CHAPTER VI
SPASHTAADHIKAARA.
DETERMINATION OF THE TRUE POSITIONS Of PLANETS,
INTRODUCTION.
The chapter known as Spashtaadhikara is
ITU.
usually^second chapter in every Hindu Astronomical
text.
This Chapter deals with the methods of
oonputing the True positions of the Planets from
their wean positions.
Former writers on the subject
of Hindu Astronomy merely presented the methods of
this Chapter as described in the texts.
They did
not explain how the methods oame to be evolved.
The Author has tried to throw full light on this
side.
The determination of the Equation of centre;
the computation of the geocentric positions of the
planets; of their stationery points in their erblts
and their retrograde notion; and the calculation of
their true dally motions are remarkable feats in
Hindu Astronomy.
A complete mathematical exposition
of these has been set forth.
149
CHAPTER VI
S PASHTAADHIKAARA.
DETLRMINATION OF THE TRUE POSITIONS OF THE PLANETS.
1*
The mean positions of the planets are readily
calculated by a knowledge of their mean motions
and the number of days (oalled Ahargana) that have
elapsed from a particular epoch at which their mean
positions are known*
The determination of the True
positions as seen from the Earth will be called the
rectification of the mean positions.
In the case
of the Sun and the Moon the rectification is arrive*
at froa a single Equation called the Mandaphala
which corresponds to the Equation oi Centre* in
modem astronoay.
The five planets known as
Taaraagrahas namely Mercury, Venus, Mars, Jupiter
and Saturn are rectified by two Equations known as
(a) the Mandaphala (Equation of Centre); and
(b) the Seeghraphala«
We shall prove later that the Seeghraphala is
the correction to reduce the heliocentric positions
150
of the planets to the geocentric
2.
The Hindu Astronomers were not aware oi th •
heliocentric revolution of the planets and of k'eoler'a
laws of planetary motion,
hut they were aware t iai
the point about which the planets revolved did n*t
coincide with the Earth's centre.
says:
In fact Bhaskara
Quotation \^ - (Refer Appendix-A).
"The eentre of the Earth is the centre of the
celestial sphere; the centre of the circle in which
the planet moves does not oolncide with the Fartk'e
centre"•
The Hindu Astronomers were able to obtain
the two oorreotlons by introducing a theory of
epicycles and a theory of eccentrics, both being
virtually the same.
Bhaskara says:
Referring to these theories
Quotation I
- (Refer Appendix-A i.
"The ancient Astronomers Introduced these epicycles
for the sake of calculating the above two corrections".
Note the word 'Kalpjtaani1 which suggests that the
151
epioyolio or tho eccentric theory was only a
postulation in order to compute the Equations.
We hare also to note that the epicycle ased for
the Equation of Centre and that used for the
Seeghraphala, second equation, are of different
radii, even for the sane planet.
3.
The eplcyollc theory: the Equation of Centre
for the Sun and the Moon.
A'
i
A
4y
£
„
V
d
c
152
In Fig. xvv:s,
E
is the Earth *» centre and
AR*B
is the aean orbit {Kakshaaaaadala) of the planet.
The aean planet is supposed to move on the circum­
ference of this circle with uniform velocity.
velocity is the aean motion.
the epicycle, the
This
he circle (B) is
Niohoochavrltta whose radius
varies froa planet to planet.
We conceive of such
a circle at each point of the mean orbit.
the apogee or Mandoccha.
\ is
The position of the
apogee was got by noting the point where the observed
True planet coincided with the computed mean planet
in the case of the Sun and the Moon.
In the case
of the five Taaraagrahas Mercury etc., their aphella
are located according to the methods to be explained
shortly.
The apogee is also the point where the True
notion Is least in the case of the Sun and the 'icon,
The angle a subtended by the arc \n at E is called
the Mandakendra or aean anomaly correspond in# to the
position of the aean planet.
By convention it
153
taken to be the excess of the longitude or the mean
planet over that of the apogee.
The True planet
is supposed to move along the epioyele in the opposite
direction with the same rate of sotion as the mean
planet does along the mean orbit.
Bhaskara gives
the following analogy in this context.
Quotation | 4. - (Refer Appendix-A).
"Here is an example; even as in the case of the
oil-machine, the vertical axis of the machine
appears to aove in a direction opposite to that
in which the bullock moves, just the same way the
True planet moves in a direction opposite to that
in whloh the wean planet moves".
When the mean planet is at its apogee A,
the True planet is at A».
The projection of
a*
on the mean orbit is A itself; and hence the True
planet appears to be at A.
Similarly in the
position C, the True planet is at C* and its
projection on the mean orbit is C.
154
The points
A* and £’ are respectively called the Ucoha and
the Neecha of the planetary orbit.
Bhaskara says:
Quotation IS
At these points,
- (ltefer Appendix-A
"The planet is farthest from the nearest to the
Earth respectively and hence presents the least
and the greatest discs respectively".
In an intermediate position, at R, say, the True
planet is at to in the epicycle where Bto is
parallel to FA.
If B* is the projection of |i on
the mean orbit, B* is the point where the True
planet appears to toe so far as direction Is concerned.
Similarly when the mean planet is at r the True
planet appears to toe at J> *.
Sinoe B* and P’ are
both nearer to A than B and P, the True planet is
nearer to the apogee than the mean planet.
The
apogee is therefore spoken of as drawing the planet
towards it as is remarked in the Suryasiddhaanta,
Quotation
\ \c
- (Refer Appendix-A),
"The planets are attracted East or West as the
155
case nay be so that the true planets are always
nearer to the apogee than the swan".
3.1
The arcs BB' or DP* are defined as the
Mandaphala or Equation of centre.
When n the mean
anomaly lies between 0* and 180s, the True planet is
behind the mean planet and so the Equation of Centre
Is negative.
When 180s ^
m
360s
the True olanet
Is In advance of the mean and hence the Equation of
Centre is positive.
3.2
Bhaskara gives the peripheries of the epicycles
of the Sun and the Moon as 13s-40* and 31s-36* res­
pectively.
Expressing the periphery of a circle as
an angle as above Is peculiar.
Tt means here that
If the periphery of the man orbit is taken to he
360s, the periphery of the epicycle will be as giver*.
In otherwords, the ratio of the peripheries of the
epicycle and the mean orbit is i3s-40‘:360s.
3.3
The method of obtaining the peripheries
of the epicycles required to calculate the Equation®
156
of Centro in the ease of the San and the '?oon can
be explained ae follows.
of the True planet.
Observe the position
Compute the difference between
this position and the aean position calculated.
Find the maximum difference thus obtained.
Taking
the Heine of this maximum difference as the radius,
calculate the corresponding circumference.
That
will be the periphery of the epicycle.
This may be further elucidated as follows*
In Fig.
j lei r and R be the radii of the epicycle
and the mean orbit respectively.
BN and bM are drawn
perpendicular to EA and EB respectively,
as 3438 as usual.
u is taken
From the similar triangles b.MH
and BHE. we have
|j|y * ^
|
.*.
bM* jjjfiN * ~Hsin m
where m is mean anomaly, the angle AEB.
The are BB*
defined as the Equation of Centre, may be taken
to be equal to the length of bM approximately.
Hence Equation of Centre * |y Hsin m.
157
The saximun value of this Is r, since the maximum
value of Bsln ■ « UsIn 90* » H.
This maxlnum value
can he obtained as stated above by observing the
naxiaun divergence between the observed position
of the True planet and the calculated position of
the nean planet.
The aaxiaua Equation of Centre in the ease
of the Sun is thus found to be 2*~il’-SO".
This is r.
Hence the periphery of the epicycle is equal to
m
4.
13|-* as stated by Bhaskara.
Method of eccentric cycles (Pratlvrlttas),
£
-r
V-
A
fc
X
h
v
,4
e v
w
x.
*
l
>
158
S
t-
.'}■ :
In Fig. XXy:,k. E is the Barth's centre and AB'B the
mean orbit of the planet (Kakahaamandala).
Cut off
EE* along EA equal to the radius of the epicycle
namely Bb where B is the position of the mean planet
and h the position of the planet in the epicycle*
Since EE* and Bb are equal and parallel, EBbF*
is a parallelogram.
Hence jBB * E'b.
The True
planet b therefore lies on the oircle with E* as
centre and radius equal to that of the circle (E•.
This circle (E*) is called the Pratlvrltta or the
ecoentrlo circle.
The point B* shere Eb cuts the
mean orbit is the apparent position of the planet.
The calculation of the Equation of Centre namely BB'
proceeds according to the same method as before.
5.
The value of the Equation of Centre of the
Earth, that is, relatively of the Sun according to
modern astronomy is given by
S e 2 sin 2 »
2 e sin ■ ♦ j
approximately,
tvhere e is the eccentricity of the Barth's orbit
equal to .01675.
The maximum Equation of Centre Is
159
therefore 2e radians which will he equal to 115'-17.
Bhaskara's value la equal to 131*-30* which exceeds
the true value by about a quarter of a degree.
