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

22
belated by the duration of time that the arc of the ecliptic
covered by the Sun's today's motion takes to rise. This
duration of time is variable on two counts; first by the
variable motion of the Sun and second by the obliquity of
the ecliptic on account of which even equal arcs of the
ecliptic will not rise in equal times. In other words the
duration of time between two consecutive Sun-rises will
not be the same. This duration of a particular day can
roughly be calculated by the rule of three as follows. Let
the Sun be in a particular Rasi, the rising time T of which
could be computed; let the Sun cover an arc of x° in that
Rasi on that day. Then the time taken by that arc to rise
is xT / 30, where a Rasi consists of 30°. This time computed
in Sidereal measure added to 60 Sideral ghatis is equal to
the length of the day. During the course of an year ie the
time taken by the Sun to move round the ecliptic starting
from the Zero-point of the zodiac and again returning to
the same point, the Sun will have made one revolution less
than the stars. Thus if ‘ R ’ the number of diurnal revolu-
tions of the Sun (where R will be not an integer) during an
year or what is the same the number of Sāvana or civil
days during an year, they are equal to R + 1 sideral days.
Hence the number of sideral days in a kalpa will be equal
to the number of civil days in a kalpa together with the
number 4320000.000 which is the number of revolutions
made by the Sun relative to the stars.
In this context we are to know the number of civil
days in a kalpa. The proof given by Bhaskara in Gaṇita-
dhyāya under verses 1-6 in Bhagaṇopapatthi is as follows.
Draw a circle on a horizontal plane and place a vertical
pole called gnomon at the centre of the circle. Observe the
point of intersection of the gnomon's shadow with the
circumference of the circle at Sun-rise on a day in the
Uttarāyaṇa ie during the course of the Sun’s north-word
journey, just at the time when his rising point is very near
the east point and also to the south thereof. Then from

that day go on counting the number of days, which will
be 365 when the Sun again rises very nearly at the same
point and just to the south of the east point. It will be
found that the Sun will rise the next day just a little to
the north of the east point, which means that the Sun has
taken 365 days and a fractional part of a day to complete
his revolution round the stars. Having noted the two
points of intersection of the gnomonic shadow with the
circle, on those two consecutive days when the Sun
happens to rise just a little to the south and then on the
next day just a little north of the east point, and having
measured the arcs in minutes between those points of
intersection and the western point of the horizontal circle
(western because the gnomonic shadow of the rising Sun
is cast towards west) then the following rule of three is
to be applied. If during 60 ghatis of the day, the sum of
the arcs in minutes is covered, what will be the time
taken by the shadow to traverse the arc between the west
point and the northern point of intersection. This added
to 365, gives the number of civil days in an year. Here
it will be noted that this year is tropical because rising in
the east signifies the Sun's position at an equinox.
Verse 8. The number of solar days in a kalpa is
equal to 1555200000000 and of the lunar days or tithis is
1602999000000.
Comm. The solar days here cited are counted at the
rate of 360 per year; and the lunar days at the rate of 30
per lunation. Tithi is defined as mentioned by us under
verse 31 of the previous section. Since there are thirty
tithis in a lunation, their enumeration is quite alright;
but there seems to be a little oddity in saying that there
are 360 solar days in an year. This kind of a solar day
is a little longer than a civil day and does not correspond
to any particular motion of the Sun, say for example the
time taken by the Sun to move a degree along the ecliptic.
The definition of solar days pertains only to a stipulation

