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

342
the necessity for knowing the value of CN at any subse-
quent moment arises. β, of course, could be computed,
knowing the hourly variation of β, which in its turn could
be known, by a knowledge of the hourly variation in λ, the
longitude of the Sapātachandra.
The magnitude of CN is calculated by the rule of
three "If by the Sparsa-Sthiti-Khanda we have initially
the initial value of CN, what shall we have for (T –t) ?"
The result is [(T–t) × CN] / T where CN is the initial value
of CN and T the Sparsa-Sthiti-Khanda. Substituting the
values of CN and T from verse 12, where
CN = √((R+r)²–β²) and T = [√((R+r)²–β²) × 60] / (m₁–s₁)
we have the required Bhuja as
[(T–t) (√(R+r)²–β²)] / [60 (√(R+r)²–β²)] × (m₁–s₁) = [(T–t) {m₁–s₁}] / 60 minutes
= (T–t) (m₁–s₁) degrees as given.
Similarly we could find the Bhuja with respect to
'totality' ie. the 'Marda-Bhuja' as it is called.
Note. One might mistake M₁ M₂ of fig. 74 (M₁ per-
taining to a subsequent moment) to be the Bhuja defined
above, which is the join of the centre of the eclipsing body
and the foot of the latitude at the middle of the eclipse.
That is why Bhāskara uses the word 'Madhya-Sarāgra-
Chihna' in the commentary, meaning thereby not the
foot of the actual latitude at the middle of the eclipse but
only the point N of fig. 77 which 'signifies' it.
Second half of verse 16 and first half of verse 17.
Taking the latitude of the Moon at a given time as
Koti, and Bhuja as the Bhuja of the moment defined
above, we have the Karṇa of the moment as √(Bhuja² + β²);
R+r–Karṇa gives the Grāsa at the moment.

378
Comm. The word 'Grāsa' at the moment stands
for AB of fig. 77, Karṇa for CM, where CN is the Bhuja
and MN is the Kōti. The 'Grāsa' at the moment of
opposition has the special name Sthagita.
Second half of verse 17 and verse 18.
To obtain the time after the moment of first contact,
knowing 'Grāsa' at the moment.
T — √((P+r —Grāsa)²—β²) / (m₁—s₁) = t; this 't' is to be rectified
by obtaining the β of the moment and again finding t and
repeating the process till an invariable magnitude is got.
Comm. This is the converse of finding the Grāsa
given the time. The method of rectification is also
evident. In the above equation considering β and t as
variables, and differenting,
δt = (1 × —βδβ) / (2 (m₁—s₁) √((P+r—g)²—β²)) = —βδβ / ((T—t) (m₁—s₁)²)
Knowing δβ, δt could be got without taking recourse
to the method of successive approximations.
Verse 19. Certain definitions.
The 'Middle of the eclipse' (or strictly speaking the
moment when the portion eclipsed is a maximum) occurs
at the moment of opposition. Sparsa or Pragraha is at
the moment of first contact and Mokṣa is at the moment
of last contact, separated from the moment of the middle
of the eclipse by times equal to Sparsa-Sthiti-Khanda and
Mokṣa-Sthiti-Khanda respectively before and after.
Similarly Sammilana and Unmilana or the moment of
the commencement of totality and the end thereof occur
before and after the moment of 'the middle of the eclipse'
by times equal to Sammilana-Marda-Khanda and Un-
milana-Marda-Khanda respectively.

374
Comm. Clear.
Verse 20. To get what is called the Valana.
The hour-angle of the eclipsed body expressed in
nādīs, multiplied by 90 and divided by half the duration
of night (if it be lunar eclipse) or half the duration of day
(if it be solar) as the case may be will give the degrees of
an angle, whose H sine being multiplied by the H sine of
the latitude and divided by (H cos δ, (where δ is the decli-
nation of the eclipsed body), gives the H sine of what is
called Ākṣavalana which is north when the hour angle
is east, and south otherwise.
Comm. This subject of Valana requires a detailed
treatment as is given in the Golādhyāya by Bhāskara.
Here only a practical formula is given to proceed with
the computation. For an understanding of this formula
we have to necessarily draw upon the treatment in
Golādhyāya.
The word ‘Valana’ means ‘deflection’. The problem
posed is at what point of the disc of the eclipsed body does
the eclipse begin and at what points it ends. Since an
observer sees the disc of the eclipsed body on the back-
ground of the spherical surface of the sky, the specification
of the point of first contact must necessarily be made with
respect to east, west, north and south. These directions
could be specified with respect great circles drawn second-
ary to the prime-vertical. But the Earth's shadow moves
along the ecliptic and the Moon is also very nearly
moving on the ecliptic at the moment of an eclipse. Thus
‘Valana’ should give the angle between the ecliptic and
the prime-vertical; rather it should be described by two
diameters of the Moon's disc, one a secondary to the
prime-vertical and one a secondary to the ecliptic. In
other words we have to get the angle subtended at the
centre of the Moon's disc between those diameters.

