महाभास्करीयम्

by भास्करः I

महाभास्करीयम् भास्करः I १९५२

HINDU ASTRONOMICAL AND MATHEMATICAL TEXTS SERIES No. 3

General Editor

RAM BALLABH, M. Sc., Ph. D.
Professor of Mathematics, Lucknow University

BHASKARA I AND HIS WORKS

PART II

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MAHA-BHASKARIYA

श्रीभास्कराचार्यविरचितम्
महाभास्करीयम्
लखनऊ विश्वविद्यालयस्य गणिताध्यापकेन एम० ए० डी० लिट्० इत्युपाधिधारिणा
श्री कृपाशंकर शुक्लेन
आङ्ग्लानुवाद - व्याख्या - टिप्पण्यादिभिः सहितं
सम्पादितम्
लखनऊ विश्वविद्यालयस्य गणित-ज्योतिष-विभागेन
प्रकाशितम्
सं० २०१७वि०

First Edition 1960

Price Rs.15/- $ వీరి • Printed at Nav Jyoti Press Pandariba, Lucknow. PREFACE The object of the "Hindu Astronomical and Mathematical Texts Series" is to bring out authoritative and critical editions of important unpublished works dealing with ancient Indian astronomy and mathe- matics. The present edition of Bhaskara I's Maha-Bhaskarlya is No. 3 of this series. The idea of bringing out the above series is due to Dr. A. N. Singh, late Professor of Mathematics, Lucknow University, who organised a scheme of research in the history of Hindu mathematics and astronomy in the Department of Mathematics, Lucknow University, with the object of collecting, studying, and editing important works on Hindu mathematics and astronomy. Under his able guidance remarkable progress was made in this direction and a number of manuscripts were acquired, studied and edited. In 1954 he submitted to the Government of the Uttar Pradesh a detailed plan for the publication of the work carried out under the above scheme of research in a series to be called the "Hindu Astro- nomical and Mathematical Texts Series". The plan was approved by the Government and a sum of Rs. 1000/- was sanctioned in the name of Bhārata Ganita Parisad to undertake the publication of the series. In the year 1955 the Government of the Uttar Pradesh was generous enough to sanction the remaining sum of Rs. 9000/- to the Department of Mathematics and Astronomy, Lucknow University, for the said publi- cation. The scheme of research in the history of Indian mathematics and astronomy referred to above has been financed by the Government of the Uttar Pradesh through the kind help of Dr. Sampurnanand, its then Education Minister, for which we express our sincere thanks to them. We are specially grateful to Dr. Sampurnanand who, a great scholar of Jyotisa as he himself is, has been taking keen interest in the progress of this research and helping us with necessary funds and encouragement from time to time. R. Ballabh Introduction Sanskrit Text प्रथमोऽध्यायः द्वितीयोऽध्यायः तृतीयोऽध्यायः चतुर्थोऽध्यायः पञ्चमोऽध्यायः षष्ठोऽध्यायः सप्तमोऽध्यायः अष्टमोऽध्यायः शब्दानुक्रमणिका छन्दानुक्रमणिका English Translation CONTENTS Chapter I. MEAN LONGITUDE OF A PLANET AND PLANETARY PULVERISER Homage to God Siva, planets and stars 1-- Appreciation of the Aryabhatiya and the pupils of Aryabhata 1-Ahargana 2-6-Mean longitude of a planet by various methods 6-7,15-- Mean longitudes of Sun and Moon 8,9– Mean longitude by simplified rules 16-28 – Correction to the mean longitudes of Moon's apogee and ascending node 28 – Planetary pulveriser 29- Methods for solving a planetary pulveriser 30-36–Week- day pulveriser 36-Time-pulveriser 38-General solution of a pulveriser 40--When the interpolator is positive 41-When the residues of two or more planets are given 43. Chapter II. THE LONGITUDE-CORRECTION Places on the Hindu prime meridian 47-Distance from the prime meridian 49,55– Criticism of two rules 51- Longi- tude in time 52, 53-Criteria for knowing whether the local place is to the east or to the west of the prime meridian 54- Longitude-correction and its application 55. Page I १ १० १२ २२ २९ ४४ ४८ ५३ 1 47 CONTENTS Chapter III. DIRECTION, PLACE AND TIME. JUNCTION- STARS OF THE ZODIACAL ASTERISMS AND CON- JUNCTION OF PLANETS WITH THEM Setting up of the gnomon 56-Finding the directions from the shadow of the gnomon 57, 59-Hypotenuse of the shadow 60-Finding the latitude of the place and the Sun's altitude 61--Sun's declination, day-radius, earthsine, and ascensional difference 62-Ascensional differences of the signs 65-Times of rising of the signs at the equator 68- Times of rising of the signs at the local place 69-Sun's meri- dian altitude, declination, and longitude 70-73,76-Latitude from the Sun's meridian zenith distance and declination 74- Sun's altitude for the given time 74, 77-78--Sun's altitude in the night 79-Time corresponding to the Sun's altitude 80-Longi- tude of the rising point of the ecliptic 81-Longitude of the setting point of the ecliptic 82-To find the time from the longitudes of the Sun and the rising point of the ecliptic 82- Sun's altitude, hour angle, and longitude when the Sun is on the prime vertical 84-88-Locus of the shadow-end 89-92-Sun's agra and sankvagra and derivation of the equinoctial midday shadow therefrom 92-93-Finding the Sun's agrā and sankv- agra by the observation of the Sun and deriving the equinoc- tial midday shadow and the local latitude therefrom 94- Finding the longitude of a planet or star with the help of that of another planet or star 95-Longitudes of the junc- tion-stars of the nakṣatras 98-Celestial latitudes of the junction-stars 102-Conjunction in longitude of a planet and a star, and distance between them 102-Occultation of certain stars by the Moon 104. Chapter IV. TRUE LONGITUDE OF A PLANET Sun's mean anomaly and its Rsine 106-Rsine of an arc 107-Sun's equation of the centre 110, 111-Sun's correction for the equation of time due to the eccentricity of the ecliptic 114-Bahu and koți due to anomaly 115- True distance of the Sun or Moon 115-True daily motion of the Sun or Moon 119-122-True distance of the Sun or Moon by the eccentric theory 122-Sun's true longitude by the eccentric theory 125-Sun's correction for the equation of time due to the eccentricity of the ecliptic Page 56 106 CONTENTS under the eccentric theory 126-An approximate formula for the Sun's declination 127-Correction due to the Sun's ascensional difference 127-Semi-durations of day and night 128 Corrections for the Moon 129-Tithi 130-Karana 131-Nakṣatra 132-Phenomena of vyatipāta 132-Manda and sighra anomalies (kendra) for the planets 134-Corrected manda and sighra epicycles 134-Calculation of the true longitudes of the planets 136-Calculation of the true longi- tudes of the planets under the eccentric theory 141-145- Mandakarna and sighrakarna 146-Direct and retrograde motions of a planet 146-True daily motion of the planets 148-Longitudes of the Sun and Moon at the end of the parva-tithi 152. Chapter V. ECLIPSES Page Distances of the Sun and Moon 153-Diameters of the Earth, Sun, Moon and the shadow 153-157-Longitude of the meridian-ecliptic point 157-The five Rsines relating to the Sun and to the Moon 158-162-Time of apparent conjunction of the Sun and Moon 162-Finding the Moon's latitude corrected for parallax (true nati) 166-Possibility of a solar eclipse 167-Calculation of an eclipse : Sparsa and mokṣa sthityardhas 167-Sparśa and mokṣa vimardār- dhas 168-Time of actual visibility of a solar eclipse 169- Projection of an eclipse: Akşa-valana 169---Ayana-valana 171 Resultant valana 172-Finding the centre of the eclips- ing body for the time of the first contact, last contact and middle of the eclipse 173-Difference of procedure in the case of a lunar eclipse 174-Measure of the eclipse 175- Path of the eclipsing body 175-Calculation and construc- .. tion of the phase of the eclipse for the given time 175, 176- Eclipse of the Moon: Diameter of the shadow 177----Sthity- ardhas and vimardärdhas 180-Direction of the Moon's latitude to be used in the projection of a lunar eclipse 180. Chapter VI. RISING, SETTING AND CONJUNCTION OF PLANETS Visibility-corrections known as akṣa-drkkarma and ayana- dekkarma 182, 184-Visibility of the Moon 186-The phase of the Moon 187-Moon's true declination 188-Elevation of the lunar horns 189-Graphical representation of the eleva- 153 182 CONTENTS tion of the lunar horns 190-193-Time of moonrise on the full moon day 193-Duration of the Moon's visibility during the night, or time of moonrise 194-196-Time of meridian passage of the Moon 197--Moon's meridian zenith distance 198-Elevation of the horns of the half-risen or half- set Moon 198-Procedure to be adopted in the case of the planets 198-Heliacal visibility of the planets 199-200--- Minimum distances of the planets from the Sun at which they become visible 199--Conjunction of planets - 204. Chapter VII. ASTRONOMICAL CONSTANTS " Constants under the sunrise day-reckoning: Revolution- numbers, intercalary months, omitted lunar days, and civil days in a yuga 205-Inclinations of orbits, longitudes of ascending nodes and apogees, and manda and sighra epicy- cles of the planets; methods for finding the celestial latitude and manda and sighra anomalies of a planet 206-Method for finding the bahuphala and koṭiphala, etc., without using the Rsine-difference table 207-Circle of the sky and orbits of the planets 210. Constants under the midnight day-reckon- ing: Civil days, omitted lunar days, and revolution-numbers of Mercury and Jupiter in a yuga; diameters of the Earth, Sun, and Moon; mean distances of the Sun and Moon; and longitudes of the apogees of the planets 211-Manda and sighra epicycles and manda and sighra pātas of the planets 212-Methods for finding the celestial latitude and the true- mean longitude of a planet; Circle of the sky and orbits of the planets 213. Chapter VIII. EXAMPLES Quotations from the Maha-Bhaskariya in later works. Glossary of terms used in the Maha-Bhaskariya Page 205 214 226 230 Short : Long : Anusvāra : Visarga : Non-aspirant : Classified : Unclassified : Compound : अ a आ a 16 क् Ic tur to प् P TRANSLITERATION VOWELS य् Y इ क्ष् kṣ ई Shar I'№ C ch S kch == in फ् ph Wa U r ū ऊ 2 , th त् थ् द् t th ग् 9 15. CONSONANTS ज् hora. for ब b र् ल् ऋ ए e ज्ञ् jn घ् gh झू jh d dh ढ् dh भ् bh व् ८ V ar व् ध् न् n n 5' 8. म् mo श् $ ओ S औ au स् S h h Ā BJ BrSpSi By Sam GL KK KKau KKu KPr LBh 'LMa MBh MSi MuCi PiSi PSi SK SiDV r Si Sa Sise Siśi SiTV SuSi TS V Si ViMā VVSi LIST OF ABBREVIATIONS Aryabhatiya Brhaj-jataka Brahma-sphuta-siddhanta Brhat-samhita Graha-laghava Khanda-khadyaka Karana-kaustubha Karana-kutuhala Karana-prakāśa Laghu-Bhaskariya Laghu-mānasa Maha-Bhaskariya Maha-siddhanta Muhurta-cintamani Pitamaha-siddhānta (of Visnudharmottara-purāṇa) Panca-siddhantikā Sarvananda-karaṇa Sisya-dhi-vrddhida Siddhanta-sarvabhauma Siddhanta-sekhara Siddhanta-siromani Siddhanta-tatva-viveka Surya-siddhanta Tantra-sangraha Vateśvara-siddhanta Vidya-Madhaviya Vyddha-Vasistha-siddhānta INTRODUCTION This Part¹ contains the text with English translation, notes, short comments, and explanations, where necessary, of the Maha-Bhaskariya ("the bigger work of Bhaskara I").² Sanskrit Text. The text has been edited on the basis of the following five manuscripts collected by the late Dr. A. N. Singh: MSS. A and B -Containing the text only; MS. C-Containing the text together with a commentary entitled Bhäsya of Govinda Svāmi; ¹ The present work has been divided into four parts: Part I- General Introduction (containing a general study of the life and works of Bhaskara I); Part II-Maha-Bhaskariya; Part III-Laghu-Bhaskariya ("The smaller work of Bhaskara I") edited with English translation, critical notes and comments, etc.; Part IV-Bhaskara I's commentary on the Aryabhatiya of Āryabhata I. 2 Bhaskara I, the author of the Maha-Bhāskariya, was a different person from his namesake of the twelfth century A. D., the author of the Siddhanta-siromani and Lilavati, etc. He lived in the seventh century of the Christian era and was a contemporary of Brahmagupta (628 A. D). He wrote three works on astronomy which were composed in the following order: (1) the Maha-Bhaskariya, (2) a commentary on the Aryabhatiya, and (3) the Laghu-Bhaskariya. His commentary on the Aryabhatiya was written in 629 A. D., i. e., one year after the completion of the Brahma-sphuṭa-siddhanta of Brahmagupta. Bhaskara I was a follower of Aryabhata I, the author of the Aryabha- tiya. His works provide us with a detailed exposition of the astronomical methods taught by Aryabhata I and throw light on the development of astronomy during the sixth and early seventh century A. D. in India. For details regarding the life and works of Bhaskara I, the reader is referred to Part I of the present work. 11 INTRODUCTION MS. D-Containing a commentary known as Prayoga-racana and also supplying the beginnings of the passages commented upon ; MS. E-Containing the commentary Karma- dipika of Parameśvara and also the beginnings of the passages commented upon. agree- MSS. A and B, which contain the text only, are in ment in so far as the extent of the two is concerned. In other respects also they are almost the same. Both of them contain 396 verses. Agreement between the two manuscripts seems to indicate that they are perhaps derived from the same source. There are other reasons also for this conjecture. Verse iii. 45, which has been misplaced in one of them, occupies the same wrong position in the other also. Moreover, six and a half verses, which have been commented upon by Govinda Svāmi (MS. C), Paramesvara (MS. E), and in the Prayoga-racanã (MS. D), and are included in the text of MS. C, are absent from both MSS. A and B. Its text has evident MS. C contains 394 and a half verses. gaps at some places, for a few verses, some of which have been actually commented upon in the commentary at their proper places, are missing from the text. These are amongst those verses which belong to MSS. A and B and have been explained in the Prayoga-racana (MS. D), and in the commentary of Paramesvara (MS. E). MSS. D and E contain commentaries of the Maha-Bhaska- riya. In these manuscripts only the beginnings of the passages commented have been given; the full text is not given. The following table will show at a glance how far the above manuscripts have differed from one another: ¹ viz, i. 12; v. 18, 26-27(i), 47-48(i); vi, 14(ii)-15(i), 58(ii Chapters and verses of the edited text Chapter I Verses 1-7 Verse 8 Verses 9-11 Verse 12 Verses 13-52 Chapter II Verses 1-10 Chapter III Verses 1-74(i) Verse 74(ii) Verse 75(i) Chapter IV Verse 1 Verse 2(i) Verses 2(ii)-12 Verse 13 Whether available in MSS. A and B? Yes Yes 99 No Yes " "" " "" " Verses 14-64 Chapter V Verses 1-17 23 Verse 18 No Verses 19-25 Yes Verses 26-27(i) No Verses 27(ii)-46 Yes Verses 47-48(i) No >> INTRODUCTION MS C Whether available in the Text? Yes No Yes " " " No No Yes No Yes No Yes " "" "" "" comm.? MS. D? Yes " "" "" "" No Yes "" "" " "" "" "" "" " Whether com- mented upon in "" Yes " " "" 29 MS. de- fective Yes 29 33 >> "" "" " "" Yes MS. E ? " "" "" "" No Yes 29 33 No, but mentioned Yes "" III E 27 "" IV Chapters and verses of the edited text Verses 48(ii)-78 Chapter VI Verses 1-14(i) Verses 14(ii)-15(i) Verses Verse 19(i) Verses 15(ii)-18 19(ii)-25 Verse 26 Verses 27-36 (i) Verse 36(ii) Verses 37-40 Verse 41 Verses 42-58(i) Verses 58(ii) Verses 59-62 Chapter VII Verses 1-35 Chapter VIII Verses 1-22 Verses 23-24 Verses 25-27 Whether available in MSS. A and B? Yes "" No Yes 29 33 "" "" " No Yes "" "3 29 INTRODUCTION Text ? Yes "> No 3 Yes No Yes No Yes No Yes 22 " " MS. C Whether available in the No Yes Comm.? MS. D? Yes "" "" "" >> "" "" "" >> "" "" "" "" Whether com- mented upon in "" No Yes Yes 22 "3 " 23 " "" "" "" " " >> " "" 39 MS. E? Yes >> 22 "2 >> "" " " "" " 29 "" " "" .99 "" "" INTRODUCTION The edited text of the Maha-Bhaskariya contains 402 and a half verses, of which 387 and a half occur in all manuscripts. The remaining fifteen verses, which occur either in MSS. A and B or in MS. C, are of the following categories: (1) Those which have been commented upon by all commentators, viz. i. 8, 12; iii. 75(i); iv. 2(i); v. 18, 26-27(i), 47-48(i); vi. 14(ii)-15(i), 19(i), 26, 36(ii), 41, and 58(ii). (2) Those which have been commented upon in one or two commentaries only, viz. iii. 74(ii); iv. 13; viii. 23-24. Inclusion in the edited text of the verses of the first category requires no explanation. Those of the second category have been taken as genuine for the following reasons : Half-verse iii. 74(ii). This has been regarded as genuine because it is relevant to the context and the subject matter contained in it has its counterpart in the author's smaller work, the Laghu-Bhaskariya. It occurs in MSS. A and B and forms part of a passage [vv. 71-75(i)] which describes the occultation of certain stars by the Moon. This description is a statement of facts and hardly requires any explanation, which explains why the half-verse in question has been left unexplained by Govinda Svami and Paramesvara. The author of the commentary Prayoga-racana has simply paraphrased it.

Verse iv. 13. The genuineness of this verse is evident from the mention of the word "athava" (meaning "or") in the beginn- ing of the next verse 14, which shows that verse 14 gives an alternative method. Verse 13 is thus indispensable, for if it is removed verse 14 would no longer be an alternative method. The author of the Prayoga-racanã, commenting on these two verses, states: VI INTRODUCTION "viskambhardha etc. Here the author finds the true daily motion." "antyajivä etc. This is an alternative method." So also, commenting upon the verse 14, Parameśvara writes: "The author sets forth the method of finding the true daily motion by another method."³ Absence of the verse in question from the text of MS. C seems to be due to the carelessness of the scribe. Verses viii. 23-24. These verses are taken as genuine because they occur in MSS. A and B and have been mentioned in MSS. D and E. Moreover, they have been mentioned and the examples contained therein have been solved in MS. C in Chapter I under verses 49 and 50 respectively. It seems that like so many other verses these also have been left out from MS. C due to the carelessness of the scribe. There is no doubt that they were composed by Bhaskara I, because they occur in his commentary on the Aryabhatiya along with several other verses containing similar examples. There is one more verse which requires consideration here. It is not included in the numbered verses of the edited text. This verse marked 24* occurs after verse 24 of Chapter VIII. It does not occur in any of the available manuscripts of the Maha-Bhaskariya, but I have taken the liberty to include it in the edited text, though I have not given it any specific number. My reasons for including this verse in the text are as follows: (1) The preceding verse 24 contains an example which. विष्कम्भार्धेनेति स्फुटभुक्ति साधयति । 2 अन्त्यजीवेत्युपायान्तरम् । 3 स्फुटभुक्त्यानयनं प्रकारान्तरेणाह । INTRODUCTION unlike other examples of that chapter, is of an abstract nature. In order to give that example a concrete form, it is necessary to supply some such data as mentioned in verse 24*. (2) Verse 24 occurs in Bhaskara I's commentary on the Aryabhatiya¹ and in Govinda Svāmi's com- mentary on the Maha-Bhaskariya. At both the places it has been followed by verse 24* and the two verses taken together are treated as form- ing one complete example. The example has been solved both by Bhaskara I and Govinda- Svāmi. Chapters Verses Counting this additional verse also, the edited text of the Maha-Bhaskariya consists of 403 and a half verses, which are distributed over the eight chapters as follows: VII V IV 742 64 78 I II III 52 10 VI VII 62 35 VIII 28 Reading-differences. The question of deciding between differing readings of the text as encountered in the manuscripts consulted by me has in general not presented much difficulty. In my choice between alternative readings I have been guided solely by the criterion of appropriateness. My task has been simplified by the fact that the readings selected and accepted should give the correct mathematical interpretation of the text as well as should fit in the metre.³ Quotations from the Maha-Bhaskariya are found in other works. I have gathered together such quotations and this has helped me to verify the edited text in a number of cases. ¹ ii. 32-33. 2 i. 52. 3 Metres used in the Maha-Bhaskariya have been arranged alphabetically at the end of the Sanskrit portion. VIII INTRODUCTION In some cases, however, alternative readings are possible and it may be that I have not given in the edited text the readings which might have occurred in the original. I have given in the foot-notes all alternative readings found in the various manuscripts consulted by me so that the interested reader may for himself decide whether the readings given in the edited text are appropriate or not. English Translation. The question of translating tech- nical material written in Sanskrit into English presents consi- derable difficulty. It requires a thorough knowledge of both languages which few can claim. My effort has been directed towards giving as far as possible a literal version in English of the text. The portions of the English translation enclosed within brackets do not occur in the text and have been given in the translation to make it understandable, and at places are explanatory. Without these portions the translation at places would appear meaningless to a reader who cannot consult the original for lack of knowledge of Sanskrit. I have tried my best to keep the spirit of the original and have as far as pos- sible not altered the sequence in the translation. Sanskrit tech- nical terms having no equivalents in English have been given as such in the translation. They are explained in the subjoined notes and the reader can always refer to the Glossary given in the end to find the meaning of such terms whenever the sub- joined note does not contain the explanation of the terms. Verses dealing with the same topic have been translated together and are preceded by an introductory heading briefly summarising their contents. In doing so I have followed the practice of the commentators. The translation is followed by short notes and comments comprising (i) elucidation of the text where necessary, (ii) rationale of the rule given in the text, (iii) critical notes, and (iv) other relevant matter, depending on the nature of the passage translated. INTRODUCTION IX Technical Terms. I have already pointed out (in Part I) the peculiarities of the language of the Maha-Bhaskariya. I have also noted that some of the technical terms used therein do not occur in other works. These technical terms are not found in Sanskrit lexicons, and I have succeeded in arriving at their correct interpretation and meaning by (i) the context, and (ii) comparison of the same or similar topics in other works available to me. The commentaries consulted by me have been of immense help, for without them quite a number of passages in the text would have remained obscure. In the foot-notes I have given references to parallel or similar pas- sages found elsewhere, so that the reader may judge for himself whether I have arrived at correct interpretations or not. At the end of this Part I have appended (1) a list of passages from the Maha-Bhaskariya quoted by later writers, and (2) a Glossary of terms used in the Maha-Bhaskariya. In the end it is my sacred duty to express my great indebtedness to the late Dr. A. N. Singh for the help and guidance that I received from him in the preparation of the present edition of the Maha-Bhaskariya. I am also under great obligation to the late Dr. Bibhutibhusan Datta (alias Swami Vidyaranya) who kindly went through the whole of this work and gave valuable suggestions and advice. My sincere thanks are due to Dr. Ram Ballabh, Professor of Mathematics, Lucknow University, for his suggestions and advice from time to time and for affording me all facilities in my researches. I am also thankful to Sri Gopal Dvivedi, Jyotisacharya, and Sri Markandeya Misra, Jyotisacharya, my Research Assistants, for the assistance rendered by them to me. My thanks are also due to the Nav Jyoti Press, Lucknow, for their unfailing courtesy and care in the printing of this book.

K. S. Shukla

श्रीमद्भास्कराचार्यप्रणीतम्
महाभास्करीयम्

प्रथमोऽध्यायः
कलां बिभति क्षणदाकरस्य यः
प्रकाशिताशां शिरसा गभस्तिभिः |
नमोऽस्तु तस्मै सुरवन्दिताङ्घ्रये
समस्त विद्याप्रभवाय शम्भवे ॥ १ ॥
जयन्ति भानोः कमलावबोधिनः
करा हिमांशोर्वनिताननस्विषः' ।
सरितारास्कुटवीर्घरश्मयो
धरासुतज्ञाकसितत्विषः पुनः ॥ २ ॥
तपोभिराप्तं स्फुटतन्त्रमाश्मक*
चिरत्वमभ्येतु' जगत्सु सद्गुणैः ।
चिरं च जीव्यासुरपेतकल्मषा'
भटस्य शिष्या जितरागशत्रवः || ३ ||
नवाद्रिरूपाग्नित महीभुजां
शकेन्द्रनाम्नां गतवर्षसङ्ग्रहम् ।
द्विषट्कनिघ्नं गतमाससंयुतं
युगाधिमासैर्गुणयेद् द्विराशितम् ॥ ४॥
३
' हिमांशोर्वनिता तत्विषः A, B. धरासुतज्ञातिसितत्विषं A, B.
'जीयासुरपेतकल्मषा A, B. • नवाग्नि-
'युर्गार्धमासँगणयेद्विराशितम् B; युगाधि-
.
'बमस्तिनि B.

• स्फुटतन्त्रमात्मकं C. 'चिरं समभ्येतु A, B.
  

रूपाग्नियुतं A, B; नावाग्निरूपाग्नियुतं C.
मास येद्विराशिकम् C,
६

२

युगार्कमासाप्तगताधिमासकै- र्युतं' तिथिघ्नं गतवासरैर्युतम्' । युगावमैस्तद् गुणयेद् द्विराशितं निशाकराहैविभजेत नित्यशः ॥ ५ ॥ १२ तिथिप्रणाशाप्तिरतो विशोधिते" भवत्यथाह्नां निचयः कलेर्गतः । वदन्ति' वारं 'दितिसुनुपूजितात् प्रवृत्तिमप्याहुरुदञ्चतो रवे: ।। ६ ।। शशाङ्कमासैरभिताडितान् हरे- दतीतमासानथवार्कसम्भवैः । दिनीकृतान् भूमिदिनैर्हतान् दिनै- विभज्य लब्धशशशिजै रहर्गणः ।। ७ ।। उदीरितान् यान् भगणान् क्षमादिनँ- र्लभामहे कान् कलियातवासरैः" । इति प्रलब्धा भगणास्ततः क्रमाद् गृहांशलिप्ताविकला: " सतत्पराः ॥ ८॥ कलीकृतं वा ससमं " दिवाकरं स्वगीतिकोक्तैर्भगणैः समाहतम् " १५ भजेत बर्षेर्युगसङङ्ख्ययोदितै विहङ्गमानां प्रवदन्ति लिप्तिकाः ॥ ९ ॥ निशाकरं वा ग्रहमुच्चमेव वा कलीकृतं " तत्सहयातमण्डलैः । [ महाभास्करीये • उदी- S '°युता: A; युत: B. The word fafe here means 30, and not 15 as usual. 'गतवासरान्वितम् A, B. * द्विराशित: A, B; द्विराशिकं C. 'विशो- ध्यते A, B. * भवन्ति A, B. " तिथिसूनुपूजितात् A, B. ' शशाब्दमासै रमिताडि- तान् C. दिनोकृताद्भुमिदिनाहृतां A; दिनीकृताद्भूमिदिनाहतान् B, C. रितान्यामगणात्क्षमादिर्नलंभामहे तान् कलियातवासरै: A, B. " ग्रहांशलिप्ता ° A, B, Missing but commented upon in C. संसमम्) means "together with years"] " पक्षैर्युगसंख्ययोदितैः A, B. " कुलीकृतं C. १३ " [ससमं ( = समाभिः सह वर्तमानं, 'समाहत: A ; समाहतो प्रथमोऽध्यायः ] यथेष्टनक्षत्रगणैर्हतं हरेत् तदीयनक्षत्रगणैस्ततः कलाः ॥१०॥ युगाधिमासैर्द्युगणं हतं हरेत् क्षमादिनैर्वा भगणादि लभ्यते । त्रयोदशघ्ने सवितर्यथ' क्षिपे- न्निशीथिनीनांपतिचारसिद्धये ॥११॥ कुमुद्धतीनां सुहृदोऽथवाऽऽगतं विशोध्य शेषस्य लवस्त्रयोदशः । स मध्यमार्को गणकैनिरूप्यते गुरुप्रसादात्प्रतिबुद्धबुद्धिभिः ।।१२।। विना चुराशेरपि चन्द्रभास्करौ प्रकुर्वतो वा विधिरेष कथ्यते । समासु मासीकृतविग्रहासु ये ह्यतीतमासा विनियोज्य तान् पुनः ॥१३॥ खरामनिघ्नान् दिवसेषु योजयेद् गतेषु मासस्य ततोऽधिमासकैः । निहत्य सर्वं विभजेत सर्वदा युगार्कमासैदिवसत्वमागतैः ॥१४॥ · भवन्ति लब्धास्त्वधिमासका: पुन स्ततोऽपनीयाशु च भागहारकम् । भजेत शेषं शशिमाससङ्ख्ययां ततोंऽशलिप्ताविकला : सतत्पराः ||१५| ततोऽधिमासान् प्रणिहत्य खाग्निभि नियोज्य ं सम्यग्गतवासरैः क्रमात् । युगावमघ्नाञ्छशिवासरैर्हरेत् तमत्र शेषं प्रवदन्ति चाह्निकम् ॥१६॥ ' सवितर्यना A; संवितयंता B. २ निशीथिनीनां पथिचारसिद्धये A, C; निशीथिनानां पथिचारसिद्धये B. च वस्त्रयोदश C. "राशेरगि

• Missing in A and B.

" भवेदशेषा: A, B. 'ततांश लिम्ताविकलाः A. A, B. 'विनियोज्यतां पुन: C. 'सुयोज्य A, B. " युगावमघ्नाच्छशिवासरर्हरेत् A, B. हत्वाऽधिमासैरवमस्य शेषं छित्त्वा धराया दिवस: प्रलब्धम् । संयोज्य नित्यं त्वधिमासशेषे का पुस्तकरणैर्यथोक्तम् ॥१७॥ युगप्रसिद्धैर्धरणीदिनैर्हरे- निहत्य षष्ट्यावमशेषमाह्निकम्' । कला विलिप्ता: क्रमशस्सतत्परा- स्त्वतीतमासा दिवसा गृहांशका : * ||१८|| त्रयोदशघ्नादपि रूपताडिता - द्विशोधयेद्यत्त्वधिमासशेषजम् । ९ निशाकराको गणकैः प्रकीर्तितौ भटप्रणीताविति बुद्धिमत्तमैः ॥१९॥ ॥ अम्बरोरुपरिधिविभाजितो भूदिनैदिवसयोजनानि तैः । सगुणय्य दिवसानथाहरेत् कक्ष्यया भगणराशयः स्वया ||२०॥ अदृष्टमन्यैरिदमाश्मकीयैः कर्म ग्रहाणां लघुतन्त्रसिद्धम् । सञ्चिन्त्य शास्त्रार्णवमाश्मकीय- मुद्घाट्यते' तत्र॰ रहस्यभूतम् ॥२१॥ रुद्रैः सहस्रहतषट्छ्कलैश्च" हत्वा मुक्त्वा A, B. "रसरामभक्तैः A, B. वर्षाणि रन्ध्रवसुवमानसङ्ख्यैः । युक्त्वा " सदा प्रविगणय्य खरामभक्ते" 1 मासा भवन्ति दिवसाश्च हृतेऽवशिष्टाः ||२२|| संहत्य रन्ध्रयमलै" रसरामभाग र्भूयोऽग्निवेदगुणितेषु हरेच्च भागम् । ' हत्वाधिमासैरपरस्य A, B. ३ कार्यों A, B. षष्ठ्यामवशेषमाह्निकम् B. ग्रहां- शकाः A, B, C. ` रूप डिता B. 'बुद्धिमत्तरै: C. भूदिनैदिवसयोजनोनितैः A, B; भूदिनैदिवसयोजनान्वितै: C. 'अदृष्टमन्यैरिदम।स्मकीयै: A, B; अदृष्टमान्येरिदमाश्म- कीय: C. उत्पाट्यते A, B. १९ सहस्रहतषट्चफलेषण B. १० ' तन्त्र रहस्यभूतम् A, B. १२ १३ १४ " हृतेवशिष्ट : B. " रन्ध्रयुमर्म: C. [ महाभारकरीबे खरामभक्त: A, B. प्रथमोऽध्यायः ] खब्योमखद्विमुनिभिः प्रलयस्तिथीनां संयोज्य भूतयमरुद्रहृते दिनानि ||२३|| तेभ्योऽधिकाहान्प्रविशोध्य शेषं पातादतीतो' ह्यवमस्य कालः । ४ यदा न शुध्येदवमं प्रगृह्य दत्वा चतुष्पष्टिमतो' विशोध्यः ||२४|| मासाधिमासकगणाद् गिरिभागशेषा- त्रिशद्गुणाद'पचयोऽयमुदीर्यतेऽतः । शैलावशिष्टकलियातमिषुप्रणिघ्नं संयोज्य हीनदिवसेषु नगावशेषः ||२५|| एकयुक्तदिवसेषु वर्षपः " कीर्तितः सितखगादि तद्विदा । हीनरात्रगतयुक्तवासराद् वेदवृन्दविहृतास्तदा "वमाः ॥२६॥ वर्षेषु रन्ध्रकृतचन्द्र समाहतेषु " षट्सप्तपञ्चविहृतेषु दिनादिलाभः । ते योजिता दशहतासु "समासु संज्ञां सम्प्राप्नुवन्ति रविजा इति निश्चयो " मे ||२७|| रविजदिवसयोज्याश्चावमा " येऽत्र" लब्धाः सततमधिकमासाञ्छोधयेत् " खाग्निनिघ्नान् । भवति यदवशिष्टं शोधनीयं समायां यदि तदधिकशुद्धं क्षेप्यमेवोपदिष्टम् ॥२८॥ ' खव्योमवद्विमुनिभिः प्रलयस्थितीनां A; खव्योमवद्विमुनिभिः प्रलयस्थितीनां B. 'तेभ्योऽधिकाहात्प्रविशोध्य A, B; तेभ्योऽधिकाहात्प्ररिणशोध्य C. # वातारदातो A, B. "कालम् B. "शुध्येदपमं B. " दत्त्वा चतुष्षष्टितमो A, B. मासाधिमासकगरणा गिरि- भागशेषाः A, B; मासाधिमासकगणान् गिरिभागशेषान् C. ‘द Missing from C 'नगावशेषम् A, B. १० कीर्तितः स्थितखगादितीद्वदा A; कीर्तितस्थितखगादिती द्वदा B. 'रन्ध्रकृतचन्द्रमसा हतेषु C. १३ समाससंज्ञां A, B; समाससंज्ञा: C. १७ सततमधिकमासाच्छोधयेत् ११ १२ अपमा: B. १४ निश्चलो A, B. १५ ० चापमा B. १६ यत्र A, C. निघ्नान् C. १८ शोधनीयां B. षष्टिशतत्रयनिघ्नो वर्षगणो ग्रहतनुः सदा कथितः । तेन' समेता विहगा' ध्रुवका इति कीर्तिताः' सद्भिः ॥२९॥ मधुसितदिवसाद्यो हीनहीनो गणोऽह्नां दिविचरहृतशिष्टो वारमाहाब्दपादिम् । अत इदमपि शोध्यं शोधनीयं समायां [ महाभास्करीये पतितसमतिरिक्तो गृह्यते नापरोऽत्र ||३०|| सप्तत्या दिवसाद्याश्शरभागा' द्विगुणिता विघटिकाश्च । तद्रहितो ग्रहदेहो रविबुधभृगवश्च निर्दिष्टाः |॥३१॥ कुमुदवनसुबन्धो रन्ध्रवर्गो द्वियुक्तो ग्रहतनुगुणकारी भागहारः प्रदिष्ट: । शरयमयमलाख्यो भागपूर्वोऽत्र लाभो 'घृणमपि शिवनिघ्ने खेन्द्रियाप्ता विलिप्ताः ॥ ३२॥ भागाः खत्रिघनांशास्त्रिरुद्रगुणिते विलिप्तिका ज्ञेयाः । षड्भिः शतैविभक्ते विंशत्यंशो रवेश्च तमः ॥ ३३॥ अचलहतनवांशा लिप्तिका रुद्रनिघ्ने 'गगनरस विभक्ते लिप्तिकास्ता विपूर्वाः । ग्रहतनुखयमांशास्तत्परा: शोधनीया दशलवसमवेत`श्चन्द्रतुङ्गः स भानोः ||३४|| भूभृद्रामहतां हरेच्छतगुणै" " रन्धैर्ग्रहाणां तनुं " भागाद्यास्फुजितो विमौरिकगणा" भागे शतेनोद्धृते । रामांशेन युतं रवेश्च सकलं द्विघ्नाद्ववेश्शोधये- त्क्षेपः सोमजसोमयोः कृतगुण: सूर्योऽथ विश्वाहतः ॥ ३५॥ व्योमशून्यनेत्रभाजिते फलं राशयोऽष्टभाजितेऽथ लिप्तिकाः । ३ वारमा- ५ ६ " तेन Missing from C. बिभागा C. ध्रुवका इति कीर्तित: B. सान्दवादि: A, B. सप्तद्या दिवसाद्याश्शरभाभागा C. ह्युगमविशिखनिघ्नो वेन्द्रि- याप्तावलिप्ता: A; ह्युणमविशिखनिष्ठो वेन्द्रियाप्ता विलिप्ता: B. खत्रिदिनाशास्त्रिरुद्र- गुणिते A; खत्रिघनांशास्त्रितरुद्रगुणिते C. ` ° समुपेत्: A, B. ° हरेच्छरगुणे A, B. मौरिकगणा A, B; भागाद्यास्फुजितोपि मौरिकगुणो C. बिलिप्तागणाः | 'लिप्तिकां विदुः A, B. ' गगनरसविभक्तलिप्तिकास्तापिपूर्वा: A, B. १९ ° ग्रहाणान्तरं C. १२ भागाद्यास्थजितोऽपि - १* विमोरिकगणा: means 0 प्रथमोऽध्यायः] ७ 'बिन्दुषड्ढ़ते विलिप्तिका विदुः सर्वमेव योज्य गण्यते बुधः ॥३६॥ । अष्टाहते शरयमाश्विहृते कलाः स्यु- देहे तथा त्रिशतभक्तविलिप्तिकाश्च । युक्त्वैतदेव'मुभयं शनिरत्र गण्य - स्त्रिशल्लवो रविभवो धनमत्र कार्यम् ॥३७॥ द्विकनिघ्ने ग्रहदेहे स्वविंशभागरहिते" तु लिप्ताद्याः । पञ्चाशदंशविकलाः क्षेप्या भौमो रवेरर्धे ॥३८॥ द्वियमघ्ने ग्रहदेहे शरनगरामोद्धृते' तु लिप्ताद्याः । सुरनाथगुरोर्नोगो रविभोगद्वादशांशयुतः ॥३९॥ राशित्रयं क्षिप निशाकरतुङ्गमध्ये "पातं निपात्य भगणात्" क्षिप राशिषट्कम् । त्रैराशिकागतदिनेषु च रूपमेकं व्यावर्णयन्ति गणका भटशास्त्रचित्ताः ॥४०॥ भूदिनेष्टगणा"न्योन्यभक्तशेषेण भाजितौ । हारभाज्यौ दृढौ स्यातां कुट्टाकारं तयोविदुः ॥४१॥ भाज्यं न्यसेदुपरि हारमधश्च तस्य खण्डयात्परस्परमधो विनिधाय लब्धम् । केनाऽऽहतोऽयमपनीय यथाऽस्य शेषं" भागं ददाति परिशुद्धमिति प्रचिन्त्यम् ॥४२॥ "आप्तां मति तां विनिधाय वल्ल्यां "नित्यं ह्यधोऽधः क्रमशश्च लब्धम् । 'विन्दुषड्ढते विलिप्तिकां विदुः सर्वमेव योज्यते A; विन्दष......... ते विलिप्तिक द्विदस्सर्वमेव योज्यते B. २ त्रिदशभक्तविलिप्तिकाश्च A, B; सदातिशतभक्तविलिप्ति- काश्च C. 'मुक्त्वैतदेतदुभयं A; मुक्त्वैतदेवमुभयं । C. गण्याः A, B. स्वविंशद्- भागरहिते A, B. 'पञ्चाशद्दशविकलाः A, B. "क्षेप्यो A. 'रवेरर्थे C. 'शरी- गरामोद्धृते C. वातं निपात्य B. १५ 'भगणान् C. १२ त्रैराशिकागतगुणेषु A. "प्रावर्त यन्ति C. "भूदिनेष्टगुणो A, B; क्ष्मादिनेष्टगणा C. वल्यः A, B. "तेनाहतो. धमपनीय A, B. १७ The reading स्वशेष in place of 'स्य शेषं has been mentioned by Govinda Svami in his commentary. “प्रचिन्त्य A, B. अवेच....."तिनां A; अर्वञ्च तिन्तां B. २० नीतं A. ४ १५ १४ मत्या हृतं स्यादुपरिस्थितं य- ल्लब्धेन युक्तं परतश्च तद्वत् ॥४३॥ हारेण भाज्यो विधिनोपरिस्थो भाज्येन नित्यं तदधः स्थितश्च । अह्नांगो स्मिन् भगणादयश्च तद्वा' भवेद्यस्य समीहितं यत् ॥४४॥ रूपमेकमपास्यापि कुट्टाकारः प्रसाध्यते । गुणकारोऽथ लब्धं च राशी स्यातामुपर्यधः ॥४५॥ इष्टेन शेषमभिहत्य भजेद् दृढाभ्यां शेषं दिनानि' भगणादि च कीर्त्यतेऽत्र | राश्यादयो निरपवर्तितवासरघ्ना राश्यादिमानभजिताः' प्रवदन्ति शेषम् ||४६|| [ महाभास्करीये भाज्योऽधिको' यदि भवेत् खलु हारराशे- स्तत्राधिकं समपनीय तथैव कर्म । तेनाधिकेन गुणितो गुणकारराशि- र्युक्तोऽधरेण स भवेत् पृथगत्र लब्धम् ॥४७॥ अपवर्तितवासरादिशेषात् क्रमशस्तानपनीय रूपपूर्वम् । अहर्गणो A, B, C. १० " दिनादि A, B. 'म B. ९ भाज्याधिको A, B. भाज्यापवर्तना C. १२ 'प्रस्तारयुक्ते वा A, कृतकुट्टनलब्धराशिमेषां गुरणकारं समुशन्ति वारहेतोः ॥४८॥ छेदभाज्यापवर्तेन" यच्छेदस्यातिरिच्यते । तेन हारं समभ्यस्य वेलाकुट्टस्तु पूर्ववत् ॥४९॥ प्रक्षिप्य भागहारं कुट्टाकारे पुनः पुनः प्राज्ञैः । योज्यं च भागलब्धं भाज्ये " प्रस्तारयुक्त्यैव " ॥५०॥ गन्तव्यमिष्टं यदि कस्यचित्स्याद्- गन्तव्ययोगादिदमेव कर्म । तष्ट्वा A; त...... B निरपवर्जितवासरघ्ना A, B. च A, B. भुजेदृढरभ्यां C. 'राश्यादिमानभजितां A, B. छ्दे भाज्यापवर्तेन A; छेद- " भाज्ये Missing from C. ११ कृत्तकुट्टनलब्ध राशिशेषां B. 'पुन: Missing from A, B. B. ४ प्रथमोऽध्यायः ] रूपेण वा योज्य विधिविचिन्त्यः' सर्वं समानं खलु लक्षणेन ॥५१॥ योगेषु तेषां भगणादियोग- विशेषितैर्वापि तथा विशेषे । अन्योन्यशेषादपि चिन्तनीय- मिष्टग्रहस्पष्टगणैविधानम् ॥५२॥ इति महाभास्करीये प्रथमोऽध्यायः । 'योन्यविधिविचिन्त्य A, B. विशेष: C. १ द्वितीयोऽध्यायः लङ्कातः खरनगरं सितोरुगेहं पाणाटो' मिसितपुरी तथा तपर्णी । उत्तुङ्गस्सितवरनामधेयशैलो लक्ष्मीवत्पुरमपि' वात्स्यगुल्मसंज्ञम् ॥ १ ॥ विख्याता वननगरी' तथा ह्यवन्ती स्थानेशो मुदितजनस्तथा च मेरुः । अध्वाख्यः' करणविधिस्तु मध्यमाना- मेतेषु प्रतिवसतां न विद्यते सः ॥ २ ॥ अक्षांशान्निगदितपत्तनांशहीनान्' संहन्यान्नवनवपक्षपुष्कराख्यैः । अष्टाभिश्शरकृतिभागहीनसङ्ख्यै श्चक्रांशैरपहृतयोजनानि कोटिः ॥ ३ ॥ कर्णाख्यः स्वगदितपत्तनान्तराध्वा" तिर्यस्थो जनपदभाषितो" जगत्याम् । तत्कृत्योविवरपदं वदन्ति केचि - दध्वानं ग्रहगणितस्य वेदितार : ” ॥ ४ ॥ अध्वानं गणितविदो भटस्य शिष्या: १४ स्थूलत्वाच्छ्रवणविधेर्न सम्यगाहुः । अक्षादेरपि च विधेरथोपपत्तिं वक्रत्वात् क्षितिपरिधेर्वदन्ति सन्तः ॥ ५ ॥ ' वाणाटो A, B. २ तथातपण्णि A, B. ३ लक्ष्मीवत्परमपि A. 'खात्यगुल्म° A, " From पाणाटो to वात्स्यगुल्मसंज्ञम् is missing from C. " परनगरी C. " तथा ह्यवन्ति B; तथाप्यवन्ती C. ' निन्ध्याख्य: A; विन्ध्याख्य: B; अधाख्य: C. १९ स्वविहितपत्तनान्त- B. १२ जनवदभाषितो B. 'स्वाक्षांशाणिगदितवित्तनाशहीनात् A, B. १° चन्द्रांश: ° A, B. रध्वा A; स्वविहितपत्तनान्तरद्धा B; स्वर्गादितपत्तनान्तराध्वा C. " गणितस्य वेदितारम् A; गणितस्य वेदिस्तारम् B. B; विधेरथाप्रवृत्ति C. १६ चक्रत्वात्क्षितिपरिधेर्भवन्ति C. ४ शिष्यान् C. १५ विधेरथोपपत्ति द्वितीयोऽध्यायः ] छायाप्तस्फुटदिवसार्धतिग्मरश्म्यो- रध्वानं विवरकलाग्रमाहुरेके । नैतत् स्यात् समपरपूर्व दिस्थितानां तुल्यत्वात् पलगणितस्य वर्णयन्ति ॥ ६ ॥ सूर्येन्द्वोरकृतसमाध्वनोविधानात् सम्प्राप्त स्थितिदलदृष्टकालयोश्च' । विश्लेष: स्फुटतर उच्यतेऽत्र कालो गोलज्ञैविदितभटप्रणीततन्त्रैः ॥ ७ ॥ ४ नित्यं वा शशिन मदेशकालसङ्ख्यं सूर्यञ्च स्फुटमुदयास्तगं घटीभिः । कृत्वैवं जलविधिना घटीश्च बुद्ध्वा "तन्त्रज्ञा विवरमुशन्ति देशकालम् ॥ ८ ॥ ग्रहोदयो यदा पूर्व स्पर्शः पश्चाच्च लक्ष्यते' । पूर्वेण समरेखाया द्रष्टा " स्यात्पश्चिमेऽन्यथा ॥ ९ ॥ कालेनाहत्य भुक्ति ग्रहरवितमसां देशजातेन नित्यं षष्ट्या हृत्वाऽथ लब्धं जलपसुरपयोदिग्गतानां धनर्णे" । लम्बेनाहत्य भूमेः सकलगुणहृते वृत्तसङ्ख्यां घटीभि र्हत्वा" देशान्तराभिर्गगनरसहृते योजनाग्रं वदन्ति " ॥१०॥ इति महाभास्करीये द्वितीयोऽध्यायः । ' छायार्धस्फुटदिवसस्थतिग्मरश्म्यो: A; छायार्थस्फुटदिवसतिग्मरश्म्यो: B; छायार्ध- २ फलगणितस्य A, B. तिग्मरश्म्यो: C.

• सम्प्राप्तस्सतिदलकालदृष्टयोश्च

६ A, B.

• गोलोक्तविदितभटप्रणीतनेत्र: A, B. " सूर्यश्च A, B.

७ 'स्फुटमुदयास्तथा A; स्फुट- मुदयास्तथङ् B; स्फुटमुदयास्तनं C. ' तत्र स्याद्विवरमुशन्ति A; तत्रस्य द्विपरमुशन्ति B. पूर्व: C. ' स्पर्शश्चैवोपलक्ष्यते A, B; स्पर्शश्चैवोपलभ्यते C. भ्रष्टा A, B. षष्ट्या हृत्वाथ लब्धा जलवसुरवयोदिग्गतानां धनर्णे A; षष्ठ्या हृत्वाथ लब्धा जलवसुरवयो दिग्गतानां धनर्णे B; षष्ट्या हत्वाथ लब्धः जलवसुरवयो दिग्गतानां धनं तत् C. हृत्वा C. ?" In MSS A and B the last two lines read as follows: लम्बेनाभ्यस्य भूमेस्सकलगुणहृतो वृत्तसंख्याघटीभिहं त्वा देशान्तराभिर्गगनरसहृतो योजनाग्रं वदन्ति । १२ १० तृतीयोऽध्यायः अद्भिः समत्वमवगम्य' धरातलस्य वृत्तं लिखेत् स्फुटतरं खलु कर्कटेन । सूत्रैश्चतुभिरवलम्बकसन्निबद्धै- 'ज्ञतार्जवोरुसमवृत्तगुरुर्नर: स्यात् ॥ १ ॥ छायाप्रवेशनिर्गमबिन्दुभ्यां' मीनमालिखेद् व्यक्तम् । "तद्वक्त्रपुच्छनिःसृतं सूत्रं याम्योत्तरं शङ्कोः ॥ २ ॥ बिन्दुभिस्त्रिभिरतुल्यकालजैः संलिखेच्छफरिके विधानतः । सूत्रयोर्मुखसमप्रयातयो– र्योगतः कुजबुधाशयोविधिः ॥ ३ ॥ नृच्छायाकृतियोगस्य मूलमाहुर्मनीषिणः । विष्कम्भार्ध स्ववृत्तस्य छायाकर्मणि सर्वदा ॥ ४॥ छायाहतं त्रिभवनस्य" गुणप्रतानं " हत्वा नरेण च पृथग्विभजेत्पदेन । अक्षावलम्बकगुगौ" विषुवत्प्रसिद्धौ छायानरौ च विपुलावपरत्र दृष्टौ ॥ ५ ॥ इष्टज्यां मुनिरन्ध्रपुष्करशशिक्षुण्णां सदा संहरेद् व्यासार्धेन भवेदपक्रमगुणस्तात्कालिकस्तत्कृतिम् । विष्कम्भार्धकृतेविंशोध्य" च पदं द्युव्यासखण्डं " विदुः १७ स्वेष्टक्रान्तिहतं पलं प्रविभजेल्लम्बेन जीवा क्षितेः ॥ ६॥ २ ' समत्वमधिगम्य A, B. अवलम्वततं निबद्धै: A; अवलम्वत तन्निवद्ध: B. ४ • जाता° C. समवृत्त is missing from A, B. "छायाप्रवेशनिर्गमवीन्दुभ्यां C. ६ ★ वृत्तम् C. " तद्वत्कृपुच्छनिःसृतं सूत्रं याम्मोत्तरं A; B.; तद्वक्त्रपुच्छनिर्गतसूत्रं याम्मोत्तरे C. १० विष्कम्भार्धा A, B. गुणने C. १४ व्यासार्धस्य B; पदं दर्व्यासखण्डं C. तद्वक्तृपुच्छनिःसृतसूत्रं याम्मोत्तरं ' मुखमाहुर्मनीषिण: C. १३ ' प्रयान्तयोः B. १२ गुणव्रतानं C. १९ त्रिभुवनस्य A, B, C. कृतेविंशोध्य C. पदाद् द्युव्यासखण्डं A; पदा १५ २६ फलं A, B; परं. C. १७ क्षितो C. अक्षावलङ्क- 'न्यासखण्डं योsध्याय: ] व्यासखण्डगुणितं' क्षितेर्गुणं संहरेद् दिवसजीवया' पुनः । काष्ठितं च यदवाप्तमत्र तु प्रोच्यते चरद सतां वरैः ॥ ७ ॥ जिना दशघ्ना यमरन्ध्रशालिनो निशाकराष्टौ गुणिताः पलाङ्गुलैः । हृताश्चतुभिः क्रिय-गो-नरान्तजा भवन्ति निःश्वासलवा: चरोद्भवाः ॥ ८ ॥ शशिकृतशशिरामैराहता राशिजीवाः स्वकदिवसगुणार्धेर्भाजिता:' काष्ठताश्च । पतितसमतिरिक्ताः पूर्वचापैर जाद्यै- " ११ १२ विषुवदुदयराशिप्राणपिण्डाः क्रियाद्याः ॥ ९ ॥ खनगरसशशाङ्काः पञ्चरन्ध्राद्रिरूपा विषयशिखिनवैकास्ते च दृष्टा विधिज्ञैः । चरदलपरिहीणा योजिता व्युत्क्रमेण प्रतिविषयसमुत्थास्तूदया मेषपूर्वा : ४ ॥१०॥ अपगमपलभागा" मेषजूकादिगोले रहितसहितसङ्ख्या मध्यसूर्यावनामः । अवनतिलवहीनः प्रोन्नतिश्चक्रपाद - स्त्ववनतिलवजीवा साप्रभा नेतरा" स्थात् ॥ ११॥ क्षितिजाद्यदलसमासो विश्लेषो वोत्तरेतरे गोले । लम्बघ्नस्त्रिज्याप्तः शङ्कदिनमध्यगे “ सूर्ये ॥१२॥ १ . व्यासखण्डगुणिता A, B. 'संहरेद् द्युदलजी° A; संहरेद्युदलजी° B. ३ काष्ठकं ५ A, B * परदलं C. कृतरन्ध्रभूमयो A, B; यम ...शालिनो C. 'नवादयस्ते गुणिता वलाङ्गुकैः A; नवादियस्तेगुणिता वलाङ्गुलै B. ' वलोद्भवाः A, B; पलोद्भवा: C. — Parameśvara_quotes the following reading of this verse in his Siddhānta-dipika: वसुत्रिदस्रा गुणरन्ध्रभूमयो नवाद्रयश्चाभिहताः पलागु लैः । हृताश्चतुर्मिः क्रियगोयमान्तजाश्चरासवः स्युः क्रमशस्तु चापिताः ॥ १० काष्ठिकाश्च B. १३ ' स्वधिदि° A, B. १२ भवन्ति A, B. रन्ध्राश्मिरूपा B. वनामम् A, B. १७ नेतरा = ना + इतरा । १४ १९ पूर्वचापैरजाद्या A; पूर्वचापैरजाद्या B.

खनगरसशशाङ्कां पञ्च-

१६ मध्यसूर्या- खनगरसशशाङ्कां पञ्चरन्ध्राश्विरूपा 'तुभयोर्वेषपूर्वा: C. १५ अपगमवलभागा A, B. १८ शङ्कुर्दीनमध्यगे B. १४ छायया समभिनीतसन्नतेः ' जायायां C. साक्षजं A, B. काष्ठतोऽधिकतरं यदा पलम् । तद्भिदः' स्फुटरवेरपक्रम - स्तिग्मरश्मिरपि चोत्तरे तदा ॥ १३॥ छायायां याम्यकाष्ठायामयुक्ता नतिः स्फुटा 1 जायन्तेऽपक्रमा भागा भास्वतोऽदक्षिणापथे ॥१४॥ अक्षतोऽधिकतरा यदा नतिः पात्यते पलमतस्तदा सदा । शिष्यतेऽपमधनुः' स्फुटं ततो भास्करोऽपि खलु याम्यगोलगः ॥१५॥ तद्गुणेन गुणितां त्रिराशिजां ज्यामपक्रमगुरणेन” संहरेत् । लब्धचाषगणिते पदक्रमाद्- भास्करस्त्रिकसषण्णवाधिकः ॥१६॥ उत्तरे संयुतिः” सूर्ये विश्लेषो दक्षिणे स्मृतः । अपक्रमनतांशानां " छायायां च पलं भवेत् ॥ १७॥ दक्षिणोत्तरगते विवस्वति राशिनिचयाद् धनक्षयौ“ । स्वाक्षजं चरदलं सदा ततो " जीवयाऽत्र गुणितं दिवागुणम् ॥१८॥ [ महाभास्करीये संहरेत् त्रिभवनस्य " जीवया तत्र लब्धनिचये क्षितेर्गुणम् । ' समभिनीतसंगते: A, B; समभिनीतसन्तते: C. ३ लवम् A, B. तद्भिद B; तत्क्षय C. ४ °क्षयोज्या C. “ This hemistich is missing from B. ६ भोगो A; भागो B. ७ अपक्रमधनुर्भागा जायन्ते दक्षिणापथे C. ' पश्यते वलमतस्तदा सता A, B. ९ ° तेवमधनु: A; ° तेवमतनु C. *° व्योमवक्रमगुणेन C. ११ लब्धचापगुणिते स्फुटं योजयेत् C. १३ अपिक्रमनतांशानां A, B; अवक्रमनतांशोनां C. १२ ततः A, B. १४ १६ १५ प्राणराशिनिचया धनक्षयौ A, B ; प्राणराशिविचयाद्धगक्षयौ C. १७ सतो A, B. १८ त्रिभुवनस्य A, B, C. तृतीयोऽध्यायः ] ९ व्यत्ययं चरदलस्य तत्कुरु १ १२ व्यासखण्डनिचयेन लभ्यते द्युव्यासलम्बकपरस्परताडितस्य छेदोऽर्धविस्तरकृतिः फलमस्य शङ्कुः ||२३|| तज्ज्याविपर्ययकृतं" द्युज्याभ्यस्तेन शकुना हत्वा । विषुवत्कर्णाभ्यस्ता त्रिज्या छेदः" फलं शङ्कः ||२४|| इष्टासुभ्यश्चराशुद्धौ” व्यत्ययः शेषजीवया | प्राग्वल्लब्धं क्षितेमौर्व्यां " हित्वा शङ्क: " स्वकर्मणा ||२५|| शर्वर्यां शङ्करर्कस्य कार्यो व्यस्तेन कर्मणा | दिनस्य " क्षयवृद्धिभ्यां रात्रौ व्यस्तेन ते यतः ॥२६॥ इष्टच्छायाप्तशङ्कु सकलगुणहतं " संहरेल्लम्बकेन प्राप्ते मेषादिगेऽर्के क्षितिजगुणकलाः शोधयेद्दक्षिणस्थे । २ ० विशेषजा A, B; °विशेषकं C. हिसा A, B. स्फुटशङ्कुरुक्तं A. ६ वा B. " इनरविकक्ष्यामध्यज्यात्रिज्या-

• विष्क-

५ कृत्योर्विशेष्यशेषपदम् A, B; रविकक्ष्यामध्यज्याकृत्योविशेषशेषपदम् C. ' प्राणैश्चरैर्यत ° C. तज्याविपर्ययगुणेन B. १० तज्ज्याविपर्ययकृता A, B. 'इष्टासभ्यश्चराशुद्धेन A, B; इष्टादिभ्यश्चराशुद्धौ C. A; शंकु B. अह्नस्तु C. सकलगुणहतां A, B. म्भभेदरहिता A, B. स्वावलम्बकहतं हरेत् पुनः ॥ १६ ॥ व्यासतत्कृतिविशेषजं पदं कथ्यते स्फुटतरा प्रभा सदा ' ||२०|| आदित्यलग्नविवरांशगुणेन हत्वा तत्काल मध्यपरिनिष्ठितलम्बकाख्यम् । विष्कम्भभेदभजितः स्फुटशङ्कुरुक्त - स्तद्गोलभेदकृतिशुद्धपदं प्रभा स्यात् ' ॥२१॥ रविकक्ष्यामध्यज्यात्रिज्याकृत्योर्विशेषशेषपदम् । तत्कालमध्यजातो गोलज्ञैर्लम्बकः कथितः ॥ २२ ॥ प्राणैश्चरैर्युतविहीनघटीगुणेन तज्ज्याविपर्ययकृतेन हृतस्य वाऽस्य | ° समुद्भवा A, B. शङ्कुरिष्टघटिकासमुद्भवः' । १५ १६ १५ ११ च्छेद: A, B, C. १४ शङ्कु १३ क्षितेमौर्व्या C. १७ मेषा गोले C. [ महाभास्करीये योज्यन्ते ताः क्रमेण त्रिगृहगुणहतं संहरेच्छेषराशि युव्यासार्धेन चापे स्वचरदललवव्यत्ययेनासुराशिः ॥ २७॥ अह्नः शेषो गतो वा रस - खरसहृतो' नाडिकाद्याः प्रदिष्टाः त्रिज्यावर्गाहतं* वा स्वमभिमतन भाजयेद् घातजेन' । द्युव्यासार्धाक्षकोट्योः स्वचरदलगुणे पूर्ववत् कल्पितस्य ७ चापे प्राणाश्चराख्याः पुनरपि विधिना व्यत्ययेनासुराशिः ॥२८॥ अक्षकर्णहतः शङ्कुर्भूयो व्यासार्धताडितः । शङ्कुघ्नद्युदलाप्तो” वा पूर्ववन्नाडिकाविधिः ||२९|| सूर्यागतसमभ्यस्ताः तद्राशीष्टासवो" हृताः । राशिभागैः कलाभिर्वा लब्धा रव्यगतासवः" ॥३०॥ इष्टासुभ्यो विशोध्यैतान् रवौ चाप्यगतं" क्षिपेत् । राशिप्राणांस्ततोऽपास्य देया भानौ च राशयः ॥३१॥ शेषं” त्रिंशत्समभ्यस्तमिष्टराश्यसुभिर्हृतम् । लब्धभागादिसंयुक्तमिष्टकालोदयं विदुः " ||३२|| १५ पूर्वलग्नं सचक्रार्धमस्तलग्नं विधीयते । उदयस्यवशादस्त मयन्ते यतः१७ राशयो ॥३३॥ उदयस्य गता "भागा: स्वोदयेन हता हृताः । त्रिंशता प्राणलब्धि: स्यात् लग्नराश्यसवः पुनः ॥३४॥ आहार्या" यावदर्कस्य राशिभिस्तूदयासवः । जायन्ते पुनरर्कस्य गन्तव्यांशासुभिर्युताः ||३५|| प्राणा दिवसशर्वर्थोविज्ञेया: षड्विभाजिताः । षष्ट्या" भूयोऽपि ये लब्धा घटीविघटिकासवः ॥३६॥ २ रसखर- ५ धुव्या- ९ १० त्रिगुणगुणहता C. द्यावासेनाप्तचापे A, B; धुव्यासार्धाप्तचापे C. सहृते C. 'स्त्रिष्ट्यौ वर्गाहतं C. समभिमतनरा A, B. ६ सातजेन C. सार्धाक्षकोट्या A, C; धुव्यासार्धाभकोट्या B. ' प्राणैश्चराख्यैः A, B. अक्षकर्णनतः: C. 'अङ्कनद्युदलाप्ताद् C. १९ समभ्यस्ता तद्रासिष्टासवो A; समभ्यस्ता तद्राशिष्टा- सवो B; समभ्यस्तात्तद्वाशिष्टासवो C. १३ विशोध्यैता रवौ चान्यगतः रव्यहतासवः C. A, B; विशोध्यैनां स्रवौ चाप्यगतं C. १४ शेष: C. १५ विधि: C. १६ पूर्वलग्ना A, B. १७ गताः C. १८ भागा सोदयेन A, B. १९ आहार्य A, B. २० दिवसशर्वर्योविज्ञेया: A, C; दिसशर्वर्योविज्ञेया B. षष्ठ्या B. १२ २१ 1 ७ तृतीयोऽध्यायः ] स्फुटरविभुजनिघ्नां यां' परां क्रान्तिजीवां हरतु समवलम्ब ज्याकलापेन भूयः । स्फुटदिवसकराग्रा सा यदाऽक्षांशहीना रविरपि यदि गोले चोत्तरे लम्बकघ्नाम् ||३७॥ अक्षज्यया हरेद् भूयः शङ्कः स्यात् सममण्डले । तद्वर्गव्यासकृत्योर्यद् विश्लेषं तत्पदं प्रभा ||३८|| भानोर्भुजामभिहतां परमापमेन युव्यासभेदभजिताप्तह्ताक्षकोटिः । अक्षज्ययाप्तकृतिशुद्धकृतेस्त्रिमौर्व्या मूलस्य काष्ठमसवो गगनावधेर्वा ||३९ || व्यासार्धताडितवपुः सममण्डलस्य दृग्ज्याऽथवा दिवसविस्तरभेदभक्ता | लब्धस्य कार्मुकविधेरसुनाडिकास्ता' ३ भास्वत्वमध्यविवरप्रभवा भवन्ति ॥४०॥ छायाभिनीतसमण्डलशकुनिघ्न- मक्षस्य 'यद्गुणमुपाहर नित्यमेव । सर्वापमेन" समवाप्तधनुर्विवस्वान् युक्त्या " त्रिराशिसहितश्च भटप्रणीतः ॥ ॥४१॥ अर्काग्राशङ्क्वग्रे छायां च यथेष्टकालिकां कृत्वा । अग्रद्वयस्य योगस्तुल्याशस्यान्यथा विवरम् " || ८२ || तेन क्षुण्णां छायां भङ्क्त्वा तद्दुग्गुणेन" यल्लब्धम् । कृतदिग्विभागकेन्द्राद् दिग्विपरीतं निधातव्यम् ॥४३॥ ' स्फुटरविभुजनिघ्ना यां A; स्फुटरविभुजनिघ्नायां B.. लम्बकघ्नाः A, B. ४ शङ्कुज्यासममण्डले A, B. तद्वर्गव्यासकृत्योर्यद्विश्लेषे यत्पदं A, B; तद्वर्गव्यासकृयो- ५ स्तविश्लेषेऽल्पपदं C. भानोर्भुजामविहतां परमावमेन A; भानोर्भुजामविहितां परमावमेन B; भानोर्भुजामतिहतां परमावमेन C. " या प्रकृति ° B. 'दृश्याथवा A, B; दृग्ज्याः C. ' काष्ठकविधेरसुनाडिकास्ता: A, B. ' यद् missing from A, B. वमेन A, C. " समवाप्ततनुविवस्वान् A, B. १० सर्वा - १२ १३ १४ युज्या A, B. 'योगतुल्यांशं नान्यथा विवर: A, B; गोलस्तुल्याशस्यान्यथा विवरम् C. A; तद्युगुणेन B. ७ १७ २ भटप्रणीतम् A, B. १५ तद्वगुणेन १८ तन्मत्स्यभेदिसूत्रं प्रागपरदिशोः प्रसार्यते दूरम् । छायाप्रमाणसूत्रं तिर्यकेन्द्रादिदं नेयम् ॥४४॥ बिन्दु विरच्यं पूर्वापरयोदिशोर्यथायोगम् । मध्यच्छायाशिरसि ज्ञेयो बिन्दुस्तृतीयोऽन्यः ॥४५॥ यो वा स्यादविदित दिग्विभागकेन्द्रो' दिक्छाया भ्रमणविधि प्रकर्तुकामः । तस्याशास्फुटलिखितोरुवृत्तनेमिं १४ न' च्छाया त्यजति यथा तथा प्रवक्ष्ये ॥४६॥ इष्टद्युतेस्तु" कृत्वा दृग्ज्यां" शङ्कं तथा च" शङ्क्वग्रम् । विश्लेषो वाऽग्रयुतिर्बाहुः कर्णोऽत्र दृग्जीवा" ॥४७॥ तत्कृतिविश्लेषपदं " दृग्ज्याकर्णस्य " कथ्यते कोटि: । छायाघ्ने कोटिभुजे दृग्ज्याभक्ते “ तयोर्माने ॥४८॥ शङ्कोस्त्वक्समवपुषस्त्वृजवो" भुजकोटिमानसङ्घटिताः" । वंशशलाकास्ताभिः कोणे कृतशङ्‌ङ्कुसञ्चारम् ॥४९॥ चतुरश्रं द्युतिकर्णं त्रिभुजं" वा कारयेत् स्फुटं यन्त्रम् । विन्यस्यैतद् भ्रमयेच्छायाकर्णानुगा यावत् ॥५०॥ आशा भुजकोटिभ्यां " छायाकर्णाग्रयोरुभौ बिन्दू | मध्यच्छायाशिरसि ज्ञेयो बिन्दुस्तृतीयोऽन्यः ॥५१॥ २४ २५ बिन्दुत्रयस्य " सकलस्य शिरोवगाहि संलिख्यते शफरिकाद्वितयेन" वृत्तम् । तन्मध्यभेदिसूत्रं A, B; तन्मध्ये भेदसूत्रं C. ९ ज्ञेयम् A, B. ••• हबिन्दुद्वयं A, ' बिन्दुस्तुतियायत् A; विन्दस्तुतियायत् B. " In MSS A and B; बिन्दुत्र्यं C. B, verse 45 occurs after verse 47. भितदिग्विभागकेन्द्रो B. स्फुटलिखितोऽत्रवृत्त ° C. १२ [ महाभास्करीये दृज्यां A, B. " स्यादभिहिततद्विभागकेन्द्रो A; स्याद- ६ " fध missing from B. " य A, B. १" च वक्ष्ये C. B. A, B. " बृज्जीवा A, B. २१ भक्ते A; दुज्याभक्ते B. जवो C. 'मान missing from A, B. श्रद्वितीयकणं त्रिभुजं B; चतुरश्रद्युतिकर्णं त्रिभुजं C. A, B. २४ शफरिकान्वितयेन C. २५ बिन्दुसूतियोऽन्यः A, B. १" च missing from B. १६ तत्कृतिविशेषपदं C. १७ ८ ● ° वृत्तनेमिः A, B; तस्याशाः इष्टद्विकेस्तु A; इष्टद्विकैस्तु १४ विश्लेषो वाग्रयुति प्राहुः ११ दृज्याकर्णस्य A, B. १८. दृज्ज्या- शङ्कोस्तत्समवपुषस्त्रिजवो A, B, शङ्कुस्त्वर्कसमवपुषस्त्व- चतुरश्रं द्वितीयकर्णं त्रिभुजं A; चतुर- २३ यत्र C. आशाभुजकोटिज्यां बिन्दुद्व यस्य C. २६ सफरिकौ द्वितयेन A, B; २२ तृतीयोऽध्यायः ] तन्मण्डलाग्रविनिवेशितमस्तकेयं छाया' प्रयाति फणिनीव हि' मन्त्ररुद्धा |॥५२॥ मूलमर्काग्रा । इष्टक्रान्तिक्षितिजावर्गसमासस्य क्षितिजा व्यासार्धहता सूर्याग्रहृता पलस्य' गुणः ॥ ५३॥ अक्षजीवाहतः शङ्कुरिष्टकालसमुद्भवः । भाजितो लम्बकेनाथ शङ्क्वग्रं नित्यदक्षिणम् ॥५४॥ शङ्क्वग्रे द्वादशाभ्यस्ते स्वेष्टशङ्कुहृते फलम् । छाया वैषुवती ज्ञेया विस्तरश्चापि कथ्यते ॥५५॥ ग्रीवासमां— भगणभागविभक्तवृत्तां कुर्यात् स्थलीं समतलां कृतदिग्विभागाम्' । तस्या जलेशदिशि मण्डलमध्यदृष्टि- विध्याद्रबिं" परिधिलग्नमनाकुलात्मा ॥५६॥ पूर्वरेखाग्रवेधस्य रविवेधस्य चान्तरम् । अर्काग्राचापनिर्माण परिधौ" भागलक्षिते ॥ ५७॥ अर्काग्रा ज्या भवेत्तस्य तन्नतिज्याविशेषजा" । लिप्ता " शङ्क्वग्रजीवाया दक्षिणे चोत्तरेऽन्यथा ॥५८॥ दक्षिणाभिमुखी छाया यदा भवति भास्वतः । नतिज्यारहितार्काग्रा शङ्क्वग्रं कथ्यते तदा ॥ ५९॥ विद्धि तेन विषुवत्प्रभां सतीं" पूर्ववच्च पललम्बकौ" पुनः । वेदितव्यविदितग्रहान्तरं नाडिकाभिरवगम्य तत्त्वतः ७ ॥६०॥ षड्गुणास्तु" घटिका लवास्तु तैः " १९ पूर्वपश्चिमदिशि स्थिते क्षयः” । १ मस्तके यच्छाया A, B; मस्तके यज्जाया C. ४ मुच्यतेऽर्काग्रा C. * सूर्याग्रं हृता B. हत: B. " वष्ठपति B. दिग्विभागाम् A, B; परितो A, B. १२ ५ फणिनि वह्नि C. °समासपद- ६ वलस्य A, B. इष्टजीवाहत: A; इष्टनीवा- २ ' दिवोसमां A, B; ग्रीवां समां C. स्थलीं समतुलां कृतदिग्विभागाम् C. °विशेषजाम् C. १३ लिप्त्वा C. १७ १६ ★ पूर्ववच्च चललम्बकी A, B; पूर्ववच्चावलम्बकौ C. A, B. २० १९ कै: A, B. क्षधी: C. १६ · • स्थलीनमत प्राकृत- १० विद्याद्रवि A, B, C. १४ केन C. १५ सती A, B. तद्गुणास्तु १८ तत्पर: A, B. २० [ महाभास्करीये उच्यते धनमपि क्रमेण त- ज्ज्ञातचारनिचयै: सदा' बुधैः ॥६१॥ एवं नक्षत्रताराणां ग्रस्ताराभिरेव च । साधितं क्षेत्र निर्माणं युक्त्या सर्वत्र सर्वदा ॥६२॥ अष्टौ भानि क्रिये, षट् स्युरेकोना' विशतिर्वृषे | द्वौ दिशो मिथुने, द्वौ च तिथयस्त्र्यष्टकं परे ॥ ६३॥ अष्टौ सार्धास्त्रिसप्तैने', वेदाः शैलोनकन्यकाः । पञ्चाद्रयेकं तुलाराशौ, द्वाविनोऽष्टादशा : परे ॥६४॥ एकोऽब्धीन्दुस्त्रिरन्ध्राणि' चापे, भूतैकपड्यमा' । मृगेऽद्रिर्द्वचूनकुम्भे॰ तु पञ्चेन्द्वन्त्येऽश्विभादितः" ॥६५॥ योगभागाः” क्रमेणैते बोद्धव्यास्तेऽश्विभादिषु" । उदगाशार्कभूतानि दक्षिणे पञ्चदिग्भवाः ॥ ॥६६॥ उदग्रसास्तथाकाशं दक्षिणे पर्वतोऽम्बरम्" । उदगर्काश्च ते " सैका दक्षिणे सप्त चाश्विनौ ॥ ६७ ॥ सप्तत्रिंशदुदग्भागा" याम्ये सार्धांशकास्त्रयः | अब्धयोऽष्ट त्रिभागाश्च" स्वरास्ते सत्रिभागकाः ||६८|| उदक् त्रिंशत् कृतिः षण्णां याम्ये लिप्तास्त्रिषट्ककाः । उदग्जिना द्विविश्वे" च विहाय : कीर्त्यतेऽपरम् ॥६९॥ विक्षेपांशाः क्रमेणोक्ता बोद्धव्यास्तेऽश्विभादिषु ” । योगभागसमाः सर्वे दृश्यन्ते योगजा ग्रहाः ॥७०॥ १४ क्रमेण ज्ञातचा निचयै: समा C. 'स्यु: एकोऽजा " सार्धा त्रिसप्तैने A, B; सार्धास्त्रिसप्तैन C. ● एकोऽपीन्दुस्त्रि- १० १९ योग- रन्ध्राणि A, C. चापभूतैकषष्ठ्युमा A, B. ' मृगेद्रिर्भूनिताः कुम्भे A, B; मृगे- द्रिद्यूनकुम्भ C. १ पञ्चेन्दिन्तेऽश्वि° A; पंचेन्द्वन्तेऽश्वि° B; ऽश्विनादय: C. भाग: B; यागभागा: C. १३ बोद्धव्याश्शशिभादिषु A; वोद्धव्याश्चाश्विनादिषु B. उदग्रसास्तथाकाश दक्षिणे

• दक्षिणे वाद्यनायनम् A; दक्षिणे वाद्यानयनं च दिग्भवाः B.

