सिद्धान्तशिरोमणि: गणिताध्याय (भास्कराचार्य - ग्रहगणित, मध्यमाधिकार व स्पष्टाधिकार सटीक)

D E D I C A T I O N English Translation of the Sanskrit verses

1. Oh ! Lord Venkateśvara ! I offer thee this work of mine as a flower at Thy feet, who art being approached day and night by throngs of people in a flow like that of the Ganga, jumping down the heights of the Himalayas, crying out full- throated one of Thy names ‘ Govinda ’ in shrill imploring voices, surging out from the depths of agonizing hearts !
2. The great work named Siddhānta Śiromaṇi of Bhāskarācārya, was indeed commented upon by a host of scholars both ancient and modern. Yet, I not a great scholar make bold to com- ment once again. Oh ! Lord Venkateśvara ! Thou alone Knowest and no mortal does, what stress and strain I underwent while working on this commentary. Hence I dedicate this at Thy feet alone not to any mortal, however great he may be !
3. This commentary has been written by me care- fully understanding the depths of both the ancient and modern systems of astronomy. If this could invoke the pleasure of well - meaning scholars, then I deem myself fortunate, and that my toil will have been well rewarded.

An accurate, line-by-line transcription of the text in pure Unicode Devanagari:

viii श्लो. 4 यो ग्रन्थो महता श्रमेण लिखितः प्रज्ञैर्विशेषैर्युतः यत्सारोऽपि विपश्चितां सुमनसां सूक्ष्मेक्षिकागोचरः । यः कश्चिद्यदि पिष्टपेषणमिदं स्यादित्यबुद्ध्या वदेत् यद्वा किञ्चिदवेक्ष्य सोऽत्र निहितान् भावान् कथं ज्ञास्यति ॥ श्लो. 5. यः सिद्धान्तशिरोमणिं न पठितः श्रीभास्करार्यस्य य- श्चऽऽत्मानं लघुपुस्तकेक्षणपरः चेत्पण्डितंमन्यते । रत्नोत्तुङ्गतरङ्गितं हिमगिरिप्रस्यन्दि गङ्गाजलम् हित्वा तेन सुपङ्किलं किल जलं स्नातुं समाश्रीयते ॥ व्याख्याता डा. धूलिपाळोल्ला अर्कसोमयाजी

ix 4. This work indeed sucked the very blood out of me. It contains many a new detail unnoticed hitherto. Only unbiased scholars could under- stand the essence of this work. If somebody pronounces unknowingly that this is all a repetition of what has been already written, simply having turned out a few pages, how could he know where I showed new aspects of the genius of Bhāskara ? 5. He, who does not study the Siddhānta Śiromaṇi of Bhāskarācārya, and feels that he is a scholar reading a few other sub-standard books on Hindu astronomy, does verily go to bathe in a dirty pond, ignoring the holy Ganga, jumping down the heights of the Himalayas in surging and dancing billows ! D. ARKASOMAYAJI

व्याख्यात्रभिजनः श्लो. 1. आन्ध्रेषु प्रथितं विराजति पुरं गोदावरीतीरगम् श्रीमद्राजमहेन्द्रमित्यभिहितं यस्यान्तिके भ्राजते । ग्रामः श्रीवलिचेरु नाम विबुधैर्देदीप्यमानस्सदा देवब्राह्मणयाबजूकविलसद्देशे जनिस्तत्र मे ॥ श्लो. 2. माता मे मङ्गमाम्बा सततमपि पतिं सेवमाना च दुर्गाम् देवी नित्यं भजन्ती निजतनूमनयत्सेव्यसेवां चरन्ती । विद्वान् बापाय्यनामा श्रुतिविहितपथे सञ्चरन् मे पिताऽऽसीत् मातापित्रो स्तयोर्मे पदभजनमहो बाल्य एव व्यरंसीत् ॥ 3. श्रीमद्वेङ्कट रामाख्यमग्रजं कर्मठं स्तुवे । प्राविक्षं वेदवेदाङ्गवाङ्मयं यत्पदान्तिके ॥ 4. द्वितीय मग्रजं वन्दे सुब्रह्मण्याह्वयं

