The 'Hill of Many Stanes'

[An extract from a conversation with a correspondent in the US, from May and June, 2020, shortly after 'The Mathematics of the Megalithic Yard' was completed.] 

On Monday, June 1, 2020, 09:31:47 AM PDT, Thomas Yaeger [....] wrote:

[....], hi. Thanks for your mail. I'm going to respond to it in separate mails, since there is a lot to say. Interleaved, as usual (bad academic habit!)

At 06:03 29/05/2020, [....]  wrote:

Hi Thomas,
Sorry I haven't responded sooner. I've been working on a response to your article (& other emails) about the Megalithic Yard and didn' t want to write again until I had made some progress. I'm probably making it into too much of a project lol. [....} So, I'll send what I have for now (including other stuff I've been putting in a draft) and get it off to you. Sorry if it doesn't do justice to your arguments. 

[....] 

Your argument is very compelling and interesting. It seems like a real breakthrough although, naturally, I'm not enough of an expert to judge!

I think it is a real breakthrough, but it took a while to make it (as I said, the article was written in about a day and a half, after thinking it through for around two years). Developments are happening very fast now, which is interfering with my writing programme.

. I understand that math as such isn't the point of your argument, it's more about what Euler's number signified, right?

Yes. It's Euler's number, what it represents, and how they calculated it in the 2nd and 1st millennia BCE. I think I've changed my mind about how much Alexander Thom actually knew. I think he knew that it was a pointer towards the idea of the infinite. But he did not know that in those ancient days the ideas of the divine, the infinite, and reality itself were regarded as coterminous, and were just different ways of speaking about the same thing (which is an understanding which still survives in Hindu thought and religion). So for Thom, he could see the mathematics, but didn't understand the idea of reality itself as a primal fulness, or a plenum, and why that would engage ancient interest.

There is in Scotland a site near John O'Groats which is known as 'the hill of many stanes', which has remained uncleared since the neolithic. In the documentary he says he is impressed by what the builders of the circles were able to do without pen and paper, and logarithms. But that without such constructions (as the 'hill of many stanes') 'you can't really do it'.

What was he talking about in this short insert into the documentary? He doesn't explain what the small stones were for, or how they were used. I think I understand now that the field of stones was used to calculate Euler's number, in the context of an engineering construction. That site needs extensive re-evaluation.

Thom's book publications are very plain and not dogged with interpretation. I think he realised that what he could do, and get away with, was to draw attention to the fact that something very interesting and mathematically disciplined was happening in the Neolithic and Early Bronze Age, but the whole thing was just too big a pill to swallow for the academic community. He held back.

One thing that interests me is people's motivations, in particular, which of their psychological needs are being served by engaging in different courses of thought and action. I assume that people have always been curious about life and the world (some more than others, of course!), but what struck me about what you wrote is people's need for or a sense of order and structure in order to feel a degree of safety in a world that is challenging to fathom.

 It depends on where you are in society. Sometimes, as now people are told convenient lies (there is no money!), or circumspect evasions. Ancient priesthoods, because of their picture of the world, understood themselves to be dealing with the nature of reality itself. Neophytes would be chosen from all levels of society, since it was necessary to put a premium on intelligence, in order to join the worlds and make the incommensurate commensurate. Reality itself was the home of all knowledge, and all possibility. You can't deal with that without intelligence. The rest of society would have to make do with what Plato described as likelihoods, because they were too far from an understanding of reality.

Thom was not a classicist or a historian, so he did not know (as most modern scholars still don't) that ancient religion was about *knowledge* (scientia). The ancient priesthoods understood themselves to be dealing with knowledge, and that their activity was a science. That's all changed, but we continue to project modern religious intellectual weaknesses into the ancient past.

Thanks for the photographs.

More later,

Best,Thomas

The Wider Scope of Ancient Mathematics (letter to an American Scholar)

 

Avebury Circle, photographed in 2001

Dear....., 

Hi. I became aware of your short book [.......................]  relatively recently. I wish I’d known it earlier.

