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

488
of the Sun and the Moon. Herein Bhāskara defines the
Koti as the sum of the Saṅkus of the Sun and the Moon
which is the vertical distance between the points on the
armillary sphere which represent the Sun and the Moon.
Bhāskara says 'I deem' to signify that he differs
from Brahmagupta in this, who defines the chord joining
the Sun and Moon on the armillary sphere which is equal
to 2 Hsine of the SM on the sphere as the Karṇa. Brahma-
gupta defines the Bhuja and Karṇa from which he deduces
the Koti as √(Karṇa² — Bhuja²). Bhāskara argues that since
the Karṇa defined by Brahmagupta is not in the vertical
plane, since the Sun and the Moon are not in the same
vertical plane, so, the Koti defined by him throught he Karṇa
will not be in the vertical plane. The Karṇa defined by
Bhāskara is not on the other hand the chord joining the
Sun and the Moon but lies in the vertical plane in which
the perpendicular from the Moon's position on the armil-
lary sphere, on the horizontal plane containing the Sun's
position on that sphere. Also the Bhuja defined both by
Brahmagupta and Bhāskara is the projection on the north-
south line of the join of the Sun's position on the sphere and
the foot of the perpendicular from the Moon's position on
the horizontal plane through the Sun's position and it is
not actually the above join.
Bhāskara is evidently
guided by the right angled
triangle SMN' which is not
strictly a spherical triangle
as per its modern definition
as the arc SN is not that
of a great circle. This
figure guided him to take
the Bhuja horizontal and
the Koti vertical so that
his Karṇa is also in a
Fig. 115
vertical plane. Bhāskara's

489
Karṇa therefore has nothing to do with the arc SM but
only has the virtue of being in a vertical plane. Sripati
adopted Brahmagupta's method. Kamalākara neither
follows Brahmagupta nor Bhāskara but follows a method
of his own. In fact the methods adopted by these ancient
Hindu astronomers are not mathematically correct because
the question of determining the cusps as well as phase of
the Moon is concerned with the actual positions of the
Sun, Moon and the earth in space and not as seen on the
sphere. Hence M. M. Sudhakara Dwivedi has written a
small book by the name Vāstava-Sṛṅgonnati following
modern methods. The modern method which gives the
truth of the matter is depicted in standar modern texts.
Verse 5. The hypotenuse or Karṇa is the square-root
of the sum of the squares of the Bhuja and Koti. The
Bhuja multiplied by 6 and divided by the Karṇa gives
what is known as the Dik-valana of the Moon. The dire-
ction of the Valana has the same direction as the Spasṭa
Bhuja defined before.
Comm. The idea is that taking the radius of the
Moon's disc to be six angulas or units, representing the
Karṇa, the magnitude of the Bhuja on the same scale
gives a measure of what is defined as Valana. Thus Valana
= 6B/K where B and K stand for the Bhuja and Karṇa
defined before. The idea of this Valana will be clarified in
the ensuing verses.
Verse 6. The Hsine of the elongation of the Moon
is to be multiplied by the radius vector of the Moon
measured in Yojanas, and divided by the radius vector of
the Sun, also measured in Yojanas ; the arc of the Hsine
so obtained is to be added to the longitude of the Moon
in the bright half of the lunation and is to be subtracted
from the same in the dark half.
62

490
Comm. This is a correction to be made in the longi-
tude of the Moon to depict graphically the phase of the
Moon. Bhāskara says that many of the prior astronomers
took that the phase of the Moon was in direct proportion
to the elongation of the Moon. Taking the radius of the
Moon's disc to be six angulas or units, and assuming that
when the elongation is 180°, the entire disc being illumi-
nated, it was thought that 12 units of the diameter
correspond to 180° of elongation so that the Śukla of the
disc measured by the central width of the illuminated disc
increases at the rate of 1 unit for 15° of elongation. (The
word Śukla may be taken to correspond to the modern
word phase. Śukla is expressed in angulas, taking the
diameter of the disc to be 12 angulas. The measure of the
portion of the diameter of the Moon's disc perpendicular
to the diameter which is the join of the extremities of
the cusps, covered by the illuminated part of the disc,
(which may be defined as the maximum width of the
illuminated part of the disc) measured in angulas is said
to be the Śukla. The word 'phase' is used to signify the
ratio of the above width to the diameter. Śukla is
expressed in angulas, whereas phase is expressed as a ratio.
The Śukla is equal to twelve times phase). Bhāskara
rightly argues that this method of measuring the Śukla
is approximate because he says that six angulas of Śukla
is had when the elongation is not 90°, but only 85°-45'.
This may be substantiated as follows from fig. 116. Let
E, M, S stand for the earth, Moon and the Sun. Let
^
EMS = 90° so that it is a moment of dichotomy ie. the
moment when the phase is half and the Śukla 6 angulas.
Let θ be then the elongation of the Moon, so that
cos θ = EM / ES . Taking the average values given for EM
and ES by Bhāskara,
cos θ = 51566 / 689377 = 19 / 254 (obtaining a convergent) = .0748.

