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

429
where that method of successive approximation is
circumvented and where by the presence of K in the place
of R, analogy is with the eccentric method of formulation
of Śīghraphala and not that of Mandaphala. Also
K² = E₂M² = E₂L² + ML² = (E₂K–LK)² + ⊙K² =
(E₂K–M⊙)² + ⊙K² = (H cos KÊ₂ ⊙ –Para)² +
H sin² KÊ₂ ⊙.
But if L be the lagna of the moment L⊙ = 90 – V⊙
so that H cos KÊ₂ ⊙ = H sin ⊙L and H sin KÊ₂ ⊙ =
H cos ⊙L ∴ K² = (H sin ⊙L–Para)² + H cos² ⊙L as
formulated in the verse.
Fig. 95 is in the plane of the Ecliptic. The parallax
in the vertical circle is projected on to the plane of the
Ecliptic by taking 4/R H cos ZV as the Para, and deriving
the parallax in longitude from this Para.
Now, the doubt arises as to why the Śīghrocca is
not taken to coincide with the Vitribha but is taken as
removed 180° therefrom.
Verse 10. H sin ZV (of fig. 92) is called the Dṛk-
kṣepa of the Sun, which is considered to be north in case
the northern declination of the Vitribha is greater than
ϕ the latitude, otherwise south.
Comm. Let in fig. 96, AV be the Ecliptic whereof
A is the ascendant or Lagna and V the Vitribhalagna.
Let EQR be the celestial Equator. Let δ be the decli-
nation of the Vitribhalagna. Then if δ > ϕ. then ZV, the
arc of the the Dṛk-kṣepa, (H sin ZV being defined as the
Dṛk-kṣepa) as well as H sin ZV are considered to be north.
Thus in fig. 96, it is north whereas in fig. 97 it is south.
(In fig. 97, r is shown outside the celestial sphere, signi-

430
fying that r is in the western hemisphere and is brought
into view for clarity).
Fig. 96 Fig. 97
Verse 11 and first half of verse 12. Then the sum
of ZV and the latitude of V assuming V to be the Moon,
or the difference of the above two, as the case may be,
according as both of them are north or of opposite
directions, gives the arc whose Hsine is the Dṛk-kṣepa
of the Moon, The Dṛk-kṣepas of the Sun and the Moon
multiplied respectively by 1/15th of their daily motions and
divided by the radius R (equal to 3438′) are the parallaxes
of the Sun and the Moon in latitude. The sum or
difference of these parallaxes according as they are of
opposite or the same direction, is the true parallax in
latitude in the context of a solar eclipse.
Comm. The true parallax in latitude sought above is
the relative parallax of the Sun and the Moon in latitude.
Suppose in fig. 92, VB is the parallax in latitude pertain-
ing to the Sun and VB′ that pertaining to the Moon; then
BB′ is the relative parallax, the difference being taken in
this case because both are of the same direction.
Parallax in latitude namely CD in fig. 92, we saw
equal to VB which is equal to (4H / R) sin ZV, In other words

431
the parallax in latitude either of the Sun or the Moon is
equal to 4/R H sin of the zenith-distance of the respective
Vitribhalagna wherever the Sun or the Moon be situated
in their orbits namely the Ecliptic or the Vimandala. Let
H sin ZV be the Dṛk-kṣepa of the Sun, V being the
Vitribhalagna pertaining to the Sun and let H sin Zv
be the Dṛk-kṣepa of the Moon where v is the Vitribha-
lagna of the Moon. (Ref. figures 98 and 99) Let K' be
the pole of the Vimandala and vv₂ the latitude of v. Since
v and V are in the proximo, the latitude of v may be
taken to be very nearly equal to the latitude of V so that
ZV ± latitude of V is very nearly equal to Zv. In fig. 98,
ZV—latitude of V is very nearly equal to Zv because both
ZV and latitude of V are of the same direction. In fig. 99
ZV+latitude of V is very namely equal to Zv because both
arc of opposite direction. Thus Zv = ZV ± latitude of V
approximately and H sin ZV and H sin Zv are the Dṛk-
kṣepas of the Sun and the Moon respectively. Having
got these Dṛk-kṣepas 4/R × Dṛk-kṣepa gives the nati in
each case ie. the parallax in latitude and the sum or
difference of these natis as mentioned in the beginning of
the commentary of this verse gives the relative parallax
of the Moon with respect to the Sun which is called the
Fig. 98 Fig. 99

