The definition of pi is the ratio of the circumference of a circle to
its diameter. To the early Greeks, it was initially a constant of
proportion in the area formula for a circle. Hippocrates (another one,
not the physician) proved in the 5th century B.C. that the areas of two
circles are in the same proportion as the squares of their diameters.
This is equivalent to proving that there exists a constant c such that:
A = c * d^2 [A=area, d=diameter]
In modern terms, c=pi/4. Substituting r=d/2 gives the familiar formula
for a circle's area in terms of its radius.
It was Archimedes in the 3rd century B.C. who established the value of
the constant c and related it to the circumference/diameter ratio. His
approximate for pi was:
(3 + 1/7) < pi < (3 + 10/71)
Is this the sort of stuff you were interested in?
Cheers,
Mike.