375
This angle between the two diameters mentioned, is,
for convenience divided into two parts namely K̂MP and
P̂MN, where K, P, and N are the poles of the ecliptic,
celestial equator and the prime-vertical and M is the centre
of the Moon's disc. K̂MP is called Āyana Valana, so
called because it depends upon the obliquity of the ecliptic
to the Equator (अयनयोः वलनं आयनं वलनम् ie. the deflection due
to the deflection of the solsticial points from the equator)
whereas the angle P̂MN is called Ākṣa Valana ie. defle-
ction of a secondary to the prime-vertical namely NM
with respect to a secondary to the celestial equator namely
PM which is due to Ākṣa or latitude of the place.
We shall first treat the subject on modern lines and
then depict Bhāskara's treatment. Let θ, ξ, η stand
respectively for the Āyana, Āṣka and total Valanas
respectively, where by 'total Valana' we mean K̂MN
which is the algebraic sum of K̂MP and P̂MN.
From the spherical triangle KMP fig. 78.
Fig. 78

376
Sin 90 ∓ α / Sin (90 - β) = sin θ / sin ω = sin (90 - λ) / sin (90 - δ)
∴ Sin θ = (sin ω cos λ) / cos δ or (sin ω cos α) / cos β I
[Fig. 79] Similarly from fig. 79, where
P = celestial pole, N = North-
point, M = centre of the Moon's
disc, Q = latitude of the place,
h = hour-angle of the Moon,
ξ = Āksha Valana and z = Arc
of the prime-vertical inter-
cepted between the zenith
and the foot of the secondary
to the prime-vertical drawn
through M, which arc goes by
the name Sama-Vṛitta-Natāṃ,
sa or zenith-distance measured along the prime-vertical-
μ = distance of M from the prime-vertical measured along
the above secondary,
Sin ξ / Sin φ = sin (180 - h) / sin (90 - μ) = sin z / sin (90 - δ)
Sin ξ = (sin φ sin h) / cos μ = (sin φ sin z) / cos δ II
In fig. 80, where (M) is the Moon's disc, AB, the
diameter of the disc extending along the ecliptic (assuming
the Moon's centre almost on the ecliptic, which is the
case at the time of an eclipse), K, P, N respectively the
pole of the ecliptic, the celestial pole and the north point
and θ, ξ the Āyana and Ākṣa Valanas defined above, the
eclipse starts at A, the eastern side of AB, called the
Krānti-Vṛitta-Prāchī, AB being perpendicular to EF a
diameter of the disc secondary to the ecliptic. An observer
with his physical eye construes the diameter CD, which
is secondary to the prime-vertical as indicating north and
south. Naturally therefore, it is required to specify the

377
Fig. 80
location of A, the point of first contact, with respect to
the diameters GH and CD, which are respectively East-
West and North-South. Suppose θ+ξ̅ = 45° = GMA, then
we say that the eclipse begins at the north-point of the
disc and so on. For this purpose, the concept of Valana
arose. We have said above that the angle KM̂N = GMA
is to be got, and that it is the algebraic sum of θ and ξ,
meaning thereby that when K comes in between P and N,
or below N, which is also likely for places of latitude less
48