432
true parallax in latitude. This true parallax in latitude
increases or decreases the latitude of the Moon at the
moment of conjunction as is going to be mentioned in the
latter half of verse 14.
In deriving the parallax in latitude from the respe-
ctive Dṛk-kṣepas, instead of using the formula 4/R Hsine
(Dṛk-kṣepa) which is an expression in time, it is sought
to express the same in arc because the latitude of the Moon
is expressed in arc and we have to take the sum or differ-
ence of the latitude and the parallax in latitude to obtain
the apparent latitude of the Moon at the moment of
conjunction. In the case of the parallax in longitude we
sought to express the same in time because the moment
of apparent conjunction was sought therefrom.
Latter half of verse 12 and first half of verse 13.
An approximate method of obtaining the relative parallax
in latitude of the Moon with respect to the Sun.
The Hsine of the zenith-distance of the nonagesimal
pertaining to the Moon or what is called the Moon's
Dṛk-kṣepa multiplied by 2 and divided by 141, gives the
relative parallax in latitude of the Moon with respect to
the Sun ; or working with the smaller table of Hsines
(where R is taken to be 120) the Moon's Dṛk-kṣepa being
multiplied by 2 and divided by 5 and the result being
increased by 1/60th of itself gives approximately the relative
parallax in latitude.
Comm. Herein, the Vitribha or the nonagesimal of
the Sun is taken to coincide with that of the Moon. In
other words the Dṛk-kṣepas (the Hsines of the zenith-
distances of the nonagesimals, of both the Sun and the
Moon are taken to be identical. Then using the following
proportion "If by a Dṛk-kṣepa equal to R, the relative
parallax in latitude is equal to 1/15th of the difference of the

433
daily motions namely 48′-46″, what will it be for an
arbitrary Dṛk-kṣepa ?" We have
(D × 48′-46″) / 3438 . Converting 48¾ / 3438 ie. 195 / 13752 ie. 65 / 4584 into a
continued fraction, this will be equal to
1/(70+) 1/(1+) 1/(1+) 1/(10+) 1/3 of which a very approximate convergent
is 1/71 as taken by Bhāskara.
If the radius be taken to be 120, the coefficient of D
will be 48¾ / 120 = 195 / 480 = 13 / 32 = 1/(2+) 1/(2+) 1/6 = 2/5 very
approximately.
Latter half of verse 13 and first half of verse 14.
An easy method to compute the parallax in longitude and
latitude.
Taking the Dṛk-ṣepa of the Moon as well as the Sun
to be the Hsine of the meridian zenith-distance of the
Vitribha and the H cosine of its meridian zenith-distance
as the Vitribha-Sanku, the parallaxes in latitude and
longitude could be got from them respectively.
Comm. Parallax in longitude is computed from the
Vitribha-Sanku, whereas parallax in latitude is computed
from the Dṛk-kṣepa or the Hsine of the zenith-distance
of the Vitribha. Thus for both the purpose the Vitribha's
position is important, whose zenith-distance and altitude
give respectively the parallax in latitude and longitude.
Since in practice it is a little cumbrous to obtain the
Vitribha's altitude and zenith-distance, an approximate
procedure is suggested. Obtaining the declination or the
Sphuṭa-krānti of the Moon taking him to coincide with
the Vitribha by the method described in verse 3 of the
Graha-cchāyādhikāra, and using the formula z+δ=ϕ, the
meridian zenith-distance of the Vitribha can be got. This
may be assumed to be the Dṛk-ṣepa approximately. The
55

