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    Re: Advancing position circles: Huxtable vs. Zevering
    From: Herman Zevering
    Date: 2010 Feb 25, 20:37 -0800

    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 north:

    (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 data
    (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 navigation.

    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 technique.

    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 technique?
    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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