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

Chapter IV

Subject Matter Classified in the

Brāhmasphuṭasiddhānta

& Khaṇḍakhādyaka

It needs no emphasis that Āryabhaṭa commanded a great
influence as an astronomer not only prior to Brahmagupta but
during his days also, and we have seen how Brahmagupta quoted
this great authority in his writings, sometimes borrowing from
him and sometimes contradicting him or improving upon his
calculations. Āryabhaṭa's great work is known as the Aryabh-
atīyam written in 499 A. D. Āryabhaṭa was born in 476 A. D.
( Kaliyuga Saṁvat 3577 ). The Aryabhaṭīyam is also known
as the Aryasiddhānta. For details about Āryabhaṭa the reader
is referred to the Chapter entitled "Aryabhaṭa lays Foundations
of Algebra" (Founders of Sciences in Ancient India, 1965, Chapter
X). This great work was written in Kusumapura, (modern Patna,
Bihar).

Divisions of Āryabhaṭīyam

The Aryabhaṭīyam is divided into four chapters called
Pāda: (i) the Gītikā Pāda, with ten verses; (ii) the Gaṇita Pāda
with 33 verses; (iii) the Kālakriyā Pāda with 25 verses and (vi)
the Gola Pāda with 50 verses.
In the Gaṇita Pāda, a chapter on mathematics, we have
such subjects : squares (varga), cubes (ghana) (verse 3),
square-root (vargamūla) (4), cube-root (ghanamūla) (5) ; area of
a triangle, volume of a prism (6), area of a circle, volume of a
sphere (7), area of a quadrilateral (viṣamacaturasra) (8); circumfe-
rence of a circle (10), Rsine (radius x sine) (Jīvā) (11); determina-
tion of the Rsine of the zenith distance, the base (bāhu) of a right-
angled triangle, and the upright (koṭi) of a right-angled triangle
(16); hypotenuse (karṇa) of a right-angled triangle and ardhajyā
(17), Reversed-sine (śara) (18), areas of series-figures (śreḍhīphala)
(19), rule of three (trairāśika) (23), reduction of fractions (savar-
ṇīkaraṇa of bhinna) (27) ; inverse-rule of three (vyasta) (28) ;

• line by line check:
  64 BRAHMASPHUṬASIDDHĀNTA & KHAṆḌAKHĀDYAKA
  evaluation of unknown values (mūlya-pradarśana of avyakta
  mūlya (30); and the theory of pulveriser (Kuṭṭaka) (32, 33). In
  this chapter, Āryabhaṭa gives the solutions of quadratic equa-
  tions, and thus he earns for himself the credit of founding the
  science of algebra.
  (iii) The Kālakriyā Pāda with 25 verses, which enumerates
  the units of time [1 year (varṣa)=12 months (māsa); 1 month=
  30 days (divasa); 1 day =60 nāḍīs; 1 nāḍī=60 vināḍī; vināḍī or
  vināḍikā is the same as vighaṭikā equivalent to our 24 seconds;
  nāḍī or nāḍikā or ghaṭī is equal to 24 minutes]¹;

DIVISIONS OF ĀRYABHAṬĪYAM 65
day should begin from the month Caitra, Śukla Pratipadā (the
first day of the brighter half of the month Caitra) (11); man-
docca (apex of slowest motion) and śīghrocca (apex of fastest
motion) (17—24).
(iv) The Gola Pāda with 50 verses. In Verse 1, there is
a reference to a point in the Sun's path, the commencement of
meṣa (meṣādi); this must have been vasanta-equinox. The ascen-
ding nodes (pāta) of planets and the shadow of the Earth move
on the path of the Sun (arka-apamaṇḍala) (2-3); the angular
difference in relation to the Sun at the appearance of Moon (12
degree or aṁśa), of the Venus (9 degrees or 9 vināḍika); of the
Jupiter (2 more than the Venus, i.e. 11 vināḍikas); the Mercury
or Budha (13 vināḍikas), of the Saturn or Śani (15 vināḍikas);
and of Mars or kuja (17 vināḍikas).
The half of the Earth, Moon, planets and stars is dark, since
those parts happen to be under their own shadow; the other
half is bright as it faces the Sun (this is not true with respect
to stars—author) (5). The Earth is surrounded with an atmos-
phere of air and water (6, 7). In the Brahma-Divasa (Brahma's
day), the sphere of the Earth is increased by one yojana and
decreased by this amount during the Brahmā's night (8). Just
as a person sitting on a moving boat sees the stationary trees
etc. on the bank of a river moving in the opposite direction,
similarly the stationary stars are seen moving from Laṅkā (or
equator) moving towards the West(9). On account of pravahavāyu
(air) the nakṣatra system and planets rise and set receding towards
the West (10). The dimensions of Sumeru Parvata (North pole)
is given to be one yojana, and it shines like a jewel (11) and in
the next verse is given the position of Sumeru and the Bāḍa-
vamukha (South pole) (12). The four cities situated at a diffe-
rence of 90° each on the equator are given (13). The distance of
Ujjayinī from Laṅkā (thus giving the latitude of Ujjayinī) is
given (14).
On account of the thickness of the Earth-sphere, the
Khagola (celestial sphere) is seen less than the hemisphere (15).
The next verse describes how moving appears the Khagola on
the North and South poles (16). Then is given the measure of day
and night of devas (gods), pitar (fathers), asura (demons) and
manuja (men) (17). Then are given a few technical definitions

