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

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

445
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

446
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

447
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

448
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.

GRAHACCHĀYĀDHIKĀRA
Verse 1. The orbital inclinations of the planets.
The inclinations of the orbits of Mars, Mercury,
Jupiter, Venus and Saturn to the ecliptic are respectively
110, 152, 76, 136 and 130 minutes of arc. The nodes of
Venus and Mercury get rectified by adding their respective
Śīghra anomalies to their values obtained originally.
Comm. The values given above are said to be the
mean values. Those given for the superior planets namely
Mars, Jupiter and Saturn approximately accord with their
modern values. Bhāskara says that these values pertain
to that moment of observation, when the Śīghra anomaly
is equal to 90 + ½ H sin⁻¹ a where a is Hsine of the
maximum Śīghraphala. This is quite in order because
when the Śīghra anomaly assumes the said value, the true
planet is at the point of intersection of the deferent and
the eccentric, which means that the planet is equidistant
from E₁ as well as E₂ (vide fig. 103). Identifying E₁ to
be the earth's centre and E₂ to be that of the Sun in the
case of the superior planets, the mean latitude of the
planet observed will be the same as that observed either
from the earth or from the Sun. Hence in the case of
the superior planets, the maximum latitudes of the planets
observed accord with the geocentric as well as heliocentric
observation. These are taken to be the mean values of
the maximum latitudes.
In the case of the inferior planets, it will be clear why
the modern values of 7° and 3°-24' for Mercury and Venus
are far higher than the Hindu values namely 152' and
136'. Since the mean planet in this case is taken to be
the Sun, the linear values of the latitude observed from
E and S, the centres of the Earth and the Sun respectively
will be in the ratio SP/ES (vide fig. 104).
57

450
Fig. 103 Fig. 104
In the case of Mercury this ratio is 4/10 and that in
the case of Venus is 7/10. Hence the values (420 × 4)/10 and
(204 × 7)/10 ie. 168′ and 142′ roughly accord with the Hindu
values. In other words the Hindu values are geocentric
and the modern heliocentric.
Bhāskara adds that the nodes of Mercury and Venus
as computed previously are to be increased by the Śīghra
anomaly to obtain the actual longitude of the node from
which the latitude is to be computed. This directive is
a beautiful example to show implied heliocentric motion.
Let the convex angle ASN be the heliocentric longitude
of the node measured negatively as the node has a negative
motion along the ecliptic. Adding the heliocentric Śīghra
anomaly to this longitude of the node means convex
angle ASN + ÛSP = 360° + N̂SP − ÂSU = N̂SP − ÂSU. Now
add the longitude of the planet ie. here ÂSU (= ÂES) to
obtain the argument from which the latitude of P the
planet is to be computed. The result is N̂SP as should be.

451
Computing the latitude as indicated in the next verse,
by the formula (H sin NSP × β × R) / (R × K) where β is the maximum
latitude cited above in the respective cases, multiplied by
R and divided by K indicates that the linear magnitude
of the latitude will be increased or decreased according as
K is smaller or greater than R.
Fig. 105 Fig. 106
(a) In this context, we are to throw light on some
moot points. Bhāskara says “मन्दस्फुटो ग्रहः स्वशीघ्र प्रतिमण्डले
भ्रमति, तत्र च तस्य पातोऽपि”, which tantamounts to saying
that the node is situated in the heliocentric orbit “स्वशीघ्र-
तिमण्डल”. The Vimandala or the planet's orbit, taking
for example the inner orbit of P in fig. 105, does not
exactly lie in the plane of the ecliptic as shown in that
figure but is in an inclined position cutting the coplanar
inner orbit in N and N', as shown by the ellipse NPN'
where NP = NP'. The argument to calculate the latitude
is NP' and the formula is (H sin NP' / R) × β to give the helio-
centric linear latitude. To obtain its geocentric linear
value, we have to multiply by R / K.

452
(b) In the case of the superior planets, Bhāskara
mentions in the Golādhyāya “पातोऽथवा शीघ्रफलं विलोमं
कृत्वा स्फुटात् तेन युतात् शरोऽनः ” verse 22 (Golabandhādhikāra).
This also clearly indicates that the orbits of the superior
planets are also heliocentric, for, otherwise, there is no
purpose of subtracting the Sīghraphala from the True
position of the planet. The purpose in doing so is to
obtain the heliocentric longitude of the planet (vide
fig. 106).
True geocentric longitude of J is A/EJ = ASJ/.
Subtracting the S'ighraphala EJS = JEJ' from ASJ',
we have AŜJ the heliocentric longitude. Adding the
retrograde longitude of N (तेन युतात् as cited above in
verse 22) means adding ASN to AŜJ which gives NŜJ.
This is the argument to calculate the latitude of J.
(c) Bhāskara alludes to a confusion in the mind of
Chaturvedāchārya in this context while commenting upon
verses 23, 24 in Golabandhādhikāra (Golādhyāya). Chatur-
vedāchārya commenting on Brahmagupta's words (verse 10
Grahayukti-adhikāra ch. 9, Brahma Sphutasiddhānta)
exclaims. "The latitude of Mercury and Venus will be the
same as what they are at the point of S'īghroccha ; correct-
ness of the result alone is proof ; no other reason could be
adduced !" Bhāskara clears his misconception in the
following words. "The number of sidereal revolutions of
the nodes of Mercury and Venus mentioned in the chapter
on mean motion, are to be increased by the number of the
sidereal revolutions of the S'īghrocchas of Mercury and
Venus, as mentioned by Mādhava in his work “ Siddhānta-
chudāmani ”. This means that the S'īghra anomaly is to
be added to the position of the node obtained by the smaller
number of revolutions given in the chapter on mean
motion.

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

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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

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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

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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