ramasubramanian - algorithms in indian astronomy (2005)

Algorithms in Indian Astronomy

K.Ramasubramanian

Cell for Indian Science and Technology in Sanskrit

Department of HSS, IIT Bombay

Powai, Mumbai 400 076, India

[email protected]

Abstract

Indian Astronomy is rich in algorithms. The algorithms pre-sented in the Indian astronomical texts have varying degrees ofcomplexities starting from the simple trairasika rule, to the treat-ment of parallax in a solar eclipse or the computation of the ele-vation of lunar cusps. In the present article we will discuss a fewalgorithms that are representative of the ingenuity and continuityof the Indian astronomical tradition. We start with the interpo-lation formula presented by Brahmagupta (c.665 AD) and thenproceed to describe a select few algorithms from Tantrasangrahaof Nilakan. t.ha composed in 1500 AD. Here we present the algo-rithm for the calculation of time from shadow measurements andthe exact algorithm for the computation of lagna and the timefor the duration of an eclipse. We also comment on the iterativeprocess known as avises.akarma which aims at circumventing theproblem of interdependencies among several variables.

1 Introduction

It is not uncommon to find words which originate with a different conno-tation and in due course pick up a completely different connotation. Theword algorithm forms a good example of this. Its origin can be tracedback to the Persian mathematician, al-Khwarazmi (800-847 AD). It isquite interesting to note the observations made by D.E. Knuth in thiscontext [1]:

183

184 K.Ramasubramanian

In the middle ages, abacists computed on the abacus andalgorists computed by algorism. Following the middle ages,the origin of this word was in doubt, and early linguists at-tempted to guess at its derivation by making combinationslike algiros [painful] + arithmos [number]; others said no, theword comes from King Algor of Castile. Finally, historiansof mathematics found the true origin of the word algorism: itcomes from the name of a famous Persian textbook author,Abu Jafar Mohammed ibn Musa al-Khwarizmi ( 825 AD)- literally, Father of Jafar, Mohammed, son of Moses, na-tive of Khwarizm. Khwarizm is today the small Soviet cityof Khiva. al-Khwarizmi wrote the celebrated book Kitab aljabr wal-muqabala (Rules of restoration and reduction);another word, algebra, stems from the title of his book,although the book wasnt really very algebraic.

Gradually the form and meaning of algorism got distorted; The changefrom algorism to algorithm is any bodys guess. The remarks bythe well-known historian C.B.Boyer in this context are also noteworthy[2].

. . . . . . when subsequently Latin translations of his (alKhwarizmis) work appeared in Europe, careless readers be-gan to attribute not only the book but also the numerationto the author. The new notation came to be known as that ofal-Khwarizmi, or more carelessly, algorismi; ultimately thescheme of numeration making use of Hindu numerals cameto be called simply algorism or algorithm.

The terms process, method, technique, procedure, routine, and so onall essentially refer to a sequence of operations to be carried out to ac-complish a given task. The word algorithm though similar, connotessomething more. For a procedure to be termed algorithm it must termi-nate after n steps and upon termination it must yield a sensible result.However, this is not true of all procedures. In this sense, all algorithmsare procedures; but all procedures are not algorithms.

Indian astronomy is essentially algorithmic in nature. The algorithmspresented are precise and fairly sophisticated [3]. Some of them are

Algorithms in Indian Astronomy 185

amazingly accurate [4]. We shall illustrate these points by consideringa few examples.

2 Brahmaguptas interpolation formula

Interpolation is the art of reading between the lines in a table. Therule of trairasika [5,6] employed in Indian astronomy is close to what isknown as the first-order interpolation in modern parlance. This tech-nique has been extensively applied to solve a variety of mathematicaland astronomical problems, beginning from the evaluation of sine func-tion to the calculation of eclipses, at least from the time of Aryabhat. ya(c.499 AD).

