page 4
The following are actual cases published by ley hunters at various intervals. Each case must be compared with its own chance hypothesis, as the number of ley lines depends heavily on the two parameters n = the number of ley points on the map in question, and x = the width of the ley corridor adopted.
The value of n is perhaps the subject of greatest debate in that authorities differ in their criteria as to what counts as a ley point and what does not. Some ley hunters count cross-roads, for instance, whilst others do not. Some count mile-posts and tree clumps, others count points associated with certain types of place names – notably those embodying the word ley itself. Major Tyler was not certain whether or not moats were to be counted as ley points, but stated that other students had found them to fall on leys. Nearly all ley hunters insist that the only churches to be counted are those built over ancient pagan sites, as indeed many were deliberately built. However, certain entirely modern churches, not thus built, have been found to be sited on leys, and some ley hunters have been known to talk of ‘subconscious siting’. To be fair, however, they are usually the first to admit that in doing so they are laying themselves open to criticism. Finally, ancient earthworks, forts, camps, barrows, tumuli, standing stones, crosses and stone circles appear to be almost unanimously counted.
The value of x too, as we have seen, is also open to debate in that it is dependent (a) on the nature of the ley points in question – e.g. moat alignments tend to entail a larger x value than tumuli alignments – and (b) on the tightness of alignment which the ley hunter requires to define a ley. It seems to be a fact that most published ley lines are of the looser alignment type of Fig. 4(b) in the Introduction.
In Appendix 1 will be found tables of the number of alignments of various orders to be expected by chance on a map 25 miles square, containing n ley points and when the corridor width for alignments is taken to be x.
For example, with 300 points on the map, and with x=0.02 mile, we can expect 749 alignments of 4 points, 65 of 5 points, and 4 of 6 points. We would write this as
With n=300, x=0.02 mile:
| N(4) | = | 749 |
| N(5) | = | 65 |
| N(6) | = | 4 |
If we allow a larger value of x, i.e. if we do not insist that the ley points be quite so tightly packed about a straight line, or if we are dealing with moat alignments for example (see Case 5 below), we have:
With n=300, x=0.03 mile:
| N(4) | = | 1240 |
| N(5) | = | 161 |
| N(6) | = | 16 |
and so on. Clearly there are more loosely packed leys than tightly packed ones.
It is to be emphasised that the numbers of alignments tabulted are the numbers of alignments to be expected on average.
We now proceed with the cases.
Case 1.Relevant to Paul Screeton’s Quicksilver Heritage, p. 43, on the South Durham leys. Here are presented four leys of order 4, four leys of order 5 and three leys of order 6. The accompanying map is an extract from the 1-inch OS map 85 covering an area of some 690 square miles. A count over the map reveals:
| Churches | 538 |
| Earthworks etc. | 24 |
| Moats | 4 |
| Miscellaneous | 30 |
| 596 |
The heading ‘miscellaneous’ includes possible ley point contenders such as Beacon Hill, West Gate, Hag Wood, crosses, tree clumps etc. The heading ‘churches’ includes all churches shown on the map, ancient and modern.
How many of the 596 possible ley point contenders are valid? It is difficult to say, but let us suppose that 400 of them are. With n=400 and x=0.02 mile, we can expect*:
page 5
| N(4) | = | 1926 |
| N(5) | = | 222 |
| N(6) | = | 19 |
| N(7) | = | 1 |
2014: Click here for update to expected alignments in Case 1
The lines discovered by Paul Screeton, therefore, are in no way remarkable.
* Strictly speaking, the tables of Appendix 1 are for use with single 1:50,000 OS sheets, 25 miles by 25 miles, predicting alignments which possibly invole the whole map. They can, however, be used safely to predict whole map alignments for 1-inch OS maps of the type used in Case 1 (about 25 miles by 28 miles). See the section “1:50,000 compared with 1-inch maps” in Appendix 1 for details.
Case 2.Salisbury Plain.
