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Calculus in Prose and Poetry: Contribution of the Kerala School ´ nkara [M¯adhava to Sa ˙ V¯ ariyar (c

- 1350-1550)]

Ramasubramanian IIT Bombay

- 2015 Seminar on Intellectual Traditions in Ancient India Jain University,

Outline

N¯ılakan

ha’s discussion of irrationality of π

kara’s discussion of the binomial series expansion San

Estimation of sums of powers of integers 1 to n for large n ¯ Derivation of the Madhava series for π

Great thinkers of all the civilizations – Hindu,

– wondered how to interpret the celestial phenomena

Nasir al-Din al-Tusi,

ZUa:nya and A:na:nta

E SSENCE OF CALCULUS ≡ Use of infinitesmals/limits2 Greeks could not do this neat little mathematical trick

they seemed to get smaller and smaller without any particular end in sight

They pondered the concept of void but rejected zero as a number,

and they toyed with the concept of infinite but refused to allow infinity – numbers 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

Viking,

Indeed,

- it embraced them

The roots of Indian mathematics are hidden by time

Our numbers (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 the Indian did not see the squares in the square numbers or the areas of rectangles when they multiplied two different values

they saw the interplay of numerals—numbers stripped of their geometric significance

63–70

It has taken more than 18 centuries (3rd BCE – 15th CE) for the numerical notation to acquire the present form

The present form seems to have got adopted ‘permanently’ with the advent of printing press in Europe

However,

there are as many as 15 different scripts used in India even today (Nagari,

Tamil (Grantha),

Malayalam,

Laplace5 while describing the contribution of Indians to mathematics observes: The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India

The idea seems so simple nowadays that its significance and profound importance is no longer appreciated

Its simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions

The importance of this invention is more readily appreciated when one considers that it was beyond the two greatest men of Antiquity,

Archimedes and Appolonius

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

Description of decimal place value system Indian philosophical literature I

sya on the Yogas¯ utra of Pata˜ njali,

we find an interesting description of the place value system:

- ya:TEa:k+a :=e
- a Za:ta:~Ta
- a:nea Za:tMa d:Za:~Ta
- a:nea d:Za O

Just as the same line in the hundreds place [means] a hundred,

- in the tens place ten,
- and one in the ones place
- kara in his BSSB (2
- 17) observes: In the same vein,
- :k+eaY:a
- sa:n,a :de:va:d
- sa:}ba:a
- ca A:pea:[ya A:nea:k+
- d:pra:tya:ya:Ba
- a:gBa:va: a
- ta – ma:nua:SyaH,
- a::NaH,
- ea:aa:yaH,
- a:nyaH,
- ta Á ya:Ta
- :k+a:a
- a (A:ñÍ*:H) öÐÅ
- a:nya:tvea:na
- va:Za:ma
- :k-d:Za-Za:ta-
- sa:h:~å:òa

d:pra:tya:ya:Bea:d:m,a A:nua:Ba:va: a

- sa:}ba:a
- n/ Da:na

The earliest comprehensive astronomical/mathematical work ¯ that is available to us today is Aryabhat

a has presented the number of revolutions made by the planets etc

clearly points to the fact that they had perfect knowledge of zero and the place value system

his algorithms for finding square-root,

- cube-root etc
- are also based on this

a is indeed unique in the whole history of written numeration

it was not made use of by anybody other than ¯ Aryabhat

a — perhaps luckily as it is too complicated to read

- x5 x3 + −

- 1 1 1 Paridhi = 4 × Vy¯ asa × 1 − + − +

3 5 7 I

The derivative of sine inverse function r i cos M dM d'h −1 r dt sin sin M = q R 2 dt R r 1 − R sin M

and many more remarkable results are found in the works of Kerala mathematicians (14th–16th cent

The quotation marks indicate the discrepancy between the commonly employed names to these series and their historical accuracy

