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

284 ARABIC AND INDIAN DIVISIONS OF THE ZODIAC
and γ Aquilae. According to Arabs, the twenty-third lunar
mansion is Balā, which consists of two stars in the left hand of
Aquarius. Here again Arabian and Hindu divisions are at
variance.
24. Dhaniṣṭhā, the twenty-fourth nakṣatra, is represented
by a drum or tabor. It comprises four stars, the westernmost
of which is placed in 36° N and according to Brahmagupta,
Śiromaṇi and the Sūrya-Siddhānta in 290° E (Grahalāghava
gives 286°). This longitude of the circle of declination and the
distance of the star on it from the ecliptic indicate the Dolphin :
and the four stars are α, β, γ and δ Dolphini. The correspond-
ing lunar mansion of Arabs is Sāud, which comprises two stars
in Aquarius (β and ζ Aquarii). Here again the two divisions
disagree completely.
24. Śatabhiṣak, the twenty fifth nakṣatra, is a cluster of 100
stars figured by a circle. The principal or the brightest has no
latitude; or only a third, or at utmost half, a degree of south
latitude; and longitude 320°. This best corresponds with λ
Aquarii. According to Arabs, the twenty-fifth lunar mansion
is known as Akhbiyah which consists of three stars only, placed
in the wrist of the right hand of Aquarius. However, it appears
from Ulugh Beg's tables, as well as from Mohammad of Tīzīn's,
that four stars are assigned to this mansion. The Indian and
Arabian systems of division differ considerably but less widely
according to some.
26. Pūrva-Bhādrapada, the twenty-sixth nakṣatra, consists
of two stars represented by a couch or bed, or else by a double
headed figure, one of which is placed in 24° N and 325° or 326°
E. The only conspicuous star nearly in that position is the bright
star in Pegasus (α Pegasi) and the other may be the nearest
considerable star in the same constellation (ζ Pegasi). The
twenty-sixth Arabian lunar mansion is Mukaddim, consisting
of two brightest stars in Pegasus (α and β). Here the Indian
and Arabian divisions show concurrence.
27. Uttar-Bhādrapada, the twenty-seventh nakṣatra, consists
of two stars, figured as a twin or a person with double face, or
else as a couch. The position of the most northerly of the two

ARABIC AND INDIAN DIVISIONS OF THE ZODIAC 285
is in 26° or 27°N and 337° E. which probably indicates the bright
star in the head of Andromeda, and the other star to be the
one in the extremity of the wing of Pegasus (γ Pegasi). This
exactly agrees with the twenty-seventh lunar mansion of Arabs
named as Muakkher. Ulugh Beg assigns those stars to it.
28. Revatī, the twenty-eighth nakṣatra, comprises thirty-
two stars figured as a tabor. The principal star is the southern-
most one, it has no latitude, and two of them assert no longitude,
but some make it ten minutes short of the origin of the ecliptic,
viz. 359° 50'. This clearly marks the star on the ecliptic in the
string of the Fishes (ζ Piscium). The ascertainment of this star is
important in regard to the adjustment of the Hindu sphere. The
Arabic name for this mansion is Risha, signifying a cord. But
the constellation as described by Jauhari and cited by Golius,
consists of a multitude of stars in the shape of a fish and termed
Betnu'lhūt; in the navel of which is the lunar mansion. Moham-
mad of Tīzīn alse makes this lunar mansion to be the same with
Betnu'lhūt, which appears, however, to be the bright star in the
girdle of Andromeda (β Andromedae) though others describe it
as the northern fish, extending, however, to the horns of Ram.
The lunar mansion and the Indian asterism, therefore, are not
reconcileable in this last instance.
I leave it to the readers to draw an inference as to the
concurrence of the divisions of zodiac in Indian and Arabian
systems. I would personally agree with Sir William Jones that
the agreements are by chance. Arabs derived the idea of
dividing zodiac in 27 or 28 mansions from Indians, or may have
got it from Greeks, and then they proceeded in their own way
for details. I do not agree with those scholars who sometimes
state that the Hindus took the hint of dividing the ecliptic from
Greeks. The Atharvaveda devotes a number of Suktas or
hymns on Nakṣatras, and I have shown elsewhere that inspired
by these hymns, Gārgya was the first Ṛṣi who detailed out
the nakṣatras. This happened much before Greeks developed
even their first notions of astronomy. While the concept of 27
nakṣatras is Vedic and most ancient and of purely Indian origin
the concept of 12 Rāśis (signs) or twelve constellations is proba-
bly inspired from Greeks. [The names Kanyā, (virgo), Tulā

