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    Re: Proportional logs, etc.
    From: Henry Halboth
    Date: 2004 Aug 25, 15:43 -0400

    Hi Jan:
    
    Many thanks for the come-back.
    
    I have been working Lunars for some 60-years, but have
    always favored the rigorous method of clearing by spherical
    trigonometry. I became academically interested in short methods
     in reading your work and that of George, Frank, and Bruce
     on the NavList.
    
    There is existent an apparently rare text, which I possess,
    entitled "The American Practical Lunarian and Seaman's Guide",
    by one Thomas Arnold, published at Philadelphia, PA in 1822,
    which sets forth a short Lunar Distance clearance method the
    writer claims to have invented, and includes the tables to which
     I have referred. As you suggest, there is little or nothing by
    way of explanation - only sparse rules for calculation as in other
    publications of this vintage. It is this method which I am attempting
    to decipher.
    
    Three tables are presented with respect to "first and second
    corrections" - all dealing respectively with correction of the Moon, Sun,
    and Star apparent altitudes. I am able to verify the Moon related
    correction, but am having the difficulty described in my previous
    posting with both the Sun and the Stars, where the parallax -
    refraction value is small, the tables proving out at a value of 2m50s
    and greater but showing an increasing difference as the
    correction becomes smaller. I have suspected the use of another
    prolog table for small values - am I to assume that the author has
    himself calculated such a table, or might there have been one
    in existence at the time. The author provides detailed comparisons
    with distances cleared by Witchell's method, both Sun and Stars,
    and his calculations seem reasonably accurate.
    
    The general formula stated by the author for all corrections
    tabulated is "The logarithm in table ____ is found by adding together
    the logarithmic sine of thirty degrees, the logarithmic cosine of the
    ________ apparent altitude, and the proportional logarithm of the
    ________ altitude, rejecting the tens in the indexes."
    
    Thanks again for your interest.
    
    On Wed, 25 Aug 2004 19:45:23 +0200 Jan Kalivoda 
    writes:
    > Hello, Henry,
    >
    > I only try to find out some solution, maybe quite wrongly.
    >
    > Maybe there is another basis for proplogs of Sun corrections than
    > for proplogs of Moon corrections. They should not necessarily be the
    > same. As Sun corrections are much smaller, it can be convenient to
    > choose another basis for their proplogs so as to allow for their
    > smaller values and to obtain more precise results. If the results of
    > your short method are correct compared with another method, there
    > must be some trick somewhere in your tables. Don't expect that an
    > introduction for the table users would explain such details at the
    > beginning of the 19th century.
    >
    > Take the basic formula for interpolation:
    >
    > T1/t = D/d
    > log T1 - log t = log D - log d
    > log T1 - log t = (log T2 - log d) - (log T2 - log D)
    >
    > In the most cases of tables and solutions, T1 = T2 and one table of
    > proplogs (proplog t = log T1 - log t) suffices. But according to
    > limits of some values (see above), it could be of advantage to use
    > two tables of proplogs, each one with another basis (T1 T1>T2). Bruce Stark used this trick in his tables for clearing
    > lunars very cleverly.
    >
    > Of course, this was only the simplest mode of usinf proplogs. The
    > use of your tables is more complex, but I cannot say anything about
    > them without studying them.
    >
    > But if your tables give wrong results anywhere in their limits,
    > forget this posting.
    >
    >
    > Jan Kalivoda
    >
    >
    > ----- Original Message -----
    > From: "Henry C. Halboth" 
    > To: 
    > Sent: Wednesday, August 25, 2004 4:48 PM
    > Subject: Proportional logs, etc.
    >
    >
    > This is intended for George, Fred, or someone otherwise familiar
    > with short methods for clearing Lunar Distances, or in the use
    > and construction of tables of Proportional Logarithms.
    >
    > I am currently working on a previously undiscussed short
    > clearing method, circa 1820, which I intend to post, however,
    > am having some difficulties is verifying the accuracy of
    > certain included tables which depend on the use of
    > proportional logarithms.
    >
    > A Table III, supposedly entered with the Sun's Apparent Altitude,
    > produces a logarithm employed in calculating a "second correction".
    > This tabular, value is stated as the summation of the log sin 30-deg
    > + log cos Apparent Altitude + proportional  log of the Altitude
    > Correction (parallax - refraction). At small values of Altitude
    > Correction, say in the order of 1-minute or less, I am unable to
    > accurately replicate the tabular values presented, while at larger
    > values acceptable coincidence can be demonstrated. For example ...
    >
    > At 30-deg Apparent Altitude, and tabulated Sun's Correction
    > as 1m-30sec, the referenced Table III produces a log of 1.7115.
    > My calculation is log sine 30 = 9.6990 + log cos 30 = 9.9375 +
    > pl 1-30 = 2.0792, for a correction log of 1.7157. This is not an
    > earth shaking difference at 30-deg, however, as the altitude
    > becomes greater, and the correction therefore less,
    > the difference becomes greater and the result questionable.
    >
    > A similar table published for the Moon, where altitude
    > corrections are significantly larger, utilizing the same deriving
    > formula, works out with amazing accuracy.
    >
    > I am using proportional logarithm tables dating back to 1828,
    > all British, i.e., Norie's, and find no difference in entries at
    > small
    > values. Were there other, perhaps American tables published,
    > circa 1820, am I having a "senior moment", are there other forms
    > of pl tables, or are proportional logs simply not accurate enough
    > at small values.
    >
    >
    > --
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    >
    
    
    

       
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