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

115
above differences in the reverse order give the H versines.
The rest of the contents of the verses is simple, the proces-
ses being based on the ‘Rule of three’.
Verse 16. Rectification of the next H sine difference
known as Bhōgya—Khanda.
The difference of the previous and the following H
sine —differences being multiplied by the remaining degrees
and divided by 20, the result is subtracted from the arith-
metic mean of the previous and following H sine-differences
to give the rectified H sine —difference, in question.
Comm. This is a formula for interpolation which
agrees with the interpolation formula given by Ball in his
spherical astronomy on page 18 in the form y = yo + x/h
(y₁ — yo) + x (x — h) / 2h² (y₂ — 2y₁ + yo). This formula is
a re-statement of the formula enunciated by Brahmagupta
in his work Brahma Sphuta Siddhānta as well as Uttara-
Khandakhādya in the form “गतभोग्यखण्डकान्तरदलविकलवधात्
शतैर्नवभिराप्तैः, तद्युतिदलं युतोनं भोग्यादूनाधिकं भोग्यम्” wherein in
the place of ten-degree-interval, a fifteen-degree-interval
ie 900'-interval was taken. Rule of three is a linear for-
mula of interpolation, whereas the above is a quadratic
formula reflecting much credit on the mathematical genius
of Brahmagupta.
We shall now see how the formula is applied and what
mathematical significance it has. Suppose it is required to
find the H sine of 24° from the previous table of H sine —
differences given for intervals of 10° — from the table H
sin 10° = 21, H sine 20 = 41, H sine 30 = 60 where R is
taken to be 120. Now to find H sin 24°, we are asked to
rectify the next H sine —difference namely 19', where the
table is 21, 20, 19 etc. As a first approximation, applying
rule of three H sin 24° = 41 + ⁴/₁₀ × 19=48·6=48'—36"

116
This is a crude approximation, the actual value being
48' — 48" — 14"'. Application of rule of three is justified
if the H sine—differences are uniform, but they are not so,
being in a decreasing order. So, the following H sine—
difference namely 19' is to be rectified so as to be applicable
at 24°. In other words we have to take such a H sine—
difference which will hold good at 24°, not at 20° or 30° —
from 10° to 20°, the H. S. d. (H Sine—difference) is 20',
and from 20° to 30° it is 19'. If that be so what will be
exactly at 24°? It should be less than 20' and greater
than 19'. Now the argument advanced by Bhāskara is that
the H Sine—difference at the mid-point of the 2nd and 3rd
differences namely 20 and 19 should be (20 + 19)/2 = 19·5.
The H sine—difference at the end of the third interval is
19. Then by the rule of three ‘If there is a decrease of
19·5 — 19 = ·5, during the course of the 10° of the third
interval, what should the difference be for 4°? (where we
have to find the H sine at 24°) The result is 4/10 × ·5 = (4 × 1)/20
which is given by the words “यातैष्ययोः खण्डकयोर्विशेषः
शेषांश निघ्नो नखहृत् ”. This decrease makes the H sine—
difference at 24°, 19·5 — 4/20 = 19·3 at 24°—Now the argu-
ment to find the H sine at 24° is ‘If for an interval of 10°,
the H sine—difference is 19·3, what should it be for 4°?’
The answer is 4/10 × 19·3 = 7·72. Hence the H sine of 24° is
21 + 20 + 7·72 = 48·72 = 48' – 43" – 12"' which is nearer the
truth 48' – 48" – 14"' than what was obtained by the crude
rule of three namely 48' – 36".
Here we have to explain Bhaskara’s words more elabo-
rately, because, Kamalākara happened to criticise Bhas-
kara’s words in this context. Bhaskara Says “The H sine
difference at the end of an interval is the arithmetic mean
of the preceding and succeeding differences, whereas the
succeeding one is that which holds good at end of the suc-
ceeding. In between, we have to apply the rule of three to

