HDTenEightyP/NetNews huggingface dataset:
From: kogeddes@daisy.uwaterloo.ca (Keith O. Geddes)
Subject: Re: function arguments type checking in MapleV
Organization: University of Waterloo
Date: Wed, 26 Aug 1992 14:07:14 GMT

In article <FMMOORE.92Aug25193637@momoney.mit.edu> fmmoore@athena.mit.edu (Fes-Mike Moore) writes:
>I'm trying to define type/procargs as described in ?type[argcheck] (page
>223 of the Maple V Library Reference manual).  This is what I'm
>entering:
>
>	> `type/expectationargs` := [indexed * function(indexed)];
>              type/expectationargs := [indexed function(indexed)]
>	> expectation := proc(expr)
>	> local i,j,k,l;
>	>   if not type([expr], expectationargs) then
>	>       ERROR(
>	>   `invalid arguments; must be in form of indexed * function(indexed)`)
>	> fi;
>	> i := op(1,op(1,expr)); 
>	> j := op(2,op(1,expr)); 
>	> k := op(1,op(1,op(2,expr))); 
>	> l := op(2,op(1,op(2,expr))); 
>	> print(i,j,k,l) 
>	> end;
>
>When I run the program, though, I get:
>
>	> testarg := z[a,b]*conjugate(z[fes,mike]);
>                   testarg := z[a, b] conjugate(z[fes, mike])
>
>	_____________________________________________________________________
>	> expectation(testarg);
>	Error, (in expectation) testing against an invalid type
>
>Aren't function and indexed valid types?  If so, why can't they be used
>in this manner?  Or am I overlooking something else?
>

Yes, function and indexed are valid types.

The problem is with the "product" type.
As one discovers from the help page  ?type[structured]  (or equivalently,
in the Maple V Library Reference Manual, Section 2.1.335 "Definition of
a structured type in Maple" -- see the table on page 252),
the type "product of terms" must be specified using  &*
and similarly the type "sum of terms" must be specified using  &+ .
An alternative in cases where all terms are of the same type, is to use
the forms  `*`(type)  or  `+`(type) .

In your case, by changing your first line to

> `type/expectationargs` := [indexed &* function(indexed)];

it will work fine.


Professor Keith Geddes
Director, Symbolic Computation Group
Department of Computer Science
University of Waterloo
Waterloo, Ontario
Canada N2L 3G1                     kogeddes@daisy.uwaterloo.ca
Newsgroups: sci.math.symbolic
From: qhuang@wiliki.eng.hawaii.edu (Qiming Huang)
Subject: ode solver, different results from different packages
Organization: University of Engineering, College of Engineering
Date: Thu, 3 Sep 1992 10:20:37 GMT

Hi, I am trying to learn these different symbolic math packages
such as mathmatica, mapleV, and macsyma. I tested one example
listed on the macsyma manual for ordinary differential equations
which writes as

	x*dy/dx +a*x*y^2+2*y+b*x=0

the solution given by macsyma is
						  
						  (2 a sqrt(b) x)/sqrt(-a)
	y=((2 %c a b x - 2 %c sqrt(-a) sqrt(b)) %e

	                  			        (2 a sqrt(b) x)/sqrt(-a)
+ sqrt(-a) a sqrt(b) x -a)/(2 %c sqrt(-a) a sqrt(b) x %e
   
   2
+ a x)
where %c is the integration constant.

the solution given by mathematica is
In:=DSolve[x*y'[x]+a*x*y^2+2*y+b*x==0, y[x],x]
Out={{y[x]--> -(bx) -axy^2 +C[1] - 2y log[x]}}

and the mapleV didn't give me any thing back.

What went wrong?

Please email me or post your opinion. Thanks very much!

Newsgroups: sci.math.symbolic
From: petry@runners.math.washington.edu (David Petry)
Subject: Re: Mma Solve[] oddity (a much shorter version)
Keywords: Solve, oddity
Organization: University of Washington, Mathematics, Seattle
Date: Sun, 23 Aug 1992 21:53:53 GMT

In article <28914@option.GBA.NYU.EDU> jcao@option.GBA.NYU.EDU (Jingbin Cao) writes:
>The following log documents some weired Solve[] oddities. They are:
>
>	1. Solve[] cannot find roots in some cases, at least not within
>	   reasonable period of time. But if you divide the equations
>	   by something, Solve[] finds some roots!

