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

453
The misconception in the mind of Chaturvedāchārya
as well as the wrong notion in the minds of Mādhava and
even Bhāskara in construing that the number of revolu-
tions of the nodes are to be increased by the number of
revolutions of the S'īghrocchas, are due to the fact that
it was overlooked that the position of S'īghrocohas with
respect to Mercury and Venus are given by their helio-
centric longitudes. In other words as is mentioned by
the verse “ ज्ञशुकयोः ग्रहः सूर्यः भवेत्तौ शीघ्रनामकाँ ” the S'īghroc-
chas of Mercury and Venus are no other than the helio-
centric positions of those planets. Thus the exclamation
of Chaturvedācharya cited above arose out of thinking
that the S'īghrocchas differ from the planets, whereas they
are the same heliocentrically, though they differ geocentri-
cally. Heliocentrically the planets longitude is (vide fig.
105) ÂSP which is equal to A/EP' geocentrically. The
geocentric longitude of the planet is on the other hand
A/EP differing from the above, though both the helio-
centric and geocentric longitudes point to the same planet.
Bhāskara, no doubt, gave a correct procedure but missed
to identify the S'īghroccha and the heliocentric position
of the planet. In this context, the reader is referred to
the author's ‘ peculiar concept of S'īghroccha in Hindu
astronomy published in the journal of Oriental Research
of the S. V. University Vol. XIV, Part 2, Dec. 1971.
Verse 2. The Hsine of the arc of the planet’s orbit
Vimandala intercepted between the nearer node and the
planet multiplied by the maximum latitude of the planet
cited, and divided by the S'īghrakarṇa gives the latitude
of the planet at the given place.
Comm. From the formula akin to that which gives
the declination of a point on the ecliptic namely sin δ =
sin λ sin ω or in the Hindu form H sin δ = (H sin λ H sin ω) / R ,

454
H sin β = (H sin λ H sin i) / R where β is the latitude required,
i the maximum latitude and λ the arc of the planetary
orbit intercepted between the nearer node and the planet.
Since β and i are small H sin β and H sin i could be taken
to be β and i. Hence we have β = (i × H sin λ) / R . Since
this value is had at the distance of the Śīghrakarṇa, its
value at a distance of R should be given by
β = (i H sin λ) / R × R / K = (i H sin λ) / K as formulated.
Verse 3. √(R² - H sin² v) is called Yaṣṭi where v is
Āyanavalana. The latitude β of the planet multiplied by
Yaṣṭi and divided by R or multiplied by H cos δ where δ
is the declination of the point whose longitude is 90 + λ,
λ being that of the planet and divided by R gives the value
of the rectified latitude which could be added to the
declination of the foot of the latitude to give what we say
the modern declination of the planet.
[Diagram: Spherical geometry diagram showing points P, P', K, R, R', B, B', M, M', S, L, N, ω, γ]
Fig. 107

455
Comm. Let rM (fig. 107) be the ecliptic and rN the
celestial equator whose poles are K and P respectively.
Let R be a celestial body whose latitude is β and whose
modern declination is RL. Let M be the foot of the
latitude circle and let δ be the declination of M. The
word Krānti in Hindu astronomy is applied to connote
the declination of a point on the ecliptic alone and not of
any other point like R. RL is called Sphuta Kranti which
is equal to R'N = R'M + MN. RM is called Vikṣepa and
R'M Sphuta Vikṣēpa. K̂MP is called the Āyanavalana at
the point M of the ecliptic. Produce MP to P' where
PP' = δ so that MK = MP = 90°. Hence P̂' is a right angle
and KP' = K̂MP' = v̂ = Āyanavalana. From the spherical
triangle KPP', cos ω = cos v cos δ. Draw perpendicular
RR' on MP so that MR' = β' = Sphuta Vikṣēpa = β cos v
= (β H cos v) / R. But H cos v = √(R² - H sin² v) = Yaṣti
∴ β' = (Yaṣti × β) / R which is to be added to δ to obtain
the Sphutakrānti R'N or RL, the modern declination.
Note (1) One Mukhopādhyāya, in his thesis ‘The
Hindu nakṣatras’ submitted to the Calcutta University,
mistook RM' to be the Sphutavikṣēpa instead of R'M and
hastily remarked that Bhāskara was wrong in making RM'
less than RM.
Note (2) v̂ shown in the fig. 107, is called Sthanīya-
valana or valana at the point M which is considered to be
place of the planet ‘Sthāna’ on the ecliptic. K̂RA, on
the other hand is called Bimbīya-valana or valana at the
bimba or disc of the planet.
Note (3) v̂ could be obtained from the spherical
triangle KMP where KM = 90°, PM = 90-δ, using the