पर्वताम्बरम् A; उदग्रसास्तथाकाश दक्षिणे पर्वतांवरम् B; उदग्रास्तास्तथाकाशा दक्षिणे १७ सप्तत्रिंश पर्वताम्बरम् C. १९ उदगर्थाश्च ते A; उदगर्थाश्चेत B; उदगार्कश्वते C. १९ उदिग्जि १८ अद्रयब्दयोऽद्रिभागाश्च A; अद्रयब्धयोद्रिभागाश्च B. दुदग्भावा A, B. नाशिवलिप्ते A, B; उदग्जीना द्विविश्वे C. कीत्यते वरा A, B. २५ °श्विनादिषु C. २० १ धनमता A, B; ऽयनमपि C. A, B.

• तिथयस्त्वष्टकं A, B,

" शैलोनिताः परे A, B. ' द्वौ विनाष्टादशा A; द्वौ विनाष्टादशा: B. ९ १५ s:/ विक्षेपांशस्तयोः साध्यमन्तरं ग्रहतारयोः । उच्य क्षिप्तिलिप्ताभिर्हन्ती दुर्दक्षिणागतः ॥७१॥ रोहिणीशकटं षष्ट्या' त्वष्टिवर्गेण तारकाम् । नवत्या साध्या चित्रां' द्विशत्या शकतारकाम् ॥७२॥ शतेन चार्धयुक्तेन मैत्रं शतभिषग्जिनैः । नवत्या द्वयूनयैन्द्राग्नं पौष्णं विक्षिप्तिवजितम् ॥७३॥ उत्तरेण शते षष्टौ" बहुलाभेद" उच्यते । उत्तरां परमां गत्वा मघामध्यस्थतारकाम्" ।।७४।। यष्टियुक्तकलास्त्वेता ग्रहैर्नक्षत्रभेदने || ७५ (i) || १* इति महाभास्करीये तृतीयोऽध्यायः । • त्वष्टवर्गेण C. .* उच्यन्ते C. भषष्ट्या: A, भषष्ठ्याः B. " पित्रां C. " विशत्या शक्रतारकाम् A, B; द्विशत्या शात्रतारकाम् C. ४ २१ तारका: A, B. 'स्वार्धयुक्तेन A, ७ ९ B. ' शतभिषा जिनै: A, B ; शतभिषग्जनै: C. ' द्वयूनयैन्द्राग्रं A, B; धूनयैन्द्राग्रं C. १३ १० विक्षिप्तवर्जितः A, B; विक्षिप्तवर्जितम् C. " शतेनाष्टौ A, B. १२ बहुलं भेद A, B. त्यक्त्वा मध्यं विश्वस्य तारका A; मत्वा मध्यं विश्वस्य तारका B. १४ उत्तरां भेदने missing from C. चतुर्थोऽध्यायः कृत्वा देशान्तरं कर्म रवेरुच्चं विशोधयेत् । शेषं' सूर्यस्य यत्केन्द्रं तस्मिन् राशित्रयं पदम् ॥१॥ जीवाः क्रमोत्क्रमाभ्यां तु ग्राह्याः केन्द्रपदक्रमात् ।' जीवानां ग्रहणोपाय : " कथ्यते विस्तरेण सः ॥ २॥ लिप्तीकृत्य हरेन्मख्या जीवा लब्धास्ततः पुनः । वर्तमानाहतं शेषं मख्या चैव विभाजयेत् ॥ ३॥ पूर्वसङ्कलिते युक्ते ज्याः क्रमेणोत्क्रमेण वा । स्वपरिध्याहतेऽशीत्या लब्धं क्षयधनं फलम् ॥ ४॥ केन्द्रात्पदविभागेन क्षयधनधनक्षयाः । देशान्तरकृते सूर्ये कुर्यात्तन्मध्यमे सदा ॥५॥ केन्द्रे क्रियादिके” चाथ फलं बाहोविंशोधयेत् | तुलादिके तु तन्नित्यं देयं स्फुटदिदृक्षुणा" ॥६॥ क्रमोत्क्रमफलाभ्यस्ता मध्या" बाहुफलेन " वा । भुक्तिश्चक्रकलालब्धं पूर्ववत्तत्प्रकल्पयेत् ॥७॥ बाहुकोटी" क्रमात्केन्द्रे कोटिबाहू " गतागते । तयोर्गुणफले प्राग्वत् कर्णार्थे परिकीर्तिते ॥८॥ आद्ये पदे चतुर्थे च व्यासार्धे कोटिसाधनम् । क्षिप्यते शोध्यते चैव शेषयोः कोटिका भवेत् ॥ ९॥ तद्वाहुवर्गयोगस्य मूलं" कर्णः प्रकीर्तितः । बाहुकोटिफलाभ्यस्ते कर्णे व्यासार्धभाजिते ॥१०॥

• जीवाया
• शिष्ट: C.

२ तत्केन्द्रं C. ' जीवाः पदक्रमात् missing from C. 'ग्रहणोपायं C. " ज्याक्रमेणोत्क्रमेण C. " स्वपरिघ्याहताशीत्या भक्तोऽक्षयधनं १० केन्द्रात्- ५ A, B. C. “ क्षयं घनघनक्षयाः A; क्षय्यं धनधनक्षया: B. 'देशान्तरीकृते A C. १३ क्रियादिके A, B. १९ केन्द्रक्रियादिगे वा फलं वाहोविंशोध्यते । तुलादिगे च तन्नित्यं देयं स्फुटदिदृक्षुभिः ॥ C. १९ क्रमोत्क्रमफलाभ्यस्तां मध्यां A, B. बाहुः फलेन A. 'भुक्ति- १५ कोटिबाहु B, C. श्चककलावाप्ता A, B. १५ बाहुकोटि A, B, C. १७ प्राग्वत्कर्णार्धा १८ परिकीर्त्यते A, B. १९ तद्बाहुवर्गसंयोगमूलं A, B. C. चतुर्थोऽध्यायः ] भुजाकोटिफले स्यातां ताभ्यां' कर्णश्च पूर्ववत् । भूयः पूर्वफलाभ्यस्ते कर्णे त्रिज्याविभाजिते ॥११॥ एवं पुनः पुनः कुर्यात्कर्णः पूर्वोतकर्मणा । यावत्तुल्यो भवेत्कर्णः पूर्वोक्तविधिनाऽमुना' ॥१२॥ विष्कम्भार्धहता भुक्तिः सूर्याचन्द्रमसोः सदा । स्वाविशेषेण कर्णेन स्फुटभुक्तिरवाप्यते ||१३|| अन्त्यजीवाऽथवा भुक्त्या गुणिता धनुषा हृता । स्वपरिध्याहतेऽशीत्या लब्धे हीनाधिके स्फुटा' ||१४|| अन्त्यजीवाधनुःखण्डं केन्द्रभोगाद्विशोधयेत् । तद्विशोध्य मखेश्शेष पात्यतेऽविषमे' ततः ॥१५॥ उच्चभुक्तिविहीनाया भुक्तेः' शीतांशुमालिनः । उत्क्रमज्या क्रमे ग्राह्या क्रमज्या चोत्क्रमस्थिते " ॥१६॥ आद्यन्तयोः फलं युक्त्वा " गुणयोश्चानुपाततः । तत्फलेन” विहीनाढ्या भुक्तिः स्फुटतरा हि सा ॥१७॥ ह्यस्तनाद्यतनयोर्यो विश्लेषोऽतीतभोगकः” । श्वस्तनाद्यतनयोश्च भावी भोग : " प्रकीर्तितः ॥ १८॥ कोट्यां पदवशाद्धित्वा युक्त्वा वाऽन्त्यफलं पुनः । तद्वर्गबाहुवर्गस्य योगात्कर्णः पदं " भवेत् ॥ १९ ॥ कर्णेनान्त्यफलं हत्वा विष्कम्भार्धेन लभ्यते । - पूर्वकोट्यां धनर्णं " स्याद्यावत् कर्ण: " समो भवेत् ॥ २०॥ भुजज्याभिहता त्रिज्या " कर्णेनाप्तधनुः क्रमात् । केन्द्रात्पदविभागेन घनं स्वोच्चे प्रकल्पयेत् ॥ २१॥ १ स्यातामाभ्यां C.

• Missing from C.

७ द्वशोधयेत् A, B. ' तद्विशुद्धमधः शेषं A, B; तद्विशोध्यमखेश्शेष: C. A, B, C. ' भक्ते A, B. ९० चेत्क्रमे° A, B. ११ °विहीनाप्ता A, B. तीतभोगतः B. १३ २ ° कर्ण A, B. ३ यावत्तुल्या भवेत्संख्या कर्णस्य विधिनामुना C. "स्त्रपरिध्याहताशीत्या लब्धा हीनाधिका स्फुटा C. " केन्द्रभागा- ‘पात्यते विषमे 'युक्त्या C. १२ तत्फलेह° C; ह्यस्तनाद्यतनयोविशेषोऽतीतभोगत: A; ह्यस्तनाद्यतनयोपिशेषो- १४ भाविभोग: A, B. पूर्वकोट्या धनर्णं A, B; पूर्वकोट्यायनर्णं C. भिहता त्रिज्या B; भुजज्याहतां त्रिज्यां C. तद्वर्गबाहुवर्गस्य यः कर्णः १७ स्याद्यावान्कर्ण: A, B. १५ २३ १८ पदं C. भुजाज्या२४ ७ [ महाभास्करीये तदेव केवलं पूर्वं चक्रार्धं तेन वर्जितम् । चक्रार्धं तच्च चक्राच्च शुद्धं स स्फुटो रविः ||२२।। प्रतिमण्डलसंसिद्धिरेषा सम्यक् प्रकीर्तिता । स्वान्त्यं फलं च सर्वेषामुच्यते प्रतिमण्डलम् ||२३|| स्फुट स्वदेशमध्यार्कविश्लेषान्तरसंगुणा । भुक्तिश्चक्रकलावाप्तं' पूर्ववत्तद्” भुजान्तरम् ।।२४।। त्रयोदशहता ' जीवा त्रिस्फुटस्य विवस्वतः | द्वांत्रिंशता हृता क्रान्तिः परिशेषस्तु पूर्ववत् ||२५|| चरप्राणहता' भुक्तिरहोरात्र सुभाजिता । उदयास्तमयोः शुद्धिः क्षेपश्चोत्तरगे रवौ ॥२६॥ व्यत्ययो दक्षिणे भानावन्येषामनुपाततः' । फलं च तद्वशाद्विद्धि" क्षेपः शोधनमेव वा ॥ २७॥ उदक्चरार्धसंयुक्तः पादोऽहोरात्रसम्भवः” । दिनार्धं दक्षिणे हीनं " राज्यधं तद्विपर्ययात् ||२८|| सूर्यबाहुहता भुक्तिर्मंध्या चक्रकलाहता" । भास्करस्य वशात् क्षेपः शुद्धिर्वापि निशाकृतः ॥ २९॥ शेषं विवस्वता तुल्यं कर्म चन्द्रस्य कीर्तितम् । भास्वद्भुजाफलेनैव शेषाणां तु प्रकल्पयेत् ||३०|| स्फुटार्कोन: " शशी छेद्यो लिप्ताभिः खद्विभूधरैः" । तिथयस्तत्र लभ्यन्ते शेषं षष्ट्या" समाहतम् ॥३१॥ छिन्द्याद् भुक्तिविशेषेण घटीविघटिकासवः । तिथेश्शेषो" गतो वापि निर्दिष्टो भास्करोदयात् ॥३२॥

• स्थान्त्यं C.

६ ' संशुद्धं चक्रोच्च: A; संयुक्तं चक्रोच्च: B; चक्राच्च संशुद्ध: C. • चक्रकलाप्राप्ता A, B. पूर्व C " त्रयो शुहता B. चरप्राणाहता C. 'भुक्तिरहोमात्रासुभाजिता B; भुक्तिरहोरात्रन्तभाजिता C. ' उदयास्तगयो: A, B; १० फलं 'भानावन्येषामनुपात्ततः A, B ; भानामन्येषामनुपातत: C. उदयास्तमययो: C. भास्त्रद्वशाद्विद्धि A; फलं च तद्वशासिद्धि C. रार्धसंयुक्तः वादोहो° B. १२ होने C. १४ स्फुटॉर्थोन: A, B. B. १७ तिथि: शेषो A, B. " " उदक्परार्धसंयुक्तः पादा हो° A; उदक्प- भुक्तिर्मध्यचन्द्रकला ° A, B; °चक्रकलामता C. षष्ट्या समं न्यसेत् A; षष्ठ्या समभ्यसेत् १५.. “खभूधरै: C. १६ चतुर्थोऽध्यायः ] तिथ्यधंहारलब्धानि करणानि बवादितः । विरूपाणि सिते पक्षे सरूपाण्यसिते विदुः ॥ ३३॥ लिप्तीकृतो ग्रहश्छेद्यः शतंरष्टाभिराप्यते । ज्योतिषां निचयो यातो भुक्त्या शेषाद् घटीविदुः ||३४|| १ सूर्येन्दुयोगे चक्रार्धे व्यतीपातोऽथ वैधृतः । चक्रे च मैत्रपर्यन्ते' विज्ञेयः सार्पमस्तकः ॥३५॥ नानायने व्यतीपातस्तुल्यापक्रमयोस्तयोः । उद्देशस्तस्य चक्रार्धं विक्षेपात्त्वधिकोनकम् ॥ ३६॥ तिग्मांशुकेन्द्रवज्जीवाः' क्रमशश्चोत्क्रमादपि । भुजाकोट्यादिसिद्धिश्च विशेषोऽतोऽभिधास्यते ॥३७॥ स्ववृत्तान्तरगुणां ज्यां" पदयोरोजयुग्मयोः । क्रमोत्क्रमात् त्रिमौर्व्याप्त परिधौ परिकल्पयेत् ॥३८॥ क्षयोऽधिकातिने" परिधि: स्यात् स्फुटो मतः" । तेनाहतेष्टकेन्द्रज्यां” छित्वाऽशीत्या " फलं विदुः ॥ ३९ ॥ स्वमन्दकेन्द्रसंप्राप्तफलचापार्धमिष्यते । पदक्रमाद्यथा भानोः स्वमध्ये तद्विधीयते ॥४०॥ शीघ्र केन्द्रफलाभ्यस्तं विष्कम्भार्धं विभज्यते । स्वकर्णेनाप्तचापा" कार्यं तस्मिन्विपर्ययात् ॥४१॥ तस्मान्मन्दफलं कृत्स्नं कार्यमिष्टं स्वमध्यमे | एवं भौमार्किजीवानां विज्ञेयाः स्फुटमध्यमाः ॥४२॥

• शेषाद्घटीं

बहूदित: C. ' शरैरष्टाभिराप्यते A, B; शकैरष्टाभिराप्यते C. विदु: A; शेषात् घटिं विदुः B; शेषघटीविंदु: C. A, B. वैधृतम् C. " चक्रे मैत्रे च पर्यन्ते " सार्वमस्तक: A, B, C. १० १२ " नानात्मने व्यतीपाते सूर्योपक्रमयोस्तयोरुद्देशस्तत्स- म्पातार्धं C. — विक्षेपादधिकोन्नतम् A; विक्षेपादधिकोनतम् B. ' तिग्मांशुः केन्द्रवज्जीवा A, B; तिग्मांशुकेन्द्रवज्जीवां C. स्त्रवृत्ता स्त्रोत्तरगुणा A, B; स्ववृत्तान्तरगुणाज्यां C. " क्रमात्क्रमात्त्रिमौर्व्याप्तां A; क्रमात्क्रमात्त्रिमौर्वाप्तं B. क्षयेऽधिकद्युती होने A; क्षयोऽधिकद्युती होने B; क्षयोऽधिकायुतिहीने C. स्यात्स्फुटा मता A, B. तैर्ग्रहेन्द्रज्यां A, B. १५ जित्वाशीत्या C. १६ मन्दकेन्द्रस्थसंप्राप्तं फलं चा° A; मन्दकेन्द्रस्य प्राप्तफल B. १७ विभाजयेत् A, B. " स्वकर्णेनाप्तस्य चा° C. १९ कार्यन्तिविपर्ययात् C. २० तस्मिन्मन्दफलं A, B. २९ भौमार्कजीवानां A. B. १४ तेनाह - २५ १३ २६ १ तद्विहीनचलोत्पन्नफलचापेन' संस्कृतः । स्फुटमध्यः स्फुटो ज्ञेयः शेषयोरुच्यते विधिः ॥४३॥ शीघ्रन्यायाप्तचापार्धयुक्तहीनो' विपर्ययात् । मन्दोच्च: स्फुटमध्यस्य कर्ता शीघ्रात् स्फुटं विदुः ॥४४॥ प्रतिमण्डलकर्मापि योज्यमत्र विपश्चिता । मन्दोच्चे पूर्ववत्कुर्याच्छीघ्रोच्चात्तद्विशोध्यते ॥४५॥ तदेव केवलं शोध्यं चक्रार्धाच्छोध्य तच्चलात् । चक्रार्धसंयुतञ्चापं चक्राच्छुद्धं च शेषयेत् ॥४६॥ स्फुटवृत्तगुणां त्रिज्यां हृत्वाऽशीत्या स्वकोटितः । त्यक्त्वा पदेषु युक्त्वा वा कर्णः प्राग्वत् प्रसाध्यते ॥४७॥ मन्दोच्चसिद्धतन्मध्यविश्लेषार्धसमन्वितः” । मन्दसिद्धेऽधिके हीने रहितो मध्यमो ग्रहः ॥४८॥ स शीघ्रोच्चात्पुनः साध्यः सिद्धयोरन्तरालजम्" । अर्धीकृत्य सकृत्सिद्धे पूर्ववत् परिकल्पयेत् ||४९।। एवं कृतस्य भूयोऽपि मन्दसिद्धिं समाचरेत् । मन्दसिद्धस्य तस्यायं विशेषोऽतोऽभिधास्यते ॥५०॥ द्विसिद्धमन्दसिद्धस्य " द्विसिद्धस्य यदन्तरम् | प्राग्वत्तन्मध्यमे कृत्वा शीघ्रसिद्धः " स्फुटो ग्रहः ।। ५१ ।। एवमाराकिजीवानामाख्यातं " प्रतिमण्डलम् । शेषयोरप्ययं सम्यगुच्यते यो विधिक्रमः " ॥५२॥ शीघ्रोच्चसिद्धतन्मध्यविश्लेषार्धसमन्वितम् । १३ १७ मध्यान्न्यूनेऽधि के हीनं मन्दोच्चं संस्कृतं विदुः ॥५३॥९ • शीघ्रस्या- स्फुटमध्यस्फुटो A, B, C. ५ वाप्तचापाधं युक्तहीनो A, B. " "शीघ्रोच्चं तद्विशोधयेत् A, B. • चक्रार्धसंयु- भङ्क्त्वा- मन्दोच्चस्फुटमध्यस्प A, B, C. ' शीघ्रस्फुटं पुन: C. ७ तदेव केवलं योज्यं पादयोः पूर्ववच्चलात् A; तदेव केवलं योज्यं पदयोः पूर्ववच्चलात् B; तदेव केवलं शोघ्यं पदयोः पूर्वयोश्चलात् C. तश्चापं चक्राच्छोध्यश्च शेषयो: A; चक्रार्धंसंयुतश्चापं चक्राच्छोद्यश्च शेषयो: B. शीत्या स्त्रकोटिता C. १० 'कृत्वा पदेषु युक्ता वा कर्णं यावत् प्रसाध्यते C. C. १२ सिद्धयोरन्तरा भुजम् A, B. १३ विश्लेषोऽतो विधास्प A, B. मन्दस्य A, B. १५ शीघ्रसिद्ध C. " एवमारार्कजी° C. १७ विधिः क्रम: B. A. १९ मध्यादूने धिके हीनं मन्दमेवं तु कर्मणा B, ११ °समन्वितम् [ महाभास्करीये तहीनफलोत्पन्नफलंचापेन A, B. ४ १४ ततो द्विसिद्ध- १८ संस्कृते चतुर्थोऽध्यायः ] ४ बुधभृग्वोः पुनस्साध्यं मान्दमेवं स्वकर्मणा' ।' तेन सिद्धौ चलाद् भूयः स्फुटावेतौ प्रकीर्तितौ ॥५४॥ कोटेरन्त्यफलं शोध्यं न शुध्येद् व्यत्ययस्तदा । कार्यः कर्णोऽसकृन्मान्द: सकृत्कर्णस्तु शीघ्रजः ॥ ५५॥ स्फुटमध्यमान्तरदलं मध्यवशादृणं धनं चले कृत्वा । वक्रातिवक्रगमने विज्ञेये तन्निवृत्तिश्च ||५६ || शीघ्रोच्चात् स्फुटखेचरो निपतितः शेषो यदा राशय- श्चत्वारो यदि वक्रगत्यभिमुखः षट् चातिव स्थितः । अष्टौ चेत्कुटिलं जहाति विहगः पन्थानमाश्वेव स त्वैष्यातीतविचारिणोविवरकं" भुक्तिर्भवेदाह्निकी" ॥५७।। मन्दान्त्यजीवागुणितां स्वभुक्तिं भूयः स्ववृत्तेन हतां विभज्य | राशेः कलाभिर्दशताडिताभि- र्भुक्तौ धनर्णं पदयुक्तितोऽर्धम् ॥५८॥ शीघ्रोच्चभुक्तेस्तदपास्य" शेषात् केन्द्रान्त्यजीवा विधिना त्रिज्याहतं कर्णविभक्तभेदं यदाप्तम् । न्यायेन" शीघ्रस्य धनर्णमिष्टम् ॥ ५९॥ तन्मन्दमौर्वीफलचापयुक्तं १८ सर्व स्वभुक्तौ " धनशोधने तु । तस्याविनष्टस्य " चलोच्चभुक्ते- जवाफलं "तुक° A, B. 'बुधभग्वोः शुध्येद्वयत्ययात्तदा साध्य: A, B. णधनं C. ¨व तातिवक्रगमने C. षट्भस्तु वक्रे B; षट्चातिवक्रो C. नेष्यातीत विचारिणोविवरकं C. ३ स्फुटापेतौ A, B. सर्वमनष्टराशौ ॥६० ॥ स्वकर्मणा missing from B. ५° मन्दः सकृत्कर्ण: स्वशीघ्रज: A, B. मध्यवंशादृ- • निपतिताच्छेषं A, B. पड्भिस्तु वक्रे A; १९ त्वेष्यातीतविचारिणोविपरकं A, B; ९ १० सन् C. भुक्तिर्भवेदाहकी C. मन्दान्त्यजीवागुणिका C. १" कर्णविभक्तभेदं न्यायेन २७ १२ १३ १४ ' वद युक्तितोऽर्धम् A,B. १५ शीघ्रोच्चभक्तेस्तदपास्य A, B. १८ A, B; कर्ण विभक्तभेद न्यायेन C. मुक्ता A; पमुक्त्वा B. ' सर्वस्स्वभुक्तौ A; सर्वस्वभुक्तौ B. १९ तस्याविनष्टं स्व C. २० चलोच्चभक्ते जीवाफलं A, B. २९ सर्व- मनिष्ट राशौ A, B. २८ एवं ग्राणां स्फुटभुक्तिरिष्टा तस्मान्न शुध्येद्यदि शीघ्रलब्धम् । तयोविशेषः स्फुटभुक्तिसंख्या वक्री ग्रहोऽसाविति' सद्भिरुक्तः ||६१|| एवं सुरेड्याकिंधरासुतानां भुक्तिः स्फुटातो भृगुसोमसून्वोः । मन्दोच्चकेन्द्रादपि संस्कृताद्धि शी तेनैव सर्वेण युतो विहीनो भोगः स्फुटोऽयं कथितो विशेषः । इत्थं" ग्रहाणां व्यवहारिकी स्याद् भुक्तिः स्फुटासन्नतरा च नित्यम्" ||६३|| गन्तव्ययाततिथिशेषहते रवीन्द्वो- र्भुक्ती क्रमेण" दिनभुक्तिविशेषभक्ते" । लब्धेन " युक्तरहितौ शशितिग्मरश्मी ज्ञेयौसमौ सकललोकविधानहेतु " || ६४ || त्यजीवाविधिना' यदाप्तम् ||६२|| इति महाभास्करीये चतुर्थोऽध्यायः । [ महाभास्करीये २ ३ वक्र ग्रहोऽसाविति A, B. * सुरेड्याकं 'भुक्तिस्फुटातो C. ५ मृगसोमसून्वोः C. युतोऽथ हीनो C. ८ " शीघ्राद्यजी A, B. ७ च लब्धम् A, B. युते विहीनौ A, B; १० १२ ' भोगः स्फुटो यः A, B; भोगस्फुटोऽयं C. इयं A, B; उक्तं C. सन्नतरा च नित्या A, B; भुक्तिः स्फुटासन्नतरा च नित्यम् C. भुक्तिक्रमेण C. १३ दिनशो भविशेषभक्ते A, B. १४ भक्तेन B. ' एवं स्फुटा स्याद् ग्रहभुक्तिरिष्टा C. घरासुतानां A; सुरेघ्वार्कधरासुतानां B. ४ " भुक्तिः स्फुटा- भुक्ति क्रमेण A, B; °हेतु: A, B. १५ पञ्चमोऽध्यायः भास्वतो ग्रहणं वाच्यमाचार्यार्यभटोदितम् । तस्य चादौ विजानीयादुपायानां विनिश्चयम् ॥ १॥ पञ्चाष्टभूतरन्ध्रेषुवेदाः कर्णो विवस्वतः ।' सप्तपर्वतरामाब्धिगुणसंख्या निशाकृतः ||२| कलाकर्णहतावेतौ विष्कम्भार्धविभाजितौ । स्फुटयोजनकर्णौ तौ सूर्याचन्द्रमसोविंदुः' ।।३।। दिग्वेदसागरा भानोरिन्दोस्तिथिहुताशनाः | योजनैरुच्यते सद्भिर्भूव्यासः खेषुपङ्क्तयः' ।।४।। विष्कम्भार्धहतौ व्यासौ स्फुटयोजनभाजितौ । भवतस्तौ कलाव्यासावुष्णशीतलतेजसोः ॥५॥ नवांशाः पञ्चभोगस्य भूतवर्गांश एव च । स्वतुरीयविलिप्ताभिर्युतहीने तनू स्फुट ॥६॥ सूर्याचन्द्रमसोर्विद्धि राहुबिम्बं च कथ्यते । पङ्क्त्यंशश्चन्द्रभोगस्य षोडशांशो विलिप्तिकाः ।।७।। अथातो मध्यलग्नस्य विधानं सम्प्रवक्ष्यते । निरक्षासुभिरुत्पत्तिर्बोद्धव्या शास्त्रकोविदैः ॥८॥ पूर्वाह्ये सूर्ययुक्तस्य गतभागासवो हि ये । गन्तव्यानां चापरा भागानामसवः स्मृताः ॥ ९॥ मध्यतिथ्यन्तरासुभ्यः शोध्या भागादि' भास्करे । शोध्या देयाश्च भूयोऽपि निरक्षासुवशाद्रवौ ॥१०॥ १ 'पञ्चाष्टरन्ध्रभूतेषुवेदाः कर्णो विवस्वतः A, B; पञ्चाष्टभूतरन्ध्राणि भूतवेदा विव- स्वतः C. २ ° सोस्सदा C. सदभिर्भुवासः खेषु पङ्क्तिभिः A, B.

• कलाख्यासा° A, B;

५ ° शीतगुतेजसोः A, B, C.

• षोटशांशोनलिप्तिका: A, B.

८ युक्तहीने तनुः स्फुटे A, B; युतहीने तनूस्फुटम् C. ' गन्तव्याणां चापराह्ने B; गन्तव्यानामपरा C. मध्यं तिथ्यन्तराशाभ्य: A; मध्यं तिथ्यन्तराभ्य: B. ` भागाश्च A, B; भागाहि C. fr ७ १० राशय: कालतत्त्वज्ञैरनुपाताप्तमेव च । मध्यलग्नमिदं स्पष्टं' श्रीमद्भटमुखोदितम् ।।११।। कक्ष्याभेदाच्छशीभान्वोर्जीवाभेदः प्रकीर्तितः । ज्ञापकं च स्वदृक्षेप इत्यादिवचनं प्रभोः ।।१२।। बाहुज्योदयलग्नस्य परक्रान्तिहता' हृता । लम्बकेनोदयज्याप्ता कक्ष्यायां भास्वतः स्फुट | ॥१३॥ सैंहिकेयविहीनस्य जीवा लग्नस्य ताडिता | तिथिभिश्चन्द्ररन्धैकैविक्षेपज्या विलग्नजा ॥ १४॥ विलग्नक्रान्तिविक्षेपधनुषोस्तुल्यदिक्कयोः । युक्तिवियुक्तिरन्यत्वे शेषकाष्ठगुणाहतम् ||१५|| उदयज्येन्दुकक्ष्यायां व्यासार्धं लम्बकोद्धृतम्" । राहूनमध्यलग्नाच्च विक्षेपज्या प्रसाध्यते ॥ १६॥ मध्यक्रान्तिविषुवज्ज्याकाष्ठयोरेकदिक्कयोः” । योगो वियोगो वाऽन्यत्वे" मध्यज्या शेषदिग्वशात् ” ||१७|| रवेरिन्दोः पलक्रान्तिविक्षेपधनुषां" वशात् । योगविश्लेषजा जीवा मध्यज्या शेषदिग्भवा ।।१८।। स्वमध्यज्योदयाभ्यासं विष्कम्भार्धाप्तवगितम् । ७ मध्यज्यावर्गतोऽपास्य" स्वदृक्क्षेप " पदं विदुः ।।१९।। गतगन्तव्यनाडीभिर्दृग्ज्या पूर्वोक्तकर्मणा । १४ साध्या रवेश्शशाङ्कस्य” विधानमिदमुच्यते ||२०|| १४ १ मध्यलग्ना A, B. कक्याभेदाच्छश्चीभावो जीवाभेद: A; कक्ष्याभेदाच्च- चीभावो जीवाभेदः B; कक्ष्याभेदाच्छशिभान्वोर्जीवाभेद: C. प्रकीर्त्यते C. ज्ञापनं च स्वदॄक्षेपौ A, B. ...त्यादिवचनं B. परत्रास्ति हता C. भास्वते स्फुट: A. B; ११ १२ भास्वतः स्फुटम् C. ' पिलग्नजा C. विलग्नकान्तिपक्षेवध B. " शेषकाष्ठगुणाज्ज्यागता A, B. लम्बकाहृतम् C. "ज्ज्याताष्टयोरेक दिक्कयो: A, B; काष्ठयोरेकद्रिक्रयोः शेर्पादिशात् B. १५ पलाक्रान्तिविक्षेपधनुषां C. १३ नानात्वे C. १६ This C. verse is missing from A, B. 'स्वमध्योदययाभ्यासाद्विष्कम्भार्वाप्तवगितात् A, B; स्वमध्यज्योदयाभ्यासविष्कम्भार्धाप्तवर्जितम् C. "° तोऽवास्य C. १९ A and B read दॄक्षेप in place of दृक्क्षेप. गतगन्तव्यनाडीभिर्भज्यात् A, B. २२ भवेच्छ- शाङ्कस्य A, B. २ २० [ महाभास्करीये १७ ३ ४ पञ्चमोऽध्यायः ] समलिप्तेन्दुविक्षेपक्रान्तिज्याचापसंयुतिः । वियुक्ति विदिशोर्जीवा शेषस्येन्दोरपक्रमः ॥२१॥ तेनाहोरात्रविष्कम्भक्षितिजीवाचरासवः । प्रसाध्यास्तैश्च दृग्जीवा गतगन्तव्यकालतः ॥२२॥ स्वदृग्दृग्क्षेपगुणयोर्वर्गविश्लेषजे पदे ।' दृग्गतिज्ये भवेतां ते भास्करामृततेजसोः ॥२३॥ स्वदृग्गतिक्षमाव्यासभेदसंवर्गसंभवम्' । पृथग्योजनकर्णाप्तं लिप्ता लम्बनं विदुः ||२४|| तद्विशेष हतः षष्ट्या स्फुटभुक्त्यन्तरोद्धृतः । घटिकादिस्तिथेः' प्रा शुद्धिः क्षेपोऽपरे मतः ||२५|| दिनार्धकालनिष्पन्नं लम्बनं शोध्यते तिथेः । उदगिन्दूदयज्यायां दीयते तत्र दक्षिणे ||२६|| एवं पुनः पुनः कर्म यावत्तदविशिष्यते ।' तिथिवच्चन्द्रतीक्ष्णांशू सञ्चार्यावेव पण्डितैः ||२७|| दृक्क्षेपज्ये त्वविशिष्टे" भूव्यासार्धहते हृते । स्फुटयोजनकरर्णाभ्यामवाप्ता लिप्तिकादयः ||२८|| अभिन्नाशयोविश्लेषस्तासां" सूर्यनिशाकृतोः । मध्यज्ययोरथान्यत्वे योगोऽवनतिलिप्तिकाः" ॥२९॥ दिक् तत्र शशिनो ग्राहया पातहीनान्निशाकृतः ज्यां खसप्ताश्विभिः क्षुण्णां कलाकर्णेन संहरेत् ||३०॥ विक्षेपः शशिनः स्पष्टस्तेन युक्ता नतिः स्फुटा । नानाशयोस्तु विश्लेष: शेषा सा नतिरुच्यते ॥३१॥४ लम्बनान्तरसंयुक्तग्रासमध्याप्तजन्मनोः" । १३ दृक्क्षेपक्षेपयोः सिद्धा मध्या चावनतिः " स्फुटा |॥ ३२ ॥ 'दृक्क्षेपगुणवर्गस्य विश्लेषपदमाहृतम् C. २ ° संवर्गसञ्चय: C. ३ लिप्तायु A, B; ६ घटिकादितिथेः लिप्ताद्या C. * तद्विश्लेषो A, B. A, B; घटिकादिस्थित: C. स्फुटभुक्त्यन्तराहृतः A, B. १७ के वो परे C. This verse is missing from A and B. • This hemistich does not occur in A and B. In C the initial word एवं is missing. “ल्पविश्लिष्टे A, B. " अभिन्नाशयो विशेषस्तासां C. १२ योगोपनतिलिप्तिका: A, B; योगावनतिलिप्तिका C. १४ तातातयोस्तु विश्लेषशेपास्मा गतिरुच्यते A, द्यवनति: C. १३ पातहीनां निशाकृत: C. १५° संयुक्तं ग्रहम° A, B. मध्या- १६ ३२ सूर्येन्दुविम्बसम्पर्कदलेन सदृशी नतिः । ग्रहणं भास्वतो न स्याद्धीनायामस्ति सम्भवः ॥ ३३॥ सम्पर्कार्धनतिवर्गविशेषपदमाहतम् ।' षष्ट्या गत्यन्तरेणाप्ताः' स्थित्यर्धघटिकाः स्मृताः' ॥३४॥ अविशेषतिथिस्ताभिविहीना सहिता सदा । ग्रासादिमोक्षकालौ स्तः' ताभ्यां जीवाविधिस्तदा ॥३५॥ ग्रासमध्यविनिष्पन्नलम्बनान्तर नाडिकाः । स्थित्यर्धे प्रक्षिपेन्नित्यं [ महाभास्करीये तिग्मदधिः || ३६॥ मध्यान्तलम्बनं स्थित्या' वर्धते मोक्षसम्भवः । स्थित्यर्धमित्यतिस्पष्टमुत्क्षिप्य भुजमुच्यते ||३७|| स्पर्श मोक्षौ यदान्यस्मिन्कपाले ग्रहमध्यतः । स्पर्शजं लम्बनं सर्वं देयं स्थित्यर्धनाडिषु ॥ ३८॥ मोक्षेऽप्येवं तदायातं " स्थित्यर्धे दीयते सदा । दिनार्धे ग्रासमध्ये च कल्प्यतेऽयं विधिक्रमः" ॥३९॥ ग्राह्यग्राहकबिम्बार्धविश्लेषक्षेपवर्गयोः । विश्लेषस्य पदं ज्ञेयं विमर्धस्य लिप्तिकाः ॥४०॥ तीक्ष्णांशुबिम्बवस्वंशलिप्तिकाकालसंयुतः । स्पर्शंकालो भवेत्सत्यो” भासुरत्वाद्विवस्वतः ॥४१॥ मध्यतिथ्यन्तरासूनामुत्क्रमज्याक्षसङ्गुणा । विष्कम्भार्धेन भक्तव्या लब्धकाष्ठस्य दिग्विधिः ॥४२॥ व्यासार्धादधिकासूनां " क्रमज्यां" त्रिज्यया युताम् । कृत्वैवमेव दिक्कल्प्या मध्ये प्राग्ग्रासवदिशा " ॥४३॥ उदग्दक्षिणतः प्राह्णे बिम्बप्राक्पश्चिमार्धयोः" । • नभसः " पश्चिमे व्यस्तमक्षस्य वलनं सदा ॥४४॥ विष्कम्भार्धनतीवर्गविश्लेषपदमागतम् A, B. २ षष्ट्याहत्यास्तरेणाप्ता A; षष्ठ्या ३

• विशेषस्थितिभिस्ताभिविहीना C.

तदा A, हत्या स्तरेणाप्त B. स्फुटा A, B. B. ६ ७ तौ C. प्रक्षिपेन्नित्यग्रहणे C. " मध्यान्तलम्बनस्थित्या A, B. देयं सर्वस्थित्यर्धनाडिषु A, B. १० सदायातं A, B. १२ १४ हर्त - १७ ग्रहमध्ये च कल्प्यतेऽयं विधिः क्रमात् A, B. " विक्षेपस्य पदाद्देयं विमर्धिस्य नाडिका: A, B. १३ भवेत्तस्य A, B. व्या A, B. १५ व्यासार्धाच्चाधिकासूनां C. १६ क्रमज्या A, B. B; प्रागासवद्दिशा C. १८ दिग्दक्षिणतः प्रा बिम्बंप्रा° A, B. प्रागासवद्दिशो: A, १९ नभसं C. " स्पर्शनं लम्बनं पञ्चमोऽध्यायः] : त्रिराशिसहितेन्द्वर्कविपरीतगुणापम:' । ॥४८॥ अयनाद्दिक्तु' पूर्वार्धे बिम्बस्यैवापरेऽन्यथा ॥४५॥ नानाशयोस्तयोविद्धि विश्लेष: धनुषोः सदा । संयोगोऽन्यत्र तज्जीवा सम्पर्काहता हृता ॥४६॥ विष्कम्भार्धेन यल्लब्धमाशासाम्ये युतं नौ । विश्लेषो व्यत्यये कार्यो वलनं तत्र शिष्यते ॥४७॥ मुखविन्यस्त सुश्लक्ष्णवतिकाङ्कुरशोभिना । अङ्किताङ्गुलतद्भागसमस्निग्धोरुमूर्तिना— ग्राह्यबिम्बार्धमानेन' कर्कटेनालिखेत्क्षितौ । ग्राह्यग्राहकबिम्बार्धसमवक्त्रेण चापरम् ।।४९।। पूर्वापरे" ततः कुर्यान्मीनेनोत्तरदक्षिणे । दक्षिणोत्तरतः केन्द्रान्नीत्वा वलनमप्यतः ॥ ॥ ५० ॥ उत्पाद्य तत्र मत्स्याङ्ङ्क" युक्त्या सूत्रं नयेद् बुधः । बाह्यमण्डलतत्सूत्रसम्पाते बिन्दुनिश्चयः ॥५१॥ बिन्दुतः केन्द्रसम्प्रापि" सूत्रं तस्मात्" प्रसार्यते । तस्यग्राह्यपरिधेश्च सम्पातो" यत्र लक्ष्यते ॥५२॥ तत्र तीक्ष्णांशुबिम्बस्य प्रदेशौ " स्पर्शमोक्षयोः । मौरिकार्धाङ्गुला" ज्ञेया यथा वा लक्ष्यते दिवि ॥ ५३॥ नत्यसंयुक्तविश्लिष्टं चलनं " ग्रहमध्यंजम् । तन्नत्योस्तुल्यकाष्ठत्वे वलनं पूर्वतो नयेत् ॥५४॥ १७ ९ °वमः A, C. 8 ५ धनुषोः सदा C. सन्वर्कार्धिहता A, B. ° साम्य- युते (C. 47-48(i) are missing from A and B. अन्तिकाङ्गुलतद्भाग° A, १० B; धोऽथ मूर्तिना C. पूर्वावरे B; पूर्वापर C. B. उत्पाद्य मत्स्याङ्क C. ११ बलमानतः C. १२ १३ • विन्दुतः केन्द्रसम्प्राप्ति A, B. " सूत्रं तस्मात्प्रसाध्यते A, B; तस्मात् सूत्रं प्रसार्यते C. १५ तस्य ग्राह्यपरिधेश्च सन्तापो A; तच्च ग्राह्यपरिधेश्च सम्पातो C. १" प्रदेश: A, B. २ २ लग्नाद्या दिक्तु A, B. विश्लेषधनुषोस्तयो: A, B; विश्लेष- ७. व्यत्यय: C. • Verses ३३ ' ग्राह्याङ्गुलार्धमानेन A, B. उत्पाटय तत्र मत्स्यान्तं A, ८ १८ मौर्वी कार्धाङ्गुला A; मौर्वीकार्द्धाङ्गुला B. नत्या संयुक्तविश्लिष्टं वलनं A, B; नत्यसंयुक्तविश्लिष्टवननं C. ३४ [ महाभास्करीये भिन्नाशयोस्तयोः केन्द्रात् पश्चिमेन प्रसार्यते । तन्नत्यनुदिशं सूत्रं ततो मत्स्येन नीयते ॥५५॥ बाह्यमण्डलतः केन्द्रमानयेत्तद्विचक्षणः' । केन्द्रात्तदनुसारेण नतिसूत्रं प्रसार्यते ।।५६।। मध्यबिन्दुस्तस्यद्वन्दू तौ' स्पर्शमोक्षयोः । दक्षिणस्यां नतौ भागे याम्ये सौम्ये विपर्ययः ।। ५७।। विधिग्रहणमध्यस्य भानोरिन्दोविपर्ययात्' । आलिखेत्तद्ग्रहे व्यक्तं ग्रासमध्यान्तसम्भवम् ॥ ५८ ॥ मध्यबिन्दुशिरोन्यस्तग्राहकार्धवपुर्धृता । ì खण्डयेत् कर्कटेनाशु निर्दिष्टस्पष्टमानतः ॥ ५९॥ ग्राह्यस्य खण्डितं यावच्छेद्यके लिखितं च यत् ग्रहमध्ये तथा सर्वं विस्पष्टमुपलक्ष्यते ||६०|| न्यस्तबिन्दुत्र्यप्रापि" मीनाभ्यां वृत्तमालिखेत् । ग्राहकस्य भवेत्पन्थाः तत्रेष्टग्रासकल्पना ||६१॥ इष्टकाल विहीनेन" स्थित्यर्धेन हृतं" हरेत् । सूर्येन्द्वोर्भुक्तिविश्लेषं षष्ट्या लब्धस्य " वर्गितम् ||६२।। प्रक्षिप्यावन तेर्वर्गे यन्मूलं रविसोमयोः । इष्टग्रासशलाका स्यात् तच्छेषो" ग्रास इष्टजः ॥६३|| सुश्लक्ष्णा वैणवी श्लाका" केन्द्रात्तिर्यक् प्रसार्यते । तस्याग्रेण तथा पन्थाः प्राप्यते" ग्राहकस्य य: " ॥६४॥ तत्र * ग्राहकमानेन खंडयेद् ग्राह्यबिम्बकम् | तावदेव तथाग्रस्तं" दृश्यते ग्राह्यमण्डलम् ||६५|| 1 १ केन्द्रमानीयेत्तद्विचक्षण: A. २ मध्यबिन्दु ° C. स्याबिन्दुस्तत् A, B. यात् A, B. " भानोरिन्दोश्च पर्ययात् A, B. ६ सम्भव: C. • ° पुर्युता C. ९ १२ मानता A, B. ग्राहास्य खण्डिता यावच्छेद्यतेऽथ यत् A; ग्राह्यस्य यावच्छेद्यतेऽथ यत् B. १० सर्वा A, B. युक्तेन A, B. " हतां A, B; नतं C. " वैर्णाविशत्कात् A, B; १४ १५ स्याच्छेषो B; "न्यस्तं बिन्दुत्र्यप्राप्ति A, B. लम्बस्य C. वैणवी शलाका C. यत्र A, B. C. २० A, B. १९ च A, B. मण्डले A, B. १७ यथाग्रेण C . यथावक्रं C. ६१ ● विपयं- ८ °स्फुट खण्डिता 2 इष्टहीनेन न्धात्तच्छेषं १८ प्रोच्यते ग्राह्य- २२ पञ्चमोऽध्यायः ] विमर्दार्धकलाहीनं यत् स्थित्यर्थं कलामितम्' । तत्कालच्छेद्यकस्तेन भास्वद्विम्बं विखण्डयेत् ||६६||' प्रदेशस्तस्य' बिम्बस्य कर्कटेनावगाह्यते । पश्चार्धे गृह्यते व्यक्तं पूर्वे चासौ प्रमुच्यते ||६७|| एवमाशामुखादर्शसकलोरुकलाभृतः । कान्तावदनसंवृत्तवपुषः' शशलक्ष्मणः ||६८|| अपि कार्यो विधिज्र्ज्यानां विशेषो यः स कथ्यते । भूच्छायाया गुणाः' कल्प्या रवेः कक्ष्यासमुद्भवाः ॥६९॥ . भागहार: शशाङ्कस्य करर्ण एव प्रकीर्तितः । शशिवल्लम्बनाल्लब्धं व्यत्ययात् क्षेपशोधने ॥७०॥ ताडितो योजनः कर्णो धात्रीव्यासेन भास्वतः । तयोर्व्यासविशेषेण भूच्छायादैर्घ्यमाप्यते ।।७१॥ पञ्चाहतो रवेः कर्णः षोडशापहृतः फलम् । भूच्छायादैर्घ्यमाख्यातमिन्दुकर्णस्ततः क्षयः ॥७२॥ भूव्यासगुणिते शेषे छायादैर्घ्यहृते फलम् । विष्कम्भार्धहतं भक्तं चन्द्रकर्णेन तत्तमः " ॥७३॥" अन्ये वदन्ति " शशिनो ग्रहणोपदेशं हीनं गुणैर्दशभिरल्पफलान्तरत्वात् । स्थित्यर्धकालमचलं विदधीत तस्मिन् ३५ आद्यन्तयोर्ग्रहणमध्यसमुत्थितं यत् ॥७४ || क्षुण्णा" स्थित्यर्धकालेन भुक्तिः षष्ट्या" समाहृता । समलिप्ते क्षय: स्पर्शे" मोक्षे क्षेपो निगद्यते ॥७५॥ विक्षेपस्तस्य तस्माच्च स्थित्यर्धं च प्रसाध्यते । एवं कर्माविशेषोऽयं विमर्धिस्य वा पुनः ॥७६॥ कलाभृत: A, C; कलाभित: B. २ तत्कालच्छेद्यकं तेन भास्वत्खण्डादि खण्डयेत् A, ६ भूच्छायायां गुणाः " अविकार्यो A, B. प्रदेशसूत्रं A, B. * °संवृत्त: वपुषः B. " धात्रिव्यासेन B. ' छायादीर्घत्वमाप्यते C. 'छाया रवे: C. १० षड्व- second half of this verse १३ भवन्ति A, B. "क्षयस्पर्शे A, B, C. १७ तस्मिंश्च C. "तत्तमः B. २ A, B; भू शापहृत: C. तत्तत: A, B. " In C the reads as follows विष्कम्भार्धहृतं चन्द्रकर्णेन " °रन्यफ° A, B. १४ क्षुण्ण: A. १५. षष्ठ्या B. कर्माविशेषोऽयं A, B. १८ us १ विक्षेपः शशिनः कल्प्यश्छेद्यकाले सतां वरैः । उत्तरो दक्षिणे नित्यं दक्षिणश्चोत्तरे तथा ॥७७॥ प्रोक्तमेतदवधूय मत्सरं ' सूर्यचन्द्रतमसां क्रमागतम् । कर्म येन विदुषा समर्जित ॐ दैवविद्भवति सर्वतन्त्रवित् ॥७८ ॥ इति महाभास्करीये पञ्चमोऽध्यायः । मत्सर: B; यत्परं C. महाभारकरीये २ समर्पितं A, B, C. षष्ठोऽध्यायः स्वेष्टदेशपलजीवया हतां क्षिप्तिमिष्टशशिजां समाहरेत् । लम्बकेन यदवाप्तमुत्तरे शोधयेदुदयगे निशाकरे ॥ १ ॥ अस्तगे धनमुशन्ति तद्विदो दक्षिणे विधिरयं' विपर्ययात् । वर्जितत्रिभवस्य शीतगो- रुत्क्रमापम विसंहति हरेत् ॥ २ ॥ व्यासवर्ग निचयेन शोधये-- च्चन्द्रतोऽयनविमण्डलाशयो: तुल्ययोर्धनमुशन्ति तद्विदो व्यत्यये शशिनि तत्फलं सदा ॥ ३ ॥ दृश्यचन्द्र इति कथ्यते बुधै- १० रेवमाकलितचारसञ्चयः । भास्करेन्दुविवरांशकोद्भव- प्राणराशिघटिकाद्वये" शशी ॥ ४ ॥ भास्करेऽस्तगिरिमूर्घगेऽम्बरे । चन्द्रबिम्बममिहत्व भाजयेत् ॥ ५ ॥ दृश्यतेऽमलनिरभ्रतारके चन्द्रभानुविवरोकमज्यया षण्णगाष्टरससंख्यया" सितं नित्यमेव गणकाः प्रजानते । २ १ " शिप्तिमिष्टशशिजां C. हेरेपुन: A, B. ता रवे: A, B. *विपर्ययः A,B. ६ रुत्क्रमावमविसंहति A; रुत्क्रमाम संहति C. १• °मालिखित ' A, B. १९ भास्क - १२ षण्णवाष्ट° A, B. १* प्रजायते " वर्जितं त्रिभुवनस्य A, B, C. " च्चन्द्र मोऽय° C. ८ व्यत्ययेन B. ९ तत्पलं A. रेन्दुविवरांशकोद्भवः प्राणराशिघटिकाद्वयैः A, B. A, B. eu चन्द्रभानुविवर' पदाधिक स्यात्तदा क्रमगुणेन युक्तया ॥ ६ ॥ त्रिज्या सितविधिविधीयते पौर्णमास्यपरतोऽसितं तथा । सूर्यचन्द्रविवरांशजीवया चोत्क्रमक्रमवशात्सितं विदुः ॥ ७ ॥ चन्द्रमोपमविकाष्ठयोर्युति- स्तुल्यगोलभवयोरतोऽन्यथा । काष्ठयोविवरतोऽमों गुण- स्तेन चन्द्रचरनाडिकाविधिः ॥ ८ ॥ भानुचन्द्रविवरासुभिः सदा ६ शङ्कुरुक्तविधिना विधीयते । अक्षचापगुणसंगुणं हरे- ल्लम्बकेन शशिकीलकं स्फुटम् ।। ९ ।। कीलकाग्रगुण आप्यते ततो नित्यदक्षिणगतोऽस्तसूत्रतः । गोलखण्डगुणितो' विभज्यते लम्बकेन शशिनोऽपमः स्फुट: " ॥१०॥ दक्षिणोत्तर दिशोनिशाकृतो" लभ्यतेऽग्रगुणसंज्ञितः" सदा । तुल्यकाष्ठगतयोस्तयोर्वृति" शुद्धिमन्यभवयोस्तयोविदुः" ॥११॥ भास्कराग्रगुण केन तस्य तु व्यत्ययेन युतिशोधने कृते । · 'भानुचन्द्रवि ° C. वदाधिकं A, B. युक्त्या B. सतोसितं A; पूर्वमानवरतोसितं B. " चन्द्रमाबदवि ° A, B; योविवरजापरो A, B; काष्ठयोविवरतोऽवमो C. — In B, been interchanged. ' गोलखण्डगुणिते C. शशिनोऽवमस्फुटा C. १९ °कृतो. A, B. " लभ्यते ग्रहण संज्ञित: C. C. ४ शुद्धि रग्रहणयोरतोऽन्यथा C. [ महाभास्करीये ४ त्रज्यया B. पूर्वमानव चन्द्रमोऽवम° C. ५ काष्ठ- 8 (ii) and 9 (i) have १० शशिनोपमा स्फुट: A, B; १२° युतिः A, B, •N

zrre?<T3^ft crm w ^rf^r^r: n^n ^w^ttt invqf ftsnn^r ^f^TT:" i

' TOm: to *m C. ' ^ C. 1 ^#to^?t: A, B. " C H The second half of verse 14 and the first half of verse 15 is missing from A and B both. * C. • ^f^r° C.

' fW: A, B. * sjlftW^ ^fhTFJr TVmfozt A, B ; jrffenWRT

• rfc<£ C. «• °5«twv^ A , B. » 3r^5 C. " The first half of verse

19 w missing from C. " rovFTfrifar: A, B. art^fronfaiR* ^ 'Tf : i w^RRsrr^rpff ;rnrftpS ^sriT ii A, B. ts <nrt*tf: A, B; ^ *k«rt*f C. " wftrct mm d *• ^TOTRrt *nitf : A, B; C. w irfirar: C, ४० पक्षान्तात्परतः ' सोमो हरिजादुपरि स्थितः । इष्टकालत्रिलग्नोत्थैः प्रमाणैः परिलिख्यते ॥२३॥ कोटि: पूर्वाग्रतः कार्या भुजा याम्योत्तरायता | अग्रे कोटिशिरःस्पृक्क" कर्णसूत्रं प्रसार्यते ||२४|| पूर्वतः कर्णसूत्रेण सितमानं प्रवेशयेत् । असितं' वा पराद्भागं शीतांशोः परिलेख ॥२५॥ अथवाऽदये कार्य तत्कालेन्द्रर्कसम्भवैः । सितासितस्य पूर्वोक्तं दृश्यकालोऽभिधास्यते ॥२६॥ षड्रराशियुक्त सूर्येन्दुवैवरादुदयासवः" । अविशिष्टाः सिते पक्षे दृश्यकालोऽप्यतः परम् ||२७|| चक्रार्धयुततीक्ष्णांशोश्च॒न्द्राच्च करणागतात्” । अविशेषान्तरप्राणैनिशीथे दृश्यते शशी ||२८|| अस्तकालविलग्नेन्द्वोरन्तर प्राणजन्मना" । कालेन लग्नशीतांशू " कृत्वा भूयोन्तरासवः ॥ २९॥ पूर्वकालेन ते योज्याश्चन्द्रे" लग्नाधिकेऽथवा "।:- विश्लेषो वाऽन्यथा यावत्तुल्यकालोदयेन्दवः ॥३०॥ इत्थं कर्मक्रमावाप्त काले नामृत तेजसः । रश्मिभिः पूरयन्नाशां निशीथे दृश्यते शशी ।।३१।। अस्ताद्रिमस्तका रूढ़तिग्मांशुंगतभागतः । स्वदेशे भोदयप्राणा“ ग्राह्या यावन्निशाकरम् ॥३२॥ तेष्वहर्मातशुद्धेषु रात्रौ व्युष्टेषु चन्द्रमाः । दृश्यते त्वविशिष्टेषु तस्मात्तानविशेषयेत्” ॥३३॥ [ महाभास्करीये, १९ कालोच्युतःपरम् C. C. १५ योज्याश्चन्द्र C. ' पक्षान्तात्परितः A, B; पक्षात्परत: C. लग्नोक्तैः C. * पूर्वंगता A, B; पूर्वाश्रतः C. 19 ७ वापरभागात् C. ८ °ऽभिधार्यते A, B. १० • वैपरादोदयासवः A, B; °वैवराभोभयासवः C. " दृश्यकायोस्यतं वरम् A, B; दृश्य- १२ चरणागतात् C. १३ विलग्नेन्दो C. १ कालोन लग्नशीतांशो: १६ केरवी A, B. १७ °येनवः A, B. देशभेदोदयाप्ररणा A, B. स्वे स्वे शोभोदयप्रारणा C १९ यावन्निशाकृत: A; यावन्निशाकृता: B. परिशुद्धेषु A. B. १९ तस्मात्तानपि शेषयेत् C. २० हरिमादुपरि A, B. • इष्टकालवि- ५ ' तदग्रे कोटिशिरःस्पृक् C. अन्वितं A, B.

• Verse 26 is missing from C.

रात्रे C.

षष्ठोऽध्यायः॥ ४१

तघटीभोगसंयुक्तसूर्येन्द्वन्तर्घटीक्रमात् ।
ततोऽहम्नसंशुद्धिः शेषभोगादिकर्मणा ।।३४॥
अकॅन्दुमध्यनाडीभ्यो' दिनमानं विवर्धते ।
तद्विशेषघटीशेषे दिवा चन्द्रोदयः स्मृतः ।।३५।
तन्नाडिकानुपाताप्तभोगहीनाकंचन्द्रयोः
अन्तरालोदयप्राणैः कल्प्यं तत्राविशेषणम् ।।३६।।'
चन्द्रादौदयिकात्प्राणा' यावानौदयिक' रविः।
ग्राह्यास्ततोऽपि विश्लेषात् कल्प्यते निश्चलकिया ।।३७।
कृताविशेषनाडीभिर्यावन्नोदेति भास्करः।
तावदाशामुखादर्शः प्राणैराक्रमते शश ।।३८।।
आसन्नौ स्वधियाऽभ्यूह्य' मध्यलग्ननिशाकरौ।
कालेन्दुमष्यलग्नानामविशेषं समाचरेत् ।३९।
नाड्योऽन्तरालजाः‘ साध्या लङ्काराश्युदयैस्तयोः'।
ऊने ' विश्लेषनाडीभ्यः" शुद्धिः क्षेपोऽधिके स्मृतः ।|४०।।
मध्यलग्नसमश्चन्द्रो जायतेऽनेन कर्मणा।
तत्क्षिप्यपक्रमायैस्तु" मध्यच्छाया प्रसाध्यते ।।४१।"
अधोदितस्य चन्द्रस्य तथाऽर्धास्तमितस्य च ।
इन्दूदयास्तलग्नाग्रेः शृङ्गस्योन्नतिकल्पना" ।।४२।।
कर्मेदं शशिनस्तस्य कुर्यादमृतदीधितेः ।
ग्रहाणामपि सर्वेषामिदं कर्म विधीयते ।।४३।
अंशैरन्तरितः सूर्यान्नवभिर्युच्यते भृगुः ।
द्वयधिकैर्युयधिकैद या ज्यौज्ञसौरिधरासुताः ॥४४॥
प्रपन्नवक्रःसद्वर्मा सितो दृश्योऽर्धपञ्चमैः '।
चतुभिर्वाशुमालित्वादंशैरंशुमतोऽन्तरैः ।।४५।

---

२ °संयुक्तः° A, B. २ अकॅन्दुमध्यनाडियो A; अकॅन्दुभोगनाडिभ्यो B. १ The
second half of verse B6 is missing from C. *चन्द्रादौदयितात्प्राणा B;
चन्द्रादौदयिकाः प्राणा C. ५ द्यावानौदयिको A, B. ‘कृतावशेषनाडी° A, B; कृताविशे-
नाडी° C. " आसन्नस्वधिया ह्यत्र A, B. ‘नाड्योन्तरालजैः A, B; नाड्योऽन्तरा भुजाः
C. ‘लङ्कोर° c. ऊना C. १९ शद्धि A, B. • तत्क्षित्याप° A, B. " Verse
41 is missing from C. " शगस्यो° A, B. " ज्योज्ञसौरिधरासुताः A, B;
शाकंसोरिधरासुतान् C. " प्रपन्नदृश्युः A, B; प्रसन्नवक्त्रः C. t" सदृश्येऽर्ध° C.
.
.१४
१५
७
१४

४२

१ द्विघ्ना A; दि .C. • पिघटिका: B. ५ ग्रह सूर्यान्तरांशघ्नात् A; ग्रहसूर्यान्तरांशघ्ना B. १० 'भुक्तवि° A, B. दिनादि लभ्यते C. ९२ कालभागा: क्रमेणैते दिग्घ्ना' विघटिकाः स्मृताः । ऐन्द्रयां तद्राशिजा ज्ञेया वारुण्यां सप्तमस्य तु ॥४६॥ ग्रह सूर्यान्तरांशघ्नं त्रिंशता स्वोदयं हरेत् । लब्धकालो निरुक्तेन यदा तुल्यस्तदोदयः ||४७ ।। मन्दोच्चकर्णगुणितं शीघ्रकर्णं विभाजयेत् । विष्कम्भार्धेन संलब्धो भागहारः प्रकीर्तितः ॥४८॥ ग्रहयोरन्तरं भाज्यं प्रतिलोमानुलोमयोः । भुक्तियोगेन शेषाणां भोगविश्लेषसंख्यया ॥४९॥ दिनादिर्लभ्यते' कालो योगिनां योगकारकः । भुक्तेरनेकरूपत्वात् स्थूलः कालोऽत्र गम्यते ॥ ५० ॥ समलिप्तौ ततो" युक्त्या कुर्यात्तन्त्रस्य वेदिता । स्वोपदेशाद्गुरोनित्यमभ्यासेनापि गम्यते ॥५१॥ पातभागविहीनस्य " समलिप्तस्य जीवया " । हत्वा सदा स्वविक्षेप " भागहारेण भाजयेत् ॥५२॥ जीवभौमार्कपुत्राणामेवं " विक्षेपकल्पना | शीघ्रोच्चाच्छेषयोश्चापि विक्षेपो दक्षिणोत्तरः ॥५३॥ भिन्नदिक्कौ तु विक्षेपौ युक्तावन्तरमिष्टयो: “ । तुल्यदिक्कौ विशेषेण" विद्याद्विवरलिप्तिकाः ॥५४॥ पादाङ्गुलकलार्द्धाद्वा यथा” वा लक्ष्यते दिवि । तदन्तरं तयोर्वाच्यं योगिनां योगकोविदः ॥५५॥ द्वात्रिंशत्पञ्चभिर्हत्वा भूयो भूयस्तदुत्तरैः । शुक्रज्यौज्ञाकिभौमानां” व्यासलिप्ताक्रमं विदुः ॥५६॥ १२ १८ २० B. A. वतो C. A, B. १७ दक्षिणोत्तरे C. " यत्ता° C. १९ C. २१ ८ १७ [ महाभास्करीये ऐन्द्रं A, B. सप्तमं स्मृतम् A, ६ ७ तुल्यस्तयोदय: A. ● ●गुणितः १९ समलिप्ता- 'सोपदेशदाद्गुरोनित्यमभ्यास वावगभ्यते C. १५ सभास्वविक्षेप A, B; सदास्य विक्षेपं C. १६ ४ स्थूलस्सम्यग्गते```C. १३ वात° A, B. १४ निश्चयात् जीवभौमार्किपुत्राणामेवं A, B. २७ 'तुल्यदिको तु विश्लिष्टौ C. विद्याद्विपरलिप्तिका वदाङ्गुलकलार्धार्धाद्यथा A, B; पादाङ्गुलकलाद्वा यथा C. किभौमानां A, B; शुक्रज्ञार्काकिंभौमानां C, शुक्रज्योग्ज्ञा- २२ षष्ठोऽध्यायः ] एतैरेव हतः कर्णश्चन्द्रयोजनज: क्रमात् । लम्बनादिषु हार: स्यान्मानयोजनमाश्रितः ।।५७।। विष्कम्भार्धेन हर्तव्या भागहारहताः स्फुटाः । भागहारहता व्यासा" व्यासार्धघ्नाश्च लिप्तिकाः ।।५८॥ शेषः शीतांशुवत्कार्यो दशजीवाविनिश्चयः || चन्द्रोदयोपदेशेन' शङ्कु: स्यात् स्वचरादिभिः ॥ ५९॥ स्वहारैर्ग्रहयोगेषु" लम्बनावनती विदुः" । ग्रहोपरागवच्छेषं स्थित्यर्धादिविधिक्रमः ||६०। इति प्रतिदिनाभ्यासविमलीकृतचेतसाम्” । गुरुप्रसादसम्प्राप्तशास्त्र सद्भावचक्षुषाम् ।।६१॥ नान्यथा जायते वाणी ग्रहचारानुयायिनी" | रम्यानुरक्त कान्तायश्चित्तवृत्तिरिवामला ||६२|| 8 इति महाभास्करीये षष्ठोऽध्यायः । एतैरपहृतः A, B; एतैरेव C. १° योजनता: A, B; योजनतां C. is missng from C. • योजन " लिप्ता C. t The second half of " शेष C. 'विनिश्चयम् C. " चन्द्रोदयो १९ बन्धनावनतिविदुः A, B; लम्बेनावनती- पाणिग्रहचारानुयायिनि A, B. ४३ भक्तव्या C. verse 58 is missing from A, B. पदेशौ तु A, B. " स्वभागहारैर्वा वारे C. १२ °चेतसः A, B. १४ गुरुप्र° B. विदुः C. १ सप्तमोऽध्यायः शतमष्टोत्तरं भानोश्चतुभिरयुतैर्हतम्' इन्दोः षट्श्यग्निरामेषुभूभृन्नगशिलीमुखाः ॥ १॥ सागरत्विषुषड्वेदशीत रश्मिसमाः' शनेः । वेदारिवद्विचतुष्षट्करामा: सूरेरसृक्तनोः ||२|| कृतद्वयष्टर्तुरन्ध्रद्वियमला" भास्करस्य ये | बुधभृग्वोस्तु शेषाणां शीघ्रोच्चभगणाः स्मृताः ||३|| इन्दूच्चस्य नवैकाश्विवस्वष्टकृतसंज्ञिताः । बुधस्य खाश्वि खाद्यग्निरन्ध्राचलनिशाकराः ॥४॥ भृगोर्वस्वष्टरामाश्विद्विखसप्तसमा — गणाः' । राहो: षड्विकनेत्राश्विविष्णुक्रमयमाः " स्मृताः ॥५॥ अङ्गपुष्कररामाग्निरन्ध्रेष्विन्दुमिताऽधिकाः । द्वादशघ्नं युगं भानोर्भागहारोऽधिकाप्तये ||६|| अवमा व्योमवस्वक्षद्वयष्टखेष्वश्विनो गणा: । खाष्टव्योमा भ्रखा भ्राग्निखाष्टयो" हार इष्यते ॥७॥ व्योमखेष्वद्रिशीतांशुरन्ध्राद्रचद्रीषु चन्द्रकाः" । युगस्य दिवसाः प्रोक्ता विक्षेपांशास्ततः परम् ॥ ८॥ बुधास्फुजिद्रविजानां " द्वावेको वचसां पतेः । सार्धोऽश: " क्षितिपुत्रस्य पातभागाः क्रमेण च ॥९॥ २ भानोश्चतुभिरयुतैहृतम् A, B. °भूभृन्नागशिलीमुखा: A; °भूभृन्नगतिलिखामुखाः 'सागरं चीषु A, B; सागरतुंषु ° C.

• शनै: C.

B; भूगून्नगशिलीमुखा: C. द्वयष्टर्तुरन्ध्राद्वियमला C. ६° संज्ञित: A, B. गोर्वस्वष्टरामाश्वि॰ A, B; १० • "नेत्राग्निविष्णु क्रममिता: A, B. द्वादशघ्नयुतं भानो भंगहारोधिकाप्तयोः B. A; अवमारव्योमस्त्रक्षद्वयष्टखेष्ठश्विनो गणा: B. A, B; खाष्टव्योमाभ्रखाभ्रानलाष्टयो C. १४ °चन्द्रखा: A. १५ बुधस्फुजिभ्रविणां च A, B. १९ सार्धांश: A ; सार्धांश: B. कृत- खभृ- गुणा: A, B, C. द्वादशघ्नयुतं भानोर्भङ्गहाराधिकाप्तयोः A ; १२ अवमाख्योऽवमस्वक्षद्वयष्टखेष्वश्विनो गरणा: १३ खाष्टव्योमाभिखाभ्राग्नि खाष्टयो 'द्यग्निरन्ध्राकरनिशाकरा: C. भृगोर्वस्वष्टराशिद्वि° C. ११ ९ ८ सप्तमोऽध्यायः ] विशतिः खरसाश्चापि शतं खाष्टौ खसागराः । उदगाशादिविक्षेपान्' पातहीनाद् विनिर्दिशेत् ॥१०॥ आशाश्विनः खरन्ध्राणि रसविष्णुक्रमाश्विनः । खाष्टेन्दवोऽष्टरुद्राश्च मन्दोच्चांशा * यथाक्रमम् ।।११।। भास्करस्य विजानीयादष्टसप्ततिमंशकान्' । स्वमन्दोच्चं ग्रहाच्छोघ्यं शीघ्राच्छोध्या ग्रहाः' सदा ॥ १२ ॥ सप्त चत्वारि रन्ध्राणि पर्वता मनवः क्रमात् । एकत्रिंशन्नवशरा नव चाष्टिस्त्रिकेषवः ||१३|| मन्दशीघ्रोच्चवृत्तानि विद्याद् विषमयोरपि । समयोः पदयोश्चापि कथ्यन्ते मन्दशीघ्रयोः ॥ १४॥ शिलीमुखाश्विनोऽग्न्येकवसवोऽष्टादशैव” च । नवाश्विनो नगशरा" वसवस्तिथयः क्रमात् ॥ १५॥ एकपच्चाशतं " चैव सूर्याचन्द्रमसोरतः" । विष्णुक्रम: क्षितिधरा " जीवा मख्यादयो मताः ।।१६।। मख्यादिरहितं कर्म " वक्ष्यते" तत्समासतः । चक्राशकसमूहाद्विशोध्या " ये भुजांशकाः ॥ १७॥ तच्छेषगुणिता " द्विष्ठाः शोध्या: खाभ्रेषुखाब्धितः" । चतुर्थांशेन शेषस्य द्विष्ठमन्त्यफलं हृतम् ||१८|| बाहुकोट्यो: ” फलं कृत्स्नं क्रमोत्क्रमगुणस्य वा । लभ्यते चन्द्रतीक्ष्णांवोस्ताराणां वापि तत्त्वतः ॥ १९॥ २३. १ इन्दोर्गणाः” खखवियद्रसवृन्दनिघ्ना व्योम्नो भवेयुरिह" वृत्तसमानसंख्याः । ३ ...विष्णुक्रमाश्विन: C. °विक्षेपात् C. २ खनवती A, B. स ५°तिरंशका: A, B. शीघ्राच्छोध्यग्रहा: A, B. चाष्टत्रिकेषवः C. ७ A, B; विशेषसमयोरपि C. पदयोः पश्चात् C. नवशरा १५ कर्म १३ सूर्यचन्द्रमसोरत: A. १२ एकपञ्चाशकं A, B. C. missing from C. १८ १७ १६ कथ्यते C. • °समूहा विशोध्या B. तच्च' गुरिणता C. "द्विष्ठाशोध्यखाभ्रेषु खाब्धितः A, B; द्विष्ठाश्शोध्या लाभ्रेष खण्डिता: C. २२ इन्दोगंणः A, B; इन्दोर्गुणा: C. २४ ० रथ A, B. 'वीषुम° A, २३ खखवियसद्रवृन्द निघ्नो B. २९ वाहुकोट्याः C. A; खखवीयसवृन्द निघ्नो B.

• मन्दोच्चांशे C.

विद्याविषमयोरपि १० °डष्टिदशैव A, B. ४५ १४ क्षितिधा C. २० ४६ [ महाभास्करोये इष्टग्रहस्य भगणैर्गगनस्य वृत्तं भङ्क्त्वऽथ तस्य परिधिं लभते समन्तात्' ।।२०। निबन्धः कर्मणां प्रोक्तो योऽसावौदयिको विधिः । अर्धरात्रे त्वयं' सर्वो यो विशेषः स कथ्यते ।।२१।। त्रिशती दिने' क्षप्या ह्यवमेभ्यो विशोध्यते । ज्ञगुर्लभंगणेभ्योऽपि' विशतिश्च ततोऽब्धयः ।।२२।। अष्टिश्शतगुणा’ व्यासो योजनानां भुवो रवेः । खाष्टाब्ध्यङ्गानि शीतांशोः शून्यवस्वब्धयस्तथा*।।२३।। वस्विन्द्रियगुणच्छिद्वस्वङ्गानि’ विभावसोः । अङ्गाङ्गष्वेकभूतानि चन्द्र कर्णः प्रकीतितः ।।२४। अष्टिरष्टौ जिना रुद्रा' विंशतिर्युयधिकाः क्रमात् । दशघ्ना गुरुशुक्राकभौमज्ञांशाः" स्वमन्दजाः ।।२५।। मन्दवृत्तानि द्वात्रिंशन्मनवः" षष्टिरेव " च। खद्यो वसुदस्राः स्युः शीघ्रवृत्तान्यथ क्रमात् ।।२६। द्वयद्रयः• खाङ्गनेत्राणि खाब्धयोऽGध्यग्निदस्रका:” । द्वयग्नीन्दवो रवेर्मन्दंॐ शुक्रवद् वृत्तमेव च ।।२७। एकत्रिंशत्क्षपाभर्तुरर्धरात्रे विधीयते । पातभागाश्च विज्ञेयाः पण्डितैः परिकल्पिताः ।२८। मन्दशीघ्रोच्चयोः क्षेप्यं चक्रार्ध बुधशुक्रयोः । राशित्रयं तु शेषाणां पात्यते" पातसिद्धये ॥२९॥ शुक्रकिदेवपूज्यानां भागौ द्वावेव संयुतौ । मन्दपाताच्च शीघ्रोच्चात " सार्धाशस्तु कुजज्ञयोः ॥३०॥ .२१ २१ ३५ परिधिर्लभते समन्तात् A, B, परिधि लभते समानात् C. २ अर्धरात्रेश्चयं A, B. दिने A, B. ५ क्षेप्याऽप्यवमेभ्यो A; क्षेप्याप्यवमे द्यो B. ५ भृगुर्वोमं गणेभ्यो Q. अद्भिश्शतगुणो A, B. ५ शन्यावस्वब्धयः सदा C. “ वस्विन्द्रीय° }; वस्वीन्द्रीय B. अह्रष्ट A,B. • द्रिघ्ना A,B; रुद्र C. १ विशतिद्वयधिकाः B, C. १२ दशघ्नं C. ९ ९भौमज्ञान्धि A, B; भौमज्यांशाः C. १४ मन्दवृत्तान्यथ किशन्मनव A, B; मन्दवृत्तान्यपित्रशन्मनवः C. ५ ~~टुरेव A, B. " वसुरुद्राः C. " द्वयग्नयः A, B, C. “ खश्योऽध्यग्निदस्रकाः A; खायोध्यग्निदस्रकाः B; खाब्धयो ह्यग्नि दस्रकाः C. ९ द्वयद्वन्दवो A, B. ३ रवेर्मन्द A, B, C. २९ ऽभिधीयते C. * वात° A; तेवाभा° B; पात missing from C. २ मन्दोच्चयोः C. २* वारयते B. २५ कीfततो A, B.

