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

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.

Chapter IV
Brahmagupta’s Originality
in the Khaṇḍakhādyaka
Sengupta in his Indroduction to the Commentary of the
Khaṇḍakhādyaka has discussed this point. We shall reproduce
here some of the points mentioned by him.
Brahmagupta’s Khaṇḍakhādyaka
(i) Brahmagupta does not accept the system of Āryabhaṭa
but has simplified it in the Khaṇḍakhādyaka proper ; and here
he has given the system which he thinks to be correct.
Uttar Khaṇḍa Khādyaka
(ii) In the Uttara-Khaṇḍakhādyaka, he has further correc-
ted some of his results, given earlier in the Khaṇḍakhādyaka
proper. In the proper Khaṇḍakhādyaka, Brahmagupta assignes
to the longitude of the Sun's apogee the value 80º, whereas in
the Uttara text he corrects it to 77º (UKK. 4) :
As the process of finding the apparent places of planets
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 (2 signs 17º=60 plus 17 degrees=77º).¹
Compare this with the value given in the Khaṇḍakhādyaka
proper (I.13)² :
The longitude of the Sun's apogee is 80º [KK.I,13] (The
Sun's apogee is 80º or two signs plus 20 degrees) inocco means

1. न स्फुटमार्य्यभटोक्तं स्पष्टीकरणं यतस्ततो वक्ष्ये ।
   भानुमतो मन्दोच्चं राशिद्वयमंशकाश्च सप्तदश ॥ UKK. IX 4
2. भागाशीतिरितिनोच्चं शशिनः पादोनकृत शरकृतोनाद् ।
   भगणादि द्विस्त्रिर्दैर्वसुशुनव यम नव गुणैः सकलम् ॥ KK. I, 13

96 BRAHMAGUPTA'S KHAṆḌAKHĀDYAKA
mandocca of the Sun). The value given in the Pañcasiddhān-
tikā, IX.7-8) is also the same.
Let us compare it with the present value. According to
the astronomical constants as given in the Conn. des Temps., the
longitude of the Sun's apogee in 499 A.D. (i.e. 1,400 years before
1900 A.D.) was
—77° 19′19.44″ according to Conn.des Temps’
equation.
—76° 40′37.22″ according to Newcomb's equation.
The mean of these two values is very nearly 77⁰ as given
by Brahmagupta in the Uttara text. Thus the value given by
Brahmagupta is more correct than the value given by Āryabhaṭa.
The Āryabhaṭīya gives the value 78⁰ which is less correct.
Brahmagupta more correct than Āryabhaṭa
(iii) Brahmagupta detected that Āryabhaṭa had made the
Moon's apogee quicker and nodes slower, than they really are.
In both the cases, Brahmagupta made rather an over-correction.
We shall give the extract from Uttara-Khaṇḍakhādyaka in this
connection :
Multiply the ahargaṇa by 110, increase the product
by 511 and divide by 30, 31; subtract the result taken
as revolutions, etc., from the mean Moon; the final
result is the Moon's apogee.¹
Evidently Brahmagupta assumes that the anomalistic
month = 3031/110 days. This convergent to the anomalistic
month was known to the author of the Vasiṣṭha Siddhānta as
summarised in the Pañcasiddhāntikā² (II-2-6).
According to Brahmagupta, the length of the anomalistic
month
= 1582236450000—4320000000 days. (BrSpSi. I.15,16,18,
───────────────────────── and 20)
577533000000—488105858
= 27.55454641 days which is for 1900 A. D.
= 27.5545502 days according to Radau.
= 27.554602 according to the Āryabhaṭīya.

1. द्युगणात् ख रुद्र गुणिताद् भवशरयुक्ताच्छशित्रिखाग्नि हृतात् ।
   भगणादि फलं शोध्यं घसचन्द्राच्छशाङ्कोच्चम् ॥ (UKK. IX. 5)

