T H E C A S T L E R I G G
P R O J E C T
THE CASTLE RIGG PROJECT
PETER STEWART
Submitted on behalf of the
Keswick School Project Team
27th April, 1976  Keswick School, Keswick, Cumbria. 
 Peter Bestley 
 Peter Hodgson 
 Philip Johnson 
 Anne Seneviratne 
 Peter Stewart 
 Alan Wylie 
 David Wylie 
 Margaret Wylie 
THE CASTLE RIGG PROJECT
OBJECTIVE
The
objective of the Castle Rigg project is to study the mathematics,
including the geometry, of megalithic stone circles, using computer
techniques and to make deductions about their design and use.
CHOICE
OF SUBJECT
This
subject was chosen for study since there is a megalithic stone circle
at Castle Rigg, near Keswick School and because the project leader, a
member of the lower sixth form, is interested in mathematics and in
the application of computer techniques to problem solving.
HISTORICAL
REVIEW
QUANTUM Broadbent
(1955) has developed statistical methods for the investigation of a
quantum hypothesis (a quantum being a fixed unit of length) and has
extended this (Broadbent l956) to the examination of a quantum
hypothesis based on a single set of data. Thom (1962) applied these
methods to his data on megalithic sites and has concluded that a
standard length, which he calls the megalithic yard (My.), is used
extensively if not exclusively. These methods are considered adequate
by Porteous (1973) to exclude a rectangular hypothesis (random
distribution) but are said not to differentiate between quanta based
on pacing, the pacing hypothesis, or on measuring with a standard
length, the exact quantum hypothesis. It is preferred here to cite
these as the pacing hypothesis and the yardstick hypothesis
respectively. Porteous (1972) exemplified his thesis with made up
data and did not produce objective evidence to support it. In the
discussion which followed the paper by Thom (1955) the possibility
that the quantum was the human pace was advanced. Kendall (1955)
suggested that the variance of an observed length was of the form an
+ b where n is the number of paces; a is a component representing the
variance of the individual and b is a component representing the
variance between individuals. GEOMETRY In his book "Megalithic
sites in Britain" Thom (1967) has surveyed and classified many
sites and further data has been provided in a subsequent book
"Megalithic Lunar Observatories" by Thom (1971). The term
"stone circle" is widely used although it has to be
qualified and the shape of some is described as a flattened circle or
as an egg shaped circle. Some ellipses are also found. The geometry
has been described for the flattened circles and for the egg shaped
circles (Thom 1967). At least 125 circular "circles" are
known with 35 flattened circles, 9 egg shaped circles and 9 ellipses.
Castle Rigg is an example of a flattened circle and Clava is an
example of an egg shaped circle. These types were illustrated in the
first report of 15.12.75.
ASTRONOMICAL ALIGNMENTS The
application of computer techniques to the analysis of all possible
alignments at megalithic sites has confirmed previously suggested
alignments at Stonehenge and Callanish (Hawkins 1963,1965) and has
disclosed others (Hawkins 1963). The application and value of
computer techniques in discovering the alignments of megalithic
astronomy has been reviewed by Hawkins (1970). The possibility of
astronomical alignments at Castle Rigg was apparently first suggested
by Morrow (190309) cited by Anderson (1915). It was suggested by
Anderson (1915) that the main alignment marked sunrise at an
important date in the Celtic calendar.
AXIS
ORIENTATION No report has been discovered which attempts to
relate axis orientation to latitude or to any other factor. Data
about axis orientation were obtained from the plans of circles
published by Thom (1967)
LEY
LINES The occurrence of ley lines was described by Watkins (l925)
in his book "The Old Straight Track". He produced evidence
of "the alignment across miles of country of a great number of
objects, or sites of objects, of prehistoric antiquity" and
concluded that there were "straight trackways in prehistoric
time in Britain". Mitchell (1969) believed in the existence of
ley lines and supported their existence with speculation and psychic
revelation. The occurrence of ley lines does not appear to have been
investigated scientifically.
INVOLVEMENT
OF PUPILS
The
project has been carried. out by a team of eight pupils from Keswick
School, Keswick, Cumbria and this report is submitted by the project
leader, Peter Stewart, on behalf of the Keswick School Project Team
(ages shown): Peter Bestley, (15); Peter Hodgson, (17); Philip
Johnson, (18); Anne Seneviratne, (16); Peter Stewart, (17); Alan
Wylie, (18); David Wylie, (12) and Margaret Wylie, (16). Some
assistance with data extraction was provided by John Bibby, (11).
Some of the discussions which took place prior to preparation of the
earlier report were attended by Richard Smith, (15); Stephen Temple,
(17) and Paul Whittaker, (15). Following discussion with advisors the
initial project was prepared. The field work, data collection and
computer programming have been carried out solely by pupils but
teachers and other advisors have been consulted for astronomical,
statistical and other advice.
DEVELOPMENT Since
the first report was submitted on 15.12.75 the main stages have been
modified and the fourth stage has been subdivided into a fourth and
fifth stage. Methods have been changed in the light of practical
experience and these changes are described under each stage. It has
been realised that the project is extremely complex. and an attempt
has been made to modify it so that results could be obtained within a
reasonable time.
