TRUE POSITIONS OF THE SUN AND THE MOON ACCORDING TO · In the present paper the procedures of...

TRUE POSITIONS OF THE SUN AND THE MOON ACCORDING TO

��

S. Balachandra Rao. Hon. Director, Gandhi Centre for Science & Human Values,

Bharatiya Vidya Bhavan, 43/1, Race Course Road, Bangalore – 560 001, India

S. K. Uma Department of Mathematics,

Sir. M. Visvesvaraya Institute of Technology,

Hunasamaranahalli, Bangalore – 562 157, India

Padmaja Venugopal Head, Department of Mathematics,

S.J.B. Institute of Technology, Uttarahalli Post, Kengeri, Bangalore – 560 060, India

and

2

Abstract :

In this paper the procedure for computations of the true positions of the Sun,

and the Moon according to the popular Sanskrit astronomical text, ��

(GL) of Gan�� ña (early 16th Century) is presented. The importance of Gan��

Daivajña lies in his simplifying mathematical computations by providing justifiable

approximations for trigonometric functions among other interesting features. The

special features of GL in such simplifications are explained. Illustrative examples are

worked out and the results are compared with those obtained according to other

traditional texts as also modern computations.

Key Words : Gan �� ña, �� True planets, Indain Astronomy,

Manda.

1. Introduction :

In the present paper the procedures of determining the true positions of the

Sun and the Moon according to the popular traditional Indian astronomical text,

��havam (GL) of Gan�� ña (epoch : 1520 AD) are explained.

The importance of GL lies in the fact that the conventional procedures are

greatly simplified by dispensing with the trigonometric functions, sine and cosine.

The author Gan�� ña has accomplished this task admirably by employing

mathematically justified approximations. The efficacy of his approximate formula

� �� 0 to 900.

The three corrections in respect of the true position of the Moon namely cara,

�� and ��, as simplified by Gan�� ña, are explained.

For establishing the validity of the simplified procedures of GL illustrative

examples of two different periods are worked out. The values of the true positions

of the Sun and the Moon for these two dates are compared with those obtained

according to other traditional texts as also with modern values.

3

2. Man�daphala (Equation of centre)

In obtaining the mean positions of the Sun and the Moon, it is assumed that

these bodies move in circular orbits round the earth with uniform angular velocities.

However, by observations it is found that the true motions are non-uniform.

The procedure for calculating the major corrections to the mean positions in

order to obtain the true positions, in the ancient and medieval Indian astronomical

texts, is related to the epicyclic theory, which is explained in the following section.

Epicyclic theory and Mandaphala1

The theory is that while the mean Sun or the Moon moves along a big circular

orbit (dotted in Fig. 1), the actual (or true) Sun or Moon moves along another smaller

circle, called epicycle, whose centre is on the bigger circle.

The bigger circle ABP with the earth E as its centre is called the kaks �avr�tta. Let

A be the position of the mean Sun when the true Sun is farthest from the earth. The

line AEP is called the apse line (or ��) and AE is the �� (radius) of this

orbit. The epicycle, with A as centre and a prescribed radius (smaller than AE) is

called the �� tta. Let the apse line PEA cut the epicycle at U and N. The two

points U and N are respectively called the mandocca (apogee) and the �� of

the Sun. Note that as the Sun moves (as seen from the earth) along the epicycle, he is

farthest from the earth when he is at U and nearest when at N.

�

U

A

N

D

E

S S’

C A’

N’

U’

/ B

P Fig.1. Sun’s manda Epicycle

4

The epicycle theory assumes that as the centre of the epicycle (i.e. mean Sun)

moves along the circle ABP in the direction of the signs (from west to east) with the

velocity of the mean Sun, the true Sun himself moves along the epicycle with the

same velocity but in the opposite direction (from east to west). Further, the time

taken by the Sun to complete one revolution along the epicycle is the same as that

taken by the mean Sun (i.e., centre of the epicycle), to complete a revolution along

the orbit.

Now, in Fig. 1, suppose the mean Sun moves from A to A’. Let A’E be joined

cutting the epicycle at U’ and N’ which are the current positions of the apogee

(mandocca) and the ��. While the mean Sun is at A’, suppose the true Sun is

at S on the epicycle so that AEUSAU ˆˆ ′=′′ . Join ES cutting the orbit (i.e., circle ABP at

S’). Then A’ is the madhyama (mean Sun) and S’ is spas �t�a (or sphut�a) Ravi (Sun). The

difference between the two positions viz, SEA ′′ ˆ (or arc A’S’) is called the equation of

centre (mandaphala).

Now, in order to obtain the true position of the Sun, it is necessary to get an

expression for the equation of centre which will have to be applied to the mean

position.

In Fig. 1, SC and A’D are drawn perpendicular to U’N’E and UNE

respectively. The arc AA’ ( or AEA ′ˆ ), the angle between the mean Sun and the

apogee is called the mean anomaly (mandakendra) of the Sun.