BIOGRAPHICAL

1. In the state known as the Andhra Pradesh, there is a famous city called Rajahmundry on the banks of the holy Godāvarī. Not far away from this City, there is a village known Velicheru, teeming with Vedic and Sanskrit scholars. There I had my birth in a family reputed for a religious conduct.
2. My mother, by name, Mangamāmbā led her life both in the worship of my father, and her favourite Deity Kanaka Durgā. My father, a Vedic scholar conducted his life on a rigorous Vedic path. Alas! It was not given to me to serve them for long!
3. I bow to my eldest brother Śri Venkata Rāma who leads a rigorously religious life, and who it was, that initiated me into a study of the Vedic and Sanskrit lore.
4. I bow to my next elder brother by name Subrahmaṇya Somayāji who has passed away having spent his life in worshipping God and godly Brahmins.
5. Alas! Though I was initiated into the Vedic and Sanskrit lore in my boyhood, I was later distracted into the secular English education only to eke out my livelihood, as if this is the Summum Bonum of Life.

xii 6. मादृशाः कति वा विप्रकुलोद्भूताः श्रुतिस्मृतीः । हित्वा हन्ताङ्ग्लविद्यायां प्रविष्टाः कलिकालतः ॥ 7. श्रीनिवासमहं सेवे योऽत्र पर्वतमूर्धनि । येन स्वपादसेवाया आनीतः स्वपदान्तिकम् ॥ 8. यत्सेवा च दिवारात्रं धर्मपत्न्या कृता मम । आवामत्र निवासार्थमानयत् स्वगृहादिव ॥ 9. स्वस्ति तस्मै भवेद्राज्ञे रक्षितुं यो व्यवस्यति । पारावारगभीरं तद्वेदवेदाङ्गवाङ्मयम् ॥ व्याख्याता धूळिपाळोळ्ळा अर्कसोमयाजी

xiii 6. How many like me, born in Brahmin families, are being proselytized into the English system of education burying in the Bay of Bengal all that is grand in the Vedic and Sanskrit lore. 7. I raise my hands in supplication to Lord Venkateśvara, who took His abode on the Seven Hills as though to look down upon this mundane and irreligious world. It was He that has brought me here, to worship Him residing at His very feet. 8. I feel it was the deep devotion of my wife to Lord Venkateśvara that has brought us here, dislodging us from our home and hearth. 9. May those administrators flourish, who exert and strive to save the ocean—like Vedic lore from the yawning mouth of Oblivion ! D. ARKASOMAYAJI

FOREWORD Bhāskarācārya, the Second (c. 1100 A.D.) was one of the greatest mathematicians and astronomers of the World. He anticipated modern theory on the convention of signs (minus × minus = plus) and KEPLER’S method of determining the surface and the volume of a sphere. Six hundred years before the calculus of LEIBNITZ and NEWTON, Bhāskara worked at the differential coefficient. He raised and solved the problem — 67 x² + 1 = y² —a problem which FERMAT resolved after 500 years. The credit goes to Bhāskarācārya for having delineated the image of Jyotiṣa in its proper contours and graces. The Siddhāntaśiromaṇi is often said to be the magnum opus of Bhāskarācārya. This monumental treatise consists of four parts : (1) Līlāvatī (arithmetic), (2) bījagaṇita- (algebra), (3) Golādhyāya- (Trigonometry including spherical trigonometry) and (4) Grahagaṇita- (Planetary motion). This work was edited with the Vāsanābhāṣya- of the author by Bapu Deva SASTRI. Muralidhar JHA brought out two commentaries —the Vāsanāvārttika- of Nṛsiṁha (1621 A.D.) and the Marīci of Munīśvara (1635 A.D.) on the first Chapter of the Gaṇitādhyāya (1917). Girija Prasad DVIVEDI’s commentaries in Sanskrit and Hindi (volumes I & II) appeared in 1911 and 1926.

