1) The "squaring of the circle" refers to operations using the Greek formula of only straightedge and compass. The Greek formula is formally a second degree equation. It is rigorously proven that a circle *cannot* be squared using only first and second degree operations .
2) A circle *can* be exactly squared if we are allowed to use higher order curves - Hippia's quadratrix, Tschirnausen's quadratix, Ozam's quadratix, the cochleoid, the archimedian spiral... but that is cheating on the Greeks.
3) Pi is pi. As it unites algebra and analytic geometry through Euler's equation, e^[(i)(pi)]=-1, you can't go screwing around with it.
4) The ratio of a circle's circumference to it diameter is only pi in plane (curvaure = zero) geometry. In curved geometries (elliptic, hyperbolic) there are additional terms in the ratio. Pi remains pi. Exactly.
5) Since sqrt(2) is an irrational number... You've been had.
6) You can transform a circle into straignt linear motion witha mechanical linkage,
nullright here, but that isn't squaring the circle by the book.
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Uncle Al
http://www.mazepath.com/uncleal/
(Toxic URL! Unsafe for children and most mammals)
http://www.mazepath.com/uncleal/qz4.htm