*
See library work addendum page 5
DATA
COLLECTION All data have been collected solely by pupils without
assistance and without supervision from teachers or advisors. In
Stage I the measurement of quanta by pace and stick has been done by
a team of 25 pupils, usually 3 or 4. In Stage II the survey of
Castle Rigg was completed after several preliminary surveys, each
involving at least 4 pupils, and all pupils have taken part in the
survey at some stage. In Stage II coordinate data from published
circles have been extracted by 4 pupils. It is intended that each set
of data extracted by one pupil should be checked by another. In Stage
III the survey of Castle Rigg to obtain stone and horizon altitude
measurement has been carried out by a team of 3 or 4 pupils. In Stage
IV data relating to axis orientation and to latitude and longitude
were taken from published circles. The data were extracted by one
pupil and were checked by another. In Stage V most of the data were
extracted by one pupil from ordinance survey maps. These data have
not yet been checked.
COMPUTER
PROGRAMMES In Cumbria a Hewlett Packard 9830A calculator model 30
is available and some schools have the use of the calculator for
approximately six weeks in each year. The calculator was available to
Keswick School from late February until April, 1976. Programmes have
been written in BASIC language and some development work for the
project has been carried out. The programming and development work
has been carried out solely by Peter Stewart and Alan Wylie without
assistance and without supervision. Programmes for Stage I and Stage
II were written by Peter Stewart and for Stage III by Alan Wylie. The
programme for Stage V was written in part by Peter Stewart and in
part by Alan Wylie. A series of small programmes to load, reload,
correct, list or store data were written by Peter Stewart or Alan
Wylie. All programmes have been jointly developed.
LIBRARY
WORK The Librarian at Keswick Public Library, Mr. Elsby, kindly
supplied a list of references relating to Castle Rigg in the
Transactions of the Cumberland and Westmorland Antiquarian and
Archaeological Society. These have been studied and some other
references have been found.
Dymond
(1880) gave an accurate plan of Castle Rigg circle and cited a
reference that there was "a second circle in the field nearer to
Keswick". There is, however, no record of the position of this
circle. Anderson (1915) also gave a plan of Castle Rigg and cited
Morrow (190809) as being the first to associate alignments with
Castle Rigg, both astronomical and geographical, the latter being an
alignment with Great Mell Fell. Anderson (1915)^{*} stated
that a second outlier was known at one time and considered that the
main outlier (now the only one) and the centre of the circle were
aligned with Fiends Fell in the Pennines. This was said to mark the
sight of sunrise on or about the 1st May, the time of the Bealtuinn
feast, an important data in the Celtic calendar. Long Meg and her
daughters and Little Meg are said to indicate the same alignment.
Anderson (1923 B) reported that the outlier now standing had been
moved when a new road was built and that it had plough marks on it
indicating that it had Fallen and had since been reerected. A claim
of Thom (1967) that this outlier makes Castle Rigg one of the most
important stone circles must, therefore, be regarded with caution.
Anderson (1923C) found a tumulus on Great Mell Fell which is circular
but with one gap in the circle. This gap aligns exactly with Castle
Rigg.
References
have been found to other stone circles in Cumbria. All those with
plans have been examined and the coordinates of each stone have been
determined from these plans on superimposed tracing graph paper. A
list of circles from which data were extracted in this way are given
in Table 1. The plan of Castle Rigg by Dymond (1880) was not used
because the binding makes it impossible to extract data accurately.
Standard
texts on statistics (Spiegel, 1972) and on astronomy (Smart, 1942)
were used for reference.
CASTLE
RIGG SITE Further examination of the site has shown that the line
of buried stones referred to in the first report of 15.12.75 is not
on the meridian, although it is fairly close to it, and probably marks
the line of an old wall. It is believed that the site of the second
circle has now been located. A tumulus has been found with
surrounding circular earthwork and close by are many large stones,
possibly displaced from the circle. The tumulus, the outlier and the
centre of Castle Rigg circle appear to be in a straight line and the
outlier is approximately equidistant from both. It is hoped that an
aerial photograph can be obtained and that it will provide
verification of this.
OTHER
CUMBRIAN CIRCLES Other circles in Cumbria have been visited
including Long Meg, Little Meg, Little Salkeld, Sunkenkirk, Burnmoor,
Elva Plain and Seascale. The cup and ring marks noted by Dymond
(1913) to be present on Long Meg and Little Meg were seen and
photographed. Seascale was regarded with particular interest because
Thom (1967) had not been able to classify it. It is hoped to survey
and to analyse it when the analysis programme is working. It consists
of 10 stones, all upright, and seemingly well preserved. The good
state of preservation raised some doubts as to its authenticity and
this was strengthened by the ordinance survey map which describes it
as "stone circle, site of". Subsequently the report by
Fletcher (1958) was found relating to Seascale stone circle. It was
reerected in 1957 by pupils from Pelham House School and data have
been extracted from the plan reported by Fletcher (1958). The
reerection might explain the difficulty which Thom (1967) had in
classifying it. Objective analysis of this circle would clearly be of
great interest.
Photographs
have been taken of a number of circles and of interesting features.
These should be available in May, 1976.
