ब्राह्मस्फुटसिद्धान्त (ब्रह्मगुप्त - शून्य, कुट्टक, बीजगणित एवं सम्पूर्ण २१ अध्याय सान्वय सटीक)

70 BRAHMASPHUṬASIDDHĀNTA & KHAṆḌAKHĀDYAKA
bhaṭa I's midnight system. " In this connection, it must be
remembered that Varāhamihira nowhere expresses his indebted-
ness to Āryabhaṭa I.
The Sūryasiddhānta undoubtedly is the most popular book
on Astronomy in this country. It has been so for the last 1000
years as is seen from the list of commentators on its text (K. S.
Sukla has given a list of 28 commentators includiug those who
wrote commentaries in the South Indian languges like Telugu and
Kannaḍa.) :
Allanarya Suri Maheśvara
Amaredya Mallikārjuna Sūri (1178 A. D.)
Bhaṭṭotpala (966 A. D.) Nārāyaṇa
Bhūdhara (1572 A. D.) Nrsinha Daivajña (b. 1586 A.)
Bhūtiviṣṇu Nṛsiṃhadeva
Caṇḍeśvara (1178) Parameśvara (1432 A. D.)
Cola Rāghava Śarmā (1592 A. D.)
Dādābhai (1719 A. D.) Rāmakṛṣṇa Ārādhya (1472 A. D.)
Devīdāsa Raṅganātha (1603 A. D.)
Kamala Kara Daneśvarah Sārvabhauma
(1618 A. D.)
Kāmabhaṭṭa Tamma Yajva (1599 A. D.)
Kṛṣṇa Daivajña Yallaya (1472 A. D.)
Madanapāla Viśvanātha (1628 A. D.)
Mādhavācārya
A large number of astronomical books in this country were
written on the basis of the Sūryasiddhānta, as the Gaṇakānanda
by Sūrya (1387-1447 A. D.), Gaṇitādarśa by Dharmapathin,
Makaranda Sāriṇī by Makaranda (1478 A D.), Grahacakra by
Kucanācārya (1299 A.D.), Viṣṇukaraṇa by Viṣṇu (1556 A.D.)
besides many others with indefinite dates such as the Sūrya-
siddhānta nayana-prakāraḥ, Sūryasiddhānta-gaṇita Sūryasid-
dhānta-saṃgraha by Viśvanātha Suri, and Sūryasiddhānta Sāriṇī
by Rāmadatta Daivajña. K. S. Sukla has given a list of this
literature in his Introduction to the Sūryasiddhānta.
(iv) Period of Bhāskara I : The author cr the compiler
ot the Sūryasiddhānta is not known nor its date of composition.
Bhāskara and Brahmagupta are the brilliant names of a contem-
porary period. Bhāskara I lived in the seventh century of the
Christian era and was a contemporary of Brahmagupta (628

DIVISION OF MAHĀBHĀSKARĪYA 71
A.D.). He wrote three works on Astronomy which were most
likely composed in the following order : (i) the Mahā-Bhāskarīya,
(ii) a commentary on the Āryabhāṭīya, and (iii) the Laghu-
Bhāskarīya, His commentary on the Āryabhāṭīya was written
in 629, i.e., only one year after the Brāhmasphuṭasiddhānta. His
commentary on the Āryabhaṭīya was written in 629 A.D., i.e.
one year after the completion of the Brāhmasphuṭasiddhānta.
Undoubtedly Bhāskara was the follower of Āryabhaṭa I. Shukla
says, that his works provide us with a detailed exposition of the
astronomical methods taught by Āryabhaṭa I and throw light on
the development of Astronomy in India during the sixth and
early seventh centuries A. D. which was the most brilliant
period in the history of Indian Astonomy-Shukla has brought a
critical edition with English translation of the Mahābhāskarīya
and the Laghu-Bhāskarīya and he proposes to bring out a volume
on the life and works of this great astronomer,
Division of Mahābhāskarīya
The Mahābhāskarīya is divided into Chapters called the
adhyāyas. The adhyāyas have not been named as some of the
Chapters of the Sūryasiddhānta or the Brāhmaspuhtasiddhānta.
The number of verses in the Mahābhāskarīya is as follows :
Subject Number of
verses
Adhyāya I Mean Longitude of a
Planet and Planetary
Pulveriser 52
Adhyāya II The Longitude Correction 10
Adhyāya III Direction, Place and Time.
Junction-stars of the
Zodiacal Asterisms and
conjunction of Planets with
them. 75
Adhyāya IV The Longitude of a Planet 64
Adhyāya V Eclipses 78
Adhyāya VI Rising, Setting and Con-
junction of Planets 62
Adhyāya VII Astronomical Constants 35
Adhyāya VIII Examples 27
These titles to the subject-matter given in the above table
have been assigned by Dr. Kripa Shanker Shukla in his critical

