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

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

91
begins simultaneously for any place of the earth, if y > x,
then he should know that he lies on the east of the primary
meridian because his Sun-rise happens to be earlier than
the Sun-rise on the primary meridian. Also the difference
y—x gives the Desāntara correction in time for his place.
The converse is the case if he happens to lie on the western
side of the primary meridian. The time at which the
eclipse takes place on the primary meridian after the Sun-
rise there which is obtained by computation is called Drik-
grahaṇa - Kāla ; whereas the local time after Sun-rise
observed by the observer is called pragrahaṇa-Kāla. Their
difference is therefore the Desāntara correction in time.
If the Desāntara is to be got in yojanas, (T × C) / 60 is the
answer, where T=y—x and C is the rectified circumference
of the earth, for, if a difference of 60 ghaṭis be there for C
yojanas, what should be the distance in yojanas in order
that the difference is T”? The answer is as given above.
Hence to obtain the positions of the Sun and the
Moon at the beginning of the eclipse at the locality we
have to add or substract as the case may be (T × δm) / 60 where
δm is the daily motion of the Sun or the Moon, and T is
y—x cited above, for, “ If in 60 ghaṭis the motion be δm,
what would it be in T ?” is the rule of three for which the
answer is as stated above.
Now the question is when the week-day begins for the
locality. It must be noted clearly, that in Hindu Astro-
nomy the moment of Sun-rise at the primary meridian
alone is to be reckoned as the beginning of the week-day
universally. This convention is adopted for convenience.
Thus the astronomical week day for any locality does not
begin from the Sun-rise of the locality, but may begin
earlier or later. This difference is given by y — x cited
above.

92
There is yet another subtlety in the commencement
of the week-day, arising out of the latitude of the place.
The former analysis pertains to the longitudinal difference.
The difference arising out of latitude between the local
Sun-rise and the Lanka-Sun-rise is given by what is called
Chara-Kāla. Since the week-day begins at Lanka Sun-rise
and the local Sun-rise differs from the Lanka Sun-rise not
merely by a longitudinal difference but also by a latitudi-
nal difference, to compute the actual beginning of the
week-day before or after the local Sun-rise, we have to take
into account both the differences cited above. In other
words, computing the local Sun-rise and also the Lanka
Sun-rise, we have to decide the beginning of the week day
before or after the local Sun-rise.
Verses 7, 8. The correction called Bijakarma for the
planetary positions.
The number of years from the beginning of the Kalpa
divided by 12000, the remainder, or the difference of the
divisor and the remainder whichever is less is to be divided
by 200. The quotient in minutes of arc, multiplied by
3, 5, 5, 15, 2 respectively is a negative correction in the
positions of the Sun, Moon, Jupiter, Venus, and the lunar
apogee and multiplied by 1, 52, 2 and 4 gives the positive
ocrrection in the positions of Mars, Mercury, the lunar
Node and the Saturn respectively.
Comm. By the phrase 'The remainder or the diffe-
rence of the remainder and the divisor', it is plain that the
corrections positive or negative increase for 6000 years and
decrease for the next 6000 years. Bhāskara gives no reason
for these corrections, but, we have to construe these correc-
tions on the following rational grounds. Bhāskara,
however, says that the corrections were accepted by him on
the basis of Āgama. This Āgama-stipulation was there
in Brahma-Sphuta-Siddhānta and was later incorporated
by Sripati also in his Siddhānta-Sekhara and as such was

98
accepted by Bhāskara also. However, in Brahma Sphuṭa
Siddhānta both as first published as an edition of M. M.
Sudhākara Dwivedi and later by the late Rāmaswarūpa
Śarmā in 1966, the verses 59, 60 of Madhyamādhikāra
suggest that the corrections are negative in the case of all
the planets; whereas both Śrīpati and Bhāskara make them
positive in the case of the latter four viz. Mars, Mercury,
the lunar Node and Saturn. By this we have to construe
that Śrīpati and Bhāskara must have had before them a
text which should have read ‘स्व’ in the place of ‘च’ in the
last pāda of verse 61. M. M. Sudhākara-Dwivedi did not
notice this anomaly of the positiveness of the correction
with respect to the latter four, but he remarked, however,
that there was a prosodial lapse in the last pāda of verse
61, for which he offered a suggestion that instead of वेदैर्,
we had better read वेदैः:—This suggestion, no doubt, rec-
tifies the prosody of the verse, but not the the anomaly
cited above which was not noticed by M. M. Sudhākara
Dwivedi. So, we have offered our own suggestion namely
that in the place of च as mentioned above if we read स्वं,
we not only rectify the prosodial error but also the anomaly
referred to. Rāmaswarūpa Śarma noted the anomaly but
did neither refer to the prosodial error nor offer a correc-
tion. It seems that Rāmaswarūpa Śarma did not verify
the corrections stipulated from the verses 91, 92, 93 of
Madhyamādhyāya of Siddhānta Śekhara. In this latter
work, there is another anomaly namely that in the case of
Mercury, the number 62 is the multiplier and not 52.
Makkibhaṭṭa, the ancient commentator had before him a
text which read 62 in the place of 52, in all probability, a
mistake of the scribe. M. M. Sudhākara Dwivedi is repor-
ted to have later pronounced that 52 must be the correct
figure when this was brought to his notice as this number
52 was found both in Brahmagupta and Bhāskara. As re-
ported by the editor of Siddhānta Śekhara Pandit Babuaji
Mishra, who mentions this latter pronouncement of Sudhā-
kara Dwivedi his teacher, also says that Sudhākara Dwi-

