Megalithic Metrology and Design

An Alternative Route to the Megalithic Yard

The Myth, or Otherwise, of the Megalithic Yard Thesis: That archaeologists may have failed to find Alexander Thom’s megalithic yard (MY) in the diameters of stone circles in Britain and Ireland for four reasons: 1. The megalithic yard may well appear on stone circle diameters as a consequence of there being a common unit present on circumferences (effectively, MY can be calculated as a circumferential length of 2.6m divided by pi). 2. Analysis suggests that this circumferential unit of 2.6m can be perceived as being divided into 16 parts, that is, 16 perimetric units, with the result that the megalithic yard (if it truly exists as a physical entity rather than as a mathematical consequence) is similarly divided. 3. It follows that a whole number of perimetric units on a circumference will frequently produce fractions of a megalithic yard on the diameter. 4. From extensive analysis, the perimetric unit has been found to vary in length by up to two percent either side of the mean MY (829mm, which is also the median and mode). Such variance would therefore distort calculations using Thom's mean megalithic yard. Hence, only infrequently will a whole number of megalithic yards be found on a stone circle diameter. That is to say, the unit is present in the diameters, but not glaringly so. By way of greater detail, the two articles presented here were first published in Northern Earth (issue 153, June 2018 pp. 22f. and ussue 155, December 2018 pp. 21-5) and are reproduced with kind permission.

An Alternative Route to the Megalithic Yard (Part 1) G.J. Bath Aubrey Burl, archaeology’s leading authority on stone circles, declares that, “...the countrywide Megalithic Yard is a chimera, a grotesque statistical misconception,” (Burl, 1979: 51). He is presumably reflecting the preconceptions of his mentors, but does not explain why he and archaeology present this as a fact rather than concede that theirs is supposition based on limited data and even more limited study. Whilst archaeology concedes that there is evidence of Thom’s megalithic yard in certain regions, it stands out against there having been a common, let alone uniform, system of measurement and unit of measure across Britain and Ireland. There is something intellectually disturbing about an academic discipline taking the negative stand that, “There is no x,” in such cases. The position might be perceived as arrogant, and certainly seems unscientific. It requires the assertion that all future data and analysis will fail to lend further support for ‘x’. How can archaeologists claim knowledge of this? Some might argue that however cogent the case for the megalithic yard, limitations in the quality and quantity of the data suggest that investigation should be abandoned. That would surely be defeatist. At the very least, archaeology should maintain the hypothesis as a possibility, and search, or wait, for new data to appear, and look for new approaches that might resolve the uncertainty. Lack of evidence is not evidence of lack. Archaeologists may choose to take a subjective view concerning the likelihood of their supposition, but the objective (statistical) evidence remains, and investigation into megalithic metrology and the uniform unit should perhaps not have been ruled out as categorically as Burl declares. Thus, a supplementary argument suggesting that the hypothesis may yet be viable, and further developed, is presented here. Background Alexander Thom’s statistical analysis of a subset of assumed reliable stone circle diameters reveals a potentially common unit of 829mm which he termed the megalithic yard. Thom also suggested that stone circles that are not truly circular fall into one of four categories - flattened circles, egg-shapes, ellipses and compound types - each with a different design mechanism. In contrast, this paper suggests that the megalith builders may well have employed a perimetric unit having a pi relationship with the megalithic yard, which unit may well have been subdivided. While a large unit might be appropriate to large circles, a small unit would be more suitable for small circles. Furthermore, it is suggested that rather than there being four different design practices for Thom’s suggested design types there could well have been just one design philosophy covering all (see Fig. 1).

Figure 1: Arc-constructs: a single concept for drawing Thom’s designs and others. The text string parameters are angles and fractions required to form the figures.