5.1
Zn the oaae of the Moon Bhaskara's value fer­
tile Equation of oentre is 302' approximately, whereas
the modern value la 370' approximately,
he shall
see in Chapter XI under "The corrections of Varfliatlon
and Eveetion in Hindu Astronomy", how Bhaskara meicee
amends for the above difference in the case of the
Let us now see to what degree the theory oi
the eccentric circles accords with the modern
elliptic theory.
In the elliptic theory E ia the focus and
the direction of the major axis.
and FA as the initial line.
ca
Take E as the pole
(Refer Eig,
Eb ** r the radius vector of the planet.
). Ptr.
The polar
equation of the ellipse is given by
„
1
r “ i-e cose
t2
aTi^e~cosoT
„ a 2d-.2l
a(l-e cob)
2
*
a(i-e ) (1-e coso )
»
a(i-e 2 ) (i+ e cos 6 + e 2 cos 2
160
) neglect.ng
con.
terns containing higher powers of e than the second.
r a a ♦ ae ooa 0
* a + ae ooa 6
ae
- ae
In the shore working
2
2
2
co* e
- ae
2
2
sin ©
0
{ ' i
is the angle bFA.
In the case of the eooentrio theory, from the
triangle EE«h
E*b2 a Eb2 ♦ EE»2 - 2 Eb x EF» Cos bFA
Putting E*b a a, Eb a r, bEA a 0
(as above)
and EE* a ae (by analogy from the ellipse wherein
the distance between the foous and the centre is ae)
we have
a2 a r2 4 a2 e 2 - 2r ae cose
2
a r
22
2~
+ a e cos ©
2 2 .2
+ a e sine
2
i.e., a2 - a2 e2 sine
r - ae cos©
•
- 2r ae cos
a (r - ae cose
2
2
a (i-e sin©
)2
1
)*
2 . 2
* a {.- 2-|Uk9)
expanding binoaially and neglecting higher powers
of e2
r a a + ae cos 0
2
- S&.
2
(2)
Comparing (l) and (2) we find the difference between
161
the values of £ computed according to the elliptic
theory and the eooentric theory is of the order
aeA .
Since £ is inversely proportional to the angular semidlameter of the disc of the planet or of the Sun or
the Moon, one should be able to measure a quantity
equal to |e
x semi-diameter, in order to distinguish
between the two theories.
In the case of the Sm
this Quantity will be equal to | x ^ x ~ x 30 ; t>0 «
approximately.
This means that the two theories
differ by such a small quantity as would require the
measurement of the angular diameter of the disc
correct to i" to ascertain which is correct and
which is not.
Thus the Hindu eccentric or the epicyclic
theory oould have worked alright so far as the compu­
tation of the Equation of Centre was concerned,
provided the maximum Equation was estimated correctly.
7.
It is interesting to note that
in
the
Suryaslddhaanta, the dimensions of the epicycles
162
used to calculate the Equation of Centre of rx<t
only the Sun and the Moon but also of the five
Taaraagrahas Mercury etc., are given to vary.
We shall show that this variation secures
ellipticity in the orbit.
In the case of the Sun the epicycle has a
periphery of 14® when
a -
0® or 180® and of 13
when a * 90® or 270® according to the Suryasiddhaanta.
At any arbitrary point, where the mean anomaly
is m , the periphery is given to be
14®
20* ilsin ■
R
The corresponding radius will be therefore
r where r *
* r ~
141
> Ilsin m (say)
•\
; K
b.
163
Let E be the Earth's centre, A the position
of the apogee, EE' * the radius of the epicycle at
A measured along EA. B any arbitrary position of the
planet on the naan orbit AB'B and b the position of
the planet in the epicycle.
Here the radius Bb
of the epicycle at B is not equal to the maximum
radius which is equal to EE* l.e., r but will be
equal to r -Xllsin a.
Take E' as the origin, 1-; *a
as the Y-axis and a perpendicular to E'A through E*
namely E*X as the positive direction of the -X-axis.
If the mean anomaly BEA be a, then the coordinates
of the True planet b are given by
x = bK = BL = II sin a
'1
y « E'K e EL ♦ LK - EE* * Hoos a + r - ' Hsin » - r
y « Hcos a -Xx
•
y ♦ Xx » llcos a
(3)
Squaring and adding (l) and (3)
x2 ♦ (y*Xx)2 * Hsin2 a + Hcos2 m * li2
x2 (1+ \2) ♦ 2 Xxy+y2 « R2
Thus the locus of b is an ellipse with E• as Centre.
164
’-I
2)
The above work shows that the data given
by the Suryaslddhaanta are equivalent to the result
that the orbit is an ellipse for any planet, since
the epicycle for every planet is given to vary in
dinensions as above.
8.
The Second Equation SEEGMUAPilALA.
rswpvMs.
PL If*? MnmHp
Mercury etc.)
Thus far we have occupied ourselves in finding
the Equation of Centre with respect to the Sun and
the Moon and thereby determining their True positions
from the mean.
In the case of the Taaraagrahas
Mercury, Venus, Mars, Jupiter and Saturn, we shall
have to answer
(a) llow were their sidereal periods obtained?
(b) Are those sidereal periods geocentric or
heliocentric?
Are there geocentric periods?
(c) How were the longitudes of their aphelia
obtained?
165
(d) Above all, how waa the divergence between the
computed aean positions and the observed True
positions, separated into two components
Mandaphala and Seeghraphala, each of which was
formulated differently with different arguments;
(e) How did the concept of S««ghroccha arise?
8.1
We shall first prove that a geocentric
sidereal period of a Superior planet differs from
it8 heliocentric sidereal period by a small variable
quantity.
S --
:
*■1
Let E^, Jj represent the positions of the iarth and
a Superior planet, say, Jupiter at a particular instant*
Then the geocentric longitude of Jupiter is given by
166
V*3
the angle AE^J^ where A le the Hindu tfewo-poiirt
of the Zodiao.
Let Eg,J2 rePr#*en^ the positions
after one revolution of Jupiter with respect to
the Earth i.e., when the geocentric longitude
A.
AE2J2 is the sane.
Thus E^J^ i® parallel to E0<l9*
But the heliocentric longitudes of Jupiter at
A
A
the two instants are ASJ^ and ASJ^ which differ
A
by the angle J^SJ».
But fvon the figure
. OSJ2 . Jtw2 - 0J2S
. OJjEj - E2J2S
- SJjEj - SJ2E2
*
We shall prove later that SJ.E, and SJnE„
are the second Equations on the two occasions.
The difference of these two cannot be expected to
be the ease at the two instants.
Hence, though,
Jupiter has nade one cosplete revolution with
respect to the Earth, it does not make exactly one
ooaplete revolution with respect to the Sun, but
a little wore or a little less the difference being
a variable quantity.
In fact, striotly speaking,
only the word 'heliocentric period* has a signi­
ficance in the case of the planets, for, the planet
167
go round the Sun directly.
The word ’geocentric
period' is not so meaningful.
Yet, we will not
be wrong if we say that a sufficiently large number
of geocentric revolutions of Jupiter has to he taken
to give us an average which will be equal to the
mean heliocentric sidereal period, for, though
there be a difference between the two periods
individually, yet, the mean geocentric period will
be also the mean heliocentric period.
8.2
planet.
Tt is not so in the case of an Inferior
The mean geocentric period of an Inferior
planet will be equal to a mean solar year, for,
an Inferior planet will be oscillating about the
mean position of the Sun.
This being so, the
question arises as to how the sidereal period of
an Inferior Planet was found by the Hindu Astronomer
8.3
In the case of the Superior planets the
sidereal periods given in the Hindu astronomy are
equal to the hellooentric sidereal periods of
nodern astronomy, though the Hindu astronomers
were not msare that the planetary motion is helio­
centric*
fhls is evident according to the statement
made above in 8*1
namely that the mean heliocentric
period of a Superior planet will be tfce same as the
mean geocentric period, if a sufficiently large
number of sidereal revolutions are taken and the
average struck.
The fact that the sidereal period
of Saturn of nearly 30 years was also correctly
estimated in Hindu astronomy signifies that
observations should have been carried on extending
over hundreds of years even in those remote times.
8*4
Or again there are two more methods which
might have been followed by the Hindu Astronomers
to calculate the sidereal periods of Superior planets
The first of them is as follows.
we have the formula
In modern astronomy
where
S
ie the synodic period, P the sidereal
period of a planet and Y the sidereal period of
the Earth or relatively of the Sun.
This formula
was known to the Hindu Astronomers though not in
the above fora.
In the case of the Moon, the
Suryasiddh&anta states that
Quotation
\rt
- (Refer Appendix-A).
i.e.,
"The Number of sidereal revolutions of the Moon
minus the number of sidereal revolutions of the
Sun
* the number of lunations".
This statement is equivalent to the formula
where L, a and Y are respectively a lunation, t.he
sidereal revolution of the Moon and the relative
sidereal revolution of the Sun.
Similarly
subtracting the sidereal revolutions of a Superior
planet from the relative sidereal revolutions of
the Sun, the synodic revolutions of the planet were
obtained from which the synodic period could be
170
calculated.
