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

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

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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.
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530
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

532
when the lunar orbit comes in between the celestial
Equator and the ecliptic. This situation, in its turn,
depends upon the position of Rāhu. Since Rāhu's sidereal
period amounts to nearly 18½ years, the lunar orbit
happens to take its position in between the celestial
Equator and the Ecliptic for sufficiently a long period as
shown below.
That Bhāskara could visualize this, reflects credit
to his genius. This is why he formulated Δδ ± Δβ
stressing upon the plus or minus sign.
Fig. 128
Bhāskara gives an example under the above verse
in the commentary to justify his finding given above.
Suppose Rāhu is at the autumnal equinox and the longi-
tudes of both the Sun and the Moon are equal to 79°.
Adding the Ayanāṁs'as 11°, their longitudes are each 90°
so that both of them are at the summer solstice ie. an
Āyana Sandhi. (Of course the Moon is not exactly at the
position of the Sun, but he being in his own orbit, his
longitude measured along the Ecliptic is 90°). Then
evidently the lunar orbit lies in between the celestial
equator and tho Ecliptic. Then the declination of the
Moon is 24°–4½°＝19½°＝1170′ and the declination of the
Sun is 24°＝1440′. Thus the Moon's declination when
it is maximum falls short of that of the Sun. After half
of the sidereal period of the Moon namely 13⅔ days, the

533
longitudes of the Sun, Moon and Rāhu are respectively
3-2-28-12, 8-19-4-26 and 6-11-43-28 (in Rāśis, degrees
etc), taking mean motions into account. When the Moon
has the maximum southern declination in the position M₁
of fig. 129 his longitude will be however 8-10-9-35 and then
his declination will be 1169. At that moment the decli-
nation of the Sun will be 1398. Even here the Moon's
declination falls short of that of the Sun. Again after
13⅔ days, we find the declination of the Moon falling short
of that of the Sun. Even after 2 months, the Moon
cannot have a declination equalling that of the Sun.
This phenomenon occurs when Rāhu of the lunar orbit
happens to be at Libra, and the Sun is at the summer
solstice approximately.
Verse 8. The definitions of Vyatīpāta and Vaidhṛti.
When the Sun and the Moon are in opposite Ayanas,
but in the same Goḷa, and if then their declinations be
equal, then that moment is said to constitute vyatīpātayoga.
If on the other hand, if both the Sun and the Moon be in
the same Ayana and opposite Goḷas, and if then their
declinations be equal, that moment is said to constitute
Vaidhṛti Yoga.
Comm. Explained before.
Verse 9 and first half of verse 10. To prognosticate
the occurrence of a pāta.
When the sum of the tropical longitudes of the Sun
and the Moon happens to be 180° or 360°, then the Vyati-
pāta and the Vaidhṛti respectively will occur or will have
occured. The number of minutes of arc by which the sum
of the tropical longitudes falls short of or exceeds 180° or
360° as the case may be, are to be divided by the sum of the
true daily motions of the Sun and the Moon, which will
give aqproximately the number of days after which or

before which the Yogas will occur or would have occured.
Compute the declinations of the Sun and the Moon for
that moment from the then true daily motions.
Comm. From fig. 124, it is clear that triangle ♈SL
and ♎ MN are congruent so that ♈S = ♎ M ∴ SS₁ = MS₁
∴ ♈S + ♈M = ♎ M + ♈M = 180°. Again triangle ♈SL
and ♈M'N' are congruent ∴ ♈S = M'♈
∴ ♈S + ♈M' = ♈M' + M'♈ = 360°. Thus in the former
case which constitutes Vyatīpāta, the sum of the tropical
longitudes of the Sun and the Moon is 180°; whereas in
the latter case, which constitutes the Vaidhṛtipāta, the
sum of those two longitudes is equal to 360°.
Hence we are asked to note when the sum of the two
longitudes is likely to amount to 180° or 360°. On a
particular day suppose the sum is 180 — θ or 360 — θ; so
that θ is to be made up by the then velocities of the Sun
and the Moon conjointly. Then θ / (u + v) where u, v are their
respective velocities gives the number of days or fractiou
of a day, by which the sum would be 180° or 360° as the
case may be. As the velocities change from moment to
moment, the above θ / (u + v) is only approximate. So, compute
again the respective positions and respective velocities.
Suppose the sum of the longitudes is 180 ± θ' or 360 ± θ'
and the velocities u' and v', Then θ' / (u' + v') gives the fraction
of a day after or before the moment in question when the
pātas occur Vyatīpāta or Vaidhṛti.
Verse 10. To know whether the Yoga is past or
future.
If the declination of the Moon situated in an odd
quadrant exceeds that of the Sun or falls short of the