We have, in the right-angled triangle A’DE,

EADAAEDAEA ′′=′=′ /ˆsinˆsin

so that

mRAARDA sin) arcsin( =′=′

(where R = A’E and m = arc AA’) is called �� of the mandakendra. From the similar

right-angled triangles SCA’ and A’DE, we have

EADA

ASSC

′′

=′

so that EA

ASDASC

′′×′

=

5

Since SA’ and A’E are respectively the radii of the epicycle and the orbit, these

are proportional to the circumferences of the two circles; that is,

orbittheofncecircumfere

epicycletheofncecircumfere=

′′

EAAS

DASC ′×

=∴

orbit of ncecircumfereepicycle of ncecircumfere

Taking the circumference of the orbit as 3600, we have

0360

majySC

of epicycle) the of ence(circumfer ×=

Now, taking SC approximately the same as A’S’, we have

Equation of centre (mandaphala)

0360

majy of epicycle) the of ence(circumfer ×=

( )mRRr

sin

=

where m = mandakendra (anomaly) and R sin (m) the “Indian sine” of the anomaly m

of the Sun. The maximum value of the equation of centre is r, the radius of the

epicycle. By observation this can be obtained as the maximum deviation of the Sun’s

true position from the calculated mean positions. Note that when the Sun is at his

apogee or perigee, the mean and true positions coincide since sin (m) is 0 when m =

00 or 1800.

The maximum equation of centre for the Sun was observed by Bh��

(b.1114) to be 031120 ′′′ (i.e., 5113 .′ )2 which is the value of r. Therefore,

Circumference of the epicycle of the Sun = 66133603438

5131 00 .. =×

��

6

The same epicycle theory is applied to the Moon also. In the case of the Moon,

��

taken the epicycles as of varying radii and not fixed.

The peripheries (paridhis) of the manda epicycles of the heavenly bodies

according to different Indian astronomical texts are compared in Table 1. The GL

does not give these explicitly. However, its author, Gan�� uses the values of

peripheries implicitly.

Table 1 : Peripheries of Epicycles of Apsis

Bodies ��m Khan�d�a

��

��

��

��

��

Ravi (Sun) 130 30’ 140 140 13040’ to 140

Candra (Moon) 310 30’ 310 310 31040’ to 320

Kuja (Mars) 63.00 to 81.00 700 700 720 to 750

Budha (Mercury) 22.50 to 31.50 280 280 280 to 300

Guru (Jupiter) 31.50 to 36.50 320 320 320 to 330

�� (Venus) 9.00 to 18.00 140 140 110 to 120

�� (Saturn) 400.5 to 58.50 600 600 480 to 490

From Table 1 we notice that the Khan�d�� 3 (628 A.D.) of Brahmagupta

and the �� (as given in Pañcasiddh�� 4, 505 A.D.) take

the epicycles as of constant periphery (and hence radius). The �� 5 and the

later �� 6 ��

I (476 A.D.) has taken constant values in the case of the Sun and the Moon.

��

The true midnight of a place differs from the mean midnight by an amount of

time called “equation of time”. The equation of time is caused by

(i) the eccentricity of the earth’s orbit; and

(ii) the obliquity of the ecliptic with the celestial equator.

7

The correction to the longitude of a planet due to the part of the equation of

time caused by eccentricity of the earth’s orbit is called ��. The other

correction caused by the obliquity of the ecliptic is called ��

While all the � �� texts have taken into consideration the ��

correction, the other correction, ��, was first introduced by �� (about

�� – II and others.

We shall discuss the �� correction which is mentioned in the ��

� ��. The eccentricity of the earth’s orbit is a part of the equation of the centre

(mandaphala) of the Sun which is converted into time at the rate of 150 per hour or 60

per ghat��. This rate of conversion is due to the fact that the earth rotates about its

axis at the rate of 3600 in 24 hours (or 60 ghat��). The resulting amount in time unit

is the equation of time caused by the eccentricity of earth’s orbit. Thus, the equation of

time (due to the eccentricity).

= [(Equation of centre of the Sun) / 15] hours.

= [(Equation of centre of the Sun) / 6] ghat��

Now, to get the �� correction for the Sun or the Moon or any other

planet, the equation of time obtained above must be multiplied by the motion of the

planet per hour or per ghat�� as the case may be. That is,

�� correction for a planet

= [ (Equation of time in hours) x (Daily motion) / 24]

= [ (Equation of centre of the Sun) / 15 ] x [Daily motion / 24]

= [ (Equation of centre of the Sun) ] x (Daily motion) / 360

where the factors in the numerator are in degrees and the daily motion is that of the

planet. If the time unit used is ghat��, then

�� correction

= [(Equation of time in ghat��) x (Daily motion / 60]

8

= [(Equation of centre of the Sun in degrees) /6]

x [(Daily motion of the planet) / 60]

= [Equation of centre of Sun in degrees]

x [Daily motion of the planet / 360]

where the daily motion of the planet is in degrees and hence the ��

correction is also in degrees.

However, if the daily motion of the planet is in minutes of arc, then ��

correction in degrees

= (Equation of centre of the Sun in degrees)

x (Daily motion of the planet) / 21600

Further, the �� correction is additive or subtractive according as the

equation of centre of the Sun is so.

For example, in the case of Moon, its mean daily motion is 130.176352 or

790’.58112.

Therefore, we have (mean) �� correction

= (Equation of centre of the Sun) 02160

58112079′

′× .

