Pop-Up glossary and paragraph links. Move mouse over text in this colour and Click.

Megalithic Mathematics 2

Megalithic Maths. 3.2.1  From a large data base of over 400 surveyed stone ring sites Alexander Thom believed he could detect exact units of measurement used in the construction plans. He has termed these units the Megalithic Yard (MY) and Megalithic Rod (MR) which equals 2.5 MY. Thom maintains that these units are identifiable in every stone ring intact enough to allow a reliable survey. 1 Megalithic Yard (MY) = 2.721ft = 0.830m 1 Megalithic Rod (MR) = 2.5MY = 6.80ft = 2.073m

A.Thom, Megalithic Sites in Britain p.85.

3.2.2  The reason why Thom was so confident in his discoveries was due to the insistence of the megalithic ring builders to adhere to integrality in as many of the vital construction lengths as possible. Throughout their ring and circle designs Thom had discovered rigorous mathematical procedures governing size and shape. These numeric and geometric imperatives bear striking resemblance to some of the early thinking of the great Greek philosopher/mathematicians although from astronomical dating Thom demonstrates that this British expertise disappeared around 1600 BC - some 1000 years before the birth of Pythagoras.

A.Thom, Megalithic Sites in Britain p.27.

'As we shall see later the builders of the circles, rings, alignments, etc., had a remarkable knowledge of practical geometry. They were intensely interested in measurements and attained a proficiency which as we shall see is only equalled today by a trained surveyor. They concentrated on geometrical figures which had as many dimensions as possible arranged to be integral multiples of their units of length. They abhorred 'incommensurable' lengths. This is fortunate for us because once we have established their unit of length we can very often unravel designs which would otherwise be meaningless.'

Mathematical imperatives in the design of megalithic stone circles and rings.

3.2.3  Integral construction lengths. Thom claims to have found, in all sites capable of supporting an accurate survey, that the main radii of the designs were usually expressed in whole units of Megalithic Yards whilst the perimeters of the rings were measures of whole Megalithic Rods.

This regime increases the number of integral circles which might be employed and, where examples have been found which are not mathematically perfect, slight adjustments are recognised in the ideal length of the diameter. These adjustments are always applied to the diameters leaving the perimeter's measurements closest to the ideal, thus reflecting the priority of integrality for the perimeters over the radii in the finished design, i.e. exact multiples in perimeter lengths were more important than multiples in the radii and diameters.

A.Thom, Megalithic Sites in Britain p.44.

3.2.4

The sizes of the circles. The erectors of Megalithic monuments were evidently interested in getting the dimensions of their structures to be multiples of certain units of length. Since they were capable of measuring to a high degree of accuracy how does it come about that many circles which seem to have been undisturbed have mean diameters which differ by appreciable amounts from what were presumably their nominal diameters? It will be shown that in a significant number of cases the discrepancy is produced by a small adjustment made by the erectors to the diameter, to bring the circumference nearer to an integer. This desire to have both dimensions integral has a further consequence in that at many sites it affects the integer chosen for the diameter.

The histogram of circle dimensions. The following histogram is Thom's analysis prepared from his data base of over 400 surveyed megalithic ring sites. Most surveys were carried out by Thom himself to a rigorous degree of proficiency. 3.2.5  Radii The peaks in the histogram represent 'clumping' of large numbers of sites around particular radii. It can be readily seen how fewer rings were established with radii expressed in fractions of, rather than whole, Megalithic Yards. The builders preferred whole yard measurements for radii giving diameters expressed in even numbers of yards though half yards were sometimes employed which still leaves the diameters integral. A.Thom, Megalithic Sites in Britain p.45. The figure is in itself a pictorial proof of the existence of the Megalithic yard, but it contains much more information. Most of the diameters are seen to lie near to an even number of yards. In other words the radii are integers. But there are also concentrations at odd numbers, so the designers frequently used half-yards for the radius. A.Thom, Megalithic Sites in Britain p.46. 3.2.6  Perimeters When the dimensions of perimeters are expressed in Megalithic Rods further coherance of numeric intent is apparent. The 5MR/2MY sequence. A favoured unit of perimeter measurement appears to be 5 Megalithic Rods, (12.5 MY), as high clumping occurances reveal the sequence; Megalithic Rods  5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70 MR in perimeter lengths. The preference for this sequence, in true circles, may be that the series of radii which comes very close to supporting these perimeters is; Megalithic Yards  2,  4,  6,  8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 MY. in radius lengths. We can see that accepting integers of 5MR for perimeters and 2MY for radii allows a long elegant sequence of possibilities for true circle designs with construction lengths in close multiples of the integers chosen. 3.2.7  Conservation of integrality. Although the 5MR/2MY series of perimeter/radius integers accounts for more than forty percent of the peaks in Thom's histogram, other peaks are evident which indicate other favoured integer values. In both these examples taken from fig.5.1 above it can be seen how the radii have been adjusted to preserve the intregality of the perimeters. This priority of maintaining whole numbers of rods in perimeters results in many other instances of 'tweaking'(t) of radii, e.g.- r=4-tMY for perimeter=10MR, r= 6-tMY for p= 15MR, r= 12-tMY for p= 30MR, r= 14-tMY for p= 35MR, r= 15+tMY for p= 38MR, r= 19+tMY for p= 48MR, r= 20-tMY for p= 50MR, r= 24-tMY for p= 60MR. Here a radius just over 5 MY gives a perimeter of 13 MR, (32.5 MY}. Here, with a radius slightly larger than 9 MY, the perimeter equals 23 MR, (57.5 MY). 3.2.8   A.Thom, Megalithic Sites in Britain p.45. In the histogram there are concentrations at radii 5, 10, 15, and 20 MY but only a few at 7.5, 12.5, and 17.5. The concentrations at 4, 8, 12, 16, 20, 24, and 28 are obvious. The reason for this last sequence becomes evident when we consider the circumferences. The circumferences. Perhaps the most striking feature of the circumferences shown in Fig. 5.1 is that large concentrations occur at 12.5, 25, 37.5, 50, 62.5, 75, and 87.5 MY, all multiples of  l2.5, (5MR). If we accept the approximation pi = 3&1/8 then a circle with radius 4 has a circumference of 12.5. So the above sequence of circumferences follows from a radius sequence of 4, 8, 12, etc. This immediately explains why there are so many circles with a radius of 4 or 8, since these have cicumferences very close to 25 (10 MR) and 50 MY (20 MR). On the following page some of Thom's original surveys are tested digitally with templates derived from the theory of integrality. Pop Up Glossary and paragraph links. Move mouse over text in this colour and Click <<<Back Megalithic Mathematics - 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10 Next>>>> Pop Up Glossary. Move mouse over text in this colour and click with comments or queries- powys@megalithicsites.co.uk