Research Paper




 The  Metrology and  Design of Cup and Ring Carvings

   Alan Davis   



 
Abstract

The characteristic 'style' evident in the shapes of many carved rings from prehistoric cup and ring sites in northern England can be explained in terms of an elementary geometry which arises directly from a method of 'diameter construction' using simple apparatus. When this model of construction is assumed, substantial evidence is found for a system of measurement based upon a unit of about 2.1 cm - very close to the 'megalithic inch' proposed by Alexander Thom - together with indications of a possible counting base of 5. Strong supporting evidence for these hypotheses is then obtained by analysing a further set of carved rings at Greenland, Dunbartonshire, suggesting the use by the prehistoric carvers of a consistent design ritual in northern England and southern Scotland. The evidence is consistent with the use of hand and/or finger widths as a measure.

***** 
   In parts of Scotland and northern England are to be found hundreds of examples of the so-called 'cup and ring' marks, carved on the surfaces of rock outcrops, slabs, and boulders. Part of the fascination of their study lies in the apparent simplicity of the basic motif - usually a single small depression or 'cup', surrounded by one or more carved 'rings'. George Tate, writing in 1865, was aware of it: '...their wide distribution, and notwithstanding differences in detail, their family resemblance prove that they had a common origin, and indicate a symbolic meaning representing some popular thought ...' (Tate 1865).

   This family resemblance is not confined to the mere repetition of the cup and ring motif. Certain repeated qualities of shape are inherent in many of the carved rings, which may not be obvious from a casual inspection of the (invariably weathered) rock surface, but which are readily seen in rubbings or tracings of the carvings. Numerous examples may be found in published illustrations (eg Hadingham 1974; Davis 1988), and those given in Figures 1 and 2 of this present article are characteristic. Evidently many - indeed most - carved rings are not truly circular, though this is not to imply that they are merely crude or botched attempts. On the contrary, their characteristic 'flattened' arcs form a recognisable style, and its repeated appearance at sites which are widely separated geographically makes it seem clear that such rings are not just bad shots at freehand-drawn circles. It seems inevitable that a geometrical origin for the shapes should be sought.

   The most detailed geometrical study of the carvings has been made by Alexander Thom (Thom 1968; 1969; Thom and Thom 1978; Thom and Thom 1988). Working from rubbings of Scottish carvings, and by analysing diameters of apparently circular rings, he first deduced the existence of a prehistoric measurement, the 'megalithic inch' (MI) of about 2.07 cm. He went on to show how non-circular rings and spiral carvings could have been designed using circular and/or elliptical arcs whose diameters were measured out in MI.

   Thom's papers amply demonstrate the feasibility of his hypotheses, but they fall short of providing any convincing proof of them. This is partly because the statistical evidence he provides for the reality of the MI is unsatisfactory (see Davis 1988, pp. 393-5). The situation in this respect has improved since Thom's original work (Davis 1983a; 1988), but even so there is no compelling reason why the particular type of geometrical construction proposed by Thom should be accepted. Thom's argument essentially hinges on the undoubted fact that every non-circular design he examined could be explained in terms of elliptical arcs based on measurement in MI, but in order to achieve a satisfactory fit to the rubbings, dimensions  are often required to have been measured out in half-MI or even quarter-MI units. These are such small values, and the rubbed images of carved grooves so wide in comparison, that one suspects that such geometries could be fitted adequately to any series of curves, whether based on a system of measurement or not.

   Thom's geometrical hypotheses remain unproven, then, but in the search for an explanation of the origin of the carved shapes, the basic notion of an underlying geometry of some sort seems almost inescapable. This does not necessarily imply that the designers of the carvings possessed sophisticated geometrical knowledge however. Those puzzling 'flattened' arcs may be just an inevitable by-product of whatever mechanical process was used to lay out the carvings. A paper by Thaddeus Cowan (Cowan 1988) offers a possible alternative approach along these lines, and forms the basis for the rest of this paper. 

Equal-width figures and 'diameter construction' 

   Cowan’s paper concerns the design of stone circles, but its principles are equally applicable to the problem of the shapes of carved rings. Essentially, Cowan considers alternative concepts underlying the practical construction of a circle. In modern times, perhaps because of early training with compasses, we tend to conceive of a circle as a figure of constant radius, set out from a fixed centre. But in certain ways the concept of diameter may be more readily apprehended than that of radius, and perhaps was indeed so apprehended in prehistory. However, as we shall see, ‘diameter construction’ (as opposed to ‘radius construction’) does not necessarily produce a true circle.