UTTARAKHAṆḌA KHĀDYAKA 97
Here also Brahmagupta is more accurate.
Again, the length of the sidereal period of the Moon's
apogee
= 1577918450000 / 488105858 days
= 3232.732048 days.
Āryabhaṭa's value of the same is 3231.987844 days, and the
modern value is 3232.3754 days. Hence Brahmagupta's result
is by 0.3566 of a day out, while Āryabhaṭa's is by 0.3876 of a
day in.
Further in the Uttara-Khaṇḍakhādyaka(IX.10) we
have : Deduct 354½ from the ahargaṇa, divide the
remainder by 6792 ; subtract the quotient that is
obtained in revolutions etc. from the circle : the
result is the longitude of the ascending node.¹ (IX.10)
Here Brahmagupta gives the approximate period of the
sidereal revolution of the Moon's node to be=6792 days. This
according to his Brāhmasphuṭasiddhānta = 1577916450000 / 232311168 days
= 6792.25396 days, which according to Lockyer would be
6793.39108 days and according to the Khaṇḍakhā

98 UTTARAKHAṆḌA KHĀDYAKA
one-sixteenth.¹
This stanza says that in 499 A.D., Mars's aphelion point
had a longitude of 127⁰ ; of Jupiter the longitude of the aphelion
was 170⁰. ( KK.II. 6²)
According to Newcomb's rule, the longitude of the aphelion
point of Mars in 499 A. D. works out to have been = 128⁰28'12''.
According to the Conn. des. Temps' rule, the same was 128⁰27'51''.
Hence Brahmagupta's determination of Mars's aphelion is correct
within 1⁰30' and is therefore, quite satisfactory. According to
the Khaṇḍakhādyaka proper it was 110⁰, and according to the
Āryabhaṭīya 118⁰.
Of planets, beginning with Mars, the degrees of longitude
of the apogees are respectively 11, 22, 16, 8 and 24, each multiplied
by 10. (KK .II. 6)
Thus the longitudes of apogees of Mars = 110 (3 signs 20⁰);
of Mercury = 220⁰ (7 signs 10⁰) ; of Jupiter = 160⁰ (5 signs 10⁰);
of Venus = 80⁰ (2 signs 20⁰) and of Saturn = 240⁰ (8 signs).
Compare these values with those given in the Pañcasiddhāntikā
XVII. 2 (the Sūrya-siddhānta).
Again according to this stanza Jupiter's aphelion had a
longitude of 170⁰ in 499 A.D.
According to Conn. des. Temps' rule the same was 170⁰25'.
Thus here too, Brahmagupta is very accurate. According to
the Khaṇḍakhādyaka proper, Jupiter's aphelion had a longitude
of 160⁰ (KK. II.6) and according to the Āryabhaṭīya, the value
was 180⁰.
Brahmagupta First to Use Second Differences
All these illustrations reproduced here very well establish
the point that the great Indian astronomers from Āryabhaṭa I
to Brahmagupta were aware of the methods of separating the
two distinct planetary inequalities, viz., that of the apsis and
of conjunction in the cases of the five ‘star’ planets (PSi. Intro-
duction Lii). In the Khaṇḍakhādyaka, Brahmagupta having
given the “sines” and the equations of the Sun and the Moon

1. सप्तदशांशैरधिकं भौमस्योच्चं गुरोर्दशभिरंशैः ।
   सितशीघ्रात् कृतमुनयो लिप्ताः शोध्याः शनेः फलं मान्द्यम् ।।
   पञ्चांशोनं शैघ्यं षोडशभागाधिकं बुधस्य फलम् ।। —UKK. IX. 11
2. मन्दांशा दशगुणिता रुद्रा द्वियमाश्व षोडशाष्टजिनाः । —KK. II. 6

BRAHMAGUPTA FIRST TO USE SECOND DIFFERENCES 99
at the interval of 15° of arc of the mean anomaly, in the Uttara
Khaṇḍakhādyaka teaches, for the first time in the history of
mathematics, the improved rules for interpolation by using the
second difference. This very important feature I am reproducing
here from the translation by Senagupta of the verse ¹
(UKK. 8) :
Multiply the residual arc left after division by 900'
(i.e by 15°), by half the difference of the tabular
difference passed over and that to be passed over and
divide by 900' (i. e. 15°) : by the result increase or
decrease, as the case may be, half the sum of the same
two tabular differences ; the result which, whether less
or greater than the tabular difference to be passed, is
the true tabular difference to be passed over. (UKK. 8)
The rule given here applies to the case of all functions
hitherto considered in the Khaṇḍakhādyaka, which are tabulated
at the difference of 15° of arc of the argument. They are :
(i) the tabular differences of the Sun's equation,
(ii) the tabular differences of the Moon's equation.
(iii) the tabular differences of the 'sines'.
Sengupta has illustrated the rule by an example belonging
to the table of sines.
Illustration—To find the 'sine' of 57°.
Brahmagupta's table of sines in the Khaṇḍakhādyaka is as
follows :
Thirty increased severally by nine, six and one; twenty-
four, fifteen and five, are the tabular differences of
sines at intervals of half-a-sign. For any arc, the 'sine'
is the sum of the parts passed over, increased by the
proportional part of the tabular difference to be passed
over.² (*KK.*I.30 ; also III.6)