535
Sun's in an even quadraut the moment of the occurence of
pāta has elapsed. Otherwise the pāta is to take place
shortly after.
Comm. This is clear because in an odd quadrant, the
declination of the Moon is on the increase. So if it be
already greater than the Sun's, in future it will be far
greater and as such cannot equal the Sun's declination.
In other words the pāta had already taken place.
Similarly in an even quadrant, the declination of the
Moon is on the decrease. As such if it be greater than
the Sun's it will become equal in future. Thus the pāta
is to take place. On the other hand if the Moon's decli-
nation is less than the Sun's, it will decrease further so
that prior to the moment concerned a pāta should have
occured.
Latter half of verse 11 and verses 12, 13, 14. To
compute the time of the occurrence of pāta through a
consideration of declinations.
Obtain the difference of the declinations if they be of
the same direction or their sum if they be of opposite
directions in the case of Vyatipāta ; obtain the sum or
difference of the declinations according as the declinations
are of the same or different directions in the case of
Vaidhṛti. Call this sum or difference 'the Ādya'.
After a lapse of an arbitrary time or before the moment
concerned, obtain the positions of the Sun, Moon and the
Rāhu ; also compute their declinations and form their
difference or sum as the case may be. Call this Anya. If
on both the occasions it is indicated that the pāta has
elapsed or is to elapse, obtain the difference of the Ādya
and Anya ; otherwise their sum. Divide the arbitrary
time taken, by the above sum or difference of the Ādya and
Anya and multiply by the Ādya. Take the result in
ghatis. Taking this as the arbitrary time, repeat the

536
process till an invariable quantity is obtained in ghatis.
This is the time by which the moment of the pāta has
elapsed or after which it is going to occur.
Comm. (1) It might be asked “why bother finding
the declinations at all, when it is defined that Vyatīpāta
happens when the sum of the longitudes of the Sun and
the Moon is equal to 180° and Vaidhṛti is defined when
the sum is equal to 360, since easily we could know when
this happens without taking recourse to declinations?”
The problem is not that simple. The above definition
in terms of the sum of the longitudes is an approximate
statement, because, the original and correct definition is
that their declinations must be equal in magnitude. If
they be equal in magnitude but opposite in direction, they
constitute Vaidhṛti Yoga. If they be both equal in
magnitude and direction, the situation constitutes Vyatī-
pātayoga. In other words, when the diurnal paths of
the Sun and the Moon coincide, the moment will be
Vyatīpāta. If on the other hand their diurnal paths are
of equal dimentions but one to the north of the celestial
equator and the other to the south, then the moment is
Vaidhṛti. Though the sum of the longitudes happens to be
180°, the diurnal paths may not coincide because the Moon
does not actually move on the ecliptic. So the situation
is more complicated.
Further the calculation of the declination of the Moon
is to be done by calculating the Asphuṭakrānti ie. the
declination of the point of the ecliptic which indicates
the longitudinal position of the Moon and his celestial
latitude ie. the Asphuṭavikṣepa from the known position
of the Node, and then by rectifying the Asphuṭavikṣepa to
obtain the Sphuṭavikṣepa and then adding the Asphuṭa-
krānti to the Sphuṭavikṣepa to obtain the Sphuṭakrānti.
(2) The latter half of verse (11) and the first half
of verse (12).

537
Let us consider the case of Vyatipāta. Suppose at
the moment concerned, the declinations are of opposite
direction. Find the sum of their numerical magnitudes.
Suppose they are of the same direction; find their differ-
ence. This sum or difference of the declinations gives the
distance between the planes of the diurnal paths of the
Sun and the Moon. This distance is to vanish in order
that the diurnal paths may coincide. So we are asked first
to find the above distance. Suppose we understand that
the Vyatipātha has elapsed by noting the deolinations.
This may be known easily by the criteria given previ-
ously. Supposing 𝑥 and 𝑦 to be the declinatious of the
Moon and the Sun and supposing that 𝑥 is on the decrease
and approaching 𝑦, then the Vyatipātha is to occur. But
suppose 𝑥 < 𝑦 and 𝑥 is on the decrease, then the Vyatipāta
has taken place. Thus knowing whether the Vyatipātha
has elapsed or is to occur, after an arbitrary time 𝑡, com-
pute the declinations of the Sun and Moon and form their
difference or sum as the case may be which gives the
distance between the diurnal paths. Let the first distance
found be called Ādya and the second distance the Anya.
Then the Anya will be less than the Ādya because we have
taken a time towards the occurrence of Vyatipāta when
the distance is to get nullified. Find the difference of the
Ādya and Anya. Then by the proportion “ If in time ‘𝑡’
taken in between the moments of the Ādya and Anya,
the distance between the diurnal paths is reduced by
Ādya — Anya, what time will be taken for the distance
Ādya to vanish ?” we have the result
T = (Ādya × 𝑡) / (Ādya — Anya) which gives approximately the time
that has to elapse for the occurrence of the pāta. This will
be approximate. After a lapse of time T from the Ādya
moment concerned, again compute the declinations and
repeat the procsss. We arrive at a particular point of
time, which gives the moment of occurrence of the pāta.
68

