Did Alexander Thom discover interesting stuff about the
British Neolithic, or was he deluded in what he thought he saw? The modern
consensus among the archaeological community is that he discovered nothing of
importance which was actually present in the evidence. This was supposedly
shown by a large scale resurvey of the stone circles conducted by Clive Ruggles
in the eighties. This resurvey was conducted with a great sensitiveness to the
possibility of selection bias. This sensitivity was taken to such extremes
however, that it would have been impossible to verify much of Thom’s surveying
and interpretation as the archaeologist Euan Mackie has indicated.
That of course, was the point. We already had some nice
models of antiquity which didn’t involve much in the way of interpretative
mathematics, there was little interest in the precision which seemed to be present
in an ancient preoccupation with the sky, and in the observation of rising and
setting points, equinoxes and solstices, and in the nineteen year metonic cycle
of the moon’s movements. The foresights which seemed to be used to indicate
something of importance to the ancient astronomers and priests were largely
ignored in the Ruggles resurvey. We liked the models we had before, and didn’t
like or understand what might be implied in a British antiquity which was
populated by mathematicians, engineers and astronomers who thought the sky was a key
object of interest, and who threw vast resources at the construction of
monuments whose purpose was hard to fathom.
After the resurvey of the monuments the archaeological
community turned away from the questions which Thom’s original surveys and
measurements had thrown up. Enough doubt had been sown to make the territory he
had explored a matter of disinterest to the community, and entirely lacking in
anything that archaeologists needed to consider. Whatever we could find out
about the megalith builders would not be found out by following Thom’s lead. There
is still interest out there in Thom’s work of course, but no-one is pursuing
similar research within an academic context.
We know from Classical writers that the study of aporia
was a matter of some interest to those interested in philosophy, mathematics
and physics, discussed in Plato’s Sophist and the Timaeus, and
also in Bk 3 of Aristotle’s Metaphysics. Pythagorean triangles are one
kind of puzzle which could be explored, and it was evidently a matter of great
interest in the Neolithic, since they used a number of the Pythagorean triangles,
and not just the basic 3.4.5. instance. The 16 basic triangles can be
enumerated as follows:
(3, 4, 5) (5, 12, 13) (8, 15, 17) (7, 24,
25)
(20, 21, 29) (12, 35, 37) (9, 40, 41) (28,
45, 53)
(11, 60, 61) (16, 63, 65) (33, 56, 65)
(48, 55, 73)
(13, 84, 85) (36, 77, 85) (39, 80, 89) (65, 72, 97)
To us, these are just geometrical figures, and we don’t ask
many questions about why these exist. But that was not the case in antiquity.
For those engaging with these figures, they were puzzles. Why did these
triangles with sides which were whole numbers meet and agree once two of the
sides were squared and the hypotenuse was squared? Their sides don’t meet and
agree when considered as triangles, yet they do when multiplied into their square
values.
We also know from classical writers that there was a great
deal of interest in the idea that things should ‘meet and agree’. Once of the
most famous stories from antiquity concerns a conversation between Solon and
Croesus, involving some bizarre mathematics to bring together the mathematics
of the cosmos and the days of the life of a man. (Herodotus).
So looking at these stone circles as forms of puzzle, with
some relation to the universe in which we live, and as objects which were
intended in some way to meet and agree with that cosmos, may provide some
answers.
I’ve written about some aspects of this before, in ‘Pythagorean
Triples and the Generation of Space’. I quote some passages from it here:
In antiquity, it was obvious to anyone interested in
number, mathematics and geometry, that there were several aspects of the
physical world that involved irrationality, long before it was possible to
provide logical proof of such irrationality. One of these irrationalities was
the relationship between the diameter and the circumference of the circle. We
know that irrationality (understood as an absence of commensuration) was a major
concern in antiquity, since the existence of it seemed to undermine the idea
that the world was rational, and constructed by the divine on rational
principles. In other words, the existence of irrational things served to
undermine the idea that the world made sense, and that it was good.
