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IIT Bombay,

for his talk explaining the History of Mathematics in I


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

Ramasubramanian IIT Bombay

August 27,



Introduction (Discoveries,

Motivation and Lineage)

Zero and Infinity – dangerous idea


ha’s discussion of irrationality of π

Sum of an infinite geometric series ´

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

Derivation of end-correction terms (Antya-sam ara)

Instantaneous velocity and derivatives

Concluding Remarks

Introduction Celestial Sphere

Great thinkers of all the civilizations – Hindu,

Arabic1 ,


– wondered how to interpret the celestial phenomena

Nasir al-Din al-Tusi,

Ibn al-Shatir,

Introduction Zero and Infinity:

ZUa:nya and A:na:nta

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

They didn’t have the concept of a limit because they didn’t believe in zero

The terms in the infinite series didn’t have a limit or a destination

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

As a result the Greeks couldn’t handle the infinite

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

One of the passages to “limit” is by summing an infinite series

Charles Seife,

Zero:The Biography of a Dangerous Idea,


Rupa & Co

Introduction Continuing further,

Charles Seife observes:4 Unlike Greece,

India never had the fear of the infinite or of the void


Indian mathematicians did more than simply accept zero

They transformed it changing its role from mere placeholder to number

The reincarnation was what gave zero its power

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

This was the birth of what we now know of algebra


Evolution of Numerals: Brahmi → Modern

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


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


Tamil (Grantha),



Ingenuity of the advent of Place value system & Zero I

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

In Vy¯ asa-bh¯ a

sya on the Yogas¯ utra of Pata˜ njali,

we find an interesting description of the place value system:

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

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

Earliest explicit use of decimal place value system Indian mathematical and astronomical texts I

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

¯ıya (499 CE)

¯ The degree of sophistication with which 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,

¯ The system developed by Aryabhat

a is indeed unique in the whole history of written numeration

Not only unique but also quite ingenious and sophisticated

Numbers of the order of 1016 can be represented by a single character


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

a — perhaps luckily as it is too complicated to read

Signal achievements of Kerala Mathematicians I

The “Newton” series sin x = x −

The “Gregory-Leibniz”6 series 

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

Sine function (jy¯ a) is ubiquitous

For instance,

In the computation of longitude of the planets,

r  sin M λ = λ0 − sin−1 R

The declincation of the Sun is computed using the formula,

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

The time of sunrise,

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

Sources and Lineage I

M¯adhava (c

Parame´svara (c

great observer and a prolofic writer



´ nkara Sa ˙ V¯ariyar (c

Acyuta Pis

Only a couple of works of M¯ adhava (Ven

v¯ aroha and Sphut

acandr¯ apti) seem to be extant


ha’s discussion of irrationality of π I

While discussing the value of π N¯ılakan

ùÁ a-

ZRa:taH Á

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

}Ba:~ya I+yMa ñÍ :pa:a=

Da:sa:Ë*É +

ùÁ a o+€+a Á ku+

ùÁ a:m,a o+tsxa

? 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


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

DaH :pua:naH

nEa:va ma

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]

Again if in terms of certain [other] measure the circumference has no [fractional] part,

then employing the same measure 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


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

va:ya:va:tvMa tua ëÐ*:ëÅÁ +a: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

A situation in which there will be no [fractional] part (i

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

However small the unit be,

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

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

AB is c¯ apa (c) as it looks like a bow

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

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

The expression given by N¯ılakan

ha is: s  1 c'≈ 1+ s2 + j 2


Da:nuaH :pra

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

Repeated halving of the arc-bit,

c¯ apa c'to get c1

Finding the corresponding semi-chords,

jy¯ a (ji ) and the Rversines,

´sara (si ) 3

Estimating the difference between the c¯ apa and jy¯ a at each step

If ∆i be the difference between the c¯ apa and jy¯ a at the i th step,

Here N¯ılakan

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

Sum of an infinite geometric series ta

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


ãúaÁ :t,a :pra:de:ZMa ga:tva

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

Sum of an infinite geometric series N¯ılakan

ha poses a very important question:

