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

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.

141
equation of centre pertaining to the planet which may be
taken to be equal to 2e as a first approximation where e is
the eccentricity of the elliptic orbit of the planet.
It may be further mentioned here that in Sūrya-
siddhānta, as well as elsewhere in this work, the circum-
ferences are given to vary continuously. This variability
curiously achieves ellipticity in the orbit as may be seen
as follows.
In the case of the Sun, the epicycle has a periphery
of 14° when m = 0 or 180° and of 13 ⅔° when m = 90° or
270 according to Sūryasiddhānta, where m is the Manda-
kendra or mean anomaly. At any arbitrary point, where
Fig. 10

142
the mean anomaly is m the periphery is given to be
14° — (20' H sin m) / R. The corresponding radius will there-
fore be r — (20' / 2π) · (H sin m / R) = r — λ H sin m (say) where r =
14° / 2π. [λ = 20' / 2πR] (Ref. fig. 10) Let E be the earth's centre, A the posi-
tion of the apogee EE₁ = the radius of the epicycle meas-
ured along EA, B any arbitrary position of the mean planet
and b the position of the planet in the epicycle. Here the
radius Bb is not equal to the max. radius equal to EE₁ i.e.
r but equal to r — λ H sin m. Take E₁ as the origin and
E₁A as the y-axis and a perpendicular to E₁A through E₁
namely E₁x as the positive direction of the X-axis. If the
mean anomaly BEA be m, then the coordinates of the true
planet are given by x = BL = H sin m (1), y = E₁L +
Bb = EL — EE₁ + r — λ H sin m = H cos m — r + (r —
λ H sin m = H cos m — λx.
∴ y + λx = H cos m (2) Squaring and adding (1)
and (2) x² + (y + λx)² = H sin²m + H cos²m = R²
∴ x² (1 + λ²) + 2λxy + y² = R² which is an ellipse
with centre E₁
Verses 23, 24, 25. The peripheries of Śīghra epicycles.
The peripheries of the epicycles of the star-planets Mars,
Mercury, Jupiter, Venus and Saturn are respectively
243°-40', 132°, 68°, 258° and 40°. The H sine of the Manda
mean anomaly of Venus being multiplied by 2 and divided
by 343, and the result being subtracted from the periphery
gives the rectified Manda periphery. The H sine of its
Śīghra mean anomaly being multiplied by 5 and divided
by R, and the result being added to the Śīghra periphery
gives the rectified periphery. The smaller of the H sine or
H cosine of the Śīghra anomaly of Mars being multiplied
by 6⅔ and divided by H sin 45° and the result in degrees
being subtracted from or added to as the case may be,

143
the aphelion gives the rectified aphelion. The S'ighra peri-
phery being reduced by the above degrees gives the rectified
S'ighra periphery in case the S'ighra anomaly is 90° < m
< 180 or 270° < m < 360°.
Comm. In the commentary Bhāskara adds that in the
case of Venus the Mandaperiphery of 11° as given is at the
end of even quadrants whereas at the end of odd quadrants
it is 9°, wherefore the enunciated rectification. Similarly
in the case of his S'ighraphala, the periphery of 245° men-
tioned is at the end of even quadrants whereas at the end
of odd quadrants it is 263°, and so the suggested rectifica-
tion. Again in the case of Mars, the aphelion as computed
is the same at the end of all quadrants whereas in the
middle of the quadrants it is to be increased or decreased
by 6⅔° when the anomaly is as stated. Also in the case of
this Mars, the S'ighra periphery mentioned is at the ends of
quadrants. In the middle of the quadrants the periphery
is to be reduced as suggested. In all these interpolations
Bhāskara accepts the Āgama as enunciated by Brahma-
gupta.
We shall deal with the geometrical nature of these
S'ighra peripheries shortly in the appropriate place.
Verse 26. To obtain what are called Bhujaphala and
Kotiphala both in the case of Mandaphala as well as S'ighra-
phala. The H sine and H cosine of the Manda or S'ighra
anomalies multiplied by the respective peripheries and
divided by 360°, or multiplied by r and divided by R gives
the Bhujaphala or Kotiphala where r and R are respectively
the radius of the Manda or S'ighra peripheries and R the
radius of the deferent taken to be 3438′. If the radius
3438′ be respectively multiplied by the Manda or S'ighra
peripheries and divided by 360°, the result will be the H sine
of the maximum Mandaphala or S'ighraphala, known as
Antyaphalajyā in either case.