24
that 360 solar days constitute a solar year and no defini-
tion is given for a single solar day. This definition of
solar days, though apparently artificial, has some signi-
ficance, namely that the difference of the solar and lunar
days defined above constitutes 159330000000 Adhikamāsas
in a kalpa at the rate of 30 tithis per month. In other
words the difference between the solar and lunar days, is
the number of tithis that the luni-solar reckoning gains
over the solar. This topic will be dealt with later.
Verse 9. The number of civil days in a kalpa is
equal to 1577916450000; the number of the diurnal revolu-
tions of the stars minus the number of sidereal revolutions
of any particular planet constitute the days of that parti-
cular planet with respect to the earth.
Comm. The civil days in a kalpa are evidently the
number of Sun-rises. These are as mentioned before the
difference of the number of diurnal revolutions of the stars
and the number of the sidereal revolutions of the Sun.
By analogy, the number of the days of a particular planet
with respect to the earth or what is the same the number
of risings of that planet in a kalpa as seen from the earth,
is the difference of the number of the diurnal revolutions
of the stars and the number of the sidereal revolutions of
that planet in a kalpa. Thus we have Saura-Ku-dināni,
Chāndra-Ku-dināni, Bhauma-Ku-dināni etc, where the
word ‘ Ku ’ means the earth. The Saura-Ku-dināni are
the civil days defined before. It will be noted that the
Chāndra-Ku-dināni are not the lunar days.
Verse 10. The number of Adhikamāsās or intercalary
months in a kalpa is equal to 1593300000 and the number
of Dina-Kshayas is 25082550000.
Comm. A luni-solar year as per Cāndramāna ie the
luni-solar reckoning consists of twelve lunations; as such
its length falls short of that of the solar year by 11 days
3 ghatis, 52 Vighatis and 30 Sukshmaghatis. If both the

25 .

solar and luni-solar years begin simultaneously this year,
by the end of the luni-solar year, it will have gained over
the solar year the above-mentioned days. This difference
which goes by the name Adhimasa-Sesha or Suddhi
accrues to the length of a lunation in about 32½ solar
months. Unless this accruing difference is set apart by some
device, and the beginnings of the two years again brought
together, the luni-Solar year looses its significance of an
year, for, it does not accord with seasons. It being a con-
vention that an year should begin with the spring, if the
luni-Solar year also is to begin with the spring, it is to be
wedded to the solar year by some device. The device
adopted in this behalf was to leave out a month in the luni-
Solar reckoning as soon as it could be seen that a month
has been gained by this reckoning over the solar. This
knowledge is had from the following fact. The zodiac is
divided into twelve equal portions called Rasi's beginning
from the first point of the Hindu Zodiac, ie. from the first
point of the asterism division called Aswini. The Sun
traverses each of these Rasis in one Solar month, and on
account of the unequal motion of the Sun, these solar
months are of unequal length. The entrance of the Sun
from one Rasi into another is called a Samkrānti, and the
moments of Samkrāntis are held to be holy for religious pur-
poses. The solar month being a little longer than a luna-
tion, normally a Samkrānti occurs in a lunation. But it
so happens that in a particular lunar month this Samkrānti
might not occur. The Suddhi which is the time between
the moment of New Moon and the subsequent Samkrānti
and which is therefore the time gained by the luni-Solar
reckoning over the Solar, having accrued to a lunation, it
is an indication that the luni-Solar reckoning has gained a
lunation over the Solar. That lunation not carrying a
Samkrānti is termed an Adhikamasa and is left out. That
it is left out is connoted by the word Adhika which means
extra, as well as by the convention that no auspicious
celebrations like a marriage etc. should not take place
4

26
during that month. The next lunar month is termed as
the Nija-māsa or the month which is the true. In this
convention however, there may occur two Saṁkrāntis in
one particular lunar month. For this to happen two cri-
teria are to be satisfied namely that, that particular lunar
month must be greater than the concurrent solar month
and secondly one of the two Saṁkrāntis must occur imme-
diately after a New-moon so as to permit the second Saṁk-
rānti to occur just before the lapse of the lunar month.
The Sun coming to his perigee roughly about the lunar
month Mārgasira, and as such having the quickest motion
at that point, he covers the length of the Rasi of 30° within
the shortest span of time making that Solar month shorter
than the corresponding lunar month. Since the months
Kārtica and pausha, one before and the other after Mārga-
sira, also being thus longer than the corresponding solar
months, two Sankrāntis could occur, if at all, in these three
lunar months. If they occur, that lunar month is termed
a Kshayamāsa. Since two lunar months are to correspond
to two Saṁkrāntis, convention has it that two lunar months
lapse simultaneously during that lunation, the previous
lunation running during the forenoon and the subsequent
lunation during the afternoon. Thus a Kshaya māsa is also
called a yugalībhūta māsa or a twin māsa, which therefore
makes one lunation virtually vanish. On account of this
vanishing’, it is called a Kshayamāsa. After what an inter-
val of time this Kshaya māsa occurs how a month precedes
it having no Saṁkrānti and how again a lunation follows
it without a Saṁkrānti will be seen in a subsequent
context. This convention pertaining to the institution
of an Adhikamāsa goes by the name ‘Inter-calation’.
It was mentioned before that on an average an
Adhikamāsa occurs once in 32½ solar months roughly–Also,
Adhikamāsas are the excess of lunations over the solar
months in a given period. There being 51840000000 solar
months in a Kalpa and 53433300000 lunations the number
of Adhikamāsas is therefore 1593300000 as mentioned,

