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    Re: Horizons, was Summary of Bowditch Table 15
    From: Fred Hebard
    Date: 2005 Jan 30, 20:20 -0500

    I can only confess that I considered the non spherical shape of the
    earth while writing, but thought to keep to one component at a time.
    George is correct in what he says.
    On Jan 30, 2005, at 5:52 PM, George Huxtable wrote:
    > Jim Thompson and Fed Hebard are making somewhat heavy weather about
    > this
    > question of the difference between local gravity (the direction of a
    > plumb-bob) and the direction of a line to the centre of the Earth.
    > These
    > directions are not the same (except at the poles and the equator)
    > because
    > the Earth is not a uniform sphere, but a non-uniform spheroid, with
    > some
    > local bulges.
    > Fred wrote-
    >> This is called deflection from the vertical, where the vertical points
    >> to the center of the earth.  It is not a significant factor until you
    >> get next to the Andes or some other huge mountain close to the sea.
    >> As
    >> I recall, the  errors are on the order of 1' of arc or so, which would
    >> make it more a problem for surveyors than navigators.
    > But in that passage Fred is referring only to local gravitational
    > distortions that are due to the local non-uniformities in the Earth's
    > crust. They exist, but there are bigger factors at work, because of the
    > non-spherical Earth.
    > You can see easily, by simply sketching an exaggerated ellipse as a
    > slice
    > through the Earth from pole to pole, that a line at right-angles to the
    > sea-surface through a point P is not in the same direction as a line
    > between P and the Earth's centre. Except, that is, when P is at a pole
    > or
    > on the equator. The biggest divergence is at a latitude of 45 degrees,
    > North or South, when the difference amounts to 11' 33".
    > The direction of bodies in the sky (the declination part, anyway) is
    > defined by its direction, up or down from the Earth's equator.
    > Similarly,
    > the geographic latitude is defined by the angle that the local vertical
    > makes with the plane of the Earth's equator. Devices that measure
    > altitudes, such as sextants, bubble-sextants, or surveyors'
    > theodolites,
    > all measure with respect to the local horizontal or sea-surface or
    > artificial horizon surface, always 90 degrees from that local vertical.
    > Astronomer's telescopes are set up with respect to that local vertical.
    > So, when we observe a body on the meridian, we can relate directly its
    > altitude to its declination and to our latitude. If our latitude had
    > been
    > defined in a different way, such as the direction of a line between P
    > and
    > the centre, that simple relation would not apply, and all sorts of
    > complications would result.
    > It's all been made rather easy for us navigators by defining latitude
    > in
    > that way. Mercator charts, beside their stretching toward the poles
    > that
    > we're all familiar with, also have a bit of extra distortion in them,
    > to
    > allow for the length of a sea-mile to vary somewhat with latitude. For
    > that's part of the price that has to be paid for the simplicity of
    > defining
    > latitude in the way we do. Because the Earth's surface is (a bit) more
    > tightly curved, in the N-S direction, near the equator than it is at
    > the
    > poles, the length of a sea-mile varies , being less near the equator,
    > greater near the poles. And the length of a sea-mile, measured in the
    > E-W
    > direction, differs from a sea-mile measured in the N-S direction. Just
    > slightly; not so much as you would notice. Doesn't affect navigation
    > much,
    > does it? It adds up to that 11' 33" divergence, around 45deg latitude.
    > In navigation, when we set a course and distance to our destination,
    > we are
    > not usually bothered by discrepancies of a few miles in distance, or a
    > fraction of a degree in the course, so for those purposes a sphere is a
    > close-enough approximation. Formulae exist that take the spheroidal
    > shape
    > into account, for both rhumb-line and great-circle navigation. Tables
    > for
    > "meridional parts" allow for the true shape. But who bothers?
    > So the result of it all, in celestial navigation, is that everything
    > that
    > we measure in the sky has to be with respect to the direction of a
    > plumb
    > line, or the plane of the horizontal (which are so closely related that
    > either one exactly defines the other).
    > George.
    > ================================================================
    > 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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