An analysis of the 1:50,000 OS map 184 revealed:
| Churches | 264 |
| Tumuli, Earthworks etc. | 682 |
| Miscellaneous | 60 |
| 1006 |
With about 1000 points and a corridor width of 0.02 mile, the number of alignments is fantastic:
| N(4) | = | 25338 |
| N(5) | = | 7297 |
| N(6) | = | 1576 |
| N(7) | = | 272 |
| N(8) | = | 39 |
| N(9) | = | 5 |
2014: Click here for update to expected alignments in Case 2
7-point leys are almost commonplace, and a 9-point ley could easily be the work of chance!
Of course, many of these 1006 points may not be valid ley points. But how many? The above figures show that we would have to discard a very large number indeed before a 6 or 7-point ley becomes a matter of design on the part of some ancient architect and not simply the work of coincidence.
(No specifically published case is involved here, though the region is one rich in ley lore. The results presented here are used later in the text, though they do a good deal towards explaining the Salisbury Plain sections of the leys presented by Alfred Watkins in The Old Straight Track pp. 104 ff.)
Case 3.Relevant to Janet and Colin Bord Mysterious Britain, p. 200, leys in the Bedford and Luton area. The map used here is the 1-inch OS sheet 147. Four leys are presented: ley 1 of order 9, ley 2 of order 7, ley 3 of order 8 and ley 4 of order 5.
A survey over the whole map revealed:
| Churches | 468 |
| Earthworks etc. | 35 |
| Moats | 97 |
| Miscellaneous | 40 |
| 640 |
With 600 points, and with x=0.02 mile, the following alignments can be expected:
| N(4) | = | 6622 | |
| N(5) | = | 1144 | |
| N(6) | = | 148 | |
| N(7) | = | 15 | |
| N(8) | = | 1 | |
| N(9) | = | 0 | .1 |
2014: Click here for update to expected alignments in Case 3
Now let us look at the actual lines on the map.
Ley 1, allegedly of order 9, includes two large points – Dray’s Ditches and an earthwork. Both are skirted. Also, Chicksands Priory is in none too good alignment with the remainder of the points. This ley, I would say, is of the looser type of Fig. 4(b), and probably, within alignment limit of 0.02 mile, only of order 6 or 7.
Ley 2, allegedly order 7, also contains two large points – Waulud’s Bank and Dray’s Ditches.
Ley 3, allegedly of order 8, is probably the best case presented. It is a good 7-point alignment and possibly (within the 0.02 mile limit) an 8-point one, the dubious contender being Arlesey Church. Once again, a ley of type Fig. 4(b).
Ley 4, allegedly order 5, is a good alignment, but one of its points, an earthwork, is a large point.
Further items of interest: Two of the points on ley 2 are hills – Galley Hill and Deacon Hill, the first of which is mentioned as having two small barrows on the summit (not shown on the map). I have not counted hills in my analysis of the map. Secondly, the authors mention that ley 3 crosses the A1 at a crossroads (“often a significant feature of a ley”), though they do not count this as a ley point. We return to the problem of crossroads later.
page 6
Meanwhile, how many points are to be discarded as ‘non-valid’ ley points out of the 640 counted? How many must be discarded to put these leys beyond the realm of chance? (Retaining x=0.02 mile, a 7-point ley on a map this size is not an extraordinary rarity even with only 300 valid ley points.) And finally, when does a hill rank as a ley point?
Case 4.Relevant to John Michell’s The View Over Atlantis, p. 40, concerning Dorchester and Weymouth 1:50,000 OS sheet 194.
Analysis of the whole map revealed:
| Churches | 258 |
| Tumuli etc. | 626 |
| Castles, Camps, Forts | 34 |
| Miscellaneous | 53 |
| 971 |
The number of mixed leys (i.e. leys passing through any or all types of ley points counted) of various orders, before any points have been discounted as non-valid, is thus roughly about the same as Case 2. That is, leys of order 8 or even 9 can be expected by chance.
But what of the leys given in this example by John Michell? These are alignments of churches only – one line being of order 5, and the other of order 6.
