Madhava of
Sangamagramma 

Born: 1350 in Sangamagramma (near Cochin),
Kerala, India Died: 1425 in India Madhava of
Sangamagramma was born near Cochin on the coast in the Kerala state
in southwestern India. It is only due to research into Keralese
mathematics over the last twentyfive years that the remarkable
contributions of Madhava have come to light. In [10] Rajagopal and
Rangachari put his achievement into context when they write:
[Madhava] took the decisive step onwards from the finite
procedures of ancient mathematics to treat their limitpassage to
infinity, which is the kernel of modern classical analysis.
All the mathematical writings of Madhava have been lost,
although some of his texts on astronomy have survived. However his
brilliant work in mathematics has been largely discovered by the
reports of other Keralese mathematicians such as Nilakantha who lived
about 100 years later. Madhava discovered the
series equivalent to the Maclaurin expansions of sin x, cos x, and
arctan x around 1400, which is over two hundred years before they
were rediscovered in Europe. Details appear in a number of works
written by his followers such as Mahajyanayana prakara which means
Method of computing the great sines. In fact this work had been
claimed by some historians such as Sarma (see for example [2]) to be
by Madhava himself but this seems highly unlikely and it is now
accepted by most historians to be a 16th century work by a follower
of Madhava. This is discussed in detail in [4]. Jyesthadeva
wrote YuktiBhasa in Malayalam, the regional language of Kerala,
around 1550. In [9] Gupta gives a translation of the text and this is
also given in [2] and a number of other sources. Jyesthadeva
describes Madhava's series as follows: 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 the 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.
This is a remarkable passage describing Madhava's series,
but remember that even this passage by Jyesthadeva was written more
than 100 years before James Gregory rediscovered this series
expansion. Perhaps we should write down in modern symbols exactly
what the series is that Madhava has found. The first thing to note is
that the Indian meaning for sine of q would be written in our
notation as r sin q and the Indian cosine of would be r cos q in our
notation, where r is the radius. Thus the series is r
q = r(r sin q)/1(r cos q)  r(r sin q)3/3r(r cos q)3 + r(r sin
q)5/5r(r cos q)5 r(r sin q)7/7r(r cos q)7 + ... putting
tan = sin/cos and cancelling r gives q = tan q 
(tan3q)/3 + (tan5q)/5  ... which is equivalent to
Gregory's series tan1q = q  q3/3 + q5/5  ...
Now Madhava put q = p/4 into his series to obtain p/4
= 1  1/3 + 1/5  ... and he also put q = p/6 into
his series to obtain p = v12(1  1/(33) + 1/(532) 
1/(733) + ... We know that Madhava obtained an
approximation for p correct to 11 decimal places when he gave
p = 3.14159265359 which can be obtained
from the last of Madhava's series above by taking 21 terms. In [5]
Gupta gives a translation of the Sanskrit text giving Madhava's
approximation of p correct to 11 places. Perhaps
even more impressive is the fact that Madhava gave a remainder term
for his series which improved the approximation. He improved the
approximation of the series for p/4 by adding a correction term Rn to
obtain p/4 = 1  1/3 + 1/5  ... 1/(2n1) Rn
Madhava gave three forms of Rn which improved the
approximation, namely Rn = 1/(4n) or Rn
= n/(4n2 + 1) or Rn = (n2 + 1)/(4n3 + 5n).
There has been a lot of work done in trying to reconstruct
how Madhava might have found his correction terms. The most
convincing is that they come as the first three convergents of a
continued fraction which can itself be derived from the standard
Indian approximation to p namely 62832/20000. Madhava
also gave a table of almost accurate values of halfsine chords for
twentyfour arcs drawn at equal intervals in a quarter of a given
circle. It is thought that the way that he found these highly
accurate tables was to use the equivalent of the series expansions
sin q = q  q3/3! + q5/5!  ... cos q = 1
 q2/2! + q4/4!  ...
Jyesthadeva in YuktiBhasa gave an explanation of how Madhava found
his series expansions around 1400 which are equivalent to these
modern versions rediscovered by Newton around 1676. Historians have
claimed that the method used by Madhava amounts to term by term
integration. Rajagopal's claim that Madhava took
the decisive step towards modern classical analysis seems very fair
given his remarkable achievements. In the same vein Joseph writes in
[1]: We may consider Madhava to have been the
founder of mathematical analysis. Some of his discoveries in this
field show him to have possessed extraordinary intuition, making him
almost the equal of the more recent intuitive genius Srinivasa
Ramanujan, who spent his childhood and youth at Kumbakonam, not far
from Madhava's birthplace. Article by: J J
O'Connor and E F Robertson Source:
www.history.mcs.standrews.ac.uk/Mathematicians