72 BRAHMASPHUṬA SIDDHĀNTA & KHAṆḌKHĀDYAKA
edition. The total number of verses in this work is 403. The
Sūryasiddhānta has, as stated above, 500 verses in all.
Laghu-bhāskarīya
The Laghu-Bhāskarīya is also divided into eight chapters,
each chapter is known as Adhyāya. The subject matter in the
book has been dealt with as follows :
Subject Number of
verses
Adhyāya I Mean longitudes of the
Planets 37
Adhyāya II True Longitudes of the
Planets 41
Adhyāya III Direction, Place and
Time from Shadow 35
Adhyāya IV The Lunar Eclipse 32
Ādhyāya V The Solar Eclipse 15
Adhyāya VI Visibility, Phases and
Rising and Setting of
the Moon 25
Adhyāya VII Visibility and Con-
junction of the Planets 10
Ādhyāya VIII Conjunction of a Pla-
net and a Star 19
The total number of verses in this text is 214. The Laghu-
Bhāskarīyam is thus an abridged edition of the Mahā-Bhāskarīyam.
From the closing stanza of this work, it is clear that the author
wrote this work for the benefit of young students with immature
mind by condensing and simplifying the contents of his larger
work, the Mahā-Bhāsarīyam (also known as Karma-nibandha).
Thus we have a little of the parallelism: Brahmagupta, after
finishing his bigger treatise, the Brahma-sphuṭasiddhānta, wrote
a minor abridged work, the Khaṇḍa-khādyaka, as a karaṇa
grantha. This latter work of Brahmagupta, however, incorporates
some original ideas not included in the earlier work. Shukla
has given an analytical table indicating the rules of the Mahā-
Bhāskarīyam incorporated in the Laghu-Bhāskarīyam also in an
abridged or modified form, and also a list of the rules which have
been omitted in the Laghu-Bhāsakrīya. There are a few rules
in the Laghu-Bhāskarīya also which have no counterpart in the

LAGHU-BHĀSKARĪYA 73

Mahā-Bhāskarīya. Shukla further says that the arrangement
of the contents of the Laghu-Bhāskarīya is more systematic and
logical than that of the Mahā-Bhāskarīya, and is, at the same
time, in keeping with the general practice followed by the other
Hindu astronomers. Numerous quotations of this work occur
in the annotative works of Sūryadeva (b. 1191 A.D.), Yallaya
(1480 A.D.), Nīlakaṇṭha (1500 A.D.), Raghunātha Rāja (1597
A.D.), Govinda Somayājī and Viṣṇu Śarmā and in the Prayoga-
racanā, an anonymous commentary on the Mahā-Bhāskarīyam.
We find the commentaries of this abridged work in Malayalam
and Tamil also. All this speaks of the great popularity of this
work.
There are circumstantial evidences to show that Bhāskara I
had associations with the countries of Aśmaka and Surāṣṭra.
His commentary on the Āryabhaṭīya was probably written in
the city of Valabhī in Surāṣṭra. Perhaps Bhāskara I was born
and educated in Aśmaka and later on he migrated to Valabhī,
where he wrote his commentary on the Āryabhaṭīya or that he
was a native of Valabhī and got his education in the Aśmaka
country. Perhaps there was a strong school of Astronomy in
the Aśmaka country, which was founded by the followers of
Āryabhaṭa, so much so that at places, Bhāskara I has also called
Āryabhaṭa as Aśmaka, his Āryabhaṭīyam by the name Aśmaka-
Tantra or the Āśmakīya and the followers of Āryabhaṭa as
Āśmakīyāḥ. This Aśmaka country or Aśmaka Janapada is men-
tioned in the Buddhist literature also. It was somewhere either
in the north-west of India, or was situated between the rivers
Narmadā and Godāvarī. Bhāskara I was evidently a resident
of the latter Aśmaka (which was between the Narmadā and
Godāvarī).
Brahamagupta and Bhāskara I were contemporaries. Both
of them developed their systems in the earlier part of the seventh
century A.D. (3700 years of Kaliyuga). Brahāmasphuṭasiddhānta
was written in 628 A.D. and the commentary on the Ārya-
bhaṭīyam by Bhāskara I was composed in 629 A. D. Bhāskara
closely followed Āryabhaṭa, but Brahmagupta had the guts to
oppose the views expressed by this great master and he not only
contradicted him at places, but also propounded many new ideas,
methods of calculation and constants of greater accuracy.