TMa :pua:naH ta

? 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

Infinite Geometric Series – tua:ya:

cCe +d:pa

Divisor – Ce +d


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 + ,

which leads to the general result,

"  2  n #  n   1 1 1 1 1 1 − + +

As we sum more terms,

the difference between 13 and sum of powers of 41 ,

What is a Limit

? 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


TMa :pua:naH ta

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


Cours d’Analyse,

A History of Mathematics,

Addison Wesley Longman,

New York 1998,

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

Consider the product a

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

 If we consider the ratio bc ,

Case i: gun

In this case we rewrite the product in the following form a

Case ii: gun

In this case we rewrite the product as c  (b − c)

Binomial series expansion In the expression a (b−c) 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 ×

If we again replace the division by the divisor b by the multiplier c,

  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

Binomial series expansion Thus,

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

‹aH Á ta:Ta


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

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

If b − c'< c,

then the successive correction terms keep decreasing

Different approximations to π I

´ The Sulba-s¯ utra-s,

give the value of π close to 3

¯ Aryabhat

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

Da:kM Za:ta:ma::gua:NMa dõ


}Ba:~ya ‘A


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


vMa:Za: a

3927 = 3

1416 1250

¯ that’s same as Aryabhat

¯ ¯ arya,

¯ 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¯

Ma :pa:a=


ÙùÅ a:mua:€+


aH Á 10 Á The values of π given by the above verses are: π=

The latter one is due to M¯ adhava




Hut¯ a´sana=3,



V¯ aran

B¯ ahu=2,

Nava-nikharva=9 × 1011

(The word nikharva represents 1011 )

Infinite series for π – as given in Yukti-d¯ıpik¯a v

Ba:h:tea Á

ùÁ a:Ba:€+

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

Again the products of the diameter and four are divided by the odd numbers like three,

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

(bh¯ utasankhy¯ ˙ a system)

→ to be subtracted and added [successively] 

Infinite series for π The triangles OPi−1 Ci and OAi−1 Bi are similar

Pi−1 Ci Ai−1 Bi = OAi−1 OPi−1

Similarly triangles Pi−1 Ci Pi and P0 OPi are similar

Pi−1 Ci OP0 = Pi−1 Pi OPi

Infinite series for π From these two relations we have,

Ai−1 Bi


Pi−1 Pi OPi−1

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

 It is nr that is refered to as khan

The text also notes that,

the Rsines can be taken as the arc-bits itself

(local approximation by linear functions i


DyMa:Za tangents/differentiation) i

Ai−1 Bi → Ai−1 Ai

Infinite series for π

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

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 (

?) 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

=M ZUa:nya:pra

Infinite series for π Thus we have,

  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 − + +

Infinite series for π If we take out the powers of bhuj¯ a-khan

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

namely ed¯ adyekottara-varga-sankalita,

˙ 12 + 22 +

ed¯ adyekottara-varga-varga-sankalita,

˙ 14 + 24 +

Kerala astronomers knew that n X i=1

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

Summation of series (sankalita) ˙ [Integral

¯ ¯ The Aryabhat ˙

¯ıya of Aryabhat

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

From these,

it is easy to estimate these sums when n is large

Yuktibh¯ a

s¯ a gives a general method of estimating the sama-gh¯ ata-sankalita ˙ (k)

Sn = 1k + 2k + · · · + nk ,

What it presents is a general method of estimation,

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

Summation of series (sankalita) ˙ Samaghata-sankalita ˙

Thus in general we have,

(n − 1)k (n − 2)k (n − 3)k + + +

Rewriting the above equation we have (k) Sn

†a:tsa:ñÍ*:öÐÅ + a

ùÁ a:‹a-


Thus we obtain the estimate (k )

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

The series for

To obtain value of π which is accurate to 4-5 decimal places we need to consider millions of terms

To circumvent this problem,

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

The expression provided to estimate the remainder terms is noted to be quite effective

Even if a consider a few terms (say 20),

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)

