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

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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

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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

520
polar longitude, using the modern Ayanāṁsas. Since in
this process, we have no definite knowledge of the then
celestial longitude ie. of the Āyanasūnyakāla, the method
of successive approximation is appealed to.
But this method of Bhāskara
may be modified to an easier pro-
cess as follows. Let rL, LS be
the polar longitude and polar lati-
tude of a star as given by Acharyas
in whose time the Hindu first
point of the zodiac coincided
with r. It is required to find rN
and NS the celestial longitude
and latitude which hold good even
today becaus rN and NS are the same as AN, AS, A being
the first point of Aświni and a celestial longitude measured
from A along the ecliptic is not subject to change on account
of precession of the equinoxes as well as the celestial lati-
tude. From the spherical triangle rLM, cot rLM = cos rL
tan ω (1) and from the triangle SNL, tan LN =
cos SL̂N × tan SL = cos rLM tan SL (4) and sin SN =
sin SL × sin SLN = sin SL × sin rLM (5). From (1)
the angle rL̂M is obtained ; substituting this value in (2)
and (3) LN and SN are obtained. Adding LN to rL we
have rN.
Thus the celestial longitude and latitude are found
far more easily and more accurately than from the
laborious method indicated which gives only approximate
results.
Here ends the Bhagrahayutyadhikāra.

PĀTĀDHYĀYA
Verse 1. Even scholars get confused while computing
the occurence of a Pāta, unable to know whether it has
already occurred or is going to occur. So, I seek to clarify
the method of computing the moment of occurrence of a
pāta.
Comm. (1) There are what are called pātās, two in
number, called Vyatipāta and Vaidhṛti. The first is
defined as occurring at that moment when the Sun and
Moon have equal declinations, being situated in opposite
Ayanas but the same goḷa. The words Uttaragoḷa and
Dakṣiṇagoḷa are used by Hindu Astronomers as the
northern and southern halves of the celestial sphere on
either side of the celestial equator, respectively; where as
the words Uttara-Ayana and Dakṣiṇa-Ayana are used to
connote the times when the Sun or Moon have tropical
longitudes (measured from r instead of Aswini, the zero
point of the Hindu Zodiac) one lying between Capricorn
and Cancer, and the other lying between Cancer and
Capricorn. Thus Vyatipāta occurs when the Sun and
Moon each lies in one of the first or second quadrants of
the ecliptic measured from r or each lies in one of the
third or fourth quadrants of the ecliptic; (both should not
lie in the same quadrant) and have equal declinations.
The second Pāta Vaidhṛti occurs when the Sun and
Moon have equal declinations, they being situated in the
same Ayana but opposite goḷas. These two moments are
considered to have malefic effects on humans and the world
of life. Though the moments are to be computed by a
knowledge of spherical astronomy, they have only an
astrological significance. A chapter, usually the last, has
been devoted to this subject in every book of Hindu
Astronomy. The method of computation was felt difficult
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522
by the ancient Hindu astronomers, before the time of
Bhāskara, for the reason that the Moon does not exactly
move in the ecliptic but in his own orbit whose inclination
to the ecliptic was taken to be 4½°. The points of intersec-
tion of the ecliptic with the lunar orbit, or what are called
Rāhu and Ketu, the nodes of the lunar orbit, have
themselves a motion backwards on the ecliptic in 18.59575
solar years as per Bhāskara. This makes the lunar orbit
oscillate about the mean position of the ecliptic, so that
sometimes the lunar orbit lies between the celestial equator
and some-times not between them. This phenomenon
makes the computation more difficult, to obtain the
declination of the Moon. Bhāskara says that even great
Ācāryas like Lalla and Brahmagupta went wrong or got
confused in computing the moments of the occurence of a
Pāta. Bearing upon Bhāskara’s statement that such Ācāryas
also got confused, some second-rate astronomer exclaimed
in the following interesting manner “त्रिस्कन्धविद्याकुशलैकमल्लो
लल्लोऽपि यत्राऽप्रतिभा बभूव, यातेऽपि किञ्चित् गणिताधिकारे पाताधिकारे
मम नाऽधिकारः” ie. “when even an unrivalled scholar like
Lalla, who was well-versed in the three branches of
Jyotiṣa betrayed his ignorance in this context of Pātādhi-
kāra, though I am a bit of a mathematician, I could have
no pretensions to any knowledge in this difficult chapter”.
In Fig. 124, Vyatīpāta occurs when SL=MN, where
S and M are the Sun and the Moon situated respectively
in Uttarāyaṇa and Dakṣi-
ṇāyana but in the same
uttaragola. The positions
of S and M could be
interchanged ie. S may be
in the second quadrant of
the ecliptic and M in the
first quadrant and if their
declinations be equal, then
Fig. 124
also Vyatīpāta occurs. Of course in this figure 124, we
have taken the Moon also situated on the ecliptic. In

