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

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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

64
to take the elapsed tithis from the previous Chaitra and
the Suddhi of the previous year. Also in this case the
Dhruvakas pertain to the commencement of the previous
Solar year.
Comm. Easy.
Verse 15. To obtain the position of the Sun.
The number of the days in the Ahargaṇa is to be
diminished by 1/60 part of itself to obtain the number of
degrees, and fractions thereof; then the Ahargaṇa multi-
plied by three and divided by 22 gives the minutes and
fractions thereof. Adding the two results we get the
position of the Sun.
Comm. The mean daily motion of the Sun is
0-59-8-10-21. Here 59' = 1° - 1' = (1 - 1/60)°; for x
days x (1 - 1/60)° = x° - x°/60 as mentioned. The remain-
ing part namely 0-0-8-10-21 = 9807'/72000; Converting this
into a continued fraction it is equal to 1/(7 +) 1/(2 +) 1/(1 +) 1/(12 +) ...
of which a good convergent is 3'/22. Hence for x days 3x/22
as stated in the verse.
Verse 16. To obtain the position of the Moon.
The number of elapsed integral tithis multiplied by
12 and added to the Sun's position in degrees, gives the
Moon's position in degrees at the ending moment of the
tithi preceeding the day at the Sun-rise of which the
planetary positions are sought. To find the position at
the Sun-rise required, ten times the Kshaya-dina-Sesha
increased by 1/7 th of itself gives the number of minutes to
be added to the position got above.

65
Comm. The first part is clear, because for every tithi,
there will be an increase of elongation of 12°. To get the
fractional part of the tithi in between the ending moment
of the tithi and the subsequent Sun-rise, the interval which
is Sāvana ie expressed in the unit of civil day has to be
converted into luni-Solar units. The rule of three is ‘ If for
63 Sāvana days there are 64 tithis, what will it be for the
above interval ?’ Here the approximate ratio of 64/63 is used
because the Ahargaṇa which is Laghu is small ie is less
than 365. The Kshaya-dina-Sesha which was got under
verses 12, 13, has a divisor of 64, and is of the form y/64 .
Hence the quantity to be added is (y × 64)/(64 × 63) = y/63 of a tithi
= y/63 × 12 × 60 minutes of arc = 8/7 × 10 y = 10 y (1 + 1/7)
minutes as given in the verse. This has to be added to
12t°, got before where t is the number of elapsed tithis.
Verse 17. Computation of Mars.
The mean daily motion of Mars is 0-31-26-28-7=0-30

• 0-0-90 minus 0-0-3-31-53. If x be the number of
  days x × 0-30' = x°/2 ; x × 0-0-90" = x/2 × 3' ; thus the
  rule prescribed is “ Half of the number of days gives the
  degrees ; half the number of days multiplied by 3 gives
  the number of minutes ” from this we have to subtract
  x × (0-0-3-31-53). The quantity within the brackets
  is approximately 1/17 of a minute for 1/17' = 0-0-3"-31"'-46"'.
  Since the Ahargaṇa is small the precision required is there.
  Thus the rule is x°/2 + x/2 × 3' + x'/17 + the Dhruvaka at the
  beginning of the Solar year.
  Verse 18. Computation of the S'īghroccha of Mercury.
  The mean daily motion of the Budha-S'īghra is
  4°-5'-32"-18"'-28"".
  9

