calculus in prose & poetry: kerala school

7/23/2019 Calculus in Prose & Poetry: Kerala School

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Calculus in Prose and Poetry:

Contribution of the Kerala School[MadhavatoSankara Variyar (c.1350-1550)]

K. RamasubramanianIIT Bombay

August 27, 2015Seminar on Intellectual Traditions in Ancient India

Jain University, Bangalore


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Outline

Introduction(Discoveries, Motivation and Lineage) Zero and Infinitydangerous idea ?

Nlakan. t.has discussion of irrationality of

Sum of an infinite geometric series

San. karas discussion of the binomial series expansion

Estimation of sums of powers of integers 1 tonfor largen

Derivation of the Madhava series for

Derivation of end-correction terms (Antya-sam.skara)

Instantaneous velocity and derivatives

Concluding Remarks


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IntroductionCelestial Sphere

Great thinkers of all the civilizations Hindu, Greek,

Arabic1, Chinese, etc. wondered how to interpret the

celestial phenomena.1Nasir al-Din al-Tusi, Ibn al-Shatir, . . .


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IntroductionZero and Infinity:and

ESSENCE OF CALCULUS Use of infinitesmals/limits2

Greeks could not do this neat little mathematical trick. They didnt

have the concept of a limit because they didnt believe in zero. The

terms in the infinite series didnt have a limit or a destination; they

seemed to get smaller and smaller without any particular end in sight.As a result the Greeks couldnt handle the infinite. They pondered the

concept of void but rejected zero as a number, and they toyed with

the concept of infinite butrefused to allow infinitynumbers that are

inifinitely small and infinitely large anywhere near the realm of

numbers. This is the biggest failure in the Greek Mathematics, and it

is the only thing that kept them from discovering calculus. 3

2One of the passages to limit is by summing an infinite series.3

Charles Seife,Zero:The Biography of a Dangerous Idea, Viking, 2000;Rupa & Co. 2008.


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Introduction

Continuing further, Charles Seife observes:4

Unlike Greece,India never had the fearof the infinite or of the void.

Indeed, it embraced them. . . . Indian mathematicians did more than

simply accept zero. They transformed itchanging its role from mere

placeholder to number.The reincarnation was what gave zero its

power. The roots of Indian mathematics are hidden by time. . . . Ournumbers (the current system) evolved from the symbols that the

Indians used; by rights they should be called Indian numerals rather

than Arabic ones. . . . Unlike the Greeks theIndian did not see the

squares in the square numbers or the areas of rectangleswhen they

multiplied two different values. Instead, they saw the interplay of

numeralsnumbers stripped of their geometric significance. This

was thebirth of what we now know of algebra.

4Ibid. pp. 6370.


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Evolution of Numerals: Brahmi Modern

It has takenmore than 18 centuries(3rd BCE 15th CE) for thenumerical notation to acquire the present form.

The present form seems to have got adopted permanently withthe advent of printing press in Europe. However, there are asmany as15 different scripts used in Indiaeven today (Nagari,

Bengali, Tamil (Grantha), Punjabi, Malayalam, etc.).


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Ingenuity of the advent of Place value system & Zero

Laplace5 while describing the contribution of Indians to

mathematics observes:

Theingenious methodof expressing every possiblenumber using a set of ten symbols(each symbolhaving a place value and an absolute value)emergedin India. The idea seems so simple nowadays that its

significance and profound importance is no longerappreciated. Its simplicity lies in the way it facilitatedcalculation andplaced arithmetic foremost amongstuseful inventions. The importance of this invention ismore readily appreciated when one considers that it

wasbeyond the two greatest men of Antiquity,Archimedes and Appolonius.

5

A renowned French Scientist of the 18th-19th century who madephenomenal contributions to the fields of mathematics and astronomy


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Description of decimal place value systemIndian philosophical literature

InVyasa-bhas. yaon theYogasutraof Patanjali, we find an

interesting description of the place value system:

;Just as the same linein the hundreds place [means] ahundred, in the tens place ten, and one in the ones

place;

In the same vein, San. kara in his BSSB (2.2.17) observes:

,,,,,, , , , ,, () ---

,

. . .


