Squaring the Circle: Sulbasutra Approximations of Pi
Ancient India's practical solutions to a classical problem
Discover how the Sulbasutras provide remarkably accurate approximations of pi through geometric constructions.
Squaring the Circle: Sulbasutra Approximations of Pi
In April 1975, in the village of Panjal in Thrissur district, Kerala, a team of Nambūdiri brahmins are laying out the groundwork for a twelve-day Agnicayana ritual. The air is hot and smells of kiln-baked brick. A square altar has already been marked in the dust. It must now be transformed into a circular altar of exactly the same area so the next stage of the yajña can proceed. The senior priest walks to the centre of the square with a coil of plaited cord in his hand. At the edge of the ritual ground, the Dutch Sanskritist Frits Staal is watching with a film crew. The priest stretches the cord along the half-diagonal of the square, takes the excess over half the side, adds one-third of that excess, and uses the result as the radius of the new circle. The construction he has just performed was written down in Sanskrit by a śulbakāra named Baudhāyana around 800 BCE. It gives an implicit value of pi within 1.7 percent of the modern figure. It has been passed from teacher to student, unbroken, for twenty-eight centuries. Frits Staal's camera records the moment.
The problem the Nambūdiri priest has just solved is called squaring the circle: given a circle, construct a square of equal area, or given a square, construct a circle of equal area. It looks trivial with modern tools. It is not. For two and a half millennia, European geometers from Anaxagoras in 5th century BCE Athens to professional mathematicians in 19th century Europe attempted to solve it exactly using only compass and straightedge. None of them succeeded, because none of them could. In 1882, Ferdinand von Lindemann proved that pi is a transcendental number, which means the exact construction they were chasing is mathematically impossible. The entire European effort, spanning generations, was a pursuit of something that could not exist.