[long transcript deleted]

Here's a simple example of what Jingbin Cao is getting at (I think):

eqs  =  {  a*x*y + b*x + c*y ==0,  d*x*y + e*x + f*y + g == 0}

Solve[ eqs, {x,y}]

Mathematica 2.  seems to be unable to solve the above equations, but if we 
divide out the "a" and "d", so that

eqs = { x*y + b*y + c*x == 0,  x*y + e*x + f*y + g == 0}

then Mathematica gets the answer right away.

A frustrating deficiency!


David Petry

Newsgroups: sci.math.symbolic
From: root@ccsoan.irkutsk.su (Ogloblin A.F.)
Subject: I need in the Reduce 3.5 or Reduce 3.4 for MS DOS,UNIX.
Keywords: Reduce
Reply-To: root@ccsoan.irkutsk.su
Organization: Computing Center SD AS
Date: Fri, 28 Aug 92 16:47:11 -0500

  I need in the Reduce 3.5 or Reduce 3.4 for MS-DOS, UNIX.
I'd like to get these systems. Help me, please.
                        Titorenko Tatyana from the Irkutsk
                                           Computer Center
From: rosentha@bldrdoc.gov (Peter Rosenthal 303-497-5844)
Newsgroups: sci.math.symbolic
Subject: random functions
Date: 31 Aug 92 20:23:23 GMT
Organization: National Institute of Standards and Technology

I would like to be able to generate random numbers that obey some
arbitrary probability distribution function.  Is there some
simple way to do this in mathematica?
For example, I would like a function that looks like this:

rand[pde_]

which might be called to generate Gaussian random variables
in this way:

rand[gaussian[mean,sigma]]

where gaussian[mean,sigma] specifies the Gaussian probability
distribution with a given mean and standard deviation.

I would like to use this function to solve various brownian
motion problems such as a physical pendulum coupled to a dissipative
bath.  If I had a way of generating the noise efficiently then I
could use the numerical differential equation solver built into
mathematica.

Thanks in advance.

Peter Rosenthal


Rosenthal@cmg.eeel.nist.gov

		 
		  
-- 
Peter Rosenthal       Email rosenthal@cmg.eeel.nist.gov
From: jtien@venice.eng.uci.edu (Joe Tien)
Subject: need math functions
Newsgroups: sci.math.symbolic
Reply-To: jtien@venice.eng.uci.edu (Joe Tien)
Organization: University of California, Irvine
Date: 3 Sep 92 20:40:53 GMT


Hello everyone,

Hope this is the right place for the question.

Does anybody know where I can get the software for computing math functions
such as bessel, erf, and matrix inversions. FTP sites would be okay too. 

Please send email to jtien@balboa.eng.uci.edu.

Thanks much,

Joe Tien
From: charles@capmkt.COM (Charles Neveu)
Newsgroups: sci.math.symbolic
Subject: Saving Mma sounds in .snd format
Summary: How do you save Mma sounds in .snd format
Keywords: Mathematica sounds .snd NeXT
Date: 26 Aug 92 01:30:38 GMT
Organization: Capital Market Technology

Is it possible to save Mathematica-generated sounds to .snd files,
esp. on the NeXT? The standard method of Displaying to a stream saves
the file in Mathematica format.

Charles Neveu
neveu@milo.berkeley.edu
UCBerkeley
From: Antoine@osd.ulaval.ca (Antoine Gautier)
Newsgroups: sci.math.symbolic
Subject: Linear programming polytopes and Mathematica
Date: 2 Sep 92 16:48:15 GMT
Reply-To: Antoine@osd.ulaval.ca    (Antoine Gautier)
Organization: Universite Laval

Does ther exist a way in MMA to draw a 3D polytope defined by a system of  
inequalities on 3 variables?
(this may be trivial, apologies if it is....)
Thks.
From: ara@zurich.ai.mit.edu (Allan Adler)
Newsgroups: sci.math.symbolic
Subject: Re: Todd-Coxeter algorithm
Date: 3 Sep 92 18:55:11 GMT
Article-I.D.: camelot.ARA.92Sep3135511
Organization: M.I.T. Artificial Intelligence Lab.


Get GAP.
Newsgroups: sci.math.symbolic,uiuc.mathematica
From: whitmore@iastate.edu (Kurt D Whitmore)
Subject: Re: MMA Version?
Organization: Iowa State University, Ames IA
Date: Thu, 27 Aug 1992 16:46:11 GMT

In <BtnGvx.6IG@helios.physics.utoronto.ca> butscher@helios.physics.utoronto.ca (Adrian Butscher) writes:

>In article <1992Aug26.103708.26823@afterlife.ncsc.mil> ricampb@afterlife.ncsc.mil (Robert Campbell) writes:

>>...Version 2.1 corrected this for the packages in the standard distribution...

