It was proven algebraically in 19th century that it is impossible to construct a square with an area equal to the area of a given circle using only a compass and straight edge. However, I once came across a remark that this pertained to constructions involving a finite number of steps.
- I was wondering if this is an accurate statement, and 2) if a procedure exists using a compass and straight edge which could square a circle with an infinite number of steps.
Rational numbers are constructible with compass and straightedge, and it's very easy to construct sequences of rational numbers that converge to $\pi$, for example: $3, 3.1, 3.14, 3.141, 3.1415, \ldots$. That solves the squaring a circle problem. The same argument proves that practically everything can be constructed with compass and straightedge if an infinite number of steps are allowed.