1. INTRODUCTION
K. H. Zevering has recently proposed a way to calculate the position of a moving observer, from the intersection of position circles, that differs from the methods hitherto accepted by marinersFootnote 1. Zevering concludes, from discrepancies between his own results and that of other methods, that those others are in error. However, the discrepancies result from incorrect assumptions in his proposed method, which give rise to erroneous answers. My comments will be restricted to the simplest case of two observations only, and will assume that the Earth is, for our purposes, spherical.
2. THE STATIONARY OBSERVER
An observer determines the altitude H1 of a celestial body such as a star S1, which at the moment of measurement is at a position (Dec1, GHA1), with a corresponding Geographical Position (GP1) below it on the Earth's surface. He must then be somewhere on a circle, centred on GP1, with a radius of (90° – H1) degrees, where a degree corresponds to 60 nautical miles measured along the Earth's surface. That circle is the locus of all possible observers who measure that same altitude at that moment, but we know nothing, yet, about where on that circle he may be.
The observer then measures the altitude H2 of another body S2, situated then at (Dec2, GHA2), which puts him on another circle, radius (90° – H2), centred on GP2, the point below the second body at the moment of the second observation. If the observer had not moved between those two observations, perhaps because they were effectively simultaneous, or perhaps because his vessel was stationary, then he must be at one of the two intersections of those two circles. Commonsense will usually decide which one. There are several methods of computing the intersection points of those circles, which are equivalent and give the same answers. There is no disagreement about any of that, I suggest; it's common ground.
3. THE TRAVELLING OBSERVER, AND ZEVERING'S PROPOSAL
A rather more complex problem occurs when the observer travels across the Earth's surface, by a known distance d and course α, the “run”, between two observations made at different times. Somehow, that travel has to be allowed for. I will address only Zevering's proposed solution to that problem, and how he adjusts GP1 to allow for the intervening travel.
Zevering, in his section 2, proposes to take the first position circle, centred on GP1, preserving its radius (90° – H1), and displaces its centre by the same distance d, and in the same direction α, as the movement of the observer between the two sights. His Table 1 shows how the position of the new centre is to be calculated. He now appears to assume that at the time of the second observation, the observer must be somewhere on that displaced position circle. We should be seeking the locus of all possible observers, who were somewhere on the first circle, and have since moved through that known course α and distance d. Then Zevering assumes that the intersections of that transferred position circle with the circle from a second observation can be calculated as before. This proposed method for adjusting the initial position circle for the intervening travel is what he terms the “GHA-Dec updating technique”, or in his abbreviation “GD – UT”.
I will show that the proposal does not work.
4. TESTING THE PROPOSED METHOD
We can test the method by applying it to a hypothetical case, in which the geometry is simple, and which has been designed to show up the defects in the procedure.
1 An observer, at position P1, measures the altitude of a star S1, at (Dec1=0°, GHA1=0°), to be 30°.
2 Then he travels due North by 60 nautical miles (=1°), to P2.
3 From there, he observes another star S2 (then at Dec2=N 1°, GHA2=W 45°) to be at an altitude of 45°. Where on Earth is he then?
To start with, we will put the Zevering proposal to one side.
Because of the simple geometry, the result is almost self-evident. There are, in general, two possible solutions for P2. One is in the Northern hemisphere, at Lat=N 46°, Long=W 45°, in which case P1 was 1° further South, at Lat=N 45°, Long=W 45°. The other possible position for P2 is at Lat=S 44°, Long=W 45°, with P1 situated, 1° further South, at Lat=S 45°, Long=W 45°.
My aim here is simply to show up the errors in the Zevering method, rather than to choose between better alternatives, so I do not propose to discuss here how that result should be derived. However, a sceptical reader can easily check that the solutions listed above are indeed exact, using computed altitude tables or spherical trig. He will find that from either position for P2, the altitude of star S2 is indeed exactly 45°, so condition 3 has been met. From either position for P1, the altitude of star S1 is exactly 30°, so condition 1 has been met.
And of course P2 is 1° North of P1, so condition 2 has been met.