117
obtain the rectified difference." What Bhāskara means is
this. At the end of an interval, to obtain the H sine, it is
enough to add the H sine difference belonging to that inter-
val to the preceding differences. But when it is required
to find the H sine in the interior of an interval, we have to
construe that the difference at the end of the previous inter-
val is the arithmetic mean of the previous and succeeding
differences. There is apparently a self-contradiction in
Bhāskara's words; for, at the end of the interval, according
to his own words, the difference is that belonging to the
previous interval and not the arithmetic mean as postula-
ted. The contradiction will not be there when we read
Bhaskara's mind that he means "When we require to find
the H sine in the interior of an interval only, the difference
at the beginning of that interval is to be taken as the arith-
metic mean of the previous and the current differences, and
that at the end of the interval the current difference holds
good."
The truth of Bhāskara's statement could be seen
analytically as follows. The arithmetic mean of the prece-
ding and succeeding H sine differences is (The context is to
rectify the third H sine difference namely 19, for, we were
finding the H sine of 24°) (AB + BC) / 2 (Ref fig. 8) where OA,
AB, BC etc are the successive differences.
(AB + BC) / 2 = (H sin 20 - H sin 10 + H sin 30 - H S 20) / 2
= (H sin 30 - H S 10) / 2 = R ((Sin 30 - S 10) / 2)
= R × Cos 20 Sin 10 = (H Cos 20 H Sin 10) / R
The numerical difference of the preceding and succeed-
ing H sine differences is (i.e. यातैष्यखण्डविशेष). AB - BC
= (H sin 20 - H S 10) - (H sin 30 - H sin 20) =

118
(2 H Cos 15 H sin 5) / R — (2 H Cos 25 H sin 5) / R = (2 H sin 5) / R
(H Cos 15 — H Cos 25) = (2 H sin 5) / R × (2 H sin 20 H sin 5) / R
Let now X be the point where we are to find the H sine
(Here let us take it as x° after the previous interval for
generalisation). Then Bhaskara's formula would give
(H Cos 20 H Sin 10) / R — (2 H sin 5) / R² × (2 H sin 20 H sin 5) / 20 × x
= (H Cos 20 H sin 10) / R — x / (10 R²) × 2 H sin 20 H sin² 5
= (2 H Cos 20 H sin 5 H Cos 5) / R² — 1 / R² × x / 10 × 2 H sin 20
H sin² 5
= (2 H sin 5) / R { (H Cos 20 H Cos 5 — x / 10 × H sin 20 H sin 5) / R }
put now successively x = 0° and 10° to get the rectified
differences at B and C respectively; then those rectified
differences would be respectively H Cos 20 H sin 10 and
(2 H sin 5) / R × H Cos 25. But (AB + BC) / 2 = (H Cos 20 H sin 10) / R
(found above) and BC = H sin 30 — H sin 20 =
(2 H Cos 25 H sin 5) / R In other words the rectified differences
at B and C are respectively what exactly has been stated by
Bhāskara. Hence Kamalākara's condemnation of Bhāskara
is quite unjustified.
Verse 17. To rectify the arcual difference to obtain
the arc for a given H sine.
Subtract as many H sine-differences as could be sub-
tracted from the given H-sine. Half of the remainder
multiplied by the difference of the preceding and succeed-
ing H sine-differences and divided by the succeeding and

119
the result being subtracted from or added to as the case
may be (added in the case of Hversines) the arithmetic
mean of the preceding and succeeding H sine-differences
gives the rectified H sine-difference while finding the arc
for a given H sine.
Comm. Let the given H sine be that of 24° found
before i.e. 48·72. We could subtract 21 and 20 from this
and the remainder is 7·72; half of this is 3·86 which
multiplied by (20-19) is 3·86. This divided by the succeed-
ing difference namely 19 is ·2 approximately. The arith-
metic mean of the preceding and succeeding si (20 + 19) / 2
= 19·5. If the above result is subtracted from this, we
have 19·5 − ·2 = 19·3. If for 19·3 we have 10° increment
what shall we have for 7·72. The answer is (10 × 7·72) / 19·3
= 10 × ·4 = 4°. Hence the required arc is 20° + 4° = 24°.
The proof is analogous to the previous proof. Having
subtracted 21 and 20, the remainder is 7·72, (Ref. fig. 8)
Fig. 8
so that BX = 7·72. Now during the course of the succeed-
ing interval of 19, there has been a decrease of 19·5 −
19 = ·5 or to put it in general terms, during the course of
y the succeeding interval BC there has been a decrease