>What is the latest version of MMA for the Macintosh?  The latest I've seen is 
>version 2.03.

>Adrian


Mac version = 2.1.0.2

kw
-- 
_______________________________________________________________________
      Kurt D. Whitmore            |        Whitmore Technical Publications
   whitmore@iastate.edu           |                1003 N. Hyland
   Iowa State University          |                Ames, Ia 50010
From: jonesm@nic.cerf.net (Matthew Jones)
Newsgroups: sci.math.symbolic
Subject: Mathematica for MS/Windows Postscript Printing Problems
Date: 26 Aug 92 18:47:23 GMT
Distribution: usa
Organization: CERFnet

Using MMA for MS/Windows I did a ListPlot and then enlarged in on the screen
when I printed it everything was fine except the text and the Points
were really small, I tried changing the DefaultFont and various other
things put nothing seems to work. Text is totally unreadable.
I print the notebook/cell to a file as Postscript and then send it
to a LaserWriter II NT. Any suggestions?

Matthew Jones
jonesm@cerf.net
Newsgroups: sci.math.symbolic
Subject: Re: mathematica solve and rules
From: campo@sunthpi3.difi.unipi.it (Massimo Campostrini)
Date: 3 Sep 92 09:13:37 +0100
Reply-To: campo@sunthpi3.difi.unipi.it
Distribution: world
Organization: Istituto Nazionale di Fisica Nucleare, Pisa, Italy

In article <BtzEo8.A36@bunyip.cc.uq.oz.au>, coates@newton.physics.uq.oz.au (Tony Coates) writes:
|> 
|> 	"Solve[{a + b = 1, a - b = -1}, {a, b}]"
|> gives
|> 	"{{ a -> 0, b -> 1}}".
|> Usually, to use these rules, I would do something like
|> 	"subs = First[Solve[...]]"
|> where the "First" is to get rid of the outer list braces, which I have
|> never understood the reason for.  If you want to assign these values to
                    --------------

The reason is that if you solve a (system of) polynomial (higher degree)
equation(s) you will get a list of solutions:

In[1]:= Solve[{x^2+x+y==3, x-y==a}, {x,y}]

                                      -2 - 2 a + 2 Sqrt[4 + a]
Out[1]= {{x -> -1 + Sqrt[4 + a], y -> ------------------------}, 
                                                 2
 
                                  -2 - 2 a - 2 Sqrt[4 + a]
>    {x -> -1 - Sqrt[4 + a], y -> ------------------------}}
                                             2
    
-- 
        Massimo Campostrini,  Theoretical Physicist,
Istituto Nazionale di Fisica Nucleare, Sezione di Pisa,
Piazza Torricelli 2, I-56126 Pisa, Italy  ||  Phone: (+39)(50)42093
Internet: campo@sunthpi3.difi.unipi.it    ||  Fax:   (+39)(50)48277
From: kuechlin@tiger.cis.ohio-state.edu (Wolfgang W Kuechlin)
Newsgroups: sci.math.symbolic
Subject: algebra on multiprocessors
Date: 31 Aug 92 15:56:49 GMT
Organization: The Ohio State University Dept. of Computer and Info. Science

The following two books give an introduction to efforts in the field.
In addition, there are pertinent papers e.g. in
Proc. AAECC'90 (LNCS 539), AAECC'91 (LNCS 539),
ISSAC'90 (ACM Press), ISSAC'91 (ACM Press), ISSAC'89,
ACPC'1 (LNCS 591).

	-wk.
Wolfgang K"uchlin     e-mail: <kuechlin@informatik.uni-tuebingen.de>
Universit"at T"ubingen		Tel.:   (+49) 7071-29.7047
Wilhelm-Schickard-Institut	Fax:    (+49) 7071-29.5958
     f"ur Informatik    
Auf dem Sand 13   
D-7400 T"ubingen, W-Germany      


