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    Sight Reduction by the Cosine Haversine Method
    From: Chuck Taylor
    Date: 2004 Oct 2, 22:38 -0700

    For many years the preferred method of reducing sights
    by the method of St. Hilaire was the "Cosine Haversine
    Method".  It required tables of the logarithms of
    trigonometric functions, plus tables of natural
    and logarthmic haversines.
    For anyone who may not be familiar with haversines,
    the haversine of an angle t is defined as
            haversine(t) = (1/2)*(1 - cosine(t))
    or, equivalently,
            haversine(t) = (sine(t/2))^2
    Haversines are merely a vehicle for simplifying the
    computations.  While sines and cosines range from -1
    to + 1, haversines range only from 0 to + 1, and the
    haversine of a negative angle is the same as the
    haversine of the absolute value of that angle.
    In the example below, I show both the intercept and
    the azimuth computed by this method. Often the azimuth
    was obtained from azimuth tables such as H.O. 71 for
    the Sun and other bodies with declination 0 to 23
    degrees, or H.O. 120 for bodies with declination
    24 to 70 degrees.
    The quadrant in which the computed azimuth angle lies
    is not always obvious.  In this case the declination
    is South and the meridian angle (t) is West, so the
    azimuth angle is S 55.8 W, or 236 degrees.  At the
    time of the sight I noted that the bearing of the Sun
    was WSW, so this checks.
    Quoting from Hosmer's "Navigation" (1926), "In the
    case the sun is about east or west and the Lat. and
    Decl. are of the same name it may be difficult to tell
    whether the bearing is from the N or the S.  To remove
    this doubt, add the log cosec. Lat to log sin Decl,
    obtaining the log sin Alt. when the sun is on the
    prime vertical, E or W.  If the observed alt. is less
    than this the sun is on the side toward the [elevated]
    pole (N in N. hemisphere)."
    Here is a worked example, which is from the same sight
    as the Time Sight example I posted recently.  This
    time I worked it both by computer and by tables.
    Again, I hope that someone will find this useful.
    Best regards to all,
    Chuck Taylor
    N of Seattle
    At anchor, position by GPS:
            Lat   48d 30.1' N
            Lon  122d 49.5' W
    25 September 2004
    Corrected UT (GMT): 23-17-12
    Corrected Ho:       24d 54.3'
    Almanac data:
            GHA Sun:  171d 27.4'
            t(W) = 171d 27.4' - 122d 49.5' = 48d 37.9'
            Dec Sun:  1d 1.67' S
            GMT:      23-17-12
                            Altitude           Azimuth
    t    048 37.9  log hav  9.22930   log sin  9.87534
    Lat  048 30.1  log cos  9.82125
    Dec  001 16.7S log cos  9.99989   log cos  9.99989
                   log hav  9.05044
                   nat hav  0.11232
    L~d  049 46.8  nat hav  0.17714
    z    065 05.8  nat hav  0.28945   log csc  0.04238
         --------                              -------
    Hc   024 54.2                     log sin  9.91761
    Ho   024 54.3                     Z    =   S 55.8 W
    a    000 00.1                     Zn   =   236
    Checking the Azimuth (Z):
    Lat   log csc    0.12553
    Dec   log sin    8.34849
    Alt   log sin    8.47402
    Alt on P.V.      1d 47.2'
    Since Hc is greater than the Alt on the Prime
    the Sun was south of the Prime Vertical, and the
    Azimuth was indeed S 55.8 W, or 236d.
    In this case it was very near the equinox, so I knew
    that the altitude at the Prime Vertical would be very
    low, but I showed the computations anyway for the sake
    of completeness.  From now until spring, the altitude
    of the Sun at the Prime Vertical will be negative
    (below the horizon).
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