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    Re: Advancing position circles: Huxtable vs. Zevering
    From: George Huxtable
    Date: 2010 Feb 26, 10:42 -0000

    My heart sank, somewhat, on reading yesterday's posting from Herman 
    Zevering. I had hoped we had put all such stuff behind us, following an 
    exchange in Journal of Navigation, back in 2006. But here it is again.
    Generally, I would read through a posting with some care, especially one 
    that criticised my own work, before responding. But in this case, I have 
    done no more than scan through it, to see the sort of stuff that's within. 
    Instead, I will take another tack, as follows-
    My piece in the Forum section of The Journal of Navigation, vol 59 no 3, 
    September 2006, "An erroneous proposal to allow for travel of an observer 
    between two celestial altitude observations", pointed to errors in a 
    contribution from Zevering, in vol 59/1, "Dependability of position 
    solutions in celestial sight-run-sight cases, Part 1. (Part 2 never 
    To test his procedure, I proposed a simple example, as follows, which 
    presumed a spherical Earth-
    1. An observer, at position P1, measures the altitude of a star S1, at (Dec1 
    = 0º, GHA1 = 0º), to be 30º.
    2. Then he travels due North by 60 nautical miles,  ( = 1º ) to P2.
    3. From there, he observes another star S2 (then at Dec2 = N 1º, GHA2 = W 
    45º) to be at an altitude of 45º.  Where on Earth is he then?
    I showed that there were two solutions for P2.
    One was in the Northern hemisphere, at Lat N 46º, Long W 45º, in which case 
    P1 was 1º further South, at Lat N 45º, Long W 45º.
    The other was in the Southern hemisphere, at Lat S 44º, Long W 45º, with P1 
    situated 1º further South, at Lat S 45º, Long W 45º.
    Navlist members can check that those solutions meet conditions 1, 2, and 3 
    of the example precisely, by computed altitude tables or by spherical 
    geometry, as they prefer.
    I then challenged Zevering to apply his proposed procedure to that problem, 
    and provide us with initial and final positions for the observer, the 
    initial altitude of star S1, and the final altitude of star S2. My own 
    estimate was that his final position would be 17' in error, in longitude, 
    after a travel of only 60 miles
    Zevering's response was printed in that same issue. It was in the same 
    rambling style that readers will recognise from yesterday's posting. But 
    there was no numerical response to that challenge.
    So I say this to Herman Zevering- Provide us with your answers to that 
    problem: then, and only then, will I take any of your writings seriously.
    contact George Huxtable, at  george@hux.me.uk
    or at +44 1865 820222 (from UK, 01865 820222)
    or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
    ----- Original Message ----- 
    From: "Herman Zevering" 
    Sent: Friday, February 26, 2010 4:37 AM
    Subject: [NavList] Re: Advancing position circles: Huxtable vs. Zevering
    This is a belated response to Philip Bailey's comments put on NAVLIST 1311 
    "Advancing Position Circles: Huxtable versus Zevering" (for the actual 
    argument see (UK) JoN, 59/Forum/p 521-529).
    To fully understand this issue and also benefit from a programmatic approach 
    and a complete introduction into spherics, just buy my book "Celestial 
    Navigation - A Modern Vade-Mecum on Determining the Position of a Vessel 
    (2008)" [KHZ-CN], 331 pages, through Boatbooks Australia.  It comes with a 
    CD containing sight reduction and sight-planning spreadsheet programs in 
    both Lotus123 and Excel.  Especially Part II-App 1 "The running fix issue", 
    App 2 "The Large-Alt case" and App 9-4 "The general equation of a position 
    circle on the chart" are directly relevant.  ANM stands for Admiralty 
    Navigation Manual, 1937.
    Rather than going over the same ground as in Forum, it is perhaps more 
    informative to highlight aspects of RFT (coastal running fix technique 
    applied in celestial navigation;Huxtable) versus GD-UT (GHA & Dec updating 
    technique; Zevering) which were only summarily or not at all covered in the 
    Forum articles, namely:
    (i) the fundamental significance of the General Equation (GE) ("the general 
    equation of the curve on the chart that represents the position circle, the 
    curve itself being defined by the values 'a' (altitude), 'd' (declination) 
    and 'X' (GHA)", ANM, Vol III, p 39-43)
    (ii) the surrogate nature of the RFT transfer
    (iii) the Museum for celestial navigation artefacts
    (iv) some background on the term 'GD-UT', on 'terrestrial transfer analogy', 
    the 'Large-Alt' case
    (v) the absence in the ANM of any demonstration of the general applicability 
    of RFT to celestial navigation
    (vi) Huxtable's rejection of the general validity of the demonstration of 
    GD-UT in the ANM, Vol III, p 43-46
    (vii) Possible reasons that GD-UT was not adopted in the ANM as generally 
    applicable transfer technique.