66 BRAHMASPHUṬASIDDHĀNTA & KHAṆḌAKHĀDYAKA
of celestial mathematics (18—21) like dṛṣṭi-sthāna (intersection of
horizontal and vertical axis—pūrvāparadiggatā rekhā and adha-
ūrdhva-diggatā rekhā). Dṛṅ-maṇḍala, dṛgkṣepamaṇḍala (dṛgkṣepa
is the zenith distance of that point of a planet's orbit which is at
the shortest distance from the zenith). Then the Bhūbhagola
instrument is described (22, 23). Then follow the formulæ for
calculating lagna (the horizon ecliptic point in the East), kāla
etc. comprising the Tripraśnādhikāra (24—33). In the next
verses, we have lambaka or Rsine of colatitude (34); dṛk-karma
(35) and ayana-dṛk-karma (36). Then follow the calculations of
lunar and solar eclipses (37--47). Verse 48 describes the coordi-
nates of the Sun which are determined by the conjunction of hori-
zon with the Sun, of the Moon by the conjunction of the Sun and
the Moon, of planets by the conjunction of the Moon and planets
or stars. Verse 49 describes how this jewel treatise has been pro-
cured out of an ocean of true and untrue knowledge with the
help of a boat of intellect. This means that the author has taken
special pains in discriminating true knowledge from falsehood
with respect to the prevalent notions of astronomy. In the last
verse he says that he has not given anything new; he has given
that very knowledge which was imparted by the Svayambhu in
the earliest times (50).
The Pañchāṅgas prepared according to the rules and for-
mulæ of the Āryabhaṭīyam are still regarded with reverence by
the Vaiṣṇavas in the South. Brahmagupta was a great critic
of Āryabhaṭa, but finally he wrote his treatise, the Khaṇḍakhād-
yaka, on the basis of this very Āryabhaṭīya (this treatise is a kara-
ṇagrantha i.e. one containing a principal element of the Indian
Calendar). The four commentaries of the Āryabhaṭīya available
in Sanskrit are of Bhāskara I, Sūryadeva Yajvā, Parameśvara and
Nīlakaṇṭha. Two English translations by P.C. Senagupta (1927)
and W.E. Clark (1930) are also available.
Major Landmarks
Indian astronomy which reached its zenith in the times of
Āryabhaṭa and Brahmagupta shows its evolution in the following
stages :

1. Rudiments of astronomy in the Vedas and Brāhmaṇa
   Books like the Taittirīya Saṁhitā and the Śatapatha
   Brāhmaṇa. This period is associated with the dis-

MAJOR LANDMARKS 67
covery of the Vedic era, some of the planets, the twenty-
four nakṣatras, cycle of seasons, concept of leap year,
the dimensions of a yuga, solar and lunar years and the
like.
II. Astronomy of the Vedāṅga, also known as the Vedāṅga
Jyotiṣ. Lagadha is the most prominent figure of this
period (1400 B.C. to 850 B.C.) ; he is the first compiler
of a text on astronomy. In his work, we for the first
time in history find a reference to the Jñeya Rāśi (the
knowable or the unknown quantity) and the Jñāna-
Rāśi (known quantity). He lays the foundations of
astronomical calculations. In his treatise we find a
mention of such subjects as : Solstices (northern and
southern journey of the Sun), increase in days and
nights in the ayanas or solstices, solstitial tithis omis-
sion of tithis, parva rāśi, acceptable and non-acceptable
parvas, addition of day, acceptable parvas, concept of
yoga (a term applied to the joint space which would
be travelled by the Sun and the Moon in a given period
of time on the presumption that these two bodies have
travelled in directions opposite to each other), method
of finding out a nakṣatra on any parva day, distinction
between parva-nakṣatra and tithi-nakṣatra, correlation
of solar and lunar dates, measure of a nāḍikā (unit of
time), nakṣatra of the Sun, yoga and its nakṣatra, parva-
bhaśeṣa and equivalent kalās, solar year, lunar revolu-
tions (risings of nakṣatras) deities of nakṣtras, lunar
and sāvana day differences (āhika māsa), divisions of
a sāvana day and length of day in two ayanas. This
author probably belonged to Kashmir.
III. The period of Siddhāntas : Varāhamihira in his well
known treatise, the Pañcasiddhāntikā, refers to five
Siddhāntas or systems of Astronomy : Paitāmaha,
Vasiṣṭha, Romaka, Pauliśa and Sūrya (Saura). As re-
gards its importance, he gives the first place to the
Sūryasiddhānta, places next the Romaka and Pauliśa,
and declares the remaining two to be definitely inferior
to the former. We do not possess the full treatise of
these Siddhāntas, except the Sūryasiddhānta. Here too,

