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

GRAHŌDAYĀSTĀDHIKĀRA
Verse 1 and first half of verse 2.
The Udayalagna of a planet is termed Prāk-Dṛk-graha
and the Astalagna is termed the Paschima-Dṛk-graha.
If the Prāk-Dṛk-graha happens to be less than the current
lagna the planet had already risen. If it be greater, the
planet is still to rise. Similarly if the Paschima-Dṛk-graha
is less than the current lagna, the planet had already set ;
otherwise is yet to set.
Comm. The words Prāk-Dṛk-graha and Paschima-
Dṛk-graha are coined to indicate the points of intersection
of the ecliptic with the eastern and western horizon
respectively when the planet is rising or setting, when the
planet has latitude. When the planet has no latitude
Prāk-Dṛk-graha coincides with the rising planet and the
Paschima-Dṛk-graha with the setting planet. When the
planet has a latitude, the foot of the latitude, which
signifies the planet’s position on the ecliptic differs from
the two Dṛk-grahas. In this case ie. when the planet has
a latitude the Prāk-Dṛk-graha’s longitude will be less than
that of the planet’s longitude whereas the Paschima-Dṛk-
graha’s longitude will be greater than that of the planet.
If x and y be the differences of the longitudes of the planet
and those of the Dṛk-grahas respectively, it is clear from a
figure that tan x = (tan β) / (tan θ) and tan y = (tan β) / (tan ϕ) where β is the
latitude of the planet and θ and ϕ the angles which the
ecliptic makes with the horizon at the planet’s rising and
setting respectively.
Since θ and ϕ are then respectively the zenith-
distances of the pole of the ecliptic in each case, and since

472
in the course of half a sidereal day, these zenith-distances
could not be equal, x and y cannot be equal.
Second half of verse 2 and verse 3. To find the side-
real time in between the given time and the time of the
planet's rising.
Find the time that has elapsed after the planet's rise,
from a knowledge of the Iṣṭalagna at a given time and
Prāk-Dṛk-graha ie. the longitude of the rising point of
the ecliptic at the time when the planet rises.
This time will be in Sāvana measure, because we have
taken the planet's position at the given moment and not
the position of the rising planet. From this time, know-
ing the daily motion of the planet of the day in question,
obtain the arc; that would have been traversed in between
the moments. Subtracting this arc from the planet's
position at the moment, the planet's position at rising
would be obtained approximately; approximately, because
as per the procedure enunciated in verse eleven, Graha-
cchāyādhikāra, the Prāk-Dṛk-graha of the rising planet,
and that obtained from the position of the planet at the
moment differ. From this approximate time again com-
pute the arc that would have been traversed by the planet
during that time. Subtracting this arc which is nearer
the truth from the planet's present position, we obtain a
more approximate position of the rising planet. Again
computing the Prāk-Dṛk-graha from this position, calcu-
late the time from this Prāk-Dṛk-graha and the lagna of
the moment. Repeating the process we will obtain
the actual sidereal time in between the given time and
the actual rising time of the planet. It is sidereal because,
we have found the time as per the procedure of verse (11)
referred to, where we have used तत्कालखेटोदय and not the
actual ,खेटोदय, i.e the rising lagna of the planet obtained
from the present position and not that of the rising planet.

473
This difference between the nature of the times was already
dealt with.
The matter of this verse appears to have been
unnecessarily complicated by Bhāskara but on careful
scrutiny it is not so ; for, the problem is to find the time
that has elapsed after the rise of the planet upto a given
time. Here the data are the given time and the corres-
ponding position of the planet not on the horizon but
elsewhere. We could not know immediately the time at
which the planet rose. To find it alone, this method of
successive approximation has to be used and there is no
other go. Had we known the position of the planet while
rising, the corresponding Prāk-Dṛk-graha could be found
exactly and as this position of the Prāk-Dṛk-graha is a
point of the ecliptic and the Iṣṭa-lagna ie. the rising point
of the ecliptic at the given moment is also a point on the
ecliptic, the time between the moments of rising of these
two points of the ecliptic could be found in sidereal
measure directly by noting the rising times of the Rāśis in
between, without an appeal to the method of successive
approximation.
Verse 4. The computation of the times of heliacal
rising and setting of a planet in contradistinction to its
rising time during a day on account of earth’s diurnal
rotation.
The times of rising and setting of a planet have been
dealt with. Now I shall tell the procedure to be adopted
in computing the times of heliacal rising and setting of a
planet. If a planet has a daily motion less than that of
the Sun, it rises heliacally in the east, and sets in the
west ; otherwise the reverse.
Comm. Take the example of the superior planets, say
that of Jupiter. Since Jupiter moves slower than the Sun,
60

