How it began
Madhava was born about 1350 at Sangamagrama, near Cochin on the coast of Kerala in southwestern India, and died in 1425. Around 1400 he discovered series equivalent to the Maclaurin expansions of sine, cosine and arctangent. All of his mathematical writings have been lost, although some of his texts on astronomy survive. His mathematical work is known from the reports of later Kerala mathematicians such as Nilakantha, who lived about 100 years later, and Jyesthadeva, who wrote the Yukti-Bhasa in Malayalam around 1550.
Before it existed
Earlier mathematics, in India and elsewhere, relied on finite procedures. C. T. Rajagopal and M. S. Rangachari wrote that Madhava took the decisive step onward from the finite procedures of ancient mathematics to treat their limit passage to infinity, which is the kernel of modern classical analysis. For the value of pi, the standard Indian approximation before him was the fraction 62832/20000.
How it works
Jyesthadeva describes the arctangent series in words: the first term is the sine times the radius divided by the cosine, and later terms come from repeatedly multiplying by the square of the sine, dividing by the square of the cosine, and dividing by the odd numbers 1, 3, 5 and so on, with the terms alternately added and subtracted. Setting the angle to a quarter of pi gives pi/4 = 1 - 1/3 + 1/5 and so on. From a faster version Madhava obtained pi correct to 11 decimal places, 3.14159265359. He also gave correction terms that improve the result when the series is cut short.
How it spread
The results were preserved and extended within Kerala by followers whose works, such as a 16th century text on computing the great sines, record the details. In Europe the arctangent series was rediscovered by James Gregory more than 100 years after Jyesthadeva wrote it down, and the sine and cosine series were rediscovered by Newton around 1676. MacTutor notes that Madhava's contributions came to light only through research into Kerala mathematics in the late 20th century.
What lasted
Madhava also produced a table of nearly accurate sine values for twenty-four arcs at equal intervals in a quarter circle, thought to have been computed from his series. Historians have claimed that his method amounts to term by term integration. The historian G. G. Joseph wrote that Madhava may be considered the founder of mathematical analysis. The arctangent series is now often called the Madhava-Gregory series.
