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

LATITUDES OF JUNCTION STARS 317

318 BRAHMAGUPTA'S A
1 | 2 | 3
Hasta | 11°S | 11°S
Citrā | 1°45'S | 2°S
Svāti | 37°N | 37°N
Viśākhā | 1°23'S | 1°30'S
Anurādhā | 1°44'S | 3°S
Jyeṣṭhā | 3°30'S | 4°S
Mūla | 8°30'S | 9°S
P-Āṣāḍhā | 5°20'S | 5°30'S
U-Āṣāḍhā | 5°S | 5°S
Śravaṇa | 30°N | 30°N
Dhaniṣṭhā | 36°N | 36°N
Śatabhiṣak | 18'S | 30'S
P-Bhādra | 24°N | 24°N
U-Bhādra | 26°N | 26°N
Revatī | 0 | 0

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320 BRAHMAGUPTA A GREAT CRITIC
backing to the west) in one day. It is sufficient to postulate the
existence of only one Sun, one Moon and twenty-seven nakṣatras
to explain the astronomical phenomena.¹
3. Brahmagupta differs from Āryabhaṭa I in the length of
the four yugas. Ārvabhaṭa regards all the four yugas of equal
lengths, i. e. 1,080,000 years; the caturyuga being of 4,320,000
years. Brahmagupta regards Kaliyuga to be of 432,000 years,
Dvāpara to be twice of it, Tretā to be thrice of it and Kṛtayuga
of four times of the length of the Kaliyuga. Both have the
caturyuga of the same length.²
4. Āryabhaṭa was not clear with respect to the number of
civil days (sāvana dina) in a yuga; in one of his treatises he
gives this number to be 1,577,917,800 and in the other 1,577,917,
500 with a difference of 300 days. though in both the treatises, Ārya-
bhaṭa I regards the number of solar years to be 4,320,000 in a
Caturyuga or Mahāyuga. Why is this difference ? asks
Brahmagupta.³
5. Āryabhaṭa regards mandocca (the apogee) and pāta
(the ascending node of the orbit on the ecliptic) as constant or
stationary; then how could he propound a sphuṭa-yuga or the
concept of true yuga with the concurrence of year, month and
day on the Caitra Śukla Pratipadā (the first day of bright half
of the month Caitra) at the same time as indicated by Āryabhaṭa
in his Laghvāryabhaṭīya Tantra.⁴
Āryabhaṭa was not clear in respect to the variance in the
pāta. In the Āryāṣṭa-śata (in the Aryabhaṭīya which has 108
Āryā verses), Aryabhaṭa states that the pāta of all the planets

1. भानिचतुष्पञ्चाशद्द्वा द्वावकैर्द्वौ जिनोक्तं यत् ।
   ध्रुवमत्स्यस्यावर्त्तो भवति यतोऽन्हा ततस्तदसत् ॥ BrSpSi. XI. 3
2. आर्यभटोयुगपादांस्त्रीन् यातानाह कलियुगादौ यत् ।
   तस्यकृतान्त्यस्मात् स्वयुगान्तौ न तत् तस्मात् ॥ —BrSpSi. XI. 4
3. युगरविभगणाः स्युर्धृति यत्प्रोक्तं तत् तयोर्युगं स्पष्टम् ।
   त्रिशती स्युदयानां तदन्तरं हेतुना केन ॥ —BrSpSi. XI. 5
4. कुजवर्षादीन् वदतांचैत्रसितादेः समं प्रवृत्तान् यत् ।
   उदसत् यतः स्फुटयुगं तत् स्थैर्यान्मन्दपातानाम् ॥ —BrSpSi. XI. 6

BRAHMAGUPTA A GREAT CRITIC 321
show variance of movement, but in the Daśagītikā (a chapter of
ten Āryā verses), he states that with the exception of the pāta of
the Moon, the pāta's of all other planets are stationary or cons-
tant. Brahmagupta points out this anomaly in the concept of
Āryabhaṭa.¹
6. Brahmagupta points out a self-contradiction in Ārya-
bhaṭa. At one place he says that the Moon covers the Sun during
the solar eclipse and similarly the shadow of the Earth covers
the Moon during the lunar eclipse (and he does not mention
Rāhu) in this connection (Āryabhaṭīya, Gola. 37). At the same
time it is said that Āryabhaṭa was familar with the movement of
"eight" planets, and thus postulating the presence of Rāhu; in fact,
the pātas of planets are responsible for their eclipse, and the
eighth planet Rāhu is not present.²
7. Āryabhaṭa gives measures to Manus, Yugas and Kalpas
different from what have been given in the recognised Smṛtis.³
8. Āryabhaṭa regards Guruvāra or Thursday to be the first
day of the Kalpa, and not Sunday, which is wrong according to
Brahmagupta.⁴
9. Brahmagupta unnecessarily criticises Āryabhaṭa in
respect to the order of days. The motion of the planets decreases
in the following order : Moon, Budha (Mercury), Śukra (Venus),
Sun, Kuja (Mars), Guru (Jupiter) and Śani (Saturn). Āryabhaṭa
in one of his verses states that starting from the Sun and pro-
ceeding in the increasing order every fourth is the dinapati or the
"lord of the day" (Āryabhaṭīya, Kāla. 16); this gives the order :
Sun, Moon, Mars, Mercury, Venus and Saturn and hence the
order of days as Ravivāra (Sunday); Candravāra (Monday,)

