise, the times taken are, in vinddis, 105.4, 107.6, 108.8, 108.8. F o r the interval, of 300 vinddis = 105.4 4- 107.6 -I- 87, there are 10° 4- 10° -I- 8° ( = 10° x 87 -f- 108.8) = 28°. (iii) A d d i n g , the meridian ecliptic point = ra.2-0-0 -I- 28° = m.2-28-0. (iv) T h e declination o f m.e.p. is 23° 5 8 ' N . (v) Sine zenith distance o f m.e.p. = sine (23° 58' - 10° 24') = sine 13° 34' = 28' 9", north (-.• declination is greater). It has been stated by us, above, i n the context of the computation o f the solar eclipse according to the Paulina, that i n order to get the parallaxes i n longitude and latitude, the nonagesimai and the sine and cosine of its zenith distance, (i.e. drgjyd and Sanku), are to be found. In the Saura, a different method is given to get the sine o f the zenith distance of the nonagesimai (z.d.n.) for which the m.e.p. and the sine o f its zenith distance are necessary, and given here. It has been said that the m.e.p. is the point o f intersection o f the meridian (NZS i n the fig.) and the ecliptic ( O r O ' i n fig.) as M i n fig. 5 is the position of the Sun at new moon, (occurring i n the 17a. D.TWI^dlMd. B . dlMddl? b. A.'Vm^;
B1.2.°T?RRH°; B S . " ! ? ^ ^ ;
D.°T=TTroTRg. B.ilTdmVII: d.
18b. B 1 . 3 . c W r a c. B 1 . 3 . % T d. A.'^n^l+ld'HSJlpT"; Bi.2.'»r^^T«#r<>; B3.^^'?ra]f*T<';
218
PANCASIDDHANTIKA
IX.19
afternoon in the example given to illustrate which the figure is drawn). H Q is the right ascension corresponding to the segment o f the ecliptic s M , which is found from the time lo or from noon, by using the ascensional differences at the equator. (In the figure the time is afternoon.) sr is the longitude of the Sun at new moon. M r the longitude of m.e.p. = sr — s M , as stated for afternoon. Clearly, for forenoon, s being east of M , M r = sr + Ms, as instructed. M Q i s the declinadon of M , (north in the fig.) Z Q i s the latitude. M Z is the zenith distance. M Z = M Q — Z Q (in the fig.), i.e. when north declination is greater than the latitude, the latitude is subtracted from it, and the zenith distance is north. If M had been between Z and Q , then the north declination would be less than the latitude, and M Z would be equal to Z Q — M Q , and south. In the case when M is south of Q?.^. when the declination of M is south, it is clear that the zenith distance, M Z = M Q + Q Z , and it is south, as stated.
It is to bring in all the above three cases that we interpreted aksnviyuta in two ways, 'subtracting the ladtude', and 'subtracdng from the ladtude'. But T S and N P take only the latter case, with the result that the former case is shut off. In verse 17, N P has corrected madhydrka as madhydhna, which is not essendal. T h e defect in the second quarter of verse 17 has been rectified here by the addition of the letter ksa as ranak(sa)rdsyudgamaih, anaksa meaning niraksa or zero latitude, to give the sen required here. T S emends it as niraksa i n the same sense and N P as rantarardsyudgamaih.
IX.2()
IX. S A U R A - S I D D H A N T A — S O L A R E C L I P S E
219
D|-k$epa of the Sun 19. F i n d the sine o f the l o n g i t u d e o f the O r i e n t E c l i p t i c P o i n t (o.e.p.) at n e w m o o n , m u l t i p l y by the sine o f m a x i m u m d e c l i n a t i o n (of the S u n , 4 8 ' 48") a n d d i v i d e by the sine o f the c o l a u t u d e . ( T h i s is sine a m p l i t u d e o f o.e.p., c a l l e d Udayajyd.) M u l t i p l y this b y the sine o f the z e n i t h distance (z.d.) o f m . e . p . already f o u n d , a n d d i v i d e b y 1 2 0 ' . S q u a r e the result, a n d subtract f r o m the s q u a r e o f the sine z . d . o f the m . e . p . 2 0 . Set the r e m a i n d e r i n two places. I n o n e place, find its s q u a r e root. T h i s is the sine o f the z e n i t h distance o f the n o n a g e s i m a i (z.d. o f n.) c a l l e d the Sun's drk-ksepa. Keep this safe aside for future work. T h e following is to be done: (i) Using the vinddis of ascensional differences of the place, the o.e.p. at new moon is to be found. (ii) Sine amplitude of o.e.p. = sine z.d. of m.e.p. X 48' 48" -r-sine colatitude. (iii) Sine (m.e.p. ~ nonagesimai) = sin z.d. of m.e.p. (ii)
120.
(iv) Square o f sine (z.d. o f n) = (sine z.d. o f m.e.p.)^ — (iii)^. (v) Sine z.d. of n = V (iv). Example 13. Continue example 10, given already lat = 10° 24' and new moon is at na. 20-40. (i) Let us take it that using the ascensional differences of the place (given by chap.IV), the o.e.p. found isra.5-27-56. (ii) Sine amplitude of o.e.p. = sine ra. 5-27-56 X 48' 48" H- sin (90° - 10° 24') = sin 2° 4"
X
48' 4 8 " ^ sin 79° 36'
= 4' 20" X 48' 48" ^ 118'.0 = 1' 48". (iii) Sine (m.e.p. ~ n) = 28' 9" X 1' 48"
120 = 25".
(iv) Sin^ (z.d. of n) = (28' 9")^ - (25" f = 792'.25. (v) Sin (z.d. of n) = V 792.25 = 28' 9".
19a. A.B.ft
b. B . ° M ; C.D.°fW. A.B.D.W^^n; C . ^ s f q . A.B.oT)^ c. A.°^^
220
IX.21
PANCASIDDHANTIKA
T h e following is the explanation of the rule: See fig. 2. There, n is the nonagesimai, i.e. o.e.p. . minus three rdsis. Z is the zenith, n.z is the zenith distance of n, and sine nz, called the 'Sun's drk-ksepa is wanted here, to get the Moon's parallax i n latitude. T h i s Siddhdnta takes the spherical triangle Z n M , right angled at n, to be approximately equal to a plane right-angled triangle, with the sines of the arcs as straight lines and fmds the drk-kssepa hy, (sinezn)^ = (sine MZ)" - (sinenM)- = sin^z.d. of m.e.p. - sin^ (m.e.p. ~ n ) . (The correct method has already been expounded by us i n C h a p . V I , when dealing with the solar eclipse according to the Paulisa.) Sin (z.d. of M.e.p.) has been got already. T h e other quandty required, viz. sin (M.e.p. ~ n), is got by the well-known formulae relating to spherical right angled triangles, (already given by us), sin (M.e.p. ~ n) = sin (z.d. o f M.e.p.) Xsin M Z n -r- 120'. But, M Z n = O ' Z W = O Z E , which is the amplitude of the o.e.p. Its sine, Udayd, given here = the declination of the o.e.p. X 120' + sine colatitude = sin longitude o f o.e.p. X 48' 48" sin colatitude, as given i n chap.IV.
Gnomon 2 1 . S u b t r a c t f r o m 14,400, the square o f sin z . d . o f n , (kept u n u s e d i n the o t h e r place i n the p r e v i o u s w o r k ) , a n d f i n d its square-root. M u l t i p l y this by the sine o f the distance b e t w e e n the S u n a n d the o.e.p., a n d d i v i d e by 120'. T h e result, w h i c h is the sine o f the S u n ' s a l t i t u d e , is c a l l e d ^afiku, i.e. the Sun's ^anku. .-.^anku = V 14,400 - sin'' (z.d. of n) X sin (o.e.p. ~ sun) got, and kept apart]
120'. [sin^ (z.d. of n) has already been
Example 14. To complete example 10. F r o m Ex.13, by (i), o.e.p. = rd. 5-27-56, and by (iii), sin^ (z.d. of n) = 792.25, from Ex.10, Sun = rd. 2-0-0. ^anku = V 14,400 - 792.25 X sin {rd. 5-27-26 - rd. 2-0-0) ^ 120' = 116' 39" X sin {rd. 3-27-26) H- 120' = 116' 39"
X
106' 7" -r 120' = 103' 10".
T h e rule is derived as follows: F r o m fig. 2 it can be seen that the Sun's altitude is 90° — Zs. .-. ^annku = Sine Sun's altitude = Cos Zs = Cos Z W x Cos ns 120' (by the well-known formula, already given) = V radius'' — sin^ Ln x sin Os -r- 120', (•.• ns is os - 90°). sin Zn is sine z.d. of n already found, and its square has already been got and kept apart for use here. .-. ^afiku = V 14,400 - sin^ z.d. of n X sin (o.e. P ~ sun) 21a. A . ^ ^ ; B . ^ ^ . B . f f c T A . ^ b. A.<:IMfsldl^. A.oW^; B.o^f^
120' as given by the author.
c. B . ^ d. B . ^ . A . & > ^ X # . B.Bfsqls^cI
1X.23
iX. S A U R A - S I D D H A N T A — S O L A R ECLIPSE
221
[rifiaRmNT:]
Parallax-corrected New Moon 2 2 - 2 3 . Subtract the square o f the Sun's sanku got above f r o m 14,400. F r o m the r e m a i n d e r subtract the square o f the S u n ' s drk-ksepa kept apart i n the previous work and find its square root, (technically called Drggati). M u l d p l y this by 18 and divide by each of the kaksds of the Sun and the M o o n . Find the respective arcs (in minutes) and get their difference. Treat this as the minutes of tithi and find the tithi-nddikds for this. Subtract the nddikds from the time of new moon if forenoon, and add, if afternoon. T h e parallax-corrected new moon (p.c.n.) is got. Repeat the operation of finding the p.c.n., dll there is no difference (in dme) i n two successive operations. This is the p.c.n. (to be used i n the subsequent work). T h o u g h there is no doubt about the idea here, it is difficult to get the idea from the words used, on account of several corrupt readings. In verse 22, a word is broken at the end o f the third foot, and there are 18 mdtrds i n the fourth, sinning against the A r y a metre. In the same verse, i n the second foot N P has emended viUesdt into visesitdt against the manuscript readings, an emendation that is not needed. In the 23rd verse, evam mrgyah kdlah is a repetition. T S have succumbed to this difficulty and give the wrong interpretadon that the difference between 14,400 and the square of the sanku, should be subtracted from the square of the drk-ksepa, unaware that this is impossible since the latter would always be less than the former. We shall show this i n the explanadon. A s for calling the Sun's sanku as 'digits oi sanku' we have seen it being technically called so i n chap. I V . The method enunciated here is as follows: (i) Drggati = Vl4,400 = sanku^ — drk-ksepa-. (ii) (a) Sin Sun's parallax i n long. = 18 x (i) ^ Sun's kaksd (b) Sin Moon's parallax in long. = 18 X (i)
Moon's kaksd.
From the two sines, the arcs should be obtained i n minutes. T h e Moon's minus the Sun's parallax is the (effective) parallax i n longitude. 22a.
B.°^1%?#
b. A.i^ldf !^llHd>l; B.!^ldd!l ( B 2 . T ^ ; B3.ci^) D.1^[cT]l^ c. A.BI.2.tMcI d. A l . B . ^ ^ ^ f e ' H M
Al.^W^lt 23a. A.ft?W%ir«f; B.C.tg^tqiM«I b. A.f!T82T«#TiRT:; D.f?r«^^fdWH:; C.f?r«^-dWWd:; D . ^ a q ^ S ^ : ^ : d. A.^^?1^; B.-a?;# B.-illcKci^bi:
222
PANCASIDDHANTIKA
IX.23
(iii) Nadis of parallax = the parallax in longitude found in (ii) X 60 ^ the motion of the tithi per day. Subtracting the nddis from new moon in the forenoon, and adding in the afternoon, the p.c.n. is got. F i n d i n g the m.e.p. etc. of the p.c.n., the work should be repeated upto getdng the nddis in (iii). These nddis are to be subtracted or added to the original new noon. A better p.c.n. is got. Using this dme the work may be further repeated for a sdll better approximation. Example 15. To continue Ex. 10 In the last example the saiiku got is 103' 10". In E x . 10, the modon per day of the Sun and the M o o n found are 5 7 ' a n d 810", the Sun's fajfoa found is 16,641 and the Moon's 1171.2. T h e square of the drk-ksepa kept apart, is 792.25. F r o m these: (i) Drggati = V 14,400 - (103 1/6)^' - 792:25 = 54' 27". (ii) (a) Sine Sun's par. i n long. = 18 X 54' 27" H- 16,641 = 3".6. (b) Sine Moon's par. i n long. = 18 x 54' 27"
1171.2 = 50".2.
T h e Sun's parallax is arc of 3".6 = 1 '.7. T h e Moon's parallax is arc of 50".2 = 24'.0. T h e parallax i n longitude = 24'.0 - 1'.7 = 22'.3. (iii) Nddis of pur. = 22'.3 x 60
(810' - 57') = 1-47.
Since new moon is afternoon, adding to the time of new moon, the p.c.n. = nd. 20-40 -I- na. 1-47 = na. 22-27. We shall repeat the operation for a better approximation. (The motions of the Sun and M o o n per day, and their kaksds need not be done again.)
T h e nddis of the p.c.n. after midday = 22-27 — 15-40 = 6-47 = 407 vinddis. T h e right ascension corresponding to the interval of 407 vinddis, using the ascensional differences of zero latitude = 10° -I- 10° + 10° -I- 10° x 85.2 108.8 (for vinddis 105.4 + 107.6 -t- 108.8 -I- 85.2) = 37° 50', after the Sun ( = ra. 2-0-2, 2' more for the 2 nddis later)..-. T h e m.e.p. = rd. 2-0-2 + rd. 1-7-50 = ra. 3-7-52. T h e declination of m.e.p. = 23° 46' north. T h e zenith distance of the point = 23° 46' - 10° 24' = 13° 22', north. Sine z.d. o f m.e.p. = 27' 44" . T h e o.e.p. at p.c.n. = ra.6-8-48 (using as before the ascensional differences of 10° 24'). Since amplitude of the point = 7' 34". Sine (m.e.p. ~ n) = 27' 44" X 7'34"-^ 120 = 1'45". Sine ^ (z.d. ofn) = (27'44")^ - ( 1 ' 4 5 " )^ = 765.89. Sine (z.d. ofn) = 27'40". ^anku = V 14,400 - 765.89 X sine (ra.4-8-46) ^ 120' = 9 1 ' 1". Drggati = V 1 4 , 4 0 0 - ( 9 1 ' 1")=^ - 765.89 = 73' 9". Sine Sun's par. i n long. = 18 x 73' 9" - 16,641 = 4".8. Parallax = 2'.2. Sine Moon's par. i n long = 18 X 73' 9" H- 1171.2 = 1' 7".5. Parallax = 32'.2. Relative parallax = 30'.0. Nddis of parallax = 30' X 60
753' = nd.2-23.
A d d i n g to time of new moon, the closer p.c.n. = na.20-40 -I- wa.2-23 = nd.2S-S.
IX.24
IX. S A U R A - S I D D H A N T A — S O L A R E C L I P S E
223
Repeating the work, the p.c.n. got will be about na. 23-20. T h e rule is thus explained: It has been shown i n the context o f the Paulisa solar eclipse that the relative total parallax is obtained by multiplying the relative horizontal parallax (sr) by sine zenith distance of the Sun (drgjyd) and dividing by the radius. In this Siddhdnta, the horizontal parallaxes of the Sun and the M o o n are got separately by dividing by their distances for the sake of exactness. But the Sun's drgjya is used for the M o o n too, since the difference is very small i n the neighbourhood of new moon, with the solar eclipse occurring. In fig. 2, drgjya = sin Zs, and the relative parallax = ss'. Its projection on the ecliptic, si, is the relative parallax i n longitude, by which (Moon — Sun) has got to be increased or decreased to get their apparent difference i n longitude. In the figure, since s is west of n, it is subtractive, and p.c.n. is later, and therefore the nddis of parallax are additive. (When the Sun is east of n and parallax is additive, clearly the nddis are subtractive.) Since (Moon — Sun) is tithi element, the relative parallax is treated like tithi, and multiplied by 60 and divided by the daily motion to get the nddis o f parallax. Now, si is found thus i n this Siddhdnta: sl^ = ss'^ - s'F. (•.• the triangle ss'l is right-angled at 1, and so small that it may be considered plane.) = ss'^ - s s ' l s i n ' 1 ss' = ss'^ - ss'^.sin'Zn s i n ' Zs (•.• triangle Zns is right angled at n) drgjya^ -irr^ s i n ' (z.d. of n) (•.• ss' =irr X drgjya = sin zs, and Z n is the z.d. o f n ) = 11^ (radius' — $anku^— s i n ' z.d. of n). (drgjya^ = radius' - iahku^) .•.sl'= BTV 120' - sanku' - sm' z.d. o f n =i7r X drggati, as given B u t . m = the Moon's horizontal parallax - the Sun's horizontal parallax. .-. the drggati is multiplied by each and then subtracted. It has been said already, i n previous two solar eclipse contexts, that the sine of the horizontal parallax is obtained by dividing the earth's radius by the respective distance. Since the author uses as the divisor not the actual distance but the respective distance divided by 43, the earth's radius also has to be taken divided by 43. T h e author takes 788yojanas as the earth's radius, adopting the value of the Aryabhatiya and multiplying it by 3/2 to express it i n the yojana measure of the Ardhardtrika etc. systems. (These systems give the earth's radius as 800yojanas.) Dividing it by 43 we get 18.3, and the author gives it as 18, corrected to the nearest unit's place. Further, since Z n is perpendicular to the ecliptic, Zs, the zenith distance of the Sun at any position on the ecliptic, is always greater than Z n , and, accordingly, their sines also, since the arcs are all less than 90°, i.e. (radius' - sartku') is always greater than sin' z.d. o f n . Therefore, the interpretation of T S that the former is to be subtracted from the latter is wrong, as mentioned already. T h e need for successive approximation by repetition o f work is plain.
3il^vlMi< ( f ^ ) ^ ' g ^ * ^
u
^^n«n^
224
IX.25
PANCASIDDHANTIKA
Parallax in latitude 2 4 . T a k e the sine z . d . o f n last got i n the successive a p p r o x i m a t i o n , m u l t i p l y by 18, a n d d i v i d e by the respective kaksds. T h e respective sine p a r a l l a x i n latitude is got. T h e arc o f t h e i r difference is the relative p a r a l l a x i n latitude a n d its d i r e c t i o n is that o f sine z . d . o f m . e . p . (i.e. o f M f r o m Z.) Since the sines are very small, it is immaterial whether the arcs are found first and their difference is taken, or whether the arc of the difference of the sines is taken, both being the same practically. But the latter will entail less work. a) Sine parallax lat. of the Sun = 18 x sin z.d. o f n H- Sun's kaksd. b) Sin parallax i n lat. o f the M o o n = 18 X sin z.d. o f n H- Moon's kaksd. (b) - (a) is the sine of the relative parallax i n lat. whose arc is to be found, and its direction is that of M from Z. Example 16. To continue example 10. T h e sine z.d. o f n , last got i n the successive approximation in the last example is 27' 26" say. (a) Sun's sine par. in lat. = 18 X 27' 26"
16,641 =1".8.
(b) Moon's par. i n lat. = 18 X 27' 26" ^ 1171.2 =25".3. Sin relative par. i n lat. = 25".3. - 1".8 = 23".5. Rel. par. in lat. = arc of 23".5 = 11'.2, north, since M is north. T h e rule is thus derived: T h e zenith distance of the nonagesimai, Z n , is the Sun's drkksepa. In the context of the solar eclipse, according to the Paulisa, it was shown how the parallax i n latitude (p.c.lat) is to be got by multiplying this drk-ksepa by the relative horizontal parallax and using it as a correction to the Moon's latitude to obtain the corrected latitude. Here, the parallax is derived separately for each of the Sun and the M o o n , for the sake of greater accuracy. Now, the direction of the nonagesimai and the sine of its zenith distance is the same as that of sine z.d. of M , and the direction o f the apparent shifting o f the S u n and the M o o n by parallax, as resolved on the line of latitude, is the same as that o f sine z.d. of n , (as Is' i n fig. 2). Therefore, the direction of the parallax correction is the direction of sine z.d. of M , as mentioned by the author. (In the fig. it is north.) Thus everything is explained.
W[T]
3RT^: ^ « 4 n W f e R t ^ g ^ ||
|
2 5 . T h e M o o n ' s l a t i t u d e at the t i m e t a k e n is to be got by u s i n g the sine (of M o o n ~ R a h u ) , a n d this is to be a d d e d to o r subtracted f r o m the p a r a l l a x c o r r e c t i o n i n latitude ( a c c o r d i n g to t h e i r d i r e c t i o n ) . T h i s is the p a r a l l a x c o r r e c t e d latitude (p.c. lat.). T h i s is to be got separately for each o f the times separately a n d f r o m t h e m the times o f total o b s c u r a t i o n a n d total d u r a t i o n are to be got. 24a. A . ^ ; B . ^ b. B . ^ ^ ^ ^ ( B 2 . 3 . < ' ^ ) d. B . ^ s q i ^
25a. B . f ^ ^ b. A.3n«T c. A . B . ? T S ^
d. A . ^ M s W ; B.^^^^^lnfiT D.t9#
1X.26
IX. S A U R A - S I D D H A N T A — S O L A R E C L I P S E
225
T h e m a x i m u m latitude is given i n verse 6 to be 270'. T h e use of sine (Moon - Rahu) has been already indicated in the context of the Romaka, and therefore only indicated here. In correcting, like directions are additive, and unlike directions subtractive, the resulting direction being that of the greater. T h e parallax-corrected latitude is to be got for each of the time of first contact, last contact, and middle, the last serving for immersion and emergence too, as these times are near enough to the middle. T h e separate computation of the corrected latitude suggests that the nddis of parallax also are to be computed separately for the different times. T h u s , the following is to be done: (i) T h e uncorrected latitude = 270' X sine (Moon - Rahu) H- 120 = sine (Moon - Rahu) X 9 4. (If M o o n — Rahu) is less than 6 rdsis, the latitude is north, otherwise south. R a h u here means the Head o f R a h u ) . (ii) Parallax-corrected latitude = latitude +^ relative parallax i n latitude (-1- i f of the same direction, and ~ if of different directions, the resulting direction being that of the greater).
Example 17. (To continue Ex. 10,) find the parallax-corrected latitude at thefinalparallax corrected ne moon, i.e. at nddi 23-20. This time is 7iddis 2-40 later than new moon. So, from the data given in Ex. 10, Rahu-head = rd. 7-29-24, and M o o n = rd. 2-0-0 -h 810' x 2 2/3 - 60 = rd. 2-0-36. M o o n - R a h u = rd. 6-I-I2. F r o m this. Moon's latitude = 9 X sine (rd. 6-I-I2)
4 = 5'.7, south. (-.• M o o n - Rahu-head) > rd. 6-0-0.)
(ii) Parallax corrected lat. = 5'.7 ~ i r . 2 = 5'.5, north (•.• of different directions, north being greater.) These rules have been explained before.
Duration of the eclipse 26. S u b t r a c t the square o f the p a r a l l a x - c o r r e c t e d l a t i t u d e f r o m the square o f the s u m o f the s e m i - d i a m e t e r s o f the S u n a n d the M o o n a n d find the square r o o t . D o u b l e this, a n d find the t i m e f o r it, t r e a t i n g it as the m o t i o n o f tithi. ( T h e d u r a t i o n o f the eclipse it got.) This verse has already occurred as verse 16 of chap. V I I I and fully explained there. T h e only difference is two mis-readings here. Example 18. To continue Ex. 10. In Ex.11, the angular diameter of the Sun has been found to be 3 1 ' , and of the M o o n , 33' and the daily motion of the tithi 753'. T h e parallax-corrected lat. has been found to be 5'.5. F r o m these. Duration inna6/f)^as = 2 X 60 x V(31 -I- 33)/2 - 5.5' ^ 753 = 2 X 60 X 31.52 - 753 = na.5-1. 26b. B 3 . ° ^ : . B.MRMRHI"l'c;d ( B 3 . ° % 1 ^ )
c. A . B . ^ i p r l d. A.B1.2.fM«f^. B.°<=l
226
PANCASIDDHANTIKA
IX.27
It has already been mentioned that half the duration subtracted from the final parallax-corrected new moon is the beginning and added to it is the end of the eclipse. Further, i f the difference of the semi-diameters is used in the work, instead of the sum, the duration of total eclipse is got. Here, if the Moon's is greater, there is actual total obscuration. If the Sun's is greater, there is annular eclipse. T h e author expects us to be conversant with these things.
27. F i n d the nddis o f p a r a l l a x f o r t h e t i m e o f the b e g i n n i n g . I f the time o f b e g i n n i n g a n d t h e n e w m o o n are b o t h i n t h e f o r e n o o n o r b o t h i n t h e aftern o o n , find the difference o f the nMis o f parallax a n d a d d it to the h a l f d u r a t i o n to get the c o r r e c t h a l f d u r a t i o n to be subtracted f r o m the t i m e o f the c o r r e c t e d n e w m o o n . I f o n e is before n o o n a n d the o t h e r a f t e r n o o n , a d d the nadis o f p a r a l l a x , a n d a d d it to the h a l f - d u r a t i o n to get the correct h a l f - d u r a t i o n (to be subtracted f r o m t h e t i m e o f p a r a l l a x - c o r r e c t e d n e w m o o n ) . D o the same f o r the t i m e o f the e n d o f the eclipse, (to find the c o r r e c t h a l f d u r a t i o n to be a d d e d to t h e p a r a l l a x - c o r r e c t e d n e w m o o n , to get t h e c o r r e c t last contact). T h e following example will make the meaning clear. Example 19. To continue Ex. 10. We have already obtained, par.c. new moon = wa. 23-20, parallax-correction for new moon = na.2-40, and the total duradon na.b-\. A p p l y i n g the half duration o n both sides of p.c.n, the approx. time o f first contact = na. 20-50, last contact - na. 25-51. T h e parallax correction in time for first contact using verses (22-23) is na. 1-50. A s both the new moon and time of first contact are in the same part of the day, i.e. afternoon, the difference between their parallax correction = na.2-40 — nd. 1-50 = na. 0-50. This is to be added to the half duration to get the first half duration. Adding, na.2-30 + na.0-50 = na. 3-20. Subtracting this from the parallax-corrected new moon, the correct time o f first contact = nd. 23-20 - nd. 3-20 = nd. 20-0, after sunrise. Next, the parallax-correction for time of approx. last contact is na. 3-30. As both new moon and last contact are i n the afternoon, subtracting the corrections, for both from each other, we have na.3-30 - na.2-40 = na.0-50. A d d i n g this to the half-duration we have, na.2-31 -I- nd.0-50 = nd.3-21, for the correct second half duration. A d d i n g this to the p.c.n, the correct time of last contact = nd.23-20 -I- nd.3-21 = nd.26-41 after sunrise. T h e following is the explanation of the rules for the correction given here and the justification for our interpretation. A t first the duration is given neglecting the effect of parallax on the time. If the parallax is taken into account, the duration will always be longer than otherwise, as we have said. This can be seen from the following consideration. Let us take the case when the end of new moon 27a. A . B . ^ b. A.B.C.D.°f^-^1%°. A.B.f^?#RT; C.D.f^?^:
c. B . ° ^ ^ d. A.^RWTt A . B . H a p l . o m o f 4
IX.27
IX. S A U R A - S I D D H A N T A — S O L A R E C L I P S E
227
is before noon. T h e first contact being earlier still, its interval from noon is greater, and therefore its nddis o f parallax too is greater than those o f the new moon. Since both are subtractive, the first half duration is lengthened, the first contact happening earlier. Therefore the difference is to be added to the duration. In the case taken, the last contact may happen before noon or after noon. If before noon, the interval from noon upto the last contact is less than that upto new moon. Therefore the nddis o f parallax of the last contact is less than those of the new moon, and both are subtracdve. Therefore the second half duration also is lengthened, the last contact happening later. So the difference is, here too, additive to the duradon. If the last contact is afternoon, the nadis o f parallax are clearly addidve to the dme o f last contact, and the last contact happens later. B u t the parallax-corrected new m o o n occurs earlier, and so the second half-duration is lengthened both ways, and so the sum o f the parallaxes is added to the duradon. T h u s i n all three possibilities o f the first case, there is only additive correction. Let us now take the second case, viz. that the new moon occurs after noon. Clearly what is said for the first contact i n the first case applies to the last contact i n the second case, and vice versa, but the additiveness and subtracdveness alone have to be interchanged. Therefore, here too, in all three possibilities the differences or sums, have to be added to the duration, as we have said i n giving the meaning o f the verse. N o t understanding the above, T S have interpreted the verse in such a way that the instruction will result i n lessening the duradon, which is contrary to facts. Now, for the readings. In the second foot o f the verse three mdtrds are missing, and to restore them we have read vislesita as vislesitoyutah i n accordance with the meaning. T h e emendation, by T S and N P , of the manuscript reading vislesitasthitydm into vislesitahsthityd does not express the intended idea fully. In the fourth foot two mdtrds are missing, and to restore them we have read the meaningless ndmoksi as ndma mauksi. T o conclude: In the introduction to this chapter we said that the S u n , M o o n , and R a h u , together with the methods of computing them are better i n the Saura than i n the Romaka. N o w , we have seen that i n the computation o f the solar eclipse also, the Saura excels. F o r instance, the mean angular diameters of the Sun and the M o o n are 30' and 34' according to the Romakas, while they are 32' and 32' according to the Saura, very near the correct 32' and 3 1 ' , respectively. C o m p u t i n g true diameters and the parallax using the distance o f the instant o f eclipse, and using the true m o d o n of the time o f eclipse for getting the duradon etc. are commendable i n the Saura. Getting the sine z.d.n. by using the sine o f the zenith distance o f m.e.p. is a better method than that used by the Romaka, as also the method o f successive approximation for various things like parallax i n time o f new moon etc. T h e abandoning o f the Romaka's faulty correction o f the Moon's position i n its own orbit, is itself praiseworthy. W i t h such good features, the Saura is easily the best ofthefive Siddhdntas.
1. C o l . A . B . D . ^ T ( B . om?fe)^q^te[I%*H^ ( B . W ? ) ^raqtsjR: I C . ^Id ^4RI^I'T) •y4jJ^'J'l HIH H=(HlS«ziPT: |
Thus ends Chapter Nine entitled ^Saura-Siddhanta: Solar Eclipse' in the Paiicasiddhantika composed by Varahamihira
Chapter Ten SAURA-SIDDHANTA — LUNAR ECLIPSE
Introduction In this chapter the method of computing the lunar eclipse according to the Saura Siddhdnta is given. Since the true Sun and the M o o n and R a h u , the true distances of the Sun and the M o o n , and the Moon's angular diameter and ladtude, have already been given in chap. IX, the angular diameter of the Shadow alone is given here, as also the computation of the times of contacts etc. T h e last three stanzas give the amount of eclipse at a desired time, as also the beginning and end of total phase of the eclipses, both o f the Sun and the M o o n .
Diameter of the Shadow l - 2 a . M u l t i p l y the M o o n ' s t r u e distances i n its o r b i t by 36, a n d d i v i d e by the S u n ' s t r u e distance m u l t i p l i e d by 9 0 a n d d i v i d e d by 2 8 6 . S u b t r a c t this result f r o m 36, m u l t i p l y by 120, d i v i d e by the M o o n ' s t r u e distance a n d get the arc o f the r e s u l t i n g sine. T h i s is the a n g u l a r d i a m e t e r o f the S h a d o w . T h e following is asked to be done: (i) 'Result' = 36 X Moon's true dist. ^ (90 X Sun's true dist.
286)
= 36 X Moon's true dist. X 286 -f- (90 X Sun's true distance). (ii) Sine angular diameter o f Shadow = (36 - 'result') 120
Moon's true distance.
O r , simplifying, this is equal to; {36/Moon's true distance - (36/Sun's true distance) X 286/90} x 120 = 4320/Moon's true distance — 13,728/Sun's true distance, l a . A . ^ C T . B.Hddl'Juil b. A . ^ . A . B 1 . 3 . ^ W n : C . W C. B1.3.i?feHPn d. A.eraHt; B1.3.el#TRT«
2a. B1.3.Rl-M<4i]"1 b. A.B.^^qraT; C . D . ^ ^ . A.^T^M??:; B1.3.d4l°tim:
x.2
X. . S A U R A - S I D D H A N T A — L U N A R E C L I P S E
229
Or, (since the arc is small, multiplying this by 3438 and dividing by 120), the angular diameter of the Shadow in minutes = 1,23,768 H- Moon's true distance - 3,93,307 Sun's true distance.
Example l.Ona certain day at the time offull moon (T) the true Sun is rd. 10-0-0, the true Moon is rd.4 Rdhu Head is rd.3-25-0, the Sun's motion per day for the time is 60', and the Moon's 780'. Compute the lunar eclipse. The The The The
Sun s true dist. = 9,48,558 ^ 60 = 15,809 (by IX.15). Moon's true disu = 9,48,680 - 780 = 1216.3 (by IX.15). Moon's angular dia. = 38,640 H- 1216.3 = 31'.77 (by 1X.16). Moon's lat. at T = 9 X sine (ra.4-0-0 - ra.3-25-0) ^ 4 = 23'.5, north.
From the true distances got above, the angular diameter of the Shadow = 1,23,768 3,93,307 - 15,809
1216.3 -
= 101'.76 - 24'.88 = 76'.9. T h e following is the explanadon of the method for finding the Moon's angular diameter: T h e actual diameter of the Shadow is the diameter of the circular section of the Shadow-cone (formed by the earth intercepdng the Sun's light,) at the Moon's orbit, at the dme of full moon. This is represented in fig. 1, below by U ' U " . Moon's
T h e angle subtended by this at the centre o f the earth, E , is the angular diameter desired to be computed here. In the figure, S is the centre o f the Sun, E , that o f the Earth, and U that of the Shadow secdon. SS' is the Sun's radius, E E ' is the Earth's, and U U ' is that of the Shadow. SE is the true distance of the earth from the Sun, and E U , that o f the M o o n from the earth. S ' E ' U ' is the direct common tangent to the orbs o f the Sun and the M o o n . As the distance are very great when compared with the radii, E E ' , SS', and U U ' are practically parallel. Draw E S " parallel to E ' S ' , and U ' E " parallel to U ' E ' . T h e triangles, ES"S and U E " E , are similar. Therefore, E " E / E U = S"S/SE.
230
X.4
PANCASIDDHANTIKA
.-. E"E = S"S X E U / S E = E U X (S'S - S' S")/SE = E U X (S'S - E ' E ) / S E . In order to get the angular diameter of the Shadow, we require the radius of the Shadow, U U ' , = E E ' - E E " = E E ' - { E U (SS' - EE')/SE} = 18 — Moon's true dist. x (Sun's radius — 18)
Sun's true dist.
= 18 - Moon's true dist. X {18 x 5,14,787 - (2 x 18 X 3438) - 18} ^ Sun's true dist. (Here, 18 is the number obtained by reducing the earth's radius by 43, and given by the author in giving the parallax, see 1X.23.) 18 X 5,14,787 (2 X 18 X 3438) is the Sun's radius, since, of two orbs, the parallax as viewed from one is the angular semi-diameter as viewed from the other. Therefore: Sun's minutes of parallax: Sun's angular semi-diameter in minutes :: 18: Sun's reduced radius. But the Sun's minutes of parallax = 18 X 3438 the Sun's true distance, and the Sun's semidiameter i n minutes = 5,14,787 (2 x t h e Sun's true distance) = 18 — Moon's true distance X 18 X 284.4 ^ (90 X Sun's true dist.) Since, sine angular diameter o f the Shadow is got by multiplying the radius by 2, and the max. tabular sine and dividing by the Moon's true distance, sine angular diameter of Shadow = {36 Moon's true dist. x 36 (90 x Sun's true dist. 284.4)} X 120 ^ M o o n ' s true distance, almost the same as the author has given, but with 284.4 instead of 286. If the number had been 17.9 instead of 18 taken as a whole number for convenience, then we shall get 286 itself, as given by the author. W e have already shown that the formula can be simplified. T h e author must have given it in the involved form for indicating the geometrical construction by way of proof.
W h e n the numbers occurring are seen to be correct i n the way shown by us, it is quite improper for T S to read sadastadasra (278) as sadasvadasra (276) and to agree with this, making the Sun's reduced diameter as 146, i n their prpof, instead of the correct 149.73, got by dividing 5,14,787 by 3438. [I^Ht^chlH:]
T ^ ^ ^ : f ^ ( 1 ^ ) ^ : ^ ^ 5 r r a r i ; | | ^ {
|
Duration of the Eclipse 2b-3. A d d the a n g u l a r d i a m e t e r s o f the M o o n a n d the S h a d o w , divideJ^y two, a n d s q u a r e it. S u b t r a c t the s q u a r e o f the M o o n ' s l a t i t u d e f r o m this, a n d find
x.4
231
X. S A U R A - S I D D H A N T A — L U N A R E C L I P S E
the square root. M u l t i p l y this by 120 a n d d i v i d e by the d i f f e r e n c e o f the m o t i o n s p e r day o f the S u n a n d the M o o n p e r t a i n i n g to the t i m e o f eclipse. T h e d u r a t i o n o f the eclipse is got i n nadikds. 4. F i n d the M o o n ' s l a t i t u d e at first contact a n d u s i n g this find a m o r e correct d u r a t i o n . R e p e a t this till there is n o d i f f e r e n c e b e t w e e n the p r e v i o u s a n d the next d u r a t i o n s . Note: T h o u g h the total duration alone is given here, we are expected to know how to find the first and last contacts from this, from previous contexts. In the successive approximation, what is said for the first contact must be taken for the last contact also. Therefore the following is asked to be done: (i) Rough duration = 120 V (half-sum of angular diameters)' — lat.' ous daily motions.
difference of instantane-
(ii) T ± half (i), are the rough first and last contacts. (iii) Using the latitude of the rough first contact and repeating (i) gives successively better first contacts. (iv) Using the latitude of the rough last contact, and repeating (i), gives successively better last contacts. (It should be noted that the shorter the duration the greater are the number of repetitions required.) Example 2. Continue Ex.1. (i) Rough duration in nddlw 120 V { (76.9 + 31.77)/2 }' - 23.5' ^ (780 - 60) = 120 V 54.34- - 23.5' -^ 720 = 120 x 49
720 - nddis 8-10.
(ii) R o u g h times first and last contacts = T — nd.4-5: T + nd.4-5. (iii) T h e M o o n at rough first contact = rd. 3-29-6.9, Rahu then = rd.3-25-0.2. M o o n - Rahu = 4" 6'.7. From this the Moon's latitude is 19'.35, north. Using this, a more correct duration for first contact = V 54.34' - 19.35' X 120 + 720 = nd.8-28. Subtracting half this from the time of full moon, the first contact is at T - nd.4-14. There is no need to repeat, since the duration is long. (iv) T h e M o o n at rough last contact is rd.4-0-53.1, and Rahu then, rd.3-24-59.8. F r o m this the Moon's lat. is 27'.65, north. Using this, a more correct duration for last contact = V 54.34' - 27.65' X 120 ^ 720 = nd. 7-48. A d d i n g half this to full moon time, the last contact is at T -f- nd. 3-54. There is no need to repeat. T h e method has been explained several times before, which need not be repeated here. As for understanding that the successive approximation is for getdng the last contact also, though men2c. A . B . # : . B 1.3.1^; B 3 . ? ^ 3b. A . B . f ^ q ^ d. A.B.«J^. A . B . C . D . t F l ^ f o r f ^ ; A.eia^T
4a. A.B.C.JH^pJ^: b. A . B . o m . i l . A . B . ^ : d. A . B . f t s R q ^ :
232
X.6
PANCASIDDHANTIKA
tioned only for the first contact, the two common statements anayd sthitir bhavati and sthityavisesa krtoydvat, indicate this.
Obscuration at any desired moment 5-6 T a k e the nddis before o r after f u l l o r n e w m o o n u p t o the times for w h i c h the a m o u n t e c l i p s e d is w a n t e d . M u l t i p l y this by the difference o f the Sun's a n d M o o n ' s d a i l y m o t i o n s , ( m e n t i o n e d above), a n d d i v i d e by 6 0 . T h e 'corresp o n d i n g m i n u t e s o f arc' are got. S q u a r e this, square the M o o n ' s latitude for the m o m e n t , a d d t h e m , a n d get the s q u a r e root. Subtract this f r o m the halfs u m o f the d i a m e t e r s o f the e c H p s i n g a n d the e c l i p s e d bodies. T h e r e m a i n d e r is the m i n u t e s o f arc e c l i p s e d , at the m o m e n t t a k e n , o f the M o o n i n the case o f the l u n a r ecJipse, a n d o f the S u n i n the case o f the solar eclipse. It is clear that by 'corresponding minutes of arc' is meant here, the distance in minutes between the M o o n and the shadow, measured along the eclipdc. F r o m the instrucdon it is clear that the nddis taken is the interval between full or new moon and the moment for which the amount of eclipse is wanted. It is clear from the context that the Shadow is meant by the word Rahu. T h o u g h from the mention o f the Shadow, and the Moon's ladtude without any mention o f parallax, this seems to be given for the lunar eclipse only, the expression arkendvoh at the end shows that this is meant for the solar eclipse also. T h e author thinks that the reader has acquired sufficient knowledge, by now, to make the necessary changes when applying the rule to the solar eclipse. Therefore, i n the caseof the solar eclipse, the amount eclipsed is got by using i n the rule, the parallax-corrected ladtude for latitude, the Sun's and the Moon's angular diameters for those of the M o o n and the Shadow, and the parallax-corrected difference o f daily modons for the mere difference of daily motions. Thus, the following is instructed to be done: A. Tofindthe amount eclipsed in the case of the Moon (i) "Corresponding minutes of arc" = difference o f instantaneous daily motions of Sun and M o o n X interval i n nddis from full moon 60.
b, B.'?lflra^ (B2.B.3;f) B.+dNMIui c. A I . ^ ; A 2 - ^ d. B I . 3 . q % ^ :
A.C.D.<=M€IHH;
x.6
X. S A U R A - S I D D H A N T A — L U N A R E C L I P S E
233
(ii) Distance in minutes between the centres of the M o o n and Shadow = V(i)' + (the Moon's latitude at the given time)'. (iii) T h e amount eclipsed in minutes = half-sum of angular diameters o f the M o o n and Shadow — (ii) B. Tofindthe amount eclipsed in the case of the Sun. (i) "Corresponding minutes of arc" = T h e minutes obtained as by A (i) x the half duration not corrected for parallax -v- the half duration corrected for parallax. (This will be a little approximate, but has been given for case of computation, since the two times are known.) (ii) Distance in minutes between the centres of the Sun and the M o o n = V(i)' -I- (Parallax-corrected lat. of time)'. (iii) T h e amount eclipsed in minutes = half sum of angular diameters o f the Sun and the M o o n - (ii)Example 3. Continuing Ex. 2, find the amount of the moon eclipsed 3 nddis after T. A. (i) Corresponding minutes o f arc = (780' - 60') X 3/60 = 36' (ii) Distance between centres = V 3 6 ' + 26.6' = 44'.76 (having found that the Moon's lat. at the moment is 26'.6). (iii) A m o u n t eclipsed = 54'.34 - 44.76 = 9'.6.
Example 4. At a certain solar eclipse the difference of Sun and Moon's motions is found to be 720', the parallax-corrected latitude, 2 nddis before the parallax-corrected new moon, is found to be 15', the sum of the semi-diam£ters is 31' .9, the un-corrected half duration is nd. 2-30, and the corrected half duration is nd. 3. Find the amount of the Sun eclipsed, at 2 nddis before the parallax corrected new moon. (i) Corresponding minutes o f arc = (720 X 2 4- 60) X nd.2 1/2 + nd.3 = 24 X 5 H- 6 =20' (nearly). (ii) Distance between centres = V 2 0 ' -I- 15' = 25'. (iii) T h e amount eclipsed = 31'.9 - 25' = 6'.9. T h e following is the explanation of the method: Let us first take the case o f the lunar eclipse. A t full moon, the M o o n anci the Shadow are i n conjunction, i.e. they have the same true longitude. Since the Shadow has the same motion as the Sun, the interval between them for any interval of time before or after full m o o n is the same as the interval i n tithi proportionate to the time interval. Therefore there is the proportion, i f for 60 nddis there is the difference of the daily modon, how much for the interval i n time. So the difference i n motion is muldplied by the given time and divided by 60. Since the motions are measured along the ecliptic, the interval i n minutes along the ecliptic is got, corresponding to the time interval. T h e distance between the centres is got thus: In fig.2, S is the centre o f the Shadow and M is that o f the M o o n . S M ' is the 'corresponding minutes' got for the interval i n time. M M ' is the Moon's ladtude at the given moment. Since M M ' is directed towards the pole of the ecliptic, the triangle S M ' M is right-angled at M ' . Since the triangle, being small, can be treated as a plane triangle, we have, by the Pythagoras Theorem, the distance between the centres, S M = V S M " 4 - M M " = Vcorres. minutes' + ladtude', as given. T h e amount echpsed i n minutes = R r = S R — Sr = SR — ( S M — M r ) = SR + M r — S M = sum of semidiameters of the Shadow and the M o o n , minus the distance between their centres.
234
PANCASIDDHANTIKA
X.6
Moon's orbit
Ecliptic
Fig, X.2
What has been proved for the lunar eclipse can be taken for the solar eclipse also, with the necessary changes. T h e difference i n modons should be here corrected for parallax and used. That this corrected difference is always less than the uncorrected will be clear, when we consider that always the parallax-corrected half duradon is always greater than the uncorrected, which we have already proved. Therefore it is clear that by multiplying the "corresponding minutes" by the uncorrected half duradon, and dividing by the parallax-corrected half duradon, will give the parallax-corrected "corresponding minutes". It is also clear that in the case of the solar eclipse we must use in the proof the parallax-corrected ladtude in the place of the uncorrected ladtude, the Sun for the M o o n , and the M o o n for the Shadow. W h e n this is done, the proof is exacdy similar to that for the lunar eclipse. Another thing is to be noted. T h e H i n d u astronomers took the amount of eclipse at full moon or corrected new moon as the maximum and called it the magnitude, igrasa-pramana), though actually this is only very nearly the m a x i m u m and the actual m a x i m u m occurs a little earlier or later. If we take this full or new moon itself for doing the present work, since the time interval is zero, and thereby the 'corresponding minutes' are also zero, the latitude itself becomes the distance between the centres. Therefore we got that the amount eclipsed in this case (i.e. the magnitude) is to be got by subtracdng the ladtude of full or corrected new moon, from the half sum of the angular diameters, as already given.
Now for the readings: vdncchitanddi ('desired time') is meant here the interval in time from the full or new moon, either before or after. B u t T S have taken the expression to mean 'the desired point o f time', and in order to get the meaning o f 'interval' that is wanted for use, have emended the already correct sthitiliptastdbhyas tat into tatsthitiliptdvivardt, which is unnecessary. Anot thing must be said here. If the reading had been tithiliptdh instead of sthitiliptdh given by the manuscripts and accepted by us, it would have been better; for this would mean the minutes of tithi, as indeed these are, being part oi a tithi by nature.
x.7
X. S A U R A S I D D H A N T A — L U N A R E C L I P S E
235
[4u!i
Time of total obscuration 7. T a k e the d i f f e r e n c e o f the a n g u l a r semi-diameters, instead o f t h e i r s u m . S q u a r e it, subtract the square o f the p a r a l l a x - c o r r e c t e d latitude (in the case o f the solar eclipse) o r o f the l a t i t u d e (in the case o f the l u n a r , ) find the square root, d o u b l e it, a n d treat it as tithi, (i.e. m u l t i p l y by 6 0 , a n d d i v i d e by the difference o f the p a r a l l a x - c o r r e c t e d daily m o t i o n s for the solar eclipse, o r o f the m e r e daily m o t i o n s i n the case o f the l u n a r ) . T h e t i m e o f total o b s c u r a t i o n is got. In short, everything done for the duradon, using the difference of the semi-diameters instead of the sum, is to be done for this. Halving this time and subtracting from or adding to the corrected new moon or full moon gives the first approximate times of immersion and emergence. In the case of the lunar eclipse, successive approximation should be done. In the solar eclipse this is not necessary, because the times of immersion and emergence are very close to the corrected new moon. T h e parallax-correction for the motion alone need be taken into accoimt and that once for all. Another point to be noted is that, in the solar eclipse, if the Sun's angular diameter is greater than the Moon's, instead of a total eclipse there will be an annular (ring-like) eclipse, since the M o o n will not be big enough to hide the Sun. T h e times got, in this case, give the beginning and end of the annular phase. Also, the given examples cannot be condnued to illustrate this section, because under the conditions got there will be no total phase, since the latitudes are greater than the difference of the semi-diameters. T h e proof of the rules given here has already been given i n connecdon with the eclipses according to the Vdsistha and Paulisa with graphical illustrations. Now for the text, and readings: T h e text does not instruct that the difference of the semi-diameters should be squared before adding to the square of the ladtude. But mathematical principles indicate it, since the addition of an unsquared quantity with a squared one is unwarranted. We have corrected visesdvavanati into visesddavanati while T S have corrected it into visesdddalanati and N P into visesdrdhabhapati. It is clear that they have taken more liberty with the text than necessary, and it is also purposeless.
1. Col. A.D.^g=5Wn^;?mt5«Tra:;
B.WI^-^^«M:
C.?f^-<4-5^il^"l Hl^ <*!l4l5S?PT:
Thus ends Chapter Ten entitled 'Saura-Siddhanta: Lunar Eclipse' in the Pancasiddhantika composed by Varahamihira
b. B].3.Rl^
c. B l . f ^
d. B . ^ H c i t ^
Chapter Eleven ECLIPSE DIAGRAM
Introduction Since the distinction among echpse-types, and various ideas mentioned therein, will not be clear without graphical representation, the author 'follows up' the chapters on eclipses with one solely devoted to this subject.
Marking the ecliptic etc. 1. U s i n g the s t i c k - i n s t r u m e n t w i t h n o t c h - m a r k s o f digits, d r a w the circle c a l l e d the ' s u m - c i r c l e ' , h a v i n g f o r its r a d i u s the h a l f s u m o f the d i a m e t e r s c o n v e r t e d i n t o d i g i t s . M a r k the east-west a n d n o r t h - s o u t h lines. ( E - W , a n d N - S , i n f i g . 1). S i m i l a r l y , u s i n g the s e m i - d i a m e t e r o f the e c l i p s e d b o d y , c o n v e r t e d i n t o digits as r a d i u s , d r a w the ' e c l i p s e d b o d y circle', c o n c e n t r i c w i t h the s u m - c i r c l e . (See fig.) 2. F i n d the versine o f the hour-angle (of the M o o n at mid-eclipse) a n d multiply this by the t a b u l a r sine o f the l a t i t u d e o f the o b s e r v e r a n d d i v i d e by 120. F i n d the arc o f degrees o f the r e s u l t i n g sine. I f the h o u r - a n g l e is east, lay the degrees n o r t h o f the east-point, i f west, s o u t h o f the east-point. T h e east-point w i t h r e f e r e n c e to the e q u a t o r is thus got. ( E ' , i n the f i g u r e . E ' — W ' is the corr e s p o n d i n g east-west.) l a . B . W I T A . B . M ' q ^ ' ] ^ : ; C.D.f^sq^erai b. Al.frT; B 1 . 3 . ^ B 3 . ^ ; C . D . f ^ :
b. B. d*il>^l«Tr3qt A.B.f^T?^ B . % r a t c. A.mfiilll^^'NI; B . ^ ( B l . 3 . ^ )
c-d C.D.°<<^<^HI4WH< d. A . ^ J l ^ ; B 1 . 3 . - ^ B 2 . ^ g r q ^ 2a.
A . ^ ^ T O T R M ; B.^^lT^Mdiii^l); D . ^ R ^
d. A . ^ H ^ l ^ ^ ' ^ M M d l : (A2. g a p f o r ^ g ^ ) ; B.^PTt^^lF^; D.'Wi|lTl<''MI«ldl
XI.3
XI. E C L I P S E D I A G R A M
237
3. A d d three rdsis to the M o o n ' s l o n g i t u d e a n d find the degrees o f d e c l i n a t i o n o f this p o i n t . I f the d e c l i t t a t i o n is n o r t h , lay the degrees n o r t h o f E ' , i f s o u t h , s o u t h o f E ' . T h i s is the east-point w i t h respect to the ecliptic ( E " i n the figure.) D r a w the straight l i n e t h r o u g h the centre, E " O W " . E " - W " is the ecliptic east-west. B y means o f circles, (i.e. by d r a w i n g the p e r p e n d i c u l a r bisector), get the ecUptic n o r t h - s o u t h , viz. N " — S".