286 ARABIC AND INDIAN DIVISIONS OF THE ZODIAC
(Libra),Vṛścika (Scorpio), Dhanu, (Sagittarius), Makara, (Capri-
corn), Kumbha (Aquarius), Mīna (pisces), Meṣa (Aries), Vṛṣa
(Taurus), Mithuna (Gemini), Karka (Cancer), and Siṁha (Leo)
were not used for Rāśis or signs in the Vedic times]. I shall con-
clude this description with a passage from Colebrooke :
The result of comparison shows, I hope satis-
factorily, that the Indian asterisms, which mark the
divisions of the ecliptic, generally consist of nearly the
same stars, which constitute the lunar mansions of
the Arabians : but in a few instances, they essentially
differ. The Hindus have likewise adopted the divi-
sion of the ecliptic and zodiac into twelve signs or
constellations, agreeing in figure and designation with
those of the Greeks; and differing merely in the place
of the constellations, which are carried on the Indian
sphere a few degrees further west than on the Grecian.
That the Hindus took the hint of this mode of
dividing the ecliptic from the Greeks, is not perha-
ps altogether improbable; but if such be the origin of it
they have not implicitly received the arrangement
suggested to them, but have reconciled and adapted
it to their own ancient distribution of the ecliptic into
twenty-seven parts.
In like manner, they may have either received or given
the hint of an armillary sphere as an instrument for
astronomical observation ; but certainly they have not
copied the instrument which was described by Ptolemy,
for the construction differs considerably.
Names, Shapes, and Number
of the Stars of the Nakṣatras
The Muhūrta-cintāmaṇi provides a list of shapes associated
to the nakṣatras (MuC. II. 59-60), In this list we are giving the
number of stars as indicated by Varāhamihira, Brahmagupta
and Lalla. The identification given here is as indicated by
E. Burgess, in his Translation of the Sūrya-Siddhānta 1935, p. 378.
(Calcutta). This table has been reproduced here from the
Mahābhāskarīya of Bhāskara I, edited by K.S. Shukla.

NAMES, SHAPES AND NUMBER OF THE STARS 287

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CHAPTER XI
Brahmagupta’s
Astronomy : Its Highlights
Beginning or Starting Point
Very often in Indian astronomy, we come across a term
ahargaṇa (literally meaning collection of days), which means
the number of mean civil days elapsed at mean Sunrise at Laṅkā
on a given lunar day (tithi), since the beginning of Kaliyuga. It
is the beginning of Kaliyuga, which is taken as the starting point
for the reckoning of ahargaṇa. This happened on Friday, Febru-
ary 18, B.C. 3102, at mean sunrise at Laṅkā, when the Sun, Moon,
and the planets are supposed to have been in conjunction at the
first point of the nakṣatra Aśvinī (which is a fixed point situated
near the star ζ-Piscium). According to Āryabhaṭa and Bhāskara
I, the duration of Kaliyuga is 1,080,000 solar years. Four times
this (4,320,000) is the duration in solar years of a bigger unit
called Mahāyuga or even yuga.
Laṅkā in Indian astronomy is a hypothetical place where
the meridian of Ujjain (latitude 23° 11′ N, longitude 75° 52′ E
from Greenwich) intersects the equator. It is one of the four
hypothetical cities on the equator called Laṅkā, Romaka,
Siddhapur and Yamakoṭi (or Yavakoṭi). The Sūrya-siddhānta
describes Laṅkā as a great city (mahāpurī) situated on an island
to the south of Bhāratavarṣa.¹ The present Ceylon is not the

1. समन्तान्मेरुमध्यात्तु तुल्यभागेषु तोयधेः ।
   द्वीपेषु दिक्षु पूर्वादिन्नगर्यो देवनिर्मिताः ।।
   भूवृत्त पादे पूर्वस्यां यवकोटीति विश्रु ता ।
   भद्राश्व वर्षे नगरी स्वर्णप्राकारतोरणा ।
   याम्यायां भारतेवर्षे लङ्का तद्वन्महापुरी ।।
   (Cont. on page 290)

290 BRAHMAGUPTA'S ASTRONOMY ITS HIGHLIGHT
astronomical Laṅkā, as it is about six degrees to the north of
equator. The astronomical Laṅkā is mentioned by Brahmagupta
in the beginning of his very first Chapter¹.
According to Brahmagupta all the four yugas of a Catur-
yuga or mahāyuga are not of the equal duration :
Kaliyuga is of 432,000 years. Dvāpara of 864,000 years,
Tretā of 1,296,000 years and Kṛtayuga of 1,728,000 years;
total of the four is 4,320,000 years. Āryabhaṭa regards
all yugas of equal duration, 1,080,000 years².
The Saka era, which is usually used in Indian astronomy
for the reckoning of years commenced 3179 years after the
beginning of Kaliyuga.
The number of lunar months in a yuga does not coincide
with the number of solar months. Thus we have the conception
of the Intercalary months : the number of intercalary months in
a yuga denotes the excess of the number of lunar months in a
yuga over the number of solar months in a yuga. Thus in a
yuga we have
Lunar months 53,433,336
Solar months 51,840,000
Intercalary months 1,593,336
Lunar days 1,603,000,080
Civil days 1,577,917,500
Omitted lunar days 25,082,580
The number of omitted lunar days in a yuga is equal to the
number of lunar days in a yuga minus the number of civil days
in a yuga.
(Cont. from page 289)
पश्चिमेकेतुमालाख्ये रोमकाख्या प्रकीर्तिता ।
उदक्सिद्धपुरी नाम कुरुवर्षे प्रतिष्ठिता ॥ --Sūrya. XII. 36-39