434
complement of the meridian-zenith-distance may be
assumed to be the Vitribha-Sanku approximately. Then
the parallaxes in latitude and longitude could be computed
respectively from the two as described before.
The following figure gives a particular nomenclature
that was in the mind of Kamalākara, the author of
Siddhāntatattvavivēka.
[चित्र: Fig. 99-A]
Madhya-Mandala
Dṛk-mandala
Garbha-chinha Mandala
Dṛk-chinha Mandala
Para-Mandala
Krāntimandala
Krāntisadrasamandala
Fig. 99-A
Latter half of verse 14. The purpose of obtaining
the parallax in latitude.

435
The apparent latitude of the Moon is equal to the
algebraic sum of his geocentric latitude and the parallax
in latitude. From this apparent latitude are to be calcu-
lated the Sthiti-khanda and Marda-khanda of the solar
eclipse (by the method described in the chapter on lunar
eclipse, taking the eclipsing body or grāhaka to be the
Moon and the eclipsed or Grāhya to be the Sun).
Comm. The geocentric parallax of the Moon has a
double effect on the occurence of a solar eclipse as men-
tioned before. If the parallax be resolved along the
Ecliptic and along a secondary to the Ecliptic, we have
respectively the parallax in longitude and that in latitude.
The parallax in longitude makes the apparent moment of
conjunction at a given place, differ from the moment of
the geocentric conjunction, whereas the parallax in latitude
makes the magnitude of apparent latitude of the Moon at
the place of observation differ from that of the geocentric.
The apparent latitude is equal to the sum or difference
of the geocentric latitude and the parallax in latitude.
Having got the apparent latitude, the computation of the
Sthiti and Marda-khandas could be done according to the
method described in the chapter on lunar eclipse.
For a point C on the surface of the Earth, a solar
eclipse occurs if the latitude of the Moon be less than MD
Fig. 100

436
(vide fig. 100) where M is the centre of the Moon and MD
the latitude of the Moon at the point of first contact
MD = MB + BD = m + BÊD where m is the semi-
diameter of the Moon's disc. But
BÊD = BEA + AÊS = CB̂E − CÂE + AÊS = P − p + s
where P and p are the parallaxes of the Moon and the Sun
and s the angular semi-diameter of the Sun. Thus in
order that a solar eclipse may be possible for some point
of the Earth, the latitude of the Moon at the moment of
conjunction must be less than P + s + m − p
= 57' + 16' + 15' = 89' approximately. The lesser the
northern latitude of the Moon at the moment of conjun-
ction, more places situated on the surface of the Earth
between C and F will have solar eclipse where F is the
sub-solar point ie. the point of the Earth which has the
Sun in the zenith at the time of conjunction. Similarly,
if the southern latitude of the Moon is less than 89' at the
moment of conjunction the places situated on the Earth
between G and F will have solar eclipse. In particular
the sub-solar point F will have solar eclipse if the latitude
of the Moon at the moment of conjunction is less than
HD ie. less than s+m ie. 33' approximately. The sub-
solar point will have no parallax, so that the terms P and p
in P+s+m−p vanish. For the other points ie. points
between F and C or G parallax will be there and the
latitude may be greater than s + m but less than
s+m+P−p to have an eclipse. A latitude of 33' corres-
ponds to a distance of (33 × 15) / 70 = 99 / 14 = 7 1°/14 of the Sun
with respect to a node. Thus if at the moment of conjun-
ction, the latitude of the Moon be less than 7°, even the
sub-solar point must have an eclipse.
Verses 15, 16, 17. To find Spars'akāla, Mokṣakāla,
Sammīlanakāla and Unmīlanakāla.