491
From tables, we find θ = 85°-43' which Bhāskara takes
to be 85°-45'.
Fig. 116
In the wake of this, Bhāskara tries to make amends
in the approximate formula prescribed to obtain the Sukla.
He prescribes addition of 4°-15' to the longitude of the
Moon in the bright half, and subtraction in the dark. In
between the moment of conjunction and the moment of
dichotomy, he derives the following formula (vide fig. 117).
Fig. 117
The deficiency of 4°-15' is had in the form of the angle
S₂ E S₁. When the Sun is at S₁ the Moon being at M₁, it

492
is the moment of conjunction. When the Sun has moved
from the point S₁ to S₃, S₁ Ê S₃ being 90°, there is a
deficiency of magnitude S₁ E S₂. In other words, when
S₂ N has assumed the position S₃ E there is a deficiency
of 4°-15′ ie. for an increase of 90° of elongation, there
is a deficiency of 4°-15′ in the longitude of the Moon.
Hence, for the Hsine to become the radius, there corres-
ponds a portion E M₂ or S₂ N, which is the Hsine of 4°-15′,
so that the following rule of three is adopted. 'If the
Sun's distance E S₃ corresponds to the radius, what does
E M₂ the distance of the Moon correspond to ?' The
result is (m / s) × R, m and s being the respective distances.
Then the following proportion is used "If by H sin ξ
equal to R, ξ being the Moon's elongation, we have mR/s,
what shall we have for an arbitrary H sin ξ ?" Thus the
answer is
(H sin ξ × mRs) / R = H sin ξ · (m / s) . H sin⁻¹ ((m / s) × H sin ξ)
where m and s are the distances of the Moon and the Sun,
is to be added to the longitude of the Moon or to be
subtracted as the case may be, to have the rectified
longitude of ths Moon from which the phase is to be
calculated according to Bhāskara.
Here we are to offer the following remarks. No doubt,
Bhāskara was correct in estimating the moment of
dichotomy to be that when ξ the Moon's elongation is not
90° but 85°-45. But the amended formula is not the
correct mathematical form. The modern formula to find
the phase is (1+ cos EMS) / 2 aud since from fig. 116, SM is
nearly equal to SE, so EMS is very nearly equal to

493
180-MES, so that (1+ cos EMS) / 2 = (1 —cos MES) / 2 which may
be taken to be an approximate truth. This ‘formula was
indeed given by Lallācārya in the following verse, long
before Bhāskara. “रविशीतकरान्तरांशजीवा विपरीता शशिखण्ड-
ताडिता च, विहृता त्रिमजीवया सितं स्यात् शशलक्ष्माङ्कवदङ्गुलानि तस्मिन् ”
verse 12 (Candra Sṛṅgonnatyadhikāra). Here विपरीता
रविशीतकरान्तरांशजीवा mean Hvers ξ = |R—H cos ξ. शशिखण्डता-
डिता = multiplied by the radius of the disc of the Moon.
विहृता त्रिमजीवया = divided by R. Thus the formula given
by Lallācārya amounts to r (R—H cos ξ) / R = d/2 (1—cos ξ)
where r gives the angulas in the radius of the Moon's disc
and d the diameter, and ξ = elongation of the Moon.
Since the definition of phase in modern terms is a ratio,
namely the ratio of the maximum width of the crescent to
the diameter, phase × d = Sukla of the Hindu astronomers
∴ (1 — cos (elongation of the Moon)) / 2 × d = Sukla
= r (1 —cos MES) as given by Lallācārya. So Lallā-
cārya's formula is quite correct. We shall trace Lallā-
cārya's steps in obtaining such an intricate correct
formula which was overlooked by Bhāskara. (Refer
fig. 118) Let E be the earth, S the Sun and M₁, M₂, M₃
etc. the positions of the Moon as the elongation
gradually increases. No doubt the Sukla increases with
elongation as known to all Hindu astronomers. But,
the question is, does it increase with the sine or versine?
We know both the sine and versine increase with the
angle. Lallācārya noticed that Sm₁, Sm₂, Sm₃ as the
Moon occupied positions M₁, M₂, M₃ etc., are the Hindu
versines which are increasing. So he postulated that the
Sukla increases with the Hindu versine. His formulation
was thus correct. But why Bhāskara overlooked this
correct formula was traceable to his getting prejudiced