120
(x + y)/2 - y = (x - y)/2 where x is the previous interval AB
and (x + y)/2 is the H sine difference at B. Hence the argu-
ment is "If for y, there has been a decrease of (x - y)/2,
what will it be for δ (Here δ = 7·72) "? The answer is
δ ((x - y)/2) × 1/y = δ/2 (x - y) × 1/y = अवशेषकार्धम् × गतैष्यान्तरम्
÷ एष्यम्. This result is to be subtracted from (x + y)/2
i.e. ·2 is to be subtracted from 19·5 to give the rectified
H sine-difference.
Verse 18. Definition of Kēndra and assignment of
sign thereto.
The excess of the longitude of the mean place over
that of the apogee or aphelion as the case may be is called
the mean anomaly. The excess of the longitude of the
point called Sīghrōccha over that of the planet rectified
by the first equation known as Manda-phala or equation
of centre is known as the Sīghra-anomaly. The equation
of centre is positive or negative according as 180 < m < 360
or 0 < m < 180 where m is the mean anomaly. The case
will be reverse in the case of the sign of Sīghraphala, the
second equation.
Comm. “चन्द्रसूर्यौ स्फुटौ स्यातां मान्देनैकेन कर्मणा” i.e.
The Moon and the Sun could be rectified by the equation
of centre alone. This means that the Moon revolving
round the Earth directly and that the Sun revolving rela-
tively round the Earth are subject to only one correction
namely the equation of centre for rectification. In the
case of the Sun, though the fact is that he is revolving
round the Earth relatively, assuming as Hindu astronomy
does that he is revolving directly round the Earth and
subjecting him to the correction of the equation of centre

121
does not alter the mathematics that goes into his rectifi-
cation. According to modern astronomy the Sun and the
Moon, one relatively and the other directly go round the
Earth, in ellipses, the Earth being in one focus whereas in
Hindu astronomy both the Sun and the Moon are taken to
be going round the Earth in eccentric circles i.e. circles
whose centres do not coincide with the centre of the
Earth. In fact, Bhāskara says in so many words “ भुमेर्मध्यं
खलु भवलयस्याऽपि मध्यं यतः स्यात् , यस्मिन् वृत्ते भ्रमति खचरो नाऽस्य
मध्यं कुमध्ये, भूस्थो द्रष्टा न हि भवलये मध्यतुल्यं प्रपश्येत्, तस्मात् तद्ज्ञैः
क्रियत इहतत् दोःफलमध्यखेटे i.e. The centre of the celestial
sphere coincides with the centre of the Earth. The centre
of the circle in which a planet goes does not coincide with
the centre of the Earth. Hence an observer on the surface
of the Earth finds the True planet’s position differing from
that of the mean planet, so that what is called the correc-
tion of Bhujaphala is to be made in the mean position of
the planet to get the True position.” Here it has to be
noted that the Bhujaphala mentioned stands both for the
equation of centre and the second equation known as
Sīghraphala as well. One may wonder how it could be so,
but it may be noted that in the formulation of both the
equations, the centre of the eccentric does not coincide
with that of the Earth and also in both the cases the equa-
tion contains the term H sine of the anomaly where the
word anomaly whether it be of the first or second equation
is known as ‘Bhuja’.
. In the case of the other planets, the fact is that they
go round the Sun in elliptic orbits, the Sun being in one
focus. In Hindu Astronomy, we shall see that the centre
of the eccentric circle in which these planets are taken to
revolve coincides with the Sun. Hence, though the ancient
Hindu Astronomy postulates geocentric motion, the mathe-
matics that goes into the formulation of the second equa-
tion, makes the Sun’s centre the centre of planetary revolu-
tion. One may wonder again how the equation of centre
16

