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Megalithic Mathematics 8

The Compound Rings. The great stone ring at Avebury.

3.8.1  Alexander Thom has claimed that the greatest achievement in organised integral arc design in Britain is the large, outer stone ring at the village of Avebury in Wiltshire.

This ring is composed of seven contiguous arcs each of which has it's length close to an integer of 2.5 MY, (1 Megalthic Rod). The sum of these arc lengths, the total perimeter, is also close to a multiple of Megalithic Rods.

According to Thom the design is based on a Pythagorean special triangle and a perfect circle. This triangle and the circle and six of the radii supporting the closing arcs are all expressed in multiples of 10 Megalithic Rods to a superb degree of precision. For megalithic ring design Avebury is unique in two features; it has no axis of symmetry and several of the arcs meet at distinct corners.

A.Thom, Megalithic Sites in Britain, Frontispiece.

A.Thom,'Megalithic Sites in Britain' p.89 Avebury outer ring. The geometrical design to which the stones in the outer ring are set out differs from anything so far discussed in that the arcs forming the ring meet at definite corners not appreciably rounded off. Without a knowledge of the exact length of the Megalithic yard and of the simpler designs it is doubtful if the construction could have been discovered.

Avebury great stone ring, Wiltshire.

	Without large scale versions of Thom's survey a close digital examination of the theoretical nominal design is not expected. The following construction set is no more accurate than +/- 0.4 of a Megalithic Yard, however, adhering closely to the nominal integers of arc radii, a clear case is presented for the design set as Thom maintains.

3.8.2   Construction set for nominal Avebury design. This is a construction procedure which cannot be followed using only rope and pegs. As shown in Avebury 3 below two of the arcs have radii of 750 MY. It is not thought that any rope would maintain it's exact length when stretched taught over such a long distance. An alternative approach must have been employed. Thom suggests base line survey methods with an understanding of the relationship between a chord of an arc and it's sagitta for the establishment of these two arcs. See paragraph 3.8.10 below.

			The design is based on the first triangle in the Pythagorean series- the 3, 4, 5. From a point within this triangle a 200 MY construction circle is drawn.

	Three 260 MY radii are centred each on an apex of the triangle and three arcs drawn.

3.8.3  The radius length ET of arc FEM is 750 Megalithic Yards, (2040 ft., 621.8 m.). Even if modern steel surveyor's chain were used to scribe these large arcs the accuracy found here could not be reliably achieved. It is difficult to understand how this precision has been achieved without considering that the ring designer's studies of the properties of circles had progressed to a knowledge of the relationship that a chord of a circle has with it's arc and sagitta. See 3.8.10  below for an alternative method of drawing large arcs.

3.8.4   A.Thom,'Megalithic Sites in Britain' p.1 The surveys must be made with the same accuracy as was used in the original setting out and it will be shown that some sites, for example Avebury, were set out with an accuracy approaching 1 in 1000. Only an experienced surveyor with good equipment is likely to attain this kind of accuracy. The differences in tension applied to an ordinary measuring tape by different individuals can produce variations in length of this amount or even more. The necessity for this kind of accuracy has not in the past been appreciated and has in fact only become apparent as the work recorded here progressed.

If the base triangle has been mirrored about side CB then point P is the mirror of point D.

Many of the stone holes are now marked with concrete posts. The eastern quadrant has few stones remaining but shows a ring of depressions.

3.8.5  In this digital re-drawing of the alleged Avebury design all construction lengths barring one, radius PK of arc JK, are measured in multiples of 20 or 25 MY to an accuracy of + or - 0.4 MY, (about 12 inches).

Sides of Pythagorean triangle = 75 MY 100 MY 125 MY Radius of construction circle = 200 MY Radii of 3 arcs drawn on triangle apices = 260 MY Radii of 2 large arcs = 750 MY Radius of closing arc JK = 245 MY

John Smout, a designer from South London, proposes an alternative construction set for the Avebury outer ring which is displayed in a Macromedia Flash presentation at;- http://www.jsdesign.co.uk/avebury2 Smout's approach moves the 200 MY construction circle a short distance to the south-west allowing the over-all perimeter to pick up on more stone positions than Thom's design.

