Abstract: Pythagorean elements detected in megalith circles
in ancient Britain have no easy explanation, and precede 1st
millennium Pythagoreanism by an extraordinary period of time. This paper
explores the idea that there is a connection between some core Pythagorean mathematical
and geometrical concerns, and ideas of divinity and Eternity. On the basis of a close examination of
Pythagorean ideas in the 1st millennium, for which we have extensive
documentation, It is suggested that this connection is a logical one. It is
therefore possible that similar conclusions were arrived at in the Late
Neolithic and in the Bronze Age.
1 The Longevity of
Ideas
We often underestimate the longevity of
patterns of ideas. Sometimes when they are linked to a religious or theological
structure, they can have a very long existence. Though much of modern knowledge
about the physical world has been developed since the European Enlightenment,
there are still ideas around which have persisted with very little change,
since the first millennium BCE. Hinduism is still much as it was for example,
as is Buddhism. Later religions such as Christianity, built as it was on the
Old Testament, preserves many aspects of Hebrew ideas [Christ is made to
paraphrase YHWH’s statement in the OT that he is ‘first and last, and beside
him there is no other god’, by characterising the divine as the ‘alpha and the
omega’].
In short, there are still religious ideas
and formulations around in the world, and contained in the human mind, which
are more than two and a half millennia old. And in some cases, much older than
that. Languages and peoples may change, but ideas are sometimes much slower to
change, and may survive alteration of language, people, and material culture.
This paper explores a hypothesis: the
hypothesis that some ideas which we habitually consider to be around two and a
half millennia old, are in fact much older than that. These ideas find powerful
expression in Pythagoreanism, written about by both Plato and the later Neoplatonists.
Looked at in the Greek context alone, this body of ideas extends over nearly
eleven hundred years (if a floruit of the mid sixth century BCE for Pythagoras
is correct), until the closure of the philosophical schools in 529 CE.
It was once conjectured, on the basis of
Alexander Thom’s surveys of megalithic circles, that there was a pythagorean
element in these constructions in the late British Neolithic, and the early
Bronze Age. This idea was later rejected (briefly discussed at the end of this
paper). If the suggestion of a pythagorean element was in fact correct, that
would push an extraordinary number of key ideas we associate with the 1st
millennium BCE back into the Neolithic.
The level of engagement with Pythagoreanism
which has been brought to bear on this question has so far not been
significant. This paper is intended to provide a more sophisticated
understanding of what Pythagoreanism implies, and how such an understanding can
inform what sense we can make of such a very distant past.
2
Pythagoreanism in 1st Millennium Britain
We have Greek and Roman sources for the
supposed origins of Pythagorean modes of thought. These point in different
directions. We have the story that Pythagoras was present at the fall of
Babylon in 539 BCE, and he is also supposed to have spent some time in Egypt,
learning from the priests. On the other hand, we have information about the
beliefs of the Gaulish priests from the mid-first century BCE, in the wake of
Julius Caesar’s campaigns in north western Europe. Caesar described the Gauls in his Commentarii de Bello Gallico, [The Gallic War], book VI.
According to Caesar, the Gaulish priests were
concerned with "divine worship, the due performance of sacrifices, private
or public, and the interpretation of ritual questions." He also said that
they played an important part in Gaulish society, being one of the two
respected classes, the other being the equites (the Roman name
for ‘’knights - members of a privileged class able to provide and equip horsemen).
They also functioned as judges in disputes. Among other interesting details,
Caesar also said that they met annually at a sacred place in the region
occupied by the Carnute tribe in Gaul, and that Britain was the home
of priestly study. Caesar also said that many young men were trained as priests,
during which time they had to learn large amounts of priestly lore by heart.
Metempsychosis
was the principal point of their doctrine: “the main object of all education
is, in their opinion, to imbue their scholars with a firm belief in the
indestructibility of the human soul, which, according to their belief, merely
passes at death from one tenement to another; for by such doctrine alone, they
say, which robs death of all its terrors, can the highest form of human courage
be developed”. He also tells us that they were concerned with "the stars
and their movements, the size of the cosmos and the earth, the world of nature,
and the powers of deities". So the components of their religious cult
involved the study of theology, cosmology, astronomy and natural philosophy.
Alexander Polyhistor described the Gaulish
priests as philosophers, and explicitly called them ‘Pythagorean’ on account of
their understanding of reality. He wrote that "The Pythagorean doctrine
prevails among the Gauls' teaching that the souls of men are immortal, and that
after a fixed number of years they will enter into another body."
Diodorus Siculus,
writing in 36 BCE, also said that the Gaulish priesthood followed "the
Pythagorean doctrine", that souls "are immortal, and after a
prescribed number of years they commence a new life in a new body."
There are other descriptive references to
the Gauls and their religion from antiquity, but it is not necessary to review
all of them here. These are the main evidential details we have for the
presence of Pythagorean ideas in Gaul and in Britain in the last two centuries
of the 1st Millennium BCE. It is likely that both Polyhistor’s
account and the account of Diodorus Siculus drew on the source used by Caesar.
3.The
Principal Sources for Pythagoreanism
The preceding descriptions are usually all
that is mentioned when religion in Gaul and in Britain before the arrival of
the Romans is discussed. This is because we do not have written records from
Gaul or Britain from earlier times. And so this is where historical discussion usually
stops. The rest of the story of these cultures becomes a matter for archaeological
investigation.
However, we need not stop here, looking at
nothing. Much of what we know about the other philosophical details of the
Pythagoreans is quite extensive, if not always consistent across the range of
sources.
There is a life of Pythagoras
by Iamblichus, and another by his pupil Porphyry. A life of Pythagoras by Diogenes Laertius
also contains useful information. Plato
and the later Platonists wrote in detail about Pythagorean doctrine, if not
always being explicit that they were referencing his ideas.