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Earliest explicit use of decimal place value systemIndian mathematical and astronomical texts

The earliest comprehensive astronomical/mathematical work

that is available to us today is Aryabhat. ya(499 CE). The degree of sophistication with which Aryabhat.ahas

presented the number of revolutions made by the planets etc.,clearly points to the fact that they hadperfect knowledgeof zeroand the place value system.

Moreover, hisalgorithms for finding square-root, cube-root etc.are also based on this.

The system developed by Aryabhat.ais indeed unique in thewhole historyof written numeration.

Not only unique but also quiteingenious and sophisticated.Numbers of the order of 1016 can be represented by asinglecharacter.

However, it was not made use of by anybody other thanAryabhat

.a perhapsluckilyas it is too complicated to read!


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Signal achievements of Kerala Mathematicians

The Newton series

sin x=x x3

3! + x

5

5! . . . , (1)

The Gregory-Leibniz6 series

Paridhi= 4 Vyasa

1

1

3 +

1

5

1

7 +. . .

(2)

The derivative of sine inverse function

d

dtsin1 r

R

sin M=rR

cos M dMdt

1

rRsin M

2

(3)

and many more remarkable results are found in the works ofKerala mathematicians (14th16th cent.)

6

The quotation marks indicate the discrepancy between the commonlyemployed names to these series and their historical accuracy.


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IntroductionMotivation for finding the precise values of Sines and Derivatives

Sine function (jya) isubiquitous. For instance, In the computation of longitude of the planets,

= 0 sin1r

Rsin M

(4)

The declincation of the Sun is computed using the formula,

sin =sin sin , (5)

where obliquity of the ecliptic and declination ofthe Sun.

The time of sunrise, sunset, the computation oflagna,muhurtaetc., heavily depend on the precise computationofjyaappearing in the above relations.

This explains the need for the computation ofprecise

valuesif thejyas.


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Sources and Lineage

Madhava(c.13401420)7 pioneer of the Kerala School

of Mathematics. Paramesvara(c. 13801460) a disciple of Madhava,

great observer and a prolofic writer.

Nlakan. t.ha Somayaj(c. 14441550) monumental

contributionsTantrasa ngrahaand Aryabhat. ya-bh as.ya.

Jyes.t.hadeva, (c. 1530) author of the celebrated

Yuktibhas. a.

Sankara Variyar (c.15001560) well known for his

commentaries.

Acyuta Pis.arat.i(c. 15501621) a disciple of

Jyes.t.hadevaand a polymath.

7

Only a couple of works of Madhava(Ven. varohaand Sphut.acandrapti)seem to be extant.


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While discussing the value of Nlakan. t.haobserves:

- . . .,

? ?The relation between the circumference and the diameter

was expressed. . . .

Approximate: This value (62,832) was stated to be nearly

the circumference of a circle having a diameter of 20,000.Why then has an approximate value been mentioned here

leaving behind the actual value? It is explained [as

follows].Because it (the exact value) cannot be stated.

Why?


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, ;

Given a certain unit of measurement (mana) in terms of which

the diameter (vyasa) specified [is just an integer and] has no[fractional] part (niravayava), the same measure whenemployed to specify the circumference (paridhi) will certainlyhave a [fractional] part (savayava) [and cannot be just aninteger]. Again if in terms of certain [other] measure the

circumference has no [fractional] part, then employing the samemeasure the diameter will certainly have a [fractional] part [and

cannot be an integer]. Thus when both [the diameter and the

circumference] are measured by the same unit,they cannot

both be specified [as integers] without [fractional] parts.

N l k h di i f i i li f


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Nlakan. t.has discussion of irrationality of What if I reduce the unit of measurement?

