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

42 ASTRONOMY IN ANCIENT NATIONS
He imagines a fixed ecliptic (in the ninth sphere) which intersects
the equator in two points (the mean equinoxes) under an angle of
23° 33' 30", and a movable ecliptic (in the eighth sphere), attached
at two diametrically opposite points to two small circles, the
centres of which are in the mean equinoxes and the radii of which
are = 4° 18' 43". The movable tropical points of Cancer and
Capricorn never leave the fixed ecliptic, but move to and fro to
the extent of 8° 37' 26", while two points on the movable ecliptic
90° from the tropical points move on the circumferences of the
small circles, so that the movable ecliptic rises and falls on the
fixed one, while the points of intersection of the equator and the
movable ecliptic advance and recede to the extent of 10° 45'
either way, This is a motoin of the eighth sphere, common to
all stars, and the Sun will, therefore, sometimes reach its
greatest declination in Cancer, sometimes in Gemini. Tabit
does not say that the obliquity of the ecliptic is variable, and
perhaps it did not occur to him that this would be a necessary
consequence of his theory; he only notices the change in direction
and amount of the motion of the equinoxes, which, he says, has
increased since the days of Ptolemy, when it was only 1° in 100
years, while later observers have found 1° in 66 years. The
erroneous value given by Ptolemy was, therefore, mainly responsi-
ble for the continuance of the imaginary theory. It is to be
observed that Tabit expresses himself with a certain reservation,
and seems to think that further observations are necessary to
decide if the theory is true or not. His younger and greater
contemporary Al Battani was even more cautious, for though he
repeats the account of the trepidation given by Theon (which he
says that Ptolemy manifeste in suo libro declarat¹) he does not
make use of it, but simply adopts 1° in 66 years (or 54".5 a year),
which he finds by a comparison between his own observations and
some made by Menelaus. In rejecting the erroneous value of
Ptolemy, which Al Fargani alone had accepted,² Al Battani was
followed by Ibn Junis, who came still nearer to the truth by
adopting 1° in 70 years or 51".2 a year, and who does not allude
to trepidation. It is greatly to the credit of several other Ara-
bian writers that they were not led astray by this imaginary phe-

1. Cap. 52, (205). Plato's translation gives the period as 84 years, but Nallino's
   ed. has 80 (p. 127).
2. c. 13, p. 49. .
3. Schjellerup, Descr. des etoiles fixes, p. 43,

ARABS ON MOTIONS OF FIXED STARS 43
nomenon; among them are Al Sūfi, the author of the only urano-
metry of the Middle Ages³, who followed Al Battani, also Abu 'l
Faraj and Jagmini,¹ while Nasir ed-din mentions it but seems
to doubt its reality.² By others it was willingly accepted, for
instance by Al Zarkali, who made the period of oscillation of 10°
either way equal to 2000 Muhammedan years (or 1940 Gregorian
years, i.e. 1° in 97 years or 37" a year). The motion is in a circle
of 10° radius; at the Hijra the movable equinox was it 40' in
increasing precession, and in A: D. 1080 at 7° 25'.³. The diminu-
tion of the inclination of the ecliptic, which the astronomers of
Al Mamun had found=23° 33', no doubt lent countenance to
the idea of trepidation, and the next step in the development
of this curious theory was the combination of progressive and
oscillatory motion. Al Betrugi, who gives a sort of history
of the theory, beginning with a mythical H mes, makes
out that Theon (or Taun Alexandrinus as he calls him) com-
bined the motion of 1° in 100 years with the oscillation³. A
century later this was actually done, and the theory received
its last development by King Alfonso or his astronomers, who⁴
perceived that the equinoxes had receded much further than
Tabit's theory allowed. The equioxes were now supposed to pass
right round the heavens in 49,000 years (annual motion=26".45),
while the period of the inequality of trepidation was 7000 years,
so that in a sort of Great Jubilee year everything was again as it
had been in the beginning.⁵ The progressive motion belongs to

1. Abu 'l Faraj, p. 12, simply says that the motion is 1° in 100 years according
   to Ptolemy, or 1° in 66 years according to others. But on p. 18 he says that
   if the ancient Chaldeans gave the tropical points a motion backwards and .
   and forwards, and if ancient astrologers adopted this, then the motion of the
   fixed stars must have been unknown to them. Jagmini (p. 229) says that
   most people adopt 1° in 66 solar years.
2. Spheres celestes, p. 347.
3. Sedillot, Memoire sur les instr. astr. des Arabes, pp. 31, 32. Abraham
   ben Chija (p. 196 of Munster's Sphaera mundi. Basle, 1546) gives the
   period as 1600 years without quoting any authority. He adds that the
   ancient Indians. Egyptians, Chaldeans, Greeks, and Latins first proposed
   the theory—Ptolemy neither approved nor disapproved of it, but Al Battani
   confuted it.
4. Alpetragius, f. 12a. He says that Al Zarkali did the same.
5. A later writer, Augustinus Ricius. De motu octave sphaere, Paris, 1521, who
   traces the theory back to Hermes, 1985 years before Ptolemy (!) credits
   (Continued on next page)

44 ASTRONOMY IN ANCIENT NATIONS
the ninth sphere; the annual precession varies between 26″.45±
28″.96, or from + 55″.41 to −2″.51.¹ It was now necessary to
assume the existence of a tenth sphere, which as primum mobile
communicated the daily rotation to all the others, while the
ninth produced the progressive and the eighth periodical
motion on the small circles, which are situated "in the conca-
vity of the ninth sphere." This was a nice and comfortable
theory on account of the long periods involved and the slow
changes it produced in the amount of annual precession; and
oblivious of the fact that the theory had no foundation except
the circumstance that the obliquity of the ecliptic was now about
20′ less than it had been stated to be by Ptolemy, and that he had
given the amount of precession as 36″ a year instead of about
50″, and often shutting their eyes to several of the necessary
consequences of it, such as the changes in the latitudes of stars
which it ought to produce², astronomers continued to accept the
theory until at last a real observer of the stars arose and wiped
it out by showing that the obliquity of the ecliptic had steadily
diminished, and that the amount of annual precession had never
varied. We have in this place only alluded to it because it in-
volved some rearrangement of the spheres and because it is
eminently characteristic of the period during which no persistent
observations were taken, and hardly an attempt was made to
improve the theories of Ptolemy. The theory of trepidatio or
titubatio, as it was sometimes called, was one attempt and it
would have been better left alone. But it forms a not uninter-
esting chapter in the history of astronomy.
(Continued from previous page)
this development to a Jew of Toledo, Isaac Hassan (see above. p. 38),
adding that Alfonso four years after the completion of the tables became
convinced of the futility of the theory by reading the book on the fixed stars
by Al Sufi. Riccioli, Almag. novum, I. p. 166.

