सिद्धान्तशिरोमणि: गणिताध्याय (भास्कराचार्य - ग्रहगणित, मध्यमाधिकार व स्पष्टाधिकार सटीक)

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but inclined, is justified here since the Kotis in all the
cases cited are all horizontal lines.
In fact, as we stated before, even Bhāskara's analysis
is not sound, for which reason, M. M. Sudhākara Dwivedi,
wrote the booklet called Vāstava Sṛngonnati based on
modern lines. As this topic could be found in books of
modern astronomy, we need not go into the treatment of
Sudhākara Dwivedi here.
Here ends the Sṛngonnatyadhikāra

GRAHAYUTYADHIKĀRA
Verse 1. The mean diameters of Mars, Mercury,
Jupiter, Venus and Saturn are respectively 4′-45″, 6′-15″,
7′-20″, 9′, 5′-20″.
Comm. Bhāskara gives us a method under verse (5)
of Chandragrahaṇādhikāra, as to how the angular diameters
of celestial bodies were being measured with an instrument
that we may call 'protractor', describing it as follows.
“यस्मिन् दिने अर्कस्य मध्यतुल्या स्फुटा गतिः स्यात् तस्मिन् दिने उदयकाले
चक्रकलाव्यासार्धमितेन यष्टिद्वितयेन मूलमिलितेन तत्रस्थदृष्ट्या तदग्राभ्यां
बिम्बप्रान्तौ विध्येत् “या यष्ट्यग्रयोरन्तरकलाः ता रविबिम्बकला भवन्ति
मध्यमाः” ie. On the day on which the true motion of the
Sun equals the mean, in the morning, observe with an
instrument having two equal rods jointed at one end (and
the other ends being connected by a flexible protractor
marked with minutes and seconds of arc) placing the eye.
at the joint of the rods, and the two rods pointing to the
extremities of a diameter of the disc. The magnitude of
the disc is then read on the protractor, which gives the
mean diameter”.
This method is alright so far as it goes. ' The measur-
ing being done at the time of morning and that too
with the naked eye might have been responsible for the
exaggerated magnitudes of the diameters of the discs of
the planets, as compared with their modern values. This
kind of exaggerate estimate is due what is called the
phenomenon of irradiation which increases the apparent
size of a brilliant body when seen at some distance. Even
Tycho Brahe, an accurate and brilliant astronomer prior
to the invention of the telescope gave estimates of these
angular diameters which are nearly the same as remarked
by Burgess in his translation of Sūryasiddhānta under
verses 13, 14 ch. VII.

502
Verse 2. These estimates being multiplied by the
difference of the radius and Śīghrakarṇa and divided by
thrice the Śīghra-antyaphalajyā are to be added or sub-
tracted from the mean values above given according as the
Śīghrakarṇa is less or greater than the radius to give the
rectified values. Three minutes of arc are to be construed
as one angula in this respect.
Comm. When the Śīghrakarṇa equals the radius we
know that the planet is situated at the mean distance. The
word planet here stands, of course, for the Mandaspaṣṭa-
graha which may be roughly taken to be the mean planet,
the equation of centre being small. If the Śīghrakarṇa
falls short of the radius, then evidently the planet is nearer
the earth than the mean position so that disc of the planet
appears to be bigger. Otherwise, the planet is further
and its disc appears to be smaller. It was noted that
approximately there was an increase or decrease ⅓ of the
magnitude of the disc by the decrease or increase of the
Śīghrakarṇa by the antyaphalajyā. Hence, in between the
two positions, rule of three is used to obtain the magni-
tudes as follows. "If by a difference of the Śīghrakarṇa
and radius equal to the antyaphalajyā, there is a difference
of ⅓ of the actual magnitude, what would it be for an
arbitrary difference?" The answer is d × ⅓ × 1/a = d/3a
where d = Śīghrakarṇa or radius and a the antyaphalajyā.
This difference is to be added or subtracted to the mean
value, as the case may be.
Verse 3 and first half of 4. To obtain the time of
the conjunction of two planets, compute the difference of
the longitudes of two planets, and divide by the difference
of their daily motions. If one of the planets be retrograde,
divide by the sum of the daily motions. The result gives
the number of days approximately after the moment of
conjunction if the slower planet has a longitude falling
short of that of the quicker. If one of the planets be

