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

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)

541
When λ approaches 90°, the duration will be very
much since cos λ → 0. This is true because the variation
in the declination when λ = 90°, is very slow even for the
Moon. The above formula holds good so long as λ does
not approach 90°.
Verse 17. It must be deemed that the declinations
will be equal so long as the difference in the declinations
is less than r + p numerically.
In the case of Vaidhṛti, we are asked to take the sum
instead of difference and difference instead of sum in
contradistinction. Why? We shall see. We know that
Vaidhṛti occurs when the declinotions of the Sun and the
Moon are of equal magnitude but of opposite directions
subject to the condition that the longitudes are such that
their sum is approximately 360°. This condition is
stipulated because when the Sun is in the first quadrant
and the Moon is in the 3rd quadrant and at the same time
their diclinations are equal, the moment does not constitute
Vaidhṛti, because the sum of the longitudes is then less
than 90° + 270° ie. less than 360°. Hence, for Vaidhṛti
one of the two celestial bodies must be in the first quadrant
and the other in the fourth quadrant, so that the sum of
the longitudes could equal 360° and at the same time the
declinations could be equal though of opposite sign. Now
compute the declinations of the Sun and the Moon at a
particular moment. Suppose they are a and b and both
are north. After a few ghatis or days compute again their
declinations, say c and d, both again being north. Suppose
then c + d < a + b, then Vaidhṛti is going to occur.
Construing northern declination to be positive and the
southern negative, for Vaidhṛti to occur a + b must be
reduced to zero ie. a and b are equal and of opposite
direction. Now the argument adduced is "If in time
t ghatis or days a̅+̅b̅ has become c̅+̅d̅ (calling a̅+̅b̅ as
Ādya and c̅+̅d̅ as Anya) there is a decrease of Ādya - Anya,
what time should lapse for a̅+̅b̅ or Adya to reduce to

542
zero ?" The answer is t (a + b) / Adya - Anya which gives the time
after which Vaidhṛti is likely to occur.
Suppose a and b the declinations observed at a particular
moment, are one north and and the other south. Compute
their difference a - b; after a few ghatis or days again from
the difference of the declinations c and d ie. c - d. If
c - d < a - b then Vaidhṛti is going to occur, otherwise it
has elapsed. This is so because the difference has to vanish
for Vaidhṛti to occur.
Viewing the situation algebraically, ie. construing
northern declination to be positive and southern negative,
we could have laid down only one criterion, instead of
separating the issues and stipulating separate criteria.
Proceeding on this basis, suppose the the Sun and the Moon
are one in the first quadrant and the other in the second.
Here both declinations are therefore positive. Compute
their difference a - b; compute again the difference of
c and d which are again northern declinations observed
after a few ghatis or days. If c - d < a - b then Vyatīpāta
is going to occur.
Now take a case when one of a and b is north and the
other south ; so also c and d. We are talking of a, c to be
in the first quadrant and b, d in the fourth quadrant. Even
now if c - d < a - b Vyatīpāta is going to occur. Thus
there is no need to stipulate "Sum or difference". Difference
alone counts now, because difference of one positive and the
other negative quantities means sum only. Thus in the
case of Vyatīpāta difference should tend to zero, whether
both the declinations are north, or one north and the other
south.
Similarly in the case of Vaidhṛti the sum should tend
to zero because the algebraic sum of one positive northern
declination and the other negative southern declination

543
shall be zero, if the declinations were to be equal and of
opposite directions.
Here ends the Pātādhyāya.
Note Bhāskara gives two examples in the course
of the commentary, one, to show the fallacy in asserting
that there would be a pāta when the Sun is in an even
quadrant, the Moon in an odd quadrant and the Moon's
declination is less than that of the Sun; and the other to
show the fallacy in asserting that the pāta has elapsed
when it is still to take place, and that it is still to occur
when it has already elapsed. He quotes the example given
in his Golādhyāya wherein the longitudes of the Sun,
Moon and Rāhu are 120°, 60° and 180° respectively, there
being no Ayanāṁśas. (Since at the time of Lalla, there
were no Ayanāṁśas, Bhāskara gives such an example),
Since the Sun is in an even quadrant and the Moon in
an odd quadrant, Bhāskara says, that as per the verse of
Lalla ‘सूर्यापमात् ओजपदोद्भवात् etc.’ there should be a
pāta. But since the longitude of Rāhu is 180°, Ketu
is at the equinoctial point so that the lunar orbit
lies between the ecliptic and the equator. Hence the
declination of the Moon remains smaller than that of
the Sun for a good length of time never equalling the
declination of the Sun. This, Bhāskara says, is a fallacy
regarding ‘भावाभावत्व’ ie. ‘Is there a pāta or not’ because
when there is actually no pāta, Lalla's criterion leads to
assert that there is one. It may be reiterated here, that
Lalla gave a general criterion ignoring the latitude of the
Moon ie. supposing the Moon to be moving on the ecliptic.
Bhāskara takes the latitude also into consideration and
proves that there is no pāta.
In the second example, which Bhāskara gives, he tries
to show the fallacy with respect to “गतेष्यत्व” ie. asserting
that a pāta has elapsed when it is still to occur, and that
it is to occur when it has already elapsed.