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Rule. If the Ahargaṇa be x days then the Śīghra will
be 4x° + (4x × 3') / 130 + Dhruvaka. Proof. For x days, the mean
motion is x × 4° + x × (0-5′-32″-18‴-28⁗). The second
part is taken to be 4x × 3° / 130 ie x × 12° / 130 = x × 6/65° for
6/65 of a degree = 0-5′-32″-18‴-"28"". The fraction 6/65 can
be seen to be a convergent of the remainder for 5′-32″-
18‴-28⁗ = 299077° / 3240000 = 1/(10 +) 1/(1 +) 1/(4 +) 1/(1 +) 1/9968 of which the
penultimate convergent is 6/65.
Verse 19. The Ahargaṇa divided by 12 and by 71
gives respectively the positive degrees and negative minutes
to be added to the Dhruvaka of Jupiter.
Comm. If x be the Ahargaṇa x° / 12 − x' / 71 + Dhruvaka =
Guru.
The mean daily motion of Jupiter is 0-4-59-9.9 = 0-5′
minus 0-0-0-50-51. Evidently 0-5′ × x = x° / 12. The
remainder 0-0-0-50-51 = 61′ / 4320 of which the continued
fraction is 1/(70 +) 1/(1 +) 1/4. A very approximate convergent
is 1/71 as given.
Verse 19 contd. To obtain the position of the Śīghra
of venus.
The Ahargaṇa multiplied by 10 and divided by 6 and
155 respectively gives degrees positive and negative to be
added to the Dhruvaka of Venus to give his position.
Comm. The formula is 10 x / 6 − 10 x / 155. The mean daily
motion of the Śīghra of Venus is 1°-36′-7″-44‴-35⁗ =

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1°-40' minus 0-3'052"-15"' approximately. For x days x ✕
1 2/3° = 5/3 x° = 10 x° / 6 which gives the first part. Also 3'-52"-
15"' = 929° / 14400 = 1 / (15+) 1 / (1+) 1 / (1+) 1 / 464 of which a very ap-
proximate convergent is 2/31 = 10/155 ie for x days 10x° / 155. Hence
the position is given by 10 x° / 6 - 10 x° / 155 + Dhruvaka.
Verses 20. The position of Saturn is given by 2x' / 5 +
2x" / 5 + Dhruvaka = Saturn's position.
Comm. The mean daily motion of Saturn is
0-2-0-22-51.
For x days 2x' + x ✕ 0"-22"'-51"". The latter part
is 137" / 360 approximately = 1 / (2+) 1 / (1+) 1 / (1+) 1 / (1+) 1 / 2 of which a near
covergent 2/5 so that for x days, we have 2x' + 2x" / 5

• Dhruvaka position of Saturn.
  Verses 20 contd. To find the position of the apogee of
  Moon.
  The Ahargaṇa divided successively by 10 and 88 and
  added gives the degrees to be added to the Dhruvaka of
  the Apogee of the Moon to obtain its position.
  Comm. The mean daily motion of the apogee of the
  Moon is 0-6-40-53-56. At the rate of 6' per day, in x
  days the number of degrees covered is x / 10 ; the remainder
  0-0-40-53-56 = 4601° / 405000 = 1 / (88+) 1 / 41 of which a very
  approximate convergent is 1/88. Hence the position is given

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by x°/10 + x°/88 + Dhruvaka as given by the verse.
Verse 21. To obtain the position of the lunar Node
Rahu.
The Ahargaṇa multiplied by 30 and divided by 566
gives the number of degrees to be added to the Dhruvaka
to give the position of Rahu.
Comm. If the number of the sidereal revolutions of
Rahu in a Kalpa multiplied by twelve divide the number
of days in a Kalpa we have very approximately 566, which
means that Rahu traverses a Rasi very nearly in 566 days.
Hence dividing the Ahargaṇa by 566, and multiplying by
30, we have the degrees covered, which being added to the
Dhruvaka gives the position of Rahu.
Verses 22, 23, 24. Alternative method of obtaining
the planetary positions.
The Ahargaṇa multiplied by 100000, and divided
successively by 101461, 151787, 190833, 24436, 1203400,
62416, 2990000, 898000, 1886800 gives the respective posi-
tions of theplanets beginning from the Sun and those of
the apogee and Node of the Moon; in the case of the
Moon, however, the result is to be multiplied by 20. The
results in degrees added to the Dhruvakas give their posi-
tions.
Comm. The degrees covered by the planets etc in
D days are (R × 360 × D°) / M where R is the number of side-
real revolutions in a Kalpa, M the number of mean solar
days in a Kalpa, and D the Ahargaṇa. Now
(R × 360 × D) / M = (R × 360 × D × 100000) / (M × 100000) = (D × 100000) / ((M × 100000) / (R × 360)) .
Then (M × 100000) / (R × 360) is found for every planet. In the case

