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    Finding position ab initio
    From: Bill Noyce
    Date: 2002 Mar 22, 10:22 -0500

    George Huxtable suggests using Chuck's Venus and Moon altitudes
    to locate his position, and shows how we could get an AP to
    start with by drawing circles on a globe:
    > For a first shot (as we have no idea yet where Chuck was observing
    > the geographical positions of Moon and Venus at that time could be
    > on a globe. Chuck's measured altitudes for these bodies should be
    > in the ordinary way (ie corrections as for altitudes, not corrections
    > for lunar distances), and then the corrected altitudes converted to
    > distance by taking the 90-degree complement. Now draw circles of that
    > radius about the two GPs on the globe. These intersect in two places,
    > you have to guess which one is most likely. This paragraph could be
    > bypassed if we had a rough idea of whereabouts Chuck lived, in
    Instead of using this procedure, we could use the method of
    "double altitudes" to find the latitude directly.  It involves
    three spherical triangles, and repeated use of the formula
        cos a = cos b * cos c + cos A * sin b * sin c
    where lower-case letters represent the length of a side, and
    upper-case letters represent the angle opposite that side.
    This is the same formula used in sight reduction, except that
    altitude, latitude, and declination are each the complement of
    a side in the navigational triangle, so sines and cosines of
    the sides get interchanged.  The formula as I've presented
    it directly gives the third side if we know two sides and
    their included angle; a little algebra rearranges it to
    give an angle if we know three sides:
        cos A = (cos a - cos b * cos c) / (sin b * sin c)
    The first spherical triangle has vertices of the pole and the
    two bodies (Venus and the Moon, in this case).  We know the
    two declinations, which gives us two of the sides, and normally
    know the difference in GHA, so we can use the formula to compute
    the distance between the two bodies.  In our example we already
    know the distance between the bodies, too, so we have all three
    sides of this triangle.  We need to use the formula once more
    to compute the azimuth angle from one body (say Venus) to the
    other -- we'll use this for solving the third triangle later on.
    The second spherical triangle has vertices of the two bodies
    and the observer's zenith.  Two of its sides are the zenith
    distances of the bodies (complements of the measured altitudes,
    after correction for IE, dip, refraction, semi-diameter, and
    parallax).  The third side is the distance between the bodies,
    either computed from the first triangle, or using the cleared
    distance from our lunar work.  We need to use the formula to
    compute the angle of this triangle at the same body as we did
    above.  So in our example, this is the angle Zenith-Venus-Moon.
    The third spherical triangle has vertices of the pole, the
    observer's zenith, and the chosen body (Venus).  Its sides are
    the body's co-declination, the measured zenith distance
    (corrected), and the observer's unknown latitude.  Further,
    we know something about the angle of this triangle at the chosen
    body.  Because we already know the angles Pole-Venus-Moon and
    Zenith-Venus-Moon, a simple addition or subtraction will give
    the angle Pole-Venus-Zenith.  (A diagram is probably needed
    here, to decide whether to add or subtract.)  From this
    angle and the other two sides, the formula gives us the third
    side - our co-latitude.
    This method of finding latitude by two observations can be
    used without a chronometer, as long as we can get the declinations
    of the bodies and their difference in GHA.  For two fixed stars,
    you hardly even need a calendar for that; for the sun it helps
    to know the approximate hour and the date.  If the two bodies
    (or even the same one) are observed at different times from
    the same location, we need a watch to determine the difference
    in GHA, but it need not be set to any particular time, and need
    not have particularly good long-term accuracy.
    Once we have latitude, we can use either (or both) of the
    observations as a "time sight".  This uses the same formula to
    give us the LHA of the observed body, given latitude,
    declination, and altitude.  Since in our case we know the
    time, and thus GHA, our longitude follows directly.
    Perhaps this is only of academic interest -- certainly the
    method that finds an approximate position first lets us
    use a more familiar method (and tables).  But I was
    fascinated when I first read about it, with pictures, here:
    http://www.northwestjournal.ca/dtnav.html and go to Article VI.

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