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

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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-

135
real period equal to the heliocentric sidereal period which
again means that the geocentric longitude of the Uccha is
the heliocentric longitude of the Planet. Thus the radius
vector to the planet from the Sun is parallel to the geocen-
tric radius vector of the Uccha. This accounts as to how
the heliocentric sidereal periods of Mercury and Venus
could be found under a geocentric concept and also as to
how the heliocentric planets are themselves spoken of
as their respective Ucchas, while their mean planet is the
same as the Sun. On this count it was mentioned by the
Hindu Astronomers ज्ञशुक्रयोः ग्रहः सूर्यो भवेत् तौ शीघ्रनाकौ ie.
The mean Planet of Mercury and Venus is the Sun himself
where as they are themselves spoken of as their Ucchas.
The phrase 'they are themselves' in the above statement
is significant as it connotes that the word 'they' stands for
the heliocentric planets, though it was uot stated in so
many words. Shortly we shall see also that the centre of
the eccentric circle coincides with the centre of the Sun
also and applying Bhāskara's statement 'यस्मिन्वृत्ते भ्रमति
खचरो नाऽस्य मध्यं कुमध्ये' i.e. the centre of the circle in
which the planet moves does not coincide with that of
the Earth', the Sun was, though unwittingly taken as the
centre of the Planetary motion. Thus we see how even
the geocentric postulation also could help computation of
the Planetary positions, the mathematics behind revealing
heliocentric motion. What Copernicus achieved was that
he identified that the point about which the planets revol-
ved which was construed by the Hindu astronomers as an
imaginary point not coinciding with the earth's centre,
was no other than the Sun himself.
In the case of Mercury and Venus the so-called
Sīghra-phala came to be discovered first and we shall pre-
sently see why their elongation was called Sīghra-phala
and how the Uccha postulated as above came to be termed
Sīghroccha. Since initially the equation was to be zero,
when the Planet and the Uccha coincided with the Sun

136
and then the elongation has to increase as the Uccha
gained over the Sun, the initial conjunction was the
modern Superior conjunction. The other position of the
Uccha when again the elongation i'e. the equation is Zero
must be therefore the Inferior conjunction. Also at the
motion of Superior conjunction, the planet must be having
the maximum daily motion, as it is clear from a heliocen-
tric figure that at that point the relative motion of the
planet with respect to a geocentric observer is the sum of
the velocities of the planet and the earth. Hence this
Uccha is spoken of as the Śīghroccha also because the
Uccha being a geocentrically moving point having helio-
centric angular motion, its velocity is always greater than
that of its planet namely the Sun. The word Uccha is
applied because at the Superior conjunction the planet is
farthest or highest from the earth. The excess of the
longitude of this Uccha over the longitude of the planet ie.
the Sun is known as the Śīghrakendra or anomaly as it is
said in the verse under commentary ‘चलोच्चं ग्रहोनं भवेत्
शीघ्रकेन्द्रम्’. Thus in the case of Mercury and Venus, the
Śīghraphala came to be discovered first. This being dis-
covered, formulated as will be shortly shown, and applied
to the mean Sun as the planet, still it was found that there
was a difference between the computed position and the
observed position. Such differences were tabulated. By
analogy from the case of the Sun and the Moon, it was
thought that there should be also a Mandoccha here also,
so that the point indicated by the position of the mean
planet after being corrected by the equation where the
above tabulated difference was zero, was identified as the
Mandoccha. From the H Sine of the maximum difference
taken as the radius of the Manda epicycle, its circumference
was then computed.
In the case of the superior planets, we have already
said that the geocentric sidereal periods accord with their
heliocentric ones. Calculating the mean position of the
planet and finding its difference from the observed true

137
position, such differences were tabulated. It was discovered
that these difference almost vanished when the planet was
in conjunction with the Sun and attained a maximum
when the elongation was nearly a right angle from an
analogy from the Manda-Karma i.e. process of obtaining
the Manda-phala. Since the differences attained their maxi-
mum value when the elongations were nearly a right angle
it could be seen that the Sun played the part of the Uccha
in this case. As the Sun has a quicker motion than the
planet and also as at conjunction the planet has the quick-
est motion relative to the Earth while it is farthest from
the Earth the Uccha ie the Sun here, is termed Śīghroccha.
The excess of the longitude of the Sun over that of the
planet is termed accordingly the Śīghra-kendra and the
Śīghra-phala the equation was formulated as will be shown.
Applying this Śīghra-phala to the mean position, the diffe-
rences still found between the position so obtained and the
observed true position were tabulated. The point indicated
by the above position where the difference was found to be
zero, was identified as the Mandoccha, and through the
maximum difference, the Mandaparidhi was formulated.
In the above discourse, we have tried to give an
account of how the originators of Hindu Astronomy
could give us a workable system. We never assumed that
an Āgama gave us the numbers of sidereal revolutions of
the planets or the measures of the epicycles Manda or
Śīghra. But in the explanation given by Bhāskara under
Bhagaṇōpapatti, one will notice that when Bhāskara gave
the proof of the Moon's sidereal revolutions, he said that
having got the true positions of the Moon on two conse-
cutive days, the mean positions were computed from the
true by an inverse process of applying the equation of
centre, and getting the mean daily motion of the Moon
from those mean positions, the number of sidereal revolu-
tions in a Kalpa were obtained. Here the Upapatti or the
proof adduced by Bhāskara was not a proof but only a
verification in as much as (1) he assumed the formulation
18