69
of the Moon, however a multiplier 20 also is used in addi-
tion to 100000, because he has such a quicker motion.
Verses 25, 26. To obtain the mean daily motion of
the planets.
The number of minutes of arc moved by a planet per
day gives the mean daily motion of that planet. Though,
however, the spatial velocity of each planet is the same
per day, in angle, the velocities differ, (on account of the
varied distances) and so we perceive slowness or fastness
in the movement of the planets.
Comm. Easy.
The assumption of equal daily spatial velocities for
all the planets has been explained before. The daily
angular velocity of a planet in minutes of arc is
R × 360 × 60'
─────────────── where R is the number of its sidereal re-
M
volutions and M the number of mean solar days in a Kalpa.
Verse 27. The reason for unequal angular velocity
of the planets.
Since the planetary orbits are all construed as com-
prising of 360° alone, a minute of arc of a smaller orbit
has a smaller spatial distance which will be covered more
rapidly, the velocity being constant, whereas of a longer
orbit, a minute of arc means a longer distance which will
be covered in a longer time at the same spatial velocity,
which means that the planet appears to be slow in
motion. Thus the Moon, the Mercury, Venus, Sun, Mars,
Jupiter, Saturn being placed at longer distances from the
earth in ascending order their orbits are longer in ascend-
ing order so that they are slower in angular motion in
that order.
Here ends the section called Pratyabda Suddhi in the
chapter Madhyādhikāra.

THE SECTION KNOWN AS ADHIMĀSĀDI-
NIRNAYA IN Ch. I
Verse 1. A special feature of the computation of
Ahargaṇa. If the Ahargaṇa is to be increased or decreased
by unity to adjust it with the week-day, the tithis also are
to be increased or decreased by unity to be adjusted with
the week-day. Then in that context of increasing or
decreasing the Ahargaṇa by a day the Adhimāsa-Sesha is
to be increased or decreased by the number of Adhimāsās
of a Kalpa and the Avama-Sesha is to be inoreased or
decreased by the Avamās or Kshayāhās of a Kalpa.
Comm. After having computed the Ahargaṇa as per
the directions given under verses 1-3 of the section of
Grahānayana, we have to test its correctness on the basis
of the week-day taking it that the Kalpa began with
Sunday. In other words, since the number of the elapsed
week-days accord with the number of Sun rises, the week-
day on the day on which the Ahargaṇa is computed should
accord with the Ahargaṇa ie dividing the Ahargaṇa by
seven, the remainder must give us the week-day of the day
in question. But it so happens that we may have to add
or subtract one from the Ahargaṇa arrived at to adjust
the Ahargaṇa with the week-day. Why does this happen ?
The reason is as follows. When we find the Ahargaṇa
for a particular number of elapsed tithis, we are unwit-
tingly taking the number of elapsed true tithis in the place
of the average tithis. The fact that we are given in the
table of constants, the number of mean units of time in
a Kalpa, for example mean solar days, mean tithis etc, and
the fact that computation proceeds only on the basis of
these mean units, means that we have taken into account
only the number of mean tithis elapsed. Hence this is
to be corrected by us. If we are computing the Ahargaṇa

