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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
latitude?


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