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Henry Halboth revisits Arnold's lunar distance method, to tie up some loose
ends.
At the foot of this posting, I will restate my earlier transcription of
Arnold's text into emailese, as quoted by Henry, but amended to correct a
few things that went wrong with his original text.

I commented on Arnold's simplifications of Mendoza, on 5th Sept., as follows-

"This seems a worthwhile simplification on Mendoza's method..

However, it comes at a price. Arnold has had to supply separate tables for
correcting Moon, Sun, and stars, though he can avoid the need to use
separate refraction and parallax correction tables.

And Arnold has lost some flexibility in this simplification. When Mendoza
calculated refraction, it was possible, if he thought fit, to apply
corrections for a non-standard atmosphere. If a lunar was taken to Venus or
Mars, which require non-standard parallax corrections, that parallax, if
known, could readily be applied. (Does anyone know when lunar distances to
planets started to appear in the Almanac?)

As I see it, however, Arnold's method, including standard refraction and
parallax in his tables I, II, and III, was inflexible in that it would have
been unable to adapt to such requirements."

And Henry has responded-

>As regards Arnold's tables being somewhat restrictive by reason
>of constraints in altitude correction, please note the use criterion
>to be the bodies apparent altitude or the correction thereto; how either
>is calculated is not mandated and may be at the user's option -
>whatever refinement in the way of temperature and latitude
>corrections for refraction and parallax may certainly be applied
>at discretion.

==================

Reply from George-

Well, I don't wish to make a "big thing" of this resulting lack of
flexibility, because I don't regard it as an important matter in practical
terms. But I question Henry's argument here.

The difficulty is that the altitude correction of the Moon (and the same
applies to the Sun or star) enters into Arnold's calculation, not once but
twice.

Taken from the transcription below is the sentence-

"Add to the apparent distance the first correction and the correction of
the sun or star's altitude, and subtract the sum of the second correction,
and the correction of moon's altitude will be the corrected distance."

I think that may perhaps be slightly garbled, but its meaning is clear
enough. Perhaps it should really have ended something like "...and subtract
the sum of the second correction and the correction of moon's altitude, and
the result will be the corrected distance." No matter.

As far as that sentence is concerned, Henry is quite correct. That
"correction of Moon's altitude" can be worked out by the navigator as he
chooses, and he may include, or else disregard, temperature / pressure
corrections to refraction (same for Sun or star).

BUT that altitude correction (which combines refraction and parallax)
enters also into Table I (and table II or III, as relevant). Arnold states-
"Enter Table I with the Moon's apparent altitude and horizontal
parallax, and take out the corresponding logarithm, which place
in the first column."

and Henry himself describes the construction of that table as-
"Tabular log = log sin (30 deg) + log cos Moon's apparent altitude +
[prop-log] of Moon's altitude corrections."

So built into table I (for the Moon, and similarly into tables II or III
for star or Sun) is its own "Moon's altitude correction" (for refraction
and parallax), which the table can deduce from its knowledge of the
altitude, and which the user has no means of tinkering with. That's what
prevents him from applying corrections to refraction for extreme climatic
conditions, or allowing for the parallax of Venus or Mars.

I wonder if Henry is convinced. As I said, it's not a big deal.

=================
Henry continues-

One cavalier treatment, not previously mentioned, is that
>Arnold advocates a standardized observed altitude correction to obtain
>the apparent altitude; under a Rule III, he advocates, across the board
>...
>
>"To the moon's observed altitude, add 12', if the lower limb be taken,
>but if the upper limb be taken, subtract 20'. to the observed altitude of
>
>the sun's lower limb add 12', and from the star's observed altitude
>subtract 4', and you will have their apparent altitudes." Of course,
>we know this to be technically incorrect - perhaps it is simply a
>reflection of the often expressed opinion that an error of a few
>minutes of arch in altitude does not materially affect the result in
>clearing the distance.

Comment-

This seems a crude approach, but it's a perfectly valid approximation for
the purpose of correcting lunars, because the altitude corrections only
vary slowly with altitude itself, and because it's those corrections,
rather than the altitudes, that are so vitally important for clearing the
lunar distance.

==============

I should point out that anyone who, like me, tries to get to the bottom of
Mendoza's approximate method, and Arnold's method, will find a description
in Cotter's "A history of nautical astronomy" , pages 227 to 231, of
Merrifield's method. This is very similar to Mendoza and Arnold, and
Cotter's analysis showed the way that the trig was done. Without that, I
couldn't have tackled Mendoza's or Arnold's recipes. Taking Cotter's
description at face value, the main difference seems to be this- Merrifield
simply disregards the final correction term, provided by Arnold's Table
VII.

George.

===================================================

"A short Method of Correcting the Apparent Distance of the Moon from the
Sun or Star.

Invented by the Author.

Rule

Add together the apparent distance and apparent altitudes, and take half
their sum;

The difference between the half sum and the sun or star's altitude, call
the first remainder.

The difference between the half sum and the moon's apparent altitude, call
the second remainder.

Set down-

The sine of the apparent distance in two columns
The secant of the half sum also in both columns
The cosecant of the first remainder in the first column
And the cosecant of the second remainder in the second column.

Enter Table I with the Moon's apparent altitude and horizontal
parallax, and take out the corresponding logarithm, which place
in the first column.

Enter Table II, if a star is used, or table III, if the sun is used, and
take out the corresponding logarithm, which place in the second column.

The sum of these four logarithms, rejecting the 10's in the indexes, in the
first column. will give a proportional logarithm of the first correction.

And the sum of the four logarithms in the second column, rejecting the 10's
in the indexes, will be the proportional logarithm of the second
correction.

Add to the apparent distance the first correction and the correction of the
sun or star's altitude, and subtract the sum of the second correction, and
the correction of moon's altitude will be the corrected distance.

Then enter Table VII, with the corrected distance at the top, with the
difference of the first correction, and the correction of the moon's
altitude in the left side column, and also in said table with the
correction of the moon's altitude in the left side column, and take out two
corresponding numbers. The difference between the two numbers is to be
added to the corrected distance when less than 90 degrees, or subtracted if
above 90 degrees."

Henry Halboth adds the following notes about the tables-

"Relevant included tables".

Table I = A table of logarithms against top entries of Moon's Apparent
Altitude and with side entries of Moon's Horizontal Parallax.

This table is constructed/calculated as follows...

Tabular log = log sin (30 deg) + log cos Moon's apparent altitude +
[prop-log] of Moon's altitude corrections.

Table II = A table of logarithms agains Star's Apparent Altitude

This table is constructed/calculated as follows...

Tabular log = log sine (30 deg) + log cos Star's apparent altitude +
[prop-log] of Star's altitude correction.

Table III = A table of logarithms against Sun's Apparent Altitude.

This table is constructed/calculated as follows...

Tabular log = log sine (30 deg) + log cos Sun's apparent altitude +
[prop-log] of Sun's altitude correction.

Table VII = A table of corrections against corrected distances across top
and the difference between the first correction and moon's altitude
correction or the moon's altitude correction alone as side entries. This
table provides a third correction to the Distance for parallax and
refraction.- it is essentially Norie's Table XXXV, presented in a slightly
different manner.

There are other tables included which are essentially as contained in any
navigational epitome, i.e., altitude corrections for the Moon, Sun, and
Stars, etc., which are unnecessary of comment at this time."

================================================================
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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