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    Re: Classification of the methods for clearing the Lunar Distances
    From: Arthur Pearson
    Date: 2003 Apr 11, 22:40 -0400

    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.
    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
    -----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
    Subject: Re: Classification of the methods for clearing the Lunar
    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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