122

formulated by Hindu Astronomy agrees with its formula-
tion in Modern Astronomy which enunciates elliptic
motion; but it will be seen that the eccentric-circle theory
also gives very approximately the same formula for the
Equation of centre. There is just one point of difference,
which does not matter. Whereas in Modern Astronomy the
mean anomaly is reckoned from the perigee or perihelion
as the case may be, it is reckoned in Hindu Astronomy
from the apogee or aphelion. The difference is made up by
prefixing the appropriate sign to the equation. The word
Mandōccha stands for the apogee in the case of the Sun
and the Moon and for the aphelion in the case of the other
planets and the word Manda kendra stands for the mean
anomaly in both the cases. In Hindu Astronomy the word
‘graha’ stands for not only the five planets Mercury to
Saturn but also for the Sun and the Moon. Why that
word is applied to the Sun and the Moon as well is, that
both the Sun and the Moon also while moving among the
stars along with the five other planets Mercury to Saturn,
wield an influence on the residents of the Earth. The
etymology of the word ‘graha’ is गृह्णातीति वा गृह्यते अनेनेति
वा ग्रहः ie. ‘that which seizes upon the fates of the resi-
dents of the Earth’, with this etymological significance
only, even the lunar orbital nodes known as Rāhu and
Kētu are also taken to be grahas in Hindu Astronomy.
Hence translating the word graha as a planet and criticis-
ing Hindu Astronomy for taking the Sun, Moon and the
lunar orbital nodes also as such is not right. In other words
the translation should be pronounced wrong. Uranus, Nep-
tune and pluto were not mentioned in Hindu Astronomy.
We shall now elucidate the eccentric and epicyclic
theories of Hindu Astronomy, which will be seen to give
identical position to the planets. How they came to be
postulated will be also elucidated. Incidentally we deal
with the ‘Bhagaṇōpapatti’ or the proof of the numbers of
sideral revolutions of the various grahas enunciated in the

123
beginning of the Madhyādhikāra, Bhagaṇādhyaya in verses
1 to 6. Even Bhāskara gave such a proof as appealed to
Āgama ie. 'Authority', which proof therefore will not be
acceptable to a student of Modern Astronomy, who is likely
to question how the Āgama came into existence.
How the Āgama came to formulate the number of
sidereal revolutions of the grahas, we shall now see.
The forefathers of Hindu Astronomy (have been re-
ported to be eighteen in number in the famous verses " सूर्यः
पितामहो व्यासो वसिष्ठोऽत्रिः पराशरः कश्यपो नारदो गर्गो मरीचिर्मनु-
रङ्गिराः लोमशः पौलिशश्चैव च्यवनो यवनो भृगुः, शौनकोऽष्टादशैतेस्युः
ज्योतिषशास्त्रप्रवर्तकाः " Of these Brahmagupta mentions
Brahma-Siddhānta, which he reports to have resuscitated.
Varāha Mihira gave a version of the old Sūryasiddhānta,
mentioning that it accorded with observations. Āryabhata
says that he revived his system from the then existing
ocean of knowledge both good and spurious. The fact that
none of these outstanding astronomers mentioned that
they had derived their systems from a foreign source, and
the reasonableness in presuming that all these three could
not be impostors, make the author of this work feel
strongly that there should have been some works in the
name of Āgamas extant long before these Acharyās. The
argument that the crude Vedānga-jyotiṣa alone should
have existed before Āryabhata, simply because, no other
work worth the name has been discovered, may not be
correct. It is quite possible that crude works could exist
side by side with advanced scientific works, just as even
nowadays we have thinkers and works of a primitive type
existing along with highly advanced thinkers as well as
scientific works).
Bhagaṇopapatti. In the first place, the forefathers of
Hindu Astronomy must have noticed very easily that the
Moon has a motion among stars, for, this could be detected

