Next message: Bruce Stark: "Gone 'till the 24th"
Sounds a lot like Zeno's paradox to me.
--- "Trevor J. Kenchington" <> wrote:
> George,
>
> I know better than to question your knowledge of the mathematics of
> navigation. I also strongly suspect that, if I asked for proof that a
> loxodrome reaches (rather than approaches) a pole, you would need to
> resort to mathematics that would not easily be represented in e-mail
> format and which I would not understand if you did send it. Hence, I
> have little option but to accept what you say at face value -- something
> that I am never comfortable doing, even when you are the author of
> whatever I am accepting.
>
> Could you, perhaps, explain the final approach of a loxodrome to the
> pole, without resorting to math? Above a latitude of, say, 89 degrees 50
> minutes, the sphericity of the Earth ought to be negligible and we
> should be able to visualize the problem as one of a rhumb line
> spiralling around the pole in a simple 2-dimensional space, while
> cutting each meridian at the same angle. I can visualize that spiral
> closing in from 10 miles out to a mile and so to a tenth of a mile. But
> it then seems to me that we could simply expand the scale by 100 times
> and see the loxodrome spiralling in from 1/10 of a mile to 1/1000,
> before expanding the scale again and repeating. What I don't see is how
> that loxodrome will eventually make the final step of reducing its
> distance from the pole to zero.
>
> You wrote:
>
> > They are both wrong, and Herbert is correct; though I'm sure he is capable
> > of defending his corner without my aid.
> >
> > The presumption is that you are steering a rhumb-line course, with a
> > Northerly component, and that all the Polar ice has melted.
> >
> > As you spiral in toward the pole on a rhumb-line course, you travel a
> > FINITE distance to get there. If you can maintain a constant speed, then
> > that can be done in a finite time. The snag is, you have to travel in
> > ever-decreasing circles, as the size of the spiral shrinks. To get exactly
> > to the pole, you have to make an infinite number of such gyrations, so the
> > vessel has to be spinning at an infinite rate. This is rather an unphysical
> > state of affairs, to say the least: a "singularity".
>
>
> Maybe it is my limited understanding of math but I don't think that the
> finite length of a loxodrome is inconsistent with it never reaching the
> pole. The mathematics of infinity has some funny properties, analogous
> to the effects of dividing by zero, and I'd not rule out a curve of
> finite length which spirals without end. (There may well be members of
> this list who can say that I am wrong. I'm only saying that I can't rule
> that out myself.)
>
> > The pole will be reached at a predictable moment, at which the ship will be
> > spinning round at infinite speed. Just after that moment, it will escape
> > from the pole, still spinning with infinite speed, and the radius of the
> > spiral then increases, and the spinning slows, until the vessel reaches its
> > original latitude. What will its longitude be then?
>
>
> While I am willing to accept that I may be wrong and a loxodrome may
> actually reach the pole, it will take a whole lot more to persuade me
> that said loxodrome ever departs from the pole again. George: I really
> think you are wrong on that point.
>
> I _know_ you are wrong on the claim that our hypothetical ship will ever
> leave the pole. You have assumed that the vessel is following a
> rhumb-line course between 270 and 090. Any course away from the North
> Pole must be 180 instantaneously and, immediately thereafter, must be
> between 090 and 270. That is: the hypothetical ship can only leave the
> pole by changing its course (perhaps to its reciprocal) and that is
> contrary to the starting assumption.
>
> > Of all the unrealistic questions we have considered on this list, this is
> > perhaps the most unrealistic of all. But why should we let that deter us
> > from playing such games?
> >
> > The picture above, of a vessel spinning infinitely fast at the pole,
> > applies to all incoming courses except 0deg and 90 deg, as Trevor points
> > out. At 90deg, the pole is never reached at all: the vessel sticks to the
> > equator.
>
>
> No. That is commonly stated in textbooks but is clearly false.
> Regardless of its starting latitude, a vessel following a rhumb-line of
> 090 or 270 will never reach either pole. It will follow a parallel of
> latitude, which could be the Equator but could be any other. (This is as
> stated in my last contribution to this thread.)
>
> > At 0deg, the vessel approaches the pole along a certain line of
> > longitude, then emerges along a line of longitude 180deg different.
>
>
> Again, the vessel cannot leave the North Pole while its course remains
> the rhumb line of 000. As I wrote last time, it can only leave by
> changing to a reciprocal course -- which it would of course do if it
> continued "straight" (meaning on a Great Circle, since this is spherical
> geometry) through the pole. We are, however, assuming rhumb-line
> courses, not reversible rhumb-line courses.
>
>
> Trevor Kenchington
>
>
> --
> Trevor J. Kenchington PhD
> Gadus Associates, Office(902) 889-9250
> R.R.#1, Musquodoboit Harbour, Fax (902) 889-9251
> Nova Scotia B0J 2L0, CANADA Home (902) 889-3555
>
> Science Serving the Fisheries
> http://home.istar.ca/~gadus
__________________________________
Do you Yahoo!?
Protect your identity with Yahoo! Mail AddressGuard
http://antispam.yahoo.com/whatsnewfree