456
formula cos ω = sin (90-δ) cos v or in the Hindu form
H cos v = (R H cos ω) / (H cos δ) . In this case β' = (β H cos ω) / (H cos δ) .
Or again noting that M K̂ P = 90-λ where λ is the
longitude of R, we could use 'Inner side Inner angle
formula' with respect to the triangle KMP, which gives
0 = sin 90 cot δ - sin (90 -λ) cot v or cot v = cot ω / cos λ
or tan v = cos λ tan ω. But this formula implies the
tangent functions which were not used by the Hindu
astronomers.
Similarly using the elements 90-λ, 90°, v, and 90-δ
of the same triangle KMP, another formula could be got
for v. Or again noting that K P̂ M = 90+α, we could yet
get more formulae where α is the Right ascention of M.
Note (4) Thus far we have used modern formulae.
Let us now see as to how Bhāskara derives his formula.
He takes the triangle MKP' (fig. 107) wherein MK = 90°,
KP' = v and P̂' = 90. From fig. 108, the Hsine of MK is
Fig. 108
KO equal to R where 'O' is the centre of the sphere and
the Hsine of KP is KN so that it is the Āyanavalanajyā,
Hence

457
is the centre of the sphere and the Hsine of KP is KN so
that it is the Āyanavalanajyā. Hence
ON² = OK² — KN² ∴ ON = Yaṣṭi = √R² — Āyanavalanajyā²
= Āyanavalanakoṭijyā. From the similarity of the
triangles OKN and GRF,
OK / ON = GR / GF ∴ GF = (GR × ON) / OK = (H sin MR × Yaṣṭi) / R
GR could be taken equal to MR, the latter being
small and GF could be taken to be equal to GR′ ie.
H sin MR′ and so equal to MR′.
∴ MR′ = (MR × Yaṣṭi) / R
Adding MR′ to the declination of M, we get the
modern declination of R′ ie. the Sphuṭakrānti of R.
Note (5) In fact the spherical triangle MKP is
just like the spherical triangle r E ♋ (fig.). In the place
of the paramakrānti E ♋, we have the Āyanavalana KP
(fig. 108), and in the place of Dyujyā ♋ L (fig. 19) we have
Yaṣṭi.
Note (6) An alternative is given in the verse for
this namely MR′ = (MR × H cos δ′) / R where δ′ is the decli-
nation of a point whose longitude is 90+λ. In other
words we have to prove that H cos v = H cos δ′ or v = δ′
(of a point whose longitude is 90+λ. Since declination
is given by the formula H sin δ = (H sin λ × H sin ω) / R
Putting 90+λ for λ, H sin δ′ = (H cos λ × H sin ω) / R I
But from triangle MKP, sin v / sin ω = sin (90 — λ) / sin (90 — δ) = cos λ / cos δ
58