    The GE
    A position circle is defined by its GHA and Dec (the coordinates of its GP), 
    and 90°-Alt (its zenith distance or radius).  Any part of it can be plotted 
    point by point on the (Mercator) chart in the neighbourhood of the DR 
    position from which its GHA, Dec and Alt were established.  This is done 
    through the operational version of the GE.  Also here I skip its 
    technicalities, equations and the inherent application of the meridional 
    parts formula (see KHZ-CN, App 9-4).  Position circles plotted on 
    small-scale charts of the globe do not appear as circles but as what I call 
    "quasi-circles", "paraboles" or "waves".  Examples are found in ANM Vol III, 
    Figure 24, p 43.  The DR position is only relevant as an indication of the 
    likely Lat-Long delimiting the area on the chart in which the observer must 
    have been and in which the plot is likely to fall.  The DR position, 
    including its presumed (in)accuracy, is irrelevant to the actual position 
    solution, whether by the GE, tradional methods or modern programs, and on 
    sail boats or ocean liners.
    In the pre-electronic era the computational capacity to implement the GE 
    approach in practical navigation was simply not available.  The Intercept 
    Method (IM) circumvents the GE's computational complexity by determining one 
    point (point J) on a position circle (PC) via an assumed (DR) position and 
    two computed parameters, Zn and p, and drawing a tangent through this point, 
    the position line (PL).  Again, the DR position is only relevant for 
    constructing point J and the PL through it;  J doesn't determine a position 
    (fix).  Assuming another DR position would only determine a tangent PL in a 
    different point J.  The curvature of the PC in the neighbourhood of J is in 
    practical terms negligible when the zenith distance is large as it normally 
    is.  Hence, the tangent through J is assumed to represent a segment of the 
    PC.  This assumption and its method of construction form part of the 
    "Assumptions made when the Position Line is Drawn" (ANM, Vol II, p 135). 
    The latter assumptions represent no more than a rather ponderous statement 
    that the entire construction proceeds in plane geometry although it relates 
    to a surface of the globe.
    Assumption (1), "The bearing of the geographical position is the same at all 
    points in the neighbourhood of J" is technically incorrect as the (computed) 
    bearing or Zn belongs uniquely to the assumed DR position.  This assumption 
    reflects a justification for the use of the tables with the assumed 
    position.  The Zn from the DR position and from the assumed position derived 
    from it may differ as much as .5°, yet the points J they define are in 
    practical terms regarded as lying on the same PL.  The ANM (Vol II) devotes 
    considerable space to this subject with worked examples, but the whole 
    subject is actually trivial as the DR position is irrelevant for determining 
    a fix, as already indicated.  The importance attached to the subject in the 
    ANM also derives from the fact that the DR position determined from the 
    bridgedeck of naval steam ships through deadreckoning is regarded as always 
    close to the actual position.
    What the above expose leads up to, however, is that regardless whether one 
    uses the GE or the IM to plot (part of) a PC on the chart this can only be 
    done correctly when its parameters are known, i.e the coordinates of its GP 
    and its Zd as circle radius.  This applies to any PC and therefore also to 
    the one transferred for a run between sights. To effect such a transfer on 
    the chart it should be possible to specify the new coordinates of the GP 
    which can only move according to the displacement of the observer on the 
    globe indicated by the run data.  As stated in so many words (see below) and 
    demonstrated in the ANM Vol III (p 43-46) with the worked example of three 
    successive Sun sights (the Large-Alt case) the new coordinates are to be 
    found by adjusting the old GP coordinates for the run of the vessel, with 
    the rhumbline formulas.  This is the GD-UT principle.  The observed zenith 
    distance (radius) is unaffected in the transfer.
    The surrogate RFT transfer
    The transfer of the PL through J parallel to itself for the run to J* with 
    RFT cannot reliably represent a segment of the transferred PC.  To begin 
    with, with the RFT construction it is actually impossible to mathematically 
    specify where the GP of the supposedly transferred PC has actually shifted. 