1. In the Alfonsine Tables the maximum took place at the birth of Christ. In
   Essler's Speculum astrologicum, p. 224 (appended to Purbach's Theoricae
   novae, Basle, 1573) the epoch is A.D. 15, diebus 137 completis. Reinhold in
   his commentary to Purbach (Paris, 1558, f. 163b) explains that 26″.45 is
   the space passed over by the Sun in 10 mins. 44 secs., by which amount
   the Alfonsine Tables made the tropical year smaller than 365¼ days.
2. Abraham ben Chija (p. 103, Schrackenfuchs) says that trepidation does
   not change the latitudes. Perhaps he refers to the earliest form of the
   notion, that described by Theon of Alexandria.

ARABS ON MOTIONS OF FIXED STARS 45
Here we finish our review of ancient astronomy. We have
omitted as not coming within our province several valuable
contributions to science which did not deal with cosmology or
planetary theory. But even with this limitation enough has
been said to show that when Europeans again began to occupy
themselves with science they found astronomy practically in the
same state in which Ptolemy had left it in the second century.
But the Arabs had put a powerful tool into their hands by alter-
ing the calculus of chords of Ptolemy into the calculus of sines or
trigonometry, and hereby they influenced the advancement of
astronomy in a most important manner.
References

1. Peter Doig : A Concise History of Astronomy London
3. J.L.E. Dreyer : A History of Astronomy from Thales to
   Kepler Dover publications, 1953 (chapter
   XI reproduced).
4. Satya Prakash : Founders of Sciences in Ancient India,
   Delhi, 1965.

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CHAPTER II
Personal References of
Brahmagupta
Sudhākara Dvivedī in his Gaṇaka Taraṅgiṇī, a small book on
biographical skethes of astronomers and astrologers of this
country gives a brief account of Brahmagupta thus :
Brahmagupta was born in 520 Śaka (655 Vikramī or 589
A.D.) in the reign of King Vyāghramukha, belonging to the Cāpa
family ; his father was Jiṣṇugupta ; and at the age of 30, he wrote
in 550 Śaka (628 A. D.) his well known treatise on Astronomy
known as the Brāhmasphuṭasiddhānta, which is corroborated by
the statement in the Viṣṇudharmottara Purāṇa, (Chapter on the
Brahma-siddhānta), His other treatise, entitled the Khaṇḍa-
Khādyaka, which is a karaṇa book, was completed in 587 Śaka (665
A. D.). According to some authorities, Brahmagupta was the
grandson of Viṣṇugupta, and the family suffix (Gupta) indicated
that he belonged to the Vaiśya family, and he was in the service
of the King of Rewah, known as Vyāghrabhaṭa.
Brahmagupta was a great critic ; he did not spare any of his
predecessors like Āryabhaṭa, Varāhamihira, Śrīṣeṇa, Viṣṇucandra
and others. Later on, his influence on the writing of the succeeding
generations has been immense. Bhāskarācārya II in his Bījagaṇita
has acknowledged him as a great authority on algebra and has
given him as the first place amongst the galaxy consisting of
Brahmagupta, Śrīdhara, Padmanābha etc. The Eighteenth Chap-
ter of the Brāhmasphuṭasiddhānta, known as Kuṭṭaka Chapter (on
Pulveriser) has been translated by H. T. Colebrooke in English
in 1817. The English translation of the Twelfth Chapter on
Gaṇita or Calculations from the Brāhmasphuṭasiddhānta is also
available in English. (See Colebrook's Algebra with Arithmetic
and Mensuration from the Sanskrit of Brahmagupta and Bhās-
kara, London, 1817).
The Vāsanā Commentary on the Brāhmasphuṭasiddhānta by
Pṛthudakasvāmī (860 A. D.) is also available though with diffi-
culty (as indicated by Sudhākara Dvivedi) ; its incorrect manus-

48 PERSONAL REFERENCES OF BRAHMAGUPTA
cript is available in the Library of the King of Banaras (Kāśīrāja)
which has the colophony at the end as :
श्री चापवंशतिलके श्री व्याघ्रमुखे नृपे शकनृपात् ,
पञ्चाशत्संयुक्तैर्वर्षशतैः पञ्चभिरतीतैः ।
ब्राह्मस्फुटसिद्धान्तः सज्जनगणितज्ञगोलवित्प्रीत्यै,
त्रिंशद्वर्षेण कृतो जिष्णुसुतब्रह्मगुप्तेन ॥
Bhāskara II has written the famous treatise Siddhāntaśiromaṇi
(1150), which is almost based on the Brāhmasphuṭasiddhānta. It
has been edited by the author's own gloss (Vāsanābhāṣya) by
Bāpu Deva Śāstrī (Vārāṇasī); by Murlidhar Jha with the com-
mentaries, Vāsanāvārttika of Nṛsiṁha (1621) and Marīci of
Munīśvara (1635), vol. I (containing chapter 1 of the Gaṇitā-
dhyāya) (Vārāṇasī, 1917); by Girija Prasad Dvivedi with original
commentaries in Sanskrit and Hindi, vols. I and II (Lucknow,
1911, 1926); English translation of the text only by Bāpu Deva
Śāstrī and Wilkinson (Calcutta, 1861).
In the very first Chapter (verse 2), Brahmagupta writes :
The old calculations dealing with planets (i.e. the old astronomy),
based on the system of Brahmā have become erroneous in course
of past ages and therefore, I, the son of Jiṣṇugupta would like to
clarify them.
Brahmagupta was not a mere theorist, he based his calcula-
tions on direct observations with the help of instruments or
devices (nalikādi yantra); he was in favour of making correc-
tions on the basis of these observations. He was himself an
expert observer. In his Khaṇḍakhādyaka also he has emphasised
the need of direct observation.
At many places, Brahmagupta has severely criticised the
Romaka and Pauliśa systems of astronomy which were introduced
in this country by Lāṭadeva and Śrīṣeṇa. There are many
passages where this criticism would be available with vehe-
mance.
Brhamagupta was opposed to the system of Āryabhaṭa I.
He never spares the school of Āryabhaṭa which was regarded as
the most authoritative then. Sudhākara Dvivedi says that as
Brahmagupta was opposed to the system of Āryabhaṭa, so the
Vaṭeśvara Siddhānta was opposed to that of Brahmagupta. The
Institute has already published the Vaṭeśvara Siddhānta and now
it has the privilege of publishing the Brahmasphuṭasiddhānta.

