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

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.

304 BRAHMAGUPTA'S ASTRONOMY : ITS HIGHLIGHTS
of the Sun and the Moon without basing on ahargaṇa has also
been given by Brahmagupta in the Khaṇḍakhādyaka¹.
Concordance of Working Rules
There has been a good deal of agreement on various rules
of astronomical constants from the time of Āryabhaṭa I (499
A. D.) to the Bhāskara II (1150 A. D.) or even later to the days
of Munīśvara (1620 A, D). Earliest concepts were formulated
during the days of the Vedāṅga-Jyautiṣa and the Siddhāntas of
Indian and the western origin, for example of Brahma, Vasiṣṭha,
Pitāmaha, Romaka and Puliśa. We in this section are giving
some important concordances which we find common in the
writings of Brahmagupta and his predecessors, contemporaries
and successors as listed below. The list is not exhaustive. Only a
few illustrations have been cited.

1. Ārya.—Āryabhaṭīya, Āryabhaṭa I, 499 A. D.
2. BrSpSi. —Brāhmasphuṭasiddhānta, Brahmagupta, 628
   A. D.
3. K. K.—Khaṇḍakhādyaka, Brahmagupta, 628 A. D.
4. KKu.—Karaṇa-kutūhala, Bhāskara II, 1150 A. D.
5. LBh.—Laghu-Bhāskarīya, Bhāskara I, 522 A. D.
   MBh.—Mahā-Bhāskarīya, Bhāskara I, 522 A. D.
   MSi.—Mahā-siddhānta, Āryabhaṭa II, 950 A. D.
   PSi.—Pañcasiddhāntikā, Varāhamihira, 505 A. D.
   ŚiDVṛ.—Śiṣyadhīvṛddhida, Lalla, 598 A. D.
   SiSā.—Siddhāntasārvabhauma, Munīśvara, 1620 A. D.
   SiŚe—Siddhāntaśekhara, Śrīpati, 1039 A. D.
   SiŚi.—Siddhānta-Śiromaṇi, Bhāskara II, 1150 A. D.
   SuSi.—Sūryasiddhānta, Modern, 6th or 7th Century.
6. Rule for finding the mean longitudes of the Sun, Mercury
   and Venus : BrSpSi. I. 44.
   Also MBh. 1. 31. MSi I. 26 ; SiŚe II. 42, 43 ; SiŚi. I i.
   (d). 15 ; SiSā I. 105 ; KKu I. 7.
7. Rule for finding the mean longitude of the Moon's
   ascending node : BrSpSi, XXV. 35.
8. दिनदलभक्तमवमावशेषभाप्त दिनादि तत्सहितान् ।
   अधिमासशेषाकाञ्च त्रिंशद् गुणितादृ‌तुखदिग्भिः ॥
   मासदिन प्रथमैक्यं पृथक् त्रयोदशगुणं द्वितीयोनौ ।
   द्वाक्येवं भृव्यौ राश्याद्यवार्कचन्द्रौ वा ॥ =KK. I. 11–12