538
In the course of the commentary under these verses
Bhāskara solves two problems and points out that when
there is a pāta according to his exposition, the statements
made by Lalla, Brahmagupta Śrīpati and Mādhava all
indicate that there would not occur a pāta.
But we feel that Bhāskara read too much into those
statements on account of the following. Lalla states
सूर्योपमादोजपदोद्गवाच्चेद्युग्मादिजः चन्द्रमसो लघीयान्,
अपक्रमःस्यान्न तदास्तिपातः तदन्यथात्वेऽपमयोः समत्वम् ।
Brahmagupta states
त्रिनवगृहेन्दुकान्तिः मेषतुलादौ दिवाकरक्रान्तेः ।
ऊना यावदभावः तावत् भावोऽन्यथा चेति ॥
and Śrīpati says
त्रिनवभवनजाता क्रान्तिरिन्दोर्यदाऽल्पा
दिनकृदपमतः स्यान्मेषजूकादिजातात् ।
न हि भवति तदा च क्रान्तिसाम्यं रवीन्द्रोः
नियतमितरथात्वे जायते सम्भवोऽस्य ॥
In fact all these three statements mean one and the
same and are intended to be approximate statements, in
the first instance, having ignored the latitude of the Moon.
They are just statements like that of Bhāskara himself
when he says that there will be a pāta when the sum of
the longitudes will be 180° or 360°. The statements pur-
port to say that if the declination of the Moon when it is
on the decrease happens to be less than that of the Sun
which is on the increase, there could be no pāta. These
statements so far as they go, ignoring the latitude of the
Moon are perfectly in order. Bhāskara brings in a detailed
analysis of two critical examples, to prove that there is a
pāta, but which is negated by the rough statements of the

539
four Ācāryas. Mādhava's statement in his Siddhānta
Cūdāmaṇi is as follows which is also in similar terms as
those of the other three.
रवेरोजपदक्रान्तेः चन्द्रयुग्मपदोद्भवा
स्वल्पाचेन्नतयोःक्रान्त्योः साम्यं स्यादन्यथा भवेत्
Verses 15 and 16. To obtain the duration of the
pāta.
The semi-sum of the diameters of the Sun and the
Moon or what is the same the sum of their angular radii
being multiplied by the Spaṣṭaghaṭis (obtained in the
estimate as per the verses 11-14) and divided by the Ādya
in that context, gives the beginning and end of the pāta
from the moment of the computed time. The process
being repeated according to the method of successive
approximations we have the correct estimate of the
duration of the pāta.
Comm. This is a convention stipulated with respect
to the duration of the Pāta. Strictly speaking, when the
diurnal paths of the centres of the discs coincide in the
case of Vyatīpāta, that will be the middle moment of the
Vyatīpāta. The pāta is said to last for such a time as the
distance between the Centres of the discs (north-south
distance) is less than r + p. This will be clear from a
figure. The situation is akin to that of an eclipse.
The time obtained for the occurrence of the pāta
under verses (11) to (14) indicates the middle of the
duration of the pāta. The duration of the pāta is defined
as the time that lasts as long as the declination of the
highest point of one disc becomes equal to that of the
lowest point of the other beginning from the moment at
which the declination of the lowest point of the one
becomes equal to that of the highest point of the second.
In other words, just like in an eclipse, so long as the
distance between the diurnal paths traced by the centres

540
of the discs is less than the sum of their angular radii, the
pāta lasts. Extending the meaning of this to Vaidhṛti
also, so long as the numerical difference of the declinations
of the Sun and the Moon ignoring their direction happens
to fall short of the sum of the angular radii, the pāta
lasts.
To obtain this duration of the pāta the argument is
“If the sum or difference of the declinations according as
they are of opposite or the same direction (which was
taken to be Ādya under verses (11) to (14) ) was reduced to
zero in the time computed that time being known as
Spaṣṭaghaṭis, what time will be taken for a difference of
declinations equal to the sum of the angular radii?”
The result is ((r + p) × T) / Ādya where r and p are the
angular radii of the Sun and the Moon, T the time
calculated formerly known as Spaṣṭaghaṭis and Ādya is as
defined above. The above result gives the duration of the
pāta.
Note. An approximate estimate of this duration could
be obtained using differentiation. We have
sin δ = sin λ sin ω so that
cos δ Δδ = sin ω cos λ Δλ
Let Δλ be the motion in longitude of the Moon with
respect to the Sun per nāḍi which will be on the average
12′ approximately.
∴ Δδ = (sin ω cos λ × 12) / cos δ
If in one nāḍi, there be a variation in the declination
equal to the above, what time will be taken for 16′ + 15′
the sum of the angular radii approximately ? The result is
(31 cos δ) / (12 sin ω cos λ)