71
for, say, the pratipat of phālguna, we automatically
take that the elapsed true tithis from the beginning of
the luni-Solar year is 11 × 30 = 330 tithis. But it may
so happen that had we taken only the number of mean
tithis elapsed, either 329 tithis or 331 tithis might have
elapsed and not 330 as construed which therefore works
an error amounting to unity in the Ahargaṇa arrived at.
At the same time the error does not exceed unity, for,
during the course of every lunation, the longer tithis are
almost compensated by tithis of shorter duration in the
same lunation. The variance in the length of a tithi being
wrought by the variable motion of the Moon, which again
depends on the distance of the Moon from his apogee, is
rounded off in every lunation, as the Moon completes a
circle with respect to the apogee in what is called an
anomalistic month.
Thus it is that the Ahargaṇa is to be rectified on the
basis of the week day. When this is done and the
Ahargaṇa happens to be increased or decreased by unity,
automatically it goes without saying, that the tithis are also
increased or decreased by one. In this context there is a
still more deeper significance as detailed below. The
Adhimāsās are computed from the following formula viz.
(A × a) / S = I + (F / S) × 30 (1) where A = Ahargaṇa, a = adhi-
masās in a Kalpa, S = solar days in a Kalpa, I = Integral
quotient obtained by division and (F / S) × 30 the Adhimasa-
Sesha-tithis. If now A is to be increased or decreased by 1
((A ± 1) a) / S = (Aa / S) ± (a / s) = I + (F / S) × 30 ± a/S from (1) = I +
(F × 30 ± a) / S ; hence the adhimāsa-sesha namely F × 30 is
increased or decreased by a ie the number or adhimasās.
Similarly the Avama-Sesha or the Kshayāha-Sesha is
to be increased or decreased by the Kshayāhas in a Kalpa.

72
The particular mention of the increase or decrease in the
Adhimāsa-Sesha or the Avama-Sesha is necessitated in the
context of computing the positions of the Sun or Moon,
given the Adhimāsa-Sesha and Avama-Sesha as mentioned
under verses 6, 7 in the section of grahānayanā.
Verse 2. Pertaining to the smaller Ahargaṇa compu-
ted from the beginning of the current Solar year, called
Laghu-Ahargaṇa.
In the case of computing the Ahargaṇa from the
beginning of the Solar year also, when the Ahargaṇa is to
be increased or decreased by unity, the tithis are to be
increased or decreased by unity. The Avama-Sesha here
is to be increased or decreased not by the Kshayāhās of
the Kalpa but only by unity because we have used the
formula 1/64 of tithis to get the Kshayāhās. If, an Adhika-
māsa happens to occur during the course of the current
year, the tithis, 30 in number of this Adhikamāsa must be
also taken into account to obtain the Ahargaṇa.
Comm. Easy.
Verse 3, 4. The Ahargaṇa is to be computed (larger
Ahargaṇa) after taking into account an Adhimāsa which
has conspicuously occured but which is not obtained by
computation or by rejecting an Adhikamāsa which has
not occured but which is obtained by calculation. The
Adhimāsa-Sesha is to be increased or decreased by the
Adhimāsas of the Kalpa; the elapsed months from the
beginning of the luni-Solar year are to be increased or
decreased by unity and then the positions of the Sun and
the Moon are to be computed from such an Adhimāsa-Sesha
and such an Avama-Sesha.
Comm. Already explained.
Verse 5. A point to be noted with respect to the
Suddhi,

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In the case of obtaining the Suddhi, if an Adhikamāsa,
which did not actually occur, is obtained by calculation,
then the Suddhi is to be increased by 30, so that the
Ahargaṇa is not affected by the un-occuring Adhikamāsa.
Comm. The computation of the Adhikamāsas or inter-
calary months proceeds under the consideration of mean
lengths. So, it is likely that an Adhikamāsa may occur
un-warranted by calculation or may not occur in spite of
its being shown by calculation. Further, an Adhikamāsa
may be delayed in occurence by the fact that though the
luni-Solar reckoning has gained over the solar by one
mean lunation, the lunation at that point may still con-
tain a Samkrānti, the preceding particular lunar month
being smaller in length than the mean. Thus the conven-
tion made with respect to the occurence of an adhikamāsa,
namely that the lunation which does not carry a Samkrānti
is to be construed as an adhikamāsa, may also delay the
occurence of the Adhikamāsa, though shown in calculation.
Similarly an Adhikamāsa may be preponed though not
warranted by computation by the same logic.
Verse 6. The criteria of an Adhikamāsa and a
Kshayamāsa.
A lunation which does not carry a Samkrānti is an
Adhikamāsa; whereas a lunation which carries two Sam-
krāntis is to be taken as a Kshayamāsa. The Kshaya-
māsa, occurs only in the course of the three lunar
months named Kārtica, Mārgasīrsha and pausha and
not during any other lunar month; when a Kshayamāsa
occurs, then during the course of that year there will be
two Adhikamāsās occuring on either side of the Kshaya-
māsa.
Comm. The institution of intercalation has been
explained to some extent under verse 10 of the Bhagaṇādh-
yāya. We shall see some more particulars of intercalation.
10

