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

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

42
planet to be computed is got by multiplying the planetary
position of a planet computed by the number of sidereal
revolutions of the planet to be computed and dividing by
the number of sidereal revolutions of the planet computed.
Verse 15. We get the Ahargaṇa by multiplying the
planetary position given in number of revolutions and
fraction of a revolution by the number of days in a Kalpa
and dividing by the number of sidereal revolutions in a
Kalpa. How by indeterminate analysis we get the same
Ahargaṇa, given the number of past sidereal revolutions
alone, or by the fractional part of a revolution alone, or
by the sum of the fractional parts in the case of more items
involved, I shall tell later.
Comm. While computing the planet we have the
formula (A × P) / M = p where A = Ahargaṇa, P = the num-
ber of sidereal revolutions of the planet, M = number of
mean solar days in a Kalpa and p = the planetary position
consisting of the number of past revolutions and also the
fraction of a revolution. From the above equation, the
Ahargaṇa A = (M × p) / P as stated. In tne case of only
the integral number of revolutions or the fraction of a
revolution alone being given, or the sum of remainders if
more items than one are involved, the method of fiinding
the Ahargaṇa is illustrated in golādhyāya under praśna-
adhyāya under verses 12—21.
Verses 16, 17. Method of getting the time in solar
years that has elapsed from the beginning of the Kalpa,
given the Ahargaṇa.
The given Ahargaṇa multiplied by the number of
Kshaya-tithis in a Kalpa and divided by the number of
civil days in a Kalpa gives the number of the elapsed Kshaya
tithis. Adding these to the Ahargaṇa we have the lunar

43
days L. These again multiplied by the number of Adhika-
masas in a Kalpa and divided by the tithis in a Kalpa gives
the elapsed number of Adhikamāsas. Multiplying this
number by thirty and subtracting from the above lunar
days L, we have the elapsed solar days. Dividing these by
thirty, we have the number of elapsed solar months, the
remainder being solar days. Dividing the solar months by
12, we have the elapsed solar years and the remainder here
are the solar months. Thus we have the solar years, solar
months and solar days corresponding to the given
Ahargaṇa.
Comm. The inverse process detailed here is quite
clear.
Verse 18. Computation of the Ahargaṇa and the
planetary positions from the beginning of the Kaliyuga.
Find the Ahargaṇa from the beginning of the Kaliyuga
either (according to the method described formerly with
respect to a Kalpa) and this Ahargaṇa begins from Friday,
Computing the mean planetary positions from this Āhar-
gaṇa and adding to their mean positions at the beginning
of the Kali which are known as Dhruvakas, we have their
planetary positions for the day concerned.
Verses 19, 20. The Dhruvakas of the planetary posi-
tions at the beginning of Kali, given in a tabular form.

44
Comm. The mean Sun and the mean Moon are taken
to be in conjunction at the zero-point of the Zodiac. The
planetary positions given above are accepted by Bhaskara
on the authority of Brahmagupta. The fact that these
positions differ from those given by Aryabhata signifies
that Brahmagupta observed the True positions in his own
time and to obtain those positions by calculation, he must
have changed the fundamental constants such as the
number of civil days, and sidereal revolutions of planets
etc in a Kalpa. Here ends the section known as grahā-
nayana.

MADHYĀDHIKĀRA - KAKSHĀDHYĀYA
Verses 1, 2. The circumference of Akāsa---Kakshā.
Astronomers say that the circumference of Akāsa-
Kaksha is 18712069200000000 yojanas. Some say it is the
circumference of the universe whereas some say that it is
the circumference of the mountain which goes by the name
Lokāloka. Those who perceive the celestial sphere as a
fruit of the emblic myrobalan, (Known as Āmalaka in Sans-
krit) placed in the palm, say that it is the circumference of
the sphere of solar radiation ie the imaginary sphere whose
volume is filled by solar light.
Comm. Bhāskara, in the course of the Commentary
makes it clear that he does not subscribe to this idea which
is only mythological. Look at his words which are signifi-
cant and testify to his rational outlook “नाऽस्माकं मतमित्यर्थः,
प्रमाण शून्यत्वात्” ie “This is not our view; because it is
baseless”. A yojana will be seen to be equal to 5 miles
approximately.
Verse 3. He gives his personal view as follows.
The universe may be bounded or unbounded; our view
is that this dimension of the circumference is no other
than the distance covered by each planet in the Kalpa.
Comm. This was an assumption made by the ancient
Hindu Astronomers, as well as another assumption that the
distance covered by every planet during a day is the same.
This we shall see later.
Verse 4. The circumference of the universe given
above divided by the number of the sidereal revolutions in
a Kalpa of any planet gives the circumference of the plane-
tary orbit, so that in a Kalpa, the total distance covered is
the circumference of the universe.