CONCORDANCE OF WORKING RULES 305
Also MBh. I. 33 ; ŚiDVṛ. I. i. 52 (ii)
3. Rule for finding the mean longitude of the Śīghrocca of
Venus and also giving the additives for the Śīghrocca of Mercury
and Moon : BrSpSi. XXV. 36.
Also MBh. I. 35 ; ŚiDVṛ. I. i. 57 (ii)
4. Rule for finding the mean longitude of the Śīghrocca
of Mercury : BrSpSi. XXV. 34.
Also MBh. I. 36 ; ŚiDVṛ. I. i. 50 (ii)
5. Rule for finding the mean longitude of Saturn : BrSpSi.
XXV. 35.
Also ŚiDVṛ. I. i. 52 (i) ; MBh. I. 37.
6. Rule for finding the mean longitude of Mars : BrSpSi.
XXV. 33.
Also ŚiDVṛ I. i. 50 (i) MBh. I. 38.
7. Rule for finding the mean longitude of Jupiter: BrSpSi.
XXX. 35.
Also MBh. I. 39 ; ŚiDVṛ. I. i. 51 (i).
8. Rule for finding the distance of a place from the prime
meridian : BrSpSi. I. 36.
Also MBh. II. 3-4 ; LBh. I. 25-26 ; ŚiDVṛ. I. 57-58 (i) ;
SiŚā. I. 143-144.
9. Rule for finding the directions : BrSpSi. III. 1.
Also MBh. III. 2 ; SūSi. III. 1-4 ; LBh. III. 1; ŚiDVṛ. I. iii.
1 ; MSi. IV. 1-2 ; SiŚe. IV 1-3 ; SiŚi. I. iii. 8-9.
Alternative rule : BrSpSi. III. 2.
Also MBh. III. 3 ; PSi. XIV. 14-16 ; ŚiDVṛ. I. iii. 2 ; SiŚe.
IV. 4.
10. Rule for finding the latitude and colatitude and the
zenith distance and altitude of the Sun : BrSpSi. III. 10.
Also MBh. III 5; SūSi III. 13-14; LBh. III 2-3 ŚiDVṛ I.
iii. 4-5; SiŚe. IV. 7; SiŚi I. iii. 18.
11. Rule for determining the declination, day—radius, earth
sine and ascensional difference (for the Sun or a point on the
ecliptic) : BrSpSi. II. 55.
Also SūSi II. 28 ; LBh. II. 16; . I. ii ŚiDVṛ. 17; SiŚe. III.

306 BRAHMAGUPTA'S ASTRONOMY : ITS HIGHLIGHTS
63-64 ; SiŚi. I. ii. 47 (ii) (For RSine of the Declination).
BrSpSi. II. 56 ; also Ārya. IV 24 ; MBh. III, 6; LBh. II.
17 ; ŚiDVṛ. I. ii. 18 ; SiŚe. III. 66 ; . SiŚi I. ii. 48 (For day-
radius).
BrSpSi. II. 57-58 ; also MBh. III. 7 ; LBh. II. 18 ;
SūSi .II. 61; ŚiDVṛ. I. ii. 18; SiŚe III. 67 ; SiŚi. I. ii. 49 (i)
(For the ascensional difference).
12. For finding the times of rising of the sāyana signs at
the equator : BrSpSi. III. 15.
Also MBh. III. 9 ; SūSi. III. 42–43 : ŚiDVṛ. I. iii. 8 ; SiŚe.
IV. 15 ; SiŚi I. 11. 51.
13. Rule for finding the ascensional differences of the
Sāyana signs Aries. Taurus and Gemini : KK. 1. 21.
Also MBh. III. 8 ; PSi. III. 10 ; ŚiDVṛ. 1. XIII. 9 ; SiŚi.
I. ii. 50-51.
14. Rule for the determination of the meridian zenith
distance and meridian altitude of the Sun with the help of the
Sun’s declination and the latitude of the place ; BrSpSi. III, 47.
Also MBh. III. 11 ; LBh. III. 27 ; ŚiDVṛ. I. iii. 16 ;SiŚe.
IV. 42.
15. Rule for determination of the latitude with the help
of the Sun’s meridian zenith distance and declination :
BrSpSi. : III. 13.
Also MBh. III. 17: LBh. III. 34; SūSi. III. 15-16; SiŚe.
IV. 51,
16. Rule for finding out the Rsine of the Sun’s altitude
for the given time in ghaṭīs : BrSpSi. III. 25-26 Also Ārya IV.
28; MBh. III. 18-20; LBh. III. 7-10: ŚiDVṛ. I. iii. 24-25; SiŚe.
IV. 32,34; SiŚi. I. iii. 53-54.
17. Rule for finding out the Sun’s altitude : RSin α=
M×day radius gnomon
───────────── × ───────────────────────────────────────
R hypotenuse of equinoctial midday shadow
where M=R Sin (given ghaṭīs∓asc. diff.) Rsin (asc. diff.), the
upper or lower sign being taken according as the Sun is the nor-
thern or southern hemisphere, α is the Sun’s altitude. BrSpSi.
III. 27 (i).

