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

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

152
But PM is the H sine of m¹ where m¹ is called Sphuta-
kendra (m is called the madhya-Kendra). Hence
H sin m¹ = R/K × M₂N = R/K × Bhujajyā (in the eccentric)
∴ m¹ = H Sin⁻¹ (R/K × Bhujajyā) = PA₁
∴ M₁P = Sīghraphala = M₁ A₁ — PA₁ = Kakshya-
mandala Bahu minus the chāpa m¹ — Here a clear under-
standing of the word Bāhu or what is the same Bhuja should
be had. The arc pertaining to the angle m in the eccentric
is known as the Bāhu in the eccentric and that to the
angle m in the deferent as the Bāhu in the deferent. When
0<m<90, m is itself spoken of as Bāhu; when 90<m<180,
180—m is spoken of as the Bāhu; when 180<m<270,
m — 180 is the Bāhu and when 270<m<360, 360—m is
the Bāhu. Thus the Bāhu is that angle whose H sine will
be H sin m numerically. When it is said in the verse
‘त्रिज्याहता कर्णहृता भुजज्या’ the word Bhuja is the arc M₂A₂ as is
mentioned in the same verse ‘ज्ञेयोऽत्र बाहुः प्रतिमण्डलस्य’. The
second part of the verse divides the eccentric circle into
such quadrants that in them S'īghraphala increases from
Zero to a maximum, decreases again from a maximum to
zero, again increases from zero to a maximum and again
decreases from a maximum to zero. Thus at A₂ of the
eccentric the S'īghraphala is zero; at a₁, it is a maximum
namely the arc b₁c₁ where H sin b₁c₁ = a₁c₁ = r; Thus in
the course of A₂a₁ the arc of the eccentric the S'īghraphala
gradually increases from zero to a maximum and in the
course of a₁a₂ the S'īghraphala decreases from a max to zero.
Again from a₂ to a₃ it increases from zero to a max and
from a₃ to A₂ it decreases from the maximum to zero. Thus
the quadrants in the case of S'īghrapbala are not of 90° but
arcs A₂a₁, a₁a₂, a₂a₃ and a₃A₂ which are respectively of mag-
nitude 90° + H Sin⁻¹r, 90 — H Sin⁻¹r, 90 — H Sin⁻¹r and
90+H Sin⁻¹r. In the case of Mandaphala also, the quadrants
should have been of the same magnitude if the so-called
Karnānupāta has been postulated i.e. reducing the Manda-

153
phala from the extremity of the Karna to the extremity of
the radius in the deferent; but as this Karnānupāta is not
adopted, the difference being negligible the quadrants are
all of equal magnitude i.e. each of 90°.
In the course of the commentary of this verse Bhāskara
mentions that for Mercury, as the maximum S'īghraphala is
21°-31'-43", the quadrants are of magnitude 3-21-31-43,
2-8-28-17, 2-8-28-17 and 3-21-31-43 respectively.
Also in the commentary Bhāskara adds that a₁ which
is the point of intersectiou of the eccentric with the hori-
zontal diameter E₁b₁ the S'īghraphala is maximum and
that at that point the mean motion is itself the true motion
“कक्षामध्यगतिर्यम्रेखाप्रतिवृत्तसम्पाते, मध्यैव गतिः स्पष्टा परं फल तत्र
खेटस्य”. That the S'īghraphala at a₁ and a₃ is maximum is
clear from the figure 13, where it is equal to the arcs b₁c₁ and
b₂c₂ whose H sine is equal to r. To prove that the mean
motion is itself the true motion, we have the equation
M₂ + E₂ = S where M₂ is the mean planet here or the
Mandasphutagraha or planet rectified for the equation of
centre, (by the equation M₁ + E₁ = M₂, M₁ being the original
mean planet and E₁ the equation of centre) so that δM₂ + δE₂
= δS where δM₂ is the mean motion here, δS the true
motion and δM₂ is the variation in the S'īghraphala; but at
a₁ and a₃ E₂ the S'īghraphala being maximum δE₂ is zero.
Hence δM₂ = δS which means that the mean motion is
itself the true motion.
Verse 34. Latter half, 35 and 36 former half.
The mean planet rectified for the equation of centre or
Manda-phala is called Mandasphuta. Then subtracting the
longitude of the Mandasphuta from that of the respective
S'īghroccha, the result will be the S'īghra anomaly from
which the S'īghra-phala is to be obtained. Rectifying the
Mandasphuta for this second equation namely S'īghraphala,
again obtaining therefrom the equation of centre effecting
20