74 BRĀHMASPHUṬA SIDDHĀNTA & KHAṆḌAKHĀDYAKA
The classification of the contents of Astronomy in adhikāras
appears to be the original concept of Brahmagupta; this system
was to some extent adopted in the modern Sūrya-Siddhānta
in the case of a few chapters. The Vaṭeśvara-Siddhānta by Vaṭeś-
varācārya (born 802 Śaka or 880 A.D. in Ānandapura city,
Punjab, son of Mahādatta) also adopts the terminology : Madhya-
madhikāra, Spaṣṭādhikāra and Tripraśnādhikāra. We owe this
type of caption-nomenclature to Brahmagupta.
Contents of the Brahmasphuṭasiddhānta
Now we shall summarise the contents of the Chapters of
this great treatise and also enumerate the number of verses in
each chapter. The author has himself given the total number of
verses in the Chapter in the ending verse of each chapter. Some-
times, the verse specifying this number itself is not taken into
account while giving the total number of verses in that chapter,
and therefore, there occurs a minor discrepancy in the actual
number and the number specified by the author himself. The
following table records both these numbers separately :
${cccc} & & Number of & Actual num-
Chapter & Title & verses indi- & ber of verses
& & cated by the &
& & author & $
PŪRVA-DAŚĀDHYĀYĪ (First Ten
Chapters)
$I$ \quad Madhyamādhikāraḥ (Madh-
\qquad yamagati-radhyāyaḥ) \qquad\qquad\qquad\quad 63 \qquad\qquad\quad 63
$II$ \quad Spaṣṭādhikāraḥ (Sphuṭagati-
\qquad radhyāyaḥ) \qquad\qquad\qquad\qquad\qquad 67 \qquad\qquad\quad 68
$III$ \quad Tripraśnādhikāraḥ (Tripraś-
\qquad nādhyāyaḥ) \qquad\qquad\qquad\qquad\quad 66 \qquad\qquad\quad 66
$IV$ \quad Candragrahaṇādhikāraḥ
\qquad (Candragrahaṇādhyāyaḥ) \qquad\qquad\quad 20 \qquad\qquad\quad 20
$V$ \quad Sūryagrahaṇādhikāraḥ
\qquad (Ārka-grahaṇam or Ravigra- \qquad\quad 26 \qquad\qquad\quad 27
\qquad haṇādhyāyaḥ)
$VI$ \quad Udayāstādhikāraḥ
\qquad (Udayāstamayādhyāyaḥ) \qquad\qquad\quad 12 \qquad\qquad\quad 13

CONTENTS OF THE BRĀHMASPHUṬASIDDHĀNTA 75
VII Candraśṛṅgonnatyadhikāraḥ 18 18
(Candraśṛṅgonnati-adhyāyaḥ)
VIII Candracchāyādhikāraḥ 9 9
(Candracchāyā-adhyāyaḥ)
IX Grahayutyadhikāraḥ 26 26
(Grahamelanādhyāyaḥ)
X Bhagrahayutyadhikāraḥ 70 70
(Bhagrahayutiḥ-adhyāyaḥ)
——— ———
Total (of Daśādhyāyī) 377* 380*
XI Tantra-Parīkṣādhyāyaḥ 63 63
(Dūṣaṇādhyāyaḥ)
XII Gaṇitādhyāyaḥ
Miśraka-vyavahāraḥ 1—16
Śreḍhī-vyavahāraḥ 17—20
Kṣetra-vyavahāraḥ 21—39
Vṛttakṣetra-gaṇitam 40—43
Khāta-vyavahāraḥ 44—46
Citi-vyavahāraḥ 47—
Kākacika-vyavahāre
karaṇasūtre 48—49
Rāśi-vyavahāraḥ 50—51
Chāyā-vyavahāraḥ 52—66 66 66
XIII Praśnādhyāyaḥ 49 48
(Madhyagatyuttarādhyāyaḥ)
XIV Sphuṭagatyuttarādhyāyaḥ 54 55
XV Tripraśnottarādhyāyaḥ 60 60
XVI Grahaṇottarādhyāyaḥ 46 47
XVII Śṛṅgonnatyuttarādhyāyaḥ 10 10
XVIII Kuṭṭādhyāyaḥ 103 102
(Kuṭṭakādhyāyaḥ)
Kuṭṭākāraḥ 1—29
Dhanarṇa-śūnyānām
saṁkalanam 30—42
Ekavarṇa.samīkaraṇa-
bījam 42—50
Anekavarṇa-samīkaraṇa-

76 BRAHMASPHUṬA SIDDHĀNTA & KHAṆḌAKHĀDYAKA
bījam 51—59
Bhāvita-bījam 60—63
Varga-prakṛtiḥ 64—74
Udāharaṇāni 75—102
XIX Śaṅkucchāyādijñanādhyāyaḥ 20 20
XX Chandaścityuttarādhyāyaḥ 20 19
XXI Golādhyāyaḥ 70 70
Sāmānya-golaprakara-
ṇam 1—16
Jyā-prakaraṇam 17—23
Sphuṭagati-vāsanā 24—30
Bimba-sādhanam 31—35
Grahaṇa-vāsanā 36—48
Golabandhādhikā-
raḥ 49—70
XXII Yantrādhyāyaḥ 53 57
XXIII Mānādhyāyaḥ 12 12
XXIV Sañjñādhyāyaḥ 13 13
Total 1016 1022
In one of the verses, Brahmagupta states that he has composed
the treatise containing 1008 verses. Sudhākara Dvivedī has
given the total as 1021, whereas he says, this number according
to Brahmagupta's own statement should be 1020. If one de-
ducts the concluding 12 verses of the Sañjñānadhyāya, the num-
ber should be 1008.
Sudhākara Dvivedī, in his addition of the Brahma-sphuṭa-
siddhānta (published in the Paṇḍita, 1901 and 1902) gives as a
supplement a small treatise of Brahmagupta known as Dhyāna-
grahaṇopadeśādhyāya or Dhyānagrahaṇādhikāraḥ which has 72
verses.
It would be worthwhile to give here the details of the
Khaṇḍakhādyaka also, a book of Brahmagupta about which we
have spoken so much. The titles to the chapters have not been
indicated in the Text ; most likely they have been assigned by
the commentator, Pṛthūdaka Svāmī known as the Khaṇḍakhā-
dyaka-vivaraṇam.