Relationships of arcs to entrance causeways. Photo montage by John Smout.

3.8.6  There are four breaks in the bank of Avebury with causeways of undisturbed chalk crossing the ditch. Three of these 'entrances' can be seen to coincide with the joining points of arcs at F, S and K. The fourth causeway, to the south near arc-point L, which is framed by the two largest stones on the site, is directly on the axis of symmetry of one of the large arcs.

Avebury southern entrance.

3.8.7  Lengths of arcs and total perimeter. Here, as in the other vital dimensions, the designers accomplished superb adherence to the fundamental dictum illustrated in most other rings in Britain. That is that all perimeter dimensions should be expressed in multiples of Megalithic Rods. (1 MR = 2.5 MY).

	Arc	Actual length	Nominal length
	ME	97.23 MY	97.5 MY = 39 MR
	EF	117.43 MY	117.5 MY = 47 MR
	FG	199.87 MY	200 MY = 80 MR
	GH	129.68 MY	130 MY = 52 MR
	HJ	150.09 MY	150 MY = 60 MR
	JKLM	608.10 MY	607.5 MY = 243 MR
	Total perimeter	1302.40 MY	1302.5 MY = 521 MR

3.8.8  The value for the Megalithic Yard employed in this digital reconstruction is 1 MY = 2.721 ft, (829.4 mm). If the value chosen had been 2.720 ft, (829.1 mm), as Thom used, then the matching fit with the surveyed stone positions would have been even closer as in Thom's original survey. 3.8.9  This is an independent illustration of the maintenance of the working accuracy of the Megalithic Yard in Bronze Age times. Whilst this obviously demands that highly experienced survey procedures have been employed it should be noted that no matter how carefully a design of this scale is laid out accuracy at this level could not be achieved unless the measuring devices used were constructed to be reliable to better than 1 part in 1500.

3.8.10  Sagitta of the chord method of laying out an arc of a circle. In view of the limited equipment available for measuring long lengths it seems likely that the Megalith builders had developed a mathematical approach to the drawing of large arcs. The sagitta of a chord method requires an understanding of the relationship between the length of a chord and the length of the perpendicular bisector, (sagitta), which meets the arc. This can be stated as:- chord length x 1/2 then sagitta length x 1/4. i.e. when chord is halved, sagitta is quartered.

Three equidistant radii must first be drawn by base-line survey. That is by laying measuring rods end-to-end over carefully leveled ground. This will establish the first three points on the perimeter, A, B, C, and give the length of the first sagitta, DB. Then the procedure is to half and half again the chords standing each calculated sagitta on the bisector of it's chord until enough points on the perimeter have been established. For an understanding of the uses of sagitta/chord relationships see Thom's survey of Kerry Pole stone egg ring and an alternative construction set for the great stone ring at Avebury.

3.8.11  Thom's survey of the Avebury stones is regarded by English Heritage as the best in existence. He concludes that the placing of most of the stones is within 12 inches, (300 mm), of their nominal intended positions. For the arcs with the 750 MY radii this equates to an engineering tolerance of better than 1 part in 2000. This precision is far beyond the requirements of most modern civil engineering projects including modern highways and motorways construction. 3.8.12  If base line surveys were employed at this site using rods measuring 2.5 Megalithic Yards, (a Megalithic Rod), then both of the rods together, (at least two are required), must have had a combined error in their lengths of less than 4 millimetre. This is a fine degree of fabrication and if it is seen, as Thom maintains, that this degree of accuracy is evident in localities as far north from Avebury as Caithness and the Outer Hebrides then serious speculation is required on where these measuring rods were made, what materials were used and how maintained, and how disseminated throughout this region with it's variety of climate. Ditch and bank. Reconstruction.

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