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

402
Latter half of verse 33. To know the magnitude of
the eclipse at any given moment during the course of the
eclipse.
Let the product of the time elapsed from the moment
of first contact and the length of the path of the eclipsing
body traced from the moment of the first contact to the
middle of the eclipse divided by the time between the
moment of first contact and the middle of the eclipse, be
x. Similarly let the product of the time before the end of
last contact and the path of the eclipsing body traced
between the middle moment of the eclipse and the moment
of last contact divided by the time between the middle
moment and the moment of last contact be y. Lay off
x and y units of length from the first and last points of
the path of the eclipsing body along the path respectively.
Then we get the points of the centre of the eclipsing body
at the required moments. With these points as centre
and radius p, if circles be drawn, they represent the
eclipsing body. The length of the diameter of the eclipsed
body shaded, gives the magnitude of the eclipse called
grāsa.
Comm. Here rule of three is applied namely “If
during time T₁ or T₂ a path equal to l₁ or l₂ in length is
traced what length will be traced in times t₁ or t₂?”,
where T₁ and T₂ are the times called Sparsa-Sthiti-Khanda
and Mokṣa-Sthiti-Khanda respectively, l₁ and l₂ are the
times elapsed from the moment of first contact or before
the moment of last contact and t₁, t₂ are the times from
the beginning of the eclipse and before the end of the
eclipse respectively. Then x and y give the points where
the centre of the eclipsing body lies.
Verse 35. Given the magnitude of the eclipse at any
time to obtain the time elapsed after the first contact.
The time taken by the centre of the eclipsing body to
move through the segment of the path of the eclipsing

408
body which lies between the position of the eclipsing body
at the moment of first contact and the point of intersection
with the path of the eclipsing body of the circle drawn
with the centre of the eclipsed body as centre and radius
equal to the difference of p+r -g where g is the magni-
tude of the eclipse (grāsa) at the moment, or similarly the
time taken by the centre of the eclipsing body to move
through a similar and equal segment of the path of the
eclipsing body on the other side, gives the time elapsed
after the moment of first contact or the time before the
moment of last contact.
Comm. This is the converse of the above problem.
The method is clear being based on rule of three as above.
Both the problems could be algebraically expressed as
follows. Let T, t, l, g, and k, stand respectively for the
Sthiti-Khanda ie. the time between the moment of first
contact to the middle of the eclipse or the time between
the middle moment to the moment of last contact ; (2).
the time elapsed after the moment of first contact or the
time before the moment of last contact, as the case may
be ; (3) the length of the Pragrahamārga or Mokṣamārga ;
(4) the grāsa which is defined as p+r-k ; (5) the Karṇa
whose expression is √(B²+β²), B being the Bhuja defined
and β the latitude of the Moon.
Then the following working is stipulated (a) If in
time T, a path of length l is traced, what will be traced in
t? The result is lt/T (b) Then B = l - lt/T (c) B² + β² = K²
(d) p+r-k=g. Thus combining all the steps
{l (T-t) / T}² + β² = (p+r-g)² ie.
l² (T-t)² + β² T² = T² (p+r-g)² I
given t, this equation gives g and given g it gives t.