ADJUDICATION 25TH MAY, 1976 It
is noted that adjudication will take place in London on Tuesday, 25th
May, 1976. Unfortunately two members of the team, Philip Johnson and
Alan Wylie will not be able to attend because they have 'A' level
examinations on this date. Alan Wylie had contributed particularly to
the computer programming, having been solely responsible for the
Stage III programme.
Description
of the method used for data collection for Stage I will be presented
by Peter Bestley.
Extraction
of coordinates from published stone circles in Stage II will be
described by Anne Seneviratne.
The
surveying methods for stone circles used in Stages II and III will be
presented by Peter Hodgson.
The
investigation of axis orientation from published circles and
correlation with latitude in Stage IV will be presented by Margaret
Wylie.
In
Stage V the extraction of data relating to ancient sites in Cumbria
will be presented by David Wylie.
All
questions relating to computer programmes will be dealt with by Peter
Stewart.
LIBRARY WORK ADDENDUM
The
paper by Morrow J., (1908), Proceedings of the University of Durham
Philosophical Society iii, 71 has now been obtained. It makes
no reference to a second outlier and the statement that there was a
second outlier appears to be based on a misreading, by Anderson
(1915) of the paper by Morrow (1908).
STAGE I: QUANTUM DETERMINATION
INTRODUCTION In
megalithic times circles may have been measured by means of a
yardstick (Thom 1955) or by pacing (Kendall 1955). The pacing
hypothesis was favoured by Porteous (1973) who used made up data for
paces which had an arbitrarily chosen mean and standard deviation.
The main purpose of this stage of the project was to repeat the
calculation of Porteous (1973) using real data instead of made up
data. It was intended to calculate the mean pace and standard
deviation not only for each individual but also to compare one step
with another and thus to provide information about variation for each
individual and between individuals. The performance of thirty
volunteers at Keswick School are being measured in respect of
comfortable paces: The Keswick Pace; in respect of paces made to
simulate the megalithic yard: the Keswick Yard and in respect of
measurements made with a stick to simulate the megalithic yard: the
Keswick Stick. At the time of writing data have been collected from
14 subjects. Data are also being collected from each subject in
respect of age, sex, standing height wearing shoes and of sitting
height. It is intended to seek correlation between these parameters
and the pace and stick measurements.
THE
KESWICK PACE Each subject was asked to take five comfortable
paces, neither overstretched nor understepped, in a sand pit. The
first pace was taken from a reference board. The subject was free to
start with either foot but the starting foot, left or right, was
noted. A plastic marker tag was placed in the sand to indicate the
limit of the heel print of each step. The position of each marker was
agreed by two independent observers. The horizontal distance of each
pace from the reference point was measured by two independent
observers and, when agreement was reached the distance was recorded.
The experiment was repeated with a further five paces but with the
first step being taken by the opposite foot.
THE
KESWICK YARD After completion of the experiment to measure 10
Keswick Paces the subject was asked to adjust his or her paces to be
equal, as nearly as possible to the megalithic yard. A practise area
approximately 1OO yards away from the sand pit was marked out in
megalithic yards and the subject was allowed to practise for about
five or ten minutes while other subjects were being measured. The
subject then returned to the sand pit and took five paces, starting
with the same foot as was used for the first set of 5 Keswick Paces.
The paces were measured and recorded. After an interval of about five
minutes, during which time further practise was not allowed, the
experiment was repeated starting with the opposite foot.
THE
KESWICK STICK Each subject was given a cane stick and a measure
cut to the length (829 mm) of the megalithic yard. The subject was
asked to mark on the stick the length of the megalithic yard, or if
preferred to cut the stick to the length of a megalithic yard. The
subject was then asked to mark out five megalithic yards on the
ground starting with one end of the stick against the reference
point. When the stick had been placed on the ground to record the
second yard a plastic marker tag was placed at the end nearest to the
start to indicate the end of the first yard. The Horizontal distance
from the start to she first and to each subsequent marker was agreed
by two independent observers and was recorded.
HORIZONTAL
MEASUREMENTS Particular care was taken to ensure that accurate
measurements were obtained of the horizontal distance between paces
or stick measurements. A reference board was used for all paces and
measurements. Two metal tape measures were placed approximately three
foot apart, parallel to each other at right angles to the reference
board and along each side of the sand pit. A loop at the end of each
tape was hooked over a metal screw fixed to the reference board. The
first step commenced from a wooden block about 6 cm from the screws
and this distance, called the end correction, has to be subtracted
from the first marker point to give the measurement of the first
pace. A straight cross piece was placed so that it exactly touches
the edge of a plastic marker. The cross piece must be parallel to the
reference board and this was ensured if the edge of the cross piece
was lying on each tape measure at an equal distance from the screws.
At least two observers were involved in the measurements and in some
instances results were more quickly obtained with up to five
observers. One observer for each tape measure ensured that the cross
piece was at the same position on each tape measure. When agreement
was reached, the measurement was recorded.
SUBJECT
DATA For each subject age and sex was recorded together with
measurements of standing height wearing shoes and of sitting height.
In each instance the measurement was agreed by two independent
observers before it was recorded.
