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

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

GNOMON USED FOR FINDING THE DIRECTIONS 357
declination gets changed. It follows. therefore, that the East-
West line in the above determination is not the true position
of the actual East-West line. Brahmagupta (628 A. D.) was
the first Hindu astronomer who prescribed the determination of
the East-West line with proper allowance for the change in the
Sun's declination. (Shukla) The details of the method intended
by him have been supplied by his commentator Pṛthūdaka
Svāmī (860 A. D.).
Bhāskara and Brahmagupta both give another
method of determining directions : (BrSpSi. III. 2 ;
MBh. III. 3) : With the three points (at the ends of the three
shadows of the gnomon) corresponding to (any three) different
times (in the day), draw two fish-figures (each with two of the
three points) in accordance with the usual method. From the
point of intersection of the lines passing through the mouth and
tail (of the two fish-figures), determine the north and south
directions. (MBh. III. 3).
Brahmagupta in his rule is more precise :
The point where the lines passing through the two
fish-figures, which are drawn by means of three shadow
ends (of the gnomon), intersect each other is for places
in the northern hemisphere, the south direction, (if
the midday shadow falls to the north of the foot of the
gnomon). If the midday shadow falls towards the
south of the foot of the gnomon, it is the north
direction. (BrSpSi. III. 2).
This rule is obviously based on the assumption that the
the locus of the end of the shadow of the gnomon is a circle.
In tact the locus for places whose latitude is less than (90°—the
obliquity of the ecliptic), this locus is a hyperbola.
Brahmagupta has made numerous uses of gnomon. He
and Bhāskara, for example, both give the rules for finding the
latitude and colatiude and the zenith distance and altitude of
the Sun by finding out the length of the shadow and the length
of the gnomon (BrSpSi. III. 10 ; MBh. III. 5) ; also rule for
the determination of the latitude with the help of the Sun's
meridian zenith distance and declination (BrSpSi. III. 13 ; MBh.

338 BRAHMAGUPTA AND ASTRONOMICAL INSTRUMENTS
III. 17) ; also rule for finding the Sun's altitude (BrSpSi. III. 27 ;
MBh. III. 24) (The Sun's altitude for the night has been called
by Brahmagupta as pātāla-śaṅku, BrSpSi. XV. 9).
Golayantra or Armillary Sphere
The first mention of the Golayantra or the armillary sphere
is in the Āryabhaṭīya (Golapāda. 22)¹ which was a uniformly
round circle made of wood or of bamboo and which
was of uniform weight or density alround. It was levelled with
mercury, oil or water. A śalākā or pin (or rod) was fixed in it
in the south-north direction. Its description from the com-
mentary Bhaṭadīpikā of Paramādiśvara is given here :
A sphere of wood. uniformly round on all sides and
with uniform density, and also light is made to revolve
round an iron axis fixed north-south without friction
(oil may be introduced to avoid friction). To the
backside of the sphere, is fixed a nālaka full of water
which has the length equal to the circumference of the
sphere: and which has a hole at the bottom.
Now a thread, connected to the hook of the wooden
ball (on the top side) passing over another small
ball (in the same axis of the wooden ball) is attached
to the mercury lobe by its other end. The mercury
lobe is placed on the level of water and water is allo-
wed to flow through the bottom hole and with water
mercury lobe also goes down. The time in which the
above hook of ball comes to bottom (180°) is noted.
The experiment is repeated with oil. The use of this
mechanism is to revolve the ball by water or oil¹

