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    Re: The flat earth notion
    From: Trevor Kenchington
    Date: 2003 Nov 7, 20:48 -0400

    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@iStar.ca
    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

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