What proof?

On several occasions I was told by people that it was all very interesting and well thought-out, but that this was of course no proof that the use of this math and these measures really had taken place.
So what am I to answer there?

The question which comes before the question of an actual valid proof is: what do you consider as a proof under these circumstances, or do you think that it is simply never possible to prove any of these things, conclusively, anyway, in which case whatever one tries, it is trivial, because it can never be a sound proof.

People who reason that way will never be satisfied, because once the predicted measure is a few centimetres off the actual data they will say that it is no proof, but once it is spot on they will say it is coincidence. This is not what they think it is: a critical scientific attitude, what it is, is: scepticism resisting evidence by all possible means.

Solid proof?

In Stone Age archaeology it is hard to find any solid proof, unless it is an artifact and in that sense especially pottery can be a proof of something, this is why pottery is so highly valued in archaeology, because a style belongs to a period and does not suddenly turn up in another period, it relates things. Crafts belong to traditions and go from one generation to the next without changing, sometimes for considerable periods, see arrowheads, another priced piece of evidence in archaeology.
Masonry is also a craft, and architecture has styles and these styles do not return some few hundred years later, they also belong to a period, a tradition, and then are superseded by something else. A building, even a stone circle, is also an artifact, something made by humans.

Certain buildings and certain circles have definite measurements, the more so when they are square or rectangular and when their measures return, especially in other buildings. A measure is practically never a straightforward thing; even in the most sophisticated scientific experiments the official data are averages of many measurements.

In megalithic building we have the additional problem of natural building blocks which are never the same and of dimensions that baffle anyone who has the slightest idea of building or weight. So measuring is difficult, especially for instance a stone circle. But what we in many cases must assume is that they definitely measured and then you need a unit of length. So even if it may never be proven conclusively some people are not daunted by that prospect because they know there must be something provable by statistical inference. The whole of quantum mechanics is built on that.

Probability in archaeological theory

So what we must do in archaeology is find as many instances as possible and then look at the probability of some specific units, so also in archaeology, when it comes to these kind of artifacts, we can do no better than measure in degrees of probability, a scale like:
highest, high, probable, possible, unlikely, improbable, impossible,
which you could then put in percentages, or the other way round.
[If such a system of probability scales were used for different archaeological theories with polls every year, or two, under professionals and under readers of archaeological journals, that would change the face of archaeology and would pull it free from the unscrupulous mercy of the media.]
I think my system has a high to the highest possible probability, given that I can add another four major megalithic works to the 6, Stenness, Brodgar, Maes Howe, Stonehenge, Avebury, Newgrange, I have already analysed. These others are:
Structure-1 at the Ness of Brodgar, an outstandingly precise building, the best one could ever hope to find and my measures prove strikingly adequate as I show in ‘Math at the Ness’, they constitute a nice proof of the Megalithic Ell, the method and the system.
Then there are the circles:
Stanton Drew near the Bristol Channel, a complex of three circles, of which one the biggest stone circle after Avebury, next the Merry Maidens, a famous circle in Cornwall and eventually, in the north-east of Scotland, the ‘original’ Clava cairns, three circles with cairns, part of a larger group of identical places with a ‘scientific ring’ about them, like the ‘recumbent stone circles’, (level flat stones) also in NE Scotland, which will be added later on. All these circles tell their own story with their own numbers.

What I show is that : a) a set of measurement units, which have a geometrical relationship in whole numbers, can succesfully be applied to the dimensions of the respective buildings and circles, b) that the ratios of the different outcomes make mathematical and/or calendrical sense and c) that these results taken together cannot be deemed coincidences on statistical grounds.

Communication of Stone Age people

So what we need to begin with is show that our data have a logical and mathematical coherence and that it is even possible to make sense of them, which implies that they have a meaning. This in turn means that we communicate in abstract terms about abstract eternal ideas, that is mathematical proportions, with people 5000 years ago. And that, I think, is extra-ordinary and could, maybe, only be achieved by logic and mathematics.

The only abstract ideas of stone age man we have are the dimensions of his buildings, be it a stone circle, its number of stones or the proportion of well planned buildings like Quoyness, Quanterness, Maeshowe and now at least Structure 1 at The Ness of Brodgar, but the idea seems to be stubbornly resisted by the professionals.

An additional problem is that mathematical evidence may be impaired, because the excavators are not looking for it.

My simple postulate (self-evident starting point) is that late Stone Age man did build according to plan and to custom and eventually according to complete design which they were capable of executing to a high degree of accuracy. All this could not have been achieved without a probably growing tradition of building with one or more standard units of length. So another postulate is that they must have had at least one standard of length to begin with, otherwise it would make no sense here trying to figure anything out, because nothing could be put into numbers. (‘meten is weten’, we say in Dutch, ‘measuring is knowing’)

Our measurement of a diameter of a stone circle cannot be decisive, because of the often huge differences in stone size and shape and because the placing is not always accurate as regards a proper circle. This easily occurs when you pin down a point and start digging there, then after a while your point is empty space and your dig may easily shift its centre. This often happened no doubt. This is one of the reasons circles are never ‘exact’ and very understandably so, given the huge holes to be dug, in shifts, and the nature of the building blocks, but still we can be sure at the big circles, the major works, that there was a measure, just because there is usually a rather proper circle.