Deriving a Perimetric Unit The megalithic yard does not only emerge from statistical analysis, it can be logically deduced. Archaeology’s declaration of no uniform unit can be challenged. Proceed by assuming that there is a megalithic peripheral unit at the Aubrey Ring at Stonehenge, in the south of England, and that there is an integer number of this unit on the circumference. This, being composed of 56 fairly evenly spaced holes, would sensibly have a circumference divisible by 56 units, because each of the presumably equal arcs (gaps) would likely have an integer number of units of measure. Then, assume that the same applies to the inner circle of Machrie Moor V, at the west of Scotland, which is equally spaced with eight arcs (gaps): that is, assume that each arc is an integer number of the exact same unit employed at the Aubrey Ring. In effect, assume that there is a uniform perimetric unit across England and Scotland. Taking the Aubrey Ring diameter as 87.05m (Cleal et al, 1995) and that of Machrie Moor V as 11.6m (Burl, 2000, Gazetteer, and by survey), each Aubrey Arc would be 4.883m and each Machrie Moor V Arc would be 4.555m. So, by combination and calculation, the favoured common unit would be 325.5mm (approximately the difference between the two arcs). The reasoning can be applied to another circle having apparently equal divisions on the circumference. Stennes, Orkney, has twelve divisions. The diameter is given as 32.3m x 30.6m (Ritchie, 1975), apparently very slightly elliptical. The average is 31.4m. This would resolve to a circumference having 300 of the above units (the nearest multiple of twelve), each gap being 25 units. One possible consequence is that it might be the presence of this unit on the perimeters of stone circles that produces the megalithic yard on the diameter. Simply consider that a circle with a circumference of this peripheral unit of 325.5mm would have a diameter of 103.6mm, being one eighth of a megalithic yard, which length has been potentially found in cup-and-ring motifs in northeast England as five of Thom’s megalithic inches (Davis, 1988: 413). Note that a circle with a circumference of eight such units would have a diameter of one megalithic yard. However, in reality, the same circumference could be described using a diameter of 2.55 peripheral units, and the megalithic yard might simply be a consequence of pi, having no existence as a unit. This seems not to have occurred to anyone following up Thom’s work. Consequences When extending the hypothesis to consider whether the gaps between the orthostats at all stone circles might be measured in this same unit, two points emerge. First, the common perimetric unit appears to be half that derived above, that is, it is 162.8mm, which might argue that the megalithic yard, if it exists, is divided into sixteen equal parts of 51.8mm. However, consider the following. A circle with a circumference of 16m has a diameter of 5.1m, and a circle with a circumference of 11m has a diameter of 3.5m. The two diameters may be perceived, in like fashion, as having a common unit of 318mm (1m / pi), as in Fig. 2, but no such unit exists in the metric system. We use a single unit, being the perimetric unit in this example.

Figure 2: Dividing the diameter into as many equal parts as there are on the circumference introduces a potentially phantom unit.

Nevertheless, this is not to say that the megalith builders did the same. They may have employed two units - a diametric unit and a perimetric unit - either because it was simpler to work with circles and arcs in this manner or to overcome the problems of the incommensurability of pi. Indeed, one might adduce artefactual support for perimetric and diametric measures from Stonehenge. The incised panel on trilithon Stone 57, if megalithic in origin, measuring 3 feet 9 inches or 1143mm (Crawford, 1954: 27), could well be a celebration of pi, being seven perimetric units of 162.8mm and, at the same time, 22 diametric units of 51.8mm. There is also support for a two unit system at a stone circle site in Germany. Second, it becomes apparent that the common unit is not as uniform across Britain and Ireland as Thom supposed. At some sites, it can vary by as much as two percent. This might help explain why the megalithic yard emerges from the data, but cannot be found in the diameters of all stone circles. Stenness, Orkney It can be seen from the excavation report that the twelve arcs (gaps) on the circumference at Stennes are not the same size. Ten arcs are 50 perimetric units, one arc is 40 units, and one is 60 units. Whether this is intentional cannot be known, but there is a possibility that the inequality results from a tallying error. Had the stone locations been determined by measurement using a length of rope ten units long (1.63m) then there could have been a tallying error at one of two points, at which the builders counted either four or six lengths instead of five (see Fig. 3).

