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

407
In the second example cited by Bhāskara the Ecliptic
coincides with the horizon, the pole of the Ecliptic being
in the zenith. Then in a Solar Eclipse the Moon eclipses
the Sun from the south showing that H sin V = R. That
H sin V = R is also evident from the fact that the Ecliptic
makes 90° with the prime-vertical, having coincided with
the horizon. But here according to Lallāchārya’s formula,
H sin ξ = Hvers 90° = R and Āyana Valanajyā is Hvers δ,
where δ is the declination of a point whose longitude is
90° more than ☉. If ☉ = 30°, 60° H sin θ =
R/2 sin ω/R or (√3/2) (R sin ω)/R ie. (sin ω)/2 or √3/2 sin ω.
Evidently the sum of the two Valanas Āyana and Ākṣa
cannot be 90° as is also vouchsafed from geometry. So,
here also, the flaw is evident.
Note 1. Śrīpatyāchārya also followed Lallāchārya
vide verses 18, 19, 20 Chandragrahaṇādhyāya, Siddhānta
Śekhara. It will be noted that the commentator of
Siddhānta Śekhara, while reiterating Bhāskara’s stand as
the correct one, himself commits a mistake in saying
“सममण्डलीय नतांशज्यास्थाने नतकाकोत्क्रमज्या गृहीता” In fact
sin ξ = (sin ϕ sin h) / cos μ = (sin ϕ sin z) / cos δ as proved by us before.
The commentator cited above overlooked that sin ξ could
be also equal to (sin ϕ sin h) / cos μ , wherein natakāla also is
implied.
Note 2. It will be noted that even Pṛthūdakāchārya,
while commenting on Brahmasphuṭa Siddhānta, ignored
Brahmagupta and followed Lallāchārya blindly.
Note 3. The formula given by Lallāchārya and
followed by Pṛthūdaka as well as by Śrīpati is very rough
besides containing the flaw cited, in as much as both μ
and δ are taken to be zero, which are not so,

SURYAGRAHANĀDHIKĀRA
Verse 1. In as much as the observer situated on the
surface of the Earth and as such elevated by the radius
of the Earth from the centre there of, perceives not the
Sun and the Moon having the same longitude at the
moment of conjunction, to be in the same line of sight,
heyt being depressed unequally having different orbits, so
I proceed to elucidate what are called Lambana and Nati
ie. parallax in longitude and latitude, on which account
they are not in the same line of sight.
Fig. 91
Comm. (Refer fig. 91) Let E be the centre of the
Earth, M and S the centres of the discs of the Moon and

the Sun. Let A be the position of an observer on the
surface of the Earth, elevated by the radius EA from E.
Let M and S be in the same line of sight as seen from E.
But as seen from A, AS and AM are respectively the lines
of sight to the Sun and the Moon. Evidently these lines
of sight differ the Moon being depressed more than the
Sun. If a line AS' be drawn which is parallel to the central
line of sight namely EMS, we find that the Sun is
depressed by the angle S'AS whereas the Moon is depressed
by the angle S'AM'. These angles differ because the orbits
of the Sun and Moon differ.
Here the angle S'AS will be very very small, its
magnitude being in truth just about 8" only. But the
angle S'AM' will be sufficiently large since the Moon is
very near the Earth compared with the Sun. Taking ES
and AS to be almost parallel due to the largeness of the
Sun's distance, the angle SÂM will be almost equal to
AM̂S so that we could consider that the Moon is depressed
from AS the line of sight to the Sun by the angle SÂM' =
AM̂E. This angle AME is called the geocentric parallax
of the Moon and the angle AŜE that of the Sun M'ÂS =
angle of depression of the Moon over and above that of
the Sun = M'AS' - M'AS = EM̂A - EŜA = geocentric
parallax of the Moon minus geocentric parallax of the Sun.
Verse 2. The presence or absence as well as the
positiveness and negativeness of the parallax in longitude.
Compute the Lagna at the moment of conjunction of
the Sun and the Moon. There will be no parallax in
longitude when the Sun is situated at the point called
Vitribha or the point whose longitude is ≡ L - 90°, L
being the longitude of the Lagna ie. the ascendant which
is the point of intersection of the Ecliptic with the
52

