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

PARVASAMBHAVĀDHIKĀRA
Investigation into the occurence of an eclipse
Verses 1–2. Multiply the number of years that have
elapsed from the beginning of the Kaliyuga by twelve and
add the number of months elapsed from the beginning of
the luni-solar year. Let the result be x. Then add
[2 x (1 - 1/898)] / 65 to x. Let the result by y. Then the longi-
tude of what is called Sapāta-Sūrya or the longitude of the
Sun with respect to a node will be x Rasis +
[(2 y + 503) (1 + 1/169)] / (3 × 30) Rasis. If this longitude be less
than 14°, then a lunar eclipse is likely to occur.
Comm. The first operation indicated above in direct-
ing x to be added to [2 x (1 - 1/898)] / 65 is intended to obtain the
lunations that have elapsed from the beginning of the
Kaliyuga. In this behalf we are asked to multiply the
elapsed years by twelve to get the number of solar months.
Here there is one subtlety to be noticed. The years that
have elapsed are not entirely solar. In fact the years
reckoned according to the luni-solar system were all
originally luni-solar; but according to the convention of
intercalary months, they were rendered solar upto the
point of the latest intercalation, for, solar months plus
intercalary months are equal to the elapsed lunations.
From the moment of the end of the latest intercalary
month, the subsequent years or year or fraction thereof
would be luni-solar only. Nonetheless, no difference will
be there in the computed Adhikamāsas in adding a few
lunar months to the solar and taking them all to be solar.
The maximum error committed in so doing will be of
the order of (no. of days in a solar month minus no.

of days in a lunar month) multiplied by 36 × ²/₆₅ × ¹/₃₀
of an adhikamāsa, assuming that an adhikamāsa would
occur at the latest in 36 solar months. (In fact, an
adhikamāsa would occur on the average in 32½ solar
months, but we have taken 36 roughly as the maximum
figure in as much as the occurence of the Adhikamāsa
might be belated on account of the convention stipulated).
Thus the error would be 36 × 2 × ²/₆₅ × ¹/₃₀ = ¹/₁₃th of an
adhikamāsa at the maximum. Hence, we are directed not
only to construe that all the years elapsed to be solar but
also the subsequent lunations of the current luni-solar year
also to be solar months. Thus getting the number of
elapsed months from the beginning of the Kaliyuga, the
computation of the Adhikamāsas is formulated as follows.
If in the course of 51840000 solar months of the Yuga
there be 1593300 Adhikamāsas then during the elapsed
solar months x, what is the number of elapsed Adhika-
māsas? The result is
x × 1593300 x × 1593300
----------- = ---------------
51840000 796650

51840000

796650
= 2 × x
--------- . Since Bhāskara knows that there will be two
65-4-21
Adhikamāsas roughly in 65 solar months, he performed the
above operation. This shows that for every 65 solar months
roughly there occur two Adhikamāsas or more accurately
a little less than two Adhikamās. So, taking, in the first
instance 2/65 as the ratio of Adhikamāsas to the number
of solar months, Bhāskara tries to find as to what quantity
is to be subtracted from 2. That is found as follows. If
there be A adhikamāsas in s solar months what will be the
number of Adhikamāsas in x solar months? The result is
A x
--- . Again if there be two Adhikamāsas roughly in 65
s
solar months, how many will be there in x solar months?
2 x
The answer is --- . But we have seen about that the
65

accurate number should be (2 x / 65) - λ ie. a little less than
(2 x / 65) . The question is now to find the value of λ. So,
equating (2 x / 65) - λ to (A x / s), λ = (2 x / 65) - (A x / s) = x ((2 / 65) - (A / s))
= x ((2 s - 65 A) / (65 s)). Substituting for 2 s - 65 A namely
2 × 51840000 - 65 × 1593300 = 115500
λ = (x × 115500) / (65 × 51840000) = (x × 2 × 57750) / (65 × 51840000) = (2 x / 65) × 1 / (51840000 / 57750)
= (2 x) / (65 × 898) ∴ (A x) / s = (2 x / 65) - λ = (2 x / 65) - (2 x) / (65 × 898)
= (2 x / 65) (1 - 1 / 898) as given.
The procedure, adopted as above, is in a way a short
cut in Hindu Astronomy to obtaining a convenient con-
vergent to a continued fraction. Let us use the method
of continued fractions; the number of Adhikamāsas in
x solar months is (A x / s) ie. x × A/s =
(x × 1593300) / 51840000 = (x × 5311) / 172800 . Converting 172800 / 5311
into a continued fraction we have
32 + 1/(1+) 1/(1+) 1/(6+) 1/2 + 1/(1+) 1/(1+) 1/(18+) 1/4 to which 65/2 is a convergent
but a good convergent is 245 / 69 . As this good convergent
is unwieldy, Bhāskara used 2/65 and made amends for the
roughness introduced by adopting it. Wherever a con-
venient convergent is not available, an easy and rough
convergent is used and amends will be made for the rough-

