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

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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,

73
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

74

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

82
N.B. It does not suffice merely to compute the S'uddhis
alone in the beginning of the years but a rigorous compu-
tation necessitates the calculation of the moments of New
Moons and Samkrāntis also during the particular years as
well, in as much as, the motion of the Moon also comes in-
to the picture and it differs from month to month on
account of the rapid motion of his apogee.
Here ends the section named Adhimāsādi-nirṇaya.

THE SECTION BHŪ-PARIDHI-MĀNĀDIKA-
CIRCUMFERENCE OF THE EARTH
Verse 1. The circumference of the earth's globe is
4967 yojanas ; its diameter 1581. A yojana is equal to
(d × 360) / (C × δϕ) where δϕ is the difference in the latitudes of two
places on the same terrestrial meridian in degrees, C the
circumference of the earth's globe given above and d the
distance between the two places.
Comm. The second half of the verse gives the method
of computing the circumference of the earth's globe, which
is mathematically correct ; for, by the rule of three "If by
a difference of 90° in latitude we have ¼ C, what shall we
have for 1° difference in latitude ? " The answer is C / 360°
Again "If by distance of d between two places on the same
meridian, we have a difference of δϕ° in latitude, what
shall we have for 1° difference in latitude ? ' The answer
is d / δϕ. Both the answers must be the same ; so equating
them, we have C / 360 = d / δϕ = number of yojanas = x
say. Hence one yojana = x / x = (d / δϕ) ÷ (C / 360) = (d × 360) / (C × δϕ)
N.B. Here it must be noted that a yojana's length is
derived from the number of yojanas contained in the
circumference as reported in the Āgama.
In the course of the commentary under this verse,
Bhāskara explains why he had recourse to this kind of
definition, which is based upon āgama and as such does
not contain a proof. He says that in as much as the defi-

84
nition of a yojana was given basing ultimately on the units
of Angulas and yavas (grains of paddy) differently by differ-
ent authorities, and in as much as the circumference of
the earth’s globe was more or less unanimously accepted,
he has chosen to define a yojana on the accepted measure
of the circumference. In some other place Bhāskara says
that a yojana is equal to 4 Krōsās, where the word Krōsa
etymologically means that distance through which the
topmost voice of a healthy person could be heard by
another healthy person having good audition.
In this context it may be recalled how Śrīpati defined
a yojana under verses 69, 70 Madhyamādhyāya of his Sid-
dhānta Śekhara. “The minute speck of dust that is seen
flying in the light of the Sun’s rays entering a house
through the windows, is called a paramāṇu (not an atom as
is being translated now). Eight such paramāṇus equal
one Reṇu. Eight Reṇus equal the breadth of the end of a
hair known as Vālāgra or Kaccha-mukha or yūka. Eight
Vālāgras equal one Likshā; Eight Likshās equal one Yava;
eight Yavas equal one Angulā; twelve Angulās make what is
called a Vitasti (i.e. the expanded length of a palm of the
tallest person). Two Vitastis equal one Hāsta. Four has-
tās make one Chāpa. Two thousand Chāpas make one
Krōsa and four Krōsās are reported to be equal to a yojana
by Astronomers”
Verse 2. Rectification of the circumference .ie. find-
ing the length of the circumference of the earth parallel to
the terrestrial equator and passing through the locality in
question. Also the definition of the primary meridian.
The equatorial circumference of the earth multiplied
by cos ϕ and divided by R or multiplied by 12 and divided
by the hypotenuse of the right angled triangle formed by
the gnomon and the equinoctial midday shadow thereof.
(Hereafter called equinoctial hypotenuse) gives the circum-
ference of the earth parallel to the equator and passing

85
through the locality (hereafter called the rectified circum-
ference). Also the primary terrestrial meridian is that
longitudinal line passing through the places (1) Lanka (2)
Ujjain (3) Kurukshetra and (4) the north pole.
Fig. 4
Comm. (Ref. fig. 4) Let ABC be the terrestrial Equa-
tor, and QST be a circle parallel to this terrestrial equator
and passing through the locality S. This QST is called the
rectified circumference of the earth at the place S. The
terrestrial equator is defined as follows. It is known that
the earth rotates about herself about the axis P₁ P₂ where
P₁ P₂ is called the polar axis or Dhruvayashti. In other
words the entire heavens appear to revolve round the earth
in such a way that any star will appear to be revolving in
a circle called its diurnal circle which is parallel to the
circle ABC. The points P₁ and P₂ are called the north and