COMPUTER
PROGRAMME OUTLINE
The
input data consisted of measurements taken from the end of the tapes
at each step. The actual length of each step was obtained by
subtraction of adjacent measurements and the length of each step was
stored. For all thirty subjects a mean length and standard deviation
will be obtained for each of five steps, starting with the left foot
and for a similar set starting with the right foot. To calculate the
significance of different means the Z statistic is calculated. It is
intended that a comparison should be made firstly of steps taken with
the left foot and of the right foot, secondly of all the paces taken
by each of thirty subjects and thirdly of the first step with
subsequent steps. The mean pace size for different subjects will be
compared with age, sex, height, sitting height and leg length.
Possible correlation between different groups will be tested by
calculating the least squares, regression line and the coefficient of
correlation. It is expected that the correlation will be found
between the mean pace of different subjects with age and leg length.
Data
in respect of the Keswick Yard and the Keswick Stick will be
investigated in the same way as the Keswick Pace. The correlation
coefficient in respect of the Keswick Yard should show to what extent
subjects can overcome factors such as leg length which may determine
the natural step size. No correlation is expected between the Keswick
Stick and the factors which may be correlated with paces.
To
establish the existence of a quantum, the Criterion C of Broadbent
(l956) will be calculated. A convenient method of doing this is
described by Thom (1967). The Criterion C will be estimated for the
Keswick Pace and the Keswick Stick to see which best fits the
hypotheses suggested by Thom (1955) with respect to the megalithic
yard and Kendall (1955) with respect to the pace.
RESULTS A
computer programme has been written to carry out a preliminary
analysis of 14 sets of data presently available. The computer
programme, data and results are given. A list of variables used is
given in the appendix. It is obvious that the data have not been
correctly recorded for Subject 12, column 1 and the measurements for
this subject will have to be repeated.
CONCLUSION The
preliminary results obtained for analysis in respect of l4 subjects
are of considerable interest. Several tentative conclusions may be
drawn but are advanced with caution, until such time as all data have
been obtained and analysed.
1.
In the Keswick Yard the first stride is significantly shorter than
the fifth stride and there is a tendency for successive strides to
become progressively longer, at least as far as the fifth stride. It
would appear that the formula proposed by Kendall (1955) may require
modification.
2.
The Keswick Stick is much more accurate than the Keswick Pace or the
Keswick Yard as is indicated by the much smaller standard deviation
of the former.
3.
The Keswick Yard at 86.32 cm ±6.74 or 33.98 inches ±2.655 has a
standard deviation which is much bigger than that arbitrarily chosen
by Porteous (1973) at 33 inches ± 0.5. This suggests that the pace
hypothesis is a much weaker contender than Porteous believed.
4.
The preliminary analysis strongly supports the existence of a
megalithic yard as suggested by Thom (1955).
STAGE
II: GEOMETRICAL ANALYSIS
In
this stage methods have been developed to classify objectively
different types of stone ring known. Objective analysis would be
particularly useful in
the
examination of difficult circles which have not previously been
classified. It would also lead to more accurate determination of the
circle centre and hence would provide more accurate information for
investigation of megalithic yard measurements. There are four main
geometrical types of stone ring: circle, flattened ring, egg shaped
ring and ellipse. In the cases of the circle there is one
construction point, the centre, and in the case of the ellipse there
are two points, the foci. In the case of the flattened and the egg
shaped rings there are several construction points from which
auxiliary arcs are drawn but in both the flattened and the egg shaped
ring there is a main centre from which an arc of at least 180° is
derived. Determination of this main arc centre is of prime importance
in all rings (for an ellipse this point would be the centroid) for
the shape can be specified with reference to this point. In all cases
classification involves determination of the centroid, the main arc
centre, the long axis and the short axis. The problem has turned out
to be much more complex than had been anticipated but it is believed
that a solution can be found and an objective method of
classification is being developed.
COLLECTION
OF DATA Data were obtained in two ways. Firstly from a survey of
the site carried out by pupils and secondly from published plans of
the various sites.
SURVEY
OF CASTLE RIGG The method used has been modified in the light of
experience. It has also been discovered that the theodolite used is
accurate only to above five minutes of arc. Measurements are now
being taken as polar coordinates. The theodolite was placed over the
estimated centre of the circle and was first sighted on the cairn on
top of Skiddaw Little Man, a prominent local land mark. Measurements
were then taken on each stone, marked by a thin rod set up at its
highest point. The final reading was taken on Skiddaw Little Man
cairn and the data was accepted if this reading was the same as the
original one. It is desirable that such measurements should be
checked by a second independent observer but this has not yet been
done. The distance on the ground from underneath the theodolite to
the left and right edges of each stone (that is tangential
measurements) at its base was made and agreed by two independent
observers. The distance from the reference point to the centre of
each stone was taken to be the arithmetic mean of the two tangential
measurements.
To
simplify the project it is intended to restrict measurements in the
vertical (Z) axis to that of the skyline above each stone. The
photographic method would be satisfactory but is extremely
timeconsuming and has only recently been abandoned. Measurements
will have to be made when there is a suitable clear day. It is
expected to complete the data early in May.