For illustration, we shall represent in the figure the lunar eclipse worked out in the examples of chap. X . T h e angular diameters of the M o o n and the Shadow got there are 31'.8 and 76'.9. T h e Moon's lat. at first and last contacts are, respectively, 19'.35 and 27'.65, both north. T h e first and second half durations are nd. 4-14 and nd. 3-54, respectively. Let us assume that at T , the hour angle, is lOnddis, i.e. 60°, west and the latitudeof the observer is 10° 24' (N). T h e Moon's longitude has already been given as rd.4-0-0. (i) T h e half sum of the angular diameters = 108'. 7 ^ 2 = 54'.35. T h i s is to be converted into digits using the formula of verse 6, below, and used as the radius of the sum-circle. It is, 54.35 (3 - 10/15) = 23.3 digits. According to the scale in the figure, 1 unit = 10 digits, this is 2".33. 3b. B.°qt?lI^g«?T c. A . M ^ M d. A . B . ^ r a ^
A.'MI'^M"^;
B.'MI«ilTl
238
PANCASIDDHANTIKA
XI.5
(ii) The semi-diameter of the ecUpsed body, (here the Moon), is 31'.8 H - 2 = 15'.9= 1 5 ^ ( 3 - 10/15) in digits, = 6.8. According to the scale used, this is represented as, "0.7, radius of eclipsed circle. (iii) For the hour angle of 60°, the tabular versine = 60', and tabular sine latitude of observer is 21'40". F r o m these, the tabular sineof the angle of deflection caused by lat. = 60' x 21'40"-^ 120' = 10' 50". T h e angle of deflection = 5° 11',south of the east point, since the hour angle is west. This is angle, E O E ' , and E ' - W is the equatorial east-west. (iv) Longitude of M o o n + rd. 3-0-0 = rd. 7-0-0. T h e declination of this point is 11° 44', south, which is the southward defection from E ' , represented as angle E ' O E". E" - W" drawn is therefore the east-west o f the ecliptic. (v) T h e perpendicular bisector of this, N " S", is the north-south direction with respect to the ecliptic.
T h e explanation that may be required here has all been given in the context of the Vdsistha lunar eclipse (chap. V I ) , in connection with the graphical representation there. It has also been shown that the methods for the deflections are rough, and that using the versine for ladtudinal deflection, as done by the early astronomers instead of the sine, is incorrect. It is also to be noted that the way of giving the representation by our author here is different from those o f others, as also very limited in its scope.
T h e readings: We have emended certain readings according to the idea intended to be conveyed. In the first stanza, the reading disam, is all right, and emending it into disah by T S and N P i unnecessary, since the singular form can give the meaning of the intended plural number, according to the sdstra. A g a i n , in verse 3, it is enough if vakdt is read as 'cakrdt,' and there is no need to eme it into matsydt, as T S have done. N P emend it as bakdt, give it the meaning Tish' (!) and add a sylla 'ca' at the end of the line to make up the metre. T h e intended meaning, viz. 'perpendicular bisector', can be got from cakrdt, following the ms lettering and meaning 'by means of circles', for it is by the intersection o f circles that the fish-figure itself is got.
Marking of points of contact etc. 4. I n the case o f the l u n a r eclipse, m a r k o n the ' e c l i p s e d b o d y c i r c l e ' the direct i o n p o i n t s i n reverse o f the p o i n t s o n the s u m - c i r c l e i n the fig. s e n w for N W S E . O n the N " - S" line, m a r k the M o o n ' s l a t i t u d e at first contact (conv e r t e d i n t o digits,) a c c o r d i n g to its d i r e c t i o n , ( p o i n t L i n the figure,) a n d take it (westward) to the sum-circle, (to f i n the fig.) J o i n this p o i n t o n the semi-circle a n d the centre w i t h a straight l i n e .
XI.5
XI. E C L I P S E D I A G R A M
239
5. W h e r e this Hne cuts the e c h p s e d c i r c l e (f i n the fig.) is the p o i n t o f first contact. T o get the p o i n t o f last contact also, s i m i l a r w o r k s h o u l d be d o n e , u s i n g he M o o n ' s l a t i t u d e at the t i m e o f last contact, m a r k i n g it o n N " - S" l i n e , a n d d r a w i n g the l i n e to the s u m - c i r c l e i n the o p p o s i t e d i r e c t i o n , (i.e. not west w a r d b u t earst-ward). ( T h e p o i n t o f last contact got is 1 i n the fig.) Note: Since the directions are asked to be marked reversed in the case of the M o o n eclipsed, we understand that they have to be marked as they are when the S u n is the eclipsed body. T h e first part of the instructions can be understood to be intended for first contact, since the second part is expressly stated to be for last contact. Since the Moon's ladtude of the moment of last contact is asked to be used to find that point, we infer that the latitude of the moment of first contact is to be used to find the first point. T h e drawing of lines to the westward r i m and eastward r i m of the sumcircle can be inferred from the known directions o f first and last contacts, keeping i n m i n d that they are marked reversed on the Moon-circle. Thus, the following is the work to be done: (i) In the case of the lunar eclipse alone, mark the directions reversed, on its r i m . Take the Moon's latitude at the time of that particular contact whose point is to be found. Measure it along N " — S" according to its own direction, and mark the point. (In the solar eclipse the latitude is to be parallax-corrected.) (In the fig. these points are L , A , for first and last contacts, respectively.) (ii) From this point, draw a line parallel to E" — W", westward or eastward, respectively, for first or last contact, to meet the rim of the sum-circle, (as i n fig, L f or Al'.) (iii) Draw f ' O or 1 ' O to intersect the r i m of the eclipsed at f or 1. These are the points of first and last contacts. For example, in the fig, we shall find these: (i) T h e points of contat for the lunar eclipse is wanted. Therefore N W S E are reversed as s e n w. T h e latitudes are 19'.35 N , and 27'.65 N , Converted into digits, these are 8.3 and 11.9. According to scale, i n the fig., these are 0".8 and 1".2. Measuring along O N " , the points marked are L , and A , respectively. (ii) T h e westward parallel for first contact drawn is L f ' , and the eastward parallel for last contact drawn is A l ' . (iii) J o i n i n g f and 1' with O , the point of first contact got is f, and the point of last contact got is 1. Thus, we find from the figure that the first contact is very near the southeast point of the Moon's rim, and the last contact is a little to the south of its west point.
4b. A . B . ^ ^ l q R I ^ C . i ^ ^ ^ D.I^^ITO^ A.C.D.I^^RI*; B . f ^ ^ c. A . B . - ^ ; C . ^ ? ^ D.^jq?)^ d.
C . ° ^ - ? g # q a i q ; D . ° ^ [tT]#'T«n^ 5b. A . ^ « ^ B.I^M4<1VI1*^:
C . ^ 3 t ^ ^ ; D.flsfW^
c. A . - ^ f W ; B. •wt^«j«zjr D.-?sif?n
A.°<-Mi^-*l'"i II B.°<-MT:^-*lt^W^MI
d. A . B . ^ e a ^ i ^ ; C.^tledc^fiif; D . ^ ^ n f ^
Really cT^W^ belongs to verse 5.
A2.R)tlldoi|l
240
XI.6
PANCASIDDHANTIKA
T h e explanation of all this has already been given. T h e author does not go beyond this i n his graphical representation. T h e corrections of the text to agree with the ideas intended to be conveyed, are easily understood.
Conversion of minutes into angles 6. So diat the graphical representation may appear as the eclipse is seen actually, the m i n u t e s o f arc are to be c o n v e r t e d i n t o digits, at 2 ' p e r d i g i t w h e n the M o o n is n e a r the h o r i z o n , a n d at 3 ' p e r d i g i t w h e n it is o n the t e n t h s i g n , i.e. meridian, and proportionately i n between. T h e proportion is as follows: In the fifteen nddikds (roughly) when the M o o n rises from the horizon to the meridian, or falls from the meridian to the horizon, there is an increase of one minute of arc from 2' to 3', or decrease of one minute of arc from 3' to 2'; what is it at a given time? T h e number of minutes thus obtained is to be represented by one digit in the graphical representation. A t this rate the minutes of latitude etc. are to be converted into digits. In the example, for the hour angle o f tenliddis, we have the rate per digit, (3 - 10/15) = 2 1/3 minutes, i.e. three digits per seven minutes. T h e following is the explanation of the conversion formula: T h e Sun and the M o o n appear to be larger at the horizon, and to become smaller and smaller as they proceed to the meridian and near the zenith. T h i s is an optical illusion, and really the size is practically the same, as can be proved by measurement, taking photographs etc. W e shall not explain the phenomenon here as it is outside the pale of astronomy proper. But we must mention that the explanation given in some works like the SiddhdntaAekhara in wrong. O u r author has taken that the magnification at the horizon is one and a half times that at the zenith, (practically the meridian i n our latitudes) and the increase i n size is proportionate to the hour angle, though this may not be strictly true. H e also thinks that the orbs, which are nearly 32', appear to be about 11 digits near the zenith, and about 16 digits at the horizon. Hence his rule, two minutes per digit at the horizon, and three on the meridian. W i t h the possibility of different people giving the actual estimate of size differendy, what the author says must be taken as only approximate. Therefore no harm will ensue by taking the meridian for the zenith, or by taking the mean value of the m a x i m u m hour angle, viz.j 15 nddikas, in finding the proportion, especially in our latitudes. 6b. A . % ^
1. Col.A.B.3qcrD^:TR^gn^#Hjm
Thus ends Chapter Eleven entitled 'Eclipse Diagram' in the Paiicasiddhantika composed by Varahamihira
Chapter Twelve PAITAMAHA SIDDHANTA
Introduction In this chapter Vardhamihira deals with the Paitdmaha siddhdnta, the other four having already been dealt with. As an astronomical work the Paitdmaha is of very little value, as the author has remarked in his Introduction, "the tithis of the other two, (meaning the Vdsistha and the Paitdmaha), are far from correct." Since this siddhdnta gives only the mean Sun and M o o n , and therefrom the mean tithis, it cannot satisfy the requirements of the Dharmasdstras. B u t it is historically important as a system that immediately followed the Veddhga Jyotisa. W e shall explain at the end of the chapter how this could have subserved religious purposes i n ancient times, and what merits it possesses as the basis of a civil calendar.
Vl^^cblH
VlMctqfuim^
Days from Epoch 1. T h e Siddhdnta o f Paitdmaha teaches that the l u n i - s o l a r ywg'a is five years. A f t e r e v e r y t h i r t y s y n o d i c m o n t h s t h e r e is a n i n t e r c a l a r y m o n t h , a n d t h e r e is a n o m i t t e d day f o r every 62 l u n a r days o r tithis. 2. Subtract two f r o m the years o f the e l a p s e d S a k a era, a n d d i v i d e o u t the r e m a i n i n g year by five. T h e 'days f r o m e p o c h ' are to be c a l c u l a t e d f o r the r e m a i n i n g years etc., the first d a y b e i n g the suklapratipad o f the m o n t h o f M a g h a . T h e naksastras o f the S u n a n d the M o o n , c a l c u l a t e d b y u s i n g the days, are f o r sunrise. 1-2 Quoted by Utpala o n B S 8.22 Ic. D.-RW:
d. A.HI^<'=IHr»IM^I'^ I; B.HlR^<<=IMfel'^y^l4 (B2.3.^iq3q^)
2a. A . B . ^ ; U . ^ b. A.1«ftli»^<^; B.Ma(^'J,^c
A.B.C.D.fqil(B.^C.D.^^7M A.C.D.d
242
PANCASIDDHANTIKA
XII.2
T h e period after which the Sun, M o o n , and the planets all meet again at the first point of the zodiac is commonly called theyuga, meaning "the period of union." Here this is meant for the Sun and Moon alone, and this Siddhdnta takes it to be five years, approximately, with a view to convenience. In the same way, the statements that there is an intercalary month after every thirty months, and an omitted day for every 62 lunar days, are approximate, and rounded off for convenience. T h e first tithi of the light fortnight of M a g h a begiris theyuga, and the year and the day begins with sunrise. T h e author has not mentioned the number of intercalary months or days in theyuga, nor has he given how to get the 'days from epoch', expecting the readers to be experienced enough, by now, to know it for themselves. H e has indicated it in 1.16, and we have explained it under 1.14-17. T h e only thing that is necessary is to know the years of the beginnings of the yuga, and this has been given here as two years after the saka epoch, and every five years thereafter. We shall first compute the number of the intercalary months etc in the Yuga. T h e number of years in theyuga is 5, given. T h e solar months are 5 x 1 2 = 60. T h e intercalary months are, 60/30 = 2. T h e lunar or synodic months are, 60 -t- 2 = 62. T h e lunar days or tithis are 62 X 30 = 1860. T h e omitted lunar days are, 1860 ^ 62 = 30. T h e (civil) days are, 1860 - 30 = 1830. T h e Moon's revolutions are, the Sun's revolutions -I- the synodic months = 5 -i- 62 = 67. T h e Vyatipdtas are, solar revolutions -I- lunar revolutions = 5 -f- 67 = 72. T h e number of days in the solar year is, 1830 ^ 5 = 366. T h e days perayana are 366 2 = 183. T h i s is enough for our purpose. T o get the 'days':(i) (Elapsed saka years — 2)
5. Take the remainder alone.
(ii) T h e solar months gone = (i) X 12-1- elapsed months from Magha. (iii) The intercalary months = (ii)
30. Take the quotient alone.
(iv) T h e lunar months gone = (ii) -I- (iii). (v) T h e lunar days gone = (iv) X 30 -I- tithis gone i n current month. (vi) Omitted days = (v) ^ 62. Take the quotient alone. (vii) T h e days from epoch are, (v) — (vi). T h e tithis gone, used in (v) are actual elapsed tithis, and not those increased by one (according to verse 4. of this chapter,) for calendrical purposes. If the latter is used, subtract the calendrical elapsed tithi from the remainder got in (vi). If this is greater than 46, lessen the days from epoch by one to get the correct days. T h e reason for this will be explained while dealing with verse 4, following. Further, since there can be no fraction of intercalary month or avama at the beginning of ayuga carried over from a previousyug-a, there is no ksepa for these, in the computation rules. As for the explanation of these rules it has already been given in chap.I, when dealing with the rules for days from epoch according to the Romaka. T h e names of the five years, (not given in the text,) are: Samvatsara, Parivatsara, Iddvatsara, Anuvatsara, and Idvatsara. In certain Ved^csdkhds, t a slight variadon in some names. As for the reading of the text, we have corrected dyunam, into dvyunam, since the former is meaningless in the context.
XII.3
243
XII. P A I T A M A H A S I D D H A N T A
Example 1. Compute the 'days from epoch' according in the Paitdmaha, for the sunrise of calendrical da eleventh of the light fortnight of Kdrttika, in the elapsed Saka year 426. (i) (426 — 2) 5 = 424 -H 5. Here the remainder is 4, the years gone, i n the currentyw^a. T h e fifth, Idvatsara is current. (ii) C o u n t i n g from Magha, 9 months have elapsed before {Kdrttika), in the current year. .-. the solar months gone = 4 X 12 4- 9 = 57. (iii) Intercalary months = 57/30, = 1 27/30 (quotient = 1) (iv) L u n a r months gone = 57 4- 1 = 58. (v) ThecalendricaUMtjgoneinthemontbare 10..-.thetotalftttwgone = 58 X 30 + 10= 1750. (vi) Omitted tithis = 1750 ^ 62 = 28 14/62. (The quotient, 28 are the omitted tithis). Since the remainder, 14, minus 10, leaves 4, which is not greater than 46, the calendrical tithi itself is the tithi. (vii) Days from epoch = 1750 - 28 = 1722.
*i?JW'«n^: fudl^^-i ? T f W M P I ^ I ^ H II ^ Tithi, Nak§atra etc. 3. A d d to the 'days' a sixty-first part o f itself. T h e total tithis are got, ( w h i c h , d i v i d e d o u t by thirty, leaves the tithis i n the m o n t h ) . M u l t i p l y the 'days' by 9, a n d d i v i d e by 122. T h e total naksatras are got, ( w h i c h , d i v i d e d o u t by 27, gives its actual naksatra, r e c k o n e d f r o m Sravisthd). M u l t i p l y the 'days' by 7 a n d d i v i d e by 610. S u b t r a c t this f r o m the 'days'. T h e r e m a i n d e r are the total naksatras o f the M o o n , ( w h i c h , d i v i d e d o u t Iby 27 a n d the r e m a i n d e r c o u n t e d f r o m Sravisthd, is the M o o n ' s naksatra). T h e following is to be done:(i) Tithi = 'days' + 'days' H- 61. (This, divided out by 30, and the remainder counted homsuklapratipad, is the tithi proper). (ii) Sun's naksatra = 'days' X 9 -f- 122. (This, divided out by 27 and the remainder counted from Sravisthd, is the Sun's naksatra). (iii) Moon's naksatra = 'days' — 'days' X 7 -H 610. (This, divided out by 27 and the remainder counted from Sravisthd, is the Moon's naksatra). Example 2. For the date of Ex. l,find the tithi etc. T h e 'days' got there are 1722. (i) Tithi = 1722 4- 1722 ^ 61 = 1722 + 28 14/61 = 1750 14/61. Divided out by 30, the remainder is 10 14/61..-. the 10th Ti^/ii of the light fortnight is gone, and 14/61 o f Ekadasi has gone at sunrise. 3. Q u o t e d b y U t p a l a o n 5 S 8 . 2 2 . 3a. A . B . ^ ^ ^ 5 5 T O ^ ( B . ^ ) % M ;
^
^ ^
^ ^ ^ ^ ^ g ^ ^ ^ .
U.°^1^°
C.'S*Ng'M^^J|iJl; D.'S+^-^^^JIul;
c. A . B . f ? ^ ; D . ^ :
u.^te?t
d. A . ^ A . s i P r m ; B2.«Rtm^
244
PANCASIDDHANTIKA
XII.4
(ii) Sunsnaksatra = 1722 X 9 1 2 2 = 127 4/122. Dividing out by 27, the remainder is 194/122. C o u n d n g from Dhanisthd, Citra is gone, and the Sun is at 4/122 of Svati. (iii) Moon's naksatra = 1722 - 1722 X 7 - 610 = 1722 - 19464/610 = 1702 146/610. Divided out by 27, the remainder is 1 146/610. Sravistha is gone, and the M o o n is at 146/610 of Satabhisaj. It is to be noted, that two of these being known, the third can be counted from them. T h e following is the explanation of the rules: U n d e r verses 1 —2, the number of days in xheyuga etc. have been got. Using them, the total tithis are got by the proportion, 1830: 'days':: 1860: Tithis. .-. Tithis = 1860 X 'days' ^ 1830 = 'days' X 62 ^ 61 = 'days' (1 + 1/61) = 'days' + 'days' ^ 6 1 .
Next, since there are five solar years i n theyuga, there are 5 x 27 = 135 solar naksatras. We have the proportion, 1830: 'days' :: 135: total Sun's naksatra. .-. Sun's naksatra = 135 X 'days' 1830 = 'days' X 9 122. Next, since there are 67 revolutions of the M o o n in theyuga, there are 67 x 27 = \ 809 naksatras. So, we have the proportion, 1830: 'days' :: 1809: Moon's naksatras. .-. Total Moon's naksatras. 1809 x 'days' ^ 1830 = 'days' x 603 610 = days (1 - 7/610) = 'days' - 'days' x 7 610. While the mss. readings tryamsatvarhce etc. are corrupt, Bhattotpala's reading saikatrimse itself wrong, since it contradicts facts, and we have emended it into saikartvamse. T S have corrected is as saikasastyarnse garie which is not proper since the correction does not follow the letters of the tex N P emends it as saikasadarnse for the number 61 that is required, but generally the said number is not found to be formed thus.
Vyatipata 4. I f the m o m e n t o f f u l l o r n e w m o o n falls before n o o n , the s e c o n d o f the two Cithis c o n n e c t e d w i t h the day is the (civil) tithi for the day, otherwise the first. M u l t i p l y the 'days' by 12 a n d d i v i d e by 3 0 5 . T h e Vyatipdtas are got.
T h e tithi of this Siddhdnta is mean tithi. Since this is less than the day, each day has parts of two tithis connected with it, and we have to fix one of them as the date of the particular day. It is this that is done by the first half o f the stanza, it seems. If others like the Srdddha-tithi are meant to be fixed here, they would be mentioned by name. T h e mere word tithi without an attribute must mean only the date. Agreeing that the date is meant to be fixed here, is it the date of the full or new M o o n alone that is fixed here, or that of any day? F r o m the -word parva used, one may think it is only the former that is sought to be fixed. But this cannot be, since fixing the date of one particular day among so many is practically useless. I f it is argued that the fixing of the day as parva or pratipad is useful to 4b. A c R l ^ R R l ^ ; B 1 . 3 . ' a ^ ^ # ^ ; B 2 . c i ^ ^ c. AB.sqrfqqrai; D.oHrdMldl d.
A.m^ A.^rara^v.T^:; B.M^aiHi^cli^l: ( B 2 . ^ ' 9 t : )
X1I.4
XII. P A I T A M A H A S I D D H A N T A
245
determine whether the anvddhdna or the isti is to be performed on that day, then the general term, tithi, need not have been used, and it would have been easier to mention the thing. Further this is a matter for the Dharmasastras to deal with, not for an astronomical work. Therefore, by fixing the date of the parva, the author means to fix all the subsequent dates following, upto the next parva, and make them convenient for civil use. If the dates are consecutive, one for each day, it will be convenient for civil reckoning. If there is ajump, omitting one date i n the middle {tithiksaya), it is plainly inconvenient. T h e instrucdon i n the verse secures that the dates follow without omission in the middle of the fortnight. For, if the moment of full or new moon is after noon, there will be no omitted tithi in the fortnight following, and the corresponding dates will follow one after another each day, beginning from prathamd, next day. But, i f the moment of full or new moon is before noon, then there will be an omitted tithi in the fortnight following, since the remainder i n getdng the omitted days will be greater than 46, increasing by one each day. Therefore, i f now the day of full or new moon itself is reckoned as the first date of the fortnight, then the reckoning can be continued to the end of the fortnight without omission. It is bearing i n m i n d this idea implied by the text, that we made a distinction between the astronomical and the civil date, i n giving the computation of the 'days from epoch'. T h e Siddhdnta is only repeating here the idea of the Veddnga Jyotisa i n : dyu heyam parva cet pdde pddas trirnsattu saikikd \ the term pdde (meaning 'quarter day') corresponding to the term ardhe in our text. T h e thirty-one parts mentioned form the measure of the pdda, i n units of 1/124 parts of a day. We have explained the vyatipdta in detail, in our commentary on III.20. W e must remember here two things that we said there: (1) Vyatipdta occurs when the Sun and the M o o n have the same declination, (both north or both south), and when one is moving northward while the other is moving southward. (2) If the Sun, M o o n , and declination are all mean, as here, and i f the first point is at the solstice, as here, mid-vyatipdta-yoga must fall when the Sun + M o o n equals 12 rdsis. In theyuga, (of 1830 days) there are 5 -I- 67 = 72 such yogas, i.e., vyatipdtas. Therefore, for given 'days', there are, 72 x 'days' 1830 = 12 x 'days' 305, vyatipdtas, as given here. T h e quotient obtained are the wyaft/xJto gone. But this knowledge is practically useless, and we must take it that the Siddhdnta intends here to give when the vyatipdta occurs, a time extremely propitious for g\hs,iapa, homa, etc. T h i s can be found easily from the remainder. Divide this by 12. T h e result is days etc. gone from the last vyatipdta. Subtracting the remainder from 305 and dividing by 12, we get the days to the middle o f the next vyatipdta. If the remainder is zero or nearly so, it is clear that the vyatipdta is on.
Example 3(a). Given the 'days' 1697, what is the tithi for that day, and the subsequent days, upto the en the fortnight? T h e tithis gone = 1697 1697 61 = 1697 + 27 50/61 = 1724 50/61. Dividing out by 30, the remainder is 14 50/61, i.e. 50/61 part ofpurnimd has gone, and 11/61 part remains. Since the tithi is equal to 61/62 day, 11/61 tithi equals, 11/61 X 61/62 = 11/62 day. T h e full M o o n ends at 11/62 day, i.e. before noon. Therefore that day itself is Prathamd, (not Purnimd). (The same conclusion follows from the remainder of the omitted day, 51, i n this case, (minus zero, for the civil day gone,) being greater than 46. After this, for ten days, the tithis are from the second to the eleventh, for civil purposes, though at sunrise the tithis are from the first to the tenth, the eleventh being the omitted tithi. O n the next day the tithi at sunrise is the twelfth, as also the civil tithi, and so on.
246
X1I.5
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Example 3(b). Given the 'days' 1722, find when the vyatipata falls. 1722 X 12 305 = 67 229/305, i.e. 67 vyatipdtas have gone, from the beginning oixheyuga. T h e remainder is 229. Dividing by 12, we get 19-5, days etc. gone from last vyatipdta. Subtracdng from 305, and dividing by 12, we get days etc. 6-20, to go for the middle of the next vyatipdta, i.e, it will be falling at 20 nddis on Kdrttika Krsria-dvitiyd. T h e correction of the reading here is easy to understand. O f T S , the latter says some farfetched thing as the meaning of the first part of this verse, which seems meaningless to me, and the former without saying anything himself, refers us to the Sanskrit commentary, evidently not understanding it himself. In explaining the Vyatipdta, their use of the one forming the seventeenth of the series Viskambha etc., can serve no purpose i n the present case. T h e same confusion is seen also in N P (see Pt Il.p.82). We have already shown that the series itself did not exist at the period of Varahamihira.
Duration of a day 5. T a k e the days g o n e i n the Uttardyaria, i.e. the n o r t h w a r d c o u r s e o f the S u n , a n d the days to g o i n the Daksiridyana, i.e. the s o u t h w a r d c o u r s e . M u l t i p l y by t w o , d i v i d e by 6 1 , a n d a d d 12. T h e d u r a t i o n o f the d a y t i m e (in muhurtas) is got. T h o u g h the text is very corrupt here, since we know that the duration of daytime is given here, and that as mentioned i n the VeddhgaJyotisa, many words occurring i n that work being recognisable here, we have succeeded i n reconstructing the verse using the letters found i n the text, though not to o u r entire satisfaction. B u t there is no doubt about the idea intended to be conveyed, viz. that of VJ (verse 22 o f the Rgveda version and verse 40 of the Yajurveda version). B u t T S and N P , i n order to retain the expression dvddasahlnam, read into the text many impossible things and make several emendations not caring even for the metre. T h e rule given is thus explained: A c c o r d i n g to this Siddhdnta the shortest day is 12 muhurtas, at the end of the southward course and the beginning of the northward at Winter solstice, and the longest is 18 muhurtas at the end of the northward course and the beginning of the southward, at Summer solstice. Each course has 183 days, and the increase or decrease o f daytime is considered uniform. Therefore, since there is an increase o f 6 muhurtas from 12 i n the 183 days of the northward course, the increase for a desired number of days (counted from the beginning) is: days gone X 6-T- 183 = days gone x 2 61. So, theduration is 12 -I- days gone x 2 -H 61. T a k i n g the southward course, it is 12 muhurtas at the end, proportionately greater, the earlier is the day, the increase being 6 i n 183 days, at the beginning of the course. Therefore, d u r i n g the southward course, the duration is 12 + the days to go i n the course X 2 H- 61. It is to be noted that the author has not said 5a. A.B.i^fcl
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XII.5
XII.
PAITAMAHA
SIDDHANTA
247
when the northward course begins, and when the southward one. (Perhaps he has said that i n this verse, and the corruption of the text has masked it.) W h e n the Sun enters Sravistha its northward course begins, and when it is at the middle o f Aslesa, its southward course begins, o r the first half of the year is the northward course, and the second half, the southward one. It can also be inferred from the rule for vyatipdta. Example 4. Given the 'days' 1722, find the daytime for the day following. 1722 H- 366 = 4 258/366. Four years have gone, and 258 days in the fifth. T h e 183 days of the northward course have gone, and 75 days have elapsed i n the southward course. T h e days to go = 1 8 3 - 7 5 = 108. T h e duradon of daytime in wM/jMrto= 1 2 + 1 0 8 x 2 - ^ 6 1 = 12 + 333/61 = 1533/61. T o conclude, it has been said i n the introducdon to this chapter that the Sun, M o o n etc. of this Siddhdnta are only mean, not true. Some think it is even worse than this, which we must investigate now. We have seen that, according to this Siddhdnta, there are 366 days i n the year, 5 years or 1830 days constitute theyuga, and there are 62 synodic months i n it. B u t actually there are about 365 days i n the year, and also 62 synodic months take 1830.8965 days. Therefore, instead o f being at the zero point of Sravistha at the beginning of each yuga, the Sun will be advancing by about three degrees per yuga. T h e synodic month being not completed, it will be really only Caturdasi then, and at the end of e\ery yuga, the tithi will be preceding by about one per yuga. T h i s is common to both this Siddhdnta and the Veddhga Jyotisa. B u t accumulation of this error is prevented by tying the Yuga to the correct Saka year, by the instruction to use the Saka year for finding the year of theyuga. (This is of the same nature as tying the lunar year to the solar.) It may be said, i n passing, that at the period when the VJ was followed, actual observation was used to find out when a correction was wanted and the same made by simply omitting an intercalation, for which, it is possible, they could even have discovered a formula i n the long r u n . W e have made this clear in our Notes to the VJ, (Indian National Se. A e . , New Delhi, 1985) and the d u r i n g the Seminar on Indian Calendar held by the Institute of Traditional Cultures, University of Madras. (Bulletin of the Institute of Traditional Cultures, Madras, 1968, Part I, page 60.) W h e n thus the accumulation of error is taken care of, giving 366 days to the year is extremely convenient for civil, calendrical, and even religious purposes, since it makes calculations easy i n the same way as we, even now, take the year to have 365 days, and the months 31 etc. days. W e get 183 days for each ayana, 122 days for each cdtarmdsya, and61 dzysiorezchrtu,allinwholenumbers. T h e r e are 6 lidt^ana months i n the ywga. Eventhejawra month has 30 days and a half, a fraction easy to work with. Further, the fortnight or paksa was the unit used then i n the place of the modern week. W e have shown, i n our Notes o n verses 2 and 4 above, how it was secured that the dates follow consecutively i n the paksa, a factor so essential for a civil calendar. T h e approximateness of the tithis etc. being mean, is of course there, but this is only an advantage i n civil reckoning. A s for religious purposes, the true tithis etc. required for rites like darsa-purna-mdsa were guessed from earlier observations, as we have said, and there are several indications i n the Vedas that such was the case. T h e abhyudayesti, enjoined to be performed i f the M o o n is observable i n the east between the anvddhdna o n the previous day and the isti on the next day, is one such indication, for, i f the visibility had been properly calculated from the true S u n and M o o n , using drkkarma etc. there would be no need for abhyudayesti at all.
Thus ends Chapter Twelve entitled Taitamaha Siddhanta' in the Paiicasiddhantika composed by Varahamihira C o l . A.D.^
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Chapter Thirteen SITUATION OF THE EARTH: COSMOGONY
t 4 H c ( H i 3?^TniTt r T ^ 5 « r :
Situation of the earth 1. T h e s p h e r i c a l e a r t h w h i c h is c o n s t i t u t e d o f the five elements, stands p o i s e d i n the r e g i o n o f space, m a r k e d by the host o f stars f o r m i n g a cage as it were. Note: T h e earth is called so because of the five elements constituting it, it is predominantly earthy. 2. T h e w h o l e earth-surface is spotted by trees, m o u n t a i n s , cities, rivers, oceans, etc. T h e M e r u m o u n t a i n , ( f o r m i n g the N o r t h pole), is the abode o f D e v a s . T h e A s u r a s , ( D e m o n s ) , are d o w n b e l o w (i.e. at the S o u t h pole.)
1-4. Q u o t e d b y U t p a l a o n B S 2 . 5 5 ; b y Suryadeva and Nilakantha on ABh. Gola. 6; 2-3 by Prthiidaka on BrSS 21.3; 3 by Suryadeva on A B / i . Gola 12. la. A . ^
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XIII.6
XIIl. SITUATION OF T H E EARTH
249
3. J u s t as the r e f l e c t i o n o f the objects o n the b u n d o f a water course is u p s i d e - d o w n , so the A s u r a s are, (with respect to the Devas). T h e A s u r a s too c o n s i d e r the Devas to be u p s i d e - d o w n . 4. J u s t as the flame o f the fire, o b s e r v e d by m e n here, flares u p w a r d s , a n d a n y t h i n g t h r o w n u p falls d o w n towards the e a r t h , the same u p w a r d flaring o f the flame, a n d the d o w n - w a r d f a l l i n g o f a heavy object is e x p e r i e n c e d by the A s u r a s , (at the a n t i - p o d a l r e g i o n ) .
l a R J R ^ ^ oJ^*4femt ^ ( S ^ S ) ^ :
Rotation of the earth 5. T h e axis o f the e a r t h extends r i g h t u p a n d r i g h t d o w n to the stellar sphere. T h e stellar s p h e r e , b o u n d by the axis to the e a r t h , rotates by the w i n d system c a l l e d Pravaha. Note: Seven wind systems are spoken of by H i n d u astronomers. T h e uppermost, blowing permanently westward in the region of the planets and stars, is the cause of their westward rotation once a day. 6. O t h e r s say that the e a r t h rotates o n its axis, l i k e a n object p l a c e d at the h u b o f a w h e e l , a n d not the stars. I f so, birds l i k e the eagle flying u p i n t o the sky, c a n n o t reach t h e i r nests back. Note. This is what is meant: D u r i n g the time they are away from the nest, they would have been carried far away by the rotating earth, from the spot o f origin o f their flight and the birds cannot reach the nest. But they do reach, and so this theory is false. 3a. A.B.dolWdHI ( B 3 . ^ ^ ) ; Cdd^^-dHI b. B1.3.^?II^ • c. Al.cTSnfrl; A 2 . < T ^ ^ ; B . c l i : ^ . B . ^ l ^ A . H a p l . o m . ^ [... (in verse 4) d. B 1 . 2 . f ^ « n ^ 4a-b. B.%3lf^?^#^; D . f W [ f e ^ ] c. B.'^ISi^
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250
XIII.13
PANCASIDDHANTIKA
V9
7. F u r t h e r , o n account o f the great speed o f the rotation o f the earth, banners, flags, etc. w i l l always be f l o w n westwards, (just as the c l o t h o f a m a n r u n n i n g eastwards i n still a i r , is b l o w n westwards). If, to obviate this objection, a very slow r o t a t i o n is p o s t u l a t e d , h o w does the r o t a t i o n (once a day), take place at all (i.e. o n e r o t a t i o n c a n n o t be c o m p l e t e d i n o n e day). 8. A r h a t , (the p r o p o u n d e r o f the J a i n r e l i g i o n ) has w r i t t e n that there are two S u n s a n d two M o o n s , e a c h r i s i n g alternately. I f so, h o w does the l i n e j o i n i n g the S u n a n d the celestial p o l e goes r o u n d exactly o n c e i n a day r o u n d the pole? Note. This observation shows that the celestial sphere movers once rouAd in a day and carries with it all heavenly bodies. So the same Sun appears each day after one rotation. Therefore, the postulation o f a second Sun is purposeless. T h e same for the M o o n .
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Situation of the Gods and Asuras 9. T h e S u n , situated at the beginning o f the sign Mesa, moves along the h o r i z o n i n the clockwise d i r e c t i o n , as seen by the D e v a s , at the N o r t h p o l e . A s seen at the equator, it moves u p w a r d s ( a l o n g the p r i m e vertical). F o r the A s u r a s at the S o u t h p o l e it moves a l o n g the h o r i z o n , i n the a n t i - c l o c k w i s e d i r e c t i o n . 10. T h e S u n at the e n d o f the s i g n G e m i n i is seen m o v i n g (by the Devas) r o u n d at a n a l t i t u d e o f 24°. O n that day, it is seen c r o s s i n g the z e n i t h at U j j a i n . {Note. V M considers that the m a x i m u m declination of the Sun is 24°, and the latitude o f Ujjain is 24°, which is only approximately correct.) 11. T h u s , o n that day, at m i d - d a y , t h e r e is n o s h a d o w cast b y the g n o m o n at U j j a i n . N o r t h o f U j j a i n the m i d - d a y s h a d o w is d i r e c t e d n o r t h , a n d f o r p e o p l e s o u t h o f U j j a i n the s h a d o w is d i r e c t e d s o u t h . 12. S o m e say, " w h e n the S u n is situated i n the t h r e e signs M e s a , Rsabha and Mithuna, it is day-time for the Devas, but when i n the signs Karkataka, Simha and Kanya, it is night-dme. I salute them, (and wish to be r i d of them since they are quite wrong.) Note. T h e authors of the Dharmasastras, followed by the generality of people, consider that the uttardyana, i.e. the northward course o f the Sun from the beginning o f Makara to the end o f Mithuna is day-time for Devas. Its southward course from the beginning of Karka^a to the end of Dhanus is considered their night-dme. It is their ignorance that is referred to here. 13. M o v i n g i n the same n o r t h latitudes w h e n i n K a r k a t a , S i r n h a a n d K a n y a , as w h e n i n M i t h u n a , R s a b h a a n d M e s a , h o w c a n the S u n be seen a n d n o t seen by the D e v a s , so that it is d a y - t i m e (in the first three m o n t h s , ) a n d n i g h t - t i m e (in the n e x t three m o n t h s ) . Note. T h e mistake o f these people lies i n thinking that uttardyaria is the day-dme o f the Devas, and daksiridyana is night-time. It is only when the Sun is north of the equator while i n the 6 signs Mesa to Kanya, it can be seen by the Devas, forming their day-time, and i n the 6 other signs, it cannot be seen, and it is night-dme. 9-13 Quoted by Utpala on 5 S 2. pp.
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II
T?^ ^ wcictiijiiKi ^ ( F ) ^ 4 l v i n v w i P j
Signs and Yojanas 14. F o r t h e D e v a s , i.e. at t h e N o r t h p o l e , w h e r e t h e z e n i t h is the n o r t h celestial p o l e , a n y visible semi-great-circle p a s s i n g t h r o u g h the z e n i t h is a m e r i d i a n o n the e a r t h . It is 3 signs distance, i.e. 9 0 ° , f r o m the z e n i t h to the h o r i z o n . T h i s is to be d i v i d e d i n t o 9 0 parts, so that e a c h d i v i s i o n is a d e g r e e . Note. These semi-circles are actually the meridians o f places having the respective meridians of longitude o n the earth. Conventionally, Ujjain longitude is the zero longitude, like Greenwich in modern astronomy. 15. T h e s e d e g r e e - d i v i s i o n s i n t h e sky a r e the zeniths o f c o r r e s p o n d i n g degrees o f latitudes o n the e a r t h a l o n g a n y m e r i d i a n , a n d t h e distance b e t w e e n e a c h d e g r e e o f l a t i t u d e is 8 8/9 yojanas. Note. Since the degrees of latitude are zero at the equator and 90 at the N.pole, and the same to the S.pole, i n the visible half of the meridian circle, the circumference of the earth is 360° equal to 360 X 8 8/9 yojanas = 3200yojanas. Aryabhata'sy^yana-measure is very nearly V M ' s . 16. T h u s f o r 9 0 ° , t h e r e a r e 800yojanas, c a l c u l a t e d . W h e n a n o b s e r v e r o n a n y m e r i d i a n sees the S u n r i s i n g , it is m i d d a y o n t h e m e r i d i a n 800yojanas (east) o f him. Note. T h e mean sunrise is meant here since the actual sunrise is affected with cara:
14-17 Q u o t e d b y U t p a l a o n B S 2 , p . 5 8 . 14a.
B.^a5^^ A.'^nairaM; B .
c. A.'?TO^'jllTi<|4|>J|; B.'?1RC^8(1"|1T1
U.^^^° d. A . B . D . 3 I c ^ ^ ; C . U . - S I ^ ^ : A.»^«Wf; B . K>^; C.U.#2W; D . * ^ :
b. A.B.ilVW^^II:; C.D.<|!^W*^i^ll:
A.D.ynt^i^
c. A . W U l ^ ; B 1 . 3 . ^ ^ t T # ; B 2 . ^ ^ s l 5 ^ d. B.
16a.
A.B.^H=it^c
B.'J.^'hlVJl; U.'J.^'t.Wl)
b.
B . ^ K I <;KIR
A.Bl.2.^:
c.
C D .
b. A . ^ ; C . D . U . % 3 i ^ ^
<4l^°
om^H
C . cTgW^lT^; D . d^MHI"II^Vl. B . ^
Xlll. S n U A l l O N OF THE EARTH
XIIl.21
253
c^lfPTT: ^ ( v l f ^ d l ^ T t r l ^ ^ m ^
Position of Lanka and Ujjayini 17. T h e m i d - d a y o f L a n k a is the same as that at U j j a i n , w h i c h is n o r t h o f L a r t k a o n the same l o n g i t u d e . B u t t h e i r d a y - t i m e d u r a t i o n s are d i f f e r e n t , except w h e n the S u n is o n the e q u a t o r .
Measures of the earth etc. 18. T h e c i r c u m f e r e n c e o f the e a r t h is 3 2 0 0 yojanas. W h e n situated o n the e q u a t o r , the S u n is visible f r o m p o l e to p o l e at a l l latitudes, ( m a k i n g the day a n d n i g h t equal). 19. T h e m i d d l e o f the e a r t h , (the N o r t h p o l e is m e a n t here), is n o r t h o f U j j a i n by 5 8 6 2/3 yojanas. It is n o r t h o f L a r t k a by 8 0 0 yojanas. Note: T h e distance in ladtude from L a i i k a to Ujjain is 24°, and from Ujjain to the N o r t h Pole is 66°. T h u s 6 6 x \ ' 8 W = 586 2/3, and 90 x 8 8/9 = 800.
17a. A.B2.3f5pRl; B 1 . 3 . 3 * # A . B 1.2.#FRr^°; B3.-?ifr^° b. B . •Hid l^dl-Mltl>l
|; B 3 . yHW:!)
c. A . B . I I W I d. A . B . f 5 3 ^ ( B 2 . 3 . - ^ ) Al.l^^cidl-q:; A2.f5r....5ra)^:; B 2 . 3 . ^ ^ : ; Bl.^^:
18-19 Q u o t e d b y U t p a l a o n B S 2 . p . 5 8 . 18c. A . c i r a ^ ; C.cl^vn# A.-^^FTSI^ d. A . B . ^ ^ R ^ t # A . B . C . f ^ l ^ ; 19a. A . o ^ q ^ B.W?lf% b. B . % R . U . d. A . B 1 . 3 . - O T T . B . ° ^ :
D . W l ^
254
XIII.23
PANCASIDDHANTI KA
Visibility of the Sun 20. A t a n y latitude, the e q u a t o r i a l S u n is b e n t so m a n y degrees south at m i d day, as the n o r t h p o l e is raised u p f r o m the n o r t h p o i n t o f the h o r i z o n . 21. G o i n g n o r t h f r o m U j j a i n , 373 1/3 yojanas, t h e stellar sphere, ( m a r k e d by the 27 asterisms o f the e c l i p t i c r i s i n g i n a n o r d e r ) becomes d i s c o n t i n u o u s , (i.e. the o r d e r i n the r i s i n g is d i s r u p t e d ) . Note. A t 66° N o r t h latitude, which is 42°, (i.e. 66° - 24°), north o f Ujjain, peculiarities occur in the rising of the signs of the eclipdc, duradon of day-time etc. 42° = 37.3 1/3 yojanas.
22. A t that latitude, the S u n c a n be visible e v e n t h r o u g h o u t a day. N o r t h a n d n o r t h o f this place, t h e S u n m a y n o t set m o r e a n d m o r e t h a n o n e day, u n t i l at the n o r t h - p o l e it w i l l n o t set f o r six m o n t h s at a stretch. 23. A t a distance greater t h a n 403 5/9 yojanas n o r t h o f U j j a i n , the signs D h a n u s a n d M a k a r a c a n n e v e r be visible. Note. T h e Dhanus and Makara segments of the ecliptic have a south declination greater than 20° 36'. In the north latitudes 90° - 20° 36' (= 69° 24') and beyond, the zenith distance of these signs becomes greater than 90°, and so they are not visible in those latitudes. 69° 24' is 45° 24' north of Ujjain, i.e. 45° 24' x 8 8/9 = 403 myojanas north.
20-29. Quoted by Utpala on
22a. A . ^ ^ ;
C . D . ^ ^ :
2,p.58-59. 20a.
b. B1.3.^fl5f^; B 2 . ^ R ^
B.f^^I^^
b. A . ^R'Ml^rai
B . f^R'MI ?*HI
B2.^'2ni c. B.i
c. A.Bl."^f|cR° 23a. B . % r r a ° A.^^mil:; U . ^ m i l ^ b. A . c . ^ p ! # m ; B.^^;qfw; D.^^qf^i^Fi^
A l . f ^ ^ ^ i ^ ^ ; B . f t ^ (B2.1t5pfiT) d.
B.^^(B3.'RIlt)
A.^u|d*HN
2Ia. Al.l5Wf!f; A 2 . i ^ ; B . f 5 r a f c T . A 2 . ^ H ^ ^ b. A . C . D . ^ f M f ^ ; B.^frat^ A2.'%^^. A . 1 ^ ^ ^ c. A.B.C.D.ftwfcT d. A . B 1 . 2 . C . D . i T ^ s q . B.WlTTtaf:
A.B.D.^: (AI.^:; A2.^:) c m B3.ti
Bl.^?^^^:
XIII.28
Xlll. S l T L A l ION OF IHF, EARTH
M^Vlifd M ^ V l d l
^sjt^ ^ ^
255
Ticf |
24. A t latitudes n o r t h o f U j j a i n greater t h a n 4 8 2 yojanas a n d a fraction Vrscika, Dhanus, Makara and K u m b h a signs will never be visible. Note. These four signs have a declination greater than 11° 44' south. Therefore latitudes 90° — I r 44' = 78° 16' North and more cannot see these signs, since their zenith distance is greater than 90°. 78° 16' is 54° 16' north or Ujjain = 54° 16' X 8 8/9 = 482 10/27yo7«ms. 25. 5 8 6 2/3 yojanas n o r t h o f U j j a i n , i.e. at the N o r t h p o l e , the s e c o n d h a l f o f the ecliptic, i.e. the signs T u l a to M i n a , c a n n o t be seen. Nole. Being situated south of the celestial equator, the zenith distance of these signs from the North pole is greater than 90°, and therefore they are not visible at the North pole. T h e distance of the North pole from Ujjain is 90° - 24° = 66° = 586 213 yojanas. 26. P e o p l e o n the e q u a t o r see the N o r t h p o l a r star o n t h e h o r i z o n . A t the N o r t h p o l e , p e o p l e observe it at the z e n i t h . I n between, p e o p l e observe it at attitudes 0° to 9 0 ° . Note. As the north latitude increases, so the latitude of the Pole star increases cfjually. This fact is mentioned elsewhere also.
24a. A.^'lRTB.°?J?ftfll b. A."?ra!wr; B1.3."?mtc?FTTi^ B 3 . ° ? I ^ ^ D.°Vldfc!4I'il
c. Al.'H^TST; B 1 . 3 . ' T ^ ; B 2 . W ? t B.^^a^ d. A.^Ttcqra^'TRq^; B . % r ^ (B2.cqT'?) ^
c. A.^^'jq^'ltcqfQ^o; B . % q g ( B 2 . 1 ) 26a. A . B . d. B 1 . 3 . ^ ? ^ U . o m 3 # f 25a. A . ^ 5 ? M ; B 1 . 3 . ^ l 5 ? M a n d B2.^S?T# in place o f ^ ^ ^
b. B.'-ffiRTl A . W f i s T ; B . W % (B2.''f) ^° A.B.^^?T|^ d. B l . c T ? ^ ; B3.-a^^^A.B1.2.\qTTcn:; B3.\qTM
256
PANCASIDDHANTIKA
XIII.32
27. F o r the p e o p l e o n M e r u , i.e. at t h e N o r t h P o l e , the S u n is visible at a stretch, w h e n it is i n the six signs M e s a to K a n y a . W h e n it is i n the next six signs, it is visible to the d e m o n s at the S o u t h p o l e , at a stretch. Note. M o v i n g i n the first six signs, the Sun's zenith distance is less than 90° at the North pole and it is visible to the Devas there. Being greater than 90° at the South pole it is invisible to the Asuras at the South pole. It is vice versa in the next six signs. T h u s the Devas and the Asuras have their day and night alternately, each for six months at a stretch. 28. F o r t h e m , the first p o i n t o f M e s a is t h e Lagna o r O r i e n t ecHptic p o i n t , p e r m a n e n t l y , M a r s is the L o r d o f the Drekkdna, Navdrhsa, Dvadasdrhsa a n d Trimsdrhsa lagnas p e r m a n e n t l y . Note. T h e first point of Mesa moves r o u n d and round there o n the horizon, and no other point rises o r sets. T h e lordships are as prescribed i n the Hordsdstra. There the L o r d o f Mesa is not only the master the Rdsi-lagna, but also of Drekkdna, Navdrhsa etc. lagnas. 29. L a i i k a is b e n e a t h the celestial e q u a t o r , i.e. the celestial e q u a t o r itself is the p r i m e vertical at L a f i k a . T h e r e t h e stellar s p h e r e is e q u a l l y d i v i d e d (into the n o r t h e r n h a l f w i t h the N . P . at its centre, a n d t h e s o u t h e r n h a l f w i t h the S.P. at its centre). T h e r e the day a n d n i g h t are always 3 0 nddis each. Note. T h i s is because all diurnal circles o f the Sun are divided into two equal halves by the equatorial horizon. Note also that what is said o f Laiika applies to all places o n the equator. [^qSRFIT:]
trf?T^ «raf?T ^
27a. B1.3.yTb^r
B.^:W^
d. B . c R ^
29a. B . t ^ ^ ^ (B2.Qrr)^Qf«RdI3l ( B 2 . ^ ) f f c. B.Bi!^HI*J1 A.B.^cRT d. A . B . C . D . U . R V H W i ^ ^ ( A . ^ ) (B1.2.'?Rr; U . f ^ )
XIII.33
XIII. SITUATION OF THE EARTH
257
Astronomical observation 30-32 Place a p l a n k i n a r a i s e d p o s i t i o n , w i t h its surface p l a n e , as e x a m i n e d by d r o p p i n g water o n it. Set it so as t o have its surface f i o r i z o n t a l a n d level w i t h the eye, a n d its p a r a l l e l sides n o r t h - s o u t h a n d east-west. A t the s o u t h e r n edge, i n the m i d d l e , h i n g e a s i g h t i n g tube (sanku) e q u a l i n l e n g t h to the n o r t h - s o u t h l e n g t h o f the p l a n k . W i t h t h e eye at the h o l e o f the r i g i d s i g h t i n g i n s t r u m e n t , at the h i n g e , l o w e r the i n s t r u m e n t so m u c h , that t h e N o r t h - p o l e - s t a r is s i g h t e d t h r o u g h t h e h o l e o f t h e i n s t r u m e n t . W h e n l o w e r e d c o m p l e t e l y , (the observation) w i l l be t o w a r d s L a r t k a : w h e n v e r t i c a l it w i l l be towards M e r u ; a n d l o w e r e d a p p r o p r i a t e l y , it w i l l b e e q u a l to t h e (local) l a t i t u d e (as read) f r o m t h e plank. Note. F r o m the next verse we can understand that V . M . implies here that the north-south length of the plank is 120 units, so that the length of the sighting instrument also is 120units = R. So taken, the perpendicular dropped on to the plank from the end o f the instrument will be equal to R sine raised angle, and the base from the foot of the perpendicular to the hinge will be R cos. raised angle.
33. W h e n so s i g h t i n g the pole-star, the p e r p e n d i c u l a r , d r o p p e d o n to the p l a n k f r o m t h e e n d o f the sight is the R . s i n e o f the l a t i t u d e o f the place. T h e base so f o r m e d is the R cosine o f the latitude o f the place. T h e R cosine Hne, i.e. the base, c o i n c i d e s w i t h the n o r t h s o u t h - d i r e c t i o n l i n e . Note. V M uses a table o f R sines, taking R = 120 units. Hence the rule.
