1. THE FIX AS INTERSECTION OF POSITION CIRCLES

A fix is determined by the intersection of at least two position circles. There is no argument when sights are simultaneous. Two transfer techniques are demonstrated in the ANM when sights are not simultaneous, the running fix technique (RFT) and the GHA-Dec updating technique (GD-UT). With RFT/LSQ*Footnote ¹ the transferred position line is thus supposed to represent a transferred position circle as the mathematical locus of all points on the original position circle transferred for a given run. This condition is met with GD-UT but as we will see it is not with RFT/LSQ*.

2. THE TERRESTRIAL ANALOGY

The model for celestial RFT is terrestrial RFT (e.g ANM Vol.II, p 62, p 191). Points A and B in Figure 1 represent landmarks with known height, but also the geographical positions (GP) of two celestial bodies with large altitudes. Assume in the terrestrial case the distance to A and to B is determined by vertical sextant angle; no bearings are used. In the terrestrial case just draw a transferred position circle with radius r_A from A′ and another position circle with radius r_B from B. In the celestial case, as r_A and r_B are short zenith distances, AA′ virtually equals d and the angle at A the course (α). The fix F is at the intersection of PC*₁ and PC₂. Had only the bearings been used the fix would also have been in F, assuming that the celestial azimuths could be observed. In the celestial case fix F is therefore found with GD-UT by transferring the 1^st sight's GP at A to A′ for the run data and plotting the relevant sections of PC*₁ and PC₂ (with the general equation in ANM, Vol.III, p40).

Figure 1.

The GD-UT principle of terrestrial RFT.

In Figure 1 the observer could be at J or J′ and JJ*=J′J′*=AA′=d. The RFT/LSQ* construction in this case is equivalent to moving a run track like JJ* until it coincides with DF, i.e the position backward-projected from the fix lies exactly on the original position circle. In this case, terrestrial RFT, celestial RFT and GD-UT are equivalent techniques. But in the general celestial case (large zenith distances) it is incorrectly presumed that DF remains sufficiently equivalent to AA′ and A′D parallel to A′F.

3. THE POSITION CIRCLE LOCUS OF TRANSFERRED POINTS

Huxtable's present due-NFootnote ² double sight example is a variation on his earlier attempt at refuting the general validity of GD-UTFootnote ³. Three vessels travel the same distance north from known points A, B, C on the same position circle (see Figure 2). The terrestrial analogy is clearly recognizable: the distances AA*, BB*, CC* are equivalent to JJ*, J′J′*, DF in Fig-2. It is evidently expected that the transferred position circle should pass through A*, B*, C*, like the transferred position circle in Figure 1 passes through J*, J′*, F, as the RFT precondition is that positions backward-projected from fixes like A*, B*, C* must lie on the original position circle. At the cardinal points A and C, GD-UT virtually achieves the correct transfer, but X*B*=59°.7564, 15′ too short: “..the Zevering procedure (i.e GD-UT) … assumes that the transferred position circle defines a locus which includes all such vessels and looks for its intersections with another circle. But that assumption is wrong.”Footnote ⁴

Figure 2.

Sketch of cases A, B and C.

But the argument remains inescapably about the correct transfer. If GD-UT is seen as failing to correctly define the transferred position circle as locus of all transferred points, can RFT/LSQ* do this? A*, B*, C* are seen as correctly transferred positions so that a transferred position circle through two such points should correctly account for any vessel's destination. B* and C* lie on a transferred position circle with GHA=0/Dec′*=0·7123/H′_o=30·0068Footnote ⁵ but it will not pass through any other point with a backward-projected position 1° to the south on the original position circle; e.g AA*=0°.7055, considerably short of 1°. As is seen, in the large-Zd case the run of 1° at B, C would imply a smaller GP transfer of 0°.7123 and a (negligible) contraction in Zd. The peculiarities of this transfer are nowhere supported in celestial navigation theory.

More revealing is the displacement at cardinal points A and C. The transferred position circle through A*, C* is GHA=0/Dec′*=1·0102/H′_o=30·0102, practically the same as the position circle transferred with GD-UT (GHA=0/Dec=1/H_o=30). The position circle transferred with RFT would be regarded as correct as the positions backward-projected for the run of 1° are exactly at respectively A and C. If the latitudes of the transferred positions were known, say Lat_A*=61 and Lat_C*=1, RFT would determine the positions at A* and C* as 61°N/0° and 1°N/60°E, GD-UT+K-Z as 61°N/0° and 1°N/60°.0101E.