138
of the Mandaphala from the Āgama without pointing out
how it was formulated and (2) he assumed the period of a
Kalpa and that at the beginning of the Kalpa the planets
were all in conjunction at the first point of Aries. Similarly
in the Upapatti adduced by Bhāskara with respect to the
Mandocchas of the planets, he assumed the formulation of
Sīghra-phala on the basis of Āgama without proving how
the concept of Sīghra-phala was arrived at by the founders
of Astronomy and how the difference between the observed
apparent positions and the computed mean positions, was
resolved into two equations the Mandaphala and the Sīghra-
phala. In the proof adduced with respect to the Sīghroccha
of Mercury and Venus also Bhāskara did not mention
anything as to how their heliocentric sidereal periods could
be obtained but simply assumed the formulation of the
Mandaphala and Sīghra-phala as already being there on
the basis of Āgama.
We shall now proceed to describe the method of formu-
lation of the Sīghra-phala with respect to the five Tāra-
grahas, star planets namely Mercury, Venus and Mars,
Jupiter, Saturn and show how so different a set of geomet-
ries of the ancients and the moderns the one geocentric
and the other heliocentric could give identical formulation
with respect to Sīghra-phala. Let us consider the case of
Mercury and Venus in the first instance.
Having taken the mean Sun to play the part of the
‘Graha’ in the case of Mercury and Venus and having
formed a concept of Sīghroccha as mentioned before, whose
geocentric period of revolution was determined, without
suspecting it to be the heliocentric sidereal period of the
planet the ancient Hindu astronomers assumed by analogy
from the case of Mandaphala with respect to the Sun and
Moon, that the centre of the circle in which these planets
revolve does not coincide with the centre of the Earth. In
other words, they continued eccentric circle theory here
also so that without suspecting heliocentric revolution

139
their mathematics led to them to make the centre of the
eccentric circle coincide with the Sun himself. On this
basis alone it was given to Copernicus to formulate helio-
centric theory, sponsored by a thought that the Heavenly
Sun could not be deemed as a satellite of the ‘ Mundane ’
Earth.
(Ref. fig. 9). The same figure 9 will also serve the pur-
pose to obtain the Sīghraphala, only M₁ M₂ will be now on
the right hand side of the Sīghrocchas A₁ A₂, for, the latter
will be taken to be in advance of the mean planets Kakshā-
Vrittīya Madhyagraha M₁ (i.e. mean planet of the deferent)
and prati-Vrittīya Madhyagraha M₂ (i.e. mean planet of
the eccentric). The points A₁ add A₂ are themselves called
the Kaksha-Vrittīya Sīghrōccha and prati-Vrittīya Sīgh-
rōccha respectively. As was shown in the case of the
Mandaphala from the eccentric figure 9, M₂ N₂ =
r/R H sin (Kendra) so that PN₁ = r/R H sin (Kendra) × R/K
= r/K H sin (Kendra) = (Antyaphalajya × Sīghrakendrajyā) / Sīghrakarna .
We shall take this for elucidation in the appropriate
context.
Verse 19. Three Rasis each of 30° constitute a quad-
rant, and there are four quadrants in a circle which are res-
pectively odd, even, odd and even. In the odd quadrants
the Kendra covered is itself called Bhuja whereas in the
even ones, the complement thereof is called Bhuja. Also,
the complement of the Bhuja is called the Koti.
Verse 20. R − H sine = Co. H versine and R −
H cosine = H versine and R − Co. H versine = H sine,
R − H versine = H cosine.
Verse 21. Also √(R² − H sine²) = H cosine,
√(R² − H cosine²) = H sine. Similarly √(R² − Krantijya²)
= Dyujya, √(R² − Dyujya²) = Krāntijya; √(R² − Drig-jyā²)

140
= Śaṅku, √(R² — Śaṅku²) = Drig-jyā. In all the cases
cited above, the radins R happens to be the hypotenuse.
Comm. The convention in verse 19 corresponds to
saying in modern trigonometry that sin 90 + θ = cos θ
sin (180 — θ) sin θ, sin 180 + θ = — sin θ, sin 270 — θ
= — cos θ sin 270 + θ = — cos θ, sin 360 — θ = — sin θ.
In Hindu trigonometry the sign is understood and not
explicitly mentioned.
Krāntijyā, Dyujyā, Drig-jyā and Śaṅku, are respecti-
vely H sin δ, H cos δ, H sin Z, H cos Z where δ is decli-
nation and Z the Zenith-distance of a celestial body.
Taking the radius of the celertial equator to be R, the
radius of the diurnal circle of a celestial would be equal
to R × cos δ = H cos δ which is called Dyujyā because it
is the radius of the diurnal circle.
Verse 22. The lengths of the circumferences of the
Manda—epicycles are respectively 13° — 40', 31° — 36',
70°, 38°, 33°, 50,* for the Sun, Moon, Mars, Mercury,
Jupiter, Venus and Saturn.
Comm. Bhāskara has given these measures reportedly
on the basis of Āgama or ancient authority. The peculiarity
of measuring the circumferences in degrees less than 360°,
is due to the idea that these circumstances are measured in
relation to that of the deferent or Kakshā Vritta taken to be
360°. In other words, circumference of the epicycle of a
planet as given above : circumference of the mean orbit ::
x : 360 = radius of the epicycle : radius of the mean orbit
where x is the measure of the circumference of the planet-
ary epicycle. It may be mentioned once again that the
radius of a planetary epicycle is the measure of the greatest

• The printed book of Brahma Sphuta Siddhānta gives in the case of
  Saturn 30° only which might have been the mistake of the scribe (Vide
  verse 36, Spashtadhikāra B. S.). In the place of शून्यरामाः it ought to
  have been शून्यबाणाः.