Saturday, 25 April 2015

In the late 1960s Professor Alexander Thom published the results of a statistical analysis of cup and ring marks - the prehistoric carvings that are found on rocks in Scotland and northern England (and indeed, all over the world), and whose meaning is still a mystery. From his study of the diameters of a selection of carved rings, he believed he had found evidence for a prehistoric unit of measurement, which he called the 'Megalithic inch' (MI), whose value was a little over 2cm. He related this to another unit, the 'Megalithic yard' (MY), derived from a study of prehistoric stone circles in Britain, and proposed the hypothesis that a formal system of measurement was used in prehistory, where 1MY=40MI.

Thom's theories were contentious among archaeologists from the start (still are), but they held an attraction for statisticians because they posed statistical questions that had no clear solution at the time. New methods of analysis (called 'quantum analysis') were developed to try to settle the issue of whether Thom's findings could reasonably be attributed to chance, and the results of these tests suggested  there was good reason to suppose that something like the Megalithic yard may have been in use, but its actual nature, and its significance, remained unsettled and controversial. (Thom believed it to be a highly accurate unit, in use across the whole of Britain in prehistoric times, but this seems to have been based on a misunderstanding of certain aspects of his statistical analyses.)
Although the megalithic yard has been widely discussed, Thom's work on cup and ring marks attracted very little attention, and when I first encountered it in about 1980, it seemed that the main problem was a lack of data. Thom's analysis was based on a collection of rubbings of 57 carved rings (made by laying a sheet of paper on the carved surface and rubbing lightly with a wax crayon in the same way as brass rubbings are made). The sample was too small to give a reliable result; the evidence for the MI unit was not statistically significant; and an undesirable subjective element entered the methodology because of the need to decide which rings were circular and suitable for measurement (when many of them clearly are not). Ilkley Moor, where many cup and ring carved rocks are to be found, was only about an hour's drive from where I lived; I decided to try to gather some extra measurements myself, and test Thom's hypothesis.

The methods I adopted evolved gradually as I discovered what the problems were. Initially  I spent a lot of time locating carvings on Ilkley Moor, and trying out various approaches. Experiments with rubbings were discouraging, and instead I developed a technique of marking out the details of the carvings in chalk, and then making measurements directly on the rock surface using a home-made adjustable trammel. It soon became clear that the decision about whether a ring could be regarded as circular was fraught with difficulties that I didn't know how to resolve, and so I extended the investigation to test the separations of  the centres of neighbouring cupmarks for evidence of a unit of measurement. From five carved rocks, I was able to gather together 39 ring diameters (of rings assumed to be circular) and 126 neighbouring cup-centre separations. I discovered that there was indeed a tendency for these values to group in multiples of an apparent unit, or quantum, very close in value to Thom's 'megalithic inch', and statistical analysis suggested that the results were unlikely to be explicable by chance. These preliminary results were published in Science and Archaeology (Davis 1983).

It was clearly worth persevering with the enquiry, but I did not know how to solve the problem of the subjective decision about the circularity of rings, and I needed more data. I changed my strategy. From here on I marked out in chalk every detail I could detect on each rock surface and then traced the whole design onto a large polythene sheet. Calibration marks, measured directly on the rock surface, made it possible to check, later, to see if the polythene sheet had shrunk or expanded, and correct for it if necessary. (Corrections of that kind were usually very small indeed.)  Subtleties of ring shape were easier to assess from the tracings (as opposed to looking at the rock surface itself), and it became clear that my earlier judgements about the circularity of some rings were probably incorrect. So for the next phase of the enquiry, I no longer measured the rings themselves (though I continued to trace them carefully, so as to have an accurate copy of the whole design) and instead concentrated on gathering together as large a sample of neighbouring cup-centre separations as possible. The enquiry was also extended to include some well-known carvings in Northumberland.