A NOTE ON BHILLAMĀLA 49
A Note on Bhillamāla
It is said that Brahmagupta completed his Brāhmasphuṭa-
siddhānta in Śaka 550, and he has come to be known as Bhilla-
mālakācārya or a teacher residing in "Bhillamālaka." In this
connection, therefore, it would be interesting to reproduce a note
on Bhillamāla from G. Bhūler's article on Gurjara Inscriptions,
No. III, published in the Indian Antiquary, July 1888, vol. 17, p.
192 :
With a single exception all the complete inscriptions call
the princes enumerated above, scions of the Gurjara race; and
Khe I. and II. highly extol the greatness and wide extent of this
family. Na. alone names the Mahārāja Karṇa as their ancestor.
With respect to this personage it is for the present impossible to
say whether the famous hero of the Mahābhārata may be meant,
or some real historical king. But the name Gurjara makes it
evident that this dynasty belonged to the great tribe which is
still found in Northern and Western India and after which two
provinces, one in the Bombay Presidency and one in the Pañjāba,
have been named. The Gurjaras or Gūjars are at present pretty
numerous in the western Himālaya, in the Pañjāba and in Eastern
Rājputānā. In Kachh and Gujarāt their number is much smal-
ler. It would, therefore, seem that they came into Western
India from the north. Their immigration must have taken place
in early times, about the beginning of our era or shortly after-
wards. In Western India they founded, besides the kingdom of
Broach, another larger state which lay some hundred miles
further north. Hiuen Tsiang mentions in his travels¹ the
kingdom of Kiu-che-lo and its capital Pi-lo-mi-lo. It has been
long known that the former word corresponds to Gurjara.
But the name of the town has been incorrectly connected
by the French scholars with Bālmer in the Jēsalmīr territory, and
this ident ification has been accepted in Mr. Beal's new transla-
tion of Siyuki. As I have stated already formerly² following
Colonel. J. Watson, Pilomilo corresponds exactly to Bhillamāla

1. Beal, Siyuki, Vol. II, p. 269f. Hiuen Tsiang assigns to the nor-
   thern Gurjara State an extent about double of that given for the kingdom of
   Broach.
2. Ante, Vol. VI. P. 63.

50 PERSONAL REFERENCES OF BRAHMAGUPTA
one of the old names of the modern Bhīnmāl or Śrīmāl¹ in
southern Mārwāḍ close to the northern frontier of Gujarāt.
Another work, which was composed a few years before Hiuen
Tsiang's visit to Gujarāt, contains likewise a notice of this
northern kingdom of the Gurjaras. The astronomer, Brahma-
gupta, who completed his Siddhānta in Śaka-Samvat 550 or 628
A.D. calls himself Bhillamālakakācārya², "the teacher residing
in Bhillamālaka and is called so by his commentator Pṛthūda-
kasvāmin. He further states that he wrote under king Vyāghra-
mukha who was 'an ornament of the Cāpa race.' This family,
whose name recurs in the Haḍḍāla grant of Dharaṇīvarāha³
prince of Vadhvān, thus seems to have been the reigning house
of Bhillamāla. It is most probably identical with the Cāuḍas,
Cāvōṭakas⁴ or Chāpōtkaṭas, who from 756 to 941 A.D. held
Aṇhilvāḍ and still possess various small districts in northern
Gujarāt. The Gurjara kingdom of Broach was without a doubt
an offshoot of the larger State in the north, and it may be that
its rulers, too, belonged to the Cāpa family.

1. Bhillamāla means etymologically 'the field of the Bhil' and Śrī-
   māla 'the field of Śrī'. The latter name must also be ancient, as the Śrī-
   mālī Brāhmaṇ as are called after it. The Jainas narrate various, of course
   incredible, legends, which explain how Śrīmāla came to be called Bhillamāla.
   Merutuṅga says that king Bhoja invented the latter name, because the
   people of Śrīmāla let the poet Māgha die of starvation. According to
   another authority, the town had a different name in each Yuga. It is in
   India very common for ancient towns to have two or even more names.
   Thus Kanauj was called, Kānyakubja, Gādhipura, and Mahodaya.
2. See Professor A. Weber, Die Sanskrit und Prakrit Handschriften
   der Berliner Bibliothek Vol. II. pp. 297-298. In the first passage the MSS.
   offers incorrectly Bhilamācārya; in the second which occurs in the commen-
   tary on the Khaṇḍakhādyaka, we have Bhillamālavakācārya, a slightly cor-
   rupt reading. This latter varia lectio occurs also in other MSS., see
   Weber, Indische Streifen. Vol. III, p. 90, and has given rise to erroneous
   suppositions regarding Brahmagupta's home. The Gujarātī Joshīs still pre-
   serve the tradition that Brahmagupta was a native of Bhinmāla.
3. Ante, Vol. XII. p. 190ff. The remark which I have made there
   that the Cāpas are not named elsewhere, of course requires correction.
4. The form Cāvoṭaka, which occurs in Dr. Bhagavanlal's grant of
   the Gujarāt Cālukya king Pulakeśin of Samvat 490, is the immediate prede-
   cessor of the word Cāuḍa. Its Sanskrit original is certainly not cāpotkaṭa
   which probably has been coined in comparatively speaking modern times,
   in order to explain the difficult Prakrit word, just as the bards of Rajputana
   have invented Rāstrauḍha as etymon for Rāṭhoḍ.