474
the Sun will have to overtake Jupiter and not vice versa.
When it is the Sun that is to overtake, taking a position
of Jupiter near the western horizon gradually the planet
gets fainter and fainter as the Sun approaches him from
west to east. Ultimately one evening the planet ceases
to be perceptible within a particular distance of the Sun.
This we call heliacal setting of the planet. Having thus
set in the west, after a few days the planet rises in the
east. This is so because, after the planet's heliacal setting,
the Sun gradually approaches the planet from west to east;
at a particular moment will have a longitude equal to that
of the planet and then gradually gains in longitude over
the planet. After a few days, in the eastern horizon, the
planet will emerge from the rays of the Sun before Sun-
rise; it is not the planet moving west from east but it is
the Sun moving from west to east so that the Sun recedes
away from the planet towards east. The planet also will
be having a motion from west to east but it being slower
than that of the Sun, the latter gains over the former in
its motion from west to east.
In the case of the inferior planets, say for example,
Mercury, the velocity of Mercury is greater than that of
the Sun; as such, Mercury will be overtaking the Sun and
not vice versa. Thus in the eastern horizon, it is Mercury
that enters the rays of the Sun, going from west to east,
so that he sets in the east. After a few days of this
heliacal setting, Mercury acquires a longitude equal to
that of the Sun, and while overtaking the Sun from west
to east, he emerges out of the rays of the Sun in the west,
which is therefore heliacal rising. But, when Mercury is
retrograde, the case is different. Some days after heliacal
rising in the west, Mercury attains his maximum elonga-
tion and then begins to retrograde. The elongation then
gradually decreases, and he will set again in the rays of
the Sun in the west itself. A few days thereafter, still
continuing retrograde, he emerges out of the Sun's rays

475
in the east, thus rising heliacally. After attaining the
maximum elongation in his retrograde motion, his motion
will then become direct, so that he again approaches the
Sun, going from west to east. Next a little before
overtaking the Sun, he sets heliacally in the east, and
thereafter, having overtaking the Sun, he rises heliacally
in the west.
Here, one point is to be mentioned. The time between
Mercury's maximum elongation in the west and again the
maximum elongation in the east (which are termed
maximum eastern elongation and maximum western
elongation with respect to the Sun) will be far less than
the time between the maximum elongation in the east
(ie. maximum western elongation with respect to the Sun)
and that in the west (ie. maximum western elongation
with respect to the Sun) in as much as when Mercury is
retrograde, and the Sun having always direct motion, the
relative velocity will be the sum of the retrograde velocity
of Mercury and the direct velocity of the Sun.
In the case of the superior planets, as mentioned
above, the superior planets set heliacally in the west and
rise heliacally in the east. They will not rise heliacally
in the west and will not set heliacally in the east which
happens only if either the superior planet has a greater
velocity than the Sun or the Sun has a retrograde motion
which is never the case.
Verse 5. Speciality with respect to Mercury and
Venus.
Mercury and Venus rise heliacally in the west in their
direct motion, (attain maximum elongation before they)
become retrograde, set heliacally there itself, then rise
heliacally in the east continuing to be retrograde, (attain
maximum elongation there before they) next become direct

476
and gradually set there (to rise again in the west) as
before.
Comm. Explained above.
Verse 6. Kālāmśas or distance in degrees from the
Sun within which the planets rise or set heliacally.
The Kālāmśas with respect to the Moon, Mars, Mer-
cury, Jupiter, Venus and Saturn, or the degrees of distance
from the Sun within which they rise or set heliacally are
12, 17, 14, 11, 10, 15 respectively. In the case of Mercury
and Venus when they are retrograde the Kālāmśas are 12,
and 8 respectively.
Comm. The Kālāmśas given above depend upon the
luminosity of the respective planets. When Mercury and
Venus happen to be retrograde, the Kālāmśas happen to be
2° less in each case because they are then nearest to the
earth and as such being most luminous as seen by us will
not set heliacally till they are very near the Sun.
Verse 7. To compute the moment when a planet
rises or sets heliacally.
If it is to be known when a planet rises or sets helia-
cally the position of the Prāk-Dṛk-graha or the Paschima-
Dṛk-graha as the case may be (Prāk-Dṛk-graha in case the
rising or setting takes place in the east or the other in the
other case) and that of the Sun also are to be computed on
a day a little before the day of rising or setting as
prognosticated by the Śīghra anomaly. In case the planet
rises or sets in the west, to obtain the lagna the position
of the Sun is to be increased by 180°.
Comm. Clear.