CONCORDANCE OF WORKING RULES 307
Also MBh. III 24; ŚiDVṛ. I. iii. 27; SiŚe IV. 37
18. Rule for finding the Sun's altitude when the Sun's
ascensional difference is greater than the given time : BrSpSi.
III. 33 :
Also MBh. III 25, LBh. III. 11; ŚiDVṛ. I. iii. 29 SiŚe. IV. 41.
19. Rule for finding the Sun's altitude in the night :
BrSpSi. III. 63.
Also MBh. III. 26; LBh. III. 11; SiŚe. IV. 89.
The Sun's altitude for the night has been called Pātāla
Śaṅku by Brahmagupta (BrSpSi. XV.9)
20. Rule for finding the longitude of the rising point of
the ecliptic with the help of (i) the instantaneous sāyana longi-
tude of the Sun and (ii) the civil time measured since sunrise,
or with the help of (i) the Sun's sāyana longitude at sunrise
and (ii) the sidereal time elapsed since sunrise : BrSpSi III. 18-20.
Also MBh. III. 30-32: LBh. III. 17-19; SūSi. III. 46-48;
ŚiDVṛ. I. iii. 11-12: SiŚe IV. 18-19 (i) : SiŚi I. iii. 2-4.
21. Rule for obtaining the civil time measured since
sunrise with the help of (i) the Sun's instantaneous sāyana longi-
tude and (ii) the sāyana longitude of the rising point of the
ecliptic, or the sidereal time elapsed since sunrise with the help
of (i) the Sun's sāyana longitude at sunrise and (ii) the sāyana
longitude of the rising point of the ecliptic: BrSpSi III. 21-23.
Also SūSi. III.50-51; MBh. III. 34-36; LBh. III. 20;
ŚiDVṛ. I. iii. 13; SiŚe IV. 19 (ii)—22 (i); SiŚi I. iii. 5–7. (i).
22. Rule for determining the R Sines of the Sun's prime
vertical altitude : BrSpSi III. 52.
Also Ārya. IV; MBh. III.37-38; LBh. III. 52.
(An error created by Āryabhaṭa has been criticised by
Brahmagupta.)
23. Construction of the locus of the end of the shadow
of a gnomon : BrSpSi. III. 2-3.
Also MBh. III 52; ŚiDVṛ. I. iii. 3; SiŚe, IV. 5;
24. Rule for finding the Sun's mean anomaly: BrSpSi,
II. 12 (i).