It is quite interesting to note that Brahmagupta introduced the secondorder interpolation formula to determine more precise value of the sinefunction, called jya , for an arbitrary angle, from the set of tabulatedvalues of sine given at fixed intervals. The following verse from hisfamous work Khan. d. akhadyaka (c.665 AD) explains the algorithm [7]:

! "

#

%$ & '( ) +*

( ,.-

/

0 213

/

1

45 ) 0 67 819:

) ;


Before we represent the content of the above verse in a mathematicalform, it would be useful to introduce the terminologies and the notationemployed. Let the equal intervals chosen be represented by . That is,

xn xn1 = xn+1 xn = = 900, (1)

where the variable x denotes the angle and xis are multiples of 900.

The term Bhogya refers to the present interval between xn and xn+1.The term khand. aka refers to the I order tabular differences in the sinevalues and we denote it by n.

n = f(xn) f(xn1). (2)

Let us suppose that the value of the function at three pointsf(xn1), f(xn), and f(xn+1) are known, and it is required to find outf(xn + ). Now, the formula given by Brahmagupta may be writtenas

f(xn + ) = f(xn) +

[(n + n+1)

2 (n n+1)

2

], (3)

where 0 < < 1. With a little algebraic manipulation, and suitablyrearranging the terms, the above equation may be rewritten as

f(xn+) = f(xn)+

n+1 +

(

1

) (n n+1)

2. (4)

Clearly, the formula given by Brahmagupta is identical with the stan-dard quadratic interpolation formula [8].

3 Algorithm for finding the time from shadow

The technique of making a fairly good estimate of the time from theshadow of an object cast by the Sun has been in vogue from time im-memorial. Different cultures and traditions across the world have de-vised simple instruments for this purpose. In the Indian astronomicaltradition the instrument used is called sanku .


E

W

rays from the sun

Circle drawn around the Sanku

Y

O

X

Sankuz

Figure 1: Zenith distance and the length of the shadow.

The sanku essentially consists of a rod of suitable thickness and height.Generally the height is taken to be 12 angulas . For performing experi-ments with sanku , it must be placed at the centre of a circle as shownin Fig.1. Here, OX represents the sanku and OY is its shadow cast bythe sun. If the longitude of the sun, is known at a given instant, thenits declination, at that instant can be calculated using the relation (seeFig.2)

sin = sin sin, (5)

where is the obliquity of the ecliptic. The procedure for obtainingaccurate values of the observers latitude, , are discussed in detail inseveral Indian astronomical texts [9].

Ecliptic

Celestial Equator

(Vernal equinox)

N

S

Figure 2: Declination of the Sun.


The formula given by Nilakan. t.ha in his celebrated work Tantrasangraha(c.1500 AD) for determining the time from shadow measurement turnsout to be essentially a function of , and z, where z is the zenithdistance of the sun. Since and are already found through shadowmeasurements, only z needs to be known. For this, consider the triangleOXY in Fig.1. It can be easily seen that

sin z =OY

XY

or z = sin1(

OY

XY

), (6)

where XY is the hypotenuse given by

122 + OY 2. Thus, the zenithdistance of the sun can also be obtained at any time by measuring theshadow cast by the sanku .

The algorithm for finding the time from shadow measurements is pre-sented by Nilakan. t.ha as follows [10]:

. ) : 6$

*

*

) (


in the first line of the quotation is used to denote R cos z.3 Carajya isR sin. With this, the the first two lines of the verse translate to therelation

R sin =

[R cos z

cos cos R sin

](7)

In the latter half of the quotation given above it is mentioned that tothe arc of the above (), the ascensional difference, which is the angularseparation between the vernal equinox and the celestial object measuredalong the celestial equator (see Fig.3), has to be applied in the reverseorder to obtain the required time t. That is,

t = . (8)

Substituting for we have

t = (R sin)1[

R cos z

cos cos R sin

]. (9)

The above result can be easily understood using the tools of sphericaltrigonometry.4 For this, consider Fig.3 where S is the Sun on its diurnalpath whose zenith distance is z, corresponding to the arc ZS. Thepoint where the sun sets in the western part of the horizon is denotedby St. The segment PW is a part of the 6 0

clock circle, and the angleZPW = 90o. Applying four-part formula to the triangle PWSt, it canbe shown that, the ascensional difference is given by the relation(well known in Indian Astronomy)

sin = tan tan . (10)

Further, it may be noted that SPSt = + is the angle to be coveredby the Sun from the given instant up to the sunset. This angle dividedby 6 and 15 give the time that is to elapse before sunset in hours andnad. ikas (time unit of approximately 24 minutes) respectively.