= 27.321674

Sun the of centre ofEquation

Note : Brahamagupta takes the denominator approximately as 27 in his

Khand �� 3. The same expression is adopted by GL.

It is important to note that to obtain the actual (and not the mean) ��

correction of a planet, we have to use the true daily motion of the planet for the

given day.

Example : Find the �� correction for the longitudes of the Sun and the Moon

given that on a certain day we have

True daily motion of the Sun : 59’.65

True daily motion of the Moon : 855’.23

Equation of centre of the Sun : + 20 7’ 32’’ = 127’.53

9

`We have

(i) True �� correction of Sun

= (Equation of centre of the Sun) x 21600

Sun the ofmotion True

= 1203521835002160

659553712 ′′′=′=′′×′

...

Since the equation of centre of the Sun is additive, the �� correction is

also additive.

(ii) True �� correction of the Moon

=(Equation of centre of the Sun) 21600

Moon the ofmotion True×

= 350494205502160

2358553712 ′′′=′=′

′×′.

..

Here also, the correction is additive since the Sun’s equation of centre is so.

4. Further corrections for the Moon

We have applied so far an important correction namely, the equation of centre

(mandaphala), to the mean position of the Moon. Besides this correction, the other

two corrections applied viz, �� and �� are mainly to get true position

of the moon at the true local midnight at the place of observation.

However, in modern astronomy, to get the actual true position of the Moon at

least two more important corrections will have to be applied, of course, ignoring a

large number of other minor corrections due to planetary perturbations. These are :

(i) Evection = )sin()sin( φ−ξ′′′=φ−ξ′

262672

4

15me

where m is the ratio of the mean daily motions of the Sun and the Moon, e is the

eccentricity of the Moon’s orbit, )( SM −=ξ , the elongation of the Moon from the

Sun and PM −=φ , the mean anomaly of the Moon (P being the Moon’s perigee).

(ii) Variation = )sin( ξ′′′ 20393

In the above formulae, S and M are respectively the mean longitudes of the

Sun and the Moon. Mañjula (932 A.D), Bh�� II (b. 1114 A.D.) and later Indian

10

astronomers have recognized the evection correction in addition to the equation of

�� hara

Simha discovered independently a fourth correction called annual equation.

��

(iii) Annual equation = 67211 .′′′ sin (Sun’s anomaly from apogee)

�� ( )67211 .′′′ is very close to the known

modern value. Tycho Brahe took the coefficient wrongly as 034 ′′′ .

The mandakendra (anomaly) of a body and the bhuja of mandakendra are

defined as follows:

Mandakendra (MK) = Mandocca of the planet – Mean body

Where mandocca is the apogee and body refers the Sun or the Moon

(i) If the mandakendra of the planet is less than 3 �� (i.e., 00 < MK � ��0), the

MK itself is the bhuja i.e., Bhuja = MK. [Note: �� sign = 300]

(ii) If the mandakendra is greater than 3 �� and less than 6 �� (i.e., 900 < MK

� !�0) then Bhuja = 6 �� – MK = 1800 – MK.

(iii) If mandakendra is greater than 6 �� and less than 9 �� (i.e., if 1800 < MK �

2700) then Bhuja = MK – 6 �� = MK – 1800.

(iv) If mandakendra is greater than 9 �� and less than 12 �� (i.e., 2700 < MK �

3600) then Bhuja = 12 �� – MK = 3600 – MK

Kot�i of MK : Subtracting bhuja from 3 �� we get kot�i

i.e., Kot�i = 3 �� – bhuja = 900 – bhuja

12 �� (or 3600) have been divided into four �� (quadrants) each containing 3

�� (900). The I and III quadrants are vis �� (odd quadrants) and II and IV are

called ��s (even quadrants).

Mandocca of the Sun = 780 = 2R 180 (taken as fixed).

5. The mandaphala (equation of centre) of the Sun

The method of determining the mandaphala is as follows7:

(i) Find the mandakendra (MK) of the Sun.

11

(ii) Find the bhuja of MK (hereafter denoted by BMK)

(iii) Subtract 9

bhujafrom 20 i.e., obtain (20

9

BMK− )

(iv) Multiply the results of step (iii) and 9

BMK

(v) Divide (iv) by 9.

(vi) Subtract step (v) from 57.

(vii) Express the results of step (vi) and step (iv) in v�� (seconds of arc) and

divide step (iv) by step (vi)

The result is the mandaphala of the Sun

i.e., Mandaphala of the Sun =

−

−

−

999

2057

9920

BMKBMK

BMKBMK

…(5.1)

Note :

(i) If the mandakendra is within 6 �� from Mes �a (i.e., 00 < MK � !�0) then the

mandaphala is additive.

(ii) If the mandakendra is within 6 �� from �� (1800 < MK � ��0) then the

mandaphala is subtractive.

6. Rationale for the mandaphala of the Sun

��

seventh century has given a very good rational approximation for sine of an angle.

In fact, it is important to note that here the angle need not be small. His formula is

θθ−−

θθ−≈θ

)()(

sin18040500

1804 … (6.1)

�� 8,9.