494
Fig. 118
against Lallācārya's some other formulae which used the
Hversine where he ought to have used the Hsine. For
example in the case of the Valana Lallācārya's mistake
is traceable to his confusion as to whether he was to choose
the Hsine or Hversine when both increase as the angle
increases.
Verse 7. Graphical depiction of the cusps.
Fig. 119

495
Let in fig. 119, ST be the Bhuja, MT the Koti and
SM the Karṇa formerly defined. As per verse (5)
6B / K = Valana where B = ST, K = SM, and 6 = MG so
that GE = 6B / K = Valana, which is in practice drawn as E'G'
from the east point e, in the form of a Hsine. CG goes by
the name Valanasūtra. Compute the Śukla in angulas
after making the prescribed correction in the longitude of
the Moon as stated by Bhāskara and dividing the elon-
gation of the Moon, thereafter obtained, by 15. Mark off
the Śukla in angulas along the Valanasūtra from G.
Suppose GD is the Śukla. Draw the diameter AB
perpendicular to the Valanasūtra passing through M.
Draw the circle circumscribing D, A, B. Its centre lies
evidently on the Valanasūtra, say C. The circle drawn
is called Parilekha Vṛtta, its radius CA is called Pari-
lekhasūtra and the point C Parilekha Vṛtta Madhya.
In the triangle CAM, which is right-angled CA is
the Karṇa, AM the Bhuja and MC the Koti. CD is
equal to the Karṇa CA so that MD = CD – CM = Karṇa
– Koti = K – k (say). We have AM² = CA² – CM² =
K² – k² = 36 = B²; Hence K + k = B² / (K – k).
Here K – k = MD is known because GD the Śukla is
known. So B² / (K – k) = K + k is known. Thus knowing
K – k and K + k, by using what is called Saṁkramagaṇita
ie. by adding K + k and K – k, we have K and by sub-
tracting we have k. Here the Koti CM is called Vibhā
and the Karṇa CD the Swabhā. The Parilekhasūtra or
the radius of the Parilekha Vṛtta being the Karṇa, the
Swabhā is thus the Parilekhasūtra.
In the wake of this exposition, the translation of
verse (7) runs as follows. "Let the compliment of the
elongation (corrected as directed in verse (6)) divided by
15 be the denominator; let the numerator be 36; take the

496
result after division and put it in two places; with this
and the complement of the elongation divided by 15, using
Samkramagaṇita, we have successively Vibhā and Swabhā''.
Comm. Here, the meaning of taking the complement
of the elongation and dividing by 15 is to obtain the
magnitude of MD directly; for, (90 - e) / 15 = 6 - e/15 =
MG - e/15 = MG - GD = MD where e is the elongation
and e/15 is the Sukla. In other words, finding the Sukla
by dividing the elongation by 15 and subtracting it from
the radius to get MD is the same as taking the complement
of the elongation and dividing it by 15 to obtain MD.
Bhāskara quotes in the course of the commentary,
the verse from his Leelāvatī, namely "भुजाद्वर्गितात् कोटिकर्णान्त
राप्तम् द्विधा कोटिकर्णान्तरेणोनयुक्तम्। तदर्धे क्रमात्कोटिकर्णौ भवेताम्,
इदं धीमताऽऽवेद्य सर्वत्र योज्यम्" which means B² / (K - k) = K + k
since K² - k² = B². This situation arises when the Bhuja
B and the difference of Koti and Karṇa ie. K - k are given
and it is required to find K and k separately.
Verses (8) and (9). Draw a circle with radius 6
angulas to represent the disc of the Moon. Mark the disc
with the Cardinal points signifying the east, west, north
and south. Compute the Valana as directed in verse (5)
(which represents E' G' in fig. 119) and mark it off as a
Hsine from the west point w in the last quarter and from
the east point e in the first quarter. From the centre M
of the Moon's disc, mark off Vibhā MC (computed as
directed) along the join of MG'. With centre C and
radius Swabhā (already computed as directed) draw a circle.
The spherical sector of the Moon's globe thus demarked by
the arc of the circle ADB namely AGBD is found to have
the elevated cusp in the opposite direction in which
the Valana has been marked.