• मन्दवाताच्च शीघ्रोच्चात् B ; मन्दपातोच्च शीघ्रोच्चात् C. " भृगुशयोः A, B सप्तमोऽध्यायः ]

विबुधानां च सर्वेषां शीघ्रपाताः प्रकीर्तिताः' । शोधयित्वा क्रमात् पातान् विक्षेपांशान्' प्रसाधयेत् ॥३१॥ योगविश्लेषनिष्पत्तिरेकानेकस्वदिग्वशात्' । विक्षेपः स स्फुटो ज्ञेयो ग्रहस्यैकस्य कीर्तितः ||३२| अन्यस्याप्येवमेव स्याच्छेषाः प्रागुक्तकल्पनाः । एतत्सर्वं समासेन तन्त्रान्तरमुदाहृतम् ||३३|| शीघ्रमन्दोच्चचापार्धसंस्कृतात्स्वीयमन्दतः । स्फुटमध्यग्रहाः' सर्वे विशेषः' परिकीर्तितः ||३४|| वेदाश्विरामगुणितान्ययुताहतानि चन्द्रस्य शून्य रहितान्यथ मण्डलानि । स्वैः स्वैर्हतानि भगणैः क्रमशो ग्रहाणां कक्ष्या भवन्ति खलु योजनमानदृष्टया ॥३५॥ इति महाभास्करीये सप्तमोऽध्यायः । च कीर्तिताः C. • °रनेक···स्वदिग्वशात् C. B. चन्द्र C. नमानदृष्टा A, B, C. १ क्रमात्पाताद्विक्षेपांशात् A, B ; क्रमापाताद्विक्षेपाशात् C. शीघ्रोम° A, B. स्फुटमध्याग्रहा: C. ६ विश्लेष: A, ५ ' स्वैस्स्वैवतानि भगणी A; स्वैस्स्वैवतानि भगणै: B. ' योजअष्टमोऽध्यायः शशिवत्सरताडिते गणेऽह्नां युगभूवासरभाजिते समादि' । अधिकाब्दगुणे तथैव विद्या- तथा A, B. दुभयोरन्तरमर्कवर्षपूर्वम् ॥ १॥ रविवर्षगणेन नास्ति कृत्यं परिशेषीकृत राशितोऽथ मासौ । धृतिसम्मितवासरान्' दिनेभ्यः रविसंहृतमर्कंकाललब्धं रात् C. तदा ... शशिवास रेषु C. विशुद्धभुजाफलं B. रीतादपि शेषविधे B. परिशोध्यैव ततो' भुजादि कार्यम् ॥२॥ शशिकेन्द्रजमप्यथाशु' चैवं विपरीतं तु धनर्णमिन्द्रहेषु । परिनिष्ठितनाडिका व्यतीताः शशिनो या दिवसस्य' षष्टिनिघ्नाः | स्फुटभोगविशेषसूर्यभागै शशिवत्तच्छशिवासरेषु' कार्यम् ॥३॥ र्भजितास्ताः स्फुटनाडिकास्तदाप्ताः ॥४॥ दिनमध्यच्छायाकदुच्चविशुद्धाद् भुजाफलं यत् स्यात्" । तत् क्षयधनविपरीतादविशेषविधे” रवेर्मध्यम् ॥५॥ ज्यासङ्कलितात्क्रमशः शोधितजीवामखिर्मखेश्शेषम् । मख्या हतमन्त्याप्तं " पूर्वयुतं तद्भवति चापः ॥६॥ २ अधिकाब्धगुणे A, B. कृत्या A, B. A, B. "शशिकेन्द्रजमप्यथात्र C. ७ “ ऽप्यतीता A, B. ' दिवसविहीनत्स्य B. कृत्वा C. धृतिसंमितवास- शशिवत्तचिवासरेषु B; विशुद्धभुजाफले A; १२ तत्क्ष्यसनविपरीतादपिशेषविधे A; तत् क्ष्यतनविप मध्याह्तमन्त्याप्तं A, B, १४ भवति तच्चापम् A, B. १० ४ अष्टमोऽध्यायः ] सार्धांशकोऽक्षोऽष्ट कलाविहीन - रछाया दिना' समभूमिभागे । पञ्चाङ्गुला' द्वादशकस्य शङ्कों- र्वदार्कमस्मिन्दिनमध्ययातम्' ||७|| अष्टौ लवा: ' षोडशलिप्तिकोना: पलप्रमाणं प्रवदन्ति यस्मिन् । छाया दिननर्धेऽर्धचतुर्थसंख्या तत्राशु वाच्य : सविता नभस्थ ः ॥८॥ १० पञ्चाधिका विंशति रक्षभागा: " त्रिंशच्च यस्मिन् विहिताः पलस्य " । छाया तयोः शङ्कुसमा दिनार्धे क्षिप्रं समाचक्ष्व” तयोः स्फुटार्कम् ।।९।। १४ अक्षांशका: * पञ्चदशैव यस्मिन् छाया रवेः पञ्चमभागयुक्ता । सार्धाङ्गुला स्यात् सममण्डलोत्था " १५ वाच्यो" विवस्वान् खलु तत्र कीदक् ॥१०॥ सप्तत्रिंशच्छाया सममण्डलजा " पलाङ्गुलास्त्रिशत्” । वाच्यो जगत्प्रदीपः सममण्डलसंस्थितः सविता ॥११॥ छाया षोडश दृष्टा प्रागपरसमायता समे देशे । सार्धा : सप्त पलांशास्तत्र " विवस्वान् कियान् वाच्यः ॥१२॥ नीता रवेर्बलवता मरुता समस्ता राश्यादयोऽत्र गणिताः सह तत्पराभिः | शेषो मया परिगतः खलु तत्पराणां सैक" शतं कथय भानुमहर्गंणं च ||१३|| ' सार्धांशकाक्षाष्ट° A; सार्धांशकोक्षाष्ट° B. ' च्छायादिनार्धे A, B. ३ पञ्चाङगुल A, B; पञ्चाङ्गुलात् C. शङ्कु C. ....स्मिन् दिनमध्ययातः A ; आत्वात्स्मिन् दिन- मध्ययातः B; र्कमस्मिन् दिनमध्ययातम् C. ६ अष्टौ लव: A; अष्टो लव: B. वल- प्रमाणं A, B; फलप्रमाणं C. ' तत्रापि वाप्य: C. नभः स्थ: A, C. • भाग C. १८ फलस्य A, B; लवस्य C. १९ चमाचक्ष्व B; त्वमाचक्ष्व C. स्फुटार्कान् A, B; स्फुटा- र्कम् C. " अक्षांशका C. १५ OSक्षा C. " वाचो C. १७ A, B. सप्तफलांशास्तत्र C. २० ' समण्डलजा A, B. " पलाङ्गुल: 'समन्ताद्राश्यादयोऽत्र C. २९ सैकां B. १९ ४ ४६ ७ ५० राशिभागसहिताः शशिलिप्ता ९ बालहस्तपरिमृष्टविनष्टाः । पञ्चवर्गविकला:' खलु दृष्टा- स्ताभिरा दिनराशि शशाङ्कौ ॥१४॥ राश्यंशका' हृता वात्या भागशेषस्त्रिसप्ततिः । वाच्यो' भौमः कियांस्तत्र कीदृशो वाऽप्यहर्गणः ॥१५॥ राशित्रयं पञ्चदशांशयुक्तं लिप्ता निशानाथ सुतस्य पञ्च | एतत्समीक्ष्याथ गतान्यहानि यातानि तस्यैव च मण्डलानि ॥ १६ ॥ मघवद्गुरुराशिभागलिप्ताः शिशुना चपलेन नाशितास्ताः । नव तत्र विलिप्तिकास्तु दृष्टा दिनराशि गुरुमध्यमं च ताभिः ॥ १७॥ मण्डलादि भृगुजस्य सलिप्तं " नष्टमत्र विकला दश दृष्टाः । सूर्यजस्य दश सप्तसमेता " ११ ब्रूहि तौ दिनगणावथ" शीघ्रम् ॥१८॥ पञ्च सप्त नव भौमशशाङ्कौ राशिपूर्वगणितौ समवेतौ । उच्यतां दिनगण: शशिभौमौ कीदृशौ च भटतन्त्रविदाशु" ||१९|| भौमशऋगुरुमध्यविशेष: * १४ पञ्चराशिगणित:" परिपूर्ण: । ' पञ्चवर्ग विकला B, C. २° रार्थ A, B; °रर्ध C. भागशेषं त्रिसप्तभिः A; वांत्या भागशेष त्रिसप्तभिः B; " पात्यो C. एतत्समीक्ष्याक्षगतान्यहानि A, B. ६ ३ कास्तु B. १" शास्तममेता C. १२ यदिगणवथ A; दिगणावथ B; विदाशु A, B. १४ °शुक्र° A, B. १५° गणितं A, B. [ महाभास्करीये रात्र्यंशका A, B. पात्या भाग शेषस्तु 9 ' नाशितास्ताम् A, B. गुरुमध्यमाक्षताभिः A, B; गुरुमध्यमात्तथाद्भि: C. ४ वात्या सप्तति: C. ● लिप्ति- १० सलिप्तौ A, B. दिगणावधि C. १२ भभतन्त्रअष्टमोऽध्यायः ] उच्यतां दिनगण: कलियातो देवमन्त्रिरुधिरौ' च कियन्तौ ॥२०॥ ४ सूर्याचन्द्रमसौ तुलाधरगतौ दृष्टौ मया तत्त्वतो भागद्वदशभिर्द्वयेन च युतौ सूर्यस्य वारोदये | लिप्ताभिः शशिशून्यसागरयुतौ जीवस्य वारे पुनः शुक्रस्थाथ शनैश्चरस्य दिवसे' तुल्यौ कियद्भिदः' ।।२१।। विलिप्ताभिरधिकोऽर्को विज्ञेयो भूधरेन्दुभिः । शोधयेच्च निशानाथाद्विलिप्ता धृतिसंमिताः ||२२|| नाडीभिः कियतीभिरभ्युपगतादह्नां गणादागत- स्तीक्ष्णांशोर्भगणादिकोऽत्र विलयं नीतोऽधुना वात्यया | दृष्ट: सप्ततिरेकरूपसहिता शेष: कलानां मया वक्तव्यो द्युगणो गतश्च सवितुः स्पष्टाश्च या नाडिकाः ॥ २३ ॥ अर्काकवासरपहृतः कश्चिद्दिनानां गणो लब्धौ तत्र न वेद्मि नैव च तयोः शेषौ मया लक्षितौ । यौ तौ मण्डलताडितावथ पुनर्भक्तौ दिनैः स्वैः' पृथक् तत्राप्तं मरुतापनीत मधुना चाग्रे तयोस्तिष्ठतः ॥२४॥ अर्कस्थाश्विनगाब्धिनागशिखिनः शेषः कुजस्योच्यते भूता व्यङ्गन भोष्टशीतकिरणक्षोणीधरक्ष्माभृतः । एताभ्यां पृथगर्कभूमिसुतयोरह्नां गणौ तद्गतं द्वयग्रं चापि तयोविंगण्य गणका व्या वर्णयध्वं क्रमात् ॥२४*॥ भास्करे" मिथुनपर्यवसाने शर्वरी त्रिगुणसप्तघटी स्यात् । अक्षचापगणितं " वद तस्मिन् लम्बकेन सहितं विगणय्य ||२५|| दिवसं C. ' देवमद्रि रुधिरौ A; देवमन्द्रिरुधिरौ B. दिनः B. धृतिसंमता: A, B. गादिकेऽत्र निचिते नीतेऽधुना A, B. ९ नेव च A ; नवे नेव च B. पनीतमधुना A, B. १२ अक्षप्रश्नास्करे C. • दिनदिनै: A; दिनद्भ- " नाडीभिरभ्युपगतादह्नां गणा भागतः । तीक्ष्णांशुर्भग- ६ ° सहिताः शेषा: A, B. " ये A, B. " निवेश- मधुना- पुनर्भक्ते स्वदृष्टे: A; पुनर्भक्ते स्वैर्दृढ़ B. " The verses 23 and 24 do not occur in C. १३

अक्षचापगुणता A, B.

[महाभास्करीय ५२
भास्करेण परिचिन्य कृतोऽयं
मन्दबुद्धिपरिबोधसमर्थः ।
सम्यगार्यभटकर्मनिबन्धः
स्पष्टवाक्यकरणैः समवेतः ।२६।
स्पष्टार्थानेककरणैः छेद्यके ग्रहणे रवेः ।
यदिहास्ति तदन्यत्र यन्नेहास्ति न तत् क्वचित् ।।२७।

इति महाभास्करीयेऽष्टमोऽध्यायः ।

इति महाभास्करीयं समाप्तम्

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१ ततो या C. २ कर्मनिबद्धः A, B. १ स्पष्टानेककरणे C. ॐ छेद्यकेन्द्रगणे A, B.

महाभास्करीये प्रयुक्तपारिभाषिक

शब्दानाम् अनुक्रमणिका अंश ( = भाग ) i. ३३-३५,३८,३६; v. = ६, ७, ४१ - ( वृत्तविभाग) i ८, १५ ; iii. २१, ३५; vi. ७, ४४, ४५, ४७; vii. ८, ९, ११, २५ —(=अक्षांश) ii. ३ अंशक i. १८; vi. ४; vii. १२, १७; viii. ७, १५ अक्ष (===अक्षांश ) ii. ५; iii. १४, १५; v. ४२; vi ४१; viii. ७ ~(=५) vii. ७ - कर्ण iii. २६ --कोटि iii. २८, ३६ - गुण iii. ५, ४१ - चाप viii. २६ - चापगुण vi. e. - जीवा iii. ५४ - ज्या iii. ३८, ३६ - भाग viii. १, ४, ६, ७, १५, २७ अग्र (===अग्रा) iii. ५३ - गुणक (= अग्राज्या) vi. १२ अग्रा iii. ३७, ५३, ५७-५६; vi. ४२ अङ्ग (=६) vii. ६, २३, २४, २७ अङ्गुल V. ४८, ५३; vi. ५५. viii. ७, १०, ११ - गुण (==अग्राज्या) vi. ११ अचल (= ७) i. ३४, vii. ४ –(=स्थिर) v. ७४ अज ( == मेष) iii. अतिवऋ iv. ५६, ५७ अद्रि (==७) i. ४, ४८; iii. १०, ६४, ६५; vii. ४, ८, २६, २७ अधिक (= अधिकमास) vii. ६ - मास i. २८ अधिकाब्द viii. १ अक्षवलन V. ४४ अक्षांश ii. ३; iii. ३७ —क viii. १० अग्नि (= ३) i. ४, १६, २३,२८; vii. अनुलोम vi. ४६ अधिकाह i. २४ अधिमास i. ४, १६, १७ —क i. ५, १४, १५,.२५, २८ – शेष i. १७, १९ अध्व (===देशान्तर) ii. २ अध्वा (==देशान्तर) ii, अध्वान ( = देशान्तर ) ii. ४-६ अनुपात iv. १७, २७ ; v. ११ ; vi. ३६ अन्त्य ( = मीन) iii. ६५ – ( = अन्त्यजीवा ) viii. ६ ---गुण iv. १७ in ५४ -जीवा iv. १४, १५, ५८, ५६, ६२ -फल iv. १६, ५५ ; vii. १८ अपक्रम iii. १३, १४, १७ ; iv. ३६ ; v. २१ ; vi. ४१ -- गुण iii. ६, १६ अपगम (=क्रान्ति) iii. ११ अपचय i. २५ अपम ( = क्रान्ति ) v. ४५ ; vi. २, ८, १० - गुण vi.s - धनु ( = क्रान्तिचाप ) iii. १५ --- संभव (मास) i. ७ अर्काग्रा iii. ४२, ५३, ५८, ५६

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चाप iii. ५७

अर्धविस्तर (= अर्धव्यास) iii. २३ अवनति (=नतांश ) iii. ११ - ( = ग्रह विक्षेप) v. ६३ ~ ( - विक्षेपलम्बन ) v. ३२ ; vi६० -- लव iii. ११ -लवजीवा iii. ११ - लिप्तिका v. २६ अवम i. ५, २४, २६, २८; vii. ७, २२ - शेष i. १७, १८ अवलम्बक (= लम्बांश ) iii. १६ गुण iii. ५ अविशिष्ट vi. २०, २७, २८, ३३ अविशेष v. २८ अपवर्त i. ४६ अब्दप i. ३० अब्धि·(=४) iii. ६५, ६८, v. २ ; vii. अश्विन् ( = २ ) iii. ६७ = १८, २२, २३, २७

• अभ्यास (= गुणन ) v. १६

अम्बर ( == ० ) iii. ६७ अम्बरोरुपरिधि (= आकाशकक्ष्या) i. २० अयन iv. ३६; v. ४५; vi. ३ अयुत vii. १, ३५ अर्क (= १२) iii. ६६, ६७ -वर्ष viii. १ -विधि viii. ५ -कर्ण iv, १३ -कर्म v. ७६ -तिथि v. ३५ नाडी vi. ३८ अविशेषण vi. ३६ [महाभास्करीये अविषम iv. १५ अश्वि ( == २ ) i. ३७; v. ३०; vii. २, ४, ५, ३५ अष्टि (= १६) vii. ७, १३, २३, २५ असकृत् iv. ५५ असित ( पक्ष) iv. ३३ – (भाग ) vi. ७, २५, २६-२७ असु iii. २५, ३० - ३२, ३४ ३६, ३६, ४० ; iv. ३२ ; v. ६, १०, ४२, ४३ ; vi. ६, २६ - राशि iii. २७, २८ अस्त ( ग्रहास्त) ii.८; vi. २, २०, २२ – काल vi. २६ - लग्न iii. ३३; vi. १८, ४२ -सूत्र (= उदयास्तसूत्र) vi. १० अहर्गण i. ७ ; viii. १३, १५ अहोरात्र iv. २८ विष्कम्भ v. २२ अह्नां गणः i. ३०, ४४; viii. १, २३, २४* अह्नां निचयः i. ६ आकाश (=o) iii. ६७ आशा (=१०) iii. ६६; vii. ११ आसन्न vi. ३६ शब्दानुक्रमणिका ] तर iv. ६३ आह्निक i. १६, १८ इन (==१२) iii. ६४ इन्दु ( = १ ) iii. ६५; v. २६; vii. ६, ११, २७; viii. २२ इन्द्रिय ( ५ ) i. ३२; vii. २४ इन्द्रहः (= चान्द्रदिन या तिथि) viii. ३ इषु (== ५) i. २५; v. २, ४, vii. १, २, कर्ण ii. ४; iii. ४७, ५०, ५१; iv. 5, १०-१२, १६-२१, ४१, ४७, ५५, ५६ ; v. २, ३, २४, ७०-७३; vi १३-१५, २४, २५, ५७; vii. २४ ६-८, १३, १८, २४ उच्च i. १०; iv. १; vii. ४ ; viii. ५ - भुक्ति iv. १६ उत्क्रम (= विपरीत क्रम ) iv. २, ४, ३७, ३८; vi. २ -(= उत्क्रमज्या ) vi. २, ७ - गुण vii. १६ - ज्या iv. १६; v. ४२; vi. ५ -फल iv. ७ कन्यका iii. ६४ कपाल ३८ करण ( = क्रिया) vi. २८ - (बवादि करण) iv. ३३ -विधि ii. २ कर्कट ( = कम्पास ) iii. १; v. ४६, ५६, ५५ -सूत्र vi. १५, २४, २५ कला i. १०, १८, ३७; iii. २७, ३०, v. ६६; vi. ५५; viii. ७, २३ कर्ण V. ३, ३० – शेष viii. २३ - व्यास V. ५ उदय (===उदयलग्न) iii. ३२-३४; vi. ३० कलि i. ६, ८; vili. २० कार्मुक (=चाप) iii. ४० कालभाग vi. ४६ - ( = उदयज्या) v. १६ —(=स्वदेशोदयासु) iii. १० - प्राण vi. ३२, ३६ - राशिप्राणपिण्ड iii & - लग्न v. १३ उदयासु iii. ३५; vi. २७ उदयास्तमय iv. २६ उन्नति vi. १७ उपराग (==ग्रहण) vi. ६० ऋतु ( = ६) vii. २, ३ ऐन ( = सिंहराशि ) iii. ६४ ऐन्द्राग्न (= विशाखा) iii. ७३ ऐन्द्री (=== पूर्व) vi, ४६ ओज ( = विषम ) iv. ३८ कुम्भ iii. ६५ कक्ष्या i. २०;iii. २२; v. १२, १३, १६, कृत (= ४) i२७, ३५; iii. १; vii. ३, ४ ६९; vii. ३५ कृति (= वर्ग ) ii. ३, ४ ; iii. ४, ६, काष्ठ (==चाप ) iii १३, ३६; v. १५, १७, ४२; vi. s - (= दिशा ) v. ५४ काष्ठा (= दिशा) iii. १४ कीलक (= शङ्कु ) vi. & कोलकाग्रगुण ( = शङक्वग्र ) vi. १० कुजाशा (= दक्षिण दिशा) iii. ३ कुटिल iv. ५७ कुट्ट i. ४६ कुट्टन i. ४८ कुट्टाकार 1. ४१, ४५, ५० i. ५६ २०-२३, ३८, ३६, ४८, ६६ केन्द्र ( === ग्रहोच्चान्तर ) iv. १, २, ५, ६, ८, १५, २१, ३७, ३६, ५६, ६२ –(== मध्यबिन्दु) iii. ४३, ४४, ४६; v. ५०, ५२, ५५, ५६, ६४ - ज्या iv. ३६ --भोग iv. १५ कोटि (समकोण त्रिभुज की कोटि) ii. ३; iii. ४८, ४६, ५१; iv. १६, २०, ५५ -- ( = ९०-भुज ) iv. ८, १६, ३७, ४७; vi. १२–१४, २४, vii. १६ का iv. -- फल iv. १०, ११ -साधन ( = कोटिफल ) iv. & कोण iii. ४६ क्रम ( == युग्मपद ) iv. १६ - गुण ( = क्रमज्या ) vi. ६; vii. १६ – ज्या iv. १६; v. ४३ - फल iv. ७ क्रान्ति iii. ६, ५३; iv. २५, १८, १८ --- जीवा iii. ३७ -ज्या v. २१ क्रिय (= मेष) iii. ८, ६, ६३; iv. ६ क्षमा - दिन i ८, ११ - व्यासभेद ( = भूत्रिज्या) v. २४ क्षिति ii. ५; v. ४६ – गुण iii. ७, १६ - जगुण iii. २७ -जा ( = कुज्या ) iii. १२ जीवा iii. ६; v. २२ -- ज्या iii. ५३ धर (= ७) vii. १६ • मौर्वी iii. २५ - [ महाभास्करीये क्षिप्ति iii. ७१; vi. १, ४१ क्षेत्रनिर्माण iii. ६२ क्षेप i. ३५ ____(=शर) v. ३२, ४० ख ( = ० ) i. १४, १६, २२, २३, २५, ३२, ३३, ३४; iii. १०, २८; iv. ३१; v. ४, ३०; vii. ४, ५, ७, ८, १०, ११, १८, २०, २३, २६, २७, – मध्य ( याम्योत्तरवृत्त) iii. ४० गगन (याम्योत्तरवृत्त) iii. ३६ ~(=०) i. ३४; ii. १० गगनस्य वृत्तं vii. २० गण (===भगण) i. ४१, ५२; vii. ५, ७, गति iv. ५७ गुण ( गुणा करना) i. २५, ३५; iii. ६१; iv. ३८, ४७; vii. २३; viii. १, २५ - (=ज्या) iii. ५, १६, २१, २३, २८, ५३,; iv. १७; v. १५, २३; vi. ८, १० – ( == ३ ) v. २; vii. २४ प्रतान ( == ज्या) iii. ५ गुणकार i. ३२, ४५, ४७, ४८ गुणफल iv. ८ गृह (= राशि) i. ८, १८ गो (= वृष) iii. ८ गोल iii. ११, ३७ २० • खण्ड (= त्रिज्या ) vi. १० भेद (=== त्रिज्या) iii. २१ ग्रह i. १०, २१, ५२ ; ii. १०; iii. ६०, ६२, ७०, ७१, ७५ (i); iv. ३४, ४८, ६१, ६३; vi. ४३,४७, ४६, ६०, ६२; vii. १२, २०, ३२, ३४, ३५ शब्दानुक्रमणिका ] - (= ग्रहण) v. ५८ - गणित ii. ४ • चार vi. ६२ - तनु i. २६, ३२, ३४, ३५ - देह i. ३१, ३८, ३६ - मध्य v. ३८, ५४, ६० योग vi. ६० ग्रहण v. १, ३३, ३६, ७४; viii. २७ मध्य v. ५८, ७४ ग्रहोदय ii. e ग्रहोपराग vi. ६० ग्रास ( ग्रहण) v. ३२, ३६, ३६, ५८, ६१, -- शलाका V. ६३ ग्राहक V. ४०, ४६, ५६, ६१,६५ ग्राह्य V. ४० ४६, ५२, ६०, ६५ परिधि v. ५२ बिम्ब V. ४६ -बिम्बक V. ६५ मण्डल V. ५१, ६५ घटिका iii. ६१; v. २५; vi. ४ घटी ii. ८, १०; iii. २३, ३६; iv. ३२, ३४; vi. ३४, ३५; viii. २५ धन i. ३३ .घात iii. २८ चक्र iv. २२, ३५, ४६ – कला (==२१६००') i. v. २४, २६ –पाद iii. ११ चक्रार्ध (=१८०° ) iii ३३; iv. २२,३५, ३६, ४६; vi.२८; vii. २६ चक्राशक vii. १७ चक्रांश ii ३ चतुरस्र iii. ५० चन्द्र (== १ ) i. २७; v. १४ चन्द्रक (= १) vii. s चन्द्रोदय vi. ३५, ५६ चर iii. ८, २५; vi. ve चरदल iii. ७, १०, १८, १६, २८ - लव iii. २७ चरप्राण iii. २३, २८ iv. २६ चरार्ध iv. २८ चरासु V. २२ चल (= शीघ्रोच्च ) iv. ४३, ४६, ५४, ५६ चलोच्च iv. ६० चाप ( = धनुराशि ) iii. ६५;iv. ४१; ४४, ६०; v. २१ – ( वृत्त चाप) iii. ६, १६, २७, २८, ५७; iv. ४०, ४१, ४३, ४६; v २१; vii. ३४; viii. ६ चार (=गति) i. ११; vi, ४, ६२ चारनिचय iii. ६१ चित्रा iii. ७२ छाया iii. २, ४, ५, १३, १४, १७, २७, ४१- ४६,४८, ५०, ५२, ५५, ५६; viii७ - १०, १२ -कर्म iii. ४ – दैर्ध्य v. ७३ – भ्रमण iii. ४६ छिद्र ( = ९ ) vii. २४ छेद (= हर) i. ४९; iii. २३, २४ छेद्यक ( = परिलेख) v. ६०; viii. २७ छेद्यकाल V. ७७ जलपदिक् (= पश्चिम) ii. १० जलविधि ( जलघटीप्रकार ) ii. s जलेशदिक् (=पश्चिम) iii. ५६ जिन (=२४) iii. ८, ६६, ७३; vii. २५. जीवा (=ज्या) iii. १८, १६, २५, ५८; iv. २, २५, ३७, ६०; v. १२, १४, १८, २१, ४६; vi. ७, ५२; viii. ६. ५८ [ महाभास्करीये - (वृत्तपाद की २४ जीवाएँ) iv, ३; vii. त्रिज्या (= व्यासार्ध, ३४३८) iii. १२, २४, १६; vii. ६ २८; iv. ११, २१, ४७, ५६; v. ४३; vi. ७ जीवाविधि v. ३५ जूक (= तुला) iii. ११ ज्ञ (==बुध) i. २; vi. ४४, ५६; vii. २२, २५, ३० ज्या ( == जीवा) iii. २३, २४, ५८; iv. ४, ३६; v. ३०, ६९ —(=खण्डज्या) viii. ६ ज्यासंकलित viii. ६ ज्योतिषां निचयः (= नक्षत्र) iv. ३४ तत्परा i ८, १५, १८, ३४; viii. १३ - शेष viii. १३ तन्त्र ii. ८; v. ७८ तम (== भूच्छाया) v. ७३ –(=पात ) vi. ३३; ii. १०, ७८ तमोमय (= राहु) vi. २१ तारक iii. ७२, ७४ तारा i. २; iii. ६२, ७१; vii. १६ तिथि (===चान्द्रदिन) iv. ३१, ३२ - ( अमा, पूर्णिमा ) iv. ६४; v. १०, २५-२७ – ( स्पर्श, मध्य, मोक्ष का समय ) V. ४२ -~(=१५) iii. ६३; v. ४, १४; vii. १५ –(=३०) i. ५ - प्रलय i. २३ तिथिप्रणाश ( तिथिक्षय) i. ६ तिथ्यन्तर v. १०, ४२ तिर्यक् ii. ४; iii. ४४. v. ६४ तुङ्ग i. ३४, ४० तुला iii. ६४; iv. ६ तुलाधर viii. २१ त्रिगृहगुण (=== त्रिज्या) iii. २७ त्रिभवन ( = ३ राशि) iii. ५, १९; vi. २ त्रिमौर्वी ( = व्यासार्ध ) iii. ३९; iv. ३८ त्रिराशिजा ज्या (= व्यासार्ध) iii. १६ त्रैराशिक i. ४० दक्षिणापथ (= दक्षिणगोल) iii. १४ दशगुण (=== दशज्या) v. ७४ - दशजीवा (== दशगुण) vi. ५६ दस्र ( = २) vii. २६, २७ दिक् (==१०) iii. ६६; v. Y; vi. ४६ दिनगण viii. १८-२०, २४ दिनमान vi. २२, ३५ दिनराशि viii. १४, १७ दिवस (=== अहर्गण) i. २० दिवसगुणार्ध ( = ज्या) iii. दिवसजीवा (=ज्या) iii. ७ दिवसयोजन ( योजनात्मिका ग्रहगति) i. २० दिवसविस्तरभेद ( – युज्या) iii. ४० दिवागुण ( = ज्या) iii. १८ दिविचर (= ७) i. ३० दिश् (= १० ) iii. ६३ दृक्क्षेप v. १२, १६, २३, ३२ गुण V. २३ - ज्या V. २८ दृग्गति v. २४ - ज्या v. २३ दृग्गुण iii. ४३; v. २३ दृग्जीवा iii. ४७; v. २२ दृग्ज्या iii. ४०, ४७; v. २०; vi. १८ - कर्ण iii. ४८ दृढ़ i. ४१, ४६ दृश्यकाल vi. २६, २७ शब्दानुक्रमणिका ] दृश्यचन्द्र vi. ४ दृष्टकाल ii. ७ देशकाल ( = देशान्तरकाल) ii. 5 देशजातकाल (=देशान्तरकाल) ii. १० देशान्तर ii. १०; iv. १ -कर्म iv. १ देह (===ग्रहदेह) i. ३७ द्युगण i. ११; viii. २३ iii. २४ द्युज्या द्युति (छाया) iii. ४७, ५० -- कर्ण iii. ५० युदल ( = चुज्या ) iii. १२, २९ धुराशि ( = अहर्गण) i. १३ धुव्यास iii. २३ - खण्ड ( == धुज्या ) iii. ६ -भेद (= धुज्या) iii ३६ व्यासा iii. २७, २८ द्रष्टा ii. द्विराशित i. ४, ५, द्वयग्र viii. २४* धनु: खण्ड (२२५' का चाप) iv. १५ धनुष् ( २२५ ' का चाप ) iv. १४ धृति (== १८) viii. २, २२ ध्रुवक i. २६ नक्षत्र iii. ६२, ७५ –गण ( == भगण ) i. १० –तारा (२७ नक्षत्रों के तारे) iii. ६२ - भेद jii. ७५ (i) नग (==७) i. २५, ३६; iii. १०; vi. vii. १, १५; viii. २४* नति (= याम्योत्तर नतांश) iii. १४, १५ - ( नतिसंस्कृत चन्द्रविक्षेप) v. ३४, ४७, ६; ५४-५७ - (विक्षेपलम्बन ) v. ३१ नतिज्या (==नतांशज्या) iii. ५८, ५६ नतांश iii. १७ नभ ( = ० ) viii. २५ – ( याम्योत्तर वृत्त) viii. s नर ( = मिथुन राशि ) iii. ८ - ( = शङ्कु) iii. १, ५ - ( = उन्नतांशज्या ) iii. २८ ना (= शङ्कु) iii. ११ नाडिका iii. २८, २९, ४०, ६०; vi. ३६; viii. ४, २३ नाडी v. २०३८; vi. २२, ३५, ३८, ४०; viii. २३ निरक्षासु V 5, १० निशाकर ( = १) iii. ८; vii. ४ निश्चलक्रिया vi. ३७ निःश्वासलव iii. s नृ (= शङ्कु) iii. ४ नेत्र (= २) i. ३६; vii. ५, २७ पक्ष ( = २) ii. ३ पद (= वृत्तचतुर्थांश) iii. १६; iv. १, २, ५, ६, १६, २१, ३८, ४०, ४७, ५८; vi. ६; vii. १४ - (= वर्गमूल ) ii. ४; iii. ५, ६, २०- २२,३८, ४८; iv. १६; v. १६, २३, ३४,४० परक्रान्ति (= परमक्रान्ति) प्र. १३ परमापम iii. ३९ परिधि (मंदशीघ्र परिधि) iv. ३६३६ परिलेख vi. २५ पर्वत (= ७) iii. ६७ ; v. २ ; vii. १३ पल ( = अक्षांश ) iii. ६, १३, १५. १७, ५३, ६० ; v. १८ ; viii. ८, ९, ११ -गणित (= अक्षांश) ii. ६ -- जीवा vi. १ ६० – भाग iii. ११ पलाङ्गुल (=पलभाङ्गुल) iii. ११ पलांश (= अक्षांश ) viii. १२ पात i. ४० ; v. ३०; vii. ६, १०, २८, भव ( = ११ ) iii. ६६ २६, ३१ - भाग vi, ५२ ; vii. ६, २८ पुष्कर (= ३) ii. ३; पूर्वलग्न jii. ३३ पूर्वापरायत vi. १३ पौष्ण iii. ७३ iii. ६; vii. ६ पङ्क्ति (=१०) v. ४ पञ्चदशी vi. २१ प्रतिमण्डल iv. २३, ४५, ५२ -कर्म iv. ४५ प्रतिलोम vi. ४६ प्रभा ( =नतांशज्या) iii. ११, २०, २१, ३८ प्रस्तार i. ५० प्राग्रास V. ४३ प्राण (==असु) iii. २३, ३१, ३४, ३६ ; vi. २०, २८, २६, ३७, ३८ - राशि iii. १८ ; vi. ४ प्रोन्नति (=उन्नतांश) iii. ११ बa iv. ३३ [ महाभास्करीये भगण i. ८, ६, १९, २०, ४४, ४६, ५२ : vii. ३, २०, २२, ३५; viii. २३ - (= १२ राशि या ३६०°) i. ४० बहुला (=कृत्तिका) iii. ७४ बाहु (समकोण त्रिभुज का आधार) iii. ४७; iv. १०, १; vi. १३ - (१०°- कोटिचाप) iv: ८ ; vii. १e - ( भुजफल ) iv. २६ – क ( समकोण त्रिभुज का आधार ) vi. १२ - ज्या V. १३ - फल iv. ६. ७, १० बुधाशा (= उत्तरदिशा) iii. ३ भ (=२७) iii. ६३ भाग (==खण्ड) i. ३१, ३८, ४२; ii. ३; viii. ४ —( भाग देना) i. २५, ३१, ३५ - ( = अंश) i. ३२, ३३ ; iii. १४, ३०, ३२, ३४, ५६, ५७, ६८; v. ६, १०; vi. ३२, ५२; vii. ६, २८, ३०; viii. १४, १५, १७, २१ --शेष (= अंशशेष) viii. १५ भागहार i. ३२, ५०; v. ७० ; vi. ४८, ५२, ५८; vii. ६ भागहारक i. १५ भाज्य i. ४१, ४२, ४४, ४६, ५० भुक्ति ii. १०; iv. १३, १४, १६, १७, २४, २६, २९, ३३, ३४, ५७-६०, ६२-६४;v. ६२, ७५; vi. ४६, ५० भुंज ( समकोण त्रिभुज का आधार ) iii. ४८, ४६, ५१ –(=६०°–कोटि चाप) iii. ३७ - ज्या iv. २१ भुजा (क्षेत्र का भुज) vi. २४ - iii. ३६; iv. ३७; viii. २ भुजान्तर iv. २४ भुजाफल iv. ११, ३० ; viii. ५ : भूच्छाया V. ६९ — दैर्ध्य v. ७१, ७२ भूत (= ५) i. २३; iii. ६५, ६६; v. २, ६; vii. २४; viii. २४* भूदिन i. २०, ४१; vii. २२ भूधर (== ७) iv. ३१; viii. २२ भूभृत् (==७) i. ३५; vii. १ शब्दानुक्रमणिका ] भूमिदिन i. ७ भेद ( = आधा ) iv. ५६; v. २४ – ( = भेदयोग ) iii. ७४, ७५, (i) भोग (=== गति) iv. १८; v. ६; vi. ३४, ३६ मखि (==२२५') iv. ३, १५; vii. १६, १७; viii. ६ मघा iii. ७४ मण्डल (= वृत्त ) iii. ५२; v. ५१, ५६, ६५ –(== भगण ) i. १०; vii. ३५; viii. १६, २४ मण्डलमध्य (= वृत्तकेन्द्र ) iii. ५६ मति i. ४३ मत्स्य v. ५१, ५५ मधु (= चैत्र) i. ३० मध्य (===बीच ) vi. १४ - ( = ग्रहणमध्य ) v. ३६, ३७, ४३,५८ - ( = वृत्तकेन्द्र) iii. ५६ = (= मध्यम ) i. ४०; iv. २४, २६, ४०, ४८, ५३, ५६; viii. ५, २० - ( = मध्यलग्न ) v. १७ - क्रान्ति ( = मध्यलग्न - कान्ति) v. १७ —च्छाया iii. ४५,५१; vi. ४१ - जातः लम्बक: iii. २२ - ज्या iii. २२; v. १७-१६, २६ 12 -परिनिष्ठितलम्बक iii. २१ - बिन्दु V. ५७, ५६ :

--- लग्न ८, ११, १६; vi. ३६, ४१ मध्यम i. १२; ii. २; iv. ५, ४८, ५१, ५६; viii. १७ मध्यमार्क i. १२ मध्यसूर्यावनाम iii. ११ मध्या भुक्ति: iv. ७ मध्यार्क iv. २४ मध्यावनति v. ३२ मनु (= १४) vii. १३, २६ मन्द (==मन्दोच्च) iv. ४८, ५०, ५४; vii. १४, ३४ ~ (परिधि) vii. २७ – केन्द्र iv. ४० - फल iv. ४२ -मौर्वीफलचाप iv. ६० - वृत्त vii. १४, २६ - सिद्ध iv. ४८, ५०, ५१ - सिद्धि iv. ५० मन्दान्त्यजीवा iv. ५८ मन्दोच्च iv. ४४, ४५, ४८, ५३; vii. ११ १२, २६, ३४ -कर्ण vi. ४८ - केन्द्र iv. ६२ —वृत्त vii. १४ - सिद्ध iv. ४८ मन्दोच्चांश vii. ११ मान्दः कर्ण: iv. ५५ मिथुन iii. ६३; viii. २५ मीन ( = मत्स्यक्षेत्र ) iii. २; v. ५०, ६१ मुनि ( =७ ) i. २३; iii. ६ मूल (= वर्गमूल) iii. ४, ३६, ५३; iv १०; v. ६३

मृग (= मकर राशि ) iii. ६५ मेष iii. १०, ११, २७. मंत्र ( = अनुराधा ) iii. ७३; iv..३५ मोक्ष v. ३७-३६, ५३, ५७, ७५ -काल v. ३५ मौरिक V. ५३ यन्त्र iii. ५० यम ( = २ ) i. २३, ३२, ३४, ३७, ३६; iii. ८, ६५; vii. ५ ६२ यमल (==२) i. ३२; vii. ३ यात i. १०; iv. ३४, ६४; viii. १६ याम्य (===दक्षिणदिशा) iii. १४, ६८, ६६; v. ५७ -गोल iii. १५ याम्योत्तर ( = दक्षिण से उत्तर ) iii. २; vi. १३ याम्योत्तरायत vi. २४ युग i. ५, ६, १८; vii. ६, ८ —भूवासर viii. १ युगाघिमास i. ४, ११ युगार्कमास i. ५, १४ युगावम i. ५, १६ युग्म (==सम) iv. ३८ योग (= संपात ) iii. ३ - ( == भेदयोग ) vi. ५०, ६० - ( = संकलन ) i. ५१, ५२. iii. ४, ४२; iv १०, १६, ३५; v. १७, १८, २६; vii. ३२ -कारक vi. ५० योगभाग iii. ६६, ७० योजन ii. ३; v. ४, २४, vii. २३, ३५ --कर्ण v. २४, ७१ योजना ii. १० रन्ध्र ( = ६) i. २२, २३, २७, ३२, ३५; iii. ६, ८, १०, ६५; v. २, १४; १; vi. ५७; vii. ३, ४, ६, ८, ११, १३ रवि ( = १२) viii. ३ -कक्ष्या iii. २२ [ महाभास्करीये ६ ; v. २; vii. १, २, ५, ६, ३५ राशि ( = सङ्ख्या ) i. ४५, ४७, ४८ —i. २०, ३६, ४०, ४६; iii. ३० - ३५, ! ४१, ६४; iv. ५७, ५८; v. ११, ४५; vi. २७, ४६; vii. २६; viii. १३-१७, २० कला iv. ५८ — जीवा iii. रविज दिवस रस ( = ६) i. २८ २३, ३४; iii. १०, २८, ६; vii. १०, ११ ६७; vi. राम (= ३) i. १४, २२, २३, ३५, ३६; iii. राहु v. ७, १६; vii. ५ रुद्र (= ११ ) i. २२, २३, ३३, ३४; vii. ११, २५ रूप (=१) i. ४, १६, ४०, ४८, ५१; iii. १० रोहिणीशकट iii. ७२ लग्न iii. २१; v. १४; vi. १६, २९, ३० लघुतन्त्र (===लघुप्रकार) i. २१ लङ्का ii. १; vi.४० -राश्युदय vi. ४० लम्ब (=== लम्बज्या) ii. १०; iii. ६, १२ लम्बक (= लम्बज्या) iii. २३, २७, ३७, ५४, ६०; v. १३, १६; vi. १, ६, १० ; viii. २५ लम्बन v. २४, २६, ३७, ३८, ७०; vi. ५७, ६० लम्बनान्तर v. ३२, ३६ -नाडिका V. ३६ लव (अंश) i. ३४, ३७; iii. ११, ६१; viii. s लिप्ता i. ८, १५, ३८, ३९; iii. ५८, ६९, ७१; iv. ३१; v. २४; vi. ५६; viii. १४, १६-१८, २१ लिप्तिका i. ६, ३४, ३६; v. २५, २६, ४०, ४१; vi. ५४, ५८; viii. ८ -विपूर्वा (== विलिप्ता) i. ३४ शब्दानुक्रमणिका ] वक्र ii. ५; iv. ५६; vi. ४५ -गति iv. ५७ - -गमन iv. ५६ वक्रीग्रह iv. ६१ वर्ग i. ३२; iii. २८, ३८, ५३, iv. १०, १६; v. ६, १६, २४, ३४, ४०, ६३; vi. ३; viii. १४ विवर (==अन्तर) ii. ४; iii. २१, ४२; vi. ४-९, १६, २०, २७, ५४ विवरकलाग्र ii. ६ विश्व ( = १३ ) i. ३५; iii. ६९ विषमपद vii. १४ विषय (= ५) iii. १० वलन V. ४४, ४७, ५०, ५४ विषुवज्ज्या V. १७ वसु (=८) i. २२; v. ४१; vii ४, ५, ७, विषुवत् (= नाडीवृत्त) iii. ५ १५, २३, २४, २६ -- कर्ण iii. २४ -- प्रभा iii. ६० वर्षy i. २६ वह्नि (३) i. २२ वारुणी (===पश्चिम) vi. ४६ वि ( == विक्षेप) vi. २ विकला i. ८, १५, ३८; viii. १४, १८ विकाष्ठ (= विक्षेपचाप ) vi विक्षिप्ति iii. ७३ विक्षेप iv. ३६; v. १५, १८, २१, ३१, ७६, ७७; vi. ५२-५४; vii. १०, ३२ ६३ विलिप्तिका i. ३३, ३६, ३७; v. ७; viii. १७ -कला V. ६६ विमौरिक (=विकला) i. ३५ वियत् (= ० ) vii. २० विलग्न v. १४, १५; vi. २३, २६ विलिप्ता i. १८, ३२; v. ६; viii. २२ विषुवदुदयराशिप्राणपिण्ड iii. c विष्कम्भभेद ( = त्रिज्या) iii. २१ विष्कम्भार्ध iii. ४, ६; iv. १३, २०, ४१; v. ३, ५, १६, ४२, ४७, ७३; vi. ४८, ५८ विष्णुक्रम ( = ३) vii. ५,११, १६ : विहायस् (=०) iii. ६९ वृन्द (==घन) i. २६ —ज्या V. १४, १६ विक्षेपांश iii. ७०, ७१; vii. ८, ३१ वृष iii.६३ विघटिका i. ३१; iii. ३६; iv. ३२; vi. वेद (= ४) i. २३, २६; iii. ६४; v. २, ४६ ४; vii. २, ३५ विदिश v. २१ विपरीतगुण V. ४५ विपुलच्छाया iii. ५ विपुलनर iii. ५ विमण्डल vi. ३ विमर्धि V. ४०, ६६, ७६ वैलाकुट्ट i. ४६ वैधृत iv. ३५ वैषुवती छाया iii. ५५ व्यतीपात iv. ३५, ३६ व्यास v. ७१; vii. २३. - ( == त्रिज्या) iii. २०३८; vi. ३ —खण्ड iii. ७ -खण्डनिचय iii. २० - लिप्ता vi. ५६ व्यासार्ध iii. ६, २९, ४०, ५३; iv. ६, १०; v. १६, ४३; vi. ५८ ६४ व्योम (===०) i. २३, ३६; vii. ७, ८ शकतारक ( == ज्येष्ठा) iii. ७२ शङ्क iii. २, १२, २०, २१, २३, २७, २६, ३८, ४७, ४६, ५४, ५५; vi. ६, १२, १६, ५६; viii. ७, ६ शङ्क्वग्र iii. ४२, ४७, ५४, ५५, ५६. - जीवा iii. ५८ शतभिषक् iii. ७३ शृङ्गनति vi. १७ शृङ्गसाम्य vi १७ शृङ्गोन्नति vi. १७, ४२ शफरिका (= मत्स्य) iii. ३, ५२ शर ( = ५ ) i. ३१, ३२, ३७, ३९; vii. शैल ( = ७ ) i. २५; iii. ६४ १३, १५ शोधनीय i. २८, ३०, ३४ शोध्य i. ३० शौक्य vi. १६ संवर्ग (= गुणन) v. २४ संस्कृत iv. ५३, ६२; vii. ३४ शशाङ्क (=१) iii. १० —मास i. ७ शशि ( == १ ) iii. ६, ६; viii. २१ — कीलक vi. c -मास i. १५

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वत्सर viii. १

-- वासर i. १६; viii. ३ -शकु vi. १२ शाली (== १) iii. ८ शिखि ( = ३ ) iii. १०; viii. २४* शिलीमुख ( = ५ ) vii. १, १५ शिव (= ११ ) i. ३२ शीघ्र (= शीघ्रोच्च ) iv. ५६, ६१, ६२; vii. १२, १४, ३०, ३४ - कर्ण iv. ५५; vi. ४८ [महाभास्करीये शीघ्रोच्च iv. ४५, ४६, ५३, ५७, ५६; vi. ५३; vii. ३, २८ - वृत्त vii. १४ शीतकिरण (= १) viii. २४* शीतरश्मि ( = १ ) vii. २ = शीतांशु (= १) vii. c = - केन्द्र iv. ४१ - केन्द्रफल iv. ४ - ज कर्ण iv. ५५ -न्यायाप्तचाप ( = शीघ्रफल ) iv. ४४ - पात vii. ३१ - वृत्त vii. २६ सिद्ध iv. ५१ शीघ्रान्त्यजीवा iv. ६२ सकलगुण ( = त्रिज्या) ii. १०; iii. २७ सकृत् iv. ५५ -- सिद्ध iv. ४६ सङ्कलित iv. ४; viii ६ सन्नति (= मध्यनतांश) iii. १३ सम (= युग्म) vii. १४ सममण्डल iii. ३८, ४०; viii. १०, ११ - जा छाया viii. ११ - शङ्कु iii. ४१ समरेखा ii. c समलिप्त v. २१, ७५; समवलम्बज्या iii. ३७ vi. ५१, ५२ समा (=वर्ष) i. ६, १३, २७, २८, ३० समाध्वनोविधान ii. ७ सम्पर्क (=सूर्यचन्द्रव्यासयोग) v. ३३, ३४, ४६ सर्वापम iii. ४१ सागर (==४) v. ४; vii. २, १०; viii. २१ शब्दानुक्रमणिका ] सार्पमस्तक (व्यतीपात) iv. ३५ सित ( = चन्द्रशुक्ल ) vi. ६, ७, २६ ( = शुक्लपक्ष ) i. ३० -पक्ष iv. ३३; vi. २७ - बिन्दु vi. १५ - - मान vi. २५ सुरपदिक् (=== पूर्वदिशा) ii. १० सूर्य (= १२) viii. ४ सूर्याग्र ( = सूर्य की अग्रा) iii. ५३ सैहिकेय V. १४ सौम्य (= उत्तर ) V.५७ स्थिति (== ग्रहणस्थिति) v. ३७ स्थितिदल ( = स्थित्यर्ध) ii. ७ स्थित्यर्ध V. ३६-३६, ६२, ६६, ७६; vi. ६० --काल V. ७४, ७५ -घटिका v. ३४ - नाडी v. ३८ स्थूल (= आसन्न) vi. ५० स्पर्श ii. c; v. ३८, ५३, ५७, ७५ --काल V. ४१ स्पष्ट i. ५२; v. ११, ३१, ३७, ५६; viii. २३ ii. ६, ८; iii. १३-१५, २१, ४६; iv. ६, १३, १४, २२, २४, ३१, स्फुट ३६, ४३, ५४, ५६, ६२, ६३; v. ६, १३, ३१, ३२; vi. १०, १२,५८ vii. ३२; viii. ४, ६ -खेचर iv. ५७ -ग्रह iv. ५१ —– तर ii. ७; iii. १, २० : iv. १७ — भुक्ति iv. १३, ६० -भुक्त्यन्तर V. २५ —भोग viii. ४ -मध्य iv. ४३, ४४; vii. ३४ - मध्यम iv. ४२ योजन (कर्ण) v. ५ -योजनकर्ण v. ३, २८ -वृत्त iv. ४७ स्वर (==७) iii. ६८ हरिज (= क्षितिज ) vi. २३ हार i. ४१, ४२, ४४, ४६; vi. ५७, ६०; vii. ७ -राशि i. ४७ हीन ( == अवमदिन) i. ३० (== न्यून) iv. ३६, ४८ - दिवस i. २५ ---रात्र i. २६ हुताशन (= ३) v. ४ महाभास्करीये प्रयुक्त- छन्दसाम् अनुक्रमणिका अनुष्टुभ् (श्लोक ) i. ४१, ४५, ४६; ii. ९; iii. ४, १४, १७, २२, २५, २६, २६-३६, ३८, ५४, ५५, ५७-५६, ६२-७५(i); iv. १-५५; v. १-७३, ७५-७७; vi. १३-६२; vii. १-१६, २१-३४; viii. १५, २२, २७ आर्या i. २६, ३१, ३३, ३८, ३६, ५०; iii. २, १२, २४, ४२-४५, ४७-५१, ५३; iv. ५६; viii. ५, ६, ११, १२ इन्द्रवज्रा i. १७, २१, २४, ४३, ४४. ५१, ५२ ; iv. ५८६३; viii. ७-१०, १६ प्रहर्षिणी ii. १-८; iii. ४६ मालभारिणी i. ४८; viii. १-४, १७ मालिनी i. २८, ३०, ३२, ३४; iii. &, १०, ११, ३७ रथोद्धता i. २०, २६; iii. ३, ७, १३, १५, १६, १८-२०, ६०, ६१; v. ७८; vi. १-१२ वंशस्थ i. १ - १६, १८, १९; iii. ८ वसन्ततिलका i. २२, २३, २५, २७, ३७, ४०, ४२, ४६, ४७; iii. १, ५, २१, २३, ३९-४१, ५२, ५६; iv. ६४; v. ७४; vii. २०, ३५; viii. १३ शार्दूलविक्रीडित i. ३५; iii. ६; iv. ५७; viii. २१, २३, २४, २४* श्थेनिका i. ३६ स्रग्धरा ii. १०; iii. २७, २८ स्वागता viii. १४, १८-२०, २५, २६ English Translation OF THE MAHĀ-BHASKARĪYA CHAPTER I MEAN LONGITUDE OF A PLANET AND PLANETARY PULVERISER Homage to God Śiva : 1. I bow to God Sambhu who bears on His forehead a digit of the Moon illumining all directions by its rays, to Him whose feet are adored by the gods and who is the source of all knowledge. Homage to planets and stars : 2. Glorious are the rays of the Sun lotus blossom forth, (and those) of the Moon which make the whose beauty is like that of the damsel's face, (as also are) the long and clear rays of the stars including Jupiter; so also is the lustre of Mars, Mercury, Saturn, and Venus. A benedictory stanza in appreciation of the Aryabhatiya and the pupils of Āryabhața I: 3. May the accurate Asmaka-tantra (asmaka m sphuta tantram), which has been acquired by penance, live long in the world for its excellent qualities. May also the pupils of (Arya)- bhata, who are free from sins and have conquered the enemies of passions, live long. The aśmakam sphuţa-tantram ("the accurate Asmaka-tantra"), accor- ding to the commentators Govinda Svāmi and Parameśvara, is the Ārya- bhatiya of Aryabhața I (born 476 A.D.). Govinda Svāmi writes : "By this (stanza) the author exhibits the greatness of the Arya- bhatiya, So also has said Parameśvara : "tapobhiḥ etc. is a benediction on the Aryabhaṭa-tantra and the pupils of Aryabhata." Exclusive references to Aryabhața I and his work at several places in the present work and the name Aryabhata-karma-nibandha ("a compen2 MEAN LONGITUDE OF A PLANET dium of the astronomical processes taught by Aryabhata") given to this work by the author¹ indeed show that aśmakaṁ sphuta-tantram is none else than the Aryabhatiya. The word aśmaka (aśmaka+an) literally means "pertaining to Aśmaka", and likewise āśmakam sphuta-tantram means "an accurate work on astronomy written, studied, or venerated in Ašmaka, or belonging to Aśmaka". This seems to suggest that Aryabhata I, the author of that work, belonged to the Aśmaka coun try.2 It is noteworthy in this connec- tion that according to Nilakantha (1500 A.D.) he was born in that country.s Reference to the Aryabhatiya in the above stanza at the beginning of this work is meant, as stated in the Prayoga-racanã and by Govinda Svāmi to indicate the school to which the present work belongs. MEAN LONGITUDE OF A PLANET A rule for calculating the ahargana: 4-6. Add 3179 to the number of elapsed years of Saka kings; then multiply (that sum) by 12; and then add the num- ber of months elapsed (since the beginning of Caitra). Put down the result at two places. At one place multiply (that) by the number of intercalary months in a yuga and divide by the number of solar months in a yuga; and add the resulting inter- calary months (omitting the fraction of a month) to the result put at the other place. Multiply tha tsum by 30 and then add the number of lunar days (tithis) elapsed (since the beginning of the current month). Set down the result (i.e., the sum obtained) at two places. At one place multiply that by the number of omitted lunar days (in a yuga) and divide by the number of lunar days (in a yuga), and subtract the resulting omitted lunar days (neglecting the fraction of a day) from the result set down at the other place. The result (thus obtained) is the number of (mean) civil days elapsed since the beginning of Kali- yuga (at mean sunrise at Lankā on the given lunar day). These ¹ See vs. 26 of Chapter VIII.

• For the Aśmaka country, see Part 1, Chapter 2.
• See Nilakantha's comm, on Ã, ii. 1, AHARGANA

3 days are said to have commenced with Friday and at sunrise (at Lankā).¹ The above rule tells us how to determine the number of mean civil days elapsed at mean sunrise at Lankā on the given lunar day (tithi) since the beginning of Kaliyuga (when all the planets were in conjunction at the first point of the asterism Aśvini). According to Aryabhata I and Bhaskara I, the duration of Kaliyuga is 10,80,000 years and it began on Friday, February 18,, B.C. 3102, at sunrise at Lanka in the beginning of the month Caitra. The Saka Era, which is usually used in Hindu astronomy for reckoning the years, commenced 3179 years after the beginning of Kaliyuga. Caitra is the first month of the year. The months in Hindu astro- nomy are reckoned from one new moon to the next. The yuga is a period of 43,20,000 years. At the beginning and end of a yuga the planets together with the Moon's apogee and ascending node are supposed to be in general conjunction. The number of intercalary months in a yuga denotes the excess of the number of lunar months in a yuga over the number of solar months in a yuga. The number of omitted lunar days in a yuga is equal to the number of lunar days in a yuga minus the number of civil days in a yuga. The number of intercalary months, omitted lunar days, etc., in a yuga are given in the seventh chapter³. In the rule stated above, the given true lunar month is treated as a mean lunar month and the given true lunar day is treated as a mean lunar day. The following is an explanation of the above rule: K A 3 L S s A Fig. 1 Let K denote the beginning of Kaliyuga; AA', the current mean lunar month; L, the beginning of the current mean lunar day; and s, the mean sunrise on that day. Also let S denote the beginning of the corres- ponding mean solar month; and s', the beginning of the mean solar day ¹ The same rule occurs also in LBh, i. 4-8 and TS, i. 23-26 (i). For other similar rules, see SuSi, i. 47-50; BrSp Si, i. 29-30, 34 (i); ŠiDV, I, i. 15-17; MSi, i. 21-22; Sise, ii. 15-17; Sisi, I, i (c), -3; SiSä, i. 44-47. Lankā is a hypothetical place where the Hindu prime meridian ("the meridian of Ujjain") intersects the equator. 3 See verses 1-8.

• That is to say, if A is the beginning of the tth mean lunar month,

then S is the beginning of the tth mean solar month. 4 MEAN LONGITUDE OF A PLANET corresponding to the current mean lunar day, so that the number of mean solar days between S and s' is equal to the number of mean lunar days between A and L. Adding 3179 to the number of elapsed years of the Saka Era, we. obtain the number of solar years elapsed since the beginning of Kaliyuga. Multiplying that sum by 12 and adding to the product the number of months elapsed since the beginning of Caitra, we get the number of mean solar months elapsed since the beginning of Kaliyuga. This number is equal to the number of mean solar months lying between K and S. (See Fig. 1.) Let us denote it by M. When we multiply M by the number of intercalary months in a yuga and divide by the number of solar months in a yuga, we obtain the number of mean intercalary months corresponding to M mean solar months. This number is, in general, made up of a whole number and a fraction. The fraction evidently denotes the fraction of a mean lunar month lying between A and S.¹ When the whole number of mean intercalary months is added to M, we get the number of mean lunar • months lying between K and A. When we multiply that by 30 and add to the product the number of lunar days (tithis) elapsed since the beginning of the current lunar month, we get the number of mean lunar days lying between K and L. Let us denote this number by T. When we multiply T by the number of omitted lunar days in a yuga and divide by the num- ber of lunar days in a yuga, we get the number of mean omitted lunar days corresponding to T mean lunar days. This also, in general, consists of a whole number and a fraction. The fraction evidently denotes the part of a mean civil day lying between L and s.³ When the whole number of mean omitted lunar days is subtracted from T, we get the number of mean civil days lying between K and s. This is, in general, the number of mean civil days elapsed at mean sunrise at Lanka on the given lunar day since the beginning of Kaliyuga. This. number, is known as "ahargana" (literally meaning "a collection of days"). In the above passage, Bhaskara I has called it by the synonym ahnam nicayaḥ. The mean lunar day (madhyama-tithi) may, however, differ from a true lunar day (spasta-tithi) by one, so that the ahargana obtained by the 1 See Sisi, II, iv. 16. 2 If we add the whole number of mean intercalary months as also the fraction, we shall get the mean lunar days lying between K and S. 3 See Sisi, II, iv. 18 (i). 4 If we subtract the whole number of mean omitted lunar days as also the fraction, we shall get the mean civil days lying between K and L. AHARGANA 5. above process may sometimes be in excess or defect by one. To test whether the ahargana (obtained by the above process) is correct, it is divided by seven and the remainder counted with Friday. If this leads to the day of calculation, the ahargana is correct; if that leads to the preceding day, the ahargana is in defect; and if that leads to the succeeding day, the ahargana is in excess. When the ahargana is found to be in defect, it is increased by one; when it is found to be in excess, it is diminished by one. Similarly, when a true intercalary month has recently occurred prior to the given lunar month or is about to occur thereafter, the true lunar month may differ from the mean lunar month by one. When a true intercalary month has occurred prior to the given month and the intercalary fraction amounts to one month approximately, then the whole number of mean intercalary months obtained in the above process is increased by one.¹ When no intercalary month has occurred prior to the given month but the intercalary fraction is small enough, the whole number of mean intercalary months obtained in the above process in diminished by one.² The word tithi in verse 4 stands for 30. It is generally used to denote the number 15. An alternative rule: 7. Or, multiply the number of (solar) months elapsed (since the beginning of Kaliyuga) by the number of lunar months (in a yuga) and divide by the number of solar months (in a yuga). Reduce the quotient to days (and add the number of lunar days elapsed since the beginning of the current lunar month); then multiply by the number of civil days (in a yuga) and divide by the number of lunar days (in a yuga); the quotient denotes the ahargana.³ The number of solar months elapsed since the beginning of Kaliyuga is equal to the number of mean solar months lying between K and S. 1 "If the current lunar month is preceded by an intercalary month, then that intercalary month should also be treated as a lunar month." (Parameśvara). ² Cf. Siśi, I, i (f). 3. ³ The same rule occurs also in BrSp.Si, xiii. 18 and Sise, ii. 3. 6 MBAN LONGITUDE OF A PLANET (See Fig. 1.) Let us denote this number by M, as before. When we multiply M by the number of lunar months in a yuga, and divide by the number of solar months in a yuga we obtain the number of mean lunar months corresponding to M mean solar days. This number is in general, made up of a whole number and a fraction. The whole, number, which is the quotient of the division, denotes the number of mean lunar months lying between K and A. Multiplying this by 30 and adding to it the number of lunar days elapsed since the beginning of the curren month, we get the number of mean lunar days lying between K and L. Let us denote this number by T as before. When we multiply T by the number civil days in a yuga and divide by the number of lunar days in a yuga, we get the number of mean civil days corresponding to T mean lunar days. If this number be a whole number, then it denotes the number of mean civil days lying between K and s and is therefore the ahargana. (This case will occur when L and s coincide). If that number is made up of a whole number and a fraction, then the whole number as increased by one will be the ahargana. This addition of one is not mentioned in the above stanza. It is stated later in verse 40.¹ Statement of the proportion used in finding the mean longitude of a planet: 8. If from the civil days (corresponding to a yuga) we get the tabulated revolutions of a planet, how many of those (revo- lutions) will we get from the civil days elapsed since the begin- ning of Kaliyuga ? Thus (i.e., by applying this proportion) are obtained the revolutions (performed by the planet), and then successively the signs, degrees, minutes, seconds, and thirds (of the planet's mean longitude).³ 1 "This ahargana has sometimes to be increased by one as the author will say later." (Parameśvara). So also says Govinda Svāmi. See also Br Sp Si, xiii. 18 and Sise, ii. 3. 2 Cf. Sūsi, i. 53; BrSp Si, i. 31; LBh, i. 15-17 (i); ŠiDVṛ, I, i. 21 (i); MSi, i. 25 (i); Sise, ii. 14; Siśi, I, i (c). 4; SiSā, i. 53. DERIVATION OF MEAN LONGITUDE That is to say, mean longitude of a planet in revolutions (revolution-number of the planet) × (ahargana) civil days in a yuga A rule for deriving the mean longitude of a planet from that of the Sun: 9. Reduce the Sun's mean longitude (given in terms of signs, degrees, and minutes) together with the years elapsed (treated as revolutions) to minutes of arc. Multiply them by the planet's own revolution-number stated in the Gitika¹ and divide (the product) by the number of (solar) years in a yuga. The result, say (the learned), is the planet's mean longitude in terms of minutes. That is to say, mean longitude of a planet in terms of minutes · 7 (Sun's mean longitude in revolutions etc. reduced to minutes) > (planet's revolution-number) Sun's revolution-number A rule for deriving the mean longitude of a planet from the mean longitude of the Moon or a planet or the ucca of a planet. 10. The (mean) longitude of the Moon, the planet, or the ucca (whichever is known) together with the revolutions per- formed should be reduced to minutes. The resulting minutes should then be multiplied by the revolution-number of the desired planet and (the product obtained should be) divided by the revolution-number of that (known) planet. The result is (the mean longitude of the desired planet) in terms of minutes.² ¹ This is the name of the first chapter of the Aryabhatiya. This rule occurs also in BrSpSi, xiii. 27; ŚiDVṛ, I, i. 30 (i); Siśe, ii, 25-26: SiŚi, I, i (c). 14 (i). MEAN LONGITUDE OF A PLANET That is to say mean longitude of the desired planet in minutes (mean longitude of the known planet in revolutions, etc., reduced to minutes) x (revolution-number of the desired planet) revolution-number of the known planet This rule and the previous one are based on the following principle. If there are two planets P and g, then revolution-number of P: revolution-number of g

(mean longitude of Pin revolutions etc.) : (mean longitude of g in

revolutions etc.). Alternative rules for deriving the mean longitude of the Moon from that of the Sun and vice vers: Il. Or, multiply the chargeta by the number of intercalary months in a ५ge am divide (the product) by the number of civil days (in a yugg) : bhe result is in terms of revolutions, etc. Add that to tbhirteen times the mean longitude of the Sun. (This is the process) to obtain the mean longitude of the Moon. 12. Or, subtract the result obtained (in revolutions etc.) from the mean longitude of the Moon and take one-thirteenth of the remainder : this is stated to be the mean longitude of the Sun by the mathematicians whose intellect has been awakened by the grace of the teacher. The following is the rationale : We know that intercalary months in a yuga = lunar months in a yuga – solar months in a yuga. But lunar months in a yuga = Moon's revolution-number – Sun's revolution-number; and solar months in a yuga = 12 (Sun's revolution number). • This rule is found also in BrspS, xi. 33; iDV१, , i. 24 (i)Sise ii. 19. MEAN LONGITUDE WITHOUT AHARGANA Therefore, we have intercalary months in a yuga = Moon's revolution-number 13 (Sun's revolution-number), giving Moon's revolution-number = intercalary months in a yuga +-13 (Sun's revolution-number). Multiplying the two sides of this equation by the ahargana and dividing by the number of civil days in a yuga, we get mean longitude of the Moon (intercalary months a yuga) × (ahargana) revolutions civil days in a yuga + 13 (Sun's mean longitude). And rearranging this equation, we have mean longitude of the Sun - (1/13) 9 (intercalary months in a yuga) x ahargana civil days in a yuga { mean longitude of the Moon revolutions (1) (2) A rule for calculating the mean longitudes of the Sun and the Moon without making use of the ahargana: 13-19. For one (desirous of) calculating the mean longitudes of the Moon and the Sun without the use of the ahargana, the following method is stated: Reduce the years (elapsed since the beginning of Kaliyuga) to months, and add to them the elapsed months (of the current year). Then multiply that (sum) by 30, and add the product to the number of (lunar) days elapsed since the beginning of the current month. Multiply that (sum) by the number of intercalary months (in a yuga) and divide by the number of solar months in a yuga reduced to days: the quotient denotes the number of intercalary months (elapsed). Delete (or rub out) the divisor 10 MEAN LONGITUDE OF A PLANET and divide the remainder (called adhimāsaseṣa, i.e., the residue of the intercalary months) by the number of lunar months (in a yuga): thus are obtained degrees, minutes, seconds, and thirds. Then multiply the (complete) intercalary months elapsed by 30 and to the product add the number of solar days (elapsed since the beginning of Kaliyuga); then multiply that (sum) by the number of omitted lunar days in a yuga and divide by the number of lunar days (in a yuga): the remainder obtained is (the avamašeşa, i.e., the residue of the omitted lunar days) called ähnika. Then multiply the avamasesa by the number of inter- calary months (in a yuga) and divide by the number of civil days (in a yuga). Add the resulting quotient to the adhimāsaseṣa and then apply the process stated above (i.e., divide by the number of lunar months in a yuga: the result is in degrees, minutes, etc. This is the total adhimasasesa). Next multiply. the avamasesa called ähnika by 60 and divide by the number of civil days in a yuga: the result is in minutes, seconds, and thirds respectively. The number of months elapsed (since the beginning of Caitra) are to be taken as signs, and the number of lunar days elapsed (of the current month) as degrees. (The sum of these signs and degrees and the minutes, seconds, etc. correspondi to the avamasesa is the grahatanu). From thirteen times and from one time that (grahatanu) severally subtract the degrees, minutes, etc. corresponding to the (total) adhimasašeşa: the remainders (thus obtained) are stated by the wise astronomers to be the mean longitudes of the Moon and and Sun (respectively) conforming to the teachings of (Ārya)bhaṭa.¹ The process described in the above rule is not in proper sequence. The direction given in verse 15 ought to have been after verse 17. Stated in proper sequence, the rule would be: ¹ This rule occurs also in BrSpSi, xiii. 20-22; KK (Sengupta), i. 11-12; and Sife, ii. 21-22. For similar rules, see SiDV, I, i. 27, 25-26; and SiSi, I, i (c). 6-7, MEAN LONGITUDE WITHOUT AHARGANA 11 "Reduce the years (elapsed since the beginning of Kaliyuga) to months, and add to them the elapsed months (of the current year). Then multiply the sum by 30, and add the product to the number of (lunar) days elapsed since the beginning of the current month. Multiply that sum by the number of intercalary months (in a yuga) and divide by the number of solar months in a yuga reduced to days: the quotient denotes the number of intercalary months (elapsed). (The remainder is the adhimasaseṣa). Multiply the (complete) intercalary months (thus obtained) by 30 and to the product add the number of solar days (elapsed since the beginning of Kaliyuga) ¹; then multiply that (sum) by the number of omitted lunar days in a yuga and divide by the number of lunar days (in a yuga); the remainder obtained is (the avamaseșa) called āhnika. Then multiply the avamasesa (called ähnika) by the number of intercalary months (in a yuga) and divide by the number of civil days (in a yuga). Add the resulting quotient to the adhimasašesa and divide the sum by the number of lunar months in a yuga: this gives degrees, etc. (This is the total adhimāsasesa.) Next multiply (again) the avamaseșa called ähnika by 60 and divide by the number of civil days in a yuga : the result is in minutes, seconds, and thirds, etc. The number of months elapsed (since the beginning of Caitra) are to be taken as signs and the number of lunar days elapsed (of the current month) as degrees. (The sum of these signs and degrees, and the minutes, seconds, etc. corresponding to the avamašeșa is the grahatanu). From thirteen times and from one time that (grahatanu) severally subtract the degrees, minutes, etc. corresponding to the (total) adhimasaseṣa: the remainders (thus obtained) are stated by the wise astronomers to be the mean longitudes of the Moon and the Sun (respectively) conforming to the teachings of (Arya)bhata." The following is the rationale of the above rule: The fraction of the intercalary month (obtained in the rule) adhimāsasesa solar days in a yuga › in mean lunar months. ¹ By the number of solar days here is meant the number obtained above by 'reducing the years elapsed since the beginning of Kaliyuga to months, then adding to them the number of months elapsed since the beginning of the current year, then multiplying the sum by 30, and then adding to the product thus obtained the number of lunar days elapsed of the current month'. 12 MEAN LONGITUDE OF A PLANET adhimāsaseṣa lunar days in a yuga The fraction of the omitted lunar day (obtained in the rule) avamasesa or ähnika lunar days in a yuga = avamasesa civil days in a yuga avamasesa × 60 civil days in a yuga' in mean solar months. in mean civil days. in mean lunar days. in mean lunar ghatis. The fraction of the intercalary month corresponding to the above fraction of the omitted lunar day (intercalary months in a yuga) × (avamasesa) (lunar days in a yuga)×(civil days in a yuga)' in mean solar months. (3) Adding (1) and (3) and multiplying by 30, the total fraction of the intercalary month adhimāsasesa lunar months in a yuga + (intercalary months in a yuga) x (avamasesa) (lunar months in a yuga) × (civil days in a yuga) in mean solar days. (4) Suppose that m lunar months and d lunar days have elapsed since the beginning of Caitra. Then, treating them as mean lunar months and mean lunar days, m months and d days denote the time elapsed since the beginning of mean Caitra up to the beginning of the current lunar day (treated as mean lunar day). As (2) is the interval, in mean lunar ghatis, between the beginning of the current lunar day and the mean sunrise on that day, therefore m months + d days + (2) MEAN LONGITUDE WITHOUT AHARGANA denotes the time in mean lunar months, days, and ghatis¹ elapsed since the beginning of mean Caitra up to the mean sunrise on the current lunar day. Likewise m months + d days + (2)-(4) denotes the time in mean solar months, days, ghatis, etc. elapsed since the beginning of the curent mean solar year up to the mean sunrise on the current lunar day. 2 Let M, D, G, V, and P denote respectively the mean solar months, mean solar days, mean solar ghatis, mean solar vighatis, and mean solar pravighatis elapsed since the beginning of the current mean solar year up to the mean sunrise on the current lunar day. Then evidently mean longitude of the Sun and mean longitude of the Moon because = M signs, D degrees, G minutes, V seconds, and P thirds. (m signs and d degrees) + [minutes, seconds, etc. corresponding to (2)]-[degrees, minutes, etc. corresponding to (4)]; 13 [m signs and d degrees + (minutes, seconds, etc. corresponding to (2))]- [degrees, minutes, etc. corresponding to (4) ], (1/12) (mean longitude of the Moon - mean longitude of the Sun) = m signs + d degrees + [minutes, seconds, etc. corresponding to (2) ].³ 13 11 hour 1 ghati 1 vighati — = 21 ghatis, 60 vighatis, 60 pravighatis. 2 Because (4) is equal to { fraction of a lunar month between the beginning of Caitra and the beginning of the current mean solar year} + {fraction of an intercalary month corresponding to the tithis elapsed up to the beginning of the current mean lunar day since the beginning of Caitra}+{fraction of an intercalary month corresponding to the avamaseșa, i.e, the lunar portion between the beginning of the current lunar date and the following sunrise }. 3 This equality is based on the fact that the left hand side denotes the mean lunar date (madhyama-tithi). Vide infra, iv. 31. 14 MEAN LONGITUDE OF A PLANET It must be noted that, unless otherwise stated, the mean lunar day and the mean sunrise, etc., correspond to Lanka¹, and that the mean longitude of a planet' means 'the mean longitude for mean sunrise at Lankä'. Sripati gives in addition to the above rule, the following two interes- ting rules also. It may be pointed out that Sripati uses a period of 4,32,00,00,000 years called a kalpa in place of a yuga. Rule 1. Moon's longitude - Sun's longitude + 12 { Sun's longitude = Moon's longitude avamasesa beginning of the current tithi+ civil days in a kalpa -} beginning of the current tithi + Rule 2. Sun's daily motion in degrees) x, tithis beginning of the current tithi) degrees. avamasesa `lunar days in a kalpa Sun's longitude = Moon's longitude 12{₁ 12 tithis elapsed up to the 1 Vide supra, p. 3 (footnote). avamasesa civil days in a kalpa Moon's longitude === Sun's longitude + 12 { beginning of the current tithi) degrees + (Moon's daily motion in degrees- elapsed up to the 12 degrees. {"

