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

LUNAR ECLIPSES
Verse 1. The ritualistic purpose served at the time
of an eclipse.
Scholars of Smṛtis and pūrāṇas declare that prayer,
charity or offerings to gods made in fire at the moment of
an eclipse conduce to much spirituality. Hence I give
hereunder the methods of computing the moment of an
eclipse (lunar or solar) in as much as such a knowledge
apart from its religious importance, is also wrought with a
beautiful mathematical treatment.
Verse 2. The initial procedure to be adopted to
compute an eclipse.
To know the occurence of a solar eclipse, find the
exact moment of the New Moon, which is indicated by the
equality of longitudes of the Sun and the Moon, and to
know the occurence of a lunar eclipse compute the exact
moment of the full Moon which is indicated by the fact
that M = S + 180° where M and S are the longitudes of
the Moon and the Sun, agreeing in degrees, minutes and
seconds, though differing in Rasis by six. Also compute
the longitude of the Node (Rāhu) for the moment as
directed.
Comm. Having ascertained the possibility for the
occurence of a solar eclipse, we are directed to compute
the positions of the Sun, the Moon and Rāhu for the day
of the New Moon. The Sun and the Moon are to be
rectified for corrections like Desāntara, Bhujāntara,
Udayāntara etc. From the elongation of the Moon the
tithi is to be computed and the method of successive
approximations called Chālana Karma is to be used to
obtain the exact moment of conjunction. This process of

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Chālana is as follows. At first knowing the elongation of
the Moon at Sunrise, by rule of three, using the then daily
motions of the Sun and Moon, the moment of the
conjunction is to be computed. Again the positions of
the Sun and Moon are to be computed for that moment
and also their daily motions are to be rectified for the
moment. With these rectified daily motions and with the
then positions of the Sun and the Moon, again the
moment of conjunction is to be computed. Repeating the
process till an invariable answer is reached, we have the
exact moment of conjunction. For that moment, the
position of the Rāhu is also to be calculated. Similar
is the procedure for a lunar eclipse also. It is to be
noted that the correction called Natakarma, which we
formerly identified to be the correction for ' Astronomical
Refraction' is also prescribed here as particularly
mentioned by Bhāskara, in the commentary. (Vide verses
68, 69 Spaṣṭādhikāra).
Verse 3. The magnitndes of the orbits and the .
orbital radii of the Sun and Moon.
The distances of the centres of the globes of the Sun
and the Moon from the centre of the Earth in yojanas are
respectively 689377 and 51566.
Comm. We saw in the Kakshādhyāya of the first
chapter as to how these distances were estimated. Some
scholars pronounced these distances are parameters; but
as per the modern estimate of the Earth's radius as
compared with that of Bhāskara, (The method indicated
by him in the Bhuparidhimānādhyāya of chapter I, is
quite mathematical) a yojana equals five modern miles
and with this correspondence, the distance of the Moon
is very near the truth. So to say that the distances
given above are mere parameters is wrong; also if one
of the parameters gives a correct value of the quan-

848
tity in question, the other also should; but because
the latter does not, to call them parameters is merely
meaningless.
It is interssting to note that in the commentary under
this verse, Bhāskara says “ If for a circumference of 3927,
the diameter will be 1250, ...” This means that Bhāskara
takes π = 3927 / 1250 = 3·1416 which compares very well with
the modern value 3·14159. The value of π adopted
by Bhāskara in this, seems to have been taken from Lallā-
chārya’s Śiṣyadhīvṛddhida, Chandragrahaṇādhikāra verse
no. 3.
“ शरयमाङ्गहता 625 भनवाग्निहृत्
ग्रहवृतिश्रवणः फलमुच्यते ”
Verse 4. Computation of what is called the ‘ Kalā-
karṇa ’.
The radius vector is to be computed even in the case
of the Equation of centre as we did in the case of Śīghra-
phala. If it be ‘ K ’, R² / (2 R - K) will be what is Kalākarṇa
both in the case of the Sun, as well as the Moon.
Comm. While obtaining the Equation of centre, the
formula used was (r sin m) / R whereas, strictly speaking, it
should have been as in the case of the Śīghraphala, after
effecting the so called ‘ Karṇānupāta ’ (r / K) sinm. While
trying to answer why this Karṇānupāta was not done there
also, Brahmagupta gave such an answer as made Bhāskara
exclaim ‘ यतो विचित्रा फलवासनाऽत्र ’ i.e. ‘ It is really curious
in this respect.’ Bhāskara was really a most rational
type of astronomer, and one will not fail to appreciate his
sense of rationality when he declares that (1) “ अस्मिन्
गणितस्कन्धे उपपत्तिमानेव आगमः प्रमाणम् ”: and when he was

