Sacred Number

Since I went to Brittany this journal has been neglected. However, some important facts emerged for me there and I have taken one of them out my writing since in the hope that someone might find it interesting (or indeed reach the end). It should be illustrated but at least I have gone over it a few times to reduce ambiguity. There is a more general work underway and I hope there are sufficient references within this text to make sense. 

Brittany or “little Britain”, is a large, nearly peninsular, land mass that forms the western-most portion of modern France. In the last two millenia it was occupied by the Breton culture, in part from Wales, to be a Celtic language area. Geologically it has similar roots to Cornwall, Dartmoor and other areas in southern England that make up the same Granite massif. Thus, it truly is like a little Britain and in Megalithic times it was arguably the strongest, concentrated region of megalithic activity, certainly comparable with the island of Britain that was once connected to France across a massive river valley where the English Channel now is.

Carnac is located on the south-facing coast of Brittany, on the same meridian as Edinburgh and has one of the most unique topographies found anywhere associated with megalithic works. The complex of monuments these form is extensive, with alignments to other sites within Brittany. As well as its meridian and topography, it is at the unique latitude on the Earth at which the solsticial sun, summer and winter, forms a perfect pythagorean triangle relative to the parallel of latitude, that is to the east-west, equinoctal axis of the site. In turn, this 3:4:5 triangle, the first of the Pythagorean triangular set and derived from the Golden Mean (see later), is implicit within the structure of the Bay of Quiberon, formed by a north south peninsular and an approximately east-west coastal region. This implicit character was clearly made explicit by the megalith builders who built a large menhir, possibly the largest ever raised, at the eastern end of this triangle as a foresight from a backsight atQuiberon, on the tip of the peninsular, evidently to observe the mid-summer sunrise.

Alexander Thom, in a matter of just a few summer seasons of work, found many further examples of such alignments so that the entire complex at Carnac (a small village at the centre) provided many sightlines to solsticial sunrises and sunsets, both for winter and summer. There is further evidence of other alignments from monuments such as from the Tumulus of St Michel on to small island longsights to both east and west. Thus, with great confidence, the Sun’s solstitial behaviour was captured there, to high accuracy, according to the geometry of the 3:4:5 triangle.

Thom also found alignments to the Moon, naturally either side of the solar ones, according to the motion of the lunar orbit that precesses over 18.6 years to give times when the Moon exceeds the ecliptic (a lunar maximum standstill). 9.3 years later is oppositely below the Sun’s path (a lunar minimum standstill). In between these extremes the Moon is closer to the ecliptic during the extent of its orbital motion seen from the Earth. These lunar standstills prove even more interesting than the Solar 3:4:5 geometry, yet they formed a much more complex challenge to megalithic astronomers than the Sun did.

Squaring Up on the Moon

Further work by Howard Crowhurst [Crowhurst, 2007], a resident with over 20 years experience of working with these monuments and alignments, has revealed a “rule of thumb” for lunar alignments for the maximum and minimum standstills at Carnac. Namely, that the maximum standstill, viewed on the horizon, closely approximates to the diagonal of a simple square and that the minimum standstill closely approximates to the diagonal of a double square. Further to this he found that some of the monuments within the complex, at the alignments, had been rendered into an elongated, east-west, design. For instance the tumuli tend to have an east-west ridge that allows the deviation from ideality to be offset through observing from different points on the flat elongated platform atop these tumuli.

The basic side length chosen by the builders can be deduced from the 3:4:5 triangle built around the topography of the Bay. I found that this was in units of 11025 feet, a figure arrived at from the extrapolation of an unknown but canonical ancient measure of 35/32 feet long, a length just one part in 8000 longer than the present meter [Heath, Richard, 2007]. This unit multiplied by the standard canonical transform of Neal and Michell yields a foot 1.1025 feet long, 10,000 of which would form a length 11025 feet long. Thus the ideal square proposed by Crowhurst for the maximum standstill alignment is 11025 by 11025 English feet in practice. Two of these squares then form a rectangle, east west, forming nearly the minimum lunar alignment across the two available diagonals.