Figure 3: Stenness (Orkney) and Balbirnie (Fife), showing possible tallying errors on the perimeter.

Machrie Moor V (Outer Ring), Arran Machrie Moor V offers an excellent test in further study because in addition to the equally-divided inner ring (used in the unit derivation above) it has an outer circle that might reasonably be supposed to have been measured in the same unit as the inner. The first thing of note is that the stones appear to be distributed in an equal pattern either side of an axis running from northwest to southeast. As this would form a segment of 45 degrees from north, it might suggest that a north-south line was perceived, and the axis would thus run from one eighth of a revolution west of north to one eighth of a revolution east of south, all segments potentially being an integer number of perimetric units (see Fig. 4). There is an obvious shortcut to calculation in such analysis. As there are as many diametric units on the diameter as there are perimetric units on the circumference it is not necessary to multiply the diameter by pi to work in the perimetric unit. As the dimensions of stone circles are measured, and given, by their diameters then use of the diametric unit is the obvious preferred option. The outer circle has a diameter of 18.2m (Burl, 2000, also by survey) which is 351 diametric units. If the circumference is divisible by eight (as suggested above) then the likely diameter is 352 units (also 18.2m). It is then possible to calculate how many units there are in each arc (gap) between the stones.

Figure 4: Machrie Moor V showing perimetric units. The two ‘pi arcs’ would be the same length as the diameter and also the inner semi-circumference were pi = 22/7.

It becomes apparent that the gaps are all multiples of seven or eleven perimetric units. The gaps either side of the axis are 22 and 14 units, and this is the ratio between the circumferences of the two circles (352:224, or 11:7, the ratio of semi-circumference to diameter.) The ratio also appears at Cullerlie, in the east of Scotland, where there are seven main inner cairn rings having perimeters of 11 stones. Drombohilly Upper, Ireland Apparent tallying errors of an integer number of perimetric units, as at Stennes, occur also at Balbirnie in Scotland, perceived from the excavation report (Ritchie, 1974) and by survey of the restored ring, and at Drombohilly in Ireland (by survey). At Balbirnie, the rope length used for tallying would have been four perimetric units (65cm), at right in Fig. 3, and at Drombohilly three units (49cm) as in Fig.5.

Figure 5: Drombohilly Upper (Oval Progressive design), showing an apparent tallying error of three units at north. A Model (left) and by Survey (right).

It should be appreciated that although archaeologists can perceive no trace of the megalithic yard in Ireland a system potentially incorporating this unit could yet be present, as it is suggested that the basic unit employed might be one sixteenth of the megalithic yard, not the yard itself. At Drombohilly, the originating circle is actually 12 megalithic yards in diameter (192 units), but the derived width and height of the finished design are both 164 diametric units (8.5m), and the perimeter is 164 perimetric units (26.7m). The axis is one-twelfth of a revolution west of south (16 units). Thus, the megalithic yard itself does not appear in the finished dimensions, though there initially, but may nevertheless exist in the measuring system used. Concluding A number of suggestions arise from the foregoing analysis. First, that a perimetric unit may have been employed which is not a multiple of the megalithic yard. Second, it is possible that there was also a diametric unit related to the perimetric by a factor of pi, and that this unit is one-sixteenth of Thom’s yard. Third, the gaps between orthostats may well have been measured rather than set by eye. Fourth, that there is an axis at all circles which is a rational (common) fraction of a revolution from one of the cardinal points of the compass. Fifth, it may be that there is symmetry about the axis which is not generally perceived due to an assumed and apparent erratic distribution of orthostats. These considerations, and more, are explored in the second part of this paper. References Burl, A. (1979) Rings of Stone, Frances Lincoln / Weidenfeld and Nicholson, London ----- (2000) The Stone Circles of Britain, Ireland and Brittany, Yale University Press, New Haven and London Cleal, R., Montague, R. & Walker, K.E (1995) Stonehenge in its Landscape: Twentieth Century Excavations, English Heritage, Arch. Report 10 Crawford, O.G.S. (1954) ‘The Symbols Carved on Stonehenge’, Antiquity 109, 1954, 25-31 Davis, Alan (1988) ‘The Metrology of Cup-and-Ring Carvings’, in Ruggles, C.L.N. (ed.), 1988, 392-422 Ritchie, J.N.G. (1974) ‘Excavation of the Stone Circle and Cairn at Balbirnie, Fife’, The Archaeological Journal 131, 1974, 1-32 ----- (1975) ‘The Stones of Stenness, Orkney’, Proceedings of the Society of Antiquaries of Scotland 107, 1975-76, 1-60 Roy A.E., McGrail, N.& Carmichael, R (1963) ‘A New Survey of the Tormore Circles’, Transactions of the Glasgow Archaeological Society 51 (2), 1963, 59-67 Ruggles, C.L.N. (ed.) (1988) Records in Stone: Papers in Memory of Alexander Thom, Cambridge University Press