@PROCEEDINGS{CAP:90,
	Title = {Computer Algebra and Parallelism},
	BookTitle = {Computer Algebra and Parallelism},
	Editor = {Richard Zippel},
	Publisher = {Springer-Verlag},
	Volume = 584,
	Series = {{LNCS}},
	Address = {New York, Heidelberg},
	Month = Mar,
	Year = 1992,
	Note = {(Proc.\ CAP'90, Ithaca, NY, May 1990)}
}
@BOOK{CAP:88,
	Editor = {Della Dora, J. and Fitch, J.},
	Title = {Computer Algebra and Parallelism},
	BookTitle = {Computer Algebra and Parallelism},
	Publisher = {Academic Press},
	Year = 1989,
	Address = {London},
	Series = {Computational Mathematics and Applications},
	Note = {(Proc.\  CAP'88,
		Grenoble, France, June 1988)}
}
-- 
Wolfgang Kuechlin,  Asst. Prof.,        NSFNet: <kuechlin@cis.ohio-state.edu>
Computer and Information Science        Office: (614) 292-5813        
The Ohio State University               Fax:    (614) 292-9021
Columbus, OHIO 43210-1277
Newsgroups: sci.math.symbolic
From: parkes@itp.ethz.ch (Andrew Parkes)
Subject: Re: Mma Solve[] oddity (a much shorter version)
Organization: Institute for Theoretical Physics, ETH, Zuerich, Switzerland
Date: Wed, 26 Aug 1992 16:32:41 GMT


David Petry writes:
> In article <28914@option.GBA.NYU.EDU> jcao@option.GBA.NYU.EDU (Jingbin Cao) writes:
> >The following log documents some weired Solve[] oddities. They are:
> >
> >	1. Solve[] cannot find roots in some cases, at least not within
> >	   reasonable period of time. But if you divide the equations
> >	   by something, Solve[] finds some roots!
> 
> [long transcript deleted]
> 
> Here's a simple example of what Jingbin Cao is getting at (I think):
> 
> eqs  =  {  a*x*y + b*x + c*y ==0,  d*x*y + e*x + f*y + g == 0}
> 
> Solve[ eqs, {x,y}]
> 
> Mathematica 2.  seems to be unable to solve the above equations, but if we 
> divide out the "a" and "d", so that
> 
> eqs = { x*y + b*y + c*x == 0,  x*y + e*x + f*y + g == 0}
> 
> then Mathematica gets the answer right away.
> 
> A frustrating deficiency!
> 
> 
> David Petry
> 
Here's an even simpler version of the above 

eq  = (a x^2 + b x + c)/(d1 + d2 x) 
answer = Solve[ eq == 0 , x]

Mma 1.2 gets the answer immediately of course.

But I find that our  Mma 2.0  gets  gets nowhere. 
I killed the job after 40mins when it got to 30MB.

If I give d1 a numerical value then the answer is again immediate.  I
haven't looked whether giving some options to Solve[] makes things
work correctly (as I don't think changing options should be necessary
to solve a quadratic equation!).

In case this is version dependent, our system is:-
$Version =  SPARC 2.0 (July 8, 1991)
$VersionNumber = 2

Andrew Parkes,              | E-mail: parkes@itp.ethz.ch 
Theoretische Physik,        | 
ETH-Honggerberg,            |                           
CH-8093 Zurich, Switzerland.|                           
Newsgroups: sci.math.symbolic
From: pete@marsh.cs.curtin.edu.au (Pete Staples)
Subject: plot on xmaple
Organization: Curtin University of Technology
Date: Thu, 3 Sep 1992 10:17:36 GMT

Is there any way plot (2d or 3d) on xmaple will plot with a 
white background?

Thanx in advance Peter.
Newsgroups: sci.math.symbolic
Subject: Re: ode solver, different results from different packages
From: a_rubin@dsg4.dse.beckman.com (Arthur Rubin)
Date: 3 Sep 92 18:24:30 GMT

In <1992Sep3.102037.20539@news.Hawaii.Edu> qhuang@wiliki.eng.hawaii.edu (Qiming Huang) writes:

>Hi, I am trying to learn these different symbolic math packages
>such as mathmatica, mapleV, and macsyma. I tested one example
>listed on the macsyma manual for ordinary differential equations
>which writes as

>	x*dy/dx +a*x*y^2+2*y+b*x=0

>the solution given by macsyma is
>						  
>						  (2 a sqrt(b) x)/sqrt(-a)
>	y=((2 %c a b x - 2 %c sqrt(-a) sqrt(b)) %e

>	                  			        (2 a sqrt(b) x)/sqrt(-a)
>+ sqrt(-a) a sqrt(b) x -a)/(2 %c sqrt(-a) a sqrt(b) x %e
>   
>   2
>+ a x)


