/home/precedings/apps/precedings/shared/storage/document/2008/2153/npre20082153-1

Pegs and Ropes: Geometry at Stonehenge

Anthony Johnson

, Alberto Pimpinelli

Research Laboratory for Archaeology and the History of Art,

University of Oxford, Oxford, UK

LASMEA, Universit´e Blaise Pascal-CNRS

63177 Aubi`

ere cedex, France

Abstract

A recent computer-aided-design investigation of the
Neolithic 56 Aubrey Hole circuit at Stonehenge has
led to the discovery of an astonishingly simple geo-
metrical construction for drawing an approximately
regular 56-sided polygon, feasible with a compass and
straightedge. In the present work, we prove analyti-
cally that the aforementioned construction yields as
a byproduct, an extremely accurate method for ap-
proximating a regular heptagon, and we quantify the
accuracy that prehistoric surveyors may have ideally
attained using simple pegs and ropes. We compare
this method with previous approximations, and argue
that it is likely to be at the same time the simplest
and most accurate. Implications of our findings are
discussed.

Introduction

Our story begins in 1666, when the antiquarian and
essayist John Aubrey, walking around the iconic
stone structure that gave Stonehenge its name, no-
ticed five small depressions just inside the bank of
the encircling earthwork, describing them as . . . cav-
ities in the ground from whence one may conjecture
stones . . . were taken. [1]. These hollows were not
mentioned by subsequent writers, and there was no
trace of them on the surface when excavations began
at Stonehenge in 1919, which is curious for, having
seemingly survived for millennia, it is difficult to ac-
count for their disappearance in the 250-year period

since Aubrey's days. The possibility must therefore
remain that small stones had once occupied these po-
sitions and that they had been removed not long be-
fore the 1600s and that Aubrey's cavities represented
relatively recent traces of their removal.

Aubrey's then unpublished manuscript lodged at

the Bodleian Library Oxford was to distract Col.
William Hawley and his colleague Robert Newall
from investigations and restoration work within the
interior of the monument, and in 1920 they set about
looking for traces of the missing holes. An initial
search was done using a steel bar (as a probe). A total
of 34 were finally excavated in a continuous arc cover-
ing the east and south-east of the circuit and a further
22 located by probing, forming a complete circuit of
56 holes, spaced at quite regular intervals of 16 ft
(4.88 m) around the geometrical henge centre. Haw-
ley and Newall called these features `Aubrey Holes' as
a compliment to our pioneer [2]. They are currently
interpreted as belonging to the most ancient phase
of Stonehenge, to which the bank-and-ditch earth-
work, and possibly the unworked Heelstone belong
[3]. While it may not be clear that these holes were
responsible for the 5 hollows recorded by Aubrey, his
sketch and notes certainly led to their discovery.

Averaging just over 1 m in width and 1 m deep

the holes were found to have been set on an accurate
circle just over 87 m in diameter (fig. 1) running just
inside the now much weathered and almost invisible
5,000 year-old chalk bank. There has been consid-
erable debate as to their potential purpose, and not
least the significance of the enigmatic number, 56.

Figure 1: Sketch of Stonehenge's plan, the Aubrey
Hole circuit evidenced as filled red dots.

A re-examination of all the archaeological evidence
in 1995 supported the view that they originally held
timber posts [4]. The timbers were removed and over
the course of a few hundred years the hollows left
by the sockets were variously treated, but eventually
most were used for the burial of cremated human re-
mains.

Whatever the number 56 implies it is reasonable

to suppose it was not a random occurrence, and that
despite the relatively small variations in the spacing
that it was the intention of the prehistoric surveyors
to create an accurate circuit of 56 holes. This then
leaves open the question as to whether the number
was desired i.e. that the surveyors were charged with
the task of setting out a predetermined array, for
whatever purpose, or whether there was something
inherent in the method of Neolithic `peg and rope'
survey that might return 56.