3Generally R cos z, which is the perpedicular distance of the sun from the horizon,is called mahasanku in order to distinguish it from the gnomon (dvadasangulasanku)used for shadow measurement. But sometimes mahasanku would be simply referredas sanku , as in the above verse, and what it refers to would be clear from the context.

4Detailed demonstrations of the algorithms enunciated in Indian astronomicaland mathematical texts are given in the famous Malayalam work Gan. ita-Yuktibhas. a(c.1530 AD) of Jyes.t.hadeva [Ref.6, Vol.I & II].


Equat

or

(Ghatik

avrtta)

(Tiryak vrtta)

6Oclock circle

Z

L M

C

P

D

W

Q

S

A

B

90

90

dirunal pathof the sun

90

S

Horizon

t

Figure 3: Determination of time from shadow measurements.

Applying the cosine formula to the spherical triangle PZS, we have

cos z = sin sin + cos cos sin ,

or,

sin =cos z

cos cos tan tan

=cos z

cos cos sin.

Hence, the time, t that is yet to elapse before sunset in angular measureis

t = + = sin1[

cosz

cos cos sin

]+ , (11)

which is the same as Eq.(9) given by Nilakan. t.ha .


The method to determine the time elapsed after sunrise is exactly sim-ilar. It is worth mentioning in this context that, in finding z, the cor-rection due to the finite size of the Sun and its parallax have also beentaken into account by Nilakan. t.ha [11]. If z

were the apparent zenithdistance, then the true zenith distance z is given by

z = z + with = ds p,

where ds is the angular semi-diameter of the Sun and p its parallax. As is small,

R cos z = R cos(z + )

R cos z (

R sin z

R

). (12)

This is precisely what is stated to be the mahasanku, R cos z, in an earlierverse [12], which is to be used in the expression (Eq.9) for determiningthe time from the shadow.

4 Algorithm for finding lagna

Lagna refers to the orient ecliptic point, that is, the point of the eclip-tic which intersects with the eastern horizon at any desired instant.Nilakan. t.ha while discussing the procedure for finding lagna, in hisTantrasangraha first presents the conventional method, which could befound in many of the earlier texts on Indian astronomy. Later, pointingout that this procedure would give only approximate results, he proceedsto give an exact algorithm for finding the lagna .

4.1 Conventional method

The following verses of Tantrasangraha describe the standard procedureused for the computation of lagna at any desired instant [13].

1 ) : 4 " ) . ) 0 ' * *

( "

) " *

" "

. (

/

:)

)


:

: " ) 1 "

67) "

( *

4 .

#

"1 &

" 1 ( ( * *

"

/

) . 1

,"

1 "

-

/


S

P

NS

E

R

R

1

2

Diurnal circle

(pole)H

h

tR1

Horizon

Equator

Ecliptic

(zenith)Z

Figure 4: Determination of praglagna (orient ecliptic point) by theconventional method

Let s be the sayana longitude5 of the Sun. Suppose the Sun is in the ith

rasi (in the Fig.4, it is shown to be in the first rasi ), whose rising timeat the observers location is given by Ti. If Ri be the angle remaining tobe covered by the Sun in that rasi (in minutes), then the time requiredfor that segment of the rasi to come above the horizon is given by

tRi =Ri Ti30 60 , (13)

where Ti is in ghat.ikas . A factor of 60 in the denominator indicatesthat the result tRi is expressed in pran. as.