477
First half of verse 8. To obtain what are called
Iṣṭa-Kālāṁśas.
The time between the rising of the planet or of the
Dṛk-graha and that of the Sun measured in ghaṭīs multi-
plied by six gives what are called Iṣṭa-Kālāṁśas.
Comm. Having found the approximate position of
the Dṛk-graha as mentioned above when the planet is
likely to rise or set heliacally, let the time in between the
rising of this Dṛk-graha and that of the Sun (if it be the
case of setting or rising in the west the position of the
Sun is to be increased by 180° because the astalagna
directed to be found in verse (1) is the point of intersection
of the ecliptic with the eastern horizon, which is removed
180° from the setting point of the Sun) be multiplied by
six. Since both the Dṛk-graha and the Sun's position are
points on the Ecliptic and since we have considered the
time in between their rising moments, which is measured
on the equator, and again since this time is measured in
ghaṭīs, the number of ghaṭīs multiplied by six give the
degrees, for, each ghaṭī corresponds to six degrees of the
equator (sixty ghaṭīs corresponding to one sidereal day).
These degrees are said to be Iṣṭa-Kālāṁśas, which means
that it gives the arc in between the feet of the declination
circles of the Dṛk-graha and the Sun at the Iṣṭa-kāla ie.
that particular time considered.
Second half of verse 8 and verses (9) and (10).
If the number of degrees so found ie. the Iṣṭa-
Kālāṁśas fall short of or exceed the number of Kālāṁśas
postulated for the rising or setting of the planet, then the
planet's rising has to take place or has already taken place
respectively and vice versa in the case of setting. The
number of minutes of the difference of the prescribed and
Iṣṭa-Kālāṁśas multiplied by 1800 and divided by the
rising time of the Rāśi expressed in Kalās and again

478
divided by the difference of the daily motions of the planet
and the Sun expressed in minutes of arc if the planet is
direct in motion or divided by the sum of those daily
motions if the planet be retrograde gives the days elapsed
after rising or to elapse for the rising to take place. Again,
compute the positions of the Dṛk-graha and the Sun for the
moment thus obtained and repeat the process till the actual
moment is obtained.
Comm. Suppose the prescribed Kālāṁśas for rising
be x and suppose the Iṣṭa-Kālāṁśas are y such that y<x;
then for the planet to rise, the arc of the equator in
between the feet of the declination circles of the Sun and
the Dṛk-graha has to increase for the planet to rise which
means that this takes some more time to happen. Similarly
if y>x, the planet has already risen. The question of the
arc decreasing does not arise in this case of rising, because
the planet's position in the east is behind that of the Sun,
and in the case of a superior planet the Sun has to advance
further for the planet to rise ie. the arc has to increase and
in the case of a retrograde inferior planet also, the arc
will be increasing. In the case of an inferior planet being
direct, the arc will be decreasing no doubt, but we have
to remember that this is a case of an inferior planet setting
and not rising which we are considering. In the case of
setting, in the east, however, of the inferior planet the
moment of setting has already elapsed and thus the condi-
tion is reverse to that of rising. The case of setting of
a superior planet in the east never happens. In the case
of setting of a superior planet in the west, if the Iṣṭa-
Kālāṁśas be less than the prescribed Kālāṁśas, ie. y<x,
setting must have already taken place, the Sun having
approached the planet from behind and already effected
heliacal setting. Thus this is also reverse to the condition
of rising as stated. In the case of an inferior planet
setting in the west if y>x, the planet should have set
already since the inferior planet is retrograde while setting
in the west.