Even if you go a long way(i.e., keep on reducing the

measure of the unit employed), the fractional part [in

specifying one of them] will only become very small. A

situation in which there will beno [fractional] part (i.e,both the diameter and circumference can be specified

in terms of integers) is impossible, and this is what is

the import [of the expressionasanna]

WhatNlakan. t.hais trying to explain is the incommensurability

of the circumference and the diameter of a circle.

However small the unit be, the two quantities willnever become

commensurate is indeed a noteworthy statement.

S f i fi i i i


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Sum of an infinite geometric seriesApproximation for the arc of circle in terms of the jya(Rsine)

In his Aryabhat. ya-bh as. ya while deriving an interestingapproximation for the arc of circle in terms of thejya(Rsine) and the sara(Rversine) Nlakan. t.hapresents adetailed demonstration of how to sum an infinite geometric

series.

The specific geometric series that arises in the above

context is:

1

4

+1

4

2

+. . .+ 1

4

n

+. . .=1

3

.

Here, we shall present an outline of Nlakan. t.has argument

that gives a cue to understand as to how thenotion of limit

was present and understood by them.

S f i fi it t i i


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AB iscapa(c) as it

looks like a bow. ADisjyardha(j) as it

half the string.

BD is sara(s) as it

looks like an arrow.

The expression given by Nlakan. t.hais:

c

1 +

1

3

s2 +j2. (6)

S f i fi it t i i


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The proof of (6) presented by Nlakan. t.hainvolves:

1. Repeated halving of the arc-bit, capa cto getc1. . . ci.

2. Finding the corresponding semi-chords, jya(ji) and theRversines, sara(si)

3. Estimating the difference between thecapaandjyaat eachstep.

Ifibe the difference between the capaandjyaat theith step,

i=ciji. (7)

HereNlakan. t.haobserves : as the size of thecapadecreasesthe differenceialso decreases.

S m of an infinite geometric series


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Generating successive values of the jis andsis is an

unending processas one can keep on dividing the capainto halfad infinitum.

It would therefore be appropriate torecognize that the

differenceiis tending to zeroand hence make anintelligent approximation, to obtain the value of the

difference betweencandj approximately.



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Nlakan. t.haposes a very important question:

?How is it that [the sum of the series]increases only

upto that[limiting value] and thatcertainly increases

upto that[limiting value]?

Proceeding to answer he first states the general result

a1

r +1

r2

+1

r3

+. . .= a

r 1 . (8)

Infinite Geometric Series

Divisor ( )



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Noting that the result is best demonstrated with r=4Nlakan. t.haobtains the sequence of results,

1

3 =

1

4+

1

(4.3) ,

1

(4.3) =

1

(4.4)+

1

(4.4.3) ,

1(4.4.3)

= 1(4.4.4)

+ 1(4.4.4.3)

, (9)

and so on, which leads to the general result,

13

14

+

14

2+. . .+

14

n=

14

n13

. (10)

As we sum more terms, the difference between 13 and sum of

powers of 14

,becomes extremely small,but never zero.

What is a Limit ?


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What is a Limit ?

Cauchys (1821) definition of limit:

If the successive values attributed to the same

variable approach indefinitely a fixed value, such that

finally they differ from it by as little as one wishes, this

latter is called the limit of all the others.8

Nlakan. t.hain his Aryabhat. ya-bh as.ya:

?How is it that [the sum of the series]increases only

upto that[limiting value] and thatcertainly increases

upto that[limiting value]?

8

Cauchy,Cours dAnalyse, cited by Victor J. Katz,A History ofMathematics, Addison Wesley Longman, New York 1998, p. 708.

Binomial series expansion


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Sankara Variyar in hisKriyakramakardiscusses as follows

Consider the producta

cb

Here,ais calledgun. ya,c thegun. akaandbthehara(these

are all assumed to be positive).