Plato is the best place to start. He had
the concept of an inner and outer knowledge, which reflects something of a
priestly understanding of both teaching and of reality. He referred to these
grades of knowledge as ta eso and ta exo In
the Theaetetus. Which means that
teaching operated at two levels – the exoteric and public level, and another
which was esoteric in nature.
Esoteric knowledge is by definition
obscure, and/or difficult to understand. Which is what the story of the
prisoners in the cave in Plato’s Republic is all about. They
see the shadows of reality on the wall before them, but not the reality itself.
When they are released with suddenness, their reason is deranged by the
experience. Instead they should have been released gradually, being shown
details of reality first, without the whole of the shocking truth of reality
being given to them all at once. Plato was engaged with both exoteric and
esoteric understandings of knowledge, but mostly what he tells us about is an
esoteric doctrine, which explains what is hidden and obscure, and relates to
the gods, and what is divine. As one might expect, the rules for the gods are
different.
4 The
Core of Pythagorean Doctrine
In the Timaeus Plato
refers to a principle of wholes, or totalities. It is later mentioned by the
Neoplatonist Porphyry as a Pythagorean doctrine, and Pythagoras is supposed to
have learned of it in a lecture in Babylon, after the fall of the city to the
Persians in 539 BCE. The doctrine is of course, very much older. It can be
detected in the Iliad, in Bk 18, where Hephaestus
makes objects which, on account of their nature, can pass into the counsel of
the gods, and return. The principle might, as Porphyry suggests, have been
brought back to the west by Pythagoras after his spell in the east, or it may
already have been part of a body of ideas already well established in Italy and
in Greece. The principle might be simply put, as ‘things which are total
participate in totality’, in the same way that Plato declared that ‘greatness
is participation in the great.’ But it is so much more important than a statement
that wholes conjoin with one another. It is the essence of the ascent from
image to image to an apprehension of the Good which Plato refers to in both
the Timaeus and the Republic.
Each of these images must represent or
embody an aspect of what Plato referred to as ‘the Good’. Each of the images
must allow the supplicant to pass from one to the other via their essential
identity. What varies between them is the degree of their participation in the
Good. Plato is very clear that the viewer of the images must be able to pass
along the chain of images in either direction. The chain of images is not
therefore purely about gaining an understanding of the Good (meaning the
divine, or Being itself), either in reality or figuratively. Passage through
the chain of images is about both the transcendence of images or forms, and
about the descent of Being into the world of generation, as a generative power.
The images are constructed in the way they are in order to reduplicate and
re-energise the power and presence of divine Being in the human world. For man,
this might be seen as an act of worship or observance of what is holy, but it
can also be understood also as a form of theurgy, even if the technical term
post-dates classical Athens by several centuries.
In the Timaeus [30a-b],
Plato speaks through Timaeus, saying:
For God desired that, so far as possible, all things should be good and
nothing evil; wherefore, when He took over all that was visible, seeing that it
was not in a state of rest but in a state of discordant and disorderly motion,
He brought it into order out of disorder, deeming that the former state is in
all ways better than the latter. For Him who is most good it neither was nor is
permissible to perform any action save what is most fair. As He reflected,
therefore, He perceived that of such creature as are by nature visible, none
that is irrational will be fairer, comparing wholes with wholes, than the
rational….
Plato, in using the phrase ‘comparing
wholes with wholes’, is referring to the principle of wholes and totalities
mentioned in Porphyry’s account of Pythagoras.
Pythagoras is said by Porphyry to have
associated with the ‘other Chaldeans,’ after he mentions his conferring with
the king of Arabia. The current academic view is that the Chaldean dynasties
were essentially Arab dynasties, and that they were in control of Babylon at
this time. This helps to confirm the reliability of some of the detail in
this important passage, written so long after the lifetime of Pythagoras.
What did Pythagoras take from his long
sojourn in Egypt, and the near-east? Is his doctrine like Plato’s? The point of
the doctrine of wholes and totalities, is to establish connection between the
divine world and secular reality. Porphyry’s account tells us that:
He cultivated philosophy, the
scope of which is to free the mind implanted within us from the impediments and
fetters within which it is confined; without whose freedom none can learn
anything sound or true, or perceive the unsoundedness in the operation of
sense. Pythagoras thought that mind alone sees and hears, while all the rest
are blind and deaf. The purified mind should be applied to the discovery of
beneficial things, which can be effected by, certain artificial ways, which by
degrees induce it to the contemplation of eternal and incorporeal things, which
never vary. This orderliness of perception should begin from consideration of
the most minute things, lest by any change the mind should be jarred and
withdraw itself, through the failure of continuousness in its subject-matter.
To summarise: the principle of wholes can be
understood as a logical modality which connects the world of the mundane with
transcendent reality. The definition of transcendent reality in Plato
(articulated by Socrates) is that it is a place beyond shape, form, size, etc.,
and occupies no place on earth. It is however the place where knowledge has its
reality (the ‘eternal and incorporeal things’ mentioned by Pythagoras).
Connection with transcendent reality is possible by the likenesses to the
transcendent which have existence on earth, such as things which are complete
and whole, which therefore participate in the completeness and wholeness of the
transcendent reality. Completeness and wholeness require (in the world of the
mundane) delineation and limits, and so the limits and the extremes of things
are also things which participate in transcendent reality.
The principle of ascent to the ‘eternal and
incorporeal things’ in the doctrines of both Plato and Pythagoras, is entirely
a mental process, which does not involve any of the senses. It proceeds via
chains of similitudes, both up and down, as a sequence of orderly perceptions.
The goal is a form of communion with that which never varies, and which is
always one and unchanging, as Plato tells us in the Sophist. The
return from the communion with the Good delivers beneficial things, because the
Good is the source of all knowledge.