PUBLISHED
DATA Data were extracted from published plans of eight Cumbrian
circles (Table 1) and will be extracted from other circles of
geometric interest and from plans published by Thom (1967) and
others. Tracing graph paper is placed over the published plan. The
centre of each stone and its reference number are marked and the
cartesian coordinates are read off.
PROPOSED
METHOD The method is first explained in outline. Three complete
rings, circle, flat and egg, each having the main arc centre o are
shown superimposed, (Figure 1). The distance from the centre to
points on the circumference may be calculated. It is obvious that the
distances from the main arc centre to the main arc circumference is
average for the circle, below average for the egg and above average
for the flat. A profile from point A on the circumference clockwise
through 360° defines the shape of each figure which is scaled such
that the main arc radius = 100. The profiles are also shown in Figure
1 for the circle, the flat and the egg. The line AB is the long axis
of the egg and the short axis of the flat. The line CD is the short
axis of the egg and the long axis of the flat. These axes define the
primary quadrants AOC, COB, BOD and DOA. The lines EF and GH are the
perpendicular bisectors of the main axis. They define the secondary
quadrants EOG, GOF, FOH, and HOE. The characteristics of each figure
may then be specified in respect of the mean distance of points (or
stones) on the perimeter from the main arc centre. The scaled radius
of 100 is accepted as standard.
1.
Circle. All eight quadrants are standard.
2.
Ellipse. The four primary quadrants are equal (and may or may
not be standard) but the four secondary quadrants are not. Opposite
quadrants in the long axis are above standard and in the short axis
are below standard.
3.
Flat. Adjacent primary quadrants on one side of the long axis
are standard and on the other side are below standard. Secondary
quadrants at opposite ends of the short axis are respectively
standard and substantially below standard. Secondary quadrants at
opposite ends of the long axis are equal and slightly below standard.
4.
Egg. Adjacent primary quadrants on one side of the short axis
are standard and on the other side are above standard. Secondary
quadrants at opposite ends of the long axis are respectively standard
and substantially above standard. Secondary quadrants at opposite
ends of the short ends are equal and slightly above standard.
Determination
of the main arc centre requires a complex set of calculations. In
preliminary studies two alternative methods were considered but were
rejected because they were not entirely suitable. The centroid may be
used to give a first approximation to the main arc centre and the
mean distance (M) from
the stone to the centroid is calculated. The centroid and the mean
calculated in his way should give fairly good results for a circle
and ellipse but for an egg shaped ring the centroid is displaced
towards the pointed end and in the flattened ring it is displaced
away from the flattened side. This method has an important
disadvantage in that correction cannot be made for displaced stones
and a very misleading position would be given by an incomplete ring.
Similar but less serious objections apply to the use of the least
squares technique. This may yet, however, be used with an adaptation
for the rejection of deviant stones. It was decided to develop a
technique based on determination of a circle centre given a chord of
three points. For each set of three adjacent stones an arc centre was
determined. No arc centre should lie outside the circle and any
centre which lay more than 1.5 times the mean distance (M)
of perimeter stones from the centroid was rejected. This step is
necessary to reject data from misplaced stones and from stones which
lie in a straight line, for these would give an arc centre at
infinity. In some types of egg shaped rings a small segment of the
perimeter does consist of stones in a straight line. The centroid of
the remaining arc centres was determined and distances from this
centroid to all stones was calculated. The mean distance (D)
was determined and the stone which was most deviant from this mean
was eliminated. The cycle of elimination of the most deviant stone is
repeated until three stones remain. It is anticipated that the method
will first of all eliminate fallen stones which are more likely to be
misplaced than standing stones. The three stones which remain lie on
the arc which gives the best estimate of the main arc centre. From
this centre the position of perfect rings, circular, flat and egg are
predicted and the positions of observed stones are measured. In
Figure l stones K and R are at distance d1 (d1 = OK) and d2 (d2 = OR)
from U. OJKLM is a radius through K and OPQRS is a radius through R.
Points L and Q are on the circle circumference at standard distance
from O (100 = OL = OQ). The deviation (E) of stone K from the circle
circumference is given by E_{k} = (d_{1}100) and the
deviation of stone R is similarly given by E_{r} = (d_{2}100).
It would be noted that E is positive, as for stone R, if it is
outside the circle and is negative as for stone K, if it is inside.
Predicted values for the deviation in respect of the flat ring are
given by (OJOL) for stone K and (OPOQ) for stone R. A method for
calculating J and P the predicted ideal positions on the
circumference of a flat ring will be developed later. Similarly for
an egg shaped ring the deviations are given by (OMOL) for stone K
and (OSOQ) for stone R. Calculation of the predicted points on the
circumference are expressed as an observed and as an expected
deviation from a true circle in which the deviation of each stone
should be zero. The data may be printed out in graphical form.
Observed and expected data may also be examined statistically for
goodness of fit with one or other of the theoretical types of stone
ring.
PRELIMINARY
RESULTS A programme has been written for the first part of the
geometrical analysis and a copy of this programme is given together
with data, results and variables used. It must be emphasised that the
results are not yet satisfactory and are presented merely to indicate
the potential of the method. A long and complex programme will be
required and the use of a computer would be necessary for its
development.
CONCLUSION It
is concluded that objective analysis of stone circle is possible. A
method of doing this is outlined.