If we consider the ratio cb, there are two possibilities: Case i: gun. aka> hara(c>b). In this case we rewrite the

product in the following form

a c

b= a+a (c b)

b . (11)

Case ii: gun. aka< hara(c


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In the expressiona(bc)

b , if we want to replace the division bybby

division byc, then we have to make a subtractive correction(sodhya-phala) which amounts to the following equation.

a(b c)

b =a

(b c)

c a

(b c)

c

(b c)

b . (13)

If we again replace the division by the divisorbby the multiplierc,

ac

b = a

a

(b c)

c a

(b c)

c

(b c)

c

c

b

= a a(b c)c

a(b c)2

c2 a(b c)

2

c2

(b c)

b(14)

The quantitya(bc)2

c2 is calleddvitya-phalaor simplydvityaand the

one subtracted from that isdvitya-sodhya-phala.



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Binomial series expansionThus, after takingm sodhya-phala-s we get

ac

b

= a a(b c)

c

+a (b c)c

2

. . .+ (1)m1a (b c)c

m1

+(1)ma

(b c)

c

m1(b c)

b . (15)

Still, if we keep including correction terms, thenthere is logically

no end to the seriesof correction terms (phala-parampara).

For achieving a given level of accuracy,we can terminate theprocesswhen the correction term becomes small enough.

Ifb c


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Different approximations to

The Sulba-sutra-s, give the value of close to 3.088.

Aryabhat.a(499 AD) gives an approximation which is correct tofour decimal places.

(100 +4) 8 +6200020000

= 6283220000

=3.1416

Then we have the verse ofLl avat9

= 3927

1250=3.1416 thats same as Aryabhat.as value.

9Llavatof Bhaskaracarya, verse 199.



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The commentaryKriyakramakarfurther proceeds to present moreaccurate values of given by different Acaryas.

10

The values of given by the above verses are:

= 2827433388233

9 1011

=3.141592653592 (correct to 11 places)

The latter one is due toMadhava.

10Vibudha=33, Netra=2, Gaja=8, Ahi=8, Hutasana=3, Trigun. a=3,

Veda=4, Bha=27, Varan.

a=8, Bahu=2, Nava-nikharva=9

10

11

. (Thewordnikharvarepresents 1011).

Infinite series for as given in Yukti-dpika


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Infinite series for as given inYukti dpik a

The diameter multiplied by four and divided by unity (is found and

stored). Again the products of the diameter and four are divided bythe odd numbers like three, five, etc., and the results are subtractedand added in order (to the earlier stored result).

vyase varidhinihate 4 Diameter (varidhi)

vis. amasankhyabhaktam Divided by odd numbers

trisar adi 3, 5, etc. (bhutasankhyasystem)

r. n. am. svam. to be subtracted and added [successively]

Paridhi= 4 Vyasa

1

1

3+

1

5

1

7+. . . . . .

Infinite series for


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Infinite series for

The trianglesOPi1Ci and

OAi1Biare similar. Hence,

Ai1Bi

OAi1=

Pi1Ci

OPi1(16)

Similarly trianglesPi1CiPiandP0OPiare similar.Hence,

Pi1CiPi1Pi

= OP0

OPi(17)

Infinite series for


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Infinite series for

From these two relations we have,

Ai1Bi = OA

i

1.OP

0.P

i

1P

iOPi1.OPi

= Pi1PiOAi1OPi1

OP0

OPi

= rn r

ki+1

r

ki

= r

n

r2kiki+1

. (18)

It is rnthat is refered to as khan. d. ain the text. The text also notes

that, when thekhan. d. a-s become small (or equivalentlynbecomeslarge), the Rsines can be taken as the arc-bits itself.

i.e., Ai1Bi Ai1Ai .

