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

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

90 BRĀHMASPHUṬASIDDHĀNTA & KHAṆḌAKHĀDYAKA
the nodes of the planets to be the same as in the Āryabhaṭīya.
All these are the same as given by Brahmagupta in the Khaṇḍa-
khādyaka.
In the stanza 33 we have rules for finding the geocentric
longitudes of planets which may be taken to be the same as in
the Khaṇḍakhādyaka¹; compare these values with those in the
Sūryasiddhānta of Varāhamihira in the Pañcasiddhāntikā, XVII.
6, but slight different from the Āryabhaṭīya.²
The last stanza of the Mahābhāskarīya (35) gives the dimen-
sions in yojanas of the orbits of planets ; these are the same as in
the modern Sūryasiddhānta.
Thus we find a great semblance in the constants as given by
the Mahābhāskarīya of Bhāskara I, of the Sūryasiddhānta as
given by the Pañcasiddhāntikā and understood by Varāhamihira,
and also the constants as given by Brahmagupta in his treatises,
specially the Khaṇḍakhādyaka. It must not be forgotten that
the same Āryabhaṭa I who is the celeberated author of the
Āryabhaṭīya is also the author of another treatise very often
referred to as the Tantra.
I shall quote Prabodha Chandra Sengupta in connection
with these similarities, and the great influence of Āryabhaṭa on
Indian Astronomy. He writes in his Introduction to the
Khaṇḍakhādyaka as follows :
We have shown that there is much resemblance in the
constants between the Sūryasiddhānta of Varāha and
the Khaṇḍakhādyaka and for the matter of that with
the Tantrāntara of Āryabhaṭa I. In my papers “Ārya-
bhaṭa and Āryabhaṭa's Lost Work”, I have establish-
ed the fact that the Sūryasiddhānta as it existed
before the time of Varāha, was made more accurate
by him by borrowing the constants from Āryabhaṭa's
ārdharātrika system. That there was a Sūryasiddhānta
१. शीघ्रफलाद्धं मध्ये मन्दफलाद्धं च मन्दशीघ्रफले ।
सकले मध्ये स्पष्टः शीघ्रं मध्योनकं केन्द्रम् ॥ —KK. II. 18
२. मन्दोच्चाच्छीघ्रोच्चादर्धमृणधनं ग्रहेषु मन्द्येषु ।
मन्दोच्चात्स्फुट मध्याच्छीघ्रोच्चाच्च स्फुटा ज्ञेयाः ॥
शीघ्रोच्चादर्धोनं कर्तव्यमृणं धनं स्वमन्दोच्चे ।
स्फुटमध्यौ तु भृगुबुधौ सिद्धान्मन्दात्स्फुटौ भवतः ॥ —Arya. III. 23-24

ĀRYABHAṬA & BRAHMAGUPTA CONTROVERSY 91
before the time of Varāha, is seen from Section 6 of
the Table on page xii given before. This point is made
clear from another consideration, viz., the star table
in the modern Sūryasiddhānta, which unmistakably
points to the conclusion that the longitudes of some
stars, e.g., Spica etc., correspond to a time much ante-
rior to that of Āryabhaṭa I. The great fame of Ārya-
bhaṭa I induced Varāha, the first maker of a neo-Sūrya-
siddhānta to use the elements of Āryabhaṭa's ardha-
rātrika system to supplant the older materials in it.
No wonder, therefore, that there is an opinion in
favour of the hypothesis that Āryabhaṭa I was the
author of the Sūryasiddhānta. If there were a shadow
of truth in it, Varāha would have admitted it. Albe-
runi indeed says that the Sūryasiddhānta was compos-
ed by Lāṭa (Alberuni's India, translated by Sachau,
Vol. I.p. 153). We now know that this Lāṭa or Lāṭa-
deva was one of the first pupils of Āryabhaṭa I. He
was the expounder of the Romaka and Paulīśa Siddhāntas
as we learn from Varāhamihira's Pañcasiddhāntikā,
(I.3). As Alberuni's statement is not corroborated by
Varāha, we are not inclined to take it as correct. None
of the earlier writers suggest that the Sūryasiddhānta
was in any way modified or changed by Ārya-
bhaṭa I.
It has now been established beyond doubt that the
same Āryabhaṭa was the author of the Aryabhaṭīya and
another Tantra which is now lost. There is reason in
support of hypothesis that this Tantra itself was the
first work of Āryabhaṭa I and that the Aryabhaṭīya was
the second work from the order in which Varāha
mentions them in the Stanza quoted earlier. If this
hypothesis be true, the stanza in the Aryabhaṭīya¹,
which was translated by me as :
“Now when sixty times sixty years and three quarter
yugas also have elapsed, twenty increased by three
years have elapsed since my birth.”

1. षष्ट्यब्दानां षष्टिर्यदा व्यतीतास्त्रयश्च युगपादाः ।
   व्यधिका विंशतिरब्दास्तदेह मम जन्मनोऽतीताः ॥ —Ārya. III. 10.

