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.