349
unable to adduce a proof he declares (2) 'उपलब्धिरेव
वासना' meaning thereby (1) 'In this branch of science, we
reckon only such an authority which has a proof behind
it ie. which could be substantiated' and (2) 'There is no
proof in this but accordance with observations alone
has to be taken as a proof'.
Though in the case of formulating the equation of
centre Karṇānupāta was not stipulated to simplify matters,
as there was not much difference, the Equation of centre
being generally small. It is to answer such contexts as
this that Bhāskara said in the Golādhyāya 'स्वल्पान्तरत्वात्,
अबहूपयोगात्, प्रसिद्धभावाच्च बहुप्रयासात्, ग्रन्थस्य तद्वैर्गुरुताभयेन
यत्त्यज्यतेऽर्थो न स दूषणाय" ie. 'If we in some particular
context do not mention certain things it should not be
condemned because in such contexts, (1) there is not
much difference or (2) no useful purpose is served to
a good extent or (3) it is too clear as does not require
to be mentioned (4) the procedure implies a lot of
cumbrous calculations and the result is after all negli-
gible and (5) Mention will make the work on hand
too unwieldy and brevity which is the soul of wit is
to be sacrificed. In the present context this procedure
of 'rectification of the Karṇa' is sought to improve
matters. Bhāskara's words 'यदा ग्रहस्य कर्ण उत्पन्नः तदा कर्णो
व्यासार्धं ग्रहकक्षायाः' seem really to imply that in the formula
r/R H sin m for the equation of centre in the place of R,
we are called upon to substitute really K. Though we had
been in default for not doing so in the context of the
Equation of centre, there is no reason why we should not
make up the deficiency in this context. So, therefore,
this rectification of Karṇa is stipulated here.

350
The procedure originally called for a rectification is
that taking K to be R, we have to compute r and again
taking the resulting K to be R, we have to compute r and
so on repeating the process till an invariable value for K is
obtained. This means that we should go on substituting
for r, r K / R . Instead of following this laborious process
of ' Asakṛt-Karma ' ie. method of successive approxi-
mations, Bhāskara gives an alternative in the verse, whieh
is as follows.
Let K be the value of the Karṇa, for a value r of r.
Since we are directed to make this K as R, ie. we have to
add R—K to K thus making it R, we add also R—K to R,
to keep the relative position of R and K to be almost the
same. In other words considering the fraction K / R , adding
R—K to both the numerator and denominator we have
R / (2 R—K) whieh means that for a radius 2 R—K, the Karṇa
will be R; that being so for a Radius R what will be the
Karṇa ? The result is R² / (2 R—K) as given.
It will be noted that the above interpolative procedure
is adopted as a short cut technique to the otherwise
laborious process. The mathematical correctness of this
procedure will be seen from the following analysis.
The problem is to change the Mandaparidhi to a
radius K of the deferent by the formula (as indicated by
Bhāskara in the course of the commentary) r' = r K / R
so that δr = r¹ — r = r K / R — r = r (K—R) / R (1) Now we
have K² = R² + r³ + 2 R r cos m construing R and m as
constants we have to find δ K for δ r

351
Differentiating 2 K δ K = 2 r δ r + 2 R cos m δ r
ie. δ K = [δ r (r + R cos m)] / K . But from fig. 65
(which is a portion of the epicyclic figure)
M̂₁ = 180—m
Ô₁ = θ = Mandaphala
M̂₂ = m—θ
Fig. 65
r = K cos m̅—̅θ̅ — R cos m so that r + R cos m = K cos (m — θ).
Substituting in the above,
δ K = [δ r × K cos m̅—̅θ̅] / K = δ r cos m̅—̅θ̅
But δ r from (1) is [r (K—R)] / R
∴ δ K = [r (K—R)] / R cos m̅—̅θ̅. But from the tri-
angle of fig. 65. K = R cos θ + r cos m̅—̅θ̅ so that
r cos m̅—̅θ̅ = K—R cos θ. But θ being small cos θ may
be taken to be unity so that r cos m̅—̅θ̅ = K—R. Again
substituting in the above
δ K = [(K—R)²] / R (2) Now as per the formulation of
Bhāskara
K¹ = R² / (2 R—K) = R² / (R + R — K) = R / [1 + (R—K)/R] = R / [1 — (K—R)/R]
= R (1 — (K—R)/R)⁻¹
Since |(K—R)/R| < 1, expanding binomially,