   For the initial laying-out of the rings, prior to carving, we must suppose the use of some elementary piece of apparatus – essentially a trammel: a length of wood with a sharpened flint fixed at each end would be adequate. The following examples illustrate possible (idealised) sequences of construction. (These manoeuvres are simple and intuitively obvious to perform, but less simple to describe): 

Method 1. Suppose the trammel is held with a flint in each hand. Let each flint be moved at the same speed, at right angles  to the stick, in (say) a clockwise direction. Then a true circle is traced out, whose diameter is simply the distance between the points of the flints. 

Method 2. Suppose instead that one flint (A) is held still while the other (B) is moved clockwise a little way. B traces out an arc whose radius is the separation of the flints, and which is centred on A. Let A now be moved clockwise while B is held still, and so on. In due course a closed figure is traced out which consists of a series of abutted circular arcs, whose radii are equal to the separation of the flints. It is not a circle, but it is, as Cowan points out, a figure of ‘equal width’. The width or diameter (equal to the separation of the flints) will be the same regardless of the direction in which we measure it. (A 50 p piece is a regular example of such a figure.)

   It is important to note at this point that neither of the above methods is being proposed as a rigid prescription for the design of a petroglyph. Methods 1 and 2 are formalised ‘ideal’ routines which may be combined or varied in numerous ways and, indeed, alternatives may be found in Cowan’s paper. Their importance lies in the fact that (a) they illustrate the general notion of ‘diameter’ (as opposed to ‘radius’) construction; and (b) that such a construction method, while not necessarily leading to the drawing of a perfect circle, does tend to produce a figure of equal width. Further, a little experimentation with an actual trammel soon shows that this is a very natural, intuitive method for constructing closed convex figures.

   It is probably impossible to prove that any particular geometrical scheme was employed in the construction of this or that carved ring, but if the ‘diameter construction’ hypothesis is to be a serious contender, then our first step is surely clear. We must demonstrate that it is capable of producing shapes typical of those we find carved on the rocks. 
Figure 1. Carved rings from the Panorama Stone, Ilkley

Figure 1 is based on an accurately reduced tracing of a series of carved rings from the Panorama Stone, at Ilkley in Yorkshire. The geometrical construction superimposed on it was obtained as follows, using a fixed diameter trammel, in ten successive movements, thus:

1. Centre A, draw arc PB
2. Centre B, draw arc AC
3. Centre C, draw arc BD
4. Centre D, draw arc CE
5. Centre E, draw arc DF
6. Centre F, draw arc EG
7. Centre G, draw arc FH
8. Centre H, draw arc GI
9. Centre I, draw arc HK (joining up with arc CA produced)
10. Centre J, draw arc IP (joining up with arc BP)


   The ring is almost (but not exactly) a figure of equal width. In order to complete it, the closing arc had to be centred on J, rather than K, for an acceptable fit. This is the sort of slipping of arc centres, in fact, which would be almost certain to occur in the actual execution of such a figure with real flints fastened on a real stick, scratching on a real rock surface. Such slippage of the trammel may also explain why the theoretical construction systematically deviates from the apparent centre of the groove in places, although there are many possible explanations for deviations of this sort. The deepest part of a groove, for instance, is not necessarily midway between the groove edges; weathering is often uneven in its effects; and it is possible that with further trials, a better fit could be found. But such conjectures are of little value and miss the point of this exercise, which is not to convince the reader on grounds of ‘goodness of fit’ that the exact original design has been recovered. We can never claim that. The important conclusion is that a simple method of diameter construction can readily produce a ring of this very typical shape. The diameter construction hypothesis is, at least, a plausible contender.

Figure 2. Carved rings from Weetwood Moor, Northumberland

A further example is offered in Figure 2, which shows part of a carving traced from Weetwood Moor, Northumberland, with a geometrical figure superimposed. This also is a type of ‘diameter construction’, consisting of arcs whose centres lie elsewhere on the perimeter of the ring (points A to G), though the sequence in which the arcs are constructed does not follow the strictly alternating pattern shown in Figure 1. Here, some of the arc centres lie somewhere in the middle of previously drawn arcs, rather than at one end. Only one arc (the one centred at H) fails to fit the pattern – and again this is the sort of error that one would expect to occur as the construction proceeded. It is not, strictly, an equal width figure – though it is approximately so.