Introduction Motivation for finding the precise values of Sines and Derivatives I

r sin M λ = λ0 − sin−1 R

- sin δ = sin sin λ,

where → obliquity of the ecliptic and δ → declination of the Sun

- sunset,
- the computation of lagna,
- muh¯ urta etc

heavily depend on the precise computation of jy¯ a appearing in the above relations

This explains the need for the computation of precise values if the jy¯ as

- 1340–1420)7 — pioneer of the Kerala School of Mathematics

- 1380–1460) — a disciple of M¯adhava,

great observer and a prolofic writer

- ha Somay¯ aj¯ı (c
- 1444–1550) — monumental ¯ contributions Tantrasangraha ˙ and Aryabhat a

- hadeva,
- 1530) — author of the celebrated Yuktibh¯ a

´ nkara Sa ˙ V¯ariyar (c

- 1500–1560) — well known for his commentaries

- 1550–1621) — a disciple of Jyes
- hadeva and a polymath

v¯ aroha and Sphut

acandr¯ apti) seem to be extant

N¯ılakan

ha’s discussion of irrationality of π I

While discussing the value of π N¯ılakan

- ha observes:
- :pa:a=
- a:sa:ya
- sa:Ë*ñÍÉ +

ùÁ a-

- sa:}ba:nDaH :pra:d:a

- a:sa:aH,

a:sa:a:ta:yEa:va A:yua:ta:dõ:ya:sa:Ë*ñÍÉ +

- a: ùÁ a

- taH :pua:naH va
- a:~ta:va
- sa:Ë*ñÍÉ +

- a:sa:Ea:va I+h
- ea:+a
- cya:tea Á ta:~ya
- a va:u+
- ma:Za:k+
- a:t,a Á ku+

? 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]

N¯ılakan

ha’s discussion of irrationality of π yea:na ma

- a:nea:na ma
- a:ya:ma
- va:ya:vaH
- tea:nEa:va ma
- a:ya:ma
- a:naH :pa:a=

- a:va:ya:va O
- a:t,a Á yea:na
- a:ya:ma
- a:naH :pa:a=
- va:ya:vaH
- tea:nEa:va ma
- a:ya:ma
- a:va:ya:va O

- a:ya:ma
- a:na:ya
- eaH o+Ba:ya
- eaH ëÐ*:ëÅÁ +a:a
- va:ya:va:tvMa

a:t,a Á Given a certain unit of measurement (m¯ ana) in terms of which the diameter (vy¯ asa) specified [is just an integer and] has no [fractional] part (niravayava),

the same measure when employed to specify the circumference (paridhi) will certainly have a [fractional] part (s¯ avayava) [and cannot be just an integer]

then employing the same measure the diameter will certainly have a [fractional] part [and cannot be an integer]

they cannot both be specified [as integers] without [fractional] parts

ha’s discussion of irrationality of π What if I reduce the unit of measurement

- a:nta:m,a A:Dva
- a:nMa ga:tva
- pa A:pa
- a:va:ya:va:tva:m,a O
- :va l+Bya:m,a Á

va:ya:va:tvMa tua ëÐ*:ëÅÁ +a:a

- pa na l+Bya:m,a I+ a

a:vaH Á Even if you go a long way (i

keep on reducing the measure of the unit employed),

the fractional part [in specifying one of them] will only become very small

both the diameter and circumference can be specified in terms of integers) is impossible,

and this is what is the import [of the expression ¯ asanna] What N¯ılakan

ha is trying to explain is the incommensurability of the circumference and the diameter of a circle

the two quantities will never become commensurate – is indeed a noteworthy statement

Sum of an infinite geometric series Approximation for the arc of circle in terms of the jy¯ a (Rsine)

¯ In his Aryabhat a

sya – while deriving an interesting

¯ıya-bh¯ approximation for the arc of circle in terms of the jy¯ a (Rsine) and the ´sara (Rversine) – N¯ılakan

ha presents a detailed demonstration of how to sum an infinite geometric series

The specific geometric series that arises in the above context is: 1 + 4

- 2 n 1 1 1 +

we shall present an outline of N¯ılakan

ha’s argument that gives a cue to understand as to how the notion of limit was present and understood by them