Fig. XIII. 1 RSin 0
Hinge
J „ ,
30-34. Quoted by Utpala on B S 2 . p . 5 9 30a. B.'^T^TtficlI^; b. B . ' i i ^ . c . D . u . ^ q s n c. A . ^ ; B.^«1T. A.'^Iff; B1.2."?T?Tf
32a. A 2 . ^ b. B . ^ I W ( T A . B . ^ ( B . ^ ) T t q ( B . ^ ) l c. A . W i l d ; B . ^ W i l d l ; C.D.^^RR!^ d. A.#aT^ngjIW^; B . ^ « l ^ ^ ^ ^ : ( B 2 . ^ 0 ;
d. Al.-SfcR; A2.'5lt^; B.3lf?m 31a. B.^^l^nf ( B 3 . o m f ) A . i g ^ ; B . ^ b. B . ^ ; A.C.D.^^nq^. A 2 . " ? f ^
33a. A . c T ^ ^ ; B . c T ^ ' ^ b.
A.B.^Bl^; B.!^|^ftc(
c.
A1.°^^;B.?I5^
c. B.5(m^<=(il d.
A.B.C.D.^RTO^
258
PANCASIDDHANTIKA
XIII.37
34. L e a r n e d m e n , o b s e r v i n g t h i n g s f o r themselves thus, d e t e r m i n e the N o r t h p o l e , the d i m e n s i o n s o f the w h o l e e a r t h , etc. as o n e w o u l d d e t e r m i n e the salty taste o f the w h o l e q u a n t i t y o f the s o l u t i o n by tasting a small q u a n t i t y o f it. Note. What is meant here is that observadon made at a small place on the earth can give us knowledge of the whole earth, by suitable reasoning.
Moon's luminosity 3 5 . T h e S u n lights u p o n e h a l f o f t h e M o o n situated b e l o w it always, (at any p o s i t i o n r o u n d the earth), a n d t h e o t h e r h a l f is d a r k b y its o w n shadow, (i.e. the M o o n o b s t r u c t i n g the s u n - l i g h t b y its o w n b o d y ) j u s t l i k e a p o t p l a c e d i n sun-light. Note. This is because the M o o n gets its light from the Sun, and is not self-luminous. 36. T h e Sun's rays, r e f l e c t e d i n t h e watery M o o n dispels the darkness o n the e a r t h , j u s t as t h e rays o f the s u n f a l l i n g o n a m i r r o r i n the i n t e r i o r o f a house, does. Note. It is the belief of the ancients that the Sun is fiery, the M o o n watery and the earth mainly earthy. 3 7 . A c c o r d i n g to the p o s i t i o n o f t h e M o o n u n d e r n e a t h the S u n , every day, the l i g h t e d u p part increases ( f r o m t h e t i m e o f n e w m o o n , as seen f r o m the earth), as the l i g h t e d p o r t i o n increases o n the pot, o n the western side, i n the afternoon. Note. Instead of the expression, after-noon a better one would be, 'as the day-time elapses, beginning from sunrise.' 34c.
B l . 3 . ^
d. A . B . D . T # T — W l P s r a t S e ^ ; ( B . T ^ ; D.T^TPT^)
XIII.41
Xlll. SITUATION OF T H E EARTH
259
38. A n y w h e r e o n the M o o n , its d e n i z e n s , (the Pitrs, in this case) see the Sun for half the time d u r i n g each fortnight, (on the whole, not seeing the Sun for a fortnight's time, and seeing it for a fortnight's time), because the visible part of the sky extends only upto 90° from the zenith.
q ^
iJt^
^^JsmogcWT:^:
The Planets and their situation 39. B e y o n d t h e m o o n are o r b i t i n g h i g h e r a n d h i g h e r . M e r c u r y , V e n u s , the S u n , M a r s , J u p i t e r a n d S a t u r n , a n d b e y o n d that t h e r e are fixed stars. A l l t h e planets ( f r o m M e r c u r y to S a t u r n ) m o v e i n t h e i r o w n i n d i v i d u a l orbits at a c o n stant speed. Note. A l l this is H i n d u theory. 4 0 . J u s t as the spokes o f the oil-press w h e e l are thick, (close to o n e a n o t h e r ) , n e a r the n a v e l , a n d t h e space b e t w e e n o n e a n o t h e r increases as t h e r i m is a p p r o a c h e d , so the l i n e a r e x t e n s i o n o f t h e rdsi increases as t h e orbits are situated h i g h e r a n d h i g h e r . 35a. Q u o t e d by Prthudaka on BrSS 21.8 end, 36 quoted by Siirydeva on
d. 37a.
B.fT^^RwW: A.C.D.°^
Abh. Gola. 5, and Makkibhatta on
b. B . U^M 7^2? A . •qM??:
Sid. Sekhara 1.1.
d. B . W l ^
35a. A2.1wm«f A . ^ : ; D . A. B . « 1 ^ « # : c. A.W^WWfHrl 36a. B.^?#leM. A.-if^"?Tf?lRl. A.f^«iqt ( A 2 . ^ ) ; B. -sra^ c. A . B . m q f %
38a. B.3TJ#R!lft?^g b. B.°#eR% c. A."^'!?!^; D.°^iqT5*r d. A . ^ R ^ : ; B . C . ^ M ( C . ° r I : ) ;
260
PANCASIDDHANTIKA
XIII.42
4 1 . S i t u a t e d near-most, t h e M o o n goes r o u n d i n t h e shortest time, its o r b i t b e i n g the shortest. B u t S a t u r n situated farther-most, i n its longest orbit, c a n n o t m o v e so fast, i.e. moves slowest. [TfRT-i^FT-g^rfflltTT:]
3 ^ s h A u i l ^ q m g w n ^ f ^ : Cm) \\^^ Lords of the Months, Days and Year 4 2 . T h e successive L o r d s o f t h e M o n t h are the successive f a r t h e r planets, b e g i n n i n g f r o m the M o o n . T h e L o r d s o f the H o r a s are t h e successive nearer a n d n e a r e r planets, b e g i n n i n g f r o m S a t u r n . T h e successive fifth i n the a s c e n d i n g o r d e r o f its distance is the successive L o r d o f the D a y . T h e sixth i n its a s c e n d i n g distance o r d e r is successively t h e L o r d o f the year. Note. T h e month meant here is the sdvana month of 30 days, and the year, the sdvana year of 360 days. W e get the lords of the hords; Saturn, Jupiter, Mars, Sun, Venus, Mercury, M o o n , Saturn etc. the lords o f the day, Sunday, Monday, etc., the lords o f the months. M o o n , Mercury, Venus, Sun, Mars, Jupiter, Saturn, M o o n etc., and the lords o f the year. M o o n , Jupiter, Sun, Mercury, Saturn, Mars, Venus, M o o n etc. M^R4<4lPd«*>NI*ic|
Thus ends Chapter Thirteen entitled 'Situation of the Earth: Cosmogony' in the Paiicasiddhantika composed by Varahamihira 39-41. Quoted by Utpala on BS, 2, pp. 42-43 39a. A . ^£Fsa^;%«lftl^(A2.1«f); B1.2.^-5^l^'Mfed; B3.^5F5I^p^ c. B . -SIMOT(B3.-SIM) A . •^icM^cll; B.-FR5(T
41a. B.Tlf^rotsR^ ( B 2 . 3 . ' ? M H ) b. B.iTST???: c. A.SJ'i^t^cn-Jiql d.
B.i^^ A.B.omW«IT; C . D . o m ^ ;
d. A.3l§T:^. B1.3.^RiWrf: 40a. B 1 . 3 . ^ e l ^ b. B.1%cR. B . ^ l « n c. B . i p ^ W c-d. U.^*^i H^^<^ yfeldllH
42a. A.B.^RraiM^#«l! ( B . - ^ : ) b. W?i c. A . 3 ^ ; A . f ^ ^ ; B.t^M^ d. A . B . W T M I : A . B . C . D . W :
d. B.^dR^dlR. A . ° ^ ; B . ° ^ l.col. A.5cii=W*H^:<
Chapter Fourteen GRAPHICAL METHODS AND ASTRONOMICAL INSTRUMENTS*
Introductory In chapter I, vss. 5-7, Varahamihira enumerated yantra and chedyaka among the topics to be dealt with i n the present work. T h e present chapter deals with these two topics. T h e compound word chedyakayantrdni is equivalent to the compound word yantracchedydni of I. 7. T h e word chedyaka means graphics or graphical methods and the wordyanZra, in the present context, means astronomical instruments. Dvivedi interpreted the word chedyakayantrdni as follows: "That which cuts or removes doubts is chedynka; the instruments which serve as chedyaka are chedyakayantrdni." If it were so, there would be no difference betweenyarifra and chedyakayantra. Chedyaka and Yantra, in fact, are two distinct topics of Indian astronomical works called Gola or Spherics. Lalla, Vatesvara and Bhaskara II, for example, have earmarked two separate chapters for their treatment i n their works on spherics. Pingree, on the other hand, translates chedyakayantrdni as "the Magical Diagrams of the (Graphical) Construcdons." But the diagrams or astronomical instruments discussed i n the present chapter bear no magical significance.
[^:]
GRAPHICAL METHODS Ascensional differences of the zodiacal signs. In III, 10-12, Varahamihira stated an approximate practical method for finding the ascensional differences of the signs for places living between the Indian Ocean and the Himalayas and prom* This chapter was left untranslated by T.S. Kuppanna Sastri. The translation given here was supplied by K.S. Shukla.
262
PANCASIDDHANTIKA
XIV.4
ised to give a method for other places in a subsequent chapter devoted to chedyaka. T h e following rule is in fulfilment of that promise. 1. C o n s t r u c t o n the g r o u n d a level circle w i t h d i a m e t e r e q u a l to 180 digits. O n its c i r c u m f e r e n c e p u t d o w n , at e q u a l distances, m a r k s s h o w i n g signs a n d degrees (etc.). A l s o p u t d o w n m a r k s s h o w i n g the d e c l i n a t i o n s (of the e n d points o f the signs A r i e s , T a u r u s a n d G e m i n i ) . ( T h r o u g h the centre o f the circle d r a w the n o r t h - s o u t h l i n e a n d at r i g h t angles to it d r a w three c h o r d s t h r o u g h the m a r k s s h o w i n g the d e c l i n a t i o n s for the ends-points o f the signs Aries, Taurus and Gemini). 2. F r o m the centre, d r a w three circles w i t h d i a m e t e r s e q u a l to the three c h o r d s w h i c h have been d r a w n t h r o u g h the d e c l i n a t i o n m a r k s at r i g h t angles to the n o r t h - s o u t h l i n e , a n d g r a d u a t e t h e m w i t h m a r k s (of signs a n d degrees) like the first circle. 3. T h e n ( f r o m the same centre) d r a w a l i n e towards the latitude (i.e. towards that p o i n t o f the first circle w h i c h m a r k s the latitude o f the place) a n d e x t e n d it u p t o the m a r k ( i n d i c a t i n g the latitude o f the place) o n the c i r c u m f e r e n c e o f the first circle. ( T h i s is the latitude-line). O n the c h o r d c o r r e s p o n d i n g to the d e s i r e d d e c h n a t i o n {i.e. d e c l i n a t i o n o f the e n d - p o i n t o f the d e s i r e d sign), m e a sure the p o r t i o n l y i n g between the latitude-line a n d the n o r t h l i n e (i.e. the l i n e d r a w n f r o m the centre to the n o r t h p o i n t ) . 4. L a y o f f the d o u b l e o f that (like a c h o r d ) o n the c o r r e s p o n d i n g circle. T e n m u l t i p l i e d by o n e - h a l f o f the degrees i n the arc s u b t e n d e d by that c h o r d are to be k n o w n as the vinddis o f ascensional d i f f e r e n c e i n the case o f the first s i g n . I n the case o f the o t h e r two signs, they are the vinddis o f the m i x e d ascensional difference (i.e., the m i x e d ascensional difference o f A r i e s a n d T a u r u s a n d the m i x e d ascensional difference o f A r i e s , T a u r u s a n d G e m i n i ) . Consider F"ig. 1. E N W S is the level circle of diameter 180 digits drawn on the ground, E, W , N and S being the east, west, north and south cardinal points, respectively. E W is the east-west line and N S the north-south line. T h e arcs E d , , Edgand Ed^ are equal to the declinations ff^, J ^ a n d c/!, of the end-points, of the signs Aries, Taurus and Gemini respectively. W d ' = E d , . W d " = E d and Wd" = E d , , d.d', d.^d" and d.,d"' are the three chords corresponding to the declinations (f^, j - ^ and cT,,, respecdvely. These are at right angles to the north-south line. Ic. A . T R ^ a 5 ^ C . ¥ 9 q f c . d. A . ^ T I ^ ; B.^mWT: 2b.
B.^for^B2.3.?FW
c. AB.°<=(<;=t)i(^; ( B . g a p f o r ^ ) d. A 2 . ^ ; B . ^ T l ^ 3b.
A . om'Jf A . ^ i q i ^ B . f^rf^JOTOiq^tn^ (B1.2. °^vqf#;o)
B.3T^
d. A . B . °Vl1-^*lKW ( B 2 . ? t e ) 4a.
B.f|[55B1.3.W^; B 2 . ' 5 r R ! r f B . ^
b. A . B . o m f ^ B.H-^imi^l+KdW (B1.«!) W : c. A . ^ ^ ; B . f ^ d. B2.%?T: B3.ftW
XIV.4
XIV. ASTRONOMICAL INSTRUMENTS
263
E
W Fig. XIV. 1
T h e arc N L is equal to the latitude 0 of the place, L being the point marking the ladtude of the place. O L is the latitude4ine drawn from 0 to meet the point L . O N is the north line. enws is the circle drawn with centre 0 and diameter d, d'. (The other two circles drawn with diameters d.^ d" and d,, d'" are not shown i n the figure), dl is the portion of the chord d, d ' lying between O L and O N . In the triangle O d l , Od
= Rsin (/•,
dl = O d . t a n 0 = Rsin cfi.tan 0 , R denoting the radius o f the circle E N W S .
(1)
XIV.6
PANCASIDDHANTIKA
264
T h e chord D M D ' of the circle enws, with its middle point at M , is equal to 2dl. Let 2c, be the angle subtended by D D ' at 0 (or, wdhat is the same thing, the arc D W D ' subtended by D D ' ) . 'I hen 1/2 D D ' = D M = O D sin c, = R c o s ^ , . sin c,. 1/2 D D ' = dl = Rsin,J',. tan 0 , from, .-.Rcos J*, sin c, = Rsin .-.sine, = tan j - , . tan 0
(1).
ei-tan0. (2).
This c, is the ascensional difference of the end point of the sign Aries, or the ascensional difference for the sign Aries. See supra, I V . 26. See also I V . 34. Since c, is equal to the number of degrees lying i n half rhe arc D w D ' , therefore the degrees in half the arc D w D ' give the ascensional difference of the sign Aries. These degrees muldplied by 10 give the corresponding vinddis. Similarly, in the case of Taurus and Gemini. But in these cases if c, and c,, be the ascensional differences for the end-points of Taurus and Gemini respectively, then ascensional difference for Taurus = c, - c, and ascensional difference for Gemini = c,, - c,^.
Rsine of the Sun's zenith distance for the given time, (and vice versa). 5. T h e nadis (elapsed since stinrise i n the f o r e n o o n o r to elapse before sunset i n the afternoon) m u l t i p l i e d by 6 are degrees; the (versed) R s i n e o f that subtracted f r o m the r a d i u s (R) a n d t h e n i n c r e a s e d by the R s i n e o f the Sun's z e n i t h distance for m i d d a y gives the R s i n e o f the Sun's z e n i t h distance (for that d m e ) . I n o r d e r to find the nadis ( f r o m the g i v e n R s i n e o f the S u n ' s z e n i t h distance) the (given) R s i n e o f the Sun's z e n i t h distance s h o u l d be d i m i n i s h e d by the R s i n e o f the Sun's z e n i t h distance for m i d d a y . 6. W h a t e v e r is the s i x t h p a r t o f the degrees o f the arc c o r r e s p o n d i n g to the (versed) R s i n e e q u a l to the d i f f e r e n c e b e t w e e n the g i v e n R s i n e o f the Sun's z e n i t h distance (as d i m i n i s h e d by the R s i n e o f the Sun's z e n i t h distance f o r m i d d a y ) a n d the r a d i u s {chdydharijdbhyantara), gives the nddis e l a p s e d since sunrise i n the f o r e n o o n o r to elapse before sunset i n the a f t e r n o o n . Let n be the nddis elapsed since sunrise i n the forenoon or to elapse before sunset i n the afternoon, and the Sun's zenith distance at midday. T h e n , according to the rule stated in verse 5 above, the Rsine of the Sun's zenith distance for that time (which we shall denote by Rsinz) is given by Rsinz = R - Rvers (6n) + Rsin z„, where R stands for the radius.
(3)
XIV.6
265
XIV. ASTRONOMICAL INSTRUMENTS
This relation is approximately correct for a place on the equator, and is accurate at an equinox. For any other place it is incorrect, for Rsin z really depends upon 0 , the latitude of the place, and the Sun's declination on the day in question, besides the time of the day elapsed since sunrise i n the forenoon or to elapse before sunset in the afternoon. A n y such formula must therefore involve these elements. Formula (3) stated above does not occur in any other work on Indian astronomy. B u t analogous formulae do occur. O f these mention may be made of the following. (i) Vasistha'sformula. Let denote the gnomonic shadow at midday and .s the gnomonic shadow at any other time. 1 hen, in the forenoon: (iag7i(i — Sun) in mins. and in the afternoon s =
i3 r— - 12+,s„ see above 11.12-13) 10800 - (lagna - Sun) in mms. " '
(ii) I'aii/isa's/ormii/a s = 6 X day-length _ day elapsed
+ s,, gnomon = V2. Sec IV. 49. "
This formula was later restated by Mahavira in the form: ,s =
i gnomon— _ ] 2 + « (gnomon = Vl.See Ganita-sara-safip-rahaAX. 18). 2 day elapsed / day-length • > /
(iii) Sridhara'sformuhi '/2gnomon day elapsed or to elapse —^ =gnomon, where d = d day-length See Trisatikd (ed. Sudhakara Dvivedi), Rule 65. V=
This may be derived from Paulisa formula by assuming .v,^ = 0. 5^ridhara's formula was restated by Narayana Paiidita in the form: 5 = { -= day-length - 1} X gnomon day elapsed or to elpase ° See Gariita-kaumudi, Rule 13, p. 207. 5a. A.q^ilT^imi; B . ^ (B2.af) ^?f«WFTT (B2.°WTT)
6a. V>2.W^ ab.
d. A . B . ^ a i # ^ ° A . ^ « n C . ° ^
B . ^ ^ T T T W ^
b. A . ^
b. A . ^ ; B I . 2 . ^ ; B3.TO
c. A.^K^faRt B . ^ 0 ^
C. A.'SIP^:; B . 3 i r ^
A.-Jiraf;
B.WT; C .
[^]
PANCASlDDHANilKA
266
Formula (3) may also be stated in the form: Rvers (6ra) = R - (Rsin z - Rsin 2^), so that 71 = one-sixth of the degrees in the arc corresponding to the versed Rsine equal to R - (Rsin 2 Rsin z ). Hence the rule stated in vs. 6.
V9
Right ascensions of the signs (defined by means of the armillary sphere) 7. T h e degrees i n the arcs o f the e q u a t o r vv^hich lies o r t h o g o n a l l y {tiryak) betw e e n the n o r t h - s o u t h d e c l i n a t i o n arcs for the ends o f the signs, m u l t i p J i e d by 10, are the vinddis o f the r i g h t ascensions o f the signs ( A r i e s , T a u r u s a n d G e m i n i ) i n t h e i r respective o r d e r .
Fig. X1V.2
In Fig. 2,S'R is the equator a n d ^ C the ecliptic. A P , B Q a n d C R are the declinations for the endpoints of the signs Aries, Taurus and Gemini, respectively. T h e n ' T P is the right ascension of Aries, P Q i s the right ascension of Taurus, and Q R is the right ascension of G e m i n i .
7a. C.\9raiT° b. A . °<|c|*HI!i'l; B . °
c. A . ^ : ; B . ^ — T R ^ d. A.f^^nfe^^" A . 3 ^ : , B . a w n
i. Chdydharijdbhyantara literally means the distance between the chdyd (i.e. Rsine of the .Sun's zenith distance) and zon. This is equal to the difference between the Rsine of the Sun's zenith distance and the radius, for the former being diminished by the Rsine of the Sun's zenith distance for midday as instructed in verse 5.
XIV. ASTRONOMICAL INSTRUMENTS
X I V . 10
267
T h e degrees of the equator multiplied by 10 are obviously vinddis. T h e vinddis o f the right ascensions of the signs could also have been defined with the help of Fig. 1. Let the arc W P (in Fig. 1) be equal to 30° (i.e., the tropical longitude of the end-point of the final sign Aries), O Q , the perpendicular dropped from P on O W intersecting the circle drawn with radious Rcos ^ a t R, and O R T , the line drawn from O through R. T h e n the number of degrees in the arc W T , multiplied by 10, are the vinddis of the right ascension of the first sign Aries. T h e vinddi-s of the right ascensions of the second and third signs may also be defined similarly.
ASTRONOMICAL INSTRUMENTS Gnomon The Sun's declination defined by means of the moving gnomon and the armillary sphere. 8. W h a t e v e r be the p o s i t i o n o f the ( m o v i n g ) g n o m o n i n the p a t h d e s c r i b e d by it (lit. i n the shadow), w h e t h e r at m i d d a y , o r w h e n towards the east o r elsewhere, the a n g u l a r distance b e t w e e n the e q u a t o r (visuvat) a n d the t h r e a d that proceeds f r o m the c e n t r e a n d passes t h r o u g h the vertex o f the g n o m o n is c a l l e d the (Sun's) d e c l i n a t i o n .
8a. A.Tnr»mtlsncT«[r; B . ^ « 2 r t WIT; b.
A . B . ^ ™ P R ( B . ^ ) tit;
c. A.'Mld^^^l; B . ' M l d ^ l : C . [ W T R T ^ ] d. A . B . t ^ ^ R R ^ W ; C.ft^Ptrwt?raSI3feTT:
C . [ i T « ^ 1 g ^ ] cT«II C . ^ S ^
A.B.1^cIcI:-?TB.?T^; D . ? I f ^
A . ^ R t ^ ^ : ; B . ^ : f e T T : ( B2. °anf^i^:)
X I V . 11
PANCASIDDHANTIKA
268
Ijocal latitude (jrom the equinoctial midday shadow and vice versa) 9-10 (a-b). L a y o f f the (equinoctial m i d d a y ) s h a d o w towards the n o r t h , a n d let a g n o m o n be caused to fall to the west (or east) f r o m the tip o f the shadow. T h e n stretch a t h r e a d f r o m the centre a l o n g the h y p o t e n u s e (of the g n o m o n triangle) u p to the c i r c u m f e r e n c e (of the circle d r a w n o n the g r o t u i d ) . T h e (angular) distance between the p o i n t thus r e a c h e d a n d the e q u i n o c t i a l p o i n t (i.e. the west o r east p o i n t , w h i c h e v e r is nearer) is the latitude o f the place. S i m i l a r l y , f r o m the latitude o n e may f i n d the ( e q u i n o c t i a l m i d d a y ) shadow. w ^ In Fig. 3, let E N W S be the circle drawn an level ground, E, N , W, and S being the east, north, west and south cardinal points. O A is the equinoctial midday shadow of the gnomon, A B the gnomon which is caused to fall to the west fiom the tip of A of the equinoctial midday shadow. O B P is the thread stretched along the hypotenuse O B up the point P on the circumference of the circle. T h e n , the angle P O N or the arc P N is equal to the alutude of the equinoctial midday Sun, i.e., the colatitude of the place, a n d the complementary angle P O W or arc P W is equal to the zenith distance of the equinoctial midday Sun, i.f., the latitude of the place. Fig. XIV.3
Sun's longitude 10 (c-d)- I I . O n the d e s i r e d day find the S u n ' s d e c l i n a t i o n , n o m a t t e r w h e t h e r it is g r e a t e r o r less t h a n the latitude. F i n d the R s i n e o f that a n d insert it betw e e n the e c l i p t i c a n d the e q u a t o r (at r i g h t angles to the latter, i n its o w n q u a d r a n t ) . T h e arc o f the e c l i p t i c m e a s u r e d eastwards f r o m the first p o i n t o f A r i e s u p to the p o i n t w h e r e the R s i n e touches the e c l i p t i c , s h o u l d be k n o w n as (the l o n g i t u d e of) the S u n .
9a. A . ^ ^ ^ ; B . f t 2 « O T l ( B 2 . l 5 R ) B . ^ f o r ^ B.^rai and hapl. om of one b. A . ^ : ; B . W : d. A . ^ ^ B 2 . ' ^ ' ? t ; B1.3.T3jT'^ 10a. A.?lti?R; B.^TflScit b. A . ^ i ^ . B . ^
10c. A.5«Z[T^; B.^S«2FR^°; D.5'^:^Pn d. Al.°TT?Rf%?* 1 l a . Al.cfHiT; A2.cFqT a-b. B.Haplom.3t-:gif^ c. B1.2.^gmT d. B.°'HH
B.Wt^
XIV.14
x i v . .Asi R ( ) N ( ) M K : A I . I N S ' I R L ' M E N I S
269
V-Shaped Ya§ti True tithi 12. O n e - t w e l f t h of the degrees i n t e r v e n i n g between the S u n a n d M o o n o b s e r v e d by m e a n s of a ( V - s h a p e d ) Y a s t i o f l e n g t h e q u a l to the s e m i - d i a m e t e r o f the level circle (with o n e a r m p o i n t e d towards the S u n a n d the o t h e r towards the M o o n ) is to be k n o w n as the t r u e tithi w h i c h b e i n g d e s t r o y e d is to be k n o w n . F r o m that (true tithi) o n e m a y d e r i v e a n o t h e r . Similar rules are found to ocxur in Lalla's Gola (VIII 42-43) and Sripati's Siddhdnta-sehhara ( X I X 26).
Moons longitude 13. W h e n the Sun's l o n g i t u d e , obtained graphically, (vide above, vv, 10 c-d-11) is a d d e d to those degrees ( i n t e r v e n i n g between the M o o n a n d the S u n ) , the result is the M o o n ' s l o n g i t u d e for that time. T h i s is h o w the M o o n ' s l o n g i t u d e is o b t a i n e d g r a p h i c a l l y .
13a. A . <=%5; B l .2. read: ^ ^ ^ ^ ( B 2 . 3Tt?T%5) 12a. A . ^ ? } ; B.t?!?!; C.%=51^
B3.^^^
c. A.frrf«T^
c. . A ^ B L ^ T ^ f l c r f F l ^
d.
d. A . ° ^ f e ^ ; B . ° ^ — ^ ( B 2 . 3 . « I ) % ^
A2.^EIF^
PANCASIODHA.Nl IKA
270
(#nT)
l ^ ' ^ c b ? l i ^ i ^ w v f ^ A ^ T^m^
XIV.18 m,^^
Cardinal directions (by means of three shadows of a gnomon) 14-16. M a r k t h r e e times (at intervals) the t i p o f the s h a d o w o f a g n o m o n set u p at the c e n t r e o f the circle ( d r a w n o n level g r o u n d ) . W i t h the h e l p o f those (three points) c o n s t r u c t two fishes. T a k i n g the p o i n t o f intersection o f the two strings p a s s i n g t h r o u g h the h e a d a n d tail o f those fishes as c e n t r e a n d a t h r e a d so l o n g as to t o u c h the t h r e e p o i n t s as r a d i u s , c o n s t r u c t a circle (passing t h r o u g h the t h r e e points). T h e t i p o f the shadow o f the g n o m o n that daymoves o n that circle w i t h o u t s w e r v i n g f r o m it. 16. T h e c e n t r e o f that circle hes to the n o r t h o r s o u t h o f the g n o m o n . Betw e e n that a n d the g n o m o n is the m i d d a y s h a d o w l y i n g to the n o r t h (or south) o f the g n o m o n . T h e midday shadow falls towards the north or south according as the Sun's northern declination cTis less or greater than the ladtude 0 of the place. W h e n the Sun's declination is south, the shadow always falls towards the north.
[TsnttH:] ^ R v j i f i l f d ^MH*4cn1 ilflTl>fMc< ^ i l t ^ V ^ ^ i - ^ t ^
The Celestial Sphere 17. ( T h e circle) w h e r e the sky a p p e a r s to meet the earth at t h e i r skirts is called the h o r i z o n (harija). T h e (vertical) circle w h i c h r u n s f r o m east to west is called 14a. A.B.^^TWRHW (62.^^^)
^ 16a. A . W ; B.iTKjm; l).iT?2ITg^
b. Bi.°TTO^ ( B 2 . ^ ^ - ^ r g a t ) c.
B.q^-fq^ct^l^
d. A . B . C . D . g ? ^ L5a. A . ^ ; B . W ^ c.
Bl.fcRfor^
b. A.%5f: a BS.'Sf^. B.^'^^wmt c. A . c T ^ ; B.dsJ-^qlqcK d. B.°^Tlfel^a
XIV.20
XIV. ASTRONOMICAL INSTRUMENTS
271
the p r i m e vertical (samamandala). S i m i l a r l y , t h e (vertical) circle w h i c h r u n s n o r t h to south is c a l l e d the m e r i d i a n (daksinottara) 18. T h e (arcual) distance b e t w e e n the n o r t h p o l e a n d the h o r i z o n is c a l l e d the latitude ( o f the place). T h e d i f f e r e n c e b e t w e e n 9 0 ° a n d the latitude is c a l l e d the c o l a t i t u d e . T h e c o l a t i t u d e is the d e p r e s s i o n ( o f the n o r t h pole) f r o m the z e n i t h . T h e d a y - d i a m e t e r is the d i a m e t e r o f the so c a l l e d d i u r n a l circle (astodaydkhya).
^Tf%FpT#?Rt ^ r f ^ ^ ^ r ^
H e m i s p h e r i c a l B o w l and its use 19. C o n s t r u c t a h e m i s p h e r i c a l b o w l o f t h e r a d i u s c h o s e n f o r o u r constructions w i t h a g n o m o n fitted at its centre. G r a d u a t e its circvdar r i m w i t h m a r k s o f c a r d i n a l d i r e c t i o n s a n d c i r c u l a r divisions (signs a n d degrees etc.). Place it i n a s m o o t h ( h e m i s p h e r i c a l ) cavity i n the g r o u n d w i t h the g n o m o n i n c l i n e d to the h o r i z o n at a n a n g l e e q u a l l o the latitude o f the place ( a n d p o i n t i n g to the n o r t h pole). 20. R e a d the degrees t r a v e r s e d since sunrise by t h e s h a d o w (of t h e g n o m o n ) w h i c h passes t h r o u g h the intersection o f the two lines {viz., east-west a n d n o r t h - s o u t h j , a n d a d d t h e m to the Sun's l o n g i t u d e . T h e s u m thus o b t a i n e d is the l o n g i t u d e o f the r i s i n g p o i n t o f t h e ecliptic. T h e degrees crossed o v e r by the s h a d o w d i v i d e d by six are the nddis o f the d a y (elapsed since sunrise). d. A . ^ ! ^ | | ; B . - ^ — I I ;
17a. A.B.^fwiilRi B.'lHc)Mcn1 b. A l . - S l ^ ^ Bl.-slf^?^Rl3 ( B 3 . ° % ) c.d. A . ' ^ ^ ^ ^ ; B . T O t ^ T f f T ^ ^ ; C . 1 I ^ T #
19a. B.°^D.°<=lc;w5+Mld b.
d. B i . 3 . ° W R T ; B2.°TRCT;
D.°TR
18a. B . t ? ^ A-B.f^^^: h. B.°f«PTffrf: c. A.B.D.erat(B.^)^Tftrf!T A . B , W Z T ; D.Wm
"n?rr
B.^(B3.^)1^[B2.1]H^^^f^°
c. B . g ( B 3 . ^ ) ^ M 5 ( B 3 . ? ) f ^ ^ ( B 3 . ^ ) d. A . B . f q f f ^ : ( B 1 . 2 . ^ B 3 . ^ ) A.B.^a^Tn^(B.f) 2()a. B2.^l3«im; D.^OTrai b. A.B.^g^TJTW; C.D.^tfii^ra c. B.
272
I'ANCASIUDHAN riKA
XIV.23
Wll^l4)*4<|jisi 3 t t y : T ^ ^ MRfM[^c<>U|
Hoop and it use 2 1 . C o n s t r u c t a c i r c u l a r h o o p w i t h d i a m e t e r e q u a l to o n e c u b i t (= 24 digits) a n d r i m h a l f a d i g i t b r o a d . G r a d u a t e its (inner) r i m e v e n l y w i t h the m a r k s o f signs a n d degrees (etc). A t o n e place i n the m i d d l e o f its b r o a d r i m , pierce a fine h o l e at r i g h t angles to the r i m . 22. W h e n it is n o o n , let a ray o f the S u n pass t h r o u g h the fine h o l e i n the r i m (and fall o n the d i a m e t r i c a l l y o p p o s i t e p o i n t o f the r i m ) . T h e degrees that lie b e t w e e n the t h r e a d h a n g i n g vertically t h r o u g h the centre a n d that (ray o f light) i n d i c a t e the Sun's m e r i d i a n z e n i t h distance. T h e word khdksa is a technical term used in Indian astronomy in the sense of "meridian zenith distance". Also see supra, I V . 21, where the same term has been used in the same sense.
Armillary Sphere 23. C o n s t r u c t a perfecdy r o u n d l i g h t ( A r m i l l a r y ) S p h e r e by means o f w o o d e n strips (or metallic spokes). O n the surface construct two circles, one r e p r e s e n t i n g the e q u a t o r (kdla-rekhd) a n d the o t h e r the ecliptic (bhoga-rekhd) b e i n g m a r k s w h e r e the S u n stops (i.e. at the two solstices).
21a. . \ . ° w n ^ ° ; B . W l i ^ ° 22a. B.T?2TI^ ( B 2 . f l ) B.^for^ b. B1.3.TPI^':pra^cTt A . ^ B . l ^ c-d. B.feS — d.
A.B.fefe
^
23a. A.TJOTH; B . ^ ( B 2 . ^ ) i T R
A.^;B.TRJ^
b. B.-?I^^(B3.^^)3raitr
b. B . ^ ^ ; D . [fspER^] c-d. A.•(45|id(rlld
B . C . D . W W ; A2.«nfT^ c. A.lsiPmi#;i1%RT; B . l D.J
XIV. ASTRONOMICAL INSTRUMENTS
XIV.26
273
24. O n e i t h e r side o f the j u n c t i o n o f Pisces a n d A r i e s , i n the n o r t h - s o u t h ( h o u r ) circles at p o i n t s d e n o t i n g the degrees o f the (Sun's) d e c l i n a t i o n , fasten, by m e a n s o f o b s e r v a t i o n , l i g h t p o i n t s w h i c h m a y i l l u m i n e the (Sun's) o b l i q u e d i u r n a l circles. 25. M o u n t the ( A r m i l l a r y ) S p h e r e i n s u c h a way that it is e l e v a t e d towards the n o r t h by a n a m o u n t e q u a l to the local l a t i t u d e . T h e n the nddts that lie ( o n the S u n ' s d i u r n a l circle) b e t w e e n the l i g h t p o i n t o n the (Sun's) o b l i q u e d i u r n a l circle a n d the h o r i z o n d e n o t e the nddis that h a v e e l a p s e d (since s u n r i s e i n the f o r e n o o n ) , e a c h nddi b e i n g m a d e u p o f six degrees.
Sun's northward and southward journeys 26. I n the cycle o f t i m e , as l o n g as the S u n is i n the six signs b e g i n n i n g w i t h C a p r i c o r n , it is U t t r a y a r i a (i.e. the p e r i o d o f the S u n ' s n o r t h w a r d j o u r n e y ) a n d t h e r e is increase o f day; i n the c o n t r a r y case (i.e., w h e n the S u n is i n the six signs b e g i n n i n g w i t h C a n c e r ) , there is decrease o f d a y . W h a t r e m a i n s to be said (here) is to be u n d e r s t o o d f r o m w h a t has a l r e a d y b e e n stated. 24a-b. B.\^n'^a<<^N^'«JI*nTOt b.
A2.^B1.3.%«PJ;C.^
unknowingly three lines :'n to"yi] ^ r t ^ , 27a.
c. A . B . D . 3 p m r °
d. A.B.-#FraT:
B.°^l?qT
26a. A.C.D.-q^aiqfil
(A2.^);
d. B.fd4'^aM°; A 2 . ^ i ^ B.^JRI^
b. A.^rmif^^o; B.-^Tf^o; C.l»HllR*° A . B . C . g ^ ; D.^[-T^]
25a-b. A.3T#8sr; B.Sl^^f'JBqt^ (B3.'iftH>l*l+-^«l<) b. B.^siRSH: c. A.'MWI*!:; B 3 . Scribe omits
c. A.^TSllrsin; B.d«lR<=ill° d. A 1 . ° M T O ° ; A2.°ftrfct-'^''; B . o m i P ^
274
PANCASIDDHANTIKA
XIV.30
[cMH«4W?>l("l]
Instruments for measuring time 27. T h e bijas (i.e. seeds o r basic necessities) o f all i n s t r u m e n t s (for m e a s u r i n g time) are f u r n i s h e d by s t r i n g , water a n d s a n d ' . B y m e a n s o f t h e m o n e m a y c o n s t r u c t i n s t r u m e n t s r e s e m b l i n g a tortoise, a m a n , o r a n y o t h e r d e s i r e d shape, a n d m o u n t t h e m o n a w o o d e n b o a r d . 28. T h e teacher s h o u l d i m p a r t (this knowledge) only to a devoted (lit. steadfast) p u p i l , a n d the p u p i l too, after h a v i n g learnt it, s h o u l d a p p l y the bfjas (string, water a n d sand) to his i n s t r u m e n t s , k e e p i n g the secret u n k n o w n to his son even. Varahamihira has not described these instruments here only to keep their mystery a secret. But these instruments have been described in some of the later works. For example, see the chapter on astronomical instruments in the works o f Lalla and Sripati. B u t there too the description is not very explicit.
^9,
Local longitude in terms of time 29. B y m e a n s o f o b s e r v a t i o n at the d e s i r e d place i n a g i v e n l a t i t u d e , p e r f o r m the Moon spurnimdntakarma (i.e. f i n d o u t h o w m u c h t i m e after o r before local sunset the M o o n rises at the local place). T h e degrees o f the ecliptic w h i c h rise d u r i n g that t i m e s h o u l d be s u b t r a c t e d f r o m o r a d d e d to the M o o n ' s l o n g i t u d e a c c o r d i n g as the M o o n rises later o r e a r l i e r t h a n sunset. 28a-b. A.^MMW; B.'Jite^Md ( B 2 . ° ^ ) — 27a. b. A.^^W^B.^I-^idlP c. B 1 . 3 . ^ ; B 2 . f ^ A2.=W1*
^
Al.°^zi5lR A . ^ B.^NI^-^II
I. When ii heavier substance than sand was required, Mercury was used.
X1V.32
XIV. ASTRONOMICAL INSTRUMENTS
275
30. D i v i d i n g as i n the case o f tithi, f i n d the local t i m e i n terms o f ghatis for the e n d o f full m o o n tithi (as r e c k o n e d f r o m local sunset). Increase o r d i m i n i s h the ghatis o b t a i n e d by the t i m e c o r r e s p o n d i n g to the Sun's ascensional difference, a c c o r d i n g as the S u n is the six signs b e g i n n i n g w i t h A r i e s o r i n the six signs b e g i n n i n g w i t h L i b r a . T h e result is the local t i m e f o r the e n d o f the full m o o n tithi.at the local e q u a t o r i a l place. ( T h e difference o f this a n d the local t i m e for the e n d o f the full m o o n tithi at Lartka) is the t r u e l o n g i t u d e for the local place. The rule is obvious, because: (a) Local time for the end of the full moon tithi at the local place (as measured since sunset at the local place) ± time corresponding to the Sun's ascensional difference = local dme for the end of the full moon tithi at the local place (as measured since sunset at the local equatorial place), -I— sign are to be taken according as the Sun is in the six signs beginning with Aries or in the six signs beginning with Libra. (b) Also, local dme for the end of the full moon tithi at the local equatorial place ~ local dme foi the end of the full moon tithi at Lanka = local longitude in terms of time. The local time for the end of the full moon tithi at the local place has to be obtained by the process of successive approximations, but the whole process has not been described in the text.
(^)
fmq}
f n s n « r f 3 F R rrm ^
^
%5
8. The N a 4 i or Ghafi 3 1 . O n e - s i x t i e t h o f the t i m e t a k e n b y water to flow o u t t h r o u g h a d e s i r e d h o l e d u r i n g a n y c h t h e m e r o n is d e f i n e d as the d u r a t i o n o f a nddi. O r , it is the t i m e o f 180 breaths o f a m a n . 29a-b.
B.^^RfSfW^RTT^^Ccrafeqf;
c. B . ^ c-d. A . B . M l f i + V ^ i ^ ^ ^ M ^ ; (B.°^g??IF?^); C. '•-lli*W
d. A . C . D . ^ ^ I ^ ; B . f ^ ^ ( B 2 . t % ' 5 c r ) ^ ^ 30a. B.tM«Tsrf«mM D.1§rfK^ b. A.-^<^MiR;-iiPqci°; B.-c|ohM
276
XIV.35
PANCASIDDHANTIKA
32. C o n s t r u c t a c o p p e r vessel r e s e m b l i n g o n e - h a l f o f a s p h e r i c a l pot a n d p i e r c e a h o l e at its b o t t o m . P u t it i n p u r e water i n a b a s i n . T h e t i m e i n w h i c h the vessel is filled u p is t h e d u r a t i o n o f a nddi. T h e h o l e at the b o t t o m o f the vessel s h o u l d be so s m a l l that o n a c c o u n t o f its s m a l l size, the vessel m a y s i n k i n t o w a t e r exactly sixty times d u r i n g n y c h t h e m e r o n . O r , it is the t i m e i n w h i c h o n e m a y recite 6 0 times a verse c o m p o s e d o f 6 0 l o n g syllables (as vs. 32 is).
Conjunction of the Moon with a Star 3 3 . H a v i n g ascertained the M o o n ' s celestial latitude a n d observed the distance o f the M o o n f r o m a star a n d h a v i n g m a d e the requisite c a l c u l a t i o n s , o n e s h o u l d p r e d i c t the t i m e o f c o n j u n c t i o n o f (the M o o n w i t h ) the star w h i c h is to take place i n the f u t u r e .
3 T ^ ^ I g ^
31a.
B . # f ? I . A.t^l^ng^; B.fsiPraRT
b. A.B.dl'!^ RaR*j(U! c. B1.3.Wrat; B2.Wrat; D . ^ i a ^ d. A.B.'WWIiilild^H. B . l ^ : 32a. B1.3.fwnfl3nt; B 2 . f » 1 T ^ B . ^ B1.3.M b. A . ^ ; B . " ? 5 ! ^ — ^ B.?ft^
II ^ ^ II
33. Quoted by Utpala o n
24.5-6.
a. U . f ) ^ B . After^;*^ occurs the passage f?f% (of X V . 8) to^: I I ( o f X V . 23),
being transferred here, apparently due to the misplacement of a folio i n the common archetype of the B mss., B 1,2 and 3.
B.'J^'llil c. D.»i
A.B.^sifS^: A l . W C T ^ ; A2.'J^?«^
B.Tif^lf^^ b. B.^=WRT^nf^ c. A . W « ^ ^ I ^ : ; B . W « ^ — ^ r s i : ;
XIV.38
277
XIV. ASTRONOMICAL INSTRUMENTS
Positions of Certain Junction-Stars 34. ( T h e j u n c t i o n - s t a r of) K r t t i k a is at the e n d o f the sixth d e g r e e ( f r o m the i n i t i a l p o i n t o f that naksatra) a n d 3 Vs cubits to the n o r t h o f the ecliptic. ( T h e j u n c t i o n - s t a r of) R o h i n i is at the e n d o f the e i g h t h d e g r e e (of that naksatra) a n d 5 V2 (cubits) to the s o u t h . 35. T h e s o u t h e r n a n d n o r t h e r n (junction-) stars o f P u n a r v a s u are at the e n d of the eighth degree (of the naksatra) a n d 8 (cubits) south a n d n o r t h (respectively). T h e (juncdon-)star o f P u s y a is at the e n d o f the f o u r t h d e g r e e (of that naksatra) a n d 3 V2 cubits to the n o r t h . 36. T h e s o u t h e r n (junction-) star o f A s l e s a is at the e n d o f the first d e g r e e (of that naksatra) a n d o n e c u b i t (to the south); so also is the n o r t h e r n ( j u n c t i o n star). T h e c o n j u n c t i o n (of the M o o n ) (with the j u n c t i o n - s t a r of) M a g h a occurs i n its o w n field at the e n d o f the s i x t h d e g r e e (of that naksatra). 37. ( T h e junction-star of) C i t r a is at the e n d o f 7 V2 degrees (of that naksatra) a n d 3 cubits to the s o u t h . T h e digits are c o u n t e d f r o m the centre o f the M o o n w h e r e the m i n u t e s o f the l a t i t u d e e n d .
f^^[
R ^ ^ > l ( d c | > U | II -^6 I
Digits between the Moon and a Star in Conjunction And Time of Conjunction 38. H a v i n g subtracted 17 f r o m the l a t i t u d e (of the star w i t h respect to the M o o n ) m u l t i p l y by 15 a n d take a t h i r t y - f o u r t h o f that; this is to be k n o w n as the n u m b e r o f digits (between the M o o n a n d the star). 34a.
B . ^ W ; D . ^ : A.TO?TI%; B . T O ^ ; B2.fm
b. A . W T I ^ ; B . W T t ^ : c. A . ^ d. A . B . ^ W S i a A . ^ s f ; B . W ^ ;
35a. B . ^ ^ S S ^ b. A . B . D . ^ # F ^ ? f ^ c. A 2 . eft f o r ^ B . f ^ ^ d. A . ° ^ P T l ^ A . ^ s ^ 36a. A . W ^ M ^ ^ t r R r a R I
d.
A . B . ^ A . ^ j B . ^
37a. A.^r^qf^PTOFt b. B . ^ f ? ^ : c. A.+dWK'J; B.<=bdldl^'J. d. A . ^ T ^ J W ; B . i ^ s n g i : ? ! ^
278
PANCASIDDHANTIKA
XIV.41
T h e t i m e (of c o n j u n c t i o n ) is to be k n o w n f r o m the distance between the star a n d the M o o n a n d the daily m o t i o n o f the M o o n . According to Varahamihira, the measure of the Moon's diameter is 34 i n terms of minutes and 15 i n terms of digits. Hence the above rule. Polar longitudes and latitudes of the Junction-stars Junction-star of Krtdka Rohini Punarvasu (southern) Punarvasu (northern) Pusya Aslesa (southern) Aslesa (northern) Magha Citra
Polar longitude 32°40' 48° 88° 88° 97° 20' 107°40' 107°40' 126° 180°50'
Polar latitude 3.5 cubits or 3° 10' 2 4 ' N 5.5 cubits or 4° 59' 12"S 8 cubits or 7° 15' 12"S 8 cubits or 7° 15° 12" N 3.5cubitsor3°10'24"N 1 cubit or 54' 24" S 1 cubit or 54'24" N 0 3 cubits or 2° 43' 12"S
^|U||vilc(Pdlii^[^V^n[dHehl *iP{<4|d^: \\>io
^
Heliacal Rising of Canopus 39. M u l t i p l y t h e square o f 5(i.e. 25) by h a l f the e q u i n o c t i a l m i d d a y shadow; (treating it as the R s i n e o f a n arc) find t h e c o r r e s p o n d i n g arc (in terms o f degrees) a n d a d d 15 (degrees) to that. M u l t i p l y that b y 10 a n d a d d 21 times the e q u i n o c t i a l m i d d a y s h a d o w . T h e s e are vinddis.
39-40 Quoted by Utpala on J55 12.21 38a. B . ° W R k
39a. A . f ^ ^ l ^ B . f l ^ ^
b. A.^HTfTTfcn
b. A.B.Q-d*clWc*dlWc1°; D . 4 ^ * R l W ^ d l cR:
c. A.oqM
c. A.^I-MH^k1<»°; B.t$MMk14> A l
A.B.^>¥c(>l ( B . ° f ^ ) ;
d. A . B . ^ I ^
XIV.41
XIV. ASTRONOMICAL INSTRUMENTS
279
4 0 - 4 1 . A s s u m i n g these vinddis as the t i m e e l a p s e d since sunrise a n d t a k i n g the S u n at the first p o i n t o f C a n c e r , calculate the l o n g i t u d e o f the r i s i n g p o i n t o f the ecliptic. W h e n (the l o n g i t u d e of) the S u n h a p p e n s to be e q u a l to that, t h e n , by v i r t u e o f the g r a p h i c a l m e t h o d s a n d i n s t r u m e n t s available to the science o f m a t h e m a t i c a l a s t r o n o m y , the sage A g a s t y a {i.e. the star C a n o p u s ) that looks like the special r e d t i l a k a - m a r k o n the f o r e h e a d o f the l a d y - l i k e s o u t h e r n d i r e c t i o n shines f o r t h a n d delights the m i n d s o f m e n . S u c h is the d i v i n e k n o w l e d g e based o n t i m e .
Consider Fig. 4. It represents the celesdal sphere for a place in ladtude 0. S E N is the horizon and Z the z e n i t h ; V R E T is the equator and P and Q are its north and south poles;VGSD is the eclipdc. 40a. Al.B.*\iim'A
(33.^^)
D.•Ml^lldl <^Pldl
41a-b.
B . ( B 2 . ^ )
b. B l . c ! ^ ; B 2 . T 1 ^ ; B 3 . c l ^ B.-H^WWll d. A . ^ ; B . ^ B . W d r + l
b. B . W W I | B. W
c. A.B.'!4I«JIWI ( B . ^ )
c. B.^sl^^Mfd
A.gf^IWT:
280
PANCASIDDHANTIKA
XIV.41
A is Canopus at the time of its heliacal rising and S the Sun at that time. P G A Q i s the hour circle of Canopus, and G the point where it intersects the ecliptic. Assuming that the celestial longitude of Canopus is 90° and the celestial latitude 75° 20' S', we have Y G = 90°, A G = 75° 20' and A A ' = 75° 20' - 24° = 51° 20'. Therefore Rsin A ' E = Rsin (asc. diff. of Canopus A ) = Rtan eT^^^ ®' ^i^e formula (2), where J"is the declination A A ' of Canopus and 0 the ladtude of the place, _ sin ^ . sin 0 ^
R
cos 0
cos tf
_ s i n 5 1 ° 2 0 ' palahha ^
120
12
cos51°20'
because, according to Varahamihira, R = 120' „_ ^^palabhd = 25 X'^ , approx. 2 .-. A ' E = arc (in terms of degrees) corresponding to Rsine equal to 25 x
palahha ', approx. 2
E G " = asc. diff. of G , approx. = 21 X palahha, vinddis, approx. because for unit pdlahhd, the ascensional difference for G (the first point of Cancer) is 21 vinddis. Also, assuming 15 to be the time-degrees for the visibility of Canopus, G'S = G"S' approx. = 15 degrees, approx. .-. A ' S ' = A ' E + E G " -f G"S' degrees = [10 ( A ' E + 15) -t- 2\palnhhd] vinddis, where A ' E -I- 15 is in degrees and palahhd in digits. Degrees muldplied by 10 are vinddis. Now G S is the arc of the ecliptic which rises above the horizon (of Laiika) in the time given by the arc A ' S ' of the equator. Hence it is obvious that Canopus A will rise heliacally when the Sun is at S, i.e., when Sun's longitude = longitude of G -I- arc G S = longitude of G {i.e., 90°) 4- arc of the eclipdc which rises (at Lahka) in the time given by the arc A ' S ' of the equator. 1. Actually, the longitude of Canopus is 85°4' and the latitude of Canopus is 75°50'S.
XIV. ASTRONOMICAL INSTRUMENTS
XIV.41
281
Hence the rule. Obviously, the rule is very crude. It was discarded by the later astronomers who replaced it by better rules.