27
Kshayāhas were explained before and their number in
a Kalpa is the difference of the civil days and the tithis.
Verses 11, 12. The numbr of solar months in a Kalpa
is 51840000000; the number of lunar months is
53433300000. The number of solar months being subtracted
from the number of lunations, we have the number of
Adhikamāsas—The number of solar days together with the
days of Adhika months are equal to the lunar days or the
tithis; or again the lunar days minus the Kshayāhas are
equal to the number of civil days or the reverse will be
had by a reverse process.
Comm. Already explained.
Verse 13. The excess of the sidereal revolutions of the
moon over the number of the Sun’s sidereal revolutions is
equal to the number of chāndramāsas or lunations.
Or again the excess of the sum of the sidereal revolu-
tions of the moon and the tithis over the sum of the luna-
tions and the diurnal revolutions of the stars is equal to the
number of Kshayāhas.
Comm. The first part is clear. Regarding the second,
let the number of lunations be x and the number of the
sidereal revolutions of the moon be y. Then y —x =z, the
number of the sidereal revolutions of the Sun because
y — z = x from the first part above. If now, the number
of the diurnal revolutions of the stars be t, then t — z = t —
(y — x) = t + x — y = civil days. Subtracting these civil
days from ‘U’ the number of Tithis, we have the number
of Kshayāhas namely U — (t + x — y) = (U + y) —
(t + x) which accords with the statement.
Verse 14. The number of Adhikamāsas is equal to the
excess of the number of sidereal revolutions of the Moon
over thirteen times the number of sidereal revolutions of
the Sun.

28
Comm. Let x and y be the numbers of sidereal revolu-
tions of the Moon and the Sun respectively. The x—y is
evidently the number of lunations as mentioned before.
Also 12 y is the number of solar months which if subtracted
from the number of lunations will give the number of
Adhimāsas ie x — y — 12 y = x — 13 y = Adhikamāsas as
mentioned.
This completes the Bhaganādhyāya of Madhyādhi-
kāra.

GRAHĀNAYANĀDHYĀYA ग्रहानयनाध्यायः
Verse 1. To Compute the Ahargaṇa the collection of
days from the beginning of Kalpa ie from the beginning
of creation. Multiply the number of sidereal solar years
from the beginning of Kalpa by 12; add the number of
elapsed lunar months ; multiply by thirty ; add the number
of elapsed tithis. Let this number be x. Then [ (x × A) / s ]
the integral number obtained by dividing the product of x
and A the number of Adhikamāsas in a Kalpa by s the
number of solar days thereof gives the number of elapsed
Adhikamāsas. Multiply this number of Adhikamāsas by
30 and add to x. The result gives the number of elapsed
tithis, Let this be y. Then [ (y × K) / T ] the integral num-
ber obtained by dividing the product of y and K the num-
ber of Kshayāhas in a Kalpa by T the number of Tithis in
a Kalpa, gives the number of Kshayāhas. Subtracting this
number from y, we have the Ahargaṇa ie the number of the
elapsed civil days from the beginning of Kalpa. This
Ahargaṇa has its beginning on Sunday and is itself consti-
tuted of mean solar days. While computing the Adhika-
māsas or Kshayāhas, the integral numbers of the quosients
alone should be taken rejecting the remainders.
Comm. The elapsed number of solar years is directed
to be multiplied by 12 in the beginning. This number
does not constitute purely solar years. Solarity was
secured upto the point of the last intercalation of an Adhi-
kamāsa and thereafter one or two luni -solar years would
have been added. But Construing these one or two luni-
solar years as mean solar years does not make a difference
while computing the Adhikamāsas, for the following reason.
Adhikamāsas normally occur once in three years. Even