1. चैत्रसितादेरुदयाद्भानोर्दिनमासवर्षयुगकल्पाः ।
   सृष्ट्यादौ लंकायां समं प्रवृत्ता दिनेऽर्कस्य ॥ —BrSpSi. I. 4
2. युगदशभागो गुणितः कृतं चतुर्भिस्त्रिभिर्गुणस्त्रेता ।
   द्विगुणो द्वापरमेकेन संगुणः कलियुगं भवति ॥
   युगपादानायैभटश्चत्वारि समानि कृतयुगादीनि ।
   यदभिहितवान् न तेषां स्मृत्युक्तसमानमेकमपि ॥ —BrSpSi. I. 8-9

UNITS OF TIME 291
Units of time
For the measurements of durations, it is necessary to have
units of time. Brahmagupta gives the following units :¹
6 prāṇas or Asus=1 Ṛkṣa-vināḍikā or nakṣatra-vighaṭikā
or one pala (24 seconds)
60 palas =1 ghaṭikā (24 minutes)
60 ghaṭikās =1 divasa or dina (day) (24 hours)
30 dinas =1 māsa (month)
12 māsas =1 varṣa or year
Similar to the divisions of time, we have the divisions of
an arc :²
Vikalā (or viliptā or viliptikā) =second of arc.
60 vikalās =1 kalā (minute of arc)
60 kalās =1 aṁśa (degree of arc)
30 aṁśas =1 rāśi
12 rāśis =1 bhagaṇa (complete circle,
360°)
Unlike Āryabhaṭa and others, who take kali, dvāpara,
tretā, and kṛta of equal number of years, Brahmagupta regards
kali consisting of 432,000 years, dvāpara twice of it, consisting
of 864,000 years, tretā thrice of Kali consisting of 1,296,000
years and kṛta four times of kali conisting of thus 1,728,000
years, all the four to be making a yuga of 4,320,000 years.³ Further
in the beginning of kṛta there is a sandhyā of 1728000/12 years
(=144,000 years), and at the end of Kṛta, there is a sandhyāṁśa of
144,000 years; similarly in the beginning of tretā, we have a

1. प्राणैर्विनाडिकाऽर्क्षी षड्भिर्घटिका षष्ट्या ।
   घटिका षष्ट्या दिवसो दिवसानां त्रिंशता मासः ॥ —BrSpSi. I. 6
2. मासा द्वादशवर्षं विकलालिप्तांशराशिभगणांतः ।
   क्षेत्र विभागस्तुल्यः कालेन विनाडिकाद्ये न ॥ —BrSpSi. I. 6
   वर्षं द्वादश मासास्त्रिंशद्दिवसो भवेत्स मासस्तु ।
   षष्टि र्नाड्यो दिवसष्षष्टिश्च विनाडिका नाडी ॥ —Ārya. III. 1
   गुर्वक्षराणि षष्टिर्विनाडिकार्क्षी षडेव वा प्राणाः ।
   एवं काल विभागः क्षेत्र विभागस्तथा भगणात् ॥ —Ārya, III. 2
3. खचतुष्टयरदवेदा रवि वर्षाणां चतुर्यु गं भवति ।
   सन्ध्या सन्ध्यांशैः सह चत्वारि पृथक् कृतादीनि ॥
   युगदशभागो गुणितः कृतं चतुर्भिस्रिभिर्गुणस्त्रे ता ।
   द्विगुणो द्वापरमेकेन संगुणः कलियुगं भवति ॥ —BrSpSi. I. 7. 8

292 BRAHMAGUPTA'S ASTRONOMY ITS HIGHLIGHT
sandhyā of 1.296.000/12; i. e. 108.000 years and at the close of
tretā a sandhyāṁśa of 108,000 years. Again, in the beginning
of dvāpara we have a sandhyā of 864.000/12, i. e. 72.000 years
and at the close of dvāpara a sandhyāṁśa of 72000 years; and
similarly at the beginning a sandhyā and at the close a
sandhyāṁśa of 432.000/ 1, i. e. of 36,000 years in the case of kali.
In this respect Brahmagupta appears to follow Manu. the first
author or giver of law. He regards further the following
divisions of time :¹
71 yugas =1 manu
14 manus =1 kalpa
Again, in the beginning, at the middle and at the close of
each manu, there are sandhis, each equal to the measure of kṛta.
Thus, taken as a whole
1 kalpa=71×14 yugas+15 sandhyā-sandhyāṁśa
=994 yugas+15×duration of kṛta
=994 yugas+15×(4×432,000) years
=994 yugas + 6 yugas =1000 yugas
= 1 Brahma-dina (Brahmā's day)
Thus Brahmā's day is regarded as 1 kalpa or one thousand
caturyugis or 1000 yugas or the same as 1000 mahāyugas).
Āryabhaṭa regards a manu to consist of 72 yugas and
therefore a kalpa according to him would be of 14×72 yugas, or
1008 yugas.² Since in the foreign Siddhāntas like Romaka, there
is no reference to yuga, manu and kalpa. Brahmagupta regards
these systems to be unauthoritative.³
We have said that our starting point was the beginning of
Kaliyuga, Friday February 18, B.C. 3102, at mean rise at Laṅkā,
when the Sun, Moon and the planets are supposed to have been
in conjunction at the first point of the Nakṣatra Aśviṇī. This
type of conjunction would again happen after a period of kalpa.