BRAHMAGUPTA'S OWN REFERENCES 51
Brahmagupta's own References
In the Twenty-fourth Chapter (Sañjñādhyāya), of the
Brāhmasphuṭasiddhānta, Brahmagupta has made a reference to his
own biography : In the reign of Vyāghramukha belonging
to the family of Cāpa, in the year 550 Śaka the treatise
Brāhmasphuṭasiddhānta was composed for the benefit of benevo-
lent astronomers by Brahmagupta, son of Jiṣṇugupta at the age
of 30. (BrSpSi. XXIV. 7. 8)
Then again he says : The Brāhmasphuṭasiddhānta has been
written by Brahmagupta, son of Jiṣṇu, in 1008 verses of Āryāch-
anda. (ibid 10 )
In the beginning of this Sañjñādhyāya, he refers to the
differences in fundamental notions created by the various existing
systems of astronomy as the Sūrya-siddhānta, Pulisa-siddhānta.
Romaka-siddhānta, Vasiṣṭha-siddhānta and other Yavana-siddhā-
ntas, which have caused anomalies in the calculations of eclipses.
He also refers to the anomalies due to the calculations based on
midnight day-reckoning and sunrise day-reckoning.
From the point of view of own references, the following
would be of interest :
Brahmagupta, son of Jiṣṇugupta (Jiṣṇusuta-Brahmagupta) :
BrSpSi. I. 2; XVI. 35. 37; XXIV. 8. 10; XXV. 73.
It is strange that in the Khaṇḍakhādyaka, Brahmagupta has
not given his name nor his father's name anywhere. At least the
reading of the Khaṇḍakhādyaka as given by Pṛthūdakasvāmī does
not contain this name. In the edition of Bhaṭṭotpala, there are three
more chapters in the Khaṇḍakhādyaka (Chapters IX, X and XI).
In the Chapter XI (known as Pātādhikāra), we have 21 verses and
in the last 21st verse we find the name of Brahmagupta,¹ son of
Jiṣṇu mentioned :
Those who are eager to have the knowledge of the motion
of stars and planets, for them and for the benefit of disciples
in this field, Brahmagupta son of Jiṣṇu has composed this
Khaṇḍakhādyaka.

1. खण्डखाद्यकमिदं तृप्त्यर्थं ग्रहगतिश्रुतात्तांनाम् ।
   शिष्याणां हितार्थं प्रोक्तं जिष्णुसुतब्रह्मगुप्तेन ॥ — KK. XI. 21

52 PERSONAL REFERENCES OF BRAHMAGUPTA
Reference to Āryabhaṭa
We have said that it appears that Brahmagupta was a bitter
opponent of Āryabhaṭa in his younger days (628 A. D.), but
later on (in 665 A. D.), he climbed down to describe and teach
one of the Āryabhaṭa's system of astronomy. Āryabhaṭa was
universally revered. and it was difficult for Brahmagupta to have
ignored him and thus he has to refer to this great authority some
times to oppose some of his views and some times to expound his
views. The following are the pasages in the Brahmasphuṭasiddha-
nta, where the author has referred to the name of Āryabhaṭa.
There are many other passages where the name "Aryabhaṭa"
does not occur but where Brahmagupta indirectly means to quote
the views of this great master.
BrSpSi. I. 9. 12. 28. 32. 60. 61 ; II. 33. 46; V. 21.25;
VI. 13; IX.10; XI. 4. 9. 10. 12. 25. 29. 33. 41.
42. 43. 44, 46, 47. 49, 62; XIII, 27; XIV. 45; XVI.
37. 46; XXI. 39.
The largest references are in Chapter XI, where Brahmagupta
has made an attempt to show the discrepancies of the Āryabhaṭī-
aya system of reckoning astronomical observations and constants.
In the Khaṇḍakhādyaka also there is a reference to the
name of Āryabhaṭa but on very few occasions. In Chapter I,
we have this reference at three places-
Having made obeisance to God Mahādeva, who is the great
cause of this world's rise (i.e. creation), existence and destruc-
tion, I shall declare the Khaṇḍakhādyaka (i.e. a short treatise
on astronomy, which is as pleasant as food prepared with
sugarcandy), which will yield the same results as the great
astronomical treatise of Āryabhaṭa. As in most cases
calculation by the great work of Āryabhaṭa, for (the know-
ledge of time and longitude of planets etc. at ) marriage,
nativity and the like, is impracticable for common use
every day, this smaller treatise is made so as to yield the
same results as that. ¹

1. प्रणिपत्य महादेवं जगदुत्पत्तिस्थितिप्रलयहेतुम् ।
   वक्ष्यामि खण्डखाद्यकमाचार्य्यार्य्यभट तुल्यफलम् ॥
   प्रायेणार्य्यभटेन व्यवहारः प्रतिदिनं यतोऽशक्यः ।
   उद्वाहजातकादिषु तत्समफललघुतरोक्तिरतः ॥
   —KK. I, 1-2

REFERENCE TO ĀRYABHAṬA 53
These two verses show that Brahmagupta was not hostile to Ārya-
bhaṭa when he wrote this Khaṇḍakhādyaka ; he merely intends
presenting the subject matter on simple lines and furnishing
the results obtained by Āryabhaṭa in a simpler way.
In the following verse of the same chapter, he refers to
Āryabhaṭa’s midnight day-reckoning system :
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 Ārya-
bhaṭa's “midnight-system”.¹
In the Appendix of the Khaṇḍakhādyaka known as Khaṇḍa-
khādyakottara, we again find a few verses where the name of
Āryabhaṭa occurs; Three of these verses have been reproduced
from the Brāhmasphuṭasiddhānta (BrSpSi I. 62, 6

