These days we take a measurement or define a length and call it a radius or diameter for a circle, using an irrational value for the ratio between a radius (or diameter) and the circle's perimeter, called PI - a Greek letter - which is 3.141592654 etc, where the fractional part has no limit of non-repeating cyclicity.
Circumferences of Ancient Circles
In the ancient world, irrational numbers like PI required one to find an accurate rational approximation, in the form of a ratio between two whole numbers (called integers). The simplest and most accurate of these, rather than 3.141592654..., was 22/7 = 3.142857 , which meant that if a diameter was considered of seven parts then the circumference produced when arcing a rope around a central peg would be 22 of the same units. Since that rope from the centre would be half the diameter then if the diameter is 14 units, the rope is half that or seven units and the circumference must be twice 22 units long or 44 units. The advantage of seeing the diameter or radius as containing seven units is that the denominator of 22/7 divides into those seven units leaving the circumference as 22 or 44 units.
Another habit of the ancient world was to analyse geometrical situations in terms of an underlying grid, so with a radius of seven units the following figure shows the situation "on the ground":
Figure 1 The Circle Radius seven for a PI of 22/7 leading to 44 units around the perimeter
The core of 22/7 is 11/7, which has been doubled, meaning that one can extend the radius by a 4 further units beyond the center and achieve a length of 11 units, which is one quarter of the circles diameter (figure 2). This length of 11 units is the length of one quarter of the circumference, called a quadrant, and since that length now exists as a linear length within the circle, one has access, according to the PI of 22/7 of the circumference as four times that length, without drawing and measuring the circumference. This is important when considering the apparently intentional lengths of ancient landforms, that they never needed to be drawn and measured. One could place "alignments" (stones, places, buildings) on them using a bearing and the radial length.
Figure 2 Negative Extension of the Radius beyond the Centre to achieve 1/4 of the circumference
Having demystified the intended circumferences of ancient circles, when 22/7 was used for PI, one can then observe that the quarter length is most naturally the side length 11, of a square. Whilst one could simply draw that square either above or below the extended radius or with the radius dividing it, far better is to center the line within the circle and move it to the bottom of the grid. However, half of eleven is 5.5 and so the grid can be doubled to being 28 units across (as below) so that the ends of the line will land on the grid. The line is now 22 units long, sitting centrally in a grid of 28. It is then easy to see how a square can be drawn within the grid of 28 that remains three units inside the outer extent of the circle since 22 + 3 + 3 = 28, as per figure 3
Figure 3 Representing the Extended Radius as a Square
The result is a square set within a circle of the same perimeter. A circle's perimeter is easy to inscribe in a single operation with a rope but hard to measure. In contrast, a square perimeter is harder to form accurately but then already known as to its perimeter length.
Applying the New Jerusalem Diagram
The above geometrical creation of such an equi-perimeter square within a circle was not that proposed by Michell when using it as the foundation of his remarkable New Jerusalem geometry and this has led to conceptual problems in understanding how ancient geometers knew the length of their perimeters. It was said, as Michell asserts, that "the squared circle of square and circle of equal perimeter ... can be constructed using by the method shown in figure 21", as below.
Figure 4 (FIGURE 21 of The Dimensions of Paradise. page 76) "The construction of the squared circle, consisting of a square and a circle with equal perimeters (if π = 22/7), begins with a Pythagorean right-angle triangle (shaded) with sides in the proportions 3, 4, 5. Multiplied by 720, these base numbers are raised to the dimensions of the New Jerusalem diagram."
In the above diagram, a square side 3 is placed atop the circle's centre, then to achieve the required side length 11 units, adding 3-4-5 triangles either side. The square then leaves 1.5 units on each cardinal direction hence the need for an underlying grid of 28 rather than 14 (the diameter). This implies that the circle is made first yet someone will make the desired radius and, as we have described, know the square of equal perimeter to that of the circle. In Michell, the square is constructed first having made a side length of 11 units but not out of the radius, making the relationship of radius to side length inexplicit as 7 + 4 = 11. I would also argue that the square does not need to be built since the extended radius of 11 units is directly one quarter of the circle's perimeter.