An Alternative Route to the Megalithic Yard (Part 2) G.J. Bath In the first part of this paper it was observed that there appears to be a perimetric unit of 162.8mm present upon the circumferences of equally-divided stone circles - and many others - and that this has a pi relationship with a diametric unit that would be one-sixteenth of a megalithic yard (51.8mm). Furthermore, that circle axes appear to be measured as a fraction of a revolution from the cardinal points of the compass, and the distribution of orthostats upon perimeters looks to be both measured and patterned. Further examples are provided in this continuation of the argument. Grange B, Lough Gur, Eire The Great Circle, Grange B, Limerick, excavated by Seán P. Ó Riordáin in 1939, consists of a continuous ring of 113 earth-fast stones set against a raised bank with the central area scooped out. There was a post pipe at the exact centre of the circle. A number of stones stand out as being taller or bulkier than the rest, but the perimeter has a particular archaeology: one stone is basalt, two are sandstone, twenty-three are volcanic breccia (22?) and those remaining (88?) are limestone. It happens that one of the sandstone blocks lies at northwest, one eighth of a revolution west of north, and the other is one eleventh of a revolution counterclockwise from it. This may be significant, because the axis (entrance) is flanked by breccia blocks at one eleventh of a revolution north of east, and by far the largest stone, a massive block of breccia, is two elevenths north. Thereafter, the stones at 4, 5, 6, 7 and 8 elevenths are also breccia. There are further stones of breccia at east and south, suggesting that these compass points are thus specifically marked, and another stone of breccia at one eleventh of a revolution east of south (Fig. 1). This accounts for half the breccia stones in the circle.

Figure 1: Grange B, the division of the circle into eighths and elevenths. Legend: S = sandstone, VB = breccia.

This suggests that the circumference could be a multiple of 88 perimetric units (dividing by both eleven and eight), but, equally, the diameter would then be a multiple of 88 diametric units - a particularly large 4.56m. By measurement, the diameter satisfies this at 45.6m (55 MY) which was potentially rounded by Ó Riordáin to 150 feet (45.7m). This is one of eleven sites identified as potentially incorporating pi as a common fraction in the layout (termed Pi Arc Circles). Machrie Moor V (see Part 1) is another. Mount Pleasant Site IV The five fairly regularly spaced concentric circles at Mount Pleasant IV, Dorset, seem not to have prompted archaeologists to wonder if these might be in an identifiable arithmetic sequence. Table I shows the maximum mean diameters of the rings from the excavation report (Wainwright, 1979:23) converted to the diametric unit with the calculated gaps between. It can be appreciated, particularly from the Average column at right in the table, that the gaps are effectively 120 units, and the rings could be multiples of this length (6.2m) in progression (see Fig. 2, left). The central Cove, 6.3m square by scale, a later Beaker addition, would contain a base ring of 120 units, perhaps hinting that something may have been present previously to define it.