What we understand as Pythagoreanism is actually a way
of approaching the world and reality on the basis of number, mathematics and
geometry. We have lost a grasp of this, particularly since the close of the
ancient world. Pythagorean ideas are not the creation of Pythagoras in the
sixth century B.C.E., but a range of ideas about the world, focussing
particularly on numbers and geometry, and the puzzles which the study of these
throws up … As such, these ideas and puzzles belong to any culture which
chooses to address the divine in terms of how the universe is constructed. As
already suggested, the Babylonians had a sense of this, though they were also
interested in the practical applications. It is also the case that the
inhabitants of Britain in the late Neolithic and the early Bronze Age had such
a sense.
…. Alexander Thom surveyed many of the megalithic
circles across Britain from the 1930s into the 1970s, and established that the
circles were constructed on the basis of a number of different Pythagorean
triangles, and that these circles were not in fact circular. The circumferences
of these circles were modified in order to make their lengths commensurate with
the length of the sides of the underlying triangles.These modifications testify
to the contemporary idea in ancient times that the incommensurate nature of
diameter and circumference shouldn’t be the case.
I’ve written elsewhere that Pythagoreanism, whether in
the sixth century or long before, was a transcendentalist view of the world.
Meaning that the world of physics and appearance in which we live, is not
reality itself, but simply a presentation of it. And the presentation of it is,
in a number of ways, crooked. So some aspects of physical reality are not rational.
This does not mean that the ancient Pythagoreans were
pitching themselves against the workings of the divine, but rather that they
were trying to understand why what they saw, experienced and understood, was
not rational. The answer was that their place of refuge was not reality itself,
but a false representation of it.
In the physical world, they could therefore not expect
rationality to be woven all through it. Thom identified the obsessive concern
of the ancient Britons with whole numbers, and as a consequence (though this
was not understood at the time he was studying the megaliths), we know that
they were looking to a world beyond the puzzles and paradoxes, in which the
relationships of one thing to another were rational in nature.
The theorem of Pythagoras, however it was articulated
in the late Neolithic and the early Bronze Age, provided the answer to this.
The relationship between the sides of a 3, 4, 5 triangle is irrational in
nature, but by squaring the sides, the result is rational and commensurate.
This would have been understood to point to a world which transcended space, in
that it indicated a one-dimensional reality.
In that world, some things which are
incommensurate here,were commensurate. Which they might have taken to indicate
that, beyond that limited form of
reality, there was another reality with no dimensions at all, in which all
irrational values existed as commensurate with one another.
Plato echoed a range of Pythagorean ideas in his work,
including that reality itself exists in no particular place, has no form or
shape or colour. He also suggested that forms existed beyond geometrical
figures existing in space, and that these were to be accessed in the mind
alone.
The Pythagoreans may have understood physical reality
to have been generated as the square root of mathematical values in a higher
reality. The resulting incommensuration would necessarily generate space. We
could not possibly live in a reality which embraced only one dimension, or even
none at all. In which case physical reality might have been understood by the
ancient Pythagoreans as a compromise of sorts, which made it possible for
mankind to live.
Alexander Thom didn’t know any of this of course. He was an
engineer and mathematician. Intellectually he was enormously bright, curious,
and industrious, but he was lacking basic information about the ancient past,
just as many archaeologists were in the 60s and 70s. He gave us a phenomenological
and statistical description of what he was seeing. He noted the obsession with
whole numbers, the construction of the stone circles using various instances of
Pythagorean triangles, and the fact that many of the circles were not in fact
circles, but were modified ellipses and egg shapes, designed to make the
circumferences commensurate with the values of the triangles used in their
construction.
We can see now that what the megalith builders were up to is
reflected in written texts from the 1st millennium B.C.E if we read
them carefully. The three things noted by Thom are all discussed – the importance
of whole numbers, the interest in the strange nature of Pythagorean triangles,
and the importance of making the incommensurate commensurate with itself.