Thus ends Chapter Fourteen entitled 'Graphical Methods and Astronomical Instruments' in the Pancasiddhantika composed by Varahamihira
1. A .
^g[^#5gTFT: 1
Chapter Fifteen SECRETS OF ASTRONOMY
TI^(TIT) ^
W T f ^ ^ t ^ ^ d ^ ^ II ^
Eclipses 1. I declare the f o l l o w i n g to those w h o possess pre-knowledge o f the relative positions o f the S u n , the M o o n , the zodiac with the p r i n c i p a l stars, a n d the e a r t h . T h e r e is always an eclipse o f the S u n visible s o m e w h e r e i n space, a c c o r d i n g to the p o s i t i o n o f the place. This is what is meant: If a solar eclipse is defined as being caused by the Moon hiding the Sun, it is obvious that at any moment some place in space will have the Siui hidden behind the Moon. 2. F o r those w h o d o not k n o w the relative positions o f the above m e n t i o n e d , e v e n k n o w l e d g e w i l l b e c o m e s i m i l a r to the m i l k i n the c o n t a i n e r with c o n c h i n it b e c o m i n g capable o f d e s t r o y i n g the teeth. This is the meaning:- Milk and (kjnch are by themselves beneficial things, espedall) conch which can strengthen the teeth on riccount of its calcium content. But milk with conch soaked in it becomes positively harmful to the teeth on account of their incompatibility. So also, even knowledge can become harmful unless the knowledge as to when, where and how to use it is also known.
T S have amended the correct diuinydm into dhdriyam, not realising that the word intended in the context is ddhdni, meaning the vessel for holding the milk, and also not uncletstanding what is meant by the verse. For similar reasons, N P have emended the word as dhydndi, and gives a meaning opposite to what is intended by V M .
la.
A.WITB.*^'
c. A . W T T
2a. B1.3.31T3lfgfe^ A.^-^HT^T; B . W R H i b. B.'^^Itfi; B3.cira%
d. B.fsr^lTO A.'^^lf^; B.^^fep^ ( B 3 . ^ )
c. A . - ^ ; B1.3.^?TTO; B 2 . ^ t ^
XV.4
XV. SECRETS OF ASTRONOMY
283
3. F o r these to w h o m the S u n is h i d d e n by the M o o n , a c c o r d i n g to the straight Une f r o m t h e m t h r o u g h the M o o n t o u c h i n g the S u n , f o r t h e m there is a solar eclipse. A n d every m o m e n t (ht. every day) there is s u c h a place somewhere. 4. T h e pitrs (Manes) o n the M o o n see the S u n ecHpsed for o n e w h o l e f o r t n i g h t , a n d not e c l i p s e d d u r i n g the o t h e r f o r t n i g h t . T h e m i d - e c l i p s e is at full m o o n .
Fig. XV. 1. (diagrammatic, not to scale) P: The
PitrS
On
the
moon M , : Position of moon at middle of dark fortnight. M,^: New moon position M ^ : Position at middle light fortnight M ^ : Posidon at full moon O n the side facing the sun, the M o o n has sunlight. O n the other side it is darkness.
Fig. XV. 1
3a. A 2 . ^ ^ A.B.^q^Ior?Tf?R ( B . ^ ) ; 4a. B.^^f^TW b. A . B . f ^ ; C.fSR^; D.cfl4^ c. B . H a p l . om of M
b. B3.ft?Kr:
d.
d. B.WTTSZTFA.TR^
A.^^^:
c. A . 3 P T F I B . ° ^ ^
PANCASIDDHANTIKA
284
XV.7
At M , position P (Pitrs) just begin to get sunlight. It is sunrise to them. A t it is midday, with Sun abovehead. A t daylight ends and darkness begins. A t M ^ , it is midnight for the Pitrs. O n the earth, M.^ is new moon, mid-light fortnight, is full moon, and M,mid-dark fortnight. Thus, full moon on earth is midnight for the Pitrs, the whole night being the eclipsed time. Note that the Pitrs are supposed to dwell in the region opposite, along the line of sight of the M o o n . Note also that the synodic month forms the whole day for the Pitrs, full moon being midnight and new moon being mid-day. Note again that the explanation above can hold good only if the M o o n shows the same face to the Sun, as it really does. T h e H i n d u astronomers held the same view, without any conception of it, for they held this view in the case of every planet. T S are confused, not keeping in m i n d clearly that the month for us is the day for the Pitrs. Here, graha means eclipse, and really there is no difficulty, which they seem to feel.
5. P e o p l e at the N o r t h - p o l a r r e g i o n n e v e r see a solar eclipse o c c u r r i n g , because o f the S u n a n d the M o o n n e v e r b e i n g h i g h e n o u g h i n the sky. 6. T h e S u n a n d the M o o n c a n n e v e r be i n a straight line w i t h the eye for people i n the N o r t h - p o l a r r e g i o n . B e i n g o n the side (as viewed f r o m the region o f l o w latitudes) they always view a gap b e t w e e n the S u n a n d the M o o n . V . M . is wrong here. Every general partial eclipse, and some total or annular eclipses also, are visible even at the pole. T h o u g h the Moon's parallax will be near m a x i m u m , with a high enough celestial latitude, the Moon can be projected on the Sun to form an eclipse. If V M had worked out an example, he would have discovered his mistake. H e seems to have been misled by the fact that eclipses visible in low latitudes like India, would not be visible in the polar region. T S too say V M is wrong. Prosody requires some additional mdtrds in (6), and I have added nam and bliavati in the second foot. TS's emendations which are adopted also by N P , do too much violence to the text.
5b.
B.y(siit|<+)SI
d. B.aga^nsf^Fli^: 6a. C . f g ^ ; D.^i|%«n b. A.B.C.D.%^:-»
(A.B.^tW; A.^?g) c. C.D.% d. B . ^ ^ :
XV.9
XV. SECRETS OF ASTRONOMY
285
7. T h o u g h the S u n is l o w near the h o r i z o n , n e a r sunrise o r sunset, the M o o n , b e i n g h i g h e r u p , can h i d e the S u n like a c l o u d . V M here answers an objector to his stand (5-6). H e is unaware that the case in (7) is similar to that (5-6). TS's emendations/;r/Jrtworm, which means 'standing high up in the sky', will make an eclipse itseli' impossible. NP's emendation candroparasamavastha does not give the sense of 'being under the M o o n ' given by them for the expression.
8. F o r p e o p l e w h o have sunset a n d for p e o p l e w h o have m i d - d a y , w h e n we have sunrise, for a l l o f us, the solar eclipse does not o c c u r at the same t i m e . 9. T h r o u g h o t i t the t i m e w h e n t h e r e is eclipse f o r the m i d - d a y p e o p l e , it is past f o r the sunrise p e o p l e by f o u r nddikds, a n d w i l l be yet to o c c u r f o r the s u n set p e o p l e by f o u r nddikds. The context is the solar eclipse and the M o o n is near the Sun, So for the sunrise people the apparent longitude of the M o o n has increased by parallax, and the circumstances of the eclipse are advanced by more than four ruulis. T h e opposite happens for the sunset people, and the circumstances are delayed by four nddis. T h e duration itself is shortened by a slow rate of change of parallax for both. Therefore when the mid-day people's eclipse begins, the morning people's eclipse has ended, and when it ends, the evening people's eclipse has not begun. So there is no overlapping of them. T h e explanation given by me is general, and several other circumstances will have to be taken into account. But VM's statement is conect in a general way. Hindu astronomers give the maximum parallax converted into time as four nddis. We must contrast this with the lunar eclipse, which begins and ends at the same moments, wherever the M o o n is visible on the earth.
8a. B . H a p l . o m o f o n e ^ 7a. A . B . ^ ( B . ^ ) ^ ;
C.Tim^
b. A.B.f=rERs}) A . °Jt?jqi^75ri?T C. A . B . ^ g ^ W T ^ ; C . ^ : W f r 5 ^ ;
b. B . 1 ^ « T : d. B.clNii^fAnyiimd 9a. B . c m d g — ^ W I T b. A.B.^tM
d. A l . ' q ^ T ^ : ; A2.^?rWTr:; B.g^(B3.-En)# In B 2 , there is transposition of folios here.
c. A . ^ d. B. WRcI^f^lH; B . gap for,'=T ||3 (of next verse)
286
PANCASIDDHANTIKA
il^UlWI i l B l d l T i
X V . 16
TT|lf^f|Rrat:
II
10. T h i s m a t t e r o f eclipses has been expatiated u p o n by m e at t h e b e g i n n i n g o f the c h a p t e r o n R a h u ' s (the N o d e ' s ) m o t i o n (in the Brhalsamhitd, 5. 8-11). A l s o the causes for the eclipse o f the S u n a n d the M o o n w i t h o u t the considera t i o n o f R a h u has been d i l a t e d u p o n . V M indicates both the nodes as Rahu (the dragon of mythology), one as the 'head' and the other as the 'tail'. Here, V M takes the correct astronomical posidon in the matter of the eclipses. In this chapter as also as in his Brhajjdtaka, V M refers to without inhibition, the incorrect views of the authors of the early astronomical samhitds.
dPwvlciiiWMil 1 % T i r a ^
r^tr^ ||
d-iJWH<(^)Ru|HH<|: q?^f%qt(S^)TT(rf:)
Situation at the Poles 11. T h e r e is no d i s d n c t i o n o f direction at the N o r t h pole, because East cannot be d e t e r m i n e d there u s i n g the S u n (rising a n d setting a n d c u l m i n a t i n g ) , for, as l o n g as the S u n stays r i s e n , it goes r o u n d a n d r o u n d the sky like a beautiful damsel. T h e idea is that there is no daily rising and setting of the Sun to determine east and west. In fact, at the N o r t h pole all directions are south. 10a. A l . 3 * ^ ; A 2 . 3 * ^ ; B . —
11a.
Bi.2.fe;fWit
b. B . W I T O W T ^ H c. Al.^^^lft?!; B.^qftPra
c. B . ^ T ^
d. A . B . t ^ l ^ ( B . ^ )
d. Al.^rralM; A2.^irat^;
B.fCRfqffRta
X V . 16
XV. SECRETS OF ASTRONOMY
287
12. I f it is a r g u e d that f r o m the p o i n t w h e r e the S u n j u s t a p p e a r s above the h o r i z o n east is d e t e r m i n e d , as the S u n sets at the same p o i n t after h a l f a year, c a n this east b e c o m e west also? What V M says in these two verses is essentially true. But his statement that it sets at the same point after half a year is not correct. It is not exactly half a year, and the S u n will disappear at any point proportionate to the fraction remaining over the full sidereal days gone between rising and setting. 13. F o r the gods at the N o r t h Pole, the day is d e t e r m i n e d by the S u n ' s d e c l i n a t i o n , ( n o r t h d e c l i n a t i o n b e i n g d a y - t i m e , a n d s o u t h d e c l i n a t i o n n i g h t ) , not like o u r s , d e p e n d i n g o n the d a i l y r o t a t i o n . It is 60 nddikds f o r us, a n d o n e year for the gods. 14. E v e r y year the day a n d the n i g h t o f the gods a n d o f the asuras (demons, at the S o u t h Pole) is o p p o s i t e , (i.e. w h e n it is d a y - t i m e f o r the gods it is n i g h t t i m e f o r the asuras, a n d vice versa); f o r the pitrs (on the M o o n ) , the d a y - n i g h t is o n e s y n o d i c m o n t h ; a n d for m e n , it is sixty nddikds. 15. T o the extent the S u n rises above the h o r i z o n t a l by two m u h i i r t a s , (i.e. 24° above the h o r i z o n ) to that extent the gods at the p o l e see the S u n r i s i n g above the h o r i z o n , a n d not m o r e t h a n that. T h e Sun spirals round and r o u n d after rising, with its altitude increasing with its north declination. As the m a x i m u m declination is 24° (according to H i n d u astronomy, and fairly correct at V M ' s dme), the aldtude never exceeds 24°. After that is spirals down. 16. T h e series o f the L o r d s o f the H o r a s a n d L o r d s o f the days d o not fit there as it does f o r us, because the sixty-nadz'Aa-day-night does not o b t a i n there. This statement fits not only the pole, but also all places in the arctic zone, when the Sun seen above the horizon exceeds 24 hours. T h e hord is one hour's time, and the Lords of the hords are successively Saturn, Jupiter, Mars, Sun, Venus, Mercury and M o o n , and again Saturn etc. T h e lord of the first hord after sunrise is the lord of the day. It can be see that the lord o f the 25th hord is the lord of the day next in the day series. So, if the day is more than 24 hours, the two series cannot fit. 12a
AB
b. A . B . ^ # M 3 ( A 2 . ° f i M ^ B . ° f t ^ ) c. B . ^ l f t ^ M d. B . ^ n f ^ ^ ^ 13a. A.^tTOWT; B . ^ ( B 1 . 2 . o m ^ ) c. d. B 3 . g a p a f t e r ^ ( B 1 . 2 . n o g a p ) 14a. B.^cl^-^fn^'l c. B . ^ ^ A . B . f * T f i n d. B.T^^pqtB.Hlfe+Mfg: u
Quotedby Utpala on BS'17.4-5. ^^a. B l . ^ ; B 2 . ^ q ^ ; BS.Wmfor^J^, b.
A.C.D.U.OTS^; B . ^ T w r ^ M r ^ (B3. ° ^ )
'
A.B^(B.omBT) A.^«^«f:; B.^«W«T: ^ ^ ^ ^ ^ ^ ^ ^
a-b. C.WTOc1^^'T«TrFII^ | D.wqn^wg^wn^ c. A.^rfFRI?); B . Ca-lM^Pw^lgl
288
PANCASIDUHANllKA
XV.21
Weekday 17. T h e d e t e r m i n a t i o n o f the weekday is not the same e v e r y w h e r e . A s no reason is g i v e n i n this m a t t e r too, e v e n astrologers disagree a m o n g thereselves. T h e commencement of the day is fixed arbitrarily, as a matter of convention, like midnight, sunrise, sunset, etc, following the custom of different peoples. 18. T h e w e e k d a y is o b t a i n e d f r o m the total days c o m m e n c i n g f r o m a stated p o i n t o f t i m e , o f a p a r t i c u l a r day at a p a r t i c u l a r place. A c a r y a L a t a d e v a has said that the day begins at the exact (mean) sunset at Y a v a n a p u r a .
This convention is that of the Romaka and the Paulisa Siddhdntas. mendoned by V M in PS I 8-1 Latadeva is said to have redacted these two siddhdntas. Yavanapma is Alexandria in Egypt, as can be fixed from the longitude correction for Ujjain in AS 111. 13. see also I. 8 and note on p. 10 above.
^Vil'drd[
17-20. Quoted by Makkibhatta on Si. Sekhara,2.\0. 17a. A.^Rsifeq^^; B . ^ r o r a f i r * ^ b. M.^FR^^fif«T^ c. A . B . C . D . M . ^
^
W
fTf?H: I
18-29. Quoted by Utpala on BS 2, pp. 31-32 18a. B.ii'l"llRH B1.2.°r^feU"lllM; ( B 3 . ° M | ° ) b. A l . ^ « n ; A 2 . ^ a ^ ; B . D . M . U . W ^ : c. A.B1.2.elMWl5 d. A . B . ' ^ ( B . ^ ) ^ ; M.^^lWt
XV. SECRETS OF ASTRONOMY
XV.27
rT?TT3fqqTS6tdc<|cHlq (^)
289
chlfac^^l^Rd
||
9^
Day-reckoning 19. S i r i i h a c a r y a has d e c l a r e d that r e c k o n i n g >.*ay-total c o m m e n c e s at a s u n rise i n L a i i k a . T h e p r e c e p t o r o f the Y a v a n a s has said that the day c o m m e n c e s for the Y a v a n a s ten muhurtas, o r twenty nddikds, i n the n i g h t , (i.e. after sunset). H i n d u siddhantas suppose Laiika to be on the equator, at the junction of the Ujjain meridian. Sirnhacarya's view is that of many later siddhantas. T h e preceptor of the Yavanas mentioned is probably the Yavanacarya of the Ydvanajdlaka, the well-known astrological work. There is a section devoted to astronomy also in that work. If the people in Greece are meant by Yavanas here, Yavanacarya perhaps tries to fit a Greek astronomical work into serviceability in Ujjain, for 20 nddts after sun-set in Greece is the moment of sun-rise at Ujjain, assuming a rough longitude correction 10 nddis. 20. A r y a b h a l a has said that the day c o m m e n c e s at m i d - n i g h t at L a i i k a . H e h i m s e l f a g a i n has said, the day c o m m e n c e s f r o m sunrise at L a i i k a . Aryabhata has written two works. One is the wellknown Aryabhatiya. H e has written another work, not extant now. It is referred to by others as the Midnight School and commences the day at midnight. Bhaskara I has given its system in chap V I I of his Kannanibandha, better known as Ma/idbhdskariya. The system given in this is the same as that of the Saurasiddhdnta of the PS. Brahmagupta professes to follow this in his Khandakhdyaka. 20a. J y , N . W T I ^ 19c. B.^cUHPirHRdP^M^^-
20. Quoted by M l a k a n t h a in his
A.D.U.°?r9TtVf^°;
Jyolirrmmdnisd. p.8,asa!soon
M.^ra^Mf^"
AB/j. Kala. 16.
b. B1 .S.-Jigfef-^; B2.M<^^ft:^'IK Bl.S.^^^J^: c.
M,N,U.^«fe^
290
XV.27
PANCASIDDHANTI KA
2 1 . I f it is a r g u e d that the d i f f e r e n t times f o r c o m m e n c i n g the day c a n be a c c o u n t e d for by c o r r e c t i o n f o r l o n g i t u d e , it does n o t a g r e e w i t h what they themselves have said i n this matter, a c c o r d i n g to the sdstras, (which is as follows). 22. ' T h e s u n r i s i n g i n B h a r a t a - v a r s a , makes at that very m o m e n t , m i d - d a y i n Bhadrasva-varsa,
sun-set
in
Uttara-kuru-varsa,
and
mid-night
in
the
Ketumala-varsa. 2 3 . W h a t is sun-rise at L a r t k a , that same m o m e n t is sun-set at S i d d h a p u r a , n o o n at Y a m a k o t i , a n d m i d - n i g h t i n the R o m a k a - p u r a . In the above two verses the early siddhantic conception of a world geography is given briefly. T h e equator is the Jambudvipa, with the N o r t h Pole at its centre. L a n k a is the point where the Ujjain meridian cuts the equator. T h e point 90° east of L a n k a is Yamakoti, also called Yavakoti. Here seems to be a vague concept of Java, called Yavadvipa, whose exact distance was not realised. Ninety degrees west of L a n k a is Romaka-pura, answering to Rome, whose exact position was not realised. T h e antipode point of Lanka is called Siddhapura. A vague notion of the Mayan and Aztec civilisation brought in by early exporters sailing the seas might have given rise to the idea. T h e astronomical idea of sun-rise, moon etc. is correct according to the conception. T h e four varsas mentioned, Bharata, Bhadrasva, K u r u , and Ketumala are supposed to be situated round the N o r t h Pole, at its south, east, beyond the pole and west, from our stand point, i n Bharatavarsa. T h e Puranas give seven divisions o f the Jambudvipa, and these are the principal four. T h e puranic concept is that o f an earlier period of a flat earth, with the mountain M e r u at the centre with J a m b i i d v i p a arranged all round, transmitted by tradition. T h e Siddhantas tried to fit whatever is possible of the Puranic geography, into the conception o f the spherical earth, refuting the rest outright or explaining them away. 24. A t the b e g i n n i n g o f the y u g a , the i n t e r c a l a r y m o n t h s , the o m i t t e d days, the p l a n e t a r y days, the l u n a r days, the first p o i n t o f M e s a , the M o o n , the S u n , half-years, rtus, a n d the s i d e r e a l days, b e g i n t o g e t h e r (and c a n be r e c k o n e d anew). 2 5 . T h e l o n g i t u d e c o r r e c t i o n r e c k o n e d f r o m the R o m a k a r e g i o n is different f r o m that f r o m Y a v a n a p u r a . R e c k o n i n g t i m e f r o m m i d - n i g h t at L a r t k a is d i f f e r e n t f r o m that f r o m sun-rise. 21a-b. B . f ^ T r a ^ ( B 3 . ° ^ ° )
23a. A . H a p . o m o f o n e ^ ;
c. B . ^ W I ^ ^ M ( B 3 . W q t ) d. A . ^ ^ B 1.2.3. gap indicated for^«Tmi^ and part ofthe next line up t o ^ ^ ^ K 3 ^ 22a. A.^IstW; C . D . U . ^ ^ I W T ^ b. A.C.D.U.omBrftg and read
c. A.°g?WxT d. B2.^'IM<4>:
B.^^cTfFlt c. A . B l . ^ q q ^ U . •Mc(*l*ji TTE^lt d.
B.%J#(^U.^^
, , ^ 24a. B . 3T'lost. B.Tlf3ra? b. B . f ^ ^ ^ I ^ A . ^ l ^ c. A . 3 m - ^ ; B . S T O ^ ^ ; D.SPR^^FlfcT
25b. d.
B.f^fCRl^ A.-^FJ^iR.'^-'^
291
XV. SECRETS OF ASTRONOMY
XV.29
26. I f we d e t e r m i n e the D a y - l o r d f r o m the half-setting o f the S u n every day, t h e r e is n e i t h e r t r a d i t i o n a l a u t h o r i t y n o r r e a s o n i n g to s u p p o r t this. 27. E v e n i n q u i t e adjacent places, i n o n e place t h e r e is sun-rise o r sun-set, a n d not i n the o t h e r , d a y - t i m e i n o n e place a n d n i g h t i n the o t h e r , a n d vice versa. T h u s , t h e r e is c o n f u s i o n a m o n g p e o p l e i n the m a t t e r o f the L o r d o f the day.
^<|c|Mf(x^)^ ^ I ^ l t T T (
28. T h e m a t t e r o f d e t e r m i n i n g the H o r a - l o r d also is i n the same mess. W h e n the D a y - l o r d is not d e t e r m i n e d , h o w c a n the H o r a - l o r d be d e t e r m i n e d ? 29. W i t h o u t g i v i n g a t h o u g h t to a l l these difficulties, p e o p l e g e n e r a l l y use the n a m e o f the D a y - l o r d i n t h e i r daily r o u t i n e , ( a n d get o n w i t h t h e i r w o r k ) . L e a r n e d a u t h o r i t i e s say that the best t h i n g w o u l d be to use the t r u e tithi, ( l u n a r day) a n d its parts f o r daily i n t e r c o u r s e (as f o r f i x i n g a d e f i n i t e p o i n t o f t i m e etc.) What is meant is as follows:- Sun-rise etc. may vary from place to place, according to the local time. But the lunar day is the same for every place on the earth. So this can fix a point of dme without any ambiguity. We learn that the ancient Babylonians used the lunar day as the unit of dme, just as we use the solar day. In the above verses V M indulges in a lot of discussion about Hora-lord, Day-lord, etc. B u t these are only matters of convention, and astrologers and governments can agree upon some convention to avoid difficulties, e.g. do we not have tfie standard mean time for our daily dealings. 26a. B.°«J5mr^ b. B . ° t ^ ^ B . f ^ « n ^ A2.1H:
27b. A . B . f ^ t ^ ^ T ^ : A2.»R»fa4
c. B . ^ s t m t
c. A . B . ^
d. A . ^ 1 ^ : ; B . C . D . ^ ^ ^ :
d. D.°Hlj.d^cj
B.°^^:;U.°^^
28a. A . ^ ^ ; B.^grTI^ ( B 3 . ° ^ ° ) ^-
B.°Tmm!\:
c. A . f ^ ; B . f % ^ 29a. U . ^ r t ^ l t ^ b. A . ^ ; B l . 2 . ^ ; B 3 . ^ : c. B.^f#f«T A . B 1 . 3 . f ^ d. B l . W i r f :
q^fan^lPHchl^i c)<|^fi4[^
Thus ends Chapter Fifteen on 'Secrets of Astronomy' in the Pancasiddhantika composed by Varahamihira f.Cof.: A . B . D . ^ ( A l . ^ ) t?T^tqft^tilR# ( B . ^ )
Chapter Sixteen SAURA SIDDHANTA : MEAN PLANETS
Introductory
Chapter X V I of the Pahcasiddhdntikd deals with the computation of the mean star-planets. Mars etc.. according to the Saitra Siddhdnta. and chapter X V I I of their true motions, with their heliacal risings and ladtudes. T h e mean planets are made true by employing the method of epicycles, as in the case of the Sun and the M o o n , in chapters X I and X . O f the xiddhdntas condensed by Varahamihira the Saura alone uses epicycles, and there is no evidence of its use in any other. So, in the originals also, only the Saura must have used epicycles, since V M follows the originals as far as necessary. T h u s the Saura is the most mature, and may be considered to begin the highest developed stage of H i n d u astronomy, represented by the Aryabhatiya, the Brdhma-sphuta-siddhdnta, the Ijiler Surya Siddhdnta etc. T h o u g h V M ' s Saura, being a karana, does not useyi/^^n-cycles for the planets, the original must have had them and they can be reconstructed from the epoch-constants given, as we have done in the case of the Sun, M o o n , Moon's apogee and nodes. These can be seen to agree with the corresponding parameters of the PaitUsa quoted by Bhattotpala in his commentary on the BrhatMimhild, and with ihe Ardhardtrika-paksa of Aryabhata, a work now lost, but reconstructible from its description given in the Mohdhhdskariya, chapt.VII, 21 -35, and from the Khandakhddyaka of Brahmagupta, which latter expressly follows the Ardhardlrika-paksa. Not only the •yMgc'-cycles, but also the yuga days, and epicycles and apogee positions and nodes agree in these. Strangely enough, the 'New' Surya Siddhdnta does not agree with the 'Old' in many things. In the matter of computing the ladtudes of the star-planets, the Saura gives the same method as the Ardhardtrika-paksa combining two types of latitudes, but the Khandakhddyaka follows the Aryabhatiya itself exactly as propounded in the Mahdbhdskariya, V I . 52-55.
As lor agreement of VM's Saura with the other siddhdntas of the period, a perusal of the table given undct X V I I . 11 will show this. B u t it must be noted that the agreement in mere number of cycles is not real agreement, because, theywg^a days being different, there will fje difference in the calculated mean values. But at the period we are considering, viz. (-.500 A . D . , the mean posidons fairly agree with one another, and also with what would be got by modern astronomy, showing thi'i the accuracN' of their observations. For example, for FS's epoch, all except the later Siii-ya Siddhdnta give nearly 236° for Rahu, including the moderns. T h i s is seen only in the Rahu of the iMler Siina Siddhdnta. (P^vidently, there is error of reading here. In 1.33 of the iMter Sur. Sid., the original should have been vasvasviyamdsvinkhidasrakdh instead of vasvagni etc. T h i s mis-reading must have occurred before the commentator Rai'iganatha, for he gives astardrndkrtirdmadvimitdh. If what I suggest is correct, 3° will be added to the 232° 29' got according to the wrong reading making Rahu = 235° 29', giving fair agreement). There is agreement in the degrees for heliacal rising and setung and the method of computing the star-planets between the Sa\ira ol the PS and the l/iler Sdrya Siddhdnta, though the epicycles differ in many ways.
XVI. 1
XVI. SAURA : MI.AN PI.ANF/l S
293
Another important matter should be mendoned. In X V I . 10-11, and X V I I . 10-1 l a , V M gives corrections, which are his own, to secure agreement with observation to make the Saura fit for correct almanac-making, which naturally will be demanded by the literate. Thus, in X V I . 10-11, certain bijas are given to correct the means of Mars, Jupiter and Saturn and the sighra of Mercury and Venus. T h e correcdons amount, in terms of yt/^a-cycles, to: Mars, -f .57; Mercury. -I- 400; Jupiter, — 33 '/2; Venus, — 150; and Saturn, + 25. These corrections are similar and approximately equal to the famous Vdghhdvona correction on the Aryabhatiya. propounded by his successors i n his school, to correct his cycles to agree with their observation. I do not suggest that V M was aware of the Vdgbhdva correction in that form, but the tendency to correct the earlier results with bijas based on observations is found everywhere, whether north or south, a healthy sign of the growth of the science. One might refer also to XVII.10-1 l a , where V M attempts to conect Mercury and Venus to secure agreement with observation. Another thing is to be noted. In (1) the Aryabhatiya. in (2) the Ardhardtrika-paksa (which means ipso facto the Khandnkhddyaka), V M ' s Saura and Bhattotpala-quoted PaulKa, and in (3) the iMicr Suryasiddhdnta, theyuga cycles are such that the mean planets are all zero at the beginning of Kali, the Moon's apogee is 90°, and the Moon's node 180°. Now the Aryabhatiya had equal yuga-jjddas, Krta, Treld, Dvdpara and Kali, i.e., they are equal in length. T h e oxher siddhdntas have unequaly?^,ij'a divisions, Krta being 4 parts, Treld 3 parts, Dvdpara 2 parts and Kali 1 part. If the other siddhdntas also postulate, like the Aryabkitiya, that the planets were created and began to move from the beginning of the Kalpa from a zero position, then the cycles should be divisible by 20. But they are not so divisible in all. This necessity is avoided by postuladng a dme later than the beginning of the Kalpa called 'the time ot creation of planets' by the Ijiter Surya-Siddhdnta, as started in the verse, graharksadevadaityddi srjato'sya cardcaram krtdhdhivedd divydbddh .iataghnd vedhaso gatdh \ \ 1.24 \ \ and by having both the number of cycles andyMg-a-cycles divisible by four. In the case of the Moon's apogee, the cycles should be odd, and in the case o f R a h u the cycles should be even, but not divisible by 4. These necessary conditions are indeed found in the Later Surya Siddhdnta and its kind. Thus, if there is any observed difference in the mean planets. Moon's apogee and nodes, they must be due to the 3600 years elapsed after Kali, for the perif)d r.499 A . D . But the observed differences should be only small, and due to error of observation. T h e cycles must have been, and have been, constructed with an eye to this also. In fact, the number of cycles have been determined by observation, and by using the Diophandne equation (kuttaka). T h e difference of just 300 days in the length of theyuga, (it does not matter much i f it is 328 days, as in the Later Sdrya Siddhdnta) to secure equality at c.499 A . D . , between the Ardhardlrika-paksa and the Aryabhatiya, which is called, for the sake of distinction, the Audayika-paksa, meaning the type beginning the day from mean sunrise at Ujjain, provided the number of cycles are the same. (See tables under X V I I . 11). There is a difference of just a quarter of a day accumulated from zero Kali to c.499 A . D . and the difference is made zero at this point of time.
X V I . 11
PANCASIDDHANTIKA
294
Mean positions of the star-planets I. T h e f o l l o w i n g is the d e t e r m i n e d p o s i t i o n o f the star-planets at m i d n i g h t at U j j a i n a c c o r d i n g to the Saura Siddhdnta. F o r t h e i r c o m p u t a t i o n , the m e a n S u n s h o u l d be t a k e n as the m e a n M e r c u r y a n d V e n u s . Note: I follow TS's emendations.
Example: Find the mean Venus at 1,20,553 days after Epoch for the star-planets, viz. 427 saka midnight at Ujjain. This is the mean Sun at 1,20,553..5 days from midday of the Saura epoch, (yide expl. under I X . 1). Therefore the mean Venus = the mean Sun = 1,20,553.5 X 800 - 442) 2,92,207 = 17° 18' 27".
3 T ^ « n ( T I T ) % ( t r ! T ) ' (sJyT)^:' ' ^ ( q ^ ) '
ar^cbldVI ^
^m:
l^lHrkicbl ^^lum^ifuidl: \\6 \\
'^g^'l^cbHI: 3rf?Tcr(it) w r R f ^ ( # )
RlfdMI^
X V I . 11
XVI. SAURA : MEAN PLANETS
295
2-9 To get mean Jupiter, m u l t i p l y the days f r o m e p o c h by 100, a n d d i v i d e by 4,33,2.S2. R e v o l u t i o n s etc. are got. D e d u c t 10"' p e r r e v o l u t i o n . A d d 8° 6' 20", the m e a n at e p o c h ( T h i s is c a l l e d ksepa.) ( A hija c o r r e c t i o n is g i v e n by V M , to this, for w h i c h see verses 10-11, below.) T0 get Mean Mars, d i v i d e the days by 6 8 7 . R e v o l u t i o n s etc. are got. A d d 14"' p e r r e v o l u t i o n . A d d 2' 15° 3 5 ' 0", the m e a n at e p o c h . (See verses 10-11, b e l o w , for bija c o r r e c t i o n . ) To get mean Saturn, m u l t i p l y the days by 1000 a n d d i v i d e by 1,07,66,066. R e v o l u t i o n s etc. are got. D e d u c t 5"' p e r r e v o l u t i o n . A d d 4 ' 2° 2 8 ' 49", the m e a n at e p o c h . (See verses 10-11, below, f o r bija c o r r e c t i o n ) . To get the Sighra of Mercury, m u l t i p l y the days by 100 a n d d i v i d e by 8 7 9 7 . R e v o l u t i o n s etc. are got. A d d 41/2'" p e r r e v o l u t i o n . A d d 4" 2 8 ° 17' 0", the Sighra at e p o c h . (See verses 10-11, b e l o w , f o r hija c o r r e c t i o n . ) To get the Sighra of Venus, m u l t i p l y the days by 10 a n d d i v i d e by 2 2 4 7 . R e v o l u tions etc. are got. A d d IOV2" p e r r e v o l u t i o n . A d d 8' 2 7 ° 3 0 ' 39", the sighra at e p o c h . (See verses, 10-11, b e l o w , f o r bija c o r r e c t i o n ) . l a A.'-^el'Sli; B.^SR^ b. A.P|'J|1'5iRH'#; B.'l^u|<+ifl)
c. B . ^ : A.^5rqf?[f«f; B.^wfclfcTf«T
c. A.H^l^l4u|; B . H w ^ i ^ I ^
d. B.R^I^ 7a.
2a. B . f e ^ B 2 . ! ^ l d l V | | y
c. B.°*ra4!cf° (B3.°Wy.
b. Ai.c.°'4iiirijR^i'r^^S'q%;
d. A . ? ^ : ; B . ^ O T I ; C . D .
B.°1^^H d.
8a. B.'iuilcl
Al.f3Al.°lg^
3a. B . repeats words from previous verse:
b. A . c. A 1.3^^4)1°; A 2 . " 3 l i ^ d. A.ftfdkll
a-b. b. A.<^<+1^-^|; B . ^ ° B . ^ : d. B1.2.D.'?1^^BqT 4a. B1.3.
9a. A.ra^W;
B.RH^<^
b. A.y\-^"i; B . ^ ^ S ^ R ^ I k l * ! c. A.i^llf^d^; B3.!^ll[tidW B.f^+dl d. A.B.WTpnci^HI: (A.cUO
om by haplography
10a. A.B.%^B.R(=b
b. B.°^IWWI: C. B . H ^ c ) : - ^ ^ ^ d. B 1 . 3 . ? A S J B 3 . ^ S T A . ° ^ B . ^ : 5a. B.°At
b. A . B . g ^ q r s q T T A . B . W r ^ 11a. A . B 1.2 4^^41; B 3 . ° I ^ : b. A . B . ° 1 ^
B.-^T(^TW:
b.
B3.°?lf?I^C.D.^
c. A . B . f t * l f d * l :
C.
B.°^«f^:
d. A . B . C . D . o m ' g : B . ^
d. A . ° « ^ H W R ^ ; B.?Fr^«TOc%^ 6a. A . ^ T M f ^ ; B.i7Fn:f?RT b. A . C . D . ^ : ; B.m-; A . ^ ^
(A2.°^°);
^Um
( B 3 . ^ . A.B.C.D.JTsqrg In D.ch. X V I I of the Mss. and of C is condnued as part of c h . X V I I . with
B.M:?t^^^;
verse numbers duly altered.
296
X V I . 11
PANCASIDDHANTIKA
Note 1. 1 follow TS's emendations, except in verse 6, where 1 have read khamaksau as khapaksau instead of their khamakso, makso being meaningless. But their meaning, 20, is all right. In 7, the word ksepa can stand, and need not be emended as done by them. Note 2. T h e word madhya with reference to Mars, Jupiter and Saturn is mean planet in modern parlance, and sighra with reference to Mercury and Venus, is mean planet according to modern terminology. Note 3. H o w to get the days from epoch has already been explained, and it should only to be brought to the mid-night following to be used here.
Example 1. Find the mean Mars at 1,20,553 days from the midnightfollowing the Romaka epoch the epoch given for star-planets. 1 , 2 0 , 5 . 5 3 6 8 7 = 175 revolutions and T h e revoludon correction = 1 7 5 x 1 4 " ' Ksepa or mean at epoch
= = =
5^ 21° 52' 40" 4-41" 2 ' 15° 35' 0"
Mean Mars at required date
=
8'
7° 28' 21"
Example 2. Find the Sighra Venus at 1,20,553 days for epoch. 1,20,553 X 10-^ 2247 = 536 revolutions and R e v o l u t i o n C o r r e c t i o n : 5 3 6 x IOV2" at epoch
= =
6'
2° 19' 23" + 1 ° 3 3 ' 48" 8"^ 27° 3 0 ' 39"
% W o f V e n u s a t 1,20,553 days
=
3^
1° 2 3 ' 5(;"
Note 4. T h e rules give to find the mean planets etc. depend on the fact that there are approximately 100 revoludons of Jupiter i n 4,33,232 days, one revoludon o f Mars i n 687 days, 1000 revolutions o f Saturn i n 1,07,66,066 days, 100 sighra (truly mean) revolutions of Mercury in 8,797 days a n d 10 o f Venus i n 2247 days. T h e revolution correcdons make these exact. T h e epoch constants are the means at epoch. Note 5. F r o m the rules given we can reconstruct theyuga cycles of the original Saurasiddhdnta of which the Saura o f the PS is a Karana, and from these the epoch constants. These we shall do now. T h e y u g a days o f the original Saura are 1,57,79,17,800, as computed from the short Saurayuga given i n 1.14, from which it can be computed that i n 1,80,000 years there are 6,57,46,575 days, since theyMgo is 43,20,000 years, being 24 times the shortywga.
We might now verify by calculadon, the yuga revolutions (yuga-paryaya) and epoch constants (ksepa) of the several planets. Jupiter: Yuga revolutions 1,57,79,17,800 X 100-^4,33,232 = 3,64,220, rev. Revolutioncorrection = 3,64,220 X 1 0 " ' .-.The number of rev. etc. i n the ywga = 3,64,220 rev.,
=
0^ 17° 2 5 ' 1" - 1 6 ° 5 1 ' 43" 0^
0° 3 1 ' 18"
X V I . 11
XVI. SAURA : MEAN PLANETS
297
T h e error i n the karana method is 3 1 ' 18" in 43,20,000 years, which is negligible when we consider that the rule is given i n a kararia, which is not intended to be used for such a long period. Theyuga revolutions 3,64,220, is indeed that given in the original, as seen from the Ardhardtrika-paksa and Bhattotpala's Paulisa and Khandakhddyaka. Epoch Constant (ksepa) for Jupiter The epoch is 427 Saka i.e. 427 + 3179 = 3606 years from zero K a l i , i.e. midnight, - 3 nddh, 9 vinddis. For 3606 years, the modon is 3,64,220 x ( 1200
+ 1200 X 600 304 rev.,
(y
8°
6'
36" — 16"
=
0'
8°
6'
20"
1,57,79,17,800 X 1 0 0 0 - M , 0 7 , 6 6 , 0 6 6 =1,46,564 rev., T h e c y c l e c o r r e c t i o n = 1,46.564x5'" =
0^
3° -3"
26' 23'
12" 34"
.-. T h e Fwg-a cycles got =
0^
0°
2'
38"
Subtracting the motion for 3 nddis. 9 vinddis, Jupiter's epoch constant got This is exactly what is given above in verse 6. Saturn: Yuga revolutions
1,46,564 rev.,
This is indeed the yuga cycles given i n the Ardhardtrika-paksa etc. neglecting the small error of 2' 38" accumuladng in 43,20,000 years, due to the kararia roughness. Epoch constant for Saturn 1,46,564C—+ ^ 1200 600 X 1 2 0 0 Deducting for 3 nddis, 9 vinddis
122rev.,
T h e epoch constant got =
A'
2°
28'
5" — 6"
4'
2°
28'
49"
6^ +4^ IT
29° 28° 27°
4' 52' 57'
59" 5" 4"
Mars: Yuga revolutions 1,57,79,17,800-687= Rev. Correction = 22,96,823 X 14'" = .•.Yuga-cycles=
22,96,823 rev. 22,96,823 rev.
= 22,96,824, i n round numbers, being short only by 2° 3', negligible i n the long period. We see agreement with the original. Epoch constant for Mars The epoch constant is 22,96,824 (—!—+ 1 ) = 1917 rev., 1200 600 X 1200
2^
15°
36'
43".2
X V I . 11
PANCASIDDHANTIKA
298
Deduction for 3 nddis, 9 vinddis Less 1917'" ^ 5 " T h e epoch constant
1' 39.5" -6" =
2'
15° 34' 58"
There is agreement. Mercury: Yuga revolutions 1,57,79,17,800 X 100 ^ 8997 = Rev.Correction = 1,79,36,99 X 4'/2'"
1,79,36,998 rev. =
T h e FMg-fl cycles = 1,79,37,000 Rev.
IV +V
21° 4 1 ' 13° 4 1 '
33" 15"
0^
5° 22'
48"
There is fair agreement with the Ardhardtrika-paksa etc. with an excess of 5° 22' 48" in theyuga, which need not be considered great in a kararia rule. 4 7/16 instead of 4'/2'" would have taken this difference also into account. Epoch comtant for Mercury 1,79,37,000 (-i— + ^ )= 1200 600 x 1200 Subtracting for the excess 3 nddis, 9 vinddis For 1/16 repeat the correction Epoch constant
14,972 rev..
=
A" 28° 30' 0 " — 2 3 ' 53" + 16" 4'
28° 17' 23"
Here the constant seems to have been given to nearest minute. Venus: Yuga revolutions 1,57,79,17,800 X 10-T-2247
Revolutioncorrection = 70,22,331 X 101/2=
70,22,331 rev.
56Rev.
Kwgacycles = 70,22,388
1'
8°
55° 5 4 "
10^ 21° 4 7 ' 5 2 " 0'^
0° 4 3 ' 4 6 "
T h e r e is a small error o f 43' 4 6 " , negligible in the long period of yuga, owing to the kararia rule. 10 85/178 would have been very correct. Epoch constant for Venus Epoch constant, 70,22,388 {-^— + ^ ) = 5861 rev. 1200 600 X 1200 'Less iov 3 nddis, 9 vinddis Extra in the correction
8^ 27° 3 5 ' 38".4 5' 2 " + 2' 4 "
Epoch correction (in full agreement)
8' 27° 30' 3 9 "
=
In the Sun, M o o n , Rahu, and Moon's apogee too we see much exact agreement with Ardhardtrika-paksa, Kharidakhddyaka and Bhattotpala-quoted Paulisa, from which we can conclude that source of V M ' s .Saura is the Old Saurasiddhdnta.
XVI. SAURA : MEAN PLANETS
X V I . 11
299
VM's Bija corrections 10-11. A d d 17" p e r year to m e a n M a r s . D e d u c t 10" p e r year f r o m m e a n J u p iter. A d d 7 ' / 2 " p e r year to m e a n S a t u r n . A d d 1 2 0 " p e r year to the sighra ('mean' a c c o r d i n g to m o d e r n parlance) o f M e r c u r y . Subtract 4 5 " p e r year f r o m the sighra ( m o d e r n 'mean') V e n u s . I n a d d i t i o n , subtract 1400" o r 2 3 ' 20", constant f r o m J u p i t e r ' s m e a n . Note.l I follow TS's corrections. Note 2. These corrections are obviously V M ' s own, to secure agreement with observation, because V M sees the Saura used widely for almanac making, (besides himself being its follower) and uses these bija corrections to the Saura. Being V M ' s own, we cannot verify the numbers used, but we can compare these corrections with those given by the followers of the Aryabhatiya belonging nearly to his time. Note how close they are, and commend the tendency to observe and correct, instead of blindly following the masters. T h e Kerala school following the Aryabhatiya gives the well-known vdgbhdva correction: vdgbhdvondc chakdbddd dhanasatalayahdn mandavailaksyardgaih prdptdbhir liptikdbhir virahitatanavas candratattungapdtdh | sobhdnirudhasarnvidgariakanarahatdn mdgardptdh kujddydh sarnyukytd jddrasaurdh suragurubhrgujau vajitau bhdnuvarjam \\ (Katapayddi notation is used here.) According to this the corrections per annum are for Mars + 11.5", for Mercury -I- 105", for Jupiter - 12", for Venus - 39", and for Saturn H- 5". See that these compare well with V M ' s . Example: Give the bija corrections for Mars and Venus at 1,20,553 days from epoch. This is 330 years. T h e correction for Mars = 330 X 17" (positive) = -I- 1° 33' 3 0 " . T h e correcdon for Venus = 330 x 45 (negative) = - 4° 7' 30". T h u s &f/a-corrected mean Mars of date is 8' 9° 1' 51", and ^)ya-corrected Venus, 2' 27° 16' 20".
Thus ends Chapter Sixteen entitled 'Saura-Siddhanta - Mean Planets' in the Pancasiddhantika composed by Varahamihira l . C o l . : A.B.D.^fSfW^^«2Tqf^-(B.^lf^forTTfir) (D. gives this as a section colophon). C. ifil ^4R1<^I'T1 fT«2Flf!RflT ^l¥#«2TR:
Chapter Seventeen SAURA-SIDDHANTA — TRUE PLANETS
As stated earlier, the computation of the Saura star-planets is contiiuied in ch. X V I I , the topics treated being, the T r u e planets, their heliacal risings and their latitudes.
iftm^S cFfs?^ •tr^R^'
^ q f i g p r t fl^jtrTT:
( ? 5 c n § [ q ^ w : ) ' 'is^'
w ^ v j i i ^ l w i H II ^ II
Epicycles of the planets 1. F o r the o t h e r planets (i.e. o t h e r t h a n M e r c u r y a n d V e n u s , viz., for M a r s , J u p i t e r a n d S a t u r n ) , the S u n is t h e i r Sighra. T h e epicycles o f e q u a t i o n o f the apsis o f M a r s etc. are twice, 3 5 ° , 14°, 16°, 7°, a n d 3 0 ° , (i.e., o f M a r s 7 0 ° , o f M e r c u r y 2 8 ° , o f J u p i t e r 3 2 ° , o f V e n u s 14° a n d o f S a t u r n 60°.)
Note 1. 1 follow TS's emendation mpahcatrimsanrnanavah. B u t I read,SMraA assvardh and not.9'a like T S because svardh is nearer the given reading surdh, and also 14° is the epicycle given in the Ardhardtrika-paksa etc. Five mdtrds are wanting i n the last foot, and it must be supplied with som such words as bhdgdh, as all numbers are already given. But T S make it sadyutds-tnrrisdh a strangely enough, translate it as 24, confusing the addition mentioned by themselves for subtraction. (However, on page X X I I I of the Introduction Thibaut gives the correct 30° x 2 = 60°.)
Note 2. T h e first foot is to be read with verses 7 and 8 of chap. X V I where the sighra of Mercury and Venus have already been given. Properly speaking, the matter i n the foot should have been given i n chap. X V I .
( ? ) WTT: ^ ^ « ! , M U ^ s * » ( ^ ) i n J n i 3 ; || ^ | 2. 6, 11, 8, 4, 12 m u l d p l i e d by 2 0 , M a r s ' s b e i n g less by 10°, (i.e. 110°, 2 2 0 ° , 160°, 8 0 ° , a n d 2 4 0 ° ) are the a p o g e e positions o f M a r s , M e r c u r y , J u p i t e r , Venus and Saturn.
la. B . # b. ° ^ 5 ^ ; D . g [TR] C. D . f l y n : ^ A.Bi^ldlH'JI
d. A . ^ O f ^ : I B.°F!T:^gcrfW?n II; C.°W: [^:] |
TRf; q f ^ d l l W : ||; D.°-CT:^iSffWr[«J] II
301
XVII. SAURA ; TRUE PLANETS
XVI.6
Note 1.1 follow TS's emendations. Mandagatinamabhdgdh does not make any sense. But the meaning is obvious, it must mean apogee positions. Some drastic emendadon of the word can be made to give this meaning, but I am against such as emendation. Note 2. These positions agree with those given i n the Ardhardtrika-pahsa etc., as also in the Aryabhatiya. T h e correct positions according to modern astronomy are 128°, 234°, 170°, 290°, and 244°, respectively. Note 3. T h e apogee 80° for Venus and the epicycle 14° are the same as given for Sun. T h e apogee position 80° given is near the perigee position of modern astronomy, so faraway. We shall explain this under verses 10-1 l a , below.
3. T h e degrees o f epicycles o f c o n j u n c t i o n o f M a r s is 2 3 4 , o f M e r c u r y 132, of J u p i t e r 72, o f V e n u s 260, a n d o f Saturn 40. Note 1. 1 have generally adopted TS's corrections. B u t the text is corrupt in the third foot, and TS's correction itself wants one mdtrd. I would read the third and fourth feet thus: paksasvards ca kham sadyamdh khakrtds ca kujddindm. This would follow the original work. Note 2. T h e values agree with the Ardhardtrika-paksa, Khandakhddyaka, and Bhattotpala-quoledPaulisa group, as to be expected.
2a.
B.WW^^^<^l
b. B.-fm
A.-^W^-; B.<^I<+^>J|«'J|I:
(B2.°WTT:; B 2 . W n : ) c. D."n#lt*PII: B . ^ e n « r ^ 1 1 ^ d. A.-?pR#!Flt°
3a.
B.^^«TO
b. B . ^ r a ^ ° A . B . C . D . W t l . B 1 . 2 . ^ c. B . om^^RW: A . ^ S S 5 S ^ 3 ^
B.l§m
d. A . B . ' R T ^ ( B 3 . ^ : ^ ) C . D . f c I I : ^ : ^ ^
302
PANCASIDDHANTIKA
XVII.7
T r u e planets Thefirststep 4. D e d u c t the m e a n f r o m the sighra. I f the r e m a i n d e r (called sighra-kendra) is w i t h i n 9 0 ° , s i n . sighra-kendra is c a l l e d bhuja, a n d sin ( 9 0 ° - sighra-kendra) is c a l l e d koti.
I f sighra-kendra is m o r e t h a n 9 0 ° a n d less t h a n 1 8 0 ° , subtract it f r o m 180°. ( T a k i n g this as the sighra-kendra, s i n . sighra-kendra is bhuja a n d sin ( 1 8 0 ° — sighra-kendra) is koti. li sighra-kendra is m o r e t h a n 1 8 0 ° a n d less t h a n 2 7 0 ° , d e d u c t 1 8 0 ° f r o m it a n d take this as sighra-kendra. S i n . sighra-kendra is bhuja a n d sin (90° - sighra-kendra) is koti. I f s i g h r a - k e n d r a is f r o m 2 7 0 ° to 360°, deduct it f r o m 3 6 0 ° a n d take its sine as the bhuja a n d sin 9 0 ° - sighra-kendra is the koti. ( T h e bhuja oi manda-kendra is to be f o u n d i n the same way using manda-kendra i n the place of sighra-kendra)^ 5-6. T h e bhuja a n d koti m u s t be m u l t i p l i e d by the planet's epicycle o f conjunct i o n a n d d i v i d e d by 3 6 0 . T h u s t r a n s f o r m e d , they are c a l l e d bhuja-result a n d koti-result p e r t a i n i n g to the e q u a t i o n o f c o n j u n c t i o n . I f the sighra-madhya is f r o m 2 7 0 ° to 9 0 ° , the koti-result is to be a d d e d to 120 (the R . o f the PS). I f sighra-madhya is f r o m 9 0 ° to 2 7 0 ° , the koti-result is to be subtracted f r o m 120. S q u a r e this a n d a d d it to the square o f the bhuja-result. F i n d its square root, a n d by this d i v i d e 120 X bhuja-result. F i n d arc-sine o f this. Subtract h a l f this f r o m the l o n g i t u d e o f apsis i f the sighra-kendra is f r o m 0° to 180°. A d d i f f r o m 180° to 3 6 0 ° .
