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Stone Circles and Measuring Wheels.

(Article by Michael Spender, 2012)

 

INTRODUCTION

This article has two parts. In the first we show that it is possible that Neolithic stone circles were laid out using a measuring wheel to set the diameter and we point out that this will lead quite naturally to integer value perimeters. When the evidence is examined in greater detail we discover that a universal wheel may have existed one with a radius producing a circumference equal to the Megalithic Yard. In the second part we look at the Avebury Circle in some detail and explore the possibility that the 98 stone outer construct is partly elliptical and partly circular and in so doing we find that there is evidence that the layout does follow the Megalithic Yard theory.

 

PART ONE: MEASURING WHEELS

BASIC THEORY

Neolithic humans have revealed extreme ingenuity in the building of their stone monuments. Some of these constructs are perfect circles while others are elliptical. It seems fairly unanimous that circular constructs were used to mark important solar events to assist in agricultural efficiency. This would be especially important where seasonality affects crop growth but not so important in regions where hours of sun shine and darkness are evenly balanced. Thus in southern climes we find elliptical constructs such as Nabta Playa in Egypt and the Mzoura Circle in Morocco but in Northern Europe, circular constructs predate ellipses. This does suggest that ellipses had different, probably additional functions when compared to circles It has been argued (1) that there was a change from circular to elliptical layouts because it was apparently easier to obtain integer value perimeters using ellipses than with circles. We believe that quite the opposite is true.


Consider using a measuring wheel to determine the radius of a stone circle. If the wheel has radius r then its circumference will be:

c = 2r.pi

And the marked out radius of the stone circle will be:

R = nc = 2nr.pi

Where n is the number of wheel rotations used to mark out the radius of the stone circle. The circumference of the stone circle will be :

C = 2pi(2nr.pi)

Now we believe that by trial and error Neolithic man would discover that by using the same measuring wheel to assess this circumference, C would be divided into a whole number of segments each of length:

L = C/c =(2.pi.2nr.pi)/(2r.pi) = 2n.pi

And since to an excellent approximation pi = 22/7, they would be able to predict quite precisely how many segments there will be as long as they restricted values of n to multiples of seven (and plausibly some fractions). Thus

n = 7/4, 7/2, 7, 14, 21, 28...........etc

Moreover they could decide how many stones they wanted to erect and how many segments would be allocated to each. Note that the segment length is independent of r so in principle any size measuring wheel could be used.

A great many theories exist about stone circles, the positions, number and significance of the stones in the constructs here we simply want to test the idea of using a measuring wheel to lay them out. The idea is testable for, if a wheel was used, the diameters of the constructs will be multiples of pi by virtue of the fact that each diameter is

4nr.pi

In table1 the data for several stone circles was collected which shows that the diameters of the circles are multiples of pi. We have rounded some numbers up or down to the nearest whole number and feel justified in doing so because of likely errors in the measurements of the diameters used. The stones are often huge leading to errors of order +/- 1 m and they may not still be in their original locations as a result of earthquakes. As an example consider the Mzoura Circle, it is listed at 57.7m in diameter and with that dimension we found that the diameter was 18.3pi, but take account of a +/- 1 m error range and we find the diameter to lie between 18pi and 18.6pi.

 

TABLE 1 PARAMETERS RELEVANT TO MULTIPLES OF pi.

Name of Circle.

Diameter (m).

Xpi Rounded.

Number of Wheel Rotations on diameter.

Number of Wheel Rotations on Radius.

Stone Spacing (MW)

           
Nabta Playa (Egypt) 3.69 1 4.4 14 0.5
Mzoura (Morocco) 57.7 18 69 217 1
Aubrey Holes (UK) 86.6 28 104 326 6
Sarsens (UK) 30 10 36 113 4
Avebury North (UK) 97.4 31 117 366 14
Avebury South (UK) 103.6 33 124 389 13
Avebury 3 103 33 123 387 12 (ref 2)
Avebury Outer (UK) 331.6 106 397 1246 13
Brodgar (UK) 104 33 124 391 7
Stanton Drew Great (UK) 113 36 135 425 1
Stanton Drew NE (UK) 34 11 41 128 11
Stanton Drew SE (UK) 40 13 48 150 19
Almendres (Port) 11.4 4 13.6 43 2
Almendres (Port) 18.8 6 23 71 3

Column 3 in table 1 shows that, the diameters, after rounding, are multiples of pi, suggesting the use of measuring wheels in the constructions. We will look in more detail at Avebury in part 2 of this note.

