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A Table of offsets, such as the one-page job which is table 4 in my 1981
Bowditch, rests on an unstated, but perfectly acceptable, approximation.

It does NOT show the deviation of a position circle from a rhumbline drawn
at a tangent to that circle. It shows the deviation of a position circle
from a GREAT CIRCLE drawn at a tangent to the circle. [If that were not the
case, the offsets would depend on the observer's nearness to a pole, and
would be asymmetrical, different when offsetting in a direction that
approached the pole, compared with the opposite direction.] It's a pretty
good approximation, except in very high latitudes, but it limits the
validity of the offset table to small distances, over which there's really
no distinction between the rhumbline and the great-circle, passing through
the same point,  and pointing there toward the same azimuth.

Because the offset table presumes a great-circle, we can apply it just the
same to any point on Earth, and it may help Gary LaPook's conceptual
difficulty if we take a simple case. On a globe, put yourself at my
favourite hypothetical spot, in the Gulf of Guinea, at lat = 0, long = 0.
Now consider celestial objects at varying declinations, all on the Greenwich
meridian. The relevant great-circle is now the equator. Draw a small-circle
around each of these objects, so that each circle just meets the equator at
a tangent, at (0,0),  and see how much each circle deviates from the equator
as you travel a few miles East or West from our origin point. That's exactly
what the offset table does. Think of it in terms of a Mercator map of a
small area centred on our origin point.

Take one of these objects to be a true pole-star, with dec. = 90 deg. It
would be just on the horizon, at altitude 0 deg, viewed from our origin.
It's obvious, isn't it, that in that case our small-circle of equal altitude
is also a great-circle, the circle of the Equator. There are no offsets at
all. Our Mercator map of the relevant bit, near the Equator, of the
zero-altitude circle round the true pole-star would be a dead straight-line,
the Equator itself. No curvature. Infinitely-great radius of curvature.

Now take another object, near the pole but not quite there at dec = 89 deg,
say, still on the Greenwich meridian. Viewed from our origin point that will
have an altitude of 1 deg. The1-deg altitude circle will touch the Equator
at (0,0), but will be indistinguishable from it until you go a long way East
or West. It will have an enormous radius of curvature on the Mercator map,
many times greater than the Earth's radius. There's nothing incongruous in
that fact.

George.

contact George Huxtable at george@huxtable.u-net.com
or at +44 1865 820222 (from UK, 01865 820222)
or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.


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