p. b. davidson. navigation in the neolithic 86 supporting paper iv mensuration 2011

7/27/2019 P. B. Davidson. Navigation in the Neolithic 86 Supporting Paper IV Mensuration 2011

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Peter Davidson 1986 - 2009

MEGALITHIC NAVIGATION 1986 - SUPPORT PAPER IV

Mensuration

Quite the most difficult question to tackle is that of the assertion that there is evidencein the Rude stone monuments for a unit of length of widespread application, of great

consistency and over a lengthy period of time; this is the Megalithic yard (MY) or

Megalithic Rod (MR) equal to 2 MY. The proposition is conditioned by the

assertion that the setting out of the Rude stone monuments is based on Pythagorean

triangles, whose sides are integers of MY or MR or, at worst, or of these; and

further conditioned by the assertion that there is a preference in choosing the integer

values in MY for circles (or rings) so that the perimeters are integers in MR.

That has caused a lot of bother to the archaeologist who sought confirmation from the

statistician. Im not sure it hasnt caused the statistician more bother than the

archaeologist! The problem is set out by Thom in the second chapter of his first Book,Thom (1967); that there is a problem to determine from data whether there is a

statistical justification for a quantum of measurement having been used. The work

published by Broadbent (1955) was used. There were two cases to consider. (1) where

there was an a priori knowledge that a quantum may exist; and case (2), where the

data itself provides the evidence for a quantum. This second case is quite difficult and

involves the use of a statistical model, a Monte Carlo solution.

The conclusion (Thom 1967) that circle diameters were based on MY, and that the

Carnac Alignments (JHA Vol.3, Pt.1, 1972) are based on the MR are derived by using

Broadbents method.

The matter aroused enormous interest from a wide variety of disciplines and led to a

debate at the Royal Society in December 1972. (Nature Vol.24O, Dec.29, 1972); an

interdisciplinary debate that was memorable at least for an exposition by Prof. Kendal

on the art of hunting quanta. A contribution that concluded that, with reservations,

there was evidence for the MY and, while it left the protagonists reluctantly

acquiescing in the proposals of Prof. Thom, it was not the end of the matter.

Prof. Atkinson, who, as an archaeologist, has been particularly active in projecting to

prehistorians the consequences of engineering and scientific knowledge, (the

engineering problems of moving the Stonehenge Bluestones, for instance) wrote inJHA. Vol.6, Pt.1, 1975, of his considered acceptance of the proposals of Thom.

Unfortunately, a simple statistical error which, while it S4.2

was picked up by Dr Freeman, did not invalidate the general substance of Atkinsons

article, did not give it the authority it deserved.

Dr Freeman, on the other hand, was working on a criticism of the work of Broadbent

and Kendal; which he presented to the Royal Statistical Society in October 1975. This

paper, A Bayesian Analysis of the Megalithic Yard is important not only for the

technique proposed by Dr Freeman and the conclusion that he reaches, but for the

comments and contributions in discussion by most of the leading protagonists.

Dr Freeman developed a statistic that could be fed with simulated and real data;

showed that the random data produced a noisy curve and the simulated data


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produced a noisy curve with more pronounced spikes at particular quantum values.

By taking Thoms (1967) data on circles and rings and subdividing them into Scotland

and England and Wales, produced curves that showed spikes in the noisy curve at

various quanta. His further use of data from Carnac alignments produced so many

spikes that he concluded there was no evidence for a quantum there; he subdivided the

circle data geographically but found nothing more and does not publish the data orresults.

The paper is worth reading, not for any enlightenment on what is, no doubt, a quite

proper statistical technique, but for the imprecise display even of a graph known to

possess a quantum. He makes it clear that, at the end, the individual must form his

own judgment. I read nothing in these statistical papers that upsets the dichotomy

suggested in my note on Jigsaw puzzles; classical statistical techniques do not use all

the data in such a case.

Let us take, to make our case, three examples and show why we believe the data

displays more information than the statisticians would accept. The examples are a) thedesign of complex rings, b) the design of Avebury, and c) the design of the Carnac

alignments.

Complex rings. We have shown (p. ) how we would determine, from a survey, the fit

of a stone ring to the shape of an ellipse. There are 15 definite ellipses tabled in

(Thom 1967). If we look at the geometry of an ellipse, the critical dimensions are the

major (2a) and minor (2b) axes and the separation of the foci (2c); the ellipse is set out

by keeping the length of a cord from the periphery and round the two foci constant so

that (2a + 2c) = k.

The dimension b is then determined as b2 = a2 - c and the perimeter P may be

calculated or measured. It is Thoms contention that the choice of a and c as integers

in MY was made with the object of obtaining b and P as integers in MY (for b) or

MR (for P).

Thom applies a similar argument to the dimensions of eggshaped rings but finds

no equivalent geometry for Type A and Type B rings. His geometry of these types is

based on the setting out of circular arcs; and, while he obtains a very good fit, he

observes that there appears to be no attempt to choose integer values of the diameter

to give integer values for the perimeter.