    This can only be done as in my Forum article when one of the three 
    parameters defining the PC can be held constant, which is the GHA in the 
    simplified (Due-North) model of Fig 2 used in the article.  When applying 
    RFT transfer it is then seen that a PC fitted through two arbitrary points 
    J* and J'* (which do not happen to lie on the constant meridian of the 
    due-North model), so that JJ* and J'J'* equal the distance of the run due 
    (a) does not have the same parameters (GHA, Dec and Ho) as the original PC
    (b) cannot specify a shift of the original GP in accordance with the run 
    (c) has a Zd which differs from the observed Zd.
    (d) cannot account for (i.e does not go through) any third point J" 
    transferred with RFT to J"* for the same run data.
    The PCs fitted through other pairs of points J* and J'* will all have 
    different parameters.  In fact, not only has the implied PC transferred with 
    RFT and going through two points J* and J'* the wrong parameters, it will 
    also not go through any other point presumed correctly transferred with RFT. 
    In addition, with the double-sight due-North model (Fig 3), a 2nd sight 
    cannot confirm the correctness of any PC transferred with RFT.  The result 
    of the distortions introduced by using RFT is of course, that the PL through 
    J transferred with it will not intersect a later sight's PC in the correct 
    spot; hence the fix will be incorrect.
    Findings like these through quantitative models also nullify another 
    'self-evident' principle traditionalists presume to exist, namely that a 
    correctly transferred PC guarantees that any point on it 
    'backward-projected' for the run must lie on the original PC.  The origin of 
    this belief is that this principle applies in coastal RFT and it is 
    therefore assumed that it also applies in celestial navigation.
    It is impossible to quantify for actual double-sight cases what the bearing 
    from the fix is to the new GP of the 1st sight supposedly transferred with 
    RFT.  It is therefore also impossible to determine where the original GP is 
    supposed to have been transferred to.  But it is possible to demonstrate 
    with actual double-sight cases that a PC fitted through the positions 
    backward-projected from the two alternative fixes cannot be the original 
    earlier sight's PC.  This is because such a PC would have a distorted Zd. 
    As we have seen, a correctly transferred PC must minimally have the same 
    (observed) Zd.   Thus with the Moon-run-Sun case in the ANM, Vol II, the 
    original PC would need to have had a Zd of 72°.8551, whereas the observed Zd 
    is 72°.5883 (see KHZ-CN, Part II: App A1-4).  In other words, the Zd would 
    have had to mysteriously expand in the transfer process.  The same problem 
    can be demonstrated for any double-sight case in which the transfer is done 
    through RFT (or LSQ*, which incorporates RFT).  It is because of the above 
    type of distortions that I say (last sentence of my Forum article): "There 
    is no theoretical explanation for this".
    This change or distortion in observed Alt or Zd  should not be confused for 
    the transfer method I denote as A-UT (altitude updating technique), which is 
    based on a 'surrogate' altitude or zenith distance (KHZ-CN, II:14-1). A-UT 
    has been applied in several versions by G. Keys and G.G. Bennett for finding 
    a fix with a double sight.  As with RFT, this transfer technique assumes 
    that a PC can be transferred by transferring a position on it for the run 
    data.  If X is the GP and J* the transferred point J on the earlier sight's 
    PC, it is then thought all right to use 90°-XJ* as (surrogate) Alt in 
    solving the double sight.  But application of A-UT introduces its own 
    position-fix anomalies.
    Incidentally, our own G.G. Bennett almost hit upon GD-UT with his A-UT 
    Method B (see Nav Newsletter no. 86, 2005), by shifting the GP of the 
    earlier sight along its parallel of declination through an arc equal to 
    d'Long.  But he curiously fails to shift the Dec of the GP in the required 
    d'Lat direction.  The manipulations nullify the validity of A-UT Method B, 
    which otherwise would have just emulated the GD-UT transfer principle (see 
    KHZ-CN, Part II:14-3.1).