Most initial measures would have been a whole number times our standard unit(s) and the question is then: can we discover a consistent system in the relationships of those uncertain data. Our uncertainty is caused by the great variety of building blocks of which never one is the same, so the only certainty we go from is that they indeed did measure their circles and their buildings and that by doing so they had certain numbers in mind and communicated them.

And this is of course what we are after, the numbers they had in their heads, the design and its use and its possible meaning, not whether they were able to perfectly execute their intent, because that is a matter of manual skill, technique and material. Ask any architect about the battles s/he has to fight with the contractor or the skilled builders who have their ways of doing things and resist doing them otherwise.
We can see that the Maeshowe culture developed architecture and eventually worked by complete design and that is a sign of the highest spiritual importance.


As a Veda scripture says, ‘The harmony of heaven is expressed in the proportions of the temple’ and ‘when there is order in the design of the temple, then there will be order in the universe’, that is an attitude we find everywhere in antiquity, so why not in Orkney. It also meant the huge responsibility to get it right because mistakes could be detrimental to Heaven.

Let me try to make clear how I came to start measuring the data, the plans and eventually the realities of the megalithic works on Orkney in the first place.

Independently I had developed what could be called a geometrical natural number logic, a concise system of whole numbers that expresses all relations between straight lines, squares, diagonals, perimeters, circles, diameters and circumferences etc. in whole numbers, which turn out to be very close and logically consistent approximations, pure arithmetics in relation to geometry, though of a special kind. The approximations never exceed the thousandths and are usually in the ten-thousandths of unit, trivial when it comes to building.

This system happened to be the basis of the geometry of the Great Pyramid as I found out to my great surprise. (You can find the proof of that elsewhere on this site, Numbers explained)
My subsequent research into the Pyramids resulted in my own keys to translating the measurements from inches, feet and metres into the ancient Royal Cubit of just over 52 centimetres (52.36cm). ( The Royal Cubit’s existence is fact, not its exact measure, its hieroglyph is the forearm with hand)
It all convinced me that the design of the Pyramids was thoroughly mathematical.

I further developed the model by applying it to mathematical and astronomical questions and found astonishing harmonies. This of course strengthened my belief in the valididty of the logic I had unearthed.

In Orkney out of curiosity I had divided the diameter of Brodgar, 103.60 m (Burl), by my measure for the Royal Cubit which was .5236 m , this resulted in 197.860.. , but it did not immediately occur to me this is very close to 198, twice 99, until I found another diameter measure by Renfrew of 103.70m (198.051…) and realized that 198 x .5236 = 103.67 m, is just in between the two data, of these unquestioned authorities.

Next I managed to make measurements in Maes Howe and found the basic square was 9 of my cubit lengths squared, 81, and this was the very heart of my mathematical model and discoveries.
From that moment I started to think the unimaginable, that is, that they shared both the unit of measurement and the mathematics with the Egyptians, possibly even as contemporaries.

This improbability, which threatened the serious status of my work, eventually worked out very fruitful once I had found that this unit of measure in Orkney could be traced back to the length of the male ulna bone of the forearm and out of this emerged an ingenious system of geometrically related bone-units. (see Neolithic Bone-measures)
The rest is no more than again and again testing the official data against my bundle of measurement units and in nearly all instances the results are resounding and produce a very coherent picture of meaningful scientific endeavour over a considerable period of time.

I dare say that the geometrical nature of the geological building blocks in Stone Age Orkney may have had a decisive influence on the mathematical outlook of the builders of the chambers, that is: working with straight parallel lines, with rectangles and eventually the discovery of the square and its peculiar relation to the circle.

The mathematical sophistication which developed in Orkney, in close relation with architecture and cosmology, found, maybe hundreds of years later, its most daring expression and statement in that temple of heavenly proportion and cosmological science, that Stonehenge was probably meant to be.

This is what we can read if we understand the mathematical message which is in many different ways expressed in the designs of the major Neolithic works in Britain.

Mathematical meaning

The strength of my arguments lies in the fact that I can show that all the different designs are based on mathematical theorems and are even interrelated, sporting the same measurements criss-cross the country.
The convincing point is that the data consistently reveal the use of crucial numerals accompanied by certain geometries and related aspects of the overall theorem; there is definite mathematical meaning in them; a geometry may express a certain numerical order without using symbols for the numbers.

The translation of the Egyptian hieroglyphs of the Rosetta stone was not the final proof of the correctness of the decoding system, that proof came when all subsequent translations proved to be consistently meaningful.
It is this meaningfulness and consistency by which we judge the power of a decoding system and translation.
Just because all the numbers and proportions that I find are meaningful and consistent, this shows me beyond reasonable doubt that I have found the keys to Stone Age arithmetics and geometry; and to its metrology.

The proof of this theorem and its Stone Age use is hidden and comes to light in their works, it cannot but be circumstantial evidence which sustains it, but this evidence can become so strong that it is ‘beyond reasonable doubt’. And even when there is doubt we owe these skillful builders ‘the benefit of the doubt’
It is true that this is not a mathematical criterion, but that is not the kind of proof we need in archaeology. There are no theories in archaeology that are sustained by such rigorous unambiguous evidence as this number-logic model and the metrology I present. For this to be refuted one has to change the data.

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