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Arithmetic question

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Jymesion

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Feb 20, 2013, 11:52:13 PM2/20/13
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This is only remotely sfnal (it grew out of stories I was writing, but
after it's fully developed, it's not going back) . . .

After teaching numerals, the next obvious step is 1+1=2.

I'm really torn, though, on what should follow.

The choice I see is between 2-1=1 and 1+2=3.

There are valid reasons for each.

The main reason for 2-1=1 is it's complimentary to 1+1=2, and a
concept is more readily understood when an opposing concept is
presented alongside.

The main reason for 1+2=3 is there is value in presenting all the
relevant ramifications of a concept (in this case, addition) before
presenting another concept.

(I can't use current human-based education systems as a guide because
this race's ability for rote memorization is weak while their recall
of concepts is very pronounced.)

Thoughts?


One aspect which might affect judgment -- addition was presented, in a
way, in teaching the numerals (combining a 1 and a 3 creates a 4, a 2
and a 3 creates a 5, etc.).

Dorothy J Heydt

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Feb 21, 2013, 10:12:39 AM2/21/13
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In article <gt5bi8li1otcg0103...@4ax.com>,
I'm not up on modern educational theory. Back when they were
teaching me arithmetic in 1950 or so, we did a good bit of
addition before we got into subtraction.

--
Dorothy J. Heydt
Vallejo, California
djheydt at gmail dot com
Should you wish to email me, you'd better use the gmail edress.
Kithrup's all spammy and hotmail's been hacked.

Jymesion

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Feb 21, 2013, 6:30:50 PM2/21/13
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On Thu, 21 Feb 2013 15:12:39 GMT, djh...@kithrup.com (Dorothy J
Heydt) wrote:

>I'm not up on modern educational theory. Back when they were
>teaching me arithmetic in 1950 or so, we did a good bit of
>addition before we got into subtraction.

That's the traditional way.

I have to wonder if someone figured out that's the best way, or if it
became standard because a teacher did it that way in 926 B.C., and
everybody ever afterwards taught it the way they were taught.

I also wonder how much it's influenced by being rote memorization.

A biggie, for me, is I have no objective way of looking at it. I
suspect I'm prejudiced against teaching all the addition first because
that's our standard, but I don't know how much that bias is/might be
affecting my logic.

One of the reasons I want to show subtraction parallel with addition
is it goes hand-in-hand with disassembling numbers.

That's important because we learn 2+2=4 by rote. In this system, it
should be looked at as 2+2=2+1+1=3+1=4 because the numeral 4 is a
combination of the numerals 1 and 3.

J.Pascal

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Feb 21, 2013, 9:16:53 PM2/21/13
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If it's a different culture/species why not just have them subtract for a long time before ever adding anything. It's just as logical to have something and then use it up as it is to get more.

Adding, I suppose is just counting, but subtracting is counting too. Maybe "what I have in whatever quantity" is always the "zero" or "one" value, no matter what the quantity is, and it's quantity is determined as it is used up or consumed or sold.

Here I have a basket of turnips. I count them as I take them away.

jl...@sofluc.co.uk

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May 10, 2013, 7:57:42 AM5/10/13
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On Thursday, 21 February 2013 23:30:50 UTC, Jymesion wrote:

> That's important because we learn 2+2=4 by rote. In this system, it
> should be looked at as 2+2=2+1+1=3+1=4 because the numeral 4 is a
> combination of the numerals 1 and 3.

I'm posting from Google groups -- some of you may remember me as J L Cunningham, but I haven't looked at rec.arts.sf.composition for literally years. I just wondered whether the discussions were still all about politics and economics. (And I was sorry to see the news about Marilee. I still have my
cat vacuuming pin.)

If you start with counting, then +1 means "the number after this", so 3+1 = 4 because four is the number after three. (One, two, three, four, oh yes, that's right.) That is, 3+1=4 by definition (the definition of what 4 is.)

2+2=4 requires two new things: what does +2 mean? and then showing that, using this definition, if you start from 2 you get 4.

One way to define +2, or +5 or whatever, is to say "keep doing +1, count how many times you do it, and stop when you've done it 2 times, or 5 times or whatever. If you'll excuse a little algebra, +k means "do +1 k times" except it would all be done in words.

Then 2+2 is 2+1+1 (count how many +1s there are: two of them).
But 2+1 is 3 by definition, so 2+2 is the same as 3+1, which is 4 (by definition).

I have now proved 2+2 = 4. (Actually, I haven't. There is a hidden assumption, or several. But that ought to be pretty convincing.)