After Alexander Thom died in 1985, I was invited by Clive Ruggles to contribute to a Festschrift he was editing, which gave me the opportunity to publish reduced copies of the detailed plans of cup and ring marks that I'd been making, and to report on the progress of the search for the megalithic inch. The evidence, such as it was at that time, was difficult to interpret. From Ilkley carvings alone I'd gathered 274 neighbouring cup-centre separations; and while there was some weak support for the existence of an MI quantum, it wasn't statistically significant, and so the findings of my earlier experiments were not confirmed. A further 113 cup-centre separations from Northumberland sites showed no evidence for Thom's MI at all. On the other hand, at some sites in both Ilkley and in Northumberland, there were quite strong indications of the frequent use of a rough measure of about 10.5 cm - that is, very close to 5MI. There was little to be done except describe what I had found, give full details of the analyses, and accept the lack of any conclusive result.

I did no further fieldwork for a while after that, but when the Thom Festschrift was published in 1988 (Records in Stone), I was intrigued by the essay that immediately preceded mine in the book, for it offered a possible solution to the problem of the carved rings. It was written by Thaddeus Cowan, who proposed a new way of thinking about the shapes of some stone circles that were clearly non-circular. Today we conceive of a circle in terms of a radius and a centre, but Cowan's point was that this is a relatively sophisticated concept. He suggested that a more primitive, intuitive approach to drawing a circular-like closed figure might involve not 'radius' construction (from a fixed centre), but 'diameter' construction: suppose Jack and Jill each hold one end of a taut piece of rope, with some means of scratching a line in the ground. Jill stands still, and Jack scratches an arc centred on Jill. Then Jack stands still, while Jill scratches an arc centred on Jack. They keep doing this alternately until Jack arrives at the point where Jill started, and Jill arrives at where Jack started, and between them they have scratched a complete figure consisting of a series of 'flattened' arcs. It is not a true circle, but it is a figure of constant width (i.e. constant diameter. A 50p piece is an example of a regular 'constant diameter' or 'equal width' figure.) I was particularly struck by the fact that the shapes of some of Cowan's equal-width figures very closely resembled the shapes of many of the rings on cup and ring carved rocks. Here below is an example, showing how a simple irregular 'equal-width' figure, superimposed upon an accurate plan of a cup and ring, roughly mimicks the characteristic 'flattened arcs':

(It's important to realise that the purpose of the above figure is not to suggest that the prehistoric carver's 'original geometry' has been found. It merely shows the kind of shape that the method produces, and demonstrates that, broadly, its characteristics resemble those of the carved ring.)

I already possessed the means to investigate Cowan's idea, by using all the tracings of cup and ring marks I'd made at Ilkley and in Northumberland.  From these 119 rings were sufficiently well-defined and complete to permit a reasonably accurate estimate of diameter to be obtained. Many of them were shaped with 'flattened' arcs like the diagram above, and by measuring the diameter of each one in a range of different directions, it was possible to determine whether it was reasonable to consider it to be an 'equal-width' figure. 8 of them certainly were not, and were discarded; but that left 111 rings whose shapes were consistent with the idea that they were conceived as constant-diameter figures.

It would have been an elementary, intuitive process for the prehistoric carvers to adopt. All they needed was a simple trammel - a stick with a pointed flint at each end: hold one flint against the rock, and scribe an arc with the other. Then hold that still, and scratch with the other flint, alternating the two until the whole figure is scratched out. With this scratched line as a starting point, the groove could be pecked out with another sharp tool. It's an appealing idea, because it provides a feasible model of ring construction that can be tested for a possible unit of measurement. The mean diameter of each ring would be an estimate (of unknown accuracy) of the distance between the flints on the trammel that was used to scratch out its initial design on the rock. And perhaps that was a distance that (as Thom would have suggested) had been measured out? It meant that all those non-circular rings, carefully traced over several years, could be analysed statistically, after all.

When I carried out a statistical analysis of the 111 ring diameters, I was shocked by the outcome. It's clear in the diagram below that some particular diameters were very strongly preferred, and that those preferred values were very often close to a multiple of Thom's 'megalithic inch'.