503
retrograde, and if its longitude be the lesser then also the
conjunction was past by the number of days computed.
In the other cases the conjunction is to take place after
the number of days computed. If, however, both the
planets be retrograde, then if the slower of them has a
longitude less than that of the quicker, then the conjun-
ction is ahead, otherwise past by the number of days.
Comm. Clear. In the case of one or both the planets
being retrograde, the word 'slower planet' means that
planet whose retrograde motion is slower and not the
planet whose mean motion is slower.
Latter half of verse 4 and verse 5. To rectify the
moment of conjunction.
Having computed the approximate time of conjun-
ction, obtain the true motions of the planets pertaining to
that day, rectify them for Āyana-Dṛk-Karma and following
the process indicated in verse (3) above, again compute
the moment of conjunction. (This will be a good approxi-
mation). This conjunction will be one on the polar
latitudinal circle. If Āyana-Dṛk-Karma be not done, then
the conjunction will be on the circle of celestial latitude.
Comm. Bhāskara says that the conjunction on the
circle of polar latitude is preferred because this could be
observed as there is a star at the celestial pole and the
movable circle of polar latitude could be moved into a
position in which the two planets could be seen situated
thereupon. The method of successive approximation is
self-explanatory. Bhāskara, however, adds that when the
conjunction is on the circle of celestial latitude, the
planets will be seen to be closer.
Verse 6 and first half of verse 7. To obtain the north-
south celestial latitudinal distance between two planets.

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Having computed the number of days by which the
celestial latitudinal conjunction was past or is going to
take place, let the common celestial longitude of the two
planets be obtained by the method of successive approxi-
mation for the moment of celestial latitudinal conjunction.
Let the celestial latitudes of the two planets be rectified
for parallax in latitude as in the case of solar eclipse. The
difference of these celestial latitudes in case they are of the
same direction or the sum if, of opposite direction, gives
the north-south distance of the planets with respect to the
ecliptic. Having known the directions of the celestial
latitudes with respect to the ecliptic, if the two celestial
latitudes happen to be both south or both north, then the
planet with lesser celestial latitude is said to be in the
opposite direction with respect to the other; that is,
suppose both have northern celestial latitudes and suppose
p₁ has a smaller northern celestial latitude than p₂, then
p₁ is said to be south of p₂. Similar is the case if both the
celestial latitudes happen to be south.
Comm. Here Bhāskara does not specify whether he
is talking of conjunction on a polar latitudinal circle or
on a celestial latitudiaal circle, though the previous
procedure indicated by him to obtain the moment of con-
junction gives preference to polar latitudinal conjunction
which is more easily observable. But, here, as he prescribes
rectification of the latitudes for parallax in celestial
latitude, we have to construe that he is speaking of
conjunction with respect to celestial longitudes alone,
because, in the context of parallax, no method was indi-
cated by him for parallzx in polar latitude.
Latter half of verse 7 and verses 8 and 9. Case of occultation.
If the north-south celestial latitudinal distance
happens to be less than the sum of the angular radii of the
two planets, an occultation occurs (what we say 'eclipse'