308 BRAHMAGUPTA' ASTRONOMY : ITS HIGHLIGHTS
Also MBh. IV. 1; SūSi. II. 29 ; ŚiDVṛ I. ii. 10; SiŚe. iii.
12; SiŚi. I. ii. 18-19 (i).
25. Rule for finding the RSine (Reversed sine) of an arc
(<90°) : BrSpSi II. 10.
Also SūSi. II. 31-32 ; MBh. IV. 3-14; LBh. II, 2 (11)-3 (i);
ŚiDVṛ. I. ii. 12; SiŚe. III. 15; SiŚi. I, ii. 10 (ii)-11.
(We shall discuss it separately in the light of Brahmagupta
formula.)
26. Rule for finding the Sun's equation of the centre:
BrSpSi.. II. 15 (ii);
Also MBh. IV. 4 (ii); . III SiŚe. 27
27. Rule for determining the Sun's true longitude: BrSpSi.
XIV. 17-18.
Also MBh. IV. 21-23; SiŚe. III. 52.
28. Rule for finding the Sun's bhujāntara correction
under the eccentric theory : BrSpSi. XIV. 19.
Also MBh.-IV. 24.
29. Rule for determining the cara-saṁskāra or cara
correction : KK. I. 22.
30. Rule for finding the semi-durations of the day and
night : BrSpSi. II. 60 ; KK. I. 23.
Also SūSi. II. 62-63 ; ŚiDVṛ. I. ii. 20-21 ; SiŚe. III. 70 ;
SiŚi. I. ii. 52.
31. Rule for calculating the tithi : BrSpSi. II. 62 ; KK.
I. 25.
Also SūSi. II. 66 ; ŚiDVṛ I. ii. 22 ; SiŚe. III. 71 ; SiŚi.
I. ii. 66.
32. Rule for calculating the Karaṇa : KK. I. 27.
Also. ŚiDVṛ. I. ii. 24 ; SiŚe. III. 77 ; SiŚi. I. ii. 66.
33. Rule for calculating nakṣatra : BrSpSi. II. 62 ; KK.
I. 24.
Also SūSi. II. 64 ; ŚiDVṛ. I. ii. 23 (i) ; SiŚe. III. 75; SiŚi.
I. ii. 67.
34. Rule pertaining to direct and retrograde motions of
a planet : BrSpSi. II. 50-51.

CONCORDANCE OF WORKING RULES 309
Also MBh. IV. 56-57 ; SiŚe. III. 59 ; ŚiDVṛ. I. ii. 42 ;
35. A rule for converting true distances known in min-
utes into true distances into yojanas : for example : Sun's true
distance in yojanas
Sun's mean distance in yojanas × Sun's true dist. in minutes
——————————————————————————————————————————
Radius
BrSpSi. XXI. 31 (ii).
Also MBh. V. 3 ; ŚiDVṛ. I. iv. 5 (i) ; LBh. IV. 3 ; SiŚe. V.
4 (ii) ; SiŚi I.v. 5 (i) ;
36. Rule for finding angular diameters of the Sun and the
Moon : BrSpSi. XXI. 34 (ii) ;
Also MBh. V. 5 : ŚiDVṛ. I. iv. 8 ; SiŚe. V. 6 ; SiŚi.
I. v. 7,
37. Formulae for the true (i. e. angular) diameters of the
Sun, the Moon and the shadow in terms of the true daily
motions of the Sun and the Moon (Here by shadow is meant
the section of the cone of the Earth's shadow at the Moon's
distance.) : BrSpSi. IV. 6 (i) : KK. IV. 2 (i).
Also MBh. V. 6-7 ; ŚiDVṛ. I. iv. 9 ; MSi. V. 5 (ii) ;
SiŚe V. 9 ; SiŚi. I. v. 8-9 ;
38. Rule for finding the spaṣṭa-valana (resultant valana)
for the circle drawn with half the sum of the diameters of the
eclipsed and eclipsing bodies as radius : BrSpS.. IV. 18 (i).
Also MBh. V. 46 ; ŚiDVṛ. I. iv. 26.
39. Method for calculating the phase of the eclipse for
the given time : BrSpSi. IV. 11-12.
Also MBh. V. 62-63 ; ŚiDVṛ. I. iv. 19-20 ; SiŚe. V. 14.
40. Rule for the determination of the diameter of the
shadow i.e., the diameter of the Section of the Earth's shadow
where the Moon crosses it : BrSpSi. XXIII. 8-9.
Also MBh. V. 71-73 ; Āryc. IV. 39-40 ; ŚiDVṛ. I. iv. 6
(ii)-7.
41. Process of successive approximations in connection
with calculations of a lunar eclipse : BrSpSi. IV. 8-9.
Also MBh. V. 75-76 ; LBh. IV. 10-12 ; ŚiDVṛ. I. iv. 14-16
SiŚe. V. 12-13; SiŚi. I, v. 12-13.

