Sacred Number

The square root of two has a massive significance in geometry, musical harmony and ancient symbolism. I have a little piece of geometry about this that happened the other day:

A unit radius has been drawn so that, using a rational Pi of 22/7, one eighth of the perimeter is 11/14 in length. We know that a very good approximation to the square root of 2 is 140/99 and that the vertical shown must be the reciprocal. On can notice therefore that to transform 11/14 into 99/140 we have multiplied by 9/10, the minor wholetone ratio (lost in our equal temperament).

This is an interesting relationship because it employs the key ancient approximations of Pi and root two with a linkage straight out of ancient tuning theory - a tradition so well developed that it was arguably the coded source of most surviving mythic material such as the Old and New Testaments, Gilgamesh, etc. [Much of what remains is then absorbed in the massive ancient speculations about precession of the equinoxes].

The Biblical source of 99/140 concerns Abraham and circumcision. As Ernest McClain shows in Meditations on the Quran,  Abraham's decision to circumcise himself at the age of 100, and similarly Ishmael at age 13, uses circumcision as a code for removing a little bit from a known rational ratio, 10/14 by removing a hundredth part to obtain 99/140. The 13th fifth in music causes the Pythagorean comma, a musical sin that must be corrected, symbolised again by removing something small "like" a foreskin. More of Ernest at www.ernestmcclain.net.

In the diagram, a geometric arrangement has achieved the same as this rule of thumb but also shown a relationship between a circle and an inscribed square. Multiplying by 14/11, the sector becomes a unit length and the vertical just 9/10, the radius 14/11. It is ideal (as shown by Neal in the Scottish Brochs) that the radius have 7 in its length. Here there is 14 and we also see the 11 as "pre-removed" to obtain the unitary circumference (of 8 units long).

It is very unlikely that the diagram above was not known by  the specialists of the Megalithic and Middle Eastern cultures as it reveals rational order between two irrationals, linked to a harmonic ratio.