144
Comm. As per the formulation.
Bhujaphala = (H sin m × c) / 360 = (H sin m × r) / R
Kotiphala = (H cos m × c) / 360 = (H cos m × r) / R
in the case of Mandaphala or Sīghraphala where c = peri-
phery of the Manda or Sīghra periphery, r=Antyaphalajyā
defined above R = 3438' and m stands for the Manda or
Sīghra anomaly. These Bhujaphala and Kotiphala will
be used in their respective contexts.
Verses 27, 28, 29. Calculation of what is known as
Sīghrakarṇa.
(H Cos m ± r)² + H Sin² m = K² (1)
(R ± Kotiphala)² + Bhujaphala² = K² (2)
R² + r² ± 2 R × Kotiphala = K² (3)
R² + r² ± 2 r × H Cos m = K² (4)
The arc of the H Sine of the equation of centre is
called the Mandaphala.
Comm. Ref. fig. 9. From triangle E₁M₁M₂ (E₁M₁ +
M₁N₂)² + M₂N₂² = E₁M₂² = K². But M₁N₂ = Kotiphala
and M₂N₂ = Bhujaphala defined previously so that we have
the second formula for K enunciated above. From the
similarity of the triangles E₁M₁Q₁ and M₁M₂N₂ we have
M₂N₂ / M₁Q₁ = M₁M₂ / E₁M₁ so that M₂N₂ = r / R × M₁Q₁ ; Since M₁Q₁ is
called the Bhuja, the corresponding M₂N₂ in the Antya-
phalajyā triangle is called the Bhujaphala. Similarly
M₁N₂ is called the Kotiphala.
Again (E₁Q₁ + Q₁Q₂)² + M₂Q₂² = E₁M₂² where E₁Q₁ =
H Cos m, Q₁Q₂ = M₁N₂ = r and M₂Q₂ = M₁Q₁ = H Sin m.
From this we have the first formula enunciated.

145
Again expanding (E₁M₁ + M₁N₂)², (E₁M₁ + M₁N₂)²

• M₂N₂² = E₁M₂² we have the third formula; similarly
  expanding (E₁Q₁ + Q₁Q₂)², (E₁Q₁ + Q₁Q₂)² + M₂Q₂² =
  E₁M₂² we have the fourth formula.
  Since the S'ighraphala has been defined to be equal to
  r/K H Sin m, we have had the necessity of knowing the
  value of K. The convention of signs mentioned in the
  formulation in the words 'योगो मृगादावथ कर्कटादौ केन्द्रेऽन्तरम्'
  is due to the fact that cosine is positive in the fourth and
  first Quadrants and that the Kotiphala becomes negative
  in the 2nd and 3rd Quadrants as could be seen by draw-
  ing the figure in those Quadrants.
  Now we shall prove what is most important, namely
  that postulating an entirely different geocentric motion
  how the Hindu Astronomers could formulate the S'ighra-
  phala which accords exactly with the heliocentric theory,
  assuming of course coplanar circular orbits. Let figures
  11 and 12 pertain to the modern heliocentric geometry,
  Fig. 11 Fig. 12
  the former with respect to the Inferior planets Mercury
  and Venus signified by V, and the latter to the superior
  planets Mars, Jupiter and Saturn signified by J. Let fig. 13
  pertain to the Hindu geocentric geometry dealing with
  both the Inferior and Superior planets as well. In the
  heliocentric figures let S = Sun, E = Earth, SA = directon to
  Aswini, the Zero-point of the Zodiac from the Sun, EA' =
  19

146
[चित्र: FIG. 13]
FIG. 13
geocentric direction towards Aswini. Let SV, EV¹, be the
heliocentric and geocentric directions of the S'īghroccha
where V is the actual planet and V¹ an imaginary point.
Draw EJ' ∥ to SJ. Let the radius of the inner and outer
heliocentric circles be respectively r and R. Let K be the
radius vector to the planet in both the figures.
Let in fig. 13, E₁=Earth's centre, E₂=the centre of
the eccentric circle which we shall presently show to be
coinciding with the Sun's position. Let M₁, M₂ represent
the mean planets in the deferent and the eccentric known as
Kaksha-Vrittīya Madhyagraba and prati-Vrittīya Madhya-
graha. Let A₁, A₂ be the S'īghrocchas in the deferent and
the eccentric. Let E₁E₂ = r known as Antyaphalajyā and
R the radius of both the deferent and the eccentric. Let K
be the radius vector to the planet known as S'īghra-Karṇa.