The correspondences are not exact, of course, nor should we expect them to be. If the prehistoric carvers really did measure out their rings prior to carving them, we have no idea how accurately they might have been working, either while scratching the initial outlines (one can easily imagine the 'fixed' trammel tip slipping on the rock surface when the other tip is making the arc), or subsequently pecking out the grooves; also the final width of the grooves introduces uncertainty in our present-day estimate of the diameter, and the effect of millenia of weathering tends to widen the grooves still further.

The question that needs to be asked is whether such a striking distribution could have arisen purely by chance. Statistical analysis of the diameters for Thom's precisely specified MI (2.0725cm) showed that the probability of finding such a striking 'quantal' distribution in a set of random data was lower than 1%. But when the analysis was extended, what was even more striking was the discovery of another unit, in the same set of diameters, of a little over 10 cm - very close indeed to 5MI. The evidence for this was so strong that even if Thom's preliminary work were discarded, it would have to be taken seriously. The probability of a set of random diameters producing such a result by chance, for any unit at all (not just Thom's), was as low as 0.2%.

It isn't clear exactly how the results should be interpreted. If there really had been a formal unit of measurement as Thom suggests, then the presence of the 5MI unit in the same data may suggest that the carvers were using a counting base of 5, which might tend to make multiples of 5MI the most common.  In fact this tendency can be seen in the histogram above, where there are noticeable heaps at 5, 10, 15, and 20 MI. Alternatively, it may be that the carvers were using hands and fingers: 1 MI could, in reality, be an average fingerwidth, and 5MI an average handwidth. It's not easy to see how these two ideas could be distinguished. However, setting aside such speculation, it was clear that above all, another, wholly independent set of data was needed - preferably from another geographical area (i.e. not in Yorkshire or Northumberland), on which to test these findings further.

It was at this point that I received a timely invitation from the archaeologist Dr Euan MacKie, who was supervising a 'rescue' recording of an extensive cup and ring site at Greenland (Auchentorlie), near Glasgow, which was in danger from quarrying activities. His invitation was to test the carvings for evidence of prehistoric measurement, and it was perfect for my purpose. There were very many carved ring systems at this one site, which has been described by Ronald Morris as 'one of the finest examples of these carved rock surfaces in Scotland'. It was extensively photographed as part of the rescue project, and some of the images can be seen in the paper published, later, in the Glasgow Archaeological Journal.

Every ring system was traced onto polythene sheets using the usual method, and the rings were analysed using exactly the same procedure as I'd used with the Ilkley and Northumberland carvings. 79 rings were complete enough to be considered for analysis. Five of them were obviously not equal-width figures and were discarded, but the remaining 74 rings were acceptable approximations to equal-width figures. The distribution of their mean diameters was startlingly similar to the distribution of the English ring diameters. They can be compared below (I've limited the diameters to those under 25cm, to make the visual comparison easier):

Distribution of diameters from the site at Greenland, near Glasgow (up to 25cm)
Distribution of diameters from English sites (up to 25cm)
The similarity between these two histograms (or 'curvigrams' as they are called) - especially those big heaps close to 4, 5, and 6MI - was the most remarkable result I had found during the 8 or 9 years I'd been working in this area. Statistical analysis confirmed what was obvious merely from looking at the diagrams: the probability of obtaining such strong evidence for Thom's MI from random data was very small indeed (well below 0.1%). It was so strong that it had to be taken seriously, independently of Thom's original work. And as with the English sites, the 5MI unit was also strongly present in the data. The coherent structure within each set of data was strikingly similar.

What does it all mean?
Much of the controversy about Thom's ideas concerning measurement in prehistoric times arose not so much because the idea of prehistoric units of measurement was implausible, but because of his insistence on a very high degree of accuracy on the part of the megalith builders and prehistoric carvers. Thom's hypothesis proposed the use of standard MY and MI measures through prehistoric Britain, with accurate yardsticks to ensure consistent precision over a wide geographical area. This was never an idea that could be supported by statistical analysis of his data, and in fact there are many indications among the data from stone cicles, stone rows, and cup and ring marks that suggest the use of a rougher measure, perhaps based on some dimension of the human body that would be broadly consistent from one place to another. But we don't need to throw the baby out with the bathwater. The crucial aspect of Thom's hypothesis from the experimental point of view is that it makes a clear and precise prediction about the quanta we might expect to find when we examine the dimensions of megalithic sites, and that prediction can be tested statistically.