505
with respect to the Sun and the Moon, holds good with
respect to occultation). In this case of occultation, we
have to rectify the time of conjunction with respect to
parallax in latitude also. For obtaining this parallax in
longitude, let the planet which is nearer the earth be
taken as the Moon and the other the Sun. But to obtain
the longitude of the Vithribha, which is necessary to
compute parallaxes in longitude and latitude, the lagna of
the moment of conjunction is to be computed from the
position of the Sun and not that of the planet assumed
to be the Sun as directed above. Having obtained the
parallax in longitude, the computed moment of conjun-
ction is to be rectified for parallax in longitude, (if
necessary by the method of successive approximation) to
obtain the actual moment of apparent conjunction ie.
occultation here. This procedure is worth-adopting only
when the occultation in question takes place above the
horizon and is observable. The north-south celestial
latitudinal distance in this case of occultation corresponds
to the celestial latitude of the Moon in the case of a solar
eclipse. The direction of this celestial latitude is to be
construed as that of the direction in which the planet near
the earth is situated with respect to the other. If the
planet which is nearer the earth happens to have a motion
lesser than that of the other, or be retrograde, then the
planet which is situated at a greater distance from the
earth will be over taking the other so that the higher
planet gets occulted in the eastern direction of its disc.
Thus the first contact is to be known to be in the east
and the last contact would be in the west; (If otherwise,
the other way).
Comm. Self-explanatory.
Here ends the Grahayutyadhikāra.
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506
Bhagrahayuti (Conjunction of a planet with respect
to a star)
The longitudes of the stars (professed to be polar).
The polar longitudes of the stars from Aswini includ-
ing Abhijit are as follows.
R d m R d m
Aswinī 0- 8- 0 Swātī 6-19- 0
Bharaṇī 0-20- 0 Visākhā 7- 2- 5
Kritticā 1- 7-18 Anūrādhā 7-14- 5
Rohiṇī 1-19-18 Jyeṣṭhā 7-19- 5
Mṛgasīrṣa 2- 3- 0 Mūlā 8- 1- 0
Ārdrā 2- 7- 0 Purvāṣādhā 8-14- 0
Punarvasū 3- 3- 0 Uttarāṣādhā 8-20- 0
Puṣyamī 3-16- 0 Abhijit 8-25- 0
Asreṣā 3-18- 0 Sravaṇam 9- 8- 0
Makhā 4- 9- 0 Dhaniṣṭhā 9-20- 0
Purvāphalgunī 4-27- 0 Satabhiṣak 10-20- 0
Uttarāphalgunī 5- 5- 0 Purvābhādrā 10-26- 0
Hasta 5-20- 0 Uttarābhādrā 11- 7- 0
Chitrā 6- 3- 0 Revatī 0- 0- 0
Comm. In the enumeration of these longitudes,
Bhāskara makes three statements which we have to note.
(1) That these longitudes are rectified for Āyana-Dṛk-
Karma ie. that they are polar longitudes, (2) That the
longitudes of Kritticā and Rohinī are less by 32′. (The
longitudes shown in the table above are those incorporating
this specified correction), (3) That the longitudes of
Visākhā, Anūrādhā and Jyeṣṭhā are to be increased by 5′.
(This correction is also incorporated in the table given
above).

507
It is to be noted that these longitudes are the same
given by Brahmagupta originally and copied by S'rīpati
as well as the corrections indicated above. But herein a
mistake was committed as clarified by Bhāskara later in
the end of the chapter namely that the polar longitudes
are subject to what is called Āyanavikāra though the
celestial longitudes are not. We say that the celestial
longitudes are not subject to such an Āyanavikāra ie. that
change due to the phenomenon known as the precession
of equinoxes, because the Hindu system is Nirayana ie.
reckons the longitudes from the first point of Aswinī
which is its zero point instead of reckoning from r. If
longitudes are measured from r, as r is preceeding, the
longitudes of stars will be steadily increasing all at the
same annual rate of precession namely about 50-25"
per year. These increasing longitudes of the modern
system go by the name Sāyana longitudes. It might
be thought that since the polar longitudes also are
measured from Aswinī and along the ecliptic like celestial
longitudes, they also don't vary like the Nirayana celestial
longitudes ; but it is not so, because the effect of precession
on the right ascension and declination of a star have
both their effect upon the polar longitude as well as polar
latitude. We cannot say that Bhāskara did not know
this but we may say that the effect on the polar longitude
and latitudes were construed by him as neglible and
would be appreciable only in the long run.
Verses 4, 5, 6. Sphutas'aras or rectified latitudes
of the stars.
Aswinī 10°- 0 north Ārdrā 11 - 0 south
Bharaṇī 12°- 0 ,, Punarvasū 6 - 0 north
Kṛtticā 4 -30 ,, Puṣyamī 0 - 0 ,,
Rohinī 4 -30 south Asreṣā 7 - 0 south
Mṛgas'irṣa 10 - 0 ,, Makhā 0 - 0 north