1. आर्य्याष्टशते पाता भ्रमन्ति दशगीतिके स्थिराः पाताः ।
   मुक्तेन्दु पातमपमण्डले भ्रमन्ति स्थिरा नान्ये ॥ —BrSpSi. XI. 8
2. आर्य्यभटो जानाति ग्रहाष्टगतिं यदुक्तवांस्तदसत् ।
   राहुकृतं न ग्रहणं तत्पातो नाष्टमो राहुः ॥ —BrSpSi. XI. 9
3. समामनुयुगकल्पाः कल्पादिगतं कृतादियातं च ।
   स्मृत्युक्तं र्रार्य्यभटो नातो जानाति मन्वादिम् ॥ —BrSpSi. XI. 10
4. ओङ्कारो दिनवारो गुरुरौदयिकोऽस्य भवति कल्पादौ ।
   न भव्यर्को यस्मादोङ्कारो विस्तरस्तस्मात् ॥ —BrSpSi. XI. 11

322 BRAHMAGUPTA A GREAT CRITIC
Maṅgalavāra (Tuesday), (Budhavāra) (Wednesday), Guruvāra
(Thursday) Śukravāra (Friday) and Śanivāra (Saturday). Thus
Āryabhaṭa gives the same order of days as other authorities and
there is no reason why he should be criticised.
Certainly what is Sunday for Laṅkā, may not be Sunday
at the same time for Siddhapura; and Āryabhaṭa should have
emphasised that the day (Monday, Tuesday etc.) is not cons-
tant for all the places.¹
10. Brahmagupta expresses surprise why Āryabhaṭa, in two
of his treatises propounds two different systems of reckoning one
from the Sunrise in Laṅkā and the other from the Midnight in
Laṅkā.² This causes a difference of one-fourth of the daily-
motion in the two reckonings of the motion of planets.³
11. Brahmagupta criticises Āryabhaṭa on the point of dia-
meter of the Earth. In the Gītikāpāda, 5 and 6, Āryabhaṭa
states that one yojana = 8,000 × puruṣa, and 1 puruṣa = 4 hasta,
thus 1 yojana = 32,000 puruṣas, and the diameter of the Earth is
1050 yojanas. Brahmagupta further says that an error in the
diameter of the Earth would cause an error in deśāntara or
longitude and thus also in the true tithi and consequently in the
calculation of eclipses also.⁴
12. Āryabhaṭa has rightly stated that the Earth is in
motion and the Bhagaṇas are stationary. Brahmagupta's objec-
tion is that if the Earth is in motion, birds would not be able to
return to their nests, and if the Earth's motion is upside-down,

1. सूर्याद्यश्चतुर्थी दिनवारा यदुवाच तदसदार्यभटः ।
   लङ्कोदयो यतोऽर्कं स्यास्तमयं प्राह सिद्धपुरे ॥ — BrSpSi. XI. 12
2. अधिकैः शतैश्चतुर्भिर्वर्षसहस्रै रचतुर्दशभिरेकः ।
   युगपाद्दिनवारान्तर मौदयिकार्ध रात्रिकयोः ॥ — BrSpSi. XI. 13.
3. औदयिकाद्दिनमुक्त गत्यंशो नार्ध रात्रिको भवत्यूनः ।
   कतरं स्फुटं न निश्चितमनयोः स्फुटमेकमपि नातः ॥ — BrSpSi. XI. 14
4. षोडशगणितोजनरिधिं प्रतिभूव्यासं पुलावदतः ।
   आत्मानं र या.पितमनिश्चयस्तनि कृतकन्द्यात् ।।
   भूव्यासस्याज्ञानाद् व्यर्थ देशान्तरं तदज्ञानात् ।
   स्फुटतिथ्यज्ञानात् तिथिनाशाद् ग्रहणयोर्नाशः ॥ — BrSpSi. XI. 15-16

BRAHMAGUPTA A GREAT CRITIC 323
then the roof and hills would come down, which is contrary
to our observation.¹ Obviously Brahmagupta is not justified in
his criticism.
13. Brahmagupta points out to the differences in his calcu-
lations and the calculations of Āryabhaṭa in the peripheries of
the manda and śīghra epicycles of planets in the odd and even
quadrants. (This difference we have shown in Table V, p. 313)²
14. Āryabhaṭa I and Bhāskara I have both given a rule for
the determination of the dṛkkṣepajyās of the Sun and the
Moon :
Take the product of the Sun's or Moon's own madhya-
jyā and udayajyā, then divide the product by the
radius and then take the square of the quotient. Sub-
tract that from the square of the own madhyajyā ;
the square-root of that difference is known as the Sun's
or Moon's dṛkkṣepajyā.³
The Sun's dṛkkṣepajyā is the Rsine of the zenith distance
of that point of the ecliptic which is at the shortest distance from
the zenith (this point is called nonagesimal or the central ecliptic
point). The Moon's dṛkkṣepajyā is the Rsine of the zenith
distance of that point of the Moon's orbit which is at the short-
est distance from the zenith. The rule given above is only
approximate and has been criticised by Brahmagupta.⁴
15. In the Āryabhaṭīya, there is a rule in the Golapāda for
finding the Rsine of the agrā of the true Sun; and also for the