479
This condition is also reverse to what has been stated
in the case of rising. The case of a superior planet rising
in the west also never happens, because it is the Sun that
overtakes the planet and also the superior planet cannot
be retrograde while near the Sun.
The case of rising which has already elapsed the Iṣṭa-
Kālāṁśas y will be evidently greater than x ie. y > x. To
obtain the time by which the rising will take place after
the moment in question, ie. the rising of a superior planet
which is direct and an inferior planet which is retrograde,
we have to take the difference of the prescribed and Iṣṭa-
Kālāṁśas and find out the time by rule of three as follows.
(1) Since we have to find the corresponding arc of
the ecliptic in minutes of arc, from (y - x) × 60' the
difference of the Kālāṁśas of the equator converted into
minutes ie. asus we have first to use the proportion. "If
by the rising time of the Rāśi t in asus in which the planet
is situated we have 1800' of the ecliptic, what will we have
by (y - x) × 60? The answer is (y - x) × 60 × 1800 / t .
Next we have to find the days when this arc of the ecliptic
is covered by the proportion. "If the Sun overtakes the
planet by (u - v) minutes of the ecliptic per day, how many
days are taken to cover the above arc?" The answer is
as stated.
The answer gives in days together with fraction of
a day when the planet is likely to rise. We have to repeat
the process because the motions of the planet as well as
the Sun differ from moment to moment and the time calcu-
lated above by rule of three taking into account their
motions for the entire day will be only approximate. The
computation in the case of setting of a planet may be
similarly considered, the setting of a superior planet in the
west or that of a direct inferior planet in the east.

480
Verses 11 and 12. If the Prāk-Dṛk-graha has a
longitude greater than that of the Sun or the Paschima-
Dṛk-graha has one less than that of the Sun, the sum of
the prescribed Kālāṁśas and Iṣṭa-Kālāṁśas converted into
minutes of arc will have to be used to compute the elapsed
days or the days after which the respective phenomenon is
going to occur.
If the Iṣṭa-Kālāṁśas y be greater than the prescribed
namely x, the reverse to what has been stated in the second
half of verse (8) and in the first line of verse (9) happens
ie. a phenomenon which has elapsed when y < x will happen
in the future and vice versa.
Comm. If the Prāk-Dṛk-graha has a longitude greater
than that of the Sun, (since the prescribed Kālāṁśas are
between the planet and the Sun, the planet being behind
the Sun, and the Iṣṭa-Kālāṁśas are now on the other side
of the Sun) the planet has advanced over its position of
heliacal setting be it a superior planet or inferior by the
sum of the prescribed Kālāṁśas and Iṣṭa-Kālāṁśas. Hence
the days that have elapsed after the heliacal setting have
to be calculated with the sum of the two Kālāṁśas. If it
be also a case of a retrograde inferior Prāk-Dṛk-graha
rising, having a longitude greater than that of the Sun,
even then the prescribed Kālāṁśas are behind the Sun and
the Iṣṭa-Kālāṁśas ahead of the Sun. So from the position
of the planet’s heliacal rising, the Sun and the planet have
advanced in opposite direction to a distance which is the
sum of the prescribed Kālāṁśas and Iṣṭa-Kālāṁśas. Hence
computation has to proceed with the sum of the two
Kālāṁśas to get the time that has elapsed after the planet’s
heliacal rising. Similar is the argument for the Paśchima-
Dṛk-graha.
We have commented on verses from the latter half of
(8) upto (10) where the Iṣṭa-Kālāṁśas y happen to be less

481
than x. If y>x, or again the Prāk-Dṛk-graha has a longi-
tude greater that of the Sun, the phenomenon of rising or
setting which happened when y<x, will have to be taken
as going to happen and that which was going to happen
when y<x, must have happened already. It is enough to
consider just one case instead of all the four cases since
the argument is similar as before. Let us take y>x and
it is an inferior planet in the east behind the Sun.
According to the latter half verse (8) the rising had to
take place y being less than x, whereas, now, y being
greater than x, rising had already taken place, in contra-
distinction to what happens when y<x.
Here ends the Udayāstādhikāra.