352
K¹ = R (1 + (K - R) / R + (K - R)² / R²) = R + K - R + (K - R)² / R
= K + (K - R)² / R ∴ K¹ - K = δ K = (K - R)² / R as found
above.
Note. Bhāskara, having formulated this, appeals to
' Dhulikarma' ie. arithmetical computation, for convin-
cing those who may not be able to follow his logio. Here
one may note also the wrong directive given by the
Samśodhaka in the text. The proof furnished by us above
gives a mathematical veracity to Bhāskara's formulation.
Verse 5. To rectify the Yōjanakarṇa or the spatial
radius Vector.
The above Kalākarṇa multiplied by the Karṇa given
in Yōjanas and divided by the Radius gives the rectified
Yōjanakarṇa.
Comm. In the formula given above δ K = (K - R)² / R
which is in units of spatial minutes (on the scale of R=3438).
(K' × y) / R where K' is the rectified Kalā-karṇa, and y the
Yōjanakarṇa given in verse 3, gives the rectified Yōjana-
·karṇa.
Second half of verse 5. The spherical radii of the
Sun and the Moon.
The spherical diameters of the Sun and the Moon are
respectively 6522 and 480 Yōjanas.
Comm. The word- ‘ Bimba ’ is used to connote the
spherical diameter. The diameter of Moon as given will
be equal to 480×5=2400 miles in modern terms which
is not far from truth. Once we accept that the method
indicated by us in the Kakshādhyāya of chapter I was that

353
followed by the ancient Hindu Astronomers to estimate
the distance of the Moon, the spherical radius the hori-
zontal parallax, the orbital radius pertaining to the Moon
could all be deduced and the magnitudes so deduced accord
with their average values in modern astronomy. The
magnitudes pertaining to the Sun however, should be
deemed as parameters.
Verse 6. e - [(S - E) Kₘ / Ks] = 2 α where e = Earth's
diameter, s = The Sun's diameter; Km = Moon's dis-
tance from the Earth's centre; Ks = The Sun's distance
from the Earth's centre, and α = radius of the Earth's
shadow cone at the lunar orbit.
Comm. In fig. 66, let CD be the radius of the Earth's
shadow cone at the lunar orbit. Required to find the
magnitude of CD. Triangles DEF and ESG are similar
Fig. 66
Fig. 67
45

354
Fig. 68
Fig. 69
Fig. 70
where DF and EG are drawn parallel to VBA, the
common tangent.
∴ EF/DE = SG/ES ie. (½e - a) / Km = (½s - ½e) / Ks

355
where ɑ = FB = CD required
∵ ɑ = ½ { e - Km/Ks (s - e) } I as formulated 2ɑ
being what is called Rāhu-Bimba or diameter of the
Earth's shadow cone at the lunar orbit which is called
Ku-bhā Vistṛti in the verse (Ku = Earth ; Bha = Shadow ;
Vistṛti = diameter).
Note (1) We shall prove that this formula accords
with the modern formula given for the radius of the
shadow cone. Divide I throughout by 2 Km, so that
ɑ/Km = e/2Km - (s - e)/2Ks II
But from fig. 67. ɑ/Km = sin Ê = Ê = angular
radius of the shadows cone expressed in radius. From fig. 68,
e/2Km = Horizontal parallax of the Moon ; s/2Ks = sin Ê
= Ê = angular radius of the Sun expressed in radians
from fig. 69 ; and e/2Ks = (from fig. 70) Horizontal
parallax of the Sun. Thus Equation II means ρ = P -
σ + P¹ III where ρ = angular radius of the shadow cone,
P = Horizontal parallax of the Moon ; P¹ = that of the
Sun and σ = angular radius of the Sun.
Note (2) If we don't divide I by 2 Km, we have the
radius of the shadow-cone in Yojanas, substituting the
values of e and s, Km and Ks.
Note (3) It is worth hearing Bhāskara in his
commentary under this verse. Observe the Sun's disc
while rising on the day when his true motion is equal to
his mean, with a compass composed of two rods hinged at
one end and carrying a protractor at the other. We get
the mean diameter of the Sun equal to 32' - 31" - 33"'.
Similarly, observe the Moon's disc on a full-moon-day

