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

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.

60
The Dinādya may be also obtained from the beginning
of the Kali, which begins with Friday. The Dhruvas
calculated for the commencement of the solar year are to
be added to the planetary positions at the beginning of the
Kali, in the event of computing the Ahargaṇa and thereby
the planetary positions from the beginning of the Kaliyuga
for the sake of convenience.
Comm. The Dinādya at the commencement of Kali
is Zero, since the number of civil days during the length
of time equal to a Kaliyuga is integral and equal to
157791645 according to Brahmagupta and Bhaskara who
follows him. The Suryasiddhānta, it may be noted here,
gives the number of days in a Kaliyuga as not integral
but equal to 157791782.8. Hence the Dinādya computed
from the beginning of the Kali is to be increased by .8 to
obtain its value according to Suryasiddhānta. The plane-
tary positions at the commencement of Kali were given
by Bhaskara already.
Verse 12. The number of what are called Kshepadinās
to obtain the Ahargaṇa.
Hereafter Bhaskara is going to obtain the planetary
positions for any day during the current solar year having
obtained the Dhruvakas or the planetary positions for the
beginning of the Solar year. In that behalf the Ahargaṇa
or the collection of days which have elapsed from the
commencement of the Solar year is to be found. This
Ahargaṇa is obtained by subtracting the number of Ksha-
yāhās from the number of tithis that have elapsed. In
finding these Kshayāhās, we have to take note that there
is a little remnant of Kshayāhās at the beginning of the
Solar year which is also to be taken into account while
computing the number of Kshayāhās during the course of
the year. In other words, the number of Kshayāhās that
are going to be computed during the course of the year,
for the elapsed part of the year will be in default of the

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actual number if we ignore the accrued fraction of Ksha-
yāhās at the commencement of the Solar year. To make
amends for that default we have to add some number to
the numerator of the improper fraction which is going to
give us the number of the elapsed Kshayāhās during the
course of the year. The formula that is going to be used
to obtain the number of Kshayāhās during the elapsed
tithis is x/64 (1 + 1/702). This formula arises out of the
fact that there are 55739 Kshayāhas during 3562220 tithis,
so that for 64 tithis the number of Kshayāhās is equal to
(64 × 55739) / 3562220 = 3567296 / 3562220 = 1 + 5076 / 3562220 = 1 +
1 / (3562220 / 5076) = 1 + 1/702. Let the fraction of Kshayāha at
the commencement of the solar year, to be taken into
account be x ghatis i.e. x/60 of a tithis (for Kshayāhās are
are computed out of tithis) i.e. (x tithis) / 60. Let y be the
number of tithis elapsed after the commencement of the
solar year. Then to compute the Kshayāhās that ensue
after the commencement of the solar year upto the day
concerned during the course of the year, the formula to
be used is y/64 (1 + 1/702). To this we have to add (x tithis) / 60
as the balance of Kshāyāha at the commencement of the
solar year to be taken into acccount
(x tithis) / 60 = (x × 64) / (64 × 60) = (x × 64/60) / 60
Now, in computing the number of tithis which have
elapsed after the commencement of the Solar year, we
subtract Śuddhi from the number of tithis that have
elapsed after the beginning of the luni-Solar year. But

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in this Śuddhi we have subtracted the fractional part of
the Kshayāhas for a different purpose so that in sub-
tracting the Suddhi, we have increased the tithis by the
fractional part of the Ksbayāhās. This increase there-
fore should be nullified, which means that the quantity
to be added is
(x × 64/60) / 64 − x/60 = (x × 63/60) / 64 = (21/20) (x/64) = x (1 + 1/20) / 64
Hence the Kshepadinas or the tithis to be added to the
number of tithis which have elapsed from the commence-
ment of the Solar year, are x (1 + 1/20). This means that
the ghaṭis x which form the fractional part of the
Kshayāhas are to be increased by one twentieth part of
themselves and are to be viewed as tithis and not ghaṭis as
mentioned.
Verse 12 (contd

next Sun-rise, T₃ the ending moment of the tithi preceding
the Sun-rise R₄ upto which point the Ahargaṇa has to
be found from the commencement of the Solar year. Thus
the Ahargaṇa to be found is SR₄ = ST₃ + T₃ R₄. Here
T₃ R₄ is the Kshayāghatis, at the Sun-rise concerned.
SR₃ is a fraction of a day that is to be there in the Ahar-
gaṇa SR₄ we are seeking. While subtracting the Śuddhi
T₁ S diminished by the Kshayāghatis T₂ R₃ from the
number of elapsed integral number of tithis from the
commencement of the luni-Solar year namely T₁ T₃ we
have T₁ T₃ — (T₁ S — T₂ R₃) = T₁ T₃ — (T₁ T₂ — SR₃) =
T₂ T₃ + SR₃. Thus instead of SR₃ + R₃ R₄ we have by
the above procedure SR₃ + T₂ T₃. Though both T₂ T₃ and
R₃ R₄ are integral numbers and should be the same if the
interval is small, they may differ by an integral number,
if the interval happens to be long. Bhaskara says that
this difference of an integral number will be rectified by
the Kshepadinas found under verse No. 12, for, from these
Kshepadinas, the Kshayāhas are found and subtracted
from the tithis. It will be noted that the difference of
an integral number between T₂ T₃ and R₃ R₄ is no other
than Kshayāhas.
Or, this may be seen in an other way. We are to find
SR₄ = T₁ R₄ — T₁ S = T₁ T₃ + T₃ R₄ — T₁ S = T₁ T₃ —
(T₁ S — T₃ R₄) = No. of elapsed tithis — (Suddhi —
Kshayaghatis at the day concerned). But instead of the
Kshayaghatis at the day concerned Bhaskara prescribes
the subtraction of the Kshayāghatis at the end of the Solar
year ie instead of subtracting T₃ R₄ it is prescribed to sub-
tract T₂ R₃. This difference, Bhaskara says is made up
by taking into account the Kshepadinas pertaining to the
Kshayāhas.
Verse 14. In case the Ahargaṇa is required for a day
preceeding the commencement of the Solar year, then the
elapsed tithis are less than the Tithis of the Suddhi; so
subtraction is not feasible. In this case, it is prescribed