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    Re: Proportional logs, etc.
    From: Jan Kalivoda
    Date: 2004 Aug 25, 19:45 +0200

    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 (T1T2). 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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