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1. Bhaskara has given that 1593300000 Adhika-
   māsās occur in a Kalpa of 4320000000 Solar years, which
   means that 15933 Adhikamāsās occur in 43200 Solar years
   ie. 5311 Adhikamāsas in 14400 solar months. Converting
   14400 / 5311 into a continued fraction, we have 2 + 1/(1+) 1/(2+)
   1/(2+) 1/(6+) 1/(1+) 1/(1+) 1/(7+) 1/(3+) 1/2. The successive convergents are
   2/1 3/1 8/3 19/7 122/45 141/52. Let us see what these convergents
   signify. (a) The convergent 19/7 means that on an average
   there are 7 Adhikamāsās per 19 years. This ratio was
   adopted in the Romaka Siddhānta of Panchasiddhāntika.
   It means that 19 × 12 = 228 Solar months are equal
   to 235 lunations. The Metonic cycle described in
   modern astronomy is based upon this equivalence. The
   recurrence of Moon's phases in 19 years ie correspondence
   of the Moon's phases or tithis with the dates of the
   English year and Meton's formula are based on this
   equivalence. Recurrence of phase means recurrence of the
   relative positions of the Sun and the Moon, which
   again means recurrence of the Suddhi, for Suddhi is no
   other than the interval between the New Moon and
   Saṁkrānti. In the next verse Bhāskara says that a
   Kshayamāsa recurs after a lapse of either 19 years or 122
   years or 141 years. The numerators of the last three
   convergents are 19, 122 and 141 which are the numbers of
   Solar years that effect recurrence of the same Suddhi and
   as is going to be mentioned shortly a Suddhi of 21 tithis
   is likely to bring in a Kshayamāsa. Hence a Kshayamāsa
   recurs either in 19 years or 121 years or 141 years.
2. Mention of the occurence of a Kshayamāsa was
   made by Sripati first and not by the preceding astro-
   nomers. However, mention of it is there in the Vedic

76
literature and it is not clear whether observance of this
Kshayamāsa was defunct for some centuries in between.
3. We shall now proceed to see how there occur two
Adhikamāsas on either side of a Kshayamāsa. A simple
argument is as follows. The convergents cited above
namely 19/7 or 122/45 or 141/52 signify that either 7 or 45 or 52
Adhikamāsās are to occur in the course of 19 or 122 or 141
Solar years normally. But when the S'uddhi happens to
be 21 days at the beginning of the Solar year, it so happens
that the S'uddhi goes on increasing for the first five
months because the Sun is in his apogee when his
longitude is 78°, and his motion being slow for three
months when he is on either side of his apogee, the
Moon gains over him in shorter intervals of time and as
a consequence the lunations are of shorter duration. This
means that the S'uddhi goes on increasing during those
months and rapidly increases from its value 21 at the
beginning to 30 by about Bhādrapada month. Under
these circumstances, a Samkramaṇa occurs generally just
before the beginning of Bhādrapada. The next Samkra-
maṇa happens just a little after the lapse of Bhādrapada,
so that the month of Bhādrapada goes without a Samkra-
maṇa and as a consequence, it becomes an Adhikamāsa.
Thus far it is alright that an Adhikamāsa has occured
as per the meaning of the convergents. But when the
Bhādrapada thus becomes an Adhikamāsa, the subsequent
months from Kārtica to Mārgasira, being of longer duration
than the corresponding Solar months, the Sun having a
quicker motion on either side of his perigee, there is every
likelihood of a Solar month being contained between two
conjunctions or New Moon days. In other words two
Samkramaṇas occur either in Kārtica or Mārgasira or
Pausha. This means that as per the convention for the
occurence of a Khayamāsa, one of the aforesaid lunations
must become a Kshayamāsa. Thus the Adhikamāsa which