For alignments of churches only, n=250, and using the tables of Appendix 1, with n=250 and x=0.02 mile, we have:
| N(4) | = | 402 | |
| N(5) | = | 29 | |
| N(6) | = | 1 | .6 |
| N(7) | = | 0 | .07 |
2014: Click here for update to expected alignments in Case 4
How many of the counted 258 churches are modern, and thus non-valid as ley points? I do not know, but quite a lot would have to be before a solitary ley, consisting only of churches, of order 6, could be deemed to be beyond the realms of coincidence.
Case 5.Again relevant to John Michell’s The View Over Atlantis, from the notes and illustrations p. xx, concerning alignments of moats in East Anglia. The large map presented by John Michell is an area of some 126 sq. km, containing 37 moats and Castle Hill – 38 points in all. It is an extract from the 1:50,000 OS sheet 155. As all the action takes place inside a rectangle 9 km by 14 km, we can consider the extract as a single entity.
The first consideration here is that of the value of the ley width, x, as the 38 ‘points’ are far from being points. On the actual OS sheet the ‘points’ have a mean width of 1.5 mm, corresponding, on the ground, to 0.075 km (about 0.047 mile).
The second consideration is that of mean ley length, L, which for a rectangle 9 km by 14 km is about 12 km (see notes in Appendix 1).
With x=0.075 km, L=12 km and A=126 sq. km, we have, for 38 ‘points’:
| N(3) | = | 92 | |
| N(4) | = | 12 | .5 |
| N(5) | = | 1 | .1 |
| N(6) | = | 0 | .08 |
2014: Click here for update to expected alignments in Case 5
On the actual map extract, John Michell produces two leys allegedly of order 6:
The first ley, therefore, is a bona fide case of order 6, whereas the second is certainly of order 5, and possibly order 6.
Does the case seriously transcend chance? Not if the second ley is really only one of order 5. If the second ley is of order 6? Well, it is not too difficult to show that the probability of obtaining two leys of order 6 on such a map is 0.003 – a level of probability which is conventionally taken as transcending chance.
For comparison, we look at the whole of the 1:50,000 OS sheet 155. It contains a total of 126 moats in its 625 sq. miles. Taking a mean moat diameter of 0.04 mile we find: N(4)=103, N(5)=7.5, N(6)=0.4, and the probability of finding two leys of order 6 is 0.05. Looked at from this point of view the event of two leys of order 6 is rather less remarkable than before, and the event of one ley of order 6 and one of order 5 is not remarkable at all.
page 7
Of course, in considering the whole map, we are finding the likelihood that leys of order 5 or 6 will occur somewhere on the map, and not necessarily within the same confined space in which John Michell found them.
(A scale model of this case was simulated with random ‘points’ instead of moats – see Graph 2. This simulation also demonstrates that several familiar ley concepts can arise in a chance model rather more easily than expected.)
Case 6.Also from notes and illustrations p. xx of John Michell’s book, being the small map there presented showing an alignment of 5 moats between Athelington and Laxfield. This particular ley can be found on the 1-inch OS map sheet 137 of the Lowestoft region, analysed as follows:
| Churches | 341 |
| Moats | 172 |
| Tumuli | 11 |
| Old Halls | 52 |
| Miscellaneous | 36 |
| 612 |
For pure moat alignments, taking x=0.03 mile, n=170, we find:
| N(4) | = | 191 | |
| N(5) | = | 14 | .5 |
| N(6) | = | 0 | .8 |
| N(7) | = | 0 | .03 |
2014: Click here for update to expected alignments in Case 6
A 5-point moat alignment, on its own, is thus in no way remarkable.
The number of mixed alignments (i.e. consisting of any mixture of ley point types in the above list) to be expected on this map can be studied in the tables of Appendix 1.