404
Again the following relation holds good between T
and l, l² = (p+r)² – β² II and
l / (m₁–s₁) = T with the nomenclature already employed which
means T = √(p+r)²–β² / (m₁–s₁) III
In the above working, the fundamental elements are
p, r, β, m₁ and s₁ with which the other elements could be
worked out. Replacing the other elements from equation
I, we have
{(p+r)²–β²} { √(p+r)²–β² / (m₁–s₁) – t }² +
β² (p+r²–β²) / (m₁–s₁)² = (p+r)²–β² / (m₁–s₁)² (p+r–g)²
ie. {(p+r)²–β²} [√(p+r)²–β² – t (m₁–s₁)]² +
β² (p+r²–β²) = (p+r²–β²) (p+r–g)²
ie. {√p+r²–β² – t (m₁–s₁)}² + β² = (p+r–g)² IV
Putting t=0 in this equation, we have
(p+r)² = (p+r–g)² ie. g=0 which means at the moment
of first contact, the grāsa is zero. Again putting
t = T ie. t = √(p+r)²–β² / (m₁–s₁) ie. t (m₁ – s₁) = √p+r²–β²
we have
β² = p+r–g² ie. g = p+r–β which gives the grāha
at the middle of the eclipse which was defined as the
Sthagita. In equation IV which we may take as a funda-
mental equation, the two unknowns are t and g one of
which being given the other could be got.
Verse 36. The colour of the eclipse.
When less than half the disc of the Moon is eclipsed,
the colour will be what is called Dhumra ie. of the colour

405
of smoke; when the disc is half eclipsed, the colour is
black; when more than half is eclipsed, the colour would
be a blend of black and red and when the entire disc is
eclipsed, the colour will be what is called pisanga or
reddish-brown.
Comm. Clear.
Verse 37. When declare the occurrence of an eclipse.
When even one-sixteenth of the diameter of the
Moon's disc is shadowed, the eclipse will not be visible
in as much as the shadowed portion is covered by the
illaminating rays of the disc. In the case of the Sun,
when even one-twelfth of the diameter is shadowed, the
eclipse will not be visible for the same reason. Hence we
shall not declare the occurence of an eclipse upto the
shadowing of the discs to the extents stated above.
Verses 38 and 39. Examples which disclose the
invalidity of construing Valana in terms of Hversine
instead of Hsine.
When the Sun is in the zenith, the Ecliptic being
vertical, the Valana is clearly seen to be the Agra of
(☉+90) where ☉ is the longitude of the Sun. If you
could show that the Valana will be the same on the basis
of Hversine-formula, then I would accept that what
Lallāchārya postulated in his work Śiṣya-Dhī-Vṛddhida
is correct.
Again, in a place of latitude 90—ω, ω being the obli-
quity of the Ecliptic (ω is taken to be 24°), when the Sun
being situated in Meṣa, Vṛṣabha, Mīna or Kumbha, the
Moon contacts him from the south at the moment of a
solar eclipse, in as much as the Ecliptic coincides with
the horizon. In this circumstance, how could the Valana
be equal to R, as made out by the Hversine-formula.

406
Comm. Lallāchārya gave the Valana in terms of the
following verses “स्पर्शादिकालजनितोत्क्रमशिञ्जिनीभिः, क्षुण्णाक्षभा
पलभवश्रवणेन भक्ता, चापानि पूर्वनतपश्चिमयोः फलानि, सौम्येतराणि
समवेहि पृथक् क्रमेण; ग्राह्यात् सराशित्रितयाद् भुजज्या व्यस्ता ततः प्राग्वदप-
क्रमज्या...” Verses 23, 25 Chandragrahaṇādhikāra, wherein
he formulated the Valana in terms of Hversine in the
place of Hsine. The reason for his slip, we have already
explained. Now Bhāskara gives two glaring examples to
substantiate his formula and to show up the flaw in
Lallāchārya's formulation.
In the first example, where the Ecliptic takes the form
of a Vertical, the Sun being in the zenith, the Spaṣṭa
Valana which is the angle between the Ecliptic and the
prime-vertical is the same as the arc between the East
point and the intersection of the Ecliptic with the horizon
known as Lagna. Since the Sun is then in the zenith,
the longitude of the Lagna is (90 + ☉) so that the said
arc is the Agra of the point whose longitude is 90 + ☉ as
stated. Hence Spaṣṭa Valanajyā = sin A = sin δ/cos ϕ
where A is the agrā (using Napier's rule from triangle
PNL where L is the Lagna N the north-point and P the
celestial pole). In the Hindu form, this is given by
H sin V = H sin A = (R H sin δ) / (H cos ϕ) where δ is the declination
of a point of the Ecliptic whose longitude is (90 + ☉). But
Lallāchārya's formula gives the Valanajyā as Hvers δ,
δ being the declination of a point of the Ecliptic whose
longitude is (90 + λ), λ being the longitude of the Eclipsed
body ignoring the latitude. In other words, in the case of the
lunar Eclipse when the Moon is in the zenith his Valanajyā
= Hvers δ (δ having the above value) the Ākṣa Valanajyā
here being zero. Since (R H sin δ) / (H cos ϕ) ≮ Hvers δ, the mistake
committed by Lallāchārya is evident even supposing
H cos ϕ = R when we ignore the latitude ie. take ϕ to
be zero.

