Table Of ContentScience and Archaeology no.25 (1983) pp. 13-30 however, does not in itself convincingly establish
the reality of the unit. D.C. Heggie, for example,
THE METROLOGY OF CUP AND has recently commented that the evidence presented
by Thom has no statistical significance (Heggie,
RING CARVINGS NEAR ILKLEY 1981). In order to appreciate such criticisms it
is necessary to examine the nature of the statis
IN YORKSHIRE tical tests which may be employed in the search
for possible units of measurement, and a brief
assessment of these is given later in this paper.
A. Davis
Important though such considerations are,
Lancaster Royal Grammar School
the prime requisite for testing Professor Thom's
hypothesis is the acquisition of more data, and
the present paper is the result of an independent
investigation carried out by the author, with the
following aims:
(1) To obtain a sufficiently large sample of
measurements for a statistical examination of the
'megalithic inch' hypothesis to be viable.
(2) To examine whether the cup-marks themselves
are positioned in random fashion, or whether their
relative positions show evidence of mensuration.
The paper is presented in three sections.
The first examines the statistical techniques
available for testing quantum hypotheses. The
second describes the results of a preliminary
study which, because of the rigorous selection
O ini ?o
procedure and objectivity of the methods employed,
O cm$ 50
it seems best to preserve as a whole. The third
Keywords section reports the results of subsequent work
which also has an important bearing on the
MEGALITHIC INCH, CUP AND RING CARVINGS, BROADBENT'S subject.
METHOD, KENDALL'S METHOD, ILKLEY MOOR
1. Techniques of quantum analysis
Abstract 1.1 Broadbent's method
1.2 Kendall's method
Professor A. Thom suggests that the existence
of a unit of measurement, the "megalithic inch", 2. Measurement and analysis of a group of
may be deduced from an analysis of cup and ring carvings
carvings. This hypothesis has been tested on a
group of carved designs in West Yorkshire, and an 2.1 Measurement technique
apparent quantum emerges which is very close to the 2.2 Analysis of the positions of the cup
value proposed by Thom. It further appears that centres
the spacing of cups show evidence of the use of 2.3 Variations in the value of the megalithic
this unit. Evidence is found in support of the inch
conjecture that cups with surrounding rings may be 2.4 The residuals
closely related to the measurements made by the pre
historic carvers, and one of the designs examined 3. Further work
shows evidence of the use of 5 megalithic inches as
a basic unit. 3.1 Alternative measureftent techniques
3.2 Pancake Ridge (2)
3.3 Cups with surrounding rings
Introduction 3.4 Weary Hill West
The debate concerning the reality and signifi
cance of Neolithic/Bronze Age units of measurement 1. Techniques of quantum analysis
has centred, understandably, on the statistical
analysis of stone circle diameters, and associated Methods for testing a set of data for the
measurements of megalithic sites. Comparatively presence of a quantum have been proposed by
little attention, however, has been bestowed upon Broadbent (1955; 1956), Kendall (1974), and
the work of Professor Thom concerning cup and ring Freeman (1976), though only the methods of
marks (Thom, 1968; Thom & Thom, 1978). Broadbent and Kendall will be considered here.
Whichever method is adopted, there are two very
Thom, using measurements taken from rubbings, different situations with which the investigator
suggests that the diameters of carved circular may be presented:
rings (mostly Scottish) show evidence of the use of
a unit of about 2.07 cm. which he calls the (A) The quantum is specified independently of the
'megalithic inch' (m.i.). He suggests that this data to be tested.
unit may be identified as one fortieth of a mega
lithic yard (2.0725 cm.), implying 100 m.i. to a (B) The investigator assumes no a priori expecta
megalithic rod. tion of a quantum, but derives a value for a
suspected quantum from the data themselves.