378
than ω, the obliquity of the ecliptic, KM̂N will be equal
to ξ - θ and θ - ξ respectively.
The formula given in the present verse is
ξ = H sin⁻¹ (H sin 90h / (D/2) × H sin ϕ / H cos δ)
or H sin ξ = H sin (90h / (D/2)) × H sin ϕ / H cos δ
where h and D/2 are measured in nādīs, h being the hour
angle of the Moon and D/2 half-the duration of the
Moon's stay above the horizon.
Evidently the formula is intended as an approximate
one, for all practical purposes considered equivalent to
formula II given above namely
sin ξ = sin ϕ sin z / cos δ or H sin ξ = H sin ϕ H sin z / H cos δ .
Thus in the place of z we are given 90h / (D/2) which means
" when z = 90°, D/2 is the hour angle measured in nādīs,
what is z when the hour angle is h?". The answer is
h × 90 / (D/2) . This formula is approximate because h and z are
not strictly in proportion though h increases or decreases
along with z.
Nonetheless, the formula serves for practical purposes
very approximately and the beauty lies in the concept of
Sama-Vritta-Natāṁśa, which means measuring hour-
angle in terms of the arc of the prime-vertical instead of
an arc of the celestial equator. The error, it will be noted
will not be much in low latitudes.
So far with respect to the commentary on the present
verse. Now we shall see how Bhāskara tackles the pro-
blem rigorously in Golādhyāya under the caption Valana
Vāsanā' ie. 'concept of Valana'.

379
We defined above that the angle KM̂P is Āyana
Valana and the angle PM̂N as the Ākṣa-Valana. These
are respectively called Bimbīya-Āyana Valana and
Bimbīya-Ākṣa-Valana being subtended at M, the centre
of the Bimba ie. the disc of the Moon. If in fig. 78, T be
the foot of the celestial latitude of the Moon, then the
respective angles KT̂P and PT̂N are called the Sthānīya-
Āyana-Valana and the Sthānīya-Ākṣa-Valana ie. the
angles subtended at the Sthāna or the construed position
of the Moon on the ecliptic.
The Āyana-Valana is zero and a minimum when
M or T lies at the solstices, and a maximum equal to ω
when those points lie at r or ♎. Similarly the Ākṣa-
Valana is a minimum equal to zero when M or T lies on
the meridian and a maximum equal to ϕ when those
points lie at the east or west points. In other words the
Āyana-Valana increases from zero to ω as M or T moves
along the ecliptic from a solstice to an equinoctial point;
and the Ākṣa-Valana increases from zero to the maxi-
mum value of ϕ as M or T moves along zE or zω from z
the zenith to E or ω along the prime-vertical. Hence
Āyana Valana is perceived to be proportional to
H sin (90+λ) where λ is the longitude of M or T, since
when λ = 90, H sin (90 + 90) = 0 and when λ = 0,
H sin (90+90)=0 and when λ=0, H sin (90+0)=R, a
maximum; similarly the Ākṣa-Valana is perceived to
be proportional to H sin z where z is the Sama-Vritta-
Natāṁśa defined before, since, when z=0, M or T is at
the zenith and the Ākṣa Valana is zero and when M or T
is at the East or West point, z=90° and H sin z=R, a
maximum.
It is worth-hearing Bhāskara, at this juncture (Ref.
verses 30-74 under the caption Valana Vāsanā pages 305-
306. Ānandāśrama edition of Golādhyāya Vol. 2. Poona).

380
“The north and the south with respect to the Equator
and Ecliptic (ie. the north-pole and south-pole) are differ-
ent at the points ♈ and ♎, being at a distance of ω from
each other. Hence the Āyana Valanajyā at those points
is equal to H sin 24° (ω taken to be equal to 24°). But at
the solstices, the north and south will be the same (mean-
ing thereby that the angle subtended by PK at the solstices
is zero, or what is the same, the directions to the respective
poles (of the Equator and the Ecliptic) at the solstitial
points are the same so that the East will be the same for
both the circles at those points. Thus there is no Valana
at the solstices ie. P ⌒ K = 0 where ♋ = cancer. In between
♈ and ♋, the Valana is found in proportion to
H sin (90 + λ) where λ is the longitude of the point and
in inverse proportion to H cos δ, where δ is the decli-
nation. Hence H sin θ = [H sin (90 + λ) / H cos δ] × H sin ω, where
θ is the Āyana Valana. Similarly at the points of inter-
section of the Equator and prime-vertical namely E and W,
the Unmandala (the Equatorial horizon) decides the
north-south direction with respect to the Equator, whereas
the horizon decides the same with respect to the prime-
vertical. These north-south directions with respect to
those two great circles namely the Equator and the prime-
vertical differ by the angle between the Unmandala and
the horizon which is equal PN = ϕ, the latitude of the
place. Hence at the East and West points the Ākṣa
Valanajya or the H sine of Ākṣa-Valana is equal to
H sin ϕ. But at the zenith, the north-south directions
of the Equator and prime-vertical coincide so that there
is no Ākṣa Valana at the zenith. Thus H sine of the
Valana is proportional to H sin ϕ in between the points
on the prime-vertical between the zenith and the East and
West points. (Roughly speaking) H sin ξ =
(H sin ϕ H sin z) / (H cos δ) where ξ = Ākṣa Valana, z = Sama-Vritta-