407
In the second example cited by Bhāskara the Ecliptic
coincides with the horizon, the pole of the Ecliptic being
in the zenith. Then in a Solar Eclipse the Moon eclipses
the Sun from the south showing that H sin V = R. That
H sin V = R is also evident from the fact that the Ecliptic
makes 90° with the prime-vertical, having coincided with
the horizon. But here according to Lallāchārya’s formula,
H sin ξ = Hvers 90° = R and Āyana Valanajyā is Hvers δ,
where δ is the declination of a point whose longitude is
90° more than ☉. If ☉ = 30°, 60° H sin θ =
R/2 sin ω/R or (√3/2) (R sin ω)/R ie. (sin ω)/2 or √3/2 sin ω.
Evidently the sum of the two Valanas Āyana and Ākṣa
cannot be 90° as is also vouchsafed from geometry. So,
here also, the flaw is evident.
Note 1. Śrīpatyāchārya also followed Lallāchārya
vide verses 18, 19, 20 Chandragrahaṇādhyāya, Siddhānta
Śekhara. It will be noted that the commentator of
Siddhānta Śekhara, while reiterating Bhāskara’s stand as
the correct one, himself commits a mistake in saying
“सममण्डलीय नतांशज्यास्थाने नतकाकोत्क्रमज्या गृहीता” In fact
sin ξ = (sin ϕ sin h) / cos μ = (sin ϕ sin z) / cos δ as proved by us before.
The commentator cited above overlooked that sin ξ could
be also equal to (sin ϕ sin h) / cos μ , wherein natakāla also is
implied.
Note 2. It will be noted that even Pṛthūdakāchārya,
while commenting on Brahmasphuṭa Siddhānta, ignored
Brahmagupta and followed Lallāchārya blindly.
Note 3. The formula given by Lallāchārya and
followed by Pṛthūdaka as well as by Śrīpati is very rough
besides containing the flaw cited, in as much as both μ
and δ are taken to be zero, which are not so,

SURYAGRAHANĀDHIKĀRA
Verse 1. In as much as the observer situated on the
surface of the Earth and as such elevated by the radius
of the Earth from the centre there of, perceives not the
Sun and the Moon having the same longitude at the
moment of conjunction, to be in the same line of sight,
heyt being depressed unequally having different orbits, so
I proceed to elucidate what are called Lambana and Nati
ie. parallax in longitude and latitude, on which account
they are not in the same line of sight.
Fig. 91
Comm. (Refer fig. 91) Let E be the centre of the
Earth, M and S the centres of the discs of the Moon and