Case 7.Relevant to John Michell The View Over Atlantis, Notes and illustrations p. xvi: Alignments of Norfolk Castles and Mounts. This region is to be found on the 1-inch OS sheet 136 of the Bury St Edmunds region, the whole map being analysed as follows:
| Churches | 333 |
| Tumuli | 70 |
| Moats | 129 |
| Old Halls | 56 |
| Miscellaneous | 26 |
| 614 |
John Michell presents the following leys:
Comments:
Ley (a) is a very good alignment to within x=0.02 mile, though the included castle (at 072925) is a relatively large point. On the map are 70 tumuli and 5 castles (included under miscellaneous) giving 75 ley points in all over the whole map. We can expect:
| N(4) | = | 5 | |
| N(5) | = | 0 | .1 |
| N(6) | = | 0 | .002 |
2014: Click here for update to expected alignments in Case 7(a)
If counted as 5-point, therefore, this ley is a minor rarity, but is within the bounds of chance if counted as 4-point (i.e. if the castle is discounted as a large point).
Ley (b) is a rather looser alignment of the 6 points. On the map we have 70 tumuli and over 300 churches – about 400 points of the two types. By the tables in Appendix 1, such a 6-point ley is no rarity – the chance hypothesis predicting about 19 such leys. This, once again, is before we have rejected those churches which are non-valid as ley points.
2014: Click here for update to expected alignments in Case 7(b)
Ley (c) is the most curious. It is an alignment of churches only, but John Michell does not specify those churches actually involved – he simply specifies the number 6. But there are 8 churches on this alignment, being as follows: Two churches at New Buckenham (089905, 086904), Benham Church (063882), Kenninghall Church (041860), Coney Weston Church (956778), The Grange Church (921742) and two churches at Troston (900723, 899722). Perhaps two of them were discarded as non-valid, I do not know. At any rate, taking n=300 and x=0.02 mile, the tables in Appendix 1 tell us that pure church alignments can be expected as follows:
page 8
| N(5) | = | 65 | |
| N(6) | = | 4 | .2 |
| N(7) | = | 0 | .22 |
| N(8) | = | 0 | .01 |
2014: Click here for update to expected alignments in Case 7(c), 0.02 mile
That is, if we were to examine 100 maps, each containing 300 churches, we could expect to find such an 8-point church ley as we have here on about 1 of the maps.
If the alignment is not quite so strict and x=0.03 mile then the tables in Appendix 1 show that we can expect:
| N(5) | = | 161 | |
| N(6) | = | 16 | |
| N(7) | = | 1 | .2 |
| N(8) | = | 0 | .08 |
2014: Click here for update to expected alignments in Case 7(c), 0.03 mile
or, in other words, we could expect to find such a ley on 8 of the hundred maps.
Case 8.Relevant to John Michell, The View Over Atlantis, pl. xxix, facing p. 45: leys through Gare Hill, Wiltshire. This is the Yeovil and Frome region, 1:50,000 OS sheet 183, which was analysed as follows:
| Churches | 416 |
| Tumuli, Castles, Forts etc. | 268 |
| Moats | 8 |
| Miscellaneous | 49 |
| 741 |
Even if only 500 of these are valid with x=0.02 mile, the tables in Appendix 1 show that leys of up to order 7 can be expected by chance. The leys presented by John Michell are:
Ley (a) of order 7, one point of which is a large point (Scratchbury Hill). Only three of these ley points, however, lie on this map, the remainder being found by continuing the ley over onto the adjacent map of the Salisbury Plain (sheet 184) – a procedure about which we shall have more to say later under the heading of the Warminster mystery. As we shall see then, there is nothing remarkable about this ley at all.
Ley (b) a 4-‘point’ ley, one of which is the cursus on Salisbury Plain – a large point. Remarks as for (a).
Ley (c) a 4-point ley, one point of which is Stonehenge. Remarks as for (a).
Ley (d) totally contained within map 183, and a 5-point ley. Since leys of up to order 7 can be expected by chance, this ley is not extraordinary.
But what of the fact that all four leys pass through the one point, Gare Hill? Is this not remarkable? Considering the sections of the leys which lie on sheet 183 only, we have two 3-point leys (a & c) and a 5-point ley (d) to contend with. Assuming 500 valid ley points on the map, the methods of Appendix 2 (vertex analysis) predict the finding of 21 leys of order 4 and 3 leys of order 5 passing through Gare Hill (or any ley point, for that matter). All this, of course without any extension of leys onto the Salisbury Plain map, a region which (see Case 2) is rich in alignment potential. (These figures being calculated with x=0.02 mile.)