381
Natāṁśa (defined before) and H sin z may be taken to be
roughly equal to 90h / (D/2) (as depicted before).
In the East the Ākṣa Valana is north, for, in fig. 80
ĜMI which gives the East of the Equator with respect to the
the East of the prime vertical, is north; whereas in the
West ĤMJ is south. (The definition of the direction of
the Valana is given as a directive to add the two kinds of
Valanas if they be of the same direction otherwise to take
the difference; in the fig. 80, the Āyana Valana ie. ÎMA
is also north, so that adding ĜMI + ÎMA = ĜMA is the
Sphuta Valana or the actual Valana). Hence Sphuta
Valana measured by GMA is had by the sum or difference
of the two angles ĜMI and ÎMA which define respectively
the Āyana and the Ākṣa Valanas.
Similarly, at the point of intersection of the Ecliptic
and the prime-vertical, the Sphuta Valana is a maximum
which is the sum or difference of the Valanas as the case
may be. At points removed 90° on either side, from the
point of intersection of the Ecliptic and the prime vertical,
in as much as the north-south directions with respect to
the Ecliptic and the prime-vertical coincide, the Sphuta
Valana is zero.
If (as Lallāchārya said) the Valana varies as the
Hversine at those points which are removed by 90° from
the points of intersection of the Ecliptic and the prime-
vertical, the Sphuta Valana will not be zero (which is
against common sense). Hence the Valanajya varies as
Hsine and not as Hversine.
We shall look at the subject from another point of
view for the sake of clarity.........Fix a circle on the sphere
with the celestial pole as centre and ω as the angular

382
radius. This circle is called Kadamba-Bhrama-Vritta or
the circle in which the pole of the Ecliptic revolves round
P (due to diurnal revolution of the Earth). In that circle
Hsine of an angle will be H sin δ............Or again draw the
great circle with the planet's position as the pole, called
the horizon of the planet. The arc intercepted on this
circle between the Ecliptic and the celestial Equator will
be Āyana Valana and that intercepted between the celestial
Equator and the horizon is the Ākṣa Valana; and the
arc intercepted between the Ecliptic and the horizon is the
Sphuṭa Valana.
Or again draw a circle with K as centre and radius
ω = 24°. This circle is called the Jina-Vritta where the
word Jina means 24. Let a secondary to the Ecliptic
passing through K and K' the poles of the Ecliptic revolve
with KK' as fixed. When this revolving circle passes
through Cancer (Sāyana) it will be passing through P.
The angle turned through by this circle from Cancer, will
be equal to the angle turned through from P. The Hsine
of that angle in the Jina Vritta will be H sin δ of a
longitude equal to that angle. This is the Āyana Valana
and it arises at the end of Dyujyā, since the north-polar
distance of the planet is (90-δ) whose Hsine is Dyujya
ie. H cos δ. The corresponding Āyana Valana in a circle
of radius R is got by multiplying by R and divided by
H cos δ.
Let us clarify Bhāskara's mind. (Ref. fig. 81) Let
PBD be the Jina Vritta drawn on the sphere with K, the
pole of the Ecliptic as centre and ω=24° as radius. Let
a revolving secondary to the Ecliptic coincide initially
with KC where C is Cancer. Let it occupy subsequently
the position KM where M is the centre of the Moon's disc
taken to be on the Ecliptic as is the case very approxi-
mately at the moment of an eclipse. Now the Āyana
Valana is the angle KMP. Let MA be the declination of
M. Produce MP to L such that MK=ML=90°. Hence

