## Introducing Neolithic (Stone) Circle Mathematics

**Introducing Neolithic (Stone-) Circle Mathematics **

**Measuring circles**

It will be difficult to convince anybody of what I am going to put forward here I suppose, because it is so completely contrary to the image we are bound to have of the people of the late Stone Age, and of their state of mind. In general we tend to think that peoples that could not write cannot be very sophisticated whatsoever. The term Stone Age itself has a ring of primitiveness about it. This whole website is an exercise in trying to overcome prejudices regarding the state of peoples knowledge and insights (also regarding my own).

What follows is a reconstruction of the state of Stone Age people’s knowledge based on the ratios they expressed in their megalithic works and architecture, and especially those of the Maeshowe Grooved Ware culture in Orkney, roughly 5000 years ago.

It will be shown that given a unit of measurement based on a bone of the male forearm, the ulna, the designs of Stenness, Brodgar and Maes Howe, can be translated into specific mathematical ratios that fit into a natural number calculus relating the geometric interaction of circles and squares. The square and its relation to the circle may have been a revolutionary new insight in those days.

Both the Stones of Stenness and the Ring(s) of Brodgar deal with the problem of the ‘irrationality’ of square root 2, the fact that the diagonal of the square did not relate to its side in a simple whole number ratio. This ‘practical’ problem Stone Age man seems to have solved brilliantly in practice by creating two material units of length which had the **ratio √2 : 1,**** besides finding the big numbers where the approximation is virtually correct (198:140=1.41428…)**, similar to what the Egyptians did at about the same time in their well established units of the **Royal Cubit ****and the**** Remen.** This ‘invention’ of two units made it possible to express all their measures in ** natural numbers**.

In the chapter **‘the Bone Measures’ **it is explained, but here it may suffice to state that a standard unit, the Megalithic Ell (ME= 0.5236m), was **2 (male)**** Megalithic Ulna **(MU=0.2618m, ulna = underarm bone) and that a (female) **Megalithic Fibula **(MF= 0.37025, fibula = calf bone)) was **1 Ulna x √2.
**

**I don’t mean to say they had a concept of square root, but in practice they could make a right angle with ulna bones and connect them with a fibula bone to make a triangle (see extensive use of this in design Maeshowe), as they could make a right angle with fibula bones and connect them with 2 ulna bones, and create the standard length of 1 Megalithic Ell = 0.5236m.**(Maybe they never had the concept of the double ulna as Ell, but that does not really matter, it is more convenient as length, Thom’s megalithic yard was also the double of an original length he had found in his statistics)

At the **Stones of Stenness **we see the equation 70 Megalithic Ell = 99 Megalithic Fibula, at the Ring of Brodgar it is 198 Megalithic Ell = 280 Megalithic Fibula.

These two ratios approximate square root 2 as: 99 : 70 = 1.4 14 28 57 14… and as 280 :198 (140:99) = 1.4 14 14 14……., whereas the irrational √2 = **1.41421356** 2 37… , which is practically ‘exactly’ in the middle of these two ratios (99/70+140/99)/2= **1.41421356** 4 213564 21…, a difference of 0.00000000184 which in a world of megaliths, or in any world except mathematics or high technology for that matter, is trivial and of no consequence. Nevertheless the inconvenient big numbers 70, 99, 140 and 198 give every reason to use such a handy unit as the ‘remen’ and to conserve its derivation in huge stone settings for posterity .

When we analyse these two ratios, which are better expressed as 99 : 70 and 140 : 99, then we see which factors they are made of and that is, indeed again, **9×11** and **7×10**, which numbers are in the **two basic ratios ****that** their whole geometry was based on, that is-> **10 : 9** and **11 : 7 **; and it is easy to see why.

By setting out circles in the field, as we very well know they did, they had as a matter of course discovered that if you divided a circle’s diameter by 7 you got a unit length which exactly fitted the arc over this basis, the half circle, in a whole number, that is: 11 times.

The next step they took, and here is where the Maeshowe culture comes in, is that they also found that if you put a square in the circle, there was no ready unit that could be found to fit both diameter and sides of squares. What they did find though is that if you divided the side of the square inscribed in the circle by 9 and you added 1 (=10), that then you get exactly the length of the arc over the side of the square, the quarter circle. So from these two empirical facts: 11 over 7 for circle and diameter and 10 over 9 for circle and square circumference, they eventually found out that these two ratios would meet where they merge, which is at 99 : 70 (9×11 : 7×10) (because** 70√2 = 98.998…., **which is virtually **99)**, so these two are the pivotal ratios of the *approximation calculus* and the geometry we found, which are based on the **first 5 primes**, since 10 = **2×5 **and 9 = **3×3**, plus **7** and **11**. (**primes: 2, 3, 5, 7, 11**)

We shall see that we will find the quintessential numbers of this natural number arithmetic cum geometry back in the ratios of the circles, because from now on it must be clear that what distinguishes a henge from a stone circle, mathematically, is that a henge has in principle rather more circles and symbolises more ratios (platform diameter, ditch width, embankment width, maybe middle ditch and embankment width + a stone circle proper, if present).

Before we can start to apply our geometry to the plans, which are drawn by the archaeologists who conducted excavations at the sites, we have to get an idea of how to deal with the data. Archaeologists differ in their assessments of diameters because it usually is an average of, in general, not-perfect circles, and there is the perennial question of the inner face or the centre of the stone as marker. One has to keep this in mind in appreciating the approximations in the calculations, absolute precision here is not attainable, there is always a kind of arbitrariness, but this may have been true also for the circle-setters themselves and it is, in a sense, inherent in the model and the methods.