The multiplier, the number of times pi divides into the diameter is X = 4nr

We investigated to determine whether there is a universal value of r, for the measuring wheel (MW). This work is shown in table 2 where it will be seen that there is some evidence for a universal wheel of radius 0.133 m, and a circumference of 0.836 m. This value has been used to calculate the wheel rotations in columns 4 and five in table 1. The stone spacings are then found by dividing the usually estimated number of stones into the perimeter wheel rotations.

If the same size measuring wheel was used for all the circle constructs, a universal wheel would show up because a common value of r will emerge when the expression:

r =X/4n where n takes values 7/4, 7/2, 7,14, 21 ...........etc.

is evaluated for all the circles and various values of n. We find the results shown in table 2.

 

TABLE 2 COMMON WHEEL RADIUS ?

Name of Circle

Values of (r) for Various Values of (n)

n =     

X/4

7 14 21 28 35 42
               
Almendres 0.9 0.128 0.064 0.042      
Nabta Playa 0.925 0.132 0.066 0.044      
Almendres 1.5 0.214 0.107 0.071      
Stanton Drew N 2.7 0.387 0.192 0.l28 0.096    
Sarsens 2.75 0.392 0.196 0.130 0.098    
Stanton Drew S 3.175   0.226 0.151 0.113 0.091  
               

n =    

  35 42 49 56 63 70
               
Mzoura 4.7 0.134 0.111 0.095      
Aubrey Holes 6.9     1.140 0.123 0.109 0.098
Avebury N 7.75     0.158 0.138 0.123 0.110
Avebury S 8.25     0.168 0.147 0.131 0.117
Avebury 3 8.19     0.167 0.146 0.130 0.117
Brodgar 8.25     0.168 0.147 0.131 0.117
               

n =    

  70 77 84 91    
               
Stanton Drew Outer 9.07 0.130 0.118 0.107 0.098    
               

n =    

  175 182 189 196 203  
               
Avebury Outer 26.375   0.145 0.139 0.134 0.129  

The process is akin to looking for a lowest common factor. The divisor can only take certain values so the numbers listed for r are discrete values not interpolations. Further analysis reveals that there are:

Five numbers in a range 0.14 to 0.149 m whose average is 0.145 m

Ten numbers in a range 0.13 to 0.139 m whose average is 0.133 m

Five numbers in a range 0.12 to 0.129 m whose average is 0.125 m

Seven numbers in a range 0.11 to 0.119 m whose average is 0.113 m

If there is a universal wheel its radius is probably between 0.13 and 0.139 m with a likely value of 0.133 m.

The circles are far apart geographically so it is unlikely that the choice of the radius is coincidental but rather the result of an oral tradition passed along from generation to generation. It is quite exciting to note that a wheel with a radius of 0.133 m would produce a perimeter of 0.836 m (2.74feet), which is only 0.7% greater than the accepted value of the Megalithic Yard. What exactly Neolithic humans based this radius on remains mysterious but we like to think it had some celestial origin in keeping with the constructions being surveyed.


 

PART TWO: THE AVEBURY CIRCLE

It seems reasonable that Neolithic communities, like all species in new environments, first looked for solutions to specific pressing problems threatening their survival. Then having identified the issues, they sought solutions in the simplest possible way. The origin of the Neolithic humans seems to be in areas near the equator where growing food is not really a seasonal problem since there, the days are evenly divided between light and dark the entire year and only water availability is an issue. Neolithic monuments in these regions are elliptical probably because the people had time to explore more interesting phenomena than events like sun rise. When Neolithic man began to move west and north into Europe, seasonality created problems associated with agriculture, for example, when to sow and how long until harvest time? Stone circles with a good view of the horizon, present a simple, effective way to determine from sun risings and settings, whether the days are lengthening out or not and when the winter is approaching. This is an event recorder and it is not surprising that this kind of structure was used extensively in Europe. Once an agricultural calendar was in place however, its builders would have the time to look at more challenging (interesting) problems. For example eclipses, irregularities in moon cycles and the rising and setting of star constellations.

The design of an observatory to study these latter types of phenomena would reflect the paths taken by the heavenly bodies through the sky and be shaped, to match the celestial equator, somewhat elliptical to make studies of spatial relationships easier.

The construction of an elliptical locus is achieved using two stakes and a sufficiently long string. It is not too far removed from the process of creating a circular locus. There are many examples of elliptical stone constructs. The earliest of these appears to be Nabta Playa in Egypt built around 4,000 BC. There is an important construct in Morocco called the Mzora ring, also an ellipse. Both track several solar features. The Cromlech dos Almendres is the oldest in Europe and shows clearly how these structures developed. The first construct there is a set of concentric circles then apparently 1000 years later a pair of concentric ellipses was added to the site. The circles predate the ellipses. It seems clear that the idea of using elliptical stone structures to track the relative movements of sun and moon and to study eclipses spread eastwards and northwards.