    The Museum
    A-UT belongs in a Museum for theoretically incorrect or at least no longer 
    warranted and outdated methods in celestnav, in company with RFT, Meridian 
    Passage, Pole Star, SAC (star altitude curves), the Baker navigation 
    machine, etc.   Next to the "world's first sextants" and the astrolabe, the 
    Museum displays other archaic methods invented in the pre-electronic era, 
    most notably historical versions of sight-reduction tables, right up to the 
    'modern' concise tables found in the Nautical Almanac and those in G.G 
    Bennett's "Celestial Navigator".  The concise tables are as curious as the 
    cuneiform characters of the Sumerians and the hieroglyphs.  One finds 
    caveats in other contexts, for angles of cut for instance to reduce certain 
    observation errors.  Visitors wonder why so many decades after powerful 
    computing electronics became virtually a household article, inventors still 
    found it necessary to subject navigators to a process of blindly extracting 
    numbers from tables and adding or subtracting them.
    Other tables on display in the Museum are Azimuth tables, Ex-Meridian 
    tables, Traverse tables and many others.  Captain Weir's Azimuth Diagram and 
    of course G.G Bennett's version adapted from it in his Celestial Navigator 
    would be there too.  The Museum's information sticker mentions that G.G. 
    Bennett considered the "Weir diagram" to be far more accurate than going 
    through the four "steps", the "rounding off" and other procedures required 
    with his own invention of Azimuth tables in "Celestial Navigator". 
    Displayed from the same work are other ingenious but crude diagrams to 
    extract data from, like from the "Lat of Sunse/Lat of Sunrise" diagrams. 
    The diagram which attracts most attention, however, is the "change in 
    altitude in 5 minutes of time".  To show how navigtors in those days were 
    given to extracting computation results from tables and diagrams the Museum 
    guide writes the formula 15'CosLatSinZn for the change in altitude per 
    minute (see ANM, Vol III, p 145) on a blackboard and enters combinations of 
    Lat and Zn on the calculator he pulls from his pocket.
    Also certain publications on what celestial navigation is all about 
    theoretically and how to go about it practically are on display in the 
    Museum.  Visitors will for instance find a copy of all the editions of an 
    iconic work like M. Blewitt's "Celestial Navigation for Yachtsmen" there. 
    As the English Museum guide explains proudly, for more than half a century 
    no one felt it necessary to revise any of this work's ideas and 
    applications, such was the awe in which the author was held all this time, 
    her last functions being Secretary of the R.O.R.C, Chairman of the R.R.C of 
    the Royal Yachting Association.
    After the author had passed away others were entrusted with the task of 
    updating the book's standard example of the seven star sights.  As the 
    Museum guide explains, this was a recurring ritual in all editions.  The 
    maximum number of star sights supported by AP 3270 Vol III for a given LHAγ 
    is seven and the navigator had to shoot them all.  As required by 
    long-standing expository tradition, one star sight and its PL had to be 
    eliminated as being "of little help".   Even as a desk-chair exercise the 
    author found the plotting "a long and tedious business ... a degree wrong in 
    plotting an azimuth makes a sizeable error if there is a big intercept". 
    The sights had to form a consistent configuration: the azimuths of all PLs 
    had to either point in to or point out from a cocked hat area for locating 
    the fix.  If the navigator didn't get this his plot was useless, but it was 
    then too late to take seven other twilight shots.
    The person entrusted with the updating of the 11th edition of 1997, reissued 
    in 2004 made a complete mess of the usual plot at Fig 35 of the book.  A 
    former Director, M.W. Richey, of the Royal Institute of Navigation paid a 
    glowing tribute in its "Forewords", one of them being by the captain who had 
    since long deceased.  Mike Rickey mentions the author's "severely practical 
    aim", a "standard work" that withstood "the test of time".  The work is a 
    curious blend of theoretical misperceptions and impractical prescriptions, 
    eminently suited to acquiring Museum-status.
    Some authors even pretended they had hands-on experience in taking seven 
    star sights at sea, getting the same contrived consistent configurations. 
    One such author is T Cunliffe.  In his booklet "Celestial Navigation", also 
    on display, he even lambasts fellow navigators: "Most people who dabble in 
    celestial navigation are under the mistaken impression that star sights are 
    a problem."
