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Topic: Re: Claim 3.141592653... is new
Replies: 2   Last Post: Mar 20, 2008 4:15 PM

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Don Cook

Posts: 101
Registered: 12/3/04
Re: Claim 3.141592653... is new
Posted: Mar 20, 2008 4:15 PM
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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



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