410
horizon. If the Sun's longitude falls short of the longitude
of the Vitribha or exceeds it, there will be parallax in
longitude which will be positive in the former case and
negative in the latter.
Fig. 92
Comm. (Ref. fig. 92) Let SN be the horizon, Z the
zenith and VA the Ecliptic. A is the ascendant or Lagna.
Let V be the point called Vitribha which is 90° behind A.
Strictly speaking V is called Vitribhalagna or lagna from
which three Rāśis or 90° are subtracted (Bha = Rāśi.
त्रिभिर्विरहितम् वित्रिभम्; वित्रिभम् च तत् लग्नम् च वित्रिभलग्नम् ie.
a point whose longitude is got by subtracting three
Rāśis from that of the Lagna). Let ZV be the vertical of
V so that ZV̂A = 90°. It will be seen that AV = 90° as
follows. Let A' be the point where the Ecliptic intersects
the horizon on the west. One will construe that the
Ecliptic is bisected by the meridian; but it is not so.
Spherical triangles AVZ and A'VZ being right-angled at

411
V are congruent because AZ = A'Z and ZV is common
∴ AV = VA'. But AV + VA' = 180° because the Ecliptic
and the horizon being two great circles, they bisect each
other. Hence AV = 90°. Then a celestial body situated at
V will be depressed along ZV the vertical, say, to a point B.
Let ☉ be any arbitrary position of the Sun; then ☉ will
be depressed along the vertical Z☉, say, to a point C.
Draw CD perpendicular on the Ecliptic. Then ☉D is the
component of the parallax ☉C along the Ecliptic where
as DC is its component perpendicular to the Ecliptic.
Thus ☉D is the parallax in longitude and DC is the
parallax in latitude. The word ‘Lambana’ means
etymologically लम्बते अनेनेति लम्बनम् ie. that amount by
which the celestial body is depressed (along the Ecliptic).
In Hindu Astronomy the word Lambana is applied to
parallax in longitude alone whereas the word Nati is
applied to parallax in latitude. Hence to translate
Lambana as parallax alone is not correct. The word
Drik-lambana is applied to mean parallax along the
vertical, and the word Sphutalambana is occasionally used
to connote parallax in longitude.
As Bhāskara rapidly comments on the verses in this
Gaṇitādhyāya, he having dealt with the subject of parallax
elaborately under the caption, Grahaṇa Vāsanā, in the
Golādhyāya, to catch up his thought, we have to treat the
subject first from the modern view point and then elucidate
what he has said in the Golādhyāya, much matter of which
is reiterated by him under the commentary here in the
Gaṇitādhyāya.
(Ref. Fig. 91) From the △EAM,
(sin EMA) / a = (sin EÂM) / d = (sin ZÂM) / d where a is the radius of
the Earth and d the distance of the celestial body (here the
Moon)

412
∴ Sin EMA = a/d sin ẐAM = ÊMA expressed in radian
measure since ÊMA is very small
∴ ÊMA (expressed in radian measure) = a/d sin z I where
z is the apparent zenith-distance of the Moon ie. zenith-
distance as seen by the observer (in contradistinction to
the geocentric zenith-distance of the Moon namely ẐEM).
In particular, when z = 90°, ÊMA = a/d which is the
maximum parallax known as the horizontal parallax ie.
the parallax when the Moon is situated on the horizon of
the observer. Also the parallax is zero when M is situated
at Z as is seen from formula I and as is rightly remarked
by Bhāskara in the words 'खमध्ये नास्ति लम्बनम्'.
In fig. 91, ÊMA is the angle by which the line of
sight of the observer namely AM is depressed from the
geocentric line of sight EM. Since the plane of the paper
represents a vertical through the Sun and the Moon, the
depression of either the Sun or the Moon or the excess of
the depression of the Moon over the Sun are all in the
vertical plane. This depression is called Drik-lambana
because it is a lambana or depression in the Drik-maṇḍala
or vertical.
This Drik-lambana varies as sin z as is seen from
formula I where a, and d may be taken to be constants.
(Both a and d vary slightly a varying slightly from place
to place on the Earth, the Earth being an oblate spheroid,
and d varying from position to position of the Moon).
The maximum horizontal parallax is given by a/d in
radian measure which is equal to, according to Bhāskara's