}

Cf. Sise, ii, 20. (tithis elapsed up to the daily motion in degrees - Sun's daily motion in degrees degrees. (tithis elapsed up to the avamašeṣa junar days in a kalpa X (Moon's Cf. Sise, ii. 24. Proof:-- + Moon's longitude at sunrise-Sun's longitude at sunrise MEAN LONGITUDE BY ORBITAL METHOD avamasesa ponding to lunar days in a kalpa = tithi at the beginning of the current tithi Moon's daily motion in degrees -Sun's daily motion in degrees 12 12 = tithi at sunrise = tithi at the beginning of the current tithi + tithi corres- = because tithi corresponding to 1 civil day Moon's daily motion in degrees X From stanza 8 above, we have mean longitude of a planet But from vii. 20 avamašesa lunar days in a kalpa Another rule for finding the mean longitude of a planet : 20. Divide the (yojanas of the circumference of the sky by the number of civil days (in a yuga): the result is the number of yojanas traversed (by a planet) per day. By those (yojanas) multiply the ahargana and then divide (the product) by the length (in yojanas) of the own orbit of the planet. From that are obtained the revolutions, signs, etc. (of the mean longitude of the planet). length of a planet's orbit, (revolution-number of the planet) x ( ahargana). civil days in a yuga 13 Sun's daily motion in degrees 12° circumference of the sky¹ revolution-number of the planet ¹ For the length of the circumference of the sky, see vii, 20, 16 so that MBAN LONGITUDE OF A PLANET revolution-number of a planet circumference of the sky length of the planet's orbit Hence we have mean longitude of a planet (circumference of the sky) X ahargana (civil days in a yuga) x (length of the planet's orbit). Introduction to the topic discussed in the succeeding eighteen stanzas: 21. After a careful study of the ocean of the Asmakiya sāstras (sāstrārnavam aśmakiyam) I reveal the planetary proce- dure, the secret there, (hitherto) unnoticed by the other follow- ers of the Asmakiya (aśmakiyaiḥ) by means of simplified rules (laghu-tantra). Asmakiyam (literally meaning a book written by one born in or belonging to the Aśmaka country') refers to the Aśmaka-tantra (i.e., Aryabhatiya) mentioned in stanza 3 above. āśmakiyāḥ means "the follo- wers of the Asmakiya", or, as the commentator Parameśvara says, "the pupils (or followers) of Aryabhata I”¹. The stanza under consideration shows that Bhäskara I, the author of the present work, was the earliest Aśmakiya ("follower of Aryabhata I") to give the method of the pratyabda-suddhi stated in the next eighteen stanzas. The method was, however, not invented by him. In his commen- tary on the Aryabhatiya he himself writes that it was already in use amongst the followers of the Romaka-siddhanta. It occurs in the Brahma- sphuta-siddhanta also. ¹ "āśmakiya āryabhaṭaśiṣyāḥ” (Parameśvara) 2 See comm, on A, iii, 10. SIMPLIFIED RULES 17 A rule for determining the number of mean lunar days lying between the beginning of mean Caitra and the beginning of the mean solar year : 22. Always having ascertained the number of; years (elap- sed since the beginning of Kaliyuga), multiply them by 11 and by 389/6000 (separately). Add the two results, and divide the sum (thus obtained) by 30. The quotient of division denotes (the mean intercalary) months, and the remainder (the mean inter- calary) days. The mean intercalary days obtained from this rule are equal to the number of mean lunar days lying between the beginning of mean Caitra and the beginning of the mean solar year. The number of mean intercalary days in a mean solar year is equal to 11+ 389/6000. Hence the above rule. A rule for finding the number of mean lunar days elap- sed at the beginning of the mean solar year since the occurrence of a mean omitted lunar day : 23. Multiply (the number of years elapsed since the beginning of Kaliyuga) by 29 as divided by 36 (i.e., by 29/36). Again multiply the same (number of years) by 43 and divide by 72000. The sum of the two quotients gives the (residual mean omitted lunar) days. (Multiply the remainder of the first division by 2000, increase the product by the remainder of the second division, and then) divide (the sum) by 1125 then are obtained (the mean lunar) days (which have elapsed at the beginning of the mean solar year since the occurrence of a mean omitted lunar day). The number of mean omitted lunar days in a mean solar year is 5+29/36+43/72000; and the number of mean lunar days between two successive mean omitted lunar days is approximately 64. Hence the above rule. 18 MEAN LONGITUDE OF A PLANET A rule for finding the number of mean lunar days elapsed at the beginning of mean Caitra since the occurrence of a mean omitted lunar day: 24. From them (i.e., from the mean lunar days elapsed at the beginning of the mean solar year since the occurrence of a mean omitted lunar day) subtract the mean intercalary days (obtained in stanza 22 above): the remainder (obtained) is the time (in terms of mean lunar days) elapsed (at the commence- ment of mean Caitra) since the fall of a (mean) omitted lunar day. In case the subtraction is not possible, add 64 (to the minuend) and then from the sum perform the subtraction. The subtraction is not possible when a mean omitted lunar day happens to fall between the beginning of mean Caitra and the beginning of the mean solar year. In such a case the omitted lunar days (obtained in stanza 23) should be diminished by one. A rule for finding the lord of the year : 25-26(i). Divide the sum of the months¹ (which have ela- psed at the beginning of the mean solar year since the beginning of Kaliyuga) and the (corresponding complete mean) intercalary months (obtained in stanza 22) by seven; and multiply the remainder by 30. Now we say what is to be subtracted from this: Divide the number of years elapsed since the beginning of Kaliyuga by seven and multiply the remainder (of the division) by five, add this product to the number of (residual mean) omitted lunar days (obtained in stanza 23) and divide the sum by seven: the remainder (of this division is the quantity to be subtracted). (Divide the difference of this quantity and the one obtained previously, by seven). The remainder increased by one counted with Friday gives the lord of the year (i.e., the planet presiding over the first day of Caitra). So has been stated by the learned. 1 These months are mean solar months and are otained by multi- plying the years elapsed since the beginning of Kaliyuga by twelve. SIMPLIFIED RULES 19 The number of mean civil days elapsed at the beginning of the mean solar year = 12 x 30 x (solar years elapsed) + (11 + 389/6000) x (solar years elapsed) (5+29/36+43/72000) x (solar years elapsed) = 30 x (solar months elapsed + mean intercalary months obtained in stanza 22) + mean intercalary days obtained in stanza 22 --- (5x solar years elapsed + residual mean omitted lunar days obtained in stanza 23). Therefore, the number of mean civil days elapsed on the first day of mean Caitra (or true Caitra)¹ = 30 x (solar months elapsed + mean intercalany months obtain- ed in stanza 22) (5xsolars years elapsed + residual mean omitted lunar days obtained in stanza 23). When these civil days are increased by one and divided by seven, the remainder of the division counted with Friday gives the day on which Caitra begins. In the above rule the author has avoided big numbers by dividing by seven at every stage. His rule is therefore easy to apply in practice. A rule for finding the the number of mean omitted lunar days occurring since the fall of the mean omitted lunar day just before the beginning of mean Caitra : 26(ii). Increase the number of (lunar) days (elapsed since the beginning of Caitra) by the number of (mean lunar) days elapsed (at the beginning of Caitra) since the fall of a mean omitted lunar day, and divide that (sum) by 64: the quotient gives the number of (mean) omitted lunar days (which have occurred since the mean omitted lunar day occurring before the beginning of Caitra) : A rule for finding the number of mean lunar days lying between the beginning of mean Caitra and the beginning of the mean solar year (called "the subtractive"): ¹ The first day of true Caitra may sometimes differ from that of mean Caitra by one day. 20 MEAN LONGITUDE OF A PLANET 27-28. Multiply the number of years (elapsed since the beginning of Kaliyuga) by 149 and then divide by 576: the quotient is in terms of days. Add these days to ten times the number of years (elapsed): thus are obtained the so called ravija days. To the ravija days add the (residual mean) omitted lunar days obtained above (in stanza 23). From the sum subtract the (complete mean) intercalary months (obtained in stanza 22) as multiplied by 30. Whatever is obtained as the remainder is "the subtractive" for the (current) year. When the subtra- hend is greater, then the difference is prescribed as "the additive". Let Y denote the number of years elapsed since the beginning of Kaliyuga. The number of mean civil days in one mean solar year 365 + 149/576. Therefore, the number of mean civil days in Y mean solar years 365Y+ (149/576)Y 365Y+ residual mean civil days. = = = The number of mean omitted lunar days in Y mean solar years (5+29/36 + 43/72000) Y ==5Y+ residual mean omitted lunar days. (2) Adding (1) and (2), the number of mean lunar days in Y mean solar years (1) 370Y+ residual mean civil days + residual mean omitted lunar days. (3) From (3) subtracting 360Y (į. e., the number of mean solar days in Y years), the number of mean intercalary days in Y solar years 10Y+ residual mean civil days + residual mean omitted lunar days. (4) Subtracting the complete mean intercalary months (elapsed since the beginning of Kaliyuga as reduced to days) from(4),we get the residual mean intercalary days. These are equal to the mean lunar days lying between the 21 beginning of mean Caitra and the beginning of the mean solar year, and constitute the so called "subtractive." SIMPLIFIED RULES The definition of the so called grahatanu for the Moon, Mars, Jupiter, and Saturn : 29. The number of years elapsed (since the commencement of Kaliyuga) multiplied by 360 is always called grahatanu. The (mean) longitudes (reduced to degrees) of the planets (Sun, Mercury, and Venus) together with the grahatanu are called dhruvaka by the learned. The term grahatanu denotes the number of mean solar days elapsed at the beginning of the mean solar year since the beginning of Kaliyuga. This grahatanu, as remarks the commentator Parameśvara, is really a part of the grahatanu. The dhruvaka (i.e., complete grahatanu) denotes the number of mean solar days elapsed on the given lunar day since the beginning of Kaliyuga. The above dhruvaka, or grahatanu, is defined for the Moon, Mars, Jupiter, and Saturn only; that for the Sun, Mercury, and Venus is defined in the next stanza. The grahatanu for the Sun, Mercury, and Venus : 30. Diminish the (lunar) days elapsed since the beginning of Caitra by the corresponding complete omitted lunar days (obtained in the second half of stanza 26) and divide (the difference) by seven: the remainder (of the division) counted with the first day of Caitra is said to give the (current) day. From that, the "subtractive" for the year (obtained in stanzas 27-28) should also be subtracted. (But it must be remembered that) the minuend of this subtraction is the difference of the previous subtraction and not the other (i.e., not the remainder of the division). (The remainder obtained by subtracting the "subtrac- tive" is the grahatanu for the Sun, Mercury, and Venus. It denotes the number of mean civil days elapsed since the beginning of the mean solar year). The number of mean civil days elapsed since the beginning of the mean solar year is generally known as laghvahargana ("smaller ahargana"). 22 MEAN LONGITUDE OF A PLANET Rules for finding the laghvahargana occur in the Brahma-sphuţa-siddhanta¹, the Śisya-dhi-vṛddhida², the Siddhanta-sśekhara, and the Siddhanta-siromani*, etc., but the process described in the above rule is slightly different from them. A rule for finding of the mean longitudes of the Sun, Mercury, and Venus: 31. Divide the grahadeha (for the Sun) by 70: the result is in days, etc. Then multiply one-fifth of the grahadeha by 2: the result is in vighatikas. These (days and vighatikas) subtracted from the grahadeha are stated to be (the degrees, minutes, etc. of) the mean longitudes of the Sun, Mercury, and Venus.³ The grahadeha is the same as grahatanu defined above. The following is the rationale of the above rule. Rationale 1. The grahadeha is in terms of mean civil days. This has to be converted into mean solar days. The difference between civil and solar days in a yuga 1555200000 1577917500 = 22717500 days. Therefore, this difference per mean civil day 22717500/1577917500 of a day 1/70 of a day-2/5 of a vighaṭikā approx. = ¹ i. 42-43. 2 I, i. 37. ³ ii. 40-41 (i).

• I, i(e). 12-13.

The same rule occurs also in ŚiDV, I, i. 39 and in GL, i. 10(i). Similar rules occur in BrSpSi, i. 44; MSi, i. 26 ; Sise, ii. 42, 43; Sisi, I, i(d). 15; SiSā, i. 105; KPr, i. 4;. KKau, i. 16; KKu, i. 7; and SK, i. 6 (i). Hence the number of mean solar days corresponding to the desired grahadeha (1-1/70) xgrahadeha days - (2/5)×(grahadeha) vighatikās. Consequently, the mean longitude of the Sun, Mercury, or Venus (1-1/70) × (grahadeha) degrees Rationale II. The mean SIMPLIFIED RULES - Hence the rule. - (2/5)×(grahadeha) seconds. - daily motion of the Sun 4320000 1577917500 of a revolution 4320000 x 12 x 30 1577917500 of a degree (1-3029/210389) of a degree (1-1/70) of a degree (1-1/70) of a degree - 1641 x 60×60 210389 × 70 of a second 2/5 of a second approx. 23 A rule for finding the mean longitude of the Moon : 32. Multiply the grahatanu for the Moon by 83 (lit. 92 + 2) and divide by 225; the result is in terms of degrees, etc. From that subtract the seconds obtained by multiplying the grahatanu by 11 and dividing by 50. (Then add the remainder to thirteen times the mean longitude of the Sun as prescribed in stanza 35 below : the sum thus obtained is the mean longitude of the Moon). ¹ ¹ Similar rules occur in SiDV, I, i. 40-41; MSi, i. 43(i); KPr, i. 5; GL, i. 10 (ii); KKu, i. 8; KKau, i. 17; and SK, i. 6(ii). Sā, i. 106; 24 Rationale I. The grahatanu for the Moon denotes the number of mean solar days elapsed since the beginning of Kaliyuga or the mean longitude of the Sun in revolutions, etc., reduced to degrees. Since therefore - MEAN LONGITUDE OF A PLANET mean longitude of the Moon in revolutions etc. mean longitude of the Sun in revolutions etc. revolution-number of the Moon revolution-number of the Sun 57753336 4320000 13 + 83/225 • 264/4320000 1383/225-11/(50×60×60), mean longitude of the Moon =13x (mean longitude of the Sun) 83 x G 225 + degrees where G is the grahatanu for the Moon. Rationale II. 57753336 4320000 The mean motion of the Moon per solar day degrees 83 -{13+ 2/3} 225 11 x G 50 degrees - {13G + 836 } = 225 + Therefore, the mean motion of the Moon for G solar days, i.c., the mean longitude of the Moon degrees 13 x Sun's mean longitude 83G 225 degrees seconds, 11G 50 11 50 of a second 11G 50 seconds seconds. A rule for finding the mean longitude of the Moon's ascending node: 33. Divide (the grahatanu) by 270: these are degrees. Multiply (the grahatanu) by 113 and divide by 600: these are seconds. These together with one-twentieth part of the (mean) longitude of the Sun (in revolutions, etc.) constitute the (mean) longitude of the Moon's ascending node. ¹ The mean motion of the Moon's ascending node per solar day 232226 4320000 (1/20+1/270) of a degree + 113/600 of a second. = = SIMPLIFIED RULES Hence the rule. = of a degree A rule for finding the mean longitude of the Moon's apogee : 34. Multiply the grahatanu by seven and divide by nine: these are minutes. Then multiply the grahatanu by 11 and divide by 60: these are seconds. Then divide the grahatanu by 20 these are thirds to be subtracted. These together with one-tenth of the Sun's (mean) longitude (in revolutions, etc.) constitute the (mean) longitude of the Moon's apogee : The mean motion of the Moon's apogee per solar day 488219 4320000 Hence the rule. 25 of a degree 1/10 of a degree + 7/9 of a minute + 11/60 of a second 1/20 of a third. ¹ Similar rules occur in BrSpSi, xxv. 35, and ŚiDVṛ, I, i. 52 (ii). 26 MEAN LONGITUDE OF A PLANET A rule for finding the mean longitude of the sighrocca of Venus, and also giving the additives for the sighrocca of Mercury and the Moon : 35. Multiply the grahatanu by 37 and divide by 900: these are the degrees, etc., (forming part) of the (mean) longitude of (the sighrocca of) Venus. Then divide the grahatanu by 100: these are seconds. Add to these one-third of the Sun's (mean) longitude (in revolutions, etc.). Then subtract the whole of that (sum) from two times the Sun's (mean) longitude. (The difference thus obtained is the mean longitude of the sighrocca of Venus). ¹ To the (mean) longitudes of (the sighrocca of) Mercury and the Moon add four times the Sun's (mean) longitude and thirteen times the Sun's (mean) longitude respectively. 2 The mean motion of the sighrocca of Venus per solar day = ! 7022388 4320000 degrees (2-1/3-37/900) degrees-1/100 of a second. Hence the rule. A rule for finding the mean longitude of the sighrocca of Mercury: 36. Divide the grahatanu by 200: the result is in terms of signs. Then divide the grahatanu by 8: these are minutes. Then divide the grahatanu by 60: these are seconds. Adding all these (and also four times the Sun's mean longitude as prescribed in stanza 35) is obtained the (mean) longitude of (the sighrocca of) Mercury. ¹ Similar rules occur in BrSp.Si, xxv. 36 and SiDVṛ, I, i. 57 (ii). 2 See stanzas 32 and 36. 3 Similar rules occur in Br.Sp.Si, xxv. 34 and ŚiDVṛ, I, i. 50 (ii). SIMPLIFIED RULES The mean motion of the sighrocca of Mercury per solar day – 17937020 degrees 4320000 1/200 of a sign+4 degrees+1/8 of a minute +1/60 of a second. Hence the rule. A rule for finding the mean longitude of Saturn: 37. Multiplying the grahatanu by 8 and dividing by 225 are obtained minutes; and dividing (the grahatanu) by 300 are obtained seconds. Adding these two together and increasing that (sum) by one-thirtieth of the Sun's (mean) longitude is obtained the (mean) longitude of Saturn.¹ The mean motion of Saturn per solar day = 27 Hence the rule. 146564 of a degree 4320000 1/30 of a degree +8/225 of a minute+1/300 of a second. Hence the rule. A rule for finding the mean longitude of Mars: 38. Multiply the grahatanu by two and subtract one- twentieth of itself from that these are minutes, etc. Then divide the grahatanu by 50: these are seconds. Add these (minutes and seconds) to half the Sun's (mean) longitude (in revolutions, etc.).: the sum is the (mean) longitude of Mars.³ The mean motion of Mars per solar day 2296824 4320000 of a degree 1/2 of a degree + (22/20) minutes + 1/50 of a second. ¹ Similar rules occur in Br.Sp.Si, xxv. 35 and Ś¡DVṛ, I, i. 52 (i ). ⠀

• Similar rules occur in BrSpSi, xxv. 33 and Ś¡DVṛ, I, i. 50 (i ). 28

MEAN LONGITUDE OF A PLANET A rule for finding the mean longitude of Jupiter : 39. Multiply the grahadeha by 22 and divide by 375: these are minutes, etc. Add them to one-twelfth of the Sun's (mean) longitude (in revolutions, etc.) : the result is the (mean) longitude of Jupiter.¹ The mean motion of Jupiter per solar day 364224 4320000 Hence the rule. - = of a degree 1/12 of a degree +22/375 of a minute. Corrections to be applied to the mean longitudes of the Moon's apogee and ascending node, and to the ahargana: 40. Add three signs to the mean longitude of the Moon's apogee. Subtract the (mean) longitude of the Moon's ascending node from 12 signs and then add 6 signs. Also (if necessary) add one to the ahargana obtained by proportion (in stanza 7 above). So say the astronomers whose hearts are devoted to Aryabhaṭa's system of astronomy (bhatasastra). At the beginning of Kaliyuga, according to Aryabhata I's system of astronomy, the mean longitude of the Moon's apogee was 3 signs and that of the Moon's ascending node. 6 signs. Hence the addition of 3 signs to the mean longitude of the Moon's apogee and of 6 signs to the mean longitude of 'the Moon's ascending node prescribed by the author. The longitude of the Moon's ascending node has to be subtracted from 12 signs, because the motion of the Moon's ascending node is retrograde. The remaining chapter deals with the solution of pulverisers (kuṭṭākāra) having reference to problems in astronomy. ¹ Similar rules occur in BrSp.Si, xxx. 35 and SiDVṛ, I, i. 51 (i). PLANETARY PULVERISER PLANETARY PULVERISER Preliminary operation to be performed on the divisor and dividend of a pulveriser: 41. The divisor (which is "the number of civil days in a yuga) and the dividend (which is "the revolution-number of the desired planet") become prime to each other on being divided by the (last non-zero) residue of the mutual division of the number of civil days in a yuga and the revolution-number of the desired planet. The operations of the pulveriser should be performed on them (i.e., on the abraded divisor and abraded dividend). So has been said. An indeterminate equation of the first degree of the type ax - C b (with x and y unknown) is known in Hindu mathematics by the name of "pulveriser (kuṭṭākāra)". In this equation, a is called the "dividend", b the "divisor", c the "interpolator", x the "multiplier", and y the "quotient". In the pulveriser contemplated in the above stanza, a = revolution-number of a planet, b- civil days in a yuga, c residue of the revolutions of the planet, 29 = y x = ahargana, and y= complete revolutions performed by the planet. The text says that as a preliminary operation to the solution of this pulveriser, a and b, i.e., civil days in a yuga and 'revolution-number of the planet, should be made prime to each other by dividing them out by their greatest common factor. That is to say, in solving a pulveriser one should always make use of abraded divisor and abraded dividend. The interpolator, i.e., the residue, should also be divided out by the same factor. This instruction is not given in the text, but it is implied that the residue should be computed for the abraded dividend and abraded divisor. 30 PLANETARY PULVERISER A rule for solving a pulveriser, when the dividend is smaller than the divisor: 42-44. Set down the dividend above and the divisor below that. Divide them mutually, and write down the quotients of divi- sion one below the other (in the form of a chain). (When an even number of quotients are obtained) think out by what number the (last) remainder be multiplied so that the product being diminish- ed by the (given) residue be exactly divisible (by the divisor corres- ponding to that remainder). Put down the chosen number (call- ed mati) below the chain and then the new quotient underneath it. Then by the chosen number multiply the number which stands just above it, and to the product add the quotient (written below the chosen number). (Replace the upper number by the resulting sum and cancel the number below). Proceed afterwards also in the same way (until only two numbers remain). Divide the upper number (called "the multiplier") by the divisor by the usual process and the lower one (called "the quotient") by the dividend: the remainders (thus obtained) will respectively be the ahargana and the revolutions, etc., or what one wants to know. We explain this rule by means of an example.¹ Example. The residue of the revolutions of Saturn is 24, find the ahargana and the revolutions performed by Saturn.² The revolution-number of Saturn is 146564, and the number of civil days in a yuga is 1577917500. In the present problem these are respectively the dividend and the divisor. Their H.C.F. is 4, so that dividing them out by 4 we get 36641 and 394479375 as the abraded dividend and the abraded divisor respectively. We have, therefore, to solve the pulveriser 36641x-24 394479375 where x denotes the ahargana and y the revolutions made by Saturn.³ ¹ For other details, the reader is referred to B. Datta and A. N. Singh, History of Hindu Mathematics, Part II, P. 87 ff. 2 Based on Bhaskara I's problem given in LBh, viii. 17. 3 We have not divided the given residue 24 by 4, because it is already computed for the abraded dividend and abraded divisor. METHOD OF SOLVING A PULVERISER Mutually dividing 36641 and 394479375, we have 36641) 394479375 (10766 394477006 2369) 36641 (15 35535 1106) 2369 (2 2212 157) 1106 (7 1099 10766 15 2 7 22 7) 157 (22 154 2 (mati) 27 1 3) 7 (2 6 We have chosen here the number 27 as the optional number (mati).¹ Writing down the quotients one below the other as prescribed in the rule, we get the chain 1x27-24-3(1 3 0 31 ¹ The mati may be chosen at any stage after an even number of quotients are obtained. . PLANETARY PULVERISER Reducing the chain, we successively get 10766 10766 10766 10766 10766 3108044439 (multiplier) 15 15 15 15 288689 288689 (quotient) 2 18665 18665 32 2 7 22 55 27 2 7 8714 8714 1237 1237 55 Dividing 3108044439 by 394479375, and 288689 by 36641, we obtain 346688814 and 32202 respectively as remainders.¹ These are the minimum values of x and y satisfying the above equation. Therefore, the required ahargana 346688814, and the revolutions performed by Saturn 32202. = General solution. The general solution of the above equation (vide stanza 50) is x=394479375d +346688814, y == 3664132202, where = 0, 1, 2, 3, An alternative rule: 45-46(i). Alternatively, the pulveriser is solved by subtrac- ting one (i.e., by assuming the residue to be unity). The upper and lower quantities (in the reduced chain) are the (correspon- ding) multiplier and quotient (respectively). By the multiplier and quotient (thus obtained) multiply the given residue, and then divide the respective products by the abraded divisor and dividend. The remainders obtained are here (in astronomy) the ahargana and the revolutions (performed respectively). This division is performed only when the multiplier and quotient are greater than the divisor and dividend respectively. FINDING THE RESIDUE OF REVOLUTIONS The pulveriser may be written as ax-c b = y ax-1 aX=1= Y, b or (2) = CY. where x = cX, and y If X = d, Y = ß is a solution of (2), then x cd, y = cß will be a solution of (1). Hence the above rule.¹ A rule for finding the residue of revolutions from the longi- tude of a planet given in signs, etc.: 46(ii). (In case the longitude of a planet is given in terms of signs, etc.,) the signs, etc., are multiplied by the abraded number of civil days (in a yuga) and the product is divided by the number of signs, etc., (in a circle). The quotient is stated to be the residue (of revolutions). When the longitude is given in terms of signs, it should be multi- plied by the abraded number of civil days in a yuga and then the product should be divided by 12. The quotient thus obtained should be used as the residue of revolutions. When the longitude is given in terms of signs and degrees, it should be reduced to degrees, then the resulting degrees should be multiplied by the abraded number of civil days in a yuga and the product should be divided by 360. The quotient should be treated as the residue of revolutions. When the longitude is given in terms of signs, degrees, and minutes, it should be reduced to minutes, then the resulting minutes should be multiplied by the abraded number of civil days in a yuga and the product should be divided by 21600. The quotient should be used as the residue of revolutions. (abraded rev.-number) x- 33 : Let x be the ahargana, y the revolutions performed by a planet, and s signs the given residue. Then (abraded revolution-number) x abraded civil days (1) -s signs = y, (abraded civil days) s 12 abraded civil days ¹ For illustration see Example under stanza 46 (ii). = y. 34: Hence the above rule. Example. "The mean (position) of the Sun has been observed by me at sunrise to be in the sign Leo in the middle of the navamāṁsa Sagittarius. Calculate the ahargana according to the (Arya)bhaṭa-sastra, and also the revolutions performed by the Sun since the beginning of Kaliyuga."1 The mean longitude of the Sun PLANETARY PULVERISER The abraded revolution-number of the Sun = 576, and the abraded number of civil days in a yuga = 210389. Hence, by the above rule, the residue of revolutions = 86688. We have, therefore, to solve the equation 4 signs 28° 20' = 8900'. 576 x 86688 210389 where x is the ahargapa and y the number of revolutions performed by the Sun. Solving this equation with unit residue, we get 94602. where d 0, 1, 2, 3, =y, y = 259. Deducing the solution for the given residue, we get x = 105345, y = 288, ... which is the minimum solution of the problem. The general solution is X = 210389+105345, y 576α + 288, A rule for solving a pulveriser when the dividend is greater than the divisor: 47. When the dividend is greater than the divisor, then, having subtracted the greatest multiple of the divisor (from ¹ Bhaskara I's example occurring in his comm. on Ā, ii. 32-33, 2 See the rule given in stanzas 45-46(i). DIVIDEND. GREATER THAN DIVISOR the dividend), apply the same process (as prescribed in stanzas 42-44 or 45-46 (i)). Multiply the multiplier (thus obtained) by that multiple and (to the product) add the quotient¹: the result will be the quotient here (required). Let the pulveriser be ax- c b = y, where a > b. Then if, a = mb + A, A ≤ b, (1) may be written as Ax c (2) b (1) where y Y + mx. If x=d, YB, be a solution of (2), then x== d,y= md +B will be a solution of (1). Hence the above rule. The abraded dividend for the Sun 576 revolutions = 576x12x30x60x60x60 thirds 44789760000 thirds. We have, therefore, to solve the equation 44789760000 x 101 210389 = 35 Example. "The signs, etc., up to the thirds of the Sun's (mean) longitude have all been carried away by the strong wind; the residue of thirds is known to me to be 101. Tell (me) the Sun's (mean) longitude and also the ahargaṇa." = y, (3) where x is the ahargana and y the thirds described by the Sun since the beginning of Kaliyuga. Since in this equation the dividend 44789760000 is greater than the divisor 210389, therefore, as directed in the above rule, we divide out the dividend by the divisor, and put the equation in the form 45790 x 101 210389 Y, ¹ The literal translation is "the lower quantity (in the reduced chain)", which means "the quotient". 2 MBh, viii. 13. 36 PLANETARY PULVERISER where Y is related to y by the relation y = 212890 x + Y.¹ Solving this equation, we get x = 106141, Y = 23101. Hence the solution of the equation (3) is X = 106141, y = 212890 x + Y 22596380591. may be written as = The required ahargana is therefore 106141; and the mean longitude of the Sun is 22596380591 thirds, i.e., 3 signs, 32 degrees, 52 minutes, 23 seconds, and 11 thirds. Alternative method. When a > b, the pulveriser ax-c by + c = y which can be solved ordinarily by applying the rule stated in stanza 51 below. A rule for solving the so called vara-kuṭṭakara (week-day pulveriser): 48. Divide the abraded number of civil days (in a yuga) by 7. Take the remainder as the dividend, and 7 as the divisor. Also take the excess 1, 2, etc., of the required day over the given day as the residue. Whatever number (i.e., multiplier) results on solving this pulveriser is the multiplier of the abraded number of civil days. The product of these added to the ¹ 212890 and 45790 are obtained as the quotient and the remainder when 44789760000 is divided by 210389. WEBK-DAY PULVERISER 37 ahargana calculated (for the given day) gives the ahargana for the required day¹. This rule will become clear by the following solved example. Example. "The mean longitude of the Sun (for sunrise) on a Wednesday is stated to be 8 signs, 25 degrees, 36 minutes, and 10 seconds. Say correctly after how much time (since the beginning of Kaliyuga) will the Sun again assume the same position (at sunrise) on a Thursday, Friday, and Wednesday." We first determine the ahargana elapsed at sunrise on Wednesday when the Sun's mean longitude 8 signs, 25 degrees, 36 minutes, and 10 seconds. Since the Sun's mean longitude = 8 signs 25° 36' 10" 956170", therefore, by stanza 46(ii), the residue of revolutions Thus we have to solve the pulveriser 576 x 155222 210389 = }, 1000, 155222. where x is the ahargana and y the revolutions performed by the Sun. Solving this equation, we obtain y = 2. Hence the ahargana for the given Wednesday <= 1000. (i) Now we find out the ahargana elapsed at sunrise on a Thursday when the Sun again occupies the same position. Let the required ahargana be 1000+A. Then in A days the Sun will describe complete revolutions. Also since Thursday is in advance of ¹ The text is a little obscure at this place. Our translation is based on the interpretations given by the commentators. It also agrees with the details of the rule supplied by the author Bhaskara I himself in his commen- tary on A, ii. 32-33. 2 Bhāskara I's example occurring in his comm, on A, ii. 32-33. 38 PLANETARY PULVERISER Wednesday by one day, the residue of the week-cycle is unity. In other words, 4-1 will be whole numbers. If we assume A to be a multiple of 210389, we have simply to determine A such that A-1 may be completely divisible by 7. Let A 210389X. Then we have to solve the pulveriser 210389X - 1 =Y, 7 or (vide stanza 47) 576A 210389 and or (vide stanza 47) 4X-1 = 7 This is what the rule prescribes. Y', where Y= 30055 X+Y'. Evidently, a solution of (2) is X = 2, Y' = 1. The corresponding solution of (1) is X= 2, Y 30055×2 + 1 = 60111. The required ahargana is therefore 1000 + A, i.e., 1000+ 210389X i.e., 1000+ 210389x2 or 421778. 2, (ii) To find the ahargana for Friday. In this case, the residue of the week-cycle is 2. Let the required ahargana be 1000+ 210389X. Then we have to solve the pulveriser 210389X2 = Y, 7 4X-2 (1) (2) = Y', (3) (4) where Y30055X+Y'. Evidently, a solution of (4) is X= 4, Y' 2. The corresponding solution of (3) is X= 4, Y- 30055X+2= 120222. The required ahargana is therefore 842556. (iii) To find the ahargana for Wednesday. As before, let the ahargana be 1000+210389X. In this case, the residue of the week-cycle is 0 and we evidently have X-7, so that the required ahargana is 1473723. A rule for the solution of the so called vela-kuttakara (time- pulveriser): 49. First make the abraded dividend and the (new) TIME-PULVERISER divisor prime to each other. Then by what remains as the (new) divisor multiply the abraded divisor (and also the residue). Thereafter the process for the time-pulveriser is the same as described before (for the ordinary pulveriser). This rule is applicable when the ahargana is not a whole number but a whole number and a fraction. That is to say, when the pulveriser is of the type a(xr/s). b (1) Let a mA, and s = mB, m being the greatest common factor of a and s. Then the equation (1) can be written as AX - Bc Bb (2) where X sx = r. The above rule tells us that whenever we have to solve an equation of the form (1), we must solve it by reducing it to the form (2). If X =d, B is a solution of equation (2), then x = (α Fr)/s, y = B will be a solution of equation (1). where X = Example 1. "The (mean) longitude of the Sun for midnight is found to be 9 signs, 15 degrees, 32 minutes, and 40 seconds. Quickly say the ahargana and the revolutions (performed by the Sun) according to the Āśmakiya." 4x -1. 39 Since the mean longitude of the Sun = 9 signs 15° 32′ 40", therefore, by stanza 46 (ii), the residue of revolutions 166876. We have, therefore, to solve the equation 576 (x1/4) - 166876 210389 = y, (3) where x-1 is the required ahargana and y the revolutions performed by the Sun. As this equation is of the form (1), we reduce it to the form (2) as prescribed in the rule. Thus we get 144X 210389 = 166876 =y, (4) ¹ The "(new) divisor" of the text is the denominator of this fraction. Bhaskara I's example, occurring in his comm, on A, ii, 32-33, 40 PLANETARY PULVERISER 7003, y == 4, giving x 1751. Solving equation (4), we get X = Hence the required ahargana is 1750, and the number of revolutions performed by the Sun is 4. Example 2. "The revolutions, etc., of the Sun's mean longitude, calculated from an ahargana plus a few nädis elapsed, have now been destroyed by the wind; 71 minutes are seen by me to remain intact. Say the ahargana, the Sun's (mean) longitude, and the correct value of the nadis (used in the calculation)." 1 Here we have to solve the equation 576x12x30×60(x+n/60) - -71 210389 = y, (5) where x is the ahargana, y the minutes traversed by the Sun since the beginning of the Kaliyuga, and n the nädis elapsed. As this equation is of the form (1), we reduce it to the form (2), and thus we get where X 60x+n. 207360X-71 210389 = y, Solving equation (6), we obtain 43203, y = 42581, whence we have x = 720, and n = 3. (6) the mean Hence the ahargana is 720, the nādis elapsed are 3, and longitude of the Sun is 42581 minutes, i.e., 11 signs, 19 degrees, and 41 minutes. A rule for getting the other solutions of a pulveriser with the help of the known minimum solution : 50. (To obtain the other solutions of the pulveriser) the intelligent (astronomer) should again and again add the divisor to the multiplier and the dividend to the quotient as in the process of prastara ("representation of combinations"). 1 Bhaskara I's example, occurring in his comm, and in MBh, viii. 23. on A, iii. 32-33, 41 That is to say, if x = d, y = ß is the minimum solution of the pulveriser WHEN THE INTERPOLATOR IS POSITIVE ax-c b = y, then the other solutions of the same pulveriser are X = mb + d y = ma + ß, where m = 1, 2, 3,.. Procedure for problems in which the given quantity is the part of the revolution to be traversed by a planet: 51. When the part (of the revolution) to be traversed by some (planet) is the given quantity, then (also) the same process should be applied, treating the part to be traversed as the addi- tive, or taking unity as the additive. All details of procedure are the same (as before). The pulveriser contemplated above is of the type ax + c b (1) According to the above rule, this is to be solved in the same way as ax- c b = y. ax 1 b = y, with the difference that wherever in solving (2) c is subtracted, in solving (1) it should be added. Or, the solution of (1) may be derived as before from the solution of = y.¹ (3) Example. "Given that 100 minutes of the eighth sign are to be traversed by the Sun, say quickly, after carefully considering, O intelligent one, if the Gunita of Äśmaka is known to you, all the years that have elapsed this day since the beginning of Kaliyuga. Also say the number of days that have elapsed since the beginning of Kaliyuga."2 ¹ It is also possible to reduce the pulveriser (1) to the form (2). For, when the part of the revolution to be traversed by a planet is given, the part traversed may be easily derived therefrom. 2 Bhaskara I's example, occurring in his comm, on Ā, ii, 32-33. 42 PLANETARY PULVERISER Here, according to Bhaskara I's interpretation, the part of the re- volution to be traversed by the Sun = 7 signs 100'. The corresponding residue of revolutions = 123707. This is positive. We have, therefore, to solve the equation 576 x 123701 210389 where x is the required ahargana, and y-1 the number of years elapsed. Mutually dividing 576 and 210389 and taking 1 for the optional number (mati) after six quotients, we get the following chain which reduces to 365 3 1 6 2 4 1 (mati) 61851 1310408037 3587617 Dividing 1310408037 by 210389 and 3587617 by 576, we obtain as remainders 105345 and 289 respectively. Therefore x = 105345, y = 289. Hence the required ahargana 105345, and the number of years elapsed 288. Note. According to Govinda Svāmi's interpretation the part of the revolution to be traversed by the Sun = 4 signs 1° 40'. The corresponding residue of the revolution = 71104. The resulting pulveriser is 576x+71104 210389 of which the solution is x = 186889, y = = y, = 512. Therefore, the required ahargaṇa 186889, and the number of years clapsed = 511, RESIDUES OF TWO OR MORE PLANETS GIVEN Rules relating to the two cases: (i) when the sum or difference of the residues (of revolutions) of any two planets is given, and (ii) when the residues for two or more planets are given separately: 52. When the sum of the residues (of revolutions of two or more planets ) is given, proceed with the sum of their revolu- tion-numbers (as the dividend); and when the difference between the residues (for any two planets) is given, proceed with the difference of their revolution-numbers ( as the dividend ). When the residues (for two or more planets) are given (separately), think out the method of solution by the help of the given residues and the true revolution-numbers of the given planets. These rules will be clear from the following solved examples. Example 1. "The sum of the (mean) longitudes of Mars and the Moon is calculated to be 5 signs, 7 degrees, 9 minutes, (9 seconds, and 6 thirds). O you, well versed in the (Arya )bhata-tantra, quickly say the ahargaṇa and also the (mean) longitudes of the Moon and Mars."¹ The revolution-number of the Moon = 57753336. The revolution-number of Mars 2296824. 60050160. 1577917500. Their sum The number of civil days in a yuga The H. C. F. of 60050160 and abraded sum of the revolution-numbers and the abraded number of civil days = = 43 1577917500 is 60. Therefore, the of the Moon and Mars 60050160 60= 1000836, =1577917500 60-26298625. The sum of the mean longitudes of the Moon and Mars =5 signs 7° 9' 9" 6" -33944946 thirds. Therefore, by stanza 46(ii), the residue of revolutions=11480265. We have, therefore, to solve the equation 11480265 1000836 x 26298625 = y, ¹ Bhaskara I's example (MBh, viii. 19) with Govinda Svami's modification. 44 where x denotes the required ahargana. PLANETARY PULVERISER The minimum solution of this equation is x= 10157490, y = 386459. The required ahargana is, therefore, 10157490. The mean longitudes of the Moon and Mars can be easily calculated from this ahargana. Example 2. "The difference between the mean longitudes of Mars and Jupiter is exactly 5 signs. Say what is the number of days elapsed since the beginning of Kaliyuga and what are the (mean ) longitudes of Jupiter and Mars." 1 The revolution-number of Mars The revolution-number of Jupiter Their difference Also the number of civil days 2296824. 364224. 1932600. =1577917500. The H. C. F. of 1932600 and 1577917500 is 300. Therefore, the abraded difference of the revolution-numbers of Mars and Jupiter 1932600 300, i.e., 6442, and the abraded number of civil days = 1577917500 300, i.e., 5259725. The difference between the mean longitudes of Mars and Jupiter = 5 signs. Therefore, by stanza 46(ii), the residue of revolutions=2191552. Hence we have to solve the equation 6442 x 2191552 5259725 where x denotes the required ahargana. 1 Mbh, viii. 20. =y, The minimum solution of this equation is X= 1133606, y = 1388. The required ahargana is therefore 1133606. The corresponding mean longitudes of Mars and Jupiter may be easily obtained. RESIDUES OF TWO OR MORE PLANETS GIVEN 45 Example 3. "Some number of days is (severally) divided by the (abraded) civil days for the Sun and for Mars. The (resulting) quotients are unknown to me; the residues, too, are not seen by me. The quotients obtain- ed by multiplying those residues by the respective (abraded) revolution- numbers and then dividing (the products) by the respective (abraded) civil days are also biown away by the wind. The remainders of the two (divisions) now exist. The remainder for the Sun is 38472, and that for Mars is 77180625. From these remainders severally calculate, O mathema- tician, the aharganas for the Sun and Mars and also the ahargana conforming to the two residues and state them in proper order."¹, The abraded revolution-number and the. abraded civil days for the Sun are 576 and 210389 respectively; the same for Mars are 191402 and 131493125 respectively. Let A be the number of days (i.e., the ahargana conforming to the two residues). Then suppose that and A 210389 A 131493125 576 a 210389 191402 b 131493125 = x + = y + =B+

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+ a 210389 The equations (2) reduce to b 131493125 38472 210389 131493125 77180625 131493125 where A, x, y, a, b, B, and λ are all unknown quantities. The problem is to find a and b and therefrom A. 576a38472 210389 191402 b-77180625 = B, (1) ¹ Bhāskara I's example, occurring in his comm. Also see MBh, viii. 24-24*. (3) on A, ii. 32-33. 46 b = Solving (3), we get a 8833, B 640000, 931. Hence the ahargana for the Sun is 8833, and that for Mars 931. The equations (1) now reduce to 210389 x 8833 == 131493125 y + 640000, i.e, A PLANETARY PULVERISEER whence we get the pulveriser Hence A - - 210389 x 631167 131493125 The minimum solution of (5) is evidently X = 628, y = 1. 24; and solving (4), we get x 3 625 132133125. (5) - The ahargana conforming to the two residues a 8833) and b ( 640000) is therefore equal to 132133125. THE CHAPTER II LONGITUDE-CORRECTION Names of certain places lying on the Hindu prime meridian: 1-2. From Lanka (towards the north, we have the follow- ing places on the prime meridian): Kharanagara, Sitorugeha, Pāṇāṭa, Misitapuri, Taparni, the lofty mountain called Sitavara, the wealthy town called Vatsyagulma, the well known Vanana- gari, Avanti, Sthanesa, and then Meru, which is inhabited by happy people. For those who reside in these places, the correction for the longitude (of the local place) does not exist. Lankā in Hindu astronomy denotes the place where the Hindu prime meridian passing through Ujjain¹ intersects the equator (i.e., the place in 0 latitude and 0 longitude). It is one of the four hypothetical cities on the equator, called Lanka, Romaka, Siddhapura, and Yamakoți. Lankā is described in the Surya-siddhanta² as a great city (mahāpuri) situated on an island (dvipa) to the south of Bhärata-varṣa (India). The island of Ceylon which bears then ame Lanka, however, is not the astronomical Lankā, as the former is about six degrees to the north of the equator. Kharanagara ("the town of Khara") is Nasik where Khara, cousin of Rāvaṇa, lived. identified. probably the place near Sitorugeha has not been Pāṇāṭa seems to have been an important place, as it has been mentioned by other astronomers also, such as Lalla, Vateśvara, and Śripati. Lalla has called it by the name Pārṇāṭā, and Sripati by the name Pānāța. We have not been able to identify this place also. ¹ Situated in latitude 23° 11' N. and longitude 75° 52' east of Greenwich.

• xii. 37, 39. 48

LONGITUDE-CORRECTION Misitapuri and Taparni, too, remain unidentified. Sankaranārāyaṇa in his commentary on the Laghu-Bhaskariya¹ pronounces Misitapura as Nisitapura, so it is difficult to say which pronunciation is correct. The Sitavara mountain ("the excellent white mountain") is the Śvetasaila of Lalla, the Sitadri of Sripati, and the Sitaparvata of Bhaskara II. According to śripati, it is the seat of the six-faced god Svämikärtikey a. It can therefore be identified with Krauñca-giri or Kumāra-parvata, situated at a distance of 3 yojanas from Śrisaila.² Vätsy agulma is the town of Vatsarāja Udayana, usually called Vatsapattana. It has been identified with Kauśämbi (modern Kosam) situated on the river Jumna at a distance of about 38 miles from Allahabad. Vananagari³ is probably Tumba-vana-nagara (modern Tumain) in Madhya Bharata. Avanti is modern Ujjain. Sthäneśa is Sthaneśvara, a place in Kurukṣetra. Meru is the north pole. From the above identification we find that the places ment oned in the text do not lie precisely on one meridian. The places mentioned by other astronomers also do not satisfy this requirement. It has not been possible to give any satisfactory explanation to this discrepancy. Probably the geographical knowledge of ancient Hindu writers was not sound in respect of places other than their own. We give below the lists of places lying on the Hindu prime meridian according to other Hindu astronomers which will be useful for comparison and reference. (i) Lalla's list. Lankā, Kumāri, Kāñci(varam), Pārṇāṭā, Kṛṣṇā (the river), Svetaśaila ("the white mountain"), Vätsyagulma, Ujjayini, Gargarāt, Aśraya (? Aśrama), Mālavanagara, Cãyuśiva (?), Rohitaka (Rohtak), Kuruksetra (the battle field of the Bharata War), Himavân (the Himalayas), and Meru. ¹ i. 23. 2 See Kalyana, Tirthanka, pp. 310 and 330. 3 In case the correct reading is Varanagari, it may be identified with on KK, i. 13, Barnagar. 4 Mentioned in Āmarāja's comm. DISTANCE FROM THE PRIME MERIDIAN (ii) Vateśvara's list. Lankā, Kumari, Kāñci, Mänaṭamaśvetapuri², Sveta Acala, Vātsyagulma, Avanti-puḥ, Gargarat, Aśrama-pattana, Mālava- nagara, Paṭṭaśiva, Rohitaka, Sthāṇviśvara (Sthāneśvara), Himavān, and Meru. 49 (iii) Śripati's list.³ Lanka, Kumāri, Kanci-nagari, Panāṭa, Şadāsya Sitadri, Sri Vatsagulma, Māhismati (modern Maheśvara situated on the north bank of river Narmada in Nimar district in Madhya Bhārata), Ujjayini, Aśrama-nagara, Pattaśiva, śri Gargarāt, Rohita (Rohtak), Sthāṇviśvara, Śitagiri (the Himalayas), and Sumeru. (iv) Bhaskara II's lists. 1. Lanka-puri, Vatsagulma, Mähismati, Ujjayini, Gargarāt, Kuruksetra, Himäcala, etc.¹ 2. Lanka, Devakanya, Kāñci, Sitaparvata, Paryali, Vatsagulma, Ujjayini-puri, Gargarat, Kurukṣetra, and Meru.5 (v) List of the Surya-siddhanta. Räkṣasālaya (i.e., Lankā), Devauka Saila (i.e., Meru), Rohitaka (modern Rohtak), Avanti, and Sannihita Sara7. A rule for finding the distance of a place from the prime meridian: 3-4. Subtract the degrees of the latitude of one of the places (lit. towns) mentioned above from the degrees of the (local) latitude; then multiply (the degrees of the difference) by 3299 minus 8/25, and divide (the product) by the number of degrees in a circle (i.e., by 360). The resulting yojanas constitute the upright (koti). The oblique distance between the local ¹ VSi, chapter I, section ix, stanzas 1-2. 2 A compound word giving the names of two places which probably correspond to Pāṇāṭa and Misitapuri mentioned in the text. 3 Sise, ii. 95-97. 4 Comm. on SiDVṛ, I, i. 55. 5 KKu, i. 14. 6 SMSi, i. 62.

• Sannihita Sara is in Kurukṣetra. See Kalyāṇa, Tirthānka, p. 79. 50

LONGITUDE-CORRECTION place and the place (on the prime meridian) chosen above, which is known in the world by the utterance of the common people, is the hypotenuse. The square root of the difference between their squares (i.e., between the squares of the hypotenuse and the upright) is defined by some astronomers to be the distance (in yojanas of the local place from the prime meridian).¹ In Fig. 2, let CD be a portion of the prime meridian and AB that of the local circle of latitude. Let L be the local place and Xa place on the prime meridian. L and X being joined, we C get the right-angled triangle XYL. The above rule tells us how to deter- A- mine the distance YL of L from the prime meridian in linear units (i.e., in yojanas). The triangle XYL is suppo- sed to be plane and sides XY, YL, and XL are taken as the upright (koti), the base (bhuja), and the hypotenuse (karna) respectively. Y = X = L Fig. 2 Subtracting the degrees of the latitude of X from those of L (or Y), we get the length XY in terms of degrees. Now the circumference of the Earth is equal to 3299 minus 8/25 yojanas in linear units and to 360 degrees in circular units; therefore, multiplying the degrees of XY by 3299 minus 8/25 and dividing the product by 360, we get the upright (koți) XY in terms of yojanas. The hypotenuse XL is assumed to be known in terms of yojanas by common usage. Hence the above rule. Earth's circumference. In the above rule, as according to Bhāskara I also, the diameter of the Earth has been taken to be equal to 1050 yojanas² and π equal to 3.1416. Therefore, the circumference of the Earth = -B 1050x3.1416 yojanas 3298.68 yojanas (3299-8/25) yojanas. ¹ This rule is found also in BrSp Si, i. 36; LBh, i. 25-26; Ś¡DVṛ, I, 57-58 (i); SiSā, i. 143-144. 2 Vide infra, chapter V, stanza 4. CRITICISM OF CERTAIN RULES FOR LONGITUDE 51 In the Laghu-Bhaskariya, the author has neglected the fraction 8/25 and has given the whole number 3299 as the yojanas of the Earth's circumference. Lalla has prescribed the more convenient number 3300. Criticism of the above rule: 5. The distance (obtained above) has been stated to be incorrect by the disciples of (Arya)bhata, who are well versed in astronomy, on the ground that the method of knowing the hypotenuse is gross. (Those) wise people further say that on account of the sphericity of the earth (also), the method used for deriving the above rule commencing with "akşa" is inaccurate. Sripati, too, has criticised the above rule for the same reasons. His commentator Makkibhaṭṭa sums up his criticism in the following words:

"The above rule is incorrect, because of the curvature of the Earth and because of uncertainty of the distances in yojanas depending on hearsay. No intelligent person has verified the popular (estimates of distances in) yojanas by actual measurement with the help of hand, staff, or rope. Therefore, in the face of plurality of popular estimates of distances, this rule is improper."4 It is noteworthy that the inaccurate rule criticised above occurs in the Brāhma-sphuṭa-siddhanta of Brahmagupta and in the Sisya-dhi- vrddhida of Lalla. Criticism of another rule: 6. Some (astronomers) say that the minutes of the difference between the true longitude of the Sun calculated from the midday shadow (of the gnomon at the local place) ¹ LBh, i. 24. 2 Ś¡DVr, I, i, 56. 3 This criticism occurs also in LBh, i. 27 and Sise, i. 104. 4 Siśe, ii. 104(1), comm. 5 i. 36. & I, i. 57-58(i). 52 and the true longitude of the Sun calculated (from the ahargana) for the middle of the day (without the application of the longitude-correction) give the longitude (correction for the Sun). This also is not so, because for people who live on the same parallel of latitude, the latitude (and therefore the shadow of the gnomon) is the same.¹ This rule has also been criticised by Sripati, who says: LONGITUDE-CORRECTION "Whatever is obtained here as the difference between the longitude of the Sun obtained from the midday shadow and that obtained by calculation (for midday, without the application of the longitude-correction) when multiplied by the (local) circumference of the Earth and divided by the (Sun's daily) motion gives the yojanas of the longitude (i.e., the distance in yojanas of the local place from the prime meridian). This is gross on account of the small change in the Sun's declination.". A rule for finding the longitude in time : 7. Those who have studied the astronomical tantra com- posed by (Arya)bhata and are well versed in Spherics state that the difference between the time of an eclipse calculated by the usual method from the longitudes of the Sun and the Moon (both) uncorrected for the longitude-correction and the time of the eclipse determined by observation is the more accurate value of the (longitude in) time.* "Choice is made, of course, of a lunar eclipse, and not of a solar, for the purpose of the determination of longitude, because its phenomena, being unaffected by parallax, are seen everywhere at the same instant of absolute time; and the moments of the total disappearance and first reappearance of the moon in a total eclipse are further selected, because 1 See also LBh, i. 28. The local circumference of the Earth is the circumference of the local circle of latitude. ³ Si Se, ii. 103. 4 Similar rules occur also in LBh, i. 29; Sise, ii. 106(i); TS, i, 31(ii)-32(i). LONGITUDE IN TIME 53 the precise instant of their occurrence is observable with more accuracy than that of the first and last contact of the moon with the shadow".¹ Thus, says Nilakantha, "whatever accrues as the difference between (the local times of) immersion (of a lunar eclipse) corresponding to the local and prime meridians is the time due to the longitude (of the local place). This may also be obtained from (the difference between the corresponding times for) the emersion of the lunar eclipse".2 The calculated time is the local time for the place lying at the intersection of the prime meridian and the local circle of latitude, while The difference the observed time is the local time for the local place. between the two is obviously the longitude in time for the local place. Another rule : 8. On any day calculate the longitude of the Sun and the Moon for sunrise or sunset without applying the longitude- correction, and therefrom find the time (since sunrise or sunset), in ghatis, of rising or setting of the Moon; and having done this, note the corresponding time in ghatis from the water-clock. The difference (between the two times), say the astronomers well versed in the tantra (composed by Aryabhata), is (the time of rising at the local place of a portion of the ecliptic equal to the motion-difference of the Sun and Moon corresponding to) the local longitude in time. (From this, the local longitude in time may be easily derived). Let Ø° N. be the latitude and λ° E. the longitude of the local place. Also let T₁ ghatis be the time of sunrise and T, ghatis the time of moonrise at the place in latitude Ø°N. and longitude 0. Then (T₂-T₁) ghatis is the time of moonrise (as measured from sunrise) as calculated from the longitudes of the Sun and Moon uncorrected for the longitude- correction. For those longitudes correspond to true sunrise at the place in latitude Ø°N. and longitude 0. 1 Burgess, E., Surya-siddhanta (English translation), Calcutta (1935), p. 47. 2 Cf. TS, i. 31(ii)-32(i). 3 Rules for finding the time of moonrise are given in Chapter VI. 54 LONGITUDE-CORRECTION = Thus the calculated time of moonrise T₂-T₁ ghatis. At the local place, the time of sunrise on that day will be (T₁-λ/6) ghatis; (1) and the time of moonrise will be (T₂/6) ghatis-(the time of rising at the local place of a portion of the ecliptic equal to > (Moon's daily motion in minutes - Sun's daily motion in minutes) 360 minutes of arc.) Therefore at the time of moonrise the time indicated by the water- clock will be. (T₂-T₁) ghatis- (the time of rising at the local place of a portion of the ecliptic equal to > (Moon's daily motion in minutes-Sun's daily motion in minutes) 360 minutes of arc). The difference between (1) and (2) gives the time of rising at the local place of a portion of the ecliptic equal to λ (Moon's daily motion in minutes - Sun's daily motion in minutes) 360 minutes of arc, which is evidently the time of rising at the local place of a portion of the ecliptic equal to the motion-difference of the Sun and Hence the Moon corresponding to the longitude of the local place. rule. To obtain the local longitude in ghatis we should multiply the above difference of (1) and (2) by 60 and divide the product (thus obtained) by the time in which an arc of the ecliptic equal to the difference between the daily motions of the Sun and Moon rises above the local horizon.¹ Criteria for knowing whether the local place is to the east or to the west of the prime meridian: 9. When the rising of a planet is observed before the comup- ted time or the first contact of an eclipse is observed after the 1 See Govinda Svāmi's commentary and the Siddhanta-dipikā. LONGITUDE-CORRECTION AND ITS APPLICATION 55 computed time, the observer is to the east of the prime meri- dian. In the contrary case, he is to the west (of the prime meridian). 2 The longitude correction and its application: 10(i). Multiply the (mean) daily motion of a planet the Sun, or the Moon's ascending node by the longitude in ghatis and divide by 60. Apply the resulting correction to the (corres- ponding) mean longitude of the planet, the Sun, or the Moon's ascending node (calculated for mean sunrise at Lankā) posi- tively or negatively according as the local place is to the west or east of the prime meridian. (Thus is obtained the mean longi- tude of the planet, the Sun, or the Moon's ascending node for mean sunrise at the svaniraksa place³).4 Rule for finding the length of the local circle of latitude and the distance of the local place from the prime meridian: 10(ii). Multiply the number of (yojanas in) the Earth's circumference by the Rsine of the colatitude and divide by the radius; (the result is the number of yojanas in the local circle of latitude). Multiply that by the longitude in ghatis and divide by 60; the result (thus obtained) is stated to be the (distance in) yojanas (of the local place from the prime meridian).5 ¹ The computed time corresponds to the place lying at the intersection of the local circle of latitude and the prime meridian. 2 Cf. LBh, i. 29; SuSi, i. 63; Sise, ii. 105(i)-106(i). 3 The svanirakṣa place is the place where the local meridian intersects the equator.

• This rule occurs also in LBh, i. 31 and Sise, ii. 106(ii).

5 Cf. SūSi, i, 60(i), 64(ii)-65, CHAPTER III DIRECTION, PLACE AND TIME. JUNCTION-STARS OF THE ZODIACAL ASTERISMS AND CONJUNCTION OF PLANETS WITH THEM them. (1) DIRECTION, PLACE AND TIME. Setting up of the gnomon : 1. After having tested the level of the ground by means of water, draw a neat circle with a pair of compasses. (At the centre of that circle, set up a vertical gnomon). The gnomon should be large, cylindrical, massive, and tested for its perpendicularity by means of four threads with plumbs tied to Bhaskara I in his commentary on the Aryabhatiya¹ tells us that there was difference of opinion amongst astronomers in his time regarding the shape and size of a gnomon (also called style). Some astronomers prescribed a gnomon with its one third in the bottom of the shape of a prism on a square base (caturaśra), one-third in the middle of the shape of a cow's tail (gopucchakāra), and one-third at the top of the shape of a spear-head (śülākāra); and some others prescribed a square prismoidal (samacaturaśra) gnomon. The followers of Aryabhata I, he informs us, prescribed the use of a broad (prthu), massive (guru), and large (dirgha) cylindrical gnomon, made of excellent timber, and free from any hole, scar, or knot on its body. In the above stanza Bhaskara I prescribes this last kind of gnomon: the other two kinds he proves in the commentary to be defective and so he rejects them. For getting the shadow-end easily and correctly, the cylindrical gnomon was surmounted by a fine cylindrical iron or wooden nail fixed vertically at the centre of the upper end. The nail was taken to be longer than the radius of the gnomon, so that its shadow was always seen on the ground. 2 ¹ ii. 14. 2 See Bhaskara I's commentary commentary on MBh, iii. 1.. Ā, ii. 13. Also see Parameśvara's FINDING THE DIRECTIONS. Certain writers, Bhāskara I tells us in the commentary, prescribed a gnomon of half a cubit (-12 angulas) in length and having twelve divisions. But, according to Bhāskara I, (although it was the usual custom) there was no such hard and fast rule. The gnomon could be of any length and any number of divisions. The gnomon should, however, be large enough, so that the rings of graduation on the gnomon may be clearly seen on the shadow. A broad and massive gnomon was preferred because it was unaffected by the wind. 57 As regards testing the level of the ground, Bhaskara I observes: "When there is no wind, place a jar (full) of water upon a tripod on the ground which has been made plane by means. of eye or thread, and bore a (fine) hole (at the bottom of the jar) so that the water may have continuous flow. Where the water falling on the ground spreads in a circle, there the ground is in perfect level; where the water accumulates after departing from the circle of water, there it is low; and where the water does not reach, there it is high."2 The same test has been prescribed by Govinda Svāmi³ and Nilakantha. After the ground was levelled, a prominently distinct circle was drawn on the ground as stated in the text. In the time of Sankaranārā- yaṇa (869 A. D.) it seems that all lines were drawn on the ground with sandal paste (candanakṣodārdra).5 The above circle having been thus drawn and coated with sandal paste, another small concentric circle was drawn with the radius of the gnomon. The gnomon was then placed vertically with the periphery of its base in coincidence with that circle. The gnomon was thus set up exactly in the middle of the bigger circle. The verticality of the gnomon was tested by means of four plumb-lines hung on the four sides of the gnomon. A rule for finding the directions: 2. With the two points where the shadow (of the gnomon) enters into and passes out of the circle, neatly draw a fish-figure (lit. fish). The thread-line which goes through the ¹ See Bhaskara I's comm. on A, ii. 14. 2 Bhaskara I's comm, on A, ii. 13. 3 In his comm. on MBh, iii. 1. 4 In his comm. on A, ii. 13. 56 Vide Sankaranārāyaṇa's comm. on LBh, iii. 1-2. 58 DIRECTION, PLACE AND TIME mouth and tail of the fish-figure indicates the north and south directions with respect to the gnomon.¹ the point where Let ENWS (See Fig. 3) be the circle drawn on the ground, and O its centre where the gnomon is set. Let W₁ be the shadow enters into the circle (in the forenoon), and E, the point where the shadow passes out of the circle (in the afternoon). Join E, and W₁. The line E₁W₁ is directed east With E, as centre and with E,W₁ as radius² draw an arc of a circle, and with W, as centre and with the same radius draw another arc cutting the former at the points N₁ and S₁. Join N₁ and S₁. The line N₁S₁ is directed north to south. to Fig. 3 Let the line N₁S₁ meet the circle in the points N and S and the line through O drawn parallel to E,W₁ in E and W. Then E, W, N, and S are respectively the east, west, north, and south directions relative to the gnomon, i.e., for an observer situated at O. E E X²₂0 W S The figure N₁E,S,W₁N₁ is called "fish or fish-figure", and the points N₁ and S, are called the mouth and tail of the fish-figure. As the Sun moves along the ecliptic, its declination changes. By the time the shadow moves from OW₁ to OE₁, the Sun traverses some distance of the ecliptic and so, theoretically speaking, its declination gets changed. It follows that EW is not the true position of the east-west-line. Brahmagupta (628 A.D.) was probably the first. Hindu astronomer who prescribed the determination of the east-west line with proper allowance for the change in the Sun's declination. The details of the method intended 1 This rule is found also in SuSi, iii. 1-4; BrSpSi, iii. 1; LBh, iii. 1; SiDV, I, iii. 1; MSi, iv. 1-2, Sise, iv. 1-3; and Sisi, I, iii. 8-9. 2 In general, as Parameśvara says in his comm. on LBh, iii-1-3 this radius may be any length greater than (1/2) E₂W₁. 3 Varahamihira calls this figure "yava (barley or barley-figuresr. also. See PSi, iv. 19. AN ALTERNATIVE RULE 59 by him have been supplied by his commentator Pṛthūdaka Svāmi (860 A.D.)¹ The method of getting the correct east-west line is found to occur also in the Siddhanta-sekhara² of Sripati (c. 1039 A.D.) the Siddhanta-siromani³ of Bhāskara II (1150 A.D.) etc. For practical purposes, however, the method given in the text is good enough. An alternative rule: 3. With the three points (at the ends of the three shadows of the gnomon) corresponding to (any three) different times (in the day), draw two fish-figures (each with two of the three points) in accordance with the usual method. From the point of intersection of the lines passing through the mouth and tail (of the two fish-figures), determine the north and south directions.5 According to this rule, the north-south line is the one joining the foot of the gnomon with the point of intersection of the mouth-tail lines of the two fish-figures. Varahamihira states this rule as follows: "Mark three times, from the centre, the end of the gnomon's shadow, and then describe two fish-figures. Thereupon describe a circle, taking for radius a string, that is fastened to the point in which the two strings issuing from the heads of the fish-figures intersect, and that is so long as to reach the three points marked. On the given day the shadow of the gnomon moves in that circle, without departing from it. "The line joining the centre of that circle and the base of the gnomon is the south-north line; and the interval in north direction (between that circle and the gnomon) is the midday shadow."6 ¹ See Sudhākara Dvivedi's comm. on BrSpSi, iii. 1. 2 iv. 14-16. s I, iii. 8. This point of intersection is the same as the centre of the circle passing through the three shadow-ends. 5 This rule is found also in PSi, xiv. 14-16; BrSp.Si, iii. 2; SiDV, I, iii. 2; and Sise, iv. 4. See G. Thibaut and S. Dvivedi, The Panca-siddhāntikā, Banaras (1889), xiv, 14-16, English translation, p. 79. 60

direction, place and time

Brahmagupta is more precise. He says :

"The point where the lines passing through the two fish-figures, which are drawn by means of three shadow-ends (of the gnomon), intersect each other is, for places in the northern hemisphere, the south direction* (if the midday shadow falls to the north of the foot of the gnomon) If the midday shadow falls towards the south of the foot of the gnomon, it is the north direction". 2

The above rule is evidently based on the assumption that the locus of the end of the shadow of the gnomon is a circle. In fact for places whose latitudes are less than 90°- Q (where Q is the obliquity of the ecliptic), this locus is a hyperbola, so the above assumption is not a correct one. The above rule will, however, give an approximately correct result if the three shadow-ends chosen are not far removed from the vertex of the hyperbola.

The method of drawing a circle through three given points by. the aid of two fish^figures is called "trisarkara-vidhana" by Bhaskara I. 3

A rule . for getting the length of the hypotenuse of the shadow :

4. The square root. of the sum of the squares of the gno- mon and its shadow (is equal to the hypotenuse of the shadow : this), say the learned (astronomers), is always the semi-diameter of its own circle in the calculations with the shadow. 4

By "the semi-diameter of its own circle" is meant "the semi-diameter of the circle of shadow".

The circle of shadow is, as Bhaskara I has said 5 , useful in the appli- cation of proportion in connection with the problems involving the shadow of the gnomon. For example, in finding out the Rsine 6 of the Sun's zenith distance from the shadow of the gnomon, the proportion is :

1 The north „ direction being indicated by the end of the midday shadow of the gnomon.

2 BrSpSi, iii. 2.

3 See LBh, vi. 16.

4 This rule is found also in A, ii. 14. 6 In his commentary on A, ii. 14.

• Rsine stands for "radius x sine". LATITUDE AND COLATITUDE, ZENITH DISTANCE AND ALTITUDE 61 "When to the radius of the circle of shadow corresponds the shadow of the gnomon, what will correspond to the radius of the celestial sphere? The result is the Rsine of the Sun's zenith distance". Rules for finding the latitude and colatitude and the zenith distance and altitude of the Sun : 5. Multiply the radius by (the length of) the shadow and (at another place) by (the length of) the gnomon. Divide (the two results) separately by the square root (obtained above). When this calculation is performed for an equinoctial midday, the (two) results denote the Rsine of the latitude and the Rsine of the colatitude (respectively); elsewhere, they denote the great shadow (i.e., the Rsine of the Sun's zenith distance) and the great gnomon (i.e., the Rsine of the Sun's altitude) (respectively).¹ That is Rsin and = -Rsin C = Rsin z = Rsin a = equinoctial midday shadow x radius hypotenuse of equinoctial midday shadow gnomon X radius hypotenuse of equinoctial midday shadow shadow x radius hypotenuse of shadow gnomon x radius hypotenuse of shadow 2 where and C denote the latitude and colatitude of the place, and z and a denote the zenith distance and altitude of the Sun.² These results are easily proved by assuming that the rays coming from the Sun are parallel. This rule is found also in SuSi, iii. 13-14; BrSpSi, iii, 10; LBh, iii. 2-3; ŚiDVṛ, I, iii. 4-5; Siśe, iv. 7; and SïŚi, I, iii. 18. 2 The equinoctial midday shadow is the shadow cast by the gnomon at midday at an equinox. 62 DIRECTION, PLACE AND TIME In fact, however, the rays coming from the Sun are not exactly parallel, so that the angle between the gnomon and the Sun's ray reaching the ground through the upper end of the gnomon is not exactly equal to the zenith distance of the Sun. Moreover, the shadow which is actually measured is the umbra (i.e., the shadow between the foot of the gnomon and the point where the ray coming from the uppermost point of the Sun's disc and passing through the upper end of the gnomon meets the ground) and not the theoretical shadow corresponding to the central ray of the Sun coming through the upper end of the gnomon. Later Hindu astronomers have, therefore, prescribed corrections to the results deter- . mined according to the rules in the above stanza.¹ 4 For practical purposes the rules stated above are good enough. The error is negligible. 1 Rules for determining the declination, day-radius, earthsine, and ascensional difference (for the Sun or a point on the ecliptic): 6-7. Multiply the Rsine of the given longitude by 1397 and always divide by the radius; the result is the Rsine of the declination for that time. Subtract the square of that (Rsine of the declination) from the square of the radius and then take the square root (of the difference); the result is called the day-radius.³ Multiply the Rsine of the latitude by (the Rsine of) the given declination and divide by (the Rsine of) the colatitude: the result is the earthsine. Multiply the earthsine by the radius and then divide (the product) by the day-radius; then reduce (the resulting Rsine) to arc. Whatever (arc) is thus obtained is termed "the ascensional difference" by the best amongst the good (astronomers).5 1 These corrections occur in KPr, iv. 2; KP, viii. 3; and TS, iii. 10(ii)-11. } 2 This rule is found also in S Si, ii. 28; BrSpSi, ii. 55; LBh, ií. 16; SiDVr, I..ii. 17; Sise, iii. 63-64; Si Si, I, ii. 47(ii). 3 This rule is found also in A, iv. 24; BrSpSi, ii. 56; LBh, ii. 17; ŚiDVŢ, I, ii. 18; Siśe, iii. 66; SiŚi, I, ii. 48.

• This rule is found also in A, iv. 26; LBh, ii. 17-18; SiSe, iii. 65.