5 Diogenes
Laertius on Pythagoreanism
Diogenes Laertius is generally
not regarded as a great historian of the philosophy of the ancient world, but
his Lives of the Philosophers is the
only general account which survives from antiquity. We get snippets from
elsewhere, but not the comprehensive sweep that he gives. He does not always
have good materials, or understand them well. But with his writing on Pythagoras, we get
something different. He is working with some very good materials indeed. His
date (actually quite uncertain) may be contemporary with the Neoplatonists who
also wrote about Pythagoras, and possibly he is using the same now long
vanished materials, since he reproduces the same sort of inconsistencies of
detail which appear in Iamblichus’s Life
of Pythagoras. These inconsistencies, mainly concerning religious observances,
may be explained by the fact that the core of Pythagoras’s doctrine isn’t about
these things at all, but about an agrapha,
or ‘unwritten doctrine’, revolving around deeper matters.
As already mentioned, it was a
popular opinion in antiquity that Pythagoras did not write any books – “There
are some who insist, absurdly enough, that Pythagoras left no writings whatever”
[D.L., Book VIII, 6], however Diogenes says that ‘Heraclitus, the physicist,
almost shouts in our ear, “Pythagoras, son of Mnesarchus, practised inquiry
beyond all other men, and in this selection of his writings made himself a
wisdom of his own, showing much learning, but poor workmanship”’. Heraclitus
therefore seems to recognise the disparate origins of the material used by
Pythagoras (in his book On Nature), and feels that it has not been
worked properly. Diogenes tells us that Pythagoras wrote three books
altogether, which were (since we no longer have them): On Education, On Statesmanship,
and On Nature. Other works were also attributed to him [D.L., Bk VIII 7].
Diogenes appears to have had
access to the three Pythagorean texts, or extracts from them, or perhaps from epitomes
of them, since he talks about the contents [D.L., Bk VIII 9-10]. He says that Pythagoras was understood to be
the first to speak of the idea of metempsychosis – he declared that “the soul,
bound now in this creature, now in that, thus goes on a round ordained of
necessity”. D. L. says that ‘so greatly was he admired that his disciples used
to be called “prophets to declare the voice of God” [D.L., Bk VIII 14].
The books seem to have been kept
secret, since Diogenes says that “Down to the time of Philolaus it was not
possible to acquire knowledge of any Pythagorean doctrine” until Philolaus “brought
out those three celebrated books.” Diogenes says that Plato sent a hundred
minas in order to purchase these texts [D.L.
Bk VIII 15]. He cites Aristoxenus in the tenth book of his Rules of Pedagogy
“where we are …. told that one of the school, Xenophilus by name, asked by
someone how he could best educate his son, replied, “By making him the citizen
of a well-governed state.”’ This is of course the clearest anticipation of
Plato’s interest in education.
Diogenes relates some details,
not always in agreement with each other, of the religious nature of Pythagoras'
philosophy: “He used to practise divination by sounds or voices, and by
auguries, never by burnt offerings, beyond frankincense. The offerings he made
were always inanimate; though some say that he would offer cocks, sucking goats
and porkers, as they are called, but lambs never. However, Aristoxenus has it
that he consented to the eating of all other animals, and only abstained from
ploughing oxen and rams” [D.L., Bk VIII 20]. Diogenes relates later that
‘Apollodorus the calculator’ says “he offered a sacrifice of oxen on finding
that in a right-angled triangle the square on the hypotenuse is equal to the
squares on the sides containing the right angle”.
There was therefore some
uncertainty in antiquity about exactly what the religious practice of
Pythagoras was – it may not have been consistent in its nature, and it follows
that it is possible that some of the practices attributed to Pythagoras,
(vegetarianism, avoidance of killing animals, the avoidance of beans, etc.) are
not in themselves of essential importance to Pythagorean doctrine, but only
seemed so to compilers and commentators in late antiquity.
If we look at some further
statements by Diogenies we can guess what the important things in Pythagorean
doctrine are. Diogenes says that Pythagoras advised his disciples to say to
themselves when entering their own doors: ‘Where did I trespass? What did I
achieve? And unfulfilled what duties did I leave?’ [D.L., Bk VIII 22]. This indicates (among other things) the importance
of the threshold or limit to Pythagoras.
Pythagoras also urged that the
memory be trained. This was also extremely important to Plato, and he regarded
the invention of letters to have been a disaster on the grounds that they
impaired the training of the memory through making its importance less clear.
There were in any case already people in Greece who held large parts of the
Homeric poems in memory, since the poems were not committed to writing until
the time of Peisistratus (some time after he first became tyrant of Athens in
560 BCE). Memory seems to have been cultivated in Egypt, and was certainly
practised (and discussed) in late antiquity in various parts of the Roman
Empire (Cicero mentions it, and it surfaces in the work of St. Augustine).
Pythagoras also said that men
should sing to the lyre and by hymns to show due gratitude to gods and to good
men. He bade men “to honour gods before demi-gods, heroes before men, and first
among men their parents”. The principal image here is the gods, who are more
important than the demi-gods, in terms of their claim on our worship and
honour. Heroes stand in the same relation before men, and our parents stand in
the same relation to us. He amplifies the importance of this metaphorical
perspective, by saying that men should ‘honour their elders, on the principle
that precedence in time gives a greater title to respect; for as in the world
sunrise comes before sunset, so in human life the beginning before the end, and
in all organic life birth precedes death’ [D. L., Bk VIII 22-4].
At one level this kind of
metaphor-making looks trivial, which is one of the reasons why little has been
made of these passages. However. Pythagoras is setting up oppositions between
extremes within defined classes (Gods and demi-gods, who are immortal, Heroes
and men, who are mortal, etc.), and making a comparison between them. He is
also establishing a line of connection between them. He isn’t just
comparing one image with another, he has created chains of images, with one end
of the chain representing the extreme of reality (the Gods), and we stand at
the other extreme.
The image of ourselves and our parents might be taken to
suggest a parallel with the relation between Gods and demigods. In terms of the
relationships implied in the image, the familial image can be understood as a
copy of sorts, more or less imperfect, of the relationships between Gods and
demigods. We are of course familiar with the Greek Gods and their shocking
personal relations with each other, which often suggest an earthly and
dysfunctional extended family.