418
estimate 1581/2 × 1/51566 × 3438 = 53′ approximately. Its
modern value is about 57′ so that the Hindu estimate is
not far from truth. Here 1581 and 51566 are the values
of a and d in Yojanas according to Bhāskara.
The Hindu astronomers do not, however, proceed
exactly as we have done in the para above to obtain the
maximum horizontal parallax. Their treatment is a little
different and is as follows. Whereas according to Modern
astronomy E M̂′ A (fig. 93) is viewed as the horizontal
[चित्र: Fig. 93]
Fig. 93
parallax, in Hindu Astronomy MÊM′ or the angular
measure of the Moon’s path equal to the radius of the
Earth is taken to be the horizontal parallax. Both,
of course, mean the same as is seen from the figure.
In as much as the Hindu astronomers knew very well
what they term कलीकरण or converting linear distances
into angular measure, converting a linear magnitude equal
to the radius of the Earth namely MM′ at the lunar orbit
into angular measure, they got
MM′/EM × 3438 = 1581/2 × 1/51566 radians = 52′-42″ as the
maximum horizontal parallax.

414
Having got this estimate, they reckoned this angular
measure in time as the time taken by the Moon to traverse
the distance MM' equal to the radius of the Earth as
follows. The Moon traverses 790′-35″, which is exactly
15 times 52′-42″. Hence they said that the maximum
horizontal parallax is 1/15th of the Moon's daily motion in
arc and expressing it in terms of time, that the maximum
horizontal parallax is 1/15 of a day or 1/15th of 60 nādis or
4 nādis.
That this horizontal parallax is 4 nādis as a maximum,
would have been also verified at the time of a solar eclipse
when the Sun was situated on the horizon at the time of
conjunction, by the fact that the eclipse occurred four nādis
in advance of the moment of geocentric conjunction
(which could be calculated very accurately by the Hindu
astronomers, as could be seen by the very correct estimate
of a lunation in Hindu Astronomy. In fact, the length
of a lunation must have been estimated correctly by noting
the time-interval between two solar eclipses or lunar and
by dividing that time by the integral number of lunations
elapsed in between the two eclipses).
The question then arises as to how the Hindu Astro-
nomers could know the distance of the Moon. From the
estimate of the horizontal parallax by actual observation,
and from the geometry of fig. 93, a correct estimate of the
distance of the Moon must have been arrived at.
Having thus known that the Moon traverses a
distance equal to the radius of the Earth in 4 nādis, his
daily linear motion was estimated to be 15 times the radius
of the Earth ie. (15 × 1581 1/6) / 2 = 11858 3/4 Yojanas.
The daily motion of the Moon having thus been
estimated almost correctly, an act of inexpedience on the
part of the Hindu astronomers was that they should have

415
presumed that all the other planets including the Sun would
be traversing the same linear distance during the course of
a day. This led to a wrong estimate of the Sun’s distance
as well as his spherical radius. Also, they supposed wrongly
that the parallax of the Sun also would be equal to 1/15th
of his daily arcual motion.
Their estimate of the spherical diameter of the Moon
was, however, very near the truth, for, they argued, that
if 790′-35″ angular motion per day corresponded to 11858¾
Yojanas in linear measure, to what linear measure did the
angular diameter of the Moon namely 32′-0″-9′′′ corres-
pond? The answer was
11858¾ × 32′ 1/400
------------------ = 480 Yojanas.
790′-35″
[चित्र: Fig. 94]
Fig. 94
It may be here pointed out that there is a relation
between the angular radii of two celestial bodies as seen
from each other and their mutual horizontal parallaxes
(Fig. 94). Let E and M bet he centres of the Earth and
the Moon respectively. E M̂ A = horizontal parallax of
the Moon = angular radius of the Earth as seen from the
Moon and B Ê M = Angular radius of the Moon = Hori-
zontal parallax of the Earth as seen from the Moon. Thus,
we see that the Earth will be seen from the Moon, as a
Moon with an angular radius equal to 57′. In other words
our Earth will be a Moon to our Moon, having nearly 16
times the area of our Moon’s disc.

