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    Clearing Lunar Distances by trigonometry
    From: Henry Halboth
    Date: 2004 May 12, 23:10 -0400

    I make one more effort this evening to get this format across straight -
    hope it works this time.
    As a matter of potential interest, the following is the haversine format
    I employed  in clearing Lunar Distances by spherical trigonometry, before
    the advent of calculators and, wherein ...
    Hs = sextant altitude..............     Ds = sextant distance.............S =
    Ha = apparent altitude............      Da = apparent distance..........        M =
    Hc = true altitude....................  Dc = calculated distance........        Z
    = angle at zenith
    1)  hav Z = sin (s - Ha M) x sin (s - Ha S) x sec Ha M x  sec Ha S
            in which ...  s = � (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~S  49-38-48        n hav   0.176241
    Dc      74-37-07        n hav   0.367378
    To afford a comparison, the altitudes 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 -
    all of 3" greater than that found by the haversine formulae stated above.

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