46
Comm. Clear.
Verse 5. The circumferences of the orbits of the Sun,
Moon and the Stars.
The circumference of the Sun's orbit is 4331497½ yoja-
nas, that of the Moon 324000 yojanas, of the stellar sphere
259889850 yojanas.
Comm. Later, we are told by Bhaskara that the cir-
cumference of the earth is 4967 yojanas and its diameter is
1581. As the method given by him in the Commentary in
that context, to measure the circumference of the earth is
correct, we may take it that the ancient Hindu Astronomers
could estimate the same correctly. If that be so, when
Bhaskara gives the circumference to be 4967 yojanas, it
means 4967 yoj = (3960 × 44) / 7 miles or 1 yojana
= (3960 × 44) / (7 × 4967) = 5.01 miles or what is the same 1581 yoja-
nas = 7920 miles ie one yojana = 5.01 miles approximately.
With this measure of a yojana, the Moon's mean distance
from the earth's centre should be (as given above)
(324000 × 7) / 44 × 5 miles = 257725 miles approximately.
This seems to be a fair estimate and we have to find
out how this estimate could be made—Indeed, there are
many elementary trigonometrical methods of [finding the
distance of the Moon. Which of them was used by the
Hindu Astronomers, we have to discuss. Bhaskara, however
takes it implicitly from Brahmagupta's version. The latter
does not mention from what source he derived it but simply
mentions that he has resuscitated the Brahma-Siddhānta,
which grew obsolete. Either he or the author of Brahma-
Siddhānta must have computed this distance using trigono-
metry. The following seems to be the simplest method, by
which the Moon's distance was originally estimated-Refer
fig. 1. Let z be the zenith-distance of the Moon as obser-

47
Fig. 1
ved from a place A on the primary meridian going through
Lanka, Ujjain, Kerukshetra etc by an instrumeut (A pro-
tractor) described by Bhaskara under verse 5 Chandragraha-
ṇādhikāra, grahagaṇita at the time of transitting. Let B be
a sublunar point on the earth at the moment of that obser-
vation, where B is also on the same primary meridian.
Since both the places happen to be on the primary meridian,
and such places were primarily known, both the observa-
tions could be made simultaneously. Knowing the distance
A B between the two places, the angle θ subtended by A B
could be easily got from the triangle A O M where O is the
earth's centre and M the Moon. As a first approximation,
d / Sin θ = (d + a) / Sin Z = a / (Sin Z − Sin θ) taking O M roughly to
be equal to (a + d). In the above equation, a being known

48
d = a Sin θ / (Sin Z - (Sin θ)). Strictly speaking d / Sin θ = a / Sin (Z - θ)
so that d = a Sin θ / Sin (Z - θ). Since (Sin Z - Sin θ) < Sin (Z - θ)
∴ the estimate of d obtained is a little greater than its
true value. We shall discuss other possible methods of
finding the Moon's distance in the chapter on lunar eclipses.
Having got the distance of the Moon as afore-said,
it was easy to obtain the angle ∝ marked in fig. 1 for,
Sin ∝ = M₁ M¹ / O M¹ = a / d = 2πa / 2πd = 4967 / 324000
Since ∝ is small Sin ∝ = ∝ radians.
∴ ∝ = (4967 × 3438) / 324000 = (4967 × 191) / 18000 = 52.7'
Since the Moon's daily average motion 790' — 35" is had
in sixty Nadis, 52.7' of motion is covered in (52.7 × 60) / 790.5
approximately = 60 / 15 = 4 Nadis. Thus the fact that we are
given the horizontal parallax as 4 Nadis in the context of
lunar eclipses is based on this.
Having obtained thus the distance of the Moon from
the centre of the earth E, and having measured the angular
diameter of the Moon's disc with the help of the protractor
mentioned above, from the triangle EAM (Ref. fig. 2) where
[चित्र: Fig. 2 - त्रिभुज EAM जिसमें E पर कोण β, भुजा EM = d, बिन्दु M चन्द्रमा का केन्द्र, AM = r' त्रिज्या, और स्पर्शरेखा EA है।]
Fig. 2