Following their excavation, the interest in the

Aubrey Holes dwindled, they being no match for the
magnificent and puzzling rings of megaliths that oc-
cupy the central part of the monument, capturing all
of the viewers' attention.

Unexpectedly, focus was suddenly brought back

to the Aubrey Holes in 1964 when an English-born
American astronomer, Gerald Hawkins, using one of
the first available computers investigated the solar
and lunar alignments and possible astronomical im-
plications of Stonehenge [5]. Hawkins was intrigued
by the number, 56. He noticed that 56/3 is very close

to the 18.61-year period of the rotation of the moon's
orbital plane. Since this period is related to eclipse
cycles--even though, admittedly, in a rather convo-
luted way--and since attention to the motions of the
sun and moon seemed to be contained in the Stone-
henge design, Hawkins devised an ingenious method
for predicting eclipses using the Aubrey Holes. He
surmised that Stonehenge, at least in its most an-
cient parts, was a "Neolithic computer", devoted to
astronomical use.

The astronomer Fred Hoyle [6] quickly joined

Hawkins in supporting this Neolithic computer idea,
although the proposal was not widely accepted by
archaeologists, with Richard Atkinson, who had re-
cently undertaken several seasons excavation and
restoration work at Stonehenge [7], being the first
to refute the astronomers position.

Stonehenge is built within latitudes where astro-

nomical alignments may be matched, to a varying
degree of approximation, with specific features of a
number of geometrical figures.

As a consequence,

one may never know whether an intended geometrical
construction casually leads to a coincidence with an
astronomically relevant direction, or if, on the con-
trary, geometry is servicing astronomy. Of course,
even if the latter is true, it does not imply that Stone-
henge was ever used as an observatory, not to speak
of a computer.

As a matter of fact, archaeologists are ill at ease

with the idea of assigning to ancient structures a
"function", at least before a full assessment of the
design of the structures themselves has been reached.
They are also acutely aware of the difficulties in col-
lating records from the disparate Stonehenge archive,
and re-creating from it a truly reliable groundplan of
such a complex monument (remembering that much
of the information comes from buried remains which
were recorded long before the advent of sophisticated
modern excavation, survey control and scientific dat-
ing methods).

It is therefore understandable that

they frown at the uncritical use of this early data
for proving specific geometrical, or astronomical, re-
lationships.

Recently, the whole problem of data collecting has

been revisited by one of us [3]. When reviewing and
critically analyzing all available surveys, one realizes

how little we know of the design of the most basic fea-
tures of Stonehenge. In particular, the most difficult
information to recover remains the precise measure-
ments of linear dimensions. However, careful gath-
ering of all available survey data has permitted the
investigation of geometric relationships to a reason-
able accuracy [3]. Amongst the most important of
all the Stonehenge datasets is the carefully measured
and annotated survey made by John Wood, the ar-
chitect of Georgian Bath, in the year 1740. It may
seem somewhat paradoxical that a survey made over
350 years ago is so valuable in an age in which elec-
tronic data measurement and laser survey is common-
place. This is because Wood's plan was made before
the collapse of the southwest trilithon in 1797 and
the extensive remedial engineering work of the 20th
century. Wood's records have been computed to re-
create the earliest accurate record of the stones. Us-
ing this data we may appreciate more precisely how
the original ground plans may have been laid out by
the prehistoric surveyors using pegs and ropes. We
are then entitled to ask and answer questions about
the methods and techniques that would have allowed
them to organize space geometrically.

In this paper, we will focus on the Aubrey Hole

circuit. We will argue that it was laid down with the
specific purpose of drawing a 56-sided polygon, and
that a geometrical construction based on the circle
and square, readily do-able with pegs and ropes, al-
lows one to trace the polygon to an extremely high
accuracy.

As a matter of fact, we will show that

the method discussed here provides the best known
approximation to such a polygon, as well as an ex-
ceedingly accurate regular heptagon.

