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 *

Since the sun rose and set (at the two solsticial extremes of Winter and Summer) away from east-west by the lesser angle of a 3-4-5 triangle the builders of Crucuno created a rectangle of standing stones whose longest sides ran exactly east-west. It was made four units long relative to three units north-south. The resulting diagonals between opposite corners were then automatically aligned to the solstitial extremes of the sun. Thom made an accurate survey of the monument which he identified as 40 by 30 megalithic yards or thereabouts, depending on how one deals with the thickness of the stones in a relatively small monument. Thom's survey of Crucuno has recently by validated by a 3D scan.

*Figure 2 The orientation and 4:3 side ratio of Crucuno defining solsticial sunrises and sets in winter and summer*

*Figure 3 Midsummer sunrise 2007, between stone 6 and 7, when above the trees. (The alignment is less accurate today)*

If one takes Thom's figure for the megalithic yard of 2.72 feet, 40 such yards equal 108.8 feet. Rounding the figure 108 feet makes it little different and, given past assumptions, 108 feet equally fits as in inscribed rectangle within the stones. This number of feet has many unique properties. For example, 100 feet of the Drusian variety (27/25 [1.08] feet) equal 108 feet, making a megalithic yard/step of 2.7 feet, which is then the root module of the Astronomical Megalithic Yard or AMY. An explicit AMY is found within 4Km east of Crucuno, marked in the end stone (C3) of Gavrinis (see figure 4), alongside the foot and the royal cubit of 12/7 feet. The foot, royal cubit and AMY were all astronomically derived since the chambered cairn of Gavrinis is oriented to the solar and lunar extremes of the south east quadrant.

*Figure 4 The left-hand end stone called C3 within which day-inches, the foot, the royal cubit and AMY are all shown, within astronomical and monumental contexts [**figure 5.10 in Sacred Number and the Lords of Time. 131.]*

The length 108 feet can be viewed in many ways and we can try a few. It is 63 (9 x 7) royal cubits and that length, having a factor of seven, could be a convenient diameter when multiplied by the pi of 22/7; the seven divides to leave a circumference of 22 x 9 (198) royal cubits. The shorter side of the Rectangle in these units is then 3 x 63/4 = 47.25 royal cubits whilst one quarter of the Circumference is 198/4 or 49.5 rotal cubits, the side length of a square of equal perimeter length. The four-by-three geometry cannot therefore square the circle with its shorter side length, being deficient by 2.25 royal cubits (3 and 6/7th feet), though *it is close to doing so, *and therefore compatible within the monument. When John Neal extended the 4 x 3 rectangle to the south he found the required the side length for the southern kerb to achieve the squaring of the circular perimeter (63 royal cubits in diameter in figure 5). This adjustment allowed five otherwise spare stones of the southern kerb, numbered 13-17 in Thom's survey, to be part of that extended geometry.

*Figure 5 Neal's extention of the Rectangle to find **five **otherwise "spare" stones of the southern kerb apparently defining the "squaring of the circle" by elongating the shorter side of the rectangle*

Figure 5 clearly shows that the southern kerb was extended southwards, from being a 4 by 3 rectangle, so as to include the similarly-sized rectangle required for the squaring of the circle, with a 4 x 3 rectangle having astronomical virtues. The stones numbered 3, 5, 8, 12, 16, and 20 could each have been part of this plan, touching as they do the circle's perimeter at significant points of crossing. Stone 8 appears to have a contour to follow the circle and a twin stone 20 opposite it [see also Neal. *All Done With Mirrors,* pp164-5].

If these two interpretations are correct, the Rectangle demonstrated the geometrical peculiarity of the solsticial sun at Carnac in 4000 BC, in rising and setting across the diagonals of Crucuno's 4 by 3 rectangle; whilst also demonstrating the squaring of the circle. Neal's metrology for Crucuno used megalithic yards varied by one part in 175 and in 440, and this may not reflect how the builders of the Rectangle chose their measures to design it.

