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Jan,

What a wonderful dialogue you have started, thanks for getting this
thread going.

I have produced a diagram to illustrate the trigonometry of the
"approximate" methods you describe in section 3 of your original
posting. It is available as a .jpg file in the Nav-L section of
www.LD-DEADLINK-com. From the home page, click on the link to Nav-L
and look for the topic "Various Methods for Clearing the Lunar
Distance". There is a link to the first posting in this thread and a
link to the diagram. A direct link to the diagram is at
http://members.verizon.net/~vze3nfrm/Nav_L_Graphs/ApproxLD.JPG. I have
included an overview diagram and an enlargement of the section showing
the distances and the perpendiculars.

In labeling the diagram, I have followed the definition of terms from
your original posting:
Z = Zenith of observer
Z' = Angle at the zenith
S = True Sun
s = Apparent Sun
M = True Moon
m = Apparent Moon
X = Point of intersection of Apparent and True Lunar Distances
MS = True Lunar Distance (LD)
ms = Apparent Lunar Distance (LD)

In illustrating the perpendiculars, I have added the following:
a = where perpendicular from point M meets the Apparent LD (ms)
t = where perpendicular from point m meets the True LD (MS)
a' = where perpendicular from point S meets the Apparent LD (ms)
t' = where perpendicular from point s meets the True LD (MS)

Now I need to study your propositions that from this diagram "we can
trigonometrically deduce approximate equation permitting to reduce
("clear") the apparent L.D. to the true L.D" and "the spherical
trigonometry alone can find the long line of always diminishing
trigonometrical terms of corrections".  I am hoping other members of the
list will help me see how this is done as I am very keen to understand
it graphically.

Section 3 of your original posting is appended below for convenience.  I
hope other list members will help puzzle out the solution.

Regards,
Arthur

Section 3 from original posting:
"Imagine the triangle in the sky with the vertices Z - zenith, S - true
Sun/star and M - true Moon. And another triangle with wertices Z -
zenith, s - apparent=observed Sun/star and m - apparent=observed Moon.
The two triangles have the common vertex (and angle) Z and their two
sides (zenith distances of the four bodies mentioned!) crossing at Z and
perpendicular to the horizon coincide for the most part of them: s lies
above S, as the daily parallax (which always lowers the
apparent=observed body below true=supposed-to-be body for an observer on
Earth's surface) of the Sun or planet (not mentioning the stars) is
always much smaller then the effect of refraction (which always raises
the apparent body above true body). On the contrary, m lies below M, as
her great daily parallax is always greater then the effect of
refraction. As a result, the third sides (apparent and true lunar
distance!) of both triangles, ms (apparent=observed L.D.) and MS (
true=cleared L.D.) cross each o!
ther at the common point X. But the sections mM and sS are very short
(half degree at most, but mostly shorter), which is essential for
further procedures.

Therefore if we drop perpendiculars from the points M and S to the side
ms (apparent L.D.) and vice versa, we can trigonometrically deduce
approximate equation permitting to reduce ("clear") the apparent L.D. to
the true L.D. (Here you can see a very remote similarity with Ageton's
and other methods for resolving the nautical triangle; but these are not
approximate in any degree, only their use of perpendiculars to triangle
sides is somewhat similar.)

(M,S,m,s are meant as centres of bodies - the limbs are measured, of
course, but applying the corrections for the semidiameters of bodies,
one obtains the values for centres. I neglect all three efects of
ellipsoidal earth's shape on clearing L.D., too; they can make a maximal
error of 13 arc-seconds in the true distance cleared, when neglected.)

The final approximate formula can be confirmed directly by the calculus
(Taylor's polynoms), too, but the spherical trigonometry alone can find
the long line of always diminishing trigonometrical terms of corrections
allowing for effects of parallax, refraction and their combinations on a
measured lunar distance. 10 (ten) terms were sometimes used for
calculation! This formula is called "approximate", as it is not derived
strictly, but only in gradually approaching steps and terms; but when
sufficient number of terms is included, its accuracy leaves nothing
open."