MRuIIUI Chi jchUS? d H j ^ ^ c t
II
Second step
7. H a l f r e c t i f y i n g the a p o g e e p o s i t i o n thus, d e d u c t it f r o m the m e a n . T h e result is to be u s e d as the a n o m a l y o f the apsis i n the s e c o n d step. A s we find the bhuja o f the a n o m a l y o f c o n j u n c t i o n (sighra-kendra) so find the bhuja o f the a n o m a l y o f apsis. M u l t i p l y the bhuja by the manda e p i c y c l e a n d d i v i d e by 360. a n d get the t r a n s f o r m e d bhuja-result o f the apsis. ( T h i s is sine e q u a t i o n o f the 4.5. QuotedbyUtpalaonBS,2.pp.44-45 4a. V.wm^ b. A ^ « ^ A B . ^ ( B 2 . 3 . ^ )
d. A l . ^ P ^ : ; B . C . D . U . ^ : A . B . ^ -^b- ^ - ^ r : , ^ - ^ ^ ^ ^ C.ir^m^^(D.^^^^)
A . B . ° ^ ^ ; U.°^?Fqi c. B.'SFtfe
' In modern usage, for all the above we can siniplv say sin. sighra-kendra is the bhuja and cos. sighrais the koti, without taking into account the sign + or —.)
XVI1.9
XVII. SAURA: TRUE PLANETS
303
centre). F i n d its arc-sine. A d d h a l f this arc to the h a l f rectified l o n g i t u d e o f a p o g e e i f the a n o m a l y o f apsis is f r o m 0° to 1 8 0 ° a n d subtract i f 180° to 3 6 0 ° . T h u s the a p o g e e is r e c t i f i e d c o m p l e t e l y .
Third sU'j) 8. Substract this rectified a p o g e e f r o m the m e a n a n d thus get the a n o m a l y o f apsis. F i n d its bhuja a n d m u l t i p l y it by the e p i c y c l e o f the apsis a n d d i v i d e by 3 6 0 ° . T h e bhuja-result, (this is the equation o f the centre), is got. F i n d the arcsine o f this, a n d subtract the w h o l e o f this arc f r o m the m e a n i f the a n o m a l y o f apsis is f r o m 0° to 180°, a n d a d d it f r o m 180° to 3 6 0 ° . T h e result is rectified mean.
Fourth step 9. D e d u c t the rectified m e a n f r o m the sighra. T h e a n o m a l y o f c o n j u n c t i o n is got. F i n d the bhuja a n d koti o f this i n the same m a n n e r as we d i d i n the first step. M u l t i p l y the bhuja by the e p i c y c l e o f c o n j u n c t i o n a n d d i v i d e by 3 6 0 ° . S i n e a n o m a l y o f conj. is got. M u l t i p l y the koti, i.e., cos. a n o m a l y o f c o n j u n c t i o n , by the e p i c y c l e o f conj. a n d d i v i d e by 3 6 0 ° . T h e r e l a t e d cosine is got. A d d this to 120 i f the a n o m a l y is f r o m 2 7 0 ° to 9 0 ° a n d subtract f r o m 120 i f f r o m 9 0 ° to 2 7 0 ° . S q u a r e this, a d d the square o f the bhuja (i.e. eqttation o f c o n j u n c t i o n ) a n d find the square root. D i v i d e the e q u a t i o n o f conj. X 120 by this square root. T h e arc sine o f this is the result. A d d this result to the rectified m e a n i f the a n o m a l y o f conj. is f r o m 0° to 1 8 0 ° . S u b t r a c t o t h e r w i s e . T h e g e o c e n t r i c t r u e p l a n e t is got. d. B.«^Rfri^:; C D .
6a. A 1 . ? I ^ ; B.cT^ratTJ b. A.^w<^5n^( A2.'5) B. « t r 3 [ ^ ^ ^ ; C. f ^ W ^ ^s^^^t ^ D. ^ i r a ^ t R i t ^ ^ B . ^ q * ^ : (B2.W;) c. A . B . W N I ( B . ^ ) ^ d. b . " ? M b . ^ w ^ 7a. B . ^ g 3 f ^ ( B l . f q , B 3 . ^ ) ^ ^ b. C t ^ ^ T t P ^ ; D . f ^ « T ^ c. B2.^TftaimB3.'5Rig^«5
23
B l . 2 . 3 . repeat the verse twice and J3-
give them two consecutive verse numbers. 8a. B.HSTIl^ A . ^ t ^ S T ; C.D.^f##KT: b. A . B . C . D . c T F n ^ ^ ^ 9a.-d.-^^^^im;c.-Rmmi b. D . ^ ^ r c. A.B.D.3Hlf<=)<|kl B . ^ d. A . B . ' q « ! m ^ ( B . ^ ) ^^WlWf; ( B . ' q w ^ : )
304
PANCASIDDHANTIKA
XVII.9
Note 1. In verse 4,1 follow TS's reading, except that I have emended sadhhydh into sadbhih, inste TS's sadbhyah, because my reading allows us subtraction or addition, as is wanted. In verse 5, I follow TS's except i n the second foot, where I give the Brihatsamhitd reading. Either reading gives the same sense. In verse 6,1 follow TS's except i n the second foot, where I have given bhdjyam for vibhajet, as being more likely. But the meaning is the same. In verse 7, the text required no emending, and TS's dhanabdni is unnecessary. In verse 8, like T S I have corrected ^ r o into punah but I have also corrected bdhum into bdhur which is required by grammar. In verse 9, I have corrected rmdhydkhydm mto rnadhydkhyam, which is the reading of some of the manuscripts. Otherwise I follo IS. Note 2. V M , here, as elsewhere i n the PS, uses his tabular sine where R is 120', as given in chap IV. So we must use his tabular values to get the R sines and R cosines. O f course, we may use the modem table, or the Siddhantic table with R = 3438. But then the R, 1 for the modern tables, and 3438 for the Siddhantic tables is to be used instead o f 120' which is instructed here. ( V M uses bhuja to mean sine, and koti to mean cosine, instead using the v/ord jyd).
Note 3. T h e method is the same as what is found in the Later Surya Siddhdnta, with some change for convenience. But i n the matter of the number or order of the steps, the Aryabhatiya and the Siddhdnta-Siromani differ. T h i s is because, correctly speaking, the first two steps are useless, and the last two steps alone are necessary. In essence, the third serves to get the true heliocentric position, and the fourth to convert the heliocentric position into geocentric. T h e earlier steps are i n the fond hope of getting correct positions agreeing with observation, while the real trouble is i n the inexact parameters followed by the Siddhdntas. Note 4. T h e second and third steps are merely akin to finding the equation of the centre and applying to the mean. T h e first and fourth steps are conversion o f heliocentric to geocentric positions, neglecting the latitude, which is small and does not affect the result much. T h e work can be illustrated thus:
Epicycle of Conjunction
Chapter XVII. Fig. 1
XVII.9
XVII. SAURA: TRUE PLANETS
305
T h e fig, is for Mercury and Venus, having the Sun as the mean planet. For Mars, Jupiter and Saturn, interchange the planet and the Sun. T h e geocentric position in both cases is mean planet plus Q . If Q i s formed below the horizontal line, it is so much negative and must be subtracted from the mean. Example 1. Find the true, i.e. geocentric, Mars at 1,20,553 days from epoch. Given: Mean Mars, already found with bija corr. 8' 9° 2' Sighra Mars = mean Sun of date = 17° 18' A p h e l i o n (apogee) of Mars assumed, y 20° (for 120° given) Epicycle o f apsis = 70°, Apsis o f conj. = 234°. First step. Anomaly of conj. = Sighra - mean = 17° 18' - 8' 9° 2' = 128° 16'. This is more than 90° and less than 180°. So, subtracting from 180°, bhujdfhsa is 51° 44', Kotiamsa = 38° 16' Bhuja = 94' 2". Koti = 74' 17" Bhuja-result = 94' 2" X 234° ^ 360° = 6 1 ' 14" Koti-result = 74' 17" X 234° ^ 360° = 48' 17" As anomaly of conj. is between 90° and 270°, this is subtractive from 120'. 120' - 48' 17" = 71' 43'^ 120' X bhuja-result 4- V 71' 43"' + bhuja-result^ = 120' x 6 1 ' 14" V 7 1 ' 4 3 " ' + 61' 14"^ = 77' 55". A r c for 7 7 ' 5 5 " = 40° 30' i/^arc = 20° 15' Subtracting from aphelion (since anomaly of conj. is from 0° to 180°), 110° - 20° 15' = 89° 5', which is the half-corrected aphelion. Second step Anomaly of apsis = 249° 2' - 89° 45' = 150° 17' This is more than 90° and less than 180°. So, Bhuja degrees = 20° 43'. Bhujd = 42' 24". Bhuja-resuU (i.e. eq. of the centre) = 42' 24" x 70°/360° = 8' 15" Arc for this = 3° 56'; Vs arc = 1° 58' This is addidve because A n . of apsis is between 0° and 180°. H a l f recdfied aphelion -h 1° 58' = 91° 43' = full recdfied aphelion. Third step T h e corrected anomaly of apsis = 249° 2' - 91° 43' = 157° 19'. T h e bhuja degrees = 22° 4 1 ' . Bhuja = 46' 17". T h e Bhuja-result = 46' 17" x 70°/360° = 9' 0". A r c of 9' 0" = 4° 18', deductive because an. of apsis is between 0° and 180°. Mean - arc = 249° 2' - 4° IS' = 244° 44' = mean corrected (for eq. centre).
306
PANCASIDDHANTIKA
XVII.9
Fourth step A n . conj = 17° 18' - 244° 44' = 132° 34'. Bhujamsa = 180° - 132° 44' = 47° 26', Kotiamsa = 42° 34'. Bhuja = 88' 20". Koti = 81' 8". Bhuja-result = 88' 20" X 234° ^ 360° = 57° 25". Koti-result = 8 1 ' 8" X 234° -r- 360° = 52' 44". As an. of conj. is in 270° to 90°, this is deductive from 120'. So, 120' - 52' 44" = 67' 16". 120' X bhuja-result 4- y/bhuja-result' + 67' 16"' = 77' 32". A r c of this taken as sine = 46° 16', additive because an. conj. is from 0° to 180°. So, corrected mean arc = 244° 44' + 40° 16' = 285° 0' = geocentric true Mars. Example 2. Find the geocentric Venus at 1,20,553 days from epoch. Given: The sighra of Venus = 2' 27° 16' 20" = 87° 16' Mean Venus = Mean Sun = 17° 18' A p h e l i o n o f Venus = 2' 20° = 80°. Epicycle of Conj. of Venus = 260°. Epicycles of the apsis = 14°. First step A n . Conj = 87° 16' - 17° 18' = 69° 58'. T h e Bhuja degrees = 69° 58'. Koti degrees = 20° 2'. Bhuja = 1 1 2 ' 42". Koti = 4 1 ' 5". ' Bhuja-result = 1 1 2 ' 42" x 260° - 360° = 8 1 ' 24". Koti-result = 4 1 ' 5" X 260° h- 360° = 29' 40". A n . Conj. is between 270° and 90°. So the Koti-result is additive to 120'. So, 120' + 29' 40" = 149' 40" Bhuja-result X 120' V 8 1 ' 2 4 " ' + 149'40'"' = 57' 20". A r c 57' 20" = 28° 33'. H a l f arc = 14° 17', subtractive to aphelion (as A n . conj. is from 0° to 180° 80° - 14° 17' = 65° 43' = H a l f corrected aphelion. Second step A n . o f apsis = mean - half cor. aphelion = 17° 18' - 65° 4 3 ' = 311° 35'. T h e Bhuja degrees are 48° 25'. • Bhuja = 89' 44". Bhuja-result = 89' 44" x 14° ^ 360° = 3' 29". A r c . 3' 29" = 1° 40'. H a l f arc = 50', subtractive, as an. conj. is from 180° to 360°. Corrected aphelion = Half-corrected aphelion - 50' = 65° 4 3 ' - 50' = 64° 53'. Third step A n . of apsis = mean-corrected aphelion = 1 7 ° 1 8 ' - 6 4 ° 5 3 ' = 312°25' Bhuja degrees = 47° 35'. Bhuja = 88' 33". Bhuja-result = 88' 33" X 14° -r 360° = 3' 27". A r c sine 3' 27" = 1° 39' , additive as an. of apsis is 180° to 360°. So, mean Venus + arc = 17° 18' -I- 1° 39' = 18° 57', is the eq. cent, corrected mean.
X V I I . 10
XVII. SAURA: TRUE PLANETS
307
Fourth step A n . conj. = .^fg-Ara-corrected mean = 87° 16' - 18° 57' = 68° 19'. MM>a degrees = 68° 19'. A:o<2 degrees = 2 1 ° 4 1 ' Bhuja = 1 1 1 ' 29". Koti = 44' 19". Bhuja-resuh = 1 1 1 ' 29" X 260° ^ 360° = 80° 31' Koti-result = 44' 19" x 260° 360° = 32' 0". As an. conj is from 270° to 90°, additive to 120°. So, 32' 0" + 120' = 152' 0". Bhuja-result X 120 4- ^Bhuja-result' + 152"' = 62' 12". A r c sine = 62' 12". H a l f arc = 3 1 ° 14', addidve as an. conj is 0° to 180°. Geocentric true Venus = 18° 57' + 31° 14' = 50° 11'. In verse 11 below, V M requires us to subtract 67' or 1° 7' constant, as blja-correction, after all work is over. So, geocentric T r u e Venus = 50° 11' - 1° 7' = 49° 4 1 ' .
Special work for M e r c u r y and Venus 10. A l l star-planets are (geocentrically) m a d e t r u e i n the above m a n n e r . B u t i n the case o f M e r c u r y , this a d d i t i o n a l w o r k is to be d o n e : S u b t r a c t its a p o g e e f r o m the sighra a n d , u s i n g the S u n ' s e p i c y c l e , find the bhuja-result a n d a p p l y it to the m e a n M e r c u r y ( w h i c h , o f course, is the same as the Sun's), w i t h the a d d i t i o n o r subtraction d o n e , as the Sun's Mw^a-result is additive o r subtractive. 11 a-b. F r o m V e n u s , subtract 6 7 ' , constant, after a l l the e a r l i e r sphuta w o r k i n s t r u c t e d has b e e n d o n e . Note 1. T h e reading ksayadhane is better as it is, and T S need not have corrected it into ksayadhanam. T h e reading budhavat is deficient by two syllables, Lalla's reading budherkavat supplies these and makes the meaning more clear. So I have adopted it. TS's emendation budhaphalavat, with phala added for the two mdtrds wanting, is not different i n meaning from budhavat. B e i n g an arbitrary rule, we cannot decide which gives the original meaning, budherkavat or budhavat. But since Lalla's reading is not defective, at least as far as the mdtrds are concerned, I have adopted it.
10a. B . o m ^ : B . ^ > f t ^ b. A2.WH?ff; B . ^ H ? ! ^ ; D-^l^"?Tt c. A 2 . ^ B . ^ d. A . B . ^ s m ; C . 1 ? T O ^ D . ^ ( ^ ) ^(t)
1 la. A . ^ : ; B . ^ b. A.B.?il«1T A ^ ^ j f e f c l ^ ; B.^^gfeR^ (B2.3.°^cR^)
XVII.11
PANCASIDDHANTIKA
308
Also, it is clear that it is not a substitute for any of the four steps because, i f so the separate epicycle for Mercury will be useless. Note 2. It is clear that the rules given here are V M ' s own, to secure, i n his opinion, better agreement with observation, because they are not given in the Ardhardtrika-paksa etc. and the original four steps are all in line with them, as also the modern Surya-Siddhdnta and the Siddhdnta Siromani. Note-3. T h e whole work of finding the true positions, especially of the star-planets is defective i n H i n d u astronomy, i n that the equation of the centre of H i n d u astronomy neglects the second, third, etc. terms, which is considerable in the case of the M o o n , Mars, Saturn and Mercury, in which last case the second term is as large as 3°. In the case of Mercury and Venus it is applicable to the Sun, instead of their Sighra which is really their mean. In the equation of conjunction, the Sun's true distance from the earth and true longitude should be used, instead of the mean distance and mean longitude, as is done in H i n d u astronomy. O n account of these defects, computation does not agree with observation, and all sorts o f hotch-potch rules are given in different astronomical works. T h e disagreement among themselves would itself show that they are beside the mark. When these defects are remedied, the third and fourth steps alone would be necessary, the third step giving the heliocentric true planet; and the fourth step converting the heliocentric position to the geocentric. Note 4. In the case of Venus, there is another kind of defect. Its m a x i m u m eq. of cent, being small, it is confused with the Sun's, and the Sun's epicycle and apogee are given to Venus also. While its aphelion position is 290°, according to modern astronomy, its apogee is given as 80°, the same as the Sun's.
Table of Heliocentric Star-planets at epoch. (For mutual comparison)
Planets
Mercury Venus Mars Jupiter Saturn
7 2 3 Modern Sidd. Siromani Later Surya Siddhdnta astronomy 151° 269° 75° 9° 122°
1481/2° 2681/2° 761/2° 9 1/2° 122 °
166°* 264° 78° 9° 1231/2°
5 6 4 Earlier Vdsistha Interpolation S. Siddhdnta Paulisa of PS in P S X V I I I oiPS 148° 267° 751/2° 8° 1221/2°
2691/2° 831/2° 12 ° 120 °
161 1/2° 2691/2° 831/2° 9 ° 118 °
For values in column (5) see their derivation in the Notes to ch. X V I I I . A l l values have been computed by me.
* This needs explanation: Perhaps the reading is sdnydsvi in Sdrya Siddhdnta 1.31, which will reduc the degrees by 12. But the commentator Raiiganatha takes it as sunyartuh.
X V I I . 12
XVII. SAURA: TRUE PLANETS
309
Table of Synodic periods of the Star-planets
Planets
Mer. 115. Venus .583. Mars. 779. J u p . 398. Sat. 378.
1 2 Mod. Astr. Sid. Sir.
3 Later Su. Siddh.
4 PS-Su. Siddh.
5 PS-Vas.Paulisa
6 Interpolated PS XVIII
7 Ptolemy
87747766 92136655 93610175 88404760 09190150
8780110 9001782 9242712 8891768 0863874
8785195 8975750 9211734 8891698 0860183
8791307 9092440 9553326 8891358 0997090
8750556 9060301 9787326 8852917 1100185
.879 584.000 .943 .886 .093
8784290 8968279 9222494 8894794 0859936
A l l values except those in column (7) have been computed by me. In column (6), the solar 'days' given have been converted into ordinary days.
Retrograde motion ] 1 c-d. T h e times f r o m the b e g i n n i n g o f the r e t r o g r a d e m o t i o n to its e n d a n d the follow u p p e r i o d c a n be f o u n d by the d a i l y m o t i o n ( b e i n g negative, d u r i n g this p e r i o d , a n d the c o n v e n t i o n r e g a r d i n g these). Note: T h e terms vakra (retrograde) and anuvakra (follow-up at the end of retrograde) are technical. They are eight in number according to the Surya Siddhdnta, given by the verse: vakrdtivakra kutild manda mandatard samdl tathd sighratard sighrd grahdndm astadlid gatihil T h e generally given reading vakrd-anuvakrd is wrong in my opinion and I have read it asvakrdatix/akrd, and atixjakrd has taken the place of anuvakrd in the verse. T h e expression yd vakrd sd'mwakragd in the next verse makes it clear. Generally the near ones are subsumed into one another. B u t in the case of Mars, V M gives all these eight and their degrees and periods. See below under X V I I I . 33-34.
[i)^
l l c - d . A.^rFng. B . ^ ? H ? R :
PANCASIDDHANTIKA
310
XVII.13
H e l i a c a l rising o f t h e p l a n e t s 12. T h e h e l i a c a l r i s i n g a n d setting o f the M o o n , M a r s , M e r c u r y , J u p i t e r , V e n u s a n d S a t u r n are w h e n t h e i r e l o n g a t i o n ( f r o m the t r u e S u n ) are 12°, 17°, 13°, 11°, 9°, a n d 15°. Note 1. I generally adopt TS's readings. B u t Sasi is extra, and evidently a mistake which has crept into the reading. T o make up for this they have removed rudra which is necessary, and this emendation has spoiled the correct agreement with other siddhdntas.
Note 2. These are time-degrees, i.e. time expressed in degrees (kdlabhdga) and are arbitrary in essence, and depend on the keennees of the observer's eyesight, as also the atmospheric conditions. T h e later Surya-Siddhdnta gives 10° and 8° for Venus at superior and inferior conjunctions, and 14° and 12° for Mercury, respectively, while the Surya-Siddhdnta here and some others give the mean of each. (The Mahdbhaskariya gives even 4° or 41/2° for Venus at inferior conj. and 8° at superior conj.)
Latitudes o f planets 13. A d d o n e e i g h t h o f itself to the R (120') sine o f ( m e a n p l a n e t - apogee), i n the case o f S a t u r n , J u p i t e r a n d V e n u s . F o r , the two others, (i.e.. M e r c u r y a n d M a r s ) , subtract o n e f o u r t h o f itself. ( T h i s is o n e p a r t o f latitude). T h e r e is a n o t h e r p a r t o f l a t i t u d e u s i n g the A n o m a l y o f c o n j u n c t i o n . 14. F r o m the R sine a n o m a l y o f c o n j u n c t i o n o f J u p i t e r , M a r s a n d V e n u s subtract o n e f o u r t h o f itself. F r o m that o f the rest, (viz., M e r c u r y a n d S a t u r n ) a d d a n e i g h t h . A d d b o t h a l g e b r a i c a l l y a n d n o t e the d i r e c t i o n , n o r t h o r south. M u l t i p l y this by R (i.e. 120') a n d d i v i d e by the h y p o t e n u s e got i n the last step. T h e l a t i t u d e is got, its d i r e c t i o n b e i n g that o f the n o t e d d i r e c t i o n . 12a. A . •^d l
c. B . w W - ^ l : ;
B.-^<(
B.^^^^:
c. A . B . 1 ^ ^ ( B . ? ; ) ^
D.fiiwJKAil: A . ^ ; B l . 2 . ^ d.
A.f^^qt^a
14a. B.'J
d. A . B . C . ^ 7 l M ? T f e A . ^ 5 ^ : ; B . ^ 6 ^ : ; C. B.W^ 13a. B.H^Idi^^l ( B 2 . ^ R ^ , B 3 . < T ^ ) b. B . M S t ? r ^ % ^ ° ; D . ^ ^ ^ ^ ^ ! { l * i u i l ' i
A.Pl-^l'l-si^Wf^^:;
X V I I . 14
XVII. SAURA: T R U E
PLANETS
311
Note 1. This is a peculiar primitive way of finding the latitude of the star-planets. It is not found in the allied Khandakhddyaka and the quoted part o f the Bhattotpala-quoted Paulisa. It is found in Aryabhata's Ardhardtrika-paksa given i n the Mahdbhdskariya (VII. 28-33). B u t there are some differences between the two, and we cannot decide which follows the original Saura here, and which has slighdy modified the original. T h e y both menuon two kinds of latitudes for each star-planet which are to be added algebraically. But there is a difference in the maximum latitudes, and in the ascending nodes to be subtracted from the mean longitudes or sighras. V M ' s Saura implies the max. latitude 90', 90', 135', 135', and 135', for Mars, Mercury, Jupiter, Venus and Saturn, respectively, to be multiplied by sine anomaly of conjunction, and 90', 135', 90', 90', and 135' to be multiplied by sine anomaly of conjunction, no separate node being given, which means that the apogee itself is the node for one kind of latitude, and the mean planet itself for the other. B u t the Ardhardtrika gives only one set of m a x i m u m latitudes for both, viz., 90', 120', 60', 120', 120°. It gives the nodes, 20°, 40°, 70°, 260°, and 150° for the former and 20°, nil, 7 0 ± , 260° and 150° for the latter, Goxdndasvdmi's Bhdsya on the Mahdbhdskariya, being meagre, does not help us. Note 2. By implication, we had better take the arguments o f the eq. cent, used i n the third step for the former, and the anomaly of conj. used and hypotenuse obtained i n the fourth step for the latter. Example: Find the latitude of Mars at 1,20,553 days from epoch. In the third step of the earlier example, the sine of the argument of eq. conj. is 42' 24". As it is Mars, deducting quarter of itself, the latitude is 3 1 ' 48", north, as this argument is between 0° and 180°. In the fourth step, the sine of the argument of conj. is 88' 20". F o r Mars one fourth is to be subtracted. So, the latitude due to this is 66' 15", again north, since the argument is from 0° to 180°. A d d i n g , 31' 48" + 66' 15" = 98' 3", north. The hypotenuse obtained there, i n the fourth step, is 88' 52". 98' 3" X 120 88' 52" = 132" north, is the true latitude o f Mars for the day. (As it is, this is far from the latitude obtained from using the later Siddhdntas,) Note 3. T h e treatment of Saura is taken up by V M i n X V I I I . 57-60, where the latitude found here is used to correct the mean elongation given i n verse 12, for the heliacal setting and rising. T h e exposition of these verses is given i n X V I I I .
Thus ends Chapter Seventeen entitled 'Saura-Siddhanta - True Planets' in the Pancasiddhantika composed by Varahamihira
C o l . : A.B.D.dKIH^^ilcti^ui bi).^!^|l5S!IFI:;
C h a p t e r
E i g h t e e n
(VASI3THA-) PAULI^A-SIDDHANTA — RISING AND SETTING OF PLANETS
Introductory
Chapter X V I I I of PS (NP's ch. X V I I ) deals primarily with the star-planets according to the Vdsistha and Paulisa siddhdntas. T h e true motion of the planets is traced from one heliacal rising to the next. T h e method of getting the true anomaly of the equation o f the centre is similar to that of the M o o n given by the Vdsistha i n ch. II and based on the same theory of the rate of motion, forming a linear zigzag. Ev.en the same technical term pada is used here. A l l these are reminiscent of the Babylonian astronomy of the Selucid period. T h e text as available too is not very pure and this too has made the interpretation of this chapter difficult for T S . In my paper on 'Some misinterpretations and omissions of Thibaut and Sudhakara Dvivedi in the PS of V M ' (VfJ.. 11 (1973) 107-18) I have indicated the errors occurring in the said publication. N P have improved upon TS's interpretation in some places, but have committed worse mistakes i n other places. T S and N P have also failed to understand how the equation of the centre has been computed and applied to the equation of conjunction of Jupiter and Saturn. Since they had to interpret, in someway or other, the related verses, they have altered the verses, i n all sorts of ways, to yield what they thought the meaning might be. Even i n the case of Venus, where computation has been made simpler by neglecting the small equation of the centre, they have committed several errors which have been pointed out in the exposition o f these verses given below.
Motion of Venus 1. S u b t r a c t i n g 174 f r o m the days f r o m e p o c h , the q u o t i e n t got by d i v i d i n g the r e m a i n d e r by 584 are the h e l i a c a l risings o f V e n u s . Its m o t i o n s d u r i n g the p e r i o d s is 7-^ 5° 3 0 ' 2 0 " each. 2. H a v i n g g o n e to 2 6 ° o f V i r g o , i.e. at, 5" 2 6 ° , V e n u s rises i n the west (for the first t i m e after e p o c h ) . A d d i n g a n e l e v e n t h o f the q u o t i e n t (given i n verse 1) to the r e m a i n i n g days, the m o t i o n s (are to be t a k e n f r o m the T a b l e g i v e n i n verses 3-5).
XVIII.3
XVIII. V A S I S T H A - P A U U S A - R I S I N G A N D
SETTING
313
174 days after epoch, Venus rises heliacally i n the west. We have emended munijala dsjalamuni because 147 will not agree with the longitude of Venus at rising given as 5' 26°. If Venus is 5' 26°, the sun must be 5' 18°, to satisfy the 8° given for Venus's heliacal risng in verse 58. But since at epoch the. sun is in the neighbourhood of 358°, it can be only near 4' 25° after 147 days, i.e., 38° from Venus. After 174 days from epoch, the Sun would be near 169° according to the VdsisthaPaulisa. (Vide my p a p e r ' T h e epoch of the Romaka etc', Indian Journal of History of Science, 13 (1978) ii. 155-58). T h i s gives the elongadon as 7° instead of the required 8°. But this small discrepancy can be put to an accumulated error in computing from the original. T S have not felt the need for this emendation because they have emended kanydmsdn into kdldrrisdn and omitted the necessary ksepa viz. the constant equal to 5' 26°, for beginning the motion. T h e y have not understood the meaning of the word kdldmsdn which they have brought i n . It is the same as the 8° mendoned above, for Venus. N P have interpreted kanydmsdn correctly. But since they have kept 147 days to be deducted intact, they find a serious discrepancy expressed by them on page 124 of Part II. However, they derive satisfacdon from the fact that the September 10th position of the sun would agree with the dme of Venus's rising and their longitudes. This at least must have shown them that the rising takes place only more than 20 days later. T h e y have made all sorts of unnecessary emendations, but they have failed to do this necessary one. We can infer from the instrucdon to add an eleventh of the quodent, that one synodic revolution takes 583 10/11 days. In this period the sun has moved 1 revolution 7' 5° 30' 20" and Venus, 2 revolutions 7' 5° 30' 20". F r o m this we can infer that the sun takes 365-15-25 days for a sidereal revolution. F r o m the methods given for the other planets also we can see that the sidereal period of the sun used is c. 365-15-30, which is an evidence for the system given here being connected with the Paulisa also, as in the case of the M o o n . T S have unnecessarily emended hhogdh into bhdgdh and taYmg sdrdhdh to mean 'together with', instead of 'with half, have given a modon of 7' 5° 20' per synodic period. T h e y are unaware that this would make the sidereal year c. 365-22 days, so wrong. T h e error of 10' 20" i n Venus would accumulate by 1° every nine years. N P interpret sdrdhdh correctly, but take gundptaih to mean 'with 1/3 degree' and given 7' 5° 50', which would make the sun's sidereal year c. 365-3-0, so very wrong. By this the error in Venus would accumulate by 1° in every 5 years. Next the days for segments of m o d o n in the synodic cycle are given by verses 3-5.
1. A . B . C . D . ^ ^ l ^ ( B . ^ ) . A . B . ^
2a. A.D.^FTO:; B . ' ^ ^ : ^ : ; . C^^ROT:
b. B . A 2 . ^ corrected tol^; B . ^ C. A . ' ? ^ ^ . C^nt^:; D.^rjTO: d. A . ^ ; B . ^ B1.3.TTsrt^; B2.M«irdHl. C.WV:;
h. A l . ' J W ; ^ ; A 2 . ^ 1 I ^ c. B . 3 ^ . C . D . ^ d. A.C.^^raW:;. D.^ra?^:
314
PANCASIDDHANTIKA
XVIII.5
N^'MiVIM^iy'dldfiltclliMidl
Days for segments o f m o t i o n i n the s y n o d i c cycle 3. I n t h r e e p e r i o d s o f 6 0 days V e n u s m o v e s 74°, 7 3 ° , 7 2 ° , respectively. I n 4 0 days it m o v e s 3 2 ° a n d i n 17 days, 5 V4° 4. F r o m h e r e r e t r o g r a d e m o t i o n (begins). I n 15 days these are 2 ° ; i n 5 days the same, i.e., 2°. T h e n , setting i n the west, it rises i n the east after ten days. V e n u s is i n f o l l o w - u p r e t r o g r a d e f o r 2 0 days, m o v i n g 4 ° . 5. T h e n c o n t i n u i n g the (direct) m o t i o n r o u n d , i n the o r d e r o f days a n d m o t i o n s r e v e r s e d , V e n u s sets i n the east. T h e n m o v i n g 7 5 ° i n 6 0 days, it b e c o m e s visible i n the west. T h e argument to be used i n the above table o f motions'are the days left over, together with the eleventh o f the quotient, as mentioned. It can be seen that i n my emendations o f some o f these I have done very little violence to the text. I have been guided i n these by the actual motion that must have been observed, putting it to observational or other error, where the numbers are clear, but deviate from the actual. T h e ratio o f Venus's distance from the sun to the earth is c. 0.72 given by all H i n d u and modern astronomy, and this I have used to compute the segments o f actual motion for comparison. (See Table on next page) Another guide is that the days and motions from the superior to the inferior conjunction must add up to half o f the whole, i.e., 292 days, and 2 8 7 % ° , since the equation of the centre has been dispensed with.
72° motion for the third 60 days is about 4° i n excess. F o r the next 40 days the motion has to be 36°, and I could have filled up the lacuna by (khaissat) instead o f (khairdvd) to get this. B u t t Siddhdnta seems to have compensated the earlier 4° excess by the 4° defect here, which of course is an error. F r o m this we can see that the emendation o f 'dvimsat' into 'trirhsat' is necessary lest motion be reduced to 26°, which is too small. 3a. A . B . C . D . * 5 P | n i B . ^ B . T W < T \%; B . W ^ ^
c. A l . s n i f e ^ f e i l f l ( A l . o m | ? T ) ;
4a. B.d^dfelldfa d. B . W r l ^ a A.feTTTlM; B . ^ C T T W ; D.fe^forfer
B. 3T%jfe^HR5;idi (B2.f%f^)
5. B . verse missing
C . srafe^^W; D . s n f e ^ l ^
5a. C . ^ t I ^ : f o r T # M
d. A.B.Mi%lf«f:^WT; C.W<1cq?lifelpT:^™^
b. Al..*l^^cJbji; A 2 . ^ ^ ^ c ^ ; C.<=tt/R'MMHWi^c4^^Hj;
XVIII. VASISTHA-PAULISA - RISING A N D
XVIII.5
SETTING
315
TABLE I M o t i o n on the synodic circle (computed) Superior conjunction 30 days Rising West 60 d. 60 d. 60 d. 271/2 d. 12i/2d. I7d. 6d.
371/2° 74 ° 73 ° 681/4° 261/4° 91/4° 7 ° 1/4°
Retrograde ends
Retrograde begins 9d. 5d. Setting West 5d. Inferior conjunction 5d. Rising East 5d. 9d.
-21/4° -21/2° -21/2°
243 d. 258 ° Setting East 30 d. 371/2° Superior conjunction Total 514 d. 5751/2°
-3° -21/2° -21/4°
Next, 11/4° for 23 days is too too small to be correct. Further, the correct total of days and degrees clearly given by the numbers will be spoiled by this. So I have given the meaning as 5 '/4° in 17 days, which fairly agrees with the actuality, by introducing a (vi) for the defect of two rmtrds and emending sapdddmsam into sapdddrdhdn. Since the motion only 15' for the 6 days near the stationary point as seen in the actual, the siddhdnta is justified in combining this with the - 2 ° 13' for the next 9 days and giving - 2 ° for 15 days. But, for the next 5 days, the motion is almost —21/2° and not —2°, and this is an observational error. For the 5 days forming die half period of invisibility till the inferior conjunction, the actual motion is about — 3° but we are constrained to make it - 2 ° for agreement with the other numbers, especially when it is left to be understood, no motion being given by the text. A glance at tHe comparative table will make everything clear. T S have made the serious mistake o f thinking that the segments given begin with the superior conjunction instead of the rising on the west (vide the scheme given i n the Sanskrit Commentary, p. 98). By the total of 610 days, they have given that the rising takes place 26 days after superior conjunction, passing 221/2°, which is absurd because it should be 371/2° i n 30 days, i.e. half the time and degrees given for the time from setting to rising. T h e y do not realise that this is the period o f the quickest motion. W i t h i n the scheme they get the 77° for 85 days, by making a drastic change in the wording of the text. Further, for 85 days i n that part, the motion would be more than 91°. Giving 11/4° for 3 days near the stationary point is wrong since it should be practically zero. - 4 ° for 5 days in retrograde is too great. B u t they have given the same - 4 ° for the 10 days i n the ativakra region where the rate should be the greatest. As for N P , they have correcdy interpreted that in the three 60-day periods after rising, the motions are 74°, 73° and 72°, that from setting i n the east and rising i n the west there are 75° for 60 days, half before the superior conjunction and half after, and that near the inferior conjunction there are 50 days of retrogression and - 1 2 ° , half o f each falling on each side of the point, as given by the text. A d d i n g these we can account for 2501/2° i n 235 days. Since we should get 2 8 7 % ° for the 292 days from the superior to the inferior conjunction, we have still to account for 371/4° in 57 days. This we must seek i n the second half of verse 3. B y some likely emendations we can secure this, as I have done. But N P have drastically changed the text as arthdstakavimsatyd vimsatyarn(sahd)stribhis sa paddmsam,
PANCASIDDHANTIKA
316
XVIII.7
also sinning against prosody, and given only 28° for the 27 V2 days, after the third sixty-day period, and 11/2° for the next 3 days, thus, not accounting for 16° and 30'/2 days. ardhdstakavimsatyd cannot mean 27 V2, besides being an un-Sanskritic formation. Further, for the 27 V2 days in that part o f the synodic circle the motion should be more than 26° and for the rwext 3 days, more than 2 V4° as can be seen by examining the actual. T h i s error o f 16° and 301/2 days is doubled for the whole cycle, and the weight of this error of 32° and 61 days has been carried by them to the 60-day period of invisibility and drawn the remark o n page 121, Part II: "a rather implausible conclusion. A t any event, the descripuon of the motion of Venus as given in our text seems incomplete." T h e footnote here is uncalled for. Jupiter and Saturn T h e computation of Jupiter and Saturn follows next to Venus. T h i s is because their treatment is next simple, o n account of their small mean motion and equation of conjunction, owing to their great distance. Both T S and N P have expressed inability to understand the part of the computation where the equation o f the centre is obtained and applied, before the application of the eq. of conjunction. Still, they have attempted to interpret the concerned verses, changing the wordings, drastically, to yield their fancied ideas. I n getting the eq. of conj., too, they have made several mistakes. As in the case of Venus, here too, the true modon is traced from one heliacal rising to the next. T h e method of getting the true anomaly of the eq. cent, is similar to that o f the moon given by the Vdsistha in chap. II, and based on the same theory of the uniform increase and decrease of the rate of motion, forming a linear zigzag. Even the same technical term, pada, is used here. A l l these are reminiscent of the Babylonian astronomy of the Selucid period, as I have stated above. As between Jupiter and Saturn, their treatment is exactly similar, so that explaining one would suffice for both. Verses 6-13, deal with Jupiter and 14-23 with Saturn. M y main aim here is to state and explain the procedure i n the computation, a thing not understood by investigators. T h e verification of the epoch constants depend mainly on comparison with other systems and modern astronomy. So this will be done separately.
35[2T=rat(?t) ^
tl4cJ4F^JH?|^(^:)
Risings of Jupiter 6. T h e days o f J u p i t e r f r o m e p o c h m i n u s 34 days, 34 nddikds d i v i d e d by 3 9 9 , give t h e n u m b e r o f risings. T h e r e m a i n i n g a r e days (after rising). 7. A d d to these days a n i n t h o f the n u m b e r o f risings. M u l t i p l y the n u m b e r o f r i s i n g s by 3 6 , a d d 18, a n d d i v i d e by 3 9 1 . T h e r e m a i n d e r h e r e me padas. We can conclude the following from these two verses: (i). A t 34 d. 34 n. from epoch, the first period from rising to rising begins.
XVIII.8
XVIII. V A S I S T H A - P A U L I S A - R I S I N G A N D
SETTING
317
(ii) T h e interval between the risings, i.e., the synodic period is 399 - 1/9 = 398 8/9 days. (iii) 391 padas make one full sidereal revolution of Jupiter, i.e., 360° o f mean motion. In one synodic period, Jupiter moves 36padas. Onepada = 55' 15". 36padas = 36° 9'. A t epochal days 34 d, 34 n, Jupiter's longitude is \8padas (= 16° 35'). But N P have taken the 18 given here as degrees. This is wrong. T h e difference o f 1° 25' is too small to show itself i n their verification. Table 32, Part II. But for Saturn this pada constant is 89, and for Mars, 85. T a k i n g these as degrees have resulted i n big differences which have puzzled them. See part II, page 124. (iv) In 391 syn. revoludons there are 391 + 36 = 427 solar sidereal revolutions = 36 Jupiter's sid. revolutions..-. one sid. rev. of Jupiter takes 4332-22-48 days, and one sid. rev. of the sun = 36515-32 days. T h e latter being very near Paulisa's 365-15-30, we conclude that, there too, as i n the M o o n , it is mixed up with the Vasistha's. T S and N P give the same interpretation, though making more than necessary emendations. In their verification, T S use the rough syn. period of 399 days instead of the correct 398 8/9, making the sid. period = 4333-35-0.
C'fmm)
^ ( ^ ) s ^ m f r f ^ : 116 I
8. O n e after a n o t h e r , m e a n a n d t r u e segments are to be a r r a n g e d . T a k i n g t h e i r difference, i f the t r u e is less t h a n the m e a n , the d i f f e r e n c e is to be a d d e d to the g r o u p o f J u p i t e r ' s days (left o v e r i n the s y n o d i c cycle as the r e m a i n i n g days). O t h e r w i s e , (i.e., i f the t r u e is m o r e ) , the d i f f e r e n c e is to be subtracted. 'True' here means 'true as corrected for the eq. of the cent'. H o w to get these true positions is given i n verses 9-11, and the segments are to be got using these. So, this verse seems to have strayed here from after verse 11. T h e mean positions are to be got by using the remaining/>ada5, extending the work done i n verse 7. Karyau is introduced to make up for the syllables wanting. Saure would mean 'pertaining to either Sun or Saturn'. But we are dealing with Jupiter Surih here. Saurya alone would mean, 'pertaining to Jupiter'. Some mss. have no dvi. C o m p u t i n g and arranging the mean and true segments against each other is to facilitate interpolation to any required day. It will also be useful to prepare an ephemeride. T h e example worked will make things clear. 6a. A l . t ^ A 2 . f ^ C . D . i r ^ . A.B.tsipr b. A . B . C . D . ^ i m t f ^ . A B . ^ :
c. B 2 . 3 . U n i n d i c a t e d o m . o f ^[^....iT)^;Mlin7d. d. A l . - g ? ^ ; A 2 . 3 ^ ; B . o m . l i n e . A . ° f i « T i ^ : 7a. A . B O T T T ; B.a-c. missing. C.D.3™?H^ b. C . D . ' g o t ^ . A . ° ^ : c. A . C . D . f e ^
d.
A.^^;B.°MT:.BI.3.^RI5^
8a. A.B.IISvTOT; C . t : ^ ^ ; D.fl;[fl?T:]a5rR]l A . H<=qta?
PANCASIDDHANTIKA
318
XVIlI.ll
T S have expressed doubts about their translation, since they have not understood verses 9-11. They have retained hi, not supplied the wanting mdtrds, and not noticed the grammatical error in Saure. N P have made three drastic emendations, quite unrelated to the lettenng of the text, mhitah:, maridalah: and tanmadhya-kharide, though generally following T S .
^ 1 3 T s r f : ch'^iVII: ^rST^^atil [ ^ d l ' M ^ u ^
(^:)
9. J u p i t e r b e i n g i n the d i m i n i s h i n g - m o t i o n - s e c t o r u p t o 180 padas, there is the constant 1456 (to w o r k w i t h , i n o r d e r to get the eq. cent-corrected-Jupiter). B e i n g i n the i n c r e a s i n g - m o t i o n - s e c t o r i n the n e x t 195 padas (i.e. 181 to 375), t h e r e is the constant 1165. 10. J u p i t e r b e i n g i n the d i m i n i s h i n g - m o t i o n - s e c t o r (again) i n the next 16 padas, t h e r e is the c o n s t a n t 1486. ( A f t e r s u b t r a c t i n g o r a d d i n g the padas for w h i c h we w a n t c o m p u t a t i o n f r o m these n u m b e r s , i n the respective sectors), m u l t i p l y i n g t h e m by the padas a n d d i v i d i n g by 24, m i n u t e s of arc are got, (as the eq. cent, c o r r e c t e d total m o t i o n i n the respective sector) at the r i s i n g i n the east ( a n d also thereafter i f w a n t e d ) . 11. T h e total o f s u c h m o t i o n o f J u p i t e r i n the first sector is 5'^ 9° 3 0 ' . I n the s e c o n d sector, it is 6 ' 4° 1 0 ' . Briefly expressed as formulae, the eq. cent, corrected Jupiter is given by: i. li padas are from 0 to 180, (1456 -padas) x padas' h- 24.
ii. li padas are in the next increasing sector, i.e. from 181 to 375, (1165 -h padas) X padas' ^24+ 9° 30', where the padas used are those given i n that sector.
9a. b.
B.m B.Hapl.om:W^[^«JcR:....^]
d. A . B . C . D . ^ : . A . B . I . 2 .
^(lOb) c. A.T^;
C.D.T^
d. A . M<^H<=lId
lib. A . B 2 . 3 . ^ ^ : c-d. B l . 2 . 3 . missing. c. A . w « ! ^ T ] i T O ; c w ^ ^ T j i r m : ;
10b. B.-?te?I% c. A.B.C.WjfnT?f; D . w f ^ F j l ^
d. A . ^ ? T ^ ; ( ' . . W [ ^ ; D . c ^ ^ ^ T ^
XVIlI.ll
XVIII. VASISTHA-PAULISA - RISING A N D
SETTING
319
iii. If the padas are in the next following sector, i.e. 376 to 391, (1486 - padas X padas' 24 + 5' 9° 30' + 6' 4° 10', where the padas used are those gone in that sector. T h o u g h the instructions are laconic, comparison with the Moon's computation makes things clear. T h e increasing-motion sector is obviously the 180° from apogee to perigee, where the rate of motion is supposed by this siddhdnta to increase uniformly from a m i n i m u m to a maximum. T h e apogee is at 180padas (= 166°) and the perigee at 576padasi= 345°). T h e last IGpadas, condnued by the first 180 padas form the diminishing half circle where the rate of motion diminishes uniformly from the perigee to the apogee. Differendadng the formula, (constant + pada) pada'/24, the increase or decrease i n the rate o f motion is found to be 2'/24 = r / 1 2 per pada. T h e r e may be a small hiatus at the junction, apogee and perigee, owing to the unequal division o f 391 into 196 and 195, to avoid halfpada. B u t the average of the rates at apogee and perigee, (1165' and 1486')/ 24 = 55'V4, agrees with the mean motion forming one/?a(ia. (Incidentally, thisjustifies our amendment of visayarasond into visayarasesdh. T h e r e are other justificauons also, as we shall show later). Further, the first sector being a condnuity of the third, the rate d u r i n g the iirstpada i n the first sector must follow next to the rate d u r i n g the \&th pada o f the third sector. Since 1486'/24 is taken as the motion of the firstpada, the motion of the 16th is (1486-30)/24 = 1456'/24. T h i s must be the commencement o f the third sector, and this is what is given. We can also see that the fastest rate, (at perigee), is 1486'/24 = 62', and the slowest, 1165'/24 = 48' 1/2, (at apogee), giving the mean value 551/4, of the pada, already found. B u t the rate for the \96th pada, ending which there is the apogee, is, (1486 - 195 X 2)'/24 = 1096'/24. B u t the m i n i m u m motion falling at apogee is given as 1165'/24. This hiatus'must also be due to the fact that the 2'/24 increase i n the rate per pada is only approximate, and the actual is a little less than 2'/24. B u t the formulae are so given that the total of the first sector is (1456 - 180) 180'/24 = 5' 9° 30', as given. T h e total o f the second sector is, (1165 + 195) 195 '/24 = 6^ 4° 10'. T h e total of the third sector is, (1486 - 16) 16 '/24 = 16° 20'. These add up to 12 rdsis, exactly, as they should. Incidentally, this justifies my emendation of gundrnsdh into yugamsdh, pdncaguriite into padagunite, and giving the meaning o f tryastaka as 3 X 8 = 24. T h e justification for correcting vidhrtayah into khadhrtayah to get 180, and rasond into rasesd to get 1165, are also reinforced by this perfect agreement found here. T S and N P also give khadhrtayah:, seeing the reason for that. TS emend rasondh into rasend (= 1265), which will give the total 6' 17° 4 3 ' , far from the correct 6' 4° 10'. T h e text itself gives 6"^ 3° 10', one degree off. T S give 6 rdsis exactly, not knowing the peculiarity of this Siddhdnta. Using cakrdrdhe thus, they are left with gundsah dasa ca kuldh. T h i s they interpret as 13° (wrongly, for it can mean only 30 or 103). E m e n d i n g dasa ca kaldh into parasakale, they say that this 13° is the total motion of the third sector. T h e y do not realise that the 16 padas of the third sector is near perigee, and the total motion must be greater than the mean motion, 14° 44'. N o t knowing the nature of the method here, they think that the total of the diird sector also should be given. It has no use, and Varahamihira has not given it. About pdncaguriite tryastakabhdjite, I have emended pdnca into pada, to delete the one mdtrd in excess, and to give the agreement already seen, tryastaka is 24, as already said. T S retain thepafica, but emend tryastaka into astaha, making it 5/8, leading nowhere. As for N P , they generally follow TS's emendations. But, for the divisor 8 they suggest the alternative 83 {tryastaka). Unlike T S , they realise that the three sectors must add upto \ 2rdsis and make their own emendation of the last part of verse 11, as dvigundrnsd dasa sadald, interpreting it as 20° 30'. N P have given the gist of verse correctly, but making a lot o f unnecessary emendations. They have wondered i n Part II, why such small units, as padas, have been taken. This is because, they
24
PANCASIDDHANTIKA
320
XVIII.13
em to think, that the three sectors are each taken wholly to get intermediate values by interpolation. A n examination of the total of each sector would show how wrong it would be. T h e true eq. cent, corrected Jupiter is given for the end of any pada we want. W e are expected to use these to get the true motion through any segmentation o f the total padas, for correct interpolation, and the ends of the segments may fall anywhere, hom pada 0 to pada 390. Therefore the smaW pada segments are used. I shall work out an example at the end to make everything clear. I shall explain the rationale of the instruction i n verse 8, of adding or subtracdng the difference. T h e eq. cent-corrected Jupiter is subtracted from the Sun to get the anomaly of conjuncuon. So, a positive eq. cent, means less anomaly o f conjunction. T h e days left over represent the anomaly of conj. with the 399 days of the synodic period, corresponding to 360° o f anomaly. So the day is taken as roughly equal to the degree of anomaly, and th e difference in degree subtracted. Vice versa for the eq. cent, corrected Jupiter, it being less then the mean. Varahamihira is too astute to confuse day and degree, as N P think. (In verses 64-81 too, there is no confusion i n the author's mind, as N P seem to think. T h e r e he has deliberately chosen the time taken by the Sun to move one degree as the unit o f time, and call it 'day', for convenience. T h i s is patent on the face of the synodic periods given, though T S have not even seen it, and are perplexed. We have reason to think that verses 6481 are by somebody else).
I «5^wfd : I 12. B y 6 0 days, ( J u p i t e r moves) 12°, b y 4 0 days 4 ° a n d by 24 days 2 ° . B e c o m i n g r e t r o g r a d e , by 56 days he moves 6° (i.e. - 6 ° ) a n d b y 60 days, 6° (i.e. - 6 ° ) . 13. F o l l o w i n g after r e t r o g r a d e , he m o v e s 12° i n 80 days, a n d 9° i n 48 days. T h e n setting, s t a y i n g so for a m o n t h p l u s o n e day, he c l e a r l y rises m o v i n g 6° 8'. E n d s J u p i t e r . T h e Scheme given Days Rising east Degrees
60 12°
40 4°
24 2°
12a. A . B . ^ ^ ^ ; C.D.^t^^im b.
A.B.C.D.^:
56 60 -6° -6°
80 12°
48 Setting wast 9°
13b. A.B.c|*1vil^+?(B.?Rf)^ b. A . B . f ^ ' ^ ;
c. B.'^yiiKi^'i d. A . B . q ^ ^ : . B . ^ ^ B . combines with the next verse.
31 Rising east = 3 9 9 6°8' = 33°8'
C . ' ^ ' ^ r ^ ; D.f^'^m^
D . ^ [^] crat c. B . f e ^ . D . f ^ d.