76
is due to occur during the course of the year, though it
has occured has been lost. So, to make amends, another
Adhikamāsa is to occur as is warranted by the convergents
cited above. It might be asked what if two Adhikamāsas
occur and why a Kshayamāsa be instituted at all. The
reason is not that a religious convention warrants it
but because the wedding of the luni-Solar year to the
Solar year has to be made on a particular principle. Nor-
mally, so long as a Samkramaṇa goes on occuring during
the course of a lunation, the two systems of reckoning
may be seen to be running parallel. But if a particular lunar
month does not contain a Samkramaṇa it is to be taken
as a warning that the luni-Solar reckoning has overtaken
the Solar by one lunation. This lunar month has to be
curtailed to make the two kinds of reckoning to proceed
side by side. This convention naturally raised the ques-
tion as to how to deal with a lunation which contains two
Saṁkrāntis. The Solar month their has to be deleted to
make the two systems run concurrently. This deletion of
a Solar month is achieved not by declaring the particular
Solar month as an Adhikamāsa but what is virtually the
same two lunar months are deemed to lapse during the
course of that solar month. This kind convention helped
the occurrence of one Adhikamāsa alone as scheduled be-
cause one of the two Adhikamāsas has been nullified by
the convention of a Kshayamāsa.
Why the proposition that a Kshayamāsa generally
occurs when the Śuddhi at the beginning of the Solar year
happens to be 21 tithis is quite evident because in such a
case generally Bhādrapada becomes an Adhikamāsa, which
again entails the occurence of two Samkramaṇas during the
course of one of the three lunations beginning with
Kārtica, which happen to be longer than the corresponding
Solar months. In other words a Kshayamāsa is expected
to occur only when Bhādrapada happens to be an Adhika-
māsa, and this in turns happens only when the Śuddhi
happens to be 21 at the beginning of the Solar year. Adhi-

77

kamāsās do occur when the Suddhi happens to be more
than 21, but in this case one of the months prior to Bhādra-
pada happens to be an Adhikamāsa and then by the time
Kārtica is reached, the Samkramaṇa occurs not at the
beginning of Kārtica but a little later, which means that a
second Samkramaṇa could not occur before the lapse of
that lunation. This therefore will not be a Kshaya month.

The Suddhi which is defined as the number of tithis in
between a New-Moon and a Samkramaṇa, is clearly the in-
terval which has been gained by the luni-Solar system over
the Solar. This S'uddhi increases at the average rate of
11 days, 3 ghatis, 52 palas and 30 Vipalas per an year as
we have seen already. For the occurrence of an Adhika-
māsa during the course of an year the Suddhi at the begin-
ning of the year must be such that it accrues to a lunation
during the course of the year. The apogee of the Sun being
almost stationary having a very slow motion the lengths of
the Solar months do not vary appreciably for a good number
of years. Taking that the Sun's apogee is roughly at 80°
longitude (78° according to Hindu Astronomy) the Sun's
motion is less than his average from the moment when he
has 350° longitude upto the moment when he has 170°, so
that the Moon gains rapidly over him during this period.
In other words the lunations during this period are of
shorter duration i.e. the luni-Solar months of chaitra upto
Sravana will be shorter in length so that S'uddhi increases
rapidly from March upto August. Thereafter it will attain
a stationary value just for a little time and then decreases
for the next six months upto February. The increase,
however, far exceeds the decrease and the balance of in-
crease per year is as mentioned above is 11-3-52-30.