It would seem, then, that the Gare Hill leys are well within the bounds of chance.
2014: Click here for update to expected alignments in Case 8
page 9
Case 9.Relevant to Alfred Watkins, The Old Straight Track, Fig. 6, of South Radnor Leys. The relevant half of the new 1:50,000 OS sheet 148 was analysed as follows:
| Churches | 115 |
| Tumuli etc. | 50 |
| Moats | 13 |
| 178 |
The methods of Appendix 1, with x=0.02 mile, L=1.61×12½=20.1 miles, A=625/2=312.5 sq. miles, and n=175 yield:
| N(4) | = | 209 | |
| N(5) | = | 16 | |
| N(6) | = | 0 | .88 |
| N(7) | = | 0 | .04 |
| N(8) | = | 0 | .001 |
2014: Click here for update to expected alignments in Case 9
The two leys introduced by Alfred Watkins are:
Both leys would seem to stretch chance at first sight, but map inspection reveals the following details:
(a) Wylfre (116521) is a hill peak (a large point) and Glascwm Hill (around 170523) is even more ill defined. Further, the line of best fit skirts the moat, labelled ‘the Camp’ in Watkins’s book, at (287520), a relatively large point 1.5–2.0 mm in diameter on the map. This ley therefore is at most a bona fide ley of order 5.
(b) The camp at (163537) is a large point. As in (a), the moat at (287520) is skirted by the line of best fit. And in addition, the line of best fit misses the tumulus at (113544) by just over 0.03 mile, and so is is not in alignment to within our adopted limit of x=0.02 mile used in the above calculations. I would say that this ley, to within our limit, is probably a bona fide ley of order at most 6.
The two leys presented here by Alfred Watkins, therefore, would seem to be within the bounds of chance.
That a ley of order 5 and one of order 6 should cross at a ley point (the moat at 287520) might have constituted a minor rarity had not this ley point been relatively large, and had it not been simply skirted by both leys.
The whole case, therefore, could easily be the work of chance, though of course, in the above we have not discounted ‘non-valid’ ley points. How many such points there are remains to be determined.
Case 10.Relevant to Major F.C. Tyler’s The Geometrical Arrangement of Ancient Sites, Fig. 2, of leys in the region of Tiverton, Devon. Here, I used an original version of the map used by Tyler – the old popular 1-inch OS edition, sheet 123. This map represents an area 27 miles by 18 miles, and was analysed as follows:
| Churches | 136 |
| Tumuli etc. | 51 |
| Old Crosses | 10 |
| Moats | 1 |
| 198 |
Using the methods of Appendix 1, with x=0.02 mile, L=1.34×18=24.12 miles, A=18×27=486 sq. miles, n=200 points, we have:
| N(4) | = | 227 | |
| N(5) | = | 15 | |
| N(6) | = | 0 | .75 |
| N(7) | = | 0 | .03 |
| N(8) | = | 0 | .001 |
2014: Click here for update to expected alignments in Case 10
Allowing for skirting of large points etc., Tyler presents in his diagram: 4 leys of order 4, 6 leys of order 5, and 1 ley of order 6. However, he states in the text that be found “no less than fifty-five such lines, each of them taking up not less than four sites of the sort which we are considering”.
The figures, before non-valid points are discounted (and there is no evidence that Tyler discounted any before drawing his ley lines), are well within the bounds of chance.
page 10
There are two ways in which the ley case might be proved. Either the ley hunter must produce a profusion of medium order leys, or he must produce a small number of extraordinarily high order leys. In either case, the ley hunter must transcend chance to a degree which leaves no doubt whatsoever that he has transcended chance.
No case treated so far can be said to have conclusively done either.
None of the cases has even approached success on the first count – the profusion of leys. It is extremely difficult and time consuming to insert all alignments of four or more points on a map which contains, say, 400 ley points. I say this from experience on a much smaller scale with 50 points, and as the time factor involved varies roughly as the square of the number of ley points, it is 64 times as tedious with 400 points as with 50!