416
The periphery of the Moon’s orbit was arrived at as
follows. “If 790′-35″ of the Moon’s angular motion
corresponds to 11858¾ Yojanas to what periphery must
360 × 60′ correspond ?” The answer is
(11858¾ × 360 × 60) / (790′-35″) = 324000 Yojanas. Reverting to the
subject of parallax on hand, the Drik-lambana or the
parallax along the vertical has the formula (4 H sin z) / R I
nādīs in Hindu Astronomy where 4 nādīs is the maximum
parallax obtained when H sin z = R.
From fig. 92, C☉² = CD² + D☉² ie.
Drik-lambana² = Nati² + Sphuṭalambana² II
CD = ☉C sin CÔD = (4 H sin z / R) ×
sin CÔD = 4 sin z sin VÔZ
= 4 sin ☉Z sin VÔZ = 4 sin ZV
= (4 H sin ZV) / R = VB III
Thus, the parallax in latitude at any point of the Ecliptic
is that at the Vitribha which is conveyed by Bhāskara
in the words “कक्षयोरन्तरं यत् स्यात् वित्रीभे सर्वतोऽपि तत्”.
Also ☉D = C☉ cos CÔD = 4 sin ☉Z cos Z☉V
= 4 cos ZV sin V☉ = (4 H cos ZV H sin V☉) / R² IV
= Maximum parallax × Vitribha-Sanku × H sine of the
arc V☉.
In the above working we proceeded in a modern way.
It is worth-hearing Bhāskara as to how these results were
arrived at elegantly and ingeneously from first principles.

417
In fig. 91, EMS is called Garbha-Sūtra whereas AS
is called Dṛṣṭi-Sūtra. दृक्सूत्रात् लम्बितश्चन्द्रः तेन तल्लम्बनं
स्मृतम् ie. In as much as the Moon is depressed from the
Dṛk-Sūtra, so this phenomenon goes by the name Lambana.
It will be noted that in Hindu Astronomy geocentric
parallax will not be treated separately for the Moon and
the Sun but dealt with simultaneously as it is called for,
in the context of a solar eclipse. They were interested
in knowing the relative depression of the Moon with
respect to the Sun rather than knowing the separate
magnitudes with respect to the Moon and Sun, for which
they had no application.
'दृग्गर्भसूत्रयोरैक्यात् खमध्ये नास्ति लम्बनम्' ie. In as much
as the Garbha-Sūtra and Dṛk-Sūtra are identical in the
direction EAZ (fig. 91) there is no parallax at the zenith.
Now consider the plane through ZV of fig. 92. Suppose
the EMS of fig. 91 is in the direction EV. Then both
the Sun and the Moon may be considered to have the
same Vitribha at that moment of conjunction. Both the
Sun and the Moon being then depressed along ZV, to V'
and V'' respectively the Ecliptic will then be a circle
parallel to VA (fig. 92) through V' and the orbit of the
Moon will be another circle parallel to VA through V'', V''
being below V'. If we neglect, for a moment, the depres-
sion of the Sun, and consider VD to be the Ecliptic on
which the Sun is situated undeflected, and BC to be the
deflected orbit of the Moon relative to the Sun, then VB
is the Nati of the Moon, which will be the same distance
between VD and BC, ie. the orbits of the Sun and the
Moon.
This fact was proved by us analytically in the modern
way showing that CD = BV.
This Nati it is that influences the latitude of the Moon,
which may cause apparent conjunction when there is no geo-
53

centric conjunction and which does not show an apparent
conjunction when there is a geocentric conjunction. In
other words parallax in latitude plays a very important part
in solar eclipses. Also ☉ D being the parallax in longitude,
the moment of apparent conjunction might be preceded or
followed by a geocentric conjunction according as the Sun
lies along VA or AV. Thus having determined the exact
moment of apparent conjunction using the magnitude of
☉ D, then we have to rectify the latitude using the
magnitude of VB. If that rectified latitude of β falls
short of R+r where R is the angular radius of the Sun and
r that of the Moon, then there will be a solar eclipse.
It will be noted that when the Sun coincides with V
at the moment of conjunction, there is no parallax in
longitude ☉ D being zero (Fig. 92) in that position. Also
there will be no parallax in latitude when the Ecliptic
assumes the position of a vertical circle passing through
the zenith, the Drik-lambana then being entirely along
the Ecliptic. In this case the Vitribhalagna V will
coincide with Z and (4 H sin V ☉) / R which is termed the
Madhyamalambana is now entirely along the Ecliptic and
as such it is the Sphutalambana in this case. We have said
above that when V coincides with Z, the Madhyamalambana
is zero at Z, and that the maximum is equal to 4 nādīs on
the horizon. In between Z and the horizon it has the formula
(4 H sin V ☉) / R . Noting further that in this case when
the H cosine of the zenith-distance of V ie. the Śanku of
V is R, the entire lambana is along the Ecliptic, the nāti
being zero, and the Madhyamalambana is itself the Sphuta-
lambana, and again when V does not coincide with Z,
H cos ZV is no longer R but has assumed Kōti-Rūpa ie.
the form of a H cosine, as well as the Sphuta-lambana
also, which assumes Kōti-Rūpa ie. of the form ☉ D of
Fig. 92, where ☉ C is the Madhyamalambana, ☉ D is the

