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    Re: Clarification of Question regarding LAN
    From: Chuck Taylor
    Date: 2004 Aug 4, 21:04 -0700

    Robert Gainer asked about how a time sight works, and
    how compass corrections fit in.
    
    Robert,
    
    The basic formula for a time sight, given in Bowditch
    (1984 and other editions), is
    
    cos(t) = ( sin(h) - sin(L)*sin(d)) / cos(L)*cos(d)
    
    where you are finding
    
    t = meridian angle
    
    as a function of
    
    h = observed altitude (corrected)
    L = latitude, assumed or obtained by other means
    d = declination
    
    A time sight can be taken at any time you can see the
    sun.  The above formula is a derivative of the Law of
    Cosines for a spherical triangle. You solve for
    meridian angle (and hence longitude) by assuming that
    the latitude is known.
    
    Notice that if your assumed latitude is off a bit,
    then your solution for the meridian angle (and hence
    for longitude) will also be off by some amount.  It so
    happens that this error is minimized if you take the
    sight at the time at which the sun crosses the prime
    vertical circle (PVC).  That is one reason why time
    sights were traditionally taken at the time of PVC
    crossing.
    
    The time of PVC crossing is usually computed, either
    by tables (such as Table 25 of Bowditch 1981 Vol 2),
    or by formula.  You don't need a magnetic compass for
    that.
    
    For an intuitive picture of PVC crossing, picture
    yourself in the northern hemisphere (say at 48 deg
    north latitude) between the spring and fall equinoxes.
     Note that the sun rises somewhat to the north of
    east, and sets somewhat to the north of west.  Yet,
    when it crosses your meridian at noon, it is to the
    south.  If it starts the day to the north of the
    east-west line, yet is to the south of the east-west
    line at noon, it follows that it must cross that
    east-west line on its way south sometime between
    sunrise and LAN.  At that time it will bear due east.
    That is the morning PVC crossing.  Similarly, sometime
    between noon and sunset, it must again cross the
    east-west line going north.  That is the afternoon PVC
    crossing, at which time the sun will bear due west.
    
    Others have pointed out that you don't always have the
    benefit of a PVC crossing.  For example, at latitude
    48 north (where I live), there are no visible PVC
    crossing between mid-September and mid-March (the
    equinoxes). As I mentioned earlier, a time sight of
    the sun can be taken at any time that the sun is
    visible.
    
    To understand the other major reason why time sights
    were traditionally taken at the time of PVC crossing,
    try to put yourself in the mindset of an 18th century
    navigator, before the time of Sumner and St. Hilaire.
    The concept of a line of position had not yet been
    introduced.  Finding your position meant finding your
    latitude and then finding your longitude as a separate
    and distinct process.  First, you found your latitude
    by observing Polaris or by observing the sun at LAN.
    Note that finding your latitude is equivalent to
    measuring the distance between you and the equator (or
    pole) in a north-south direction. Naturally, your
    observation is of a body that is due north or due
    south of you.  Similarly, finding your longitude was
    thought of as equivalent to finding the distance
    between you and the prime meridian (Greenwich for the
    English-speaking world).  What better way to measure
    an east-west direction than by observing the sun when
    it bore due east or due west?  Given the historical
    context, this made perfect sense to navigators of the
    time.
    
    After the introduction of the line of position methods
    developed by Sumner and St. Hilaire in the 19th
    century (although adoption and acceptance of these
    methods lagged well into the 20th century), navigators
    recognized that it was no longer necessary to take
    separate sights for latitude and for longitude, and
    that it wasn't necessary to shoot a sight at the time
    of PVC crossing (or at LAN for that matter).  Two LOPs
    obtained at any time are at least as useful as
    separate latitude and longitude sights.  That is a
    natural concept for us today, but it would have been
    incomprehensible to most (if not to all) 18th century
    navigators.
    
    I hope this helps.
    
    Best regards,
    
    Chuck Taylor
    North of Seattle
    
    --- Robert Gainer  wrote:
    
    > Chuck said,
    > >The more traditional way of determining longitude
    > was
    > >to use a time sight at the time the sun crosses the
    > >Prime Vertical (i.e., the time at which the sun is
    > due
    > >east or due west of you). ...
    >
    > Chuck,
    > I don?t understand how that will work. The magnetic
    > variation and the
    > latitude must be problems in that method. If you are
    > at 23 degrees north
    > latitude or greater the sun is never due east or
    > west.  If you do not know
    > the magnetite variation with some degree of accuracy
    > wont that have a very
    > large effect on the method? Is this practical at
    > all?
    > All the best,
    > Robert Gainer
    >
    
    
    
    
    
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