Better and better corrections
For 1 − 1/3 + 1/5 − … (a quarter of π) the Kerala texts give three corrections after n terms, each better than the last: 1/(4n), n/(4n² + 1) and (n² + 1)/(4n³ + 5n). This lesson uses the second.
38 / ANTIQUITY TO THE RENAISSANCE
Madhava of Sangamagrama
How long is the rim of a well 1 m across? Madhava’s rule builds the answer from pieces of 4, 4/3, 4/5, 4/7, … metres, added and taken back in turn.
Explore the mathematics ↓
01 / AN ENDLESS SUM
Can endless corrections settle on one length?
Lay pieces of 4, 4/3, 4/5, … metres back and forth along a measuring line, and watch where the rope’s end settles.
A fixed ×5 close-up shows the end swinging past the rim and back, a little less each time.
Stop anywhere and add Madhava’s correction. After only four pieces the end lands within a fifth of a millimetre of the rim.
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THE IDEAThe rim of a circle 1 unit across is 4 − 4/3 + 4/5 − 4/7 + … units long: π. The endless sum settles slowly, but the rim always lies inside its last swing, and Madhava’s correction after the odd number p, 4 × ((p + 1) ÷ 2) ÷ ((p + 1)² + 1), lands almost exactly on it.
The person behind the ideas
- Founded the Kerala school of astronomy and mathematics.
- His mathematical writings are lost; his results on infinite series survive through later Kerala authors such as Nilakantha and Jyeshthadeva.
- Gave π correct to eleven decimal places: 3.14159265359.
More of the mathematics
Series for sine and cosine
Madhava is also credited with infinite series for the sine, the cosine and the inverse tangent, centuries before they appeared in Europe.