the Sun. Let A be the position of an observer on the
surface of the Earth, elevated by the radius EA from E.
Let M and S be in the same line of sight as seen from E.
But as seen from A, AS and AM are respectively the lines
of sight to the Sun and the Moon. Evidently these lines
of sight differ the Moon being depressed more than the
Sun. If a line AS' be drawn which is parallel to the central
line of sight namely EMS, we find that the Sun is
depressed by the angle S'AS whereas the Moon is depressed
by the angle S'AM'. These angles differ because the orbits
of the Sun and Moon differ.
Here the angle S'AS will be very very small, its
magnitude being in truth just about 8" only. But the
angle S'AM' will be sufficiently large since the Moon is
very near the Earth compared with the Sun. Taking ES
and AS to be almost parallel due to the largeness of the
Sun's distance, the angle SÂM will be almost equal to
AM̂S so that we could consider that the Moon is depressed
from AS the line of sight to the Sun by the angle SÂM' =
AM̂E. This angle AME is called the geocentric parallax
of the Moon and the angle AŜE that of the Sun M'ÂS =
angle of depression of the Moon over and above that of
the Sun = M'AS' - M'AS = EM̂A - EŜA = geocentric
parallax of the Moon minus geocentric parallax of the Sun.
Verse 2. The presence or absence as well as the
positiveness and negativeness of the parallax in longitude.
Compute the Lagna at the moment of conjunction of
the Sun and the Moon. There will be no parallax in
longitude when the Sun is situated at the point called
Vitribha or the point whose longitude is ≡ L - 90°, L
being the longitude of the Lagna ie. the ascendant which
is the point of intersection of the Ecliptic with the
52

410
horizon. If the Sun's longitude falls short of the longitude
of the Vitribha or exceeds it, there will be parallax in
longitude which will be positive in the former case and
negative in the latter.
Fig. 92
Comm. (Ref. fig. 92) Let SN be the horizon, Z the
zenith and VA the Ecliptic. A is the ascendant or Lagna.
Let V be the point called Vitribha which is 90° behind A.
Strictly speaking V is called Vitribhalagna or lagna from
which three Rāśis or 90° are subtracted (Bha = Rāśi.
त्रिभिर्विरहितम् वित्रिभम्; वित्रिभम् च तत् लग्नम् च वित्रिभलग्नम् ie.
a point whose longitude is got by subtracting three
Rāśis from that of the Lagna). Let ZV be the vertical of
V so that ZV̂A = 90°. It will be seen that AV = 90° as
follows. Let A' be the point where the Ecliptic intersects
the horizon on the west. One will construe that the
Ecliptic is bisected by the meridian; but it is not so.
Spherical triangles AVZ and A'VZ being right-angled at

411
V are congruent because AZ = A'Z and ZV is common
∴ AV = VA'. But AV + VA' = 180° because the Ecliptic
and the horizon being two great circles, they bisect each
other. Hence AV = 90°. Then a celestial body situated at
V will be depressed along ZV the vertical, say, to a point B.
Let ☉ be any arbitrary position of the Sun; then ☉ will
be depressed along the vertical Z☉, say, to a point C.
Draw CD perpendicular on the Ecliptic. Then ☉D is the
component of the parallax ☉C along the Ecliptic where
as DC is its component perpendicular to the Ecliptic.
Thus ☉D is the parallax in longitude and DC is the
parallax in latitude. The word ‘Lambana’ means
etymologically लम्बते अनेनेति लम्बनम् ie. that amount by
which the celestial body is depressed (along the Ecliptic).
In Hindu Astronomy the word Lambana is applied to
parallax in longitude alone whereas the word Nati is
applied to parallax in latitude. Hence to translate
Lambana as parallax alone is not correct. The word
Drik-lambana is applied to mean parallax along the
vertical, and the word Sphutalambana is occasionally used
to connote parallax in longitude.
As Bhāskara rapidly comments on the verses in this
Gaṇitādhyāya, he having dealt with the subject of parallax
elaborately under the caption, Grahaṇa Vāsanā, in the
Golādhyāya, to catch up his thought, we have to treat the
subject first from the modern view point and then elucidate
what he has said in the Golādhyāya, much matter of which
is reiterated by him under the commentary here in the
Gaṇitādhyāya.
(Ref. Fig. 91) From the △EAM,
(sin EMA) / a = (sin EÂM) / d = (sin ZÂM) / d where a is the radius of
the Earth and d the distance of the celestial body (here the
Moon)