CONTENTS OF THE BRĀHMASPHUṬASIDDHĀNTA 77
Chapter Title Number of
verses
I Tithi-nakṣatrādhikārādhyāyaḥ 32
(On tithis, nakṣatras etc.)
II Grahagatyadhyāyaḥ 19
(On the mean and true places
of 'star' planets)
III Tripraśnādhyāyaḥ 16
(On the three problems relat-
ing to diurnal motion)
IV Candragrahaṇādhyāyaḥ 7
(On lunar eclipses)
V Suryagrahaṇādhyāyaḥ 6
(On solar eclipses)
VI Udayāstādhikāraḥ 7
(On the rising and setting of
planets)
VII Candraśṛṅgonnatyadhyāyaḥ 4
(On the position of the
Moon's cusps)
VIII Samāgamādhyāyaḥ 6
(On conjunction of planets)
UTTARA KHAṆḌAKHĀDYAKA—APPENDIX
IX Corrections and new methods 14
X On conjunction of stars and planets 16
Total 127
Bhaṭṭotpala, in his commentary on the Khaṇḍakhādyaka has
given several additional verses in the main or proper treatise
and also in its Uttara portion or the Appendix. P.C. Sengupta's
edition (Sanskrit Text 1941) has given at the end of this pub-
lication the account of these additional verses. The English

78 BRAHMASPHUṬA SIDDHĀNTA & KHAṆḌAKHĀDYAKA
edition (1934) classifies the Uttara-Khaṇḍakhādyaka into two
chapters (which the author calls as Chapters IX and X), the
Sanskrit edition gives 3 verses in Chapter IX, 21 verses in
Chapter X and 24½ verses in Chapter XI. Of these three
chapters, the Chapter X has been given the title "Pātādhikāra"
and Chapter XI the title "Parilekhādhyāya" by Bhaṭṭotpala.
T A B L E
Arrangement of contents in different treatises

ĀRYABHAṬA AND BRAHMAGUPTA CONTROVERSY 79

1. प्रणिपत्य महादेवं जगदुत्पत्तिस्थितिप्रलयहेतुम् ।
   वक्ष्यामि खण्डखाद्यकमाचार्यार्य्यभटतुल्यफलम् ॥

60 BRAHMASPHUṬA SIDDHĀNTA & KHAṆḌAKHĀDYAKA
treatise (i.e. Khaṇḍakhādyaka, literally meaning food
prepared from sugar-candy) is made so as to yield the
same results as that.¹
The mean saturn diminished by 3 seconds, the Śīghrocca
of Mercury diminished by 22 seconds, the mean Mars
increased by 2 seconds and the mean Jupiter increased
by 4 seconds are equal to the respective mean planets
of Āryabhaṭa's midnight system.²
In the Brāhmasphuṭasiddhānta, Brahmagupta accepts the
astronomical day to begin with the sunrise at Laṅkā, and the
calculations of days, months, years, Yugas, and Kalpas all begin
from Caitra Śukla Pratipadā (the first tithi of the month Caitra
in the bright-half of the Moon) and the first day is regarded as
Sunday.³
Varāhamihira in his epicyclic cast to the Sūryasiddhānta
in his Pañcasiddhāntikā adopts the ardharātrika system or the
system of reckoning days from midnight. The question why
Brahmagupta who was so bitter an opponent of Āryabhaṭa I
in his younger days (628 A.D.) claimbed down to describe and
teach one of the systems of Āryabhaṭa's astronomy in his sixty-
seventh year (665 A.D.) is difficult to explain. In fact so great
was Āryabhaṭa's reputation and fame that in spite of Brahma-
gupta's severe ricticisms of the former in Chapter XI of the
Brāhmasphuṭasiddhānta, it perhaps was undiminished and it
was Āryabhaṭa who continued to be universally followed.
Some authorities have thus expressed the view that to meet
the popular demand Brahmagupta in the Khaṇḍakhādyaka took
upon himself the task of simplifying Āryabhaṭa's ardharātrika
system and in this task he became eminently successful. But it
has been supposed that in this task he could not be a mere
simplifier or expounder.
प्रायेणार्यभटेन व्यवहारः प्रतिदिनं यतोऽशक्यः ।
उद्वाहजातकादिषु तत्समफल लघूक्तरोक्ति रतः ॥
तिसृभिः शनिज्ञैशीघ्रं द्वाविंशत्या कुजोऽधिको द्वाभ्याम् ।
चतसृभिरधिको जीवोऽर्द्धरात्रिकार्य्यभट मध्य समाः ॥
—KK. I. 1,2,7
☛ चैत्रसितादेरुदयाद्भानोर्दिनमासवर्षयुगकल्पाः ।
सृष्ट्यादौ लंकायां समं प्रवृत्तः दिनेऽर्कस्य ॥
—BrSpSi I.4