419
Kōti = Sphuta-lambana, and DC = Bhuja = Nāti, it is
argued that the Sphuta-lambana is proportional to
H cos ZV, assuming a maximum value when V coincides
with Z (☉ not being at Z).
Verses 3 and 4. Parallax in longitude based on two
proportions.
Compute the H cosine of ZV, by calculating the
rising time of AV, the Kujyā, Dyujyā and Antyā pertain-
ing to V, as was formulated in the Tripraśnādhikāra, then
H sin V☉, multiplied by 4 and divided by R, and again
multiplied by H cos ZV and divided by R again gives the
parallax in longitude.
Comm. As per the above formula, parallax in longi-
tude equal to ☉D of Fig. 92 is equal to
(4 H sin V☉ × H cos ZV) / R² . This is evidently derived out
of two proportions that the parallax in longitude is
proportional to H sin V☉ as well as H cos ZV. This we
have already derived through modern methods as formula
IV.
Under verse 2. The two proportions are (1) V coin-
ciding with Z, if by H sin V☉ equal to R, we have 4 nādis
as the maximum lambana on the horizon, what shall we
have by an arbitrary H sin V☉? The result is
(4 H sin V☉) / R and (2) V not coinciding with Z, if by
H cos ZV equal to R we have (4 H sin V☉) / R as the
Madhyamalambana, what shall we have for an arbitrary
H cos ZV? The result is
(4 H sin V☉) / R × (H cos ZV) / R as formulated.
First half of verse 5. Alternate method of rectifying
lambana. The Madhyamalambana multiplied by 12 and

420
divided by the Chāyākarṇa of the Vitribha will also give
the Sphuta-lambana.
Comm. From Tripras'nādhikāra, we have
12 / K = (H cos Z) / R so that in the formula cited above instead of
(H cos ZV) / R we are asked to use 12/K.
Latter half of verse 5 and first half of verse 6.
Dṛk-nati² = H cos² ZV − H cos² Z☉
= H sin² Z☉ − H sin² ZV (fig. 92)
(4 Dṛk-nati) / R = Sphutalambana.
Comm. We shall first prove this on modern lines.
Cos Z☉ = cos ZV cos Z☉
∴ cos² ZV − cos² Z☉ = cos² ZV (1 − cos² V☉)
= cos² ZV sin² V☉ = (H cos² ZV H sin² V☉) / R⁴
Also cos² ZV − cos² Z☉
= sin² Z☉ − sin² ZV =
(H cos² ZV − H cos² Z☉) / R² = (H sin² Z☉ − H sin² ZV) / R²
∴ H cos² ZV − H cos² Z☉ = H sin² Z☉ − H sin² ZV
= (H cos² ZV H sin² V☉) / R⁴
∴ Dṛk-nati defined above = √(H cos² ZV − H cos² Z☉) =
√(H sin² Z☉ − H sin² ZV) = (H cos ZV H siu V☉) / R
∴ (4 Dṛk-nati) / R = (4 H cos ZV H sin V☉) / R²
= Sphutalambana.