154
this in the original mean planet and again correcting for
Sīghraphala and repeating the process till a constant value
is obtained, the true planet is had with respect to the star-
planets other than Mars. But with respect to Mars, let
first the mean planet be corrected for half of the equation
of centre. Then make half of the correction of Sīghraphala.
Take the resulting planet to be the mean planet and again
finding the equation of centre, make this whole correction
in the original mean planet. Again taking the resulting
planet to be the Mandasphuta effect the entire Sīghraphala.
Then we have the true planet.
Comm. The Suryasiddhānta stipulates the same kind
of correction in the case of all the star-planets. "शैघ्र्यं
मान्दं पुनः मान्दं शैघ्र्यं चेति चातुर्विधं, कर्तव्यं हि कुजादीनां स्फुटत्वे कर्म
सूरिभिः। शैघ्र्यं फलार्धं प्रथमं ततो मन्दार्धमेव च, पश्चान्मान्दफलं सर्वं तद्वत्
शीघ्रफलं ग्रहे ॥" i.e. In the first place half of the Sīghra-
phala is to be effected in the mean planet; taking that to
be the mean planet and computing the equation of centre
half of it is administered; taking the resulting to be the
mean planet and computing the equation of centre, the
entire equation of centre is now to be administered in the
original mean planet; taking the result to be the Manda-
sphutagraha, and computing the Sīghraphala, it is to be
administered in full in the Mandasphuta. Then we have
the true planet.
In modern astronomy, the equation of centre is first
done and the result will be the planet in its heliocentric
elliptic orbit. To reduce it to the geocentric position, the
second correction is made which corresponds to the Hindu
Sīghraphala. Thus the two corrections administered suc-
cessively gives the apparent or true geocentric planet.
Though the mutual relationship as conceived between
the equation of centre and the Sīghraphala in Hindu Astro-
nomy is deemed irrational by modern interpreters of Hindu
Astronomy, there is some rationale in the process as ex-

155
plained by this author in his work ‘A critical study of
Ancient Hindu Astronomy’ (published by the Karnatak
University) page 98.
Verse. Cited from Golādhyāya.
The equation of centre is to be applied to the mean
planet to obtain the centre of the Śīghraepicycle; then to
obtain the true position the Śīghraphala is to applied to the
Mandasphutagraha; hence the two equations are mutually
related so that the true position is obtained after repeated
application of the two equations.
Comm. Explained above.
Verse 36 latter half and 37. The true daily motion
of the planet is the excess of the longitude of the true planet
of the next day over that of the true planet of the previous
day.
The Kotiphala being multiplied by the daily motion of
the Manda mean anomaly and divided by the radius, and
the result being added to or subtracted from the mean
motion, gives what is called Mandasphutagati.
Comm. Already explained before. If M₁, M₂ and S be
the mean planet, Mandasphutagraha, and the true planet
respectively, and if E₁ and E₂ be respectively the two equa-
tions, then
M₁ + E₁ = M₂, M₂ + E₂ = S so that
δM₁ + δE₁ = δM₂ and δM₂ + SE₂ = δS
Here δM₁ = mean daily motion, δE₁ = daily variation
in the Mandaphala, δM₂ = daily motion of the Mandasphuta
graha or what is the same Mandasphutagati, δE₂ = daily
variation in the second equation and δS = True daily
motion.