Sum of an infinite geometric series I

AD is jy¯ ardha (j) as it half the string

BD is ´sara (s) as it looks like an arrow

ha is: s 1 c'≈ 1+ s2 + j 2

yMa:Za

- d:Sua:va:ga
- a:va:ga
- a:t,a :pa:dM

Da:nuaH :pra

- a:yaH Á

Sum of an infinite geometric series The proof of (6) presented by N¯ılakan

- ha involves: 1

jy¯ a (ji ) and the Rversines,

´sara (si ) 3

- ∆ i = c'i − ji

ha observes : “as the size of the c¯ apa decreases the difference ∆i also decreases

- a:pa:ya

~ya :pua:naH :pua:naH nyUa:na:tvMa

- a:pa:pa:a=
- a:pa:tva:kÒ+
- mea:Nea: a

a:d:DRa:

- a:m,a A:DRa
- a:ya:ma
- a na ëÐ*:ëÅÁ +
- ca:d:a
- pa :pa:yRa:va:~ya: a
- a:na:ntya
- a:ga:~ya Á ta:taH
- ya:nta: a*

- a:pa:~ya
- a:(ãÉa A:pa
- a:ya:~tva:m,a A
- a:dùÅ
- ca ZUa:nya:pra
- a:yMa l+b
- a :pua:na
- pya:ma
- a:na:ma:nta
- m,a A:tya:pa:ma:a
- pa k+Ea:Za:l
- a:t,a ¼ea:ya:m,a Á I

Generating successive values of the ji s'and si s'is an “unending” process as one can keep on dividing the c¯ apa into half ad infinitum

It would therefore be appropriate to recognize that the difference ∆i is tending to zero and hence make an “intelligent approximation”,

to obtain the value of the difference between c'and j approximately

ha poses a very important question:

- a:va:de:va va:DRa:tea ta
- a:va:dõ:DRa:tea

? How is it that [the sum of the series] increases only upto that [limiting value] and that certainly increases upto that [limiting value]

? Proceeding to answer he first states the general result " # 3 1 1 2 1 a a

- = r r r r −1

Infinite Geometric Series – tua:ya:

- a:ga:pa

Divisor – Ce +d

- a:tea A:nea:nea: a
- ta – k+=
- yua:tpa:aa)

Sum of an infinite geometric series Noting that the result is best demonstrated with r = 4 N¯ılakan

ha obtains the sequence of results,

1 3 1 (4

3) 1 (4

1 1 + ,

3) 1 1 + ,

3) 1 1 + ,

- and so on,

which leads to the general result,

" 2 n # n 1 1 1 1 1 1 − + +

- 3 4 4 4 4 3

As we sum more terms,

the difference between 13 and sum of powers of 41 ,

- becomes extremely small,
- but never zero

? Cauchy’s (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 ¯ N¯ılakan
- ha in his Aryabhat a

¯ıya-bh¯

- a:va:de:va va:DRa:tea ta
- a:va:dõ:DRa:tea

? How is it that [the sum of the series] increases only upto that [limiting value] and that certainly increases upto that [limiting value]

Cauchy,

Cours d’Analyse,

- cited by Victor J

Binomial series expansion ´ nkara Sa ˙ V¯ariyar in his Kriy¯ akramakar¯ı discusses as follows c'b

- a is called gun
- c the gun

aka and b the h¯ ara (these are all assumed to be positive)

If we consider the ratio bc ,

- there are two possibilities:

Case i: gun

- aka > h¯ ara (c > b)

- (c − b)

Case ii: gun

- aka < h¯ ara (c < b)

- (12) a =a−a b b

if we want to replace the division by b by division by c,

then we have to make a subtractive correction (´sodhya-phala) which amounts to the following equation