1. काष्ठमयं समवृत्तं समन्ततस्सम गुरु लघु गोलम् ।
   पारतं तैलजलैस्तं भ्रमयेत्स्वधिया च कालसमम् ॥ Arya. IV. 22
   काष्ठमयं वंशादि काष्ठेन निर्मितं समवृत्तं सर्वतोवृत्तं समन्ततस्सम गुरुं सर्वावयवेषु समं गुरुत्वं
   यथा भवति तथा कृतं । लघुमगुरुं एवं भूतं गोलं कृत्वा पारतादिभिस्तं स्वधिया च कालसमं भ्रमयेत् ।
   अयमर्थः । भूमिष्ठ दक्षिणोत्तरस्तम्भयोरुपरि गोलप्रोतायश्शलाकाया अग्रे स्थापयेत् । गोलदक्षिणोत्तरा-
   च्छिद्रे च तैलेन सिञ्चेत् यथा निस्सङ्गो गोलो भ्रमति । गोलस्यापरतो गोलपरिधिसंमित दैर्ध्यं साधश्छिद्रं
   जलपूर्णं नलकं निदध्यात् ततो गोलस्यापरस्वस्तिक कीलकं विधाय तस्मिन्सूत्रस्यैक मग्रं बद्ध्वादौ
   विषुवन्मण्डलपृष्ठेन प्राङ्मुखं नीत्वा तदग्रबद्धं पारतपूर्णमलावु जलपूर्णे नलके निदध्यात् ततो नलकस्याध
   [Cont. on Page 337]

GOLAYANTRA OR ARMILLARY SPHERE 339
In the Arabic epitome of the Almagest entitled Tahriru'l
mejesti. the armillary sphere, Za ul halk, is thus described :
Two equal circles are placed at right angles, the one
representing the ecliptic, the other the solstitial colure.
Two pins pass throught the poles of the ecliptic and
two other pins are placed on the poles of the equator.
On the first two pins are suspended a couple of circles
moving, the one within, the other without, the
first mentioned circles, and representing two seconda-
ries of the ecliptic. On the two other pins a circle
is placed, which encompasses the whole instrument,
and within which the different circles turn; it repre-
sents the meridian. Within the inner secondary of
the ecliptic, a circle is fitted to it, in the same plane
and turning in it. This is adapted to measure latitudes.
To this internal circle, two apertures or sights, opposite
to each other, and without its plane are adapted like
the sights of an instrument for altitudes. The armil-
lary sphere is complete when consisting of these six
circles. The ecliptic and secondaries are to be gra-
duated as minutely as may be practicable. It is best
to place both secondaries, as by some directed, within
the ecliptic (instead of placing one of them without it),
that the complete revolution of the outer secondary
may not be obstructed by the pins at the poles of the
equator. The meridian likewise should be doubled,
or made to consist of two circles; the external one gra-
duated and the internal one moving within it. Thus the
pole may be adjusted at its proper elevation above the
horizon of any place. The instrument so constructed
consists of seven circles.
It is remarked that when the circle representing the
meridian, is placed in the plane of the true meridian, so
छिद्रं विकृतं कुर्यात् तेन जलं निस्स्रवति । नलकाच्च जलमधो गच्छति । तद्वशाच्च तदस्थमलावु
पारतपूर्त्या गुरुत्वाज्जलेन सहाधो गच्छद् गोलं प्रत्यङ्मुखमाकर्षति । एवं त्रिंशद् घटिकाभिर्व्वसंमितं यथा
जलं भवति गोलस्य चार्धं भ्रमति तथा स्वबुद्ध्या जलनिस्स्रावो योज्यः । इति ।
अमृतस्रावयोगेन कालभ्रमण साधनम् ।
गुणबीजसमाकृष्टं गोलयंत्रं प्रकल्पयेत् ॥ SuSi. XIII, 16-17