Like Plato, Pythagoras had an agrapha, since some Pythagoreans “used
to say that not all his doctrines were for all men to hear” – which is perhaps
why it was so difficult to acquire knowledge of Pythagorean doctrine until the
indiscretion of Philolaus [D.L., VIII 15-6]. Diogenes authority for this is the
tenth book of Aristoxenus’ Rules of Pedagogy. Diogenes draws details of
the Pythagorean philosophy from another lost author – Alexander, author of Successions
of Philosophers, who claimed to find the following in the Pythagorean
memoirs:
The principle of all things is
the monad or unit; arising from this monad the undefined dyad or two serve as
material substratum to the monad, which is cause... [D.L., VIII 25].
This is very like the conception of the Neoplatonists,
who argued that in order that the good should remain untainted with generation
and change, a copy came into being, which did participate in creation:
from the monad and the undefined
dyad [the ‘undefined dyad’ may also be translated as ‘unlimited dyad’, or
‘unbounded dyad’ (the Greek term is ‘aoriston’) spring numbers; from numbers,
points; from points, lines; from lines, plane figures; from plane figures,
solid figures; from solid figures, sensible bodies, the elements of which are
four, fire, water, earth and air. These elements interchange and turn into one
another completely, and combine to produce a universe animate, intelligent,
spherical, with the earth at its centre… [D.L., VIII 25].
Once again we have a chain of
images: the monad and the undefined dyad, numbers, points, lines, plane
figures, solid figures, sensible bodies, the four elements. Note that we don’t
have the monad included in his sequence – the first image in the sequence is
the monad and the undefined dyad, which “serves as material substratum
to the monad, which is cause…”
Each of these conceptions is an
imaging of the properties of the monad, one leading to the other, increasing in
definition, attributes and properties until we reach the sensible bodies with
the properties of fire, water, earth and air, all of which can interchange into
one another completely. This chain represents an order of generation, rather
than an order of perception.
That each of the sensible bodies
can interchange into one another completely is a corollary of the fact that
they arise from the undefined dyad. They are differentiations of the undefined
dyad, and their fundamental identity resides there. Thus, fire may stand
metaphorically for water, earth for air, and so on. The interchange occurs with
reference to the monad and undefined dyad, since the combination of the monad
and undefined dyad is the fount of all cause, generation and change.
The monad itself is an image, but
can have no definition beyond the ‘One’.
The numbers did not arise from the monad itself: for that to happen, something
else was necessary. That he does not say that numbers arise from the monad is
an important clue towards understanding that not only is Pythagoras speaking in
terms of images, but that these images are related to an account of another
level of reality, referred to, but not articulated. This level of reality is
entirely without rational form, and which has an esoteric nature. Such an
account of reality also informs Hesiod’s version of the creation, in which there
is an ultimate reality which does not conform to the categories of our
understanding (i.e., ‘chaos’).
In addition, there is a strong
ethical element to Pythagoras’ life and philosophy, and it is clearly
associated with the properties of the threshold, where one thing conjoins with,
or turns into another. It would not be unreasonable to describe Pythagoras’
philosophy as theurgic in nature, since he is concerned that man should control
his own destiny, and not trust to the gods alone. He also laid down precepts
for religious practice and religious discipline, meaning that there was
understood to be an efficacious ritual element in the transformation of the
human soul.
For example, Pythagoras urged
that victims ought not to be brought for sacrifice to the gods, and that
worship should be conducted “only at the altar unstained with blood”. In
addition, he stipulated that the gods should not be called to witness, “man’s
duty being rather to strive to make his own word carry conviction”. He also said that men should avoid excess of
flesh, and that they should respect all divination. Abstention from beans was
recommended “because they are flatulent and partake most of the breath of life;
and besides, it is better for the stomach if they are not taken, and this again
will make our dreams in sleep smooth and untroubled” [D.L., Bk VIII 23-4].
That they ‘partake most of the
breath of life’ is an objection to beans seems a little odd, unless by this
Pythagoras is indicating that the target of the transformation of the
individual through discipline, ritual and understanding is a condition which
does not partake of the breath of life. We recall that Socrates in his
final moments asked that a cock be sacrificed to Asclepius, which was to mark a
return to health – in this case a healing from the trials and tribulations of
life.
Pythagoras seems to have shared this view of earthly life. In vol 2, Bk.
VII on Zeno (333-261 BCE, of Citium in Cyprus, a city which ‘had received
Phoenician settlers’), we are told by Diogenes that the author Hecato, and also
Apollonius of Tyre, in his first book on Zeno, that Zeno consulted the oracle
(presumably Delphi) ‘to know what he should do to attain the best life’. The
response of the god was that ‘he should take on the complexion of the dead’.
Diogenes Laertius takes from his sources that Zeno’s interpretation of this is
that ‘he should study ancient authors’. We can see that the true meaning of the
oracle was both much more straightforward and much more profound than that.
6 Pythagorean Thought in Italy
Pythagoras was creating an
eclectic doctrine, by syncretising elements from different sources. This is
what Heraclitus means by saying that the collections of information in the
three books are ‘poorly worked’. Not much interest has been shown in what
Pythagoras, a long-time resident in Italy, might have drawn from Italian
sources. In fact, it would seem that much of what was later passed off as
Pythagorean in origin, actually has its origin among the Latins.
For example, the Romans also had a tradition
of veneration of the boundaries and limits of things. Oskar Seyffert
says of the god Janus that “even the
ancients were by no means clear as to his special significance; he was,
however, regarded as one of the oldest, holiest, and most exalted of gods”.
Of course, if the special
significance of Janus was close to the heart of Roman religion, an absence
of discussion might, rather than signifying a lack of clarity about his
special significance, mean quite the opposite, and that the written tradition
is quite misleading as to the Roman understanding of Janus, at
least within the priestly community.
