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

Megalithic Mathematics 3 True circles. The largest class of megalithic rings are the simple or true circles.

4.3.1  The true circles are drawn in a straight forward way with compasses, or rope and pegs, from a single centre with a fixed radius. The resulting perimeter length will not automatically be a simple multiple of the diameter as the constant pi will govern the ratio between the two lengths. Approximate value for pi is 3 & 1/7, consequently if a draughtsman or geometer wishes to draw circles with diameters and perimeters expressed in whole multiples of a basic unit the number of circles allowable is limited to a sequence of diameters times multiples of 7 as;-                           for diameters =   7, 14, 21, 28,   35,   42,   49, & etc units                       then perimeters = 22, 44, 66, 88, 110, 132, 154, & etc units This sequence can be described as diameter/perimeter = 7 units/22 units. 4.3.2  We have seen in the previous page how Thom's histogram Fig. 5.1 showed the preference for the sequence diameter/perimeter = 4 Megalithic Yards/5 Megalithic Rods. As 1 Megalithic Rod = 2.5 Megalithic Yards then this sequence is 4 MY/12.5 MY. Further examination of Fig. 5.1 shows that radii in multiples of 1 MY were most often used with some at 0.5 MY resulting in diameters which are multiples of 2 and 1 MY. With the the possibility that 2.5 MY might be the required integer for perimeter lengths then most true circle designs appear to conform close to the rule - diameter integer = 1 MY with perimeter integer = 2.5 MY = 1 MR. 4.3.3  We can see that this choice of integers allows a very much greater number of integral circle plans than the sequence constrained to simple multiples of pi e.g: -            for circles up to 56 units of diameter there are only 8 circles with near integral perimeters;                              for diameters =   7, 14, 21, 28,   35,   42,   49,   56.                           then perimeters = 22, 44, 66, 88, 110, 132, 154, 176.            whilst the integers chosen by the stone circle builders support 29 acceptable plans in the same range of circle sizes. 4.3.4  In the highly rhythmic sequence - diameter 4 MY/perimeter 5 MR - there are 14 possible plans;                               for diameters = 4,   8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56 MY.                           then perimeters = 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70 MR . and, allowing for the 'tweaking' of diameters apparent in the 4 MY/5 MR sequence 15 other potential designs satisfying integrality are possible and apparent in more clumping of data in Thom's fig.5.1. These are circles with diameters not in the above main sequence and diameters in odd numbers of MY, denoting the choice of half yards in the radii.     on or near nominal diameters = 5, 7,   9, 13, 17, 18, 21, 22, 26, 30, 33, 34, 38, 42, 54. MY.                            then perimeters = 6, 9, 11, 16, 22, 23, 26, 27, 33, 38, 41, 42, 48, 53, 68 MR.

4.3.5   From A.Thom, Megalithic Sites in Britain p.46.

Digital Testing. Following are examinations of several of Alexander Thom's original stone circle surveys. In each case a test template of a precise nominal construction set has been made, derived from theory, and layered over Thom's surveys.

4.3.6  Arquorthies, Manar. B1/6. A test template showing five circles with perimeters of 27, 28, 29, 30 and 31 MR has been layered precisely over Thom's original survey of the recumbent stone circle near Easter Aquorthies, Manar, Nairn.

None of the four circles drawn in red at 27, 28, 30, and 31 have diameters close to multiples of one MY. It can be seen that the the 29 MR perimeter circle drawn in black is the ring which accords closest to integrality of 1 MY in the diameter.

A.Thom, Megalithic Sites in Britain p.143.

With only 0.08 MY, (2.6 ins, 65 mm), deviance from the ideal diameter of 23 whole MY the perimeter of this circle passes through the centres of most of the stones though not the recumbent itself. It is a common feature of recumbent rings that the large stone laid on it's side, with it's upright 'jambs' is not placed on the perimeter but just to one side.

4.3.7  Aquorthies, Kingausie. B3/1. Here, at a much disturbed multiple ring site, another Arquorthies near Kingausie, Aberdeenshire, we can see how the most apt solutions to the formula of integral diameters in Megalithic Yards with integral perimeters in Megalithic Rods are again the closest circles to the rings of stone at present surviving.

For the outer circle the diameter of the black coloured ring is 27.056 MY which supports a perimeter of precisely 34 MR, (85 MY). The discrepancy in the diameter length is 0.056 MY = 1.82 ins = 46.4 mm.

A.Thom, Megalithic Sites in Britain p.147.

With the inner circle we see how the black 24 MR perimeter lies just outside the present positions of most of the stones, which have fallen, but it has the most satisfying diameter at 19.098 MY. The discrepancy or 'tweak' in the diameter is plus 0.098 MY = 3.2 ins = 81 mm. The three small isolated stones near the centre may show evidence for the first of the most favoured series of diameters - diameter 4MY/perimeter 5MR. The actual diameter which supports a 5 MR perimeter is 3.979 MY. The difference - 0.021 MY - from the nominal diameter 4 MY is 0.69 inches, (17.5 mm).

	4.3.8  Rollright Stone Circle, Oxfordshire. S 6/1. This much visited circle exhibits a good example of a circle with the required numeric formula for diameter/perimeter lengths. If the perimeter was intended to be precisely 48 Megalithic Rods, (120 MY), then the diameter, at 37.197 MY, is within 6 inches, (15 cm), of an ideal integer. For astronomical data on this site see html page Rollright stone circle, Oxfordshire

4.3.9  Mains of Gask double circles. B 7/15. With the inner circle of the Mains of Gask site near Inverness there is a close fit of a highly satisfactory 39 MR ring on the remaining standing stones. It is constructed with a diameter at 31.04 MY - within 0.04 MY of a perfect multiple of one Megalithic Yard.

However the ruined remains of the outer circle cannot be easily related to either of the nearest two fully intregal rings. When the 54 and 59 MR rings are drawn they fall on either side of most of the stones. There are no circles between these two which support the required intregality of diameter.

4.3.10  The above digital examinations of half a dozen of Thom's stone circle surveys do not in themselves prove the theory of integral construction lengths in megalithic stone circles. However, they do serve to illustrate that finding rings with both diameter and perimeter laid out in exact multiples of chosen units is not an easy task and the clumping of circles around certain radii as discovered in Fig. 5.1 must indicate preferences for those dimensions. The apparent reason for these preferred nominal radii is that they support perimeters laid in nominal multiples of the integer 2.5 MY. With this ratio of integers for diameter/perimeter, ( 1 MY/2.5 MY), many more completely integral circles can be drawn within a reasonable scale than by simply using the 7 units/22 units ratio dictated by pi. 4.3.11  In Britain, sometime before 2000 BC, the idea evolved that numbers might be juggled to provide a greater quantity of harmonious solutions to the design of stone circles. Abstraction in mathematical thought is believed to have been the invention of the Ionian philosophers of the Mediterranean and like these early Greeks, the British ring designers dealt with integers, rational and irrational numbers and incommensurables. In the next pages further digital examination of Thom's surveys will test his claims that, in a strikingly similar manner to the great Greek mathematicians, the British ring builders had a comparable facility with the properties of arcs, ellipses and right angled triangles.

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>>>>

with comments or queries- powys@megalithicsites.co.uk