The statistical analysis of these diameters,
Science and Archaeology no.25 (1983)
13
The procedures appropriate to Type A and Type some compromises had to be accepted. Thus, since
B situations are very different and have been dis neither very small nor very large values of 1/d
cussed in depth elsewhere (Broadbent, 1956; Kendall, were relevant to the megalithic inch investigation,
1974; Freeman, 1976). In the case of Thom's the 'search range' was restricted to the interval
examination of cup and ring marks, a Type A signifi 5<l/d<60.
cance level is obtained of about 27. (Thom & Thom,
1978) for the megalithic inch of 2.0725 cm. But Figure 1(a) presents the results of 100 sets
this value was suggested by the data themselves each for n = 20 and n = 50 (Only maxima of C
(even though in retrospect it is seen to be related greater than 0.8 are plotted here, in order to
in a comparatively simple way to the megalithic avoid an unnecessary congestion of points which
yard), and a Type B procedure is therefore appro have no bearing on the matter in hand). An
priate; on these terms the megalithic inch has no examination of the diagram confirms Broadbent's
statistical significance. conjecture: that values of C greater than unity
represent grounds for rejection of the rectangular,
In this present investigation, however, since non-quantal hypothesis at about the 1% level.
the quantum is specified in advance, a Type A analy
sis of independent data would seem to be admissable Figure 1(b) presents maxima of C for 100
- although both types of analysis will in fact be sets of n • 100, and 50 sets of n = 150. It is
employed in this paper. clear, even from this limited number of simulations,
that the criterion C>1 is not appropriate for such
1.1 Broadbent's method large values of n. C>1.1 appears to be a minimum
requirement for rejection of the rectangular
The quantum hypothesis may be represented by hypothesis.
the relation: y » 2md + e , where y corresponds to
an individual measurement, m is an integer, 2d the However, as will become apparent, the data
suspected quantum, and e an error term. In presented later in this paper clearly do not arise
Broadbent's method the m values are estimated by from a rectangular distribution, regardless of any
taking the nearest integer multiple of 2d for each apparently quantal nature they may display, and so
y value. The residuals (e = y - 2md) are formed, a further series of simulations were performed.
and the 'lumped variance', s2=Ee2/n calculated, 200 sets of n = 100 were produced where the random
where n is the number of measurements in the sample. numbers forming each set were chosen such that 50
Finally, s2/d2 is calculated. are distributed uniformly between 0.1 and 0.4, 45
between 0.4 and 0.8, and 5 between 0.8 and 1
If the data arise from a non-quantal rec (including exactly 1), in an attempt to simulate,
tangular distribution, then s2/d2 has expectation very roughly, the distribution of one of the sam
1/3 and variance 4/45n for a given quantum 2d. A ples of cup and ring measurements presented later.
Type A test is obtained by determining whether the Figure 2 gives maxima of C for these simulations,
value of s2/d2 for the data is significantly below and while it is necessary to approach this diagram
1/3. Thom provides a useful chart for determining with circumspection in view of the crudity of the
significance levels (Thom, 1967). data distribution, there are indications here that
the test may be somewhat sensitive to the distribu
For a Type B test (where the investigator is tion from which the data arise. In the range
free to choose the quantum that best fits the data), 5^1/d^5O, for example, maxima of C exceeded 1 on
Monte Carlo methods are necessary. Broadbent, in seven occasions, suggesting that the criterion
his 1956 paper, offers the results of 200 simula C^l provides sufficient grounds for a tentative
tions, and suggests that a rough guide for rejecting rejection of a non-quantal hypothesis (at about the
the rectangular hypothesis is that the quantity 4% level) , in this range and for n = 100.
C =/n(l/3 - s2/d2) should be greater than unity.
These simulations, however, are not particularly From what we have seen above, it is clear
numerous; neither do they test samples for which n that the results of a Broadbent Type B analysis
exceeds 50. Further, data raising from non- must be interpreted very cautiously. The apparent
rectangular distributions are not considered. dependence of significance levels on both the size
and distribution of the sample implies that values
In view of these difficulties, inferences of C in any particular case may be used only as a
based on Broadbent's Type B test are necessarily very rough indicator of the significance of a
somewhat uncertain, especially when large samples of derived quantum.
data are involved. The author has therefore attemp
ted to cast extra light on the method by means of a 1.2 Kendall's method
further series of simulations. The necessary compu
tations were performed using a BBC Microcomputer. Using the same notation as above, Kendall's
method employs as the test statistic the function:
Following Broadbent, for each sample of size
0 (T) = /(2/n) ?cos(2ryT)
n, a set of (n-1) random numbers between 0 and 1
were generated, and the number 1 included to com
plete the set. As Broadbent points out, the distri where T = l/2d. For a given T, 0(T) is normally
bution of C in these circumstances will be directly distributed with expectation zero and unit
comparable to that for a rectangular distribution of variance (on the rectangular hypothesis), and this
data from 0 to some maximum value, a, other than 1. fact may be used as the basis for a Type A
For each set of simulated data, successive absolute analysis - though we shall not be concerned with
maxima of C were computed as 1/d increased over a this aspect here.
pre-selected range.