1. मनुरेकसप्ततियुगः कल्पो मनवश्चतुर्दश मनूनाम् ।
   आद्यन्तरान्त सन्धिषु कृतकालोऽस्माद्युग सहस्रम् ॥ —BrSpSi. I. 10
2. दिव्यं वर्ष सहस्रम् ग्रहसामान्यं युगं द्विषट्क गुणम् ।
   अष्टोत्तरसहस्रं ब्राह्मो दिवसो ग्रहयुगानाम् ॥ —Ārya. III. 8
3. युगमन्वन्तरकल्पाः कालपरिच्छेदकाः स्मृतावुक्ताः ।
   यस्मान्नरोमके ते स्मृतिबाह्यो रोमकस्तस्मात् ॥ —BrSpSi. I- 13

UNITS OF TIME 293
We shall have the same type of conjunction of grahocca, mandocca
śīghrocca and pāta after a complete cycle of kalpa as we had in
the beginning of creation. This is a natural observable cycle
which is recognised in Indian astronomy and in no other foreign
system; and hence only the Indian system recognises the time
measure of kalpa.
Ucca or apex is of two kinds : mandocca (apex of the
slowest motion) and śīghrocca (apex of the fastest motion). The
mandocca is that point of a planet's orbit which is at the remotest
distance and where the motion of the planet is slowest. In the
case of the Sun and the Moon, it is the apogee; and in the case of
other planets, it is the apogee or aphelion, the geocentric longi-
tude of the apogee being equal to the helio-centric longitude of
the aphelion. The śīghrocca of a superior planet (Mars, Jupiter
and Saturn) is defined as the mean Sun; that of an inferior planet
Mercury or Venus) śīghrocca is an imaginary body which is
supposed to move in such a way that its direction from the earth
is always approximately the same as that of the actual planet
from the Sun.
The bhagaṇas or the revolution numbers of a planet have
been given by Āryabhaṭa and Brahmagupta¹ both; they mean the
number of revolutions that a planet performs in a certain period,
say a kalpa of 4,320,000,000 years.
The bhagaṇas of planets are given as follows :

1. कल्पेऽर्कबुधसितानां भगणाः शून्यानि सप्त रदवेदाः ।
   प्राग्ब्रजता कुजगुरुशनि शीघ्रोच्चानां स्वकक्षासु ॥
   पञ्चाम्बराणि गुण-गुण पञ्चमुनि स्वरैर्मिताः शशिनः ।
   भौमस्य द्वियमशराष्टपञ्चवसूरसनवद्वियमाः ॥
   कृतवसुखवाष्टनन्द-नवषट्‌त्रि नवागेन्द्वो ज्ञशीघ्रस्य ।
   जीवस्य शरेष्वद्रि षट्पञ्चद्वि कृतरसरामाः ॥
   सितशीघ्रस्य यमलगोवेदनवाष्टाग्निपञ्चयमखनगाः ।
   अष्टनवपञ्च मुनि रसशररस मनवोऽर्कपुत्रस्य ॥
   खाष्टाब्धयो वसुशरवसुपञ्चखचन्द्रक्सुवसुसमुद्राः ।
   द्विनवयमा द्वित्रिगुणाः शरेषु वसवस्त्रि पञ्चरसाः ॥
   शशिवेदा मन्दानामर्कादीनां विलोमपातानाम् ।
   वसूरसरुद्रेन्दुगुणा द्वित्रियमाः सप्तरसपञ्चाः ॥
   शशियमाशरा गुणरसास्त्रिनन्दवसवः समुद्रवसु विषयाः ।
   चन्द्रादीनां पश्चात् व्रजतोऽश्विन्यादि भगणस्य ॥ BrSpSi. I. 15-21

294 BRAHMAGUPTA'S ASTRONOMY ITS : HIGHLIGHTS
| Planet or a body | Bhagaṇas |
| Ravi or Sun | 4,320,000,000 |
| Budha or Mercury | 4,320,000,000 |
| Śukra or Venus | 4,320,000,000 |
| Candra or Moon | 57,753,300,000 |
| Kuja or Bhauma or Mars | 2,296,828,522 |
| Budha-śīghrocca | 17,936,998,984 |
| Bṛhaspati or Jupiter | 364,226,455 |
| Śukra-śīghrocca | 7,022,389,492 |
| Śani or Saturn | 146,567,298 |
| Arka or Ravi-mandocca | 480 |
| Candra mandocca | 488,105,858 |
| Kuja or Bhauma mandocca | 292 |
| Budha-mandocca | 332 |
| Bṛhaspati or Jīva-mandocca | 855 |
| Śukra-mandocca | 653 |
| Śani-mandocca | 41 |
| Candra-pāta | 232,311,168 |
| Kuja or Bhauma-pāta | 267 |
| Budha-pāta | 521 |
| Bṛhaspati or Guru-pāta | 63 |
| Śukra-pāta | 893 |
| Śani-pāta | 584 |
By pāta is meant the ascending node of a planet's orbit (on
the ecliptic).
In a kalpa, the number of bha-bhramas (sidereal days) or
also known as bha-parivartas is 51,040,000,000. If we subtract out
from this number the bhagaṇa of the Sun, we get what is known
as ku-dinas or Savana days or the solar or sacrificial days.
(51,040,000,000—4,320,000,000=46,720,000,000 Sāvana days or
kudinas).
In a kalpa, the number of Ravi-bhagaṇas also correspond to
the number of solar years (Saura-varṣas), i.e., 4,320,000,000; this
number multiplied by 12 gives the number (i.e. 51,840,000,000) of
solar months.
The difference between the candra-bhagaṇas and the Ravi-
bhagaṇas in a kalpa gives the number of lunar months (Cāndra-