The differences in the data of the monuments we focus on here, are often not more than a few centimetres, but still these can provide ammunition for great debate when a serious mathematical or other claim is made. This is notoriously the case with the data of the Pyramids in Egypt, and will inevitably play a role in the judgement of the model that is proposed here. The question is what margin of precision is reasonable to demand and in character with the material, before the theorem is judged as:** unlikely, possible, highly probable, or: proven.**

Elements of this mathematical model have been successfully applied to the **geometries of Giza **(see impressive work by** John Legon**) but it has not been applied by anyone to the other geometries, that of the stone circles and the megalithic chambers, as far as I know.

Well founded claims as these made here are bound not to be taken seriously by the scientific community, partly because of the notorious nonsense that has gathered around the pyramids over time, but mostly because **that**** level of sophistication in antiquity **is just not contemplated by most scientists (they just don’t believe it, period).

This is the same with the Pyramids; serious surveyors as Flinders Petrie have from the beginning pointed out that the 11:7 ratio was involved and later also the 10:9 ratio was highlighted by Lauer, but it is still not accepted by the archaeological, let alone the wider scientific, establishment.

On a public lecture I attended on the history of Pi, the Pyramids were not even mentioned, and, on questioning, treated as a possible footnote, whereas probably one of the main incentives to build the pyramids as they are built was to preserve the knowledge of Pi as 22/7 in the Great Pyramid, the Pythagorean theorem 3:4:5 in the Second Pyramid and the ratio 280: 198 (140:99) as square root 2 in the Third Pyramid.

When people in antiquity have gone to such great lengths to preserve such precious mathematical knowledge and you choose to ignore it with your ‘sceptical critical modern mindset’ as a scientist then in my view you are in an important sense not up to your job.

It is ridiculous, but any proof in this archaeological area can be rejected, although the present day obsession with precision cannot be demanded; and then, when there **is precision**, it is rejected as **accidental**. The point here of course is, and I say this again and again, that what matters is what they intended, not the precision of the execution. When we know as fact that the Egyptians had standard units of measurement, the Royal Cubit (RC) and the Remen, and when we know that the sides of the Great Pyramid come to within centimetres of 440RC and the height to 280RC so a ratio of 11:7, and you know that the perimeter is equal to a circle of circumference of 1760RC (4×440) which has a radius of 280RC, then you must be an incurable blockhead not wanting to admit that to all probability this mathematical theorem is meant by such a design, sorry. Do you want it in black on white, then? Not in stone?

I would like to emphasize here again that, whether this model has indeed been used by megalithic man, as I tend to believe and think to having proven in several places, this is of no consequence to the validity of the model itself, because it is a logically rigorous mathematics in its own right, whether this is accepted by established mathematics or not, let alone by archaeologists.

There is nothing fabricated about the model, **once the ***natural number rule*** is accepted**, because then everything follows clearly from the numbers themselves, that is the great thing about it and that is why I believe they were capable of finding this out in the field, experimentally, empirically, and started to put things together. I think this is how mathematics began in the Stone Age, with whole numbers and as an abstraction from the empirical data, and, maybe more importantly, I believe that they could find this* because I could*; because when I can do it, they could do it.

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**Brodgar and Stenness**

Here we set out to describe how the model works in practice, because it is a model originating and well suited to the work with in the field.

The crucial data in this approach are diameters, because they define the circles and may also define the numbers of stones, given a supposed mathematical theorem.

One of the best spaced stone circles ever is the Stones of Stenness in Orkney, but its circle is flawed and one of the most enigmatic is the Ring of Brodgar, a stone throw away, where the spacing is irregular, but I believe that in the specific returning width of different spacings there is mathematical regularity, anyway here the circularity is perfect.

The pivotal diameter on which the whole argument is based is the diameter of Brodgar and on that basis we lay down the rules of how to apply and interpret the data.

The Ring of Brodgar has been measured by many, but the most authoritative surveyors are Aubrey Burl (103.60m), Colin Renfrew (103.70m) and Alexander Thom (103.625m = 125 Megalithic Yard)

We see that these experts agree within 10cm (103.60 – 103.70) and this means for us that any value in between is acceptable. In our model the value hovers between 103.65 and 103.68 , which depends on the type of factor which is chosen, but because there are no uses of **√2 **and** ****π**, everything is calculated with 22/7 for Pi and the other rational numbers that govern this model, as **10/9 – 11/7 – 14/11 **and the specific unit-values. This entails that there are a few centimetres difference in our calculations, but in reality they are of no consequence because of the dimensions of the large, to very large, stones.

When for whatever reason a huge stone got set deeper or not exactly at centre, there was little they could do to correct it, what counted was the precision of the mathematical principle and the intended design, its order and its practicality, not the kind of precision of execution demanded or expected by some unreasonable critics 5000 years later. For them mathematics was not something of absolute precision as it is today, to its own detriment as will be shown elsewhere, no, it was in the first place a tool and a wonderment, a gift from Heaven or as some say a ‘sacred geometry’.

There seems circumstantial evidence that large ‘imprecision’ of the settings means it was done on purpose and would have mathematical or acoustic reasons, as is probably the case at both the Stenness and Brodgar circles and the standing stones inside Maeshowe.

The point is that when more and more features sustain the proposed model, then there comes a moment that one must infer that mathematically something can be expected in terms of recurring ratios and it can be shown that indeed the mathematical model fits the most important of the megalithic structures once we have accepted that minor approximations are part of the heart of the model.

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See for a further discussion and explanation ‘The Neolithic Numbers’ and ‘Neolithic Numbers Explained’ and Neolithic Bone Measures.