6 Subtracting this time tRifrom the time elapsed since sunrise h, we have

h = h tRi .From h the time required for the subsequent rasis to rise, Ti+1, Ti+2,etc., are subtracted till the remainder r remains positive. That is,

r = h Ti+1 Ti+2 . . . Ti+j1 (r + ve)5This refers to the longitude measured from the vernal equinox. Here, it may be

noted that in Indian astronomy, nirayan. a longitudes are more commonly used. Theyrefer to the longitude measured from a fixed point, which is generally taken to be thebeginning point of star called Asvini. It is also referred to as mes. adi as in the lastline of the Sanskrit quotation given above.

6Ghat.ika is a measure which is close to 24 minutes and pran. a 4 seconds.


Suppose we are in the i + j th rasi whose rising time is Ti+j. Then,the portion of Ri+j , which would have come above the horizon in theremaining time r is given by

Ei+j =r 30Ti+j

(in deg.). (14)

Now, the longitude of the praglagna (L) is given by

L = s + Ri + 30 + 30 + . . . + Ei+j . (15)

If so, then the astalagna is given by

astalagna = praglagna + 180, (16)

because the horizon divides the ecliptic exactly into two parts. Thelagnas obtained by the above procedure are sayana lagnas. To obtainthe nirayan. a ones, one needs to subtract the ayanam. sa.

After describing the above method, Nilakan. t.ha remarks that it is onlyapproximate [14]:

; "

#

/

;:


: ) $7

/

(


the zenith distance of the vitribhalagna (R sinZV ). When is on thehorizon, K is on the meridian. It can be easily seen that the kalalagnais the hour angle of K, or L = ZPK. Now,

ZK = KV ZV = 90 ZV,

as K is the pole of the ecliptic. Also, KP = and ZP = 90 . Then,using the cosine formula we obtain the following expression for dr. ks.epa

R sinZV = R cos ZK = R cos sin + R cos sin cos L. (18)

S

Z

P

NS

(zenith)

(pole)

E

V

H

K

L

EclipticHorizon

Equator

Figure 6: Determination of praglagna (orient ecliptic point) - exactmethod.

This is essentially what is stated in the following verse [16].

)() ,%-

/

,%-


is within the 6 signs beginning from karkat.aka or mr. ga Capri-corn.

Here Nilakan. t.ha defines two intermediate quantities x and y whose sumor difference gives the expression for the sine of the zenith distance ofthe the vitribhalagna . They are given by

x =antyadyujya aks.a

trijya=

R cos R sinR

,

and,

y =lambaka kot.i of kalalagna

8452=

R cos R cos L sin R

.

It may be noted in the above equation that sin R

= sin 243438

is taken to be1

8452. Now, dr. ks.epa (= R sinZV , refer Fig.6) is given to be

dr. ks.epa = x y

Substituting for x and y we have,

R sinZV = R cos sinR cos cos L sin , (19)

which is same as Eq.(18). From this, cos ZV is to be calculated. Withthis, the lagna is to be found as follows [17]:

; 67

2 ) ) " !

. *

) ) 8 ) ) * *

)

)

. %$ )


result deficient from carajya is found. [This is the procedurefor the northern hemisphere]. For the southern hemispherethe operation is reversed.

Here it is essentially stated that

R sin(l s) =R(cos cos [cos H + sin])

cosZV

=R(cos cos cos H + sin sin )

cos ZV,

where l and s are the lagna and the Suns longitude respectively. Fromthe above equation, taking the inverse sine we get ls. To this, if weadd the longitude of the Sun we get the desired longitude that is lagna(l). The rationale behind the above equation can be easily understoodwith the help of Fig.6.

Applying the cosine formula to the spherical triangle PZS we have,

cos ZS = sin sin + cos cos cos H.

Similarly from the triangle ZVS we get,

cos V S =cos ZS

cos ZV.

Now, V S + SL = V S + l s = 90. Therefore, cos V S = sin(l s),which results in the above expression for sin(l s).