1. प्राणेनैति कलां भूर्यदि तर्हि कुतो ब्रजेत् कमध्वानम् ।
   आवर्त्तनमुर्व्याश्चेन्न पतन्ति समुच्छ्रयाः कस्मात् ॥ —BrSpSi. XI. 17
2. औदयिको यः परिधिर्विषमेऽन्योमेऽन्यः समे भुजस्य गुणः ।
   तदसद्विषमान्तफलं यतो न युग्मादि फलतुल्यम् ॥
   विषमेऽन्योऽन्यो युग्मे परिधिर्गुणकः क्रमोत्क्रमज्यानाम् ।
   चक्रार्धे फलनाशो न भवति यस्मादसत् तदपि । —BrSpSi. XI. 18-19
3. स्वमध्यज्योदयाभ्यासं विष्कम्भार्धाप्तवर्गितम् ॥
   मध्यज्यावर्गंतोऽपास्य स्वदृक् क्षेपं पदं विदुः ॥ —Mbh. V. 19
4. वित्रिभलग्ने दृक्क्षेपमण्डलं तदपमण्डलयुतौ ज्या ।
   मध्याद्दक्क्षेपज्या नार्थभटोक्ताऽनया तुल्या ॥
   दृक्क्षेपज्याऽतोऽसव तन्नाशादवनतेर्नांशः ।
   अवनतिनाशात् ग्रासस्योनाधिकता रविग्रहणे ॥ —BrSpSi. XI. 29-30

324 BRAHMAGUPTA A GREAT CRITIC
Rsine of the Sun's prime vertical altitude.¹ Bhāskara also gives
the rule in his Mahābhāskarīya :
Multiply the Rsine of the (Sun's) greatest declination
by the Rsine of the Sun's true (sāyana) longitude;
then divide (the product) by the Rsine of the colati-
tude, the result is (the Rsine of) the agrā of the true
Sun. When that (agrā) is less than the latitude and
when the Sun is also in the northern hemisphere,
multiply (the Rsine of the Sun's agrā) by (the Rsine
of) the colatitude; the result is the Rsine of the Sun's
prime vertical altitude.²
The condition laid down in the rule that the "Sun's agrā
should be less than the latitude" is incorrect. The error was
originally comitted by Āryabhaṭa and Bhāskara followed it.
This error was noticed by Brahmagupta³. Bhāskara I, however,
corrected the error in the Laghu-Bhāskarīya. There he gives the
correct conditions. It is not the agrā that should be less than
the latitude, it is the Sun's declination (or rather the Rsine of
the Sun's northern declination as we have in the Laghu-
Bhāskarīya), which should be less than the latitude (or rather
the Rsine of the latitude). This condition is necessary for the
existence of the prime vertical shadow of the gnomon.
It may be pointed out that the commentators of Āryabhaṭa I
have interpreted the rule given by Āryabhaṭa I as conveying
the correct meaning; they say that Āryabhaṭa also meant
declination when he used the term agrā.

1. परमापक्रमजीवामिष्टज्यार्हतां ततो विभजेत् ।
   ज्यालम्बकेन लब्धार्काग्रा पूर्वापरे क्षित्तिजे ॥
   सा विषुवज्ज्योना चेद् विषुवदुदग्लम्बकेन संगुणिता ।
   विषुवज्ज्या विभक्ता लब्धः पूर्वापरे शंकुः ॥ — Ārya. V. 30-31
2. स्फुटरविभुजनिघ्नां यां परां क्रांतिजीवां ।
   हरतु समलम्बज्या कलापेन भूयः ॥
   स्फुट दिवस कराग्रा सा यदाऽक्षांशहीना ।
   रविरपि षडगोले चोत्तरे लम्बकृघ्नाम् ॥
   अक्षज्यया हरेद् भूयः शंकुः स्यात् सममण्डले ।
   तद् वर्गं व्यास कृत्योर्यद् विश्लेषं तत्पदं प्रभा ॥ — MBh. III. 37-38
3. उत्तरगोलेऽग्रायां विषुवज्ज्यातो यदुक्तमूनायाम् ।
   सममण्डलमस्तदसत् क्रान्तिज्यायां यतो भवति ॥ — BrSpSi. XI. 22