UNITS OF TIME 295
māsa) in a kalpa (57,753,300,000-4,320,000,000=53,433,300,000 lunar
months).
By subtracting the number of solar months from the
number of lunar months in a kalpa, one gets the number of
adhi-māsas (additional-months) : 53,433,300,000-51,840,000,000=
1,593,300,000 adhimāsas. This multiplied by 30 gives the number
of lunar days (śaśi-divasa) in a kalpa ; 53,433,300,000×30=
1,602,999,000,000 lunar days. The difference between the lunar
days and kudinas in a kalpa gives the number of avama-dinas
in a kalpa : 1,602,999,000,000-45,720,000,000=1,556,279,000,000.¹
Brahmagupta calculates out the sṛṣṭi-saṁvatsara or the
Creation Era during his year of composition of the Treatise. He
says : Six manus have gone in the kalpa ; the seventh manu is
now running of which have lapsed 27 caturyugis ; of the twenty
eighth caturyugī, the three yugas, kṛta, dvāpara and tretā have
gone by and also of the present kaliyuga 3179 years have
lapsed. The total period thus lapsed on calculation comes to
be 1,972,947,179 years :²
Total Period=6 manus+7 manu-sandhis+27 yugas
+kṛta+dvāpara+tretā+3179 years of kali.
=(6×71×4,320,000 years) +(7×4×432,000 years)+
(27×4,320,000 years)+(1,728,000+1,296,000+864,000)+
3,179=1,972,947,179 years.
=1,840,320,000+12,096,000+116,640,000+3,888,000+
3,179=1,972,947,179 years.
Calculation of Ahargaṇa : The method of calculating ahar-
gaṇa (number of days elapsed since the beginning of kaliyuga)

1. परिवर्त्ताः स्वचतुष्टयशराश्वि रसगुणयमद्विवसुतिथयः ।
   रवि भगणोना भानोः सावनदिवसाः कुदिवसास्ते ॥
   रवि भगणाख्यब्दा द्वादशगुणिता भवन्ति रविमासाः ।
   भगणान्तरं रवीन्द्वोः शशिमासाः सूर्यमासोनाः ॥
   अधिमासाः शशिमासास्त्रिंशद्गुणिता भवन्ति शशिदिवसाः ।
   शशिसावनदिवसान्तरमवमानि तिथिः शशांकदिनम् ॥ —BrSpSi. I. 22-24
2. कल्पपरार्द्धं मनवः षट् कस्य गताश्चतुर्युगत्रिघनाः ।
   त्रीणिकृतादीनिकलेर्गोऽङ्गैक गुणाः शकान्तेऽब्दाः ॥
   नवनगशशि मुनिकृत नव यमनगनन्देन्दवः शकनृपान्ते ।
   सार्धमतीतमनूनां सन्धिभिराद्यन्तरान्तर्गैः ॥ BrSpSi. I. 26-27

: is it a footnote mark, or is the print literally:
- खराऽष्टेषु?
Wait, look at line 39:
-खराऽष्टेषुद्वयष्टशून्यशरारिभिः ।
Wait, look at the first character:
संगुण्याख्या?
No, look at line 39 again very carefully:
First word: संगुण्याख्या?
Wait! Could it be:
संगुण्याख्या-
No, look at the image:
संगुण्याख्या-बराऽष्टेषुद्वयष्टशून्यशरारिभिः ।
Wait, why बरा?
Could it be ऽबरा?
Wait, look at the English text above:
"25,082,580"
Why would the Sanskrit have those characters?
Wait, this is an Indian printed book from around 1970-1980 (looks like Indian National Science Academy, INSA, "BRAHMAGUPTA'S ASTRONOMY ITS : HIGHLIGHTS", page 296).
In these INSA publications (often edited by K. S. Shukla or similar scholars), let's look at how the typesetters set Sanskrit text.
Often the typesetter made typos, or used specific ligatures!
Let's transcribe what is VISUALLY PRINTED on the page, because the user wants: "Transcribe all text on the ancient Indian