ARYABHAṬA & BRAHMAGUPTA CONTROVERSY 81
The minor work of Brahmagupta known as Khaṇḍakhādyaka
has two distinct parts : Khaṇḍakhādyaka proper and the Uttara
Khaṇḍakhādyaka. In the first part the astronomical constants
are the same as those of Āryabhaṭa's ardharātrika system, but
the methods of spherical astronomy, calculation of eclipses and
other topics are almost the same as in the Brāhmasphuṭasidd-
hānta. The corrections for parallax in calculating a solar eclipse
is here an important illustration.¹
In the Uttarakhaṇḍakhādyaka, Brahmagupta gives correc-
tions to the Khaṇḍakhādyaka proper. In it are to be found the
neat and original methods of interpolation and correction to the
longitudes of the aphelia, as also to the dimensions to the
epicycles of apsis of the Sun and the Moon.² while a few addi-
tional chapters supply what else is necessary to the first seven
chapters of the first part, to make the whole a complete treatise
on Indian scientific astronomy. It was perhaps through the in-
fluence of this supplementary part of the Khaṇḍakhādyaka that
Brahmagupta's great work, the Brāhmasphuṭasiddhānta came to
be valued among a distinct school of Indian astronomers. For
long in this country India, this Siddhānta of Brahmagupta has
been forming the basis for the calculation of almanacs by astro-
nomers of the orthodox school of Rājasthān, Bombay and others.
We might at this stage take up the question : Was
Āryabhaṭa the author of two distinct systems of astronomy ?
Undoubtedly he was. Several authors have written on this
subject. I may specially mention the name of Prabodha
Chandra Sengupta (Journal of the Department of Letters,
Calcutta University, vol. XVIII; Bulletin, Calcutta Mathemati-
cal Society, vol. XXII. Nos. 2 and 3). The reasons advanced by
him may be restated in slight details thus. In his Brāhma-

1. व्यासार्धेन विभक्ता दृग्गतिजीवा चतुर्गुणा लब्धम् ।
   लम्बननाड्यः पञ्चदश गुणिताया त्रिज्यया भक्ता ॥
   दृक्क्षेपज्या मुहुत्यन्तरा हता लब्धमवनतिर्भवति ।
   स्फुटयोजनकर्णाभ्यां भू-यासेन च विना स्फ०टे ॥
   आर्यभटेनास्मिन् सति लघुनि किमर्थं महत् कृतं कर्म ।
   गणिताज्ञानाज् जाड्यं विजानता यदि ततः सुतराम् ॥
   —BrSpSi. XI. 23-25, also KK. V.
2. See UKK. 9.

82 BRAHMASPHUṬA SIDDHĀNTA & KHAṆḌAKHĀDYAKA
sphuṭasiddhānta, Brahmagupta thus speaks of the two works of
Āryabhaṭa:
As in both the works the number of the Sun's revolutions
is spoken of as 432,000 years, their planetary cycle is
clear, i. e., of 4,320,000 years. Why then is there
difference of 300 civil days in the same cycle of the
two books ?¹
Again, he says :
In 14,400 years elapsed of the Mahāyuga, there is produced
a difference of one day in the audayika and ardharāt-
rika systems.²
Varāhamihira in the Pañcasiddhāntikā writes -
Āryabhaṭa maintains that the beginning of the day is to
be reckoned from midnight at Laṅkā; and the same teacher
again says that the day begins from sunrise at Laṅkā. ³
Thus from the writings of Brahmagupta and Varāhamihira, it
is clear that Āryabhaṭa I was the author of both the audayika
and ardharātrika systems of astronomy. In Varāhamihira's
verse the phrase sa eva (स एव), meaning "he undoubtedly" is of
special significance. It removes the least doubt as to Ārya-
bhaṭa's authorship of both these systems. The audayika and
ardharātrika astronomical constants are respectively to be found
from the Āryabhaṭīya and may be deduced from the Khaṇḍa-
khādyaka as well. The following is the comparative view of the
constants of Varāhamihira and of the present day Sūrya-
siddhānta.
TABLE I
Planetary revolutions in a mahāyuga of 4,320,000 years,
according to various authorities.

