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

opposition or conjunction as the case may be is called
Sparśa-Sthiti-Khaṇḍa.
(2) The time between the middle of the eclipse and
the moment of last contact is called the Mokṣa-Sthiti-
Khaṇḍa.
(3) The time between the commencement of total
eclipse and the middle of the eclipse is called the Sammi-
lana-Marda-Khaṇḍa.
(4) The time between the middle of the eclipse and
the end of total eclipse is called Unmilana-Marda-Khaṇḍa.
The suffix Khaṇḍa meaning ‘half’ is generally omit-
ted while refering to these phases. In the above verse
we are given formulae for Sthiti-Khaṇḍa and Marda-
Khaṇḍa only without specifying whether they pertain to
Sparśa or Mokṣa. Though the same formulae serve
for both the Sparśa phase as well as the Mokṣa phase
under the supposition that β does not vary, it will be
noted that the Sparśa-Sthiti-Khaṇḍa will not be equal to
Mokṣa-Sthiti-Khaṇḍa and that the Sparśa-Manda-
Khaṇḍa will not be equal to the Mokṣa-Marda Khaṇḍa
in as much as β changes from moment to moment.
In fig. 73, let C₁ be the position of the eclipsing body
at the moment of first contact and C₂ its position at the
moment of last contact. In the figure is shown only one
position of the Moon's disc signifying that we may con-
sider the motion of the eclipsing body keeping the eclipsed
body fixed (or what is the same relative to the position of
the eclipsed body). It is evident from the fig. that C₁M =
C₂M = P + r so that C₁M C₂ is an Isosceles triangle. Let
MN be the ⊥ᵃʳ dropped from M on C₁ C₂. C₁ C₂ is the
ecliptic because the centre of the shadow will be moving
along the ecliptic, for, in fig. 66, SE the ecliptic passes
through D the centre of the cross-section of the shadow-
cone, as well as through the vertex V of the shadow-cone.

868
Fig. 73
MN is therefore the latitude of the Moon. Since the lati-
tude is not the same at the moment of first contact and
that of the last contact, the figure drawn does not
represent the true figure but only a figure drawn on the
supposition that β remains the same and C₁ Moves relative
to M. From the figure C₁N² = (P+r)² − β² = C₂N² I
The Sthiti-Khanda defined in this verse is the time
taken by C₁ to reach the position N ie. the position at
the moment of opposition, and again from the position N
to the position C₂. The velocity of C₁ relative to M is
no other than the excess of the velocity of, the Moon over
that of the Sun. (The velocity of the Earth is the relative
velocity of the Sun with respect to the Earth and this is
equal to the velocity of the shadow moving along the
ecliptic). So, the time taken by C₁ to reach the position
of N relative to the Moon is equal to [ 60 × √(P + r² − β²) ] / [ m₁ − s₁ ]
Similarly the time taken by the centre of the shadow from
N to C₂ ie. from the point of opposition to the moment

386
of last contact has also the same formula where in each
case β is the latitude at the moment of opposition. The
path taken by the centre of the shadow is called 'ग्राहकमार्ग'
ie. the path of the eclipsing body. The actual case when
both C and M are both moving and when β is considered
as a non-changing quantity is shown in fig. 74. In this
Fig. 74
case, three positions are shown, (1) that at the first con-
tact (2) that at opposition and (3) that at last contact,
where C₁, M₁, C₂, M₂ and C₃, M₃ give the positions of the
centre of the shadow and that of the Moon's disc respe-
ctively, both the centres being shown as moving. Since
the Moon moves faster than C and as such overtakes C,
the path of M from M₁ M₃ which synchronizes with the
path of C from C₁ to C₃, is shown to be longer. But, one
may wonder, how C₁ N₁ and C₃ N₃ represent the Sparśa-
Sthiti-Khanda and Mokṣa-Sthiti-Khanda respectively.
The distance overtaken by M with respect to C from the
point of first contact to the point of opposition is M₁ M₂ —
C₁ C₂ = C₁ N₁. Hence we compute C₁ N₁ by the formula
C₁ N₁² = (P+r)² — β². Similarly from the point of
opposition to the point of last contact M overtakes C by
the distance M₂ M₃ — C₂ C₃ = C₃ N₃ = √(P+r)² — β².
Fig. 75 shows the situation when β changes as is the
actuality. When the opposition takes place after the
Moon crosses the node, then β₃ > β₂ > β₁, whereas if

367
[चित्र: Fig. 75]
Fig. 75
the opposition precedes the Moon’s position at the node
β₃ < β₂ < β₁. Also, when β changes, M₁ M₂ does not
exceed C₁ C₂ exactly by C₁ N₁. So, on both the counts,
the formulae, given in verse 12 are approximate. What
is done in practice is that β is computed for the moment
of opposition and estimating the Sparsa-Sthiti-Khanda by
the formula given above, and subtracting it from the time
of opposition the moment of first contact is got. Then β is
computed for that time and again the formula is applied
to get the Sparsa-Sthiti-Khanda. Repeating the process,
we rectify the Sparsa-Sthiti-Khanda. Even then, we do
not have the actual value of the Sparsa-Sthiti-Khanda,
because M₁ M₂ does not exceed C₁ C₂ exactly by C₁ N₁.
A more correct procedure would be to compute the
time between the moment of first contact and the moment
of opposition and by that time, to compute the length of
M₁ M₂ and take [m' (β₁ ~ β₂) / (M₁ M₂)] in the place of m' and use the
formula of verse 12. This nicety, however, need not be
attended to with respect to the duration of totality, for,
it does not make much difference.
Another way of obtaining a better value for T, the
Sparsa-Sthiti-Khanda is to take average values for β₁ and
β₂, m₁ and m₂, s₁ and s₂ where m₁ and m₂ are the values of
the Moon’s daily motion and s₁ and s₂ are those of the