It vlll be noted here that tanking the
sidereal revolutions of the planets to the geo­
centric as done by the Hindu Astronomers, does
not Offset the calculation, for the reason already
stated.
The synodic period of a Superior Planet
is an easily observable quantity being less than
an year.
It is equal to the interval between two
conjunctions of the planet or two heliacal risings
or two heliaoal settings of the planet.
A correct
estimate of the mean synodic period could be gut
by carrying observations during the course of a
small number of years.
Then from the equation
x - y *
where
x
z
and £ are respectively the number of the
relative sidereal revolutions of the bun and the
number of the sidereal revolutions of the planet;
and z is the number of the synodic revolutions of
the planet.
Knowing
x
and
z,
£ could be got.
From this, the sidereal period of a planet coi^ld be
calculated correctly.
171
8.5
Yet one sore method might have been used to
calculate the sidereal period of a Superior planet.
Let two heliaoal risings of the planet be noted.
The interval between the two observations is the
synodic period, whose estimate has been arrived at
already.
Let the difference of the longitudes of the
planet at the two observations be X*.
evidently 360S
X
Then
gives the sidereal period of the
planet, say equal to P.
By carrying observations
for a good number of synodic periods, the mean
value of P could be correctly estimated.
This
method might have been plausibly adopted by the Hindu
astronomers in estimating the sidereal periods of
Superior planets, for, in the first place, even
a dozen observations need not take more than ten
years and secondly the retrograde motion of tie
planet affects each synodic period to the same extent
so that the average is not vitiated.
above formula 360S
constant.
v
The X in the
will be seem to be almost a
8.6
Next ve shall see how the sidereal periods
of the Inferior planets were found and how the
concept of Seeghrocoha arises incidentally,
have stated above that the geocentric eiderea
period of an Inferior planet coincides with that
of the Sun.
Uenoe the first and the third methods
described above to obtain the sidereal period of
a Superior planet do not apply here.
method alone could be used.
i
S
i
The secend
Tn the Equation
1
P ~ Y
if S and Y be known JP could be calculated.
We
may take it that the synodic period of an Inferior
planet also oould be correctly arrived at by ebservinf
the Interval between two conjunctions Inferior or
Superior.
So, the number of synodic revolutions in
a Yuga could be calculated.
Then from the equation
x - y ** z
where x and £ stand for the sidereal periods «f the
Inferior planet and the Earth (or of the Sun relatively
respectively and z stands for the number of synodic
revolutions, knowing £ and *,
8.7
5
could be obtained,
But this does not sews to be the method
followed by the Hindu Astronomers in the case of
the Inferior planets as will be seen presently,
Sinoe the geooentrio sidereal period of an Inferior
planet coincides with that of the Sun, wo the
Hindu Astronomers took the mean Sun as the mean
planet, Mercury or Venus
m
i.e., the longitude
of the mean planet was taken as equal to the longi­
tude of the mean Sun.
Quotation
This is stated in the words
>3 - (Hefer Appendix-A),
i.e.,
"The mean planet for Mercury and Venus is the
mean Sun”.
When the Sun is thus taken as the mean planet, the
geooentrio position of the planet is got by adding
to or subtracting from the longitude of the mean
Sun, the elongation of the planet.
Since by
definition
mean planet + equation * True planet,
the Bean planet coinciding with the mean Sun, the
elongation becomes here the I quation.
Thus In the
ease of an Inferior planet, the elongation is the
Seeghraphala or second equation.
Nov postulate a point M satisfying the
criteria
(a) M is a point taken to go in a geocentric circle;
(b) it coincides with the Sun when the Inferior
Superior
planet is in
conjunction;
(o) it has a longitude (90+Sj)# when the planet has
West
the maximum elongation in the mufe
being the
longitude of the Sun;
(d) it has a longitude (180+S2)° when the planet
Is in Inferior conjunction, s2 being the then
longitude of the Sun;
(e) it has a longitude (270+S3)° when the planet
has the maximum eastern elongation (S^ * Sun’s
longitude now);
(f) it completes a revolution with respect to the sur
i.e., lias a longitude (360+S4)° ,
being the
longitude of the Sun at this nonent.
(Note:- The worde Inferior conjunction and
Superior conjunction* have been borrowed from the
modern Astronomy.
These words were not used by
the iiindu Astronomers).
We perceive that the point M overtakes the Sun
by 360* as the True Planet aakes an oscillation
about the Sun.
as True Planet*
Thus it has the same synodic period
But the synodic period could be
observed as indicated before and its mean value
found.
Let this be equal to 120 days (say) in the
case of Mercury.
Let the daily mean notion of m
be mm
n° which is to be found.
Since the Sun has a
daily mean notion roughly equal to i®, M overtakes the
Sun by (n-i)° per day.
Hence the tine taken to
overtake the Sun by 360* is
days.
this number to 120, we have w*4*.
sidereal period of M is
176
Equating
Hence the
» 90 days (or 87 days
to bo nearer the truth).
Seeghroceha point,
M Is called the
in the case of Mercury it
is called Budha Seeghra and in the case of Venus,,
Sukraaecghra.
Thus the actual heliocentric
sidereal periods of Mercury and Venus are
asoribed to the hypothetical points called
Seeehras. which are construed to hare a geo­
centric circular motion and which have the same
synodic periods as the planets.
It will he seen
from the nature of the postulation above, v
has only a mean motion like the Pynamical mean
Sun, postulated with respect to the concept of
Equation of time in Modern Astronomy.
This
y
is postulated as a means to calculate the Second
Equation, Seeghraphala.
8.8
In computing the Seeghraphala, it is
necessary to calculate the geocentric longitude
of this M at any given moment.
as follows.
This can be done
The mean motion is known as mentioned
above.
So, we have to know its position at any
particular epoch.
This information is riven by
the fact that at the beginning of the Kalpa all
the planets and planetary points are at the Hindu
Zero-point,
In fact, the Kalpa is the period In
which all the planets and planetary points make
an integral number of revolutions.
Since M Is
also a planetary point, its sidereal period also
should have been taken into account to work out
the magnitude of the Kalpa.
So, therefore, the
mean motion as well as the position of K at a
particular epoch are known.
Hence its geocentric
longitude at a given moment can be computed.
It oan be proved that this geocentric
longitude of M in the Hindu Astronomy is equal to
the heliocentric longitude of the Inferior Planet
in Modern Astronomy.
Presuming that M was in
conjunction with the Sun at the beginning of the
Kalpa amounts to saying that the Inferior planet
was in Superior conjunction, (this follows from
the postulation of M ).
Hence the geocentric
longitude of M was equal to the heliooentric
longitude of the Inferior planet at that epoch.
C(TV\Atn,u£<^.
Since theaean geocentric notion is equal to the
mean heliocentric notion of the planet, the hello centric longitude of the planet at
subsequent
nonent will be equal to the geocentric longitude
of M. It will be noted, that M does not coincide
in position with the inferior planet.
9.
We have stated before (8.7) that the
elongation of the Inferior planet is the Second
Equation or £>eeghraphala.
elongation.
Let 0
Taka Hain0 ■ a
SiftMfcrft.YlMM,
be the maximum
as the radius of tae
or the epicycle required
to caloulate the Seeghraphala.
iron this radius,
the periphery is calculated.
9,1
In the heliocentric figure when
179
>
7-'
is
maxima , Sin t
a
b
and R being the orbital radii
- i
O
*
V ‘
6
'
'
of the Inferior planet and the barth; hence
a
*
RSinf * HSin
So, the radius of the epicycle above,
is the
radius of the inner circle here i.e., the orbital
radius of the Inferior planet, where the harth’s
orbit or the Sun's relative orbit is taken as
the Kakshaamandala of the epicyclic figure.
10.
The Seeghraphala or the Second Equation
is formulatsd as follows.
n
t
L.
*
£
180
*s
»
Let E, S, M be the posit lone of the Earth, the
Sun and the Seeghra at a given moment.
let
MP*S he the mean orbit or Kakshaamandala taken
ae the relative orbit of the Sun, since the mean
Sun is taken as the mean planet.
Let p be the
planet in the epicycle, P* the apparent geocentric
position of the planet, i,e., the true planet.
SM the excess of the longitude of the Seeghra
over that of the mean planet i.e., the
un, is
called the Seeghrakendra or -eegbra anomaly m.
Let PH and SL be drawn perpendiculars on ES ami
EM respectively.
Then the triangles PSN and SEX.
are similar.
,
* •
PN
SL
PS * W
,
m
PS
SL
a
* * PN " SE x
s H
H‘sln E*
Now the arc SP* is defined as the Seeghraphala,
Taking SP* as a straight segment approximately
„wt
SP* u
EP* PN
R
PN
gp x
* k X
where EP * K is oalled the Seeghra Karna.
Hence
SP* *= j| x ~ US in m = ~ HSin m
substituting for PN from the above*
181
10.1
Let u« compare this formula with that
derived from the heliocentric figure.
s
v r
K
\ ^ -Xxk in
t-
t
Let S, £, V represent the Sun, the Earth and
the Inferior Planet respectively.
The inner
circle is the orbit of the Inferior planet arwi
the outer that of the Barth.
a and K respectively.