508
Purvāphalguni 12 - 0 north Purvāṣādhā 5 -20 south
Uttarāphalguni 13 0 ,, Uttarāṣādhā 5 - 0 ,,
Hasta 11 - 0 south Sravaṇam 30 - 0 north
Chitrā 1°-45' ,, Abhijit 62 - 0 ,,
Swātī 37°- 0 north Dhaniṣṭha 36 - 0 ,,
Visākhā 1 -20' south Śatabhiṣak 0 -20 south
Anūrādhā 1 -45 ,, Purvābhādrā 24 - 0 north
Jyeṣṭhā 3 -30 ,, Uttarābhādrā 26 - 0 ,,
Mulā 8 -30 ,, Revatī 0 - 0 ,,
(a) These are also what were given by Brahmagupta
and copied by Śrīpati, (b) Bhāskara mentions in the
Golādhyāya "नक्षत्राणां स्फुटा एव स्थिरत्वात् पठिताः शराः, दृक्कर्मणा-
ऽऽयनेनैषां संस्कृताश्च तथा ध्रुवाः" ie. In as much as the stars
are fixed we have given their latitudes and longitudes
rectified. But here, there is a point to be noted as
Bhāskara has put us in a doubt namely that the rectified
latitudes which he speaks of elsewhere under verse (3)
Grahacchāyādhikāra, Gaṇitādhyaya are not exactly the
polar latitudes spoken of here. (Ref. fig. 120). Let ♈MR
be the ecliptic and ♈N be the celes-
tial equator. Let S be a star, k be
the pole of the ecliptic, and p be
the celestial pole. Then ♈R is the
celestial longitude, SR the celestial
latitude which are known as the
Dhruvaka and S'ara (Asphuta-
s'ara). ♈M is the polar longitude
[Fig. 120] and SM the polar latitude. MR
is the arc connoting the Āyana-
Dṛk-Karma correction, so that celestial longitude ± Āyana-
Dṛk-Karma correction = polar longitude (plus or minus
according as the longitude lies in the 2nd and 4th
quadrants or 1st and 3rd quadrants). If the longitude just
equals 0°, 90°, 180° or 270°, the correction of Āyana--Dṛk

: 509
Karma vanishes. It must be noted here that when the
longitude is 0° or 180° Āyanavalana is maximum but
Āyana-Dṛk-Karma vanishes. There is no ambignity here
in this polar longitude because Bhāskara is unequivocal in
defining this. But under verse (3) Grahacchāyādhikāra,
Bhāskara means by Sphuṭaśara SL and not SM but by
Sphuṭaśara here he means SM. It is ridiculous to suppose
that Bhāskara did not know that B √(R²—Āyanavalana is
less than β. This B √(R²—a²)/R (a=Āyanavalanajyā) he
defined as Sphuṭaśara there under verse (3) cited. SR is
defined by him as Asphuṭaśara or simply Śara. Now here
in the commentary he makes us believe that Sphuṭaśara
is SM which is the polar latitude. This confusion created
by Bhāskara leads Ramaswarup the editor of Brahmagupta
Siddhānta as well as one Mukhopadhyaya the author of the
thesis, ‘Hindu Nakshatras’ to suppose that Bhāskara was
wrong in supposing that (B/R) √(R²—a²) > β. Bhāskara could
not evidently commit such a silly mistake but we must
infer that in that context he called SL as the Sphuṭaśara
which being added to Rn the Krānti or what is the same
LN gives SN the Sphuṭakrānti or the modern declination.
Rn is called by him as Asphuṭakrānti. In this context he
calls SM as the Sphuṭaśara since it is Dhruvābhimukha ie.
directed towards the pole. Of course, Bhāskara should
not have called both SL and SM as Sphuṭaśaras but since
he was deliberately defining the Sphuṭaśara as SL previously
and now as SM, and since he could not commit such
a glaring mistake as to construe (B/R) √(R²—a²) as greater
than β, we should not rush to pronounce that Bhāskara
was wrong. Only we could say that he is inconsistent
to that extent.
Verse 7. The polar longitudes of Agastya (Canopus)
is 87° and his polar latitude is 77 south. The polar