This actually means that the world of the megalith builders
is (in theory at least) intellectually accessible to us, though the last
circles were built in the 14th century BCE, or thereabouts, and the builders left
no written records about anything, never mind the construction of their
circles. Those three things we know for sure, are huge clues to what they understood
about what they were doing.
In ‘Patterns of Thought in Neolithic and Early Bronze Age Britain’ I wrote that:
…. the syncretism of Pythagoras draws on mathematical
and geometrical ideas, as well as religious ideas. We normally choose to keep
these separate. We imagine that they are separate. However, it is …. clear
that they perceived the necessary impact of the various puzzles and paradoxes
which investigation of mathematics and geometry had on their view of reality.
These were not parlour games.
Pythagoras was putting together a new religion, rather
than a secular philosophy. It is unlikely to have occurred to him that a
secular philosophy was possible, or for him to imagine what that would mean. We
think of Pythagoras as a philosopher, because of how we understand what came
after Pythagoras and his school. It is possible for us to so distinguish
religion and philosophy, because we have lost sight of some very important
aspects of how the gods were understood in antiquity. Pythagoras was well aware
of the importance of the mathematical and geometrical aspects of religion,
which is why he included them with the materials that we more naturally
understand as religious ideas.
….
Much of what we
think we understand of ancient religion is the product of a more or less modern
view, which sees a continuity between the religion of the common era and
antiquity. So, since ’rational belief’ concerning the divine, rather than
actual knowledge of the divine was (and is) of great importance in the major
religions of the common era, it is assumed that ancient religions drew their
strength from the same source, and are qualitatively similar phenomena. Modern
scholarship is able to hold this view because, since the Enlightenment, we see
the phenomenon of religion as irrational. The behaviour which supported ancient
cultic life (sacrifice, divination by entrails, the worship of statues, etc.)
is clearly more irrational than medieval religious practice, so there is little
about it which demands the application of modern critical thought.
If belief is what is important in ancient religion,
then we have missed nothing. If however there is a technical substrate to
ancient religious thought, a substructure which depends on a combination of
logical analysis, number theory, mathematics and geometry, then we have missed
almost everything. Such a substructure does exist, and Pythagoras was aware of
it, which is why religious precepts, number theory, mathematics and geometry
were all present in the three books of Pythagoras.
It is possible to make a list of things which are part
of this technical substructure in the religions of the ancient
world. These are:
Extremity, the Mean, Totality, Perfection, Completion,
Invariance, Integral (whole) numbers, the Incorruptible, the Commensurate,
Greatness, Rising, Setting, Beginning, Ending, Duration, Periodicity, Points of
transformation. And so on.
This list illustrates some of the things have
exemplars on both the earth, and in the sky. These characteristics would,
within this conceptual model of Pythagoras, have been understood to provide
points of contact, and a bridge to the divine.
Why would Pythagoras want to create a synthesis of key
components of ancient religions? There are many possible reasons, but the most
important may be the intention to restore the technical level of religious
thought and practice, then experiencing a long slow decline, so that number,
mathematics and geometry might serve again, to make sense of the transcendental
understanding of reality.
Can we apply the content of this discussion to the
Late Neolithic and the early Bronze age in Britain? If, for the purposes of
argument, we make the assumption that just as Pythagorean number, mathematics,
geometry, and the transcendentalist outlook were, in the mind of Pythagoras,
necessarily connected with each other, these four things would also be present
in megalithic culture in Britain and Gaul, for the reason that the missing
piece in the record, the philosophical transcendentalism, is the necessary
logical consequence of an understanding of number, mathematics, and geometry.
As we know from the studies made by Alexander Thom,
the stone circles were built on the basis of various sizes of pythagorean
right-angled triangles, and laid out with ropes of precise length. There has
been some critical discussion of the ubiquity of the measure he described as
the megalithic yard, which measured 2.72 feet, which he established by
statistical analysis. However, if Thom identified different standard pythagorean
triangles in the construction of different megalithic circles, all of which
were based on the measure, then the presence and use of the measure is
confirmed. It need not however, have been the only standard measure.