UNITS OF TIME 297
Addendum : The mean lunar day (madhyama tithi) may,
however, differ from a true lunar day (spaṣṭa tithi) by one, so that
the ahargaṇa obtained by the above process may sometimes be in
excess or defect by one. To test whether the ahargaṇa (obtained
by the above process) is correct, it is divided by seven and the
remainder counted with Friday. If this leads to the day of
calculation, the ahargaṇa is correct; if it leads to the preceding
day, the ahargaṇa is in defect; and if that leads to the succeeding
day, the ahargaṇa is in excess. When the ahargaṇa is found to
be in defect, it is increased by one; when it is found to be in
excess, it is diminished by one. (K.S. Shukla : MBh. p. 4-5)
Example—Calculate the ahargaṇa on October 1,1965.
From Indian Calendar we find that October 1,1965 falls
on Friday. 7th lunar day (tithi) in the light half of the 7th
month Āśvina in the Saka year 1887 (elapsed). Let us proceed
as follows :
Adding 3,179 to 1,887. we get 5,066. (1)
Multiplying this by 12 and adding 6 (i. e. the number
of lunar months elapsed since the beginning of Caitra)
we get 60,798. ... ... (2).
Multiplying this by 1,593,336 and dividing the product
by 51,840,000, we get 1,868 as quotient. (The remainder
is discarded as unnecessary) (3)
Adding this number (i.e. 1,868) to the previous one (i.e.)
60,798) we get 62,666. (4)
Multiplying this by 30 and adding 6 (i.e. the number
of lunar days elapsed since the beginning of the current
month) to the product, we get 1,879,986. (5)
Multiplying this by 25,082,580 and dividing the product
by 1,603,000,080, we get 29,416 as the quotient. (The
remainder is discarded as not necessary). (6)
Subtracting this number (i.e. 29,416) from the previous
one (i.e. 1,879,986) we get 1,850,570. (7)
This is the required ahargaṇa. Since division by 7 leaves

298 BRAHMAGUPTA'S ASTRONOMY: ITS HIGHLIGHTS
1 as the remainder. we subtract one from it, and get
1,850,569 as the correct ahargaṇa for the day.
An Alternative Rule for Ahargaṇa
Both Bhāskara I and Brahmagupta give an alternative rule
for calculating out ahargaṇa¹ :
Multiply the number of (solar months) elapsed since
the beginning of kaliyuga by the number of lunar
months (in a yuga) and divide by the number of solar
months (in a yuga). Reduce the quotient to days (and
add the number of lunar days elapsed since the begin-
ning of the current lunar month); then multiply by
the number of civil days (in a yuga) and divide by
the number of lunar days (in a yuga); the quotient
denotes the ahargaṇa.
Mean Longitude of a Planet
(i) The mean longitude of a planet in revolutions is given
by the expression : (Brahmagupta² and also Bhāskara³).

1. शशांकमासैरभिताडितान् हरेदतीतमासानथ वार्कसम्भवैः ।
   दिनीकृतान् भूमिदिनैर्हतान् दिनैर्विभज्य लब्धश्शशिजैर्हरर्गणः ॥ MBh. I. 7
   युगगतशशिमासवधाद्रविमासाप्तं दिनीकृतं सदिनम् ।
   भूदिनगुणितं शशिदिनहृतमाप्तमहर्गणः सैकः ॥ —BrSpSi. XIII. 18
2. इष्टग्रह भगण गुणादहर्गणात् कल्पसावन द्यु हृतात् ।
   भगणादि फलं मध्यो लंकायां भास्करौदयिकः ॥ BrSpSi. I. 31
3. उद्दारितान् यान् भगणान् क्षमादिनैर्लभामहे कान् कलियातवासरैः ।
   इति प्रलब्धा भगणास्ततः क्रमाद् गृहाशलिप्ता विकलाः सतत्पराः । —MBh. I. 8
   पर्यायाहर्गणाभ्यासो ह्रियते भूदिनैस्ततः ।
   लभ्यन्ते पर्यायाः शेषाद्राशि भागकलादयः ॥
   भास्करै स्त्रिंशता षष्ट्या सङ् गुणय्य पृथक् पृथक् ।
   तेनैव भागहारेण लभ्यन्तेऽर्कोदयावधेः ॥ —LBh. I. 15-17
   (Divide the product of the revolution-number of a planet and the
   ahargaṇa by the (number of) civil days (in a yuga); thus are obtained the
   (number of) revolutions (performed by that planet). From the (successive
   remainders multiplied respectively by 12,30 and 60 and divided by the same
   divisor (i.e. the number of civil days in yuga) are obtained the signs, degrees
   and minutes etc. (of the mean longitude of that planet) for (mean) sunrise
   (at Laṅkā).

MEAN LONGITUDE OF PLANET 299
revolution number of planet × ahargaṇa
Mean longitude = ——————————————————————————————————————
civil days in a yuga
Similar expression is given by more recent Indian astro-
nomers also.
(ii) Mean longitude of desired planets in minutes
(mean longitude of the known planet in revolu-
tions etc. reduced to minutes) × (revolution number
of the desired planet)
= ————————————————————————————————————————————————
revolution number of the known planet.
This rule is common to Brahmagupta¹ and Bhāskara I².
(iii) An alternative rule for deriving the mean longitude of
the Moon from that of the Sun and vice versa has been given
by Bhāsakara I and Brahmagupta both.
Multiply the ahargaṇa by the number of intercalary
months in a yuga and divide (the product) by the
number of civil days (in a yuga) : the result is in the
terms of revolutions etc. Add that to thirteen times
the mean longitude of the Sun. (This is the process)
to obtain the mean longitude of the Moon³.
Mean longitude of the Moon
(intercalary months in a yuga) × ahargaṇa
= ————————————————————————————————————————— revolutions
civil days in a yuga