(b − c) (b − c) (b − c) (b − c) =a −a ×

- b c'c b

c'(b − c) (b − c) (b − c) c'a = a− a −a × × b c'c c'b 2 (b − c) (b − c)2 (b − c) (b − c) = a− a − a − a × (14) c'c2 c2 b 2

The quantity a (b−c) is called dvit¯ıya-phala or simply dvit¯ıya and the c2 one subtracted from that is dvit¯ıya-´sodhya-phala

after taking m ´sodhya-phala-s we get 2 m−1 c'(b − c) (b − c) (b − c) m−1 a = a−a +a −

+ (−1) a b c'c c' m−1 (b − c) (b − c) +(−1)m a

- (15) c'b
- :vMa mua:huH :P+l
- a:na:ya:nea kx+
- teaY:a
- pa yua: a
- taH ëÐ*:ëÅÁ +a:a

- a:va:d:pea:[Ma

- a:dùÅ
- a:(ãÉa
- a:nyua:pea:[ya :P+l
- a:na:ya:nMa
- a:pa:na
- a:ya:m,a Á I+h

Ma nyUa:na:tvMa tua gua:Na:h

- =:a:nta:=e gua:Na:k+a
- =:a:yUa:na O
- a:t,a Á I

if we keep including correction terms,

then there is logically no end to the series of correction terms (phala-parampar¯ a)

For achieving a given level of accuracy,

we can terminate the process when the correction term becomes small enough

then the successive correction terms keep decreasing

Different approximations to π I

give the value of π close to 3

¯ Aryabhat

a (499 AD) gives an approximation which is correct to four decimal places

- a:Sa:a

- sa:h:~å:òa
- a:m,a Á A:yua:ta:dõ:ya:a

- a:sa:a
- ea’ vxa

- a:hH Á Á π≈ I
- 62832 (100 + 4) × 8 + 62000 = = 3
- 1416 20000 20000

Then we have the verse of L¯ıl¯ avat¯ı9

- a:sea Ba:na:nd

çîå+

- a:É h:tea
- va:Ba:e Ka:ba
- a:Na:sUa:yERaH :pa:a=
- sua:sUa:[maH Á ÈîåeÁ
- va:&+teaY:Ta ZEa:lEH
- eaY:Ta:va
- ya:va:h
- ea:gyaH Á Á dõ

3927 = 3

1416 1250

- a’s value

¯ L¯ıl¯ avat¯ı of Bhaskar ac verse 199

Different approximations to π The commentary Kriy¯ akramakar¯ı further proceeds to present more ¯ aryas

accurate values of π given by different Ac¯

- a:Da:va
- a:yRaH :pua:naH A:ta
- a:sa:a:ta:ma

- a:n,a –
- va:bua:Da:nea
- h:hu:ta

a:Za:na:aa:gua:Na:vea:d:Ba:va

- a:h:vaH Á na:va: a
- na:Ka:vRa: a
- ma:tea vxa: a
- va:~ta:=e :pa:a=
- a:na: a
- ja:ga:du:bRua:Da

- 2827433388233 9 × 1011
- 141592653592
- (correct to 11 places)

The latter one is due to M¯ adhava

Veda=4,

Bha=27,

B¯ ahu=2,

(The word nikharva represents 1011 )

- a:sea va
- na:h:tea
- +pa:&+tea v
- a:sa:sa

- aa:Za
- va:Sa:ma:sa:Ë*ñÍÉ +

- m,a +NMa
- ~vMa :pxa:Ta:k,
- a:t,a ku+

Ra:t,a Á Á The diameter multiplied by four and divided by unity (is found and stored)

and the results are subtracted and added in order (to the earlier stored result)

vy¯ ase v¯ aridhinihate → 4 × Diameter (v¯ aridhi)

amasankhy¯ ˙ abhaktam → Divided by odd numbers

- tri´sar¯ adi → 3,

(bh¯ utasankhy¯ ˙ a system)

→ to be subtracted and added [successively]