In short, we don't need to accept the full range of Thom's interpretations in order to find his hypotheses useful. We can still test for the presence of his 'MI' in a set of carved ring measurements (with its mean value of 2.0725 cm known in advance), while still being appropriately sceptical about Thom's own interpretation of it as part of a coherent measurement system related to the megalithic yard.

In fact there is a good statistical method for investigating the accuracy with which the unit was used. If we were dealing with rough measures involving, say, hand or finger widths, then errors would accumulate when larger distances were measured out, and this would make the presence of the unit less noticeable when we try to detect it. We can check this by dividing the diameters into two groups according to size, and finding out where the greatest evidence for the quantum lies. It turns out that the evidence for the MI lies overwhelmingly in the 92 smallest ring diameters. The 93 largest diameters provide very little evidence for the MI. In other words the smaller diameters are very closely grouped around multiples of the MI, and the larger diameters not very much at all. We can't be certain about our conclusion because the sample size of 185 diameters is not a large one, and splitting it in two reduces the sample size still further - but the indications, such as they are, are not favourable to the Thom hypothesis of an accurate unit. To interpret the MI as a fingerwidth (and the 5MI unit as a hand width, or as an indicator of a counting base of 5), seems to be the most obvious option available to us.

There are other interpretations worth bearing in mind, however. The statistical tests compare the support for a quantum offered by actual ring diameters with the results we would expect from random diameters. The tests tell us that the diameters are not random, and they are not random in a way that supports the idea of a unit of measurement. But this doesn't prove unequivocally that such a quantum really existed. For example, looking at the histograms for the Scottish and English diameters, we see that by far the most impressive peaks are at about 8 cm, 10 cm, and 12 cm (which may be interpreted as 4, 5, and 6 MI respectively). The bulk of the evidence for the quantum comes from these diameters at about 12 cm and under. Now, an argument could be put forward that three 'special' distances were in use: a 4-finger distance (handwidth-without-thumb); a full handwidth (including thumb); and a 6-finger distance. These would account for the three obvious, biggest peaks in the curvigrams. I suspect (though I haven't formally tested the idea) that a set of diameters created on such a basis, mixed with a collection of larger random diameters, could quite easily 'fool' the statistical test into falsely declaring the existence of a 1MI quantum.

Whether such an idea seems contrived or 'natural' is a matter for debate, and in any case there are not enough data at present to distinguish between these hypotheses. Still, whether one supposes a formal system of measurement to have been in use, or whether one favours a rougher, intuitive ritual of marking out with hands and fingers, one conclusion seems overwhelmingly probable: the same kind of ritual was in use at the Ilkley sites, the Northumberland sites, and at the Greenland site, near Glasgow. The diameters of the carved rings are not random, and their distinctive 'flattened' shapes are characteristic of the method of diameter construction; they are not merely the results of poor freehand attempts at circles.

During the course of working on cup and ring marks, I've been fortunate to have the help and encouragement of many people, and I'm grateful to them all. But steadily, throughout the whole of this work spread over years, I had continual encouragement and invaluable help from Richard Atkinson, Douglas Heggie, Euan MacKie, and Archie Thom. I couldn't have done it without them.


This work was completed at the end of the 1980s. I thought it shed important light on the study of cup and ring marks, and I submitted a detailed research article to the archaeological journal Antiquity. Two years passed, during which I made occasional enquiries and was reassured that it was under consideration. Finally I received word that although its content had been favourably refereed, it was not considered appropriate for the readership of the journal (a conclusion that, one might suppose, could have been reached in less than two years.) My personal circumstances had changed during those two years, and I was not in a position to pursue the matter further. I put the material in a cupboard, where it has remained for over twenty years. But a friend recently pointed out to me that it may as well be available on the internet, as shut away in a cupboard. Hence this blog. The original research paper (slightly modified in the light of further thought), which contains full mathematical details of the analysis, can be found here.