147
We shall prove that m the Sīghra anomaly of the Hindu
figure will be the same as 'm' as marked in the heliocentric
figures. Sīghra anomaly is defined as longitude of Sīgh-
roccha - longitude of the planet = a Ê₁ A₁ - a E₁ M₁ = a¹ Ê₂ A₂ -
a¹ Ê₂ M₂ = m(fig. 13.) In the heliocentric fig. 11, since A¹EV¹
is the longitude of the Sīghroccha, and A¹ Ê S the longitude
of the Sun treated as the Madhyagraha of the Inferior
planet A¹ÊV¹ - A¹ÊS = V¹ÊS = V Ŝ S¹ = m = the Sīghra ano-
maly. In fig 12, m = S¹ŜJ = SÊJ = A¹ÊS - A¹EJ¹ = A¹ES -
A Ŝ J = Longitude of the Sun treated as the Sīghroccha
of the Superior planet minus longitude of the heliocentric
planet known as Mandasphutagraha or the planet rectified
for the Mandaphala or equation of centre = Sīghra anomaly
In the case of the Inferior planet the heliocentric direction
of the planet is equal to the Sīghroccha. Now consider the
triangles ESV, JSE, E₁M₁M₂ of the three figures. Evidently
E Ŝ V = J Ŝ E = E₁ M̂₁ M₂ = 180 - m = Supplement of
Sīghra anomaly. If, further it is shown and (it will be
shown subsequently) SV / SE = SE / SJ = M₁M₂ / E₁M₁ the similarity
of the triangle E₁M₁M geocentric figure separately with
ESV and JSV will have been established. Taking this
similarity to have been established, M₁ Ê M₂ known as
Sīghraphala will be equal to SÊV in fig. 11 and SĴE in
fig. 12. In fig. 13, M₂ the prativritta Madhagraha is also
known as the pāramārthikagraha or the actual planet where
as p its geocentric position on the Kakshāmandala is taken
to be the true planet or apparent position of the planet. In
figures 11 and 12, EV and EJ are the directions to the
true planets V and J so that the angles between the Sphuta-
graha and the Madhyagraha (ie the Manda Sphutagraha =

148
A¹ÊV - A¹ES = SÊV (in fig 11) and = A¹ÊJ - AŜJ = S Ĵ V̂ᴱ =
Sighraphala. Once the similarity of the triangles fig. 12 is
established, the equality of the Sighraphala will be establi-
shed. Also due to the similarity mentioned above the for-
mulae for K as given in Hindu Astronomy should also
accord with that in the heliocentric figures. In fact in the
heliocentric figures K² = R² + r² + 2Rr cosm = R² + r² +
2RH cosm which is indentical with the four formulae given
before as per verses 27, 28, 29. It will be seen that the
epicycle (M₁) with radius M₁M₂ will be identical with the
inner circles in the heliocentric circles, whereas the
Kakshamandal (E₁) with radius R will be identical with the
outer circles of the heliocentric figures.
Before we proceed further, we shall annex the table
wherein the ratio r/R as given in Hindu Astronomy will be
seen to accord with that in modern astronomy.