    Although of more recent vintage, also G.G. Bennett's valiant Celestial 
    Navigator is well on its way to acquire iconic status, if only as a reward 
    for the author's outstanding efforts in avoiding the use of the simplest 
    electronic computations, all for the benefit of the dummies in celestial 
    Some background
    Transferring the GP of a PC is the easiest way to transfer a circle in plane 
    geometry, by transferring its centre and using its known radius with a pair 
    of compasses to draw the new circle.  It resembles the spherical 
    construction principle I called GD-UT.   If the Zd is very small (very large 
    altitude), GD-UT and RFT become practically equivalent methods, which I 
    later presented as and called 'the terrestrial transfer analogy'.   At the 
    time, Huxtable simply avoided commenting on the drawing concerned and its 
    obvious implications, claiming in email to the editor of the Navigator's 
    Newsletter that his computer wouldn't open my drawing, but that all my ideas 
    were erroneous anyway.
    Meanwhile I had discovered that what I had called GD-UT is actually 
    demonstrated in the Large-Alt case in the ANM, Vol III.  As discussed below, 
    Huxtable in his increasingly quixotic attempts to stymie the use of GD-UT 
    dismissed the general principle of the demonstration in the ANM of GD-UT.
    GD-UT versus RFT is in fact a pseudo-argument, brought about by a 
    tradionalist Klan that still relies on sight-reduction tables and 
    voluntarily do turns at the Museum.  This pseudo-argument can only be 
    resolved through a review by specialists who are no longer shackled by the 
    pre-electronic lack of computational capacity and have an open mind. 
    Equally needed is a thorough, specialist review and revision of some archaic 
    stuff contained in the ANM itself.
    The UK RIN obviously doesn't see itself in any kind of role as custodian of 
    the body of knowledge contained in the ANM and apparently lacks any 
    (in-house) expertise for such a role.  Its present Director, David Broughton 
    has so far steadfastedly avoided even to acknowledge the receipt of a free 
    copy of my book.  This is perhaps related to my irreverent attitude towards 
    Mary Pera.  Who knows what he has done with it.  So before you wander into 
    the UK RIN at Kensigton Gore in London and ask the librarian to have a look 
    at this copy, ring them first.
    The presumed general applicability of RFT in the ANM
    I come back below to the question why GD-UT wasn't applied in the past in 
    all sight-run-sight cases.  In the pre-electronic era this approach had its 
    own difficulties and the ANM authors must have cast about for a short-cut 
    transfer method and settled for RFT.  It is hardly imaginable that the 
    composers of ANM were not aware of RFT's implications and theoretical 
    shortcomings.  The ANM abounds with detailed proofs and demonstrations of 
    the methods it covers.  An exception is the section "Accuracy of Almanacs 
    Tabulated to One Minute of Arc" (ANM Vol II, p 122; see KHZ-CN, A11-2).  The 
    other exception is RFT applied to celestial navigation.  The entire manual 
    only includes two apodictic statements, both just stressing the the general 
    applicability of RFT (Vol II, bottom p 62 and bottom p 191). 
    Traditionalists  can therefore hardly be blamed for rigidly following this 
    Even in the pre-electronic era it should have been possible to prove 
    minimally that in the general case (large Zd) the GP of the PC transferred 
    with RFT does not transfer in accordance with the run data, which in itself 
    refutes the RFT's general applicability.  Curious in this respect is that 
    the application of the IM is surrounded with (basically trivial) 
    "assumptions". One finds caveats in other contexts too, for angles of cut 
    for instance to reduce certain observation errors.  But there are no 
    assumptions to hedge or cushion the general applicability of RFT.
    Although this would have been extremely laborious, it is very well possible 
    that the worked example of Moon-run-Sun case in Vol II (p 191-201), already 
    mentioned, is chosen (doctored!) so that the RFT application yields a fix 
    which is close to the fix that would have been obtained had the relevant 
    segments of the transferred Moon's PC and the Sun's PC been plotted with the 
    GE.  Thus we get in the ANM for the fix in the Northern Hemisphere 50°.4917 
    N/13°.8383 W, with LSQ* which incorporates RFT 50°.4918 N/13°.8418 W.  The 
    LSQ* fix and the ANM fix should be practically the same.  With GD-UT+K-Z it 
    is 50°.5117 N/13°.8323.  The discrepancies are relatively small.  But for 
    the alternative fix in the Southern Hemisphere the discrepancy between the 
    LSQ* fix, 59°,0949 S/85°.8293E and the GD-UT+K-Z fix, 59°.3735 W/85°.1076 E, 
    is very large.  This test with GD-UT+K-Z which can also be done by applying 
    the GE shows that RFT is not generally valid.