“In Rome the king, and in later times the rex sacrōrum, sacrificed
to him. At every sacrifice he was remembered first; in every prayer he was the first
invoked, being mentioned even before Jupiter”. Which is indication of high
status. If we recall the remarks of Pythagoras on what comes first and why, we
can see that the significance of Janus is extremely important indeed. This is
further emphasised by the fact that “in the songs of the Salii (‘jumpers’ or
dancers) he was called the good creator, and the god of gods; he is elsewhere
named the oldest of the gods and the beginning of all things.” The Salii were
an old Italian college of priests of Mars, said to have been originally
introduced at Rome by Numa Pompilius, the legendary
2nd king of Rome. He was said to be a native of Cures in the Sabine country,
and was elected king a year after the death of Romulus.
William Smith says that Numa Pompilius “was renowned for his wisdom and piety;
and it was generally believed that he derived his knowledge from Pythagoras”. Given
that the foundation of Rome is traditionally 753 BCE, this is impossible, since
Numa and Pythagoras would have been two centuries apart. However, the fact that
later the institutions of Numa were associated with Pythagorean influence
suggests that there was a perception of a relationship between the doctrines of
Pythagoras and the foundation of Roman religion. Smith continues: “…he devoted
his chief care to the establishment of religion among his rude subjects”, and
to giving them appropriate forms of worship. He was instructed by the Camena
Egeria (Aegeria), one of the twelve nymphs in Roman mythology. Numa later
dedicated the grove in which he had his interviews with the goddess, in which a
well gushed forth from a dark recess, to the Camenae.
Seyffert continues regarding Janus: “It would appear that originally he was a
god of the light and of the sun, who opened the gates of heaven on going forth
in the morning and closed them on returning at evening”. Rather, Janus, being
the divinity associated with boundaries, is associated with gates, crossings,
risings and settings, beginnings and endings, and the daily movement of the sun
is the most important visible instance of beginnings and endings. In course of
time (Seyffert suggests) “he became the god of all going out and coming in, to
whom all places of entrance and passage, all doors and gates were holy”
[my italics]. He continues:
In Rome all doors and covered
passages were suggestive of his name. The former were called ianuae; over the
latter, the arches which spanned the streets were called iani.
Many of these were expressly
dedicated to him, especially those “which were situated in markets and
frequented streets, or at crossroads”. In the case of crossroads, Seyffert
tells us that “they were adorned with his image, and the double arch became a
temple with two doors, or the two double arches a temple with four”. The way
Janus was generally represented was “as a porter with a staff and a key in his
hands, and with two bearded faces placed back to back and looking in opposite
directions.”
Further, he is also the god of entrance into a new division of time, and was
therefore saluted every morning as the god of the breaking day (pater
matutinus); the beginnings of all the months (the calends) were sacred to him,
as well as to Juno; and, among the months, the first of the natural year, which
derived from him, Ianuarius. For sacrifices on the calends twelve altars were
dedicated to him; his chief festival, however, was the 1st of January,
especially as in B.C. 153 this was made the official beginning of the new year.
On this day he was invoked as the god of good beginnings, and was honoured with
cakes of meal called ianuae; every disturbance, every quarrel, was carefully
avoided, and no more work was done than necessary to make a lucky beginning of
the daily business of the year; mutual good wishes were exchanged, and people
made presents of sweets to one another as a good omen that the new year might
bring nothing but that which was sweet and pleasant in its train.
For the Romans, this juncture of the year,
like every other juncture over which Janus presided, was a region in which
change was more possible, more likely, than at any other time. Therefore, any
immoderate behaviour, any departure from the normal daily pattern of life,
whether through a quarrel or some other unpleasantness, might easily have taken
root, and they might have found their whole lives dislocated as a result.
Seyffert continues that:
the origin of all organic life,
and especially all human life, was referred to him; he was therefore called
consivius (‘sower’). From him sprang all wells, rivers, and streams; in this
relation he was called the spouse of Juturna, the goddess of springs, and
father of Fontus, the god of fountains.
7 The existence of Irrational numbers
It is generally supposed that the
Pythagoreans understood the world to be rational in nature, and it had long
been argued that rational numbers were the product of ratios of other numbers. Their
belief in rational whole numbers seems to have been a principal concern,
possibly because whole numbers are often commensurable. The ancient assumption
that the world was a rational creation, was maintained at least at the level of
open public discussion.
There is however a famous story about the discovery of
irrational numbers by the Pythagoreans, and their utter horror at the
discovery. The discoverer of irrational
numbers was supposedly drowned at sea, perhaps in consequence of this discovery.
In fact, the story is likely to have a quite different meaning at an esoteric
level, which I will discuss at the close of this paper.
So how was the Pythagorean proof
of the existence of irrational numbers achieved? We should remember that
The Eleatic school (home to Parmenides and Zeno, the former of which
argued for the One and unmoving reality transcending the world of
appearances) attacked Pythagorean doctrine by assuming their opponents'
tenets, using the reductio ad
absurdum technique to examine their credibility. The effect of such arguments was to reinforce
the importance of the incommensurate in the world of number.
The Greeks attempted to extricate
themselves from these difficulties by distinguishing between things which they
would have preferred to have been commensurable (numbers and magnitudes),
thereby rendering them incomparable. So the diagonal of a square could be
regarded as a magnitude rather than as a length equal to the ratio of two
numbers. By this means, irrational numbers could be largely ignored (a similar
convenient fiction to one devised by Aristotle in connection with infinity, in
which he subverted the difficulty of the infinite by dividing it into two: a potential infinite, and the actual infinite,
which could be ignored).
From Thomas Heath:
We mentioned... the dictum
of Proclus... that Pythagoras discovered the theory or study
of irrationals. This subject was regarded by the Greeks as
belonging to geometry rather than arithmetic. The irrationals in Euclid,
Book X, are straight lines or areas, and Proclus mentions as special topics in
geometry matters relating (1) to positions (for numbers have
no position) (2) to contacts (for tangency is between continuous things),
and (3) to irrational straight lines (for where there is
division ad infinitum, there also is the irrational).