94
vedi suggested the reading द्विशर in the place of द्विरस of
verse 93 of Siddhānta Śekhara. The fact that Makkibhaṭṭa
commented द्विरससङ्गुणं as द्विषष्टिसङगुणं shows that he did
not consult Brahmasphuta Siddhānta in this place; also,
he must have had a manuscript before him which scribed
द्विरस in the place of द्विशर. Using श ष, स, indiscretely
is not uncommon in many books of North India, from a
long time and the scribe of the manuscript probably having
used स in the place of श and then by an oversight a
latter scribe having inverted सर as रस, Makkibhatta must
have commented like that.
Incidentally a remark may be made here about Makki-
bhatta. He was evidently a keralite because he used letters
to signify numbers as was a common practice among the
Kerala Astronomers, and as he also commented upon Brihad-
Bhāskarīya. Further, it is interesting to note that he
wrote in his commentary under verse 39 of the Sādhanā-
dhyāya of Siddhānta Śekhara viz. “भभ्रमोऽर्कमण्डलान्तरं
सावनानि कुदिनानि तानिवा”, “भूमेः प्राङ्मुखी भ्रमति” etc”. This
idea shows that he accepted Āryabhaṭa's verse “अनुलोम गतिः
etc” implying that the earth is rotating.
Bhāskara says that the Bīja correction mentioned was
purely based on Āgama and Upalabdhi (meaning authority
and observation'). M. M. Sudhākara Dwivedi seems to
have reiterated the same as reported by Babuaji Mishra, in
a foot-note. Kamalākara, condemned this Bījakarma as
it was unwarranted and had no proof.
A rational explanation as to why this Bīja-Karma was
prescribed either by Brahmagupta himself or some autho-
rity which he seems to have accepted may be given as
follows. The small differences in the numbers of sidereal
revolutions or what is the same the minute differences in
the accepted daily motions of the planets and the assump-
tion of a conjunction of all the planets and planetary
points at the beginning of Kalpa, which is beyond proof,

95
resulted in a difference between the computed planetary
positions and their observed positions. So, the originator
of this Bīja-Saṁskāra, noting the differences in his own
time devised a formula, which could account for those
differences. But this formulation was bound to go wrong in
later times as long as the daily motions are not corrected
to the minutest extent possible and as long as the funda-
mental basis of the conjunction of all the planets and pla-
netary points is not proved. This seems to be the reason
why so may texts were written incorporating small diffe-
rences in different times as reported by Gaṇēśa (1507 A.D.)
in his work Bṛihat-Tithi-Chintāmaṇi in the words "The
calculations of planetary positions according to the
methods indicated by Brahma, Vasishtha and Kasyapa
Siddhantas held good in their own times, but grew obsolete
later; Then Maya, the demon at the end of Krita obtained
the science from the Sun God, which again grew obsolete
in this Kaliyuga wherein parāśara began to hold the ground
for a good length of time. Then Āryabhata rectified the
methods; when even those methods grew obsolete, Durga-
Simha, Varāha Mihira and others set them right. Again
Brahmagupta came into the picture to rectify the methods
by his own observations. Then Came Kēśava (Gaṇēsa's
father) who rectified further. After a lapse of sixty years,
his son Gaṇesa has now to correct the Science. If this
also grows obsolete (as it is bound to) in course of time, let
others again rectify it by observing conjunctions of the
Moon and planets with the asterisms."
Obsoleteness arises out of two contexts, one a justifi-
able situation and the other based upon a wrong premise.
The first is as follows. Suppose as a first approximation
we take the length of an year as 365 days. We will have
committed an error nearly ¼ of a day, so that the error
accures to a day in 4 years. Thus the convention of the
leap year arose so that during four years we give a day
more to February. Here again we have overestimated the
error by nearly 1/100th of a day. Hence in 400 years the