The sine values according to (6.1) of Bh�skara I are compared with the actual

values (obtained from calculators or tables) correct to 3 decimal places for angles

from 00 to 900 at intervals of 100 in Table 2.

12

Table 2 Sine values of angles

Angle �� Angle �� Actual � �� Actual 00 0.000 0.000 500 0.765 0.766

100 0.175 0.174 600 0.865 0.866

200 0.343 0.342 700 0.939 0.940

300 0.500 0.500 800 0.985 0.985

400 0.642 0.643 900 1.000 1.000

!�"��#� � $� �� mandakendra MK and multiplying equation (6.1) both sides by

R (=120), we get

�� = ( )MK

MKMKMK

4

18010125

120180−−

×− )( … (6.2 )

where MK stands for the bhuja of mandakendra. This expression has been used by

��"�� (about 11th Cent.). We have equation (6.2):

�� (MK) = MKMK

MKMK)(

)(−−

×−18040500

480180

99

180

99

40500

48099

180

MKMK

MKMK

−−

×

×

−

= (dividing by 9 x 9)

9920500

48099

20

MKMK

MKMK

−−

×

−

= …(6.3)

Now, according to the ��,

The parama mandaphala (i.e., maximum mandaphala) of Sun = 4311257

125 00

′′′= .

∴ Mandaphala of the Sun = 12057

125 ajyamandakendr×

13

=

−−

×

−

××

9920500

48099

20

12057

125MKMK

MKMK

using (6.3)

9920500

499

2057

125

MKMK

MKMK

−−

×

−

= =

9920500

9920

57

500

MKMK

MKMK

−−

−

−

−

−

=

57

50099

20

57

500500

9920

MKMK

MKMK

=

7719298899

2057

9920

.

MKMK

MKMK

−

−

−

i.e., Mandaphala of the Sun

999

2057

9920

MKMK

MKMK

−

−

−

≈

which is the same as (5.1) and MK means BMK. Here Gan �� ña has

approximated 8.7719298 to 9 in the denominator.

Note : The usual formula for the mandaphala of the Sun is Ra

sin m

where R = 3600, a the periphery of the manda epicycle (degrees) and m is the Sun’s

anomaly (from the apogee, mandocca).

When m = 900, sin m = 1 ∴ parama mandaphala = Ra

According to the ��

= radianin

Ra

Ra

57

125 ⇒ degrees

57

125180 =π

×Ra

⇒57

1251

2=

π×a

(taking R = 3600) ⇒ 57

1252 ×π=a deg. ⇒ 778915130.=a

14

According to the book, Ancient Indian Astronomy10 by S. Balachandra Rao, the

value of a for the Sun is between 110.80781 and 120.31284, based on the eccentricity of

the earth’s orbit.

Note : In the modern expression for the equation of centre, considering only the

leading term of the series, as a first approximation, we have

Equation of Centre ≈ (2e) sin m.

The coefficient 2e corresponds to Ra

of the classical Indian astronomers.

Example : Mandocca of the Sun = 2R 180 (fixed)

Mean Sun = 243141 0 ′′′R

Mandakendra of the Sun (MK) = Mandocca – Mean Sun

= 816443243141182 000 ′′′=′′′− RR

Since 0 < MK = 00 90816443 <′′′ ,

Bhuja of MK = 8164430 ′′′ denoted by BMK.

∴ Mandaphala of the Sun =

−

−

−

999

2057

9920

BMKBMK

BMKBMK

= 820315079062182038248

61655873 00 ′′′== ...

.

Mandaphala is additive because 00 < MK = 8164430 ′′′ <1800.

Note : According to the formula, using the sine function, we have

Mandaphala of the Sun mRa

sin= (where a = 130.778915, R = 3600, m 8164430 ′′′= )

= 0264779.0)816443sin(360

778915.13 0 =′′′ radian = 47.11310 ′′′ .

15

We note that the mandaphala of the Sun, without the sine-function, is very close to

that obtained using the trigonometric function. The error is only about 33 seconds of

arc.

Now, we have

Mandaphala corrected Sun = Mean Sun + Mandaphala

= 82031243141 00 ′′′+′′′R = 014451 0 ′′′R .

After the manda correction, the Grahal�� adopts the cara correction (explained

later in section 10).

7.1 Mandaphala (Equation of centre) of the Moon

The method is as follows :

(i) Find the Mandakendra (MK) of the Moon

(ii) Subtract 6

MK from 30

(iii) Multiply step (ii) by 6

MK

(iv) Divide the result of step (iii) by 20 and subtract the quotient from 56

(v) Divide the result of step (iii) by that of step (iv).

The result gives the mandaphala of the Moon

i.e, Mandaphala of the Moon =

−

−

−

2066

3056

6630

MKMK

MKMK

Here, MK is actually the bhuja of the mandakendra.

Note : In the case of the Moon, the Grahal�� gives three corrections in the

order, cara, �� and ��. After obtaining the Moon with these three

corrections, finally the manda correction (i.e., the equation of centre) is applied.