412
∴ Sin EMA = a/d sin ẐAM = ÊMA expressed in radian
measure since ÊMA is very small
∴ ÊMA (expressed in radian measure) = a/d sin z I where
z is the apparent zenith-distance of the Moon ie. zenith-
distance as seen by the observer (in contradistinction to
the geocentric zenith-distance of the Moon namely ẐEM).
In particular, when z = 90°, ÊMA = a/d which is the
maximum parallax known as the horizontal parallax ie.
the parallax when the Moon is situated on the horizon of
the observer. Also the parallax is zero when M is situated
at Z as is seen from formula I and as is rightly remarked
by Bhāskara in the words 'खमध्ये नास्ति लम्बनम्'.
In fig. 91, ÊMA is the angle by which the line of
sight of the observer namely AM is depressed from the
geocentric line of sight EM. Since the plane of the paper
represents a vertical through the Sun and the Moon, the
depression of either the Sun or the Moon or the excess of
the depression of the Moon over the Sun are all in the
vertical plane. This depression is called Drik-lambana
because it is a lambana or depression in the Drik-maṇḍala
or vertical.
This Drik-lambana varies as sin z as is seen from
formula I where a, and d may be taken to be constants.
(Both a and d vary slightly a varying slightly from place
to place on the Earth, the Earth being an oblate spheroid,
and d varying from position to position of the Moon).
The maximum horizontal parallax is given by a/d in
radian measure which is equal to, according to Bhāskara's

418
estimate 1581/2 × 1/51566 × 3438 = 53′ approximately. Its
modern value is about 57′ so that the Hindu estimate is
not far from truth. Here 1581 and 51566 are the values
of a and d in Yojanas according to Bhāskara.
The Hindu astronomers do not, however, proceed
exactly as we have done in the para above to obtain the
maximum horizontal parallax. Their treatment is a little
different and is as follows. Whereas according to Modern
astronomy E M̂′ A (fig. 93) is viewed as the horizontal
[चित्र: Fig. 93]
Fig. 93
parallax, in Hindu Astronomy MÊM′ or the angular
measure of the Moon’s path equal to the radius of the
Earth is taken to be the horizontal parallax. Both,
of course, mean the same as is seen from the figure.
In as much as the Hindu astronomers knew very well
what they term कलीकरण or converting linear distances
into angular measure, converting a linear magnitude equal
to the radius of the Earth namely MM′ at the lunar orbit
into angular measure, they got
MM′/EM × 3438 = 1581/2 × 1/51566 radians = 52′-42″ as the
maximum horizontal parallax.

414
Having got this estimate, they reckoned this angular
measure in time as the time taken by the Moon to traverse
the distance MM' equal to the radius of the Earth as
follows. The Moon traverses 790′-35″, which is exactly
15 times 52′-42″. Hence they said that the maximum
horizontal parallax is 1/15th of the Moon's daily motion in
arc and expressing it in terms of time, that the maximum
horizontal parallax is 1/15 of a day or 1/15th of 60 nādis or
4 nādis.
That this horizontal parallax is 4 nādis as a maximum,
would have been also verified at the time of a solar eclipse
when the Sun was situated on the horizon at the time of
conjunction, by the fact that the eclipse occurred four nādis
in advance of the moment of geocentric conjunction
(which could be calculated very accurately by the Hindu
astronomers, as could be seen by the very correct estimate
of a lunation in Hindu Astronomy. In fact, the length
of a lunation must have been estimated correctly by noting
the time-interval between two solar eclipses or lunar and
by dividing that time by the integral number of lunations
elapsed in between the two eclipses).
The question then arises as to how the Hindu Astro-
nomers could know the distance of the Moon. From the
estimate of the horizontal parallax by actual observation,
and from the geometry of fig. 93, a correct estimate of the
distance of the Moon must have been arrived at.
Having thus known that the Moon traverses a
distance equal to the radius of the Earth in 4 nādis, his
daily linear motion was estimated to be 15 times the radius
of the Earth ie. (15 × 1581 1/6) / 2 = 11858 3/4 Yojanas.
The daily motion of the Moon having thus been
estimated almost correctly, an act of inexpedience on the
part of the Hindu astronomers was that they should have