The construction process was designed and executed in
such a way that the circumference of the circles, whether elliptical,
egg-shaped, or flattened, would always be an integral number of the units used.
This interest in integral numbers appears to have been universal among the
builders of the circles. The connectivity the integral numbers opened to
transcendent Being is the reason why this was important.
This transcendent reality, understood to lie behind
the physical world of appearances and its paradoxes (such as the essential
identity of commensurate and incommensurate values), would be the principle
focus of the megalith builders interest, and the design of the megalithic
structures would have been understood to serve the function of strengthening
the connections between the two worlds. The transcendent world contains what is
perfect, and the world of phenomena contains only approximations to such perfections.
As Robin Heath pointed out in his account of Thom’s work, Cracking the Stone
Age Code, the phenomenal world would have seemed to the megalith builders
to be something of a crooked universe.
Looked at from this point of view, we can discern a
significant motive in the geometrical construction of the major circles which
Thom surveyed and analysed in detail. We can also begin to understand why there
were different approaches to the construction of the circles, rather than a
single standard design. In a crooked universe, there could be no universal
answer to the problems they were trying to resolve. This universe is full of
irrationality, simply because it is not the transcendent reality, but an imperfect
representation of it. The irrationality could however be overcome in the
physical world in specific instances of geometrical construction. In
one case, by creating a design utilising an ellipse which measured precisely a
specific multiple of the units employed in the pythagorean triangle used as the
basis of the structure. In another, by making the structure egg-shaped, again
with the same intention. The circle might also be flattened, in order to make
the circumference commensurate with the units of the underlying triangle.
But there is also the astronomical function of
megalithic circles. As Thom identified, some are connected with the sun and its
movements throughout the year. Others are keyed to the complex movements of the
moon. For the later Pythagoreans, and for Plato, the heavens represented a
moving image of eternity. For these earlier pythagoreans, the heavens would be
understood in the same way, and for the same reason. A megalithic circle might
therefore be conceived as a representation, in an abstracted form, of some the
properties and attributes of Eternity. Eternity is something which is whole and
complete, and returns into itself.
It therefore made sense to mark the extreme points of
the movement of the heavenly bodies (which have their existence in the moving
image of eternity), as a further embodiment of the connection between the
worlds. These were constructed using only integral values, derived where
possible, proportionately, from the movement of the heavens in relation to the
earth. Heavenly cycles would be explored and represented in the structure where
possible, together with indications of their periods. The motive for building
the circles was performative, meaning that the structures served a set of
religious functions on account of their existence and nature.
One of the objections made by the archaeologist
Jaquetta Hawkes in the Chronicle documentary on Alexander Thom, made by the BBC
in 1970, was that since the megalith builders did not have writing, there was
no way of handing information on to succeeding generations. She also suggested
that the inhabitants of the island during the period of megalithic culture were
‘simple farming communities... nomadic even’. But we know that the later Pythagoreans cultivated
memory. We are also told in Caesar’s account that becoming a priest in the
late 1st millennium involved many years of study (around
twenty), during which time a vast amount of information was committed to memory.
So Hawkes suggestion that there was an absence of a means of handing on
information is likely to be false. The cultivation of memory is built into the
pythagorean view of reality, since what exists in the mind was understood to be
more real than what could be understood by the senses.
Alexander Thom reported that the
standard measure used in the construction
of the megalithic circles was 2.72 feet. This was established on the
basis of a statistical analysis of the data from his surveys. What is perhaps peculiar
to us now about this value is that it is expressed in terms of English feet. But
Thom had been doing his surveys for forty years or so, beginning long before
the UK chose to use the metric system, so that is how he had been analysing
data since he began. It was based on the traditional measures used in the UK as
far back as anyone knew. We don’t know the origin of the English foot.