16

7.2 Rationale for the mandaphala of the Moon

We have �� = MKMK

MKMK)(

)(−−

×−18040500

480180

according to �� t �a ��

Dividing the numerator and the denominator by 6 x 6,

66

180

66

405006

480

6

180

MKMK

MKMK

MKajy

−−

×

×

−

=)( …(i)

According to the �� we have the maximum mandaphala,

Parama mandaphala of the Moon = 50

∴ Mandaphala of the Moon = 120

5 ajyamandakendr×

=

−−×

×

−×

66301125120

48066

305

MKMK

MKMK

using (i)

=

66301125

663020

66301125

6630

120

2400

MKMK

MKMK

MKMK

MKMK

−−

−

=

−−

−

=

2066

30

20

1125

6630

MKMK

MKMK

−

−

−

=

−

−

−

2066

302556

6630

MKMK

MKMK

.

i.e., Mandaphala of the Moon

2066

3056

6630

MKMK

MKMK

−

−

−

≈

This is the expression given by the Grahal��7 (Ch. 2, �� 3).

17

Note : The usual formula for the mandaphala of the Moon is mRa

sin .

When m = 900, sin m = 1.

Now, according to the ��, the maximum mandaphala is R

a(radian).

It is given that Parama mandaphala of the Moon = 50

i.e., 415927.31525180

3605

180 0≈⇒×π=⇒=π

×⇒=π

× aaa

R

a.

According to the book Ancient Indian Astronomy10 by S. Balachandra Rao the

value of a for the Moon lies between 360.8085 and 420.22932 based on the eccentricity

of the Moon’s orbit.

8. True daily motions of the Sun and the Moon

GL gives the method (Ch.2, �l. 4) as follows :

I. True daily motion of the Sun :

We first find the ‘gatiphalam’ as follows :

(i) Find the bhuja of the mandakendra (MK) of the Sun in degrees

(ii) Find kot�i of MK

(iii) Divide kot�� by 20

(iv) Subtract the result of step (iii) from 11

(v) Multiply the results of step (iii) and step (iv)

(vi) Divide the result of step (v) by 13. This gives the ‘gatiphalam’ of the Sun in

�� (minutes of arc)

i.e., Gatiphalam of the Sun = 13

202011

− kotikoti

If the mandakendra of the Sun is within 6 �� from Karka (i.e., if 900 < MK <

2700) then add ‘gatiphalam’ to the mean daily motion of the Sun to get true daily

motion.

18

If mandakendra of the Sun is within 6 �� from Makara (i.e., If MK is in IV or I

quadrant), subtract ‘gatiphalam’ from the mean daily motion to get the true daily

motion of the Sun.

Example :

Mandakendra (MK) of the Sun = 8164430 ′′′ ∴ bhuja = 8164430 ′′′

Kot�� of MK = 900 – bhuja = 24314681644390 000 ′′′=′′′−

∴ Gatiphalam of the Sun = 13

202011

kotikoti

−

43142315448413113

20

243146

20

24314611

00

.. ′′′′′′=′=

′′′

′′′−

= .

Since MK of Ravi is in the first quadrant (i.e., within 6 �� from Makara) we have to

subtract ‘gatiphalam’ from the mean daily motion of the Sun.

∴ True daily motion of the Sun = Mean motion – ‘gatiphalam’

= 4314231895 .′′′′′′−′′′ = 57815375 .′′′′′′

Remark : The usual trigonometric formula for the correction to get the true daily

motion of the Sun according to the classical Sanskrit texts is given by

∆∆−=∆

tM

MRb

n cos

where b = 14, R = 360 and the mean daily motion of the Sun, 8095 ′′′=∆

∆t

M

∴ ( )[ ] ( ) 46.38318095816443cos360

14 0 ′′′′′′−=′′′′′′−=∆n

Remark : The gatiphalam of the Sun, obtained without the use of trigonometric

function differs from that using cos M only by about 5 ′′ of arc.

19

II. True daily motion of the Moon

We first find ‘gatiphalam’ for the Moon as follows :

(i) Find the mandakendra MK of the Moon and its bhuja

(ii) Find kot�i of the mandakendra (MK); kot�i = 900 – bhuja

(iii) Divide kot�i by 20

(iv) Subtract the result of step (iii) from 11

(v) Multiply the results of steps (iv) and (iii)

(vi) Multiply the result of step (v) by 2

(vii) Divide the result of step (vi) by 6

(viii) Add the results of steps (vi) and (vii)

i.e., Gatiphalam of Moon =

+

−

6

22

202011

kotikoti and

True daily motion of the Moon = Mean motion of the Moon ± gatiphalam

If MK is within 6 �� from Karka (i.e., if the mandakendra (MK) is in II and III

quadrant) add gatiphalam to the mean daily motion.

If MK is within 6 �� from Makara (i.e., if MK is in I and IV quadrant), subtract

gatiphalam from the mean daily motion.