86
south pole respectively. These two points evidently do
not move, though the earth is herself rotating in the
clockwise direction. If O be the centre of the earth, OP₁
produced meets the skies at a point called the north celes-
tial pole and OP₂ produced meets the skies in the point
called the south pole. It so happens that the north celes-
tial pole is very near a star which is called the pole star.
This star is known as the Dhruva-Tāra in as much as it does
not appear to move at all while all the heavens (i.e. all the
stars of the sky) appear to be revolving round the earth
rising and setting as seen at any place. (The word Dhruva
means fixed). Aryabhaṭāchārya mentioned in so many
words that it is the earth that really rotates and so the
stars which are themselves fixed appear to be going round
the earth in circles parallel to the circle ABC. अनुलोमगतिः
नौस्थः पश्यत्यचलं विलोमगं यद्वत्। अचलानि भानि तद्वत्समपश्चिमगानि
लङ्कायाम्॥ (Explained under verse 7 Bhagaṇādhyāya). The
celestial Equator is the great circle which is the circle
of intersection of a plane perpendicular to the polar axis
and passing through the earth's centre with the celestial
sphere (celestial sphere is the sphere-like surface which shape
the Sky takes and on which the stars and the planets appear
to be studded.) Similarly the terrestrial equator ABC is
the circle of intersection of the earth's globe with the same
plane. Thus the terrestrial and the celestial equators are
concentric coplanar circles with the earth's centre as the
common centre. A great circle of a sphere is a circle whose
plane passes through its centre. Thus ABC is a great circle
on the earth's surface, because its plane passes through the
earth's centre. Similarly the celestial equator is a great
circle of the skies. The circle QST is called a small circle,
just as the diurnal circle traced by any star in its diurnal
rotation is also a small circle parallel to the celestial equa-
tor. Thus small circles, an infinity of them can be drawn
parallel to the terrestrial equator ABC and they will be in
decreasing dimension as we proceed towards the pole.
Hence the Sphutaparidhi or the rectified circumference

87
QST at the locality S is a small circle whose circumference
is smaller than that of ABC. The problem is now to find
the length of this circle QST. Evidently QST / ABC = 2 π r / 2 π R
= r / R where r and R are respectively the radii of the two
circles. But r / R = Cos ϕ from the triangle OSO', where
Ô'₁SO = ŜOA = latitude of the place S. ∴ QST =
ABC × Cos ϕ = (ABC × H Cos ϕ) / R I where H Cos ϕ is
the Hindu cosine of ϕ known as lambajyā and stands for
OS₁ = O'₁S = r. This lambajyā is also called Dyujya if
the small circle is the diurnal circle of a Star. The diurnal
circle of a Star is called Dyujya-Vritta and its radius is
called Dyujyā-Equation I proves the first statement of the
Fig. 5
verse. In fig. 5 Gn N is called the fundamental gnomonic
triangle where Gn is the vertical gnomon pointing to the
Zenith Z of the celestial sphere and is considered to be of
12 units (Angulas as they are called), GN the midday-
shadow of the gnomon cast on an equinoctial day when the

88
Sun is at the point Q where Q is the point of intersection
of the celestial equator with the meridian of the place. If
the equinoctial shadow be denoted by ' s ' and the hypo-
tenuse of the triangle nGN namely nN be denoted by K
(called the Vishuvatkarṇa or the equinoctial hypotenuse)
then s / K = Sin ϕ and 12 / K = Cosϕ. If H Sin ϕ be the
Hindu sine of ϕ called the Akshajyā H Sin ϕ = R Sin ϕ
so that s / K = H Sin ϕ / R and 12 / K = HCosϕ / R = Lambajyā / Trijyā .
Substituting for H Cos ϕ / R the value 12 / K in equation I
above, we have
QST = Sphuṭa - paridhi = (Bhū - paridhi × 12) / K II
which proves the second statement given in the verse.
Regarding the third statement, which defines the Bhu-
Madhya-Rekhā (In modern text books of geography the
terrestrial equator is spoken of as the Bhu-Madhya-Rekha.
The terrestrial equator is called Niraksha-Rekha in Hindu
Astronomy which means the circle of Zero-latitude), it is
the primary meridian taken by Hindu Astronomers. In
modern astronomy the primary meriodian is taken as the
Greenwich meridian. Bhāskara has given four places locat-
ed on the Hindu primary meridian, but, Śrīpati gives many
more places located on this primary meridian under verse
96 Madhyamādhyāya namely (1) Laṅkā (2) Kanyākumārī
(3) Kāñchī (4) Pannāta (5) the six-faced white mountain
(6) Sri-Valma-gulmam (7) Māhishmatī (8) Ujjain (9) An
Āsrama (10) Pattasiva, a town (11) Sri Gargarāna (12)
Sthānviswara known also as purohita (13) Sītagiri and (14)
Sumeru. Some of these places cannot be properly identi-
fied but the following remarks may be made (a) Pannāta is
one of the fifty-six small countries into which India was
divided in ancient times according to the purāṇic literature