Then you can define subtraction as what happens if you ask, "What do I have to add to 3 to get 7?" (In almost-algebra, which hasn't been invented yet, 3+? = 7). Solving that kind of puzzle gives you a definition of subtraction.

That's sort of the argument you'll find in "History of Mathematics" text books, but I've heavily put my own slant on it.

It could be fun to invent an arithmetic starting with counting down, or subtraction (as Julie suggests).

But the real reason for my post is to say that the definition of addition, above, is not only not obvious, it isn't what the ancient Romans used.

For them, 3+4 = 6.

As a very famous example, Christ was crucified on a Friday, and on the third day he rose again. On Monday? No. Three days after Friday gets you to Sunday,
because they counted Friday, Saturday, Sunday. Two days after today is tomorrow. One day after today is just today. They didn't have a zero, remember?

I started inventing an arithmetic based on this kind of addition, and with a strange notion of multiplying, which didn't give you areas as we know them (3 metres by 3 metres isn't 9 square metres) but the number of columns required to support a roof: measuring from the centre, 3 by 3 in all four directions comes out to 13.

I got stuck when I tried to prove that 2 times 2 equals 5 (which it does, in this arithmetic). Something like that, it's a while since I looked at it.

Jonathan

Jacey Bedford

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May 10, 2013, 10:16:41 AM5/10/13
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In message <5e457c28-1661-4dd8...@googlegroups.com>,
"chromo...@googlemail.com" <jl...@sofluc.co.uk> writes
>I'm posting from Google groups -- some of you may remember me as J L
>Cunningham, but I haven't looked at rec.arts.sf.composition for
>literally years. I just wondered whether the discussions were still all
>about politics and economics. (And I was sorry to see the news about
>Marilee. I still have my cat vacuuming pin.)

Hello, Jonathan, how are you? Nice to hear from you again. I've been
away for a while but back for a few months now, however I'm changing
ISPs soon and I'm not sure whether my new on (BT) provides usenet
access, so I might have to check in via Eternal September.

Or I could try google groups. I haven't explored them yet.

Jacey
--
Jacey Bedford

John F. Eldredge

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May 10, 2013, 6:07:43 PM5/10/13
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I recommend news.individual.net, run by the Free University of Berlin.
It is inexpensive (10 Euros per year, about $13 US at current exchange
rates), and they filter out nearly all spam. The only downside is that
they don't carry binary groups, only text groups.

--
John F. Eldredge -- jo...@jfeldredge.com
"Reserve your right to think, for even to think wrongly
is better than not to think at all." -- Hypatia of Alexandria

Jymesion

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May 12, 2013, 7:46:58 PM5/12/13
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Sorry to take so long to reply to this -- a health issue took me away
from my computer for a while.

On Fri, 10 May 2013 04:57:42 -0700 (PDT),
"chromo...@googlemail.com" <jl...@sofluc.co.uk> wrote:

>On Thursday, 21 February 2013 23:30:50 UTC, Jymesion wrote:
>> That's important because we learn 2+2=4 by rote. In this system, it
>> should be looked at as 2+2=2+1+1=3+1=4 because the numeral 4 is a
>> combination of the numerals 1 and 3.
>
>some of you may remember me as J L Cunningham

I remember you! (Not sure that's always a good thing . . . ;) )

>If you start with counting, then +1 means "the number after this",
>so 3+1 = 4because four is the number after three. (One, two, three,
>four, oh yes, that's right.)

That's using rote memory.

>That is, 3+1=4 by definition (the definition of what 4 is.)

Definitions are (by definition) the antithesis of my project. :)

I posit a race with very little capacity for learning by rote. Their
brains are wired more towards recognizing processes and understanding
relationships.

>One way to define +2, or +5 or whatever, is to say "keep doing +1,
>count how many times you do it, and stop when you've done it 2 times,
>or 5 times or whatever. If you'll excuse a little algebra, +k means
>"do +1 k times" except it would all be done in words.

Partway into my playing with this project, it dawned on me that
certain elements of the process mimicked computers' +k method since
they don't have addition tables in memory.

>Then 2+2 is 2+1+1 (count how many +1s there are: two of them).
>But 2+1 is 3 by definition, so 2+2 is the same as 3+1, which is 4 (by definition).

Again with the definitions! :)

In my project, you don't have to remember (or define) 1+3=4, 2+3=5,
1+6=7, or 2+6=8 because it's simply a matter of combining the symbols
(positional notation within duplex digits).

