From Chapter 2. Models of Time and Space
When we picture a geometrical shape in our minds it is pre-realized, as if in the world of space, because our minds have learned to visualize, using an imagination based upon our seeing in the physical world. This power of imagination, like the power of memory, has had a strong influence upon the world. The angels can be compared to this power and, indeed, angels prefigured this imaginative power in their work of making a place exist where beings such as ourselves could come about. It is in this sense that humans can enter the world of the angelic mind and hence come to see what must have been done to approximate a high state of order in our world through sacred geometry and sacred number.
As already stated, this sharing of mind-stuff with angels developed the human mind’s capacity for rational thinking and numeracy and fed into the civilizations largely responsible for monuments that expressed the angelic models and techniques for arriving at them. . . .
Circles and Squares
The subject we call sacred geometry had practical roots during its early developmental history. It would have been a way of thinking and a type of language. Experiment and learning were happening in a world without pre-existing rules and without the preconceptions of later epochs. Sacred geometry is not functionally useful unless you want to build a monument. The acts of angelic geometry considered here were enduring acts of Will that could explain the World, without having or using our function-based physics or high-level mathematics. The only human capacity available in megalithic times for understanding geometry was the visual imagination that can make geometrical and numerical representations of phenomena, an imagination shared by the angelic world.
The geometrical models of this chapter, while part of sacred geometry, were more importantly involved in the sun-moon-earth’s development as a system. They are relics from before the invention of sacredness as a term, but they are often but unaccountably present throughout the built heritage of sacred buildings. We are the minds the earth developed, but recently our culture decided to move away from admitting that higher minds than our own first conceived the geometrical models built into sacred buildings.
The Squaring of the Circle by Perimeter Length
This geometry was crucially re-discovered by John Michell through his work on Stonehenge and the Great Pyramid of Giza, which both express it.
Many circular monuments of the last 2,400 years, such as domes, appear to exploit a very simple geometrical property, visible as two concentric circles whose diameters, in the same units, are 11 and 14 units. The reason for this is that these circles and their out-squares and in-squares have very special simple numeric relationships to one another. The out-square for the 11-unit long diameter circle is 4 × 11 = 44 units in perimeter, while the circumference of the circle 14 units in diameter is also 44 units (14 × 22/7). The out-square and the outer circle are therefore of the same length when pi is taken to be 22/7, as in figure 1.7. This arrangement is one type of squaring of the (outer) circle by the out-square of the inner circle--our equal perimeter model.
That a circle of diameter 14 has a circumference equal to the perimeter of a square with side-length 11 is the key point (connected to pi as 22/7). The initial circle of diameter 11 might be considered irrelevant, except for the fact that the relative sizes of the earth and the moon appear to be accurately shown by this geometrical model (figure 1.8).
The difference between 14 and 11 is 3 units, and the relative sizes of the earth and moon are accurately 11 to 3. So the inner circle of 11 is the mean size of the earth and the small circle of diameter 3 between the 11 and 14 circles is the mean size of the moon.
The traditional symbol of the earth is a square and the traditional symbol of the celestial world a circle. The square is a symbol of the solidity of objects on the earth. The stars, planets, and Sun travel around the earth at an apparently equal radius and the numbers of this model arise from the cosmos, from the outside in. The outer cosmic radius is 14, the natural diameter regulated by the ratio pi of 22/7. A diameter 14 has radius 7 and an outer circumference of 44 (2 × 22/7 × 7). Each quadrant (quarter-circumference) of the circle is therefore 11, and a square of equal perimeter (44) would have a side length of 11. This model of a square of equal perimeter to the outer circle is primary. Only after calculating it can the circle of diameter 11, the square’s in-circle (the mean earth), can be added, which has a circumference of 11 × 22/7 (= 242/7).
The square’s perimeter of 44 minus the in-circle mean-earth perimeter of 242/7 leaves 66/7 which, divided by 22/7, results in 3 units, the diameter of the moon, as we saw earlier. Therefore the circumference of the Moon is the difference between the perimeter of the out-square of the mean earth and the circumference of the mean earth.
It is extraordinary that the earth-moon system, in an uncanny act of cosmic accounting, had the combined circumferences equal to the out-square of the mean earth and relative diameters of 11 to 3. The low numbers involved make the geometry simple while the earth and moon achieved this 11/3 configuration billions of years ago. And megalithic astronomy, the earliest culture with the known level of numeracy to infer it, reached the symbolic assumption that the outer 3-diameter circle was the moon and the 11-diameter circle the earth’s mean size.