386
when his true motion equals the mean. It will be 32'-
0'' - 9'''.
Note (4) Substituting the values of P, P' and σ in
III ρ = 52'-42'' + 3'-57'' - 16'-16'' = 40'-23'' = Angular
radius of the shadow-cone at the lunar orbit which almost
accords with the modern value 41'-49''.
Note (5) One may wonder as to how, taking wrong
values for s and Ks, such a correct value could be obtained
for P. In equation II, α, Km, e are all near the truth so
that the terms effected are s/K and e/Ks; but both s and
Ks being parameters s/Ks comes off alright, which is the
angular radius of the Sun's disc which could be measured.
The only vitiating term is e/Ks which is the horizontal
parallax of the Sun which was overestimated unwittingly
by a wrong supposition as indicated in the Kakshādhyāya.
However e/Ks comes to be 3'-57'' and this overestimate is
mitigated to some extent that the Earth has an atmos-
phere which boosts the angular radius of the shadow cone
by about 1' and the remainder of the overestimate makes
amends for the smaller value of P taken.
Verse 7. To convert spatial measures into angular
measure. The diameters of the Sun, the Moon, and Rāhu
in Yōjanas multiplied by R = 3438', and divided
respectively by Ks, Km and Km give their angular measures.
Comm. From fig. 69, (s/2) / Ks = sin Ê = Ê expressed
in radians = E' / 3438 ∴ E' = (3438 × s) / (2 Ks) which means that
the angular radius of the Sun is got by multiplying s/2
ie. the spherical radius of the Sun by R = 3438 and divid-
ing by Ks as mentioned. Similar is the case with respect
to the other two.
Note. - The word Kalākaraṇa is used to signify To
convert into angular measure.

367
Verse 8. An alternative method of obtaining the
angular radii.
The daily motion of the Sun increased by one-tenth
of its value and halved, gives the angular diameter of the
Sun. The Moon's daily motion multiplied by 3 and
divided by 71, gives the angular diameter of the Moon.
Or the daily motion of the Moon being decreased by 715
and divided by 25 and the result being added to 29 gives
the angular diameter of the Moon.
Comm. This method gives in an easy way the true
angular radii. The formulae given are s' = ½s₁ (1+¹⁄₁₀) ;
and m' = (3 m₁) / 74 = (m₁ - 715) / 25 + 29. This may be eluci-
dated as follows. The argument used is "If the spherical
diameter of 6522 Yojanas corresponds to a spatial daily
motion of 11858¾ Yojanas, what angular diameter corres-
ponds to the angular daily motion s₁?". The proportion-
ality is clear and the result is
(s₁ × 6522) / 11858¾ = 26088 / 47435 . Converting 26088 / 47435 , into a continued
fraction, it is
1/(1+) 1/(1+) 1/(4+) 1/(1+) 1/(1+) 1/94 . The penultimate convergent is
11/20 = ½ (1+¹⁄₁₀). The formula follows.
Similar calculation gives m'.
Note. The advantage of these formulae is that they
are not only easy but also adopting the true daily motion
we have the true angular radii. This procedure was
adopted by Bhāskara from Brahmagupta. The latter, how-
ever, prescribes a nearer convergent namely 10/247 but
actually 17/420 is the nearest convergent.
The next formula namely m' = (m₁ - 715) / 29 + 29 is
approximate. This may be elucidated as follows. Let the