1. ज्ञातभगणादिभुक्तं सविकलमिष्टयुग भगणसंगुणितम् ।
   ज्ञात युगभगणभक्तं मध्यो भगणादि फलमिष्टः ॥ — BrSpSi. XIII. 27
2. निशाकरं वाग्रहमुच्चमेव वा कलीकृतं तत्सहयातमण्डलैः ।
   यथेष्ट नक्षत्रगणैर्हतं हरेत् तदीयनक्षत्र गणैस्ततः कला ॥ MBh. I. 10
   The (mean) longitude of the Moon, the planet, or the Ucca
   (whichever is known) together with the revolutions performed
   should be reduced to minutes. The resulting minutes should
   then be multiplied by the revolution-number of the desired
   planet and (the product obtained should be) divided by the
   revolution-number of that (known) planet. The result is
   (the mean longitude of the desired planet) in minutes.
3. द्युगणं युगाधिमासैर्गुणितं युग भूदिनैर्हृतेऽलब्धम् ।
   भगणादि मध्यमार्के त्रयोदश गुणार्धिकं चन्द्रः ॥ —BrSpSi. XIII. 33
   युगाधिमासैर्धुगणं हतं हरेत् क्ष्मादिनैर्वां भगणादि लभ्यते ।
   त्रयोदशघ्ने सवितर्यथा क्षिपेन्निशीथिनीनां पतिचारसिद्धये ॥ —MBh. I. 11

300 BRAHMAGUPTA'S ASTRONOMY : ITS HIGHLIGHTS
+13 (Sun's mean longitude)
This expression may be rearranged to get the mean
longitude of the Sun from the mean longitude of the Moon¹.
Mean longitude of the Sun
= 1/13 [mean longitude of the Moon

• ((intercalary months in a yuga) × ahargaṇa / civil days in a yuga) revolutions]
  Calculating the Mean
  Longitudes of the Sun and
  the Moon without using Ahargaṇa
  Bhāskara I follows the method of Āryabhaṭa I and the
  same method more or less has been adopted by Brahmagupta in
  calculating the mean longitudes of the Moon and the Sun
  without the use of ahargaṇa. The method may be described
  thus :
  Reduce the years elapsed since the beginning of kali-
  yuga to months and add to them elapsed months of the
  current year. Then multiply the sum by 30 and add
  the product to the number of lunar days elapsed since
  the beginning of the current month. Multiply that
  sum by the number of intercalary months in a yuga
  and divide by the number of solar months in a yuga
  reduced to days; the quotient denotes the number of
  intercalary months elapsed. The remainder is the
  adhimāsaśeṣa. Multiply the complete intercalary
  months thus obtained by 30 and to the product add the
  number of solar days elapsed since the beginning of
  kaliyuga² : then multiply that sum by the number of
  omitted lunar days in a yuga and divide by the number
  of lunar days in a yuga ; the remainder obtained is the
  avamaśeṣa called āhnika. Then multiply the avamaśeṣa

1. कुमुदतीनां सुहृदोऽधवाऽऽगतं विशोध्य शेषस्य लवस्त्रयोदशः ।
   स मध्यमार्को गणकैर्निरूप्यते गुरुप्रसादात्प्रति बुद्ध बुद्धिभिः ॥ MBh. I. 11-12
2. By the number of solar days here is meant the number obtained above
   by reducing the years elapsed since the beginning of kaliyuga to
   months, then adding to them the number of months elapsed since the
   beginning of the current year, then multiplying the sum by 30, and then
   adding to the product thus obtained the number of lunar days elapsed
   of the current month.

MEAN LONGITUDES OF THE SUN AND THE MOON 301
(also called āhnika) by the number of intercalary months
in a yuga and divide by the number of civil days (in a
yuga). Add the resulting quotient to the adhimāsaśeṣa
and divide the sum by the number of lunar months in
a yuga : this gives degrees etc. (This is the total adhi-
māsaśeṣa). Next multiply again the avamaśeṣa called
āhanika by 60 and divide by the number of civil days
in a yuga : the result is in minutes, seconds, thirds etc.
The number of months elapsed (since the beginning of
Caitra) are to be taken as signs and the number of
lunar days elapsed of the current month as degrees.
The sum of these signs and degrees and the minutes,
seconds etc. corresponding to the avamaśeṣa is the
grahatanu. From thirteen times and from one time that
grahatanu severally subtract the degrees, minutes etc.
corresponding to the total adhimāsaśeṣa : the remain-
ders thus obtained are the mean longitudes of the
Moon and the Sun respectively¹.