30
supposing that the number of the elapsed years contain
three luni-solar years after the year carrying the last inter-
calary month, the error in construing them to be solar will
be roughly minus one solar month. In other words the
difference between three mean solar years and three luni-
solar years will be roughly one solar month. Since one
Adhikamāsa occurs roughly in 32½ solar months, so the
number of Adhikamasas obtained by construing three luni-
solar years as mean Solar years will be in default by roughly
1/32½ or 2/63. As we are counting only the integral number
of Adhikamāsas obtained as a quotient rejecting the
remainder, the above default of 2/63 should not normally
effect the quotient. Most rarely, however, it might effect
the quotient by one, for which provision is made by Bhas-
kara in verse 3 under Adhimāsādinirṇaya section, Madhya-
mādhikara namely स्पष्टोऽधिमासः पतितोपलब्धः etc. which
means that if the quotient is in default by one, where it
is definitely known that one more Adhikamāsa did occur,
we are directed to add one and if we certainly know that
the quotient contains one more Adhikamāsa, when the
Adhikamāsa is shortly to occur and has not occrued we
are directed to substract one from the quotient. This
principle of सैकत्वनिरेकत्व is to be observed even in the
context of Kshayāhas ie we are directed to add one or
subtract one from the number of civil days by adjusting
the Ahargaṇa to the week-day on which the Ahargaṇa is
sought to be found.
Thus construing the elapsed number of years to be
solar, multiplying them by twelve we have the elapsed
solar months. Adding to this the number of the elapsed
luni-Solar months construing these to be solar, which fact
also does not affect normally the number of Adhikamāsas to
be obtained (for the same reason above) and multiplying by
30 and adding the elapsed tithis we have the elapsed num-
ber of solar days (This number may not be exactly the

31
elapsed number of solar days, for, we have construed the
one, two or three luni-Solar years as mean solar as well as
the elapsed lunations of the present year as solar months;
but this difference does not affect the computation of Adhika-
māsas as mentioned above). Then, as we are given that
15933,00000 Adhikamāsas occur in 1555200000000 solar
days of the Kalpa, if x be the number of solar days found
above, (x ✕ 1593300000) / 1555200000000 will give the number of the elapsed
Adhikamāsas. Adding these Adhikamāsas multiplied by 30,
to x the solar days obtained above, we have the Tithis elap-
sed upto the day in question. If now, we subtract the
Kshayāhās from these Tithis, we shall have the number of
Sāvanāhas or the Ahargaṇa required. Since in 1602999000000
Tithis of the Kalpa there will be 25082550000 Kshayāhas, if
y be the Tithis above obtained, (y ✕ 25082550000) / 1602999000000 will be
the Kshayāhās from the beginning of the Kalpa upto the
day in question ; subtracting these from y, we have the
required Ahargaṇa.
Verse 4. Computation of the planetary positions.
The Ahargaṇa multiplied by the number of sidereal
revolutions of a planet and divided by the number of civil
days in a Kalpa gives the planet ie its number of revolu-
tions upto the day concerned both integral and fractional.
Comm. Applying ‘Rule of three’, if in C the
number of civil days in the Kalpa, the planet makes P
sidereal revolutions how many revolutions would have been
made in A the Ahargaṇa ? The answer is (A ✕ P) / C. In this,
the integral quotient gives the number of complete revolu-
tions made; the remainder, multiplied by 12 and divided by
C again, gives the number of Rasis covered by the planet
from the Zero-point of the Zodiac, and again the remainder
multiplied by 30 and divided by C gives the number of deg-