124
even by a lay man during the course of a single night. So,
the period of a single sidereal revolution, could be roughly
recognized by noticing the conjunction of the Moon with
a luminous star. Having thus observed a good number of
sidereal revolutions, which could be done even with the
naked eye, the average period could be arrived at with
sufficient accuracy within the course of a few years.
Having thus obtained almost accurately the average of a
sidereal revolution of the Moon, the sidereal revolution of
the Sun could have been arrived at as follows. The
moment of an eclipse solar or lunar could be observed with
the naked eye.-Observing a good number of eclipses within
the course of a few years, the average of a lunation could
be easily arrived at very accurately, for, in between the
eclipses of the same nature an integral number of lunations
elapse. That the Sun also has a motion among stars must
have been noticed clearly during the course of a few
months, for, observing at Sunset the star that was rising,
it should have been noticed gradually even during the
course of a month, that the Sun must have been approach-
ing the star or vice versa and as the stars were found to
keep the distances amongst them constant, it was the Sun
that was approaching the star and not the star it was that
was approaching the Sun. Having decided thus that the
Sun was moving among stars from west to East, the
approximate period that the Sun took to complete a side-
real revolution was arrived at. Then, as both the Sun and
the Moon were having east ward motion, and as the Moon
has a more rapid motion, the arc by which the Moon over-
takes the Sun during a day was roughly noticed. Thus
arriving at a rough estimate of a lunation within which
a conjunction of the Moon with the Sun recurs, it was
noticed that the excess of the sidereal revolutions of the
Moon over those of the Sun gave the number of lunations.
Since a correct estimate of both a sidereal revolution of the
Moon as well as that of a lunation were previously arrived
at, the number of sidereal revolutions of the Sun during

125
the course of a certain period were computed whereby a
correct estimate of a sidereal solar year was arrived at.
This period could also be checked simultaneously by obser-
ving the interval between the heliacal risings or settings of
a particular star of the Zodiac as well. Thus far, we have
seen how the sidereal periods of the Sun and the Moon
were determined very accurately. It may be noted that
these periods as determined by the Hindu astronomers
were correct to a good number of decimal places.
When once the Sun and the Moon were found to be
having eastward motion among stars, and when it was
discovered that there were other luminous bodies like the
Jupiter and Venus etc. moving among stars, it was
attempted to determine their sidereal periods. It must
have been done as follows. In the first place, it was
noticed that these other luminous bodies which were five in
number, namely Mars, Mercury, Jupiter, Venus and
Saturn, were found to be having retrograde motion also
unlike the Sun and the Moon. As these five bodies were
looking like stars they were named Tārā-grahas i.e. grahas
looking like stars. Also a distrinction could be drawn very
easily between Mercury and Venus on the one hand and
the other three on the other, for, the former were always
found oscillating about the Sun, never parting from him
through long distances. Thus during the course of a suffi-
ciently long interval, the geocentric sidereal periods of
Mercury and Venus coincided with that of the Sun. In
other words, it was taken that the geocentric sidereal
periods of Mercury and Venus also were taken to be an
year. This is clear from the statements made by all the
Siddhāntas that in a Kalpa of 4320000000 years the Sun, the
Mercury and Venus all the three make 4320000000 sidereal
revolutions. Then with respect to the other three planets
Mars, Jupiter and Saturn, it was noticed that they were
having a pre-dominantly longer period of direct motion,
though there was a retrograde motion for some time. This

126
gave the clue to arrive at an approximate estimate of their
sidereal revolutions. But the correct estimates were arri-
ved at not by observing their conjunctions with stars, for,
that would take a very long period of observation in the
case of Saturn, but, by observing a good number of their
heliacal risings or settings. The interval between two
consecutive heliacal risings or settings being a little grea-
ter than an year, ten or fifteen observations could be done
very easily by a single person. Then the aforesaid argu-
ment given in the case of finding the sidereal revolution of
the Sun, was also advanced in the case of these three
planets Mars, Jupiter and Saturn. Let x° be the arc that
the planet covers during the course of a day. Let a° be the
arc covered by the Sun during the same period which was
previously known and that correctly. So, during the
course of a day the Sun overtakes the planet by (a - x)°.
Hence to overtake 360°, the period S was computed. This
period was observed as the interval between two consecu-
tive heliacal risings or settings and known as the synodic
period. So, from the equation 360 / (a - x) = S, the value of x
could be arrived at, wherefrom p the sidereal period was
determined. This sidereal period could be also determined
in another way. Noting the distance covered among stars
by a planet during the course of a synodic period, using
rule of three, the sidereal period could also be arrived at
with a good accuracy, for, the retrograde motion affects
equally each synodic period. An average of such determi-
nations made in two ways could give the sidereal periods
of Mars, Jupiter and Saturn with a good amount of accu-
racy. It will be noted here, that the average of a good
number of geocentric sidereal periods in the case of these
planets (called Superior) is also the heliocentric sidereal
period (for a proof of this statement reference may be made
to page 80 of the author's 'A critical study of the ancient
Hindu astronomy, published by the Karnatak University
Dharwar). It is why the sidereal periods of these planets