Nea kx+


ùÁ a ta:‰

lM gua:Na

Ga:nea [ea:pa O

DaH :pa:a=

sUa:[maH ba:hu:kx+

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

j¯ amitay¯ a → being bored (due to slow-convergence)   Remainder term = 

→ 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,

TMa :pua:na

vRa:Sa:ma:sa:Ë*ñÍÉ +

ùÁ a:h

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

a:sa:Ša:tva:m,a A:ntya:sMa:~k+a:=e


Nea kx+


Ra:t,a Á A:Ta ta:du


Ra:t,a Á O

Ea :pa:a=

Ea Ba:va:taH ta:a


sUa:[ma I+ a


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

instead of repeatedly dividing by

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

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

1 ap−2

if the correction term denominator p,

π 1 1 1 1 1 1 = 1 − + −

If the correction terms are exact,

then both should yield the same result

That is,

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

End-correction in the infinite series for π Optimal choice for error-minimizaion

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,

We can,

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”,

it can be seen right away that,

the condition for accuracy is not exactly satisfied

The measure of inaccuracy (sthaulya) E(p) is introduced,

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

End-correction in the infinite series for π Optimal choice for error-minimizaion

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

It can be shown that among all possible correction divisors of the type ap = 2p + m,

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)

End-correction in the infinite series for π Optimal choice for error-minimizaion

If we take the correction divisor to be ap = 2p + 2 + inaccuracy is found to be E(p)

4 2p − 2

4 (2p+2) ,

1 2p + 2 +

4 2p + 2


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

It can be shown that if we take any other correction divisor m ap = 2p + 2 + (2p+2) ,

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

Error-minimization in the evaluation of Pi

Construction of the Sine-table I

A quadrant is divided into 24 equal parts,

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

A procedure for finding R sin iα,

Pi Ni are known

The R sines of the intermediate angles are determined by interpolation (I order or II order)

Recursion relation for the construction of sine-table ¯ Aryabhat

¯ıya’s algorithm for constructing of sine-table I

¯ The content of the verse in Aryabhat

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

R sin iα

R sin α

In fact,

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

The exact recursion relation for the Rsine differences is:

R sin(i +1)α−R sin iα = R sin iα−R sin(i −1)α−R sin iα 2(1−cos α)

¯ Approximation used by Aryabhat


In the recursion relation provided by N¯ılakan



¯ 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,

” – Prix prize citation 1789

Historie de l’Astronomie Ancienne,

Paris 1817,

Datta and A

Hindu Trigonometry,

IJHS 18,

Infinite series for the sine function I

The verses giving the ∞ series for the sine function is13 –


N0 = Rθ

D0 = 1 (Rθ)2

Ni+1 = Ni × (Rθ)2

N1 = Rθ ×

D1 = R 2 (2 + 22 )

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

Di = Di−1 × R 2 (2i + (2i)2 )

N3 N2 1 − [N D1 − ( D2 − { D3 −

¯ Yuktid¯ıpik¯ a (16th cent) and attributed to Madhava (14th cent

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 −

Rθ × (Rθ)2 × (Rθ)2 Rθ × (Rθ)2 + −

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

Simplifying the above we have – J¯ıv¯ a = Rθ −

(Rθ)3 (Rθ)5 (Rθ)7 + − +

R 2 × 6 R 4 × 6 × 20 R 6 × 6 × 20 × 42

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

Thus the given expression ≡ well known sine series

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

P0 – mean planet

P – true planet

θ0 – mean longitude

θMS – true longitude called the manda-sphut

A (direction of mandocca)

P (planet)

Q θ0 − ϖ

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,

takes care of the eccentricity of the planetary orbit

Instantaneous velocity of a planet Derivative of sin−1 function

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

ha in his Tantrasangraha ˙ as follows:



ˆa k+ea: a


Ma :ke+

h ya:ƒa 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 ,

According to Acyuta,

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   ,

which is nothing but the derivative of (

Concluding Remarks I

It is clear that major discoveries in the foundations of calculus,

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

Besides arriving at the infinite series,

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

While the procedure by which they arrived at many of these results are evident,

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

Many of these achievements are attributed to M¯adhava,

who lived in the 14th century (his works

Whether some of these results came to be known to the European mathematicians