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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.

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