544
In this example the declinations of the Sun and Moon
are respectively 1416′, 1324′, when their tropical longitudes
are 80° and 100°. The Moon is in an even quadrant,
and his declination is less than that of the Sun. As per
the criterion of Lalla given in the verse “सूर्यापमात् etc.”
the pāta is not there at all. But Bhāskara computes and
shows that the pāta has occured 70 nādīs before (Rāhu
having a reverse tropical longitude of 100°). We shall
show here, how using differential calculus we could cut
short the process. We have
sin δ = sin λ sin ω ; differentiating
cos δ Δδ = cos λ δλ sin ω or
Δδ = (sin ω cos λ / cos δ) Δλ
= Āyanavalana × Variation in Bhuja.
This could be seen graphically also from fig. 129. Let
P and Q be two contiguous positions of the Sun (say) on
the ecliptic when arc pQ = Δλ. Let the increment in his
Fig. 129
declination in going from P to Q be NQ equal to Δδ.
Taking PNQ to be a plane Δ Δδ = Δλ sin v where
v̂ = QP̂N. We have named this angle to be v, because it
is no other than the Āyanavalana which was formulated
to be equal to (sin ω cos λ / cos δ) . Thus Δδ = Δλ × Āyana-
valanajyā.

APPENDIX
LIST OF TECHNICAL TERMS

1. Adhikamāsa : A month gained by the lunar reckoning
   over the solar. This is located in that
   lunar month which does not contain a
   Saṁkramaṇa,
2. Agrajyā : The Hindu sine of the arc of the horizon
   in between the rising point of the Sun
   and the east point.
3. Akṣa (or) Pala : Latitude (Terrestrial)
4. Ākṣa Dṛkkarma : The arc of the ecliptic between the point
   of intersection of the ecliptic with a
   secondary through the star to the prime
   vertical and the point of intersection of
   the ecliptic with the star's declination
   circle.
5. Akṣakarṇa : The hypotenuse of the gnomonic triangle
   when its shadow is equal to what is called
   Viṣuvat-chāyā
6. Akṣa-Kṣetra : A right-angled spherical triangle one of
   whose sides may be the arc of a small
   circle namely the diurnal circle but one of
   whose angles is a right angle and another
   equal to the terrestrial latitude.
7. Akṣavalanam : The angle at the point of the star in
   between the declination circle of the star
   and a secondary to the prime vertical
   through the star.
8. Antyā : The Hindu sine of an arc of the celestial
   equator corresponding to Hṛti.
   69

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9 Asta : Setting or heliacal setting.
10. Ayanabindu : Solstice.
11. Ayana-Dṛkkarma: The arc of the ecliptic intercepted between
its point of intersection with the star's
declination circle and the secondary to the
ecliptic through the star.
12. Ayanāṁśam : The arc of the ecliptic in between the
vernal equinoctial point and the Hindu
zero of the ecliptic ie the first point of
the Zodiacal sign called Aśvinī.
13. Ayanavalanam : The angle at the point of a star, between
its declination circle and the secondary to
the ecliptic through the star.
14. Bārhaspatyamāna: The time taken by Jupiter to reside in a
Rāśi, on the average, is called a jovian
year. This falls short of a solar year.
15. Bhāga : A degree
16. Cāpa or : Arc.
Kārmuka
17. Carajyā . : The Hindu sine of the arc intercepted
between the east point and the declination
circle of a rising star or planet or the
Sun.
18. Cāndra-māsa : The time between two consecutive full-
moons or New moons.
19. Chāyā or Bhā : Shadow cast by the gnomon.
20. Chāyābhuja : The projection of the shadow on the east-
west line.
21. Chāyākarṇa or : The hypotenuse of the gnomonic triangle
Bhākarṇa whose two sides are the gnomon and its
shadow.