Where:
Rote(x) = something that must be memorized
Process(x) = something you can figure out if you remember the
preceding Rotes.
Rote/Process(x) = something you should be able to figure out, but it
helps if you remember something about the process

Rote(1): 1 is represented by a certain symbol (a stylized stroke)
Rote(2): 2 is represented by a different symbol (stylized pair of
strokes)
Rote(3): 3 is represented by an inverted 1 (to move it into the second
position) because it represents 1 group of 3 items.
Process(1): 6 is represented by an inverted 2 (2 groups of 3).
Rote/Process(1): The symbols for 4, 5, 7, and 8 are made by combining
1 and 3, 2 and 3, 1 and 6, and 2 and 6, respectively, within the
digit.
Rote(4): 1+1=2
Process(2): 3+3=6 (1+1=2 in the second position).
Rote/Process(2): 2=1+1
Process(3): 6=3+3 (Process(2) and Rote/Process(2)).
Rote(5): 1+2=3
Process(4): 3=1+2 (Rote/Process(2) and Rote(5))
Rote(6): 1+8=11 (Positional notation, equivalent to decimal's 1+9=10,
in Base8 without a zero (which I've always felt is highly overrated).)
Process(5): 1+88=111, 1+888=1111, etc.

Memorizing only 6 things and mostly remembering 2 allows all addition
(and I'd be willing to argue that Rote(6)(above) isn't completely rote
because position value has already been established, so moving into a
new digit should be a logical step).

I think that compares favorably with "Arabic numerals/decimal with
zero" system's 92 required learning by rote memorizations.

Your Latinate system sounds interesting, but I won't explore it
because I can only wrap my mind around a certain amount of weirdness
at one time. :)

Jacey Bedford

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May 12, 2013, 6:59:59 PM5/12/13
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In message <av59df...@mid.individual.net>, John F. Eldredge
<jo...@jfeldredge.com> writes
>On Fri, 10 May 2013 15:16:41 +0100, Jacey Bedford wrote:
>
>> In message <5e457c28-1661-4dd8...@googlegroups.com>,
>> "chromo...@googlemail.com" <jl...@sofluc.co.uk> writes
>>>I'm posting from Google groups -- some of you may remember me as J L
>>>Cunningham, but I haven't looked at rec.arts.sf.composition for
>>>literally years. I just wondered whether the discussions were still all
>>>about politics and economics. (And I was sorry to see the news about
>>>Marilee. I still have my cat vacuuming pin.)
>>
>> Hello, Jonathan, how are you? Nice to hear from you again. I've been
>> away for a while but back for a few months now, however I'm changing
>> ISPs soon and I'm not sure whether my new on (BT) provides usenet
>> access, so I might have to check in via Eternal September.
>>
>> Or I could try google groups. I haven't explored them yet.
>>
>> Jacey
>
>I recommend news.individual.net, run by the Free University of Berlin.
>It is inexpensive (10 Euros per year, about $13 US at current exchange
>rates), and they filter out nearly all spam. The only downside is that
>they don't carry binary groups, only text groups.
>
Thanks, John.

Jacey
--
Jacey Bedford

Dan Goodman

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May 14, 2013, 4:22:12 PM5/14/13
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On Sun, 12 May 2013 17:46:58 -0600, Jymesion wrote:

> Your Latinate system sounds interesting, but I won't explore it because
> I can only wrap my mind around a certain amount of weirdness at one
> time. :)

Okay; then you probably won't want to know about the ways some (human)
synesthetes do arithmetic. (And some syns are utterly hopeless about
math, but that's another matter.)



--
Dan Goodman

jl...@sofluc.co.uk

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May 15, 2013, 9:54:52 AM5/15/13
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On Sunday, 12 May 2013 23:46:35 UTC+1, Jymesion wrote:
> Sorry to take so long to reply to this -- a health issue took me away
> from my computer for a while.

> >If you start with counting, then +1 means "the number after this",
> >so 3+1 = 4because four is the number after three. (One, two, three,
> >four, oh yes, that's right.)
>
> That's using rote memory.

Well, yes. There's no obvious connection between the word "three" and the
word "four" if you don't know what they mean.

> >That is, 3+1=4 by definition (the definition of what 4 is.)
>
> Definitions are (by definition) the antithesis of my project. :)

>
> I posit a race with very little capacity for learning by rote. Their

So could your aliens learn what "apple" meant? (The word "apple" - not
the physical fruit (or mega-corporation).)

There are lots of different kinds of learning -- like learning where the
bathroom is, in your own home, is a kind of learning. Otherwise you'd need
to explore every time you wanted to go to the bathroom. Learning to count is
a bit like learning a song. "One, two, buckle my shoe; three, four, knock on the door...")