The Avebury Circle is newer than the Cromlech and is well documented. The earliest parts of Avebury are three perfect stone circles. Much later, the circles were surrounded by a ditch and bank henge, the largest in Europe. At the same time 98 large stones were set up inside the henge. The pattern of construction follows that of the Cromlech, first the circles then- the complex shape of the Avebury construct. It seemed worthwhile to see whether the locus of the Avebury stones could be found. The detailed map shown in the article “Avebury-A Present from the Past” (3) was used as follows:

An axis was drawn from stone 24 to stone 73. because literature suggested that it was an important solar line of sight and we then found that an orthogonal axis, drawn through the mid point of 24-73 passed through the Ringstone so this was aesthetically pleasing and anyway, after considerable trial and error we discovered that this was the only pair of orthogonal axes that intersected at their mid points. On the scaled map the 24-73 axis measured 11cm which translated to 343.75m. The published diameter of the Avebury Circle is 331.6m but its shape is quite irregular and the diameter can be as large as 343.9m and as small as 321.9m. The axis as drawn divided the stones into two groups, 48 south of the line comprising the stones numbered 74 to 98 plus stones1 through 23, and 50 stones north of the axis comprising the stones numbered 24 to 73. The point of intersection of the axes fell close to stone 213 on the “North Circle”. Using the point of intersection as centre, a circle of radius 5.5cm (171.9m) was drawn. It passed through all the stones from 74 to 98 in the SE sector but otherwise the fit was erratic, sometimes within and sometimes outside the standing stones. We concluded that although the Avebury construction is not a circle, the stones in the SE segment do lie on a circular locus.

An ellipse seemed more likely to represent the locus of the stones in the NE and NW sections of the henge. The axis perpendicular to the 24-73 axis was produced to the northern perimeter of the stones. It terminated midway between stones 48 and 49. This is slightly east of the north entrance way. When produced to the southern perimeter this line passed through the Ringstone and terminated just east of the south entrance, between stones 98 and 97. Thus the semi minor axis (b) and semi major axis (a) of the prospective ellipse could be found and hence its foci calculated. The semi major axis was 5.5cm (171.9m) and the semi minor axis measured 5.1cm translating to159.4m.

The distances of the foci from the centre of such an ellipse are found from the formula:

f = +/- ( a² -b ²)

The foci were at +/- 2.05cm translating to +/-64.0m.

Only original standing stones were used to test the criteria for their positions on the locus of an ellipse using the formula

S(n)f(1) + S(n)f(2)= 2 a

Where S(n) is a particular stone f(1) and f(2) are the foci, Sf is the distance from the particular stone to one of the foci of the ellipse, and a is the semi major axis. Stones 30, 40, 44, 68, 73 in the north section and stones 6, 10,14, 16 in the south section were tested. Numerical results are shown in table3.


 

TABLE 3 MEASUREMENTS (Taken from map (Ref: 3)

Stone Number

Sf (1)  p Sf (2) q p + q
       
68 3.8 7.5 11.3
73 3.8 7.5 11.3
30 7.3 3.9 11.2
40 6.6 4.8 11.4
44 6.0 4.8 11.3
16 6.9 3.4 11.3
14 6.6 3.7 11.3
10 6.8 4.5 11.3
6 6.4 5.0 11.4
       
    Sum: 101.6
    Average: 11.3
       
77 3.3 7.3 10.6
78 3.4 7.2 10.6
98 5.6 5.5 11.1

The measurements indicated a value for (a) of 176.9m about 5m, 3% greater than expected. We tested stones 77, 78, and 98 in the southwest sector. The results are also shown in Table 3 but they do not support the idea of an elliptical locus.

As the location of the original axis was somewhat arbitrary, the average of the length of the chosen semi axis and the semi axis indicated by the measurements and calculation above was used in further calculations, namely 174.4m. We now consider the geometry and stone placements using the idea of a measuring wheel with circumference 0.836 m. We note first that the major axis will be 348.8 m or111 pi and the minor axis at about 318.8 m will be 101 pi. Again we see the idea of multiples of pi entering into the dimensions. We can convert the dimensions of the axes into MW and find the major axis is 417 MW and the minor axis 381 MW. The circumference of an ellipse is not a simple calculation but to a first approximation may be written:

C= pi (a+b)

Where a is the semi major axis and b the semi minor axis.