5 Instantaneous velocity of the true planet

Both in the modern as well as ancient Indian astronomy, the true po-sition of the planet7 is obtained from the mean position by applyinga correction to it. The correction term is known as mandaphala , inIndian astronomy, while it is known as equation of centre in modernastronomy.

In Fig.7, A represents the direction of the mandocca (apside) and itslongitude OA = $. P0 is the mean planet whose longitude called

7Though the treatment is general, in this section planet refers to either the Sunor the Moon.


A

O

P

0 MS

0

(direction of mandocca)(planet)

P

0

Q

Figure 7: The manda-samskara or the equation of centre correction.

madhyamagraha (mean longitude) is given by 0 = OP0. The circle ofradius r with P0 as centre, is the epicycle on which the mandasphut.a(true planet) is located. By construction, PP0 is parallel to OA. Thelongitude of the mandasphut.a is given by MS = OP . It can be easilyseen that

sin(0 MS) =PQ

OP=

r sin(0 $)K

,

where

K = [(R + r cos(0 $))2 + r2 sin2(0 $)]1

2 ,

is the mandakarn. a. As per the formulation in Tantrasangraha , r also

varies such thatr

K=

r

R,

is a constant. Here r is a given parameter. Then,

sin(MS 0) = r

Rsin(0 $). (20)

The difference between the true (mandasphut.a ) and the mean position,is called mandaphala, whereas the difference between the mandocca and


mean position is called mandakendra. Denoting them by and Mrespectively, the mandaphala is given by

= sin1( r

RsinM

). (21)

Thus the mandasphut.a (true longitude) of the planet is given by

MS = 0 + . (22)

It may be noted that the calculation of true position of the planet in-volves sine inverse function ( Eq.(22)). Hence, if one needs to find theinstantaneous velocity of the planet called tatkalikagati, one would haveto find the time derivative of this function. It is indeed remarkable thatan exact formula for the derivative of sine inverse function is given inTantrasangraha as follows [18]:

)

/

$ "

0 67 ) 0

) 1

(


This verse appears in the context of finding the true rate of motion ofthe Moon (instantaneous velocity) from its average rate of motion (meanvelocity). The term gati refers to the rate of change of the longitude ofthe planet.

Recalling the expression for the true longitude of the planet, Eq.(22),the rate of change of it is

d

dtMS =

d

dt0

d

dt. (24)

Here, the first term in the RHS represents the mean velocity of the planetand the second term the change in the mandaphala given by Eq.(23).In the following section dealing with the computation of the duration ofan eclipse, the daily motion of the Moon denoted by dm is nothing butthe derivative of mandasphut.a given by Eq.(23),(24).

6 Algorithm for finding eclipse duration

In Indian astronomy, the total duration of an eclipse is found by deter-mining the first and the second half durations separately. More oftenthan not, there will be significant difference between the two durations.For obtaining accurate values, they are calculated using an iterative pro-cedure, called avises.akarma, as will be explained in this section. Thesum of the two durations gives the total duration of the eclipse. For in-stance, if T1 and T2 be the first and second half durations of the eclipse,then the total duration of the eclipse is given by

T = T1 + T2,

where both T1 and T2 are found iteratively.

6.1 Half duration of the eclipse

The time interval between the Moon entering the umbral portion of theshadow and the instant of opposition is the first half duration, T1, andthat between the instant of opposition and the exit of Moon out of theumbral region is the second half duration T2. Naively, one may think


that these two durations would be equal. However, this is not truebecause of the continuous change in the velocities of both the Sun andthe Moon.

In Fig.6(a), AX = and OX = S represent the latitude of the Moon andthe sum of the semi-diameters of the shadow and the Moon respectively.If dm ds refers to the difference in the instantaneous daily motion ofthe Sun and the Moon, then the first half duration (T1) is given by

T1 =OA

Diff. in daily motion=

OX2 AX2dm ds

=

S2 2

dm ds(25)

O

O .

.X

A

M

MY

.

.