340 BRAHMAGUPTA AND ASTRONOMICAL INSTRUMENTS
that it cuts the plane of the horizon at right angles,
and one of the poles of the equator is elevated above
the horizon conformably with the latitude of the
place; then the motions of all the circles round the
poles represent the motions of the universe.
After rectifying the meridian, if it be wished to
observe the Sun and Moon together, the outer secon-
dary of the ecliptic must be made to intersect the
ecliptic at the Sun's place for that time: and the
solstitial colure must be moved until the place of
intersection be opposite to the Sun. Both circles are
thus adjusted to their true places; or if any object but
the Sun, be observed, the colure is turned until the
object be seen in its proper place, on that secondary
referred to the ecliptic: the circle representing the
ecliptic being at the same time in the plane of the
true ecliptic and in its proper situation. Afterwards,
the inner secondary is turned towards the Moon (or to
any star intended to be observed), and the smaller
circle within it, bearing the two sights is turned, until
the Moon, (or to any star intended to be observed),
and the smaller circle within it, bearing the two sights,
is turned, until the Moon be seen in the line of the
apertures. The intersection of the secondary circle and
ecliptic is the place of the Moon in longitude: and the
arc of the secondary, between the aperture and the
ecliptic, is the latitude of the Moon on either side
(North or South). (From Colebrooke's Miscellaneous
Essays).
The same instrument, as described by Montucla from the
text of Ptolemy (1. 3. c. 2) consists of six circles: first a large
circle representing the meridian; next four circles united toge-
ther, representing the equator, ecliptic and two colures, and
turning within the first circle on the poles of the equator, lastly
a circle turning on the poles of the ecliptic, furnished with sights
and nearly touching, on its concave side, the circumference of
the ecliptic.
The armillary sphere described by the Arabian epitomiser,
differs, therefore; from Ptolemy's in omitting the equator and

GOLAYANTRA OR ARMILLARY SPHERE 341
equinoctial colure, and adding an inner secondary of the ecliptic,
which as well as the meridian, is doubled.
According to Lalande, the astrolobe of Ptolemy, from which
Tycho Brahe derived his equatorial armillary, consisted only of
four circles: two placed at right angles to represent the ecliptic
and solstitial colure; a third turning on the poles of the ecliptic
and serving to mark longitudes; and a fourth, within the other
three, furnished with sights to observe celestial objects and
measure their latitudes and longitudes.
Whether the ancient Greeks had any more complicated
instrument formed on similar principles, and applicable to astro-
nomical observations, is perhaps uncertain. We have no detailed
description of the instrument which Archimedes is said to have
devised to represent the phenomena and motions of the heavenly
bodies; nor any sufficient hint of its construction; nor does
Cicero’s account of the sphere exhibited by Posidonius suggest a
distinct notion of its structure.
Among the Arabs, no addition is at present known to have
been made to the armillary sphere; between the period when the
Almagest was translated and the time of Alhazen, who wrote a
treatise of optics, in which a more complicated instrument than
that of Ptolemy, is described; Alhazen’s armillary sphere is stated
to have been the prototype of Tycho Brahe’s; but neither the
original treatise, nor the Latin translation of it, are procurable
and one is therefore unable to ascertain whether the sphere,
mentioned by the Arabian author, resembled that described by
Indian astronomers. At all events, says Colebrooke, he is more
modern than the oldest of the Hindu writers.
Here we give the literal translation of the passage on armi-
llary sphere or Golayantra occurring in the Sūrya-Siddhānta :
Let the astronomer frame the surprising structure of
the terrestial and celestial spheres.
Having caused a wooden globe to be made (of such
size) as he pleases; to represent the Earth : with a
staff for the axis passing through the centre, and
exceeding the globe at both ends; let him place the
supporting hooks, as also the equinoctial circle.
Three circles must be prepared, (divided for signs and
degrees), the radius of which must agree with the

