HDTenEightyP/NetNews huggingface dataset:
From: myrberger@e.kth.se (Johan Myrberger)
Newsgroups: sci.math
Subject: BCH error correcting code parity check matrix
Date: 22 Jan 93 16:11:20 GMT
Reply-To: myrberger@e.kth.se (Johan Myrberger)
Organization: School of EE, Royal Institute of Technolgy

Hello,

is there someone out there who can help me with this?

I'm trying to construct a parity check matrix for a BCH-code.
The code is a BCH(31,21) with g(x)=(x^5+x^2+1)(x^5+x^4+x^3+x^2+1), and
has a dmin of 5.

I understand that one way to construct the parity check matrix (and the way
I'm interested in) is to use a primitive element of GF(32), alpha.
The matrix would look like this:

   __                                               __
   ! alpha^0 alpha^1 alpha^2 ..... alpha^29 alpha^30 !
H= !                                                 !
   ! alpha^0 alpha^3 alpha^6 ..... alpha^87 alpha^90 !
   --                                               --

After reducing the powers of alpha using the fact that alpha^31 = alpha^0
and replacing alpha^a with the binary representation
I get the following matrix:

   __                                                               __ 
   !  1 0 0 0 0 1 0 0 1 0 1 1 0 0 1 1 1 1 1 0 0 0 1 1 0 1 1 1 0 1 0  !
   !  0 1 0 0 0 0 1 0 0 1 0 1 1 0 0 1 1 1 1 1 0 0 0 1 1 0 1 1 1 0 1  !
   !  0 0 1 0 0 1 0 1 1 0 0 1 1 1 1 1 0 0 0 1 1 0 1 1 1 0 1 0 1 0 0  !
   !  0 0 0 1 0 0 1 0 1 1 0 0 1 1 1 1 1 0 0 0 1 1 0 1 1 1 0 1 0 1 0  !
   !  0 0 0 0 1 0 0 1 0 1 1 0 0 1 1 1 1 1 0 0 0 1 1 0 1 1 1 0 1 0 1  !
H= !                                                                 !
   !  1 0 0 0 0 1 1 0 0 1 0 0 1 1 1 1 1 0 1 1 1 0 0 0 1 0 1 0 1 1 0  !
   !  0 0 1 1 1 1 1 0 1 1 1 0 0 0 1 0 1 0 1 1 0 1 0 0 0 0 1 1 0 0 1  !
   !  0 0 0 0 1 1 0 0 1 0 0 1 1 1 1 1 0 1 1 1 0 0 0 1 0 1 0 1 1 0 1  !
   !  0 1 1 1 1 1 0 1 1 1 0 0 0 1 0 1 0 1 1 0 1 0 0 0 0 1 1 0 0 1 0  !
   !  0 0 0 1 0 1 0 1 1 0 1 0 0 0 0 1 1 0 0 1 0 0 1 1 1 1 1 0 1 1 1  !
   --                                                               --

                                                     T
The problem  is that with this matrix the relation cH  = 0 don't hold!
(c being a codeword)

Why is this? What am I missing?

Perhaps there's something wrong with the binary representation of alpha^i?

If someone can give me a hint please MAIL me!

Johan Myrberger

MAIL: myrberger@e.kth.se
Newsgroups: sci.math,sci.physics
From: platt@watson.ibm.com (Daniel E. Platt)
Subject: Re: Vectors Question
Date: Tue, 26 Jan 1993 14:58:42 GMT
Disclaimer: This posting represents the poster's views, not necessarily those of IBM
Organization: IBM T.J. Watson Research Center

In article <93025.223136A54SI@CUNYVM.BITNET>, <A54SI@CUNYVM.BITNET> writes:
|> I'm reading Principles of Electrodynamics by Schwartz. In it he discusses
|>  two types of vectors, polar (where the components negate under pure
|>  reflective transformation) and axial (where the components are invariant
|>  for pure reflective transformations).
|> 
|> Either my thinking is wrong or he contradicts himself :
|> 
|> Let us set c = a cross b. In my mind, if a and b are both polar or they are
|>  both axial then c is axial. I do this by picturing reflections performed on
|>  the vectors, and I am fairly certain of this result. If we try to cross two
|>  vectors of different types, then the c vectors before and after the
|>  reflection (c and c') differ in magnitude as well as direction and so
|>  I feel that it's impossible to cross different type vectors. Be that as it
|>  it may you won't get a polar vector by crossing a polar with and axial
|>  (Unless my mental pictures are totally wrong.)
|> 
|> From Coulomb's law he gets that force is a polar vector. Fine.
|> 
|> Now from magnetism we know that force F = qv cross B. Velocity v is polar.
|>  He's already assumed that charge q is scalar (totally invariant). From all
|>  this he thinks this tells us that B is axial!