...it is certain that the
incommensurability of the diagonal of a square with its side, that is, the
irrationality of {\displaystyle
{\sqrt {2}}}root 2, was discovered in the school of Pythagoras... the
traditional proof of the fact depends on the elementary theory of numbers,
and... the Pythagoreans invented a method of obtaining an infinite series of
arithmetical ratios approaching more and more closely to the value of {\displaystyle {\sqrt {2}}}roordroot
2.
Thomas Heath was writing at a
time (1921) when classicists had very little knowledge of what was coming out
of the ground in Mesopotamia and elsewhere, so his certainty that the school of
Pythagoras ‘discovered’ the incommensurability of root 2 is a product of that
time. He writes:
The actual method by which the
Pythagoreans proved the fact that {\displaystyle
{\sqrt {2}}}rorrrrrrrrkkk root 2 is incommensurable with 1 was
doubtless that indicated by Aristotle, a reductio ad absurdum showing that, if the diagonal of a square
is commensurable with its side, it will follow that the same number is both odd
and even. This is evidently the proof interpolated in the texts of Euclid as X. 117... [Heath,
T. (1921) Vol. 1 pp. 90-91].
It is a proof based on the law of
non-contradiction. However, it is the consequence of the properties of
Pythagorean triangles as they are represented to our understanding. The point
of this demonstration is that: how things are represented to us is not the same as how they actually are.
Or how they are in what we might term ‘transcendent space’.
Heath continues:
We have first the passage of
the Theaetetus recording
that Theodoras proved the incommensurability of root 3, root 5….
Root 17, {\displaystyle {\sqrt
{3}},{\sqrt {5}}...{\sqrt {17}}}after which Theaetetus generalized the
theory of such 'roots.'... The subject of incommensurables comes up again in
the Laws, where Plato inveighs against the ignorance
prevailing among the Greeks of his time of the fact that lengths, breadths, and
depths may be incommensurable as well as commensurable with one another, and
appears to imply that he himself had not learnt the fact till late, so that he
was ashamed for himself as well as for his countrymen in general.
This is interpretation about what
Plato is saying which isn’t warranted. Plato was quite plain elsewhere (Republic) that all things may pass into
one another, and hence are in some way commensurate. He says this in connection
with the Forms. As a general statement, it implies that the same is true for
both commensurable and incommensurable numbers. We find ourselves in a strange
place where the incommensurate may also be commensurate. Heath continues:
But the irrationals known to
Plato included more than mere 'surds' or the sides of non-squares; in one place
he says that, just as an even number may be the sum of either two odd or two
even numbers, the sum of two irrationals may be either rational or irrational.
An obvious illustration of the former case is afforded by a rational straight
line divided 'in extreme and mean ratio' (Euclid XIII. 6) proves that each of
the segments is a particular kind of irrational straight line called by him in
Book X an apotome; and to suppose that the irrationality of the two
segments was already known to Plato is natural enough if we are correct in
supposing that 'the theorems which' (in the words of Proclus) 'Plato
originated regarding
the section' were theorems about what
came to be called the 'golden section', namely the division of a straight line
in extreme and mean ratio as in Euclid. II. 11, and VI. 30. The appearance of the latter problem in Book II, the content of which
is probably all Pythagorean, suggests that the incommensurability of the
segments with the whole line was discovered before Plato's time, if not as
early as the irrationality of
{\displaystyle
{\sqrt {2}}}root 2
[Heath, T. (1921) Vol. 1 pp
. 304-305].
8 Religious aspects of Pythagoreanism
Pythagorean thought is therefore
a species of transcendentalism. It is a pattern of thought which understands
reality itself (whatever that may be) as a principal concern, and as something
which, as it is, transcends mundane earthly reality.
Within this pattern of thought
however, earthly reality has properties and characteristics which
have counterparts in the divine world. If 'God is Great’ for example,
there are earthly examples of greatness, and so greatness is understood to
be a property held in common between the worlds. What is held in common
was understood by those of a transcendentalist persuasion to offer a connection
between the worlds.
In essence the transcendentalist
outlook holds that Being, or the ultimate reality, is both transcendent, and
also present in the physical world. It is hard to imagine how such a view could
arise except as the result of sophisticated logical discussion of the nature of
reality. The idea defies common sense, and is counter intuitive.
This view of the world represents
a paradoxical understanding of reality, in that the divine both transcends
mundane reality, but is also at the same time present in every aspect of
that reality. The connections between reality itself and earthly reality
are not obvious, and often not easy to discover. The difficulty in discovering
the connections is an index of the distance between the worlds. Yet it is
possible to discover these connections. Reading the mind of the divine was of
course a major concern in antiquity, since interpreting divine intention
conferred knowledge and earthly power.
Working on or with the gods
(theurgy) is often thought of these days as some obscure form
of theological lunacy practised by the Neoplatonists and a few other
groups in the dying days of the Roman empire. It is however a very old
idea, based on the understanding that the sacred and profane worlds are
connected with each other. It is also
built into Plato’s account of the creation of the cosmos (in the Timaeus). The practice of theurgy is a
corollary of the transcendentalist outlook, since if reality is transcendent,
but we are also paradoxically part of it, then human will and intention are
important to the way in which the world works. Physical reality does not
represent a copy of transcendent reality, which Plato labelled as a likelihood
only, but rather a subjective understanding
of that reality *[note 1].
According to this way of
thinking, there are processes which we can use to enhance what we have
in common with divinity. Most often this was expressed in terms of gaining
knowledge of the divine, since the supreme divinity was necessarily the
fount of all knowledge (both Plato and the Mesopotamians concur on this
point). If theurgy is an important component in early religious practice, it
tells us something about how much knowledge was prized at the time, and also something
of the scope of that knowledge in antiquity. Unlike homeopathic magic, which
theurgy sometimes resembles, the practice of theurgy is entirely dependent
on an understanding of Being for its effective use.