Of course, the correct answer is:

if a=0, y = -b x /3 + %c / x^2

if a!=0, y = -1/(a x) + l/a produces the equation

dl/dx + l^2 + a b = 0,

which Mathematica also cannot solve, but "clearly" has solutions

(a b < 0) sqrt(-a b) tanh ( sqrt(-a b) (x + %c) )   (including c=+/- infinity)
(a b = 0) 1/(x + %c)
(a b > 0) sqrt(a b) tan ( sqrt(a b) (x + %c) )      
--
Arthur L. Rubin: a_rubin@dsg4.dse.beckman.com (work) Beckman Instruments/Brea
216-5888@mcimail.com 70707.453@compuserve.com arthur@pnet01.cts.com (personal)
My opinions are my own, and do not represent those of my employer.
My interaction with our news system is unstable; if you want to be sure I see a post, mail it.
From: ara@zurich.ai.mit.edu (Allan Adler)
Newsgroups: sci.math.symbolic
Subject: algebra on multiprocessors
Date: 31 Aug 92 06:46:17 GMT
Distribution: sci
Organization: M.I.T. Artificial Intelligence Lab.


I would like to know about literature dealing with computer algebra on
multiprocessors, including algorithms that take advantage of the many
processors, snags that defeat these advantages and how to avoid them,
implementations, performance, etc.

Allan Adler
ara@altdorf.ai.mit.edu
From: sherod@newton.Colorado.EDU (Scott Herod)
Newsgroups: sci.math.symbolic
Subject: Work around for bug with Dt in Mma
Date: 26 Aug 92 02:31:23 GMT
Organization: University of Colorado, Boulder

I posted something about this bug a while ago and then changed code
to avoid it.  Now I must deal with it again and could use some
help if others have found a way to correct it.

The problem is in pattern matching with the Dt command.

In version 2.0 for Suns

In[1] := r = Dt[u,{x,c_}] :> y[c]

Out[1] = 0 :> y[c]

(From memory since I'm can't run windows here.)

The problem seems to be that Dt evaluates to zero if the space where
"c" is is anything but an integer.  Can someone suggest a fix?

Scott Herod
(sherod@newton.colorado.edu)
From: ma1q+@andrew.cmu.edu (Michael Albert)
Newsgroups: sci.math.symbolic
Subject: Re: Todd-Coxeter algorithm
Date: 4 Sep 92 05:35:53 GMT
Article-I.D.: andrew.YedqMtq00iUxQ1xF5h
Organization: Mathematics, Carnegie Mellon, Pittsburgh, PA

Thanks to all who replied to my query about public domain software 
incorporating the Todd-Coxeter algorithm. The universal recommendation
was a system called

GAP (groups, algorithms, and programming)

which is available from a number of sites:

dimacs.rutgers.edu
math.ucla.edu
samson.math.rwth-aachen.de

From the last three days of experimentation with the system I am very
impressed. It includes comprehensive (overwhelming?) documentation,
including many examples and easily handled the problems which were 
choking up Maple.

Once again, thanks to all who took the time to reply.

Michael Albert

PS: I must confess to having heard about GAP before, I don't know
how it slipped my mind.
From: d3a132@pnl.gov (Carl P Baker)
Newsgroups: sci.math.symbolic
Subject: Re: Mma Solve[] oddity (a much shorter version)
Date: 26 Aug 92 20:13:32 GMT
Article-I.D.: oracle.1992Aug26.201332.11648
Reply-To: d3a132@pnl.gov
Organization: Battelle Pacific Northwest Laboratories

In article 92Aug26173241@ptah.itp.ethz.ch, parkes@itp.ethz.ch (Andrew Parkes) writes:
>eq  = (a x^2 + b x + c)/(d1 + d2 x) 
>answer = Solve[ eq == 0 , x]
>
>Mma 1.2 gets the answer immediately of course.
>
>But I find that our  Mma 2.0  gets  gets nowhere. 
>I killed the job after 40mins when it got to 30MB.

Eliminate, however, works quite well.  Of course, you have to
decipher the output, which is pretty messy, but quite rigorous.
The problem is that Solve tries to allow the case of the numerator
being zero, and doesn't cope well with it.