Jackson calls polar vectors 'vectors' and axial vectors 'pseudovectors.'  He
identifies location, velocity, momentum, force, electric field, etc as vectors
(polar), and angular momentum and magnetic field as pseudovectors (axial).

You should not think of the transformation as a reflection about some plane.
It is a transformation x -> x' = -x for all components.  For functions of
location, it goes like B -> B'(x, t) = +B(-x, t), E -> E'(x, t) = -E(-x, t).

Cross products of even with even and odd with odd are even (Angular momentum
is axial), while odd with even are odd (polar).  F = v X B  with F and v odd
implies B must be even.

Dan

-- 
-------------------------------------------------------------------------------
Daniel E. Platt                                      platt@watson.ibm.com
The views expressed here do not necessarily reflect those of my employer.
-------------------------------------------------------------------------------
From: scavo@cie.uoregon.edu (Tom Scavo)
Newsgroups: sci.math
Subject: Re: Geometer's Sketchpad
Keywords: software Geometer's Sketchpad
Date: 24 Jan 93 03:01:41 GMT
Article-I.D.: pith.1jt0qlINNevr
Organization: University of Oregon Campus Information Exchange
NNTP-Posting-Host: cie.uoregon.edu

In article <C1A02n.4E4@news.udel.edu> dsacher@ravel.udel.edu (Dick Sacher) writes:
>
>Does anyone have any information on the program Geometer's Sketchpad,,
>such as how to obtain a copy?   Thanks for your help.

See Vol. 4, No. 2 (May 1992) of the _Trends_.  The closing paragraph
of a nice review article on p.2 reads

	The software, which runs on Macintosh computers, was devel-
	oped by Nicholas Jackiw of Key Curriculum Press, under the
	auspices of the Visual Geometry Project, a National Science
	Foundation project under the leadership of Eugene Klotz
	(Swarthmore College) and Doris Schattschneider (Moravian
	College).  The single-user price is $169.95, a laboratory
	ten-pack is $399.95, and a network/site license is $899.95. 
	Contact Key Curriculum Press, P.O. Box 2304, Berkeley, CA,
	94702, telephone (800) 338-7638, fax (510) 548-0755.

-- 
Tom Scavo
scavo@cie.uoregon.edu
From: isaacs@hpcc01.corp.hp.com (Stan Isaacs)
Date: Wed, 27 Jan 1993 19:25:16 GMT
Subject: Re: Why algebra and what is important?
Organization: the HP Corporate notes server
Newsgroups: sci.math


> I like an analogy with carpentry.  Our current math teaching is like
> giving someone a pile of nails and wood, making them drive nails for a
> few years until they learn to do it perfectly, and only then letting
> them do something interesting, i.e. build something.  How boring!  I
> think its much better to let them build something fun (and kids do
> find conceptual math fun, if taught right) even if its a little
> rickety because they are not great at driving nails.  In the process
> they not only learn the basics, but learn why they should care about
> the basics.
> --
> Jeff


Mine favorite analogies were/are the answers to a question posed by
Scot Kim at the annual meeting of the California NCTM this year: What
would it be like if music were taught like mathematics?

My favorite answer was:  You wouldn't hear your first symphony until
graduate school.

  -- Stan Isaacs
Newsgroups: sci.math
From: gore@inferno.rutgers.edu (Bittu)
Newsgroups: sci.math
Subject: Matrices (problem)
Keywords: Rank
Date: 25 Jan 93 03:14:32 GMT
Organization: Recreation Center, Busch Campus

A simple question about the rank of a special matrix (this is not a
homework problem; I already know a solution, want something better).
In fact, this was probably on the Math Olympiad a few years ago.

Let A be a matrix with the following properties:

1. A is 2n+1 x 2n+1 for a positive integer n.

2. A has 0's on the diagonal.

3. The remaining entries are either 1's or -1's.

4. Every row has exactly n 1's and n -1's.

We want to show that the rank of this matrix is 2n (over complex
numbers, say). Clearly the rank is <= 2n (all columns sum to the zero
vector). 