547
22. Chāyākoṭi : The perpendicular from the extremity of
a shadow on the east-west line.
23. Dhruva : The star near the celestial pole or the
celestial pole itself.
24. Dhruvaka : The celestial longitude.
25. Dhruva- : The declination circle.
protavṛttam
26. Digjyā : The Hindu sine of the azimuth measured
by the angle between the prime vertical
and the vertical of a star or a planet.
27. Dorjyā or : Hindu sine of celestial longitude.
Bhujajyā
28. Dṛgjyā : The Hindu sine of the Zenith distance.
29. Dṛg-lambana : Total parallax.
30. Dvāparayuga : Twice the period of a kaliyuga.
31. Dyujyā : The Hindu cosine of declination or the
radius of the diurnal circle taking the
radius of the celestial equator to be R
equal to 3438 units.
32. Dyujyā-vṛtta or : The diurnal circle of a star or a planet.
Ahorātra-vṛtta
33. Ghaṭi or Nāḍi : An interval of time equal to 24.' (minutes)
34. Grahaṇa : Eclipse.
35. Hṛti or Iṣṭahṛti : The Hindu sine of the arc of the diurnal
circle from a point of the same upto the
plane of the horizon.
36. Kadamba : Pole of the ecliptic.
37. Kadamba- : A secondary to the ecliptic through a star
protavṛtta or planet.

548
38. Kakṣāmaṇḍala : The deferent of a planet or the circle with
the earth as centre and radius equal to 3438
units.
39. Kalā : The Hindu sine in the diurnal circle
corresponding to the Sūtra (given below).
40. Kalā or Liptā : A minute of angle.
41. Kaliyuga : The period consisting of 4,32,000 mean
solar years.
42. Kalpa : Dvāparayuga is twice Kaliyuga; Tretāyuga
thrice and Kṛta four times. All these put
together constitute a Mahāyuga. 71 Mahā-
yugas make one Manvantara. Fourteen
Manvantaras with what are called Sandhi
periods on either side equal to a Kṛtayuga
or thousand Mahāyugas make a Kalpa. The
creation is supposed to last one Kalpa,
which is said to constitute the day time of
Brahma, the Creator. His night also is of the
same duration when there is no creation. It
is held that we are now in what is called
Śveta-Varāha Kalpa, wherein the seventh
Manvantara called Vaivasvata manvantara
is current. In this Manvantara, twenty seven
Mahāyugas are supposed to have elapsed
and in the twenty-eighth Mahāyuga, kṛta
Treta and Dvāparayugas have elapsed and
that we are now in the Kaliyuga. In this
Kaliyuga which began on the day when
Lord Kṛṣṇa gave up his mortal coil, 5080
mean solar years have elapsed by about
21 March, 1979.
43. Kramajyā or : Hindu sine of an angle.
simply jyā or
Jīvā or guṇa
44. Karaṇa : Half of the duration of a tithi.

549
45. Karṇāgrajyā or : The Hindu sine Agrajyā multiplied by K,
simply Karṇāgrā and divided by R where K is the hypotenuse
of the gnomonic triangle whose two sides
are the gnomon and its shadow and R the
radius of the celestial sphere taken to be
3438 units.
46. Kendra : The centre of a circle.
47. Ketu : The diametrically opposite point of Rāhu.
Rāhu also means the circular section of the
earth’s shadow at the Moon.
48. Kha-Svastika : The Zenith at a place.
49. Koṭijyā : Hindu Cosine of an angle.
50. Krānti or Apama: The declination of a point on the ecliptic.
51. Krānti-Vṛttam : Ecliptic.
52. Kṛtayuga : Four times the period of a Kaliyuga.
53. Kṣayamāsa : That lunar month in which there are two
Saṁkramaṇas.
54. Kṣitija : The Horizon at a place.
55. Kujyā : The Hindu sine measured in the diurnal
circle corresponding to the Carajyā defined
above.
56. Lagna : The Rāśi which rises at any moment or the
rising point of the ecliptic.
57. Lambajyā : The Hindu Cosine of the latitude.
58. Lambana : Parallax in longitude.
59. Mahāyuga : The Sum of the four yugas mentioned above.
60. Manvantara : A period equal to 71 Mahāyugas.