5 This rule is found also in üsi, ii. 61; BrSp.Si, ii. 57-58; LBh, ii. 18; SiDV, I, ii. 18; SiSe, iii. 67 (i); SiŚi, I, ii. 49 (i). .....: IV.. DECLINATION, DAY-RADIUS, EARTHSINE AND ASCENSIONAL DIFFERENCE That is (1) Rsin 8 = 1397 x Rsin > R (Rsin 8)² x Rsin 8 Rsin (90°-+) (2) Day-radius=√√ R²- Rsin (3) Earthsine earthsine X radius day-radius 63 (4) Rsin (ascensional difference) where R is the radius, is the latitude of the place, and and are the sayana longitude and declination respectively. Definitions. The day-radius is the radius of the small circle parallel to the celestial equator. A small circle parallel to the celestial equator is called a diurnal circle (ahoratra-vṛtta). In particular, the Sun's diurnal circle is the small circle parallel to the equator which the Sun describes in the course of a day. The earthsine is the Rsine of the arc of a diurnal circle intercepted between the local horizon and the six o'clock circle.2 In Hindu astronomy the six o'clock circle is called the equatorial horizon (niraksa-ksitija) as it is the horizon of a places on the equator. The ascensional difference is defined by the arc of the celestial equator lying between (i) the equatorial horizon and (ii) the secondary to the equator passing through the intersection of the diurnal circle and the (eastern or western) horizon. It is measured in time (i.e., in asus). The ascensional difference of the Sun thus denotes the difference between the times of rising of the Sun on the local and equatorial horizons. ¹ By the sayana longitude is meant the celestial longitude measured from the moving vernal equinox. 2 The six o'clock circle is the great circle of the celestial sphere which passes through the east and west points of the celestial horizon and the poles of the celestial equator. 3. This place lies at the intersection of the local meridian and the equator. 4 One așu corresponds to one minute of arc of the celestial equtaro Thus one asu= 4 seconds of sidereal time. 64 Rationale of the above rules. (1) In Fig. 4, let S be the Sun (or a point on the ecliptic), SL the perpendicular from S on the plane of the celestial equator, and SM the perpendicular from S on the line joining the centre of the celestial sphere with the first point of Aries. Then in the plane triangle SLM, we have SL = Rsin 8, SM Rsin λ, LSML €, and SLM = 90°. Therefore, or DIRECTION, PLACE AND TIMB = SL/SM Rsin > = and LKAB = $. = KA = Rsin 8, KB = earthsine, LKBA = 90⁰ - $, = M Rsin €/Rsin 90°, Rsin Ex Rsin > R 1397 x Rsin A R Fig. 4 (2) The arcual distance of the diurnal circle from the north pole of the celestial equator-90°-8. Therefore, the day-radius-Rsin (90°-8), or √R²(Rsin 8 )². (3) In Fig. 5, let K be the point of intersection of the diurnal circle and the six o'clock circle, KB the perpendicular from K on the rising-setting line,² and KA the perpendicular from K on the east-west line. Then in the plane triangle KAB, we have A S for Rsin E=1397'.¹ Fig. 5. 1 See LBh, ii. 16. 2. The rising-setting line of a heavenly body is the line joining the point where the heavenly body rises on the eastern horizon with the point where it sets on the western horizon. In other words, it is the line of intersection of the planes of the celestial horizon and the diurnal circle. ASCENSIONAL DIFFERENCES OF THE SIGNS Therefore, we have Therefore earthsine Rsin S or earthsine (4) By definition Rsin (asc. diff.) earthsine Rsin (asc. diff.) Rsin Rsin (90⁰- Rsin x Rsin 8 Rsin (90⁰) radius day-radius earthsine x radius day-radius 65 A rule for finding the ascensional differences of the (sayana) signs Aries, Taurus, and Gemini : 8. Twenty-four multiplied by ten (i.e., 240), 192, and 81--these when (successively) multiplied by the angulas of the equinoctial midday shadow and (the products thus obtained) divided by four become the asus of the ascensional differences corresponding to Aries, Taurus, and Gemini respectively. ¹ 1 The numbers 240, 192, and 81 given above are four times the ascensional differences in asus of the signs, Aries, Taurus, and Gemini respectively for a place having one angula for the equinoctial midday shadow. We have seen that the ascensional difference of the Sun is the difference between the times of rising of the Sun on the local and equatorial horizons. The ascensional difference of the sign Aries is the difference between the times that the sign Aries takes in rising above the local and equatorial horizons. Since the first point of Aries rises simultaneously at both the horizons, therefore the ascensional difference of Aries is equal to the ascensional difference of the last point of Aries (for which >=30º). Similarly, the ascensional difference of Aries and Taurus (taken together) is equal to the ascensional difference of the last point of Taurus (for which λ=60⁰). The ascensional difference of Taurus is equal to the ascensional difference of Aries and Taurus minus the ascensional difference of Aries. ¹ Similar rules occur also in PSi, iii. 10; KK (Sengupta), i. 21; KK (Babua Misra), iii. 1; ŚiDVṛ, I, xiii. 9; SiŚi, I, ii. 50-51, 66 DIRECTION, PLACE AND TIME That is to say, it is equal to the ascensional difference of the last point of Taurus minus the ascensional difference of the first point of Taurus. The ascensional difference of Gemini, similarly, is equal to the ascensional difference of the last point of Gemini minus the ascensional difference of the first point of Gemini. The following is the rationale of the above rule: From stanza 7, we have Rsin x Rsin 8 Rsin (asc. diff.) Rsin (90⁰-$) But from stanza 5, assuming gnomon = 12 angulas, equinoctial midday shadow 12 Rsin Rsin (90°-) Therefore Rsin (asc. diff.) - (equinoctial midday shadow) x Rsin 8 x radius 12 x Rcos 8 Hence for a place having one angula for the equinoctial midday shadow Rsin (asc. diff.) - Rsin 8 x R 12 x Rcos 8 radius day-radius X 1397× Rsin λ R R 12 x Rcos & (using stanza 6) where is the sayana longitude, 8 the declination, and R the radius (=3438'). Now we will calculate the ascensional differences of Aries, Taurus, and Gemini for a place having one angula for the equinoctial midday shadow. (1) Calculation of the ascensional difference of Aries. At the last point of Aries, λ = 30°, so that Rsin λ = R/2. Therefore, Rsin 8′=1397/2 = 698' 5 and Rcos 8 = 3366'. ASCENSIONAL DIFFERENCES OF THE SIGNS Hence Rsin (asc. diff. of Aries) mately. 60' approx. Therefore, the ascensional difference of Aries is 60 asus = 698.5 x 3438 3366 x 12 Therefore, Rsin & and Rcos d 2401443 40392 (2) Calculation of the ascensional difference of Taurus. At the last point of Taurus, λ = 60⁰, so that Rsin λ = 2977'. = 1210' approx. 3218' approx. Hence Rsin (asc. diff. of Aries and Taurus) 1210 x 3438 12 x 3218 = 108' approx. Therefore, the ascensional difference of Aries and Taurus is 108 asus. Subtracting from it the ascensional difference of Aries, the ascensional difference of Taurus comes out to be 48 asus. (3) Calculatlon of the ascensional difference of Gemini. = At the R. last point of Gemini, λ = 90⁰, so that Rsin λ = Therefore, Rsin 8 = 1397' and Rcos & = 3141'. Hence Rsin (asc. diff. of Aries, Taurus, and Gemini) = 1397 X R 12 × 3141 67 4802886 37692 approxi- 128' approx. 68 DIRECTION, PLACE AND TIMÉ It follows that the ascensional difference of Aries, Taurus, and Gemini (taken as a whole) is 128 asus. Subtracting from it the ascen- sional difference of Aries and Taurus, we get 20 asus. This is the ascensional difference of Gemini.¹ The formula for the ascensional difference may now be written as Rşin (asc. diff.) Rsin (asc. diff. for unit equinoctial midday shadow)x (equinoctial midday shadow). [4 x (asc. diff. for unit equinoctial midday shadow)] X (equinoctial midday shadow) or asc. diff. approximately. Hence the above rule. A rule for finding the times of rising of the (sayana) signs at the equator: = 9. (Severally) multiply the Rsines of (one, two, and three) signs by 3141 and divide (each of the products) by the corresponding day-radius. Reduce the resulting Rsines to the corresponding arcs, and then diminish each are by the preceding arc (if any). The residues obtained after subtraction are the times (in asus) of rising of the signs Aries, Taurus, and Gemini at the equator.² ¹ We have taken above the approximate values of the ascensional differences of the last points of Aries, Taurus, and Gemini. Better values are 59-45, 107.77 and 127-4 asus. If these values are taken, then the ascensional differences of Aries, Taurus, and Gemini would come out to be 59.45, 48.32, and 19-63 asus. Four times of these are 238, 193 and 79 asus approx. Hence some astronomers (see Parameśvara's Siddhanta- dipika) give the following reading of the text: वसुत्रिदस्रा (238) गुणरन्ध्रभूमयो (193) नवाद्रव ( 79 ) श्चाभिहताः पलाङ्गुलैः । क्रियगोयमान्तजा- dragin: श्चरासवः स्युः क्रमशस्तु चापिताः || 2 This rule is found also in SüSi, iii. 42-43; BrSpSi, iii. 15; ŠiDVṛ, I, iii. 8; Sise, iv. 15; SiŚi, I, ii. 51. TIMES OF RISING OF THE SIGNS If A, B, and C be the last points of the signs Aries, Taurus, and Gemini respectively, then the time of rising of Aries at the equator is equal to the right ascension of A, the time of rising of Taurus at the equator is equal to the right ascension of B minus the right ascension of A, and the time of rising of Gemini at the equator is equal to the right ascension of C minus the right ascension of B. If λ and 8 be the sayana longitude and declination of a point on the ecliptic, then the right ascention of that point is given by the formula:¹ Rsin d - Rcos x Rsin > Rcos & where is the obliquity of the ecliptic. 69 , But, according to Bhāskara I, € = 24°; ² therefore we have Rsin d 3141 x Rsin > day-radius Hence the above rule. Times of rising of the (sayana) signs, Aries, Taurus, and Gemini at the equator and a rule for finding the times of rising of the (sayana) signs at the local place : 10. Those who know astronomical methods have found them (i.e., the times of rising of Aries, Taurus, and Gemini at the equator) to be 1670, 1795, and 1935 (asus respectively). These respectively diminished and the same reversed and increased by the corresponding ascensional differences are the times (in asus) of rising of the six signs beginning with Aries at the local place. (The same in the inverse order are the times of rising of the six signs beginning with Libra at the local place.)³ If a, b, and c denote the ascensional differences of Aries, Taurus and Gemini respectively, then the times of rising of the signs at the local place are given by the following table : ¹ This formula occurs in A, iv. 25. For its rationale see Part I, Chapter IX. 2 See LBh, ii. 16, where Rsin has been stated to be equal to 1397'. 8 This rule occurs also in SüSi, iii. 43-45; LBh, iii. 5-6; SiDV, I, iii. 9; Sise, iv. 17, 15(ii); SiŚi, I, iii. 58-59(i). 70 Sign 1 Aries 2 Taurus 3 Gemini 4 Cancer 5 Leo 6 Virgo DIRECTION, PLACE AND TIME Times of Rising of the Signs at the Local Place Time of rising at the local place, in asus 1670 - a 1795 b 1935-c 1935 + c 1795 + b 1670 + a Sign 12 Pisces 11 Aquarius 10 Capricorn 9 Sagittarius 8 Scorpio 7 Libra A rule for the determination of the meridian zenith distance. and meridian altitude of the Sun with the help of the Sun's declination and the latitude of the place : 11. The difference or the sum of the Sun's declination and the latitude (of the place) according as the Sun is in the six signs beginning with Aries or in the six signs beginning with Libra is the Sun's meridian zenith distance (i.e., the zenith distance of the idday Sun).¹ 90 degrees (literally, a quadrant of a circle) minus the degrees of the (Sun's) meridian zenith distance is the (Sun's) meridian altitude.² The Rsine of the degrees of the Sun's meridian zenith distance is the great shadow; and the other (i.e., the Rsine of the Sun's meridian altitude) is the great gnomon. An alternative rule for finding the Sun's meridian altitude : 12. Or, take the Sum or difference of the earthsine and the day-radius according as the Sun is in the northern or southern ¹ This rule is found also in BrSpSi, iii. 47; LBh, iii. 27; ŚiDVṛ, I, iii. 16; SiSe, iv. 42(i). 2 This rule is found also in Br.Sp.Si, iii. 47(ii); SiŚe, iv. 42(ii). SUN'S DECLINATION AND LOCAL LATITUDE hemisphere; then multiply that (sum or difference) by (the Rsine of) the colatitude and divide by the radius. The result thus obtained is the Rsine of the Sun's altitude at midday. Let S (See Fig. 6) be the position of the midday Sun on the celestial sphere. Also let SA and SB be the perpendiculars dropped from S on the plane of the celestial horizon and the Sun's rising-setting line respec- tively. Then in the plane triangle SAB, we have S SA = Rsin a SB = day-radius earthsine, LSBA = 90° - $, and SAB = 90°, where a denotes the Sun's altitude, and the latitude of the place. Therefore SA/SB or Rsin a = A Rsin SBA / Rsin SAB, 71 Fig. 6 (day-radius earthsine) x Rsin (90°-) R B + or - sign being taken according as the Sun is to the north or south of the equator. A rule for determining the Sun's declination with the help of the Sun's meridian zenith distance and the latitude of the local place, when the latitude is greater than the Sun's meridian zenith distance: 1 Vide supra, p. 64 (footnote). 2 This rule is found also in LBh, iii. 30. 13. When the latitude is greater than the arc of the Sun's meridian zenith distance derived from the (midday) shadow (of the gnomon), their difference is the declination of the apparent Sun. The Sun is also, in that case, in the northern hemisphere.2 This rule relates to the case when the midday shadow of the gnomon falls to the north of the gnomon. 72 DIRECTION, PLACE AND TIME A rule for determining the Sun's declination with the help of the Sun's meridian zenith distance and the latitude of the local place, when the midday shadow of the gnomon falls to the south of the gnomon : 14. When the (midday) shadow (of the gnomon) falls to the south (of the gnomon), then the sum of the latitude and the Sun's true meridian zenith distance gives the declination of the Sun lying in the northern hemisphere.¹ A rule for finding the Sun's declination with the help of the latitude and the Sun's meridian zenith distance, when the Sun's meridian zenith distance is greater than the latitude and the shadow of the gnomon falls towards the north of the gnomon: 15. When the Sun's meridian zenith distance is greater than the latitude, then the latitude is always subtracted from that (i. e., from the Sun's meridian zenith distance): the remain- der (obtained) after subtraction denotes the Sun's true decli- nation. The Sun is also, in that case, undoubtedly in the southern hemisphere.² The rules in the above three stanzas may be summarised as follows: (1) When the midday shadow of the gnomon falls towards the north of the gnomon, then Sun's declination latitude Sun's meridian zenith distance, the Sun's declination being north or south according as the latitude is greater or less than the Sun's meridian zenith distance. (2) When the midday shadow of the gnomon falls towards the south of the gnomon, then Sun's declination latitude + Sun's meridian zenith distance, the Sun's declination in this case being always north. 1 This rule is found also in LBh, iii. 30.

• This rule is found also in LBh, iii. 31. SUN'S LONGITUDE FROM DECLINATION

A rule for the determination of the Sun's longitude from its declination : 73 16. The radius multiplied by the Rsine of that (Sun's declination) should be divided by the Rsine of the Sun's great- est declination. The resulting Rsine reduced to arc, or (90° minus that arc) increased by three signs, or that (arc) increased by six signs, or (90° minus that arc) increased by nine signs, according as the Sun is in the first, second, third, or fourth quadrant, is the Sun's longitude.¹ The longitude thus obtained is sāyana. In the above rule a knowledge of the Sun's quadrant is assumed, but nowhere in the present work are we told how to know the Sun's quadrant. From other works on Indian astronomy we learn that it was known from the nature of the midday shadow. In the Pitāmaha-siddhanta² we are given the following criteria for knowing whether the Sun is in the first, second, third, or fourth quadrant: "(When the Sun is) in the first quadrant, the (midday) shadow of the trees is smaller than the equinoctial midday shadow and also decreasing (day to day); in the second quadrant, it is smaller (than the same) but increasing; in the third quadrant, it is greater and also increasing; and in the fourth, it is greater but decreasing". So also says Sripatis, but (for places below the Tropic of Cancer) he adds: "If the (midday) shadow fall towards the south and be on the increase, even then the quadrant is the first. Similarly, if you see that the (midday) shadow (falling towards the south) is on the decrease, you must understand that the quadrant is the second."* 1 This rule occurs also in SuSi, iii. 18-19; and LBh, iii. 32-33. This work is in prose and was edited along with a few other siddhantas in the Joytisa-siddhanta-sangraha by Pandit Vindhyeshwari Prasad Dwivedi, Banaras (1912). 3 Siśe, iv. 70. • Sise, iv. 71. 74 DIRECTION, PLACE AND TIME Kamalakara has followed Sripati.¹ Parameśvara has also made a similar statement. He says: "The Sun's northern or southern hemisphere (gola) and the Sun's northerly or southerly course (ayana) should be determined from the point where the Sun rises and from the decrease or increase of the (midday) shadow (of a gnomon) on two consecutive days".³ Certain astronomers decided the Sun's quadrant on the basis of the seasons. For example, Bhāskara II writes : "The quarters of the year are known from the characteristics of the seasons, so I will describe them afterwards".4 A rule for the determination of the latitude with the help of the Sun's meridian zenith distance and declination : 17. When the Sun is in the northern hemisphere (and the shadow of the gnomon falls towards the north), add the (Sun's) declination and the (Sun's) meridian zenith distance; when the Sun is in the southern hemisphere, or when the (midday) shadow (of the gnomon) falls towards the south (of the gnomon), take their difference: the sum or difference thus obtained is the latitude.5 A rule for finding the Rsine of the Sun's altitude or zenith distance from the time elapsed since sunrise in the forenoon or from the time to elapse before sunset in the afternoon : 18-20. Add the (Sun's) ascensional difference derived from the local latitude to or subtract that from the asus (elapsed since sunrise in the forenoon or to elapse before sunset in the afternoon) according as the Sun is in the southern or northern 1 See SiTV, iii. 192-193. 2 अर्कस्योदयस्थानवशात् दिनद्वयोत्थयोश्छाययोर्न्यूनाधिकभावशाच्चाऽत्र गोलायने वेद्ये । See Parameśvara's comm. on SuSi, ii. 19. 4 Sisi, II, xi. 38. 5 This rule is found also in SüSi, iii, 15-16; BrSpSi, iii. 13; LBh, iii. 34; Sise, iv.13. LOCAL LATITUDE 75 hemisphere. By the Rsine of that (sum or difference) multiply the day-radius and then divide (the product) by the radius. In the resulting quantity apply the earthsine reversely to the application of the ascensional difference (i.e., subtract the earth- sine when the Sun is in the southern hemisphere and add the earthsine when the Sun is in the nothern hemisphere). Then multiply that (i.e., the resulting difference or sum) by the Rsine of the colatitude of the local place and then divide (the product) by the radius again. Thus is obtained the Rsine of the Sun's altitude for the given time in ghatis.¹ The square root of the difference between the squares of the radius and that (Rsine of the Sun's altitude) is known as the (great) shadow.2 The asus elapsed since sunrise in the forenoon or to elapse before sunset in the afternoon are obtained by multiplying the given ghatis³ by 360. These are equivalent to the number of minutes in the arc of the celestial equator lying between the hour circles passing through the Sun and through the Sun's position on the horizon at sunrise or sunset. When these asus are diminished or increased by the asus of the Sun's ascensional difference (according as the Sun is in the northern or southern hemisphere), the asus of the difference or sum correspond to the minutes lying in the arc of the celestial equator intercepted between the Sun's hour circle and the six o'clock circle. The Rsine of that multiplied by the day-radius and divided by the radius gives the distance in minutes of the Sun from the line joining the points of intersection of the six o'clock circle and the Sun's diurnal circle. That increased or diminished by the earthsine (accord- ing as the Sun is in the northern or southern hemisphere) gives the distance of the Sun from the rising-setting line. In Fig. 7, let S be the position of the Sun on the celestial sphere, SA the perpendicular from S on the plane of the horizon, and SB the perpendicular from S on the rising-setting line. Then in the plane triangle SAB, we have ¹ This rule is found also in A, iv. 28; BrSpSi, iii. 25-26; LBh, iii. 7-10; SiDVṛ, I, iii. 24-25; Siśe, iv. 32, 34; Siśi, I, iii. 53-54. 2 This rule is found also in BrSp.Si, iii. 27(ii); and Siśe, iv. 34. 76 SA SB = Rsin a. Rsin (given time in asus asc. diff.) × day-radius LSBA and SAB DIRECTION, PLACE AND TIME 90º-, 90°, where a is the Sun's altitude, and the latitude of the place. Therefore, we have Rsin a = Hence the rule. = SBX Rcos R radius Rsin a = S A Fig. 7 An approximate rule for finding the Sun's altitude: 21. Multiply "the upright due to the instantaneous meridian-ecliptic point" by the Rsine of the degrees intervening between the Sun and the rising point of the ecliptic and then divide (the product) by the radius: the result is the Rsine of the Sun's true altitude. The square root of the difference between the squares of that and the radius is the Rsine of the Sun's zenith distance. 1 See the next stánza. "The upright due to the meridian-ecliptic point" is the Rsine of the altitude of the meridian-ecliptic point. Therefore, the rule stated above may be expressed as + earthsine, Rsin x Rsin (L- -S). R B 2 where a is the Sun's altitude, the altitude of the meridian-ecliptic point, L the longitude of the rising point of the ecliptic, S the longitude of the Sun, and R the radius of the celestial sphere (= 3438'). The rising point of the ecliptic is that point of the ecliptic which lies on the eastern horizon. This formula is approximate, because the distance between the rising point of the ecliptic and the meridian-ecliptic point is not always exactly equal to 90° as assumed there. SUN'S ALTITUDE The correct formula is Rsin a Rsin x Rsin (L-S) R where denotes the altitude of the central ecliptic point.¹ 77 The author does not prescribe this correct formula, because the value of Rsin has not been accurately determined by him. In Chapter V he gives only an approximate formula for it." For practical purposes the approximate formula is good enough. Definition of "the upright due to the meridian-ecliptic point:" 22. The square root of the difference between the squares of the Rsine of the zenith distance of the meridian-ecliptic point and of the radius (ravi-kaksya) is called "the upright due to the meridian-ecliptic point" by those who are well versed in Spherics. Thus we see that "the upright due to the meridian-ecliptic point" is the Rsine of the altitude of the meridian-ecliptic point. It is usually called madhya-śanku. The word ravi-kakṣyā, literally meaning "the Sun's orbit", is used in the text in the sense of "the radius (of the Sun's orbit)". Two alternative rules for finding the Sun's altitude : 23-24. Increase or diminish the ghatis (elapsed since sunrise in the forenoon or to elapse before sunset in the after- noon) by the asus of the (Sun's) ascensional difference (according as the Sun is in the southern or northern hemis- phere). To the Rsine of that apply the Rsine of the (Sun's) ascensional difference reversely to the above. By what ¹ The central ecliptic point (also called the "nonagesimal") is that point of the ecliptic which is 90 degrees behind the rising point of the ecliptic and 90 degrees ahead of the setting point of the ecliptic. This point of the ecliptic is at the shortest distance from the zenith and is the central point of that part of the ecliptic which lies above the horizon. 2 See infra, Chapter V, verse 19. 78 DIRECTION, PLACE AND TIME is thus obtained multiply the product of the day-radius and (the Rsine of) the colatitude and then divide (the resulting product) by the square of the radius. The result of this (operation) is the Rsine of the (Sun's) altitude. Or, multiply the result obtained by the inverse applica- tion of Rsine of the (Sun's) ascensional difference (in the above process) by the product of (the length of) the gnomon and the day-radius and then divide by the product of (the length of the hypotenuse of the equinoctial midday shadow and the radius; the result is the Rsine of the (Sun's) altitude.¹ That is to say, (1) MX day-radius Rcos R R where M = Rsin (given ghatisFasc. diff)+Rsin (asc. diff.), the upper or lower sign being taken according as the Sun is in the northern or southern hemisphere. a andare, as usual, the Sun's altitude and the latitude of the place respectively. Or, Rsin a = Rsin a = Mx day-radius R X '" gnomon hypotenuse of equinoctial midday shadow "M multiplied by day-radius and divided by radius" represents in the celestial sphere the perpendicular distance of the Sun from the rising-setting line. A rule for finding the Sun's altitude when the Sun's ascensional difference is greater than the given time: 25. When the (Sun's) ascensional difference (is greater than and) cannot be subtracted from the given asus (elapsed since sunrise in the forenoon or to elapse before sunset in the after- noon), subtract the latter from the former and with the Rsine of the remainder proceed as before (i.e., multiply that by the day- radius and divide by the radius); then subtract the resulting ¹ This rule occurs also in BrSpSi, iii. 27(i); ŚiDVṛ, I, iii. 27; Sise, iv. 37. SUN'S ALTITUDE 79 Rsine from the earthsine and then performing the usual process (i.e., multiplying that by the Rsine of the colatitude and dividing the product by the radius) determine the Rsine of the (Sun's) altitude.¹ The case contemplated here arises when the Sun lies between the equatorial and local horizons, i.e., shortly after sunrise or before sunset. A rule for finding the Sun's altitude in the night: 26. In the night, the Rsine of the Sun's altitude is to be obtained by applying the operations (of addition and subtraction) inversely, because the (laws of) addition and subtraction (of the Sun's ascensional difference and earthsine) in the night are contrary to those in the day.² The Rsine of the Sun's altitude in the night is required (i) in the calculation of the elevation of lunar horns,³ and (ii) in the calculation of the solar eclipse.4 The details of the method indicated in the above stanza have been explained by Parameśvara as follows: "(When the Sun is) in the northern hemisphere, having calculated the Rsine of the given nocturnal asus (i.e., those elapsed since sunset in the first half of the night or those to elapse before sunrise in the second half of the night) as increased by the (Sun's) ascensional difference, (then) multiplying (that) by the day-radius and dividing by the radius, (then) from the (resulting) quotient subtracting the earthsine, and (finally) multiplying the remainder by the Rsine of the colatitude and dividing by the radius is obtained the Rsine of the Sun's altitude. (When the Sun is) in the southern hemisphere, the (Sun's) ascensional difference and the earthsine are (respectively) subtractive and additive: this is the difference."5 1 This rule is found also in BrSp.Si. iii, 33; LBh, iii. 11; ŚiDVṛ, I, iii. 29; Sise, iv. 41. 2 This rule is found also in BrSp.Si, iii. 63: LBh, iii. 11; Sise, iv. 89. ³ See BrSpSi, iii. 63. 4 See Parameśvara's comm. on LBh, iii, 11. 5 Parameśvara's comm. on LBh. iii. 11. 80 DIRECTION, PLACE AND TIME The Sun's altitude for the night has been called patala-sanku by Brahmagupta.¹ Three rules for finding the time elapsed since sunrise in the forenoon or to elapse before sunset in the afternoon when the Sun has a given altitude : 27-29. Multiply the Rsine of the Sun's altitude derived from the given shadow (of the gnomon) by the radius and divide (the product) by (the Rsine of) the colatitude. Then subtract the minutes of the earthsine from or add them to the resulting quantity according as the Sun is in the six signs beginning with Aries or in the southern hemisphere. Multiply the resulting quantity by the radius and divide (the product) by the day-radius. To the corresponding arc apply the ascensional difference contrarily to the above: thus is obtained the number of asus (elapsed since sunrise in the forenoon or to elapse before sunset in the afternoon). These very same asus corresponding to the day to elapse (before sunset in the afternoon) or the day elapsed (since sunrise in the forenoon),. when divided by 360, are declared to be the nadis, etc., (of the required time).² Or, multiply the given Rsine of the Sun's altitude by the square of the radius and divide by the product of the day-radius and (the Rsine of) the colatitude. To the result apply the Rsine of the (Sun's) ascensional difference as before (i.e., sub- tract or add according as the Sun is in the northern or southern hemisphere). Then to the corresponding arc reversely apply the asus of the ascensional difference: the result obtained is again the number of asus (elapsed since sunrise in the forenoon or to elapse before sunset in the afternoon.) Or, multiply the Rsine of the Sun's altitude by the hypotenuse of the equinoctial midday shadow and again by the ¹ See BrSp.Si, xv. 9. 2 This rule is found also in LBh, iii. 12-15 and in Sise, iv. 51-52, RISING POINT OF THE ECLIPTIC radius and then divide (the resulting product) by the day- radius as multiplied by (the length of) the gnomon. From that the process for the determination of the (desired time in) nādis is the same as before. 81 The first rule is the converse of the rule given in stanzas 18-20; the second rule is the converse of that given in stanza 23; and the third rule is the converse of that stated in stanza 24. A rule for finding the longitude of the rising point of the ecliptic with the help of (i) the instantaneous sayana longitude of the Sun and (ii) the civil time measured since sunrise, or with the help of (i) the Sun's sayana longitude at sunrise and (ii) the sidereal time elapsed since sunrise : 30-32. Multiply by the untraversed portion of (the sign occupied by) the Sun, the asus of the oblique ascension (i.e., the time, in asus, of rising at the local place) of that sign and divide (that product) by the number of degrees or minutes in a sign (i.e., by 30 or 1800) (according as the untraversed portion of the Sun's sign is taken in degrees or minutes). Thus are obtained the asus of the oblique ascension of the untraversed. portion of the sign occupied by the Sun. Subtract these from the given (time as reduced to) asus; and add the untraversed portion of the Sun's sine to the Sun's longitude (which is given). Then from the remaining asus subtract the asus of the oblique ascension of as many (succeeding) signs as possible; and add the same number of signs to the Sun's longitude. Then multiply the outstanding residue of the given (time in) asus by 30 and divide (the product) by the asus of the oblique ascension of the next sign. Add the resulting degrees, etc., to the Sun's longitude (obtained above). The resulting longitude is stated to be the (sayana) longitude of the rising point of the ecliptic.¹ ¹ This rule in found also in SuSi, iii. 46-48; BrSp.Si, iii. 18-20; L.Bh, iii. 17-19; SiDV, I, iii. 11-12; Siśe, iv. 18-19(i); Sisi, I, iii. 2-4. The rising point of the ecliptic is that point of the ecliptic eastern horizon. lies on the 82 DIRECTION, PLACE AND TIME It may be pointed out that the civil time elapsed since sunrise is equal to the sidereal time measured by the arc of the celestial equator lying between the hour circles which pass through the instantaneous position of the Sun and through the rising point of the ecliptic. For greater accuracy in the result, Aryabhata II (c. 950 A. D.) and Bhaskara II (1150 A. D.) prescribed the use of oblique ascensions of every 10 degrees of the ecliptic in place of those for every 30 degrees. Later on Muniśvara (1646 A.D.) and Kamalakara (1658 A. D.) prescribed the use of oblique ascensions of every degree of the ecliptic. Theoreti- cally accurate method for getting the longitude the rising point of the ecliptic was given earlier by the Kerala mathematician Nilakaṇṭha (1500 A. D.).¹ A rule for the determination of the longitude of the setting point of the ecliptic : 33. The longitude of the horizon-ecliptic point in the east² increased by half a circle (i.e., by 180°) is the longitude of the setting point of the ecliptic³. For, the time of setting of a sign is equal to the time taken in rising by the then rising sign. Bhaskara II also says: "The time in which a sign rises (above the horizon), is the same as that in which the seventh sign sets (below the horizon)." . A rule for obtaining the civil time measured since sunrise with the help of (i) the Sun's instantaneous sayana longitude and (ii) the sayana longitude of the rising point of the ecliptic, or the sidereal time elapsed since sunrise with the help of (i) the Sun's sayana longitude at sunrise and (ii) the sayana longitude of the rising point of the ecliptic: 34-36. Multiply the degrees of the traversed portion of the sign occupied by the rising point of the ecliptic by the ""5 1 His method occurs in TS, iii. 155(ii)-164(i). 2 i. e., the rising point of the ecliptic. 3 The setting point of the ecliptic is that point of the eclipti which lies on the western horizon. 4 Cf. ŚiDV, I, iii. 13(i); SiŚi, I, ii. 59(ii). 5 Siśi, I, ii. 59 (ii). TIME ELAPSED SINCE SUNRISE 83 oblique ascension of that sign and divide by 30, the resulting asus denote the asus of the oblique ascension of the traversed portion of the sign occupied by the rising point of the ecliptic. (Also subtract the traversed portion of the sign occupied by the rising point of the ecliptic from the longitude of the same point). Then from the resulting longitude subtract as many preceding signs as there are up to the Sun; and find out the asus of the oblique ascensions of those signs. These asus together with those (obtained above) when added with the asus of the oblique ascension of the untraversed portion of the Sun's sign give the asus (elapsed since sunrise) in the day and (enable us to know) the asus (elapsed since sunset) in the night. Dividing them by 6 and then by 60 are obtained the ghatis, vighatis, and asus (of the time elapsed during the day or night).¹ This rule is the converse of that given in stanzas 30-32. It actually gives the method for finding the time elapsed since sunrise. When, therefore, the Sun's longitude is given for sometime in the night and it is required to find the time elapsed since sunset, then the time elapsed since sunrise obtained according to the above rule should be diminished by the length of that day, or, as says the commentator Parameśvara, the given longitude of the Sun should be increased by six signs and then should be applied the above rule. In case the given quantities be (i) the longitude of the rising point of the ecliptic and (ii) the Sun's longitude at sunrise and the problem be to find out the civil time elapsed since sunrise, the above rule should be applied by treating the Sun's longitude for sunrise as the first approxi- ¹This rule is found also in SuSi, iii. 50-51; BrSpSi, iii. 21-23; LBh, iii. 20; Ś¡DVṛ, I, iii. 13; Siśe, iv. 19(ii)-22(i); SiŚi, I, iii. 5-7(i). The ghațis, vighatis, and asus are related by the following relations: 1 asu 4 seconds, 6 asus = 1 vighați (=24 seconds), 60 vighatis = 1 ghați (=24 minutes), 84 DIRECTION, PLACE AND TIME mation to the instantaneous longitude of the Sun. The process should then be repeated over and over again with the help of the successive approximations to the Sun's instantaneous longitude calculated after each round of the operation. The process should be continued until two successive approximations to the required time agree to asus.¹ It must be noted that the longitudes used in this and the preceding rule (stated in stanzas 30-32) are all sayana. A rule for the determination of the Rsines of the Sun's prime vertical altitude and zenith distance : 37-38. Multiply the Rsine of the (Sun's) greatest decli- nation by the Rsine of the Sun's true (sayana) longitude; then divide (the product) by the Rsine of the colatitude: the result is (the Rsine of) the agra of the true Sun. When that (agrā) is less than the latitude and when the Sun is also in the northern hemisphere, multiply (the Rsine of the Sun's agrā) by (the Rsine of) the colatitude and then divide (the product) by the Rsine of the latitude: the result is the Rsine of the Sun's prime vertical altitude. The square root of the difference between the squares of that and the radius is the Rsine of the Sun's (prime vertical) zenith distance. The Sun's agrā is the amplitude of the Sun at rising or setting. It. is defined to be the arc of the celestial horizon lying between the east-west and rising-setting lines. That is to say, it is the arc of the celestial horizon lying between the east point and the point where the Sun rises or between the west point and the point where the Sun sets. The Rsine of the Sun's agrā is equal to the distance between the east-west and rising-setting lines. In Fig. 8, let S be the Sun on the prime vertical, SA and SB the on the east-west and rising-setting lines perpendiculars from S ¹ Cf. BrSp.Si, iii. 21-23; Sisi, 1, iii. 6(ii).

• This rule occurs also in A, iv, 30; LBh, iii. 21; Sise, iv. 58.

3 This rule occurs also in A, iv. 31; BrSpSi, iii. 52; LBh, iii. 52, 4 Cf. LBh, iii. 22, SUN'S PRIME VERTICAL ALTITUDE AND ZENITH DISTANCE respectively, and C the point where SB meets the line joining the points of intersection of the Sun's diurnal circle and the six o'clock circle. Naturally, SB is perpendicular to that line. In the triangle ACB, we have LABC and LACB Therefore AC = Rsin 8, AB= where is the declination the latitude of the place. Therefore = Rsin (Sun's agrā), 90° - $, 90°, of the Sun, and Rsin (Sun's agrā) But (vide stanza 6 above) Rsin = Therefore Rsin 8 x R Rcos Rsin λx 1397 R Rsin (Sun's agrā) where is the Sun's sayana longitude. and LSBA - 90°- $. Now in the triangle SAB, right-angled at A, we have SA Rsin a, AB= Rsin (Sun's agrā), S A Rsina x 1397 Rcos Rsin (Sun's agrā) x Rcos Rsin 85 Fig. 8 (1) (2) B Rsin a = where a is the Sun's altitude. The necessary conditions for the existence of the prime vertical altitude of the Sun are: (i) that the Sun should be in the northern hemi- sphere, and (ii) that the Sun's declination should be less than the latitude of the place. The condition laid down in the above rule that the Sun's agrā should be less than the latitude is incorrect. This error is, in fact, due to Aryabhata I, whom the author is following in this work. But the 1 See A, iv. 30-31, 86 author seems to be unaware of the error. For, in his later work, the Laghu-Bhaskariya¹, he has rectified the error and has stated the condition correctly. The above error of Aryabhata I was noticed and criticised by Brahmagupta. The commentators of Aryabhata I, however, have inter- preted the above rule as conveying the correct meaning. DIRECTION, PLACE AND TIME A rule for the determination of the time in asus to elapse before or elapsed since midday when the Sun is on the prime vertical, i.e., the time taken by the Sun in going from the prime vertical to the local meridian or vice versa: 39. Multiply the Rsine of the Sun's longitude (when the Sun is on the prime vertical) by the Rsine of the (Sun's) greatest declination and divide (the product) by the day-radius: by the result thus obtained multiply the Rsine of the colatitude and (then) divide by the Rsine of the latitude. Subtract the square of the resulting quantity from the square of the radius: the are corresponding to the square root of that gives the asus (measured on the equator from the Sun's hour circle) up to the meridian. Let m denote the minutes in the arc of the Sun's diurnal circle intercepted between the Sun and the six o'clock circle, and r (= Rcos 8) the radius of the Sun's diurnal circle. Then in Fig. 8, SC = rsin m; so that from the triangle SCA, right-angled at C, we have or, SC CA rsin m Rsin SAC Rsin CSA Rsin 8 x Rcos Rsin Rsin x Rsin € R Rcos ¹ iii. 22. 2 See BrSpSi, xi. 22. Rsin 2 (using stanza 6) where λ, & denote the Sun's sayana longitude and declination respectively, and denotes the latitude of the place; € is the Sun's greatest declination. 87 TIME TO ELAPSE BEFORE OR ELAPSED SINCE MIDDAY Therefore, the Rsine of the corresponding arc of the equator, i. e., Rsin 2 x Rsin € Rsin m X. r so that Therefore, = Rsin 2 x Rsin € day-radius X₁ Rsin (90⁰ m) 1 R² (Rsin m)². 5400' -m= Rsini [Rsin (90° — m)], which gives the required time in asus, because 5400'- m is the number of minutes in the arc of the equator lying between the Sun's hour circle and the local meridian. Rcos Rsin An alternative rule: 40. Or, multiply the Rsine of the Sun's prime vertical zenith distance by the radius and divide by the day-radius. Then applying the method of finding out the arc (correspond. ing to a given Rsine), convert the resulting Rsine into (the corresponding) arc. Then reduce the asus (thus obtained) to nadis, etc. These lie between the (prime vertical) Sun and the meridian. SA = Rsin z, SB = LSAB = 90°, and LSBA= the angle between the Sun's day-radius, Rcos Rsin In Fig. 9, let S be the Sun on the prime vertical, B the centre of the Sun's diurnal circle, and A the point where the plane of the Sun's diurnal circle intersects the zenith-nadir line. Then in the plane triangle SAB, we have Rsin LSBA= hour circle and the local meridian, z being the Sun's zenith distance. Rsin z X R day-radius A Fig. 9 = arc of the equator intervening between the Sun's hour circle the local meridian. Therefore, we have B 88 DIRECTION, PLACE AND TIME A rule for finding the Sun's longitude when the the prime vertical: 41. Multiply the Rsine of the latitude by the Rsine of the prime vertical altitude derived from the shadow (of the gnomon) and always divide the product by the Rsine of the Sun's greatest declination; (then reduce the resulting Rsine to the corresponding arc). The arc thus obtained, or(the complement of) that are increased by three signs is the longitude of the (prime vertical) Sun (according as the Sun is in the first or the second quadrant). This is in accordance with what (Arya)bhata has written.¹ That is to say, ifλ and a be respectively the sayana longitude and altitude of the Sun when it is on the prime vertical, and the latitude of the place, then Rsin a X Rsin Rsin E where is the Sun's greatest declination. Rsin λ === Therefore The rationale of this rule is as follows: Referring to Fig. 8, we see that in the triangle SAC SA Rsin a, AC = Rsin 8, LACS = 90°, and LASC = 4. Rsin 8 Rsin a X Rsin R Also from stanza 6, we have Rsin λ = " Rsin 8 x R Rsin € Sun is on Hence, eliminating Rsin 8 between (1) and (2), we get Rsin λ = Rsin ax Rsin Rsin € (1) (2) The next eleven stanzas relate to the determination of the locus of the end of the shadow of the gnomon. ¹ Cf. LBh, iii. 24-25. THREE POINTS ON THE LOCUS OF THE SHADOW-END 89 To determine at any given time three points lying on the locus of the shadow-end : 42-45. Determine the agrā and sankvagra of the Sun and also the shadow (of the gnomon) for the desired time. Then take the sum of the two agras (i.e., of agra and sankvagra) provided that they are of like direction¹; in the contrary case, take their difference. (This sum or difference is called the bhuja and its direction corresponds to that of the Sun from the prime verti- cal.) By that (sum or difference) multiply the shadow and divide (the product) by the Rsine of the Sun's zenith distance (for that time). Whatever is thus obtained should be laid off from the centre in the contrary direction (i.e., contrary to the direction of the Sun from the prime vertical) within a circle (drawn on level ground and) marked with the directions. Through the fish-figure drawn about the point thus obtained stretch out a thread in the east and west directions (bothways) to a distance. Then take a thread equal in length to the shadow (of the gnomon) and lay it off from the centre obliquely (to meet the other thread). Mark the two points where the end of this thread meets (the other thread stretched out) in the east and west directions. At the end of the midday shadow take a third point. The sum or difference of the agrā and śaḥkvagra² of the Sun denotes the distance of the Sun's projection on the plane of the horizon from the east-west line. This is called the bhuja and its direction is the same as that of the Sun from the prime vertical. This multiplied by the length of the shadow of the gnomon and divided by the Rsine of the Sun's zenith distance gives the distance of the end of the shadow of the gnomon from the east-west line. Suppose, for example, that the sun is towards the south of the prime vertical, and that the end of the shadow of the gnomon is at a distance d from the east-west line. Let the circle in Fig. 10 be the one constructed on level ground with ¹ The agra is north or south according as it is towards the north or south of the east-west line; the sankvagra is always south. 2 For agrā see supra stanzas 37-38, and for sankvagra see infra stanza 54, 90 DIRECTION, PLACE AND TIME respectively. the foot of the gnomon as centre and any arbitrary radius. Let EW and NS be the east-west and north-south lines Then the construction stated in the text is as follows: From O, along ON, measure a distance d and mark there a point, say X. Through X stretch a thread (or draw a line) parallel to EW. Let this be LM. Now from centre O and radius equal to the shadow of the gnomon draw an arc of a circle cutting LM at the points A and B. Then OA and OB are the positions of the threads, equal in length to the shadow of the gnomon, laid off obli- quely from the centre. The two points thus obtained are A and B. If OC be the midday shadow of the gnomon, then the thirdp oint is C. The three points thus determined are A, B, and C. E Fig. 10 A rule for determining the same three points when the directions are not known: S W M A L N 46-51. For one who does not know the directions with regard to the centre but wants to determine the directions and the locus of (the end of) the shadow (of the gnomon), I state the method such that (the end of) the shadow (cf the gnomon) may not leave the periphery of the large circle distinctly drawn amidst the directions (representing the path of the shadow-end). Calculate the Rsine of the Sun's zenith distance, the Rsine of the Sun's altitude, and the sankvagra corresponding to the desired shadow; then take the difference or sum of the two agras (i.e., of the sankvagra and the agra) (according as they are of unlike or like directions). This (difference or sum) is the base, the Rsine of the Sun's zenith distance is the hypotenuse, and the square root of the difference between their squares is called the upright corresponding to the hypotenuse equal to the Rsine of the Sun's zenith distance. On multiplying these, upright and base, (severally) by the (length of the) shadow and dividing by the Rsine the Sun's zenith distance are obtained LOCUS OF THE SHADOW-END 91 the values of the upright and the base (corresponding to the hypotenuse equal to the shadow of the gnomon). Now take straight bamboo scales of breadth equal to the diameter of the gnomon and of lengths equal to the base and the upright (obtain- ed above) (and one equal to the shadow); and with their help construct an accurate framework instrument in the form of a rectangle or a (right-angled) triangle having the shadow (of the gnomon) for the hypotenuse. At the corner (where the base and the hypotenuse meet) fix a gnomon. Then put the instru- ment (on level ground) and rotate it (about the gnomon) until the shadow (of the gnomon) falls along the hypotenuse. Then having determined the directions (north and south, east and west) as indicated by the base and the upright, mark two points at the end of the shadow-hypotenuse (one towards the east and the other towards the west). At the end of the midday shadow, take the third point. The above stanzas give the method for finding the points A, B, and C (see Fig. 10), when the lines EW and NS are not known. The method. is as follows: Construct a triangular frame of bamboo scales having its sides equal to those of the triangle AXO or a rectangular frame having its sides equal to those of the rectangle OXAY, and let a gnomon (having its diameter equal to the breadth of the bamboo scales) be fitted at O vertically to the frame. Place the frame on level ground with the point O at the centre of a circle drawn on the ground and with the hypotenuse OA in coincidence with the shadow of the gnomon. Then OX is directed south to north and AX west to east. Draw the lines OX and AX on the ground and produce them bothways. With centre O and radius equal to OA, the shadow of the gnomon, draw an arc of a circle cutting the line AX at two points. These are the points A and B. The point C is obtained as before. Construction of the locus of the end of the shadow of a gnomon: 52. Through all the three points (thus obtained) a circle is (then) drawn with the help of two fish-figures. The shadow (of the 92 DIRECTION, PLACE AND TIME gnomon) moves, like a spell-bound serpent, with its head (i.e., end) kept upon the periphery of that circle.¹ Lalla says: "With the point of intersection of the head-tail lines (of the two fish-figures) as centre, draw a circle passing through the ends of the three shadows. (The end of) the shadow (of the gnomon) does not leave the periphery of that circle in the same way as a lady born in a noble family does not leave the traditions of her family." Sripati (c. 1039 A. D.) writes: "The (end of the) shadow of the gnomon erected at the centre of the circle does not leave the periphery of the circle drawn with the common point of the head and tail lines (of the two fish-figures) and passing through the three shadow-ends in the same way as a noble-minded person does not leave the path of righteousness." The above rule was later criticised by Bhaskara II (1150 A. D.), who made the following remark : "It is declared (by some astronomers) that the shadow of the gnomon moves on the circle passing through the ends of the three shadows made by the same gnomon (placed in the centre of the horizon), but this is wrong, and consequently the east-west and north-south lines, the latitude, etc., found by the aid of the circle just mentioned are also wrong".4 Bhaskara II's criticism is proper. The locus of the end of the shadow of a gnomon will not be a circle as stated in the text unless the latitude of the place is 90°. The locus will be, in general, a conic section. For places whose latitudes are less than € ( being the obliquity of the ecliptic), in particular, this locus will be a hyperbola. Alternative rules for finding the Sun's agra and the latitude of the place: 53. The squareroot of the sum of the squares of the 1 This rule occurs also in BrSpSi, iii. 2-3; ŚiDVṛ, I, iii. 3; Siśe, iv. 5; TS, iii. 42(ii)-47. 2 SiDV, I, iii. 3. 3 Sise, iv. 5.

• L. Wilkinson, Translation of the Siddhanta-siromani (Goladhyaya,

xi. 38(ii), Calcutta (1861), p.221. We have replaced the word "revolves" by "moves". SANKVAGRA AND EQUINOCTIAL MIDDAY SHADOW Rsine of the (Sun's) declination and the earthsine for the desired time is (the Rsine of) the Sun's agrā (for that time). The earthsine multiplied by the radius and divided by (the Rsine of) the Sun's agrā is the Rsine of the latitude. That is Rsin (Sun's agrā) and Rsin = - ✓(Rsin 8) + ( earthsine )2, earthsine x R Rsin (Sun's agrā) where is the Sun's declination and the latitude of the place. These results easily follow from the triangle ACB in Fig. 8. A rule for the determination of the samkvagra: 54. The Rsine of the altitude for the desired time multi- plied by the Rsine of the latitude and divided by (the Rsine of) the colatitude is the sankvagra, which is always south (of the rising-setting line).¹ SA = Rsin a, AB 93 The sankvagra (of a heavenly body) is the distance of the projection of the heavenly body on the plane of the horizon from its rising- setting line. In Fig. 8, the line AB denotes the Sun's śaḥkvagra. The sank vagra is measured from the rising-setting line and is always to the south of that line. It is more commonly known a sankutala. In Fig. 8, we have sankvagra, śankvagra Rsin a Rcos which gives the formula stated in the text. (1) (2) LSAB 90⁰, and SBA 90⁰, where a denotes the altitude, and the latitude of the place. Therefore from the triangle SAB, we have Rsin 5 A rule for finding the equinoctial midday shadow: 55. Multiply the samkvagra by 12 and divide by the Rsine of the altitude: the result is (the length of) the equinoctial 1 This rule occurs also in A, iv. 29 and in LBh, iii. 16. 94 DIRECTION, PLACE AND TIME (midday) shadow (of the gnomon). The details are being stated below. We have proved above (stanza 54) that sankvagra Rsin a Therefore - Rsin Rcos Also, from stanza 5 above, we have Rsin Rcos . equinoctial midday shadow 12 sankvagra x 12 Rsin a equinoctial midday shadow=: Method of finding the Sun's agra by observation and deriving therefrom the sankvagra and then the equinoctial midday shadow of the gnomon and the latitude or colatitude of the place: 56-60(i). One should erect a (circular) platform, as high as one's neck, with its floor in the same level, and its circumference graduated with the divisions of signs, degrees, etc., and bearing the marks of the directions. (Then standing) on the western side thereof, one, having undisturbed state of mind, should, with the line of sight passing through the centre of the circular base, make the observation of the Sun when (at sunrise) it appears clinging to the circunference, (and mark there a point). The (arcual) distance, measured along the circumference graduated with the marks of degrees, between the end of the line drawn eastwards (i.e., the east point) and the point where the Sun is observed is the arc of the Sun's agrā. The Rsine of that (arc) is (the Rsine of) the Sun's agra. The minutes of the difference between that (Rsine of the Sun's agra) and the Rsine of the Sun' meridian zenith distance are the minutes of the sankvagra, provided that the Sun is in the southern hemisphere; when the Sun is in the northern hemi- phere (and the shadow of the gnomon falls towards the north), the process is otherwise (i.e., the addition of the two). When, however, (the Sun being in the northern hemisphere) the PROMINENT STARS OF THE NAKSATRAS shadow (of the gnomon) due to the Sun falls towards the south, the Sun's agra minus the Rsine of the Sun's meridian zenith distance is stated to be (the value of) the sankvagra. From that (samkvagra) determine the true value of the equinoctial midday shadow (of the gnomon),' and then calculate as before the lati- tude and colatitude (for the place). A rule for finding the longitude of an unknown planet with the help of (i) the longitude of a known planet and (ii) the difference between the times of rising or setting of the known and unknown planets: 95 60(ii)-61. Having correctly ascertained in terms of nāḍikās (i.e., ghatis) the difference between (the times of rising or setting or culmination of) the planets to be known and known, multiply those ghatis by six. Thus are obtained the degrees (of the difference between the longitudes of the two planets). By those degrees diminish or increase the longitude of the known planet according as it is to the east or west of the planet to be known. This is stated by the learned people well versed in planetary motions (to be the method for getting the the longitude of the planet to be known). This rule is approximate, as the inclinations of the orbits of the two planets to the ecliptic have been neglected and the ecliptic has been suppos- ed to rise above the horizon uniformly at the rate of six degrees per ghati (i.e., 15 degrees per hour). (2) JUNCTION-STARS OF THE ZODIACAL ASTERISMS AND CONJUNCTION OF PLANETS WITH THEM. Longitudes of the prominent stars of the naksatras: 62. In this way from (the known longitudes of) the planets or stars have been determined, at all places and at all times, the celestial longitudes of the (prominent) stars of the nakṣatras. The nakṣatras are the twenty-seven zodiacal asterisms. In the following table we give a few details regarding these nakṣatras. ¹ Vide supra stanza 55, 96 Name Aśvini Bharani Kṛttikā Rohini JUNCTION-STARS AND CONJUNCTION OF PLANETS WITH THEM Names, Shapes, and Number of the Stars of the Nakṣatras Mṛgaśirā Ārdrā Punarvasu Pusya Äśleṣā Magha Pūrvā Phalguni Shape¹ Head of a horse Yoni Razor Cart Head of a deer Jewel House Arrow-head Wheel House Manca² Uttara Phâlguni Cot Hasta Hand Number of stars as given by Varāha- mihira 3 3 10 6 5 3 1 5 3 6 5 8 2 5 Brahma- guptas ¹ Muci, ii. 59-60. BrSam, xcvii. 1-2. KK (Sengupta's edition), x. 1-2. 2 3 6 5 3 1 2 1 6 6 2 2 5 Lalla 3 3 6 5 3 1 4 3 5 5 2 2 5 Identification d, ß, y Arietis 35, 39, 41 Arietis n Tauri, etc. (Pleiades) d, 0, 7, 8, E, Tauri (Hyades) λ, ₁, ₂ Orionis d Orionis ß, di, v, " Geminorum 0, 8, 7 Cancri €, 8, o, n, d, n, Y, , 8, 0 Leonis Hydrae , E Leonis B, 93 Leonis 8, 7, E, d. ß Corvi (Corvus) • Ratna-Kosa.

• E. Burgess, Translation of the Surya-Siddhanta, Calcutta (1935), p. 378.

i.e., the femal organ of generation. 7 Mañca means an elevated platform resting on columns, Name Citrā Svāti NAMES, SHAPES, AND NUMBER OF STARS OF THE NAKSATRAS Višākhā Anuradhā Jyesthā Mula Pūrvā- sādha Shape Pearl Coral bead An arched doorway Heaps of offering to gods Ear- pendant Tail of a lion Tusk of an elephant Uttarā- ṣādha Śravaṇa Dhanisthā Drum Satabhisak Circle Manca Three feet Pūrva- Bhadrapada Manca Uttara- Bhadrapada Pair Revati Drum Number of stars as given by Varaha- mihira 1 1 5 4 3 11 2 3 3 5 100 2 8 32 Brahma- gupta 1 1 2 4 3 2 4 4 3 5 1 2 2 1 Lalla 1 1 4 4. 3 11 2 2 3 4 100 2 2 32 Identification o Virginis (Spica) o Bootis (Arcturus) i, Y, ß, & Librac 8, ß, Scorpionis 97 a, o, Scorpionis d, v, K, 6, 8, n, K, , E Scorpionis 8, € Sagittarii o, & Sagittarii d, ß, y Aquila ß, d, y, 8 Delphini > Aquarii, etc. d, ß Pegasi y Pegasi, o Andromedae Piscium, etc. 98 JUNCTION-STARS AND CONJUNCTION OF PLANETS WITH THEM Positions of the junction-stars of the asterisms (naksatras) in the twelve signs, Aries, etc: 63-66(i). In Aries, eight, twenty-seven; in Taurus, six, nineteen; in Gemini, two, ten; in the next sign (i.e., Cancer), two, fifteen, twenty-four; in Leo, eight and a half, twenty-one; in Virgo, four, twenty-three; in Libra, five, seventeen; in the next sign (i.e., Scorpio), two, twelve, eighteen; in Sagittarius, one, fourteen, twenty-seven; in Capricorn, fifteen, twenty-six; in Aquarius, seven, twenty-eight; in the last sign (i.e., Pisces), fifteen, thirty-these are the degrees of the positions (with reference to the signs) of the junction-stars of the naksatras beginning with Ašvini. · The prominent stars of the nakṣatras which were used in the study of the conjunction of the planets, especially the Moon, with them are called junction-stars (yoga-tārā). The study of the conjunction of the planets with the junction-stars was originally meant to verify the computed longitudes of the planets with a view to test the accuracy of and to make improvements, if necessary, in the astronomical theories on which those computations were based. The longitudes of the junction-stars which have been enumerated in the text are, in some cases, slightly at variance with those given by the author in his subsequent smaller work, the Laghu-Bhaskariya. The differences are exhibited by the following table: 1 "Of the stars in each naksatra, those that are seen to be the brightest are the junction-stars." KK (Sengupta's edition), x. 3(i). The author of the Surya-siddhanta gives the relative position of the junction- star in each nakṣatra, which facilitates their identification. See SuSi, viii. 16-21. LONGITUDES OF THE JUNCTION-STARS Differences between the Longitudes of the Junction-Stars in the Two Works of Bhaskara I Junction-star of Asvini Bharani Kṛttikā Rohini Mrgasirā Ārdrā Punarvasu Pusya Āśleṣā Maghā Pūrvā Phlaguni Uttarā Phalguni Hasta Citrā Svāti Visakhā Anuradhā Jyesthä Müla Pūrvāsādha Uttarāṣādha Maha-Bhaskariya Laghu-Bhaskariya 80 8° Longitude given in 270 1S 60 1S 19⁰ 25 20 2⁹ 10⁰ 3S 2⁰ 3s 15⁰ 3S 24⁰ 45 8° 30' 4 21⁰ 58 58 40 23° 68 6° 17° 7° 2° 7° 12° 7s 18⁰ 5° 8 1⁰ 85 14° 8S 27° 26° 30' 1° 6° 1° 20° 2° 2° 2° 10° 38 2° 3 15⁰ 3⁹ 24° 45 8° 30° 4° 21° 5 4° 23° 65 50 6⁹ 17⁰ 7° 2° 7⁹ 12° 7⁹ 18⁰ 8$ 1° 30' 99 8 14° 30' 85 26° 30' Difference - 30'. +1° + 30' + 30' - 30' 100 JUNCTION-STARS AND CONJUNCTION OF PLANETS WITH THEM Junction-star of Śravaṇa Dhanisthā Satabhisak Pūrva-Bhadrapada Uttara-Bhadrapada Revati Junction-star of Aśvini Bharani Kṛttikā Rohini Mṛgaśirā Ārdrā Punarvasu Maha-Bhaskariya Laghu-Bhaskariya Longitude given in 9$ 15⁰ 9s 26° 10⁹ 7⁰ 10⁹ 28° 11° 15° 12⁰ MBh 8° 27° 36° 49° 62° 70° 92° Longitude according to The following table gives the longitudes of the junction-stars of the nakṣatras as stated in the Maha-Bhāskariya, the Laghu-Bhaskariya, the Sisya-dhi-vṛddhida, the Surya-siddhanta, the Siddhanta-sekhara, and the Siddhanta-siromani : Longitudes of the Junction-Stars according to various Hindu Authorities LBh 8⁰ 8° 26°30' 20° 36° 50° 62° 70° 92° 9s 36° SiDVr 49° 62° 98 70° 92° 14° 30' 10⁹ 7⁰ 10⁹ 28° 11$ 15° 12⁹ 25° 30' Difference Susi - 30' 89 20° 37°30' 49°30' 63° 67°20' 93° <-30' (Polar) longitude according to Br SpSi, KK, Sise Sisi 8° 20° 37°28' 49°28' 63° 67° 93° Junction-star of Pusya Āśleṣā Citrā Svāti Visakhā Anuradhā Jyestha Müla LONGITUDES OF THE JUNCTION-STARS. Purvasadha Uttaraṣaḍha Śravaṇa Dhanistha Satabhisak Longitude according to Magha Pūrvā Phālguni 141° Uttarā Phälguni 154° Hasta 173° Uttara-Bhadra- pada Revati MBh 105° 105° 114° 114° 128°30' 128°30' 141° 154° 154° 173⁰ 173° 185⁰ 184°20' 197° 197° 197° 212° 212° 212° 222° 222° 222° 228° 228° 228° 241° 241°30' 241° 254° 254°30' 254° 267° 266°30' 267°20' 285° 284°30' 283°10' 280° 296° 307° Pūrva-Bhadra- 328° pada 185° LBh SiDVr 360° 105° 114° 128°

360° 345° 345° 335°20' (Polar) longitude according to SuSi 106° 109° 129° 139°20' 144° 155° 170° 180° 199° 213° 224° 229° 241° 254° 359° 295°30° 296°20. 2900° 307° 313°20' 320° 328° 327° 326° 260° 337° 359°50' Br.Sp.Si, Sise, Sisi 106° 108° 129° 147° 155° 170° 183° 199° 212°5' 224°5' 229°5' 241° 254° 260° 278° 290: 320° . 326° 337⁰: 101 0 1.02 JUNCTION-STARS AND CONJUNCTION OF PLANETS WIT HTHEM The longitudes given in the Maha-Bhaskariya, the Laghu-Bhaskariya, and the Sisya-dhi-vrddhida are the usual celestial longitudes, whereas those given in the Surya-siddhanta, the Brahma-sphuta-siddhanta, the Siddhanta- sekhara and the Siddhanta-siromani are the polar longitudes.¹ This explains why there are significant differences between them, but, as should be expected, the celestial longitudes as also the polar longitudes exhibit general agreement amongst themselves. The minor differences that occur in a few cases are probably due to the errors of observation or to the different methods used. Celestial latitudes of the junction-stars, definition of the conjunction of a planet with a star, and a rule for determining the distance between a planet and a star when they are in conjunction: 66(ii)-71(i). North, ten, twelve, five; south, five, ten, .eleven; north, six, zero; south, seven, zero; north, twelve, thirteen; south, seven, two; north, thirty-seven; south, one and a half, three, four, eight plus one-third, seven, seven plus one- third; north, thirty, thirty-six; south, eighteen minutes; north, twenty-four, twenty-six, zero-these have been stated by the learned to be the degrees of the celestial latitudes of the junction- stars of the nakṣatras beginning with Aśvini. All planets having their longitudes equal to those of the junction-stars are seen in conjunction with them. The distance between a planet and a star (when they are in conjunction) is determined from their celestial latitudes.² The celestial latitudes of the junction-stars stated above are exhibited in the following table. We also give the celestial latitudes stated in the other important works on Hindu astronomy. 1 That is, the longitudes of those points where the secondaries to the equater passing through the junction-stars intersect the ecliptic. 2 See MBh, vi. 54, Celestial Latitudes of the Junction-Stars according to various Hindu Authorities Junction-star of Aśvini Bharani Kṛttikā Rohini Mrgasira Ārdră Punarvasu Pusya Āśleṣā Maghā Purva-Phälguni Uttara-Phälguni Hasta Citrā Svāti Višākhā Anurādhā Jyesthā Müla LATITUDES OF THE JUNCTION-STARS Pūrvāsādha Uttarāsādha Latitude given in MBh 10° N 12° N 5° N 5° S 10° S 11° S 6° N 0 7° S 0 12° N 13° N LBh SiDVr 7° S 10° N 7°20' S 12° N 5° N 5° S 10° S; 11° S 6° N 0 7° S 0 12° N 13° N 7° S 2° S 10° N 12° N 4° S 5° N 5° S 7° S 2° S 37° N 37° N 37° N 1°30' S 1°30' S 1°30' S 3° S 7° S 7° S 10° S 11° S 6° N 0 7° S 0 12° N 13° N 3° S 4° S 8°20' S 8°30' S 8°30' S 5°20' S 5° S 8° S 2° S -3° S 4° S (Polar) latitude given in Br Sp Si, KK, Sise Sisi Susi 10° N 12° N 5° N 5° S 10° S 9° S 6° N 0 7° S 0 12° N 13° N 11° S 2° S 37° N 1°30' S 3° S 4° S 9° S 5°30' S 5° S 10° N 12° N 4°31' N 4°33' S 10° S 11° S 6° N 0 7° S 0 12° N 13° N 11° S 1°45' S 37° N 1°23' S 1°44′' S 3°30' S 103 8°30' S 5°20' S 5° S 10° N 12° N 4°30' N 4°30' S 10° S 11° S 6° N 0 7° S 0 12° N 13° N 11° S 1°45' S 37° N 1°20' S 1°45' S 3°30' S 8°30' S 5°20'S 5° S 104 JUNCTION-STARS AND CONJUNCTION OF PLANETS WITH THEM Junction-star of Śravaṇa Dhanisthá Satabhisak Pūrva-Bhadra- pada Uttara-Bhadra- pada Revati Latitude Given in MBh 30° N 36° N 18' S 24° N 26° N 0 LBh 30° N 36° N 18' S 24° N 26° N 0 SiDV 30° N 36° N 20' S 24° N 26° N 0 (Polar) latitude give in BrSpSi, KK, SiSe Sisi Susi 30° N 36° N 30' S 24° N 26° N 0 30° N 36° N 18' S 24° N 26° N 0 30° N 36° N 20' S [24° N 26° N 0 Celestial latitudes of the Moon when she occults some of the prominent stars of the zodiac: 71 (ii)-75(i). It is stated that the Moon, moving towards the south of the ecliptic, obliterates (i.e., occults) the cart of Rohini (i. e., the constellation of Hyades), when her latitude amounts to 60 minutes; the junction-star (of Rohini) (i. e., Aldebaran), when her latitude amounts to 256 minutes; (the junction-star of) Citra (i.e., Spica), when her latitude amounts to 95 minutes; (the junction-star of) Jyestha (i.e., Antares), when her latitude amounts to 200 minutes; (the junction-star of) Anuradha' when her latitude amounts to 150 minutes; (the junction-star of) Satabhisak (i. e., ^ Aquarii), when her latitude amounts to 24 minutes; (the junction-star of) Viśākhā², when ¹ According to H. T. Colebrooke and E. Burgess, it is & Scorpii. According to Bentley, it is ß Scorpii.

• According to Colebrooke, it is d or * Librae; according to

Whitney and Burgess, it is 24 Librae, MOON'S LATITUDE WHEN SHE OCCULTS CERTAIN STARS 105 Piscium), her latitude amounts to 88 minutes; and (the junction-star of) Revati (i.e., when her latitude vanishes. When she moves towards the north (of the ecliptic), she occults the nakṣatra Kṛttikā (i.e., the Pleiades), when her latitude amounts to 160 minutes; and the central star of the naksatra Magha, when she assumes the greatest northern latitude. These minutes (of the Moon's latitude) which have been stated (here) in connection with the occultation of a star by the planet (Moon) are based on actual observation made by means of the instrument (called) Yasti.¹ ¹ The occultations of the stars of the nakṣatras are stated also in Susi, viii. 13; BrSpSi, x. 11-12; KK, x. 15-16; LBh, viii. 11-16; ŚiDV, I, xi. 11; Sise, xii. 8-9. CHAPTER IV TRUE LONGITUDE OF A PLANET Definition of the Sun's mean anomaly: 1. Having applied the correction for the (local) longitude to the mean longitude of the Sun, subtract (therefrom) the longitude of the Sun's apogee (ucca): the remainder is the Sun's (mean) anomaly. In that (anomaly), three signs form a quadrant. That is to say, Sun's mean anomaly (mean longitude of the Sun)- (longitude of the Sun's apogee). The Sun's apogee is the remotest point of the Sun's apparent orbit. Beginning with the Sun's apogee, in Hindu astronomy, the kakṣāvṛtta (deferent or concentric) for the Sun is divided into four equal parts called anomalistic quadrants (pada or päda) and twelve equal parts called ano- malistic signs (rāśi). Thus there are three anomalistic signs in an anomalistic quadrant. The anomalistic signs are given the same names as the signs of the zodiac, viz., Aries (mesa). Taurus (vrsa), Gemini (mithuna), etc. When the mean anomaly of the Sun is between 0⁰ and 180°, the Sun is said to be in the six signs (or in the half-orbit) beginning with the anomalistic sign Aries; and when the mean anomaly of the Sun is between 90⁰ and 270°, the Sun is said to be in the six signs (or in the half-orbit) commencing with the anomalistic sign Cancer; and so on. The same is true also for the planets, Mars, etc. A rule relating to the Rsine of the Sun's mean anomaly: 2. (Of the parts of the Sun's mean anomaly lying) in the odd quadrants, calculate the Rsine; and (of the parts lying) in the even quadrants, calculate the Rversed-sine. The method ¹ This rule occurs also in SuSi, ii. 29; Br.Sp.Si, ii. 12(i); ŚiDVy, I, ii. 10; Sise, iii. 12; Siśi, I, ii. 18-19(i). 2 Or, that the mean anomaly is in the six signs (or in the half-orbit) beginning with the sign Aries. 107 for finding the Rsines (i.e., Rsines and Rversed-sines) is being told in detail (below). FINDING THE RSINE OF AN ARC For example, if the Sun's mean anomaly is 140°, calculate the Rsine of 90° and the Rversed-sine of 50%; if the Sun's mean anomaly is 240⁰, calculate the Rsine of 90°, the Rversed-sine of 90°, and the Rsine of 600; and if the Sun's mean anomaly be 300°, calculate the Rsine of 90⁰, the Rversed-sine of 90°, again the Rsine of 90°, and the Rversed- sine of 30⁰. The above passage shows that in the time of Bhaskara I one of the methods used for finding the Rsine of an arc (>90°) was to apply the following formulae; Rsin (90°+0) = Rsin 90°- Rversin 0. Rsin (180⁰+0) = Rsin 90 Rversin 90°- Rsin 0 =Rsin 0. Rsin (270⁰ +0): = Rsin 90 Rversin 90°- Rsin 90⁰ + Rversin ( Rsin 90⁰ - Rversin 0, - where <90⁰. A rule for finding the Rsine (or Rversed-sine) of an arc (<90⁰): 3-4 (1). Reduce the arc to minutes and then divide by 225: the quotient denotes the number of (tabulated) Rsine- differences (or Rversed-sine-differences) to be taken com- pletely. Then multiply the remainder by the next (or current) Rsine-difference (or Rversed-sine-difference) and divide (the product) by 225. Add the quotient (thus obtained) to the sum of the (tabulated) Rsine-differences (or Rversed-sine-differences) obtained before. The sum thus obtained is the Rsine (or Rversed-sine) of the given arc.¹ This rule gives a method for calculating the Rsine or Rversed-sine of an arc by the help of the following table of Rsine-differences given by Aryabhata I in his Aryabhatiya². ¹ This rule occurs also in SuSi, ii. 31-32; BrSpSi, ii. 10; LBh, ii. 2 (ii)-3(i); ŠiDVṛ, I, ii. 12; Sise, iii. 15; Siśi, I, ii. 10(ii)-11. 2 i. 12. This table has been referred to in MBh, vii. 13, 108 Serial No. 2 !Serial No. 6 3 4 Rsine- difference in minutes 5 Rversed-sine- difference in minutes 7 TRUE LONGITUDE OF A PLANET 65 79 Serial No. 8 9 10 11 12 Serial No. Table of Rsine-differences 8 9 Rsine- difference in minutes 10 205 199 191 183 174 164 11 12 This table gives the Rsine-differences corresponding to the twenty- four elementary arcs in which a quadrant of a circle is divided, each elementary are being equal to 225 minutes. The same table reversed becomes the table of Rversed-sine-differences. Rversed-sine- difference in minutes 93 106 119 131 Serial No. 143 13 Table of Rversed-sine-differences 154 14 15 16 17 18 Serial No. 13 14 15 16 Rsine- difference in minutes 17 154 143 131 119 106 93 18 Rversed-sine- difference in minutes 164 174 199 Serial No. 205 19 Serial No. 21 22 23 Rsine- difference in minutes 24 22 Rversed-sine- difference in minutes 210 215 219 222 224 225 FINDING THE RSINE OF AN ARC 109 The method for calculating the Rsine or Rversed-sine of an arc, as stated in the text, may be explained by means of an example as follows: Example. Calculate Rsin 32º and Rversin 32º. Reducing 32 degrees to minutes, we get 1920'. Dividing this by 225, we get 8 as the quotient and 120 as the remainder. (1) The sum of the first 8 Rsine-differences is 1719'. Multiplying the remainder 120 by the 9th Rsine-difference (viz. 191') and dividing the product by 225, we get 101' 52". Adding this to the previous sum, we get 1820' 52" or 1821' approx. This is the value of Rsin 32º. (2) The sum of the first 8 Rversed-sine-differences is 460'. Multi- plying the remainder 120 by the 9th Rversed-sine-difference (viz. 119') and dividing that product by 225, we get 63' 28". Adding this to the previous sum, we get 523' 28" or 523' approx. This is the value of Rversin 32°.¹ The above method for finding the Rsine of a given arc is evidently based on the simplest law of interpolation, viz. that of proportion. In later works, we come across more elegant methods of interpolation. We state here two of them. 1. Brahmagupta's formula. If <225' and t be an integer, then Rsin (225't+0)=sum of t Rsine-differences + 1 0' 225 { tth Rsine-diff. +(t+1)th Rsine-diff, 2 tth Rsine-diff. - (t+1)th Rsine-diff. 2 0' 225 = sum of t Rsine-differences 0' + •{(t+1)th Rsine-difference} 225 -}. 1 +7.225 (225-1) ((t+1)th Rsine-difference -tth Rsine-difference}. (2) Using modern four-figure tables and assuming that one radian= 206265", we get Rsine 32° 30° 21' 43" approx. and Rversin 32°=8°42′ 22" approx. This shows that the values derived from Aryabhaţa I's table give fairly good approximations to the Rsines and Rversed-sines up to minutes of arc. 110 TRUE LONGITUDE OF A PLANET Form (1) occurs in P. C. Sengupta's edition of the Khandakhadyaka¹ of Brahmagupta and also in the Siddhanta-siromani of Bhaskara II. Form (2) is found to occur in Parameśvara's commentary on the Laghu-Bhas- kariya³. This formula agrees with Newton's interpolation formula for equidistant knots. 2. Madhava's formula. If t be a positive integer and 0<225', then Rsin (225' t+0')=sum of t Rsine-differences + This formula is ascribed to Madhava by Nilakantha in his commen- tary on the Aryabhatiya. It occurs also in the Tantra-sangraha.5 In Chapter VII of the present work, Bhāskara I gives a very interes- ting method for finding the Rsine of a given arc without the use of a table.6 A rule for finding the Sun's equation of the centre: 4(ii). The Rsines and Rversed-sines (of the parts of the Sun's mean anomaly lying in the odd and even quadrants res- pectively) should be (severally) multiplied by the (Sun's) own epicycle and divided by 80: the resulting quantities should be subtracted and added (in the manner prescribed below).' 0x[Rcos (225'(t+1)} + Rcos (225't)] 2R Application of the Sun's equation of the centre: 5. The resulting quantities due to the first, second, third and fourth anomalistic quadrants should always be respectively subtracted from, added to, added to, and subtracted from the Sun's mean longitude corrected for the (local) longitude.Ⓡ 1 ix. 8. 2 3 4 5 8 I, ii, 16. ii. 2(ii)-3(i). ii. 12. ii. 10-13(i). See infra chapter VII, stanzas 17-19.