366
daily motion be 715 ; then as per the previous formula the
angular diameter should be 3/74 × 715 = 2145/74 = 28 73/74 = 29'
very approximately. The mean daily motion is 790 which
corresponds to 32' of angular diameter. Taking advant-
age of this arithmetical correlation namely that the excess
of 3' over 29' corresponds to 75' of daily motion. Bhāskara
gives the formula m' = (m₁ - 715)/25 + 29. This formula
correctly holds good when m₁ = 740, for, equating
3x/74 = (x - 715)/25 + 29 = (x + 10)/25 , x will be equal to 740.
For other values between 715 and above it holds very
approximately. Thus, when m₁ = 715, m' = 28 73/74 ie. 29
when m₁ = 740, m' = 30, when m₁ = 765, m' = 31 1/75
(error 1/75) when m₁ = 790, m' = 32 1/37 (error 1/37) and so on.
Verse 9. An alternative method of finding the
angular diameter of the shadow cone.
2 ρ = 2/15 m₁ - 5/12 s, where m₁ and s₁ are the daily
motions of the Moon and the Sun respectively.
Comm. In the previous verse, we had formulae to
compute the diameters of the discs of the Sun and Moon,
knowing their daily motions. Since in practice we have
these daily motions computed for every day, so the compu-
tation based upon those daily motions conduces to ease in
the matter of calculation. Now in this verse, the radius
of the shadow cone is also calculated in terms of the daily
motions of the Sun and the Moon, which is more an
ingenious device adopted in practice. The elucidation of the
formula depends on the following technique as conceived
by the Hindu astronomers. In as much as the Sun's
sphere is far bigger than that of the Earth, the shadow of
the Earth assumes the form of a cone. From a knowledge
of the decrease in the diameter, as we proceed from the

309
Sun to the Earth and a knowledge of the Sun's distance,
we can compute similar decrease as we proceed from the
Earth to the lunar orbit knowing the distance of the
Moon. Such a decrease measured in Yojanas is termed by
Bhāskara as 'अपचययोजनानि' ie. Yojanas of decrease in
diameter. It was this concept that led to the formulation
of 2a in Yojanas in the form 2a = e — ((s — e) Km) / Ks of verse
6 and is indeed based upon the similarity of triangles as
proved by us in that context. Dividing the above equation
by 2 Km, we have
a/Km = e / (2 Km) — 1/2 ((s — e) / Ks) I
Dividing a, e and (s—e) thus by the distances Km and Ks is
termed 'Kalā-Karaṇa' ie. converting spatial distance
into angular measure. Thus dividing a by Km is con-
verting the radius of the shadow cone into angular measure
at the lunar orbit ; dividing 1/2e by Km is estimating the
angular measure of the earth's radius as seen from the
Moon's distance or what is the same the horizontal parallax
of the Moon, whereas dividing 1/2s by Ks is getting the
angular radius of the Sun's disc as seen from the Earth
and dividing 1/2e by Ks is getting the angular radius of the .
Earth's disc as seen from the Sun or what is the same the
horizontal parallax of the Sun.
Equation I which gives a/Km the angular radius of
the shadow cone, may also be interpreted as follows. e / (2Km)
= Horizontal parallax of the Moon as mentioned above
which is equal to the angle (fig. 66) EĈB, for, 1/2e = EB ;
Km = EC so that e/2Km = EB/EC = sin EĈB = EĈB
expressed in radian measure. Also
1/2 (s—e) / Ks = (SA—GA) / SE = SG / SE = sin SÊG = SÊG = EṼB

360
(alternate angle) = Semivertical angle of the shadow cone
(say θ). Now EĈB − ÊVB = CÊD = angular measure
of CD ie. the angular radius of the shadow cone (expressed
in radian measure).
Converting ½e/Km into angular measure, the propor-
tion used by Bhāskara is "If by the daily spatial motion
of 11859¾ Yojanas of the Moon, we have its daily motion
in arc, what shall we have for e = 1581 Yojanas?" The
result is 1581/11859¾ m₁. Converting the coefficient into
a continued fraction we have 1/(7+) 1/(1+) 1/(1+) 1/175 ......... The
penultimate convergent is ²/₁₅. Hence e/Km = ²/₁₅ m₁.
Converting ½ ((s − e)/Ks) into arc, the proportion used is
"If by the daily spatial motion of 11859¾ Yojanas, we have
the daily motion of s₁, what shall we have for s − e =
6522 − 1581 = 4941 Yojanas?" The result is 4941/11859¾ s₁.
Converting the coefficient into a continued fraction, we
have 1/(2+) 1/(2+) 1/(2+) 1/146 . The penultimate convergent is ⁵/₁₂.
Hence, the result is ⁵/₁₂ s₁.
Note (1) It might be asked whether the Hindu
astronomers used the theory of continued fractions. The
answer is, they did though they did not write the con-
tinued form in the form we do now. They arranged the
successive quotients in a vertical line and called the
column as a 'Valli' or 'creeper'. One may refer to the
chapter in Bhāskara's Bijaganita on 'Kuttaka' in this
context.
Note (2) The formula derived above to obtain the
Rahu-Bimba or diameter of the shadow-cone at the lunar
orbit, is one which could be conveniently used in practice,