The Avebury construction is, as we have seen, roughly one quarter circle and three quarters ellipse, so the actual elliptical circumference is:

3C/4 = 3pi (a+b)/4

Evaluated for the Avebury ellipse this becomes 787.5 m or 942 MW. There are 73 stones on the elliptical section so the stones are placed on centres 13 MW apart.

The circumference of the quarter circle is 270.1 m which translates to 325 MW and since there are 25 stones on the circular locus they are placed on centres 13 Wheels apart, the same as those on the ellipse. So we find that the Avebury stones do conform to Thom's theory, provided their loci are correctly identified.


 

DISCUSSION:

It appears the two older Avebury circles were used in two ways; to set up the required agricultural calendar for the region and, to record full moon risings and settings. In total the circles have 57 stones. This seems similar to the Cromlech dos Almendres. There, the older circles are concentric and one set is clearly marked with perfect circles. The number of stones in the Avebury circles, 57 is similar to the number of Aubrey Holes so we think they measure the same events. The same number of stones features in the Cromlech elliptical construct, basically two, one month moon cycles.

Consideration of the assessment of spatial relationships leads us to suggest the following. The obelisk stone could be used to sight the movement of the moon by an observer standing at the elliptical stones. The observer moved clockwise around the ellipse as the moon moved east to west. Simultaneously, a second observer standing at the elliptical stones would follow the shadow of the obelisk cast by the sun, noting to which stone the shadow pointed. Elongation would be measured by the number of ellipse stones separating the sun and moon at some agreed moment.

Sightings on rising star constellations could be made from the north entrance of the henge at sun set. Avebury bears an uncanny resemblance to the Nabta Circle in Egypt even to the placement of three stones in the south circle perhaps representing “Orion's Belt” and since his feet and Sirius are missing, this would be the sky at the beginning of spring.

Declination measurement seems problematical without the introduction of another tool, a stave perhaps, something that could be marked according to directions given by an observer stationed in the ditch. One person holds the stave upright and slides his hand upwards until the moon is obscured from the observer in the ditch. The position of the mark would then give a relative measure of declination. A daily repetition over several years would eventually provide the information the users needed to predict moon swings, stand stills and so on. Another and particularly attractive idea, is the use of the Celtic Cross the “Wheeled Cross” to measure declination. According to Crichton Miller this type of instrument is extremely ancient predating the building of the pyramids for example, yet quite capable of measuring angles of declination with great precision.

The Neolithic people left no written records, so how all these their data would be remembered and passed down through generations is a mystery even greater than the stones themselves. But if some credence is given to the ideas discussed above then one more mystery is how a Megalithic Wheel came into existence. We are inclined to think that the idea is originally Middle-Eastern, that it would bear some relationship to the moon, the planets, the stars and the sun, just because of what was to be built, temples to these celestial objects. We suggest for consideration, the apparent separation of the Moon and Venus at the first new moon after the vernal equinox.

 

CONCLUSIONS:

1. There is strong evidence that true Neolithic circles were set out using measuring wheels and that the diameters of true circles were deliberately chosen to be multiples of seven wheel rotations so that the resulting circle had a perimeter equal to an integral number of rotations of the same wheel.

2. That Thom's Megalithic Yard distance probably results from the use of a measuring wheel with a perimeter of 0.83 m. Such a wheel would have a radius of 0.132 m. Our calculations showed up a measuring wheel with radius 0.133 m or circumference 0.836 m, about 0.7% larger than the Megalithic Yard.

3. That 3/4 of the the Avebury Henge was deliberately constructed on an elliptical locus with semi major axis of 174.4 +/- 5m, a semi minor axis of 159.4 +/- 5m, and eccentricity e= 0.37.

4. That the SE quadrant was constructed on a circular locus with a radius of 171.9+/- 5m.

5. The major axis of the ellipse is orientated south west to north east and the minor axis south east to north west and passing through the Ringstone.

6. Avebury was conceived as a multipurpose facility which why its layout is complex.

7. Avebury was built with a view to studying relationships like elongation and declination Using the elliptically located stones in the NE, NW, and SW sectors and events like the rising of constellations using the stones located on the circular segment in the SE sector.

 

(Contact the Author: michaelrspender@gmail.com)

 

(More about Avebury)

(More about Stone Circles)

(More abut the Megalithic Yard)

 

References:

1). Alexander Thom Megalithic Sites in Britain Oxford University Press 1967
2).  http://www.ancient-wisdom.com
3). Avebury - A present from the past: http://www.avebury-web.co.uk
 

 

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