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    Re: AP terminology, WAS: 2-Body Fix -- take three
    From: Tom Sult
    Date: 2009 Nov 13, 19:57 -0600

    Thanks... I think that is what I tried to say a few days ago ; )
    Thomas A. Sult, MD
    IntegraCare Clinic
    On Nov 13, 2009, at 7:05 PM, douglas.denny@btopenworld.com wrote:
    > 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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