- 1 1 1 Paridhi = 4 × Vy¯ asa × 1 − + − +

Infinite series for π From these two relations we have,

OAi−1

OPi OAi−1 OP0 = Pi−1 Pi × × OPi−1 OPi r r r = × × n ki+1 ki r r2 =

- n ki ki+1 =

It is nr that is refered to as khan

- a in the text

The text also notes that,

- when the khan
- a-s become small (or equivalently n becomes large),

the Rsines can be taken as the arc-bits itself

(local approximation by linear functions i

- :pa:a=

Da:Ka:Nq+~ya

- a → :pa:a=

(Error estimate) Though the value of 18 th of the circumference has been obtained as r r 2 r 2 r 2 r2 C = + + + ··· + ,

- (19) 8 n k0 k1 k1 k2 k2 k3 kn−1 kn

there may not be much difference in approximating it by either of the following expressions: "

!# r r2 r2 r2 C r2 = + + + ··· + (20) 2 8 n k02 k12 k22 kn−1 2 r r 2 r 2 r 2 r C = + + + · · · + (21) or 8 n k12 k22 k32 kn2 The difference between (

- ?) and (

?) will be r r 2 r 2 r 1 − = 1 − n n 2 k02 kn2 r 1 = n 2

Ka:Nq+~ya A:pa:tva:va:Za

- a:t,a ta:d:nta

- a:ya:mea:va Á
- ( k02 ,
- kn2 = r 2 ,
- 2r 2 ) (22)

n X r r2 = summming up/integration n ki2 i=1 " # 2 n X r r ki2 − r 2 r ki2 − r 2 = − + −

n n r2 n r2 i=1 r = [1 + 1 +

+ 1] n # " r 1 r 2 2r 2 nr 2 + − +

+ n r2 n n n " # 4 r 1 r 4 nr 4 2r + + +

+ n r4 n n n " # 6 r 1 r 6 nr 6 2r − + +

- + n r6 n n n +

the summations involved are that of even powers of the natural numbers,

namely ed¯ adyekottara-varga-sankalita,

ed¯ adyekottara-varga-varga-sankalita,

- and so on

Kerala astronomers knew that n X i=1

we arrive at the result C 1 1 1 = r 1 − + − + ··· ,

- 8 3 5 7 which is given in the form 1 1 1 Paridhi = 4 × Vy¯ asa × 1 − + − + · · · · · · 3 5 7

Summation of series (sankalita) ˙ [Integral

- ?] Background

a has the formula for the sankalita-s (1)

n(n + 1) 2 n(n + 1)(2n + 1) = 12 + 22 + · · · + n2 = 6 2 n(n + 1) 3 3 3 = 1 + 2 + ··· + n = 2

- = 1 + 2 + ··· + n =

From these,

it is easy to estimate these sums when n is large

- when n is large

which does make use of the actual value of the sum

the argument is repeated even for k = 1,

although the result of summation is well known in these cases

Thus in general we have,

- (k −1)
- ≈ ≈

(n − 1)k (n − 2)k (n − 3)k + + +

- k k k 1 (k) Sn

- (k −1) nSn
- 1 (k) Sn
- sa:ñÍ*:öÐÅ + a
- a:na:ya:na
- a:ya ta

- l+ta:~ya v
- a:DRa:gua:Na:na:m,a O
- sa:Ë*ñÍÉ +

ùÁ a:a-

- ea:Da:nMa
- ca k+a:yRa:m,a I+ a
- ~/ /Ta:ta:m,a Á )

Thus we obtain the estimate (k )

- (k + 1)

End-correction in the infinite series for π Need for the end-correction terms π 4

M¯ adhava seems to have found an ingenious way called “antya-sam ara”

sk¯ It essentially consists of –

is an extremely slowly convergent series

Terminating the series are a particular term if you get boredom (j¯ amitay¯ a)

Make an estimate of the remainder terms in the series Apply it (+vely/-vely) to the value obtained by summation after termination

we are able to get π values accurate to 8-9 decimal places

End-correction in the infinite series for π Expression for the “remainder” terms (Antyasam ara)