SṚṄGONNATYADHIKĀRA
Verse 1. Either in the last quarter of a lunation, or
in the first quarter, on the day when the elevation of the
cusps of the crescent Moon is to be determined, then either
at the moment of Moon-rise or Moon-set or (for the matter
of that during any part of the night) the Hsine of the
altitude of the Moon is to be computed by noting the time
from the moment of Moon-rise.
Comm. Either in the first quarter or the last quarter
of the lunation, ie. when the phase of the Moon according
to the definition of phase in modern terms is less than
half, the Moon will be a crescent. Also, generally, on the
back-ground of the horizon, we notice that one of the
cusps is more elevated than the other. This elevation
goes by the name Sṛṅgonnati.
Even in the middle half of the lunation, Bhāskara
mentions that Brahmagupta and some others (meaning
Sripati whom he closely follows) attempted at finding the
elevation (strictly speaking elevation during the second
quarter and depression during the third quarter) of the
dark horns. Bhāskara does not appreciate this since,
nobody would think about this as it does not appeal to the
eye at all.
Verse 2. To find the Hsine of the altitude of the Sun.
The Hsine of the altitude of the Sun is to be
com-puted, assuming the rising Sun to be in the opposite
hemisphere, south or north (ie. if he be originally in the
northern, assume him to be in the south) and using the
formula given in verse 54 of Tripras'nādhyāya "अथोन्नता-
दूनयुताच्चरेण" given the time measured in asus that has
elapsed after Sunset.

483
Fig. 112
Comm. During the early part of the night or during
the latter part thereof, when the Sun is below the horizon,
the Sun will be occupying symmetrical positions with
respect to the horizon at times which are equally removed
from Sunset and the next Sun-rise. This is clear from the
figure 112. Let A and B be two such symmetrical positions
where AM and BN are the Hsines of the altitudes.
Evidently AM = BN, C P̂ A = C P̂ B. Since the rising eastern
hour-angle of the Sun equals he setting western hour-
angle, and since C P̂ A = C P̂ B, the time elapsed after
Sunset when the Sun is at B, will be equal to the time
before Sun-rise in the position A. Hence by congruence
AM = BN = Hsine of the altitudes in the two symmetric
positions. Using the modern formula from the triangle
PZS, cos z = sin ϕ sin δ + cos ϕ cos δ cos h, when
C P̂ A = C P̂ B = h cos z will be the same in the two positions
A and B. Putting z = 90 + θ, cos z = - sin θ will be
the same ie. H sin θ will be the same in the two positions
ie. the Śaṅkus will be the same. In the formula

484
cos z = sin ϕ sin δ + cos ϕ cos δ cos h
Put z = 90 + θ, h = 90 + H, δ = δ in the positions A, B
and z = 90−θ, h=90−H. δ=−δ in the positions A', B'
We have then − sin θ = sin ϕ sin δ − cos ϕ cos δ sin H
and sin θ = − sin ϕ sin δ + cos ϕ cos δ sin H
which are identical. This means that the altitude θ below
the horizon with +δ in the positions A, B is computable
with −δ in the positions A', B'. This accounts for the
statement made ‘गोलविपर्ययेण’.
The second statement of the latter half of verse (2)
says Śankutala = Śanku × ˢ/₁₂ which we have proved in
Tripraśnādhyāya.
Verse (3) and first half of (4). To obtain the Bhuja
of the Sun.
The Śankutala of the inverse altitude or the altitude
below the horizon, is north (in contradistinction to what
it is above the horizon). The sum or difference of the
Agrā and Śankutala according as they are of the same
direction or of opposite directions, is the Bhuja. The sum
or difference of the Bhujas of the Sun and Moon, according
as they are of opposite or the same directions is what is
called the Spaṣṭa Bhuja whose direction is to be construed
as that of the Moon. If the Bhuja of the Moon falls short
of that of the Sun, then the direction of the Spaṣṭa Bhuja
is that opposite to that of the Moon.
Comm. In fig. 113, we have shown five different
positions of the Sun S₁ to S₅ the feet of the Śankus being
B₁ to B. From these feet of the Śankus draw perpendiculars
on the Udayāstasutras as well as on the East-west line their
points of intersection being respectively A₁ to A₅ and
C₁ to C₅. The perpendiculars from the feet of the Śankus
on the East-west line go by the name Bhujas, the perpendi-

485
Fig. 113
cular distances between the Udayāstasutras and the East-
west line are called Agrās and the perpendiculars from the
feet of the S'ankus on the Udayāstasutras are called
S'ankutalas which were all defined in the course of the
Tripras'nādhyāya. We have from the spherical triangle
PZS, sin δ = sin ϕ cos z + cos ϕ sin z sin a so that
sin δ / cos ϕ = tan ϕ cos z + sin z sin a which was shown as
A=S+B in the Tripras'nādhyāya ie. Agrā = Sankutala +
Bhuja. Changing δ into —δ and a into —a we have different
formulae, the standard form being A=S+B. We don't
propose to change ϕ into —ϕ, because in India this case
does not arise. Thus our latitude being north, the S'anku-
talas defined above as the distances from the feet of the
Sankus from the Udayāstasutras are always deemed as
south. It need not be reiterated that the Udayāstasutras