Those are the conditions that were specified. We have two exact solutions, then, and the observer has to choose the correct one from other evidence, which should present no difficulty.
5. THE ZEVERING ALTERNATIVE
This is set out in his section 4, which states:
“GD-UT as explained finds the locus of the transferred position circle by moving X (=GP) to X* for the magnitude and direction of the displacement (d and α) and by projecting the circle from X* with its given radius (Zd=90° – H0)=XZ=X*Z*”
So, start with a circle, centred on Geographic Position GP1, directly below S1, at Lat=0°, Long=0°, which corresponds to his position X. That circle, with radius 60°, is the locus of all points at which the altitude of S1 is 30°. The observer's initial position, at P1 (Z, in his notation) must lie somewhere on that circle.
Next (and this is the false step), displace the centre of that circle Northward by 1°, to correspond to the observer's travel between observations. The new circle still has a diameter of 60°, corresponding to an altitude of 30°, but its centre is now at Lat=N 1°, W 0° (position X*). Zevering presumes that the final position P2 (his Z*) must be somewhere on that displaced circle.
Next, find the two intersections between that displaced circle and a new circle, centred on GP2, directly below S2, at Lat=N 1°, Long=W 45°, with a radius corresponding to 45° altitude. As Zevering states, those intersections can be found algebraically.
The Northern result of this calculation of P2 (or Z*) is Lat=N 45·998°, Long=W 45·429°.
The Southern alternative is at Lat=S 43·999°, Long=W 44·600°.
We can see that the latitudes are very close to the exact values found above, but the longitudes are seriously in error, with displacements from the true value of the order of 0·4°, which amounts to about 17 miles at that latitude. And this, after a run of only 60 miles.
6. WHAT HAS GONE WRONG?
We can back-calculate to check, as before. The calculated altitude of star S2, from either position of P2, is exactly 45°, as it should be, so condition 3 has been met precisely. P1 (or Z), the observer's position prior to the Northerly displacement, must have been exactly 1° further south than P2, to meet condition 2. Therefore, the Northerly solution must place the initial position P1 at Lat=N 44·998°, Long=W 45·429°, and the Southerly solution for P1 at Lat=S 44·999°, Long=W 44·600°.
If we back-calculate from that Northern alternative for P1, then for the altitude of the star S1, which according to the specification of the problem should be at an altitude of 30°, we find instead that it is at 29·753°. Similarly, from the Southern alternative for P1, we find an altitude of 30·231°, instead of the specified 30°. These are very serious discrepancies. Condition 1 has not been met, by a long way.
The errors result from the incorrect assumption that XZ=X*Z*. That would be true in plane geometry, in which case, when the centre of a circle is displaced through a certain distance and direction, points on its periphery are displaced through the same distance and direction. On a sphere, however, that is not the case, except for circles that are small compared with the size of the sphere. On a sphere, if all points on a circle are displaced through the same distance and direction, the result is not a circle at all, but will have been distorted into a sort-of egg-shape. Therefore, the next step, calculating the intersections of two circles, cannot apply.
Section 5 states- “… It is argued that the correct method (GHA-Dec Updating Technique; GD-UT) for transferring the position circle of an earlier sight is to transfer the coordinates of its GP (GHA and Dec) for the run data, i.e. distance (d) and course (α)”. However, that has not been argued; just asserted. Section 5 continues, strangely- “The effect is that an observer's position on this position circle will not be transferred according to the distance d and course α … .” Exactly so! What is strange is that, in making that statement, Zevering fatally undermines his own proposal. The observer must move through a distance d with a course α; it's one of the preconditions of the problem.
7. CHALLENGE
I challenge Zevering to apply his procedure to the problem as set out above, to provide us with initial and final positions for the observer, the initial altitude of star S1, and the final altitude of star S2.
This is not the first airing of Zevering's proposal, which has previously appeared, at considerable length, in other journalsFootnote 2, Footnote 3. I have criticised that proposal, in similar terms to those aboveFootnote 4, but direct answers to my questions have been avoided. I hope they will fare better in this Journal.