127
as given by Hindu astronomy tally with the heliocentric
sidereal periods given by modern astronomy. This is also
one of the reasons why heliocentric motion of the planets
could not be detected by Hindu astronomy, and also why a
statement was made that "In the case of Mercury and
Venus the Sun was the planet. and they are termed as
S'īghrōcchas, whereas in the case of Mars, Jupiter and
Saturn, they are the planets while the Sun plays the part
of S'īghrōccha "कुजजीवशनीनां तु रविः शीघ्रोच्चनामकः, ज्ञशुक्रयोः
ग्रहः सूर्योभवेत्, तौ शीघ्रनामकौ" (An elucidation of this state-
ment will be given shortly)
In the case of the planets Mercury and Venus (Inferior
planets) one may wonder how under the geocentric theory,
their heliocentric periods could be arrived at, though they
were not recognized as such but were pronounced as the
"geocentric periods (not considered as heliocentric)" of
two points known as their S'īghrōcchas. Here we come
across the peculiar concept of a S'īghrōccha which arose
out of the fact of postulating a geocentric system in the
place of the heliocentric. This concept is to be elaborated,
in as much as confusion is there in the minds of many
interpreters of Hindu astronomy in this behalf.
In the first place let us consider as to how the rectifi-
cation of the Sun and the Moon, known as sphutikaraṇa
was achieved. Having got their sidereal periods, their
mean daily motions were calculated. Also a period was
conceived, during which this Sun and the Moon would per-
form an integral number of revolutions. This period was
termed as a Mahāyuga (or simply a yuga as we hereafter
name it) whose duration was estimated as 4320000 solar
years. That the yuga is an integral L.C.M. so to say of
the sidereal periods of the Sun and the Moon (also of the
other planets as we shall see shortly) could be seen from
the statement of the Surya Siddhānta युगे सूर्यज्ञशुक्राणां

128
खचतुष्करदार्णवाः, कुजार्किगुरुशीघ्राणां भगणाः पूर्वयायिनाम्, इन्‍दो
रसाग्नित्रित्रीषुसप्त भूधरमार्गणाः '' i,e. ‘In a yuga, the Sun the
Mercury and Venus perform 432000 sidereal revolutions as
well as the Śīghrōcchas of Mars, Jupiter and Saturn,
whereas the Moon performs 57753336 revolutions' (It may
be recalled here that the Mercury and Venus are oscillating
about the mean position of the Sun; also it will be noticed
that the Sun playing the part of the Śīghrōcchas in the
case of Mars, Jupiter and Saturn, their Śīghrōcchas are
also deemed as making the same number of revolutions as
the Sun. In as much as the Śīghrōcchas in the case of
Mercury and Venus are looked upon as different from the
planets, so in the case of Mars, Jupiter and Saturn also the
Sighrocchas are taken as different Divine entities though
coinciding with the Sun in position). However smaller
periods could be conceived as integral L.C.M's of the side-
real revolutions of the Sun and the Moon, but a presump-
tion sponsored by a sense of orderliness in the Cosmos,
that the planets should all have been started from the Zero
point of the Zodiac, made the integral L.C.M. to be of
such a dimension as 4320000 solar years in which period
the other planets also would have made an integral num-
ber of sidereal revolutions. Here in this point the tradi-
tional Hindu astronomers place their faith in the Āgama,
which said that the planets were all started at the Zero-
point of the Zodiac in the beginning of the yuga and were
ordained to return to the same point at the close of the
yuga. Even a rational astronomer like Bhāskara, appa-
rently placing faith in the Āgama, while adducing a proof
in the name of Bhagaṇōpapatti, states that after obtaining
the mean daily motions of the planets, calculates them
for the period of a Kalpa taking it on trust that the planets
were started at the initial point of the Zodiac in the begin-