Let I_m be the m x m identity matrix and J_m be the all 1's matrix.
The solution I know involves showing that the matrix J_m - I_m has
rank m if m is even and m-1 if m is odd (over GF[2]; done by showing
that its determinant is (-1)^(m-1) . (m-1)). We are only interested
in the case where m is odd. We now use the well-known result that if
the rank is m-1 over GF[2], it is m-1 over rationals, reals, complexes
etc. Note that J_{2n+1} - I_{2n+1} is the same as A over GF[2].

I am sure there is a much easier and elegant proof of this (for the
field of complexes). Any takers?

--Bittu
From: loeb@greco-prog.fr (Daniel LOEB)
Newsgroups: sci.math
Subject: Re: Combinatorial Problems
Date: 27 Jan 93 21:34:13 GMT
Reply-To: loeb@geocub.UUCP (Daniel LOEB)
Organization: Greco Programmation du CNRS

n article <1993Jan27.141536.28877@greco-prog.fr> I wrote:
>>In article <93022.114846DCC117@psuvm.psu.edu> <DCC117@psuvm.psu.edu>
>>posed the following problem: 
>>   A 6-by-6 chessboard is perfectly covered with 18 dominoes.
>> Prove that it is possible to cut it either horizontally or
>> vertically into two non-empty pieces without cutting through
>> a domino, that is, prove there must be a fault-line.
>
>Gabor Megyesi gave a proof.
>His proof also shows that there are no 2xn, or 4xn, or 6x1, or 6x3, or
>fault-less rectangles. 
>
>We can now ask the following question. For what i (even) and what j is
>there a fault-less rectangle of size i x j.
>
Since this afternoon I've not that if you take a fault-less rectangle
and cut a line across it (breaking a few dominos) there is a canonical
way to repair the damage and lengthen the rectangle by 2 in the
process. 

I've already given examples of 5x6 rectangles and 6x8 rectangles.
Thus there exist ixj fault-less rectangles if and only if
either ixj are both even and one is at least 8 while the other is at
least 6, or one is odd and at least 5 while the other is even and at
least 6.

Now let me pose yet another problem thqt I haven't thought much about
yet. For what i,jk does there exist an ixjxk fault-less block of 1x1x2
dominos.

-- 

Yours, Daniel Loeb				loeb@geocub.greco-prog.fr
HOME 	150, cours Victor-Hugo; Appt D45; 33000 Bordeaux France
WORK	LABRI; Universite de Bordeaux I; 33405 Talence Cedex France
Newsgroups: sci.math
From: solan@smaug.uio.no (Svein Olav G. Nyberg)
Subject: Non-Standard Analysis and philosophy
Reply-To: solan@smaug.uio.no (Svein Olav G. Nyberg)
Organization: University of Oslo, Norway
Date: Thu, 21 Jan 1993 18:03:59 GMT


Under which philosophies of mathematics is non-standard
analysis possible, and to what extent?


Thanks in advance,
                Solan
Newsgroups: sci.math
From: jab@cs.brown.edu (James M. Borger)
Subject: Lattice points in balls
Organization: Brown Computer Science Dept.
Date: Mon, 25 Jan 1993 21:04:31 GMT

Does anyone know anything about the number of lattice points in an n-dimensional
ball centered at a lattice point?  Of course, it is O(r^n), but is anything known
about local jumps.

Thanks,

Jim Borger
From: cxm7@po.CWRU.Edu (Colin Mclarty)
Newsgroups: sci.math
Subject: Re: Non-Standard Analysis and philosophy
Date: 21 Jan 1993 22:53:44 GMT
Organization: Case Western Reserve University, Cleveland, OH (USA)
Reply-To: cxm7@po.CWRU.Edu (Colin Mclarty)
NNTP-Posting-Host: slc5.ins.cwru.edu


In a previous article, solan@smaug.uio.no (Svein Olav G. Nyberg) says:

>In article <TORKEL.93Jan21204806@bast.sics.se>, torkel@sics.se (Torkel
>Franzen) writes:
>|> In article <1993Jan21.180359.21766@ulrik.uio.no> solan@smaug.uio.no 
>|> (Svein Olav G. Nyberg) writes:
>|> 
>|>    >Under which philosophies of mathematics is non-standard
>|>    >analysis possible, and to what extent?
>|> 
>|>   Your question makes no apparent sense. Non-standard analysis is simply a
>|> field of mathematics that exists, hence is possible.
>
>I am serious, Torkel. Surely finitist mathematicians will not
>allow for infinitely large integers. And rumors have it that
>Martin-Lof's attempts to fit non-standard analysis into the
>Constructive Mathematics programme (mathematics as possible
>under a certain philosophy of mathematics) yields a weaker sort
>of non-standard analysis than the standard non-standard analysis.
>
>My question stands.
>
>
>Solan
>

	Okay.  Every philosophy of mathematics that allows standard
analysis in its usual form allows non-standard analysis in its usual
form.  Every philosophy of mathematics that allows standard analysis
only in weakened forms will allow non-standard analysis only in weakened
forms.