342 BRAHMAGUPTA AND ASTRONOMICAL INSTRUMENTS
respective diurnal circles, in proportion to the equino-
ctial : the three circles should be placed for the Ram
(Meṣa) and following signs, respectively, at the proper
declination in degrees N. or S. ; the same answer
contrariwise for the Crab (Karkaṭa) and other signs.
In like manner three circles are placed in the southern
hemisphere, for the Balance (Tulā) and the rest, and
contrariwise for Capricorn (Mṛga) and remaining
signs. Circles are similarly placed on both hoops for
the asterisms in both hemispheres, as also for Abhijit
and for the Seven Ṛṣis, Agastya, Brahmahṛdaya and
other stars.
In the middle of all these circles is placed the equinoctial.
At the intersection of that and supporting hoops, the
distant from each other half the signs, the two equino-
xes should be determined; and the two solstices, at the
degrees of obliquity from the equinoctial; and the
the places of the Ram (Meṣa) and the rest, in the
order of the signs, should be adjusted by the strings of
the curve. Another circle thus passing from equinox
to equinox, is named the ecliptic; and by this path, the
Sun illuminating worlds, for ever travels. The Moon
and other planets are seen deviating from their nodes
in the ecliptic to the extent of their respective greatest
latitudes (within the zodiac).¹

1. भूलोकस्य रचनां कुर्यादाश्चर्यकारिणीम् ।
   अभीष्टं पृथिवीगोलं कारयित्वा तु दारवम् ॥
   दण्डं तन्मध्यगं मेरोरुभयत्र विनिर्गतम् ।
   आधारकक्षाद्वितयं कक्ष्यां वैषुवतीं तथा ॥
   भगणांगुलैः कार्या दलितास्तिस्र एव ताः ।
   स्वाहोरात्रार्धकर्णैश्च तत्प्रमाणानुपातः ॥
   क्रान्तिविक्षेपभागैश्च दलिता दक्षिणोत्तरा ।
   स्वैस्स्वैरपक्रमैः कार्या मेषाद् नामपक्रमात् ।
   कक्षाः प्रकल्पयेत्ताश्च कर्क्यादीनां विपर्ययात् ।
   तद्वत्तिस्रस्तुलादीनां मृगादीनां विलोमतः ॥
   याम्यगोलाश्रिताः कुर्यात् कक्ष्यावारद्वयोपरि ।
   याम्योद्ग्भाग संस्थानां भानामभिजितस्तथा ॥
   सप्तर्षीणामगस्त्यस्य ब्रह्महृदीनां प्रकल्पयेत् ।
   मध्ये वैषुवती कक्ष्या सर्वासामेव संस्थिता । [Cont. on page 341]

HOW TO OBSERVE PLACES OF STARS 343
The author of the Sūrya-Siddhānta then proceeds to notice
the relation of the great circles before mentioned to the horizon,
and observes that whatever place be assumed for the apex of
the sphere, the middle of the heaven for that place is its horizon.
He concludes by showing, that the instrument may be made to
revolve with regularity, by means of a current of water; and
hints, that the appearance of spontaneous motion may be given,
by a concealed mechanism, for which quicksilver is to be
employed. There is a hint of secrecy also in one of the lines,
and it has, therefore, been stated that the construction and the
mechanism of working should be learnt under the guidance of a
teacher.
How to Observe Places of Stars
Details are not available in this connection. The Sūrya-
Siddhānta only hints "that the astronomer should frame a sphere
and examine the apparent longitude and latitude (sphuṭavikṣepa
and sphuṭadhruvaka)". The commentators, however, describe
the manner of making the observation. They direct a spherical
instrument (Golayantra) to be constructed as described above.
This instrument is very much similar to the armillary sphere.
An additional circle graduated for degrees and minutes, is direct-
ed to be suspended on the pins of the axis as pivots. It is named as
Vedhavalaya or intersecting circle, and appears to be a circle of
declination. After noticing this addition to the instrument,
the instructions proceed to the rectifying of the Golayantra or
armillary sphere which is said to be placed, so that the axis shall
point to the pole: and the horizon be true by a water level.
The instrument being thus placed, the observer is instructed
to look at the star Revatī through a sight fitted to an orifice at
the centre of the sphere; and having found the star, to adjust by
it the end of the sign Pisces on the ecliptic. The observer is
then to look through the sight, at the yoga star of Aśvinī, or at
तदाधारयुतेः भार्धमयने विषुवद्वये ।
अयनाद्वयने चैव कक्ष्या तिर्यक्संस्थाऽपरा ॥
क्रान्तिसंज्ञा तया सूर्यः सदा सदापर्य्येति भासयन् ।
चन्द्राद्याश्च स्वरकैः पातैरपमण्डलमाश्रितैः ॥
ततोऽपकृष्टादृश्यन्ते विक्षेपाग्रे स्वपक्रमात् । SuSi. XIII. 3-12