   It should now be clear that the typical shapes encountered among the carved rings are readily explicable in terms of a design method involving ‘diameter construction’, and this has consequences. If such a design method had indeed been used, we should expect the majority of prehistoric carved rings to be, at least to a sensible approximation, figures of constant width. Here, then, we have a prediction which is potentially testable.

The diameters of carved rings

   Over a number of years during the 1980s, I  made accurate tracings of cup and ring carvings from sites in Yorkshire and Northumberland, and small-scale copies of these have been published (Davis 1988). From this collection, 119 carved rings are sufficiently well-preserved and complete to make a useful contribution to the present discussion. The tracings were made with felt-tipped pens on transparent sheets, after a preliminary chalking-in of the design on the rock surface. Lines were drawn along estimated centres of carved grooves, and also along their edges. This means that for all the carvings it is possible to estimate not only the diameter of a ring, but also the uncertainty in that estimate due to the width of the groove.


   Details of the 119 carved rings are given in Table 1. The various identifying letters refer to the plans published in Davis (1988), and in the majority of cases these correspond to labelled cupmarks around which the rings are carved. In a few cases no identifying letter is given, for reasons which will be obvious from inspection of the plans; spiral carvings are not included. For each ring, diameters were measured in each of 6 directions, at 30 degree intervals, except in cases where the ring was incomplete as a result of damage or weathering. A criterion was established at the outset that to qualify for inclusion in this study, a ring must be sufficiently complete for at least three such diameters to be measurable. Table 1 lists the mean diameter D, the mean groove width w, and the maximum and minimum measured diameter, D(max) and D(min), for each ring.


 We require to know whether the data presented in Table 1 are consistent with the hypothesis of 'diameter construction' - which implies that we should be dealing with ring shapes of at least approximately constant diameter. But of course whenever we measure a diameter, our estimate of it is uncertain for two reasons:

1. We are dealing with grooves of finite width, produced by whatever abrasive or pecking process was used by the prehistoric carvers. Presumably the grooves were carved following some preliminary marking, but of course those markings would be lost as carving proceeded. We do not know how accurately the carvers managed to follow the intended lines.
2. Weathering has widened the grooves still further in the majority of cases.

   Basically, we must approach each ring with the following question: can any variations in diameter be reasonably attributed to uncertainties due to groove width? It seems reasonable to assume that if the maximum variation in measured diameter for a given ring is less than twice the groove width, then such a ring is not inconsistent with the 'equal width' hypothesis. The ratio (D(max)-D(min))/w, therefore, should be less than 2.00 if the ring is to be acceptable. (This is not an arbitrary choice: by adopting the figure 2.00, we ensure that we eliminate all rings which are definitely not equal width figures.)

Inspection of Table 1 shows that 111 rings satisfy this criterion, and 8 do not. These 8 rings will not be considered further. Indeed, we can say little about them: they may have been scribed freehand, or by some other geometrically-based process; or they may just be badly executed 'equal width' designs. But the remaining 111 rings are not inconsistent with the 'equal width' hypothesis: clearly the hypothesis fits the majority of the facts tolerably well, within sensible limits. Again, this does not constitute proof that such design methods were actually used in prehistory, but it does provide us with a promising realistic model which can be investigated further. This has important consequences for metrological study of the carvings. Alexander Thom, in the statistical analysis from which he derived his 'megalithic inch' (MI) hypothesis, selected only those carved rings which he supposed to be circular. Further testing of Thom's hypothesis by myself was disadvantaged by similar subjective choices (Davis 1983a), and in an attempt to overcome the problem, subsequent work concentrated purely on the analysis of the distances between cup-mark centres (Davis 1988). But if the 'diameter construction' hypothesis is correct, then the mean diameter of a ring, as given in Table 1, should be a useful estimate of the original separation of the flints in the trammel used by the designer to lay it out. If this separation were initially a measured distance, perhaps in MI or some other unit, then statistical analysis may be capable of demonstrating this. If on the other hand the rings were drawn freehand, we should expect their mean diameters to be more or less randomly distributed. (This may also be our expectation even if the non-circular rings were designed by sophisticated methods such as those suggested by Thom, involving combinations of circular and elliptical arcs; in such cases the 'mean diameter' would be a quantity of no clear significance except in the case of a true circle.) The question arises, therefore, as to whether there is any evidence of mensuration in the data of Table 1.