	Any philosophy of mathematics that allows the compactness theorem
for first order logic (and thus for axiomatized higher order logic) has
an easy proof that standard and non-standard analysis are equally 
possible.

Colin McLarty
From: zou@ccs.northeastern.edu (Chendong Zou)
Newsgroups: sci.math
Subject: Help needed!
Date: 26 Jan 93 21:59:53 GMT
Organization: College of CS, Northeastern U

Hi:

Does anyone know if there is a  answer book for exercises in 
John E. Hopcroft, J.D. Ullman's classic book "Introduction
to Automata theory, Languages, and Computation" ? 

Please email me if you know. Thanks.

-zou
Newsgroups: sci.math
From: tim@iss.nus.sg (Tim Poston)
Subject: Re: Eliminant?
Organization: Institute of Systems Science, NUS, Singapore
X-Newsreader: Tin 1.1 PL4
Date: Thu, 21 Jan 1993 04:03:24 GMT

hertz@ecf.toronto.edu (HERTZ ROGER BARRY) writes:
: Hi,
: 
: I've come across the term 'eliminant'.  Could someone refer me to a
: reference it is discussed?
Your example fits the `resultant', as described in for instance
  Keith Kendig
  Elementary Algebraic Geometry
  Springer Graduate Texts in Mathematics, 1977
Nice book.

Tim
Newsgroups: sci.math
From: gjm11@cus.cam.ac.uk (G.J. McCaughan)
Subject: Re: why is pi irrational
Organization: U of Cambridge, England
Date: Thu, 21 Jan 1993 13:36:23 GMT

In article <pete.03vb@bignode.equinox.gen.nz> pete@bignode.equinox.gen.nz writes:

>>So the Bible is accurate, just not very precise.  :-)
>
>When it says the Earth is flat, however, it is inaccurate, rather than
>imprecise.

Where does the Bible say that the earth is flat? I do not deny that most
of the writers of the Bible thought the earth to be flat (as did everyone
else at the time), but did they say so in the writings that now form the
Bible?

(There are a few bits of decidedly funny cosmology in the book of Job, but
Job is very clearly a story anyway, and I don't believe its writer ever
actually wanted anyone to believe God had said the things put in his mouth
there.)

-- 
Gareth McCaughan     Dept. of Pure Mathematics & Mathematical Statistics,
gjm11@cus.cam.ac.uk  Cambridge University, England.    [Research student]
From: bruck@mathj.usc.edu (Ronald Bruck)
Newsgroups: sci.math
Subject: Re: Integers
Date: 22 Jan 1993 09:15:05 -0800
Organization: University of Southern California, Los Angeles, CA
NNTP-Posting-Host: mathj.usc.edu

In article <1993Jan22.094318.10326@aristo.tau.ac.il> jhusdhui@math.tau.ac.il (Gurel-gurevitch Ori) writes:
>
> This is not true.
> the correct Peano's Postulate are:
> 1)there exist an integer called 0(sometimes 1).
> 2)every integer n has a successor S(n).
> 3)for every intgers n and m S(n+m)=S(n)+m
> 4)if A is a setof integers 0 belongs to A and for every n which
> belongs
>   to A S(n) belongs to A too then A is N(the set of all integers).
>
(Stuff deleted)

This is _still_ not Peano's axioms; indeed, axiom 3 doesn't make any
sense, since + is undefined in your axiom system.  You also need:  0
is not the successor of anything; and you need S(n) = S(m) => n = m.
Addition and multiplication are then _defined_ in terms of the
induction axiom.