B

(a)

(b)

Ecliptic

Ecliptic

Moons orb

it

Moons orb

it

Figure 8: (a) The Earths shadow and the Moon just before the begin-ning of the eclipse and just after the release. (b) The Earths shadowand the Moon just after the beginning of total eclipse and just beforeits release.

6.2 The need for iteration

In Eq.(25), is the Moons latitude at the beginning of the eclipse. Thedenominator represents the difference in the daily motion of the Sun and


the Moon. If m and s are the longitudes of the Sun and the Moon,this difference in their daily motion called gatyantara or bhuktyantara isgiven by

gatyantara = D(t) =d

dt(m s).

Initially, and D(t) are calculated at the instant of opposition. IfMoons latitude and the rate of motion of the Sun and the Moon wereto be constant, then Eq.(25) would at once give the correct half dura-tion of the eclipse and there would be no need for an iterative procedure.However, they are continuously varying quantities. Hence, determiningT1, using the latitude and gatyantara computed at the instant of opposi-tion is only approximate and to get more accurate values avises.akarma,a special kind of iterative procedure, is prescribed.

6.3 Concept of Avises.akarma

Avises.akarma refers to an iterative process that is to be carried outwhenever there is an interdependency (anyonyasraya) between the quan-tity to be calculated and the parameters which are involved in the cal-culation. For instance, in eclipse calculation, without knowing the lati-tude at the end of the eclipse, we will not be able to give the exact halfduration, and without knowing the half-duration it is not possible tofind the exact latitude of the Moon. To get over this tricky situation,avises.akarma is recommended. The need for it is succinctly explainedSankara Variyar in his Yuktidpika as follows [19]:

0 1 (


fore, the half-duration may be determined through the avises.aprocess.

The word vises.a means distinction; therefore avises.a is without dis-tinction. Though the meanings of the words vises.a and avises.a areopposed to each other, the latter should not be taken to refer to tulyaor completely identical. In the context of mathematical calculations,it only means without distinction to a desired degree of accuracy. Inother words, in avises.akarma, the iterative process needs to be carriedout only up to a point wherein the two successive values of the resultsare without distinction for a desired degree of accuracy. Once thisaccuracy is reached the process should be terminated.

6.4 The iterative process for half duration

Now, we illustrate the avises.a process by considering the example offinding the half duration of an eclipse. Let tm be the time of oppositionor madhyakala of a lunar eclipse, and t0 be the zeroth order approx-imation of the half duration of eclipse determined with the parametervalues obtained at tm. That is,

t0 =

S2(tm) 2(tm)

D(tm)

To get the first approximation, the sum of the semi diameters S, thelatitude of the Moon and the difference in daily motion D are thendetermined at tm t0. With them the first approximation to the halfduration is obtained. It is given by

t1 =

S2(tm t0) 2(tm t0)

D(tm t0)To get the second approximation, S, and D are determined at tmt1.With these values the second approximation to the half duration is

t2 =

S2(tm t1) 2(tm t1)

D(tm t1)


Similarly, determining S, and D at tmt2, the third approximationis found.

t3 =

S2(tm t2) 2(tm t2)

D(tm t2)

This process will be continued until,

tn tn1 < , (26)

where is the desired degree of accuracy. At this stage, since t hasconverged to the desired accuracy, the iteration is terminated. In termsof the notation used earlier, the first half duration T1 = tn. Therefore,the instant of the commencement of the eclipse, known as sparsakala, isgiven by

tb = tm tn. (27)

A similar procedure is to be adopted for the determination of moks.akala(te), with the only difference that, instead of subtracting, the half du-ration t from tm, we need to add to it. If t

i be the second halfduration of the eclipse obtained after the ith iteration, then it is givenby

ti =

S2(tm + ti1) 2(tm + ti1)

D(tm + ti1)

As in the case of sparsa, here again the process of iteration has to becontinued till ti converges. That is,

tr tr1 < . (28)

At this stage, the second half duration of the eclipse and the moks.akalaare given by

T2 = t

r

te = tm + t

r (29)

Now, the total duration of the eclipse is te tb = T1 + T2.