32
rees covered in the next Rasi; proceeding thus, the planetary
position could be had next in minutes and then in seconds
of arc also. This position is the mean position of the
planet, at the time when the mean Sun is very nearly at
the Eastern horizon at Lanka. Why it is said "very
nearly at the horizon" will be clear in the context of uda-
yāntara Samskāra to be dealt with later in Spaṣṭādhikāra.
To obtain the planetary position at the moment when the
mean Sun is exactly on the horizon, we have to apply what
is called the Udayāntara correction and again to obtain the
position at the moment of True Sun-rise we have to apply
what is called the Bhujāntara correction, both of which will
be dealt with in Spaṣṭādhikara. Having thus got the
mean planetary position at True Sun-rise, we have to apply
one or two as the case may be, corrections to obtain the
True planetary position, besides effecting two more correc-
tions known as Desāntara and chara to obtain the True
planetary position at the time of True Sun-rise not at
Lanka but at the place concerned.
Verse 5. To obtain the position of the mean Moon,
when the mean Sun is known from what is called Avama-
Seṣa.
The Avama Seṣa divided by 131490000000 in degrees
is to be added to twelve times the elapsed tithis and the
result added to the Sun's position gives the position of the
Moon. Conversely the position of the Sun can be had from
the position of the Moon.
Comm. The moon's longitude minus the Sun's longi-
tude known as elongation is called Vyarkēndu, which
divided by twelve gives the number of elapsed tithis; a
lunation which is the time in which the Moon overtakes
the Sun by 360°, contains thirty tithis and a tithi is defined
as the time, in which the Moon overtakes the Sun by 12°,
beginning from the moment of conjunction ie Amāvāsyā.
Hence if the Sun's longitude be x°, aud y be the number of

33
elapsed tithis, integral or fractional (x + 12y)° will be the
longitude of the Moon. If, however, we consider y only as
the integral number of the elapsed tithis, we will have
obtained the Moon’s position from the formula (x + 12 y)°
at the ending moment of the tithi on the previous day.
The time in between this ending moment of the tithi and
the Sun-rise of the day concerned is known as Avama-
Sesha, since the Avama-days or Kshayāhas are the differ-
ence of days between the number of tithis in a given
period and the number of civil days during the same
period. In other words, the excess of a civil day over
a tithi which falls short of it, is the part of a civil day
that contributes towards the number of Kshayāhās in a
given period. Thus we have to compute the increase in
the Moon's longitude during the aforesaid Avama-Sesha
to obtain his longitude at the Sun-rise of the day concern-
ed. But this Avama-Sesha is of the form F { (t × K) / T }
where ‘ t ’ is the number of elapsed tithis upto the Sun-rise
of the day concerned K the number of Kshayāhas in a Kalpa,
T is the number of tithis in a Kalpa and F { (t × K) / T }
signifies the fractional part called Avama-Sesha, the integral
quotient having given the number of elapsed Kshayāhas.
Hence writing F { (t × K) / T } as R / T where R is the remainder
obtained by dividing (t × K) by T, the Avama-Sesha is of
the form R / T. This Avama-Sesha being mean solar. it has
to be rendered luni-Solar, which process is known as
चान्द्रीकरणम्. The rule of three used in this behalf is
‘ If for C civil days we have T tithis of the Kalpa, what
shall we have for R" / T ? The answer is (R / T) × (T / C) = R / C. This
then is the balance fractional part of a tithi that is there in
between the end of the elapsed tithi and the Sun-rise of the
day concerned. This part of a tithi is to be multiplied by
5