Example :

Mandakendra of the Moon = 71211157121253 00 ′′′=′′′R

Bhuja of MK = BMK≡′′′=′′′− 3474647121115180 000

∴ Kot�i of MK = 900 – BMK = 71212534746490 000 ′′′=′′′−

∴ Gatiphalam of the Moon =

+

−

6

22

202011

kotikoti

= 428382640272826

22

20

712125

20

71212511

00

′′′′′′=′=

+

′′′

′′′− . … (8.1)

Since MK = 71211150 ′′′ is in the II quadrant, the gatiphalam is additive. Therefore, we

have

20

True daily motion of Moon = Mean motion of the Moon + gatiphalam

= 3198142838253079 ′′′≈′′′′′′+′′′

Remark : From the usual trigonometric formula for the Moon, we have

∆∆−

∆∆=

∆∆

ta

tM

MRb

tn

cos

where a = Moon’s apogee and its mean daily motion 146 ′′′=∆∆

ta

,

b = 310, R = 3600, 53079 ′′′=∆

∆t

M, the Moon’s mean daily motion.

We have the bhuja of the mandakendra,

M = BMK = 3474640 ′′′

∴ 64448214653079347464360

31 0 ′′′′′′=′′′−′′′′′′=∆∆

])[cos(tn

…(8.2)

We observe that even in the difference between (8.1) and (8.2) is less than 6 ′′ .

9. �� and carakhan�d�as of a given place

(Gnomon’s shadow and ascensional differences)

I. To find �� :

The day on which the true longitude of the �� (tropical) Sun becomes

0000 ′′′ (i.e., on the equinoctial days) determine the shadow of a 12 an ��gula long cone

(��ku) placed on a plane surface at the noon. This shadow length is called �� or

aks ��. In other words, it is the shadow of the ��ku at the equinoctial noon.

II. To find carakhan�d�as :

Multiply �� by 10, 8 and 3

10 respectively. We get three khan�d�as (set of

values). They are called carakhan�d�as which are in �� (seconds of arc).

Example : �� of Almora [Long. 790 E 40’ ; Lat. 290 N 36’ ] = 6|49 a¡gulas

The three carakhan�d�as are

(6|49) x 10, (6|49) x 8 and (6|49) x3

10

= 68.166, 54.533 and 22.722

� 86 ′′ , 55 ′′ and 32 ′′

21

Remark : The latitude φ of a place can be obtained from the �� of the place as

follows :

In the right-angled �le ABC (Fig. 2), we have

12

apalabhABBC

kunasapalabh

===φ��

tan

∴ �� = 12 tan φ an��gula

From this we have

=φ −

121 aPalabh

tan

For example, the �� of Almora ( φ = 29036’) is

12 tan φ = 12 tan (29036’) = 6ang49prat.

Note : 1 an��gula = 60 pratyan��gulas.

10. Cara correction

The GL7 gives the method of finding cara as follows (Ch.2, ��)

(i) Find �� Ravi (tropical Sun)

(ii) Find bhuja of the �� Sun. Express it in ��, ��, �� and ��

(iii) The number which represents �� gives the number of elapsed khan�d�as

(iv) Find the bhogya khan�d�a (i.e., khan�d�a to be covered) and multiply it by the

remaining �� etc.

(v) Divide the result of (iv) by 30

(vi) Add the result of (v) to the sum of the elapsed khan�d�as.

This give the cara.

Note : During the day time

(i) If �� Ravi (SR) is within 6 �� from Mes �� (i.e., 00 < SR < 1800), then cara is

negative.

(ii) If �� Ravi (SR) is within 6 �� from Tula (i.e., 1800 < SR < 3600) cara is

positive.

A

B C Shadow (��)

Fig. 2

φ

12 gna�

��¡ku

22

During the night time

(i) cara is positive if 00 < SR < 1800.

(ii) cara is negative if 1800 < SR < 3600.

Example : �� an��gulas 45 pratyan��gula = 5|45 a¡gulas.

The three carakhan�d�as are

(5|45) x 10, (5|45) x 8 and (5|45) x 3

10

= 75 ′′ , 64 ′′ and 91 ′′

Suppose �� Ravi = 0145231 0 ′′′R , bhuja = 0145231 0 ′′′R .

The number 1 in the �� position implies that the number of elapsed khan�d�as = 1.

∴ Elapsed khan�d�a = 75 ′′

Bhogya khan�d�a = 64 ′′

Remaining �� etc. of �� Ravi = 0145230 ′′′

Now, 6330

109964

30

0145230

0

′′≈=′′×′′′

∴ Cara = Elapsed khan�d�a + 63 ′′ = 396375 ′′=′′+′′

Since �� Ravi is within 6 �� from ��, the cara is negative (during the day

time).

∴ Cara corrected nirayan�a Sun = 73245139014451 00 ′′′=′′−′′′ RR

Remark : The latitude of �� φ = 250 N 19’.

Therefore, �� = 12 tan φ = 12 tan 250 19’ = 5|40 a¡gulas

�� (early 17th Cent.) has taken its value as 5|45

a¡gulas

23

The carakhan�d�as are

(5|40) x 10, (5|40) x 8 and (5|40) x3

10

i.e., 76665 .′′ , 4154 .′′ and 9281 .′′

The elapsed khan �� = 76665 .′′ , Bhogyakhan �� = 4154 .′′

R

Remaining �� etc. = 0145230 ′′′ . Now, 18163415430

0145230

.. ′′=′′×′′′

∴ Cara = Elapsed khan �� + 18163 .′′

= 7765 .′′ + 18163 .′′ � 9529 .′′

Therefore, cara corrected nirayan�a Sun

= 7324519529014451 00 ′′′≈′′−′′′ RR . .