383
Fig. 81
PL = δ since PA = 90° and LM = 90°. The Āyana Valana
K̂MP is measured by the arc ML where ML is an arc of
the Grahakṣitija or the horizon of the planet M (ie. the
circle with M as centre and 90° as radius drawn on the
sphere or what is the same the great circle whose pole is
M). PB is an arc of the small circle parallel to KL which
is an arc of a great circle. Then in the Jina Vritta
sin PB = sin PK × sin P̂KB = sin ω sin (90 - λ) = sin ω cos λ
∴ sin KL = sin ω cos λ/cos PL = (sin ω cos λ) / (cos δ) = sin PMK.

384
Here sin ω cos λ is called Sa-thribha-graha-ja-kranti or
the declination of a point whose longitude is 90+λ where λ
is the longitude of M. As we have the formula sin δ =
sin ω sin λ, sine of the declination of such a point is equal
to sin ω sin (90+λ)=sin ω cos λ. When δ is very small
sin PMK may be taken to be sin ω cos λ or what is the
same Sa-thribha-graha-ja-krānti as is formulated by Sūrya-
siddhānta.
It may be doubted how sin PB=sin PK sin 90-λ.
(Ref. fig. 82). Let K′ be the centre of the circle PBD,
Fig. 82
K′ being in the plane of PBD. K′P and K′B are radii
of this circle. Since the arc PB stands for 90-λ PK′B=
90-λ. Draw the H sine of arc PB, which is PB′. Now
K′P=H sin ω, as PK′ is ⊥ar drawn on OK
∴ PB′=PK′ sin PK′B
=H sin ω ✕ sin PK′B = (H sin ω ✕ H sin PK′B) / R =
(H sin ω H cos λ) / R
∴ H sin KL = [(H sin ω H cos λ) / R] ✕ H cos δ =
(H sin ω H cos λ) / (H cos δ) as given.

385
[चित्र: Fig. 83 — Equator, Ecliptic, Prime vertical]
Fig. 83
Note 1. Bhāskara says PB' (fig. 82) is Krānti-Sinjanī; so
it is because PB' = PK' sin (90 - λ) = (H sin ω H cos λ) / R
= H sine of the declination of a point whose longitude is
90 + λ = Satribha-grahaja-krānti as is mentioned by
Bhāskara and Sūrya-siddhānta.
Note 2. If M be the centre of the Moon's disc and
ABC its horizon defined above, the arc AB intercepted
between the Ecliptic and the Equator is Āyana Valana, the
arc BC intercepted between the Equator and the prime-
vertical is Ākṣa Valana and the arc AC intercepted
between the Ecliptic and the prime-vertical is Sphuta-
Valana.
Note 3. The analysis of Ākṣa Valana proceeds on
similar lines, only we have P and N in the place of K
and P.
Note 4. The mistake of Lallāchārya alluded to by
Bhāskara is as follows. The Āyana Valana, we have seen
is zero at the Ayanas ie. the solstices and maximum at
♈ and ♎ ie. the equincotial points removed by 90° from
the Ayanas. Now Hversine = R — H cosine so that when
90 — λ = 0 ie. λ = 90°, Hversine (90 — λ) = R — H cos (90 — λ)
= R — H sin 90° = R — R = 0 and when 90 — λ = 90° ie.
49