STAGE
III: ASTRONOMICAL ALIGNMENTS It has already been mentioned that
this stage has been simplified, in order that the project might be
completed in a reasonable time. It is intended that all possible
alignments from the centre of the circle at Castle Rigg to points on
the horizon over each stone should be determined. They should be
correlated with the positions of the sun and moon and bright stars
related to the period 2000 B.C. to 1600 B.C. At the present time sun,
moon and bright star positions are given in astronomical tables.
These figures must be corrected for the latitude of Castle Rigg and
this is done using a standard formula of spherical trigonometry:
sin δ = sin Φ sin h + cos Φ
cos h cos Az
where δ = declination, h=
horizon altitude (true) Φ = latitude, Az = azimuth.
Correction
to the years 2000 B.C. to 16OO B.C. could be obtained using
DeSitter's formula but it has been discovered that the accuracy
obtainable with the present theodolite does not warrant this
correction.
Correction
for refraction and temperature is important for points near the
horizon but for a mountainous skyline, as it is in Keswick, this is
not necessary. Alignments from the centre are determined by
calculating the equation for a line joining two points, one of which
is the centre (X_{o}Y_{o}Z_{o})
and the other, the skyline above a stone (X_{1}Y_{1}Z_{1})
as viewed from the centre of the circle. Alignments which fall within
a chosen margin of error (for example 1° of arc) would be calculated
and displayed in the printout.
RESULTS A
preliminary programme has been written but has not yet been run or
tested. It does, however, indicate some subroutines and a format
which might be used and a copy of provided.
STAGE
IV: AXIS ORIENTATION In this stage of the investigation published
data on stone circles have been examined for correlation between axis
orientation and latitude. All circles published by Thom (1967) have
been examined and data extracted in respect of latitude and axis
orientation. The data have been plotted and do not appear to be
correlated. This is confirmed by calculating the correlation
coefficient, which is not significant. It is intended that the
correlation coefficient should be worked out by computer and that
other factors including longitude should also be examined. In
plotting the circle data it was noted that many were in hilly areas
and a search for possible correlation with height above sea level
would be of interest. It might be that many circles have been
destroyed by agricultural needs and that fields in arable areas have
been cleared for this purpose.
STAGE
V: LEY LINES
The
claim of Watkins (1925) that there is a widespread network of leys or
Lines linking up ancient sites dues not appear to have been
investigated objectively.
DATA
COLLECTION The one inch ordinance survey map of Cumbria was
scrutinised and features of prehistoric interest were marked. There
were found 89 sites in all: twelve stone circles, coded as type 1;
two cairn circles, coded as type 2; fortyone cairns, coded as type 3
(it should be noted that only cairns indicated by gothic type are
included); three settlements, coded as type 4; twentythree tumuli,
coded as type 5, and eight enclosures or other types of ancient site,
coded as type 6. The national grid coordinates were estimated for
each site and information was tabulated as follows:
1.
Reference number.
2.
National grid coordinate in X axis.
3.
National grid coordinate on Y axis.
4.
Code type of each site.
Cumbria
is covered by national grid squares NX, NY, SC, SD and the reference
letters are coded by means of an additional figure in the X and Y
axes (NY = 1,000X and 1,00OY, NX = 1,000Y, SD = 1,OOOX). Thus Castle
Rigg which is situated on sheet NY at national grid reference 293/237
is recorded as having an X coordinate of 1293 and a Y coordinate of
1237. The national grid coordinate of the reference cairn on Skiddaw
Little Man is NY 267/279. Examples of typical sites all from sheet
NY, are identified in table II to facilitate checking from the
ordinance survey map.
All
possible combinations of three sites were considered and each set of
three was regarded as a triangle having coordinates (X_{1}Y_{1})
(X_{2}Y_{2})
(X_{3}Y_{3}).
The slope of each side was than calculated (θ being the angle of
inclination to grid north). A test was first made of the Y
coordinate. If Y1 = Y2 then θ = 90°. If not θ is calculated from
the formula: θ = Tan^{1
}(X_{1}X_{2})/(Y_{1}Y_{2})
The Slope of all three sides θ1, θ2 and θ3 were compared and the
difference between each pair was measured, A = (θ1θ2), B =
(θ2θ3), c = (θ1θ3). The three points, X1, Y1; X2, Y2; and X3, Y3
were considered to be in alignment if the slope of adjacent sides
(ABC) was not greater than l (A≤1°;B≤1°;C≤1°) Data for each
set of three features in alignment were printed out. When the
programme was run originally cairns were included but a very large
number of alignments were found. Cairns were then deleted and the
programme was run again with 48 sites.
The
results did not appear to be different from what might be expected by
chance. A programme was, therefore, written to generate random
numbers. They were inserted instead of actual X and Y coordinates
and the programme was run again.
RESULTS The
programme, written to determine ley line alignments, is given
together with data for 89 sites. The cancellation marks at the right
hand side of the printout should be ignored. They were inserted when
there was a shortage of paper, to reuse paper and cancel a previous
printout. In all 209 alignments were found and are given in the
printout. A copy of the programme to generate random numbers is also
given together with a printout of results: 144 alignments were found.
The actual number of 209 alignments found is not significantly
different from the 144 obtained with random numbers.