Mathematica 2.0 for SPARC
Copyright 1988-91 Wolfram Research, Inc.
 -- X11 windows graphics initialized -- 
 
In[1]:= eq  = (a x^2 + b x + c)/(d1 + d2 x) 
                     2
        c + b x + a x
Out[1]= --------------
          d1 + d2 x
 
In[2]:= Eliminate[eq==0,x]
 
Out[2]= d2 != 0 && a != 0 || d1 != 0 && a != 0 && d2 == 0 || 
 
          c d2
>    d1 - ---- != 0 && b != 0 && a == 0 || d2 != 0 && b == 0 && c == 0 && a == 0 || 
           b
 
>    d1 != 0 && b == 0 && c == 0 && a == 0 && d2 == 0
 
In[3]:= $Version
 
Out[3]= SPARC 2.0 (August 26, 1991)
 
In[4]:= $VersionNumber
 
Out[4]= 2.


The results from Eliminate were immediate.


-- carl

Carl P Baker
cp_baker@pnl.gov
509-375-2724

Disclaimer: I am not the official spokesperson for anyone.  For which
organizations that use spokespersons are profoundly grateful.
Newsgroups: sci.math
From: cj@isy.liu.se (Christian Jonsson)
Subject: Generating Group for Johnson Space/Scheme/Graph
Organization: Dept of EE, University of Linkoping
Date: 31 Aug 92 06:27:59 GMT

Space/Scheme:	Given the set of points in the Hamming space of
		length n and constant weight w.

			J(n,w)= x in H(n): w(x)=w

		Is there a (permutation) group in Sn of size
		n choose w, that generates the whole space from
		an (arbitrary) starting point (in J(n,w))?

Graph:		Given the distance regular graph; the collection
		of all w-subset of an n-set X.

		Is there a (permutation) group in Sn of size
		n choose w, that generates the graph from
		an (arbitrary) starting vertex?
		

Follow up question:

		Given a generating group for the Johnson space,
		is the cyclic subgroup of w-shifts a normal subgroup?


If necessary, assume w divides n!
--
____________________________________________________________________________
Christian Jnsson                                E-mail:       cj@isy.liu.se
Division of Data Transmission                    Telephone: (+46) 13 28 2653
Department of Electrical Engineering             Telefax:   (+46) 13 28 1339
Linkping University                             SWEDEN                     
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
From: wilson@web.ctron.com
Newsgroups: sci.math
Subject: Re: Parallel Postulate
Date: 28 Aug 92 17:52:44 GMT
Reply-To: wilson@web.ctron.com ()
Organization: Cabletron Systems INc.


    In <1992Aug27.115903.10390@waikato.ac.nz>, maj@waikato.ac.nz
    (Murray A. Jorgensen) writes:

> I have always felt that Euclid's form is superior because it contains
> no mention of the mysterious undefined concept 'parallel'. And how
> are we supposed to have any intuition of what 'parallel' means in the
> various possible geometries that may exist without the axiom. I don't
> think Playfairs Axiom plays fair in the least.

    While Playfair's Postulate (PP) tacitly uses the concept of
    "parallel lines," Euclid's Parallel Postulate (EPP) tacitly uses
    the concept of "side of a line."  Their definitions are as follows
    (note "coplanar" in both):

	parallel lines:  Two lines are parallel if they are coplanar
	and nonintersecting.

	side of a line:  Two points are on the same side of a line if
	the points and line are all coplanar and the segment between
	the points does not intersect the line.

    Which concept is more "intuitive," which more "mysterious?"

> It has always struck me as question begging.

    For either PP or EPP to be meaningful in a geometry, "coplanar"
    must be meaningful in that geometry.  In planar absolute geometry,
    "coplanar" is implicit.  In higher-dimensional geometries, for
    example, we must define "coplanar" in order to use either axiom.
    So both axioms beg the question "What is a plane?"

    Given that we know what a plane is, PP becomes:

	For any point P off line l, there exists a unique line
	passing through P, coplanar with l, and not intersecting l.

    Compare this little gem with EPP.  Even given "coplanar," the
    precise statement of EPP still requires defining "the sum of the
    adjacent interior angles is less than two right angles," with
    its incumbent definitions of "side of a line," "sum of angles,"
    etc.  For this reason, I find EPP inferior to PP.

    At any rate, it would be nice to a generalized version of the
    Parallel Postulate which does not need to know about planes.  Is
    this possible?

    
-- 
David W. Wilson (wilson@ctron.com)

Disclaimer: "Truth is just truth...You can't have opinions about truth."
- Peter Schikele, introduction to P.D.Q. Bach's oratorio "The Seasonings."