--Ron Bruck
bruck@mtha.usc.edu preferred


Newsgroups: sci.math,rec.arts.books,sci.philosophy.meta
From: bhanafee@picasso.ads.com (Brian Hanafee)
Subject: Re: Geometry = the mind of God.
Organization: Advanced Decision Systems, Mountain View, CA 94043, +1 (415)
	960-7300
	<7HmiXB1w165w@kalki33.lakes.trenton.sc.us>
	<1993Jan18.152951.1557@eff.org>
Date: Sat, 23 Jan 1993 03:45:31 GMT


In article <1993Jan18.152951.1557@eff.org> mnemonic@eff.org (Mike Godwin) writes:
>In article <7HmiXB1w165w@kalki33.lakes.trenton.sc.us> system@kalki33.lakes.trenton.sc.us (Kalki Dasa) writes:
>>mnemonic@eff.org (Mike Godwin) writes:
>>
>>> In article <C0y8CB.67J@ccu.umanitoba.ca> feld@ccu.umanitoba.ca (Michael Feld)
>>> >"Absence makes the heart grow fonder."
>>> 
>>> I thought it was abstinence.
>>
>>Abstinence makes the head grow clearer.
>
>This depends on that from which one is abstaining.

Absinthe, presumably.


--
Brian Hanafee                         Advanced Decision Systems
bhanafee@ads.com                      1500 Plymouth Street
(415) 960-7300                        Mountain View, CA 94043-1230
Newsgroups: sci.math
From: hanche@ams.sunysb.edu (Harald Hanche-Olsen)
Subject: Re: Frobenius Thm. on real dision algebras
Organization: University at Stony Brook, NY
Date: Wed, 27 Jan 1993 19:19:30 GMT

>>>>> In article <93025.140941E62802@TRMETU.BITNET>, Melih Sener <E62802@TRMETU.BITNET> writes:

Melih>   As far as I know there is a theorem due to Frobenius, which
Melih> states that the only associative divison algebras over real numbers
Melih> are complex numbers and quaternions (if you drop the requirement
Melih> of associativity octonions are also added to this list.)

Melih>  Here are my questions...
Melih> 1- Where can I find an explicit proof of this and related
Melih> theorems?  (In Porteus' (Topological Geometry) , there is some
Melih> info about the subject but if there other sources, I'd like to
Melih> know them.)

begin unashamed plug alert:

 Author:         Hanche-Olsen, Harald and Erling Stoermer.
 Title:          Jordan operator algebras
 Published:      Boston : Pitman Advanced Pub. Program, 1984.

 Look at Theorem 2.2.6.

end unashamed plug alert.

Melih> 2- Is this theorem valid only for finite dimensional case?
Melih> (i.e. Is there an infinite dimensional real divison algebra?)

The rational functions form a field.  It's clearly infinite dimensional.

- Harald
Xref: sparky sci.math:18686 misc.education:6098
From: elg@elgamy.jpunix.com (Eric Lee Green)
Date: 21 Jan 93 21:38:16 CDT
Newsgroups: sci.math,misc.education
Subject: Re: Why algebra and what is important?
Distribution: world
Organization: Eric's Amiga 2000 @ Home

From article <C105z9.J6t@mentor.cc.purdue.edu>, by hrubin@pop.stat.purdue.edu (Herman Rubin):
> The failure of people to use mathematics intelligently is, IMO, due to
> this failure to use symbols.  It is similar to the problems in using
> natural languages to communicate.  This use of symbolism, however, should
> not wait until high school, but should be taught as early as possible.
> There is no point in teaching physics or chemistry or biology, beyond
> memorization and carrying out routine procedures under the direction of
> someone who understands, if the use of symbols is not understood.

Anybody who thinks Algebra is irrelevant to real life has never tried to
solve a system of linear equations to detirmine the best values for
price/volume tradeoffs...

Anywho: I agree about symbols. I have a bunch of problem-solving decks that
I toss out to my brighter kids. Simple stuff, mostly. Anyhow, a kid handed
me a card that stumped him. I said, hmm, let me see, we have an unknown X
and unkown Y, , 3 * X + Y = 12, X + Y = 6 (just a couple of swift equations
off the top of my head), let's see... X and Y are natural numbers since
we're talking number of beans in this problem... so X is somewhere between
0 and 6, and Y is somewhere between 0 and 6, and you can use "try and
check" to brute-force it since 4th graders don't know how to solve a system
(X=3,Y=3, by the way)... but then he asks, "What's all this X and Y
stuff?!". Whoops!

So much just doesn't make sense if you don't know how to formulate a
real-world problem as symbols. I don't know how they expected a 4th grader
to solve this problem without being able to write equations.

