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    Lunar Distances - trigonometric clearing
    From: Henry Halboth
    Date: 2004 May 12, 21:05 -0400

    As a matter of potential interest, the following is the haversine format
    I employed in clearing Lunar Distances before the advent of calculators
    and, wherein ...
    
            Hs = sextant altitude           Ds = sextant distance
            Ha = apparent altitude          Da = apparent distance
            Hc = true altitude                      Dc = cleared distance
            M  = moon                       S   = sun or star
            Z  = angle at zenith
    
            1) hav Z = (s - sin Ha M) x sin (s - Ha S) x sec Ha M x sec Ha S,
                    in which ... s = 1/2 (Ha M + Ha S + Da)
            2) hav Dc = hav (Hc M ~ Hc S) + cos Hc M x cos Hc S x hav Z,
             therefore ...
    
            Ha M              75-07-00      l sec   0.590318
            Ha S          25-45-03          l sec   0.045424
            Da            74-46-17
            2s =        175-38-20
            s   =         87-49-10
            s - Ha M              12-42-10  l sin   9.342213
            s - Ha S              62-04-07          l sin   9.946211
            Z  =                    l hav  9.924166
                                             /
                                    l hav  9.924166
            Hc M          75-22-00          l cos  9.402489
            Hc S          25-43-12          l cos  9.954689
                                    l hav  9.281344
                                            /
                                    n hav  0.191137
            Hc M ~ Hc S  49-38-48           n hav  0.176241
            Dc            74-37-07          n hav  0.367378
    
    To afford a comparison, altitudes here used, both apparent and true, are
    as employed in an example of Borda's method, set forth on page 417 of
    Norie's 1889 edition, where the cleared distance is found to be 74-37-10.
    
    I hope that this transmits without becoming hopelessly screwed up, as it
    has been most difficult to set it up so as to even print with any degree
    of accuracy.
    
    
    

       
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