421
Bhāskara's proof proceeds in two stages from first
principles. In the first place when V coincides with z,
Lambana is seen to be equal to 4 nādis on the horizon and
zero at Z ie. it is zero when H sin Z☉ = 0 and 4 nādis,
a maximum when H sin Z☉ = R. So, it is meet that
Lambana should be taken to be proportional to H sin Z☉
ie. proportional to natajyā. In this context the lambana
termed as Madhyamalambana is entirely along the ecliptic.
It is taken to be in the form of Karṇa, because in the
position of ☉C also it is in the form of a Karṇa.
Then let the Ecliptic be deflected from the zenith
(deflected = क्षिप्त). Vitribhalagna then being deflected
from the position of Z, occupies the position of V (fig. 92).
So ZV is called Dṛk-kshepa since the Ecliptic which was
in the form of a Dṛk-mandala is deflected from that posi-
tion. Also the circle ZV is called Dṛk-kshepa-mandala
because V is deflected along that circle. Now consider
the △ whose sides are H sin ZV, H cos ZV and R.
H cos ZV equal to R and as such in the form of a Karṇa
corresponds to the Madhyamalambana which is also in the
form of a Karṇa ; when this Vitribha-Sanku assumed the
form H cos ZV, ie. rendered a Kōti from its form of a
Karṇa, R, the Sphutalambana is also rendered a Kōti in
the form of ☉ D so that
Madhyamalambana / R = Sphutalambana / H cos ZV
∴ Sphutalambana = (H cos ZV / R) Madhyamalambana
= (H cos ZV / R) × (4 H sin Z☉ / R) .
Then Bhāskara says that we could look at this, from
another angle in the words “यदेव स्फुटलम्बनस्य कोटिरूपत्व-
मुपपन्नम् etc. ”.

422
In fig. 92, H sin Z☉ is in the form of a Karṇa;
H sin ZV is in the form of the corresponding Bhuja. This
triangle formed by these two as sides may be taken to be
similar to the triangle ☉DC, both being called parallax
Δs. This plane triangle ☉DC is like the plane triangle
which has for its sides H sin λ, H sin δ, where λ and δ
are the longitude and declination of a point of the Ecliptic.
In the Triprasnādhikara, we had occasion to deal
with this triangle and there we had R
√(H sin² λ - H sin² δ)
--------------------- H sin α where α is the right ascen-
H cos δ
√(H sin² Z☉ - H sin² ZV)
sion of the point. Similarly R ------------------------
H cos ZV
= H sin V☉. In other words Dṛk-nati is H sin V☉
projected into a circle of radius H cos ZV from a circle
of radius R. We have the proportion
∴ C☉ CD D☉
-------- = ------------- = -------------------------
H sin Z☉ ☉ sin ZV √(H sin² Z☉ - H sin² ZV)
D☉ = Sphutalambana
= -------------------------
√(H sin² Z☉ - H sin² ZV)
The quantity under the radical in the denominator
is called Dṛk-nati for the following reasons.
When V coincides with Z, V ☉ is the Dṛk-mandala-
nata, so that when V is deflected also, we continue
to view the Dṛk-nati placed along V☉. Since Madhyama-
lambana ☉C is in the form of a Karṇa in the Δ ☉DC,
we perceive it to be in the form of a Karṇa even when
V coincides with Z. This Madhyamalambana being equal
to Sphutalambana when V coincides with Z, Sphuta-
lambana is also in the form of a Karṇa then. Now in the
position ☉DC, Sphutalambana has assumed the position
of a Kōti ie. the Sphutalambana which, in the form of a
Karṇa, being placed along Dṛk-mandala natāmsa, is now

423
rendered a Kōti and is placed along V ☉, So the quantity
√(H sin² Z☉ - H sin² ZV) which is the Kōti of the △
formed by H sin Z☉ and H sin ZV, corresponds to the
Kōti of Sphuta-lambana ☉D. So we call
√(H sin² Z☉ - H sin² ZV) as Dṛk-nati, in as much as the
Sphutalambana being placed along V☉ in the form of a
Karṇa when V☉ is Dṛk-mandala-nata, continues to be
placed along V☉ in the deflected position also and
becomes a Kōti corresponding to the Kōti of the triangle
formed by H sin Z☉ and H sin ZV ie. corresponding to
the quantity √(H sin² Z☉ - H sin² ZV). The Sphuta-
lambana should be construed as being associated with
Dṛk-mandala-nata which term is now abbreviated to the
term Dṛk-nati.
At A of fig. 92, the Dṛk-nati = √(R² - H sin² ZV)
= H cos ZV = H cos ZV = Vitribha-lagna-Śanku. Hence
the proportion proceeds in accordance with this Dṛk-nati.
Verse 6 (latter half) and first half of verse 7.
Alternative method of obtaining parallax in longitude.
√(((H cos ZV) / (R/4))² - ((H cos Z☉) / (R/4))²)
or √(((H sin Z☉) / (R/4))² - ((H sin ZV) / (R/4))²)
gives the parallax in longitude expressed in nādīs.
Comm. These formulae just constitude another mode
of expressing the parallax in longitude and the equivalence
of the formulae with the formula (4/R) Dṛk-nati is evident.
Latter half of verse 7. Use of the parallax in
longitude.
The time of the ending moment of New Moon ie. the
moment of geocentric conjunction is to be rectified by this