VST « a
Let their radii be
Put EV » K
where ES is produced to T.
and angle
Then from
the A ESV
SJ^SEV
.StaESV
» | Sin ■
^
S1„ SEV . | sln (1S0-.)
HSln SEV * | IIS in
k.
The angle SEV is the elongation of the planet,
which has been pointed out to be the Second
Equation.
Thus the Equation has the same
182
fora as in the eployole figure.
a * y SV - yST «
where
Here
YSV -
sES
Y Is the first point of Aries.
Hut
SV
is the heliooentrie longitude of V which we h&ve
proved to be equal to the geooentric longitude of *».
YES is the longitude of the Sun.
Hence a in the
heliooentrie figure is equal to the angle SBM of
the epicyclic figure, where it will be noted that
the difference of two tropical longitudes will
be equal to the difference of the corresponding
Hindu longitudes.
Also K the Seeghra Kama is the
geocentric radius vector of the planet in both the
figures.
We shall later prove that the value of this
K derived froa the epioyolio figure accords with its
value obtained froa the heliocentric figure.
We
have particularly to note that the inner circle In
the heliocentric figure plays the part of the epi­
cycle in the Hindu figure.
This will be seen to be
the case also with respect to Superior planets.
Thus we have seen that the method of finding
183
the geocentric position of an Inferior planet 1 run
the Hindu epicyclic figure agrees with the same from
the heliocentric figure in all particulars.
Incidentally, it will be notioed that the True
planet is always displaced towards M from the mean
position S.
This agrees with the ooncept of an
Ucoha. either a Mandoocha or a Seeghroocha as
attracting the planet towards it.
Quotation H
Appendix-A),
Bhaskara says
i.e.,
BAn Ucoha is sueh a planetary point as would
attract the planet towards it”.
10,2
In the oase of the five Taaraagrahas, vise.,
Mercury etc., the Second Equation is of far greater
magnitude than the First Equation, i.e., the Equation
of Centre,
So, it is reasonable to assert that the
Second Equation was first defected.
This was found
to be Zero when the planet was in conjunction and
was noticed to increase with the elongation.
184
when
this Second Equation came to be formulated as
described before, it was next possible to formulate
the First Equation*
In this behalf, the aphella
have to be located.
This is done as follows in the
case of the Inferior planets.
We have stated above that the mean helio*
centric longitude of the Inferior planet is the same
as t|ie computed geocentric longitude of fc.
Hence
in the position when the computed geocentric longitude
of M is equal to the longitude of the rean ;;un, i.e.,
when the Second Equation or Seeghraphala is Eero,
though it should happen that the True planet should
be also in Superior conjunction from the nature of
the postulation of M, it so happens that it will not
be i.e., the True planet will not be exactly in
Superior Conjunction.
This happens on account of
the presence of the Equation of Centre in the he! ipcentric orbit.
Note the difference oi the longitudes
of the True planet at that moment and the longitude
of the a*an Sun.
Tabulate
6
>,
Let this difference be
for a number of Superior conjunctions.
Its maximum value gives the maximum Equation of
Centre.
Let this be ©
•
Put HSinf
* aj
a
gives the radius of the epicycle pertaining te the
Equation of Centre, from which the periphery could
be calculated.
When ©,
happens to be Zero, the
Equation of Centre vanishes and the longitude of
the mean Sun in that position gives the longitude
of the aphelion.
We shall now consider the case of a Superior
planet.
In the course of a synodic period the
difference between the computed mean position and
the observed True position of the planet was noted
to be almost a minimum when the mean planet coincided
with the mean Sun and almost a maximum when their
longitudes differed by 90°.
It was also noted that
the True planet was always nearer the Sun tha* the
mean.
It was ssen therefore that the Sun was playing
186
a part with regard to the Second Equation, which was
analogous to the part played by the apogee (Vandoccha)
in the First Equation i.e., the Equation of Centre,
Hence for a Superior planet the Sun came to be named
as the Seeghroccha.
Quotation
2j0
~ (Refer Appendix-A) i.e.,
"The Sun is called the Seeghroccha with respect to
Mars, Jupiter and Saturn".
12.
ou
Formulation of the Equations with respect to
Superior planet:-
When the True longitudes of the Sun and the
planet differ by 90*, i.e,, when the planet is In
quadrature, compute the mean longitude of the planet
for the moment.
Let the difference between the
computed mean longitude and the observed True longitude
be
Q 2*
Q 2 is evidently the algebraic sum
of the First and the Second Equations.
values of these Equations be
for a number of quadratures.
187
and
Let the maximum
.
Tabulate
The maximum and minimum
9
values of
0 2 will be equal to
respectively.
and
Adding then, we get the maxiaua
Second Equation
First Equation
7 ♦(
and subtracting the maxima
B
.
Put HSin 0 * aj a gives the
radius of the epicycle pertaining to the Equation
of Centre.
Putting A » HSin t
, A is taken as the
radius of the epieyole pertaining to the Second
Equation.
12.1
¥e shall prove later that the Second
Equation in the case of a Superior planet in the
heliocentric orbit is the angle SJE
\where S, 6
and E are respectively the Sun, the planet ant the
Earth) and not the elongation as in the case el the
Inferior planets.
This angle SJE will be a lanxiaun
when the planet is in quadrature i.e., when UJ
is tangential to the Earth*s orbit.
Then Sin
= |r
where a and R are the orbital radii of the orbits
of the Earth and the planet respectively.
K Sin
0
* a
i.e., HSin ,
188
■ a
Hence
Thus in the ease of a Superior planet also the
inner orhlt plays the part of the epicycle, and
the outer the part of the Kakshaamandala, as mentioned
before.
12*2
The Second Equation is then formulated as
follows in the epicycllc figure.
£
■
T
?
T1
L •
i
H KWTI
Let E, £, M be the Earth, the Sun and the mean planet
(Superior), Let P be the actual planet in the
epioyole.
Let MP*S be the Kakshaamandala or mean orbit,
S is here the seeghroceha as mentioned before.
Let
JP* be the projection of P on the mean orbit so that
P* is the apparent geocentrlo position of the planet
or the True planet.
Draw perpendiculars PN and ML on
189
EM and ES respectively,
MP*
is the Seeghraphala
or the Second Equation by definition.
As an approxi­
mation, we way take MP* as a segment of a straight
line at right angles to EM,
MP* EP*
R
Ifp * f? * K
seeghra Kama,
Then
Ep
,*,
*
K is called the
MP* » NP x |
how Triangles PNH and MLE are similar.
.
• •
PN
PM
ET * EK
• •
....
™ ‘
PM
em
*
Ml.
a
“ if
Ilu4
Hbln ">
where angle MEL « arc MS = m *= Seeghra Kendra or
Seeghra anomaly = Excess of the longitude of the
Sceghroccha, here the Sun, over that of the mean
planet
M.
U»* « | US in m x | « |
HSin m
as in the case of the Inferior planets.
12.3
Let us compare this with the formula derived
from the heliocentric figure.
Let E, J, S
be the
Earth, the Superior planet and the Sun respectively.
Let the inner Circle and the outer be respectively
190
r
5
'
0L
/C
£
/1r ..
the orbits of the Earth and the Superior planet
Let their orbital radii be a and E respectively.
Put EJ c k
JST « a*
Sin Sjfe
•
Produce
ES to T.
Let angle
Then from the Triangle BSJ i
*
Sin Ms
* Sin (180-a)
Therefore Sin SJE * | Sin a
*
Sin m
HSin SJF * | HSin a.
Now from the geoaetry of the figure
Y EJ * > Sd
+ 5 JE
,A
1'EJ is the geocentric longitude of the :~up< rioi
planet or the True planet,
longitude*
s
VSJ is the heliocentric
Je ie the angle therefore to be added
to the heliocentric longitude to get the True planet,
Let us take J to be the position of the Superior
Planet after it is rectified for the Equation
of
Centre in the hellooentric orbit, i.e., for the first
Equation*
Its heliooentrlo longitude
then equal to
SJ is
M + Ej where M stands for the mean
heliocentric longitude and E1 is the First Equation.
Hence the longitude of the True planet
191
A
"Y EJ *
M
+
Eg
where we put
SJF =r
Eg i* therefore the Seeghraphala or the Second
Equation.
It la given by the formula,
/V,
n
HSln SJE » " USin m, fro* the derivation
above.
The angle
ySJ
whioh is equal to
m
♦
ie said to be the longitude of the Mandasphutagraha
or mean planet rectified for the ‘andaphala or
Equation of Centre.
Thue the formulae for the
Second Equation derived from the epicvclic figure
and heliocentric figure have the same form namely
j£ HSin M, where K and m have the same sig.nif ieanco
In both the figures while a is the radius of the
epicycle in the enioyclic figure and the radius o‘
the inner circle in the heliocentric figure.
12.3
Now, the question arises as to how t.he
aphelion of a Superior planet is located wherefrom
the Equation of Centre is to be calculated.
At a
given moment the mean longitudes of the Sun and the
planet are to bo calculated.
