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

338
elapsed after the Saṃkramaṇa day (ie. the day on which
the Sun has left one Rāśi and entered another); obtain
the hour-angle in nāḍīs of the Sun at the ending moment
of the Amāvāsyā ie. at the moment of New Moon; add or
subtract one-fourth thereof in Rāśis from the position of
the Sun according as the Sun is in the Western or Eastern
hemisphere; then finding the declination of that point
and from the sum or difference of the declination and
latitude of the place, obtain the zenith-distance of the
culminating point of the ecliptic; taking that point to be
roughly the Vitribha ie. the point of the ecliptic which is
90° behind the Sun on the ecliptic, find one-sixth of the
zenith-distance; taking the sum or difference of the result
and the longitude of the Sun with respect to the node (got
in the beginning by adding half a Rāśi to his position at
full-Moon) if the result happens to fall short of 7°, then
we could expect a solar eclipse.
If there be no eclipse at the current New Moon, then
go on adding 1 Rāśi - 0° - 40′ - 15″ to the longitude of
the Sun with respect to the Node (which will be his longi-
tude for the moment of the next New Moon) and repeating
the procedure indicated, the occurence of an eclipse or
otherwise could be known. If occurence be indicated then
compute the actual positions of the Sun, Moon and Rāhu
and following the procedure to be indicated in the chapter
on solar eclipses, the moment of the occurence of the
eclipse and other relevant details could be computed.
Comm. In the case of a lunar eclipse, the Manaik-
yārdha ie. half the sum of the diameters of the eclipsing
and eclipsed bodies (namely the cross-section of the Earth's
shadow at the lunar orbit and the Moon) is 56′. This is
the maximum limit to the celestial latitude of the Moon
if an eclipse were to occur, and this latitude will be there
if the longitude of the Moon with respect to the nearer node
is 12°. Since a lunar eclipse occurs at the moment of a full
Moon, the distance of the Sun then from the opposite node

339

should be also 12° for the occurence of an eclipse. Since
it is customary to check the occurence of an eclipse
through Sapātasūrya ie. longitude of the Sun with respect
to the node (the prefix Sa is to signify that the sum of the
Sun's longitude and that of the node should be taken, as
the longitude of a node is measured in the opposite dire-
ction from the zero-point of the ecliptic) it is stipulated
that the Sapātasūrya should be 12° for the occurence of a
lunar eclipse. But, as the True Sun might differ from
the Mean by about 2°, and as we are concerned with the
True Sun only, the limit is increased by 2°, so that a lunar
eclipse may occur if the Sapātasūrya happens to be less
than 14°. Thus there is no more complication with res-
pect to the occurence of a lunar eclipse than requiring the
longitude of the Sapātasūrya to be less than 14° for the
occurence of a lunar eclipse. If this condition be satisfied,
there will be an eclipse and that will be visible at all
places, where there is night, since a body in shadow will
not be seen from any place whatsoever.
But, there is a complication with respect to the occur-
ence of a solar eclipse namely that it is not a question of
the Sun entering a shadow. The Sun could never be
shadowed. A solar eclipse occurs when the disc. of the
Moon comes in between the Sun and an observer and
obstructs a vision of the Sun's disc. The Moon being very
near us compared with the Sun, its coming in between the
Sun and an observer may well be compared with a cloud
obstructing the vision of the Sun. Just as, when a cloud
obstructs the vision of the Sun for an observer, it could
not do so with respect to another who is situated at a
distance, so also, if the Moon effects a solar eclipse for a
particular place, it could not do so for all places. This is
said to be due to parallax or Lambana as it is called (Refer
fig. 64). It is so called because, when there is an eclipse
of the Sun for an imaginary observer at the centre C of
the Earth, the Moon intersecting in the line of sight to

340
Fig. 64
the Sun, for an observer 'O' on the surface of the Earth,
the Moon is not in the line of sight namely OS but hangs
down that line (लम्बते अनेनेति लम्बनम् = That phenomenon
by which the Moon hangs down the line of sight). Hence
it is not sufficient to say that the Sapātasūrya has a
longitude of 7° to conclude the occurence of a solar
eclipse for a place. If the Sapātasūrya be less than 7°,
certainly there will be a solar eclipse for some place on
the earth but not for all places. So, to decide whether
there will be a solar eclipse for a given place, we are
to take into account the phenomenon of parallax. At
every New Moon, the longitudes of the Sun and the
Moon will be no doubt equal; yet the Moon may not
obstruct a vision of the Sun, not being situated in the
ecliptic plane. He may be above the ecliptic plane or
below it and if he be within 32' from the plane, a part of
the Moon's globe may hide a part of the Sun's from the
vision of certain observers who are situated about the point
o' of fig. 64. But suppose an observer is at O. For him.
there is no eclipse at all as could be seen from the figure,
As the observer moves away and away from O' towards O,
the effect of parallax will be greater and greater in longi-
tude, whereas, as the observer moves away and away from
O' towards O₁ (where O₁ is the geocentric pole of the circle