458
∴ sin ν = (sin ω cos λ) / (cos δ) or H sin ν = (H sin ω × H cos λ) / (H cos δ) II
Comparing I and II, ν would be equal to δ′ provided
H cos δ = R. This is accepted as an approximation, as
δ is generally small.
This approximate formula is given by Sūryasiddhānta
in the context of Āyana Dṛk-karma in verse 10, ch. 7,
where ν is assumed to be equal to δ′ defined above.
Verses 4 and 6. The process called Āyana Dṛk-karma.
The Āyana Dṛk-karma correction measured in minutes
is obtained by multiplying Āyanavalana by the unrectified
celestial latitude, and divided by H cos δ and then
multiplied by 1800 and divided by the rising time of the
Rāśi which is occupied by the planet. Or again it is
obtained approximately by the product of the Āyana-
valana and the unrectified celestial latitude divided by the
Yaṣṭi.
This Āyana Dṛk-karma correction is negative if the
hemisphere and the latitude have the same direction.
On symbols, Āyana Dṛk-karma correction =
(Āyanavalana × β × 1800) / (H cos δ × T) or approximately equal to
(β × Āyanavalana) / (Yaṣṭi (= H cos ν)) .
Comm. Let G be the position of the planet when the
foot of the planet’s secondary (called Grahasthāna) to the
Ecliptic namely A is rising, where rCA is the Ecliptic.
Let rML be the celestial Equator. Let P and K be the
poles of the celestial Equator and Ecliptic respectively.
The arc AC of the Ecliptic intercepted between A and the
declination circle of G expressed in minutes represents the
आयनकलोः formulated in the verse. To find the magnitude
of AC, the first formula mentioned above envisages finding

459
Fig. 109
it through the magnitude of ML the corresponding
Equatorial arc which is itself found through finding the
magnitude of AB. AB expressed in minutes will be equal
to the number of asus taken by the declination circle of G
namely PGB to have traversed from the position PDA to
PGB which is the time taken by the planet G to be at its
position from the moment of its rising at the Equatorial
horizon namely PDA, D being then its position. When
G was at D, K, the pole of the Ecliptic should have been
itself rising on the Equatorial horizon. Thus it is sought
to find the time expressed in asus taken by K from its
moment of rising on the Equatorial horizon to its present
position at K. When Bhāskara mentions in the commen-
tary under the verse that ‘‘क्षितिजकदम्बयोरन्तरं तदेवोत्तरमायनं
वलनम्’’ by the word क्षितिज, he had at the back of his mind
लङ्काक्षितिज ie. Equatorial horizon and not the horizon of
the place shown in the figure. Also as was mentioned
before in a previous context the time expressed in asus
taken by an equatorial arc to rise is equal to the number
of minutes in that arc. Since the time taken by the
planet to traverse the arc of the diurnal circle namely DG
is the same as the time taken by the equatorial arc LM
to rise, Bhāskara seeks to find the number of minutes in
AB, then compute, the number of Asus taken by LM to rise

460
through which the number of minutes in AC could be
computed.
AB = AG sin AĜB =
(AG × H sin AĜB) / R = (β × Āyanavalanajyā) / R where AG = β.
In the formula given the word Āyanavalana is used for
Āyanavalanajyā ie. the Hindu sine of Āyanavalana for
brevity. Also, since generally the angle AĜB defined as
Āyanavalana happens to be small, its Hsine will be almost
equal to it. In this sense also the term Valana is used
where Valanajyā is to be used.
From AB we pass on to the magnitude of LM. From
Hindu spherical Trigonometry
LM = (AB × R) / (H cos AL) = (AB × R) / (H cos δ) = (β × Āyanavalanajyā) / R ×
R / (H cos δ) = (β × Āyanavalanajyā) / (H cos δ) . (H cos δ is called Dyujyā
which is connoted by the term Dyu-guna in the verse, the
words Jyā and Guna being synonymous).
Thus LM = (β × Āyanavalanajyā) / Dyu-guna
From LM we pass on to the corresponding arc of the
Ecliptic namely AC, which does not imply the latitude of
the place. So we have to use what are called Niraksha-
Udayas of Rāśis ie. the rising times of the Rāśis at
Equatorial places. Rule of three is used here to find AC
from LM. Locating the particular Rāśi in which the
planet is situated, and using the proportion. “If a Rāśi
of 30° ie. 1800′ of the Ecliptic rises at an Equatorial place
in say x asus, what arc in minutes corresponds to the
number of minutes or what is the same the number of Asus