xvi Bapu Deva SASTRI and WILKINSON published an English translation of the text in 1861. Yet the mathe- matical aspect of the Siddhāntaśiromaṇi has remained a terra incognita to students of Hindu Astronomy. While Phalita-Jyotiṣa-continues to be studied in traditional Sanskrit institutions, the Gaṇita-side of Astronomy is reduced to a secondary position. Realising that specialists in the twin fields of Mathematics and Astronomy have been diminishing day by day, Kendriya Sanskrit Vidyapeetha at Tirupati has started a project entitled, "Coordination of Sanskrit and Ancient Indian Sciences". Under this scheme, Dr. Arka Somayaji has now come forward to give an exposition and annotation of the Siddhāntaśiromaṇi in simple English based on the language of modern Astronomy for the benefit of the students and scholars interested in the Khagola-śāstram. Dr. Arka Somayaji is an eminent scholar in Mathe- matics and Dynamic or Spherical Astronomy. He hails from a family of "Siddhāntin-s". To whet his appetite in learning the mnemonical methodology for compiling an almanac, he studied Mathematics and Jyotiṣa (includ- ing Spherical Astronomy). He learnt the Taittirīya- saṁhitā under his brother, Dhulipala Venkatarama Avadhani of Rajahmundry. He wrote a thesis on "A Critical Study of the Ancient Hindu Astronomy", which was published in 1972. He has to his credit eight works including the Jyotirvijñānam (1964) and two Sanskrit poetical compositions (Brahmañjali and the Hanumat- vijaya-). He won the President's award in 1974.

xvii In order that critical editions of rare and valuable texts on Jyotiṣa be brought out, the Tirupati Vidyapeetha appointed Dr. Arka Somayaji as Reader in Hindu Astronomy. Accordingly the Vidyapeetha has undertaken the publication of his English annotation of the Siddhāntaśiromaṇi. I trust this treatise will go a long way not only to project India's image in the World of Mathematics and Astronomy, but also inspire scholars from the transoceanic distance to listen to the jungle roar of the ancient Indian Wisdom. KENDRIYA SANSKRIT VIDYAPEETHA } M. D. BALASUBRAHMANYAM TIRUPATI October 21, 1980 Principal

PREFACE From the Mathematician's point of view, Bhāskarā- cārya's Siddhānta Śiromaṇi contains all that was beautiful in the Ancient Hindu Astronomy. I am aware that a translation of this work into English was done long ago by M. M. Bapudeva Śastry and Wilkinson; but I did not have the good fortune of having a copy in my hands all these years. I did go through the translation once long ago, but it gave me the impression that all the beauties there that appeal to a Mathematician were not brought out fully. There are, however, a good number of Sanskrit commentaries both ancient and modern but even they, in my humble opinion, have not done full justice to the elucidation of Bhāskara's mathematical genius. Further misinterpretations are not infrequent in some of those books, as will be pointed out in the course of this book. What has sponsored me to undertake to write an English commentary, (I may add that a Sanskrit commentary also has been written by me on this work and has been awaiting printing) is essentially that innumer- able modern professors of Mathematics complain many a time that there is no such a presentation of Siddhānta Śiromaṇi in English as will enable them to assess the Mathematical content of it in the light and language of modern Astronomy. Hence, I have sought to produce a fresh commentary, which I hope will meet the desire of such professors and students of Mathematics. I may add here that this work of mine seeks to present only the mathematical side of Siddhānta Śiromaṇi. It is no history of Hindu Astronomy, where the originality of the Ancient Hindu Astronomers is sought to be evaluated. I may confess that I am no historian.

xx In the course of this book, I shall have occasion to quote from most of the Astronomers of ancient India like Aryabhaṭa I, Varāhamihira, Lalla, Aryabhaṭa II, Brahmagupta, Bhāskara I, Muñjāla, Vateśvara and Śrīpati not to speak of some others, whose books are available in print or manuscript. One thought that lurks in my mind, I make bold to present in this preface. I say ‘I make bold’ because, there are some historians of Hindu Astronomy, who are too ready to attack me if I claim that there must have been not a primitive astronomical activity in ancient India prior to Aryabhaṭa I. The reason for their attack is that prior to Aryabhaṭa’s work, only one crude Vedāṅga Jyotiṣa has come to light. Also some of these historians have a strong impression that the galaxy of Hindu Astronomers ranging from Aryabhaṭa derived an incentive from a foreign source especially the Greek. If such historians agree to keep an open mind, I make bold to present my thought as follows. In my humble opinion there must have been considerable astronomical activity in ancient India even prior to Aryabhaṭa. This is borne out by the following expressions of Aryabhaṭa and others. Aryabhaṭa says¹ that he dived into the then extant astro- nomical lore, which got mixed up with mathematical and non-mathematical (mythological or otherwise) knowledge, and by his intellect and the grace of his Goddess, brought- out the truly mathematical. Varāhamihira² says that he was codifying the then extant five Siddhāntas, out of