241
(c) shown in the figure) along the circle of intersection of
the Earth with a plane through CO' perpendicular to the
plane of the paper, where O₁ is a point on the earth such
that SĈO=90° the effect of parallax will be greater and
greater in latitude. In other words the parallax has both
an effect in longitude as well as in latitude. When it has
an effect in longitude only it is called Lambana, whereas,
when it has an effect in latitude, it is called Nati. (Thus the
translation of 'parallax' as Lambana alone is not fully
correct, though at times the parallax may have its com-
plete effect in longitude only or in latitude only). For the
observer who moves in the ecliptic plane only as the one
moving from O' towards O, the parallax will have its entire
effect in longitude only and for the observer moving in the
perpendicular plane from O' to O₁ mentioned before, the
parallax will have its entire effect in latitude only. For
observes other than the two above, it will have effect both
in longitude and latitude also. When parallax effects
longitude the time of conjunction is preponed or post-
poned, whereas when it effects latitude, the latitude of the
Moon appears to have increased or decreased. When it
increases, no eclipse occurs and when it decreases an
eclipse does occur. At O', the latitude will be exactly what
has been computed; at the point of intersection O'' of the
join of the centres of the Sun and Moon with the surface
of the Earth, parallax nullifies the latitude and on the great
circle O' O'' there will be parallax in latitude only effecting
the magnitude of the latitude.
It will be seen that for the point O'', the Sun and the
Moon are in the zenith, so that neither will suffer from
parallax. For the point O the Sun will be on the
horizon and the Moon being depressed below the horizon
though he is in geocentric conjunction the parallax in
longitude or lambana is maximum and the occurence of
the New Moon had already elapsed 4 nādis ago. Further

342

it will be seen that at the points O₁ and O₂ where O₁ is the
other geocentric pole of the circle (c) drawn there cannot
be an eclipse, the latitude being increased (as per Hindu
astronomy) by 48' – 46''. In fact, there will be eclipse
for the places on either side of O'' (the point of intersection
of the join of the centres of the Sun and Moon with the
Earth's surface) to such a distance as will increase the
latitude to 32' only. For the other places on O' O'' beyond
these points, the latitude of the Moon exceeds this limit
and so there will be no eclipse. Further clarification of
Lambana and Nati will be given later.
The above analysis underlies our investigation for the
occurence of a solar eclipse. Sapātasūrya might be less
than 7°, but it does not mean that every place will enjoy
an eclipse. So, for the place concerned, we have to see
that, even after taking into account the parallax in
latitude ie. in Nati, still the latitude will be less than 32'.
To obtain this parallax in latitude, the method adopted is
to find it at the point called Vithribha, (nonagesimal)
ie. the point which is behind the Lagna the rising
point of the Ecliptic by 90°; for, as we shall see in the
Chapter on Solar eclipses, the parallax in latitude at the
Vitribha will be equal to the parallax in latitude at any
point of the ecliptic. In other words, wherever be the Sun
and Moon on the ecliptic (Moon also being very near the
node may be taken roughly to lie on the ecliptic) to
compute the amount by which the latitude is increased, we
compute it for the Vitribha, and this will hold good for
the arbitrary position of the Moon, for, there also the
latitude will be increased by the same amount. The
procedure given in verse 4 is to locate the Vitribha from
the position of the Sun and to find its zenith-distance to
compute the Nati; or rather, it is to locate the culminating
point and taking it roughly to be Vitribha, to compute
the influence of Nati on the latitude or Śara. If it were
only to find the Vitribha, it could be computed from the

