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Thanks for Rod Deyo's helpful contribution to this discussion. I have a few
comments-

>     You can determine the circles of constant
>altitude for three or more celestial bodies (assuming a spherical earth -
>something quite reasonable for practical navigation) and find their best-fit
>intersection numerically.

In practice, only two bodies are required, not three. Two bodies will give
two position circles, which cross at two places. One of these crossings
will be the position of the observer, and the other can be ignored as
irrelevant. As long as the two bodies are well separated in the sky, it is
obvious which of the two is the relevant crossing because they are so far
apart on the Earth's surface.

Instead of two bodies, two positions of the same body can be used just as
well, if a time interval has allowed the position of that body in the sky
to change significantly, and allowance has been made for the travel of the
vessel in the interim.

Incidentally, there's no need to assume a spherical Earth. Latitudes have
been defined in such a way that the ellipsoidal shape is allowed for, at
the expense of the length of a degree of latitude being slightly
inconstant.

>A problem arises if you need to plot the circles
>of constant altitude on a Mercator chart to find the intersection since they
>are not simple circles or even ellipses, but rather complex figures (this
>does simplify if the GP of the body is near your actual position, the "large
>altitude" case, where circles are used as good approximations). Then you
>really do want something like the Marcq St. Hilarie intercept method or even
>the older noon sight along with the morning/evening "time sight".

Well that problem almost ALWAYS arises, doesn't it, whenever a navigator
wishes to plot position on some sort of flat chart, on which the circles of
constant amplitude will be distorted. A useful way of demonstrating such
navigation in principle is to draw such circles on a globe, which is how
much early navigation was done: an ancient example of the analogue
calculator.

George Huxtable.



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