    The fix constructed with RFT in the Polaris-Mirfak-Venus case, which is 
    another non-simultaneous case in the ANM (Vol II, Fig 135, p 201) would 
    strike the traditionalist as self-evident, but it can be demonstrated 
    mathematically that the PL of Polaris does not transfer north as assumed in 
    this construction.  The fix constructed with RFT is some 3' too far North. 
    (see KHZ-CN, A1-7).  The Moon-run-Sun and Polaris-Mirfak-Venus examples are 
    part of the archaic baggage found in the ANM.
    Huxtable's dismissal of the general validity of GD-UT
    G. Huxtable dismissed the GD-UT transfer procedure demonstrated in the ANM 
    with the Large-Alt case as being a "special" case only: .   With his usual 
    aplomb he wrote:
    "Yes, under some circumstances, when you shift a position circle over the 
    globe, every point on the periphery of that circle will move approximately 
    through the same course and distance as its centre does.  Those special 
    circumstances are spelled out in the ANM:
    -The radius of the circle must be small
    -The observer and the body must be in reasonably low latitudes.
    The example on pages 43-45 of the ANM (Vol III) has been chosen with these 
    points in mind."
    To start with, the worked example in Vol III is connected with the earlier 
    mentioned "Assumptions" (Vol II, p 135): the ANM specifically excepts the 
    Large-Alt case from the general applicability of the IM construction because 
    the relevant position circle segments would be curved (on the chart!) and 
    thus can no longer be represented on the chart as straight-line tangents.
    Remarkable is that Huxtable obviously considers the Large-Alt case "special" 
    a.o because the GP of the PC moves practically in the same direction for the 
    same distance as any point on its periphery.  This is of course the 
    'terrestrial transfer analogy' again.  Indeed, it is a limiting case where 
    RFT happens to apply in practical terms.  The conclusion should have been 
    that this is the only instance in which RFT may be substituted for GD-UT!
    It is true to say that there is no mention anywhere in the ANM that GD-UT 
    and RFT as transfer techniques in practical terms converge when Zd becomes 
    large.  To obtain a fix with the three successive Large-Alt Sun sights it is 
    in fact unnecessary to apply the GD-UT construction demonstrated in the ANM, 
    as the fix in this case would be practically identical if the IM would be 
    applied with RFT.  The seemingly inordinate attention given to the Large-Alt 
    case in the ANM indeed stands in sharp contrast to the total absence of any 
    theoretical justification for the (incorrect) application of the RFT 
    transfer in the general case when zenith distances are large.  Taking 
    Large-Alt sights stands also in contrast to the skepticism and caveat 
    expressed in the ANM itself regarding the observation by sextant of meridian 
    passage (Vol II, p 167).  In fact,  there are many good reasons for 
    recommending that successive Sun sights should be taken before and after 
    noon at advantageous azimuth angles between them.
    It is therefore quite obvious that the Large-Alt case dealt with in the ANM 
    is foremost intended to demonstrate the GD-UT principle when the 
    (approximate) application of the GE is within reach of the traditional 
    navigator.  The method of constructing the Large-Alt cocked hat through 
    GD-UT is only special in this latter sense and also in representing the 
    intersecting postion circles as circles on the chart.  They are at best 
    "quasi-circles" (ellipses).
    However, Huxtable misses the important points made in the Section "Error 
    Introduced by taking the Position Circle as a circle" (ANM, Vol III, p 
    45-46).  First,  the reference to "low latitude" refers to the fact that a 
    Large-Alt case can only occur whenever the Sun's declination and the 
    latitude of the observer practically coincide, for in this situation the Sun 
    is nearly overhead.  As indicated in Vol III, the distortion introduced by 
    representing a (Large-Alt) PC as a circle on the chart increases with the 
    latitude (Dec) of its GP!  As the construction with a pair of compasses is 
    the only realistic application, the ANM is at pains to demonstrate that it 
    can be used in all Large-Alt cases without introducing signficant errors. 
    Thus, for an altitude of 88.5° at the Sun's maximum declination of 23.5° the 
    distortion of the PC drawn as a circle is said to be only 0'.6 along its N-S 
    axis.  For the same reasons it is stated further that when the Zd is 
    measured as radius on the chart's latitude scale midway between the 
    declination of its GP and the latitude of the DR position, even for 
    altitudes of the Sun between 82° to 86° and maximum declination, the 
    distortion is reckoned to be as little as between 0'.7 and 0'.1.