BRAHMAGUPTA A GREAT CRITIC 325
Usually by agra, we mean the arc of the celestial horizon
lying between the east point and the point where a heavenly
body rises ; or between the west point and the point where a
heavenly body sets. Declination is krānti.
16. Brahmagupta has criticised Āryabhaṭa and his group
for their expressions for determining lambana (i. e. the difference
of the parallaxes, in longitude, of the Sun and the Moon), and
the rule for determining the avanati or nati (i.e. the difference in
parallaxes, in latitude of the Sun and Moon).¹
Lambana is obtained with the help of the five Rsines : (i)
madhya-jyā, (ii) udaya-jyā, (iii) dṛk-kṣepa-jyā, (iv) dṛg-jyā, and
(v) dṛg-gati-jyā.
(i) The madhyajyā is the Rsine of the zenith distance of
meridian ecliptic point :
madhyajyā=Rsin (ϕ± declination of the meridian ecliptic
point).²
In this expression ϕ is the latitude of the place, and by
R sin is meant R × sine, R being the radius of the celestial sphere.
(ii) The udayajyā is the Rsine of the arc of the horizon
intervening between the equator and the ecliptic, and is given
by :
udayajyā = (R sin L × R sin ε) / (R cos ϕ)
where L is the longitude of the horizon ecliptic point in the east.
and ε the obliquity of the ecliptic.
(iii) The dṛkksepajyā is the R sine of the zenith distance of
the central ecliptic point³, and is given by :

1. व्यासार्धेन विभक्ता दृग्गति जीवा चतुर्गुणा लब्धम् ।
   लम्बननाड्यः पञ्चदशगुणिताया त्रिज्यया भक्ता ॥
   दृक्क्षेपज्या भुक्त्यन्तराहता लब्धमवनतिर्भवति ।
   स्फुटयोजनकर्णाभ्यां भूव्यासेन च विना स्पष्टे ॥
   आर्यभटेनास्मिन् सति लघुनि किमर्थं महत् कृतं कर्म ।
   गणित ज्ञानाज्जाड्यं विजानता यदि ततः सुतराम् ॥ —BrSpSi. XI. 23-25
2. The meridian ecliptic point is the point of the ecliptic on the meridian.
3. The central ecliptic point is the central point of the portion of the eclip-
   tic lying above the horizon.

326 BRAHMAGUPTA A GREAT CRITIC
dṛkkṣepajyā
= [ (madhyajyā)² - { udayajyā × madhyajyā / R }² ]^(1/2)
where R is the radius of the celestial sphere.
(iv) The dṛgjyā is the Rsine of the zenith distance (of the
Sun) and is given by :
dṛgjyā = [ R² - { dṛggatijyā × R sin (L - θ) / R }² ]^(1/2)
where L is the longitude of the horizon ecliptic in the east and
θ the longitude of the Sun.
(v) The dṛggatijyā is the Rsine of the altitude of the central
ecliptic point, and is given by :
dṛggatijyā = [R² - (dṛkkṣepajyā)²]^(1/2)
where R is the radius of the celestial sphere.
In the Mahābhāskarīya¹, the expression for the Sun's
dṛggatijyā is :
(Sun's dṛggatijyā)²
= (Sun's dṛgjyā)² - (Sun's dṛkkṣepajyā)²
and similar is the expression for the the Moon's dṛggatijyā.
Now lambana, which is the difference of the parallaxes,
in longitude, of the Sun and the Moon, is given by the expres-
sion :
Lambana = Moon's lambana - Sun's lambana.
Sun's lambana
= Sun's dṛggatijyā × Earth's semidiameter / Sun's true distance in yojanas
Moon's lambana
= Moon's dṛggatijyā × Earth's semi-diameter / Moon's true distance in yojanas
These lambanas are in terms of minutes of arc etc².

1. स्वदृग्दृक्क्षेप गुणयोर्वर्गे विश्लेषजे पदे ।
   दृग्गतिज्ये भवेतां ते भास्करामृत तेजसः ॥ -MBh. V, 23
2. स्वदृग्गतिज्यया व्यास भेद संवर्ग संभवम् ।
   पृथग्योजन कर्णाप्तं लिप्ताद्यं लम्बनं विदुः ॥
   (Cont. on Page 225}

BRAHMAGUPTA A GREAT CRITIC 327
Thus lambana is given by subtracting the Sun's lambana
from Moon's lambana.
This lambana is also expressed in the following way .
Lambana
[ {(dṛgjyā)²–(dṛkkṣepajyā)²}¹/² × 18
= [ ────────────────────────────────── —
[ Moon's true distance
{(dṛgjyā)²–(dṛkkṣepajyā)²}¹/² × 18 ]
— ────────────────────────────────── ]
Sun's true distance ]
in minutes.
60
= ── × ( lambana calculated in minutes ) is the
d
lambana in ghaṭīs, where d denotes the difference
between the daily motions of the Sun and the
Moon.
Āryabhaṭa I has given his description of the determination
of lambana and avanati in the Golapāda 33, 34 of the
Āryabhaṭīya¹, and Bhāskara has followed his rules in the
Mahābhāskarīya². Brahmagupta criticises them in his Brāhma-
sphuṭasiddhānta³.
(vii) We shall now take up nati or avanati (both the terms
mean the same). Nati is the difference of the parallaxes in
latitude, of the Sun and the Moon and is given by :
[ dṛkkṣepajyā × 18 dṛkkṣepajyā × 18 ]
nati = [ ───────────────── — ───────────────── ]
[ Moon's true dist. Sun's true dist. ]
minutes.
─────────────────────────────────────────────────────────────────
(Cont. from Page 324)
तद्विशेषो हतः षष्ट्या स्फुटभुक्त्यन्तरोद्धृतः ।
घटिकादिस्तिथेः प्राह्णे शुद्धिः छेदोऽपरे मतः ॥
दिनार्धकालनिष्पन्नं लम्बनं शोध्यते तिथेः ।
उदगिन्दूदयज्यायां दीयते तत्र दक्षिणे ॥
एवं पुनः पुनः कर्म यावत्तदविशिष्यते ।
तिथिवच्चन्द्रतीक्ष्णांशू सञ्चार्यावेव पण्डितैः ॥ — MBh. V. 24-27