96
above correction leads to an error of a day. So, it is
that we pronounced that out of the years 2000, 2100,
2200, 2300 A.D., the year 2000 A.D. alone is a leap year
and not the remaining, the convention being that the
number of the century, here 20, must be also a multiple
of four. On this back-ground, suppose we prepare a
manual called a Karaṇa grantha taking the length of the
year to be 365·25 days. It works alright for some time
but in the course of 400 years the error will have reached
to as much as one day. Thus a manual like the above
works only for a short time and the approximation made
gradually brings in a divergence on account of which such
a manual grows obsolete. That is why one Narasimha who
happened to prepare a manual in 1333 Saka year (1411AD)
opens his work with the words “तिथिचक्रं यत्प्रणीतं मल्लिका-
र्जुनसूरिणा, कालेन महता तस्मिन् खिलीभूते तदादरात्, नौपुरीसिङ्गयार्यस्य
नरसिंहेनसूनुना एतदेव स्फुटतरं क्रियते सौरसम्मतम्” i.e. “In as much
as a manual named Tithi-Cakra prepared by one Mallikār-
juna Sūri long ago, based upon the Sūryasiddhānta has now
diverged far from the Sūryasiddhānta (on account of the
approximations made transcending the limits of negligibi-
lity) I, the son of one Singaya belonging to a place named
Nau-puri (probably Vada-palle of the East Godavary Dt.)
am rectifying it and bringing it to accord with the Sūrya
Siddhānta again.”
This kind of obsoleteness arising out of inevitable ap-
proximations that have to be made in the preparation of
manuals is permissible. But Suppose the premise of the
manuals itself is incorrect, then the rectification of the
manuals is no good so long as the data given in the pre-
mise are not corrected. There are two fundamental
defects in the ancient works according to a modern analysis
namely (1) The Supposition that all the planets were in
conjunction at the Zero-point of the Zodiac in the beginn-
ing of a Mahāyuga (2) Small variations in the constants
like the daily motion of the planets and the like. Accord-
ing to the modern interpreters of Hindu Astronomy the

97
first premise was not correct. According to them, some
astronomers having observed the daily motions of the pla-
nets or what is the same the sidereal periods of the planets
to a sufficiently good approximation calculated back or
extrapolated a date on which these planets should have
been in conjunction at the Zero-point of the Zodiac. The
extra-polated date was naturally wrong to some extent be-
cause the sidereal periods found could not but be correct
only to a particular degree of approximation. Thus a little
alteration in the number of sidereal revolutions alone or
the number of days in a Mahayuga made to suit the obser-
ved positions at a particular epoch would be only a tinker-
ing of the problem and not a true solution. Thus Hindu
Astronomy could be saved and its methods could still be
followed provided instead of trying to presume a date at
which all the planets were in conjunction (No doubt in the
long bosom of time, such a presumption also could not be
ruled out) correctly observed positions of the planets by the
help of modern instruments were taken as the basis of an
epoch and thereafter using more correct values of the
constants such as the sidereal revolutions, maximum equa-
tions of centre and maximum Sīghraphala, obliquity of the
ecliptic etc. The second defect cited above thus being
removed, and the original premise being changed, the
methods of calculation still hold good and there would be
no necessity to be going on with tinkerings of the problem.
The Bīja-correction which we are commenting upon
was rightly criticised by Kamalākara as irrational though he
himself fanatically tried to uphold Surya Siddhānta. Even
today there are a good number of the traditional Hindu
Astronomers who do hold that the Sūrya Siddhānta was
revealed to Maya at the end of Kṛtayuga in spite of the
fact that scholars like M. M. Sudhākara Dvivedi pronoun-
ced that it was an extra-polated work shortly after the time
of Brahma-Guptāchārya. It is interesting to note that
Bhāskara, a very rational astronomer, had before him
the verse “त्रिंशत्कृत्वो युगे भानां चक्रं प्राक् परिलम्बते” of the
13