1. गुणिताद्यु गाधिमासैर्यु गभूदिवसैर्ह तादवमशेषात् ।
   फलयुक्तमधिकमासकशेषं मध्यावतोऽर्केन्दू ॥
   अधिमासावमशेषे युगशशि भूदिनहृते पृथग्लब्धेः ।
   मासदिनाद्ये स्थाप्ये गतमासदिनानि चैत्रादेः ।।
   अवमशेषलब्ध्या सहितानि पृथक् त्रयोदश गुणानि ।
   अधिमास शेषलब्ध्या हीनानि पृथक् रविशशांकौ । —BrSpSi. XIII. 20-22
   विनाद्यु राशेरपि चन्द्रभास्करौ प्रकुर्व्वतो वा विधिरेष क थ्यते ।
   समास मासिकृतविग्रहासु ये ह्यतीतमासा विनियोज्य तान् पुनः ।।
   खरामनिघ्नान् दिवसेषु योजयेद् गतेषु मासस्य ततोऽधिमासकैः ।
   निहत्य सर्व विभजेत् सर्वदा युगार्कमासैर्दिवसत्वमागतैः ॥
   भवन्ति लब्धास्त्वधिमासकाः पुनस्ततोऽपनीयाशु च भागहारकम् ।
   भजेत शेषं शशिमास संख्यया ततोऽशलिप्ता विकलाः स तत्पराः ।।
   ततोऽधिमासान् प्रणिहत्य खाग्निभिर्नियोज्य सम्यग्गतवासरैः क्रमात् ।
   युगावमघ्नाञ्छशिवासरैर्हरेत् तमत्र शेषं प्रवदन्ति चान्हिकम् ॥
   हत्वाऽधिमासैरवमस्य शेषं छित्वा धराया दिवसैः प्रलब्धम् ।
   संयोज्य नित्यं त्वधिमासशेषे कार्यं पुनस्तत् करणैर्यथोक्तम् ॥
   युगप्रसिद्धैर्धरणी दिनैर्हरेन्निहत्य षष्ट्यावमशेषमान्हिकम् ।
   कलाविलिप्ताः क्रमशस्तत्परास्ततोऽवमास्ता दिवसा गृहांशकाः ॥
   त्रयोदशघ्नादपि रूपतादिताद्विशोधयेत्त्वधिमासशेषजम् ।
   निशाकरार्कौ गणकैः प्रकीर्त्तितौ भट प्रणीताविति बुद्धिमत्तमैः । —MBh. I. 13-19
   (Here verse 17 should follow verse 15—K.S. Shukla)

302 BRAHMAGUPTA'S ASTRONOMY : ITS HIGHLIGHTS
K. S. Shukla has provided the following rationale to the
rule cited above :
The fraction of the intercalary month (obtained in the
rule) = (adhimāsaśeṣa) / (solar days in a yuga), in mean lunar months.
= (adhimāsaśeṣa) / (lunar days in a yuga), in mean solar months. (i)
The fraction of the omitted lunar day (obtained in the
rule)
= (avamaśeṣa or āhnika) / (lunar days in a yuga), in mean civil days.
= (avamaśeṣa) / (civil days in a yuga), in mean lunar days.
= (avmaśeṣa × 60) / (civil days in a yuga), in mean lunar ghaṭīs. (ii)
The fraction of the intercalary month corresponding to the
above fraction of the omitted lunar day
= [(intercalary months in a yuga) × (avamaśeṣa)] / [(lunar days in a yuga) × (civil days in a yuga)]
in mean solar months. (iii)
Adding (i) and (iii) and multiplying by 30, the total
fraction of the intercalary month
= { (adhimāsaśeṣa) / (lunar months in a yuga)

• [(intercalary months in a yuga) × (avamaśeṣa)] / [(lunar months in a yuga) × (civil days in a yuga)] }
  in mean solar days. (iv)
  Suppose that m lunar months and d lunar days have
  elapsed since the beginning of Caitra. Then, treating them as
  mean lunar months and mean lunar days, m months and d days
  denote the time elapsed since the beginning of mean Caitra up
  to the beginning of the current lunar day (treated as mean lunar
  day). As (ii) is the interval, in mean lunar ghaṭīs, between the
  beginning of the current lunar day and the mean sunrise on that
  day, therefore
  m months + d days + (ii)
  denotes the time in mean lunar months, days, ghaṭīs¹ elapsed

1. 1 hour = 2½ ghaṭīs; 1 ghaṭī = 60 vighaṭīs; 1 vighaṭī = 60 pravighaṭīs.

CONCORDANCE OF WORKING RULES 303
since the beginning of mean Caitra up to the mean sunrise on
the current lunar day.
Like wise
m months + d days + (ii) - (iv)
denotes the time in mean solar months, days, ghaṭīs etc.
elapsed since the beginning of the current mean solar
year up to the mean sunrise on the current lunar
day¹.
Let M, D, G, V, and P denote respectively the mean solar
months, mean solar days, mean solar ghaṭīs mean solar vighaṭīs
and mean solar pravighaṭīs elapsed since the beginning of the
current mean solar year up to the mean sunrise on the current
lunar day. Then evidently mean longitude of the Sun
= M signs, D degrees, G minutes, V seconds and
P thirds.
= (m signs and d degrees) + [minutes, seconds etc.
corresponding to (ii)] - [degrees, minutes etc.
corresponding to (iv)].
and mean longitude of the Moon
= 13 [m signs and d degrees + (minutes, seconds, etc.
corresponding to (ii)] - [degrees, minutes etc.
corresponding to (iv)]
because
[(1/12) mean longitude of the Moon - mean longitude
of the Sun.
= m signs + d degrees + (minutes, seconds etc.
corresponding to (ii)]
(This equality is based on the fact that the left hand side
denotes the mean lunar date also known as madhyama tithi).
A similar rule of these calculations of the mean longitude

1. Because (iv) is equal to fraction of a lunar month between the beginning
   of Caitra and the beginning of the current mean solar year fraction of an
   intercalary month corresponding to the tithis elapsed up to the beginn-
   ing of the current mean lunar day since the beginning of Caitra fraction
   of an intercalary month corresponding to the avamaśeṣa, i.e., the lunar
   portion between the beginning of the current lunar date and the follow-
   ing sunrise.