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    Re: Poor St. Hilaire
    From: John Karl
    Date: 2007 Oct 16, 09:42 -0700

    John Karl wrote:
    | In my English translation of his
    | 1873 publication...
    George H. replied:
    > When he says "my" translation, does that imply that he has translated it? Or
    > is it John referring to his copy of  a translation, perhaps the one by
    > Michel Vanvaerenbergh, in the book he wrote with Peter Ifland, "Line of
    > Position Navigation".
    I'm referring to the translation that I have, which is the above
    Vanvaerenbergh & Ifland's.
    JK wrote:
    | Therefore no sight
    | reduction method uses estimations or assumptions.  Again, it's the
    | terminology that screws up the concepts.
    GH replied:
    > I don't understand why that follows. There are all sorts of sight reduction
    > methods making all sorts of estimations and assumptions.
    I'm speaking of a priori estimations and assumptions, such as the
    "assumed position", which, in fact, is not an assumption at all.
    Rather it simply specifies a region of interest where the navigator
    wishes to compute a point on the celestial LOP. This point on the LOP
    is computed exactly, with no estimations, no assumptions.  He can
    compute as many of these points as he wishes.  But if instead he
    decides to draw a straight line tangent to the LOP at this point, he's
    making an approximation for computational convenience (with known
    errors, in fact).  OK, we can also call that approximation an
    estimation, but that's in a different sense; it's not a required
    estimation of some quantity peculiar to that sight.  So maybe I
    shouldn't attempt to use "estimation" in too different senses; but in
    any case, there are no assumptions in the process.
    Because the LOP is determined exactly from the observed altitude and
    the body's GP, no assumptions are necessary for any sight reduction.
    And I known of none that makes assumptions.  Approximations, yes.
    JK wrote:
    | There are several ways to determine the exact LOP by directly
    | computing and plotting latitudes and longitudes of as many points
    | we wish (Sumner's being one such method).  But St. Hilaire is
    | different -- it doesn't compute Lats and Lons at all.
    GH replied:
    > That's exactly what his 1873 paper, "Note sur la determination du point", IS
    > doing! In section VI, he assumes a nunber of different lats (or longs), and
    > deduces the corresponding longs (or lats), using the formulae in section
    > VII. If there are enough such points, he can draw an exact curve through
    > them. If there are only two, he can draw a chord between them. And if
    > there's only one point, obtained by using his estimated lat (or estimated
    > long) he can work out, using the formulae in section VII, the corresponding
    > long (or lat) and the azimuth, the position line being taken as the line
    > through that position at right angles to that azimuth. And the precision of
    > that approach depends on how good is his initial estimate, for one of those
    > coordinates.
    I'm sorry. I used the shorten "St. Hilaire" for the St. Hilaire
    intercept method.  It does not compute lats and Longs, but determines
    them graphically from the computed intercept distance and azimuth.
    The method George describes above of specifying a latitude and
    computing the corresponding longitude is not the St. Hilaire intercept
    method.  This method does (obviously) compute the lat and lon of a
    point on the LOP.  Now here's the confusion wrought by terminology:
    The terminology traps George (and many others) into thinking that the
    "estimated" latitude generates an approximation whose error depends on
    how good the estimate is.  Not so.  The point's location is exactly on
    the LOP, no matter what initial latitude is specified (providing
    there's a solution at all).  When it comes to approximating the LOP
    with a straight line, the error depends only on the curvature of the
    true LOP relative to the distance along the straight-line LOP.  The
    error does not depend on the initial latitude.
    So it seems to me that it would be better to call this "estimated"
    latitude something else, maybe a specified latitude, or a reference
    latitude.  Likewise for the "assumed" position.
    JK wrote:
    | Rather, it
    | specifies a point's latitude and longitude which I'll call the
    | reference point, RP, even though conventionally it's called the
    | "assumed" position -- a misnomer.  St. Hilaire calculates the
    | circle distance and azimuth from this RP to the nearest point on
    | LOP.  This point is exactly on the LOP; let's call it the St.
    | point (SHP).
    GH replied:
    > It's rather strange: John and I seem to be referring to different papers
    > here. I just can't find any of that in the 1873 paper I have referred to.
    > Perhaps John could give section and page numbers. If he is working from the
    > same translation as I have, it follows the original pagination.
    It starts in the section called "Calculation of a Single Observation"
    on page B2-345. In the second line he introduces the "estimated
    position" which is called the assumed position in all the modern
    treatments I've seen.
    John Karl
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