Ancient civilizations knew that there was a fixed ratio of circumference
to diameter that was approximately equal to three. The Greeks refined
the process and Archimedes is credited with the first theoretical
calculation of Pi.
In 1761 Lambert proved that Pi was irrational, that is, that it can't be written as a ratio of integer numbers.
In 1882 Lindeman proved that Pi was transcendental, that is, that Pi is
not the root of any algebraic equation with rational coefficients. This
discovery proved that you can't "square a circle", which was a problem
that occupied many mathematicians up to that time. (More information on
squaring the circle.)