7 This correction is found also in BrSpSi,i i. 15(ii) and Sise, iii, 27. This correction is found also in BrSpSi, ii. 16(i). Also see Sise, iii. 28(i). SUN'S EQUATION OF THE CENTRE Alternative rule for the determination and application of the Sun's equation of the centre (called bahuphala): 6. Or, (find the bahuphala and) subtract the bahuphala when the (Sun's mean) anomaly is in the half-orbit beginning with Aries; and add that when (the Sun's mean anomaly is) in the half-orbit beginning with Libra. This correction should always be performed by one who seeks the true longitude (of the Sun).¹ Bāhu ( due to a planet's mean anomaly) is defined in stanza 8 below. It is the arcual distance of a planet from its apogee or perigee, whichever is nearer. The Sun's bahuphala is obtained by the following formula: (Sun's tabulated epicycle) In the adjoining figure, the bigger circle UMN, centred at the Earth E, is the Sun's mean orbit called kakṣāvṛtta (deferent); the small circles are nicoccavṛttas (epicycles); and U is the Sun's ucca (apogee). Under the epicyclic theory, the mean Sun is supposed to move on the deferent, and the true Sun is supposed to move on its epicycle (centred at the mean planet) with the same angular velocity as the mean Sun has relative to the apogee but in the opposite sense. (See the arrows). Sun's bahuphala- The Sun's tabulated epicycle is 3.² The Sun's bahuphala corresponds to the Sun's equation of the centre, which is shown by means of the Hindu epicyclic theory as follows: U1 80 Rsin (bahu due to the Sun's mean anomaly) U . 111 C E N Fig. 11 2 ¹ This rule is found to occur also in SuSi, ii. 39; SiDV, I, ii. 14; Si se, iii. 26(i). 2 Vide infra, vii. 16. It must be noted that the epicycles have been tabulated after abrading them by 41. 112 TRUB LONGITUDE OF A PLANET Initially, when the mean Sun is at U, the true Sun is at U₁. Subsequ- ently, when the mean Sun is at M, the true Sun is at T₁, such that MT, is parallel to EU. Since both the mean Sun and the true Sun have the same angular velocity relative to the apogee, the line MT, will always be parallel to EU. According to the Hindu astronomers, the tabulated (manda) epicycles are the mean epicycles (i.e., the epicycles corresponding to the mean dis- tances of the planets) whereas the true epicycles (on which the planets are supposed to move) are those which correspond to their true distances.¹ That is to say, the point T₁ in the above figure is not the position of the true Sun. According to the Hindu theory the true epicycle and the true position of the Sun, when the mean Sun is at M, are obtained as follows: Let C be a point in EU such that EC-UU₁. Join CT₁ and let it intersect the deferent at S. Produce ES and MT, to meet at T. Then MT is the radius of the true epicycle at M and T the true position of the Sun. Obviously, the epicycle varies from point to point. If denote the first point of Aries. Then Sun's mean longitude = arc TUM, and Sun's true longitude-arc TUS. The difference between the two, i.e., arc SM, is the Sun's equation of the centre. Let MA be perpendicular to EU and T,B, and SB be perpendi- culars to EM or EM produced. T₂B₁ is called bahuphala or bhujaphala and B₂M is called kotiphala. The triangles B₂MT₁ and MAE are similar. T₂B₁ T₁M MA EM' or, T₁B₁, i.e., Sun's bahuphala - T,MX MA EM Therefore, radius of Sun's mean epicycle )× Rsin (arc MU) R (Sun's tabulated epicycle) x Rsin (bahu due to the Sun's mean anomaly). 80 ¹ See BrSpSi, xxi. 29; Sise, xvi. 24; Sisi, II, v. 36-37. Also see Bhas- kara II's comm. on Siśi, II, v. 36-37; and the extract from the Adityapra- tāpa-siddhanta quoted by Amarāja in his comm. on KK, page 33. SUN'S EQUATION OF THE CENTRE But T₁B₁=SB-arc SM approx. Hence it follows that the Sun's bahuphala is the same as the Sun's equation of the centre. Sun If 0 dénote the bahu due to the Sun's mean anomaly, then according to Aryabhata I and Bháskara I Sun's bahuphala p 3x3438'xsin ( 80 128'-9 sin 0 = '0375 sin 0 radians. = 3x Rsin 0 80 = (1) According to Ptolemy, the greatest equation of the centre for the 2° 23'. Therefore, according to him, Sun's equation of the centre = = According to modern astronomy, Sun's equation of the centre - 113: 2°23' sin 143' sin 0 143 sin 0 3438 0416 sin radians radians. 2e sin , where e= .0167 0334 sin 0. 9 (2) (3) neglecting e and higher powers of e. Comparison of the results (1), (2), and (3) shows that the Hindu approximation for the Sun's equation of the centre in good enough and much better than that of Ptolemy. Addition and subtraction of the bahuphala. When the Sun's mean anomaly is less than 180°, the true Sun is behind the mean Sun; and when the Sun's mean anomaly is greater than 180⁰, the true Sun is in advance of the mean Sun. Hence the Sun's bahuphala is subtractive or additive according as the Sun's mean anomaly is in the half-orbit beginning with Aries or in the half-orbit beginning with Libra. The Sun's bahuphala correction is applied to the Sun's mean longi- tude as corrected for the longitude-correction (i.e., to the Sun's mean longitude for mean sunrise at the svanirakṣa place). And after the bahuphala correction has been made, we obtain the Sun's true longitude for mean sunrise at the svanirakṣa place, 114 Sun's correction for the equation of time due to the eccentricity of the ecliptic (called bhujantara or bhujavivara correction): 7. Multiply the mean daily motion (of the Sun) by the (Sun's) equation (of the centre derived from the Rsines and Rversed-sines of the parts of the Sun's mean anomaly lying in the odd ond even quadrants respectively) or by the (Sun's) bahuphala (i.e., the Sun's equation of the centre derived from the bahu) and then divide the product by the number af minutes in a circle (i.e., by 21600); apply that (as correction, positive or negative, to the Sun's mean longitude corrected for the local longitude and for the Sun's equation of the centre) as before. TRUB LONGITUDE OF A PLANET The bhujantara correction is the third correction to be applied to the Sun's mean longitude. By this correction "allowance is made for that part of the equation of time, or of the difference between mean and apparent solar time, which is due to the difference between the Sun's mean and true places". This correction having been applied to the Sun's longitude, we obtain the Sun's true longitude for true sunrise at the svanirakṣa place. In the case of the Moon and other planets, according to the com- mentator Parameśvara, this correction is applied to the mean longitude as corrected for the longitude-correction. The bhujantara correction is subtractive when the Sun's mean anomaly is less than 180° and additive when the Sun's mean anomaly is greater than 180°, because in the former case true sunrise occurs earlier than mean sunrise and in the latter case true sunrise occurs later than mean sunrise. The correction for "the equation of time due to the obliquity of the ecliptic" occurs for the first time in the Siddhanta-sekhara of Sripati (c.1039 ¹ Vide supra stanza 4(ii). & Vide stanzas 5 and 6. This rule occurs also in Susi, ii. 46; BrSp.Si, ii. 29(i); SiDV, I, ii. 16. E. Burgess, English Translation of the Sürya-siddhanta, Calcutta (1935), page 87. Vide his comm, on MBh, iv. 29-30. TRUE DISTANCE OF THE SUN OR MOON 115 A. D.) under the name of udayantara-saṁskāra.¹ It reappears in the works of Bhaskara II (1150 A. D.) and Nilakantha (1500 A. D.). Definitions of the bahu and koti (due to a planet's mean anomaly): 8. The portions (of the mean anomalistic quadrant) traversed and to be traversed (by a planet) are called bahu and koti or koti and bahu, according as the mean anomalistic quadrant (occupied by the planet) is odd or even. The bahuphala and kotiphala are obtained as before for the determination of the hypotenuse (i.e., the distance of the planet). A rule for the determination of the true distance in minutes of the Sun or Moon: 9-12. (When the Sun or Moon is) in the first or fourth (mean anomalistic) quadrant, add the kotiphala to the radius; (when) in the remaining (quadrants), subtract that from the radius: the resulting sum or difference is the upright. The square root of the sum of the squares of that and the bahuphala is called the hypotenuse. Multiply that hypotenuse (severally) by the bahuphala and kotiphala and divide (each product) by the radius: the results are (again) the bahuphala and kotiphala. From them obtain the hypotenuse (again) as before. Again multiply this hypotenuse (severally) by the initial bahuphala and kotiphala and divide (each product) by the radius. In this way, proceeding as above, obtain the hypotenuse again and again until two successive values of the hypotenuse agree (to minutes). (Thus is obtained the nearest approximation to the true distance in minutes of the Sun or Moon)." 1 Sise, xi. 1, 2 SiSi, 1, ii. 62-63. 3 TS, ii. 30. This definition is found also in SuSi, ii. 30; BrSpSi, ii. 12(ii); ŚiDVṛ, I, ii. 10-11; SiŚe, iii. 13(i); SiŚi, 1, ii. 19. 5 Cf. LBh, ii. 6-7. 116 TRUE LONGITUDE OF A PLANET Fig. 12 is a reproduction of the previous figure. As in the previous figure, M is the mean Sun and T the true Sun.. ET is the true distance of the Sun. The above rule relates to the determination of ET. The method used is the method of successive approximations. In the triangle ESD, where SD is parallel to EU, we have ES = R, and SD = T',M, the radius of the Sun's mean epicycle. If the value of TM, the radius of the Sun's true epicycle, were known, the Sun's true distance ET could be easily derived from a comparison of the similar triangles ESD and ETM. But the value of TM is unknown and is itself dependent on that of ET. Hence the necessity of the method of successive approximations (usakṛtkarma). B²B M1/D₂ D U C Fig. 12 With centre E and radius ET, draw an arc of a circle cutting ET at S₁; through S₁ draw a line S₁D, parallel to EU and a line S,T, parallel to EM meeting MT, produced at T₂; and from T₂ draw a line T₂B₂ perpendicular to EM produced. Again with centre E and radius ET, draw an arc of a circle cutting ET at S₂; through S, draw a line S₂D₂ parallel to EU and

1. line S₂T, parallel to EM meeting MT, produced at T3; and from T, draw

a line T,B, perpendicular to EM produced. Continue this process succes- sively. The sequence of points S₁, S₁, S,... and also that of points T₁, T₂, TRUE DISTANCE OF THE SUN OR MOON 117 T,...will converge to T. This is the basis of the method used. The details are as follows: MT, is taken as the first approximation r, to the radius of the Sun's true epicycle and likewise ES₁, which is equal to¹ R+kotiphala)2+(bahuphala), is taken as the first approximation H, to the Sun's true distance.² Now from the similar triangles S₁D₂E and SDE, SDXH₁_XH₁ S₂D₁ R But MT, S,D₁. Therefore, LXH₁ MT₂= R Again from the similar triangles T,B,M and MAE, we have T₂B₂= Similarly, B.M Therefore, MAXT,M R R = MAX₁, H₂ R bahuphalax H₁ R kotiphalax H₁ R ET,= R+MB₂)+T₂B₂³, where MB, and T₂B, are given by (3) and (2) respectively. (1) (2) (3). MT, is taken as the second approximation r₂ to the radius of the Sun's true epicycle and likewise ES, (=ET, ) is taken as the second approximation H, to the Sun's true distance. Since H, > R, therefore from (1) 1₂1₁; and consequently, H₂> H₂. ¹ For, ES₁=ET,; and from the right-angled triangle T₁B₁E, we have ET, EB,+B₁T₂³=(EM+MB₂)²+B₂T₁³, where EM-R, MB, is the kotiphala and B₂T, is the bahuphala. In the right-angled triangle EB₂T, BT, is called the base, EB, is called the upright, and ET, is called the hypotenuse. 118 TRUB LONGITUDE OF A PLANET Similarly, MT, MT, ,... are the next successive approximations 13, r₁, ... to the radius of the Sun's true epicycle, and ET3, ET4, ... are the next successive approximations H₂, H₁, ... to the Sun's true distance. As before, it can be easily shown that In < 'n+1 and H <H Moreover, from the method of construction 1₁, 1₂, 13, ... are each n less than MT, which is the upper bound of the sequence { }, and H₁, H₂, Hą, ... are each less than ET, which is the upper bound of the sequence {H}. Hence it follows that n+1. 1₁ < 1₂ < 1₂ <. <<... < MT. H₁ <H₂ <H₂ < ... <H₂ < ... < ET. and The sequences {n} and {H} are each monotonic and therefore convergent. The first converges to MT, the radius of the Sun's true epicycle, and the second to ET, the Sun's true distance. It may be seen that the sequences {n} and {H} converge rapid- ly so that the third or fourth approximation will give the result correct to the minute. In actual practice, however, the process of successive approxi- mations is carried on until two successive approximations are the same to minutes of arc. For this reason, this process is sometimes called aviseşakarma ("the process of reducing the difference to zero").¹ The method explained above is applicable to the Sun as also to the Moon. If in the above figure DM (which is equal to ST₂) be assumed to be equal to SS₂, i. e., H₁-R, we shall have ED=R-(H₁-R) =2R-H₁, and likewise, from the similar triangles SDE and TME, we shall get. ET = ESX EM ED R² 2R-H₂' 1 In the above discussion, we have assumed the Sun to be in the first anomalistic quadrant as shown in the figure. When the Sun is in any other quadrant, the procedure is similar. 119 which is Bhaskara II's approximation for the true distance of the Sun or Moon.¹ TRUB DAILY MOTION OF THE SUN OR MOON It may be pointed out here that the method of successive approxi- mations, which has been used for finding the true distance of the Sun or Moon in the stanzas under consideration, was used by Lalla for determi- ning the radii of the true epicycles of the Sun and Moon.² A rule for finding the true daily motion (called karnabhukti) of the Sun or Moon: 13. Always multiply the (mean) daily motion of the Sun or Moon by the radius and (then) divide (the product) by the hypotenuse (i.e., the true distance) determined by the method of successive approximations; the result is the true daily motion (of the Sun or Moon). That is Sun's true daily motion Sun's mean daily motion xR Sun's true distance in minutes Moon's mean daily motion X R Moon's true distance in minutes' and Moon's true daily motion= where R is the standard radius (-3438').³ ¹ See Si Si, I, v. 4. 2 See Ś¡DVṛ, I, ii. 44. 3 The object here is to obtain the angular velocity expressed in minutes, which will correspond also to the linear velocity in a circle of radius R. Parameśvara in his comm. on LBh, ii. 8 writes that, in place of the true distance in the above formulae, certain astronomers make use of the expression (kotiphala )² R RF kotiphala F (1) where - or + sign is taken according as the planet is in the half-orbit beginning with the anomalistic sign Cancer or in that beginning with the anomalistic sign Capricorn. t may be pointed out that the expression (1) is an approximate ue of H,, the second approximation to the pe distance. 120 TRUE LONGITUDE OF A PLANET The true daily motion determined from the above formulae is called karnabhukti ("the motion derived from the hypotenuse"). A rule for finding the true daily motion (called jivabhukti) of the Sun: 14. Or, multiply the current Rsine-difference by the (mean) daily motion (of the Sun) and divide by 225. Then multiply that by the (Sun's) own (tabulated) epicycle and divide by 80: the result thus obtained subtracted from or added to the (Sun's) mean daily motion (according as the Sun is in the half-orbit beginning with the anomalistic sign Capricorn or in that begin- ning with the anomalistic sign Cancer) gives the true daily mo- tion (of the Sun).¹ Let M and M' be the mean longitudes and S and S' the true longi- tudes of the Sun at sunrise yesterday and today respectively. Also let 0 and be the corresponding values of the bahu (due to the Sun's mean anomaly). Then, we have and S=MF S'=M'F. Rsin 0x1₁ 80 S'-S=(M'-M) F

m

Rsin 0' x 1₂ 80 where r, is the Sun's tabulated epicycle, or + sign being taken accord- ing as the Sun's mean anomaly is less than 180° or greater. Therefore, " (Rsin 0 Rsin {)Xr, 80 Rsin)XI, (Rsin' 80 where m denotes the Sun's mean daily motion, - or + sign being taken according as the Sun's mean anomaly is in the half-orbit beginning with the sign Capricorn or in that beginning with Cancer. ¹ This rule occurs also in ŚDVe, I, ii. 38 and Sise, iii. 40-41.

• Vide supra, stanza 8. 121

Neglecting the motion of the Sun's apogee and assuming that the Rsines vary uniformly, we have Rsin' Rsin Therefore, TRUE DAILY MOTION OF THE MOON S' Sm F (current Rsine-difference) xm 225 approx. (current Rsine-difference) > mxr₁ 225× 80 approx. Hence the above rule. Since the Sun's true daily motion has been obtained here with the help of the Rsines (jivā), therefore it is generally called jivābhukti.. A rule for finding the true daily motion (called jivabhukti) of the Moon: 15-17. (When the Moon is in the odd quadrant) subtract the part of the bahu due to her mean anomaly lying in the elementary are corresponding to the current Rsine-difference (antyajivādhanus- khanda) from the daily motion of the (Moon's) mean anomaly; (when the Moon is) in the even quadrant, subtract the remain- der obtained by subtracting that (part) from 225. Then take (as many) Rsine-differences in the reverse order, (if the Moon is) in the odd quadrant, or in the serial order, if the Moon is in the even quadrant, as correspond to (the above residue of) the motion of the (Moon's) mean anomaly (literally, the mean daily motion of the moon diminished by that of its apogee). (To the sum of those Rsine-differences) add the Rsine- differences (phala) corresponding to the arcs (of the daily motion of the Moon's mean anomaly which lie) in the first and last elementary arcs which are to be determined by proportion with the Rsine-differences (corresponding to those elementary arcs). Then calculate the (Moon's) equation of the centre (phala) co- rresponding to that (i.e., multiply that by the Moon's tabulated epicycle and divide by 80). The (Moon's) mean daily motion, when diminished or increased by that equation (according as the ¹ The assumption that the Rsines vary uniformly is false. 122 TRUE LONGITUDE OF A PLANET Moon is in the half-orbit beginning with the anomalistic sign Capricorn or in that beginning with the anomalistic sign Cancer), becomes truer than the true. The rationale of this rule is exactly similar to that of the previous rule. The difference is that the motion of the Moon's apogee is also taken into account in this case. For a criticism of the jivabhukti, see LBh, ii. 14-15(i). Another rule for finding the motion of the Sun or Moon for the day elapsed or for the day to elapse: 18. The difference between the longitudes (of the Sun or Moon) computed for (sunrise) today and for (sunrise)yesterday is the motion (of the Sun or Moon) which has taken place(on the day elapsed). The difference between the longitudes (of the Sun or Moon) computed for (sunrise) tomorrow and for (sunrise) today is stated to be the motion (of the Sun or Moon) which will take place (today). A rule for determining the true distance in minutes of the Sun or Moon on the basis of the eccentric (pratimandala) theory: 19-20. Subtract (the Rsine of) the greatest equation of the centre from add that to (the Rsine of) the koti (due to mean anomaly) depending on the anomalistic quadrant (i.e., according as the Sun or Moon is in the second and third or first and fourth anomalistic quadrants). The square root of the sum of the squares of that and (the Rsine of) the bahu (due to mean anomaly) is the hypotenuse. By that hypotenuse multiply (the Rsine of) the greatest equation of the centre, and then divide (the product) by the radius: add this result to or subtract that from the previous (Rsine of the) koti (as before). Continue this process until two successive approximations for the hypotenuse are the same (up to minutes). (Thus is ob- tained the true distance of the Sun or Moon). 1 This rule occurs also in SiSi, I, ii. 36(ii). 123: The author takes up now the Hindu eccentric theory. Here the mean Sun (or Moon) is supposed to move on a circle centred at the earth called the concentric (kakṣāvṛtta), whereas the true Sun (or Moon) is supposed to move on another equal circle called the eccentric (pratimaṇḍala) with the same angular velocity as the mean Sun (or Moon) has. The centre of the eccentric is supposed to be deviated from the Earth towards the Sun's (or Moon's) apogee by an amount equal to the Rsine of the Sun's (or Moon's) greatest equation of the centre. TRUE DISTANCE OF THE SUN OR MOON In Fig. 13, the circle UMNY centred at E, the Earth, is the concentric and the circle U₁T₁L centred at C is the eccentric for the Sun. When the mean Sun is at U, the true Sun is at U₁(the apogee). Subsequently, when the mean Sun is at M, the true Sun is at T₁. Since the mean Sun and the true Sun have the same angular velocity relative to the apogee, the line MT, will always be parallel to the apse line EU. Let XY be perpendicular to EU through E, and T₂B perpendicular to XY. Then in the triangle T,BE, right- angled at B, we have BE=MA=Rsin (arc MU), and T₂B T₂M + MB = EC+ Rsin (arc MX), where the arc MX is the koți and EC is the Rsine of the greatest equation of the centre for the Sun. = L X U1 W U K A N Fig.13. 8 Y Had the length EC been cqual to the radius of the Sun's true epicycle for the mean sun at M,T,E would have been the Sun's true distance, but EC corresponds to the radius of the Sun's tabulated epicycle, which is mean and not true, therefore T₁ is not the true position of the Sun and likewise T,E is not the true distance of the Sun. The true distance is determined as follows: Join T,C and let it intersect the concentric at S. Produce MT₁ and ES to meet at T. Then MT denotes the true distance between the centres of the concentric and the eccentric, T the position of the true Sun, and ET the true distance of the sun. 124 The method It is based on the explained as follows: TRUB LONGITUDE OF A PLANET stated in the text relates to the determination of ET. process of successive approximations and may be With centre E and radius ET, draw an arc of a circle cutting ET at the point S₁, and through S, draw a line parallel to EM meeting MT at T₂. Again with centre E and radius ET, draw an arc of a circle cutting ET at S₂, and through Są draw a line parallel to EM to meet MT at Tg. Continue this process repeatedly. Also let SD, S₁D₁, S₂D₂,... be parallel to EU. The method begins with assuming MT, as the first approximation r₂ to MT and likewise ET, is taken as the first approximation H, to ET. Now from the similar triangles S₁D₂E and SDE, we have SDXES₁_₁XH₁. S₂D₁=: But S₂D₁=MT₂. Therefore, MT, =XH₂ R Therefore, from the triangle T,BE, right-angled at B, we have ET₂ = MB+MT₂)³+BE², where MT₂ is given by (1), MT₂ is taken as the second approximation r₂ to MT, and ET, as the second approximation H₂ to ET. H₂, Similarly, MT3, MT4, ... to MT, and ET3, ET4, to ET. Obviously, R Since H, R, therefore r₂ > 1₁; and consequently, H₂> H₁. and R are the next successive approximations r3, are the next successive approximations H₂, rn <¹+1 and H₁ <Hn+1. n ... are each Moreover, from the construction it is clear that r₁, T₂, T3, less than MT, which is the upper bound of the sequence { }; and H₁, H₂, H3, are each less than ET, which is the upper bound of the sequ- ence {H}. n n Hence it follows that (1) 1₁ < 1g < 1₂ << H₂ <H₂ <H₂ < <H < < ... < MT <ET. SUN'S TRUE LONGITUDE UNDER THE ECCENTRIC THEORY 125 Obviously, the sequences {n} and {H} are convergent. The former converges to MT and the latter to ET. It will be noticed that the convergence is rapid, so that the third or fourth approximation will give the required distance correct to the minute. In practice, the process is repeated until two successive approximations agree to minutes.¹ The process of finding the true distance of the Moon is similar. It may be added that the position and distance of the Sun or Moon derived from the eccentric theory are the same as derived from the epicy- clic theory. A rule for the determination of the Sun's true longitude (for mean sunrise at the svaniraksa place) under the eccentric theory: 21-23. Multiply the radius by the Rsine of the bhuja (due to the Sun's mean anomaly) and divide (the product) by the (Sun's true) distance. Add the arc corresponding to that (re- sult) to the longitude of the (Sun's) own apogee depending on the anomalistic quadrant (occupied by the Sun) (as follows): (When the Sun is in the first anomalistic quadrant, add) that are itself, (when the Sun is in the second anomalistic quad- rant, add) half a circle (i.e., 180°) as diminished by that arc, (when the Sun is in the third anomalistic quadrant, add) half a circle as increased by that arc, and (when the Sun is in the fourth anomalistic quadrant, add) a circle as diminished by that arc: the result is the true longitude of the Sun (for mean sunrise at the place where the local meridian intersects the equator).² This is stated to be the determination (of the Sun's true longitude) under the eccentric theory. The greatest equation ¹ In the above discussion we have assumed that the Sun is in the first anomalistic quadrant as shown in the figure. When the Sun is in the other quadrants, the process is similar. This rule occurs also in BrSpSi, xiv. 17-18 and Sise, iii. 52. 126 TRUE LONGITUDE OF A PLANET of the centre for each individual planet determines its own eccentric (pratimandala). In Fig. 13, SK and TW are perpendiculars to the apse line EU. TW (which is equal to MA) is the Rsine of the bhuja (or bahu) MU, and SK is the Rsine of the true bhuja (called spasta-bhuja) SU. Comparing the similar triangles SKE and TWE, we have SK SE SK = giving or Rsin (arc SU) Therefore, arc SU Rsin-¹ TW TE TWXSE TE Rsin (bhuja)xR Sun's true distance = "" Rsin (bhuja) xR Sun's true distance (1) Now let be the first point of Aries. (See Fig. 13). Then, if the Sun is in the first anomalistic quadrant (as in the figure), Sun's true longitude arc Ts arc TU + arc SU. longitude of the Sun's apogee + arc SU. When the Sun (i.e., the true Sun) is in the second quadrant, say at Q, the expression on the right hand side of (1) turns out to be the value of arc QN. Hence, in this case. Sun's true longitude arc TQ = arc TU + (180⁰. - arc QN). Similarly, in the remaining quadrants. The method for finding the Moon's true longitude is similar. A rule for finding the Sun's bhujantara correction under the eccentric theory: 24. The (mean) daily motion (of the Sun) multiplied by the difference between the (Sun's) true and mean longitudes computed for the local place¹ and (the product then) divided by the number of minutes in a circle (i.e., by 21600) gives, as before, the (Sun's) bhujantara.² ¹ What is meant here is the svanirakṣa place, i.e., the place where intersects the equator. the local 2 This rule occurs also in BrSpSi, xiv. 19. 127 The bhujantara correction is, as stated before, the correction for the equation of time due to the eccentricity of the ecliptic. An approximate formula for finding the Rsine of the Sun's declination: 25. The Rsine of the Sun's longitude corrected for the three corrections (viz. desantara, bahuphala, and bhujantara), as multiplied by 13 and divided by 32, is (the Rsine of) the Sun's declination. The remaining determinations (such as the calculation of the day-radius, etc.) should be made as before. CORRECTION UE TO SUN'S ASCENSIONAL DIFFERENCE From iii. 6-7, we have (1) Rsin & 1397 x Rsin > 3438 where and are the Sun's sayana longitude and declination respec- tively. giving Now 1397 3438 - 1 1 1 1. 1 1 1 2 + 2+5+1+9+1+9⁹ 1 2 11 13 2' 5'27' 32 as the successive approximations of 9 Writing for 1397 3438 13 its fourth approximation 32 (1) reduces to , Rsin S 1 13 x Rsin > 32 1397 3438 which is the formula stated in the text. A rule relating to the determination and application of the correction due to the ascensional difference of the Sun (called cara-samskara or cara correction): 26-27. The (mean) daily motion (of the Sun) multiplied by the asus of the (Sun's) ascensional difference and divided by the number of asus in a day and night (i.e., by 21600) should be subtracted from or added to the (Sun's) longitude computed for sunrise or sunset respectively, provided that the Sun is in the] 128 TRUE LONGITUDE OF A PLANET northern hemishere; if the Sun is in the southern hemisphere, it should be applied reversely.¹ In the case of other planets, this correction is determined by proportion (with the Sun's ascensional difference and the planet's mean daily motion); the law for its addition or sub- traction (to the planet's true longitude) is the same as in the case of the Sun. The correction due to the Sun's ascensional difference is the fourth and last correction to be applied to the Sun. By this correction llowance is made for the difference between the times of sunrise (or sunset) at the local and the svanirakṣa places. This is applied to the Sun's true longi- tude for true sunrise at the svanirakṣa place to get the Sun's true longitude for true sunrise at the local place. When the Sun is in the northern hemisphere, sunrise at the local place occurs earlier than that at the svanirakṣa place, and sunset at the local place occurs later than that at the svaniraksa place. When the Sun is in the southern hemisphere, it is just the contrary. Hence the law of correction stated in the text. The general formula for the cara correction is³ cara correction (Sun's asc. diff. in asus) x (planet's mean daily motion) 21600 A rule for finding the semi-durations of the day and night: 28. (When the Sun is) in the northern hemisphere, one- fourth of the total duration of the day and night increased by the (Sun's) ascensional difference, and (when the Sun is) in the southern hemisphere, one-fourth of the total duration of the day and night diminished by the (Sun's) ascensional difference is ¹ This rule occurs also in KK (Sengupta's edition ), i. 22; ŠiDV, I, ii. 19; Sisi, I, ii. 53; etc. The local place has been always assumed to be in the northern hemisphere, i.e., to the north of the equator. 8 Sise, iii. 69. 129 the measure of half the day. The measure of half the night is obtained contrarily.¹ This can be easily seen to be true from the celestial sphere. Rules relating to the corrections for the Moon: 29-30. Multiply the (Moon's) mean daily motion by the Sun's equation of the centre and then divide (the product) by the number of minutes in a circle (i.e., by 21600): (the result is the bhujantara correction for the Moon). Add it to or sub- tract it from the Moon's (mean) longitude (corrected for the longitude of the local place) in the same way as in the case of the Sun.

. All remaining corrections for the Moon are prescribed as in the case of the Sun. CORRECTIONS FOR THE MOON (The bhujantara correction) for the remaining planets also is calculated from the Sun's equation of the centre. The general formula for the bhujantara correction is: bhujantara correction (Sun's equation of the centre) x (planet's mean daily motion) 21600 The formula for the bhujantara correction for the Moon, stated in the text, is a particular case of this. A We have seen above that in the case of the Sun four corrections are applied in the following order: (1) the longitude correction, (2) the bhujaphala correction (i. e., the equation of the centre), (3) the bhujantara correction (i. e., the correction due to the Sun's equation of the centre). (4) the correction due to the Sun's ascensional difference. In the case of the Moon, the same four corrections are applied in A This rule is found also in SuSi, ii. 62-63; BrSpSi, ii. 60; KK (Sengupta's edition), i. 23; SiDV, I, ii. 20-21; Sise, iii. 70; Sisi, 1, ii. 52.. 130 the following order : TRUE LONGITUDE OF A PLANET (1) the longitude correction. (2) the bhujantara correction (i.e., the correction due to the Sun's equation of the centre). (3) the bhujaphala correction (i.e., the Moon's equation of the centre). (4) the correction due to the Sun's ascensional difference. In later works, two more corrections are prescribed for the Moon. The one is equivalent to (i) the deficit of the equation of the centre of the Moon plus (ii) the evection¹, and the other is what is now called "the variation". The former occurs for the first time in the Vatesvara-siddhanta of Vateśvara (904 A.D.) and the Laghu-manasa of Manjula (932 A. D.) and the latter in the Bijopanaya of Bhaskara II (1150 A.D.). The next six stanzas relate to the calculation of tithi, karaṇa, and nakṣatra, which are three out of the five principal elements of the Hindu Calendar, the other two being yoga and vāra (week-day), and to the pheno- mena of vyatipāta. The calculations for tithi, karana and naksatra are made for sunrise. Calculation of the tithi: 31-32. Divide the true longitude of the Moon as diminished by that of the Sun by 720 minutes (of arc); the quotient (obtai- ned) denotes the number of tithis (elapsed). Multiply the re- mainder by 60 and divide (the product) by the difference bet- ween the (true) daily motions of the Sun and the Moon: then are obtained the ghatis, vighatis, and asus (elapsed of the current tithi). (The time in ghatis, vighatis, etc. of) the current tithi to elapse or elapsed is measured from sunrise. A lunar (or synodic) month is defined in Hindu astronomy from one new moon to the next. There are thirty tithis (lunar days) in a lunar month. The first tithi begins at new moon (when the Sun and the Moon have the same longitude) and continues till the Moon, due to her rapid motion, is 12° (or 720') in advance of the Sun; the second tithi then begins and continues till the Moon is 24° in advance of the Sun; the third tithi ¹ See my paper entitled "The evection and the deficit of the equation of the centre of the Moon in Hindu Astronomy" in Bull. Banaras Math, Soc., Vol. VII, No. 2, 1945. This rule occurs also in Susi, 66; BrSpSi, ii. 62; KK (Sengupta's edition), i. 25; SiDV, I, ii. 22; Sise, iii. 71; SiSi, I, i. 66. KARANA then begins and continues till the Moon is 36° in advance of the Sun; and so on. 131 A lunar month is divided also into two halves, the light half and the dark half. The light half begins at new moon and continues till full moon, and the dark half begins at full moon and continues till new moon. Evidently there are fifteen tithis in each half. The tithis falling in the two halves are numbered 1, 2, 3, ... The text gives the method for finding the number of tithis elapsed since new moon, and the time elapsed at sunrise since the beginning of the current tithi. Calculation of the karana: 33. The karanas (elapsed) are obtained by taking "half the measure of a titki (i.e., 360 minutes)" for the diviser, and are counted with Bava. But the number of karanas elapsed in the light half of the month should be diminished by one, where- as those elapsed in the dark half of the month should be increa- sed by one. This is what has been stated.¹ The karana is half a tithi, so that there are 60 karanas in a lunar month. These karanas are divided into 8 cycles of 7 movable karanas, bearing the names Bava, Bālava, Kaulava, Taitila, Gara, Vanija, and Visti respectively, and 4 immovable karanas, bearing the names sakuni, Catus- pada, Nāga, and Kimstughna respectively. 1 That is to say: If it is the light half of the month, divide the true longitude of the Moon as diminished by that of the Sun, reduced to minutes, by 360. The quotient diminished by one should be divided by seven and the remainder obtained should be counted with Bava. This gives the karaṇa elapsed before sunrise. If it is the dark half of the month, subtract the longitude of the Sun from that of the Moon, and diminish that difference by six signs. Reduce it to minutes and divide by 360. The quotient increased by one should be divided by seven and the remainder obtained should be counted with Bava. This gives the karana elapsed before sunrise. The time elapsed at sunrise since the beginning of the current karaṇa should be determined from the remainder obtained in the division by 360, as in the case of the tithi. This rule is found to occur also in KK (Sengupta's edition), i. 27; SiDV, I, ii. 24; Sise, iii. 77; Siśi, I, ii. 66. 132 TRUE LONGITUDE OF A PLANET The first round of the movable karanas begins with the second half of the first tithi in the light half of the month, and the eighth round ends. on the first half of the fourteenth tithi in the dark half of the month. Thus in the light half of the month, the second karana is Bava, the third karana is Bālava, the fourth karana is Kaulava, and so on; and in the dark half of the month, the first karana is Bālava, the second karapa is, Kaulava, and so on. The four immovable karanas occur in succession after the eighth round of the cycle of the seven movable karanas. Calculation of the naksatra: 34. The true longitude of a planet reduced to minutes and then divided by 800 gives the number of nakṣatras passed over (by the planet). From the remainder (multiplied by 60 and) divided by the (planet's true daily) motion are obtained the ghatis elapsed (since the planet's entrance into the current nakṣatra).¹ We have seen that in Hindu astronomy the stars lying near the eclip- tic are divided into 27 groups called naksatras. Beginning with the first point of the nakṣatra Asvini,³ the ecliptic is divided into 27 equal parts, each of 800 minutes (of arc). These divisions of the ecliptic also are called nakṣatra and given the same names as the twenty-seven star-groups, i. e., Aśvini, etc. The nakṣatras referred to in the above stanza are these divi- sions of the ecliptic. Given the longitude of a planet the above rule enables us to de- termine the number of nakṣatras passed over by the planet and the time elapsed since it crossed into the current nakṣatra. The phenomena of vyatipāta: 35-36. When the sum of the (true) longitudes of the Sun and the Moon amounts to half a circle (i.e., 180°), the pheno- ¹ This rule is found also in SüSi, ii. 64; BrSpSi, ii. 61; KK (Sengupta's edition), i. 24; SiDVṛ, I, ii. 23(i); Siśe, iii. 75; SiSi, I, ii. 67. 2 Vide supra, Chapter III, stanzas 62-75(i). 3 The first point of the nakṣatra Aśvini (also called the first point of Aries) is the fixed point from which the longitudes of the planets are measured in Hindu astronomy. This point coincides with the junction-star of the nakṣatra Revati, i.e. with y Piscium. 133 menon is called (Tata) vyatipāta; when that (sum) amounts to a circle (i.e., 360°), the phenomenon in called vaidhṛta (vyatipata); and when that (sum) extends to the end of the naksatra Anuradha (i.e., when that sum amounts to 7 signs, 16 degrees, and 40 minutes), the phenomenon is called sārpamastaka (vyatipāta). VYATIPATA The (Tata) vyatipata occurs when the Sun and Moon are in different courses of motion (ayana)¹ and their (true) declina- tions are equal. Its region is half a circle, but due to the Moon's latitude it may be more or less.² The sarpamastaka vyatipăta corresponds to the yoga known by the name vyatipāta. That the region of the (lata) vyatipāta is half a circle means that the (lāṭa) vyatipāta takes place when the Sun and Moon are within half a circle measured from the first point of nakṣatra Asvini. The phenomena of vyatipāta (usually called pata or mahāpāta) are treated in detail in later works. In the Surya-siddhānta, Brāhma-sphuta- siddhānta, Siddhānta-šekhara, and Siddhānta-śiromaņi, etc., a whole chapter is devoted to that subject. In modern Hindu Calendars (called Pancanga) are given the tithi, karana, Moon's nakṣatra, and yoga current at sunrise for every day of the year and also the times when they end and the next ones begin. The yoga has not been treated by Bhaskara I, but it forms one of the five important elements of the Hindu Calendar. Like the nakṣatras, the number of these yogas is also twenty-seven. The method of finding the yoga passed over and the time elapsed at sunrise since the commencement of the current yoga is similar to that prescribed for the naksatra. The difference is that in the case of the yoga calculation is made with the sum of the longitudes of the Sun and the Moon, whereas in the case of the naksatra calculation is made with the help of the longitude of the Moon only. The first yoga (called Viskambha) begins when the sum of the longitudes of the Sun and moon is zero, the second yoga (called Priti) begins when ¹ That is to say, when one has northward motion and the other has southward motion. ² For details, see Sisi, I, xii. 134 that sum amounts to 13°20', the third yoga (called Ayusman) begins when that sum amounts to 26°40', and so on.¹ The remaining chapter relates to the planets, Mars, Mercury, Jupiter, Venus, and Saturn. TRUE LONGITUDE OF A PLANET Calculations relating to the planets, Mars, etc.: 37. (In the case of the planets, Mars, etc.) the determina- tion of the direct and inverse Rsines (relating to the kendra, i.e., mandakendra and sighrakendra) as also the calculation of the bhuja and koti etc. is to be made as in the case of the Sun. The differences (in the case of Mars, etc.) will now be stated. The mandakendra and the sighrakendra are defined by the following equations: ● mandakendra = longitude of the mean planet -- longitude of the planet's mandocca (apogee). śighrakendra longitude of the planet's fighrocca-longitude of the true- mean planet.² - A rule for finding the (planet's) corrected epicycle: 38-39(i). Multiply the Rsine or Rversed-sine (of the part of the kendra lying in the current quadrant³), according as the (current) quadrant is odd or even, by the difference between ¹ The names of the 27 yogas are: (1) Viskambha, (2) Priti, (3) Ayuṣmān, (4) Saubhagya, (5) Sobhana, (6) Atiganda, (7) Sukarmā, (8) Dhrti, (9) Śūla, (10) Ganda, (11) Vṛddhi, (12) Dhruva, (13) Vyāghāta, (14) Harṣaṇa, (15) Vajra, (16) Siddhi, (17) Vyatipāta, (18) Variyan, (19) Parigha, (20) Śiva, (21) Siddha, (22) Sadhya, (23) Subha, (24) Sukla, (25) Brahmā, (26) Indra, and (27) Vaidhṛta. There is another system of twenty-eight yogas, beginning with Ananda. In some Hindu calendars yogas of this system are also given for each day of the month. But these yogas are only of astrological interest. 2 See infra, chapter vii, stanza 12. 3 The kendra is said to be in the first quadrant if it is less than 90°, in the second quadrant if it is between 90° and 180°, and so on. .. 135 the (planet's) own epicycles (for the beginnings of the odd and even quadrants) and then divide (the product) by the radius; and apply the result (thus obtained) to the (planet's) epicycle (for the beginning of the current quadrant). Subtract (that result), when the epicycle for (the beginning of) the current quadrant is greater; add (that result), when the epicycle for (the beginning of) the current quadrant is smaller. Thus is obtained the (planet's) corrected epicycle.¹ In the case of the Sun and the Moon, which move around the Earth, we have seen that only one epicycle is contemplated which is meant to account for the eccentricity of the orbit. In the case of the planets, Mars, Mercury, Jupiter, Venus, and Saturn, which revolve round the Sun, two kinds of epicycles are envisaged, (i) manda, and (ii) ghra. We shall presently see how these epicycles are utilized to explain the motion of the planets. CORRECTED EPICYCLE Unlike the mean epicycle for the Sun or Moon, the manda and sighra epicycles for Mars, etc., are supposed to vary from place to place. Their values at the beginnings of the odd and even quadrants are given in the seventh chapter. Those for any other point of the. orbit are determined by the method taught in the stanza under consideration. Let and be the epicycles (manda or sighra) of a planet for the beginnings of the odd and even quadrants respectively. Then (1) if the planet be in the first quadrant (of the kendra), say at P, and its kendra be 0, epicycle at P - (B a + -α) x Rsin 0 R (dB) x Rsin () R (when d<B) (when α>B) and (2) if the planet be in the second quadrant (of the kendra), say at Q, and its kendra be 90° + +, ¹ This rule occurs also in Susi, ii. 38 and Sise, iii. 22. a Stanzas 13-16. 136 2 TRUE LONGITUDE OF A PLANET epicycle at Q-8- That is, mandakendraphala (B-d) x Rversin R and sighrakendraphala =B+ (α-8)x Rversin R Similarly, in the third and fourth quadrants. Rule for finding the kendraphala (i.e., mandakendra-phala or sighrakendra-phala): 39(ii). By that (corrected epicycle) multiply the Rsine of the kendra of the desired planet and then divide (the product obtained) by 80; this is known as the (kendra) phala. (whena <B). (when d>B). (corrected manda epicycle) > Rsin (mandakendra) 80 (corrected sighra epicycle) Rsin (sighrakendra) 80 By the mandakendraphala is meant the bahuphala derived from the planet's corrected manda epicycle, and by the sighrakendraphala is meant the bahuphala derived from the planet's corrected sighra epicycle. The method of finding the bahuphala is the same as taught in the case of the Sun. In what follows we shall see how the mandakendraphala and the sighrakendraphala are used in finding the true geocentric longitudes of the planets. Their significance will also then become clear. Procedure to be adopted for finding the true geocentric longitude in the case of Mars, Jupiter, Saturn, Mercury and Venus: 40-44. Calculate half the arc corresponding to the (planet's) mandakendraphala and apply that to the (planet's) mean longi- tude depending on the quadrant (of the planet's kendra) as in the case of the Sun. TRUB LONGITUDE OF MARS, BETC., BY EPICYCLIC THEORY 137 (Then calculate the sighrakendraphala). Multiply the ra- dius by the sighrakendraphala and divide (the product) by the (planet's) sighrakarna; then reduce that to arc. Apply half of that are to the longitude obtained above, reversely (i.e., add when the sighrakendra is in the half orbit beginning with the sign Aries and subtract when the sighrakendra is in the half orbit beginning with Libra). Therefrom calculate (the arc corresponding to) the manda- (kendra)phala and apply the whole of that to the mean longitude of the planet. Thus are obtained the true-mean longitudes of Mars, Saturn, and Jupiter. The true-mean longitude corrected for the are derived from the sighrakendraphala (literally, the arc corresponding to the result derived from the longitude of the sighrocca minus the true- mean longitude of the planet) is to be known as the true longi- tude. The method (to be used) for the remaining planets (i.e., Mercury and Venus) is now being told. The longitude of the (planet's) mandocca (i.e., apogee) reversely increased or decreased by half the arc derived from the sighrakendraphala determines the true-mean longitude (of the planet). And that (true-mean longitude) corrected for the arc derived from the sighra(kendraphala) is known as the true longitude. ¹ 'The arc derived from the sighrakendraphala' is obtained by first multiplying the radius by the sighrakendraphala and dividing by the sighrakarna and then reducing that to arc. See stanza 41. The rule given in stanzas 40-43 is found also in A, iii. 23; LBh, ii. 32- 35; ŚiDV, I, ii. 30-34; TS, ii. 55-65, 66(1). 2 What is meant here may be explained as follows: Subtract the mean longitude of the planet from the longitude of the planet's sighrocca. Multiply the Rsine of that by the planet's corrected sighra epicycle and divide by 80: the result is the planet's sighrakendraphala. Multiply that by the radius and divide by the planet's sighrakarna, and reduce the resul- ting Rsine to the corresponding arc. Add half of it to or subtract that from the longitude of the planet's mandocca according as the sighrakendra of the planet is in the half orbit beginning with the sign Libra or Aries. Treat this 138 TRUE LONGITUDE OF A PLANET We explain below the motion of the planets, Mars, Mercury, Jupiter Venus, and Saturn, according to the Hindu epicyclic theory. U Consider Fig. 14. E is the Earth. The bigger circle USM centred at E is the deferent (kakṣāvṛtta). The point U on the deferent is the planet's mandocca (apogee). M is the position of the mean planet which is supposed to move on the deferent with mean velocity from west to east (in the anticlockwise direction indicated by an arrow). The small circle centred at M is the planet's manda epicycle corresponding to the position M of the mean planet: this manda epicycle is determined as taught in stanzas 38-39(i). EC is equal to MM₁, M₂ and C are joined by a line which intersects the deferent at the point S. MM, and ES are produced to meet at the point T'. This point T' is, according to the Hindu astronomers, the position of the true-mean planet. The so called true-mean planet is assumed to move on the periphery of the true epicycle of radius MT' centred at M with the same velocity as the mean planet has relative to the apogee but in the opposite sense (i.e., clock- wise). The point S denotes the position of the true-mean planet on the deferent. C E N Fig. 14 sum or difference as the correct longitude of the planet's mandocea. There- from calculate the arc corresponding to the planet's mandakendraphala and apply that to the planet's mean longitude. Thus is obtained the planet's true-mean longitude. Then calculate the arc derived from the planet's sighrakendraphala and apply that to the true-mean longitude of the planet. Then is obtained the true longitude of the planet (Mercury or Venus). The rule stated in stanza 44 occurs also in Ā, iii. 24; LBh, ii. 37-38; and SiDV, I, li. 35. The method prescribed here for finding the true longitudes of Mercury and Venus has been prescribed for all the planets in the Karana-prakāśa (li (b). 3, 4), the Graha-läghava (ili. 10), the Ravi- siddhanta-manjart (ii. 1), and the Karana-kaustubha (iii. 19), etc., all of them being calendrical works, TRUB LONGITUDE OF MARS, ETC., BY EPICYCLIC THEORY If T be the first point of Aries, then L MET (or are MUT) is the mean longitude of the planet, and LSET (or arc SUT) is the true-mean longitude of the planet. The arc MS by which the true-mean longitude of the planet differs from the mean longitude of the planet is obtained as follows: Let MA be the perpendicular from M on EU, M,B, and SD the per- pendiculars from M, and S on EM. Then comparing the triangles M₂B₂M and MAE, we have M₁B₁, i.e., SD = MA X MM₂ EM 139 Rsin (bahu due to mandakendra) × (radius of corrected manda epicycle) R (corrected manda epicycle) × Rsin 80 where denotes the bahu due to mandakendra.¹ Reducing the right-hand side of (1) to the corresponding are, we get the arc MS. This arc MS has been referred to by Bhaskara 1 as the arc corres- ponding to the mandakendraphala, because it corresponds to M₂B, which denotes the mandakendraphala. Generally it is known as mandaphala. It is. subtracted from or added to the mean longitude of the planet, according as the mandakendra is less than or greater than 180°, as in the case of the Sun and Moon. Thus true-mean longitude-mean longitude-mandaphala, according as the mendakendra is less than or greater than 180°. Now consider Fig. 15. Here also E is the Earth and the bigger circle centred at E is the deferent (kaksävṛtta); U is the planet's mandocca ("apogee") and V the planet's fighrocca. S is the position of the true-mean planet on the deferent. The small circle centred at S is the planet's sighra epicycle: it is derived as taught in stanzas 38-39(i). ST is drawn parallel to EV. Then T denotes the position of the true planet. ET is called the sighrakarna. ¹ The bahu due to mandakendra is derived in the same way as in the case of the Sun. The corrected manda epicycle used in this last result is that divided by 4).

• It may be pointed out that in the case of the Sun and Moon the

mandaphala is the equation of the centre, called bahuphala by Bhāskara 1. 140 TRUE LONGITUDE OF A PLANET The true planet is assu- med to move on the sighra epicycle with the same angu- lar velocity as the true-mean planet appears to have in the deferent with respect to the sighrocca. Whereas the true- mean planet appears to move on the deferent (in the clock- wise direction) away from the sighrocca, the true pla- net is supposed to move on the epicycle, centred at the true-mean planet, in the anti- clock-wise direction, so that the line ST is always parallel to EV. F E Fig. 15 BP 3 But and G U Let the line ET intersect the deferent at R. Then R denotes the true position of the planet on the deferent. If T be the first point of Aries, then L SET (or arc SUT) is the true-mean longitude of the planet, and RET (or arc RUT) is the true longitude of the planet. The arc RS of the deferent, by which the true longitude of the planet differs from the true-mean longitude, denotes the planet's sighra correction (usually called sighraphala). It is derived as follows: Let SF, TG, and SH be the perpendiculars drawn from S, T, and S. on, EV, ES produced and ET respectively. Now arc VS TV-Ts, i.e., ŝighrakendra = (longitude of fighrocca)- (longitude of true- mean planet). Let this sighrakendra (reduced to bahu, if necessary) be denoted by . Then comparing the similar triangles TGS and SFE, we get SFXST TG = ES SF == ST Rsin , sighra epicycle 360 corrected sighra epicycle¹ ES 80 ¹ As pointed out earlier, this corrected epicycle is divided by 41. TRUB LONGITUDE OF MARS, ETC., BY ECCENTRIC THEORY Rsin x (corrected sighra epicycle) 80 Again from the similar triangles ESH and ETG, we have TG X ES SH TET Therefore i.e., Rsin (arc RS) TG - >. 141 Rsin + x (corrected sighra epicycle) > R 80 x śighrakarṇa Therefore arc RS or sighraphala is the arc corresponding to the right hand side of this equality. This sighraphala is ided to or subtracted from the true-mean longi- tude of the planet according as the sighrakendra is less than 180° or grea- ter than 180°, because in the former case the true planet is in advance of the true-mean planet and in the latter case the true planet is behind the true- mean planet. Thus true longitude = true-mean longitude + śighraphala, according as the sighrakendra is less than or greater than 180°. The true longitude of the planet thus obtained was found to differ from the actual longitude determined from observation. Aryabhata I supposed that the error was due to the inaccuracy of the mandakendra (manda ano- maly). So to get rid of the error, in the case of the superior planets (Mars, Jupiter and Saturn), he applied in succession (i) half the planet's mandaphala, and (ii) half the planet's sighraphala to the manda- kendra of the planet; and in the case of the inferior planets (Mercury and Venus), he applied half the planet's sighraphala to the mandakendra of the planet From the mandakendra thus corrected, he calculated and applied in succession the mandaphala and the sighraphala corrections to the mean longitude of the planet. The same procedure has been follo- wed by the pupils and followers of Aryabhata I. Hence the rules stated in stanzas 40-44.

The device contemplated by Aryabhata I continued to be used by his followers, but it was never very successful. Naturally, it never came into general use. Several other devices were attempted from time to time by later astronomers. A rule relating to the eccentric theory: 45-46. The wise (astronomer) should apply the eccentric theory here (i. e., in the case of the planets Mars, etc.) also. (Under this theory the mandocca and sighrocca operations are as follows:) 142 TRUB LONGITUDE OF A PLANET To the longitude of the mandocca ("apogee"), apply (the spasta-bhuja due to the mandakendra, as a positive correction) in the manner prescribed above (in stanza 22). From the longitude of the sighrocca subtract the spasta-bhuja (due to the sighra- kendra) (as follows) : (When the sighrakendra is) in the first and second quad- rants, subtract from the longitude of the sighrocca the spasta- bhuja itself and that subtracted from half a circle (i.e., 180°) respectively; (when the sighrakendra is) in the remaining quad- rants (i.e., third and fourth), subtract that (spasta-bhuja) as increased by half a circle and that (spasta-bhuja) subtracted from a circle respectively. In Fig. 16, let the circle UMN centred at E, the Earth, be the con- cyclic (kakṣāvṛtta), the circle centred at. C the manda eccentric (manda- prativṛtta), U the planet's mandocca (apogee), and M the mean position of the planet. Let MM, be parallel to EU; and let S be the point where CM, intersects the concyclic, and T' the point where MM, and ES produced meet. Then T is the position of the true-mean planet and S the position of the true-mean planet on the concyclic. If be the first point of Aries, then are US is the true- mean longitude of the planet. When the mean planet is in the first quadrant begin- ning with U, as shown in the figure, arc TUS-TU+US, i. e., true-mean longitude = longitude of the planet's apogee + spasta-bhuja.¹ M U When the mean planet is in the second anomalistic quadrant, the spasta-bhuja is the arcual distance of the true mean planet from the perigee M. Thus, in this case N Fig. 16 ¹ As in the case of the Sun, arc MU is the bahu or bhuja (due planet's mandakendra) and arc SU is the spaṣṭa-bhuja. TRUE LONGITUDE OF MARS, ETC., BY ECCENTRIC THEORY true-mean longitude longitude of planet's apogee + (180°-spasta-bhuja). Similarly, when the mean planet is in the third anomalistic quadrant, true-mean longitude = iongitude of the planet's apogee + (180° + spasta-bhuja); and when the mean planet is in the fourth anomalistic quadrant true-mean longitude = longitude of the planet's apogee +(360° spasta-bhuja). The spasta bhuja is obtained by the the following formula as in the case of the Sun: Rsin (spasta-bhuja SU)= MA X ES ET Rsin x R H where in the bahu or bhuja (due to the planet's mandakendra), R is the radius, and H the planet's distance ET' which is called mandakarna and determined by the method of successive approximations as in the case of the Sun. (See stanza 55) When the true-mean planet is in the first quadrant beginning with V and measu- red in the clockwise direction as shown in the figure, Now consider Fig. 17. The circle VSU, centred at E, is the con- cyclic, V is the sighrocca, and S is the true-mean planet. The circle centred at C, is the fighra eccentric. The point T, where the line through S drawn parallel to EV meets the eccentric, is the true planet. R is the point where ET intersects the concyclic. Tis the first point of Aries. true longitude=arcTSR=arcSv - arc VR D B V R longitude of the sighrocca-spasta-bhuja. 143 T S G Fig. 17 U 144 TRUB LONGITUDE OF A PLANET When the true-mean planet is in the second quadrant, true longitude longitude of the sighrocca (180° - spasta-bhuja). When the true-mean planet is in the third quadrant, true longitude=longitude of the sighrocca - (180°+spasta-bhuja). When the true-mean planet is in the fourth quadrant, true longitude-longitude of the sighrocca - (360° - spasta-bhuja). The spasta-bhuja due to the sighrocca is determined by the formula: SB X ER Rsin (spasta-bhuja)= ET Rsin X R H' - where is the bahu (due to the sighrakendra), R the radius, and H' the distance ET of the true planet, called sighrakarna. A rule for finding the mandakarna and sighrakarna: 47. Multiply the radius by the (planet's) corrected epicycle and then divide (the product) by 80; then subtract the quotient from or add that to the Rsine of the corresponding koti (due to the kendra) in accordance with the quadrant (of the kendra) : and then calculate the (planet's) karna as before. This method is analogous to that stated for the Sun and Moon in stanzas 19-20. The important thing to be noted is that in finding the manda- karna we have to apply the method of successive approximations as in the case of the Sun and Moon, whereas in finding the sighrakarna we have to apply the method only once.¹ The reason for this difference must have become clear to the reader from the epicyclic and eccentric theories, which have been explained above in detail. Procedure to be adopted for finding the true longitude of the planets under the eccentric theory: 48-54. Add half the difference between the (mean) planet corrected by the mandocca operation and the mean planet to or subtract that from the mean planet according as the (mean) planet as corrected for the mandocca operation is greater or less (than the mean planet). (The planet thus obtained is call- ¹ See infra, stanza $5... TRUE LONGITUDE OF MARS, ETC., BY ECCENTRIC THEORY ed the once-corrected planet). Then correct it by the sighrocca operation. (The planet thus obtained is called the twice-corrected planet). Then find the difference between the two planets thus obtained (i.e., the once-corrected and twice-corrected pla- nets); divide that by two; and apply it to the once-corrected planet, as before. Whatever is thus obtained should be again corrected by the mandocca operation. Next calculate the diffe- rence between the twice-corrected planet, as corrected by the mandocca operation, and that (twice-corrected planet). Apply whatever be the difference between the twice-corrected planet as corrected by the mandocca operation and the twice-corrected planet to the mean longitude of the planet, as before. That (i.e., the resulting longitude) corrected by the sighrocca operation is the true longitude of the planet. 145 Thus has been stated the method for finding (the true lon- gitudes of) Mars, Saturn, and Jupiter under the eccentric theory. Now is described the procedure to be adopted in the case of the remaining planets (viz. Mercury and Venus). (First of all obtain the mean planet as corrected by the sighrocca operation). Then add half the difference between the mean planet corrected by the sighrocca operation and the mean planet to or subtract that from the planet's mandocca, according as the mean planet corrected by the sighrocca operation is less or greater (than the mean planet). Thus is obtained the true mandocca. Then find out, by the method under the eccen- tric theory,' the correction due to the true mandocca for Mercury as well as for Venus. The mean longitudes of Mercury and Venus each corrected for that and thereafter for the correction due to the sighrocca are known as true longitudes of the planets. The procedure for finding the true longitudes of the superior and inferior planets stated in stanzas 40-44 according to the epicyclic theory has been translated in the above stanzas into the eccentric theory. The results in both cases are the same. 1 The method is to find the difference between (i) the mean planet corrected by the mandocca operation and (ii) the mean planet. 146 TRUE LONGITUDE OF A PLANET Further instructions relating to mandakarna and sighrakarna : 55. When the Rsine of the greatest correction (antyaphala) is to be subtracted from the Rsine of the koti (due to the kendra), but subtraction is not possible, then subtract reversely (i. e., the latter from the former). Determine the mandakarna by the method of successive approximations (as in the case of the Sun or Moon) and the sighrakarna by a single application of the process (as taught in stanza 47). In the case of the mandakarna, the Rsine of the greatest correction is equal to the radius of the corrected manda epicycle, i.e., to (corrected manda epicycle) x R 80 and in the case of the sighrakarna, the greatest correction is equal to the radius of the corrected sighra epicycle, i.e., to (corrected sighra epicycle) XR 80 A rule pertaining to the direct and retrograde motions of a planet : 56-57. Having applied to the longitude of the sighrocca half the difference between the true and mean longitudes (of a planet) positively or negatively, depending upon (whether) the mean longitude (of the planet is greater or less than the true longitude), determine whether the motion of the planet is vakra or ativakra or whether it is the end of the vakra motion. The true longitude of the planet having been subtracted from the longitude of the (corrected) sighrocca, when the diffe- rence is 4 signs, the planet is about to take up vakra (retrograde) motion; when 6 signs, it is in ativakra (maximum retrograde) motion; and when ³8 signs, it soon abandons the regressive path.* The difference between the true longitudes of a planet computed for (sunrise ohj the day to'elapse (i.e., today) and ¹ Reference is to the rule given in stanza 47.

• This rule is found also in Sise, iii. 59 BrSpSi, ii, 50-51; ŠiDV, I,

'ii. -42. DIRECT AND RETROGRADE MOTION for (sunrise on) the day elapsed (i.e., yesterday) is the (true) daily motion (of the planet for the day elapsed). Hindu astronomers have recognised eight kinds of motion of the planets. According to the Surya-siddhanta, these are: (1) vakra (begin- ning of regression), (2) ativakra (maximum regression), (3) kutila (end of regression and beginning of direct motion), (4) manda (slow), (5) manda- tara (slower), (6) sama (mean), (7) śighra (fast), and (8) sighratara (faster). Of these, says the author of the Surya-siddhanta, the first three are the different kinds of retrograde motion and the last five the various forms of direct motion. The above stanzas 56 and 57 deal with the three varieties of retrograde motion. The details of the five varieties of direct motion are given by Sripati in his Siddhanta-sekhara.³ According to him, the motion is said to be "very fast", when the planet (measured from its fighrocca) is in the beginning of the sign Aries or Pisces; "fast", when in the beginning of Taurus or Aquarius; "mean" when in the beginning of Gemini or Capricorn; "slow", when in the first half of Cancer or in the last half of Sagittarius; and "very slow", when in the first half of Sagit- tarius or in the last half of Cancer. The following table gives the sighrakendras of the planets when they take up, retrograde motion according to various Hindu authorities: Sighrakendras of the planets when they take up retrograde motion. Sighrakendra Planet Mars Mercury Jupiter Venus Saturn Br SpSi (ii.48), SiDV (Iii. 47), KPT (iii. 8), Sise (iii. 58), Si Si (1, ii. 41) 1630 1450 1250 1650 1130 KKau (ii.23), GLā (iii.15) 163⁰ 1450 1250 . 1670 1130 MSi (iii.31) 1 ii. 12. Susi, ii.13. iii. 60. !~ 147 1630 1450 1250.. 166⁰ 1130 VVSi (ii.30), SuSi (ii.53-54) 1640 1440 130° 1630. 1150 PiSi, KK (iii. 8-17) 1640 1460 1250 1658! 133⁰ 148 A rule for finding the true daily motion (called jivabhukti) of the planets : TRUB LONGITUDE OF A PLANET 58-63. Multiply the (planet's) own (mean) daily motion by the current Rsine-difference relating to the mandocca (i.e., the current Rsine-difference corresponding to the mandakendra of the planet) and again by the (planet's) own (corrected manda) epicycle; then divide (the product) by the number of minutes in a sign as multiplied by 10 (i.e., by 18000¹). Add half of that to or subtract that from the (planet's) mean daily motion according to (the law of addition and subtraction in) the (four) quadrants. (Thus is obtained. the once-corrected daily motion). Subtract that from the daily motion of the sighrocca. Multiply whatever is obtained (i.e., sighrakendrajyagatiphala) by proceeding with the remainder in accordance with the rule "kendrāntyajiva etc." (stated in the previous stanza)³ by the radius and divide by the sighrakarna (of the planet); (and reduce the resulting Rsine to the corresponding arc). Add half of that (arc) to or subtract that from (the once-corrected daily motion) in accordance with the law (of addition and subtraction) for the correction due to sighrocca. (Thus is obtained the twice-corrected daily motion). Then add the entire of the mandakendrajyagatiphala (derived from the current Rsine-difference corresponding to the ¹ This number is the product of 225 and 80. 2 See supra stanza 5. 3 That is to say, multiply the remainder by the current Rsine-diffe- rence corresponding to the sighrakendra of the once-corrected planet and also by the planet's corrected sighra epicycle and divide the product by 18000: the result is the sighrakendrajyagatiphala.

• Multiply the twice-corrected daily motion by the current Rsine-

difference corresponding to the mandakendra of the twice-corrected planet and also by the planet's corrected manda epicycle and divide the product by 18000: the result is the mandakendrajyagatiphala. TRUB DAILY MOTION OF MARS, ETC. mandakendra of the twice-corrected planet) to or subtract that from the (planet's) mean daily motion (according to the law of addition and subtraction in the four quadrants). Set down the result at two places. At one place (subtract that from the daily motion of the sighrocca and then) calculate (the arc corresponding to) the sighrakendrajyagatiphala. Add the entire of that (arc) to or subtract that from the result kept at the other place (according to the law of addition and subtraction in the four quadrants). Thus is obtained the desired true daily motion (of the planet). 149 When the result derived from the sighra operation (i.e., the arc corresponding to the sighrakendrajyagatiphala) cannot be subtracted from that, the difference between the two then denotes the value of the true daily motion and the planet is said by the learned to be retrograde.³ This is the method for finding the true daily motion in the case of Jupiter, Saturn, and Mars. Now is being des-. cribed the method for Venus and Mercury. Increase or diminish (as usual) the mean daily motion of Venus or Mercury by the entire motion-correction (i. e., manda- kendrajyagatiphala) determined from the corrected mandakendra and also by that obtained by proceeding according to the rule 1 The process is: Multiply the difference obtained after subtraction from the daily motion of the sighrocca by the current Rsine-difference corresponding to the sighrakendra and also by the corrected sighra epicy- cle and divide that product by 18000: the result is the sighrakendra- jyagatiphala. Multiply that by the radius and divide by the sighrakarna, and reduce that to the corresponding arc.

• That is, the result kept at the other place.

3 This rule is found also in SiDV, I, ii. 38-40. 150 TRUB LONGITUDE OF A PLANET "sighrantyajiva etc.". This (sum or difference) is the true daily motion (of Venus or Mercury). Thus has been stated the difference of procedure (in the case of the superior and inferior planets). The daily motion thus obtained is always very nearly equal to the true daily motion and should be made use of in practical calculations. The above rules relate to the determination of the true daily motion of the planets, Mars, Mercury, Jupiter, Venus, and Saturn, and seem to have been inspired by the rule of Aryabhata I's midnight day-reckoning which has been adopted by Brahmagupta in his Khanda-khadyaka. They are different from the analogous rules found in the other works on Hindu astronomy. ³ They have been derived by taking the difference between the longi- tudes of the planets for two consecutive days as in the case of the Sun and Moon. The rules differ in the case of superior and inferior planets be- cause the methods of finding the true longitudes in the two cases differ. The daily motion obtained by the application of the above rules is known as jivabhukti as in the case of the Sun and Moon. It is probably ¹ That is to say: Multiply the mean daily motion by the current Rsine- difference corresponding to the mandakendra determined from the correct- ed mandocca (see stanza 53) and also by the corrected manda epicycle and divide the product by 18000: the result is the mandakendrajyagati- phala. Add it to or subtract it from the mean daily motion of the planet (as necessary). Subtract the sum or difference thus obtained from the daily motion of the sighrocca. Multiply the difference by the current Rsine- difference corresponding to the sighrakendra and also by the corrected sighra epicycle and divide the product by 18000. Multiply that by the radius and divide by the sighrakarna, and reduce that to the correspond- ing arc. Add it to or subtract it from the mean daily motion already. corrected for the mandakendrajyagatiphala. 2 See KK (Sengupta), ii. 19; KK (Babua Misra), ii. 26. Also see Pṛthua daka's comm, on this stanza. 3 e. g. by Lalla, SiDVṛ, I, ii. 40.

• See supra stanzas 14-17. TRUE DAILY MOTION OF MARS, ETC

151 because the rules depend upon the use of the table of Rsine-differences. It is assumed that the Rsines very uniformly, so the results obtained from the above rules are only approximate as the author himself admits. The true daily motion is required for the purpose of computing dis- placements of the planets. In the case of the Sun and the star-planets (Mars, Mercury, etc.), the daily motion is small enough and not much error is introduced by using the jivabhukti but in the case of the Moon the daily motion is so much that the use of the jivabhukti in computing displace- mets of the Moon may cause serious error. This thing was noticed by the author of the present work himself who in his smaller work, the Laghu-Bhās- kariya, has criticised the jivabhukti and has preferred the karnabhukti, i.e., the daily motion derived by the use of the instantaneous distance of the planet. Lalla, the author of the Sisya-dhi-vṛddhida, also has pointed out the above-mentioned discrepancy of the jivabhukti. In the works of Brahma- gupta, Lalla, and Sripati, the rules of finding the true daily motion of the planets are based on the distances of the planets.³ The object of devising rules depending on the instantaneous distances of the planets was essentially to obtain the instantaneous velocities of the planets, but the aim was not wholly achieved by the rules given by Brahma- gupta, Lalla, and Sripati. For the velocity obtained by their methods turned out to be the same as the mean daily motion of the planet at the intersection of the concentric and the eccentric, which was wrong. An accurate rule for the instantaneous velocity of a planet was given by Man- jula (932 A. D.) and by Aryabhata II (c. 950 A.D.). The method was fully explained by Bhaskara II (1150 A. D.). More accurate rules occur in the Tantra-sangraha of Nilakantha,5 'Instantaneous velocity is known in Hindu astronomy by the terms tātkāliki-gati, tatkala-gati, tatkṣaṇa-gati, tatsamayajā-gati, and velā-bhukti. ¹ ii. 14-15. 2 SiDV, I, ii. 43. 3 The rules are given in Br.Sp.Si, ii: 41-42; Ś¡DVṛ, I ii. 45-46; and Sise, iii. 42-43 respectively.

• See ŚiDVṛ, II, i. 13(ii) and also SiŚi, II, v. 39.

5 The rules of Mañjula, Āryabhaṭa II, Bhāskara II, and Nilakantha are given in LMā, ii. 4(ii), MSi, iii. 15 (ii). 27; SiSi, I, ii. 37, 39; and TS, ii. 51-52 respectively. 152 TRUB LONGITUDE OF A PLANET A rule for finding the longitudes of the Sun and the Moon at the end of the parva-tithi: 64. Multiply the unelapsed, part of the (parva) tithi or the elapsed part of the (next) tithi by the (true) daily motions of the Sun and the Moon and divide (each product) by the diffe- rence between the (true) daily motions (of the Sun and Moon). The longitudes of the Sun and the Moon increased or dimini- shed (in the two cases respectively) by the quotients (thus ob- tained) should be known as the longitudes agreeing to minutes of the Sun and Moon-the causes of the performances of the world. By the parva-tithi is meant the fifteenth tithi called purnima (i.e., full moon day) or the thirtieth tithi called amāvāsyā (i.e., new moon day). The end of the former occurs when the Sun and Moon are in opposition in longitude, and the end of the latter occurs when the Sun and Moon are in conjunction in longitude. At the time of opposition of the Sun and Moon, the longitudes of the Sun and Moon differ by six signs but otherwise agree to minuts of arc. At the time of conjunction of the Sun and Moon, the longitudes of the Sun and Moon are exactly the same and therefore agree to minutes of arc. Hence the above rule. ¹ For sunrise on the parva-tithi in the first case and for sunrise on next tithi in the second case. 2 The same rule is stated in SuSi, iv. 8; PSi, vi. 1; MSi, v. 4(ii)-5 (i); Sise, ii. 84; and Sisi, I, ii. 70; Nilakantha (TS, iv. 1-8) and Kamalā- kara (SiTV, ix. 1) have prescribed a successive repetition of the above rule. The method given in LBh, iv. 1 is approximate and simpler. CHAPTER V ECLIPSES (1) ECLIPSE OF THE SUN Introductory stanza: 1. Now shall be given the solar eclipse as taught by Ācārya Aryabhaṭa. At the beginning of that one should know the determination of the elements (to be used). Mean distances in yojanas the Sun and Moon : 2. (The mean distance) of the Sun is 459585 (yojanas); that of the Moon is 34377 (yojanas). A rule for converting true distances known in minutes into true distances in yojanas: 3. These (severally) multiplied by their true distances in minutes (as determined before)¹ and divided by the radius (i.e., 3438 minutes) are known as the true distances in yojanas of the Sun and the Moon.2 That is, Sun's true distance in yojanas Sun's mean distance in yojanas x Sun's true distance in minutes Radius and Moon's true distance in yojanas Moon's mean distance in yojanas x Moon's true distance in minutes Radius Diameters of the Earth, the Sun, and the Moon in terms of yojanas: 4. The diameter, in terms of yojanas, of the Earth has been stated by the learned to be 1050; of the Sun 4410; and of the Moon, 315. ¹ Vide supra, Chapter IV, stanzas 9-12 and 19-20. 2 This rule is found to occur also in BrSpSi, xxi. 31(ii); ŚiDVṛ, I, iv. 5(i) ; LBh, iv. 3 ; Siśe, v. 4 (ii) ; Siśi, I, v. 5(i); TS, iv, 10(ii)-11, 154 The following table gives the diameters and distances of the Sun and the Moon and their ratios according to Bhaskara I, Śripati, Bhāskara II, and also according to modern astronomers. Comparative table of diameters and distances of the Sun and Moon. Sun's diameter in yojanas Sun's distance in yojanas Ratio

Moon's diameter in yojanas Moon's distance in yojanas Ratio Bhāskara I Śripati Bháskara II ● 4410 459585

• 009596

ECLIPSES 315 34377

• 009163

6522 A rule for finding the angular Moon: 684870

• 009523

480 51566

• 009308

6522 480 Modern, in miles 689377 92900000 009461 51566

• 009308

864000

• 0093

2160 238900 •009 This table shows that, although the values of the diameters and mean distances of the Sun and the Moon given by different authorities differ, their ratios are practically the same. It may be pointed out that it is these ratios and not the diameters or distances that are used in the calculation of the eclipses-a fact which is partly responsible for the great accuracy attained by Hindu astronomers in the prediction of the eclipses. diameters of the Sun and the

5. The diameters of the Sun and the Moon when (seve- rally) multiplied by the radius and divided by their true distances in yojanas become the (angular) diameters in minutes.¹ 1 This rule is found also in BrSpSi, xxi. 34(ii) ; ŚiDVṛ, I, iv, 8; Siśe, v. 6; and SiSi I, v .7. ANGULAR DIAMETERS OF SUN, MOON AND SHADOW That is, Sun's diameter in minutes = and Moon's diameter in minutes = Formulae for the true (i.e., angular) diameters of the Sun, the Moon, and the shadow' in terms of the true daily motions of the Sun and the Moon : That is to say, (1) Sun's true diameter 6-7. Five-ninths of the (minutes of the Sun's true) daily motion and one-twenty-fifth of the (minutes of the Moon's true) daily motion (treated as minutes) respectively increased and diminished by the seconds equal to one-fourths of themselves are to be known as the true diameters of the Sun and Moon (respectively). One-tenth of (the minutes of) the moon's true daily motion (treated as minutes) plus one-sixteenth of the same treated as seconds is stated to be the (true) diameter of the shadow.2 = (2) Moon's true diameter Sun's diameter in yojanas x R Sun's true distance in yojanas Moon's diameter in yojanas x R Moon's true distance in yojanas 9 + 5 + 36 (Sun's true daily motion in minutes) seconds; (Sun's true daily motion in minutes) minutes tss (3) True diameter of the shadow Moon's true daily motion in minutes 25 minutes seconds; Moon's true daily motion in minutes 100 Moon's true daily motion in minutes minutes. 10 Moon's true daily motion in minutes 16 seconds. ¹ By the shadow is meant in this chapter the section of the cone of the Earth's shadow at the Moon's distance. 2 Similar rules occur also in Br.Sp.Si, iv. 6(i); KK (Sengupta's edition), iv. 2(i) ; ŚiDVṛ, I, iv. 9; MSi, v. 5(ii); SiŚe, v. 9; Siśi, I, v. 8-9 ; KPr, v. 2. 156 The following is the rationale of the above formulae: From stanza 5, we have Sun's true diameter = But R Sun's true distance Therefore, Sun's true diameter Sun's diameter in yojanasXR minutes. Sun's true distance in yojanas ECLIPSES in yojanas Sun's true daily motion in minutes Sun's mean daily motion in yojanas (Sun's diameter in yojanas)×(Sun's true daily motion in minutes) Sun's mean daily motion in yojanas minutes. travels through Now, according to Bhaskara I, a planet 1,24,74,72,05,76,000 yojanas in 1,57,79,17,500 mean civil days, therefore the mean daily motion of a planet comes out to be 7905-8 yojanas. Hence, 4410 (Sun's true daily motion in minutes) Sun's true diameter = 7905.8 + minutes. 5 (Sun's true daily motion in minutes), minutes 9 161 (Sun's true daily motion in minutes) > 60 7905.8 + Similarly, Moon's true diameter seconds. 5 (Sun's true daily motion in minutes) minutes. 9 5 (Sun's true daily motion in minutes), seconds. 36 - 315 (Moon's true daily motion in minutes) 7905-8 minutes. Moon's true daily motion in minutes 25 minutes. 308 (Moon's true daily motion in minutes) x 60 7905-8X25 Moon's true daily motion in minutes 25 Moon's true daily motion in minutes 100 seconds. minutes seconds. MERIDIAN-ECLIPTIC POINT FOR GEOCENTRIC CONJUNCTION 157 The formula for the true diameter of the shadow given in the text depends entirely upon the true daily motion of the Moon whereas it ought to depend upon the true daily motions of the Sun and Moon both. The author obviously takes the mean diameter of the shadow to be that which corresponds to the mean distances of the Sun and the Moon and derives the value of the true diameter of the shadow therefrom by apply- ing the usual process. The rationale seems to be as follows: Mean diameter of the shadow Earth's diameter __(Sun's diameter. Earth's diameter) × (Moon's mean ace) Sun's mean distance =1050-(4410-1050) × 34377 459585 -1050- 3360x34377 459585 -1050-251-3-798-7 yojanas. Therefore, True diameter of the shadow 798-7XR minutes, Sun's true distance in yajanas 798-7X(Moon's true daily motion in minutes) minutes. Moon's mean daily motion in yojanas 798 7x (Moon's true daily motion in minutes) minutes 7905-8 Moon's true daily motion in minutes minutes, 10 + Moon's true daily motion in minutes 16 seconds. A rule for the determination of the (sayana) longitude of the meridian-ecliptic point for the time of geocentric conjunction of the Sun and Moon: 8-11. Now is stated the method for (finding the longitude of) the meridian-ecliptic point. Those proficient in the (astro- nomical) science should know that the determination (of that) is made with the asus due to right ascension (i.e., with the times in asus of rising of the signs at the equator). 158 ECLIPSES From the asus intervening between midday and the tithyanta ("the time of geocentric conjunction of the Sun and Moon") one should subtract in the forenoon the asus correspon- ding to the degrees traversed of the sign occupied by the Sun (at the tithyanta) and in the afternoon the asus corresponding to the degrees to be traversed. The degrees (traversed or to be traversed) should be (respectively) subtracted from or added to the longitude of the Sun (for the tithyanta). The complete signs determined with the help of the asus of the right ascensions of the signs and whatever (fraction of a sign) is obtained by pro- portion should also be (respectively) subtracted or added by those who know the true principles of the science of time. This (i.e., the longitude thus obtained) is the true (sayana) longitude of the meridian-ecliptic point. So has come out of the mouth of the illustrious (Acarya Ārya)bhaṭa.¹ The five Rsines relating to the Sun and the Moon: 12. The orbits of the Sun and the Moon being different, the (five) Rsines for them. are said to differ. This (difference) is indicated by the words "svadrkksepa etc." of the Master (Āryabhata I).³ The five Rsines contemplated here are the so called udayajyā, madhyajya, drkkṣepajyā, dṛgjyā and dṛggatijyā. Rules for finding these are gives in the next eleven stanzas. A rule for finding the Sun's udayaja: 13. Multiply the Rsine of the bahu due to the (sayana) longitude of the rising point of the ecliptic by (the Rsine of) the (Sun's) greatest declination and then divide (the product) by. (the Rsine of) the colatitude: the quotient is the Sun's true udayajyā.* 1 The Sun's longitude to be used in this rule must be sayana. 2 Lalla in his Śisya-dhi-vṛddhida takes for simplicity the five Rsines for the Moon to be the same as those for the Sun. 8 Vide A, iv. 33.