1. सदसत्ज्ञानसमुद्रात् समुद्धृतं देवताप्रसादेन सद्ज्ञानोत्तमरत्नं मया निमग्नं स्वमतिनावा Verse 49—Golapāda
2. पूर्वाचार्यमतेभ्यो यद्यत् श्रेष्ठं लघुस्फुटं बीजम् तत्तदिहाऽविकलमहं रहस्यमभ्युद्यतो वक्तुम् Verse 2—Ch. I. पोलिशकृतः स्फुतोऽसौ तस्यासन्नस्तु रोमकप्रोक्तः स्पष्टतरः सावित्रः परिशेषौ दूरविभ्रष्टौ Verse 4—Ch. I.

which the ‘Sāvitra’ was more accurate. Brahmagupta³ says, again, that he was giving a clear presentation of the ancient Brahma Siddhānta which got obsolete by a long lapse of time. In the wake of these statements and a number of others, is it not right to construe that there did exist some ancient astronomical texts which are lost to us. If, as asserted by a host of modern interpreters, we think that there was only the crude Vedāṅgajyotiṣa, before Aryabhaṭa, does it not tantamount to saying that these three great astronomers (leave others) were impostors, who, having derived their knowledge from a foreign source, simply claimed that there were existent before them, Saura, Brāhma and some other Siddhāntas which dealt with Graha-gaṇita. It is uncharitable to say that three rational astronomers were sueh impostors. So, we must conclude that there did exist some astronomical activity, which we have no right to call primitive like the Vedāṅgajyotiṣa. Some might think that there might be existing some other texts in between the times of Vedāṅgajyotiṣa and Aryabha but that at the time of the Vedāṅgajyotiṣa⁴ (roughly 1180 B.C.) astronomy in India was that crude. Even this conclusion need not be

“I am going to tell clearly what has been, the best, what has been kept secret and briefly-worded in a nut shell from out of the works of ancient Acāryas”. “The Siddhānta of pauliśa is allright, the Romaka is very nearly the same, but the Sūryasiddhānta is more accurate, whereas the two others Vāsiṣṭha and paitāmaha are crude”. 3. ब्रह्मोक्तं ग्रहगणितं महता कालेन यत् खिलीभूतम् अभिधीयते स्फुटं तत् जिष्णुसुतब्रह्मगुप्तेन । Verse 2—Ch. I. 4. As per statement आश्लेषाधर्धात् दक्षिणमुत्तरमयनं रवेर्धनिष्ठाद्यम् of Varāhamihira quoting the meaning of the verse. प्रपद्येते श्रविष्ठादौ सूर्याचन्द्रमसावुदक् सार्पार्धे दक्षिणा......Verse 7—Vedāṅgajyotiṣa, the vernal equinoctial point was at the beginning of Dhaniṣṭha, which means that the time was 1180 B.C. approximately.

xxii correct, for the simple reason that, even today, crude works exist side by side with advanced works. The simple fact that a Vedāṅgajyautiṣa belonging to 1180 B.C. has been unearthed and nothing else, is not a complete proof that there did not exist more advanced texts some where else in such a big country like India , especially when locomotion or transport was difficult. Just one more thought, I place before the learned historians, which is pertinent to this context. According to geology, biology and some other similar modern sciences, the first man came into the cosmic picture some millions of years ago. If that be so, how is it that we say that man remained stupid all these years and happened suddenly to blossom into a genius one fine morning round about Eighteen fifties (1850 A.D.) whereafter came all Scientific discoveries in a series, as if by the waving of a magic wand ? It could not be so, as if, we are the chosen few of God. Civilizations must have been there, which got buried in the bosom of the earth, as has been revealed by the Mohenzadaro excavations. Hence we are not right in saying that the ancient Hindu civilization was so primitive at 1150 B.C. as is reflected in the Vedāṅga Jyotiṣa. This is another and more important reason that this author makes bold to place before the historians as to why he (the author) does have faith in the statements of Aryabhaṭa, Varāhamihira and Brahmagupta who said that they were peacing out the knowledge contained in the then extant works signifying at the same time that many more books were lost even to them. In this commentary of mine, I have chosen to keep the commentary away from the Sanskrit text and Bhās- kara’s own Vāsanā Bhāṣya, for, otherwise the book grows bulkily unwieldy. I have chosen to keep silent over passages which do not call for a mathematical elucidation. Here and there I have chosen to present what modern astronomy presents in some particular contexts, so that,