1. मध्यज्योदयजीवासंवर्गे व्यासदलहृते यत् स्यात् ।
   तन्मध्यज्याकृत्योर्विशेषमूलं स्वदृक् चे पः ॥
   दृग्दृक्क्षेप कृति विशेषितस्यमूलं स्वदृग्गतिः कुवशभून् ।
   क्षितिजे स्व दृक् छाया भूव्यासार्धं नभोमध्यात् ॥ —Arya. IV. 33-34
2. loc. cit. MBh. V. 24-27
3. loc. cit. BrSpSi. XI. 23-25

328 BRAHMAGUPTA A GREAT CRITIC
(viii) Moon’s true latitude = Moon’s latitude × nati.
The present Sūryasiddhānta and Brahmagupta both utilise
the following expressions for lambana and nati which give more
accurate values :
lambana = [Rsin (M - ☉) × dṛggatijyā] / {Rsin (30°)}² ghaṭis
where M and ☉ denote the longitudes of the meridian ecliptic
and the Sun respectively.
nati = (dṛkkṣepajyā × d) / (15 × R)
where R is the radius of the celestial sphere and d denotes the
difference between the daily motions of the Sun and the
Moon¹.
Brahmagupta has raised objections to the Āryabhaṭa system
regarding lambana (XI. 26-28), dṛkkṣepa (XI. 30-31) ayanadṛk-
karma (XI. 35), elevation of Moon’s horns (śṛṅgonnati) (XI. 39)
and similar several other points. He is so vehemently opposed
to Āryabhaṭa that finally he declares :
“It is beyond my capacity to enumerate all the defects
of Āryabhaṭa. Only a few have been given here as
illustration. Intelligent people can easily find out
others².”
He also says :
Āryabhaṭa is neither conversant with the Gaṇita
(mathematics), nor Kāla (time calculations) nor Gola
(celestial or spherical calculations). It is difficult to
enumerate separately the fallacies committed by him
in the respective chapters of the Gaṇitapāda, Kālakri-
yapāda and Golapāda³.

1. loc. cit. BrSpSi. XI. 23.
2. आर्यभटदूषणानां संख्या वक्तुं न शक्यते यस्मात् ।
   तस्मादयमुद्देशो बुद्धिमताऽन्यानि योज्यानि ॥ —BrSpSi. XI. 44
3. जानात्येकमपि यतो नार्यभटो गणितकालगोलानाम् ।
   न मया प्रोक्तानि ततः पृथक् पृथग् दूषणान्येषाम् ॥ —BrSpSi. XI. 43.

BRAHMAGUPTA AND ŚRĪṢEṆA 329
Brahmagupta and Śrīṣeṇa
In Varāhamihira's Pañcasiddhāntikā we have a critical
review of the five Siddhāntas or five systems of astronomical
study : Puliśa Siddhānta, Romaka Siddhānta, Vasiṣṭha Siddhānta,
Sūrya Siddhānta and Brāhma Siddhānta. Colebrooke in his
Paper “On the notion of the Hindu Astronomers concerning the
precession of the equinoxes and motions of the planets”, publi-
shed in the Asiatic Researches, vol. xii. p. 209-250, Calcutta.
1816,4 to, reproduced in the Miscellaneous Essays. Vol. II. 1872,
says the following in regards to the authorship of these schools
of astronomy :
All these books are frequently cited in the astrono-
mical compilations and are occasionally referred to
their real or supposed authors. The first is everywhere
assigned to Puliśa, whose name it bears. The Romaka
Siddhānta is ascribed by the scholiast of Brahmagupta
and by a commentator of the Sūrya Siddhānta to
Śrīṣeṇa. The Vāsiṣṭha Siddhānta is by the same
authority given to Viṣṇucandra. Both these authors
are repeatedly mentioned with censure by Brahma-
gupta ; and it is acknowledged that they are entitled
to no particular deference.
The Brāhma Siddhānta, which is the basis of Brahma-
gupta’s work, is not anywhere attributed to a known
author ; but referred to in all quotations of it which
have fallen under observation, either to the Viṣṇudhar-
mottara Purāṇa, of which it is considered as forming
a part, or to Brahmā (also called Pitāmaha) who is
introduced into it as the speaker in a dialogue with
Bhṛgu, or it is acknowledged to be the work of some
unknown person. The true author it may be now
impracticable to discover, and would be vain to con-
jecture.
The Sūrya Siddhānta (if the same which we now pos-
sess) is in the like manner ascribed to no certain author
unless in the passage cited by my colleague Mr. Bently
(Asiatic Researches, vol. vi. p. 572) who says that “in
the commentary of the Bhāsvatī, it is declared, that