510
longitude of Lubdhaka (Sirius) 86° and its polar latitude
is 40° south.
Comm. Clear. The star Agastya is considered to be
important in Indian literature because the heliacal rising
and setting of Agastya are directed to be noted and in
fact are being noted in every panchanga even today from
times.immemorial. Even Kālidāsa alludes to this heliacal
rising of Agastya as inaugurating the Sarat-kāla or
autumn which was reiterated by Varaha Mihira in his
Bṛhat-Saṁhitā under the verse “भगवति जलधरपक्ष्मक्षपाकरा-
केंक्षणे कमलनाभे, उन्मीलयति तुरङ्गमकरिरथनीराजनं कुर्यात्” ie. when
Lord Vishnu whose eyes are supposed to be the Sun and
the Moon and his eyelids the clouds, opens his eyes ie. on
Kārtica Ekādasi the 11th day of the bright half of the
lunar month of Kārtica then Kings are directed to perform
Nīrājanavidhi for his horses, elephants and chariots (to
start on an expedition for war). It is to be noted that this
was so long ago in times of yore that Agastya used to rise
at the beginning of autum. Now the star is rising about
22nd August long in advance even in the rainy season.
This is on account of the effect of precession of equinoxes.
Verse 8. The Iṣṭanādis for Agastya are said to be
two, for Lubdhaka 2⅙, for other stars which are next in
size, 2⅓ and for still smaller ones the Iṣṭanādis are to be
taken still more.
Comm. Iṣṭanādis ie. the time in nādis (where a nādi
is equal to 24′ of time) giving the time in between the
rising moments of the star and the Sun, when the star rises
heliacally. In other words, let the star Agastya rise at a
particular time t. If the Sun rises at t+48′, then it is
the time for Agastya to rise heliacally. This again means
that if the Sun sets at time T and the star at T+48′ in
its diurnal motion, it is time for the star to set heliacally
in the west. Similarly for the other stars. It will be
noted here that stars set heliacally in the west and rise

511
heliacally in the east. This phenomenon has been long
in the notice of even the most illiterate people of India
from times immemorial, as they were used to get up from
beds round about the time when a brilliant star rose
heliacally and stood in the eastern horizon, shining for a
good length of time before Sun-rise. Each star of first
magnitude thus played the part of a morning star for some
time, though perhaps the illiterate folk mistake them to
be the same star. Especially the brightest stars of the
zodiac thus play the part of morning and evening stars.
The case with respect to Agastya and Lubdhaka is different
in that even though they are far away from the zodiac,
yet they were noticed to be morning and evening stars by
virtue of their being of the first magnitude at a spot of the
sky in the vicinity of which no other such brilliant stars
are there. This is the reason why panchangā—Computers
have been in the habit of recording in the panchangā even
to-date the heliacal rising and setting of Agastya, also
because the heliacal rising of this star synchronized with
the setting in of S'arat-kāla or autumn. Since in India
the lunar Kārtica Ekādasi, ie. the 11th day of the bright
half of the lunar month roughly synchronized with 15th
Nov., when the Sun rose far in the south of the horizon,
this Agastya, though it be in the far south, happens to
play the part of a morning star and about the day when
it rose heliacally, there were no more rains and waters of
the rivers stood crystal-clear as described by many a
Sanskrit poet like Kālidāsa (vide the famous verse of
Kālidāsa प्रसादोदयादम्भः कुम्भयोने र्महौजसः 4th canto Raghu-
vams'a).
The star Lubdhaka or Sirius, the dog-star as it is called
in English parlance also was conspicuous as a morning
and evening star, whose heliacal rising was noticed and
recorded by English poets like Shakespeare and Milton.
Since one nādi corresponds to 6°, two nādis correspond
to 12°, degrees, which are spoken as Kālāṁs'as for the