344 BRAHMAGUPTA AND ASTRONOMICAL INSTRUMENTS
some other proposed object; and to bring the moveable circle of
declination over it. The distance in degrees, from the inter-
section of this circle and ecliptic, to the end of Mīna or Pisces,
is its longitude (dhruvaka) in degrees; and the number of degrees
on the moveable circle of declination, from the same intersection
to the place of the star, is its latitude (vikṣepa) North or
South.
The commentators have rightly remarked that the 'latitude
so found is sphuṭa or apparent, being the place intercepted bet-
ween the star and the ecliptic, on a circle passing through the
poles; but the true latitude (asphuṭa) is found on a circle hung
upon the poles of the celestial sphere as directed in another
place". (From Colebrooke's Paper on the Indian and Arabian
Divisions of the Zodiac. Miscellaneous Essays, Vol. II, 324-326).
For the details of the Golayantra, readers are requested to
refer to the description in the Siddhānta-Śiromaṇi of Bhās-
kara II.
—: o :—
Reference
Brahmagupta : yantrādhyāya in —BrSpSi.
K.S. Shukla : The Mahābhāskarīya.
Bhāskara I : Commentary on the Āryabhaṭīya.
Paramādīśvara : Bhaṭadīpikā, a commentary on the Āryabhaṭīya,
H.T. Colebrooke : Miscellaneous Essays, Vol. II., 1872.

१
॥ श्रीगणेशायनमः ॥
अथ ब्राह्मस्फुट सिद्धान्तस्य
पूर्वावशाध्यायां मध्यमाधिकारः
जयति प्रणतसुरासुरकिरीटरत्नप्रभाछुरितपादः ।
कर्ता जगदुत्पत्तिस्थितिविलयानां महादेवः ॥१॥
ब्रह्मणोक्तं ग्रहगणितं महता कालेन यत् खिलीभूतम् ।
अभिधीयते स्फुटं तज्जिष्णुसुतब्रह्मगुप्तेन ॥ २ ॥
ध्रुवताराप्रतिबद्धं ज्योतिश्चक्रं प्रतिक्षणगमादौ ।
पौष्णाश्विन्यन्तस्थैः সহ ग्रहैर्ब्रह्मणा सृष्टम् ॥ ३ ॥
चैत्रसितादेरुदयाद्भानोर्दिनमासवर्ष युगकल्पाः ।
सृष्ट्यादौ लंकायां समं प्रवृत्ता दिनेऽर्कस्य ॥ ४ ॥
प्राणैर्विनाडिकाक्षैः षड्भिर्घटिका विनाडिका षष्ट्या ।
घटिका षष्ट्या दिवसो दिवसानां त्रिशता मासाः ॥ ५ ॥
(ग) १. श्री नमः ।। परमात्मने । श्री रामा

( २ )
मासा^१ द्वादशवर्षं विकलालिप्तांशराशिभगणांतः^२
क्षेत्रविभागस्तुल्यः कालेन विनाडिकाद्येन^३ ॥ ६ ॥
स्वचतुष्टयरदवेदा^१ रविवर्षाणां चतुर्युगं भवति । ४३२००००^२
सन्ध्या सन्ध्यांशै^३ सह चत्वारि पृथक्कृतादीनि ॥७॥
युगदशभागो गुणितः कृतं चतुर्भि^२स्त्रिभिर्गुणास्त्रेता^३ ।
द्विगुणो^४ द्वापरमेकेन संगुणः कलियुगं भवति ॥ ८ ॥
१७२८०००।१२९६०००।८३४०००^५।४३२०००
युगपादानार्य^१भटश्चत्वारि समानि^२ कृतयुगादीनि । १०८००००^३
यदभिहितवान्^४ न तेषां स्मृत्युक्तसमानमेकमपि^५ ॥ ६ ॥