(local approximation bylinear functionsi.e.,tangents/differentiation)

Infinite series for (Error estimate)


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( )

Though the value of 18

th of the circumference has been obtained as

C

8 =

r

n r2

k0

k1 +

r2

k1

k2 +

r2

k2

k3 + +

r2

kn

1k

n , (19)

there may not be much difference in approximating it by either of thefollowing expressions:

C

8

= r

n

r2

k

2

0+

r2

k

2

1+

r2

k

2

2+ +

r2

k

2

n1 (20)

or C

8 =

rn

r2k21

+

r2

k22

+

r2

k23

+ +

r2

k2n

(21)

The difference between (??)and (??)will bern

r2

k20

r2

k2n

=

rn

1

12

(k20 , k

2n =r

2, 2r2)

= r

n

12

(22)

Infinite series for


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Thus we have,

C

8

=n

i=1

r

n r

2

k2

i summming up/integration

=

ni=1

r

n

r

n

k2i r

2

r2

+

r

n

k2i r

2

r2

2 . . .

= rn [1 +1 +. . .+1]

r

n

1r2

rn

2+

2r

n

2+. . .+

nrn

2

+r

n

1

r4

r

n

4

+ 2r

n

4

+. . .+ nr

n

4

r

n

1r6

rn

6+

2r

n

6+. . .+

nrn

6

+ . . . . (23)

Infinite series for


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If we take out the powers of bhuja-khan. d. a rn

, the summations involvedare that of even powers of the natural numbers, namelyedadyekottara-varga-sankalita, 12 +22 +...+n2,edadyekottara-varga-varga-sankalita, 14 +24 +...+n4, and so on.Kerala astronomers knew that

n

i=1ik

nk+1

k+1. (24)

Thus, we arrive at the result

C

8 =r

1

1

3+

1

5

1

7+

, (25)

which is given in the form

Paridhi= 4 Vyasa

1

1

3+

1

5

1

7+

Summation of series(sa nkalita)[Integral ?]


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( ) [ g ]Background

The Aryabhat.yaof Aryabhat.ahas the formula for the sa nkalita-s

S(1)n = 1 +2 + +n= n(n+1)2

S(2)n = 1

2 +22 + +n2 = n(n+1)(2n+1)

6

S(3)n = 1

3 +23 + +n3 = n(n+1)2

2

(26)

From these, it is easy to estimate these sumswhennis large.Yuktibhas. agives a general method of estimating thesama-ghata-sankalita

S(k)n =1k +2k + +nk, (27)

whennis large. What it presents is a general method of estimation,which does make use of the actual value of the sum. So, theargument is repeated even for k=1, 2, 3,although the result of

summation is well known in these cases.

Summation of series(sa nkalita)


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Samaghata-sa nkalita

Thus in general we have,

nS(k1)n S

(k)n

(n 1)k

k +(n 2)k

k +(n 3)k

k +. . .

1

k

S(k)n . (28)

Rewriting the above equation we have

S(k)n nS

(k1)n

1

k

S(k)n . (29)

(

- )

Thus we obtain the estimate

S(k)n

nk+1

(k+1). (30)

End-correction in the infinite series for


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Need for the end-correction terms

The series for 4 is anextremely slowlyconvergent series.

To obtain value of which is accurate to 4-5 decimalplaces we need toconsider millions of terms.

To circumvent this problem, Madhavaseems to have found

aningenious waycalled antya-sam.skara

It essentially consists of Terminating the series are a particular term if you get

boredom (jamitaya). Make anestimate of the remainder termsin the series Apply it (+vely/-vely) to the value obtained by summation

after termination.

The expression provided to estimate the remainder terms

is noted to bequite effective.

Even if a consider a few terms (say 20), we are able to get

valuesaccurate to 8-9 decimal places.



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Expression for the remainder terms (Antyasam.skara)

yatsankhyayatra haran. e Dividing by a certain number (p)

nivr. tta hr. tistu if the division is stopped

jamitaya being bored (due to slow-convergence)

Remainder term=

p+12

p+12

2+1

labdhah. paridhih. suks. mah. the circumference obtained would

be quite accurate



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When does the end-correction give exact result ?