415
presumed that all the other planets including the Sun would
be traversing the same linear distance during the course of
a day. This led to a wrong estimate of the Sun’s distance
as well as his spherical radius. Also, they supposed wrongly
that the parallax of the Sun also would be equal to 1/15th
of his daily arcual motion.
Their estimate of the spherical diameter of the Moon
was, however, very near the truth, for, they argued, that
if 790′-35″ angular motion per day corresponded to 11858¾
Yojanas in linear measure, to what linear measure did the
angular diameter of the Moon namely 32′-0″-9′′′ corres-
pond? The answer was
11858¾ × 32′ 1/400
------------------ = 480 Yojanas.
790′-35″
[चित्र: Fig. 94]
Fig. 94
It may be here pointed out that there is a relation
between the angular radii of two celestial bodies as seen
from each other and their mutual horizontal parallaxes
(Fig. 94). Let E and M bet he centres of the Earth and
the Moon respectively. E M̂ A = horizontal parallax of
the Moon = angular radius of the Earth as seen from the
Moon and B Ê M = Angular radius of the Moon = Hori-
zontal parallax of the Earth as seen from the Moon. Thus,
we see that the Earth will be seen from the Moon, as a
Moon with an angular radius equal to 57′. In other words
our Earth will be a Moon to our Moon, having nearly 16
times the area of our Moon’s disc.

416
The periphery of the Moon’s orbit was arrived at as
follows. “If 790′-35″ of the Moon’s angular motion
corresponds to 11858¾ Yojanas to what periphery must
360 × 60′ correspond ?” The answer is
(11858¾ × 360 × 60) / (790′-35″) = 324000 Yojanas. Reverting to the
subject of parallax on hand, the Drik-lambana or the
parallax along the vertical has the formula (4 H sin z) / R I
nādīs in Hindu Astronomy where 4 nādīs is the maximum
parallax obtained when H sin z = R.
From fig. 92, C☉² = CD² + D☉² ie.
Drik-lambana² = Nati² + Sphuṭalambana² II
CD = ☉C sin CÔD = (4 H sin z / R) ×
sin CÔD = 4 sin z sin VÔZ
= 4 sin ☉Z sin VÔZ = 4 sin ZV
= (4 H sin ZV) / R = VB III
Thus, the parallax in latitude at any point of the Ecliptic
is that at the Vitribha which is conveyed by Bhāskara
in the words “कक्षयोरन्तरं यत् स्यात् वित्रीभे सर्वतोऽपि तत्”.
Also ☉D = C☉ cos CÔD = 4 sin ☉Z cos Z☉V
= 4 cos ZV sin V☉ = (4 H cos ZV H sin V☉) / R² IV
= Maximum parallax × Vitribha-Sanku × H sine of the
arc V☉.
In the above working we proceeded in a modern way.
It is worth-hearing Bhāskara as to how these results were
arrived at elegantly and ingeneously from first principles.