The metrology of the carved ring diameters

   We shall use Kendall's method to perform a quantum analysis of the data (Kendall 1974). The use of this method has been described in detail elsewhere by the author (Davis 1988), and here it is sufficient merely to outline the method. It employs the test statistic Φ, where 
Here N is the size of the sample, T=1/q (where q is the quantum under test), and the values of y are the individual measurements in the sample. Two types of analysis are possible:
Type A. This is appropriate where we wish to test a set of independent data for a quantum whose value has been determined in advance. On the null hypothesis that the data are random, Φ is expected to be normally distributed with mean zero and unit standard deviation.
Type B. Here, adopting no prior assumptions about any particular quantum, we examine the way in which Φ varies over some sensible range of T values. The resulting graphical display of Φ against T is called a 'cosine quantogram' or CQG, and evidence for a quantum is implied by the presence of a significantly large peak at some particular value of T. 

   Since none of the carvings in the present sample were included in Thom's original paper (Thom 1968) where the MI hypothesis was proposed, it is permissable to perform a Type A test upon the 111 diameters obtained from Table 1, for the 1MI quantum of 2.0725 cm. We find that Φ(1MI)=2.69. In other words, the hypothesis that the diameters are randomly distributed is rejected at a probability level of 0.4%. The best estimate of the value of the quantum is 2.080 cm, where Φ=2.75.

Clearly there is substantial support for Thom's megalithic inch hypothesis among the present data, but a Type B analysis is required to see whether evidence for any other quantum exists. The question of a suitable 'search range' for possible quanta is discussed in Davis (1988), and it is both appropriate and convenient to adopt the same limits here: upper and lower limits for the 'q' values are 12.5cm and 1.72cm respectively (corresponding to values of 0.08 and 0.58 for T).


Figure 3. Cosine quantogram for 111 ring diameters from Yorkshire and Northumberland

 Figure 3 presents the CQG over this range for the 111 ring diameters, and we see that two peaks dominate all others. The one on the right of the diagram corresponds to 1 MI (marked by an arrow) and is 2.751 units high. The peak on the left, corresponding to a quantum of 10.09 cm, is substantially higher at 4.161 units. A convenient chart for the estimation of significance levels for Type B tests is given in Davis (1986), and in order to use this we need the root mean square value (rms(y)) of the diameters, which in this case is 20.85cm. The corresponding probability level for this highest peak is so small that it is impossible to estimate it accurately, but it appears to be about 0.2%.

   Obviously these diameters are far from being randomly distributed, and when it is realised that the quantum indicated by the largest peak in the CQG does not differ significantly from 5MI, a coherent picture begins to emerge. It is possible to explain the presence of these two quanta in the same data set in terms of a basic unit of 1MI, and a counting base of 5. In other words here, where an appreciable number of diameters are approximately integer multiples of 1 MI, diameters which are multiples of 5MI predominate. The effect is demonstrated clearly by inspection of figures 4 and 5. 

Figure 4. Curvigram for 111 diameters from Yorkshire and Northumberland

   Figure 4 displays the distribution of the diameters as a sort of probability histogram, or 'curvigram', where the tendency for the data is to pile into humps at MI intervals is evident - particularly so for the smaller diameters. The predominance of 5 MI muliples is less easy to see, and is better demonstrated by Figure 5.
Figure 5. Curvigram of residuals from 5MI multiples for the 111 diameters from Yorkshire and Northumberland
Figure 5 is also of curvigram type, but here each data item is the residual from the next lowest multiple of 5MI. Thus a diameter corresponding to, say, 8.1 MI would be plotted on the diagram as 3.1MI; 14.2MI would be plotted as 4.2MI, and so on. The heaps at left and right make the preference for 5 MI multiples very obvious.

   This is not, in fact, the first appearance of a 5MI quantum. Significant evidence for its existence has been found previously in various analyses of cup-mark separations (Davis 1983a; 1988). The fact that statistically significant evidence for the same quantum should be found in two data samples which are effectively independent (ring diameters and cup separations) provides a strong argument for the reality of the quantum.