For example, the east-west length might have been chosen (for numerical reasons) to be 108 feet, being 4 times 27 feet. When divided by the royal foot of 8/7 feet, (108 x 7/8) x 22/7 causes seven to cancel, as too the four in 108 leaving 27 x 11 = 297 royal feet as the circumference and the equal perimeter square. The square is an important structure since its dimensions are linear rather than radial, enabling easy definition of ropes around the perimeter, each 297 royal feet long, whilst then also being 125 Astronomical megalithic yards (AMY) of 2.71542857 feet. We can see that the AMY is then 176/175 of the Drusian step of 2.7 feet, as the key to this geometry involving implicit cancellations where the Drusian module is transformed into its root canonical form (x176/175), the AMY. This was the AMY John Neal recognised on Lundy Island, in 2001, as being the *176/175 variation* of the root Drusian module. Robin Heath had found this unit some years before from measuring British monuments after calculating that the astronomical excess of the solar year over the lunar year equalled one English foot, when counting lunar months using these astronomical megalithic yards to represent them.

*Figure 6 The key metrological transformation at Crucuno*

The 40 Drusian steps of the 108 foot Rectangle lead to a circle whose circumference is 125 AMY, the AMY being 176/175 longer than the Drusian step of 2.7 feet. The formula for 176/175 is 4/25 x 44/7, the cross-multiplication of two different approximations as a ratio Pi_{1}/Pi_{2 (}where Pi1 = 22/7 andPi2 = 25/8.) We actually use the more accurate 22/7 as pi whilst the 125/40 of circumference to diameter has produced a pi of 3.125, as the *effective* pi preserving integer units that differ by 176/175. This is a general property of the ratio of 176/175 which is why it was so useful in the megalithic when dealing with circular geometries governed by pi. However, the lunar month (29.53) divided by the lunar orbit (27.32166) is, at 1.08085, closely 27/25, the Drusian foot and this makes this particularly good unit for demonstrating the close affinity of the lunar orbit, lunar month and lunar years and the solar year.

The squaring of the circle naturally allows integer units to describe the diameter, circumference and equivalent square perimeter when one uses the same unit enlarged by 176/175 to measure these perimeters. The circle is actually theoretical whilst the square perimeter is practical since it can be both deduced from the diameter and measured for the formation of longer lengths. 125 AMY is half of a Drusian stade of 625 such feet and one 16th of a Drusian mile of 5000 of those feet. (n.b. The stade was the unit of field measure found by archaic Greece and the mile is traditionally eight stades equalling 5,000 feet, the English stade being the furlong of 660 ft x 8 = 5280, based then of 600 Saxon feet of 11/10 ft)

Conclusions

It is perhaps no coincidence that John Michell created a numerical cosmology on the squaring of the circle whilst his collaborator in recovering ancient metrology from its historical debris was John Neal, who finds that geometry at Crucuno. As I proposed in Sacred Number and the Lords of Time, metrology is more probably innovated by the megalithic than by an unknown "civilisation X" like Atlantis, unless that myth is about the megalithic - about whom there are concrete facts due to their monumentalism. What Thom called a "symbolic observatory", what Michell anticipated in his New Jerusalem geometry, which Neal found expressed at Crucuno, using the ancient metrology restored by Neal and Michell, we find here to be also a lesson in astronomy and metrology and a general-purpose geometry for manufacturing units of length which are 176/175 greater than the units used to define in its diameter. In particular, astronomical megalithic yards could be manufactured from a length of 40 Drusian steps (108 feet). Since the 4 x 3 geometry for solstice sun events on the horizon was very similar to that required for squaring the circle: The three of 4 x 3 becomes 3.142857 or pi as 22/7. The ratio of 4 to 22/7 is the ratio of a square to its in-circle.

*Figure 7 The squaring of the circle, prefigured in the 1/4 circumference of a circle and the side length of its out-circle. This ratio of 22/7 (3.142) to a side length of 4 is the ratio of Crucuno's smaller side length to its longer dimension, clearly four units in order to make the 3-4-5 triangles as the diagonals aligned to solstice events. Far from theoretical, the Crucuno rectangle, in its full height, could manufacture 125 AMY from a length of 40 Drusian steps.*