512
heliacal rising of Agastya. They are so termed, because
they indicate the Kāla or the time in between the rising
of the star and the Sun which signifies the moment of its
heliacal rising. Indirectly therefore these Kālāmsas indi-
cate which star is of which magnitude. A star which has
12° as Kālāmsas is therefore of first magnitude ; and as
the Kālāmsas increase, the magnitude also increases. It
will be rembered that the higher the magnitude of a star,
the fainter it will be and not the brighter as is likely to be
misconstrued by lay people.
Verse 9. To compute the moment of conjunction of
a planet and a star.
The Āyana-Dṛk-Karma is to be done as mentioned
before (with respect to the planet) and the Sphutasara is
to be computed to know the time of (polar latitudinal)
conjunction.
Comm. We are directed to use the polar longitude
and polar latitude with respect to the planet because this
kind of conjunction will be more conspicuous than a
celestial latitudinal conjunction because the Ecliptic is
far more inclined than the Equator with respect to the
horizon.
Verses 10, 11.
The difference in the longitudes of the planet and the
star, divided by the daily motion of the planet, gives the
number of days approximately after or before the moment
of conjunction.
If the planet be retrograde, the conjunction past or
future will be in the reverse ie. future or past.
Comm. Let x and y be the polar longitudes of the
planet and the star and let x<y. Since y is constant as
the star has no motion, x has to increase to the extent of

513
y to be in a polar latitudinal conjunction. Hence (y—x)
is to be covered by the planet as per its daily motion say
m. So the number of days that is to elapse for conjun-
ction is (y—x)/m. If x>y, then the conjunction was past by
x—y/m. If, the planet be retrograde and x<y, the con-
junction was past by (y—x)/m days and if x>y, the planet
being retrograde, the conjunction is to take place in (x—y)/m.
Note. Though Bhāskara does not mention here
Asakṛt-Karma ie. method of successive approximation, it
is implied because the motion of the planet differs from
moment to moment as well as its Sphutas'ara. Hence
having obtained the approximate moment of conjunction,
compute again the true motion of the planet at that
instant, as well as its Sphutas'ara and Āyana-Dṛk-Karma.
The latter ie. Āyana-Dṛk-Karma is to find the polar
longitude from the celestial and the Sphutas'ara is the polar
latitude ie. SM of fig. 120. The Sphutas'ara is required
for the purpose of finding the distance in between the
planet and the star on a polar latitudinal circle and not to
compute the moment of conjunction.
Verses 12, 13, 14. To compute the moments of
heliacal rising and heliacal setting of a star.
Compute the Udayalagna and the Astalagna of Agastya
and Lubdhaka doing Ākṣa-Dṛk-Karma alone. Assuming
the Udayalagna to be the Sun, compute the lagna for the
Iṣṭa-kāla nāḍīs (ie. after a lapse of Iṣṭanādīs after the
moment) given before (namely 2 nāḍīs for Agastya and
2⅙ for Lubdhaka) which will be the longitude of the Sun,
when the star (Agastya or Lubdhaka or whatever it be)
rises heliacally. Thus the Udayārka is a point of the
ecliptic which rises when the star rises heliacally.
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514
Having computed the Asta-lagna of the star, taking
it to be the Sun, compute the lagna in the reverse direction
ie. the lagna which preceeds it by the Iṣṭa-kāla given. If
this lagna be decreased by 180°, it will give the longitude
of the setting Sun at the time of the heliacal setting of the
star. Or again find the longitude of the point of the
ecliptic which is ahead of the Astalagna which takes (60—
Iṣṭanādis) to rise after the Astalagna; this longitude
decreased by 180° gives the longitude of the setting Sun
at the time of the heliacal setting of the star. The heliacal
rising or setting takes place when the longitude of the Sun
equals the longitude of the point of the ecliptic which is
technically called the Udayārka or the Astārka respectively.
The difference between the longitude of the Sun and that
of the Udayārka or the Astārka divided by the daily motion
of the Sun gives approximately the number of days that
have elapsed or are to elapse for the heliacal rising or
setting as the case may be,
Comm. Here we are to carefully differentiate between
the technical words (1) Udayalagna of the star, (2) Asta-
lagna of the star, (3) Udayārka and (4) Astārka of the
star. The word Udayalagna means that point of the
ecliptic which rises simultaneously with the star. The
word Astalagna means that point of the ecliptic which is
rising when the star is setting. The word Udayārka means
that point of the ecliptic which rises when the star rises
heliacally. The word Astārka means that point of the
ecliptic which is setting while the star sets heliacally. Thus
the four points are only points of the ecliptic.
Let A be the Udayalagna of a star on the circle CAB
which is the ecliptic. We know that the position of the
star should be above the eastern horizon, to rise heliacally,
by such a distance that the time in between the rising of
the Udayalagna and that point of the ecliptic which will
be rising when the star rises heliacally must be the Iṣṭa-
kāla nādis. Let B be a point ahead of A on the ecliptic