• This rule occurs also in ŠiDV₁, I, v. 4. That is

THE TEN RSINES Rsin x Rsin € Rcos 159 Sun's udayajyā where is the sayana longitude of the rising point of the ecliptic, € the greatest declination of the Sun, and the latitude of the place.¹ The Sun's udayajyā is the Rsine of that part of the local horizon which lies between the east point and the rising point of the ecliptic. It is equal to the Rsine of the agrā of the rising point of the ecliptic and is, therefore, also known as udayalagnāgrā or simply lagnagrā. A rule for finding the Moon's udayajya: 14-16(i). The Rsine of (the longitude of) the rising point of the ecliptic minus (the longitude of) the Moon's ascending node, multiplied by 15 and divided by 191, is the Rsine of the (Moon's) latitude corresponding to the rising point of the ecliptic. When the declination and (Moon's) latitude corresponding to the rising point of the ecliptic are of like direction, take their sum; in the contrary case, take their difference. The radius multiplied by the Rsine of the resulting arc (of the sum or difference) and then divided by (the Rsine of) the colatitude gives the the Moon's udayajyā. The Moon's udayajya is the Rsine of that part of the local horizon which lies between the east point and the rising point of the Moon's orbit. Rules for finding the madhyajyas of the Sun and the Moon: 16(i)-18. Calculate the Rsine of the celestial latitude (of the Moon) from the longitude of the meridian-ecliptic point minus the longitude of the Moon's ascending node. 1 The rationale of this rule is similar to that of the Sun's agrā. See stanza 37 of Chapter III. 2 This rule is approximate as the declination of the rising point of the Moon's orbit is not exactly equal to the sum or difference of the declination and Moon's latitude corresponding to the rising point of the ecliptic. 160 BCLIPSES When the declination of the meridian-ecliptic point and the local latitude are of like direction', take their sum; in the contrary case, take their difference; (and determine the Rsine of that sum or difference). This is the Sun's madhyajyā which has the same direction as the above sum or difference. In the case of the Moon, take the sum or difference of the local latitude, the declination (of the meridian-ecliptic point), and the (Moon's) latitude (corresponding to the meridian -ecliptic point) on the basis of likeness or unlikeness of direc- tion; and then determine the Rsine of the resulting arc. This is the (Moon's) madhyajyā, which has the same direction as the resulting arc.² The Sun's madhyajya is the Rsine of the zenith distance of the meri- dian-ecliptic point. The Moon's madhyajya is the Rsine of the zenith distance of the meridian point of the Moon's orbit. A rule for the determination of the drkksepajyas of the Sun and the Moon : 19. Take the product of (the Sun's or Moon's) own madhyajya and udayajya, then divide (the product) by the radius, and then take the square (of the quotient). Subtract that from the square of the (own) madhyajya : the square root of that (difference) is known as (the Sun's or Moon's) drkkṣepajya.³ 1 The direction of the local latitude is always south; the direction of the declination of the meridian-ecliptic point is north or south according as the meridian-ecliptic point is to the north or south of the equator. The rule for the Moon's madhyajya is approximate, because the arcual distance between the points where the meridian intersects the ecliptic and the Moon's orbit is not equal to the Moon's latitude corres- ponding to the meridian-ecliptic point. 3 This rule too is approximate Brahmagupta. (See Br.Sp.Si, xi. 29, 30). as follows: Let Z be the zenith, M the and has been criticised by The rationale of the ru is meridian point of the ecliptic 161 The Sun's drkkṣepajya is the Rsine of the zenith distance of that point of the ecliptic which is at the shortest distance from the zenith.¹ The Moon's drkkṣepajya is the Rsine of the zenith distance of that point of the Moon's orbit which is at the shortest distance from the zenith. THE TEN RSINES A rule for finding the drgjyas (i.e., the Rsines of the zenith distances) of the Sun and the Moon : 20-22. Calculate the Rsine of the Sun's zenith distance (drgjya) from the nadis elapsed (since sunrise in the forenoon) or to elapse (before sunset in the afternoon) in accordance with the method stated before. The method for (finding the Rsine of the zenith distance of) the Moon is now being described. Take the sum or difference of the celestial latitude and declination of the Moon for the time of geocentric conjunction (of the Sun and Moon) according as they are of like or unlike direction. The Rsine of the resulting sum or difference is (the Rsine of) the Moon's (true) declination. From that calculate the day-radius, the earthsine, and the asus of the ascensional difference. With the help of these and the nädis elapsed (since sunrise in the forenoon) or to elapse (before sunset in the afternoon) obtain the Rsine of the zenith distance. (This is the Rsine of the Moon's zenith distance).³ (or Moon's orbit) and C the point of the ecliptic (or Moon's orbit) which is at the shortest distance from Z. Then in the triangle ZCM, Rsin ZM = madhyajya, LZCM 90°, and Rsin MZC = udayajya. Therefore Rsin (arc MC)=(madhyajyā × udayajyā)/R. The final result is obtained by treating the triangle formed of the Rsines of the sides of the triangle ZCM as a plane right-angled triangle (which assumption is however incorrect). The same rule occurs also in SiDV, I, v. 5. ¹.The point of the ecliptic which is at the shortest distance from the zenith is called the nonagesimal or the central ecliptic point. 2 Vide supra, chapter III, stanzas 18-25. This rule for the Rsine of the Moon's zenith distance is evidently approximate, 162 ECLIPSES A rule for finding the drggatijyās of the Sun and the Moon : 23. Obtain the difference between the squares of the (Sun's as also of the Moon's) own drgya and drkkṣepajyā, and then take their square-roots. These (square-roots) are the drggatijyās of the Sun and the Moon.¹ The Sun's drggatijya is the distance of the zenith from the plane of the secondary to the ecliptic passing through the Sun. The Moon's drggatijyā is the distance of the zenith from the plane of the secondary to the Moon's orbit passing through the Moon. In later astronomical literature, the dṛggatijyä is used to mean the Rsine of the altitude of the central ecliptic point (i. e., the point of the ecliptic nearest from the zenith); and the distance of the zenith from the plane of the secondary to the ecliptic is denoted by the term drinatijyā. The rationale of the above rule is as follows: In Fig. 18,³ CS is the ecliptic and K its pole; S is the Sun and Z the zenith; KZC and KS are secondaries to the ecliptic; and ZA is perpendicular to KS. Since the arcs ZC and ZA are perpendicular to CS and AS respectively, therefore (Rsin ZS)³ - (Rsin ZC)², (Rsin ZA)² i.e., (Sun's drggatijyā)² (Sun's dṛgjyā)² - (Sun's drkksepajyā). Similarly, in the case of the Moon. = A rule for finding the time of apparent conjunction of the Sun and Moon: 24-27. Severally multiply the own drggatijyas (of the Sun and the Moon) by the Earth's semi-diameter and divide the products by the respective true distances in yojanas. The quotients (thus obtained) are known as the lambanas (of the Sun and the Moon) in terms of minutes (of arc), etc. Multiply their difference by 60 and divide that by the difference between the true daily motions of the Sun and the Moon. Thus are obtained the ghatis etc. (of the lambana). In the forenoon, subtract them from, and in the afternoon, add ¹ This rule occurs also in SiDV, I, v. 6(i). 2 See infra, p. 165, TIME OF APPARENT CONJUNCTION OF THE SUN AND MooN 163 them to the time of geocentric conjunction of the Sun and Moon. (Then is obtained the first approximation to the time of apparent conjunction).

The lambama computed for the middle of the day is subtracted from the time of geocentric conjunction when the Moon's udg¢jya is north and added when south.

Repeat this process until the nearest approximation (to the lambama for the time of apparent conjunction) is arrived at:

The corresponding displacements should be given by the learned to (the longitudes of) the Sun and the Moon, as in the case of the tithi (i.e., the time of conjunction of the Sun and the Moon ). The term lambama is the technical term for “parallax in longitude". When used alone in connection with a solar eclipse it generally stands for the difference between the parallaxes in longitude of the Sun and the Moon. The above stanzas aim at finding out the time of apparent conjunc tion of the Sun and Moon. This involves a knowledge of the lambana for that timeFor, time of apparent conjunction = time of geocentric conjunction + lambama in time for the time of apparent conjunction, where + or – sign is taken according as the Sun and the Moon at the time of apparent conjunction lie to the west or east of the central ecliptic point. The lambama for the time of apparent conjunction depends on the time of apparent conjunction itself. But as the time of apparent conjune tion is unknown, the corresponding lambama cannot be obtained directly and recourse is taken to the method of successive approximations stated in the text. 1 The literal translation would run as follows: Repeat this process until the time of apparent conjunction is fixed. A What is meant is that after the first approximation to the time of apparent conjunction is obtained, the corresponding longitudes of the Sun and the Moon should be calculated and the process repeated. 164 To begin with, the time of geocentric conjunction is taken as the first approximation to the time of apparent conjunction and the corres- ponding lambana is obtained by the following formula: lambana = Moon's drggatijyä x Earth's semi-diameter Moon's true distance in yojanas Sun's drggatijyä x Earth's semi-diameter minutes. Sun's true distance in yojanas This formula may be derived as follows: Consider Fig.18.CS is the ecliptic, C and S being the central ecliptic point and the Sun at the time of geocentric conjunction (treated as the time of apparent conjunction). K is the pole of the ecliptic and Z the zenith. S' is the position of the Sun as observed from the local place. ZA and S'B are perpendiculars on the secondary to the ecliptic passing through S' (i.e., on KS); S'D is perpendicular to the ecliptic. But ECLIPSES But Rsin (arc BS')= Sun's lambana- Fig. 18 In the triangle S'DS right-angled at D, SS' denotes the Sun's parallax in zenith distance and SD denotes the Sun's parallax in longitude (lambana). Hence From the triangles SBS' and SAZ, right-angled at B and A respectively, we have Rsin (arc BS')= Z Rsin (arc SS)= Rsin (arc AZ) x Rsin (arc SS') Rsin (arc ZS) BS' or SD approximately. Therefore, (Sun's drggatijyä) x Rsin (arc SS') Rsin (arc ZS) Sun's lambana== approx. Earth's semi-diameter x Rsin (arc ZS) Sun's true distance in yojanas K Sun's drggatijyä x Earth's semi-diameter Sun's true distance in yojanas (1) TIME OF APPARENT CONJUNCTION OF THE SUN AND MOON Similarly, Moon's lambana= Moon's drggatijyā × Earth's semi-diameter Moon's true distance in yojanas 165 (2) Subtracting (1) from (2), we obtain the required formula. The lambana obtained by the above formula is in terms of minutes of arc. When this is multiplied by 60 and divided by the difference between the true daily motions of the Sun and the Moon, it is reduced to the corres- ponding ghatis. The lambana in ghatis according to the text is to be subtracted from or added to the time of geocentric conjunction according as it occurs in the forenoon or afternoon. In fact, subtraction or addition should be made according as the conjunction occurs to the east or to the west of the central ecliptic point. The law of addition and subtraction given in the text is, however, more convenient in practice. The rule in stanza 26 shows that the author was aware that at noon the lambana was different from zero. Still the rule prescribed for the application of that lambana in that stanza shows that the author did not know the correct law for the addition or subtraction of that lambana.¹ The application of the lambana for the time of apparent conjunction having been thus made to the time of geocentric conjunction, we obtain the second approximation to the time of apparent conjunction. The Sun and the Moon are then calculated for that time and the method is repeated again and again until the nearest approximation to the time of apparent conjunction is obtained. 1 The commentators have, however, tried to interpret the stanza as conveying the desired meening. For example, Parameśvara writes: "The word indu here stands for the madhyajya. Therefore, subtraction is to be made when the madhyajyā and the udayajyā are of like direction. When they are of unlike directions, addition is to be made. This is what has been stated here". The word indu means "Moon" and it cannot be interpreted to mean "madhyajya". There is no scope for such a meaning. More- over, Parameśvara says that subtraction or addition is to be made accord- ing as the madhyajya and udayajya are of like or unlike directions. In fact, there is no reference to any directions, like or unlike. The words used are udak and dakṣine which mean "in the north" and "in the south" respectively and not "like direction" and "unlike directions" as Parames- vara has supposed. 166 A rule for finding the true nati: 28-32. Multiply the drkksepajyas (of the Sun and the Moon), obtained by the method of successive approximations, (severally). by the Earth's semi-diameter, and divide (the resulting products) by the true distances in yojanas (of the Sun and the Moon respectively): the quotients are in minutes of arc (the parallaxes in latitude of the Sun and the Moon). Take their difference, provided that the madhyajyas of the Sun and the Moon are of like direction; in the contrary case, take their sum: thus are obtained the minutes of the avanati (or nati). (As regards the direction of the nati) take the direction of the Moon's (madhyajyā). ECLIPSES Multiply the Rsine of the longitude of the Moon minus the longitude of the Moon's ascending node by 270, and then divide that product by the Moon's true distance in minutes. Thus is obtained the true celestial latitude of the Moon. This increased by that (nati) (provided the two are of like direction) is the true nati. In case they are of unlike directions, take their difference. The difference is then called the (true) nati. Thus is obtained the true avanati (or true nati) for the middle of the eclipse as determined from the drkksepa and the true latitude of the Sun and the Moon for the time of apparent conjunction (literally, the time of geocentric conjunc- tion corrected for the lambana-difference). The term nati or avanati means "parallax in latitude". When used alone in connection with a solar eclipse, it denotes the difference betweeu the parallaxes in latitude of the Sun and the Moon. The true nati is the Moon's latitude corrected for the nati (i.e., Moon's apparent latitude). It denotes the arcual distance of the Moon from the Sun's apparent orbit due to parallax. The formula given for the nati for the time of apparent conjunction of the Sun and the moon is nati = Moon's dṛkksepajya X Earth's semi-diameter Moon's true distance in yojanas Sun's drkkṣepajyā × Earth's semi-diameter Sun's true distance in yojanas SPARSA-STHITYARDHA AND MOKSA-STHITYARDHA The rationale of this formula is as follows: Refer to the previous figure. From the triangles S'DS and ZCS, right-angled at D and C respectively, we have Rsin (arc S'D) or arc S'D = But Rsin (arc SS') Therefore, Sun's nati Similarly, Moon's nati Rsin (arc ZC) x Rsin (arc SS') Rsin (arc ZS) = Earth's semi-diameter x Rsin (arc ZS) Sun's true distance in yojanas Rsin (arc ZC) x Earth's semi-diameter Sun's true distance in yojanas Sun's drkkṣepajya x Earth's semi-diameter Sun's true distance in yojanas Moon's drkksepajya X Earth's semi-diameter Moon's true distance in yojanas Subtracting (1) from (2), we get the required formula. 167 6 On the possibility of a solar eclipse : (1) Like the lambana for the time of apparent conjunction, the nati too for that time is determined by the method of successive approximations. 33. An eclipse of the Sun will not occur if the (true) nati is equal to (or greater than) half the sum of the diameters of the Sun and the Moon. It is possible when it (i.e., the true nati) is less (than that). A rule for the determination of the sparsa-sthityardha and mokşa- sthityardha : 34-39. Multiply the square root of the difference detween the squares of half the sum of the diameters of the Sun and Moon and of the (true) nati by 60 and then divide (the pro- duct) by the motion-difference (of the Sun and the Moon): thus are obtained the ghatis of the sthityardha. By these ghatis 168 ECLIPSES diminish and increase the time of apparent conjunction as obtained by the method of successive approximations. Then are obtained the (approximate) times for the first and last contacts respectively. Proceeding with them, calculate the (ten) Rsines (for the Sun and the Moon), etc., (and obtain the nearest approximations to the lambanas for the times of the first and last contacts). Always add, in the case of a solar eclipse, the nadis of the difference between the lambanas for the first con- tact and the apparent conjunction to the sthityardha : (the result is the sparsa-sthityardha). Also add the (nadis of the difference between the) lambanas for the apparent conjunction and the end of the eclipse to the sthityardha : the result is the mokṣa-sthityardha. The sthityardhas thus obtained are very accurate I say this raising my hands aloft (i.e., with firm determination). When the first contact and the apparent conjunction occur in different halves (eastern and western) of the celestial sphere, then the entire lambana (in nadis) for the time of the first contact is added to the 'sthityardha. Similarly, when the last contact and the apparent conjunction occur in different halves of the celestial sphere, the entire lambana (in nadis) for the time of the last contact is always added to the sthityardha. The same procedure is also adopted when the apparent conjunction occurs at noon. The term sthityardha means "half the duration (of an eclipse)", The sparsa-sthityardha is the interval of time between the first contact and the apparent conjunction. The mokṣa-sthityardha is the time-interval bet- ween the apparent conjunction and the last contact. A rule for the determination of the vimardardha : 40. The nadis of the vimardärdha are to be determined from the square root of the difference between the squares of (i) the difference between the semi-diameters of the eclipsed and eclipsing bodies and (ii) the Moon's latitude (corrected for the nati). 169 The term vimardärdha means "half the duration of the totality of an eclipse", i.e., the time-interval between the immersion and the apparent conjunction or between the apparent conjunction and the emersion. The time-interval between the immersion and the apparent conjunction is called the sparsa-vimardardha and that between the apparent conjunction and the emersion is called the mokṣa-vimardärdha. AKBA-VALANA The above stanza gives the method for finding the first approxima- tion to the vimardärdha in minutes of arc. The corresponding nadis are obtained by multiplying that by 60 and dividing by the difference between the true daily motions of the Sun and the Moon. The nearest approximations to the sparsa- and mokṣa-vimardärdhas are obtained as in the case of the sthityardhas. A rule for knowing the time of actual visibility of the first contact in the case of a solar eclipse : 41. On account of the brightness of the Sun, the time of (actual visibility of) the first contact (in the case of a solar eclipse) is the (computed) time of the first contact plus the time corresponding to the minutes of arc amounting to one-eighth of the Sun's diameter. Aryabhata I says: "When the moon eclipses the Sun, even though one-eighth part of the Sun is eclipsed this is not perceptible because of the brightness of the Sun and the transparency of the Moon's circumference."¹ A rule for finding the magnitude and direction of the akşa- valana: 42-44. Multiply the Rversed-sine of the asus intervening between midday and the tithi (i.e., the time of the first contact, the middle of the eclipse, or of the last contact) by (the Rsine of) the (local) latitude and divide that (product) by the radius. Reduce the resulting Rsine to the corresponding arc (called akşa-valana) and determine its direction. When the above asus exceed (those corresponding to) a quadrant, add the Rsine of the excess to the radius and operate as before and then find the direction.

1 A, iv. 47. 170 ECLIPSES The direction (of the akşa-valana) for the middle of the ecliptic is the same as that for the first contact. In the forenoon, the directions (of the akşa-valana) in the eastern and western halves of the disc (of the eclipsed body) are north and south respectively. To the west of the sky (i.e., in the afternoon), the akşa-valana is always of the contrary direction.¹ The great circle of the celestial sphere which has the centre of the eclipsed body as one of its poles is called the horizon of the eclipsed body. Suppose that the prime vertical, the equator, and the ecliptic intersect the horizon of the eclipsed body at the points E₁, T₁, and Y, respectively towards the east of the eclipsed body. Then the point E₁ is called the east point of the horizon of the eclipsed body; the arc E₁T₁ (which denotes the deflection of the equator from the prime vertical on the horizon of the eclipsed body) is called the akşa-valana; and the arc T₁Y₁ (which denotes the deflection of the ecliptic from the equator along the same circle) is called the ayana-valana. The formula for the akşa-valana stated in the text is Rversin H x Rsin R Rsin (akṣa-valana) = = where H denotes the hour angle (nata-kāla) and the local latitude. This formula is based on inference. Early Hindu astronomers noted that when the eclipsed body was at the intersection of the meridian and the equator, the Rversed-sine of the hour angle was zero and the Rsine of the akṣa-valana was also zero; and that with the increase of the Rversed-sine of the hour angle the Rsine of the akṣa-valana also increased; and further that when the eclipsed body was at the intersection of the horizon and the equator, the Rversed-sine of the hour angle was equal to its maximum value R and the Rsine of the akşa-valana was also maximum and equal to the Rsine of the latitude. They, therefore, supposed that the Rsine of the akşa-valana varied as the Rversed-sine of the hour angle, and to obtain the Rsine of the akṣa-valana for the desired time they made use of the The same rule is found also in ŚiDVṛ, I, iv. 23 and Sise, v. 18. By the eclipsed body here is meant the position of the eclipsed body on the ecliptic. AYANA-VALANA 171 proportion: When the Rversed-sine of the hour angle amounts to R, the Rsine of the aksa-valana equals the Rsine of the latitude, what then would be the value of the Rsine of the akṣa-valana corresponding to the Rversed- sine of the desired hour angle? Hence the above formula. The above formula can be easily seen to be incorrect. It was first modified by Brahmagupta, who replaced Rversin H by Rsin H.¹ Better and accurate formulae were given by Bhaskara II.³ The rules for the direction of the akṣa-valana can be seen to be true by means of a diagram. A rule for the determination of the magnitude and direction of the ayana-valana: 45. The (Sun's) declination determined from the Rversed- sine of the longitude of the Sun or Moon as increased by three signs (treated as the Rsine of the bhuja) (is the ayana-valana). Its direction in the eastern half (of the disc of the eclipsed body) is the same as that of the ayana (of the Sun or Moon); ³ in the other half, it is contrary to that.* That is, Rsin (ayana-valana) Rsin Ex Rversin (λ+90°) R where e denotes the obliquity of the ecliptic and the sayana longitude of the eclipsed body (the Sun or Moon). This formula also is based on inference. The proportion used is the following: "When Rversin (+90°) is equal to R, the Rsine of the ayana-válana is equal to the Rsine of the obliquity of the ecliptic, what then would be the Rsine of the ayana-valana corresponding to the desired value of the Rversed-sine ?" 1. See Br Sp.Si, iv. 16. 2 See SiSi, I, v. 20-21(i); II, viii. 68; and II, viii. 66(ii)-67. 3 The ayana of a planet is north or south according as it is in the half-orbit beginning with the (sāyana) sign Capricorn or in that beginning with the (sayana) sign Cancer. ..

• This rule occurs also in ŚiDVṛ, I, iv. 25 and Sise, v. 20. 172

ECLIPSES This formula also is incorrect. It was modified by Brahmagupta,¹ who replaced Rversin (+90°) in the formula by Rsin (+90°). An accurate expression for the ayana-valana was given by Bhaskara II (1150 A. D.).³ A rule for finding the value of the resultant valana (spasta-valana) for the circle drawn with half the sum of the diameters of the eclipsed and eclipsing bodies as radius: 46-47. When they (i.e., the akşa-valana and the ayana- valana) are of unlike directions, take the difference of their arcs; in the contrary case, take their sum.³ Multiply the Rsine of that (sum or difference) by half the sum of the diameters of the eclipsed and eclipsing bodies and divide (the product) by the radius. Add whatever is thus obtained to the (Moon's true) nati, provided that they are of like directions; in the contrary case, take their difference: the resulting sum or difference is the valana. The sum or difference of the akşa-valana and the ayana-valana accord- ing as they are of like or unlike directions gives the so called spasta-valana, i.e., the amount of deflection of the ecliptic from the prime vertical on the horizon of the eclipsed body. When the Rsine of that is multiplied by half the sum of the diameters of the eclipsed and eclipsing bodies and the product divided by radius, we the corresponding deflection on the circumference of the circle drawn with half the sum of the diameters of eclipsed and eclipsing bodies as radius. The sum or difference of this and the Moon's true nati according as the two are of like or unlike direc- tions gives the distance of the centre of the eclipsing body from the east- west line passing through the centre of the eclipsed body. So has been assumed in the above rule. The direction in the rule for adding or taking the difference of the reduced spasta-valana and the Moon's true nati is wrong. The two quanti- ties should be kept separately and laid off properly one after the other (in the projected figure). ¹ See BrSp.Si, iv. 17. & See Sisi, I, v. 21(ii)-22(i). s This rule occurs also in BrSpSi, iv. 18 (i) and SiDV, I, iv. 26. 173 PROJECTION OF ECLIPSES The next twenty stanzas ralate to the projection (i.e., graphical representation) of an eclipse. A method for ascertaining the centre of the eclipsing body for the times of the first and last contacts (called the sparsa-bindu and the mokşa-bindu respectively): 48-53. By means of a pair of compasses, whose smooth and large body is graduated with angulas and subdivisions there- of and which is embellished by the pointed end of a smoothened chalk-stick placed into its mouth, construct on the ground a circle with half the measure, in angulas, of the eclipsed body as radius, and another (concentric circle) with half the sum of the diameters of the eclipsed and eclipsing bodies as radius. (Through the common centre) then draw the east-west line and, with the help of a fish-figure, the north-south line. From the centre then lay off the valana (for the first or last contact) to- wards the north or south (according to its direction); draw a fish-figure there; and (through its head and tail) carefully¹ draw a line to meet the outer circle. At the meeting point of the outer circle and that line set a point. (This is the centre of the eclipsing body for the time of the first or last contact). From that point stretch out a line to reach the centre. Where this line is seen to intersect the circumference of the eclipsed body, lies the point of contact or separation of the Sun's disc. One minute of arc should be taken as equivalent to one half of an angula or as it appears in the sky. A method for determining the centre of the eclipsing body for the middle of the (solar) eclipse (called the madhya- bindu): 54-57. The valana for the middle of the eclipse is taken ¹ The word "carefully" indicates that when the valana corresponds to the first contact, the line should be drawn towards the west or east according as the eclipse is solar or lunar; but when the valana corresponds to the last contact, the line should be drawn in the contrary direction. 174 ECLIPSES without the addition or subtraction of the nati. When that and the nati are of like direction, the (madhya)valana should be laid off towards the east; when they are of unlike directions, it is stretched out from the centre towards the west. With the help of a fish-figure (drawn about the point thus obtained), a line should then be drawn in the direction of the nati. From the meeting point of that with the outer circle, the intelligent should then draw a line to reach the centre (of the circle).¹ The nati should then be laid off from the centre along that line. At the end of that lies the centre of the eclipsing body at the middle of the eclipse (i.e., the madhya-bindu). The two points (already marked) are the centres of the eclipsing body for the times of the first and last contacts (i.e., the sparsa-bindu and (mokṣa-bindu). When the nati (for the middle of the eclipse) is of south direction, it should be, laid off towards the south; when it is of north direction, it is laid off towards the north. Points of difference of procedure in the case of a lunar eclipse: 58. This (i.e., the previous rule) is the method for the middle of the eclipse in the case of a solar eclipse. In the case of a lunar eclipse, the points of the first contact, the middle of the eclipse, and the last contact should be clearly indicated reversely. That is, in the case of a lunar eclipse, the following procedure should be adopted: To begin with, the madhya-valana should be laid off towards the west or east, according as the madhya-valana and the nati are of like or unlike directions. With the help of a fish-figure drawn about the point thus obtained a line should then be drawn through that point in the direc- tion contrary to the direction of the Moon's latitude. From the meeting point of that line with the outer circle a line should then be drawn to reach ¹ This line is prependicular to the ecliptic at the time of the middle of the eclipse. PROJECTION OF ECLIPSES the centre of the circle. The Moon's latitude for the middle of the eclipse should then be laid off from the centre along that line. 175 Construction of the phase of the eclipse for the time of the middle of the eclipse : 59-60. Quickly cut off the eclipsed body by means of a pair of compasses (one leg of) which is placed at the madhya- bindu and (the other leg of) which is streched out by half the specified true measure of the eclipsing body. The portion thus cut off (in case the eclipse is partial), or the entire disc of the eclipsed body drawn (likhitam) on the projection (in case the eclipse is total)-all of that is clearly seen (in the sky) in that way at the time of the middle of the eclipse. By the world likhita, says Parameśvara, is meant a total eclipse, or, in case the Moon's latitude is zero and the disc of the eclipsed body is larger, an annular eclipse. Construction of the path of the eclipsing body: 61. Draw (an arc of) a circle passing through the three points set down above with the help of two fish-figures: this is the path of the eclipsing body. The phase of the eclipse for the given time is ascertained (by determining the position of the eclipsing body on that path and drawing its disc with its centre) there.¹ A method for calculating the phase of the eclipse for the given time : 1 62-63. Multiply the difference between the (true) daily motions of the Sun and the Moon by the sthityardha-ghatis* minus the given time (isṭakala) and divide the product by sixty. Add the square of the quotient to the square of the Moon's nati (corrected latitude) (for the given time) and then take the square root of that (sum). This (square root) is (the length of) 1 This rule occurs also in ŠiDVṛ, 1, iv. 34. I, That is, ghafis corresponding to the sthityardha, 176 the needle joining the centres of the eclipsed and eclipsing bodies at the given time in the case of solar and lunar eclipses. (Subtract that from half the sum of the diameters of the eclipsed and eclipsing bodies). The remainder is the phase of the eclipse for the given time.¹ ECLIPSES By the given time is meant the time elapsed since the first contact or the time to elapse before the last contact. Construction of the phase of an eclipse for the given time: 64-65. Stretch out a fine bamboo needle (equal in length to that joining the centres of the eclipsed and eclipsing bodies at the given time) obliquely from the centre in such a way that its end may fall on the so called path of the eclipsing body. Taking the centre at that point, cut off the eclipsed body by means of a circle drawn with half the diameter of the eclipsing body as radius. As much portion is thus cut off, so much of the eclipsed body is seen to be eclipsed (in the sky). Construction of the phase of a (solar) eclipse for the time of immersion or emersion: 66-67. The sthityardha in terms of minutes of arc minus the minutes of the vimardardha is the means for projecting the phase of the eclipse for that time (i.e., immersion or emersion). The Sun's disc should be cut off with the help of that (i.e., that should be laid off from the sparsa or moksa bindu along the path of the eclipsing body towards the centre and the point thus obtained should be treated as the centre of the eclipsing body for that time). The disc of the Sun should be cut off by means of a pair of compasses. The seizure (of the Sun) occurs on the western side of the disc and the separation on the eastern side. The remaining chapter deals with the lunar eclipse. ¹ This rule is found also in BrSpSi, iv. 11-12; SiDV, I, iv. 19-20; Sise, v. 14. 177 THE DIAMETER OF THE SHADOW (2) ECLIPSE OF THE MOON Points of difference of procedure in the case of a lunar eclipse: 68-70. Similarly, in the case of the Moon, which is the mirror for the face of the directions and exhibits (or bears) all excellent phases and whose round body looks like the face of a damsel, too, the (ten) Rsines should be found out. The points of difference (in the procedure) are being stated. The (five) Rsines relating to the shadow should be deter- mined as arising from the Sun's orbit. (In plac of the Sun's distance) the Moon's distance is stated to be the divisor. The lambana, determined as in the case of the Sun, should be added or subtracted reversely. Use of parallax in a lunar eclipse prescribed in the above stanzas is obviously wrong. Parameśvara comments: "Thus in the case of a lunar eclipse also, the use of parallax is stated here. This, say the profi- cients in Spherics, is improper". It must be mentioned that the application of parallax in the case of a lunar eclipse has not been prescribed in any other work on Hindu astronomy, not even in the smaller work of the present author. A rule for the determination of the diameter of the shadow, i.e., the diameter of the section of the Earth's shadow where the Moon crosses it : 71-73. Multiply the Sun's (true) distance in yojanas by the Earth's diameter and divide by the difference of their diameters: thus is obtained the length of the Earth's shadow. Or, multiply the Sun's (true) distance in yojanas by 5 and divide by 16: the result is called the length of the Earth's shadow, From that (length of the Earth's shadow) subtract the Moon's distance. Multiply the remainder by the Earth's dia- meter and divide (the product) by the length of the (Earth's) shadow. Multiply the resulting quotient by the radius and divide (the product) by the Moon's distance (in yojanas): this is (the diameter of) the shadow (in minutes of arc). 178 That is, the diameter of the shadow where length of Earth's shadow { length of Earth's shadow - Moon's distance} S ECLIPSES length of Earth's shadow This result is approximate and is the same as that given by Aryabhaṭa I. It is usually derived by the following method (called "the lamp and shadow method"): Therefore Sun's distance X Earth's diameter Sun's diameter - - Earth's diameter Consider Fig. 19. S is the centre of the Sun and E that of the Earth. SA and EB are drawn perpendicular to SE and denote the semi-diameters of the Sun and the Earth respectively. BL is parallel to ES. O is the point where SE and AB produced meet each other. E Fig. 19 Hindu astronomers compare SA with a lamp-post, EB with a gno- mon, and EO with the length of the shadow cast by the gnomon due to the light of the lamp. Consequently, they call EO "the length of the shadow". EO x Earth's diameter The triangles BEO and ALB are similar, therefore EO BL BE LA = B i.e., length of Earth's shadow= SE SA-EB SEX BE SEX 2BE = SA EB 2SA 2EB Sun's distance x Earth's diameter Sun's diameter - Earth's diameter = ¹ See A, iv. 39-40. This rule is found also in BrSpSi, xxiii. 8-9; and ŚiDV, I, iv. 6 (ii)-7. THE DIAMETER OF THE SHADOW 179 Now consider Fig. 20. AC and BD are the diameters of the Sun and the Earth. BOD is the shadow-cone¹, O its vertex. S and E are the centres of the Sun and the Earth. M is the point where the Moon crosses the shadow cone. MN is perpendicular to the axis of the shadow- C Therefore, B MN BD KO EO MN= - = Fig. 20 cone and denotes the diameter of the section of the shadow cone where the Moon crosses it. It is called the diameter of the shadow. The triangles MON and BOD are similar, so that KO X BD EO (EO M (EO K N - EK) X BD EO EM) X BD EO approx. O i.e., the diameter of the shadow _{length of Earth's shadow - Moon's distance } x Earth's diameter length of Earth's shadow The approximations made in the above procedure are obvious.² The diameter thus obtained is in yojanas. To reduce it to minutes of arc we have to multiply it by the radius (i.e., 3438') and divide by the Moon's distance in yojanas. ¹ Approximately. The formula for the diameter of the shadow stated above was modified and refined by Muniśvara (1646 A. D.) and Kamaläkara (1658 A. D.). The latter astronomer gave an accurate expression for the diameter of the shadow (in SiTV, ix. 29-33). 180 BCLIPSES Views of other astronomers regarding the calculation of a lunar eclipse: 74. Others give instruction in the lunar eclipse without the use of the ten Rsines, because it causes little difference in the result (and is simpler). There (i.e., in the rule stated by them) the sparsa- and mokṣa-sthityardhas arising from (the Moon's latitude for) the time of opposition of the Sun and Moon (lit. middle of the eclipse) should be operated upon by the method of successive approximations. Details of the process of successive approximations referred to above: 75-76. Multiply the (true) daily motion (of the Moon) by the time (in ghatis) of the sthityardha and divide the product by 60. Subtract the quotient from or add that to the Moon's (true) longitude for the time of opposition (of the Sun and Moon), according as it is the first or last contact. From that find out the Moon's latitude; and therefrom (again) calculate the sthityardha. (In this way repeat the above process again and again until two successive approximations agree). This is the process of successive approximations. Similar again is the process of (determining) the (sparsa- and mokṣa-) vimar- dardhas.¹ A rule relating to the direction of the Moon's latitude to be taken in the projection of a lunar eclipse: 77. While projecting an eclipse (of the Moon), the best amongst the learned should take the direction of the Moon's latitude to be north when it is south, and south when it is north. ¹ For details see our notes on Lbh, iv. 10-12. The rule stated here is found also in BrSpSi, iv. 8-9; SiDV, I, iv. 14-16; Siśe, v. 12-13; Sisi, I, v. 12-13. 181 Concluding stanza in praise of the methods for calculating and projecting an eclipse that have been stated above: CONCLUDING STANZA 78. This procedure regarding the Sun, the Moon, and the shadow, which has come down (to us) by tradition, has been stated here having cast off pride and jealousy. A learned person who acquires a mastery of this (procedure) shall become a (proficient) astronomer well versed in all astronomical methods. . CHAPTER VI RISING, SETTING AND CONJUNCTION OF PLANETS. A rule relating to the visibility-correction known as aksa- drkkarma : 1-2(i). Multiply the Moon's latitude for the desired time by the Rsine of latitude of the local place, and divide (the product) by the Rsine of the colatitude; whatever is thus obtained, say the learned, should be subtracted (from the Moon's longitude) in the case of rising of the Moon (i.e., in the eastern hemisphere) and added (to the Moon's longitude) in the case of setting of the Moon (i.e., in the western hemisphere), provided that the Moon is to the north of the ecliptic (i.e., if the Moon's latitude is north). When the Moon is to the south of the ecliptic, the law (of addition and subtraction) is the reverse.¹ The correction stated in the first three stanzas of this chapter is called "the visibility-correction (drk-karma)". When we apply this correc- tion to the true longitude of the Moon, we obtain the longitude of that point of the ecliptic which rises or sets with the apparent Moon. The visibility-correction is generally broken up into two compo- nents: (1) the visibility-correction due to the latitude of the local place (akşa-drkkarma), and (2) the visibility-correction due to the Sun's northward or southward course (i.e., ecliptic-deviation) (ayana-dṛkkarma). Let Fig. 21 represent the celestial sphere for the local place. SEN is the eastern horizon and Z the zenith; TE is the equator and P its north pole; TT is the ecliptic and K its north pole. Suppose that the Moon is rising at the point M' on the horizon. Let M be the point where the secondary to the ecliptic (kadambaprota-vṛtta) drawn through M' intersects the ecliptic, L the point where the hour circle (dhruvaprota-vṛtta) ¹ The same rule is found to occur in BrSpSi, vi. 4; Ś¡DVŢ, I, vii. 3 (ii); M.Si, vii. 4 ; Sise, ix. 7. AKSA-DRKKARMA drawn through M' intersects the ecliptic¹, and T the point where the horizon intersects the ecliptic. Then the arc MT of the ecliptic denotes the total visi- bility-correction; the arc ML denotes the visibility correction due to ecliptic-deviation (ayana-dṛkkarma); and the arc LT denotes the visibility correction due to the latitude of the local place (akṣa- dṛkkarma). The visi- bility-correction for a planet is defined in the same manner. - S E Z = T Fig. 21 The correction stated in the above stanzas is the akşa-dṛkkarma for the Moon. The formula stated is Rsin x Moon's latitude Rcos p akṣa-dṛkkarma where is the latitude of the local place. This formula is approximate. Let A be the point where the diurnal circle through M intersects the hour circle through M', B the point where the diurnal circle through M intersects the horizon, and C the point where the hour circle through B intersects the diurnal circle through M'. Then proceeding as for finding the earthsine, it can be easily shown that arc CM' Rsin x Rsin (arc BC) Rcos Rsin x Moon's latitude Rcos 183 approx.

• T is called dig-graha. See ŚiDV, I, vii. 4.

approx. ¹ The point L is called ayana-graha or ayana-graha. See ŚiDVṛ, 1. vii. 2,4. 184 RISING, SETTING AND CONJUNCTION OF PLANETS It follows that the formula given in the text actually gives an approximate value of the arc CM' or AB.¹ The rule stated in the text has been generally used in the cases where the latitude of the body concerned is small. In the cases of fixed stars whose latitudes may be considerable, a more accurate rule is pres- cribed.2 When the Moon's latitude is north, the longitude of the point T is smaller or greater than the longitude of the point L according as the Moon M' is rising or setting; and when the Moon's latitude is south, the longitude of the point T is respectively greater or smaller; hence the rule of addition and subtraction stated in the text. A rule relating to the visibility correction known as ayana- drkkarma : 2(ii)-3. Divide the product of the Rversed-sine of the Moon's longitude diminished by three signs, the Rsine of the Sun's greatest declination, and the Moon's latitude by the square of the radius. Whatever is thus obtained, say the learned, should be subtracted from the Moon's longitude provided that her ayana and latitude are of like direction; in the contrary case, that result should always be added to the Moon's longitude. ¹That this formula gives an approximate value of arc AB may be demonstrated as follows: Since MM' is small, we may treat the triangle M'BA as plane. Then from the triangle M'BA, we have Rsin BM'AXM'A Rsin M'BA Rsin L, BM'AXM'M AB = Rsin M'BA Rsin x Moon's latitude Rsin approx. approx. See E. Burgess, SuSi, vii. 7-12, notes, 2 See BrSpSi, x. 18-19; ŚiDVṛ, I, xi. 12-13; and Siśi, I, vii. 6. Bhāskara II has given a slightly modified formula for small latitudes also. See SiSi, I. vii. 7. The most accurate formula for the aksa-drkkarma occurs in SiTV, vii. 103-104.

• This rule occurs also in SiDVṛ, I, vii. 2-3(i) and Siśe, ix. 4, 5.

The same rule in a modified form occurs in BrSpSi, vi. 3; x. 17 and in MSi, vii. 2, 3. More accurate rules occur in SiSi, I, vii, 4, 5 and in SITV, vii, 77-80.

AYANA-DRKKARMA That is ayana-dṛkkarma Rversin (M-90°) x Rsin € x Moon's latitude RX R where M denotes the Moon's (sayana) longitude and the Sun's greatest declination. arc MA This formula is also approximate. Referring to the previous figure, we have Rsin LMM'A x Rsin (arc MM') R Rsin LKMP x Rsin (arc MM') R ayana-valana x Moon's latitude R approx. 185 approx. approx. Rversin (M-90°) × Rsin Ex Moon's latitude RX R on substituting the value of the ayana-valana.¹ The formula stated in the text, therefore, is an approximate value of the arc MA, or ML, which is the ayana-drkkarma. When the ayana and latitude of the Moon are of like directions, the longitude of the point L is smaller than the longitude of the point M; and when the ayana and latitude of the Moon are of unlike directions, the longitude of the point L is greater than the longitude of the point M; hence the rule of addition and subtraction stated in the text. ¹ Vide supra, chapter V, stanza 45, p. 171, 2 Vide supra, p. 171 (footnote). The visibility-corrections should be applied as follows. The true longitude of the Moon (which corresponds to the longitude of the point M of the ecliptic) should be first corrected for the 'ayana-dṛkkarma: the resulting longitude corresponds to that of the point L of the ecliptic. This is technically called the polar longitude of the Moon. This polar longitude should then be corrected for the aksa-drkkarma : the longitude thus obtained corresponds to that of the point T of the ecliptic, which rises (or sets) with the Moon's disc. This is technically called the longi- tude of the visible Moon (dṛśya-candra). In the text the order of the corrections is reversed. The difference is negligible. 186 A rule relating to the visibility of the Moon: 4-5(i). The Moon's longitude, which is obtained in this way after the application of the above-mentioned (visibility) corrections, is stated by the learned to be the longitude of the visible Moon (i.e., the longitude of that point of the ecliptic which rises with the Moon). RISING, SETTING AND CONJUNCTION OF PLANETS When the pranas¹ (of the oblique ascension) due to the degrees intervening between the Sun and the (visible) Moon,³ reduced to ghatis, amount to two, then the Moon is seen to rise in the clear, cloudless, starry sky after sunset.³ The latter part of the above passage relates to the visibility of the Moon, or, in other words, the heliacal rising of the Moon. On the fifteenth lunar day of the dark half of the month, the Moon comes near the Sun from behind and is lost in his splendour. After about two days it is beyond the limit of invisibility and is again seen in the sky after sunset, being in advance of the Sun. In order to see whether the Moon will be visible on the first or second lunar day of the light half of the month, one should calculate the (sayana) longitude of the Sun for sunset on that day and also the (sayana) longitude of the Moon corrected for the visibility corrections for the same time. If the oblique ascension of the part of the ecliptic lying between the Sun and the Moon thus obtained is equal to or greater than two ghatis, the Moon will be visible (after sunset that day, otherwise not. Similarly, in order to test whether the Moon will be visible in the night just (before she sets heliacally) on the fourteenth or fifteenth lunar day of the dark half of the lunar month, one should calculate the (sāyana) longitude of the Sun for sunrise following that night and also the (sayana) longitude of the Moon corrected for the visibility corrections for the same time. If the oblique ascension of the part of the ecliptic lying between the Sun and the Moon thus obtained is equal to or greater than two ghatis, the Moon will be seen before sunrise, otherwise not. 1 Prāṇa is the same as asu. • Both sayana. 3 This rule is found to occur also in PSi, v. 3; BrSpSi, vi. 6; x. 32; SiDV, 1, vii. 5; Siśe, ix. 8(i), 13, THE PHASE OF THE MOON A rule for calculating the phase of the Moon: 5(ii)-7. (In the light half of the month) multiply (the diameter of) the Moon's disc by the Rversed-sine of the difference between the longitudes of the Moon and the Sun (when less than a quadrant) and divide (the product) by the number 6876: the result is always taken by the astronomers to be the measure of the illuminated part (of the Moon). When the difference between the Moon and the Sun exceeds a quadrant, then the Moon's illuminated part is calculated from the Rsine of that excess increased by the radius.¹ After full Moon (i.e., in the dark half of the month) the unilluminated part of the Moon is determined from the Rversed- sine or Rsine of (the excess over six or nine signs respectively of) the difference between the longitudes of the Moon and the Sun in the same way as the illuminated part is determined (in the light half of the month). Let the longitude of the Moon minus the longitude of the Sun be denoted by D. Then according to the above rule- (1) In the light half of the month, the illuminated part of the Moon Rversin D X Moon's diameter 6876 if D <3 signs, i.e., if it is the first quarter of the month; and [R+Rsin (D-90°)] x Moon's diameter 6876 if D>3 signs, i.e., if it is the second quarter of the month. 187 if D> 6 signs, i.e., if it is the third quarter of the month; and [R+Rsin (D-270°)] x Moon's diameter (2) In the dark half of the month, the unilluminated part of the Moon Rversin (D-180°) x Moon's diameter 6876 = , 6876 if D> 9 signs, i.e., if it is the last quarter of the month. 1 This rule is found to occur also in BrSpSi, vii. 11(ii)-12 and SiDV, I, ix. 12. 188 RISING, SETTING AND CONJUNCTION OF PLANETS Consider Fig. 22. The sphere centred at M is the Moon's globe, and E is the centre of the Earth. The lines MS' and ES (which are approximately parallel to each other) are directed towards the Sun. Half the globe of the Moon bounded by the circle ABCD and lying towards the Sun is illuminated by the rays of the Sun, and half the globe bounded by the circle LBTD and lying towards E is visible from the Earth. An observer on the Earth will, therefore, see only that part of the Moon's illuminated surface which lies between the semicircles BCD and BTD. In fact, he will see the projection of that on the plane of the circle LBTD. Let BZD be the projection of the semicircle BCD on the plane LBTD. Then the observer will see that part of the Moon's disc illuminated by the Sun which lies between BTD and BZD. This illumi- nated part of the Moon's disc is measured by the length ZT of the Moon's diameter. From the figure, it is evident that Rcos LCMZ X MC R LCMZ = L FMS' = L MES, so that from the plane triangle CZM right-angled at Z, we have MZ- Therefore ZT =MT-MZ F (R-Rcos L. CMZ) x MC R Rversin CMZ x MC R Rversin MES X MC R E Fig. 22 TO SUN TO SUN Hence the rule. A rule for the determination of the Moon's true declination, i.e., the declination of the centre of the Moon's disc: 8. Take the sum of the arcs of the Moon's declination and (celestial) latitude when they are of like directions; in the ELEVATION OF THE LUNAR HORNS 189 contrary case, take their difference. Then take the Rsine of that (sum or difference). (This is the Rsine of the Moon's true declina- tion).¹ From that calculate the nādis of the ascensional diffe- rence of the Moon. The Moon's true declination is used in finding the radius of the Moon's diurnal circle and the Moon's ascensional difference. The pro- cess is the same as that for the Sun. A rule for the determination of the base (bahu) and upright (koti) to be used in the graphical representation of the eleva- tion of the Moon's horns, when the calculation is made in the first quarter of the month for sunset: 9-12. (Calculate the longitudes of the Sun and the visible Moon for sunset on the day of calculation). By the help of the asus intervening between the Sun and the (visible) Moon always find out, in the manner stated before, the Rsine of the Moon's altitude. Then divide the product of the Rsine of Moon's true altitude (thus obtained) and the Rsine of the (local) latitude by the Rsine of the colatitude: thus is obtained the Moon sankvagra, which is always to the south of the Moon's rising-setting line. Then multiply (the Rsine of) the Moon's true declination by the radius and divide (the resulting product) by the Rsine of the colatitude: thus is ained the so called Rsine of the agrā of the (apparent) Moon lying to the north or south (of the ecliptic). Take their sum (i.e., the sum of the Rsines of the Moon's ankvagra and agrā) when they are of like directions, and the difference when they are of unlike direc- tions. Then reversely add or subtract the Rsine of the Sun's agra. Then is obtained the true value of the Moon's base (bahu). Then Rsine of the Moon's altitude is the upright (koti). The asus intervening between the Sun and the visible Moon are the asus of the oblique ascension of that part of the ecliptic which lies between ¹ This rule occurs also in BrSpSi, vii. 5; ŠiDVṛ, I, viii. 2; Sise, x. 7. It is obviously approximate. A better and more accurate rule occurs in Sisi, I, vii. 3 and 13. 190 RISING, SETTING AND CONJUNCTION OF PLANETS the Sun and the visible Moon. These correspond to the time to elapse before moonset. The Moon's agra and sankvagra are defined in the same way as in the case of the Sun.¹ The base (bahu or bhuja) for a planet is the distance of the foot of the perpendicular dropped from the planet on the plane of the horizon from the east-west line. It is equal to śankvagra + Rsin (agrā), + or -sign being taken acccording as the sankvagra and the Rsine of the agră are of like or unlike directions. The true value of the Moon's base (usually called spasta-bahu or spasta-bhuja) denotes the north-south distance between the projections of the Sun and the Moon on the plane of the horizon. It is equal to Moon's base + Sun's base, + or - sign being taken according as the bases of the Sun and the Moon are of unlike or like directions. When the Sun is on the horizon as in the case contemplated in the above stanzas, is is equal to Moon's base + Sun's agrā.² The upright denotes the difference between the Rsines of the alti- tudes of the Moon and the Sun. In the present case, the Sun's altitude is zero, so that the upright is equal to the Rsine of the Moon altitude. The base and the upright defined above will be required in the stanzas below. Method for the graphical representation of the elevation of the lunar horns in the first quarter of the month at sunset: 13-17. To the north or south of the Sun is (to be laid off) the (true) base (according to its direction); and to the easts (of the point thus obtained) is (to be laid off) the upright; the line which joins the ends of the base and the upright is called the hypotenuse. (Taking the centre) at the meeting point of the hypotenuse and the upright, draw the Moon's disc. The ¹ Vide supra, chapter III, stanzas 37 and 54, pp. 84 and 93. 2 The Sun's base in this case is the same as the Sun's agră. 3 The upright is laid off towards the east because in the light half of the month the Moon is towards the east of the Sun. 191 ELEVATION OF THE LUNAR HORNS hypotenuse is the east-west line of that (Moon's disc); through the middle of that (i.e., through the centre of the Moon's disc) draw the north-south line. At the extremities of the north-south line mark two points on the periphery of the Moon's disc. (Then lay off the Moon's illuminated part) along the hypotenuse (from the west point) towards the interior of the Moon's disc and mark there the point of illumination. Thereafter always draw a circle passing through the (above-mentioned) three points. The portion lying between that (circle) and the (periphery of the) Moon's disc (lying towards the Sun) is called the illumi- nated part (of the Moon). The elevation, depression, or horizon- talness of the Moon's horns, in whatever unit be it measured (in the figure), is clearly perceived in the sky (as in the figure).¹ In Fig. 23, AB is the base and MA the upright. Then, according to Bhaskara I, B denotes the Sun's centre and M the Moon's centre. The circle NESW centred at M is the Moon's disc, the points N, E, S, and W being the north, east, south, and west points on its periphery. The measure of the illuminated part of the Moon, IW, is laid off from W to- wards M and an arc of a circle is drawn through N, I, and S.² The shaded portion lying bet- ween this arc and the arc NWS is the illuminated part of the Moon's disc. The cusps at N and S are called the Moon's horns. The figure shows that in the present case the case the northern horn is higher than the southern one, E Fig. 23 Representation of the elevation of the lunar horns at any other time instead of sunset in the first quarter of the month : 18. This (above) method is to be followed at sunset. At (any other) given time, all calculations such as the determination ¹ This rule is approximate. For other rules, see BrSpSi, vii. 7-10; SiDVṛ, I, ix; Sisi. I, ix.

• The point I is called "the point of illumination" (sitabindu). 192

RISING, SETTING AND CONJUNCTION OF PLANETS of the Rsine of the zenith distance of the Moon for that time are prescribed to be made with the setting point of the ecliptic (taken for the Sun). The only difference in this case is that the time to elapse before moonset, instead of being found out from the asus intervening between the Moon and the Sun (as was done in the previous case), should be found out in this case from the asus intervening between the Moon and the setting point of the ecliptic, or, as the commentator Parameśvara says, from the asus intervening between the rising point of the ecliptic and the point six signs in advance of the Moon. The asus correspond to oblique ascension as in the previous case. Representation of the elevation of the lunar horns in the second quarter of the month : 19. (When the calculation relating to the elevation of the Moon's horns is made) after the eighth lunar day, the rising point of the ecliptic itself should be regarded as the Sun. And under that assumption should be made the calculation of the Rsine of the Moon's altitude, etc., with the exception of the calculation of the measure of the illuminated part (of the Moon's disc). A rule for the determination of the Rsine of the Moon's altitude to be used in connection with the elevation of the lunar horns : 20. The Rsine of the Moon's altitude should be calcula- ted from the asus intervening between the Sun and the Moon, or between the rising or setting point of the ecliptic and the Moon subject to the time of calculation, the asus being those obtained by applying the rule once and not successively. A rule telling that the above calculations pertaining to the elevation of the lunar horns relate to the first half of the month only: 21. In this manner, at sunset or any other time, with the help of the longitudes of the Sun, Moon and the Moon's ascending node, should be made this calculation relating to the Moon till the fifteenth lunar day: so has been said, ELEVATION OF THE LUNAR HORNS 193 A rule telling how many nadis before or after sunset will the Moon be seen to rise on the full Moon day: 22. Diminish the nadis (due to the oblique ascension of the part of the ecliptic) intervening between the Sun and the (visible) Moon (at sunset on the full moon day) (from or) by (the nadis of) the length of the day: so many nadis before or after sunset is the Moon seen (to rise on the full moon day).¹ A rule relating to the representation of the elevation of the lunar horns in the dark half of the month: 23-25. After the end of the (light) fortnight, the Moon, lying above the horizon, is drawn by using the measure (of the Rsine of the Moon's altitude) computed from (the asus due to the oblique ascension of the part of the ecliptic intervening between the visible Moon and) the rising point of the ecliptic at the given time. The upright is laid off towards the west³; the base is laid off along the north-south line (in its proper direction); and the hypotenuse-line is stretched out from the end of that (base) to meet the end of the upright. Then from the east point (of the Moon's figure) lay off the measure of the (Moon's) illuminated part along the hypotenuse-line within the figure of the Moon; or, from the west point (of the Moon's figure), lay off the (measure of the Moon's) unilluminated part. A rule telling how to do the same at sunrise in the dark half of the month : 26. Or, perform the operation, stated above, concerning the Moon's illuminated or unilluminated part at sunrise with the asus intervening between the (visible) Moon and the Sun at that time. The time of moonrise will now be told. 1 This rule gives an approximate time of moonrise. In order to obtain the nearest approximation to the correct time of moonrise use should be made of the method of successive approximations. See infra s tanzas 31-35.

• Because in the dark fortnight the Moon is to the west of the Sun, 194

A rule for getting the duration of the Moon's visibility at night in the light half of the month (I quarter): 27. In the light fortnight, find out the asus due to oblique ascension (of the part of the ecliptic) intervening between the Sun and the (visible) Moon (at moonset) both increased by six signs, by the method of successive approximations. These give the duration of visibility of the Moon (at night) (or, in other words, the time of moonset).¹ RISING, SETTING AND CONJUNCTION OF PLANETS The process of successive approximations may be explained as follows: Compute the (sayana) longitudes of the visible Moon and the Sun for sunset and increase both of them by six signs. Then find out the asus (4₁) due to the oblique ascension of the part of the ecliptic lying between the two positions thus obtained. Then A₁ asus denote the first approximation to the duration of the Moon's visibility at night. Then calculate the displacements of the Moon and the Sun for A₁ asus and add them res- pectively to the longitudes of the visible Moon and the Sun for sunset and increase the resulting longitudes by six signs; and then find out the asus (A₂) due to the oblique ascension of the part of the ecliptic lying between the two positions thus obtained. Then A₂ asus denote the second approximation to the duration of the Moon's visibility at night. Repeat the above process successively until the successive approximations to the duration of the Moon's visibility agree to vighatis. The time thus obtained is in terms of civil reckoning. If, however, the use of the Moon's displacement alone be made at every stage, the time obtained will be in terms of sidereal reckoning. A rule for finding the time of moonrise in the dark half of the month (III quarter): 28. Thereafter (i.e., in the dark half of the month), the Moon is seen (to rise) at night (at the time) determined by the asus (due to oblique ascension) derived by the method of succes- sive approximations from the part of the ecliptic intervening bet- ween the Sun as increased by six signs and the (visible) Moon as obtained by computation, (the Sun and the Moon both being those calculated for sunset).* ¹ Cf. Susi, x. 2-4. Cf. Susi, x. 5. TIME OF MOONRISE IN THE DARK FORTNIGHT In the night, the time is measured since sunset. 195 Details of the method of successive approximations contemplated in the above rule: 29-31. Determine the time in asus (due to the oblique ascension of the part of the ecliptic) intervening between the rising point of the ecliptic and the (visible) Moon computed for sunset. (This is the first approximation to the required time). Now calculate the positions of the rising point of the ecliptic and the (visible) Moon for that time; and then deter- mine the asus intervening between those positions again. In case the longitude of the (visible) Moon is greater than that of the rising point of the ecliptic, add these asus to the time obtained above; in the contrary case, subtract them. (This is the second approximation to the required time). Repeat this process successively until the successive approximations to the time, the longitude of the rising point of the ecliptic, and the longitude of the (visible) Moon are. (severally) equal (up to vighatis or minutes). At the time ascertained by this proce- dure for the Moon, the Moon is seen (to rise) in the night filling (the space in) all the directions with her rays. The above rule is based on the fact that at moonrise the longitudes of the visible Moon and the rising point of the ecliptic are the same. An alternative rule for finding the time of moonrise in the dark half of the month (III quarter): 32-33. Find out the asus due to the oblique ascension of the part of the ecliptic lying from the setting Sun up to the (visible) Moon; and therefrom subtract the length of the day. (This approximately gives the time of moonrise as measured since sunset). Since the Moon is seen (to rise) at night when so much time, corrected by method of successive approximations, is elapsed, therefore the asus obtained above should be operat- ed upon by the method of successive approximations. " 196 RISING, SETTING AND CONJUNCTION OF PLANETS Details of the method of successive approximations contem- plated in the above rule: 34. Find out the displacements of the Sun and the Moon for the ghatis (corresponding to the approximate time) obtained above and add them to the longitudes of the Sun and the (visible) Moon respectively; then determine the ghatis (due to the oblique ascension of the part of the ecliptic) intervening between them; and then from those (ghatis) subtract the length of the day. (Thus is obtained the second approximation to the required time). Then find out the the displacements of the Sun and the Moon corresponding to (the ghatis of) the remain- der (and proceed as above again and again until the successive approximations agree to vighatis). An analogous rule for finding the time of moonrise in the light half of the month (II quarter): 35-36. (In the light half of the month) when the measure of the day exceeds the nadis (due to the oblique ascension of the part of the ecliptic) lying between the Sun and the (visible) Moon (computed for sunset), the moonrise is said to occur in the day when the residue of the day (i.e., time to elapse before sunset) is equal to the ghaiis of their difference. (In this case) the longitudes of the Sun and the (visible) Moon should be diminished by their displacements determined by proportion from the nadis (of the residue); and then should be obtained the asus (due to the oblique ascension of the part of the eclip- tic) between the Sun and the (visible) Moon (thus obtained). These asus should then be operated upon by the method of successive approximations. Another rule for getting the time of moonrise in the dark half of the month (IV quarter): 37-38. Determine the asus. (due to the oblique ascension of the part of the ecliptic lying) from the (visible) Moon at sunrise up to the rising Sun; then subtract the corresponding displacements (of the Moon and the Sun) from them (i.e., from 197 the longitudes of the visible Moon and the Sun computed for sunrise); and on them apply the method of successive approxi- mations (to obtain the nearest approximation to the time between the visible Moon and the Sun computed for moonrise, i.e., between the risings of the Moon and the Sun). The Moon, who is like a looking glass for the face of the directions, rises as many asus before sunrise as correspond to the nādis obtained by the method of successive approximations. MERIDIAN PASSAGE OF THE MOON A rule for the determination of the time of the meridian passage of the Moon, and the longitudes of the Moon and the meridian- ecliptic point at that time : 39. Infer by your intellect the time when the meridian- ecliptic point and the Moon are together. Then, by the method of successive approximations, find out the nearest approxi- mations for that time, the longitude of the Moon, and the longitude of the meridian-ecliptic point (for that time). Assuming that the rising point of the ecliptic is three signs in advance of the meridian-ecliptic point, the time to elapse before or elapsed since the Moon is on the meridian is the same as the time to elapse before or elapsed since the point of the ecliptic three signs in advance of the Moon is on the horizon. Therefore in order to get an approxi- mate time when the Moon occupies the meridian-ecliptic point one may proceed as follows: First calculate the longitudes of the Sun, the Moon, and the rising point of the ecliptic with the help of the given time. Then increase the longitude of the Moon by three signs and find the time due to the oblique ascension of the part of the ecliptic lying between the rising point and the Moon as increased by three signs. The time thus obtained is the approximate time to elapse before or elapsed since the meridian passage of the Moon. From this calculate the time of the meridian passage of the Moon. Details of the method of successive approximations contemplat- ed in the above rule: 40. Determine the nadis intervening between the Moon and the meridian-ecliptic point (for the time determined by inference) with the help of the times of rising of the signs at Lanka. When (the longitude of the Moon is) less (than the 198 RISING, SETTING AND CONJUNCTION OF PLANETS longitude of the meridian-ecliptic point), subtract the resulting nadis from those corresponding to the inferred time; when greater, addition is prescribed. (For the time thus obtained calculate the longitudes of the Moon and the meridan-ecliptic point and find the nadis intervening between them with the help of the right ascensions of the signs as before; and then repeat the above process successively until the nearest approxi- mation to the time of meridian passage of the Moon is obtained). A rule for the determination of the Rsine of the Moon's meri- dian zenith distance: 41. By this process is obtained the Moon when she is on the meridian (lit. when her longitude is equal to that of the meridian-ecliptic point). From her celestial latitude and decli- nation, and from the (local) latitude is determined the Rsine of her meridian zenith distance. First obtain the Moon's true declination by rule 8 above; then apply the following formula: Moon's meridian zenith distance local latitude true declination, + or-sign being taken according as the Moon is to the south or to the north of the equator. A rule regarding the elevation of the horns of the half-risen or half-set Moon: 42. The determination of the elevation of the horns of the half-risen or half-set Moon is made with the help of the agra of the rising or setting point of the Moon's orbit.

When the Moon is rising or setting, its base is obviously equal 10 the agra of the rising or setting point of the Moon's orbit. The text does not say anything about the base and depth of the Sun lying below the horizon but these elements have to be calculated and made use of in the above-mentioned determination. Procedure to be adopted in the case of the planets : 43. This (above-mentioned) procedure should be adopted in the case of the nector-rayed Moon; the same process is pres- cribed for all the planets also. LIMITS OF VISIBILITY OF THE PLANETS The remainder of this chapter deals exclusively with the planets. Minimum distances of the planets from the Sun when they are visible: 199 44. Venus is visible when it is 9 degrees away from the Sun; Jupiter, Mercury, Saturn, and Mars are observed when they are respectively further away by two degrees in succession (i.e., when they are respectively 11°, 13°, 15°, and 17° away from the Sun). 45. Ve. as, which moves in its proper orbit but. appears retrograde, is visible, due to profusion of its rays, when it is (only) 4 or 4 degrees away from the Sun. The degrees above are "the degrees of time" (called kalabhāga). One degree of time is equivalent to 60 asus. Thus Venus is visible in the east if the time taken by the part of the ecliptic between the Sun and Venus to rise above the eastern horizon amounts at least to 9x 60 asus; and in the west, if the time taken by the part of the ecliptic between the Sun and Venus to set below the western horizon, or the time taken by the diametrically opposite part of the ecliptic to rise above the eastern hori- zon, amounts at least to 9x60 asus. It may be pointed out that a sign sets in the same time that the seventh, i.e., the diametrically opposite sign, takes to rise. According to Lalla, Mercury and Venus, when in regression, are visible when they are respectively 12 and 8 degrees distant from the Sun. According to Sripati, Mercury and Venus, when near their apogees, are visible when they are respectively 14 and 10 degrees distant from the Sun; and when in regression, they are visible when respectively 12 and 8 degrees distant from the Sun.5 1 The same degrees of visibility are prescribed also in BrSpSi, vi. 6; x. 32 ; SIDVE, I, vii. 5 (i); and Sise, ix. 8 (i), 12. 2 When Mercury or Venus is in retrograde motion, it is nearer to the Earth and so its size is a little enlarged. 3 Venus in both the cases being that corrected for the visibility corrections. 4 SiDV, I, vii. 5 (ii). 5 Sise, ix. 9. The Greek astronomer Ptolemy (150 A. D.) also considered the heliacal rising and setting of the planets and defined the limits of visibility of each planet when in the sign Cancer (i.e., when the equator and the ecliptic are nearly parallel). His limits are; for Saturn, 14°; for Jupiter, 12°45'; for Mars, 14°30'; and for Venus and Mercury, in the west, 5°40' and 11°30' respectively. Vide Syntaxis, xiii. 7-9. RISING, SETTING AND CONJUNCTION OF PLANETS A rule telling us (1) how to convert the degrees of time into vighatis, and (2) how to determine the degrees of time between the Sun and a planet: 200 46-47. These degrees of time when multiplied by ten are called vighatis. (When the planet is seen) in the east, they are determined from (the oblique ascension of) the sign occu- pied by the Sun and the planet; (when the planet is visible) in the west, they are determined from (the oblique ascension of) the seventh sign (as measured from the sign occupied by the Sun and the planet).¹ (The process is as follows): Divide the oblique ascension of the sign occupied by the Sun and the planet (or of the seventh sign, as the case may be) as multiplied by the degrees of the difference between the longitudes of the planet and the Sun by 30. If the resulting time is equal to (or greater than) that stated (for that planet), the planet will be seen to rise (heliacally). The longitude of the planet is that corrected for the visibility- corrections. CONJUNCTION OF PLANETS Definition of the "divisor" to be used later : 48. The sighra-karna as multiplied by the mandocca- karna (or manda-karna) should be divided by the radius: the result thus obtained is called the "divisor".2 This "divisor" denotes the distance between the Earth and a planet (bhu-taragraha-vivara). The above rule for the divisor was pro- bably derived by proportion as follows: "If the radius of the sighra concentric denotes the manda-karna, what will the sighra-karṇa stand for? The result is the geocentric distance of the planet, the so called "divisor". Cf. SIDV I, vii. 5 (iv).

• Cf. SidV, I, x. 1. THE CELESTIAL LATITUDE OF A PLANET

A rule relating to the determination of the time and the com- mon longitude of two planets when they are in conjunction in longitude: 201 49-51. If one planet is retrograde and the other direct, divide the difference of their longitudes by the sum of their daily motions; otherwise (i.e., if both of them are either retrograde or direct), divide that by the difference of their daily motions: thus is obtained the time in terms of days, etc., after or before which the two planets are in conjunction (in longitude). The velocity of the planets being different (literally, manifold) (from time to time), the time thus obtained is gross (i.e., approximate). One, proficient in astronomical science, should, therefore, apply some method to make the longitudes of the two planets agree to minutes. Such a method is possible from the teachings of the precepter or by day to day practice (of the astronomical science).¹ The method to be used here is obviously the method of successive approximations. A rule relating to the computation of the celestial latitude of a planet when it is in conjunction with another planet: 52-53. Diminish the longitude of the planet in conjunc- tion with another planet by the degrees of (the longitude of) the ascending node (of that planet); by the Rsine of that multiply the greatest latitude (of the planet) and divide (the product) always by the (corresponding) "divisor" (defined in stanza 48): thus is obtained the latitude of Jupiter, Mars, or Saturn. In order to find the latitudes, north or south, of the remaining planets (Mercury and Venus), subtraction (of the degrees of the longitude of the ascending node) should be made from the longitude of the planet's sighrocca. The longitude of the planet to be used in the above rule is really the heliocentric longitude and not the geocentric longitude. Brahmagupta ¹ Cf. Br SpSi, ix. 5-6 ; ŚiDVṛ, I, x. 7-9 (i) ; Siśe, xi. 12-13.

• This rule occurs also in SiDV, 1, x. 10, 9 (ii). 202

RISING, SETTING AND CONJUNCTION OF PLANETS (628 A. D.) has therefore prescribed the use of the true-mean longitude of the planet in the case of Mars, Jupiter and Saturn, and that of the longitude of the planet's sighrocca as corrected for the planet's mandaphala in the case of Mercury and Venus.¹ A rule relating to the distance between two planets which are in conjunction in longitude: 54-55. When the latitudes of the two planets (in conjunction) are of unlike directions, their sum is the (angular) distance between them. When their latitudes are of like direc- tions, the minutes of the distance between them are obtained by taking their difference.² The (linear) distance between the two planets (in conjunc- tion) should be announced by those proficient in the processes of planetary conjunction by taking a minute as equivalent to one-fourth or one-half of an angula, whichever agrees with the phenomenon observed in the sky. According to the commentator Parameśvara, one minute of the distance between the two planets is equal to one-half or one-fourth of an angula, according as the two planets are or are not near the horizon. Brahmagupta has criticised the longitudinal conjunction of the planets. He favours horizontal conjunction which occurs when the two planets are on the same secondary to the prime vertical, because it can be easily observed. Diameters of the planets in minutes of arc: 56. Having (first) divided 32 by 5, divide the same number (i.e., 32) again and again by the same (5) as increased by itself in succession (i.e., by 10, 15, 20, and 25): the results thus obtained are known as the minutes of the diameters of Venus, Jupiter, Mercury, Saturn, and Mars respectively. 1 See Br Sp.Si, ix. 9. Also see SuSi, ii. 56-57;, Sise, xi. 15; and Sisi, II, vi. 20-25(i).

• This rule occurs also in SiDVṛ, I. x. 11; KPr, vii. 8; and has been

quoted by Brahmagupta in BrSpSi, ix 11, and by Sripati in Sise, xi. 18. 3 In BrSpSi, ix. 11-12. Also see Sise, ix. 19-20, THE SO CALLED "DIVIDING NUMBERS" FOR THE PLANETS 203 According to Aryabhata I,' the diameter of the Moon is 315 yojanas, and the diameters of Venus, Jupiter, Mercury, Saturn, and Mars are respec- tively one-fifth, one-tenth, one-fifteenth, one-twentieth, and one-twenty- fifth of the diameter of the Moon (at the Moon's mean distance). It follows that the diameter of a planet in minutes diameter of the planet in yojanas x R Moon's mean distance in yojanas diameter of the planet in yojanas 10 diameter of the Moon in yojanas 10 X D where D = 5, 10, 15, 20, or 25, according as the planet is Venus, Jupiter, Mercury, Saturn, or Mars. 32 D ¹ A, i. 7. approx. approx. approx., Hence the rule in the text. Definition of the "dividing numbers" for the planets : 57 (Severally) multiply the yojanas of the Moon's (mean) distance by the same numbers (i.e., by 5, 10, 15, 20 and 25): the result, in each case, is the "dividing number", in terms of yojanas, used in the determination of the lambana and nati (for the respective planets). The "dividing numbers" denote the mean distances of the planets under the assumption that the diameter of each of them is 315 yojanas (the same as that of the Moon). A rule for finding the true distance of a planet, assuming 315 yojanas for its diameter, in terms of yojanas : 58 (i). These (above-mentioned) "dividing numbers" be- come accurate when multiplied by the "divisor" and divided by the radius. 204 RISING, SETTING AND CONJUNCTION OF PLANETS The obvious proportion is: When the radius (i.e., 3438') corresponds to the planet's mean distance in yojanas, what will the divisor (i.e., the true distance of the planet in minutes) correspond to ? The result is the true distance of the planet in terms of yojanas. A rule relating to the determination of the true values of the diameters of the planets in minutes : 58 (ii). So also become the minutes of the diameters when divided by the "divisor" and multiplied by the radius.¹ The remaining processes concerning the occultation of one planet by another: 59-60. The determination of the ten Rsines (viz. madhyajyā, drkksepajya, drggatijyā, drgjyā and udayajya-five for each of the two planets concerned) and other remaining determinations should be made as in the case of the Moon. The Rsine of the altitude (for each planet) is to be calculated from the (planet's) own ascensional difference, etc., as taught in connection with the Moon's rising. In the case of conjunction of the planets (i.e., occul- tation of one planet by another), the lambana and nati are prescribed to be found out (by proportion) from the (planets') own "dividing numbers". The remaining processes such as the calculation of the sthityardha (i.e., the semi-duration of occultation) are the same as in the case of a (solar) eclipse. Concluding stanzas : 61-62. The predictions of those (astronomers) whose minds are purified by day to day practice and who have acquir- ed by the grace of the teacher the eye of true conception of the astronomical science, (always) agree with the planetary phenomena and do not go astray as the pure thoughts (desires) of a lovely and devoted wife (do not go astray). 1 The same rule occurs in SiDVE, I, x. 4. CHAPTER VII ASTRONOMICAL CONSTANTS Revolutions performed by the planets around the Earth in a period of 43,20,000 solar years (called a yuga): 1-5. The revolution-number (bhagana) of the Sun is 43,20,000; of the Moon, 5,77,53,336; of Saturn, 1,46,564; of Jupiter, 3,64,224; of Mars, 22,96,824 ; of Mercury and Venus and of the sighroccas of the other planets, the same as that of the Sun; of the Moon's apogee, 4,88,219; of (the sighrocca of) Mercury, 1,79,37,020; of (the sighrocca of) Venus, 70,22,388; and of the Moon's ascending node, 2,32,226. Intercalary months, omitted lunar days, and civil days in a yuga: 6-8. The number of intercalary months (in a yuga) is 15,93,336. For the determination of the number of intercalary months elapsed, (this is the multiplier): the diviser is twelve times the number of solar years in a yuga (i.e., 5,18,40,000). The number of omitted lunar days (in a yuga) is 2,50,82,580. (For finding the number of omitted lunar days elapsed, this is the multiplier): the divisor is 1,60,30,00,080 (which is the number of lunar days in a yuga). The number of civil days in a yuga is stated to be 1,57,79,17,500. Intercalary months in a yuga = (lunar months in a yuga) · (solar months in a yuga) = {(revolution-number of the Moon) (revolution-num- ber of the Sun)}- 12×(revolution-number of the Sun) = (revolution-number of the Moon)- 13x (revolution- number of the Sun) = 15,93,336. Omitted lunar days in a yuga (lunar days in a yuga) - (civil days in a yuga) = 2,50,82,580. 206 ASTRONOMICAL CONSTANTS Inclinations of the orbits of the planets to the ecliptic : 9. The degrees of the greatest celestial latitudes of Mercury, Venus, and Saturn are each 2; of Jupiter, 1 ; and of Mars, 13.¹ Longitudes of the ascending nodes of the planets and a rule for finding the celestial latitude of a planet: 10. The degrees of the longitudes of the ascending nodes (of the same planets) àre 20, 60, 100, 80, and 40 respectively. The celestial latitude, north or south, (of a planet) should be given out after calculation from the longitude of the planet minus the longitude of its ascending node. Longitudes of the apogees of the planets and the method for finding the manda and sighra anomalies: 11-12. The degrees of the longitudes of the apogees (of the same planets) are respectively 210, 90, 236, 180, and 118. Of the Sun they are to be known as 78.³ A (In order to get the manda anomaly) subtract the longi- tude of the apogee (of the planet) from the longitude of the planet; and (in order to obtain the sighra anomaly) always subtract the longitude of the planet from the longitude of the sighrocca (of the planet). Manda and sighra epicycles of the planets, and Rsine-differences corresponding to the twenty-four elements of a quadrant; 13-16. In (the beginnings of) the odd quadrants the manda and sighra epicycles (of Mercury, Venus, Saturn, Jupiter, and Mars) are 7, 4, 9, 7, 14 and 31, 59, 9, 16, 53 (respectively); and in (the beginnings of) the even quadrants they are stated to be 5, 2, 13, 8, 18 and 29, 57, 8, 15, 51 (respectively). Of the Sun and the Moon, the epicycles are 3 and 7 (respectively). 1 The same values are found in A, i. 8 and ŚiDV, I, x. 5(i). 2 The same longitudes are given in A, i. 9(i) and SiDV, I, x. 5(ii). 3 These longitudes are the same as found in A, i. 9(ii) and SiDV, I, ii. 28(i), 9(iv). AN APPROXIMATE FORMULA FOR RSIN 207 The Rsine-differences (corresponding to the twenty-four elements of a quadrant) are 225 (makhi), etc. (as stated by Āryabhata I).² The dimensions of the epicycles have been stated after dividing them by 4. They are the same as given by Aryabhata I. In fact all the astro- nomical constants stated in the above stanzas are the same as found in the Āryabhatiya. It must be pointed out that, according to Bhāskara I, the manda and sighra epicycles stated above from the Aryabhațiya of Aryabhaṭa I, correspond to the beginnings of the respective quadrants as is evident from the rule stated in MBh, iv. 38-39(i) and LBh, ii. 31-32 and explained in Bhaskara I's commentary on A, iii. 22. Govinda Svāmi, however, is of opinion that the sighra epicycles stated by Aryabhata I correspond to the beginnings of the respective quadrants, but the manda epicycles stated by Aryabhata I, correspond to the endings of the respective quadrants.³ Consequently, he has replaced the rule referred to above by another rule which has been quoted by Udaya-divākara in his commentary on LBh, ii. 31-32. This controversy is due to the fact that Aryabhata I himself does not say whether the epicycles given by him correspond to the begin- nings or endings of the quadrants. A rule for finding the bahuphala and kotiphala, etc., without the use of the Rsine-difference table: 17-19. (Now) I briefly state the rule (for finding the bhujaphala and kotiphala etc.) without making use of the Rsine-differences, 225, etc. Subtract the degrees of the bhuja (or koti) from the deg- rees of half a circle (i.e., 180°). Then multiply the remainder by the degrees of the bhuja (or koti) and put down the result at two places. At one place subtract the result from 40500. By one-fourth of the remainder (thus obtained) divide the result 2 For the table of Rsine-differences referred to here, see supra, p. 108. 8 See Govinda Svāmi's comm, on MBh, iv. 38-39(i). 208 at the other place as multiplied by the antyaphala (i.e., the epicyclic radius)¹. Thus is obtained the entire bahuphala (or kotiphala) for the Sun, Moon, or the star-planets. So also are obtained the direct and inverse Rsines.² This rule is based on the formula: sin 0 Therefore, ASTRONOMICAL CONSTANTS so that where is in degrees. The following is a rationale of this approximate formula : In Fig. 24, let CA be the diameter of a circle of radius R, where arc AB is equal to 0 degrees and BD-Rsin 0. Then Area ABC = Also area ABC = Let 1 BD = 4(1800)0/{40500-(180-0)0}, AB. BC. AC. BD. AC AB. BC Putting 30°, 1 AC BD (arc AB) × (arc BC) 1 BD so that Rsin (= R= i.e., 2xR +4500 y x. AC (arc AB)x(arc BC) +y, 0 (180--0) 2xR+0 (180-0) y = 2 xR 0 (180-0) 30 x 150 2xR+30 x 150 y 9000 R Fig. 24 + y 1 इष्टोच्चनीचव्यासार्धं परमफलजीवा | BrSpSi, xiv. 24 (ii).