149
Formula for Sīghraphala from the heliocentric figures
In fig. 11, r / K = sin SÊV / sin EŜV so that sin SÊV =
r / K × sin m
In fig. 12 r / K = sin SĴE / sin EŜJ so that sin SĴE = r / K sin m
Both these accord with the Hindu formula.
It will be interesting to point out here that in fig 11,
keeping the earth constant and supposing the Sun S to go
in a circle with centre E and radius ES, the orbit of the
Inferior planet V will play the part of the epicycle of
Hindu Astronomy. Thus in the case of the Inferior planets,
the epicyclic theory is only a different version of the helio-
centric theory. In the case of the Superior planets, how-
ever, (fig. 12) cut off EJ¹=SJ along EJ" parallel to SJ; then
J¹J will be parallel to ES just as M₁M₂ is parallel to E₁E₂ in
fig. 13. Then the circle with E as centre and EJ¹ as radius
corresponds to the deferent of fig 13, whereas the circle (J¹)
with centre J¹ and radius J¹J corresponds to the epicycle.
The circle with S as centre and radius SJ corresponds to
the eccentric.
Verse 30. The equation of centre pertaining to the
Sun and the Moon using a simpler table of H sines where
the radius = 120 units. The H sines of the mean anomaly
as found from the simpler H sine table where radius = 120,
multiplied by 20, and divided by 1103 and 477 respectively
gives the equation of centre of the Sun and the Moon in
degrees.
Comm. The maximum equation of centre with respect
to the Sun is 2°-10'-31". Then the argument is "If by the
H sine of the anomaly equal to the radius 120, we have the
above max. equation what shall we have for H sin m?".

160
The answer is (H sin m × 2°-10'-31") / 120 = (2 21/120°) / 120 × H sin m
very approximately = 261 / 14400 H sin m = 20 / 1103 H sin m
Similarly in the case of the Moon, the maximum equation
of centre is 5°-2'-8". By the same argument as above we
have (H sin m × 1133) / (225 × 120) = (H sin m × 20) / (54000 / 1133) = (H sin m × 20) / 477
Verse 31. Rectification of the mean daily motion of
the Sun and the Moon.
The H cosine of the mean anomaly divided by 54 in
the case of the Sun and in the case of the Moon multiplied
by 4 and divided by 7 gives the increment or decrement in
the respective mean motions according as 90 < m < 270 or
270 < m < 360 + 90.
Comm. We have Equation of centre = r / R H sin m =
E (say) so that differentiating δE = r / R H cosm δm / R . But
r / R H cosm is called kotiphala and δm is called Kendra
gati so that δE = Kotiphala × Kendragati / R . Since Koti-
phala is negative when 90 < m < 270 δE is negative but in
Hindu Astronomy we measure the Kendra not from perigee
as in modern astronomy but from aphelion so that the equ-
ation of centre is strictly - r / R H sin m if sign is also
taken into consideration. Hence δE must be + ve. Since
M + E = S where M is the mean planet, E the equation of
centre and S the true planet δS=δm+δE so that the true
motion is equal to the mean motion plus δE. As δE is +ve
when 90 < m < 270 as mentioned above we have to add
this to the mean motion to get the true motion. This δE

151
is called gatiphala or what is to be added to the mean
motion to give the true motion. Incidentally we have
commented on the contents of verse 37.
Verse 32. The Śīghra phala with respect to the Star-
planets.
(H sin m × r) / R being multiplied by the radius R or
the product of H sin m and r being divided by K the arc of
the result gives the Śīghraphala.
Comm. In the verse it is mentioned that the product
of the Bhujaphala and the radius is divided by K so that the
formula for the Bhujaphala being (H sin m × r) / R the Śīghra-
phalajyā will be (H sin m × r) / K which is stated in the alter-
native. We have already derived this formula before where
we got H sin E₂ = (H sin m × r) / K. The arc of this will be
E₂ i.e. the Śīghra-phala, (E₂ because we take E₁ as the
Mandaphala).
Verses 33 and 34. An alternative formulation of
Śīghra-phala.
H sin m being multiplied by R and divided by K
and the difference between the arc of the result and
H sin m will be the Śīghra-phala. Here H sin m belongs
to the eccentric. The arc of the maximum Śīghra-phalajyā
added to or subtracted from 90° will give respectively the
Quadrants and the H sine will have to be taken of the
elapsed Kendra or its Koti according as the Quadrant is odd
or even.
Comm. Ref. fig. 13. From the similarity of the tri-
angles E₁ M₂ N and E₁ PM, PM / M₂N = E₁P / E₁M₂ = R / K ∴ PM
= (R / K) × M₂N