386
λ = 0 Hversin (90 — λ) = R — H cos (90 — λ) = R — H sin λ
= R — H sin 0° = R — 0 = R. Hence Lallācharya took by
mistake that the Āyana Valana varies as Hvers (90 — λ)
instead of H sin (90 — λ) since both Hversine and H sine
of 90 — λ̄ are zero at the Ayanas and maximum at r and ♎.
The same mistake was committed by Lallācharya in the
context of the Moon’s phase also as criticised by Bhāskara
as we shall see later. In fact, this latter criticism is not
so justified as the former, as will be shown in that context.
Note 5. If instead of taking the Āyana Valana to
vary as H sin (90 — λ) we happen to take according to
Lallācharya that it varies as Hversine, then in places (Ref.
verses 38, 39 Valana Vāsanā, Golādhyāya) removed by
90° from the points of intersection of the Ecliptic and
the prime-vertical, where there should be no Sphuta-
Valana, we do get that there is some Sphuta Valana there,
since the value of Hversine differs from Hsine, though
these two functions happen to be zero simultaneously and
maximum simultaneously.
Bhāskara continues in verses 66-68 (Ibid) “ I shall
now depict Ākṣa Valana by means of the hour-angle.
Take the sum or difference of S'anku-Agrā and S'anku-tala
according as they are of the same direction or not ; compute
√(R² — B²) where B is the result ; then H sin ϕ H sin h
───────────────
is equal to H sin ξ where ξ is the Ākṣa Valana ”. √(R² — B²)
Comm. We saw before in the Tripraśnādhyāya that
A = S + B where A = S'anku-Agrā, S = S'anku-tala, and B =
S'anku-bhuja = H sin µ where µ is represented in fig. 79.
Hence √(R² — B²) = H cos µ so that the above formula gives
H sin ϕ × H sin h
H sin ξ = ───────────────────── which is the same as got by
H cos µ
the modern formula in Equation II.

887
Note. A small circle parallel to the prime-vertical
is called Upa-Vṛtta. Also secondaries drawn to the
Ecliptic, Equator and the prime-vertical are called
Kadamba-Sūtra, Dhruva-Sūtra and Sama-Sūtra. They are
also called occasionally as Kadamba-prota-Vṛtta, Dhruva-
prota-Vṛtta, and Sama-prota-Vṛtta.
Bhāskara proceeds to find the Āyana Valana in a very
ingenious way in verses 69-74. We shall first give it a
modern treatment so that we may better appreciate his
genius. Let (S) be the Sun's disc. (It does not matter
whether we take the Sun or the Moon). EQ is its diameter
along the diurnal circle, and CL along the Ecliptic. LM
is the difference of the declination of L and S. Let
SL = Δ λ and LM = Δ δ. We have
sin δ = sin λ sin ω ;
differentiating cos δ Δ δ = sin ω cos λ Δ λ
∴ Δ δ = (sin ω cos λ / cos δ) × Δ λ ; put Δ λ = b, the angular
radius of the disc
∴ Δ δ = (b × sin ω cos λ / cos δ) = (b H sin ω × H cos λ / R × H cos δ) .
This gives the Valana in the disc of radius b. If that
be so, what will it be on the sphere of radius R? The result
is (b H sin ω × H cos λ / R × H cos δ) × R / b = (H sin ω × H cos λ / H cos δ)
as got before.
Let us hear Bhāskara, " put the disc of the Sun at
the point of intersection of the Ecliptic and the diurnal
circle. The Valana (LM of fig. 84) at the periphery of
the disc is the difference of the declinations of L and S.
To get the value of this let us first get the value. SL in
terms of λ, the longitude of S. It is (b × B / 225) ; so that LM

388
Fig. 84
will be equal to (b × B) / 225 × (H sin ω) / R where B is the Bhogya-
khanda of λ. To obtain the value of the above for a circle
of radius R from a circle of radius b, we have to multiply
by R/b. So, the result is
(b × B) / 225 × (H sin ω) / R × R / b = (B H sin ω) / 225 . But the value of B
is got as follows. 'If for H cos λ equal to R we have the
first Bhogyakhanda equal to 225, what shall we have for
H cos λ?' The result is (225 × H cos λ) / R . Substituting for
B, we have (H sin ω) / 225 × (225 × H cos λ) / R = (H sin ω × H cos λ) / R
Now, on account of declination, the Sun's disc is inclined
like an umbrella. So LM of fig. 84 will take a position
like L'M as shown in fig. 85 where the triangle MLL' is
similar to SMO, S being the centre of the Sun's disc, O
the centre of the sphere. Hence
(L'M) / LM = R / (H cos δ) ∴ L'M = R / (H cos δ) × (H sin ω H cos λ) / R
= (H sin ω × H cos λ) / (H cos δ) as got before '.