192
\
£\
The excess of the
f
longitude of the mean Sun over that of the mean
planet is the Seeghra anomaly a .
From this, the
Seeghraphala & KSin a, can be computed, K being
computed according to the formula which will he
given shortly.
Let this Seeghraphala be Eg.
Observe the longitude of the True planet which
is equal to M + Ej, + Eg where M is the mean helio­
centric longitude, already computed.
Eg being known,
can be obtained.
this Ej for every day.
Hence v and
Tabulate
Let its maximum value be
Put USin& « a; a gives the radius of the epicycle
pertaining to the Equation of Centre.
day when
Note the
« 0; then the mean heliocentric longitude
of the planet will be the heliocentric longitude of
the aphelion.
The position of the aphelion being
known, the Equation of Centre has the formula
j HSin a where a is the radius of the epicycle, h
is the radius of the Kakshaamandala or mean orbit,
and is taken to be 3438,w and mm
a is the vandakendm
193
or the mean anomaly pertaining to the First Equation,
and is equal to the excess of the longitude of the
mean planet over that of the aphelion.
12.4
It is Important to note here two points.
(a) The longitude of the aphelion is to he taken as
only heliocentric.
If it is to he construed as geo­
centric, it will he different for different positions
of the Earth.
In otherwords when a planet appears
to be at its aphelion for a heliocentric observer,
and has a particular longitude, say, L, in the next
revolustion also, the planet will he at the aphelion
when it has the longitude L.
But, for a geocentric
observer, if in one revolution the planet appears if
he at the aphelion and has a longitude 1_, in another
revolution, it will he seen to he at the aphelion
when the longitude is P and not l,
(b) The mean longitude of the planet is only heliocentric,
for, the planet goes round the Sun and not round the Earth,
Though both these statements are not made explicitly in
Hindu Astronomy, we have to understand them as such.
194
13*
Thus far we hare answered the questions
posed at the beginning of Section 8 of this Chapter.
It appears from the above analysis, that the Second
Equation, Seeghraphala came to be formulated first,
as arising out of the planet's elongation.
The
difference between the computed mean position of the
planet and the observed True position was perceived
to be very great in the case of the planets, which
was not so in the case of the Sun and the boon.
In
trying to find the rationale of this big divergence,
it was noticed that the divergence was almost a
minimum at the moment of the conjunction of the
planet and almost a maximum in its quadrature.
Naturally it was formulated that a major portion
of the divergence increased as the elongation
increased.
This part of the divergence was named
the Seeghraphala,
When the Seeghraphala was thus
formulated, there remained a residue of the
divergence which was comparable with the Equation
of Centre (in magnitude) in the case of the Sun
195
and the Moon.
Thie residue was found to vanish
when the planet was at a particular point of the
Zodiao and attain Maxima value when the planet was
situated at 90* iron that point.
So, this was fort-m­
inted as the Equation of Centre as in the case of the
Sun and the Moon and that particular point of ti e
Zodiac was teraed the Mandoecha, the aphelion (we
hare translated this word as apogee in the case of
the Sun and the Moon, and as aphelion in the case
of the planets for obvious reason, though the Hindu
Astronomical term is the sane for both).
Thus,
when it was not possible to account for the divergence
between the computed mean longitude of the planet,
and the observed True longitude, by a single Equation,
it was separated into two components as described
above, which were named the Mandaphala and -eeghraphala, the First and the Second Equations and a
separate formulation was given to them depending
upon different arguments viz., ‘'anda Kendra and
Seoghra Kendra, (which we have translated as the mu it
195
anomaly and Seeghra anomaly respectively).
Even as
the point share the First Equation vanished, was
tanned Mandocoha, the point where the Second Equation
vanished was tensed Seeghrocoha.
The prefixes ?ianda
and Seeghra have an ntynological significance.
The
apogees or aphella have a very slow notion compared
with the planets.
to then.
So the prefix Manda sras applied
The Sun has a quicker motion than the
Superior planets, with respect to which it plays the
part of a Seeghrocoha.
applied to it.
So the prefix Seeghra was
In the case of the Inferior planets
also, the point M, which is the Seeghra, has a
quicker notion than the Sun, which is taken to he
the mean planet thereof.
Seeghra is nsaningful.
So, hore also, the prefix
The suffix Uccha is there
in both the terns Mandoeoha and Seeghrocoha because,
as quoted already fro at Bhaskara, the Uechas appear
as though they are attracting the planets i.e., the
True planets are found to be nearer the Ucchas
than the nean.
197
14.
The Seeghraphala has different connotation
In the case of the Inferior and Superior planets*
In the case of the former the elongation SEV Is the
Seeghraphala, whereas in the case of the latter the
A
angle SJE is the Seeghraphala.
The reason for this
difference of treatment has been already mentioned.
15.
The explanation given in Sections $ to 13
Is our own*
We oould not accept Bhaakara *s
explanation in this behalf (Vide his commentary
under Verses 1 to 6 - Madhyaadhikaara, hrahaganlta
as he based his explanation on the authority of
the Aagama i*e.f eeienee handed down from times
immemorial, wherefrom he presumed a knowledge of
the positions of the aphelia*
lienee we have ted
to so te the root of the problem as to how the
aphelia could be looated at all.
ills explanation*
however, which goes to a certain extent is worthy
of attention, and might be accepted to that extent,
provided, the Second Equation is formulated without
a knowledge of the First Equation as depleted hy us,
Hie explanation ie as follows.
Quotation Z>\ - (Refer Appendix-A) i.e.,
"There, by observation, obtain the position of the
True planet.
Take it as a first approximation to be
the wean planet rectified for the Equation of
centre.
The position of this planet is called
Mandasphuta.
Compute the Second Equation.
Applying
this correction inversely to the position of the
True planet, the next approximation to the ’ a«da~
sphuta is obtained.
Compute the Second Equation
for this Mandasphuta and proceed as before.
the proosss«
Repeat
In this way by successive approximations
we obtain an Invariable quantity which is the True
Mandasphuta.
By observing this position on different
days, we mark the day when the Mandasphuta coincides
with the position of the mean planet (i.e., the day
on which the Equation of Centre is Zero).
The
longitude of the mean planet on that day is the
longitude of the aphelion".
ExplanationThe True planet la observed.
The
position of the naan planet is computed fro* the
sidereal period whose determination has been expia mod
previously.
known as yet.
The position of the aphelion is not
So, we are not in a position to
separate the two components of the divergence between
the positions of the mean and True planets,
we know the position of the Seeghroccha.
nut
This point
in the case of the Inferior planets is the point M
while it Is the Sun itself in the case of the
Superior planets.
The normal process of getting
the Second Equation would have been to oorrect the
mean planet for the Equation of Centre and get the
-Mandasphuta and correct this again for the Seeghraphala i.e., the Second Equation and thus arrive at
the position of the True planet.
Hut we cannot
proceed in this order since the aphelion is not
known and has to be known.
the reverse order.
Hence we have to follow
Taking the True planet itself as
the .Mandasphuta in the first instance, we calculate
200
roughly the Seeghraphala and apply it inversely
to the True planet,
A better approximation to the
Mandasphuta is thus obtained.
We repeat this process
till we finally arrive at a quantity which is nearly
a constant.
This gives the True Mandasphuta.
By
calculating this on successive days, we inark the day
when the yiandasphuta coincides with the position of
the mean planet.
This means that on that day the
Equation of Centre is Zero and the corresponding
longitude of the mean planet determines the aphelion*
Here also tabulating the divergences between
the mean planet and the computed Mandasphuta on
successive days, the maximum Equation of Centre
could be obtained wherefrom the dimension of tic
epicycle could be calculated,
16,
In the case of the l-'irat Equation (Refer
3,3 of this Chapter) the are BB* is defined as
the Equation of Centre but we stopped short with
taking its value as Bfe,
But in the ease of the
Seeghraphala (Refer Sections 10 and 12,2) we have
not stopped with getting the value PN but miltipiled
this by
to get the Seeghraphala defined as SF' in
(10) and MP* in (12*2).
This mil tipi icat ion by
is called Karnaanubaata. Bhaskara explains why
this is done only in the case of the Second Equation
and not in the case of the First Equation,
ie says
that the difference is negligible in the case
of the First Equation, but not so in the nose of
the Second,
His statement is justified because
the dimensions of the epicycles of the Hrst
Equation are far smaller than those of the second
so that the Karna l.e., the radius vector to the
planet in the epicycle is almost equal to K in the
First Equation whereas in the Second Equation it
differs very much from R, where R is the radius
tho FJUcshaaraandala.
17,
We are told in the Hindu Astronomical Texts
that the First Equation is positive if ra lies between
0* and 180* and negative if m lies between ISO* and 360®
Whereas in the case of the Second Equation it is
21)2
vice versa.
The reason is that in the case of the
First £quation( the argument
d
is
L - 1 where L
and 1^ are respectively the longitudes of the planet
and the Uccha (Mandoccha) vhile in the Second
Equation the argument a is 1/- L ’ siteri 1. * and
L' are respectively the longitudes of the Uccha
(Seeghroccha) and the planet.
