## The Megalithic Yard: A Consequence of Circumferential Distribution?

 Developing the Hypotheses It requires but little thought to appreciate that any circle with an integer number of metres on the circumference will have that same number of an apparent unit of 1m ÷ π (= 318.3mm) on the diameter, despite such a unit not having been introduced there, as at Figure 1. Figure 1: A Phantom Diametric Unit? The Megalithic Yard In the same way, any circle having an integer number of a unit with a length of 2.6m on the circumference will be found to have that same number of Thom’s Megalithic Yard (2.6m ÷ π = 829mm) on the diameter. The Megalithic Yard would thus be present on the diameters, but, in the same way as Figure 1 not necessarily consciously introduced. Sizing the Megalithic Yard Thom assigned a length of 2.72 feet or 32.64 inches (829mm) to the Megalithic Yard and suggested that the builders employed a unit of 2.5 Megalithic Yards on circumferences to overcome problems associated with the incommensurability of π. Though having rejected the Megalithic Yard as a common unit, archaeologists have not assessed the possibility of there having been a circumferential unit related to it and, also, this being potentially sub-divided and subject to a greater variance than Thom suggested. Such a potential circumferential unit of 2.6m (8.5 feet) might be appropriate for large circles but would be totally unsuited to small circles and would thus ideally be subdivided. It appears that the likely number of sub-divisions can be inferred logically from a subset of well-defined stone circles. A guide to the length of such a circumferential or perimetric unit can be deduced from a number of equally divided circles across the UK by assuming that all gaps are a multiple of a common unit. The determination is simpler by assuming that the Megalithic Yard is present on the diameters of these circles.

 Table 1: Size of Common Circumferential Unit Assuming the Existence of the Megalithic Yard Site Region Div. Diam. (m) MY Gap Units Size(mm) Stones of Stenness Scotland N 12 31.1 37.5 3.125 25 325.7 Cullerlie Scotland E 8 10 12 1.5 12 327.2 Balbirnie Scotland E 10 14.5 17.5 1.75 14 325.4 Machrie Moor V Inner Scotland W 8 11.6 14 1.75 14 325.4 Aubrey Ring England S 56 87.05 105 1.875 15 325.6

 In Table 1, ‘Gap’ is ‘MY’ (Megalithic Yards) divided by ‘Div.’ the number of divisions its unit length being Diam. ÷ MY × π. ‘Units’ renders the Gap in whole numbers by applying a multiplier of eight. Simply assume that the Aubrey Ring at Stonehenge has an integer number of perimetric units in each of its 56 arcs (gaps). The diameter of 87.05m (Cleal et al) is 105 Megalithic Yards, so there would be 105 corresponding perimetric units of 2.6m on the circumference. By simple division, each arc would be 1.875 such units. However, to make this an integer it should be multiplied by 8 to make 15. Thus, there would be 840 perimetric units of 325.5mm on the circumference of the Aubrey Ring and the Megalithic Yard in this case would have been subdivided into eight units of 103.625mm on the diameter which would be 840 such diametric units in length. Figure 2: Machrie Moor V, Arran. Machrie Moor V on the Isle of Arran, Scotland, has two concentric circles both with diameters in Megalithic Yards. The ratio between them is, perhaps significantly, 11:7 with the inner circle having eight equal divisions. The diameter of this circle at 11.6m would be 112 of the above derived diametric units and the circumference would thus be 8 x 14 corresponding perimetric units. Further, assuming that the outer circle uses the same unit of measure results in there being a half unit on the circumference, suggesting that the site Megalithic Yard is divided into 16 units of 51.8mm (2.04 inches) with a corresponding perimetric unit of 162.8mm (6.4 inches) as at right in Figure 3. Analysis of stone circles further afield, including Ireland, seems to bear this out. Figure 3: The Stones of Stenness and Balbirnie. The existence of such a perimetric unit is supported by the circles at Stenness (diameter 31.1m, 37.5MY) and Balbirnie (14.5m, 17.5MY) both in Scotland. The excavation reports suggest that these are not properly equally divided, but it can be appreciated that at each there appears to be a potential tallying error in a multiple of the suggested perimetric unit, as at A and B in Figure 3. At Stenness, the gaps are in multiples of ten units, at Balbirnie they are in multiples of four. It has been observed above that, given a circumferential unit, the Megalithic Yard need not actually have been introduced on the diameter in such cases - which is to say that it was not necessarily a megalithic unit of measure. Nevertheless, the converse could be true as there may be physical evidence in the form of a measuring rod to support its existence. Whether or not such was actually the case, a Bi-Metric Hypothesis proves useful in analysing stone circle dimensions and designs. It is therefore assumed to hold true for the purposes of this work. In all, over 300 stone circles and associated sites have been visited and surveyed, but not all are included in the descriptive statistics. The key consideration is that the distribution of orthostats on the perimeter might be used to indicate the multiple of the unit present on the diameter. Figure 4: An Example of Rational Distribution. Figure 4 provides an illustration of the process used throughout the study. At left is a token circle of 16.5m diameter with a circumference appearing to be unsystematically divided. However, when the gaps are expressed in degrees close inspection shows this to be split with balance about an axis. As the gaps are all multiples of 9 degrees the circle divides into 40 equal parts as illustrated at centre. The circumference thus being a multiple of 40 perimetric units the diameter is thereby a multiple of 40 diametric units. As the diameter is given as 16.5m (318.5 diametric units) the nearest multiple of 40 is 320 units (20 Megalithic Yards) making the site Megalithic Yard 825mm not 829mm. This can be further appreciated from an actual example at Figure 5. Figure 5: Loanhead of Daviot: Rational Distribution. At left, the gaps between the orthostats are shown in degrees suggesting that the circumference is divided into 48 equal parts (all multiples of 7.5 degrees) as illustrated in the diagram at centre. So, the diameter would be a multiple of this same number of diametric units which is 2.5 metres. The diameter was surveyed at 20.2 metres which is 390 diametric units suggesting 384 units (24 Megalithic Yards) as the nearest multiple of 48. Thus, the Megalithic Yard at this site would be 842mm not 829mm. The key feature of analysis is the assumption that orthostats are positioned on perimeters with pattern and balance about an axis and that the gaps are a whole number of the common circumferential unit, as seen at right in the figure. In this case, the axis formed by the distribution bears from 285 to 105 degrees, offset by one-twenty-fourth of a revolution (16 perimetric units). Despite an initial semblance of irregularity, nearly all stone circles appear to have a balanced Axial Frame (frequently consisting of five points) with the gaps sub-divided according to pattern. It is the sub-division of these balanced segments that produces the apparent lack of order, but it should be appreciated that the angles formed by the gaps on the circumference would still disclose the base division of the circle.

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