361
for, as mentioned before, the daily motions of the Moon
and the Sun are ready computed for every day. Also the
advantage in using this formula is that besides the fact
that we need to deal only with small quantities instead of
the big numbers of Yojanas, the true value for the day of
eclipse is got by using the true daily motions. Using the
mean daily motions, however, we have for the mean
diameter, ²/₁₅ m₁ — ⁵/₁₂ s₁ = ²/₁₅ × 790′–35″ — ⁵/₁₂ × 59′–8″
= 105 ⅔ — 24 ⅔ = 81 very approximately.
Note (3) In obtaining a convergent for (s − e)/Ks , since
the values of s and Ks are parameters s/Ks is got alright,
where e/Ks is more exaggerated as the value of the hori-
zontal solar parallax. By this term the result is increased
to an extent of 3′ out of which 1′ is mitigated by the fact
that we have to take the earth's atmosphere also into

362
the Moon with respect to the ecliptic where λ is the longi-
tude of the Moon with respect to the nearer node and
270' or 4½° is taken to be the inclination of the lunar
orbit to the ecliptic or what is the same, the maximum
latitude of the Moon.
Comm. The formula is evident and similar to that
for calculating the declination of the Sun. Thus
H sin β = (H sin λ × H sin 4½°) / R . But since β is small
and also 4½°, we can take β = (H sin λ × 270') / R .
The argument used by Bhāskara, however, is ‘ If by
a H sin λ equal to R, we have the maximum latitude of
270', what shall we have for H sin λ?’ The result is as
given.
Note. The modern value for i the inclination of the
lunar orbit to the ecliptic is given to vary between 4°-58'
and 5°-18'.
Verse 11. The definition of the magnitude of a lunar
eclipse.
Sthagita or the magnitude of an eclipse is defined
as P+r+β where P and r are respectively the radii of
the eclipsing and eclipsed bodies and β is the latitude of
the Moon. If the Sthagita is greater than 2r, then the
eclipse is total.
[Diagram: Two intersecting circles representing the Moon (center M, radius r) and Earth's shadow (center C, radius P), along lines labeled "LUNAR ORBIT" and "ECLIPTIC"]
Fig. 71

363
Comm. In figure 71, the eclipsing body is just con-
tacting the eclipsed body. Taking the case of a lunar
eclipse, the latitude then of the Moon is evidently P+r
ie. β=P+r holds good at the moment of first con-
tact. (C) is the cross-section of the shadow-cone at the
lunar orbit and (M) is the Moon.
(fig. 72), CB+AM-CM=CB+AB+
BM-CM=AB+(CB+BM)-CM=
AB+CM-CM=AB
∴ P+r-β=AB=Sthagita. Thus
Sthagita gives the portion of the
diameter of the eclipsed body which
is shadowed. The eclipsed body is
termed the Chādya, the eclipsing
body as the Chādaka and P+r as
the Manaikya-ardha ie. half the
sum ofthe diameters of the eclipsing
and eclipsed bodies.
Fig. 72
When the Sthagita exceeds the diameter of the
eclipsed body the eclipse is evidently total ie., when
P+r-β>2r ie. P-r>β.
Verse 12. Duration of the eclipse and duration of
its totality.
Sthiti-Khanda = √(P+r)²-β² × 60 = ½ Duration of
—————————————— the eclipse
m₁-s₁
Marda-Khanda = √(P-r)²-β² × 60 = ½ Duration of
—————————————— totality
m₁-s₁
where P is the radius of the shadow-cone, r the radius of
the Moon's disc, β its latitude taken to be constant during
the eclipse, m₁ and s₁ the daily motions of the Moon and
Sun respectively.
Comm. (1) The time between the moment of first
contact and the middle of the eclipse or the moment of

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