Hence the First
Equation is proportional to Sin a whereas the
Second Equation is proportional to -Sin a.
Hence we have the above convention of signs.
18.
The peripheries of the Seeghra epicycles of
the five planets Mercury, Venus, Mars, Jupiter and
Saturn are respectively 132*, 258*, 243^°, 68°
and 40* as given by Bhaskara (we have explained
before the peculiarity of expressing the peripheries
in degrees as above).
These peripheries throw
light on the dimensions of the orbits of the
planets.
The following table compares these
dimensions as given in the Hindu and Modern
Astronomies, taking the Earth's orbital radius
203
to be unity
Periphery of
the Seeghra
epicycle.
Mercury
132*
Venus
258*
.Mars
OO
243|
Jupiter.
68*
Oe
Saturn
Value of the
orbital
radlus.
HI
fH
Value in
odern
Astronoras,
.387
* -37
.723
360 ■ -716
2«3§' 1<5
360 ^
r
o
360
0
"40 * 9
1.52
5.2
A. 5
Earth*e orbital radius « 1.
19.
We have defined Seeghraphala as the arc
i
In (12.2) and the arc SP* In (10) and Its value is
|r liSin n.
In Figy.xy.wwe define
Bhu.laphala • PN s | HSln *
Kotlphala
* MK * | IiCos m
The value of K, the Seeghra Karnam can be expressed
in terns of these magnitudes as given by Bhaskare 1*
verses 27-28 Spashtaadhikaara, Grahaganlta.
K2*IISin2® + (a ♦ HCos n)2«(R
+
is svmhni
KotlPhala) 2 + Bhujaphala“
** a2 + il2 +_ 2R Kotlphala » a2+R2 + 2a HCos m
These fornulae are easily derivable from the epievoiir
figure as follows (Refer Pigvy).
K2 * EP2 * EN2 4 PN2 = (EM+MN)2 4 PN2
* (R + Kotiphala)2 + Bhujaphala2
(1
Where EM ■ R, MN * Kotiphala and PN ** Bhujaphala.
Again from the above,
K2 * (EM+MN)2 + PN2 s Ell2 + (MN2 + PN2)+2EM X MN
But MN2 4 PN2 = MP2 * a2
K2 * R2 + a2 ♦ 2R i Kotiphala.
(2)
or again drawing PI perpendicular on ES
K2 « El2 4 PI2 * (EL+Ll )2 4 MI,2
(3 )
e(IICoa ■ 4 a)2 ♦ HSin2 a
o
* HCoa" a + a
« R
2
16.1
4 a
2
o
°
+ 2aIIcoa a 4 IlSin" m
+ 2a Ucos m
ii}
These formulae are borne out even frosi the
heliocentric figure, for,
K2 » EJ2 « ES2 + SJ2 - 2ES zSJz Cos (iBO-m)
« a
* a
2
2
+ R
+ R
2
2
+ 2aRCos a
* a2 4 K2 4 2R x
«* a
2
(4
+ 2a HCos a
2
-a
+ R. + 2R x Kotiphala
205
(2
Again iron (4)
K2 * a2 ♦ R2
2a HCos ■
■
a2 + HSin2 n + HCos2m 4 2a HCos m
*
HSin2* ♦ (a+HCos m)2
and fro* (2)
K2 * a2 + R2 ♦ 2R x Kotlphala
* a2(USIn2* + HCos2*) + R2 + 2R x * HtoS B
R
R§
-------- --------
*>
a2 HSln2* . (R + a HCos a)2
----RZ
»— +
--- R
s----
o Bhujaphala 2 + (R ♦ Kotlphala) 2
19.2
i 1 i
The exact accordance between the formulae
derived from the eploycllc and heliocentric figures
is note-worthy and confirms the correctness of otr
interpretation.
20.
In Modern Astronomy we have stated that
gives the True planet where
v
is the
longitude of the mean planet and t , F
X
First and Second Equations.
are the
«i
Hindu Astronomy, however,
prescribes a peculiar procedure in the process of
rectification as follows.
21)8
Let M be the mean planet..
Let Eg be the Second Equation, derived from the
formula
Eg ** jr HSin (S-M)
mite re S
and M
are the longitudes of the Seeghroccha and the planet
Eo
respectively.
Take M +rjp
as the longitude of the sear;
planet and derive E^, the Equation of Centre.
Take
again M + E 1 as the actual mean planet. To this
2
mean planet apply the First and Second Equations
successively.
We get the True planet.
All interpreters of Hindu Astronomy pronosnced
this procedure as peculiar and irrational.
In fact
Bhaskara himself exclaims
Quotation
cLqL (Refer Appendix-A) i.e.,
"It is really curious; tradition has it; so it must
be respected; accordance with observations alone
is proofI *•
We shall prove that the procedure does
have a meaning.
Let M be the mean planet (consider a Superior
planet for example). Let Eg be the Second Equation.
E2 . A HS1„ (S-M)
. A_.HS.1I (S*)
approxlllBtely
* A Sin (S-M) where A is the maximum Second Equation
21)7
and S and M the longitudes of the mean Sun and
mean planet.
M+| * M ♦ | Sin (S - M)
Taking this M «► — as the man planet and eoinput inf
the First Equation
E« * ~ HSin (M + | - L) o a Sin(M-l -4 Sin
Where a is the maximum Equation of Centre, and i
the longitude of the aphelion.
E% * a Sin ( M - L) ♦ ~ Cos (m - I.) Sin {
-
Since ~ Sin (S - M) is small
Taking M ♦
as the actual mean planet and
computing again the Equation of Centre, we have
«
T±' m a Sin(M ♦ | Sin
Cos WT, x Sin
* a Sin (M-L + | Sin MkL +
-
? )
Cos SWL x Sin S^T)
* a Sin M-L x i + a Cos (M-L) (!| Sin M-L +
Cos M~L x Sin CT)
Since | Sin (M-L) + ~ Cos (M-L) Sin (S-M) is
of a small magnitude.
2
E1* * a sin (M-L) + |
2
Sin(M-L) Cos (M-L) + a t er m
containing a A of a negligible magnitude.
208
Et* * a Sin (M-L) ♦ j
Sin 2 (M-L)
Bat we have seen in the context of explaining the
Equation of Centre that a * 2e
.*.
Et* ■ 2e Sin (M-L) +
2
sin 2(M-L)
Hence the result of the first three operations
amounts to taking the Equation of Centre as given
above.
Comparing this with the Equation of Centro
5
2
given in Modern Astronomy, viz,, 2e Sin i + | e Mn 2
e.2
we find that in the place of j
we have got ~
.
Thus it appears that the three operations are
designed to secure the Second term in the Equation
4e2
of Centre namely
Sin 2m which is not there
originally.
In other words we are nearer the Truth
by performing the three operations than by applying
the Equation of Centre only once.
21,
Computation of Tlthl.
Tithi has been defined in 2.5 Chapter T13..
In the Hindu chronology Tithi plays an important part
as rnuoh as the 'Date* in the modern chronology
The computation of Tlthi is effected as
follows*
Let S and M be the longitudes of the True
Sun and the True Moon at the Sunrise on a particular
day.
Then M-S gives the elongation of the Moon.
Divide (M-S) by 12; the quotient gives the number
of the elapsed Tithis.
Let the remainder be r".
Let the True daily motions of the ~un and the Moon
»
at the Sunrise of the day concerned be s and m.
Then, it is argued as follows.
M3f the : oon
overtakes the Sun by w-a degrees on that day, shat
time is taken to overtake r*?n.
The result is
r/v„
- ghatis where 60 ghatls are equal to the length
of the day.
This result gives the ending moment of
the Tlthi on that day.
22.
Computation of Nakshatra.
Nakshatra has been defined in 2.3 Chapter
in.
The Computation of the Nakshatra on a particular day
is as much important in Hindu Chronology as that of
the Tithi.
It is done as follows.
210
Let M be the True
longitude of the Moon at the Sunrise and
True dally notion on that day.
m
its
Divide M by
13rr.
The quotient gives the number of elapsed \Takshatras
beginning from the Hindu Zero-point of the ecliptic,
Let the remainder be r••
follows.
Then it is argued as
"If in a day of 60 ghatis the Moon
traverses an arc of m*f what time does it take to
f" f\ii«
traverse r°?",
The result is
m
.
This gives
the ending moment of the Nakshatra on that day.
23, '
Vg&kr,
the..jLqg. of. Centre.
The mean daily motions of the planets are
known from their mean sidereal periods.
Let it be
required to find the True daily notions.
In the
case of the Sun and the Moon, the mean motion has
to be corrected for the Equation of Centre only,
while in the case of the planets it has to be rectified
for the centre as well as for Seeghraphala.
23.1
Correction for the Equation of Centre.
We first consider the correction arising out
21
of th« Equation of Cento.
This correction is called
Gat1-Mandaphala i.e,, 'Equation of Centre in the
mean daily motion'•
The mean motion thus corrected
is called Mandasphutagatl.
In Modern Astronomy,
a
taking the case of the Sun, the True longitude
is given in terms of the mean u longitude L by the
formula
0
« L + 2e Sin a +
anomaly.