६. (ख) १. मास for (मासा)
[L1

( ३ )
मनुरेक सप्ततियुगः² कल्पो³ मनवश्चतुर्दश मनूनाम् ।
आद्यंतरांतसंधिषु⁵ कृतकालो⁴ऽस्माद्द्युगसहस्रम्⁶ ॥ १० ॥
४३२०००००००¹
आद्यंतरातसंधिषु⁴ कल्पमनूनां⁵ कृताब्दसमकालम्⁶ ।
नेच्छंति ये षडूनं¹ तेषां कल्पो युगसहस्रम्² ॥ ११ ॥
४२६४०८००००
मनुसंधियुग⁵मिच्छत्यार्यभटस्तन्मनुर्यतस्त्वयुगः ।
कल्पश्चतुर्युगानां² सहस्रमष्टाधिकं³ तस्य ॥१२॥
४३५४५६००००⁴

१०. (घ) २. ४३२०००००० for (४३२०००००००) (क) संख्या लुप्त है। टीका में
अंकित है।
(ग) २ + ७१ + (च)
३ कल्पो नवचतुर्दश १४ (च) for (कल्पोमनवश्चतुर्दश

( ४ )
युग्मन्वन्तरकल्पाः कालपरिच्छेदकाः² स्मृतावुक्ताः ।
यस्मान्न¹ रोमके ते स्मृति³ बाह्यो रोमकस्तस्मात् ॥ १३ ॥
कालर्क्ष¹देशयोगाद्भू²यो ग्रहमन्दशीघ्रपातानाम्⁵ ।
कल्पेन यतो योगस्ततः³ स्फुटं ग्रहयुगं⁴ कल्पः ॥ १४ ॥
कल्पेऽर्कबुध¹ सितानां भगणाः⁴ शून्यानि सप्तरदवेदाः ।
प्राग्व्रजता² कुजगुरुशनिशीघ्रोच्चानां स्वकक्षासु⁵ ॥ १५ ॥
४३२०००००००³
१३. (घ) १. न रोमके ते (च) न रौनुके ते for (न्न रोमके ते)
(ख) २. कलापरिच्छेदकाः for (कालपरिच्छेदकाः)
(च) ३. वि० इस प्रति में “……परिच्छे” के पश्चात् लेखक ने भूल से फिर “मनुरेक
सप्तति युग” से लिखना आरंभ कर दिया । इस प्रकार ५½ पंक्तियां पुनः
लिखी गईं ! उसके पश्चात् फिर क्रमशः लिखता गया ।
४. स्मृतबाह्यो for (स्मृतिबाह्यो)
१४. (घ) १. कालक्ष्यं (ख) कालक्ष (च) कालक्ष्यं for (कालर्क्ष)
२. दूद्भूयो (च) for (द्भूयो)
३. तत (च) for (ततः)
४. ग्रहयुगकल्पः (ग) ग्रहयुतं कल्पः for (ग्रहयुगं कल्पः)
(ग) ५. ग्रहशीघ्रमंद पातानां (क) (ख) for (ग्रहमन्द शीघ्रपातानां)
(च) ४. ग्रहयुग कल्पः for (ग्रहयुगं कल्प.)
१५. (घ) १. बुध (च) for (बुध)
२. व्रजनां (ग) व्रजती (ख) व्रजतां for (व्रजता)
(क) २. प्राग्व्रजतां for (प्राग्व्रजता)
३. संख्या लुप्त है । टीका में अंकित है ।
(ख) ४. भगराः for (भगणाः)
५. सा for (सु) (च) स कक्षासु for (स्व कक्षासु)