    Furthermore, it is impossible to construe the statement in the ANM ("If the 
    observer is in a ship and there is a run between sights, the first position 
    circle must be transferred for the run.  This can be done by transferring 
    the geographical position and then drawing the circle.") as anything else 
    than a statement of the general (GD-UT) transfer principle.  The worked 
    example of the Large-Alt case in the ANM shows that this principle is based 
    on transferring the GP of an earlier sight for the run data with the 
    rhumbline formulas.  Moreover, this statement appears in the context of 
    Chapter IV (THE MERCATOR CHART) in Vol III and more specifically in the 
    context of the derivation of the GE.  In this connection it is also 
    significant that the reference found in the Index of Vol. III is to 
    "Transference of a position circle on a Mercator chart".  There is not a 
    word to suggest that this transference (which is based on the GD-UT transfer 
    principle) is applicable only in the Large-Alt case!
    We thus have here in fact two generally valid principles demonstrated in the 
    same context, the GD-UT principle and the principle mentioned in the 
    preceding statement in the same section: "When two observations of this kind 
    are taken, two position circles may be drawn, and the observer's position is 
    at one of their two points of intersection".  Again, these are undeniably 
    references to intersecting position circles as the basis of celestial 
    navigation, whether two sights are simultaneous or non-simultaneous.  The 
    words "of this kind", wrongly assumed to mean "special", only make a 
    provision for the representation in the Large-Alt case of the PC constructed 
    on the chart as a circle (which is an approximation)!  The error is rather 
    fastidiously shown as indicated above as being practically negligible in the 
    section "Error Introduced by taking the Position Circle as a Circle".
    Why was GD-UT not adopted in the ANM as generally applicable transfer 
    One can only guess, but given the completeness of celestial navigation 
    theory in the ANM it is simply impossible to maintain that this was done 
    because RFT is the correct transfer technique or that the authors of the ANM 
    believed it to be so.  The GD-UT transfer is possible by applying the GE, 
    but the lack of computational capacity in the pre-electroninc era would have 
    made this extremely cumbersome and impractical.  For the same reason the ANM 
    authors were basically bereft of the necessary computational capacity to 
    study or test the effects on the fix by applying RFT instead of GD-UT, for 
    example in the manner I have done, using case analysis.  It was just assumed 
    without any theoretical justification that the errors introduced in the fix 
    by using RFT would be acceptable.
    Using the traverse table for transferring the GP of an earlier sight for the 
    run data was in principle possible.  Had this been a practicable method, 
    every sight-run-sight case could in the past have been converted first into 
    an equivalent simultaneous sights case prior to applying the traditional 
    sight reduction methods with the tables.  But also this approach had rather 
    severe limitations.  The traverse table was quite possibly not considered 
    for this purpose because it is both too inaccurate and complicated.  For 
    instance to determine d'Lat and Dep, Inman's traverse table could only 
    handle distances in whole minutes and courses in whole degrees and would 
    have required rather cumbersome interpolations of the navigator.  To apply 
    Inman's Tables to the transfer problem it is necessary to find d'Long after 
    first computing Dep (a simple procedure with the rhumbline equation and a 
    calculator), a process described in the ANM as:
    "..it is solved by searching until, against a course-angle equal to the mean 
    latitude, the number equal to departure is found in the cosine part of the 
    distance column..(this) distance is d'Long." (Vol II, p 26).
    There would have been two further problems with the Inman's Tables.  One is 
    that the compass course (rounded to a whole degree) must be specified for 
    the relevant quadrant first.  For instance, a course of 115° would be 65° SE 
    (or S65°E), so that d'Lat (S) is algebraically negative and has to be 
    subtracted; Dep (E) is positive so that d'Long has to be added.  But to 
    express d'Long as a change in GHA its sign has to be reversed.  The 
    navigator must remember to deduct a positive d'Long from GHA and vice versa. 
    It is not difficult to see that the steps required in the past would not 
    only have been involved but also prone to inaccuracies and error, even if 
    the GHA of only one earlier sight needed to be updated for the run in this 
    manner.  There is also the theoretical complication of the half-convergency 
    adjustment of the run distance.  As I have shown, the effect of not allowing 
    for it is negligible but again, in the pre-electronic era this was not easy 
    to assess.
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