9 The Pattern of Eternity
That Socrates says at Phaedrus 247c that he dares to speak the truth concerning the nature of the region above
heaven implies strongly that it is dangerous to do so - and after all, one of
the charges against him was that he made theological innovations. Xenophon
suggests that, though he was not formally charged with disbelief in the gods per se, Socrates was suspected of a form
of atheism ["And how could he, who trusted the Gods, think that there were
no Gods?" Memorabilia Bk 1 ch.1.5]. To hold the ultimate reality to be virtually
indistinguishable at root from chaos, a place devoid of justice, beauty, order,
etc., (and without location in time or space) except in potential, would be
indistinguishable to the ordinary citizen from atheism. No wonder therefore
that Plato writes the ironical words at Tim
40d:
Concerning the other divinities,
to discover and declare their origin is too great a task for us, and we must
trust to those who have declared it aforetime, they being, as they affirmed,
descendants of gods and knowing well, no doubt, their own forefathers.
And at Tim 29a, concerning the model after which the universe was
patterned, Timaeus asks:
Was it after that which is
self-identical and uniform, or after that which had come into existence?
The latter implies change and
disorder; therefore
if so be that this Cosmos is
beautiful and its Constructor good, it is plain that he fixed his gaze on the
Eternal, but if otherwise (which is an impious supposition), his gaze was on
that which has come into existence.
Which is no more than an appeal
to common sense. The nature of the arguments which might be adduced in
antiquity to explain the world of appearance are, as the Sophist shows, much more complex.
Conditioned therefore both by the
difficulty of the subject matter, and the social impracticability of the
doctrine, we are forced to work out the doctrine for ourselves. That the method
employed to convey the doctrine sometimes created unnecessary difficulties for
the understanding, quite apart from its inherent difficulty is shown by the
remark at 48c where some matters are not explained:
solely for this reason, that it
is difficult for us to explain our views while keeping to our present method of
exposition.
Nevertheless, the description of
the Receptacle at Tim 50-51 is
possibly the clearest exposition in Plato of the Real:
... it is right that the
substance which is to receive within itself all the kinds should be void of all
forms... that the substance which is to be fitted to receive frequently over
its whole extent the copies of all things intelligible and eternal should
itself, of its own nature, be void of all the forms... a Kind invisible and
unshaped, all--receptive, and in some most perplexing and most baffling way
partaking of the intelligible...
If then, Plato's unwritten
doctrine (agrapha) placed chaos at
the heart of Being, his conclusion would not be out of place among Greek
speculations in general as to the nature of the arche: the difference is simply that he underpinned this conclusion
with philosophical argument [Compare for example lines 116-128 of Hesiod's Theogony]. These
we do not have for the earlier speculations, and therefore it is easy to
conclude that they did not in fact exist; that the early speculations were not
supported by cogent argument, and that the idea of chaos as the root or
beginning of things never was any more than a concrete image of disorder. But
Plato himself, putting the argument into mouth the of Timaeus at Tim 30a,
uses such a concrete image, saying that God
took over all that was visible,
seeing that it was not in a state of rest but in a state of discordant and
disorderly motion, He brought it into order out of disorder...
We have virtually the whole of
the Platonic corpus: of the earlier philosophers we have fragments like the one
above. We should be cautious in presuming the absence of clear reasoning behind
images simply because we have no direct access to such reasoning: that we do
read concrete conceptions into the concrete images of the Presocratics is
partly due to the fact that this was often the practice among the ancients
themselves, and partly because, building upon this fragmentary and distorted
evidence, we can frame a satisfying scheme in which there is a beginning,
middle and notional end to the history of ideas, starting with concrete images
and working up to pure abstraction.
10 Pythagorean
Syncretism
It is important to recognise 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 from the discussion of Pythagorean mathematics, number and geometry,
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.
We know that Pythagoras drew on
many sources for what became known as Pythagoreanism. He is likely to have
drawn on both Italian and Greek ideas, and he travelled in Egypt, talking with
the priests of the various cults (though we are told that most of them were not
much interested in answering his questions); and also in the Levant, Arabia,
and Babylon. He borrowed from them too.
We are accustomed to thinking
that the intellectual life of these disparate cultures must have been as distinct
as their iconography, their mythologies, languages, and systems of writing. But
it is not necessarily so. 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.
11. Transcendentalism in the Late Neolithic and early Bronze Age in Britain
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.
12 Walking back the insight into Ancient Mind
Robin Heath has documented the
archaeological community’s turning away from engagement with Thom’s work. This
happened for a number of reasons. One of the reasons was that there was little
that archaeologists could do with the information which he presented, and they had
no idea at all what it might have meant. Thom surveyed and studied the stone
circles, and inferred various properties, such as the apparent obsession with
whole number in their construction, their use in eclipse prediction, and their
connection with foresights in the surrounding landscape. Beyond this, Thom himself
was largely unable to supply useful interpretative context to the phenomenon of
megalith building.
Thom was also in conflict with a
key assumption of archaeology, which is that man (in general) has been
progressing, generation by generation, since the earliest times. The suggestion
that there was a profound body of mathematical and astronomical knowledge so
far back in time just didn’t fit with this paradigm. In the end it was argued
that, despite the undoubted quality of his surveying of the monuments, he was
seeing something that wasn’t actually
there.
A significant part of his work was resurveyed by the mathematician
Clive Ruggles in order to determine the case. Ruggles’ approach involved
avoiding any concern for the exact orientation and location of the foresights
which Thom had identified, and so necessarily made that part of the evidence meaningless
for the interpretation of the function of the sites. As a consequence of this
approach, as far as the discipline of archaeology was concerned in the late
eighties, there now was no longer a puzzle to be addressed.
The real problem for the
interpretation of the stone circles however is the absence of any understanding
in the modern world of a necessary connection between ideas of number,
mathematics and geometry, and thought concerning the divine. As suggested, we read ancient religions as
analogues of modern religions, which we understand without reference to a
technical substrate (though these substrates are sometimes still present in
vestigial form). For us, religion is
about bodies of belief. So instead of an understanding of religion in terms of
series of responses to fundamental philosophical questions, it is understood in
terms of collective belief in socially useful behaviours, ritual, and myth. We
understand ancient religion (as far as is possible) in terms of sociological,
ideological, and sometimes pathological functions.
