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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
Vertical,
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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