If the transfer with GD-UT at any position circle cardinal points is correct, would GD-UT not account in a general way for the correct transfer? The position circle transferred with RFT (GHA=0, Dec′*=1·0102 and H′_o=30·0102) cannot account for the displacement from B to B* on Long=45°E as BB* would be 1°.4197, contradicting the RFT precondition that the distance BB* must be equal to the run distance (1°).

In fact, a 2^nd sight cannot confirm the correctness of any position circle transferred with RFT. A 2^nd sight is say GHA=280/Dec=35/H_o=58 with LSQ* fixes 49·6674N/40·7923E and 8·9502N/59·6785E. The transferred position circle through these two fixes is GHA=0/Dec′*=0·8218/H_o′*=30·0586. If the transferred position circle were correct, the vessel going north along 45°E from B arrives at Lat_B*=46·0607 and BB*=1°.0607, not 1° as postulated. For the vessel at A, Lat_A*=60·7632 and AA′*=0°.7632. AA* was presumed to be 1° with either RFT or GD-UT, regardless of the properties of a 2^nd sight.

Therefore, not only do RFT/LSQ* imply a transfer of the 1^st sight's GP that is incompatible with the run data but the implied transferred position circle cannot account for any other positions presumed correctly transferred based on the backward-projected position criterion.

Already indicated is that at the cardinal points A and C the position circle transferred with either RFT or GD-UT is virtually equivalent. If the transferred position circle through A* and C* is run together with the same 2^nd sight by means of K-Z, the position solution differences with GD-UT+K-Z are negligible. We would argue here that if the transfer for the run at the two cardinal points A and C is basically considered correct according to both RFT and GD-UT, the whole 1^st sight's position circle should be accepted as correctly transferred with GD-UT.

To demonstrate that terrestrial RFT has in fact served as model for RFT/LSQ*, we trace the effect of decreasing Zd in Table 1 for the transferred position circle through B* and C*. The mer.diff. of point B is consistently taken at 75% of Zd; cardinal point C is on the equator at zenith distance east of Greenwich. Point A is of course the other cardinal point. In each case the transferred position circle passes through B* and C* and BB* and CC* are exactly 1°.

Table 1.

Transfer conformity between terrestrial RFT and RFT/LSQ* when Zd decreases.

As H_o increases from 60 to 88, Dec′* increases from 0·93 to almost 1·00. In other words, the GP transfer increasingly approaches the run distance as in the terrestrial analogy (Figure 1). When H_o=30, the transferred position circle falls way short of passing through A*. As H_o increases the distance AA* begins to conform to the run distance of 1° too. Meanwhile the required adjustment to altitude remains practically insignificant, the difference between H′_o and H_o decreasing steadily, from 0·0068 when H_o=30 to 0·0003 when H_o=88.

4. HUXTABLE'S DUE-NORTH DOUBLE SIGHT

In Huxtable's present thought experiment the position at J (our Figure 3) is the western equivalent of position B in Figure 2, but also provided for is a 2^nd sight to confirm the fix at J* so that its GP must lie on the same meridian as J and J*. The configuration is contrived but nonetheless relevant. Mindful of the requirements, Huxtable also provided for two possible position solutions, while each fix must meet the backward-projected position criterion of RFT. As the positions J and J′ on the same meridian are postulated as 45°N and 45°S, the 2^nd sight is GHA=45/Dec=1/H_o=45.

Figure 3.

The due-N double sight thought experiment.

Huxtable relies on simple geometry, but the fixes at J* and J′* are obtainable with LSQ*, showing that the argument implicitly supports RFT. J* and J′* are obviously regarded as being on the correctly transferred position circle, which in fact is GHA=0/Dec′*=0·7071/H′_o=30·0025. AA* measured with the transferred position circle is 0°.7071. Thus again, the implied transfer with RFT/LSQ* cannot account for the correct transfer at cardinal point A! The correctness of positions J* and J′* based merely on the presumed certainty of the backward-projected positions J and J' respectively is illusory.

Also in this case will RFT/LSQ* begin to conform to terrestrial RFT when Zd decreases. This can be shown when the 1^st sight's H_o is allowed to increase, while the 2^nd sight's Dec remains at 1°N and its GHA at 75% of the 1^st sight's Zd. To obtain the 2^nd sight's altitude compatible with the northern RFT/LSQ* fix at J* (so that JJ*=d=1°), apply the sine formula followed by the Lat-finding quadratic to triangle JPX. As terrestrial RFT is essentially based on GD-UT we may compare the fix discrepancies. With Huxtable's case in Figure 3, the northern fix discrepancy with GD-UT is 17′.88. When the 1^st sight's altitude increases from 30° to 60° this discrepancy decreases to 4′.35 and to a mere 0′.03 when altitude increases to 88°. In other words, RFT/LSQ* hold when Zd is small but not in the general case when Zd is large.