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    Determination of longitude and the prime vertical
    From: Joel Silverberg
    Date: 2007 Apr 08, 04:18 -0700

    Thomas Sumner in his "New and Accurate Method of Finding a Ship's
    Position at Sea" (1843) states repeatedly that there is but one proper
    moment in the day when a single observation can determine the Latitude
    of the ship, unless the apparent time at the ship is accurately
    known.   And when the Latitude is uncertain, ther are only two proper
    instants per day (when the sun bears exactly East or exactly West)
    when the solar altitude can be used to find the Longitude by
    Chronometer with accuracy.  If one makes an observation other than at
    those times, he stresses, the unavoidable errors in measurement may
    cause frequent errors which are very great.  He also mentions more
    than once that in his latitudes, the sun in not observable in the
    proper E and W points for seven months of the year (taking into
    account that the altitude must exceed 6 or 7 degrees to be reliable.
    My question is :   what are the crucial issues around the sun being on
    the prime vertical (that is, either due E or due W) when his (her?)
    altitude is taken?   Am I correct in thinking that the hour angle
    between the observer's meridian and the sun's meridian would be
    exactly 90 degrees when the sun bears true E or W ?   Are we dealing
    with a Pole-Sun-Zenith triangle with a right angle at the Zenith?
    If so, could we not calculate not only the hour angle (at P) but also
    the Latitude?  {  sin(D) = sin(L) sin(h)    would give us L and
    cos(D) sin(HA) = cos(h)  would give us HA , where D is the
    declination, L the latitude, h the altitude, and HA the hour angle .}
    Or is it more a question of sensitivity to errors in Latitude when the
    zenith angle is close to 90 degrees?   Did people calculate longitude
    when the sun was on the prime vertical without using any asumptions as
    to latitude, or did they always (never say "always" or "never") use an
    oblique triangle together with their best estimate or guess for
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