54 PERSONAL REFERENCES OF BRAHMAGUPTA
In the Appendix of the Khaṇḍakhādyaka, there is another verse
which also speaks in the same strain against Āryabhaṭa (this
verse does not occur in the Brāhmasphuṭasiddhānta) :
As the process of finding the apparent places of pla-
nets as given by Āryabhaṭa does not make them agree with
observation, I shall, therefore, speak of this process. Of the
Sun the apogee is at two signs and seventeen degrees (KK.
IX. 4)¹
Brahmagupta opposed to Śrīṣeṇa—
Viṣṇucandra, Lāṭa, Vijayanandī and others
Brahmagupta was a great critic; he did not spare Āryabhaṭa,
and along with him he was vehemently opposed to the doctrine
of Śrīṣeṇa, Viṣṇucandra, Lāṭadeva, Vijayanandī and Pradyumna
also. He was opposed to the Romaka and Pauliśa Siddhāntas,
which were the systems of foreign astronomy, derived from
Greece, Babylonia and other centres of learning. He did his best
to resist the foreign influences on astronomy.
The following are the verses in the Brāhmasphuṭasiddhānta,
where Brahmagupta expressed his note of discord against the sys-
tems or notions of Āryabhaṭa, Śrīṣeṇa and Viṣṇucandra :
BrSpSi. I.60; II 46, 47; X 13, 62; XI.31, 46, 47, 48-50, 55,
XVI. 36, 46; XXI.39; XXII. 2.
In two of the verses, he refers to Lāṭasiṃha :
BrSpSi. XI. 46, 48.
In the following verse, he refers to Aṅkaciti, Vijayanandī, Prad-
yumna and others: BrSpSi. XI. 58.
In the Khaṇḍakhādyaka we do not find the names of these
adversaries of Brahmagupta ; we, however, have a reference of
Śrīṣeṇa, Āryabhaṭa and Viṣṇucandra in the verses already quoted,
occurring in the Appendix of the Khaṇḍakhādyaka (Khaṇḍa-
khādyakottara), Chapter IX. 2 and 3. As we have said before,
these verses of the Appendix have been reproduced from the
Brāhmasphuṭasiddhānta. (I.62 and II.47).
Reference to Romaka and Pauliśa systems
Brahmagupta speaks of his system as if expounded by Brahmā
himself for the first time, later on deteriorated, and then revived
by Brahmagupta himself. The very second verse of the Brāhma-

1. न स्फुटमार्यभटोक्तं स्पष्टीकरणं यतस्ततो वक्ष्ये ।
   भानुमतो मन्दोच्चं राशिद्वयमंशकांश्च सप्तदश ॥ —KK. IX. 4.

REFERENCE TO ROMAKA AND PAULIŚA SYSTEMS 55
sphuṭasiddhānta (I. 2) substantiates this view :
The science of astronomy (or the calculations of heavenly
bodies) in course of long duration became ineffective or erro-
neous; this was revived by Brahmagupta, son of Jiṣṇu.¹ (BrSpSi.
I.2).
The system of astronomy which goes by the name Brahma
(Brahmasiddhānta) has been handed down to us in three forms :
(i) one is as treated by the Śākalya Saṁhitā, (ii) one as described
in prose in the Viṣṇudharmottara Purāṇa, and (iii) the one des-
cribed by Varāhamihira in the Pañcasiddhāntikā, which re-
cognises the yuga of the duration of five years. Which of these
three was accepted by Brahmagupta is not clear. But from the
measures of the number of revolutions performed by a planet in
a given period (graha-bhagaṇa) etc., it is clear that Brahmagupta
acknowledges the system as propounded in the Viṣṇudharmottara
Purāṇa. In his chapter "Tantra-parīkṣādhyāya or Dūṣaṇā-
dhyāya, he contradicts the notions of the Vedāṅga Jyotiṣa which
accepts the yuga of five years.
We may further emphasize the fact that Brahmagupta has
not clearly detailed out the errors to which the Brahma-siddhānta
succumbed in course of time, and how these errors were eradi-
cated by him. During the days of Brahmagupta, Romaka and
Pauliśa systems were getting currency in this country. Reference
to these two are found in the Brāhmasphuṭasiddhānta at several
places as follows :
ROMAK : I.13; XI.50; XXIV.3
PAULIŚA : XIV.45; XXIV.3
In fact, BrSpSi. XXIV.3, we find the line "Sūryendu-Puliśa-
Romaka-Vasiṣṭha-Yavanādyaiḥ", where we have a reference to
all the then existing systems : Sūrya-siddhānta, Indu-siddhānta,
Puliśa-siddhānta, Vasiṣṭha-siddhānta, and other Yavana-siddhā-
ntas. Just as the Sun is one, so the astronomical system is also
one ; this is a different thing that calculations in different systems
may vary according to different sunrises in different places.
Brahmagupta refers at one place to Varāhamihira in BrSpSi.
XXI.39, where he has been spoken of in connection with a list of

१. ब्रह्मोक्तं ग्रहगणितं महता कालेन यत् खिलीभूतम् ।
अभिधीयते स्फुटं तज्जिष्णुसुतब्रह्मगुप्तेन ॥ — BrSpSi. I. 2.