368
Sun's at the point of first contact and the moment of con-
junction respectively.
We may also use calculus to obtain δT, the variation
in time for a variation of δβ in β and a variation of
δm₁ in m₁ ignoring the small variation in s₁, as follows.
T² = ((P + r)² - β²) / (m₁ - s₁)
∴ 2T δT = [(m₁ - s₁) × -2β δβ - ((P + r)² - β²) δm₁] / (m₁ - s₁)²
∴ δT = - β δβ / [T (m₁ - s₁)] - δm₁ ((P + r)² - β²) / [T (m₁ - s₁)²]
The first term on the Right hand side gives the variation
for δβ and the second for δm₁.
Fig. 76 shows the case of totality.
[Diagram: Fig. 76]
Fig. 76
M₁ M₂ = N₁ C₁ + C₁ C₂ ∴ The Moon has to over-
take C from the moment of the beginning of totality to
the moment of opposition by the distance C₁ N₁ with a
relative velocity of m₁ - s₁. Hence the time of Sammilana-
Marda-Khanda is equal to
[√(C₁M₁² - β²) × 60] / (m₁ - s₁) = [√((R - r)² - β²) × 60] / (m₁ - s₁)
as given, taking β to be constant. Similarly the Un-
milana-Marda-Khanda from the position (M₂ C₂) to the
position (M₃ C₃) will also be the same, taking β to be
constant.

369
Rectification of this time, when β is considered as
varying will proceed on the same lines as before.
Verse 13. Rectification of the times of Sparsa-Sthiti-
Khanda and Mōksha-Sthiti-Khanda.
From the position of the Moon and that of the Node
obtained for the moment of opposition, have to be com-
puted their positions for the moment of first contact and
those for the moment of last contact. For this, (T × v) / 60
is to be subtracted and added respectively to the positions
at the moment of opposition of the Moon and Node, where
T is the time of the Sparsa-Sthiti-Khanda, and v the daily
motion (of the Moon or the Node as the case may be).
From these positions β has to be computed for the moment
of first contact and that of last contact, and from this β
the time of Sparsa-Sthiti-Khanda and Mōksha-Sthiti-
Khanda have to be rectified by the method of successive
approximation.
Comm. From fig. 75, C₁ N₁ and C₂ N₂ are the dist-
ances gained by the Moon over the centre of the shadow
so that to get their correct values β₁ and β₂ are to be used
and not β. Hence β₁ and β₂ are to be computed using T
the time of Sparsa-Sthiti-Khanda and that of the Mōksha-
Sthiti-Khanda which are taken to be equal in the first
instance. Since T is the time taken as a first approxi-
mation, β₁, β₂ are also approximate in the first instance.
From these β₁, β₂ T is to be rectified and in this rectifi-
cation, we have T₁ and T₂ differing, as the times of Sparsa-
Sthiti-Khanda and Mōksha-Sthiti-Khanda. From these
rectified times again β₁, β₂ are further to be rectified and
from them again T₁, T₂ are to be further rectified. This
procedure is to be continued till constant values are
obtained for T₁ and T₂.
47

370
Note. T₂ will be less than or greater than T₁ accord-
ing as β₂ ≷ β₁.
Verse 14. Rectification of the Sammilana-Marda-
Khanda and Unmilana-Marda-Khanda.
Proceeding on the same lines as above and obtaining
β₃ and β₄ the rectified latitudes of the Moon for the
moments of the commencement and end of totality of the
eclipse, the Sammilana-Marda-Khanda and Unmilana-
Marda-Khanda, T₃ and T₄ are to be rectified.
Note. We have the formula sin β = sin λ sin i so
that by differentiating. we have cos βδβ = cos λδλ × sin i
sin i cos λ Δλ
∴ δβ = ——————————————
cos β
This formula gives in one stroke the rectified latitudes
of the Moon at the respective moments from which the
respective rectified times could be got.
Verse 15 and the first half of verse 16. The definition
of Bhuja and the method of finding it at an intermediate
point of time.
The word ‘Iṣṭa’ is used to connote ‘At any given
time’. The word ‘Spārśika-Iṣṭa’ means ‘At a given
time after the moment of first contact’; similarly the
word ‘Maukṣika Iṣṭa’ means ‘At a given time before
the moment of last contact’. (T–t) (m₁—s₁) where
(m₁—s₁) is in degrees (m₁ and s₁ of course being expressed
in minutes); T stands for the Sthiti-Khanda (Spārśika
or Maukṣika) and t stands for the Iṣṭa (Spārśika or
Maukṣika) gives the Bhuja. Similarly with respect to
obtaining the Marda-Bhuja. (The former is called Sthiti-
Bhuja).
Comm. In fig. 77, let C and M be the centres of the
Rāhu (cross-section of the shadow-cone at the lunar orbit)

271
Fig. 77
and the Moon respectively; let MN be the perpendicular
from M on the Grāhaka-mārga or the path of the eclipsing
body (ie. the ecliptic). Then CN is called the Bhuja at
the time.
At the moment of first contact, the value of CN is
√((P+r)²—β²) where β is the latitude of the Moon at that
moment. At any subsequent moment, from fig. 77, CN
is equal to √((P+r—AB)²—β²) where β is the latitude at
the subsequent moment and AB the portion of the radius
of the eclipsed body shaded. Hence we could obtain the
Bhuja at any subsequent moment, by computing the
latitude at that moment and the value of AB. But AB
could be computed only by knowledge of CN and β. Hence

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.