34
12 to give in degrees the increase of Moon's longitude from
the ending moment of the tithi on the previous day. Thus
this increase is R/C × 12 ; but C = 1577916450000
∴ R/C × 12 = R / (1577916450000 / 12) = R / 131490000000
approximately.
Thus this increase is to be added to (x + 12 y)° where
x° is the longitude of the Sun and y the integral part of
the elapsed tithis so that the Moon's longitude is
(x + 12 y)° + R° / 131490000000˙
Verses 6, 7. Computation of the positions of the Sun
and the Moon from the Adhimāsa-Sesha and Avama-Sesha.
The Avama-Sesha divided by 27110000000 is termed an
additive constant in minutes of arc to the Sun’s position ;
the same Avama-Sesha multiplied by 13 and divided by 35
is termed such an additive constant to the position of the
Moon ; Construe that the Sun's position is given by as
many degrees as there are elapsed tithis after the beginning
of Chaitra and that the Moon's position is given by Thir-
teen times the same. Let these positions of the Sun and
the Moon be diminished by a number of degrees equal to
what is obtained by dividing the Adhimāsa-Sesha by the
number of lunations in a Kalpa. Then add the respective
additive constants to the positions of the Sun and the Moon
so obtained. The results will be the positions of the Mean
Sun and the Mean Moon.
Comm. Here the data are the Adhimāsa-Sesha and
the Avama-Sesha and nothing else and the problem set is
to find the Mean positions of the Sun and Moon. The
Adhimāsa-Sesha is of the form (R' × 30) / S where R' is the
remainder obtained while finding the elapsed Adhimāsas

35
by dividing the product of the number of lunations in
a Kalpa and (12 x + y) 30 + t where x is the number
of elapsed years, y the number of elapsed lunations t
the number of elapsed tithis in the current lunar month
at the time when the Ahargaṇa is being computed and
S the number of Solar days in a Kalpa. The form
of the Avama-Sesha was formerly stated as R / T. The
procedure given is as follows. In the first place, we are
asked to assume that the number of elapsed solar days is
equal to the number of elapsed tithis from the beginning of
Chaitra ie the beginning of the luni-Solar year, since we do
not know when the Solar year began; so, at the rate of 1°
per a Solar day (A Solar day is not a mean solar day,
Vide definition given before under verse 8. Bhagaṇādh-
yāya) the Sun’s position is given by as many degrees as
there are elapsed tithis; and the Moon’s position must be
13 times the same since each tithi means an increase of
12° in the elougation and if x° be the Sun’s position, the
Moon’s position must be 12 x° ahead of the Sun ie 13 x°.
Having got thus approximate positions of the Sun and the
Moon, we have to make amends for the roughness of the
assumption made. In assuming that the longitude of the
Sun is equal to the number of elapsed tithis, we have over-
estimated the longitude since it has to begin from the
beginning of the solar year. The time in between the
beginnings of the luni-Solar year and the solar year is
known as the Adhimāsa-Sesha at the time of the beginning
of the solar year, measured in tithis; also, we have com-
mitted an error in assuming the Sun’s rate to be 1° per tithi.
This also is due to Adhimāsa-Sesha subsequent to the
beginning of the Solar year upto the day concerned- Thus
the entire error committed by assuming the Sun’s longitude
to be as many degrees as there are elapsed tithis from the
beginning of the luni-Solar year. is no other than the Adhi-
māsa-Sesha at the day concerned, But this Adhimāsa-
Sesha is of the form (R′ × 30) / S as mentioned and is in tithis.

36
This has to be converted into solar days to give us the
number of degrees of the error. The conversion is effected
by the rule of three as “If T tithis of a Kalpa constitute S
solar days of the Kalpa, what number of solar days corres-
ponds to (R' × 30) / S ?” The answer is ((R' × 30) / S) × (S / T) =
(R' × 30) / T = R' / (T/30) = R' / L where L is the number of
lunations in a Kalpa. This R' / L being the number of solar
days corresponding to the Adhimāsa-Sesha upto the day
concerned, the corresponding longitude of the Sun namely
R'° / L must be subtracted from the Sun’s longitude. Now the
question arises whether we have to multiply this R''° / L by 13
to be subtracted from the longitude of the Moon; not
necessary, enough to subtract R'° / L only, for, the Adhimāsa-
sesha extends from the preceeding Amāvāsya only when
the Moon’s longitude was equal to the Sun’s longitude.
We have thus got the mean positions of the Sun and the
Moon at the ending moment of the tithi on the previous
day. To get their mean positions at the Sun-rise of the
concerned day, we have to make amends for the time in
between, which is no other than Avama-Sesha as a fraction
of a mean solar day. Here Bhaskara makes an ingenious
approximation. The maximum Avama-Sesha could be a
tithi only and the Sun moves roughly by his daily mean
motion during a tithi. So, the rule of three adopted is
‘If for one tithi, the Sun’s daily motion is to be reckoned,
what for the Avama-Sesha? The answer is Avama-Sesha
multiplied by the Sun’s daily motion. The Avama-Sesha
being of the form R / T, (R / T) × m' = (R / T) × 59 (8' / 60) =

