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

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

49
εA is a tangent to the Moon's disc and M the centre of
the Moon, Sin B = r' / d where B is the angular semi-dia-
meter of the Moon's disc, r' = the spherical radius of
the Moon's disc measured in yojanas, so that since
Sin B = B' in Hindu trigonometry when the angle is small,
2πr' / 2πd = 2πr' / 324000 = 16 / 3438 ∴ 2πr' = (324000 × 16) / 3438 ; hence
r' = (18000 × 16) / 191 × 7 / 44 = 504000 / 2101 = 240 yojanas.
Bhaskara gives the semi-diameter of the Moon's disc to be
16'-0"-9"' so that the above r' is very approximately 240
yojanas as given by Bhaskara.
Now with these constants pertaining to the Moon's
distance, and his spherical radius, it was sought to find the
spatial distance traversed by the Moon during a day. The
rule of three used was " If 16'-0"-19"' of the Moon's angu-
lar semi-diameter pertains to a spatial distance of 240 yoja-
nas at his orbit, what does the mean daily motion of
790'-35" pertain to ? " The answer is 683064000 / 57609 = 11858¾
yojanas as given by Bhaskara elsewhere or taking the
Moon's mean semi-angular diameter to be 16' only, and
using the rule of three " If 16' at the lunar orbit correspond
to 240 yojanas, what do 790'-35" correspond to " we have
(790'-35" × 240) / 16' = 15 × 790 7/12 = 5/4 × 9487 = 9487 + 2371¾
= 11858¾ yojanas as given by Bhaskara.
Thus having obtained the daily spatial motion, the
Hindu Astronomers assumed all the other planets (includ-
ing the Sun) to have the same daily mean spatial motion.
On this assumption since the, Moon's orbit will be
27.3217 × 11858¾ = 324000 yojanas, and the circumference
of the universe will be 324000 × 57753300000
= 18712069200000000 yojanas the Sun's orbit will be
7

50
circumference of the universe
—————————————————————————————— = 4331497½ yojanas
Number of sidereal revolutions
(because (324000 × 57753300000) / 4320000000 = (3240 / 432) × 577533
= (15 / 2) × 577533 = 8662995 / 2 = 4331497½).
Also, the circumference of the stellar universe was
presumed to be sixty times that of the Sun’s orbit.
With the constants obtained for the Moon’s diameter
and distance and with assumption that all the planets have
the same daily motion, the constants pertaining to the Sun
and the other planets were obtained. We shall resume this
topic in another context.
Verse 6. The mean daily motion of the planets.
The circumference of the universe divided by the num-
ber of days in the Kalpa, gives the daily spatial motion of
a planet. The planets move thus a distance of 11858¾
yojanas in a day.
Comm. Already explained.
Verse 7, and half the verse 8. The Ahargaṇa multiplied
by 11859 decreased by the quotient obtained by dividing
the product of the Ahargaṇa and 9921 by 35419 gives the
distance covered by a planet in yojanas. These yojanas
divided by the circumference of the planet’s orbit gives the
fraction of a revolution and the integral number of revolu-
tions made.
Comm. Let the Ahargaṇa be A; every planet should
have described a space equal to A × D where D is the dis-
tance traversed per day and is just a little less than 11859,
or more correctly should have described a space equal to
A × (C / c) where C is the afore-said circumference of the

51
universe, and c the number of civil days or mean solar
days in a Kalpa. ' For the sake of an easy computation
Bhaskara gives here an interpolation. In the first place,
we are asked to multiply A by 11859 by which, an excess is
there in the result; then this excess is sought to be removed.
The excess is A × 11859 - (A × C) / c because A × C / c is the
correct distance described. In other words, the distance
described by any planet is
(A × C) / c = A × 11859 - (A × 11859 - (A × C) / c) I
But C / c = 18712069200000000 / 1577916450000 = 420024000 / 35419 dividing by
the common factor 44550000
∴ The space described by any planet is from I
A × 11859 - A { (c × 11859 - C) / c }
= A × 11859 - A { (35419 × 11859 - 420024000) / 35419 }
But the numerator within the brackets is 9921
∴ Space described = A × 11859 - (A × 9921) / 35419
as stated. This distance divided by the individual orbital
lengths, we have the integral number of revolutions made
by each planet, rejecting which we have the fractional part
of a revolution which gives the position of the planet.
Verses 8, 9. The orbit of the planet itself is no doubt
the orbit of the Mandoccha (apogee with respect to the Sun
and the Moon and aphelion with respect to the other Star-
planets ie Mars, Mercury, Jupiter, Venus and Saturn) and
of the node (point of intersection of the orbits of the Star-
planets with the ecliptic); but while Computing the posi-
tions of these Mandocchas and the Nodes, as per the
method indicated above ie according to the method of