The Aubrey Hole circuit

Although not unique, for over 100 timber circles are
known from Britain and Ireland [8], few are known to
be as old or anywhere near so large or as accurately
surveyed as the Aubrey Holes, although a number are
clearly complex concentric structures which appear
also to employ geometric constructions in both the
number of posts and spacing of concentric elements.
The number of posts utilised is also variable, none of

the others having 56

Most of the questioning about the 56 holes that go

round close to the inner edge of the boundary bank,
concerns their possible use. Ranging from Hawkins'
and Hoyle's Neolithic computer, in which the num-
ber 56 plays a very important role, through offer-
ing pits for communicating with the dead [10], to a
more prosaic posthole ring, many hypotheses have
been formulated. Rather than speculating upon why
there were 56 pits at Stonehenge, a more productive
avenue might be to ask how?. Is there a way to ar-
range 56 holes regularly spaced in a ring using only
ropes and pegs?

The question deserves attention,

because 56 divides by seven, so that drawing a 56
sided polygon--to which the ring of Aubrey Holes of
course corresponds--implies drawing (or at least by
default including) a heptagon. Since Gauss' proof
[11] of the impossibility of constructing the latter
using only a straightedge and a compass, we know
that tracing a regular heptagon--and a fortiori a 56
sided polygon--is not possible with pegs and ropes,
unless one makes use of a graduated, or marked,
ruler, realizing what is called in the jargon a neusis
construction[12].

How was the 56-sided polygon at Stonehenge

worked out, then?

In the absence of written evi-

dences, we cannot answer this question, either. We
are in fact left with the following choice: either we
make a number of assumptions about what we be-
lieve the Stonehenge surveyors did, or we simply lay
down a set of rules, and try to discover whether a
construction can be made according to those rules.

Compass

and

straightedge

geometry

These are our requirements.

First, we want a method which is readily repro-

ducible at all scales, based only on geometric propor-
tions, not on measurements.

The numbers are 8, 10, 13, 14 and 44 [10].

Note that

Neolithic surveyors seemed to have a preference for non-
constructible polygons, since polygons with 13, 14 or 44 sides
cannot be drawn using just straightedge and compass.

Note that this first requirement is just a rephrasing

of the basic rule of the Euclidean construction, which
only permitted the use of compass and straightedge.
In fact, it is easy to conjecture that the classical rules
of Greek geometry may well have been a direct conse-
quence of the peg-and-rope surveying techniques. We
may also assume that small-scale experiments and
`scale drawings' were made. In the Stonehenge re-
gion chalk plaques have been found that display ge-
ometric, and in one case mirrored geometric designs
[3]. We surmise that strict geometric design was also
behind the Aubrey Holes circuit

Second, we want a method which is easily remem-

bered and executed.

Here, we run into problems,

because we have to define what easy may mean.

The best definition we may think of is in terms

of operational complexity. One construction may be
deemed easier than another one, if the former needs
the execution of a smaller number of different actions
than the latter.

Third, we require that the construction be suffi-

ciently accurate.

The Aubrey Hole circuit has a diameter very close

to 87 m. The 56 holes are thus 4.88 m apart, on
average, and the perimeter of the polygon is approx-
imately 273.18 m. In general, such geometrical con-
structions start from a circle, and produce a chord
which is the length of the side of the sought-after
polygon [13]. The chord must then be transferred an
appropriate number of times around the circle to gen-
erate the whole polygon. Assume that the construc-
tion allows for the first side with a 1 cm accuracy--
i.e. a 0.2 % accuracy in our case. After transferring
it 55 more times around the circuit, one has spanned
a length of 272.72 m, 46 cm less than the "ideal"
perimeter. Given the unavoidable errors introduced
by the peg-and-rope technique--although the latter
are random, non-systematic errors, with a tendency
to averaging out--about 50 cm is likely to be an ac-

As a matter of fact, one may draw a 56 sided polygon by

first drawing an octagon, and then subdividing each side into
seven parts by stretching a piece of rope. If the rope turns
out to be too long, one just has to alter it until the correct
one-seventh length is found by trial and error. Even though
one would indeed be able to trace a 56 sided polygon this way,
the method is clearly cumbersome.

ceptable inaccuracy. However, 10 times more, a mere
10 cm over a 5 m length, would give rise to an inaccu-
racy of about 5 m--a whole side!--at the end of the
process, which would definitely render the construc-
tion useless. Note, by the way, that if one wants to
reach an accuracy of the order of 0.1 % for a 56 sided
polygon, one has consistently to reach a (relative)
accuracy about ten times larger for the heptagon.