437
First compute the time called Sthiti-khanda (as
mentioned in the chapter on lunar eclipses). The ending
moment of local Amāvāsyā or what is called the moment
of local conjunction is known as the Madhya-Grahakāla
or the moment of the middle of the eclipse. Subtract the
Sthiti-khanda from the computed time of Geocentric
conjunction ; the result will be the approximate Sparśa-
kāla. This has to be rectified for parallax in longitude
as well as the approximate Madhyagrahakāla of geocentric
conjunction to obtain the local Sparśakāla and the local
Madhyagrahakāla ; Similarly the Mokṣakāla, the Sammī-
lana and the Un-mīlanakālas are to be rectified for
parallax in longitude. But while effecting this correction
for the parallax in longitude, the Moon's latitude also
differs for the corrected time which in turn effects the
durations of Sthiti-khanda, Mokṣa-khanda etc. Correcting
the first computed Sthiti-khanda. Mokṣa-khanda etc. for
this variation in the latitude, and subtracting the Sthiti-
khanda from the time of Madhya-graha, we have a better
approximation for the Sparśakāla. In as much as parallax
in longitude, that in latitude, and the Moon's latitude vary
from time to time, and the times of Sparśa, Madhyagraha
etc. are effected by them, the process of computation
proceeds by the method of successive approximation.
Subtracting the rectified Marda-khanda from the rectified
Madhyagrahakāla, we have the true Sammīlanakāla ;
similarly adding the former to the latter we have the true
Un-mīlanakāla.
If, as mentioned before in verse 9, the parallax in
longitude is found without using the method of successive
approximation, the Sparśa-kāla and the Mokṣa-kāla are
had at once. But the latitude of the Moon and the parallax
in latitude are to be computed using the then longitudes
of the Moon and the non-agesimal.
Comm. Clear.

438
Verses 18, 19. To obtain the true values of the Bhuja
and the Iṣṭakāla.
The remaining work proceeds on the lines indicated
in the chapter on ‘lunar eclipses’ (ie. the computation of
the Bimbavalana, Bhuja, Koti and the like is to be done
as indicated there). The Bhuja will be rectified by multi-
plying it by the Sthiti-khanda obtained by adopting the
latitude of the Moon effected by parallax in latitude and
divided by the Sthiti-khanda rectified for parallax in
longitude. Similarly given the grāsa ie. the magnitude
of the eclipse, the result found before by verse 15 in the
chapter of lunar eclipses, is to be multiplied by the Sthiti-
khanda rectified for parallax in longitude and divided by
that obtained adopting the latitude of the Moon effected
by parallax in latitude, and the result so obtained being
subtracted from the Sthiti-khanda, we get the Iṣṭa-kāla.
Comm.

439
parallax in latitude, so that it is a variable. We are to
make a correction for this variability, both in the com-
putation of Sthiti-khanda, Bhuja and Iṣṭa-kāla. The
formula for the Sthiti-khanda is √(R+r)² - β² / (m₁ - s₁) where β
is the latitude of the Moon at the moment of conjunction.
The value of MN at the moment of conjunction will not
be equal to its value at any intermediate point because
parallax in latitude differs from position to position of the
Moon. In other words β is variable. The formulae given
for the rectification of the Bhuja B or the Iṣṭakāla I are
B' = (B × T') / T and I' = T' - √(R+r - g)² - β² / (m₁ - s₁) × T / T' where
T' is the Sthiti-khanda rectified for the variability of β,
B' is the Bhuja rectified for the same whereas T and B are
the values of the Sthiti-khanda and Bhuja computed
taking the effect of parallax in longitude above.
The effect of parallax in longitude is to prepone or
postpone the moment of first contact as well as that of
conjunction. The verse under commentary uses two terms
Sphuta-Sthiti-khanda and Sphuteshuja-Sthiti-khanda.
The former is the Sthiti-khanda rectified for parallax in
longitude whereas the latter is that rectified for parallax
in latitude ie. by adopting β' instead of β in the formula
√(R+r² - β²) / (m₁ - s₁) where β' = β ± effect of parallax in latitude.
Suppose on account of parallax in longitude the
moment of first contact t₁ becomes t₁ + δt₁, and let the
moment of conjunction t₂ become t₂ + δt₂. Then the
Sthitikhanda unrectified for parallax in longitude will
be (t₂ - t₁) whereas that rectified for parallax will be (t₂ - t₁)