52
Kakshādhyāya, their orbits are taken to differ from those
of the planets. (because slow-moving, points will have
longer orbits as per the assumption made namely that the
circumference of the universe divided by the number of the
sidereal revolutions gives the length of the orbit). Simi-
larly the orbit of the Sun itself will be the orbit of Mercury
and Venus, and the orbits of their S'īghrocchas are their
real orbits wherein Mercury and Venus are taken to move
with the velocity of the Sun.
Comm. The prescription of the computation of a
planetary position as per the method of Kakshādhyāya, has
brought in an awkward situation. Let us consider the
Computation of the positions of Mercury and Venus. These
two planets oscillate about the mean position of the Sun,
because their orbits happen to lie within the earth's orbit.
Hence their mean sidereal periods coincide with that of the
Sun, which means that their numbers of sidereal revolu-
tions coincide with the number of sidereal revolutions of
the Sun. Hence the Kakshādhyāya method of Computing
a planetary position brings in the idea that the orbits of
Mercury and Venus coincide with the orbit of the Sun.
Bhāskara perceived the awkwardness of this situation and
says therefore, the above coincidence of the orbits must not
be taken to be a reality but is intended only for the sake of
computation. The actual planets Mercury and Venus in
fact revolve says Bhaskara in the orbits of their Sīghroc-
chas, with the velocity of the Sun-Even this supposition
that the actual planets move with the velocity of the Sun
sounds odd, but this will be clarified later in the spashta-
adhikāra, wherein we propose to explain the peculiar con-
cept of S'īghroccha at length.
Here ends the Grahānayanādhyāya according to the
Kakshādhyāya method.

THE ADHYĀYA KNOWN AS PRATYABDA
S'UDDHI IN MADHYĀDHIKĀRA
Verse 1. The number of years which have elapsed from
the beginning of the Kalpa, respectively multiplied by 2, 4
and 3 and divided by 8, gives what is called Dinādya in
days, ghatis and Vighatis respectively. If this be added to
the number of years and divided by seven, the remainder
gives the Abdapa or the lord of the year, (under whose name
the week-day of the commencement of the year stands).
Comm. One mean solar year consists of 365 days, 15
ghatis, 30 palas, and 22½ Vipalas where the units are all
mean solar and one mean solar day is equal to 60 mean solar
ghatis, one mean solar ghati is equal to 60 mean solar palas,
one mean solar pala is equal to 60 mean solar Vipalas and
so on in sexagesimal sub-division. The fraction of the day
over and above 365 days, namely 0-15-30-22-30 multiplied
by 8 gives 2 days, 4 ghatis and 3 palas so that by the rule
of three i.e. ‘If in 8 mean solar years, the fraction accrues
to 2 days, 4 ghatis and 3 palas, what will it accrue to in x
elapsed mean solar years from the beginning of the Kalpa ?
we have the answer (x × 2)/8 days, (x × 4)/8 ghatis and (x × 3)/8
palas. If this is added to the number of the elapsed years,
and the result divided by seven, the remainder gives the
week-day of the commencement of the concerned year, be-
cause the remainder got by dividing 365 by 7 is one, and
the week day advances at the rate of one per year. Also
the Kalpa began on Sun-day.
Verse 2. Alternate method.
Half the number of elapsed years added to 1/60 of itself,
then divided by 60 and added to 1/4 of the elapsed years,
gives the Dinādya.

54
Comm. Since the Dinādya per year is 0-15-30-22-30,
in x elapsed years it accrues to x × 0-15-30-22-30
= (x × 15)/60 days + (x × 30)/60 ghatis + x × 22½ palas = x/4 days +
x/2 g + (x × 45)/(2 × 60) palas = x/4 d + x/2 g + 3x/8 × 1/60 g = x/4 d +
x/2 (1 + 1/80) g = x/4 d + (x/2 (1 + 1/80))/60 d which is the given
formula.
Verse 2. Alternative method.
The number of elapsed years divided by respectively 4,
120, and 9600 and the Sum taken gives the Dinādya.
Comm. Let x be the number of elapsed years. The
Dinādya as before is x × 0-15-30-22-30 = x/4 d + (x × 30)/(60 × 60) d