497
Comm. The directions are clear in the wake of the
previous commentary. In marking off the Valana, Bhās-
kara says ‘from the west in the last quarter and from the
east in the first quarter’. In the first quarter, the Moon
will be visible immediately after Sunset in the western
sky and the illuminated part of the Moon will be towards
the western side of the disc in which direction the Sun is
situated. The eastern point and the western point of the
disc are easily discernible on the background of the
horizon. Drawing first the diameter ‘WE’ of the disc
along the vertical circle in case the Moon is due west or
else along a small circle parallel to the prime-vertical called
Upa-Vṛtta, we have the east point of the disc namely e
vertically above the disc in the first quarter. (The figure
119 is shown for the first quarter). Then we are directed
to mark off the Valana E′ G′ in the form of a Hsine'.
Since the Valana is already computed as directed in verse
(5), to mark off this Valana we have to lay the scale
perpendicular to WE. Such a point G′ on the circum-
ference of the disc is to be marked from which the Hsine
equals the Valana computed. It is to be noted here that
it is not marking G′ E′ from G′, for, G′ is not known,
nor E′ for the matter of that, but locating G′ knowing
the magnitude of the Valana. Though we happened to
mention before that GE is the Valana, in the light of
what Bhāskara mentions in verse (8) we have to under-
stand that in the first quarter, the practice is to mark
off Valana from the point e marked on the disc before hand.
Though it does not matter when we take EG to be the
Valana E′ G′, we have to notice the practice followed.
The direction given in the course of the commentary
“केन्द्राद्वलनोपरिवृत्ताद् बहिरपि खटिकया सूत्रमुच्छाद्यम्” meaning
thereby that from the point M, we have to mark off
along MG′ where G′ is the point pertaining to Valana, a
line with the help of a chalk and a fine thread, namely
MC. This practice is still in vogue adopted by masons to
draw straight lines.
63

498
It is interesting to note Bhāskara waxing poetic in
the course of the commentary under these verses describing
how the illumination of the Moon's disc by the rays of
the Sun is effected. The description is quite apt, inter-
esting and scientific reminding us of Varāhamihirācārya's
description in the verse 'सलिलमये शशिनि etc.' in Bṛhat-
Saṁhitā. Of course, following Varāhamihira Bhāskara
also takes the Moon's globe to contain water within its
bosom which Modern Science has yet to confirm.
Verses 10, 11, 12. The Koti and Karṇa defined by
Brahmagupta, do not lead us to accordance between
computation and observation to locate the cusps. I
request good mathematicians to verify this carefully. In
a place where the latitude is (90—ω) = (90—24) = 66°,
when the ecliptic coincides with the horizon, and
when the Sun is in the beginning of Meṣa which is
then rising in the east, and the Moon in the beginning
of Makara, then the Moon is dichotomized by the
meridian and the illuminated part of the Moon's disc is
towards the east. This does not hold good according to
Brahmagupta's definition of Koti, because then the Bhuja
as well as Koti according to his definition is equal to R.
When the Bhuja is zero, the cusps will be horizontal, and
when the Koti is zero, they will be vertical. Brahma-
gupta's Bhuja and Koti both being equal to R, the cusps
therefore cannot be vertical which is against truth as
stated above. Why should I make this statement? May
my homage be to those great.
Comm. When the latitude of a place be 90—ω, and
when the ecliptic coincides with the horizon, it is clear
that in whatever positions be the Sun and the Moon
(assuming that the Moon is also approximately on the
horizon) the cusps of the Moon are always vertical, there
being Bhuja only and no Koti. But according to Brahma-

499

gupta's definition of Karṇa, it will be in this particular
case. √(R² + R²) = R√2 so that the Koṭi will be
√(2 R² — R²) = √R² = R, where the Bhuja also is R
(projection of the line joining the Sun and the Moon on
the north-south line. Though in the verse (11) above one
particular position of the Sun and one of the Moon, are
contemplated, the same argument holds good, says
Bhāskara in the course of the commentary, whatever
positions are occupied by the Sun and the Moon on the
ecliptic. In this case there is only Bhuja existing and
no Koṭi, so that the cusps will be vertical. But in all
these cases, there is Koṭi aecording to Brahmagupta's
definition, so that the ousps will not be vertical according
to him, which is clearly against truth.
Here we have to note that in the example cited by
Bhāskara the Bhuja equal to R is along the north-south
direction, and even though the Koṭi according to Bhās-
karajs definition is conceived to be vertical, the Karṇa
according to Brahmagupta's conception being ES where
E and S are the east and south points, his Koṭi will be
horizontal coinciding with OE, O being the centre of the
horizontal ecliptic. Further according to Brahmagupta,
the join of the ousps will be perpendioular to the Karṇa
whioh does not therefore go against truth. Similarly in
all the cases wherever be the Sun and the Moon on the
horizon, Brahmagupta's Karṇa being a line joining the
centres of the disos of the Sun and the Moon, it will be a
horizontal line and so the ousps could be vertical. It is
not clear whether or not Bhāskara recognized this namely
that the Karṇa and Koṭi of Brahmagupta in the cases
cited above are all horizontal lines so that the cusps could
be vertical even according to Brahmagupta. That is why
he pays homage to Brahmagupta in the last line of (12).
Probably Bhāskara expected that Brahmagupta's Koṭi also
should have been vertical as his. The fact that he
recognized that Brahmagupta's Koṭi will not be vertical

500
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.

504
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.
64

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