89
(b) It is not clear what places are indicated by (5) and (6)
cited above (c) In some works Māhishmatī and Ujjain are
used synonymously. (8) is not clear. Regarding (9) there
is one pattasīva near Rajahmundry but S'rī pati does not
seem to have meant it. Again (10) is not clear, Regard-
ing (11) it is to be noted that the place is now pronounced
(probably mis-pronounced) as Sthānes'wara. If (12) means
the Himalaya mountain, there is not much meaning to say
that it lies on the primary meridian; only a cross-section
of it could lie there upon. The entire mountain extends
from west to East over more than a thousand miles.
Verse 3. To find the correction known as Desāntara.
The distance between two places on the same latitude
multiplied by the daily motion of a planet and divided by
the rectified circumference is a correction subtractive in
the east and additive in the west of the primary meridian
in the planetary position obtained.
Comm. In Hindu Astronomy the mean planetary
positions are first calculated for the Sun-rise at the primary
meridian. Now suppose a place lies to the east of this meri-
dian. Then the Sun-rise at the place happens to occur
earlier than on the primary meridian. Hence the correc-
tion in the mean computed position of the planet is nega-
tive if the position were to be calculated for the local Sun-
rise. If the place happens to be on the western side of the
primary meridian the reverse holds good i.e. the correction
is to be additive. The amount of the correction is the
amount of the motion of the planet in between the two
Sun-rises. Let the planet move an arc equal to δm per day
i.e. it moves δm when the earth rotates once about her
axis. The time between the two Sun-rises above is the
time by which the local meridian is carried through the
distance between the locality and the primary meridian's
point of inter-section with the latitudinal line or what
is the same through the arc of the rectified circumfer-
12

90
ence of the earth pertaining to the locality. If d be this
distance then the rule of three to be used is “ If the
length of the rectified circumference viz. C rotates by the
time the planet moves a distance δm, what is the arc
traversed through by the planet if an arc ‘d’ of C rotates
through ? ’’ The answer is (d × δm) / C which is the cor-
rection required.
Verses 4, 5, 6. The Correction Desāntara expressed
in time.
The eclipse of the Moon occurs at a place situated on
the east of the primary meridian later than on the primary
meridian and vice versa. The time in between the two
moments is the Deśāntara expressed in time. The distance
of Desāntara ie. the distance of the locality from the pri-
mary meridian measured along a parallel to the terrestrial
equator or Niraksha Rekhā is obtained by multiplying the
rectified circumference by the Desāntara measured as above
in ghatis and dividing by 60. Also the above time in
ghatis multiplied by the planets’ daily motion and divided
by 60, gives the correction in arc in the computed mean
planetary motion.
Further the week-day begins after or before the local
Sun-rise by that Desāntara expressed in time according as
the locality is on the east or west of the primary meridian.
Also the week-day begins after or before the local Sunrise
by the ghatis of the correction known as chara according
as the Sun is in the northern or Southern hemisphere.
Comm. An eclipse is first computed for the primary
meridian. If an observer wants to know whether he lies
east or west of the primary meridian and to know the
Desāntara correction in time, the following procedure is to
be adopted. Let a lunar eclipse begin x ghatis after the
Sun-rise of the primary meridian. Let the observer note
the time y ghatis which have elapsed after Sun-rise at his
own place when the eclipse begins. Since a lunar eclipse