• (δt₂ - δt₁). This rectified Sthitikhanda is called
  Sphuta-Sthiti-khanda. Now the verse under commentary
  gives a procedure to rectify the Bhuja, and Iṣṭakāla in the

440
wake of β being effected by parallax in longitude. The
formula for Bhuja is √(R+r -g² — β²) / (m₁ — s₁) wherein all quantities
except β may be taken to be constant. Suppose β effected
by parallax be comes β'. If β' > β, Bhuja will decrease ;
also the Sthiti-khanda whose formula is √(R+r)² — β² / (m₁ — s₁)
decreases if β' > β. Hence if T' be the new value of T
the Sthiti-khanda, T' < T. In other words when β' > β,
B' < B and T' < T. Also if β' < β, both B' and T' will
be greater than B and T respectively. Hence as a rough
measure B and T are taken to vary together positively or
negatively and as such proportionally. Though both
increase or both decrease together, strictly speaking the
concept of proportionality is there ; but roughly speaking
they are taken to vary proportionally which means
B' = (B × T') / T . The fact that proportionality is not there
could be seen in two ways. B = (T — I) (m₁ — s₁) (1) with
usual notation so that taking B and T to vary on account
of the variation in β, δB = δT (m₁ — s₁), I not varying so
that B + δB = B' = (T + δT) (m₁ — s₁) — I (m₁ — s₁) =
T' (m₁ — s₁) — I (m₁ — s₁) = (T' — I) (m₁ — s₁) (2).
Dividing (1) by (2) B/B' = (T — I) / (T' — I) which will be
approximately equal to T/T' provided I is very small
compared with T. Assuming so, B/B, could be taken to
be equal to T/T', which means B' = (B × T') / T as mentioned
in the verse. Or again, considering the formulae for B
and T and differentiating them with respect to β and
getting B' and T', we shall have the following working.
B² = ((R+r—g)² — β²) / (m₁ — s₁) so that 2 BδB = (-2 βδβ) / (m₁ — s₁)

441
or δB = -βδβ / B (m₁ - s₁) ; similarly T² = (R + r)² - β² / m₁ - s₁ so that
2T δT = -2βδβ / m₁ - s₁ so that δT = -βδβ / T (m₁ - s₁)
∴ B¹ = B + δB = B - βδβ / B (m₁ - s₁) and T¹ = T + δT =
T - βδβ / T (m₁ - s₁)
∴ B¹ / T¹ = (B² - βδβ / B (m₁ - s₁)) × (T (m₁ - s₁) / T² - βδβ) = (B² - βδβ) T / B (T² - βδβ)
= (B - βδβ / B) / (T - βδβ / T) . Since βδβ / B ± βδβ / T
B¹ / T¹ ± B / T .
Regarding the finding of Iṣṭakāla when g the grāsa
is given, we have the formula
T - I = B / m₁ - s₁ = √(R+r-g)² - β² / m₁ - s₁ so that putting T - I = t
t = √R+r-g² - β² / m₁ - s₁ ∴ As β increases t decreases so that
T - t increases ie. I increases. Whereas as β increases T
decreases. This means in a way that as T decreases. I
increases so that I is taken to be inversely proportional
to T ie. I' is taken to be (I × T) / T' as given.
Here also, it could be seen that the inverse propor-
tionality is not strictly there ; for β increasing both t and
T decrease so that T - t ie. I will increase only if the
decrease in t is greater that in T. But t²
56