98
Sūrya Siddhānta (verse 9 ch. 3). He did not give it the
interpretation that was later put upon it through the
two subsequent lines "तद्दोः त्रिघ्ना" etc., which lines were
not there evidently in Bhāskara's time. Without these
two latter lines the rate of precession was too small to be
accepted by Bhāskara and so he chose to follow Muñjāla
rather than the Sūrya Siddhānta. Our Traditional astro-
nomers today have no reservation to accept the greatness
of Bhāskara and worship him though they do not question
what necessity Bhāskara had to write another treatise and
that too basing it upon the Āgama accepted by Brahma-
gupta and not Sūrya Siddhānta, when there existed Sūrya
Siddhānta before him and from which he had no objection
to quote verses like "अदृश्वरूपाः कालस्य मूर्तयः" etc. (verse I
ch. 2.)
The Bīja-correction first incorporated by Brahma-
gupta and later followed by a good number of astronomers
because Śrīpati and Bhāskara accepted it, will not be accep-
table to modern astronomers, though it might have worked
well at the time of Brahmagupta and for some years later.
The reason is that it is construed only as a tinkering of the
defect as explained before.
It is also to be noted that the originator of this Bīja-
correction did not make it secular i.e. valid for all time
increasing without a limit, for, then, the respective
corrections transcend all limits and render the corrections
meaningless. So, he said that the corrections would be
increasing for 6000 years and thereafter begin to decrease
to nothing. They were Zero at the beginning of the Kalī
because all the yugas are multiples of 12000 years. Also
the maximum correction is in the case of Mercury
6000/200 × 52 = 1560' = 26°. Let us see how far this is justifi-
able. The daily mean motion of Mercury as given by
Bhāskara is 4° - 5' - 32" - 18'" - 28"" whereas as per modern
astronomy it is 4° - 5' - 37⅞" approxly. So there is a posi-
tive error of 5" ²³⁄₄₀ which will accrue to 18' - 35" in the

99
course of 200 years. But as per the Bīja-correction it
should be 52'. Hence it is a fact that there is a positive
error but not so much as indicated. But it must be noted
that Mercury's orbit has the highest eccentricity of as
much as .2, and the observer who stipulated the correction
must have observed when Mercury was near its perihelion,
where the error could have been as much as indicated and
even more. Similarly on close analysis it could be proved
that the Bīja-correction should have been as indicated,
say, roughly about 3600 Kali era, which might be roughly
the date of its stipulation.
Verses 9, 10. Concluding verses of the Madhyādhikāra.
If the work is made more voluminous by describing
various methods which are easy and interesting to un-
intelligent people, learned men look down upon such a work
as indulging in unnecessary verbosity. Hence the volume
of a work does not add to its greatness; So I have made my
work neither voluminous nor brief-worded. The reason is
that both the intelligent as well as the unintelligent people
are to be enlightened.
For the sake of clarity of exposition, different ingeni-
ous methods being used in such a way that the work does
not exceed the normal limits of the previous works, and in
corporating as far as possible unit numerators, fractions
having numerators and denominators mutually prime, using
methods of interpolation and reduction, making use of
different kinds of denominators and numerators in many
ways, this kind of treatment must be given to a work of
this nature by an intelligent man.
Comm. Easy.
Before we proceed to the next chapter, we shall add
here tables of astronomical constants as given by different
authorities, which will help comparison and appreciation of
the work.

100
TABLE 1
Number of sidereal revolutions of planets and planetary points in a Kalpa

TABLE 2

101
[L

SPAṢṬĀDHIKĀRA — RECTIFICATION
OF PLANETS
Introduction. In the Bhagaṇādhyāya section of the
previous chapter Bhāskara gave under the Caption Bhaga-
ṇopapatti his proofs as to how the ancient is might have
obtained the number of sidereal revolutions of the planets
and the planetary points called apogees or aphelia and
Nodes. But in trying to give those proofs, he was aware
and he confessed also in so many words that some of his
proofs at least were obsessed by what is called Itarēta-
rāśraya-Doṣa i.e. “answer begging the question” It is
worth hearing his words in his commentary under verses
1–6 of the section cited above—“That the planets, and the
planetary points perform so many revolutions in a Kalpa,
is essentially conveyed by the Āgama i.e. the Śāstra (which
is to be taken on faith). That Āgama, got diversified i.e.
there are many versions of that Science, due to the defects
of scribes, the teachers and the students and due to a long
lapse of time from the originators of the Āgama. That
being so, the question arises as to which of the versions is
to be trusted as the right authority. If it be said so, in
mathematics only an āgama which could be proved also
should be taken as authority. Such a number of revolu-
tions as are obtained by proof, is to be accepted. Even
that could not be (a proof); for, a great scholar could just
understand the proof and by that proof alone, it is not
possible to know the exact number of revolutions (in a
kalpa), for, a man's longevity is not much. In the proof
that could possibly be given, the planet's position is to be
observed and noted every day, during the entire course of
its revolution. Thus Saturn Completes its sidereal re-
volution in about 30 years. The apogee of the Sun and the
aphelia of the planets have their revolutions running into
hundreds of years. Hence the observation of one complete
revolution (of such a planetary point) is beyond the capa-