486

are the straight lines joining the rising and setting points
of the celestial body. As such, these Udayāstasutras are all
parallel to the East-west line. In fig. 114, in the horizontal
plane containing the points
A's, B's and C's in the
respective cases, according
to the Hindu convention,
we talk of Agrā and Bhuja
as being Uttara Agrā,
Dakṣiṇāgrā, Uttara Bhuja
and Dakṣiṇa Bhuja.
Śaṅkutalam is always
taken to be south except in
the fifth case shown in figs.
113 and 114 when the altitude happens to be below the
horizon. In this case the Śaṅkutala is spoken as north.
Thus the general formula A=S+B assumes the following
various forms, in the respective cases.
(1) Uttarāgrā—Dakṣiṇa Śaṅkutala = Uttara Bhuja.
This corresponds to A, B, C of fig. 114 where AC, is
Uttarāgrā, BA, Dakshina Śaṅkutala and BC, Uttara
Bhuja.
(2) In the case of A₃ B₃ C₃, Śaṅkutala—Agrā =
Dakṣiṇa Bhuja since B₃ A₃—C₃ A₃=B₃ C₃. This occurs
after the Sun's diurnal path has crossed the prime-vertical
and the Sun has a position on the south of the prime-
vertical, having a northern declination. In the case of
A₄ B₄ C₄, B₄ C₄ = B₄ A₄ + A₄ C₄ ie. Dakṣiṇa Bhuja =
Dakṣiṇa Śaṅkutala + Dakṣiṇāgrā. In the fifth case
when the Sun's altitude is below the horizon, B₅ C₅ =
B₅ A₅ + C₅ A₅ ie. Uttara Bhuja = Uttara Śaṅkutala +
Uttarāgrā. In the sixth case when the Sun has a southern
declination and an altitude below the horizon, B₆ C₆ =
A₆ C₆ + A₆ B₆ ie. Dakṣiṇa Bhuja = Dakṣiṇāgrā minus
Uttara Śaṅkutala. In the case of A₂ B₂ C₂, Uttarāgrā =

487
Dakṣina Sankutala + Uttara Bhuja, since A₂ C₂ =
B₂ A₂ + B₂ C₂. In the absence of a unifying convention,
all these formulae are loose and apt to create confu-
sion. So we shall unify all these formulae into the
standard form A=S+B which holds good universally, if
we have the conventions.
(1) Agrā shall be deemed positive if it be north and
negative if it be south ;
(2) Similarly with respect to the Bhuja. But, with
respect to the Śankutala, we shall deem it positive if it be
south and negative if north. These conventions compre-
hend all the cases and unify them into the standard form
A=S+B. The corresponding conventions with respect
to δ, a are that +δ corresponds to Uttarāgrā and +a
corresponds to Uttara Bhuja.
Having the above conventions, and finding the Bhujas
of the Moon above the horizon and the Sun below the
horizon, the Spaṣṭa Bhuja required in finding the Śṛṅgon-
nati in the present chapter, is defined as the difference of
the Bhujas of the Sun and the Moon and the sum thereof
according as they are of the same direction north or south
or of different directions. Also this Spaṣṭa Bhuja is by
convention said to have the same direction as that of the
Moon. If, however, the Bhuja of the Moon falls short of
that of the Sun, then the Spaṣṭa Bhuja is said to have the
direction opposite to that of the Moon.
Verse (4). Definition of Koti.
I deem that the Koti should be taken as the sum
of the Śankus of the Sun and the Moon, the one being
below the horizon and the other above respectively.
Comm. We have defined above the Spaṣṭa Bhuja as
the north-south distance between the feet of the Śankus

