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

RASHTRIYA SANSKRIT SANSTHAN, NEW DELHI
(Kendriya Sanskrit Vidyapeetha, Tirupati Series No. 29)
SIDDHĀNTA ŚIROMAṆI
OF
BHĀSKARĀCĀRYA
English Exposition and Annotation in the
light and language of modern Astronomy
By
Dr. D. ARKASOMAYAJI
Ex-Principal, D. N. R. College, Bhimavaram, A.P.,
Ex-Reader in Astronomy, Kendriya Sanskrit Vidyapeetha, Tirupati,
Recipient of President's Award.
योऽनूचानः स नो महान्
राष्ट्रियसंस्कृतसंस्थानम्
KENDRIYA SANSKRIT VIDYAPEETHA
TIRUPATI
1980

First Edition 1980
© All Rights reserved
Price Rs. 52/-
Published for the
RASHTRIYA SANSKRIT SANSTHAN
Ministry of Education, S. W. & Culture.,
Government of India,
New Delhi
By
Dr. M. D. BALASUBRAHMANYAM,
Principal, Kendriya Sanskrit Vidyapeetha,
TIRUPATI (A.P.)
Printed by the Rathnam Press, Madras-600 001.

राष्ट्रियसंस्कृतसंस्थानम्, नवदेहली
[ केन्द्रीयसंस्कृतविद्यापीठग्रन्थमाला, संख्या-२९ ]
भास्कराचार्यविरचितः
सिद्धान्तशिरोमणिः
डा० धूलिपाल० अर्कसोमयाजिकृतया आधुनिकखगोलशास्त्रानुसारिण्या
आङ्ग्लभाषाव्याख्यया अलंकृतः
संपादक :
डा० धूलिपाल० अर्कसोमयाजी
[प्रतीकचिह्नम् : योऽनूचानः स नो महान् / राष्ट्रियसंस्कृतसंस्थानम्]
केन्द्रीयसंस्कृतविद्यापीठम्, तिरुपतिः
1980

समर्पणम्
श्लो. 1. त्वत्पादाब्जदिदृक्षया 'हिमगिरिप्रस्यन्दिगङ्गाझरी
सङ्काशप्रवहज्जनावलिरहोरात्रं च जोघुष्यते ।
गोविन्देति विमुक्तकण्ठमसकृत् श्री वेङ्कटेशप्रभो !
त्वत्पादाब्जसमर्पिता कृतिरियं भक्त्या प्रसूनायताम् ॥
श्लो. 2. यः सिद्धान्तशिरोमणि र्विजयते श्रीभास्कराचार्यस्य सः
व्याख्यातो बहुभिस्तथाऽपि मयका व्याख्यायतेऽसौ पुनः ।
व्याख्येयं मणिदीधितिर्विरचिता यादृक्श्रमेण प्रभो !
जानीषे भगवन् ! त्वमेव तमिमां त्वत्पादयो र्पये ॥
श्लो. 3. पुराचार्यैः प्रोक्तं यदपि च नवीनै स्तदखिलम्
विचार्यैव व्याख्या व्यरचि भगवन् ! भास्करकृतेः ।
यदीयं व्याख्या मे सहृदयबुधाह्लादनकरी
तदाऽहं धन्यः स्यामियमपि कृषिर्मे सफलति ॥

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