- ya:tsa:Ë*ñÍÉ +
- a: ùÁ ya

- ta:~tua
- ma:ta:ya
- a Á ta:~ya

- sa:ma:sa:Ë*ñÍÉ +

- eaY:ntea
- a:t,a Á Á ta:dõ:ga
- +pa:yua:ta
- a:ta:taH :pra
- a:gva:t,a Á ta
- ~va:mxa:Nea kx+

- :va k+=
- a:yaH Á Á l+b

DaH :pa:a=

- ta:sUa:[maH

a:t,a Á Á yatsankhyay¯ ˙ atra haran

e → Dividing by a certain number (p) I nivr a hr

tistu → if the division is stopped

- labdhah
- paridhih
- s¯ uks

→ the circumference obtained would be quite accurate

End-correction in the infinite series for π When does the end-correction give exact result

´ nkara The discussion by Sa ˙ V¯ ariyar is almost in the form of a engaging dialogue between the teacher and the taught and commences with the question,

- how do you ensure accuracy

TMa :pua:na

- =:a mua:hu:a

Nea:na l+Bya:~ya :pa:a=

a:sa:a:tva:m,a A:ntya:sMa:~k+a:=e

- a:dùÅ
- cya:tea Á ta
- a:va:du:++pa:ssMa:~k+a

- ea na :vea: a
- ta :pra:Ta:mMa
- +pa:Na
- a:ya:m,a Á ta:d:Ta ya:ya
- va:Sa:ma:sMa:K

- tea :pxa:Ta:k,

sMa:~k+a

- va:Sa:ma:sMa:K
- a:na:nta
- ca :pxa:Ta:k,

sMa:~k+a

- :vMa kx+
- tea l+b

Ea :pa:a=

- a ya:a
- d tua:ya

sUa:[ma I+ a

- a:m,a Á k+

? How is it that you get the value close to the circumference by using antya-sam ara,

instead of repeatedly dividing by

- sk¯ odd numbers
- ? This is being explained

The argument is as follows: If the correction term odd denominator p − 2 (with

- p−1 2

1 ap−2

- is applied after
- is odd),
- 1 1 1 1 1 π = 1 − + −
- 4 3 5 7 p − 2 ap−2 On the other hand,

if the correction term denominator p,

- is applied after the odd

- − + −
- 4 3 5 7 p − 2 p ap

If the correction terms are exact,

then both should yield the same result

That is,

- 1 1 1 = − ap−2 p ap
- 1 1 1 + = ,
- ap−2 ap p

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

It is first observed that we cannot satisfy this condition trivially by taking ap−2 = ap = 2p

the correction has to follow a uniform rule of application and thus,

- if ap−2 = 2p,
- then ap = 2(p + 2)

We can,

- however,

have both ap−2 and ap close to 2p as possible

as first (order) estimate one tries with,

“double the even number above the last odd-number divisor p”,

- ap = 2(p + 1)

it can be seen right away that,

the condition for accuracy is not exactly satisfied

and is estimated 1 1 1 + −

E(p) = ap−2 ap p The objective is to find the correction denominators ap such that the inaccuracy E(p) is minimised

When we set ap = 2(p + 1),

- the inaccuracy will be E(p)
- 1 1 1 + − (2p − 2) (2p + 2) p 4 3 (4p − 4p) 1
- (p3 − p)

- where m is an integer,

the choice of m = 2 is optimal,

as in all other cases there will arise a term proportional to p in the numerator of the inaccuracy E(p)

- 1 2p − 2 +

4 2p − 2

4 (2p+2) ,

1 2p + 2 +

4 2p + 2

- then the
- 1 − p
- (p5 + 4p)

Clearly,

the sthaulya with this (second order) correction divisor has improved considerably,

in that it is now proportional to the inverse fifth power of the odd number

- where m is an integer,

we will end up having a contribution proportional to p2 in the numerator of the inaccuracy E(p),