However, as stated, the alignments are slightly different from this very simple situation and the question is by how much. The reader and the researcher need to know exactly which type of phenomenon on the horizon to use, a full disk visible on the horizon, half a disk or just any little sector of the Moon showing above the horizon. The angles involved in azimuth (clockwise from north) are shown in the table below for Summer sunrise and Moonrise either side of this, at the lunat maximum and minimum standstill:

Another issue is how the alignment is then being measured, not as an angle from the viewing position but instead as a triangle with a side running north=south and the other running east-west, obviously a right angled triangle forming half of the square or rectangle involved. This way of measuring angles and of expressing an angle has known roots in the mathematics of Egypt that has fortunately survived on Papyrii, notably the Rhind Papyrus. In Egypt angles were often just the tangent fraction, opposite length over the adjacent length, rather than what we call degrees, namely the division of a circle into 360 parts, unless one was measuring angles in the sky, especially along the ecliptic or celestial equator – where one had difficulty building a large, free form triangle.

Using metrology for the creation of such tangent triangles, to achieve landscape alignments of high accuracy, appears to be a very likely yet new-to-us application for the ancient metrology. When I visited the Carnac area on the summer solstice of 2007, Crowhurst’s book was also just published and I was able to start calculations to see how metrology would work to define the slightly varied-from-square alignments found there by him.

The Lunar Maximum Standstill Alignment

Using Table 1 above, the half disk would appear on the horizon at lunar maximum at an angle to east or west of 45.59 degrees. If the north-south side is 11025 feet long then the east-west side of the triangle would be 11025/ tan(45.59) = 10799.4 feet long, effectively 10800 feet. Amazingly, this rather perfect result was the first attempt at trying to define what such a triangle might measure on its east-west side. It is amazing because the Drusian foot is just 1.08 feet in length, making the east-west side for this alignment 10,000 of these whilst the north-south side is, as stated above, 11025 feet long, 10,000 of the 1.1025 foot measure already identified as present at Carnac and used to create the basic grid and 3:4:5 triangle of the Bay alignment to the solsticial Sun.

Another measure found at Carnac, by Alexander Thom, was the megalithic yard he deduced from his surveys of many of Britain’s stone circles and other megalithic monuments. What is little known about the megalithic yard is that it derives most naturally from the Drusian foot [1] . However, unknown to Thom, the megalithic yard was actually not a yard of three feet but a “step” of 2 ½ feet. When so considered it is one part in 175 different from the step of Drusian foot (length 27/25 feet). This reveals that the Carnac “module” of measures has a simple rational relationship to the Drusian “module” and that this relationship is embodied in the sides of the triangle for the Moon’s maximum solsticial alignment, that is in the tangent of this angle.

The difference between the two sides is 225 feet (11025 – 225 = 10800). If this difference is divided into 10800 the result is 48 and hence 11025 has 49 of these units contained within it. The triangle for this alignment, as calculated, is therefore a 48:49 triangle with 48 units east-west (adjacent side) and 49 units north-south (opposite side).

The Carnac foot of 1.1025 feet is also 49/48 of the Drusian foot of 1.08 feet, and in this one can see that the Carnac foot contains 49 or 7 squared in its formula, which can be written as 441/400. 441 is 3 squared times 7 squared and so the square of seven can be removed, whereupon 48, which is 3 times 2⁴ will add to 3² above and remove the 2⁴ in 400, leaving 25, resulting in 27/25, the Drusian foot.

Finally it seems probable that the sides were not viewed as 10,000 feet long but instead as 4,000 steps of 2 ½ feet in length given that the megalithic yard is in fact 2 ½ Drusian feet long and was found in, for example, the alignments of stone rows and other buildings in the Carnac megalithic complex.

[1] *This root foot, like all ancient measures, is a rational fraction of the English foot, namely 27/25 feet long and it has the fomula of three cubed over five squared, or 3³/5².