To “square the circle” is to find the length of the
sides of a square with the same area as a circle of a given
radius. The area of a circle of radius
, so success depends on the nature of π. The proofs of Lambert
and Lindemann mean that we know that it is impossible for anyone
to construct on a flat surface a square with the same area as a
given circle using a compass and ruler.
The expression “squaring the circle” means
trying to do the impossible.
Archimedes used the “method of exhaustion” to
approximate the value of π. The work of both Archimedes and Liu
Hui prefigured calculus; Archimedes wrote of infinitesimals and
Liu Hui described the use of the limit.
See also in The book of science:
Readings on wikipedia: