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With Frank Reed's help, from his "Easy Lunars" and later postings, I'm
starting to understand more about the "approximate" methods for clearing
lunars.

Armed with the recipe, in Norie's, for using Mendoza's method, I have been
using some "reverse engineering" to discover the basic trig. that underlies
it.

Cotter's "History of Nautical Astronomy doesn't tackle Mendoza's method,
but does analyse Merrifield's method of many years later. This is an
approximate method appearing to have many similarities to Mendoza.

Cotter tells us that in 1884, Merrifield "described a method for clearing
lunar distances which he had invented...   ...The method is direct in its
application, requires no special tables, and is claimed to be a very close
approximation, well adapted for sea use...."

However, those claims seem to be somewhat overstated, if Cotter's account
of his method is to be accepted at face value. We have to be a bit careful
here, because Cotter has proved to be very error-prone in other contexts,
in his book.

As far as my understanding goes, Merrifield's clearing method is only
trivially different from Mendoza's, with one important exception which we
will address in the next paragraph. Mendoza corrects his observed lunar
distance by adding two terms, which Frank has notated as dh_Moon*A and
dh_Sun*B, which make the assumption that the little right-angled triangles
that are used to add and subtract corrections are, as near as dammit, plane
triangles. That is what Merrifield does, too, to give exactly the same
result.

But then Mendoza goes a step further, adding an additional small
correction: a quadratic term that Frank refers to as "Q", with numbers
taken from Table XXXV in Norie's. And that step is omitted completely by
Merrifield. That omission appears to be the basis of Merrifield's
"invention"! That's why his method "requires no special tables", because he
simply neglects that part of the correction that would call for them. In
doing so, he ignores a correction term that could amount to half an
arc-minute or more under some circumstances; a serious matter for a lunar.

So, it seems to me, in the 70 years or so between Mendoza's publication and
Merrifield's, lunar-distance navigation had taken a serious step backwards.
Such is progress!

George.

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