For a Type B analysis, Monte Carlo methods
Because of the limited speed capability of the are again necessary. Kendall gives the results of
machine available, and the need to use very fine 600 simulations which are specific to his analysis
increments in 1/d in order to determine the maxima of stone circle diameters, but offers a means of
of C precisely, the simulations involved several assessing significance levels for situations where
weeks of more or less continuous computing time, and the effective "search range" for the quantum and
Science and Archaeology no.25 (1983)
/.2-i
(a)
• n-rj
r n—!
1.0- X X
•
x • • >
X X X
X X X
0.9- X •
« ’
• X
X X , K X •
X X
M • *
'c 0.8-i----------------i---------------r -5-“-----------------r-“---------------- ~1 ~i1 M «-■-- ---X--- ----------X-------fr—~*-—-----------------1-----•-------------------
ie O IO 2 0 !/d 30 40 50 6 0
n
c
e
a
n Figure 1. Broadbent's method. Successive
d
absolute maxima of C for rectangular
A
r distribution of data:
ch (a) n = 20 (100 sets) and n = 50 (100 sets).
ae (b) n = 100 (100 sets) and n = 150 (50 sets)
o
lo
g
y
n
o
.2
5
(
1
9
8
3
)
1.4-1
O
1.3-
O
(b)
• n=IOO
1.2- x n = !5O
x
X
1.1 -
X
X
X
X
••
X X X
X X X
• X x X x‘x
X
X
X X
X x X X
M • X
±
T
I
Figure 2. Broadbent's method. Successive absolute
maxima of C for 200 sets of n = 100 with
distribution: 50 between .1 and .4;
45 between .4 and .8; 5 between .8 and 1.
the root mean square value of y are different. In Table 1
order to facilitate the estimation of significance
levels, limiting values of 0(T) for 5, 2, and 1Z O.S. Ref. No. of No. of
significance have been calculated using the graphs cups circular
of simulated data given in Kendall's 1974 paper rings
(Kendall's Figures A9 and A10), and these are dis
played in Figure 3 for a range of values of Weary Hill East SE107465 10 4
(Tj - T0).rms(y) (Here Tj is the reciprocal of the
Weary Hill West SE106466 11 4
smallest quantum contemplated, and Tq the reciprocal
of the largest. Thus (Tj - To) is the effective Pancake Ridge SE131463 14 11
"search range" for T).
Panorama Stone(l) SE114473 7 7
II
The great advantage of Kendall's method is Panorama Stone(2) 10 5
that it is (a) independent of the size of the
Backstone Beck SE 127463 not 3
sample, and (b) insensitive to differing distribu
recorded
tions of the data (see "Discussion" following
Freeman's 1976 paper).
Both Broadbent's and Kendall's methods will
patterns and "ladders"; these have been ignored in
be extensively applied in the analyses which
the context of this investigation, and are there
follow.
fore omitted from Figure 4 in the interests of
clarity, but sketches of two of the more interes
ting examples are given in full in Figure 5.
2. Measurement and analysis of a group of
carvings
It should be noted that the data for the
Panorama Stone refer to two apparently distinct
As a preliminary study, a group of carved
designs at opposite corners of the rock. Many of
rocks near Ilkley, in West Yorkshire, was selected.
the rings on this stone are clearly not circular,
Selection was made solely on the basis that each
and these are not included in the following
design should possess relatively few cups, so as to
analysis, nor listed in Table 1.
be amenable to the simple kind of analysis subse
quently employed (but decided upon in advance). No
2.2 Analysis of the positions of the cup
measurements were made prior to selection. In
centres
order to avoid the uncertainties inherent in the
use of rubbings, all measurements were made at the
The analysis was confined to the distances
sites, directly from the rocks.
between the centres of neighbouring cups.