92 BRĀHMASPHUṬASIDDHĀNTA & KHAṆḌAKHĀDYAKA
should now be translated thus :
"In this Mahāyuga when sixty times sixty years and
three quarter yugas also had passed, twenty increased
by three years had elapsed since my birth."
Now Bhāskara I the author of the Mahābhāskarīya and the
Lagubhāskarīya, wrote a commentary on the Āryabhaṭīya. The
author commenting on this stanza observes that :
"Or this was addressed by Āryabhaṭa when expound-
ing the science to Pāṇḍuraṅgasvāmin, Lāṭadeva
Niḥśaṅku and other pupils."¹
This direct pupil of Āryabhaṭa I also says that this stanza
does not show that the Āryabhaṭīya was composed when Ārya-
bhaṭa I was only 23 years old, but refers to the time when he
probably began his career as a teacher of Astronomy.
Senagupta out of his discussion concludes that we are not
justified in accepting that the Āryabhaṭīya was composed when
Āryabhaṭa was only 23 years of age. This treatise as it exists in
the present form must have been the composition of a mature
age; it is a treatise highly finished in form; the date mentioned in
this great work refers to a date when its author became a
reputed guru or teacher.
Alberuni and Brahmagupta
Dr. E.C. Sachau in his translation of Alberuni's India (vol.
II, p. 304) speaks of Brahmagupta in the following words :
Brahmagupta holds a remarkable place in the history
of Eastern civilization. It was he who taught the
Arabs astronomy before they became acquainted with
Ptolemy; for the famous Sindhind of Arabian litera-
ture, frequently mentioned but not yet brought to
light, is a translation of his Brahmasiddhānta; and the
only other book on Indian astronomy, called Atarkand,
which they knew, was a translation of his Khaṇḍa-
khādyaka.
Brahmagupta, the celebrated author of the Brāhmasphuṭa-
siddhānta, has another great work as we have said before to his
credit which goes by the name Khaṇḍakhādyaka. This has

1. एतदेवाचार्य्यार्य्यभटस्य शास्त्रव्याख्यान समये वा पाण्डुरङ्गस्वामिलाटदेवनिःशङ्कुप्रभृतिभ्यः
   प्रोवाच ।

ĀRYABHAṬA & BRAHMAGUPTA CONTROVERSY 93
already been said that perhaps to meet the popular demand,
Brahmagupta in this treatise took upon himself the task of sim-
plifying Āryabhaṭa's ardharātrika system or the system of
midnight day reckoning. Alberuni, the author of the Indika has
made several references or quotations from the Khaṇḍakhādyaka
proper and also its supplement, known as Uttara-Khaṇḍa-
khādyaka.
(a) There is a reference to the accepted circumference of
the Earth, as given in the Khaṇḍakhādyaka (Sachau's
Alberuni, Vol. I, p. 312)
Multiply the difference in longitude (from Ujjayinī) by
the (mean) daily motion of a planet (in minutes) and
divide by 4,800; apply the quotient taken as minutes
negatively in places east of the meridian line of Ujja-
yinī and positively in places lying west.¹
(b) The rules for finding the ahargaṇa as given in the
Khaṇḍakhādyaka in I. 3-5 (Sachau's Alberuni, Vol.
II, 46-47), to which Dr. Schram adds a valuable anno-
tation, the constants being taken from the later
Pauliśa Tantra as known to Bhaṭṭotpala. This Pauliśa
astronomy is derived from Āryabhaṭa I's ardharātrika
system.²
(c) A quotation from the Uttara Khaṇḍakhādyaka
(Sachau's Alberuni, Vol. II, pp. 84-86) which Sengupta
has given in his translation, Chapter X, pp. 148-152.
(d) A quotation also probably from the Uttara Khaṇḍakhā-
dyaka (Sachau's Alberuni, Vol. II, p. 87). These stanzas
are found in the Brāhmasphuṭasiddhānta, XIV, 47-52,
also quoted by Bhaṭṭotpala as occurring in the Brahma
Siddhānta in his commentary on the Bṛhat-Saṁhitā,
IV, 7. The manuscripts which Sengupta used did not
show them as occurring in the Uttara Khaṇḍakhādyaka.
These relate to the dimensions of the nakṣatras as seen,
as distinguished from the same as calculated.

1. उज्जयिनी-याम्योत्तर-रेखायाः प्रागृणं धनं पश्चात् ।
   देशान्तर भुक्तिवधात् ख खाष्टवेदैः कलाद्याप्तम् ॥

• KK. I, 15.

2. • KK, Pt. B. Misra's edition, p. 145.

94 BRĀHMASPHUṬASIDDHĀNTA & KHAṆḌAKHĀDYAKA
(e) Two quotations from the Uttara Khaṇḍakhādyaka rel-
ating to the celestial co-ordinates of Canopus and Sirius
(Sachau's Alberuni, Vol. II, p. 91). Present manus-
cripts do not show these stanzas, which are probably
the same as stanzas 35-36 and 40 of Chapter X of the
Brāhmasphuṭasiddhānta.
(f) Two quotations from the Khaṇḍakhādyaka proper as
alleged by Alberuni (Sachau's Alberuni, Vol.II, p. 116).
According to Āmarāja, the first is a couple of stanzas of
which the author is Bhaṭṭotpala and not Brahmagupa.
The second quotation cannot be traced. These relate to
finding the possibility of an eclipse whether of the Sun
or of the Moon.
(g) Two quotations from the Khaṇḍakhādyaka proper as
asserted by Alberuni (Sachau's Alberuni, Vol. II. p.
119). These relate to finding the Lords of the year and
of the month. According to Āmarāja the rules in
question were given by Bhaṭṭotpala and not by Brah-
magupta. Pṛthūdaka in his commentary on the first
chapter at its concluding portion says ;
“In this work the Khaṇḍakhādyaka, the teacher
(Brahmagupta) has not given the rules for finding
the Lords of the year and the month¹.”

1. अथाऽत्र खण्डखाद्यके वर्षाधिपमासाधिपा नयनमाचार्येण नाभिहितम् ।
   —: ० :—
   Reference
   P.C. Sengupta : The Khaṇḍakhādyaka, 1934.
   K.S. Shukla : Mahābhāskarīya, 1960.
   K.S. Shukla : Sūrya-Siddhānta, 1957.