In the Appendix, we will provide two examples:

a construction which is simple--according to our
definition--but not accurate enough, and another one
which is very accurate, but not simple.

It may seem very unlikely that a geometrical con-

struction yielding both a simple and very accurate
56-sided polygon capable of being laid out on the
ground using pegs and ropes, exists. Astonishingly,
it does.

Figure 2: Step 1: Inscribe in a circle a square, then a
second one rotated by 45

and standing on a corner.

A story of pegs and ropes

The construction in question has been discovered by
one of us empirically, with the help of a CAD pro-
gramme, and published in his very recent book on
Stonehenge [3]. Here, we will recall how the construc-
tion works, and give an explicit, analytical evaluation
of the accuracy of the approximation it provides to
the heptagon and the 56-sided polygon. To begin
with, we draw a circle, and inscribe two squares ro-
tated by 45

one with the respect to the other (Fig.

2). We might note that a line between any two oppos-
ing corners of a square may be considered a potential

Figure 3: Step 2: Drop a diameter from the topmost
vertex A.

surveyors base-line, i.e. a practical origin for the sur-
vey passing through the exact centre of our intended
array. It is not difficult find several groups of oppos-
ing Aubrey Holes within the known (excavated) series
that mark corners of quite accurate squares, however
it is also apparent that some within the array deviate
from ideal positions [3].

The eight points where the squares and the circle

intersect define an octagon, as well as eight vertices
of the 56-sided polygon. Draw a diameter from point
A (Fig. 3) then set a peg in point B, as in figure 4,
and pull a rope between B and C, the latter being the
point where the diameter through A cuts the side of
the square opposite to A (Fig. 4).

Figure 4: Step 3: Placing the pin of the compass at
point B, open it to join point C, where the diameter
through A cuts the side of the first square.

Using BC as the radius, draw a circle centred on

B. Call D

and D

the points where this circle cuts

the original circle (Fig. 5.) Perform the same action,

using the point symmetrical to B with respect to the
axis AC as pivot (Fig. 6.) The heptagon's side is the
segment D

. But we are not yet done. Repeat the

same construction from the other three points equiv-
alent to B. We obtain figure 7, in which 24 vertices of
the 56-sided polygon are found. In fact, each pair of
points between two successive vertices of the squares
delimits one side of the 56-sided polygon.

Figure 5: Step 4: Using BC as the radius, draw an
arc and mark the points D

and D

where the arc

cuts the initial circle.

An analytical proof

In this section, we will prove that the preceding geo-
metrical construction yields an approximation for the
heptagon as well as for 56-sided polygon we are after,
and we will explicitly compute both polygons' sides.

We will do it using an analytical geometry ap-

proach.

Figure 6: Step 5: the same as in step 4, from the
point mirror symmetric of B.

Figure 7: Step 6: Same as steps 4 and 5 from the re-
maining points equivalent to B. The 24 marked points
are vertices of the 56-sided polygon.

Let us choose the Cartesian axis in such a way that

the circle's centre is at the origin, and its radius is
unity. Then, points A and B have coordinates (0,1)
and (

2/2, 1 -

2/2), respectively. The side of the

hexagon is then given by twice the abscissa of point
D

, which is the intersection point of the unit circle at

the origin, and of the circle centred in B and having
radius R = BC =

3/2. Lengthy but straightforward

algebra yields

- R

[(R + 1)

- d

] [d

- (1 - R)

]

(1)

where d

= x

+ y

, and (x

, y

) are the coordinates

of point B.