156
Lallāchārya formulates the Mandasphutagati in a
different manner which is equally correct (Ref. verse 45
Spaṣtādhikāra Śiṣyadhī Vriddhida “त्रिज्याहता ग्रहगतिः
मृदुकर्णभक्ता मन्दस्फुटा भवति ” i. e. (δm × R) / K = Mandasphuta-
gati where δm = Madhyagati or mean motion and K is the
Manda Karṇa equal to √(R² + r² ± 2 R r cos m)
∴ Mandasphutagati =
(δm × R) / √(1 ± 2 (r / R) cos m + r² / R²) = δm ( 1 ± (2r / R) cos m )⁻¹⁄₂
= δm ( 1 ∓ (r / R) cos m ) neglecting the smaller term
− r² / (2R²) within brackets
= δm ∓ (r H cos m δm) / R² = δm ∓ (r H cos m / R) × (δm / R)
= δm ∓ (Kotiphala × Mandakendragati) / — as given by
Bhāskara.
Verse 38. In the case of the Moon, obtaining the true
Moon for a particular moment and his daily motion for the
day, the ending moment of the tithi near at hand is to be
computed with that daily motion, and the method of suc-
cessive approximations is to be used to rectify the ending
moment. In the case of the ending moment of the tithi be-
ing sufficiently far away, then it does not matter even if
the above daily motion is applied to get the approximate
ending moment. In as much as the Moon’s daily motion is
great and varies from moment to moment the motion
at the moment is to be used.
Comm. Strictly speaking the ending moment of every
tithi is to be computed by the method of successive approxi-
mation. That is why in the computation of eclipses,

157
Moon's hourly motion is given in modern almanacs. Since
it is very cumbrous to use the method of successive approxi-
mation to determine the ending moment of every tithi, the
Hindu almanac-makers generally compute the ending
moment of a tithi using the daily motion of the moon
computed for the moment of Sun-rise of the day. Only for
ritual purposes, the method of successive approximation is
used and also in the computation of eclipses.
Verse 39. Computation of the Sīghragatiphala.
δl − [H sin (90−E₂) × δm] / K = δS where δl is the daily
motion of the Sīghrōcca, E₂ = Sīghraphala, δm = daily
motion in the Sīghra mean anomaly, K = Sīghrakarṇa,
and δS the true motion of the planet. If δS is negative the
planet is retrograde.
Comm. This is mathematically an important verse,
and the proof given by Bhāskara really reflects his
genius. Before we attend to his proof, we shall give this a
modern treatment. (Ref. figs. 14, 15). SA and EA' are the
Fig. 14
heliocentric and geocentric directions to Aswini the Hindu
Zero-point of the Zodiac; S = Sun, E = Earth; V = In-
ferior planet Venus or Mercury; J = Superior planet; E =
Sīghra-phala, K = Sīghra Karṇa; M = Sīghra anomaly.

188
[चित्र: Fig. 15]
Fig. 15
True motion of the planets = δ (A'EV) or δ (A'EJ)
But δ (A'EV) = δ (A'EV'–n̂) and δ (A'EJ) = δ (A'ES–n̂)
In the case of the Inferior planet δ (A'EV') = δ (ASV)
= Sīghrōcchagati and δn is Sphutakendragati where n is
called Sphutakendra, m being called Madhyakendra. In
the case of the Superior planet δ (A'ES) = S'īghrōcchagati
because the Sun plays the part of S'īghrōccha in the case of
a Superior planet and δn = Sphutakendragati as before.
Hence in both the cases, Sphutagati = S'īghragati – Sphuta-
kendragati. We have now to find Sphutakendragati to
obtain Sphutagati, as Bhāskara remarks rightly “महामति-
द्भिः केन्द्रगतिरेव स्पष्टीकृता”. In other words we have to find
δn. From the figures K cosn – R cos m = r (i) Differentia-
ting this we have – K sin n δn + cos n δK + R sin m δm
= 0 (ii). But K² = R² + r² + 2Rr cos m so that 2δK × K
= – 2Rr sin mδ m (3) Eliminating δK between (2) and
(3) – K sin nδ n – (Rr sin mδ m × cos n) / K + R sin mδm = 0
i.e. K sin nδn = R sin mδm ( 1 – r / K cos n )
= (R sin mδm / K) (K – r cos n)

