In Sacred Number and the Roots of Civilization, chapter three, based on John Michell's reconstruction of the ancient model of the Earth I resolved the mean earth radius (in feet) as being only made up of powers of prime numbers 2, 3 and 7: the primes two and three are factors of twelve, 22 times 3 equalling 12, and Michell's figure of 20901888 feet for the radius is then 126 times 7 feet. The utility of this came, not least, in the ability to form the mean earth circumference as 126 times 44 feet, using a 2 x pi value of 44/7.

Figure 1 The Ancient Model of the Earth

Considering the Oakleaf Decagon of south western England and Wales, we ignore here the other value of 2 x pi (63/10) relating the mean circumference to the polare radius. Top right the rational 44 units around the circumference is also shown in the geometry of a square of equal perimeter length having sides eleven units, squaring the circle in one sense and generating the preferred symbol of the element earth as the square.

Whenever a model of the earth's mean radius is formed, this has to be through a scaling down by some factor into a smaller length. If this was done arbitrarily, without carefully choosing a desireable scaling, then the resulting length would not be a rational model but instead, any length could be said to model anything. The principle of rationality within the model of the earth requires that the scaled down length of a model also be rational in a meaningful way.

In the Oakleaf (perpetual choirs) Decagon, in line with John Michell's finding that perimeters of sacred stuctures were (a) related to the mean earth and (b) employ in some fashion a perimeter of 3168 equal units of length. In our last article, we showed that the scaling of this decagon's radius relative to the mean earth radius was 63.36, so it is important to understand why and how this scaling was chosen (through prime factorization) and achieved (through metrological manipulation) to make the decagon's radius,

Of particular interest is the fractional part of 63.36. As a rational fraction this 0.36 equals 9/25. Dividing 0.36 into 63.36 yields an integer significant to metrology, namely 176 so that 176 times 0.36 equals 63.36.

There is another interesting clue in 0.36 in that the Assyrian foot of 9/10 (0.9) feet as a double foot (1.8 feet) was seen as having 60 parts called shu-si whose length was 0.36 inches**.  However,  the mean earth radius is 126 times 7 feet when divided by 63.36 feet (the scaling factor) becomes the decagon's radius of 329890.9091 - not rational in feet but ideally suited to the approximation to the golden mean of 160/99 (1.6162) found between the radius and the side lengths of a regular decagon. If that radius is viewed as 512 units long then each side length is 316.8 units long so that the ten sides of the decagon will be 3168 units in all, conforming to Michell's ancient norm for sacred spaces.

**Marduk's ziggurat (the Etemenanki)in Babylon can be seen to have been built within a rational cube of 4320 shu-si or 2400 double Assyrian feet.

To see how the decagon was conceived, one needs to see the mean earth radius as a more complex factorization than 126 times 7 feet, since 20901888 feet can also be arrived at by multiplying: 9/25 (=0.36) x 176 (=63.36, the scaling factor) x 175  x 12/11 x 1728 (123) = 20901888. This factorization reveals the need, to maintain rationality, for the root Sumerian foot of 12/11 feet to measure the octagon model of the mean earth's radius, when it is scaled down by 1/63.36. The decagon radius then becomes a rational 302400 Sumerian feet long whilst the side lengths become 187110 Sumerian feet long - the length of 204120 English feet identified by Michell.