528
actual computation, we should not take the Moon as such.
The pāta named Vaidhṛti occurs when in the same Fig.
124, SL=M¹N¹. Herein both the Sun and Moon are in
the same Ayana namely Uttarāyaṇa but in opposite goḷas.
If we assume the Moon to be moving on the ecliptic,
Vyatīpāta occurs when the sum of the tropical longitudes
of the Sun and Moon equals 180° and Vaidhṛti occurs
when the sum of those longitudes is equal to 360.
Verse 2. Definitions of Goḷa-Sandhi and Ayana-
Sandhi with respect to the Sun.
When the tropical longitude of the Sun ie. his
longitude measured from ♈ along the ecliptic is equal to
0° or 180°, then he is said to be at his goḷa Sandhi. In
other words, when the Sun who moves along the ecliptic
comes to the celestial equator, he will be at his Goḷa
sandhi. Similarly when his longitude is 90° or 270°, he is
said to be at the his Ayana-Sandhi.
Comm. This means that when the Sun is about to
pass from the Dakṣiṇa goḷa to the uttaragoḷa or from the
uttaragoḷa to the Dakṣiṇagoḷa he is said to be at a Goḷa
Sandhi. Similarly when he is about to go south or when
he is about to go north, he is said to be at Ayana Sandhi.
The word 'Sandhi' means junction. Thus the points ♈ and
♎ (Libra) are said to be goḷa Sandhis whereas the points
denoting Cancer and Capricorn are Ayana-Sandhis. Since
in Bhāskara's time the Ayanamsas were 11°, ie. the point
♈ was behind the zero point of the Hindu Zodiac by 11°,
therefore Bhāskara gives the Goḷa Sandhis as the points
having longitudes 349° and 169° respectively. Similarly
the Ayana Sandhis were the points having longitudes 79°
and 259° respectively.
Bhāskara gives the method of locating these goḷa
Sandhis or Ayana Sandhis as follows. Erect a gnomon

524
vertically with the help of a plumb-line. Draw a circle on
the horizontal plane having the foot of the gnomon as the
centre and any arbitrary radius. Draw the East-West and
North - South diameters of the circle. The longitudes of
the Sun when the shadow of the gnomon lies along the
East - West diameter give the gola Sandhis. Note the
direction of the shadow daily after the day when his
shadow is along the East - West line. If the Sun rises
thereafter a little towards North of the East point, then he
is said to be in the uttara gola; his shadow will be a little
south of the East - West line. The point at which the
Sun thus begins to rise north of the east point, is the Meṣa
gola Sandhi or the vernal equinox. Gradually the Sun
goes on rising at points which are farther and farther away
from the East point. When he has reached the extreme
north point, ie when the gnomon's shadow is extreme
south, he is at the Cancer. Similarly when he rises at the
extreme South point on the horizon, he is at Capricorn.
Bhāskara actually noted these four points and the longi-
tudes cited above were given by him as the Gola Sandhis
and Ayana Sandhis. Indirectly, he could know that
the Ayanāṁśas at the time of his writing the book,
were 11°.
Verses 3 to 6. Speciality with respect to the
Moon.
Let the Hsine and Hcosine of the tropical longitude of
the pāta (Rāhu the ascending node of the lunar orbit) as
per the smaller table of Hsines when the radius is taken to
be 120', be respectively multiplied by 123 and 7 and
divided respectively by 4 and 12. The results are known
as the Bāhuphala and Koṭiphala respectively. According
as the tropical longitude of Rāhu ie λ be such that as
270° < λ < 90° or 90° < λ < 270°, the Bāhuphala is to be
divided by 362 ± Kotiphala. Subtract the result from
the Gola Sandhis and Ayana Sandhis of the Sun, to get
those of the Moon.

525
Comm. from the Hindu Astronomical point of view
this is an intricate procedure which made that particular
half-learned astronomer declare "पाताधिकारे मम नाऽधिकारः"
ie. "I have no pretensions to have understood the
chapter known as Pātādhikārā", We perceive herein
Bhāskara's mathematical understanding of the problem.
His procedure is approximate because he uses (1) the
smaller crude table of Hsines taking the radius to be 120'
instead of 3438. (2) also because he gives the Sun's
declinations for longitudes 15°, 30° etc. as well as Moon's
celestial latitudes for his longitudes of 15°, 30° etc.
relative to the node in his orbit. Of course, his mathema-
tical procedure was correct.
He exemplifies his mathematical procedure by solving
a problem given in the prasna-Adhyāya of his book
Golādhyāya. The problem set by him is as follows.
युक्तायनांशोऽशशतं शशी चेत्
अशीतिरर्को द्विशती विपातः
चन्द्रः, तदानीं वद पातमाशु
धीवृद्धिदं त्वं यदि बोबुधीषि ।
ie. If the tropical longitude of the Moon is 100°, that of
the Sun 80°, and the longitude of the Moon measured in
his orbit from the Node Rāhu is 200°, compute the moment
of the occurence of the pāta, if you know what was said by
Lalla in his work Siṣya dhīvṛddhida. We shall first
understand Bhāskara's mind and subsequently we shall
give a modern procedure.