401
in the arc LM?" We get the magnitude of the arc AC in
minutes.
The computation of this arc AC is intended to know
the difference between the times of rising of the point A
and the point G the former being the point of the ecliptic
signifying the position of the planet and the latter being
the planet itself. This difference of the times of rising is
itself required to know the actual time of rising of the
planet by a knowledge of the time of rising of the point
which signifies the planet’s position on the Ecliptic by its
longitude.
In verse 5, Bhāskara gives an alternate and easy
method of obtaining the magnitude of AC construing AGC
to be roughly a plane triangle which does not involve any
appreciable error.
AC = AG tan AĜC = (β × H sin AĜC) / (H cos AĜC)
= (β × Āyanavalanajyā) / Yaṣṭi since Yaṣṭi is defined as
H cos AGC in verse 3.
Note. The point C is called the Kṛta-Āyana-Dṛk-
Karma Sthāna ie. the point of the Ecliptic signifying the
planetary position rectified for the Samskāra or correction
called Āyana-Dṛk-karma. The longitude of C thus got
is called the polar longitude of the planet which is defined
as the longitude of the planet as measured on the Ecliptic
upto the point of intersection of the planet’s declination
circle with the Ecliptic. Bhāskara mentions elsewhere,
“ नक्षत्राणां स्फुटा एव स्थिरत्वात् पठिताः शराः
दृक्कर्मणाऽऽयनेनैषां संस्कृताश्च तथा ध्रुवाः ”
ie. in the case of stars which are fixed, the Sphuta-saras
or polar latitudes (given by GC in the above case) are

462
given and longitudes rectified for Āyana Dṛk-karma are
given ie. polar longitudes. Bhāskara made this as an
approximate statement, for, we know, even in the case of
stars, though they be fixed, their polar latitudes also do
change by the precession of equinoxes be it just a little;
whereas their polar longitudes do not change by the same
constant quantity as their celestial longitudes.
Verses 6, 7 and 8. Obtain the Carakhandas or
ascentional differences of the Sphuta and Asphuta Krāntis.
If they be of the same direction their difference is to be
taken, if of opposite direction, their sum is to be taken.
The result in asus gives the Ākṣa Dṛk-karma correction if
the celestial latitude is of appreciable magnitude. If it
be not appreciably large, it (the celestial latitude) is to be
multiplied by the Ākṣavalana, then divid by H cos ϕ or
what is the same, multiplied by the equinoctial shadow
and divided by 12. The result is to be multiplied by the
radius and divided by H cos δ. Then we have the Ākṣa
Dṛk-karma correction in asus. Assuming the planet’s
position corrected for Āyana Dṛk-karma to be the Sun,
obtain the lagna using the asus of the Ākṣa-Dṛk-karma.
If the planet has a southern latitude let the lagna be found
in the positive direction; otherwise in the negative dire-
ction. Then we have the rising lagna of the planet or
what is the same the longitude of the rising point of the
planet. Again assuming the planet’s position increased
by 180° to be the Sun, and using the asus of Ākṣa Dṛk-
karma, obtain the lagna in the positive direction with
respect to a northern latitude of the planet or else in the
negative direction. The result gives the longitude of the
setting planet or what is called the Asta-lagna.
Comm. (Refer to fig. 110) Let p be the planet whose
Graha-sthāna or foot of the latitude is p′. Let q be what
is called the Kṛta-Āyana-Dṛk-Karmaka-Sthāna or the
planet’s position corrected for the correction of Āyana