You could make a distinction between abstract and concrete thoughts?

> Again with the definitions! :)

That just means knowing what words mean! :-/

> In my project, you don't have to remember (or define) 1+3=4, 2+3=5,
> 1+6=7, or 2+6=8 because it's simply a matter of combining the symbols
> (positional notation within duplex digits).

Not sure what you mean by "duplex" but I get a flavour of what you mean
from your following explanation/examples which I snipped.

I think that's actually quite sophisticated: not that it matters, but I
wonder how creatures like that could have evolved? (I'm slightly reminded
of Vinge's skrode(sp?) riders.)

In my copious "ideas" folder I had almost the opposite idea: aliens with
very little processing ability, but very, very long lifespans (millions of
years) and very good eidetic memories. When playing chess, they make
moves almost randomly unless they've seen a position before. To start
with, any three year old child can beat them, but after a few thousand years
(and millions of games) they are very strong in the opening and end-game!

Jonathan

jl...@sofluc.co.uk

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May 15, 2013, 10:06:27 AM5/15/13
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On Friday, 10 May 2013 15:16:41 UTC+1, Jacey Bedford wrote:

> Hello, Jonathan, how are you? Nice to hear from you again. I've been
> away for a while but back for a few months now, however I'm changing
> ISPs soon and I'm not sure whether my new on (BT) provides usenet
> access, so I might have to check in via Eternal September.

Hi Jacey!

I'm with BT, and was using the BT news server, but that was on my old
computer. I have yet to install proper news reading software on this machine
(which isn't that new any more either) and haven't checked whether BT still
run a newsserver.

> Or I could try google groups. I haven't explored them yet.

The interface is better than last time I tried it (also a few years ago). Since I uses Google mail, it knows who I am and so I can get to rec.arts.sf.composition easily enough. Reading isn't really a problem any more: it's fast enough. But replying via google groups is still a bit of a pain. (Might encouragement me to think before deciding to write a post! :-/ )

But I also note John Eldrege's suggestion. If the BT newsserver no longer
exists (I can check by switching on my old computer and running the
newsreader to see if it still downloads stuff) then it depends whether I
think I would read news enough. I no longer (and haven't) read any newsgroups
for some years now.

Jonathan

Jymesion

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May 16, 2013, 2:12:21 AM5/16/13
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On Wed, 15 May 2013 06:54:52 -0700 (PDT),
"chromo...@googlemail.com" <jl...@sofluc.co.uk> wrote:
>On Sunday, 12 May 2013 23:46:35 UTC+1, Jymesion wrote:
>> I posit a race with very little capacity for learning by rote.
>So could your aliens learn what "apple" meant? (The word "apple" - not
>the physical fruit (or mega-corporation).)

I see their words being built up, based on concepts. (I hope you'll
excuse me, this is difficult for me to explain without drawing
pictures . . .)

Example:
Where:
1 as the first letter indicates human.
The second letter indicates quantity.
The third letter indicates gender.

'Man' is 111.
'Woman' is 112.
'Pregnant woman' is 113.
'A pair of men' is 121
'A group of women' is 132. ('3' here is not a distinct number -- like
'some' is more than 2 and less than a crowd.)
'Crowd' is 14.
etc.

'Apple' would, thus, be 413xyz...
4 indicates flora
1 indicates singular
3 indicates it carries fertile seeds
the following letters further define it.

I haven't delved into this too deeply because it needs a focused and
cohesive effort (and I keep getting carried away on tangents, like
'trouble' must be 224 (112+112)).

>You could make a distinction between abstract and concrete thoughts?

I find the lines between them blurry even in human psychology. To
posit something for an alien race would be incredibly muddled.

>> In my project, you don't have to remember (or define) 1+3=4, 2+3=5,
>> 1+6=7, or 2+6=8 because it's simply a matter of combining the symbols
>> (positional notation within duplex digits).
>Not sure what you mean by "duplex" but I get a flavour of what you mean
>from your following explanation/examples which I snipped.

A rough draft of a grade-school primer is at:
http://rukalen.jymes.com/wp/?page_id=11

In the pages towards the bottom, you can see how I tried to maintain
the 'more concept than rote' idea -- the + sign is a stylized hook (to
pull things together), = is a stylized hand (to present the result),
and - is a stylized axe to chop off part of what you have.

>I wonder how creatures like that could have evolved?

I believe it could be the result of a chaotic environment.

Take for example berries. We can learn by trial and error which ones
are safe to eat and teach this to our children.