1. युगरविभगणाः रव्यूधृति यत् प्रोक्तं तत् तयोर्यु गं स्पष्टम् ।
   त्रिशती रव्युदयानां तदन्तरं हेतुना केन ॥
   —BrSpSi XI. 5
2. अधिकैः शतैश्चतुर्भिर्वर्ष सहस्रैश्चतुर्दशभिरेकः ।
   युगायातैर्दिनवारान्तर मौदयिकार्ध रात्रिकयोः ॥
   —BrSpSi XI. 13
3. लङ्कार्द्धरात्रसमये दिनप्रवृत्तिं जगाद चार्य्यभटः ।
   भूयः स एव सूर्योदयात् प्रभृत्याह लङ्कायाम् ॥
   —PSi. XV. 20

ĀRYABHAṬA & BRAHMAGUPTA CONTROVERSY 83
Planet BrSpSi Āryabhaṭīya Khaṇḍa- Varāha- Later or
khādyaka Sūrya- modern
siddhānta Sūrya-
(PSi.) siddhānta
Moon 57,753,300 57,753,336 57,753,336 57,753,336 57,753,336
Sun 4,320,000 4,320,000 4,320,000 4,320,000 3,320,000
Mars 2,296,828.522 2,296,824 2,296,824 2,296,824 2,296,832
Jupiter 364,226.455 364,224 364,220 364,220 364,220
Saturn 146,567.298 146,564 146,564 146,564 146,568
Moon's
apogee — 488,219 488,219 488,219 488,203
Venus 7,022,389.492 7,022,338 7,022,388 7,022,388 7,022,376
Mercury 17,936,998.984 17,937,020 17937,000 17,937,000 17,937.060
Moon's
nodes 232,311.168 232,226 232,226 232,226 232,238
TABLE II
Longitudes of the apogees of the orbits of Planets.
Āryabhaṭīya Khaṇḍa- Varāha Modern
Planets khādyaka Sūrya-sid- Sūrya-siddh-
dhānta. ānta
Sun 78° 80° 80° 77° 07′
Mercury 210° 220° 220° 220° 26′
Venus 90° 80° 80° 79° 49′
Mars 118° 110° 110° 130° 00′
Jupiter 180° 160° 160° 171° 16′
Saturn 236° 240° 240° 236° 37′
TABLE III
Dimensions of the epicycles of Apsis
Planets Ārybhatīya Khaṇḍa- Varāha. Modern
khādyaka Sūrya- Sūrya-
siddhānta siddhānta
Sun 13°-30′ 14° 14° 13½°-14°
Moon 31°-30′ 31° 31° 31½°-32°
Mercury 22½°-31½° 28° 28° 28°-30°
Venus 9°-18° 14° 14° 11°-12°
Mars 63°-81° 70° 70° 72°-75°
Jupiter 31½-36½° 32° 32° 32°-33°
Saturn 40½-58½ 60° 60° 48°-49°

84 BRĀHMASPHUṬA SIDDHĀNTA & KHAṆḌAKHĀDYAKA
Table IV
Dimensions of the Śīghra epicycles (i.e. conjunctions)
Planet Āryabhaṭīya Khaṇḍa- Varāha Modern
Khādyaka Sūrya- Sūrya-
siddhānta siddhānta
Saturn 36½°- 40° 40° 40° 39°- 40°
Jupiter 67½°- 72° 72° 72° 70°- 72°
Mars 229½°-239½° 234° 234° 232°-235°
Venus 256½°-265½° 260° 260° 262°-262°
Mercury 130½°-139½° 132° 132° 132°-133°
Table V
Longitudes of the nodes of the orbits of planets
Planets Āryabhaṭīya Khaṇḍa- Varāha Modern
khādyaka Sūrya- Sūrya-
siddhānta siddhānta siddhānta
Saturn 40° 40° Not stated Have to be
Jupiter 20° 20° in the calculated
Mars 80° 80° Text from the
Venus 60° 60° data of the
Mercury 100° 100° text
Table VI
Orbital inclinations (geocentric) to the ecliptic
Planets Āryabhaṭīya Khaṇḍa- Varāha Modern
khādyaka Sūrya- Sūrya-
siddhānta siddhānta
Mars 90' 90' 10' 90'
Mercury 120' 120' 135' 120'
Jupiter 60' 60' 101' 60'
Venus 120' 120' 101' 120'
Saturn 120' 120' 135' 100'
The Mahābhāskarīya of Bhāskara I (522 A. D.) contains
a passage which corroborates the fact that Āryabhaṭa I was the
author of both the audayika and the ardharātrika systems of
Indian Astronomy. According to Pṛthūdakasvāmin, whose

ĀRYABHAṬA & BRAHMAGUPTA CONTROVERSY 85
commentary on the Brāhmasphuṭasiddhānta we have the
privilege of presenting to the public, it is clear that in certain
respects Bhāskara and others may be wrong but the Āryabhaṭa's
authenticity cannot be questioned. Pṛthūdakasvāmin while
commenting on the Brāhmasphuṭasiddhānta, XI. 26 writes:
Such a mistake may have been made by Bhāskara and
others; they have not understood his (Āryabhaṭa's)
intention.
The passage in the Mahābhāskarīya giving constants of the
ardharātrika system runs as follows (we are giving the translation
from Kripa Shankar Shukla's edition on the Mahābhāskarīya) :
The astronomical processes which have been set forth
above come under the sunrise day - reckoning (audayika
system). In the midnight day-reckoning (ardharātrika
system) too, all this is found to occur: the difference that
exists is being stated (below). ¹
The next fourteen stanzas relate to the midnight day-
reckoning of Āryabhaṭa I.
(i) Civil days and omitted lunar days in a yuga and
revolution numbers of Mercury and Jupiter are thus
given :
(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 śīghrocca of) Mercury and Jupiter
subtract 20 and 4 respectively. ²
Thus according to the midnight day-reckoning, we get
civil days in a yuga = 1,577,917,800
omitted lunar days in a yuga = 25,082,280
revolution number of the śīghrocca of
Mercury = 17,937,000
revolution number of Jupiter = 364,220

