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I'll attempt an answer to this question:

" ....No one has addressed my question of why the St Hilaire method
calculates an altitude at a location our ship is NOT at, when we've
just measured the altitude where our ship IS at...."
---------

Answer: Because you are not comparing altitudes with the St Hilaire method but 
you are comparing zenith distances.

You -do- know your zenith distance to an astronomical body because you have 
just measured it with a sextant, but you do not have enough information to 
work out the other parameters you want i.e. lat or long from that alone.

What you can do however is imagine you are at position you _do_ know and work 
out a zenith distance for that position. Then you can compare the difference 
between them because they will be close together in value.

If you go back to basics and consider what you DO know and what do NOT know, 
and what you have available and not available with the PZX spherical 
triangle, it should be clearer,(see diagrams).

You have:

The geographical position of the body from the almanac. Hence:-

1) co-declination of the body (from Dec in the almanac:-i.e. 90-dec).
and 
2)the GHA of the body.  (This not enough to know the local hour angle in the 
PZX triangle because you do not know your longitude).

You also have from your sextant sight the altitude of the body to your local 
horizon, which means you have immediately:-
3) the 'true' zenith distance from yourself to the body. (i.e. the side ZX of PZX).

And therein lies the problem.  You do not have enough information about the PZX triangle to solve it. 
Consider:

a)If you knew the longitude accurately you would know what the angle ZPX is 
(i.e.the local hour angle) and could work out the latitude accurately.

b)If you knew the latitude accurately, you would know the side PZ accurately - 
the co-latitude and you could work out the local hour angle and hence 
longitude. 

But you do not have either.

Sumner stumbled on the position line principle by making an assumption as if 
he did know the latitude and worked out a longitude. He did this three times 
with three different latitudes and suddenly realised that what he had was a 
line of position where the sun could have been along that line found with the 
three results at that instant of the time of the sight - it was the line of 
equal altitude from the body,  and a certain fixed zenith distance from the 
body to the position line is implied, it is an equal zenith distance - a 
circle surrounding the geographical position of the body.  (see diagram 
"position line X at pole).

St Hilare's brilliance is in the realisation that it is not necessary to know 
either either lat or long singly with precision to solve the pzx triangle, as 
you can work out an accurate zenith distance if you use an _assumed position_ 
  (this is the correct use of this term here) which is _near_  where you are, 
and using an exact lat and long for that assumed position.

 .... and then compare the accurate (or 'true') zenith distance you have 
measured with your sextant with the calculated zenith distance for the 
assumed position nearby (which must be similar in value to your sextant sight 
value) - and close by as the assumed position is close by.  And hence find 
the difference.

That difference (the intercept) is easily marked off on the chart from the 
assumed position. You then have a position line along where you must lie. 
(neglecting errors).

In other words you are supplying yourself with the (accurate) necessary 
information of latitude and longitude to work out the PZX triangle that you 
cannot work out normally with insufficiently accurate informaation, then 
using a comparison method with your sextant (measured) information you do 
have in terms of zenith distance.

That in a nutshell is the Mark St Hilaire's (or intercept) method.

Douglas Denny.
Chichester. England.


=====================
Original Posting:-

No one has addressed my question of why the St Hilaire method
calculates an altitude at a location our ship is NOT at, when we've
just measured the altitude where our ship IS at.  (For politically
correct reasons, I'm not using the name of this location.)

Now lets go back to Sumner's 1837 calculation, where he picked three
different longitudes and calculated three points on the circular LOP.
This calculation is exact, and the equation for each point is the same
as the one of the two necessary in the St Hilaire method (thus each
Sumner point is half the work of a St Hilaire reduction).  And he
could calculate as many exact points as he wished.

So I'll put my question yet another way:  Why is the St Hilaire method
superior to Sumner's and consequently the only one used today??

I claim that the answer to this question has been made confusing
because of the conventional name (names?) used for the location of the
St Hilaire altitude calculation.  As evidence of this confusion I note
that some authors write that we need to assume some point because the
distance between the GP and the LOP is too great to plot, that there's
insufficient information to plot the LOP, or that iterations are
required to get exact points on the LOP.  The Sumner calculation
demonstrates that none of this is correct.

JK
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