Using the values above, and choosing the positive

root, one finds the heptagon side S

= 2x

- 1 + 15 - 4

0.8677173844,

(2)

whence the heptagon angle

= 2 arcsin(S

/2) 51.4253858

(3)

to be compared with the exact values

h-exact

= 360

/7 51.4285714

(4)

h-exact

= 2 sin(360

/14) 0.867767478,

(5)

which means a relative angular error of 6.2 × 10

-5

or 0.0062 %.

Since 1/56 = 1/7 - 1/8, once one has the angle

of the heptagon, subtracting the angle of the octagon
yields the angle of the 56 sided polygon. Indeed, each
arc is drawn starting from peg positions at 45

from

each other. Thus,

- 45

(6)

a relation that we will explicitly check in the Ap-
pendix. Then, one finds

6.425385806

(7)

so that the side of the 56-sided polygon is

= 2 sin

0.11208538,

(8)

the exact result being

56-exact

= 2 sin(360

/112) 0.112140894,

(9)

a 0.05 % relative difference.

An alternative calculation of

will be presented

in the Appendix.

As we recalled above, the 56 Aubrey Holes span a

polygon inscribed in a circle of diameter 87 m. Thus,
the side of the polygon measures 4.876 m. This must
be compared with the exact average value 4.878 m.
The difference, as we said, is about 0.05 %, or 2.4
mm. Over the whole circuit, the difference between
the "ideal" and approximate perimeter would be 13.5
cm within the present approximation, obviously im-
material for practical purposes.

Discussion and Conclusions

A shrine erected:
a holy shrine it is, its interior is like a maze;
a shrine whose interior is a twisted thread,
a thing unknown to man,
a shrine whose lower station is the roving iku-
constellation,
a holy shrine whose upper station moves toward the
chariot-constellation,
a turbulent flood-wave...
its melam is awesome. [16]

--------

Let . . . the child of the sun-god Utu,
light up for him the netherworld, the place of dark-
ness!
Let him set up the threshold there (as bright) as the
moon
(for) all mankind, whatever their names be,
(for) those whose statues were fashioned in days of
yore,
(for) the heroes, the young men and the . . . !
From there the strong and the mighty will march out.
Without him no light would be there during the
month ne-IZI-gar, during the festi[val of the gh]osts.
[17]

--------

[These are they who] hold the measuring cord in
Ament, and they go over therewith the fields of the
KHU (i.e., the beatified spirits). [Ra saith to them]:
`Take ye the cord, draw it tight, and mark out the
limit (or, passage) of the fields of Amentet, the KHU
whereof are in your abodes, and the gods whereof
are on your thrones.' The KHU of NETERTI are
in the Field of Peace, [and] each KHU hath been
judged by him that is in the cord. Righteousness is to
those who are (i.e., who exist), and unrighteousness
to those who are not. Ra saith unto them:`What is
right is the cord in Ament, and Ra is content with
the stretching (or, drawing) of the same. Your pos-
sessions are yours, O ye gods, your homesteads are
yours, O ye KHU. Behold ye, Ra maketh (or, wor-
keth) your fields, and he commandeth on your behalf
that there may be sand (?) with you.' [18]

--------

the fourfold siding, fourfold cornering,
measuring, fourfold stacking,
halving the cord, stretching the cord
in the sky, on the earth,
the four sides, the four corners, as it is said. [19]

------
Atkinson wrote in 1956 that `one thing upon which

it is agreed is that it [Stonehenge] is primarily a `tem-
ple', a structure in which it was possible for man to
establish contact and communication with extramun-
dane forces or beings. [9]' The preceding excerpts,
written down by different cultures in different epochs
all show that ancient societies considered the foun-
dation of a `temple' as a sacred action, inasmuch as

it was a repetition of the divine creation of the Cos-
mos. As such, the laying down of the foundations
was a ritualized act, to be performed according to
the primeval gestures of the gods.

Creation is the act of turning Chaos into Order.