526
Fig. 125
Refer to fig. 125. Gr NK is the celestial equator. RrL
is the ecliptic and RGAM the Moon's orbit where R is the
Rāhu or ascending node of the lunar orbit. Let the
obliquity of the ecliptic be ω (omega) and the inclination
of the lunar orbit to the ecliptic be i. ω was taken to be
24° and i 4½° by Bhāskara. Let the lunar orbit RGAM
cut the celestial equator in G which is called the Moon's
Gola Sandhi. r is the Sun's Gola Sandhi. Bhāskara first
wants us to locate G ie. to find rQ which gives its longitude.
Let A be the position of the Moon when his celestial
longitude is zero ie rA is perpendicular to the ecliptic.
Let M be the position of the Moon when his celestial
longitude is 15° (Here the figure is not drawn to scale but
a little exaggerated for the sake of clarity). Let ML be
the celestial latitude of the Moon in its position. M. If
LK be drawn perpendicular to the equator, LK is called
the Asphuṭa - krānti of the Moon. ML is called the
Asphuṭa-Vikṣepa of the Moon. MN drawn perpendicular
to the celestial equator ie. the declination of the Moon is
called Sphuṭakrānti of the Moon. Draw perpendicular LS
on MN. Then MS is called the Sphuṭa Vikṣepa of the
Moon. Since MN = Ms + LK Sphuṭakrānti = sphuṭa
Vikṣepa + Asphuṭakrānti. Let Ap be the declination of
the Moon when his longitude is zero.

527
[Fig. 126]
Fig. 126
Now refer to fig. 126. Let NS be the ecliptic and
NM the celestial equator. The quadrant NS equal to 90°
is divided into six equal parts at S₁, S₂ etc. The successive
declinations of S₁, S₂ etc. are given by Brahmagupta as
362, 703, 1002, 1238, 1388, 1440 where the radius is given
to be 3438'. In other words 1440 = H Sin ω = H Sin 24°.
These can be easily verified from the formula
H Sin δ = (H sin λ × H sin ω) / R
putting λ = 15°, 30° etc. and R = 3438'.
[Fig. 127]
Fig. 127
Now refer to fig. 127. R M is the lunar orbit, R being
Rāhu. RL is the ecliptic. Using the formula.

528
H sin β = (H Sin λ × H Sin i) / R or β = (H Sin λ × i) / 120
since β and i are small and R is taken to be 120′, we have
the successive values of M₁L₁, M₂L₂ etc. of the celestial
latitudes of the Moon for arcs RM₁, RM₂ etc. successively
equal to 15°, 30° etc. given by 70, 135, 191, 234, 261,
270. In other words the maximum celestial latitude is
270′ = 4½°.
Since when λ = 15°, taking R = 120 β = 70′, the first
Sarakhanda ie the celestial latitude for λ = 15°, is 70′
(H cos λ × 70) / 120 Bhāskara Calls Koṭiphala.
It will be noted here that the first Sarakhanda is
taken to mean the increase in the celestial latitude
from zero to 70′ when the longitude increases from
0° to 15°. The argument adduced by Bhāskara in
such a context is as follows: “If for a H Cosine λ
equal to R equal to 120′ (when λ = 0) we have the first
Sarakhanda namely 70′, what shall we have for an
arbitrary H Cosine λ”. The result is (H cos λ × 70) / 120 =
⁷⁄₁₂ H cos λ. Since the word Cosine means Koṭijyā,
Bhāskara calls this Koṭiphala.
Now in fig. 125, let rL = 15°, then LK = 362′ as given
by Brahmagupta. Bhāskara takes this 362 as Δ (δ) where
δ is the declination of the foot of the celestial latitude of
the Moon. Then if β be the celestial latitude of the Moon,
Bhāskara construes that Δ β is given by the formula ⁷⁄₁₂
H Cos λ, which he calls Koṭiphala. Taking Δ δ ± Δ β as
the joint variation of δ and β which is roughly equated
with the variation in the modern declination MN of the
Moon, Bhāskara's argument is “If for a longitude r M = 15°
of the Moon corresponds Δ δ ± Δ β what should be the
longitude corresponding to the declination Ap of the Moon