-----Original Message-----
From: Navigation Mailing List
[mailto:NAVIGATION-L@LISTSERV.WEBKAHUNA.COM] On Behalf Of Jan Kalivoda
Sent: Friday, April 11, 2003 2:31 PM
To: NAVIGATION-L@LISTSERV.WEBKAHUNA.COM
Subject: Re: Classification of the methods for clearing the Lunar
Distances

Hello, All,

I am sorry for my late answer. I spent some days at my mother, without a
wire into the world.

Thank you for your approval and comments. I will try to react to them
one by one.




To Bill Noyce:

Yes, the effort to escape antilogs could promote the Borda's method. But
the Dunthone's formula requires three tables: of natural cosines, of
"logarithmic difference" and of logarithms of numbers. In Borda's
method, one had to browse through three tables, too: log cos, log sin
and "logarithmic difference". And the number of arithmetical operations
was twelve in it; it falls to number nine in the variant of this method
that Herbert Prinz has sent into the list in another reaction to my
text. Dunthorne's method required five operations.

------------------------

To George Huxtable:

You are right, George. The amount of Mm is the difference between the
Moon's parallax in altitude and its refraction in altitude. This value
is greatest in 14 degrees of altitude: 55.5 arc-minutes, when the
horizontal parallax of the moon is greatest (61 arc-minutes) and 48
arc-minutes, when H.P. is minimal (53 arc-minutes). Above and below 14
degrees this difference decreases, as the refraction slims down more
quickly with the growing altitude than the parallax.

--------------------------

To Bruce Stark:

Your identification of the mysterious Thomson with the Canadian surveyor
David Thompson is very interesting. But according to the last edition of
Encyclopedia Britannica, this David Thompson died at 1857 and there was
a foreword in the edition of the Thomson's tables from 1845 (written by
Boulter J. Bell), which speaks about Thomson as about a dead man. And
G.Coleman (the editor of Norie's "Epitome of practical Navigation" after
Norie's death) said before 1849 that he was in contact with "Captain
Thomson". And thirdly, the Thomson's tables had been edited by the
publisher W. Allen & Co. from the first edition, exactly by the same
publisher , who edited the 67th edition from 1880 you have in your
library. I doubt, whether the publisher's mystification replacing the
surveyor Thompson for a captain Thomson could survive such long decades?
Maybe it could?

Thank you for that information about the 67th edition - it is
fascinating to see such a long life for a set of nautical tables; only
Inman, Norie and Bowditch can beat Thomson, isn't it?

As for the year 1824 of the first edition, you are right, I have checked
it now in the online catalogue of British Library. My source was in
error and I took it over wrongly.

I am very interested in your tables for clearing L.D.'s, even more after
reading praise of them in the list.

----------------------------

To Arthur Pearson:

Thank you for your assessment, Arthur. But don't ask graphics from me,
please! My teacher of drawing would be very angry, as he was in my
childhood. For that matter, the picture No.6 on your nice website (in
PDF-file) is exactly the proper one for illustrating my text about
approximative and rigorous methods; only the scale can be enlarged and
the symbols M,m,S,s added. It would suffice.

I will try to send a more detailed description of approximative formulas
for clearing the L.D.'s in the next hours or days, as you wanted.

----------------------------

To Herbert Prinz and Fred Hebart:

Yes, I had stressed that my analogies between the methods of clearing
the L.D.'s and resolving the LOP's are only very rough. The intercept
method for Marcq St. Hilaire problem is an approximative solution in the
core, the rigorous one would be to resolve the system of two circle
equations on the surface of the Earth's sphere (even ellipsoid!). But
Fred said it for me - the cosine-haversine equation is strictly deduced
from basic trigonometric formulas for spherical triangles as in the
rigorous methods for clearing the L.D.'s. This was my argument in
comparing these two sets of procedures.

As for Chauvenet, it was my greatest blunder. I saw him cited in an old
German article as a French astronomer and under the French title of his
work - the French title, the "French" name, the French attribution - I
didn't verify it once more. I blush and thank you for the correction.



Thank you for all comments, once more.


Jan Kalivoda

   

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