A.B.^?53l^WrR(B.crR)
XVIII. V A S I S T H A - P A U U S A - RISING A N D
X V I I I . 13
SETTING
321
These values agree well with actualities, considering that whole days and whole degrees are given, excepting the last 6° 8', given to complete the value for the synodic cycle. 6° 12' would be better at that region and for the whole number, 399 days. 56 days for — 6°, and 60 days for the same - 6° must be explained by the intention to give whole degrees and segmentation. Vargdk is an obvious mistake for bhdgdn, and so corrected. T S have interpreted saptdsiakena to mean 15, which such an expression never means. It can mean either 56 or 87. T h e y understand another 60 days by the word ca used. A l l this, to make u p the wrong scheme used by them, based on the mistaken idea that the statement of motions here begins with conjunction and ends with the rising i n the east after the next conjunction. T h e following is their scheme: Days Conjunction Degrees
60 12°
40 4°
24 2°
15 0°
60 60 -6° -6°
80 12°
45 Setdngwest 30 Rising east 9° " (15°)
=414 =42'
ddinardhamatena is emended by T S into dhyundrdhasatena but how can this word mean their 45? As for the last part, sthitvd saikam mdsarn, they have taken it to mean 30 days instead of the correct 31 days. Let that be. T h e y have not given any motion for it i n their interpretation. It cannot be left to be guessed and completed by an ordinary computer. T h e y , who can be expected to know, have guessed, quite wrongly, 15° motion for 30 days, not realising that it can be only 6° and a few minutes more. For the 414 days from conj. to the rising after the conjunction, the total can only be about, 33° 9' + 3° = 36° 9', and not the 42° given by them. As for N P , they have emended ddinardhamatena into dindrdhasatena to mean 50 days. Since they take 30 days for the setting i.e. one day less, they make the total o f days, 400. They give 7° m o d o n for the 30 days (which they make even 29 days in the last part). T h e y have changed the wording to some ununderstandable form here, dyavantye mdsasya. Further, the 7° is far too much for 30 days. But there is no 7° in the text. T h e y have corrected the text saikam into svam, thinking that asvam in bhutasarikhyd means 7°. Incidentally, one other matter may be considered here, viz., the degrees of heliacal rising, for Jupiter. D u r i n g the set-period of 31 days, the sun moves about 30 V2 degrees, and Jupiter, about 6° 8', and the relative motion is 3 0 ¥ 2 ° — 6° 8' = about 24°, from setting to rising. This gives about 12°, for the heliacal rising of Jupiter, which is fairly accurate, especially for very high latitudes. (Classical H i n d u astronomy gives 11°). Verse X V I I I . 58 gives the Vasistha-Paulisa's degrees o f heliacal rising as 12°^ 14°, 12°, 15°, 8°, 15° from M o o n onwards, by candrddindm dvddasamanuravitithyastaiithisamkhyaih:. 15° for Jupiter given here is too much, and 14° for Mars is too low. (Classical H i n d u astronomy gives 17° for Mars). So, the scribe seems to have made a small change i n the order, and the correct order is "candrddindm dvddaiatithimanuravyastatithisafikhyaih:", 12°, 15°, 14°, 12°, 8°, 15°, with only one change of place. Example: Find the True Jupiter at 2415 days from epoch. (i) T h e beginning of the first cycle after rising next to the epoch is 34-34 days later. The days after this, required to find the number o f cycles gone = 2415 - 34-34 = 2380-26. Dividing by 399, cycles gone = 2380 - 26/399 = 5, with 385-26 remainder. A d d i n g 5 X 1/9 days, (= 0-33), we have 385-59 days left over after 5 cycles gone. (ii) T h e padas at 5 cycles gone = 18 -I- 5 X 36 = 198. Mean Jupiter = 198 padas = 198 X 360°/391 = 6'^ 2° 18'. True Jupiter:F o r t h e 198 padas, 180/>adas forming the first sector has gone and 18/?a
PANCASIDDHANTIKA
322
XVIII.15
. - . T r u e j u p i t e r = 5'^9°30' + (1165 + 18) 18724 = 5^ 9° 30' + 14° 47' = 5^24° 17'. E q . cent. = T r u e - Mean = 5-^4° 17' - 6'^ 2° 18' = - 8 ° 1'. (iii) T h e padas at 399 days i n the cycle, i.e., the beginning of 6 cycles gone = 198 + 36 = 234 = 180 + 54. Mean Jupiter = 234 X 360 - 391 = 7'-5° 27', T r u e Jupiter = 5^ 9° 30' + (1165 + 54) 54'/24 = 6' 25° 13' T r u e - mean = E q . cent. = 10° 14'. Eq. cent, at 0 day o f 6th cycle = - 8 ° 1' Eq. cent, at 399 days of 6th cycle = - 8° 1' E q . cent, at remaining days (385-59) = = (385-59) X - 2 ° 13' -r 399 -f - 8 ° 1' = - 1 0 ° 10'. (iv) Trwejup. is less than Mean J u p . by 10° 10'..-. days of Anomaly of Conj. = 385-59-I- 10-10 = 396-9. (v) T r u e an. of conj. = for 60 days for 40 days for 24 days for 56 days for 60 days for, 80 days for 48 days Total 368 days for 28-9 days 396-9
+ 12° + 4° -1- 2° - 6° - 6° + 12° -f- 9° + 27° 5° 32' 32° 32'
28-9 = J j ^ x 6 ° 8 ' = 5° 32'
(vi) T r u e J u p . = Mean J u p . at 0 day o f A n . of conj. -I- eq. cent. -I- true ano. of conj. = 6^ 2° 18' - 10° 10' + 32° 32' = 6'^ 24° 40'. Note 1: T h e need for interpolating the eq. cent, to the remaining days in the cycle can be seen by working for 399 days o f the 6th cycle and 0 day of the 7th cycle and comparing. T h e y must be the same. Note 2: T h e eq. cent, is computed for 0 day of each cycle, i.e., for intervals of 36 padas = 33° 9'. Interpolation using these, as we have done, can be only rough. T o get better interpolations, we can divide the 36 padas into desired segments, find the eq. cent, of each, and use. We can form an ephemeride, giving the values at the ends o f the day segments given, 60, 40, 24, etc. and use for interpolation. A l l these logically follow from the instructions, though not specifically stated. [VllH-dK:]
XVIII. V A S I S T H A - P A U L I S A - R I S I N G A N D
X V I I I . 18
SETTING
323
M o t i o n o f Saturn As already has been said, the treatment o f Saturn is similar to that of Jupiter. So there will be little need for fresh explanations. 14. R e g a r d i n g S a t u r n , 150-20 days are to be s u b t r a c t e d f r o m the days f r o m e p o c h . T h e s e b e i n g d i v i d e d by 3 7 8 , the r e m a i n d e r are the days f r o m the r i s i n g g o n e , the q u o t i e n t b e i n g the n u m b e r o f risings g o n e . 15. O n e t e n t h o f the r i s i n g s , (i.e., the q u o t i e n t ) , i n days, is to be s u b t r a c t e d f r o m the r e m a i n d e r . T h e n u m b e r o f r i s i n g s got is to be m u h i p l i e d by 9, a n d d i v i d e d o u t by 2 5 6 . T h e r e m a i n d e r / ? Z i « 89padas f o r m (the padas r e q u i r e d f o r u s i n g i n the c o m p u t a t i o n ) . ( T h e i d e a is that 8 9 is to be a d d e d to ( q u o t i e n t X 9), a n d t h e n d i v i d e d by 2 5 6 , to find the padas f o r use). In (15), I have emended safigunad and rudayat into safigundn and rudaydn to agree with bhajet requiring accusatives; so also N P . B u t T S have kept them. In NP's emendation dinddydptam, dptam does not agree with the w o r d sthitd and, the meaning also is redundant. Both T S and N P have emendedpadaih intopade, thinking that navdsitih is degrees. Even this they doubt, as seen in the translation, because as mentioned by them i n Part II, page 124, it had led to disagreement. Padaih, as it is, clearly, says that the 89 is padas. So is the 18 of Jupiter and the 85 of Mars. We understand from the instructions that the synodic revolution o f Saturn takes 378710 days, that in one synodic revolution Saturn moves 9padas, that 2.b6padas make nine sidereal revolutions of Saturn, that there are 256 -I- 9 = 265 sidereal revolutions o f the sun i n 256 synodic periods of Saturn, and that at 150-20 days from epoch, Saturn's mean longitude is 89padas. ( N P give i n their translation, "89°?", as mentioned already. Therefore, one sidereal revolution of Saturn takes 3787 10 x 256 9 = 10754.84 days. O n e sid. revolution o f the sun = 378710 x 256 + 265 = 365-15-32. A g a i n , the Sun's sid. period got is Paulisa's. Onepada = 3607256 = 84' 22".5. T h e motion i n one synodic revolution = 9 X 84' 22".5 = 12° 39' 22".5. Mean Saturn at 150-20 days after epoch = 89 x 8 4 ' 22".5 = 125° 9'.4.
^ ( ^ )
% o m ^ 2 F ^ : ? T ( ^ ) ^ :
||
\\
^ t e y r ^ t w ^ q r m ^
14a. A . ' ^ M ; o m . the two letters.
15a. A.B.^31111^. B . ^
b. B."mMH
b. B . ^ ° . A . B . C R ^ ^ ^ ^
c. A.B.JJuil*^*^^?:
d. C . D . ^ ^ . A . ^ R f s i R t a ; B.iRlclRftcqi
d. A.B.D.ft«m. A.B.^^TOTO;
D.RhI€JM
PANCASIDDHANTIKA
324
XVIII.18
16. R e g a r d i n g S a t u r n , there is a n increase (of the rate o f m o d o n ) f o r t h i r t y padas, f r o m 2 4 1 6 . T h e n , t h e r e is a decrease f o r 127 padas f r o m 2 5 1 9 . 17. N e x t t h e r e is a n increase ior 99 padai f r o m 2 0 3 7 . T h e a m o u n t o f decrease a n d increase are by the padas m u l t i p l i e d by 2. T h e d i v i s o r o f the total m i n u t e s is 2 7 , its m u l t i p l i e r b e i n g o n e . 18. T h e total o f the first sector is T 15° 5 1 ' a n d the total o f the m i d d l e sector is 5^ 2 7 ° 3 4 ' . Note: T h e multiplication by one is unnecessary, but given to clear the doubt that may arise by the instruction to multiply the padas by two for subtraction and additions coming before. T h e meaning is clear, and no material change has been needed. I shall give what is given in the form of formulae:
The total motion upto any pada i n the first sector, viz., (1-30) = (2416 -I- 2 X padas) padas -r- 27, in minutes. That in the second sector, viz., (31-157) = (2519 - 2 X padas)padas 4- 27, in minutes. That in the third sector, viz., (158-256) = (2037 -I- 2 x padas) padas -^ 27, in minutes. T h e total of the whole of first sector given. T 15° 5 1 ' can be verified thus: (2416 + 2 X 30) 30 - 27 = 2751' = r 15° 5 1 ' given. T h e t o t a l o f the whole second sector = (2519 - 2 X 127) 127 + 27 = 10654' = 5^ 27° 34', given. Being unnecessary, the total of the third sector is not given. B u t we can calculate it and use it to see i f all those add up to 12 rasis, as necessary, and this will verify every instrucdon given. The total o f the third sector = (2037 -I- 2 X 99) X 99' -h 27 = 8195' = 4^ 16° 35'. N o w , V 15° 5 1 ' + 5'^ 27° 34' 16° 35' = 12^ E x a m i n i n g the constants, we find that the m a x i m u m motion per pada is 2519' 4- 27 = 93'.3, T h e m i n i m u m rate is 2037' 4- 27 = 75'.4. T h e mean rate is = 84' .35 as already found, as the mean motion equal to the pada. Differentiating as before, the increase or decrease i n the rate is 4'/27. Actually it is slightly less than this, the mukiplier being slightly less than 2, given. (2037 -I- 4 X 98) = 2416 shows this. T h e perigee falls at end oiZQpadas, i.e., V 14°, and the apogee, 127padas later, at 7'1.3°. T h e instruction how to use the result o f these verses has not been given, because it is the same as that given in verse 8 for Jupiter. Indeed, the un-emended reading 50t^re there means, "with reference to Saturn".
c. 16a.
B . ^ ^ . A.D.^W
c. A . C . D . W f l ; B-m^ d. A . B . C . D . ^ T ^ : 17a.
A.B.C.D.-smi
b. A.WRRlfeER; B.HciHcidfelNH
A.^;C.D.f^:
c-d. C D . fejjui^d»*^*'ju|y: d. A . B . " ? ! ^ ^ : 18a. B . ^ W b. A.^?3^:; B.Tsm c. A.B.C.D.151^:. A . B . t ^ T :
XVIII.20
XVIII. V A S I S T H A - P A U U S A - RISING A N D
SETTING
325
As in the case o f Jupiter, here too T S and N P have not understood what exactly is given in these verses, how it is got by applying the three formulae, how the eq. cent, is got, and why the instrucdon to apply this to the days remaining is given, i n the manner said. So, their emendations o f the readings, done without knowing the subject matter, need not be taken seriously. T S have emended the correct dvigwnapadaih: into dvigunahrtah, meaning "divided by 32", applied to the risings and not to the number got i n the formulae. N P have kept the reading, but given the translation as, "There is a subtraction or addition o f 12 degrees and minutes, (i.e., 12° 12'). Multiply by 31 and divide (the product) by 32 (or by 32padas). (The result is) Saturn's rising." Where is 12° 12' mentioned? T h e y take the 32, not as a number, but as a segment o f longitude equal to 32 padas, i.e., 45°. A g a i n , how can this give the risings? A n d the risings have already been given i n verse 14. A l l these show that they do not understand what is said.
19. S a t u r n (moves) 3 ° i n 36 days, 3 5 ' i n 7 days, 8 0 ' i n 16 days, a n d 2 2 4 ' i n 5 6 days. 20. T h e n b e c o m i n g r e t r o g r a d e , h e m o v e s 3° i n 55 days, a n d 4 ° i n 6 0 days. T h e n , f o l l o w i n g u p direct, h e m o v e s 8° i n 112 days, a n d setting, h e m o v e s 3 ° i n 3 6 days i n t h e set p e r i o d , (i.e., rises i n t h e east after that). Ends Saturn. This is the scheme given Days Rising east Degrees
36 7 16 56 Retrograde 3° 35' r20'3°44'
55 60 Direct -3°-4°
112 Setting west 8°
36 Rising east 3°
=378 = 12°39'
I shall now discuss the values given, justifying the three emendations. 1 have made. T h e corrupt sadhratdstrirmrnsdn has to be emended as 3° for 36 days, considering the position, and the fact that it must practically be equal to the rate between setting and rising, 3° for 36 days. T h e days must add upto 378 days from rising to rising, also as from conjunction to conjunction. A l l the numbers for 20a. B . ^ « p 19a.
A.B.^e?cn:; C.^a'iSR^. D.qR#n:
a-b. C^aiKI^^I^Ndid:
b. A.B.^#iiwrj; c.f^^Rn. D .
A.I?R?IPi; B.f|:T?n^-Ai.fcng;ilk:;
c. A . B . D . f ^ T ^ H a g ^ :
B1.3.fK5Rk:
c. A . B . a i * ; c . a w n i H d. A . B . C . D . N^
c. A . - ? ! ^ ; B l . - ? I c ( i # ( B 2 . ^ ) ; D.-?T^ d.
A1.2.'t^||B1.2.^?^^ref%^i
XVIII.20
PANCASIDDHANTIKA
326
days are clear. Therefore, the days for the second segment must be 7. So I have emended raanM into muni. T h e modon given there, 28', gives the rate 2', too absurd for that position, i f the original 14 days are accepted, and it cannot be that the Siddhdnta does not know the absurdity. Even for the emended 7 days, it is too low, being only 4' rate, while the rate on both sides is 5', and also consistent with facts. Therefore, scaturgund is emended into scesugund. N o w , these three segments can be combined into 4° 55' for 59 days, without affecting the result. I do not know why the Siddhdnta has broken it into such bits. Next, the total for the 378 days must be the mean motion for the period, i.e., 9padas, equal to 12° 39'.4, roughly taken by the Siddhdnta as 12° 39'. Therefore the motion for the fourth segment, 56 days, must be 224'. So I have emended dviguna into vedayama.
In the case of Saturn, too, as i n the case of Jupiter, T S and N P have thought that the unnecessary total motion for the third sector has been given. F i n d i n g no wording answering to that, they have changed drastically the first half of verse 19, and obliterated the first two segments of days and motion. T h e y have emended the half verse into khandentye simhasdmunayo liptdscaturgundssapt if they are writing their own book. This means, in the last sector the total is 4' 7° 28'. But even diis does not help to get 12 rdsis, the total coming to only IV 20° 53'. W i t h the other half and the next verse, they make up the whole scheme as:days 16 56 55 68 60 105 36 =396 motion 4-3° -H232' +4° -3° -4° +8° +3° = 15° changing astii into arrmgnin; not giving any word for the motion of 4° in 55 days, but simply putting the motion there; trinamsdn into astarasaistrin, newly introducing 68 days, and giving it the retrograde motion —3°; and arkasatena into arthasatena to mean 105 days. As i n the case of Jupiter, they trace the motion from conjunction to the rising after the next conjunction, taking 396 days. But the total motions must then be, 12° 39' -I- 1° 30' = 14° 9' and not 15° given. T h e y must know that 3°.for 16 days, giving the rate 11V4' per day is very much wrong, when the rate is only 5' for the nearer segment got from the motion 3° for 36 days.
As for N P , they emended the first half of verse 19, asparihindh strikhdrnsd manubhirliptdscesugun sapta meaning "Zero degree of V i r g o diminished by 14°, plus 35' ", i.e., 4' 16° 35'. T h e y are here better than T S because they have seen that the aim should be to get the total o f 12 rdsis for the three sectors combined. T h e y have also kept closer to the lettering o f the text, though the manner in which they have got their total for the third sector is far-fetched. After this, they follow the text without changing it. O n l y at the end they interpret that the motion o f 3° for 36 days comes before the setting, and leave the period set without any days or motion given. T h u s , their scheme is: days Rising east motion
16 Retro80 grade
56 55 60 Direct 232' - 3 ° - 4 °
112 36 Setting 8° 3° west
? Rising ?east
Total 378 Total 12°39'
T o make u p the totals, a motion of 3° 27' for 43 days has to be given. B u t it must be at least 3° 35'. For the 43 days of the set period, the sun's motion is 42° 20'. Therefore, the degrees o f Saturn for heliacal rising comes to (42° 20' - 3° 27') + 2 = 1 9 ° 26'. T h i s is far greater than the 15° given in verse 58, and also in all Siddhdntas. Further, the opposition must occur at the middle of the period from rising to setting and also the middle o f the retrograde period. T h e one falls 160 days after rising, and the other 130 days after, as great as 30 days off. I am sure N P have noted all these discrepancies, but have given them as they understood the wording, just to mark time.
XVIII.20
XVIII, V A S I S T H A - P A U L I S A - R I S I N G A N D
SETTING
327
I shall now give an example, to make the method clear. Example: Find true Saturn at 5000 days gone from epoch.
i. Days T o be subtracted Divided by
5000 150-20 378
4849-40 313-40
Days to be deducted — : 12
(12 = full cycles gone) = (Remaining days)
1-12
312-28
(corrected remainder)
89 -f 1 2 x 9 ii. Padas at 0 day o f the 13th cycle: — = 197 remainder Mean longitude = 197 padas = 9' 7° 2' 197 = 30-1-127 + 40 (in the third sector) Eq. cent, corrected mean longitude:= V 15° 5 1 ' + 5"" 27° 34' + (2037 + 2 x 4 0 ) 4 0 ' = r 1 5 ° 5 1 ' + 5'^27° 34' + r 2 2 ° 16' = 9 ' 5 ° 4 1 ' Eq.cent. = 9^ 5° 4 1 ' - 9"^ 7° 2' = - 1 ° 2 1 '
27
iii. Padas at 378 days gone in the cycle = 197 + 9 = 206 Mean longitude = 2()Q padas = 9' 19° 41' 206 padas = 3 0 + 127 + 49 (in the third sector) E q . cent, corrected mean longitude = V 15° 5 1 ' + 5^ 27° 34' + (2037 +2 X 47) 47'/27 = 9"^ 18° 0' Eq. cent. = 9 M 8 ° 0' - 9 M 9 ° 4 1 ' = - 1 ° 4 1 ' Interpolated for days 312-28, the eq. cent = - 1 = 2 1 ' - 0 ° 17' = - 1 ° 3 8 ' . iv. Correcung the remaining days 312-28 by this, 312-28 + 1-38 = 314-6 days, to be used to find anomaly of conjunction. V. 36 days 7 ... 16 ... 56 ... 55 ... 60 ... Remaining 84-6 314-6
+3° +0°35' + r20' +3°42' -3° -4° +6°1' + 7°40'
M e a n Sat. at 0 day 9-^ 7° 2' Eq.cent - r38' An.ofconj. +7°40' T r u e Saturn =
9 ^ 3 ° 4'
As per Ephemeris: 9' 11 °.7 (sdyana)
Mars As indicated earlier. Mars, like Mercury, needed elaborate treatment owing to certain peculiarities about it, and so had been reserved by V M to the end of PS. T h e synodic period of
PANCASIDDHANTIKA
328
XVIII.24
Mars, on which the equation of conjunction depends, is 780 days, d u r i n g which there are more than two revolutions of the Sun, and one revolution of Mars, so that one full anomalistic period of Mars is contained within this period. This, with the large equation of the centre, and the large equation of conjunction causes large variations i n its motion from sign to sign, and even in the same sign, according to the different types of motion governed by the anomaly of conjunction, like, fast, slow, retrograde etc. Hence is the need for detailed treatment. Further we have reason to think that the various motions given are all empirical, based on long observation, synodic period after synodic period. T h e separation into the equation of the centre, and the equation of conjunction is yet to come, it seems, unlike the cases of Jupiter and Saturn, where it is easy, and done. T h i s would explain discrepancies found i n the values given. Regarding the constants given, some can be verified by mutual comparison, and corrected where necessary, when there is a doubt about the reading itself. But some, like the epoch constants, which are peculiar to the Siddhdnta itself, cannot be so verified and corrected when there is a doubt. Only in such cases, where we can argue that no Siddhdnta is likely to give such wrong values, and when these values are so far from the real, that we can make some plausible corrections. TS and N P have not understood the nature of the motion of Mars, Jul . as they have not understood Jupiter and Saturn. While T S have not even attempted translating some verses, wrongly interpreting those attempted, N P have attempted translating all, but many wrongly. I shall point out these after my own translation and discussion o f the verses, step by step.
< l ( V I ^ * 4 I U | d i S ( ^ ) , ^ ( W i g ) W T ^ ^ II
I
Motion of Mars 21. 780,
S u b t r a c t i n g 256-40-0 days f r o m the days f r o m E p o c h , a n d d i v i d i n g by the s y n o d i c r i s i n g s o f M a r s i n the East are got.
22-23. (157 plus 4) vinddis, m u l t i p l i e d by the risings got, are to be a d d e d to the r e m a i n i n g days. M u l t i p l y the risings got by 18, a n d a d d i n g 8 5 , d i v i d e by 133. T h e r e m a i n d e r , c o n v e r t e d i n t o rdsis is M a r s at r i s i n g . A c c o r d i n g to the w h o l e or p o r t i o n s oi rdsis, the t r u e m o t i o n s are to be t a k e n o n e after a n o t h e r , a n d p i e c e d together.
XVIII, V A , S I S T H A - P A U L I S A - R I S I N G A N D
XVIII.24
SETTING
329
24. T h e difference b e t w e e n the m e a n a n d t r u e degrees s h o u l d be a d d e d (to the r e m a i n i n g days got i n 21), i f the m e a n is greater. I f the m e a n is less, the difference s h o u l d be subtracted f r o m the r e m a i n i n g days. T h i s d o n e , I s h a l l give the t r u e m o t i o n s , a c c o r d i n g to each type o f m o t i o n : We learn from the verses the following: 1. 256-40-0 days from Epoch, Mars rises in the East, after which the counting of risings begin. 2. One synodic revolution takes 780 days — 161 vinadis, (i.e. 779-57-19 days). T h e addition of vinadis multiplied by revolutions, is for taking the synodic period as 780 days approximately. 3. Forthisperiodof779-57-19days,weget 1 -I- 18/133siderealrevolutionofMars,and2+.,18/133 sidereal revolutions of the Sun. So, i n one synodic period. Mars moves 408° 43'.3 4. In 133 syn. periods = 1,03,734-3-7 days, there are 151 sid. rev. of Mars, and 284 sid, rev, of the Sur, From this, the Sun's sid. period got is 365-15-38 days and Mars's 686-58-50 days. T h e Sun's period is 38 vinadis more than that given for it by the Vdsistha, and near the 365-15-30 of the Paulisa. Therefore, like the M o o n , Venus, Jupiter and Saturn, Mars also is common toPaulisa. 5. A t the first rising when calculation commences, mean Mars = 85/133 rev. = 7' 20° 4'.5 6. T h e addidon or subtraction of the difference from the remaining days has been already explained with reference to Jupiter and Saturn. But, here, no method is given to find the equation of the centre. Now the true motion is affected by both the eq. of the centre and eq. of conjunction. T h e segments of motion given in verses 25-26, below, are as affected by the eq, of conjunction alone. B y making the days given for true motion i n verses 27-35 conform to the segments, we can get the degrees, and through that the days affected by the eq. centre alone. This can be of use only for the remainder of days. But the eq. of centre at the beginning of each cycle is necessary. It has not been given by any rule. Since its period is about 687 days and it has its own rise and fall of about 11° from perigee to apogee and back, it cannot be associated with the synodic period of 780 days. So this is an omission. I have corrected the corrupt tdstraydmridubhih into tdim tryagnimlubhih to mean 133, T h i s is necessary for agreement with the actuals, and the effect of my emendation is seen in my discussion (3) above, T S have made it hdnendubhih. H o w can bdna, with such different lettering, come in here? Further, this will give 18/15 rev. = 432° as the mean motion of Mars in one syn. period, about 23° wrong per period. They have made paficdsiti krtvd into pancdmsonam krtvd and thus shut out the position constant of Mars on the first day where reckoning begins, viz. the point of time 256-40-0 days from epoch, (It will be remembered that in the cases of M o o n , Venus, Jupiter and Saturn also, they have made this mistake). By this emendadon they reduce the motion by 86° 24', and make the 21a. C.D.^nnn^ A , B . ^ ^ W 1 ( B 3 , ^ ) . D.^el5^F^
b. C,D,-5ltTOl?r; B . i P ^ : . A.B.aiim
b. B.^^nfe^
c. A.B.yH|u|dl'?r
d. B . m
d. A , B . C , ^ ' 3 m ^ ° ; D , ^ 5 5 f e n ^ °
22b. A . ' l t s t n f ^ ; B.'WVlPcldl: c. A . B . # R n w f ( B . # ) ^ ; C . D . ° ^ l f & I c I I ^ ^T^l^ d. A 2 . ? ^ ; A , B , f e r a t ^ W : ; C.D.f^ScTtSW^ 23a. A . B . W * ; C D .
B.a^f^lfg; D . ^ [ l ^ ] 5^=1?^ 24a. A.B.f^^qt b. A . ° ? T ^ f ^ B . C . D . T I ^ f ^ c, A . B . C . D . d. A . " f T M « I M ^ ° ; B,7TfcmtW5rRilf«T=; C . T | f ^ 5 # W T f i l ; D.^lR)<1l6uiNKi
PANCASIDDHANTIKA
330
xvni.26
mean motion of Mars 345° 36' per syn. period of 780 days! T h e y have also wrongly emended pratirdsya to pratirdsyam. As for N P , they have made the correct emendation tryagnindubhih, giving correctly 151 revolutions of mean Mars i n 133 syn. periods, and identified it with that given by the Babylonian astronomy of the Selucid period.
But N P have not seen that pancdsitim meaning 85, is correct as it is, and give the constant 7" 20° 4' .5 at 256-40-0 days from epoch (see item 5 of discussion). T h e y think it is the constant i n degrees, though no word meaning degrees is found here. (This kind of mistake they have made i n the case of Jupiter and Saturn also, as we have shown). B u t 85° would not do, so they have substituted satrirdsim for pratirdsya and made it 175°. B u t even this would not do, and therefore they hav changed the days from E p o c h itself into 216-40-0, by emendingjatkam vayamdn into satkaikayamd But this has led to other troubles, leading to their remark, " F o r Mars this would mean a longitude of 175° (instead o f 194° derived on the basis of ao in table 32). T h i s longitude would correspond to September 27, and a solar position at 186°, hence to an elongation of 11°" (p. 124, Part II). 11° for the first visibility of Mars is given by nobody. It is i n the range o f 14° to 17°.
3 T g M ^ ? e ? M [ W d ) ^ w ( M d ) ^ ^ T ^ ] II
I
M [ d r ( | < ( < ( ^ ) ^ ^ ; 1 1 ^ ^ II
25-26. A f t e r r i s i n g i n the East, M a r s m o v e s 1 4 6 ° ( i n q u i c k m o t i o n ) a n d t h e n 18° e a c h o f (slow m o t i o n ) , r e t r o g r a d e a n d " f o l l o w u p after" r e t r o g r a d e (anuvakra), a n d after that 150° o f q u i c k (sighra) m o t i o n . T h e n setting, it reaches c o n j u n c t i o n (nirarnsagatah) i n 60 days m o v i n g 13 p l u s 30 (= 43) degrees. T h e n it rises, ( m o v i n g the same d e g r e e i n the same n u m b e r o f days). B e g i n n i n g f r o m h e r e , I s h a l l m e n t i o n the series o f m o t i o n s w i t h t h e i r days. T h e scheme is Moves 146° 18° -18° 18° 150° 43° 43°
Total
400'
Rises East type of motion I II III & I V V VI Sets i n the West V I I in 60 days Conjunction V I I I in 60 days Rises i n the East
(slghmgati) (Mandagati) (Vakaragati) (Ativakragati) (Anuvakragati) (Sighragati)
(Atisighragati) (
"
)
XVIII.26
XVIII. VASISTHA-PAULISA - RISING A N D
SETTING
331
T h e numbers 1 have given i n the scheme are practically what are found i n the text, without emendation excepting three. In verse 25,1 have emended captasteka into catvdryeka to get 146° the most plausible value. N P have made Itsastdstaika to get 186° which is too large. See discussion below, 150° is given by adhyardham ca satam where tatah is emended into satam. This is necessary to make up the total 410° motion i n 780 days. Secondly, 43° motion for 60 days given from setting to conjunction is required to agree with the 17° usually given for heliacal rising. T h i s is made u p by emending vimsatam into trimsatam with the 13° given by dasatriyutam added. I shall now show that the motion of Mars is near 43° in 60 days, in the region o f the conjunction. For its distance, nearly 1.53 that of the Sun, given by modern astronomy and also as computed from H i n d u astronomy, the equation of conjunction at this region is 11' per day, (as can be verified) w h i c h , t h e daily mean motion of 31'.4 gives 42'.4 per day, making 42°.4 i n 60 days, roughly 43°. This also agrees with the angle for heliacal rising of Mars, nearly 17°, given by H i n d u astronomy. (In 60 days the Sun moves 59.1°. So, the elongation is 59.1 - 42.4 = 16.7°, nearly 17°). If vimsatim is taken as it is, we get 20° + 13° = 33°, which is 9.4° short of the actual 42.4 and which also gives the angle for heliacal rising as great as 26°, so far from the 14°-17° given by all. In the mean, the motion from setting to conjunction must be equal to the motion from conj. to rising. That is why it is not given by the text separately. That the motion segments given i n the two verses is mean is also clear, since no position of Mars from its apogee is taken into account. So the total motion must be equal to 409°. B u t the total got by adding the segments is 400°. This must be due to the defective method o f the original or the empirical nature of the motions, and rounding off to whole degrees, as seen from 43° being given for 42.4°. T h e opposition must fall at the midpoint of the retrograde motion, - 18°, and divide it into - 9 , - 9 . T h e total motion from conj. to opposition must be equal to that from opposition to conj. B u t what we actually get is 43° -(- 146° -t18° - 9° = 198°, and - 9 ° -I- 18° -1- 150° + 43° = 202°. It may be that the angle segments given are empirical and also there are errors i n the apparently correct numbers giving the segments, needing emendation. It is only i n the case of Mars, does V M give these eight types of motion. In II. 12-13 of the Later Surya Siddhdnta, a set of 8 types o f motion is given. B u t they cannot be equated to these each to each. So we have only to guess when i n doubt. T h e days on the synodic cycles to pass each type of motion must be nearly equal to the average o f the days given i n verses 27-35 for that type of motion. This has been used to check the degrees of each type. But the synodic period, as also the mean motion o f 1 -I- 18/133 revolution in that period are very nearly correct and they must have been got by analysis o f the observed motions. So the Siddhdnta must have known that the motions and times are half and half on both sides of the opposition. Beginning from rising type I is sighra (quick) motion. II is manda (slow) motion. T h e distinction seems to be faster or slower than the mean motion. So, the dividing point must be where the tangent from the earth touches the synodic circle. Since the mean distance of Mars is 1.53 times that of earth from Sun, this point falls about 189.5° from conjunction. Subtracting 43.5° motion from d. A . B . C . ^ ( B . M ) HI#Wdlw(Hd:; 25a. A.M<^k)«l*°; B.^I^^'^I^"; D.t|<'»l^'»; C . M^^m*
D.tlsrat&crats^clft^: 26a. A.B.C.%cTT; D.Bl^cIT^
b. A . B . "(iVWWIWd^ C . °HKW!ifmm'IWdl;
b. A.B.C.Pi<^ici1 R^ita (B.^?lfcT)
B.^jD.-sra;: c. A . B . C . 3 ? ^ ^ M : - ? f t ? n ^ . D.^4'gcra:?Ttwg,
c. B.3^i|gqq# d. A . B . C . ^ ( B . W ) HmfeWdlwftd:;
PANCASIDDHANTIKA
332
XVIII.27
conjunction to rising (given as type VIII), 146° is left for type I. T h i s segment extends upto the point where retrogression begins. As the planet is stationary, here a small error of observation can make this lesser or greater than the actuals. T h e text seems to give it as 18°. Types III and I V form the retrograde motion. I l l is called vakra (retrograde) and IV ativakra (faster retrograde). T h e text is defective here, and we cannot fix the exact extent o f the two segments separately. B u t III and I V seem to be divided i n the ratio 5:7 of the total. V is anuvakra (follow up after iia^ra). In the detailed motions given this is the suiii of III and I V but direct motion. This anuvakra must be the counterpart of II. T y p e V I is sighra and so the counterpart o f I. Its extent is given as 150°. T y p e V I I is the very quick motion from setting to conjunction and given as 43° i n 60 days. T y p e V I I I is the counterpart of V I I from conjunction to rising. These divisions are mostly based on convention. But as these are given only i n the case o f Mars and classical astronomy does not give them, we have only to guess regarding them. T o add to the difficulty the text is corrupt i n the places giving the numbers. T S have expressed inability to understand verse 25. Still, they have made some emendations which do not give any cogent meaning. N o translation is given. T h e r e is only a question mark. In verse 26, they give 20° motion from conjunction to rising. T h i s can give only 26 days, as against the 60 days given by the text. By this the elongation for heliacal rising would be 7 V 2 ° , so absurdly low. As for N P , i n both verses, they have needlessly emended correct forms, wrongly emended the corrupt ones, some i n faulty Sanskrit and given an untenable scheme. T h e following is their scheme: Rising east / 186° motion / 18° retrograde motion / 180° motion / setting / 30° motion / Conj. / 30° motion / rising east. T h e y have made the emendations and substitutions with their eye on the total motion o f 4 0 9 ° i n the synodic period. T h e y make the total 408°, nearly correct. B u t they do not identify the vestiges of the different types of motion found i n these verses. Further, 30° motion from setting to conj. and then from conj. to rising, is short by 12 V 2 ° from the actual 421/^°. T h e time required to move 30° is 42.2 days and the S u n would move 41.5° d u r i n g this time, giving an elongation of 11.5° for heliacal rising, far short o f the actual, especially for such high latitudes as the Vdsistha-Paulisa envisages.
27. I n the I t y p e m o t i o n , t h e r e are 4 0 -f- 1 (= 41), 4 0 + 7 (= 47), 4 0 + 7 (= 47), 4 0 -H 8 (= 48), 4 0 + 2 (= 42), 4 0 - 2 (= 38), days p e r m o r i o n o f 3 0 ° each respectively i n e a c h m o n t h o f the d i a d o f rdsis b e g i n n i n g f r o m M i n a , (i.e. Pisces). The above means, that for 30° o f motion, the time taken is 41 days in the rdsis M l n a (Pisces) and Mesa (Aries), 47 days i n Rsabha (Taurus) and Mithuna (Gemini), 47 days i n Kataka (Cancer) and Sirnha (Leo), 48 days i n Kanya (Virgo) and T u l a (Libra), 42 days i n Vrscika (Scorpio) and Dhanus (Saggittarius) and 38 days i n Makara (Capricorn) and K u m b h a (Aquarius).
27a-b. A.B.C.^^lfolfilH (B.g)
(B.om.sq) ^°
C.D.'SI«lH'Id1^<4^Ro|yi; ( D . 1 ^ ^ ) d.
D . ^
XVIII. VASISTHA-PAULISA - RISING A N D
XVIII.28
SETTING
333
A n examination of the rate shows that the perigee is situated at the end of the Makara (Capricorn) and the apogee at the end of Kataka (Cancer), which both fairly agree with the actual. T S say that they do not understand this verse, and no translation is given; its place being taken by a question mark. N P translate thus: "In the first gad 240 plus 28 minus V2 (= 267 V2) (days). O n e should calculate days for every two signs from Pisces." It is clear that they do not see that this verse gives the detailed rate of motion of the T y p e I gati i n the diads of rdsis from Pisces, as affected by the equation of the centre. T h e y think that the first motion given i n verse 25, which 186° according to them, takes 267 V2 days, as given by them here. If so, what use is the instruction to calculate for "every two signs from Pisces"?
28. T h e I I type m o t i o n , i n the same o r d e r , (i.e. f o r each m o n t h o f the diads, P i s c e s - A r i e s , etc.) f o r the 18° take 5 x 1 0 + 7 , 6 x 1 0 + 1 , 7 x 1 0 + 2 , 6 x 1 0 + 6, 6 X 10 + 1, a n d 5 X 10 + 1 days. This gives 57 days each to move i n the signs Pisces-Aries, 61 days for each of Taurus-Gemini, 75 days for each of Cancer-Leo, 66 days for each o f V i r g o - L i b r a , 61 days for each of Scorpio-Sagittarius, and 51 days for each of Capricorn-Aquarius. F r o m the days given, it can be seen that there is a slight tilt i n the apogee towards Leo, and i n the perigee towards Aquarius. T h i s small difference from the findings i n verse 27 shows that the values are empirical. As for the readings, rasa has been inserted because we want six numbers for the six diads, and one is wanting. Symmetry requires that it must be rasa (= 6) there. Also two mdtrds are wanting, sapta is emended into rasa because, 76 for V i r g o - L i b r a , with 72 o n one side, and 61 on the other side, will take the apogee to the end of Virgo, 60° off from its place. The average for II type motion is 18° for 81 days which is about the average of the mean rate and 0 (stationary). T h u s I type motion is faster than the mean, and the II type slower, as we have surmised. T S have expressed inability to interpret this verse also, and not translated it. Yet they have made an emendation which need not be taken seriously, since it has been done without understanding. N P have interpreted the verse as giving 57, 71, 72, 66,61 and 51 by inserting rfw as the fourth. But symmetry shows that the second number 71 is wrong, and it must be 61, to avoid the j u m p from 57 to 72. A t any rate, read with their interpretation of verse 27, we can see that they do not understand the use of this series of numbers. T h e y do not even say these are days.
28a.
A.B.C.D.&WW'H'
b. A.B.M-^'hK'^llf llfe°|i|<11 (B.°feqTT#) C.Ml+l<^l'J,"ll-^fe<=i'l
PANCASIDDHANTI KA
334
^i^P^«i^d1fc^c*>-^l(^Ti^f^l^A:
( ^ ) ^ ; '
XVIII.32
^ ) t t ' ^ w 3 :
- ^ ^ ^ ^ (9ra[|)rTr?T(^)' ( ^ )
29. I n t h e signs, Pisces, S c o r p i o , A r i e s a n d Sagittarius, M a r s m o v e s 7° i n 4 2 days w h e n r e t r o g r a d e (vakra) a n d 9 ° i n 4 2 days w h e n e x t r a - r e t r o g r a d e (ativakra). I n t h e f o l l o w - u p after r e t r o g r a d e (anuvakra) M a r s m o v e s 16° i n 6 0 days. 3 0 . I n t h e signs T a u r u s , G e m i n i , L i b r a a n d V i r g o , M a r s m o v e s 7° i n 4 3 days r e t r o g r a d e , 10° i n 4 3 days e x t r a - r e t r o g r a d e , a n d 17° i n 6 3 days i n t h e followu p after r e t r o g r a d e . 3 1 . I n C a n c e r a n d L e o , M a r s m o v e s 7° i n 4 4 days, 11° i n 4 6 days, a n d 18° i n 6 6 days, respectively, i n t h e t h r e e types r e t r o g r a d e etc. 3 2 . I n C a p r i c o r n a n d A q u a r i u s , M a r s m o v e s 6° i n 37 days, 9 ° i n 3 9 days, a n d 15° i n 57 days, respectively, i n the t h r e e types o f m o t i o n .
29a.
B . ^ . A . ^ ; B . ^
b. A . B . ^ C . D . ^ ^ . B . W ^
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A.B.C.D.o^fR^:
b. A . B . ^ ( B . f ) <^'ldl i^l^ci of (w. ^ ) ; 30a. B . ^ H ^ b. A . B . o g ; ^ : W m i F i ; C.^^srfara^:
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XVIII.33
XVIII. V A S I S T H A - P A U L I S A - R I S I N G A N D
SETTING
335
Types III, I V and V , called, respectively, retrograde (vakra), extra-retrograde (ativakra) and follow-up after retrograde (anuvakra) are given i n these verses. T h e first two are actual retrograde motion, and the third is the slow direct m o d o n following. T h e y are shown hereunder i n a tabular form.
Signs
Pis-Ari
Tau-Gem
Can-Leo
Vir-Lib
Scor-Sag
Cap-Aq
T y p e III
-7742 d
-7743 d
-7744d
-7743 d
-7742d
-6737d
Type I V
-9742d
-10743 d
-11746d
-10743 d
-9742d
-9739d
-M6760d
-Hi 7763 d
+ 18766d
-1-17763 d
+ 16760d
+ 15757d
TypeV
The division into the three types is arbitrary, based o n some convention. B y examining the table we can see two things note-worthy. T h e total of arcs o f III and I V is equal to V , though V is positive. The days for III and I V are the same?, except for Cancer-Leo, and Capri-Aquarius. T h e r e is symmetry o n both sides of these sets. G u i d e d by the above, I have emended certain numbers which glaringly go against these points. In verse 29, nava for vakra is corrected into naga since it must be less than the 9° given for ativakra, and both equal to 16°, clearly given for anuvakra. In verse 30, the corrupt stara is changed into svara to make up the total 17°. T h e corrupt nuvdsanaih is amended into agnisdgaraih, guided by symmetry. Khakrteh is emended into trikrteh since the number should be greater than 42, by symmetry. In ver. 31 the corrupt sapta khdrnavaisca divasdn: is corrected into rasdmavaih sivdn arnsdn because 11° required to make u p the total 18° for anuvakra. sapta is a repetidon, khdrnavaisca divasdn = 40 days, does not fit, since the m a x i m u m number o f days is required there, and 46 fits eminently. In verse 32, yama is corrected into naga sinceyatnadahanaih (= 32) is too short a period, and far from the 42 days on both sides, and the number should also be a little less than 39. reva ca, corrupt, is emended into nava ca, which will make u p the total 15° oianuvakra.
As for verse 33, the words i n it are all perfect, without any corruption. B u t they do not make any sense. It seems that some rules are given here for the division into the three types with their days, and the proportion is roughly 5:7:12, of the degrees of all three combined. A t any rate, this instruction does not seem to serve any purpose. Ativakra represents the faster retrograde motion near opposition/)/ias the slower vakra motion o n the other side. That is why it is greater and faster. B u t why exactly the same number of days? T h i s seems to be a convention. B u t this is against logic. For, only i n Cancer-Leo, and Capricorn33b. A . ^ ^
25
D.^1S|P?I^^o B . m ^ : ;
c.
D. M ^ ^ :
d. B.<=I^H:iM^'^Vi; D.°*lfil
Al.3^fira%;B.3?fil^B2.1»
PANCASIDDHANTIKA
336
XVIII.34
Aquarius, there is a small excess of days for ativakra, but even this is far too small. T h e sum of vakra and ativakra is 18° and a m a x i m u m at Cancer-Leo, and m i n i m u m 15°, at Capricorn-Aquarius, and fairly evenly distributed i n between. B u t actually, at Capricorn-Aquarius, the sum is near 9°, as a comparison with the motion o f Mars given i n the Vdkyakarana will show (vide A p p . I l l , Kujavakra, where retrograde occurs i n the signs Capricorn-Aquarius).
T S have translated verses 29-32, omitting verse 33 as obscure. B u t they think that all three types are retrograde motion, (while actually only III and I V are retrograde, and V is direct motion). This would make the range o f the retrograde motion from 36° i n Cancer-Leo to 30° in CapricornAquarius, while the range given is 18° to 15°, actually the latter is even as small as 9°. Also, they do not see that the I V type should be greater than the III. So they give the numbers as they got from the words, instead o f emending them appropriately. So, i n verse 29, navabhdgam as emended by them should be navabhagdn. nagdn vakri should be navdtivakri. In verse 30, their emendation abdhisamudraih should be agnisdgaraih Khakrtaih should be trikrtaih. In verse 31, their khdrn divasaih giving no degrees at all for the days, and defective in mdtrds should have been emended into rasdrnavaih sivdnarnsdn. In verse 32, symmetry requires our emendation of yamadahanaih in nagadahanaih, while T S have kept it. As for N P , they have understood that type III gives retrograde and type I V , extreme retrograde, though the numbers they give for degrees and days are untenable i n many cases. Seeing that the degrees o f V are the sums o f those o f III and I V , they think that V is the total of the retrograde motions, while actually V is direct motion. T h e y do not see that i f the degrees o f V are the total o f III and I V , the days too must be the sum o f the days, and therefore V is not the total retrograde. T h e y have translated verse 33, but this does not give any sense.
34. I n the q u i c k m o t i o n follow^ing sighragati (= t y p e V I ) , t h e r e are the days, 4 0 4- 1 , 4 0 + 5 , 4 0 + 8 , 4 0 + 1 1 , 4 0 + 1 4 , 4 0 + 1 4 , 4 0 + 1 1 , 4 0 + 9 , 4 0 + 5, 4 0 - 1, 4 0 - 4 a n d 4 0 - 4 f o r every 3 0 degrees. Obviously, these days are given for each one o f the signs beginning from Pisces. T h e numbers are almost perfectly symmetrical o n both sides o f the inter-secdon o f Cancer-Leo, and CapricornAquarius, re-inforcing the conclusion that the former intersection i n apogee, and the latter perigee.
manu for Leo is a glaring omission and is inserted. Symmetry requires 45 for Scorpio, and sopaksa is emended intopanca. T h e number-words forming a 'dvandva' compound, trivargam is wrong fo trivarga, and so has been emended. -
A . B . C . D . H a p l o m . of o n e ^
b. A . B . C . D . ' q ^ ( B . o m ^ ) T ^ ( B 2 . ^ ) l(A2.B1.2.f D . g ^
A.B.C.^^wg^
D . W ^
c. D . ^ i # d. A . o t J T ^ B I . S . c T t T ^ : ; B2.cT?|5lt
XVIII.35
XVIII. VASISIHA-PAULISA-RISINC; AND S E I l l N G
337
N P have emended the already correct pancdstakam, meaning 40, into pancasastm which they think would mean 5 X 60 = 300. pancasastim would mean only 65. T o mean 5 + 60, the form should be parwasastayah (nominauve) or pancasastih (accusative). T h e trouble is that they have not understood that the days given are for every 30°, everywhere except in types III, I V and V . T h a t is why they make such un-authoiised corrections and give wrong translations.
35. I n the V I I type, (i.e. atisighra), f o r every 3 0 ° o f the diads Pisces-Aries, T a u r u s - G e m i n i , etc. there are the days, 36 + 3, 36 + 9, 36 + 12, 36 + 9, 36 + 3, 36 -I- 0. T h e m o t i o n as g i v e n f o r the V I I type is f o r the V I I I type too. Here too the symmetry on both sides of the apogee, and perigee is necessary, and the corrupt dvikaldhvdrkd is corrected into hyanaldnkdrka, corresponding to trivargaguna. This forms the motion from setting to conjunction. As the V I I I type, forming the motion from conjunction to rising is exactly the counterpart of the V I I in the synodic cycle, it follows that type V I I I is the same. T S have emended dvikaldhvdrkd into dvyanaldnkdrka, resulting in 7 quantities of days, while only 6 are wanted for the diads of signs. N P , here too, as i n verse 34, have made an unauthorised correction under the same mis-apprehension. T h e y have substituted sadbhis ca for the correct sat-trimsat. They do not see the coniradiction this leads to. From setting to conj. or from conj. to rising the days they give are not less than 60, 66, on the average. For this, the average motion would be about 46° in this region of the synodic cycle. But they have interpreted that it is 30°, in verse 26. T h e y have not seen the contradiction. As 1 have done l o r Jupiter and Saturn, here too, I shall work out an example, to make my explanations clear. Before closing, I wish to say something about Mercury. T h e case of Mercury is as involved as that of Mars, but in a different way. In one sidereal year Mercury traverses three full synodic cycles and more. But it moves in the zodiacal circle together with the Sun i n the mean and its equation o f the centre depends on its position i n the 12 signs. So the motion types vary from synodic cycle to synodic cycle in the same year, needing a large number of day-groups, and these are given by V M . But it is these numbers that are spoiled by scribes most and the reconstruction is a tedious job. (The methods, though sometimes peculiar, can be guessed and explained). I have neither the leisure nor the equanimity of m i n d to undertake this work at present, and hope someone else would do it.
35a.
A.B.t^iR^IW^*!; D. [^fef^S]^%W
b. A . B . f l 3 ; ( B . ^ ) W ( B . f r ) - ^ ;
A.B.^^I^J^-^: d. A . ' ^ ^ ; B . ^ ^ ( B 2 . ' ^ ) A.^igF=!)[t
PANCASIDDHANTIKA
338
XVIII.35
Example: Find true Mars at 800 days from Epoch. Days from Epoch Subtract days at first rising
8000-0 256 - 40 - 0 543 - 20 - 0
, . No. of revolutions gone
543-20 —i^^O—• = 0
Remainder Correction for revolutions gone
543 - 20 - 0 00-0
Days after rising
543 - 20 - 0
Mars at rising =
^33^
revoludons = 230°, i.e. 20° i n Scorpio.
Starting from this point, i.e. 20° i n Scorpio at rising, true Mars moves i n 543-20 days, as per the scheme given i n verses 24-35, to 29° 47' i n Cancer as per details given below: T y p e I: 146° Total days = 543-20 Balance 10° i n Scorpio 30° 30° 30° 30°
i n Sagittarius i n Capricorn in Aquarius in Pisces
16°in Aries Total
14 days 42 38 38 41
x 42)
" " " "
21 + 52
x 41)
194^ + 52
Balance N o . of days
348.28
II: 18° 14° i n Aries
44-20
4° in Taurus
13-33
(i|x57) 18 4_ (—x6]) 18
57-53 Balance N o . of days
290-35
III: 7° (Retrograde) - 4 in Taurus
24-34
(^x43)
- 3 i n Aries
18-00
{~x42)
42-34 Balance N o . of days
248-1
XVIII.36
XVIll.
V A S I S T H A - P A U U S A - RISING A N D
I V : 9° (Fast Retrograde) - 9 i n Aries
SETTING
42-00
Balance N o . of days
V : 16°
12° i n Aries 4°inTaurus
339
206-1
45-00 14-49
16 ( J | X 60) (^ X 63)
59-49
V I : 85° 4 7 '
26° i n T a u r u s
41-36
30° i n G e m i n i
51-00
29° 4 7 ' i n Cancer
53-36
(|§X48) 53-36 ( ~ 5 4 ~ X 30)
146-12 Therefore, T r u e motion i n 543-20 days from 20° to 29° 4 7 ' Cancer Mean motion i n 543-20 days: 408 - 43 X 543 1/3 /780
= 249° 47 = 284° 42'
Therefore days to be added
=
34-55
T r u e motion for 34-55 to be added In Cancer 0-13' for 0-23, and for the balance 34-32 days. In Leo 30 X
= 19° I T
Therefore T r u e Mars: 19° 11' i n L e o .