56 PERSONAL REFERENCES OF BRAHMAGUPTA
anti-authoritative versions of astronomical systems :
Evam Varāhamihira-Śrīṣeṇācāryabhaṭa-Viṣṇucandrādyaiḥ
Lokaviruddhamabhihitam Veda-smṛtisaṃhitābāhyam.
At one place, he mentions the difference between the calculations
based on the system of Āryabhaṭa and the Pañca-siddhāntas
(Five systems) : Pauliśa, Romaka, Vasiṣṭha, Saura, and Paitāmaha.
(BrSpSi. XIV.46).
Brahmagupta was also familiar with the Jaina systems of
astronomy; for example, at one place he uses the term "Jinoktam"
(i.e., one propounded by the Jainas) : He repudiates the concept
of two Suns and two Moons "Dvāvarkaindavau" (do canda do sujja)
(BrSpSi. XI.3) as enunciated by the Jainas.
At several places we find a reference to the Vasiṣṭha-sid-
dhānta (BrSpSi. XI.49, 50; XXIV.3)
Wherever, Brahmagupta has to press for his views in pre-
ference to the views of others, he uses the words Brāhma or
Brahmokta : BrSpSi.
Brāhma : I.32; X.62; XI, 61; XVI.37
Brahmokta : II.31, 33; X.63, 69; XV.59; XVI.33
Reference
Sudhākara Dvivedi : Gaṇaka Taraṅgiṇī
G. Bühler : Gurjara Inscriptions, Indian Anti-
quary, 1888.
P. C. Sengupta : Khaṇḍakhādyaka, Calcutta, 1934.
— . o : —

CHAPTER III
Manuscripts of the
Brāhmasphuṭa Siddhānta
Sudhākara Dvivedī has given an account of some of the
manuscripts of the Brāhmasphuṭasiddhānta in his Bhūmikā or
Introduction appended to the edition published in the PAṆḌITA,
Vol.XXIV, 1902 (New Series) : (i) available in the Library of the
Government College, Kashi (Vārāṇasī) i. e. Kāśīka-Rājakīya-Pāṭha-
laya,(ii) Dr.Thibaut's Manuscript,(iii) the Manuscript in possession
with Yajñadatta Śarmā, the Chief Astronomer attached to the
Prince of Ayodhyā. It is further mentioned that Dr. Thibaut's
Manuscript was a copy of a Manuscript available in the Deccan
College, Poona. The Manuscripts (ii) and (iii) were identical.
The Manuscripts (iii) was very faulty and incorrect.
The Manuscript from which Colebrooke translated out in
English the Twelfth and the Eighteenth Chapters on Gaṇita or
Mathematics and Kuṭṭaka (the Chapter on Pulveriser) respectively,
appeared to be different from the three Manuscripts described
above. The readings differed considerably. The Kuṭṭaka-
Chapter of that book is, writes Sudhākara Dvivedī, still available
in the India Office Library. (See Catalogue of the Sanskrit¹
Manuscripts in the Library of India Office, Part V.p. 995).¹

1. एतत्कृतस्यास्य सिद्धान्तग्रन्थस्यैका प्रतिः काशिकराजकीय पाठालयतो द्वितीया डा० थिबौ साहिब
   महाशयतस्तृतीया चायोध्यानरेशप्रधानज्योतिर्विच्छ्री यज्ञदत्तशर्मणो मया लब्धा । डा० थिबौमहा-
   शयस्य पुस्तकं कस्यचिद्दक्षिणदेशीय (Deccan College, Poona ) डेकन पुस्तकस्य
   प्रत्यन्तरम् । इदं पुस्तकं तथा पं० श्री यज्ञदत्त पुस्तकं चैकमातृकमेव । इदं पुस्तकत्रयमतीवाशुद्धं
   बहु च स्खलितं चास्ति ।
   यत्पुस्तकानुसारेण व्यक्ताव्यक्तयोर्द्वादशाष्टादशसंख्ययोरंगलभाषायामनुवादः कोलब्रूकसा-
   हिबेन कृतस्तत्पुस्तकमेतत्त्रयतो भिन्नमित्यसंशयं विभाति पाठविभेदात् । तत्पुस्तकस्य कुट्टकाध्यायः
   संप्रति इण्डिया-आफिस-सरस्वती भवने वर्तते (See Catalogue of the Sanskrit
   Manuscripts in the Library of India Office, Part V, Page 995)
   पृथूदककृताऽस्य सिद्धान्तस्य टीका या कोलब्रूकसाहिबेन भारतवर्षे ह्युपलब्धा सा च
   संप्रति लण्डननगरे इण्डिया-आफिस-सरस्वती भवने वर्तते । तस्या एका प्रतिर्मद्द्वारेण श्री डा०
   थिबौ साहिबमहाशयैरुत्पादिता सापि संप्रति मन्निकटेऽस्ति । अहो इण्डिया-आफिस-सरस्वती
   (Continued on next page)

58 MANUSCRIPTS OF THE BRĀHMASPHUṬASIDDHĀNTA
Colebroke procured a Manuscript of the Pṛthūdaka Svāmī's
Annotation on the Brāhmasphuṭasiddhānta from somewhere in
India. This is also available in the India office Library, London.
Undoubtedly it appears that this Manuscript is a copy from one
written in the Maithila script. Dvivedī describes a few character-
istics of this Manuscript. For example, on the Folio 11. 7 (७) is
written instead of 9 (९). At a few other places also, the same has
happened. Following a visarga, (kva क्व) is inscribed instead of
'ka' (क), for example refer to Folio 12, line 6. At some places
instead of visarga (:) we find ṣa (ष) inscribed, for example on Folio
12, line 1. Sometimes we find a sandhi at the virāma or end of
a sententence ; e. g. in the Golādhyāya, Folio 21, line 2: sarva-
mupapannamuktamakhaṇḍena (सर्वमुपपन्नमुक्तमखण्डेन). At some places
we find nū (नू) written in place of nta (न्त). In one of the Folios,
Śrī Gaṇeśāya namaḥ. श्रीगणेशायनमः is written in Maithila script.¹
Sudhākara Dvivedī prepared a copy of this Manuscript for Dr.
Thibaut, and this copy was available with Sudhākara Dvivedī
when he published his commentary with Text in the Paṇḍita.
During the course of binding, on account of carelessness, many
of the Folios got misarranged, and many of them got fragmented.
Sudhākara Dvivedī emphasises in his Introduction the need
of careful research on the arrangement of these Folios and their
readings. (See Catalogue of the Sanskrit Manuscripts in the Lib-
rary of India Office, Part V,p. 993-995).
Sudhākara Dvivedī with considerable efforts could rearrange
the Text :
Golādhyaya-Bhāṣyam 1 to 45, mutilated at the beginning
Madhyamādhikāra-Bhāṣyam 45 to 59
(Continued from previous page)
भवने अस्य पुस्तकस्य पुटकबन्धनकालेऽनवधानतया पत्राण्यसंगतानि जातानि, बहुत्र खण्डितानि
च सन्ति । तानि कदाचिदनुपयुक्तपत्राणां मध्ये स्युरिति तेषां सम्यगन्वेषणं समुचितम् । (See
Catalogue of the Sanskrit Manuscripts in the Library of India
Office Part, V, Page 993-995)