xxiii mathematics students who do not happen to study modern astronomy may have a better perspective of the ancient Hindu astronomy, presented in Juxtaposition to the corresponding modern treatment. However, I do not propose to enter into the intricacies of modern astronomy which are not called for to elucidate the text. Before concluding, it is my sacred duty to thank the Rāṣṭrīya Sanskrit Samsthan, under the Ministry of Education, Social Welfare and Culture, Government of India, for having accepted my work under the publication series of K. S. Vidyapeetha, Tirupati and the late- lamented Dr. M. Ananthasayanam Ayyangar, former Chairman of KSV and Ex-Governor of Bihar, who encouraged me in my studies on Hindu Astronomy. Dr. M. D. Balasubrahmanyam deserves my thanks for the encouraging interest he has shown in the publication of this annotation, D. ARKASOMAYAJI

C O N T E N T S Page Foreword ... xv Preface ... xix

1. सिद्धान्तशिरोमणौ गणिताध्याये मध्याधिकारे कालमानाध्यायः ... 1
2. Madhyādhikāra – Section II Bhagaṇādhyāya ... 18
3. Grahānayanādhyāya ... 29
4. Madhyādhikāra - Kakṣādhyāya ... 45
5. The Adhyāya known as Pratyabda Śuddhi in Madhyādhikāra ... 53
6. The Section known as Adhimāsādi- nirṇaya in Ch. I ... 70
7. The Section Bhū-Paridhi-Mānādikara — Circumference of the Earth ... 83
8. Spaṣṭādhikāra - Rectification of Planets ... 102
9. The Tripraśnādhikāra ... 223
10. Parvasambhavādhikāra ... 331
11. Lunar Eclipses ... 346
12. Grahacchāyādhikāra ... 449
13. Grahodayāstādhikāra ... 471
14. Sṛṅgonnatyadhikāra ... 482
15. Grahayutyadhikāra ... 501
16. Pātādhyāya ... 521 Appendix ... 545

ओं श्रीमहागणपतये नमः, श्रीगुरुभ्यो नमः सिद्धान्तशिरोमणौ गणिताध्याये मध्याधिकारे कालमानाध्यायः Verse 1. May the Sun at once give expression to our tongue in meaning ful words-that Sun, who rises to protect this world, whose duty it is to expel darkness, who is reported to be the Spouse of the lotus-creeper, who purges out the sins of all those that supplicate him, and on whose rising take place Vedic sacrifices and thereby the gods in heaven headed by Indra get feasted. Commentary (Comm.). Needs no Commentary. Verse 2. Excels that blessed Brahmagupta, the son of Jishnu, who is hailed as the crest-jewel of Mathemati- cians ; excel also those like Varāhamihira who were the authors of well known works, who were adepts in reason- ing, and u age of beautiful expression and on a study of whose works, one like me even of lesser intellect, will be able to produce monumental works. Comm. Needless to comment. However, we may state one thing. Bhaskara’s quoting the name of Brahma- gupta at the very outset, and that too with reverence, informs us that he is going to accept the Āgama of Brahmagupta in preference to that of others like Arya- bhata. This means that he is going to adopt the number of revolutions of planets in a Yuga and such other astro- nomical constants as were adopted by Brahmagupta. This point will be clarified later. Verse 3 This blessed author Bhaskara is now writing the work named Siddhānta-Siromani, the crest-jewel of all astronomical works, for the pleasure of good minded astronomers, after having bowed to the lotus-feet of his