Here
•*.
&Q
motion.
C;Q *
... where mmm
m is the mean
M. ♦ 2e Cos m
m
is the True daily motion and
(a)
L mean
Bhoskara gives the method of rectification in
the Verses 36-38, Spashtaadhlkaara, Grahaganitha, which
is given by the formula
m
True daily notion * Mean daily motion +
where Kotlphala * ~ UCos ra (defined before and
the dally motion in mean anomaly.
m
Both the formulae
(a) and (b) are the same If it be noted that
a * 2e;
” « Cob b.
The ♦ sign arises because
Coe m may be positive or negative.
We shall hear Bhaskara's words which we put
(b
in modern terns.
Let L be the longitude of a planet
Let E^ be the Equation
at Sunrise on a particular day.
of Centre at that aonent so that the True longitude
Is L ♦ Ej.
L ♦
Let the nean longitude of the planet be
hh at the next Sunrise, where
notion.
L Is the wean daily
Let Eg he the Equation of Centre then.
So
hh + Eg
the True longitude Is now L ♦
Then the daily notion rectified for the Equation of
Centre l.e., the Mandasphutagatl is
(L ♦ $>L + E2) - (L +E1) *
But
* || HSin n
and Eg «
!, + (E2 - E1)
HSin (n +
n)
Eg - Ei s | (HSin n V l n - HSin n)
This Bhaskara gives as j| HCos m
s= Kotlphala x-2-
• • Mandasphutagati * Mean notion
L+Kotiphala x -r
as given above.
The above formula Implies that
Bhaskara used formula
.
(
d (HSin n ) as HCos n ~ which corresponds to in
c
modern notation
6(Sin n) = Cos m
n
(for the proof of this formula Hefer Chapter on
Hindu Trigonometry).
213
24.
In the oase of the computation of Tithl
Bbaekara comments ae foil owe, (Vide verse 38 Ibid).
Suppose at a particular sunrise the Moon has a
longitude Just a little less than that of the Hun.
By the next Sunrise the Moon will he ahead of the
Sun*
Hence conjunction will have taken place on
the first day*
This moment of conjunction is usually
calculated by taking the longitudes of the Sun and
the Moon and their True daily notions at the
first Sunrise*
If on the other hand, hourly varia­
tion in the vrue notion is also taken into account,
the New-Moon occurs a little earlier or a little later,
Bhaskara therefore directs us to effect the calculation
taking that variation into consideration so as to
find the exact ending moment of the Tithi.
The
precision suggested by Bhaekara is, however, generally
not being worked out by the Calender-Computers except
with regard to the occurrence of an eclipse and a
few other religiously important occasions.
25.
In the case of the Sun and the Moon
214
hhaskara gives an ea»y practical method of rectifying
the mean notion in Verse 31 Ibid, which says "The
H Cosine of the nean anomaly taken out of the smaller
table of H Sines taking R * 120, divided by 54 gives
in minutes of are the Gatlphala or the Correction
to be added to the mean daily motion to get the
True daily notion of the Sun, whereas for the "oon
4
HCos m multiplied by the fraction
correction".
j
gives the required
This may be elucidated as follows.
From the formula for Gatlphala proved previously viz.,
Gatlphala ** j| HCos m ~ its maximum value is
.
But a * 2" ~i0*-3i* in the case of the sun.
o m = dally motion in mean anomaly «= mean daily
motion of the Sun - mean daily motion of the apogee 0*—59*«-8" — 0*—6*-41" a* 0#»52 *-27".
Gatlphala « 130* - 31" x
Hence maximum
* 2'-14".
Similarly for the Moon it will be 08*-48”.
Hence the general value of the Gatlphala is
for tb. Son and
HCo. g
{or the
2,-14" HCos ra
fi
Taking R * 120, then® beooae
2g~
-rj4
HCos a
and
ikO
2-1
t>7
°°K HCoa a respectively. The coefficient 30 ■
120“
l^T
i
1 approximately. In the case of
3600
54
“W”
aj i
iO
the Moon, the coefficient ia ^
^ *
1+ T+
1^+
*
Tlkkin8 the convergent before the
big partial quotient 10, it is
j
.
Hence Bhaskara
A
writes y HCoa a as an approximate value.
Bhaskara
reveals here and elsewhere a knowledge of continued
fractions and their use towards approximation.
Regarding the Sign of the Gatlphala, since it
contains HCos a, it is negative when a lies between
90* and 270*.
26.
The variation of the Second Equation.
Seeghranhala ~ Gatl-Seeghraphala.
We have seen that the longitude of the True
planet vis., P * M ♦ Ej
♦ Eg where W is the longitude
of the aean planet and E^, Eg the hirst and the second
Equations.
Then we have
8 P **<>M +
+
Eg i.e.,
the True dally motion of the planet is equal to the
sun of the mean daily motion and the dally variations
in the two Equations.
Gatlaandaphala.
We have found above
We have now to find
ealled Gati-Seeshranhala.
context
6 e2
^uetatton
- u - (P-VJ. HC°.
F ^ tt*e
Eg which is
Bhaskara states in this
(Refer Appettdix-A)
i.e. t
s 1B the B8an
motion of the Seeghroccha, V is that of the planet,
Q
is the Seeghraphala or Second Equation, K is the
geocentric radius vector of the planet called
Seeghrakarna•
(U-V) is thus mean motion of the
Seeghra anomaly which is defined as the excess of
the longitude of Seeghroccha over the longitude of the
planet.
We shall establish this result first by
modern methods.
It is required to find the True
motion of an Inferior planet M or of a superior
planet J.
'Y
This motion in the case of M (ilefer Figs, below),
!
;
>n ■
#■ -
K
s
*
91i. 7*
t
is equal to
( V EM) where
"V
is the first
point of Aries.
3! ( > EM) « 3! (
VSM - SME)
In the case of J, this notion is equal to
•j! ( YEJ) ■ «g|
( fES - SEJ)
The first tern namely
(
vSM) or ^
(
vs)
as the case may he, gives the notion of the Seeghra,
called Seeghragati, for, we have seen that in
the case of an Inferior planet, the planet Itself
is the Seeghra while for the Superior, the sum is
the Seeghra.
Let the second angle SME or SEJ as
the case may he denoted by
/
in the above is then
From the figures above,
we hare K Cos
• K
Sinfjffi♦
y
Cos
.
- H Cos ra = a
f
The second term
Hence differentiating
H ♦ E sin m || ** 0
(in
Next, differentiating K2 « a2 + R2 + 2aK Cos
2k
m
- 2aR Sin ■
2}
Eliminating || from (1) and (2) we obtain,
RSin a (K -
a
Cos
v)
K2 sinY
da
dt
now k - a Cosy> ■ R 00s 0
213
and
K
Sin m
llettce
Here
dV
.. H CO®0
TT d*
4 n
i*e*» 0 "
.
H cos
-- “
m
we have denoted by (U-V)
Thws we have proved that the True motion of the
planet = U - ..g°*-
(U-V)
Bhaskara naturally would have proved it from
the epicyclic figure.
Though the method is long
and a little cumbersome, we reproduce it here
with
our symbology) as it will give an insight into the
genius of the great Mathematician.
26.1
Let P^, Pg be the positions of the planet
(Refer fig. below) on a particular day in the mean
D
''l
w
% ' •
h'
b
•F,
219
. /K
orbit and the eccentric circle respectively.
Join OjPg to out the mean orbit in P.
the position of the True planet.
then £ is
Now lay out arcs
PjQ^ and PgQg in both the circles equal to the daily
motion in the Seeghra anomaly.
mean orbit in Q,
Join O^Q,, to out the
Then AjP^ and A^P are the neai*
and True Seeghra anomalies of the first day, while
and AjQ are those of the next.
Thus PQ is the
True daily motion of the Seeghra anomaly called
Sphutakendragati.
Bhaskara points out that ?>;■
should not be construed as giving the notion of the
True planet, misconstruing P and Q as the True planets
on the two consecutive days; for, with respect to
both Aj and
will have moved by the next day m the
anti-clockwise direction so that v is not the actual
position of the True planet on the next day.
required to find the magnitude of PQ,
It is
Take a point l
on the eccentric circle such that Pgl = P^P * Seeghraphnla; then 0gL is parallel io
^
p^ because
A-
Ai °1 Pi * A2 °2 P2 *
220
POi Pi * h02 12
A-
and substractlng
0^ P * a2 o2 L
Let PgM be the nsine of the arc PgL i.e., Seeghraphala; let QgN be the HSlne of the arc q2L * q?p0+p0l
* mean notion of Seeghra anomaly ♦ Seoghraphala.
These two ohords PgM and QgN are perpendicular to OgL
and so perpendicular to O^Pg also.
which we shall find first.
Their dlffereneewQ^B
Taking Q2E>2 a® an i««re»e»t
in the arc PgL (The Seeghraphala will be generally
greater than the daily motion in seeghra anomaly,,
though perhaps In the figure it does not appear
bo),
Q2tt will be the increment in PgM i.e,, HSin
where
is the Seeghraphala,
is
The increment in HSin
b (HSin 6 ) = ilcosG
II cos G x
since we
have taken P2Q2 as *he increw€nt in BgL.