1. इदं पुस्तकं कस्यचिन्मिथिलाक्षरैर्लिखितस्य पुस्तकस्य प्रत्यन्तरमिति निःसंशयं प्रतिभाति । ११ पत्रे,
   ९ इत्यस्य स्थाने ७ इति लेखात् । एवमन्यत्रापि । विसर्गात् परतः ‘क’ स्थाने ‘क्व’ इति लेखात् ।
   यथा १२ पत्रे, ६ पंक्तौ । तथा विसर्गस्थाने ‘ष’ इति लेखात् , यथा १२ पत्रे, १ पंक्तौ । वाक्य-
   विरामेपि सन्धिकरणात् । यथा गोलाध्यायस्य २१ पत्रे, २ पंक्तौ सर्वमुपपन्नमुक्तमखण्डेन ।
   कुत्रचित् ‘न्त’ स्थाने ‘नू’ इति लेखात् । एकस्मिन् पत्रे मैथिलाक्षरैः श्री गणेशायनमः इति
   लेखाच्च ।

MANUSCRIPTS OF THE BRĀHMASPHUṬASIDDHĀNTA · 59
In this, the commentary is up to verse 31. In the
Spaṣṭādhikāra, the commentary begins from verse 29. Folio 60.
At the top of the Folio, we have the old numbering 1 and 115.
After this, 68 folios are mis-arranged, and there the old
numbering is marked 9.
After that, up to 87 folio, we have in the bound volume a
commentary up to the verse 6 of Tripraśnādhikāra. Here the old
numbering is marked 28. Then we have the commentary up to
verse 27 of the Tripraśnādhikāra. Here the numbering of folios
is marked 1 and 159 (old numbering) and 218 new. After this,
then we have in the bound volume, commentary up to verse 33
of the Tripraśnādhikāra. On the last folio, the old numbering is
marked 5 and 163, the new numbering 222. After this begins the
commentary on the Candragrahaṇādhikāra from verse 4, old
numbering 1 and 297 and new numbering 257. Then follows
the commentary on Sūryagrahaṇa up to verse 23. The last old
folio numbering is 36 and 232, the new numbering 292. This is
the last folio-numbering of the volume in the India Office Library.
After this, begins the commentary on Grahayutyadhikāra, verse 11,
old numbering 1 and 164 and the new numbering 223.
Then follows the Madhyagatyuttarādhyāya up to verse 40,
old folio numbering 119, and new numbering 178. Then we have
the commentary of the Madhyagatyuttarādhyāya beginning from
the verse 45; the old folio numbering 120 and the new numbering
179. Then we have the commentary on Tripraśnottarādhyāya
up to verse 56. Here the last old folio number is 48 and 158,
and the new numbering 217.
In this way, we have the commentary on : the mutilated
Golādhyāya, mutilated Madhyamādhikāra, mutilated Spaṣṭā-
dhikāra, mutilated Tripraśnādhikāra, mutilated Candra-graha-
ṇādhikāra, mutilated Sūrya-grahaṇādhikāra, mutilated Grahayū-
tyadhikāra, Bhagrahayutyadhikāra, Tantra-parīkṣādhyāya (also
known as Dūṣaṇādhikāra), Gaṇitādhyāya (Arithmetic), mutilated
Madhyagatyuttarādhyāya, Sphuṭagatyuttarādhyāya, mutilated
Tripraśnottarādhyāya.¹

1. मया महताऽऽयासेन तत्प्रति-पाठक्रम एवं नियोजितः ।
   गोलाध्यायभाष्यम् १-४५ आदौ खण्डितम् ।
   मध्यमाधिकारभाष्यम् ४५-५१ ।
   (Continued on next page)