Table I: Mount Pleasant Ring Analysis.

Figure 2: At left, Mount Pleasant Site IV after Wainwright (1979), and at right The Sanctuary after Cunnington (1931). Order and pattern (number and geometry) appear to be present.

The north and south corridors at Mount Pleasant IV appear to be 60 diametric units wide, converting to 19 perimetric units (note 60/19 x 19/6, pi = SQR10), and the splays of the east and west corridors appear to increment one unit per ring per quadrant. Three so-called ‘replacement’ posts (marked ‘R’) are taken here to be points on a circle enclosing a quadrant of Ring E and passing through the centre of the design. Ring E would be 240 units in diameter, so the ‘replacement’ circle would be 340 units were the square root of two taken to be 17/12. Seahenge Since publishing my work on stone circle design and measurement, I have had further thoughts about Seahenge. A coppiced pole of 839mm found between timbers 32 and 33 (Brennan and Taylor, 2003: 14) may well be the megalithic yard of 16 diametric units used at this site. The rod had been stripped of bark and was charred, potentially to dry and harden it, and, presumably buried upright, was very likely deliberately placed. The diameter of the originating circle would be seven such yards (5.9m). This is 112 diametric units producing a circumference of 112 perimetric units, averaging two such units per post given a count of 56 timbers. The count is 54 on the perimeter and two outside, though post 43 is a particularly small timber. By construction, the axis is 3/56ths of a revolution east of south.

Figure 3: Seahenge as a triple-arc design. The innermost circle is three-eighths of the originating circle. The origins of the arcs AB, AC and BC are O, O1 and O2 respectively. The numbers identify the timbers.

Timber 30 stands out, as it is the only orthostat positioned with its freshly-split side facing outwards. This may be a significant pointer to the geometrical design of this ring, as shown in the plan (Fig. 3, right). Note that archaeologists would declare the design to be a badly formed circle. To my mind, this makes little sense given the fit of this particular design to the timbers. In the diagram, the line from timber 30 (unique, as above) through the centre terminates precisely at the eastward extent of the northern arc (upon a diameter of the outer circle). The excavation report suggests that the line from timber 36 (at the ‘entrance’) through the centre of the design to timber 10 was intended to be a solstitial alignment. In the design shown here, the bearing is 1/7th of a revolution west of south, at az. 231.43 degrees. The coppiced pole was found at 1/7th of a revolution south of west. One cannot know for sure at sites like this whether the builders set their solstitial axial alignments on the summer or winter solstice. Nor might one know if the favoured point lay at first or last glimmer, the point the sun was bisected by the horizon or when it was sitting upon it. If this suggested alignment at Seahenge be solstitial then it would seem that last glimmer of the winter sunset is being celebrated (based on astronomy software Stellarium, on the shortest day 2050 BC to 2048 BC *). The Recumbent Angle There is a subset of Scottish recumbent stone circles (RSC) typified by having one of the flankers (usually at east) on a path diverging from the circumference defined by the other stones. The reason may be that the misaligned flanker lies upon a secondary arc defined by the flankers and a point on the opposite side of the circle, in some cases marked by a stone. Analysis of three such RSCs - Loanhead of Daviot, Easter Aquhorthies and Sunhoney - is presented at Figures 4 and 5. In all cases, the centre of the primary circle is at O and that of the secondary circle is at P. The values shown inside the circles are the lengths in perimetric units of the arcs forming the gaps. At Loanhead the gaps (angles) are all multiples of 7.5 degrees, so the diameter would be a multiple of 48 units (360 ÷ 7.5), 2.5m, making the unit 52.6mm. At Easter Aquhorthies and Sunhoney the gaps are multiples of six degrees, so the diameters would be multiples of 60 units (3.1m), the unit thus being 51.8mm and 52.5mm respectively.

Figure 4: Loanhead of Daviot (left) and Easter Aquhorthies, by survey, showing primary and secondary circles and pentagonal frames. Not to same scale.