330 BRAHMAGUPTA A GREAT CRITIC
Varāha was the author of the Sūrya Siddhānata¹, and
who adds, that “Satānanda, the author of the Bhāsvatī
was a pupil of Varāha under whose directions, he
himself acknowledges, he wrote that work”.
This concluding remark alludes to the following verse
of the Bhāsvatī Karaṇa : “Next I will propound
succinctly., from Mihira’s instruction, (the system)
equal to the Sūrya Siddhānta¹, (Miscellaneous Essays,
p. 388-90) (The word ‘Mihira’ has double meaning : it
might be an abbreviation of Varāhamihira, or it may
mean sun or Sūrya).
Thus on the authority of Colebrooke, Śrīṣeṇa may by re-
garded as the initiator of the Romaka system. Brahmagupta him-
self mentions in one of his passages the name of Śrīṣeṇa in connec-
tion with the Romaka system, and further the conceptions of the
Romaka system came down as Vāsiṣṭha system through Viṣṇucan-
dra,² Lāṭadeva also derived from Śrīṣeṇa the concepts of the mean
motions of the Sun, the Moon, the Moon’s apogee and her node
and the mean motions of Mars, Mercury’s. Śīghra, Jupiter, Venus’
śīghra and Saturn. I have indicated elsewhere, which is also
the view of Sankara Bālakṛṣṇa Dīkṣita, that the original Romaka
and Paulīśa Siddhāntas were introduced to Indians by Lāṭadeva,
and the latter Romaka Siddhānta by Śrīṣeṇa (Original Romaka
Siddhānta was prevalent before Śaka 427 and this is the one
which is mentioned by Varāhamihira who makes no reference to
Śrīṣeṇa and Viṣṇucandra in the Pañcasidhāntikā, and the latter
Romaka Siddhānta was introduced by Śrīṣeṇa as is indicated by
Brahmagupta. Thus we have two Vāsiṣṭha Siddhāntas and two
Romaka Siddhāntas). My personal view is that Lāṭadeva,
Śrīṣeṇa and possibly Viṣṇucandra also, were naturalised Greeks,
settled in India and they had adopted themselve to Indian life.
They were conversant in Greek and Indian Astronomy both and
had contributed substantially to Indian astronomy. Brahmagupta
was opposed to any of these f oreign influences dominating Indian

1. अथ प्रवक्ष्ये मिहिरोपदेशात् तत्सूर्यसिद्धान्त समं समासात् ॥
   —Bhāsvatī Karaṇa
2. श्रीषेणेन गृहीत्वा रचोन्त्यरोमकः कृतः कन्था ।
   एतानेव गृहीत्वा वासिष्ठो विष्णुचन्द्रेण ॥ —BrSpSi. XI. 50

BRAHMAGUPTA AND ŚRĪṢEṆA 331
systems, and he very much resented such interferences in pure
academic life of this country. He was opposed to Āryabhaṭa
for a different reason. Āryabhaṭa was universally regarded as
an authority in this country, and the conservatism was so deep
that even where it could be shown by direct observation or
on valid theoretical grounds, that a particular concept was
erroneous or less accurate, people still chose to adhere to it,
since they had the backing of Āryabhaṭa's authority. Brahma-
gupta was against this nonscientific attitude. Needless to say,
Brahmagupta was not always fair to Āryabhaṭa in his criticism ;
he overdid in enumerating the shortcomings of Āryabhaṭa's
system, as if he was personally jealous of his wide popularity.
Brahmagupta's feelings against Lāṭadeva, Śrīṣeṇa, Viṣṇu-
candra and others would be seen from the following passage in
the Brāhmasphuṭasiddhānta :
From the fact that Śrīṣeṇa, Viṣṇucandra, Pradyumna,
Āryabhaṭa, Lāṭa, and Siṁha contradict one another
regarding eclipses and similar topics, their ignorance
is proved daily. The criticisms which I have passed
on Āryabhaṭa are, with the requisite modifications, to
be applied to the doctrines of each of these teachers as
well. I will, however, make some further critical re-
marks on Śrīṣeṇa and others.
Śrīṣeṇa took from Lāṭa the rules concerning the mean
motions of the Sun, and the Moon, the Moon's apogee
and her node, and the mean motions of Mars, Mercu-
ry's Śīghra, Jupiter, Venus's Śīghra, and Saturn ; he
took elapsed years and the revolutions of yuga (yuga-
yāta-varṣa-bhagaṇa) from Vasiṣṭha and the Padakaraṇa
of Vijayānandi; further took from Āryabhaṭa the rules
concerning the apogee, epicycles and nodes, and
those referring to the true motions of the planets and
thus the Romaka Siddhānta which was (or is) a heap
of jewels (as it were) has, by Śrīṣeṇa, been made into
a patched rag (as it were)¹.

