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    Re: The flat earth notion
    From: George Huxtable
    Date: 2003 Nov 9, 13:03 +0000

    Zorbec Legras said-
    >By definition a loxodromic track is a logarithmic spiral.
    Response from George-
    I rather doubt that. It becomes close to a logarithmic spiral near to the
    pole, when we can approximate the Earth's surface by a plane, but not
    otherwise. Where does Zorbec's "definition" come from?
    Zorbec continues-
    >By definition a logarithmic spiral never reaches the assymptote point
    >which, in this case, the up pole is.
    Another "definition". Can we have a reference, please?
    We have to be careful here. It depends on the meanings of the words we are
    using. What does "never" mean in this context?
    If we are following a decreasing logarithmic spiral down to its end-point
    at the pole at a constant rate of change of longitude (each turn around the
    pole taking the same time), then never, ever, will we reach that end-point,
    because an infinite number of turns is required, and there is not enough
    time available in the Universe for that to be completed. In that case,
    Zorbec's "never" would be justified.
    On the other hand, the original problem specified, not a constant rate of
    change of angle, but a constant velocity along the ever-decreasing spirals.
    In that case, each decreasing spiral takes a shorter and shorter time. And
    because the integrated path length along a (decreasing) logarithmic spiral
    is FINITE, then the end-point will be reached in a finite time, but after
    an infinite number of turns.
    Zorbec continued-
    >By the way how many loxodroms are there to join 2 points ?
    >Hint : more than 2...
    This topic has been aired on this list before, under the threadname "The
    pedant's rhumb-line", commencing 10 Oct 2002.
    The answer is- As long as the latitudes of those points differ, as many as
    you like, depending on how many times you are prepared to circumnavigate
    the Earth, Eastward and Westward, along the way.
    contact George Huxtable by email at george@huxtable.u-net.com, by phone at
    01865 820222 (from outside UK, +44 1865 820222), or by mail at 1 Sandy
    Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.

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