- unless m = 4

Error-minimization in the evaluation of Pi

Construction of the Sine-table I

so that each arc bit ◦ 0 0 α = 90 24 = 3 45 = 225

- 24 is explicitly given

Recursion relation for the construction of sine-table ¯ Aryabhat

¯ıya translates to: R sin(i + 1)α − R sin iα = R sin iα − R sin(i − 1)α −

In fact,

the values of the 24 Rsines themselves are explicitly noted in another verse

- a is 2(1 − cos α) =
- 2(1 − cos α) = 0

0042822,

- ha we find 1 1 225 → 233

0042827)

00444444

¯ Comment on Aryabhat

a’s Method (Delambre) ¯ Commenting upon the method of Aryabhat

a in his monumental 11 work Delambre 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 was the square of the chord

Here then is a method which the Indians possessed and which is found neither amongst the Greeks nor amongst the Arabs

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

- 12 Delambre,

- cited from B

- na:h:tya
- a:pa:va:geRa:Na
- a:pMa ta

- ca Á h:=e
- sa:mUa:l+yua:gva:gERaH
- a:va:gRa:h:tEaH kÒ+
- a:t,a Á Á
- a:pMa :P+l
- ea nya:~ya
- ea:pa:yRua:pa:a= tya
- jea:t,a Á
- a:yEa,
- sa:ñÍ*:çÅ"Å +h
- eaY:~yEa:va
- taH Á Á I

N0 = Rθ

D0 = 1 (Rθ)2

a:yEa = For obtaining the j¯ıva (Rsine)

Di = Di−1 × R 2 (2i + (2i)2 )

N3 N2 1 − [N D1 − ( D2 − { D3 −

Infinite series for the sine function I

Expressing the series using modern notation as described as described in the above verse – J¯ıv¯ a = Rθ −

R 2 (2 + 22 ) R 2 (2 + 22 ) R 2 (4 + 42 )

(Rθ)3 (Rθ)5 (Rθ)7 + − +

Further simplifying – θ3 θ5 θ7 J¯ıv¯ a=R θ− + − +

- = R sin θ 3

Instantaneous velocity of a planet The mandaphala or “equation of centre” correction I

P0 – mean planet

θMS – true longitude called the manda-sphut

A (direction of mandocca)

P (planet)

The true longitude of the planet is given by r sin M θ = θ0 ± sin−1 R where M (manda-kendra) = θ0 − longitude of apogee

The second term in the RHS,

- known as manda-phala,

takes care of the eccentricity of the planetary orbit

The instantaneous velocity of the planet called t¯ atk¯ alikagati is given by N¯ılakan

ha in his Tantrasangraha ˙ as follows:

- ca:ndÒ
- a:hu:P+l+va:gRa:Za

Da:ta:aa

- $ya:k+a:kx+
- ta:pa:de:na

- t,a Á ta

a k+ea: a

- f:P+l+ a

h ya:a l+Bya:tea Á Á If M be the manda-kendra,

then the content of the above verse can be expressed as r dM i cos M d'h −1 r dt sin sin M = qR 2 dt R r 1 − R sin M

Instantaneous velocity of a planet Derivative of the ratio of two functions

Some of the astronomers in the Indian tradition including Munj¯ ala had proposed the expression for mandaphala to be r sin M R ,

- ∆θ = r 1 − cos M R

the correction to the mean velocity of a planet to obtain its instantaneous velocity in this case is given by 2 r sin M R

! cos M + r R 1− cos M dM R ,

- r dt 1 − cos M R

which is nothing but the derivative of (

- mathematical analysis,

did take place in Kerala School (14-16 century)

that the Kerala astronomers could manipulate with them to obtain several forms of rapidly convergent series is indeed remarkable

there are still certain grey areas (derivative of sine inverse function,

- ratio of two functions)

Many of these achievements are attributed to M¯adhava,

who lived in the 14th century (his works

T HANK YOU

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