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George is absolutely correct - in transcribing Arnold's method,
I omitted a paragraph to follow the statement "And the cosecant
of the second remainder in the second column.", which should read ...

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

The next paragraph should then read ... Enter Table II if a star is
used, or Table III if the sun is used ... etc

My apologies for the error - I simply lost track of my place while
trying to preserve this old book

I certainly do not question George's analysis and comparison with
Brother Mendoza's method - the similarity is readily apparent. It is,
however, interesting to note Arnold's effort at simplification in the
preparation of tables suited to the working. I don't believe these
tables to be existent elsewhere, and the effort expended in their
preparation cannot be judged insignificant - they must have
been useful to those seeking to avoid to the extent possible
mathematical involvement. After all, were not the short methods
developed to avoid the rigorous spherical trigonometrical approach.

There is no intent to enter into a discussion of who came first,
but it is of interest that two men working at opposite sides of
the ocean at roughly the same time should come up with such
similar methods - but then again, as they say, there can be only
so many ways to skin a cat.

My real purpose in posting this item was to underscore the
diversity of effort that apparently went into the development
of these short methods, and to further invite attention to
Arnold's work and book. At least, in my limited exposure, he
treats the subject in far greater depth than so either Bowditch
or Norie, and includes an interesting explanation of graphical
methods which, if not overly practical, is very instructive.

On Sun, 5 Sep 2004 00:35:29 +0100 George Huxtable
 writes:
> 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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