• (x × 45)/(2 × 60) × d/(60 × 60) = (x/4 + x/120 + x/9600) d as given.
  Verse 3. To obtain what is known as Kshayāhādya.
  The number of elapsed Kshayāhās from the beginning of
  Kalpa upto the commencement of the year, is obtained as
  follows. Let x be the number of elapsed years; then
  x − (x (1 + 1/80) + 30x)/160 = Kshayāhās.
  Comm. The number of Kshayāhās in a Kalpa of
  4320000000 solar years is 25082550000 so that per year their
  number is 5-48-22-7-30. In this 0-48-22-7-30 is said to
  be Kshayāhādya per year = 1 − (0-11-37-52-30) putting
  the quantity within the brackets into a fraction, 52½ vipalas
  = 105/2 × 1/60 = 7/8 palas; (37 + 7/8) palas = 101/8 × 1/60
  ghatis = 101/160 ghatis ; 11 101/160 ghatis = 1861/(160 × 60) days

55
= 1/160 of 1861/60 days = 1/160 of 31 days, one ghati. Hence
per year the Kshayāhādya is 1 − 1/160 of (31ᵈ - 1ᵍ) so that
for x years it would be {x − x/160 (31ᵈ - 1ᵍ)} d
= x − x/160 (31 1/60) = x − 1/160 { 30x + x + x/60 }
= x − 1/160 { 30x + x (1 + 1/60) } which is the formula
given.
Verse 4. Alternative method.
The Dinādya obtained before multiplied by three, is to
be diminished by 1/400th of the number of years; the result
increased by 1/30th of the number of years gives the number
of Kshayāhās.
Comm. The Dinādya pertaining to one year is
0-15-30-22-30 and the Kshayāhādya is 0-48-22-7-30.
Multiply the former by 3 and subtract from the latter; we
have 0-1-51 = 0-1-51/60 = 0-1-17/20 = 0-37/20 = 37/20 × 1/60 = 37/1200
day Hence K−3D = 37/1200 day where K = Kshayāhādya and
D = Dinādya ∴ K = 3D + 37/1200 hence per x years it
will be 3D × x + 37x/1200 = 3D × x + (40 − 3) x / 1200
= 3D × x + x/30 − x/400 which is the formula given.
Verse 4 contd. Alternative method.
Or else K = K/160 (1 − 1/60) + x(1 − 1/5)
Comm. The Kshayāhādya for an year is 0-48-22-7-30
48 ghatis = (1 − 1/5) day; hence for x years x (1 − 1/5).

56
The remaining fraction = 0-0-22-7½ = 0-0-(22 + 15/2 × 1/60)
= 0-0-22⅛ = 0 — 177/8 × 1/60 = 0 — 59/160 = (60 — 1) / (160 × 60)
day = 1/160 — 1/(60 × 160) = 1/160 (1 — 1/60) ; hence for x
years x/160 (1 — 1/60)
Adding the two we have the required formula.
Verse 5. To obtain the elapsed number of Adhikamā-
sās and what is called Suddhi.
The sum of the Dinādya, Kshayāhādya and ten times
the number oi elapsed years divided by 30, gives the num-
ber of the elapsed Adhikamāsas; the remainder is known as
Suddhi if diminished by the fraction of the Kshayāhas.
Comm. The number of mean solar days in an year is
365-15-30-22-30; the Kshayāhās in an year are 5-48-22-7-30;
adding the two we have the number of tithis in an year
equal to 371-3-52-30. The number of solar days being 360,
the number of tithis which constitute the Adhimāsas is
equal to 11-3-52-30. The sum of the Dinādya and Ksha-
yāhādya in an year = 0-15-30-22-30 + 0-48-22-7-30 =
1-3-52-30. Hence the above number of tithis which cons-
titute the Adhimāsas namely 11-3-52-30 = 10 + Sum of
Dinādya and Kshayāhādya pertaining to an year. Hence
for x years, the Adhimāsa days are equal to x × 10 + Sum of
Dinādya and Kshāhādya for x years. These Adhimāsa days
divided by 30 give the Adhimāsas, and the remainder is
called Suddhi, so called because while finding the Abargaṇa
from the beginning of a solar year, this remainder has to be
subtracted. Why the fraction of Kshayahas, which is there
in this Suddhi is prescribed to be subtracted, will be explai-
ned in another context.
Verse 6. An alternative method to find the Adhimā-
sas. The number of years divided separately by 32 and 30,