QgH * SPPy.Q,.. B
where
b a
ifenc*
is the daily motion in
Seeghra anomaly equal to ?2q2.
Now from the similarity
of triangles O^QgR, and O^QS,
OS
li2R
°l‘i
H
QjQj -
K
•
•
ns
. S .
Q,R
Qs *
x
*
*
M
^
P
QS being small, may be taken to be equal to PQ, which
is sought to be found.
But, it can be proved that
Man notion of the Seeghra - True notion of the
Seeghra anomaly * True notion of the planet from
the following figure •
F's ZTxTI
Let S, Sj be the positions of the Seeghra on
two eonseoutiwe days where SS1 is the mean motlea
of the Seeghra, (i.e., S and
are the mean posi­
tions of the Seeghra on the two days),
let P awi f .
be the positions of the True planet on those twe days*
Then SSj - (SjPj - SP) * SSj-^SjS ♦ SPj- 5P ♦
« PP4 ** True notion of the planet.
Here S^p
is the True notion of the Seeghra anonaly.
)
- >P
Hense
if U be the mean notion of the Seeghra
V - PQ * True notion of the planet * V where
^n
» -*
is the nean notion of the Seeghra aneaaiv.
We may put £n
« U-V where V is the mean motion »?
the planet.
. *.
True notion of the planet * U -
99 l
■
as proved by the Modern Method.
27.
Retrograde notion.
•■MMMlftHMMlWMMMMMMM
Bhaskara saye in the end of the Verse 313
quoted above - Quotation
"If U - iyryj.j.cps.6
retrograde".
(Refer Appendix~A) i.e
l8 negative, the planet is
This means that the planet will
appear stationary as seen from the Earth when
n
(U-V) H cosk
. .
„
Jt----- *
*•••» wh®n JJ *
Vlicos
We shall prove the truth of this statement by
the method of modern Astronomy.
’7e shall consider
the case of a Superior planet J, for example.
Let u* and v’ be the linear velocities of the fart
D i f
223
and the Superior planet J respectively.
Resolving
these perpendicular to the radius vector K, as
shown in the Figure, we say that J appears static miry
as seen iron the Earth when
Ut
u* oos E 4- v* cos© * o i.e., when
.
QQg.
= ' cos"™
But from the triangle SEJt it cos e + a cos F =■ K
Eliminating oos E between (1) and (2)
- cosQ
«-a cose
k-lt cose
k-R cose
a
If u, v are the angular velocities of the Earth am)
u*
v*
the planet u’ « au; and v* * Rv.
Substituting in (3) for u' and v'
au
Rv
•
• •
cos©
k-R COSi
oos ©
R cos6 -k
hi
v R cos©
v B cose
* r cos© -k * H cos© -k
as given by Bhaskara*
27,1
This may be further reduoed to the modern
formula for stationary points namely,
_____
marnmm
where
__ 0
cos
r-j.
<*6 is the angle
i
4
x b‘f-----------*a -...y
a-a‘b* 4 b
ESJ and b * It.
We have from the triangle
SEJ
i
k cos 0
Eliminating
VlKb,
u *
+ a oos v*
oos Q
'r \
* R = b
between (4) and (5)
a CO.S6 )
by X (b - « CO*
h- u> - » co»tn=c ’b (b - a co’
)
>-k*
K
But K2 ■ a2 ♦ b2 - 2 ab oos0
,*.
b2 - ab cos 0
-k2 * - a 2 + ab cos
Substituting this value in the denominator above
bv (b - a cost/
""6
-a + ab cos 0
)
cos
a 2u
.2
+ b u
aW (u + v
The standard formula of modern Astronomy viz.,
cos 0
m
i 4-^.—
ft.—
follows from (6), by
a - aab2 + b
applying Kepler’s third law which is equivalent
/ ~§
to saying
uiv **
/ b
.
In the absence of a
knowledge of Kepler's third law, the contribution
of Hindu Astronomy stops at formula (6).
28.
Dhaskara states in Verse 41, that the
planets Mars, Mercury, Jupiter, Venus and Saturn
being to retrograde when the respective Seeghra
anomalies are equal to 163°, 145*, 125°, 165* ami
113* respectively.
from the formula
These values can be obtained
oos m
« -
99
2
^aR~fu‘
+'2 v)
6
which corresponds to equation (6) given above,
taking a as the supplement of
and H « b.
The Hindu fora of the equation is
„__ __
(a 2u ♦ R 2 v)
H COS B ® “
■■■ .... ’
a (u+v)
Substituting the values of u and v
where they are the velocities of the planet and
the Sun in the case of the Inferior planet and in
the case of a Superior planet they are the velocities
of the Earth (relatively of the Sun) and the planet.
the respective values of a can be computed.
29.
Then it is mentioned in Verse 41, Spastaadbikaara,
Gr&haganitha, that retrograde notion ends for the
five Taaraagrahas when the Seeghra anomaly equals
(360-r) degrees where r stands for the values of the
anomaly when retrograde motion begins in each case.
This may be seen as follows, from the following figure.
e‘
*
•
*
t
9 f> 5
cf’
IMS WU v/
We know that the points J and J* where the Superior
planet begins to retrograde and ceases to retrograde,
are symmetrically situated with respect to SE»
Thus
the values of the Seeghra anomaly namely JSE* and
the oonvex angle .PSE* (where ES is produced to E* i
at the two moments are such that their sum is equal
to 360®,
Hence if r be the value of the Seeghra
anomaly when retrograde motion begins (360-r^
will be its value when the planet's motion becomes
direct.
30.
In verses 42, 43* Bhaakara mentions that
Mars, Jupiter and Saturn, rise heliaeally in the last
when the Seeghra anomaly equals 28°, 14*, and 174
respectively and set heliaeally in the West when
the anomaly ie equal to (360*-28*), (360®-14*
and (360*-17#) respectively.
from the following figure.
This may be elucidated
When the s«eghra anomaly m of Mars la equal to 28®,
the Seeghraphala 0
will be equal to 11°; hence the
elongation SEM « * - 6
« 28-11 * IT®.
of 17® goes by the name Kaalaamsas.
This arc
By actual
observation It mas found that Mars becomes Invisible
In the rays of the Sun mhen the elongation equals 1?®*
Similarly for Jupiter, the Seeghraphala, when the
anomaly is 14® is equal to 3®; so that the elongation
will he then 14 - 3 ■= 11®,
Jupiter was fount! to
beoome invisible in the rays of the Sun at that
distance.
In the case of Saturn, the Seeghraphala
amounts to 2® when the anomaly is 17®; so that the
elongation will be then 16*.
This arc gives the
Kaalaamsas for Saturn, which is its distance fro®
the Sun to become Invisible.
Regarding the heliacal rising, we know that
the points M and M* (Refer figure on next page), the
points where the Superior planet rises and sets neiiacally
are symmetrically situated with respect to ES,
223
V-? A - f 1 l_
Hence the anomalies MSE* and the convex angle
M*SE* are such that their sure Is equal to 360®*
Hence If a Superior planet rises hellacally when
the anomaly is equal to r*, It sets hellacally
when the anomaly is equal to 360-r*.
In the case of the Inferior planets, we are
told that Mercury rises in the West when the
anomaly Is 50*, sets in the West when it is 155*,
rises in the ^ast when the anomaly is 205* ant
sets In the East when It is 310*.
As for Venus,
the respective anomalies are given as 24*, 17?°,
183*, 336*.
This may he elucidated as follows.
223
I
H_
'» .11, *i,
r
I
'*
I.'
v
: '
We know that an Inferior planet is retrograde
at its Inferior conjunction.
nearest to the Barth.
Also it is then
The Kaalaaasas for ^ercur.v
and Venus at Inferior Conjunction ar© given to he
15* and 8* respectively.
At Superior Conjunction
they are 17° and 10° respectively.
These arc their
respective elongations when they rise or set heliacally
as found by actual observation.
Since the planets
are nearest to the Barth at the Inferior conjunction,
they are brightest so that the t.'aalaamtas then are
given to bo 2° less than what they are at Superior
Conjunction.
At Mj the Inferior planet sets
heliacally in the East, at Mg rises heliacally in
the West at Mg sets heliacally in the West and
230
at
riaaa heliaoally in the East.
Since
, m;,
and Mg, M4 are symmetrically situated with respect
to ES, the mean anomalies of
such that their
sums
are
and Mg,
are equal to 360*.
Hence the
mean anomalies of hellaoal setting In the west and
heliacal rising in the East are given such that
their sum is 360* (156 ♦ 205 = 360* etc),
similarly
the anomalies for setting in the East and rising in
the West are such that their sum is 360° (50®+3i0**360°
Further it will be noted that in the case of
the Inferior planets the Seeghraphala is itself
the elongation.
If it is stated that
iercury rises
in the West when the mean anomaly is 50°, it means
that the Seeghraphala then is 13*.
This being the
elongation equal to the Kaalaamsas for ' ereury,
rises then heliaoally.
The case is similar with
respect to the remaining situations.
it