Note that at Loanhead of Daviot (Fig. 4, left) the centres O and P lie on a distribution axis connecting stones 3 and 8, and that five stones (2, 4, 6, 8, 10) may be seen to form a balanced pentagonal frame about this, the sides spanning 3 x 80 units and 2 x 72 units (such frames being common to many stone circles). The inner kerb (cairn) is potentially geometrically defined within this frame. Easter Aquhorthies (Fig. 4, right) may be seen to have a balanced pentagonal frame (2 x 72, 2 x 60 and 96) defined by stones 3, 5, 7, 9, 11.

Figure 5: Sunhoney by survey (left) and without an apparent tallying error (right).

Sunhoney, on the other hand, appears to be anomalous in this respect (Fig. 5, left), but this might be resolved by suggesting that there may have been a tallying error of a length of eight perimetric units (1.3m) at stone 10. The circle is divided geometrically by the line connecting stones 8 and 2 with P at its centre. Tallying may have begun at stone 2 working clockwise correctly to stone 9. Stone 10 should then have been placed at five lengths (40 units) not six (48), and thereafter the counts are correct (the gap between stones 1 and 2 being simply a remainder). Were this so, the line from stone 5 to the point opposite, marked X and thence midway between stones 10 and 11 (Fig. 5, right), would pass through O and P at a right angle to the line connecting stones 8 and 2. This would permit a balanced pentagonal frame about the distribution axis spanning 4 x 88 units and 128 units (locations 1, 3, 5, 7, 9). Concluding Analysis such as the preceding has been applied to many stone. timber, pit and stake circles across Britain and Ireland in various states with Thom’s megalithic yard, if it existed (as at Seahenge), indicated by extension over a range of 813mm to 845mm, though with an altogether different unit on perimeters. Such development of Thom’s work may be of interest to some, but has been of no interest to archaeologists for fifty years. The dismissive response of twentieth century archaeology to potential traces of a uniform unit and design philosophy in megalithic ritual architecture is perhaps understandable. It may have arisen from the Victorian attitudes of those teaching archaeologists of the time, and, reflecting an emotive response to outside interference, their failure to apply fundamental precepts of inference. Archaeologists of that day did not have access to current scientific advances, or data from recent excavations, and it might be hoped that archaeologists of the twenty-first century will not continue to espouse the preconceptions of their predecessors. This is not to claim that what is presented here is necessarily correct or complete, but archaeologists may have missed something by declaring so emphatically against the proposition. It may well be that the megalithic yard was based upon a pace, but the structured perimetric divisions argued here suggest that such a pace may have been standardized and subdivided – thus introducing a system of measurement. The inference is that there may yet be something to Thom’s metrological and design hypotheses, and some evidence to justify considering these on a continuing basis. Hopefully, the topic will cease to be taboo in the minds and hands of incoming and rising archaeologists, untainted by past prejudice, who might finally feel at liberty to explore megalithic design and metrology in academic papers and excavation reports without risking their careers and reputations. References Brennan, Mark & Taylor, Maisie (2003) ‘The Survey and Excavation of a Bronze Age Timber Circle at Holme-next-the-Sea, Norfolk 1998-1999’, Proceedings of the Prehistoric Society 69, 2003, 1-84 Cunnington, Maud (1931) ‘The “Sanctuary” on Overton Hill, near Avebury [excavations 1930]’, Wiltshire Archaeological and Natural History Magazine 45, 1931, 300-35 Ó Ríordáin, Seán P. (1951) ‘Lough Gur Excavations: The Great Stone Circle (B) in Grange Townland’, Proceedings of the Royal Irish Academy 54, 1951, C, No. 2, 37-74 Wainwright, G.J. (1979) Mount Pleasant, Dorset: Excavations 1970-1971, Thames and Hudson Ltd. *Note: Stellarium astronomy software is available as a free download. © G.J. Bath