Bhāskara mentions in the course of the commentary
that (1) the average length of a lunation is 29 days, 31
ghatis and 50 palas; (2) the average length of a Solar
month is 30-26-17 ; (3) when the daily motion of the Sun
happens to be 61', then the Solar month will have a length

of 29-30 and as such falls short of a lunation and (4) that
the minimum length of a Solar month is 29-20-40 only.
Then he gives a general and broad explanation as to how a
Kshayamāsa occurs on the following lines. Let us call NS
as the interval between a New Moon and the subsequent
Saṁkrānti. In other words it is the Suddhi. Suppose
Bhādrapada goes without a Saṁkrānti thus becoming an
Adhikamāsa. Then in Āswayuja NS will have a small
value, because the Saṁkrānti S which should have occured
at the general rate of one per a lunation, has not occured
before the lapse of Bhādrapada and being belated a little
must have occured close on the heels of the New Moon of
Bhādrapada. Let this interval NS have a particular value
say x. Thereafter the Solar months grow gradually smaller
i.e. the Saṁkrāntis occur earlier ; so in Kārtica the value
of NS decreases. It might decrease to such an extent that
the next Saṁkrānti might occur before the next New Moon.
Suppose this does not happen in Kārtica ; then in Marga-
sira the value of this NS is still smaller and there is a
greater likelihood of another Saṁkrānti occuring during this
Mārgasira. Then the value of NS being the smallest in
the month of Pausha, another Saṁkrānti is bound to take
place during this month at the latest. In other words S
now occurs before the next N i.e. the Saṁkrānti occurs just
a little before the next New Moon. Let now this SN be a
small quantity say y; or what is the same the next NS or
the Suddhi will be nearly a lunation. There after the
Solar months begin to gain in length i.e. S occurs later and
later. This means NS gains in length. Being nearly of
a length equal to a lunation, and now gradually increasing
it is bound to exceed a lunation shortly thereafter. This
again means that S occurs later than the next New Moon,
which is to say that one more lunation goes without a
Saṁkrānti. Thus it is that another Adhikamāsa occurs
shortly after the occurence of a Kshayamāsa. Hence in
the course of one year there occur two Adhikamāsās and
one Kshayamāsa which means again that the balance of an

79
Adhikamāsa is there in that year. In other words, the
occurence of a Kshayamāsa during the course of an year
as per a stipulated convention does not preclude the average
occurence of an Adhikamāsa which is normally due after
an average lapse of 32½ Solar months. That the occurence
of a Kshayamāsa entails the occurrence of two Adhikamāsās
on either side may be also seen in another way. Twelve
lunations put together have a length of 354⅓ days approxi-
mately during the course of which only eleven Saṁkrāntis
could occur in case the initial Śuddhi is 21 days, for, sub-
tracting this 21 from 354⅓, there remain 333⅓ days only
which could contain only ten Solar months and not eleven.
Ten Solar months are contained if there be only eleven
Saṁkrāntis. Out of these eleven, two are consumed in a
single lunation which happens to be a Kshayamāsa. Hence
the remaining eleven lunations have to contain only nine
Saṁkrāntis which means that two lunations have to go
without Saṁkrāntis. In other words there are to be two
Adhikamāsās during that year. Further both these Adhi-
kamāsās could not occur on one side of the Kshayamāsa
for the following reason. Since a Kshayamāsa could occur
only during the course of Kārtica or Mārgasirsha or
Pansha which alone are longer than the corresponding solar
months, in the event of both the Adhikamāsās taking their
place on one side of these months, they are to take place in
nine lunations which are on one side of the three months
beginning with Kārtica. This means that only seven
Saṁkrāntis are to occur during the course of 9 × 29½ days
= 265½ days. Even seven Solar months fall short of this
period so that it is impossible that only seven Saṁkrāntis
should occur in this period. Hence, the two Adhikamāsās
which are to occur during the year containing a Kshaya-
māsa have to take their place on either side of the Kshaya-
māsa.
Bhāskara mentions that when the daily motion of the
Sun equals 61', the length of the Solar month will be 1800/61
= 29–30 approximately. He also states that the minimum