1. निबन्धः कर्मणां प्रोक्तो योऽसावौदयिको विधिः ।
   अर्धरात्रे त्वयं सर्वो यो विशेषः स कथ्यते ॥ —MBh. VII. 21
2. त्रिशती भूर्दिने क्षेप्या ह्यवमेभ्यो विशोध्यते ।
   ज्ञ गुर्वोर्भगणेभ्योऽपि विंशतिश्च ततोऽब्धयः ॥ —MBh. VII. 22

86 BRAHMASPHUṬASIDDHĀNTA & KHAṆḌAKHĀDYAKA
(ii) Diameters of the Earth, the Sun and the Moon are
thus given :
(In the midnight day-reckoning) the diameter of
the Earth is ( stated to be ) 1,600 yojanas; of the Sun
6,480 (yojanas) and of the Moon, 480 (yojanas)¹.
(iii) Mean distances of the Sun and the Moon are as
follows :
The (mean) distance of the Sun is stated to be 689,358
(yojanas), and of the Moon 51,566 (yojanas).²
(iv) Longitudes of the apogees of the planets are as follows:
160, 80, 240, 110, and 220 are in degrees the longitudes
of the apogees of Jupiter, Venus, Saturn, Mars and
Mercury respectively.³
(v) Manda and Śīghra epicycles of the planets are as
follows :
The Manda epicycles ( of the same planets ) are 32, 14,
60, 70, and 28 (degrees) respectively; and the Śīghra
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). ⁴
(vi) The positions of the so called manda and śīghra pātas
of the planets are given below:
( The following directions for) the degrees of the
(manda and śīghra) pātas of the planets as devised

1. अष्टिशतगुणा व्यासो योजनानां भुवो रवेः ।
   खाष्टाब्ध्यक्ष्यङ्गानि शीतांशोः शून्यवस्वब्धयस्तथा ॥ —23
2. वस्विन्द्रिय गुणच्छिद्रवस्वङ्गानि विभावसोः ।
   अङ्गाङ्गेष्वेक भूतानि चन्द्रकर्णः प्रकीर्तितः ॥ —24
3. अष्टिरष्टौ जिनारुद्रा विंशतिर्द्वयधिकाः क्रमात् ।
   दशघ्ना गुरुशुकार्कि भौमज्ञांशाः स्वमन्दजाः ॥ —25
4. मन्दवृत्तानि द्वात्रिंशन्मनवः षष्टिरेव च ।
   खाद्रयो वसुदस्त्राः स्युः शीघ्रवृत्तान्यथ क्रमात् ॥ —26
   द्वयद्वयः खाङ्कनेत्राणि खाब्धयोऽब्ध्यग्निदस्रकाः ।
   द्वयग्नीन्दवो रवेर्मन्दं शुक्रवद् वृत्तमेव च ॥ —27
   एकत्रिंशत्क्षपाभर्तु र्धरात्रे विधीयते ।
   —MBh. VII 26-28 (i)

ĀRYABHAṬA & BRAHMAGUPTA CONTROVERSY 87
(under the midnight day-reckoning) should be
noted carefully by learned scholars :
Add 180° to the longitudes of the mandoccas
(apogees) apd śīghroccas of Mercury and Venus, and
subtract 3 signs from the mandoccas (apogees) and
śīghroccas of the remaining planets. Then are
obtained the longitudes of the manda and śīghra pātas
of the planets. (Also) add 2 degrees to the longitudes
of the manda pātas and śīghroccas of Venus, Saturn
and Jupiter, and 1½ degrees to those of Mars and
Mercury. (It should be noted that) the śīghra pātas
have been stated for all the planets excepting Mercury.
(Mercury does not have a śīghra pāta).¹
That is to say, the longitudes of the manda pātas of Mars,
Mercury, Jupiter, Venus, and Saturn are 21.5, 41.5, 72, 262 and
152 degrees respectively; and the longitudes of the śīghra pātas of
Mars, Jupiter, Venus and Saturn are (śīghrocca--88.5°), (śīghrocca
—88°), (śīghrocca+182°) and (śīghrocca--88°) respectively.
(vii) A rule for finding the celestial latitude of a planet is
as follows :
(From the longitude of a planet severally) subtract the
longitudes of its (manda and śīghra) pātas and there-
from calculate (as usual) the corresponding celestial
latitude of that planet. Add them or take their differ-
ence 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-reckon-