1. गतभोग्य खण्डकान्तरदलविकलवधात् शतैर्नवभिराप्त्या ।
   तद्युतिदलं युतोनं भोग्यादूनाधिकं भोग्यम् ॥ —UKK. IX. 8
2. त्रिंशत् सन्नवरसेन्दुजिन (तिथि) विषयागृहार्द्धं चापानां ।
   भर्द्ध्ज्या खण्डानि ज्या भुक्तैक्यं सभोग्यफलम् ॥ —KK. I. 30; also III. 6

100 BRAHMAGUPTA'S KHAṆḌAKHĀDYAKA
This can be shown in the tabular form thus :

Now 57° = 3420 minutes = 900' × 3 + 720'. Thus three of the
tabular differences are considered as passed over ; the last one
being 31 and the one to be passed over is 24.
The true tabular difference by the rule, for arc 57°,
= (31 + 24) / 2 - (720 / 900) × (31 - 24) / 2
Hence the 'sine' of 57°
= 39 + 36 + 31 + (720 / 900) [ (31 + 24) / 2 - (720 / 900) × (31 - 24) / 2 ]
= 125.76
As worked out from the logarithm tables the same comes
out to be 125.80.
Again 'sine' of 57° from Brahmagupta's formula
= 106 + (720 / 900) × 24 + ((31 - 24) / 2) × (720 / 900) - [(720 / 900)] × (31 - 24) / 2
= 106 + (720 / 900) × 24 + (720 / 900) { (720 / 900) - 1 } × (24 - 31) / 2
This in fact is the modern form the interpolation equation
up to the term containing the second difference. Brahmagupta thus
takes a decidedly improved step here and is undoubtedly the first
man in the history of mathematics who has done this. One should
also remember that in the case where the function is not tabula-
ted at a constant interval, Brahmagupta's rule is remarkable.

BRAHMAGUPTA AND SINE RULE 101
Brahmagupta First to Introduce
Sine Rule in Indian Plane Trigonometry
In this connection, we shall reproduce the following verse
from the Khaṇḍakhādyaka :
Multiply the ‘sine’ of the (Śīghra) anomaly by the ‘sine’
of the maximum Śīghra equation and divide by the
‘sine’ of the corresponding Śīghra equation, the result
is the ‘Śīghra hypotenuse’ when the (Śīghra) anomaly
is half a circle, this śīghra hypotenuse is equal to the
radius diminished by the ‘sine’ of the maximum
equation ; when the anomaly is equal to the whole
circle, the same is equal to the radius increased by the
same ‘sine’ of the maximum equation.¹
Let S, E and P be the positions of the Sun, the Earth and the
planet, say Mars, respecti-
vely. Complete the paralle-
logram SEMP ; with M as
centre and MP as radius,
describe a circle. This circle
is the epicycle of conjunction
of Mars. Produce EM to cut
this circle at K. The ∠PMK
= ∠S′SP (the point S′ is on
[Fig. 4] ES produced ), the angle
gained by the Earth over Mars since the preceding conjunction.
The ∠PMK is called the śīghra anomaly or anomaly of conjunc-
tion. We take EM=360, and MP=234. The ∠PEM, which is
equal to ∠PES, the annual parallax of Mars, is called the śīghra
equation. The ∠MPE is equal to the ∠SEP, the elongation.
The ∠PMK is given, and PM and ME are also given. Hence in
the triangle MPE, we have
tan ½ (P−E) = (EM−MP)/(EM+MP) tan ½ PMK
= 126/594 tan ½ PMK
∴ L tan ½(P−E) = log [126/594] + L tan ½ PMK
We have also ½ (P+E) = ½ ∠PMK

1. केन्द्रज्याऽन्त्यफलज्या गुणिता फलजीवया हृताकर्णः ।
   त्रिज्यास्य फलज्योना चक्रार्द्धे संयुता चक्रे ॥ KK. VI. 1.