129
ning of the Kalpa, A modern astronomer, however, ques-
tions the assumption that the planets were all started at the
first point of the Zodiac, and even though they might all
have been in conjunction at that point in some remote
past, whether it was the initial point of the reported Kalpa.
Proceeding on the basis of the reported initial conjunction
of all the planets at the first point of the Zodiac, and calcu-
lating the number of days that have elapsed from the begin-
ning of the yuga, the mean positions of the Sun and the
Moon were computed. Noticing that these mean positions
did not exactly accord with the true observed positions, the
ancient astronomers tabulated the differences between those
mean and true positions. These differences were found to
be zero at two diametrically opposite points, and maximum
roughly at two points differing by a quadrant from them.
To account for these differences, the thought that occured
to their minds was that probably the Sun and Moon did
not move in a circle whose centre coincided with that of
the Earth but were moving in an eccentric circle i.e. a
circle whose centre is at some other point than the Earth's
centre. This surmise could be made because unequal
motion was accountable only on varying distance from the
Earth's centre and a celestial body appearing to move
fastest must be nearest whereas the same appearing to
move slowest must be farthest. Thus in the first place
seeing no reason for non-circular motions and also expec-
ting the celestial bodies to move only in circles, for, a
circular motion appealed to them as the most ideal motion,
the ancient astronomers later postulated an eccentric
circular motion with respect to the Sun and the Moon.
This postulation appeared to give good results as seen
below.
17

130
Fig. 9
Eccentric circle theory.—Let E₁ be the earth's centre;
let M₁PA₁ be the circular orbit in which the planet (here
the Sun or the Moon) moves with a uniform motion. This
planet is termed the Madhyagraha or the mean planet.
Let M₂A₂P₂ be the actual orbit of the planet whose centre
E₂ is removed a little away from E₁. Since the centre E₂
is moved in a vertical direction away from E₁, every point
of the eccentric circle (E₂) will be vertically over the
corresponding point of the mean circle. Thus M₂ will be
the position of the actual planet where M₂ is vertically
above M₁ the mean planet. Join E₁M₂ to cut the mean
circle in P. Since A₂ is the position of the actual planet
farthest from E₁ the Earth's centre, the planet should have

131

the slowest motion there. So this point A₂ is termed
Mandoccha, Manda because it is point where the planet is
slowest and Uccha because it is the highest or the farthest
point from the Earth’s centre. Corresponding to this
Mandoccha in the eccentric circle A₁ is termed the
Uccha in the mean circle. Also p the point where E₁M₂
the line joining the Earth’s centre to the actual planet
and called the Mandakarna, cuts the mean orbit is taken
to be the position where the apparent planet is situ-
ated. Thus ‘p’ is seen to be deflected from the mean
planet M₁ towards the Mandoccha on which account
the Mandoccha is considered to be attracting the planet
“ उच्चोद्याःकर्षको भवति ” as Bhāskara puts it. The angle
A₂Ê₂M₂ is spoken of as the Manda-Kendra or the mean
anomaly and it is equal to Z₂Ê₂M₂ – Z₂Ê₂A₂ = longi-
tude of the planet minus the longitude of the Mandoccha,
where E₁Z₁ and the parallel E₂Z₂ are directions towards
the Zero-Point of the Zodiac. This accounts for the
statement ‘मन्दूच्चेन हीनो ग्रहो मन्दकेन्द्रम्’ (of the verse under
elucidation) i.e. the excess of the longitude of the planet
over that of the Mandoccha is termed Mandakendra.
While M₁ is termed the Madhya-graha in the mean
orbit, M₂ is termed the prativritta-Madhyagraha, and
not spaṣṭagraha as might be deemed, while p is spoken
of as the spaṣṭagraha or the True planet or apparent
position of the planet. The word Prativritta stands for the
eccentric circle. Now M₁P the difference between the
mean and True positions is spoken of as the Mandaphala
which corresponds to