550
61. Nakṣatra : A star. Also the time which elapses as
the longitude of the Moon increases by
13⅓ degrees starting from the zero point of
Aśvini.
62. Nākṣatra-Dina : The time that elapses between two consecutive
risings of a star.
63. Nākṣatra-māsa : The time taken by the Moon to go from
Aśvini again to Aśvini.
64. Natakāla : Hour angle measured in ghaṭis.
65. Natāṁśajyā : Hindu sine of the Zenith distance.
66. Nati : Parallax in latitude.
67. Nīcoccavṛtta : Epicycle.
68. Parama-Antyā : The Antyā when the celestial body is at
the point of intersection of the celestial
equator and the meridian or what is the
same at the point of culmination.
69. Paramakrānti : Obliquity of the ecliptic taken by the Hindu
astronomers to be 24°.
70. Pāta : The point of time when the declinations
of the Sun and the Moon are equal and
of the same sign or opposite sign. Also it
means the point of intersection of two great
circles.
71. Prācī : East point.
72. Prācyaparā : East-west line.
73. Prativṛtta : The circle in which the planet moves and
whose centre is at a distance from the
centre of the Kakṣāmaṇḍala or the orbit
of the mean planet. This is the same as
the orbit of the the true planet supposed
to be a circle.

551
74. Rāhu : The point of intersection of the Moon's
path with the ecliptic (ascending point of
the Moon's path). Also it means the
circular section of the earth's shadow at
the Moon.
75. Rāśi : An arc equal to 30° (on the ecliptic).
76. Samabindu or : North point.
Udagbindu
77. Samamaṇḍala : Prime vertical.
78. Sama-Śaṅku : The Hindu cosine of the altitude when the
Sun is on the Prime Vertical.
79. Saṁkramaṇa : The point of time when the Sun enters
from one Rāśi to another of the twelve
Rāśis, Meṣa etc. measured from the zero-
point of the Hindu Zodiac.
80. Śaṅku : Gnomon.
81. Śaṅku : The Hindu sine of the Altitude.
(Also means)
82. Śaṅku-cchāyā : The shadow cast by the gnomon.
83. Saura-māsa : The time when the Sun occupies one Rāśi.
84. Śara : The Hindu versed-sine of the hour-angle.
Also it means celestial latitude.
85. Sāvanāha : The time between two consecutive Sun-rises.
86. Sūtra : The Hindu sine of the complement of the
hour-angle.
87. Taddhṛti : The Hṛti of a celestial body when it is
on the prime vertical.
88. Tithi : The time taken by the elongation of the
moon to increase by 12° starting from
zero.

552
89. Tithi-kṣaya : Since a Tithi falls short of a Sāvanāha
or civil day, in course of time, it so
happens, that a tithi begins after sun-rise
and does not last upto the next sun-rise.
Such a tithi is said to be lost and goes
by the name Tithi kṣaya. One such tithi
kṣaya occurs out of 64 tithis approximately
or eleven out of 703 more approximately.
90. Tretāyuga : Thrice the period of a Kaliyuga.
91. Trijyā : The Hindu sine of three Rasis or 90°
equal to R or 3438 units.
92. Udaya : Rising or heliacal rising.
93. Unmaṇḍala or : The great circle through the celestial pole
Udvṛtta and the east point.
94. Unmaṇḍala : The Hindu cosine of the altitude of a
Śaṅku celestial body situated on the unmaṇḍala.
95. Upavṛtta : A small circle parallel to the prime vertical
through a star or a planet.
96. Vighaṭi or Vināḍi: One sixtieth of a ghaṭī.
97. Vikṣepa or śara : Celestial latitude.
or Viśikha
98. Viṣuvatbindu : Equinoctial point.
99. Viṣuvat-chāyā : The shadow cast by the gnomon when the
Sun is at the point of intersection of the
celestial equator and the meridian on an
equinoctial day.
100. Viṣuvat-Vṛtta : Celestial equator.
101. Vṛtta or Maṇḍala: A circle.

553
102. Yāmyottara- : The Hindu sine of the Altitude of a
saṁku or Celestial body situated on the meridian.
Dinārdha-Śaṅku
103. Yāmyottaravṛtta : The meridian at a place.
104. Yaṣṭi : R² - Ayanavalanajā²
Yaṣṭi has also another meaning namely
the length of the perpendicular from a
point on the diurnal circle on the plane
parallel to the plane of the horizon through
the point of intersection of the diurnal
circle with the unmaṇḍala.
105. Yoga : The time which elapses when the sum of
the longitudes of the Sun and the Moon
to increase by 13¹⁄₃º starting from zero.
106. Yuti : Conjunction.