334
ness resulting as follows. Let M/N be a fraction to which
m/n is a convergent having small numbers as numerator and
denominator, so that M/N is taken to be equal to m/n (1 + 1/λ).
Thus M/N = m/n (1 + 1/λ) or Mn = Nm (1 + 1/λ)
∴ Mn − Nm = Nm/λ or λ = Nm / (Mn − Nm). In the present
case M is the number of Adhikamāsas, and N the number
of solar months. m/n if taken to be 2/65
− 2 × Solar months
∴ In this case λ = --------------------------------------------------
65 × Adhikamāsas − 2 × Solar months
which is indicated in the commentary by Bhāskara.
In this context, it may be mentioned that a Karaṇa-
grantha named Nārasiṁha based upon Sūryasiddhānta
(A Karaṇagrantha is a manual according which the Hindu
calendar is computed with easy numbers without under-
going the laborious process indicated in the treatises called
Siddhāntas like the present Siddhānta Śiromaṇi. In
these Karaṇagranthas, instead of taking the beginning of
the Kalpa or Mahāyuga or the Yuga, as the epoch, a recent
date ie. the date of the author of the Karaṇagrantha is
taken as the epoch, and processes using approximations
are adopted for the sake of ease. Naturally therefore these
Karaṇas (as they are also called) get easily obsolete
within the course of a few hnudreds of years so that a fresh
Karaṇa is called for preparation, if the calculations were
to accord with the Siddhāntas which those Karaṇas pro-
fess to follow. In fact, the present Karaṇa of Nārasiṁha
written in 1333 Śaka year ie. in 1411 A.D. declares that
a previous Karaṇa named Tithicakra reported to have
been written by one Mallikārjuna Suri grew obsolete and

"?
Let's zoom in on that glyph:
It has:

• vertical line
• a loop to the left
  Wait, does it have a loop?
  Yes! A closed oval loop on the left of the vertical stem!
  Wait! An oval loop on the left of a vertical stem with a flat top is... 'व'!
  Wait, does 'च' have an oval loop? No, 'च' has an open curve!
  Wait, but look at the letter 'व' in "एतदेव":
  In "एतदेव", the 'व' has the EXACT same shape!
  Wait, could it be a misprint in the original book? Yes, old Indian presses often confused च and व.
  Wait, let's look at the letter again:
  Does it have an open top or closed loop?
  Looking at:
  त ित िथ [letter] क ं
  Wait, between 'थि' and 'क', look at the glyph:
  Wait, is the loop closed or open?
  It is completely closed, like 'व'!
  Wait, but wait! What if it's 'च'? In some fonts, ink fill makes 'च' look closed, or the typesetter mistakenly picked 'व' instead of 'च'.
  Wait, what about the 'क'?
  Does it have a repha or r-stroke?
  Wait! Look at "तिथिवक्रं" vs "तिथिचक्रं":
  Wait, is it "तिथिव्रतं"? No, the letter after

336
and s and m the angular semidiameters of the Sun and
the Moon. (This formnla we shall see later). This higher
limit comes to 88.5'. The limit of 56' for the occurence
of a lunar eclipse is the value of
pm + ps — s + m as we shall see later.
The latitude of 56' of the Moon arises out of a longi-
tude of 12° of the Moon with respect to a node, whereas
the latitude of 32' arises out of a longitude of 7° with
respect to the node. Since at an eclipse solar or lunar,
the longitude of the Moon with respect to a node, is the
same as the longitude of the Sun with respect to the same
or opposite node, the latter must be 12° for the occurence
of a lunar eclipse. But as the difference between the
mean and true Suns is about 2°, the longitude is stipulated
as 14°. In other words, for the occurence of a lunar
eclipse, the longitude of the Sun on the full-Moon day
with respect to the nearer node shall be less than 14°. To
compute this longitude of the Sun with respect to the
nearer node on a full-Moon day, we are given the subse-
quent procedure indicated in the verse. In 53433500000
lunations of the Kalpa, the sum of the sidereal revolutions
of the Sun and the Node (Rāhu) (Sum because Rāhu has
a retrograde motion) is equal to 455231168 which is equal
to 455231168 × 12 = 54627734016 Rasis. Then in one
lunation what will be the increase of the longitude with
respect to the Node? The result is
54627734016 / 53433300000 = 1 Rasi + 3583302048° / 5343330000 (= 74652126 / 111319375)
dividing by 48 both the numerator and denominator.
Taking the first two digits in the numerator and denomi-
nator of the fraction the fraction is approximately equal
to 74/111 or 2/3. Taking this as a convergent we make
amends for the roughness as follows.
74652126 / 111319375 = 2/3 (1 + 1/λ) ∴ λ = (2 × 111319375) / (3 × 74652126 — 2 × 111319375)

387
= 222638750 / 1317628 = 169 approximately. Hence the increase
of the Sun's longitude with respect to a node is
1 Rāśi + 2/3 (1 + 1/169)° I. In the beginning of the Kali-
yuga, the longitude of the node was 5 Rāśis—3°—13′ and
the arc moved by the Sun with respect to the node during
the course of half a lunation is 0—15—20, so that their sum
is 5 —18 —33. Here we have added for half a lunation
because the context is a lunar eclipse and the beginning
of the Kaliyuga was a New Moon day. Also, at the begin-
ning of the Kali, the Mean Sun being at the zero-point
of the zodiac, the negative longitude of the node only is
the longitude of the Sun with respect to the node. Hence
we have to add the above longitude of 5 —18 —33 to the
longitude obtained through the above formulation I, which
means 168°—33′ is to be added to 2x/3 (1 + 1/169) where x
is the elapsed number of lunations. Taking 168°—33′ as
nearly equal to 168°—40′, 2x/3 (1 + 1/169) + 168 2/3°
= 2x/3 (1 + 1/169) + 506/3 = 2x/3 (1 + 1/169) + 503/3 (1 + 1/169)
approximately = (2x + 503)/3 (1 + 1/169) as formulated.
Thus for x lunations, the longitude of the Sun with respect
to the node is x Rāśis + (2x + 503)/3 (1 + 1/169)°. If this
longitude falls short of 14°, we could except a lunar eclipse.
Latter half of verse 3 and verses 4, 5. Particularity
with respect to a solar eclipse.
Add half a Rāśi to the longitude previously obtained;
find out on which side the Sun lies, north or south; com-
pute the longitude of the Sun from the number of days
43

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

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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