442
= (R+r-g)²-β² / (m₁-s₁)² so that δt = -βδβ / (t (m₁-s₁)²)
and T² = (R+r)²-β² / (m₁-s₁)² so that δT = -βδβ / (T (m₁-s₁)²)
Thus we see that as β increases both t and T decrease
on account of the minus signs in δt and δT. Also if I
were to increase, βδβ / (t (m₁-s₁)²) should be greater than
βδβ / (T (m₁-s₁)²) ie. 1/t must be greater than 1/T ie.
T should be greater than t and this is so. Hence as
β increases t and T decrease but T-t increases ie. as T
decreases T-t increases ie. I increases. So, roughly I is
taken to be inversely proportional to T so that I' = (I × T) / T'
If strict inverse proportionality is there, since,
t + δt = t' = t - βδβ / (t (m₁-s₁)²)
and T + δT = T' = T - βδβ / (T (m₁-s₁)²) and therefore
t'/T' = (t/T) · { 1 - βδβ / (t² (m₁ - s₁)²) } / { 1 - βδβ / (T² (m₁ - s₁)²) } = (t/T) · { t² - βδβ / (m₁ - s₁)² } / { T² - βδβ / (m₁ - s₁)² }
t² must be roughly equal to T² which means T - t = I
must be small. In other words the formula of the verse
holds good only for small I's or Iṣṭakālas.
In fact this process of rectification of the Bhuja was
first given by Brahmagupta in verses 18 and 19 of Sūrya-
grahaṇādhikāra and accepted by Śrīpati in verse 14.
The verse 19 under Lunar eclipses in the Sūrya-
siddhānta also seems to indicate this process but uses the
words Madhya-Sthiti-khaṇḍa and Spaṣṭa-Sthiti-khaṇḍa in

443
an ambiguous sense. Ranganātha in his commentary
takes the Madhya-Sthiti-khanda to mean that rectified
for variation in β, and Spaṣṭa-Sthiti-khanda to mean that
rectified for parallax. Some modern traditional commen-
tators, for example pandit Sitha Ramajha commenting on
the verse of the Sūryasiddhānta and the commentator of
the verses of Brahma Sphutasiddhānta published under
the editorship of Acharya Rama Swarupa Sharma, have
ignored the variation in β and confined themselves only
to the effect of parallax in longitude.
Yet one more proof could be adduced in this behalf.
Let D, be the moment of geocentric conjunction, t the
time for a given grāsa in between D and T the moment of
first geocentric contact. Then the times D, t and T are
to be corrected both for parallax in longitude and that
in latitude. Take the geocentric Sthiti-khanda, and
geocentric Bhuja as the mean-values T and B and those
corrected first for parallax in longitude as the True values
T' and B'. Also take the time t corrected for parallax in
longitude to be t'. We could write T' - T = δT, t' - t = δt
and B' - B = δB. In the above, consider B expressed in
time. Then D - T = S the mean Sthiti-khanda, D - t = B
the mean Bhuja, D' - T' = S', the true Sthiti-khanda recti-
fied for parallax in longitude, D' - t' = B' the Bhuja also
rectified for the same. The δB = δD - δt, δI = δD - δT.
Then using the proportion "If for D - T we have δD - δT
as the variation what shall we have for D - t?" The result
is ((D - t) (δD - δT)) / (D - T) = δB
∵ B + δB = B' = D - t + ((D - t) (δD - δT)) / (D - T)
= (D - t) (1 + (δD - δT) / (D - T)) = (D - t) ((D + δD) - (T + δT)) / (D - T)
= (D - t) (D' - T') / (D - T) = (Mean Bhuja × True Sthiti-khanda) / (Mean Sthiti-khanda)

