From: AOUM
Newsgroups: sci.math
Subject: Re: Problems from Greek Antiquity
Date: 17 Feb 1996 00:12:29 GMT
kochman@tang.ccr-p.ida.org (Fred Kochman) wrote:
>In article <4fmk6t$93r@digital.netvoyage.net> mjcohen@digital.netvoyage.net writes:
>
> In article <4eio7b$qpp@news.ox.ac.uk>,
> Gerald McLarnon wrote:
> >Does anyone know what the definitive treatment is on the three problems
> >that caused the ancient Greeks so much trouble, namely
> >
> >1. Squaring the circle
> >2. Doubling the cube
> >3. Trisecting the angle
> >
> >using straight edge and compass alone (although I know there is some
> >debate as to whether or not the Greeks placed such stringent requirements
> >on the solutions to the problems). I have a vague notion about a
> >Russian paper c1880 which proves (1) can't be done, but would be interested
> >if anyone had definite dates and authors for proofs (I do not
> >need to know actual references).
> >
> >The FAQ on trisecting the angle provides no reference (at least the version
> >that I have).
> >
> >Thanks in advance.
> >
> >Gerald McLarnon
(snip snip ...)
Apollonius from Perga give us the following beutifull construction to 3. :
*
*
A*********************************o***************M
*|
* |
* |
* |
* |
* |
* o
* |
O********U********************************************M
* Given the Angel AOM draw a parallel AM to OM (A is arbitrary).
* Draw the perpendicular AU on OM.
* Now, take your compass opened twice the distance AO and let it slide
on your ruler.
Letting the ruler pass through the point O,
pass the distance 2*AO (opening of the ruler) in between the lines
AU and AM.
Your compass now points to the points o and o.
* The Angel AOUM is now trisected by the Angel ooOUM.
==============================================================================
From: Roman Karawatzki
Newsgroups: sci.math
Subject: Re: Trisecting the Angle !
Date: 18 Feb 1996 03:22:16 GMT
AOUM (whatever or whoever he is) wrote :
> Apollonius from Perga give us the following beutifull construction to 3. :
>
> *
> *
> A*********************************o***************M
> *|
> * |
> * |
> * |
> * |
> * |
> * o
> * |
> O********U********************************************M
>
> * Given the Angel AOM draw a parallel AM to OM (A is arbitrary).
> * Draw the perpendicular AU on OM.
> * Now, take your compass opened twice the distance AO and let it slide
> on your ruler.
> Letting the ruler pass through the point O,
> pass the distance 2*AO (opening of the ruler) in between the lines
> AU and AM.
> Your compass now points to the points o and o.
> * The Angel AOUM is now trisected by the Angel ooOUM.
Yes, that's wonderful indeed !
There is even a simple geometric proof of that :
*
*
A*********************************o***************M
*|---- ----|
* | ---- ---- |
* | --- --- |
* | --u-- |
* | ---- ---- |
* | ----- ----|
* -o---------------------------------m
* ---- |
O---*****U********************************************M
Drawing the rectangular AomoA.
By construction, the distance OA equals the distances Au and ou.
The triangles AOu and Aou are therefore equilateral.
Now, the angles (Angels :-) yes, funny)