THE CASTLE RIGG PROJECT
OBJECTIVE The objective
of this 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 (frontispiece) 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 In his
book "Megalithic sites in Britain" Thom (1967) has surveyed
and classified many such 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 (figure 1) is an example of a flattened circle
and Clava (figure 2) is an example of an egg shaped circle. These
figures are taken from Thom (1967). It is of interest that Thom
(1961) was unable to classify according to type a stone circle at
Seascale in Cumbria.
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 1956) 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 but did not produce
objective data to support it.
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).
INVOLVEMENT OF PUPILS The
project is being carried out by a team of pupils, so far ten in
number, from Keswick School, Keswick, Cumbria and is submitted by the
project leader Peter Stewart on behalf of the Keswick School Project
Team (ages shown): Peter Bestley, (15); Philip Johnson, (18); Anne
Seneviratne, (16); Richard Smith, (15); Peter Stewart, (17); Stephen
Temple, (17); Paul Whittaker, (15); Alan Wylie, (18); David Wylie,
(12) and Margaret Wylie, (16). Following discussion with advisors a
four stage project has been prepared. It is intended that the field
work should be carried out solely by the pupils but teachers and
other advisors are being consulted for statistical and other advice.
Pupils within the project team are dealing with particular aspects of
the project including photography, surveying, astronomy and computer
programming.
Preliminary searchwork in
Keswick library has revealed local knowledge in the Transactions of
the Cumberland and Westmorland Antiquarian and Archaeological Society
which was apparently unknown to Thom (1967) when he described Castle
Rigg as one of the most important sites in respect of the outlier.
Marks on the outlier would appear to have been caused by a plough
share when the stone had been lying flat and partly buried. It is not
know when it was reerected. Moreover the outlier was moved some
years ago when a wall was rebuilt. A second circle mentioned in the
Transactions has not yet been identified in the field. If facilities
were available an aerial photograph might be helpful. Field work at
Castle Rigg site by the project team has revealed a stone near to the
outlier, which has not apparently been described and which may be of
significance. Also there is a line of partially buried stones,
apparently not previously described, to the North of the outlier
which might have significance. They might have been used to determine
the meridian and this possibility is being investigated. The
preliminary survey has also revealed inaccuracies in the survey
methods used by the team and these are being corrected. The azimuth
of sunset from Castle Rigg was obtained on 8.11.75 and 25.11.75.
Weather conditions did not permit a sighting closer than this date to
the recent total eclipse of the moon on 18.11.75. A reading on this
date would have been useful since the circle may have been used for
eclipse prediction.
STAGE 1 In this stage of
the project, data is being collected to be used in a comparison of
the pacing and yardstick hypotheses. Each of 30 subjects is to make
10 paces in a sand pit and the measurement of each pace is to be
checked by two independent observers. A mean and standard deviation
for the average pace would be obtained and internal examination of
the data would involve comparison of the first step with others and
of one person with another. The experiment would be repeated using a
yardstick instead of paces and the result would be presented as a
histogram. The results would be compared using the statistical
methods described by Broadbent (1955, 1956).
STAGE II In this stage,
methods are being developed to classify objectively the different
types of stone circle known. A survey is being made of Castle Rigg
stone circle using a theodolite for measurement of angles and a metal
tape measure for measurement of distances. Three reference points
within the circle are used and the position of each stone, marked by
a thin rod set up at the highest point, is determined from each of
these points. The results are computed to give the cartesian
coordinates of each stone. If the three readings do not agree, that
is if the plot shows a triangle of uncertainty, the position is
measured again. The cairn on top of Skiddaw Lower Man, a prominent
local landmark, approximately three miles away, was used as a
reference point. After taking bearings on several stones the
theodolite reading was checked against the original reference point.
The direction of the compass points true north, south, east and west
are shown relative to the circle. Similar surveys of other stone
circles in Cumbria are planned and the data obtained from published
surveys may also be used for computer analysis.