529
when his longitude is equal to zero?" The result is
(Ap × 15) / (Δδ ± Δβ) = II
where rQ is looked upon as the longitude of the Moon at G
called the Goḷa Sandhi of the Moon.
But AP = the Sphuṭa Vikṣepa of the Moon when his
longitude is equal to zero where rA is the Asphuṭa Vikṣepa
at that point. Using his method of calculating AP from
Ar, Ap = (Ar × H cos δ) / R where
H sin δ = (H sin ω × H sin (90 + λ)) / R .
Here AP pertains to the longitude λ equal to zero, so
that H sin δ = (H sin 90° × H sin ω) / R = H sin ω.
∴ H cos δ = H cos ω = H cos 24° = 110 when R = 120'
Hence Ap = (Ar × 110) / 120 = (11 / 12) Ar.
But Ar is the celestial latitude of the Moon to be
calculated from RA taken to be λ
∴ rA = (H sin λ × 270) / 120 = (9 / 4) H sin λ (where 270' = 4½° = i)
taking R = 120 ∴ Ap = (9 / 4) H sin λ × (11 / 12). Hence
rQ from I = (9 / 4) H sin λ × (11 / 12) × 15 / (Δδ ± Δβ)
But (H sin λ / 4) × (135 × 11) / 12 = (123¾ / 4) H sin λ. This has to
be divided by Δδ ± Δβ. In the problem given. λ = 100°
and Δδ is taken as 362. To obtain Δβ Bhāskara adduces
the argument "If at λ = 0, the first Śarakhaṇḍa of 70
corresponds to H cos λ = 120' what amount of Śara-
khaṇḍa corresponds to an arbitray H cos λ?" The result
is (H cos λ × 70) / 120 = (7 / 12) H cos λ. This he calls Kotiphala
because it is based upon H cos λ which is the Kotijyā.
67

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Thus in the given problem where l = 100°,
H cos 100° = 7/12 × 21 (since H cos 100 = - H sin 10 =
120 × .1735 = - 21 approximately where R = 120')
= - 12′-15″ ∴ Δδ ± Δβ = 362 - 12′-15 = 349-45.
Now 123/4 H sin λ to be divided above = (123 × 118) / 4
= 3628′-30″,
so that 123/4 H sin λ × 1 / (Δδ ± Δβ) = (3628′-30″) / (349-45)
= 10°-22′-28″ =
= rQ. Now r's longitude from the zero point of the
Hindu zodiac is 11 Rāśis 19° so that Q's longitude =
11 R-19° minus 10°-22′-28″ = 11 Rāśis 8°-37′-32″. This
gives us the longitude of G which is called the Moon's
Goḷa Sandhi. Adding 3, 6, 9 Rāśis, we get successively
the first Āyana Sandhi, the other Goḷa Sandhi and the
second Āyana Sandhi of the Moon.
(1) Bhāskara overlooked a crudeness in his proce-
dure namely that Δβ is perpendicular to the ecliptic
whereas Δδ is perpendicular to the celestial Equator, but,
since Δβ is small, he overlooked the nicety. Strictly
speaking Δβ should have been corrected for Āyanavalana.
(2) There is also crudeness in computing Δδ and
Δβ for arcs of 15°, whereas they should have been done
for increase of every degree in the longitude. He could
have done that, because at the end of the Goḷādhyāya he
gave the method or computing the H sines for every degree
under the caption प्रतिभागज्यकाविधि.
We shall now give a modern method of computing
the value of rQ. From the spherical triangle RrA, where
⁀rR = 100°, R̂ = i = 4½°, we have
sin 100 = (tan rA) / (tan 4½)

531
∴ log tan rA = log cos 10 + log tan 4½°
= 9·9934 + 8·8960 = 8·8894 ∴ rA = 4°-26ʹ.
From the spherical triangle rAG,
cos 90—ω = tan rA / tau Gr
log tan Gr = log tan rA —
log sin ω = 8·8894 — 9·6093
= 9·2801. Again from the spherical triangle rQG,
cos ω = tan rQ / tan Gr
∴ log tan rQ = log tan Gr + log cos ω
= 9·2801 + 9·9697 = 9·2498
∴ rQ = 10°-5ʹ whereas Bhāskara got 10°-22ʹ-28″.
This shows how Bhāskara was correct mathematically.
The small difference there is, due to his taking Hsines for
arcs of 15° instead of for smaller arcs.
Note. In the commentary called Sikhā of one Sri
Kedāra Datta Joṣī (page 357) we find a mistake committed
namely that he subtracted 3 R-10° the longitude of the
pāta which is in Rāśis and degrees from the Sphuṭakrānti
349ʹ-45″. What Bhāskara meant was, that since the
longitude 100° exceeds 90°, the cosine will be negative
which therefore entails △β to be subtracted from △δ.
Verse 7. How to know the occurrence of a pāta.
If the Sphuṭakrānti of the Moon, when it is maximum
be less than that of the Sun, then there could be no
occasion for their declinations to become equal in the
near future.
Comm. The situation in which the maximum decli-
nation of the Moon falls short of that of the Sun, arises