## Early Numeracy at Table des Marchands

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Table des Marchands, a dolmen at Lochmariaquer, can explain how the Megalithic came to factorise 945 days as 32 lunar months, by looking at the properties of the numbers three, four and five. At that latitude, the solstice angle of the sun on the horizon shone along the 5-side of a 3-4-5 triangle to east and west, seen clearly at the Crucuno Rectangle. But in the dolmen's carving these numbers could have been put to another good use.

Before numbers were individually notated (as with our 3, 4 and 5 rather than |||, |||| and |||||) and given positional notation (like our decimal seen in 945 and 27), numbers were lengths or marks and, when marks are compared to accurately measured lengths measured out in inches, feet, yards, etc. then each vertical mark would naturally have represented a single unit of length. This has not been appreciated as having been behind marks like the cuneiform for ONE; that it probably meant "one unit of length".

*Figure 1 The end and cap stone inside the dolmen Table des Marchands in which the elementary numbers in columns and rows perhaps inspired its local attribution to the accounts of merchants*

In the carvings of the end stone (C4) of Table des Marchands, groups of crook shaped lines were created in which the crooks point away from the central axis of a stone whose top section is an oblate isosclese ("equal legged") triangle whose base angles are 60 degrees, within which the array of grouped crooks appear. The outer oblate edges are carved in a grooved border which looks like a long count. Below the triangular section is a rectangular panel of detailed drawings and/or symbols whose meaning is now probably lost forever.

## Comparing Erdeven and Le Menec Alignments

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In the previous article, the 7,500 foot-long Erdevan alignments were seen to have been a long count of the Saros period of 19 eclipse years versus the distance to Mane Groh dolmen of 19 solar years, this probably conceptualised as an 18-19-6 near-Pythagorean triangle, whose inner angle is the bearing from east of Mané Groh. However, the path directly east caused the actual alignments, counting the Saros, to veer south to miss the hill of Mané Bras.

It has been remarked that the form of the **northern** alignments of Edeven were similar to those starting at Le Menec's egg-shaped stone circle 4.25 miles away, at a bearing 45 degrees southeast. Whilst huge gaps have been caused in those of Edeven by agriculture, the iconic Le Menec alignments seem to have fared better than the alignments of Kermario, Kerlescan and Petit Menec which follow it east, these being known as the Carnac Alignments above the town of that name.

One similarity between alignments is the idea of starting and terminating them with ancillary structures such as cromlechs (stone kerb monuments), such as the Le Menec egg and, despite road incursion, a 3-4-5 structure similar to Crucuno, aligned to the midsummer sunset by a length 235 feet long. This is the number of lunar months in the 19 year Metonic period and is factored 5 times 47. Another similarity may be seen in Cambray's 1805 drawing of these Kerzerho alignments, at the head of ten stone rows marching east (figure 1).

*Figure 1 Cambrey's 1805 engraving of Kerzerho's western extremity of the Erdeven alignments showing the stone rows now lost to agriculture.*

## Erdeven Alignment's counting of Metonic and Saros Periods

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The word Alignment is used in France to describe its stone rows. Their interpretation has been various, from being an army turned to stone (a local myth) to their use like graph paper, for extrapolation of values (Thom). That stone rows were alignments to horizon events gives a partial but useful explanation, since menhirs (or standing stones) do form a web of horizon alignments to solstice sun and to the moon's extreme rising and setting event, at maximum and minimum standstill. At Carnac the solstice sun was aligned to the diagonal of the 4 by 3 rectangle and maximum and minimum standstill moon aligned to the diagonal of a single or double square, respectively.

It seems quite clear today that stone rows at least represented the counting of important astronomical time periods. We have seen at Crocuno that eclipse periods, exceeding the solar year, are accompanied by some rectalinear structures (Le Manio, Crucuno, Kerzerho) which embody counting in miniature, as if to record it, and it has been observed that cromlechs (or large stone kerb monuments) were built at the ends of the long stone rows of Carnac and Erdeven. Sometimes, a cromlech initiated a longer count, with or without stone rows, that ended with a rectangle (Crucuno). The focus on counting time naturally reveals a vernacular quite unique to this region and epoch. We have seen that the Kerzerho alignments were at least a 4 by 3 rectangle which recorded the 235 lunar months in feet along its diagonal to midsummer solstice sunset. After that rectangle there follows a massive Alignment of stone rows to the east, ending after 2.3 km having gradually changed their bearing to 15 degrees south of east. Just above the alignments lies a hillock with multiple dolmens and a north-south stone row (Mané Braz) whilst below its eastern extremity lies the tumulus and dolmen, "T-shaped passage-grave" (Burl. 196) called Mané Groh.

Read more: Erdeven Alignment's counting of Metonic and Saros Periods

## Another Rectangle near Kerzerho

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In 1973, Alexander Thom found the Crucuno rectangle to have been "accurately placed east and west" by its megalithic builders, and "built round a rectangle 30 MY [megalithic yards] by 40 MY" and that "only at the latitude of Crucuno could the diagonals of a 3, 4, 5 rectangle indicate at both solstices the azimuth of the sun rising and setting when it appears to rest on the horizon." In a recent article I found metrology was used between the Crucuno dolmen (within Crucuno) and the rectangle in the east to count 47 lunar months, since this closely approximates 4 eclipse years (of 346.62 days) which is the shortest eclipse prediction period available to early astronomers.