I have a strong interest in the idea and function of the concept of limit in antiquity. My main object of study at UCL was ancient  Assyria (mostly the text corpus). Like the Greeks, they had a strong interest in the idea of limit, which is illustrated on the walls of their buildings, and is also represented in their images of the sacred tree. Limit also serves an important function in setting up their gods in heaven (I’ve written about both Assyrian and Babylonian rituals for this).

This tells us something of the actual basis of Mesopotamian religion, which has an origin which is quite different from what we imagine. 

Essentially ancient religions are transcendentalist in nature. In other words, they have their origins in a focus on abstract conceptions (limit, infinity, infinite series,completion, totality, etc). Which makes a nonsense of the idea that the Greeks were the first to grapple with sophisticated abstract thought. Clement of Alexandria created a list of civilizations which practised philosophy, and added the Greeks as the* last* to adopt the practice of philosophy.

Since you might be interested in the wider scope of ancient mathematics, I am writing to you to point you at a couple of articles which illustrate that these concerns were a feature of building projects in Neolithic Britain also. The Horus numbers are there, as the basis of establishing Euler’s number via a geometric construction. Euler’s number being the final result of a convergent infinite series.

Did they get their mathematics from Egypt, or did they develop them themselves? I have no idea. Why Euler’s number? It’s a mathematical stand-in for the extreme limit, which is infinity.


‘At Reality’s Edge’

https://shrineinthesea.blogspot.com/2020/12/at-realitys-edge.html?spref=tw%20%20# (Short article)

‘The Mathematical Origins of the Megalithic Yard’

http://shrineinthesea.blogspot.com/2020/02/the-mathematical-origins-of-megalithic.html (Long  article)

Best regards,

 

Thomas Yaeger

Mathematics and Calculation in Antiquity (letter to a Cambridge Scholar)

Date: Sun, 06 Dec 2020 12:56:04 +0000
To: .............cam.ac.uk
From: thomas yaeger 
Subject: Mathematics and Calculation in Antiquity

Dear........,
 

I’m supplying here the address of an article which may be of interest to you, since a) you are interested in early examples of sophisticated human cognition, and also b) in examples of ideas, languages and cultures being transmitted west to east in deep antiquity. This article addresses both of these areas.

The article took seven years before it assumed its current form. It started off as a relatively minor component in a project on the presence of abstract ideas in the ANE and the Levant before the Greeks, which resulted in my book, ‘The Sacred History of Being’ (2015).

What the article argues is that the mathematics which can be found in the vast majority of megalithic rings in Britain, France and elsewhere, show that builders had a grasp of infinite series and Euler’s number from very early on (late 4^th mill. BCE onwards, up until around 1400 BCE, which is when they seem to have stopped constructing them).

The pattern of their distribution around Europe and the Mediterranean suggests the original builders travelled westwards, and then north to Britain.

One of the reasons why no-one has considered the presence of Euler’s number in these structures (2.72, supposedly first discovered by Bernoulli), is of course, why would they know this number? It is also assumed that the number would have been too hard to calculate in such ancient times, even if they did have a loose grasp of infinite series.

This is not actually the case – it can be established geometrically with a relatively small number of iterations (less than a dozen). Interestingly the procedure for doing this can be found in the Rhind Papyrus, which dates from around the 17^th century BCE, but was originally compiled earlier. In a publication issued by the British Museum in the late eighties, Gay Robins and her husband identified that the Egyptians were working with an understanding of infinite series. And showed the Egyptian diagram, illustrating how it was done.

The geometric process for establishing Euler’s number can be done on the ground, using small stones. I explored the Avebury complex pretty thoroughly in 2001 and 2002, and noticed  brickish sized stones collected together, on the edge of one of the circles, almost lost in the grass. I had no idea why they might be there at the time, but they may have been what they used in the geometric construction . Effectively, the small stones are telling us what the whole structure is for.

 

The article is at:

http://shrineinthesea.blogspot.com/2020/02/the-mathematical-origins-of-megalithic.html

The short book on the Rhind Papyrus is at:

https://www.amazon.co.uk/Rhind-Mathematical-Papyrus-Ancient-Egyptian/dp/0714109444

My book is available from CUL (and elsewhere) in eBook format. 

[text correction, December23, 2020]  

 

Best regards, Thomas Yaeger.