488
of the Sun and the Moon. Herein Bhāskara defines the
Koti as the sum of the Saṅkus of the Sun and the Moon
which is the vertical distance between the points on the
armillary sphere which represent the Sun and the Moon.
Bhāskara says 'I deem' to signify that he differs
from Brahmagupta in this, who defines the chord joining
the Sun and Moon on the armillary sphere which is equal
to 2 Hsine of the SM on the sphere as the Karṇa. Brahma-
gupta defines the Bhuja and Karṇa from which he deduces
the Koti as √(Karṇa² — Bhuja²). Bhāskara argues that since
the Karṇa defined by Brahmagupta is not in the vertical
plane, since the Sun and the Moon are not in the same
vertical plane, so, the Koti defined by him throught he Karṇa
will not be in the vertical plane. The Karṇa defined by
Bhāskara is not on the other hand the chord joining the
Sun and the Moon but lies in the vertical plane in which
the perpendicular from the Moon's position on the armil-
lary sphere, on the horizontal plane containing the Sun's
position on that sphere. Also the Bhuja defined both by
Brahmagupta and Bhāskara is the projection on the north-
south line of the join of the Sun's position on the sphere and
the foot of the perpendicular from the Moon's position on
the horizontal plane through the Sun's position and it is
not actually the above join.
Bhāskara is evidently
guided by the right angled
triangle SMN' which is not
strictly a spherical triangle
as per its modern definition
as the arc SN is not that
of a great circle. This
figure guided him to take
the Bhuja horizontal and
the Koti vertical so that
his Karṇa is also in a
Fig. 115
vertical plane. Bhāskara's

489
Karṇa therefore has nothing to do with the arc SM but
only has the virtue of being in a vertical plane. Sripati
adopted Brahmagupta's method. Kamalākara neither
follows Brahmagupta nor Bhāskara but follows a method
of his own. In fact the methods adopted by these ancient
Hindu astronomers are not mathematically correct because
the question of determining the cusps as well as phase of
the Moon is concerned with the actual positions of the
Sun, Moon and the earth in space and not as seen on the
sphere. Hence M. M. Sudhakara Dwivedi has written a
small book by the name Vāstava-Sṛṅgonnati following
modern methods. The modern method which gives the
truth of the matter is depicted in standar modern texts.
Verse 5. The hypotenuse or Karṇa is the square-root
of the sum of the squares of the Bhuja and Koti. The
Bhuja multiplied by 6 and divided by the Karṇa gives
what is known as the Dik-valana of the Moon. The dire-
ction of the Valana has the same direction as the Spasṭa
Bhuja defined before.
Comm. The idea is that taking the radius of the
Moon's disc to be six angulas or units, representing the
Karṇa, the magnitude of the Bhuja on the same scale
gives a measure of what is defined as Valana. Thus Valana
= 6B/K where B and K stand for the Bhuja and Karṇa
defined before. The idea of this Valana will be clarified in
the ensuing verses.
Verse 6. The Hsine of the elongation of the Moon
is to be multiplied by the radius vector of the Moon
measured in Yojanas, and divided by the radius vector of
the Sun, also measured in Yojanas ; the arc of the Hsine
so obtained is to be added to the longitude of the Moon
in the bright half of the lunation and is to be subtracted
from the same in the dark half.
62

490
Comm. This is a correction to be made in the longi-
tude of the Moon to depict graphically the phase of the
Moon. Bhāskara says that many of the prior astronomers
took that the phase of the Moon was in direct proportion
to the elongation of the Moon. Taking the radius of the
Moon's disc to be six angulas or units, and assuming that
when the elongation is 180°, the entire disc being illumi-
nated, it was thought that 12 units of the diameter
correspond to 180° of elongation so that the Śukla of the
disc measured by the central width of the illuminated disc
increases at the rate of 1 unit for 15° of elongation. (The
word Śukla may be taken to correspond to the modern
word phase. Śukla is expressed in angulas, taking the
diameter of the disc to be 12 angulas. The measure of the
portion of the diameter of the Moon's disc perpendicular
to the diameter which is the join of the extremities of
the cusps, covered by the illuminated part of the disc,
(which may be defined as the maximum width of the
illuminated part of the disc) measured in angulas is said
to be the Śukla. The word 'phase' is used to signify the
ratio of the above width to the diameter. Śukla is
expressed in angulas, whereas phase is expressed as a ratio.
The Śukla is equal to twelve times phase). Bhāskara
rightly argues that this method of measuring the Śukla
is approximate because he says that six angulas of Śukla
is had when the elongation is not 90°, but only 85°-45'.
This may be substantiated as follows from fig. 116. Let
E, M, S stand for the earth, Moon and the Sun. Let
^
EMS = 90° so that it is a moment of dichotomy ie. the
moment when the phase is half and the Śukla 6 angulas.
Let θ be then the elongation of the Moon, so that
cos θ = EM / ES . Taking the average values given for EM
and ES by Bhāskara,
cos θ = 51566 / 689377 = 19 / 254 (obtaining a convergent) = .0748.