Other interpretations of the quantum tests

Two alternative avenues of thought are worth pursuing at this stage, and the first concerns whether we are, in fact, forced to accept the reality of both the 5MI and 1MI units. The 5MI peak in the CQG is much higher than the 1MI peak, and although this is not surprising in itself because the smaller quantum is more easily drowned out by measuring errors, it does raise the following question: given that the data are indeed grouped around multiples of 5MI, could the additional grouping around 1MI multiples be explicable purely in terms of this? In other words, although the 1MI quantum is statistically significant when compared against random data, does it remain significant when judged against data which are approximate multiples of 5MI?

The problem can be readily tackled through computer simulations as follows. The observed value of Φ (4.16) for the 5MI quantum corresponds to a standard deviation (s.d.) of 2.57cm for the individual diameters. For each diameter we calculate the nearest exact multiple, m, of the 5MI quantum. We then generate a set of 111 artificial diameters, constructed according to the formula y=mq+e, where q=10.09cm and e is a pseudorandom error with zero mean and standard deviation 2.57cm. Using this data set we calculate Φ for the 1MI quantum.

100 such simulations were performed, and the largest three values of Φ(1MI) thereby obtained were 3.113, 1.873, and 1.801. Only the first of these exceeds the value obtained for the real data (2.691), and so even when viewed against this background, the 1MI quantum remains significant. The hypothesis that it arises by chance from a distribution which is only roughly based on a 5MI quantum is rejected at the 1% level.

Our second alternative arises because the present data are not, in a sense, wholly 'new'. 36 of these rings were included in a previously published analysis (Davis 1983a) and in that paper the rings were assumed to be circular (although their diameters were at that time estimated by a less satisfactory method). We should check, at this stage, the effect of the inclusion of these 36 (nominally circular) rings in the current larger set of 111 (equal width) diameters. The simplest approach is to omit the 36 'assumed circular' rings and repeat the quantum test on the remaining data. This leaves 75 diameters, the CQG for which is little different from the full sample of 111 diameters. With the 'circular' diameters excluded, we have Φ(1MI)=2.588, and Φ(5MI)=3.449, compared with 2.691 and 3.928 respectively, for the whole set. Φ, then, is reduced slightly for both quanta, but this is not surprising because of the reduction in sample size. It seems clear that neither the inclusion nor exclusion of the 36 'circular' diameters has any significant bearing on our conclusions.

Extending the data base: the carvings at Greenland

 Shortly after the foregoing analyses were completed, I was invited by Dr E.W. MacKie to contribute to an investigation of a carved site at Greenland, a few miles west of Glasgow, which was in danger due to quarrying. (The site is recorded by Morris (1981) as 'Greenland 1'.) This provided an opportunity for testing the above hypotheses which was ideal on several counts:
1. It would extend the investigation into Scotland.
2. None of Thom's data were taken from this site, so that Type A tests for the 1 MI quantum may be used.
3. The site is extensively carved, and capable of providing a substantial body of data.
4. Morris considers it 'one of the finest examples of these carved rock surfaces in Scotland'.

   For the metrological study, the carved rings were traced onto transparent polythene sheets in the usual way. The diameters and other details of these rings are given in MacKie and Davis (1989), where plans, photographs, and a comprehensive description of the whole site may also be found. Using selection criteria identical to those already discussed, 79 rings proved suitable for analysis, i.e. were sufficiently complete for at least three diameters to be measurable out of a maximum of six taken at 30 degree intervals. The ratio (D(max)-D(min))/w was found to be less than 2.00 for 74 of the 79 rings. In other words, 74 rings possess shapes which are not inconsistent with the 'equal width' hypothesis. As before, the other 5 ring diameters were discarded.