37
R' R' R'
——— = ——————————————————————— = ——————————— very
T 1602999000000 × 15 27110000000
———— ——————————————————
59 2/15 887
approximately.
This must be added to the position of the Sun to get
his position at the Sun-rise concerned. In the case of the
Moon the above additive constant of the Sun, is to be
multiplied by 13 13/33 because the Moon's daily motion is so
many times that of the Sun. Hence the additive constant
in the case of the Moon is x × 13 13/33 = 13 × (1 + 1/33)
where x is the additive constant in the case of the Sun.
This agrees with what Bhaskara has stated.
Verses 8, 9. Another way of computing the planetary
positions.
The mean position of the Sun in R

38
we have f' − (A × G) / M = I' + F where I' is some integer other
than I. But (A × D) / M which gives us the number of
diurnal rotations of the stars upto the day concerned is
equal to the Ahargaṇa plus the number of revolutions of
the Sun upto the day concerned, because Ahargaṇa +
Revolutions of the Sun = diurnal rotations of the stars
upto the day concerned. Omitting the integral Ahargaṇa
and the integral number of revolutions of the Sun we have
that the fractional part of (A × D) / M ie f' is no other than
the Sun's position. Hence.
(A × P) / M = I + F = f' − (A × G) / M and (A × G) / M =
(A × G) / 1577916450000 revolutions = (12 R) / 1577916450000 Rasis =
R / 131493037500 Rasis. Thus the planetary position is
equal to the Sun's position minus (A × G) / 131493037500 where the
integral parts on either side could be ignored.
Here there is a peculiarity in this method. The planet
could right away be obtained from the formula (A × P) / M
where A is the Ahargaṇa, P the number of sidereal revolu-
tions of the planet and M the number of mean Solar days
in a Kalpa. Though the given method is more cumbrous
than finding through the above formula, Bhaskara delibe-
rately gives it to show the equivalence of various proce-
dures at the same time giving us a beautiful technique as
mentioned in his commentary. Thus in the above equation
(A × P) / M = (A × D) / M − (A × G) / M the first term on the right
hand side could be termed the Bha-bhrama-graha and the
second term (A × G) / M the graha-Sāvana-Dina-graha. We

39
have seen above that the first term is no other than the
Mean Sun ignoring the integral number.
Verses 10, 11. Proof of other methods of computing
planetary position. Even as the sums or differences of
two or more of the numbers of Adhimāsas, Kshayāhas,
lunations etc give the number of sidereal revolutions of the
planets the sums or differences of two or more of the posi-
tions of the imaginary planets which go by the names
Adhimāsa-graha, Kshayāhagraha etc computed out of the
numbers of those Adhimāsas Kshayāhas etc. give the
respective planetary positions.
Comm. This interesting concept is based upon the
following principle. Suppose P to be the number of side-
real revolutions of a planet; then (A × P) / M gives the planetary
position, where A is the Ahargaṇa, and M the number of
mean solar days in a Kalpa. Now suppose P = x ± y ± z
where x, y, z are the numbers of Adhimāsās etc in a Kalpa,
then
(A × P) / M = (A × x) / M ± (A × y) / M ± (A × z) / M •
The terms on the right-hand-side may be construed to
be the Adhimāsa-graha etc, which are the positions of
imaginary planets and their sums or differences give there-
fore the position of the planet, as could be seen from the
above equation. Thus for example, we have the equation
P₁ — 13P₂ = a where P₁ is the number of the sidereal
revolutions of the Moon and P₂ the number of the sidereal
revolutions of the Sun, a stands for the Adhimāsas because
Chāndramāsas — Sauramāsas = Adhimāsas; but Chāndra-
māsas = P₁ — P₂ and Sauramāsas = 12 P₂ so that
P₁ — P₂ — 12P₂ = a ie P₁ — 13P₂ = a. From this equation
P₁ = a + 13P₂ ∴ (A × P₁) / M = (a × A) / M + (13P₂ × A) / M •

