Re: sin x / x tends to 1...

On Sun, 4 Sep 2005 20:09:00 +0000 (UTC), Darren J Wilkinson
<d.j.wilkinson@xxxxxxxxx> wrote:

>David C. Ullrich <ullrich@xxxxxxxxxxxxxxxx> wrote:
>> On Sun, 4 Sep 2005 16:07:44 +0000 (UTC), Darren J Wilkinson
>> >> We've been going around in circles a bit, so maybe you're
>> >> sort of on a different page. But honest, if we start by
>> >> defining the circumference as the limit of the length of
>> >> those polygons then everything works with no circularity.
>> >
>> >I've possibly lost the plot by now, but if you inscribe a regular
>> >N-agon, the perimeter is 2Nsin(pi/N), and you are telling me that
>> >_by definition_ this tends to 2pi as N increases...
>>
>> No, that's not what I said, although I suppose that it's
>> not unreasonable for you to think that that's what I meant.
>> The fact that 2Nsin(pi/N) -> 1 _is_ going to fall out
>> of all this very simply, but I'm certainly not asserting
>> that _that_ fact is true by definition.
>
><huge snip>
>
>I know we've been going around the houses a bit, but I think that we may
>be getting to the root of my problem. Ignoring the regular/irregular
>polygon stuff, which we can regard as mere detail, you (and one or two
>other people...) are choosing to _define_ arc length as the sup over
>inscribed polygon lengths, knowing that the sin x / x limit immediately
>follows.

Yes, the fact that sin(x)/x -> 1 follows immediately. But that's
not the reason "I" "chose" to define arc length that way. That
_is_ the _standard_ definition of arc length!

How _should_ arc length be defined, in your opinion?

>What I am saying is that this is essentially the same as
>_assuming_ the sin x / x limit, because you _believe_ it to be true. If
>you _didn't_ believe in the limit, you would define arc length
>differently.

That is simply not so. (In fact, although I've been saying
all along that everything follows immediately if we use
this definition of arc length, I was surprised yesterday
when I wrote it all down to see _how_ immediately the limit
of sin(x)/x followed. Realized later why that was so. You're
saying I chose that definition because it would give the
desired limit - how you can _think_ you know this about
why I did something I can't imagine, but you're wrong
about that. I didn't "choose" that definition at all,
it's the definition I _learned_.)

Two questions for you: How _should_ I define arc length,
and how, in your opinion, would I have defined it if
I hadn't been determined to justify that limit?

>Now don't get me wrong - _I_ believe the limit too, you
>just haven't (yet) convinced me that it is anything other than an unwritten
>axiom of geometry.

The definition of arc length that you somehow conclude I "chose"
is not unwritten. Look for the definition of "rectifiable curve".

Hmm. Most google hits assume we know what a rectifiable curve
is and say something about them. Look at the section "Length
of curves" in

http://encyclopedia.laborlawtalk.com/Curve

You see exactly the definition I gave for the length
of a curve, with the statement that a curve is
rectifiable if it has finite length. You don't see
anything there about sin(x)/x.

What _is_ the proper definition of arc length?

>Regards,


************************

David C. Ullrich
.

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