Land measures have been
associated with kings since time immemorial, but the reasons for this have long
been lost. Archaeologists do two things when discussing this question: they
acknowledge the association with kings, but then reify, and argue that the
value of the measure is literally the length of a king’s foot. This is ridiculous
of course, since all kings are human beings, and of different proportions. But
it enables them to argue that there is no universally agreed measure on
lengths, and consequently no understanding of standard measures. The whole
history of metrology argues against this, but it is convenient to dispose of
such arguments by reference to the length of a particular king’s foot.
Archaeologists like the model of
progress, which implies (actually requires) that the further back you go, the
less rational and intelligent people were, and that they were on a long hike to
where we are now. This might be true. But they assume that it is true,
which is why Thom’s work was more or less anathema to the profession.
Part of the problem is that we
have a different model of what rational thought is, from what was understood to
be rational thought in antiquity. There is a large grey zone between the two
which is mostly unknown to both archaeologists and the historians of ideas, and
issues relating to that zone are very rarely discussed. Archaeologists also
like to go looking for what they expect to find, rather than what they ‘know’
isn’t there. So things which don’t fit, don’t get a lot of attention. But you
don’t know what doesn’t exist by not being open to evidence which might not
support your argument. Impossibilities need to be considered, even if only to
decisively rule them out. Alexander Thom provided one of those impossibilities,
and it is still on the table, even if most of the archaeological profession is
ignoring it.
Did Alexander Thom discover the
megalithic yard in his surveys of the megalithic circles? Certainly the measure
was there, looked at from a phenomenological and statistical point of view. But
he didn’t have any inside understanding of how the megalith builders thought
about their constructions. He reported what he saw. But what you see is not necessarily
what is there.
As a mathematician, he knew
logarithms, and knew about Euler’s number. That number, rounded up very
slightly, in terms of the convention, is 2.72. Exactly the number which Thom identified
as the basis of the megalithic yard, expressed in terms of English feet. There
is, as far as I know, no evidence that the knowledge of this ‘coincidence’ gave
him a moment’s pause. But it gave me pause.
I think that what Thom actually
discovered in the British Neolithic, was the early presence of the English
foot. And that that measure was disguised by its multiplication by Euler’s
number. The reason for that disguise of the basic unit of measure will be
explained in the course of what follows.
First, we need to explore how ancient
priests in Britain might have known about Euler’s number.
Let’s look at that number, and
its significance. I’ve borrowed from two Wikipedia articles – the first on
Euler’s number a), and the second b) on logarithms:
a) The number e is a mathematical
constant that is the base of the natural logarithm: the unique number
whose natural logarithm is equal to one. It is approximately equal to 2.71828, and
is the limit of (1 + 1/n)n as n approaches
infinity, an expression that arises in the study of compound interest. It
can also be calculated as the sum of the infinite series..
The constant can be characterized in many different
ways. For example, it can be defined as the unique positive number a such
that the graph of the function y = ax has
unit slope at x = 0. The function f(x)
= ex is called the (natural) exponential
function, and is the unique exponential function equal to its own derivative.
The natural logarithm, or logarithm to base e, is the inverse
function to the natural exponential function. The natural logarithm of a
number k > 1 can be defined directly as the area
under the curve y = 1/x between x =
1 and x = k, in which case e is
the value of k for which this area equals one. ….
e is sometimes called Euler's
number after the Swiss mathematician Leonhard Euler …. Euler's
choice of the symbol e is said to have been retained in his
honour. The
constant was discovered by the Swiss mathematician Jacob Bernoulli while
studying compound interest.
The number e has eminent importance
in mathematics, alongside 0, 1, π, and i.
All five of these numbers play important and recurring roles across
mathematics, and these five constants appear in one formulation of Euler's
identity. Like the constant π, e is also irrational,
(i.e. it cannot be represented as ratio of integers) and transcendental,
(i.e. it is not a root of any non-zero polynomial with rational
coefficients). The numerical value of e truncated to 50
decimal places is
2.71828182845904523536028747135266249775724709369995...