December 6, 2020.

Meaning and Function in the British Neolithic (Writing to Paul Devereux)

Date: Fri, 14 Feb 2020 20:23
To: PAUL DEVEREUX 
From: Thomas Yaeger 
Subject: The Mathematical Origins of the Megalithic Yard

Dear Paul,

Hi. You might be interested in the following blogpost, which looks at why the supposed 'megalithic yard' has the dimensions it has. It takes an entirely different approach to both Thom's surveys and Ruggles later efforts (not statistical analysis, which doesn't do much except expose the general parameters of something which might exist), and which avoids (to a large extent at least), the risk of selection bias. These seem to be the main complaints.

What I've done is to take an entirely new approach, which looks at the megalithic yard as something which serves a function in the context of megalithic structures, and which has a strict mathematical relation to what we already know about these structures (the focus on whole numbers, the use of pythagorean triangles in their construction, and the fact that they are often deformed in various ways, in order to achieve commensuration between the sides of the triangles and the circumference of the circles).

There is a view of reality buried in pythagoreanism, which emerges from the mathematics. This is true both for the later Pythagoreanism of the sixth century BCE, and for the earlier proto-pythagoreanism, since the mathematics are the same, and the interests in the mathematics are essentially the same. That's where the megalithic yard comes from, and I describe this in the post.

I'm afraid the text is as dense as in the paper I submitted to 'Time and Mind' a couple of years ago (it is a tricky subject), but I've kept the necessary mathematics to the bare minimum. It is just under 5k words, so you will need about an hour to digest it.

....

The post is 'The Mathematical Origins of the Megalithic Yard', and is at: https://t.co/BiLRKVq5O1

Hope you are well!

Best regards, Thomas Yaeger

Answers to Questions (Writing to Euan MacKie)

(Photo by Simon Ledingham, May 2005)

Date: Wed, 11 Mar 2020 20:35

To: Euan.MacKie
From: Thomas Yaeger
Subject: The Mathematical Origins of the Megalithic Yard

Euan,

Hi. You might be interested in looking at this article, 'The Mathematical Origins of the Megalithic Yard'  http://shrineinthesea.blogspot.com/2020/02/the-mathematical-origins-of-megalithic.html  

Which I think may be the definitive answer to a number of questions about the construction and purpose of megalithic circles. Obviously this article is subject to criticism, which is fine, and I would be grateful for any comments you may care to make. 

I got to this point over seven years of rumination, and several articles on the Neolithic and patterns of thought in the Neolithic, in so far as they might be inferred from both the archaeological remains, and what ancient writers said about Britain before the Romans arrived.

I was given a classical education at school in Edinburgh (minus Greek literature), and a wider education at UCL later, where I studied Rome, Greece, and the Greek language. As well as Mesopotamia, Egypt and other cultures. My particular interest has always been Greek philosophy. Eventually I found my way back to an interest in British prehistory. I was struck by some of the things which Alexander Thom found through a phenomenological analysis, about the mindset of the Neolithic architects, because they echoed ideas which are commonplace in later Greek philosophy (the importance of the idea that reality itself is necessarily unchanging, meaning the idea of the 'One'; and of Totality, and the importance of commensurate values, and the significance of the fact that commensurate values are sometimes lacking in the physical world, etc.). I've written extensively about the Pythagoreanism of the 1st millennium BCE. Much of which came from the ANE, during Pythagoras's travels. Mainly, but not exclusively from Egypt. It is a technical substrate of Egyptian religion, which Pythagoras imported into his view of the world, after (reputedly, according to the neoplatonists) twenty years of study in Egypt. Meaning that the pythagorean perspective is older than Pythagoras himself, and possibly of immense age.

What we have in the stone circles of the British Isles, is just such a technical substrate of ancient religion, written in mathematics and geometry. Personally, I think most religions got started this way, though we are a long way off from being able to say this for sure. It is not however an argument that is considered at all at the moment in archaeological circles. I think it should be considered, even if only to finally eliminate it for rational consideration.

[Other materials relevant to this article can be found by using the search box on my blog ["neolithic" will pick most of them up].

Best wishes, Thomas