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I think there may be a bit of misunderstanding about Douwes of Amsterdam,
and the problem to which he provided the solution.

Antoine, and presumably others, have devised an analytic solution to the
question of two intersecting circles of equal altitude, drawn around two
known geographical positions, and the resulting ambiguous-pair of
intersections. These geographical positions could be of a single body at two
times, allowing for vessel's motion in between, or of two bodies at the same
time. However, Cornelius Douwes was working around 1740, long before
chronometers had arrived on the scene, and without knowing the (Greenwich)
time, those geographical positions could not be known.

Douwes was just trying to find the latitude. There was no way longitude
could be established. Anyone can find latitude from a Sun altitude if it's
taken at local noon, but what if the Sun happened to be obscured then?
Douwes problem was to find latitude from a pair of Sun altitudes, neither of
which was taken at noon, in the days when GMT was unavailable. But
navigators, although lacking chronometers, might well have a hack-watch,
good enough to assess the time INTERVAL between those Sun observations. So
that was the problem that Douwes tackled; given two Sun altitudes at unknown
times, and the time interval between them, what was the latitude?

By the way, Cotter states that "Douwes's method requires the use of an
estimated latitude".

Douwes wasn't the first to tackle this question, though. In 1728, Facio
Duillier, of Swiss origin, who had worked with Isaac Newton (and, by the
way, invented jewelling for timepieces), had published a method which was
considered particularly unwieldy. Several other methods followed that of
Douwes; Dunn, Ivory, Ainsley, Riddle, all feature in Cotter's "History of
Nautical Astronomy".

Don't expect me to expound on any of this, though. I've just quoted, without
trying to take it all in.

As for modern papers, deriving a pair of intersections of the position
circles in latitude and longitude, at known times, there are several papers.

For example, G G Bennett, "General Conventions and solutions- their use in
Celestial Navigation" Navigation, 26 (4) 1979-80
Torben Kjer, "Unambiguous two-body fix methods derived from crystallographic
principles", Navigation, 28 (1) Spring 1981.
I seem to recall that Andres has noted his procedure for such a calculation
on Navlist.

Peter Hakel added-

"My response:
Such tedium can be bypassed to some degree if the problem can be reduced to
some standardized form.  Then its solutions can be calculated once and
tabulated for practical use on a sufficiently fine grid.  So I suppose
nobody has printed any "Douwes sight-reduction tables."".

On the contrary, Douwes devised solar tables for use with his method, which
were referred to in 1760, and improved by Maskelyne in the second edition of
his "Tables requisite", 1781.

George.

contact George Huxtable, at  george@hux.me.uk
or at +44 1865 820222 (from UK, 01865 820222)
or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.


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