1. श्रीषेणविष्णुचन्द्र प्रद्युम्नार्यभटलाटसिंहानाम् ।
   ग्रहणादिविसंवादात् प्रतिदिवसं द्विगुणमज्ञत्वम् ॥ [Cont. on Page 330]

332 BRAHMAGUPTA A GREAT CRITIC
Brahmagupta very emphatically says about his system that
so long as people would be finding concordance between the
observed and theoretical results (dṛggaṇitaikyam) in respect of
solar and lunar eclipses, his Brāhma Siddhānta would be held
in esteem¹.
In other systems, whatever concordance appears to be bet-
ween the observation and calculation, of eclipses etc., it is,
Brahmagupta says, merely accidental or by chance, as the maxim
of letters bored by an insect in wood or paper².
युक्त्याऽऽर्यभटोक्तानि प्रत्येकं दूषणानि योज्यानि । [Cont. from Page 329]
श्रीषेणप्रभृतीनां कानि चिदन्यानि वक्ष्यामि ॥
लाटात् सूर्यशशांकौ मध्याबिन्दूच्च चन्द्रपातौ च ।
कुजबुधशीघ्रबृहस्पति सितशीघ्र शनैश्चरान् मध्यान् ॥
युगयातवर्षभगणान् वासिष्ठाद्विजयनन्दि कृतपादात् ।
मन्दोच्च परिधिपातस्पष्टीकरणाद्यमार्यभटात् ॥
श्रीषेणेन गृहीत्वा रत्नोच्चयरोमकः कृतः कन्था ।
एतानेव गृहीत्वा वासिष्ठो विष्णुचन्द्रेण ॥ —BrSpSi. XII. 46-50
2. चन्द्ररवि ग्रहणेन्दुच्छायादिषु सर्वदा यतो ब्राह्मे ।
दृग्गणितैक्यं भवति स्फुटसिद्धान्तस्ततो ब्राह्मः ॥ —BrSpSi. XI. 61
3. अन्योर्न कदाचिदपि ग्रहणादिषु भवति दृष्टिगणितैक्यम् ।
यद्भवति तद् घुणाक्षरमतोऽस्फुटभ्यां किमेताभ्याम् ॥ —BrSpSi. XI, 51.
—: 0 :—
Reference
Brahmagupta : Tantraparīkṣādhyāya in BrSpSi.
K.S. Shukla : The Mahābhāskarīya and the Laghubhāskarīya.
H.T. Colebrooke : Miscellaneous Essays, Vol. II., 1872.
G. Thibaut and Sudhākara Dvivedi : The Pañcasiddhāntikā,
Preface, 1889.

CHAPTER XIII
Brahmagupta and
Astronomical Instruments
The Twenty-second Chapter of the Brāhmasphuṭasiddhānta
is known as the Yantrādhyāya or a chapter on instruments.
There is a description of seventeen types of time-reckoning instru-
ments (Kāla-yantra)¹ :

1. Dhanuryantra—Bow instrument.
2. Turyagolaka yantra—Quadrant (one-fourth sphere)
3. Cakra yantra=wheel or circle.
4. Yaṣṭi yantra—a pole or staff instrument.
5. Śaṅku yantra—Gnomon.
6. Ghaṭikā yantra—a clock or pot instrument.
7. Kapāla yantra—Bowl or potsherd instrument.
8. Karttarī yantra—Scissor or knife ; cutter.
9. Pīṭha yantra=Pedastal or seat instrument.
10. Salila yantra—Water-leveller.
11. Brahma or Śāṇa yantra—For describing circles.
12. Avalamba Sūtra—Threads with plumbs (Plumb
    lines).
13. Karṇa or chāyā-karṇa—A set of squares for
    diagonals.
14. Chāyā or śaṅku-chāyā—Sundial.
    ──────────────────────────────────────────────────
15. सप्तदश कालयन्त्राण्यतो धनुस्तुर्यगोलकं चक्रम् ।
    यष्टिः शंकुर्घटिका कपालकं कर्त्तरी पीठम् ॥
    सलिलं भ्रमीऽवलम्बः कर्णश्छाया दिनार्थमर्कोच्चः ।
    नतकालज्ञानार्थं तेषां संसाधनान्यष्टौ ॥
    —BrSpSi. XXIII, 5-6

334 BRAHMAGUPTA AND ASTRONOMICAL INSTRUMENTS
15. Dinārdha yantra—Midday measure instrument.
16. Arka yantra—Sun-instrument.
17. Akṣa or Palāñśa yantra—Small degree measure arc
instrument.
Salila yantra is used for levelling; since a liquid such as
water seeks its own level, it can be utilised to know whether a
surface has been levelled or not.¹ Bhrama or Śāṇa is used for
drawing circles. Avalambaka or plambline is used for
adjusting vertical line. Karṇa is used in connection with angles
and diagonals. From Salila (no. 10) to the last (no. 17); these
eight are used for adjustments and are basically important.
The dhanuryantra is used for nata and unnata kāla
ghaṭikās.
On the paridhi or the circumference of the cakra-yantra
are indicated the twelve rāśis, ending up to Mīna (XXII. 18).
Brahmagupta has described the yaṣṭi yantra and shown how it
could be used to give time at different parts of the day, and
from its shadow dṛgjyā and other characteristics can be calculat-
ed. This instrument can also be used for ascertaining the solar-
lunar differences, and for fixing up the directions. It can
be used for determining various heights and altitudes.
The karttarī yantra is of the shape of a pair of scissors with
two semi-circular blades, fastended to a string at the centre; at
the centre is fixed a pin or a pole which casts shadows.
Setting up of the Gnomon
Here it would be interesting to describe the
setting of a gnomon, which K.S. Shukla has given in details while
commenting on the Mahābhāskarīya (IV. 1) :
After having tested the level of the ground by means
of water, draw a neat circle with a pair of compasses
(karkaṭa) (At the centre of that circle, set up a vertical
gnomon). The gnomon should be large, cylindrical,
massive, and tested for its perpendicularity by means
of four threads with plumbs (avalmbaka) tied to
them.