   Analysis of the 74 rings reveals three areas of consistency with the English data, as follows:
1. A Type A test for the 1MI quantum yields the result Φ(1MI)=2.16. The hypothesis that the diameters are randomly distributed may therefore be rejected at a probability level of 1.5%. The optimum value of the quantum is 2.086cm, for which Φ=2.23. Clearly there is substantial support here for the use of the same 1MI quantum at Greenland as we find in the English data.
2. Testing the 74 Greenland diameters for the 5MI quantum, we find Q(5MI)=2.83 (the optimum value of this quantum being 10.34cm, or 4.99MI). It appears that at Greenland, as at the English sites, a counting base of 5 was in use.
3. There is an astonishing similarity between the overall distribution of the Greenland diameters and that of the English diameters. Figure 6 gives the curvigram for the Greenland diameters up to 25cm, and Figure 7 shows the equivalent diagram for the English diameters (which is an expansion of the left hand half of Figure 4). Comparison of these two diagrams provides almost inescapable evidence that the same design-and-measurement ritual was in use at Greenland as at the majority of English sites.
Figure 6. Curvigram for diameters from Greenland (up to 25cm)
Figure 7. Curvigram for all English diameters (up to 25cm)
It is crucial in a statistical study such as this for the full sequence of events to be transparent and free from selective bias, and to that end the formulation of the original hypothesis based on the English sites, and its subsequent testing on the Greenland data, have been described in detail. Having done this, however, we may now usefully combine the two sets of data to obtain a single large sample of 185 diameters. A Type A test of this set for the 1MI quantum yields Φ(1MI)=3.45, corresponding to a probablity level of about 0.03%. A Type B test is also instructive. Figure 8 (below) gives the CQG for these data, and it is clear that the familiar peaks at 1MI and 5MI are overwhelmingly dominant. The 1MI peak occurs at 2.081cm, where Φ=3.53, indicating, by reference to the appropriate diagram in Davis (1986), a Type B probability level of about 1.5%. In other words, the present data alone provide statistically significant evidence in support of the 1MI quantum, independently of Thom's original work, and also of the material presented in Davis (1988).

Figure 8. Cosine quantogram for 185 diameters (Greenland + English data)

Analysis of the rings with the 'best' shapes

It will be recalled that our criterion for acceptance of a ring as a possible 'equal width' figure was that the ratio (D(max)-D(min))/w should be less than 2.00. It is of interest, however, to examine the effect of reducing this acceptance criterion - say from 2.00 to 1.00 - in order to study just those rings which most closely approximate to equal width figures. 135 rings satisfy this new criterion. Testing these data for the 1MI quantum, we find Φ(1MI)=3.73 (the optimum value of the quantum being 2.076cm, with Φ=3.76). Now if the quantum is real, we would expect Φ to be proportional to the square root of the sample size; and so, on the basis of the reduction in sample size from 185 to 135, we would expect the value of Φ(1MI) to fall to about 3.45 x √(135/185), or 2.95. In fact we observe that Φ has risen to 3.76, and the meaning of this result is clear. The evidence for the 1MI quantum resides most strongly in those rings which have the 'best' shape; i.e. in those rings which were either more carefully executed, or which have suffered least from weathering, or both. This may be due to chance, but it is nevertheless an eminently sensible finding, and a gratifying demonstration of the internal coherence of the data. 

The nature of the quantum

 We can gain some insight into the nature of the quantum by dividing the entire set of 185 diameters into two groups - the 92 smallest diameters, and the 93 largest - and then testing each group for the 1MI quantum. We thus obtain:
92 smallest diameters: Φ(1MI)=3.69
93 largest diameters: Φ(1MI)=1.19
 The first result corresponds to a probability level of about 0.01% - a result of overwhelming significance. The result for the largest diameters, however, has no statistical significance. In other words, almost all the evidence for the 1MI quantum resides in the smallest diameters. Of course we must beware of deducing too much. Our subdivision reduces the sample sizes, and random fluctuations are to be expected. But as far as it goes, the implication of this result for the meaning of the MI is clear: this is the behaviour one would expect of a rough quantum (Davis 1983b). To interpret the 5MI quantum as an average handwidth, and the 1MI quantum as an average fingerwidth, would then be an obvious first step, as suggested in Davis (1988). Such use of hands and fingers may be all that we need to explain the indications of an apparent counting base of 5, although it seems impossible to distinguish between these alternatives on the basis of present data.

