Contribution

2.  Irinnattapilli Madhavan Nampudiri known as
Madhava of Sangamagrama.
 [c.1340-c.1425] born near Cochin
on the coast in Kerala.
 Mathematician and Astronomer.
 Started a school in Kerala called
“The Kerala School of Mathematics
And Astronomy.”
 Made significant contributions in
Calculus, geometry, infinite series,
algebra and trigonometry.

3.  First Mathematician who has applied the endless
series in trigonometric functions like sin, cos and tan.
 Most of his books are lost, the books which are found
have been used by today’s mathematicians and
researchers to research Mathematics
 Worked as an Astronomer-Mathematician till his last
breadth.

4.  Aganita-grahacara
 Chandravakyani
 Golavada
 Lagnaprakarana
 Madhyamanayanaprakara
 Mahajyanayanaprakara
 Sphutacandrapti
 Venvaroka

6.  Among his many contributions, he discovered infinite series
for the trigonometric functions of sine, cosine, arctangent, and
many methods for calculating the circumference of a circle.
 In the text ‘Yuktibhasa’, ‘Jyesthadeva’ describes the series in the
following manner:
“The first term is the product of the given sine and radius of the
desired arc divided by the cosine of the arc. The succeeding
terms are obtained by a process of iteration when the first term
is repeatedly multiplied by the square of the sine and divided
by the square of the cosine. All the terms are then divided by
odd numbers 1,3,5,… The arc is obtained by adding and
subtracting respectively the terms of odd rank and those of
even rank. It is laid down that the sine of the arc or that of its
complement whichever is the smaller should be taken here as
the given sine. Otherwise the terms obtained by this above
iteration will not tend to the vanishing magnitude.”

7.  This yeilds
Or equivalently,
 This series is known as ‘ ’.
 It’s has been attributed to Gottfried Wilhelm Leibniz
[1646-1716] and James Gregory [1638-1675].
 The series was known in Kerala more than two
centuries before its European discoveries were born.

8.  Madhava composed an accurate table of sines.
 Marking a quarter circle at twenty-four equal intervals,
he gave the lengths of the half-chord (sines)
corresponding to each of them. It is believed that he
may have computed these values based on the series
expansions:

9.  Madhava’s work on the value of the mathematical
constant pi is cited in the ‘Mahajyanayanaprakara’
(“Methods for the great sines”).
 While some scholars such as Sarma feel that this book
may have been composed by Madhava himself, it is
more likely the work of a 16th -century successor.
 This text attributes most of the expansions to Madhava,
and gives the following infinite series expansions of ∏,
now known as the Madhava-Leibniz series:
 which he obtained from the power series expansion of
the arctangent function.

10.  He also gave a correction term Rn for the error after
computing the sum up to n terms, namely;
where the third correction leads to highly accurate
computation of ∏.
It has long been speculated how Madhava found these
correction terms.

11.  They are the first three convergents of a finite
continued fraction, which when combined with the
original Madhava’s series evaluated to n terms, yields
about 3n/2 correct digits:

12.  The absolute value of the correction term in next higher
order is
 He also gave a more rapidly converging series by
transforming the original infinite series of ∏, obtaining
the infinte series,
 By using the first 21 terms to compute an approximation
of ∏, he obtains a value correct to 11 decimal places
[3.14159265359].
 The value of 3.1415926535898, correct to 13 decimals, is
sometimes attributed to Madhava, but may be due to one
of his followers.
 These were the most accurate approximations of ∏ given,
since the 5th century

13.  Madhava developed the power series expansion for some
trigonometry functions which were further developed by
his successors at the Kerala school of astronomy and
mathematics.
 Madhava also extended some results found in earlier
works, including those of Bhaskara II.
 However, they did not combine many differing ideas
under the two unifying themes of the derivative and the
integral, show the connection between the two, or turn
calculus into the powerful problem-solving tool we have
today.

14.  The Kerala school of astronomy and mathematics flourished
for at least two centuries beyond Madhava. In Jyesthadeva we
find the notion of integration, termed sankalitam, (lit.
collection), as in the statement:
‘ekadyekothara pada sankalitam samam padavargathinte pakuti’.
which translates as the integral of a variable (pada) equals half
that variable squared (varga), i.e. The integral of x dx is equal
to x2/2. This is clearly a start to the process of integral calculus.
 A related result states that the area under a curve is its
integral. Most of these results pre-date similar results in
Europe by several centuries. In many senses, Jyeshthadeva’s
Yuktibhasa may be considered the world’s first calculus text.

15.  The group also did much other work in astronomy;
indeed many more pages are developed to astronomical
computations than are for discussing analysis related
results.
 The Kerala school also contributed much to linguistics
(the relation between language and mathematics is an
ancient Indian tradition).
 The ayurvedic and poetic traditions of Kerala can also be
traced back to this school. The famous poem,
Narayaneeyam, was composed by Narayana Bhattathiri, a
prominent scholar of this school.