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From: mccaugh@uiucdcs.UUCP
Newsgroups: net.math
Subject: Re: Transcendental Pi
Message-ID: <28200047@uiucdcs.UUCP>
Date: Sun, 30-Dec-84 03:57:00 EST
Article-I.D.: uiucdcs.28200047
Posted: Sun Dec 30 03:57:00 1984
Date-Received: Thu, 3-Jan-85 00:57:34 EST
References: <2228@garfield.UUCP>
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Nf-ID: #R:garfield:-222800:uiucdcs:28200047:000:1668
Nf-From: uiucdcs!mccaugh    Dec 30 02:57:00 1984


  To: robertj
  Re: Impossibility of squaring the circle:

  I am certainly not the best source for responding to this problem, but on
  seeing that no response had not yet been made, here goes:
  i] The construction constraint pertains to straightedge and compass, NOT
     straightedge and ruler (in fact it is fairly trivial to trisect an
     arbitrary angle given a ruler, but impossible without);
 ii] The "squaring" problem--if soluble--would amount to the ability to con-
     struct a square with area equal to that of a circle with radius 1, i.e.,
     a square satisfying: side**2 = pi, and so a line-segment (the side of
     the square) could be constructed of length side = pi**0.5, which is
     also transcendental, since pi is, but:
iii] the only constructible line-segments must have as lengths algebraic
     numbers; in fact the constraints are far stricter--e.g., cube-root of 2
     is not constructible (otherwise it is easy to show that an arbitrary
     angle can be trisected).

 Towards the end of his famous text: "Algebra", the noted authority Jacobson
 (whom some say is to Algebra what Feller was to Probability) discusses the
 subject of constructibility (but not in depth) just before his launch into
 Galois Theory...Altgeld Hall's Math Library here at Urbana has several mono-
 graphs on the subject, including one on all the famous unsolvable problems
 (which includes an interesting--if laborious--complete construction of the
 17-gon using Gauss's original construction and verification). I will be most
 happy to elaborate upon the "squaring" problem (and related problems) if
 sufficient interest accrues.

 --uiucmsl!mccaugh