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Hernry Halboth posted an interesting message about "Arnold's lunarian",
with his method of clearing lunar distances.

>I have completed a preliminary review of the short Lunar Distance
>clearing method proposed by Thomas Arnold, which I shall henceforth
>entitle Arnold's Method, as contained in his "The American Practical
>Lunarian and Seaman's Guide", published at Philadelphia, PA, in 1822.
>An abstract is attached, providing the rules for working this method,
>as well as a description of the tables utilized by the author to
>facilitate
>the working thereof. Taken as a whole, Arnold's book is more
>descriptive of the Lunar Distance method than are either Bowditch
>or Norie, although the theory on which the actual solution is based
>remains rather secretive - it is unfortunate that this book seems rather
>rare and that the author has received little or no recognition for his
>apparently unique and perhaps purely American contribution to the art.
>
>According to himself, Arnold was an American Shipmaster of some
>40-years experience at sea, including 28-years as Master. It is during
>this time that he claims to have developed the method presented, and
>claims also to have utilized it for 5-years at sea in manuscript prior to
>publication. He, by the way, states an expected accuracy in the use
>of Lunars of from 10 to 15 miles. At one point in time he operated a
>school of navigation at Philadelphia for the purpose of teaching the
>Lunar method.
>
>I have prepared a work format for Arnold's method and will shortly
>try it for accuracy and convenience against other methods, of which
>there are perhaps over a hundred, although few have actually been
>published.

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

Arnold's method seems to be very similar to Mendoza's approximate method. I
will transcribe below the appropriate part of Mendoza's method, as
described by Norie, and follow that by a transcription of Arnold's, so the
two can be readily compared.

1. Mendoza.

The long-winded title is "Formulae for finding the longitude, in which a
method invented by Mendoza Rios is used for clearing the observed distances
from the effects of refraction and parallax, with rules for working the
observations. By J.W.Norie, teacher of navigation and nautical astronomy,
published by J.W.Norie and Co., London, 1816.". This text fills the
introductory pages of what appears to be a pad of blank forms for making
the calculation. It was kindly photocopied for me by Ian Jackson from the
Scoresby Papers, in Whitby (Yorks., UK) Museum.

The first part, which we will omit, is "To find the apparent Altitudes and
Distances."

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

The second part, which interests us, is "To find the True Distance."

1. Add together the apparent distance and apparent altitudes, and take half
of their sum; the difference between the half sum and the Sun or Star's
apparent altitude call the first remainder: and the difference between the
half sum and the Moon's apparent altitude call the second remainder.

2. Add together the log sine of the apparent distance; the log co-sine of
the Moon's apparent altitude: the log secant of the half sum; the log.
co-secant of the first remainder; the proportional logarithm of the Moon's
correction ([table] XXX) and the constant logarithm 9.6990: their sum,
rejecting the tens in the index, will be the proportional logarithm of the
first correction.

3. Add together, the log. sine of the apparent distance (already found;)
the log. co-sine of the Sun or Star's apparent altitude; the log. secant of
the half sum (already found;) the log. co-secant of the second remainder;
the proportional logarithm of the Sun or Star's correction [see footnote];
and the constant logarithm 9.6990: their sum, rejecting the tens in the
index, will be the proportional logarithm of the second correction.

[ the footnote reads "The Sun's correction is the difference of the
refraction and parallax in altitude. ([table] IV, VI) The star's correction
is the refraction in altitude ([table] IV)."

4. The difference between the first correction and the correction of the
Moon's altitude, call the difference of corrections.

Enter Table XXXV. with the apparent distance at the top, and the Moon's
correction in the side column, the corresponding number will be the third
correction; in the same column, and opposite the difference of corrections,
will be found the fourth correction.

5. Subtract the sum of the Moon's correction, and the second and fourth
corrections from the apparent distance; to the remainder add the Sun or
Star's correction,and the first and third corrections; their sum will be
the true distance."

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

The text continues with- "Having the true Distance, to find the apparent
Time at Greenwich", "To find the apparent Time at the Ship by an altitude
of the Sun",  "To find the apparent Time at the Ship by the Altitude of a
Star", and finally "The Apparent Times being known, to find the Longitude".
Interesting in themselves, these are irrelevant to our present purpose.

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

Now for Arnold's Method, from the transcribed text from "Arnold's Lunarian,
abstracted by Henry Halboth."

"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.

[Here I suspect Henry may have missed out a line in these rules, which I
would expect to state something like- "Enter Table I, and take out the
corresponding logarithm, which place in the first column." Otherwise, there
would only be three logarithms, not four, to sum together in the first
column, and Arnold's Table I would languish unused. George]

Enter Table I, if a star is used, [should this be Table II, for a star?
George] 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."

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

Comment from George Huxtable.

It can be seen by comparing the two transcripts above that the Mendoza and
Arnold methods are exactly the same, except in one respect, as follows-

Where Mendoza sums six terms, in each of the logarithmic calculations in
his paragraphs 2 and 3, Arnold has managed to reduce that sum to just four
terms.

Three of the terms are exactly common to both methods. These are-

In Mendoza's paragraph 2, and in Arnold's second column-

the log sine of the apparent distance; the log secant of the half sum; the
log. co-secant of the first remainder

In Mendoza's paragraph 3, and in Arnold's first column-

the log sine of the apparent distance; the log secant of the half sum; the
log. co-secant of the second remainder.

What Arnold has done is to shrink the remaining 3 terms of Mendoza's method
into a single term.

Firstly he has got rid of Mendoza's constant log, of 9.6990, by
incorporating it in his tables I, II, and III. Percipient readers will
recognise 9.6990 as being simply log (one-half), so adding that constant
simply halved the result of the log calculation. Arnold has built it in to
his tables by including a term log sin (30 deg), which those same
percipient readers will recognise as being exactly the same, because sin
(30 deg) is exactly one-half. That was a useful simplification.

Secondly, Arnold takes Mendoza's remaining terms, and combines them into
his table I, II, or III, as appropriate. For example, in his para 2,
Mendoza requires the navigator to work out the correction, for parallax and
refraction, of the Moon, from his table XXX, then take the prop-log, then
sum with that prop-log the log cos of the Moon's apparent altitude: these
being two of the six terms in his log calculation. Arnold has seen that
from the apparent altitude (with the horizontal parallax given) a single
table (his table I for the Moon) can deduce and combine the Moon's parallax
and refraction, multiply by cos alt, and provide the prop-log, all in one
go. Similarly for the Sun or a star. 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.

The final steps in the clearing process, using table XXXV in Mendoza's
case, or Arnold's Table VII, appear to be identical, from Henry's account.


George.


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