The clue is perhaps provided by Angell (1978) where he suggests that Type A and

Type B circles may be drawn by a construction employing three pegs and a rope of

constant length. We have done some calculations using such an Angell shape for

the example he quotes for Black Marsh, Shropshire, D2/2, and we find ourselves with

a construction similar to that for ellipses. The triangle of pegs has sides of 5 MR, 3.5

MR and 3.5 MR, and a cord length of 17 MR. Employing a cusum technique to both

Thoms design and between Thoms design and the Angell shape, we find that there is

little to choose in the support for either design. To relate this geometry to the integer

length of perimeter we would need to work afresh from the site survey.

However, we have strengthened the proposition that noncircular rings were set outby stretching a cord round two or three pegs; and that the triangle of pegs and the cord

were integers in MY or MR; and that the choice of dimensions attempted to make the

triangle of pegs Pythagorean or the perimeter an integer in MR or both.


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Avebury. The enormous circle at Avebury is of a shape quite unlike any other; it is

also much damaged. However, from the 44 or more stones (or whose position is

known) Thom has prepared an accurate survey and proposed a complex curve to fit

them. This work is described in (Thom 1967) but a further survey, confirmation of the

proposed curve and a precise calculation of MY is described in (JHA, Vol.7, Pt.3,No.10, 1976).

The construction of the ring is based on the setting out of a Pythagorean triangle of

sides 30, 40, 50 MR, and radii of arcs 80, 104, 300. Seven circular arcs are thus

defined and 44 stones lie on them 2 ft (or thereabouts, which will do for our present

assessment). Now this is a very close analogue of the support for the fit of two pieces

of Jigsaw.

We need five dimensions to define the arc of a circle; we have 44 stones defined by

88 dimensions. The support for the stones fitting the curve is therefore;

S = (88 35) Ln 2 = 36

for 8 stones S = (16 5) Ln 2 = 7.8

for 16 stones S = (32 5) Ln 2 = 19

for 10 stones S = (20 5) Ln 2 = 10.5

for 7 stones S = (14 5) Ln 2 = 6.5

If we take the four principal arcs, they are defined by 8, 16, 10 and 7 stones and the

individual supports would be

S = (16-5) Ln 2, (32-5) Ln 2, (20-5) Ln 2 and (14-5) Ln 2 :

or 7.8, 19 10.5 and 6.5

The other three arcs being represented by only three extant stones (but by the remains

of 5 burning pits not included in the calculation).

From all this we might say that we can support the proposition: - That using a

construction involving Pythagorean triangle and dimension of MR (1 MR = 2.5 MY

= 2.5 x 2.72 ft) the curve of seven arcs are all integral in MR and the 44 stones lie on

this curve.

I am not sure how much extra support we should claim for this conditioning of theproposition, something like S = 7; but the support for the proposition is very strong.

That is particularly so when we take into account that the curve is defined

independently of the stones and that a value for MY of 2.73 ft (instead of 2.72) means

that the curve fits none of the stones.

Carnac Alignments. There are several locations around Carnac where we find

hundreds of large stones arranged in long rows; their purpose is not known, they are

all much disturbed and some were. restored in 19th century, the reerected stones

being marked. It is the stones referred to variously as the Carnac alignments or le

Menec with which we are particularly concerned; although the nearby alignments ofKermario appear to have similar characteristics but to be incomplete.


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The geometry of the rows at Le Menec you will find in the supporting paper on Tide

predictors (Fig. V) and you will also find there why I think them important and how

they fit into the whole pattern.

The analysis of the layout of the stones is given in Thom & Thorn 1978 and in JHA,

Vol.3, Pt.1, 1972, No.6. There are substantial gaps to the East of the knee but Thomwas able to analyse substantial numbers in each row West of the knee (between A and

B) and at the east end. In each case, S4.5

and for the stones as a whole, he applied Broadbents Case I (that there was an a priori

value, 1 MR to be considered); in each case he found a close agreement with 1 MR =

6.80 ft.

Freemans analysis is based on the assumption that no a priori value exists: he

analyses row by row and his spiky curve produces a plethora of values. The

argument must rest, I think, on whether Thom is right to presume an a priori value;

there are several reasons why he should.

a) The geometry of the alignments. At each end is an eggshaped ring, based on

a Pythagorean triangle of sides 12, 16, 20 MR. The lines are not parallel but their

spacing and the taper can be described in integers of MY and the geometry of the

knee can be described as a pair of Pythagorean triangles (also integer in MR).

b) Within the rows there are sections where the stones are relatively complete, if

disturbed, where a dozen or more will average about 6.8 ft. If there is any overall

quantum that gives prior data.

c) Thoms results. For each row West and at the East where stones still exist the

same quantum fits the data, so that we can contemplate the support for some such

proposition that if (in these alignments) we have a substantial row of stones, they will

have a quantum of 6.80 ft and get a support of S = 20 Ln 2 = 14.

We can extend the proposition to include Kermario, where a similar spacing occurs at

the West end; but we should have to be careful because of the confused lines at the

East.


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p. b. davidson. navigation in the neolithic 86 supporting paper iv mensuration 2011

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