463
Fig. 110
Dṛk-karma. (This correction is given by the arc p'q).
The difference of the rising times of the planet p and q
expressed in asus is what is sought here. The planet p
rose at a and has covered the arc ap of the diurnal circle
after rising. So, we have to compute ap and therefrom
the corresponding arc of the Equator namely AC. If the
planet has no latitude ie. is situated on the ecliptic at p',
the arc p'q or Āyana Dṛk-karma vanishes, for, the Āyana
Dṛk-karma is no other than the projection of the latitude
of the planet on the ecliptic taking P or celestial pole as
the vertex of projection. Also if there were no Akṣa ie.
if the place be equatorial, p and q rise simultaneously ie.
the planet will rise along with the point called Kṛta-
Āyana-Dṛk-Karmaka-Sthāna q, mentioned above. In fact
the Dṛk-karma corrections Āyana and Ākṣa are contem-
plated to obtain the difference in the rising times of p and
p' ie. the planet and its position on the ecliptic. This
time is resolved into two parts (i) the difference of the
rising times of p' and q and (ii) that of the rising times
of q and p. The former is given by the equatorial arc
BC which goes by the name Āyana-Dṛk-karma and the
latter by the arc AC which goes by the name Ākṣa-Dṛk-
karma. The algebraic sum of these two corrections gives
the difference of the rising time of p and p' ie. the time
given by the arc AB. Here AB=AC-BC, In verses

464
4, 5 we have seen how BC is computed. Here we are
seeking to compute AC through the corresponding arc ap
of the diurnal circle.
The verse uses the word Sphuta-Krānti, Asphuta-
Krānti, and their Cara-khandas. Here in the figure pC
is the Sphuta-Krānti and p'B is the Asphuta-Krānti or
simply Krānti. Also pp' is called Asphuta-Vikṣēpa and
bp' Sphuta-Vikṣēpa. Sphuta-Krānti = pc = bB = bp' + p'B =
Sphuta-Vikṣēpa + Asphuta-Krānti. In other words the
declination of p, the planet, is called Sphuta-Krānti and
that of its longitudinal position on the ecliptic namely p'
is called Asphuta-Krānti or simply Krānti. We have
obtained Sphuta-Krānti from Krānti by adding to the
latter the Sphuta-Vikṣēpa. The method of obtaining
Sphuta-Vikṣēpa from the Vikṣēpa, otherwise called Vikṣēpa
Sphuta-karama was dealt with in verse 3 above.
Now we have to define the Cara-khandas pertaining
to the Sphuta-Krānti pC and the Asphuta-Krānti p'B.
The former is defined as AE and the latter by CE very
approximately. We say very approximately because when
p' comes to the horizon, the Cara-khanda will not be
exactly EC but a little more or less than EC since
Bp' ≠ Cq. In other words, the arc of the equator between
the declination circle of q while rising and the equatorial
horizon ie. AE is defined as the Cara-khanda of pC ;
similarly CE is that of p'B. Their difference is
AE—CE = AC. If we use the modern notation, we have
to take their algebraical difference. If that be done, the
alternative case of adding the Cara-khandas when the
Sphuta and the Asphuta-Krānti are of opposite directions,
will be automatically implied. The asus pertaining to
this arc AC will be the correction of Ākṣa-Dṛk-karma as
mentioned in verse (6) when the latitude of the planet is
appreciable. These asus will be equal to the time taken by
p the planet to cover the arc ap or what is the same, the