• Cf. Br.Sp.Si, xiv. 23-24; Sise, iii. 17.

(1) 0 (2) A Putting 0-90°, From (2) and (3), AN APPROXIMATE FORMULA FOR RSIN Therefore, from (1) 2xR+8100y=" or or giving y and 2xR= Therefore, Rsin=. The same result may also be derived as follows: Let sinλ= where is in radians and corresponds to 0 degrees. Putting λ=0, a=0. 8100 R Putting λ=t, b+c=0. Therefore, c-- Thus sinλ 40(180-0)R 40500-0(180-0) Since sinλ=sin(->), therefore bλ(π-^)/π bλ(π-2)/T A+BA+CA³ A+B(T-A)+C(π-λ)³ Putting in this λ=tn, 1 4R 40500 4R H sin λ= a+bx+cλ² A+BX+Cx²² C=- A+BX+CX=A+B(x-2)+C(n-1)², B(21-T)=CT(T-21), ba(n-λ)/T A+BX+Cx² - B bλ (π-λ) AT+BN(T-λ) AT + Bn(n-1)=2b. ln(n-1), 5x²B_10x¹b 36 36 or AT + = 209 (3) 210 Also putting λ=1a, From (4) and (5), Therefore, ASTRONOMICAL CONSTANTS AT +B.(-x)=b.ln(n-1), or An +³B-1³b. . B = -tb, Sub 16 and A = sin λ = 16x (T-λ) 5² 42(T-2)² where λ =

1. 0/180.

The length of the so called circle of the sky and the rule for deriving the length of the orbit of a planet: 20. Multiply the revolution-number of the Moon by 216000 then is obtained the length of the circle of the sky (in terms of yojanas). When the circle of the sky is divided by the revolution-number of any given planet, the quotient denotes the length of the circular orbit of that planet. Thus we get (1) length of the circle of the sky = 1,24,74,72,05,76,000 yojanas, (2) length of the Sun's orbit = 28,87,666 yojanas, (3) length of the Moon's orbit (4) length of Mars' orbit -4 = = 2,16,000 yojanas, 132027 54,31,291287103 373277 6,95,473896851 3,42,50,133, (5) length of Mercury's orbit (6) length of Jupiter's orbit (7) length of Venus's orbit (8) length of Saturn's orbit Midnight day-reckoning of Aryabhaṭa I: 21. The astronomical processes which have been set forth above come under the sunrise day-reckoning. In the midnight 699 1897 255221 585199 17,76,421; yojanas, yojanas, yojanas, = 8,51,14,493; (5) yojanas, 5987 36641 yojanas. 211 MIDNIGHT DAY-RECKONING day-reckoning too, all this is found to occur: the difference that exists is being stated (below). The next fourteen stanzas relate to the midnight day-reckoning of Aryabhata I. Civil days and omitted lunar days in a yuga and revolution- numbers of Mercury and Jupiter : 22. (To get the corresponding elements of the midnight day-reckoning) add 300 to the number of civil days (in a yuga) and subtract the same (number) from the number of omitted lunar days (in a yuga); and from the revolution-numbers of (the sighrocca of) Mercury and Jupiter subtract 20 and 4 respectively. Thus according to the midnight day-reckoning, civil days in a yuga= 1,57,79,17,800, omitted lunar days in a yuga- 2,50,82,280, revolution-number of the sighrocca of Mercury 1,79,37,000, revolution-number of Jupiter= 3,64,220.. Diameters of the Earth, the Sun, and the Moon: 23. (In the midnight day-reckoning) the diameter of the Earth is (stated to be) 1600 yojanas; of the Sun, 6480 (yojanas); and of the Moon, 480 (yojanas). Mean distances of the Sun and the Moon : 24. The (mean) distance of the Sun is stated to be 689358 (yojanas); and of the Moon, 51566 (yojanas). Longitudes of the apogees of the planets : 25. 160, 80, 240, 110, and 220 are in degrees the longi- tudes of the apogees of Jupiter, Venus, Saturn, Mars, and Mercury respectively ¹. ¹ Cf. KK (Sengupta's edition), ii. 6(i). 212 ASTRONOMICAL CONSTANTS Manda and sighra epicycles of the planets : 26-28(i). The manda epicycles (of the same planets) are 32, 14, 60, 70, and 28 (degrees) respectively; and the sighra epicycles are 72, 260, 40, 234, and 132 (degrees) respectively. The Sun's apogee and epicycle are the same as those of Venus (i.e., 80° and 14° respectively). The Moon's epicycle in the midnight day-reckoning is stated to be 31 (degrees). Positions of the so called manda and sighra pātas of the planets : 28(ii)-31(i). (The following directions for) the degrees of the (manda and sighra) patas of the planets as devised (under the midnight day-reckoning) should be noted carefully by learned scholars. Add 180° to the longitudes of the mandoccas (apogees) and sighroccas of Mercury and Venus, and subtract 3 signs from the mandoccas (apogees) and sighroccas of the remaining planets. Then are obtained the longitudes of the manda and sighra patas of the planets. (Also) add 2 degrees to the longitudes of the manda pātas and sighroccas of Venus, Satura, and Jupiter; and 1 degrees to those of Mars and Mercury. (It should be noted that) the sighra patas have been stated for all the planets excepting Mercury. (Mercury does not have a sighra pāta).¹ That is to say, the longitudes of the manda-patas of Mars, Mercury, Jupiter, Venus, and Saturn are 21°5, 41°-5, 12°, 262°, and 152° respectively; and the longitudes of the sighra-patas of Mars, Jupiter, Venus, and Saturn are (sighrocca - 88°-5), (sighrocca - 88°), (śighrocca + 182°) and (sighrocca - 88°) respectively. 1 This passage represents rather old ideas of Hindu astronomy. The conception of manda and sighra pātas does not occur in any other work. Our translation agrees with the interpretations given in the various commentaries. The translation given by P. C. Sengupta in the intro- duction to his Khaṇḍa-khādyaka is wrong. 213 MIDNIGHT DAY-RECKONING A rule for finding the celestial latitude of a planet : 31 (ii)-33. (From the longitude of a planet severally) subtract the longitudes of its (manda and sighra) pātas, and therefrom calculate (as usual) the corresponding celestial latitudes of that planet. Add them or take their difference according as they are of like or unlike directions. Then is obtained the true celestial latitude of that particular planet. The true celestial latitude of any other planet is also obtained in the same way. The remaining (astronomical) determinations are the same as stated before. This all is in brief the difference of the other tantra (embodying the midnight day-reckoning of Aryabhața I). A rule for finding the longitude of the true-mean planet according to the midnight day-reckoning: 34. Apply half the sighraphala and (then) half the mandaphala to the longitude of the planet's own mandocca (reversely). From the resulting longitude of the planet's mandocca calculate (the mandaphala and apply it to the the mean longitude of the planet: the resulting longitude of planet is stated to be) the true-mean longitude of the planet. This is stated to be another difference (of the midnight day-reckoning).¹ Length of the circle of the sky and derivation of the lengths of the orbits of the planets : 35. Multiply the revolutions of the Moon (in a yuga) by 32,40,000 and then discard the zero in the unit's place: (this is the length of the circle of the sky in terms of yojanas). (Severally) divide that by the revolutions of the planets (in a yuga) thus are obtained the lengths of the orbits of the respective planets in terms of yojanas. From stanzas 20 and 35 it is evident that one yojana of the sunrise day-reckoning is one and a half times that of the midnight day-reckoning. 1 This rule is the same as found in PSi, xvii. 4-9; SuSi, ii, 44; MSi, iii. 28; and SiTV, ii. 247. CHAPTER VIII EXAMPLES (Solved Examples) To find the true lunar day (tithi) and the ghatis elapsed at sunrise since the beginning of the current lunar day without the knowledge of the true longitudes of the Sun and Moon: 1-4. Multiply the ahargana by the number of lunar years¹ (in a yuga) and divide by the number of civil days² (in a yuga). Then are obtained the mean lunar years, etc. (corresponding to the ahargana). Also multiply the ahargana by the number of intercalary lunar years³ (in a yuga) and divide by the number of civil days (in a yuga). (Thus are obtained the mean interca- lary years, etc., corresponding to the ahargana). The difference between the two denotes the (mean) solar years, etc. (i.e., years, months, days, and ghatis) (corresponding to the ahargana). The solar years are not required (so they are to be omitted). From the remaining quantity (in months, days, and ghatis) subtract two months and eighteen days. Then (treating the months, days, etc., of the remainder thus obtained as the signs, degrees, etc., of the Sun's mean anomaly) calculate the (Sun's) equation of the centre. ¹ i.e., 44,52,778. 3 i.e., 1,57,79,17,500. s i.e., 1,32,778. Divide the (corresponding) solar time by 12 and apply that to the (mean) lunar days (and ghatis) (obtained from the ahargana) contrarily (i.e., add when the Sun's equation of the centre is subtractive and subtract when the Sun's equation of the centre is additive). Also apply one-twelfth of the time corres215 SOLVED EXAMPLES ponding to the Moon's equation of the centre¹ to the resulting lunar days (and ghatis) in the same way as it is applied to the Moon's longitude. (The lunar days thus obtained are the true lunar days which have elapsed at sunrise since the beginning of the current month). The ghatis obtained above denote the elapsed portion of the current lunar day in terms of ghatis. Multiply those ghatis by 60 and divide by one-twelfth of the difference between the true daily motions of the Sun and Moon, in degrees : the quotient denotes the true time in ghatis (which has elapsed at sunrise since the beginning of the current lunar day).³ Let the mean lunar years, etc., elapsed at sunrise be a years, b months, c days, and d ghatis; and let the mean intercalary years, etc., elapsed at sunrise be a' years, b' months, c' days, and d' ghatis. Then the mean solar years, etc., elapsed at sunrise are (a-a') years, (b-b') months, (c-c') days, and (d-d') ghațis. The mean longitude of the Sun is therefore equal to (a-a') revolu- tions, (b-b') signs, (c-c') degrees, and (d-d') minutes. The longitude of the Sun's apogee being 2°18', the Sun's mean anomaly is equal to (b-b') signs, (c-c'-2) degrees, and (d-d'-18) minutes. Suppose that the Sun's equation of the centre derived from the above Sun's mean anomaly is m minutes. Then we subtract m/12 ghatis from or add the same amount to c days and d ghatis obtained above, according as the Sun's equation of the centre is additive or subtractive. Thus we get c days and (dm/12) ghatis. 1 To obtain the Moon's equation of the centre, the mean longitude of the Moon may be obtained as follows: Multiply the mean lunar days, etc., (corresponding to the ahargana) by 12, convert the resulting days etc. into months, etc., and add to them the mean solar months, etc., (corres- ponding to the ahargana); treat the months, etc., thus obtained as the signs, etc., of the mean longitude of the Moon. 2 A similar rule occurs in Br.Sp.Si, xiii. 23-25 and ȘïŠe, iii, 72-74. 216 Also suppose that the Moon's equation of the centre is n minutes. Then we add n/12 ghatis to or subtract the same from c days and (dFm/12) ghatis. Thus we obtain c days, (dm/12+n/12) ghatis. c days in this result shows that c complete frue lunar days have elapsed at sunrise; and (dm/12+n/12) ghatis shows that part of the current lunar day amounting to so many ghatis has also elapsed at sunrise. Multiplying (dm/12+n/12) by 60 and dividing the product by 1/12 of the degrees of difference between the true daily motions of the Sun and Moon are obtained the ghatis elapsed at sunrise since the beginning of the current lunar day. The following is the rationale of the above rule: True lunar day (tithi) (true longitude of the Moon-true longitude of the Sun)/12. {(mean longitude of the Moon Moon's equation of the centre)(mean longitude of the Sun Sun's equation of the centre)}/12. - EXAMPLES = - = {(mean longitude of the Moon- mean longitude of the Sun) F (Sun's equation of the centre) +(Moon's equation of the centre)}/12. . aharganax (Moon's rev.-number-Sun's rev.-number)¹ civil days in a yugax 12 (Sun's equation of the centre)/12 (Moon's equation of the centre)/12 ahargana x (lunar years in a yuga) civil days in a yuga (Sun's equation of the centre)/12 ± (Moon's equation of the centre)/12 mean lunar years, etc., elapsed at sunrise (Sun's equation of the centre)/12 (Moon's equation of the centre)/12. ¹ Rev.-number denotes revolution-number, = SOLVED EXAMPLES mean lunar days and ghatis elapsed at sunrise (Sun's equation of the centre)/12 (Moon's equation of the centre)/12, 217 where the signs are chosen appropriately. Lunar months and years are discarded as they are not required. To obtain the Sun's mean longitude from the Sun's true longitude derived from the midday shadow of the gnomon: 5. Subtract the longitude of the Sun's apogee from the Sun's true longitude derived from the midday shadow (of the gnomon) and (then treating the remainder as the Sun's mean anomaly) calculate the Sun's equation of the centre. Apply that (equation of the centre) to the Sun's true longitude con- trarily to the usual law for its subtraction and addition. (Treating this result as the mean longitude of the Sun, calculate the Sun's equation of the centre afresh and apply that to the Sun's true longitude as before.) Repeat the same process again and again (until two successive results agree to minutes). Thus is obtained the mean longitude of the Sun.¹ The method used here is evidently the method of successive approximations. To find the arc corresponding to a given Rsine : 6. From the given Rsine subtract in serial order (as many tabulated Rsine-differences as possible): multiply the number of the Rsine-differences subtracted by 225. Then multiply the residue (of the given Rsine) by 225 and divide by the current Rsine-difference. Add this result to the pre- vious one. Thus is obtained the arc (corresponding to the given Rsine in terms of minutes).³ 1 This rule occurs also in. BrSp.Si, xiv. 28; iii. 61-62; Siśi, I, ii. 45. 2 i.e., the tabulated Rsine-difference which is next to those sub- tracted. 8 This rule is found also in SūSi, ii. 33; BrSpSi, ii. 11; ŚIDVţ, I, ii. 13; SiSe, iii. 16; ȘiŚi, I, ii. 11(ii)-12(i). 218 EXAMPLES This rule is the converse of that stated in chapter IV, stanzas 3-4(i). EXAMPLES Six examples on the shadow of the gnomon : 7. The latitude (of a place) is one and a half degrees minus eight minutes (i.e., 1° 22'); and the midday shadow of the gnomon of 12 añgulas on level ground is 5 angulas. Give out the sun's longitude at noon on that day.¹ 8. Quickly say the longitude of the meridian Sun for the place where the latitude is stated to be 8 degrees minus 16 minutes (i. e., 7° 44') and the midday shadow of the gnomon, 3 and a half (angulas). 9. Quickly say the true longitudes of the Sun for the places where the latitudes are stated to be 25 and 30 degrees respectively and the lengths of the midday shadows (of the gnomon) are equal to the gnomon (itself). 10. Say what is the longitude of the Sun at the place where the latitude is 15 degrees and the prime vertical shadow of (the gnomon due to) the Sun, one and a half angulas to- gether with one-fifth of an angula (i.e., 17/10 angulas).² 11. The prime vertical shadow (of the gnomon) is 37 angulas and the equinoctial midday shadow is 30 angulas. Say the longitude of the universal lamp, the Sun, for its position on the prime vertical. 12. The east-west shadow (of the gnomon) is seen on level ground to be 16 (angulas). The latitude of the place is seven and a half degrees. Say what is the Sun's longitude therë. ¹ The examples set in this and the next two stanzas are based on the rules stated before in chapter III, stanzas 5 and 13-15. The examples set in this and the next two stanzas are based on the rules given before in chapter III, stanzas 5 and 41, EXAMPLES Eleven examples on the pulveriser (kuttākāra): 13. The signs, etc., up to the thirds of the Sun's (mean) longitude have all been carried away by the strong wind; the residue of thirds is known to me to be 101. Tell (me) the Sun's (mean) longitude and also the ahargana.¹ 14. The minutes together with the signs and degrees of the Moon's (mean) longitude have been destroyed being rubbed out by the hands of a child; twenty-five seconds are seen to remain (undestroyed). Calculate from them, O you of noble descent, the ahargana and the (mean) longitude of the Moon. The following is the solution of this example: Revolution-number of the Moon reduced to minutes 68167872 civil days in a yuga 86225 By the usual process³, we get x = 70091, y = 55412633'. Multiplying 25 by 86225 and dividing the product by 60, the quotient is 35927. This is the residue of the minutes (Kalašeṣa).² We have, therefore, to solve the pulveriser 68167872 x 35927 86225 = y, where x denotes the required ahargana and y the Moon's mean longitude in terms of minutes. Hence the required ahargana longitude 4 signs 23° 53′ 25". 219 = H 70091 days, and the Moon's mean 15. The signs and degrees (of the mean longitude of Mars) have been carried away by the hurricane; the residue of the ¹ Answer: Ahargana -106141; Sun's mean longitude 52'23"11". For complete solution see supra, pp. 35-36. 2 Vide supra, chapter I, stanza 46(ii), p. 33. 8 Vide supra, chapter I, stanzas 42-44, p. 30. 3 signs 32° 220 degrees is 73. Say what is the (mean) longitude of Mars and also what is the ahargana. The following is the solution of this example: Revolution-number of Mars reduced to degrees civil days in a yuga EXAMPLES The pulveriser to be solved is therefore 13780944x73 26298625 = y, where x denotes the required ahargana and y the mean longitude of Mars in terms of degrees. Solving the pulveriser by the usual process, we get x= 17420617 y = 9128711. Hence the required ahargana-17420617 days, and the mean longitude of Mars 9128711°, i.e., 25357 revs., 6 signs, 11 degrees. 16. The (mean) longitude of Mercury is 3 signs, 15 degrees, and 5 minutes. Considering this give out the days elapsed (i.e., the ahargana) and also the revolutions performed by him. The following is the solution of this example: Longitude of Mercury- 3 signs 15° 5' = 6305'. Now 13780944 26298625* revolution-number of Mercury civil days in a yuga 896851 78895875* Therefore multiplying 6305 by 78895875 and then dividing the pro- duct by 21600, we get 23029559 as the quotient. This is the residue of the revolutions. Thus we have to solve the pulveriser 896851x23029559 78895875 where x denotes the ahargana and y the required revolutions. = y, EXAMPLES By the usual method, we get x = 74350409 and y 845180. = 17. The signs, degrees, and minutes of the (mean) longi- tude of Jupiter have been destroyed by a mischievous child; nine seconds are seen to remain (undisturbed). Say therefrom the ahargana and the mean longitude of Jupiter. The following is the solution of this example: Revolution-number of Jupiter reduced to minutes civil days in a yuga - 26224128x788958 5259725 26224128 5259725 221 Multiplying 9 by 5259725 and dividing the product by 60, we obtain 788958 as the quotient. This is the residue of the minutes. Then we have to solve the pulveriser where x denotes the ahargana and y the longitude of Jupiter in terms of minutes. Solving the pulveriser, we get x = 2269811, y = 11316906', i. e., 523 revs., 11 signs, 5 degrees, 6 minutes. The required ahargana is there- fore 2269811 days, and the mean longitude of Jupiter is 11 signs 5° 6′ 9″.. 18. The revolutions, etc., up to the minutes of the (mean) longitude of Venus are destroyed; 10 seconds are seen to re- main intact. Of Saturn, 17 seconds are found to remain in- tact. Quickly say (the mean longitudes of) them and also the ahargana (in the two cases). The following is the solution of this example: Revolution-number of Venus reduced to minutes civil days in a yuga 505611936 5259725 Multiplying 10 by 5259725 and dividing the product by 60, the quotient is 876620: this is the residue of the minutes. The resulting pulveriser is 505611936 x - 876620 5259725 where x is the ahargana and y the corresponding 18ngitude of Venus in 222 392318660. minutes. Solving this pulveriser, we get x = 4081170, y = Hence the ahargana-4081170 days, and the corresponding mean longitude of Venus = 10 signs 24° 20' 10". Again EXAMPLES revolution-number of Saturn reduced to minutes civil days in a yuga Multiplying 17 by 5259725 and dividing the product by 60, the quotient is 1490255: this is the residue of the minutes. The resulting pulveriser is = 10552608 x 1490255 5259725 where x denotes the ahargana and y the mean longitude of Saturn in minutes. Solving this pulveriser, we get x = 3308510, y = 6637877. 3308510, and the mean longitude of Hence the required ahargana Saturn 3 signs 21° 17' 17". y, = 10552608 5259725 19. The sum of the (mean) longitudes of Mars and the Moon is calculated to be 5 signs, 7 degrees, and 9 minutes. O you, well versed in the (Arya) bhata-tantra, quickly say the ahargana and also the (mean) longitudes of the Moon and Mars.¹ 11. The following is the solution of this example: Mean longitude of Mars + mean longitude of the Moon Also sum of the revolution-numbers of Mars and the Moon civil days in a yuga y, 5 signs 7° 9' = 9429'. Multiplying 9429 by 26298625 and dividing the product by 21600, the quotient is 11480080: this is the residue of the revolutions. We have therefore to solve the pulveriser 1000836 x 11480080 26298625 1000836 26298625 where x is the required ahargana. Solving the pulveriser, we get x = 5646655. From this ahargana we can easily calculate the mean longitudes of Mars and the Moon. ¹ For Govinda Svāmi's modification of this example, see supra, chapter I, under stanza 52. 223 20. The difference between the mean longitudes of Mars and Jupiter is exactly 5 signs. Say what is the number of days elapsed since the beginning of Kaliyuga and what are the (mean) longitudes of Jupiter and Mars.¹ EXAMPLES 21-22. The Sun and Moon on a Sunday at sunrise are carefully seen by me in (the sign) Libra. The degrees of their (mean) longitudes are 12 and 2 respectively; the minutes are 1 and 40 respectively. After how many days will they assume the same longitudes again (at sunrise) on a Thursday, Friday, and Saturday respectively? (It is also given that) the (mean) longitude of the Sun is in excess by 17 seconds (over that given above); whereas from the (mean) longitude of the Moon (given above 18 seconds have to be subtracted.2 That is to say, the Sun's longitude-6 signs, 12 degrees, 1 minute, and 17 seconds; and the Moon's longitude-6 signs, 2 degrees, 39 minutes, and 42 seconds. Calculation will show that the Sun and Moon assume these longi- tudes on a Sunday 7500 days after the commencement of Kaliyuga. The problem now is to find out the aharganas when the Sun and Moon again assume the same longitudes at sunrise on a Thursday, Friday, and Saturday respectively. This is done as follows: (i) Ahargana for Thursday. Let the corresponding ahargana be 7500 +A. Obviously, in A days the Sun and Moon will describe complete revo- lutions. Also since Thursday is four days in advance of Sunday, therefore A-4 will be perfectly divisible by seven. In other words, 576A 78898A 210389' 2155625². will be whole numbers. If we assume A and A-4 7 131493125X, the first two ¹ This example has been solved in chapter I under stanza 52. See supra, p. 44. • Bhāskara I's example, occurring in his comm. on A, ii. 32-33. 8 This number is the L.C.M. of 210389 and 2155625. 224 fractions will obviously be whole numbers, and we have only to make (131493125 X-4)/7 a whole number. Let or EXAMPLES 131493125X-4 7 X-4 7 = Y, Z, where Y= 18784732X+Z. Solving this equation, we find that X = 4 makes (131493125X-4)/7 a whole number. Therefore, the required ahargaṇa = 7500+ A 7500++-131493125X = 7500+131493125x4 = 525980000 days. (ii) Ahargaṇa for Friday. In this case, the required ahargana is obviously equal to 7500+131493125×5, i.e., 657473125 days. (iii) Ahargana for Saturday. In this case, the required ahargana 7500+131493125x6 = 788966250 days. 23. The revolutions, etc., of the Sun's (mean) longitude, calculated from an ahargana plus a few naḍis elapsed, have now been destroyed by the wind; 71 minutes are seen by me to remain intact. Say the ahargana, the sun's (mean) longitude, and the correct value of the nadis (used in the calculation).¹ 24-24*. Some number of days is (severally) divided by the (abraded) civil days for the Sun and for Mars. The (resulting) quotients are unknown to me; the residues, too, are not seen by me. The quotients obtained by multiplying those residues by the respective (abraded) revolution-numbers and then divid- ing (the products) by the respective (abraded) civil days are also blown away by the wind. The remainders of the two (divisions) now exist. The remainder for the Sun is 38472; that for Mars, 77180625. From these remainders severally ¹ This example has been solved in Chapter I under stanza 49. See supra, p. 40. AUTHORSHIP AND APPRECIATION OF THE WORK 225 calculate, 0 mathematician, the aharganas for the Sun and Mars and also the ahargana conforming to the two residues and state them in proper order.¹ One example on the determination of the latitude: 25. The Sun being at the end of (the sign) Gemini, the length of the night is 21 ghatis. Calculate and give out the latitude and also the colatitude of the place.² Authorship and appreciation of the work: 26. This Aryabhata-karma-nibandha ("a compendium of astronomical processes based on the teachings of Aryabhata I"), which has clear expressions and simple methods (of calculation) and which can be comprehended even by those with lesser intellect, is written by Bhaskara after full deliberation. 27. Whatever occurs in this work regarding the projec- tion and calculation of solar eclipses (etcetra), which are dealt with by giving numerous rules with clear meaning, also finds place elsewhere; and what does not find place here does not occur anywhere else. ¹This example has been solved in Chapter I under stanza 52. See supra, p. 45-46.

• Assuming the obliquity of the ecliptic to be 23°30', the required

latitude comes out to be 61°45' approx.... QUOTATIONS from the Maha-Bhaskariya in Later Works Passages from the Maha-Bhaskariya occur as quotations in the following commentaries : (1) Sankaranārāyaṇa's commentary (869 A. D.) on the Laghu- Bhaskariya. (2) Udayadivākara's commentary (1073 A. D.) on the Laghu- Bhaskariya. (3) Sūryadeva's commentaries on the Aryabhatiya and the Laghu-mānasa. (4) Makkibhaṭṭa's commentary (1377 A. D.) on the Siddhanta- sekhara. (5) Parameśvara's commentary (1408 A. D.) on the Laghu- Bhaskariya. (6) Nilakantha's commentary (c. 1500 A. D.) on the Aryabhatiya. (7) Govinda Somayaji's commentary, entitled Dafadhyayi, on the Brhajjataka of Varāḥamihira. (8) Visņu Sarma's commentary (c. 1363) on the Vidya-madha- viya. Below we refer to these passages and to the places where they occur as quotations. 1. Passages quoted by Sankaranarayaṇa. Passages quoted MBh, i. 8 MBh, iii. 1(ii) MBh, iii. 33 (ii) MBh, iii. 63-65 MBh, iv. 33 MBh, iv. 55 MBh, v. 14 MBh, vi. 44 1 Quoted under LBh, i. 15-16. LBh, iii. 1-3. LBh, vi. 23-25. LBh, viii. 4. LBh, ii. 27. LBh, ii. 7. LBh, v. 4. LBh, vii. 1-2. 2. Passages quoted by Udayadivākara. QUOTATIONS Passages quoted MBh, ii. 2(ii) MBh, ii. 4(i) MBh, iii. 25 MBh, iii. 26 MBh, iv. 4(ii) MBh, iv. 18 MBh, v. 12(i) MBh, v. 33 MBh, v. 43(ii) MBh, v. 77 MBh, vi. 56-58 MBh, vii. 20(i) Passages quoted 3. Passages quoted by Suryadeva. MBh, i. 4-6 MBh, i. 41 MBh, v. 33-35 MBh, vi. 48, 52, and 53 Quoted under MBh, v. 75-76 MBh, vii. 23(iv). LBh, i. 23 LBh, i. 27 LBh, iii. 7-10 LBh, iii. 7-10 LBh, i. 24 LBh, ii. 41 LBh, v. 15 LBh, v. 15 LBh, iv. 17 LBh, iv. 21 LBh, vii. 12 LBh, iv. 2 Quoted under A, iii. 6 A, ii. 33 LMa, iv. 14 LMa, iv. 6-7 LMa, iv. 14 LMā, iv. 3 227 228 QUOTATIONS 4. Passages quoted by Makkibhaṭṭa. Passages quoted MBh, i. 25(1) MBh, i. 40(ii) MBh, iv. 23(ii) Passages quoted 5. Passages quoted by Parameśvara. MBh, iii. 1(ii) MBh, v. 14 MBh, v. 15-16(i) MBh, vi. 45 Quoted under Passages quoted Sise, i. 40-41(i) Sise, i. 3 Sise, ii. 49 MBh, i. 41 MBh, i. 42 (ii)-43 (i) MBh, i. 45, 46, 47 MBh, i. 50 MBh, i. 51(i) MBh, iv. 2-6 Quoted under 6. Passages quoted by Nilakantha. LBh, iii. 3 LBh, v. 11-12 LBh, v. 11-12 LBh, vii. 1-2(i) Quoted under A, ii. 32-33 "} 29 39 "" Ā, iii. 22-25 QUOTATIONS Passages quoted MBh, iv. 35 MBh, v. 6 MBh, vii. 1 (i) MBh, vii. 19, 20 MBh, vii. 35 Quoted under A, iii. 3- Ā, iii. 22-25 Ā, ii. 32-33 Ā, iii. 12-15 "" 7. Passages quoted in the Dasadhyayı.¹ Passage quoted: MBh, iii. 9. Quoted under BJ, i. 19. 8. Passage quoted by Visnu Sarmā. Passage quoted: MBh, vii, 20 (ii). Quoted under: ViMā, i. 19. ¹ Vide Sudhākara Dvivedi, Gaṇaka-tarangini, p. 14. 229 GLOSSARY of Terms used in the Maha-Bhaskariya Amsa (3i) (1) Part, fraction. (2) Degree (°). Amsaka () Degree. Amsumat (³) Sun. Akşa (TT) (1) Latitude. (2) Five. Akşakarna () The hypote- nuse of the equinoctial midday shadow (of the gnomon). Akşakoti (erife) Colatitude. Also, sometimes, the Rsine of colatitude. Akşaguna (TTT) The Rsine of latitude. Akşacapa () The arc of latitude, or simply latitude. Aksacāpaguna (अक्षचापगुण) The Rsine of latitude. Aksarjiva (अक्षजीवा) The Rsine of latitude. Akṣajya (3) The Rsine of latitude. Akşabhāga (³) The degrees of latitude. Akşavalana (³) See Valana. Akşasya valana (अक्षस्य वलनम्) Akşavalana. See Valana. Akṣāmsa (³TTTT) The latitude of a place, terrestrial lati- tude, or simply latitude. Akşamsaka (3) Same as Akṣāmsa. Akşonnati (fa) Inclination of the (Earth's) axis, i.e., the latitude of the place. Agata (3¹) Untraversed por- tion; portion to be travers- ed. Agni (af) Three. Agra (3) (1) End. (2) Resi- due, remainder (3) Agrā. Agraguna (³) The Rsine of Agrā. Agra (T) The arc of the celes- tial horizon lying between the east point and the point where a heavenly body rises, or between the west point and the point where a heavenly body sets. Anga (³¹) Six. Angāraka (³) Mars. Angula () Finger-breadth. A unit of linear measureGLOSSARY ment defined by the breadth of eight barley corns. Acala (³) (1) Seven. (2) Fixed. To make acala in astronomy means to apply the method of successive approxima- tions. Aja () The sign Aries. Ativakra (f) A planet is said to be ativakra when it is in the middle of its retrograde motion. Adri (af) Seven. Adhika-masa (3) Inter- calary month. The Inter- calary months denote the excess of the lunar (synodic) months over the solar months in a certain period. Thus intercalary months in a yuga lunar months in a yuga-solar months in a yuga. A true intercalary month is one in which the Sun does not pass from one sign into the next. Adhikabda (fu) Intercalary year, i.e., a collection of twelve intercalary months. See Adhikamāsa. Adhikāha (³) Intercalary day, i.e., intercalary tithi. 231 Adhimāsa (अधिमास) Intercalary month. See Adhikamasa. Adhimāsaka (अधिमासक) Same as Adhimāsa. Adhimāsaśesa (अधिमासशेष) The residue of the intercalary months. Adhva (ra) The distance of a place from the prime meri- dian. Adhva (ET) Same as Adhva. Adhvana (अध्वान ) Same as Adhva. Anudis (fr) Parallel. Anupata () Proportion. Anuloma (³) Direct. A planet is said to be anuloma when its motion is direct, i.e., from west to east. Antyaguna (T) See Antya- jivā. Antyajiva (अन्त्यजीवा) The cur- rent Rsine-difference, i.e., the Rsine-difference corres- ponding to the elementary arc occupied by a planet. In Hindu trigonometry a quadrant of a circle is divi- ded into 24 equal parts, called elementary arcs. Antyajya (1) Same as Antyajiva. 232 Antyaphala (अन्त्यफल) Maxi- mum to GLOSSARY correction due mandocca or maximum correction due to fighrocca. The former is equal to the radius of the manda epicycle and the latter is equal to the radius of the sighra epicycle. Apakrama (³) Declination. Apakramaguna (अपक्रमगुण ) The Rsine of declination. • Aparamajya (अपक्रमज्या ) The Rsine of declination. Apakramadhana (अपक्रमधनु) The arc of declination, or simply declination. Apagama (³4) Declination. Apacaya (3) Decrease. Apama (3) Declination. Apamadhanu (अपमधनु ) The arc of declination, i.e., declina- tion. • Apamo gunah (अपमो गुण:) The Rsine of declination. Apavarta (³¶¶) The greatest common divisor; abrader. Abdapa (1) The lord of the year, i.e., the planet which is the regent of the first day of the year. 1 भिधीयते । ( गोविन्दस्वामी) + Abdhi (fe) Four. Abhyasa (³) Multiplication. Abhra (³) Zero. Amrtatejas () Moon. Amrdadidhiti (alfafa) Moon. Ambara (3) Zero. Ambaroruparidhi (अम्ब्ररोरुपरिधि) The word ambara means, according to Hindu astro- nomers, "that part of the sky which is illuminated by the rays of the Sun." "The word ambaroruparidhi like- wise means "the periphery of the illuminated sphere of the sky". a (4) The northward of southward course planet, particularly the Sun. The ayana of a planet is north or south according as the planet lies in the half-orbit beginning with the sayana sign Capri- corn or in that beginning with the sayana sign Cancer. Ayuta (3) Ten thousand. Arka (3) (1) Sun. (2) Twelve. Arkaputra (ar) Saturn. Arkavarşa (a) Solar year. Arkasambhava (mūs) ( मास ) Solar month. यावन्तमाकाशप्रदेशं वेर्मयूखा: अभिद्योतयन्ति तावानिह प्रदेशोऽम्बरशब्देना- Ayana or Ardhacaturtha (अर्धचतुर्थ ) Three and a half (31). Lite- rally, four minus half. Ardhapancaka (अर्धपञ्चक) or Ardhapancama (अर्धपञ्चम) Four and a half (4). Literally, five minus half. Ardhavistara (अर्धविस्तर) Semi- diameter, radius, or 3438'. Ardhastamita (अर्धास्तमित) Half- set. GLOSSARY Avanati (अवनति ) (1) Meridian zenith distance. (2) Celestial latitude. (3) Parallax in celestial latitude. Avanatilava (अवनतिलव) The degrees of meridian zenith distance. Avanatilavajvā (अवनतिलवजीवा) The Rsine of avanatilava. Avanatiliptikā (अवनतिलिप्तिका) Avanati in minutes of arc. Avanāma (अवनाम) Zenith- distance. Avama (3) Omitted lunar days or omitted tithis. Avamarātraseṣa (अवमरात्रशेष ) The residue of the omitted lunar days. Avamasesa (अवमशेष ) The residue of the omitted lunar days. 233 ( अवलम्बक) (1) Plumb. (2) The Rsine of colatitude. Avalambaka Avalambakaguna (अवलम्बकगुण) The Rsine of colatitude. Avasesa () Remainder. Avisista (अविशिष्ट) Obtained by applying the method of successive approximations. Avisesakarma (अविशेषकर्ण ) The distance (lit. hypotenuse) obtained by the method of successive approxi- mations. Avisesakarma (अविशेषकर्म) The method of successive approximations. Avisesana (अविशेषण) To perform Aviseşakarma. Avisesatithi (अविशेषतिथि) The tithi (i.e., the time of apparent conjunction of the sun and Moon) obtained by the method of succes- sive approximations. Avisesanādi (अविशेषनाडी) The nadis obtained by the method of successive appro- ximations. Avisesarvidhi (अविशेषविधि) See Aviseşakarma. 234 Avislista (faf) Same as Avisista. Avisama (³) Even. Asvi ( अश्वि) Two. Asvin (अश्विन्) Two. Asti (3) Sixteen. GLOSSARY Asakrt () Repeatedly, or by using the method of successive approximations. Asita (fr) (1) Dark. (2) The unilluminated part of the Moon's disc. (3) Saturn. Asita-pakṣa (af) The dark half of a lunar month. Asu (3rg) A unit of time equi- valent to 4 seconds. Asṛktanu (³) Mars. Mars is called asṛktanu (asṛk= blood, tanu-body) because it is red in colour. Asta (3) Setting. Astakāla (अस्तकाल) Time of setting. Astalagna (³) The setting point of the ecliptic, i.e. that point of the ecliptic which lies on the western horizon. Astasutra (³) The rising- setting line (udayastasutra). Astodayāgrarekha (ar) The rising-setting line. Ahargana (3) The number of days elapsed since the beginning of Kaliyuga (or any other epoch). Aharmana (3) The length of day. Ahoratra (T) A day and night, a nychthemeron. Ahoratra-viskambha (- C) Day-radius. Ahmān ganah (अह्नां गण:) Same as Ahargana. Ahnāin nicayah (अह्नां निचयः ) Same as Ahargana. Akasa (³) Zero. Apyа (²) Purvāṣāḍha, the twentieth nakṣatra. Ayama (³¶¶) Length. Āra (³) Mars. Ārki (3) Saturn. Asa (T) (1) Direction. (2) Ten. Asanna () Approximate. Astamika (आस्तमिक) Pertaining to sunset. Asphujit (af) Venus. Ahnika (3) (1) Pertaining to day. (2) A special astronomical term used by Bhaskara I. See MBh, i. 16-18. Ina () (1) Sun. (2) Twelve.. Indu () (1) Moon. (2) One. GLOSSARY Inducca () Moon's apogee, i.e., the remotest point of the Moon's orbit. Indriya (fa) Five. Indvahah () Lunar day or tithi. Isu () Five. Ista (2) Given, or desired, or chosen at pleasure. Ucca () The ucca is the apex of a planet's orbit. It is of two kinds: (1) mandocca, i.e., the of apex slow motion, and (2) sighrocca, i.e., the apex of fast motion. In Hindu astronomy, the mandocca is defined to be the remo- test point of the planet's orbit where the planet appears smallest¹. It is therefore the same as the "apogee" of modern astro- nomy. The sighrocca of the superior planets is an imagi- nary body which remains in the same direction as the mean Sun; that of an in- ferior planet lies approxi- mately in the same direction from the Earth as the actual planet is from the Sun. 235 Uccabhukti (उच्चभुक्ति ) (Daily) motion of the ucca; apsidal motion. Utkrama (3) Reverse order. Utlcramaguna (उत्क्रमगुण ) ) Same as Utkramajya. Utleramajzva (उत्क्रमजीवा) Same as Utkramajyā. Utkramajyā (उत्क्रमज्या) Rversed- sine (Radius X versed- sine). Uttara (3) North. Uttargola (3) Northern hemisphere, i.e. the hemi- sphere lying to the north of of the equator. Udak () North. Udayajya (3) The agra of the rising point of a planet's orbit. Udayaprāna (उदयप्राण ) Times of rising of the signs measured in asus. Udayarāśiprāṇapinda (Gutifu- प्राणपिड) The time in asus of rising of the rising sign. Udayalagna () The rising point of the ecliptic, i.e., the horizon-ecliptic point in the east. 1 यत्र ग्रहाः सूक्ष्मा लक्ष्यन्ते कर्णस्य महत्वात् स आकाशप्रदेश उच्चसंज्ञितः । See Bhaskara I's comm. on A, iii 4(i). 236 Udayāsu (3) Times of rising (of the signs) in asus. Udyāstamaya (उदयास्तमय ) Rising and setting. Unnati (af) Elevation. Uparāga (3) Eclipse. Usnatejas (उष्णतेजस्) Sun. Usnadādhiti (उष्णदीधिति) Sun. Rtu (g) Six. Aindragna GLOSSARY (ऐन्द्राग्न) The name of the sixteenth naksatra Viśākhā. Aindri () East. Oja (³) Odd. Kakubh (4) Ten. Kakṣya(T) Orbit of a planet. Kanyaka(¹) The sign Virgo. Kapala () Hemisphere. Karaṇa (*) (1)Process; work- ing. (2) The name of one of the principal elements of the Hindu Calendar. Karkata() (1)A pair of com- passes. (2) The sign Cancer. Karna (o)(1) The hypotenuse of a right-angled triangle. (2) The distance of a planet. Karnasutra (T) Hypotenuse- line. Kala () Minute of arc. Kalakarna () The true distance of a planet in minutes of arc. Kalanam seṣaḥ (:) The residue of the minutes (kalāseṣa). Kaliyuga (T) According to Bhaskara I, Kaliyuga is a period of 1080000 solar years. The current Kali- yuga began on Friday, February 18, B. C. 3102, at sunrise at Lanka. Karmuka () Arc. Kalabhaga (TTT) Degrees of time. A degree of time is equivalent to 60 asus or 10 vinādis. Kastha (3) (1) Arc. Direction. Kāṣṭhā () Direction. Kilaka () Gnomon. (2) Kilakagraguna (कोलकाग्रगुण) Same as Sankvagra. Kuja () Mars. Kujāsā () South. Kutila () Retrograde. Kuṭṭa (3) Pulveriser. See Kuṭṭākāra. Kuttana () The Process of solving a pulveriser (Kuṭṭā- kāra). Kuṭṭākāra () Pulveriser. Equations of the type 237 (2) (1) Kotiphala () The result obtained by multiplying the Rsine of koti due to a planet's kendra by the epicy- cle and dividing the result- ting product by 360. Kotisadhana (fr) Same as Kotiphala. GLOSSARY N=ax+r-by+s or ax-c-by are called in Hindu mathe- matics by the name kutta- kāra. A kuṭṭakara (pul- veriser) is called sagra (residual) or niragra (non- residual) according as it is of the type (1) or (2). Kumbha()The sign Aquarius. Kulira (F) The sign Cancer. Krta (a) Four. Krti (a) Square. Kritika (f) The name of the third naksatra. Kendra () (1) Anomaly. The kendra is of two kinds : manda-kendra and sighra- kendra. The manda-kendra of a planet is equal to "the longitude of the planet minus the longitude of the planet's mandocca (apogee)," and the sighra-kendra of a planet is equal to the "longi- tude of the planet's sighrocca minus the longitude of the planet." (2) Centre. Kendrajya (1) The Rsine of kendra. Koti (fe) See Bāhu. Kotikā (fr) Same as Koti. Krama () (1) Serial order. (2) Odd quadrant. Kramaguna ( क्रमगुण) Same as Kramajya. Kramajyā (क्रमज्या) Rsine ( Radius X sine). Kranti (f) Declination. Kriya (f) The sign Aries. Kşamādina (f) Civil day. Ksitiguna (fea) Same as Kşitijya. Ksitijaguna (ff) Same as Kşitiguna. Kṣitija (farfa) A corrupt form of Kṣitijya. Ksitija (fafafar) Same Kṣitijyā. Ksitijga ( क्षितिज्या) as Earthsine. The distance between the rising-setting line and the line joining the points of intersection of the diurnal circle and the six o'clock circle. 238 Kşitidhara (farfa) Seven. Kşitiputra (ferfaga) Mars. Ksitimaurvi ( क्षितिमौर्वी) Same as Kṣitijyā. Kṣşipti (ff) Celestial latitude. Kṣetranirmāṇa (ff) Celes- tial longitude. Kṣepa () Quantity to be ad- ded. Ksonidhara (ofta) Seven. Kha () Zero. Khamadhya () Meridian. Khecara () Planet. Gagana (T) (1) Meridian. (2) Zero. Gaganasya vrttam (गगनस्य वृत्तं ) The circumference of the sky. See Ambaroruparidhi. Gana (TT) Used in the sense of Bhagana. Gata () Traversed, elapsed. Gati (fa) Motion. Generally used in the sense of "daily motion". Gantavya () To be traver- sed. Guna (T) (1) Multiple or multi- plication.(2) Rsine. (3)Three. Gunakara (T) Multiplier, coefficient. Gunapratāna (T) Rsine. Gunapkala () Bahuphala GLOSSARY and kotiphala. Guru (T) Jupiter. Grha (TE) Sign. Go () The sign Taurus. Gola (m) Hemisphere. Golakhaṇḍa (s) The semi- diameter of the (celestial) sphere. ( गोल = sphere, खण्ड =half) Golabheda (गोलभेद ) Same as Golakhanda. (भेद = {===खण्ड = half) Graha () (1) Planet. (2) Eclipse. Grahaganita (f) Astrono- my. Grahacara (g) Motion of a planet. Grahana () Eclipse. Grahatanu (¹) A special term used by Bhaskara I. See MBh, i. 29. Grahadeha () Same as Grahatanu. Grahayoga () Conjunction of planets. Grahaņam tamuh ( ग्रहाणां तनु:) Grahatanu. Grahoparāga (¶¶¶¶) Eclipse. Grasa () (1) Eclipse. (2) Measure of an eclipse. (2) Beginning of an eclipse. GLOSSARY Grāsamadhya ( ग्रासमध्य ) The middle of an eclipse. Grāsasalaka (ग्रासशलाका ) A needle (or line) of length equal to the portion of the diameter eclipsed. . Grasadi (fa) The beginning of an eclipse. Grahaka () The eclipsing body, the eclipser. Grahya (T) The eclipsed body. Ghaṭika (fe) Same as Ghati. Ghati () A unit of time equivalent to 24 minutes. Ghana (+) Cube. Ghata () Product, multipli- cation. Cakra () Circle, twelve signs, or 360°. Caturasra (T) Quadrilateral. Candra () (1) Moon. (2) One. Candraka () Same as Candra. Cara () Ascensional differ- ence. It is defined by the are of the celestial equator lying between the six- o'clock circle and the hour circle of a heavenly body at rising. 239 Caradala () Ascensional difference. See Cara. Cala () Sighrocca. Calocca () Sighrocca. Capa (1) (1) The sign Sagit- tarius. (2) Arc. Cara () Motion or daily motion. Citra (f) The name of the fourteenth nakṣatra. Caitra (7) The name of the first month of the year. Chaya (or) (1) Shadow. (2) The Rsine of the zenith dis- tance. Chāyādairghya (छायादैर्ध्य) Same as Bhucchāyādairghya. Chāyābhramana (छायाभ्रमण) The path of the end of the shadow (of the gnomon). Chidra (fo) Nine. Cheda (a) Divisor or denomi- nator. Chedyaka () Projection, or graphical representation. Jaladhara () Four. Jaladhi (f) Four. Jalapadilk (जलपदिक्) West. Jalesadik (af) West. Jina (fa) Twenty-four. Jiva (a) Jupiter. Jivadina (af) Thursday. 240 Jiva (ar) (1)Rsine (= Radius. X sine). (2) The Rsine- differences corresponding to the twenty-four divisions of the quadrant. Jivabhukti (ta) True daily motion derived with the help of the table of Rsine- differences. Juka () The sign Libra. Jña () The planet Mercury. Jya (T) Same as Jivā. Jyasankalita (ज्यांसंकलित) Used in the sense of "the given Jya". Jyau () Jupiter. Tatparā () Third of arc, i.e., one-sixtieth of a second of arc. Tattva (a) Twenty-five.. Tantra () Principle, doctrine, theory, rule, method. Also, class of astronomical works. a Tama () (1) The shadow of the Earth), particularly, the section of the shadow cone where the Moon crosses it, by a plane perpendicular to the axis of the shadow (2) The Moon's as- cending node, con GLOSSARY Tamomaya () The Moon's ascending node. Taraka (ar) Star. Tara (¹) Star. Taragraha () The star- planets. The planets Mars, Mercury, Jupiter, Venus, and Saturn are called star- planets (tārägraha) in Hindu astronomy. Tigmāinsu (तिग्मांशु ) (1) The Sun. (2) Twelve. Tithi (fafa) (1) Lunar day (called tithi). (2) Time of conjunction or opposition of the Sun and Moon. (3) Time of beginning, middle, or end of eclipse. (4) Fifteen. (5) Thirty. Tithipranāśa (तिथिप्रणाश) Omit- ted tithis. Tithyanta (f) Time of con- junction or opposition of the Sun and Moon. Tiryak (fi) (1) Oblique. (2) Transverse. Tunga () Same as Ucca. Turiya () One-fourth. Tula () The sign Libra. Tuladhara () The sign Libra. Trigrhaguna (त्रिगृहगुण) The Rsine of three signs, i.e., GLOSSARY the radius or 3438'. Trijya (f) Radius or 3438'. Tribhavana (f) Three signs. Trimaurvi (ff) Radius. Trarāsika (त्रैराशिक) Rule of Three. Dala () Half. Dasaguna (दशगुण ) The ten Rsines, viz. Sun's udayajya, Sun's madhyajya, Sun's drkkṣepajya, Sun's drgjya, Sun's drggatijya, Moon's udayajya, Moon's madhya- jya, Moon's drkkṣepajya, Moon's drgjya, and Moon's drggatijyā. Dasasjivā (दशजीवा) Same Dasaguna. Dasra () Two. Dahana () Three. Dik (fa) (1) Direction. (2) Ten. as Dikka (f) Direction. Ditisunupujita ( दितिसूनुपूजित ) Venus. Dina (fa) (1) Day. (2) Fifteen. Dinagana (दिनगण ) Same as Ahargana Dinamana (f) Measure (or length) of the day. Dinarasi (fa) Ahargana. 241 Dinanam gamah (दिनानां गणः) Same as Ahargana. Divasa (f) (1) Day. (2) Ahargana. Divasagumārdha (दिवसगुणार्ध) The day radius. Divasajiva (fa) The day radius. Divasayojana ( दिवसयोजन) The number of yojanas that a planet traverses in a day. Divasavistarabheda (दिवसविस्तर - a) The day radius. Diväguna (feaT) The day ra- dius. Divicara (दिविचर ) (1) Seven. (2) Planet. Dis. (fe) (1) Direction. (2) Ten. Drkkṣepa () The dykksepa is the shortest arcual dis- tance of the planet's orbit from the zenith. It is also used for the Rsine of that distance. Drggati (fa) Arc correspond- ing to the Drggatijyā. Drggatijya (fa) The drgga- tijya is the distance from the zenith of the plane of a planet's circle of celestial longitude, or the Rsine of the shortest distance from 242 GLOSSARY the zenith of a planet's circle of celestial longitude. Drgguna () The Rsine of zenith distance. Drgjiva (tar) The Rsine of zenith distance. Drgjya () The Rsine of zenith distance. Ordha (6) Prime. Drsya-candra (T)The longi- tude of the Moon corrected for the visibility correc- tions. Devapujya (y) Jupiter. Devamantri (fa) Jupiter. Desakala () Used in the sense of desantara-kala, i.e., the longitude-correction in terms of time. Desajata-kala () See Deśakāla. Desantara () The longi. tude of the place. That is, either the distance of the local place from the prime meridian, or the difference between the local and stan- dard times. Deśāntara-karma ( देशान्तर-कर्म) Correction for the longitude of the place, the longitude- correction. Deśāntara-ghaṭi (-) De- santara in ghatis. Deha () Used in the sense of grahadeha. See Grahadeha. Dyugana (T) Ahargana. Dyujya () The day radius. Dyuti (af) Shadow (meaning "the shadow of the gno- mon"). Dyuti-karna (fa) The hypo- tenuse of the shadow (of the gnomon). Dyudala () The day radius. Dyurasi (af) Ahargana. Dyuvyasa () The day radius. Dyuvyāsakhaṇḍa (³) The day radius. Dyuvyāsa-bheda () The day radius. Dywāysārdha (धुव्यासार्ध) The day radius. Dvyagra () A sagra kuṭṭā- kara (residual pulveriser) with two residues. Dhana (7) Addition. Dhanistha (af) The name of the twenty-third nakṣatra. Dhanuh ( धनु:) (1) Arc. ( 2 ) The sign Sagittarius. Dhanuh-khanda (7:que) In Hindu astronomy, the quadrant of a circle is divided into twenty-four equal parts and these parts are known as kaştha, dhanu, dhanuḥkha- nda, dhanurbhaga, etc. Dhanurbhāga (धनुर्भाग) 225'. Dhanus (धनुष्) (1) Arc. ( 2 ) 225. Dhanvin () The sign Sagittarius. Dharanidina (धरणीदिन) Civil day. Dharādivasa day. Dharasuta (a) Mars. $ (धरादिवस ) GLOSSARY Civil Dhatridhara () Seven. Dhrti (fa) Eighteen. Dhruvaka (T) A technical term. See MBh, i. 29. Nakṣatra (T)(1)Star. (2)Aster- ism. (3) Twenty-seven. Nakṣatragana (¹) Sam as Bhagana. Nakha () Twenty. Naga () Seven. Nakṣatra-bheda () Occul- tation of stars. Natabhaga () The degrees (bhaga) of zenith distance (nata). Natāmsa (at) Zenith distance. Nati (नति) (1) The meridian zenith distance. (2) Celestial latitude as corrected for parallax in latitude. (3) Parallax in latitude. Natijya (f) The Rsine of the meridian zenith dista- nce. 243 Nanda () Nine. Nabha () Zero. Nara () (1) The sign Gemini. (2) Gnomon. (3) The Rsine of altitude. Na (n) The Rsine of altitude. Nadika (fs) A unit of time equivalent to 24 minutes. Nadi (ast) See Nadika. Nirakṣa (f) Equator. Nirakṣāsu (fr) Asus of the right ascension, i.e., the time in asus of rising at the the equator. Nirapavartita (af) Una- braded, unabridged. Nisakara (F) (1) Moon. (2) One. Niscalakriya (निश्चलक्रिया) Method of successive approxima- tions. Nihśvāsalava (निःश्वासलव) Asus. Nr. (7) Gnomon. 244 GLOSSARY Netra () Two. Nemi (f) Circumference, peri- phery. Pakṣa (T) (1) Lunar fortnight, period from new moon to full moon, or from full moon to new moon. (2) Two Pada (7) (1) Quadrant. (2) Square root. Parakrānti ( परक्रान्ति) (Sun's) greatest declination, or ob- liquity of the ecliptic. Paramāpama (परमापम) Same as Parakranti. Paridhi (ff) (1) Circumfere- nce, periphery. (2) Epicycle. Parilekha (fa) Projection, graphical representation. Parvata (a) Seven. Parvamadhya (1) The mid- dle of the eclipse. Pala () Latitude. Palajiva () The Rsine of the latitude. Palajya () The Rsine of the latitude. Palabhaga (TTT) The degrees of the latitude. Patangula (पलाङ्गुल) Used in the sense of 'palabhangula', i. e., the angulas of the equinoctial midday shadow. Palamsa (i) The degrees of the latitude. Pascardha (f) The western half. Pata (a) The ascending node of a planet's orbit (on the ecliptic). Patabhāga (¹) The degrees of the longitude of the asc- ending node. Puşkara (go) Three. Purvalagna () The hori- zon-ecliptic point in the east. Pankti (if) Ten. Pratimandḍala (4) Ecc- entric. Pratimandala-karma ( प्रतिमण्ड- लकर्म) Processes under the eccentric theory. Prabhā (H) (1) The shadow of the gnomon. (2) The Rsine of the zenith distance. -Pralayastithinām (प्रलयास्तिथीनां ) Omitted lunar days. Prastāra ( प्रस्तार ) The state- ment of possible combina- tions in a serial order. Praggrasa () The first contact in an eclipse. Prana () Same as Asu. Prāleyarasmi (प्रालेयरश्मि) One. GLOSSARY Pronnati (fa) Altitude.. Phala () (1) Result. (2) Cor- rection. Bava () The first of the 245 Bahujya (ar) The Rsine of the bahu (of a planet's seven movable karanas. The karana is one of the five important elements of the Hindu Calendar. Bahula (बहुल) The nakşatra Kṛttikā. Bahu (g) (1) The base (of a right-angled triangle). The upright of a right-angled triangle is called koti. (2) The bahu corresponding to a planet's anomaly. This is the arcual distance of the planet from its apogee or perigee whichever is nearer. Suppose that 0 is the anomaly of a planet (or any arc, whatever). If is less than x/2, 0 itself is the bahu; if 0 is greater than T/2 but less than 7, (0) is the bahu; if 0 is greater then म but less than 3x/2, (0-T) is the bahu; and if is greater than 3/2, (2-0) is the bahu. The complement of the bahu is called koti. Bahuka (g) Same as Bahu. anomaly). Bahuphala () See notes on MBh, iv. 6. Bahoh phalan (बाहो: फलं) Same as Bahuphala. Bimba (fa) The disc of a planet. Bimbardha (f) The semi- diameter of the disc. Budhāśā (ar) North. Bham () Twenty-seven. Bhaga (T) The nakṣatra Purvā-phālguni, the regent of which is Bhaga. Bhagana (TT) (1) The revolu- tion-number of a planet, i.e., the number of revolutions that a planet performs in a certain period. The revolutions given by Bhaskara I correspond to a yuga, i.e., to a period of 43,20,000 years. (2) The nakṣatras. (3) Twelve signs (or 360°). Bhava (₁) Eleven. Bhavana () Sign. Bhaga (T) (1) Part, fraction. (2) Division. (3) Degree. Bhagalabdha (*) Quotient. 246 GLOSSARY Bhagašeşa (T) The residue of the degrees. Bhagahāra () Divisor. Bhagaharaka (भागहारक) Same as Bhagahāra. Bhajya () Dividend Bhargava (ia) Venus. Bhidah (f:) Half. Bhukti (f) Motion, or daily motion. Bhuja () Same as Bahu. Bhuja (n) Same as Bhuja. Bhujantara () Correction for the equation of time due to the eccentricity of the ecliptic. Bhujaphala () The equa- tion of the centre. Bhucchaya (T) The Earth's shadow. Bhucchayādairghya (भूच्छायादैर्घ्य ) The length of the Earth's shadow, i.e., the distance of the vertex of the shadow cone from the Earth's centre. • Bhujya (T) See Kṣitijya. Bhūta () Five. Bhudina (f) Civil day. Bhudicasa (fa) Civil day. Bhudhara (a) Seven. Bhubhṛt () Seven. Bhumidina (ff) Civil day. Bhrgu (T) Venus. Bhrguja () Venus. Bheda (F) (1) Half. (2) Occul- tation of a heavenly body. Bhoga (T) Motion. Bhauma () Mars. Maghavadguru (मघवद्गुरु ) Jupi- ter. Magha (T) Name of the tenth nakṣatra. Mandala (F) Circle; a col- lection of 12 signs. Mandalamadhya (मण्डलमध्य) The centre of a circle. Mati (f) An optional num- ber. Matsya () Fish-figure. Madhu () Caitra, the first month of the year. Madhyakranti (f) The declination of the meridian- ecliptic point. Madhyacchāyā (मध्यच्छाया) The midday shadow (of the gno- mon). Madhyajatah lambakah (मध्यजात: लम्बक:) The upright due to the meridian-ecliptic point, GLOSSARY i.e., the Rsine of the alti- tude of the meridian ecliptic point. Madhyajya (H) The Raine of the zenith-distance of the meridian-ecliptic point; the meridian-sine. Madhyaparinişthitalambaka (मध्यपरिनिष्ठितलम्बक) Same as Madhyajataḥ lambakah. Madhyalagna (F) Meridian- ecliptic point. Madhyasūryāvanāma (मध्यसूर्याव- नाम) The zenith distance of the midday Sun, or the meridian zenith distance of the Sun. Madhyāvanati (मध्यावनति ) The zenith distance of the midday Sun. Mandakendra () Manda anomaly (=longitude of the planet minus longitude of the planet's apogee). Mandapata (G) See MBh, vii. 30. Mandaphala () Correction due to the planet's man- docca. In the case of the Sun and Moon, the equation of the centre. 1 247 Mandavrtta (मन्दवृत्त) Manda epicycle. Mandasiddha (Hrafura) Cor- rected for the mandaphala. Mandasiddhi (मन्दसिद्धि ) Correction (of a planet) for the manda- phala. Mandantyajivā (मन्दान्त्यजीवा) The present corresponding to Rsine-difference the mandakendra i.e., the Rsine- difference of the elementary arc in which the planet lies. Mandocca () The apogee of a planet. See Ucca. Mandoccakarņa (मन्दोच्चकर्ण) See Mandakarna. Mandamaurvīphalacāpa ( मन्द- मौर्वीफलचाप) Same as Mand- aphala. Mandoccakendra (मन्दोच्चकेन्द्र) See Mandakendra. ( मन्दोच्चवृत्त ) Mandoccavṛtta Manda epicycle. Mithuna (मिथुन ) Gemini. Mina (मीन) The (1) Fish-figure, (2) the sign Pisces. Muni () Seven. sign Mula () Square root. Mrga (¹) The sign Capricorn. Meşa () The sign Aries, 248 GLOSSARY Maitra () The nakṣatra Anuradha, the regent of which is Mitra. Moksa () The separation of the eclipsed body after an eclipse. Maurika (f) Minute of arc. Maurvi (f) Rsine. Yantra Instrument. Yama () (1) Saturn. (2) Name of the second naksatra Bharani, whose divinity is Yama. (3) Two. Yamala (TH) Two. Yamya () South. Yamyagola. ( याम्यगोल ) The southern hemisphere, i.e., the hemisphere lying to the south of the equator. Yamyottara (T) South- north. Yamyottarayata (याम्योत्तरायत) Directed south-to-north. Yugala () Two. Yugma () Even. Yoga () (1) Addition. (2) Conjunction of two heaven- ly bodies. Yogatara (T) Junction- stars. These are those prominent stars of the twenty-seven nakṣatras which were used by the Hindu astronomers for the study of the conjunction of the planets, especially the Moon, with them. Yogabhāga (योगभाग ) The degrees of the longitudes of the junction-stars. Yojana () The yojana is a measure of distance. The length of a yojana has differed at different places and at different times. The yojana of Āryabhata Bhaskara I is roughly equivalent to 7 miles. I and Randhra (T) Nine. Ravi (fa) (1) Sun. (2) Twelve. Ravija (f) (1) Saturn. (2) A special term used by Bhaskara I. See MBh, i. 27. Ravijadivasa A special term used by Bhaskara I. See MBh, i. 28. Rasa () Six. Rama (4) Three. Rasi (राशि) ( 1 ) Quantity. (2) Sign. Rashijvā (राशिजीवा) The Rsine of one sign, i.e., Rsin (30°). Rahu (g) The Moon's ascend- ing node. GLOSSARY Rudra () Eleven. Rudhira (fr) Mars. Rupa (T) One. Lagna (लग्न) The horizon- ecliptic point in the east. Laghutantra () Short or simplified method. Lanka () A place in 0 latitude and 0 longitude. Also see supra, p. 47. Lankārāsyudaya (लङ्काराश्युदय) Times of rising of the signs at Lanka, or right ascen- sions of the signs. Lankodaya () Times of rising (of the signs) at Lankā, or right ascensions (of the signs). Labdha (a) Quotient. Lamba () The Rsine of the colatitude (of the place). Lambaka () The Rsine of the colatitude. Lambakaguna (लम्बकगुण) Same as Lambaka. Lambana (+) Parallax in longitude; or, in parti- cular, the difference be- tween the parallaxes in longitude of the Sun and Moon, Lambanalipta (लम्बनलिप्ता) Lambana in longitude in terms of minutes of arc. Lambanāntara ( लम्बनान्तर) The lambana-difference. Lambanāntaranādika (लम्बनान्तर- नाडिका) The nādis of the lambana-difference, or the lambana-difference terms of nadis. in 249 Lava () (1) Part, portion, fraction. (2) Degree. Lipta (fr) Minute of arc. Liptika (ff) Same as Liptā. Liptikka vipwrvad (लिप्तिका विपूर्वा) Viliptikā; second of arc. Vakra (*) Retrograde. Vakragati (f) Retrograde motion. Vakragamana () Retro- grade motion. Vakrigraha () A planet in retrograde motion. Vacasām patiḥ ( वचसां पतिः) Jupiter. Vatsara () Year. Vapu (g) The body (globe or disc) of a planet. Varga (a) Square. Vartamana () Present, current, 250 GLOSSARY Vartamānaguna (वर्तमानगुण) The present (or current) Rsine- difference, i.e., the Rsine- difference of the element- ary are occupied by a planet. Varşapa (a) The lord of the year, i.e., the planet after whose name the first day of the bears its name. year Valana () Deflection. Val- ana relates to an eclipsed body. It is the angle subten- ded at the body by the are joining the north point of the celestial horizon and the north pole of the eclip- tic. Valana is generally divided into two compon- ents, (i) Akşavalana and (ii) Ayanavalana. The Akşa- valana is the angle subten- ded at the body by the arc joining the north point of the celestial horizon and the north pole of the equ- ator. The Ayanavalana is the angle subtended at the body by the arc joining the north poles of the equator and the ecliptic. The Valana is also defined as follows: The great cle of which the eclipsed body is the pole is called the hori- zon of the eclipsed body. Suppose that the prime vertical, equator, and the ecliptic intersect the horizon of the eclipsed body at the points A, B and C towards the east of the eclipsed body. Then arc AB is called the Aksavalana, arc BC is called the Ayana- valana and arc AC is called Valana. Valana is also called spasta- valana. Vasu (a) Eight. Vahni (af) Three. Varun (areºît) West. Vi (fa) Celestial latitude. Evi- dently, Vi is an abbre- viated form of vikṣepa. Vikala (fr) Second of arc. Vikastha (fr) The are of celestial latitude. Vikṣipti (fafafa) Celestial lati- tude. Vikṣepa (fata) Celestial lati- tude. Vikṣepajya (faqar) The Rsine of celestial latitude. Vikṣepamsa (fata) The deg- rees of celestial latitude. GLOSSARY Vighatika (ff) A unit of time, equivalent to 24 se- conds. Vidis (fafa) Contrary direc- tion. Vinādilka (विनाडिका) Same as Vighatikā. Vinadi (fast) Same as Vinā- dikā. Viparitaguna (विपरीतगुण ) Rver- sed-sine. Vipulacchayā (विपुलच्छाया) Great shadow, meaning "the Rsi- ne of the zenith distance". Vipulanara (विपुलनर) Great gnomon, meaning "the Rsine of altitude." Vimandala (F) The orbit of a planet. Vimardärdha (faf) Half the duration of totality of an eclipse. Vimawwrika (विमौरिक) Second of arc. Viyat (faa) Zero. Vilagna (fa) The horizon- ecliptic point in the east. Vilipta (fafan) Second of arc. Viliptika (fafafan) Same as Viliptā. 251 Vivara (faaz) Difference, inter- vening space. Visakhā (fagar) Name of the sixteenth nakşatra. Visesa (fa) Difference. Vislesa (fa) Difference. Visva (fara) Thirteen. Vişama (f¶) Odd. Vişaya (faqa) Five. Visuwajyā (विषुवज्या) The Rsine of the latitude (of a place). Visuvat (faqaa) The equator. Visuvatkarna (विषुवत्कर्ण ) The hypotenuse of the equi- noctial midday shadow. Visuvatprabha ( विषुवत्प्रभा) The equinoctial midday sha- dow. Visuvadudayarāśiprānapinda (विषुवदुदयराशिप्राणपिंड) Time in asus of rising of the signs at the equator, i.e., right ascension of the signs in terms of asus. Viskambha (f) Diameter. Vişnukrama (f) Three. Vistara (fa) Same as Vistara. Vistāra (विस्तार ) (1) Diameter. "Vyasa, viskambha, and vistāra are synonyms", says Bhaskara I. (2) Length, 252 GLOSSARY breadth, etc. "Ayāma, vis- tara, and dairghya are synonyms," says Bhaskara Vrttasanchyā (वृत्तसंख्या) The length of the circumference of a circle. Vrnda (a) Cube. Vrsa () The sign Taurus. Veda () Four. I. Vihangama (fa) Planet. Vihaga (fa) Planet. Vihāgas (विहायस्) Zero. Śakraguru (*) Jupiter. Vṛtta () (1) Circle. (2) Epi- Sanku (v) (1) Gnomon. (2) cycle. The Rsine of altitude (of a heavenly body). Velakutia () Time-pul- veriser. See notes on MBh, i. 49. Vaidhṛta (a) An astronomi- cal phenomenon. See MBh, iv. 35. Vaisuvati chayā (वैषुवती छाया)- The equinoctial midday shadow. Vyatipāta (¶¶) An astro- nomical phenomenon. See MBh, iv. 35. Vyāsārdha () Radius or 3438". Vyasa () Diameter. Vyasakhanda (4) Radius. Vyasakhandanicaya ( व्यासखण्ड- f¶¶¶) same as Vyāsakhaṇḍa. Vyoma () Zero. Śakratāraka ( शकतारक ) The nakṣatra Jyeṣṭha, whose regent is Indra. Sankragra (4) The dis- tance of the projection of a heavenly body on the plane of the celestial hori- zon from the planet's ris- ing-setting line. Šankvagrajivā ( शङ्कवग्रजीवा) Same as Sankvagra. Satabhisak (f¶¶) The nak- satra Satabhikhā. Sapharika (शफरिका) A fish- Sara () (1) Rversed-sine... figure. (2) Five. Śaši (f) (1) The Moon. (2) One. Sasija (af) Lunar. Salin (शालिन्) One. Sikhi (fafa) Three. Śilimukha (fî¶) Five. GLOSSARY Śiva (fa) Eleven. Śighra (¹) Same as Sighroc- ca. 253 Śitarasmi (f) (1) Moon. (2) One. Sitainsu (शीतांशु) (1) Moon. (2) One. Śighrakarṇa (¹) The dis- tance of a planet obtained by the sighrocca process. Sighrakendra (शीघ्रकेन्द्र) The sighra anomaly. See Kendra. Sighrakendraphala (शीघ्रकेन्द्रफल) Saila () Seven. Sighraphala, i.e., correction due to the sighrocca. Sodhana (a) Subtraction. Sighrajah karnah (शीघ्रजः कर्णः) = Sodhaniya (शोधनीय ) A subtrac- Sighrakarna. tive quantity technically called sodhaniya or sodhya. See MBh, i. 28. Sighranyāyāptacāpa (शीघ्रन्याया- प्तचाप) = Sighraphala. Sighraparidhi (reff) Śighra epicycle. Sighrapāta () See MBh, vii. 31. Sighravṛtta () Śighra epi- cycle. Śighrasiddha (af) Correc- ted for the sighraphala. Sighrantyajivā (शीघ्रान्त्यजीवा) The present Rsine-difference re- lating to the fighra (ken- dra). Śighrocca () See Ucca. Śighroccavṛtta (ga) Śighra epicycle. Sitakirana (f) (1) Moon. (2) One. Šukla () The illuminated part of the Moon's dise; the phase of the Moon. Srigonnati (af) The ele- vation of the Moon's horns. Śodhya (T) See Sodhaniya. Šauklya () The illuminated part of the Moon's disc. Śravaṇa (*) (1) Name of the 22nd nakṣatra. (2) The Hy- potenuse (of a right-angled triangle). Samskrta (a) Corrected... Sakalaguma (सकलगुण) Radius or 3438'. Sankalita (f) Sum, total. Sannati (fa) Meridian zenith distance. Sama () Even. Samamaṇḍala () The prime vertical. 254 GLOSSARY Samamandalaja chayā (सममण्डल- जा छाया) The prime vertical shadow. Samamandalasanlcu (सममण्डलशंकु) The Rsine of the prime vertical altitude. Samarekha () The prime meridian. Samalipta (समलिप्त ) Two planets are said to be samalipta ( समलिप्त ) when their longi - tudes are equal up to minutes. Samavalambajya (समवलम्बज्या ) The Rsine of the colati- tude. Sarvāpama ( सर्वापम) The grea- test declination (of the Sun), i. e., the obliquity of the ecliptic. Sagara () Four. Sārpamastaka (सार्पमस्तक) astronomical phenomenon. See MBh, iv. 35. Simha (f) The sign Leo. Sita (fa) (1) The illuminated part of the Moons's disc; the phase of the Moon. (2) The light half of the month. (3) Venus. Sitakhagá (¹) Venus. Sitapakṣa (f) The light half of a lunar month, light fortnight. Sitabindu (ff) That point of the Moon's diameter which lies at the end of the illuminated part of the Moon. Sitamāna (सितमान) The mea- sure of the illuminated part of the Moon's disc. Suranāthaguru (सुरनाथगुरु) Jupi- ter. Surapadilk (सुरपदिक्) East. Suredya () Jupiter. Suri (f) Jupiter. Surya () (1) Sun. (2) Twelve. Sūryakakṣyā (71) The orbit of the Sun, the ecliptic. Suryaja () Saturn. Saimhikeya (f) The ascen- ding node of the Moon's orbit. (Saimhikeya* liter- ally means Rahu, son of Simhikā). Somaja () Mercury. Somasunu () Mercury. Saumya () (1) North. (2) The nakṣatra Mrgasirā. (3) Mercury. Sauri (f) Saturn. Sthiti (ff) Duration (of an eclipse). Sthitidala (ff) Half the duration (of an eclipse). GLOSSARY Sthityardha (स्थिरपर्ध) Half the duration (of an eclipse). Sthula () Gross, approxi- mate. Sparsa (FT) Contact. Sparsakala (FT) Time of the first contact (in an eclipse). Spasta () True, corrected. Sphuta () True. Sphutamadhya () True- mean; the true-mean pla- net. Sphrutamadhyama (स्फुटमध्यम) Same as sphutamadhya. Sphutayojana (स्फुटयोजन) Used in the sense of sphutayojana- karna. 255 Sphutayojanakarna (स्फुटयोजनकर्ण) The true distance (of a planet) in terms of yojanas. Sphutavrtta (F) True or corrected epicycle. Svara (F) Seven. Harija () Horizon. Hara (gr) Divisor. Hararadi (हारराशि) Divisor. Himansu (हिमांशु ) ( 1 ) Moon. ( 2 ) One. Hina () (1) Less. (2) Omit- ted lunar day (hinadivasa). Hinadivasa ( हीनदिवस) Omitted lunar day. Hinaratra (T) Same as Hinadivasa. Hutāśana () Three.

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