417
In fig. 91, EMS is called Garbha-Sūtra whereas AS
is called Dṛṣṭi-Sūtra. दृक्सूत्रात् लम्बितश्चन्द्रः तेन तल्लम्बनं
स्मृतम् ie. In as much as the Moon is depressed from the
Dṛk-Sūtra, so this phenomenon goes by the name Lambana.
It will be noted that in Hindu Astronomy geocentric
parallax will not be treated separately for the Moon and
the Sun but dealt with simultaneously as it is called for,
in the context of a solar eclipse. They were interested
in knowing the relative depression of the Moon with
respect to the Sun rather than knowing the separate
magnitudes with respect to the Moon and Sun, for which
they had no application.
'दृग्गर्भसूत्रयोरैक्यात् खमध्ये नास्ति लम्बनम्' ie. In as much
as the Garbha-Sūtra and Dṛk-Sūtra are identical in the
direction EAZ (fig. 91) there is no parallax at the zenith.
Now consider the plane through ZV of fig. 92. Suppose
the EMS of fig. 91 is in the direction EV. Then both
the Sun and the Moon may be considered to have the
same Vitribha at that moment of conjunction. Both the
Sun and the Moon being then depressed along ZV, to V'
and V'' respectively the Ecliptic will then be a circle
parallel to VA (fig. 92) through V' and the orbit of the
Moon will be another circle parallel to VA through V'', V''
being below V'. If we neglect, for a moment, the depres-
sion of the Sun, and consider VD to be the Ecliptic on
which the Sun is situated undeflected, and BC to be the
deflected orbit of the Moon relative to the Sun, then VB
is the Nati of the Moon, which will be the same distance
between VD and BC, ie. the orbits of the Sun and the
Moon.
This fact was proved by us analytically in the modern
way showing that CD = BV.
This Nati it is that influences the latitude of the Moon,
which may cause apparent conjunction when there is no geo-
53

centric conjunction and which does not show an apparent
conjunction when there is a geocentric conjunction. In
other words parallax in latitude plays a very important part
in solar eclipses. Also ☉ D being the parallax in longitude,
the moment of apparent conjunction might be preceded or
followed by a geocentric conjunction according as the Sun
lies along VA or AV. Thus having determined the exact
moment of apparent conjunction using the magnitude of
☉ D, then we have to rectify the latitude using the
magnitude of VB. If that rectified latitude of β falls
short of R+r where R is the angular radius of the Sun and
r that of the Moon, then there will be a solar eclipse.
It will be noted that when the Sun coincides with V
at the moment of conjunction, there is no parallax in
longitude ☉ D being zero (Fig. 92) in that position. Also
there will be no parallax in latitude when the Ecliptic
assumes the position of a vertical circle passing through
the zenith, the Drik-lambana then being entirely along
the Ecliptic. In this case the Vitribhalagna V will
coincide with Z and (4 H sin V ☉) / R which is termed the
Madhyamalambana is now entirely along the Ecliptic and
as such it is the Sphutalambana in this case. We have said
above that when V coincides with Z, the Madhyamalambana
is zero at Z, and that the maximum is equal to 4 nādīs on
the horizon. In between Z and the horizon it has the formula
(4 H sin V ☉) / R . Noting further that in this case when
the H cosine of the zenith-distance of V ie. the Śanku of
V is R, the entire lambana is along the Ecliptic, the nāti
being zero, and the Madhyamalambana is itself the Sphuta-
lambana, and again when V does not coincide with Z,
H cos ZV is no longer R but has assumed Kōti-Rūpa ie.
the form of a H cosine, as well as the Sphuta-lambana
also, which assumes Kōti-Rūpa ie. of the form ☉ D of
Fig. 92, where ☉ C is the Madhyamalambana, ☉ D is the

419
Kōti = Sphuta-lambana, and DC = Bhuja = Nāti, it is
argued that the Sphuta-lambana is proportional to
H cos ZV, assuming a maximum value when V coincides
with Z (☉ not being at Z).
Verses 3 and 4. Parallax in longitude based on two
proportions.
Compute the H cosine of ZV, by calculating the
rising time of AV, the Kujyā, Dyujyā and Antyā pertain-
ing to V, as was formulated in the Tripraśnādhikāra, then
H sin V☉, multiplied by 4 and divided by R, and again
multiplied by H cos ZV and divided by R again gives the
parallax in longitude.
Comm. As per the above formula, parallax in longi-
tude equal to ☉D of Fig. 92 is equal to
(4 H sin V☉ × H cos ZV) / R² . This is evidently derived out
of two proportions that the parallax in longitude is
proportional to H sin V☉ as well as H cos ZV. This we
have already derived through modern methods as formula
IV.
Under verse 2. The two proportions are (1) V coin-
ciding with Z, if by H sin V☉ equal to R, we have 4 nādis
as the maximum lambana on the horizon, what shall we
have by an arbitrary H sin V☉? The result is
(4 H sin V☉) / R and (2) V not coinciding with Z, if by
H cos ZV equal to R we have (4 H sin V☉) / R as the
Madhyamalambana, what shall we have for an arbitrary
H cos ZV? The result is
(4 H sin V☉) / R × (H cos ZV) / R as formulated.
First half of verse 5. Alternate method of rectifying
lambana. The Madhyamalambana multiplied by 12 and