lagna of the moment of New Moon. Computation of the
zenith-distance of the Vitribha is a little cumbrous, so
that, for brevity, it is sought to compute the culminating
point, obtain its declination and thereby its zenith-distance
which could be taken to be the zenith-distance of the
Vitribha also, from which the Nati is calculated.
We are directed to obtain first the Nata or the hour
angle of the Sun for the moment of New Moon. We know
on that particular New Moon day how long Amāvāsya
will last after Sun-rise ie. we know when the actual
moment of New Moon occurs on that day. We also know
the duration of day time on that day so that subtracting
the time of occurence of the New Moon after Sun-rise
from half the duration of day, we obtain the hour angle of
the Sun (Nata) in nādis. Now the Moon's longitude is
effected by parallax, the effect being depression of the
Moon. It is roughly estimated that the hour angle ex-
pressed in nādis is increased by ¼ of its value on account
of this. Strictly speaking the effect of parallax is far more
on the position of the Moon than on the Sun. But the
Hindu procedure apparently treats the Sun alone for
parallax. The reason is that at the moment of geocentric
conjunction of the Sun and the Moon, when we consider
the combined effect of parallax on the Moon and the Sun
at once, for a given place, we may as well compute the
relative position of the Sun effected by parallax. Let the
hour-angle of the Sun in nādis be x. Then effected hour-
angle will be x (1 + ¼) = 5x/4 nādis. But each Rāsi being
taken roughly to rise in 5 nādis, the hour-angle in Rāsis
will be 5x/4 ÷ 5 Rāsis x/4. Hence we are directed to
divide the hour-angle of the Sun in nādis to divide by 4.
This x/4 being substracted from the longitude of the Sun,
we get the longitude of the culminating point. Then we
are directed to obtain the declination of the culminating
point from the formula H sin δ = H sin ω × H sin λ ÷ R,

344
This declination of the culminating point being known,
and the latitude being known, its meridian zenith-distance
could be got. Take that to be roughly the zenith-distance
of the Vitribha. Then an approximate estimate of the
effect in latitude is obtained as follows. Let the zenith-
distance of the Vitribha be x. Then if we have for
H sin z = R, the maximum effect of 48′ — 46″ in the
latitude, what shall we have for H sin 45° ? The result is
(H sin 45 × 48′ — 46″) / 3438 = 2431 / 3438 × 48′ — 46″ = 34′ — 30″.
Then again another approximate estimate is made as
follows. Let the Sapātasūrya (Sapātasūrya = longitude
of the Sun or what is the same of the Moon with respect
to the Node) be λ. Then for λ = 15°, we have a latitude
of 70′. That being so, for a variation of 34′ — 30″ in the
latitude, what will be the corresponding variation in the
longitude λ ?
The result is = (69 / 2) × (15 / 70) = 207 / 28 = 7° 11/28 which is
roughly one-sixth of 45°. Hence if the zenith-distance of
the culminating point taken to be the Vitribha be 45°
the variation in the Sapāta-sūrya (= Sapāta chandra)
will be one-sixth thereof. Hence we are asked to increase
or decrease as the may be, the Sapāta-sūrya by 1/6th.
If the resulting Sapātasūrya be less than 7°. there may be
an eclipse. When the lunar orbit lies north of the ecliptic,
the culminating point of ecliptic being south of the zenith,
the latitude of the Moon is decreased by parallax, so that,
we have to decrease the Sapātasūrya; whereas when the
lunar orbit is then south of the ecliptic, the effect of
parallax is to increase the latitude and consequently, we
have to increase the Sapātasūrya. Or again when the
culminating point of the ecliptic is north of the zenith and
the lunar orbit is north of the ecliptic, parallax appears
to increase so that we have to increase the Sapātasūrya ;
and when at that moment the lunar orbit is south of

345
the ecliptic, parallax appears to decrease the latitude so
that we have to decrease the Sapātasūrya.
The proportion that 70' of latitude correspond to 15°
of Sapāta-chandra is due to the formula
H sin 15° ✕ H sin 4½
—————————— = H sin β. Using logarithms,
R
log sin β = 9·4130 + 8·8946 = 8·3076 so that
β = 1° - 10' = 70'. This proportion could be used
because the Moon is within 15° of the Node. Hence when
we are investigating the occurence of a solar eclipse,
application of rule of three is not unjustified or crude.
Herein, Bhāskara assumed the zenith-distance of
the nonagesimal to be round about 45° and drew a
conclusion that the effect in the longitude on account of
parallax in latitude is one-sixth of the zenith-distance.
Instead if the zenith-distance be assumed to be z, the
result would be (48' - 43" ✕ sin z ✕ 15) / 70 = 21/2 sin z which
may be taken as a better approximation. If, on the other
hand the modern value of 57' of the lunar parallax be
taken, the result would be (57 sin z ✕ 15) / 70 = 12 sin z
approximately which is a better value.
In this calculation, it is better to compute the zenith-
distance of the nonagesinal using modern methods instead
of assuming the nonagesimal to be on the meridian.
44

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

347
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