1. पातभागाश्च विज्ञेयाः पण्डितैः परिकल्पिताः ॥ —28
   मन्द शीघ्रोच्चयोः क्षेप्यं चक्रार्धं बुधशुक्रयोः ।
   राशित्रयं तु शेषाणां पात्यते पातसिद्धये ॥ —29
   शुक्राऽर्किदेव पूज्यानां भागौ द्वावेव संयुतौ ।
   मन्दपाताच्च शीघ्रोच्चात् सार्धांशस्तु कुजज्ञयोः ॥ —30
   विबुधानां च सर्वेषां शीघ्रपाताः प्रकीर्तिताः । MBh. VII. 28 (ii) 31 (i)

88 BRĀHMASPHUṬASIDDHĀNTA & KHAṆḌAKHĀDYAKA
ing of Āryabhaṭa I).¹
(viii) A rule for finding the longitude of the true mean planet
according to midnight-day reckoning is as follows :
Apply half the Śīghraphala and (then) half the
mandaphala to the longitude of the planet's own
mandocca (reversely).
From the resulting longitude of the planet's mandocca cal-
culate the (mandaphala and apply it to 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) ²
(ix) Length of the circle of the sky and derivation of the
lengths of the orbits of the planets are given as
follows :
Multiply the revolutions of the Moon (in a yuga) by
3, 240,000 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.³
From these stanzas (from 20-35), it is evident that one yojana
of the sunrise day-reckoning is one and a half times that of the
midnight day-reckoning.
Now from stanza 22, it appears that 300 is to be added to
the number of civil days in a Mahāyuga. According to the
Āryabhaṭīya. the number of civil days in this cycle is 1,577,917,
500, which increased by 300, becomes 1,577,917,800, the number of

1. शोधयित्वा क्रमात् पातान् विक्षेपार्शान् प्रसाधयेत् । —31
   योगविश्लेषणोत्पत्तिरेकानेकस्वदिग्वशात् ।
   विक्षेपः स स्फुटो ज्ञेयो ग्रहस्यैकस्य कीर्तितः ॥ —32
   अन्यस्याप्येवमेव स्याच्छेषाः प्रागुक्त कल्पनाः ।
   एतत्सर्वं समासेन तन्त्रान्तरमुदाहृतम् ॥ —MBh. VII. 31 (ii)–33
2. शीघ्रमन्दोच्चचापाधैसंस्कृता त् स्वीयमन्दतः ।
   स्फुटमध्यग्रहः स धै विशेषः परिकीर्तितः ॥ —MBh. VII. 34
3. वेदाश्विरामगुणितान्य युताहतानि ।
   चन्द्रस्य शून्यरहितान्यथ मण्डलानि ॥
   र : स्वैर्हृतानि भगणैः क्रमश्रो ग्रहाणां ।
   कक्ष्या भवन्ति खलु योजनमण्ड्रस्था ॥ —MBh. VII. 35

civil days in a Mahāyuga according to Brahmagupta as referred
to in the Khaṇḍakhādyaka.
Again, the same stanza tells us to subtract 20 and 4 respec-
tively from the revolutions of Mercury and Jupiter, and we arrive
at the figures 17,937,000 and 364, 220, which are the revolutions
of Mercury and Jupiter in a Mahāyuga according to Brahma-
gupta as given in the Khaṇḍakhādyaka.
Again we can compare the figures for the diameters of
the Earth, the Sun and the Moon given in the Mahābhāskarīya,
in the present day Sūrya-siddhānta and the Āryabhaṭīya.
Diameter Mahābhāskarīya Modern. Āryabhaṭīya
of Sūrya-siddhānta
Earth 1,600 yojanas 1,600 yojanas 1,050, yojanas
Sun 6,480 6,500 4,410
Moon 480 480 315
Then in stanza 24, we are given the distances of the Sun and
the Moon as 689, 358 and 51, 566 yojanas respectively. The
same figures are worked out by Lalla according to the
Āryabhaṭīya and quoted in the Śiṣyadhwṛddhida (IV.3.4) and they
come to be 459,585 and 34,377.
The stanza 25 states the longitudes of the aphelia of planets
these figures tally with the corresponding figures given by Brah-
magupta in the Khaṇḍakhādyaka :
Longitude of aphelion of Jupiter 160°, of Venus 80°, of
Saturn 240°, of Mars 110° and of Mercury 220°.
Similarly the stanza 26 gives the peripheries of planets’
epicycle of apsis, which also is in concordance with the values
given by Brahmagupta :
Periphery of epicycle of apsis of Jupiter 32°, of Venus
14°, of Saturn 60°, of Mars 70° and of Merury 28°.
In the stanza 27 of the Mahābhāskarīya, we have the
dimensions of the epicycles of conjunction for planets; these
figures are also the same as given by Brahmagupta :
Epicycles of conjuction for Jupiter 72°, for Venus 260°,
for Saturn 40°, for Mars 234° and for Mercury 132°.
In the stanza 28, we have the Sun’s epicycle having a
periphery of 14° and the Moon’s epicycle 31° ; the longitudes of