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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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