420
divided by the Chāyākarṇa of the Vitribha will also give
the Sphuta-lambana.
Comm. From Tripras'nādhikāra, we have
12 / K = (H cos Z) / R so that in the formula cited above instead of
(H cos ZV) / R we are asked to use 12/K.
Latter half of verse 5 and first half of verse 6.
Dṛk-nati² = H cos² ZV − H cos² Z☉
= H sin² Z☉ − H sin² ZV (fig. 92)
(4 Dṛk-nati) / R = Sphutalambana.
Comm. We shall first prove this on modern lines.
Cos Z☉ = cos ZV cos Z☉
∴ cos² ZV − cos² Z☉ = cos² ZV (1 − cos² V☉)
= cos² ZV sin² V☉ = (H cos² ZV H sin² V☉) / R⁴
Also cos² ZV − cos² Z☉
= sin² Z☉ − sin² ZV =
(H cos² ZV − H cos² Z☉) / R² = (H sin² Z☉ − H sin² ZV) / R²
∴ H cos² ZV − H cos² Z☉ = H sin² Z☉ − H sin² ZV
= (H cos² ZV H sin² V☉) / R⁴
∴ Dṛk-nati defined above = √(H cos² ZV − H cos² Z☉) =
√(H sin² Z☉ − H sin² ZV) = (H cos ZV H siu V☉) / R
∴ (4 Dṛk-nati) / R = (4 H cos ZV H sin V☉) / R²
= Sphutalambana.

421
Bhāskara's proof proceeds in two stages from first
principles. In the first place when V coincides with z,
Lambana is seen to be equal to 4 nādis on the horizon and
zero at Z ie. it is zero when H sin Z☉ = 0 and 4 nādis,
a maximum when H sin Z☉ = R. So, it is meet that
Lambana should be taken to be proportional to H sin Z☉
ie. proportional to natajyā. In this context the lambana
termed as Madhyamalambana is entirely along the ecliptic.
It is taken to be in the form of Karṇa, because in the
position of ☉C also it is in the form of a Karṇa.
Then let the Ecliptic be deflected from the zenith
(deflected = क्षिप्त). Vitribhalagna then being deflected
from the position of Z, occupies the position of V (fig. 92).
So ZV is called Dṛk-kshepa since the Ecliptic which was
in the form of a Dṛk-mandala is deflected from that posi-
tion. Also the circle ZV is called Dṛk-kshepa-mandala
because V is deflected along that circle. Now consider
the △ whose sides are H sin ZV, H cos ZV and R.
H cos ZV equal to R and as such in the form of a Karṇa
corresponds to the Madhyamalambana which is also in the
form of a Karṇa ; when this Vitribha-Sanku assumed the
form H cos ZV, ie. rendered a Kōti from its form of a
Karṇa, R, the Sphutalambana is also rendered a Kōti in
the form of ☉ D so that
Madhyamalambana / R = Sphutalambana / H cos ZV
∴ Sphutalambana = (H cos ZV / R) Madhyamalambana
= (H cos ZV / R) × (4 H sin Z☉ / R) .
Then Bhāskara says that we could look at this, from
another angle in the words “यदेव स्फुटलम्बनस्य कोटिरूपत्व-
मुपपन्नम् etc. ”.