We notice how the traditional astronomers were remarkably correct in

determining the �� and hence the latitude of a place.

11. Applying the cara, �� and �� corrections to the Moon

(1) Cara correction for the Moon :

Subtract

×

9

2 Cara�� (minutes of arc) from the mean position of the Moon

to get the cara corrected Moon. Here, cara must be taken in seconds (“) of arc.

i.e., Cara corrected Moon = Mean Moon ′

×−

92 Cara

(2) �� correction for the Moon (using mandaphala of the Sun) :

Divide the mandaphala of the Sun by 27. The result will be in degrees, minutes of

arc. Add this to the cara corrected Moon. That is

�� corrected Moon = Cara corrected Moon + 27

Sun the of mandaphala

(3) �� correction for the Moon :

Find the yojanas �� ). Divide it by 6 to

get the �� correction in �� (minutes of arc).

24

Note : In modern astronomy, the equivalent of �� is given by ( )

3600λ−λ

part of

the day where λ and λ 0 are respectively the longitudes of a place and of the central

meridian (in degrees). Let φ be the terrestrial latitude of a place so that the R sine of

its co-latitude CP is 3438 sin (900 - φ). The diameter of the earth is taken as 1600

yojanas (i.e., about 8000 miles). The maximum (equatorial) circumference,

MC = 2 π(800) = 1600 π yojanas (Note: 1 yojanas � 5 miles).

The corrected circumference CC at a place is given by

8343 ′⋅= CPMC

CC

From these, we get

�� correction = ( )

CCDMCPMC ⋅⋅⋅

λ−λ34383600

00

= ( )

⋅

⋅⋅⋅λ−λ

343834383600

00

CPMCDMCPMC

= ( )

DM⋅λ−λ

360

00

where DM is the daily motion of a planet.

�� gives the expression for the �� correction in �� as one-

sixth of the distance in yojanas �� e commentator

�� from the central meridian as 64 yojanas.

Let r = 4000 miles � �� yojanas (Fig. 3)

We have rr1=φcos

∴ )cos( φ= 8001r yojanas.

The circuference of the small circle through a place A is given by

O E Q

C r1

r

r

/ A

�

Fig. 3

25

φπ=π cos)(16002 1r yojanas.

Taking the circumference of the small circle as 3600, we have

( )φπcos1600 yojanas = 3600

∴ x yojanas = φπcos1600

360x

φ

≈cos

).( x07160 degrees

φ=

cos).(

15

07160 xin hours or

φcos).(

6

07160 x in ghat��

� � ��

�� correction = 91160

53079

63256

64071600

′′′≈′′′

×′

.)cos(

).( … (1)

Now, according to the �� .

�� correction = 6

1(distance in yojanas)

= 0401646

1 ′′′≈× … (2)

The difference between (1) and (2) is just about 92 ′′ .

Example : Mean Moon = 4201206 0 ′′′R ,

Moon’s apogee, Candrocca = 34451410 0 ′′′R

Cara = 39 ′′

(i) Cara correction for the Moon :

Cara corrected Moon = Mean Moon 9

2 Cara×−

= ′

×−′′′

9

9324201206 0R 04024201206 0 ′′′−′′′= R 4494196 0 ′′′= R

(ii) �� correction for the Moon :

Mandaphala of the Sun = 820310 ′′′ (additive)

�� corrected Moon = Cara corrected Moon + 27

Sun the of mandaphala

26

= 4494196 0 ′′′R

27

820310 ′′′+ = 4494196 0 ′′′R + 12300 ′′′ = 5035196 0 ′′′R

(iii) �� correction for the Moon :

�� = 64 yojanas.

∴ �� correction = 04016

64 ′′′=′

∴ �� corrected Moon

= �� corrected Moon + �� correction

= 5035196 0 ′′′R 0401 ′′′− = 5224196 0 ′′′R

Now, we have

the Moon after all the three corrections = 5224196 0 ′′′R .

Manda correction for the Moon :

Candrocca = 34451410 0 ′′′R , the above corrected Moon = 5224196 0 ′′′R .

MK � Mandakendra = 8121253 0 ′′′R

Bhuja of MK = 1800 8121253 0 ′′′− R = 2474640 ′′′

Mandaphala of the Moon =

20

6630

56

6630

−

−

−

MKMK

MKMK

[from sec. 7.1]

20

6

247464

6

24746430

56

6

247464

6

24746430

00

00

′′′

′′′−

−

′′′

′′′−

= = 833340 ′′′

Since MK = 8121253 0 ′′′R < 1800, the mandaphala is additive.

27

∴ True Moon = Mean Moon (after three corrections) + mandaphala

= 5224196 0 ′′′R + 833340 ′′′

= 3061246 0 ′′′R = 30612040 ′′′ .

12. Comparison of true positions according to different texts

We compare the true positions of the Sun and the Moon obtained from different

texts viz., Brahmagupta’s Khan�d�� 3, �� 6 (later version),

Karan��11 � �� 7 with the modern values for two

different dates in Table 3.