So Euler’s number is intimately
related to the idea of ‘one’, and is in a sense another representation of it.
But instead of the representation being a rational whole number, this constant
number is irrational in nature, and cannot be expressed in terms of a ratio of rational
numbers. It is also a limit to which an infinite series tends, and it reaches
that limit at infinity.
The history of our knowledge of logarithmic
functions is relatively modern:
b) Logarithms were introduced by John Napier in
1614 as a means to simplify calculations. They were rapidly adopted by
navigators, scientists, engineers, surveyors and others to perform
high-accuracy computations more easily. Using logarithm tables, tedious
multi-digit multiplication steps can be replaced by table look-ups and simpler
addition. This is possible because of the fact—important in its own right—that
the logarithm of a product is the sum of the logarithms of
the factors.
I am not claiming that
logarithms were known or used in the British Neolithic. I am suggesting however
that those responsible for the stone circles were interested in the idea of
infinite series, and knew, as consequence of that interest, the fact that such
series tend towards a limit. That limit can be rounded up to 2.72. And that is
what we now call Euler’s number.
So, if this hypothesis is
correct (and to some extent I’m attempting to enter the souls of the priests
here), why would they multiply these two numbers together, to arrive at what
Thom called ‘the megalithic yard’? As already mentioned, making things ‘meet and
agree’ is an interest arising out of the consideration of natural puzzles,
where not everything is commensurable. The ancient priests and their scholars
had a notion, arising out of the nature of some natural puzzles,that the
natural world is full of irrational numbers, which by definition are not
commensurate with each other. They also had the notion that these numbers are somehow
commensurate with each other in some other place. Not necessarily somewhere conceived
of as a physical space. The pythagorean triangles are an instance of this, in
that, when subjected to a standard operation such as the squaring of their
sides, they meet and agree. They can be represented meeting and agreeing in
physical space also, but without representing one of the principal
characteristics of triangles, which is that they enclose space.
The ‘some other place’ where
incommensurate things are commensurate with one another cannot be seen, because
it is not actually a place. Plato was careful to define the Heavens not as
Eternity itself, but as a moving image of Eternity. I think it likely
that the same notion was entertained in the British Neolithic. The heavens,
however, as some kind of representation of Eternity, could be studied for clues
about the nature of reality, hence the ancient interest in the Heavens.
But if Eternity is in no
physical space, then it must be present all through the physical world.
Not easily detectable, but often aspects of it could be manifest in physical
instances. Some of which could be understood to meet and agree, even if represented
in what is essentially a crooked representation of Eternity .I listed these
things – abstract concepts – earlier in this essay. Hence the importance of
wholes and totalities, and what is complete.
It follows that if Eternity
itself is all through the world, then the nature of reality is necessarily
two fold. Eternity is infinite, and physical reality is finite. But in fact
the two are essentially the same thing, just viewed from different
perspectives. We cannot see the infinite, but we can know that it is there, and
that it is something which stands behind all sense experience. In which case,
religious observance, expressed through the building of the monuments, and through
ritual action, was about both honouring the underlying identity between the
worlds, and healing the rift which exists between them. The major preoccupation
of the priests was to bring more of eternal reality into the physical world.
Eternity is one, whole, the
totality of what is possible, and is complete in itself. On earth we can
identify wholes as things which also belong to Eternity. In Eternity it is possible
for all things which on earth are incommensurate, to be commensurate. On earth,
wholes can be understood as things which are not irrational (such as whole
numbers). But we can also represent a whole with an irrational number, which is
the number which we know as Euler’s number. In Eternity both these numbers are
commensurate with each other.
In multiplying these two numbers
together, one rational, and the other entirely irrational, into the measure we know
as the ‘megalithic yard’, they were attempting to represent in their monuments a
state which properly exists in Eternity alone. A state in which all things meet and agree.
Thomas Yaeger, February 13-14,
2020.