515
[चित्रम् : Fig. 121] such that the time in bet-
ween the rising of B and
the rising of A is equal to
the Iṣṭa-nādis. By the time
B rises, A will have gone
up a little above the eastern
horizon, as well as the star
that has risen along with A
and now the position of the
star is such that the time
in between its rising and
the rising of B is equal
to the Iṣṭa-nādis. Hence
B, which is a point of the ecliptic is termed Udayārka
signifying thereby that when the Sun coincides with B,
the star rises heliacally. Thus the Udayārka B is ahead
of the Udayalagna A and the time in between their risings
is equal to Iṣṭa-nādis.
Let now A' be the Astalagna ie. the point which rises
when the star sets. This point will not be, generally,
diametrically opposite to A because, the time between the
rising and setting of a star which is given by double the
rising hour-angle will be far more than half a sidereal day
when the star has a large northern latitude and in the case
of one having a large southern latitude, the time will be
far less than half a sidereal day.
From A' find C' which is behind A' such that the time
in between the rising of A' and C' is equal to Iṣṭa-nādis.
Then the point C diametrically opposite to C' namely C is
called Astārka ie. when the Sun is at C the star sets helia-
cally in the west. It is so because, when the Sun is on
the western horizon setting at C, C' will be the lagna and
when the star is setting A' is the lagna and as these two
lagnas .differ by Iṣṭa-nādis, the time between the Sun’s
setting and the star’s setting is also equal to Iṣṭa-nādis.

516
In other words the star is within the Iṣṭa-nāḍi distance
from the Sun and as C is behind A by Iṣṭa-nāḍis, the
setting Sun at C will be behind (ie. has a lesser longitude)
the setting star by Iṣṭa-nāḍis. Hence the star sets then
heliacally. Thus we see that the Udayārka and Astārka
given by the points B and C are respectively in advance
(ie. has a greater longitude) and behind (ie. has a lesser
longitude) of the Udayalagna and Astalagna removed by
an Iṣṭa-nāḍī distance. It will be noted that the arcs AC
and AB are not equal though the equatorial arcs corres-
ponding to them are equal.
Instead of finding C' from A' by the Vilomalagna
method and taking its diametrically opposite point, Bhās-
kara gives an alternative namely to find C from A' by the
Kramalagna method the time being (60-Iṣṭa-nāḍikas).
This is evident.
Bhāskara prescribes only Ākṣa-Dṛk-Karma to be per-
formed here in finding the Udayalagna and Astalagna of
the star because the star's polar longitude is already one in
which the Āyana-Dṛk-Karma is contained.
Thus from fig. 121, when the Sun-rises at A, the star
rises, when the Suns rises at B the star rises heliacally
and when the Sun sets at C, the star sets heliacally.
Verse 15. If in the case of a star, the Astārka
happens to have a longitude greater than the Udayārka on
account of a very big northern latitude, that star does not
set heliacally (and so the question of heliacal rising does
not arise).
Comm. In Fig. 121, the Astārka C happens to have
a longitude less than that of B (because CAB is the dire-
ction of increasing longitude ie. positive direction). At
times it so happens that C lies towards the positive
direction of B ie. it has a longitude greater than that of B.

