Journal of Geomancy vol. 2 no. 2, January 1978
{45}
Continuing its brief as open forum for all comment on geomantic research, the Journal of Geomancy presents more comment on ley and landscape geometry statistics by Robert Forrest.
I was much amused by Tony’s observation that my F-unit was close to 9 Stades in length. I had not realised this myself, for the simple reason that I had not even considered the Stade in my investigations, so that the thing is a rather unfortunate freak of chance. (I had realised that 1·661 was close to 12/3, with the numbers 1, 2 and 3 playing a merry little game with each other, a sort of numerical ring-a-ring-a-roses etc., but I let it stand, as I had conjured 2√3−1 ‘at random’, as it were. I reasoned roughly that as the ancients didn’t use the km*, they were hardly likely to use 12/3 km. This was before I knew that the Stade had crept up impishly behind me, of course …). So, the F-unit = 9 Stades – a natty little coincidence, I think, that belongs in the first half of my article in J. Geomancy 1/4 alongside π metres = 6 Royal Cubits.
The question arises: was I chasing the Stade in my experiment with the 20 churches? In that experiment four distances came out as close multiples of the F-unit (I discount here the use of ½ F-units). But as I said in the article, the F-unit ‘uses’ there are no more than chance would predict amongst a set of 20 random points. After hearing of Tony’s observation, I did an experiment with 20 randomly-chosen modern buildings, over much the same area, and this yielded 3 such integral F-unit ‘uses’. So that if the churches were using the F-unit (and Stade), then so too are the modern buildings. This is fairly conclusive evidence for the non-use of these units, I think.
Numerical coincidence is a fascinating phenomenon – the point of the first half of my article being that metrology is riddled, or can be riddled, – with it. What I did not realise was that my F-unit, unbeknown to me, was demonstrating this very fact. I think this must be the first time I have ever inadvertently demonstrated a point by well and truly putting my foot in it!
Denote the class of ‘shown-on-the-map’ sites by S, and the class of ‘not-shown-on-the-map’ sides by N. A general ley will consist of a mixture of S-sites and N-sites, and by virtue of the fact that the ley hunter frequently starts with mapwork prior to fieldwork, we should normally expect any ley to contain at the very least 3 S-sites, and more likely at least 4 or 5 S-sites.
Any six-pointer must fall into one of the following categories:–
| Combination of: | Probability of this combination for a given six-pointer (a) | |
|---|---|---|
| S-sites | N-sites | |
| 0 | 6 | ·0156 |
| 1 | 5 | ·0938 |
| 2 | 4 | ·2344 |
| 3 | 3 | ·3123 |
| 4 | 2 | ·2344 |
| 5 | 1 | ·0938 |
| 6 | 0 | ·0156 |
(a) NOTE: probabilities calculated on assumption that S and N sites equal in number, as in Behrend’s hypothetical example. {46} One of the differences between the significance tests of Raison and Forrest is this: Raison asks the probability of finding a six-pointer consisting of exactly 3 S-sites and 3 N-sites. Forrest says what is the expected number of six-pointers regardless of the S–N combination. Forrest is thus allowing in his expected 1·6 lines all the possible S–N combinations represented in the above table. Raison is restricting himself to the specifically (3S, 3N) case. There are 1·6 of any type (Forrest), but only 0·3125 x 1·6 = 0·3 of the (3S ,3N) type (Raison). The Forrest argument thus allows alignments of order 6 discovered totally in the field – i.e. all the six aligned sites are N-sites. Presumably this could happen, and presumably, with proper classes of ley marker points, it would count as a ‘valid ley’.
Perhaps the model for the ley hunter who first studies a map and then goes out into the field to do fieldwork should read something like this: amongst the 200 points (S & N combined), we expect 1·6 six-pointers of which 1·6 × ( 1 − ·0156 − ·0938 − ·2344) = 1·05 will contain at least 3 S-sites. This is the expected total of six-pointers (each containing at least 3 S-sites) to be exceeded in order to justify a claim of beyond-chance alignments.
The 1% level claimed by the Raison argument is rather suspect. In my opinion this 1% level is telling not about the probability of the alignment itself being chance or otherwise, but the probability of Raison finding it, at a given attempt, whether or not it is a chance alignment.
Consider the following hypothetical experiment with a computer picking out alignments of random points on a piece of graph paper. There are 50 points on the piece of paper, and, to a given alignment limit, the computer finds (say) 10 aligned triads.
The Raison style of argument could be applied here thus: there are 50C2 = 1225 dyads; of these 10 × 3C2= 30 give rise to triads. Thus, when, on a given go, the computer finds an aligned triad, it is a 30/1223 = 0·024 probability event – ‘beyond chance’ if a 5% level is adopted. The point is of course that the probability here is not telling us about the probability of the alignment being other than chance, it is telling us about the probability of hitting upon a lucky-dyad (i.e. one which leads to a triad), amongst the morass of possible dyads.
If the expected frequency of 1·05 lines is not exceeded, it might indicate the chance nature of the alignments found. On the other hand, it might only indicate that the ley hunter has not yet found all the valid six-pointers in his area of study. An area must be thoroughly searched for all possible alignments before reliable comparison with chance expectation can be made.
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*EDITOR’S NOTE: The question of the use of the kilometre in ancient times is not cut-and-dried. For other opinion, see Josef Heinsch’s Principles of Prehistoric Sacred Geography (FENRIS-WOLF) available from IGR @ 50p + 10p postage/handling.