80
length of a Solar month will be 29-20-40. This will be so
when the average daily motion during the course of a
month is 61-20-30.
Verse 7. A mention of the years past and future that
had and will have Kshayamāsās.
A Kshayamāsa occured in the Saka year 974 and will
occur in the years 1115, 1256, 1378. Thus generally it
occurs once in 141 years or even 19 years.
Comm. We have seen before that according to the
convergents given above a Kshayamāsa generally occurs in
19 years or 122 years or 141 years. In the course of the
commentary Bhāskara says "19 years before or after" so that
141-19=122 years was also meant by him. He gives how
in the course of 141 years or 19 years the Suddhi at the be-
ginning of the year happens to be approximately 21 days.
In this behalf, he invokes his formulation of the Adhika-
māsās in the verse 6 under Pratyabdasuddhi. As per that
formula the number of Adhikamāsās in 19 years will be
7-13-37-30 taken by Bhāskara as 7-13-40. In other words
the Suddhi increases by 13 ghatis and 38 palas from what
it was 19 years ago; thus practically the value of the
Suddhi recurs in periods of 19 years which means that if a
Kshayamāsa occurred this year, there is a likelihood of its
recurrence after nineteen years. Of course the recurrence
would not be certain because there is an excess of 13 ghatis
Suddhi which might preclude its occurrence. Similarly in
122 years and 141 years the numbers of Adhikamāsās would
be respectively 45 minus 7-15 ghatis and 52 plus 6 ghatis.
22 palas and 30 Vipalas, the latter being taken by Bhaskara
to be 52 Adhikamāsās plus 6 ghatis and 20 palas. Thus
Kshayamāsas are more and still more likely to recur in
intervals of 122 years and 141 years respectively, the
Suddhis at the beginning of the year almost recurring and
assuming the original value of 21 days for a Kshayamāsa
to occur.

81
Bhāskara adds that a Kshayamāsa occured in 974 Saka
i.e. 62 years before his birth which knowledge he must have
derived from hearsay or even by calculation. He then
predicted its recurrence in the Saka years 1115, 1256 and
1378 which were at intervals of 141 years, 141 years and
122 years respectively. In the year 974 Saka i.e. 4153
Kali year, applying the above formula Dwidhābdāḥ the
number of Adhikamāsās elapsed from the beginning of the
Kali were 1531-21-12-52-30 i.e. the S'uddhi at the begin-
ning of that year was 21-12-52-30. This naturally entailed
the occurence of a Kshayamāsa. Thereafter during the
years 1115, 1256 and 1378, the S'uddhis in the beginning of
the years should be as per the above analysis 21-12-52-30
plus 68-22-30 i.e. 21-19-15; 21-25-37-30 and 21-25-37-30
and 21-18-23 respectively so that Bhāskara could forecast
the occurrence of the Kshayamāsās in those years also. In
this context it is worth-noting that Gaṇeśa notified in a
verse that Kshayamāsās would recur in the years 1462,
1603, 1744, 1885, 2167, 2232, 2373, 2392, 2524, 2533, 2655,
2674, 2796, 2815 according to Sūrya Siddhānta and accord-
ing to Āryabhaṭīya during the years 1482, 1793, 1904, 2129,
2186, 2251. These years also may be verified by computing
the S'uddhis as said before and also with respect to Ārya-
bhaṭīya according to which the number of Adhimāsas
during a yuga differs by a little and as such effects a differ-
ence in the sequence of the Kshayamāsa years.
Verse 8. Tell me, how, when and in the course of
how many years do two Adhikamāsas occur as mentioned
by the Rishis? Questioned accordingly by an expert in
questioning, if a mathematician could know the answer, I
would reckon him as no other than Bhāskara (either the
Sun-god or he himself i.e. Bhāskarāchārya) who could make
the lotus-buds of mathematicians blossom.
Comm. Probably taking up this challenge alone
Gaṇeśa answered the question and gave the years cited
above which should bring in a recurrence of Kshayamāsās.
11