Table 3 : True positions of the Sun and the Moon

May 14, 1612 A.D. (G) August 11,1998 at Varanasi at Bangalore

Sun Moon Sun Moon Texts

D M S D M S D M S D M S 35 51 42 204 45 50 113 55 58 331 52 14

35 49 19 204 45 40 113 51 12 332 19 35

35 58 50 205 05 46 114 09 57 333 00 17

35 43 00 204 16 03 113 50 36 331 52 37

Khan�d��

��

Karan��

��

Modern 35 13 00 204 35 00 113 35 14 332 51 14

From the values in Table 3, we observe that the true positions according to different

texts are very close. However, between GL and the modern values, the difference

exists due to the Daivajña’s dispensing with sine and cosine and also the need to

update the parameters.

13. Conclusion :

In the present paper we have discussed the procedure of obtaining the true

positions of the Sun and the Moon according to the popular Sanskrit text

�� (GL) of Gan��a Daivajña (early 16th cent.). His justifiable

approximation for the trigonometric function sine (even when the argument is not

small) is explained in respect of determining the mandaphala (equation of centre) of

the Sun and the Moon. Despite these approximations, dispensing with

trigonometric functions, the results obtained from the GL procedures are reasonably

28

good. The true daily motions of the Sun and the Moon by the simplified procedure

of GL are shown to be close to those obtained by the conventional procedure. The

three main corrections to obtain the true Moon viz., cara, �� and ��,

besides the mandaphala (equation of centre) are explained. Illustrative examples are

worked out according to the GL procedure and the results thus obtained are

compared with those according to other traditional texts as also with the modern

procedure. We notice that the results according to GL are reasonably satisfactory.

In the light of the trigonometric approximations and the fact that Gan ��

Daivajña lived more than five centuries ago, the small differences between his results

and the modern ones are ignorable. However, there is the need to update his

parameters.

In fact, Gan �� ña’s procedures of computing the true positions of the

Sun and the Moon as also the geometrical model are truly good. The veracity of the

same in the computations of lunar eclipse12 and the mean planetary positions13 was

highlighted by the present authors in their earlier papers. The English exposition of

the entire text of Grahalaghavam by the present authors is published by INSA14.

Acknowledgement :

We acknowledge our indebtedness to the History of Science Division, INSA,

New Delhi, for sponsoring our research project under which the present paper is

prepared.

29

References

1. Balacahandra Rao, S., Indian Astronomy – An Introduction, Universities Press, Hyderabad, 2000, pp.87-90.

2. �� i � �� , ed. Sudhakara Dvivedi, Kashi Skt. Series, No.72, Benaras, 1929. (ii) with Eng. Exposition by D.Arkasomayaji, Kendriya Skt.Vidyapeetha, Tirupati, 1980.

3. Khan�d�� of Brahmagupta (628 A.D.), (i) with Com. Pr�� ed. & tr. by P. C. Sengupta, Calcutta, 1934, 1941 . (ii) with Bhattotpala’s Com., ed. & tr. Bina Chatterjee, Parts I & II, New Delhi, 1970.

4. Pañcasiddh�� Dvivedi (Reprint), Motilal Banarsidass, 1930. (ii) with tr. and notes of T. S. Kuppanna Sastri, Cr. Ed. by K. V. Sarma, PPST Foundation, Madras, 1990.

5. �� a I (b.476 A.D.), (i) Cr.ed. & tr. by K. S. Shukla & K. V. �� t ��

Trivandrum, 1977.

6. �� (i) Tr. by Rev. E. Burgess, ed. by Phanindralal Gangooly & Intn. By P.C. Sengupta, Motilal Banarsidass, Delhi, 1989. (ii) with com. of ��!�� "� #�� $� �� %$�� &��

7. �� of Gan�� ña �� Hindi com. by Kedranatha Joshi, Motilal Banarsidass, 1981; (ii) with com. of �� Skt. Series, Varanasi, 1994.

8. �� ! �"#$ %�&�� '�� '�(� )�� *� +� +�,��Lucknow, 1960, Ch.7, ��. 17-19.

9. Bag, A. K., Mathematics in Ancient and Medieval India, Chowkhambha Orientalia, Varanasi, 1979, pp.231.

10. Balachandra Rao, S. Ancient Indian Astronomy – Planetary Positions and Eclipses, B. R. Publishing Corp. Ltd., Delhi, 2000, pp 174.

11. �� !! �� +,�� +,��

Dvivedi and Hindi tr. by Dr.Satyendra Mishra, Varanasi, 1991.

12. Balachandra Rao, S., Uma, S.K. and Padmaja Venugopal Lunar Eclipse Computation in Indian Astronomy With Special Reference �� , Ind.I. of His. of Sc. (IJHS), 38.3,255-271, New Delhi, 2003.

13. Balachandra Rao, S., Uma, S. K. and Padmaja Venugopal, Mean Planetary �� ghavam, Ind.J. of His. of Sc. (IJHS), 39.4 (2004), 441-466.

14. Balachandra Ra, S., and Uma S.K., Grahalaghavam of Ganesh Daivajna – English Exposition, Mathematical Explanation, Derivations, Examples, Tables and Diagrams, Ind.J. of His of Sc, Ind.Nat.Sc.Ac; New Delhi, 2006.

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��

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