13 Pythagoreanism and the Deep
Returning to the question of the
Pythagorean disciple who drowned at sea, we are told that the drowning occurred
because, either he had discovered irrational numbers, or because he had divulged
the fact that they exist (the sources for the story are inconsistent, which is
often a pointer to glossed interpretation). In the 1st and 2nd
millennia BCE, Ocean was an image which referenced the idea of Being. Like
Being itself, ocean seems without limit, and to be without form, shape and
colour. It was an idea which was common to the Greeks and to Near Eastern
cultures.
In Mesopotamia, there was an important story
which told how man was first educated in the sciences, agriculture, and
land-measure, by an amphibious creature (the sage Oannes) who emerged from the
sea in the daytime and conversed with men, before disappearing back into the
deep in the evening. As a creature of the ocean, and a sage of Being itself, he
had access to all knowledge.
The idea of this is reduplicated in the more
famous Mesopotamian story of Gilgamesh, which opens with the protagonist diving
down to the depths of the sea. This makes sense once it is understood that the
poem was known to the Mesopotamians as: ‘He who saw the deep’, meaning that
Gilgamesh had access to knowledge of divine things. Perhaps the real meaning of the story of the
drowning of the Pythagorean disciple is that, in understanding the fact that there
are such things as irrational numbers, and that both irrational and rational
numbers can be commensurate with each other, he was in possession of an
esoteric understanding of the divine, which lay at the heart of the unwritten
doctrine of the Pythagoreans.
Notes
1. In William Sullivan’s The Secret of the Incas, it is argued
that the Incas were attempting to turn back the precession of the equinoxes, in
order to preserve a heavenly bridge that they imagined gave them access to the
divine world. The subtitle of the book is: myth,
astronomy and the war against time. They came to the view that they could
turn back time because of a transcendentalist understanding of reality, and the
place of the Incas within it. It is a completely counter-intuitive outlook.
References:
Caesar, Julius, Commentarii de Bello Gallico, [The
Gallic War], book VI. Harvard, Loeb Classical Library, 1917.
Chronicle (BBC), Cracking the Stone Age Code, 1970. The
documentary film is available from the BBC Archive.
Cory, Isaac P., Cory’s Ancient Fragments, [contains the account
of Berossus concerning the encounter with the sage Oannes, and passages from
Alexander Polyhistor and Diodorus Siculus], London, 1828.
Diodorus Siculus, Bibliotheca
historica, Harvard, Loeb Classical
Library, 1989.
Diogenes Laertius, Lives and Opinions of Eminent Philosophers.
Harvard, Loeb Classical Library,
1925.
Euclid, The Elements: Books I–XIII. Translated
by Sir Thomas Heath, Barnes & Noble,
2006.
George, A.R., The Babylonian Gilgamesh epic: introduction, critical edition and
cuneiform texts. Vol.1. OUP, 2003.
Guthrie, K. G., The Pythagorean sourcebook and library: an
anthology of ancient writings which relate to Pythagoras and Pythagorean
philosophy. Phanes Press, 1987.
Heath, Thomas L., A
History of Greek Mathematics, 2 vols., OUP, 1921.
Heath, Robin, Alexander Thom: Cracking the Stone Age Code. Bluestone Press,
2007.
Hesiod, Theogony; Works and days. Harvard, Loeb Classical Library, 2006.
Homer, Iliad, Harvard, Loeb Classical Library Iliad. Books 1-24, 2nd ed.
1999.
Iamblichus,
On the Mysteries and Life of Pythagoras. Works of Thomas Taylor, Vol. XVII. Prometheus Trust, 2004
.
Plato, Timaeus, Republic, Theaetetus, Phaedrus,
etc. [twelve
volumes]. Harvard, Loeb Classical Library, 1929.
Ruggles, Clive, Records in Stone, CUP, 1988 and 2003
Ruggles, Clive, Astronomy in Prehistoric Britain and Ireland,
Yale University Press, 1999.
Seyffert, Oskar, Dictionary of Classical Antiquities. Revised and Edited, with Additions by Nettleship and Sandys. Article: ‘Janus’. Swan
Sonnenschein, Macmillan, 1906.
Smith, William, A Smaller Classical Dictionary. Article: ‘Janus’.
John Murray, 1891.
Sullivan, William, The Secret of the Incas: myth, astronomy and the war against time. Three Rivers Press, 1996.
Thom, Alexander, Megalithic Sites in Britain, OUP, 1967.
Thom, Alexander, Megalithic Lunar Observatories OUP, 1971
(repr. 1973 with corrections).
Xenophon,
Memorabilia, Harvard,
Loeb Classical Library, 2013.
Materials
There is a significant collection of papers and reviews concerning
Thom's work available from The SAO/NASA Astrophysics Data System (ADS) which is a Digital Library portal for researchers in Astronomy and Physics, operated by the Smithsonian Astrophysical Observatory (SAO) under a NASA grant. It is possible to select the files you want, and to download them as a collection.
The BBC Chronicle episode from October 1970,
Cracking the Stone Age Code, which is available from the BBC Archive. The file uses Flash. If it doesn't work first time, try reloading the page. It is also available on Youtube:
https://www.youtube.com/watch?v=WafRqdOQK30&t=128s
The archaeologist and anthropologist Euan MacKie gave a
lecture (British Archaeology and Alexander Thom) to a lay audience in 2013, in which he discussed both Thom's work and the subsequent rejection of Thom's findings by the archaeological community. He also discusses the problematic nature of Clive Ruggles methodology in resurveying some of the megalithic monuments in Scotland which Thom examined. Essentially Ruggles methodology, conceived as a way of excluding subjective bias from the survey process, necessarily makes it virtually impossible to reproduce Thom's results.
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