χ^{2
}= 6.04 0.025>P>0.01
CONCLUSION Although
results have been obtained with this programme it should not be
regarded as final and in particular alterations may be required to
ensure that the areas covered by the actual and random generated
sites are equal. Nevertheless, tentative conclusions may be made.
1.
There is no objective evidence of a network of alignments linking up
ancient sites in Cumbria.
2.
The first conclusion is strengthened when actual alignments are
examined. For example there is an alignment which includes Castle
Rigg, site number ll, and long Rigg, site number 50. These sites are
fourteen miles apart and the line joining them passes through one of
the most mountainous areas of the Lake District: between Great Gable
and Scafell Pike. It appears likely that this alignment has occurred
by chance rather than by design.
3.
The general conclusions stated above should not be taken to exclude
the possibility of significant alignments in individual cases. In may
be for instance that the apparent alignment of Castle Rigg and Long
Meg with Fiends Fell in the Pennines is real. This possibility could
he investigated at other sites when the astronomical alignments
programme is working.
SUMMARY AND CONCLUSIONS
In
the first stage of the project statistical data on measurement by
pace or stick have been obtained. The data strongly favour the
existence of megalithic yard and the measurement of sites with a
yardstick. The measurement of paces also show a significant
difference between the first and subsequent paces and indicate that a
formula proposed for the variance of paces may require revision.
In
the second stage of the project preliminary studies suggest that
objective classification of stone circles by computer analysis is
feasible.
In
the third stage of the project a possible outline programme for part
of the stage has been proposed.
In
the fourth stage of the project no correlation has been found between
axis orientation and latitude.
In
the fifth stage of the project no objective evidence has been found
for the existence of a network of ley lines between ancient sites in
Cumbria.
A C K N O W L E D G E M E N T
S
Grateful
acknowledgement is made to the Headmaster, Mr. J. E. Thompson, M.C.,
M.A., V.R.D., J.P. for providing facilities; to Mr. A. Rothwell,
B.Sc. for general advice; to Dr. Thompson Ph.D for advice on
astronomy; to Mr. J. Stewart B.Sc., for suggesting Stage I of the
investigation, for discussions of methods and for advice on the
presentation of results; to Mr. Rothwell and Mr. Stewart for
organising visits to other circles in Cumbria. Thanks are due also to
Mr. Elsby, Librarian, Keswick Public Library for providing
references; to Mr. K. R. Bull, Librarian, PostGraduate Institute,
Wigan for obtaining references, and to Mrs. Pilkington for typing
this paper.
TABLE 1 CUMBRIAN STONE CIRCLES
CIRCLE  NUMBER OF STONES  REFERENCE 
Sunkenkirk (Swinside)  54  Dymond (1902) 
Long Meg  67  Dymond (1913) 
Little Meg  10  Dymcnd (1913) 
Elva Plain  16  Anderson (1923 A) 
Shap  34  Spence (1935) 
Lacra  6  Dixon & Fell (1948) 
Seascale  10  Fletcher (1958) 
TABLE 2 INFORMATION ON SOME
TYPICAL ANCIENT SITES IN CUMBRIA
No.  X  Y  GRID  NAME  CODE  TYPE 
11  1293  1237  293/237  Castle Rigg  1  Circle 
50  1173  1027  173/027  Longrigg  2  Cairn circles 
51  1217  1015  217/015  Hardknot  3  Cairn 
55  1330  1232  330/242*  Threlkeld  4  Settlement 
56  1398  1254  398/254  Great Mell Fell  5  Tumulus 
67  1178  1317  178/317  Elva Plain  1  Circle 
64  1288  1394  288/394  Thistle Bottom  6  Enclosure 
*
Note error in computer input data.