Motion of Mercury' As in the case of other planets, the epoch date is so selected that Mercury rises heliacally i n the west on that date. Since, for Mercury, the elongation required for heliacal rising is given as twelve degrees, on the epoch day its longitude is 12 degrees ahead o f the S u n . T h e n , the number of days elapsed from the epoch is obtained and divided by the sidereal period o f Mercury, so that the resulting quodent represents the number o f risings that had taken place since the epoch, and the remainder the number of days elapsed i n the current sidereal cycle. T h e movement of Mercury i n one synodic period being known, (given by the text), the total longitude moved d u r i n g the elapsed number o f risings is also known and adding to this the epoch constant, that is, the position o f Mercury on the epoch day, its longitude on the day o f the current rising is obtained. However, these figures are on the assumpuon that the motion o f Mercury is uniform, which is not the case. So, what we have got is only the mean position of Mercury and not its true position. Here, as also in the text, 'mean position' does not mean the mean heliocentric position as we now understand, but the mean geocentric position on the assumption of uniform modon. T h e true position varies from the mean by quite a few degrees. T h e date o f rising also is only a mean date and the true date may be several dates behind or ahead. 1. Mercury : Translation and Notes by S. Hariharan, Bangalore.
PANCASIDDHANTIKA
340
XVIII.37
T h e text then gives a table from which for a given mean longitude the true position on the true date can be obtained. Since the Sun's position has to be 12 degrees behind the true position of Mercury, the same correction applied to Mercury has to be applied to the Sun. T h a t means, keeping i n mind that the Sun moves one degree i n one day approximately, the true date is ahead of the mean date by as many days as the number of degrees in the correction, i f the correction is positive, and vice versa. Consequently, the number of days elapsed since the (true) rising is also modified by the same number of days, reduced, if the correction is positive, and vice versa. H a v i n g got the true Mercury as on the true date of rising and the number of days elapsed since rising, we have now to get the degrees moved by Mercury d u r i n g these remaining days i n the synodic cycle. For this purpose the synodic period is divided into several sections just as in the case of other planets. In the case of Mercury the division is made into four sections or gati-s. T h e first is from rising to starting of retrogression, the second is the period of retrogression, the third is from end of retrogression or anuvakra to setting, while the fourth is from setting in the east to rising in the west. T h e number of days for each section and the movement are not constant but vary depending upon the position of Mercury in the zodiac. Accordingly, tables giving these values for each of the twelve signs are given in the text. W i t h the help o f these tables we can trace the movement of Mercury d u r i n g the remaining days and finally arrive at its position. What is new in the case of Mercury is the method of interpolation within these tables. For other planets no specific method of interpolation is suggested, with the result we assume linear interpolation. But here a second degree interpolation is suggested for certain sections. T S and N P do not appear to have understood the procedure nor the radonale expounded by these verses, as could be gauged from the emendadons they make and explanations they give to the verses.
Mean Mercury 36-.38a A d d to the ahargaria seven d m e s f o u r , i.e., 28, a n d a t h i r d . M u l t i p l y by eight a n d d i v i d e by 9 2 7 . T h e q u o t i e n t is the risings o f M e r c u r y . T a k e the e i g h t h part o f the r e m a i n d e r a n d d e d u c t t h e r e f r o m , i n nddikds, o n e f o u r t h o f the risings, a n d the result is the ( r e m a i n i n g ) days o f M e r c u r y . M u l t i p l y the ris36a. A.?gTB.'HMi»l-j, b. B . ? m t A . ^ : ; B.>W B.iymFT: D . ' ^ :
D.RHISIill: 37a. A . B . f c ^ b. A.'gKTPf; B.^SR^T
c. A . ' ^
c. A.B.BKi^I'MHWI-i; D . [ 3 T t 5 ^ ? m ] w n ^
d. A.B.Ttl^RW: C . T t f l ^ ; D.TtfeiT:-g: A . ^ ; B.im; D.-^art
B . ° ^ d. B . W I ^ g f ^ a i ^ D . W S ^ r a f ^ f e : ^
xvm.37
XVIII. V A S I S T H A - P A U L I S A - R I S I N G A N D
SETTING
341
ings by 2 6 3 , d e d u c t 43 a n d d i v i d e by 829 a n d the result is the m e a n M e r c u r y (in revohttinns). These will lead to the following: i. 28 1/3 diivs before the epoch, Mercury rises in the west, after which the countings of the risings begin. ii. One synodic revolution takes (927/8 + 1/240) days. T h e deduction of V4th nMls or l/240th dav per rising is for taking the synodic period as approximately 927/8 davs. T h e balance remaining after division by 927 is the number of eighth parts of a day elapsed since the last rising. So. dividing this by eight, the niunber r>f full days are obtained. D u r i n g a synodic period the motion of mean Mercury is equal to 263/829 revolutions (which is the same for Sun). Hence for one full revolution Sun will take, (927/8 + 1/240) X 829/263 days, or, 365.261708 days, i.e., 365-15-42. iii. At the epoch date, that is 28 1/3 days before 20-3-505 A D . , Sun's longitude is 357-37 mmus 27-55 or 329-12. iv. O n this date. Mercury's longitude is given by: (No. of risings (zero) X 263 - 43)/829 or - 43/829 revolutions. This reduces to - (18-40), or .34120, which is ahead of Siui by 11-38 as against 12 needed, and therefore acceptable. In verse, 36c, we, also others, have emended the ms. reading muniva.nanaca as muniyamanava to get the coiTect figure 927. In 37a krtvd has been made as hrtvd. The meaning and rationale of these verses is clear. But N P have made medhd i n 36d assodhyo and have translated it as "Subtract an eighth part of a day (for every synodic period)" etc. and comment, "We find in X V I I , 36 a subtraction of 1/8 of a day and a division of the number of risings by 4. We cannot explain these steps which seem in excess of the normal procedure." It is rather surprising that N P mi.ssed the elementary step of dividing the remainder by 8 to get the number of days, particularly when T S have correctly interpreted it. Without the correct number of days (elapsed since the last rising) the further processing does not make sense. Hence we feel that N P have not understood the process at all. In the next step of finding the longitude of Mercury at the time of the last rising, the figures in the verses have been badly corrupted. In 37b tridasayama has been emended as tridivasapa by T S , and in 38a navavnsmamu as nn.vavasnrdma. so as to give them a multiplier of 123 and divisor of 389. But this would give the Sun's period as 366.479641 days which cannot be. N P have made tridasayama as adridasayama; rdmdrnava as pdndava and navavasuydma as navavasurasa. They have also changed the meaning so as to subtract/^an^/ava (5) from the divisor 689 rather than to make the deduction from the product. A l l these they have done just to make these constants agree with those of Babylonian and then claim that "Table 24 reveals exact numerical agreement for the outer planets and Mercury" (!). T h e y forgot that in this process they have obliterated the initial epoch constant of the longitude of Mercury and comment "For Mercury we have no epoch constant giving us direcdy a longitude." We have emended tridasayama as trirasayama, and navavasuyama as navayamavasu. As shown above, these numbers give the sidereal period of Sun as 365-15-42, which is within limits. O f
PA5JCASIDDHANTIKA
342
XVIII.40
course, the constants as emended by N P also will give the correct sidereal period, namely, 365-1535, as it should be, but the emendments made are too violent.
t^^RVlt*!! ^ a ^ c h l Rviid*^^
||>Jo
True Mercury 3 8 c - 4 0 . F o r the r i s i n g d e g r e e s i n o r d e r , we h a v e for 3 0 + 5 35 degrees 9+ 60 69 8 + 80 88 1 0 0 - 12 3+ 30 33 30+ 7 4 + 100 104 1 0 0 + 12 5+ 26 31 3 + 30
30 degrees 60 88 37 112 33
Total
360
360
T h u s true Mercury.
We have made some minor emendations like navakrtyd sastim to navasastyd sastih, ca tiksndms to vittksnamsuh, trimsadbhufikte for trimsadbhakta, to get the total as 360 for both sides as they shou be. W i t h this table corresponding to the mean mercury obtained as per the previous verse the true mercury can be got. T h i s is for the true date of rising. Now, the true date has lo be obtained or, what is the same as getting the true remaining days passed after the true rising date. T h i s is dealt with in the next verse.
38a. A . B . ^ 1 5 r a i ! m G ? ( B . r e p . W ? G T ) ^ :
39a. A . B . ' W f ? ' ? ! ^ ^ : ; C.D.4c'ir<^# b. A . B . ^ a W f t c q i . A.B.'?I^^#«fr?il:;
b. A.^sCTi32^«; B.VI*MH! D. [^5St?TPja>RR?^a] 38c. B 3 . H a p l . o m . ' # [I^^Tlfe]^-' d. A.B.I5lVlifttt>*»iKiH5!ll<
c . D . V M wdla^ni^ c. c.D.'sn^. A.T«gfti^ d. A . B . f t ? i ^ « n ^ c.D.1?m^sii5ni^ 40a. Al.'gg^f*!^. A . B . C . D . T I c R f t f ^ c. A.B.C.D.5lf«Wi d. A.B.C.D.1?SvifclJ|ai
XVIII. V A S I S T H A - P A U U S A - RISING A N D
XVIII.40
SETTING
343
These verses actually give the true longitudes at the dme of rising corresponding to mean longitudes. T h e mean longitude is obtained as per the previous verse. T S and N P think that, i n verses 38b-40, the true m o d o n of Mercury is given, for certain days certain degrees, and so on. They have made some emendments of their own and both of them get the total degrees moved as 360 but as for the number of days T S get 389 while N P get 388. T h e 389 of T S is the same as the divisor of the previous verse and so they think that corresponding to the remaining days which will be upto 389 these verses give the true sighra-sphuta. B u t their idea is vague and they are not able to expand it properly. N P think that what is given is an eightfold division of the ecliptic but they are not able to explain anything further. T h e y comment, " T h e text calls n l 'days' which, i n any case, is meaningless". In fact there seems to be no reference to days at all in these verses.
41. D e d u c t f r o m the days the degrees o f d i f f e r e n c e o f these two, (i.e., o f the m e a n a n d t r u e places o f M e r c u r y ) , i n case the t r u e place is i n excess o f the m e a n place, a n d a d d i f the m e a n is i n excess o f the t r u e . T h e n the c o u r s e o f true M e r c u r y is as follows. T S and N P have not been able to appreciate the rationale for this operation. T S have not given any rationale. N P think that the correction applied to the days is a mistake and it should beapplied to the longitude; and the correction to the days should be applied by dividing the correction (in degrees) by the synodic motion per day. T h i s however is not the correct position. Since this relates to the rising position of Mercury on the true date the Sun will be 12 degrees less than the true Mercury. So the Sun also will have to be corrected by the difference between the true and mean positions of Mercury. Since the motion of the Sun can be taken as one degree per day for small durations, to that extent o f the difference the rising date will shift. If the true Sun is in excess, the date will get advanced, and, consequently, the remaining days will get reduced, and vice versa. Obviously the number of days will be equal to the number o f degrees.
Note on Verses 42-53 These verses give the gati-s of Mercury relating to the twelve signs, Aries to Pisces. I n each verse, the first quarter gives the degrees (days) o f motion from the rising of the planet to the commencement of retrogression, the second quarter for the period of retrogression, the third for the period from the end of retrogression to the setting o f the planet and the fourth from setting to its rising. In order to check the accuracy o f the figures, they have been calculated by m o d e m methods. T h e
variants, asi'Sig^felJW^IrRf^l^ and 41a.
A.B.o^
Tm\r
b. A."?!)*!^^. C.ft!%'^.
c.
A.B.^lHj^W
c. A . B . repeat after b, verse 40, with a few
d. A.B.<;aWK ( B I . 2 . ^ : )
PANCASIDDHAN'I IKA
344
XVIII.43
textual figures might be seen to agree with the calculated figures in several cases with minor differences, of course on account of the constants used in Paulisa and Vdsistha, and due to observational errors. Where they differed widely, it has been examined whether there could be due to defective readings and suitable emendadons have been suggested in consonance with the reading of the manuscripts, as far as possible. Neither T S nor N P appear to have understood the purport of these verses and have suggested all sorts of emendations. T S do not translated at all verse 54.
M e r c u r y ' s gatis for A r i e s 4 2 . T h e ^^a/Knn A r i e s are: Degrees
1. 2. 3. 4.
Rise to R e t r o . Retro. R e t r o , to Set. Set. to Rise
Minutes
Textual
25 3 X 4=12 6 X 7=42 3^X5=45
25 25 36 22
Degrees
Minutes
Textual
4 X 8-h0=32 2 X 8 + 8 = 24 4 X 8 + 3 = 35 2 X 8 + 3=19
4 4 - 11=33 44-33=11 4 4 - 7 = 37 4 4 - 1=43
3 6 - 11 = 25 3 6 - 11=25 360=36 3 6 - 14 = 22
25 12 42 45
Calculated
23 24 39 22
26 11 44 45
[^:] ]
4 3 . I n T a u r u s , the gati-s are:
I. 2. 3. 4.
Rise to R e t r o . Retro. R e t r o , to Set. Set. to Rise
42a. A.B.2.3.fcqt; B l . ' s r s ^ b. A.B.-9TO; C . D . ^ a ^ ( D . T m ) W . A.B.«(FT:; C D . c. A.B.f^ff^; C f e f e ;
32 24 35 19
43a. A . ^ ; B.^.
33 11 37 43
Calculated
31 24 33 20
A l . ^ ; A2-^.
A . f ^ ; B.lMt; C.D.^d.-MHfe"*^: b. A . B . t l ^ : ; C D . t ^ : .
A.B.C.D.Bl^kl*. A . B . ^ W ^ (B.^ie^)
c. A.B.?TcTT*R#; C D . w J i l i f t :
7]f&ra(B.Tlf&RT); C.M
d. A.B.W^ST; C . ^ ? T ; D . ^ ^ . C. •H'
34 9 36 43
XVIII.46
XVin.
VASISTHA-PAULISA-RISING
AND
SETTING
345
44. I n G e m i n i , the gati-s are:
1. 2. 3. 4.
Rise to R e t r o . Retro. R e t r o , to Set. Set. to R i s e
Degrees
Minutes
20 + 2 5 = 4 5 2 0 + 3 = 23 20+ 6=26 2 0 + 3 = 23
50 - 7 = 4 3 3 3^ =27 45
Textual
45 23 26 23
Calculated
43 3 27 45
38 23 27 21
42 8 29 45
[^:]
C % ^ )
' ^ 1 ^ ^ '
( ' # ^ ' )
'M^cbc<4|fpc(dl'^^||>{
4 5 . I n C a n c e r , the gati-s are:
1. 2. 3. 4.
Rise to R e t r o . Retro. R e t r o , to Set. Set. to Rise
Degrees
Minutes
8 X 4+8=40 8 x 2 + 8=24 8 x 3 + 0=24
40+ 7=47 24 X 1/2=12 24+ 1=25 22 + 2 5 = 4 7
8 X 2+6=22
Textual
40 24 24 22
Calculated o /
47 12 25 47
42 24 22 23
48 10 24 47
44b. A . B . C . D . WWHlPdd A . B . ^ ^ ^ c. B.?m; C . D . m A . B . ^ ; C . D . ^ d. A . m^S;
B. f w m g and indicated
omission iipto^Kiig of verse 47c below;
45a. A . g ^ f ^ . A . B . 1 ^ : b. A.C.D.td^lPvi'Jui^^: -Hfechia^J^-^^^: c. A . C . D . ^ ^ . A . # § d. A . c . D . W R S R f f ^ g m r i ;
346
PANCASIDDHANTIKA
XVIII.48
46. I n L e o , the gati-s are:
Degrees
Minutes
Textual o
1. 2. 3. 4.
Rise to R e t r o . Retro. R e t r o , to Set. Set. to R i s e
5 X 8+4=44 3 x 8 + 1 = 25 2x8+2=18 3x8+5=29
44+
7 = 51
44
25x1/^= 1 2 2 5 18+ 4=22 18 29 + 2 5 = 5 4 29
Calculated
/ 51 12V2 22 54
44 25
51 12
19 28
22 52
] ch-MwiH''^i^(^«^')-'?ugeF''15ra^[1?T]'-
4 7 . I n V i r g o , the gati-s are:
L R i s e to R e t r o . 2. R e t r o . 3. R e t r o , to Set. 4. Set. to R i s e
Degrees
Minutes
46 24 20 35
33 X 2 = 54
46
54
44
9 x 5=45 3 x 8=24 50 + 9 = 5 9
24 20 35
45 24 59
23 19 35
46a. A.C.D.Jj"l-^
Textual
Calculated 52 14 23 59
47a, A.(:.D.<»-^WI'Jd:j»^l b. A.KHVWill d<4R^:; C . 8l3«(lcll d
A.C.D.^(A.B.^) at the commencement of the line. a.(;.d.?t'
d. A . C . D . - ? ? ^ ' ' (A.^?1^)
c. B1.2.3. commerce again with«k1l84) •?I?n4, after the indicated gap. A.B.'H"<1ll!=b
XVIII.50
XVIII. V A S I S T H A - P A U L I S A - RISING A N D
SETTING
347
4 8 . I n L i b r a , the gati-s are:
Degrees
I. Rise to R e t i o . 2. R e t r o . 3. R e t r o , to Set. 4. Set. to Rise.
3#1^ W
:
(21 (21 (21 (21
0)x2=42 - 10)x2=22 8)x2=26 - 0)x2=42
Minutes
Textual
42+ 3=45 228=14 26+ 1 = 27 42 + 2 3 = 6 5
[ ^ ] - W - ( ' W S ' ) -'fKlT:' (
W
42 22 26 43
"
45 14 27 5
) "xifgr'-(
Calculated
39 21 24 42
47 15 27 5
)
49. I n Scorpio, the gati-s are:
Degrees
1. Rise to Retro. 2. Retro. 3. Retro, to Set. 4. Set. to Rise
10x3+ 6=36 1 0 x 1 + 11=21 10x3+ 4=34 1 0 x 4 + 2=42
Minutes
36+ 2134+
2=38 7=14 3=37
42 + 2 6 = 6 8
Textual
36 21 34 43
38 14 37 8
Calculated
35 21 33 43
39 16 38 6
49a. A.^?ml; B . ^ W b. A . B . C . D . ^ir^l+d (B.om.cf, D.^lRflfefcl) fort^RUf^T. A . B . ^ : q f O T o k ? g c 1 T : ; 48a. A . B . > ^ b. A.B.aqqgraif?rf»rt5H^9sr; c.#Tr
C.^g^^-Woks^:; D . ^ : c. A.%?II; B 1 . 3 . M A.B.D.^JW5ll#1T; C.^WfTf^
d. A . B . C . D . ? ! ^ ( A . - S t ^ ) t t ? I ^ | g
d.
A.B.^^(B.^)
PANCASIDDHANTIKA
348
XVIII.52
5 0 . I n Sagittarius, the gati-s are:
Degrees 1. 2. 3. 4.
Rise to R e t r o . Retro. R e t r o , to Set. Set. to Rise
2x10+
8=28 20=20 7 =42 7 - 4=38
6x 6x
Minutes
Textual
28+ 5=23 2 0 - 5=15 42+ 7=49 38 + 3 0 = 6 8
28 20 42 39
33 15 49 8
Calculated
28 20 42 39
32 16 50 3
[ W : ]
(3t^
51.
' u f g ' - ^ f ^ )
( ' # r T ' ) - * g t c ^ ' ^ s [ ij
I n C a p r i c o r n , the gati-s are:
Degrees 1. 2. 3. 4.
Rise to R e t r o . 20+ 3=23 Retro. 20+ 0=20 R e t r o , to Set. 2 0 + 7 + 1 8 = 4 5 Set. to Rise 20+12=32
Minutes 23 + 4 = 2 7 20-4=16 45 + 7 = 5 2 32 +26=58
^lf^Vlfd
50a. A.«lP^; B.«RPr. A . B . f ^ ^ « ! I ^ ( B . ^ ) ;
Textual
23 20 45 32
27 16 52 58
Calculated
23 20 46 33
II
51a.
A.B.^.
b. A . B . C . # # 7 ; D . [ ^ J ^ ] b. A.B.C.D.^fe?! (A.^IS^) f o r f ^ A . B . C . D . ^ . A . ^ for^ c. A l . % ^ : ; A 2 . B . % ^ : . A . B . C . D . (A.B.4) d.
A.B.C.D.^^^(A.^)?i^.
c.
A.B.C.D.^^^^TOTftl:
d. A . B . C . D . ^ ^ ^ s f e f i T
28 16 54 56
XVni.53
XVII!. v . 4 s i s t h . v p . ' \ l ' l 1 ! 5 a - r i s i n c ;
and
setting
349
52. In Aquariu.s, the gati-s are:
Degrees
1. 2. 3. 4.
= 20 1 = 21 =43 2 X 12 = 24
Rise to R e t r o . Retro. R e t r o , to Set. Set. to Rise.
20+
*7ErggFT: ' f j f w '
C
53. I n Pisces, the
W
Rise to R e t r o . Retro. R e t r o , to Set. Set. to Rise
20 21 43 25
22 15 55 60
22 15 55 00
Calculated
20 21 46 27
22 15 55 51
'?mT«f' ^ ' c h l d M V I I :
18+ 18+
1 = 19 5=23 43=43 3X 8=24
Minutes
20 13 50 49
52a. A . f % . Al.f5rwn; D.f^fcqi CD.fe^c^
(D.f^) 1^^: d. A . B . C . W ^ « 1 # W F ^ : (C.Ttq«lfegR): D. 4^=1'1
Textual O ' 19 23 43 24
(3^)d^i'd
b. A.B.^^fk^
Textual
are:
Degrees
1. 2. 3. 4.
Minutes
20 13 50 49
Calculated O f 19 23 43 24
19 14 52 47
i
53b. A . B . C . D . ^ =^a=HM^i b. forWRSBT;, A l . W ; Bl.'?T?te, B2."?irafW; B3."?r?fft; CD.^tgtOT c. A.B.C.D.:^a=h'+Rlfii;ic^l (D.ft?T?Tr:;) d. B2.?Hl4. A.B.TTgT^:
PANCASIDDHANTIKA
350
XVIII.56
SpeciHcations o f the gati-s o f M e r c u r y
54. F r o m setting t o r i s i n g t h e degrees a n d days a r e g i v e n by t h e f o u r t h gati. F r o m r i s i n g to vakra ( b e g i n n i n g ) is (given by) t h e first gati. T h e t h i r d gati is (from) anuvakra (i.e., e n d o f r e t r o g r e s s i o n ) to setting.
It will be seen that this verse explains what the (our gati-s indicate a fact which has not been mentioned earlier. Actually, there is no direct reference to the second gati. B u t it can be inferred that this gives the retrogression period and the degrees. T S have made some emendations but have not translated them but say that they are not able to comprehend the meaning of these verses. N P have adopted the emendations of T S and have just translated them as they are and that does not make any sense. T h e y comment: "We cannot connect any rational procedure with the verses X V I I , 54-56".
5 5 . B y t h e square o f the d i f f e r e n c e i n gati-s (days) m u l t i p l y t h e degrees a n d d i v i d e b y t h e square o f t h e gati-s. T h e result is to be subtracted f r o m the degrees a n d that gives the degrees g o n e i n the first g-ai? a n d i n the s e c o n d h a l f o f vakra-gati. 56. I n t h e first h a l f o f vakra-gati a n d i n t h e t h i r d gati m u l t i p l y t h e square (of the days passed) b y t h e degrees a n d d i v i d e by t h e square o f the gati-s (days). F o r t h e f o u r t h gati t h e r e s u l t is o b t a i n e d b y (direct) p r o p o r t i o n [since the m o t i o n i n this section is p r a c t i c a l l y u n i f o r m . ]
54a. A.aiSt;
D.arait. A . B . C . t R O T J ,
b. A.B.C.D.fs^HagsS^Tcm occurs as c. T h e y have as b , ^ « W ^ r a m (A.B.'+MiS!!)
55a. B.°ttT?T° c. B . ^ d. A.TTT^. A.^SiqaiSJ; B.^sCTSM;.
WiifeFRp^(A.B.TRflig, D.^Tcft^) d. A . B . C . D . 3 ? ^ l 3 W 5 I # T % ( ^ ( A . % a ? ^ B . % ^ ) .
56a. A.=l'lf*J|d1; B.-M*J|d1 b. A.C.Tlc^fl^; D . w g A.-^^;
B.cT^. C.-^pmi D. [^Id
c. A.B.^Tf?1ff?T(B.a#). B l . ' q ^ ; B 2 . 3 . ^ d. D . ° ^ 9 ^ . A.B.°«?WHt ( B 2 . 3 . ^ )
XVIII. V A S I S T H A - P A U U S A - RISING A N D
XVIII.58
SETTING
351
Correction for mean elongation i n heliacal rising 5 7 . F i n d the R sine o f the latitude o f the b o d y . M u l t i p l y the ( m a x i m u m ) halfcara (i.e. the h a l f d i f f e r e n c e b e t w e e n the h a l f d a y - t i m e o r n i g h t - d m e f r o m 15 (nddis.) i n vinddis, by this R sine, a n d d i v i d e by 4 8 0 . A d d this o r subtract this (to o r f r o m the M o o n o r star-planet) i f the latitude is n o r t h o r south, a c c o r d i n g to the p r o p e r d i r e c t i o n , (i.e. a c c o r d i n g as the p h e n o m e n o n o f setting o r rising, takes place i n the west o r east, respectively). 58a-b. W h e n "this is d o n e , t h e i r setting o r r i s i n g h a p p e n s a c c o r d i n g to the i n t e r v a l i n degrees (between the S u n , a n d the p l a n e t g i v e n i n X V I I . 12, o r the Moon). Note 1. There is a lot of lacunae i n verse 57. T h e \i-d\i-cara-vina4is meant is the m a x i m u m for the place, yathdkaksam is meaningless here and is corrected intoyathdkdstham, i.e. according to the direction, but the direcdon is not mentioned. If north latitude, the addition is for the M o o n or planet in the west. Also i f north latitude, the subtraction from the M o o n or star planet is to be done for the east. If south latitude, the subtracdon is for the west, and the addition for the east. These things can be got by a little reflection. Note 2. T h e amount of degrees to be applied can simply be got by multiplying the degrees o f latitude of the body by the tangent of the latitude of the place. (The equinoctial mid-day shadow of the 12" gnomon 12, is tan. latitude o f a place). T h e amount got is very rough. T h e degrees wanted = Sin. half cara (i.e., tan latitude o f place x tan. declination o f the Sun) X sin (90° - angle for heliacal rising), nearly. T h e last term is neglected here, tan declination is roughly taken as 4 8 ' , half-cara is converted into degrees by division by 10, and the conversion into sine-function is applied to the latitude of the planet instead of the half-cara, as roughly equal. Note 3. This application is what is technically called Aksa-drkkarma. T h e Ayana-drkkarma is neglected. Note 4. T h e heliacal rising of the M o o n , and of Venus and Mercury when retrograde, takfes place in the west. T h e heliacal setting o f the M o o n and retrograde Venus and retrograde Mercury takes place i n the east. Otherwise, all star-planets set i n the west and rise i n the east. (This Siddhdnta does not envisage the setting or rising of Venus and Mercury when retrograde, no separate degree for -hat being given). d. A . t R ; B.^m". C . ^ . D . [ ^ ^ ]
57a. B1.3.^iqi1^«lfarfKr b.
A B . ^ ^ A . ° ^ ; B
(B2.^fiMf^)mi^ c. A.^imif^. A . ^ m
.
^
A.B.C.D.l^WI#^:
B.°t»qo
PANCASIDDHANTIKA
352
XVIII.60
58c-d. T h e setting a n d r i s i n g ( m e n t i o n e d above i n verse 58a) is by 12°, 14°, 12°, 15°, 8° a n d 15° f o r the M o o n etc.
Note 1. This has to be taken with verse 58a. T h e degrees given here separately are according to the Vdsistha-Paulisa, which do not instruct the correction due to the latitude of the planet or for even the latitude of place (dksa-valana). T h e result will therefore be very rough.
Note 2. These degrees are necessarily arbitrary as mentioned already, and the correctness of the numbers cannot be verified i n the absence of the ovi^naX siddhdntas which are now lost. B u t we can guess the probable values as we are sure of the relative luminosities of the planets. T h e numbers seem to have been misplaced. T h e y should he dvddasa, tithi, manu, ravi, asta, tithi (12°, 15°, 14° 8°, 15° for M o o n etc.) A l l siddhantas give 17° for Mars instead of 15°. T h e rest are nearly correctly given, according to one siddhdnta or other.
d c s y i v i * M m u i i ^ < i i l 5 ^ ^ g r a n ? ; || w< \ Conversion of time-degrees into distance-degrees 59. M u l t i p l y the degrees by 3 0 0 a n d d i v i d e by the vinddis o f o b l i q u e ascens i o n a l d i f f e r e n c e o f the sign, r i s i n g at that m o m e n t , (near sunset o r sunrise, as the case m a y be), a n d get the respective degrees. W h e n the distance betw e e n the s u n a n d p l a n e t is that m u c h , the respective setting o r r i s i n g takes place.
Note 1. This work is what is known as the conversion o f time-degrees {kdlabhdga) into degrees o distance on the ecliptic (ksetrabhdga). Since the rule has to apply commonly to Saura on the one han and the Vdsistha-Paulisa on the other, it has been placed last. Note 2. Since the positions of the Sun, M o o n and planets are given only on the ecliptic, this conversion is necessary to measure distances.
^TTc^i^^MKI^VimMId
II
I
6 0 . ( T h e r i s i n g takes place i n the east w h e n ) M e r c u r y , V e n u s , M a r s , J u p i t e r a n d S a t u r n are less i n l o n g i t u d e t h a n the S u n , a n d the S u n is less t h a n the M o o n i n the opposite d i r e c t i o n , (i.e., west). M a k i n g the c o m p u t a t i o n a c c o r d i n g to the i n s t r u c t i o n g i v e n above u s i n g the l a t i t u d e etc, the p h e n o m e n o n s h o u l d be p r e d i c t e d . 60a, A,B,C,MlchTHI: 59a, A , t W ; B , ^ ^ . B . f e l ^ b. A, *=I<^HI^1; B , ac;:«|His1'
b. A.B.•a^l'^llHI; C.^amT^; D . [ W T U ] B.WFRT^
XVIII.63
XVIII. VASISTHA-PAL'LI.'^A-RISING AND SF.ITING
3.53
Note. T h e verse is very corrupt. But knowing what it is aboiit we can give tlic meaning, making possible corrections. T h e rising is mentioned here as it is more important for application lodhnnuasdstra etc. But rising also envisages setting with the word 'less' taken for'more', and inorc' I'or'less'.
Example 1. The latitude of the moon is 3° 45' N. The maximum half-cara of the place is 150 vinddis. The oblique ascensional difference of the rising sign is 280 vinddis near sunset. Find the ecliptic distance betw the Sun and Moon, for the heliacal rising of the Moon. T h e time-degrees for the moon is 12°. T h e heliacal rising of the M o o n lakes place in the evening. R sin 3 ° 4 5 ' x 150-^480 = 7 38/60 X 150 ^ 480 = 2° 23''. This is additive since the Moon's latitude is north, and the phenomena pertains to the west. Therefore, the corrected time-degrees = 14°. 14° X 300 -i- 280 = 15° is the distance on the ecliptic between the Sun and the M o o n , required.
Example 2. For the same place, (i.e., max. half-cara 150 vinddis) find the ecliptic distance required fo heliacal rising, given: the latitude of Mars 1° 15' N, and the oblique ascensional difference near sunrise at th time is 330 vinddis. T h e latitude correction to the time degrees (17° for Mars) = R sin 1° 15' X 150 ^ 480 = 2' 33 " X 150 480 taken as degrees = 48'. As the latitude is north, and the phenomenon pertains to the east, (since it is the rising o f Mars that is considered), it is subtractive. .-. 1 7 ° - 4 8 ' = 16° 12'is the corrected time-degrees. 16° 12' X 300-^-330= 14° 44', is the distance on the ecliptic required.
M^y"ii(Mdd^ ^ ( % ) (^s«rf^)vJiimRt^ ^ ( w « r a t ) [ c ? n ? : ] ^ d f M < « n r r T r ^ II
1 1 ^ ^ II 6 1 . F o r the g o o d o f his disciples, V a r a h a m i h i r a , Joelonging to the A v a n t i c o u n t r y (Ujjain region), wrote this section d e a l i n g w i t h the star-planets, briefly but w i t h the constants a g r e e i n g w i t h the o r i g i n a l {siddhdntas). 62. A l e a r n e r , d i s c o u r a g e d b y the c o m p u t a t i o n o f M a r s by the a s t r o n o m e r P r a d y u m n a , the c o m p u t a t i o n o f J u p i t e r a c c o r d i n g to the Saura siddhanta, a n d
PANCASIDDHANTIKA
354
XVIII.63
the c o m p u l a t i o n o f M e r c u r y by V i j a y a n a n d i , c a n have recourse to this section o f the m a n u a l . 63. B y V a r a h a m i h i r a has been seen, (i.e., written) (this karana) easy to u n d e r stand, Note 1. Verses 61 and 62 clearly close the section dealing with the star-planets. Since V M says that he has improved on the earlier authors, he must be referring to Chapter X V I and X V I I , dealing with the Saura. His reference to his improvement on the Saura itself in the case of Jupiter must refer to the btja correction made by h i m in X V I . Indeed, his dissatisfaction with the Jupiter of the Saura is reflected i n his formula for computing Jupiter to give the years of the sixty-year Jovian cycle, given i n his Brhatsarnhitd, i n the chapter dealing with the motion of Brhaspati (Jupiter). As for chap. X V I I I , he could not have meant the Vdsistha-Paulisa star-planets there as an improvement, they being crude.
Note 2. Verse 63 evidently closes the Pdncasiddhdntikd, as indicated by the Vasantatilakd metre o the verse instead of the regular dryd metre. B u t unfortunately the last three feet are missing. Perhaps it is a purposely done 'black-out' by a later astronomer-scribe, to add his spurious verses 64-81 i n condnuation (see below) and, unfortunately, only his manuscript has survived as the archetype of the few extant manuscripts.
61a.
B.3Tra^:^aJTWr:
b. A.fecq^o. B . M s ^ H " ; B 2 . 3 . f e ^ 1 ? ° A.B.°sldH^'l^°; C . ^ [TO] D.°«J [f^q^TRgst?!^] c. 62a. b.
A.B.fe
d. C . M ^ e f i l i o m . ^ . A . B . W ; D. drops the word. 63a.
A.B.C.ft?toT. B.^rat^n
b-d. A . B . C . gap indicated for the three quarters of the verse. D. Ignores the expression •y
A.3I?ra. B.gpT
in this verse and constitutes a new
A.B.^(B3.4t%)
half verse for the previous verse,
A . ^l^«c||cfNijC; B . ^ ^ J p f ^
c(l II In consequence, the
a s : ^ *TJf: ^ f H <
'T5if!r?;5
numbering of verses than in A is one c-d. A . ^ ^ ^ T O R f S ; B . f ^ ^ ^ T O R J J ; D . ^ « R T :
less in D, hereforward.
Thus ends Chapter Eighteen entitled '(Vasistha-) Paulisa-Siddhanta - Rising and Setting of Planets', in the Pancasiddhantika composed by Varahamihira
Chapter Eighteen 64-81 (A Spurious Supplement) Computation of the Star-Planets in brief
Spuriousness of this Section That verses 64-81 of Pdnrosiddhdntikd form only an appendage to a manuscript of the work and not a composition of Varahamihira, is evident from its occurring after the work has closed in the customary way with a concluding colophonic verse, with its metre changed to Vasantatilakd from the dryd metre in which all the previous verses of the chapter had been couched, and with the author speaking about himself in this concluding verse. It is also to be noted that the set of verse 64 to 81 begin with a new salutation. H a d these verses really belonged to the PS. the customary colophonic verse must have come at their end, and significantly there is no such colophonic verses at the end. Further, in verse 6.5 it is said that the author considers this as a superior set containing a previous method or matter and thathe was giving it out, with a liberal mind, to the generality of astronomers without hiding it from them. But actually it is inferior stuff, and can give only very rough results since the equation of the centre is dispensed with, only the equadon of conjuncdon being given, which makes it valueless. T h e author boasts here that he has made things easy, and takes credit for this which only a novice could have done. Fancy V M speaking thus, when in verse 62 he is so intent on accuracy that he says. "Let people who have been dissatisfied with the inaccuracy of astronomers like Pradyumna, Vijayanandi etc., have recourse to his treatment of the Saura." Further, there are mistakes in the computation of Venus and Mercury, unpardonable in any astronomer. (See below, Note 2 under verses 70-72). T h e above-mentioned aspects of verses 64-8) have escaped the nonce both of T S and N P , who both take them as genuine compositions of V M . Moreover, N P obliterate verse 63, the concluding genuine verse oiPS. by not including it in their text, though a few words thereof are present in all the manuscripts of the work and in some of the manuscripts it bears a serial number; thus, verses 64 to 81 are numbered in N P as 63 to 80. These spurious verses are dealt with below for the sake of completeness of the text as found in the parent manuscript of the PS from which all its manuscripts available now have been derived.
Salutation 64. S a l u t a t i o n to the g o o d p e o p l e , ever interested i n the welfare o f others, w h o e v e n w h e n k n o w i n g the faults o f others, a n d e v e n w h e n t h e r e is a n o p p o r t u n i t y , d o not m e n t i o n t h e i r faults, b u t p r o c l a i m t h e i r g o o d qualities.
356
PANCASIDDHANTIKA
XVIII.66
6 5 . In e i g h t e e n arya verses, V a r a h a m i h i r a , w i t h o u t feeUng any j e a l o u s y , gives this i n a n u a l to the w o r l d , e n d i n g w i t h the t r e a t m e n t o f the planets, t h i n k i n g that it is g o o d . Note 1. T h e emendations are TS's also. Note 2. Verse 64 is a paranomasia and means also, "Salutation to the good science of astronomy called technically parahita-ganita (prevalent i n Kerala i n South India), which at the beginning deals only with mean motions, though knowing its defective nature as not being true motions, and which furpisjies tabular values o f equations going by the means, mandajyd (R sine table of the equation of the centre), karkijyd (R sine table of the equation of conjunction for the anomaly 90° to 270°) and makarajyd (R sine table of the equation of conjunction for the anomaly 270° to 90°)." T h i s meaning being not obvious to the ordinary reader, I give the phrase by phrase meaning:
prastdve (granthdrambhe), paroksasya (asphutagrahasya) dosdn (asphuiatvddi-dosdn) jdnann a pakti (jdnann api na vadati, madhyagater eva prakrtatvdt) gundn (mandajyd, karkijyd, makarajy gunasabda-vdcyd jydh) prathayati (prakatikaroti, gariayitvd likhatitt ydvat), tasmai sujandya sobhanajanmane) parahitdya (parahitaganitdya) namah (namo 'stu). Note 3. These two verses also form part of the 18 verses mentioned. So, actually there are only 16 verses (66 to 81), giving the computation.
'Sun's day' (Sauradina) 6 6 . F r o m the e p o c h , to t h e time o f c o m p u t a t i o n o f the p l a n e t , find the Sun's degrees passed. T h e s e are to be t e c h n i c a l l y c a l l e d 'days', ( a n d u s e d i n the c o m p u t a t i o n ) . F i n d the r e m a i n d e r after d i v i d i n g by the cycle n u m b e r g i v e n f o r the respective star-planet. T a k e the 'days' o f m o t i o n c o r r e s p o n d i n g to the set o f m o t i o n s g i v e n to the respective p l a n e t . T h e s e are degrees o f plarietary m o t i o n . A d d this to the S u n ' s l o n g i t u d e . T h e t r u e p l a n e t is got. Note 1. T h e days' mentioned here is only what is called sauradina ('Sun's day') as distinguished from the sdvana or civil day, and are actually degrees. (This is like the word 'light-year', which is used as a unit of distance), T h i s instruction is given with respect to all planets. 64a. h.
A . B . ^ : A . - ^ H t I I P j ^ ; B.-^HHIPm
c. A.^Tr#T?^Tr?ftt|% (A2.°ftf?r?^); ( B . ^ ° ) .
A.^a^ c. B.^«Wlf?T. A.TiynacR^; B.'iUIIW^ tl. A.^gSRWTO; B . ^ g ^ T O . B.MRr^cii^i 65a. A . ° ^ ; B.°^«rf b. B.^raif^:
d. B.fro^R: 66b. A.f53^a
B. ° ^ 5 I ^
XVIII.69
XVIII. SI AR-PLANF.TS
Note 2. T h e cycle given for each planet is only the period of the planet's synodic revolution converted into the solar days, i.e., it is the synodic period X 360° 365-15-30, nearly. W h e n so converted, we have: The regular synodic days: Converted into solar days:
Mercury 115-52-45 114 6/29
Venus 583-55 5751/2
Mars 779-57 768-45
Jup398-53 393 1/7
Saturn 378-6 372 2/3
The second set is given for the respective planet, saying that the synodic period is so many 'days'. Not knowing this T S have remarked that they do not understand why there is so much difference in the periods from the regular days generally known. Also, they say they carmot dismiss them as wrong, since the numbers given are checked in the computation itself. (See pages Ixm-lx/t' of Introduction in TS's edition). N P have understood that the cycles are i n solar 'days'. But, they have remarked that ' V M ' has confused the days and degrees, not realising that there is a purpose in giving the cycles i n the solar day units. These units have been used because, now, the 'days' and the 'degrees' will have the same meaning, and they can be combined without, at every point, instructing it. T h u s , ultimately, the combined value is the degrees of the true planet.
Example 1. Days from epoch 1,20,553. Find the 'days', and assuming the Sun at epoch as zero, find the Su at the end of 1,20,553 days. 1,20,.553 X 360 ^ 365-15-30=
1,18,187.5'days'.
This plus zero, and divided out by 360° = 17°.5, Sun's longitude.
[ITSI^:]
Motion of Mars
67. S u b t r a c t 6 3 2 9 f r o m the 'days'. M u l t i p l y the r e m a i n d e r by 4 a n d d i v i d e out jby b 6 (i.e., 56 u8 enh -c3 'days' 6it09 nithe 7.o d5nA by . (i.e., lags T fhta6 ek rl0ie°a56 15° the ,cthe 6a'days' 0lba°setting ern , eh9om i0m n°it aad,ilgoes 5ny a0dends). n°oed,fr 7 bgoes conj.) a0en°hdi,Innrespectively. didnifv188, to the oirdceM o Sbn au 108, yrjnsu4. .by n cT73, T t 15°, iho henes6.ena8nare ,itd220, sets becomes theheliacally, 'days' 'days'observable, M after a rasncdlags on i n-
PANCASIDDHANTIKA
358
XVIIL69
Note 1. I generally agree with TS's emendations. B u t i n verse 68,1 give sadvisayaih for sadtrinLsat savaih. which latter is both meaningless and has one-mdtrd extra. TS'ssadvargaih does not agree with the last saptdsiakena, for the numbers should agree or at least nearly agree, saptatyd dvyadhikayd h been emended by me into saptatyd tradhikayd, and khdbdhi into khdsvi. These will not only make t total correct, but also bring about agreement with the Siddhdntas, which all generally agree with the actual as given by modern astronomy.
1 Degrees moving behind 'Days' given Actual 'days'
3
2
4
5
- 15° asta..
- 60°
- 60°
- 90° vakra
56 54
188 188
108 106
73 72
•
- 50°
6
7
- 70°
- 15° asta.
220 220
56 54
68 75
Total - 360°
769 769
T h e great difference i n 'days' between the 68 given and 75 actual must be explained by their following next to the retrograde period, where even a large number of days can produce a very small difference in degrees. So, correction to whole degrees can produce this difference in days. Note 2. Verse 67 means that 6329 'days' after epoch, there is conjunction, which repeats after each synodic cycle. T h e cycle for Mars is 7 6 8 % 'days'. So instead of dividing by 7 6 8 % , we are asked to multiply by 4 and divide by 3075. T o get back the true remainder, the remainder here is divided by 4. Note 3. T h e following points deserve to be noted; (a) I n the case of all star-planets the total 'days' should be equal to the days o f the respective cycle. (b) In the case of the superior planets, viz. Mars, Jupiter and Saturn, the degrees are all negative and add upto — 360°. W h e n the given degrees is numerically greater than the corresponding 'days', (for e.g., - 90° for 73 'days' here) the planet is retrograde. (c) T h e heliacal setting and rising are at the beginning and end of the cycle for all. But, for the two inferior planets, viz. Mercury and Venus, there is another setting and rising at inferior conjunction when the two are retrograde. 67a. A . ! P ^ ;
B.
b. A . f ^ ; B.fcflf^. B . f s ^ . B.^ki^rfl
69a. A.^?3te; B . ^
c. A . B . ' J T l ? ^ . B . ' ^ f ^
a-b. A.'HcT^; B.'HWci
d. A . M ^ ° ; B . M h 1 ^ ° ; . B 3 . H ^ 1 ^ ^ I
68a. A.%B(<^^Rd«J^; B.M^Iel^Rdaiq^; CI-^cI'lRd*^; D.N4R!^lRd»T?TI b. B . 5 ? l . A . B . ' j f ^ i O T J T ^ :
C. A . ^ T ^ T ^ ^ :
d. A.B.'^°Mrycb|A||
(B."^^:)
b. A . B . C . D . ^ i a r f s ^ : (B.-f^l^:) A.B1.2.Wirc;yi:; C . D . ^ f ^ : c. B3.3?Mft(Tl c-d. B . W ^ d. A.B.fwmfH:; C . D . M ? n # :
359
XVIII. STAR-PLANETS
XVIII.72
(d) T h e rising and setting are given by observation at different regions and different conditions of the atmosphere, and therefore vary among the siddhdntas. (e) In the case of the inferior planets the degrees should add upto zero. W h e n the degrees are positive and greater than the days, the planet is gaining upon the Sun, and the total gain is its elongation. W h e n the degrees are less than the 'days', the planet is lagging behind. W h e n the degrees are negative and numerically greater than the 'days', the planet is retrograde and comes at the middle o f the cycle, i f the cycle begins and ends at superior conjunction. Example 2. Compute Mars at 1,20,553 days from epoch. T h e solar days are, 1,20,553 X 360 -r- 365-15-30 = 1,18,817.5, and the Sun is 17°.5, taking the sun at epoch as zero, which it nearly is as already shown. 1,18,817.5 - 6329 = 1,12,488.5 1,12,488.5 x 4 3075 leaves the remainder 1004. 1004-r 4 = 251, real remainder of'days'. D u r i n g this period we get the movement: - 15° 7 'days' remaining, total - 79°.
in 56 'days', - 60° i n 188 'days', and - 4° for the
A d d i n g - 79° to the Sun, 17°.5, T r u e Mars = 298.5°.
jpichK^^
HcJMI'^^iP
Vndiii>|4,?(^
||V9o
I
Motion of Mercury 70. S u b t r a c t 14,681 f r o m the 'days'. M u l t i p l y the r e m a i n d e r by 29, a n d d i v i d e o u t b y 3 3 1 2 . T a k e the r e m a i n d e r a n d d i v i d e by 2 9 . T h e 'days' f o r M e r c u r y i n the cycle is got. 70a. A.ft^lftr. A . B . 7 % ; D . b.
A.B.1.2.HclHctJjiJ|^. A.S.1.2.°TPfm%
c. c-d.
71a. A.cJ?lf^°. A . B . # n : b. A.^nPlPTFWaRTT:; B.T^=Tmte=raat?TT:; CR^Cf^^l^^aWT:]; D . ^ [f^T^WaWTI:]
A.B.^ B1.2.°^TPT-gap-c)RJ°;
c. A . T R ^ ; B . ° f ^ : - g a p - ^ ; D . [°f^T*#]
B3.
d.
°^i;if?I- gap^cii^°
A . ° ^ . A . B . W r a : (B.W:); C. [ ^ f ™ : ] ; D . [ ^ W T I : ]
PANCASIDDHANTIKA
360
XVIII.72
71. I n 10 'days' M e r c u r y falls back b y 12°, a n d rises i n the east. I n 14 'days' m o r e it lags by 9 ° . T h e n i n 18 'days' it gains 9 ° . T h e n it sets, a n d i n 30 'days' gains 2 5 ° a n d rises heliacally. 72. T h e n i n 18 'days' it gains 9°. I n 16 'days' it lags 12°, a n d sets i n the west. T h e n i n 8 'days' i t lags 9 ° , a n d gets i n t o c o n j u n c t i o n . Note 1. I n verses 71 and 72 my corrections are based on the need to conform as nearly a« possible to reality, and they are also, as far as possible, kept close to the text, sardsvih, 25, may also hejaldsvih, 24. T h e values are:
1 Given degrees
- 12°
Given 'days' Correct 'days'
vakrasta 10 8
2
3
- 9°
+ 9°
vakra 14 16
18 18
4 + 25° (? + 24) asta 30 30(?28)
5
6
+ 9°
- 12°
18 18
vakra 16 18
7 - 9° vakrasta 8 6(?8)
Total 0°
114 114
In columns 6 and 7 it should be - 9° and - 12°, or adeast - 10° - 11° though the text letters are unmistakable. Note 2. T h e constant for subtracdon, 14,681 shows that the planet is i n superior conjunction, since the epoch position of the planet must agree with that given by modern astronomy and other siddhantas, atleast within a few degrees. (See table appended). If so, the Table of cycle motions given should begin and close with superior conjunction. B u t i n the Table given, the cycle begins and closes with the inferior conj. as can be seen from the retrograde motion with which the Table begins and ends, and the most rapid motion (aticdra) coming at the middle. T h e astronomer o f very inferior calibre who has made this interpolation, has been misled by the two sets o f heliacal rising and setting i n the case of the inferior planets. Mercury and Venus. H e has wanted to begin the motions with the rising i n the east and setting i n the west, to fall i n line with others, not realising that this occurs d u r i n g its retrograde motion which falls at inferior conjunction coming i n the middle. T h i s is another proof that V M cannot be the author of this set of dealing with the star-planets. (This author has committed the same mistake i n the similar case of Venus, where the jnistake can be seen glaringly when the true Venus got is compared with that o f the other siddhdntas or modern astronomy). I f he does want to begin with rising in the east and end with setting i n the west, he must begin and end his cycle table with the inferior conj. and to this he must add to the days to be subtracted half the cycle days, equal to 57 3/29 days. (The cycle days = 3312- 29 = 114 6/29). 72a.
A.315[Rf^:^; B.315TS:?Tfil:q^:;
b.
A.B.C.D.^te^lf^aiS^i^
d.
A.ITOT; B.1TOT.
A.C.D.1#T^
XVIII.75
W i l l . STAR-PLANKIS
Motion of Jupiter 73. D e d u c t 16,552 f r o m the 'days'. M u l t i p l y the r e m a i n d e r by 7 a n d d i v i d e by 2 7 5 2 . D i v i d e t h e r e m a i n d e r h e r e by 7. T h e 'days' f r o m c o n j u n c t i o n a r e got. 74-75. A H degrees g i v e n are to be s u b t r a c t e d f r o m t h e s u n . I n 16 'days' it moves 12° a n d rises i n t h e east. T h e n i n 54, 70, 4 9 , 88, 4 0 days it moves 4 4 ° , 64°, 120°, 7 6 ° , 3 2 ° . T h e n it sets i n the west, moves 12° i n 16 days a n d joins the Sun. Note 1. T h e first foot of verse 73 is faulty containing 3 mdtrds extra, and corrupt. So it has been corrected. T h e rest o f verses 73 and 74 are TS's emendations. I n 75, all emendations are TS's, excepting those for grammar. Note 2. T h e cycle is 2752 ^ 7 = 393 1/7 days. T h e days and degrees are: 1 Degrees Given days Near correct'days"
- ]'2° asta 16 16
2 - 44°
3 - 64°
54 •54
70 70
4
5
- 120° vakra 109 109
73a. A.B.C.Tf?%5gf|°; D . T ^ ^ o A.B.C.D."?RlfEf«T: b. A . m B . f | ^ A . ^ ^ ^ ( A 2 . a ) Tif^f ( A i . ^ ) B.fsmgnft?^ c. A . B . ' g ^ d. A.R)
- 76°
b. A.5n^
- 32°
88 88
40 40
- 12° asta 16 16
c. B . Hapl.om.of+dftt^^: d. C . D . ' ^ l M ^ ^ : 75a. B . ^ t ^ : a-b.
A.B.C.D.^J'?rafeRteT(B.°*gI'=)
b. A.B.C.Wrarf^:; D . W # F : c.