The discussion by Sankara Variyaris almost in the form of a

engaging dialogue between the teacher and the taught and

commences with the question, how do you ensure accuracy.

? ?

How is it that you getthe value close to the circumferenceby usingantya-sam.skara, instead of repeatedly dividing byodd numbers? This is being explained.



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When does the end-correction give exact result ?

The argument is as follows: If the correction term 1ap2

is applied after

odd denominatorp 2 (with p1

2

is odd), then

4 =1

1

3+

1

5

1

7. . .

1

p 2+

1

ap2. (31)

On the other hand, if the correction term lap

, is applied after the odd

denominatorp, then

4 =1

1

3+

1

5

1

7. . .

1

p 2+

1

p

1

ap. (32)

If the correction terms are exact, then both should yield the sameresult. That is,

1

ap2=

1

p

1

apor

1

ap2+

1

ap=

1

p, (33)

is the condition for the end-correction to lead to the exact result.



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Optimal choice for error-minimizaion ?

It is first observed that we cannot satisfy this condition trivially by

takingap

2 =ap=2p. For, the correction has to follow a uniform ruleof application and thus,if ap2 =2p, thenap=2(p+2);

We can, however,have bothap2 andapclose to 2pas possible.Hence, as first (order) estimate one tries with, double the evennumber above the last odd-number divisor p,ap=2(p+1).

But, it can be seen right away that, the condition for accuracy is notexactly satisfied. The measure of inaccuracy (sthaulya)E(p)isintroduced, and is estimated

E(p) =

1ap2

+ 1ap

1

p .

The objective is to find the correction denominators apsuch that the

inaccuracyE(p)is minimised.

End-correction in the infinite series for O i l h i f i i i i ?


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When we setap=2(p+1), the inaccuracy will be

E(p) = 1(2p 2)

+ 1(2p+2)

1p

= 4

(4p3 4p)

= 1

(p3 p) .

It can be shown that among all possible correction divisors of the type

ap=2p+m,

wheremis an integer,the choice ofm=2 is optimal, as in all other

cases there will arise a term proportional to pin the numerator of the

inaccuracyE(p).

End-correction in the infinite series for O ti l h i f i i i i ?


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If we take the correction divisor to be ap=2p+2 + 4(2p+2) ,then the

inaccuracy is found to be

E(p) = 1

2p 2 + 4

2p 2

+ 1

2p+2 + 4

2p+2

1

p

=

4

(p5 +4p) .

Clearly, thesthaulyawith this (second order) correction divisor hasimproved considerably, in that it is now proportional to theinverse fifthpower of the odd number.

It can be shown that if we take any other correction divisor

ap=2p+2 + m(2p+2) , wheremis an integer, we will end up having a

contribution proportional top2 in the numerator of the inaccuracy

E(p),unlessm=4.

Error-minimization in the evaluation of Pi


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Construction of the Sine-table


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A quadrant is divided into24 equal parts, so that each arc bit= 90

24=345 =225.

A procedure for finding Rsin i, i=1, 2, . . . 24 is explicitlygiven. PiNiare known.

The R sines of the intermediate angles are determined byinterpolation (I order or II order).

Recursion relation for the construction of sine-tableAryabhatyas algorithm for constructing of sine table


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Aryabhat.ya s algorithm for constructing of sine-table

The content of the verse in Aryabhat.yatranslates to:

Rsin(i+1) Rsin i= Rsin i Rsin(i 1) Rsin i

Rsin .

In fact, the values of the 24Rsinesthemselves are explicitlynoted in another verse.

Theexact recursion relationfor the Rsine differences is:

Rsin(i+1)Rsin i= Rsin iRsin(i1)Rsin i 2(1cos ).

Approximation used by Aryabhat.ais 2(1 cos ) = 1225

.

While, 2(1 cos ) =0.0042822, 1225

=0.00444444.

In the recursion relation provided byNlakan. t.hawe find1

225 1

233.5(=0.0042827).