If plants didn't breed true, two plants side by side might have
berries which look identical, but one's are good to eat and the
other's are poisonous. So you develop tests. Crush a berry and rub it
against your gum. If it stings, it's probably bad. If it passes that
test, spit on it. If there's a color change, it's probably bad for a
different reason. Etc.

Rather than remembering color, shape, location, etc. of hundreds of
different plants, you think of a few basic ways to test them.

This would give 'using concepts is more important than memorizing
facts' a kickstart.

As a side effect, I see this as giving them an impetus towards
agriculture. Someone pulls a branch off a bush, takes it home, shoves
it in the ground to keep it handy for snacks, and it takes root.

When the migration brings him back to that spot the next year, he
finds its berries are good and remembers putting it there, so pulls
branches off of it and shoves them into the ground. Before long,
there's a thicket, and he expands it with twigs from other plants.

When the gathering part of hunter-gatherer is actually: find, test,
gather, then having a known good supply is important.

Re: newsgroup access. Many years ago, when my regular server became
temporarily undependable, I established an account with teranews.
There's a one-time set-up fee (now $3.95) and a limit of 50MB a day.
http://www.teranews.com/
They also offer unlimited accounts for a monthly fee, but 50Mb is an
awful lot of text!

david.sh...@ymail.com

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May 16, 2013, 12:42:57 PM5/16/13
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How do you write 16 in your system?

david.sh...@ymail.com

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May 16, 2013, 12:47:08 PM5/16/13
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On May 16, 12:42 pm, david.shallcr...@ymail.com wrote:
> How do you write 16 in your system?

Never mind, 16 isn't a problem, it has to be 1x8 + 8.

Jymesion

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May 17, 2013, 1:42:58 AM5/17/13
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On Thu, 16 May 2013 09:47:08 -0700 (PDT), david.sh...@ymail.com
wrote:

>On May 16, 12:42�pm, david.shallcr...@ymail.com wrote:
>> How do you write 16 in your system?
>
>Never mind, 16 isn't a problem, it has to be 1x8 + 8.

Exactly! :)

I don't know how unique it is to have a BaseX wherein X is also a
digit.

Bill Swears

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May 18, 2013, 3:18:17 AM5/18/13
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On 5/15/2013 6:06 AM, chromo...@googlemail.com wrote:
> On Friday, 10 May 2013 15:16:41 UTC+1, Jacey Bedford wrote:
>
>> Hello, Jonathan, how are you? Nice to hear from you again. I've been
>> away for a while but back for a few months now, however I'm changing
>> ISPs soon and I'm not sure whether my new on (BT) provides usenet
>> access, so I might have to check in via Eternal September.
>
> Hi Jacey!
>
> I'm with BT, and was using the BT news server, but that was on my old
> computer. I have yet to install proper news reading software on this machine
> (which isn't that new any more either) and haven't checked whether BT still
> run a newsserver.

I'm using Eternal September, and it melds quite nicely with Thunderbird.
It was surprisingly easy to join and set up. I figured I'd have some
sort of advertizing to put up with, but nope.

Bill


--
Bill Swears
http://www.billswears.com/
Zook Country - http://twilighttimesbooks.com/ZookCountry_ch1.html
Also at Amazon, Barnes and Noble, and other fine ebook emporia.
Puppies - http://www.mtaonline.net/~wswears/
Opinions - http://wswears.livejournal.com/

Jacey Bedford

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May 19, 2013, 2:51:45 PM5/19/13
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In message <kn79o9$d18$1...@dont-email.me>, Bill Swears <wsw...@gci.net>
writes
>On 5/15/2013 6:06 AM, chromo...@googlemail.com wrote:
>> On Friday, 10 May 2013 15:16:41 UTC+1, Jacey Bedford wrote:
>>
>>> Hello, Jonathan, how are you? Nice to hear from you again. I've been
>>> away for a while but back for a few months now, however I'm changing
>>> ISPs soon and I'm not sure whether my new on (BT) provides usenet
>>> access, so I might have to check in via Eternal September.
>>
>> Hi Jacey!
>>
>> I'm with BT, and was using the BT news server, but that was on my old
>> computer. I have yet to install proper news reading software on this machine
>> (which isn't that new any more either) and haven't checked whether BT still
>> run a newsserver.
>
>I'm using Eternal September, and it melds quite nicely with
>Thunderbird. It was surprisingly easy to join and set up. I figured
>I'd have some sort of advertizing to put up with, but nope.
>
>Bill
>
>
Good to know, Bill, thanks.

Jacey
--
Jacey Bedford
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