A.B.°ff?tlf°
d. A . B . ° ^ ( A 2 . 3 ; B l . 2 . ^ ) A.wn
74a. A.^51%ig; B . ^ H i ^
7
6
( B l . 2 . ^ B3.?n)
A.°?TP^T^ifg; B.°?nf^1^; c.D.°^*r^
Total - 360° 393 393
P A N C A S I D U H A N 1 IKA
362
'c<^«^^r
*f?f(8Z[jT:)'
'<^dira[^:'^(M^cblR?ivlrt)
XVIII.78
||V9V9
^6
Motion of Venus 76. D e d u c t 1,18,122 f r o m the 'days'. M u l d p l y by 2 a n d d i v i d e by 1151. T a k e the r e m a i n d e r a n d d i v i d e by 2. W e have the 'days' f r o m the c o n j u n c t i o n o f Venus. 77-78. I n 5 'days' V e n u s lags by 9°, a n d rises i n the east. I n 15 days it lags 21 °. I n 64 days it lags 15°. I n 164 days it gains by 3.5°. T h e n it sets i n the east. T h e n i n 4 0 days it gains 10°, a n d j o i n s the S i m . T h e n , m o v i n g i n a c c o r d a n c e w i t h the r e v e r s e d o r d e r o f the days for cycles g i v e n , it rises, after the days given f r o m setting to c o n j u n c t i o n (i.e., 4 0 days), i n the west, a n d moves till it reaches the setting i n r e t r o g r a d e ( a n d g e t t i n g i n t o the i n f e r i o r c o n j u n c t i o n ) section.
Nnte 1. I have correctedTOU!"nf/uinto dhrtfndu for agreement with the Sun in superior conjunction which alone fits. TS's correction (also NP's) mahindu does not bring agreement with the Sun either at superior conj. or inferior conj. There can be another possible correction matindu (mati is 8). In the 64 days, flanking the retrograde, the days may be a little more or less, since a small error of observation can produce a difference of a large number of days. T h e lacunae is filled by me with sapancakmtrirmat, meaning 35°, to fit the number of degrees wanted to make up the total zero, and fitdng the number of days given. TS's emendation, krtdstabhih will be far from fitting the total.
76a.
A.B.°^ftf!t^; C . D . ^ ^ ^ R f ^
b.
A.B.^^;^:^^(B2.«r)%
78a.
A.B.C.^(B.^)B5^^B^(B.^);
C. A I . ^ T : . Al.
b. A.RWdldl; B.P)<^ldld"1
d.
c.
77a.
B.°^^: B3.^:ig^
b. A.B.1%f8^3m ( B . ^ ) #T; c. A.f?rs^ d. A . B . °f|[f^:^, rest of verse missing: D . ^ : f ^ (t] M ^ : ^ C.°f^: [^g,
c-d.
A.H^^fctfs^ftt?]^^; B.H^'ifMfelf^TTW^ C.3?^!%f=R?mcH,^;
d. A . B . ^ (B.«raf?T)
W
A . B . f ^ « n # : ; D. [1^=7nT«T]"fTf?T:
XVIII.79
363
XVIII. STAR-PLANETS
Moreover, their filling the lacuna by sesuh, meaning 5° is quite inadequate to make up zero. I have emended sastdslakena into pancdstakena, meaning 5 x 8 = 40, which will fit the number ot days. Also, 10° synodic motion there requires 40 days and it is also the period from setting to going into superior conjunction, sasta is patently wrong spelling, and sastdslakena is meaningless. But T S and N P keep it, which is wrong. That this is the segment of heliacal setting to conj. can be inferred from vdayati given for the next segment, and 10° for heliacal setting and rising at superior conj. is given by many siddhdntas. T h e other minor emendations are TS's. Note 2. 1,18,122 seems to be a very large subtractive constant, equal to more than 300 years, while all others are very near V M ' s time. But I cannot think of any other number to fit. Note 3. T h e maximum elongadon is seen to be 45°, correcdy. (Cf. Table).
1 Degrees passed Given days Correct days
2
3
- 9° - 21° - 15° vakrasta vakra 64 5 15 5 15 64
4
5
6
7
8
+ 35° -1- 10° + 10° -1- 35° - 15° asta asta 164 40 40 164 64 164 40 40 164 64
9 - 21° asta 15 15
10
lota!
- 9° 0 vakrasta 5 576 5 576
Note 4. T h e remark about Mercury, that the cycles begin and end with the superior conjuncdon according to the subtractive constant given, but the motions in the cycles begin and end with the inferior conjunction, holds in the case of Venus also, showing thereby that the author is an ignorant imposter, and cannot be V M . T o correct the fault, 2 8 7 % 'days' should be added or subtracted from the subtractive constant. Example 3. Compute Venus at 1,20,553 days from epoch. If the subtracdve constant given i n t h e text is used, 1,18,817.5 (already found in the example in M a r s ) - 1,18,122 = 695.5. This X 2
1151 leaves the remainder 240.
This divided by 2 gives 120 'days' gone i n the cycle. W e have for the first 5 days — 9°, and the next I S d a y s - 21° and the next 64 days - 15° and the remaining 36 days, 36 X 35 ^ 164 = 7° 40', totally — 37° 20'. A d d i n g the Sun 17°.5 already found i n the example for Mars, the true longitude of Venus is - 20°, i.e. 340°. (The example i n the Saurasiddhdnta for the same date has given 46°.) T h e error in Venus, in using this method here, is 66°. O n the other hand, let us use the cycle order re-arranged to begin from superior conjunction. It is 10° for 40 days, 35° for 164 days etc. We have 10° for 40 days, and for the remaining 80 days, 80 x 35 164 = 17°, total 27°. A d d i n g the Sun, 17°.5, we have, true Venus, 44°.5. T h i s is close to the correct 46°. T h i s exposes the ignorance of the imposter.
[VlPl^d:]
364
PANCASIDDHANTIKA
XVIII.81
Motion of Saturn 79. S u b t r a c t 16,518 f r o m the 'days', m u l d p l y by 3 a n d d i v i d e by 1118. T a k e the r e m a i n d e r h e r e a n d d i v i d e by 3. T h e days left o v e r i n the cycle are got. I n 18 days S a t u r n lags b e h i n d by 16 V2°, a n d rises i n the east. 8 0 - 8 1 . I n 9 8 , 14, 113, 98, a n d 13 'days', it falls b e h i n d 9 0 V 2 ° , 13°, 120°, 9 1 ° , a n d 12V2°, respectively. T h e n S a t u r n sets i n the west, a n d j o i n s the S u n p a s s i n g 16 V2° i n 19 days.
Note 1. In verse 79, satkavarka is patently extra, forming syllables not required for the foot, and has been deleted by me as also by T S . astdbhih is corrected into astih by me, as also by T S , to conform to grammar and facts. In the rest of the m i n o r corrections there is no difference between our corrections.
In verse 80, I have retained the dvi i n dvyunena, while T S have made it dyu, meaning one day, which is not necessary, and which leads into trouble later, needing further correction. T h e minor correcdons are common to both. In 81, I have filled up the lacuna by {tyd sdrdha), while T S have made it {tibhirardhd). T h e word is atijagati, and not atijagatih, which alone can justify TS's tib Also, only sdrdhdrka can mean 12 V 2 , but their ardhdrka can mean only 6.1 have corrected navati int navaku keeping the nava. But T S have corrected it into atidhrti, making unnecessary changes in the lettering, though both of us mean the same. The rest of the corrections are minor, and are acceptable.
79a. A . B . C . ° T m H ^ ^ ( B . ^ ) ? R T I ^ b. B.^Hl^dlPJI (B3.1w) c. A.^^lkFt. A . B . t ^ f w i f ^ (B1.2.f«r:) d. A . C . ^ « # ; B.^arej%; D . W « l f ^ 80a. A . l ^ i f ^ ^ o ; C.D.^T^rfri^:
81a.
Al.WT—^
a-b.
A.1.3.°^feT(B.^)^?Wt(B3.^); C.°<44HWi^c^dl; D . [ ° ^ W # r e T ^ ]
b. Al.B.Hc|fd(^l^i!^'l; C.5f!T«p%TJ?; D . ^ [ ^ ] M % F I
b. A l . B . C . D . 1 ^ :
c. A l . ' f f l ^ i y k ; B.^araJ^k
c. B2.^?J=iRif
d. A1.T%:
d. C D . [ ^ ] ; Ms. A 2 . breaks off here. Al.TTlf?!
XVIII.81
X V l l l . SIAR-HIANETS
365
Note 2. T h e ioUowiiig is ihe table of motions:
1 Degrees passed Given days Correct days
16'/2°
(ista. 18 18
2
3
4
5
- 90 ' / 2 °
- 13°
- 120° vakra 113 113
- 91°
98 98
14 14
98 98
6 -
12'/2°
7 -
161/2°
asta 19
13 13 1/2
181/2
Total - 360° 373 373
Note 3. T h e supplementary verses end abruptly without the usual verses giving details about the author, his parentage, date o f writing etc. Note 4. T h e colophon is simply "The star-planets of the Paulisa siddhdnta ends." B u t after this is found details about the scribe, his lineage, his time o f writing, viz. 1673 Vikrama Sam vat, and 1538 Saka, equal to 1616 A D , and the purpose of his copying the work, ('for his own reading and helping others', i.e., other astronomers).
C o l . A I . B . D . # 7 ? l f e ^ ( B 3 ^ ) -mm- ( B addsiJ^^ j
Post-C^olophonic statements:
qfecT ^
I
4fed'llft-<: I dWIcH^^H t'ikM ^^^^^^^^ Wm
I
B1.2.3. ^^Wl4c|
(dRsldl 11 SHH^MK-PlcflfH-ll s|-«M^fctij_w4iidRsld II ?fl II C . IRt c|<|^rHRl mM\ I D. 4c4Ml4c|<|^iMr^{4)dl 4-^f^^lPd*l ^Hlkll ||
Thus ends Chapter XVIII 61-81: A Spurious Supplement entitled 'Computation of the Star Planets in Brief, to the Pancasiddhantika of Varahamihira
APPENDIX I INDEX OF VERSES 3mmteiT9.7
3
^ F J i r a r ^ ^ 18.47
3 T ^ f ? ^ # s n 14.3
31l*
^?5marnj?5fKT i s . 4 7
3T^tfe((yiwi<;q> 14.25
3TRRH^: 18.61
*Hcii
3{UjMN
3{IV(rlMl^ki'«l
*4.
3rf«mra%5i.i3
$8thlf^-^m 4.39
chftlui ^ n t : 18.45
3Tfijm«*lH 15.24
?gi?lfSTjOTtq 4.5
* M l i l ^ 3.36
3T«zrsf?TH 18.14
?gr
^ng?Pn*4.53
3Tqqt1^?^qt9lH, 18.41
^sfftt^^Esn
3i^etl* ^ft "Hc^ 18.5
jai^{
3rt?rim?n 1 8 . 8 I
cbAdRj^-ti) 18.31
^4.50 14.10c
fa-rc>T<(i!^l^1^: 6.5 IpBTOiHBR rlTlR 14.32
ST^g^S^ft^n 18.13
Ip^Sflt
3 i ^ l t l 4 i l : 10.7
StFST ^^^\H\
*HUUHjrl2.1
S P r a r q ^ ^ : 13.7
aFOSrteF 15.25
*d<'ftc^i
^drratTq^ 10.6
3{HMH<(cl^ 5.5
*rta8.12
fK^T Til^iyK^I'd^ 18.37 fn.
15.10
18.52
amRTOT^rPt 5.1
W^JH^dlJ? 4.22
g ^ T T ^ T j ^ M 7.3
^ • x ^ M i ^ l 9.13
31M
^l^-MliRlldl 18.22
gW#TT«ZT: 18.8
3=lf«PCra^^ 14.29
3^?T:Rrmi8.7
ar^^^lfg 15.6
g?^n^5njf?r^ 8.10
aii-^-yi* 10.5
3 ? q t # d ^ W i 15.23
^4^^
^551^16.10 l§reR5WS»pT 8.4 IslRrtHW)
343>'^4l«W^ 3.20 arif-iild^^ 8.18
^ r a ^ 5.10
4.34
sR^ (ciVfiaVWl 3.16
«4cHi(|:7Tf?r8.7
13.31
13.8
3?f?^cRT5rr: 18.49
3.2
'I'UH^fd 13.4
3T5Rf?ra48.16 ; 9.26
Ti^if^?ra^ 18.34
MfuidfiH"UlM(rl«ar 14.41
3?f5En^ 15.29
^^vcfifs^lt 13.15
MrlMiWI^idl 12.5
34(c»l^t1WW 15.2
^ f ^ " n ? r a 18.58
'lldl^V'^Htld 18.55
3lRlVl*tl
Ti|5^
TRIT^
^ H U ' J I f^WilVli 3.28
XrggHeit
•nfefgfrn8.43
o!RBfe5Sr 5.9
18.80
'Q;^ftr9n^ 16.1
27
TJorf^lfePJtnrfTl 3.17 Tjawfertqi^ 14.27
3
T f ^ ^ ^ 18.43 fn. m/KflHcJcf)! 4.10
3
2.5
«h'*
*i*."*id4<4l 17.14
PANC.ASIDDHAN IIKA
368
dteHlcH^frh^m 9.14
^ISTR^Wg^SFTi; 18.36
^llfa^Wd^fa^ 18.30
d«^irlP:J4l 4.44
c5!Ft
VTi^Pif: 10.4
d^l
2!!?lf*rsi^ 18.71
dsHeJHM*) 13.33
dlVIM^cbia* 3.3
dWUii^ TpT^if 11.5
l^e^rwidR4Mti 1.6
flqR«in^1^
f<'J.VMI:Tn? 1.11
m q l ^ i ^ l ^ d 2.3
4.47
d
«rnTll 16.4
(^ioq
rl^^ppBtfrf 4.46 d4^«f>(d 17.6
ir«MM 5.3
y^lRVIs^lMI 4.9
HI»fJ1tn##: 7.6
f^^T^TssTRHafrar 8.9
^FgB^6.13
Tlfg^cnrR 14.10
IsFmsgiTOsrnzfT 7.i
t4.j;,r»M>°Mm 10.20
H'HUi(4M'ai^14.16
(<4i4W!Ami*l 7.4
g ^ S T ^ s t ^ 13.39
HHMtiili^Hy 9.18
Rl4c)KJ|(di(Rl 15.17
rB5n5RPHT11.2
r l W ' H ' K I 13.2
l<4H
«R^(3i^^t9T 4.27
dWI^ol lOT^TTr^ 13.24
^g^Mid ^j1(SJIId,9.21
^T3!cT5BM 4.31
rlftjR.l2R^4.33
^ 9 ^ t : ) * W I ^ 13.14
nl<4liT*M 4.51
Htft MM9(Wf^dl 1.18
f g ^ T l ? 18.63
I^RTFsrfgWFt 14.37
HTfiT: chAdehl^l 14.40 t?rf«R[?tW 1.12
^^IW<«!iiCi 15.21
»I<4I&I
1?rfiH^I5f^ 3.32
^<|U||f<4 15.18
1?fftTO^9I8.3
^»Tilf ^ftqT«lf«l% 1.23
frfsrafS*'^ 14.30
^[TOtSeFfsg 9.1
grar^w^'sn 4.55
^aERfferRT 9.19
^ n t i t ^ ? q i r 18.21
STan?fen«Pr^ 14.6
%8ZRT^^3.27
^(4Vl(«
d®ra2;iil*MlcHI 14.19
%8ZF^U?tfS!i 6.9A
?J3;9Tf«r: flT^I^: 2.11
# a t « k f g 14.12
feasra^
(ft'^UI Mf«l4 ^
WT^18-15 vsftciw^rai 16.2
9.27
14.4
ffl4A
l 5 ^ ^ f ? r 8 Z m : 5.4
^ f t ^ T ^ t W H 4.28
H S ^ ^ f l W 18.44
??TT Jai<4>Vft 4.40
^jsq^ S^I^-AehH 12.2
^jftdlSEIsf 4.16
^MimJ**
^PHdl^v^l 18.60
^ I H « * > t : < * ^ 13.40
qlNf^RcJffl 18.50 fn.
t:anl5rf«RTf^^9.25
R V W t ^ W??T:
q f ^ f ^ ^ I T 18.50
^ssrrfgfsri^^ 18.57
|5lidyi< ^ 3 ^ 3 . 1 4
«jfd
BlVUVM^ 9.5
SjeigR^Jlfqcti 14.18
?l^l^%gfn 18.29
1.19
BlVldfamj) 18.59 Bl!|lldf (sUHldld 13.21
flsgvsan Ri4il
?Machtiiui^ 8.8
d<*M^
fslMfy 3.7 ^T«RT* 18.76
ffjjlviill •fejtTRI 4.42 d><^IHl ?lf?T 8.13
^ « » > < | R « ( m 15.5
c^UIHKI
14.36
<
q ^ q q ^
3.33
Ht|
369
AI'P.I: INDEX O F VERSES
•3.19
Hc(
a f ^ q g i ^ r r r f 14.34
Hc|(dlfe|4idl 4 . 1 8 ^yfnWidVWl
9.11
14.33
4clRfi«?: 18.75 ^IFRfg^qXTIgt 3.31
^ r a w j o i 18.67 qeJiJMel^fir: 18.38 ^ c j c i ^ l l ^ l l a j 18.38 f n .
13.13
^frsR^mrPf
•qsqrmRT 8.15
q T « n : •^ifBp^rm
14.5
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APPENDIX II INDEX O F P A N C A S I D D H A N T I K A VERSES Q U O T E D BY L A T E R ASTRONOMERS 1. Makkibhatta's C o m . on the Siddhdntasehhara of Sripati : PS X1II.36; X V . 17-20 (5 verses) 2. Nilakaritha Somayaji in his Jyotirmimamsd : PS 1.3, 4; X V . 2 0 (3 verses) 3. Nilakaritha Somayaji in \\K Bhdsya on the Aryabhatiya : PS 111.21; X l l l . l , 10; X V . 2 0 (4 verses) NTlakantha quotes two more verses from the Pancasiddhdntika, which, however are not to be found in the present edition of the work : See Nilakaritha on ABh IV.IO :
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4. Paramesvara's C o m . on the Aryabhatiya : PS X l l l . 12(1 verse) 5. Prthudakasvamin's C o m . on the Khandakhddyaka of Brahmagupta : PS. X l l I . 2 - 3 , 5-6, 9, 12, 27, 35 (8 verses) 6. Suryadevayajvan's C o m . on the Aryabhatiya : X I I I . 1, 3, 36. 7. Utpala's C o m . on the Brhatsarnhitd of Varahamihira : PS. I . l , 8-10, 16-22; 11.13; III.I, 10, 21, 25; IV.20-23, 27-28. 30-33, 35-36, 38, 41-44; 48-49; V . l - I O ; VI.9-10, 15; V I I I . l , 9-18; I X . l ; X I I . 1 - 3 ; XIII.1-34, 39-42; X I V . 3 3 , 39-40; X V . 1 5 , 18-29; X V I I . 4 - 5 (117 verses)
APPENDIX III B H t J T A S A N K H Y A (OBJECT NUMERALS) Used in the Paiicasiddhantika
aksa aksi agni anka atijagati adri anala abdhi amara ambara arka arnava artha asva asvi akasa asa ina indu indriya Isvara utkrd rtu kara ku krta krti kha gagana guna ghana candra
5 2 3 9 13 7 3 4 33 0 12 4 5 7 2 0 10 12 1 5 11 26 6 2 1 4 20 0 0 3 4 1
carana Jala jaladhi jina tanu Uthi tiksnaiiisu darsa dahana dik dinanatha divapa divakara dhrti nakha nayana naraka naga paksa bana bindu bhava bhaskara bhu mandala manu mahi muni murcchana yama yamala yuga
4 4 4 24 8 15 12 2 3 10 12 12 12 18 20 2 40 7 2 5 0 11 12 1 12 14 1 7 21 2 2 4
randhra ravi rasa rama rudra rupa lavanoda vasu vahni viyat visva visaya veda vyoman sara sasaiika sasi sikhin siva sitakara sitarasmi sitamsu sunya samudra sagara svara svargesa himarhsu hutabhuj hutasa hutasana hotr
9 12 6 3 11 1 4 8 3 0 13 5 4 0 5 1 1 3 11 1 I I 0 4 4 7 11 1 3 3 3 3
APPENDIX IV INDEX O F PLACES, PERSONS A N D TEXTS Cited in Pancasiddhantika Arkasiddhanta : See Saurasiddhanta Arhat, XIII.8 Avand, X V I I I . 6 1 Aryabhata, X V . 2 0 Avantyaka : See Varahamihira U j j a y i n i . X I I I . l ? , 21 Kukaranakara, III..37 Kuru, XV.22 Ketumala, X V . 2 2 Jayanandi, X V I I I . 6 2 Padaditya, III.3.S, (34), PQrvacarya-s, 1.2, Paitamaha-siddhanta, 1.3; X I I . 1 Paulisa-siddhanta, 1.3, 4, 10, 11 Pradyumna, X V I I I . 6 2 (Brhat)Samhita,XV.10 Bhadravisnu, III.32 Bhadrasva, IV.22
Bharatavarsa, X V . 2 2 M e r u , XIII..5, 12, 18, 21, 27; X V . 5 , 6, 11 Yamakoti, X V . 2 3 Yavana, 111.13; X V . 19 Yavanapura, 1.8; III. 13; X V . 18, 25 Romaka, X V . 2 3 , 25, Romakavisaya, X V . 2 3 , 25 Romaka-.siddhanta, 1.3, 4, 10, 15; III.35; VIII.l Latika,XIII.9,17,19.26,29,32; X V . 19,20,23,25 Latadeva, Latacarya, 1.3; X V . 8 Varahamihira, Avantyaka, X V I I I . 6 1 , 63, 65 Varanasi. III. 13 Saka, Sakendra, 1.8; X I I . 2 Savitra : see Saurasiddhanta Siddhapura, X V . 2 3 Saura-siddhanta. Arkasiddhanta, Savitra. 1.3,4, 14; I X . l ; X V I . I H i m a d r i . III. 12 Horatantra, 1.22
APPENDIX V BIBLIOGRAPHY Aryabha^, C. Nilakanjlia Somayaji The Aryabhatiya of Aryabhatacarya with the Bhasya of Gargya-Kerala-Nilakaritha Somasutvan, T r i v a n d r u m , 3 vols., 1930, 1931, 1957. (Triv.Skt.Ser. Nos. 101, 110, 185) Aryabha^, C. Paramesvara Aryabhatiya with the com. Bhatadlpika o f Paramadisvara, ed. by H . K e r n , Leiden, 1874 Aryabha^, C. Suryadeva Yajvan Aryabhatiya of Aryabhata with the com. of Siiryadeva Yajvan, E d . by K . V . Sarma, New Delhi, I N S A , 1976 Brahmagupta, C. Amaraja Khandakhadyakam by Brahmagupta, with the commentary called Vasanabhasya by Amaraja, E d . by Babua Misra, Calcutta, Univ. o f Calcutta, 1925 Fleet. J.F. T h e Topographical list o f the Brihat Sariihita, E d . (with notes) by Kalyan K u m a r Dasgupta, Calcutta, Pub. Semushi, 1973 Nilakap^ha Somayaji Jyotirmimamsa (Investigations o n astronomical theories) by NTlakantha Somayaji, E d . K . V . Sarma, Hosliiarpur, V V B I S I S , 1977 Prthuyasas, C. Utpala Prthuyasas' Satparicasika with Bhattotpala's commentary, Bombay, 1864 Shastri, Ajay Mitra India as seen in the Brhatsaiiihita o f Varahamihira, D e l h i , M L B D , 1969 ( Varahamihira and his times, Kusumanjali Prakashan, J o d h p u r , 1991 Sastry, T.S. Kuppanna Collected Papers on Jyodsha, T i r u p a t i , Kendriya Vidyapeetha, 1989 ^ripati, C. Makki Bhatfa T h e Siddhantasekhara ... E d . with the com. o f Makkibhatta .... by Babuaji Misra, Calcutta, Univ. o f Calcutta, 2 vols., 1932, 1947 Vakyakaraijia with the com. Laghuprakasika by Sundararaja, E d . T . S . K . Sastry and K . V . Sarma, Madras, K . S . R . Institute, 1962
BIBLIOGRAPHY
375
Varahamihira T h e Pancasiddhantika, the astronomical work of Varahamihira. T h e text ed. with an original com. in Skt. and an E n g . trans, by T . G . Thibaut and M M Sudhakara Dvivedi, Rep. C h . Skt. Series Office, Varanasi, 1968 T h e Pancasiddhantika, o f Varahamihira, E d . and T r . by O . Neugebauer and D . Pingree, Kobenhavn, Munksgaard, 1970, 2 vols., Samasasarhhita known through quotations. Tikanika-yatra of Varahamihira, ed. by V . K . Pandit, J / . of the Bombay Univ., 20 (Arts 26) (1951) 40-63 Vatakanika known through quotations Vivahapatala, Mss. Nos. 35161,35162, Des. Catal. of Skt. Mss. Vol. IX, Sanskrit University, Varanasi. Die Yogayatra des Varahamihira [Ed. and T r . by H . Kern] Indische Studien 10 (1868) 161-212; 14(1876)312-58; 15(1878) 167-84 [Text in Roman characters and Tr.] Yogayatra ... E d . ... by Jagadish L a i Shastri, Lahore, 1944
Varahamihira, C. Rudra (Brhajjatakam) Horasastram, with the com. Horavivarana by Rudra, E d . S. Kunjan Pillai, 2nd edn., T r i v a n d r u m , 1958 (Triv. Skt. Ser., N o . 91)
Varahamihira, C. Utpala Brhadyatra of Varahamihira, ed. (with a fragment of Utpala's com.) by David Pingree, Madras, Govt. O r . Mss. Library, 1972
Varahamihira, C. Utpala Varahamihira's Brhajjataka with Bhattotpala's com. Jagaccandrika, Bombay, 1874
Varahamihira, C. Utpala Brhat Sariihita by Varahamihira, with the com. of Bhattotpala, E d . by Avadha V i h a r i Tripathi, Varanasi, 1968, Varanaseya Skt. Vishvavidyalaya, 2 vols.
Varahamihira, C. Utpala Laghujatakam, with com. of Utpala and a Bengali T r . by Rajanikanta Acharya, Calcutta, 1910
APPENDIX VI SUBJECT INDEX Abhra, 24 Adar, 24 Adhikamasa, 17 Adityadasa, xxviii Agastya, xxix Agastyodaya, xxix, 278-81 Agni, 24 Agra, 113-14, 126, 131, 133,135,136 Ahargana, xxvii, 241, 243 Aharmana, xxvii, 34-35, 51-52, 246-47 Ahurmazd, 24 Aja, 24 Afcjadrkkarma, 143, 148, 150,351 Akfak^etras, 129 Ak§amsa, 267-68 Ak^avalana, 184, 185 Alexandria, 10, 288 Almagest, 10 Amordad, 24 Amplitude, 114, 126, 131 Anala, 24 Aneran, 24 Artgiras, 4 Anna, 24 Anuvakra, 334, 335 Anuvarnana, 240 Anuvatsara, 242 Apamandala, 236-38 Ardhakapalayantra, 271 Ardharatrika (ratra)-pak$a, 4, 17, 197, 200. 209, 210, 289, 293, 294,311 Ardibahesht, 24 Arhat, 250 Arkagra, 131 Armillary sphere, 272-73 Aruna, 4 Aryabhata, xxvii, 17, 92, 196, 197, 198, 200, 210, 252,289,311
Aryabhajiya, xxvii, xxviii, 3, 5, 10, 26, 42, 44, 59, 60, 61, 62, 196, 198, 205, 210,223,289,292, 299,371 Aryabhatiya-bha$ya, xviii Ascending node, 67 Ascensional differences, 261-62 Ashisvang, 24 Ashtad, 24 Asman, 24 Astronomer, bad, 74; qualiRcations of, 112 Astronomical observation, 257-58 Asura, 248, 249, 251 Ativakra, 334, 335 Atri, 4 Audayika-pak$a, 293 Avan, 24 Avanti, xxi, 353. See also Ujjain Ayana-drkkarma, 143, 150,351 Ayanamsa, 68, 73 Ayana-valana, 164 Babylonian astronomy, 316 Babylonians, 291 'Bahu', 126 Baladeva, 24 Balava, 57 Banaras, 51, 52, 53 See also Varanasi 'Base', 126, 132, 135 Baudhayana, 139 Bava, 57 Behram, 24 Bhadrasva-var^a, 290 Bhadravisnu, 71 Bhadra (vi^ji), 57 Bhagola, 85-87
Bharata-var$a, 290 Bhaskara I, 4, 62, 129 Bhaskara II, 118,129,207 261 Bhattotpala, 4, 10, 179, 187, 244, 292, 293, 301,310. 5ee also Utpala Bhau Daji, 10 Bhava, 24 Bhoga-rekha, 272 Bhrgu, 4 Bhubhramana, 249-50 'Bhuja', 131, 132, 133, 143,144, 145, 146 Bhujantara, 54, 209 Bhumi, 24 BhiUasaAkhya, 372 Bija correction of VM for planets, 299 Bimbamana, 189-90, 215-16 Brahma, 4, 5, 24 Brahmagupta, xxvii, xxviii, 4, 5, 196, 197, 289 Brahman, 24 Brahmasiddhanta, 5 Brahmasphutasiddhanta, xxvii, xxviii. 3, 5, 60, 70, 293 Bfhadyatra, xxix Bfhadyogayatra, xxix Brhajjataka, xxviii, xxix, 286 Brhatsamhita, xxviii, xxix, 3, 4, 10, 17, 22,62 68, 179, 293, 354, 371 Budhacara, 340 Budhasphuta, 342-60 Burgess, J., xix Caitra-pak^a, 44 Cakrayantra, 272 Calendar reform, 73 Candracchaya, 123-25
I'ANCASinnHANTlKA
Candragrahana, xxvii 5ee also Eclipse, Lunar eclipse Candramsa, 269 Candrasphuta, xxviii, 27 Candrasrrtgonnati, 137-51 Canopus, 278-81 Cara, 48-49, 57,98 ff., 261 Cara-karana, 57 Carakhanda, 48, 49 Cara-vinadi, 49, 50, 100,105 Cardinal directions, 269-70 Catui^pada, 57 Celestial equator, 86 Celestial sphere, 270-71 Chaya, 284 Chedyaka, 267-81 Chedyakayantrani, 267-81 Christian Calendar, 73 Comet, 69 Conjunction of the Moon with a Star, 276-78 Correction for the Equation of the centre, 54 Cosmogony, 248-61 Cyavana, 4 Daily motion of the Sun and the Moon, 185 Dak^inayana, 60, 176, 177,248 Dak§inottara, 271 Day, duration of, 246-47 Day, lord of, 21-23, 287,291 Day-diameter, 96-98 Day-reckoning, 289 Days from Epoch, 5, 8,10, 11,25, 26, 197, 241-43 Day-time, 34-35, 51-52 Declination, 85-90 Declination of the Moon 87-90
Declination of the Sun, 87-90 Demons, 248, 249 Depadar, 24 Depdin, 24 Depmohr, 24 Desantara, 51-53, 209, 274 Desantara-nadi, 51 Descending Node, 67, 69 Devas, 248, 249 Dhana, 78 Dhanada, 24 Dharma-sastra, 10, 69, 74, 139,251 Dhatri, 24 Dhruva, 286-87 Dhruvakarani, 77 Dhiimaketu, 68 Dig-jya, 113-15 Dikshit, S.B., xix, 10 Din,24 Dinaganana, 288-91 Dina-vyasa, 96-98 Diophantine equation, 293 Directions from shadow, 126-32 Directions, problems of, 76-130 Dragon's head, 67 Dragon's tail, 67 Drekkana, 258 Drk?epa, 187-89 Dj-k§epa of the Sun, 219-20 Druma, 24 Duration of a day, 246-47 Duration of the Eclipse, 230-35 Duryavana, 4 Dvadasamsa, 258 Dvapara, 293 Dvivedi, See also 5, 19, 20; Sudhakara Dvivedi, TS Dyauh, 24 Dyugana, 7
.'577
Earth, 248 Earth, measures of, 253 Earth, situation of, 248 Earth's shadow, 228-30 Earth-sine, 118-21 Eclipse diagram, 236-40 Eclipse, duration of, 225-27 Eclipses, 195-96, 232-86 Ecliptic, xxvi, 236-38 Egypt 10, 288 Epicycles of the planets, 300-1 Epicyclic theory, 206-9 Epoch, 7, 30, 59 Epoch constant (k^epa) for Jupiter, 297 Epoch constant for Mars, 297 Epoch constant for Mercury, 298 Epoch constant for Saturn, 297 Epoch constant for Venus, 298 Equation of the centre, 47-48 Farwardin, 24 First point of Aries, 73 First sine, 118-21 Five Schools of Astronomy, 4 Five siddhantas, comparative study, xxvi-vii Ganitakaumudi, 265 Garga, 4 Gargya, 4 Gati, 27, 312 passim. Gavam-ayana-satra, 60 Ghana, 13, 27, 28, 30, 31 Ghati-yantra, 275-76 Gnomon, 91-220 ff., 267-68 Gnomonic shadow, 35 ff. 90-136, 113 Go, 24 God,250
378
Gola, 261 Golabandha, 272-73 Gosh, 24 Govad, 24 Govindasvami, 61, 91, 311 Grahanaparilekha, 195-96 Grahastodaya, xxvii Graha-vik^epa, 310 Grahodaya, 309-10 Graphical mediods, 261-67 Grasa-pramana, 234 Great gnomon, 110-12,130 Greece, 289 Greek, 17 Greenwich, 209 Gri^ma, 65 Guha, 24 Gurucara, 316-22 Gurusphuta, 361 Hali, 24 Kara, 24 Head of Rahu, 68 Heliacal rising of the planets, 310 Hemanta, 65 Hemispherical Bowl, 271 Herodotus, 17 Hoop, 272 Hora, xxviii, 144 Hora, lord of, 20-23, 291 Hora, lords of, 287, 291 Horasastra, xxix Hora-Tantra, 22 Horoscopy, xxviii 'Hypotenuse', 6, 143, 144,145, 146 I^avatsara, 242, 243 Idvatsara, 242, 243 India as seen by Varahamihira, xxviii Indra, 24 Instruments, 267-81 Instruments for measiuing rime, 274 Intercalary month, 8, 17, 241-42, 290
S U B J E C T INDEX
I$a, 65 Istakalacchaya, 115-23 I§takalagrasa, 232-34 Istantya, 118 Jain religion, 250 Jambudvipa, 290 Jataka texts, xxviii Jews, 10 Junction-Stars, 277-79 Jupiter, xxi, 316-22; -ksepa of, 297;-mean, 294-99;-motion of, 361;rising of, 316-22;-yuga revolutions, 296 Jyauti§opani§at, 282-91 Jyotirmimamsa, xviii, 3, 4, 5, 289, 371 Kak§a, 214-I5;-ofthe Moon, 214-15;-ofthe Sun, 214-15 Kala, 24 Kalabhaga, 309-10, 352-53 Kalalokaprakasa, 59 Kalamanayantra, 274 Kala-rekha, 272 Kali, 293 Kali year, 9 Kalidasa, 138 KamalodbhaVa, 24 Kapitthaka, xxviii Karana, xxvi, 5, 40, 57-58 Karana-grantha, xxvii Karanavatara, 3 Karani, 77-80 Karmanibandha, 289 Ka^tha, 63 Kasyapa, 4 Kaulava, 57 Kendra, 40 Kepler, 206 Ketu, 68, 69 Ketumala-var^a, 290 Kha, 24 Khagola, 92-94, 270-71 Khagolardha, 271
Khandakhadyaka, 4, 17, 197, 198, 200, 205, 209, 210,289, 293, 310 Kharega, M.P., xix Khurdad, 24 Khurshed, 24 Kimstughna, 57 'Koti', 126, 143, 144, 145,146 Kranti, 85-90 Krantivrtta, 86 Krta, 293 K§epa, 8, 15, 30 Kuhu, 139 Kujacara, 328-39 Kujasphu^a, 357-59 Kuttaka, 293 Laghubhaskariya, 61 Laghujataka, xxix Laghusaitihita, xxix Lagna, 150, 256;-from shadow, 37-39 Lalla, 261 Lambajya, 96-98 Lambana 172-75, 187 Lambitaparvanta, 221-23 Laftka, 253, 289 LatU(odayamana, 216 LaAkodaya-rasimana, 102-4 Lafadeva, (Latacarya) 4, 5, 288 Later : See also Modem 'Later' Surya-Siddhanta, 4,293,294, 331 Latitude,-from cara, 100-101 ;-from shadow, 91-92; of planets, 310 Latitude-caused triangles, 129 Local, latitude, 268;longitude, 274 Lord, of the day, 20,21;of the degree, 23;-of the month, 19-23 Lords,-of the day, 297;of the horas, 287; - of the
PANCASIDDHANTIKA
months, 260;-of the weekdays, 21-22; of the year, 18, 19,22 Lunar, 228-35;-days, 290; -eclipse, 152, 228-33 Madhava, 65 Madhu, 65 Madhyajya, 216-18 Madhyalagna, 217 Magadhadvija, xxviii Magan calendar, 23 Magas, 23 Mah, 24 Mahabhaskariya, 17, 26, 49, 52, 54, 60, 91, 131, 143,293,310,311 Mahabhaskariya-bha§ya, 59 Mahado^ah, 60 Mahalayapak^a, 64 Mahavyatipata, 60, 61, 62 Mahayatra, xxix Mahayuga, 201 Makkibhatta, xxviii, 259, 288, 371 Manda-Vrtta, 206 Manes, 64, 283-84 Manu, 4 Marespand, 24 Marici, 4 Mars, 309;-mean, 294-99; -motion of, 328-29, 357-59; - yuga revolutions, 297 Masadhipa, 20 Mean, Moon, 198-201; -planets, 293;-Sun, 26, 197-98 Meher, 24 Mercury, xxi, 307-9;k§epa of, 298; -motion of, 337, 339-50, 359-60;-Sighra of, 294-99;-True, 342-60; yuga revolution, 298 Meridian, 271
Meru, 248, 290 Mid-day shadow 251 Minutes into angles, 240 Modern : 5ee also Later Modem Surya-Siddhanta, 4,5, 16,210,216, 293 Month, lord of,,20-23 Moon, daily motion, 185-86;-daily rising and setting, 147-51;- true motion, 45-47. 5ee atso Mean Moon, True Moon Moon-day time, 148, 149, 151 Moonrise, 138 ff., 148, 149,150,151 Moons, two, 250 Moon's,-anamoly, 211;ayana, 141 ff.,-cusps, 137-51 ;-declination, 87-90;-kak5a, 214-15;latitude, 69-71;-longitude, 269;-luminosity, 258-59;-nak?atra, 243;orb, 191-93;-shadow, 123-25;-visibility, 137-45 Moonset, 138 ff., 148,149, 150 Muhurta, 33 Mula-dhruva, 30 Muslim, 10, 139 Nabha, 65 Nabhasya, 65 Na^iyantra, 275-76 Naga, 57 Nak$atra, xxvi, 5, 33-34, 40, 55, 245-46;-Computation, 54-55 Nak^atranayana, 54-55 Nak§atrasthana, 277-79 Nak^atra-tithi, xxvii Narada, 4 Naraka-caturdasi, 64 Nati, 175 ff., 189-90, 223-25 Natural astrology, xxviii
379
Navamsa, 256 Naa-ratnas, xxviii Naugebauer, xiv, xx, 3, 5 5ee also NP Newton, Isac, 206 Nilakantha Somayaji, xviii, 3, 4, 248, 289, 371 Niraya, 24 Nirayana, 104 Niyati,24 Node, 67-71 Nonagesimal, 188-89 North-polar region, 284 North pole, 251,255, 256, 257,258 NP, xiv, X V , xxii, xxvii, 5,6,10,11,15,19,20,21, 23, 33, 41, 54, 63, 67, 101, 110, 125, 142, 147, 151, 154, 156, 160, 162, 165, 167, 168, 178, 179, 189, 191, 200, 216, 261, 282, 285, 312, 313, 315, 316,317,318,319, 320, 321, 323, 325, 326, 328, 330,331,332,333,336, 337, 340, 341, 343, 350, 355, 357, 362, 363 Oblique ascension, 48-49 Obscuration of Moon, 232,34 Omitted days, 290 Orbs, 215, 16 Orient Ecliptic Point, 37 Original Saura, 210, 216. 5ee also Saura 'Pada', 27, 28, 30, 46 Padaditya, 71 Paitamaha, (Siddhanta), xiii, 4, 5, 25, 34, 61,152; 241,47; - Content analysis, xxvi Paiicasiddhantika, Content analysis, xxii-xxv
380
Panca-siddhantas, 4 5ee also Five schools, Five Siddhantas Panchanga, 73 Parahita, 355-65 Parah puru^ah, 24 Parallax, 172, 190 Parallax-corrected New Moon, 221-23 Parallax correction and orbital diameter, 189-90 Parallax in latitude, 176, 223-25 Parallax of longitude, 172-75, 187 Paramesvara, xviii, 3, 91,251,371 Parasara, 4 Parasara-Siddhanta, 4 Parivatsara, 242 Paulisa Siddhanta, xiii, xxi, 4, 10, 11, 12, 13, 14, 15, 26, 42, 66, 68, 70, 76 137-51, 152, 172-80, 196, 197, 205, 217, 224, 235, 265, 288, 292, 293, 310, 312-53;-content analysis, xxv Paulisa-k^epa, 11 Paulisa-moon, 14 Paulisa-yuga, 16, 17, 18 Paurukutsa, 4 'Perpendicular', 126, 131, 133 Pi, 76-77 Pingree, D., xiv, xvii, xviii, xxi, xxvii-3, 5, 261 See also NP Pitamaha,-Siddhanta, 4, 5, See Paitamaha-Siddhanta Pitr, 24 Place, problems of, 76-130 Planetary days, 290 Planets,-epicycles of, 300-l;-their situation, 259-60 Poles, 286-87
subjec:t
index
Prabhakara, 60 Pradyumna, xxi, 353, 355 Prajesa, 24 Pravaha, 249 Precession of the equinoxes, 65 Prime vertical, 106, 113, 271 Pfthudaka, xviii, 3, 60, 248, 249, 251, 259, 371 Prthuyasas, xxviii Ptolemy II, 17, 196, 205 Rahu, 40, 67-71, 152, 186-87, 201-3, 286, 293 Rahu's head, 67 Raivata Pak^a, 44 Ram, 24 Rama, 72 Ramayaiia, 72 Ranganatha, 293, 308 Rashna, 24 R Sines, 86-87 R sine of the Sun's Zenith distance, 264,66 Rashtriya Panchang, 73 Rasi division, 252 Rasyudaya, 104-6, 266 Ravi, 24 See also Sun Ravidrk§epa, 218-20 Ravigrahana, xxvii Ravisphufa, xxvii Reduction to the equator, 54 Rekhamsa, 268 Retrograde motion, 309 Rgveda, 69, 246 Right ascensional difference, 102-4 Right ascensions of the signs, 266-67 Rising and setting of Planets, 312-53 Rising signs, 104-6 Romaka (Siddhanta), xiii, xix, 4, 5, 7, 8, 10, II, 14, 26,31,32,42,43,44, 66,
70, 73, 74, 152, 181-197, 205,213, 225, 288,313;Content analysis xxv Romaka-k§;epa, 11 Romaka-pura, 290 Romaka-Yuga, 8, 16, 17, 18, 184, 195 Romasa, 4 Rome, 290 Rotation of the Earth, 249-50 l^a^asitimukha, 64 ^a^asiti-punyakala, 64-65 Sahas, 65 Sahasya, 65 iSaka era, 9 Sakayear, 8, 11 :§akuni, 57 Samakala, Samalipta, 152 IF. Samaman^ala, 113, 271 Samasasaiphita, xxix Sama-saAku, 106, 110-12 Samhita, xxviii Samirana, 24 Samvatsara, 242 l^anaiscara, 69 Sanicara, 322-28 l^anisphu^a, 363-65 l^afikaranarayana, 61 SaAkasya, xxviii Sankramapa, 66 SaAkrantikala, 65-66 Sartku, 220 ff., 267-68 l^aAkucchaya, 35, 90-136, 113 Sankulipta, 118 :§ankutala, 126, 129, 130 ^aAkvagra, 126, 129, 131 §arad, 65 Sarosh, 24 Sarpamastaka, 59 ^asi, 24 l^asitarayoga, 276 Sastra, 24 Sastry, T.S.K., xix, 25,261 $atpaiicasika, xxviii
pancasiddhAn
Saturn, xxi, 316;-k$epa of, 297;-mean, 294-99;motion of 322-28, 363-65;-yuga revolutions, 297 ^aunaka, 4 Saura (Siddhanta), xiii, xix, 4, 10, 11, 12, 16, 197-235, 17, 31, 32, 40, 44, 66, 76, 152, 198, 228,235, 289, 293, 296, 300,311,353,354; Content analysis, xxvi Sauradina, 356-57 Saura-Saitihita, xiii, xix Saura-yuga, 198, 201 Savitr, 24 Sayana, 65, 104, 105 Seasons, 65 Secrets of Astronomy, 282-91 Shadow, 3, 8, 115;-at desired time, 115-23;diameter of, 228-30;from time, 122-23;gnomonic, 35ff.; 90-136 See also Gnomonic shadow Shadow-hypotenuse, 126, 129, 130, 135 Shahrivar, 24 Shastri, Ajay Mitra, xxviii, xxix Shukla, K.S., xix, 3, 20, 23, 25, 75, 261 Siddhantasekhara, xviii, 3, 240, 371 Siddhanta I^iromani, 32, 43, 44, 49, 63,124,130, 143, 185, 189, 207, 304, 308 Siddhapura, 290 l^ighra-kendra, 302 Sighra-vrtta, 206 Sign, 252 Siihhacarya, 289 Sine amplitude, 113-15
FIKA
Sine Co-latitude, 96-98 Sine Zenith distance of meridian pt., 95, 216-18 Sinivali, 139 l^iiijini, 78 l§isira, 65 Sky-sphere, 92-94 Solar-day, 64;-eclipse, 152, 172-227 Solstices, 65 Soma, 4 South pole, 251 Spandarmad, 24 Spheric, 261 Sphujagraha, 301 Sphu^akarma, 300-1 Sphutatithi, 269 Spurious Supplement, 355-65 ^raddha-tithi, 244 Sri, 24 l^ridhara, 265 iSrIpati, xviii, 3, 60, 371 Star-planets, Computation of, 355-65 Stellar sphere, 85-87 Sthira-karana, 57 Suci, 65 Sudhakara Dwivedi, xiii, xvii, xviii, xx, xxvi, 68, 312. 5ee a/so TS. Sukla-Yajurveda, 65 Sukra, 65 :§ukracara, 312-16 Sukrasphuja, 362 Suk^majataka, xxix Summer Solstice, 65 Sun, daily motion, 56, 185-86;-declination of, 87-90;-drk?epa of, 219-20; from shadow, 132-36; kak§a of, 21415;-longitude of, 268; northward and southward journey, 273; orb of, 191-92;-visibility of, 253-56. See also Mean
381
Sun, True Sun, Solar Suns, two, 250 Sunset, 121-22 Surya, 4 Siiryadeva Yajvan, xviii, 248, 259, 371 Suryagra, 129, 130, 131, 133,135,136 Suryaprajiiapti, 59 Suryasiddhanta, 4, 10, 16, 17,41,43,44, 52, 53, 54, 60, 61, 64, 182, 205, 308, 309,310 'Siitram', 118 Svalpajataka, xxix Svalpavivahapa^la, xxix Svalpayatra, xxix Svarga, 24 Synodic Cycle, 314-16 Taddhrti,130 Taitila, 57 Tamobimbamana, 228-30 Tantra, xxviii Tapas, 65 Tapasya, 65 Taru, 24 Thibaut, xiii, xvii, xviii, xix, X X , xxvi, 3, 5, 80, 165, 300, 312; 5ee a/so TS Three problems-Time, Place, and Direction, 76-130 Tikanikayatra, xxix Time, after sunrise, 11821;-for sunset, 121-22;measuring, 274-75;problems of, 76-130 Tir, 24 Tithi, xxvi, 5, 14, 32, 3334, 40, 55, 245-46 Tithi-nadika, 221 Total obscuration, 235 Treta, 293 Tribhonalagna, 187-89 Tridinasprg-yoga, 66-67
382
Trijya, 78 Triipsaijisa, 256 Trisatika, 265 True, Moon, 27-33,183-85, 203-6;-niotion of Moon, 211;-motion of Sun, 21213;-planets, 300, 302;Sun, 25-27, 203-6;-tithi, 269 TS, xiii, xiv, xv, xvi, xx, xxii, 5, 6, 10, 15, 19, 20, 21, 25, 28, 32, 33, 34, 39, 41,44, 47,54, 59, 63, 67, 68, 70, 71, 80, 91, 101, 110, 117, 125, 136,142, 147,151, 153,156,160, 162, 165, 167, 168,178, 179, 189, 191, 200, 201, 214,216,218, 221,223, 230, 234, 235, 244, 246, 282,284,285,294, 296, 299, 300, 301, 304, 307, 310,312,313,315,316, 317,318,319, 320, 321, 323, 325, 328, 329, 332, 333, 336, 337, 340, 341, 343, 350, 355, 356, 357, 358, 361, 362, 363, 364. See also Dvivedi; Sudhakara Dvivedi; Thibaut Udayantara, 54, 209 Ujjain, 10, 26, 32, 33, 36, 51, 52, 53, 184, 195,196, 197, 198, 200, 209, 210, 212, 251, 252, 253, 254, 255,288,289,294, 353. Unma^dala, 93, 99, 105, 106,108,150 tjrja, 85 Utpala, xviii, xxviii, xxix, 3, 16, 19,20,21,22,41, 48, 62, 65, 92, 102, 111, 115, 121, 137, 147, 166, 191, 197, 241, 243, 248, 249, 251, 252, 254, 255,
S U B J E C T INDEX
257, 260, 276, 288, 371. 5ee also Bha^otpala Uttara-kuru-var§a, 290 Uttarayana, 60, 61, 65, 176,177,180, 246 Vagbhava correction, 299 Vaidhrti, 40, 58-63 Vak, 24 Vakra, 309, 334 Vakyakaraiia, 26, 31, 49, 52, 54,77 Vanijya (Vanija), 57 Vara, xxvi, 5 Varahamihira, Life and works, xxvii-xxix Varajnana, 288 Varanasi, 51 Var§a, 65 Var§adhipa, 18, 19 Varu^a, 24 Vasa, 24 Vasanta, 65 Vasistha, 4, 265 Vasistha, Siddhanta, 4, 5, 10, 13, 15,30,31, 32, 39, 40, 152, 179, 184, 196, 235, 312-53, Contents analysis, xxvi Vasistha-moon, 14, 32 Vasistha-Paulisa, 184 Vasistha-samasaSiddhanta, 39Vasisth^'S^hita, xiii, xix, xxi Vatakanika, xxix Vatesvara, 60, 91, 261 Vatesvara-Siddhanta, 23, 52, 70 Vayu, 24 Veda, 24 Vedaftga-Jyoti§a, 34, 59, 62, 65, 74, 246, 247 Venus, xxi, 307-9, 310 Venus, ksepa of, 298; motion of 312-16, 362-63;sighra of, 294-99
Vijayanandi, xxi, 354, 3 Vikrama, xxviii Vimardakala, 225-27, 230-35 Viskambha, 59, 246 Visnu, 4 Visnudharmottara, 5 Visti (Bhadra) 57 Vivahapatala, xxviii Vyatipata, 40, 58-63, 244146 Vyatipata-vaidhj-ti, xxv Weekday, 287 Winter, solstice, 65
Yajurveda, 246 Yajusa-Jyotisa, 65 Yaksyesvamedhika-yatr xxix Yama, 24 Yamakoti, 290 Yantras, 267-81 Yasti, 269-70 Yavana, 4, 289 Yavanajataka, 289 Yavanapura, 7, 9, 10, 26 32,33,51,52,53,54,18198,288 Yoga, xxvi, 5 Yojana, 252 Yojana-measure, 252 Yogayatra, xxviii, xxix Yuan Chwang, xxviii Yuga, 8, 16, 290 Yuga of five years, 241-4 Yuga-cycles, 293 Yuga-pada, 293 Zamvad, 2^ Zodiac, 55