The sacred space is the earthly image of the Cosmos,
of the orderly Universe moulded from the disordered
matter. But above all, a sacred enclosure is an open-
ing by which communication between the world of
the gods and the world of humans is made possible.
Historian of religions Mircea Eliade wrote: "Every
sacred space implies a hierophany, an irruption of
the sacred that results in detaching a territory from
the surrounding milieu and making it qualitatively
different." [20]

Geometry is what ensures that the sacred space is

detached from the surrounding. The first act, draw-
ing a circle, makes the detachment a reality.

In-

scribing the square fixes the four cardinal directions
all sacred space must be oriented with respect to
heaven, because all the architectonic models come
from heaven.

Known examples from palaeolithic and megalithic

engravings show that symmetry was the main feature
associated with geometry. The spirals and lozenges,
clearly exhibit a deep interest for figures produced
with geometric constructions in which the same shape
could be reproduced at will at different scales, by
mean of repeated expansions or contractions

. As

Paolo Zellini noticed [21], one of the strongest drives
for scientific research, the search for unity in multi-
plicity, found a powerful tool in the geometrical and
mathematical procedures of similarity transforma-
tions. For instance, a square constructed on the di-
agonal of a smaller square will have double area, and
the construction can be iterated ad infinitum, pro-
ducing a spiraling structure made of area-doubling
squares (Fig. 8).

The power of geometry reveals itself in its ability

to reproduce ever changing objects of identical shape
through the repetition of simple operations. That
is also the main characteristic of the construction of
an approximate regular 56-sided polygon, and of an

Cf. Ref. [3] for a geometrical construction of the Bush

Barrow lozenge

Figure 8: Infinite spiral generated by area-doubling
squares.

approximate heptagon as a byproduct, discussed in
this work.

We will never be able to prove that the construc-

tion proposed in [3], and discussed here, was actually
used for crafting the template of the Aubrey Hole cir-
cuit. We will never know whether, as Hawkins sug-
gested, the 56-sided polygon inside the bank of Stone-
henge has anything to do with the polygon that, ac-
cording to Plutarch, the Pythagoreans said belonged
to Typhon [22], although the simple fact that this
complex polygon was mentioned in Classical antiq-
uity suggests that the knowledge of its construction
belongs to even earlier times which may even hark
back to the dawn of the pantheon of Indo-European
tradition.

The scenario discussed here, with its deep geomet-

ric elegance and fascinating mixture of extreme sim-
plicity and accuracy, is in full agreement with all we
know of the ways our ancestors felt about building a
temple, as well as of their attitude towards using ge-
ometry for rationalizing reality. Of course it may not
be true but, given the accepted uncertainties of our
knowledge of prehistoric cosmology, we do have here a
tangible and credible possibility; we should certainly
like to imagine the foundation of Stonehenge in this
way.

Appendix

Let us first discuss an alternative computation of the
angle

subtending the side of the 56-sided polygon.

The angle

is half the difference between the angles

of the larger and smaller triangles in fig. 9.

Figure 9: Computation of the angle

subtending

the side of the 56-sided polygon.

These angles are respectively equal to the ap-

proximate heptagonal angle

= 2 arcsin(S

/2),

and to

= 2 arcsin(/2), where = 2x

2 +

- 4

2 , D

being as in fig.

above. Then

= arcsin

- 1 + 15 - 4

- arcsin

2 +

- 4

25.712692903

-19.287307097

6.425385806

as found previously. The agreement is due to the
relation

= 90

, which is easily verified.

As we mentioned in the text, drawing a perfect

regular heptagon is impossible with pegs and cords.
However, various methods can be devised for drawing
an approximate heptagon, at varying complexity and
approximation levels.

We will quickly describe two methods for approxi-

mating a heptagon. The first one is rather simple, in
the sense that few actions have to be performed to ob-
tain the result. However, the approximation obtained
would, in our opinion, make it useless for Stonehenge.
The second one, on the contrary, attains a very good
approximation, but requires a rather involved geo-
metric construction, with a large number of different
actions.