159
But K — r cos n = R cos E.
∴ δn = (R sin mδm × R cos E) / (K² sin n) But R sin m = K sin n
Cancelling δn = (R cos Eδm) / K = (H cos E δm) / K as given by
Bhāskara. In the formula H sin E = (r H sin m) / K, Bhāskara
perceived the variability of both H sin m and K on
the right hand side and he exclaims “ न हि केन्द्रगतिजमेव
फलयोरन्तरं स्यात्, किन्त्वन्यथाऽपि अद्यतनभुजफलश्वस्तनभुजफलान्तरे
त्रिज्यागुणे अद्यतनकर्णहृते यादृशं फलं न तादृशं श्वस्तनकर्णहृते, स्वल्पा-
न्तरेऽपि कर्णे भाज्यस्य बहुत्वात् बह्वन्तरं स्यादित्येतदानयनं हित्वा अन्यत्
महामतिमद्भिः कल्पितम्, तद्यथा केन्द्रगतिरेव स्पष्टीकृता ” i.e. “ The
variation in the Sīghraphala is not entirely constituted
by the variation in m but also by that in K......So
leaving the method of seeking δE through the formula
H sin E = (r H sin m) / K the great intelligent astronomers
used the formula.
Sphutabhukti = Sīghra Bhukti — Sphuta Kendra-
bhukti (Bhukti means gati) wherein it was'sought to obtain
the variation in Sphutakendra i.e. n̂ in the figures.
We have given a proof of Bhāskara’s formula, which
circumvented finding Sīghragatiphala but which sought
directly Sphutabhukti, by using the modern heliocentric
figures. We shall now see how Bhāskara could deal with
such a tough problem. Refer fig 16. Let P₁, P₂ be two
positions of the planet on two consecutive days relative to
the Sīghra A₂, so that P₁E₁P₂ is the Sphuta Kendragati
spoken of. It will be noted that it is not Sphutagati be-
cause P₁, P₂ are positions of the planet relative to A, which
is itself moving (as rightly remarked by Bhāskara). It is
Sphuta Kendragati because A₂E₁P₁ and A₂E₁P₂ are the

160
Kendras on two consecutive
days whereas Madhyakendragati
^
is P₁E₂P₂. Also, we have the
equation.
Sīghra — Sphutagraha =
Sphutakendra so that Sīghra-
gati — Sphutagati = Sphuta-
kendragati. Hence Sphutagati
= Sīghragati — Sphutakendra-
Fig. 16 gati. So we have now to seek
the value of P₁Ê₂P₂. Let P₁a stand for the Sīghra-
phala of the first day which is equal to e f, f being the true
planet of the first day. E₂ d will be parallel to E₁ f because
P₁ e being parallel to E₁ E₂ and e f being equal to P₁ d, d f
11 P₁ e (parallels to E₁ E₂ cut off equal arcs on the two
circles). This may be seen also as follows. Since P₁ d is
taken to be equal to e f, e being the mean planet and f the
true on the first day e Ê₁ f = P₁ Ê₂ d. But E₁ e 11 E₂ P₁
∴ e E₁ f = E₁ P̂₁ E₂ ∴ P₁ Ê₂ d = E₁ P̂₁ E₂ and alter-
nate angles being equal E₁ P₁ 11 E₂d. P₁ a is the H sine
of P₁ d where P₂ b is the H sine P₂ d. Looking upon P₁P₂
as an increment in P₁ d i.e. looking upon the Kendragati
P₁ Ê₂ P₂ as an increment in Sīghraphala, Bhāskara uses
the method of Bhogyakhanda sphuti Karana to obtain the
Sphutakendragati. From the figure.
P₂c = P₂b — P₁a = H sin (G + δm) — H sin E
H sin E H cos δm + H cos E H sin δm
= ————————————————————————————————————— — H sin E
R