424
parallax in longitude to get the moment of apparent
conjunction by the method of successive approximation.
Comm. In as much as the moment of apparent
conjunction for an observer situated somewhere on the
surface of the Earth precedes or follows the moment of
geocentric conjunction being preponed or belated by the
parallax in longitude, we have got to take this parallax
in longitude into account and compute the moment of
apparent conjunction. This computation has to proceed
according to the method of successive approximation since
the hourly motions of the Sun and the Moon vary as well
as the parallax in longitude. When the Sun is in advance
of V, the Sphutalambana advances the Moon more than
the Sun so that the moment of apparent conjunction is
past. Hence the correction is negative and vice versa.
Verses 8 and 9. Computation of the parallax in
longitude without an appeal to the method of successive
approximations.
Let Para = (13 / 32) H cos ZV ; {Para ~ H sin ☉ L}² +
H cos² ☉ L = K² H sin⁻¹ { (H cos ☉ L × Para) / K } =
parallax in longitude.
Ref. fig. 95, E₁ E₂ is taken to be what is termed Para
equal to (4 / R) H cos ZV, H cos ZV being the Vitribha-Sanku.
Since 4 / R could be written as (H sin 24 / R), since 4 ghatis =
(4 × 360) / 60 = 24°, 60 ghatis being equivalent to 360°,
(4 / R) H cos ZV = (H sin 24 / R) × H cos ZV. Imagining for a
moment H cos ZV has come in the place of H sin d

425
pertaining to the formula H sin δ = (H sin λ × H sin 24) / R ,
(4/R) H cos ZV = Para = the H sine of the declination of
that point whose longitude is equal to Vitribha-Sanku.
In other words Para is termed as the Vitribha-Sanku-
Rūpa-Krānti-Vṛttiya-Bhujajyājanita-Krāntijyā.
Now take E₁ E₂ = Para defined above. Draw circles
of equal radii with E₁ and E₂ as centres. Call (E₁) and
(E₂) as the Chandra-Kakshāmandala and Ravi-Kaksha
Mandala. Para by its formulation as (4/R) H cos ZV, is equal
to the maximum parallax in longitude for a given
H cos ZV ie. for a given position of V with respect to Z.
This being so, the parallax in longitude for an arbitrary
position of ☉ with respect to V will be
[Para × H sin (☉ - v)] / R according to the previous formu-
lation thereof. This form of the formula by its similarity
with the formula (a/R) H sin m, pertaining to the eccentric-
circle-theory, suggested to Bhāskara that the parallax in
longitude could be derived from the theory of the eccen-
trics or Prati-Vṛtta-Bhangi. In fig. 95, it will be noted
that E₁, E₂ are not the centre of the Earth and the position
of the observer on the surface of the Earth but such
points as E₁ E₂ is made equal to (4/R) H cos ZV or (a/d) H cos ZV
of the modern figure 4/R being equal to a/d, so that E₁ E₂
is of a variable magnitude varying with H cos ZV.
Comm. When H cos ZV = R ie. when V coincides
with Z, we have the maximum parallax. What then will
54

426
[चित्र: Fig. 95]
Fig. 95
be had for an arbitrary H cos ZV ? The result is
(H cos ZV / 3438) × H sin 24 since 4 nādīs correspond to 24°,
60 nādīs corresponding to 360°. Hence the result is
(H cos ZV × 1397) / 3438 . Converting 3438 / 1397 into a continued
fraction we have 2 + 1/(2+) 1/(5+) 1/(1+) 1/9 ......of which the con-
vergents are 2/1, 5/2, 27/11, 32/13 and 32/13 is a very good convergent
preceding a large quotient namely 9. So the result may
be written as (13 H cos ZV) / 32 which is symbolized as Para.
Now parallax in longitude = (Para × H siu ( ☉ − v )) / R .
When H sin ( ☉ − v ) = R, the parallax will be equal to