465
time in between the moment of rising of p and that of q,
the position of the Kṛta-Āyana-Dṛk-Karmakagraha.
When the latitude is very small, an approximate
estimate of this correction is given in asus as
(β' × H sin ξ) / (H cos ϕ) × R / (H cos δ) or (β' × H sin ϕ) / (H cos ϕ) × R / (H cos δ) .
That this formula is (β'=qp) only approximate is seen
from the fact that H sin ξ ≉ H sin ϕ. This second
formula can be proved as follows.
Ākṣavalana in asus = (β' × H sin ϕ) / (H cos ϕ) × R / (H cos δ) where
β' is the Sphuṭa-Vikṣēpa. Taking qpa as a plane triangle
pa = qp tan p̂qa = qp × tan P̂En approximately
= Sphuṭa-Vikṣēpa × (H sin ϕ) / (H cos ϕ) . Hence
AC = (β' × H sin ϕ) / (H cos ϕ) × R / (H cos δ) . With respect to the first
formula ie.
Akṣavalana in asus = (β' × H sin ξ) / (H cos ϕ) × R / (H cos δ) '
pa = qp tan p̂qa = (qp × H sin p̂qa) / (H cos p̂qa) . Here instead
H cos p̂qa which is Yaṣṭi previously defined, H cos PEn
ie. H cos ϕ, an approximate value is substituted, because
the latitude is small. Having obtained the value of pa,
it is then reduced to the Equator by multiplying by R and
dividing by H cos δ. The convention with respect to
the sign is clear. The idea of Kramalagna and Vilōma-
lagna may be elucidated as follows. In fig. 110, q, is on
the horizon rising whereas p had already arisen. So to
· 59

466
obtain the rising time of the planet p, which occured
before that of q, Vilōmalagna or an anterior point of time
is to be computed, and this is done by subtracting the asus
of Akṣa-Dṛk-karma from the rising time of q. The rest
follows on these lines of elucidation.
Verse 9. The planet rises at the time known as its
Udayalagna (which is to be computed as aforesaid) and
it sets at the time known as its Astalagna (which is also
to be computed as mentioned before).
Comm. Clear.
Verse. 10. During the night, at a given moment, the
computed Udayalagna of the planet is less than the
particular lagna of the moment, ie, if the longitude of the
Udayalagna is less than that of the current lagna, and
also if the Astalagna of the planet computed is greater
than that of the then current lagna ie. if the longitude of
the Astalagna is greater than that of the current lagna,
the planet is visible ie. above the horizon. In the case
of the Moon, however, if he is not eclipsed by the rays of
the Sun, he may be visible even shortly after Sunrise or a
little before Sunset in contradistinction to a planet which
could not be seen at all during day time (except perhaps
Venus). When a planet is visible, his gnomonic shadow
could be computed. (This does not mean that the gnomon
casts a shadow of the planet but means that its zenith-
distance could be computed according to the methods
described in Tripraśnādhikāra.
Comm. Clear.
Verse 11. To obtain the shadow, the time that has
elapsed after the rise of the planet is to be known.
If it be required to find the gnomonic shadow of the
planet, then the current lagna and the Udayalagna of the

467
planet at the moment are to be computed. The time in
between the two lagnas, which will be in Sāvana measure
pertaining to the planet gives the time that has elapsed
after the rise of the planet.
Comm. According to computation, the difference of
times of the current lagna and the Udayalagna as com-
puted taking the present position of the planet will be the
time measured along the diurnal path of the planet and as
such is in Sānava measure.
Bhāskara mentions here a very subtle point. In the
analysis the latitude of the planet is taken into account
while finding the Udayalagna. Time measured in this
case is called Kṣetrātmika and not Kālātmika for the
following reason. Suppose A is the Udayalagna of the
planet and B the current lagna both points being on the
ecliptic. Let us say that the current lagna is posterior to
the Udayalagna (there is no loss of generality in supposing
this). By the time of the current lagna, the planet is no
more at A but will have moved a little towards B. Let the
present position of the planet on the ecliptic be A'. The
corresponding arc of AB on the equator gives the sidereal
measure of the time that has elapsed after the rise of the
planet whereas the corresponding arc of A'B on the
equator gives the Sāvana measure of the same time. Here
this Sāvana is that pertaining to the planet and not that
pertaining to the Sun which is Saura Sāvana. In other
words it is the Sāvana pertaining to the particular planet,
because the arc AA' is traversed by the planet in question
as per its own velocity.
The stress is here on the word Tat-Kāla-graha, ie.
the longitude of the planet at the moment at which the
shadow or the zenith-distance of the planet is sought.
Taking this as the position of the planet and taking the
latitude of the planet also into account compute the
Udayalagna. This will be A' cited above. To obtain the