132
generally a small quantity M₂N₂ is taken to be equal to
PN₁. If, however, this approximation is not made, PN₁=
(M₁N₂ × E₁P) / (E₁ M₂) (by the similarity of the triangles E₁ PN₁ and
E₁ M₂ N₂) so that the actual equation of centre is (r / R) H sin
E₂ × (R / K) = (r / R) H Sin E₂ where K = Mandakarna E₁ M₂. As
M₂ moves from A₂ to P₂, the equation of centre as given by
(1) gradually increases from Zero to a maximum r when
E₂ = 90° and decreases from this maximum to Zero when
E₂ = 180°. Thus from what was noticed from the tabulated
differences between the computed mean positions and
observed true positions, the fact that those differences
vanish at A₂ as well as P₂ the diametrically opposite point
of the Mandoccha (not called Sīghroccha, for this word
Sīghroccha will be seen to have altogether a different
connotation) was verified. The maximum

133
contradistinction to the word Sīghra which we shall
shortly deal with.
In modern astronomy the equation of centre is given
approximately to be equal to 2e sin m where ‘m’ stands for
the mandakendra so that r = 2e. It will be shortly seen
from a subsequent table that this formulation of the equa-
tion of centre gives results which closely accord with their
modern values. The true or apparent positions of the
Sun and the Moon could be obtained fairly well from the
above formulation, so that it is stated that चन्द्रसूर्यौ स्फुटौ
स्यातो मान्देनैकेनकर्मणा i.e., the Moon and the Sun could be re-
ctified by the equation of centre alone.” This is quite in
order for, the Sun and the Moon may be taken to be going
round the Earth in ellipses, with the earth in one focus,
the former relatively and the latter directly.
After having formulated the method of rectification in
the case of the Sun and the Moon, the next question was
with respect to the Tārā grahas i.e. Mercury, Venus and
Mars, Jupiter and Saturn. As these are going round the
Sun and the Sun going round the earth relatively, the
process of rectification got complicated. In the first ins-
tance, the ancient astronomers must have tabulated the
differences between the mean computed positions and the
observed true positions. In the case of Mercury and Venus,
the case appealed different from what it was in the case of
the other three planets, for the simple reason that the
mean positions of the former were taken to coincide with
the mean Sun. This meant that for rectification, the
elongation had to be computed and added to or subtracted
from the mean position of the Sun to get the apparent
geocentric positions of Mercury and Venus. The analogy
of the method of the formulation of the Mandaphala is
taken here also by imagining (1) eccentric circular motion
and (2) postulating an Uccha. In the case of the Manda-
phala, the equation was zero when the mean planet coinci-
ded with the Mandoccha. Here the equation is zero when

134
elongation is zero, i.e. when the apparent geocentric posi-
tion of the planet coincides with the Sun, who is taken to
be the mean planet. Naturally therefore the Uccha is
taken to coincide with the Sun the mean planet, when the
planet is in conjunction with the Sun. The maximum
equation was had in the case of the Mandaphala when the
arc between the Uccha and the mean planet was a right
angle. So, here also, the maximum equation i.e. the maxi-
mum elongation should be had when the Uccha is a at right
angle from the Sun. Thus an Uccha was postulated with
the following criteria namely (1) It should be a point
moving in a geocentric circle (2) It should coincide in
direction with the Sun when the planet is in conjunction
with the Sun (3) It should be removed by a right angle
from the Sun when the elongation is maximum (3) It
should be removed from the Sun by 180°, again when the
planet coincides in direction with the Sun (4) It should
have a longitude exceeding that of the Sun by 270 when
again the elongation is a maximum on the other side and
finally (5) It should complete a circle with respect to the
Sun when again the planet coincides in direction with the
Sun.
When such a point was conceived it is clear that this
Uccha is not the same as the planet, as some have mis
construed, because while the planet oscillates about the
Sun by a particular angle (29° in the case of Mercury and
45° in that of Venus) in Uccha completes a circle with
respect to the Sun and further as Hindu Astronomy postu-
lated geocentric motion, the Uccha is a point construed as
going in a geocentric circle. By the above postulation the
synodic period of the Uccha is equal to the period of oscil-
lation of the planet about the Sun. But the latter period
is no other than the synodic period of the planet so that
the synodic periods of the Uccha and the planet coinciding
their sidereal periods should be equal. In other words the
Uccha so conceived is a point other than the planet going
round in a geocentric circle and having a geocentric side-