There are 4 main geometrical
types of stone ring: circle, flattened ring,
eggshaped ring and ellipse. In
some instances formulae are well known:
circle: x^{2 }+ y^{2
}= r^{2}
ellipse: nx^{2 }+ my^{2
}= k
For the other two geometrical
shapes formulae are being derived. Objective computer investigation
of the geometry in all cases involves calculation of the
centroid:
;
and also determination of the
axis of symmetry by calculation of the regression line. In the case
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 eggshaped rings there are several construction
points from which auxiliary arcs are drawn and formulae are being
derived for the determination of these. Examples of the flattened
ring and eggshaped ring are shown in figures l and 2.
For each ring the axis of
symmetry will be determined in relation to true north and the ring
will be classified according to type with determination of
construction points.
STAGE III In this stage,
all possible alignments at Castle Rigg are determined and correlated
with the positions of the sun, moon, planets and bright stars related
to the period 2,000 BC and 1,600 BC.
The survey data in respect of
Castle Rigg described in Stage II are extended to include the heights
of each stone and are expressed as cartesian coordinates in the z
axis. Since Keswick is a mountainous area the declinations of the
actual horizon, as changed due to mountains, are determined by
readings of the skyline at 1° intervals. The data will be checked
photographically with photographs at 20° intervals taken from the
centre of the circle.
At the present time, sun, moon,
planet and bright star positions are given in astronomical tables.
These figures must be corrected for the longitude and latitude of
Castle Rigg and to its time of construction:
(1)  Correction for latitude and longitude is made using standard formulae 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. 
(2)  Correction to the years, 2,000 BC to 1,600 BC may be obtained using De Sitter's formula: 
ε = 20° 27' 8".24  47".08t  0".0059t^{2} + 0".00186t^{3} 
where ε is the obliquity of the ecliptic and t is measure in centuries forward from 1900 AD. Alternatively tables published by Hawkins (1970) may be used. 
(3)  Correction for refraction and temperature may also be required. 
Computer analysis will permit
examination of all possible alignments and those which fall within a
chosen margin of error, (for example, one minute of arc) will be
calculated and displayed in the printout.
STAGE IV In this stage of
the investigation published data on stone circles are examined for
correlations between orientation and geographical position. Recent
discussion on television about the possible use of stone circles as
signposts for the determination of "leys" or straight
tracks is relevant. Preliminary study has shown that Castle Rigg in
Keswick is almost exactly in line with Long Meg and Little Meg near
Penrith. It is of interest that they are also all in line with
Fiend's Fell in the Pennines.
Published data on stone circles
will be analysed using the methods developed in the previous stages
II and III and derived data will include the alignment of the main
axis relative to true north, latitude, longitude, geometrical type of
ring and astronomical positions.
Calculations will be made
between all possible sets of data and correlation coefficients will
be worked out. The mathematical significance of correlations will be
established using standard statistical techniques and the data will
be presented, for correlations at a chosen significance level, in the
form of a computer printout.
SUMMARY AND CONCLUSION In
the first stage of the project statistical data are collected from 30
independent observers for distances measured by pacing or by using a
standard "yardstick". The results will be compared with
data obtained from surveys of stone circles and may provide
information about the methods of measurement in megalithic time. In
the second stage, stone circles are classified objectively by
computer analysis of measured or published data. In the third stage a
survey of Castle Rigg stone circle is subjected to computer analysis
and alignments of possible astronomical significance are determined.
In the final stage of the investigation computer techniques developed
in earlier stages are used to analyse objectively published data and
to look for possible correlations between circle axis orientation,
geographical position and astronomical significance.
FIGURE
1 CASTLE RIGG
From
Thom, (1967)
FIGURE
2 CLAVA
From
Thom, (1967)
R E F E R E N C E S
BROADBENT, S.R. (1955)
Biometrika, 42, 45
BROADBENT, S.R. (1956)
Biometrika, 43, 32
HAWKINS, G.S. (1963) Nature, 200,
1258
HAWKINS, G.S. (1965) Science,
147, 127
HAWKINS, G.S. (1970) Vistas in
Astronomy, 12, 45
PORTEOUS, H.L. (1973) J.H.A., 4,
22
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.
F I G U R E S
Frontispiece Castle Rigg,
Keswick.
Photograph by Peter Bestley.
Figure 1. Castle Rigg (From
Thom 1967)
Figure 2. Clava (From Thom
1967)
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. and Mr. J. Stewart,
B.Sc., for advice. Thanks are due also to Mrs. Pilkington for typing
this paper.