The method proposed here allows one to define a perimeter length by forming one quarter of it, dividing that into eleven parts. For instance a double fathom of Saxon feet of 1.1 (a unit used since the megalithic for obvious reasons) is 11 feet, allowing that foot to demark a chosen numerical length and then use English feet to divide that into eleven parts. The English mile is 480 x 11 feet long and the furlong 660 feet enabling, in the same way, easy division by eleven under some circumstances. The model of the Earth naturally figures eleven in its mean circumference because its radius is 7 times 126 feet long, making its meridian circle length 44 times 126 feet. Hence the utility of this diagram as a basis for creating sacred spaces within circular and square boundaries which are 3168 units in length, since 3168 = 288 times 11. Sacred boundaries are revealed as being a microcosm if the macrocosmic Earth, created by a humanity who could know about connecting the world on the surface of a planet with the planetary whole and planetary system beyond. Adopting PI as 22/7 seems to be more than a human decision, a decision in which rationality could be maintained (rational in the sense of knowable as a fraction).
If one wishes, despite all this, to generate the square of equal perimeter to the circle, one can draw the quadrature points of the inner circle radius 4 units and arc a rope whose length is to the opposite quadrature point of the outer circle. Each corner of the square is defined by the opposite two quadrature points on the inner circle as per the figure 5 below.
Figure 5 Drawing the square of equal perimeter from the quadrature points of two concentric circles radius 4 and 7
The resulting pattern, the basis of Michell's New Jerusalem geometry, reminds of the compound stone circles of the British Isles such as Gunnarkeld, but out of the BAR surveys of Thom only Gunnarkeld appears to come close to this ratio.
Figure 6 Placing the pattern over the site plan of Gunnerkeld which has an inner ring touching the inner circle.
Britain's most famous stone circle is Stonehenge, also clearly involving concentric rings. John Michell's NJ template adds one further ring, inscribing the square of equal perimeter to the outer circle, which he found applies well to the Sarsen Circle complex, built around 2,500 BC, as part of its intended design as below.
Figure 7 Placing the NJ pattern over the Sarsen site plan.
The outer circle runs through the center of the lintel ring whilst the incircle of the square defines the ring of sixty bluestones from West Wales. The inner ring radius 4 corresponds with the horseshoe trilithons. The outer circle is 316.8 feet long, corresponding with the notion that sacred spaces should have a perimeter corresponding to the number 3168, emblematic of the mean earth as perfect sphere.
To be continued in Part Two
Addendum: PI at Gavrinis Stone R8
There can be little doubt that PI as 22/7 was held in the system of metrology used to realise megalithic monuments. There is even a foot measure exactly 22/21 feet long in the Manx module in which the microvariations of 176/175 and 441/440 were probably first derived (see my Sacred Number and the Lords of Time, page 170) and these microvariations are themselves cross multiplications of the three types of PI employed in the megalithic model of the Earth, namely 22/7, 63/10 and 25/8 (see my Sacred Number and the Origins of Civilization, [on the Model] chapter three and [on the ratios] 41-42). In the early megalithic at Gavrinis cairn, a set of engraved stone panels were reset to form a chambered "tomb", showing art related to the monuments around Carnac, Brittany, these often displaying counts, counted time periods and metrological measures. Stone R8 at Gavrinis displays in its upper metrological section megalithic yards, royal cubits and a larger measure a 22/7 foot PI yard equal to 22/7 feet, whose foot is 22/21, the root canonical Manx foot.
Figure Stone R8 at Gavrinis with overlay showing three panels. top:
of Metrological constants, here showing only those related to PI;
of the Saros period, see this page; of the Nodal year, see this page.
The Manx module was pointed to in my Sacred Number and the Lords of Time page 170 as encapsulating the system of modules in ancient metrology. the root is 25/24 and the root canonical 22/21, a "pi foot" for the PI yard, and the standard canonical is 21/20 (aka the Persian foot) which has seven in the denominator allowing its easy use within the radius or diameter measure of a circle - as above. The module is shown below, see also this article.
Figure The Manx Module, archetypal of ancient metrology's module system and pointing to the importance of PI in describing metrology's primary subject, the shape and dimensions of the Earth. The Manx foot has been found historically by Paul Quayle, contained within Isle of Man measuring rods [Quayle, Paul. Manx linear measures in; Isle of Man Studies Vol. XV: Proceedings of the Isle of Manx Natural History and Antiquarian Society.]