Some objections 

It is not unreasonable to ask whether the entire notion of diameter construction and measurement is an unnecessary complication - that, after all, it would be simpler merely to assume that the rings were carved freehand. Indeed, some of them may have been carved freehand: it is a limitation of a statistical treatment that it can show up only the overall trends in a set of data, and can say nothing about individual cases. However, as a general explanation of the ring shapes, the 'freehand' hypothesis suffers from two serious problems:
1. It does not satisfactorily explain how the characteristic and highly recognisable cup and ring 'style' arises, involving those typical 'flattened' arcs that were discussed at the beginning of this paper. 
2. It does not account for the fact that the mean diameters of the carved rings are distributed in a highly non-random fashion, in a way that strongly favours the quantum hypothesis. The mean diameter of a freehand ring would be, one presumes, a fairly meaningless quantity, and the distribution of such diameters would be, if not strictly random, at least irregular and largely unpredictable. In short, it is very difficult indeed to reconcile the 'freehand' hypothesis with Figures 5, 6, 7, and 8.

   There is another, perhaps more cogent, objection which we should consider. Many of the ring systems are concentric, and it is arguable whether diameters obtained from a set of concentric rings may be considered to be independent of each other. Even if no measurements were made in laying them out, in a concentric system the diameters of the rings cannot strictly be random because the choice of position and diameter of one ring in such a set will to some (unknown) extent influence the positions and diameters of the others during the original layout of the design. This inter-dependence may in some way bias the analysis, giving an apparent but misleading boost to the evidence for the quantum. Fortunately we can readily estimate the extent of any such bias by eliminating all but the inner ring of concentric systems from our analysis. This leaves us with 116 rings, for which Φ(1MI)=3.50. Now, if the inclusion of concentric ring systems were indeed enhancing the evidence for the quantum in a spurious way, we should expect their omission to produce a substantial decrease in Φ - and this clearly does not occur. We may conclude with some confidence that the inclusion of concentric ring systems is not seriously misleading us about the statistical significance of the quantum.


Conclusions

We may summarise our findings as follows: 

1. In a sample of 119 carved rings from Yorkshire and Northumberland, 111 are acceptable approximations to 'equal width' figures. They are consistent with the hypothesis that they were designed by a simple method of diameter construction, such as the use of an elementary form of trammel.
2. On the above hypothesis, the mean diameter of a carved ring should be an acceptable estimate of the distance between the tips of the original carver's trammel, and therefore a plausible quantity for quantum analysis. Strong evidence is obtained from these diameters for two quanta: a small quantum of mean value about 2.08cm (which invites identification with Thom's MI) and a larger quantum close to 5MI. This may imply the use of a counting base of 5 by a large proportion of the carvers (but also see below).
3. Further testing of these hypotheses on a fresh, independent body of carved rings from a Scottish site, Greenland 1, substantially confirms all these findings. Combining these additional data with the English data, we obtain evidence for the existence of a quantum close to 1MI which is statistically significant at very low probability levels. There are strong indications that this was not actually the basis for a formal system of measurement such as that proposed by Thom, but that it was a rough measure such as an average finger-width. (The 5MI quantum then either becomes identifiable as a hand-width, or appears as the by-product of a counting base of 5). It may be that the prehistoric carvers did not conceive the process as involving measurement, in the strict, Thomian sense, at all. Rather, the quantum may be a natural by-product of a process involving the intuitive use of hands and fingers during the making of the trammels.
 4. There is substantial coherence within the body of data:
(a) It provides evidence not only for two quanta (1MI and 5MI), but for  quanta which make intuitive sense in terms of human body measures, regardless of whether we identify the 5MI quantum as a handwidth, or as the consequence of an elementary intuitive counting system based on the number 5.
(b) The rings with the best shapes provide the best evidence for the quantum. 
(c) Figures 6 and 7 alone, regardless of any statistical analysis, provide compelling evidence that a consistent design ritual was in use over a wide geographical area.

A summary of the various quantum analyses is given in Tables 2 and 3.

In conclusion, then, by combining the geometrical hypothesis of Cowan with the metrological hypothesis of Thom, a coherent model for the design of cup and ring marks is obtained, which explains many of their markedly non-random features. An intuitive design ritual based on hand or fingerwidths, combined with an elementary method of diameter construction, is suggested as the most plausible explanation for these findings.

Acknowledgements

The material on which this paper is based was collected and analysed in the late 1980s and the paper has remained unpublished since then. Thanks are due to Douglas Heggie, who made helpful suggestions on an earlier draft of it, and on much else besides; also for the essential help and encouragement, in various ways, provided by Euan MacKie, Archie Thom, and Thaddeus Cowan (these last two, sadly, no longer with us).