1. सलिलेन समं साध्यं भ्रमेण वृत्तमवलम्बकेनोर्ध्वम् ।
   तिर्यक् कर्णेनान्यैः कथितैश्च नव प्रवक्ष्यामि ॥ —BrSpSi XXII. 7.

SETTING UP OF THE GNOMON 335
Bhāskara I in his commentary on the Āryabhaṭīya tells us
that there was a difference of opinion amongst astronomers in
his time regarding the shape and size of gnomon (also called style).
Some astronomers prescribed a gnomon with its one-third in the
bottom of the shape of a prism on a square base (caturasra), one-
third in the middle of the shape of a cow's tail (go-pucchākāra).
and one-third at the top of the shape of a spear-head (Śūlākāra)
and some others prescribed a square prismoidal (samacaturasra).
gnomon. The followers of Āryabhaṭa I, he informs us, prescrib-
ed the use of a broad (pṛthu), massive (guru), and large (dīrgha
cylindrical gnomon, made of excellent timber and free from any
hole, a scar or knot on its body. In the above stanza, Bhāskara I
prescribes this last kind of gnomon : the other two kinds he
proves in the commentary to be defective and so he rejects them.
For getting the shadow'end easily and correctly the cylindri-
cal gnomon was surmounted by a fine cylindrical iron or wooden
nail fixed vertically at the centre of the upper end. The nail
was taken to be longer than the radius of the gnomon, so that
its shadow was always seen on the ground.
Certain writers, Bhāskara I tells us in the commentary,
prescribed a gnomon of half a cubit (=12 aṅgulas) in length
and having twelve divisions. But according to Bhāskara I
(although it was the usual custom) there was no such hard and
fast rule. The gnomon could be of any length and any number
of divisions. The gnomon should, however, be large enough, so
that the rings of graduation on the gnomon may be clearly seen
on the shadow. A broad and massive gnomon was preferred
because it was unaffected by the wind.
Brahmagupta describes gnomon which at the bottom is
two aṅgulas wide, pointed as a needle., 12 aṅgulas in length, and
full of holes from the basic circular part to the pointed
extremity. (BrSpSi. XXII. 39).
As regards testing the level of the ground, Bhāskara I
observes :
When there is no wind, place a jar (full) of water
upon a tripod on the ground which has been made
plane by means of eye or thread, and bore a (fine)
hole (at the bottom of the jar) so that the water may

336 BRAHMAGUPTA AND ASTRONOMICAL INSTRUMENTS
have continuous flow. Where the water falling on
the ground spreads in a circle, there the ground is in
perfect level; where the water accumulates after
departing from the circle of water, it is low; and where
the water does not reach, there it is high. (Bhāskara's
Commentary on the Āryabhaṭīya, II. 13).
After the ground was levelled, a prominently distinct circle
was drawn on the ground as stated in the text (MBh. III. 1).
In the time of Śaṅkaranārāyaṇa (869 A.D.), there it seems that all
lines were drawn on the ground with sandal paste (candana-
kṣodārdra). The above circle having been thus drawn and coated
with sandal paste, another small concentric circle was drawn
with the radius of the gnomon. The gnomon was then placed
vertically with the periphery of its base in coincidence with
that circle. The gnomon was thus set up exactly in the middle of
the bigger circle. The verticality of the gnomon was tested by
means of four plumb lines hung on the four sides of the
gnomon
Gnomon Used for Finding the Directions
The rule in this connection has been described by
Brahmagupta in BrSpSi. III. 1. The same rule in other words
has been described by Bhāskara I in MBh. III. 2. In the Vāsanā
Bhāṣya, Pṛthūdaka Svāmī describes the details of determining
the directions. The level of the ground is ascertained by means
of water and a gnomon of 12 aṅgulas is set up. Find out two
points where the shadow of the gnomon enters into and passes
out of the circle. Bhāskara prescribes drawing out a fish figure
with these points. The thread line which goes through the
mouth and tail of the fish figure indicates the north and south
directions with respect to the gnomon. Brahmagupta says that
if the Sun is on the eastern side, then where the shadow-point
enters circle (in the forenoon) that point would be the west, and
the point where it emerges out (in the afternoon) is the east.
As the Sun moves along the ecliptic, its declination
changes. By the the time the shadow moves between the
forenoon and afternoon points as given above, the Sun , traverses
some distance of the ecliptic and, so, theoretically speaking, its