BTW: Herman notes in another message that he discounts the "basics". Well
-- unfortunately, all mathematics taught at higher grade levels assumes
that the kid can do basic arithmetic without the aid of a calculator. As
much as I wish otherwise, basic arithmetic is thus what I have to teach.
I play with math with the kid who can play math well, but otherwise,
well... how can a kid pass the TAAS (required to go to the next grade
level) if he can't add, subtract, multiply, or divide? Answer: He can't.

--
 Eric Lee Green  elg@elgamy.jpunix.com   Dodson Elementary
 (713) 664-6446                             Houston, TX
                "Kids are kids, no matter what"
Newsgroups: sci.math
From: carl@crazyman.med.ge.com (Carl Crawford)
Subject: pyramid volume
Reply-To: carl@crazyman.med.ge.com (Carl Crawford)
Organization: GE Medical Systems, Milwaukee,  WI
Date: Thu, 21 Jan 93 14:04:02 GMT


how do show that the volume of a pyramid is 

	1/3 * area of base * altitude 
	
without using calculus?
--
Carl R. Crawford	   Voice: 414-521-6572
GE Medical Systems	   Fax: 414-521-6549
P.O. Box 414, W-875	   Internet: crawfordc@med.ge.com
Milwaukee, WI 53201	   
Xref: sparky sci.math:18610 sci.math.num-analysis:3922
Organization: Penn State University
Date: Thu, 21 Jan 1993 17:55:47 EST
From: <NMR103@psuvm.psu.edu>
Newsgroups: sci.math,sci.math.num-analysis
Subject: LOOKING FOR: Spline Info

Can anybody tell me where to find a good book or so from
which I can learn about splines and spline fitting? If
so please reply via email, as I do not read this group
every day.

Thanks in advance,
Nicole.
From: dy@shire.math.columbia.edu (Deane Yang)
Newsgroups: sci.math
Subject: Re: linear algebra problem
Date: 24 Jan 93 17:51:39 GMT
Organization: Mathematics Department, Columbia University
X-Posted-From: shire.math.columbia.edu
NNTP-Posting-Host: sol.ctr.columbia.edu

In article <1993Jan24.025831.8516@galois.mit.edu> jbaez@riesz.mit.edu (John C. Baez) writes:
>Someone once told me about the solution to the following problem.
>Given two finite-dimensional vector spaces (over the complex numbers,
>say) X and Y, classify all *pairs* of linear maps S,T: X -> Y up
>to transformations of the pair (S,T) -> (gSh, gTh) where g and h are
>invertible linear transformations of Y and X respectively.  Does
>anyone recall the history of this problem and what its solution is?

In other words, given two rectangular matrices of the same dimension,
you want to put them into some kind of simultaneous normal form.
Right?

This can be done and is called the Kronecker pencil lemma. The only
reference I know is Gantmacher's book on matrices.

It is also known as Grothendieck's splitting theorem, which states that
any holomorphic vector bundle over the complex projective line
(i.e. the 2-sphere) splits into a direct sum of line bundles.
You can find a proof of this in a book by Okonek, et al on vector bundles
over complex projective space.

The latter proof is, I believe, easier to follow than the former,
assuming you know a little machinery (sheaves and all that).

I actually used this result in my thesis which was on solving overdetermined
systems of PDE's. It would be useful to understand larger dimensional families
of matrices. Again, this is equivalent to understanding vector bundles over
complex projective space, which is a fairly involved subject.

Deane Yang
Polytechnic University
Newsgroups: sci.math
From: davidc@panix.com (David B. Chorlian)
Subject: Re: why is pi irrational
Organization: PANIX Public Access Unix, NYC
Date: Sat, 23 Jan 1993 01:28:35 GMT

In <1993Jan21.133623.20957@infodev.cam.ac.uk> gjm11@cus.cam.ac.uk (G.J. McCaughan) writes:


>Where does the Bible say that the earth is flat? I do not deny that most
>of the writers of the Bible thought the earth to be flat (as did everyone
>else at the time), but did they say so in the writings that now form the
>Bible?

At the time of the writing of the New Testament, it was known to
educated Greeks that the world was spherical.  Some Church Father of
the 3rd or 4th century, whose name begins with an 'L', believed
the world to be flat, from scriptural evidence, but this was 
politely dismissed by the better informed.

>-- 
>Gareth McCaughan     Dept. of Pure Mathematics & Mathematical Statistics,
>gjm11@cus.cam.ac.uk  Cambridge University, England.    [Research student]
-- 
David B. Chorlian
Neurodynamics Lab
SUNY/HSCB 
davidc@panix.com