Comment on Aryabhat.as Method(Delambre)


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Commenting upon the method of Aryabhat.ain his monumental

work Delambre11

observes:

The method is curious: it indicates a method of calculating the

table of sines by means of their second differences. . . . The

differential process has not up to now been employed except by

Briggs, who himself did not know that the constant factor wasthe square of the chord . . . Here then is a method which the

Indians possessed and which isfound neither amongst the

Greeks nor amongst the Arabs.12

11. . . an astronomer of wisdom and fortitude, able to review 130 years of

astronomical observations, assess their inadequacies, and extract their

value. Prix prize citation 1789.12Delambre,Historie de lAstronomie Ancienne, t 1, Paris 1817, p.457;

cited from B. Datta and A. N. Singh, Hindu Trigonometry, IJHS 18, 1983, p.77.

Infinite series for the sine function


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The verses giving the series for the sine function is13

,

N0=R D0 =1

N1=R (R)2 Ni+1 =Ni (R)

2

D1

=R2(2 +22) Di=D

i

1 R2(2i+ (2i)2)

= N0D0 [N1D1 ( N2D2 {

N3D3 . . . })]

=For obtaining thejva(Rsine)

13Yuktidpik a(16th cent) and attributed to Madhava (14th cent. AD).

Infinite series for the sine function


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Expressing the seriesusing modern notationas described

as described in the above verse

Jv a= R R (R)2

R2(2 +22) +

R (R)2 (R)2

R2(2 +22)R2(4 +42) . . .

Simplifying the above we have

Jv a= R (R)3

R2 6+

(R)5

R4 6 20

(R)7

R6 6 20 42+ . . .

Further simplifying

Jv a= R

3

3!+

5

5!

7

7!+. . .

= Rsin

Thus thegiven expression well known sine series.

Instantaneous velocity of a planetThe mandaphala or equation of centre correction

AP

(direction ofmandocca)(planet)


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Themandaphalaor equation of centre correction

P0 mean planet

P true planet 0 mean longitude

MS true longitudecalled themanda-sphut

.a.

O

0

P0

Q

0

p

ms

The true longitude of the planet is given by

=0 sin

1r

Rsin M

whereM(manda-kendra) =0 longitude of apogee

The second term in the RHS, known asmanda-phala, takes careof the eccentricity of the planetary orbit.

Instantaneous velocity of a planetDerivative of sin1 function


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Derivative of sin function

The instantaneous velocity of the planet called tatkalikagatiis

given byNlakan. t.hain hisTantrasa ngrahaas follows:

IfMbe themanda-kendra, then the content of the above versecan be expressed as

ddt

sin1

rR

sin M

=

r

Rcos M

dM

dt1

rRsin M

2 (34)

Instantaneous velocity of a planetDerivative of the ratio of two functions


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Some of the astronomers in the Indian tradition includingMunjalahadproposed the expression formandaphalato be

=

r

Rsin M

1 r

Rcos M

, (35)

According to Acyuta, the correction to the mean velocity of a planet toobtain its instantaneous velocity in this case is given by

r

Rcos M +

r

Rsin M

2

1

r

Rcos M

1 r

Rcos M

dMdt

, (36)

which is nothing but the derivative of (??).

Concluding Remarks


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It is clear that major discoveries in thefoundations of calculus,

mathematical analysis, etc., did take place in Kerala School(14-16 century).

Besides arriving at the infinite series, that the Keralaastronomers could manipulate with them to obtain several formsofrapidly convergent seriesis indeed remarkable.

While the procedure by which they arrived at many of theseresults are evident, there are still certaingrey areas(derivativeof sine inverse function, ratio of two functions)

Many of these achievements are attributed to Madhava, who

lived in the14th century(his works ?). Whether some of these results came to be known to the

European mathematicians ? ? . . . .

Thanks!


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THANK YOU !

calculus in prose & poetry: kerala school

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