468
sidereal measure of the same time we have to take A and
not A'.
Verse 12. Sāvana measure alone is to be employed while
finding the gnomonic shadow ie. while the zenith-distance
of the planet is to be computed, because the arc of the
diurnal circle of the planet indicates only Sāvana measure.
Suppose the Udayalagna falls short of the current lagna,
then the Bhōgyakāla ie. the remaining rising time of the
Rāśi in which the planet is situated added to the elapsed
time of the Rāśi of the current lagna together with the
sum of the rising times of the Rāśis in between gives the
difference of the Udayalagna of the planet and the current
lagna.
Comm. (Ref. fig. 111). Let p be the planet's place
on the Ecliptic at the moment in question when the lagna
is L (or the foot of the latitude of the planet
in case it is not on the ecliptic). Let PA be
the remainder of the Rāśi in which the
planet is situated. Then the time of rising
of the arc PA is here termed Bhōgyakāla.
Let DL be the arc of the Rāśi in which the
current lagna L is situated, which has arisen.
The time of rising of this arc DL is called
Bhuktakāla. The times of risings of the
Rāśis in between namely AB, BC, CD are
called Madhyōdayās. The sum of the rising times of
PA, AB, BC, CD, DL gives the time in between the rise
of P and that of L which is the time that has elapsed after
the planet has arisen upto the current Lagna ie. the rising
time of L.
Verse 13. The method of computing the gnomonic
shadow of the planet or what is the same the zenith-
distance of the planet, as per the method of computing
that of the Sun (extended to the case of the planet) after

469
finding the Sphutakrānti (ie. modern declination) of the
planet.
The Krānti of the planet or the declination of the foot
of the latitude of the planet added to the latitude rectified
called Sphuṭasāra, gives what is called Spaṣṭakrānti of the
planet and its Hsine is called Spaṣṭa-Krāntijyā. From
this H sin δ, H cos δ etc. is to be computed (as mentioned
in Triprasnādhikāra in the context of finding the zenith-
distance of the Sun). From the time that has elapsed
after the rise of the planet called the Unnata, the shadow
is to be computed as in the case of the Sun’s shadow.
Having thus computed the shadow or what is the same the
zenith-distance of the Moon or that of the stars, the instru-
ment called Nalaka could be pointed to the spot where
that celestial body is situated.
Comm. The words Krānti, Spaṣṭakrānti were
explained before. Also the method of calculating the
zenith-distance from a knowledge of the declination was
described before in Triprasnādhikāra.
Verses 14, 15. Consideration of horizontal parallax
in the case of visibility of the Moon or planets.
H cos z or what is called the Mahāśaṅku of the Moon
or planet is to be reduced by 1/15th of the respective
daily motion gives the visible Sanku when the radius is
taken to be 3438. If the radius is taken as 120, 1/450th of
the daily motion is to be subtracted. If H cos z is less
than 1/15th (or 1/450th) of the daily motion, then the Moon
is not visible. This applies to the other planets as well,
but as this is a neglible matter, [the earlier Āchāryas did
not suggest this.
Comm. In the context of the subject of parallax, we
mentioned that in the case of the Moon, the horizontal
parallax happens to be 1/15 of the Moon’s daily motion