310 BRAHMAGUPTA'S ASTRONOMY : ITS HIGHLIGHTS
42. Rule relating to the visibility-correction known as
akṣa-dṛkkarma : BrSpSi. VI. 4.
Also MBh. VI. 1-2; ŚiDVṛ. I. vii. 3 (ii); MSi. VII. 4;
SiŚe. IX. 7.
43. Rule relating to the visibility correction known as
ayanadṛkkarma : BrSpSi. VI. 3; X. 17.
Slightly modified in MBh. VI. 2 (ii)-3; ŚiDVṛ. I. vii.
2-3 (i) · SiŚe. IX. 4-5; similar in MSi. VII. 2-3; more
accurate in SiŚi. I. viii. 4-5.
44. Rule relating to the visibilily of moon : BrSpSi. VI.
6 ; X. 32.
Also MBh; VI. 4-5 (i) PSi. V. 3 : ŚiDVṛ. I. vii. 5; SiŚe.
IX. 8 (i). 13.
45. Rule for calculating the phase of the Moon : BrSpSi.
VII. 11 (ii)-12.
Also MBh. VI. 5 (ii)-7; ŚiDVṛ. I. ix. 12.
46. Rule for the determination of the Moon's true
declination (i.e. the declination of the centre of the Moon's
disc) : BrSpSi. VII. 5.
Also MBh. VI. 8; ŚiDVṛ. 1. viii. 2: SiŚe. X. 7. (these
are approximate rules; a more accurate rule occurs in SiŚi. I.
vii. 3 and 13).
47. Graphical representation of the elevation of the
lunar horns in the first quarter of the month at sunset : BrSpSi.
VII. 7-10.
Also MBh. VI. 13-17; ŚiDV. I. ix; SiŚi. I. ix.
48. Minimum distances of the planets from the Sun when
they are visible : BrSpSi. vi 6 ; X. 32.
Also MBh. VI. 44; ŚiDVṛ. I. vii. 5 (i); SiŚe. IX. 8 (i).
12.
49. Rule relating to the determination of the time and
the common longitude of two planets when they are in conjunc-
tion in longitude : BrSpSi. IX. 5-6.
Also MBh. VI. 49-51; ŚiDVṛ. I. x. 7-9 (i); SiŚe. XI.
12-12.
50. Rule relating to the distance between two planets

CONCORDANCE OF WORKING RULES 311
which are in conjunction in longitude : BrSpSi. IX. 11.
Also MBh. VI. 54 ; ŚiDVṛ. I. x. 11 ; SiŚe. XI. 10.
51. Rule For finding the Bhujaphala and Koṭiphala etc.
without the use of the RSine-difference table : BrSpSi. XIV.
23-24.
Also MBh. VII. 17-19 ; SiŚe. III. 17.
52. To obtain the Sun's mean true longitude derived from
the midday shadow of the gnomon : BrSpSi. XIV. 28; III.
61-62.
Also MBh. VIII. 5; SiŚi I. ii. 45.
53. Rule to find the arc corresponding to a given RSine :
BrSpSi II. 11.
Also MBh. VIII. 6; SūSi. II. 33, ŚiDVṛ. I. ii. 13, SiŚe. III.
16, SiŚi. I. ii. 11 (ii)—12 (i).
For the concordance given here we express our indebted-
ness to the work of K.S. Shukla on the Mahābhāskarīya.
Tables of Constants
Some of the Tables of Constants have been given in an
earlier chapter. We here give a few more tables which would
indicate how far Brahmagupta introduced new concepts in eva-
luating these constants of greater accuracy and refinement.
TABLE I
Position of Planets for the Beginning of kaliyuga
In this Table are given the positions of the planets, includ-
ing the Moon's apogee and ascending node, for the beginning of
Kaliyuga. The calculations of Brahmagupta are different from
those of the Sūryasiddhānta and of Āryabhaṭa I,

312 BRAHMAGUPTA'S ASTRONOMY : ITS HIGHLIGHTS