68 BRĀHMSPHUṬA SIDDHĀNTA & KHAṆḌAKHĀDAYAK
we have difficulty. The Sūryasiddhānta, as summarised
by Varāhamihira in his Pañcasiddhāntikā, in many
essential features differs from the system prescribed by
the Text of the Sūryasiddhānta now available. So we
have two versions, the one with which Varāhmihira was
familiar and the modern one.
The present Sūryasiddhānta comprises of fourteen chapters
called adhyāyas. We have an authentic commetary on it by
Parameśvara. The first two chapters of the Sūryasiddhānta have
no special name. The classification of chapters is as follows :
Modern Sūrya Siddhānta
Chapter Name of the Number
Chapter of verses Subject
I — 69 Mean longitude of
planets.
II — 68 True motion and true
longitude of planet ;
and elements of Pañ-
cāṅga.
III Tripraśnādhyāya 50 Directions, place
and time.
IV Candra-grahaṇā- 29 Lunar eclipse.
dhyāya
V Sūrya-grahaṇādh- 17 Solar eclipse.
yāya
VI Chedyakādhyāya 24 Projection of eclipse
on a plane surface.
VII Graha-samāgama- 24 Conjunction of one
yuddhādhyāya planet with another.
VIII Tārāviṣayodhyāya 21 Conjunction of a
Planet with the junc-
tion star of a nakṣatra.
IX Udayāstamaya-vi- 18 Heliacal rising and
ṣayodhyāya setting of planets.
X Candrāstamayādi 16 Moon rise and eleva-
viṣayāḥ tion of Moon-horn.
XI Vyatīpāta-viṣayāḥ 23 Pāta (vyatīpāta)
XII — 87 Cosmogony and geo-
graphy

MODERN SŪRYA SIDDHĀNTA 69
Chapter Name of the Number
Chapter of verses Subject
XIII Vyatīpāta-viṣayāḥ 26 Armillary sphere and
astronomical instru-
ments
XIV — 28 Modes of reckoning
time
Total 500
From the contents of the Modern Sūryasiddhānta, it appe-
ars that its great author (or rather compiler, since this Siddhānta
has been existing much before he compiled this treatise) intends
giving a complete but brief account of the entire Indian astron-
omy. He has been careful enough to avoid controversies, and
he has omitted the methods occurring in other Siddhāntas and
Mahā-tantras as alternative rules. He describes only those
which are most important and of general nature. Part II of
the book dealing with cosmogony and geography etc.. is very
brief , a nd the entire subject has been summarised in three chap-
ters (Chapters XII to XIV). The Chapter on Astronomical
instruments (XIII) is sketchy and rather incomplete; the author
has given the names only of the instruments, as if the purpose
was to keep the details secret and guarded. The whole text of
the Sūryasiddhānta is finished in 500 verses (compare this with
the length of other Indian books on Astronomy: the Brāhmasph-
uṭasiddhānta 1008 verses, the Siddhānta-śekhara 890 verses and
Siddhānta-śiromaṇi proper 962 verses).
The Modern Sūryasiddhānta is a composition or compila-
tion of a period when the Āryabhaṭīyam and the Brāhmasphuṭa-
siddhānta had already become popular. This Sūryasiddhānta is
indebted to both these systems, though it has nowhere acknowledged
them in the Text. We shall refer to this matter later at an appro-
priate stage of our discussion.
It may casually be mentioned here that Alberuni has
ascribed the authorship of the Sūryasiddhānta to Lāṭadeva while
Munīśvara (603 A. D.) has ascribed it to Āryabhaṭa I. Though
there is little or no support to these views, it is not improbable,
that the works of Āryabhaṭa I and Lāṭadeva which followed mid-
night-day-reckoning were based on the Sūryasiddhānta prevalent
at the time. P. C. Senagupta on the other hand is of the opinion
that “ the old Sūryasiddhānta was made up-to-date by Varāhami-
hira by replacing the old constants in it by new ones from Ārya-

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