470
approximately. The same is extended to the case of the
planets as well. To get the proportion in the case of
taking the radius to be 120 only instead of 3438, rule of
three is applied as follows. “If the radius be 3438, 𝑣/15
is parallax, what will it be if the radius be 120?” The
answer is
𝑣/15 × 120/3438 = 𝑣 / (3438/8) = 𝑣/430 very approximately as stated.
If H cos z be less than this 1/15 th or 1/430 th of the daily
motion, the parallax does not allow the Moon to be seen.
Bhāskara specifically talks about the Moon only because,
the earlier Āchāryas did not apply the correction of
parallax to the case of the planets, because it is not
appreciable.
Verse 16. If an operation is neglected because its
effect is not appreciable, or is not of much use, or because
it is apparent, or if it implies great labour, or again if
it implies a lot of exposition that would make the text
unduly voluminous, ignoring the necessity of that oper-
ation should not be treated as wrong.
Comm. Here Bhāskara upholds the earlier Āchāryas
not stipulating parallax in the case of planets, because it
is not appreciable.
Here ends Graha-cchāyādhikāra.

GRAHŌDAYĀSTĀDHIKĀRA
Verse 1 and first half of verse 2.
The Udayalagna of a planet is termed Prāk-Dṛk-graha
and the Astalagna is termed the Paschima-Dṛk-graha.
If the Prāk-Dṛk-graha happens to be less than the current
lagna the planet had already risen. If it be greater, the
planet is still to rise. Similarly if the Paschima-Dṛk-graha
is less than the current lagna, the planet had already set ;
otherwise is yet to set.
Comm. The words Prāk-Dṛk-graha and Paschima-
Dṛk-graha are coined to indicate the points of intersection
of the ecliptic with the eastern and western horizon
respectively when the planet is rising or setting, when the
planet has latitude. When the planet has no latitude
Prāk-Dṛk-graha coincides with the rising planet and the
Paschima-Dṛk-graha with the setting planet. When the
planet has a latitude, the foot of the latitude, which
signifies the planet’s position on the ecliptic differs from
the two Dṛk-grahas. In this case ie. when the planet has
a latitude the Prāk-Dṛk-graha’s longitude will be less than
that of the planet’s longitude whereas the Paschima-Dṛk-
graha’s longitude will be greater than that of the planet.
If x and y be the differences of the longitudes of the planet
and those of the Dṛk-grahas respectively, it is clear from a
figure that tan x = (tan β) / (tan θ) and tan y = (tan β) / (tan ϕ) where β is the
latitude of the planet and θ and ϕ the angles which the
ecliptic makes with the horizon at the planet’s rising and
setting respectively.
Since θ and ϕ are then respectively the zenith-
distances of the pole of the ecliptic in each case, and since

472
in the course of half a sidereal day, these zenith-distances
could not be equal, x and y cannot be equal.
Second half of verse 2 and verse 3. To find the side-
real time in between the given time and the time of the
planet's rising.
Find the time that has elapsed after the planet's rise,
from a knowledge of the Iṣṭalagna at a given time and
Prāk-Dṛk-graha ie. the longitude of the rising point of
the ecliptic at the time when the planet rises.
This time will be in Sāvana measure, because we have
taken the planet's position at the given moment and not
the position of the rising planet. From this time, know-
ing the daily motion of the planet of the day in question,
obtain the arc; that would have been traversed in between
the moments. Subtracting this arc from the planet's
position at the moment, the planet's position at rising
would be obtained approximately; approximately, because
as per the procedure enunciated in verse eleven, Graha-
cchāyādhikāra, the Prāk-Dṛk-graha of the rising planet,
and that obtained from the position of the planet at the
moment differ. From this approximate time again com-
pute the arc that would have been traversed by the planet
during that time. Subtracting this arc which is nearer
the truth from the planet's present position, we obtain a
more approximate position of the rising planet. Again
computing the Prāk-Dṛk-graha from this position, calcu-
late the time from this Prāk-Dṛk-graha and the lagna of
the moment. Repeating the process we will obtain
the actual sidereal time in between the given time and
the actual rising time of the planet. It is sidereal because,
we have found the time as per the procedure of verse (11)
referred to, where we have used तत्कालखेटोदय and not the
actual ,खेटोदय, i.e the rising lagna of the planet obtained
from the present position and not that of the rising planet.