444
In other words to obtain the Bhuja rectified for
parallax in longitude, we have to multiply the mean Bhuja
by the Sthiti-khanda rectified for parallax and divide by
the Mean Sthiti-khanda. Similarly to obtain the Sthiti-
khanda rectified for the variation in latitude also, take
the Sthiti khanda rectified for parallax as the mean and
that rectified for variation in β as the True and then by
the same procedure, Bhuja rectified for parallax in latitude
will be equal to Sthiti-khanda rectified for parallax in
latitude multiplied by the Bhuja rectified for parallax in
longitude divided by the Sthiti-khanda rectified for parallax
in longitude Bhāskara gives the names Sphuta Sthiti-
khanda and Sphuta Saraja Sthiti-khanda to the Sthiti-
khanda rectified for parallax in longitude and that rectified
further for parallax in latitude. The proof adduced above
accords with the statements of Brahmagupta, Śrīpati and
Bhāskara ; Ranganātha bearing in mind Bhāskara's
version puts a correct interpretation on verse 19, Lunar
eclipses of Sūryasiddhānta. But the usage of the words
Madhya and Sphuta in that verse, misled the modern
traditional scholars including Burgess. If is to be men-
tioned here that the word Koti translated as perpendicular
by Burgess is misinterpreted by him (see his commentary
under verse 19, lunar eclipses) as ‘ The perpendicular is
furnished us in time and the rule supposes it to be stated
in the form of the interval between the given moment and
that of contact or separation’. This translation goes
against the definition of Koti contained in verse 18 just
above, since the Koti of Sūryasiddhānta is the Bhuja of
Bhāskara and vice versa.
We shall now see why there arose confusion in the
minds of many modern commentators including Burgess.
The problem mooted by the Sūryasiddhānta in calling for
a rectification of the Koti, is to be noted as that when the
grāsa g is given and not the Iṣtakāla I. The two formulae

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for Koti are √(R+r - g²- β²) / (u - v) and (T - I) (u - v). When I
is given we have to use the latter formula and take T" in
the place of T where T, T' and T" are respectively the
Sthityardhas (1) Mean, (2) Mean rectified for parallax in
latitude and (3) Mean rectified for parallax in longitude.
Of course here to arrive at T", method of successive
approximation is to be used as β goes on changing from
time to time and there is an inter-play between the
simultaneous effects of parallax in longitude and that in
latitude.
On the other hand when g is given we have to use the
former formula for the Koti and the Koti thus obtained
is to be rectified for the variation in β. So Sūryasiddhānta
proposes the formula
√(R+r - g)² - β² / (u - v) × T' / T" meaning thereby that the correction
for the variation in β is more important because the
formula is in terms of β and not the other formula. This
formulation is approximate but adopted for the sake of
ease. Otherwise from the grāsa, I is to be obtained and
the other formula could be used which method is more
laborious.
Bhāskara's Correction of Brahmagupta's Statement
Verses 1, 2 & 3. The statement of Brahmagupta
namely that the arc of the Moon's Dṛk-kṣepa will be
obtained by the sum or difference of that of the Sun with
the latitude of the Vitribha, I (Bhāskara) do not accept.
I shall give the reason why. In a place where the
latitude is 24°, when the longitudes of the Sun, the Moon
and the Node are all 180°, at the time of Sun-rise, the
ecliptic occupies the position of the prime-vertical. The
Moon will not leave the ecliptic even though depressed by
parallax in longitude. Thus there is no parallax in