7 Concluding Remarks

Many of the algorithms presented in the paper, barring some refine-ments, can be found even in the celebrated text Aryabhat. ya , composedby Aryabhat.a as early as 499 AD. The Kerala school of astronomy andmathematics which is well known for its pioneering work in mathemat-ical analysis and many innovations in the Indian astronomical traditionhas tried to perfect these algorithms, the culmination of which can beseen in the works of Nilakan. t.ha Somayaji. Particularly, the exact al-gorithm for the computation of lagna and the formula for the instan-taneous velocity of the planet presented by Nilakan. t.ha , are indicativeof how in the Indian astronomical tradition there has been a continu-ous endeavour to improvise and achieve better and better accuracy inall computations. As regards the avises.akarma, it will be interesting tostudy the convergence properties of this iterative process as employedin different contexts in Indian Astronomy.

Acknowledgement

The author would like to express gratitude to Prof. M.D.Srinivas ofCentre for Policy Studies and Prof. M.S.Sriram of University of Madrasfor many useful suggestions and discussions on this topic.

References:

1. Donald E.Knuth, The Art of Computer Programming, Vol. 1,Addison Wesley, 1973, p 1-2.

2. C.B.Boyer, A History of Mathematics, John Wiley and Sons, 1989,p 256.

3. See for instance, 500 years of Tantrasangraha : A Landmark in theHistory of Astronomy, Ed. by M.S.Sriram, K.Ramasubramanianand M.D.Srinivas, Indian Institute of Advanced Study, Shimla,2002.

4. (i) The verses attributed to Madhava (c.14th century) by SankaraVariyar in his commentary to Tantrasangraha (Chap 2, verses 437,438) beginning with

2 )


obtaining the sine and cosine values for any desired angle, yieldresults correct up to 7 decimal places. This is indeed a remarkableresult which may be considered far ahead of his times. (ii) Formore details and mathematical exposition of the above the readermay refer to the article by M.S. Sriram published in the presentvolume.

5. Llavat of Bhaskaracarya with the commentary of Sankara andNarayan. a, Ed. by K.V.Sarma, VVRI, Hoshiarpur, 1975, ver.73, p178.

6. Gan. ita-yuktibhas. a, Ed. with English Translation by K.V.Sarma,with Explanatory Notes by K.Ramasubramanian, M.D.Srinivasand M.S.Sriram (in press), Vol. I, Chap 4.

7. Khan. d. akhadyaka of Brahmagupta, Ed. and Tr. by Bina Chat-terjee, Motilal Banarsidass, 1970, Vol. 2, Uttarakhan. d. akhadyaka,Chap 1, ver.4.

8. Erwin Kreyszig, Advanced Engineering Mathematics, John Wiley& Sons, 1983, p 774.

9. See for instance, Siddhantasiroman. i , Chap 3, verses 12-14.

10. For a detailed exposition on this, see K.Ramasubramanianand M.S.Sriram, Corrections to the terrestrial latitude inTantrasangraha , Indian Journal of History of Science, 38.2, 2003,p 129-144.

11. Tantrasangraha of Nilakan. t.ha with the prose commentaryLaghuvivr. tti of Sankara Variyar, Ed. by Surnad Kunjan Pillai,Trivandrum Sanskrit Series no. 188, Trivandrum 1958, Chapter 3,verses 23-25.

12. ibid., verses 19-21.

13. ibid., verses 95-100.

14. ibid., verses 100-101.

15. ibid., verse 102.

16. ibid., verses 104-105.


17. ibid., verses 107-109.

18. ibid., Chapter 2, verses 53-54.

19. Tantrasangraha of Nilakan. t.ha with Yuktidpika , commentary inthe form of verses by Sankara Variyar, Ed. by K.V.Sarma, VVBIS,Punjab University, 1977, Chap 4, verses 79 and 87, p 261-62.

ramasubramanian - algorithms in indian astronomy (2005)

Documents