Figure 10: Sketch of the geometrical construction
yielding a regular heptagon to a low approximation.

The first technique requires that a hexagon be in-

scribed in the unit circle.

Then (Fig. 10) a chord of length 2L is traced be-

tween any pair of alternate vertices of the hexagon.
The length L =

3/2, and transferring it along the

circle as shown in fig. 10, one can construct an ap-
proximate heptagon, since

3/2 0.866, and the

side of the regular heptagon is 0.8678, the difference
being about 0.2 %. Hence this construction, extended
to yield a 56 sided polygon, would imply an error of
about 1.6 % for the side of the latter, or 8 cm, which,
after reporting this length 56 times, would be shorter
than the "ideal" perimeter by 4.6 m, a whole inter-
hole distance! The second geometrical construction
yields a much better accuracy, at the price of a con-
siderable complexity.

The construction requires drawing a pentagon, as

well as its incircle and circumcircle. Let us call

and R

, respectively, their radii.

Then, the ratio

= 2/ 1.23607, where = (

5 + 1)/2

is the Golden Mean. Drawing a pentagon is readily
feasible with pegs and cords.

Next, selecting the point A (Fig. 11) as the centre,

draw a circle of radius AB = a = R

- R

. From the

centre O of the structure, a third circle is drawn, of
radius R

= R

- a. An equilateral triangle is then

Figure 11: Sketch of the geometrical construction
yielding a regular heptagon to a high approximation.

inscribed into this circle.

A fourth circle, of radius R

= R

+ 2a has to be

drawn, and the base of the equilateral triangle is ex-
tended until it cuts the latter circle. The intersection
of the base with the circles yields two vertices of the
heptagon.

Finally, the angle

of the approximate heptagon

can be computed from 2

= 90

+ arcsin[1/(r

)],

where r

= 2R

. Letting = 2R

, yields

= 51.46048

, to be compared with the exact

value

h-exact

= 360

/7 51.42857

The dif-

ference is about 0.062%, a very good approxima-
tion (even though nowhere as good as the one dis-
cussed in this work), yielding a heptagon side equal
to S

0.86827, as well as a 56 sided polygon of side

approximately 0.11270. At Stonehenge, this means
that the Aubrey holes spacing would measure 4.902
m, and the total perimeter 274.54 m, about 1.5 m
longer than the "ideal" value. Of course, the larger
this discrepancy, the more numerous the corrections
needed, and the less effective the tracing method.

The approximate construction of the heptagon just

described allows for a high enough accuracy, though

being less accurate than the one discussed in the body
of the text, and clearly operationally more complex.

The most accurate construction of the regular hep-

tagon's side is due to R¨

ober, and has been reported

by Hamilton [14]. Hamilton states that R¨

ober's dia-

gram is not very complex, but since Hamilton's pa-
per does not contain any diagram, understanding the
construction is essentially impossible. In note 1082 of
the Mathematical Gazette, Youngman [15] provides
another accurate approximation. He compares his
with R¨

ober's, claiming that the latter's approxima-

tion "needs rather elaborate drawing".

References

[1] Monumenta Britannica. The manuscript is

housed in the Bodleian Library, Oxford. Al-
though the manuscript was widely circulated
amongst antiquarians and scholars, it was not
published in full until 1981: J. Fowles (ed.) Mon-
umenta Britannica or A Miscellany of British
Antiquities by John Aubrey (Little, Brown and
Company, 1981). See p.80.

[2] W. Hawley, The Antiquaries Journal

1, 19-41

(1921)

[3] A. Johnson, Solving Stonehenge (Thames &

Hudson, 2008)

[4] R.M.J. Cleal, K.E. Wlaker and R. Montague,

Stonehenge in its landscape. Twentieth-century
excavations, English Heritage Archaeological
Report

10 (English Heritage, 1995)

[5] G.S. Hawkins, Stonehenge Decoded (Doubleday,

1965)

[6] F. Hoyle, On Stonehenge (W.H Freeman &

Company, 1977)