40
The first term on the right hand side is termed as the
Adhimāsa-graha, and the second term is evidently 13 times
the position of the Sun whereas the term on the left-hand-
side is the Moon’s position. Hence, ignoring the integral
number of revolutions, we have Moon’s position = Adhi-
māsa — graha + 13 times Sun’s position (Ignoring the
number of integral revolutions means subtracting integral
revolutions or adding them if necessary).
Verses 12, 13. A few more examples on the afore-
said principle. The planetary position obtained by the
sum of the sidereal revolutions of two planets, added to or
Subtracted from another planetary position obtained by
the difference of the sidereal revolutions of two planets and
divided by two gives the positions of the two planets res-
pectively, the quicker of the two in the first case and the
slower in the second. Similarly the planetary position com-
puted from the difference of the sidereal revolutions of two
planets subtracted from the planetary position of the quicker
of the two gives the position of the slower whereas the
former planetary position added to the position of the
slower gives the quicker.
Comm. We have [(P₁ + P₂) + (P₁ − P₂)] / 2 = P₁ (1) and
[(P₁ + P₂) − (P₁ − P₂)] / 2 = P₂ (2) where P₁ is the number
of sidereal revolutions of a quick-moving planet and P₂
that of a slow-moving one. Multiplying the above equa-
tions by A/M with the former notation,
[A/M (P₁ + P₂) + A/M (P₁ − P₂)] / 2 = (P₁ × A) / M (3) and
[A/M (P₁ + P₂) − A/M (P₁ − P₂)] / 2 = A/M × P₂ (4)
Equations (3) and (4) mean what has been stated in verse
(12). Again we have the equations P₁ − (

41
and P₂ + (P₁ − P₂) = P₁ where P₁ and P₂ are the numbers
of sidereal revolutions of a quick and slow moving planets
respectively. Following the same principle as above we
could obtain their positions by multiplying the equations
through out by A/M and calling planets on the left-hand-
side as (1) Dwiparyayāntarodbhava-graha subtracted from
the quick-moving one and (2) the slow-moving planet
increased by the Dwiparyayāntara-graha respectively.
Verse 14. The difference of the Śīghra and Śīghra-
Kendra as well as the Sum of the Mandoccha and the Manda
Kendra give the planet to be computed. Or again a com-
puted planet multiplied by the number of sidereal revolu-
tions of a planet to be computed and divided by the number
of sidereal revolutions of the computed gives the planet to
be computed.
Comm. U₁ − P = K₁ and P − U₂ = K₂ where U₁,
P, U₂, K₁ and K₂ are respectively the number of Sidereal
revolutions of the Śīghroccha, planet, the Mandoccha, the
Śīghra-Kendra and the Manda Kendra ; hence, we have
P = U₁ − K₁ = U₂ + K₂ from these equations also
by multiplying througout by A/M , we have the planetary
position as the difference of the Śīghroccha-graha and
Śīghra Kendra-graha or the sum of the Mandoccha-graha
and Manda-Kendra-graha.
Again, if P₁, P₂ be the numbers of sidereal revolutions
of a computed planet and one to be computed respectively
and if p₁, p₂ be the computed planet and the one to be
computed, then
P₁ × A/M = p₁, P₂ × A/M = p₂ so that P₁/P₂ = p₁/p₂
∴ (P₁ × p₂)/p₁ = P₂ which means that the position of the
6