On the other hand, from the point of view of high order leys, we have only seen leys produced on maps where the number of similar alignments to be expected by chance is of the order of 0.05 – that is, in investigating maps containing an equal number of points, we should expect to find, on average, such a ley on about 1 map out of every 20 maps tested. But this degree of chance is not a serious challenge to the chance hypothesis. After all, a gambler tends to relate those stories of his most spectacular wins, and, likewise, the ley hunter tends to present us with the most spectacular results of his research. The odd 1 in 20 chance is hardly conclusive in such a context. (Incidentally, the ten cases just presented were the first ten cases I investigated in detail. The reader must take my word for it that I have not left out any case which I found ‘inconveniently awkward’ to my case.) The odd ‘1 in 1000 maps’ case would be food for thought – even if as few as half a dozen could be found over the whole of the British Isles. But have such leys been found? There are certainly none such in the ten cases just analysed.
“Just as I expected,” will say the ley sceptic, “and you would find exactly the same if you were fool enough to apply your formulae to every single so-called ley line on the face of the earth!”
On the other hand, the ley line advocate will accuse me of testing a mere ten cases, even assuming he accepts my treatment of those ten cases. (I can quite foresee, for example, those who will accuse me of cooking the statistics to fit my own preconceived notions: “All this talk of ‘large points’ and this variation in the value of x whenever he feels like it …”)
Very well, I am a fool who is interested by ley lines and who hopes that he has not (unconsciously) cooked the books.
But I do believe that at the very least, the treatment of the foregoing ten cases does show that chance should never be underestimated, and that each case of ley hunting virtually requires its own chance hypothesis. And that further, ley hunting is far from being rigorous in its approach to its problems. If it hopes ever to influence the opinion of orthodox archaeology, it must rectify this problem.
For myself, on the basis of these ten cases, and the four further studies which follow, I do not at present believe the ley hypothesis to be any more than a chance effect.
The major problem, it would seem, that the ley hunters need to settle is exactly what constitutes a valid ley point and what does not. An awful lot depends upon this decision. It is not sufficient that we be told that such and such a region is rich in leys, some of orders 8 or 9. We must be told (a) exactly what categories of points are accepted as ley points by the ley hunter in question, and (b) how many such points there are over the whole region considered. A chance alignment of order 7, for example, might be a rare occurrence in some regions and a fairly commonplace one in others, dependent upon the ley point population of the region in question. This much we have seen clearly in the foregoing cases – compare, for example, Cases 2 and 5. One cannot therefore, fix a ‘least order for valid leys’ for general use. Most ley hunters discard alignments of order 3, and begin to accept as leys only those alignments of order 4 or more. But in some regions, a ley of order 4, or even 5 or 6, can be a commonplace chance event on a par with the 3-point alignments in less populated regions. Let me repeat again, each case requires virtually its own chance hypothesis.
page 11
Lastly, a word about ‘fieldwork’. The ley hunter, say, finds his 6 or 7-point ley line on the maps and out he goes into the field to investigate his ley in situ. Lo and behold, he discovers sections of track, odd stones, clumps of trees etc., not shown on his map, and this strengthens his belief that this ley of his is no chance effect. But he has altered the chance hypothesis by his fieldwork. How many such sections of track etc. are there which are not on his ley? That is, over the region he considered in the first place, before his fieldwork, how many ‘not-shown-on-the-map’ ley points are there? It is that figure from which the new chance hypothesis springs. The old chance hypothesis in now no longer valid, except in its pre-fieldwork context.
Similarly, Major Tyler cautioned against outright rejection of the 3-point ley (p. 15). Continuation of the ley from one map to another might reveal more points, he said. Of course it might, but again the chance hypothesis model has changed. The model for the use of two or more maps is an entirely different kettle of fish to the model for the single map. The limit of chance changes as the ley is produced from map to map.
It is these facts which are at the root of two of the following four further studies. The remaining two studies show once again the amazing and almost unexpected range of chance and coincidence.
It is to those that we now turn. They are all, as Major Tyler put it, ‘developments of the straight track theory’.