517
This happens when the star has a long northern latitude
and therefore the Ākṣa-Dṛk-Karma correction will be
sufficiently large and the star has a small north polar
distance. This may be substantiated as follows.
Fig. 122
We have the formula cos h = — tan ϕ tan δ which
gives the rising hour-angle of the star. When δ is nearly
equal to 90—ϕ, then cos h will be nearly equal to ‘—1’
which means that the rising hour angle is nearly equal to
180°, ie. the duration of the star’s stay below the horizon
will be very small. This means A' approaches A very
nearly (fig. 121), for example when it is in the position A₁'.
Then let C₁' be the point which is behind A' by Iṣṭa-nāḍīs
so that C₁' the diametrically opposite point C₁ is far ahead
of B in stead of preceeding it. This means that the star
which should first set heliacally and then after a few days
rise heliacally, is now in such a position that it is to rise
heliacally even before setting heliacally. This is an un-
tenable position. That this is not tenable may be seen
otherwise. The diametrically opposite point of Astalagna
ie. the point C of the ecliptic which should set along with
the planet should have a longitude less than that of the

518
Astārka C₁′; but C₁′ will have now a longitude less than C
which is incongruous.
In such a situation, Bhāskara says, there is no question
of the star setting heliacally at all. This is indeed, an
ingenious mathematical presentation of a physical pheno-
menon, which reflects credit to Bhāskara's genius.
Verse 16. Circumpolar stars or Sadōdita stars.
If a star has a northern declination greater than
90–ϕ (ie. ϕ > 90–δ) it will be always above the horizon ;
also if the southern declination is greater than 90–ϕ such
a star will never be seen in a northern latitude, be it
Lubdhaka or Agastya or even a planet for the matter of
that.
Comm. (Vide fig. 122) Let S₁ be a star such that
PS₁′ < PN ie. 90–δ < ϕ ie. δ > 90–ϕ. It is clear from the
figure that its diurnal path is entirely above the horizon.
It is called a Sadōdita star or circumpolar.
Take the case of S₂. Its southern declination ie. QS₂′
is greater than the lamba (90–ϕ) ie. QS. It is evident
from the figure that its diurnal path is entirely below
the horizon.
Bhāskara gives two examples here namely (1) where
the latitude is greater than 37°, there Agastya will not be
visible (having a great north-polar distance), (2) where
the latitude is greater than 52°, there Abhijit is always
above the horizon (having a small north-polar distance).
Bhāskara adds, ‘even a planet’. This will be true,
for example, in a high latitude say 89°. Suppose the
southern declination of a planet is greater than 1°. On
that day and for some more days also, the planet’s diurnal
paths will be below the horizon as will be clear from a
figure.

519
Verses 17 to 21. Ancient astronomers happened to
give a list of polar longitudes and polar latitudes at a time
when there were no Ayanāṁsas ie. when the zero-point of
the ecliptic as taken by the Hindu Astronomers namely
Aświnī coincided with the modern zero-point namely r.
In fact these polar longitudes and latitudes do change
if there be Ayanāṁsas. Here in this case from the polar
latitudes the celestial latitudes are to be computed in a
reverse process; with the half of these celestial latitudes,
the Āyana-Dṛk-Karma is to be effected in the reverse
process to obtain the celestial longitudes. After having
obtained the correct celestial longitudes and celestial
latitudes, now, bringing the Ayanāṁsas into the picture,
compute the correct Dṛk-Karma and also rectify the
celestial longitudes, to obtain the correct polar longitudes
and polar latitudes to compute the moment of polar
latitudinal conjunction. This case may be taken when the
Ayanāṁsas are large, otherwise, a small difference there
will be, (which does not matter).
Comm. Bhāskara·gives the process in the course of
the commentary. We have the formula
Asphuṭa Vikṣepa × Yaṣṭi
Sphuṭa Vikṣepa = ———————————————————————
Radius
Here we know Sphuṭa Vikṣepa. At once, we could not
Sphuṭa Vikṣepa × Radius
say Asphuṭa Vikṣepa = ———————————————————————
Yaṣṭi
computing Yaṣṭi from the modern Ayanāṁsas. Yaṣṭi is
a function of declination too, because the Āyanavalana is
a function of declination. We know, the declinations
change in the wake of precession of the equinoxes. So,
there is no point in taking the value of the present Yaṣṭi.
We should compute it for the Āyanaśūnyakāla or for the
time when Aświnī coincided with r. Then the formula
could be applied. Then with this celestial latitude obtained,
Āyana-Dṛk-Karma is to be effected to obtain the present