Clearly an objective criterion is required in
2.1 Measurement technique
order to determine which cups should be accepted
as neighbours. To this end a plan of the design
At each site the rock surface was carefully
was constructed to 1/4 scale. All sensible pos
inspected and the design chalked in. For rings, a
sible candidates for neighbours were joined by
line was drawn along the estimated bottom of the
straight lines, and the perpendicular bisectors
groove; for cups, a line was drawn along the rim
of all such lines constructed. In this way a
and the centre marked. There seems to be no alter
series of "zones of influence" (Thiessen polygons)
native to achieving this last except by eye:
is obtained. Figure 6 shows an example.
weathering of the rim frequently does not occur
Wherever two zones share a common boundary, their
symmetrically, and often the deepest point of a cup
corresponding cups were accepted as neighbours,
seemed an appropriate choice.
and the distance between their centres included
in the subsequent analysis. In almost all cases
Measurements were then made between the chalk
this was a distance idiich had been measured
marks using trammels. This has the advantage of
directly from the rock, but in a few cases it was
removing the parallax errors inherent in the use of
obtained from the scale drawing.
a tape only (some cups being several centimetres
deep). Further - and perhaps more importantly at
this stage of the investigation - by adjusting the Collecting the neighbouring cup separations
trammel to the marks and then reading the distance together from all the rocks produced 83 measure
from a separate scale (to the nearest mm. or jmm.) ments. A histogram of these is given in Figure 7,
one has no knowledge of the actual value of the where each measurement is represented by a shape
measurement whilst the adjustment is taking place. roughly approximating to a gaussian; the shaded
sections represent the cup separations and the
Where rings were judged to be circular, two unshaded portions represent the ring diameters.
diameters were measured at right angles to each Inspection of this histogram reveals the tendency
other, and the mean accepted as the best estimate. for the data to collect in clumps with a spacing
The positions of the cup centres were measured from of about 2.1 cm.
three suitably spaced fixed points to enable a
scale drawing to be subsequently constructed. In Testing the cup separations by a Broadbent
addition, distances between adjacent cup centres Type A analysis for the hypothetical megalithic
were measured, so that the dependence of the final inch of 2.0725 cm. yields the following result:
result on the accuracy of a scale drawing was
reduced to a minimum. n = 83; s2/d2 = 0.257, corresponding to a 1Z
probability level.
Details of the rocks measured are given in If the ring diameters are included, we
Table 1, and rough sketches of the designs in obtain:
Figure 4. These sketches are included here merely n = 117; s2/d2 = 0.260, corresponding to a
to provide terms of reference for the data; the probability level of 0.4Z. These results clearly
presence of rings is indicated, but not their shape offer considerable support for the megalithic inch
or size. Some of the designs have complex groove hypothesis.
Science and Archaeology no.25 (1983)
16
• H • • F
{ • : A • G
0« *C
• E
Weary Hill E
Weary Hill W
S
c
ie
n
c A
e
• E
a
n
d
A • F •Ï‘C
r
c
h
a
e
o
lo G
g • >)B
y B (•;
no • C P • • Û
.2
5
(1 Panorama Stone 1 Panorama Stone 2
9
8
3
)
M •
• L 1 ii; *J ■* F
•K S;B
H <•?
Pancake Ridge
<»;C
Q ('•) ÍS’A ;$G
D
<•: E
• cup
0 40cm (approx.)
\ J ring (not to scale) «- ■ ■
Figure 4. Sketches of the cup and ring designs
t
!
! ;t
\
s I
t ,-■
/ t
1i l• 1'
t's
•t \ \ »
« . A I I 1 !
\ K ' ' '
Panorama Stone 1
Pancake Ridge
O 40 cm (approx.)
Figure 5. Examples of "ladders" and groove systems
/
/
Mean ring diameter.
-5k /
I
Approx cup diameter A + I
I
I
*
I
I +
A
X
y A
y X
Zone boundary __ "s
•" ''A
\\
\ X
' X
+
Cup centre \
+
\
\ +
\ / +
\
\ +
\ 4-
\
\
+ \
\
\
\
0 20cm
Figure 6. Thiessen polygons. Weary Hill East.
Science and Archaeology no.25 (1983)
19
S
c
ie
n
c
e
a
n
d
A
r
c
'n
a
e
o
lo
a
y
n
o
.2
5
(
1
9
8
3
)
1 cup separation
1 ring diameter
Figure 7. Histogram of cup separations and ring
diameters from the initial sample
X 1%
® 5%
0 1%
4.2
4.0
si
3.8
3.6
3.4
3.2
3.0 i
0 10 20 30 40
(T, - T0).RMS(y)
Figure 3. Kendall's method: 5, 2, and 1 %
significance levels
Science and Archaeology no.25 (1983)