2 father, from whom he has derived his knowledge; this work, containing good metres, will be easy to understand, besides being flawless and clear, and will enable intelligent readers to develop their ability to understand things. Comm. Not neeessary. Verse 4. Ancient astronomers did write, of course, works abounding in intelligent expression; nonethe- less, this work is started to give expression to some lacunae in their works. I am going to make amends for the deficiencies of the older works and these improvements will be found here and there in their respective places; So, I beseech the good-minded mathematicians to go through this entire work of mine also (for, otherwise, they may not locate my own contribution). Comm. Not necessary. Verse 5. May the good people be pleased with the particulars of my contribution! May the ill-minded people also derive pleasure out of ridiculing me with ignorance unable to understand my contribution! Comm. Not necessary. Verse 6. A Siddhānta work ie an astronomical trea- tise is such a one which deals with the various measures of time ranging from a Tṛti (to be explained shortly) upto the duration of a Kalpa which culminates in a deluge; planetary theory, arithmatical computations as well as algebraical processes, Questions with respect to intricate ideas and their answers, location of the earth, the star and the planets, and description and usage of instruments Comm. Not necessary. Verse 7. Though one knows astrology and that par of the science of Jyotisha which is known as Samhita (an which deals with various subjects like Muhurtas i.e. auspi

8 cious moments to be prescribed for various functions, Desarishtas ie Calamitous occurences to the countries etc.) which form a part of the Science, he cannot answer so many intricate problems pertaining to Astronomy. Such a person, who does not know the astronomical part of the science, which abounds in innumerable reasonings, is one like a king depicted in a drawing, or a lion fast tied to a pole. Comm. Not necessary. Verse 8. The Science of Jyautisha without Astronomy, is like a king's army without roaring elephants though ex- celling in horses etc; is like a garden without mango trees, or like a lake without water, or again like a lady parted with her newly married lover. Comm. Not necessary. Verse 9. The Vedic lore prescribes Sacrifices to be performed; these sacrifices are based upon a knowledge of appropriate time to perform them. This science of astronomy gives a knowledge of time; hence it has been reckoned as one of the six Vedāngas or limbs of the Veda. Comm. Not necessary. Verse 10. (Out of the six Vedangas) The science of grammar is like the face of the person of the Veda, the science of Jyautisha takes the place of the eyes, the Nirukta that of the ears; the Kalpa that of the hands; the Siksha that of the nose and the Chandas the place of the feet. Comm. Not necessary. Verse 11. This science of Jyautisha being depicted as the very eyes of the Person of the Veda, so it has been acclaimed as the most important of the six Angas or limbs of the Veda, in as much as, even if a person be endowed

24 with limbs like the ears, nose etc, if he be devoid of vision, he could not do anything. Comm. Not necessary. Verse 12. (Since this Science of Astronomy has been declared as the most important of the Vedangas) hence this has to be studied by the Dwijas (ie Brahmins Kshatri- yas and the Vaishyas who form the three higher castes), also because it is sacred, secret and the best discipline. By so doing they would acquire Dharma, Artha, Kāma as well as fame (Life is depicted as having a four-fold purpose out of which, Dharma, Artha and Kāma form the first trio, Moksha being the ultimate goal of life). Comm. Not neceasary. Verses 13 and 14. The creator having created the stellar circle along with the planets, placed the latter at the beginning of the circle, put them in constant revolution, at the same time putting the extreme two stars (on either side) in a fixed position. Comm. Bhaskara's own commentary Vāsanā Bhasha under this verse mentions the following points, which are to be noted. The twenty seven stars known as Aswini, Bharani etc. occupy positions roughly at equal distances along the Zodiac, arranged from west to east. The planets were all placed in the beginning of the stellar circle in such a way that they were in a straight line the moon, Mercury, Venus, Sun, Mars, Jupiter and Saturn occupying consecutive positions from the earth in increasing dist- ances, not of uniform measure. The circle of stars known as the Zodiac lies far behind all the planets. There is a wind known as pravaha which keeps the entire Zodiac, as well as the planets below going round and round in the Westerly direction. At the same time, the planets while participating in this westerly motion, have themselves individual motion towards the east. Two stars, are placed