389
Comm. Bhāskara terms SL as the Dorjyāntara ' or
the variation in H sin λ, which he knows to (H cos λ Δλ) / R.
[Fig. 85: Triangle L'LM] [Fig. 86: Triangle SMO, Hypotenuse = R, SM = H Cos δ, MO = H Sin δ]
Fig. 85 Fig. 86
But proceeding from first principles, as he always does,
he asks us to consider the Bhogyakhanda at λ namely B.
If this be for an interval of 225', what will be it be for ‘b’?
The result is (b × B) / 225 . Then to rectify B, the proportion
used is as used above. Bhāskara says many a time that
the variation in Hsine is proportional to Hcosine. This
concept he might have derived by looking at the Hsine
table of 90 Hsines. Hence the argument advanced by him
to rectify B is ‘If for Hcosine equal to R (at zero-value
of the argument) the initial Bhogyakhanda is 225, what
will it be for an arbitary H cos λ? The result is
(H cos λ) / R × 225. Substituting this for B in the above, we
have (b H cos λ) / 225 × 225 = (b H cos λ) / R . This expression we
perceive as no other than (H cos λ Δλ) / R as equal to Δ (H sin λ),
for b is to be taken as Δλ. This (H cos λ × b) / R is called by
Bhāskara as Dorjyāntara meaning thereby Δ (H sin λ).

390
Then the next argument is 'If for H sin λ equal to R we
have the declination equal to H sin ω, what shall we have
for the above Dorjyāntara. The result is
(H sin ω / R) × (H cos λ × b / R). Since this is in modern terms
sin ω cos λ × b and b = Δλ, we perceive that this expres-
sion is Δ (H sin δ) where δ stands for LM of fig. 84, ie.
the difference of the declinations of the points S and L of
the disc of fig. 84. The next argument advanced by
Bhāskara, namely that on account of declination, the disc
is slanted and LM gets thereby enlarged into L'M of
fig. 84 and adduces proportionality from fig. 86. But this
argument seems to be faulty. In fact, the magnitude of
LM is got for the diurnal circle of radius H cos δ. To
get its value for radius R, the result would be
[(H sin ω H cos λ × b) / R²] × [R / (H cos δ)] = (b. H sin ω H cos λ) / (R. H cos δ) .
Then the argument is 'If in the disc of radius b, we have
Δδ equal to the above what will it be for radius R? The
result would be (H sin ω H cos λ) / (H cos δ) .
Note. Our argument is based on the idea that lines
of the small circle namely the diurnal circle get enlarged
for a circle of radius R in the porportion R: H cos δ.
Bhāskara's concept of enlargement on account of slanting
does not seem to be plausible because, on account of decli-
nation, the disc may occupy an overhead position when
the Equator is itself inclined. Thus slanting does not
arise out of declination.
The question might be asked as to how Bhāskara got
the right answer by such an argument. He got the answer
up his sleeves through the other methods he gave and he
adduced this argument to get at that answer.

391
Second half of verse 21 and first half of verse 22.
H cos λ × H sin ω
───────────────── = H sin θ where θ is the Āyana Valana
H cos δ
The direction of this Valana is that of the hemisphere
north or south in which the Moon lies.
Comm. This is the formula we have already derived.
Regarding the direction of the Āyana Valana, the conven-
tion is that it is to be considered north, if the Moon be in
the northern hemisphere, otherwise south. The reason is
that at the time of a lunar eclipse, the Moon being in opposi-
tion, if he be north, the Sun will be south of the Equator,
and the line BA of fig. 80 representing the Ecliptic which
is roughly the join of the Sun and the Moon will be north
of the line JI which is parallel to the equator. Thus the
direction of the angle IMA gives the direction of the
Āyana Valana.
Latter half of verse 22 and verse 23. Sphuta Valana.
The Hsine of the sum or difference of the two Valanas
according as they are of the same or opposite directions,
multiplied by the sum of the angular radii of the Moon
and Rāhu, and divided by the radius gives the Hsine of
the Sphuta Valana. Those who said that the Valana is
proportional to the Hversine, do not know spherical
geometry properly.
Comm. The direction of the Akṣa Valana, was defined
in verse 20 that it is north if the hour angle is east, other-
wise south. The meaning of this convention is that the
diameter of the Moon's disc parallel to the Equator when
the hour angle is East, is north of the diameter which is
parallel to the prime-vertical. Thus combining the two
conventions regarding the directions of the Valanas, it is
clear that if both the Valanas are north, the line MI is
north of MG, and MA is north of MI (fig. 80) so that the