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latitude. The Vitribha then being in the zenith, and its
latitude being 4½°, the Dṛk-kṣepa of the Moon according
to Brahmagupta's formula will be 4½° and therefore the
parallax in latitude obtained by the Dṛk kṣepa will be
(790'-35 / 15) × (H sin 4½° / 3438) = (52'-42'' / 3438) 270 = 4'-8" which is not
the case actually.
Comm. Having thus shown the flaw in Brahma-
gupta's approximate formula, Bhāskara proceeds to show
how that formula could be justified in a particular way.
Brahmagupta assumed the Moon's orbit to be the ecliptic
because at the moment of an eclipse, the latitude of the
Moon is very small so that he might be taken to be on the
ecliptic. In figure 101 let the Ecliptic coincide with the
prime-vertical ZE. Let EV be the Moon's orbit where
V is the Vitribha of the Moon's orbit. The arc of the
Sun's Dṛk-kṣepa is here zero and that of the Moon's
Dṛk-kṣepa is ZV which is the sum of the arc of the Sun's
Dṛk kṣepa namely zero and the latitude of the Moon's
Vitribha namely ZV, since the pole of the ecliptic now
coincides with the south point S, which means that ZV is
the latitude of V.
Let VV' be the nati which is equal to AB, since V'A
is the so-called Vikṣepa Sadṛśamaṇḍala or the deflected
position of VE on account of parallax in latitude. The
four minutes of parallax in latitude is now VV'=AB. This
parallax is obtained because, Bhāskara argues, the nati
obtained by Brahmagupta is there because he took VE to
be the Ecliptic and so obtained that nati with respect to
VE. Now, Bhāskara says, this is to be corrected by the
difference of the Moon's latitudes the original one and
that obtained after the Moon is deflected by the parallax
in longitude. This difference is 0-AB=BA. Correcting
AB with BA, the result is AB+BA=0 so that ultimately

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there is no parallax in latitude ie. no nati at all, as should
be the case.
Fig. 101 Fig. 102
Or, Bhāskara's argument could be better illustrated
from figure 102. Let VS be the ecliptic where V is the
Sun's Vitribha and S, the Sun. Let V'M be the Moon's
orbit called Vikṣepamandala where V' is taken to be the
Moon's Vitribha and M the Moon. Let S', M' be the
deflected positions of the Sun and the Moon on account of
parallax. ZV' = ZV + VV' = the arc of the Sun's Dṛk-
kṣepa + the latitude at the point V called Vitribhalagna-
Bāna, as formulated by Brahmagupta that ZV' is roughly
equal to the Moon's Dṛk-kṣepa-Dhanus. (Strictly speaking
ZV' ought to be perpendicular to the Moon's orbit V'M,
if V' were to be the Vitribha of the Moon; but, as the
latitude is small, the error is negligible). Let LS' and
L'M' be the so-called Krānti-Sadṛśa-mandala and Vikṣepa
Sadṛśamandala or parallel drawn to the ecliptic and the
Moon's orbit through the deflected positions S' and M' of
the Sun and the Moon. S'n is the Sun's parallax in latitude
ie. Nati. Similarly Brahmagupta took M'n' to be the
nati of the Moon as stated by Bhāskara. The error com-
mitted will be therefore M'n'— S'n = (M'x' + x'n') —
(S'x + xn) = M'x' — xn) cancelling x'n' and S'n which are
roughly equal. Also xn could be roughly taken to be equal

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to MS. Hence the relative parallax is equal to M'x'—MS =
Difference of the latitudes of the Moon in his original
and deflected positions respectively. Hence
S'n = M'n' — (M'x' — MS) = Brahmagupta's nati+correction
of the difference of the latitudes reversely effected, as
stated by Bhāskara.
In fact, Bhāskara has misread Brahmagupta's correct
procedure, since the latter sought the relative parallax of
the Sun and the Moon.
Instead of adding the latitude at V to the zenith-
distance of V (the arc of the Sun's Dṛk-kṣepa) which
implies additional computation of that latitude, as an
approximate procedure, MS is added to ZV to get ZV' as
an alternate procedure as mentioned by Bhāskara in the
course of the commentary.
Whereas Brahmagupta was seeking relative parallax,
Bhāskara misread that Brahmagupta took M'n' as the nati
and did not effect the correction of (M'x—MS) to obtain
S'n, which Bhāskara took to be the Sphuta-nati. Not
effecting the above correction is interpreted as neglecting
it since the Moon's latitude during the course of an eclipse
is small.