60 MANUSCRIPTS OF THE BRAHMASPHUṬASIDDHĀNTA
Sudhākara Dvivedī says that nowhere in the Brahma-
Sphuṭasiddhānta, Madhyamādhikāra, is found the verse:
Saṁsādhya spaṣṭataraṁ bījaṁ nalikādiyantreṇa.
Tat-saṁsṛtagrahebhyaḥ kartavyau nirṇayādeśau.
(This verse has been quoted by Dvivedī in the Gaṇaka-
Taraṅgiṇī, page 19).¹
This verse has been quoted by Dvivedī from the Trans-
lation of Grahālāghava by Mallāri.
· Dvivedī further says that in the Manuscripts available,
there is mentioned a Twenty-fifth Chapter under the title "Dhyā-
nagrahopadeśādhyāya". Dvivedī thinks that this Chapter does not
constitute the Brahmasphuṭasiddhānta proper, which ends in fact
with twenty-four chapters.² In his commentary and Edition in
the Paṇḍita, he has published it as a separate treatise of Brahma-
gupta. Thus he has named his Edition as:
अत्र मध्यमाधिकारे ३१ श्लोकपर्यन्तमेव भाष्यम् ।
अतः स्पष्टाधिकारस्य २६ श्लोकतष्टीकारब्धा ५० पत्रतोऽत्र प्राचीन संख्या पत्रोपरि १, तथा
११५ ।
अग्रे ऽच ६८ पत्रमसंगतम् , यत्र प्राचीन संख्या ६ ।
ततः ८७ पत्रपर्यन्त संलग्नग्रन्थस्त्रिप्रश्नाधिकारस्य ६ श्लोकपर्यन्तं टीका । अत्र प्राचीनपत्रसंख्या
२८। ततस्त्रिप्रश्नाधिकारस्य २७ श्लोकतष्टीका, पत्रसंख्या प्राचीना १ तथा १५६, नवाना २१८ ।
अग्रे संलग्न ग्रंथस्य त्रिप्रश्नाधिकारस्य ३३ श्लोकपर्यन्तं टीकान्तिमपत्रप्राचीनसंख्या ५ तथा १६३,
नवीना संख्या च २२२ । ततश्चन्द्रग्रहणाधिकारस्य ४ श्लोक टीकारब्धा प्राचीनपत्रसंख्या १ तथा
२६७ । नवीना संख्या च २५७ । ततः सूर्यग्रहणास्य २३ श्लोकपर्यन्तं टीका । अन्तिम प्राचीन पत्र-
संख्या ३६ तथा २३२, नवीना संख्या च २६२ । इण्डिया-आफिस पुस्तकपुटके चेयमन्तिमपत्रसंख्या ।
ततो ग्रहद्युत्यधिकारस्य ११ श्लोकतष्टीकारब्धा यत्र प्राचीनपत्रसंख्या १ तथा १६४, नवीना संख्या
२२३ । ततो मध्यमगत्युत्तराध्यायस्य ।
४० श्लोकपर्यन्तं संलग्नग्रन्थौ यत्रान्तिमप्राचीनपत्र संख्या ११६ नवीना संख्या च १७८ ।
ततो मध्यगत्युत्तराध्यायस्य ४५ श्लोकतष्टीकारब्धा । अत्र प्राचीनपत्रसंख्या १२० नवीनासंख्या च १७६ ।
ततस्त्रिप्रश्नोत्तराध्यायस्य ५६ श्लोकपर्यन्तं टीका । अत्रान्तिमप्राचीनपत्रसंख्या ४६ तथा १५८, नवीना
संख्या च २१७ ।
खण्डितगोलाध्यायस्य । खण्डितमध्यमाधिकारस्य । खण्डितस्पष्टाधिकारस्य । खण्डित-
त्रिप्रश्नाधिकारस्य । खण्डितचन्द्रग्रहणाधिकारस्य । खण्डितसूर्यग्रहणाधिकारस्य । खण्डितग्रहद्युत्यधि-
कारस्य । भगृहद्युत्यधिकारस्य । तन्त्रपरीक्षाध्यायस्य (दूषणाधिकारस्य) । गणिताध्यायस्य पाटीग-
णितस्य) । खण्डितमध्यगत्युत्तराध्यायस्य । स्फुटगत्युत्तराध्यायस्य । खण्डितत्रिप्रश्नोत्तराध्यायस्य च
टीका वर्त्तते ।

1. सर्वेष्वपि पुस्तकेषु 'संसाध्य स्पष्टतरं बीजं नलिकादियंत्रेण' इत्यादिश्लोको मध्यमा-
   धिकारे नास्ति । मया गणकतरंगिण्यां मल्लारिकृतग्रहलाघवटीकातो व्यलेखि (द्रष्टव्या गणकतरंगिण्यो
   पृ० -१८।१९) ।

MANUSCRIPTS OF THE BRAHMASPHUṬASIDDHĀNTA 61
Brāhmasphuṭasiddhānto Dhyānagrahopadeśādhyāyaśca or
Brāhmaphuṭasiddhānta and Dhyānagrahopadeśādhyāya by Brahma-
gupta (1902)¹
The manuscript of this small treatise was also mutilated,
and Dvivedī took special pains in editing it, and he revised the
calculations also incorporated in this treatise.
The small treatise, Dhyānagrahopadeśādhyāya must have
been composed prior to the Brāhmasphuṭasiddhānta, since we
find a verse in the last Chapter (the 24th Chapter also known as
the Sañjñādhyāya of the Brāhmasphuṭasiddhānta), verse 9, a
reference to this book :
How could this result be obtained in a simple way has been
shown by me in the Dhyānagrahopadeśādhyāya of 72 Āryā verses,
and therefore, it is not repeated here. BrSpSi. XXIV. 9)
In the Dhyanagrahopadeśādhyāya, we have a verse 61,
which is also found in the Khaṇḍakhādyaka (KK. I. 21) :
Navatithayaḥ (159) divided by aṣṭi (16), pañcarasāḥ (65)
divided by vasu (8), 10 divided by 3, each multiplied by the
equinoctial shadow are the (tabular differences of) ascensional
difference expressed in vināḍīs. (KK. I. 21; DhGr. 61)
This verse then indicates that the Dhyānagrahopadeśādhyāya
has been composed after the Khaṇḍakhādyaka. It may also be
possible that the Brāhmasphuṭasiddhānta and the Dhyānagraho-
padeśādhyāya were simultaneously written, and the above verse
(DhGr. 61) was repeated again in the Khaṇḍakhādyaka.
Sudhākara Dvivedī has taken the help from the commentary
of Pṛthūdaka Svāmī in the Chapters on Pāṭīgaṇita (Arithmetic),
and has quoted the examples from this commentary. At many
places he has corrected the readings which were mutilated in
manuscripts.

1. उपलब्धमूलपुस्तकत्रये पंचविंशतितमेऽध्याये वस्तुतो ब्रह्मगुप्तकृतो ध्यानग्रहोपदेशाध्यायो
   वर्त्ततेऽतो मयायं पृथक्त्वेन तन्नाम्ना मुद्रितः । अत्र बहुत्र स्खलितानि पदानि तानि गणितेन संशोध्य
   मुद्रितान्यपि सुधीभिर्भृशं विचिन्त्यानि ।
   अस्य सिद्धान्तस्य चतुर्विंशतितमेऽन्तिमे संज्ञाध्याये (पृ० ४०८)
   गणितेन फले सिद्धिर्ब्राह्मे ध्यानग्रहे यतोऽध्याये ।
   ध्यानग्रहो द्विसप्ततिरायां न लिखितोऽत्र मया ॥
   इति नवमश्लोकेन ध्यानग्रहोपदेशाध्यायस्य रचनेतरसिद्धान्तरचनातः पूर्वं विभाति परन्तु
   तत्राहर्गणसाधनप्रकारेण सिद्धान्तरचनाकाल एवास्य रचना सिध्यति तथा 'नवतथ्योष्टिविभक्ता' इत्यादि-
   पद्येन खण्डखाद्यरचनातः पश्चात् सिध्यति ।