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 1/7th 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.

Figure 2 The essential mystery of how 40 units as diameter becomes 125 of same units (but 176/175 larger!) on the circumference or, more conveniently, the square of same perimeter length. The rectangle presented at Crucuno is of the diameter across and an approxiamtion to Pi in height of 3 and 1/7th, or 22/7

For me it was hard to see how this "magic" happens. Eventually I saw the diameter as being the whole of the circle (once created) and the radius as being the making of a circle through metrology and geometry.

To make a circle we define a centre, the length of the radius and, using a rope, we "arc" the radius around the centre to inscribe the circle; whilst the horizon is a ready-made circle with us as its centre, as if someone else has previously made it; but coming upon a circle on the ground we can measure its diameter as a whole, to define its size.

The diameter view is given by the extended Crucuno rectangle but the radius view is also at work within metrology which enables the circumference and the diameter to remain integer in similar units. It is little appreciated that ancient metrology used a different approximation of Pi of 25/8 = 3.125, for the radius view.

The use of megalithic yards of 2.5 Drusian feet (two and a half feet of 27/25 feet equalling 2.7 feet), and multiplying by a further 2.5 gives a total of 6.25 Drusian feet (a Pi approximation of 25/8). Each Drusian foot in the radius leads to 6.25 megalithic yards on the circumference, in which rods the 6.25 factor is Pi as 25/8, whilst the 22/7 is the Pi approximation taken as Pi itself. The metrological "magic" is that a cross-multiplication of these two types of Pi: 22/7 times 8/25 equaling 176/175 means that: the megalithic rods on the circumference need to be made of Drusian feet one 175th part larger than the Drusian feet of the radius. The same will be true whatever type of foot was used in the radius, the circumference will be 6.25 x 176/175 the feet in the radius.

Figure 3 Two Views of Crucuno Rectangle, according to the measurement of the circle (Pi = 22/7) or its creation, a metrological circle (Pi = 25/8) using a rod (as per Alexander Thom) of 6.25 feet which number is an approximation ot 2*Pi of 25/4 of a foot (as per 2*Pi*r); thus linking radius to circumference in an integer fashion if units on circumference are 176/175 those on the radius.

In the Crucuno rectangle, a radius of 20 Drusian megalithic yards (54 feet) leads to a circumference of (20 x 25) / 4 = 125 Drusian megalithic yards 176/175 larger, these being the Astronomical Megalithic Yard or AMY found widely at megalithic sites. The 125 AMY reduce to 50 AM rods, 2.5 more in number than the Drusian yards in the radius - making the circumference 6.25 longer than the radius. theThis approximation to Pi as cross-multiple implies that ancient metrology was based upon the megalithic rod of 6.25 Drusian feet and hence the 2.5 foot intermediary we call the megalithic yard; as the significance of John Neal's grid constant 176/175 in maintaining integers when dealing with metrological circles. Thom noted a love for imteger solutions in his Megalithic Sites in Britain but, without the restoration of ancient metrology, Thom could only observe whole number if they were megalithic yards, for example in Pythagorean triangles, and the apparent manipulations of Pi in flattened circle perimeters, for example to make a Pi equal to 3.

Another important note, relating to Figure 4, is the use of the square's diagonals to indicate both the quadrant arc-length of the circle and the side of the square which must be of equal length. I have previously proposed that circles (belonging to Eternity and essentially abstract) are not easily measured so why bother, when the side of the square of equal length gives the sector's arc-length? The circumference can be derived, as in this geometry, as a quarter length then doubled twice.

Figure 4 The diagonals of the square give the maximum alignments of the moon rising but most significantly setting.

In figure 4 the diagonals of the square cut the sector's arc-length as well as the squares side-length, which are equal in length. It is then arresting that the square's diagonals at Carnac would have been aligned to the maximum standstill of the Moon. It therefore seems likely that the (more useful) alignments of its setting, to north and south, were marked by stones 2 (still upright) and 17 (not still standing). The utility of setting alignments is that one can see the moon approach the horizon, whilst rising alignments require some degree of prediction as to when the moon will rise. It seems very appropriate therefore that the monument signified the square equal in length to the circumference in this Pi monument, alongside the 4 by 3 rectangle representing the diameter whilst enabling the solar alignments at the solar maximums within the year to north and south.

In three articles I have shown that Crucuno is another crucial monument for understanding the megalithic in its early form, showing elegance, intelligence and capability in their application of number sciences to astronomy.