*Figure 1 Two key features of Crucuno's Rectangle*

## Learning Metrology's approximations to Pi at Crucuno Rectangle

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The megalithic viewed time, like the time-counts they made, as a line; whilst the circle of the horizon, and the events upon it, was viewed as a cyclic domain of "eternity". The line and the circle presented in monuments as counted lengths and alignments to key events appear harmonised in the Crucuno rectangle in which the southern stones extend the 4 by 3 rectangle implied by solstice sun events on the horizon aligned to its diagonals. The southern stones change the north south dimension of the rectangle from 3 units to 22/7 = 3 plus 1/7^{th }units.

*Figure 1 left: The countable line of time seen on the circle of eternal events right: transposed into the rectangle at Crucuno*

The 22/7 value approximates π (Pi) at around 3.14 quite well (about one part in 2500), and π needs approximation since (a) its fractional part is indefinitely extensive whilst (b) the megalithic only used rational fractions like 22/7. In previous articles I have developed the term **Proximation** in explaining how skilled the megalithic were at approaching otherwise impossible problems given their simple use of numbers as lengths. Lengths in the real world are measured in specific units of measure and this duality of unit and measurement enabled changes of units, themselves rational to one another, to create alternative measures of the same thing hence **inventing metrology as a calculational device**. Since Pi is irrational, transfinite and unmeasurable, in how circles relate to their radius (or diameter), then the megalithic developed many approximations to Pi and even combined them to good effect within the micro-variations of their rationally connected modules of units, their growing "tool kit" of measures and geometrical methods. I noted in a previous article that: the square of equal perimeter to the circle, made by the rectangle's 4 side as diameter, contains 125 megalithic yards that are 176/175 larger than the 40 x 2.7 foot Drusian steps (a.k.a. megalithic yards) of the diameter - hence relating the diameter's straight line to its circumference, maintaining an integer measurement of both in slightly different units. *This can only be achieved using two Proximations to Pi which can "bracket" the irrational using two rational approximations*. By seeing how this was achieved, the guiding principles behind ancient metrology become clearer, especially regarding the microvariations within its modules, and Crucuno is the perfect exemplar of metrological cunning.

Read more: Learning Metrology's approximations to Pi at Crucuno Rectangle

## Lunar Counting from Crucuno Dolmen to its Rectangle

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*Figure 1 The entrance of Crucuno's cromlech, which opens to the south-east [Summer Solstice, 2007]*

It is not immediately obvious that the Crucuno dolmen (figure 1) faces the Crucuno rectangle about 1100 feet to the east. Dolmens appear to have been used to mark the beginning significant time counts. At Carnac's Alignments there are large cromlechs initiating and terminating the stone rows which, more explicitly, appear like counts. The only (surviving) intermediate stone lies 216 feet from the dolmen, within a garden and hard-up to another building, as with the dolmen (see figure 2). This length is interesting since it is twice the longest inner dimension of the Crucuno rectangle, implying that lessons learned in interpreting the rectangle might usefully apply when interpreting the distance at which this outlier was placed from the dolmen. Most obviously, the rectangle is 4 x 27 feet wide and so the outlier is 8 x 27 feet from the dolmen.

*Figure 2 The in-town outlier to the east of the Crucuno dolmen. [photo: Robin Heath, 2007]*

Momentarily at least, one can consider the two lengths of (210-) 216 feet relating to 108 feet of the rectangle and I believe dolmen to centre of rectangle to be about 1105 feet. The combined monument is then as figure 3.

*Figure 3 The combined monument seen within satellite data, showning two key dimensions in white, the section above magnifying the in-town components.*

### The Metrological Key to Crucuno

Read more: Lunar Counting from Crucuno Dolmen to its Rectangle

## Educating megalith builders at Crucuno rectangle

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The area around Carnac in Brittany is peppered with uniquely-formed megalithic designs. In contrast, Great Britain's surviving monuments are largely standing stones and stone circles. One might explain this as early experimentation at Carnac followed by a well-organised set of methods and means in Britain. What these experiments near Carnac were concerned with is contentious, there being no appetite, in many parts of society, for a prehistory of high-achieving geometers and exact scientists. Part of the problem is that pioneers interpreting monuments are themselves hampered by their own preferences. Once Alexander Thom had found the megalithic yard as a likely building unit, he tended to use that measure in isolation to the exclusion of other known metrological systems (see A.E. Berriman's Historical Metrology.) Similarly, John Neal's breakthrough in All Done With Mirrors, having found the foot we still use to be the cornerstone of ancient metrology led to an ambivalent relationship to the megalithic yard. For example, Neal's interpretation of Crucuno rectangle employs a highly variable set of megalithic yards and has perhaps missed the simpler point which supports his foot-based metrology as implicit within the Crucuno rectangle; this monument was said by Thom to be a "symbolic observatory" of the sun: that is an educational device, whilst Neal found the geometry of squaring the circle which, we see later, was the Rectangle's main metrological meaning.

*Figure 1 Alexander Thom's survey of Crucuno Rectangle by Alexander Thom, see MRBB, 1978, 19 & 175-176 *