57
the sum increased by the number of years multiplied by
eleven and the result divided by 30 gives the number of
elapsed Adhimāsas. The remainder diminished by the
fraction of Kshayāhās as mentioned before, is the S'uddhi.
Comm. Every year, the number of Tithis that consti-
tute the Adhimāsās accruing in the course of years, is
11-3-52-30. The fractional part of this namely 0-3-52-30
= 0-3-52½ = 0 — 3 + 105/(2×60) = 0 — 3⅞ = 0 — 31/8
= 31/8 × 1/60 = 31/480 = (16+15)/480 = 1/30 + 1/32 converted
into days. Hence for x years x/30 + x/32 . Adding the
number of days 11x, we have 11x + x/30 + x/32 as stated.
Dividing these days by 30, we have the number of elapsed
Adhimāsas and the remainder is S'uddhi, if the fraction
of the Kshayāhās is subtracted therefrom as indicated.
Verse 7. Method of finding the lord of the year with-
out a knowledge of Dinādya.
The week-day at the Commencement of the Solar
year, which pertains to the lord of the year, is the remain-
der got by dividing by seven the S'uddhi which is itself
diminished by the remainders got by dividing by seven
separately firstly double the excess of the elapsed years over
the elapsed Adhikamāsās and secondly the elapsed Ksha-
yāhās. This may be put in symbols as R₇ { S—R₇ (2y—A)
— R₇. K. } = R₇ { S' — R₇ (2y — 2A + K) } where R₇ signi-
fies the remainder got by dividing by seven the quantity
which follows, S signifies S'uddhi, y means the elapsed
Solar years A the elapsed Adhikamāsās and K the elapsed
Kshayāhās.
8

58
Comm. The lord of the year is the lord of the week-
day at the Commencement of the solar year. To get this
week-day we have if the number of Tithis elapsed upto the
beginning of the luni-solar year be T, the Suddhi S', the
Kshayāhās upto the Commencement of the solar year be K,
then since the number of civil days is equal to T + K, R₇
(civil days) = R₇ (T + S — K). But T = 360y + 30A where
y is the number of elapsed solar years and A the number of
elapsed Adhikamāsās. ∴ R₇ (T) = R₇ (360y + 30A) = R₇
(3y + 2A) since the remainder got by dividing 360 and 30
by seven are respectively 3 and 2. Hence R₇ (civil days =
Ahargaṇa) = R₇ (3y + 2A) + R₇S' — R₇K. The number of
Kshayāhās K = y (5-48-22-7-30) = 5y + y (0-48-22-7-30).
But y (0-48-22-7-30) = K where K is the previously calcu-
lated Kshayāhādya. Hence R₇K = R₇5y + R₇K. Substi-
tuting this in the above R₇ (civil days) = R₇ (3y + 2A) +
R₇S' — R₇ (5y + K) = R₇S' — R₇ (5y — 3y) + R₇ (2A) —
R₇K = R₇ [S' — R₇ (2y — 2A) — R₇K ] which is the given
formula = R₇ [S — R₇ (2y — 2A + K) ].
Verse 8. To obtain the fractional part of the elapsed
Kshayāhās even without a knowledge of the number of
Kshayāhās.
The excess of the ghatis of the Adhimāsa Sesha ie the
remainder got by dividing the sum of the Dinādya, Ksha-
yāhādya and ten times the number of elapsed years by 30 as
indicated in verse 5, over the ghatis of the Dinādya, obtai-
ned under verse 1 gives the fractional part of the Kshayāhās.
Comm. The ghatis or the fractional part of the Adhi-
māsa Sesha is the-sum of the fractional parts of Dinādha
and the Kshayāhādya so that the [excess of the ghatis of
the Adhimāsa Sesha over the ghatis or the fractional part
of the Dinādya gives the ghatis or [the fractional part of
the Kshayāhās.
Verse 9. The planetary positions at the end of the
elapsed solar year.

59
The number of the elapsed solar years multiplied by
the number of sidereal revolutions of the respective planets
in a Kalpa and divided by the number of Solar years in
a Kalpa gives the planetary positions at the end of the last
Solar year.
Comm. This is a simple rule of three. The plane-
tary positions so got, leaving out the integral numbers of
revolutions made which are not required, are called the
Dhruvakas of the respective planets for the ensuing solar
year. With respect to the apogee of the Sun and the aphelia
and nodes of the star—planets these Dhruvakas themselves
give their positions for the whole ensuing year, as their
motion is very very slow.
Verse 10. An alternative method of obtaining the
Dhruvaka of the Moon.
The Adhimāsa Sesha multiplied by 12 gives the posi-
tion of the Moon at the Commencement of the Solar year.
Comm. Since the position of the Sun is at the Zero-
point of the Zodiac at the Commencement of the solar year,
and since the Adhimāsa Sesha is the difference of the solar
and luni-Solar systems of reckoning, or what is the same,
the arc gained by the Moon over the Sun, which is no
other than the elongation of the Moon measured in Tithis
each tithi being of 12° gain of elongation, so the Adhimāsa
Sesha at the Commencement of the solar year in Tithis
multiplied by 12 gives the longitude of the Moon at that
Commencement. (Note - Bhaskara waxes into poetic
eloquence in the second half of the verse, having given the
procedure in the first half).
Verse 11. Procedure prescribed in the event of com-
puting planetary positions from the beginning of the
Kaliyuga for the sake of convenience.