Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Posting-Version: Notesfiles $Revision: 1.6.2.17 $; site uiucdcs.UUCP Path: utzoo!watmath!clyde!cbosgd!ihnp4!inuxc!pur-ee!uiucdcs!mccaugh 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> Lines: 31 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