R E F E R E N C E S
ANDERSON, W.D  (1915)  TCWAAS (NS) 15, 98 
ANDERSON, W.D  (1923A)  TCWAAS (NS) 23, 29 
ANDERSON, W.D  (1923B)  TCWAAS (NS) 23, 109 
ANDERSON, W.D  (1923C)  TCWAAS (NS) 23, 112 
BROADBENT, S.R.  (1955)  Biometrika, 42, 45 
BROADBENT, S.R.  (1956)  Biometrika, 43, 32 
DIXON, J.A., FELL, C,I.  (1948)  TCWAAS (NS) 48, 1 
DYMOND, C.W.  (1880)  TCWAAS (OS) 5, 39 
DYMOND, C.W.  (1902)  TCWAAS (NS) 2, 53 
DYMOND, C.W.  (1913)  TCWAAS (NS) 13, 406 
FLETCHER, W.  (1958)  TCWAAS (NS) 58, 1 
HAWKINS, G.S.  (1963)  Nature, 200, 1258 
HAWKINS, G.S.  (1965)  Science, 147, 127 
HAWKINS, G.S.  (1970)  Vistas in Astronomy, 12, 45 
KENDALL, M.G.  (1955)  J. Roy, Stat. Soc, A, 118, 291 
MITCHELL, J.  (1969)  The View Over Atlantis, Garnstone, London 
PORTEOUS, H.L.  (1973)  J.H.A., 4, 22 
SMART, W.M.  (1942)  Foundations of Astronomy, Langmans, London 
SPENCE, J.E.  (1935)  TCWAAS (NS) 35, 69 
SPIEGEL, M.R.  (1972)  Theory and Problems of Statistics. McGraw Hill, London 
THOM, A.  (1955)  J. Roy. Stat. Soc., A, 118, 275 
THOM, A.  (1961)  Mathematical Gazette, 45, 83 
THOM, A.  (1962)  J. Roy, Stat. soc., A, 125, 243 
THOM, A.  (1967)  Megalithic Sites in Britain. Clarendon, Oxford 
THOM, A.  (1971)  Megalithic Lunar Observatories. Clarendon, Oxford 
WATKINS, A.  (1925, republished 1970)  The Old Straight Track, Garnstone, London 
Appendix A
STAGE I P A S D A PROGRAMME:
EMPLOYMENT OF VARIABLES
ARRAY
VARIABLES
N (1)  Number of subjects, not more than 30 
L (25,30)  Length of each of 25 measurements for 30 subjects 
M (5)  5 mean distances to be printed simultaneously 
S (5)  5 standard deviations to be printed simultaneously 
Line 20 W$ (80)  Vertical script 
Line 80 X$ (25) 

O (5,5)  means  ) from measurement  group plane ) used to determine Z statistic 
T (5,5)  standard deviations 
SIMPLE
VARIABLES
F1) F2)  Cassette file numbers 
A) B) C) 
FOR loops 
M  Summation of lengths for mean and standard deviation 
S  Summation of (lengths)^{2} for standard deviation 
L  Letter of W$ or X$ to be printed 
Z  The Z statistic, significance of differences 
P  Probability level at which difference is significant. 
Appendix B
STAGE II S C I D PROGRAMME:
EMPLOYMENT OF VARIABLES
ARRAY
VARIABLES
P(2,70)  Polar coordinates R, θ calculated from centre of ring for up to 70 stones P (1, ...) = distance P (2, ...) = θ 
R (3,70)  Coordinates and code number of each of 70 stones R (1, ...) = ordinate R (2, ...) = abscissa R (3, ...) = number 
C (2)  Coordinates of centre calculated for 3 stones by solving equation of circle 
N (l)  Number of stone coordinates on tape 
M (2,2)  Coordinates of best centre so far (M(1,1), M(2,1)) Coordinates of new centre (M(l,2) M(2,2)) 
OI (70)  The order on the circle, as seen from the best centre of the stones 
S (2,70)  A copy of stone coordinates 
GI (5)  Used only to GI (3) for the actual numbers of stones, inside, outside and on the main arc radius 
H (5,2)  The expected ratios of stones inside and outside the radius of the main arc. 
SIMPLE
VARIABLES
A) B) C) 
FOR statements and loop generations 
D  determinant used in solving 3 equations of circle to give the coordinates of the centre 
F1) F2)  Cassette data files 
N  Number of stones 
N1  Number of stones remaining after eliminations 
T line 300) S line 360)  X and Y summation for calculation of centroid 
T 550  Subroutine count centres used in centroid of centres. 
T 2140  onward  calculation of X^{2 } 
W  = O unless there are no centres suitable for centroid of centres in which case W → 1  this inhibits certain actions at a later stage 
B1) L1)  Length of biggest and smallest radii 
B2) L2)  Stones with biggest and smallest radii 
P2  Main radius 
P1) D1)  Mean and standard deviation of radii 
Z  Stone selected to be eliminated 
Appendix C
STAGE III SKYLINE PROGRAMME:
EMPLOYMENT OF VARIABLES
ARRAY
VARIABLES
A (100)  Azimuth 
B (100)  Azimuth in error check 
D (100)  Error marker 
E (100)  Elevation 
F (100)  Elevation in error check 
J (100)  Declination 
K (100)  Hour angle 
SIMPLE
VARIABLES
G  Azimuth in error correction 
H  Elevation in error correction 
I  Correction for north 
L  Latitude 
C  Counter in FOR statements 
N  Number of readings 
P  Used in DEF FNA 
Q  Used in DEF FNA 
R  Used in DEF FNA 
S  SIN (SIN J) 
T  Number of errors 
Z  Used in DEF FNA 
Appendix D
STAGE V LEY LINES PROGRAMME:
EMPLOYMENT OF VARIABLES
ARRAY
VARIABLES
X (100)  Ordinate ) 
Y (100)  Abscissa ) of each of up to 100 sites 
T (100)  Type ) 
E (100)  Takes in numbers, compares them and stores numbers of sites for which an error has been made 
P (100) 

Q (100)  Used to check date in input programme 
R (100) 

L$ (25)  Script 
A$ (25) 

SIMPLE
VARIABLES
N  Number of site 
D  Maximum error 
J  Number of possible alignments 
P)  Angles from grid north of 2 of the stones from the third 
Q) 

E  Number of errors in data 
F  Number of errors when correcting errors 
Appendix E
FORMULAE USED
MEAN 

STANDARD DEVIATION 

CENTROID 

DISTANCE 

SIGNIFICANCE 

Z STATISTIC 

INCLINATION 

CONVERSION 

DECLINATION 

