On Mar 20, 2008, at 4:44 AM, Franz Gnaedinger wrote:
Every rational approximation of an irrational number can be constructed using ruler and compass. Most of these constructions are very complicated, and hardly of any use. Mathematically interesting are those constructions that can be iterated in order to get ever better approximations. Otherwise they are hardly of interest to mathematicians. Because they don't tell us anything about the essence of the irrational number involved.
I am reviewing Fibonacci's "De Practica Geometrie". In it he shows how Archimedes the Philosopher found that the ratio of the circumference of a circle to its diameter is 3 1/7. The proof proceeds by inscribed and circumscribed polygons. The demonstration takes five pages. Along the way he says, "Greater precision is not possible because irrational numbers lack rational roots." Then he goes on and on and finally shows that he is within 1/11 of 3 1/7. He then concludes, "The wise men of antiquity held that the [circumference of] a circle is thrice and a seventh of its diameter. And this is what I wanted to show."
My conclusion is the same a Franz's. Since the XIII century we have rationally approximated Pi by ruler and compass, never achieving it. Peace, Don